Datasets:

aluncstokes

/

alg-geom.temp

like 0

9708

alg-geom/9708002

en

https://arxiv.org/abs/alg-geom/9708002

alg-geom/9708002

James A. Carlson

James A. Carlson and Domingo Toledo

Discriminant Complements and Kernels of Monodromy Representations

20 page dvi file available at http://www.math.utah.edu/~carlson/eprints.html Minor changes for final version to appear in Duke J. Math

We show that the kernel of the monodromy representation for hypersurfaces of degree d and dimension n is large for d at least three with the exception of the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large" we mean a group that admits a homomorphism to a semisimple Lie group of noncompact type with Zariski-dense image. By the Tits alternative a large group contains a free subgroup of rank two.

alg-geom

\section{Introduction} \secref{introsection} A hypersurface of degree $d$ in a complex projective space $\P^{n+1}$ is defined by an equation of the form $$ F(x) = \sum a_L x^L = 0, \eqn \eqref{universalhypersurface} $$ where $x^L = x_0^{L_0} \cdots x_{n+1}^{L_{n+1}}$ is a monomial of degree $d$ and where the $a_L$ are arbitrary complex numbers, not all zero. Viewed as an equation in both the $a$'s and the $x$'s, \eqrefer{universalhypersurface} defines a hypersurface ${\bf X}$ in $\P^N\times\P^{n+1}$, where $N+1$ is the dimension of the space of homogeneous polynomials of degree $d$ in $n+2$ variables, and where the projection $p$ onto the first factor makes ${\bf X}$ into a family with fibers $X_a = p^{-1}(a)$. This is the universal family of hypersurfaces of degree $d$ and dimension $n$. Let $\Delta$ be the set of points $a$ in $\P^N$ such that the corresponding fiber is singular. This is the {\sl discriminant locus}; it is well-known to be irreducible and of codimension one. Our aim is to study the fundamental group of its complement, which we write as $$ \Phi = \pi_1(\P^N - \Delta). $$ When we need to make precise statements we will sometimes write $ \Phi_{d,n} = \pi_1(U_{d,n}, o), $ where $d$ and $n$ are as above, $U_{d,n} = \P^N - \Delta$, and $o$ is a base point. The groups $\Phi$ are almost always nontrivial and in fact are almost always {\sl large}. By this we mean that there is a homomorphism of $\Phi$ to a non-compact semi-simple real algebraic group which has Zariski-dense image. Large groups are infinite, and, moreover, always contain a free group of rank two. This follows from the Tits alternative \cite{Tits}, which states that in characteristic zero a linear group either has a solvable subgroup of finite index or contains a free group of rank two. To show that $\Phi = \Phi_{d,n}$ is large we consider the image $\Gamma = \Gamma_{d,n}$ of the monodromy representation $$ \rho: \Phi \map G. \eqn \eqref{monodrep} $$ Here and throughout this paper $G = G_{d,n}$ denotes the group of automorphisms of the primitive cohomology $H^n(X_o,\R)_o$ which preserve the cup product. When $n$ is odd the primitive cohomology is the same as the cohomology, and when $n$ is even it is the orthogonal complement of $h^{n/2}$, where $h$ is the hyperplane class. Thus $G$ is either a symplectic or an orthogonal group, depending on the parity of $n$, and is an almost simple real algebraic group. About the image of the monodromy representation, much is known. Using results of Ebeling \cite{Ebeling} and Janssen \cite{Janssen}, Beauville in \cite{BeauvilleLattice} established the following: \proclaim{Theorem}. Let $G_\Z$ be the subgroup of $G$ which preserves the integral cohomology. Then the monodromy group $\Gamma_{d,n}$ is of finite index in $G_\Z$. Thus it is an arithmetic subgroup. \endproclaim \procref{Beauvillethm} \noindent The result in \cite{BeauvilleLattice} is much more precise: it identifies $\Gamma$ as a specific subgroup of finite (and small) index in $G_\Z$. Now suppose that $d > 2$ and that $(d,n) \ne (3,2)$. Then $G$ is noncompact, and the results of Borel \cite{BorelDT} and Borel-Harish-Chandra \cite{BHC} apply to show that $\Gamma$ is (a) Zariski-dense and (b) a lattice. Thus (a) the smallest algebraic subgroup of $G$ which contains $\Gamma$ is $G$ itself and (b) $G/\Gamma$ has finite volume. Consider now the kernel of the monodromy representation, which we denote by $K$ and which fits in the exact sequence $$ 1 \map K \map \Phi \mapright{\rho} \Gamma \map 1. \eqn \eqref{KPhiGamma} $$ The purpose of this paper is to show that in almost all cases it is also large: \proclaim{Theorem.} The kernel of the monodromy representation \eqrefer{monodrep} is large if $d > 2$ and $(d,n) \ne (3,1),\ (3,0)$. \endproclaim \procref{maintheorem} The theorem is sharp in the sense that the remaining groups are finite. When $d = 2$, the case of quadrics, $\Phi$ is finite cyclic. When $(d,n) = (3,0)$, the configuration space $U$ parametrizes unordered sets of three distinct points in the projective line and so $\Phi$ is the braid group for three strands in the sphere. It has order 12 and can be faithfully represented by symmetries of a regular hexagon. When $(d,n) = (3,1)$ the configuration space $U$ parametrizes smooth cubic plane curves and the above sequence can be written as $$ 1 \map K \map \Phi_{3,1} \mapright{\rho} SL(2,\Z) \map 1, $$ where $K$ is the three-dimensional Heisenberg group over the field $\Z/3$, a finite group of order 27. Moreover, $\Phi_{3,1}$ is a semi-direct product, where $SL(2,\Z )$ acts on $K$ in the natural way. This result, due to Dolgachev and Libgober \cite{DolgLib}, is to our knowledge the only one which determines the exact sequence \eqrefer{KPhiGamma} for hypersurfaces of positive dimension and degree larger than two. Note that in this case $\Phi$ is large but $K$ is finite. Note also that there are two kinds of groups for which the natural monodromy representation has finite image but large kernel. These are the braid groups $\Phi_{d,0}$ for $d > 3$ and the group $\Phi_{3,2}$ for the space of cubic surfaces. Thus all of them are large. For the braid groups this result is classical, but for $\Phi_{3,2}$ it is new. Since $\Phi_{3,2}$ is large it is infinite, a fact which answers a question left open by Libgober in \cite{Lib}. Concerning the proof of Theorem \xref{maintheorem}, we would like to say first of all that it depends, like anything else in this subject, on the Picard-Lefschetz formulas. We illustrate their importance by sketching how they imply the non-triviality of the monodromy representation \eqrefer{monodrep}. Consider a smooth point $c$ of the discriminant locus. For these $X_c$ has a exactly one node: an isolated singularity defined in suitable local coordinates by a nondegenerate sum of squares. Consider also a loop $\gamma = \gamma_c$ defined by following a path $\alpha$ from the base point to the edge of a complex disk normal to $\Delta$ and centered at $c$, traveling once around the circle bounding this disk, and then returning to the base point along $\alpha$ reversed. By analogy with the case of knots, we call these loops (and also their homotopy classes) the {\sl meridians} of $\Delta$. Then $T = \rho(\gamma)$ is a {\sl Picard-Lefschetz} transformation, given by the formula $$ T(x) = x \pm (x,\delta) \delta . \eqn \eqref{plformula} $$ Here $(x,y)$ is the cup product and $\delta$ is the {\sl vanishing cycle} associated to $\gamma$. When $n$ is odd, $(\delta,\delta) = 0$ and the sign in \eqrefer{plformula} is $-$. When $n$ is even and $(\delta,\delta) = \pm 2$, the sign in \eqrefer{plformula} is $\mp$ (see \cite{DeWeOne}, paragraph 4.1). Thus when $n$ is even $\delta$ is automatically nonhomologous to zero, and so $T$ must be nontrivial. Since vanishing cycles exist whenever the hypersurface $X_o$ can degenerate to a variety with a node, we conclude that $\rho$ is nontrivial for $n$ even and $d > 1$. Slightly less elementary arguments show that the homology class of the vanishing cycle, and hence the monodromy representation, is nontrivial for all $d > 1$ except for the case $(d,n) = (2,1)$. The proofs of theorem \xref{Beauvillethm}, an earlier result of Deligne asserting the Zariski density of $\Gamma_{d,n}$, and the main result of this paper are based on the Picard-Lefschetz formulas \eqrefer{plformula}. Our proof begins with the construction of a universal family of cyclic covers of $\P^{n+1}$ branched along the hypersurfaces $X$. From it we define a second monodromy representation $\bar \rho'$ of $\Phi$. Suitable versions of the Picard-Lefschetz formulas and Deligne's theorem apply to show that $\bar \rho'$ has Zariski-dense image. Finally, we apply Margulis' super-rigidity theorem to show that $\bar\rho'(K)$, where $K$ is the kernel of the natural monodromy representation, is Zariski-dense. Thus $K$ is large. We mention the paper \cite{Mag} as an example of the use of an associated family of cyclic covers to construct representations (in this case for the braid groups of the sphere). We also note the related results of the article \cite{DOZ} which we learned of while preparing the final version of this manuscript. The main theorem is that the complement of the dual $\widehat C$ of an immersed curve $C$ of genus at least one, or of an immersed rational curve of degee at least four, is {\sl big} in the sense that it contains a free group of rank two. When $C$ is smooth, imbedded, and of even degree at least four this follows from a construction of Griffiths \cite{GriffHyperbolic}: consider the family of hyperelliptic curves obtained as double covers of a line $L$ not tangent to $C$ which is branched at the points $L\cap C$. It defines a monodromy representation of $\Phi = \pi_1(\widehat{\P}^2 - \widehat{C})$ with Zariski-dense image. Consequently $\Phi$ is large, and, {\sl a fortiori}, big. Such constructions have inspired the present paper. By using cyclic covers of higher degree one can treat the case of odd degree greater than four in the same way. The authors would like to thank Herb Clemens and Carlos Simpson for very helpful discussions. \section{Outline of the proof} \secref{outlinesection} As noted above, the proof of the main theorem is based on the construction of an auxiliary representation $\rho'$ defined via a family of cyclic covers $Y$ of $\P^{n+1}$ branched along the hypersurfaces $X$. To describe it, let $k$ be a divisor of $d$ and consider the equation $$ F(a,x) = y^k + \sum a_L x^L = 0, \eqn \eqref{universalcyclic} $$ which for the moment we view as defining a set $\widehat {\bf Y}$ in $(\C^{N+1} - \set{0})\times \C^{n+3}$ with coordinates $a_L$ for $\C^{N+1}$ and coordinates $x_0,\cdots ,x_{n+2}$ and $y$ for $\C^{n+3}$. Construct an action of $\C^*$ on it by multiplying the coordinates $x_i$ by $t$ and by multiplying $y$ by $t^{d/k}$. View the quotient ${\bf Y}$ in $(\C^{N+1} - \set{0})\times \P^{n+2}$, where we use $\P^{n+2}$ to denote the weighted projective space for which the $x_i$ have weight one and for which $y$ has weight $d/k$. The resulting universal family of cyclic covers ${\bf Y}$ is defined on $\C^{N+1} - \set{0}$ and has smooth fibers over $\widetilde U = \C^{N+1} - \widetilde \Delta$, where $\widetilde \Delta$ is the pre-image of $\Delta$. Since $\C^{N+1} - \set{0}$ is a principal $\C^*$ bundle over $\P^N$, the same holds over $\widetilde U$ and $\widetilde \Delta$. It follows that one has a central extension $$ 0 \map \Z \map \widetilde \Phi \map \Phi \map 1, $$ where $\widetilde \Phi = \pi_1(\widetilde U)$. We introduce $\widetilde U$ and $\widetilde\Phi$ purely for the technical reason that the universal family of cyclic branched covers need not be defined over $U$ itself. The family ${\bf Y} |\widetilde U$ has a monodromy representation which we denote by $\tilde\rho$ and which takes values in a real algebraic group $\widetilde G$ of automorphisms of $H^{n+1}(Y_{\tilde o},\C )$ which commute with the cyclic group of covering transformations (and which preserve the hyperplane class and the cup product). Here $\tilde o$ is a base point in $\widetilde U$ which lies above the previously chosen base point $o$ of $U$, and $Y_{\tilde o}$ denotes the $k$-fold cyclic cover of $\P^{n+1}$ branched over $X_o$. The group $\widetilde G$ is semisimple but in general has more than one simple factor. Let $G'$ be one of these and let $$ \rho ' :\widetilde\Phi\map G', $$ denote the composition of $\tilde\rho$ with the projection to $G'$. Then we must establish the following: \proclaim{Technical point}. The factor $G'$ can be chosen to be a non-compact almost simple real algebraic group. The image of $\rho '$ is Zariski-dense in $G'$. \endproclaim \procref{technicalpoint} Suppose that this is true. Then we can argue as follows. First, the group of matrices which commute with $\rho'(\Z)$ contains a Zariski-dense group. Consequently $\rho'(\Z)$ lies in the center of $G'$. Therefore there is a quotient representation $$ \bar \rho': \Phi\map \bar G', $$ where $\bar G' $ is the adjoint group of $G'$ (that is, $G'$ modulo its center). Moreover, the representation $\bar\rho'$ also has Zariski-dense image. Now consider our original representation \eqrefer{monodrep}. Replacing $\Phi$ by a normal subgroup of finite index we may assume that the image of $\rho$ lies in the identity component of $G$ in the analytic topology and that the image of $\bar\rho '$ lies in the identity component of $\bar G'$ in the Zariski topology. Let $\bar G$ denote the identity component (in the analytic topology) of $G$ modulo its center, and let $\bar\rho :\Phi\map \bar G$ denote the resulting representation. We still have that $\bar\rho (\Phi)$ is a lattice in $\bar G$ and that $\bar \rho '(\Phi)$ is Zariski-dense in $\bar G'$. Now let $\bar K$ be the kernel of $\bar\rho$, and let $L$ be the Zariski-closure of $\bar\rho'(\bar K)$. Since $\bar K$ is normal in $\Phi$ and $\bar\rho'(\Phi)$ is Zariski-dense in $\bar G'$, $L$ is normal in $\bar G'$. Since $\bar G'$ is a {\sl simple} algebraic group, either $L = \bar G'$ or $L = \set{1}$. If the first of the two alternatives holds, then $\bar\rho'(K)$ is Zariski dense, and so $K$ is large. This is because $K$ has finite index in $\bar K$ and so $\bar\rho'(K)$ and $\bar\rho'(\bar K)$ have the same Zariski closure. We now show that the second alternative leads to a contradiction, from which it follows that $K$ must be large. Indeed, if $\bar \rho'(\bar K) = \set{1}$, then the expression $\bar \rho'\circ\bar \rho^{-1}$ defines a homomorphism from the lattice $\bar \rho(\Phi)$ in $\bar G$ to the Zariski-dense subgroup $\bar \rho'(\Phi)$ in $\bar G'$. If the real rank of $\bar G$ is at least two, the Margulis rigidity theorem \cite{MargulisRigidity}, \cite{Zimmer} Theorem 5.1.2, applies to give an extension of $\bar \rho'$ to a homomorphism of $\bar G$ to $\bar G'$. Since $\bar\rho'(\Phi)$ is Zariski-dense, the extension is surjective. Since $\bar G$ is simple, it is an isomorphism. Thus the complexified lie algebras $\g_\C, \g'_\C$ must be isomorphic. However, one easily shows that $\g_\C \not\cong \g'_\C$, and this contradiction completes the proof. We carry out the details separately in two cases. First, for the simpler case where $d$ is even and $(d,n)\ne (4,1)$, we use double covers ($k = 2$). Then $G'$ is the full group of automorphisms of the primitive (or anti-invariant) part of $H^{n+1}(Y,\R)$ and so is again an orthogonal or symplectic group. The technical point \xref{technicalpoint} follows from a density result of Deligne that we recall in section \xref{zardensitysection}. Deligne's result gives an alternative between Zariski density and finite image, and the possibility of finite image is excluded in section \xref{ratdiffsection}. Finally the Lie algebras $\g_\C$ and $\g'_\C$ are not isomorphic, since when one of them is symplectic (type $C_\ell$), the other is orthogonal (type $B_\ell$ or $D_\ell$). By lemma \xref{rankboundslemma} the rank $\ell$ is at least three, so there are no accidental isomorphisms, e.g., $B_2 \cong C_2$. For the remaining cases, namely $d$ odd or $(d,n) = (4,1)$ we use $d$-fold covers, i.e., $k = d$. For these we must identify the group $\widetilde G$ of automorphisms of $H^{n+1} (Y,\R)_0$ which preserve the cup product and which commute with the cyclic automorphism $\sigma$. This is the natural group in which the monodromy representation $\tilde \rho$ takes its values. Now a linear map commutes with $\sigma$ if and only if it preserves the eigenspace decomposition of $\sigma$, which we write as $$ H^{n+1} (Y,\C)_0 = \bigoplus_{\mu \ne 1} H(\mu). $$ As noted in \eqrefer{eigenspacedimform}, the dimension of $H(\mu)$ is independent of $\mu$. Now let $\widetilde G(\mu)$ be subgroup of $\widetilde G$ which acts by the identity on $H(\lambda)$ for $\lambda \ne \mu,\; \bar\mu$. It can be viewed as a group of transformations of $H(\mu) + H(\bar\mu)$. Thus there is a decomposition $$ \widetilde G = \prod_{ \mu \in S} \widetilde G(\mu), \eqn \eqref{tildeGdecomp} $$ where $$ S = \sett{ \mu }{ \mu^k =1,\ \mu \ne 1,\ \Im \mu \ge 0 }. $$ When $\mu$ is non-real, $\widetilde G(\mu)$ can be identified via the projection $H(\mu) \oplus H(\bar\mu) \map H(\mu)$ with the group of transformations of $H(\mu)$ which are unitary with respect to the hermitian form $h(x,y) = i^{n+1}(x,\bar y)$, where $(x,y)$ is the cup product. This form may be (and usually is) indefinite. When $\mu = -1$, $\widetilde G(\mu)$ is the group of transformations of $H(-1)$ which preserve the cup product. It is therefore an orthogonal or symplectic group. We will show that at least one of the components $\widetilde\rho_\mu(\Phi)\subset \widetilde G(\mu)$ is Zariski-dense, and we will take $G' = \widetilde G(\mu)$. The necessary Zariski density result, which is a straightforward adaptation of Deligne's, is proved in section \xref{unitarydensitysection} after some preliminary work on complex reflections in section \xref{complexreflectionsection}. Again, the possibility of finite image has to be excluded, and the argument for this is in section \xref{cycliccoversection}. Finally, to prove that $\g_\C$ and $\g'_\C$ are not isomorphic one observes that $\g_\C$ is of type $B_\ell, C_\ell$ or $D_\ell$ while $\g'_\C$ is of type $A_\ell$ (since $G'$ is of type $SU(r,s)$. One only needs to avoid the isomorphism $D_3\cong A_3$, which follows from the lower bound of the rank of $\g_\C$ in lemma \xref{rankboundslemma}. In order to apply Margulis' theorem we also need to verify that the real rank of $G$ is at least two. This is done in section \xref{rankbounds}. To summarize, we have established the following general criterion, and our proof of Theorem \xref{maintheorem} is an application of it. \proclaim{Criterion.} The kernel $K$ of a linear representation $\rho: \Phi \map G$ is large if \list \i $\rho(\Phi) \subset G$ is a lattice in a simple Lie group $G$ of real rank at least two. \i There exist a non-compact, almost simple real algebraic group $G'$, a central extension $\widetilde\Phi$ of $\Phi$ and a linear representation $\rho': \widetilde \Phi \map G'$ with Zariski-dense image. \i $G$ and $G'$ are not locally isomorphic. \endlist \endproclaim \procref{largekernelcriterion} \noindent An immediate consequence is the following: \proclaim{Corollary.} Let $\Phi$ be a group which admits a representation $\rho: \Phi \map G$ to a simple Lie group of real rank greater than 1 with image a lattice. Suppose further that there exist an almost simple real algebraic group $G'$, a central extension $\widetilde \Phi$ of $\Phi$, and a representation $\rho': \widetilde\Phi \map G'$ with Zariski-dense image. Suppose in addition that $G$ and $G'$ are not locally isomorphic. Then $\Phi$ is not isomorphic to a lattice in any simple Lie group of real rank greater than 1. \endproclaim \noindent{\bf Proof:\ } Suppose that $\tau: \Phi \map \Sigma$ is an isomorphism of $\Phi$ with a lattice $\Sigma$ in a Lie group $H$ of real rank greater than one. If $H$ is not locally isomorphic to $G'$, then apply the criterion with $\tau$ in place of $\rho$ to conclude that $\tau$ has large kernel, hence cannot be an isomorphism. Suppose next that $H$ is locally isomorphic to $G'$. Apply the criterion with $\tau$ in place of $\rho$ and with $\rho$ in place of $\rho'$ to conclude as before that the kernel of $\tau$ is large. For most families of hypersurfaces the natural monodromy representation and the representation for the associated family of cyclic covers satisfy the hypotheses of the corollary to give the following: \proclaim{Theorem.} If $d > 2$, $n>0$, and $(d,n) \ne (3,1),\; (3,2)$, the group $\Phi_{d,n}$ is not isomorphic to a lattice in a simple Lie group of real rank greater than one. \endproclaim It seems reasonable that the preceding theorem holds with ``semisimple'' in place of ``simple.'' However, we are unable show that this is the case. Indeed, our results so far are compatible with an isomorphism $\Phi \cong \Gamma\times\Gamma'$. We can exclude this in certain cases (see section \xref{remarkssection}), but not for an arbitrary subgroup of finite index, which is what one expects. \proclaim{Remarks.} \procref{remarkMeridiansInfiniteOrder} \rm (a) Suppose that $d \ge 3$ and let $\gamma$ be a meridian of $\Phi_{d,n}$. When $n$ is odd, $\rho(\gamma)$ is a nontrivial symplectic transvection. Since it is of infinite order, so is the meridian $\gamma$. When $n$ is even, $\rho(\gamma)$ is a reflection, hence of order two. Now suppose that $d$ is even and consider the monodromy representation of the central extension $\widetilde \Phi$ constructed from double covers. Let $\tilde\gamma$ be a lift of $\gamma$ to an element of $\widetilde\Phi$. Then $\rho'(\tilde\gamma)$ is a nontrivial symplectic transvection, no power of which is central. Thus $\bar\rho'(\gamma)$ is of infinite order, and, once again, we conclude that $\gamma$ is of infinite order. (b) M. Kontsevich informs us that he can prove that for any $d>2$ (and at least for $n=2$) the local monodromy corresponding to a meridian is of infinite order in the group of connected components of the symplectomorphism group of $X_o$. This implies that the meridians are of infinite order for all $d>2$, not necessarily even as above. The symplectic nature of the monodromy for a meridian (for $n=2$) is studied in great detail by P. Seidel in his thesis \cite{Seidel}. (c) For the case of double covers the image $\Gamma'$ of the fundamental group under the second monodromy representation $\rho'(\widetilde\Phi)$ is a lattice. This follows from the argument given by Beauville to prove theorem \xref{Beauvillethm}. It is enough to be able to degenerate the branch locus $X$ to a variety which has an isolated singularity of the form $x^3 + y^3 + z^4 + \hbox{a sum of squares} = 0$. Then the roles of the kernels $K$ and $K'$ are symmetric and one concludes that $K'$ is also large. \endproclaim \section{Zariski Density} \secref{zardensitysection} The question of Zariski-density for monodromy groups of Lefschetz pencils was settled by Deligne in \cite{DeWeOne} and \cite{DeWeTwo}. We review these results here in a form convenient for the proof of the main theorem in the case of even degree and also for the proof of a density theorem for unitary groups (section \xref{unitarydensitysection}). To begin, we have the following purely group-theoretic fact: \cite{DeWeTwo}(4.4): \proclaim{Theorem. (Deligne)} Let $V$ be a vector space (over $\C$) with a non-degenerate bilinear form $(\ ,\ )$ which is either symmetric or skew-symmetric. Let $\Gamma$ be a group of linear transformations of $V$ which preserves the bilinear form. Assume the existence of a subset $E\subset V$ such that $\Gamma$ is generated by the Picard-Lefschetz transformations \eqrefer{plformula} with $\delta\in E$. Suppose that $E$ consists of a single $\Gamma$-orbit and that it spans $V$. Then $\Gamma$ is either finite or Zariski-dense. \endproclaim \procref{delignedensity} To apply this theorem in a geometric setting, consider a family of $n$-dimensional varieties $p: {\bf X} \map S$ with discriminant locus $\Delta$ and monodromy representation $\rho:\pi_1(S - \Delta) \map \hbox{Aut}(H^n(X_o))$. Assume that $S$ is either $\C^{N+1} - \{ 0\} , N\ge 1$ or $\P^N$, so that $S$ is simply connected and hence that $\pi_1 (S-\Delta)$ is generated by {\sl meridians} (cf. \S \xref{introsection} for the definition). Assume also that for each meridian there is a class $\delta\in H^n (X_o )$ such that the corresponding monodromy transformation is given by the Picard-Lefschetz formula \eqrefer{plformula}. Let $E$ denote the set of these classes (called the {\sl vanishing cycles}). Let $V^n(X_o)\subset H^n (X_o)$ be the span of $E$, called the {\sl vanishing cohomology}. A cycle orthogonal to $V = V^n(X_o)$ is invariant under all Picard-Lefschetz transformations, hence is invariant under the action of monodromy. Consequently its orthogonal complement $V\perp$ is the space of invariant cycles. The image of $H^n({\bf X})$ in $H^n(X_o)$ also consists of invariant cycles. By theorem 4.1.1 (or corollary (4.1.2)) of \cite{DeHodgeTwo}, this inclusion is an equality. One concludes that $V\perp$ is the same as the image of $H^n({\bf X})$, which is a sub-Hodge structure, and so the bilinear form restricted to it is nondegenerate. Therefore the bilinear form restricted to $V = V^n(X_o)$ is also nondegenerate. Consequently $V^n(X_o)$ is an orthogonal or symplectic space, and the monodromy group acts on $V^n(X_o)$ by orthogonal or symplectic transformations. When the discriminant locus is irreducible the argument of Zariski \cite{Zar} or \cite{DeWeOne}, paragraph preceding Corollary 5.5, shows that the meridians of $\pi_1(S - \Delta)$ are mutually conjugate. Writing down a conjugacy $\gamma' = \kappa^{-1}\gamma\kappa$ and applying it to \eqrefer{plformula}, one concludes that $\delta' = \rho(\kappa^{-1})(\delta)$. Thus the vanishing cycles constitute a single orbit. To summarize, we have the following, (c.f. \cite{DeWeOne}, Proposition 5.3, Theorem 5.4, and \cite{DeWeTwo}, Lemma 4.4.2): \proclaim{Theorem.} Let ${\bf X} \map S$, with $S = \C^{N+1} - \{ 0\}$ or $\P^N$ and $N \ge 1$, be a family with irreducible discriminant locus and such that the monodromy transformations of meridians are Picard-Lefschetz transformations. Then the monodromy group is either finite or is a Zariski-dense subgroup of the (orthogonal or symplectic) group of automorphisms of the vanishing cohomology. \endproclaim To decide which of the two alternatives holds, consider the period mapping $$ f : U \map D/\Gamma, $$ where $D$ is the space \cite{GriffPerDom} which classifies the Hodge structures $V^n(X_a)$ and where $\Gamma$ is the monodromy group. Then one has the following well-known principle: \proclaim{Lemma.} If the monodromy group is finite, then the period map is constant. \endproclaim \procref{finiteimagelemma} \noindent{\bf Proof:\ } Let $f$ be the period map and suppose that the monodromy representation is finite. Then there is an unramified cover $\widetilde S$ of the domain of $f$ for which the monodromy representation is trivial. Consequently there is lift $\tilde f$ to $\widetilde S$ which takes values in the period domain $D$. Let $\bar S$ be a smooth compactification of $\widetilde S$. Since $D$ acts like a bounded domain for horizontal holomorphic maps, $\tilde f$ extends to a holomorphic map of $\bar S$ to $D$. Any such map with compact domain is constant \cite{GS}. As a consequence of the previous lemma and theorem, we have a practical density criterion: \proclaim{Theorem.} Let ${\bf X}$ be a family of varieties over $\C^{N+1} - \{ 0\}$ or $\P^N$, $N\ge 1$, whose monodromy group is generated by Picard-Lefschetz transformations \eqrefer{plformula}, which has irreducible discriminant locus, and whose period map has nonzero derivative at one point. Then the monodromy group is Zariski-dense in the (orthogonal or symplectic) automorphism group of the vanishing cohomology. \endproclaim \procref{practicaldensitycriterion} Irreducibility of the discriminant locus for hypersurfaces is well known, and can be proved as follows. Consider the Veronese imbedding $v$ of $\P^{n+1}$ in $\P^N$. This is the map which sends the homogeneous coordinate vector $[ x_0 \commadots x_{n+1} ]$ to $[ x^{M_0} \commadots x^{M_N} ]$ where the $x^{M_i}$ are an ordered basis for the monomials of degree $d$ in the $x_i$. If $H$ is a hyperplane in $\P^N$, then $v^{-1}(H)$ is a hypersurface of degree $d$ in $\P^{n+1}$. All hypersurfaces are obtained in this way, so the dual projective space $\widehat \P^N$ parametrizes the universal family. A hypersurface is singular if and only if $H$ is tangent to the Veronese manifold $\VV = v(\P^{n+1})$. Thus the discriminant is the variety $\widehat \VV$ dual to $\VV$. Since the variety dual to an irreducible variety is also irreducible, it follows that the discriminant is irreducible. Finally, we observe that in the situations considered in this paper, vanishing cohomology and primitive cohomology coincide. This can easily be checked by computing the invariant cohomology using a suitable compactification and appealing to (4.1.1) of \cite{DeHodgeTwo}. Since this is not essential to our arguments we omit further details. \section{Rational differentials and the Griffiths residue calculus} \secref{ratdiffsection} Griffiths' local Torelli theorem \cite{GriffPerRat} tells us that the period map for hypersurfaces of degree $d$ and dimension $n$ is is nontrivial for $d > 2$ and $n > 1$ with the exception of the case $(d,n) = (3,2)$. In fact, it says more: the kernel of the differential is the tangent space to the orbit of the natural action of the projective linear group. The proof is based on the residue calculus for rational differential forms and some simple commutative algebra (Macaulay's theorem). What we require here is a weak (but sharp) version of Griffiths' result for the variations of Hodge structures defined by families of cyclic covers of hypersurfaces. For double covers this is straightforward, since such covers can be viewed as hypersurfaces in a weighted projective space \cite{Dolgachev}. For higher cyclic covers the variations of Hodge structure are complex, and in general the symmetry of Hodge numbers, $h^{p,q} = h^{q,p}$ is broken. Nonetheless, the residue calculus still gives the needed result. Since this last part is nonstandard, we sketch recall the basics of the residue calculus, how it applies to the case of double covers, and how it extends to the case of higher cyclic covers. To begin, consider weighted projective space $\P^{n+1}$ where the weights of $x_i$ are $w_i$. Fix a weighted homogeneous polynomial $P(x)$ and let $X$ be the variety which it defines. We assume that it is smooth. Now take a meromorphic differential $\nu$ on $\P^{n+1}$ which has a pole of order $q+1$ on $X$. Its residue is the cohomology class on $X$ defined by the formula $$ \int_\gamma \hbox{res}\, \nu = { 1 \over 2 \pi } \int_{\partial T(\gamma)} \nu, $$ where $T(\gamma)$ is a tubular neighborhood of an $n$-cycle $\gamma$. The integrand can be written as $$ \nu(A,P,q) = { A\Omega \over P^{q+1} }. \eqn\eqref{ratdiff} $$ where $$ \Omega = \sum (-1)^i\;w_i x_i \;dx_0 \wedges \widehat{dx_i} \wedges dx_{n+1}. $$ The ``volume form'' $\Omega$ has weight $w_0 + \cdots + w_{n+1}$ and the degree of $A$, which we write as $a(q)$, is such that $\nu$ is of weight zero. The primitive cohomology of $X$ is spanned by Poincar\'e residues of rational differentials, and the space of residues with a pole of order $q+1$ is precisely $F^{n-q}H^n_o(X)$, the $(n-q)$-th level of the Hodge filtration on the primitive cohomology. When the numerator polynomial is a linear combination of the partial derivatives of $P$, the residue is cohomologous in $\P^{n+1} - X$ to a differential with a pole of order one lower. Let $J = (\partial P/\partial x_0 \commadots \partial P/\partial x_{n+1})$ be the Jacobian ideal and let $R = \C[x_0 \commadots x_{n+1}]/J$ be the quotient ring, which we note is graded. Then the residue maps $R^{a(q)}$ to $F^q/F^{q+1}$. By a theorem of Griffiths \cite{GriffPerRat}, this map is an isomorphism. For a smooth variety the ``Jacobian ring'' $R$ is finite-dimensional, and so there is a least integer $$ t = (n+2)(d-2) \eqn \eqref{topJ} $$ such that $R^i = 0$ for $i > t$. Moreover, and $R^t$ is one-dimensional and the bilinear map $$ R^i\times R^{t-i} \map R^t \cong \C. $$ is a perfect pairing (Macaulay's theorem). When $R^i$ and $R^{t-i}$ correspond to graded quotients of the Hodge filtration, the pairing corresponds to the cup product \cite{CG}. The derivative of the period map is given by formal differentiation of the expressions (\xref{ratdiff}). Thus, if $P_t = P + tQ + \cdots$ represents a family of hypersurfaces and $\omega = \hbox{res}\,(A\Omega/F^\ell)$ represents a family of cohomology classes on them, then $$ { d \over dt} \hbox{res}\,{ A\Omega \over P^{q+1} } = -(q+1) \hbox{res}\, { QA\Omega \over P^{q+2} }. $$ To show that the derivative of the period map is nonzero, it suffices to exhibit an $A$ and a $Q$ which are nonzero in $R$ and such that the product $QA$ is also nonzero. Here we implicitly use the identification $ T \cong R^d $ of tangent vectors to the moduli space with the component of the Jacobian ring in degree $d$. Thus the natural components of the differential of the period map, $$ T \map \Hom(H^{p,q}(X),H^{p-1,q+1}(X)), $$ can be identified with the multiplication homomorphism $$ R^d \map \Hom(R^a,R^{a+d}), $$ where $a$ is the degree of the numerator polynomial used in the residues of the forms (\xref{ratdiff}). All of these results, discovered first by Griffiths in the case of hypersurfaces, hold for weighted hypersurfaces by the results described in \cite{Dolgachev} and \cite{Tu}. Consider now a double cover $Y$ of a hypersurface $X$ of even degree $d$. If $X$ is defined by $P(x_0 \commadots x_n) = 0$ then $Y$ is defined by $y^2 +P(x_0 \commadots x_n) = 0$, where $y$ has weight $d/2$ and where the $x$'s have weight one. This last equation is homogeneous of degree $d$ with respect to the given weighting, and $\Omega$ has weight $d/2 + n + 2$. Thus $\nu(A,y^2 + P,q)$ is of weight zero if $a(q) = (q + 1/2)d - (n+2)$. Since $y$ is in the Jacobian ideal, we may choose $A$ to be a polynomial in the $x$'s, and we may consider it modulo the Jacobian ideal of $P$. Thus the classical considerations of the residue calculus apply. If we choose $a(q)$ maximal subject to the constraints $p > q$ and $a \ge 0$ then $$ q = \left\{ { n + 1 \over 2 } \right\}, $$ where $\set{ x }$ is the greatest integer {\sl strictly less} than $x$. Both conditions are satisfied for $d \ge 4$ except that for $n = 1$ we require $d \ge 6$. Thus we have excluded the case $(d,n) = (4,1)$ in which the resulting double cover is rational and the period map is constant. Now let $A$ be a polynomial of degree $a$ which is nonzero modulo the Jacobian ideal. We must exhibit a polynomial $Q$ of degree $d$ such that $AQ$ nonzero modulo $J$. By Macaulay's theorem there is a polynomial $B$ such that $AB$ is congruent to a generator of $R^t$, hence satisfies $AB \not\equiv 0 \hbox{ mod $J$}$. Write $B$ as a linear combination of monomials $B_i$ and observe that there is an $i$ such that $AB_i \not\equiv 0$. If $B_i$ is of degree at least $d$, we can factor it as $QB_i'$ with $Q$ of degree $d$. Since $AQB_i' \not\equiv 0$, $AQ \not\equiv 0$, as required. The condition that $B$ have degree at least $d$ reads $a + d \le t$. Using the formulas \eqrefer{topJ} for $t$ and the optimal choice for $a$, we see that this inequality is satisfied for the range of $d$ and $n$ considered. This computation completes the proof of the main theorem in the case $d$ even, $d \ge 4$, except for the case $(d,n) = (4,1)$. \section{Rational differentials for higher cyclic covers } \secref{cycliccoversection} To complete the proof of the main theorem we must consider arbitrary cyclic covers of $\P^{n+1}$ branched along a smooth hypersurface of degree $d$. Since the fundamental group of the complement of $X$ is cyclic of order $d$, the number of sheets $k$ must be a divisor of $d$. As mentioned in the outline of the proof, there is an automorphism $\sigma$ of order $k$ which operates on the universal family ${\bf Y}$ of such covers. Consequently the local system $\H$ of vanishing cohomology (cf. \S \xref{zardensitysection}) splits over $\C$ into eigensystems $\H(\mu)$, where $\mu \ne 1$ is a $k$-th root of unity. Therefore the monodromy representation, which we now denote by $\rho$, splits as a sum of representations $\rho_\mu$ with values in the groups $\widetilde G(\mu)$ introduced in \eqrefer{tildeGdecomp}. As noted there we can view $\rho_\mu$ as taking values in a group of linear automorphisms of $H(\mu)$. This group is unitary for the hermitian form $h(x,y) = i^{n+1}(x,\bar y)$ if $\mu$ is non-real, and that is the case that we will consider here. Although the decomposition of $\H$ is over the complex numbers, important Hodge-theoretic data survive. The hermitian form $h(x,y)$ is nondegenerate and there is an induced Hodge decomposition, although $h^{p,q}(\mu) = h^{q,p}(\mu)$ may not hold. However, Griffiths' infinitesimal period relation, $$ { d \over dt } F^p(\mu) \subset F^{p-1}(\mu) $$ remains true. Thus each $\H(\mu)$ is a {\sl complex variation of Hodge structure}, c.f. \cite{DeMo}, \cite{SimpsonHiggs}. The associated period domains are homogeneous for the groups $\widetilde G(\mu)$. To extend the arguments given above to the unitary representations $\rho_\mu$ we must extend Deligne's density theorem to this case. The essential point is that the monodromy groups $\Gamma(\mu)$ are generated not by Picard-Lefschetz transformations, but by their unitary analogue, which is a {\sl complex reflection} \cite{Pham}, \cite{Givental}, \cite{Mostow}. These are linear maps of the form $$ T(x) = x \pm (\lambda - 1)h(x,\delta)\delta, $$ where $h$ is the hermitian inner product defined above, $h(\delta,\delta) = \pm 1$, where $\pm$ is the same sign as that of $h(\delta ,\delta )$, and where $\lambda \ne 1$ is a root of unity. The vector $\delta$ is an eigenvector of $T$ with eigenvalue $\lambda$ and $T$ acts by the identity on the hyperplane perpendicular to $\delta$. It turns out that the eigenvalue $\lambda$ of $T$ is, up to a fixed sign that depends only on the dimension of $Y$, equal to the eigenvalue $\mu$ of $\sigma$. In section \xref{unitarydensitysection} we will prove an analogue of Deligne's theorem \eqrefer{delignedensity} for groups of complex reflections. It gives the usual dichotomy: either the monodromy group is finite, or it is Zariski-dense. In section \xref{complexreflectionsection} we will show that the monodromy groups $\Gamma(\mu)$ are indeed generated by complex reflections. It remains to show that the derivative of the period map for the complex variations of Hodge structure $\H(\mu)$ are nonzero given appropriate conditions on $d$, $k$, $n$, and $\mu$. For the computation fix $\zeta = e^{2\pi i/k}$ as a primitive $k$-th {\sl root of unity} and let the cyclic action on the universal family \eqrefer{universalcyclic} be given by $y\circ \sigma = \zeta y$. Then the ``volume form'' $\Omega(x,y)$ is an eigenvector with eigenvalue $\zeta$ and the rational differential $$ { y^{i-1} A(x) \Omega(x,y) \over ( y^k + P(x) )^{q+1} } \eqn \eqref{iratdiff} $$ has eigenvalue $\mu = \zeta^i$, as does its residue. Thus we will sometimes write $\H(i)$ for $\H(\zeta^i)$ and will use the corresponding notations $\widetilde G(i)$, $\tilde\rho_i$, etc. Residues with numerator $y^{i-1}A(x)$ and denominator $(y^k + P(x))^{q+1}$ span the spaces $H^{p,q}_0(i)$, where $i$ ranges from 1 to $k-1$. Moreover, the corresponding space of numerator polynomials, taken modulo the Jacobian ideal of $P$, is isomorphic via the residue map to $H^{p,q}_0(i)$. Since $P$ varies by addition of a polynomial in the $x$'s, the standard unweighted theory applies to computation of the derivative map. Let us illustrate the relevant techniques by computing the Hodge numbers and period map for triple covers of $\P^3$ branched along a smooth cubic surface. (This period map is studied in more detail in \cite{ACT}.) A triple cover of the kind considered is a cubic hypersurface in $\P^4$, and the usual computations with rational differentials show that $h^{3,0} = 0$, $h^{2,1} = 5$. The eigenspace $H^{2,1}(i)$ is spanned by residues of differentials with numerator $ A(x)\Omega(x,y) $ and denominator $(y^3 + P(x))^2$. Since the degree of $\Omega(x_0,x_1,x_2,x_3, y)$ is 5, $A$ is must be linear in the variables $x_i$. Thus $h^{2,1}(1) = 4$. The space $H^{1,2}(1)$ is spanned by residues of differentials with numerator $ A(x)\Omega(x,y) $ and denominator $(y^3 + P(x))^3$. Thus the numerator is of degree four, but must be viewed modulo the Jacobian ideal. For dimension counts it is enough to consider the Fermat cubic, whose Jacobian ideal is generated by squares of variables. The only square-free quartic in four variables is $x_0x_1x_2x_3$, so $h^{1,2}(1) = 1$. Similar computations show that the remaining Hodge numbers for $H^3(1)$ are zero and yield in addition the numbers for $H^3(2)$. One can also argue that $H^3(1)\oplus H^3(2)$ is defined over $\R$, since the eigenvalues are conjugate. A Hodge structure defined over $\R$ satisfies $h^{p,q} = h^{q,p}$. From this one deduces that $h^{2,1}(2) = 1$, $h^{1,2}(2) = 4$. Since there is just one conjugate pair of eigenvalues of $\sigma$, there is just one component in the decomposition \eqrefer{tildeGdecomp}, $\widetilde G = \widetilde G(\zeta)$, and this group is isomorphic to $U(1,4)$. Since the coefficients of the monodromy matrices lie in the ring $\Z[\zeta]$, where $\zeta$ is a primitive cube root of unity, the representation $\tilde\rho$ takes values in a discrete subgroup of $\widetilde G$. Therefore the complex variation of Hodge structures define period mappings $$ p : U_{3,2} \map B_4/\Gamma', $$ where $B_4$ is the unit ball in complex 4-space and $\Gamma'$ a discrete group acting on it. To show that the period map $p_i$ is nonconstant it suffices to show that its differential is nonzero at a single point. We do this for the Fermat variety. A basis for $H^{2,1}(1)$ is given by the linear forms $x_i$, and a basis for $H^{1,2}$ is given by their product $x_0x_1x_2x_3$. Let $m_i$ be the product of all the $x_k$ except $x_i$. These forms constitute a basis for the tangent space to moduli. Since $m_ix_i = x_0x_1x_2x_3$, multiplication by $m_i$ defines a nonzero homomorphism from $H^{2,1}(1)$ to $H^{1,2}(1)$. Thus the differential of the period map is nonzero at the Fermat. In fact it is of rank four, since the homomorphisms defined by the $m_i$ are linearly independent. Similar considerations show that the period map for $\H(2)$ is of rank four. The relevant bases are $\set{ y }$ for $H^{2,1}(2)$ and $\set{ ym_0, ym_1, ym_2, ym_3 }$ for $H^{1,2}(2)$. For the general case it will be enough to establish the following. \proclaim{Proposition.} Let ${\bf Y}$ be the universal family of $d$-sheeted covers of $\P^{n+1}$ branched over smooth hypersurfaces of degree $d$. The derivative of the period map for $\H^{n+1}(1)$ is nontrivial if $n \ge 2$ and $d \ge 3$ or if $n = 1$ and $d \ge 4$. \endproclaim \procref{PropositionDerivNonTrivialNgeTwo} \noindent{\bf Proof:\ } Elements of $H^{p,q}(1)$ with $p+q = n+1$ are given by rational differential forms with numerator $ A(x) \Omega(x,y) $ and denominator $( y^d + P(x) )^{q+1}$. The numerator must have degree $ a = (q + 1)d - (n+3) $. As before choose $q$ so that $a$ is maximized subject to the constraints $p > q$ and $a \ge 0$. Then $q = \set{ {n / 2} + { 1 / d } }$. If $n \ge 2$ and $d \ge 3$ or if $n = 1$ and $d \ge 4$, then $a \ge 0$. Thus numerator polynomials $A(x)$ which are nonzero modulo the Jacobian ideal exist. One establishes the existence of a polynomial $Q(x)$ of degree $d$ such that $QA$ is nonzero modulo the Jacobian ideal using the same argument as in the case of double covers. A different component of the period map is required if the branch locus is a finite set of points, which is the case for the braid group of $\P^1$: \proclaim{Proposition.} For $n=0$ the period map for $\H^1(i)$ is non-constant if $d \ge 4$ and $i \ge 2$. \endproclaim \noindent{\bf Proof:\ } An element of $H^{1,0}(i)$ is the residue of a rational differential with numerator $ y^{i-1}A(x_0,x_1)\Omega $ and denominator $ y^d + P(x_0,x_1) $. The degree of $A$ is $a = d - 2 - i$. The top degree for the Jacobian ideal is $2d-4$. Thus we require $a + d \le 2d - 4$, which is satisfied if $i \ge 2$. Since $a \ge 0$, one must also require $d \ge 4$. We observe that the local systems which occur as constituents for $k$-sheeted covers, where $k$ divides $d$, also occur as constituents of $d$-sheeted covers. \proclaim{Remark.} Let $\H({k,\mu})$ be the complex variation of Hodge structure associated to a $k$-sheeted cyclic cover of $\P^{n+1}$ branched along a hypersurface of degree $d$, belonging to the eigenvalue $\mu$, where $k$ is a divisor of $d$. Then $\H({k,\mu})$ is isomorphic to $\H({d,\mu})$. \endproclaim \noindent{\bf Proof:\ } Consider the substitution $y = z^{d/k}$ which effects the transformation $$ { y^i A(x) \Omega(x,y) \over (y^k + P(x) )^{q+1} } \mapsto { (d / k) }{ z^{(i+1)(d/k) -1} A(x) \Omega(x,z) \over ( z^d + P(x) )^{q+1} } . $$ These differentials are eigenvectors with the same eigenvalue. The map which sends residues of the first kind of rational differential to residues of the second defines the required isomorphism. \section{Complex Reflections} \secref{complexreflectionsection} We now review some known facts on how complex reflections arise for degenerations of cyclic covers. When the branch locus acquires a node, the local equation is $$ y^k + x_1^2 + \cdots + x_{n+1}^2 = t, \eqn \eqref{kdoublept} $$ which is a special case of the situation studied by Pham in \cite{Pham}, where the left-hand side is a sum of powers. Our discussion is based on Chapter 9 of \cite{Milnor} and Chapter 2 of \cite{Arnold}. Consider first the case $y^k = t$. It is a family of zero-dimensional varieties $\set{ \xi_1(t) \commadots \xi_k(t)}$ whose vanishing cycles are successive differences of roots, $$ \xi_1 - \xi_2, \ \ldots,\ \xi_{k-1} - \xi_k, \eqn \eqref{stdvanishingbasis} $$ and whose monodromy is given by cyclically shifting indices to the right: $$ T( \xi_i - \xi_{i+1} ) = \xi_{i+1} - \xi_{i+2}, $$ where $i$ is taken modulo $k$. Thus $T$ acts on the $(k-1)$-dimensional space of vanishing cycles as a transformation of order $k$. Over the complex numbers it is diagonalizable, and the eigenvalues are the $k$-th roots of unity $\mu \ne 1$. Note that $T = \sigma_0$ where $\sigma_0$ is the generator for the automorphism group of the cyclic cover $y^k = t$ given by $y\map \zeta y$, where $\zeta = e^{2\pi i/k}$ is our chosen primitive $k$-th root of unity. The intersection product $B$ defines a possibly degenerate bilinear form on the space of vanishing cycles. For the singularity $y^k = t$ it is $(\xi_i,\xi_j) = \delta_{ij}$, so relative to the basis \eqrefer{stdvanishingbasis} it is the negative of the matrix for the Dynkin diagram $A_{k-1}$ --- the positive-definite matrix with two's along the diagonal, one's immediately above and below the diagonal, and zeroes elsewhere. Now suppose that $f(x) = t$ and $g(y) = t$ are families which acquire an isolated singularity at $t = 0$. Then $f(x) + g(y) = t$ is a family of the same kind; we denote it by $f \oplus g$. The theorem of Sebastiani and Thom \cite{ST}, or \cite{Arnold}, cf. Theorem 2.1.3, asserts that vanishing cycles for the sum of two singularities are given as the join of vanishing cycles for $f$ and $g$. Thus, if $a$ and $b$ are vanishing cycles of dimensions $m$ and $n$, then the join $a*b$ is a vanishing cycle of dimension $m+n+1$, and, moreover, the monodromy acts by $ T(a * b) = T(a)*T(b). $ {}From an algebraic standpoint the join is a tensor product, so one can write $V(f\oplus g) = V(f)\otimes V(g)$ where $V(f)$ is the space of vanishing cycles for $f$, and one can write the monodromy operator as $ T_{f \oplus g} = T_f\otimes T_g. $ The {\sl suspension} of a singularity $f(x) = t$ is by definition the singularity $y^2 + f(x) = t$ obtained by adding a single square. If $a$ is a vanishing cycle for $f$ then $(y_0 - y_1)\otimes a$ is a vanishing cycle for the suspension, and the suspended monodromy is given by $$ T( (y_0 - y_1)\otimes a ) = - (y_0 - y_1)\otimes T(a) . $$ In particular, the local monodromy of a singularity and its double suspension are isomorphic. The intersection matrix $B'$ of a suspended singularity (relative to the same canonical basis) is a function of the intersection matrix $B$ for the given singularity, cf. Theorem 2.14 of \cite{Arnold}. When the bilinear form for $B$ is symmetric, the rule for producing $B'$ from $B$ is: make the diagonal entries zero and change the sign of the above-diagonal entries. When $B'$ has an even number of rows of columns, the determinant is one, and when the number of rows and columns is odd, it is zero. Thus the intersection matrix for $x^2 + y^k = t$ is nondegenerate if and only if $k$ is odd. In addition, the intersection matrix of a double suspension is the negative of the given matrix. Thus the matrix of any suspension of $y^k = t$ is determined. It is nondegenerate if the dimension of the cyclic cover \eqrefer{kdoublept} is even or if the dimension is odd and $k$ is also odd. Otherwise it is degenerate. It follows from our discussion that the space of vanishing cycles $V$ for the singularity \eqrefer{kdoublept} is $(k-1)$-dimensional and that the local monodromy transformation is $ T = \sigma_0\otimes(-1)\otimes\cdots\otimes(-1) $ where $\sigma_0$ is the covering automorphism $y\map \zeta y$ for $y^k = t$. Thus $T$ is a cyclic transformation of order $k$ or $2k$, depending on whether the dimension of the cyclic cover is even or odd. In any case, $T$ is diagonalizable with eigenvectors $\eta_i$ and eigenvalues $\lambda_i$, where $\lambda_i = \pm \mu_i$ with $\mu_i = \zeta^i$ where $\zeta$ is our fixed primitive $k$-th root of unity and $i = 1,\cdots , k-1$. Note that the cyclic automorphism $\sigma$ of the universal family \eqrefer{universalcyclic}, given by $y\mapsto\zeta y$ acts as $\sigma_0\otimes(+1)\otimes\cdots\otimes(+1)$ on the vanishing homology of \eqrefer{kdoublept}. Thus the eigenspaces of $\sigma$ and $T$ coincide, and their respective eigenvalues differ by the fixed sign $(-1)^{n+1}$. Since the eigenvalues $\mu_i$ are distinct, the eigenvectors $\eta_i$ are orthogonal with respect to the hermitian form. Thus $h(\eta_i,\eta_i) \ne 0$. Moreover the sign of $h(\eta_i,\eta_i)$ depends only on the index $i$, globally determined on \eqrefer{universalcyclic}, independently of the particular smooth point on the discriminant locus whose choice is implicit in \eqrefer{kdoublept}. We conclude that on the space of vanishing cycles, $$ T(x) = \sum_{i = 1}^{k-1} \lambda_i{h(x,\eta_i) \over h(\eta_i,\eta_i)}\eta_i, \eqn \eqref{TVcxreflectionformula} $$ where $\lambda_i = (-1)^{n+1} \mu_i$. Now consider a cycle $x$ in $H^{n+1}(Y_{\tilde o})$, and {\sl suppose that} $k$ {\sl is odd}. Then the intersection form on the space $V$ of local vanishing cycles for the degeneration \eqrefer{kdoublept} is {\sl nondegenerate}. Consequently $H^{n+1}(Y_{\tilde o})$ splits orthogonally as $V \oplus V\perp$. The action on $H^{n+1}(Y_{\tilde o})$ of the monodromy transformation $T$ for the meridian corresponding to the degeneration \eqrefer{kdoublept} is given by \eqrefer{TVcxreflectionformula} on $V$ and by the identity on $V\perp$. Thus it is given for arbitrary $x$ by the formula $$ T(x) = x + \sum_{i=1}^{k-1} (\lambda_i-1){h(x,\eta_i) \over h(\eta_i,\eta_i)}\eta_i . \eqn \eqref{Tcxreflectionformula} $$ Finally, for each $i = 1,\cdots ,k-1$ we can normalize the eigenvector $\eta_i$ to an eigenvector $\delta_i$ satisfying $h(\delta_i , \delta_i ) = \epsilon_i = \pm 1$. To summarize, we have proved the following: \proclaim{Proposition.} Consider the family \eqrefer{universalcyclic} of $k$-fold cyclic covers of $\P^{n+1}$ branched over a smooth hypersurface of degree $d$, where both $k$ and $d$ are odd. Let $T$ be the monodromy corresponding to a generic degeneration of the branch locus, as in \eqrefer{kdoublept}. Then $T$ acts on the $i$-th eigenspace of the cyclic automorphism $\sigma$ (defined by $y\mapsto\zeta y$ in \eqrefer{universalcyclic}) by a complex reflection with eigenvalue $\lambda_i = (-1)^{n+1} \zeta^i$. Thus $$ T(x) = x + \epsilon_i (\lambda_i - 1) h(x,\delta_i)\delta_i $$ holds for all $x\in\H (i)$. \endproclaim \proclaim{Remark.} \rm In remark \xref{remarkMeridiansInfiniteOrder}.a we observed that the meridians of $\Phi_{d,n}$ are of infinite order for $n$ odd and for $n$ even, $d \ge 4$ even. Consider now the case in which $n$ is even and $d$ is odd, let $\zeta = \exp(2\pi i/d)$, and let $\bar\rho'$ be the corresponding representation, in which meridians of $\widetilde \Delta$ correspond to complex reflections of order $2d$. These complex reflections and their powers different from the identity are non-central if the $\zeta$ eigenspace has dimension at least two, which is always the case for $d \ge 3$, $n \ge 2$. Thus $\bar\rho'(\gamma)$ has order $2d$. By this simple argument we conclude that in the stated range of $(n,d)$, meridians always have order greater than two. However, our argument does not give the stronger result \xref{remarkMeridiansInfiniteOrder}.b asserted by Kontsevich. \endproclaim \section{Density of unitary monodromy groups} \secref{unitarydensitysection} We now show how the argument Deligne used in \cite{DeWeTwo}, section 4.4, to prove Theorem \xref{delignedensity} can be adapted to establish a density theorem for groups generated by complex reflections on a space $\C(p,q)$ endowed with a hermitian form $h$ of signature $(p,q)$. If $A$ is a subset of $\C(p,q)$ or of $U(p,q)$, we use $PA$ to denote its projection in $\P(\C(p,q))$ or $PU(p,q)$. \proclaim{Theorem.} Let $\epsilon = \pm 1$ be fixed, and let $\Delta$ be a set of vectors in a hermitian space $\C(p,q)$ which lie in the unit quadric $h(\delta,\delta) = \epsilon$. Fix a root of unity $\lambda \ne \pm 1$ and let $\Gamma$ be the subgroup of $U(p,q)$ generated by the complex reflections $s_\delta(x) = x + \epsilon ( \lambda - 1 )h(x,\delta)\delta$ for all $\delta$ in $\Delta$. Suppose that $p+q >1$, that $\Delta$ consists of a single $\Gamma$-orbit, and that $\Delta$ spans $\C(p,q)$. Then either $\Gamma$ is finite or $P\Gamma$ Zariski-dense in $PU(p,q)$. \endproclaim \procref{udensitytheo} Let $\bar\Gamma$ be the Zariski closure of a subgroup $\Gamma$ of $U(p,q)$ which contains the $\lambda$-reflections for all vectors $\delta$ in a set $\Delta$. Then $\bar\Gamma$ also contains the $\lambda$-reflections for the set $R = \bar\Gamma\Delta$. Indeed, if $g$ is an element of $\bar\Gamma$, then $$ g^{-1}s_\delta g = s_{g^{-1}(\delta)}. \eqn\eqref{reflectionconjugacy} $$ Thus it is enough to establish the following result in order to prove our density theorem: \proclaim{Theorem.} Let $\epsilon = \pm 1$ be fixed, and let $R$ be a set of vectors in a hermitian space $\C(p,q)$ which lie in the unit quadric $h(\delta,\delta) = \epsilon$. Fix a root of unity $\lambda \ne \pm 1$ and let $M$ be the smallest algebraic subgroup of $U(p,q)$ which contains the complex reflections $s_\delta(x) = x + \epsilon ( \lambda - 1 )h(x,\delta)\delta$ for all $\delta$ in $R$. Suppose that $p+q >1$, that $R$ consists of a single $M$-orbit, and that $R$ spans $\C(p,q)$. Then either $M$ is finite or $PM = PU(p,q)$. \endproclaim \procref{udensityprop} We begin with a special case of the theorem for groups generated by a pair of complex reflections. \proclaim{Lemma.} Let $\lambda \ne \pm 1$ be a root of unity, and let $U$ be the unitary group of a nondegenerate hermitian form on $\C^2$. Let $\delta_1$ and $\delta_2$ be independent vectors with nonzero inner product, and let $\Gamma$ be the group generated by complex reflections with common eigenvalue $\lambda$. Then either $\Gamma$ is finite or its image in the projective unitary group is Zariski-dense. In the positive-definite case $\Gamma$ is finite if and only if the inner products $(\delta_1,\delta_2)$ lie in a fixed finite set $S$ which depends only on $\lambda$ and $h$. In the indefinite case $\Gamma$ is never finite. \endproclaim \procref{Ulemma} We treat the definite case first. To begin, note that the group $U$ acts on the Riemann sphere $\P^1$ via the natural map $U \map PU$, where $PU$ is the projectivized unitary group. Let $PR$ be the image of $R \subset \C^2$ in $\P^1$. Since $\lambda$ is a root of unity, the projection $P\Gamma$ is a finite group if and only if $\Gamma$ is. The finite subgroups of rotations of the sphere are well known. There are two infinite series: the cyclic groups, where the vectors $\delta$ are all proportional, and the dihedral groups where $\lambda = -1$. There are three additional groups, given by the symmetries of the five platonic solids, and $S$ is the set of possible values of $h(\delta_1,\delta_2)$ that can arise for these three groups. We suppose that $(\delta_1,\delta_2)$ lies outside $S$, so that $P\Gamma$ is infinite. Then its Zariski closure $PM$ is either $PU$ or a group whose identity component is a circle. In this case $PR$ contains a great circle $\alpha$. However, $PR$ is stable under the action of $PM$, hence under the rotations corresponding to axes in $PR$. Since $\lambda \ne \pm 1$, the orbit $PR$ contains additional great circles which meet $\alpha$ in an angle $0 < \phi \le \pi/2$. The union of these, one for each point of the given circle, forms a band about the equator, hence has nonempty interior. Such a set is Zariski-dense in the Riemann sphere viewed as a real algebraic variety. Since $PR$ is a closed real algebraic set, $PR = S^2$. Since $PR \cong PM/H$, where $H$ is the isotropy group of a point on the sphere, $PM = PU$. In the case of an indefinite hermitian form, the group $U = U(1,1)$, acts on the hyperbolic plane via the projection to $PU$, and $P\Gamma$ is a group generated by a pair of elliptic elements of equal order but with distinct fixed points. One elliptic element moves the fixed point of the other, and so their commutator $\gamma$ is hyperbolic (c.f. Theorem 7.39.2 of \cite{Beardon}). The Zariski closure of the cyclic group $\set{\gamma^n}$ is a one-parameter subgroup of $PU$. Consequently the orbit $PR$ contains a geodesic $\alpha$ through one of the elliptic fixed points. By \eqrefer{reflectionconjugacy} the other points of $\alpha$ are fixed points of other elliptic transformations in $PM$. Now the orbit $PR$ contains the image of $\alpha$ under each of these transformations, and so $PR$ contains an open set of the hyperbolic plane. This implies that either $PM = PU$ or $PM$ is contained in a parabolic subgroup. Since $PM$ contains non-trivial elliptic elements that last possibility cannot occur, and so $PM = PU$. Next we show that if the set $R$ which defines the reflections is large, then so is the group containing those reflections. \proclaim{Lemma.} Fix a root of unity $\lambda\ne\pm 1$ and $\epsilon = \pm 1$. Let $R$ be a semi-algebraic subset of the unit quadric $h(\delta,\delta) = \epsilon$. Let $M$ be the smallest algebraic subgroup of $U(p,q)$ containing the complex reflections $s_\delta(x) = x + \epsilon (\lambda - 1) h(x,\delta)\delta$, $\delta\in R$. If $p+q >1$ and if $PR$ is Zariski-dense in $\P(\C(p,q))$, then $M =PU(p,q)$. \endproclaim The proof is by induction on $n = p+q$. For $n = 2$ the result follows from the proof of lemma \xref{Ulemma}. Let $n>2$ and assume $p\le q$. Then $q\ge 2$. Fix a codimension two subspace of $\C (p,q)$ of signature $(p,q-2)$ and let $W_t$ be the pencil of hyperplanes of $\C (p,q)$ containing this codimension two subspace. Then the restriction of $h$ to each $W_t$ is a non-degenerate form of signature $(p,q-1)$. Consider a subgroup $M$ of $U(p,q)$ which satisfies the hypotheses of the lemma, and let $R_t = R\cap W_t$. Since $PR$, respectively $PR_t$ is semi-algebraic in $\P (\C (p,q))$, respectively in $PW_t$, it is Zariski dense if and only if it has non-empty interior in the analytic topology. Thus $R$ has non-empty interior in $\P(\C(p,q))$, and so for dimension reasons $PR_t$ has non-empty interior in $PW_t$ for generic $t$. Thus $PR_t$ is Zariski dense in $PW_t$ for generic $t$. Fix one such value of $t$, let $W = W_t$ and let $M'(R\cap W)\subset M$ denote the smallest algebraic subgroup of $M$ containing $R\cap W$. Let $M(R\cap W)$ denote the set of restrictions of elements of $M'(R\cap W)$ to $W$. Then $R\cap W$ and $M(R\cap W)$ satisfy the induction hypothesis, thus $PM(R\cap W) = PU(W)$. Now the orthogonal complement of $W$ is a Zariski closed set, as is $W \cup W\perp$. Since $R$ is Zariski-dense there is a $\delta$ in $R - W$ and a $\delta'$ in $W$ such that $h(\delta,\delta') \ne 0$. Consider the function $f_\delta(x) = h(x,\delta')$. If it is constant on the Zariski closure $C$ of $R\cap W$, then the derivative $df_\delta$ vanishes on $C$. Therefore $C$ lies in the intersection of the hyperplane $df(x) = 0$ with $W$, which is a proper algebraic subset of $W$. Consequently $R \cap W$ is not Zariski-dense, a contradiction. Thus $f_\delta$ is nonconstant and so we can choose $\delta$ in $R \cap W$ such that $h(\delta',\delta)$ lies outside the fixed set $S$. Then lemma \xref{Ulemma} implies that the unitary group of the plane $F$ spanned by $\delta$ and $\delta'$ is contained in $M$. But $U(W)$ and $U(F)$ generate $U(p,q)$ and the proof of the lemma is complete. To complete the proof of Theorem (\xref{udensityprop}) we must show that either $R$ is sufficiently large or that $M$ is finite. Observe that since $R$ is an $M$-orbit, it is a semi-algebraic set. Let $W$ be a subspace of $\C(p,q)$ which is maximal with respect to the property ``$W \cap R$ is Zariski-dense in the unit quadric of $W$.'' Our aim is to show that either $W = \C(p,q)$ or that $M$ is finite. Consider first the case $W = 0$. Then the inner products $h(\delta,\delta')$ for any pair of elements in $R$ lie in the fixed finite set $S$ of lemma \xref{Ulemma}. Now let $\delta_1 \commadots \delta_n$ be a basis of $\C(p,q)$ whose elements are chosen from $R$. Then the inner products $h(\delta,\delta_i)$ lie in $S$ for all $\delta$ and $i$. Consequently $R$ is a finite set and $M$, which is faithfully represented as a group of permutations on $R$, is finite as well. Henceforth we assume that $W$ is nonzero. If it is not maximal there is a vector $\delta$ in $R - W$ and we may consider the function $f_\delta(x) = h(x,\delta)$ on the set $R \cap W$. If $f_\delta$ is identically zero for all $\delta$ in $R - W$, then $R \subset W \cup W\perp$. Therefore $\C(p,q) = W + W\perp$, from which one concludes that $W = W \oplus W\perp$ and so $M$ is a subgroup of $U(W)\times U(W\perp)$. But $R$ consists of a single $M$-orbit and contains a point of $W$, which implies that $R \subset W$, a contradiction. We can now assume that there is a $\delta \in R - W$ such that the function $f_\delta$ is not identically zero. If one of these functions is not locally constant, then it must take values outside the set $S$. Then the inner product $(x,\delta)$ lies outside $S$ for an open dense set of $x$ in $R \cap W$. For each such $x$, $R$ is dense in the span of $x$ and $\delta$. We conclude that $R$ is dense in $W + \C\delta$. Thus $W$ is not maximal, a contradiction. At this point we are reduced to the case in which all the functions $f_\delta$ are locally constant, with at least one which is not identically zero. To say that $f_\delta$ is locally constant on a dense subset of the unit quadric in $W$ is to say that its derivative is zero on that quadric. Equivalently, tangent spaces to the quadric are contained in the kernel of $df_\delta$, that is, in the hyperplane $\delta\perp$. But if all tangent spaces to the quadric are contained in that hyperplane, then so is the quadric itself. Then the function in question is identically zero, contrary to hypothesis. The proof is now complete. To apply the density theorem we need to show that the ``complex vanishing cycles'' contain a basis for the vanishing cohomology and form a single orbit. These cycles are by definition the eigencomponents of ordinary vanishing cycles. Consider now a generalized Picard-Lefschetz transformation given by \eqrefer{Tcxreflectionformula}. It can be rewritten as $$ \rho(\gamma)(x) = x + \sum \epsilon_i(\lambda_i-1) h(x,\delta_i) \delta_i, $$ where the $\delta_i$ are complex vanishing cycles and the $\lambda_i$ are suitable complex numbers. Let $$ \rho(\gamma')(x) = x + \sum\epsilon_i (\lambda_i-1) h(x,\delta'_i) \delta'_i $$ be another generalized Picard-Lefschetz tranformation. If $\gamma' = \kappa^{-1}\gamma\kappa$ then the two preceding equations yield $$ \sum \epsilon_i(\lambda_i - 1) h(x,\delta'_i) \delta'_i = \sum \epsilon_i(\lambda_i - 1 ) h(\kappa.x,\delta_i) \kappa^{-1}.\delta_i , $$ where $\kappa.x$ stands for $\rho(\kappa)(x)$. Comparing eigencomponents on each side we find $$ \delta'_i = \kappa^{-1}.\delta_i, $$ as required. By the same argument as used in \S \xref{zardensitysection}, one sees that the complex vanishing cycles span $H(i)$. \section{Bounds on the real and complex rank} \secref{rankbounds} In this section we derive lower bounds for the complex and real ranks of the groups $G_{d,n}$ of automorphisms of the primitive cohomology $H^n_o(X_{d,n},\R)$ where $X_{d,n}$ is a hypersurface of degree $d$ and dimension $n$. Recall that for a field $k$, the $k$-rank is the dimension of the largest subgroup that can be diagonalized over $k$. These bounds complete the outline of proof. We also show that all the eigenspaces of the cyclic automorphism $\sigma$ have the same dimension. The main result is the following: \proclaim{Lemma}. The complex rank of $G_{d,n}$ is at least five for $d \ge 3$, $n \ge 1$, with the exception of $(d,n) = (3,1)$, for which it is one, and $(d,n) = (4,1), (3,2)$ for which it is three. Under the same conditions the real rank is at least two with the exception of the cases $(d,n) = (3,1),\ (3,2)$ for which the real ranks are one and zero, respectively. \endproclaim \procref{rankboundslemma} To prove the first assertion we note that the complex rank is given by $\rank_\C G_{d,n} = [ B_{d,n} / 2 ]$ where $[x]$ is the greatest integer in $x$ and where $ B_{d,n} = \dim H^n_o(X_{d,n})$ is the primitive middle Betti number. To compute it we compute the Euler characteristic $\chi_{d,n}$ recursively using the fact that a $d$-fold cyclic cover of $\P^{n}$ branched along a hypersurface of degree $d$ is a hypersurface of degree $d$ in $\P^{n+1}$. Thus, mimicking the proof of Hurwitz's formula for Riemann surfaces, we have $$ \chi_{d,n} = d\,\chi(\P^n - B) + \chi(B) = d(n+1) + (1-d)\chi_{d,n-1} . $$ Since $\chi_{d,0} = d$, the Euler characteristics of all hypersurfaces are determined. Rewriting this recursion relation in terms of the $n$-th primitive Betti number we obtain $$ B_{d,n} = (d-1)\left(B_{d,n-1} + (-1)^n\right), \eqn \eqref{bettinumberrecursion} $$ {}From it we deduce an expression in closed form: $$ B_{d,n} = (d-1)^n \,(d-2) + { ( d-1 )^n - (-1)^n \over d } + (-1)^n . \eqn \eqref{bettinumberformul} $$ {}The preceding two formulas imply that $B_{d,n}$ is an increasing function of $n$ and of $d$. Now assume $d \ge 3$, $n \ge 1$. Then $d+n\ge 4$. If $d+n \le 6$, then $(d,n) = (3,3), (4,2), (5,1)$ and $B_{3,3} = 10 , B_{4,2} = 21, B_{5,1} = 12$. Thus $B_{d,n} \ge 10$ except when $d+n = 4$ or $5$. These are the cases $(d,n) = (3,1), (3,2), (4,1)$ where $B_{d,n} = 2,6,6$ respectively. The inequalities on the complex rank are now established. Let us now turn to the proof of the second assertion of the lemma. For $n$ odd the group $G_{d,n}$ is a real symplectic group. Its real and complex ranks are the same, and so the bound follows from the first assertion. For $n$ even the group $G_{d,n}$ is the orthogonal group of the cup product on the primitive cohomology. This bilinear form has signature $(r,s)$, and the real rank of $G$ is the minimum of $r$ and $s$. The signature is computed from the Hodge decomposition: $r$, the number of positive eigenvalues, is the sum of the $h^{p,q}$ for $p$ even, while $s$ is the sum for $p$ odd. According to the first inequality of lemma \xref{hodgeinequalities}, the Hodge numbers $h^{p,q}(d,n)$ of $X_{d,n}$ satisfy $h^{p,q}(d+1,n) > h^{p,q}(d,n)$. Thus the real rank is an increasing function of the degree. Consequently it is enough to show that it is at least two for quartic surfaces and for cubic hypersurfaces of dimension four or more. For quartic hypersurfaces $h^{2,0} = 1$ and $h^{1,1} = 19$, so $(r,s) = (2,19)$. For cubic hypersurfaces there is a greatest integer $p \le n$ such that $h^{p,q} \ne 0$, where $p+q = n$. We will compute this ``first'' Hodge number and see that under the hypotheses of the lemma, $p > q$. Since $n$ is even, $h^{p,q}$ and $h^{q,p}$ have the same parity. Thus one of $r$, $s$ is at least two. According to the second inequality of lemma \xref{hodgeinequalities}, $h^{p-1,q+1}(d,n) > h^{p,q}(d,n)$ if $p > q$. Thus $h^{p-1,q+1}(d,n) > h^{p,q}(d,n) > 0$. We conclude that the other component of the signature, $s$ or $r$, must be at least two. For the Hodge numbers of cubic hypersurfaces of dimension $n = 3k + r$ where $r = 0$, 1, or 2, one uses the calculus of \cite{GriffPerRat} to show the following: (a) if $n \equiv 0 \hbox{ mod } 3$ then the first Hodge number is $h^{2k,k} = n+2$, (b) if $n \equiv 1 \hbox{ mod } 3$ then it is $h^{2k+1,k} = 1$, (c) if $n \equiv 2 \hbox{ mod } 3$ then it is $h^{2k+1,k+1} = (n+1)(n+2)/2$. When $k > 0$ these Hodge numbers satisfy $p > q$, and so the proof of the lemma is complete. \proclaim{Lemma} Let $h^{p,q}(d,n)$ be the dimension of $H^{p,q}_o(X_{d,n})$. Then the inequalities below hold: $$ \eqalign{ & h^{p,q}(d+1,n) > h^{p,q}(d,n) \cr & h^{p,q}(d,n) > h^{p+1,q-1}(d,n) \hbox{ if $p \ge q$} \cr } $$ \endproclaim \procref{hodgeinequalities} \noindent{\bf Proof:\ }{} It is enough to prove the inequalities when $X_{d,n}$ is the Fermat hypersurface defined by $F_d(x) = x_0^d + \cdots + x_{n+1}^d = 0$. Because of the symmetry $h^{p,q} = h^{q,p}$, it is also enough to prove the inequalities for $p \ge q$. To this end recall that $h^{p,q} = \dim R^a$, where $R$ is the Jacobian ring for $F_d$ and where $a = (q+1)d - (n+2)$ is the degree of the adjoint polynomial in the numerator of the expression $$ \hbox{res}\, { A \Omega \over F_d^{q+1} }. $$ Now there is a map $\mu: R^{a(q,d)}(F_d) \map R^{a(q,d+1)}(F_{d+1})$ defined by $\mu(P) = (x_0 \cdots x_q) P$. This makes sense because $q \le n$. We claim that that resulting map from $H^{p,q}(X_{d,n})$ to $H^{p,q}(X_{d+1,n})$ is injective but not surjective. To prove the claim, observe that the Jacobian ideal is generated by the powers $x_i^{d-1}$ and so has a vector space basis consisting of monomials $x^M$. The same is true of the quotient ring $R(F_d)$. Indeed, a basis is given by (the classes of) those monomials not divisible by $x_i^{d-1}$ for any $i$. Now consider a polynomial which represents an element of the kernel of $\mu$. It can be be chosen to be a linear combination of monomials $x^M$ which are not divisible by $x_i^{d-1}$ for any $i$. Its image is represented by a linear combination of monomials $(x_0 \cdots x_q)x^M$. Each of these is divisible by some $x_i^d$. Thus either $x^M$ is divisible by $x_i^d$, $i > q$, a contradiction, or by $x_i^{d-1}$, $i \le q$, also a contradiction. Thus injectivity part the claim is established. For the surjectivity part note that image of the map $\mu$ has a basis of monomials $x^M$ which are divisible by $x_i$ for $i = 0 \commadots q$. Thus, to show that $\mu$ is not surjective it suffices to show that there is a monomial for $R^{a(q,d+1)}(F_{d+1})$ that is not divisible by $x_0$. Such a monomial has the form $x_1^{M_1} \cdots x_{n+2}^{M_{n+2}}$ where $M_i \le d-1$. It exists if $a(q,d+1) \le (n+1)(d-1)$. The largest relevant values of $q$ and $a(q,d+1)$ are $n/2$ and $(n/2 + 1)d - (n+2)$. For these the preceding inequality holds and so the first inequality of the lemma holds strictly. For the second inequality we use the fact that basis elements for the Jacobian ring of $F_d$ correspond to lattice points of the cube in $(n+2)$-space defined by the inequalities $0 \le m_i \le d-2$. A basis for $R^a$ corresponds to the set of lattice points which lie on the convex subset $C(a)$ of the cube obtained by slicing it with the hyperplane $m_0 + \cdots + m_{n+1} = a$. The volume of $C(a)$ is a strictly increasing function of $a$ for $0 \le a \le t/2$, where $t = (n+2)(d-2)$. For $t/2 \le a \le t$ the volume function $V(a)$ is strictly decreasing, and in general its graph is symmetric around $a = t/2$. Let $L(a)$ be the number of lattice points in $C(a)$. If $L(a)$ satisfies the same monotonicity properties as does $V(a)$, then the second inequality follows. To show this, we prove the following result. \proclaim{Lemma.} Let $L_{d,n}(k)$ be the number of points in the set $\LL_{d,n}(k) = \sett{ x \in \Z^n }{ 0 \le x_i \le d,\ x_1 + \cdots + x_n = k }$. Assume that $n > 1$. Then $L_{d,n}(k)$ is a strictly increasing function of $k$ for $k < dn/2$ and is symmetric around $k = dn/2$. \endproclaim \noindent{\bf Proof:\ }{} Symmetry follows from the bijection $\LL_{d,n}(k) \map \LL_{d,n}(dn - k)$ given by $x \mapsto \delta - x$ where $\delta = (d \commadots d)$. We shall say that these two sets are dual to eachother. For the inequality we argue by induction, noting first that $L_{d,2}(k) = k + 1$ for $k \le d$. Now observe that $\LL_{d,n}(k)$ can be written as a disjoint union of sets $S_i = \sett{ x \in \LL_{d,n}(i) }{ x_n = k-i }$ where $i$ ranges from $k-d$ to $k$. Thus $$ L_{d,n}(k) = \sum_{i = k-d}^k L_{d,n-1}(i) . $$ Consequently $$ L_{d,n}(k+1) - L_{d,n}(k) = L_{d,n-1}(k+1) - L_{d,n-1}(k-d) . $$ By the induction hypothesis the right-hand side is positive if $k-d < (n-1)d/2$ and if $k+1$ is not greater than the index dual to $k-d$, namely $(n-1)d - (k-d)$. Thus we require also that $k+1 \le (n-1)d - (k-d)$. Both inequalities hold if $k < nd/2$, which is what we assume. Thus the proof is complete. \subheading{Dimension of the eigenspaces.} We close this section by noting that the eigenspaces $H^n(X)(\lambda)$ for $\lambda\ne 1$ all have the same dimension, explaining why the primitive middle Betti number is divisible by $d-1$, where $d$ is the degree. Indeed, we have the following, $$ \dim H^n(X,\C)(\lambda) = \dim H^n(X,\C)(\mu) = \dim H^n(\P^n - B,\C) + (-1)^n . \eqn \eqref{eigenspacedimform} $$ When the degree is prime there is a short proof: consider the field $k = \Q[\omega]$ where $\omega$ is a primitive $d$-th root of unity and observe that its Galois group permutes the factors $H^n(X,k)(\lambda)$ for $\lambda \ne 1$. For the general case let $p: X \map \P^n$ be the projection and note that $H^n(X,\C) = H^n(\P^n, p_*\C)$. The group of $d$-th roots of unity acts on $p_*\C$ and decomposes it into eigensheaves $\C_\lambda$, where $\lambda^d = 1$. Thus the $\lambda$-th eigenspace of $H^n(X,\C)$ can be identified with $H^n(\P^n,\C_\lambda)$. The component for $\lambda = 1$ is one-dimensional and is spanned by the hyperplane class. For $\lambda \ne 1$ the sheaf $\C_\lambda$ is isomorphic to the extension by zero of its restriction to $\P^n - B$. Thus the eigenspace can be identified with $H^n(\P^n - B, \C_\lambda)$. By the argument of lecture 8 in \cite{CKM} used in the proof of vanishing theorems, the groups $H^i(\P^n - B, \C_\lambda)$ vanish for $i \ne n$, $\lambda \ne 1$. Thus $\dim H^n(\P^n - B,\C_\lambda) = (-1)^n \chi(\lambda)$, where $ \chi(\lambda)$ is the Euler characteristic of $\C_\lambda$. Fix a suitable open tubular neighborhood $U$ of $B$ and a good finite cell decomposition $K$ of $\P^n - U$. Then $\chi(\lambda)$ is the Euler characteristic of the complex of $\C_\lambda$-valued cochains on $K$, which depends only on the number of cells in each dimension, not on $\lambda$. This establishes the first equality above. For the second use $\chi(\lambda) = \chi(1)$ and the vanishing of $H^i(\P^n - B,\C)$ for $i \ne n, 0$. \section{Remarks and open questions} \secref{remarkssection} We close with some remarks on (a) the possiblity of an isomorphism $\Phi \cong \Gamma\times\Gamma'$, (b) the impossibility of producing additional representations by iterating the suspension (globally), and (c) generalizations of the main theorem. \subheading{(A) Products} So far everything that has been said is consistent with an isomorphism between $\Phi$ and the product $\Gamma\times\Gamma'$, where $\Gamma'$ is the monodromy group $\bar \rho'(\Phi)$. This, however, is not the case, at least for surfaces, for we can show that {\sl if $k$ is a divisor of $d$ and $d$ is odd, then $\Phi_{d,2}$ and $\Gamma\times\Gamma'$ are not isomorphic}. The argument is based on the fact that the abelianization of $\Phi$ is a cyclic group of order equal to the degree of the discriminant, which we denote by $r$. This is because (a) the generators $g_1 \commadots g_r$ of $\Phi$ are mutually conjugate, hence equal in the abelianization, (b) $g_1 \cdots g_r = 1$, (c) the additional relations are trivial when abelianized. See \cite{Zar}. For the last point note that $\Phi$ is also the fundamental group of the complement of a generic plane section $\Delta'$ of $\Delta$. This complement has nodes and cusps as its only singularities. The nodes yield relations of the form $gg' = g'g$ where $g$ and $g'$ are conjugates of the given generators. The cusps yield braid relations $gg'g = g'gg'$. Both are trivial in the abelianization. Thus the abelianization is generated by a single element with relation $g^r = 1$. The degree of the discriminant is given in \cite{DolgLib}, page 6, line 2: $$ r = \hbox{deg}(\Delta) = 4(d-1)^3. $$ If $\Phi$ is isomorphic to the Cartesian product, then there is a corresponding isomorphism of abelianizations. Let us therefore compute what we can of the abelianizations of $\Gamma$ and $\Gamma'$. For $\Gamma$ we note that the generators are the elements $g_i$ as above satisfying additional relations which include $g_i^2 = 1$. Therefore $\Gamma$ abelianized is a quotient of $\Z/2$. Consider next the case of $\Gamma'$ for cyclic covers of degree $k$. Then $\Gamma'$ is a product of groups $\Gamma'(i)$ for $i = 1 \commadots k-1$. Generators and relations are as in the previous case except that among the additional relations are $g_i^{2k} = 1$ instead of $g_i^2 = 1$. Therefore the abelianization is a quotient of $\Z/{2k}$. Consequently the abelianization of the product $\Gamma\times \Gamma'$ is a quotient of the product of $\Z/2$ with a product of $\Z/2k$'s. But the largest the order of an element in such a quotient can be is $2k$, which is always less than the degree of the discriminant, provided that $d > 2$, which is the case. \subheading{(B) Suspensions} Since $\Phi$ is not in general isomorphic to $\Gamma\times\Gamma'$ it is natural to ask whether there are further representations with large kernels. One potential construction of new representations is given by iterating the suspension. By this we mean that we take repeated double covers. Unfortunately, this produces nothing new, since it turns out that the global suspension is periodic of period two. To make a precise statement, let $P(x)$ be a polynomial of degree $2d$ which defines a smooth hypersurface $X$ in $\P^n$. Let $X(2)$ be the hypersurface defined by $$ P(x) + y_1^2 + y_2^2 $$ in a weighted projective space $\P^{n+2}$ where the $x_i$ have weight one and the $y_i$ have weight $d$. {\sl Then there is an isomorphism $$ H^n_o(X)\otimes T \map H^{n+2}_o(X(2)), $$ where $T$ is a trivial Hodge structure of dimension one and type $(1,1)$ and where the subscript denotes primitive cohomology}. For the proof we note that the map $$ { A\Omega(x) \over P^{q+1} } \mapsto { A\Omega(x, y_1, y_2) \over ( y_1^2 + y_2^2 +P )^{q+2} } $$ is well-defined and via the residue provides an isomorphism compatible with the Hodge filtrations which is defined over the complex numbers. However, it can be defined geometrically and so is defined over the integers. To see why, consider first the trivial case $g(x) = f(x) + y_1^2 + y_2^2 = 0$ in affine coordinates, where $x$ is a scalar variable and $f$ has degree $2d$. Thus $f(x) = 0$ defines a finite point set, and $g(x) = 0$ is its double suspension. Let $p$ be one point of the given finite set. Then $f(p) = 0$, so the locus $\sett{ (p,y_1,y_2) }{ y_1^2 + y_2^2 = 0 }$ lies on the double suspension. This locus is a pair of lines meeting in a point, and the statement remains true in projective coordinates. Thus we may associate to $p$ a difference of lines $\ell_p - \ell'_p$. This map induces an isomorphism $H_0(X,\Z) \map H_2(X(2),\Z)$ which is in fact a morphism of Hodge structures. For the general case we parametrize the construction just made. The map in cohomology which corresponds to the previous construction is the dual of the inverse of the map in homology. \subheading{(C) Generalizations} The main theorem \xref{maintheorem} can be generalized in a number of ways. First, using the techniques of \cite{Tu}, it is certainly possible to get sharp results for various kinds of weighted hypersurfaces, just as we have obtained sharp results for standard hypersurfaces. Second, one can prove a quite general (but not sharp) result that reflects the fairly weak hypothesis of criterion \xref{largekernelcriterion}: \proclaim{Theorem} Let $L$ be a positive line bundle on a projective algebraic manifold $M$ of dimension at least three. Let $P$ be the projectivization of the space of sections of $L^d$, and let $\Delta$ be the discrimant locus defined by sections of $L^d$ whose zero set $Z$ is singular. Then for $d$ sufficiently large the kernel of the monodromy representation of $\Phi = \pi_1(P - \Delta)$ is large and its image is a lattice. \endproclaim The monodromy representation has the primitive cohomology $$ H^{m-1}(Z)_0 = \mathop{kernel}{\left[ H^{m-1}(Z) \mapright{Gysin} H^{m+1}(M) \right]} $$ as underlying vector space. The results needed for the proof are all in the literature. First, note that the condition that a section $s$ of $L^d$ have a singularity of type ``$x^3 + y^3 + z^4 + \hbox{sum of squares}$'' at a given point is set of linear conditions and so can be satisfied for $d$ sufficiently large. Consequently by the Beauville-Ebeling-Janssen argument, the image of the natural monodromy representation is a lattice. Second, by the results of Green \cite{GreenPM}, the local Torelli theorem for cyclic covers holds for $d$ sufficiently large, so some component of the second monodromy representation has nonzero differential. The standard argument used just following Theorem \xref{practicaldensitycriterion} proves that the discriminant locus is irreducible, and so Theorem \xref{udensityprop} applies to give Zariski-density for the second monodromy representation. Finally, the Hodge numbers, like the standard case of projective hypersurfaces, are polynomials in $d$ with positive leading coefficient and of degree equal to the dimension of $M$. Consequently they are large for $d$ large, and therefore both the real and complex rank of the relevant algebraic groups can be assumed sufficiently large by taking $d$ large enough. Thus the hypotheses of criterion \xref{largekernelcriterion} are satisfied. For a quick proof of the statement on the behavior of the Hodge numbers, consider first the Poincar\'e residue sequence $$ 0 \map \Omega^m_M \map \Omega^m_M(L^d) \map \Omega^{m-1}_Z \map 0, $$ where $Z$ is a smooth divisor of $L^d$ and $m$ is the dimension of $M$. From the Kodaira vanishing theorem we have $$ H^0(\Omega^{m-1}_Z)_0 \cong \mathop{cokernel}{ \left[ H^0(\Omega^m_M) \map H^0(\Omega^m_M(L^d)) \right] }. $$ By the Riemann-Roch theorem the dimension of the right-most term is a polynomial with leading coefficient $Cd^m$, while the dimension of the middle term is constant as a function of $d$. Therefore the Hodge number in question is a polynomial in $d$ of the required form. For the other Hodge numbers we use the identification $$ H^q(\Omega^p_Z)_0 \cong \mathop{cokernel}{ \left[ H^0(\Omega^{m-1}_Z\otimes\Theta_M\otimes N_Z^{q-1} ) \map H^0(\Omega^{m-1}_Z\otimes N_Z^q) \right] }, $$ where $N$ is the normal bundle of $Z$ in $M$, where $\Theta_M$ is the holomorphic tangent bundle of $M$, and where $p+q=m-1$. See Proposition 6.2, \cite{JC}, a consequence of Green's Koszul cohomology formula (Theorem 4.f.1 in \cite{GreenKC}) for $d$ sufficiently large. Now tensor the Poincar\'e residue sequence with $L^{qd}$ to get $$ 0 \map \Omega^m_M(L^{qd}) \map \Omega^m_M(L^{(q+1)d}) \map \Omega^{m-1}_Z \otimes N_Z^q \map 0 . $$ From the Kodaira vanishing theorem and the Riemann-Roch formula one finds that the dimension of the right-hand part of the cokernel formula is a polynomial with leading term $C((q+1)d)^m$, where $C = c_1(L)^m/m!$. A similar argument shows that the dimension of the left-hand part is a polynomial with leading coefficient $C(qd)^m$. Thus the leading term of $\dim H^q(\Omega^p)_0$ is bounded below by a positive constant depending on $L$, $q$, and $M$, times $d^m$. \subheading{(D) Questions.} We close with some open questions. The main problem is, of course, to understand the nature of the groups $\Phi_{d,n}$. Are they linear? Are they residually finite? It seems reasonable to conjecture that in general they are not linear groups, and, in particular, are not lattices in Lie groups. We settle this last question for $\Phi_{3,2}$ in the note \cite{ACT}. The structure of $\Phi_{d,n}$ is closely related to the structure of the kernel $K$ of the natural monodromy representation. For $n=0$, $K$ is the pure $d$-strand braid group the sphere and so is finitely generated. For $(d,n) = (3,2)$, the case of cubic surfaces, $K$ is not finitely generated (see \cite{ACT}). It is therefore natural to ask when $K$ is finitely generated and when it is not. \bibliography \parskip=2pt \baselineskip=10.5pt \bi{ACT} D. Allcock, J. Carlson, and D. Toledo, Complex Hyperbolic Structures for Moduli of Cubic Surfaces, C. R. Acad. Sci. Paris {\bf 326}, ser I, pp 49-54 (1998) (alg-geom/970916) \bi{Arnold} V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Differentiable Maps, vol II, Birkhauser, Boston, 1988. \bi{Beardon} A.F. Beardon, The Geometry of Discrete Groups, Springer-Verlag (1983), pp 333. \bi{BeauvilleLattice} A. Beauville, Le groupe de monodromie d'hypersurfaces et d'intersections compl\`etes. Springer Lecture Notes in Mathematics, {\bf 1194} (1986) 8--18. \bi{BorelDT} A. Borel, Density and maximality of arithmetic groups, J. Reine Andgew. Math. {\bf 224} (1966) 78--89. \bi{BHC} A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math. (2) {\bf 75} (1962) 485--535. \bi{CKM} H. Clemens, J. Koll\'ar, and S. Mori, editors, Higher Dimensional Complex Geometry, Ast\'erisque {\bf 166} (1988). \bi{JC} J. Carlson, Hypersurface Variations are Maximal, II, Trans. Am. Math. Soc. {\bf 323} (1991), 177--196. \bi{CG} J. Carlson and P.A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, Journ\'ees de G\'eometrie Alg\'ebrique D'Angers 1979, Sijthoff \& Nordhoff, Alphen an den Rijn, the Netherlands (1980), 51--76. \bi{DeMo} P. Deligne, Un th\'eoreme de finitude pour la monodromie, in Discrete Groups in Geometry and Analysis, R. Howe (ed) Birkh\"auser, Boston. \bi{DeWeOne} P. Deligne, La Conjecture de Weil, I, Pub. Math. I.H.E.S. {\bf 43}, 273 -- 307 (1974). \bi{DeWeTwo} P. Deligne, La Conjecture de Weil, II, Pub. Math. I.H.E.S. {\bf 52}, 137 -- 252 (1980). \bi{DeHodgeTwo} P. Deligne, Theorie de Hodge II, Publ. Math. IHES {\bf 40} (1971), 5--58. \bi{DOZ} G. Detloff, S. Orekov, and M. Zaidenberg, Plane curves with a big fundamental group of the complement, Preprint 1996. \bi{Dolgachev} I. Dolgachev, Weighted projective varieties, in Group Actions and Vector Fields, Proceedings 1981, Lecture Notes in Math. {\bf 956}, Springer-Verlag, New York (1982). \bi{DolgLib} I. Dolgachev and A. Libgober, On the fundamental group to the complement of a discriminant variety, Springer Lecture Notes in Mathematics {\bf 862} (1981), pp 1--25. \bi{Ebeling} W. Ebeling, An arithmetic characterisation of the symmetric monodromy groups of singularities, Invent. Math. {\bf 77} (1984) 85-99. \bi{Givental} A. B. Givental, Twisted Picard-Lefschetz Formulas, Funct. Analysis Appl. {\bf 22} (1988) 10-18. \bi{GreenKC} M. Green, Koszul cohomology and the geometry of projective varieties. J. Differential Geom. {\bf 19} (1984), no. 1, 125--171. \bi{GreenPM} M. Green, The period map for hypersurface sections of high degree of an arbitrary variety. Compositio Math. {\bf 55} (1985), no. 2, 135--156. \bi{GriffHyperbolic} P.A. Griffiths, oral communication 1969. \bi{GriffPerRat} P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. {\bf 90} (1969) 460-541. \bi{GriffPerDom} P.A. Griffiths, Periods of integrals of algebraic manifolds, III, Pub. Math. I.H.E.S. {\bf 38} (1970), 125--180. \bi{GS} P.A. Griffiths, and W. Schmid, Locally homogenous complex manifolds, Acta Math. {\bf 123} (1969), 253--302. \bi{Janssen} W. A. M. Janssen, Skew-symmetric vanishing lattices and their monodromy groups, Math. Annalen {\bf 266} (1983) 115-133, {\bf 272} (1985) 17-22. \bi{Lib} A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. {\bf 162} (1978) 63--67. \bi{Mag} W. Magnus and A. Peluso, On a Theorem of V.I. Arnol'd, Comm. of Pure and App. Math. {\bf 22}, 683--692 (1969). \bi{MargulisRigidity} G.A. Margulis, Discrete groups of motions of manifolds of non-positive curvature, Amer. Math. Soc. Translations {\bf 109} (1977), 33-45. \bi{Milnor} J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies {\bf 61}, Princeton University Press (1968), pp 122. \bi{Mostow} G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific Journal of Mathematics {\bf 80} (1980), 171 -- 276. \bi{Pham} F. Pham, Formules de Picard-Lefschetz g\'en\'eralis\'ees et ramification des int\'egrales, Bull. Soc. Math. France {\bf 93}, 1965, 333-367. Reprinted in Homology and Feynman Integrals, Rudolph C. Hwa and Vidgor L. Teplitz, W. A. Benjamin 1966, pp. 331. \bi{Seidel} P. Seidel, Floer homology and the symplectic isotopy problem, thesis, Oxford University, 1997. \bi{ST} M. Sebastiani and R. Thom, Un r\'esultat sur la monodromie, Invent. Math. {\bf 13} (1971), 90--96. \bi{SimpsonHiggs} C. T. Simpson, Higgs bundles and local systems, Publ. Math. IHES {\bf 75} (1992), 5--95. \bi{Tits} J. Tits, Free subgroups in linear groups, J. Algebra {\bf 20}, 250--270 (1972). \bi{Tu} L. Tu, Macaulay's theorem and local Torelli for weighted hypersurfaces, Compositio Math. {\bf 60} (1986), 33--44. \bi{Zar} O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. {\bf 51} (1929) 305--328. \bi{Zimmer} R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkh\"auser 1984, pp 209. \endbibliography \bigskip \begingroup \obeylines \parskip=0pt \parindent1cm \baselineskip=10pt \def\vskip7pt{\vskip7pt} Department of Mathematics University of Utah Salt Lake City, Utah 84112 \vskip7pt [email protected] $\qquad$ [email protected] \vskip7pt http://www.math.utah.edu/$\sim$carlson http://xxx.lanl.gov --- alg-geom/9708002 To appear in Duke J. Math. \endgroup \enddoc \end

9708

alg-geom/9708012

en

https://arxiv.org/abs/alg-geom/9708012

alg-geom/9708012

Lothar Goettsche

Barbara Fantechi, Lothar G\"ottsche, Duco van Straten

Euler number of the compactified Jacobian and multiplicity of rational curves

LaTeX, 16 pages with 1 figure

We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $\delta$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.

alg-geom

\section{Introduction} Let $C$ be a reduced and irreducible projective curve with singular set $\Sigma \subset C$ and let $n: \widetilde{C} \longrightarrow C$ be its normalisation. The generalised Jacobian $JC$ of $C$ is an extension of $J\widetilde{C}$ by an affine commutative group of dimension $$\delta:=\dim H^0(n_*({\cal O}_{\widetilde{C}})/{\cal O}_C)=\sum_{p \in \Sigma} \delta(C,p)$$ so that $\dim JC=\dim J\widetilde{C} +\delta=g(\widetilde{C})+\delta$ is equal to the arithmetic genus $g_a(C)$ of $C$. The non-compact space $JC$ is naturally an open subset of the {\em compactified Jacobian} ${\overline{J}} C$ of $C$, whose points correspond to isomorphism classes of rank one torsion free sheaves ${\cal F}$ of degree zero (i.e. $\chi({\cal F})=1-g_a(C)$) on $C$. The space ${\overline{J}} C$ is irreducible if and only if $C$ has planar singularities; then ${\overline{J}} C$ is in fact a compactification of $JC$, i.e., $JC$ is dense in ${\overline{J}} C$. If moreover $C$ is rational and unibranch, then ${\overline{J}} C$ is topologically the product of compact spaces $M(C,p)$ for every $p\in \Sigma$. The space $M(C,p)$ only depends on the analytic singularity $(C,p)$; it can be defined as ${\overline{J}} D$ for any rational curve $D$ having $(C,p)$ as unique singularity.\\ Let $B=B(C,p)$ be the base of a semi-universal deformation of the singularity $(C,p)$. Inside $B$ let $B^\delta=B^\delta(C,p)$ be the locus of points for which $\delta$ remains constant. This means that $$t \in B^\delta \Leftrightarrow \sum_{p \in C_t} \delta(C_t,p) =\delta(C).$$ The codimension of $B^\delta$ is $\delta(C,p)$; its multiplicity $m(C,p)$ at $[(C,p)]$ is by definition equal to the number of intersection points with a generic $\delta$-dimensional smooth subspace of $B$. The $\delta$-constant stratum can be defined in a similar way for a semi-universal deformation of a projective curve with only planar singularities. In this paper we show the following theorem. {\bf Theorem 1.} {\sl Let $(C,p)$ be a reduced plane curve singularity. Then the Euler number of $M(C,p)$ is equal to the multiplicity of the $\delta$-constant stratum: $$e(M(C,p))=m(C,p).$$ Let $C$ be a projective, reduced rational curve with only planar singularities. Then $e({\overline{J}} C)= m(C)$, the multiplicity of the $\delta$-constant stratum $B^\delta$ at $0$.} Note that this gives an independent proof of the following result of Beauville: Let $C$ be an irreducible and reduced rational curve with planar singularities. Then $e({\overline{J}} C)$ can be written as a product over the singularities of $C$ of a number only depending on the type of the singularity, and it is the same for $C$ and its minimal unibranch partial normalisation. Theorem 1 has an application in the following situation. Let $X$ be a (smooth) $K3$ surface with a complete (hence $g$-dimensional) linear system of curves of genus $g$. Under the assumption that all curves in the system are irreducible and reduced, it was shown in \cite{Y-Z} and \cite{B} that the ``number'' $n(g)$ of rational curves occuring in the linear system, is equal to the $g^{\rm th}$ coefficient of the $24^{\rm th}$ power of the partition function, i.e: $$\sum_{g \ge 0}n(g)q^g=\frac{q}{\Delta(q)}$$ where $\Delta(q)=q\prod_{n \ge 1}(1-q^n)^{24}$. In this counting, a rational curve $C$ in the linear system contributes $e({\overline{J}} C)$ to $n(g)$: $$n(g)=\sum_{C} e({\overline{J}} C).$$ If $C$ is a rational curve with only nodes as singularities, then $e({\overline{J}} C) =1$, so that $e( {\overline{J}} C)$ seems to be a reasonable notion of multiplicity. Theorem 1 implies that $e({\overline{J}} C)$ is always positive, and in principle allows an explicit computation of it (see section {G}).\\ In fact, we prove a more precise statement. For any projective scheme $Y$ and $d \in H_2(Y,{\bf Z})$ let $M_{0,0}(Y, d)$ be the moduli space of genus zero stable maps $f: {\bf P}^1 \longrightarrow Y$ with $f_*([{\bf P}^1])=d$. Under the above assumptions on the K3 surface $X$ and the linear system corresponding to $d$, the space $M_{0,0}(X,d)$ is a zero-dimensional scheme. If $C \stackrel{i}{\hookrightarrow} X$ is a rational curve in $X$ (always assumed to be irreducible and reduced), $n:{\bf P}^1 \longrightarrow C$ its normalisation, then $f=i \circ n:{\bf P}^1 \longrightarrow X$ is a point of $M_{0,0}(X,d)$. The moduli space $M_{0,0}(X,d)$ contains naturally as a closed subscheme $M_{0,0}(C,[C])$, the submoduli space of maps whose scheme theoretic image is $C$; the latter scheme is of course defined for any projective reduced curve $C$, and it is zero-dimensional if the curve is rational. More generally, $M_{g,0}(C,[C])$ is zero dimensional, where $g$ denotes the genus of the normalisation of $C$. The following theorem gives another interpretation of $e({\overline{J}} C)$ in terms of the length of such zero-dimensional schemes.\\ {\bf Theorem 2.} {\sl Let $C$ be a reduced, irreducible projective curve with only planar singularities, and let $g$ be the genus of its normalisation. Then $m(C)=l(M_{g,0}(C,[C]))$. If moreover $C$ is rational and contained in a smooth $K3$ surface $X$, then $e({\overline{J}} C)= l(M_{0,0}(X,d),f)$ (length of the zero-dimensional component supported at $f$).} We now sketch briefly the idea of the proof of Theorem 1. Let ${\cal C}\to B$ be a semi-universal family of deformations of a curve $C$ with planar singularities. We prove that the relative compactified Jacobian $\bar J {\cal C}$ is smooth; moreover, given any deformation ${\cal C}'\to S$ of $C$ with a smooth base, $\bar J{\cal C}'$ is smooth if and only if the image of $TS$ is transversal in $TB$ to the $\delta$-codimensional vector space $V$, the support of the tangent cone to the $\delta$-constant stratum $B^\delta$. Assume now $C$ is rational and has $p$ as unique singularity. We have to show that $e(\bar JC)=m(C,p)$. Choose a one-parameter family $W_t$ of smooth $\delta$-dimensional subspaces of $B$ such that $0\in W_0$, $T_{W_0,0}\cap V=\{0\}$, and for general $t$ the intersection $W_t\cap B^\delta$ is a set of $m(C,p)$ distinct points corresponding to nodal curves. Let ${\cal C}_t\to W_t$ be the induced families. Then $\bar J{\cal C}_t$ is a family of smooth compact varieties, hence $e(\bar J{\cal C}_t)$ does not depend on $t$. Arguing as in \cite{Y-Z} and \cite{B}, we prove that $e(\bar J{\cal C}_0)=e({\overline{J}} C)$, while $e({\overline{J}} {\cal C}_t)=m(C,p)$ for $t$ general. {\bf Conventions.} In this paper we will always work over the complex numbers, and open will mean open in the strong (euclidean) topology (unless of course we specify Zariski open). {\bf Preliminaries.} We will use the language of deformation functors; we recall a few facts about them for the reader's convenience. A deformation functor $D$ will always be a covariant functor from local artinian ${\bf C}$-algebras to sets, satisfying Schlessinger's conditions $(H1)$, $(H2)$, $(H3)$, hence admitting a hull (see \cite{Sch}). In particular $D$ admits a finite-dimensional tangent space, which we denote by $TD$, functorial in $D$. A functor is smooth if its hull is. The dimension of the functor will be equal to the dimension of the hull. We will need the following elementary result. {\bf Lemma.} {\sl Let $X\to Y$ and $Z\to Y$ be morphisms of smooth deformation functors. Then $X\times_YZ$ is smooth of dimension $\dim X+\dim Z-\dim Y$ if and only if the images of $TX$ and $TZ$ span $TY$.}\\ {\bf Proof.} Base change considerations reduce the problem to the case of prorepresentable functors, where it is obvious.\hfill $\Diamond$ It would be possible to replace deformation functors with contravariant functors on the category of germs of complex spaces, and the hull with the base of a semi-universal family of deformations. The two viewpoints correspond to working with formal versus convergent power series. {\bf Acknowledgements.} This paper was written at the Mittag-Leffler Institute in Stockholm, during a special year on Enumerative Geometry. The authors are grateful for the support received and for making our collaboration possible. The first author is a member of GNSAGA of CNR. \section{A. Deformations of curves and sheaves.} Let $C$ be a reduced projective curve, with singular set $\Sigma$. Any deformation ${\cal C} \longrightarrow S$ of $C$ over a base $S$ induces a deformation of its singularities. More precisely, one can introduce the {\em functor of local deformations} by letting $D^{loc}(C)(T)$ be the set of isomorphism classes of data $(U_i,U_i^T)_{i\in I}$, where $(U_i)_{i\in I}$ is an affine open cover of $C$ and, for each $i$, $U_i^T$ is a deformation of $U_i$ over $T$; we require that the induced deformations of $U_{ij}:=U_i\cap U_j$ be the same. There is a natural transformation of functors $\hbox{\sl loc}: D(C) \longrightarrow D^{loc}(C)$; the induced map of tangent spaces can be identified with the edge homomorphism $${\bf T}_C^1 \longrightarrow H^0({\cal T}_C^1)$$ of the local-to-global spectral sequence for the ${\cal T}^i$. The kernel of this map is $H^1(\Theta_C)$, the cokernel injects in $H^2(\Theta_C)$ which is zero. The obstruction space ${\bf T}_C^2$ sits in an exact sequence $$0 \longrightarrow H^1({\cal T}_C^1) \longrightarrow {\bf T}_C^2 \longrightarrow H^0({\cal T}_C^2) \longrightarrow 0.$$ As $C$ is reduced, ${\cal T}_C^1$ is supported on a finite set of points, hence $H^1({\cal T}_C^1)=0$. If $C$ has locally complete intersection singularities, then also ${\cal T}_C^2 =0$, so that in that case ${\bf T}_C^2=0$. Hence in such a situation, and in particular when $C$ is a reduced curve with only planar singularities, the functors $D(C)$ and $D^{loc}(C)$ are smooth and $\hbox{\sl loc}$ is a smooth map. Let ${\cal F}$ be a torsion free coherent sheaf on $C$. Analogously , we denote by $D(C,{\cal F})$ the functor of deformations of the pair, and define the functor of local deformations by letting $D^{loc}(C,{\cal F})(T)$ be the set of isomorphism classes of data $(U_i,U_i^T,F_i^T)_{i\in I}$ where $(U_i)_{i\in I}$ is an affine open cover of $C$, and for each $i$, $(U_i^T,F_i^T)$ is a $T$-deformation of $(U_i,{\cal F}|_{U_i})$ such that the induced deformations on $U_{ij}$ are the same. Again we have a localisation map $D(C,{\cal F})\to D^{loc}(C,{\cal F})$. The four functors introduced sit in a natural commutative diagram $$ \begin{array}{ccc} D(C,{\cal F}) & \longrightarrow & D^{loc}(C,{\cal F})\\ \downarrow&&\downarrow\\ D(C) & \longrightarrow & D^{loc}(C)\\ \end{array} $$ with horizontally localisation maps and vertically forget maps. Note that this diagram in general is {\em not} cartesian. {\bf Proposition A.1.} {\sl The canonical map $$ D(C,{\cal F}) \longrightarrow D(C) \times_{D^{loc}(C)}D^{loc}(C,{\cal F})$$ is smooth.} {\bf Proof.} We have to show the following: Let ${\cal F}_T$, $C_T$ be flat deformations of $C$ and ${\cal F}$ over $T$, $\xi_T \in D^{loc}(C,{\cal F})(T)$ the induced local deformation. If we are given lifts $C_{T'}$ and $\xi_{T'}$ over a small extension $T'$ of $T$, then we can lift ${\cal F}_T$ to a deformation ${\cal F}_{T'}$ of ${\cal F}$ over $C_{T'}$ inducing $\xi_{T'}$. This can be done as follows: choose an affine open cover $U_i$ of $C$ such that $\xi_{T'}$ is defined by coherent sheaves $F_i'$ on the induced cover $U_{i,T'}$ of $C_{T'}$. Assume also that $U_{ij}:=U_i\cap U_j$ is smooth for every $i\ne j$. Let $F_i$ be the restriction of $F_i'$ to $U_{i,T}$. The fact that ${\cal F}$ induces $\xi_T$ means that we can find isomorphisms $\phi_i:{\cal F}_T|_{U_{i,T}}\to F_i$. The $\phi_i$ induce isomorphisms $\phi_{ij}:F_i\to F_j$ over $U_{ij,T}$, satisfying the cocycle condition. What we need to prove is that the $\phi_{ij}$ can be lifted to $\phi_{ij}':F'_i\to F'_j$, again satisfying the cocycle condition; then the $\phi_{ij}'$ can be used to glue together the $F_i'$'s to a coherent sheaf ${\cal F}_{T'}$ as required. But on $U_{ij}$ all the sheaves under consideration are line bundles, hence the obstruction to the existence of such a lifting is an element in $H^2(C_,{\cal O}_{C})$, which is zero as $C$ has dimension $1$. \hfill $\Diamond$ If $R$ is a ring and $M$ is an $R$-module, we denote by $D(R)$, respectively $D(R,M)$ the corresponding deformation functors. {\bf Lemma A.2}. {\sl Let $C$ be a reduced projective curve, ${\cal F}$ a torsion free module on $C$. Let $\Sigma$ denote the singular locus. Then the natural morphisms of functors $$ D^{loc}(C)\to \prod_{p\in \Sigma}D({\cal O}_{C,p})\qquad \hbox{\sl and}\qquad D^{loc}(C,{\cal F})\to \prod_{p\in \Sigma}D({\cal O}_{C,p},{\cal F}_p)$$ are isomorphisms.} {\bf Proof.} Both morphisms are clearly injective. On the other hand, surjectivity is obvious since on the smooth open locus, every infinitesimal deformation is locally trivial and every torsion free sheaf is locally free. \hfill $\Diamond$ {\bf Proposition A.3}. {\sl Let $P$ be a regular local ring of dimension $2$, $f\in P$ a nonzero element, and $R=P/(f)$; assume that $R$ is reduced. Let $M$ be a finitely generated, torsion free $R$-module of rank $1$. Then $D(R,M)$ is a smooth functor.} {\bf Proof.} As it is torsion free, the module $M$ has depth $1$. By the Auslander-Buchsbaum theorem (see e.g. \cite{Ma}), $M$ has a free resolution of length $1$ as a $P$-module, so is represented as the cokernel of some $n \times n$ matrix $A$ with entries from $P$: $$ 0 \longrightarrow P^n \stackrel{A}{\longrightarrow}P^n \longrightarrow M \longrightarrow 0$$ As $M$ is an $R$-module of rank $1$, the determinant ideal $(det(A))$ is equal to $(f)$.\\ Any flat deformation $M_T$ of $M$ over $T$ (as $P$-module) is obtained by deforming the matrix $A$ to a matrix $A_T$ with entries from $P_T:=T \otimes_{{\bf C}} P$, so that $M_T$ has a presentation $$0 \longrightarrow P_T^n \stackrel{A_T}{\longrightarrow}P_T^n \longrightarrow M_T \longrightarrow 0.$$ There is a unique deformation $R_T$ of $R$ over $T$ such that $M_T$ is a flat $R_T$-module, given by the ideal $(det(A_T))$. It follows that the natural transformation $$D(A) \longrightarrow D(R,M)\qquad A_T \mapsto (P_T/det(A_T),Coker(A_T))$$ is {\em smooth}. As $D(A)$, the functor of deformations of the matrix $A$, is clearly smooth, the functor $D(R,M)$ is also smooth. \hfill $\Diamond$ Note that in the assumption of A.3, although both functors $D(R,M)$ and $D(R)$ are smooth, the forgetful morphism $D(R,M)\to D(R)$ is not smooth in general. {\bf Remark A.4.} Let $R$ be a one-dimensional local ${\bf C}$-algebra, and let $M$ be a finitely generated torsion free $R$-module. Let $\hat R$ be the completion of $R$, and $\hat M=M\otimes_R\hat R$. The natural morphism $D(R,M)\to D(\hat R, \hat M)$ is smooth and induces an isomorphism on tangent spaces, and the same is true for $D(R)\to D(\hat R)$. In fact it is easy to see that the induced morphisms of tangent and obstruction spaces are isomorphisms. \section{B. Relative compactified Jacobians.} For any flat projective family of curves ${\cal C}\to S$ we let ${\overline{J}} {\cal C}\to S$ be the relative compactified Jacobian (see \cite{R}). For every closed point $s \in S$ the fiber over $s$ of ${\overline{J}} {\cal C}$ is canonically isomorphic to the compactified Jacobian ${\overline{J}} C_s$; in particular, its points correspond to isomorphism classes of torsion free rank $1$ degree zero sheaves on $C_s$. Fix a point ${\cal F} \in {\overline{J}} {\cal C}$ over $s\in S$, and denote again by $({\overline{J}}{\cal C},{\cal F})$ and $(S,s)$ the deformation functors induced by the respective germs of complex spaces. Let $C=C_s$. Remark that if ${\cal C} \longrightarrow S$ is a semi-universal family of deformations of $C$, then we have an isomorphism of functors $$ ({\overline{J}}{\cal C}, {\cal F}) \simeq D(C,{\cal F}) $$ For a general flat family one has a natural commutative diagram $$ \begin{array}{ccc} ({\overline{J}}{\cal C},{\cal F}) & \longrightarrow & D^{loc}(C,{\cal F})\\ \downarrow&&\downarrow\\ (S,s) & \longrightarrow & D^{loc}(C)\\ \end{array} $$ and analogous to {\bf A.1.} one has: {\bf Proposition B.1.} {\sl The canonical map $$ ({\overline{J}}{\cal C},{\cal F}) \longrightarrow (S,s) \times_{D^{loc}(C)} D^{loc}(C,{\cal F})$$ is smooth.} We omit the proof, which is almost identical to that of {\bf A.1.} {\bf Corollary B.2.} {\sl Let $C$ be a reduced curve with only plane curve singularities. If ${\cal C}\to S$ is a versal family of deformations of $C$, then ${\overline{J}} {\cal C}$ is smooth along ${\overline{J}} C$, and ${\overline{J}} C$ has local complete intersection singularities.} {\bf Proof.} The family is versal if and only if the natural map $S\to D(C)$ is smooth. This in turn implies that $S\to D^{loc}(C)$ is smooth, hence the first claim follows from Proposition B.1. On the other hand, all fibres of ${\overline{J}} {\cal C}\to S$ have the same dimension $g_a(C)$, therefore each of them has local complete intersection singularities. \hfill $\Diamond$ {\bf Corollary B.3.} {\sl With the same assumptions as B.2, let ${\cal C}'\to S'$ be any deformation of $C$ with smooth base $S'$. Let ${\cal F}$ be a torsion free rank $1$ degree zero coherent sheaf on $C$. Then the relative compactified Jacobian ${\overline{J}} {\cal C}'$ is smooth at $[{\cal F}]$ if and only if the image of $TS'$ in $TD^{loc}(C)$ is transversal to the image of $TD^{loc}(C,{\cal F})$.} {\bf Proof.} We keep the notation of B.2. The dimension of $\bar J{\cal C}'$ is equal to $\dim S'+g_a(C)$. As ${\overline{J}} {\cal C}'$ is equal to the fibred product of ${\overline{J}} {\cal C}$ and $S'$ over $S$, it follows that ${\overline{J}} {\cal C}'$ is smooth at $[{\cal F}]$ if and only if the image of $TS'$ in $TS$ is transversal to that of $T({\overline{J}} {\cal C}, {\cal F})$. Proposition B.1 implies that the image of $T({\overline{J}} {\cal C}, {\cal F})$ is the inverse image of the image of $TD^{loc}(C,{\cal F})$ in $TD^{loc}(C)$.\hfill $\Diamond$ \section{C. The canonical sub-space $V$} Let $C$ be a reduced curve with only planar singularities. In this section we study the map $$D^{loc}(C,{\cal F})\to D^{loc}(C)$$ at the level of tangent spaces. As both functors are products corresponding to the singularities of $C$ (Lemma A.2) and the tangent spaces only depend on the formal structure of the singularity (Remark A.4), it suffices to analyse what happens for $$ D(R,M)\to D(R)$$ where $P={\bf C}[[x,y]]$, $R=P/(f)$, $f$ a non-zero element of the maximal ideal such that $R$ is reduced, and $M$ a torsion free rank one $R$-module given by a presentation $$ 0 \longrightarrow P^n \stackrel{A}{\longrightarrow}P^n \longrightarrow M \longrightarrow 0.$$ {\bf Proposition C.1.} {\sl The image of the map $ TD(R,M)\to TD(R)$ is the image of the first Fitting ideal $F_1(M)$ in the quotient ring $TD(R)=P/(f,\partial_xf,\partial_yf)$.}\\ {\bf Proof.} Let $E_{i,j}$ be the $n \times n$ matrix which has entry $(i,j)$ equal to $1$ and all other entries equal to zero. If $\epsilon^2=0$, then $\det(A+\epsilon.E_{i,j})=\det(A)+\epsilon \wedge^{n-1}(A)_{i,j}$, therefore we see that by perturbing the matrix $A$ to first order, we generate precisely the ideal of $(n-1) \times (n-1)$ minors of the matrix $A$ as first order perturbations of $f$. This is by definition the first Fitting ideal of $F_1(M)$. \hfill $\Diamond$ Another description of the ideal $F_1(M)$ is the following {\bf Proposition C.2.} {\sl $F_1(M)$ is the set of elements $r\in R$ such that $r=\varphi(m)$ for some $m\in M$, $\varphi\in Hom_R(M,R)$. }\\ {\bf Proof.} As $M$ is maximal Cohen-Macaulay, a resolution of $M$ as an $R$-module will be $2$-periodic of the form $$ \dots \longrightarrow R^n \stackrel{\bar B}{\longrightarrow} R^n \stackrel{\bar A} {\longrightarrow} R^n \longrightarrow M \longrightarrow 0$$ for some $n\times n$ matrix with $P$-coefficients $B$ with the property that $$ AB=BA=f{\bf 1}$$ and $\bar A$, $\bar B$ are the induced matrices with $R$ coefficients (see \cite{E} or \cite{Yo}). From the $2$-periodicity it follows that there is an exact sequence $$ 0\longrightarrow M \longrightarrow R^n \stackrel{\bar A}{\longrightarrow} R^n \longrightarrow M\longrightarrow 0,$$ where $M=\ker A=\hbox{\rm im}\, B$. We split this sequence into $$ \begin{array}{c} 0\longrightarrow M\longrightarrow R^n \longrightarrow N\longrightarrow 0\\ 0\longrightarrow N \longrightarrow R^n \longrightarrow M \longrightarrow 0.\end{array}$$ As $N$ is also torsion free and $R$ is Gorenstein, $Ext^1_R(N,R)=0$ by local duality. Hence we see from the first sequence that the map $Hom_R(R^n,R)\longrightarrow Hom_R(M,R)$ is surjective. From this it follows that the ideal obtained by evaluating all homomorphisms $\phi\in Hom_R(M,R)$ on all elements of $M$ is the same as the ideal generated by the entries of the matrix $\bar B$. As $M$ has rank $1$, it follows that $det(A)=f$, and hence the matrix $B$ is the Cramer matrix $(\Lambda^{n-1}A)^{tr}$ of $A$. The claim follows. \hfill $\Diamond$ Locally, the normalisation $\widetilde C \longrightarrow C$ corresponds to the inclusion of $R$ in its integral closure $\overline R$ $$R \hookrightarrow \overline R.$$ Recall that the conductor is the ideal $I=Hom_R(\overline R,R)$. One has $$I\subset R\subset \overline R$$ and $dim(R/I)=dim(\overline R/R)=\delta(C,p)$. As an important corollary of {Proposition C.2} we have {\bf Corollary C.3.} $F_1(M)\supset I$.\\ {\bf Proof.} Write $\overline R=\oplus \overline R_i$, with $\overline R_i$ a domain isomorphic to ${\bf C}[[t]]$. Let $Q(\overline R_i)$ be the quotient field of $\overline R_i$, and let $Q(R)=\oplus Q(\overline R_i)$ be the total quotient ring of $R$. As $M$ has rank $1$, $M\otimes_RQ(R)$ is isomorphic to $Q(R)$; as it is torsion-free, the natural map $M\to M\otimes_RQ(R)$ is injective. Hence up to isomorphism we can assume that $M$ is a submodule of $Q(R)$. Let $m\in M$ be an element of minimal valuation (it exists as $M$ is finitely generated). Then multiplication by $m^{-1}$, an isomorphism of $Q(R)$ as an $R$-module, sends $M$ to a submodule of $\overline R$ containing $1$. So we can assume that $R\subset M\subset \overline R$. Let $c$ be any element of $I$. Multiplication by $c$ defines a homomorpism $\phi\in Hom_R(M,R)$ with $\phi(1)=c$ (note that $1\in R\subset M$). Hence $$\bigl\{\phi(m)\bigm| m\in M, \ \phi\in Hom(M,R)\bigr\}\supset I.$$ \hfill $\Diamond$ {\bf Remark C.4.} From the above description one also sees that $F_1(\overline R)=I$. Hence the differential of the map $D(R,M)\to D(R)$ has minimal rank for $M=\overline R$. Let $C$ be a reduced projective curve with only planar singularities, $\Sigma$ its singular locus. For $p\in \Sigma$, let $V_p$ be the subspace of codimension $\delta(C,p)$ in $TD(C,p)$ generated by the conductor, and put $$V^{loc}=\prod_{p \in \Sigma} V_p \subset TD^{loc}(C) =\prod_{p \in \Sigma}TD(C,p).$$ Let $V$ be the inverse image of $V^{loc}$ in $TD(C)$; note that $V$ is a linear subspace of codimension $\delta(C)$. If $B$ is the base space of a semi-universal family of deformations of $C$, then $TB$ is identified with $TD(C)$. {\bf Proposition C.5.} {\sl Let ${\cal C}\to B$ be a semi-universal family of deformations of $C$. Then for any ${\cal F}\in \bar JC$ the image of the tangent map ${\overline{J}} {\cal C}\to B$ at ${\cal F}$ contains the subspace $V$, and there exists at least one such ${\cal F}$ for which the image is exactly $V$. } \\ {\bf Proof.} The first statement follows immediately from Proposition C.1 and Corollary C.3, by applying Proposition B.1 and Lemma A.2. The second statement follows in the same way from Remark C.4; e.g., we can take ${\cal F}=n_*({\cal O}_{\tilde C})$, where $n:\tilde C\to C$ is the normalisation map. \hfill $\Diamond$ \section{D. The $\delta$-constant stratum.} Let $C$ be a reduced curve with only planar singularities. We denote by $B$ an appropriate representative of the semi-universal deformation of $C$. The stratum $B^\delta$ is defined as the set of points where the geometric genus of the fibres is constant. This amounts to saying that $$\sum_{x\in C_t} \delta(C_t,x)$$ is constant for $t\in B^\delta$ and equal to $g_a(C)-g(\widetilde C)$, hence the name. The analytic set $B^\delta$ (we give it the reduced induced structure) is very singular in general, but its properties can be related directly to the local $\delta$-constant strata $$B^\delta(C,p).$$ To be more precise, $B^\delta$ is the pullback of $B^{\delta,loc}=\prod B^\delta(C,p)$ under the smooth map $B\longrightarrow B^{loc}$. So let $(C,p)\subset ({\bf C}^2,0)$ be a reduced plane curve singularity, with normalisation $$(\widetilde C,q)\stackrel{n}{\longrightarrow}(C,p),\qquad q=n^{-1}(p).$$ Note that in general $q$ will be a finite set of distinct points, one for each branch of $C$ at $p$. We denote for brevity by $D(n)$ the functor of deformations of $n:(\widetilde C,q)\to (C,p)$ (that is, we are allowed to deform $C$ and $\tilde C$ as well as the map). {\bf Lemma D.1.} {\sl $D(n)$ is smooth.}\\ {\bf Proof.} The morphism $D((C,p)\to ({\bf C}^2,0)) \longrightarrow D(n)$ (given by taking the image of the deformation of the map) is smooth. Hence it is enough to verify that $D((C,p)\to ({\bf C}^2,0)$ is smooth, and this is obvious.\hfill $\Diamond$ {{\bf Theorem D.2.} (\cite{T}, \cite{D-H}).} {\sl Let $(C,p)\subset ({\bf C}^2,0)$ be a reduced plane curve singularity, $n:(\widetilde C,q) \longrightarrow(C,p) $ its normalisation. Let $B(C,p)$ be a semi-universal family for $D(C,p)$ and $$B^{\delta}(C,p)\subset B(C,p)$$ the $\delta$-constant stratum. Then one has:\\ {(1)} The normalisation $\widetilde B^{\delta}(C,p)$ of $B^\delta(C,p)$ is a smooth space. \\ {(2)} The pullback of the semi-universal family to $\tilde B^\delta$ admits a simultaneous resolution of singularities. This makes $\widetilde B^{\delta}(C,p)$ into a semi-universal family for $D(n)$.\\ {(3)} The codimension of $B^\delta\subset B$ is $\delta(C,p)$. Over the generic point $p\in B^{\delta}$, the curve $C_p$ has precisely $\delta(C,p)$ double points as its only singularities.\\ (4) The tangent cone to the $\delta$-constant stratum is supported on $V_p$, the vector subspace generated by the conductor ideal. \hfill $\Diamond$} The second half of (2) is in fact not explicitly stated in either of \cite{T}, \cite{D-H}; however it follows easily from Lemma D.1. A similar argument is presented in the proof of Proposition F.2, so we don't repeat it here. \section{E. Proof of Theorem 1.} Let $C$ be a reduced projective rational curve with only planar singularities. We want to show that $e(\bar JC)=m(C)$. In particular let $(C,p)$ be a reduced plane curve singularity. Let $C$ be a projective rational curve that has $(C,p)$ as its only singular point. Then it follows that $e(\bar JC)=m(C,p)$. Let $\Phi:{\cal C}\longrightarrow B$ be a semi-universal family of deformations of $C$; we denote its fibres by $C_s=\Phi^{-1}(s)$, $C_0=C$. Let $\pi:{\overline{J}}{\cal C} \longrightarrow B$ be the corresponding family of compactified Jacobians. We always assume that we have chosen discs as representatives for the corresponding germs. We may also assume that the induced morphism $j:B\longrightarrow B^{loc}$ is smooth and has contractible fibres. We choose a section $\sigma:B^{loc}\longrightarrow B$ of $j$ with $\sigma(0)=0$. We will denote $\overline B:=\sigma(B^{loc})$, $\overline B^\delta:=\sigma(B^\delta)$ and $\overline V:=\sigma(V)$. Let $(W,0)\subset (\overline B,0)$ be a smooth subspace of dimension $\delta+1$ containing the point $(0,0)$ together with a smooth map $\lambda:(W,0)\longrightarrow (T,0)$ to a disc $(T,0)\subset ({\bf C},0)$. $W$ is a one parameter family of $\delta$-dimensional subspaces $W_t=\lambda^{-1}(t)\subset \overline B$. We require in addition that $W_0$ is transverse to $V$. \bigskip \epsfxsize 4cm \epsfysize 4cm \centerline{\epsfbox{del.eps}} \bigskip By {Theorem D.2} we can choose $W$ in such a way that for $t\ne 0$ the fibre $W_t$ intersects $\overline B^\delta$ in $mult(B^{\delta})$ points, and for $s\in W_t\cap \overline B^\delta$ the corresponding curve $C_s$ has precisely $\delta$ nodes as singularities. For $s\in W_t \setminus \overline B^\delta$ the curve $C_s$ will have positive genus. Let $\bar \Delta\subset B$ be a closed disc, and let $Z =W\cap \bar\Delta$. We define the family $\rho:{\overline{J}}{\cal C}_Z\longrightarrow T$ by pullback: \def\mapd#1{\Big\downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} $$\begin{array}{ccccc} &&{\overline{J}}{\cal C}_Z&\longrightarrow&{\overline{J}}{\cal C}\\ &\stackrel{\rho}{\swarrow}&\mapd{\pi}&&\mapd{\pi}\\ T&\stackrel{\lambda}{\longleftarrow}&Z&\longrightarrow&B \end{array} $$ As we have chosen $W_0$ to be transversal to $V$, Proposition C.5 implies that $\rho$ is smooth along $\pi^{-1}(0)$; by making $\bar\Delta$ and $T$ smaller we can assume that $\rho$ is smooth. As $\rho$ is also proper, all the fibres $\rho^{-1}(t)$ are diffeomorphic, in particular they all have the same Euler number. The space $\rho^{-1}(t)$ is the union, for $s\in W_t$, of $\bar JC_s$. We know that if $C_s$ has positive geometric genus, then $e(\bar JC_s)$ is zero; arguing as in \cite{B}, we obtain that $$ e(\rho^{-1}(t))=\sum_{s\in W_t\cap \bar B^\delta}e(\bar JC_s)$$ (note that if $s\in W_t$, then $C_s$ is rational if and only if $s\in \bar B^\delta$).\\ The intersection of $W_0\subset \overline B$ with $\overline B^\delta$ consists only of the point $0$ corresponding to the curve $C$. Therefore $e(\rho^{-1}(0))=e({\overline{J}} C)$.\\ On the other hand, for $t\ne 0$, $W_t$ intersects $\overline B^{\delta}$ in $mult( B^\delta)$ points and for $s\in \overline B^{\delta}\cap W_t$ the curve $C_s$ has precisely $\delta$ nodes as singularities. As for a nodal rational curve $C_s$, the Euler number $e({\overline{J}} C_s)$ is equal to $1$, we obtain $$e(\rho^{-1}(t))= \sum_{s\in W_t}e({\overline{J}} C_s)=\sum_{s\in W_t\cap \overline B^{\delta}}1= mult(B^\delta).$$ So we get $$e({\overline{J}} C)=e(\rho^{-1}(0))=e(\rho^{-1}(t))=mult(B^\delta).$$ \hfill$\Diamond$ \section{F. The invariant as length of moduli of stable maps} Let $C$ be a reduced projective curve with only plane curve singularities; let $n:\tilde C\to C$ be its normalisation, and $g$ the genus of $\tilde C$. Let $m(C)=\prod m(C,p)$. The scheme $\overline M_{g,0}(C,[C])$ parametrizing stable birational maps from a genus $g$ curve to $C$ contains only one point, namely the normalisation of $C$. The aim of this section is to prove that its length is equal to $m(C)$. Note that if $C$ is an isolated rational curve inside a smooth manifold $Y$, $\overline M_{g,0}(C,[C])$ is naturally a closed subscheme of $\overline M_{g,0}(Y,[C])$; in particular, $m(C)$ is a lower bound for the length of the corresponding component of $M_{g,0}(Y,[C])$ (in case this scheme also has dimension zero). Denote by $D(n)$ the deformation functor of the triple $(n:\widetilde C\to C)$, and by $D^{loc}(n)$ the corresponding local deformation functor. As before $D^{loc}(n)$ is the product over the singular points $p$ of $C$ of $D(n,p)$, the deformation functor of the triple $n:(\tilde C, n^{-1}(p))\to (C,p)$. If $(C,p)$ is the germ of a planar reduced curve singularity, then $D(n,p)$ is a smooth functor (see section D). {\bf Lemma F.1.} {\sl The natural morphism of functors $D(n)\to D^{loc}(n)\times_{D^{loc}(C)}D(C)$ is an isomorphism.} {\bf Proof.} Let $C_T$ be an infinitesimal deformation of $C$, and let $U_i$ be an open cover of $C$ such that $U_{ij}$ is smooth for each $i\ne j$. Let $V_i=n^{-1}(U_i)$. Let $U_{i,T}$ be the deformation of $U_i$ induced by $C_T$, and assume we are given a deformation $n_{i,T}:V_{i,T}\to U_{i,T}$ of $n_i:=n|_{V_i}$. Then to lift $(C_T,n_{i,T})$ to a deformation of $n$ we must choose gluing isomorphisms $\psi_{ij}:V_{ij,T}\to V_{ji,T}$ satisfying the cocycle condition and compatible with the other data, namely the maps $n_{i,T}$ and the gluing isomorphisms $\phi_{ij}:U_{ij,T}\to U_{ji,T}$ induced by $C_T$. But $U_{ij}$ is smooth, so that $n|_{V_{ij}}$ is an isomorphism for each $i\ne j$; hence the $\psi_{ij}$ are univocally determined by the $\phi_{ij}$ and automatically satisfy the cocycle condition. \hfill $\Diamond$ Let us now denote by $B(\cdot)$ the germ of complex space being a hull for the functor $D(\cdot)$. Note that Lemma F.1 implies that there is a cartesian diagram $$ \cdia{B(n)}{B(C)}{B^{loc}(n)}{B^{loc}(C).}$$ {\bf Proposition F.2.} {\sl Let $C$ be a reduced projective curve with planar singularities, $n:\tilde C\to C$ be the normalisation, $g=g(\tilde C)$. Let $\pi:{\cal C}\to B(C)$ be a semi-universal deformation of $C$. Denote by $M= M_{g,0}({\cal C},[C])$; then $M$ is smooth at $n$, and the natural map $M\to B^\delta:=B^\delta(C)$ is the normalisation map.} {\bf Proof.} Write $M$ for the germ of $M$ at $n$. As the domain of $n$ is a smooth curve, the same is true for all stable maps in a neighborhood of $n$. Hence $M$ is isomorphic to $B(n)$. By Lemma F.1, together with Lemma D.1, we deduce that $B(n)$ is smooth. By the definition of $B^\delta$ the natural map $M\to B(C)$ factors via $B^\delta$, hence, as $M$ is smooth, via its normalisation $\tilde B^\delta$. On the other hand, we know that the family $\tilde {\cal C}\to \tilde B^\delta$ gotten by pullback admits a very weak simultaneous resolution of singularities \cite{T}, inducing a morphism $\tilde B^\delta\to M$. It is easy to check pointwise that these two morphisms are inverse to each other (both $\tilde B^\delta$ and $M$ just parametrize the normalisation maps of the fibres of $\pi$). As both $\tilde B^\delta$ and $M$ are smooth, a bijective morphism must be an isomorphism. \hfill $\Diamond$ {\bf Proof of Theorem 2}. The scheme $M_{g,0}(C,[C])$ is the fibre over the point $[C]$ of the morphism $\tilde B^\delta\to B^\delta$; this is the multiplicity of $B^\delta$ at $[C]$, as $\tilde B^\delta$ is smooth. This proves the first equality. Let now $X$ be a smooth projective surface, $C\subset X$ a reduced irreducible curve, $n:\tilde C\to C$ the normalisation, $g=g(\tilde C)$. Assume that $n$ is an isolated point of $\overline{M}_{g,0}(X,[C])$, and let $M_n$ be the connected component of $n$. $M_n$ contains $M_{g,0}(C,[C])$ as a closed subscheme, so we always have an inequality $$ l(M_n)\ge l(M_{g,0}(C,[C]))=m(C).$$ This inequality is an equality if and only the natural morphism $M_n\to Hilb(X)$ sending each map to its image factors scheme theoretically (and not only set-theoretically) via $C$. Hence to complete the proof of Theorem 2, it is enough to show that this is the case if $C$ is rational and $X$ is a $K3$ surface. Let $S$ be the complete linear system defined by $C$ on $X$, and let ${\cal C}\to S$ be the universal curve. It is known that $\bar J{\cal C}$ is smooth, see \cite{Mu}; but this means precisely that $S$ maps transverse to the $\delta$-constant stratum in $B(C)$, and we are done in view of Corollary B.3.\hfill $\Diamond$ \section{G. Examples.} \def\ell{\ell} \defM{M} \def\alpha{\alpha} \def{\bf A}{{\bf A}} \def{\bf P}{{\bf P}} {\bf Example 1 (Beauville):} {\sl Let $(C,o)$ be the singularity of equation $x^q=y^p$, with $p<q$ and $(p,q)=1$. Then $$m(C,o)= {{1}\over{p+q}}{p+q\choose p}.$$} {\bf Proof.} We write for simplicity $\overline M(X,\beta)$ instead of $\overline M_{0,0}(X,\beta)$; if $X$ is a curve and $\beta=[X]$ we omit it. Let $C$ be the plane curve of equation $y^pz^{q-p}=x^q$. $C$ is a rational curve with two singular points, $o=(0,0,1)$ and $\infty=(1,0,0)$. Let $\alpha:C'\to C$ be the partial normalisation of $C$ at $\infty$. By Theorem 2, it is enough to prove that $$l(\overline M(C'))={{1}\over{p+q}}{p+q\choose p}=:N(p,q).$$ The natural map $\overline M(C')\to \overline M(C)$ given by $\mu\mapsto \alpha\circ \mu$ is a closed embedding, and the closed subscheme $\overline M(C')$ is identified by requiring the deformation of the normalisation morphism to be locally trivial near $\infty$. On the other hand $\overline M(C)$ is naturally a closed subset of $\overline M({\bf P}^2,q\ell)$, where $\ell$ is the class of a line. Let $n:{\bf P}^1\to C$ be the normalisation map, and choose coordinates on ${\bf P}^1$ such that $n(s,t)=(t^ps^{q-p},t^q,s^q)$. A morphism in $\overline M({\bf P}^2,q\ell)$ near $n$ has equations $$(t^ps^{q-p}+x,t^q+y,s^q+z),$$ for suitable homogeneous polynomials $x,y,z$ of degree $q$. We impose the conditions that the image of the map be contained in $C$ and that the deformation be locally trivial at $\infty$. Then we eliminate the indeterminacy generated by a reparametrization of ${\bf P}^1$ and a rescaling of the coordinates on ${\bf P}^2$. We get that all deformations of $n$ in $\overline M(C)$ must be (in affine coordinates where $z=1$) of the form $$ t\mapsto (t^p+\textstyle\sum\limits_{i=0}^p x_i t^i,t^q+ \textstyle\sum\limits_{i=0}^qy_it^i).$$ Hence we are now left with the following problem: compute the length of the ${\bf C}$--algebra with generators $x_0,\ldots,x_{p-2},y_0,\ldots,y_{q-2}$ and relations given by the coefficients of the polynomial $f^q-g^p$, where $f=t^p+\sum x_it^i$ and $g=t^q+\sum y_it^i$. It is easy to check that the equation $f^q=g^p$ is equivalent to $qf'g=pg'f$ by taking $d/dt\circ\log$ on both sides. The $t$-degree of $qf'g-pg'f$ is $p+q-1$, however we only get $p+q-2$ equations as the coefficients of $t^{p+q-1}$ and $t^{p+q-2}$ are zero anyway. Moreover, if we consider the variables $x_i$ (resp.\ $y_i$) as having degree $p-i$ (resp.\ $q-i$), the equations we obtain are homogeneous of degree $2,\ldots,p+q-1$. Now we recall the weighted B\'ezout theorem, which says that if we have a zero-dimensional algebra given by $N$ homogeneous equations of degrees $e_j$ in $N$ weighted variables of degree $d_j$, then the length of the algebra is $\prod e_j/\prod d_j$. Applying the formula in our case, with $N=p+q-2$, $(d_j)=(2,3,\ldots,p,2,3,\ldots,q)$ and $e_j=(2,3,\ldots,p+q-1)$ gives $$ N(p,q)={\prod e_j\over \prod d_j}={(p+q-1)!\over p!q!} ={1\over p+q}{p+q\choose p}.$$ {\bf Example 2.} We would like to outline an algorithm for the computation of $m(C,p)$ for a planar, reduced and irreducible curve singularity $(C,p)$. Assume we know how to realize $(C,p)$ as singularity of a rational curve. It is then easy to realize it as singularity of a plane rational curve $C$, whose other singularities are only nodes. Let $d$ be the degree of the curve, $F(x,y,z)=0$ its equation, and $\bar n=(\bar x,\bar y,\bar z)$ an explicit normalisation given by homogeneous polynomials of degree $d$ in $s,t$. Assume without loss of generality that $\bar z$ contains the monomial $s^d$ with nonzero coefficient. Then we can describe the scheme $M_{0,0}(C,[C])$ explicitly as follows. Choose three points $p_i$ ($i=1,2,3$) in ${\bf P}^1$ mapping via $n$ to smooth points of $C$; let $L_i\subset {\bf P}^2$ be a line transversal to $C$ at $n(p_i)$. Choose variables $x_i$, $y_i$ and $z_i$ for $i=0,\ldots d$, and let $x$ be the polynomial $\bar x+\sum_i x_i s^it^{d-i}$; define $y$ and $z$ in a similar way. Then $M_{0,0}(C,[C])$ is naturally isomorphic the subscheme of $ \hbox{\sl Spec}\, {\bf C} [x_i,y_i,z_i]$ defined by the equations $$\begin{array}{c} z_d=0,\\ (x,y,z)(p_i)\in L_i \qquad i=1,2,3,\\ F(x,y,z)=0. \end{array}$$ In fact, all deformations of $\bar n$ are again morphisms of degree $d$ from ${\bf P}^1$ to ${\bf P}^2$, hence are given by polynomials of degree $d$. The first four equations, defining a linear subspace, correspond to choosing local coordinates near $\bar n$ on $M_{0,0}({\bf P}^2,d)$; the last one, which is a system of $d^2$ equations, imposes the condition that the scheme-theoretic image of the morphism be contained in $C$.\\

9708

alg-geom/9708007

en

https://arxiv.org/abs/alg-geom/9708007

alg-geom/9708007

Yuri G. Zarhin

Yuri G. Zarhin

Torsion of abelian varieties, Weil classes and cyclotomic extensions

LaTeX 2e 17 pages

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that under certain conditions on $X$ and $K$ the existence of infinitely many L-rational points of finite order on $X$ implies that the intersection of $L$ and $K(c)$ has infinite degree over $K$.

alg-geom

\section{Main construction} Let $F$ be the center of $\mathrm{End}_K(X)\otimes{\mathbf Q}$, $R_F=F\bigcap \mathrm{End}_K(X)$ the center of $\mathrm{End}_K(X)$. We put $$V_{{\mathbf Z}}=V_{{\mathbf Z}}(X)=H_1(X({\mathbf C}),{\mathbf Z}), \quad V=V(X)=H_1(X({\mathbf C}),{\mathbf Q})= V_{{\mathbf Z}}\otimes{\mathbf Q}.$$ For each nonnegative integer $m$ one may naturally identify the $m$th rational cohomology group $H^m(X({\mathbf C}),{\mathbf Q})$ of $X({\mathbf C})$ with $\mathrm{Hom}_{{\mathbf Q}}(\Lambda^m_{{\mathbf Q}}(V(X),{\mathbf Q})$. For each prime $\ell$ there are natural identifications $$X_{\ell}=V_{{\mathbf Z}}/\ell V_{Z}, T_{\ell}(X)=V_{Z}\otimes{\mathbf Z}_{\ell}, V_{\ell}(X)=V(X)\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}=V_{{\mathbf Z}}\otimes{\mathbf Q}_{\ell}.$$ There is a natural Galois action $$\rho_{\ell}=\rho_{\ell,X}:G(K)\to \mathrm{Aut}_{{\mathbf Z}_{\ell}}(T_{\ell}(X)) \subset \mathrm{Aut}_{{\mathbf Q}_{\ell}}(V_{\ell}(X)),$$ induced by the Galois action on the torsion points of $X$ \cite{Serre}. One may naturally identify the $m$th $\ell-$adic cohomology group $H^m(X_a,{\mathbf Q}_{\ell})$ of $X_a=X\times K(a)$ with $$\mathrm{Hom}_{{\mathbf Q}_{\ell}}(\Lambda^m_{{\mathbf Q}_{\ell}}(V_{\ell}(X),{\mathbf Q}_{\ell})=\mathrm{Hom}_{{\mathbf Q}}(\Lambda^m_{{\mathbf Q}}(V(X),{\mathbf Q}))\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}).$$ This identification is an isomorphism of the Galois modules. Assume now that $F$ is a number field, i.e., $X$ is either simple or is isogenous over $K$ to a self-product of a simple abelian variety. Let $O_F$ be the ring of integers in $F$. It is well-known that $R_F$ is a subgroup of finite index in $O_F$. Recall that for each prime $\ell$ there is a splitting $F\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}=\oplus F_{\lambda}$ where $\lambda$ runs through the set of prime ideals dividing $\ell$ in $O_F$ and $F_{\lambda}$ is the completion of $F$ with respect to $\lambda-$adic topology. There is a natural splitting $V_{\ell}(X)=\oplus V_{\lambda}(X)$ where $$V_{\lambda}(X)=F_{\lambda} V_{\ell}(X) =V(X)\otimes_F F_{\lambda}.$$ It is well-known that all $V_{\lambda}(X)$ are $G(K)-$invariant $F_{\lambda}-$vector spaces of dimension $2\mathrm{dim}(X)/[F:{\mathbf Q}]$. We write $\rho_{\lambda,X}$ for the corresponding $\lambda-$adic representation $$\rho_{\lambda,X}:G(K) \to\mathrm{Aut}_{F_{\lambda}}V_{\lambda}(X)$$ of $G(K)$ \cite{Serre},\cite{RibetA}. Similarly, for all but finitely many $\ell$ $$R_F/\ell R_F=O_F/\ell O_F = \oplus_{\lambda\mid\ell} O_F/\lambda$$ is a direct sum of finite fields $O_F/\lambda$ of characteristic $\ell$. Also, $X_{\ell}=V_{{\mathbf Z}}/\ell V_{{\mathbf Z}}$ is a free $R_F/\ell R_F=O_F/\ell O_F-$module of rank $2\mathrm{dim}(X)/[F:{\mathbf Q}]$ and there is a natural splitting $$X_{\ell}=V_{{\mathbf Z}}/\ell V_{{\mathbf Z}}=\oplus_{\lambda\mid\ell} X_{\lambda}$$ where $X_{\lambda}=(O_F/\lambda) \cdot X_{\ell}.$ Clearly, each $X_{\lambda}$ is a $G(K)-$invariant $O_F/\lambda-$vector space of dimension $2\mathrm{dim}(X)/[F:{\mathbf Q}]$. We write $\bar{\rho}_{\lambda,X}$ for the corresponding modular representation $$\bar{\rho}_{\lambda,X}:G(K) \to\mathrm{Aut}_{O_F/\lambda}X_{\lambda}$$ of $G(K)$ \cite{RibetA}. Let $d$ be a positive integer and assume that there exists a non-zero $2d-$linear form $\psi \in \mathrm{Hom}_{{\mathbf Q}}(\otimes^{2d}_{{\mathbf Q}} V(X),{\mathbf Q})$, enjoying the following properties. \begin{enumerate} \item For all $f\in F; v_1, \ldots v_{2d}\in V(X)$ $$\psi(f v_1,v_2,\ldots ,v_{2d})=\psi(v_1,fv_2,\ldots ,v_{2d})=\cdots = \psi(v_1,v_2,\ldots ,fv_{2d}).$$ \item For any prime $\ell$ let us extend $\psi$ by ${\mathbf Q}_{\ell}-$linearity to the non-zero multilinear form $\psi_{\ell} \in \mathrm{Hom}_{{\mathbf Q}_{\ell}}(\otimes^{2d}_{{\mathbf Q}_{\ell}} V_{\ell}(X),{\mathbf Q}_{\ell})$. Then for all $\sigma \in G(K); v_1, \ldots v_{2d}\in V_{\ell}(X)$ $$\psi_{\ell} (\sigma(v_1),\sigma(v_2),\ldots,\sigma(v_{2d}))= \chi_{\ell}^d(\sigma)\psi_{\ell}(v_1,v_2,\ldots ,v_{2d}).$$ \end{enumerate} We call such a form {\sl admissible} or $d-${\sl admissible}. \vskip .5cm {\bf Example.} Let us assume that $F$ is a {\sl totally real} number field. If $\mathcal L$ is an invertible sheaf on $X$ defined over $K$ and algebraically non-equivalent to zero then one may associate to $\mathcal L$ its first Chern class $$c_1({\mathcal L})\in H^2(X({\mathbf C}),{\mathbf Q})=\mathrm{Hom}_{{\mathbf Q}}(\Lambda^2_{{\mathbf Q}}(V(X),{\mathbf Q}).$$ The well-known properties of Rosati involutions and Weil pairings imply that $c_1({\mathcal L})$ is $1-$admissible (see p.~237 of \cite{MumfordAV}, especially the last sentence and Section 2 of \cite{SZMathZ}). \vskip .5cm There exists a unique $F-2d-$linear form $\psi_F\in \mathrm{Hom}_F(\otimes^{2d}_F V(X),F)$ such that $$\psi={\mathbf{Tr}}_{F/{\mathbf Q}}(\psi_F).$$ Multiplying $\psi$ by a sufficiently divisible positive integer, we may and will assume that the restriction of $\psi_F$ to $V_{{\mathbf Z}}\times \cdots V_{{\mathbf Z}}$ takes on values in $R_F$. Let $\mathrm{Im}(\psi_F)$ be the additive subgroup of $R_F$ generated by values of $\psi_F$ on $V_{{\mathbf Z}}\times \cdots V_{{\mathbf Z}}$ takes on values in $R_F$. Let $\mathrm{Im}(\psi_F)$ be the additive subgroup of $R_F$ generated by values of $\psi_F$ on $V_{{\mathbf Z}}\times\cdots V_{{\mathbf Z}}$. Clearly, $\mathrm{Im}(R_F)$ is a subgroup of finite index in $R_F$ that is an ideal. Notice that for all but finitely many primes $\ell$ $$O_F=R_F/\ell R_F, \mathrm{Im}(\psi_F)=R_F/\ell R_F.$$ Let us extend $\psi_F$ by $F_{\lambda}-$linearity to the {\sl non-zero} multilinear form $$\psi_{F,\lambda} \in \mathrm{Hom}_{F_{\ell}}(\otimes^{2d}_{F_{\lambda}} V_{\lambda}(X),F_{\lambda}).$$ Then $$\psi_{F,\lambda}(\sigma(v_1),\sigma(v_2),\ldots,\sigma(v_{2d}))= \chi_{\ell}^d(\sigma)\psi_{F,\lambda}(v_1,v_2,\ldots ,v_{2d})$$ for all $\sigma \in G(K); v_1, \ldots v_{2d}\in V_{\lambda}(X)$. Similarly, for all but finitely many $\ell$ the form $\psi_F$ induces a non-zero multilinear form $$\psi_{F}^{(\ell)} \in \mathrm{Hom}_{R_F/\ell R_F}(\otimes^{2d}_{R_F/\ell R_F} X_{\ell},R_F/\ell R_F)$$ enjoying the following properties: \begin{itemize} \item The subgroup of $R_F/\ell R_F$ generated by all the values of $\psi_{F}^{(\ell)}$ coincides with $R_F/\ell R_F$; \item For all $\sigma \in G(K); v_1, \ldots v_{2d}\in X_{\ell}$ $$\psi_{F}^{(\ell)}(\sigma(v_1),\sigma(v_2),\ldots,\sigma(v_{2d}))= \bar{\chi}_{\ell}^d(\sigma)\psi_{F}^{(\ell)}(v_1,v_2,\ldots ,v_{2d}).$$ \end{itemize} This implies that for all but finitely many $\ell$ the restriction of $\psi_{F}^{\ell}$ to $X_{\lambda}$ defines a {\sl non-zero} multilinear form $$\psi_{F}^{(\lambda)} \in \mathrm{Hom}_{O_F/\lambda}(\otimes^{2d}_{O_F/\lambda} X_{\ell},O_F/\lambda)$$ enjoying the following property: $$\psi_{F}^{(\lambda)}(\sigma(v_1),\sigma(v_2),\ldots,\sigma(v_{2d}))= \bar{\chi}_{\ell}^d(\sigma)\psi_{F}^{(\lambda)}(v_1,v_2,\ldots ,v_{2d})$$ for all $\sigma \in G(K); v_1, \ldots v_{2d}\in X_{\lambda}$. \begin{rem} Using the K\"unneth formula for $X_a^{2d}$, one may view $\psi_{\ell}$ as a Tate cohomology class on $X_a^{2d}$. If $\psi$ is skew-symmetric then $\psi_{\ell}$ is a Tate cohomology class on $X_a$. \end{rem} \begin{thm} Assume that the center $F$ of $\mathrm{End}^0 X$ is a field and there is a $d-$admissible form $\psi$ on $X$. Let $\ell$ be a prime and assume that the $\ell-$ torsion in $X(L)$ is infinite. If $L^{(\ell)}$ is the intersection of $L$ and $K(\ell)$ then the field extension $K(\ell)/L^{(\ell)}$ has finite degree dividing $(d,\ell-1)$ if $\ell$ is odd and dividing $2$ if $\ell=2$ respectively. In addition, $L$ contains ${\mathbf Q}(\ell)'$. \end{thm} \begin{proof} As explained in (\cite{ZarhinMA}, 0.8, 0.11) the assumption that the $\ell-$torsion in $X(L)$ is infinite means that there exists a place $\lambda$ of F, dividing $\ell$ such that the Galois group $G(L)$ of $L$ acts trivially on $V_{\lambda}(X)$. Since $\psi_{F,\lambda}$ is not identically zero, we conclude that $$\chi_{\ell}^d(\sigma)=1 \quad \forall \sigma \in G(L) \subset G(K).$$ We write $G'$ for the kernel of $\chi_{\ell}^d$. We have $G(L)\subset G'\subset G(K)$. Recall that the kernel of $\chi_{\ell}:G(K) \to {\mathbf Z}_\ell^*$ coincides with the Galois group $G(K(\ell))$ of $K(\ell)$ and $\chi_{\ell}$ identifies $\mathrm{Gal}(K(\ell)/K)$ with a subgroup of ${\mathbf Z}_\ell^*=\mathrm{Gal}({\mathbf Q}(\ell)/{\mathbf Q})$. Since the torsion subgroup of ${\mathbf Z}_\ell^*$ is the cyclic group $\mu({\mathbf Z}_{\ell})$ of order $\ell-1$ if $\ell$ is odd and of order $2$ if $\ell=2$, $G'$ coincides with the kernel of $(\chi_{\ell})^{d'}$ with $d'=(d,\ell-1)$ if $\ell$ is odd and $d'=(d,2)$ if $\ell=2$ respectively. This implies that the field $K'=K(a)^{G'}$ of $G'-$invariants is a subfield of $K(\ell)$ and $[K(\ell):K']$ divides $d'$, since $\chi_{\ell}$ establishes an isomorphism between $\mathrm{Gal}(K(\ell)/K')$ and $$\{s \in \mathrm{Im}(\chi_{\ell})\subset {\mathbf Z}_{\ell}^*\mid s^{d'}=1\} \subset \{s \in \mu({\mathbf Z}_{\ell})\mid s^{d'}=1\}.$$ Now it is clear that $K'\subset L$, since $G(L) \subset G'=G(K')$. It is also clear that $K(\ell)/K'$ is a cyclic extension of degree dividing $d'$. In order to prove the last assertion of the theorem, notice that $\mathrm{Gal}(K(\ell)/K) \subset \mathrm{Gal}({\mathbf Q}(\ell)/{\mathbf Q})={\mathbf Z}_{\ell}^*$ and the finite subgroup $\mathrm{Gal}(K(\ell)/K')$ of $\mathrm{Gal}(K(\ell)/K)$ sits in $\mu({\mathbf Z}_{\ell})\subset{\mathbf Z}_{\ell}^*$. Since $\mu({\mathbf Z}_{\ell})=\mathrm{Gal}({\mathbf Q}(\ell)/{\mathbf Q}(\ell)')$, ${\mathbf Q}(\ell)'\subset K'$. Since $K'\subset L$, ${\mathbf Q}(\ell)'\subset L$. \end{proof} \begin{thm} Assume that the center $F$ of $\mathrm{End}^0 X$ is a field and there is a $d-$ admissible form $\psi$ on $X$. Let $S$ be an infinite set of primes $\ell$ such that for all but finitely many $\ell\in S$ the $\ell-$torsion in $X(L)$ is not zero. Then for all but finitely many $\ell\in S$ the field extension $K(\mu_{\ell})/K(\mu_{\ell})\bigcap L$ has degree dividing $(d,\ell-1)$. \end{thm} \begin{proof} For all but finitely many $\ell$ the $G(K)-$module $X_{\ell}$ is semisimple and the centralizer of $G(K)$ in $\mathrm{End}(X_{\ell})$ coincides with $\mathrm{End}_K(X)\otimes {\mathbf Z}/\ell{\mathbf Z}$. This assertion was proven in \cite{ZarhinInv} for number fields $K$; the proof is based on results of Faltings \cite{Faltings1}. (See \cite{MW} for an effective version.) However, the same proof works for arbitrary finitely generated fields $K$, if one uses results of \cite{Faltings2}, generalizing the results of \cite{Faltings1}. Clearly, for all but finitely many $\ell$ the center of $\mathrm{End}_K(X)\otimes {\mathbf Z}/\ell{\mathbf Z}$ coincides with $R_F/\ell R_F=O_F/\ell O_F$. Applying Theorem 5f of \cite{ZarhinDuke} to $G=G(K), G'=G(L), H=X_{\ell}, D=\mathrm{End}_K(X)\otimes {\mathbf Z}/\ell{\mathbf Z}, R=F_F/\ell R_F$, we conclude that for all but finitely many $\ell \in S$ there exists $\lambda\mid\ell$ such that $G(L)$ acts trivially on $X_{\lambda}$. Using the Galois equivariance of the non-zero form $\psi_{F}^{(\lambda)}$, we conclude that for all but finitely many $\ell\in S$ the character $\bar{\chi}_{\ell}^d$ kills $G(L)$. We write $G'$ for the kernel of $\bar{\chi}_{\ell}^d$. We have $G(L)\subset G'\subset G(K)$. Recall that the kernel of $\bar{\chi}_{\ell}:G(K) \to ({\mathbf Z}/\ell {\mathbf Z})^*$ coincides with $G(K(\mu_{\ell}))$ and $({\mathbf Z}/\ell {\mathbf Z})^*$ is a cyclic group of order $\ell-1$. This implies that the field $K'=K(a)^{G'}$ of $G'-$invariants is a subfield of $K(\mu_{\ell})$ and $[K(\mu_{\ell}):K']$ divides $(\ell-1,d)$, since $\bar{\chi}_{\ell}$ establishes an isomorphism between $\mathrm{Gal}(K(\mu_{\ell})/K')$ and $\{s \in \mathrm{Im}(\bar{\chi}_{\ell})\subset ({\mathbf Z}/\ell {\mathbf Z})^*\mid s^d=1\}$. One has only to notice that $K'\subset L$, since $G(L) \subset G'=G(K')$. \end{proof} \begin{cor} Assume that the torsion subgroup of $X(L)$ is infinite. Then the intersection of $L$ and $K(c)$ has infinite degree over $K$. \end{cor} \begin{proof} Indeed, either there is a prime $\ell$ such that the $\ell-$torsion in $X(L)$ is infinite or for infinitely many primes $\ell$ the $\ell-torsion$ in $X(L)$ is not zero. Now, one has only to apply the previous two theorems. \end{proof} \section{Weil classes and admissible forms} Suppose $A$ is an abelian variety defined over $K$, $k$ is a CM-field, $\iota : k \hookrightarrow \mathrm{End}_K^0(A)$ is an embedding, and $C$ is an algebraically closed field containing $K$ (for instance, $C={\mathbf C}$ or $C=\bar{{\mathbf Q}}$). Let $\mathrm{Lie}(A)$ be the tangent space of $A$ at the origin, an $K$-vector space. If $\sigma$ is an embedding of $k$ into $C$, let $$n_\sigma = \mathrm{dim}_C\{t \in \mathrm{Lie}(A)\otimes_K C : \iota(\alpha)t = \sigma(\alpha)t {\text{ for all }} \alpha \in k\}.$$ Write ${\bar \sigma}$ for the composition of $\sigma$ with the involution complex conjugation of $k$. Recall that a triple $(A,k,\iota)$ is {\em of Weil type} if $A$ is an abelian variety over an algebraically closed field $C$ of characteristic zero, $k$ is a CM-field, and $\iota : k \hookrightarrow \mathrm{End}^0(A)$ is an embedding, such that $n_\sigma = n_{\bar \sigma}$ for all embeddings $\sigma$ of $k$ into $C$. It is known \cite{SZMathZ} that $(A,k,\iota)$ is of Weil type if and only if $\iota$ makes $\mathrm{Lie}(A) \otimes_K C$ into a free $k \otimes_{\mathbf Q} C$-module (see p.~525 of \cite{Ribet} for the case where $k$ is an imaginary quadratic field). Now, assume that $A=X$ and the image $\iota(k)$ contains the center $F$ of $\mathrm{End}_K(X)\otimes{\mathbf Q}$ (for instance, $F=k$). Notice that in the case of Weil type the degree $[k:{\mathbf Q}]$ divides $\mathrm{dim}(A)$. In particular, $\mathrm{dim}(A)$ is even. Our goal is to construct an admissible form, using a triple $(A,k,\iota)$ of Weil type. Recall that the degree $[k:{\mathbf Q}]$ divides $\mathrm{dim}(A)$, put $d=\mathrm{dim}(X)/[k:{\mathbf Q}]$ and consider the space of Weil classes (\cite{WeilHodge}, \cite{Deligne}, \cite{SZMathZ}) $$W_{k,X}=\mathrm{Hom}_k(\Lambda_k^{2d} V(X),{\mathbf Q}(d)) \hookrightarrow \mathrm{Hom}_{{\mathbf Q}}(\Lambda_{{\mathbf Q}}^{2d}V(X),{\mathbf Q}(d))=H^{2d}(X({\mathbf C}),{\mathbf Q})(d).$$ Clearly, $W_{k,X}$ carries a natural structure of one-dimensional $k-$vector space. If fix an isomorphism of one-dimensional ${\mathbf Q}-$vector spaces ${\mathbf Q} \cong {\mathbf Q}(2d)$ then one may naturally identify $\mathrm{Hom}_{{\mathbf Q}}(\Lambda_{{\mathbf Q}}^{2d}V(X),{\mathbf Q}(d))$ with $\mathrm{Hom}_{{\mathbf Q}}(\Lambda_{{\mathbf Q}}^{2d}V(X),{\mathbf Q})$ and $W_{k,X}$ can be described as the space of all $2d-$linear skew-symmetric form $\psi \in \mathrm{Hom}_{{\mathbf Q}}(\Lambda^{2d}_{{\mathbf Q}} V,{\mathbf Q})$ with $$\psi(f v_1,v_2,\ldots ,v_{2d})=\psi(v_1,fv_2,\ldots ,v_{2d})=\cdots = \psi(v_1,v_2,\ldots ,fv_{2d})$$ for all $f\in F; v_1, \ldots v_{2d}\in V(X)$. Since $(X,k.\iota)$ is of Weil type, all elements of $W_k$ are Hodge classes by Proposition 4.4 of \cite{Deligne}. Therefore, by Theorem 2.11 of \cite{Deligne} they must be also {\sl absolute Hodge cycles}; cf. \cite{Deligne}. \begin{lem} Let $\mu_k$ be the finite multiplicative group of all roots of unity in $k$. There is a continuous character $\chi_{X,k}:G(K) \to \mu_k \subset k^*,$ enjoying the following properties: For each prime $\ell$ the subgroup $$W_k \subset W_{k,X}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}\subset H^{2d}(X({\mathbf C}),{\mathbf Q})(d)\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell} =H^{2d}(X_a,{\mathbf Q}_{\ell})(d)$$ is $G(K)-$stable and the action of $G(K)$ on $W_k$ is defined via the character $$\chi_{X,k}:G(K) \to \mu_k \subset k^* =\mathrm{Aut}_k(W_{k,X}).$$ \end{lem} \begin{proof} Since all elements of $k$ are endomorphisms of $X$ defined over $K$, it follows easily that $W_{k,X}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ is $G(K)-$stable and $G(K)$ acts on $W_{k,X}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ via a certain character $\chi_{X,k,\ell}:G(K) \to [k\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}]^*= \Pi_{\lambda\mid \ell}k_{\lambda}^*.$ Let us consider the ${\mathbf Q}-$vector subspace $$C^d_{\mathrm{AH}}(X)\subset H^{2d}(X({\mathbf C}),{\mathbf Q})(d)\subset H^{2d}(X_a,{\mathbf Q}_{\ell})(d)$$ of absolute Hodge classes. Then $C^d_{\mathrm{AH}}(X)$ is $G(K)-$stable and the action of $G(K)$ on $C^d_{\mathrm{AH}}(X)$ does not depend on $\ell$ and factors through a finite quotient; cf. \cite{Deligne}, Prop. 2.9b. Since $W_{k,X}$ consists of Hodge classes and $X$ is an abelian variety, all Weil classes are absolute Hodge classes, i.e, $W_{k,X}\subset C^d_{\mathrm{AH}}(X),$ \cite{Deligne}, Th. 2.11. This implies easily that the subgroup $\mathrm{Im}(\chi_{X,k,\ell})$ is finite and contained in $k^*$, since the intersection of $W_{k,X}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ and $C^d_{\mathrm{AH}}(X)$ coincides with $W_{k,X}$. (In fact, $W_{k,X}$ coincides even with the intersection of $W_{k,X}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ and $H^{2d}(X({\mathbf C}),{\mathbf Q})(d)$.) This implies also that $\chi_{X,k,\ell}$ does not depend on the choice of $\ell$. So, we may view $\chi_{X,k,\ell}$ as the continuous homomorphism $$\chi_{X,k}:=\chi_{X,k,\ell}:G(K) \to \mu_k \subset k^*,$$ which does not depend on the choice of $\ell$. \end{proof} Let $r$ be the order of the finite group $\mathrm{Im}(\chi_{X,k})$. Clearly, $r$ divides the order of $\mu_k$. Let us put $Y=X^r$ and consider the K\"unneth chunk $$H^{2d}(X({\mathbf C}),{\mathbf Q})(d)^{\otimes r} \subset H^{2dr}(X({\mathbf C})^r,{\mathbf Q})(dr)=H^{2dr}(Y({\mathbf C}),{\mathbf Q})(dr)$$ of the $2dr$th rational cohomology group of $Y$. One may easily check that the tensor power $$W_{k,X}^{\otimes r}\subset H^{2d}(X({\mathbf C}),{\mathbf Q})(d)^{\otimes r} \subset H^{2dr}(X({\mathbf C})^r,{\mathbf Q})(dr)=H^{2dr}(Y({\mathbf C}),{\mathbf Q})(dr)$$ coincides with the space $W_{k,Y}$ of Weil classes on $Y$ attached to the ``diagonal" embedding $$k \to \mathrm{End}^0(X) \subset \mathrm{End}^0(X^r)=\mathrm{End}^0(Y).$$ Since the centers of $\mathrm{End}^0(X)$ and $\mathrm{End}^0(X^r)$ coincide, the image of $k$ in $\mathrm{End}^0(Y)$ does contain the center of $\mathrm{End}^0(Y)$. One may easily check that $G(K)$ acts on $W_{k,Y}=W_{k,X}^{\otimes r}$ via the character $\chi_{X,k}^r$, which is trivial, i.e., $W_{k,Y}$ consists of $G(K)-$invariants. Let us fix an isomorphism of one-dimensional ${\mathbf Q}-$vector spaces ${\mathbf Q} \cong {\mathbf Q}(2dr)$ and choose a {\sl non-zero} element $$\psi \in W_{k,Y} \subset H^{2dr}(Y,{\mathbf Q})(dr)=\mathrm{Hom}_{{\mathbf Q}}(\Lambda^{2dr}_{{\mathbf Q}}(V(Y),{\mathbf Q}).$$ Then a skew-symmetric $2dr-$linear form $\psi$ is admissible. Applying to $\psi$ the theorems of the previous section, we obtain the following statement, which implies the case 4 (in the hypothesis (H)) of Theorem \ref{Theorem 1}. \begin{thm} Assume that the center $F$ of $\mathrm{End}^0 X$ is a CM-field and $(X,F, \mathrm{id})$ is of Weil type. Let us put $d= \#(\mu_F) \times 2 \mathrm{dim}(X)/[F:{\mathbf Q}] \in {\mathbf Z}_{+}.$ Let $L$ be an infinite Galois extension of $K$. \begin{enumerate} \item Let $\ell$ be a prime such that the $\ell-$torsion in $X(L)$ is infinite. Let $L^{(\ell)}$ be the intersection of $L$ and $K(\ell)$. Then the field extension $K(\ell)/L^{(\ell)}$ has finite degree dividing $(d,\ell-1)$ if $\ell$ is odd and dividing $2$ if $\ell=2$ respectively. In addition, $L$ contains ${\mathbf Q}(\ell)'$. \item Let $S$ be the set of primes $\ell$ such that $X(L)$ contains a point of order $\ell$ and assume that $S$ is infinite. Then for all but finitely many $\ell\in S$ the field extension $K(\mu_{\ell})/K(\mu_{\ell})\bigcap L$ has degree dividing $(d,\ell-1)$. \end{enumerate} \end{thm} \begin{rem} Since $[F:{\mathbf Q}]$ divides $2\mathrm{dim}(X)=2g$, one may easily find an explicit positive integer $M=M(g)$, depending only on $g$ and divisible by $\#(\mu_F) \times 2 \mathrm{dim}(X)/[F:{\mathbf Q}]$ \end{rem} \section{Proof of Theorem \ref{Theorem 1}} We may and will assume that $X$ is $K-$simple. Then the center $F$ of $\mathrm{End}^0 X$ is either a totally real number field or a CM-field. If $F$ is totally real then the assertion of Theorem \ref{Theorem 1} is proven in \cite{ZarhinDuke} with $N=1$. So, further we assume that $F$ is a CM-field. We also know that the assertion of Theorem \ref{Theorem 1} is true when $(X,F)$ is of Weil type (Case 4 of Hypothesis (H)). \subsection{ Cases 1 and 3 of Hypothesis (H)} Enlarging $K$ if necessary, we may and will assume that $X$ is absolutely simple and has semistable reduction. Then, the results of \cite{SZ} imply that in both cases $\mathrm{Hdg}_X$ is semisimple. This means that $(X,F,\mathrm{id})$ is of Weil type (cf. for instance \cite{SZ}). Now, one has only to apply the result of the previous section with $d=\#(\mu_F) \times 2\mathrm{dim}(X)/[F:{\mathbf Q}]$ and get the assertion of Theorem \ref{Theorem 1} with $N=M(g)$. \subsection{Case 2 of Hypothesis (H)} We know that the assertion of the theorem is true if $(X,F,\mathrm{id})$ is of Weil type. So, we may assume that $(X,F,\mathrm{id})$ is not of Weil type. Let us consider the trace map $$\mathbf{Tr}_{\mathrm{Lie}(X)}: F \subset \mathrm{End}^0(X)\hookrightarrow\mathrm{End}_K(\mathrm{Lie}(X)) \to K\subset {\mathbf C}.$$ Our assumption means that the image $\mathbf{Tr}_{\mathrm{Lie}(X)}(F)$ is not contained in ${\mathbf R}$. On the other hand, let us fix an embedding of $F$ into ${\mathbf C}$ and let $L$ be the normal closure of $F$ into ${\mathbf C}$. Clearly, $L$ is a CM-field, containing $\mathbf{Tr}_{\mathrm{Lie}(X)}(F)$. Since $\mathbf{Tr}_{\mathrm{Lie}(X)}(F)\subset K$, the intersection $L\bigcap K$ contains an element, which is not totally real. Since any subfield of a CM-field is either totally real or CM, the field $L\bigcap K$ is a CM-subfield of $K$. \begin{rem} If $K$ is a number field not containing a CM-field, one may give another proof, using theory of abelian $\lambda-$adic representations \cite{Serre} instead of Weil/Hodge classes. The crucial point is that in this case the Serre's tori $T_{\mathfrak m}$ are isomorphic to the multiplicative group ${\mathbf G}_m$ \cite{Serre}, Sect. 3.4. \end{rem} \begin{cor} Let $X$ be a $K-$simple abelian variety of odd dimension. Assume that $K$ does not contain a CM-subfield (e.g., $K\subset {\mathbf R}$). If $X(L)$ contains infinitely many points of finite order then $L$ contains infinitely many roots of unity. \end{cor} \begin{proof} In the case of the totally real center $F$ this assertion is proven in (\cite{ZarhinDuke}, Th.6, p. 142) without restrictions on the dimension. So, in order to prove Corollary, it suffices to check that $F$ is not a CM-field. Assume that $F$ is a CM-field. Since $\mathrm{dim}(X)$ is odd, $(X,F,\mathrm{id})$ is not of Weil type. Now, the arguments, used in the proof of Case 2 imply that $K$ contains a CM-subfield. This gives us a desired contradiction. \end{proof} \begin{rem} The assertion of Corollary cannot be extended to the even-dimensional case. In Section \ref{roots} we give an explicit counterexample. \end{rem} \begin{rem} Let $X$ be a $g-$dimensional abelian variety that is not necessarily $K-$simple and let $F$ be the center of $\mathrm{End}^0(X)$. Assume that $$\mathbf{Tr}_{\mathrm{Lie}(X)}(F)\subset{\mathbf R}.$$ Then the assertion of Theorem \ref{Theorem 1} holds true for $X$. Indeed, if $Y$ is a $K-$simple abelian subvariety of $X$ and $F_Y$ is the center of $\mathrm{End}^0(Y)$ then one may easily check that either $F_Y$ is a totally real number field or $(Y,F_Y,\mathrm{id})$ is of Weil type. \end{rem} \section{Example} \label{roots} In this section we construct an abelian surface $X$ over ${\mathbf Q}$ and a Galois extension $L$ of ${\mathbf Q}$ such that $L$ contains only finitely many roots of unity but $X(L)$ contains infinitely many points of finite order. Of course, the intersection of $L$ and ${\mathbf Q}(c)$ is of infinite degree over ${\mathbf Q}$. \subsection{} Let $E$ be an elliptic curve over ${\mathbf Q}$ without complex multiplication (e. g., $j(E)$ is not an integer). Let us put $$Y=\{(e_1,e_2,e_3) \in E^3\mid e_1+e_2+e_3=0\}.$$ Clearly, $Y$ is an abelian surface over ${\mathbf Q}$ isomorphic to $E^2$. Denote by $s$ an automorphism of $Y$ induced by the cyclic permutation of factors of $E^3$, i.e., $$s(e_1,e_2,e_3)=(e_3,e_1,e_2) \quad \forall\ (e_1,e_2,e_3) \in Y.$$ Let $C$ be the cyclic subgroup in $\mathrm{Aut}(X)$ of order $3$ generated by $s$. By a theorem of Serre \cite{Serre2} the homomorphism $$\rho_{\ell,E}: G({\mathbf Q}) \to \mathrm{Aut}_{{\mathbf Z}_{\ell}}(T_{\ell}(E)) \cong \mathrm{GL}(2,{\mathbf Z}_{\ell})$$ is {\sl surjective} for all but finitely many $\ell$. Notice, that the composition $$\mathrm{det} \rho_{\ell,E}: G({\mathbf Q}) \to \mathrm{GL}(2,{\mathbf Z}_{\ell}) \to {\mathbf Z}_{\ell}^*$$ coincides with $\chi_{\ell}: G(K)\to {\mathbf Z}_{\ell}^*$ \cite{Serre2}. In particular, if ${\mathbf Q}(E(\ell^{\infty}))$ is the field of definition of all points on $E$ of $\ell$-power order then ${\mathbf Q}(E(\ell^{\infty}))/{\mathbf Q}$ is the Galois extension with the Galois group $\mathrm{GL}(2,{\mathbf Z}_{\ell})$. In addition, the cyclotomic field ${\mathbf Q}(\ell)$ is the {\sl maximal abelian} subextension of ${\mathbf Q}(E(\ell^{\infty}))$ and the subgroup $\mathrm{Gal}({\mathbf Q}(E(\ell^{\infty}))/{\mathbf Q}(\ell)) \subset \mathrm{Gal}({\mathbf Q}(E(\ell^{\infty}))/{\mathbf Q})$ coincides with $\mathrm{SL}(2,{\mathbf Z}_{\ell})$. Let us fix such an $\ell$, assuming in addition that $\ell-1$ is divisible by $3$ but not divisible by $9$. Let $\mu_{3,\ell}$ be the group of cubic roots of unity in ${\mathbf Z}_{\ell}^*$. Then there exists a continuous surjective homomorphism $\mathrm{pr}_3: {\mathbf Z}_{\ell}^* \to \mu_{3,\ell},$ coinciding with the identity map on $\mu_{3,\ell}$. These properties determine $\mathrm{pr}_3$ uniquely. Let us define field $L$ as a subextension of ${\mathbf Q}(E(\ell^{\infty}))$ such that ${\mathbf Q}(E(\ell^{\infty}))/L$ is a cubic extension, whose Galois (sub)group coincides with $$\mu_{3,\ell}\cdot\mathrm{id}=\{\gamma \cdot \mathrm{id}\mid \gamma \in \mu_{3,\ell}\} \subset \mathrm{GL}(2,{\mathbf Z}_{\ell}).$$ It follows immediately that $L$ is a Galois extension of ${\mathbf Q}$ and it does not contain a primitive $\ell$th root of unity. This implies that $1$ and $-1$ are the only roots of unity in $L$. Let us choose a {\sl primitive} cubic root of unity $\gamma \in \mu_{3,\ell}$ and let $\iota: \mu_{3,\ell} \to C$ be the isomorphism, which sends $\gamma$ to $s$. Now, let us define $X$ as the twist of $Y$ via the cubic character $$\kappa:=\iota\mathrm{pr}_3 \chi_{\ell} =\iota \mathrm{pr}_3 \mathrm{det} \rho_{\ell,E} : G({\mathbf Q})\to \mu_{3,\ell} \to C \subset \mathrm{Aut}(Y).$$ The Galois module $T_{\ell}(X)$ is the twist of $T_{\ell}(E)^2$ via $\kappa$. Namely, $$T_{\ell}(X)=\{(v_1,v_2,v_3)\in T_{\ell}(E)\oplus T_{\ell}(E)\oplus T_{\ell}(E)\mid v_1+v_2+v_3=0\}$$ as the ${\mathbf Z}_{\ell}-$module but $$\rho_{\ell,X}(\sigma)(v_1,v_2,v_3)=\kappa(\sigma)(\rho_{\ell,E}(\sigma)(v_1),\rho_{\ell,X}(\sigma)(v_2),\rho_{\ell,X}(\sigma)(v_3))$$ for all $\sigma \in G({\mathbf Q})$. Now, we construct explicitly $G(L)-$invariant elements of $T_{\ell}(X).$ Starting with any $v \in T_{\ell}(E)$, put $$w=(\gamma^{-1}v,\gamma v,v)=(\gamma^2 v,\gamma v, v)\in T_{\ell}(E)\oplus T_{\ell}(E)\oplus T_{\ell}(E).$$ Clearly, $w \in T_{\ell}(X);\quad sw =\gamma w.$ Let us check that $w$ is $G(L)-$invariant. Clearly, $$G(L)=\{\sigma\in G({\mathbf Q})\mid \rho_{\ell,E}(\sigma) \in \mu_{3,\ell}\cdot \mathrm{id}\}.$$ Let $\sigma \in G(L)$ with $\rho_{\ell,E}(\sigma)=\zeta\mathrm{id}, \quad \zeta \in \mu_{3,\ell}.$ If $\zeta=1$ , i.e., $\rho_{\ell,E}(\sigma)=\mathrm{id}$ then all elements of $V_{\ell}(X)$ are $\sigma-$invariant. Since $\mu_{3,\ell}=\{1,\gamma,\gamma^{-1}\}$, we may assume that $\zeta=\gamma$, i.e., $\rho_{\ell,E}(\sigma)=\gamma\cdot \mathrm{id}$ and therefore $\mathrm{det} \rho_{\ell,E}(\sigma)=\gamma^2=\gamma^{-1}.$ Then $$\rho_{\ell,X}(\sigma)(w)=$$ $$\iota(\mathrm{pr}_3(\mathrm{det} \rho_{\ell,E}(\sigma)))(\rho_{\ell,E}(\sigma)(\gamma^2 v),\rho_{\ell,E}(\sigma)(\gamma v), \rho_{\ell,E}(\sigma)( v))=\iota(\gamma^2)(\gamma w)=$$ $$s^2(\gamma w)=\gamma s^2 w=\gamma\gamma^{2} w=w.$$ This proves that $w$ is $G(L)-$invariant. Now, I claim that $X(L)$ contains infinitely many points, whose order is a power of $\ell$. Indeed, starting with a non-divisible element $v \in T_{\ell}(E)$ and identifying the group $X_{\ell^n}$ with the quotient $T_{\ell}(X)/\ell^n T_{\ell}(X)$, we get a $L-$rational point $(\gamma^2 v,\gamma v,v)\mod\ell^n T_{\ell}(X) \in T_{\ell}(X)/\ell^n T_{\ell}(X)=X_{\ell^n}$ of order $\ell^n$. \section{Another Example} Let $K$ be an imaginary quadratic field with class number $1$ and let $E$ be an elliptic curve over ${\mathbf Q}$ such that $\mathrm{End}_K(E)=O_K$ is the ring of integers in $K$. In this section we construct a Galois extension $L$ of $K$ such that $E(L)$ contains infinitely many points of finite order but the intersection of $L$ and $K(c)$ is of finite degree over $K$ (even coincides with $K$). We write $\iota:{\mathbf C} \to {\mathbf C}$ for the complex conjugation $z\mapsto \bar{z}$. We write $R$ for $O_K$. Clearly, $\mathrm{End}_{{\mathbf Q}}(E)={\mathbf Z}\ne R$. It follows easily that $$\iota(ux)=\bar{u}(\iota(x))\quad \forall x \in E({\mathbf C}), u\in R.$$ Notice that $K$ is abelian over ${\mathbf Q}$. Since ${\mathbf Q}(c)={\mathbf Q}(ab)$, $K \subset {\mathbf Q}(c)$ and therefore $$K(c)={\mathbf Q}(c).$$ \subsection{} Let $\ell$ be a prime number. We write $R_{\ell}$ for $R \otimes {\mathbf Z}_{\ell}$. It is well-known that $T_{\ell}(E)$ is a free $R \otimes {\mathbf Z}_{\ell}$-module of rank $1$ and therefore $$\mathrm{End}_{R_{\ell}}(T_{\ell}(E))=R_{\ell},\quad \mathrm{Aut}_{R_{\ell}}(T_{\ell}(E))=R_{\ell}^*.$$ Let us consider the corresponding $\ell$-adic representation $$\rho_{\ell,E}: G({\mathbf Q}) \to \mathrm{Aut}_{{\mathbf Z}_{\ell}}(T_{\ell}(E)) \cong \mathrm{GL}(2,{\mathbf Z}_{\ell}).$$ Clearly, $G_{\ell}:=\rho_{\ell,E}(G({\mathbf Q}))$ is not a subgroup of $R_{\ell}^*=\mathrm{Aut}_{R_{\ell}}(T_{\ell}(E))$ but $$H_{\ell}:=\rho_{\ell,E}(G(K))\subseteq R_{\ell}^*.$$ It is also known (\cite{Serre2}, Sect. 4.5) that $$H_{\ell}=R_{\ell}^*$$ for all but finitely many primes $\ell$. Let us fix such an $\ell$, assuming in addition that $\ell$ is unramified and splits in $K$. This implies that $\ell={\mathfrak q}\bar{{\mathfrak q}}$ for some ${\mathfrak q}\in K$ and $$O_K={\mathfrak q}\cdot O_K+\bar{{\mathfrak q}}\cdot O_K,\quad R_{\ell}=R_{{\mathfrak q}}\oplus R_{\bar{{\mathfrak q}}}, \quad R_{{\mathfrak q}}={\mathbf Z}_{\ell}, R_{\bar{{\mathfrak q}}}= {\mathbf Z}_{\ell},$$ $${\mathfrak q} R_{\ell}=\ell R_{{\mathfrak q}}\oplus R_{\bar{{\mathfrak q}}}= \ell {\mathbf Z}_{\ell}\oplus{\mathbf Z}_{\ell}\subset {\mathbf Z}_{\ell}\oplus{\mathbf Z}_{\ell}= R_{\ell},$$ $$\bar{{\mathfrak q}} R_{\ell}= R_{{\mathfrak q}}\oplus \ell R_{\bar{{\mathfrak q}}}= {\mathbf Z}_{\ell}\oplus\ell{\mathbf Z}_{\ell}\subset {\mathbf Z}_{\ell}\oplus{\mathbf Z}_{\ell}= R_{\ell},$$ $$R_{\ell}^*=R_{{\mathfrak q}}^*\times R_{\bar{{\mathfrak q}}}^*,\quad R_{{\mathfrak q}}^*={\mathbf Z}_{\ell}^*, R_{\bar{{\mathfrak q}}}^*= {\mathbf Z}_{\ell}^*.$$ We also have $$T_{\ell}(E)=T_{{\mathfrak q}}(E)\oplus T_{\bar{{\mathfrak q}}}(E)$$ where $$T_{{\mathfrak q}}(E):=R_{{\mathfrak q}}\cdot T_{\ell}(E),\quad T_{\bar{{\mathfrak q}}}(E):=R_{\bar{{\mathfrak q}}}\cdot T_{\ell}(E)$$ are free ${\mathbf Z}_{\ell}$-modules of rank $1$. This implies that for each positive integer $i$ $${\mathfrak q}^i T_{{\mathfrak q}}(E)=\ell^i T_{{\mathfrak q}}(E), \quad \bar{{\mathfrak q}}^i T_{{\mathfrak q}}(E)=T_{{\mathfrak q}}(E),$$ $$\bar{q}^i T_{\bar{{\mathfrak q}}}(E)=\ell^i T_{\bar{{\mathfrak q}}}(E),\quad {\mathfrak q}^i T_{\bar{{\mathfrak q}}}(E)=T_{\bar{{\mathfrak q}}}(E)$$ and therefore $$T_{\ell}(E)/\ell^i T_{\ell}(E)= T_{{\mathfrak q}}(E)/\ell^i T_{{\mathfrak q}}(E)\oplus T_{\bar{{\mathfrak q}}}(E)/\ell^i T_{\bar{{\mathfrak q}}}(E)= T_{{\mathfrak q}}(E)/{\mathfrak q}^i T_{{\mathfrak q}}(E)\oplus T_{\bar{{\mathfrak q}}}(E)/\bar{{\mathfrak q}}^i T_{\bar{{\mathfrak q}}}(E).$$ It follows easily that a point $x \in E_{\ell^i}=T_{\ell}(E)/\ell^i T_{\ell}(E)$ satisfies ${\mathfrak q}^i x=0$ (respectively $\bar{{\mathfrak q}}^i x=0$) if and only if $x \in T_{{\mathfrak q}}(E)/\ell^i T_{{\mathfrak q}}(E)$ (respectively $x \in T_{\bar{{\mathfrak q}}}(E)/\ell^i T_{\bar{{\mathfrak q}}}(E)$). Let us put $$\tau:=\rho_{\ell,E}(\iota) \in G_{\ell} \subset \mathrm{Aut}_{{\mathbf Z}_{\ell}}(T_{\ell}(E)).$$ Then $\tau^2=\mathrm{id}$ and $$\tau( R_{{\mathfrak q}}^*\times\{1\}) \tau^{-1}=\{1\}\times R_{\bar{{\mathfrak q}}}^*\subset R_{\ell}^* ,\quad \tau(\{1\}\times R_{\bar{{\mathfrak q}}}^*) \tau^{-1}=R_{{\mathfrak q}}^*\times\{1\})\subset R_{\ell}^*.$$ It is also clear that $$\tau(T_{{\mathfrak q}}(E))=T_{\bar{{\mathfrak q}}}(E), \quad \tau(T_{\bar{{\mathfrak q}}}(E))=T_{{\mathfrak q}}(E).$$ Let us consider the field $K(E(\ell^{\infty}))$ of definition of all points on $E$ of $\ell$-power order. It is the Galois extension of $K$ with the Galois group $R_{\ell}^*=R_{{\mathfrak q}}^*\times R_{\bar{{\mathfrak q}}}^*.$ It is also normal over ${\mathbf Q}$ and $\mathrm{Gal}(K(E(\ell^{\infty}))/{\mathbf Q})=G_{\ell}$, since $E$ is defined over ${\mathbf Q}$ and $K$ is normal over ${\mathbf Q}$. Let us define $L$ as a subextension of $K(E(\ell^{\infty}))/K$ such that $$\mathrm{Gal}(K(E(\ell^{\infty}))/L)=\{1\}\times R_{\bar{{\mathfrak q}}}^*\subset R_{{\mathfrak q}}^*\times R_{\bar{{\mathfrak q}}}^*=R_{\ell}^*=\mathrm{Gal}(K(E(\ell^{\infty}))/K).$$ One may easily check that $L$ coincides with the field $K(E({\mathfrak q}^{\infty}))$ of definition of all torsion points on $E$ which are killed by a power of ${\mathfrak q}$. In particular, $E(L)$ contains infinitely many points, whose order is a power of $\ell$. Let us consider the field $L'=\iota(L)$. Clearly, $K\subset L'\subset K(E(\ell^{\infty}))$ and $L'$ coincides with the field $K(E(\bar{{\mathfrak q}}^{\infty}))$ of definition of all torsion points on $E$ which are killed by a power of $\bar{{\mathfrak q}}$. It is also clear that $$\mathrm{Gal}(K(E(\ell^{\infty}))/L)=\tau(\{1\}\times R_{\bar{{\mathfrak q}}}^*)\tau^{-1}=R_{{\mathfrak q}}^*\times\{1\}\subset R_{{\mathfrak q}}^*\times R_{\bar{{\mathfrak q}}}^*=R_{\ell}^*=\mathrm{Gal}(K(E(\ell^{\infty}).$$ Since the subgroups $\{1\}\times R_{\bar{{\mathfrak q}}}^*$ and $R_{{\mathfrak q}}^*\times\{1\}$ generate the whole group $R_{\ell}^*=\mathrm{Gal}(K(E(\ell^{\infty}))/K)$, $$L\bigcap \iota(L)=L\bigcap L'=K.$$ It follows that if $M/K$ is a subextension of $L/K$ such that $M$ is normal over ${\mathbf Q}$ then $M=K$. Since $K(c)={\mathbf Q}(c)$, $L\bigcap K(c)=L\bigcap {\mathbf Q}(c)$ is a subfield of ${\mathbf Q}(c)$ and therefore is normal (even abelian) over ${\mathbf Q}$. It follows that $$L\bigcap K(c)=K.$$ \section{Abelian subextensions} The following statement may be viewed as a variant of Theorem \ref{Theorem 1} for arbitrary abelian varieties over number fields. \begin{thm} \label{abelian} \sl Let $X$ be an abelian variety over a number field $K$. Then: \begin{enumerate} \item If for some prime $\ell$ the $\ell-$primary part of $\mathrm{TORS}(X(L))$ is infinite then $L$ contains an abelian infinite subextension $E\subset L$ such that $\mathrm{Gal}(E/K)\cong {\mathbf Z}_{\ell}$ and $E/K$ is ramified only at divisors of $\ell$. \item Let $P=P(X,L)$ be the set of primes $\ell$ such that $X(L)$ contains a point of order $\ell$. If $P$ is infinite then for all but finitely many primes $\ell \in P$ there exist a finite subextension $E^{(\ell)}\subset L$ such that $E^{(\ell)}/K$ is a ramified abelian extension which is unramified outside divisors of $\ell$. In addition, the degree $[E^{(\ell)}:K]$ is prime to $\ell$ and degree $[E^{(\ell)}:K]$ tends to infinity while $\ell$ tends to infinity. \end{enumerate} \end{thm} \begin{cor}[Theorem of Bogomolov] If $\mathrm{TORS}(X(L))$ is infinite then $L$ contains an infinite abelian subextension of $K$. \end{cor} \begin{proof}[Proof of Theorem \ref{abelian}] First, we may and will assume that $X$ is $K-$simple, i.e., the center $F$ of the endomorphism algebra of $X$ is a number field. Second, there is a positive integer $d$, enjoying the following property: If $m$ is a positive integer such that $\varphi(m) \le 2g=2\mathrm{dim}(X)$ then $d$ is divisible by $m$. Third, let $\lambda$ be a prime ideal in $O_F$ dividing a prime number $\ell$. Then, in the notations of Section 1 the following statement is true. \begin{lem} \begin{enumerate} \item The composition $$\pi_{\lambda}:=(\mathrm{det}_{F_{\lambda}} \rho_{\lambda,X})^d:G(K) \to \mathrm{Aut}_{F_{\lambda}}V_{\lambda}(X) \to {F_{\lambda}}^* \to {F_{\lambda}}^*$$ is an abelian representation of $G(K)$ unramified outside divisors of $\ell$. \item For all but finitely many $\lambda$ the composition $$\bar{\pi}_{\lambda}:=(\mathrm{det}_{F_{\lambda}}\bar{\rho}_{\lambda,X})^d:G(K) \to \mathrm{Aut}_{O_F/\lambda}X_{\lambda} \to (O_F/\lambda)^* \to (O_F/\lambda)^*$$ is an abelian representation of $G(K)$ unramified outside divisors of $\ell$. \end{enumerate} \end{lem} We will prove Lemma at the end of this section. Now, let us finish the proof of Theorem, assuming validity of Lemma. First, notice that the ratio $$e=2\mathrm{dim}(X)/[F:{\mathbf Q}]$$ is a positive integer. Second, for all but finitely many primes $p$ there exists a finite collection of {\sl Weil numbers}, i.e., certain algebraic integers $\{\alpha_1,\ldots \alpha_{e}\} \subset F(a)$, enjoying the following properties: \begin{itemize} \item (Weil's condition) There is a positive integer $q>1$ such that $q$ is an integral power of $p$ and all $\mid\alpha_i\mid^2=q$ for all embeddings $F(a)\subset{\mathbf C}$. \item For all $\ell\ne p$ and $\lambda\mid\ell$ there is a subset $S_{\lambda}\subset \{1,\ldots e\}$ such that $(\prod_{i\in S_{\lambda}}\alpha_i)\in O_F$ and the group $\mathrm{Im}(\pi_{\lambda})$ contains $\prod_{i\in S_{\lambda}}\alpha_i$. \item For all but finitely many $\lambda$ the subgroup $\mathrm{Im}(\bar{\pi}_{\lambda})$ contains $(\prod_{i\in S_{\lambda}}\alpha_i)\mod \lambda \in (O_F/\lambda)^*$. \end{itemize} Indeed, let us choose a prime ideal $\mathbf v$ in the ring $O_K$ of all algebraic integers in $K$ such that $X$ has good reduction at $\mathbf v$. Let $$Fr_{\mathbf v} \in \mathrm{Im}(\rho_{\lambda,X}) \subset \mathrm{Aut}_{F_{\lambda}}V_{\lambda}(X)$$ be {\sl Frobenius element} $Fr_{\mathbf v}$ at $\mathbf v$ (defined up to conjugacy)\cite{Serre},\cite{RibetA}. Then the set of its eigenvalues belongs to $F(a)$, does not depend on the choice of $\lambda$ and satisfies all the desired properties with $p$ the residual characteristic of $\mathbf v$ and $q=\#(O_K/{\mathbf v})$(\cite{Shimura}, Ch. 7, Prop. 7.21 and proof of Prop. 7.23). \begin{proof}[Proof of assertion 1] We know that there exists $\lambda$ dividing $\ell$ such that the subspace $V_{\lambda}(X)$ consists of $G(L)-$invariants. This means that $G(L)$ lies in the kernel of $\pi_{\lambda}$. This implies that the field $E'$ of $\ker(\pi_{\lambda})-$invariants is an abelian subextension of L, unramified outside divisors of $\ell$ and $\mathrm{Gal}(E'/K)$ is isomorphic to $\mathrm{Im}(\pi_{\lambda})$. Choosing a collection of Weil numbers attached to prime $p\ne\ell$, we easily conclude that $\mathrm{Im}(\pi_{\lambda})$ is an {\sl infinite} commutative $\ell-$adic Lie group \cite{Serre} and therefore, there is a continuous quotient of $\mathrm{Im}(\pi_{\lambda})$, isomorphic to ${\mathbf Z}_{\ell}$. One has to take as $E$ the subextension of $E'$ corresponding to this quotient. \end{proof} \begin{proof}[Proof of assertion 2] We know that for all but finitely many $\ell \in P$ there exists $\lambda$ dividing $\ell$ such that $X_{\lambda}$ consists of $G(L)-$invariants. This means that the field $E^{(\ell)}$ of $\ker(\bar{\pi}_{\lambda})-$invariants is an abelian subextension of L, unramified outside divisors of $\ell$ and $\mathrm{Gal}(E^{(\ell)}/K)$ is isomorphic to $\mathrm{Im}(\pi_{\lambda})$. In order to prove that $[E^{(\ell)}:K]$ tends to infinity, let us assume that there exist an infinite subset $P'\subset P$ and a positive integer $D$ such that $\#(\mathrm{Gal}(E^{(\ell)}/K))=[E^{(\ell)}:K]$ divides $D$ for all $\ell\in P'$. This means that $$\bar{\pi}_{\lambda}^D: G(K)\to (O_F/\lambda)^*$$ is a trivial homomorphism for {\sl infinitely many} $\lambda$. In order to get a contradiction, let us choose a collection of Weil numbers $\{\alpha_1,\ldots \alpha_{e}\}$ enjoying the properties described above. Clearly. for any non-empty subset $ S \subset \{1,\ldots e\}$ the product $\alpha_S:=\prod_{i\in S} \alpha_i$ is not a root of unity. In addition, if $\alpha_S\in O_F$ then there only finitely many $\lambda$ such that $\alpha_S^D-1$ is an element of $\lambda$. Since there are only finitely many subsets of $\{1,\ldots2g\}$, for all but finitely many $\lambda$ the group $$\mathrm{Im}((\bar{\pi}_{\lambda})^D) \subset (O_F/\lambda)^*$$ contains an element of type $\alpha_S^D\mod \lambda$ different from 1. This implies that $(\bar{\pi}_{\lambda})^D$ is a non-trivial homomorphism for all but finitely many $\lambda$. This gives the desired contradiction. \end{proof} \begin{proof}[Proof of Lemma] Let $\mathbf v$ be a prime ideal in the ring $O_K$ of all algebraic integers in $K$. We write $I_{\mathbf v} \subset G(K)$ for the corresponding inertia subgroup defined up to conjugacy. Assume that the residual characteristic of $\mathbf v$ is different from $\ell$. It is known \cite{SGA} that for any $ \sigma \in I_{\mathbf v}$ there exists a positive integer $m$ such that $\rho_{\ell,X}(\sigma)^m$ is an unipotent operator in $V_{\ell}(X)$ and its characteristic polynomial has coefficients in ${\mathbf Z}$. This implies that if $m$ is the smallest integer enjoying this property then the characteristic polynomial is divisible by the $m$th cyclotomic polynomial. This implies that $2g\ge \varphi(m)$ and therefore $m$ divides $d$. Since $V_{\lambda}(X)$ is a Galois-invariant subspace of $V_{\ell}(X)$ and (for all but finitely many $\ell$) $X_{\lambda}$ is a Galois-invariant subspace of $T_{\ell}(X)/\ell T_{\ell}(X)$, a Galois automorphism $\sigma^d$ acts as an unipotent operator in $V_{\lambda}(X)$ and (for all but finitely many $\lambda$) in $X_{\lambda}$. One has only to recall that the determinant of an unipotent operator is always $1$. \end{proof} \end{proof}

9708

alg-geom/9708020

en

https://arxiv.org/abs/alg-geom/9708020

alg-geom/9708020

Gunnar Floystad

Gunnar Floystad

A property deducible from the generic initial ideal

Completely revised compared to earlier hardcopy versions. AMS-Latex v1.2, 13 pages

Journal of Pure and Applied Algebra, 136 (1999), no.2, p.127-140

10.1016/S0022-4049(97)00165-5

Let $S_d$ be the vector space of monomials of degree $d$ in the variables $x_1, ..., x_s$. For a subspace $V \sus S_d$ which is in general coordinates, consider the subspace $\gin V \sus S_d$ generated by initial monomials of polynomials in $V$ for the revlex order. We address the question of what properties of $V$ may be deduced from $\gin V$. % This is an approach for understanding what algebraic or geometric properties of a homogeneous ideal $I \sus k[x_1, ..., x_s]$ that may be deduced from its generic initial ideal $\gin I$.

alg-geom

\section*{Introduction} During the recent years the generic initial ideal of a homogeneous ideal has attracted some attention as an invariant. An intriguing problem is what algebraic or geometric properties of the original ideal can be deduced from the generic initial ideal. In this paper we take perhaps the most elementary approach possible. Let $S = k[x_1, \ldots, x_s]$ and let $>$ be the reverse lexicographic order of the monomials in $S$. Denote by $S_d$ the graded piece of degree $d$ in $S$. Suppose $V \subseteq S_d$ is a subspace. Denote by $\gin V$ the subspace of $S_d$ generated by initial monomials of polynomials of the subspace of $S_d$ obtained from $V$ by performing a general change of coordinates. Then one may ask what properties of $V$ may be deduced from $\gin V$? The following result gives an insight in this direction. Let $W = (x_1, \ldots, x_r) \subseteq S_1$ which is a linear space. Suppose that $s \geq r \geq 3$. \begin{theoremho} Let $V \subseteq S_{n+m}$ be a linear space such that \[ \gin V = W^n x_1^m \subseteq S_{n+m}. \] Then there exists a polynomial $p \in S_{m}$ and a linear subspace $W_n \subseteq S_n$ such that $V = W_n p.$ \end{theoremho} Note that if $s=r$ then $W^n x_1^m$ are the largest monomials in $S_{n+m}$ for the lexicographic order. Thus if $>$ had been the lexicographic order and $\gin V = W^n x_1^m$ then we could deduce virtually nothing about $V$. \vskip 2mm The general idea of the proof is inspired by Green and worth attention because of its seeming naturality in dealing with problems of this kind. The idea in its vaguest and most generally applicable form is the following. Suppose $\gin V$ has a given form, and suppose $V$ is in general coordinates so $\ii V = \gin V$. The given form of $\ii V$ implies some algebraic or geometric property of $V$. Let now $g : S_1 \rightarrow S_1$ be a general change of coordinates. Then $\ii {g^{-1}.V} = \gin V$ also. Thus $g^{-1}.V$ will also have this property. Then this property may be translated back to a property of $V$. This gives a continuous set of properties that $V$ will satisfy. From this one may proceed making deductions about what $V$ may look like. \vskip 2mm In this paper this is applied concretely as follows. In the case $r=s$ the given form of $\ii V = \gin V$ implies that there is a $p_1 $ in $S_m$ such that $ x_r^n \cdot p_1 \in V$. The fact that $\ii {g^{-1}.V} = \gin V$ also implies that there is a $p_{g^{-1}}$ in $S_m$ such that $ x_r^n \cdot p_{g^{-1}} \in g^{-1}.V$. Translating this property back to $V$ we get \begin{equation} (g.x_r)^n \cdot g.p_{g^{-1}} \in V. \label{hnpz} \end{equation} Now for the family of linear forms $h = \sum t_ix_i$ one may choose a general family of $g$'s depending on $h$ such that $g.x_r = h$. Then equation (\ref{hnpz}) may be written as \begin{equation} h^n p \in V \label{hnp0} \end{equation} where $p$ is a form of degree $m$ depending on $h$. \vskip 2mm The second technique, specifically suggested by Green, is to {\it differentiate} this equation with respect to the $t_i$. All the derivatives will still be in $V$. (This is just the fact that when a vector varies in a vector space its derivative is also in the vector space.) Letting $V_{|h=0}$ be the image by the composition $V \rightarrow S \rightarrow S/(h)$ this enables us to show that the forms in $V_{|h=0}$ have a common factor of degree $m$. \vskip 2mm The third basic ingredient is now proposition 3.4 which says that if the $V_{|h=0}$ have a common factor of degree $m$, then $V$ has a common factor of degree $m$. Having proven the case $s=r$, the case $s > r$ may now be proven by an induction process. \vskip 3mm The organization of the paper is as follows. In the first three sections we develop general theory which does not presuppose {\it anything} about what $\gin V$ actually is. In section 1 we give some basic definitions and notions. In section 2 we define the generical initial space of a subspace $V$ of $S$ by using a {\it generic} coordinate change on $V$. We also give some basic theory for this setting which will be used in sections 4 and 5. Section 3 presents the framework in which we will work. Instead of considering a continuously varying form $h= \sum_{i=1}^s t_i x_i$ in $k[x_1, \ldots, x_s]$, we consider $h$ as a linear form in $K[x_1, \ldots, x_s]$ where $K = k(t_1, \ldots, t_s)$, the field of rational functions of the $t_i$'s. If now $V \subseteq k[x_1, \ldots, x_s]_d$ is a subspace let $V_K = V \otimes_k K \subseteq K[x_1, \ldots, x_s]$. The main result here, proposition 3.4, says that if the forms in $V_{K|h=0}$ have a common factor of degree $m$ then the forms in $V$ have a common factor of degree $m$. This is proven using differentiation of forms with respect to the $t_i$. Only {\it from now on} do we assume that $\gin V$ has the special form given in the main theorem. In section 4 we prove the case $s=r$ in the main theorem. Section 5 proves the case $s > r$ of the main theorem. In section 6 we give an application of the main theorem. The example originated in discussions with Green and was what triggered this paper. Consider the complete intersection of three quadratic forms in ${\bf P}^3$. Let $I \subseteq k[x_1,x_2,x_3,x_4]$ be its homogeneous ideal. By standard theory one may deduce that there are two candidates for $\gin I$ : \begin{eqnarray*} J^{(1)} & = & (x_1^2, x_1x_2, x_2^2, x_1x_3^2, x_2x_3^2, x_3^4), \\ J^{(2)} & = & (x_1^2, x_1x_2, x_1x_3, x_2^3, x_2^2x_3, x_2x_3^2, x_3^4). \end{eqnarray*} By the main theorem, if $\gin I = J^{(2)}$ then the quadratic forms in $I_2 \subseteq S_2$ would have to have a common factor. Impossible. Thus $\gin I = J^{(1)}$. \vskip 2mm Throughout the article all fields have characteristic zero. \section{Basic definitions and notions} \parg Let $S = k[x_1, \ldots, x_s]$. The graded piece of degree $d$ is denoted by $S_d$. If $I = (i_1, i_2, \ldots, i_s)$ we use the notation \[ {\mbox{\boldmath $x$}}^I = x_1^{i_1}\cdots x_s^{i_s}. \] It has degree $|I| = \sum i_j$. Suppose now we have given a total order on the monomials. For a homogeneous polynomial $ f = \sum a_I{\mbox{\boldmath $x$}}^I$ in $S$ (henceforth often referred to as a form) let the initial monomial be \[ \ii f = \max \{ {\mbox{\boldmath $x$}}^I \, | \, a_I \neq 0 \}. \] For a homogeneous vector subspace $V \subseteq S$ let the {\it initial subspace} be \[ \ini V = ( \{ \ini f \, | \, f \in V\} ) \] the homogeneous vector subspace of $S$ generated by the initial monomials of forms in $V$. Sometimes we wish to consider another polynomial ring $R[x_1, \ldots, x_r]$ where $R$ is a commutative ring. Denote this by $S_R$. The initial monomials $\ii f$ for $f \in V$ may equally well be considered as elements of $S_R$. We may thus speak of $\ii V$ over $R$ (when $V \subseteq S$) which is the free $R$-module in $S_R$ generated by $\{ \ini f \, | \, f \in V\}$. \parg The monomial order we shall be concerned with in sections 4 and 5 is the reverse lexicographic order. Then the monomials of a given degree is ordered by ${\mbox{\boldmath $x$}}^I > {\mbox{\boldmath $x$}}^J$ if $i_r < j_r$ where $r$ is the greatest number with $i_r \neq j_r$. Intuitively ${\mbox{\boldmath $x$}}^J$ is "dragged down" by having a large "weight in the rear". \parg For a linear form $l \in S_1$ denote by $V|_{l=0}$ the image of the composition \[ V \longrightarrow S \longrightarrow S/(l). \] The following basic fact for the revlex order, proposition 15.12 a. in \cite{Ei}, will be used several times \[ \ii {V_{|x_s = 0}} = \ii V _{|x_s = 0}. \] \section{The generic initial space} The following section contains the definition of the generic initial space and some general theory related to it. The things presented here are certainly in the background knowledge of people but due to a lack of suitable references for a proper algebraic treatment we develop the theory here. The most important things are proposition 2.9 and paragraph 2.11. \parg We identify $S = k[x_1, \ldots, x_s]$ as the affine coordinate ring of ${\bf A}^s$. Let $G = GL(S_1^{\vee})$. There is a natural action \[ {\bf A}^s \times G \longrightarrow {\bf A}^s \] given by $(a,g) \mapsto g^{-1}.a$. This gives a $k$-algebra homomorphism \[ \gamma : k[x_1, \ldots, x_s] \longrightarrow k[x_1, \ldots, x_s] \otimes_k k[G]. \] If $R$ is a $k[G]$-algebra, we also by composition obtain a $k$-algebra homomorphism \[ \gamma_R : k[x_1, \ldots, x_s] \longrightarrow k[x_1, \ldots, x_s] \otimes_k k[G] \longrightarrow R[x_1, \ldots, x_s]. \] Note that if $R = k(g)$ for a point $g \in G$, then $\gamma_{k(g)}$ is just the action of $g$ on $k[x_1, \ldots, x_s]$. Let $K_G$ be the function field of $G$. The image of a homogeneous subspace $V \subseteq S$ by $\gamma_{K_G}$ generates a homogeneous subspace $(\gamma_{K_G}(V))$ of the same dimension as $V$. Suppose now a total monomial order is given. The initial monomials of $(\gamma_{K_G}(V))$ generate a linear subspace over $k$ (or over $K_G$), which is called the {\it generic initial subspace} of $V$ over $k$ (or over $K_G$) and is denoted $\gin V$. Henceforth we shall drop the outer paranthesis of $(\gamma_{K_G}(V))$ and write this as $\gamma_{K_G}(V)$. \parg Let $\gin V = (m_1, \ldots, m_t)$ for some monomials $m_i$. Let $b_i \in \gamma_{K_G}(V)$ be such that \[ b_i = m_i + b_{i0} \] where $b_{i0}$ consists of monomials less than $m_i$ for the given order. Now there is an open subset $U \subseteq G$ such that all the $b_i$ lift to elements of ${\mathcal O}(U) [x_1, \ldots, x_s]$. Now we immediately get. \begin{prop} There is an open subset $U \subseteq G$ (take the one above) such that for $g \in U$ then \[ \ii {(\gamma_{k(g)}(V))} = \gin V \, (\mbox{over } k(g)).\] \end{prop} (The original reference for this is \cite{Ga}.) \parg Now choose a $g_0 \in G$ such that $k(g_0) = k$. There is then a diagram \[ \begin{CD} {\bf A}^s_{K_G} @>\alpha_{g_0}>> {\bf A}^s_{K_G} \\ @VVV @VVV \\ {\bf A}^s \times G @>{\cdot g_0}>> {\bf A}^s \times G \\ @VVV @VVV \\ {\bf A}^s @>{g_0^{-1}.}>> {\bf A}^s. \end{CD} \] The lower horizontal map is the natural action. The middle map is given by $(a,g) \mapsto (a, g g_0)$ and the lower vertical maps are just the action of $G$. The upper horizontal map is the map induced by the middle map. From the commutativity of the diagram we see that \[ \gamma_{K_G}(g_0.V) = \alpha_{g_0}^*( \gamma_{K_G}(V)) \] where $\alpha_{g_0}^*$ is the automorphism of $K_G[x_1, \ldots, x_s]$ induced by $\alpha_{g_0}$. Note that $\alpha_{g_0}^*$ comes from an automorphism of $K_G$. So it does not affect the variables $x_i$. Thus we see that the $\alpha_{g_0}^*(b_i) = m_i + \alpha_{g_0}^*(b_{i0})$ are a basis for $\gamma_{K_G}(g_0.V)$, where the monomials in $\alpha_{g_0}^*(b_{i0})$ are less then $m_i$ for the given order. Also note that the $\alpha_{g_0}^*(b_i)$ lift to the open subset $U.g_0^{-1} \subseteq G$. Thus we have proven the following. \begin{lemma} Given $g \in G$, by replacing the subspace $V$ by $g_0.V$ and the open subset $U$ by $U.g_0^{-1}$ for a suitable $g_0$, we may assume that $g$ is in the open subset from proposition 2.3. \end{lemma} \parg Now let $\phi : X \longrightarrow G$ be a morphism. We get a morphism \[ {\bf A}^s \times X \longrightarrow {\bf A}^s \] and thus a $k$-algebra morphism \[ \gamma_{K_X} : k[x_1, \ldots, x_s] \longrightarrow K_X[x_1, \ldots, x_s] \] where $K_X$ is the function field of $X$. We get a homogeneous subspace $( \gamma_{K_X}(V))$ and also here we shall henceforth drop the outer paranthesis. By performing a suitable coordinate change of $V$ we may assume (by lemma 2.5) that $\phi(X) \cap U \neq \emptyset$. The following is now immediate from the results above. \begin{lemma} \begin{itemize} \item [1.] For $x$ in the open subset $\phi^{-1}(U) \subseteq X$ we have \[ \ii {\gamma_{k(x)}(V)} = \gin V \, (\mbox{over } k(x)). \] \item [2.] $\ii {\gamma_{K_X} (V)} = \gin V \, (\mbox{over } K_X).$ \item [3.] Given $x \in X$ then we may assume that $\phi(x) \in U$. \end{itemize} \end{lemma} \parg By 1.3 we have \[ \ii {V_{|x_s = 0}} = \ii V _{|x_s = 0}. \] We would like to have a suitable version of this for generic subspaces. The version we need is 2. in the following. It is used most importantly in the proof of lemma 5.2 \begin{prop} \begin{itemize} \item [1.] $ \gin {\gamma_{K_X}(V)} = \gin V \, (\mbox{over } K_X). $ \item [2.] $ \gin { \gamma_{K_X}(V)_{|x_s = 0}} = \gin V _{|x_s = 0} \, (\mbox{over } K_X).$ \end{itemize} \end{prop} \begin{proof} We prove 2. The proof of 1. is analogous and easier. Besides we will not need 1. We just state it for completeness. \vskip 2mm a) Let $S_1^{\circ} = (x_1, \ldots, x_{s-1})$ and $G^{\circ} = GL(S_1^{\circ \vee})$. Let $k \rightarrow K$ be a homomorphism of fields and let $G_K^{\circ} = GL(S_1^{\circ \vee} \otimes_k K)$. Due to the naturally split inclusion $S_1^\circ \subseteq S_1$, there is a diagram \[ \begin{CD} {\bf A}^{s-1}_K \times G_K^\circ @>>> {\bf A}^{s-1}_K \\ @VVV @VVV \\ {\bf A}^s_K \times G_K^\circ @>>> {\bf A}^s_K \end{CD} \] where the upper action is given by $(a,g) \mapsto g^{-1}.a$. The lower map gives a $K$-algebra homomorphism \[ \gamma^{\circ} : K[x_1, \ldots, x_s] \longrightarrow K[G_K^\circ] \otimes_K K[x_1, \ldots, x_s]. \] The upper map gives a $K$-algebra homomorphism \[ \gamma^{\circ}_{|x_s = 0} : K[x_1, \ldots, x_{s-1}] \longrightarrow K[G_K^\circ] \otimes_K K[x_1, \ldots, x_{s-1}]. \] For a homogeneous subspace $W \subseteq K[x_1, \ldots, x_s]$ we now see that \[ \gamma_{|x_s = 0}^{\circ}(W_{|x_s = 0}) = \gamma^{\circ}(W)_{|x_s = 0}. \] The initial space of the former is by definition $\gin {W_{|x_s = 0}}$. By 1.3 applied to the latter initial space we then get \begin{equation} \gin {W_{|x_s = 0}} = \ii { \gamma^{\circ}(W)}_{|x_s = 0}. \label{li1} \end{equation} \vskip 3mm b) Now there is a diagram \[ \begin{CD} {\bf A}^s \times G^\circ \times G @>>> {\bf A}^s \times G \\ @VVV @VVV \\ {\bf A}^s \times G @>>> {\bf A}^s. \end{CD} \] The upper horizontal map is given by $(a,h,g) \mapsto (h^{-1}.a,g)$. The lower horizontal map and the right vertical map are the actions. Lastly, the left vertical map is given by $(a,h,g) \mapsto (a,hg)$. It induces a diagram \begin{equation} \begin{CD} {\bf A}^s \times G^\circ \times X @>>> {\bf A}^s \times X \\ @VVV @VVV \\ {\bf A}^s \times G @>>> {\bf A}^s. \end{CD} \label{li2} \end{equation} Apply lemma 2.7. Then $\gamma_{K_X}(V)$ has initial space $\gin V$. Also applying 2.7 to the composition ${\bf A}^s \times G^\circ \times X \rightarrow {\bf A}^s \times G \rightarrow {\bf A}^s$ (from the diagram), gives that $\gamma_{K_{G^\circ \times X}} (V)$ has initial ideal $\gin V$ over $K_{G^\circ \times X}$. Now go back to part a) of this proof and put $K = K_X$ and $W = \gamma_{K_X} (V)$. By the commutativity of the diagram (\ref{li2}) we see that \[ \gamma^{\circ} (W) = \gamma_{K_{G^\circ \times X}} (V). \] Thus \[ \ii {\gamma^{\circ} (W)} = \ii {\gamma_{K_{G^\circ \times X}} (V)} = \gin V \, (\mbox{over } K_{G^\circ \times X}). \] Putting this together with (\ref{li1}) we get \[ \gin { \gamma_{K_X}(V)_{|x_s = 0}} = \gin V _ {|x_s = 0} \, (\mbox{over } K_X). \] \end{proof} \parg Now, there is of course also a natural action \[ G \times {\bf A}^s \longrightarrow {\bf A}^s \] given by \[ (g,a) \mapsto g.a. \] The morphism \[ \rho : G \times {\bf A}^s \longrightarrow {\bf A}^s \times G \] given by \[ (g,a) \mapsto (g.a, g)\] is an isomorphism and its inverse $\rho^{-1}$ is given by \[ (b,g) \mapsto (g, g^{-1}.b),\] The morphism $\rho$ induces a $k[G]$-algebra isomorhism \[ \Gamma : k[x_1, \ldots, x_s] \otimes_k k[G] \longrightarrow k[G] \otimes_k k[x_1, \ldots, x_s].\] Note that $\Gamma^{-1}$ is the $k[G]$-algebra isomorphism induced by $\rho^{-1}$. For any $k[G]$-algebra $R$ we get an $R$-algebra isomorphism \[ \Gamma_R : R[x_1, \ldots, x_s] \longrightarrow R[x_1, \ldots, x_s]. \] The homogeneous subspace $V \subseteq S$ induces an $R$-submodule \[ V_R = V \otimes_k R \subseteq R[x_1, \ldots, x_s] \] and so we get a free $R$-module \[ \Gamma_R^{-1}(V_R) \subseteq R[x_1, \ldots, x_s] \] which is in fact just $\gamma_R(V)$. \parg For a morphism $\phi : X \rightarrow G$ with $\phi(X) \cap U \neq 0$ we now see that \[ \ii { \Gamma_{K_X}^{-1}(V_{K_X})} = \ii {\gamma_{K_X} (V)} = \gin V \, (\mbox{over } K). \] Now consider $G = GL(S_1^{\vee})$ to be an open subset of ${\bf A}^{s^2}$ with coordinate functions $u_{ij}$ for $i,j = 1, \ldots, r$. Let the $u_{ij}$ take general values of $k$ for $i < r$ and let $u_{rj} = t_j$. Let $D$ be the determinant of the matrix thus obtained and let $T = k[t_1, \ldots, t_s]_D$. The situation to which we will apply the above is to the situation where $X = $ Spec $T$. For the rest of the paper let $K = K_X = k(t_1, \ldots, t_s)$ the field of rational functions in the $t_i$. Finally, if we let $h = \sum_{i=1}^s t_i x_i$, note the following will be used repeatedly in sections 4 and 5 : $\Gamma_K (x_s) = h$. \section{ Derivatives of forms} Given a form $p$ in $S_K = K[x_1, \ldots, x_s]$. One may then differentiate it with respect to the $t_i$ and obtain partial derivatives $\partial^{|I|}p / \partial t^I$ where $I = (i_1, \ldots, i_r)$. More generally for a homogeneous form $s({\mbox{\boldmath $t$}}) = \sum \alpha_I t^I$ of degree $d$ we get the directional derivative $\partial^d p / \partial s({\mbox{\boldmath $t$}}) = \sum \alpha_I \partial^d p/ \partial t^I$. For a form $f$ in $S_K$ let $\ba f$ be its image in $S_K / (h)$. Now consider a specific form $p$. Let $l({\mbox{\boldmath $t$}}) = \sum \alpha_i t_i$ be such that $\ba {l({\mbox{\boldmath $x$}})} = \sum \alpha_i \ba {x_i}$ is not a factor of $\ba p$. \begin{lemma} Suppose $\partial^k p/ \partial l^k = \alpha_k p$ for $k \geq 0$. If $f$ is a form such that $\ba {\partial^k f / \partial l^k}$ has $\ba p$ as a factor for all $k \geq 0$, then $f$ has $p$ as a factor. \end{lemma} \begin{proof} We have \begin{equation} f = u_1 p + h a_1 \label{a1} \end{equation} for some $u_1$ and $a_1$. Differentiating this gives \[ \partial f/ \partial l = \partial u_1 / \partial l \cdot p + u_1 \partial p / \partial l + l({\mbox{\boldmath $x$}}) a_1 + h \partial a_1 / \partial l. \] Thus $\ba p$ divides $\ba a_1$. So $a_1 = v_1 p + h a_2$ for some $v_1$ and $a_2$. Inserting this in (\ref{a1}) gives \[ f = u_2 p + h^2 a_2 \] where $u_2 = u_1 + h v_1. $ Now differentiate twice with respect to $l$. We may conclude that \[ a_2 = v_2 p + h a_3 \] for som $v_2$ and $a_3$. Continuing we get in the end that $f = up$. \end{proof} The following result is proposition 10 in \cite{Co} and is due to Green. It is assumed there that the field $k = {\bf C}$ but the proof is readily seen to work for any field of characteristic zero. Given a form $\ba p$ in $S_K / (h)$ it gives a criterion for it to lift to a form in $S_K$ which is essentially a form in $S$. \begin{prop} Let $p \in S_K$ be a form such that \[ x_i \ba{ \partial p / \partial t_j} \equiv x_j \ba{ \partial p / \partial t_i} \pmod {\ba p} \] for all $i$ and $j$. Then $p = \alpha p_0 + h R$ where $p_0 \in S$ and $\alpha \in K$. \end{prop} Consider now a form $f \in S \subseteq S_K$. It gives a hypersurface in $\bf P^{s-1}$. The following says that if all hyperplane sections of this hypersurface are reducible with a component of a given degree then the same is true for the hypersurface defined by $f$. \begin{kor} Suppose $\ba f = \ba u \cdot \ba p$ in $S_K / (h)$, where $\ba u$ and $\ba p$ do not have a common factor. Then $\ba p$ lifts to a form $\alpha p_0$ where $p_0 \in S$. Furthermore $p_0$ is a factor of $f$. \end{kor} \begin{proof} Let $u$ and $p$ in $S_K$ be liftings of $\ba u$ and $\ba p$. We get \[ f = up + hR. \] Differentiating with respect to $\partial / \partial t_i$ gives \[ 0 = \partial u / \partial t_i \cdot p + u \partial p / \partial t_i + x_i R + h \partial R / \partial t_i. \] Thus we get \[ \ba u ( x_j {\ba {\partial p/ \partial t_i}} - x_i \ba {\partial p / \partial t_j}) \equiv 0 \pmod {\ba p}. \] Then by proposition 3.2 we conclude that $\ba p$ has a lifting $\alpha p_0$ where $p_0 \in S$. By lemma 3.1 we conclude that $p_0$ is a factor of $f$ since the $\partial^k f / \partial l^k = 0$ for $k \geq 1$. \end{proof} Now suppose $V \subseteq S_{n+m}$ is a subspace so we get a subspace $V_K = V \otimes_k K \subseteq S_{K,n+m}$ and $V_{K|h=0} \subseteq S_K / (h)$. \begin{prop} Suppose the forms of $V_{K|h=0}$ have a common factor $\ba p$ where $\ba p$ is a common factor of maximal degree $m$. Then $V$ has a common factor $p_0$ of degree $m$ such that $\ba p = \alpha \ba p_0$ for some $\alpha \in K$. \end{prop} \begin{proof} We may choose an $f_0 \in V$ such that \[ \ba f_0 = \ba u_0 \ba p \] where $\ba u_0$ and $\ba p$ are relatively prime. This is seen as follows. Let $\ba p = {\ba a_1}^{e_1} \cdots {\ba a_r}^{e_r}$ be a factorization where the $\ba a_i$ are distinct irreducible factors. It is easily seen that the set of $f$ in $V$ where $\ba f$ has ${\ba a_i}^{e_i + 1}$ as a factor, is a linear subspace $V_i$ of $V$. On the other hand if $f$ varies all over $V$ the restrictions $\ba f$ generate $V_{K|h=0}$. Thus we cannot have $V_i = V$ for any $i$. But since char $k= 0$ the field $k$ is infinite, so there must be an $f_0$ in $V - \cup V_i$. By corollary 3.3, $\ba p$ lifts to $\alpha p_0$ where $p_0 \in S$. Choose now any $f$ in $V \subseteq V_K$. Then \[ \ba f = \ba u \cdot \ba {\alpha p_0}. \] By lemma 3.1 we may conclude that $p_0$ is a factor of $f$ and thus a common factor of $V$. \end{proof} \section{ The case when $s = r$} Now we are ready for the specific work in proving the Main Theorem. Consider $S = k[x_1, \ldots, x_r]$. Let $W = (x_1, \ldots, x_r) = S_1$ which is a linear space. Use the notation $W^n = S_n$. (This will make our statements more unified in form.) Let the monomial order be the revlex order. In this section we prove the following (which is the case $s=r$ of the Main Theorem.) \begin{theorem} Let $V \subseteq S_{n+m}$ be a linear space such that \[ \gin V = W^n x_1^m \subseteq S_{n+m}. \] Then there exists a polynomial $p \in S_{m}$ such that $V = W^n p.$ \end{theorem} We assume $V$ to be in general coordinates so 2.7 applies. \begin{lemma} There is a form $p$ in $S_{K,m}$ such that \[ h^n p \in V_K. \] \end{lemma} \begin{proof} From 2.11 we have $\ii {\Gamma_K^{-1}(V_K)} = \gin V$ over $K$. Thus there exists a $q_0$ in $\Gamma_K^{-1}(V_K)$ such that \[ q_0 = x_r^n x_1^m + \mbox{terms with smaller monomials}.\] By the property of the revlex order, $x_r^n$ will divide all terms of $q_0$ so there exists a $p_0 \in S_{K,m}$ such that \[ q_0 = x_r^n p_0. \] Let $p = \Gamma_K (p_0)$. Then we get \[ h^n p = \Gamma_K(x_r)^n \Gamma_K(p_0) = \Gamma_K(q_0) \in V_K. \] \end{proof} From $V_K \subseteq S_K$ we obtain the subspace \[ V_{K|h=0} \subseteq S_K / (h). \] Let $\ba p$ be the image of $p$ in $V_{K|h=0}$. \begin{lemma} The elements in $V_{K|h=0}$ have $\ba p$ as a common factor. Furthermore it is a common factor of maximal degree. \end{lemma} \begin{proof} We first find the dimension of the space $V_{K|h=0}$. The map $\Gamma_K$ gives an isomorphism \[ \ba {\Gamma_K} : K[x_1, \ldots, x_r]/ (x_r) \longrightarrow K[x_1, \ldots, x_r]/ (h). \] Thus $\ba {\Gamma_K}^{-1}(V_{K|h=0}) = \Gamma_K^{-1}(V_K)_{|x_r = 0}$. Since $\Gamma_K^{-1}(V_K)$ has initial space \[ (x_1, \ldots, x_r)^n \cdot x_1^m, \] we get by 1.3 that $\Gamma_K^{-1}(V_K)_{|x_r = 0}$ has initial space \[ (x_1, \ldots, x_{r-1})^n \cdot x_1^m. \] Hence the dimension of $V_{K|h=0}$ is equal to the dimension of this space. Now differentiate the equation \[ h^n p \in V_K \] with respect to $ \partial^{|I|}/ \partial t^I$ where $I = (i_1, \ldots, i_{r-1})$ and $|I| = n$. The derivative will also be in $V_K$. This is essentially the fact that when a vector varies in a vector space the derivatives will also be in that vector space. We thus get \[ {\mbox{\boldmath $x$}}^I p + h R_I \in V_K \] for some $R_I$. Thus \begin{equation} {\mbox{\boldmath $x$}}^I \ba p \in V_{K|h=0}. \label{bap} \end{equation} But when $I$ varies, all these forms are linearly independent since $h$ does not divide any linear combination of the ${\mbox{\boldmath $x$}}^I$. By our statement about the dimension of $V_{K|h=0}$, the forms (\ref{bap}) must generate $V_{K|h=0}$, thus proving the lemma. \end{proof} By corollary 3.4 we may now conclude that $V$ has a maximal common factor $p_0$ of degree $m$. Thus proving 4.1. \section {The case when $s > r$} Now we assume $S = k[x_1, \ldots, x_s]$. As before $W = (x_1, \ldots, x_r) \subseteq S_1$, a linear subspace and assume $s > r$. The monomial order is revlex. In this section we prove the following by induction on $s$. \begin{theorem} Let $V \subseteq S_{n+m}$ be a linear space such that \[ \gin V = W^n x_1^m \subseteq S_{n+m}. \] Then there exists a polynomial $p \in S_{m}$ and a linear subspace $W_n \subseteq S_n$ such that $V = W_n p.$ \end{theorem} Assume $V$ to be in general coordinates. Let $g : S_1 \rightarrow S_1$ be a general coordinate change. Since $\ii {g^{-1}.V} = (x_1, \ldots, x_r)^n \cdot x_1^m$, by 1.3 it follows that $ \ii {g^{-1}.V_{|x_s = 0}} = (x_1, \ldots, x_r)^n \cdot x_1^m$ also. By induction $g^{-1}.V_{|x_s = 0}$ has a common factor. By translating back, $V_{|g.x_s = 0}$ also has a common factor (depending on $g$). The following expresses this in the algebraic language we use. \begin{lemma} There is a form $p$ in $S_{K,m}$ such that $\ba p$ in $S_{K|h=0}$ is a common factor of $V_{K|h=0}$. Furthermore it is a common factor of maximal degree. \end{lemma} \begin{proof} By 2.9.2 the generic initial ideal of $\Gamma_K^{-1}(V_K)_{|x_s = 0} = \gamma_K(V)_{|x_s = 0}$ is $\gin V _{|x_s = 0}$ (over $K$). The latter is, by 1.3, seen to be \[ (x_1, \ldots, x_r)^n \cdot x_1^m. \] By induction there is a form $\overline{p_1}$ in $S_{K,m|x_s = 0}$ which is a common factor of $\Gamma_K^{-1}(V_K)_{|x_s = 0}$. Now $x_1^m$ is a common factor of $\ii {\Gamma_K^{-1}(V_K)_{|x_s = 0}}$ of maximal degree. Then $\ba p_1$ must also have maximal degree as a common factor of $ \Gamma_K^{-1}(V_K)_{|x_s = 0}$. Lift this to a form $p_1$ in $S_{K,m}$. Then $p= \Gamma_K(p_1)$ is the required form. \end{proof} By corollary 3.4 we may now conclude that $V$ has a maximal common factor $p_0$ of degree $m$. Thus proving 5.1. \section {An example} Consider the complete intersection of three quadratic forms in ${\bf P}^3$. Let $I \subseteq k[x_1,x_2,x_3,x_4]$ be its homogeneous ideal. We have the following facts. \begin{itemize} \item [1.] $I$ and $\gin I$ have the same Hilbert functions. \item [2.] $\gin I$ is Borel-fixed. ( See proposition 15.20 in \cite{Ei}.) \item [3.] Since $I$ is saturated, by proposition 2.21 in \cite{Gr} we have $\gin I : x_4 = \gin I$. This is really just the fact that $\ii {I : x_4} = \ii I : x_4$ for the revlex order (proposition 15.12 b. in \cite{Ei}), and that if $I$ is in general coordinates and saturated then $I : x_4 = I$. \end{itemize} These three facts imply that there are two possible candidates for $\gin I$~: \begin{eqnarray*} J^{(1)} & = & (x_1^2, x_1x_2, x_2^2, x_1x_3^2, x_2x_3^2, x_3^4), \\ J^{(2)} & = & (x_1^2, x_1x_2, x_1x_3, x_2^3, x_2^2x_3, x_2x_3^2, x_3^4). \end{eqnarray*} However, by the theorem above if $\gin I = J^{(2)}$ then the quadratic forms in $I_2 \subseteq S_2$ would have to have a common factor. Impossible. Thus $\gin I = J^{(1)}$. On the other hand, if $I$ is an ideal with $\gin I = J^{(2)}$ then since the quadratic forms in $I_2$ would have a common factor it must be the ideal of seven points in a plane pluss one extra point not in the plane. Note also the following. Let $>_1$ be the ordering of the monomials which is lexicographic in the three first variables, and then refined with the reverse lexicographic order with respect to the last variable. I.e. \[ x_1^{a_1}x_2^{a_2}x_3^{a_3}x_4^{a_4} > x_1^{b_1}x_2^{b_2}x_3^{b_3}x_4^{b_4} \] if $a_4 < b_4$, or $a_4 = b_4$ and \[ x_1^{a_1}x_2^{a_2}x_3^{a_3} > x_1^{b_1}x_2^{b_2}x_3^{b_3} \] for the lexicographic order. Then if the three forms are general it is easily seen that $\gin I = J^{(2)}$. In fact it is not difficult to argue that one will always have $\gin I = J^{(2)}$ if you have a complete intersection of three forms and this order. Thus both $J^{(1)}$ and $J^{(2)}$ are in fact specialisations of $I$. Furthermore it is not difficult to give an example of a complete intersection of three forms such that in$(I) = J^{(2)}$ for the reverse lexicographic order. Thus the fact that one can read some interesting algebraic or geometric information from the initial ideal depends on the fact that you are looking at the {\it generic initial ideal}. To sum up, $J^{(2)}$ is a specialisation of the ideal $I$ of a complete intersection of three quadratic forms in general coordinates through the order $>_1$ given above. It is also the specialisation of an ideal $I$ of a complete intersection of three quadratic forms through the revlex order, but it is {\em never} a specialisation of the ideal $I$ of a complete intersection of three quadratic forms through the revlex order when the forms are in general coordinates.

9708

alg-geom/9708019

en

https://arxiv.org/abs/alg-geom/9708019

alg-geom/9708019

Alexander A. Voronov

Alexander A. Voronov (RIMS and M.I.T.)

Stability of the Rational Homotopy Type of Moduli Spaces

7 pages, 1 figure

RIMS-1157

We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational homotopy type of M_{g,n} as g goes to infinity and the formality of it.

alg-geom

\section*{Introduction} The description of the algebraic topology of the moduli space $\mgn{g}$ of compact complex algebraic curves has long been a tantalizing problem. The idea of ``stable cohomology '' of $\mgn{g}$ as the genus $g \to \infty$, brought in by J.~L. Harer and D.~Mumford, suggested a more graspable object to study as a first step. Mumford's Conjecture \cite{mumford}, stating that the stable cohomology\ is the polynomial algebra on the so-called Mumford-Morita-Miller classes, has become a new tantalizing problem since then. Recently M.~Pikaart \cite{pikaart} has proven that the Hodge structure on the stable cohomology\ is pure, thus providing more evidence to the conjecture, which easily implies purity. Our purpose in this note is to show that the rational homotopy type of the moduli space $\mgn{g}$ stabilizes as $g \to \infty$ and prove that the moduli space is formal in the stable limit. If the stable cohomology\ of the moduli space were just known to be the cohomology\ of a certain space $\mgn{\infty}$, then a theorem of E.~Miller and S.~Morita \cite{miller} asserting that $H^\bullet (\mgn{\infty})$ is a free graded commutative algebra would immediately imply the formality of $\mgn{\infty}$. Thus, the essential result of this paper is showing that the limit of the rational homotopy types of $\mgn{g}$ as $g \to \infty$ exists. After this is done, the formality of the limit is automatic. Here are few words about the structure of the paper. Sections 1 and 2 are dedicated to the description of the (nonstable) rational homotopy type of the moduli space $\mgn{g}$. Section~\ref{infty}, where the stable rational homotopy type of $\mgn{g}$ is studied, is independent of the first two sections. It may be worth noting how this paper relates to Jim Stasheff's work. Stasheff polyhedra $K_n$ are known for many remarkable properties. One of them, that $K_n$ is a connected component of the real compactification of the moduli space of real projective lines with $n+1$ punctures, was pointed out by M.~Kontsevich in \cite{maxim}. Thus, Stasheff polyhedra are real analogues of the compactified moduli spaces $\mgnb{g}$. Or, perhaps, the moduli spaces $\mgnb{g}$ are complex analogues of Stasheff polyhedra, the chicken and the egg problem. The topology of $K_n$ is trivial ($K_n$ is contractible), and the topology of $\mgn{g}$ and $\mgnb{g}$ is just the opposite. It is rather the combinatorics of Stasheff polyhedra which makes them very useful in the topology of loop spaces \cite{jim}. The combinatorics of the genus zero spaces $\mgn{0}$ has similarly proven to be useful in studying the topology of double loop spaces \cite{gj} and the algebraic structure of 2d quantum field theory \cite{ksv1,ksv2,kvz}. But what is the topology of these complex analogues of Stasheff polyhedra, especially for a high genus? We also implicitly use a result of S.~Halperin and Stasheff \cite{halperin-stasheff} throughout the paper: formality over any field of characteristic zero implies that over the rationals. \begin{ack} I am very grateful to Pierre Deligne, Dick Hain, Eduard Looijenga, and Martin Pikaart for helpful discussions. I would like to thank the Institute for Advanced Study in Princeton and the Research Institute for Mathematical Sciences in Kyoto for their hospitality during the summer 1997, when the essential part of the work on this paper was done. \end{ack} \section{The rational homotopy type of a complex smooth quasi-projective variety} Here we recall J.~W. Morgan's description \cite{morgan} of a model of the rational homotopy type of the complement $U$ of a normal crossings divisor $D= \bigcup_{i=1}^r D_i$ in a compact complex manifold $X$. The rational homotopy type will be understood in the sense of D.~Sullivan \cite{sullivan}, see also P.~A. Griffiths and Morgan \cite{griffiths-morgan} and D.~Lehmann \cite{lehmann}. It determines the usual rational homotopy type of a topological space if it is simply connected. In general it determines the rational nilpotent completion thereof. The rational homotopy type of $U$ is determined by the differential graded (DG) commutative algebra that is nothing but the first term $\mathcal{A}^\bullet = E_1^\bullet$ of the spectral sequence associated with the weight filtration on the log-complex $\Omega^{\bullet,\bullet}_X(\log D)$ of the smooth $(p,q)$-forms on $U$ with logarithmic singularities along $D$. Otherwise, one can describe the same DG algebra as the second term of the Leray spectral sequence of inclusion $U \hookrightarrow X$. In any case, the model of $U$ is given by the DG algebra $\mathcal{A}^\bullet$, where \begin{align*} \mathcal{A}^k & = \bigoplus_{p+q=k} \mathcal{A}^{p,q}, & \mathcal{A}^{p,q} & = \bigoplus_{ \substack{S \subset \{1,2,\dots,r\}\\ \abs{S} = -p} } H^{2p+q}(D_S, \mathbb{C}),\\ p & \le 0, \; q \ge 0, & D_S & = \bigcap_{i \in S} D_i. \end{align*} The multiplication structure is given by \[ a \cdot a' = \begin{cases} (-1)^{pq'+\epsilon} (a|_{D_S \cap D_{S'}}) \cup (a'|_{D_S \cap D_{S'}}) & \text{if $S \cap S' = \emptyset$},\\ 0 & \text{otherwise} \end{cases} \] for $a \in H^{2p+q}(D_S,\mathbb{C})$ and $a' \in H^{2p'+q'}(D_{S'}, \mathbb{C})$, where $\epsilon$ is the sign of the shuffle putting the set $S \cap S'$ in an increasing order, assuming each of the subsets $S$ and $S'$ to be already in an increasing order. Note that this multiplication law makes $\mathcal{A}^{\bullet,\bullet}$ into a bigraded commutative algebra. Finally, the differential $d: \mathcal{A}^{p,q} \to \mathcal{A}^{p+1,q}$ can be described as \[ da = \sum_{j=1}^{-p} (-1)^{j-1} (\iota_j)_* a, \] where $a \in H^{2p+q}(D_S, \mathbb{C})$, $S = \{i_1, \dots, i_{-p}\}$, $\iota_j$ is the natural embedding $D_S \subset D_{S\setminus i_j}$, and $(\iota_j)_*: H^{2p+q}(D_S, \mathbb{C}) \to H^{2p+q+2}(D_{S\setminus i_j},\mathbb{C})$ is the Gysin map. P.~Deligne \cite{del:hodge} proved that the spectral sequence $E_1^{p,q} = \mathcal{A}^{p,q}$ degenerates at $E_2$, that is, the cohomology\ of the DG algebra $\mathcal{A}^\bullet$ is equal to $H^\bullet(U,\mathbb{C})$. The very DG algebra $\mathcal{A}^\bullet$ describes the rational homotopy type of $U$, according to Morgan's theorem \cite{morgan}. \section{The rational homotopy type of $\mgn{g}$} \label{mgn} Results of the previous section apply to the moduli space $\mgn{g}$, which is the complement of a normal crossing divisor in the Deligne-Knudsen-Mumford compactification $\mgnb{g}$. The problem that the space in question is a stack rather than variety does not arise, because we work with the \textbf{complex coefficients}, as we will assume throughout the paper. From the combinatorics of Deligne-Knudsen-Mumford's construction, we can say more specifically, cf.\ \cite{ksv2}, that \[ \mathcal{A}^{p,q} = \bigoplus_G \; \left( \bigotimes_{v \in G} H^{\bullet} (\overline{\mathcal{M}}_{g(v), n(v)}) \right)_{2p+q}^{\operatorname{Aut}(G)}, \] the summation running over all stable labeled $n$-graphs $G$ of genus $g(G) = g$ and $v(G) = - p +1$ vertices. Here we refer to \emph{graphs} of the following kind. Each graph is connected and has \emph{$n$ enumerated exterior edges}, edges which are incident with only one vertex of the graph. Each vertex $v$ of the graph is \emph{labeled} by a nonnegative integer $g(v)$, called the genus of a vertex. The \emph{stability} condition means that any vertex $v$ labeled by $g(v)=1$ should be incident with at least one edge (i.e., be at least of valence one) and each vertex $v$ with $g(v) = 0$ should be at least of valence three. The \emph{genus} $g(G)$ of a graph $G$ is given by the formula $g(G) = b_1(G) + \sum_v g(v)$, where $b_1(G)$ is the first Betti number of the graph. The \emph{number $n(v)$} is the valence of the vertex $v$ and $\operatorname{Aut}(G)$ is the \emph{automorphism group of a graph $G$} (bijections on vertices and edges, preserving the exterior edges, the labels of vertices and the incidence relation). The subscript $2p+q$ in the formula refers to taking the homogeneous component of this degree. The differential $d: \mathcal{A}^{p,q} \to \mathcal{A}^{p+1,q}$ is induced by contracting interior edges in $G$, which corresponds to replacing a neighborhood of a double point on a curve by a cylinder. Our goal here is to look at the stable (as $g \to \infty$) cohomology\ and rational homotopy type of the moduli space $\mgn{g}$. The following result of Pikaart \cite{pikaart} gives a certain clue on what is going on within a ``stable range''. \begin{thm}[Pikaart] \label{pikaart} The restriction mapping $H^k(\mgnb{g}) \linebreak[1] \to \linebreak[0] H^k(\mgn{g})$ is surjective for $k \le (g-1)/2$. \end{thm} \begin{cor} For $p+q \le (g-1)/2$, the cohomology\ of $\mathcal{A}^{p,q}$ is nonzero only for $p=0$, see Figure~$\ref{graph}$. \end{cor} \begin{figure}[tb] \centerline{\epsfxsize=1.5in \epsfbox{graph.eps}} \caption{The algebra $\mathcal{A}^{p,q}$. The shaded region is the ``stable range'', where the cohomology\ is concentrated along the fat line.} \label{graph} \end{figure} \begin{proof} By our construction of the spectral sequence, the natural composite mapping $\mathcal{A}^{0,k} \to H^{0,k} (\mathcal{A}^{\bullet,\bullet}, d) \hookrightarrow H^k (\mathcal{A}^{\bullet,\bullet}, d)$ is the same as the restriction mapping $H^k(\mgnb{g}) \to H^k(\mgn{g})$. Therefore, the $p \ne 0$ columns of $\mathcal{A}^{p,q}$ do not contribute to the cohomology\ of the DG algebra $\mathcal{A}$ in the stable range. \end{proof} \section{The stable limit} \label{infty} The stable limit in cohomology\ of the moduli spaces $\mgn{g}$ is achieved, roughly speaking, by gluing more and more handles to the complex curve. This yields an inductive system (of isomorphisms) on the level of cohomology. But since there is no natural mapping between the moduli spaces of different genera, one cannot speak of a limiting rational homotopy type. The question of taking the limit of the DG algebras $\mathcal{A}$ of Section~\ref{mgn} for $\mgn{g}$ may not be so obviously resolved, either, because these algebras are constructed out of the cohomology\ of $\mgnb{g'}$, $g'$ running between 0 and $g$. When $g\to \infty$, $g'$ does not, and on top of that, taking the stable limit of cohomology\ of $\mgnb{g}$ requires a finer tuning, cf.\ Pikaart \cite{pikaart}. Our plan here is to show that a $k$-minimal model of $\mgn{g}$ is independent of $k$ as long as $g \ge 2k+3$. In particular, the limit of $k$-minimal models exists and may be called a ``$k$-minimal model of $\mgn{\infty}$'', continuing the abuse of notation adopted for cohomology. Since a minimal model of a space may be obtained as a union of $k$-minimal models, we call this union a ``minimal model of $\mgn{\infty}$''. Each of these $k$-minimal models is $k$-formal in a natural sense, see below, and the formality of $\mgn{\infty}$ follows. First of all, we recall basic notions on $k$-minimal models, see \cite{griffiths-morgan,lehmann,morgan}. From now on we will assume that our spaces and algebras are connected and simply connected, {i.e.}, their $H^0 = \mathbb{C}$ and $H^1 = 0$. In application to moduli spaces, this is the case as long as $g \ge 1$, see Harer \cite{harer2}. A DG algebra $\mathfrak{M}$ is called \emph{minimal} if it is free as a DG commutative algebra, $\mathfrak{M}^1 = 0$, and $d(\mathfrak{M}) \subset \mathfrak{M}^+ \cdot \mathfrak{M}^+$, where $\mathfrak{M}^+ = \bigoplus_{i >0} \mathfrak{M}^i$. A \emph{minimal model} of a DG algebra $\mathcal{A}$ is a minimal DG algebra $\mathfrak{M}$ along with a \emph{quasi-isomorphism} $\mathfrak{M} \to \mathcal{A}$, a morphism of DG algebras inducing an isomorphism on cohomology. Every DG algebra $\mathcal{A}$ has a minimal model, unique up to an isomorphism, which is in its turn unique up to homotopy. Let $k \ge 0$ be an integer. A $k$-\emph{minimal model} of a DG algebra $\mathcal{A}$ is a minimal algebra $\mathfrak{M}(k)$ generated by elements in degrees $\le k$ along with a morphism $\mathfrak{M}(k) \to \mathcal{A}$ inducing an isomorphism on cohomology\ in degrees $\le k$ and an injection in degree $k+1$. A $k$-minimal model is unique up to an isomorphism uniquely defined up to homotopy. If one has an increasing sequence of embeddings \[ \mathfrak{M}(0) \subset \mathfrak{M}(1) \subset \mathfrak{M}(2) \subset \dots \] together with morphisms $\mathfrak{M}(k) \to \mathcal{A}$ compatible with each other, so that $\mathfrak{M}(k)$ is a $k$-minimal model of $\mathcal{A}$, then the union $\mathfrak{M} = \bigcup_k \mathfrak{M}(k)$ along with the natural morphism $\mathfrak{M} \to \mathcal{A}$ is a minimal model of $\mathcal{A}$. We will call a DG algebra $\mathcal{A}$ (or a space whose rational homotopy type we consider) \emph{formal} if a minimal model $\mathfrak{M}$ of $\mathcal{A}$ is isomorphic to a minimal model of its cohomology\ $H^\bullet(\mathcal{A})$ taken with the zero differential. This is equivalent to saying that there is a quasi-isomorphism $\mathfrak{M} \to H^\bullet(\mathcal{A})$. In this case the rational homotopy type of $\mathcal{A}$ (or the space) is determined by its rational cohomology\ ring. Formality implies that all Massey products are zero. If the cohomology\ algebra $H^\bullet(\mathcal{A})$ is free as a graded commutative algebra, then $\mathcal{A}$ is formal; see Proposition~\ref{k-formal} for a $k$-version of this statement. Another example of a formal space is any compact K\"ahler manifold, a famous result of Griffiths, Deligne, Morgan, and Sullivan \cite{dgms}. We will similarly call a DG algebra $\mathcal{A}$ or a space $k$-\emph{formal}, if $k$-minimal models of $\mathcal{A}$ and its cohomology\ are isomorphic. We also need to prove the following proposition establishing a particular case of $k$-formality. We say that a graded commutative algebra $\mathcal{C}$ is \emph{$k$-free} if there exists a graded vector space $V = \bigoplus_{i=0}^k V^k$ and a mapping $V \to \mathcal{C}$ of graded vector spaces defining a morphism $S(V) \to \mathcal{C}$ of graded algebras, where $S(V)$ is the free DG commutative algebra on $V$, which is an isomorphism in degrees $\le k$ and an injection in degree $k+1$. \begin{prop} \label{k-formal} Suppose that the cohomology\ $H^\bullet(\mathcal{A})$ of a DG algebra $\mathcal{A}$ is $k$-free, based on a graded vector space $V$. Then $S(V)$ is a $k$-minimal model of $\mathcal{A}$, and $\mathcal{A}$ is $k$-formal. \end{prop} \begin{proof} If $H^\bullet(\mathcal{A})$ is $k$-free, then there exists a linear injection $\phi: V \hookrightarrow H^\bullet(\mathcal{A})$. Pick a linear mapping $V \to \mathcal{A}$ which takes each element $v$ of $V$ to a cocycle representing the cohomology\ class $\phi(v)$. Then the natural morphism of graded commutative algebras $S(V) \to \mathcal{A}$ obviously respects the differentials and satisfies the axioms of a $k$-minimal model of $\mathcal{A}$. By assumption, $S(V)$ is at the same time a $k$-minimal model of $H^\bullet(\mathcal{A})$, whence $k$-formality of $\mathcal{A}$. \end{proof} Now we are ready to present the main result of the paper. \begin{thm} \begin{enumerate} \item The moduli space $\mgn{g}$ is $k$-formal for $g \ge 2k+3$. \item The subalgebra $H^\bullet (\mgn{\infty}) (k)$ of $H^\bullet (\mgn{\infty})$ generated in degrees $\le k$ is a $k$-minimal model of $\mgn{g}$ for $g \ge 2k+3$. \item A $k$-minimal model of $\mgn{g}$ is independent of $g$ as long as $g \ge 2k+3$. We will call it a $k$-\emph{minimal model of} $\mgn{\infty}$. \item The $k$-minimal models of $\mgn{\infty}$ form am increasing sequence of embeddings. The union, a \emph{minimal model of} $\mgn{\infty}$, is isomorphic to its cohomology\ $H^\bullet(\mgn{\infty})$. In particular, $\mgn{\infty}$ is formal. \end{enumerate} \end{thm} \begin{proof} 1. The stable cohomology\ $H^\bullet(\mgn{\infty})$ is a free graded commutative algebra, \emph{i.e.}, isomorphic to $S(V)$ for a graded vector space $V$, according to Miller-Morita's theorem \cite{miller} for $n=0$ and Looijenga's handling \cite{looijenga} of the $n \ge 0$ case. Moreover, $H^0(\mgn{\infty}) = \mathbb{C}$ and $H^1(\mgn{\infty}) = 0$ for $g \ge 1$, see Harer \cite{harer2}. Given a nonnegative integer $k$, the groups $H^k(\mgn{g})$ are known to stabilize as soon as $g \ge 2k+1$; this is the Harer-Ivanov Stability Theorem \cite{harer:stab,ivanov}. Therefore, $H^\bullet(\mgn{g})$ is $k$-free for $g \ge 2k+3$: the mapping $V^{\le k} \to H^\bullet (\mgn{g})$ makes $H^\bullet (\mgn{g})$ a $k$-free graded algebra. Proposition~\ref{k-formal} then implies that $\mgn{g}$ is $k$-formal for $g \ge 2k+3$, $S(V^{\le k})$ being a $k$-minimal model of it. 2 and 3. Notice that since $H^\bullet(\mgn{\infty})$ is free, the subalgebra $H^\bullet(\mgn{\infty})(k)$ generated in degrees $\le k$ can be identified with $S(V^{\le k})$, which we have just seen to be a $k$-minimal model of $\mgn{g}$ for $g \ge 2k+3$. 4. The subalgebras \[ H^\bullet (\mgn{\infty}) (0) \subset H^\bullet (\mgn{\infty}) (1) \subset H^\bullet (\mgn{\infty}) (2) \subset \dots \] form an increasing sequence of $k$-minimal models of $\mgn{\infty}$, therefore, their union, $H^\bullet (\mgn{\infty})$, is a minimal model of $\mgn{\infty}$, and thereby $\mgn{\infty}$ is formal. \end{proof} In view of this result, Mumford's Conjecture \cite{mumford}, if true, implies the following refinement: \emph{the polynomial algebra on the Mumford-Morita-Miller classes $\kappa_i$, $i=1, 2, \dots,$ and the first Chern classes $c_1(T_i)$ of the ``tangent at the $i$th puncture'' bundles, $i = 1, 2 , \dots, n$, with a zero differential is the stable minimal model of the moduli space $\mgn{g}$ as $g \to \infty$}. \bibliographystyle{alpha}

9708

alg-geom/9708010

en

https://arxiv.org/abs/alg-geom/9708010

alg-geom/9708010

Carlos Simpson

Carlos Simpson (CNRS, Universit\'e Paul Sabatier, Toulouse, France)

Limits in $n$-categories

Approximately 90 pages

We define notions of direct and inverse limits in an $n$-category. We prove that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for $n$-categories, the notion of $n$-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.

alg-geom

\section*{Limits in $n$-categories} Carlos Simpson\newline CNRS, UMR 5580, Universit\'e Paul Sabatier, 31062 Toulouse CEDEX, France. \bigskip \numero{Introduction} One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of ``homotopy limit'' commonly called ``holim'' for short. The purpose of the this paper is to begin to develop the notion of limit for $n$-categories, which should be a bridge between the categorical notion of limit and the homotopical notion of holim. We treat Tamsamani's notion of $n$-category \cite{Tamsamani}, but similar arguments and results should hold for the Baez-Dolan approach \cite{BaezDolanLetter}, \cite{BaezDolanIII}, or the Batanin approach \cite{Batanin}, \cite{Batanin2}. We define the notions of direct and inverse limits in an arbitrary (fibrant cf \cite{nCAT}) $n$-category $C$. Suppose $A$ is an $n$-category, and suppose $\varphi : A\rightarrow C$ is a morphism, which we think of as a family of objects of $C$ indexed by $A$. For any object $U\in C$ we can define the $(n-1)$-category $Hom(\varphi , U)$ of morphisms from $\varphi$ to $U$. We say that a morphism $\epsilon : \varphi \rightarrow U$ (i.e. an object of this $(n-1)$-category) is a {\em direct limit of $\varphi$} (cf \ref{defdirect} below) if, for every other object $V\in C$ the (weakly defined) composition with $\epsilon$ induces an equivalence of $n-1$-categories from $Hom _C(U,V)$ to $Hom (\varphi , V)$. An analogous definition holds for saying that a morphism $U\rightarrow \varphi$ is an {\em inverse limit of $\varphi$} (cf \ref{definverse} below). The main theorems concern the case where $C$ is the $n+1$-category $nCAT'$ (fibrant replacement of that of) of $n$-categories. \begin{Theorem} {\rm (\ref{inverse} \ref{direct})} The $n+1$-category $nCAT'$ admits arbitrary inverse and direct limits. \end{Theorem} This is the analogue of the classical statement that the category $Sets$ admits inverse and direct limits---which is the case $n=0$ of our theorem. The fact that we work in an $n$-category means that we automatically keep track of ``higher homotopies'' and the like. This brings the ideas much closer to the relatively simple notion of limits in a category. I first learned of the notion of ``2-limit'' from the paper of Deligne and Mumford \cite{DeligneMumford}, where it appears at the beginning with very little explanation. Unfortunately at the writing of the present paper I have not been able to investigate the history of the notion of $n$-limits, and I apologize in advance for any references left out. At the end of the paper we propose many applications of the notion of limit. Most of these are as of yet in an embryonic stage of development and we don't pretend to give complete proofs. \begin{center} {\em Organization} \end{center} The paper is organized as follows: we start in \S 2 with some preliminary remarks recalling the notion of $n$-category from \cite{Tamsamani} and the closed model structure from \cite{nCAT}. At the end of \S 2 we define and discuss one of our main technical tools, the $n+1$-precat $\Upsilon ^k(E_1,\ldots , E_k)$ which can be seen as a $k$-simplex with $n$-precats $E_1,\ldots , E_k$ attached to the principal edges. In \S 3 we give the basic definitions of inverse and direct limits, and treat some general properties such as invariance under equivalence, and variation with parameters. In \S 4 we start into the main result of the paper which is the existence of inverse limits in $nCAT'$. Here, the construction is relatively straightforward: if $\varphi : A\rightarrow nCAT'$ is a morphism then the inverse limit of $\varphi$ is just the $n$-category $$ \lambda = Hom _{\underline{Hom}(A,C)}(\ast , \varphi ). $$ This is in perfect accord with the usual situation for inverse limits of families of sets. Our only problem is to prove that this satisfies the definition of being an inverse limit. Thus the reader could read up to here and then skip the proof and move on to direct limits. We treat direct limits in $nCAT'$ by a trick in \S 5: given $\psi : A \rightarrow nCAT'$ we construct an $n+1$-category $D$ parametrizing all morphisms $\psi \rightarrow B$ to objects of $nCAT'$, and then construct the direct limit $U$ as the inverse limit of the functor $D\rightarrow nCAT'$. The main problem here is that, because of set-theoretic considerations, we must restrict to a category $D_{\alpha}$ of morphisms to objects $B$ with cardinality bounded by $\alpha$. We mimic a possible construction of direct limits in $Sets$ and encounter a few of the same difficulties as with inverse limits. Again, the reader might want to just look at the proof for $Sets$ and skip the difficulties encountered in extending this to $nCAT'$. At the end we discuss some proposed applications: \newline ---First, the notions of homotopy coproduct and fiber product, and their relation to the usual notions which can be calculated using the closed model structure. \newline ---Then we discuss representable functors, give a conjectural criterion for when a functor should be representable, and apply it to the problem of finding internal $\underline{Hom}$. \newline ---The next subsection concerns $n$-stacks, defined using certain inverse limits. \newline ---We give a very general discussion of the notion of stack in any setting where one knows what limits mean. \newline ---We discuss direct images of families of $n$-categories by functors of the underlying $n+1$-categories, and apply this to give a notion of ``realization''. \newline ---Finally, we use limits to propose a notion of {\em relative Malcev completion of the higher homotopy type}. In all of the above applications except the first, most of the statements which we need are left as conjectures. Thus, this discussion of applications is still only at a highly speculative stage. One recurring theme is that the argument given in \S 5 should work in a fairly general range of situations. I would like to thank A. Hirschowitz, for numerous discussions about stacks which contributed to the development of the ideas in this paper. I would like to thank J. Tapia and J. Pradines for a helpful discussion concerning the argument in \S 5. \bigskip \numero{Preliminary remarks} \subnumero{$n$-categories} \begin{parag} \label{catnerve} We begin by recalling the correspondence between categories and their nerves. Let $\Delta$ denote the simplicial category whose objects are finite ordered sets $p= \{ 0,\ldots , p\}$ and morphisms are order-preserving maps. If $C$ is a category then its nerve is the simplicial set (i.e. a functor $A:\Delta ^o\rightarrow Sets$) defined by setting $A_p$ equal to the set of composable $p$-uples of arrows in $C$. This satisfies the property that the ``Segal maps'' (cf the discussion of Segal's delooping machine \cite{Segal} in \cite{Adams} for the origin of this terminology) $$ A_p \rightarrow A_1 \times _{A_0} \ldots \times _{A_0} A_1 $$ are isomorphisms. To be precise this map is given by the $p$-uple of face maps $1\rightarrow p$ which take $0$ to $i$ and $1$ to $i+1$ for $i=0,\ldots , p-1$. Conversely, given a simplicial set $A$ such that the Segal maps are isomorphisms we obtain a category $C$ by taking $$ Ob (C) := A_0 $$ and $$ Hom _C(x,y):= A_1(x,y) $$ with the latter defined as the inverse image of $(x,y)$ under the map (given by the pair of face maps) $A_1\rightarrow A_0 \times A_0$. The condition on the Segal maps implies that (with a similar notation) $$ A_2(x,y,z)\stackrel{\cong}{\rightarrow} A_1(x,y)\times A_1(y,z) $$ and the third face map $A_2(x,y,z)\rightarrow A_1(x,z)$ thus provides the composition of morphisms for $C$. By looking at $A_3(x,y,z,w)$ one sees that the composition is associative and the degeneracy maps in the simplicial set provide the identity elements. \end{parag} \begin{parag} \label{ncatsdef0} The notion of weak $n$-category of Tamsamani \cite{Tamsamani} is a generalization of the above point of view on categories. We present the definition in a highly recursive way, using the notion of $n-1$-category in the definition of $n$-category. See \cite{Tamsamani} for a more direct approach. This definition is based on Segal's delooping machine \cite{Segal} \cite{Adams}. \end{parag} \begin{parag} \label{notstrict} Note that Tamsamani uses the terminology {\em $n$-nerve} for what we will call ``$n$-category'' since he needed to distinguish this from the notion of strict $n$-category. In the present paper we will never speak of strict $n$-categories and our terminology ``$n$-category'' means weak $n$-category or $n$-nerve in the sense of \cite{Tamsamani}. \end{parag} \begin{parag} \label{ncatsdef1} An {\em $n$-category} \cite{Tamsamani} is a functor $A$ from $\Delta ^o$ to the category of $n-1$-categories denoted $$ p\mapsto A_{p/} $$ such that $0$ is mapped to a set \footnote{ Recursively an $n$-category which is a set is a constant functor where the $A_{p/}$ are all the same set---considered as $n-1$-categories.} $A_0$ and such that the {\em Segal maps} $$ A_{p/} \rightarrow A_{1/} \times _{A_0} \ldots \times _{A_0}A_{1/} $$ are equivalences of $n-1$-categories (cf \ref{defequiv1} below). \end{parag} \begin{parag} \label{multisimplicial} The {\em category of $n$-categories} denoted $n-Cat$ is just the category whose objects are as above and whose morphisms are the morphisms strictly preserving the structure. It is a subcategory of $Hom (\Delta ^o, (n-1)-Cat)$. Working this out inductively we find in the end that $n-Cat$ is a subcategory of $Hom ((\Delta ^n)^o, Sets)$, in other words an $n$-category is a certain kind of multisimplicial set. The multisimplicial set is denoted $$ (p_1,\ldots , p_n )\mapsto A_{p_1,\ldots p_n} $$ and the $(n-1)$-category $A_{p/}$ itself considered as a multisimplicial set has the expression $$ A_{p/} = \left( (q_1,\ldots , q_{n-1})\mapsto A_{p,q_1,\ldots , q_{n-1}}\right) . $$ \end{parag} \begin{parag} \label{theta} The condition that $A_0$ be a set yields by induction the condition that if $p_i=0$ then the functor $A_{p_1,\ldots , p_n}$ is independent of the $p_{i+1}, \ldots , p_n$. We call this the {\em constancy condition}. In \cite{nCAT} we introduce the category $\Theta ^n$ which is the quotient of $\Delta ^n$ having the property that functors $(\Theta ^n)^o\rightarrow Sets$ correspond to functors on $\Delta ^n$ having the above constancy property. Now $n-Cat$ is a subcategory of the category of presheaves of sets on $\Theta ^n$. \end{parag} \begin{parag} Before discussing the notion of equivalence which enters into the above definition we take note of the relationship with \ref{catnerve}. If $A$ is an $n$-category then its {\em set of objects} is the set $A_0$. The face maps give a morphism from $n-1$-categories to sets $$ A_{p/}\rightarrow A_0 \times \ldots \times A_0 $$ and we denote by $A_{p/}(x_0,\ldots , x_p)$ the $n-1$-category inverse image of $(x_0,\ldots , x_p)$ under this map. For two objects $x,y\in A_0$ the $n-1$-category $A_{1/}(x,y)$ is the {\em $n-1$-category of morphisms from $x$ to $y$}. This is the essential part of the structure which corresponds, in the case of categories, to the $Hom$ sets. One could adopt the notation $$ Hom _A(x,y):= A_{1/}(x,y). $$ The condition that the Segal maps are equivalences of $n-1$-categories says that the $A_{p/}(x_0,\ldots , x_p)$ are determined up to equivalence by the $A_{1/}(x,y)$. The role of the higher $A_{p/}(x_0,\ldots , x_p)$ is to provide the composition (in the case $p=2$) and to keep track of the higher homotopies of associativity $(p\geq 3$). Contrary to the case of $1$-categories, here we need to go beyond \footnote{One might conjecture that it suffices to stop at $p=n+2$.} $p=3$. \end{parag} \begin{parag} \label{defequiv1} In order for the recursive definition of $n$-category given in \ref{ncatsdef1} to make sense, we need to know what an {\em equivalence} of $n$-categories is. For this we generalize the usual notion for categories: an equivalence of categories is a morphism which is (1) fully faithful and (2) essentially surjective. We would like to define what it means for a functor between $n$-categories $f:A\rightarrow B$ to be an equivalence. The generalization of the fully faithful condition is immediate: we require that for any objects $x,y\in A_0$ the morphism $$ f: A_{1/} (x,y) \rightarrow B_{1/}(f(x), f(y)) $$ be an equivalence of $n-1$-categories (and we are supposed to know what that means by recurrence). \end{parag} \begin{parag} \label{essentialsurjectivity} The remaining question is how to define the notion of essential surjectivity. Tamsamani does this by defining a truncation operation $T$ from $n$-categories to $n-1$-categories (a generalization of the truncation of topological spaces used in the Postnikov tower). Applying this $n$ times to an $n$-category $A$ we obtain a set $T^nA$ which can also be denoted $\tau _{\leq 0}A$. This set is the set of ``objects of $A$ up to equivalence'' where equivalence of objects is thought of in the $n$-categorical sense. We say that $f:A\rightarrow B$ is {\em essentially surjective} if the induced map $$ \tau _{\leq 0} (f) : \tau _{\leq 0} A \rightarrow \tau _{\leq 0} B $$ is a surjection of sets. One has in fact that if $f$ is an equivalence according to the above definition then $\tau _{\leq 0} f$ is an isomorphism. \end{parag} \begin{parag} \label{anotherapproach} Another way to approach the definition of $\tau _{\leq 0}A$ is by induction in the following way. Suppose we know what $\tau _{\leq 0}$ means for $n-1$-categories. Then for an $n$-category $A$ the simplicial set $p\mapsto \tau _{\leq 0} (A_{p/})$ satisfies the condition that the Segal maps are isomorphisms, so it is the nerve of a $1$-category. This category may be denoted $\tau _{\leq 1} A$. We then define $\tau _{\leq 0}A$ to be the set of isomorphism classes of objects in the $1$-category $\tau _{\leq 1} A$. \end{parag} The above definition is highly recursive. One must check that everything is well defined and available when it is needed. This is done in \cite{Tamsamani} although the approach there avoids some of the inductive definitions above. \bigskip \subnumero{The closed model structure} An $n$-category is a presheaf of sets on $\Theta ^n$ (\ref{theta}) satisfying certain conditions as described above. Unfortunately $n-Cat$ considered as a subcategory of the category of presheaves, is not closed under pushout or fiber product. This remark is the starting point for \cite{nCAT}. There, one considers the full category of presheaves of sets on $\Theta$ (these presheaves are called {\em $n$-precats}) and \cite{nCAT} provides a closed model structure (cf \cite{Quillen} \cite{QuillenAnnals} \cite{Jardine}) on the category $nPC$ of $n$-precats, corresponding to the homotopy theory of $n$-categories. In this section we briefly recall how this works. \begin{parag} \label{theta2} It is more convenient for the purposes of the closed model structure to work with presheaves over the category $\Theta ^n$ (cf \ref{theta} above), defined be the quotient of the cartesian product $\Delta ^n$ obtained by identifying all of the objects $(M, 0, M')$ for fixed $M = (m_1,\ldots , m_k)$ and variable $M'= (m'_1, \ldots , m'_{n-k-1})$. The object of $\Theta ^n$ corresponding to the class of $(M,0,M')$ with all $m_i >0$ will be denoted $M$. Two morphisms from $M$ to $M'$ in $\Delta ^n$ are identified if they both factor through something of the form $(u_1,\ldots , u_i, 0, u_{i+2}, \ldots , u_n)$ and if their first $i$ components are the same. \end{parag} \begin{parag} \label{precat} An {\em $n$-precat} is defined to be a presheaf on the category $\Theta ^n$. This corresponds to an $n$-simplicial set $(\Delta ^n)^o\rightarrow Sets$ which satisfies the constancy condition (cf \ref{theta}). The category $nPC$ of $n$-precats (with morphisms being the morphisms of presheaves) is to be given a closed model structure. \end{parag} \begin{parag} \label{prelims} Note for a start that $nPC$ is closed under arbitrary products and coproducts, what is more (and eventually important for our purposes) it admits an internal $\underline{Hom}(A,B)$. These statements come simply from the fact that $nPC$ is a category of presheaves over something. We denote the coproduct or pushout of $A\rightarrow B$ and $A\rightarrow C$ by $B\cup ^AC$. We denote fiber products by the usual notation. \end{parag} \begin{parag} \label{cofibs} {\em Cofibrations:} A morphism $A\rightarrow B$ of $n$-precats is a {\em cofibration} if the morphisms $A_M \rightarrow B_M$ are injective whenever $M\in \Theta ^n$ is an object of non-maximal length, i.e. $M= (m_1,\ldots , m_k, 0,\ldots , 0)$ for $k< n$. The case of sets ($n=0$) shows that we can't require injectivity at the top level $n$, nor do we need to. We often use the notation $A\hookrightarrow B$ for a cofibration, not meaning to imply injectivity at the top level. \end{parag} \begin{parag} \label{we} {\em Weak equivalences:} In order to say when a morphism $A\rightarrow B$ of $n$-precats is a ``weak equivalence'' we have to do some work. In \cite{Tamsamani} was defined the notion of equivalence between $n$-categories (cf \ref{defequiv1} above), but an $n$-precat is not yet an $n$-category. We need an operation which specifies the intended relationship between our $n$-precats and $n$-categories. This is the operation $A\mapsto Cat(A)$ which to any $n$-precat associates an $n$-category together with morphism of precats $A\rightarrow Cat(A)$, basically by throwing onto $A$ in a minimal way all of the elements which are needed in order to satisfy the definition of being an $n$-category. See \cite{nCAT} \S 2 for the details of this. Now we say that a morphism $$ A\rightarrow B $$ of $n$-precats is a {\em weak equivalence} if the induced morphism of $n$-categories $$ Cat(A)\rightarrow Cat(B) $$ is an equivalence as defined in \cite{Tamsamani}---described in \ref{defequiv1} and \ref{essentialsurjectivity} above. \end{parag} \begin{parag} \label{trivcofibs} {\em Trivial cofibrations:} A morphism $A\rightarrow B$ is said to be a {\em trivial cofibration} if it is a cofibration and a weak equivalence. \end{parag} \begin{parag} \label{fibs} {\em Fibrations:} A morphism $A\rightarrow B$ of $n$-precats is said to be a {\em fibration} if it satisfies the following lifting property: for every trivial cofibration $E'\hookrightarrow E$ and every morphism $E\rightarrow B$ provided with a lifting over $E'$ to a morphism $E'\rightarrow A$, there exists an extension of this to a lifting $E\rightarrow A$. An $n$-precat $A$ is said to be {\em fibrant} if the canonical (unique) morphism $A\rightarrow \ast$ to the constant presheaf with values one point, is a fibration. A fibrant $n$-precat is, in particular, an $n$-category. This is because the elements which need to exist to give an $n$-category may be obtained as liftings of certain standard trivial cofibrations (those denoted $\Sigma \rightarrow h$ in \cite{nCAT}). \end{parag} \begin{theorem} \label{cmc} {\rm (\cite{nCAT} Theorem 3.1)} The category $nPC$ of $n$-precats with the above classes of cofibrations, weak equivalences and fibrations, is a closed model category. \end{theorem} The basic ``yoga'' of the situation is that when we want to look at coproducts, one of the morphisms should be a cofibration; when we want to look at fiber products, one of the morphisms should be a fibration; and when we want look at the space of morphisms from $A$ to $B$, the first object $A$ should be cofibrant (in our case all objects are cofibrant) and the second object $B$ should be fibrant. \begin{parag} \label{explaincmc} We explain more precisely what information is contained in the above theorem, by explaining the axioms for a closed model category structure (CM1--CM5 of \cite{QuillenAnnals}). These are proved as such in \cite{nCAT}. \noindent {\em CM1}---This says that $nPC$ is closed under finite (and in our case, arbitrary) direct and inverse limits (\ref{prelims}). \noindent {\em CM2}---Given composable morphisms $$ X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z, $$ if any two of $f$ or $g$ or $g\circ f$ are weak equivalences then the third is also a weak equivalence. \noindent {\em CM3}---The classes of cofibrations, fibrations and weak equivalences are closed under retracts. We don't explicitly use this condition (however it is the basis for the property \ref{pushoutA} below). \noindent {\em CM4}---This says that a pair of a cofibration $E'\rightarrow E$ and a fibration $A\rightarrow B$ have the lifting property (as in the definition of fibration \ref{fibs}) if either one of the morphisms is a weak equivalence. Note that the lifting property when $E'\rightarrow E$ is a weak equivalence (i.e. trivial cofibration) is just the definition that $A\rightarrow B$ be fibrant \ref{fibs}. The other half, the lifting property for an arbitrary cofibration when $A\rightarrow B$ is a weak equivalence, comes from what Jardine calls ``Joyal's trick'' \cite{Jardine}. \noindent {\em CM5}---This says that any morphism $f$ may be factored as a composition $f= p\circ i$ of a cofibration followed by fibration, and either one of $p$ or $i$ may be assumed to be a weak equivalence. \end{parag} \begin{parag} \label{pushoutA} Another axiom in Quillen's original point of view (Axiom M3 on page 1.1 of \cite{Quillen}) is that if $A\rightarrow B$ is a trivial cofibration and $A\rightarrow C$ is any morphism then $C\rightarrow B\cup ^AC$ is again a trivial cofibration; and similarly the dual condition for fibrant weak equivalences and fiber products. In the closed model category setting this becomes a consequence of the axioms CM1--CM5, see \cite{QuillenAnnals}. In the proof of \cite{nCAT} (modelled on that of \cite{Jardine}) the main step which is done first (\cite{nCAT} Lemma 3.2) is to prove this property of preservation of trivial cofibrations by coproducts. . (On the other hand, note that with our definition \ref{fibs} of fibrations, the preservation by fiber products is obvious). \end{parag} \begin{parag} \label{perspective1} We now try to put these properties in perspective in view of how we will use them. If $A$ is any $n$-precat then applying {\em CM5} to the morphism $A\rightarrow \ast$ we obtain a factorization $$ A\rightarrow A' \rightarrow \ast $$ with the first morphism a trivial cofibration, and the second morphism a fibration. Thus $A'$ is a fibrant object. We call such a trivial cofibration to a fibrant object $A\hookrightarrow A'$ a {\em fibrant replacement for $A$}. In the constructions of \cite{Jardine}, \cite{nCAT} one obtains the fibrant replacement by adding onto $A$ the pushouts by ``all possible'' trivial cofibrations, making use of \ref{pushoutA}. The notion of ``all possible'' has to be refined in order to avoid set-theoretical problems: actually one looks at $\omega$-bounded cofibrations. The number of them is bounded by the maximum of $2^{\omega}$ or the cardinality of $A$. When looking at morphisms into an $n$-category $C$ it is important that $C$ be fibrant, for then we obtain extension properties along trivial cofibrations. In particular, we will only define what it means for limits to exist in $n$-categories $C$ which are fibrant. When we finally get to our definition of the $n+1$-category $nCAT$ below, it will not be fibrant. Thus one of the main steps is to choose a fibrant replacement $nCAT \hookrightarrow nCAT'$. \end{parag} \begin{parag} \label{interval} There is a nice ``interval'' in our closed model category (in contrast with the general situation envisioned by Quillen in \cite{Quillen}). Let $\overline{I}$ denote the $1$-category with two objects $0,1$ and with unique morphisms going in either direction between them, whose compositions are the identity. Without changing notation, we can consider $\overline{I}$ as an $n$-category (pull back by the obvious morphism $\Theta ^n\rightarrow \Theta ^1 = \Delta $). \noindent {\em Claim:} Suppose $C$ is a fibrant $n$-category. Then two objects $x,y\in C_0$ are equivalent (i.e. project to the same thing in $\tau _{\leq 0} C$ cf \ref{anotherapproach}) if and only if there exists a morphism $\overline{I}\rightarrow C$ sending $0$ to $x$ and $1$ to $y$. To prove this note that one direction is obvious: if there exists such a morphism then by functoriality of $\tau _{\leq 0}$ $x$ and $y$ are equivalent (because $\tau _{\leq 0}(\overline{I})=\ast$). For the other direction, suppose $x$ and $y$ are equivalent. Use Proposition 6.5 of \cite{nCAT} which says that there is an $n$-category $K$ with objects $0,1$ such that $K \rightarrow \ast$ is a weak equivalence, and there is a morphism $K\rightarrow C$ sending $0$ to $x$ and $1$ to $y$. Applying the factorization statement {\em CM5} to the morphism $$ K\cup ^{\{ 0 , 1\} } \overline{I} \rightarrow \ast $$ we obtain a cofibration $$ K\cup ^{\{ 0 , 1\} } \overline{I}\hookrightarrow A $$ such that $A\rightarrow \ast$ is a weak equivalence. It follows from {\em CM2} applied to $K\rightarrow A\rightarrow \ast$ that $K\rightarrow A$ is a weak equivalence, thus it is a trivial cofibration. Now the fibrant property of $C$ implies that our morphism $K\rightarrow C$ extends to a morphism $A\rightarrow C$. This morphism restricted to $\overline{I}\hookrightarrow A$ provides a morphism $\overline{I}\rightarrow C$ sending $0$ to $x$ and $1$ to $y$. This proves the other direction of the claimed statement. \end{parag} \begin{parag} \label{surje} As a corollary of the above construction, suppose $f:A\rightarrow B$ is a fibrant morphism of fibrant $n$-categories. Suppose that $a\in A_0$ and $b\in B_0$ are objects such that $f(a)$ is equivalent to $b$ (i.e. $f(a)$ is equal to $b$ in $\tau _{\leq 0} B$). Then there is a different object $a'\in A_0$ equivalent to $a$ such that $f(a')=b$ in $B_0$. To prove this, note that the equivalence between $f(a)$ and $b$ corresponds by \ref{interval} to a morphism $\overline{I}\rightarrow B$ sending $0$ to $f(a)$ and $1$ to $b$. We have a lifting $a$ over $\{ 0\}$. The inclusion $$ \{ 0\} \subset \overline{I} $$ is a trivial cofibration, so the fibrant property of $f$ means that there is a lifting to a morphism $\overline{I} \rightarrow A$. The image of $1$ by this map is an object $a'$ equivalent to $a$ and projecting to $b$. A variant says that if $f:A\rightarrow B$ is a fibrant morphism between fibrant $n$-categories and if $f$ is an equivalence then $f$ is surjective on objects. To obtain this note that essential surjectivity of $f$ means that every object $b$ is equivalent to some $f(a)$, then apply the previous statement. \end{parag} \begin{parag} \label{internal1} One of the main advantages to using a category of presheaves $nPC$ as underlying category is that we obtain an internal $\underline{Hom}(A,B)$ between two $n$-precats. This represents a functor: a map $$ E\rightarrow \underline{Hom}(A,B) $$ is the same thing as a morphism $A\times E \rightarrow B$. Of course for arbitrary $A$ and $B$, the internal $\underline{Hom}(A,B)$ will not have any reasonable properties, for example it will not transform equivalences of the $A$ or $B$ into equivalences. This situation is rectified by imposing the hypothesis that $B$ should be fibrant. \end{parag} \begin{parag} \label{internal2} We describe some of the results saying that the internal $\underline{Hom}(A,B)$ works nicely when $B$ is fibrant. The following paragraphs are Theorem 7.1 and Lemma 7.2 of \cite{nCAT}. Suppose $A$ is an $n$-precat and $B$ is a fibrant $n$-precat. Then the internal $ \underline{Hom}(A,B)$ of presheaves over $\Theta ^n$ is a fibrant $n$-category. Furthermore if $B'\rightarrow B$ is a fibrant morphism then $\underline{Hom} (A, B')\rightarrow \underline{Hom} (A, B)$ is fibrant. Similarly if $A\hookrightarrow A'$ is a cofibration and if $B$ is fibrant then $\underline{Hom}(A', B)\rightarrow \underline{Hom}(A,B)$ is fibrant. Suppose $A\rightarrow A'$ is a weak equivalence, and $B$ fibrant. Then $$ \underline{Hom} (A', B)\rightarrow \underline{Hom} (A, B) $$ is an equivalence of $n$-categories. If $B\rightarrow B'$ is an equivalence of fibrant $n$-precats then $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ is an equivalence. Suppose $A\rightarrow B$ and $A\rightarrow C$ are cofibrations. Then $$ \underline{Hom} (B\cup ^AC, D) = \underline{Hom} (B, D) \times _{\underline{Hom}(A,D)}\underline{Hom}(C,D). $$ \end{parag} \begin{parag} \label{homotopic1} We can relate several different versions of the notion of two morphisms being homotopic. Suppose $A$ and $B$ are $n$-precats with $B$ fibrant. According to Quillen's definition \cite{Quillen}, two maps $f: A\rightarrow B$ are {\em homotopic} if there is a diagram $$ A\stackrel{\displaystyle \rightarrow}{\rightarrow} A' \rightarrow A $$ such that all morphisms are weak equivalences, the first two morphisms are cofibrations, and such that the compositions are the identity of $A$, plus a morphism $A'\rightarrow B$ inducing $f$ and $g$ on the two copies of $A$. In our situation, if $B$ is fibrant then $\underline{Hom}(A,B)$ is a fibrant $n$-category whose objects are the morphisms $A\rightarrow B$. Two morphisms are equivalent objects in this $n$-category (cf \ref{essentialsurjectivity} above) if and only if they are homotopic in Quillen's sense (this is \cite{nCAT} Lemma 7.3). \end{parag} \begin{parag} \label{homotopic2} In the above situation apply the claim of \ref{interval}. Two objects $f,g$ of the fibrant $n$-category $\underline{Hom}(A,B)$ are equivalent if and only if there is a morphism $$ \overline{I}\rightarrow \underline{Hom}(A,B) $$ sending $0$ to $f$ and $1$ to $g$. Such a morphism corresponds to a map $$ A\times \overline{I} \rightarrow B; $$ so we can finish up by saying that two morphisms $f,g:A\rightarrow B$ are homotopic if and only if there exists a map $$ A\times \overline{I}\rightarrow B $$ restricting to $f$ on $A\times \{ 0\} $ and to $g$ on $A\times \{ 1\}$. \end{parag} \begin{parag} \label{charequiv} We obtain from CM4 the following characterization of fibrant weak equivalences. A morphism $f: A\rightarrow B$ is a fibrant weak equivalence if and only if it satisfies the lifting property for any cofibration $E'\hookrightarrow E$. To prove this, note that CM4 shows that a fibrant weak equivalence has this property. If $f$ has this property then it is fibrant (the case of $E'\hookrightarrow E$ a trivial cofibration). The morphisms of $n-1$-categories $$ A_{1/} (x,y)\rightarrow B_{1/}(f(x) , f(y)) $$ also have the same property (one can see this using the construction $\Upsilon (E)$ below) and $f$ is surjective on objects (by the case $\emptyset \hookrightarrow \ast$). Therefore $f$ is an equivalence. We can give the following variant characterizing when a morphism is an equivalence (not necessarily fibrant). We say that a morphism $f: A\rightarrow B$ between fibrant $n$-categories has the {\em homotopical lifting property for $E'\hookrightarrow E$} if, given a morphism $v:E\rightarrow B$ and a lifting $u':E'\rightarrow A$, there is a homotopy from $v$ to a new morphism $v_1$, a lifting $u_1$ of $v_1$, and a homotopy from $u'$ to $u'_1$ (the restriction of $u_1$ to $E'$) lifting the homotopy from $v'$ to $v'_1$ (restriction of our first homotopy to $E'$). In this definition we can use any of the equivalent notions of homotopy \ref{homotopic1}, \ref{homotopic2} above. {\em Claim:} A morphism $f: A\rightarrow B$ between two fibrant $n$-categories is an equivalence if and only if it satisfies the homotopical lifting property for all $E'\hookrightarrow E$. To prove this, use CM5 to factor $A\rightarrow A'\rightarrow B$ with the first morphism a trivial cofibration and the second morphism fibrant. Note that $A'$ is again fibrant. The statement being a homotopical one, the same hypothesis holds for $A'\rightarrow B$. If we can prove that $A'\rightarrow B$ is an equivalence then the composition with the trivial cofibration $A\rightarrow A'$ will be a weak equivalence. Thus we may reduce to the case where $A\rightarrow B$ is a fibrant morphism. Now given $E'\hookrightarrow E$ with $E\rightarrow B$ lifting to $E'\rightarrow A$, choose homotopies $$ E\times \overline{I}\rightarrow B $$ and lifting $$ E'\times \overline{I}\rightarrow A $$ compatible with a lifting $E\times \{ 1\} \rightarrow A$ as in the definition of the homotopical lifting property. These give a lifting $$ E'\times \overline{I}\cup ^{E'\times \{ 1\} } E \times \{ 1\} \rightarrow A, $$ and the morphism $$ E'\times \overline{I}\cup ^{E'\times \{ 1\} } E \times \{ 1\} \rightarrow E\times \overline{I} $$ is a trivial cofibration, so by the fibrant property of $A\rightarrow B$ (which we are now assuming) there is a lifting $$ E\times \overline{I} \rightarrow A. $$ The restriction to $E\times \{ 0\}$ gives the desired lifting of the original morphism $E\rightarrow B$, coinciding with the given lifting on $E'$. This proves that $A\rightarrow B$ satisfies the lifting criterion given above so it is a fibrant weak equivalence. This completes the proof of one direction of the claim. A similar argument (using CM2) gives the other direction. \end{parag} \bigskip \subnumero{Families of $n$-categories} \begin{parag} \label{ncat1} Using the internal $\underline{Hom}(A,B)$ of \ref{internal} between fibrant $n$-categories, we define the $n+1$-category $nCAT$ of all fibrant $n$-categories (cf \cite{nCAT} \S 7). This is the ``right'' category of $n$-categories, and is not to be confused with the first approximation $n-Cat$ as defined in \ref{multisimplicial} above. The objects of $nCAT$ are the fibrant $n$-categories. Between any two objects we have an $n$-category of morphisms $\underline{Hom}(A,B)$. Composition of morphisms gives a morphism of $n$-categories $$ \underline{Hom}(A,B)\times \underline{Hom}(B,C)\rightarrow \underline{Hom}(A,C), $$ which is strictly associative and has a unit element, the identity morphism. Using this we obtain an $n+1$-category $nCAT$: to be precise, if $A_0,\ldots , A_p$ are objects then $$ nCAT_{p/}(A_0,\ldots , A_p):= \underline{Hom}(A_0,A_1)\times \ldots \times \underline{Hom}(A_{p-1},A_p) $$ which organizes into a simplicial collection using the projections or, where necessary, the composition morphisms. The Segal maps are actually isomorphisms here so this is an $n+1$-category. \end{parag} \begin{parag} \label{ncat2} Unfortunately, $nCAT$ is not a fibrant $n+1$-category, although it does have the property that the $nCAT_{p/}$ are fibrant. Because of this, we must choose a fibrant replacement $$ nCAT \hookrightarrow nCAT'. $$ \end{parag} \begin{parag} \label{families1} Basic to the present paper is the notion of {\em family of $n$-categories indexed by an $n+1$-category $A$}, which is defined using our fibrant replacement (\ref{ncat2}) to be a morphism $A\rightarrow nCAT'$. The {\em $n+1$-category of all families indexed by $A$} is the $n+1$-category $\underline{Hom}(A, nCAT')$. \end{parag} \begin{parag} \label{families2} Suppose $\psi, \psi ' : A\rightarrow nCAT'$ are families. A {\em morphism} from $\psi$ to $\psi '$ is an object of the $n$-category $\underline{Hom}(A, nCAT')_{1/}(\psi , \psi ')$. Let $I$ be the category with two objects $0,1$ and a morphism from $0$ to $1$ (in our notations below this will also be the same as what we will call $\Upsilon (\ast )$). The set of objects of $\underline{Hom}(A, nCAT')_{1/}(\psi , \psi ')$ is equal to the set of morphisms $$ I\rightarrow \underline{Hom}(A, nCAT') $$ sending $0$ to $\psi$ and $1$ to $\psi '$. In view of the definition of internal $\underline{Hom}$ this is the same thing as a morphism $$ A \times I \rightarrow nCAT' $$ restricting on $A\times \{ 0\}$ to $\psi $ and on $A\times \{ 1\}$ to $\psi '$. \end{parag} \bigskip \subnumero{The construction $\Upsilon$} We will now introduce some of our main tools for the present paper. The basic idea is that we often would like to talk about the basic $n$-category with two objects (denoted $0$ and $1$) and with a given $n-1$-category $E$ of morphisms from $0$ to $1$ (but no morphisms in the other direction and only identity endomorphisms of $0$ and $1$). We call this $\Upsilon (E)$. To be more precise we do this on the level of precats: if $E$ is an $n-1$-precat then we obtain an $n$-precat $\Upsilon (E)$. The main property of this construction is that if $A$ is any $n$-category then a morphism of $n$-precats $$ f:\Upsilon (E) \rightarrow A $$ corresponds exactly to a choice of two objects $x=f(0)$ and $y=f(1)$ together with a morphism of $n-1$-precats $E\rightarrow A_{1/}(x,y)$. One can see $\Upsilon (E)$ as the universal $n$-precat $A$ with two objects $x,y$ and a map $E\rightarrow A_{1/}(x,y)$. \begin{parag} \label{upsilon} We also need more general things of the form $\Upsilon ^2(E,F)$ having objects $0,1,2$ and similarly a $\Upsilon ^3$. (These will not have quite so simple an interpretation as universal objects.) Thus we present the definition in a general way. Suppose $E_1,\ldots , E_k$ are $n-1$-precats. Then we define the $n$-precat $$ \Upsilon ^k(E_1,\ldots , E_k) $$ in the following way. Its object set is the set with $k+1$ elements denoted $$ \Upsilon ^k(E_1,\ldots , E_k)_0 = \{ 0,\ldots , k\} . $$ Then $$ \Upsilon ^k(E_1,\ldots , E_k)_{p/}(y_0, \ldots , y_p) $$ is defined to be empty if any $y_i > y_j$ for $i<j$, equal to $\ast$ if $y_0=\ldots = y_p$, and otherwise $$ \Upsilon ^k(E_1,\ldots , E_k)_{p/}(y_0, \ldots , y_p):= E_{y_0} \times \ldots \times E_{y_p}. $$ \end{parag} \begin{parag} For example when $k=1$ (and we drop the superscript $k$ in this case) $\Upsilon E$ is the $n$-precat with two objects $0,1$ and with $n-1$-precat of morphisms from $0$ to $1$ equal to $E$. Similarly $\Upsilon ^2(E,F)$ has objects $0,1,2$ and morphisms $E$ from $0$ to $1$, $F$ from $1$ to $2$ and $E\times F$ from $0$ to $2$. We picture $\Upsilon ^k(E_1,\ldots , E_k)$ as a $k$-gon (an edge for $k=1$, a triangle for $k=2$, a tetrahedron for $k=3$). The edges are labeled with single $E_i$, or products $E_i \times \ldots , E_j$. \end{parag} \begin{parag} There are inclusions of these $\Upsilon^k$ according to the faces of the $k$-gon. The principal faces give inclusions $$ \Upsilon ^{k-1}(E_1,\ldots , E_{k-1})\hookrightarrow \Upsilon ^k(E_1, \ldots , E_k), $$ $$ \Upsilon ^{k-1}(E_2,\ldots , E_{k})\hookrightarrow \Upsilon ^k(E_1, \ldots , E_k), $$ and $$ \Upsilon ^{k-1}(E_1,\ldots , E_i\times E_{i+1}, \ldots , E_{k})\hookrightarrow \Upsilon ^k(E_1, \ldots , E_k). $$ The inclusions of lower levels are deduced from these by induction. Note that these faces $\Upsilon ^{k-1}$ intersect along appropriate $\Upsilon ^{k-2}$. \end{parag} \begin{remark} \label{upsistar} $\Upsilon (\ast )= I$ is the category with objects $0,1$ and with a unique morphism from $0$ to $1$. A map $\Upsilon (\ast )\rightarrow A$ is the same thing as a pair of objects $x,y$ and a $1$-morphism from $x$ to $y$, i.e. an object of $A_{1/}(x,y)$. \end{remark} \bigskip Another way of constructing the $\Upsilon ^k$ is given in the following remarks \ref{interpupsilon1}--\ref{interpupsilon3}. \begin{parag} \label{interpupsilon1} For an $n-1$-precat $E$, denote by $[p](E)$ the universal $n$-precat $A$ with objects $x_0,\ldots ,x_p$ and with a morphism $E\rightarrow A_{p/}(x_0,\ldots , x_p)$. This can be described explicitly by saying that $[p](E)$ has objects $0,1,\ldots , p$, and for a sequence of objects $i_1,\ldots , i_k$ the $n-1$-precat $[p](E)_{k/}(i_1,\ldots , i_k)$ is empty if some $i_j < i_{j+1}$, is equal to $\ast$ if all $i_j$ are equal, and is equal to $E$ if some $i_j < i_{j+1}$. \end{parag} \begin{parag} \label{interpupsilon2} One has $[1]E= \Upsilon (E)$. The construction of the higher $\Upsilon ^k$ may be described inductively as follows: we will construct $\Upsilon ^k(E_1,\ldots , E_k)$ together with a morphism $$ in:[k](E_1\times \ldots \times E_k)\rightarrow \Upsilon ^k(E_1,\ldots , E_k). $$ Suppose we have constructed these maps up to $k-1$. Note that the first and last face morphisms coupled with the projections onto the first and last $k-1$ factors give a map $$ [k-1](E_1\times \ldots \times E_{k}) \cup ^{[k-2](E_1\times \ldots \times E_{k})} [k-1](E_1\times \ldots \times E_{k}) $$ $$ \stackrel{\alpha}{\rightarrow} [k](E_1,\ldots , E_k), $$ but on the other hand the projections onto subsets of factors of the product $E_1\times \ldots \times E_k$ together with the maps $in$ in our inductive construction for $k-1$ and $k-2$ give a map $$ [k-1](E_1\times \ldots \times E_{k}) \cup ^{[k-2](E_1\times \ldots \times E_{k})} [k-1](E_1\times \ldots \times E_{k}) $$ $$ \stackrel{\beta}{\rightarrow} \Upsilon ^{k-1}(E_1,\ldots , E_{k-1})\cup ^{ \Upsilon ^{k-2}(E_2,\ldots , E_{k-1})} \Upsilon ^{k-1}(E_2,\ldots , E_{k}). $$ Finally, $\Upsilon ^{k}(E_1,\ldots , E_k)$ is the coproduct of the maps $\alpha$ and $\beta$. We can think of this as saying that $\Upsilon ^{k}(E_1,\ldots , E_k)$ is obtained by adding on the cell $[k](E_1\times \ldots \times E_k)$ to the coproduct $$ \Upsilon ^{k-1}(E_1,\ldots , E_{k-1})\cup ^{ \Upsilon ^{k-2}(E_2,\ldots , E_{k-1})} \Upsilon ^{k-1}(E_2,\ldots , E_{k}) $$ of the earlier things we have inductively constructed. \end{parag} \begin{parag} \label{interpupsilon3} The case $k=2$ is simpler to write down and is worth mentioning separately. Recall that for $k=1$ we just had $\Upsilon (E)= [1](E)$. The next step is $$ \Upsilon ^2(E,F) = [2](E\times F) \cup ^{[1](E\times F)\cup ^{\ast } [1](E\times F)} ([1](E) \cup ^{\ast} [1](F)). $$ \end{parag} \begin{parag} \label{trivinclusions} One thing which we often will need to know below is when an inclusion from a union of faces, into the whole $\Upsilon ^k$, is a trivial cofibration. For $k=2$ the only inclusion which is a trivial cofibration is $$ \Upsilon (E_1)\cup ^{\{ 1\} } \Upsilon (E_2) \hookrightarrow \Upsilon ^2(E_1,E_2). $$ For $k=3$ we denote our inclusions in shorthand notation where $0,1,2,3$ refer to the vertices. To fix notations, the above inclusion for $k=2$ would be noted $$ (01) + (12) \subset (012). $$ Now for $k=3$ the inclusions which are trivial cofibrations are: $$ (01) + (12) + (23) \subset (0123) $$ (which is the standard one, coming basically from the definition of $n$-category); and then some others which we obtain from this standard one by adding in triangles on the right, keeping equivalence with $(01) + (12) + (23)$ according to the result for $k=2$: $$ (01) + (123)\subset (0123), $$ $$ (012) + (23) \subset (0123), $$ $$ (012) + (123)\subset (0123), $$ $$ (012) + (023)\subset (0123), $$ $$ (013) + (123)\subset (0123), $$ $$ (012) + (013) + (123)\subset (0123), $$ $$ (012) + (023) + (123)\subset (0123). $$ Our main examples of inclusions which are {\em not} trivial cofibrations are when we leave out the first or the last faces: $$ (012) + (023) + (013)\subset (0123) \;\;\;\; \mbox{not a t.c.}; $$ $$ (013) + (023) + (123)\subset (0123) \;\;\;\; \mbox{not a t.c.}. $$ We call these the {\em left and right shells}. We shall meet both of them and denote the left shell as $$ (012) + (023) + (013) = Shell\Upsilon ^3(E_1,E_2,E_3), $$ and the right shell as $$ (013) + (023) + (123) = Shelr\Upsilon ^3(E_1,E_2,E_3). $$ The main parts of our arguments for limits will consist of saying that under certain circumstances we have an extension property for morphisms with respect to these cofibrations which are not trivial. \end{parag} \begin{parag} \label{trivcofibs2} One of the main technical problems which will be encountered by the reader is deciding when a morphism between $n$ or $n+1$-precats is a trivial cofibration: we use this all the time in order to use the fibrant property of the domain of morphisms we are trying to extend. It is not possible to give all the details each time that this question occurs, as that would be much too long. The general principles at work are: to be aware of the examples given in \ref{trivinclusions}; to use the fact that the coproduct of a trivial cofibration with something else again yields a trivial cofibration (\ref{pushoutA}); and to use the fact that if a composable sequence of morphisms $$ \cdot \stackrel{f}{\rightarrow} \cdot \stackrel{g}{\rightarrow} \cdot $$ has composition being a weak equivalence, and one of $f$ or $g$ being a weak equivalence, then so is the other (\ref{explaincmc} CM2). And of course to use any available hypotheses that are in effect saying that certain morphisms are trivial cofibrations or equivalences. All of the cases where we need to know that something is a trivial cofibration, can be obtained using these principles. \end{parag} \begin{parag} \label{notationsr} We will often be considering morphisms of the form $$ f: \Upsilon ^k(E_1,\ldots , E_k) \rightarrow C. $$ When we would like to restrict this to a face (or higher order face such as an edge) then, denoting the face by $i_1,\ldots , i_j$ we denote the restriction of $f$ to the face by $$ r_{i_1\ldots i_j}(f). $$ For example when $k=2$ the restriction of $$ f: \Upsilon ^2(E,F)\rightarrow C $$ to the edge $(02)$ (which is a $\Upsilon (E\times F)$) would be denoted $r_{02}(f)$. The object $f(1)$ could also be denoted $r_1(f)$. We make the same convention for restricting maps of the form $$ A\times \Upsilon ^k(E_1,\ldots , E_k) \rightarrow C, $$ to maps on $A$ times some face of the $\Upsilon ^k(E_1,\ldots , E_k)$. \end{parag} \bigskip \subnumero{Inverting equivalences} In preparation for \ref{obbyobequiv} we need the following result. It says that a morphism which is an equivalence has an inverse which is essentially unique, if the notion of ``inverse'' is defined in the right way. It is an $n$-category version of the theorem of \cite{flexible} which gives a canonical inverse for a homotopy equivalence of spaces. \begin{theorem} \label{resttoIff} For any fibrant $n$-category $C$ the morphism restriction from $\overline{I}$ to $I$: $$ r:\underline{Hom}(\overline{I}, C) \rightarrow \underline{Hom}(I,C) $$ is fully faithful, so $\underline{Hom}(\overline{I}, C)$ is equivalent to the full sub-$n$-category of invertible elements of $\underline{Hom}(I,C)$. \end{theorem} {\em Proof:} We first construct some trivial cofibrations. \begin{parag} \label{ff1} Recall (\ref{upsistar}) that $I= \Upsilon (\ast )$. The morphism $$ \Upsilon (E)\cup ^{\{ 1\} } I\rightarrow \Upsilon ^2(E,\ast ) $$ is a trivial cofibration (\ref{trivinclusions}), so by \ref{pushoutA} the coproduct with $$ \Upsilon (E)\cup ^{\{ 1\} } I\rightarrow \Upsilon (E)\cup ^{\{ 1\} } \overline{I} $$ gives a trivial cofibration $$ \Upsilon (E)\cup ^{\{ 1\} } \overline{I} \rightarrow \Upsilon ^2(E,\ast )\cup ^{I} \overline{I} . $$ The morphism $$ \Upsilon (E) \rightarrow \Upsilon (E)\cup ^{\{ 1\} } \overline{I} $$ is a weak equivalence (again by \ref{pushoutA} because it is pushout of the trivial cofibration $\ast \rightarrow \overline{I}$). Therefore the composed morphism $$ i_{01}:\Upsilon (E)\rightarrow \Upsilon ^2(E,\ast )\cup ^{I} \overline{I} $$ corresponding to the edge $(01)$ is an equivalence. Thus the projection $$ \Upsilon ^2(E,\ast )\cup ^{I} \overline{I} \rightarrow \Upsilon (E) $$ is an equivalence (by \ref{explaincmc} CM2). This in turn implies that the morphism corresponding to the edge $(02)$ $$ i_{02}:\Upsilon (E)\rightarrow \Upsilon ^2(E,\ast )\cup ^{I} \overline{I} $$ is a trivial cofibration. \end{parag} \begin{parag} \label{ff2} A similar argument shows that $$ i_{02}:\Upsilon (E)\rightarrow \Upsilon ^2(\ast , E)\cup ^{I} \overline{I} $$ is a trivial cofibration. \end{parag} \begin{parag} \label{ff3} Next, note that $$ \Upsilon (E) \times I = \Upsilon ^2(E, \ast ) \cup ^{\Upsilon (E)} \Upsilon ^2(\ast ,E) $$ (the square decomposes as a union of two triangles). The morphisms in the coproduct are both $i_{02}$. Thus if we attach $\overline{I}$ to each of the intervals $I$ on the two opposite sides of this square, the result $$ \Upsilon (E) \times I\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}) $$ can be seen as a coproduct of the two objects considered in \ref{ff1} and \ref{ff2} (we don't write this coproduct out). Combining with the results of those paragraphs, the morphism from the diagonal $$ \Upsilon (E)\rightarrow \Upsilon (E) \times I\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}) $$ is an equivalence, which in turn implies that the projection $$ \Upsilon (E) \times I\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}) \rightarrow \Upsilon (E) $$ is an equivalence or, equally well, that the inclusion $$ (\Upsilon (E) \times I)\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}) \hookrightarrow \Upsilon (E) \times \overline{I} $$ is a trivial cofibration. \end{parag} \begin{parag} \label{ff4} Suppose now that $E'\subset E$. Let ${\bf G}$ denote the pushout of $$ (\Upsilon (E) \times I)\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}) $$ and $\Upsilon (E')\times \overline{I}$ along $$ (\Upsilon (E') \times I)\cup ^{\{ 0,1\} \times I}(\{ 0,1\} \times \overline{I}). $$ Paragraph \ref{ff3} and the usual (\ref{pushoutA}) and '\ref{explaincmc} CM2) imply that the morphism $$ {\bf G} \hookrightarrow \Upsilon (E)\times \overline{I} $$ is a trivial cofibration. Note, however, the simpler expression $$ {\bf G} = (\Upsilon (E) \times I)\cup ^{\Upsilon (E') \times I} (\Upsilon (E')\times \overline{I}). $$ \end{parag} \begin{parag} \label{ff5} We are now ready to prove the theorem. Fix $u,v$ objects of $\underline{Hom}(\overline{I},C)$. Suppose $E'\hookrightarrow E$ is any cofibration, and suppose given a morphism $$ E \rightarrow \underline{Hom}(I,C)_{1/}(r(u),r(v)) $$ provided with lifting $$ E'\rightarrow \underline{Hom}(\overline{I},C)_{1/}(u,v). $$ These correspond exactly to a morphism $$ {\bf G} \rightarrow C, $$ which since $C$ is fibrant extends along the trivial cofibration of (\ref{ff4}) to a morphism $$ \Upsilon (E)\times \overline{I}. $$ This is exactly the lifting to a map $$ E'\rightarrow \underline{Hom}(\overline{I},C)_{1/}(u,v) $$ needed to establish the statement that the morphism induced by $r$ $$ \underline{Hom}(\overline{I},C)_{1/}(u,v) \rightarrow \underline{Hom}(I,C)_{1/}(r(u),r(v)) $$ is an equivalence. This proves the theorem. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \end{parag} \begin{corollary} Suppose $f: U\rightarrow V$ is a morphism in a fibrant $n$-category $C$. Then the $n$-category of morphisms $\overline{I}\rightarrow C$ restricting on $I\subset \overline{I}$ to $f$ is contractible. \end{corollary} {\em Proof:} The $n$-category in question is just the fiber of the morphism in the theorem, over the object $f\in \underline{Hom}(I,C)$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} The next corollary says that equivalences may be inverted with dependence on parameters. \begin{corollary} \label{obbyobequiv} Suppose $C$ is a fibrant $n$-category. Suppose $\psi, \psi ' : A\rightarrow C$ are two morphisms and suppose $f$ is a morphism from $\psi$ to $\psi '$. Suppose that for every object $a\in A$ the induced morphism $f_a: \psi (a)\rightarrow \psi '(a)$ is an equivalence in $C$. Then $f$ is an equivalence considered as a $1$-morphism in $\underline{Hom}(A,C)$. \end{corollary} {\em Proof:} The morphism $f$ is a map $$ f: A\times I \rightarrow C, $$ which we can think of as a map $$ f_1:A\rightarrow \underline{Hom}(I,C). $$ From Theorem \ref{resttoIff} the morphism $$ \underline{Hom}(\overline{I},C)\rightarrow \underline{Hom}(I,C) $$ is a fibrant equivalence onto the full subcategory of invertible objects. The hypothesis of the corollary says exactly that the morphism $f_1$ lands in this full subcategory. Therefore it lifts to a morphism $$ g: A\rightarrow \underline{Hom}(\overline{I}, C), $$ in other words to $$ A\times \overline{I} \rightarrow C $$ or equally well $$ \overline{I}\rightarrow \underline{Hom}(A,C). $$ This shows that $f$ was an equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \bigskip \numero{The definitions of direct and inverse limits} One of the most useful tools in homotopy theory is the notion of homotopy limit or ``holim''. This can mean either direct or inverse limit and one of the two is called a ``colimit'' but I don't know which one! So we'll call both ``limits'' and specify which one in context. Our purpose is to define the notions of inverse and direct limit in an $n$-category. We always suppose that the target category $C$ is fibrant. When this is not the case we first have to take a fibrant replacement (\ref{perspective1}). \bigskip \subnumero{Inverse limits} Suppose $C$ is a fibrant $n$-category, and suppose $A$ is an $n$-category. Suppose $\varphi : A \rightarrow C$ is a morphism. If $U \in C$ is an object then we define $$ Hom (U , \varphi ):= Hom (A, C)_{1/}(U_A, \varphi ) $$ where $U_A$ denotes the constant morphism with value $U$. If $V$ is another object of $C$ then we have a morphism $$ C_{1/}(V , U )\rightarrow Hom (A, C)_{1/}(V_A, U_A) $$ and we use this to define $$ Hom (V, U , \varphi ):= Hom (A, C)_{2/} (V_A,U_A, \varphi )\times _{ Hom (A, C)_{1/}(V_A, U_A)} C_{1/}(V , U ) $$ or more generally if $V ^0,\ldots , V ^p \in C_0$ we define $$ Hom (V ^0, \ldots , V ^p, \varphi ):= $$ $$ Hom (A, C)_{(p+1)/} (V^0_A,\ldots , V ^p_A, \varphi )\times _{ Hom (A, C)_{p/} (V^0_A,\ldots , V ^p_A)} C_{p/}(V ^0, \ldots , V ^p ). $$ However we won't need this beyond $p=2$. Notice now that since $C$ is fibrant, $Hom (A,C)$ is fibrant and in particular an $n$-category, thus we get that the morphism $$ Hom (V , U , \varphi )\rightarrow C_{1/}(V , U ) \times Hom (U , \varphi ) $$ is an equivalence. On the other hand we have a projection $$ Hom (V , U , \varphi )\rightarrow Hom (V , \varphi ). $$ It is in this sense that we have a ``weak morphism'' from $C_{1/}(V , U ) \times Hom (U , \varphi )$ to $Hom (V , \varphi )$. \begin{definition} \label{definverse} We say that an object $U\in C_0$ together with element $f\in Hom (U , \varphi )_0$ is an {\em inverse limit of $\varphi$} if for any $V\in C_0$ the resulting weak morphism from $C_{1/}(V , U )$ to $Hom (V , \varphi )$ is an equivalence. To say this more precisely this means that the morphism $$ Hom (V ,U , \varphi ) \times _{Hom (U , \varphi )} \{ f \} \rightarrow Hom (V , \varphi ) $$ should be an equivalence. If such an inverse limit exists we say that {\em $\varphi$ admits an inverse limit} (we will discuss uniqueness below). If any morphism $\varphi : A \rightarrow C$ from any $n$-category $A$ to $C$ admits an inverse limit then we say that {\em $C$ admits inverse limits}. \end{definition} \begin{parag} \label{uniquenessinverse} {\em Uniqueness:} Suppose $f\in Hom (U , \varphi ) $ and $g \in Hom (V , \varphi )$ are two different inverse limits of $\varphi$. Then the inverse image of $g$ for the morphism $$ Hom (V , U , \varphi ) \times _{Hom (U , \varphi )} \{ f \} \rightarrow Hom (V , \varphi ) $$ is contractible. This gives a contractible $n$-category mapping to $Hom (V , U )$. We also have a contractible $n$-category mapping to $Hom (U , V )$. A similar argument with $p=3$ gives a contractible $n$-category mapping to $Hom (V , U , V )$ which maps into the contractible things for $V ,U$, for $U , V$ and for $V , V$. The image at the end includes the identity. This shows that the composition of the morphisms in the two directions is the identity. The same works in the other direction. This shows that the essentially well defined morphisms $U \rightarrow V$ and $V\rightarrow U$ are equivalences. (The reader is challenged to find a nicer way of saying this!) \end{parag} \begin{parag} \label{upsiloninverse1} The condition of being an inverse limit may also be interpreted in terms of the construction $\Upsilon$ described in the previous section. To do this, start by noting that for an $n$-precat $E$ a morphism $$ E\rightarrow Hom (U,\varphi ) $$ is the same thing as a morphism $$ f:A\times \Upsilon (E)\rightarrow C $$ such that $r_0(u)= U_A$ and $r_1(u)=\varphi $. \end{parag} \begin{parag} \label{upsiloninverse2} In view of the discussion \ref{interpupsilon1}--\ref{interpupsilon3}, a morphism $$ E\rightarrow Hom (V,U,\varphi ) $$ is the same thing as a morphism $$ g:A\times [2](E)\rightarrow C $$ with $r_0(g)=V_A$, $r_1(g)=U_A$ and $r_2(g)=\varphi$ and such that $r_{01}(g)$ comes from a morphism $\Upsilon (E)\rightarrow C$. To see this, use the definition of $[2](E)$ by universal property \ref{interpupsilon1}. In a similar way using the description \ref{interpupsilon3}, a morphism $$ E\rightarrow Hom (V , U , \varphi ) \times _{Hom (U , \varphi )} \{ f \} $$ is the same thing as a morphism $$ g: A\times \Upsilon ^2(E, \ast )\rightarrow C $$ such that $r_2(g)= \varphi$ and $r_{01}(g)$ comes from a morphism $g_{01}:\Upsilon (E)\rightarrow C$ with $r_0(g_{01})=V$ and $r_1(g_{01})=U$. \end{parag} \begin{parag} \label{upsiloninverse3} Noting that the morphism $$ Hom (V , U , \varphi ) \times _{Hom (U , \varphi )} \{ f \} \rightarrow Hom (V , \varphi ) $$ is fibrant, it is an equivalence if and only if it satisfies the lifting property for all cofibrations $E'\subset E$ (\ref{charequiv}). \end{parag} \begin{parag} \label{upsiloninverse4} Using the above descriptions we can describe explicitly the lifting property of the previous paragraph and thus obtain the following characterization. A morphism $f\in Hom (U,\varphi )$ is an inverse limit if and only if for every morphism $$ v:A\times \Upsilon (E) \rightarrow C $$ with $r_0(v)= V_A$ for $V\in C_0$ and $r_1(v)= \varphi$, and for every extension over $A\times \Upsilon (E')$ to a morphism $$ w': A\times \Upsilon ^2(E', \ast )\rightarrow C $$ with $r_{12}(w')= f$ and $r_{01}(w')$ coming from a morphism $z':\Upsilon (E')\rightarrow C$ with $r_0(z')= V$ and $r_1(z')=U$, there exists a common extension of these two: a morphism $$ w: A\times \Upsilon ^2(E, \ast )\rightarrow C $$ with $r_{12}(w)= f$ and $r_{01}(w)$ coming from a morphism $z:\Upsilon (E')\rightarrow C$ with $r_0(z)= V$ and $r_1(z)=U$; such that the restriction of $w$ to $A\times \Upsilon ^2(E', \ast )$ is equal to $w'$; and such that $r_{02}(w)=v$. This is the characterization we shall use in our proofs. \end{parag} \bigskip \subnumero{Direct limits} We obtain the notion of direct limit by ``reversing the arrows'' in the above discussion. Suppose $C$ is a fibrant $n$-category, and suppose $A$ is an $n$-category. Suppose $\varphi : A \rightarrow C$ is a morphism. If $U \in C_0$ is an object then we define $$ Hom (\varphi , U ):= Hom (A, C)_{1/}(\varphi , U_A) $$ where again $U_A$ denotes the constant morphism with value $U$. If $V$ is another object of $C$ then we have a morphism $$ C_{1/}(U ,V )\rightarrow Hom (A, C)_{1/}( {U}_A, {V}_A) $$ and we use this to define $$ Hom (\varphi , U , V ):= Hom (A, C)_{2/} (\varphi , {U }_A,{V }_A, \varphi )\times _{ Hom (A, C)_{1/}({U }_A, {V }_A)} C_{1/}( U ,V ) $$ or more generally if $V ^0,\ldots , V ^p \in C_0$ we define $$ Hom (\varphi , V ^0, \ldots , V ^p):= $$ $$ Hom (A, C)_{(p+1)/} (\varphi , {V ^0}_A,\ldots , {V ^p}_A)\times _{ Hom (A, C)_{p/} ({V ^0}_A,\ldots , {V ^p}_A)} C_{p/}(V ^0, \ldots , V ^p ). $$ Again we won't need this beyond $p=2$. Notice now that since $C$ is fibrant, $Hom (A,C)$ is fibrant and in particular an $n$-category, thus we get that the morphism $$ Hom (\varphi , U , V )\rightarrow C_{1/}(U , V ) \times Hom (\varphi , U ) $$ is an equivalence. On the other hand we have a projection $$ Hom (\varphi ,U,V)\rightarrow Hom (V , \varphi ). $$ It is in this sense that we have a ``weak morphism'' from $C_{1/}(U,V ) \times Hom ( \varphi ,U)$ to $Hom (\varphi ,V)$. \begin{definition} \label{defdirect} We say that an element $f\in Hom (U , \varphi )_0$ is a {\em direct limit of $\varphi$} if for any $V \in C_0$ the resulting weak morphism from $C_{1/}(U ,V )$ to $Hom (\varphi ,V )$ is an equivalence. To say this more precisely this means that the morphism $$ Hom (\varphi , U , V ) \times _{Hom (\varphi , U )} \{ f \} \rightarrow Hom ( \varphi ,V ) $$ should be an equivalence. If such a direct limit exists we say that {\em $\varphi$ admits an inverse limit}. Exactly the same discussion of uniqueness as above (\ref{uniquenessinverse}) holds here too. If any morphism $\varphi : A \rightarrow C$ from any $n$-category $A$ to $C$ admits a direct limit then we say that {\em $C$ admits direct limits}. \end{definition} \begin{parag} \label{upsilondirect} We have the following characterization analogue to \ref{upsiloninverse4}. Again, this is the characterization which we shall use in the proofs. It comes from considerations identical to \ref{upsiloninverse1}--\ref{upsiloninverse3} which we omit here. A morphism $f\in Hom (\varphi ,U)$ is a direct limit if and only if for every morphism $$ v:A\times \Upsilon (E) \rightarrow C $$ with $r_0(v)= \varphi$ and $r_1(v)= V_A$ for $V\in C_0$, and for every extension over $A\times \Upsilon (E')$ to a morphism $$ w': A\times \Upsilon ^2(\ast , E')\rightarrow C $$ with $r_{01}(w')= f$ and $r_{12}(w')$ coming from a morphism $z':\Upsilon (E')\rightarrow C$ with $r_1(z')= U$ and $r_2(z')=V$, there exists a common extension of these two: a morphism $$ w: A\times \Upsilon ^2(\ast , E )\rightarrow C $$ with $r_{01}(w)= f$ and $r_{12}(w)$ coming from a morphism $z:\Upsilon (E')\rightarrow C$ with $r_1(z)= U$ and $r_2(z)=V$; such that the restriction of $w$ to $A\times \Upsilon ^2(\ast ,E')$ is equal to $w'$; and such that $r_{02}(w)=v$. \end{parag} \bigskip \subnumero{Invariance properties} \begin{proposition} \label{invariance} Suppose $f:A'\rightarrow A$ is an equivalence of $n$-categories and suppose $C$ is a fibrant $n$-category. Suppose $\varphi : A\rightarrow C$ is a morphism. Then the inverse (resp. direct) limit of $\varphi$ exists if and only if the inverse (resp. direct) limit of $\varphi \circ f$ exists. Suppose $\varphi$ and $\psi$ are morphisms from $A$ to $C$, and suppose they are equivalent in $\underline{Hom}(A, C)$. Then the inverse (resp. direct) limit of $\varphi$ exists if and only if the inverse (resp. direct) limit of $\psi$ exists. Finally suppose $g: C\rightarrow C'$ is an equivalence between fibrant $n$-categories. Then the inverse (resp. direct) limit of $\varphi : A\rightarrow C$ exists if and only if the inverse (resp. direct) limit of $g\circ \varphi$ exists. In particular (combining with the previous paragraph) $C$ admits inverse (resp. direct) limits if and only if $C'$ does. \end{proposition} {\em Proof:} There are several statements to prove so we divide the proof into several paragraphs \ref{invariance1}--\ref{invariance9}. \begin{parag} \label{invariance1} Suppose $f: A' \hookrightarrow A$ is a cofibrant equivalence of $n$-categories. Suppose that $\varphi : A\rightarrow C$ is a morphism and that $u:\varphi \rightarrow U$ is a morphism from $\varphi$ to $U\in C$ which is a direct limit. This corresponds to a diagram $$ \epsilon : A \times \Upsilon (\ast )\rightarrow C $$ and pullback by $f$ gives a diagram $$ \epsilon ': A' \times \Upsilon (\ast )\rightarrow C. $$ We claim that $\epsilon '$ is a direct limit (note that $\epsilon '$ is a morphism from $\varphi \circ f$ to $U$). Suppose we are given $$ u: A' \times \Upsilon (E) \rightarrow C $$ and an extension over $E'\subset E$ to a diagram $$ v_1: A' \times \Upsilon ^2(\ast , E') \rightarrow C $$ whose restriction to the edge $(01)$ is $\epsilon '$ and whose restriction to the edge $(02)$ is $u$. Then we can first extend $v_1$ to a diagram $$ A \times \Upsilon (\ast , E') \rightarrow C $$ because $$ A' \times \Upsilon ^2(\ast , E') \hookrightarrow A \times \Upsilon ^2(\ast , E') $$ is a trivial cofibration (and note also that we can assume that the extension satisfies the relevant properties as in the definition of limit); then we can also extend our above morphism $u$ to a diagram $$ A \times \Upsilon (E) \rightarrow C $$ compatibly with the extension of $v_1$, because the inclusion from the coproduct of $A \times \Upsilon ( E')$ and $A' \times \Upsilon (E)$ over $A' \times \Upsilon (E')$, into $A \times \Upsilon ( E)$ is a trivial cofibration. Now we apply the limit property of $\epsilon$ to conclude that there is an extension to a diagram $$ v: A \times \Upsilon ^2(\ast , E) \rightarrow C. $$ This restricts over $A'$ to a diagram of the form we would like, showing that $\epsilon '$ is a direct limit. \end{parag} \begin{parag} \label{invariance2} Suppose that $f: A' \hookrightarrow A$ is a trivial cofibration and suppose $\varphi: A\rightarrow C$ is a morphism to a fibrant $n$-category $C$, and suppose now that we know that $\varphi \circ f$ has a limit $$ \epsilon ': \varphi \circ f \rightarrow U $$ for an object $U\in C$. We claim that $\varphi$ has a limit. The morphism $\epsilon '$ may be considered as a diagram $$ \epsilon ' : A' \times \Upsilon (\ast ) \rightarrow C. $$ This extends along $A' \times \{ 0\}$ to $$ \varphi : A \times \{ 0\} \rightarrow C $$ and it extends along $A'\times \{ 1\} $ to $$ U_A: A \times \{ 1\} \rightarrow C. $$ Putting these all together we obtain a morphism $$ A\times \{ 0 \} \cup ^{A'\times \{ 0\} } A' \times \Upsilon (\ast ) \cup ^{A' \times \{ 1\} } A \times \{ 1\} \rightarrow C. $$ Since $A'\subset A$ is a trivial cofibration the morphism $$ A\times \{ 0 \} \cup ^{A'\times \{ 0\} } A' \times \Upsilon (\ast ) \cup ^{A' \times \{ 1\} } A \times \{ 1\} \rightarrow A \times \Upsilon (\ast ) $$ is a trivial cofibration (applying \ref{explaincmc}, first part of CM4, two times), so by the fibrant property of $C$ our morphism extends to a morphism $$ \epsilon : A \times \Upsilon (\ast )\rightarrow C $$ with the required properties of being constant along $A\times \{ 1\}$ and restricting to $\varphi$ along $A\times \{ 0\}$. Thus we may write $\epsilon : \varphi \rightarrow U$. We claim that this map is a direct limit of $\varphi$. Given a diagram $$ u:A\times \Upsilon (E)\rightarrow C $$ going from $\varphi$ to a constant object $B$, the restriction $u'$ to $A'\times \Upsilon (E)$ admits (by the hypothesis that $\epsilon '$ is a direct limit) an extension to $$ v': A'\times \Upsilon (\ast , E)\rightarrow C $$ restricting along the edge $(01)$ to $\epsilon '$ and restricting along the edge $(12)$ to the pullback of a diagram $\Upsilon (E)\rightarrow C$. Then using as usual the fibrant property of $C$ and the fact that $A'\rightarrow A$ is a trivial cofibration, we can extend $v'$ to a morphism $$ v: A \times \Upsilon (\ast , E) \rightarrow C $$ again restricting along the edge $(01)$ to $\epsilon $, restricting along the edge $(02)$ to our given diagram $u$, and restricting along the edge $(12)$ to the pullback of a diagram $\Upsilon (E)\rightarrow C$. If $E'\subset E$ and we are already given an extension $v_{E'}$ over $A \times \Upsilon (\ast , E')$ then (as before, using the fibrant property of $C$ applied to an appropriate cofibration) we can assume that our extension $v$ above restricts to $v_{E'}$. This completes the proof that $\epsilon$ is a direct limit, and hence the proof of the statement claimed for \ref{invariance2}. \end{parag} \begin{parag} \label{invariance3} Now suppose $p:A' \rightarrow A$ is a trivial fibration. Then there exists a section $s:A\rightarrow A'$ (with $ps = 1_A$). Note that $s$ is a trivial cofibration. If $\varphi : A \rightarrow C$ is a morphism then $$ (\varphi \circ p)\circ s = \varphi $$ so applying the previous two paragraphs \ref{invariance1} and \ref{invariance2} to the morphism $s$ we conclude that $\varphi$ admits a limit if and only if $\varphi \circ p$ admits a limit. \end{parag} \begin{parag} \label{invariance4} Now suppose $f: A' \rightarrow A$ is any equivalence between $n$-categories. Decomposing $f= p\circ j$ into a composition of a trivial cofibration followed by a trivial fibration and applying \ref{invariance1}, \ref{invariance2} to $j$ and \ref{invariance3} to $p$ we conclude that a functor $\varphi : A\rightarrow C$ admits a direct limit if and only if $\varphi \circ f$ admits a direct limit. This proves the first paragraph of Proposition \ref{invariance} for direct limits. \end{parag} \begin{parag} \label{invariance5} The proof of the first paragraph of \ref{invariance} for inverse limits is exactly the same as the above. \end{parag} \begin{parag} \label{invariance6} Now we prove the second paragraph of the proposition. If $f,g: A\rightarrow C$ are two morphisms which are equivalent in $\underline{Hom}(A,C)$ (i.e. they are homotopic) then there exists a morphism $\varphi : A\times \overline{I}\rightarrow C$ restricting to $f$ on $A\times \{ 0\}$ and restricting to $g$ on $A\times \{ 1\}$. Applying the first paragraph of the proposition (for either direct or inverse limits) to the two inclusions $A\rightarrow A\times \overline{I}$ we find that $f$ admits a limit if and only if $\varphi$ admits a limit and similarly $g$ admits a limit if and only if $\varphi$ does---therefore $f$ admits a limit if and only if $g$ admits a limit. \end{parag} \begin{parag} \label{lifting} Suppose $C\rightarrow C'$ is an equivalence between fibrant $n$-categories. In general if $F'\subset F$ is any cofibration and if $F'\rightarrow C$ is a morphism, then there exists an extension to $F\rightarrow C$ if and only if the composed morphism $F'\rightarrow C'$ extends over $F$. To see this, look at the (exactly commutative) diagram $$ \begin{array}{ccc} \underline{Hom}(F,C) & \rightarrow & \underline{Hom}(F',C) \\ \downarrow && \downarrow \\ \underline{Hom}(F,C') & \rightarrow & \underline{Hom}(F',C') \end{array} . $$ The horizontal arrows are fibrations and the vertical arrows are equivalences. If an element $a\in \underline{Hom}(F',C)$ maps to something $b$ which is hit from $c\in \underline{Hom}(F,C')$ then there is $d\in \underline{Hom}(F,C) $ mapping to something equivalent to $c$; thus the image $e$ of $d$ in $\underline{Hom}(F',C)$ maps to something equivalent to $b$. This implies (since the right vertical arrow is an equivalence) that $e$ is equivalent to $a$. Since the top morphism is fibrant, there is another element $d'\in \underline{Hom}(F,C) $ which maps directly to $a$. \end{parag} \begin{parag} \label{invariance7} Suppose still that $C\rightarrow C'$ is an equivalence between fibrant $n$-categories. Using the general lifting principle \ref{lifting} and the fact that the property of being a limit is expressed in terms of extending morphisms across certain cofibrtions $F'\subset F$, we conclude that a functor $A\rightarrow C$ has a (direct or inverse) limit if and only if the composition $A\rightarrow C'$ does. This proves the first sentence of the last paragraph of the proposition. \end{parag} \begin{parag} \label{invariance8} If $f: C\rightarrow C'$ is an equivalence between fibrant $n$-categories and if $C'$ admits direct (resp. inverse) limits then any functor $A\rightarrow C$ admits a direct (resp. inverse) limit by \ref{invariance7}. \end{parag} \begin{parag} \label{invariance9} Suppose on the other hand that we know that $C$ admits direct (resp. inverse) limits. Suppose that $\varphi : A \rightarrow C'$ is a functor. Since $f$ induces an equivalence from $\underline{Hom}(A,C)$ to $\underline{Hom}(A,C')$ there is a morphism $\psi : A \rightarrow C$ such that $f\circ \psi $ is equivalent to $\varphi$ in $\underline{Hom}(A,C')$. By the second paragraph of the proposition (proved in \ref{invariance6} above), $\varphi$ admits direct (resp. inverse) limits if and only if $f \circ \psi$ does. Then by the first part of the last paragraph proved in \ref{invariance7} above, $f\circ \psi$ admits direct (resp. inverse) limits if and only if $\psi$ does. Now by hypothesis $\psi$ has a direct (resp. inverse) limit, so $\varphi$ does too. This shows that $C'$ admits direct (resp. inverse) limits. \end{parag} We have now completed the proof of Proposition \ref{invariance}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{parag} \label{variance1} We now start to look at variance properties in other situations. Suppose $h:C\rightarrow C'$ is a morphism between fibrant $n$-categories, and suppose $\varphi : A \rightarrow C$ is a morphism. If $$ u: \varphi \rightarrow U $$ is a direct limit then $h(u): \varphi \circ h \rightarrow h(U)$ is a morphism. Suppose that $\varphi \circ h$ admits a direct limit $$ v: \varphi \circ h \rightarrow V. $$ Then by the limit property there is a factorization i.e. a diagram $$ [v, w]:\varphi \circ h \rightarrow V \rightarrow h(U) $$ whose third edge $(02)$ is $h(u)$. We say that {\em the morphism $h$ commutes with the direct limit of $\varphi$} if the direct limit of $\varphi \circ h$ exists and if the factorization morphism $w: V\rightarrow h(U)$ is an equivalence. Suppose that $C$ and $C'$ admit direct limits. We say that {\em the morphism $h$ commutes with direct limits} if $h$ commutes with the direct limit of any $\varphi : A \rightarrow C$ in the previous sense. \end{parag} \begin{parag} \label{variance2} We have similar definitions for inverse limits, which we repeat for the record. Suppose again that $h:C\rightarrow C'$ is a morphism between fibrant $n$-categories, and suppose $\varphi : A \rightarrow C$ is a morphism. If $$ u: U\rightarrow \varphi $$ is an inverse limit then $h(u): h(U)\rightarrow \varphi \circ h $ is a morphism. Suppose that $\varphi \circ h$ admits an inverse limit $$ v: V\rightarrow \varphi \circ h . $$ Then by the limit property there is a factorization i.e. a diagram $$ [w,v]:h(U)\rightarrow V\rightarrow \varphi \circ h $$ whose third edge $(02)$ is $h(u)$. We say that {\em the morphism $h$ commutes with the inverse limit of $\varphi$} if the inverse limit of $\varphi \circ h$ exists and if the factorization morphism $w:h(U)\rightarrow V$ is an equivalence. Suppose that $C$ and $C'$ admit inverse limits. We say that {\em the morphism $h$ commutes with inverse limits} if $h$ commutes with the inverse limit of any $\varphi : A \rightarrow C$ in the above sense. \end{parag} \bigskip \subnumero{Behavior under certain precat inverse limits of $C$} We will now study certain situations of what happens when we take fiber products or other inverse limits (here we mean inverse limits in the category of $n$-precats) of the target $n$-category $C$. We study what happens to inverse limits in $C$. We could also say the same things about direct limits in $C$ but the inverse limit case is the one we need, so we state things there and leave it to the reader to make the corresponding statements for direct limits. \begin{lemma} \label{dirprod} Suppose $\{ C_i\} _{i\in S}$ is a collection of fibrant $n$-categories indexed by a set $S$. Let $C= \prod _{i\in S} C_i$ and suppose $\varphi = \{ \varphi _i\}$ is a morphism from $A$ to $C$. Suppose that the $\varphi _i$ admit inverse limits $$ u_i : U_i \rightarrow \varphi _i $$ in $C_i$. Then $U= \{ U_i\}$ is an object of $C$ and we have a morphism $$ u: U\rightarrow \varphi $$ composed of the factors $u_i$. This morphism is an inverse limit of $\varphi$ in $C$. \end{lemma} {\em Proof:} The property that $u$ be an inverse limit consists of a collection of extension properties that have to be satisfied. The morphisms $u_i$ admit the corresponding extensions and putting these together we get the required extensions for $u$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{fiprod} Suppose $f:C\rightarrow D$ and $g:E\rightarrow D$ are morphisms of fibrant $n$-categories with $f$ fibrant. Suppose that $\varphi : A \rightarrow C\times _DE$ is a morphism such that the component morphisms $\varphi _C: A\rightarrow C$, $\varphi _D:A\rightarrow D$ and $\varphi _E: A\rightarrow E$ have inverse limits $\lambda _C$, $\lambda _D$ and $\lambda _E$ respectively. Suppose furthermore that $f$ and $g$ preserve these inverse limits, which means that the projections of $\lambda _C$ and $\lambda _E$ into $D$ are equivalent (as objects with morphisms to $\varphi _D$) to $\lambda _D$. Then we may (by changing the $\lambda _C, \lambda _D, \lambda _E$ by equivalences) assume that $\lambda _C$ and $\lambda _E$ project to $\lambda _D$; and the resulting object $\lambda \in C\times _DE$ is an inverse limit of $\varphi$. \end{lemma} {\em Proof:} Set $\lambda '_E:=\lambda _E$ and let $\lambda '_D:= g(\lambda _E)$ be the projection to $D$. Note that by hypothesis $\lambda '_D$ is an inverse limit of $\varphi _D$. Now $\lambda _C$ (considered as an object with morphism to $\varphi _C$) projects in $D$ to something equivalent to $\lambda _D$ and hence equivalent to $\lambda '_D$ (equivalence of the diagrams including the morphism to $\varphi _D$). Since $f$ is a fibrant morphism, we can modify $\lambda _C$ by an equivalence, to obtain $\lambda ' _C$ projecting directly to $\lambda ' _D$. Note that the equivalent $\lambda '_C$ is again an inverse limit of $\varphi _C$. Together these give an element $\lambda \in C\times DE$ with a map $$ u: \lambda \rightarrow \varphi , $$ and we claim that $u$ is an inverse limit. Suppose $F'\subset F$ is a cofibration of $n-1$-precats and suppose $$ v: V \stackrel{F}{\rightarrow} \varphi $$ is any $F$-morphism (i.e. a diagram $$ A\times \Upsilon (F) \rightarrow C\times _DE $$ restricting on $A\times \{ 0\}$ to the constant $V_A$ and restricting on $A\times \{ 1\}$ to $\varphi$) provided with an extension over $F'$ to a diagram $$ w': A\times \Upsilon ^2(F, \ast )\rightarrow C\times _DE $$ restricting on $(02)$ to $v'$ (the restriction of $v$ to $F'$) and on $(12)$ to $u$. We look for an extension of $w'$ to a diagram $$ w: A\times \Upsilon ^2(F, \ast )\rightarrow C\times _DE $$ restricting on $(02)$ to $v$ and on $(12)$ to $u$. Denoting with subscripts the components in $C$, $D$ and $E$, we have that the pairs $(v_C,w'_C)$ and $(v_E, w'_E)$ admit extensions $w_C$ and $w_E$ respectively. The projections of these extensions in $D$ give diagrams which we denote $$ w_{C/D}, w_{E/D} : A\times \Upsilon ^2(F, \ast )\rightarrow D, $$ both restricting on $(02)$ to $v_D$ and on $(12)$ to $u_D$, and extending $w'_D$. Applying again the limit property for $u_D$ to the cofibration $$ F\times \{ 0\} \cup ^{F' \times \{ 0\} } F' \times \overline{I} \cup ^{ F' \times \{ 1\} } F \times \{ 1\} \hookrightarrow F\times \overline{I} $$ we find that there is a diagram $$ z_D: A \times \Upsilon ^2(F\times \overline{I} , \ast ) \rightarrow D $$ giving a homotopy between $w_{C/D}$ and $w_{E/D}$. Notice that this is a homotopy in Quillen's sense \cite{Quillen} because the diagram $$ \Upsilon ^2(F, \ast )\stackrel{\displaystyle \rightarrow}{\rightarrow} \Upsilon ^2(F\times \overline{I}, \ast ) \rightarrow \Upsilon ^2(F, \ast ) $$ is of the form used by Quillen cf \ref{homotopic1}. Such a homotopy can be changed into one of the more classical form $$ A\times \Upsilon ^2(F, \ast )\times \overline{I} \rightarrow D $$ because the relation of homotopy in Quillen's sense is the same as the relation of equivalence of morphisms, which in turn is the same as existence of a homotopy for the ``interval'' $\overline{I}$. In fact we don't use this remark here but we use it with $D$ replaced by $C$, below. Now apply the lifting property for the morphism $C\rightarrow D$, for the above map $z_D$, with respect to the trivial cofibration $$ A\times \Upsilon ^2( F\times \{ 0\} \cup ^{F' \times \{ 0\} } F' \times \overline{I}, \ast ) $$ $$ \rightarrow A\times \Upsilon ^2(F\times \overline{I} , \ast ). $$ We get a morphism $$ z_C: A \times \Upsilon ^2(F\times \overline{I} , \ast ) \rightarrow C $$ providing a homotopy between $w_C$ and a new morphism $w^n_C$ which projects into $D$ to $w_{E/D}$. The new $w^n_C$ is again a solution of the required extension problem (or to be more precise we can impose conditions on our lifting $z_C$ to insure that this is the case. The fact that it projects to $w_{E/D}$ means that the pair $w=(w^n_C, w_E)$ is a solution of the required extension problem to show that $u$ is an inverse limit. This completes the proof. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{invlim} Suppose $C_i$ is a collection of fibrant $n$-categories for $i=0,1,2,\ldots$ and suppose that $f_i : C_i \rightarrow C_{i-1}$ are fibrant morphisms. Let $C$ be the inverse limit of this system of $n$-precats. Then $C$ is a fibrant $n$-category. Suppose that we have $\varphi : A \rightarrow C$ projecting to the $\varphi _i : A \rightarrow C_i$ and suppose that the $\varphi _i$ admit inverse limits $u_i : U_i \rightarrow \varphi _i$. Suppose finally that the $f_i$ commute with the inverse limits of the $\varphi _i$. Then $\varphi$ admits an inverse limit and the projections $C\rightarrow C_i$ commute with the inverse limit of $\varphi$. \end{lemma} {\em Proof:} The fibrant property of $C$ may be directly checked by producing liftings of trivial cofibrations. First we construct a morphism $u: U\rightarrow \varphi$ projecting in each $C_i$ to an inverse limit of $\varphi _i$. To do this, note by \ref{invariance} that it suffices to have $u$ project to a morphism equivalent to $u_i$. On the other hand, by the hypothesis that the $f_i$ commute with the inverse limits $u_i$, we have that $f_i (u_i)$ is an inverse limit of $\varphi _{i-1}$. In particular $f_i(u_i)$ is equivalent to $u_{i-1}$ as a diagram from $A\times \Upsilon (\ast )$ to $C_{i-1}$. The morphism from such diagrams in $C_i$, to such diagrams in $C_{i-1}$, is fibrant (since it comes from $f_i$ which is fibrant). Therefore we can change $u_i$ to an equivalent diagram with $f_i(u_i)=u_{i-1}$. Do this successively for $i=1,2,\ldots$, yielding a system of morphisms $u_i$ with $f_i(u_i)=u_{i-1}$. These now form a morphism $$ u: U\rightarrow \varphi . $$ We claim that $u$ is an inverse limit of $\varphi$. Suppose $E'\subset E$ is an inclusion of $n-1$-precats and suppose $$ w:W\stackrel{E}{\rightarrow}\varphi $$ is an $E$-morphism (i.e. a diagram of the form $$ A\times \Upsilon (E)\rightarrow C $$ being constant equal to $W$ on $A\times \{ 0\}$), provided over $E'$ with an extension to a diagram $$ [v',u]:W\stackrel{E'}{\rightarrow} U \rightarrow \varphi $$ (i.e. a morphism $$ A\times \Upsilon ^2 (E,\ast ) \rightarrow C $$ restricting to $u$ on the second edge $(12)$ and restricting to $w|_{E'}$ on the third edge $(02)$). We would like to extend this to a diagram $[v,u]$ giving $w$ on the third edge. Let $w_i$ (resp. $v'_i$) be the projections of these diagrams in $C_i$. These admit extensions $v_i$. The projection of $v_i$ to $C_{i-1}$ is an extension of the desired sort for $w_{i-1}$ and $v'_{i-1}$. The extensions $v_i$ are unique up to equivalence---which means a diagram $$ A\times \Upsilon ^2 (E,\ast )\times \overline{I} \rightarrow C_i $$ satisfying appropriate boundary conditions---and from this and our usual sort of argument constructing a trivial cofibration (this occurs several times below) then making use of the fibrant property of $f_i$, we conclude that $v_i$ may be modified by an equivalence so that it projects to $v_{i-1}$. As before, do this successively for $i=1,2,\ldots $ to obtain a system of extensions $v_i$ with $f_i(v_i)=v_{i-1}$. This system corresponds to an extension $v$ of the desired sort for $w$ and $v'$. This shows that the morphism $u$ is an inverse limit. Note from our construction the projection of the inverse limit $u$ to $C_i$ is an inverse limit $u_i$ for $\varphi _i$ so the projections $C\rightarrow C_i$ commute with the inverse limit of $\varphi$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} The application of the above results which we have in mind is the following theorem. \begin{theorem} \label{variation} Suppose $C$ is a fibrant $n$-category and $B$ is an $n$-precat. Then if $C$ admits inverse (resp. direct) limits, so does the fibrant $n$-category $\underline{Hom}(B,C)$. The morphisms of functoriality for $B'\rightarrow B$ commute with inverse (resp. direct) limits. \end{theorem} {\em Proof:} Suppose $\varphi : A\rightarrow \underline{Hom}(B, C)$. We will construct an inverse limit $\lambda \in \underline{Hom}(B, C)$ such that for any $b\in B$ the restriction $\lambda (b)$ is equivalent (via the natural morphism) to the inverse limit of $\varphi (b): A\rightarrow C$. This condition implies that the restriction morphism for any $B'\rightarrow B$ commutes with the inverse limit. In effect, there is a morphism from the inverse limit over $B$ (pulled back to $B'$) to the inverse limit over $B'$, and this morphism is an equivalence over every object in $B'$ by the condition, which implies that it is an equivalence by Lemma \ref{obbyobequiv}. \begin{parag} \label{pfvar1} The first remark is that if $B\subset B'$ and $B\subset B''$ are cofibrations of $n$-precats such that $\underline{Hom}(B, C)$, $\underline{Hom}(B', C)$, and $\underline{Hom}(B'', C)$ admit inverse limits complying with the above condition, then $\underline{Hom}(B'\cup ^B B'', C)$ admits inverse limits again complying with the above condition. To see this we apply Lemma \ref{fiprod}. The only thing that we need to know is that the restriction of the inverse limits is again equivalent to the inverse limit in $\underline{Hom}(B, C)$. This follows from the fact that there is a morphism which, thanks to the condition given at the start of the proof, is an equivalence for each object of $B$---therefore it is an equivalence. \end{parag} \begin{parag} \label{pfvar2} The next remark is that weak equivalences of $n$-precats $B$ are turned into equivalences of the $\underline{Hom}(B,C)$. The morphisms $$ h(1, M') \cup ^{\ast} \ldots \cup ^{\ast }h(1,M')\rightarrow h(m,M') $$ are weak equivalences. By the previous remarks if we know the theorem (with the condition of the first paragraph of the proof) for $h(1,M')$ then we get it for any $h(m,M')$ (again with the condition of the first paragraph). As pointed out at the start of the proof, morphisms of restriction between any $B$'s for which we know that the limits exist (and satisfying the condition of the first paragraph), commute with the limits. In particular when we apply \ref{fiprod} and \ref{invlim} the hypotheses about commutation with the limits will hold. Now any $n$-precat may be expressed as a direct union of pushouts of the $h(m,M')$. The pushouts in question may be organized into a countable direct union of pushouts each of which is adding a disjoint direct sum; and the addition will be of a direct sum of things of the form $h(m,M')$ added along their boundary. Taking $\underline{Hom}$ into $C$ transforms this expression into an inverse limit indexed by the natural numbers, of fiber products of terms which are direct products of things of the form $\underline{Hom}(h(m,M'), C)$. Applying Lemmas \ref{dirprod}, \ref{fiprod} and \ref{invlim} we find that if we know the existence of limits for $h(1,M')$ (hence $h(m,M')$)---as always with the additional condition of the first paragraph of the proof---then we get existence of limits for $\underline{Hom}(B,C)$ for any $n$-precat $B$. \end{parag} \begin{parag} \label{pfvar3} Therefore it suffices to prove the theorem for $B= h(1,M')$. This is more generally of the form $\Upsilon E$ (in this case $E=h(M')$). Thus it now suffices to prove the theorem for $B=\Upsilon E$. Suppose we have a morphism $\varphi : A\times \Upsilon E \rightarrow C$. Let $\varphi (0)$ (resp. $\varphi (1)$) denote the restriction of $\varphi$ to $A\times \{ 0\} $ (resp. $A\times \{ 1\} $). Let $(\lambda (0),\epsilon (0))$ and $(\lambda (1),\epsilon (1))$ denote inverse limits of $\varphi (0)$ and $\varphi (1)$. The morphism $\varphi$ corresponds to a morphism $$ E\rightarrow \underline{Hom}(A,C)_{1/}(\varphi (0), \varphi (1)). $$ We can lift this together with $\epsilon (0)$ to a morphism $$ E\rightarrow \underline{Hom}(A,C)_{2/}(\lambda (0), \varphi (0), \varphi (1)). $$ The resulting $E\rightarrow \underline{Hom}(A,C)_{1/}(\lambda (0), \varphi (1))$ exends to $$ E\rightarrow \underline{Hom}(A,C)_{2/}(\lambda (0)_A, \lambda (1)_A, \varphi (1)) $$ projecting to $\epsilon (1)$ on the second edge, by the limit property for $\epsilon (1)$. The first edge of this comes from a morphism $\lambda : \Upsilon E \rightarrow C$. Noting that the product $\Upsilon E \times I$ is a pushout of two triangles the above morphisms glue together to give a morphism $$ \Upsilon E \times I \rightarrow \underline{Hom}(A,C), $$ in other words we get a morphism $\epsilon$ from $\lambda$ to $\varphi$ considered as families over $A$ with values in $\underline{Hom}(\Upsilon E, C)$. \end{parag} \begin{parag} \label{pfvar4} To finish the proof we just have to prove that $\epsilon$ is an inverse limit of $\varphi$. Suppose we have $\mu \in \underline{Hom}(\Upsilon E, C)$ and suppose given a morphism $f: \Upsilon F \times A \times \Upsilon E\rightarrow C$ restricting over $0^F$ to $\mu _A$ and over $1^F$ to $\varphi$ (here $0^F$ and $1^F$ denote the endpoints of $\Upsilon F$ and we will use similar notation for $E$). This extends to morphisms $$ f'(0^E): \Upsilon ^2(F, \ast )\times A \rightarrow C, $$ $$ f'(1^E): \Upsilon ^2(F, \ast )\times A \rightarrow C, $$ by the limit properties of $\lambda (0)$ and $\lambda (1)$ (the above morphisms restricting to $\epsilon (0)$ and $\epsilon (1)$ on the second edges). Now we try to extend to a morphism on all of $\Upsilon ^2(F, \ast )\times A \times \Upsilon E\rightarrow C$. For this we use notation of the form $(i,j)$ for the objects of $\Upsilon ^2(F,\ast )\times \Upsilon E$, where $i=0,1,2$ (objects in $\Upsilon ^2(F, \ast )$ and $j= 0,1$ (objects in $\Upsilon E$). We are already given maps defined over the triangles $(012, 0)$ and $(012,1)$ (these are $f'(0^E)$ and $f'(1^E)$), as well as overthe squares $(02, 01)$ (our given map $f$) and $(12, 01)$ (the map $\epsilon$). First extend using the fibrant property of $C$ to a map on the tetrahedron $$ (0,0)(1,0)(2,0)(2,1). $$ Then extend again using the fibrant property of $C$ to a map on the tetrahedron $$ (0,0)(1,0)(1,1)(2,1). $$ Here note that on the face $(0,0)(1,0)(1,1)$ the map is chosen first as coming from a map $\Upsilon ^2(F,E)\rightarrow C$. Finally we have to find an extension over the tetrahedron $$ (0,0)(0,1)(1,1)(2,1). $$ Again we require that the map on the first face $(0,0)(0,1)(1,1)$ come from a map $$ \Upsilon ^2(E,F)\rightarrow C. $$ Our problem at this stage is that the map is already specified on all of the other faces, so we can't do this using the fibrant property of $C$ (the face that is missing is not of the right kind). Instead we have to use the limit property of $\epsilon (1^E)$. The limit condition on $\lambda (1^E)$ means that the morphism $$ \underline{Hom}^{\epsilon (1^E)}(\mu (0^E), \mu (1^E), \lambda (1^E), \varphi ) \rightarrow \underline{Hom}(\mu (0^E), \mu (1^E), \varphi ) $$ is an equivalence. The morphisms $$ \underline{Hom}^{\epsilon (1^E)}(\mu (0^E), \lambda (1^E), \varphi ) \rightarrow \underline{Hom}(\mu (0^E), \varphi ) $$ and $$ \underline{Hom}^{\epsilon (1^E)}(\mu (1^E), \lambda (1^E), \varphi ) \rightarrow \underline{Hom}(\mu (1^E), \varphi ) $$ are equivalences too. This implies (in view of the fact that the edges containing $\epsilon (1^E)$ are fixed) that the morphism $$ \underline{Hom}^{\epsilon (1^E)}(\mu (0^E), \mu (1^E), \lambda (1^E), \varphi ) \rightarrow $$ $$ \underline{Hom}^{\epsilon (1^E)}(\mu (0^E), \lambda (1^E), \varphi ) \times _{ \underline{Hom}(\mu (0^E), \varphi )} $$ $$ \underline{Hom}(\mu (0^E), \mu (1^E), \varphi )\times _{ \underline{Hom}(\mu (1^E), \varphi )} $$ $$ \underline{Hom}^{\epsilon (1^E)}(\mu (1^E), \lambda (1^E), \varphi ) $$ is an equivalence. This exactly implies that the restriction to the shell that we are interested in is an equivalence. The fact that this equivalence is a fibration implies that it is surjective on objects, giving finally the extension that we need. In the relative case where we are already given an extension over $F'\subset F$, one can choose our extension in a compatible way (adding on the part concerning $F'$ in the above argument doesn't change the properties of the relevant morphisms being trivial cofibrations). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \end{parag} \begin{corollary} \label{constant} Suppose $\varphi : A\rightarrow C$ is a morphism from an $n$-precat to a fibrant $n$-category $C$ admitting inverse limits, and suppose that $B$ is an $n$-precat. Let $\varphi _B: A\rightarrow \underline{Hom}(B,C)$ denote the morphism constant along $B$. Suppose that $\varphi$ admits an inverse limit $u:U\rightarrow \varphi$. Then $u_B: U_B \rightarrow \varphi _B$ (the pullback of $u$ along $B\rightarrow \ast$) is an inverse limit of $\varphi _B$. \end{corollary} {\em Proof:} This is just commutativity for pullbacks for the morphism $B\rightarrow \ast$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{parag} \label{usevar} We can use the result of the previous theorem to obtain the variation of the limit depending on the family. Suppose $C$ is a fibrant $n$-category in which inverse limits exist, and suppose $A$ is an $n$-precat. Let $B= \underline{Hom}(A, C)$. We have a tautological morphism $$ \zeta : A\rightarrow \underline{Hom}(B,C). $$ By the previous theorem, limits exist in $\underline{Hom}(B,C)$. Thus we obtain the limit of $\zeta$ which is an element of $\underline{Hom}(B,C)$: it is a morphism $\lambda $ from $B=\underline{Hom}(A,C)$ to $C$, which is the morphism which to $\varphi \in \underline{Hom}(A,C)$ associates $\lambda (\varphi )$ which is the limit of $\varphi$. The same remark holds for direct limits. \end{parag} \begin{theorem} \label{commute} Suppose $A$, $B$ and $C$ are $n$-categories. Suppose $F: A\times B \rightarrow C$ is a functor. Then letting $\psi : A\rightarrow \underline{Hom}(B,C)$ denote the corresponding functor, suppose that $\psi$ admits an inverse limit $\lambda \in \underline{Hom}(B,C)$. Suppose now that $\lambda$ (considered as a morphism $B\rightarrow C$) admits an inverse limit $\mu \in C$. Then $\mu$ is an inverse limit of $F: A\times B\rightarrow C$. In particular if the intermediate limits exist going in the other direction then the composed limits are canonically equivalent. Thus if $C$ admits inverse limits then inverse limits commute with each other. \end{theorem} {\em Proof:} The general proof is left to the reader. In the case $C=nCAT'$ this will be easy to see from our explicit construction of the limits. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Limits in $nCAT'$} Let $nCAT \hookrightarrow nCAT'$ be a trivial cofibration to a fibrant $n+1$-category. \begin{theorem} \label{inverse} The fibrant $n+1$-category $nCAT'$ admits inverse limits. \end{theorem} The rest of this section is devoted to the proof. As a preliminary remark notice that by \ref{invariance} the statement doesn't depend on which choice of $nCAT'$ we make. We also remark, in the realm of set-theoretic niceties, that the statement means that $nCAT'$ (an $n+1$-category composed of classes) admits inverse limits indexed by any $n+1$-category composed of sets. To be more precise our proof will show that if we restrict to a subcategory of $nCAT'$ of $n$-categories represented in a certain set of fixed cardinality $\alpha$, then the inverse limit indexed by $A$ exists if $\alpha$ is infinite and at least equal to $2^{\# A}$, also at least equal to what is needed for making the fibrant replacement $nCAT\subset nCAT'$. \bigskip \subnumero{Construction of the limit} \begin{parag} \label{subscriptA} Suppose $A$ is an $n+1$-category. If $B$ is a fibrant $n$-category we have denoted by $B_A$ the constant morphism $A\rightarrow nCAT$ with value $B$, considered as a morphism $A\rightarrow nCAT'$. \end{parag} \begin{parag} \label{lambda} We now give the construction of the inverse limit. Suppose $\varphi : A \rightarrow nCAT'$ is a morphism. We define an $n$-category $$ \lambda := \underline{Hom}(A, nCAT')_{1/}(\ast _A, \varphi ). $$ This has the universal property that for any $n$-precat $B$, $$ Hom (B, \lambda )= Hom ^{\ast _A, \varphi }(A\times \Upsilon B, nCAT'). $$ The notation on the right means the fiber of the map $$ (r_0,r_1): Hom (A\times \Upsilon B, nCAT')\rightarrow Hom (A, nCAT')\times Hom(A, nCAT') $$ over $(\ast _A, \varphi )$. The problem below will be to prove that $\lambda$ is an inverse limit of $\varphi$. \end{parag} \bigskip \subnumero{Diagrams} \begin{parag} \label{diag1} Suppose $C$ is an $n+1$-precat. Then for $n$-precats $E_1,\ldots , E_k$ we define $$ Diag (E_1,\ldots , \underline{E_i},\ldots , E_k; C) $$ to be the $n$-precat which represents the functor $$ F \mapsto Hom (\Upsilon ^k (E_1,\ldots , E_i\times F,\ldots , E_k), C). $$ We establish some properties. \end{parag} \begin{parag} \label{diag1.5} The first remark is that $Diag (E_1,\ldots , \underline{E_i},\ldots , E_k; C)$ decomposes as a disjoint union over all pairs $(a,b)$ where $$ a: \Upsilon ^{i-1}(E_1,\ldots , E_{i-1})\rightarrow C $$ and $$ b: \Upsilon ^{k-1-i}(E_{i+1}, \ldots , E_k)\rightarrow C $$ are the restrictions to the first and last faces separated by the $i$-th edge. Employ the notation $$ Diag^{a,b} (E_1,\ldots , \underline{E_i},\ldots , E_k; C) $$ for the subobject restricting to a given $a$ and $b$. If we don't wish to specify $b$ for example, then denote this by the superscript $Diag^{a,\cdot}$. In particular note that we can decompose into a disjoint union over the $k+1$-tuples of objects which are the images of the vertices $0,\ldots , k$ (these objects are all specified either as a part of $a$ or as a part of $b$). \end{parag} \begin{parag} \label{diag1.6} In case $C=nCAT$ we have $$ Diag ^{a,b}(E_1,\ldots , \underline{E_i},\ldots , E_k; nCAT) = \underline{Hom}(U_{i-1} \times E_i, U_i) $$ where $U_j$ are the fibrant $n$-categories which are the images of the vertices $$ j\in \Upsilon ^k(E_1,\ldots , E_k) $$ by the maps $a$ (if $j\leq i-1$) or $b$ (if $j\geq i$). \end{parag} \begin{parag} When we are only interested in the set of objects, it doesn't matter which $E_i$ is underlined and we denote by $$ Diag (E_1,\ldots , E_k; C) = Hom (\Upsilon ^k (E_1,\ldots , E_k), C) $$ this set of objects. We can put a superscript $Diag ^{a,b}$ here if we want (with the obvious meaning as above). The edge $i$ dividing between $a$ and $b$ should be understood from the data of $a$ and $b$. \end{parag} \begin{parag} \label{quasifib} We need a way of understanding the statement that $nCAT'$ is a fibrant replacement for $nCAT$. In order to do this we will use the following property of $nCAT$ which shows that in some sense it is close to being fibrant. We say that an $n+1$-category $C$ is {\em quasifibrant} if for any sequence of objects $x_0, \ldots , x_p$ the morphism $$ C_{p/}(x_0,\ldots , x_p)\rightarrow C_{(p-1)/}(x_0,\ldots , x_{p-1})\times _{C_{(p-1)/}(x_1,\ldots , x_p)} C_{(p-2)/}(x_1,\ldots , x_{p-1}) $$ is a fibration of $n$-categories. Note inductively that the morphisms involved in the fiber product here are themselves fibrations, and we get that the projections $$ C_{p/}(x_0,\ldots , x_p)\rightarrow C_{(p-1)/}(x_0,\ldots , x_{p-1}) $$ and $$ C_{p/}(x_0,\ldots , x_p)\rightarrow C_{(p-1)/}(x_1,\ldots , x_{p}) $$ are fibrations. \end{parag} \begin{parag} \label{quasifib1.4} The condition that $C$ is an $n+1$-category implies that the morphism in the definition of quasifibrant, is an equivalence whenever $p\geq 2$. Thus if $C$ is quasifibrant, the morphism in question is actually a fibrant equivalence. \end{parag} \begin{parag} \label{quasifib1.5} If $C'$ is a fibrant $n+1$-category then it is quasifibrant. This is because the morphisms (in the notation of \ref{interpupsilon1}) $$ [p-1](E)\cup ^{[p-2](E)}[p-1](E)\rightarrow [p](E) $$ are trivial cofibrations. \end{parag} \begin{parag} \label{quasifib2} The $n+1$-category $nCAT$ is easily seen to be quasifibrant: the morphisms in question are actually isomorphisms for $p\geq 2$ and for $p=1$ they are just projections from the $\underline{Hom}(A_0,A_1)$---which are fibrant---to $\ast$. \end{parag} \medskip We now have two claims which allow us to pass between something quasifibrant such as $nCAT$ and its fibrant completion. \begin{parag} \label{diag2} First of all, if $C$ is quasi-fibrant (\ref{quasifib}) then $Diag (E_1,\ldots , \underline{E_i},\ldots , E_k; C)$ is fibrant. Furthermore in this case for cofibrations $E'_j\hookrightarrow E_j$ the morphism $$ Diag (E'_1,\ldots , \underline{E'_i},\ldots , E'_k; C) \rightarrow Diag (E_1,\ldots , \underline{E_i},\ldots , E_k; C) $$ is fibrant. \end{parag} \begin{parag} \label{diag3} Secondly, if $C$ quasifibrant (\ref{quasifib}) and if $C\rightarrow C'$ is an equivalence to a fibrant $C'$ then the morphism $$ Diag ^{a,b}(E_1,\ldots , \underline{E_i},\ldots , E_k; C) \rightarrow Diag ^{a',b'}(E_1,\ldots , \underline{E_i},\ldots , E_k; C') $$ is an equivalence of fibrant $n$-categories. Here $a,b$ are fixed as in \ref{diag1.5}, and $a',b'$ denote the images in $C'$. {\em Caution:} it is essential to restrict to the components for a fixed $a,b$ coming from $C$. \end{parag} \begin{parag} \label{diag3.5} Before getting to the proofs of \ref{diag2} and \ref{diag3}, we discuss diagrams in a quasifibrant $C$. A morphism $$ u:\Upsilon ^k(E_1,\ldots , E_k)\rightarrow C $$ may be described inductively as triple $u=(\tilde{u}, u^-, u^+)$ where $$ u^-= \Upsilon ^{k-1}(E_2,\ldots , E_k)\rightarrow C $$ and $$ u^+:\Upsilon ^{k-1}(E_1,\ldots , E_{k-1})\rightarrow C, $$ are morphisms which agree on $\Upsilon ^{k-1}(E_1,\ldots , E_{k-1})$, and where $$ \tilde{u}: E_1\times \ldots \times E_k \rightarrow C_{k/}(x_0,\ldots , x_k) $$ is a lifting of the morphism $(\tilde{u}^- ,\tilde{u}^+)$ (these are the components of $u^-$ and $u^+$ analogous to the component $\tilde{u}$ of $u$) along the morphism $$ C_{k/}(x_0,\ldots , x_k)\rightarrow C_{(k-1)/}(x_0,\ldots , x_{k-1})\times _{C_{(k-1)/}(x_1,\ldots , x_k)} C_{(k-2)/}(x_1,\ldots , x_{k-1}). $$ \end{parag} \begin{parag} \label{diag3.6} If $C$ is quasifibrant then the morphisms involved in the previous description are fibrations. We obtain the following result: that if $E'_i\subset E_i$ are trivial cofibrations and $C$ is quasifibrant then any diagram $$ \Upsilon ^k(E'_1,\ldots , E'_k)\rightarrow C $$ extends to a diagram $$ \Upsilon ^k(E'_1,\ldots , E'_k)\rightarrow C. $$ We can prove this by induction on $k$, and we are reduced exactly to the lifting property for the trivial cofibration $$ E'_1\times \ldots \times E'_k \hookrightarrow E_1\times \ldots E_k $$ along the morphism $$ C_{k/}(x_0,\ldots , x_k)\rightarrow C_{(k-1)/}(x_0,\ldots , x_{k-1})\times _{C_{(k-1)/}(x_1,\ldots , x_k)} C_{(k-2)/}(x_1,\ldots , x_{k-1}). $$ This morphism being fibrant by hypothesis, the lifting property holds. \end{parag} \begin{parag} \label{diag3.7} Suppose $C$ is quasifibrant. Then for $a,b$ fixed as in \ref{diag1.5} the morphisms $$ Diag ^{a,b}(E_1,\ldots , \underline{E_i},\ldots , E_k; C) \rightarrow Diag ^{a(i-1), b(i)}(\underline{E_i}; C) $$ are fibrant weak equivalences, where $a(i-1)$ and $b(i)$ are the images by $a$ and $b$ of the $i-1$-st and $i$-th vertices. To prove this we use the description \ref{diag3.5}, inductively reducing $k$. The remark \ref{quasifib1.4} says that for any $k\geq 2$ the choice of lifting $\tilde{u}$ doesn't change the equivalence type of the $Diag$ $n$-category. This reduces down to the case $k=1$, which gives exactly that the restriction to the $i$-th edge is an equivalence (the restrictions to the other edges are fixed and don't contribute anything because we fix $a,b$). \end{parag} \begin{parag} \label{diag4} {\em Proofs of \ref{diag2} and \ref{diag3}:} The statements of \ref{diag2} are direct consequences of the lifting property \ref{diag3.6}. To prove \ref{diag3}, in view of \ref{diag3.7} it suffices to consider the case $k=1$. Now $$ Diag ^{U,V}(\underline{E}; C) = \underline{Hom}(E, C_{1/}(U,V)). $$ Therefore if $C\rightarrow C'$ is any morphism of quasifibrant $n+1$-categories which is ``fully faithful'' i.e. induces equivalences of fibrant $n$-categories $$ C_{1/}(U,V)\rightarrow C'_{1/}(U,V), $$ then $$ Diag ^{U,V}(\underline{E}; C)\rightarrow Diag ^{U,V}(\underline{E}; C') $$ are equivalences by \ref{internal}. (Note by \ref{quasifib1.5} that the equivalence $C\rightarrow C'$ to a fibrant $C'$ that occurs in the hypothesis of \ref{diag3} is, in particular, a fully faithful morphism of quasifibrant $n+1$-categories.) This proves \ref{diag3} for $k=1$ and hence by \ref{diag3.7} for any $k$. \end{parag} \begin{parag} \label{diag5} The hypotheses on $C\rightarrow C'$ used in \ref{diag2} and \ref{diag3} are satisfied by $nCAT \rightarrow nCAT'$, cf \ref{quasifib2}. Therefore we may apply the results \ref{diag2} and \ref{diag3} to $nCAT \rightarrow nCAT'$. Fix $a,b$ as in \ref{diag1.5} for the following $Diag$'s, and suppose that the restriction of $a$ to $\Upsilon (B)$ is equal to $1_B$. From \ref{diag2} and \ref{diag3}, the morphism $$ Diag ^{a,b}(B,E_2,\ldots , \underline{E_i}, \ldots , E_k; nCAT) \rightarrow Diag ^{a,b}(B,E_2,\ldots , \underline{E_i}, \ldots , E_k; nCAT') $$ is an equivalence between fibrant $n$-categories. \end{parag} \bigskip \subnumero{Some extensions} \begin{parag} \label{k23} In the following preliminary statements we fix $k\geq 2$. We will only use these statements for $k=2,3$. \end{parag} \begin{parag} \label{def1B} We now describe what will be our main technical tool. Suppose $B$ is a fibrant $n$-category. We have a natural morphism $$ 1_B \in Hom ^{\ast , B}(\Upsilon B, nCAT') $$ coming from the identity morphism $\ast \times B \rightarrow B$ in $nCAT$ (which is then considered as a morphism in $nCAT'$). \end{parag} \begin{parag} \label{shl} For any $E_1,E_2, \ldots , E_k$ let $$ Shell\Upsilon ^k(E_1, \ldots , E_k):= $$ $$ \Upsilon ^{k-1}(E_1,\ldots , E_{k-1}) \cup \bigcup _{i=1}^{k-1}\Upsilon ^{k-1}(\ldots , E_i\times E_{i+1}, \ldots ) $$ (thus it consists of all of the ``faces'' except the one $\Upsilon ^{k-1}(E_2,\ldots , E_k)$). We have a cofibration $$ Shell\Upsilon ^k(E_1,\ldots , E_k)\rightarrow \Upsilon ^k(E_1,\ldots , E_k). $$ \end{parag} \begin{parag} Now set $E_1=B$ and let $\underline{Hom}^{1_B}(\Upsilon ^k(B, E_2,\ldots , E_k), nCAT')$ denote the fiber of $$ \underline{Hom}(\Upsilon ^k(1_B, E_2,\ldots , E_k),nCAT')\rightarrow \underline{Hom}(\Upsilon B, nCAT') $$ over $1_B$ and let $\underline{Hom}^{1_B}(Shell\Upsilon ^k(B, E_2,\ldots , E_k), nCAT')$ denote the fiber of $$ \underline{Hom}(Shell\Upsilon ^k(1_B, E_2,\ldots , E_k),nCAT')\rightarrow \underline{Hom}(\Upsilon B, nCAT') $$ over $1_B$. \end{parag} \begin{lemma} \label{extension1} Suppose $B$ is a fibrant $n$-category and $E$ an $n$-precat, and $U$ an object of $nCAT'$ (it is also an object of $nCAT$). The morphism $$ Diag ^{1_B,U}(B,\underline{E}; nCAT')\rightarrow Diag ^{\ast ,U}(\underline{B\times E}, nCAT') $$ is an equivalence of $n$-categories. \end{lemma} {\em Proof:} In view of \ref{diag2} and \ref{diag3} it suffices to prove the same thing for diagrams in $nCAT$. In this case, use the calculation of \ref{diag1.6}: both sides become equal to $\underline{Hom}(B\times E, U)$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{remark} \label{itsfibrant} Since $nCAT'$ is a fibrant $n+1$-category, the morphism $$ Diag ^{1_B,U}(B,\underline{E}; nCAT')\rightarrow Diag ^{\ast ,U}(\underline{B\times E}, nCAT') $$ is fibrant. One checks directly the lifting property for a trivial cofibration $F'\hookrightarrow F$, using the fibrant property of $nCAT'$. \end{remark} \begin{corollary} \label{extension2} The morphism $$ Hom ^{1_B}(\Upsilon ^2(B,E),nCAT')\rightarrow Hom ^{\ast}(\Upsilon (B\times E), nCAT') $$ is surjective. \end{corollary} {\em Proof:} We can fix an object $U$ for the image of the last vertex. The morphism $$ Diag ^{1_B,U}(B,\underline{E}; nCAT')\rightarrow Diag ^{\ast ,U}(\underline{B\times E}, nCAT') $$ is fibrant by the above remark \ref{itsfibrant}, and it is an equivalence by \ref{extension1}. This implies that it is surjective on objects (\ref{surje}). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{extension3} Suppose $E'\subset E$ is a cofibration of $n$-precats. Suppose we are given an object of $$ Hom ^{1_B}(\Upsilon ^2(B, E'), nCAT'). $$ and an extension over the shell to an object of $$ Hom ^{1_B}(Shell \Upsilon ^2(B,E), nCAT'). $$ Then these two have a common extension to an element of $$ Hom ^{1_B}(\Upsilon ^2(B, E), nCAT'). $$ \end{corollary} {\em Proof:} Again we can fix $U$. By Lemma \ref{extension1} and remark \ref{itsfibrant}, the morphism $$ Diag ^{1_B,U}(B,\underline{\ast}; nCAT')\rightarrow Diag ^{\ast ,U}(\underline{B}, nCAT') $$ is a trivial fibration. Therefore it has the lifting property with respect to any cofibration $E'\subset E$. This lifting property gives exactly what we want to show---this is because a morphism $$ E\rightarrow Diag ^{1_B,U}(B,\underline{\ast}; nCAT') $$ is the same thing as an object of $$ Diag ^{1_B,U}(B,E; nCAT') $$ or equivalently an element of $Hom ^{1_B}(\Upsilon ^2(B, E), nCAT')$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} Now we treat a similar type of extension problem for shells with $k=3$. \begin{parag} \label{extension4} Now suppose we have an object $b\in Diag (F; nCAT')$. Let $$ Diag ^{1_B, b}_{\rm Shell}(B,\underline{E}, F; nCAT') $$ be the $n$-precat representing the functor $$ G\mapsto Hom ^{1_B, b} (Shell \Upsilon ^3(B,E\times G,F), nCAT') $$ where the superscript on the $Hom$ has the obvious meaning that we look only at morphisms restricting to $1_B$ on the edge $01$ and to $b$ on the edge $23$. The shell $Shell \Upsilon ^3(B,E,F\times G)$ has three faces. We call the faces $(013)$ and $(023)$ the {\em last faces} and the face $(012)$ the {\em first face}. Restriction to the last faces (which meet along the edge $(03)$) gives a map $$ Diag ^{1_B, b}_{\rm Shell}(B,\underline{E}, F; nCAT') \rightarrow $$ $$ Diag ^{1_B, b_3}(B, \underline{E\times F}; nCAT') \times _{Diag ^{\ast , b_3} (\underline{B\times E \times F}; nCAT')} Diag ^{\ast , b}(\underline{B\times E}, F; nCAT'), $$ where $b_3$ denotes the object image of $3$ under the map $b$; a similar definition will hold for $b_2$ below---and recall that the image of $0$ under the map $1_B$ is $\ast$. \end{parag} \begin{parag} \label{extension5} {\em Claim:} that the map at the end of the previous paragraph is a fibrant equivalence. Call the object on the right in this morphism $D$. Restriction to the edge $(02)$ is a map $$ D\rightarrow Diag ^{\ast , b_2} (\underline{B\times E}; nCAT'). $$ We have an isomorphism $$ Diag ^{1_B, b}_{\rm Shell}(B,\underline{E}, F; nCAT') \stackrel{\cong}{\rightarrow} D \times _{Diag ^{\ast , b_2} (\underline{B\times E}; nCAT')} Diag ^{1_B, b_2}(B, \underline{E}; nCAT'). $$ However, the second morphism in this fiber product is $$ Diag ^{1_B, b_2}(B, \underline{E}; nCAT') \rightarrow Diag ^{\ast , b_2} (\underline{B\times E}; nCAT') $$ which is a fibrant equivalence by Lemma \ref{extension1}. It follows that the morphism $$ Diag ^{1_B, b}_{\rm Shell}(B,\underline{E}, F; nCAT') \rightarrow D $$ is a weak equivalence (note also that it is fibrant). This proves the claim. \end{parag} \begin{corollary} \label{extension6} The morphism $$ Diag ^{1_B, b}(B,\underline{E}, F; nCAT') \rightarrow Diag ^{1_B, b}_{\rm Shell}(B,\underline{E}, F; nCAT') $$ is a fibrant equivalence. \end{corollary} {\em Proof:} It is fibrant because $nCAT'$ is fibrant. In view of the claim \ref{extension5} it suffices to note that the map $$ Diag ^{1_B, b}(B,\underline{E}, F; nCAT') \rightarrow D $$ is an equivalence, and this by the fibrant property of $nCAT'$ (the union of the faces $(013)$ and $(023)$ is one of the admissible ones in our list of \ref{trivinclusions}). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{extension7} Suppose $E'\subset E$ is a cofibration of $n$-precats. Then for any morphism $$ \Upsilon ^3(B, E', F)\rightarrow nCAT' $$ sending the edge $(01)$ to $1_B$, and any extension of this over the shell to a morphism $$ Shell \Upsilon ^3(B, E, F)\rightarrow nCAT' $$ again restricting to $1_B$ on the edge $(01)$, there exists a common extension to a morphism $$ \Upsilon ^3(B, E, F)\rightarrow nCAT'. $$ \end{corollary} {\em Proof:} By the previous Corollary \ref{extension6} the morphism $$ Diag ^{1_B, b}(B,\underline{\ast}, F; nCAT') \rightarrow Diag ^{1_B, b}_{\rm Shell}(B,\underline{\ast}, F; nCAT'), $$ is a fibrant equivalence. Therefore it satisfies the lifting property for any cofibration $E' \hookrightarrow E$, and as before (\ref{extension3}) a map from $E$ into $Diag ^{1_B, b}(B,\underline{\ast}, F; nCAT')$ is the same thing as an object of $Diag ^{1_B, b}(B,\underline{E}, F; nCAT')$ (and the same things for $E'$ and for $Diag ^{1_B, b}_{\rm Shell}$). This gives the required statement. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \bigskip \subnumero{Extension properties for internal $\underline{Hom}$} Now we take the above extension properties and recast them in terms of internal $\underline{Hom}$. This is because we will need them for products of our precats $\Upsilon$ with an arbitrary $A$. Note that there is a difference between the internal $\underline{Hom}$ refered to in this section (which are $n+1$-categories) and the $Diag$ $n$-categories above. We state the following lemma for any value of $k$ but we will only use $k=2$ and $k=3$; and we give the proofs only in these cases, leaving it to the reader to fill in the combinatorial details for arbitrary $k$. \begin{lemma} \label{main} For any $n$-precats $E_2,\ldots , E_k$, the morphism $$ \underline{Hom}^{1_B}(\Upsilon ^k(B, E_2,\ldots , E_k), nCAT') \rightarrow \underline{Hom}^{1_B}(Shell\Upsilon ^k(B, E_2,\ldots , E_k), nCAT') $$ is an equivalence of $n+1$-categories. \end{lemma} {\em Proof:} The morphism in question is fibrant---cf \ref{internal}. The proof is divided into several paragraphs. In \ref{main1}---\ref{main3} we give the proof for $k=2$. Then in \ref{main4} we give the proof for $k=3$. \begin{parag} \label{main1} We begin the proof for $k=2$. Corollary \ref{extension2} implies that the morphism in question $$ \underline{Hom}^{1_B}(\Upsilon ^2(B, E_2), nCAT')\rightarrow \underline{Hom}^{\ast}(\Upsilon (B\times E_2), nCAT') $$ is surjective on objects. \end{parag} \begin{parag} \label{main2} Now we have to prove that our morphism induces equivalences between the morphism $n$-categories. Suppose $$ f,g: \Upsilon ^2(B, E_2)\rightarrow nCAT' $$ are two morphisms (with the appropriate behavior on $(01)$). Then the $n$-category of morphisms between them represents the functor $$ F\mapsto Hom ^{f,g; 1_B}(\Upsilon F \times \Upsilon ^2(B, E_2), nCAT') $$ where the superscript means morphisms restricting to $f$ and $g$ over $0,1\in \Upsilon F$ and restricting to $1_B$ over $\Upsilon F \times \Upsilon B$. This maps (by restricting to the edge $02$) to the functor $$ F\mapsto Hom ^{f,g; \ast}(\Upsilon F \times \Upsilon (B\times E_2), nCAT'). $$ We would like to prove that this restriction map of functors is an equivalence. In order to prove this it suffices to prove that it has the lifting property for any cofibrations $F'\subset F$. Thus we suppose that we have a morphism $$ \eta : \Upsilon F \times \Upsilon (B\times E_2)\rightarrow nCAT' $$ (restricting appropriately to $f$ and $g$ and to $\ast$), as well as a morphism $$ \zeta ' : \Upsilon F' \times \Upsilon ^2(B, E_2)\rightarrow nCAT', $$ restricting appropriately to $f$, $g$ and $1_B$. We would like to extend this latter to a map defined on $F$ and compatible with the previous one. This extension will complete the proof for $k=2$. \end{parag} \begin{parag} \label{main3} We now prove the extension statement claimed above. As in \ref{pfvar4} we consider the diagram as the product of an interval $(01)$ and a triangle $(012)$ and we denote the points by $(i,j)$ for $i=0,1$ and $j=0,1,2$. More generally for example $(ab,cd)$ denotes the square which is the edge $(ab)$ crossed with the edge $(cd)$. We are provided with maps on the end triangles $f$ on $(0, 012)$ and $g$ on $(1,012)$ as well as $\eta $ on the top square $(01, 02)$. We fix the map on the square $(01,01)$ (which is $\Upsilon (F) \times \Upsilon (B)$ pullback of $1_B$, and call this again $1_B$. We are also provided with a map $\zeta '$ defined on the whole diagram with respect to $F'$ and we would like to extend this all to $\zeta$ defined on the whole diagram. Note that we can write $\Upsilon (F) \times \Upsilon ^2(B,E)$ as the coproduct of three tetrahedra which we denote $$ (0,0)\;\; (1,0) \;\; (1,1) \;\; (1,2) \;\; i.e. \;\; \Upsilon ^3(F,B,E), $$ $$ (0,0)\;\; (0,1) \;\; (0,2) \;\; (1,2) \;\; i.e. \;\; \Upsilon ^3(B,E, F), $$ $$ (0,0)\;\; (0,1) \;\; (1,1) \;\; (1,2) \;\; i.e. \;\; \Upsilon ^3(B,F,E). $$ The first step is to use the fibrant property of $nCAT'$ to extend our given morphisms $g$, the restriction of $1_B$ to the triangle $(0,0)$, $(1,0)$, $(1,1)$, and the restriction of $\eta $ to the triangle $(0,0)$, $(1,0)$, $(1,2)$, to a map on the tetrahedron $$ (0,0)\;\; (1,0) \;\; (1,1) \;\; (1,2). $$ We can do this in a way which extends the map $\zeta '$. Next we again use the fibrant property of $nCAT'$ to extend across the tetrahedron $$ (0,0)\;\; (0,1) \;\; (0,2) \;\; (1,2). $$ Note that we are provided with the map $f$ on the triangle $(0,012)$, and the restriction of the $\eta$ on the triangle $(0,0), (0,2), (1,2)$. We can find our extension again in a way extending the given map $\zeta '$. Finally we come to the tetrahedron $$ (0,0)\;\; (0,1) \;\; (1,1) \;\; (1,2), $$ which is of the form $\Upsilon ^3(B, F,E)$. Here we are given maps on all of the faces except the last one, i.e. on the shell of this tetrahedron, and we would like to extend it. The given maps are the pullback of $1_B$ on the first face, and the maps coming from the two previous paragraphs on the other two faces. Furthermore we already have a map $\zeta '$ over the tetrahedron $\Upsilon ^3(B, F',E)$. The given map on the shell restricts on the first edge to $1_B$, so this is an extension problem of the type which we have already treated in Corollary \ref{extension7} above. (N.B. the notations $E$ and $F$ are interchanged between \ref{extension7} and the present situation.) Thus Corollary \ref{extension7} provides the extension we are looking for, and we have finished making our extension across the three tetrahedra. This completes the proof of ``fully faithfulness'' so the morphism in the lemma is an equivalence in the case $k=2$. \end{parag} \begin{parag} \label{main4} Here is the proof for $k=3$. First of all, the morphism $$ \underline{Hom}^{1_B}(\Upsilon ^3(B, E_2,E_3), nCAT')\rightarrow $$ $$ \underline{Hom}^{1_B}(\Upsilon ^{2}(B, E_2), nCAT') \times_{\underline{Hom}(\Upsilon (B\times E_2), nCAT')} \underline{Hom}(\Upsilon ^2(B\times E_2, E_3), nCAT') $$ is an equivalence, by the fibrant property for $nCAT'$. By the case $k=2$ (\ref{main1}--\ref{main3}) applied to the face $012$, the morphism $$ \underline{Hom}^{1_B}(Shell\Upsilon ^3(B, E_2,E_3), nCAT')\rightarrow $$ $$ \underline{Hom}^{1_B}(\Upsilon ^{2}(B, E_2), nCAT') \times_{\underline{Hom}(\Upsilon (B\times E_2), nCAT')} \underline{Hom}(\Upsilon ^2(B\times E_2, E_3), nCAT') $$ is an equivalence. This implies that the morphism $$ \underline{Hom}^{1_B}(\Upsilon ^3(B, E_2,E_3), nCAT')\rightarrow \underline{Hom}^{1_B}(Shell\Upsilon ^3(B, E_2,E_3), nCAT') $$ is an equivalence. This completes the case $k=3$. \end{parag} This completes the proof of the lemma (as far as we are going). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{remark} \label{main5} One might think that we have a simple argument for the case $k=3$, and the only difficult part of the argument for $k=2$ was the part where we used $k=3$. However one cannot simplify the proof: the simple argument for $k=3$ is based upon the use of internal $\underline{Hom}$ and to get $k=2$ for internal $\underline{Hom}$ we need a statement like the case of $k=3$---the statement which in the above proof is provided by Corollary \ref{extension7}. This is why we were obliged to do all of the stuff in the previous subsection. \end{remark} We will only use the subsequent corollaries in the cases $k=2$ and $k=3$, so the proof we have given of \ref{main} is sufficient. Again the reader is invited to treat the case of any $k$. \begin{corollary} \label{main6} Suppose $A$ is an $n+1$-precat and suppose $ E_i$ are $n$-precats for $i=2,\ldots , k$. Suppose we are given a morphism $$ A\times Shell\Upsilon ^k(B,E_2,\ldots , E_k)\rightarrow nCAT' $$ restricting to $1_B$ on $A\times \Upsilon B$. Then there is an extension to a morphism $$ A\times \Upsilon ^k(B,E_2,\ldots , E_k)\rightarrow nCAT'. $$ \end{corollary} {\em Proof:} The restriction morphism on the $\underline{Hom}$ is fibrant and an equivalence by the previous lemma, therefore it is surjective on objects. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} What we really need to know is a relative version of this for cofibrations $E'_i\subset E_i$. \begin{corollary} \label{main7} Suppose $A$ is an $n$-category and suppose $E'_i\subset E_i$ are cofibrations of $n$-precats for $i=2,\ldots , k$. Suppose we are given a morphism $$ A\times Shell\Upsilon ^k(B,E_2,\ldots , E_k)\rightarrow nCAT' $$ restricting to $1_B$ on $A\times \Upsilon B$, together with a filling-in $$ A\times \Upsilon ^k(B,E'_2,\ldots , E'_k)\rightarrow nCAT' , $$ then there is an extension of all of this to a morphism $$ A\times \Upsilon ^k(B,E_2,\ldots , E_k)\rightarrow nCAT'. $$ \end{corollary} {\em Proof:} Let $H_E$ denote the $\underline{Hom}$ for the full $\Upsilon$ and let $H^{Sh}_E$ denote the $\underline{Hom}$ for $Shell\Upsilon$. The morphism $$ H_E\rightarrow H^{Sh}_E \times _{H^{Sh}_{E'}}H_{E'} $$ is an equivalence (as is seen by applying the lemma for both $E$ and $E'$) and it is fibrant (since it comes from $\underline{Hom}$ applied to a cofibration). Therefore it is surjective on objects, which exactly means that we have the above extension property. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \bigskip \subnumero{Proof of Theorem \ref{inverse}} \begin{parag} \label{epsilon} Recall that $\lambda$ was defined in \ref{lambda}. We first apply the above statements to find our morphism $\epsilon :\lambda _A\rightarrow \varphi$. The universal property of $\lambda$ (\ref{lambda}) applied to the identity map $\lambda \rightarrow \lambda$ gives a morphism $$ \eta : A\times \Upsilon (\lambda ) \rightarrow nCAT' $$ sending $A\times \{ 0\} $ to $\ast _A$ and sending $A\times \{ 1\}$ to $\varphi$. By Corollary \ref{main6} (for $k=2$) there is a morphism $$ \epsilon ^{(2)} :A\times \Upsilon ^2(\lambda , \ast ) \rightarrow nCAT' $$ such that $$ r_{02}(\epsilon ^{(2)}) = \eta $$ and $$ r_{01}(\epsilon ^{(2)})=1_{\lambda} . $$ Note that $r_{12}(\epsilon ^{(2)})$ is a morphism from $A\times \Upsilon \ast = A\times I$ into $nCAT'$ restricting to $\lambda _A$ and $\varphi$, which by definition means a morphism $\lambda \rightarrow \varphi$. Call this morphism $\epsilon$. \end{parag} \begin{parag} \label{proofclaim} {\em Claim:} that for any fibrant $n$-category $B$ and any morphism $$ f:A\times \Upsilon E \rightarrow nCAT' $$ with $$ r_0(f)= B_A,\;\;\; r_1(f)=\varphi $$ there is a morphism $$ f':\Upsilon ^2(E,\ast ) \rightarrow nCAT' $$ with $$ r_{02}(f')= f,\;\;\; r_{12}(f')=\epsilon . $$ This almost gives the required property to show that $\lambda \stackrel{\epsilon}{\rightarrow }\varphi$ is an inverse limit. Technically speaking we also will have to show the above claim in the relative situation of $E'\subset E$. This we will do below (\ref{proofF}--\ref{proofEnd}) after first going through the argument in the absolute case (\ref{proofA}--\ref{proofE}). \end{parag} \begin{parag} \label{proofA} The basic idea is to use what we know up until now to construct a morphism $$ F: A\times \Upsilon ^3(B, E,\ast )\rightarrow nCAT' $$ with $$ r_{01}(F)= 1_B,\;\;\; r_{13}(F)= f, \;\;\; r_{23}(F) = \epsilon . $$ Setting $f'= r_{123}(F)$ we will obtain the morphism asked for in the previous paragraph. In order to follow the construction the reader is urged to draw a tetrahedron with vertices labeled $0,1,2,3$, putting respectively $B$, $E$, $\ast$, $B\times E$, $E$, $B\times E$ along the edges $01$, $12$, $23$, $02$, $13$, $03$; then putting in $\ast _A$, $B_A$, $\lambda _A$ and $\varphi$ at the vertices $0,1,2,3$ respectively. And finally putting in $1_B$ along edge $01$, $f$ along edge $13$ and $\epsilon$ along edge $23$. Our strategy is to fill in all of the faces except $123$, then call upon Corollary \ref{main6} to fill in the tetrahedron thus getting face $123$. \end{parag} \begin{parag} \label{proofB} The first step is the face $013$. This we fill in using simply the fact that $nCAT'$ is a fibrant $n+1$-category. The edges $01$ and $13$ are specified so we can fill in to a morphism $A\times \Upsilon ^2(B,E)\rightarrow nCAT'$ (restricting to $1_B$ and $f$ on the edges $01$ and $13$). Now the restriction of this morphism to edge $03$ provides a morphism $g:A\times \Upsilon (B\times E)\rightarrow nCAT'$ restricting to $\ast _A$ and $\varphi$. \end{parag} \begin{parag} \label{proofC} The next step is to notice that by the universal property \ref{lambda} of $\lambda$ there is a morphism $B\times E\rightarrow \lambda$ such that $g$ is deduced from $\eta$ by pullback via $\Upsilon (E\times B)\rightarrow \Upsilon (\lambda )$. This same morphism yields $$ \Upsilon ^2(E\times B, \ast )\rightarrow \Upsilon ^2(\lambda , \ast ), $$ and we can use this to pull back the above morphism $\epsilon ^{(2)}$. This gives a morphism $$ h: A \times \Upsilon ^2(E\times B, \ast ) \rightarrow nCAT' $$ where (adopting exceptionally for obvious reasons here the notations $0$, $2$ and $3$ for the vertices of this $\Upsilon ^2$) $$ r_{03}(h)= g, \;\;\; r_{23}(h)=\epsilon . $$ This treats the face $023$. \end{parag} \begin{parag} \label{proofD} Finally, for the face $012$ we have a morphism $$ r_{02}(h):A\times \Upsilon (E\times B)\rightarrow nCAT' $$ restricting to $\ast _A$ and $\lambda _A$. By Corollary \ref{main6} applied with $k=2$ (for the map $$ A\times Shell\Upsilon ^2(B, E)\rightarrow nCAT' $$ given by $1_B$ and $h$) we get a morphism $$ m: \Upsilon ^2(B,E)\rightarrow nCAT' $$ with $r_{01}(m)= 1_B$ and $r_{02}(m)= r_{02}(h)$. \end{parag} \begin{parag} \label{proofE} Putting all of these together we obtain a morphism $$ F' : A\times Shell \Upsilon ^3(B,E, \ast )\rightarrow nCAT' $$ restricting to $1_B$ on edge $01$ and restricting to $f$ on edge $13$ and $\epsilon$ on edge $23$. Corollary \ref{main6} applied with $k=3$ gives an extension over the tetrahedron to a morphism $$ F : A\times \Upsilon ^3(B,E, \ast )\rightarrow nCAT' $$ again restricting to $1_B$ on edge $01$ and restricting to $f$ on edge $13$ and $\epsilon$ on edge $23$. The restriction to the last face $r_{123}$ yields the filling-in desired. \end{parag} \begin{parag} \label{proofF} We now treat the case where $E'\subset E$ is a cofibration and where we already have a filling-in of the face $123$ for $E'$. We would like to obtain a filling-in of this face for $E$. Basically the only difficulty is that we don't yet know that the filling-in of face $123$ for $E'$ comes from a filling-in of the whole tetrahedron compatible with the above process. In particular this causes a problem at the step where we fill in face $023$. \end{parag} \begin{parag} \label{proofG} Before getting started we use the fibrant property of $nCAT'$ to obtain a morphism $$ A\times \Upsilon ^3(B,E', \ast )\rightarrow nCAT' $$ restricting to our given morphism on the face $123$, and restricting to $1_B$ on the edge $01$. Actually we would like to insure that the restriction to the face $012$ comes from a morphism $$ \Upsilon ^2(B,E')\rightarrow nCAT' $$ by pulling back along the projection $A\rightarrow \ast$. In order to do this notice that the restriction of the given map to the edge $12$ comes from $\Upsilon (E')\rightarrow nCAT'$. Thus we can first extend this map combined with $1_B$ to a morphism $\Upsilon ^2(B,E')\rightarrow nCAT'$. Now the morphism $$ (A\times \Upsilon ^2(B,E'))\cup ^{A\times \Upsilon E'} A\times \Upsilon ^2(E', \ast )\rightarrow \Upsilon ^3(B,E', \ast ) $$ is a trivial cofibration so we can extend from here to obtain $A\times \Upsilon ^3(B,E', \ast )\rightarrow nCAT'$ with restriction to the face $012$ coming from $\Upsilon ^2(B,E')\rightarrow nCAT'$. This is our point of departure for the rest of the argument. \end{parag} \begin{parag} \label{proofH} The first step following the previous outline is to fill in the face $013$. We note that the morphism $\Upsilon B \cup ^{\{ 1\} } \Upsilon E \rightarrow \Upsilon ^2(B,E)$ is a trivial cofibration. Thus also the morphism $$ (\Upsilon B \cup ^{\{ 1\} } \Upsilon E)\cup ^{\Upsilon B \cup ^{\{ 1\} } \Upsilon E'} \Upsilon ^2(B,E') \rightarrow \Upsilon ^2(B,E) $$ is a trivial cofibration, so given the edges $01$ and $13$ (for $E$) with filling-in over the face $013$ with respect to $E'$, we can fill in $013$ with respect to $E$. \end{parag} \subsubnumero{The face $023$} Now we treat the face $023$. Let $$ g: A\times \Upsilon (E\times B)\rightarrow nCAT' $$ be the restriction of the map obtained in \ref{proofH} to the edge $03$. Let $g'$ denote its restriction to $A\times \Upsilon (E'\times B)$. The map given in \ref{proofG} restricts on $(023)$ to a morphism $$ h': A \times \Upsilon ^2(E'\times B, \ast ) \rightarrow nCAT' $$ where (using as above the notations $0$, $2$ and $3$ for the vertices of this $\Upsilon ^2$) $$ r_{03}(h')= g', \;\;\; r_{23}(h')=\epsilon . $$ Let $a'=r_{02}(h')$. It is a morphism $$ a'_A:A\times \Upsilon (E'\times B)\rightarrow nCAT' $$ with $r_0(a'_A)= \ast _A$ and $r_2(a'_A)= \lambda _A$. By hypothesis on our map over the full tetrahedron for $E'$ (cf \ref{proofF}), $a'_A$ comes from a map $$ a':\Upsilon (E'\times B)\rightarrow nCAT' $$ again with values $\ast$ and $\lambda$ on the endpoints. This map corresponds to $$ E'\times B\rightarrow nCAT'_{1/}(\ast , \lambda ). $$ The morphism $nCAT\rightarrow nCAT'$ is an equivalence so our morphism is equivalent to a different morphism $b':E'\times B\rightarrow nCAT_{1/}(\ast , \lambda )$. These two resulting morphisms $\Upsilon (E'\times B)\rightarrow nCAT'$ are equivalent so by \ref{homotopic1}, \ref{homotopic2} there is a morphism $$ \overline{I}\times \Upsilon (E'\times B)\rightarrow nCAT' $$ sending the endpoints $0,1\in \overline{I}$ to $a'$ and $b'$. Using this different morphism $b'$ (which is now the same thing as a map $E'\times B\rightarrow \lambda )$ we pull back our standard $$ \eta \in \underline{Hom}(A\times \Upsilon ^2(\lambda , \ast ), nCAT') $$ to get a morphism $$ A\times \Upsilon ^2(E'\times B, \ast )\rightarrow nCAT' $$ restricting on the edges to $b'$ and $\epsilon$ respectively. Now we have a map from $$ \left( A\times \Upsilon ^2(E'\times B, \ast ) \right) \cup \left( A\times \overline{I}\times [\Upsilon (E'\times B)\cup \Upsilon (\ast )]\right) \cup \left( A \times \Upsilon ^2(E'\times B, \ast )\right) $$ to $nCAT'$, where the first term is glued to the second term along $1\in \overline{I}$ and the last term is glued to the second term along $0\in \overline{I}$ (we have omitted in the notation the $n+1$-precats along which the glueing takes place, the reader may fill them in as an exercise!). The morphism from the above domain to $$ A\times \overline{I} \times \Upsilon ^2(E'\times B, \ast ) $$ is a trivial cofibration, so since $nCAT'$ is fibrant there exists an extension of the above to a morphism $$ A\times \overline{I} \times \Upsilon ^2(E'\times B, \ast )\rightarrow nCAT'. $$ This morphism is a standard one coming from $b': E'\times B \rightarrow \lambda$ on the end $1\in \overline{I}$, and it is our given $h'$ on the end $0\in \overline{I}$. We now go to the edge $03$ of the triangle $023$. We are also given an extension of $g'$ to $g: A\times \Upsilon (E\times B)\rightarrow nCAT'$ along the edge $03$ of the triangle and $0$ of the interval $\overline{I}$. Thus, using the face $(03)\times \overline{I}$, we have a morphism $$ (A\times \Upsilon (E\times B))\cup ^{A\times \Upsilon (E'\times B)} (A\times \overline{I}\times \Upsilon (E'\times B)) \rightarrow nCAT'. $$ Fill this in along the trivial cofibration $$ (A\times \Upsilon (E\times B))\cup ^{A\times \Upsilon (E'\times B)} (A\times \overline{I}\times \Upsilon (E'\times B)) \hookrightarrow A\times \overline{I}\times \Upsilon (E\times B), $$ to give on the whole a morphism $$ A\times \overline{I}\times \Upsilon (E\times B) \cup ^{ A\times \overline{I}\times \Upsilon (E'\times B)} (A\times \overline{I}\times \Upsilon ^2(E'\times B , \ast ) \rightarrow nCAT', $$ where the morphism $\Upsilon (E'\times B)\rightarrow \Upsilon ^2(E'\times B,\ast )$ in question is the one coming from the edge $03$. Next we extend down along the triangle $023$ times the end $1\in \overline{I}$. To do this, notice that our extension from the previous paragraph gives an extension of the morphism $b': \Upsilon (E'\times B)\rightarrow nCAT'$ to a morphism $b: \Upsilon (E\times B)\rightarrow nCAT'$. By the universal property of $\lambda$ this corresponds to an extension $E\times B\rightarrow \lambda$. Now the morphism that we already have on the end $1\in \overline{I}$ comes by pulling back the standard $\eta : A\times \Upsilon ^2(\lambda , \ast )\rightarrow nCAT'$ via the map $E'\times B\rightarrow \lambda$ so our extension allows us to pull back $\eta$ to get a map $b: A\times \Upsilon ^2(E\times B, \ast )\rightarrow nCAT'$ extending the previous $b'$. Now we have our map $$ A\times \overline{I} \times \Upsilon ^2(E'\times B , \ast )\rightarrow nCAT' $$ which is provided with an extension from $E'$ to $E$, over the faces $(03)\times \overline{I}$ and $023 \times 1$ of the product of the triangle with the interval. Another small step is to notice that along the face $(02)\times \overline{I}$ the morphism is pulled back along $A\rightarrow \ast$ from a morphism $\overline{I} \times \Upsilon (E'\times B )\rightarrow nCAT'$. On the other hand at the edge $(02)\times \{ 1\}$ the extension from $E'$ to $E$ again comes from a morphism $\Upsilon (E\times B)\rightarrow nCAT'$. We get a morphism $$ \overline{I} \times \Upsilon (E'\times B) \times ^{\{ 1\} \times \Upsilon (E'\times B)} \Upsilon (E\times B) \rightarrow nCAT', $$ which can be extended along the trivial cofibration $$ \overline{I} \times \Upsilon (E'\times B) \times ^{\{ 1\} \times \Upsilon (E'\times B)} \Upsilon (E\times B) \hookrightarrow \overline{I}\times \Upsilon (E\times B) $$ to give a map $$ \overline{I}\times \Upsilon (E\times B) \rightarrow nCAT'. $$ Similarly we note that the map on the face $(23) \times \overline{I}$ is pulled back from our map $\epsilon : \Upsilon (\ast )\rightarrow nCAT'$. All together on the triangular icosahedron $(023) \times \overline{I}$ we have a morphism defined for $E'$ plus, along the faces $$ (03)\times \overline{I}, \;\; (02)\times \overline{I}, \;\; (23)\times \overline{I}, \;\; (023)\times \{ 1\} $$ extensions from $E'$ to $E$ (all compatible on intersections of the faces and having the required properties along $02$ and $23$). The inclusion of this $n+1$-precat (which we will call ${\bf G}$ for ``gory'' instead of writing it out) into $$ A\times \overline{I} \times \Upsilon ^2(E\times B , \ast ) $$ is a trivial cofibration. Indeed ${\bf G}$ comes by attaching to the end $$ A \times \{ 1\} \times \Upsilon ^2(E\times B , \ast ), $$ something of the form $$ A\times \partial \Upsilon ^2(E\times B, \ast ) \cup ^{A\times \partial \Upsilon ^2(E'\times B, \ast )} A\times \Upsilon ^2(E'\times B, \ast ) $$ where $\partial \Upsilon ^2(E\times B, \ast )$ denotes the coproduct of the three ``edges'' $\Upsilon (E\times B)$ (two times) and $\Upsilon (\ast )$. The inclusion of the end $A \times \{ 1\} \times \Upsilon ^2(E\times B , \ast )$ into ${\bf G}$ is an equivalence, as is the inclusion of this end into the full product $$ A\times \overline{I} \times \Upsilon ^2(E\times B , \ast ), $$ which proves that the map in question (from ${\bf G}$ to the above full product) is a weak equivalence (and it is obviously a cofibration). Now we again make use of the fibrant property to extend our map from ${\bf G}$ to a morphism $$ A\times \overline{I} \times \Upsilon ^2(E\times B , \ast )\rightarrow nCAT'. $$ When restricted to $A\times \{ 0\} \times \Upsilon ^2(E\times B , \ast )$ this gives the extension $h$ desired in order to complete our treatment of the face $023$. \begin{parag} For the face $012$ the argument is the same as in the previous case but we apply Corollary \ref{main7} rather than \ref{main6} in view of our relative situation $E'\subset E$. \end{parag} \begin{parag} \label{proofEnd} {\em End of the proof of \ref{inverse}} We have constructed a morphism $$ F' : A\times Shell \Upsilon ^3(B,E, \ast )\rightarrow nCAT' $$ restricting to $1_B$ on edge $01$ and restricting to $f$ on edge $13$ and $\epsilon$ on edge $23$. Furthermore, by construction it restricts to our already-given morphism over $E'$. Corollary \ref{main7} applied with $k=3$ gives an extension over the tetrahedron to a morphism $$ F : A\times \Upsilon ^3(B,E, \ast )\rightarrow nCAT' $$ again restricting to $1_B$ on edge $01$ and restricting to $f$ on edge $13$ and $\epsilon$ on edge $23$, and restricting to the already-given morphism over $E'$. The restriction to the last face $r_{123}(F)$ yields the filling-in desired. This completes the proof that $\lambda \stackrel{\epsilon}{\rightarrow }\varphi$ is an inverse limit, finishing the proof of Theorem \ref{inverse}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \end{parag} \begin{corollary} \label{commutencat} If $F:A\times B\rightarrow nCAT'$ is a functor from the product of two $n+1$-categories, then taking the inverse limits first in one direction and then in the other, is independent of which direction is chosen first. \end{corollary} {\em Proof:} This is a consequence of Theorem \ref{commute} but can also be seen directly from the construction \ref{lambda} of the limit. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Direct limits} \begin{theorem} \label{direct} The $n+1$-category $nCAT'$ admits direct limits. \end{theorem} One should probably be able to construct these direct limits in much the same way as in the topological case, roughly speaking by replacing a family by an equivalent one in which the morphisms are cofibrations (some type of telescope construction) and then taking the direct limit of $n$-precats in the usual sense. This seems a bit complicated to put into practice so we will avoid doing so by a trick. \begin{parag} \label{arg1cat} {\em The argument for a $1$-category:} Consider the following argument which shows that if $C$ is a category in which all inverse limits exist and in which projectors are effective, then $C$ admits direct limits. For a functor $\psi :A\rightarrow C$ let $D$ be the category whose objects are pairs $(c,u)$ where $c$ is an object of $C$ and $u: \psi \rightarrow c$ is a morphism. There is a forgetful functor $f:D\rightarrow C$. Let $\delta \in C$ be the inverse limit of $f$. Then for any $a\in A$ there is a unique morphism $\psi (a)\rightarrow f$. By the inverse limit property this yields a morphism $\psi (a)\rightarrow \delta$ and uniqueness implies that it is functorial in $a$. Thus we get a morphism $v: \psi \rightarrow \delta$ and $(\delta , v)$ is in $D$. As an object of $D$, $\delta$ has a morphism $$ p:=v(\delta ): \delta \rightarrow \delta. $$ This is itself a morphism in $D$, so we get $p\circ p = p$ from naturality of $v$. Thus $p$ is a projector. Let $t$ be the direct factor of $\delta$ given by $p$. Composition $\psi \rightarrow \delta \rightarrow t$ gives a map $\psi \rightarrow t$ and we get a factorization $\psi \rightarrow t \rightarrow \delta$. Now $t$ is seen to be an initial object of $D$, hence $\psi \rightarrow t$ is a direct limit. \end{parag} \begin{parag} \label{problem} The only problem with this argument is a set-theoretic one. Namely, when one speaks of ``limits'' it is presupposed that the indexing category $A$ is small, i.e. is a set of some cardinality rather than a class. However our category $C$ is likely to be a class. Thus, in the above argument, $D$ is not small and we are not allowed to take the inverse limit over $D$. \end{parag} \begin{parag} \label{fixup} Let's see how to fix this up in the case $C=Set$ is the category of sets. Suppose we have a functor $\psi : A\rightarrow Set$ from a small category $A$. Let $\alpha$ be a cardinal number bigger than $|A|$ and bigger than the cardinal of any set in the image of $\psi$. Let $D_{\alpha}$ be the category of pairs $(c,u)$ as above where $c$ is contained in a fixed set of cardinality $\alpha$. Note that $D_{\alpha}$ has cardinality $\leq 2^{\alpha}$. Let $(\delta , v)$ be as above. The only hitch is that (since we know an expression of $\delta$ as a subset of certain types of functions on $D_{\alpha}$ with values in the parametrizing sets which themselves have cardinality $\leq \alpha$) the cardinality of $\delta$ seems {\em a priori} only to be bounded by $2^{2^{\alpha}}$. Let $\delta ' \subset \delta $ be the smallest subset through which the map $v: \psi \rightarrow \delta$ factors. Note that the cardinality of $\delta '$ cannot be bigger than the sum of the cardinals of the $\psi (a)$ over $a\in A$, in particular $\delta '$ has cardinal $\leq \alpha$. But now the universal property of $\delta$ implies that $\delta = \delta '$, for it is easy to see that $\delta '\rightarrow f$ is again an inverse limit. Thus by actually counting we see that the cardinality of $\delta$ is really $\leq \alpha$ and up to isomorphism we may assume that $(\delta , v)\in D_{\alpha}$. This argument actually shows that the cardinality of $\delta$ is bounded independantly of the choice of $\alpha$. Thus $\delta$ satisfies the universal property of a direct limit for morphisms to a set of any cardinality, so $\delta$ is the direct limit of $A$. More generally, in the situation of \ref{arg1cat} if we can define the $D_{\alpha}$ and if we know for some reason that every object $B\in D$ admits a map $B'\rightarrow D$ from an object $D'\in D_{\alpha}$ then we can fix up the argument. \end{parag} We would like to do the same thing for limits in $nCAT'$, namely show that direct limits exist just using a general argument working from the existence of inverse limits. In order to do this we first need to discuss cardinality questions for $n$-categories. \bigskip \subnumero{Cardinality} Suppose $A$ is an $n$-category. We define the {\em cardinal of $A$}, denoted $\# A$ in the following way. Choose for every $y\in \pi _0(A)=\tau _{\leq 0}(A)$ (the set of equivalence classes of objects) a lifting to an object $\tilde{y}\in A_0$. Then $$ \# A := \sum _{y,z\in \pi _0(A)}\# A_{1/}(\tilde{y},\tilde{z}). $$ The sum of cardinals is of course the cardinal of the disjoint union of representing sets. This definition is recursive, as what goes into the formula is the cardinal of the $n-1$-category $A_{1/}(\tilde{y},\tilde{z})$. At the start we define the cardinal of a $0$-category (i.e. a set) in the usual way. \begin{lemma} The above definition of $\# A$ doesn't depend on choice of representatives. If $A\rightarrow B$ is an equivalence of $n$-categories then $\# A = \# B$. \end{lemma} {\em Proof:} Left to the reader. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} An easier and more obvious notion is the {\em precardinality of $A$}. If $A$ is any $n$-precat we define (with the notations of \cite{nCAT}) $$ \# ^{\rm pre}A:= \sum _{M\in \Theta ^n} \# (A_M). $$ For infinite cardinalities the precardinal of $A$ is also the maximum of the cardinalities of the sets $A_M$. In any case note that the precardinality is infinite unless $A$ is empty. \begin{remark} Let $A\mapsto Cat(A)$ denote the operation of replacing an $n$-precat by the associated $n$-category. Then the precardinal of $Cat(A)$ is bounded by the maximum of $\omega$ and the precardinal of $A$. Similarly by the argument of (\cite{nCAT} \S 6, proof of CM5(1)), for any $n$-precat $A$ there is a replacement by a fibrant $n$-category $A\hookrightarrow A'$ with $$ \# ^{\rm pre}A ' \leq max (\omega , \# ^{\rm pre}A). $$ Actually since $ \# ^{\rm pre}A\geq \omega$ we can write more simply that $\# ^{\rm pre}A ' = \# ^{\rm pre}A$. \end{remark} Note trivially that $$ \# A \leq \# ^{\rm pre}A . $$ The following lemma gives a converse up to equivalence. \begin{lemma} Suppose $A$ is an $n$-category with $\# A\leq \alpha$ for an infinite cardinal $\alpha$. Then $A$ is equivalent to an $n$-category $A'$ of precardinality $\leq \alpha$. \end{lemma} {\em Proof:} Left to the reader. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \bigskip \subnumero{A criterion for direct limits in $nCAT'$} Before getting to the application of the theory of cardinality we give a criterion which simplifies the problem of finding direct limits in $nCAT'$. \begin{parag} \label{criterion1} For this section we need another type of universal morphism. Suppose $E$ and $B$ are $n$-precats, with $B$ fibrant. Then $\underline{Hom}(B,E)$ is fibrant and we have a canonical morphism $$ \underline{Hom}(B,E)\times E \rightarrow B. $$ This may be interpreted as an object $$ \nu \in Diag ^{\underline{Hom}(B,E), B}(E; nCAT) $$ which yields by composition with $nCAT \rightarrow nCAT'$ the element which we denote by the same symbol $$ \nu \in Diag ^{\underline{Hom}(B,E), B}(E; nCAT') $$ \end{parag} \begin{parag} \label{criterion2} The element $\nu$ has the following universal property: for any $n$-precat $F$ the morphism $$ Diag ^{U , \nu }(\underline{F}, E; nCAT') \rightarrow Diag ^{U , B }(\underline{F\times E}; nCAT') $$ is a fibrant equivalence of fibrant $n$-categories. To prove this note that the fibrant property comes from the fact that $nCAT'$ is fibrant. Note that both sides are fibrant by \ref{diag2}. The fact that it is an equivalence may be checked using diagrams in $nCAT$ rather than diagrams in $nCAT'$, according to \ref{diag3}. Using \ref{diag1.6} for diagrams in $nCAT$, both sides are equal to $$ \underline{Hom}(U\times F\times E, B) $$ where $U$ is the image of the first object $0\in \Upsilon ^2(F,E)$. This shows that the morphism is an equivalence. \end{parag} \begin{parag} \label{criterion3} As a corollary of the above, given a morphism $$ f:\Upsilon(F\times E) \rightarrow nCAT' $$ with image of the last vertex equal to $B$, there is an extension to a morphism $$ g:\Upsilon ^2(F , E)\rightarrow nCAT' $$ such that $r_{02}(g)=f$ and $r_{12}(f) = \nu$. Similarly if $E'\subset E$ and we are already given the extension $g'$ for $E'$ then we can assume that $g$ is compatible with $g'$. \end{parag} \begin{parag} \label{criterion3.5} We also have a version of this universal property for shell-extension in higher degree. This concerns the {\em right shell} $Shelr \Upsilon ^k$ (cf \ref{shl}). Suppose we are given a morphism $$ f: Shelr \Upsilon ^k(F_1,\ldots , F_{k-1}, E)\rightarrow nCAT' $$ such that $f$ restricts on the last edge to $\nu$. Then there is a filling-in to a morphism $$ g: \Upsilon ^k(F_1,\ldots , F_{k-1}, E)\rightarrow nCAT'. $$ If $g'$ is already given over $F'_1,\ldots , F'_{k-1}, E'$ then we can assume that $g$ is compatible with $g'$. This is the analogue of \ref{extension7} and the like. \end{parag} \begin{parag} \label{criterion4} The above property also works in a family. Given a morphism $$ f:A\times \Upsilon(F\times E) \rightarrow nCAT' $$ sending the last vertex to the constant object $B_A$, there is an extension to a morphism $$ g:A\times \Upsilon ^2(F , E)\rightarrow nCAT' $$ such that $r_{02}(g)=f$ and $r_{12}(f) = \nu _A$ is the morphism pulled back from $\nu$. Again if an extension $g'$ is already given on $E'\subset E$ then $g$ may be chosen compatibly with $g'$. Similarly there is a shell-extension property as in \ref{criterion3.5} in a family. For the proof one has to go through a procedure analogous to the passage from diagrams to internal $\underline{Hom}$ in \ref{main1}--\ref{main7}. This discussion of the universal morphism $\nu$ is parallel to the discussion of the discussion of the universal $1_B$, but with ``arrows reversed''. \end{parag} We now come to our simplified criterion for limits in $nCAT'$. \begin{parag} \label{caution} {\em Caution:} Note that the following lemma only applies as such to limits taken in $nCAT'$ and not in general to limits in an arbitrary $n+1$-category ${\cal C}$. The proof uses in an essential way the fact that the morphism objects for the ``category'' $nCAT'$ are $n$-categories which are also basically the same thing as the objects of $nCAT'$. Of course it is possible that the same techniques of proof might work in a limited other range of circumstances which are closely related to these. \end{parag} \begin{lemma} \label{easydirect} Suppose $A$ is an $n+1$-precat and $\psi : A \rightarrow nCAT'$ is a morphism. Suppose that $\epsilon : \psi \rightarrow \delta$ is a morphism to an object $\delta \in nCAT'$ having the following weak limit-like property: for any other morphism $f:\psi \rightarrow \mu$ there exists a morphism $g:\delta \rightarrow \mu$ such that the composition $g\epsilon $ (well defined up to homotopy) is homotopic to $f$; and furthermore that such a factorization is unique up to a (not necessarily unique) homotopy of the factorization. Then $\psi \stackrel{\epsilon}{\rightarrow} \delta $ is a direct limit. \end{lemma} {\em Proof:} First we explain more precisely what the existence and uniqueness of the factorization mean. Given an element $f\in Hom (\psi , \mu )$ there exists an element $g'\in Hom ^{\epsilon } (\psi , \delta , \mu )$ projecting via $r_{02}$ to a morphism equivalent to $f$. This equivalence may be measured in the $n$-category $Hom (\psi , \mu )$. Note that since $nCAT'$ is fibrant the projection $$ Hom ^{\epsilon } (\psi , \delta , \mu ) \rightarrow Hom (\psi , \mu ) $$ is fibrant, so if an object equivalent to $f$ is in the image then $f$ is in the image. Thus we can restate the criterion as saying simply that there exists an element $g'$ projecting via $r_{02}$ to $f$. Suppose given two such factorizations $g'_1$ and $g'_2$. By ``homotopy of the factorization'' we mean a homotopy between $r_{12}(g'_1)$ and $r_{12}(g'_2)$ such that the resulting homotopy between $f$ and itself (this homotopy being well defined up to $2$-homotopy) is $2$-homotopic to the identity $1_f$. Again using the fibrant condition of $nCAT'$ we obtain that this condition implies the simpler statement that there exists a morphism $$ A\times \Upsilon ^2(\ast , \ast ) \times \overline{I}\rightarrow nCAT' $$ restricting to $g'_1$ and $g'_2$ on the two endpoints $0,1\in \overline{I}$; restricting to the pullback $\epsilon$ on the edge $(01)$ of the $\Upsilon ^2$, this edge being $A\times \Upsilon (\ast )\times \overline{I}$; and restricting to the pullback of $f$ on the edge $(02)$ which is $A\times \Upsilon (\ast ) \times \overline{I}$. \begin{parag} \label{existen} {\em Simple factorization} We start by showing the simple version of the factorization property necessary to show that $\epsilon$ is an inverse limit; we will treat the relative case for $E'\subset E$ below.So for now, suppose that we are given a morphism $$ u:A \times \Upsilon (E) \rightarrow nCAT' $$ restricting to $\psi$ on $A\times \{ 0\} $ and restricting to a constant object $B\in nCAT'$ (i.e. to the pullback $B_A$) on $A\times \{ 1\}$. We would like to extend this to a morphism $$ v: A\times \Upsilon ^2(\ast , E) \rightarrow nCAT' $$ restricting to our given morphism on the edge $(02)$, and restricting to $\epsilon$ on the edge $(01)$. Our given morphism corresponds by \ref{criterion4} to a morphism $w:A\times \Upsilon (\ast )\rightarrow nCAT'$ restricting to $\psi$ on $A\times \{ 0\}$ and restricting to the constant object $\underline{Hom}(E,B)$ (pulled back to $A$) on $A\times \{ 1\}$. More precisely there is a morphism $$ w': A\times \Upsilon ^2(\ast , E)\rightarrow nCAT ' $$ restricting to $u$ over the edge $(02)$, and restricting to the universal morphism $\nu$ (cf \ref{criterion1}) over the edge $(12)$. The restriction to the edge $(01)$ is the morphism $w$. Now $w$ is an element of $Hom (\psi , B)$, so by hypothesis there is a diagram $$ g: A\times \Upsilon ^2(\ast , \ast )\rightarrow nCAT' $$ sending the edge $(01)$ to $\epsilon$ and sending the edge $(02)$ to $w$. Putting this together with the diagram $w'$ and using the fibrant property of $nCAT'$ (i.e. composing these together) we obtain existence of a diagram $$ A\times \Upsilon ^3(\ast , \ast , E)\rightarrow nCAT' $$ restricting to $g$ on the face $(012)$ and restricting to $w'$ on the face $(023)$. The face $(013)$ yields a diagram $$ A\times \Upsilon ^2(\ast , E)\rightarrow nCAT' $$ restricting to $\epsilon$ on the first edge and restricting to our original morphism $u$ on the edge $(03)$: this is the morphism $v$ we are looking for. \end{parag} \begin{parag} \label{uniquen} {\em Uniqueness of these factorizations} The homotopy uniqueness property for factorization of morphisms implies a similar property for the factorizations of $E$-morphisms obtained in the previous paragraph. Suppose that we are given a morphism $$ u:A \times \Upsilon (E) \rightarrow nCAT' $$ as above, and suppose that we are given two extensions $$ v_1, v_2: A\times \Upsilon ^2(\ast , E) \rightarrow nCAT' $$ restricting to our given morphism on the edge $(02)$, and restricting to $\epsilon$ on the edge $(01)$. We can complete the $v_i$ to diagrams $$ z_i: A\times \Upsilon ^3(\ast , \ast , E)\rightarrow nCAT' $$ restricting to $v_i$ on the faces $(013)$ and restricting to the universal morphism $\nu$ of \ref{criterion1} on the edge $(23)$. To do this, use the universal property of $\nu$ (cf \ref{criterion4}) to fill in the faces $(023)$ and $(123)$; then we have a map defined on the shell and by the universal property of $\nu$ which gives shell extension (\ref{criterion3.5}, \ref{criterion4}) we can extend to the whole tetrahedron. Note furthermore that we can assume that the restrictions to the faces $(023)$ are the same for $z_1$ and $z_2$ (because we have chosen these faces using only the map $u$ and not refering to the $v_i$). Call these $r_{023}(z)$. In particular the restrictions to $(02)$ give the same map $w:\psi \rightarrow \underline{Hom}(E,B)$. Now the restrictions of the above diagrams $z_i$ to the faces $(012)$ give two different factorizations of this map $w$ so by hypothesis there is a homotopy between these factorizations: it is a morphism $$ A \times \Upsilon ^2(\ast , \ast ) \times \overline{I} \rightarrow nCAT' $$ restricting to $r_{012}(z_i)$ on the endpoints $i=0,1$ of $\overline{I}$, restricting to the pullback of $\epsilon$ along $(01)\times \overline{I}$ and restricting to the pullback of our morphism $w$ along $(02)\times \overline{I}$. We can attach this homotopy to the constant homotopy which is the pullback of $r_{023}(z)$ from $A\times \Upsilon ^2(\ast , E)$ to $A\times \Upsilon ^2(\ast , E)\times \overline{I}$. We obtain a homotopy defined on the union of the faces $(012)$ and $(023)$ and going between $z_0$ and $z_1$. Using the fact that the inclusion of this union of faces into the tetrahedron is a trivial cofibration (see the list \ref{trivinclusions} above) we get that the inclusion (written in an obvious shorthand notation where $(0123)$ stands for $A \times \Upsilon ^3(\ast , \ast , E)$ and $(012 + 023)$ for the union of the two faces) $$ (0123) \times \{ 0\} \cup ^{(012 + 023)\times \{ 0\} } (012 + 023)\times \overline{I} \cup ^{(012 + 023)\times \{ 1\} } (0123) \times \{ 1\} \hookrightarrow (0123)\times \overline{I} $$ is a trivial cofibration. We have a map from the left side into $nCAT'$ so it extends to a map $$ A \times \Upsilon ^3(\ast , \ast , E) \times \overline{I} \rightarrow nCAT'. $$ The restriction of this map to the face $(013)$ is a homotopy $$ A \times \Upsilon ^2(\ast , E) \times \overline{I} \rightarrow nCAT' $$ between our factorizations $v_1$ and $v_2$. \end{parag} \subsubnumero{The relative case} To actually prove the lemma, we need to obtain a factorization property as above in the relative situation $E'\subset E$ where we already have the factorization over $E'$ and we would like to extend to $E$. This is where we use the homotopy uniqueness of factorization which was in the hypothesis of the lemma (we use it in the form given in the previous paragraph \ref{uniquen}). Suppose we are given $$ v': A\times \Upsilon ^2(\ast , E') \rightarrow nCAT' $$ restricting to $\epsilon$ on the first edge, and suppose we are given $$ u:A \times \Upsilon (E) \rightarrow nCAT' $$ restricting to $\psi$ on $A\times \{ 0\} $ and restricting to a constant object $B\in nCAT'$ (i.e. to the pullback $B_A$) on $A\times \{ 1\}$. Suppose that the restriction of $u$ to $A\times \Upsilon (E')$ is equal to the restriction of $v'$ to the edge $(02)$. By \ref{existen} there exists an extension $$ v_0: A\times \Upsilon ^2(\ast , E) \rightarrow nCAT' $$ which restricts to $\epsilon$ on the first edge and to $u$ on the edge $(02)$. Let $v'_0$ denote the restriction of $v_0$ to $A\times \Upsilon ^2(\ast , E')$. By the uniqueness statement \ref{uniquen} there exists a homotopy $$ A\rightarrow \Upsilon ^2(\ast , E') \times \overline{I} \rightarrow nCAT' $$ between $v'_0$ and $v'$, constant along edges $(01)$ and $(02)$. Let $D$ be the coproduct of $\Upsilon ^2(\ast , E')$ and $\Upsilon (E)$ with the latter attached along the edge $(02)$ (i.e. the coproduct is taken over the copy of $\Upsilon (E')\subset \Upsilon ^2(\ast , E')$ corresponding to the edge $(02)$). Our homotopy glues with the constant map $u$ to give a morphism $$ A \times D \times \overline{I} \rightarrow nCAT', $$ and this glues with $u$ to obtain $$ A\times \Upsilon ^2(\ast , E)\times \{ 0\} \cup ^{A \times D \times \{ 0\} } A\times D \times I \rightarrow nCAT'. $$ The inclusion $$ A \times D \times \{ 0\} \hookrightarrow A\times D \times I $$ is a trivial cofibration so the inclusion $$ A\times \Upsilon ^2(\ast , E)\times \{ 0\} \cup ^{A \times D \times \{ 0\} } A\times D \times I \hookrightarrow A\times \Upsilon ^2(\ast , E)\times \overline{I} $$ is a trivial cofibration and by the fibrant property of $nCAT'$ there exists an extension of the above morphism to a morphism $$ A\times \Upsilon ^2(\ast , E)\times \overline{I} \rightarrow nCAT'. $$ The value of this over $1\in \overline{I}$ is the extension $$ v: A\times \Upsilon ^2(\ast , E) \rightarrow nCAT' $$ we are looking for: it restricts to $\epsilon$ on the edge $(01)$, it restricts to $u$ on the edge $(02)$, and it restricts to $v'$ over $A\times \Upsilon ^2(\ast , E')$. This completes the proof of Lemma \ref{easydirect}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{remark} \label{easyinverse} One also obtains a criterion similar to \ref{easydirect} for inverse limits in $nCAT'$. The proof is the same as above but using the universal diagram $$ B \stackrel{E}{\rightarrow} B\times E $$ in the place of $\underline{Hom}(E,B)\stackrel{E}{\rightarrow} B$. We did not choose to use this in the proof of \ref{inverse} because it didn't seem to make any substantial savings (and in fact probably would have complicated the notation in many places). \end{remark} \begin{parag} \label{alpha} We now improve the above criterion with a view toward applying this in the argument \ref{fixup} given above for the case of sets. Fix a functor $\psi : A\rightarrow nCAT'$ of $n+1$-categories. Suppose $\alpha$ is an infinite cardinal number such that $$ \# A \leq \alpha $$ and $$ \# \psi (a) \leq \alpha $$ for all $a\in A$. \end{parag} \begin{parag} \label{factorization} Suppose $B\in nCAT'$ and suppose $u: \psi \rightarrow B$ is a morphism. Then we claim that there is $B'\in nCAT'$ with $\#^{\rm pre} B' \leq \alpha$, and with a factorization $\psi \rightarrow B' \rightarrow B$. To prove this notice that $u$ is a morphism $$ u:A\times I \rightarrow nCAT', $$ and the image of $u$ is contained in an $\alpha$-bounded set of additions of trivial cofibrations to $nCAT$ (recall that $nCAT'$ was constructed by adding pushouts along trivial cofibrations to $nCAT$ \ref{perspective1}). We can take $B'\subset B$ to be a sub-precat containing all of the objects necessary for the morphisms involved in the trivial cofibrations which are added in the previous paragraph, as well as the morphisms involved in $u$ given that we fix $\psi$, all of the objects necessary for the structural morphisms of a precat, and finally add on what is necessary to get $B'$ fibrant. This has cardinality $\# ^{\rm pre}B' \leq \alpha$. \end{parag} \bigskip \subnumero{A construction} \begin{parag} \label{hypothesis} {\em Hypothesis---} With the above notations, suppose we have an object $U\in nCAT'$ together with a morphism $a: \psi \rightarrow U$ provided with the following data: \newline (A)---for every morphism $\psi \rightarrow V$ where $\# ^{\rm pre} V\leq \alpha$, a factorization which we call the {\em official factorization} $$ \psi \stackrel{a}{\rightarrow} U \rightarrow V $$ (in other words a diagram $$ A\times \Upsilon ^2(\ast , \ast ) \rightarrow nCAT' $$ restricting to $a$ on the edge $(01)$ and restricting to our given morphism on the edge $(02)$); \newline (B)---for every diagram $$ \psi \rightarrow V\rightarrow V', $$ a completion of this and the official factorization diagrams $$ \psi \rightarrow U\rightarrow V,\;\; \psi \rightarrow U\rightarrow V' $$ to a diagram (the {\em official commutativity diagram}) $$ \psi \rightarrow U \rightarrow V \rightarrow V' $$ (which again means a morphism $$ A\times \Upsilon ^3(\ast , \ast ,\ast )\rightarrow nCAT' $$ restricting to our given diagrams on the faces $(023)$, $(012)$, and $(013)$). \end{parag} \begin{parag} \label{construction1} Keep the above hypothesis \ref{hypothesis}. Let $$ [b,i]:\psi \stackrel{b}{\rightarrow} U' \stackrel{i}{\rightarrow} U $$ be a factorization of the morphism $a$ as above (\ref{factorization}) with $\# U' \leq \alpha$. This means a diagram whose third edge $(02)$ is equal to $a$. Unfortunately at this point we have no control over the choice of $U'$, so the ``real'' $U'$ which we would like to choose to satisfy the criterion of \ref{easydirect} may be a direct factor of this $U'$. To explain this notice that by hypothesis \ref{hypothesis} (A) there is a morphism $$ q:U\rightarrow U' $$ giving a factorization $$ [a,q]:\psi \rightarrow U \rightarrow U'. $$ Let $b$ be the edge $(02)$ of this diagram, so we can write $b\sim qa$. Using the fibrant property of $nCAT'$ we can glue the diagrams $[b,i]$ and $[a,q]$ together to give a diagram $$ [b,i,q]: \psi \rightarrow U' \rightarrow U \rightarrow U', $$ in other words a morphism $$ A\times \Upsilon ^3(\ast , \ast , \ast ) \rightarrow nCAT' $$ restricting to $[b,i]$ on the face $(012)$ and restricting to $[a,q]$ on the face $(023)$ (and satisfying the usual condition that the restriction to the face $(123)$ be constant in the $A$ direction). Denote by $p$ the restriction to the edge $(13)$, and denote by $[b,p]$ the restriction to the face $(013)$. Thus $$ [b,p]: \psi \rightarrow U' \rightarrow U' $$ is a diagram whose restrictions to the edges $(01)$ and $(02)$ are both equal to the morphism $b$. Restriction to the face $(123)$ is a diagram $[i,q]$ with third edge equal to $p$, in other words we can write $p \sim q\circ i $. The official commutativity diagram for $[b,p]$ is a diagram of the form $$ [a,q,p]: \psi \rightarrow U \rightarrow U' \rightarrow U'. $$ The restriction of this diagram to the face $(023)$ is the diagram $[b,p]$. The restrictions to $(012)$ and $(013)$ are both equal (by hypothesis (B)) to the official factorization diagram $[a,q]$. In particular, the face $(123)$ gives a diagram $$ [q,p]: U\rightarrow U' \rightarrow U' $$ whose third edge (which we should here denote $(13)$) is again the morphism $p$. Homotopically we get an equation $$ p \circ q \sim q. $$ In view of the fact that $p\sim q \circ i$ we get $$ p\circ p \sim p. $$ This equation says that, up to homotopy, $p$ is a projector. It is the projector onto the answer that we are looking for. \end{parag} \begin{parag} \label{construction2} {\em Construction---} Continuing with hypothesis \ref{hypothesis} and the notations of \ref{construction1}, we will construct the object corresponding to the ``image'' of the homotopy projector $p$. To do this we will take the ``mapping telescope'' of the sequence $$ U' \stackrel{p}{\rightarrow } U' \stackrel{p}{\rightarrow } U' \stackrel{p}{\rightarrow}\ldots . $$ In the present setting of $n$-categories we do this as follows (which is basically just the mapping telescope in the closed model category structure of \cite{nCAT}). Recall that $\overline{I}$ is the $1$-category with two objects $0,1$ and two morphisms inverse to each other between the objects. We consider it as an $n$-category. Glue together the $n$-precats $U' \times \overline{I}$, one for each natural number, by attaching $U'\times \{ 1\}$ in the $i-1$-st copy to $U' \times \{ 0\}$ in the $i$-th copy via the map $p: U'\times \{ 1\} \rightarrow U' \times \{ 0\}$. Denote by $T'$ the resulting $n$-precat and by $T'\hookrightarrow T$ a fibrant replacement. Inclusion of $U' \times \{ 0\}$ in the first copy gives a morphism $$ j: U' \rightarrow T. $$ On the other hand, using the projection $p$ in each variable and the homotopy $p\circ p\sim p$ gives a morphism $$ r : T\rightarrow U' $$ (which comes by extension from a map $r':T'\rightarrow U'$) and we have $rj=p$. \end{parag} \begin{parag} \label{construction3} {\em Claim:} The morphism $jr : T \rightarrow T$ is homotopic to the identity, via a homotopy compatible with the homotopy $p\circ p \sim p$. This is by a classical construction that works in any closed model category with ``interval object'' such as $\overline{I}$. As a sketch of proof, let $T^{m}$ denote the subobject of $T'$ obtained by taking only the first $m$ copies of $U'\times \overline{I}$. Then $T^m$ retracts to the last copy of $U'$, so the restriction of $r'$ to $T^m$ is homotopic (via this retraction) to $p$. On the other hand, the inclusion $T^m \hookrightarrow T^{m+1}$ is also homotopic to $p$ (via the retractions to the end copies of $U'$). Thus we may choose a homotopy (in Quillen's sense cf \ref{homotopic1}) between the restriction of $r'$ to $T^m$, and the inclusion $T^m \hookrightarrow T^{m+1}$. We can make this into a homotopy between $$ r', 1_{T^m} : T^m \stackrel{\displaystyle \rightarrow}{\rightarrow} T, $$ and since $T$ is fibrant we can do this with a homotopy using the interval $\overline{I}$. Again because $T$ is fibrant we can assume that these homotopies are compatible for all $m$, so they glue together to give a homotopy between the two maps $$ r', 1_{T'} : T'\stackrel{\displaystyle \rightarrow}{\rightarrow} T. $$ Then extend from $T'$ to $T$. \end{parag} \begin{parag} \label{construction4} We wrap things up by pointing out how $T$ fits in with the situation of \ref{construction1}. Consider the sequence of morphisms $$ \psi \rightarrow U' \rightarrow T \rightarrow U' \rightarrow T. $$ The composition of the first two gives a morphism $jb:\psi \rightarrow T$. The composition of the first three morphisms is equal to $rjb\sim pb$ which has a homotopy to the usual morphism $b:\psi \rightarrow U'$. Thus the morphism $b$ factors through $T$. Finally from our claim \ref{construction3} the composition of the last two arrows is homotopic to the identity on $T$. Our original morphism $\psi \rightarrow U$ factors through $U'$ hence it factors through $T$: the composition $$ \psi \rightarrow T \rightarrow U' \rightarrow U $$ is equal to the original morphism $a: \psi \rightarrow U$. We have the morphism $$ jq: U\rightarrow T $$ providing a factorization $$ \psi \rightarrow U \rightarrow T. $$ The composition $T\rightarrow U \rightarrow T$ is homotopic to the identity on $T$ by claim \ref{construction3}. \end{parag} \begin{lemma} \label{construction5} Under hypothesis \ref{hypothesis} and with the above notations, the morphism $\psi \rightarrow T$ has the unique homotopy factorization property of \ref{easydirect} with respect to any morphism $\psi \rightarrow B$ (without bound on the cardinality of $B$). \end{lemma} {\em Proof:} This is really only a statement about $1$-categories. We can consider the $1$-category $M$ which is the truncation of the $n+1$-category of objects under $\psi$ (cf \ref{lazy} below). Our objects $U,U',T$ and so on togther with maps from $\psi$ may be considered as objects in the category $M$. The result of \ref{construction1} says that $p: U'\rightarrow U'$ is a projector in the category $M$, and in \ref{construction2}, \ref{construction3} and \ref{construction4} we show that the object $T$ corresponding to this projector exists. The criterion of \ref{easydirect} asks simply that $T$ be an initial object in $M$. What we know from hypothesis \ref{hypothesis} is that $T$ is provided with a collection of morphisms $T\rightarrow B'$ to every $\alpha$-bounded object of $M$, in such a way that these form a natural transformation from the constant functor $T$ to the identity functor $M_{\alpha}\rightarrow M$ (where $M_{\alpha}$ is the full subcategory of objects having cardinality bounded by $\alpha$). The fact that $T$ is the object corresponding to the projector $p$ (and that $p$ was the projector defined by the natural transformation for $U'$) means that the value of this natural transformation on $T$ itself is the identity. Suppose $\psi \rightarrow B$ is an object of $M$. Then there is a factorization through $\psi \rightarrow B'\rightarrow B$ with $\# ^{\rm pre}B' \leq \alpha$. This just says that every object of $M$ has a morphism from an object in $M_{\alpha}$. It is worth mentioning that if $B\in M$ and if $B'\rightarrow B$ and $B'' \rightarrow B$ are two morphisms from objects in $M_{\alpha}$ then they both factor through a common morphism $B'''\rightarrow B$ from an object in $M_{\alpha}$. Using the above formal properties, we show that $T$ is an initial object of $M$ to prove the lemma. Suppose $B$ is an object of $M$. There exists a morphism $B'\rightarrow B$ from an object of $M_{\alpha}$ so applying our natural transformation, there exists a morphism $T\rightarrow B'$ and hence a morphism $T\rightarrow B$. Suppose that $T\stackrel{\displaystyle \rightarrow}{\rightarrow} B$ is a pair of morphisms. These factor through a common object of $M_{\alpha}$ $$ T\stackrel{\displaystyle \rightarrow}{\rightarrow} B' \rightarrow B, $$ and applying our natural transformation we obtain that the compositions of the two morphisms $$ T\rightarrow T\stackrel{\displaystyle \rightarrow}{\rightarrow} B' $$ are equal to the given morphism $T\rightarrow B'$; however, since the natural transformation $T\rightarrow T$ is the identity, this implies that our two morphisms $T\stackrel{\displaystyle \rightarrow}{\rightarrow} B'$ were equal and hence that the two original morphisms $T\stackrel{\displaystyle \rightarrow}{\rightarrow} B$ were equal. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{construction6} In the situation of Lemma \ref{construction5} the map $\psi \rightarrow T$ is a direct limit. \end{corollary} {\em Proof:} By \ref{construction5} it satisfies the condition of \ref{easydirect} so by the latter, it is a direct limit. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \bigskip \subnumero{Proof of Theorem \ref{direct}} \begin{parag} \label{lazy} {\em Objects under $\psi$:} In order to replicate the proof that was given above for the category of sets, we need to know what the category of ``objects under $\psi$'' is. Suppose $C$ is an $n+1$-category and $A$ another $n+1$-category and suppose $\psi : A\rightarrow C$ is a morphism. We define the $n+1$-category $\psi /C$ of {\em objects under $\psi$} to be the category of morphisms $$ (A\times I)\cup ^{A\times \{ 1\} }\{ 1\}\rightarrow C $$ restricting to $\psi$ on $A\times \{ 0\}$. In other words, it is the fiber of the morphism $$ \underline{Hom}((A\times I)\cup ^{A\times \{ 1\} }\{ 1\},C)\rightarrow \underline{Hom}(A,C) $$ over $\psi$. \end{parag} \begin{parag} Let $(\psi /C)_{\alpha}$ denote the category of objects under $\psi$ which are (set-theoretically speaking) contained in a given fixed set of cardinality $\alpha$. It has cardinality $\leq 2^{\alpha}$. It is a full subcategory of $\psi/C$. \end{parag} \begin{parag} \label{reeasydirect} We can restate the criterion of \ref{easydirect} in terms of the above definition. Let $\tau _{\leq 1}(\psi /C)$ denote the $1$-truncation of the category of objects under $\psi$ defined in \ref{lazy}. It is a $1$-category. The criterion says that if $u: \psi \rightarrow U$ is an initial object in this category then it is (the image under the truncation operation of) a direct limit of $\psi$. Definition \ref{lazy} and the present remark were used in the proof of \ref{construction5} already, where we denoted $\psi /C$ by $M$. \end{parag} {\em Proof of Theorem \ref{direct}:} Suppose $\psi : A\rightarrow nCAT'$. Fix a cardinal $\alpha$ bounding $(A,\psi )$ as above. Let $M_{\alpha}:= (\psi /C)_{\alpha}$ denote the $n+1$-category of objects under $\psi$, of cardinality bounded by $\alpha$. Let $U$ be the inverse limit of the forgetful functor $f:M_{\alpha}\rightarrow nCAT'$, given by Theorem \ref{inverse}. By Corollary \ref{constant}, the pullback of $U$ to a constant family $U_A$ over $A$ is again an inverse limit of the functor $$ f_A: A\times M_{\alpha}\rightarrow nCAT' $$ ($f$ pulled back along the second projection to $M_{\alpha}$). We have a morphism of families over $A\times M_{\alpha}$, from $\psi$ to $f_A$, which thus factorizes into $$ \psi \rightarrow U_A \rightarrow f_A. $$ The morphism $\psi \rightarrow U_A$ is automatically provided with the data required for Hypothesis \ref{hypothesis}. Apply the above construction \ref{construction1}--\ref{construction4} to obtain $\psi \rightarrow T_A$, and Lemma \ref{construction5} and Corollary \ref{construction6} show that $\psi \rightarrow T_A$ is a direct limit of $\psi$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Applications} We will discuss several different possible applications for the notions of inverse and direct limit in $n$-categories in general, and of the existence of limits in $nCAT'$ in particular. Many of these applications are only proposed as conjectural. Only in the first section do we give full proofs. The conjectures are for the most part supposed to be possible to do with the present techniques, except possibly \ref{mts}. \bigskip \subnumero{Coproducts and fiber products} \begin{parag} \label{fiberproducts} Taking $A$ to be the category with three objects $a$, $b$ and $c$ and morphisms $a\rightarrow b$ and $c\rightarrow b$, a functor $A\rightarrow nCAT$ is just a triple of $n$-categories $X,Y,Z$ with maps $u:X\rightarrow Y$ and $v:Z\rightarrow Y$. The inverse limit of the projection into $nCAT'$ is the {\em homotopy fiber product} denoted $X\times ^{\rm ho} _YZ$. \end{parag} \begin{lemma} \label{calcinverse} Suppose $A$ is as above and $\varphi : A\rightarrow nCAT$ is a morphism corresponding to a pair of maps $u:X\rightarrow Y$ and $v:Z\rightarrow Y$ of $n$-categories such that $u$ is fibrant. Then the usual fiber product $X\times _YZ$ is a limit of $\varphi$ so we can write $$ X\times ^{\rm ho} _YZ = X\times _YZ. $$ \end{lemma} {\em Proof:} One way to prove this is to use our explicit construction of the inverse limit (\ref{lambda}). The second way is to show that $U:=X\times _YZ$ satisfies the required universal property as follows. First of all note that the commutative square $$ \begin{array}{ccc} U&\rightarrow & X \\ \downarrow && \downarrow \\ Z &\rightarrow & Y \end{array} $$ corresponds to a map $I\times I\rightarrow nCAT$ which we can project into $I\times I\rightarrow nCAT'$. Then combine this with the projection $$ A \times I \rightarrow I \times I $$ which sends $A\times \{ 0\}$ to $(0,0)$ and sends $A\times \{ 1\}$ to the copy of $A\subset I\times I$ corresponding to the sides $(1,01)$ and $(01,1)$ of the square. We get a map $A\times I \rightarrow nCAT'$ having the required constancy property to give an element $\epsilon \in Hom (U, \varphi )$. This is the map which we claim is an inverse limit. In passing note that since $X,Y,Z$ are elements of $nCAT$ they are by definition fibrant, and since by hypothesis the map $X\rightarrow Y$ is fibrant, the map $U\rightarrow Z$ is fibrant too, and so $U$ is fibrant. We now fix a fibrant $n$-category $V$ and study the functor which to an $n$-precat $F$ associates the set of morphisms $$ g:A\times \Upsilon (F)\rightarrow nCAT' $$ with $r_0(g)=V_A$ and $r_1(g)= \varphi $. This is of course just the functor represented by $Hom (V, \varphi )$. Recalling that $\underline{Hom}(V,U)$ is the morphism set in $nCAT$, we obtain by composition with $\epsilon $ a morphism $$ C_{\epsilon}: \underline{Hom}(V,U)\rightarrow Hom (V,\varphi ). $$ In this case, since $\epsilon$ comes from $nCAT$ in which the composition at the first stage is strict, the morphism $C_{\epsilon}$ is strictly well defined rather than being a weak morphism as usual in the notion of limit. We would like to show that $C_{\epsilon}$ is an equivalence (which would prove the lemma). A morphism $g:A\times \Upsilon (F)\rightarrow nCAT'$ decomposes as a pair of morphisms $(g_1,g_2)$ with $$ g_i : I \times \Upsilon (F) \rightarrow nCAT'; $$ in turn these decompose as pairs $g_i^+$ and $g_i^-$ where $$ g_i ^+ : \Upsilon (\ast , F)\rightarrow nCAT', $$ $$ g_i^- : \Upsilon (F,\ast )\rightarrow nCAT' . $$ (Decompose the square $I \times \Upsilon (F)$ into two triangles, drawing the edge $I$ vertically with vertex $0$ on top.) The conditions on everything to correspond to a morphism $g$ are that $$ r_{12}(g_i^+) = r_{01}(g_i^-) $$ and $$ r_{02}(g_1^-)= r_{02}(g_2^-). $$ The endpoint conditions on $g$ correspond to the conditions $$ r_{12}(g_1^-)= u,\;\;\;\; r_{12}(g_2^-) = v, $$ and $$ r_{01}(g_i^+)=1_V. $$ Putting these all together we see that our functor of $F$ is of the form a fiber product of four diagram $n$-categories \ref{diag1}. More precisely, put $$ M_u :=Diag ^{1_V, X}(\ast ,\underline{\ast}; nCAT') \times _{Diag ^{V, X}(\underline{\ast} ; nCAT')} Diag ^{V, u}(\underline{\ast}, \ast ; nCAT') $$ where the morphisms in the fiber product are $r_{12}$ then $r_{01}$; and define $M_v$ similarly. Then $$ Hom (V, \varphi )= M_u \times _{Diag ^{V,Y}(\underline{\ast} ; nCAT')} M_v, $$ where here the morphisms in the fiber product are the restrictions $r_{02}$ on the second factors of the $M$. Refer now to the calculation of \ref{diag1.6} in view of the comparison result \ref{diag3} (applied to $nCAT\rightarrow nCAT'$). By this calculation the restriction morphism $$ r_{12}: Diag ^{1_V, X}(\underline{\ast}, \ast ; nCAT') \rightarrow Diag ^{V,Z}(\underline{\ast} ; nCAT') $$ is a fibrant equivalence. Therefore the second projections are equivalences $$ M_u \rightarrow Diag ^{V,u}(\ast , \underline{\ast}; nCAT') $$ and similarly for $v$. Using these second projections in each of the factors $M$ we get an equivalence $$ Hom (V, \varphi )\rightarrow $$ $$ Diag ^{V,u}( \underline{\ast},\ast ; nCAT') \times _{Diag ^{V,Y}(\underline{\ast} ; nCAT')} Diag ^{V,u}(\underline{\ast},\ast ; nCAT') $$ where the morphisms in the fiber product are $r_{02}$. There is a morphism from the same fiber product taken with respect to $nCAT$, into here. In the case of the fiber product taken with respect to $nCAT$ the calculation of \ref{diag1.6} gives directly that it is equal to $$ \underline{Hom}(V,X)\times _{\underline{Hom}(V,Y)}\underline{Hom}(V,Z) $$ which is just $\underline{Hom}(V,U)$. The morphism $$ \underline{Hom}(V,X) = Diag ^{V,u}( \underline{\ast},\ast ; nCAT) \rightarrow Diag ^{V,u}( \underline{\ast},\ast ; nCAT') $$ is an equivalence by \ref{diag3}, and similarly for the other factors in the fiber product. Now we are in the general situation that we have equivalences of fibrant $n$-precats $P\rightarrow P'$, $Q\rightarrow Q'$ and $R\rightarrow R'$ compatible with diagrams $$ P\rightarrow Q\leftarrow R, \;\;\; P'\rightarrow Q'\leftarrow R'. $$ If we know that the morphisms $P\rightarrow Q$ and $P'\rightarrow Q'$ are fibrant then we can conclude that these induce an equivalence $$ P\times _QR\rightarrow P'\times _{Q'} R'. $$ Prove this in several steps using \ref{pushoutA} and \cite{nCAT} Theorem 6.7: $$ P'\times _{Q'}R' \stackrel{\sim}{\rightarrow} P' \times _{Q'}R = (P'\times _{Q'}Q)\times _Q R $$ and $$ P'\times _{Q'}Q \stackrel{\sim}{\rightarrow} P' $$ so $$ P \stackrel{\sim}{\rightarrow} P' \times _{Q'}Q $$ giving finally $$ P \times _Q R \stackrel{\sim}{\rightarrow} (P'\times _{Q'}Q)\times _Q R; $$ then apply (\ref{explaincmc}, CM2). Applying this general fact to the previous situation gives that the morphism $$ \underline{Hom}(V,X)\times _{\underline{Hom}(V,Y)}\underline{Hom}(V,Z) \rightarrow $$ $$ Diag ^{V,u}( \underline{\ast},\ast ; nCAT') \times _{Diag ^{V,Y}(\underline{\ast} ; nCAT')} Diag ^{V,u}(\underline{\ast},\ast ; nCAT') $$ is an equivalence. By (\ref{explaincmc}, CM2) this implies that $$ C_{\epsilon} : \underline{Hom}(U,V)\rightarrow Hom (V,\varphi ) $$ is an equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} Lemma \ref{calcinverse} basically says that for calculating homotopy fiber products we can forget about the whole limit machinery and go back to our usual way of assuming that one of the morphisms is fibrant. \begin{parag} \label{coproducts} Taking $A$ to be the opposite of the category in the previous paragraph, a functor $A\rightarrow nCAT$ is a triple $U,V,W$ with morphisms $f:V\rightarrow U$ and $g:V\rightarrow W$. The direct limit is the {\em homotopy pushout} of $U$ and $W$ over $V$, denoted $U\cup ^V_{\rm ho}W$. \end{parag} \begin{lemma} \label{calcdirect} Suppose that $f$ is cofibrant. Let $P$ denote a fibrant replacement $$ U\cup ^VW \hookrightarrow P. $$ Then there is a natural morphism $u: \varphi \rightarrow P$ which is a direct limit. Thus we can say that the morphism $$ U\cup ^VW \rightarrow U\cup ^V_{\rm ho}W $$ is a weak equivalence, or equivalently that the morphism of $n$-categories $$ Cat(U\cup ^VW )\rightarrow U\cup ^V_{\rm ho}W $$ is an equivalence. \end{lemma} The proof is similar to the proof of \ref{calcinverse} and is left as an exercise. This lemma provides justification {\em a posteriori} \footnote{And therefore running a certain risk of being circular\ldots } for having said in \cite{nCAT} that $Cat(U\cup ^VW )$ is the ``categorical pushout of $U$ and $W$ over $V$''. It also shows that this pushout, which occurs in the generalized Seifert-Van Kampen theorem of \cite{nCAT}, is the same as the homotopy pushout. \bigskip \subnumero{Representable functors and internal $Hom$} Suppose $A$ is an $n+1$-category. Recall that $A^o$ is the first opposed category obtained by switching the directions of the $1$-arrows but not the rest (this comes from the inversion automorphism on the first simplicial factor of $\Delta ^{n+1}$). The ``arrow family'' is a family $$ Arr(A): A^o\times A \rightarrow nCAT', $$ associating to $X\in A^o$ and $Y\in A$ the $n$-category $A_{1/}(X,Y)$. We will not discuss here the existence and uniqueness of this family (there is not actually a natural way to define this family in Tamsamani's point of view on $n$-categories, so it must be done by constructing the family by hand making choices of various morphisms when necessary). The arrow family gives two functors $$ \alpha : A \rightarrow \underline{Hom}(A^o, nCAT') $$ and $$ \beta : A^o \rightarrow \underline{Hom}(A, nCAT'). $$ \begin{conjecture} That $\alpha$ and $\beta$ are fully faithful (as is the case for $n=0$). \end{conjecture} We say that an object of $\underline{Hom}(A^o, nCAT')$ (resp. $\underline{Hom}(A, nCAT')$ is {\em representable} if it comes from an object of $A$ (resp. $A^o$). Note that such objects are themselves functors $A\rightarrow nCAT'$ or $A^o\rightarrow nCAT'$, and we call them {\em representable functors}. \begin{conjecture} \label{representable} Suppose that an $n+1$-category $A$ admits arbitrary direct and inverse limits. Then a functor $h: A^o\rightarrow nCAT'$ is representable by an object of $A$ if and only it transforms direct limits into inverse limits. A functor $g: A \rightarrow nCAT'$ is representable by an object of $A^o$ if and only if it transforms inverse limits into inverse limits. \end{conjecture} \begin{parag} \label{internal} If this conjecture is true we would obtain the following corollary: that if an $n+1$-category $A$ admits arbitrary direct and inverse limits, then $A$ has an internal $\underline{Hom}$. To see this, fix objects $x,y\in A$. Denote by $\times $ the functor $A\times A\rightarrow A$ which associates to $(u,v)$ the direct product of $u$ and $v$ (considered as an inverse limit). This functor comes from Theorem \ref{variation} as described in (\ref{usevar}). Now the functor $u\mapsto Arr (A)(x\times u, y)$ from $A^o$ to $nCAT$ transforms direct limits to inverse limits (this uses one direction of Conjecture \ref{representable}, and I suppose without proof that the functor $u\mapsto x\times u$ is known to preserve direct limits). Therefore by (the other direction of) Conjecture \ref{representable}, the functor $u\mapsto Arr (A)(x\times u, y)$ is representable by an object $\underline{Hom}_A(x,y)$. \end{parag} \begin{parag} \label{topos} We are obviously going toward some sort of theory of {\em $n$-topoi}: an {\em $n$-topos} would be an $n$-category admitting arbitrary direct and inverse limits (indexed by small $n$-categories). There may be some other conditions that one would have to impose... \end{parag} \bigskip \subnumero{$n$-stacks} Suppose ${\cal X}$ is a site. Consider the underlying category as an $n+1$-category. An {\em $n$-stack over ${\cal X}$} is a morphism $F: {\cal X} \rightarrow nCAT'$ such that for every object $X\in {\cal X}$ and every sieve ${\cal B} \subset {\cal X} /X$ the morphism $$ \Gamma ({\cal X} /X, F|_{{\cal X} /X})\rightarrow \Gamma ({\cal B} , F|_{{\cal B}}) $$ is an equivalence of $n$-categories, where $\Gamma ({\cal B} , F|_{{\cal B}})$ denotes the inverse limit of $F|_{{\cal B}}$ and the same for $\Gamma ({\cal X} /X, F|_{{\cal X} /X})$. We define $nSTACK /{\cal X} $ to be the full subcategory of the (already fibrant) $n+1$-category $\underline{Hom}({\cal X} , nCAT')$ whose objects are the morphisms $F$ satisfying the above criterion. The $n+1$-category $nSTACK /{\cal X}$ admits inverse limits---since the only thing involved in its definition is an inverse limit and inverse limits commute with each other. In particular we may speak of {\em homotopy fiber products} of $n$-stacks. \begin{conjecture} \label{projeffstacks} Homotopy projectors are effective for $n$-stacks, in other words given an $n$-stack $U'$ with endomorphism $p$ such that $p\circ p \sim p$, the ``telescope construction'' $T$ of \S 5 is again an $n$-stack. \end{conjecture} Assuming this conjecture, the same argument as in \S 5 would work to show that $nSTACK /{\cal X}$ admits direct limits. A {\em $n$-prestack over ${\cal X}$} is just a morphism $F: {\cal X} \rightarrow nCAT'$ without any other condition (this makes sense for any category ${\cal X}$ and in fact for any $n+1$-category, it is just our notion of family of $n$-categories indexed by ${\cal X}$). We can adopt the notation $$ nPRESTACK /{\cal X} := \underline{Hom}({\cal X} , nCAT'). $$ Suppose $F$ is an $n$-prestack. We define the {\em associated stack} denoted $st(F)$ to be the universal $n$-stack to which $F$ maps. Assuming Conjecture \ref{projeffstacks}, the associated stack $st(F)$ exists again by copying the argument of \S 5 above. \begin{remark} The inverse limit of a family of stacks is the same as the inverse limit of the underlying family of prestacks. However this is not true for direct limits. \end{remark} \begin{parag} By \ref{internal} which is based on Conjecture \ref{representable}, the $n+1$-category $nSTACK /{\cal X}$ admits internal $\underline{Hom}$. Using this (or alternatively using a direct construction which associates to any $X\in{\cal X}$ the $n+1$-category $nSTACK/({\cal X} /X )$) we should be able to construct the $n+1$-stack $n\underline{STACK}/{\cal X}$. \end{parag} \begin{parag} \label{geometric1} Now that we have a notion of $n$-stack not necessarily of groupoids, one can ask how to generalize the definition of {\em geometricity} given in \cite{geometricN}, to the case where the values may not be groupoids. If $A$ is an $n$-category and $X$, $Y$ are sets with maps $a:X\rightarrow A$ and $b:Y\rightarrow A$ then the pullback $$ (a^o, b)^{\ast} (Arr (A)) =:X \times Y\rightarrow (n-1)CAT' $$ may be considered as an $(n-1)$-category (taking the union over all of the points of $X\times Y$) which we denote by $Hom _A(a,b)$. However, it is no longer the same thing as the fiber product $X\times _AY$. Both of these still satisfy the recurrence-enabling fact that they are $n-1$-categories. Thus we can still employ the same type of definition as in \cite{geometricN}. However, as many common examples quickly show, the smoothness condition should only be imposed on the fiber product, not the arrows. Thus we say that $A$ is {\em geometric} (resp. {\em locally geometric}) if: \newline (GS1) for any two morphisms from schemes $a:X\rightarrow A$ and $b:Y\rightarrow A$, the arrow $n-1$-stack $Hom _A(a,b)\rightarrow X\times Y$ and the product $X\times _AY$ are both geometric (resp. locally geometric); and \newline (GS2) there exists a smooth morphism from a finite type scheme (resp. locally finite type scheme) $X\rightarrow A$ surjective on the truncations to $0$-stacks; \newline where the morphism $X\rightarrow A$ is said to be {\em smooth} if for any morphism from a scheme of finite type $Y\rightarrow A$, the locally geometric $n-1$-stack $X\times _AY$ is actually geometric and is smooth, this latter condition meaning that the smooth surjection to it from GS2 comes from a smooth scheme of finite type. \end{parag} \begin{parag} \label{geometric2} Here is an example to show what we are thinking of (this type of example---even if relatively unknown on ``alg-geom''---apparently comes up very often on ``q-alg''). The stack of vector bundles on a given variety, for example, is locally geometric. It has an additional operation, tensor product, which allows it to be considered as a monoidal (or braided or symmetric) monoidal $1$-stack, thus allowing us to consider it as a $2$, $3$ or $4$-stack. In these cases there are only one object (locally speaking) and in the $3$ and $4$ cases, only one morphism (in the $4$ case only one $2$-morphism). The original stack comes back as an arrow stack (possibly after iterating). In this example, if we want a tensor product we are forced to consider things not of finite type, so the arrow stacks should often be allowed to be only locally geometric (also one readily sees that the arrow stacks will not necessarily be smooth). On the other hand the finite type and smoothness conditions in GS2 correspond in this example to the smoothness and finite type conditions for the Picard scheme. \end{parag} \begin{parag} \label{locallyP} Suppose ${\cal P}$ is a property of $n$-stacks of groupoids. Then we say that an $n$-stack $A$ is {\em locally ${\cal P}$} (and we call this property $loc {\cal P}$) if $F=\tau _{\leq 0}A $ is an filtered inductive limit of open subsheaves $F_i\subset F$ (the openness condition means that for any scheme $X\rightarrow F$, $X\times _FF_i$ is an open subset of $X$) such that $A\times _{F}F_i$ has property ${\cal P}$. In particular we obtain notions of {\em locally presentable} and {\em locally very presentable} $n$-stacks of groupoids. We claim that for ${\cal P} = $ ``geometric'' the above definition gives the same definition as the previous definition of locally geometric. Suppose that $A$ is locally ${\cal P}$. Let $A_i := A\times _FF_i$. This is an open substack of $A_i$. Let $X_i\rightarrow A_i$ be the smooth surjections from schemes of finite type. Then $X_i\rightarrow A$ is smooth (for example by the formal criterion for smoothness). Thus the morphism from the disjoint union of the $X_i$ to $A$ is a smooth surjection proving that $A$ is locally geometric according to the old definition. Suppose now that $A$ is locally geometric for the old definition, and let $X_i \rightarrow A$ be the smooth morphisms from schemes of finite type which together cover $A$. Let $F=\tau {\leq 0} A$ and let $F_i \subset F$ be the images of $X_i$. Let $A_i = A\times _FF_i$. It is clear that $X_i$ maps to $A_i$ by a map which is, on the one hand, smooth by the formal criterion, and on the other hand surjective on the level of $\pi _0$ by definition. Thus the $A_i$ are geometric, i.e. have property ${\cal P}$. It is clear that the union of the $A_i$ is $A$. Finally, the $F_i$ are open subsheaves of $F$, using smoothness of $X_i \rightarrow A$ plus Artin approximation. \end{parag} \begin{definition} \label{extendingproperties} If ${\cal P}$ is a property of $n$-stacks of groupoids (say, independent of $n$...) then we can extend ${\cal P}$ to a property of $n$-stacks in a minimal way such that the following conditions hold: \newline (A)\,\,\, If $A$ has property ${\cal P}$ then so does the interior groupoid $A^{\rm int}$; \newline (B)\,\,\, If $A$ has property ${\cal P}$ and $a: X\rightarrow A$ and $b: Y\rightarrow A$ are morphisms from schemes of finite type then $Hom _A(a,b)$ has property ${\cal P}$. That such a minimal extension exists is obvious by induction. \end{definition} \begin{parag} Taking the property ${\cal P}$ in the above definition \ref{extendingproperties} to be ``locally presentable'' or ``locally very presentable'' or ``locally geometric'' we obtain reasonable properties for $n$-stacks not necessarily of groupoids. The use of the locality properties is natural here since the composition operation will often be something like tensor product, which does not preserve any substack of finite type. \end{parag} \bigskip \subnumero{The notion of stack, in general} We give here a very general discussion of the notion of ``stack''. This was called ``homotopy-sheaf'' in \cite{kobe} (cf also \cite{flexible} which predates \cite{kobe} but which was made available much later), however that was not the first time that such objects were encountered---the condition of being a homotopy sheaf is the essential part of the condition of being a fibrant (or ``flasque'') simplicial presheaf \cite{Brown} \cite{Jardine} \cite{Joyal}. Suppose ${\cal C}$ is some type of category-like object (such as an $n$-category or $\infty$-category or other such thing). Suppose that we have a notion of {\em inverse limit} of a family of objects of ${\cal C}$ indexed by a category ${\cal B}$. If we call the family $F: {\cal B} ^o\rightarrow {\cal C}$ (contravariant on ${\cal B}$, for our purposes) then we denote this limit---if it exists---by $\Gamma ({\cal B} , F)\in {\cal C}$. This should be sufficiently functorial in that if we have a functor ${\cal B} \rightarrow {\cal B} '$ and $F$ is the pullback of a family $F'$ on ${\cal B} '$ (denoted $F=F'|_{{\cal B}}$) then we should obtain a morphism of functoriality (i.e. an arrow in ${\cal C}$) $$ \Gamma ({\cal B} ', F') \rightarrow \Gamma ({\cal B} , F), $$ possibly only well-defined up to some type of homotopy in ${\cal C}$. Similarly if ${\cal B}$ has a final object $b$ (initial for our functoriality which is contravariant) then the morphism (obtained from above for the inclusion $\{ b\} \rightarrow {\cal B}$) $$ \Gamma ({\cal B} , F) \rightarrow F(b) $$ should be an ``equivalence'' in ${\cal C}$ (one has to know what that means). With all this in hand (and note that we do not assume the existence of arbitrary limits, only existence of limits indexed by categories with final objects) we can define the notion of {\em stack over a site ${\cal X}$ with coefficients in ${\cal C}$}. This is to be a family $F$ of objects of ${\cal C}$ indexed by ${\cal X}$ (i.e. a morphism ${\cal X} ^o\rightarrow {\cal C}$) which satisfies the following property: for every object $X\in {\cal X}$ and every sieve ${\cal B} \subset {\cal X} /X$ the morphism $$ \Gamma ({\cal X} /X, F|_{{\cal X} /X}) \rightarrow \Gamma ({\cal B} , F|_{{\cal B}}) $$ is an equivalence in ${\cal C}$, meaning that the limit on the right exists. (Note that since ${\cal X} /X$ has a final object $X$, the morphism $$ \Gamma ({\cal X} /X, F|_{{\cal X} /X}) \rightarrow F(X) $$ is assumed to exist and to be an equivalence.) Taking the inverse limit of a family of stacks will again be a family of stacks because inverse limits should (when that notion is defined) commute with each other. If ${\cal C}$ admits arbitrary (set-theoretically reasonable) inverse limits then taking the inverse limit of a family of stacks gives again a stack. Using this we can define the {\em stack associated to a prestack}. A prestack is just any family $F: {\cal X} ^o \rightarrow {\cal C}$ not necessarily satisfying the stack condition. The {\em associated stack} is defined to be the inverse limit of all stacks $G$ to which $F$ maps. Of course this needs to be investigated some more in any specific case, in order to get useful information. When ${\cal C}$ is the $2$-category of categories we obtain the classical notion of stack \cite{LMB} \cite{ArtinInventiones}. When ${\cal C}$ is the $\infty$-category of simplicial sets we obtain the notion of ``homotopy sheaf'' which is equivalent in Jardine's terminology to a simplicial presheaf which is flasque with respect to each object of the underlying site. In particular, fibrant simplicial presheaves satisfy this condition, and the condition is just that of being object-by-object weak equivalent to a fibrant simplicial presheaf. The process of going from a prestack to the associated stack is basically the process of going from a simplicial presheaf to an equivalent fibrant simplicial presheaf. The case where ${\cal C}$ is the $n+1$-category $nCAT'$ of $n$-categories yields the notion of {\em $n$-stack} described above. \bigskip \subnumero{Localization} \begin{parag} \label{universal1} {\em Universal morphisms with certain properties} We often encounter the following situation. Suppose $X\in nCAT'$ and suppose and suppose ${\cal P}$ is a property of morphisms $X \rightarrow B$ in $nCAT'$. Then we can look for a universal morphism $\nu: X \rightarrow U$ with property ${\cal P}$. The ``universal'' property can be written out in terms of our construction $\Upsilon$: it means that for any cofibration of $n$-precats $E'\hookrightarrow E$ and any morphism (from an edge labeled $(02)$) $$ f:\Upsilon (E)\rightarrow nCAT' $$ with $f(0)= X$, $f(2)=B$ sending $\Upsilon (E_0)$ to a collection of morphisms having property ${\cal P}$, together with an extension along $E'$ to a morphism $$ g':\Upsilon ^2(\ast , E')\rightarrow nCAT' $$ with $r_{01}(g')= \nu$ and $g'(2)= B$, there exists $$ g:\Upsilon ^2(\ast , E)\rightarrow nCAT' $$ extending $g'$ and with $r_{01}(g)=\nu $ and $r_{02}(g)= f$. \end{parag} \begin{parag} \label{universal1.2} Suppose $\psi : A\rightarrow nCAT'$ is a functor and ${\cal P}$ a property of morphisms $\psi \rightarrow B$ to objects $B\in nCAT'$. Then we can make a similar definition of ``universal morphism'' $\nu : \psi \rightarrow U$ having property ${\cal P}$. In this case, it also makes sense to ask for a morphism $\nu : \psi \rightarrow U$ to an object of $nCAT'$, ``universal for morphisms with property ${\cal P}$'' (the definition is the same as above but we don't require $\nu$ to have property ${\cal P}$). Note that this definition in the case of one object $X$ is vacuous: the answer would just be the identity morphism $1_X:X\rightarrow X$. \end{parag} \begin{parag} \label{universal2} To construct $\nu$ we can try to follow the argument of \S 5, taking the full subcategory $M({\cal P} )\subset \psi /nCAT'$ of objects under $\psi$ having property ${\cal P}$. As before we consider the subcategory $M({\cal P} )_{\alpha}$ of objects of $\alpha$-bounded cardinality, and let $U$ be the inverse limit of the forgetful functor $M({\cal P} )_{\alpha}\rightarrow nCAT'$. We now need to know four things: \newline \ref{universal2}(i) that the morphism $\psi \rightarrow U$ again has property ${\cal P}$ (preservation of ${\cal P}$ by inverse limits); \newline \ref{universal2}(ii) that there is a factorization $\psi \rightarrow U' \rightarrow U$ with $\psi \rightarrow U'$ again having property ${\cal P}$ and $\# ^{\rm pre}U'\leq \alpha$ (for $\alpha$ chosen appropriately); \newline \ref{universal2}(iii) that the ``telescope'' construction of (\ref{construction2}) preserves property ${\cal P}$; and \newline \ref{universal2}(iv) that if $f:\psi \rightarrow \underline{Hom}(E,B)$ is a morphism which, when restricted to every object of $E_0$ gives a morphism $\psi \rightarrow B$ with property ${\cal P}$, then $f$ has property ${\cal P}$ (this is so that a criterion analogue to \ref{easydirect} applies). \end{parag} \begin{conjecture} \label{universal3} If we know these four things then the argument of \S 5 works to construct a universal $\nu : \psi \rightarrow T$ with property ${\cal P}$. \end{conjecture} \begin{parag} \label{localization} {\em Localization:} If $X$ is an $n$-category then we denote $Fl^i(X)$ the set of $i$-morphisms, which is the same as $X_{1,\ldots , 1}$. Suppose we are given a collection of subsets $S = \{ S^i \subset Fl ^i(X)\}$. Then we can define $S^{-1}X$ to be the universal $n$-category with map $X\rightarrow S^{-1}X$ sending the elements of $S^i$ to $i$-morphisms in $S^{-1}X$ which are invertible up to equivalence (i.e. morphisms which are invertible in $\tau _{\leq i}(S^{-1}X)$). To construct $S^{-1}X$, let ${\cal P}$ be the property of a map $X\rightarrow B$ that the arrows in $S_i$ become invertible in $B$. One has to verify the properties \ref{universal2}(i)--\ref{universal2}(iv), and then apply Conjecture \ref{universal3}. To verify the properties (i)--(iv) use Theorem \ref{resttoIff}. This is the $n$-categorical analogue of \cite{GabrielZisman}. \end{parag} {\em Caution:} If $A$ is an $m$-category considered as an $n$-category then $S^{-1}A$ may not be an $m$-category. In particular, note that by taking the group completion (see below) of $1$-categories one gets all homotopy types of $n$-groupoids. (This fact, which seems to be due to Quillen, was discussed at length in \cite{Grothendieck}...). \begin{parag} \label{gc} {\em Group completion:} The theory of $n$-categories which are not groupoids actually has a long history in homotopy theory, in the form of the study of topological monoids. In Adams' book \cite{Adams} the chapter after the one on loop-space machinery, concerns the notion of ``group completion'', namely how to go from a topological monoid to a homotopy-theoretic group ($H$-space). This is a special example of going from an $n$-category to an $n$-groupoid by ``formally inverting all arrows''. Taking $S$ to be all of the arrows in a fibrant $n$-category $X$, the localization $S^{-1}X$ is the {\em group completion of $X$} denoted $X^{\rm gc}$. It is the universal $n$-groupoid to which $X$ maps. This may also be constructed by a topological approach (which has the merit of not depending on Conjecture \ref{universal3}), as $$ X^{\rm gc} = \Pi _n (| X | ), $$ using Tamsamani's realization $|X|$ and Poincar\'e $n$-groupoid $\Pi _n$ constructions \cite{Tamsamani}. \end{parag} As an example, \ref{resttoIff} allows us to describe the group completion of $I$ which is contractible, as one might expect. \begin{corollary} The morphism $I\rightarrow \overline{I}$ is the group completion in the context of $n$-categories. \end{corollary} {\em Proof:} This follows immediately from Theorem \ref{resttoIff}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{gcCommutesWithCoprod} Group completion commutes with coproduct. More precisely, suppose $B\leftarrow A \rightarrow C$ are morphisms of $n$-precats. Then the morphism $$ (B\cup ^A C)^{\rm gp} \rightarrow B^{\rm gp} \cup ^{A^{\rm gp}} C^{\rm gp} $$ is an equivalence. \end{lemma} {\em Proof:} This can be seen directly from the topological definition $X^{\rm gc} = \Pi _n (| X | )$ using the results of \cite{nCAT} \S 9. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{parag} \label{interiorg} {\em Interior groupoid} We can do a similar type of definition as \ref{universal1} for universal maps from $B$ to $X$ having certain properties. Applying this again to the property that all $i$-morphisms become invertible, we get the following definition. If $X$ is a fibrant $n$-category then its {\em interior groupoid} $X^{\rm int}$ is the universal map $X^{\rm int}\rightarrow X$ for this property. It is an $n$-groupoid, and may be seen as the ``largest $n$-groupoid inside $X$''. Without refering to conjectures, we can construct $X^{\rm int}\subset X$ explicitly as follows. Assume that $X$ is an $n$-category. First we define $X^{k\rm -int}\subset X$ with the same objects as $X$, by setting $$ X^{1\rm -int} _{p/}(x_0,\ldots , x_p):= X_{p/}(x_0,\ldots , x_p)^{\rm int} $$ (note that we use inductively the definition of $Y^{\rm int}\subset Y$ for $n-1$-categories as well as the fact that this construction takes equivalences to equivalences). Now let $$ X^{\rm int}_{1/}(x,y)\subset X^{1\rm -int}_{1/}(x,y) $$ be the full sub-$(n-1)$-category of objects corresponding to morphisms which are invertible up to equivalence. Let $X^{\rm int}_{p/}(x_0,\ldots , x_p)$ be the full sub-$(n-1)$-category of $X^{1\rm -int}_{p/}(x_0,\ldots , x_p)$ consisting of objects which project to elements of $X^{\rm int}_{1/}(x_{i-1}, x_i)$ on the principal edges. Another way of saying this is to note that there is a morphism $$ X^{1\rm -int}\rightarrow \tau _{\leq 1}(X) $$ (cf the notation of \ref{anotherapproach}). Then define the ``interior $1$-groupoid'' of the $1$-category $\tau _{\leq 1}(X)$ to be the subcategory consisting only of invertible morphisms, and set $X^{\rm int}$ to be the fiber product of $X^{1\rm -int}\rightarrow \tau _{\leq 1}(X)$ and interior $1$-groupoid of $\tau _{\leq 1}(X)$, over $\tau _{\leq 1}(X)$. \end{parag} \bigskip \subsubnumero{$k$-groupic completion and interior} More generally we say that an $n$-category $B$ is {\em $k$-groupic} for $0\leq k \leq n$ if the $n-k$-categories $B_{m_1,\ldots , m_k/}$ are groupoids. In other words this says that the $n-k$-category whose objects are the $k$-morphisms of $B$ should be an $n-k$-groupoid. Note that being $O$-groupic means that $B$ is an $n$-groupoid, and the condition of being $n$-groupic is void of content. We can define the {\em $k$-groupic completion} $X^{k{\rm -gp}}$ as the universal $k$-groupic $n$-category to which $X$ maps. We can define the {\em $k$-groupic interior} $X^{k{\rm -int}}\subset X$ to be the universal $k$-groupic $n$-category mapping to $X$. For $k=0$ these reduce to the group completion and interior groupoid. For the $k$-groupic interior, we have the following formula whenever $k\geq 1$: $$ X^{k{\rm -int}}_{p/}(x_0,\ldots , x_p)= X_{p/}(x_0,\ldots , x_p)^{(k-1){\rm -int}}, $$ which gives an inductive construction. \bigskip \subnumero{Direct images and realizations} Suppose $F:A\rightarrow B$ is a morphism of $n+1$-categories and suppose $\varphi : A \rightarrow nCAT'$ is a family of $n$-categories over $A$. Then we can look for a universal family $\psi : B\rightarrow nCAT'$ together with morphism $\varphi \rightarrow F^{\ast}(\psi )$. If it exists, we call $\psi$ the {\em direct image} and denote it by $F_{\ast}(\varphi )$. \begin{conjecture} \label{directimage} The direct image $F_{\ast}(\varphi )$ always exists, and is essentially unique. \end{conjecture} Again, the argument of \S 5 should work to give the construction of $F_{\ast}(\varphi )$, with several things to verify analogous to \ref{universal2}(i-iv). \begin{parag} {\em Caution:} the notations ``direct image'' $F_{\ast}$ and ``inverse image'' $F^{\ast}$ are switched from the usual notations for functoriality for ``morphisms of sites''. \end{parag} \begin{parag} \label{realization1} {\em Realization:} Suppose $A$ is an $n+1$-category and suppose $$ \varphi : A \rightarrow nCAT' $$ is a family of $n$-categories, and $$ \psi : A^o \rightarrow nCAT' $$ is a contravariant family of $n$-categories. Then we define the {\em realization} of this pair, denoted $\langle \varphi , \psi \rangle$, as follows. The arrow family for $A$ corresponds to a morphism $$ \alpha : A \rightarrow \underline{Hom}(A^o, nCAT'). $$ The direct image $\alpha _{\ast}(\varphi )$ is therefore a morphism $$ \alpha _{\ast}(\varphi )\underline{Hom}(A^o, nCAT') \rightarrow nCAT'. $$ Put $$ \langle \varphi , \psi \rangle := \alpha _{\ast}(\varphi )(\psi ). $$ \end{parag} \begin{parag} \label{realization2} An example of this is when $A={\cal X}$ is a site, and when $\varphi$ and $\psi$ are families of $n$-groupoids. Then $\langle \varphi , \psi \rangle $ is an $n$-groupoid, and we conjecture that it corresponds to the topological space given as realization of the two functors as defined in \cite{realization}. \end{parag} \begin{parag} \label{realization3} In the main example of \cite{realization} one took ${\cal X}$ to be the site of schemes over $Spec ({\bf C} )$ and one took $\varphi$ to be the functor associating to each scheme the $n$-truncation of the homotopy type of the underlying topological space. Then for any presheaf $\psi$ of $n$-truncated topological spaces one obtained the ``topological realization'' of $\psi$. \end{parag} \begin{parag} \label{realization4} One can do the operation of \ref{realization1} in the other order, using the arrow family considered as a morphism $$ \beta : A^o \rightarrow \underline{Hom}(A, nCAT') $$ and looking at $\beta _{\ast}(\psi ) (\varphi )$. \newline {\em Conjecture---}that these two ways of defining $\langle \varphi , \psi \rangle$ give the same answer. \end{parag} \begin{parag} \label{triplecombo} The above construction is a special case of the more general phenomenon which we call ``triple combination''. Suppose $A$ and $B$ are $(n+1)$-categories and suppose that we have functors $$ F: A\rightarrow nCAT', $$ $$ G: B\rightarrow nCAT', $$ and $$ H: A\times B\rightarrow nCAT'. $$ Then we can consider $H$ as a functor $$ H:A\rightarrow \underline{Hom}(B, nCAT') $$ and define $$ H(F,G):= H_{\ast}(F)(G). $$ As above, one conjectures that $H(F,G)= H^{\sigma}(G,F)$ (applying the symmetry $\sigma : A\times B\cong B \times A$). The previous construction is just $$ \langle \varphi , \psi \rangle = Arr (A)(\varphi , \psi ). $$ The same definition of triple combination works for functors $F,G,H$ in any fibrant $n$-category $C$ which admits limits as does $nCAT'$. \end{parag} \bigskip \subnumero{Relative Malcev completion} An example which gets more to the point of my motivation for doing all of this type of thing is the following generalization of relative Malcev completion \cite{Hain} to higher homotopy. \begin{parag} \label{malcev1} Fix a ${\bf Q}$-algebraic group $G$. Fix an $n$-groupoid $X$ with base-object $x$ (which is the same thing as an $n$-truncated pointed homotopy type). Fix a representation $\rho :\pi _1(X,x)\rightarrow G$. Let ${\cal C}$ be the $n+1$-category of quadruples $(R,r, p, f)$ where $R$ is a connected $n$-groupoid, $r$ is an object, $p: R\rightarrow BG$ is a morphism sending $r$ to the base-object $o$, and $f: X\rightarrow R$ is a morphism sending $x$ to $r$ such that the induced morphism $\pi _1(X,x) \rightarrow G$ is equal to $\rho$. Let ${\cal C} ^{\rm uni}$ denote the subset of objects satisfying the following properties: that $\pi _1(R)$ is a ${\bf Q}$-algebraic group and $p: \pi _1(R)\rightarrow G$ is a surjection with unipotent kernel; and that $\pi _1(R)$ acts algebraically on the higher homotopy groups $\pi _i(R)$ which are themselves assumed to be finite dimensional ${\bf Q}$-vector spaces. \end{parag} \begin{parag} \label{malcev2} Inverse limits exist in ${\cal C}$. To see this, note that ${\cal C}$ is an $n+1$-category of morphisms $V\rightarrow nCAT'$ where $V$ is the category with objects $v_R$, $v_r$, $v_{BG}$, $v_X$ and morphisms $v_r\rightarrow v_R$, $v_R\rightarrow v_{BG}$, $v_X\rightarrow v_R$, $v_r\rightarrow v_X$. The $n+1$-category ${\cal C}$ is the subcategory of morphisms $V\rightarrow nCAT'$ which send $v_r$ to $\ast$, send $v_{BG}$ to $BG$ and send $v_X$ to $X$, and which send the maps $v_r\rightarrow v_X$ to the basepoint $\ast \rightarrow X$, similarly for the map $v_r\rightarrow v_{BG}$, and which send $v_X\rightarrow v_{BG}$ to the map induced by $\rho$. Our Theorem \ref{inverse} as well as \ref{variation} and Lemma \ref{fiprod} imply that ${\cal C}$ admits inverse limits. Of course ${\cal C} ^{\rm uni}$ is not closed under inverse limits. However we can still take the inverse limit in ${\cal C}$ of all the objects in ${\cal C} ^{\rm uni}$. We call this the {\em relative Malcev completion of the homotopy type of $X$ at $\rho$}, and denote it by $Malc(X, \rho)$ (technically this is the notation for the underlying $n$-groupoid which is the inverse limit of the $R$'s). \end{parag} \begin{parag} \label{malcev3} We have, for example, that $\pi _1(Malc(X, \rho ), \ast )$ is equal to the relative Malcev completion of the fundamental group $\pi _1(X)$ at $\rho$. For this statement we fall back into the realm of $1$-categories, where our Malcev completion coincides with the usual notion \cite{Hain}. \end{parag} \begin{parag} \label{malcev4} We can do the same thing with stacks. For a field $k$ (of characteristic zero, say) an algebraic group $G$ over $k$ and a representation $\rho : \pi _1(X,x)\rightarrow G$, let ${\cal C} (X, \rho )/k$ be the $n+1$-category of quadruples $(R,r, p,f)$ where $R$ is a connected $n$-stack of groupoids on $Sch/k$, $r$ is a basepoint, $p: R\rightarrow BG$, and $f: \underline{X}\rightarrow R$ are as above. Here $\underline{X}$ is the constant stack with values $X$. Let ${\cal C} ^{\rm uni}(X,\rho )/k$ be the subcategory of objects such that $\pi _1(R)$ is an algebraic group surjecting onto $G$ and where the $\pi _i(R)$ are linear finite dimensional representations of $\pi _1(R)$. Again inverse limits will exist in ${\cal C} (X,\rho )/k$ and we can take the inverse limit here of the objects of ${\cal C} ^{\rm uni}(X, \rho )/k$. Call this $Malc(X, \rho )/k$. Note that $Malc(X,\rho )/{\bf Q} )$ is an $n$-stack on $Sch /{\bf Q} $ whose $n$-groupoid of global sections is $Malc(X,\rho )$. \end{parag} \begin{parag} \label{malcev5} Suppose $X$ is a variety and let $X_B$ be the $n$-groupoid truncation of the homotopy type of $X^{\rm top}$. Fix a representation $\rho$. Then we obtain the ``Betti'' Malcev completion $Malc(X_B, \rho )/{\bf C} $. On the other hand suppose $P$ is the principal $G$-bundle with integrable connection (with regular singularities at infinity) corresponding to $\rho$, then we can define in a similar way $Malc (X_{DR}, P)/{\bf C} $. The GAGA results imply that these two are naturally equivalent: $$ Malc(X_B, \rho )/{\bf C} \cong Malc(X_{DR}, P) /{\bf C} . $$ SImilarly we can define, for a principal Higgs bundle $Q$ with vanishing Chern classes, $Malc (X_{Dol}, Q)/{\bf C}$, and (in the case $X$ smooth projective) if $Q$ corresponds to $\rho$ then $$ Malc(X_B, \rho )/{\bf C} \cong Malc(X_{Dol}, Q) /{\bf C} . $$ Finally, suppose $\rho$ is an ${\bf R}$-variation of Hodge structure and $Q$ the corresponding system of Hodge bundles. Then ${\bf C} ^{\ast }$ acts on $Malc(X_{Dol}, Q) /{\bf C} $ giving rise to a ``weight filtration'' and ``Hodge filtration''. We conjecture that these (together with the ${\bf R}$-rational structure $Malc(X_B,\rho )/{\bf R} $) define a ``mixed Hodge structure'' on $Malc(X_B, \rho)/{\bf R}$. (One has to give this definition, specially in view of the infinite size of $Malc(X_B, \rho)/{\bf R}$). \end{parag} More generally we have the following conjecture. \begin{conjecture} \label{mts} Suppose $\rho$ is a reductive representation of the fundamental group of a projective variety $X$ (we assume it is reductive when restricted to the fundamental group of the normalization). Then the relative Malcev completion of the higher homotopy type $Malc(X_B, \rho)/{\bf C}$ defined above carries a natural mixed twistor structure (cf \cite{twistor}). \end{conjecture} There should also be a statement for quasiprojective varieties, but in this case one probably needs some additional hypotheses on the behavior of $\rho$ at infinity. \bigskip

9708

alg-geom/9708014

en

https://arxiv.org/abs/alg-geom/9708014

alg-geom/9708014

Leticia B. Paz

L. Brambila-Paz and H. Lange

A stratification of the moduli space of vector bundles on curves

Latex, Permanent e-mail L. Brambila-Paz: [email protected] Classification: 14D, 14F

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k\deg E-r\max\deg F.$$ where the maximum is taken over all subbundles $F$ of rank $k$ of $E$. The ${s}_k$ gives a stratification of the moduli space ${\cal M}(r,d)$ of stable vector bundles of rank $r$ and degree on $d$ on $C$ into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of $s$ and $k$. There is a component ${\cal M}^0(r,d,k,s)$ of ${\cal M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\cal M}^0(r,d,k,s)$ admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it For $g\ge \frac{r+1}{2}$ and $0<s\leq k(r-k)(g-1) +(r+1)$, $s\equiv kd \mod r,$ ${\cal M}^0(r,d,k,s)$ is non-empty,and its component ${\cal M}^0(r,d,k,s)$ is of dimension} $$\dim {\cal M}^0(r,d,k,s)=\left\{\begin{array}{lcl} (r^2+k^2-rk)(g-1)+s-1& &s<k(r-k)(g-1) &{\rm if}& r^2(g-1)+1& & s\ge k(r-k)(g-1)\end{array}\right.$$

alg-geom

\section{The invariants ${ {}{\mbox{\euf s}_k}}(E)$} Let $C$ be a smooth projective curve of genus $g\ge 2$ over an algebraically closed field $K$ of arbitrary characteristic. and let $E$ denote a vector bundle of rank $r\ge 2$ over $C$. For any integer $k$ with $1\le k\le r-1$ let ${}{Sb_k}(E)$ denote the {\it set of subbundles of rank $k$ of $E$}. If we denote by $\xi$ the generic point of the curve $C$, then it is easy to see that there is a canonical bijection between ${}{Sb_k}(E)$ and the set of $k$-dimensional subvector spaces of the $K(\xi)$-vector space $E(\xi)$. For any subbundle $F\in {}{Sb_k}(E)$ define the integer ${\mbox{\euf s}_k}(E,F)$ by $${\mbox{\euf s}_k}(E,F):= k\deg E-r\deg \;F.$$ The vector bundle $E$ does not admit subbundles of arbitrarily high degree. Hence $${}{\mbox{\euf s}_k}(E):=\mathop{\min}_{F\in {}{Sb_k}(E)}{\mbox{\euf s}_k}(E,F)$$ is a well defined integer depending only on $E$ and $k$. \begin{rem} \label{1.1} \begin{em} The slope of a vector bundle $F$ on $C$ is defined as $\mu(F)={\deg\; F\over {\rm rk} F}$. If $F$ is a subbundle of rank $k$ of $E$ then $${\mbox{\euf s}_k}(E,F)=k(r-k)\left(\mu(E/F)-\mu(F)\right).$$ \end{em} \end{rem} In particular $${}{\mbox{\euf s}_k}(E)=k(r-k)\cdot \mathop{\min}_{F\in {}{Sb_k}(E)}\left(\mu(E/F)-\mu(F)\right)$$ So instead of the invariant ${}{\mbox{\euf s}_k}(E)$ one could also work with the invariant \par \noindent{$\min_{F\in {}{Sb_k(E)}}\left(\mu(E/F)-\mu(F)\right).$} However, for some proofs it is more convenient to work with integers. Note that there is also a geometric interpretation of the invariant ${}{\mbox{\euf s}_k}(E)$ in terms of intersection numbers on the associated projective bundle $\mbox{\cj P}(E)$ (see \cite{l3}). \begin{rem} \label{1.2} \begin{em} The following properties of the invariant ${}{\mbox{\euf s}_k}(E)$ are easy to see (see \cite{l1})\begin{itemize} \item[(a)] ${}{\mbox{\euf s}_k}(E\otimes L)={}{\mbox{\euf s}_k}(E)$ for all $L\in Pic(C)$. \item[(b)] ${}{\mbox{\euf s}_k}(E)={}{\mbox{\euf s}_{r-k}}(E^*)$. \item[(c)] $E$ is stable (respectively semistable) if and only if ${}{\mbox{\euf s}_k}(E)>0$ (respectively ${}{\mbox{\euf s}_k}(E)\ge 0$) for all $1\le k\le r-1$. \item[(d)] Let $T$ be an algebraic scheme over $K$ and ${\cal E}$ a vector bundle of rank $r$ on $C\times T$. For any point $t\in T,$ let $\bar{t}$ denote a geometric point over $t$. The function ${}{s_k}:T\to\mbox{\cj Z} $ defined as $t\mapsto{}{\mbox{\euf s}}_k({\cal E} |_{C\times\{\bar{t}\}})$ is well defined and lower semicontinuous. \end{itemize} \end{em} \end{rem} Whereas the function ${}{s_k}$ may take arbitrarily negative values (for suitable direct sums of line bundles). However, it is shown in \cite{ms} and \cite{l1} that ${}{\mbox{\euf s}_k}(E)\le k(r-k)g$. Hirschowitz gives in \cite{h1} the better bound, $${}{\mbox{\euf s}_k}(E)\le k(r-k)(g-1)+(r-1).$$ We want to study the behaviour of the invariant ${}{\mbox{\euf s}_k}(E)$ under an elementary transformation of the vector bundle $E$. Recall that an {\it elementary transformation} $E'$ of $E$ is defined by an exact sequence $$0\to E'\to E\mathop{\to}^{\ell}K(x)\to 0\eqno(1)$$ where $K(x)$ denotes the skyscraper sheaf with support $x\in C$ and fibre $K$. Since $\ell$ factorizes uniquenly via a $k$-linear form $E(x)\stackrel{\ell}\rightarrow K(x)$, also denoted by $\ell$, the set of elementary transformations of $E$ is parametrized by pairs $(x,\ell)$ where $x$ is a closed point of $C$ and $\ell$ is a linear form on the vector space $E(x)$. So the set of elementary transformations of $E$, which we denote by $elm(E)$, forms a vector bundle of rank $r$ over the curve $C$. Note that for any $E'\in\; elm(E)$ $${\rm rk} (E')={\rm rk}(E)\quad{\rm and}\quad\deg E'=\deg E-1.$$ \begin{lemma} \label{1.3} For any $E'\in elm(E)$ the map $\varphi:{}{Sb_k}(E)\to {}{Sb_k}(E') $ defined by $F\mapsto F\cap E'$ is a bijection. \end{lemma} \noindent{\bf Proof.} For the proof only note that the inverse map is given as follows: Suppose $F'\in {}{Sb_k}(E')$. Consider $F'$ as a subsheaf of $E$ and let $F$ denote the subbundle of $E$ generated by $F'$. The map $F'\mapsto F$ is inverse to $\varphi$.\hspace{\fill}$\Box$ Now consider a subbundle $F\in {}{Sb_k}(E)$ and denote $F'=\varphi(F)\in {}{Sb_k}(E')$. In order to compute the number ${\mbox{\euf s}_k}(E',F')$ we have to distinguish two cases. We say that the subbundle $F$ is of {\it type I} with respect to $E'$ if $F\subseteq E'$ and $F$ is of {\it type II} with respect to $E'$ otherwise. Let $(x,\ell)\in elm(E)$ denote the pair defining the elementary transformation $E'$ of $E$. We obviously have \begin{lemma} \label{1.4} The subbundle $F\in {}{Sb_k}(E)$ is of type I with respect to $E'$ if and only if the linear form $\ell :E(x)\to K(x)$ vanishes on the subvector space $F(x)$ of $E(x)$. \end{lemma} \begin{lemma} \label{1.5} If $E'$ is an elementary transformation of $E$, $F\in {}{Sb_k}(E)$ and $F'=\varphi(F)$ then, \begin{itemize} \item[(i)] ${\mbox{\euf s}}_k(E',F') ={\mbox{\euf s}}_k(E,F)-k$ if $F$ is of type I with respect to $E'$ \item[(ii)] ${\mbox{\euf s}}_k(E',F')={\mbox{\euf s}}_k(E,F)+(r-k)$ if $F$ is of type II with respect to $E'$. \end{itemize} \end{lemma} \noindent{\bf Proof.} If $F$ is of type I, we have the following diagram $$\begin{array}{ccccccccc} & & & & 0 & & 0 & & \\ & & & &\downarrow & &\downarrow & & \\ 0 & \to & F'&\to & E' &\to & E'/F' &\to & 0 \\ & & \|& &\downarrow & &\downarrow & & \\ 0 & \to & F &\to & E &\to & E/F &\to & 0 \\ & & & &\downarrow & &\downarrow & & \\ & & & &K(x) & = &K(x) & & \\ & & & &\downarrow & &\downarrow & & \\ & & & & 0 & & 0 & & \\ \end{array}$$ Hence ${\mbox{\euf s}}_k(E',F')=k(\deg E-1)-r\deg F={\mbox{\euf s}}_k(E,F)-k$. If $F$ is type II we have the following diagram $$\begin{array}{ccccccccc} & & 0 & & 0 & & & & \\ & & \downarrow & &\downarrow & & & & \\ 0 & \to & F' &\to & E' &\to & E'/F' &\to & 0 \\ & &\downarrow & &\downarrow & &\| && \\ 0 & \to & F &\to & E &\to & E/F &\to & 0 \\ & & \downarrow & &\downarrow & & & & \\ & & K(x) & = & K(x) & & & & \\ & &\downarrow & &\downarrow & & & & \\ & & 0 & & 0 & & & & \\ \end{array}$$ Hence ${\mbox{\euf s}}_k(E',F')=k(\deg E-1)-r(\deg F-1)= {\mbox{\euf s}}_k(E,F)+(r-k).$\hspace{\fill}$\Box$ A {\it maximal subbundle} $F\in {}{Sb_k}(E)$ is by definition a subbundle of rank $k$ of maximal degree of $E$. Note that $F\in {}{Sb_k}(E)$ is a maximal subbundle if and only if $${}{\mbox{\euf s}_k}(E)={\mbox{\euf s}}_k(E,F).$$ An elementary transformation $E'$ of $E$ will be called {\it of $k$-type I} if $E$ admits a maximal subbundle of rank $k$ which is of type I with respect to $E'$. Otherwise $E'$ will be called {\it of $k$-type II}. \begin{propn} \label{1.6} If $E'$ is an elementary transformation of $E$, then $${}{\mbox{\euf s}_k}(E')=\left\{\begin{array}{lcl} {}{\mbox{\euf s}_k}(E)-k& &E'\quad\mbox{is of $k$-type I} \\ &{\rm if} & \\ {}{\mbox{\euf s}_k}(E)+(r-k)& &E'\quad \mbox{is of $k$-type II. }\end{array}\right. $$ \end{propn} \noindent{\bf Proof.} Let $F\in {}{Sb_k}(E)$ and $F'=\varphi(F)\in {}{Sb_k}(E')$. Suppose first $E'$ is of $k$-type I. If $F$ is maximal and of type I with respect to $E'$, then ${\mbox{\euf s}_k}(E',F')={}{\mbox{\euf s}_k}(E)-k$. If $F$ is maximal and of type II, then ${\mbox{\euf s}_k}(E',F')={}{\mbox{\euf s}_k}(E)+(r-k)$. If $F$ is not maximal, then $\deg \; F\le{1\over r}(k\deg E-{}{\mbox{\euf s}_k}(E))-1$ and so $${\mbox{\euf s}}_k(E,F)\ge k\deg E- r\deg F\ge{}{\mbox{\euf s}_k}(E)+r\eqno(2)$$ Hence ${\mbox{\euf s}}_k(E',F')\ge{\mbox{\euf s}}_k(E,F)-k\ge{}{\mbox{\euf s}_k}(E)+(r-k)$. This implies the assertion if $E'$ is of $k$-type I. Suppose now $E'$ is of $k$-type II. Any maximal subbundle $F \subset E$ is by assumption of type II with respect to $E'$. Hence according to Lemma 1.5 ${\mbox{\euf s}}_k(E',F')={}{\mbox{\euf s}_k}(E)+(r-k)$. If $F$ is not maximal, then (2) and Lemma 1.5 imply $${\mbox{\euf s}}_k(E',F')\ge{\mbox{\euf s}}_k(E,F)-k\ge{}{\mbox{\euf s}_k}(E)+(r-k).$$\hspace{\fill}$\Box$ \begin{rem} \label{1.7} \begin{em} One inmediately deduces from the proof of Proposition 1.6:\begin{itemize} \item[(i)] If $E'$ is of $k$-type I, then the maximal subbundles of rank $k$ of $E'$ are exactly the maximal subbundles of rank $k$ of $E$ which are of type I with respect to $E'$. \item[(ii)] If $E'$ is of $k$-type II, then the maximal subbundles of rank $k$ of $E'$ are exactly the subbundles $F'=\varphi(F)$, where $F\in {}{Sb_k}(E)$ is either maximal or of degree one less than the degree of a maximal subbundle and of type I with respect to $E'$. \end{itemize} \end{em}\end{rem} Dualizing the exact sequence (1) we obtain an exact sequence $$0\to E^*\to E'^*\to K(x)\to 0$$ Hence $E^*$ is an elementary transformation of $E'^*$, called the {\it dual elementary transformation}. \begin{cor} \label{1.8} For an elementary transformation $E'$ of $E$ the following conditions are equivalent\begin{itemize} \item[(i)] $E'$ is of $k$-type I. \item[(ii)] The dual elementary transformation $E^*$ of $E'^*$ is of $(r-k)$-type II. \end{itemize} \end{cor} \noindent{\bf Proof:} According to Proposition 1.6 and Remark 1.7, (i) holds if and only if ${}{\mbox{\euf s}_k}(E')={}{\mbox{\euf s}_k}(E)-k$. But ${}{\mbox{\euf s}_{r-k}}(E^*)={}{\mbox{\euf s}_k}(E)$ (see Remark 1.2, (b)). Hence (i) holds if and only if ${}{\mbox{\euf s}_{r-k}}(E'^*)={}{\mbox{\euf s}_{r-k}}(E^*)-k$ i.e. if and only if ${}{\mbox{\euf s}_{r-k}}(E^*) ={}{\mbox{\euf s}_{r-k}}(E'^*)+ r-(r-k)$. Applying Proposition 1.6 again gives the assertion.\hspace{\fill}$\Box$ \section{Maximal subbundles} Let $E$ denote a vector bundle of rank $r$ and degree $d$ on the curve $C$. In this section we study the set ${}{M_k}(E)$ of maximal subbundles of rank $k$ of $E$. Let $d_k$ denote the common degree of the maximal subbundles of rank $k$ of $E$. The following lemma shows that ${}{M_k}(E)$ admits a natural structure of a projective scheme over $K$. Denote by $Q:={\rm Quot}^{r-k, d-d_k}_E$ the Quot scheme of coherent quotients of rank $r-k$ and degree $d-d_k$ of $E$. \begin{lemma} \label{2.1} There is a canonical identification of ${}{M_k}(E)$ with the set of closed points of $Q$. \end{lemma} \noindent{\bf Proof.} If $F\in {}{M_k}(E)$, then $E\to E/F$ gives a closed point of $Q$. On the other hand if $\dps E\mathop{\to}^p G\to 0$ corresponds to a closed point of $Q$, then $F=\mbox{\rm ker} \; p\in {}{M_k}(E).$\hspace{\fill}$\Box$ Let ${\cal G}$ denote the universal quotient sheaf on $C\times Q$. The maximality condition implies that ${\cal G}$ is locally free. Hence if $\dps Grass_{r-k}(E)\mathop{\to}^p C$ denotes the Grassmanian scheme of $(r-k)$-dimensional quotient vector spaces of the fibres $E(x)$ and $p^*E\to{\cal U}\to 0$ the universal quotient on $Grass_{r-k}(E)$, then any $F\in {}{M_k}(E)$ corresponds on the one hand to a closed point $t$ of $Q$ and on the other hand to a section $\sigma_t: C\to Grass_{r-k}(E)$. This leads to a morphism $$\phi :\left\{\begin{array}{l} C\times Q\to Grass_{r-k}(E)\\ (x,t)\mapsto \sigma_t(x)\end{array}\right. $$ with the property that ${\cal G}=\phi^*{\cal U}$. \begin{lemma} \label{2.2} The morphism $\phi :C\times Q\to Grass_{r-k}(E)$ is finite. \end{lemma} For a proof we refers to \cite{ms} or \cite{l3}, Lemma 3.9. The geometric interpretation of Lemma 2.2 is \begin{cor} \label{2.3} Let $x\in C$ and $V\subset E(x)$ and $k$-dimensional subvector space. There are most finitely many maximal subbundles $F$ of rank $k$ of $E$ such that $F(x)=V$. \end{cor} \begin{cor} \label{2.4} dim ${}{M_k}(E)\le k(r-k).$ \end{cor} \noindent{\bf Proof:} From Lemma 2.1 there is a canonical identification of ${}{M_k}(E)$ with the set of closed points of $Q$. Since $\phi$ is finite, we have that dim $Q\le {\rm dim}\ Grass_{r-k}(E)-$dim$ C=k(r-k)+1-1.$\hspace{\fill}$\Box$ Assume now that dim ${}{M_k}(E)=n$, where $n\le k(r-k)$ according to Corollary 2.4, and let $E'$ be an elementary transformation of $E$. We want to estimate dim ${}{M_k}(E')$. Suppose that $E'$ corresponds to the pair $(x,\ell)$ with exact sequence (1) of Section 1. For any $(r-1)$-dimensional subvector space $V$ of the vector space $E(x)$ consider the Schubert cycle $$\sigma_k(V):=\{F\in Grass_{r-k}(E(x))| F\subset V\}.$$ \begin{propn} \label{2.5}If $E'$ is of $k$-type I, then dim ${}{M_k}(E')\ge {\rm dim} {}{M_k}(E)-k$. \end{propn} \noindent{\bf Proof.} Denote $V=\mbox{\rm ker}(\ell :E(x)\to K)$. According to Remark 1.7, there is a canonical identification $${}{M_k}(E')=\{F\in {}{M_k}(E)|F(x)\subseteq V\}.$$ Defining $${}{M_k}(E)(x):=\{F(x)|F\in {}{M_k}(E)\},$$ we have from Corollary 2.3 \begin{eqnarray*} \dim {}{M_k}(E')&=&\dim {}{M_k}(E')(x)\\ &=&\dim ({}{M_k}(E)(x)\cap\sigma_k(V))\\ &\ge&\dim {}{M_k}(E)(x)+\dim\sigma_k(V)-\dim Grass_{r-k}(E(x))\\ &=& n+k(r-1-k)-k(r-k)=n-k. \hspace{5cm} \Box \end{eqnarray*} This completes the proof of the proposition.\hspace{\fill}$\Box$ \begin{propn} \label{2.6} If $E'\in elm(E)$ is of $k$-type II, then {\rm dim} ${}{M_k}(E')\le ${\rm dim} ${}{M_k}(E)+r-k$. \end{propn} \noindent{\bf Proof.} According to Corollary 1.8, the dual elementary transformation $E^*$ of $E'^*$ is of $(r-k)$-type I. Hence by Proposition 2.5 $$\dim {}{M_{r-k}}(E^*)\ge\dim {}{M_{r-k}}(E'^*)-(r-k).$$ But dualizing induces a canonical isomorphism $\dps {}{M_k}(E)\mathop{\to}^{\sim} {}{M_{r-k}}(E^*)$ and similarly for $E'$. This completes the assertion.\hspace{\fill}$\Box$ \section{Stable extensions} Suppose now that the genus $g$ of the curve $C$ is $\ge \frac{r+1}{2}$. Let $r,d,k$ and $s$ be integers with $r\ge 2$, $1\le k\le r-1$, $0<s\leq k(r-k)(g-1)+(r+1)$ and $s\equiv kd\; \mbox{\rm mod}\; r$. The aim of this section is the proof of the following theorem. \begin{thm} \label{3.1} There exists an extension $0\to F\to E\to G\to 0$ of vector bundles on the curve $C$ with the following properties \begin{itemize} \item[(i)] rk $E=r$, deg $E=d$. \item[(ii)] $F$ is a maximal subbundle of rank $k$ of $E$ with ${\mbox{\euf s}}_k(E,F)=s$. \item[(iii)] $E,F$ and $G$ are stable. \end{itemize} \end{thm} Let $d_1$ be the unique integer with $s=kd -rd_1$ and $d_2=d-d_1$. According to \cite{nr} Proposition 2.4 there are finite \'etale coverings $$\pi_1:{\widetilde{M_1}}\to {\cal M}(k,d_1) \ \ {\rm and} \ \ \pi_2:\widetilde{M_2}\to {\cal M}(r-k, d_2)$$ such that there are vector bundles ${\cal F}_i$ on $C\times\widetilde{M}_i$ whose classifying map is just $id \times \pi_i$ for $i=1,2$. Let $p_{ij}$ denote the canonical projections of $C\times\widetilde{M}_1\times \widetilde{M}_2$ for $i,j=0,1,2$. According to \cite{l2}, Lemma 4.1 the sheaf $R^1p_{12*}(p^*_{02}{\cal F}^*_2\otimes p_{01}^*{\cal F}_1)$ is locally free of rank $k(r-k)(g-1)+s$ on $\widetilde{M}_1\times\widetilde{M}_2$. Let $$\pi :\mbox{\cj P}:=\mbox{\cj P}(R^1p_{12*}(p^*_{02}{\cal F}_2^*\otimes p_{01}^*{\cal F}_1)^*)\to\widetilde{M}_1\times\widetilde{M}_2$$ denote the corresponding projective bundle. According to \cite{l2}, Corollary 4.5 there is an exact sequence $$0\to\pi^*p^*_{01}{\cal F}_1\otimes {\cal O}_{\mbox{\cj P}}(1)\to{\cal E}\to\pi^*p^*_{02} {\cal F}_2\to 0\eqno(3)$$ on $C\times\mbox{\cj P} $, universal in a sense which is outlined in that paper. In particular this means that for every closed point $q\in \mbox{\cj P}$ the restriction of the exact sequence (3) to $C\times\{q\}$ is just the extension of ${\cal F}_2|_{C\times\{p_2(q)\}}$ by ${\cal F}_1|_{C\times\{p_1(q)\}}$ modulo $K^*$, which is represented by the point $q$. Here $p_i: \mbox{\cj P}\to \widetilde{M}_i$ denotes the canonical map. With $r,k,d$ and $s$ as above consider the set $$U(r,d,k,s):=\{q\in\mbox{\cj P}:{\cal E} |_{C\times\{q\}}\;\;\mbox{is stable with}\;\;{}{\mbox{\euf s}}_k({\cal E}|_{C\times\{q\}})=s\}$$ From the lower semicontinuity of the function ${}{\mbox{\euf s}}_k$ and stability being an open conditon we deduce that the set $U(r,d,k,s)$ is an open subset of $\mbox{\cj P}$. Hence, Theorem 3.1 is equivalent to the following theorem. \bigskip \begin{thm} \label{3.2} For any $r,k,d$ and $s$ as above the set $U(r,d,k,s)$ is nonempty. \end{thm} \noindent{\bf Proof.} It suffices to show that the set $$U(r,d,k,s,i) :=\{q\in \mbox{\cj P} :s_i({\cal E} _{|_{C\times \{q\}}}) >0 \ \ \mbox{ and} \ \ s_k({\cal E} _{|_{C\times \{q\}}})=s \}$$ is nonempty for any $i=1,...,r-1, i\not= k$, since $$U(r,d,k,s) = \bigcap_{\stackrel{i=1}{i\not=k}}^{r-1} U(r,d,k,s,i) $$ and the function $s_i$ is lower semicontinuous. According to Remark 1.2 (b) dualization gives a canonical bijection $$U(r,d,k,s,i) \stackrel{\sim}{\rightarrow} U(r,-d,r-k,s,r-i).$$ Hence it suffices to show that $$ U(r,d,k,s,i) \not= \emptyset,$$ for all $r,d,k,s $ as above and all $1\leq i\leq k-1.$ Choose a positive integer $N_k$ such that $$k(r-k)(g-1) \leq s +N_kk\leq k(r-k)(g-1) +r-1 \eqno(4)$$ and denote $\widetilde{d} :=d+N_k.$ We call a vector bundle $E$ out of the moduli space ${\cal M}(r,\widetilde{d} )$ {\it general}, if for all $0<j< r$ the number ${}{\mbox{\euf s}_j}(E)$ takes a maximal value, say ${\mbox{\euf s}_{j,\max}}$. By the semicontinuity of the function ${}{\mbox{\euf s}_j}$ the set of general vector bundles is open and dense in $M(r,\widetilde{d})$. According to a theorem of Hirschowitz (see \cite{h1}, Th\'eor\`eme p. 153): $${\mbox{\euf s}}_{j,\max}=j(r-j)(g-1)+\epsilon_j$$ where $\epsilon_j$ is the unique integer with $0\le\epsilon_j\le r-1$ such that $j(r-j)(g-1)+\epsilon_j\equiv j{\widetilde{d}}\;\mbox{\rm mod}\; r$. Moreover, it is shown in \cite{l1} (p. 458), that $U(r,\widetilde{d},k,s_{k,\max})$ is non-empty and its image is open and dense in $M(r,\widetilde{d})$ for all $r,\widetilde{d}$ and $k$. Let $0\rightarrow F_0 \rightarrow E_0 \rightarrow G_0 \rightarrow 0 $ be an exact sequence corresponding to a general point in $U(r,\widetilde{d},k,s_{k,\max}).$ Then $E_0, F_0$ and $G_0$ respectively are general vector bundles in ${\cal M}(r,\widetilde{d}), {\cal M}(k,d_k)$ and ${\cal M}(r-k,\widetilde{d} -d_k)$ respectively, with $d_k =\frac{1}{r}(k\widetilde{d} -s_{k,max})$ and $s_k(E_0,F_0) = s_{k,max}.$ Choose inductively for any $\nu =1,...,N_k$ an elementary transformation $E_{\nu}$ of $k$-type $I$ of $E_{\nu -1}.$ In order to complete the proof of Theorem 3.2 it sufficies to show that $$E_{N_k} \in U(r,d,k,s,i)$$ But $$s_k(E_{N_k}) =s_k(E_0) -N_kk = s_{k,max} -N_kk=s$$ and $$ \begin{array}{lll} s_i(E_{N_k})& \geq &s_i(E_0) -N_ki \ \ \ \ \ \ \ \mbox{ (by Proposition 1.6)}\\ &\geq&i(r-i)(g-1)-\frac{i}{k}(k(r-k)(g-1)-s+r-1)\\ &&\ \ \ \ \ \ \ \mbox{ (since $E_0$ is general and using (4).)}\\ &\geq &i(k-i)(g-1)-\frac{i}{k}(r-2)\ \ \ \ \mbox{(since $s\geq 1$)}\\ &>&0 \ \ \ \ \ \ \ \ \ \mbox{(since $g\geq \frac{r+1}{2} $ by assumption)} \end{array} $$ \hspace{\fill}$\Box$ \begin{rem} \label{3.3} \begin{em} \begin{itemize} \item[(a)] The assumption on the genus $g$ in Theorems 3.1 and 3.2 is imposed by the last line in the proof of Theorem 3.2. The bound $g\geq \frac{r+1}{2}$ works for any $r$, for all $k, 1\leq k\leq r-1$ simultaneously. If one fixes also $k$, the bound is slightly better. In fact, if $k=1$ or $r-1,$ the proof shows that Theorems 3.1 and 3.2 are valid for any $g\geq 2.$ (For the proof note that in both cases using duality one only has to check that $i(r-1-i)(g-1) - \frac{i}{r-1} (r-2) >0$ for all $1 \leq i \leq r -2$. But this is valid for all $g \geq 2$.) For $2\leq k\leq r-2$ denote $k=\frac{r\pm n}{2}$ with $0\leq n\leq r-4.$ Then the Theorems are valid for any $g\geq 3+2\frac{n-1}{r-n}.$ \item[(b)] There is a modification of the proof, for which the bound for $g$ is also slightly better. The duality can also be used to reduce the proof to the case $k\geq \frac{r}{2}.$ Then one has also to show that $s_i(E_{N_k}) >0$ for $k<i<r.$ For this one has to choose the sequence of bundles $E_0,E_1,...,E_{N_k}$ more carefully: Whenever possible one should use an elementary transformation of $k$-type $I$ which is of $i$-type $II.$ \end{itemize} \end{em} \end{rem} \section{Stratification of ${\cal M}(r,d)$ according to the invariant ${{\mbox{\euf s}}_k}$} The function ${}{\mbox{\euf s}_k}: {\cal M}(r,d)\to\mbox{\cj Z}$ defined by $E\mapsto {}{\mbox{\euf s}_k}(E)$ is lower semicontinuous and this induces a stratification of the moduli space ${\cal M}(r,d)$ into locally closed subvarieties $${\cal M}(r,d,k,s):= \{ E\in {\cal M}(r,d) : s_k(E)=s \}$$ according to the value $s$ of $s_k$. It is not clear (to us) whether ${\cal M}(r,d,k,s)$ is irreducible or consists of several components. Consider the natural map $$ \phi :U(r,d,k,s) \rightarrow {\cal M}(r,d,k,s) \subset {\cal M}(r,d).$$ As an image of an irreducible variety $Im\phi $ is irreducible. Let ${\cal M}^0(r,d,k,s)$ denote the Zariski closure of $Im\phi $ in ${\cal M}(r,d,k,s).$ \begin{lemma}\label{4.1} ${\cal M}^0(r,d,k,s)$ is an irreducible component of ${\cal M}(r,d,k,s),$ if is nonempty. \end{lemma} \noindent{\bf Proof:} According to \cite{nr}, Proposition 2.4 there is a finite \'etale covering $\pi : \widetilde{M} \rightarrow {\cal M}(r,d,k,s) \subset {\cal M}(r,d)$ and a vector bundle ${\cal E} $ on $C\times \widetilde{M}$ such that $id \times \pi $ is just the classifing map. Let $Q_{\cal E}$ denote the Quot scheme of ${\cal E}$ and $$0\rightarrow {\cal F} \rightarrow p^*{\cal E} \rightarrow {\cal G} \rightarrow 0 \eqno(5)$$ the universal exact sequence on $C\times \widetilde{M} \times Q_{\cal E}$. Here $p:C\times \widetilde{M} \times Q_{\cal E} \rightarrow C\times \widetilde{M}$ denotes the projection map. Certainly there are finitely many components of $Q_{\cal E}$, the union of which we denote by $Q_{\cal E}^{r-k,d-d_k}, $ such that for all closed points $(e,x) \in \widetilde{M} \times Q_{\cal E}^{r-k,d-d_k}$ the restriction ${\cal F}_{|_{C\times \{(e,x)\}}}$ is a maximal subbundle of rank $k$ and degree $d_k =\frac{1}{r}(s-kd) $ of $E={\cal E}_{|_{C\times \{(e,x)\} }}$ and moreover every maximal subbundle occurs as a restriction of the exact sequence $(5)$ to $C\times \{(e,x)\} $ for some $(e,x) \in \widetilde{M} \times Q_{\cal E}^{r-k,d-d_k}.$ As in section 2, the maximality condition implies that ${\cal F}$ and ${\cal G}$ are vector bundles. Since stableness is an open condition, it follows that the set of points $(e,x) \in \widetilde{M} \times Q_{\cal E}^{r-k,d-d_k}$ such that ${\cal F}_{|_{C\times \{(e,x)\}}}$ and ${\cal G}_{|_{C\times \{(e,x)\}}}$ are stable consists of whole components of $\widetilde{M} \times Q_{\cal E}^{r-k,d-d_k}.$ Hence the closure of the set of points $E\in {\cal M}(r,d,k,s)$ which admits a stable maximal subbundle $F$ of rank $k$ such that $E/F$ is also stable, consists also of whole components of ${\cal M}(r,d,k,s)$. But since the set of such points $E$ of ${\cal M}(r,d,k,s)$ is just the irreducible set $Im\Phi$, this implies the assertion. \hspace{\fill}$\Box$ It would be interesting to give an example for which ${\cal M}^0(r,d,k,s)\not={\cal M}(r,d,k,s).$ For an example where $Im\phi \not= {\cal M}(r,d,k,s)$ see Remark 4.5 below. The following theorem gives us the dimension of ${\cal M}^0(r,d,k,s).$ \begin{thm} \label{4.2} Let $r,d,k$, and $s$ be integers with $r\ge 2 $, $1\le k\le r-1$, $1\geq s\geq k(r-k)(g-1 +(r-2)$ and $s\equiv kd\;\mbox{\rm mod}\; r.$ Suppose the genus of $C$ is $g\ge \frac{r+1}{2}$. Then ${\cal M}^0(r,d,k,s)$ is a non-empty algebraic variety with $$\dim {\cal M}^0(r,d,k, s)=\left\{\begin{array}{lcl} (r^2+k^2-rk)(g-1)+s+1 & & s<k(r-k)(g-1)\\ &{\rm if}& \\ r^2(g-1)+1 & & s\ge k(r-k)(g-1)\end{array}\right. $$ \end{thm} \noindent{\bf Proof:} If $s\ge k(r-k)(g-1)$, then $s =k(r-k)(g-1)+\epsilon_k$ where $\epsilon_k$ is the unique integer with $0\le\epsilon_k\leq r-1$ and $s\equiv kd\;\mbox{\rm mod}\; r$. Then $Im\phi$ is open and dense in ${\cal M}(r,d)$, which gives the assertion in this case (see [6]). So we may assume that $s <k(r-k)(g-1)$. Consider the open set $U(r,d,k,s)$ in the variety $\mbox{\cj P} =\mbox{\cj P}(R^1_{p_{12_*}}(p^*_{02}{\cal F}^*_2\otimes p^*_{01}{\cal F}_1)^*)$ of Section 3. According to Theorem 3.2 $U(r,d,k,s)$ is non-empty, open and dense in $\mbox{\cj P}$. According to the definitions of $U(r,d,k,s)$ and ${\mbox{\euf s}_k}$ the natural map $$\phi:U(r,d,k,s)\longrightarrow {\cal M}^0(r,d,k,s)\subseteq {\cal M}(r,d)$$ is dominant. We have to compute the dimension of ${\cal M}(r,d,k,s)$. Let $q\in U(r,d,k,s)$ be a general closed point. If $0\to F\to E\to G\to 0$ denotes the corresponding exact sequence, then $\phi(q)=E$ and \cite{l1}, Lemma 4.2 says that $$\dim\phi^{-1}(E)\le h^0(F^*\otimes G).$$ On the other hand, $F$ and $G$ are general vector bundles in their corresponding moduli spaces. Hence according to \cite{h2} Th\'eor\`em 4.6, the vector bundle $F^*\otimes G$ is non-special, implying $$h^0(F^*\otimes G)=0$$ since $\deg \ (F^*\otimes G)=s < k(r-k)(g-1)={\rm rk}(F^*\otimes G)(g-1)$. Hence the generic fibre of $\phi$ is finite and thus \begin{eqnarray*} \dim {\cal M}^0(r,d,k,s)&=&\dim U(r,d,k,s)\\ &=&\dim \mbox{\cj P}(R^1p_{12*}(p^*_{02}{\cal F}^*_2\otimes p^*_{01}{\cal F}_1)^*)\\ &=&\dim {\cal M}(k,d_1)+\dim {\cal M}(r-k,d-d_1)+k(r-k)(g-1)+s-1 \end{eqnarray*} with $d_1=\deg F={1\over r}(kd -s)$. Note that $\mbox{\cj P}(R^1p_{12*}(p^*_{02}{\cal F}^*_2\otimes p^*_{01}{\cal F}_1)^*)$ is a projective bundle of rank $k(r-k)(g-1)+s-1$ over a finite covering of ${\cal M}(k,d_1)\times {\cal M}(r-k, d-d_1)$ (see \cite{l1}, p. 455). Hence \begin{eqnarray*} \dim {\cal M}^0(r,d,k,s)&=&k^2(g-1)+1+(r-k)^2(g-1)+1+k(r-k)(g-1)+s-1\\ &=&(r^2+k^2-rk)(g-1)+s +1. \hspace{5,2cm} \Box \end{eqnarray*} The fibres $\phi^{-1}(E)$ of the map $\phi:U(r,d,k,s)\to {\cal M}^0(r,d,k,s)$ of the proof of Theorem 4.2 are exactly the sets of stable maximal subbundles of $E$, whose quotient is also stabile. However, maximal subbundles are not necessarily stable. Noting that a maximal subbundle of a maximal subbundle of $E$ is also a subbundle of $E$ (and similarly for quotient bundles) one easily shows \bigskip \begin{propn} \label{4.3} Suppose $E\in {\cal M}(r,d)$ with ${\mbox{\euf s}_k}(E)=s$ for some $1\le k\le r-1$. Let $F$ be a maximal subbundle of rank $k$ of $E$. Then \begin{itemize} \item[(i)] ${\mbox{\euf s}_{\nu }}(F)\ge{1\over r}(k-\nu s)$ for all $1\le \nu \le k-1$ \item[(ii)] ${\mbox{\euf s}_{\nu }}(E/F)\ge{1\over r}((r-k)-(r-k-\nu )s)$ for all $1\le \nu \le r-k-1$. \end{itemize} \end{propn} So in particular for highs ${\mbox{\euf s}_{\nu}}(F)$ or ${\mbox{\euf s}_{\nu }}(E/F)$ might be very negative. According to Lemma 2.1 there is a canonical identification of the set ${}{M_k}(E)$ of maximal subbundles of $E$ with the Quot-scheme $Q = Quot _E ^{r-k,{1\over r}((r-k)d+s)}$. Hence there is a universal subbundle ${\cal F}$ of $p^*E$ on the scheme $C\times {}{M_k}(E)$, where $p:C\times {}{M_k}(E)\to C$ denotes the projection map. Denote $${\widetilde{M_k}}(E):=\{ F\in {}{M_k}(E)|F\;\;{\rm and}\;\; E/F\;{\rm stable}\}.$$ and by $\widehat{{M_k}}(E)$ the Zariski closure of ${\widetilde{M_k}}(E)$ in ${}{M_k}(E)$. Applying the openness of stability to the universal subbundle ${\cal F}$ and the universal quotient bundle $p^*E/{\cal F}$ of $p^*E$ one deduces that $\widehat{M_k}(E)$ consists of whole irreducibility components of ${M_k}(E)$, namely exactly of those components of ${{}{M_k}}(E)$ which contain a stable subbundle $F$ of $E$ such that $E/F$ is also stable. Moreover $\widetilde{M_k}(E)$ is open in $\widehat{M_k}(E)$. By definition we may canonically identify $${\widetilde{M_k}}(E)\to \phi^{-1}(E).$$ This implies that $$\dim{\widehat{M_k}}(E)=\dim\phi^{-1}(E).\eqno(6)$$ and we may use the map $\phi$ to compute the dimension of $\widehat{{M_k}}(E).$ Let $r,d,k$ and $s$ be integers as above and $g(C)\geq \frac{r+1}{2}$. According to Theorem 4.2 the variety ${\cal M}^0(r,d,k,s)$ is non empty. \bigskip \begin{thm} \label{4.4} For a general vector bundle $E$ in ${\cal M}^0(r,d,k,s)$ we have $\dim {\widehat{M_k}}(E)=\max(s -k(r-k)(g-1),0).$ \end{thm} \bigskip In particular, if $E$ is general in ${\cal M}(r,d)$ there is a unique integer $\epsilon_k$ with $0\le\epsilon_k\le r-1$ and $k(r-k)(g-1)+\epsilon_k\equiv kd\;\mbox{\rm mod}\; r$ and we have $$\dim {\widehat{M_k}}(E)=\epsilon_k$$ If $s$ is not maximal value, i.e. $\mbox{\euf s} <k(r-k)(g-1)$, then a general vector bundle in ${\cal M}^0(r,d,k,s)$ admits only finitely many stable maximal subbundles such that $E/F$ is also stable. \noindent{\bf Proof of Theorem 4.4:} The natural map $\phi :U(r,d,k,s)\to {\cal M}^0(r,d,k,s)$ is a dominant morphism of algebraic varieties by the definition of $U(r,d,k,s)$ and ${\cal M}^0(r,d,k,s)$. Let $q\in U(r,d,k,s)$ be a general point and $0\to F\to E\to G\to 0$ be the corresponding exact sequence. According to \cite{l1} Lemma 4.2 and Hirschowitz' Theorem (see \cite{h2} Th\'eor\`eme 4.6) we have \begin{eqnarray*} \dim\phi^{-1}(E)&\le& h^0(F^*\otimes G)\\ &\le&\max(s-k(r-k)(g-1),0) \end{eqnarray*} So equation (6) implies the assertion for $s\le k(r-k)(g-1)$. For $s>k(r-k)(g-1)$ it suffices to show that the local dimension of $\phi^{-1}(E)$ at $q$ is equal to $ s-k(r-k)(g-1)$. But Mori showed in \cite{mo} that $$h^0(F^*\otimes G)-h^1(F^*\otimes G)\le \dim_qQ_{\cal E}$$ Again by \cite{h1} Th\'eor\`eme 4.6 the vector bundle $F^*\otimes G$ is non-special implying $h^1(F^*\otimes G)=0$ and thus $$\dim\widehat{M_k}(E)=\dim\phi^{-1}(E)=h^0(F^*\otimes G)=s-k(r-k)(g-1). \eqno\Box $$ \begin{rem}\label{4.5}\begin{em} Take $r=3$, $d=1$, $k=2$, $s=2$ and $g\geq 2.$ In \cite{bgn} it was proved that there are extensions $0\rightarrow {\cal O}^2 \rightarrow E\rightarrow L\rightarrow 0 $ of a line bundle $L$ of degree $1$ by the trivial bundle ${\cal O}^2$ such that $E$ is stable. Actually, such bundle $E$ is in ${\cal M}(3,1,2,2)$, since $\mu (E) <1$ and hence $s_2(E) =2.$ However, for such bundles there is no stable subbundles of degree $0$ and hence $E\notin Im\phi .$ Such bundles $E$ are in the Brill-Noether locus ${\cal W}^k_{r,d}.$ An interesting problem is to study the relation between the Brill-Noether loci ${\cal W}^k_{r,d}$ and the ${\cal M}^0(r,d,k,s)$ varieties. \end{em}\end{rem}

9708

alg-geom/9708011

en

https://arxiv.org/abs/alg-geom/9708011

alg-geom/9708011

Balazs Szendroi

Balazs Szendroi

Some finiteness results for Calabi-Yau threefolds

15 pages LaTex, uses amstex, amscd. New title, paper completely rewritten, results same as in previous versions

We investigate the moduli theory of Calabi--Yau threefolds, and using Griffiths' work on the period map, we derive some finiteness results. In particular, we confirm a prediction of Morrison's Cone Conjecture.

alg-geom

\section*{Introduction} If $X$ is a smooth complex projective $n$-fold, Hodge--Lefschetz theory provides a filtration on the primitive cohomology $H^n_0(X,{\mathbb C})$ by complex subspaces, satisfying certain compatibility conditions with a bilinear form $Q$ on cohomology. This gives a map called the {\it period map}, from a suitably defined moduli space containing $X$ to a complex analytic space $\bd / \Gamma$, the study of which was initiated by Griffiths. He showed in particular that if $X$ has trivial canonical bundle, then this map is locally injective on the Kuranishi family of $X$; further, if the global moduli theory is well-behaved, then the map can be extended to a proper map and so finiteness results can be derived. This paper considers {\it Calabi--Yau\ threefold s}. A complex projective manifold $X$ is Calabi--Yau, if it has trivial canonical bundle and satisfies $H^i(X,{\bigo}_X)=0$ for $0<i<\dim (X)$. In Section 1 we recall a theorem about their Hilbert schemes, in Section 2 we investigate the moduli theory. Then we specialize to threefolds, recall some of Griffiths' results in Section 3, which will enable us to deduce the crucial finiteness statement Theorem~\ref{maintheorem}: the period point determines the threefolds up to finitely many choices among those with bounded polarization. This will imply Corollaries 4.3-4.5, which constitute the main results of this paper. In particular, we confirm the following consequence of Morrison's Cone Conjecture: \vspace{0.1in} \noindent {\bf Corollary} {\it Let $X$ be a smooth Calabi--Yau\ threefold, fix a positive integer $\kappa$. Up to the action of ${\mathop{\rm Aut}\nolimits}\, (X)$, there are finitely many ample divisor classes $L$ on $X$ with $L^3\leq \kappa$. In particular, if the automorphism group is finite, there are finitely many such classes.} } \noindent {\bf Conventions} \,\, All schemes and varieties are assumed to be defined over ${\mathbb C}$, points of schemes are ${\mathbb C}$-valued points. By a polarized variety $(X, L)$ we mean a projective variety with a choice of an ample invertible sheaf, $(X,L)\cong (X', L')$ if there is an isomorphism $\phi:X\stackrel{\sim}{\longrightarrow} X'$ with $\phi^*(L')\sim L$. The highest self-intersection of $L$ is denoted by $L^n$. A family of polarized varieties is a flat proper morphism or holomorphic map $\bx \rightarrow S$, with an invertible sheaf ${\mathcal L}$ on $\bx$ whose restriction to every fibre is ample. \section{The Hilbert scheme of Calabi--Yau\ manifolds} \label{hilbert} First we recall the Unobstructedness Theorem for manifolds with trivial canonical bundle (${\mathcal T}_X$ is the holomorphic tangent bundle of $X$): \begin{theorem} {\rm (Bogomolov, Tian~\cite{tian}, Todorov~\cite{todorov} in the complex case, Ran~\cite{ran}, Kawamata~\cite{kawa_unobs} in the algebraic case)} Let $X$ be a smooth projective $n$-fold with trivial canonical bundle, then it has a versal deformation space $\bx \rightarrow S$ over a complex germ or spectrum of a complete Noetherian local ${\mathbb C}$-algebra $S$ with $0\in S$, $\bx_0\cong X$, and $S$ smooth. If $H^0(X, {\mathcal T}_X)=0$, this deformation is universal. The tangent space of $S$ at $0$ is canonically isomorphic to $\kod{X}$. \end{theorem} \vspace{-.2in \noindent Using standard arguments, eg. \cite{huy} Appendix A, one obtains \begin{propos} Let $n\geq 3$, $X$ as above and assume further that $H^2(X,{\mathcal O}_X)=0$. The family $\bx \rightarrow S$ is also a versal family for invertible sheaves on $X$, that is given any sheaf $L$ on $X$, there is a sheaf ${\mathcal L}$ on $\bx$ restricting to $L$ on the central fibre, with the obvious versal property. \label{univ_pol} \end{propos} \vspace{-.2in Let now $(X,L)$ be a polarized\ Calabi--Yau\ $n$-fold\ with Hilbert polynomial $p$. Matsusaka's Big Theorem~\cite{matsusaka} gives us an integer $m$ with the following property: for any polarized algebraic manifold $(X_1,L_1)$ with Hilbert polynomial $p$, the sheaf $L_1^{\otimes m}$ is very ample and has no higher cohomology. Put $N=h^0(L_1^{\otimes m})-1=p(m)-1$, then we have embeddings $\phi_{\lin{L_1^{\otimes m}}}: X_1\rightarrow {\mathbb P}^N$ and in particular embeddings $\phi_{\lin{L^{\otimes m}}}: X\rightarrow {\mathbb P}^N$, depending on the choice of a basis of $H^0(X,L^{\otimes m})$. So for a fixed choice of basis, we get a point $x\in \it Hilb_{\bp^N}^{p}$, where $\it Hilb_{\bp^N}^{p}$ is the fine projective moduli scheme representing the Hilbert functor of ${\mathbb P}^N$ with polynomial $p$. \begin{theorem} {\rm (cf. \cite{jorgtodorov})} Assume that $n\geq 3$. If $(X,L)$ is a polarized\ Calabi--Yau\ $n$-fold, then the scheme $\it Hilb_{\bp^N}^{p}$ is smooth at $x$. \label{smoothhilbert} \end{theorem} \vspace{.05in Using the Euler sequence of ${\mathbb P}^N$ restricted to $X$ and Kodaira vanishing, one obtains $H^1(X,{\mathcal T}_{{\mathbb P}^N}\restr{X})=0$. The normal bundle sequence now gives that $H^1(X, {\mathcal N}_{X/{\mathbb P}^N})\rightarrow H^2(X,{\mathcal T}_X)$ is injective, whereas $H^0(X, {\mathcal N}_{X/{\mathbb P}^N})\rightarrow\kod{X}$ is surjective. Hence by Unobstructedness the deformations of $X\subset{\mathbb P}^N$ are also unobstructed, and all deformations of $X$ can be realized in ${\mathbb P}^N$. The Theorem follows. \hspace*{\fill The following Lemma is also standard: \begin{lemma} Let $\bx\rightarrow S$ be a flat family of projective varieties with smooth fibres over the base $S$. If for $0\in S$ the fibre $\bx_0$ is Calabi--Yau, then all fibres are Calabi--Yau s. \label{allcy} \end{lemma} \hspace*{\fill Let $\bx_{Hilb}\rightarrow\it Hilb_{\bp^N}^{p}$ be the a universal family over the Hilbert scheme with the universal relatively ample invertible sheaf ${\mathcal L}_{Hilb}$ over $\bx_{Hilb}$, $H^{\prime}$ the open subset of $\it Hilb_{\bp^N}^{p}$ over which this family has smooth fibres. The quasi-projective scheme $H^{\prime}$ has several irreducible components, fix one component $H$ which contains a point corresponding to a smooth polarized\ Calabi--Yau\ fibre $(X,L)$. By the Lemma, all fibres of the pullback family $\bx_H\rightarrow H$ are polarized\ Calabi--Yau\ manifolds, so $H$ is a smooth quasi-projective variety. Let now $G=SL(N+1,{\mathbb C})$. As usual, there is an action of $G$ on $\it Hilb_{\bp^N}^{p}$. From the definition of $H$ and connectedness of $G$, it follows that there is an induced action $\sigma: G \times H \rightarrow H$. By the universal property, the action extends to an action of $G$ on $\bx_H$. The action $\sigma$ is proper, this follows from `separatedness of the moduli problem': since fibres are never ruled by Lemma~\ref{allcy}, an isomorphism of polarized families over the generic point of the spectrum of a DVR specialises to an isomorphism over the special fibre (Matsusaka and Mumford~\cite{matsumum}). Any $h\in H$ has reduced finite automorphism group. \section{The moduli space} \label{moduli} \begin{propos} The quotient $H/G$ is a separated algebraic space of finite type over ${\mathbb C}$. \end{propos} \vspace{.05in This follows from \cite{popp} II, Theorem 1.4. For an algebraic proof, see~\cite{keel_mori}. \hspace*{\fill The aim of this section is to prove \begin{theorem} There exists a quasiprojective scheme $Z$ with the following properties: \begin{enumerate} \item There exists a family $\bx_Z \rightarrow Z$ of smooth Calabi--Yau\ varieties over $Z$, polarized by an invertible sheaf ${\mathcal L}_Z$ on $\bx_Z$. \item The classifying map of the family $\bx_Z \rightarrow Z$ is a finite surjective map of algebraic spaces $Z\rightarrow H/G$. \item For $t\in Z$, let $\bx_t$ be the fibre of the family. Then the spectrum of the completion of the local ring ${\mathcal O}_{Z,t}$ together with the induced family is the (algebraic) versal family of $\bx_t$. In particular, by Unobstructedness, $Z$ is smooth. \end{enumerate} \label{moduli_theorem} \end{theorem} This theorem can be proved in two different ways. The proof we give below consists of two steps: first one builds an \'etale cover $H^{et}\rightarrow H$ directly, with a free $G$-action, using a rigidification construction; then $Z$ exists as an algebraic space, and $i\,$ of the Theorem together with results of Viehweg~\cite{viehweg} implies that $Z$ is quasiprojective. An alternative way was pointed out to the author by Alessio Corti: $H/G$ is the coarse moduli space representing the stack ${\mathcal Z}$ whose category of sections over a ${\mathbb C}$-scheme $S$ is the set of polarized families of Calabi--Yau\ $n$-fold s over $S$ as objects, with isomorphisms over $S$ as morphisms. ${\mathcal Z}$ is in fact a Deligne-Mumford stack, and one can show the existence of $Z$ as a finite union of affine schemes satisfying {\it i-iii} using Artin's method as follows: consider algebraizations of formal versal families of individual varieties~\cite{artin_algebraization}, and use openness of versality~\cite{artin} to show that a finite union of them covers $H/G$ and $iii$ is satisfied. The author decided to give the proof below because he feels that it is natural in the context and it is more concrete. \begin{propos} Condition iii follows from \vspace{0.1in} \noindent $iii^{\prime}\,$. $\,Z\cong H^{et}/G$ where $H^{et}\rightarrow H$ is a finite \'etale cover and $G$ acts freely on $H^{et}$. \end{propos} \vspace{.05in Let $\by\rightarrow S$ be the versal deformation space of $\bx_t$ over the spectrum of a complete local ${\mathbb C}$-algebra, with $\by_0\cong \bx_t$. The variety $\bx_t$ comes equipped with a distinguished ample sheaf ${\mathcal L}_t$ over it. By Proposition~\ref{univ_pol}, there is a relatively ample sheaf ${\mathcal L}$ over $\by$ extending ${\mathcal L}_t$, and we can think of $S$ as the base space of versal deformations of $(\bx_t, {\mathcal L}_t)$. Let $U={\mathop{\rm Spec}\nolimits}\,(\hat {\mathcal O}_{Z,t})$ with closed point still denoted by $t$, then the pullback family $\bx_U\rightarrow U$ is a flat polarized deformation of $\bx_t$. So by the definition of the versal family, there is a morphism $\epsilon: U\rightarrow S$ mapping $t$ to $0$ such that the family over $U$ is a pullback by $\epsilon$. On the other hand, we may assume that ${\mathcal L}$ can be trivialized by $N+1$ sections over $S$. From the universal property of the Hilbert scheme, this determines a morphism $S\rightarrow {\mathop{\rm Spec}\nolimits}\, (\widehat {\mathcal O}_{H,h})$, so a morphism $S\rightarrow {\mathop{\rm Spec}\nolimits}\, (\widehat {\mathcal O}_{H^{et},h'})$, where $h^\prime$ is chosen so that the composition with the morphism coming from $H^{et} \rightarrow Z$ gives a map $\tau:S\rightarrow U$, mapping $0$ to $t$. The composite $\tau \circ \epsilon : U\rightarrow U$ fixes $t$ and pulls back the family over $U$ to itself, so as the action of $G$ on $H^{et}$ is free, it is the identity. Similarly, $\epsilon\circ\tau:S \rightarrow S$ fixes $0$ and pulls back the polarized family over $S$ to itself, so by universality it has finite order (it can be nontrivial, giving an automorphism of the polarized central fibre). So $\epsilon$ and $\tau$ are isomorphisms, and $iii$ follows. \hspace*{\fill \vspace{0.05in} \noindent {\sc Proof of Theorem~\ref{moduli_theorem}}\hspace{.05in} First we construct $H^{et}$, using a method which is best known in the case of curves, and was applied in the higher dimensional case by Popp~\cite{popp} I, followed by a direct construction. Let us consider a smooth polarized family $\by \rightarrow S$ of complex projective $n$-folds such that the automorphisms of fibres are finite, let $(X, L)$ be a fixed fibre. Denote $H_{\mathbb Z}(\by_s)=H^n(\by_s, {\mathbb Z})/\mathop{\rm torsion}\nolimits$, a free ${\mathbb Z}$-module with a pairing $Q_s$; consider the map \begin{equation} \theta_s : {\mathop{\rm Aut}\nolimits}\,(\by_s, {\mathcal L}_s) \rightarrow {\mathop{\rm Aut}\nolimits}\,(H_{\mathbb Z}(\by_s), Q_s), \label{image} \end{equation} with image $\Theta_s$. For any $s\in S$, $H_{\mathbb Z}(\by_s)\cong H_{\mathbb Z}(X)$, as the family is locally topologically trivial. Let ${\rm I}\, (s)$ be the set of minimal orthonormal or symplectic generating systems of $H_{\mathbb Z}(\by_s)$, then $\Lambda={\mathop{\rm Aut}\nolimits}\,(H_{\mathbb Z}(\by_s), Q_s)$, a group of matrices over ${\mathbb Z}$, acts transitively on ${\rm I}\,(s)$. Consider the disjoint union $\tilde{S} = \cup_{s\in S}\,{\rm I}\,(s)$, then there is a map $\gamma: \tilde{S} \rightarrow S$ which allows one to put a topology on $\tilde{S}$ in a standard way, such that $\gamma$ is a topological covering with covering group $\Lambda$. Now recall the following result: \begin{lemma} {\rm (Serre~\cite{serre})} Let $l\geq 3$ be an integer, $\alpha \in GL_m ({\mathbb Z})$ an invertible matrix of finite order satisfying $\alpha \equiv I_m \mbox{ (mod $l$)}$. Then $\alpha = I_m$ the identity. \end{lemma} By assumption, $\Theta_s$ is a finite subgroup of $\Lambda$ for any $s$, so all its elements have finite order. For any integer $l\geq 3$, let $\levell{\Lambda}$ be the {\it l-th congruence subgroup} of $\Lambda$. Applying the above Lemma, the intersection of $\levell{\Lambda}$ and any $\Theta_s$ will be trivial. Let $\levell{\bar{\Lambda}}$ be the quotient of $\Lambda$ by $\levell{\Lambda}$, let $\levell{S}$ be the finite unramified covering of $S$ corresponding to this finite group. $\levell{S}$ is called the {\it level-l cover} of $S$. It is naturally a complex analytic space, so by the generalized Riemann existence theorem it has the structure of a complex scheme such that $\levell{S}\rightarrow S$ is an \'etale morphism. Consider the finite unramified cover $\levell{H}\rightarrow H$, together with the family $\bx_{\levell{H}} \rightarrow \levell{H}$ of polarized\ Calabi--Yau s pulled back from the Hilbert family $\bx_H \rightarrow H$. \begin{lemma} There is a proper action of the group $G=SL(N+1,{\mathbb C})$ on $\levell{H}$, \[ \rho: G \times \levell{H} \rightarrow \levell{H}\] and the map $\levell{H}\rightarrow H$ is a $G$-morphism. $G$ also acts on $\bx_{\levell{H}}$. \end{lemma} \vspace{.05in \cite{popp} I, 2.19. \hspace*{\fill \noindent The stabilizers under the action $\rho$ are the kernels of the maps~(\ref{image}), which are of very special type: \begin{lemma} Let $(X,L)$ be a polarized\ Calabi--Yau, $\alpha\in \ker\,(\theta)\subset{\mathop{\rm Aut}\nolimits}\,(X,L)$, i.e. assume that $\alpha$ acts trivially on the n-th cohomology. Let $\bx\rightarrow S$ be a small polarized deformation of $(X,L)$, then $\alpha$ extends to give an automorphism of the family $\bx$ over $S$ leaving $S$ fixed and also fixing the polarization. \label{action} \end{lemma} \vspace{.05in Once $\alpha$ extends to $\bx$ fixing $S$, it fixes the polarization also, since it fixes $L$ and the Picard group of a Calabi--Yau\ is discrete. We may also assume that $\bx\rightarrow S$ is an fact the Kuranishi family. Then by universality, $\alpha$ gives an automorphism $\tilde \alpha$ of $\bx$ over $S$. Assume that $\tilde \alpha$ acts nontrivially on $S$. Then it must also act nontrivially on the tangent space to $S$ at $X$, i.e. $\kod{X}$. This is however a direct summand of $H^n(X,{\mathbb C})$, so $\tilde\alpha$ acts nontrivially on that and hence also on $H_{\mathbb Z}(X)$. This is a contradiction. \hspace*{\fill \begin{lemma} There exists a cover $\rhostr{H} \rightarrow \levell{H}$ with a finite covering group $K$, which becomes a finite unramified map when we give $\rhostr{H}$ the induced scheme structure. There is an induced action of $G$ on $\rhostr{H}$, which is proper and free. The action extends to an action on the pullback family $\bx_{\rhostr{H}}\rightarrow \rhostr{H}$. \end{lemma} \vspace{.05in For any $h\in \levell{H}$, denote by $K_h$ the set of automorphisms of $(\bx_h, {\mathcal L}_h)$ that extend to the Kuranishi family fixing the base. For any $h$, this is isomorphic to the group $K$ of generic isomorphisms of the family. Let $\rhostr{H} = \cup_{h\in \levell{H}} K_h$, then as before, $\rhostr{H}$ can be given an induced scheme structure such that we get an unramified covering. It is easy to check that there is a proper action of $G$ on $\rhostr{H}$, and Lemma~\ref{action} will imply that the action is free. \hspace*{\fill $\rhostr{H} \rightarrow H$ is the cover $H^{et}\rightarrow H$ in $iii^{\prime}$ of the Proposition. \begin{lemma} The quotient $Z$ of $\rhostr{H}$ by $G$ exists as a quasi-projective scheme, and there is a polarized family $\bx_Z\rightarrow Z$ with smooth fibres over it. \label{quasiprojective} \end{lemma} \vspace{.05in As before, the quotient $Z$ exists as an algebraic space of finite type, and \cite{popp} III, 1.4 shows that there is a polarized family $\bx_Z\rightarrow Z$ with smooth fibres. The total space $\bx_Z$ is at the moment also an algebraic space only, but it is quasi-projective if $Z$ is. Also, from the construction and the Proposition above we obtain that $Z$ is smooth. Using smoothness of $\rhostr{H}$ and the assumptions about the action of $G$ on it, by Seshadri's Theorem \cite{seshadri}, 6.1 we have a diagram \[ \begin{CD} V @>p>> \rhostr{H} @>>> H \\ @VqVV @VVV \\ T @>>> Z. \end{CD}\] Here $T$, $V$ are normal schemes, $G$ acts on $V$ with geometric quotient $T$, and a finite group $F$ also acts on $V$ with quotient $\rhostr{H}$ such that the actions of $G$ and $F$ on $V$ commute. In particular, the map $T \rightarrow Z$ is finite. Let us pull back the family $\bx_Z\rightarrow Z$ to $T$, denote ${\mathcal L}={\mathcal L}_{T}$, $\omega=\omega_{\bx_T / T}$. We will use the deep results due to Viehweg~\cite{viehweg} to show that the scheme $T$ is quasi-projective, we refer for terminology and results to \cite{viehweg}. Let $m$ be an integer such that for $t\in T$, ${\mathcal L}_t^{\otimes m}$ is very ample on $\bx_t$ and has no higher cohomology. Choosing an integer $l>({\mathcal L}_t^{\otimes m})^n+1$, all the conditions of the Weak Positivity Criterion [ibid] 6.24 are satisfied (the dualizing sheaf is trivial on fibres), in particular $\tau_* ({\mathcal L}^{\otimes m})$ is locally free of rank $r$ on $T$, and we obtain a weakly positive sheaf \[ (\bigotimes^{rm} \tau_*({\mathcal L}^{\otimes rm}\otimes \omega^{\otimes lrm}) ) \otimes (\det\,(\tau_* {\mathcal L}^{\otimes m}))^{-rm} \] over $T$. Using [ibid] 2.16d the sheaf \[ {\mathcal A}=\tau_*({\mathcal L}^{\otimes rm}\otimes \omega^{\otimes lrm}) \otimes (\det\,(\tau_* {\mathcal L}^{\otimes m}))^{-1} \] is also weakly positive over $T$. Then for integers $\mu>1$, denote \[ {\mathcal Q} = \tau_*({\mathcal L}^{\otimes rm\mu}\otimes \omega^{\otimes lrm\mu}) \otimes (\det\,(\tau_* {\mathcal L}^{\otimes m}))^{-\mu} \] and look at the multiplication map \[ S^{\mu} ({\mathcal A}) \rightarrow {\mathcal Q}. \] For $\mu$ large enough, exactly as in the argument on [ibid] p.304, the kernel of this map has maximal variation (it is here that we use the fact that every polarized fibre occurs only finitely many times by construction). The Ampleness Criterion [ibid] 4.33 therefore applies, so for suitable (large) integers $a,b$ the sheaf \[ {\mathcal B} = \det({\mathcal A})^{\otimes a} \otimes \det({\mathcal Q})^{\otimes b} \] is ample on $T$. Hence $T$ is quasi-projective. To finish the proof, we use \begin{lemma} Assume that $\delta: Y^\prime\rightarrow Y$ is a finite surjective map from a scheme $Y^\prime$ to a normal algebraic space $Y$, let $L$ be an invertible sheaf on $Y$. If $\delta^*(L)$ is ample on $Y^\prime$, $L$ is ample on $Y$. \end{lemma} \vspace{.05in This follows from \cite{ega3}, 2.6.2, noting that the proof given there carries over to the case when $Y$ an algebraic space. \hspace*{\fill If we construct the sheaves ${\mathcal A}_Z, {\mathcal Q}_Z, {\mathcal B}_Z$ using the relative dualizing sheaf and polarization of the family over $Z$ exactly as for $T$, they pull back to the sheaves ${\mathcal A}, {\mathcal Q}, {\mathcal B}$ on $T$ via the finite surjective map $T\rightarrow Z$. By the Lemma, ${\mathcal B}_Z$ is ample on $Z$, so the proof of Lemma~\ref{quasiprojective}, and therefore also the proof of Theorem~\ref{moduli_theorem}, are complete. \hspace*{\fill \section{The period map} \label{period} From now on, let us assume that $n=3$. If $X$ is a Calabi--Yau\ threefold, Serre duality gives $H^5(X,{\mathbb C})=0$, so the whole third cohomology is primitive for topological reasons. Fix a non-negative integer $b$, let $V_{\mathbb Z}$ be the unique $(2 b+2)$--dimensional lattice with a unimodular alternating form $Q$ (the fact that this lattice is unique is proved in \cite{adkins} 6.2.36). Let $V = V_{\mathbb Z}\otimes {\mathbb C}$. The period map for Calabi--Yau\ threefold s $X$ with $H^3 (X,{\mathbb C})\cong V$ takes values in the domain \[ {\mathcal D} = \left\{ \mbox{flags } V=F^0\supset F^1 \supset F^2 \supset F^3 \mbox{ with } \dim F^p=f_p, \mbox{ satisfying (R)} \right\}, \] where $f_0=2b+2, f_1=2b+1, f_2=b+1, f_3=1$ and (R) are the Riemann bilinear relations $Q\,(F^p, F^{4-p})= 0$, $(-1)^{p+1}\,i Q(\xi, \bar \xi) > 0$ for nonzero $\xi \in F^p \cap {\bar F}^{3-p}$. The arithmetic monodromy group $\Gamma={\mathop{\rm Aut}\nolimits}\,(V_{\mathbb Z}, Q)$ acts on ${\mathcal D}$, it is well known \cite{griffiths1} that the action is proper and discontinuous, and $\bd / \Gamma$ is a separated complex analytic space. Let us define the set \[ {\mathcal C}_b = \{ X \mid X \mbox{ \rm a Calabi--Yau\ threefold\ with } b_3(X)=2 b+2\} / \cong. \] We have isomorphisms $V\cong H^3(X, {\mathbb C})$ for any $X\in{\mathcal C}_b$ well-defined up to elements of $\Gamma$, so there is a map (the `period map') \[ \phi:{\mathcal C}_b \rightarrow \bd / \Gamma \\ \] mapping $X$ to the filtration on the primitive cohomology. This is only a map between sets. However, assume that $\pi: \bx\rightarrow S$ is a smooth complex analytic family of Calabi--Yau\ threefold s with $b_3 (\bx_s)=2 b+2$ over a smooth contractible complex base. Fixing a point $0\in S$, the fibre $X$ over $0$ and a marking of the cohomology $V_{\mathbb Z} \cong H^3(X, {\mathbb Z})$, we can define the map \[ \psi: S \rightarrow {\mathcal D} \\ \] using the Leray cohomology sheaf ${\mathcal E} = R^3 {\pi}_* {\mathbb C}$ on $S$, equipped with the Gauss-Manin connection, and the bilinear form $Q: {\mathcal E} \times {\mathcal E} \rightarrow {\mathcal O}_S$ defined by integrating over the fibres the wedge product of two $3$-forms. Griffiths~\cite{griffiths1} proved that the map $\psi$ is holomorphic, and if $\pi: \bx\rightarrow S$ is the Kuranishi family of $X$, the derivative of $\psi$ is injective, so it is locally an embedding (`Infinitesimal Torelli' holds for $X$). To discuss global properties of $\psi$, assume that the base $S$ is quasi-projective, not necessarily contractible, and $\bx \rightarrow S$ is a smooth polarized algebraic family. There is a smooth compactification \[ \begin{CD} \bx @>i>> {\bar \bx} \\ @V{\pi}VV @VV{\bar{\pi}}V \\ S @>>j>{\bar S}, \end{CD} \] where $i,j$ are inclusions, $\bar{\pi}:{\bar \bx} \rightarrow \bar{S}$ is a proper map between smooth projective varieties with connected fibres and the boundary divisor $D={\bar S} \setminus S$ has simple normal crossings. Using the Gauss-Manin connection again, we can define a map \[ \psi: S \rightarrow \bd / \Gamma. \] This map is in fact well-defined if one quotients ${\mathcal D}$ by the image $\Gamma_0$ of the fundamental group $\pi_1(S)$ under the monodromy representation, but we want a map whose range does not depend on $S$. Let $X$ be a fixed fibre; for any irreducible component $\Delta_i$ of the boundary divisor $D$, there is a quasi-unipotent transformation \[ T_i : H^3(X, {\mathbb Z}) \rightarrow H^3(X, {\mathbb Z}),\] the Picard-Lefschetz transformation. If $D=\cup_i \Delta_i$ is the decomposition into irreducible components, we may assume that for $i=1\ldots k$, $T_i\in \Gamma$ is of finite order, and for $i\geq k+1$ it is of infinite order. \begin{theorem} {\rm (Griffiths~\cite{griffiths3})} The map $\psi$ has a holomorphic extension (not necessarily locally liftable) \[ \tilde{\psi}: \bar S \setminus \bigcup_{i>k} \Delta_i \rightarrow \bd / \Gamma \] such that the map $\tilde{\psi}$ is proper onto its image. \label{extend} \end{theorem} \vspace{.05in In the language of~\cite{griffiths3}, the map $\psi$ is holomophic, locally liftable and horizontal. Hence the statements follow from [ibid] 9.10, 9.11, noting that [ibid] 9.11 remains valid if $\Gamma$ is not the monodromy group $\Gamma_0$, the image of $\pi_1(S)$ under the monodromy representation, but the full arithmetic monodromy group we use. \hspace*{\fill \section{Finiteness results} \label{main} Now we can put everything together. Fix the lattice $V_{\mathbb Z}$ together with the bilinear form $Q$, $V= V_{\mathbb Z} \otimes {\mathbb C}$ as before. Let $\bd / \Gamma$ be the appropriate period domain. \begin{lemma} For any positive integer $\kappa$, there is a finite set of polynomials $p_1, \ldots, p_k$ with the following property: if $(X,L)$ is a pair consisting of a Calabi--Yau\ threefold\ $X$ and an ample $L$ on $X$ with $L^3\leq \kappa$, there exists $1\leq i \leq k$ such that the Hilbert polynomial of $(X,L)$ equals $p_i$. \end{lemma} \vspace{.05in By assumption, the leading coefficient of the Hilbert polynomial can only assume finitely many values, and the next coefficent is $0$ as $c_1=0$. The conclusion now follows from~\cite{kollarmatsu}. \hspace*{\fill \noindent Let \[ {\mathcal C}_{b,\kappa} = \{ (X,L) \mid X \in {\mathcal C}_b,\mbox{ {\it L} an ample invertible sheaf on {\it X} with } L^3\leq \kappa \}/\cong, \] where the equivalence relation is now given by polarized isomorphisms, and let $\phi_\kappa$ be the restriction of the period map $\phi$ to ${\mathcal C}_{b,\kappa}$. (The reason for including the ample sheaf here will become clear in \ref{cone}.) \begin{theorem} Fix a positive integer $\kappa$ such that the set ${\mathcal C}_{b,\kappa}$ is nonempty. The image $\Phi_\kappa=\phi_\kappa\,({\mathcal C}_{b,\kappa})$ is a locally closed analytic subspace of the complex analytic space $\bd / \Gamma$. For any point $x\in \Phi_\kappa$, there are finitely many $(X,L)\in {\mathcal C}_{b,\kappa}$ satisfying $\phi_\kappa(X,L) = x$. \label{maintheorem} \end{theorem} \vspace{.05in The previous Lemma gives us polynomials $p_1, \ldots, p_k$ as possible Hilbert polynomials. Choose an $m$ such that $L^{\otimes m}$ is very ample and has no higher cohomology for any $(X,L)$ with Hilbert polynomial in the above set, and consider the corresponding Hilbert schemes ${\rm Hilb}^{p_i}_{{\mathbb P}^{N_i}}$. Look at the open subsets over which the fibres of the universal families are smooth, and pick those irreducible components which contain Calabi--Yau\ threefold\ fibres with $b_3=2 b+2$. (The Hilbert scheme may contain components where the fibres are Calabi--Yau\ threefold s with different $b_3$, but these components are irrelevant for our discussion.) We obtain a finite set of smooth quasi-projective varieties $H_1, \ldots, H_d$ with polarized families $\bx_{H_j} \rightarrow H_j$. A group $SL(N_j+1, {\mathbb C})$ acts on $H_j$ for every $j$, and as proved in Section~\ref{moduli}, choosing an integer $l\geq 3$ and taking finite covers \[\rhostr{H_j} \rightarrow \levell{H_j} \rightarrow H_j\] we obtain a finite number of quotient families $\pi_j: \bx_{Z_j}\rightarrow Z_j$ over smooth quasi-projective bases. By construction, every $(X,L)\in {\mathcal C}_{b,\kappa}$ appears at least once as fibre. We may assume that each $Z_j$ is embedded in a smooth projective variety $\bar{Z_j}$ as the complement of a normal crossing divisor $D_j$. Corresponding to the families over $Z_j$, there are period maps \[\psi_j : Z_j \rightarrow \bd / \Gamma.\] As discussed in the previous Section, every $\psi_j$ has a proper extension \[ \tilde{\psi}_j: \tilde{Z_j} \rightarrow \bd / \Gamma,\] where $\tilde{Z_j} = \bar{Z_j} \setminus E_j$, $E_j$ is a union of some components of $D_j$. (Notice that all monodromies of $R^3 \pi_{j*} {\mathbb C}$ of finite order are trivial, this follows from Serre's lemma and the construction. So in fact, these extensions remain locally liftable.) By the Proper Mapping Theorem, $\tilde{\psi}_j (\tilde{Z_j})$ is a closed analytic subspace of $\bd / \Gamma$. $\psi_j(Z_j)$ is relatively open in this set, so it is locally closed in $\bd / \Gamma$. Then \[ \Phi_\kappa = \bigcup_{j=1}^d \psi_j(Z_j)\] so it is also locally closed. Further, since the action of $\Gamma$ is discontinuous on ${\mathcal D}$, the maps $\psi_j$ do not have positive dimensional fibres by Infinitesimal Torelli, and they have proper extensions $\tilde{\psi_j}$ as above. For $x\in \Phi_\kappa$ the sets $\psi_j^{-1}(x) = \tilde{\psi_j}^{-1}(x) \cap Z_j$ are therefore discrete (perhaps empty), and they have only finitely many components from the properness of $\tilde{\psi_j}$. So these sets are finite, which implies the finiteness of $\phi_\kappa^{-1} (x)$. \hspace*{\fill We now recall a definition. A projective surface $E$ is called an {\it elliptic quasi-ruled surface} if there is a map $E\rightarrow C$ exhibiting $E$ either as a smooth ${\mathbb P}^1$-bundle over the smooth elliptic curve $C$, or a conic bundle over such a $C$ all of whose fibres are line pairs. \begin{corollary} Let $X$ be a smooth Calabi--Yau\ threefold such that no deformation of $X$ contains an elliptic quasi-ruled surface. (This holds e.g. if $b_2(X)=1$.) Then the period point determines the manifolds among complex deformations of $X$ up to finitely many possibilities. \end{corollary} \vspace{.05in Let $Y$ be a (large) deformation of $X$, then by the main result of Wilson~\cite{wilson}, any ample class $L$ on $X$ deforms to a class $M$ on $Y$ which is ample. So any $Y$ possesses an ample class with self-intersection $\kappa=L^3$ and the result follows. \hspace*{\fill The recent result of Voisin~\cite{voisin} for quintic threefolds in ${\mathbb P}^4$ is of course much stronger than this, namely in that case the period point determines the generic threefold up to automorphisms (`Weak Global Torelli' holds). No similar result is known for other classes of Calabi--Yau\ threefold s. Using results of~\cite{wilson_elliptic}, one can formulate various conditions on $X$ which ensure the existence of ample classes with bounded self-intersection in the presence of elliptic quasi-ruled surfaces as well. This is left to the reader. We can also deduce a corollary for birationally equivalent threefolds: \begin{corollary} For any positive integer $\kappa$, the number of minimal models (up to isomorphism) of a smooth Calabi--Yau\ threefold $X$, which possess an ample sheaf $L$ with $L^3\leq \kappa$, is finite. \end{corollary} \vspace{.05in By Kawamata~\cite{crepant}, different minimal (i.e. ${\mathbb Q}$-factorial terminal) models of $X$ are related by birational maps called {\it flops}. According to Koll\'ar~\cite{flops}, these different models are all smooth and have isomorphic third cohomology, the isomorphisms respecting Hodge structure and polarization (which comes from Poincar\'e duality). Hence the statement follows from Theorem~\ref{maintheorem}. \hspace*{\fill We remark here that the unconditional finiteness of the number of minimal models up to isomorphism has recently been proved by Kawamata~\cite{kawamata_cy} for {\it relative Calabi--Yau\ models}, i.e. fibre spaces $X\rightarrow S$ with relatively (numerically) trivial canonical sheaf $K_X$, $\dim X =3$, $\dim S\geq 1$. The absolute case of Calabi--Yau\ threefolds is however unknown. Finally we would like to point out a connection to Morrison's Cone Conjecture~\cite{morrison}, which arose from string theoretic considerations leading to the phenomenon called Mirror Symmetry: \begin{corollary} Let $X$ be a smooth Calabi--Yau\ 3-fold, fix a positive integer $\kappa$. Up to the action of ${\mathop{\rm Aut}\nolimits}\, (X)$, there are finitely many ample divisor classes $L$ on $X$ with $L^3\leq \kappa$. In particular, if the automorphism group is finite, there are finitely many such classes. \label{cone} \end{corollary} \vspace{.05in By construction, every pair $(X, L)$ with $L^3 \leq \kappa$ appears as a fibre of some $\bx_{Z_j}\rightarrow Z_j$. On the other hand, the period point does not depend on the choice of the ample sheaf, hence under the period map, pairs $(X, L^{\otimes m})$ map to the same point of $\bd / \Gamma$. By Theorem~\ref{maintheorem}, there are finitely many such pairs up to the action of ${\mathop{\rm Aut}\nolimits}\,(X)$. Considering $m$-torsion as well, we get finitely many pairs $(X, L)$ up to the action of the automorphism group. \hspace*{\fill \noindent The statement certainly follows from the Cone Conjecture, but seems to have been unknown otherwise. \section*{Acknowledgements} The author wishes to thank P.M.H. Wilson for suggesting the problem, his numerous comments and help throughout, N.I. Shepherd-Barron and A. Corti for helpful suggestions, and the referee for pointing out the short proof of Theorem~\ref{smoothhilbert} given above. This work was supported by an Eastern European Research Bursary from Trinity College, Cambridge and an ORS Award from the British Government.

9708

alg-geom/9708022

en

https://arxiv.org/abs/alg-geom/9708022

alg-geom/9708022

Uwe Nagel

J. C. Migliore, U. Nagel, C. Peterson

Buchsbaum-Rim sheaves and their multiple sections

27 pages, AMS-LaTeX

This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on $Z = \Proj R$ where $R$ is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum-Rim sheaf $\cBf$, i.e, we consider morphisms $\psi: \cP \to \cBf$ of sheaves on $Z$ dropping rank in the expected codimension, where $H^0_*(Z,\cP)$ is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus $S$ of $\psi$. It turns out that $S$ is often not equidimensional. Let $X$ denote the top-dimensional part of $S$. In this paper we measure the ``difference'' between $X$ and $S$, compute their cohomology modules and describe ring-theoretic properties of their coordinate rings. Moreover, we produce graded free resolutions of $X$ (and $S$) which are in general minimal. Among the applications we show how one can embed a subscheme into an arithmetically Gorenstein subscheme of the same dimension and prove that zero-loci of sections of the dual of a null correlation bundle are arithmetically Buchsbaum.

alg-geom

\section{Introduction} A fundamental method for constructing algebraic varieties is to consider the degeneracy locus of a morphism between a pair of coherent sheaves. By varying the morphism one obtains families of varieties. By placing various restrictions on the coherent sheaves one can force the degeneracy locus to have special properties. These extra restrictions may also provide for more tools with which to study the degeneracy locus. If many restrictions are placed on the sheaves then a great deal of precise information can be extracted but at the expense of generality. If one puts no restrictions on either of the sheaves then, of course, very little information can be extracted concerning the degeneracy locus. In this paper we take the middle road between these two extremes. We consider a class of sheaves, which while quite general, behave well enough that a substantial amount of information can be obtained with respect to their degeneracy loci. The only restriction placed on the sheaves is that they arise as the sheafification of the kernels of sufficiently general maps between free $R$-modules, where $R$ is a Gorenstein $K$-algebra. We will refer to the sheaves constructed in this manner as Buchsbaum-Rim sheaves. The purpose of this paper is to introduce the class of Buchsbaum-Rim sheaves, to elicit their main properties, and to make a systematic and detailed study of the degeneracy loci obtained by taking multiple sections of these sheaves. Although Buchsbaum-Rim sheaves are necessarily reflexive (they are the sheafification of a second syzygy module) they will not, in general, be locally free. A number of tools are used to manipulate and control these objects. Certainly techniques developed by Eagon-Northcott, Buchsbaum-Rim, Buchsbaum-Eisenbud, Kirby and Kempf play an important role. These are combined with the methods of local cohomology and several homological techniques to produce the main results. The paper opens with a brief section providing preliminary background information of use in later sections of the paper. Sections three, four and five form the technical heart and the final section closes out the paper with three applications. Section three introduces Buchsbaum-Rim sheaves and Buchsbaum-Rim modules. To begin with we should make clear the definition of a Buchsbaum-Rim sheaf. In the following $R$ will always denote a graded Gorenstein $K$-algebra of dimension $n+1$ where $K$ is an infinite field. Furthermore, the scheme $Z$ will be the projective spectrum of $R$. \begin{definition} Let ${\mathcal F}$ and ${\mathcal G}$ be locally free sheaves of ranks $f$ and $g$ respectively on $Z$. Let $\varphi: {\mathcal F} \to {\mathcal G}$ be a generically surjective morphism. Suppose that the degeneracy locus of $\varphi$ has codimension $f-g+1$ and that the modules $F = H^0_*(Z,{\mathcal F})$ and $G = H^0_*(Z,{\mathcal G})$ are free $R$-modules. We call the kernel of $\varphi$ a {\it Buchsbaum-Rim sheaf} and denote it by ${\mathcal B}_{\varphi}$. By abuse of notation we denote the homomorphism $F \to G$ induced by $\varphi$ again by $\varphi$. Moreover, we put $B_{\varphi} = H^0_*(Z,{\mathcal B}_{\varphi})$ and $M_{\varphi} = \coker \varphi$, so that we have an exact sequence $$ 0 \to B_{\varphi} \to F \stackrel{\varphi}{\longrightarrow} G \to M_{\varphi} \to 0. $$ We call $B_{\varphi}$ a {\it Buchsbaum-Rim module}. \end{definition} Thus the cotangent bundle of projective space is a Buchsbaum-Rim sheaf. The cohomology of its exterior powers is given by the Bott formula. Letting $r(R)$ denote the index of regularity of a graded ring we have the following lemma which generalizes the Bott formula to arbitrary Buchsbaum-Rim sheaves. \begin{lemma} Let $B_{\varphi}$ be a Buchsbaum-Rim module of rank $r$. Then it holds: \begin{itemize} \item[(a)] For $i = 0,\ldots r$ there are isomorphisms $$ \wedge^{r-i} {\mathcal B}_{\varphi}^* \otimes \wedge^f {\mathcal F} \otimes \wedge^g {\mathcal G}^* \cong (\wedge^i {\mathcal B}_{\varphi}^*)^*. $$ \item[(b)] For $i < r$ we have $$ H^j_*(Z, \wedge^i {\mathcal B}_{\varphi}^*) \cong \left \{ \begin{array}{ll} 0 & \mbox{if} ~ 1 \leq j \leq n \; \mbox{and} \; j \neq n-i \\ S_{i}(M_{\varphi})^{\vee}(1 - r(R)) & \mbox{if} ~ j = n-i. \end{array} \right. $$ \end{itemize} \end{lemma} The value of this lemma will become clear when we construct the generalized Koszul complexes associated to taking multiple sections of Buchsbaum-Rim sheaves. The lemma also suggests a relationship to Eilenberg-MacLane sheaves. Recall that an $R$-module $E$ is called an {\it Eilenberg-MacLane module} of depth~$t$, $0 \leq t \leq n+1$ if $H_{\mathfrak m}^j(E) = 0 \quad \mbox{for all} \; j \neq t \; \mbox{where} \; 0 \leq j \leq n.$ Similarly, a sheaf ${\mathcal E}$ on $Z$ is said to be an {\it Eilenberg-MacLane sheaf} if $H^0_*({\mathcal E})$ is an Eilenberg-MacLane module. A cohomological characterization of Buchsbaum-Rim sheaves can then be stated as follows. \begin{proposition} A sheaf ${\mathcal E}$ on $Z$ is a Buchsbaum-Rim sheaf if and only if $E = H^0_*({\mathcal E})$ is a reflexive Eilenberg-MacLane module with finite projective dimension and rank $r \leq n$ such that $H_{\mathfrak m}^{n-r+1}(E)^{\vee}$ is a perfect $R$-module of dimension $n-r$ if $r \geq 2$. \end{proposition} In section four we make a detailed study of the cohomology of the degeneracy locus of multiple sections of Buchsbaum-Rim sheaves. Consider a morphism $\psi: {\mathcal P} \to {\mathcal B}_{\varphi}$ of sheaves of rank $t$ and $r$ respectively on $Z = Proj (R)$, where ${\mathcal B}_{\varphi}$ is a Buchsbaum-Rim sheaf and $H^0_*(Z, {\mathcal P})$ is a free $R$-module. If $t = 1$ then $\psi$ is just a section of some twist of ${\mathcal B}_{\varphi}$. For arbitrary $t<r$ we say that $\psi$ corresponds to taking multiple sections of ${\mathcal B}_{\varphi}$. We always suppose that the degeneracy locus $S$ of $\psi$ has (the expected) codimension $r - t + 1 \geq 2$ in $Z$ (if $t=1$ then $S$ is just the zero-locus of a regular section of a Buchsbaum-Rim sheaf). An Eagon-Northcott complex involving ${\mathcal B}_{\varphi}$ will play an essential role. Our approach will be algebraic and uses local cohomology. Taking global sections we obtain an $R$-homomorphism $$ \psi : P \to B_{\varphi} $$ where $P$ is a free $R$-module of rank $t, 1 \leq t < r$. Then there is an Eagon-Northcott complex $$ E_{\bullet} \colon \quad 0 \to E_r \stackrel{\delta_r}{\longrightarrow} E_{r-1} \stackrel{\delta_{r-1}}{\longrightarrow} \cdots \to E_t \stackrel{\delta_t}{\longrightarrow} I(\psi) \otimes \wedge^t P^* \to 0 $$ where $$ E_i = \wedge^i B_{\varphi}^* \otimes S_{i - t}(P) $$ and $I(\psi)$ is the ideal defined by the image of $\delta_t = \wedge^t \psi^*$. The saturation of $I(\psi)$ is the homogeneous ideal of the degeneracy locus $S$. With the help of this Eagon-Northcott complex and the first lemma we get the following formula for the cohomology modules of $S$. \begin{proposition} Let $I = I(\psi)$. Then it holds for $j\neq \dim R/I = n+t-r$ $$ H^j_m (R/I) \cong \left\{ \begin{array}{ll} S_i(M_{\varphi})^\vee \otimes S_{i-t} (P) \otimes \wedge^t P (1-r(R)) & \mbox{if } j=n+t-2i \quad \mbox{where } \max \{t, \frac{r+1}{2}\} \leq i \leq \left \lfloor \frac{r+t}{2} \right \rfloor \\ 0 & \mbox{otherwise}. \end{array} \right. $$ \end{proposition} This proposition allows us to decide if $S$ is equidimensional. It will often turn out that this is not the case. Thus we are also interested in the top-dimensional component $X$ of $S$, i.e. the union of the highest-dimensional components of $S$. Let $J = J(\psi)$ denote the homogeneous ideal of $X$. We need a measure of the failure of $I$ to be equidimensional and we need a close relation between the cohomology of the schemes $X$ and $S$. This is provided in the following result. \begin{proposition} Letting $I$ and $J$ denote the ideals associated to $\psi$ we have: \begin{itemize} \item[(a)] $I$ is unmixed if and only if $r+t$ is odd. \item[(b)] If $r+t$ is even then $I$ has a primary component of codimension $r+1$. Let $Q$ be the intersection of all such components. Then we have $I = J \cap Q$ and $$ H^j_m(R/J) \cong \left\{ \begin{array}{ll} H^j_m (R/I) & \mbox{if } j\neq n-r \\ 0 & \mbox{if } j=n-r \end{array}\right. $$ and $$ J/I \cong S_{\frac{r-t}{2}} (M_{\varphi}) \otimes S_{\frac{r-t}{2}}(P) \otimes \wedge^f F^* \otimes \wedge^g G \otimes \wedge^t P. $$ \end{itemize} \end{proposition} Combining these propositions in the proper manner we are in a position to prove the main theorem of section four. \begin{theorem} With the notation above we have: \begin{itemize} \item[(a)] If $r = n$ then $S$ is equidimensional and locally Cohen-Macaulay. \item[(b)] $S$ is equidimensional if and only if $r+t$ is odd or $r = n$. Moreover, if $r < n$ then $X$ is locally Cohen-Macaulay if and only if $X$ is arithmetically Cohen-Macaulay. \item[(c)] If $r+t$ is odd then $X = S$ is arithmetically Cohen-Macaulay if and only if $t=1$. In this case $S$ has Cohen-Macaulay type $\leq 1 + \binom{\frac{r}{2}+g-1}{g-1}$. \item[(d)] Let $r+t$ be even. Then \begin{itemize} \item[(i)] $X$ is arithmetically Cohen-Macaulay if and only if $1 \leq t \leq 2$. If $t =1$ then $X$ is arithmetically Gorenstein. If $t = 2$ then $X$ has Cohen-Macaulay type $\leq r - 1 + \binom{\frac{r}{2}+g-1}{g-1} \cdot (\frac{r}{2} - 1)$. \item[(ii)] If in addition $r < n$ then the components of $S$ have either codimension $r-t+1$ or codimension $r+1$. \end{itemize} \end{itemize} \end{theorem} Contained in the theorem is the following surprising conclusion. Let ${\mathcal B}_{\varphi}$ be an odd rank Buchsbaum-Rim sheaf and let $X$ denote the top dimensional component of the zero-locus of any regular section of ${\mathcal B}_{\varphi}$. Then $X$ is arithmetically Gorenstein. This generalizes the main theorem of \cite{Mig-P_gorenstein}, where the case $r=3$ was considered. \smallskip Section five treats the problem of finding free resolutions of the degeneracy loci. In order to do this it is important to understand the homology modules of the Eagon-Northcott complex $E_{\bullet}$ associated to $\psi$. We show that the homology can be summarized as follows. \begin{proposition} The homology modules of the Eagon-Northcott $E_{\bullet}$ complex are: $$ H_i (E_{\bullet}) \cong \left\{ \begin{array}{ll} S_j(M_{\varphi}) \otimes S_{r-t-j}(P) \otimes \wedge^f F^* \otimes \wedge^g G & \mbox{if} ~ i=r-1-2j \mbox{ where } j \in \ZZ, \; t \leq i \leq r-3 \\ 0 & \mbox{otherwise}. \end{array}\right. $$ \end{proposition} This result allows us to conclude that $X$ and $S$ have free resolutions of finite length. However, it does not, in general, provide enough information to compute a minimal free resolution. To do this we need several ingredients. First we need to understand the cohomology of the dual of the Eagon-Northcott complex. This needs to be mixed with knowledge of how the canonical modules of $S$ and $X$ relate. The cohomology of the dual of the Eagon-Northcott complex is summarized in the following lemma, where $K_{R/I}$ denotes the canonical module of $R/I(\psi)$. \begin{lemma} The dual of the Eagon-Northcott complex $E_{\bullet}$ provides a complex $$ 0 \to \wedge^t P \to E_t^* \stackrel{\delta^*_{t+1}}{\longrightarrow} \ldots \to E_{r-1}^* \stackrel{\delta^*_{r}}{\longrightarrow} E_r^* \stackrel{\gamma}{\longrightarrow} K_{R/I} \otimes \wedge^t P (1 - r(R)) \to 0 $$ which we denote (by slight abuse of notation) by $E_{\bullet}^*$. Its (co)homology modules are given by $$ H^i(E_{\bullet}^*) \cong \left \{ \begin{array}{ll} S_j(M_{\varphi}) \otimes S_{j-t}(P)^* & \mbox{if} ~ 2t + 1 \leq i = 2 j + 1 \leq r+1 \\ 0 & \mbox{otherwise} \end{array} \right. $$ In particular, $E_{\bullet}^*$ is exact if $t \geq \frac{r+1}{2}$. \end{lemma} To utilize these results on the Eagon-Northcott complex one still needs to know when certain terms in a free resolution can be split off. To do this we prove the following result which, although rather technical in appearance, provides a substantial generalization to an often-used result of Rao. It can be applied in situations far removed from those addressed in this paper and can even be utilized when the ring $R$ is not a Gorenstein ring but is only a Cohen-Macaulay ring. \begin{proposition} Let $N$ be a finitely generated graded torsion $R$-module which has projective dimension~$s$. Then it holds for all integers $j \geq 0$ that $\operatorname{Tor}^R_{s-j}(N,K)^{\vee}$ is a direct summand of $$ \oplus_{i=0}^j \operatorname{Tor}^R_{j-i}(\operatorname{Ext}_R^{s-i}(N,R),K). $$ Moreover, we have $\operatorname{Tor}^R_{s}(N,K)^{\vee} \cong \operatorname{Tor}^R_{0}(\operatorname{Ext}_R^{s}(N,R),K)$ and that $\operatorname{Tor}^R_{1}(\operatorname{Ext}_R^{s}(N,R),K)$ is a direct summand of $\operatorname{Tor}^R_{s-1}(N,K)^{\vee}$. \end{proposition} Together these results allow us to write down a free resolution of the degeneracy locus which is in general minimal. Thus the main theorem of section five is the following, which gives the free resolution for the degeneracy locus of a morphism $\psi: {\mathcal P} \to {\mathcal B}_{\varphi}$, where ${\mathcal P}$ has rank $t$ and ${\mathcal B}_{\varphi}$ has rank $r$. \begin{theorem} Consider the following modules where we use the conventions that $i$ and $j$ are non-negative integers and that a sum is trivial if it has no summand: $$ A_k = \bigoplus_{\begin{array}{c} {\scriptstyle i+2j = k + t -1}\\ [-4pt] {\scriptstyle t \leq i+j \leq \frac{r+t-1}{2}} \end{array}} \wedge^i F^* \otimes S_j(G)^* \otimes S_{i+j-t}(P), $$ $$ C_k = \bigoplus_{\begin{array}{c} {\scriptstyle i+2j = r+1-t-k}\\ [-4pt] {\scriptstyle i+j \leq \frac{r-t}{2}} \end{array}} \wedge^i F \otimes S_j(G) \otimes S_{r-t-i-j}(P) \otimes \wedge^f F^* \otimes \wedge^g G. $$ Observe that it holds: $A_r = 0$ if and only if $r+t$ is even, $C_1 = 0$ if and only if $r+t$ is odd, $C_k = 0$ if $k \geq r+2-t$ and $C_{r+1-t} = S_{r-t}(P) \otimes \wedge^f F^* \otimes \wedge^g G$. \\ Then the homogeneous ideal $I_X = J(\psi)$ of the top-dimensional part $X$ of the degeneracy locus $S$ has a graded free resolution of the form $$ 0 \to A_r \oplus C_r \to \ldots \to A_1 \oplus C_1 \to I_X \otimes \wedge^t P^* \to 0. $$ \end{theorem} Note that previously minimal free resolutions were known only for a few classes besides the determinantal ideals. A number of examples of particular interest round out the section and illustrate the theorem. \smallskip The final section gives several additional applications that may be of independent interest. We show how Buchsbaum-Rim sheaves can be used to situate arbitrary equidimensional schemes of arbitrary codimension into arithmetically Gorenstein schemes. This will be of relevance when one considers the problem of linkage by arithmetically Gorenstein ideals as opposed to complete intersection linkage theory. We also show how to utilize Buchsbaum-Rim sheaves to produce interesting new examples of $k$-Buchsbaum sheaves as well as of arithmetically Buchsbaum schemes. Finally we construct new vector bundles of rank $n-1$ on $\PP^n$ if $n$ is odd. We call them generalized null correlation bundles and show that our results apply to multiple sections of their duals. \section{Preliminaries} \label{preliminaries} Let $R$ be a ring. If $R = \oplus_{i \in \mathbb{N}} R_i$ is graded then the irrelevant maximal ideal $ \oplus_{i > 0} R_i$ of $R$ is denoted by ${\mathfrak m}_R$ or simply ${\mathfrak m}$. It is always assumed that $R_0$ is an infinite field $K$ and that the $K$-algebra $R$ is generated by the elements of $R_1$. Hence $(R,{\mathfrak m})$ is $^*$local in the sense of \cite{Bruns-Herzog}. If $M$ is a module over the graded ring $R$ it is always assumed to be $\mathbb{Z}$-graded. The set of its homogeneous elements of degree $i$ is denoted by $M_i$ or $[M]_i$. All homomorphisms between graded $R$-modules will be morphisms in the category of graded $R$-modules, i.e., will be graded of degree zero. If $M$ is assumed to be a graded $R$-module it is always understood that $R$ is a graded $K$-algebra as above. We refer to the context just described as the graded situation. Although we are mainly interested in graded objects we note that our results hold also true (with the usual modifications) in a local situation. Then $(R,{\mathfrak m})$ will denote a local ring with maximal ideal ${\mathfrak m}$. If $M$ is an $R$-module, $\dim M$ denotes the Krull dimension of $M$. The symbols $\rank_R$ or simply $\rank$ are reserved to denote the rank of $M$ in case it has one. For a $K$-module, $\rankk$ just denotes the vector space dimension over the field $K$. There are two types of duals of an $R$-module $M$ we are going to use. The $R$-dual of $M$ is $M^* = \Hom_R(M,R)$. If $M$ is graded then $M^*$ is graded, too. If $R$ is a graded $K$-algebra then $M$ is also a $K$-module and the $K$-dual $M^{\vee}$ of $M$ is defined to be the graded module $\Hom_K(M,K)$ where $K$ is considered as a graded module concentrated in degree zero. Note that $R^{\vee}$ is the injective hull of $K^{\vee} \cong K \cong R/{\mathfrak m}$ in the category of graded $R$-modules. If $\rankk [M]_i < \infty$ for all integers $i$ then there is a canonical isomorphism $M \cong M^{\vee \vee}$. Now let $Z$ be a projective scheme over $K$. This means $Z = Proj (R)$ where $R$ is a graded $K$-algebra. For any sheaf ${\mathcal F}$ on $Z$, we define $H^i_*(Z,{{\mathcal F}})=\bigoplus_{t\in \ZZ}H^i(Z,{{\mathcal F}}(t))$. In this paper we will use ``vector bundle'' and ``locally free sheaf'' interchangeably. Let $X$ be a non-empty projective subscheme of $Z$ with homogeneous coordinate ring $A = R/I_X$. Then $I_X$ is a saturated ideal of $R$. Recall that a homogeneous ideal $I$ in $R$ is {\em saturated} if $I = \bigcup_{d \in {\ZZ}^+} [I:{{\mathfrak m}}^d ]$, where ${{\mathfrak m}} = (x_0 ,x_1 ,\dots,x_n)$. Equivalently, $I$ is saturated if and only if $I = H^0_* (Z, {{\mathcal J}})$, where ${\mathcal J}$ is the sheafification of $I$. \bigskip \noindent {\it Generalized Koszul complexes} \medskip For more details with respect to the following discussion we refer to \cite{BV} and \cite{Eisenbud-Buch}. The differences between these presentations and ours stem from the fact that we want to have all homomorphisms graded (of degree zero). Let $R$ be a graded $K$-algebra and let $\varphi: F \to G$ be a homomorphism of finitely generated graded $R$-modules. Then there are (generalized) Koszul complexes ${\mathcal C}_i(\varphi)$: $$ 0 \to \wedge^i F \otimes S_0(G) \to \wedge^{i-1} F \otimes S_1(G) \to \ldots \to \wedge^0 F \otimes S_i(G) \to 0. $$ Let ${\mathcal C}_i(\varphi)^*$ be the $R$-dual of ${\mathcal C}_i(\varphi)$. Suppose now that $F$ is a free $R$-module of rank $f$. Then there are graded isomorphisms $$ \wedge^f F \otimes (\wedge^j F)^* \cong \wedge^{f-j} F. $$ Thus we can rewrite ${\mathcal C}_i(\varphi)^* \otimes \wedge^f F$ as follows: $$ 0 \to \wedge^f F \otimes S_i(G)^* \to \wedge^{f-1} F \otimes S_{i-1}(G)^* \to \ldots \to \wedge^{f-i} F \otimes S_0(G)^* \to 0. $$ Note that $S_j(G)^*$ is the $j$th graded component of the divided power algebra of $G^*$, but we won't need this fact. Let's assume that also $G$ is a free $R$-module of rank, say, $g$ where $g < f$. Then $\varphi^*$ induces graded homomorphisms $$ \nu_i: \wedge^{g+i} F \otimes \wedge^g G^* \to \wedge^i F. $$ Put $r = f-g$. It turns out that for $i = 0,\ldots,r$ the complexes ${\mathcal C}_{r-i}(\varphi)^* \otimes \wedge^f F \otimes \wedge^g G^*$ and ${\mathcal C}_i(\varphi)$ can be spliced via $\nu_i$ to a complex ${\mathcal D}_i(\varphi)$: $$ 0 \to \wedge^f F \otimes S_{r-i}(G)^* \otimes \wedge^g G^* \to \wedge^{f-1} F \otimes S_{r-i-1}(G)^* \otimes \wedge^g G^* \to \ldots $$ $$ \to \wedge^{g+i} F \otimes S_0(G)^* \otimes \wedge^g G^* \stackrel{\nu_i}{\longrightarrow} \wedge^i F \otimes S_0(G) \to \wedge^{i-1} F \otimes S_1(G) \to \ldots \to \wedge^0 F \otimes S_i(G) \to 0. $$ The complex ${\mathcal D}_0(\varphi)$ is called the Eagon-Northcott complex and ${\mathcal D}_1(\varphi)$ is called the Buchsbaum-Rim complex. If we fix bases of $F$ and $G$ the map $\varphi$ can be described by a matrix whose maximal minors generate an ideal which equals the image of $\nu_0$. We denote this ideal by $I(\varphi)$. Its grade is at most $f-g+1$. If $\varphi$ is general enough the complexes above have good properties. \begin{proposition} \label{EN-complexes_are_exact} Suppose $grade\, I(\varphi) = f-g+1$. Then it holds: \begin{itemize} \item[(a)] ${\mathcal D}_i(\varphi)$ is acyclic where $i = 0,\ldots,f-g = r$. \item[(b)] If $\varphi$ is a minimal homomorphism, i.e.\ $\im \varphi \subset {\mathfrak m} \cdot G$, then ${\mathcal D}_0(\varphi)$ is a minimal free graded resolution of $R/I(\varphi)$ and ${\mathcal D}_i(\varphi)$ is a minimal free graded resolution of $S_i(\coker \varphi)$, $1 \leq i \leq r$. \end{itemize} \end{proposition} The minimality of the resolutions in (b) follows by analyzing the maps described above. \bigskip \noindent {\it Gorenstein rings and schemes} \medskip A graded $K$-algebra $R$ is said to be Gorenstein if it has finite injective dimension (cf.\ \cite{Bruns-Herzog}, Definition 3.1.18). Over a Gorenstein ring duality theory is particularly simple. We denote the index of regularity of a graded ring by $r(R)$. If $R$ is just the polynomial ring $K[x_0, \dots , x_n]$ then $r(R) = -n$. We will use the following duality result (cf., for example, \cite{SV2}, Theorem 0.4.14). \begin{lemma} \label{duality} Let $M$ be a graded $R$-module where $R$ is a Gorenstein ring of dimension $n$. Then we have for all $i \in \mathbb{Z}$ natural isomorphisms of graded $R$-modules $$ H^i_{{\mathfrak m}} (M)^{\vee} \cong \Ext^{n-i}_R(M,R)(r(R)-1). $$ \end{lemma} Let $M$ be a graded $R$-module where $n = \dim R$ and $d = \dim M$. Then $$ K_M = \Ext^{n-d}_R(M,R) (r(R)-1) $$ is said to be the {\itshape canonical module} of $M$. Usually the canonical module is defined as the module representing the functor $H_{\mathfrak m}^d(M \otimes_R \__{})^{\vee}$ if such a module exists. If $R$ is Gorenstein it does and is just the module defined above (cf.\ \cite{S}). We say that $M$ has cohomology of finite length if the cohomology modules $H_{\mathfrak m}^i(M)$ have finite length for all $i < \dim M$. It is well-known that $M$ has cohomology of finite length if and only if $M$ is equidimensional and locally Cohen-Macaulay. Let now $Z = Proj (R)$ be a projective scheme over $K$. Then $Z$ is said to be {\itshape arithmetically Gorenstein} and {\itshape arithmetically Cohen-Macaulay} respectively if the homogeneous coordinate ring $R$ of $Z$ is Gorenstein and Cohen-Macaulay respectively. For a closed subscheme $X$ of $Z$, with homogeneous coordinate ring $A = R/I_X$ we will refer to the canonical module of $A$ also as the canonical module of $X$. Moreover, we say that $X$ has finite projective dimension if $A$ has finite projective dimension as an $R$-module. Assume that $Z$ is arithmetically Gorenstein. One of the things we shall be interested in is to describe when certain subschemes $X$ of $Z$ are arithmetically Gorenstein, too. To do this, it is enough to show that $X$ is arithmetically Cohen-Macaulay, with Cohen-Macaulay type $1$ provided $X$ has finite projective dimension. In this case $X$ is defined by a Gorenstein ideal $I = I_X \subset R$. Recall that the {\em Cohen-Macaulay type} of an arithmetically Cohen-Macaulay projective scheme $X$ with finite projective dimension can be defined to be the rank of the last free module occurring in a minimal free resolution of the saturated ideal of $X$. It is equal to the number of minimal generators of the canonical module of $X$. \section{Buchsbaum-Rim sheaves} From now on we will always assume that $Z$ is a projective arithmetically Gorenstein scheme over the field $K$. We denote its dimension by $n$ and its homogeneous coordinate ring by $R$. Let ${\mathcal F}$ and ${\mathcal G}$ be locally free sheaves of ranks $f$ and $g$ respectively on $Z$. Let $\varphi: {\mathcal F} \to {\mathcal G}$ be a generically surjective morphism. Since the construction of the generalized Koszul complexes as described in the previous section globalizes, we can associate to $\varphi$ several complexes. The most familiar are the Eagon-Northcott complex $$ 0 \to \wedge^f {\mathcal F} \otimes S_{f-g}({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \to \wedge^{f-1} {\mathcal F} \otimes S_{f-g-1}({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \to \ldots $$ $$ \to \wedge^{g} {\mathcal F} \otimes S_0({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \stackrel{\wedge^g \varphi}{\longrightarrow} {\mathcal O}_Z \to 0 $$ and the Buchsbaum-Rim complex $$ 0 \to \wedge^f {\mathcal F} \otimes S_{f-g-1}({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \to \wedge^{f-1} {\mathcal F} \otimes S_{f-g-2}({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \to \ldots $$ $$ \to \wedge^{g+1} {\mathcal F} \otimes S_0({\mathcal G})^* \otimes \wedge^g {\mathcal G}^* \to {\mathcal F} \stackrel{\varphi}{\longrightarrow} {\mathcal G} \to 0. $$ Moreover, Proposition~\ref{EN-complexes_are_exact} implies that these complexes are acyclic if the degeneracy locus of $\varphi$ has the expected codimension $f-g+1$ in $Z$. This lead us to the following definition. \begin{definition} With the notation above suppose that the degeneracy locus of $\varphi$ has codimension $f-g+1$. Then we call the cokernel of the map between $\wedge^{g+1+i} {\mathcal F} \otimes S_i({\mathcal G})^* \otimes \wedge^g {\mathcal G}^*$ and $\wedge^{g+i} {\mathcal F} \otimes S_{i-1}({\mathcal G})^* \otimes \wedge^g {\mathcal G}^*$ an {\it $i^{th}$ local Buchsbaum-Rim sheaf} ($1 \leq i \leq f-g$) and denote it by ${\mathcal B}^{\varphi}_i$. \end{definition} Note that the $i^{th}$ local Buchsbaum-Rim sheaf associated to $\varphi$ is just the $(i+1)^{st}$ syzygy sheaf of $\coker \varphi$. The following result is a generalization of Proposition 2.10 of \cite{KMNP}. Thanks to our set-up the proof given there works here, too. \begin{proposition} Let ${\mathcal B}$ be a first local Buchsbaum-Rim sheaf associated to a morphism $\varphi$. Let $X$ denote the degeneracy locus of $\varphi$. Let $S$ be the zero-locus of a section $s \in H^0(Z,{\mathcal B})$ and let $T$ be the zero-locus of a section $t \in H^0(Z,{\mathcal B}^*)$. Then it holds $X \subset S$ and $X \subset T$. \end{proposition} Now we put stronger assumptions on the sheaves ${\mathcal F}$ and ${\mathcal G}$. \begin{definition} \label{BR-sheaf} Suppose in addition that the modules $F = H^0_*(Z,{\mathcal F})$ and $G = H^0_*(Z,{\mathcal G})$ are free $R$-modules. Then the sheaf ${\mathcal B}^{\varphi}_i$ is called an {\it $i^{th}$ Buchsbaum-Rim sheaf}. For simplicity a first Buchsbaum-Rim sheaf is just called a {\it Buchsbaum-Rim sheaf} and denoted by ${\mathcal B}_{\varphi}$. By abuse of notation we denote the homomorphism $F \to G$ induced by $\varphi$ again by $\varphi$. Moreover, we put $B_{\varphi} = H^0_*(Z,{\mathcal B}_{\varphi})$ and $M_{\varphi} = \coker \varphi$, so that we have an exact sequence $$ 0 \to B_{\varphi} \to F \stackrel{\varphi}{\longrightarrow} G \to M_{\varphi} \to 0. $$ We call $B_{\varphi}$ a {\it Buchsbaum-Rim module}. \end{definition} \begin{remark} In this paper we will consider some degeneracy loci associated to Buchsbaum-Rim sheaves. These investigations were motivated by the work of the first and the third author in \cite{Mig-P_gorenstein}. Note that in \cite{KMNP} zero-loci of regular sections of the {\it dual} of a Buchsbaum-Rim sheaf over projective space have been characterized as determinantal subschemes which are generically complete intersections. If $g=1$ then the sheaf ${\mathcal B}^{\varphi}_{f-1}$ is the dual of a Buchsbaum-Rim sheaf. Thus it seems to be rewarding to study higher Buchsbaum-Rim sheaves, too. \end{remark} \begin{remark} \label{properties_of_BR-sheaf} (i) With the notation above the Buchsbaum-Rim sheaf ${\mathcal B}_{\varphi}$ has rank $r = f - g$. Our assumptions imply that $r \leq n = \dim Z$. Moreover, ${\mathcal B}_{\varphi}$ is locally free if and only if $n = r$. (ii) As a second syzygy sheaf over an arithmetically Gorenstein scheme a Buchsbaum-Rim sheaf ${\mathcal B}_{\varphi}$ is reflexive, i.e., the natural map ${\mathcal B}_{\varphi} \otimes {\mathcal B}_{\varphi}^* \to {\mathcal O}_Z$ induces an isomorphism ${\mathcal B}_{\varphi} \cong {\mathcal B}_{\varphi}^{**}$. Similarly, a Buchsbaum-Rim module is a reflexive $R$-module. \end{remark} The following result will become important later on. \begin{lemma} \label{prop-of-exteriour-powers} With the above notation let $B_{\varphi}$ be a Buchsbaum-Rim module of rank $r$. Then it holds: \begin{itemize} \item[(a)] For $i = 1,\ldots r$ the module $\wedge^i B_{\varphi}^*$ is a $(r-i+1)$-syzygy of the perfect module $S_{r-i}(M_{\varphi}) \otimes \wedge^f F^* \otimes \wedge^g G$ and is resolved by ${\mathcal C}_i(\varphi)^*$. \item[(b)] For $i = 0,\ldots r$ there are isomorphisms $$ \wedge^{r-i} {\mathcal B}_{\varphi}^* \otimes \wedge^f F \otimes \wedge^g G^* \cong (\wedge^i {\mathcal B}_{\varphi}^*)^*. $$ \item[(c)] For $i < r$ we have $$ H_{\mathfrak m}^j(\wedge^i B_{\varphi}^*) \cong \left \{ \begin{array}{ll} 0 & \mbox{if} ~ j \leq n \; \mbox{and} \; j \neq n+1-i \\ S_{i}(M_{\varphi})^{\vee}(1 - r(R)) & \mbox{if} ~ j = n+1-i. \end{array} \right. $$ \end{itemize} \end{lemma} \begin{proof} The assumption $\codim I(\varphi) = r+1$ ensures that $\wedge^i B_{\varphi}^*$ is resolved by ${\mathcal C}_i(\varphi)^*$ (cf.\ \cite{BV}, Remark 2.19). Hence the first claim follows by Proposition~\ref{EN-complexes_are_exact}. The latter result also implies the second claim. The isomorphism is induced by the map $\nu_i$ (cf.\ \cite{BV}, Remark 2.19). In order to prove the third claim we observe that by Lemma~\ref{duality} $$ H_{\mathfrak m}^{n+1-j}(\wedge^i B_{\varphi}^*) \cong \Ext^j(\wedge^i B_{\varphi}^*,R)^{\vee}(1-r(R)). $$ But we know already that $\Ext^j(\wedge^i B_{\varphi}^*,R)$ can be computed as the $(i-j)^{th}$ homology module of the complex ${\mathcal C}_{i}(\varphi)$ which is part of the acyclic complex ${\mathcal D}_{i}(\varphi)$. Now our assertion follows. \end{proof} \begin{remark} \label{Bott-formula} (i) The previous result implies for the Buchsbaum-Rim sheaf ${\mathcal B}_{\varphi}$ that it is just the sheafification $\widetilde{B_{\varphi}}$ of the Buchsbaum-Rim module $B_{\varphi}$ and $$ H_*^j(\wedge^i {\mathcal B}_{\varphi}^*) \cong \left \{ \begin{array}{ll} 0 & \mbox{if} ~ 1 \leq j < n \; \mbox{and} \; j \neq n-i \\ S_{i}(M_{\varphi})^{\vee}(1 - r(R)) & \mbox{if} ~ j = n-i. \end{array} \right. $$ (ii) Let us consider the example where $R = K[x_0,\ldots,x_n]$ is the polynomial ring, $F = R(-1)^{n+1}, G = R$ and $\varphi: F \to G$ a general map. Then $M_{\varphi} \cong K$ and $B_{\varphi} = \ker \varphi$ is a Buchsbaum-Rim module and the corresponding Buchsbaum-Rim sheaf is just the cotangent bundle on $\PP^n$. Via Serre duality we see that in this case Lemma~\ref{prop-of-exteriour-powers}(c) is just the dual version of the Bott formula for the cohomology of $\cOP^j = \wedge^j {\mathcal B}_{\varphi}$. \end{remark} According to our explicit description of the complexes ${\mathcal D}_{i}(\varphi)$, Proposition~\ref{EN-complexes_are_exact} implies. \begin{lemma} \label{can-module-of-symm-powers} For $i = 1,\ldots r-1$ there are isomorphisms $$ \Ext^{r+1}(S_i(M_{\varphi}),R) \cong S_{r-i}(M_{\varphi}) \otimes \wedge^f F^* \otimes \wedge^g G. $$ \end{lemma} Finally, we want to derive a cohomological characterization of Buchsbaum-Rim sheaves. It turns out that they are particular Eilenberg-MacLane sheaves. Recall that an $R$-module $E$ is called an {\it Eilenberg-MacLane module} of depth $t$, $0 \leq t \leq n+1$ if $$ H_{\mathfrak m}^j(E) = 0 \quad \mbox{for all} \; j \neq t \; \mbox{where} \; 0 \leq j \leq n. $$ Similarly, a sheaf ${\mathcal E}$ on $Z$ is said to be an {\it Eilenberg-MacLane sheaf} if $H^0_*({\mathcal E})$ is an Eilenberg-MacLane module. We will need the following result which is shown in \cite{habil} as Theorem I.3.9. \begin{lemma} \label{Eilenberg-MacLane} Let $E$ be a reflexive module of depth $t \leq n$. Then $E$ is an Eilenberg-MacLane module with finite projective dimension if and only if $E^*$ is an $(n+2-t)$-syzygy of a module $M$ of dimension $\leq t-2$. In this case it holds $$ M \cong H_{\mathfrak m}^t(E)^{\vee}(1-r(R)). $$ \end{lemma} Now we are ready for our cohomological description of Buchsbaum-Rim sheaves. \begin{proposition} \label{BR-sheaf-characterization} A sheaf ${\mathcal E}$ on $Z$ is a Buchsbaum-Rim sheaf if and only if $E = H^0_*({\mathcal E})$ is a reflexive Eilenberg-MacLane module with finite projective dimension and rank $r \leq n$ such that $H_{\mathfrak m}^{n-r+2}(E)^{\vee}$ is a perfect $R$-module of dimension $n-r$ if $r \geq 2$. \end{proposition} \begin{proof} First let us assume that ${\mathcal E}$ is a Buchsbaum-Rim sheaf. Then we have by definition for $E = H^0_*({\mathcal E})$ that it has a rank, say $r$, and sits in an exact sequence $$ 0 \to E \to F \stackrel{\varphi}{\longrightarrow} G \to M_{\varphi} \to 0 $$ where $F$ and $G$ are free modules and $I(\varphi)$ has the expected codimension $r+1$. Due to Remark~\ref{properties_of_BR-sheaf} $E$ is reflexive. Furthermore, $E$ has finite projective dimension since $M_{\varphi}$ does by Lemma~\ref{prop-of-exteriour-powers}. If $r=1$ it follows that $E$ is just $R(m)$ for some integer $m$. Let $r \geq 2$. Then the exact sequence above and the Cohen-Macaulayness of $M_{\varphi}$ imply that $E$ is an Eilenberg-Maclane module of depth $n-r+2$ and $$ H_{\mathfrak m}^{n-r+2}(E)^{\vee} \cong H_{\mathfrak m}^{n-r}(M_{\varphi})^{\vee} \cong \Ext^{r+1}(M_{\varphi},R)(r(R)-1) \cong S_{r-1}(M_{\varphi}) \otimes \wedge^f F^* \otimes \wedge^g G(r(R) - 1) $$ where the latter isomorphisms are due to Lemma~\ref{duality} and Lemma~\ref{can-module-of-symm-powers}. Since $S_{r-1}(M_{\varphi})$ is a perfect module of dimension $n-r$ by Lemma~\ref{prop-of-exteriour-powers} we have shown that the conditions in the statement are necessary. Now we want to show sufficiency. Since a reflexive module of rank $1$ with finite projective dimension must be free we are done if $r=1$. Now let $r \geq 2$. By assumption $M = H_{\mathfrak m}^{n-r+2}(E)^{\vee}$ is a perfect module of dimension $n-r$. Since $E^*$ is an $r$-syzygy of $M$ due to Lemma~\ref{Eilenberg-MacLane} we obtain that $E^*$ is a reflexive Eilenberg-MacLane module of depth $n$ with finite projective dimension and $$ H_{\mathfrak m}^n(E^*) \cong H_{\mathfrak m}^{n-r}(M). $$ Therefore Lemma~\ref{Eilenberg-MacLane} applies also to $E^*$ and says that $E^{**} \cong E$ is a $2$-syzygy of $H_{\mathfrak m}^{n-r}(M)^{\vee}(1-r(R))$. This means that there is an exact sequence $$ 0 \to E \to F \stackrel{\varphi}{\longrightarrow} G \to K_M(1-r(R)) \to 0 $$ where $F$ and $G$ are free modules with $\rank F = \rank G + r$. Since $\operatorname{Rad} I(\varphi) = \operatorname{Rad} \Ann_R K_M$ and $K_M$ has dimension $n-r$ it follows that $I(\varphi)$ has the maximal codimension $r+1$. Thus $E$ is a Buchsbaum-Rim module completing the proof. \end{proof} Since any module over a regular ring has finite projective dimension the last result takes a simpler form for sheaves on $\PP^n$. \begin{corollary} \label{Bu-Rim-sheafs-on-proj-space} A sheaf ${\mathcal E}$ on $\PP^n$ is a Buchsbaum-Rim sheaf if and only if ${\mathcal E}$ is an reflexive Eilenberg-MacLane sheaf of rank $r \leq n$ such that $H^i_*(\PP^n,{\mathcal E}) = 0$ if $i \neq 0, n-r+1, n+1$ and $H^{n-r+1}_*({\mathcal E})^{\vee}$ is a Cohen-Macaulay module of dimension $n-r$. \end{corollary} From this result we see again that the cotangent bundle on projective space is a Buchsbaum-Rim sheaf. \section{The cohomology of the degeneracy loci} Consider a morphism $\psi: {\mathcal P} \to {\mathcal B}_{\varphi}$ of sheaves of rank $t$ and $r$ respectively on the arithmetically Gorenstein scheme $Z = Proj (R)$ where ${\mathcal B}_{\varphi}$ is a Buchsbaum-Rim sheaf and $H^0_*(Z, {\mathcal P})$ is a free $R$-module. If $t = 1$ then $\psi$ is just a section of some twist of ${\mathcal B}_{\varphi}$. Thus we refer to $\psi$ as multiple sections of ${\mathcal B}_{\varphi}$. Throughout this paper we suppose that the ground field $K$ is infinite and that the degeneracy locus $S$ of $\psi$ has (the expected) codimension $r - t + 1 \geq 2$ in $Z$. If $t=1$ then $S$ is just the zero-locus of a regular section of a Buchsbaum-Rim sheaf. It will turn out that $S$ is often not equidimensional. Thus we are also interested in the top-dimensional part $X$ of $S$, i.e. the union of the highest-dimensional components of $S$. The aim of this section is to compute the cohomology modules of $S$ and $X$ respectively. An Eagon-Northcott complex involving ${\mathcal B}_{\varphi}$ will play an essential role. Observe that in contrast to the situation in the previous section where ${\mathcal F}$ and ${\mathcal G}$ were locally free the sheaf ${\mathcal B}_{\varphi}$ is in general not locally free. Our approach will be algebraic. Taking global sections we obtain an $R$-homomorphism $$ \psi : P \to B_{\varphi} $$ where $P$ is a free $R$-module of rank $t, 1 \leq t < r$. The first aim is to derive the complex mentioned above. We follow the approach described in Section~\ref{preliminaries}. The $R$-dual of the Koszul complex $\mathcal{C}_{r-t}(\psi^*)$ is $$ 0 \to (\wedge^0 B_{\varphi}^* \otimes S_{r-t}(P^*))^* \to (B_{\varphi}^* \otimes S_{r-t-1}(P^*))^* \to \ldots \to (\wedge^{r-t-1} B_{\varphi}^* \otimes P^*)^* \to (\wedge^{r-t} B_{\varphi}^* \otimes S_0(P^*))^*. $$ Using the isomorphisms in Lemma~\ref{prop-of-exteriour-powers} and $S_j(P^*)^* \cong S_j(P)$ we can rewrite $\mathcal{C}_{r-t}(\psi^*)^* \otimes \wedge^f F^* \otimes \wedge^g G$ as follows: $$ 0 \to \wedge^r B_{\varphi}^* \otimes S_{r-t}(P) \to \wedge^{r-1} B_{\varphi}^* \otimes S_{r-t-1}(P) \to \ldots \to \wedge^{t+1} B_{\varphi}^* \otimes P \to \wedge^{t} B_{\varphi}^* \otimes S_0(P). $$ The image of the map $\wedge^{t} \psi^* : \wedge^t B_{\varphi}^* \to \wedge^t P^*$ is (up to degree shift) an ideal of $R$ which we denote by $I(\psi)$ or just $I$, i.e.\ $\im \wedge^{t} \psi^* = I \otimes \wedge^t P^*$. Note that the saturation of $I$ is the homogeneous ideal $I_S$ defining the degeneracy locus $S$. Thus, using $\wedge^{t} \psi^*$ we can continue the complex above on the right-hand side and obtain the desired Eagon-Northcott complex \begin{eqnarray} \lefteqn{E_{\bullet} : \quad 0 \to \wedge^r B_{\varphi}^* \otimes S_{r-t}(P) \to \wedge^{r-1} B_{\varphi}^* \otimes S_{r-t-1}(P) \to \ldots} \\ & & \to \wedge^{t+1} B_{\varphi}^* \otimes P \to \wedge^{t} B_{\varphi}^* \to I \otimes \wedge^t P^* \to 0. \nonumber \end{eqnarray} The next result shows that the first cohomology modules of $R/I(\psi)$ vanish. \begin{lemma} \label{depth-estimate} The depth of $R/I$ is at least $n-r$. \end{lemma} \begin{proof} Let $r=2$. Then $B_{\varphi}$ has depth $n$ and $t=1$. Thus we have an exact sequence $$ 0 \to R(a) \stackrel{\psi}{\longrightarrow} B_{\varphi} \to I(b) \to 0 $$ where $a, b$ are integers. It provides the claim. Now let $r \geq 3$. We choose a sufficiently general linear form $l \in R$. For short we denote the functor $\_\_\otimes_R \overline{R}$ by $^-$ where $\overline{R} = R/l R$. Let $\alpha$ be the map $\Hom_{\overline{R}}(\overline{\psi},\overline{R}) : \Hom_{\overline{R}}(\overline{B_{\varphi}},\overline{R}) \to \Hom_{\overline{R}}(\overline{P},\overline{R})$ and define the homogeneous ideal $J \subset \overline{R}$ by $J \otimes \wedge^t \overline{P}^* = \im \wedge_{\overline{R}}^t \alpha$. Our first claim is that $\overline{I} = J$ provided $n > r$. Multiplication by $l$ provides the commutative diagram $$ \begin{array}{ccccccc} 0 \to & P(-1) & \to & P & \to & \overline{P} & \to 0 \\ & \downarrow\!{\scriptstyle \psi} & & \downarrow\!{\scriptstyle \psi} & & \downarrow\!{\scriptstyle \overline{\psi}} &\\ 0 \to & B_{\varphi}(-1) & \to & B_{\varphi} & \to & \overline{B_{\varphi}} & \to 0 \\ \end{array} $$ Dualizing gives the commutative diagram $$ \begin{array}{ccccccc} 0 \to & B_{\varphi}^* & \to & B_{\varphi}^*(1) & \to & \operatorname{Ext}_R^1(\overline{B_{\varphi}},R) & \to \operatorname{Ext}_R^1(B_{\varphi},R) = 0 \\ & \downarrow\!{\scriptstyle \psi^*} & & \downarrow\!{\scriptstyle \psi^*} & & \downarrow\!{\scriptstyle \beta} &\\ 0 \to & P^* & \to & P^*(1) & \to & \operatorname{Ext}_R^1(\overline{P},R) & \to \operatorname{Ext}_R^1(P,R) = 0 \\ \end{array} $$ where the vanishings on the right-hand side are due to duality and the fact that $B_{\varphi}$ is an Eilenberg-MacLane module of depth $n-r+2 \neq n$. Using $\operatorname{Ext}_R^1(\overline{B_{\varphi}},R)(-1) \cong \Hom_{\overline{R}}(\overline{B_{\varphi}},\overline{R})$ and $\operatorname{Ext}_R^1(\overline{P},R)(-1) \cong \Hom_{\overline{R}}(\overline{P},\overline{R})$ (cf., for example, \cite{Bruns-Herzog}, Lemma 3.1.16) we see that $\beta$ can be identified with $\alpha$ as well as $\psi^* \otimes \overline{R}$. It follows $$ J \otimes \wedge^t {\overline{P}}^* = \im \wedge_{\overline{R}}^t \alpha = \im \wedge_{\overline{R}}^t (\psi^* \otimes \overline{R}) \cong \im (\wedge_R^t \psi^* \otimes \overline{R}) \cong (\im \wedge_R^t \psi^*) \otimes \overline{R} = I \otimes \wedge^t P^* \otimes \overline{R} $$ and thus $J = \overline{I}$ as we wanted to show. The second commutative diagram also provides $B_{\varphi}^* \otimes \overline{R} \cong \Hom_{\overline{R}}(\overline{B_{\varphi}},\overline{R})$. It follows $$ \wedge^t_{\overline{R}} \Hom_{\overline{R}}(\overline{B_{\varphi}},\overline{R}) \cong \wedge^t_{\overline{R}} (B_{\varphi}^* \otimes \overline{R}) \cong (\wedge^t_R B_{\varphi}^*) \otimes \overline{R}. $$ Thus we have an exact commutative diagram $$ \begin{array}{ccccccc} & 0 & & 0 & & 0 &\\ & \downarrow & & \downarrow & & \downarrow &\\ 0 \to & C(-1) & \to & \wedge^t B_{\varphi}^*(-1) & \to & I(-1) \otimes \wedge^t P^* & \to 0 \\ & \downarrow\!{\scriptstyle l} & & \downarrow\!{\scriptstyle l} & & \downarrow\!{\scriptstyle l} &\\ 0 \to & C & \to & \wedge^t B_{\varphi}^* & \to & I \otimes \wedge^t P^* & \to 0 \\ & & & \downarrow & & &\\ 0 \to & L & \to & \wedge^t_{\overline{R}} \Hom_{\overline{R}}(\overline{B_{\varphi}},\overline{R}) & \to & J \otimes \wedge^t {\overline{P}}^* & \to 0 \\ & & & \downarrow & & &\\ & & & 0 & & & \end{array} $$ where $C = \ker \wedge_R^t \psi^*$ and $L = \ker \wedge_{\overline{R}}^t \alpha$. Since $\wedge^t B_{\varphi}^*$ has depth $n+1-t \geq n+2-r$ due to Lemma~\ref{prop-of-exteriour-powers} the first row shows that our assertion is equivalent to $\depth C \geq n+2-r$. In order to show this we induct on $n-r$. If $n = r$ the claim follows by the exact cohomology sequence induced by the top-line of the previous diagram and $\depth \wedge^t B_{\varphi}^* \geq 2$. Let $n > r$. Then our first claim and the Snake lemma applied to the diagram above imply $\overline{C} \cong L$. The induction hypothesis applies to $L$ and we obtain $$ 0 < n+1-r \leq \depth L < \depth C $$ completing the proof. \end{proof} For computing the other cohomology modules of $R/I(\psi)$ we use the Eagon-Northcott complex above. In order to ease notation let us write $E_{\bullet}$ as $$ E_{\bullet} \colon \quad 0 \to E_r \stackrel{\delta_r}{\longrightarrow} E_{r-1} \stackrel{\delta_{r-1}}{\longrightarrow} \cdots \to E_t \stackrel{\delta_t}{\longrightarrow} I(\psi) (p) \to 0 $$ where $$ E_i = \wedge^i B_{\varphi}^* \otimes S_{i - t}(P) \quad \mbox{and} \quad R(p) \cong \wedge^t P^*. $$ The number of minimal generators of an $R$-module $N$ is denoted by $\mu(N)$. \begin{proposition} \label{cohomology-of locus} Put $I = I(\psi)$. Then we have: \begin{itemize} \item[(a)] For $j\neq \dim R/I = n+t-r$ it holds $$ H^j_m (R/I) \cong \left\{ \begin{array}{ll} S_i(M_{\varphi})^\vee \otimes S_{i-t} (P) \otimes \wedge^t P (1-r(R)) & \mbox{if } j=n+t-2i \quad \mbox{where } \max \{t, \frac{r+1}{2}\} \leq i \leq \left \lfloor \frac{r+t}{2} \right \rfloor \\ 0 & \mbox{otherwise}. \end{array} \right. $$ \item[(b)] The canonical module satisfies $$ \mu(K_{R/I}) \leq \binom{r-1}{t-1} + \left\{ \begin{array}{ll} \binom{\frac{r}{2}+g-1}{g-1} \cdot \binom{\frac{r}{2} - 1}{t-1} & \mbox{if $r$ is even and } 1\leq t \leq \frac{r}{2} \\ 0 & \mbox{otherwise}. \end{array}\right. $$ \end{itemize} \end{proposition} \begin{proof} We consider the Eagon-Northcott complex $E_{\bullet}$ above. According to the Lemma~\ref{prop-of-exteriour-powers} $B_{\varphi}^*$ is an $r$-syzygy. Therefore $B_{\varphi}^*$ is locally free in codimension $r$. It follows that the Eagon-Northcott complex $E_{\bullet}$ is exact in codimension $r$. Therefore its homology modules $H_i(E_{\bullet})$ have dimension $\leq n-r$. Thus the exact sequence $$ 0 \to \im \delta_{i+1} \to \ker \delta_i \to H_i(E_{\bullet}) \to 0 $$ implies $$ H_{\mathfrak m}^j(\ker \delta_i) \cong H_{\mathfrak m}^j(\im \delta_{i+1}) \quad \mbox{if} ~ j \geq n-r+2. \leqno(1) $$ Moreover, there are exact sequences $$ 0 \to \ker \delta_i \to E_i \to \im \delta_i \to 0 $$ inducing exact sequences $$ H_{\mathfrak m}^j(E_i) \to H_{\mathfrak m}^j(\im \delta_i) \to H_{\mathfrak m}^{j+1}(\ker \delta_i) \to H_{\mathfrak m}^{j+1}(E_i) \leqno(2) $$ where the injectivity or surjectivity of the map in the middle can be checked by means of Lemma~\ref{prop-of-exteriour-powers}. This map is an isomorphism if $j \neq n-i, n+1-i, n, n+1$. Let us consider the map $\delta_r$. Due to our assumption the map $\psi^*: B_{\varphi}^* \to P^*$ is generically surjective. Thus the same applies to the Koszul map $B_{\varphi}^* \otimes S_{r-t-1}(P^*) \to S_{r-t}(P^*)$ which is induced by $\psi$. It follows that the $R$-dual of this map is injective. But the latter is (up to a degree shift) just $\delta_r$. Hence we have seen that $\im \delta_r \cong E_r$. According to Lemma~\ref{depth-estimate} it suffices to consider $H_{\mathfrak m}^j(R/I)$ where $n-r \leq j \leq n+t-r = \dim R/I$. For this we distinguish several cases. \\ {\it Case 1}: Let us assume that $n-t < j \leq n+t-r$. This can occur if and only if $t \geq \frac{r+1}{2}$. Using (2) and (1) in alternating order we get \begin{eqnarray*} \lefteqn{H_{\mathfrak m}^j(R/I)(p) \cong H_{\mathfrak m}^{j+1}(\im \delta_t) \cong H_{\mathfrak m}^{j+2}(\ker \delta_t) \cong H_{\mathfrak m}^{j+2}(\im \delta_{t+1}) \cong \ldots} \\ & & \cong H_{\mathfrak m}^{r+j-t}(\im \delta_{r-1}) \hookrightarrow H_{\mathfrak m}^{r+j+1-t}(\ker \delta_{r-1}) \cong H_{\mathfrak m}^{r+j+1-t}(\im \delta_{r}) \cong H_{\mathfrak m}^{r+j+1-t}(E_r) \end{eqnarray*} where the injection holds true because we have by our assumptions $n+3-r \leq n+2-t \leq j+1 \leq r+j-t \leq n$. It follows by Lemma~\ref{prop-of-exteriour-powers} and Lemma~\ref{duality} that $$ H_{\mathfrak m}^j(R/I) = 0 \quad \mbox{if} ~ n-t < j < n+t-r $$ and in case $j = \dim R/I$ for the canonical module $$ \mu(K_{R/I}) \leq \mu(H_{\mathfrak m}^{n+1}(E_r)^\vee) = \mu(E_r) = \mu(S_{r-t}(P)) = \binom{r-1}{t-1}. $$ {\it Case 2}: Let us assume that $n-r \leq j= n+t-1-2i \leq \min \{n+t-r, n-t \}$, i.e.\ $\max \{t, \frac{r-1}{2} \} \leq i \leq \frac{r+t-1}{2}$. Using (2) and (1) again we obtain \begin{eqnarray*} \lefteqn{H_{\mathfrak m}^{n+t-1-2i}(R/I)(p) \cong H_{\mathfrak m}^{n+t-2i}(\im \delta_t) \cong H_{\mathfrak m}^{n+t+1-2i}(\ker \delta_t) \cong H_{\mathfrak m}^{n+t+1-2i}(\im \delta_{t+1}) \cong \ldots} \\ & \cong & H_{\mathfrak m}^{n-i}(\im \delta_i) \hookrightarrow H_{\mathfrak m}^{n+1-i}(\ker \delta_i) \cong \ldots \\ & \cong & H_{\mathfrak m}^{n+r-1-2i}(\im \delta_{r-1}) \hookrightarrow H_{\mathfrak m}^{n+r-2i}(\ker \delta_{r-1}) \cong H_{\mathfrak m}^{n+r-2i}(\im \delta_{r}) \cong H_{\mathfrak m}^{n+r-2i}(E_r). \end{eqnarray*} Since $E_r$ is free we get $$ H_{\mathfrak m}^{n+t-1-2i}(R/I) = 0 \quad \mbox{if} ~ i \geq \frac{r}{2} $$ and in case $i = \frac{r-1}{2}$ we obtain the same bound for the number of minimal generators of the canonical module as in Case 1. \\ {\it Case 3}: Let us assume that $n-r \leq j= n+t-2i \leq \min \{n+t-r, n-t \}$, i.e.\ $\max \{t, \frac{r}{2} \} \leq i \leq \frac{r+t}{2}$. If follows similarly as before \begin{eqnarray*} \lefteqn{H_{\mathfrak m}^{n+t-2i}(R/I)(p) \cong H_{\mathfrak m}^{n+t+1-2i}(\im \delta_t)} \\ & \cong & H_{\mathfrak m}^{n+t+2-2i}(\ker \delta_t) \cong H_{\mathfrak m}^{n+t+2-2i}(\im \delta_{t+1}) \cong \ldots \\ & \cong & H_{\mathfrak m}^{n+1-i}(\im \delta_i). \end{eqnarray*} Now we look at the exact sequence $$ H_{\mathfrak m}^{n+1-i}(\ker \delta_i) \to H_{\mathfrak m}^{n+1-i}(E_i) \to H_{\mathfrak m}^{n+1-i}(\im \delta_i) \to H_{\mathfrak m}^{n+2-i}(\ker \delta_i). \leqno(3) $$ We use again (1) and (2) in order to obtain information on the modules on the left-hand and on the right-hand side. This provides $$ H_{\mathfrak m}^{n+1-i}(\ker \delta_i) \cong H_{\mathfrak m}^{n+1-i}(\im \delta_{i+1}) \cong \ldots \cong H_{\mathfrak m}^{n+r-2i}(\im \delta_r) \cong H_{\mathfrak m}^{n+r-2i}(E_r) = 0 $$ because $n+r-2i \leq n$ and \begin{eqnarray*} \lefteqn{H_{\mathfrak m}^{n+2-i}(\ker \delta_i) \cong H_{\mathfrak m}^{n+2-i}(\im \delta_{i+1}) \cong \ldots } \\ & & \cong H_{\mathfrak m}^{n+r-2i}(\im \delta_{r-1}) \hookrightarrow H_{\mathfrak m}^{n+r-2i}(\ker \delta_r) \cong H_{\mathfrak m}^{n+r+1-2i}(\im \delta_r) \cong H_{\mathfrak m}^{n+r+1-2i}(E_r) \end{eqnarray*} where the last module vanishes if and only if $i \neq \frac{r}{2}$. Therefore (3) yields if $i \neq \frac{r}{2}$ $$ H_{\mathfrak m}^{n+t-2i}(R/I)(p) \cong H_{\mathfrak m}^{n+1-i}(E_i) \cong H_{\mathfrak m}^{n+1-i}(\wedge^iB_{\varphi}^*) \otimes S_{{\frac{r}{2}}-t}(P) \cong S_{\frac{r}{2}}(M_{\varphi})^{\vee} \otimes S_{{\frac{r}{2}}-t}(P) (1 - r(R)). $$ In case $i = \frac{r}{2}$ taking $K$-duals of (3) furnishes the exact sequence $$ E_r^* \to \Ext^{r-t+1}(R/I,R)(-p) \to S_{\frac{r}{2}}(M_{\varphi}) \otimes S_{{\frac{r}{2}}-t}(P). $$ It follows for the canonical module $$ \begin{array}{lcl} \mu(K_{R/I}) & \leq & \mu(S_{\frac{r}{2}} (M_{\varphi})) \otimes S_{\frac{r}{2}-t} (P) + \mu(E_r) \\ & = & \binom{\frac{r}{2}+g-1}{g-1} \cdot \binom{\frac{r}{2}-1}{t-1} + \binom{r-1}{t-1}. \end{array} $$ Our assertions are now a consequence of the results in the three cases above. \end{proof} \begin{corollary} \label{depth-formula} For the depth of the coordinate ring we have $$ \depth R/I(\psi) = \left \{ \begin{array}{ll} n-r & \mbox{if} ~ r+t \; \mbox{is even} \\ n-r+1 & \mbox{if} ~ r+t \; \mbox{is odd}. \end{array} \right. $$ \end{corollary} Put $e = \depth R/I(\psi)$. Then the only non-vanishing cohomology modules of $R/I(\psi)$ besides $H_{\mathfrak m}^{n+t-r}(R/I(\psi))$ are $H_{\mathfrak m}^{e+2k}(R/I(\psi))$ where $k$ is an integer with $0 \leq k \leq \frac{1}{2} [\min \{ n-t,n+t-r-1 \} - e]$. \smallskip It has already been observed in \cite{Mig-P_gorenstein} that $I(\psi)$ is not always an unmixed ideal. This gives rise to consider the ideal $J(\psi)$ which is defined as the intersection of the primary components of $I(\psi)$ having maximal dimension. We denote by $X$ the subscheme of $Z$ defined by $J(\psi)$ and call it the top-dimensional part of the degeneracy locus $S$. Our next aim is to clarify the relationship between $S$ and $X$. For this we need a cohomological criterion for unmixedness stated as Lemma III.2.3 in \cite{habil}. \begin{lemma} \label{unmixedness-criterion} Let $I \subset R$ be a homogeneous ideal. Then $I$ is unmixed if and only if $$ \dim \Supp(H_{\mathfrak m}^i(R/I)) < i \quad \mbox{for all} \; i < \dim R/I $$ where we put $\dim \Supp(M) = - \infty$ if $M = 0$. \end{lemma} Now we can show. \begin{proposition} \label{coho_of_top-dimensional} Let $I = I(\psi)$ and $J = J(\psi)$. Then it holds: \begin{itemize} \item[(a)] $I$ is unmixed if and only if $r+t$ is odd. \item[(b)] If $r+t$ is even then $I$ has a primary component of codimension $r+1$. Let $Q$ be the intersection of all those components. Then we have $I = J \cap Q$ and $$ H^j_m(R/J) \cong \left\{ \begin{array}{ll} H^j_m (R/I) & \mbox{if } j\neq n-r \\ 0 & \mbox{if } j=n-r \end{array}\right. $$ and $$ J/I \cong S_{\frac{r-t}{2}} (M_{\varphi}) \otimes S_{\frac{r-t}{2}}(P) \otimes \wedge^f F^* \otimes \wedge^g G \otimes \wedge^t P. $$ \end{itemize} \end{proposition} \begin{proof} For $i = 0,\ldots,r$ the module $S_i(M_{\varphi})$ is a perfect module of dimension $n-r$. Hence claim (a) follows by Proposition~\ref{cohomology-of locus} and Lemma~\ref{unmixedness-criterion}. In order to show (b) we note first that the maximal codimension of a component of I is $r+1$ because $\depth R/I = n-r$. Let $Q$ be the intersection of all these components and let $J$ be the intersection of the remaining ones. Then $I=J\cap Q$ and the components of $J$ have codimension $\leq r$. We have to show that $J$ is unmixed. As a first step we will prove that $\depth R/J > n-r$. We induct on $n-r\geq 0$. If $r=n$ the claim is clear since $J$ is saturated by construction. Let $r=n-1$. It follows that $\dim R/I = 1 + t \geq2$ and that $R/Q$ and $S_{\frac{r+t}{2}}(M_{\varphi})$ have positive dimension. Now we look at the exact sequence $$ 0 \to R/I \to R/J \oplus R/Q \to R/ (J + Q) \to 0. $$ We can find an $l \in [R]_1$ which is a parameter on $R/I, R/J, R/Q, S_{\frac{r+t}{2}}(M_{\varphi})$ and also on $R/(J+Q)$ if it has positive dimension. Using $H_{\mathfrak m}^2(R/I) \cong H_{\mathfrak m}^2(R/J)$ we obtain a commutative diagram with exact rows $$ \begin{array}{cccccccc} H_{\mathfrak m}^1(R/I)(-1) & \to & H_{\mathfrak m}^1(R/J)(-1) & \oplus & H_{\mathfrak m}^1(R/Q)(-1) & \to & H_{\mathfrak m}^1(R/(J+Q))(-1) & \to 0 \\ \downarrow\!{\scriptstyle \beta_1} & & \downarrow\!{\scriptstyle \beta_2} & & \downarrow\!{\scriptstyle \beta_3} & & \downarrow\!{\scriptstyle \beta_4} & \\ H_{\mathfrak m}^1(R/I) & \to & H_{\mathfrak m}^1(R/J) & \oplus & H_{\mathfrak m}^1(R/Q) & \to & H_{\mathfrak m}^1(R/J+Q) & \to 0 \end{array} $$ where the vertical maps are multiplication by $l$. Due to our choice of $l$ it holds $H_{\mathfrak m}^1(R/(Q + l R)) = H_{\mathfrak m}^1(R/(J+Q+l R)) = 0$. Thus $\beta_3$ and $\beta_4$ are surjective. Since $l$ is a parameter of the Cohen-Macaulay module $S_{\frac{r+t}{2}}(M)$ the multiplication map $S_{\frac{r+t}{2}}(M)(-1) \stackrel{l}{\longrightarrow} S_{\frac{r+t}{2}}(M)$ is injective, thus the dual map $S_{\frac{r+t}{2}}(M)^{\vee} \stackrel{l}{\longrightarrow} S_{\frac{r+t}{2}}(M)^{\vee}(1)$ is an epimorphism. Therefore $\beta_1$ is surjective due to Proposition~\ref{cohomology-of locus}. The same is true for $\beta_2$ by the commutative diagram above. Since $R/J$ is unmixed Lemma~\ref{unmixedness-criterion} implies that $H_{\mathfrak m}^1(R/J)$ is finitely generated, hence it must be zero by Nakayama's lemma. Finally, let $r \leq n-2$. We consider the commutative diagram $$ \begin{array}{ccccccccc} 0 \to & R/I(-1) & \to & R/J(-1) & \oplus & R/Q(-1) & \to & R/(J+Q)(-1) & \to 0 \\ & \downarrow & & \downarrow & & \downarrow\ & & \downarrow & \\ 0 \to & R/I & \to & R/J & \oplus & R/Q & \to & R/(J+Q) & \to 0 \end{array} $$ where the vertical maps are multiplication by $l$. By Lemma~\ref{depth-formula} $\depth R/I \geq 2$ and by assumption on $r$ $R/J$ and $R/Q$ have positive depth. Hence the cohomology sequence induced by the bottom line provides $H_{\mathfrak m}^0(R/(J+Q)) = 0$. It follows that we may choose $l$ as non-zero divisor on $R/I, R/J, R/Q$ and $R/(J+Q)$, i.e., the vertical maps in the diagram above are all injective. Thus the Snake lemma implies the exact sequence $$ 0 \to \overline{R}/\overline{I} \to \overline{R}/\overline{J} \oplus \overline{R}/\overline{Q} \to \overline{R}/\overline{J} + \overline{Q} \to 0 $$ where we denote by $^-$ again the functor $\_\_ \otimes_{R} R/l R$. By Bertini's theorem $\overline{Q}$ is unmixed of codimension $r+1$ in $\overline{R}$ (possibly) up to a component associated to the irrelevant ideal of $\overline{R}$. Moreover, we have seen in the proof of Lemma~\ref{depth-estimate} that the induction hypothesis applies to $\overline{I}$. But the last exact sequence implies $\overline{I} = \overline{J} \cap \overline{Q}$. It follows that $\overline{J}$ is the intersection of the top-dimensional components of $\overline{I}$. Hence we get by induction $\depth \overline{R}/\overline{J} \geq n-r$. But $l$ was a non-zero divisor on $R/J$ thus we obtain $$ \depth R/J > \depth \overline{R/J} = \depth \overline{R}/\overline{J} $$ completing our induction. Next we consider the exact sequence $$ 0 \to J/I \to R/I \to R/J \to 0. \leqno(*) $$ Note that $J/I\cong (J+Q)/Q$ has dimension $\leq n-r$. Moreover we have just shown $\depth R/J > n-r$. Therefore the induced cohomology sequence yields: $$ H_{\mathfrak m}^i(R/J) \cong H_{\mathfrak m}^i(R/I) \quad \mbox{if} ~ i > n-r, $$ $$ H_{\mathfrak m}^{n-r}(J/I) \cong H_{\mathfrak m}^{n-r} (R/I) \cong S_{\frac{r+t}{2}} (M_{\varphi})^\vee\otimes S_{\frac{r-t}{2}} (P) \otimes \wedge^t P (1-r(R)) $$ and $J/I$ is Cohen-Macaulay of dimension $n-r$. The first isomorphisms, Proposition~\ref{cohomology-of locus} and Lemma~\ref{unmixedness-criterion} imply now that $J$ must be unmixed. Now we use that for a Cohen-Macaulay module $M$ it holds $K_{K_M} \cong M$. Thus we get by duality and Lemma~\ref{can-module-of-symm-powers} \begin{eqnarray*} J/I & \cong & H_{\mathfrak m}^{n-r}( H_{\mathfrak m}^{n-r}(J/I)^{\vee})^{\vee} \\ & \cong & H_{\mathfrak m}^{n-r}(S_{\frac{r+t}{2}} (M_{\varphi}))^\vee \otimes S_{\frac{r-t}{2}} (P) \otimes \wedge^t P (1 - r(R)) \\ & \cong & \Ext^{r+1}(S_{\frac{r+t}{2}} (M_{\varphi}),R) \otimes S_{\frac{r-t}{2}} (P) \otimes \wedge^t P \\ & \cong & S_{\frac{r-t}{2}} (M_{\varphi}) \otimes S_{\frac{r-t}{2}} (P) \otimes \wedge^f F^* \otimes \wedge^g G. \otimes \wedge^t P. \end{eqnarray*} This finishes the proof. \end{proof} \begin{remark} The arguments in the previous proof also provide that $R/Q$ is Cohen-Macaulay of dimension $n-r$ and $$ n-r-1 \leq \depth R/(J+Q) \leq \dim R/(J+Q) \leq n-r. $$ \end{remark} Our results with respect to ring-theoretic properties can be summarized as follows. Recall that $X$ denotes the top-dimensional part of the degeneracy locus $S$ of $\psi$. \begin{theorem} \label{summary_for_degeneracy_loci} With the notation above we have: \begin{itemize} \item[(a)] If $r = n$ then $S$ is equidimensional and locally Cohen-Macaulay. \item[(b)] $S$ is equidimensional if and only if $r+t$ is odd or $r = n$. Moreover, if $r < n$ then $X$ is locally Cohen-Macaulay if and only if $X$ is arithmetically Cohen-Macaulay. \item[(c)] If $r+t$ is odd then $X = S$ is arithmetically Cohen-Macaulay if and only if $t=1$. In this case $S$ has Cohen-Macaulay type $\leq 1 + \binom{\frac{r}{2}+g-1}{g-1}$. \item[(d)] Let $r+t$ be even. Then \begin{itemize} \item[(i)] $X$ is arithmetically Cohen-Macaulay if and only if $1 \leq t \leq 2$. If $t =1$ then $X$ is arithmetically Gorenstein. If $t = 2$ then $X$ has Cohen-Macaulay type $\leq r - 1 + \binom{\frac{r}{2}+g-1}{g-1} \cdot (\frac{r}{2} - 1)$. \item[(ii)] If in addition $r < n$ then the components of $S$ have either codimension $r-t+1$ or codimension $r+1$. \end{itemize} \end{itemize} \end{theorem} \begin{proof} (a) If $r = n$ then $\dim M_{\varphi} = 0$. Hence it follows by Proposition~\ref{cohomology-of locus} that the modules $H^i_*(Z,{\mathcal J}_S)$ have finite length if $i \leq \dim S$ which is equivalent to $S$ being equidimensional and locally Cohen-Macaulay. (b) If $r+t$ is odd then $S$ is equidimensional due to Proposition~\ref{coho_of_top-dimensional}. If $r+t$ is even then Proposition~\ref{coho_of_top-dimensional} shows that $S$ is equidimensional if and only if $r=n$. Moreover, $r<n$ implies that $M_{\varphi}$ does not have finite length. Therefore Proposition~\ref{coho_of_top-dimensional} furnishes that $R/J(\psi)$ has cohomology of finite length if and only if $R/J(\psi)$ is Cohen-Macaulay. Claim (b) follows. (c) If $r+t$ is odd then we have by Proposition~\ref{coho_of_top-dimensional} that $S$ is equidimensional and by Corollary~\ref{depth-formula} that $\depth R/I(\psi) = n-r+1$. The claim follows because of $\dim R/I(\psi) = n-r+t$. (d) If $r+t$ is even we get $r \leq t-2$. Hence the claim is a consequence of Propositions~\ref{cohomology-of locus} and \ref{coho_of_top-dimensional}. \end{proof} \begin{remark} (i) If we specialize the previous result to $Z = \PP^n, r=3, t=1$ then we get the main result of \cite{Mig-P_gorenstein}. \\ (ii) In case $t=1$ and $r$ is even the result has been first proved by the first and third author who communicated it to Kustin. Subsequently, Kustin \cite{Kustin} strengthened it by removing almost all the assumptions on the ring $R$ and computing a free resolution (cf.\ also Remark \ref{discussion_of_minimality} and Corollary \ref{one-sect-even-case}). \\ (iii) Note that in the above theorem the term equidimensional is used in the scheme-theoretic sense. Thus $S$ being equidimensional does not automatically imply that $I(\psi)$ is saturated. However, due to Corollary 4.3 $I(\psi)$ is not saturated if and only if $r = n$ and $r+t$ is even. We use this fact in Section~\ref{resolution_general}. \end{remark} The next result generalizes \cite{Mig-P_gorenstein}, Corollary 1.2. \begin{corollary} Let ${\mathcal E}$ be a vector bundle on $\PP^n$ of rank $n$ where $n$ is odd such that $H^i_*(\PP^n,{\mathcal E}) = 0$ if $2 \leq i \leq n-1$. Let $s$ be a section of ${\mathcal E}$ vanishing on a scheme $X$ of codimension $n$. Then $X$ is arithmetically Gorenstein. \end{corollary} \begin{proof} According to Corollary~\ref{Bu-Rim-sheafs-on-proj-space}, ${\mathcal E}$ is a Buchsbaum-Rim sheaf. Therefore Theorem~\ref{summary_for_degeneracy_loci} shows the claim. \end{proof} \section{The resolutions of the loci} \label{resolution_general} In this section we show how free resolutions can be obtained for the schemes described in this paper. In most cases we expect these resolutions to be minimal. The main tools are the Eagon-Northcott complex $E_{\bullet}$ in $(4.1)$, its dual and a general result for comparing the resolution of the scheme with its cohomology modules. We begin by considering the Eagon-Northcott complex again. The interested reader will have observed that this complex is {\it not} exact in general. Fortunately, we are able to compute its homology. \begin{proposition} \label{homology_of_EN} The homology modules of the Eagon-Northcott $E_{\bullet}$ complex are: $$ H_i (E_{\bullet}) \cong \left\{ \begin{array}{ll} S_j(M_{\varphi}) \otimes S_{r-t-j}(P) \otimes \wedge^f F^* \otimes \wedge^g G & \mbox{if} ~ i=r-1-2j \mbox{ where } j \in \ZZ, \; t \leq i \leq r-3 \\ 0 & \mbox{otherwise}. \end{array}\right. $$ \end{proposition} \begin{proof} According to Lemma~\ref{prop-of-exteriour-powers} and Lemma~\ref{depth-estimate} all non-trivial modules occurring in $E_{\bullet}$ have depth $\geq n-r+1$. Thus using $n-r$ general linear forms in $R$ and arguments as in Proposition~\ref{depth-estimate} we see that it suffices to consider the case where $R$ has dimension $r+1$, i.e., we may and will assume that $n=r$. Then we already know that the homology modules of $E_{\bullet}$ have finite length. As for the cohomology of $R/I$ one computes (cf.\ Proposition~\ref{cohomology-of locus}) $$ H^1_m(\im \delta_i) \cong \left\{ \begin{array}{ll} S_{r-j} (M_{\varphi})^\vee \otimes S_{r-j-t}(P)(1- r(R)) & \mbox{if} ~ t \leq i = r-2j < r \\ 0 & \mbox{if} ~ r+i \mbox{ is odd} \end{array} \right. $$ Since $r = n$ the module $M_{\varphi}$ has finite length. It follows \begin{eqnarray*} S_{r-j}(M_{\varphi})^{\vee}(1-r(R)) & \cong & H_{\mathfrak m}^0(S_{r-j}(M_{\varphi}))^{\vee}(1-r(R)) \\ & \cong & \operatorname{Ext}_R^{r+1}(S_{r-j}(M_{\varphi}),R) \quad \mbox{by \ref{duality}} \\ & \cong & S_j(M_{\varphi}) \otimes \wedge^f F^* \otimes \wedge^g G \quad \mbox{by \ref{can-module-of-symm-powers}.} \end{eqnarray*} Thus we have $$ H^1_m(\im \delta_i) \cong \left\{ \begin{array}{ll} S_{j} (M_{\varphi}) \otimes S_{r-j-t}(P) \otimes \wedge^f F^* \otimes \wedge^g G & \mbox{if} ~ t \leq i = r-2j < r \\ 0 & \mbox{if} ~ r+i \mbox{ is odd} \end{array} \right. \leqno(+) $$ The modules $\im \delta_i$ and $\ker \delta_i$ are submodules of the reflexive modules $E_{i+1}$ and $E_i$, respectively. Hence both have positive depth. Since $\depth E_i = n+1-i \geq 2$ if $i<r=n$ the exact sequence $$ 0 \to \ker \delta_i \to E_i \to \im \delta_i \to 0 $$ shows $\depth \ker \delta_i \geq 2$ for all $i=t, \ldots, r$. Using the finite length of $H_i(E_{\bullet})$ the exact sequence $$ 0 \to \im \delta_{i+1} \to \ker \delta_i \to H_i (E_{\bullet} ) \to 0 $$ provides $H^1_m (\im \delta_{i+1}) \cong H^0_m (H_i(E_{\bullet})) \cong H_i(E_{\bullet})$. Hence $(+)$ proves our assertion because $\delta_t$ is surjective by the definition of the ideal $I$. \end{proof} \begin{remark} \label{remark_resolution_in_general} Now we can compute a free resolution of $I = I(\psi)$ as follows: Proposition~\ref{homology_of_EN} provides exact sequences $$ 0 \to \ker \delta_{r+1-2j} \to E_{r+1-2j} \to E_{r-2j} \to \im \delta_{r-2j} \to 0, \quad t \leq r-2j<r, \leqno(1) $$ and if $r+t$ is odd additionally $$ 0 \to \ker \delta_t \to E_t \to I(p) \to 0, \leqno(1') $$ $$ 0 \to \im \delta_{r-2j} \to \ker \delta_{r-2j-1} \to S_j(M_{\varphi}) \otimes S_{r-t-j}(P) \otimes \wedge^f F^* \otimes \wedge^g G \to 0, \quad t<r-2j<r. \leqno(2) $$ According to Proposition~\ref{EN-complexes_are_exact} and Proposition~\ref{prop-of-exteriour-powers} we know the minimal resolution of $E_i$ and $S_j(M_{\varphi})$, respectively. Thus, in order to get a resolution of $I$ we compute successively resolutions of $\im \delta_{r-2}, \ker \delta_{r-3}, \\ \im \delta_{r-4}, \ldots, \im \delta_t = I(p)$ using the exact sequences (1) and (2). If (1) is used we just apply the mapping cone procedure twice. If we use (2) we apply the Horseshoe lemma. \end{remark} Following the procedure just described we certainly do not obtain a minimal free resolution of $I$. Thus we need some results which allow us to split off redundant terms. The idea is to compare the resolution of $I$ with those of its cohomology modules. In particular, this requires information on the canonical module $K_S = \operatorname{Ext}_R^{r-t}(R/I(\psi),R) (r(R) - 1)$ of our degeneracy locus $S$ which we derive first. \begin{lemma} \label{can_module_depth_estimate} The depth of the canonical module of $X$ is at least $\min \{n-r+t, n-r+2\}$. In particular, it is Cohen-Macaulay if $1 \leq t \leq 2$. \end{lemma} \begin{proof} Let $A = R/I(\psi)$. We induct on $n - r$. Since the canonical module always satisfies $\depth K_A \geq \min \{\dim A, 2\}$ the claim is clear if $n = r$. Let $n > r$ and let $l \in R$ be a general linear form. We want to show $$ K_A/ l K_A \cong K_{A/l A}. $$ Due to Corollary \ref{depth-formula} there is an exact sequence induced by multiplication $$ 0 \to A(-1) \stackrel{l}{\longrightarrow} A \to A/l A \to 0. $$ It provides the long exact sequence $$ 0 \to \operatorname{Ext}_R^{r-t}(A,R) \stackrel{l}{\longrightarrow} \operatorname{Ext}_R^{r-t}(A,R)(1) \to \operatorname{Ext}_R^{r-t+1}(A/l A,R) \to \operatorname{Ext}_R^{r-t+1}(A,R) \stackrel{l}{\longrightarrow} \operatorname{Ext}_R^{r-t+1}(A,R)(1) \to \ldots. $$ Using $\operatorname{Ext}_R^{r-t-1}(A/l A,R)(r(R) - 1) \cong K_{A/l A}$ we can rewrite the last sequence as $$ 0 \to K_A(-1) \stackrel{l}{\longrightarrow} K_A \to K_{A/l A} \to \operatorname{Ext}_R^{r-t+1}(A,R)(r(R)-2) \stackrel{l}{\longrightarrow} \operatorname{Ext}_R^{r-t+1}(A,R)(r(R)-1) \to \ldots. $$ We claim that the multiplication map on the right-hand side is injective. Indeed, if $r+1$ is odd or $2t \geq r+1$ then we have by duality and Proposition \ref{cohomology-of locus} that $\operatorname{Ext}_R^{r-t+1}(A,R) = 0$. Otherwise the multiplication map is (up to degree shift) essentially given by $$ S_{\frac{r+1}{2}}(M_{\varphi}) \otimes S_{\frac{r+1}{2}-t} (P)^* (-1) \stackrel{l}{\longrightarrow} S_{\frac{r+1}{2}}(M_{\varphi}) \otimes S_{\frac{r+1}{2}-t} (P)^*. $$ But the module $S_{\frac{r+1}{2}}(M_{\varphi})$ is Cohen-Macaulay of dimension $n-r > 0$ according to Lemma \ref{prop-of-exteriour-powers}. Thus we can choose $l$ as a regular element with respect to $S_{\frac{r+1}{2}}(M_{\varphi})$ and the claim follows. Hence the exact sequence above implies $$ K_A/ l K_A \cong K_{A/l A} \quad \mbox{and} \quad \depth K_A > \depth K_{A/l A}. $$ Applying the induction hypothesis to $K_{A/l A}$ completes the proof. \end{proof} \begin{remark} \label{X_and_S_have_same_can_module} The canonical modules of $S$ and its top-dimensional part $X$ are isomorphic. This follows from the corresponding (slightly stronger) result for $R/I(\psi)$ and $R/J(\psi)$. Indeed, according to Proposition \ref{coho_of_top-dimensional} there is an exact sequence $$ 0 \to S_{\frac{r-t}{2}}(M_{\varphi}) \otimes S_{\frac{r-t}{2}}(P) \otimes \wedge^f F^* \otimes \wedge^g G \otimes \wedge^t P \to R/I(\psi) \to R/J(\psi) \to 0. $$ Since $\dim S_{\frac{r-t}{2}}(M_{\varphi}) = n-r < n-r+t = \dim R/I(\psi) = \dim R/J(\psi)$ the claim follows by the long exact cohomology sequence. \end{remark} \begin{lemma} \label{dual_of_EN_complex} The dual of the Eagon-Northcott complex $E_{\bullet}$ provides a complex $$ 0 \to \wedge^t P \to E_t^* \stackrel{\delta^*_{t+1}}{\longrightarrow} \ldots \to E_{r-1}^* \stackrel{\delta^*_{r}}{\longrightarrow} E_r^* \stackrel{\gamma}{\longrightarrow} K_{R/I} \otimes \wedge^t P (1 - r(R)) \to 0 $$ which we denote (by slight abuse of notation) by $E_{\bullet}^*$. Its (co)homology modules are given by $$ H^i(E_{\bullet}^*) \cong \left \{ \begin{array}{ll} S_j(M_{\varphi}) \otimes S_{j-t}(P)^* & \mbox{if} ~ 2t + 1 \leq i = 2 j + 1 \leq r+1 \\ 0 & \mbox{otherwise} \end{array} \right. $$ In particular, $E_{\bullet}^*$ is exact if $t \geq \frac{r+1}{2}$. \end{lemma} \begin{proof} If $r = 2$ (and thus $t=1$) the claim follows immediately by dualizing because $E_{\bullet}$ is exact in this case. Now let $r \geq 3$. We begin by verifying that $E_{\bullet}^*$ is indeed a complex. This is only an issue in the beginning of this sequence. Consider the exact sequence $$ 0 \to E_r \stackrel{\delta_r}{\longrightarrow} E_{r-1} \to \im \delta_{r-1} \to 0. $$ Dualizing provides the exact sequence $$ E_{r-1}^* \stackrel{\delta^*_{r}}{\longrightarrow} E_r^* \stackrel{\alpha}{\longrightarrow} \operatorname{Ext}_R^1(\im \delta_{r-1}, R) \to \operatorname{Ext}_R^1(E_{r-1},R). \leqno(+) $$ The module on the right-hand side vanishes since $H_{\mathfrak m}^n(\wedge^{r-1}B_{\varphi}^*) = 0$ if $r \neq 2$. Hence the map $\gamma$ is surjective. In order to compare its image with $K_{R/I}$ we look at the proof of Proposition \ref{cohomology-of locus} and use its notation. If $t \geq \frac{r+1}{2}$ we have (cf.\ Case 1) $$ H_{\mathfrak m}^{n+t-r}(R/I)(p) \cong H_{\mathfrak m}^n(\im \delta_{r-1}). $$ Thus $(+)$ provides the exact sequence $$ E_{r-1}^* \stackrel{\delta^*_{r}}{\longrightarrow} E_r^* \stackrel{\alpha}{\longrightarrow} K_{R/I}(1 - r(R) -p) \to 0. $$ Putting $\gamma = \alpha$ we get the desired complex and $$ H^{r+1}(E_{\bullet}^*) = H^{r}(E_{\bullet}^*) = 0. $$ Now let $t \leq \frac{r+1}{2}$ and let $r$ be odd. Then we have (cf.\ Case 2) an embedding $$ \beta: H_{\mathfrak m}^{n+t-r}(R/I)(p) \hookrightarrow H_{\mathfrak m}^n(\im \delta_{r-1}). $$ We define $\gamma$ as the composition of $\alpha$ and $\beta^{\vee}$ (with the appropriate shift). Thus $\gamma$ is surjective, i.e. $$ H^{r+1}(E_{\bullet}^*) = 0. $$ Finally, let $t \leq \frac{r+1}{2}$ and let $r$ be even. Then we have (cf.\ Case 3) an exact sequence $$ 0 \to H_{\mathfrak m}^{n+1-\frac{r}{2}}(E_{\frac{r}{2}}) \to H_{\mathfrak m}^{n+1-\frac{r}{2}}(\im \delta_{\frac{r}{2}}) \to H_{\mathfrak m}^{n+2-\frac{r}{2}}(\ker \delta_{\frac{r}{2}}) \to H_{\mathfrak m}^{n+2-\frac{r}{2}}(E_{\frac{r}{2}}) = 0 $$ and isomorphisms $$ H_{\mathfrak m}^{n+1-\frac{r}{2}}(\im \delta_{\frac{r}{2}}) \cong H_{\mathfrak m}^{n-r+t}(R/I)(p), \quad H_{\mathfrak m}^{n+2-\frac{r}{2}}(\ker \delta_{\frac{r}{2}}) \cong H_{\mathfrak m}^n(\im \delta_{r-1}). $$ Thus we can conclude with the help of $(+)$ that there is an exact sequence $$ E_{r-1}^* \stackrel{\delta^*_{r}}{\longrightarrow} E_r^* \stackrel{\gamma}{\longrightarrow} K_{R/I} (1 - p - r(R)) \to H_{\mathfrak m}^{n+1-\frac{r}{2}}(E_{\frac{r}{2}})^{\vee}(1-r(R)) \to 0. $$ Using Lemma \ref{prop-of-exteriour-powers} it follows $$ H^{r+1}(E_{\bullet}^*) \cong H_{\mathfrak m}^{n+1-\frac{r}{2}}(E_{\frac{r}{2}})^{\vee}(1-r(R)) \cong S_{\frac{r}{2}}(M_{\varphi}) \otimes S_{\frac{r}{2}-t}(P)^*. $$ Hence we have shown in all cases that $E_{\bullet}^*$ is a complex and we have computed $H^{r+1}(E_{\bullet}^*)$. In order to compute the other cohomology modules we proceed as in the proofs of Proposition \ref{cohomology-of locus} and Proposition \ref{homology_of_EN}. Indeed, we just have to use Lemma \ref{can_module_depth_estimate} as replacement of Lemma \ref{depth-estimate} and $E_{\bullet}^*$ instead of $E_{\bullet}$. The details are tedious but straightforward. We omit them. \end{proof} The next result is also interesting in its own right. It relates the minimal free resolution of a module to those of (the duals) of its cohomology modules. Observe that the result as it is stated remains valid even if $R$ is not a Gorenstein but just a Cohen-Macaulay ring, though we won't use this fact here. \begin{proposition} \label{resolution_via_cohomology} Let $N$ be a finitely generated graded torsion $R$-module which has projective dimension $s$. Then it holds for all integers $j \geq 0$ that $\operatorname{Tor}^R_{s-j}(N,K)^{\vee}$ is a direct summand of $$ \oplus_{i=0}^j \operatorname{Tor}^R_{j-i}(\operatorname{Ext}_R^{s-i}(N,R),K). $$ Moreover, we have $\operatorname{Tor}^R_{s}(N,K)^{\vee} \cong \operatorname{Tor}^R_{0}(\operatorname{Ext}_R^{s}(N,R),K)$ and that $\operatorname{Tor}^R_{1}(\operatorname{Ext}_R^{s}(N,R),K)$ is a direct summand of $\operatorname{Tor}^R_{s-1}(N,K)^{\vee}$. \end{proposition} \begin{proof} For the purpose of this proof we write $M \ds N$ in order to express that the submodule $M$ is a direct summand of the module $N$. Consider a minimal free resolution of $N$: $$ 0 \to F_s \to \ldots \to F_1 \to F_0 \to N \to 0. $$ Dualizing with respect to $R$ provides the complex: $$ 0 \to F_0^* \stackrel{\alpha_0}{\longrightarrow} F_1^* \to \ldots \to F_{s-1}^* \stackrel{\alpha_{s-1}}{\longrightarrow} F_s^* \to \operatorname{Ext}_R^s(N.R) \to 0. $$ Since the maps $\alpha_j$ are duals of minimal maps we can write $$ \ker \alpha_j \cong G_j \oplus M_j $$ where $M_j$ does not have a free $R$-module as direct summand and $G_j$ is a free $R$-module being a direct summand of $F_j^*$ (but possibly trivial). Moreover, there are exact sequences $$ 0 \to \im \alpha_{j-1} \to \ker \alpha_j \to \operatorname{Ext}_R^j(N,R) \to 0. $$ Since $\alpha_{j-1}$ is a minimal homomorphism it holds $\im \alpha_{j-1} \subset {\mathfrak m} \cdot F_j^*$. This shows: \\ (1) The minimal generators of $G_j$ give rise to minimal generators of $\operatorname{Ext}_R^j(N,R)$. Now we consider the diagram $$ \begin{array}{ccccccc} 0 \to & \im \alpha_{j-1} & \to & \ker \alpha_j & \to & \operatorname{Ext}_R^j(N,R) & \to 0\\ & \downarrow & & \downarrow & & & \\ & F^*_j & = & F^*_j & & & \end{array} $$ where the vertical maps are the natural embeddings. Thus it is commutative and we obtain an exact sequence $$ 0 \to \operatorname{Ext}_R^j(N,R) \to F_j^*/\im \alpha_{j-1} \to F_j^*/ \ker \alpha_j \to 0. $$ The Horseshoe lemma yields a free resolution of the middle module as direct sum of the resolutions of the outer modules. After splitting off redundant terms we get a minimal free resolution, i.e, using $\operatorname{Tor}^R_i(\im \alpha_{j-1},K) \cong \operatorname{Tor}^R_{i+1}(F_j^*/\im \alpha_{j-1},K)$ we obtain $$ \operatorname{Tor}^R_i(\im \alpha_{j-1},K) \ds \operatorname{Tor}^R_{i+1}(\operatorname{Ext}_R^j(N,R),K) \oplus \operatorname{Tor}^R_i(\ker \alpha_j,K) \quad (i \geq 0). \leqno(2) $$ Now we are ready to show by induction on $j \geq 1$: $$ \operatorname{Tor}^R_i(\im \alpha_{s-j}, K) \ds \oplus_{k=0}^{j-1} \operatorname{Tor}^R_{i+j-k}(\operatorname{Ext}_R^{s-k}(N,R), K) \quad (i \geq 0) \leqno(*) $$ and $$ \operatorname{Tor}^R_{s-j+1}(N, K)^{\vee} \ds \oplus_{k=0}^{j-1} \operatorname{Tor}^R_{j-1-k}(\operatorname{Ext}_R^{s-k}(N,R), K). \leqno(**) $$ Let $j = 1$. Since $\alpha_{s-1}$ is a minimal homomorphism the exact sequence $$ 0 \to \im \alpha_{s-1} \to F_s^* \to \operatorname{Ext}_R^s(N,R) \to 0 $$ implies $$ \operatorname{Tor}^R_{s}(N,K)^{\vee} \cong \operatorname{Tor}^R_{0}(\operatorname{Ext}_R^{s}(N,R),K), $$ thus in particular $(**)$, and $$ \operatorname{Tor}^R_{i}(\im \alpha_{s-1}, K) \cong \operatorname{Tor}^R_{i+1}( \operatorname{Ext}_R^s(N,R), K) \quad (i \geq 0) $$ which shows $(*)$. Let $j \geq 2$. Consider the exact sequence $$ 0 \to \ker \alpha_{s-j+1} \to F_{s-j+1}^* \to \im \alpha_{s-j+1} \to 0. \leqno(3) $$ We obtain by the definition of $G_{s-j+1}$ \begin{eqnarray*} F_{s-j+1}^* \otimes K & \cong & (G_{s-j+1} \otimes K) \oplus \operatorname{Tor}^R_0(\im \alpha_{s-j+1}, K) \\[3pt] & \ds & \operatorname{Tor}^R_0(\operatorname{Ext}_R^{s-j+1}(N,R), K) \oplus \bigoplus_{k=0}^{j-2} \operatorname{Tor}^R_{j-1-k}(\operatorname{Ext}_R^{s-k}(N,R), K) \quad \mbox{(by (1) and induction)} \\[3pt] & = & \bigoplus_{k=0}^{j-1} \operatorname{Tor}^R_{j-1-k}(\operatorname{Ext}_R^{s-k}(N,R), K) \end{eqnarray*} which shows $(**)$. Furthermore, (3) provides \begin{eqnarray*} \operatorname{Tor}^R_i(\ker \alpha_{s-j+1}, K) & \ds & \operatorname{Tor}^R_{i+1}(\im \alpha_{s-j+1}, K) \\[3pt] & \ds & \bigoplus_{k=0}^{j-2} \operatorname{Tor}^R_{i+j-k}(\operatorname{Ext}_R^{s-k}(N,R), K) \quad \mbox{(by induction).} \end{eqnarray*} Thus we conclude with the help of (2): \begin{eqnarray*} \operatorname{Tor}^R_i(\im \alpha_{s-j}, K) & \ds & \operatorname{Tor}^R_{i+1}(\operatorname{Ext}_R^{s-j+1}(N,R), K) \oplus \bigoplus_{k=0}^{j-2} \operatorname{Tor}^R_{i+j-k}(\operatorname{Ext}_R^{s-k}(N,R), K) \\[3pt] & = & \bigoplus_{k=0}^{j-1} \operatorname{Tor}^R_{i+j-k}(\operatorname{Ext}_R^{s-k}(N,R), K). \end{eqnarray*} This proves $(*)$, i.e., the induction step is complete. It only remains to verify the very last assertion. Looking at (3) with $j=2$ we get $$ \operatorname{Tor}^R_0(\im \alpha_{s-1}, K) \ds F_{s-1}^* \otimes K. $$ But at the beginning of the induction we have seen that $\operatorname{Tor}^R_0(\im \alpha_{s-1}, K) \cong \operatorname{Tor}^R_1(\operatorname{Ext}_R^s(N,R), K)$. Our claim follows \end{proof} \begin{remark} \label{comparison_with_Rao} In order to explain how the last result can be used we compare it with a well-known statement of Rao. For this let $R = K[x_0,\ldots,x_3]$ and let $I_C \subset R$ be the homogeneous ideal of a curve $C \subset \PP^3$. Put $A = R/I_C$ and consider the following minimal free resolutions: \[ \begin{array}{c} 0 \rightarrow F_3 \rightarrow F_2 \rightarrow F_1 \rightarrow R \rightarrow A \rightarrow 0 \\ \\ 0 \rightarrow G_4 \rightarrow G_3 \rightarrow G_2 \rightarrow G_1 \rightarrow G_0 \rightarrow \operatorname{Ext}_R^3(A,R) \rightarrow 0 \\ \\ 0 \rightarrow D_2 \rightarrow D_{1} \rightarrow D_{0} \rightarrow \operatorname{Ext}_R^2(A,R) \rightarrow 0. \end{array} \] Then the additions to our general observation in the lemma above yield $$ G_0 \cong F_3^* \quad \mbox{and} \quad G_1 \ds F_2^*. $$ This is precisely the content of Rao's Theorem 2.5 in \cite{R1}. Our Lemma \ref{resolution_via_cohomology} gives in addition $$ G_1 \ds F_2^* \ds G_1 \oplus D_0 $$ and $$ F_1^* \ds G_2 \oplus D_1 $$ because $\operatorname{Ext}_R^1(A,R) = 0$. \end{remark} Now we have all the tools for establishing the main result of this section. \begin{theorem} \label{resolution_of_top-dim_part} Consider the following modules where we use the conventions that $i$ and $j$ are non-negative integers and that a sum is trivial if it has no summand: $$ A_k = \bigoplus_{\begin{array}{c} {\scriptstyle i+2j = k + t -1}\\ [-4pt] {\scriptstyle t \leq i+j \leq \frac{r+t-1}{2}} \end{array}} \wedge^i F^* \otimes S_j(G)^* \otimes S_{i+j-t}(P), $$ $$ C_k = \bigoplus_{\begin{array}{c} {\scriptstyle i+2j = r+1-t-k}\\ [-4pt] {\scriptstyle i+j \leq \frac{r-t}{2}} \end{array}} \wedge^i F \otimes S_j(G) \otimes S_{r-t-i-j}(P) \otimes \wedge^f F^* \otimes \wedge^g G. $$ Observe that it holds: $A_r = 0$ if and only if $r+t$ is even, $C_1 = 0$ if and only if $r+t$ is odd, $C_k = 0$ if $k \geq r+2-t$ and $C_{r+1-t} = S_{r-t}(P) \otimes \wedge^f F^* \otimes \wedge^g G$. \\ Then the homogeneous ideal $I_X = J(\psi)$ of the top-dimensional part $X$ of the degeneracy locus $S$ has a graded free resolution of the form $$ 0 \to A_r \oplus C_r \to \ldots \to A_1 \oplus C_1 \to I_X \otimes \wedge^t P^* \to 0. $$ \end{theorem} \begin{proof} Following the procedure described in Remark \ref{remark_resolution_in_general} we get a resolution of $I(\psi)$. According to Proposition \ref{coho_of_top-dimensional} it holds $I(\psi) = J(\psi)$ if and only if $r+t$ is odd. If $r+t$ is even we have by the same result an exact sequence $$ 0 \to I(\psi) \to J(\psi) \to S_{\frac{r-t}{2}}(M_{\varphi}) \otimes S_{\frac{r-t}{2}}(P) \otimes \wedge^f F^* \otimes \wedge^g G \otimes \wedge^t P^*. $$ Thus, using the Horseshoe lemma we get a finite free resolution of $J(\psi)$ in any case. It is not minimal. In order to split off redundant terms we proceed as follows: Since the resolution is finite the Auslander-Buchsbaum formula and Proposition \ref{coho_of_top-dimensional} yield the projective dimension of $J(\psi)$. Thus, in a first step we can split off all the terms in the resolution of $J(\psi)$ occurring past its projective dimension. Next, we use Lemma \ref{dual_of_EN_complex} (cf.\ Remark \ref{X_and_S_have_same_can_module}) in order to obtain a free resolution of $\operatorname{Ext}_R^{r-t+1}(R/J(\psi),R)$. For $j \neq r-t+1$ we know a free resolution of $\operatorname{Ext}_R^{j}(R/J(\psi),R)$ by Proposition \ref{coho_of_top-dimensional} and Proposition \ref{EN-complexes_are_exact}. Hence, in a second step we can split off further terms in the resolution of $J(\psi)$ by applying Proposition \ref{resolution_via_cohomology}. This provides the resolution as claimed. The details are very tedious but straightforward. We omit them. \end{proof} Since the above proof is somewhat sketchy, we illustrate it by deriving the resolution for $t = 2$ and $r = 6,7$ (cf.\ also the next corollaries). \begin{example} \label{t=2_r=6,7} Using our standard notation we define integers $c, p$ by $$ R(c) \cong \wedge^f F \otimes \wedge^g G^* \quad \mbox{and} \quad R(p) = \wedge^t P^*. $$ (i) Let $t=2$ and $r=7$. Then we know that $I = I(\psi)$ is unmixed and that $X$ is not arithmetically Cohen-Macaulay. Using the Eagon-Northcott complex $E_{\bullet}$ we see that a free resolution of $I$ begins as follows: $$ \begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c} &&&&& \downarrow \\ \wedge^2 F^* \otimes S_2(G)^* \otimes S_2(P) & \oplus & S_3(G)^* \otimes P & & & \oplus & & & \wedge^3 F^* \otimes S_2(P) \\ &&&&& \downarrow \\ \wedge^3 F^* \otimes G^* \otimes S_2(P) & \oplus & F^* \otimes S_2(G)^* \otimes P & & & \oplus & S_3(P)(-c) & \oplus & \wedge^4 F^* \otimes S_2(P) \\ &&&&& \downarrow \\ \wedge^4 F^* \otimes S_2(P) & \oplus & \wedge^2 F^* \otimes G^* \otimes P & \oplus & S_2(G)^* & \oplus & & & F(-c) \otimes S_2(P) \\ &&&&& \downarrow \\ && \wedge^3 F^* \otimes P & \oplus & F^* \otimes G^* & \oplus & & & G(-c) \otimes S_2(P) \\ &&&&& \downarrow \\ &&&& \wedge^2 F^*\\ &&&&& \downarrow \\ &&&&& I(p) \\ &&&&& \downarrow \\ &&&&& 0. \end{array} $$ With the help of $E_{\bullet}^*$ we get the following beginning for the resolution of $\operatorname{Ext}_R^4(R/I,R)(-p)$: $$ \begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c} &&& \downarrow \\ \wedge^2 F^*(c) \otimes P^* & \oplus & G^*(c) \otimes S_2(P)^* & \oplus & F \otimes G \\ &&& \downarrow \\ && F^*(c) \otimes S_2(P)^* & \oplus & S_2(G) \\ &&& \downarrow \\ &&& S_3(P)^*(c) \\ &&& \downarrow \\ &&& \operatorname{Ext}_R^4(R/I,R)(-p) \\ &&& \downarrow \\ &&& 0. \end{array} $$ Now we apply Proposition \ref{resolution_via_cohomology} and conclude that in the top row of the resolution of $I(p)$ only the term $S_3 (G)^* \otimes P$ remains in the minimal resolution because $S_3 (G)^* \otimes P$ surjects minimally onto $\operatorname{Ext}_R^5(R/I,R)(-p)$. Continuing in this fashion we obtain the following resolution: $$ \begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c} &&& 0 \\ &&& \downarrow \\ S_3(G)^* \otimes P & & & \oplus & & & \\ &&& \downarrow \\ F^* \otimes S_2(G)^* \otimes P & & & \oplus & S_3(P)(-c) & & \\ &&& \downarrow \\ \wedge^2 F^* \otimes G^* \otimes P & \oplus & S_2(G)^* & \oplus & & & F(-c) \otimes S_2(P) \\ &&& \downarrow \\ && \wedge^3 F^* \otimes P & \oplus & F^* \otimes G^* & \oplus & G(-c) \otimes S_2(P) \\ &&& \downarrow \\ && \wedge^2 F^*\\ &&& \downarrow \\ &&& I(p) \\ &&& \downarrow \\ &&& 0. \end{array} $$ (ii) Let $t = 2$ and $r=6$. Then $J = J(\psi \neq I(\psi)$ defines an arithmetically Cohen-Macaulay scheme. Thus we get as in the previous case but slightly easier a resolution \[ \begin{array}{cccccccccc} &&& 0 \\ &&& \downarrow \\ S_3 (G)^* \otimes P & \oplus && S_4(P)(-c_1 ) \\ &&& \downarrow \\ F^* \otimes S_2 (G)^* \otimes P &&& \oplus & F(-c_1 ) \otimes S_3 (P) \\ &&& \downarrow \\ \bigwedge^2 F^* \otimes G^* \otimes P & \oplus & S_2 (G)^* & \oplus & G(-c_1 ) \otimes S_3 (P) & \oplus & \bigwedge^2 F (-c_1 ) \otimes S_2 (P) \\ &&& \downarrow \\ \bigwedge^3 F^* \otimes P & \oplus & F^* \otimes G^* & \oplus & &&F \otimes G(-c_1 ) \otimes S_2 (P) \\ &&& \downarrow \\ && \bigwedge^2 F^* & \oplus &&& S_2 (G) (-c_1 ) \otimes S_2 (P) \\ &&& \downarrow \\ &&& J(p) \\ &&& \downarrow \\ &&& 0 \end{array} \] \end{example} \begin{remark} \label{discussion_of_minimality} We want to discuss the minimality of the resolution described in Theorem \ref{resolution_of_top-dim_part}: \\ (i) By looking at the twists of the free summands occurring in the resolution above it is clear that for suitable choices of $F, G, P$ and sufficiently general maps $\varphi, \psi$ no further cancellation is possible, i.e., the resolution is minimal. \\ (ii) Let $r$ be even and let $t=1$. This case was also studied by Kustin \cite{Kustin}; his main result gives (up to the degree shifts) the same resolution as our Theorem \ref{resolution_of_top-dim_part} (cf.\ also Corollary \ref{one-sect-even-case}). His techniques are completely different from ours, and while they are more complicated, they in fact give the maps in the resolution while our techniques use the Horseshoe Lemma and hence do not easily give the maps. Kustin has proved that his resolution is minimal in the homogeneous case if the section does not correspond to a minimal generator of the Buchsbaum-Rim module. The latter assumption cannot be removed. Indeed, the resolution predicts that the homogeneous ideal of $X = S$ has $r+1$ generators, i.e., that $X$ is an almost complete intersection because $\codim X = r$. Now consider the cotangent bundle of $\PP^2$. It has a global section whose zero locus is a point in $\PP^2$ being a complete intersection. \\ (iii) We suspect that the phenomenon just described is the only instance that prevents our resolution from being minimal. That means, we hope that in case the resolution of $X$ described above gives the correct number of minimal generators of $I_X$ then the {\it whole} resolution is minimal. \end{remark} In the theorem above our focus has been on $X$ rather than on the degeneracy locus $S$ itself. The interested reader will observe that the same methods provide a resolution for $S$. According to Theorem \ref{summary_for_degeneracy_loci} we know when $X$ is arithmetically Gorenstein. In this case its minimal free resolution is self-dual. In order to make this duality transparent we rewrite the resolution above as follows. \begin{corollary} Let ${\mathcal B}_{\varphi}$ be a Buchsbaum-Rim sheaf of odd rank $r$ and first Chern class $c_1$. Let $X$ be the top-dimensional part of a regular section of ${\mathcal B}_{\varphi}$. Make the following definitions: \begin{quote} \noindent If $i$ is odd, let $\displaystyle \ell = \min \left \{ {\frac{i-1}{2}} , {\frac{r-i-2}{2}} \right \}$, and define \[ A_i = \bigoplus_{j=0}^\ell \bigwedge^{2j+1} F^* \otimes S_{\frac{i-1-2j}{2}} (G)^* \] \noindent If $i$ is even, let $\displaystyle \ell = \min \left \{ {\frac{i}{2}} , {\frac{r-i-1}{2}} \right \}$, and define \[ A_i = \bigoplus_{j=0}^\ell \bigwedge^{2j} F^* \otimes S_{\frac{i-2j}{2}} (G)^* \] \end{quote} Then $X$ is arithmetically Gorenstein and has a free resolution of the form $$ 0 \rightarrow R(-c_1 ) \rightarrow A_{r-1} \oplus A_1^* (-c_1 ) \rightarrow A_{r-2} \oplus A_2^* (-c_1 ) \rightarrow \dots \rightarrow A_1 \oplus A_{r-1}^* (-c_1 ) \rightarrow I_X \rightarrow 0. $$ \end{corollary} \begin{proof} It follows by Lemma \ref{prop-of-exteriour-powers} that $c_1 = c_1({\mathcal B}_{\varphi}) = - c_1({\mathcal B}_{\varphi}^*)$ is the integer satisfying $$ R(-c_1) \cong \wedge^f F^* \otimes \wedge^g G. $$ Thus Theorem \ref{resolution_of_top-dim_part} provides the claim. \end{proof} \begin{example} (i) If the rank of ${\mathcal B}_{\varphi}$ is 3, this was treated in \cite{Mig-P_gorenstein}, where the following resolution was obtained: \[ \begin{array}{ccccccccccccccc} & 0 \\ & \downarrow \\ & R(-c_1 ) \\ & \downarrow \\ G^* & \oplus & F(-c_1 ) \\ & \downarrow \\ F^* & \oplus & G(-c_1 ) \\ & \downarrow \\ & I_X \\ & \downarrow \\ & 0 \end{array} \] (ii) If the rank of ${\mathcal B}_{\varphi}$ is five, then the corollary gives the following resolution \[ \begin{array}{cccccccccccccc} &&& 0 \\ &&& \downarrow \\ &&& R(-c_1 ) \\ &&& \downarrow \\ S_2 (G)^* && & \oplus & F(-c_1 ) \\ &&& \downarrow \\ F^* \otimes G^* && & \oplus & G(-c_1 ) & \oplus & \bigwedge^2 F(-c_1 ) \\ &&& \downarrow \\ \bigwedge^2 F^* & \oplus & G^* & \oplus &&& F \otimes G(-c_1 ) \\ &&& \downarrow \\ && F^* & \oplus & && S_2 (G)(-c_1 ) \\ &&& \downarrow \\ &&& I_X \\ &&& \downarrow \\ &&& 0 \end{array} \] This resolution has been conjectured in \cite{Mig-P_gorenstein}. \end{example} If we consider a regular section of an even rank Buchsbaum-Rim sheaf we cannot expect to get an arithmetically Gorenstein subscheme as zero locus. Still the resolution has some symmetry and looks very much like the corresponding one for the Gorenstein case. \begin{corollary} \label{one-sect-even-case} Let ${\mathcal B}_{\varphi}$ be a Buchsbaum-Rim sheaf of even rank $r$ and first Chern class $c_1$. Let $S$ be the zero locus of a regular section of ${\mathcal B}_{\varphi}$. Make the following definitions: \begin{quote} If $i$ is odd, let $\displaystyle \ell = \min \left \{ {\frac{i-1}{2}} , {\frac{r-i-1}{2}} \right \}$, and $\displaystyle \ell' = \min \left \{ \frac{i-1}{2} , \frac{r-i-3}{2} \right \}$. Define \[ \begin{array}{c} A_i = \displaystyle \bigoplus_{j=0}^\ell \bigwedge^{2j+1} F^* \otimes S_{\frac{i-1-2j}{2}} (G)^* \\ \\ B_i = \displaystyle \bigoplus_{j=0}^{\ell'} \bigwedge^{2j+1} F^* \otimes S_{\frac{i-1-2j}{2}} (G)^* \end{array} \] \par \noindent If $i$ is even, let $\displaystyle \ell = \min \left \{ {\frac{i}{2}} , {\frac{r-i}{2}} \right \}$, and $\displaystyle \ell' = \min \left \{ \frac{i}{2} , \frac{r-i-2}{2} \right \}$. Define \[ \begin{array}{c} A_i = \displaystyle \bigoplus_{j=0}^\ell \bigwedge^{2j} F^* \otimes S_{\frac{i-2j}{2}} (G)^* \\ \\ B_i = \displaystyle \bigoplus_{j=0}^{\ell'} \bigwedge^{2j} F^* \otimes S_{\frac{i-2j}{2}} (G)^* \end{array} \] \end{quote} Then $S$ is arithmetically Cohen-Macaulay and has a free resolution of the form $$ 0 \rightarrow R(-c_1 ) \oplus A_{r} \rightarrow B_1^* (-c_1 ) \oplus A_{r-1} \rightarrow B_2^* (-c_1 ) \oplus A_{r-2} \rightarrow \dots \rightarrow B_{r-2}^* (-c_1 ) \oplus A_2 \rightarrow A_1 \rightarrow I_S \rightarrow 0. $$ \end{corollary} \section{Some applications} In the previous section we have seen how much the properties of our degeneracy loci depend on the properties of the Buchsbaum-Rim sheaf. Now we will show how this information can be used to construct schemes with prescribed properties. Moreover, we will explain how sections of the dual of a generalized null correlation bundle can be studied with the help of our results. \bigskip \noindent {\it Construction of arithmetically Gorenstein subschemes containing a given scheme} \medskip Let $X \subset \PP^n$ be an equidimensional projective subscheme of codimension $\geq 3$. It is rather easy to find a complete intersection $Y$ such that $Y$ contains $X$ and both have the same dimension. The analogous problem where one requires $Y$ to be arithmetically Gorenstein but not a complete intersection is much more difficult. This is relevant if one wants to study linkage with respect to arithmetically Gorenstein subschemes rather than complete intersections. We want to explain a solution to this problem. Suppose $X$ has codimension $r$. Let us assume that $r$ is odd. Then we choose a Buchsbaum-Rim sheaf $B_{\varphi}$ of rank $r$ on $\PP^n = \operatorname{Proj} R$ given by an exact sequence $$ 0 \to {\mathcal B}_{\varphi} \to {\mathcal F} \to {\mathcal G} \to M_{\varphi} \to 0. $$ For example, we can take the sheafification of the first syzygy module of an ideal which is generated by $R$-regular sequence of length $r+1$. Next we choose a regular section $s \in H^0({\mathcal B}_{\varphi}(j))$ which also belongs to $H^0_*({\mathcal J}_X \otimes {\mathcal F})$. This is possible if $j$ is sufficiently large. Let $S$ be the zero-locus of $s$. Then the top-dimensional part $Y$ of $S$ is arithmetically Gorenstein due to Theorem \ref{summary_for_degeneracy_loci}. Furthermore, $s \in H^0_*({\mathcal J}_X \otimes {\mathcal F})$ ensures that $s$ vanishes on $X$. It follows that $X \subset Y$ because both are equidimensional schemes of the same dimension. For example, it was shown in \cite{Mig-P_gorenstein} that if ${\mathcal B}_{\varphi}$ is the cotangent bundle on $\PP^3$, twisted by 3, and X is a set of four distinct points, then a section of ${\mathcal B}_{\varphi}$ can be found vanishing on X and giving a Gorenstein scheme, Y, of degree 5, whereas the smallest complete intersection containing X has degree 8. Now assume that $\codim X = r$ is even. Then we take a hypersurface containing $X$ which is defined by, say $f \in R' = K[x_0,\ldots,x_n]$. Put $R = R'/f R'$ and choose a Buchsbaum-Rim sheaf ${\mathcal B}_{\varphi}$ of rank $r-1$ on $Z = \operatorname{Proj} R$. $X$ has codimension $r-1$ as a subscheme of $Z$. Thus we can find as in the previous case an arithmetically Gorenstein subscheme $Y \subset Z$ containing $X$. We can consider $Y$ also as a subscheme of $\PP^n$. As such it is still arithmetically Gorenstein, i.e., it has all the properties we wanted. \bigskip \noindent {\it Construction of $k$-Buchsbaum schemes} \medskip A projective subscheme $X \subset \PP^n$ is said to be $k$-Buchsbaum for some non-negative integer $k$ if $$ (x_0,\ldots,x_n)^k \cdot H^i_*({\mathcal J}_X) = 0 \quad \mbox{for all} \; i \leq \dim X. $$ Note that any equidimensional locally Cohen-Macaulay subscheme is $k$-Buchsbaum for some $k$. In fact, one should view the notion of a $k$-Buchsbaum scheme as a refinement of the notion of an equidimensional locally Cohen-Macaulay subscheme. The idea is to develop for such schemes a theory generalizing the one for arithmetically Buchsbaum schemes (cf., for example, \cite{Mig-MM}, \cite{NS}). However, so far there are not many examples available where one knows that they are $k$-Buchsbaum but not $(k-1)$-Buchsbaum. We are going to construct new examples now. Let $R = K[x_0,\ldots,x_n]$ and let $\varphi: R^{n+k}(-1) \to R^k$ be a homomorphism such that $I(\varphi) = (x_0,\ldots,x_n)^k$. This is true if $\varphi$ is chosen general enough. A particular choice of such a map is described in \cite{BV}, p.\ 15. Let us denote the corresponding Buchsbaum-Rim sheaf by ${\mathcal B}_k$. Observe that ${\mathcal B}_1$ is just the cotangent bundle of $\PP^n$. \begin{proposition} \label{k-Buchsbaum_schemes} Let $S$ be the degeneracy locus of a morphism $\psi: {\mathcal P} \to {\mathcal B}_k$. If $X$ has codimension $n-t+1$ then it holds: If $t = 1$ or $t=2$ and $n$ is even then $S$ is arithmetically Cohen-Macaulay; otherwise $S$ is $k$-Buchsbaum but not $(k-1)$-Buchsbaum. \end{proposition} \begin{proof} The first assertion follows by Theorem \ref{summary_for_degeneracy_loci}. According to \cite{Buchs-Eisenbud_annihilator} it holds $$ \Ann_R S_j(M_{\varphi}) = (x_0,\ldots,x_n)^k \quad \mbox{for all} \; j \geq 1. $$ Hence the second claim is a consequence of Proposition \ref{coho_of_top-dimensional}. \end{proof} In case $k = 1$ even more is true. To this end recall that any arithmetically Buchsbaum subscheme is $1$-Buchsbaum. But the converse is not true in general. Let $N$ be a finitely generated $R$-module. Then the embedding $0 :_N {\mathfrak m} \hookrightarrow H_{\mathfrak m}^0(N)$ induces natural homomorphisms of derived functors $$ \varphi^i_N: \operatorname{Ext}_R^i(K,N) \to H_{\mathfrak m}^i(N). $$ Due to \cite{SV2}, Theorem I.2.10 $N$ is a Buchsbaum module if and only if the maps $\varphi^i_N$ are surjective for all $i \neq \dim N$. A subscheme $X \subset \PP^n$ is called arithmetically Buchsbaum if its homogeneous coordinate ring $R/I_X$ is Buchsbaum. Now we can show the announced strengthening of the previous result in case $k=1$. \begin{proposition} \label{aBM-schmes} Let $S$ be the degeneracy locus of a morphism $\psi: {\mathcal P} \to \cOP$. If $S$ has codimension $n-t+1$ then it is arithmetically Buchsbaum. \end{proposition} \begin{proof} We will use again the Eagon-Northcott $E_{\bullet}$ complex associated to $\psi$. Let $B = H^0_*(\cOP)$ and put $A = R/I_S$. We want to show that $\varphi^j_A$ is surjective if $j \neq t = \dim A$. This is clear if $H_{\mathfrak m}^j(A)$ vanishes. Let $j = n+t- 2i < t$ be an integer such that $H_{\mathfrak m}^j(A) \neq 0$. Using the exact sequences in the proof of Proposition \ref{cohomology-of locus} we get diagrams $$ \begin{array}{ccc} \operatorname{Ext}_R^{n+t-2i}(K,A)(p) & \to & \operatorname{Ext}_R^{n+1-i}(K,\im \delta_i) \\[3pt] \downarrow {\scriptstyle \varphi^{n+t-2i}_A} & & \downarrow {\scriptstyle \varphi^{n+1-i}_{\im \delta_i}}\\[3pt] H_{\mathfrak m}^{n+t-2i}(A)(p) & \to & H_{\mathfrak m}^{n+1-i}(\im \delta_i) \end{array} $$ and $$ \begin{array}{ccc} \operatorname{Ext}_R^{n+1-i}(K,E_i) & \to & \operatorname{Ext}_R^{n+1-i}(K,\im \delta_i) \\[3pt] \downarrow {\scriptstyle \varphi^{n+1-i}_{E_i}} & & \downarrow {\scriptstyle \varphi^{n+1-i}_{\im \delta_i}}\\[3pt] H_{\mathfrak m}^{n+1-i}(E_i) & \to & H_{\mathfrak m}^{n+1-i}(\im \delta_i). \end{array} $$ They are commutative because the vertical maps are canonical. Moreover, we have seen in the proof of Proposition \ref{cohomology-of locus} that the lower horizontal maps are isomorphisms. Now it is well-known that the modules $\wedge^q B^*$ are Buchsbaum modules if $1 \leq q \leq n$. Thus the modules $E_q$ are Buchsbaum, too. Hence the diagrams show that the surjectivity of $\varphi^{n+1-i}_{E_i}$ implies this property first for $\varphi^{n+1-i}_{\im \delta_i}$ and then for $\varphi^{n+t-2i}_A$. It follows that $A$ is Buchsbaum. \end{proof} In the special case that ${\mathcal P}$ has rank $t = n-1$ the last result is also contained in \cite{Chang-Diff-Geom}. If $t < n-1$ our result is a little surprising. In fact, the main result of \cite{Chang-Diff-Geom} has been generalized in \cite{habil}, Corollary II.3.3. It says that arithmetically Buchsbaum subschemes of arbitrary codimension can be characterized by means of a particular locally free resolution. As a consequence, every arithmetically Buchsbaum subscheme of $\PP^n$ is the zero-locus of a global section of a vector bundle which is the direct sum of exterior powers of the cotangent bundle. \bigskip \noindent {\it Some vector bundles of low rank and their sections} \medskip Let $R$ be again a graded Gorenstein $K$-algebra of dimension $n+1$. We assume that $n \geq 3$ is an odd integer. The aim of this subsection is to show that a vector bundle arising from a Buchsbaum-Rim sheaf by quotienting out non-vanishing sections can be studied by means of our results. Then we construct vector bundles of rank $n-1$ on $Z = \operatorname{Proj} R$ and apply this principle to sections of them. Let ${\mathcal B}_{\varphi}$ be a Buchsbaum-Rim sheaf on $Z$ having global non-vanishing sections such there is an exact sequence $$ 0 \to {\mathcal Q} \stackrel{\gamma}{\longrightarrow} {\mathcal B}_{\varphi} \to {\mathcal E} \to 0 $$ where $H^0_*(Z,{\mathcal Q})$ is a free $R$-module of rank $u$ and ${\mathcal E}$ a vector bundle on $Z$ of rank $r-u$. Now, we want to consider a morphism $\psi: {\mathcal P} \to {\mathcal E}$ dropping rank in the expected codimension $r-u-t+1$. This morphism can be lifted to a morphism $\beta: {\mathcal P} \to {\mathcal B}_{\varphi}$ which provides a morphism $\alpha = (\beta,\gamma): {\mathcal P} \oplus {\mathcal Q} \to {\mathcal B}_{\varphi}$. Since the degeneracy locus of $\gamma$ is empty, ${\mathcal B}_{\varphi}^*$ is locally the direct sum of ${\mathcal E}^*$ and ${\mathcal Q}^*$. It follows that the images of $\wedge^t \psi^*$ and $\wedge^{t+u} \alpha^*$ are locally isomorphic. Hence the degeneracy locus $S$ of $\psi$ and the degeneracy locus of $\alpha$ agree. Thus $\alpha$ drops rank in the expected codimension too and we can apply our previous results. Next, we construct a class of vector bundles which contains the duals of null correlation bundles. To this end let $I = (f_0,\ldots,f_n) \subset R$ be a complete intersection. Let $d_i = \deg f_i$. The first syzygy module of $I$ defines a Buchsbaum-Rim module $B_{\varphi}$ which fits into the exact sequence $$ 0 \to B_{\varphi} \to \oplus_{i=0}^n R(-d_i) \stackrel{\varphi}{\longrightarrow} R \to R/I \to 0. $$ The Buchsbaum-Rim sheaf ${\mathcal B}_{\varphi} = \widetilde{B_{\varphi}}$ can often be used to construct a vector bundle of rank $n-1$ on $Z$. \begin{proposition} \label{new_bundles} Suppose there is an integer $c$ such that the degrees satisfy $$ c = d_0 + d_1 = d_2 + d_3 = \ldots = d_{n-1} + d_n. $$ Then ${\mathcal B}_{\varphi}(c)$ admits a non-vanishing global section $s$ which gives rise to an exact sequence $$ 0 \to {\mathcal O}_{\mathbb{P}^n}(-c) \stackrel{s}{\longrightarrow} {\mathcal B}_{\varphi} \to {\mathcal N} \to 0 $$ where ${\mathcal N}$ is a vector bundle of rank $n-1$ on $Z$. \end{proposition} \begin{proof} The Koszul relation of the generators $f_i$ and $f_{i+1}$ of $I$ gives rise to a global section of ${\mathcal B}_{\varphi}(-d_i - d_{i+1})$. Taking the sum over these sections with even $i$ yields a section $s$ which does not vanish on $Z$ since $I$ is an ${\mathfrak m}$-primary ideal. \end{proof} In case $Z = \PP^n$ and $d_0 = \ldots = d_n = 1$ the bundle ${\mathcal B}_{\varphi}$ is the cotangent bundle of $\PP^n$ and ${\mathcal N}^*$ is called {\it null correlation bundle} in \cite{OSS} where on p.\ 79 it is constructed in a slightly different way. If $n=3$ then ${\mathcal N}$ is self-dual. Thus we call the dual of a vector bundle constructed as in the proposition above {\it generalized null correlation bundle}. Due to the principle described above our previous results apply to multiple sections of the dual of a generalized null correlation bundle. We obtain, for example. \begin{corollary} The degeneracy locus of a multiple section of the dual of a null correlation bundle is arithmetically Buchsbaum but not arithmetically Cohen-Macaulay. \end{corollary} This result is well-known if $n=3$. In fact, in this case the null correlation bundle can be constructed (via the Serre correspondence) as an extension $$ 0 \to {\mathcal O}_{\mathbb{P}^n}(-2) \to {\mathcal N} \to {\mathcal J}_X \to 0 $$ where ${\mathcal J}_X$ denotes the ideal sheaf of two skew lines in $\PP^3$ (cf.\ \cite{Barth-1977}, p. 145 or \cite{Ellia-Fiorentini}).

9708

alg-geom/9708006

en

https://arxiv.org/abs/alg-geom/9708006

alg-geom/9708006

Joseph Lipman

Leovigildo Alonso, Ana Jeremias, Joseph Lipman

Duality and flat base change on formal schemes

89 pages. Change from published version: in section 2.5, about dualizing complexes on formal schemes, a weakening of one flawed Lemma is proved, and shown adequate for the several applications made of the original. For another correction, see math.AG/0106239

Contemporary Math. 244 (1999), 3-90

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.

alg-geom

\section{Preliminaries and main theorems.} \label{S:prelim} First we need some notation and terminology. Let $X$ be a ringed space,\index{ringed space} i.e., a topological space together with a sheaf of commutative rings ${\mathcal O}_{\<\<X}$.% \index{ ${\mathbf R}$@${\mathcal O}_{\<\<\<X}$ (structure sheaf of ringed space $X$)} Let ${\mathcal A}(X)$\index{ ${\mathcal A}$ (module category)} be the \hbox{category} of ${\mathcal O}_{\<\<X}$-modules, and $\A_{\qc}\<(X)$\index{ ${\mathcal A}$ (module category)!$\A_{\qc}$} (resp.\ $\A_{\mathrm c}(X)$,\index{ ${\mathcal A}$ (module category)!$\A_{\mathrm c}$} resp.~ $\A_{\vec {\mathrm c}}(X)$)\index{ ${\mathcal A}$ (module category)!$\A_{\mathrm c}$@$\A_{\vec {\mathrm c}}$} the full subcategory of~ ${\mathcal A}(X)$ whose objects are the quasi-coherent (resp.\;coherent, resp.\;\smash{$\dirlm{}\!\!$}'s\index{lim@\smash{$\subdirlm{}\mkern-4mu$}} of coherent) ${\mathcal O}_{\<\<X}$-modules.% \footnote{% ``\smash{$\subdirlm{}\!\!$}" always denotes a direct limit over a small ordered index set in which any two elements have an upper bound. More general direct limits will be referred to as \emph{colimits}.% } Let~ ${\mathbf K}(X)$\index{ ${\mathbf K}$ (homotopy category)} be the homotopy category of ${\mathcal A}(X)$-complexes, and let~${\mathbf D}(X)$\index{ ${\mathbf D}$ (derived category)} be the corresponding derived category, obtained from~${\mathbf K}(X)$ by adjoining\- an inverse for every quasi-isomorphism (=\:homotopy class of maps of complexes inducing homology isomorphisms).\index{quasi-isomorphism} \penalty-1000 For any full subcategory\- ${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$ of~${\mathcal A}(X)$, denote by ${\mathbf D}_{\scriptscriptstyle{\ldots}}(X)$ the full subcategory of~$\>{\mathbf D}(X)$ whose objects are those complexes whose homology sheaves all lie in~ ${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$, and by ${\mathbf D}^+_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}(X)$ \index{ ${\mathbf D}$ (derived category)!${\mathbf D}^\pm_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}\>$} (resp.~${\mathbf D}^-_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}(X)$) the full subcategory of~${\mathbf D}_{\scriptscriptstyle{\ldots}}(X)$ whose objects are those complexes~${\mathcal F}\in{\mathbf D}_{\scriptscriptstyle{\ldots}}(X)$ such that the homology $H^m({\mathcal F}\>)$ vanishes for all $m\ll0$ (resp.~$m\gg0$). The full subcategory ${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$ of ${\mathcal A}(X)$ is \emph{plump}\index{plump} if it contains 0 and for every exact sequence ${\mathcal M}_1\to{\mathcal M}_2\to{\mathcal M}\to{\mathcal M}_3\to {\mathcal M}_4$ in~${\mathcal A}(X)$ with ${\mathcal M}_1$, ${\mathcal M}_2$, ${\mathcal M}_3$ and~ ${\mathcal M}_4$ in~${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$, ${\mathcal M}$ is in~${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$ too. If ${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)$ is plump then it is abelian, and has a derived category ${\mathbf D}({\mathcal A}_{\scriptscriptstyle{\ldots}}(X))$. For~example, $\A_{\mathrm c}(X)$ is plump \cite[p.\,113,~(5.3.5)]{GD}. If ${\mathscr X}$ is a locally noetherian formal\index{formal scheme} scheme \footnote {Basic properties of formal schemes can be found in \cite[Chap.\,1, \S10]{GD}.% } then $\A_{\vec {\mathrm c}}({\mathscr X})\subset\A_{\qc}({\mathscr X})$ (\Cref{C:vec-c is qc})---with equality when ${\mathscr X}$ is an ordinary scheme, i.e., when ${\mathcal O}_{\mathscr X}$ has discrete topology \cite[p.\,319, (6.9.9)]{GD}---and both of these are plump subcategories of~${\mathcal A}({\mathscr X})$, see \Pref{(3.2.2)}. \smallskip Let ${\mathbf K}_1$, ${\mathbf K}_2$ be triangulated categories\index{triangulated category} with respective translation functors $T_1\>,\,T_2$ \cite[p.~20]{H1}. A (covariant) \emph{$\Delta$-functor}\index{Delta fun@$\Delta$-functors (on triangulated categories)} is a pair $(F, \Theta)$ consisting of an additive functor $F\colon {\mathbf K}_1\to{\mathbf K}_2$ together with an isomorphism of~functors $ \Theta:FT_1\iso T_2F $ such that for every triangle $ A\stackrel{u}{\longrightarrow}B\stackrel{v}{\longrightarrow} C\stackrel{w}{\longrightarrow} T_1A $ in ${\mathbf K}_1\mspace{.6mu}$, the diagram $$ FA\xrightarrow{\ Fu\ } FB\xrightarrow{\ Fv\ } FC \xrightarrow{\Theta\<\smcirc\< Fw\>} T_2FA $$ is a triangle in ${\mathbf K}_2\mspace{.6mu}$. Explicit reference to $\Theta$ is often suppressed---but one should keep it in mind. (For example, if ${\mathcal A}_{\scriptscriptstyle{\ldots}}(X)\subset{\mathcal A}(X)$ is plump, then each of ${\mathbf D}_{\scriptscriptstyle{\ldots}}(X)$ and~ ${\mathbf D}^\pm_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}(X)$ carries a unique triangulation for which the translation is the restriction of that on~${\mathbf D}(X)$ and such that inclusion into ${\mathbf D}(X)$ together with $\Theta\!:=$identity is a $\Delta$-functor; in other words, they are all \emph{triangulated subcategories}\index{triangulated category!triangulated subcategory} of~${\mathbf D}(X)$. See e.g., \Pref{P:Rhom} for the usefulness of this remark.) Compositions\index{Delta fun@$\Delta$-functors (on triangulated categories)!composition of} of $\Delta$-functors, and morphisms between\index{Delta fun@$\Delta$-functors (on triangulated categories)!morphism between} $\Delta$-functors, are defined in the natural way.% \footnote{% See also \cite[\S0,\,\S1]{De} for the multivariate case, where signs come into play---and $\Delta$-functors are called ``exact functors."% } A $\Delta$-functor $(G,\Psi)\colon{\mathbf K}_2\to{\mathbf K}_1$ is a \emph{right $\Delta$-adjoint}\index{Delta adj@$\Delta$-adjoint} of~$(F,\Theta)$ if $G$~is a right adjoint of~$F$ and the resulting functorial map $FG\to \mathbf 1$ (or equivalently, $\mathbf 1\to GF$) is a morphism of $\Delta$-functors. We use ${\mathbf R}$\index{ ${\mathbf R}$ (right-derived functor)}\index{derived functor} to denote right-derived functors, constructed e.g., via K-injective\index{K-injective resolution} resolutions (which exist for all ${\mathcal A}(X)$-complexes \cite[p.\,138, Thm.~4.5]{Sp}).% \footnote {A complex $F$ in an abelian category~${\mathcal A}$ is K-injective if for each exact ${\mathcal A}$-complex~$G$ the abelian-group complex ${\mathrm {Hom}}^{\bullet}_{\mathcal A}(G,F)$ is again exact. In particular, any bounded-below complex of injectives is K-injective. If every ${\mathcal A}$-complex~$E$ admits a K-injective resolution $E\to I(E)$ (i.e., a quasi-isomorphism into a K-injective complex~$I(E)$), then every additive functor~$\Gamma\colon{\mathcal A}\to{\mathcal A}'$ (${\mathcal A}'$~abelian) has a right-derived functor~${\mathbf R}\Gamma\colon{\mathbf D}({\mathcal A})\to{\mathbf D}({\mathcal A}')$ which satisfies ${\mathbf R}\Gamma(E)=\Gamma(I(E))$. For~example, ${\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathcal A}(E_1,E_2)={\mathrm {Hom}}^{\bullet}_{\mathcal A}(E_1,I(E_2))$.% } For a map $f\colon X\to Y$\index{ringed space!ringed-space map} of ringed spaces (i.e., a continuous map $f\colon X\to Y$ together with a ring-homomorphism ${\mathcal O}_Y\to f_{\!*}{\mathcal O}_{\<\<X}$), ${\mathbf L} f^*$% \index{ $\mathbf L$@${\mathbf L}$ (left-derived functor)} denotes the left-derived functor of $f^*\<$, constructed via K-flat resolutions \cite[p.\,147, 6.7]{Sp}. Each derived functor in this paper comes equipped, implicitly, with a $\Theta$ making it into a $\Delta$-functor (modulo obvious modifications for contravariance), cf.~\cite[Example~(2.2.4)]{Derived categories}.% \footnote {We do not know, for instance, whether ${\mathbf L} f^*$---which is defined only up to isomorphism---can always be chosen so as to commute with translation, i.e., so that $\Theta={}$Identity will do } Conscientious readers may verify that such morphisms between derived functors as occur in this paper are in fact morphisms of $\Delta$-functors. \begin{parag}\setcounter{sth}{0} Our \textbf{first main result,} global Grothendieck Duality\index{Grothendieck Duality!global} for a map \mbox{$f\colon{\mathscr X}\to{\mathscr Y}$} of quasi-compact formal schemes with ${\mathscr X}$ noetherian, is that, ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ being the derived category of~$\A_{\vec {\mathrm c}}({\mathscr X})$ and ${\boldsymbol j} \colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to{\mathbf D}({\mathscr X})$ being\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} the natural functor, \emph{the $\Delta$-functor\/~${\mathbf R f_{\!*}}\<\<\smcirc\<{\boldsymbol j}$ has a right\/ $\Delta$-adjoint.} \vspace{1pt} A more elaborate---but readily shown equivalent---statement is: \begin{thespecial} \label{Th1} Let\/ $f\colon\<{\mathscr X} \to {\mathscr Y}$ be a map of quasi-compact formal schemes, with ${\mathscr X}$ noetherian, and let\/~${\boldsymbol j}\colon\<{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X})) \to{\mathbf D}({\mathscr X})$ be the natural functor. Then there exists a\/ $\Delta$-functor% \index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$f^\times\<\<$} $f^{\times}\<\colon{\mathbf D}({\mathscr Y}) \to{\mathbf D}\left({\A_{\vec {\mathrm c}}({\mathscr X})}\right)$\vadjust{\kern.3pt} together with a morphism of\/ $\Delta$-functors $\tau:{\mathbf R} f_{\!*}\>\>{\boldsymbol j}\> f^{\times}\to{\bf 1}$\index{ {}$\tau$ (trace map)} such that for all\/ ${\mathcal G}\in{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ and\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y}),$\ the composed map $($in the derived category of abelian groups\/$)$ \begin{align*} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\A_{\vec {\mathrm c}}({\mathscr X})\!}({\mathcal G},\>f^\times\<\<{\mathcal F}\>) &\xrightarrow{\<\<\mathrm{natural}\,} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{{\mathcal A}({\mathscr Y})\!}({\mathbf R} f_{\!*}\>{\mathcal G},{\mathbf R} f_{\!*} f^\times\<\<{\mathcal F}\>)\\ &\xrightarrow{\;\>\mathrm{via}\ \tau\ } {\mathbf R}{\mathrm {Hom}}^{\bullet}_{{\mathcal A}({\mathscr Y})\!}({\mathbf R} f_{\!*}\>{\mathcal G},{\mathcal F}\>) \end{align*} is an isomorphism. \end{thespecial} Here we think of the $\A_{\vec {\mathrm c}}({\mathscr X})$-complexes ${\mathcal G}$ and $f^\times\<\<{\mathcal F}$ as objects in both ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ and~${\mathbf D}({\mathscr X})$. But as far as we know, the natural map ${\mathrm {Hom}}_{{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))}\to{\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}$ need not always be an isomorphism. It \emph{is} when ${\mathscr X}$ is \emph{properly algebraic,}\index{properly algebraic} i.e., the $J$-adic completion of a proper $B$-scheme with $B$ a noetherian ring and $J$ a $B$-ideal: then ${\boldsymbol j}$ induces an equivalence of categories ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$, see \Cref{corollary}. So for properly algebraic ${\mathscr X}$, we can replace ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ in~Theorem~1 by~$\D_{\<\vc}({\mathscr X})$, and let~${\mathcal G}$ be any ${\mathcal A}({\mathscr X})$-complex with $\A_{\vec {\mathrm c}}({\mathscr X})$-homology. We prove \Tref{Th1} (=\:\Tref{prop-duality}) in \S\ref{sec-th-duality}, adapting the argument of Deligne\index{Deligne, Pierre} in \cite[Appendix]{H1} (see also \cite[\S1.1.12]{De}) to the category~$\A_{\vec {\mathrm c}}({\mathscr X})$, which presents itself as an appropriate generalization to formal schemes of the category of quasi-coherent sheaves on an ordinary noetherian scheme. For this adaptation what is needed, mainly, is the plumpness of $\A_{\vec {\mathrm c}}({\mathscr X})$ in ${\mathcal A}({\mathscr X})$, a non-obvious fact mentioned above. In addition, we need some facts on ``boundedness" of certain derived functors in order to extend the argument to unbounded complexes. (See section~\ref{SS:bounded}, which makes use of techniques from~\cite{Sp}.)% \footnote{A $\Delta$-functor $\phi$ is \emph{bounded above} if there is an integer $b$ such that for any $n$ and any complex~${\mathcal E}$ such that $H^i{\mathcal E}=0$ for all $i\le n$ it holds that $H^j\<(\phi{\mathcal E})=0$ for all $j<n+b$. \emph{Bounded below} and \emph{bounded} (above and below) are defined analogously. Boundedness (way-outness)\index{boundedness (way-outness) of $\Delta$-functors} is what makes the very useful ``way-out Lemma" \cite[p.\,68, 7.1]{H1} applicable.} In Deligne's approach\index{Deligne, Pierre} the ``Special Adjoint Functor Theorem"\index{Special Adjoint Functor Theorem} is used to get right adjoints for certain functors on $\A_{\qc}(X)$, and then these right adjoints are applied to injective resolutions of complexes\dots There is now a neater approach to duality on a quasi-compact separated ordinary scheme~$X\<$, due to Neeman\index{Neeman, Amnon} \cite{N1}, in which ``Brown Representability"\index{Brown Representability} shows directly that a $\Delta$-functor~$F$ on~${\mathbf D}(\A_{\qc}(X))$ has a right adjoint if and only if $F$ commutes with coproducts. Both approaches need a small set of category-generators: coherent sheaves for $\A_{\qc}(X)$ in Deligne's, and perfect complexes for ${\mathbf D}(\A_{\qc}(X))$ in Neeman's. Lack of knowledge about perfect\- complexes over formal schemes discouraged us from pursuing Neeman's strategy. Recently however (after this paper was essentially written), Franke showed in~\cite{BR} that Brown Representability\index{Brown Representability} holds for the derived category of an arbitrary Grothen\-dieck category~${\mathcal A}$ \footnote{So does the closely-related existence of K-injective resolutions for all ${\mathcal A}$-complexes. (See also \cite[\S5]{AJS}.)} % Consequently \Tref{Th1} also follows from the fact that $\A_{\vec {\mathrm c}}({\mathscr X})$ is a Grothendieck category (straightforward to see once we know it---by plumpness in~${\mathcal A}({\mathscr X})$---to be abelian) together with the fact that ${\mathbf R f_{\!*}}\<\smcirc{\boldsymbol j}$ commutes with coproducts (\Pref{P:coprod}). \end{parag} \medskip \begin{parag}\label{Gamma'} Two other, probably more useful, generalizations---from ordinary schemes to formal schemes---of global Grothendieck Duality are stated below in \Tref{Th2} and treated in detail in \S\ref{S:t-duality}. To describe them, and related results, we need some preliminaries about \emph{torsion functors}. \smallskip \begin{sparag}\label{Gamma'1} Once again let $(X, {\mathcal O}_{\<\<X})$ be a ringed space. For any ${\mathcal O}_{\<\<X}$-ideal~${\mathcal J}\<$, set $$ \iG{\<{\mathcal J}\>}{{\mathcal M}} := \dirlm{n>0\,\,\>} {{\mathcal H}}om_{{\mathcal O}_{\<\<X}\!}({\mathcal O}_{\<\<X}/{\mathcal J}^{n},\, {\mathcal M})\qquad\bigl({\mathcal M}\in{\mathcal A}(X)\bigr), $$ and regard~$\iG{\<{\mathcal J}\>}$ as a subfunctor of the identity functor on ${\mathcal O}_{\<\<X}$-modules. If ${\mathcal N}\subset {\mathcal M}$ then $\iG{\<{\mathcal J}\>}{{\mathcal N}}=\iG{\<{\mathcal J}\>}{{\mathcal M}}\cap{\mathcal N}\>$; and it follows formally that the functor $\iG{\<{\mathcal J}\>}$ is idempotent ($\iG{\<{\mathcal J}\>}\iG{\<{\mathcal J}\>}{\mathcal M}=\iG{\<{\mathcal J}\>}{\mathcal M}$) and left exact \cite[p.\,138, Proposition~1.7\kern.5pt]{Stenstrom}.\vadjust{\kern.7pt} Set ${\mathcal A}_{\mathcal J}(X)\!:=\iG{\<{\mathcal J}\>}({\mathcal A}(X))$,\index{ ${\mathcal A}$ (module category)!$\A_{\mathrm c}$@${\mathcal A}_{\mathcal J}$} the full subcategory of~${\mathcal A}(X)$ whose objects are the \emph{${\mathcal J}\!$-torsion sheaves,}\index{torsion sheaf} i.e., the ${\mathcal O}_{\<\<X}$-modules~${\mathcal M}$ such that $\iG{\<{\mathcal J}\>}{{\mathcal M}}={\mathcal M}$. Since $\iG{\<{\mathcal J}\>}$ is an idempotent subfunctor of the identity functor, therefore it is right-adjoint to the inclusion $i=i_{\mathcal J}\colon{\mathcal A}_{\mathcal J}(X)\hookrightarrow{\mathcal A}(X)$. Moreover, ${\mathcal A}_{\mathcal J}(X)$ is closed under ${\mathcal A}(X)$-colimits: if $F$ is any functor into ${\mathcal A}_{\mathcal J}(X)$ such that $iF\/$ has a colimit ${\mathcal M}\in{\mathcal A}(X)$, then, since $i$ and $\iG{\<{\mathcal J}}$ are adjoint, the corresponding functorial map from $iF$ to the constant functor with value~${\mathcal M}$ factors via a functorial map from $iF$ to the constant functor with value $\iG{\<{\mathcal J}\>}{{\mathcal M}}$, and from the definition of colimits it follows that the monomorphism $\iG{\<{\mathcal J}\>}{{\mathcal M}}\hookrightarrow{\mathcal M}$ has a right inverse,\vspace{.6pt} so that it is an isomorphism, and thus ${\mathcal M}\in{\mathcal A}_{\mathcal J}(X)$. In particular, if the domain of a functor~$G$ into ${\mathcal A}_{\mathcal J}(X)$ is a small category, then $iG$ does have a colimit, which is also a colimit of $G$; and so ${\mathcal A}_{\mathcal J}(X)$ has small colimits, i.e., it is small-cocomplete. Submodules and quotient modules of~${\mathcal J}\!$-torsion sheaves are ${\mathcal J}\!$-torsion sheaves. If ${\mathcal J}$ is \emph{finitely-generated} (locally) and if ${\mathcal N}\subset{\mathcal M}$ are ${\mathcal O}_{\<\<X}$-modules such that ${\mathcal N}$ and~${\mathcal M}/{\mathcal N}$ are ${\mathcal J}\!$-torsion sheaves then ${\mathcal M}$ is a ${\mathcal J}\!$-torsion sheaf too; and hence ${\mathcal A}_{\mathcal J}(\<X)$~is plump in~${\mathcal A}(\<X)$.% \footnote{Thus the subcategory ${\mathcal A}_{{\mathcal J}}(X)$ is a \emph{hereditary torsion class} in ${\mathcal A}(X)$, in the sense of Dickson, see \cite[pp.\;139--141]{Stenstrom}. } In this case, the stalk of~$\iG{\<{\mathcal J}\>}{\mathcal M}$ at $x\in X$ is% \index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)} $$ (\iG{\<{\mathcal J}\>}{\mathcal M})_x=\dirlm{n>0\,\,\>} \mathrm{Hom}_{{\mathcal O}_{\!X\!,\>x}}\<({\mathcal O}_{\!X\!,\>x}/{\mathcal J}_x^{n},\, {\mathcal M}_x). $$ Let $X$ be a locally noetherian scheme and $Z\subset X$ a closed subset, the support of~${\mathcal O}_{\<\<X}/{\mathcal J}$ for some quasi-coherent ${\mathcal O}_{\<\<X}$-ideal ${\mathcal J}\<$. The functor $\iGp{Z}\!:=\iG{\<{\mathcal J}\>}$ \index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!$\varGamma'_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$} does not depend on the quasi-coherent ideal~${\mathcal J}$ determining $Z$. It is a subfunctor of the left-exact functor $\iG{Z}^{\phantom{.}}$ which associates to each ${\mathcal O}_{\<\<X}$-module~${\mathcal M}$ its subsheaf of sections supported in~$Z$. If ${\mathcal M}$ is quasi-coherent, then \mbox{$\iGp{Z}({\mathcal M}) =\iG{Z}({\mathcal M})$}. \pagebreak[3] More generally, for any complex ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}\<(X)$, the ${\mathbf D}(X)$-map ${\mathbf R}\iGp{Z}{\mathcal E}\to{\mathbf R}\iG{Z}{\mathcal E}$ induced by the inclusion $\iGp{Z}\hookrightarrow \iG Z$ is an isomorphism \cite[p.\,25, Corollary (3.2.4)]{AJL}; so for such~${\mathcal E}$ we usually identify ${\mathbf R}\iGp{Z}{\mathcal E}$ and ${\mathbf R}\iG{Z}{\mathcal E}$. Set ${\mathcal A}_Z(X)\!:= {\mathcal A}_{\mathcal J}(X)$,\index{ ${\mathcal A}$ (module category)!${\mathcal A}_Z$} the plump subcategory of~${\mathcal A}(X)$ whose objects are the \mbox{\emph{$Z$-torsion sheaves,}} that is, the ${\mathcal O}_{\<\<X}$-modules~${\mathcal M}$ such that \hbox{$\iGp{Z}{{\mathcal M}}={\mathcal M}\>$;} and set \hbox{$\A_{{\qc}Z}(X)\!:=\A_{\qc}(X)\cap{\mathcal A}_Z(X)$,}\index{ ${\mathcal A}$ (module category)!$\A_{{\qc}Z}$} the plump subcategory of~${\mathcal A}(X)$ whose objects are the quasi-coherent ${\mathcal O}_{\<\<X}$-modules supported in~$Z$. \enlargethispage{-\baselineskip} \smallskip \pagebreak[3] For a locally noetherian formal scheme ${\mathscr X}$ with ideal of definition~${\mathscr J}$, set $\iGp{{\mathscr X}}\!:=\iG{{\mathscr J}}\mspace{-.5mu}$,% \index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!$\varGamma'_{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$} a left-exact functor depending only on the sheaf of topological rings ${\mathcal O}_{{\mathscr X}}\>$, not on the choice of~${\mathscr J}$---for ${\mathcal M}\in{\mathcal A}({\mathscr X})$, $\iGp{{\mathscr X}}{\mathcal M}\subset{\mathcal M}$ is the submodule whose sections are those of~${\mathcal M}$ annihilated locally by an open ideal. Say that ${\mathcal M}$ is a \emph{torsion~sheaf}\index{torsion sheaf} if \mbox{$\iGp{{\mathscr X}}{\mathcal M}={\mathcal M}$}. Let $\A_{\mathrm t}\<({\mathscr X})\!:={\mathcal A}_{\mathscr J}({\mathscr X})$\index{ ${\mathcal A}$ (module category)!$\A_{\mathrm t}\<\>$} be the plump subcategory of~${\mathcal A}({\mathscr X})$ whose objects\- are all the torsion sheaves; and set $\A_{\mathrm {qct}}\<({\mathscr X})\!:=\A_{\qc}\<({\mathscr X})\cap\A_{\mathrm t}\<({\mathscr X})$,\index{ ${\mathcal A}$ (module category)!$\A_{\mathrm {qct}}\<$} the full (in fact plump, see \Cref{qct=plump}) subcategory of~${\mathcal A}({\mathscr X})$ whose objects are the quasi-coherent torsion sheaves. It holds that $\A_{\mathrm {qct}}\<({\mathscr X})\subset\A_{\vec {\mathrm c}}({\mathscr X})$, see \Cref{Gamma'+qc}. If ${\mathscr X}$ is an ordinary locally noetherian scheme (i.e., ${\mathscr J}=0$), then $\A_{\mathrm t}\<({\mathscr X})={\mathcal A}({\mathscr X})$ and~$\A_{\mathrm {qct}}\<({\mathscr X})=\A_{\qc}\<({\mathscr X})=\A_{\vec {\mathrm c}}({\mathscr X})$. \end{sparag} \begin{sparag}\label{maptypes} For any map $f\colon{\mathscr X} \to {\mathscr Y}$ of locally noetherian formal schemes there are ideals of definition ${\mathscr I}\subset {\mathcal O}_{{\mathscr Y}}$ and ${\mathscr J}\subset{\mathcal O}_{{\mathscr X}\>}$ such that ${\mathscr I}{\mathcal O}_{{\mathscr X}}\subset {\mathscr J}$ \cite[p.\,416, (10.6.10)]{GD}; and correspondingly there is a map of ordinary schemes ($=\:$formal schemes having~(0) as ideal of definition) $({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J})\to({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I})$ \cite[p.\,410, (10.5.6)]{GD}. We~say that $f$~is \emph{separated}% \index{formal-scheme map!separated} (resp.~\emph{affine,}% \index{formal-scheme map!affine} resp.~\emph{pseudo\kern.6pt-proper, \index{formal-scheme map!pseudo\kern.6pt-proper} resp.~\emph{pseudo\kern.6pt-finite,}% \index{formal-scheme map!pseudo\kern.6pt-finite} resp.~\emph{of pseudo\kern.6pt-finite type}\kern-1pt \index{formal-scheme map!of pseudo\kern.6pt-finite type} if for some---and hence any---such~${\mathscr I},{\mathscr J}$ the corresponding scheme-map is separated (resp.~affine, resp.~proper, resp.~finite, resp.~of finite type), see \cite[\S\S10.15--10.16, p.\,444\:\emph{ff.}]{GD}, keeping in mind \cite[p.\,416, (10.6.10)(ii)]{GD}.% \footnote {In \cite[Definition 1.14]{Ye}, pseudo-finite-type maps are called ``maps of formally finite type." The proof of Prop.\,1.4 in \cite{Ye} (with $A'=A$) yields the following characterization of pseudo-finite-type maps of affine formal schemes (cf.~\cite[p.\,439, Prop.\,(10.13.1)]{GD}): The map\/ $f\colon {\mathrm {Spf}}(B) \to {\mathrm {Spf}}(A)$ corresponding to a continuous homomorphism~$h\colon A\to B$ of noetherian adic rings is of pseudo-finite type $\Leftrightarrow$ for any ideal of definition\/ $I$ of\/ $A,$ there exists\vadjust{\kern.8pt} an\/ $A$-algebra of finite type\/~$A'\<,$ an\/ $A'\<$-ideal\/~ $I' \supset IA',$ and an\/ $A$-algebra homomorphism\/ \mbox{$A'\to B$} inducing an adic surjective map\/ $\widehat {A'}\twoheadrightarrow B$ where\/ $\widehat{A'}$ is the\/ $I'$-adic completion of\/~$A'$.} Any affine map is separated. Any pseudo\kern.6pt-proper map is separated and of pseudo\kern.6pt-finite type. The map~$f$ is pseudo\kern.6pt-finite $\Leftrightarrow$ it is pseudo\kern.6pt-proper and affine $\Leftrightarrow$ it is pseudo\kern.6pt-proper and has finite fibers \cite[p.\,136, (4.4.2)]{EGA}. We say that $f$ is \emph{adic}% \index{formal-scheme map!adic} if for some---and hence any---ideal of definition ${\mathscr I}\subset{\mathcal O}_{\mathscr Y}\>$, ${\mathscr I}{\mathcal O}_{{\mathscr X}}$~is an ideal of definition of~${\mathscr X}$ \cite[p.\,436, (10.12.1)]{GD}. We say that $f$ is \emph{proper}% \index{formal-scheme map!proper} (resp.~\emph{finite,}% \index{formal-scheme map!finite} resp.~\emph{of finite type}\kern-1pt)% \index{formal-scheme map!of finite type} if $f$ is pseudo\kern.6pt-proper (resp.~pseudo\kern.6pt-finite, resp.~of pseudo\kern.6pt-finite type) and adic, see \cite[p.\,119, (3.4.1)]{EGA}, \cite[p.\,148, (4.8.11)]{EGA} and \cite[p.\,440, (10.13.3)]{GD}. \end{sparag} \begin{sparag} Here is our \textbf{second main result}, Torsion Duality\index{Grothendieck Duality!Torsion (global)} for formal schemes. (See \Tref{T:qct-duality} and \Cref{C:f*gam-duality} for more elaborate statements.) In the assertion, $\wDqc({\mathscr X})\!:={\mathbf R}\iGp{\mathscr X}{}^{-1}(\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}))$% \index{ ${\mathbf D}$ (derived category)!z@${ \widetilde {\vbox to5pt{\vss\hbox{$\mathbf D$}}}_{\mkern-1.5mu\mathrm {qc}} }$} is the least $\Delta$-subcategory of~${\mathbf D}({\mathscr X})$ containing both $\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})$\vspace{.6pt} and ${\mathbf R}\iGp{\mathscr X}{}^{-1}(0)$ (\Dref{D:Dtilde}, Remarks~\ref{R:Dtilde}, (1) and~(2)).\vspace{-1.3pt} For example, when ${\mathscr X}$ is an ordinary scheme then $\wDqc({\mathscr X})=\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})$. \begin{thespecial} \label{Th2} Let $f\colon {\mathscr X}\to{\mathscr Y}$ be a map of noetherian formal schemes. Assume either that\/ $f$ is separated or that\/ ${\mathscr X}$ has finite Krull dimension, or else restrict~to bounded-below complexes. \smallskip \textup{(a)} The restriction of\/ ${\mathbf R f_{\!*}}\colon{\mathbf D}({\mathscr X})\to{\mathbf D}({\mathscr Y})$ takes\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ to\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}),$ and it has a right\/ $\Delta$-adjoint\/ $ f_{\mathrm t}^\times\colon{\mathbf D}({\mathscr Y})\to\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}).\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$f_{\mathrm t}^\times\<\<$} $ \smallskip \textup{(b)} The restriction of\/ ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}$ takes\/ $\wDqc({\mathscr X})$ to\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\subset\wDqc({\mathscr Y}),$ and it has a~right\/ $\Delta$-adjoint\/ $ \ush f\colon{\mathbf D}({\mathscr Y})\to\wDqc({\mathscr X}).\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\ush f$} $ \end{thespecial} \penalty -1000 \begin{srems}\label{R:Th2} (1) The ``homology localization" functor $$ {\boldsymbol\Lambda}^{}_{\mathscr X}(-)\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,-)\index{ $\mathbf {La}$@${\boldsymbol\Lambda}$ (homology localization)} $$ is right-adjoint to ${\mathbf R}\iGp{\mathscr X}$, and ${\boldsymbol\Lambda}_{\mathscr X}^{-1}(0)={\mathbf R}\iGp{\mathscr X}{}^{-1}(0)$ (Remarks~\ref{R:Gamma-Lambda}). The $\Delta$-functors $\ush f$ and~$f_{\mathrm t}^\times$ are connected thus (Corollaries~\ref{C:f*gam-duality} and~\ref{C:identities}(a)): $$ \ush f={\boldsymbol\Lambda}^{}_{\mathscr X}f_{\mathrm t}^\times\<, \qquad f_{\mathrm t}^\times={\mathbf R}\iGp{\mathscr X}\ush f. $$ \smallskip (2) In the footnote on page \pageref{C:completion-proper} it is indicated that ${\mathbf R}\iGp{\mathscr X}{}^{-1}(0)$ admits a ``Bousfield colocalization"\index{Bousfield colocalization} in ${\mathbf D}({\mathscr X})$, with associated ``cohomology colocalization" functor ${\mathbf R}\iGp{\mathscr X}\>$; and in \Rref{R:Gamma-Lambda}(3), \Tref{Th2} is interpreted as duality\- with coefficients in the corresponding quotient $\wDqc({\mathscr X})/{\mathbf R}\iGp{\mathscr X}{}^{-1}(0)\cong \D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})/\bigl(\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})\cap{\mathbf R}\iGp{\mathscr X}{}^{-1}(0)\bigr)$. \smallskip (3) The proof of \Tref{Th2} is similar to that of \Tref{Th1}, at least when the formal scheme~${\mathscr X}$ is separated (i.e., the unique formal-scheme map ${\mathscr X}\to{\mathrm {Spec}}(\mathbb Z)$ is separated) or finite-dimensional, in which case there is an \emph{equivalence of categories} ${\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X}))\to \D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (\Pref{1!}). (As mentioned before, we know the corresponding result with ``$\mspace{2mu}\vec{\mathrm c}\:$" in place of ``qct" only for \emph{properly algebraic} formal schemes.) In addition, replacing separatedness of~${\mathscr X}$ by separatedness of~$f$ takes a technical pasting argument. \smallskip (4) For an ordinary scheme~$X$ (having $(0)$ as ideal of definition), $\iGp X$ is just the identity functor of ${\mathcal A}(X)$, and $\D_{\mkern-1.5mu\mathrm{qct}}(X)=\D_{\mkern-1.5mu\mathrm {qc}}(X)$. In this case, Theorems~\ref{Th1}~and~\ref{Th2} both reduce to the usual global (non-sheafified) version of Grothendieck Duality. In \S\ref{S:apps} we will describe how \Tref{Th2} generalizes and ties together various strands in the literature on local, formal, and global duality. In particular, the behavior of \Tref{Th2} vis-\`a-vis variable $f$ gives compatibility of local and global duality, at least on an abstract level---i.e., without the involvement of differentials, residues,~etc. (See \Cref{C:kappa-f^times-tors}.) \end{srems} \end{sparag} \end{parag} \begin{parag}\label{culminate} As in the classic paper \cite{f!} of Verdier,% \index{Verdier, Jean-Louis} the \textbf{culminating results} devolve from flat-base-change isomorphisms, established here for the functors $f_{\mathrm t}^\times$ and $\ush f$ of \mbox{\Tref{Th2},} with $f$ \emph{pseudo\kern.6pt-proper}---in which case we denote $f_{\mathrm t}^\times$ by $f^!\<$.\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\mathstrut f^!\<$} \begin{thespecial}\label{Th3}\index{base-change isomorphism} Let\/ ${\mathscr X},$ ${\mathscr Y}$ and\/ ${\mathscr U}$ be noetherian formal schemes, let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map, and let\/ $u\colon {\mathscr U}\to{\mathscr Y}$ be flat, so that in the natural diagram $$ \begin{CD} {\mathscr X}\times_{\mathscr Y}{\mathscr U}=:\>@.{\mathscr V}@>v>>{\mathscr X} \\ @. @VgVV @VVfV \\ @.{\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ the formal scheme\/ ${\mathscr V}$ is noetherian, $g$ is pseudo\kern.6pt-proper, and $v$ is flat \textup(\Pref{P:basechange}\kern.5pt\textup). \pagebreak[3] Then for all\/ ${\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\!:=\wDqc({\mathscr Y})\cap{\mathbf D}^+({\mathscr Y})$ the base-change map\/~$\beta_{\mathcal F}$ of~\Dref{D:basechange} is an isomorphism $$ \beta_{\mathcal F}\colon{\mathbf R}\iGp{\mathscr V}\>v^*\<f^!{\mathcal F} \iso g^!\>\>{\mathbf R}\iGp{\mathscr U} u^*\<{\mathcal F}\ \underset{\textup{\ref{C:identities}(b)}}\cong g^!\<u^*\<{\mathcal F}. $$ In particular, if\/ $u$ is \emph{adic} then we have a functorial isomorphism\/ $v^*\<\<f^!{\mathcal F} \iso g^! u^*\<{\mathcal F}\<.$ \end{thespecial} This theorem is proved in \S7 (\Tref{T:basechange}). The functor ${\mathbf R}\iGp{\mathscr V}$ has a right adjoint~${\boldsymbol\Lambda}_{\<{\mathscr V}}$, see \eqref{adj}. \Tref{Th3} leads quickly to the corresponding result for~$\ush f$ (see \Tref{T:sharp-basechange} and \Cref{C:coh-basechange}): \penalty -1000 \begin{thespecial}\label{Th4}\index{base-change isomorphism} Under the preceding conditions, let $$ \ush{\beta_{\<\<{\mathcal F}}}\colon v^*\<\<\ush f\<{\mathcal F}\to\ush g u^*\<{\mathcal F}\qquad\bigl({\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr) $$ be the map adjoint to the natural composition $$ {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\>v^*\<\<\ush f\<{\mathcal F} \underset{\textup{Thm.\,\ref{Th3}}}\iso {\mathbf R} g_*g^!u^*\<{\mathcal F}\to u^*\<{\mathcal F}. $$ Then the map ${\boldsymbol\Lambda}_{\<{\mathscr V}}(\ush{\beta_{\<\<{\mathcal F}}})$ is an \emph{isomorphism} $$ {\boldsymbol\Lambda}_{\<{\mathscr V}}(\ush{\beta_{\<\<{\mathcal F}}})\colon{\boldsymbol\Lambda}_{\<{\mathscr V}}\> v^*\<\<\ush f\<{\mathcal F}\iso {\boldsymbol\Lambda}_{\<{\mathscr V}}\>\ush g u^*\<{\mathcal F} \underset{\textup{\ref{C:identities}(a)}}\cong \ush g u^*\<{\mathcal F}. $$ Moreover, if\/ $u$ is an open immersion, or if\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y}),$ then\/ $\ush{\beta_{\<\<{\mathcal F}}}$ itself is an isomorphism. \end{thespecial} The special case of Theorems~\ref{Th3} and~\ref{Th4} when $u$ is an open immersion is equivalent to what may be properly referred to as Grothendieck Duality (unqualified by the prefix ``global"), namely the following \emph{sheafified} version of \Tref{Th2} (see \Tref{T:sheafify}): \begin{thespecial}\label{Th5}\index{Grothendieck Duality!Torsion (sheafified)} Let\/ ${\mathscr X}$ and\/ ${\mathscr Y}$ be noetherian formal schemes and let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map. Then the following natural compositions are \emph{isomorphisms:} \begin{align*} {\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>\ush f\<{\mathcal F}\>) &\to\< {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\ush f\<{\mathcal F}\>) \\ &\to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathcal F}\>)\vspace{-4pt} \quad\ \bigl({\mathcal G}\in\wDqc({\mathscr X}),\;{\mathcal F}\>\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr); \end{align*} \vspace{-6pt} \noindent $ \quad{\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>f^!{\mathcal F}\>) \to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathbf R f_{\!*}} f^!{\mathcal F}\>) \to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathcal F}\>)\vspace{3pt} $ \rightline{$\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),\;{\mathcal F}\>\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr).$} \end{thespecial} Finally, if $f$ is \emph{proper} and ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$, then $\ush f\colon\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\to\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ \emph{is right-adjoint to ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})\to\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$, and ${\mathbf R}\iGp{\mathscr X}$ in \Tref{Th5} can be deleted,} see \Tref{T:properdual}. \smallskip In this---and several other results about complexes with coherent homology---an essential ingredient is \Pref{formal-GM}, deduced here from Greenlees-May duality for ordinary affine schemes, see \cite{AJL}: \smallskip \emph{% Let\/ ${\mathscr X}$ be a locally noetherian formal scheme, and let\/ ${\mathcal E}\in{\mathbf D}({\mathscr X})$. Then for all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$ the natural map\/ ${\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\to {\mathcal E}$ induces an isomorphism% } $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal E}, \>{\mathcal F}\>) \iso {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{{\mathscr X}}{\mathcal E},\>{\mathcal F}\>). $$ \begin{comment} This proposition also follows from Greenlees-May Duality for formal schemes, \Tref{GM}, which asserts that for any separated noetherian formal scheme\/ ${\mathscr X}$ and any coherent\/ ${\mathcal O}_{\mathscr X}$-ideal\/~${\mathcal I}$, $$ {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iG{\mathcal I}{\mathcal O}_{\mathscr X},-)\colon\D_{\<\vc}({\mathscr X})\to{\mathbf D}({\mathscr X}) $$ \emph{is a left-derived functor of the completion functor $\Lambda_{\mathcal I}\colon\K_{\vc}({\mathscr X})\to{\mathbf K}({\mathscr X})\colon$} $$ \Lambda_{\mathcal I}({\mathcal E})\!:= \inlm{n}\!\bigr(({\mathcal O}_{\mathscr X}/{\mathcal I}^n)\otimes\>{\mathcal E}\bigl) \qquad\bigl({\mathcal E}\in\K_{\vc}({\mathscr X})\bigr). $$ The proof occupies most of the Appendix (\S9). \end{comment} \end{parag} \medskip In closing this introductory section, we wish to express our appreciation for illuminating interchanges with Amnon Neeman\index{Neeman, Amnon} and Amnon Yekutieli\index{Yekutieli, Amnon}. \section{Applications and examples.} \label{S:apps} Again, \Tref{Th2} generalizes global Grothendieck Duality on ordinary schemes. This section illustrates further how \Tref{Th2} provides a common home for a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes\dots\!\!). For a quick example, see \Rref{R:d-vein}. \Sref{bf (a)} reviews several forms of local duality. In section~\ref{sheafify} we sheafify these results, and connect them to \Tref{Th2}. In particular, \Pref{(2.2)} is an abstract version of the Local Duality theorem of~\cite[p.\,73, Theorem 3.4]{Integration}; and \Tref{T:pf-duality} (Pseudo-finite Duality)\index{Duality!Pseudo-finite} globalizes it to formal schemes.\looseness=-1 \Sref{residue thm} relates Theorems~\ref{Th1} and~\ref{Th2} to the central ``Residue Theorems"\index{Residue theorems} in~\cite{Asterisque} and~\cite{HS} (but does not subsume those results). \Sref{bf (d)} indicates how both the Formal Duality\index{Duality!Formal} theorem of \cite[p.\,48, Proposition~(5.2)]{De-Rham-cohomology} and the Local-Global Duality\index{Duality!Local-Global} theorem in \cite[p.\,188]{Desingularization} can be deduced from \Tref{Th2}. \Sref{bf (e)}, building on work of Yekutieli \cite[\S5]{Ye},\index{Yekutieli, Amnon} treats \emph{dualizing complexes} on formal schemes, and their associated dualizing functors. For a pseudo\kern.6pt-proper map~$f\<$, the functor~$\ush f$ of \Tref{Th2} lifts dualizing complexes to dualizing complexes (\Pref{P:twisted inverse}). For any map $f\colon{\mathscr X}\to{\mathscr Y}$ of noetherian formal schemes, there is natural isomorphism $$ {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*{\mathcal F}\<,\ush f{\mathcal G})\iso\ush f{\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}({\mathcal F}\<,{\mathcal G}) \qquad \bigl({\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^-({\mathscr Y}),\;{\mathcal G}\in{\mathbf D}^+({\mathscr Y})\bigr), $$ (\Pref{P:Hom!}). For pseudo\kern.6pt-proper $f\<$, if ${\mathscr Y}$ has a dualizing complex~${\mathcal R}$, so that $\ush f{\mathcal R}$ is a dualizing complex on~${\mathscr X}$, and if ${\mathcal D}^{\mathscr Y}\!:={\mathbf R}\cH{om}^{\bullet}(-,{\mathcal R})$ and ${\mathcal D}^{\mathscr X}$ are the corresponding dualizing functors, one deduces a natural isomorphism (well-known for ordinary schemes) $$ \ush f\<{\mathcal E}\cong{\mathcal D}^{\mathscr X}{\mathbf L} f^*{\mathcal D}^{\mathscr Y}{\mathcal E}\qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\bigr), $$ see \Pref{P:Dual!}. There are corresponding results for~$f_{\mathrm t}^\times\<$ as well. \smallskip \def\GG#1{\Gamma_{\!\!#1}^{\phantom{.}}} \def\ush{\varphi_{\<\!J}}{\ush{\varphi_{\<\!J}}} \begin{parag} \label{bf (a)} \renewcommand{\theequation}{\theparag.\arabic{numb}} (Local Duality.)\index{Duality!Local|(} All rings are commutative, unless otherwise specified. Let $\varphi\colon R\to S$ be a ring homomorphism with $S$ noetherian, let $J$ be an $S$-ideal, and let $\GG {J}$ be the functor taking any $S$-module to its submodule of elements which are annihilated by some power of~$J$. Let $E$ and $E'$ be complexes in~${\mathbf D}(S)$, the derived category of $S$-modules, and let $F\in{\mathbf D}(R)$. With \smash{$\Otimes$}\vadjust{\kern.7pt} denoting derived tensor product in~${\mathbf D}(S)$ (defined via K-flat resolutions \cite[p.\,147, Proposition 6.5]{Sp}), there is a natural isomorphism \smash{$E \Otimes {\mathbf R} \GG {\<J} E' \iso {\mathbf R} \GG {J} (E \Otimes E')$},\vadjust{\kern.7pt} see~e.g., \cite[p.\,20, Corollary(3.1.2)]{AJL}. Also, viewing ${\mathbf R}{\mathrm {Hom}}_R^\bullet(E'\<, F)$ as a functor from ${\mathbf D}(S)^{\rm{op}}\<\times{\mathbf D}(R)$ to~${\mathbf D}(S)$, one has a canonical ${\mathbf D}(S)$-isomorphism $$ {\mathbf R}{\mathrm {Hom}}^{\bullet}_R( E \Otimes E'\<, F )\iso {\mathbf R}{\mathrm {Hom}}^{\bullet}_S\bigl(E, {\mathbf R}{\mathrm {Hom}}_R^\bullet(E'\<, F)\bigr),\ $$ see \cite [p.\,147; 6.6]{Sp}. Thus, with $\ush{\varphi_{\<\!J}}\colon{\mathbf D}(R)\to {\mathbf D}(S)$ the functor given by $$ \ush{\varphi_{\<\!J}}(-)\!:={\mathbf R}{\mathrm {Hom}}^{\bullet}_R({\mathbf R}\GG {\mspace{-.5mu}J} S,-) \cong {\mathbf R}{\mathrm {Hom}}^{\bullet}_S\bigl({\mathbf R}\GG {\mspace{-.5mu}J}S,{\mathbf R}{\mathrm {Hom}}^{\bullet}_R(S,-)\bigr), $$ there is a composed isomorphism $$ {\mathbf R}{\mathrm {Hom}}^{\bullet}_S(E,\ush{\varphi_{\<\!J}}\<F) \iso {\mathbf R}{\mathrm {Hom}}^{\bullet}_R(E\Otimes {\mathbf R}\GG {\mspace{-.5mu}J} S, F ) \iso {\mathbf R}{\mathrm {Hom}}^{\bullet}_R ({\mathbf R}\GG {\mspace{-.5mu}J} E, F). $$ Application of homology $H^0$ yields the (rather trivial) {\it local duality isomorphism} \stepcounter{numb} \begin{equation} \label{(2.1)} {\mathrm {Hom}}_{{\mathbf D}(S)}\<\<(E,\ush{\varphi_{\<\!J}}\<F)\iso {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG {\mspace{-.5mu}J}E, F). \end{equation} ``Non-trivial" versions of~\eqref{(2.1)} include more information about $\ush{\varphi_{\<\!J}}$. For example, Greenlees-May duality\index{Greenlees-May Duality} \cite[p.\,4, $(0.3)_{\text{aff}}$]{AJL} gives a canonical isomorphism \stepcounter{numb} \begin{equation}\label{2.1.1} \ush{\varphi_{\<\!J}}\<F \cong{\mathbf L}\Lambda_J{\mathbf R}{\mathrm {Hom}}_R^\bullet(S,F), \end{equation} where $\Lambda_J$ is the {\it $J\<$-adic completion functor,} and ${\bf L}$ denotes ``\kern.5pt left-derived.'' In particular, if $R$ is noetherian, $S$ is a finite $R$-module, and $F\in\D_{\mkern-1.5mu\mathrm c}(R)$ (i.e., each homology module of~$F$ is finitely generated), then as in~\cite[p.\,6, Proposition (0.4.1)]{AJL}, \stepcounter{numb} \begin{equation} \ush{\varphi_{\<\!J}}\<F={\mathbf R}{\mathrm {Hom}}_R^\bullet(S,F)\otimes_S{\hat S} \qquad ({\hat S}=J \mbox{-adic completion of}\, S). \label{2.1.2} \end{equation} More particularly, for $S=R$ and $\varphi=\text{id}$ (the identity map) we get $$ \ush{\text{id}_{\<\<J}}F= F\otimes_R {\hat R} \qquad \bigl(F\in\D_{\mkern-1.5mu\mathrm c}(R)\bigr). $$ Hence, {\it classical local duality} \cite[p.~278 (modulo Matlis dualization)]{H1} is just~(\ref{(2.1)}) when $R$ is local, $\varphi=\text{id}$, $J$ is the maximal ideal of~$R$, and $F$ is a normalized dualizing complex---so that, as in ~\Cref{C:Hom-Rgamma}, and by~\cite[p.\,276, Proposition~6.1]{H1}, $$ {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG {\mspace{-.5mu}J}E, F)= {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG {\mspace{-.5mu}J}E, {\mathbf R}\GG {\mspace{-.5mu}J}F)= {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG {\mspace{-.5mu}J}E, I) $$ where $I$ is an $R$-injective hull of the residue field~$R/\<J$. (See also \Lref{L:dualizing}.)\looseness=-1 \smallskip For another example, let $S=R[[{\bf t}]]$ where $\mathbf t\!:= (t_1,\dots,t_d)$ is a sequence of variables, and set $J\!:= \mathbf tS$. The standard calculation (via Koszul complexes) gives an isomorphism ${\mathbf R}\Gamma_ {\!\! J}S\cong\nu[-d\>]$ where $\nu$ is the free $R$-submodule of the localization~$S_{t_1\dots t_d}$ generated by those monomials~$t_1^{n_1}\!\dots t_d^{n_d}$ with all exponents~$n_i<0$, the $S$-module structure being induced by that of $S_{t_1\<\dots\> t_d}/S\supset \nu\>$. The \emph{relative canonical module} $\omega_{R[[\mathbf t]]/R}\!:={\mathrm {Hom}}_R(\nu,R)$ is a \emph{free, rank one, $S$-module.} There result, for finitely-generated \mbox{$R$-modules}~$F$, functorial isomorphisms \stepcounter{numb} \begin{equation}\label{2.1.3} \ush {\varphi_{\mathbf tR[[\mathbf t]]}}F\cong {\mathrm {Hom}}_R(\nu[-d\>],F)\cong \omega_{R[[\mathbf t]]/R}^{\phantom{.}}[d\>]\otimes_R F\cong R[[\mathbf t]]\otimes_RF[d\>]; \end{equation} and when $R$ is noetherian, the usual way-out argument \cite[p.\,69, (ii)]{H1} yields the same for any $F\in\D_{\mkern-1.5mu\mathrm c}^+\<(\<R)$. \smallskip Next, we give a commutative-algebra analogue of \Tref{Th2} in \S\ref{S:prelim}, in the form of a ``torsion" variant of the duality isomorphism~\eqref{(2.1)}. \Pref{P:affine} will clarify the relation between the algebraic and formal-scheme contexts. With $\varphi\colon R\to S$ and $J$ an $S$-ideal as before, let ${\mathcal A}_J(S)$ be the category of \mbox{$J$-torsion} $S$-modules, i.e., \mbox{$S$-modules}~$M$ such that $$ M=\GG{\<J}M:=\{\,m\in M\mid J^nm=0\textup{ for some }n>0\,\}. $$ The derived category of ${\mathcal A}_J(S)$ is equivalent to the full subcategory~${\mathbf D}_{\!J\<}(S)$ of~${\mathbf D}(S)$ with objects those $S$-complexes~$E$ whose homology lies in~${\mathcal A}_J(S)$, (or equivalently, such that the natural map \smash{${\mathbf R}\GG{\<J}E\to E$} is an isomorphism), and the functor~${\mathbf R}\GG{\<J}$ is right-adjoint to~the inclusion ${\mathbf D}_{\!J\<}(S)\hookrightarrow{\mathbf D}(S)$ (cf.~\Pref{Gamma'(qc)} and its proof). \goodbreak \noindent Hence the functor~$\varphi_{\!J}^\times\colon{\mathbf D}(R)\to {\mathbf D}_{\!J\<}(S)$ defined by $$ \varphi_{\!J}^\times(-)\!:={\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}^{\bullet}_R(S, -) \cong \smash{{\mathbf R}\GG{\<J}S\Otimes {\mathbf R}{\mathrm {Hom}}^{\bullet}_R(S, -)} $$ is right-adjoint to the natural composition ${\mathbf D}_{\!J\<}(S)\hookrightarrow{\mathbf D}(S)\to{\mathbf D}(R)$: in fact, for $E\in{\mathbf D}_{\!J\<}(S)$ and $F\in{\mathbf D}(R)$ there are natural isomorphisms \stepcounter{numb} \begin{equation}\label{2.2.1} {\mathbf R}{\mathrm {Hom}}^{\bullet}_S(E,\varphi_{\!J}^\times\<F)\iso {\mathbf R}{\mathrm {Hom}}^{\bullet}_S(E,{\mathbf R}{\mathrm {Hom}}^{\bullet}_R(S, F))\iso {\mathbf R}{\mathrm {Hom}}^{\bullet}_R(E,F). \end{equation} Here is another interpretation of $\varphi_{\!J}^\times\<F$. For $S$-modules~$A$ and $R$-modules~$B$~set $$ {\mathrm {Hom}}_{R,J}(A, B)\!:=\GG{\<J}{\mathrm {Hom}}_R(A,B), $$ the $S$-module of $R$-homomorphisms $\alpha$ vanishing on $J^n\<A$ for some $n$ (depending on~$\alpha$), i.e., \emph{continuous} when $A$ is $J$-adically topologized and $B$ is discrete. If $E$~is a K-flat $S$-complex and $F$ is a K-injective $R$-complex, then ${\mathrm {Hom}}^{\bullet}_R(E,F)$ is a K-injective $S$-complex; and it follows for all $E\in{\mathbf D}(S)$ and $F\in{\mathbf D}(R)$ that $$ {\mathbf R}{\mathrm {Hom}}_{R,J}^\bullet(E,F)\cong{\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}_R^\bullet(E,F). $$ Thus,\vspace{-1pt} $$ \varphi_{\!J}^\times\<F={\mathbf R}{\mathrm {Hom}}_{R,J}^\bullet(S,F). $$ \penalty -1000 A \emph{torsion version of local duality} is the isomorphism, derived from~\eqref{2.2.1}: $$ {\mathrm {Hom}}_{{\mathbf D}_{\!J\<}(S)}\<\<\bigl(E,\>{\mathbf R}{\mathrm {Hom}}_{R,J}^\bullet(S,F)\bigr) \iso {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<(E,F) \quad\ \bigl(E\in{\mathbf D}_{\!J\<}(S),\;F\in{\mathbf D}(R)\bigr). $$ \begin{small} Apropos of \Rref{R:Th2}(1), the functors $\varphi_{\!J}^\times$ and $\ush{\varphi_{\<\!J}}$ are related by \begin{alignat*}{2} {\mathbf L}\Lambda_J{\mathbf R}{\mathrm {Hom}}_R^\bullet(S,F)\; &\underset{\lower.5ex\hbox to0pt{\hss\scriptsize\eqref{2.1.1}\hss}}{\cong}\;\ush{\varphi_{\<\!J}}\<F &&\cong{\mathbf L}\Lambda_J\varphi_{\!J}^\times\<F, \\ {\mathbf R}\GG{\<J}\>{\mathbf R}{\mathrm {Hom}}^{\bullet}_R(S,F)\, &=\,\varphi_{\!J}^\times\<F && \cong{\mathbf R}\GG{\>J}\ush{\varphi_{\<\!J}}\<F. \end{alignat*} The first relation is the case $E={\mathbf R}\GG{\<J}\<S$ of \eqref{2.2.1}, followed by Greenlees-May duality. The second results, e.g., from the sequence of natural isomorphisms, holding for \mbox{$G\in{\mathbf D}_{\!J\<}(S)$}, \mbox{$E\in{\mathbf D}(S)$}, and $F\in{\mathbf D}(R)$: \begin{align*} {\mathrm {Hom}}_{{\mathbf D}(S)}\<\<\bigl(G,\> {\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}_R^\bullet(E,F)\bigr) &\cong {\mathrm {Hom}}_{{\mathbf D}(S)}\<\<\bigl(G,\> {\mathbf R}{\mathrm {Hom}}_R^\bullet(E,F)\bigr)\\ &\cong \smash{{\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG{\<J}S\Otimes_{\!\!S}\,G \Otimes_{\!\!S}\>\>E,F)}\\ &\cong {\mathrm {Hom}}_{{\mathbf D}(S)}\<\<\bigl(G,\> {\mathbf R}{\mathrm {Hom}}_R^\bullet({\mathbf R}\GG{\<J}E,F)\bigr)\\ &\cong {\mathrm {Hom}}_{{\mathbf D}(S)}\<\<\bigl(G, {\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}_R^\bullet({\mathbf R}\GG{\<J}E,F)\bigr), \end{align*} which entail that the natural map is an isomorphism $$ {\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}_R^\bullet(E,F)\iso {\mathbf R}\GG{\<J}{\mathbf R}{\mathrm {Hom}}_R^\bullet({\mathbf R}\GG{\mspace{-.5mu}J}E,F). $$ \end{small} \setcounter{sth}{\value{numb}} Local Duality theorems are often formulated, as in (c) of the following, in terms of modules and local cohomology (${\mathrm H}_{\<\<J}^\bullet\!:={\mathrm H}^\bullet{\mathbf R}\GG {\mspace{-.5mu}J}$) rather than derived categories. \begin{sprop} \label{(2.2)} Let\/ $\varphi:R\to S$ be a homomorphism of noetherian rings, let\/ $J$~ be an\/ $S$-ideal, and suppose that there exists a sequence\/ ${\bf u}=(u_1,\dots,u_d)$ in\/ $J$ such that\/ $S/{\bf u}S$ is\/ $R$-finite. Then for any\/ $R$-finite module\/ $F$\textup{:} \smallskip \textup{(a)} ${\mathrm H}^n\ush{\varphi_{\<\!J}}\<F=0$ for all\/ $n<-d,$ \ so that there is a natural\/ ${\mathbf D}(S)$-map $$ h\colon ({\mathrm H}^{-d}\ush{\varphi_{\<\!J}}\<F)[d\>]\to \ush{\varphi_{\<\!J}}\<F. $$ \smallskip\pagebreak[2] \textup{(b)} If\/ $\tau_F^{\phantom{.}}\colon {\mathbf R}\GG {\mspace{-.5mu}J}\ush{\varphi_{\<\!J}}\<F\to F$ \vspace{1pt} corresponds in\/ {\rm (\ref{(2.1)})} to the identity map of\/~$\ush{\varphi_{\<\!J}}\<F,\,$% \footnote {$\,\tau_F^{\phantom{.}}$ may be thought of as $\text{``evaluation at 1"}\<\colon {\mathbf R}{\mathrm {Hom}}_{R,J}^\bullet(S,F)\to F$.% } and\/ $\int=\int_{\varphi\<,J}^d(F)$ is the composed map $$ {\mathbf R}\GG {\mspace{-.5mu}J}({\mathrm H}^{-d}\ush{\varphi_{\<\!J}}\<F)[d\>] \xrightarrow{\!{\mathbf R}\GG {\mspace{-.5mu}J}\<(h)\>} {\mathbf R}\GG {\mspace{-.5mu}J}\ush{\varphi_{\<\!J}}\<F \xrightarrow{\,\tau_F^{\phantom{.}}\>} F, $$ then\/ $\bigl({\mathrm H}^{-d}\ush{\varphi_{\<\!J}}\<F, \int \bigr)$ represents the functor\/ ${\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\GG {\mspace{-.5mu}J}E[d\>],F)$ of\/ $S$-modules~$E$. \smallskip \textup{(c)} If\/ $J\subset \,\root\of{{\bf u}S}\>$ then there is a bifunctorial isomorphism \textup(with $E$, $F$ as before\textup{):} $$ {\mathrm {Hom}}_S(E,{\mathrm H}^{-d}\ush{\varphi_{\<\!J}}\<F) \iso {\mathrm {Hom}}_R({\mathrm H}_{\<\<J}^dE,F). $$ \end{sprop} \begin{proof} If $\mspace{6.5mu}\hat{}\mspace{-6.5mu}\varphi$ is the obvious map from $R$ to the ${\mathbf u}$-adic completion $\hat S$ of~$S$, then in~${\mathbf D}(S)$, \mbox{$\ush{\varphi_{\<\!J}} F =\mspace{6.5mu} \hat{}\mspace{-6.5mu}\ush{\varphi_{\<\!J}} F$} since ${\mathbf R}\GG {\mspace{-.5mu}J}S={\mathbf R}\GG {\mspace{-.5mu}J}\hat S$. In proving (a), therefore, we may assume\vadjust{\kern.5pt} that $S$ is ${\bf u}$-adically complete, so that $\varphi$ factors as \smash{$R\stackrel{\psi\>}{\to}R[[{\bf t}]]\stackrel{\chi\>\>}{\to} S$} with ${\bf t}=(t_1,\dots,t_d)$ a sequence of indeterminates and $S$ finite over~$R[[{\bf t}]]$. ($\psi$ is the natural map, and $\chi(t_i)=u_i\>$.) In view of the easily-verified relation $\ush{\varphi_{\<\!J}} = \ush{\chi_{\<\<J}}\smcirc\ush{\psi_{{\bf t}R[[{\bf t}]]}}$, (\ref{2.1.2}) and~(\ref{2.1.3}) yield (a). Then (b) results from the natural isomorphisms $$ {\mathrm {Hom}}_S(E, {\mathrm H}^{-d}\ush{\varphi_{\<\!J}}\<F) \underset{\text{via } h }{\iso} {\mathrm {Hom}}_{{\mathbf D}(S) }(E[d\>], \ush{\varphi_{\<\!J}}\<F) \underset{\text{\eqref{(2.1)}}}{\iso} {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\Gamma_{\!\! J}^{\phantom{.}}E[d\>], F). $$ Finally, (c) follows from (b) because ${\mathrm H}^i_{\<\<J}E={\mathrm H}^i_{\<\mathbf uS}E=0$ for all $i>d$ (as one sees from the usual calculation of ${\mathrm H}^i_{\<\mathbf uS}E$ via Koszul complexes), so that the natural map is an isomorphism $ {\mathrm {Hom}}_{{\mathbf D}(R)}\<\<({\mathbf R}\Gamma_{\!\! J}^{\phantom{.}}E[d\>], F) \iso{\mathrm {Hom}}_R({\mathrm H}_{\<\<J}^dE,F). $ \end{proof} \begin{sparag}\label{HuK} Under the hypotheses of \Pref{(2.2)}(c), the functor~${\mathrm {Hom}}_R({\mathrm H}^d_{\<\<J}E,R)$ of $S$-modules~$E$ is representable. Under suitable extra conditions (for example, $\hat S$~a generic local complete intersection over~$R[[{\bf t}]]$, H\"ubl and Kunz represent this functor by a \emph{canonical} pair described explicitly via differential forms, residues, and certain trace maps \cite[p.~73, Theorem~3.4]{Integration}. For example, with \mbox{$S=R[[{\bf t}]]$,} $J=\mathbf tS$, and $\nu$~as in~(\ref{2.1.3}), the $S$-homomorphism from the module~\smash{$\widehat\Omega_{S/R}^d$} of universally finite \hbox{$d$-forms} to the relative canonical module $\>\omega_{R[[\mathbf t]]/R}^{\phantom{.}}\<=\<{\mathrm {Hom}}_R(\nu,R)$ sending the form $dt_1\dots dt_d$ to the $R$-homomorphism $\nu\to R$ which takes the monomial~ $t_1^{-1}\! \dots t_d^{-1}$ to~$1$ and all other monomials $t_1^{n_1}\!\dots t_d^{n_d}$ to $0$, is clearly an isomorphism; and the resulting isomorphism~$\smash{\widehat\Omega_{S/R}^d}[d\>]\iso\ush{\varphi_{\<\!J}} R$ \emph{does not depend on the $d$-element sequence\/~$\mathbf t$ generating} $J$---it corresponds under~(\ref{(2.1)}) to the {\it residue map } $$ {\mathbf R}\GG {\mspace{-.5mu}J}\widehat\Omega_{S/R}^d[d\>]= {\mathrm H}^d_{\!J}\>\widehat\Omega_{S/R}^d\to R $$ (see, e.g., \cite[\S2.7]{Hochschild}). Thus ${\mathrm {Hom}}_R({\mathrm H}_{\<\<J}^dE,R)$ \emph{is represented by\/ \smash{$\widehat\Omega_{S/R}^d$} together with the residue map.} The general case reduces to this one via traces of differential forms.\index{Duality!Local|)} \end{sparag} \end{parag} \smallskip \begin{parag} \label{sheafify}\index{Duality!Local, sheafified|(} (Formal sheafification of Local Duality). For $f\colon{\mathscr X}\to{\mathscr Y}$ as in \Tref{Th2} in \S\ref{S:prelim}, there is a right $\Delta$-adjoint~$f_{\mathrm t}^\times\<$ for the functor\/ ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\to\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})$. Furthermore, with $\>{\boldsymbol j}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} the canonical functor, we have $$ {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}{\boldsymbol j}\>{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\! \!\underset{\mathstrut(\text{\ref{C:vec-c is qc})}}\subset\! \!{\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})\! \!\underset{\mathstrut(\text{\ref{Gamma'(qc)})}}\subset\! \!{\mathbf R f_{\!*}}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\! \!\underset{\mathstrut(\text{\ref{Rf-*(qct)})}}\subset\! \!\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\! \!\underset{\mathstrut(\text{\ref{C:limsub})}}\subset\! \!\D_{\<\vc}({\mathscr Y}).\vspace{-1pt} $$ It results from \eqref{adj} and \Pref{A(vec-c)-A} that ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr Y})$ has the right $\Delta$-adjoint~${\mathbf R} Q_{\mathscr X}^{}\ush f\!:={\mathbf R} Q_{\mathscr X}^{}{\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,f_{\mathrm t}^\times$). \pagebreak[4] If, moreover, ${\mathscr X}$ is \emph{properly algebraic} (\Dref{D:propalg})---in particular, if ${\mathscr X}$ is affine---then ${\boldsymbol j}$ is an equivalence of categories (\Cref{corollary}), and so the functor ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\colon\D_{\<\vc}({\mathscr X})\to\D_{\<\vc}({\mathscr Y})$ has a right $\Delta$-adjoint. For \emph{affine $f\<$}, these results are closely related to the Local Duality isomor\-phisms \eqref{2.2.1} and ~\eqref{(2.1)}. Recall that an \emph{adic ring}\index{adic ring} is a pair $(R,I)$ with $R$ a ring and $I$ an $R$-ideal such that with respect to the $I\<$-adic topology $R$ is Hausdorff and complete. The topology on~$R$ having been specified, the corresponding affine formal scheme is denoted ${\mathrm {Spf}}(\<R)$. \begin{sprop}\label{P:affine} Let\/ $\varphi\colon(\<R,I\>)\to(S,J\>)$ be a continuous homomorphism of noetherian adic rings, and let\/ ${\mathscr X}\!:={\mathrm {Spf}}(S)\;\smash{\stackrel{f}{\to}}\;{\mathrm {Spf}}(\<R)=:\!{\mathscr Y}$ be the corresponding \textup(affine\/\textup) formal-scheme map. Let\/ $\kappa_{\mathscr X}^{\phantom{.}}\colon{\mathscr X}\to X\!:={\mathrm {Spec}}(S),$ $\kappa_{\mathscr Y}^{\phantom{.}}\colon{\mathscr Y}\to Y\!:={\mathrm {Spec}}(R)$ be the completion maps, and let $^\sim={}^{\sim_S}$ denote the standard exact functor from $S$-modules to quasi-coherent ${\mathcal O}_{\<\<X}$-modules. Then: \smallskip \textup{(a)} The restriction of\/ ${\mathbf R} f_{\!*}$ takes\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ to\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}),$\ and this restricted functor has a right adjoint $f_{\mathrm t}^\times\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\to\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ given by $$ f_{\mathrm t}^\times\<{\mathcal F} \!:=\kappa_{\mathscr X}^* \bigl(\varphi_{\!J}^\times\<{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>)\bigr)^\sim \<= \kappa_{\mathscr X}^*\bigl({\mathbf R}{\mathrm {Hom}}_{R,J}^\bullet\bigl(S, \>{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>)\bigr)\bigr)^\sim \qquad\bigl({\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\bigr). $$ \textup{(b)} The restriction of\/ ${\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}$ takes\/ $\D_{\<\vc}({\mathscr X})$ to\/ $\D_{\<\vc}({\mathscr Y}),$\ and this restricted functor has a right adjoint $\ush{f_{\<\vec{\mathrm c}}}\colon\D_{\<\vc}({\mathscr Y})\to\D_{\<\vc}({\mathscr X})$ given by $$ \ush{f_{\<\vec{\mathrm c}}}{\mathcal F} \!:=\kappa_{\mathscr X}^*\bigl(\ush{\varphi_{\<\!J}}{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>)\bigr)^\sim \<= \kappa_{\mathscr X}^*\bigl({\mathbf R}{\mathrm {Hom}}_R^\bullet\bigl({\mathbf R}\GG {\mspace{-.5mu}J}S, \>{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>)\bigr)\bigr)^\sim \qquad\bigl({\mathcal F}\in\D_{\<\vc}({\mathscr Y})\bigr). $$ \textup{(c)} There are natural isomorphisms \begin{align*} {\mathbf R}\Gamma({\mathscr X},f_{\mathrm t}^\times\<{\mathcal F}\>)&\iso \varphi_{\!J}^\times{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>) \qquad\bigl({\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\bigr),\\ {\mathbf R}\Gamma({\mathscr X},\ush{f_{\<\vec{\mathrm c}}}{\mathcal F}\>)&\iso \ush{\varphi_{\<\!J}}\>{\mathbf R}\Gamma({\mathscr Y},\>{\mathcal F}\>) \qquad\bigl({\mathcal F}\in\D_{\<\vc}({\mathscr Y})\bigr). \end{align*} \end{sprop} \begin{proof} The functor~${}^\sim$ induces an equivalence of categories ${\mathbf D}(S)\to\D_{\mkern-1.5mu\mathrm {qc}}(X)$, with quasi-inverse ${\mathbf R}\GG X\!:={\mathbf R}\Gamma(X,-)$ (\cite[p.\,225, Thm.\,5.1]{BN}, \cite[p.\,12, Proposition~(1.3)]{AJL}); and \Pref{c-erator} below implies that $\kappa_{\mathscr X}^*\colon\D_{\mkern-1.5mu\mathrm {qc}}(X)\to\D_{\<\vc}({\mathscr X})$ is~an equivalence, with quasi-inverse $({\mathbf R}\GG X\kappa_{{\mathscr X}*}^{\phantom{.}}-)^\sim=({\mathbf R}\GG{\mathscr X}-)^\sim\<$.\kern2pt \footnote{In checking this note that $\kappa_{{\mathscr X}\<*}^{\phantom{.}}$ has an exact left adjoint, hence preserves K-injectivity.% } It follows that \emph{the functor taking\/ \mbox{$G\in{\mathbf D}(S)$} to\/ $\kappa_{\mathscr X}^*\widetilde G$ is an equivalence, with quasi-inverse \smash{${\mathbf R}\GG{\mathscr X}\colon\D_{\<\vc}({\mathscr X})\to{\mathbf D}(S)$}}, and similarly for~${\mathscr Y}$~and~$R$. Moreover, there is an induced equivalence between ${\mathbf D}_{\!J\<}(S)$ and $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (see \Pref{Gammas'+kappas}). In particular, (c) follows from (a) and~(b).\looseness =-1 Corresponding to~\eqref{2.2.1} and ~\eqref{(2.1)} there are then functorial isomorphisms \begin{gather*} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<(\<{\mathcal E}\<,f_{\mathrm t}^\times\<{\mathcal F}\>)\!\<\iso\!\< {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<\<(\<\kappa_{\mathscr Y}^*({\mathbf R}\GG{\mathscr X}{\mathcal E})^{\sim_R}\<\<,\>{\mathcal F}\>) \qquad\bigl(\<{\mathcal E}\<\in\<\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),\;{\mathcal F}\<\in\<\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\bigr)\<,\\ {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<(\<{\mathcal E}\<,\ush{f_{\<\vec{\mathrm c}}}{\mathcal F}\>)\!\<\iso\!\< {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<\<(\<\kappa_{\mathscr Y}^*({\mathbf R}\GG{\<J}{\mathbf R}\GG{\mathscr X}{\mathcal E})^{\sim_R}\<\<,\>{\mathcal F}\>) \qquad\bigl(\<{\mathcal E}\<\in\<\D_{\<\vc}({\mathscr X}),\medspace \>{\mathcal F}\<\in\<\D_{\<\vc}({\mathscr Y})\bigr)\<; \end{gather*} and it remains to demonstrate functorial isomorphisms \begin{alignat*}{2} \kappa_{\mathscr Y}^*({\mathbf R}\GG{\mathscr X}{\mathcal E})^{\sim_R} &\iso {\mathbf R} f_{\!*}{\mathcal E} &&\qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr), \\ \kappa_{\mathscr Y}^*({\mathbf R}\GG {\mspace{-.5mu}J}{\mathbf R}\GG{\mathscr X}{\mathcal E})^{\sim_R} &\iso {\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}{\mathcal E} &&\qquad\bigl({\mathcal E}\in\D_{\<\vc}({\mathscr X})\bigr), \end{alignat*} the first a special case of the second. \pagebreak[3] To prove the second, let $E\!:={\mathbf R}\GG{\mathscr X}{\mathcal E}$, let $Z\!:={\mathrm {Spec}}(S/J\>)\subset X$, and let $f_0\colon X\to Y$ be the scheme-map corresponding to~$\varphi$. The desired isomorphism comes from the sequence of natural isomorphisms \begin{align*} {\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}{\mathcal E} &\cong {\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*\widetilde E \\ &\cong {\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathbf R}\iG Z\widetilde E &&\qquad(\textup{\Pref{Gammas'+kappas}(c)})\\ &\cong \kappa_{\mathscr Y}^*{\mathbf R} f_{0*}{\mathbf R}\iG Z\widetilde E &&\qquad(\textup{\Cref{C:kappa-f*t}})\\ &\cong \kappa_{\mathscr Y}^*{\mathbf R} f_{0*}({\mathbf R}\GG {\mspace{-.5mu}J}E)^\sim &&\qquad(\textup{\cite[p.\,9, (0.4.5)]{AJL}})\\ &\cong \kappa_{\mathscr Y}^*({\mathbf R}\GG {\mspace{-.5mu}J}E)^{\sim_R}. \end{align*} (The last isomorphism---well-known for bounded-below~$E$---can be checked via~the equivalences ${\mathbf R}\GG X$ and ${\mathbf R}\GG Y\>$, which satisfy ${\mathbf R}\GG Y{\mathbf R} f_{0*}\cong{\mathbf R}\GG X$ (see \cite[pp.\:142--143, 5.15(b) and~5.17]{Sp}). \end{proof} \medskip \deff_{\<\<\X*}'{f_{\<\<{\mathscr X}*}'} \Tref{T:pf-duality} below globalizes \Pref{(2.2)}.\index{Duality!Pseudo-finite|(} But first some preparatory remarks are needed. Recall from~\ref{maptypes} that a map $f\colon{\mathscr X}\to{\mathscr Y}$ of noetherian formal schemes is \emph{pseudo\kern.6pt-finite}\index{formal-scheme map!pseudo\kern.6pt-finite} if it is pseudo\kern.5pt-proper and has finite fibers, or equivalently, if $f$~is pseudo\kern.5pt-proper and \emph{affine}. Such an~$f$ corresponds locally to a homomorphism \mbox{$\varphi\colon(R,I\>)\to (S,J\>)$} of noetherian adic rings such that $\varphi(I)\subset J$ and $S/\<J$ is a finite \hbox{$R$-module.} This $\varphi$ can be extended to a homomorphism from a power series ring $R[[\mathbf t]]\!:= R[[t_1, t_2,\dots,t_e]]$ such that the images of the variables~$t_i$ together with~$\varphi(I\>)$ generate~$J\<$, and thereby $S$~becomes a finite $R[[\mathbf t]]$-module. Pseudo-finiteness is preserved under arbitrary (noetherian) base change. We say that a pseudo\kern.6pt-finite map $f\colon{\mathscr X}\to{\mathscr Y}$ of noetherian formal schemes has relative dimension $\le d$ if each $y\in{\mathscr Y}$ has an affine neighborhood~${\mathscr U}$ such that the map~ $\varphi_{\mathscr U}\colon R\to S$ of adic rings corresponding to $f^{-1}{\mathscr U}\to{\mathscr U}$ has a continuous extension $R[[t_1,\dots,t_d]]\to S$ making $S$ into a finite $R[[t_1,\dots,t_d]]$-module, or equivalently, there is a topologically nilpotent sequence $\mathbf u=(u_1,\dots,u_d)$ in~$S$ (i.e., $\lim_{n\to\infty}u_i^n=0\ (1\le i\le d)$) such that $S/\mathbf uS$ is finitely generated as an $R$-module. The \emph{relative dimension}~$\dim f$ is defined to be the least among the integers~$d$ such that $f$ has relative dimension $\le d$. For any pseudo\kern.6pt-proper map $f\colon{\mathscr X}\to{\mathscr Y}$ of noetherian formal schemes, we have the functor~$\ush f\colon{\mathbf D}({\mathscr Y})\to{\mathbf D}({\mathscr X})$ of \Cref{C:f*gam-duality}, commuting with open base change on~${\mathscr Y}$ (\Tref{Th4}). The next Lemma is a special case of \Pref{P:coherence}. \begin{slem}\label{L:coh} For a pseudo\kern.6pt-finite map \/ $f\<\colon\!{\mathscr X}\to\<{\mathscr Y}$ of noetherian formal schemes and for any\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y}),$\ it holds that $\ush f\<{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$. \end{slem} \begin{proof} Since $\ush f$ commutes with open base change, the question is local, so we may assume that $f$ corresponds to \mbox{$\varphi\colon(R,I\>)\to (S,J\>)$} as above. Moreover, the isomorphism $\ush{(gf)}\cong\ush f \<\ush g$ in \Cref{C:f*gam-duality} allows us to assume that \emph{either} \mbox{$S=R[[t_1,\dots,t_d]]$} and $\varphi$ is the natural map \emph{or} $S$ is a finite $R$-module and $J=IS$. In either case $f$ is obtained by completing a proper map $f_0\colon X \to {\mathrm {Spec}}(R)$ along a closed subscheme $Z\subset f_0^{-1}{\mathrm {Spec}}(R/I\>)$. (In the first case, take $X$ to be the projective space~$\mathbb P_{\!\!R}^{\mspace{.5mu}d}\supset{\mathrm {Spec}}(R[t_1,\dots,t_d])$, and $Z\!:={\mathrm {Spec}}(R[t_1,\dots,t_d]/(I,t_1,\dots,t_d))$.) The conclusion is given then by \Cref{C:completion-proper}. \end{proof} \pagebreak[3] \begin{sth}[\textup{Pseudo-finite Duality}]\label{T:pf-duality}% \index{Duality!Pseudo-finite|)} Let\/ $f\colon {\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.5pt-finite map of noetherian formal schemes, and let\/ ${\mathcal F}$ be a coherent\/ ${\mathcal O} _{\mathscr Y}$-module. Then: \textup{(a)} $H^n\ush f\<{\mathcal F}=0$ for all\/ $n<-\dim f$. \textup{(b)} If\/ $\dim f\le d$ and\/ ${\mathscr X}$ is covered by affine open subsets with\/ \hbox{$d$-generated} defining ideals, then with\/ $f_{\<\<\X*}'\!:= f_{\!*}\iGp{\mathscr X}$ and, for\/ $i\in\mathbb Z$ and\/ ${\mathscr J}$ a defining ideal of~${\mathscr X},$ $$ R^i\<\<f_{\<\<\X*}'\!:= H^i{\mathbf R}f_{\<\<\X*}'= H^i{\mathbf R f_{\!*}}\>{\mathbf R}\iGp{\mathscr X} =\smash{\dirlm{n} H^i{\mathbf R f_{\!*}}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n,-)},\footnotemark $$% \footnotetext {The equalities hold because ${\mathscr X}$ being noetherian, any \smash{$\subdirlm{}\!\!$}\vspace{.8pt} of flasque sheaves (for example, \smash{$\subdirlm{}\<\cH{om}({\mathcal O}_{\mathscr X}/{\mathscr J}^n,{\mathcal E})$}\vspace{.8pt} with ${\mathcal E}$ an injective ${\mathcal O}_{\mathscr X}$-module) is $f_{\!*}$-acyclic, and \smash{$\subdirlm{}\!\!$} commutes with~$f_{\!*}$. (For an additive functor $\phi\colon{\mathcal A}({\mathscr X})\to{\mathcal A}({\mathscr Y})$, an ${\mathcal A}({\mathscr X})$-complex ${\mathcal F}$ is \mbox{\emph{$\phi$-acyclic}}\index{acyclic} if the natural map $\phi{\mathcal F}\to{\mathbf R}\phi{\mathcal F}$ is a ${\mathbf D}({\mathscr Y})$-isomorphism. Using a standard spectral sequence, or otherwise (cf.~\cite[(2.7.2)]{Derived categories}), one sees that any bounded-below complex of $\phi$-acyclic ${\mathcal O}_{\mathscr X}$-modules is $\phi$-acyclic. }% there is, for quasi-coherent ${\mathcal O} _{\mathscr X}$-modules\/~${\mathcal E}\<,$ a functorial isomorphism $$ \postdisplaypenalty 10000 f_{\!*}\cH{om}_{{\mathscr X}}({\mathcal E}\<, \>H^{-d}\ush f \<{\mathcal F}\>)\iso \cH{om}_{{\mathscr Y}}(R^d\<\<f_{\<\<\X*}'{\mathcal E}\<,\>{\mathcal F}\>). $$ \textup(Here $H^{-d}\ush f\<{\mathcal F}$ is coherent \textup(\Lref{L:coh}\kern1pt\textup{),} and by \textup{(a),} vanishes unless $d=\dim f$.\textup) \end{sth} \begin{proof} Since $\ush f$ commutes with open base change we may assume that ${\mathscr Y}$ is affine and that $f$ corresponds to a map $\varphi\colon(R,I\>)\to(S,J\>)$ as in \Pref{(2.2)}. Then there is an isomorphism of functors $$ {\boldsymbol j}{\mathbf R} Q_{\mathscr X}^{}\ush f\cong \kappa_{\mathscr X}^*\bigl(\ush{\varphi_{\<\!J}}{\mathbf R}\Gamma({\mathscr Y},-)\bigr)^\sim\<, $$ both of these functors being right-adjoint to ${\mathbf R f_{\!*}}\>{\mathbf R}\iGp{\mathscr X}\colon\D_{\<\vc}({\mathscr X})\to\D_{\<\vc}({\mathscr Y})$ (\Pref{P:affine}(b) and remarks about right adjoints preceding it). Since \mbox{$\ush f\<{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$} (\Lref{L:coh}), therefore, by \Cref{corollary}, the natural map is an isomorphism ${\boldsymbol j}{\mathbf R} Q_{\mathscr X}^{}\ush f\<{\mathcal F}\iso\ush f\<{\mathcal F}\<$; and so, since $\kappa_{\mathscr X}^*$ is exact, \Pref{(2.2)} gives~(a). \smallskip\enlargethispage{-\baselineskip} Next, consider the presheaf map associating to each open ${\mathscr U}\subset{\mathscr Y}$ the natural composition (with ${\mathscr V}\!:= f^{-1}{\mathscr U}$):\vspace{1.5pt} \begin{align*} \smash{{\mathrm {Hom}}_{\mathscr V}({\mathcal E}\<, \>H^{-d}\ush f \<{\mathcal F}\>) \underset{\textup{by (a)}}{\iso} {\mathrm {Hom}}_{{\mathbf D}({\mathscr V})}\<\<({\mathcal E}[d\>], \>\ush f \<{\mathcal F}\>)} &\underset{\ref{C:f*gam-duality}}\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr U})}\<\<({\mathbf R f_{\!*}}\>{\mathbf R}\iGp{\mathscr X}{\mathcal E}[d\>], {\mathcal F}\>) \vspace{1.5pt}\\ &\,\longrightarrow\, {\mathrm {Hom}}_{{\mathscr U}}(R^d\<\<f_{\<\<\X*}'{\mathcal E}\<,\>{\mathcal F}\>). \end{align*} \penalty -1000 \noindent To prove (b) by showing that the resulting sheaf map $$ f_{\!*}\cH{om}_{{\mathscr X}}({\mathcal E}\<, \>H^{-d}\ush f\<{\mathcal F}\>)\to \cH{om}_{{\mathscr Y}}(R^d\<\<f_{\<\<\X*}'{\mathcal E}\<,\>{\mathcal F}\>) $$ is an isomorphism, it suffices to show that $R^i\<\<f_{\<\<\X*}'{\mathcal E}=0$ for all~$i>d$, a local problem for which we can (and do) assume that $f$ corresponds to $\varphi\colon R\to S$ as above. Now ${\mathbf R}\iGp{\mathscr X}{\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (\Pref{Gamma'(qc)}), so \Pref{Gammas'+kappas} for~\mbox{$X\!:={\mathrm {Spec}}(S)$} and $Z\!:={\mathrm {Spec}}(S/J)$ gives ${\mathbf R}\iGp{\mathscr X}{\mathcal E}\cong\kappa_{\mathscr X}^*{\mathcal E}_0$ with ${\mathcal E}_0\!:=\kappa_{{\mathscr X}*}^{\phantom{.}}{\mathbf R}\iGp{\mathscr X}{\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}^+(X)$.\vadjust{\kern.5pt} Since ${\mathscr X}$ has, locally, a $d$-generated defining ideal, we can represent~${\mathbf R}\iGp{\mathscr X}{\mathcal E}$ locally by a \smash{$\dirlm{}\!\!$}\vadjust{\kern1pt} of Koszul complexes on $d$ elements \cite[p.\,18, Lemma 3.1.1]{AJL}, whence $H^i{\mathbf R}\iGp{\mathscr X}{\mathcal E}=0$ for all~$i>d\>$, and so, $\kappa_{{\mathscr X}*}^{\phantom{.}}$ being exact, $H^i{\mathcal E}_0=0$. Since the map $f_0\!:={\mathrm {Spec}}(\varphi)$ is affine, it follows that $H^i{\mathbf R} f_{0*}{\mathcal E}_0=0$, whereupon, $\kappa_{\mathscr Y}$ being flat, \Cref{C:kappa-f*t} yields\looseness=-1 $$ \postdisplaypenalty 10000 R^i\<\<f_{\<\<\X*}'{\mathcal E} \cong H^i{\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathcal E}_0 \cong H^i\kappa_{\mathscr Y}^*{\mathbf R} f_{0*}{\mathcal E}_0 \cong \kappa_{\mathscr Y}^*H^i{\mathbf R} f_{0*}{\mathcal E}_0=0\qquad(i>d), $$ as desired. (Alternatively, use Lemmas~\ref{affine-maps} and~\ref{Gamma'+qc}.) \end{proof} \end{parag}\index{Duality!Local, sheafified|)} \smallskip \begin{parag}\label{residue thm}\index{Residue theorems|(} Our results provide a framework for ``Residue Theorems" such as those appearing in \cite[pp.~87--88]{Asterisque} and \cite[pp.~750-752]{HS} (central theorems in those papers): roughly speaking, Theorems~\ref{Th1} and~\ref{Th2} in section~\ref{S:prelim} include both local and global duality, and \Cref{C:kappa-f^times-tors} expresses the compatibility between these dualities. But the dualizing objects we deal with are determined \vadjust{\penalty-1000}only up to isomorphism. The Residue Theorems run deeper in that they include a \emph{canonical realization} of dualizing data, via differential forms. (See the above remarks on the H\"ubl-Kunz treatment of local duality.) This extra dimension belongs properly to a theory of the ``Fundamental Class" of a morphism, a canonical map from relative differential forms to the relative dualizing complex, which will be pursued in a separate paper. \begin{sparag} Let us be more explicit, starting with some remarks about ``Grothendieck Duality with supports" for a map $f\colon X\to Y$ of noetherian separated schemes with respective closed subschemes $W\subset Y$ and $Z\subset f^{-1}W$. Via the natural equivalence of categories ${\mathbf D}(\A_{\qc}(X))\to\D_{\mkern-1.5mu\mathrm {qc}}(X)$ (see \S\ref{SS:Dvc-and-Dqc}), we regard the functor~$f^\times\colon{\mathbf D}(Y)\to{\mathbf D}(\A_{\vec {\mathrm c}}(X))={\mathbf D}(\A_{\qc}(X))$ of \Tref{Th1} as being right-adjoint to ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm {qc}}(X)\to{\mathbf D}(Y)$.% \footnote {For ordinary schemes, this functor $f^\times$ is well-known, and usually denoted $f^!$ when $f$ is proper. When $f$ is an open immersion, the functors $f^\times$ and $f^! (=f^*)$ need not agree.% } The functor ${\mathbf R}\iGp Z$ can be regarded as being right-adjoint to the inclusion ${\mathbf D}_{\!Z}(X)\hookrightarrow{\mathbf D}(X)$ (cf.~\Pref{Gamma'(qc)}(c)); and its restriction to~$\D_{\mkern-1.5mu\mathrm {qc}}(X)$ agrees naturally with that of~${\mathbf R}\iG Z\>$, both restrictions being \hbox{right-adjoint} to the inclusion $\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)\hookrightarrow\D_{\mkern-1.5mu\mathrm {qc}}(X)$. Similar statements hold for $W\subset Y\<$. Since ${\mathbf R f_{\!*}}(\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X))\subset {\mathbf D}_W\<(Y)$ (cf.~ proof of \Pref{Rf-*(qct)}), we find that the functors ${\mathbf R}\iG Zf^\times$ and~${\mathbf R}\iG Zf^\times{\mathbf R}\iGp W$ are both right-adjoint to ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)\to {\mathbf D}(Y)$, so are isomorphic. We define the \emph{local integral} (a generalized residue map, cf.~\cite[\S4]{Integration}) $$ \rho({\mathcal G})\colon{\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G}\to {\mathbf R}\iGp W{\mathcal G} \qquad \bigl({\mathcal G}\in{\mathbf D}(Y)\bigr) $$ to be the natural composition $$ {\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G} \iso {\mathbf R f_{\!*}}\>{\mathbf R}\iG Zf^\times{\mathbf R}\iGp W{\mathcal G} \to {\mathbf R f_{\!*}} f^\times{\mathbf R}\iGp W{\mathcal G} \to {\mathbf R}\iGp W{\mathcal G}. $$ Noting that for ${\mathcal F}\in{\mathbf D}_W\<(Y)$ there is a canonical isomorphism ${\mathbf R}\iGp W{\mathcal F}\iso{\mathcal F}$ (proof similar to that of \Pref{Gamma'(qc)}(a)), we have then: \begin{sprop}[Duality with supports]\label{P:dual/supports}% \index{Grothendieck Duality!with supports} For\/ ${\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(\<X),$ ${\mathcal F}\<\in\<{\mathbf D}_W\<(Y),$ the natural composition \begin{align*} {\mathrm {Hom}}_{\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)}\<\<({\mathcal E}\<, {\mathbf R}\iG Z f^\times\<{\mathcal F}\>)\: &\smash{\xrightarrow[\phantom{\rho({\mathcal F}\>)}]{}} \:{\mathrm {Hom}}_{{\mathbf D}_W\<(Y)}\<\<({\mathbf R f_{\!*}}{\mathcal E}\<, {\mathbf R f_{\!*}}\>{\mathbf R}\iG Zf^\times{\mathcal F}\>)\\ &\smash{\xrightarrow[\rho({\mathcal F}\>)]{}} \:{\mathrm {Hom}}_{{\mathbf D}_W\<(Y)}\<\<({\mathbf R f_{\!*}}{\mathcal E}\<, \>{\mathcal F}\>)\vspace{-3pt} \end{align*} is an isomorphism. \end{sprop} This follows from adjointness of ${\mathbf R f_{\!*}}$ and $f^\times\<$, via the natural diagram $$ \begin{CD} {\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G} @>>>{\mathbf R f_{\!*}} f^\times{\mathcal G}\\ @V\rho({\mathcal G}) VV @VVV \\ {\mathbf R}\iGp W{\mathcal G} @>>>{\mathcal G} \end{CD} \qquad\bigl({\mathcal G}\in{\mathbf D}(Y)\bigr)\<, $$ whose commutativity is a cheap version of the Residue Theorem \cite[pp.~750-752]{HS}. Again, however, to be worthy of the name a Residue Theorem should involve \emph{canonical realizations} of dualizing objects. For instance, when $V$ is a proper \hbox{$d$-dimensional} variety over a field $k$ and $v\in V$ is a closed point, taking $X=V\<$, $Z=\{v\}$, $W=Y={\mathrm {Spec}}(k)$, ${\mathcal G}=k$, and setting $\omega_V\!:=H^{-d}f^\times k$, we get an ${\mathcal O}_{V\!,\>v}$-module~$\omega_{V\!,\>v}$ (commonly called ``canonical", though defined only up to isomorphism) together with the $k$-linear map induced by $\rho(k)$: \enlargethispage*{\baselineskip} $$ H_{\<v}^d(\omega_{V\!,\>v})\to k, $$ a map whose truly-canonical realization via differentials and residues is indicated in \cite[p.\,86,~(9.5)]{Asterisque}. \pagebreak[3] \end{sparag} \newcommand{{\R\hat f_{\!*}}}{{{\mathbf R}\hat f_{\!*}}} \begin{sparag} \label{completion} With preceding notation, consider the completion diagram $$ \begin{CD} X_{/Z}=:\,@.{\mathscr X} @>\kappa_{\mathscr X}^{\phantom{.}}>> X \\ @. @V \hat f VV @VV f V \\ Y_{\</W}=:\,@.{\mathscr Y} @>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle\kappa_{\mathscr Y}^{\phantom{.}}$}\vss}> Y \end{CD} $$ \vskip-3pt Duality with supports can be regarded\vspace{.6pt} more intrinsically---via $\hat f\<$ rather than~$f$---as a special case of the\index{Grothendieck Duality!Torsion (global)} Torsion-Duality \Tref{T:qct-duality} ($\>\cong\:$\Tref{Th2} of \S1) for~$\hat f$: First of all, the local integral $\rho$ is completely determined by $\kappa_{\mathscr Y}^*(\rho)$: for ${\mathcal G}\in{\mathbf D}({\mathscr Y})$, the natural map ${\mathbf R}\iGp W{\mathcal G}\to \kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*{\mathbf R}\iGp W{\mathcal G}$ is an isomorphism (\Pref{Gammas'+kappas}); and the same holds for ${\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G}\to \kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*{\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G}$ since as above, $$ {\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G}\in{\mathbf R f_{\!*}}(\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X))\subset{\mathbf D}_W\<(Y) $$ ---and so $\rho=\kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*(\rho)$. Furthermore, $\kappa_{\mathscr Y}^*(\rho)$ is determined by the ``trace" map $\tau_{\<\mathrm t}^{}\colon {\R\hat f_{\!*}}\hatf_{\mathrm t}^\times\to\mathbf 1$, \index{ {}$\tau$ (trace map)!$\tau_{\<\mathrm t}$} as per the following natural commutative diagram, whose rows are isomorphisms: \begin{small} $$ \minCDarrowwidth=18pt \begin{CD} \kappa_{\mathscr Y}^*{\mathbf R f_{\!*}}\>{\mathbf R}\iG Z f^\times{\mathcal G}@>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\ref{C:kappa-f*t} > {\R\hat f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG Z f^\times \kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*{\mathcal G} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\ref{C:kappa-f^times-tors}> {\R\hat f_{\!*}}\hatf_{\mathrm t}^\times\kappa_{\mathscr Y}^*{\mathcal G} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\hbox to 16pt{$\scriptstyle\hss\ref{C:identities}\textup{(b\kern-.6pt)}\hss$}> {\R\hat f_{\!*}}\hatf_{\mathrm t}^\times{\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*{\mathcal G}\\ \vspace{-21pt}\\ @V\kappa_{\mathscr Y}^*(\rho) VV @. @. @VV\tau_{\<\mathrm t}^{\phantom{.}} V\\ \kappa_{\mathscr Y}^*{\mathbf R}\iGp W{\mathcal G} @. \hbox to 0pt{\kern75.5 pt \hss$ \overset{\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}} {\underset{\ref{Gammas'+kappas}} {\hbox to 215.5 pt{\leftarrowfill}} } $\hss } @. @. {\mathbf R}\iGp {\mathscr Y}\kappa_{\mathscr Y}^*{\mathcal G} \end{CD} $$ \end{small} (To see that the natural map ${\mathbf R}\iG Zf^\times{\mathcal G}\to{\mathbf R}\iG Zf^\times\kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*{\mathcal G}$ is an isomorphism, replace~ ${\mathbf R}\iG Zf^\times$ by the isomorphic functor~${\mathbf R}\iG Zf^\times{\mathbf R}\iGp W$ and apply \Pref{Gammas'+kappas}.) Finally, we have isomorphisms (for ${\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$, ${\mathcal F}\in{\mathbf D}_W\<(Y)$), \begin{alignat*}{2} {\mathrm {Hom}}_{{\mathbf D}(X)}\<\<({\mathcal E}\<, {\mathbf R}\iG Z f^\times\<{\mathcal F}\>) &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<(\kappa_{\mathscr X}^*{\mathcal E}\<, \>\kappa_{\mathscr X}^*{\mathbf R}\iG Z f^\times\kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*{\mathcal F}\>)\qquad &&(\ref{Gammas'+kappas})\\ &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<(\kappa_{\mathscr X}^*{\mathcal E}\<,\> \smash{\hatf_{\mathrm t}^\times\<} \kappa_{\mathscr Y}^*{\mathcal F}\>)\qquad &&(\ref{C:kappa-f^times-tors}) \\ &\iso{\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<\<({\mathbf R}\smash{\hat f_{\!*}}\kappa_{\mathscr X}^*{\mathcal E}\<,\> \kappa_{\mathscr Y}^*{\mathcal F}\>)\qquad &&(\ref{T:qct-duality}) \\ &\iso{\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<\<(\kappa_{\mathscr Y}^*{\mathbf R f_{\!*}}{\mathcal E}\<,\> \kappa_{\mathscr Y}^*{\mathcal F}\>)\qquad &&(\ref{C:kappa-f*t}) \\ &\iso{\mathrm {Hom}}_{{\mathbf D}(Y)}\<\<({\mathbf R f_{\!*}}{\mathcal E}\<,\>{\mathcal F}\>)\qquad &&(\ref{Gammas'+kappas}), \end{alignat*} whose composition can be checked, via the preceding diagram, to be the same as the isomorphism of \Pref{P:dual/supports}. \end{sparag} \def\Hp#1#2{{\mathrm H}_{#1}^{\<\prime\>#2}} \begin{sparag}\label{consequences} \Pref{P:astrix10.2} expresses some homological consequences of the foregoing dualities, and furnishes a general context for \cite[pp.~87--88, Theorem (10.2)]{Asterisque}.\looseness=-1 For any noetherian formal scheme~${\mathscr X}$, ${\mathcal E}\in{\mathbf D}({\mathscr X})$, and $n\in\mathbb Z$, set $$ \Hp{\mathscr X} n({\mathcal E})\!:={\mathrm H}^n{\mathbf R}\Gamma({\mathscr X},{\mathbf R}\iGp{\mathscr X}{\mathcal E}). $$ For instance, if ${\mathscr X}=X_{/Z}\xrightarrow{\,\kappa\,} X$ is the completion of a noetherian scheme~$X$ along a closed~$Z\subset X$, then for ${\mathcal F}\in{\mathbf D}(X)$, \Pref{Gammas'+kappas} yields natural isomorphisms \begin{align*} {\mathbf R}\Gamma({\mathscr X},{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F}\>) &= {\mathbf R}\Gamma(X,\kappa_*{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F}\>) \\ &\cong {\mathbf R}\Gamma(X,\kappa_*\kappa^*{\mathbf R}\iGp Z{\mathcal F}\>) \cong {\mathbf R}\Gamma(X,{\mathbf R}\iGp Z{\mathcal F}\>), \end{align*} and so if ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$, then with ${\mathrm H}_Z^\bullet$ the usual cohomology with supports in~$Z$, $$ \Hp{\mathscr X} n(\kappa^*\<{\mathcal F}\>)\cong{\mathrm H}_Z^n({\mathcal F}\>). $$ Let ${\mathscr J}\subset{\mathcal O}_{\mathscr X}$ be an ideal of definition. Writing $\Gamma_{\!{\mathscr X}}^{}$ for the functor $\Gamma({\mathscr X},-)$, we have a functorial map $$ \gamma({\mathcal E})\colon{\mathbf R}(\Gamma_{\!{\mathscr X}}^{}\<\smcirc\iGp{\mathscr X}){\mathcal E}\to {\mathbf R}\Gamma_{\!{\mathscr X}}^{}\<\smcirc{\mathbf R}\iGp{\mathscr X}\>{\mathcal E}\qquad \bigl({\mathcal E}\in{\mathbf D}({\mathscr X})\bigr), $$ which is an \emph{isomorphism} when ${\mathcal E}$ is bounded-below, since for any injective ${\mathcal O}_{\mathscr X}$-module ${\mathcal I}$, \smash{$\dirlm{}\!\<_i$} of the flasque modules $\cH{om}({\mathcal O}_{\mathscr X}/{\mathscr J}^i,{\mathcal I}\>)$ is $\Gamma_{\!{\mathscr X}}^{}$-acyclic. Whenever $\gamma({\mathcal E})$ is an isomorphism, the induced homology maps are isomorphisms $$ \dirlm{i}\text{Ext}^n({\mathcal O}_{\mathscr X}/{\mathscr J}^i\<,\>{\mathcal E}) \iso\Hp{\mathscr X} n({\mathcal E}). $$ \smallskip If ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})$, then ${\mathbf R}\iGp{\mathscr X}{\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (\Pref{Gamma'(qc)}). For any map $g\colon{\mathscr X}\to{\mathscr Y}$ satisfying the hypotheses of \Tref{T:qct-duality}, for ${\mathcal G}\in{\mathbf D}({\mathscr Y})$, and with $R\!:={\mathrm H}^0({\mathscr Y},{\mathcal O}_{\mathscr Y})$, there~are natural maps \begin{equation}\label{map} \begin{aligned} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,g_{\mathrm t}^{\<\times}\<{\mathcal G}) &\iso\< {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,g_{\mathrm t}^{\<\times}{\mathbf R}\iGp{\mathscr Y}{\mathcal G}) &&\qquad(\ref{C:identities}(\textup b)) \\ &\iso\< {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<\<({\mathbf R} g_*{\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,{\mathbf R}\iGp{\mathscr Y}{\mathcal G})\\ &\,\longrightarrow\mspace{1.2mu} {\mathrm {Hom}}_R(\Hp{\mathscr X} n{\mathcal E}, \Hp{\mathscr Y} n{\mathcal G}) && \end{aligned} \end{equation} where the last map arises via the functor ${\mathrm H}^n{\mathbf R}\Gamma({\mathscr Y},-)\ (n\in\mathbb Z)$. In particular, if $g=\hat f$ in the completion situation of \S\ref{completion}, and if ${\mathcal E}\!:=\kappa_{\mathscr X}^*{\mathcal E}_0\>$, ${\mathcal G}=\kappa_{\mathscr Y}^*{\mathcal G}_0\ ({\mathcal E}_0\in\D_{\mkern-1.5mu\mathrm {qc}}(X),\;{\mathcal G}_0\in\D_{\mkern-1.5mu\mathrm {qc}}(Y))$, then preceding considerations show that this composed map operates via Duality with Supports for $f$ (\Pref{P:dual/supports}), i.e., it can be identified with the natural composition \begin{align*} {\mathrm {Hom}}_{{\mathbf D}(X)}\<\<({\mathbf R}\iG Z{\mathcal E}_0, {\mathbf R}\iG Zf^\times{\mathcal G}_0) &\underset{\ref{P:dual/supports}}{\iso}\< {\mathrm {Hom}}_{{\mathbf D}(Y)}\<\<({\mathbf R f_{\!*}}\>{\mathbf R}\iG Z{\mathcal E}_0, {\mathbf R}\iG W{\mathcal G}_0) \\ &\,\longrightarrow\mspace{1.2mu} {\mathrm {Hom}}_{{\mathrm H}^0(Y,{\mathcal O}_Y)\<} ({\mathrm H}_Z^n{\mathcal E}_0, {\mathrm H}^n_W{\mathcal G}_0). \end{align*} \penalty -1500 \begin{sparag} Next, let $R$ be a complete noetherian local ring topologized as usual by its maximal ideal~$I$, let $(S,J)$ be a noetherian adic ring, let $\varphi\colon (R,I)\to (S,J)$ be a continuous homomorphism, and let $$ {\mathscr Y}\!:={\mathrm {Spf}}(S)\xrightarrow{\,f\,}{\mathrm {Spf}}(R)=:{\mathscr V} $$ be the corresponding formal-scheme map. As before, $g\colon{\mathscr X}\to{\mathscr Y}$ is a map as in \Tref{T:qct-duality}, and we set $h\!:= fg$. Since the underlying space of~${\mathscr V}$ is a single point, at which the stalk of~${\mathcal O}_{\mathscr V}$ is just~$R$, therefore the categories of ${\mathcal O}_{\<{\mathscr V}}$-modules and of $R$-modules are identical, and accordingly, for any ${\mathcal E}\in {\mathbf D}({\mathscr X})$ we can identify ${\mathbf R} h_*{\mathcal E}$ with ${\mathbf R}\Gamma({\mathscr X},{\mathcal E})\in{\mathbf D}(R)$. Let $K$ be an injective $R$-module, and ${\mathcal K}$ the corresponding injective ${\mathcal O}_{\<{\mathscr V}}$-module. There exist integers $r$, $s$ such that $H^i(\ush f {\mathcal K})=0$ for all $i<-r$ (resp.~$H^i(\ush h {\mathcal K})=0$ for all $i<-s$) (\Cref{C:f*gam-duality}). Set $\omega_{\mathscr Y}\!:= H^{-r}(\ush f {\mathcal K})$ (resp.~$\omega_{\mathscr X}\!:= H^{-s}(\ush h {\mathcal K})$). \end{sparag} \begin{sprop}\label{P:astrix10.2} In the preceding situation\/ $\omega_{\mathscr X}$ represents---via \eqref{map}---the functor\/ ${\mathrm {Hom}}_S(\Hp{\mathscr X} {\<\raisebox{.1ex}{$\scriptstyle s$}}{\mathcal E}\<,\Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptscriptstyle 0$}}(\ush f{\mathcal K})) $ of quasi-coherent ${\mathcal O}_{\mathscr X}$-modules~${\mathcal E}\<$. If\/~$\omega_{\mathscr Y}$ is the only non-zero homology of\/~$\ush f{\mathcal K},$\ this functor is isomorphic to\/ ${\mathrm {Hom}}_S(\Hp{\mathscr X} {\<\raisebox{.1ex}{$\scriptstyle s$}}{\mathcal E}\<,\Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptstyle r$}}\omega_{\mathscr Y})$. \looseness=2 \end{sprop} \emph{Proof.} There are natural maps $$ \postdisplaypenalty10000 \Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptstyle r$}}(\omega_{\mathscr Y})=\Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptscriptstyle 0$}}(\omega_{\mathscr Y}[r])\xrightarrow{\ h\ } \Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptscriptstyle 0$}}(\ush f{\mathcal K})\iso{\mathrm {Hom}}_{R,J}(S,K) $$ where the last isomorphism results from \Pref{P:affine}(a), in view of the identity \mbox{${\mathbf R}\iGp{\mathscr Y}\ush f=f_{\mathrm t}^\times$} (\Cref{C:identities}(a)) and the natural isomorphisms $$ \postdisplaypenalty 10000 {\mathbf R}\Gamma({\mathscr Y}, \kappa_{\mathscr Y}^*\widetilde G)\iso {\mathbf R}\Gamma(Y, \kappa_{{\mathscr Y}*}^{\phantom{.}}\kappa_{\mathscr Y}^*\widetilde G) \underset{\ref{Gammas'+kappas}}\iso {\mathbf R}\Gamma(Y, \widetilde G)\iso G \qquad\bigl(G\in{\mathbf D}_{\<J}^+(S)\bigr), $$ for $G\!:={\mathbf R}{\mathrm {Hom}}^{\bullet}_{R,J}(S,{\mathbf R}\Gamma({\mathscr V},{\mathcal K}))$. (In fact ${\mathbf R}\Gamma({\mathscr Y}, \kappa_{\mathscr Y}^*\widetilde G)\cong G$ for any \mbox{$G\in{\mathbf D}(S)$,} see \Cref{(3.2.3)} and the beginning of \S\ref{SS:Dvc-and-Dqc}.) In case $\omega_{\mathscr Y}$ is the only non-vanishing homology of $\ush f{\mathcal K}$, then $h$ is an isomorphism too. The assertions follow from the (easily checked) commutativity, for any quasi-coherent ${\mathcal O}_{\mathscr X}$-module~${\mathcal E}\<$, of the diagram $$ \minCDarrowwidth=22pt \mkern110mu \begin{CD} \hbox to0pt{\hss ${\mathrm {Hom}}_{\mathscr X}({\mathcal E}\<,\omega_{\mathscr X})=\!\!\<=\:$} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<({\mathcal E}[s],\>\ush g\<\ush f{\mathcal K}) @>\<\!\!\!\textup{~\ref{C:identities}(a)}\mkern.5mu >> {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<\<({\mathbf R}\iGp{\mathscr X}{\mathcal E}[s],\>{g_{\mathrm t}^\times}\!\ush f{\mathcal K})\\ @V\simeq VV @VV\eqref{map}V \\ {\mathrm {Hom}}_{{\mathbf D}({\mathscr V})}\<\<({\mathbf R} h_*{\mathbf R}\iGp{\mathscr X}{\mathcal E}[s],{\mathcal K}) @.{\mathrm {Hom}}_S\bigl(\Hp{\mathscr X} {\raisebox{.1ex} {$\scriptscriptstyle 0$}}({\mathcal E}[s]),\Hp{\mathscr Y} {\raisebox{.1ex}{$\scriptscriptstyle 0$}}(\ush f{\mathcal K})\bigr)\\ @| @VV\simeq V \\ {\mathrm {Hom}}_R(\Hp{\mathscr X} {\<\raisebox{.1ex}{$\scriptstyle s$}}{\mathcal E}\<,K) @>\;\ \vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}\ \;>> {\mathrm {Hom}}_S\bigl(\Hp{\mathscr X} {\<\raisebox{.1ex}{$\scriptstyle s$}}{\mathcal E}\<,{\mathrm {Hom}}_{R,J}(S,K)\bigr) \end{CD} $$ \end{sparag} \begin{sparag} Now let us fit \cite[pp.~87--88, Theorem (10.2)]{Asterisque} into the preceding setup. The cited Theorem has both local and global components. The first deals with maps $\varphi\colon R\to S$ of local domains essentially of finite type over a perfect field~$k$, with residue fields finite over~$k$. To each such ring~$T$ one associates the canonical module $\omega_T$ of ``regular" $k$-differentials of degree~$\dim T$. Under mild restrictions on~$\varphi$, the assertion is that the functor $$ \postdisplaypenalty 10000 {\mathrm {Hom}}_R\bigl(\textup H_{m\<\<_{\hat S}}^{\dim\mkern-1.5mu S}G,\> \>\textup H_{m\<_{\lower.3ex\hbox{$\scriptscriptstyle\<R$}}}^ {\dim\mkern-1.5mu R}\> \omega_{\<R}\bigr) \qquad(m\!:=\text{maximal ideal}) $$ of $\hat S$-modules~$G$ is represented by the completion $\widehat{\omega_S}$ together with a canonical~map, the {\it relative residue} $$ \rho_\varphi\colon \textup H_{m\<\<_{\hat S}}^{\dim\mkern-1.5mu S}\widehat{\omega_S\>} =\textup H_{m\<_{S}}^{\dim\mkern-1.5mu S\>}\omega_S\to \textup H_{m\<_{R}}^{\dim\mkern-1.5mu R\>}\>\omega_{\<R}. $$ This may be viewed as a consequence of {\it concrete\/} local duality over~$k$ (\S\ref{HuK}). The global aspect concerns a proper map of irreducible $k$-varieties $g\colon V\to W$ of respective dimensions $s$ and $r$ with all fibers over codimension~1 points of~$W$ having dimension $s-r$, a closed point $w\in W\<$, the fiber $E\!:= g^{-1}(w)$, and the completion $\widehat V\!:= V_{/E}$. The assertion is that the functor $$ \postdisplaypenalty 10000 {\mathrm {Hom}}_R\bigl(\textup H_{\widehat V}^{\prime s}\>{\mathcal G},\mkern1.5mu \textup H_{m\<\<_R}^{r}\omega_{\<R}\bigr)\qquad (R\!:= {\mathcal O}_{W\mkern-1.5mu,\>w}) $$ of coherent ${\mathcal O}_{\widehat V}$-modules~${\mathcal G}$ is represented by the completion~$\widehat{\omega_V}$ along~$E$ of the canonical sheaf $\omega_V$ of regular differentials, together with a canonical map $$ \theta\colon \textup H_{\widehat V}^{\prime s}\>\widehat{\omega_V}= \textup H_{\<E}^{s}\>\omega_V\to \textup H_{m_{\<\<R}}^{r}\omega_{\<R}\>. $$ Moreover, the local and global representations are {\it compatible\/} in the sense that if $v\in E$ is any closed point and $\varphi_v\colon R\to S\!:= {\mathcal O}_{V\!,v}$ is the canonical map, then the residue $\rho_v\!:=\rho_{\varphi_v}$ factors as the natural map $\textup H_{m_{\<S}}^s\omega_S\to \textup H_{\<E}^{s}\>\omega_V$ followed by~$\theta$. This compatibility determines $\theta$ uniquely if the $\rho_v\ (v\in E)$ are given \cite[p.\,95, (10.6)]{Asterisque}; and of course conversely. Basically, all this---\emph{without the explicit description of the $\omega\<$'s and the maps\/~$\rho_v$ via differentials and residues}---is contained in \Pref{P:astrix10.2}, as follows. In the completion situation of \S\ref{completion}, take $X$ and $Y$ to be finite-type separated schemes over an artinian local ring~$R$, of respective pure dimensions $s$ and $r$, let $W=\{w\}$ with $w$ a closed point of~$Y\<$, write $g$ in place of~$f$, and assume that $Z\subset g^{-1}W$ is proper over~$R$ (which is so, e.g., if $g$ is proper and $Z$ is closed). Let~$K$ be an injective hull of the residue field of~$R$, and let ${\mathcal K}$ be the corresponding injective sheaf on ${\mathrm {Spec}}(R)={\mathrm {Spf}}(R)$. With $f\colon Y\to {\mathrm {Spec}}(R)$ the canonical map, and $h=fg$, define the \emph{dualizing sheaves} $$ \omega_X\!:=H^{-s}h^!{\mathcal K}, \qquad \omega_Y\!:=H^{-r}f^!{\mathcal K}, $$ where $h^!$ is the Grothendieck duality functor (compatible with open immersions, and equal to $h^\times$ when $h$ is proper), and similarly for~$f^!\<$. It is well-known (for example via a local factorization of $h$ as $\text{smooth}\smcirc\text{finite}$) that $h^!{\mathcal K}$ has coherent homology, vanishing in all degrees $<-s\>$; and similarly $f^!{\mathcal K}$ has coherent homology, vanishing in all degrees $<-r$. Let $$ \hat f\colon{\mathscr Y}\!:={\mathrm {Spf}}(\widehat{{\mathcal O}_{W\<,\>w}})\to{\mathrm {Spf}}(R)=:{\mathscr V} $$ be the completion of $f$. We may assume, after compactifying $f$ and $g$---which\vadjust{\kern.7pt} does not affect $\hat f$ or~$\hat g$ (see~\cite{Lu}), that $f$ and $g$ are proper maps.\vadjust{\kern.4pt} Then \Cref{C:completion-proper} shows that $\ush {\hat h}{\mathcal K}=\kappa_{\mathscr X}^*h^!{\mathcal K}$, and so $\kappa_{\mathscr X}$ being flat, we see that \begin{equation}\label{omega} \kappa_{\mathscr X}^*\>\omega_X=\omega_{\mathscr X} \end{equation} where $\omega_{\mathscr X}$ is as in \Pref{P:astrix10.2}; and similarly $\kappa_{\mathscr Y}^*\omega_Y=\omega_{\mathscr Y}\>$. Once again, some form of the theory of the Fundamental Class will enable us to represent $\omega_X$ by means of regular differential forms; and then both the local and global components of the cited Theorem~(10.2) become special cases of \Pref{P:astrix10.2} (modulo some technicalities \cite[p.\,89, Lemma (10.3)]{Asterisque} which allow a weakening of the condition that $\omega_{\mathscr Y}$ be the only non-vanishing homology of~$\ush{\hat f}{\mathcal K}$). As for the local-global compatibility, consider quite generally a pair of maps $$ {\mathscr X}_1\xrightarrow{q\,}{\mathscr X}\xrightarrow{p\,}{\mathscr Y} $$ of noetherian formal schemes. In the above situation, for instance, we could take~$p$ to be $\hat g$, ${\mathscr X}_1$ to be the completion of~$X$ at a closed point $v\in Z$, and $q$ to be the natural map. \Tref{Th2} gives us the adjunction\vadjust{\kern1pt} $$ \lower.3ex\hbox{$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$}\ \, \vbox to0pt{ \vss \hbox{$\xrightarrow{\<{\mathbf R} p_*\>}$} \vspace{-7pt} \hbox{$\xleftarrow[\smash{\;{p_{\mathrm t}^{\<\times}}\;}]{}$} \vss} \ \,\lower.3ex\hbox{$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})$}. $$ \medskip\vspace{1pt} \noindent The natural isomorphism ${\mathbf R}(pq)_*\iso{\mathbf R} p_*{\mathbf R} q_*$ gives rise then to an adjoint isomorphism $q_{\mathrm t}^{\<\times}p_{\mathrm t}^{\<\times}\iso (pq)_{\mathrm t}^{\<\times}$; and for ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})$ the natural map ${\mathbf R}(pq)_*(pq)_{\mathrm t}^{\<\times}{\mathcal E}\to{\mathcal E}$ factors as $$ {\mathbf R}(pq)_*(pq)_{\mathrm t}^{\<\times}{\mathcal E}\iso {\mathbf R} p_*{\mathbf R} q_*q_{\mathrm t}^{\<\times}p_{\mathrm t}^{\<\times}{\mathcal E}\to {\mathbf R} p_*p_{\mathrm t}^{\<\times}{\mathcal E}\to{\mathcal E}. $$ This factorization contains the compatibility between the above maps $\theta$ and $\rho_v\>$, as one sees by interpreting them as homological derivatives of maps of the type ${\mathbf R} p_*p_{\mathrm t}^{\<\times}{\mathcal E}\to{\mathcal E}$ (with ${\mathcal E}\!:={\mathbf R}\iGp{\mathscr Y}\ush{\hat f}{\mathcal K})$. Details are left to the reader. \end{sparag} \begin{srem}\label{R:d-vein} In the preceding situation, suppose further that $Y={\mathrm {Spec}}(R)$ (with $R$~artinian) and $f=\text{identity}$, so that $h=g\colon X\to Y$ is a finite-type separated map, $X$ being of pure dimension $s$, and $\kappa_{\mathscr X}\colon{\mathscr X}\to X$ is the completion of~$X$ along a closed subset~$Z$ proper over~$Y\<$. Again, $K$ is an injective $R$-module, ${\mathcal K}$ is the corresponding ${\mathcal O}_Y$-module, and $\omega_X\!:= H^{-s}g^!{\mathcal K}$ is a ``dualizing sheaf\kern1.5pt" on~$X$. Now~\Pref{P:astrix10.2} is just the instance $i=s$ of the canonical isomorphisms, for~${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X}),\;i\in\mathbb Z$ (and with $\Hp{\mathscr X}\bullet\!:= {\mathrm H}{}^\bullet{\mathbf R}\Gamma({\mathscr X},{\mathbf R}\iGp{\mathscr X})$, see \S\ref{consequences}, and $\hat g\!:= g\smcirc\kappa_{\mathscr X}$): $$ \!\!{\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<({\mathcal E}[i],\ush {\hat g}{\mathcal K}) \underset{\textup{Thm.\,\ref{Th2}}}{\iso} {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R}\hat g_{\<*}{\mathbf R}\iGp{\mathscr X}{\mathcal E}[i],{\mathcal K})\<\iso\< {\mathrm {Hom}}_R(\Hp{\mathscr X} i\>{\mathcal E}\<,K) =: \!(\Hp{\mathscr X} i\>{\mathcal E})\mspace{.5mu}\check{}\>\>. $$ If $X$ is Cohen-Macaulay then all the homology of $g^!{\mathcal K}$ other than $\omega_X$ vanishes, so all the homology of $\ush{\hat g}{\mathcal K}\cong \kappa_{\mathscr X}^*g^!{\mathcal K}$ other than $\omega_{\mathscr X}=\kappa_{\mathscr X}^*\omega_X$ vanishes (see \eqref{omega}), and the preceding composed isomorphism becomes $$ \text{Ext}_{\mathscr X}^{s-i}({\mathcal E}\<,\omega_{\mathscr X})\iso (\Hp{\mathscr X} i\>{\mathcal E})\mspace{.5mu}\check{}\>\>. $$ In particular, when $Z=X$ (so that ${\mathscr X}=X$) this is the usual duality isomorphism $$ \text{Ext}_X^{s-i}({\mathcal E}\<,\omega_X)\iso {\mathrm H}^i(X,{\mathcal E})\mspace{.5mu}\check{}\>\>. $$ If $X$ is Gorenstein and ${\mathcal F}$ is a locally free ${\mathcal O}_{\mathscr X}$-module of finite rank, then $\omega_X$~is invertible; and taking ${\mathcal E}\!:=\cH{om}_{\mathscr X}({\mathcal F},\omega_{\mathscr X})=\check {\mathcal F}\otimes\omega_{\mathscr X}$ we get the isomorphism $$ {\mathrm H}^{s-i}({\mathscr X}, {\mathcal F}\>) \iso \bigl(\Hp{\mathscr X} i(\check{\mathcal F}\otimes\omega_{\mathscr X})\bigr) \mspace{.8mu}\check{}\>\>, $$ which generalizes the Formal Duality theorem\index{Duality!Formal|(} \cite[p.\,48, Proposition~(5.2)]{De-Rham-cohomology}. \end{srem} \end{parag}\index{Residue theorems|)} \smallskip \penalty -1200 \begin{parag} \label{bf (d)} Both \cite[p.\,48; Proposition~(5.2)]{De-Rham-cohomology} (Formal Duality) and the Theorem in \cite[p.\,188]{Desingularization} (Local-Global Duality)\index{Duality!Local-Global} are contained in \Pref{(2.8)}, see \cite[\S5.3]{AJL}. Let $R$ be a noetherian ring, discretely topologized, and set $$ Y\!:= {\mathrm {Spec}}(R)={\mathrm {Spf}} (R)=:\<{\mathscr Y}. $$ Let $g\colon X\to Y$ be a finite-type separated map, let $Z\subset X$ be \emph{proper} over~$Y\<$, let $\kappa\colon{\mathscr X}=X_{/Z}\to X$ be the completion of~$X$ along~$Z$, and set $\hat g\!:= g\smcirc\kappa\colon{\mathscr X}\to {\mathscr Y}$. Assume that $R$ has a \emph{residual complex} ${\mathcal R}$ \cite[p.\,304]{H1}. Then the corresponding quasi-coherent ${\mathcal O}_Y$-complex \smash{${{\mathcal R}}_Y\!:= {\widetilde {{\mathcal R}}}$} is a \emph{dualizing complex,} and ${{\mathcal R}}_X\!:= g^!{{\mathcal R}}_Y$ is a dualizing complex on~$X$ \cite[p.~396, Corollary~3]{f!}. For any ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}(X)$ set $$ {\mathcal F}^{\>\prime}\!:={\mathbf R}\cH{om}^{\bullet}_ X({\mathcal F},{{\mathcal R}}_X)\in\D_{\mkern-1.5mu\mathrm c}(X), $$ so that ${\mathcal F}\cong{\mathcal F}^{\>\prime}{}'={\mathbf R}\cH{om}^{\bullet}_ X({\mathcal F}^{\>\prime}\<, {{\mathcal R}}_X)$. \begin{sprop}\label{(2.8)}\index{Duality!Formal|)} In the preceding situation, with\/ $\Gamma_{\<\! Z}(-)\!:=\Gamma(X,\iG Z(-))$ there is a functorial isomorphism $$ {\mathbf R}\Gamma({\mathscr X},\kappa^*\<{\mathcal F}\>)\cong {\mathbf R}{\mathrm {Hom}}_R^\bullet({\mathbf R}\Gamma_{\<\! Z}\>{\mathcal F}^{\>\prime}\< ,\>{{\mathcal R}}) \qquad \bigl({\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}(X)\bigr). $$ \end{sprop} \begin{proof} Replacing $g$ by a compactification (\cite{Lu}) doesn't affect~${\mathscr X}$ or ${\mathbf R}\Gamma_{\<\!Z}$, so assume that $g$ is proper. Then \Cref{C:completion-proper} gives an isomorphism \mbox{$\kappa^*{\mathcal R}_X\cong\ush{\hat g}{\mathcal R}_Y$.} Now just compose the chain of functorial isomorphisms \begin{align*} {\mathbf R}\Gamma({\mathscr X}, \kappa^*\<{\mathcal F} \>) &\cong {\mathbf R}\Gamma\bigl({\mathscr X},\kappa^*{\mathbf R}\cH{om}^{\bullet}_ X({\mathcal F}^{\>\prime}\<, {{\mathcal R}}_X)\bigr) & &\textup{(see above)}\\ &\cong {\mathbf R}\Gamma\bigl({\mathscr X}, {\mathbf R}\cH{om}^{\bullet}_{ {\mathscr X}}(\kappa^*\<{\mathcal F}^{\>\prime}\<, \>\kappa^*{\mathcal R}_X)\bigr)& &\textup{(\Lref{L:kappa*Ext})}\\ &\cong {\mathbf R}{\mathrm {Hom}}_{\mathscr X}^\bullet(\kappa^*\<{\mathcal F}^{\>\prime}\<, \>\ush{\hat g}{{\mathcal R}}_Y))& &\textup{(see above)}\\ &\cong {\mathbf R}{\mathrm {Hom}}_{\mathscr Y}^\bullet({\mathbf R}\hat g_*{\mathbf R}\iGp {\mathscr X}\kappa^*\<{\mathcal F}^{\>\prime}\<, {{\mathcal R}}_Y) & &\textup{(\Tref{Th2})}\\ &\cong {\mathbf R}{\mathrm {Hom}}_Y^\bullet({\mathbf R} g_*{\mathbf R}\iG {Z_{\mathstrut}}{\mathcal F}^{\>\prime}\<, {\mathcal R}_Y)& &\textup{(\Pref{Gammas'+kappas})}\\ &\cong {\mathbf R}{\mathrm {Hom}}_Y^\bullet(\smash{\widetilde{{\mathbf R}\Gamma_{\<\! Z}{\mathcal F}}{}^{\>\prime}\<}, \>{\mathcal R}_{Y_{\mathstrut}})& &\textup{\cite[footnote, \S5.3]{AJL}}\\ &\cong {\mathbf R}{\mathrm {Hom}}_R^\bullet({\mathbf R}\Gamma_{\<\! Z}{\mathcal F}^{\>\prime}\< ,{{\mathcal R}})& &\textup{\cite[p.\,9, (0.4.4)]{AJL}}. \end{align*} \vskip-3.7ex \end{proof} \begin{slem}\label{L:kappa*Ext} Let\/ $X$ be a locally noetherian scheme, and let\/ $\kappa\colon{\mathscr X}\to X$ be its completion along some closed subset\/~$Z$. Then for\/~${\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$ of finite injective dimension and for\/~${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}(X),$ the natural map is an isomorphism $$ \kappa^*{\mathbf R}\cH{om}^{\bullet}_ X({\mathcal F},{\mathcal G})\iso {\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}}(\kappa^*\<{\mathcal F},\kappa^*{\mathcal G}). $$ \end{slem} \begin{proof} By \cite[p.\,134, Proposition~7.20]{H1} we may assume that ${\mathcal G}$ is a bounded complex of quasi-coherent injective ${\mathcal O}_{\<\<X}$-modules, vanishing, say, in all degrees $>n$. When ${\mathcal F}$ is bounded-above the (well-known) assertion is proved by localizing to the affine case and applying \cite[p.\,68, Proposition~7.1]{H1} to reduce to the trivial case ${\mathcal F}={\mathcal O}_{\<\<X}^m\ (0<m\in\mathbb Z)$. To do the same for unbounded~${\mathcal F}$ we must first show, for fixed~${\mathcal G}$, that the contravariant functor~${\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}}(\kappa^*\<{\mathcal F}\<,\kappa^*{\mathcal G})$ is bounded-above. In fact we will show that if $H^i{\mathcal F}=0$ for all $i<i_0$ then for all $j>n-i_0\>$, $$ H^j{\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}}(\kappa^*\<{\mathcal F}\<,\kappa^*{\mathcal G})= H^j\<\kappa_*{\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}}(\kappa^*\<{\mathcal F}\<,\kappa^*{\mathcal G})= H^j{\mathbf R}\cH{om}^{\bullet}_{X}({\mathcal F}\<,\kappa_*\kappa^*{\mathcal G})=0. $$ The homology in question is the sheaf associated to the presheaf which assigns $$ {\mathrm {Hom}}_{{\mathbf D}(U)}\<\bigl({\mathcal F}|_{\lower.3ex\hbox{$\scriptstyle U$}}[-j], (\kappa_*\kappa^*{\mathcal G})|_{\lower.3ex\hbox{$\scriptstyle U$}}\bigr)= {\mathrm {Hom}}_{{\mathbf D}(U)}\<\bigl({\mathcal F}|_{\lower.3ex\hbox{$\scriptstyle U$}}[-j], {\mathbf R} Q_{\lower.3ex\hbox{$\scriptstyle\<\<U$}} (\kappa_*\kappa^*{\mathcal G})|_{\lower.3ex\hbox{$\scriptstyle U$}}\bigr) $$ to each affine open subset $U={\mathrm {Spec}}(A)$ in~$X\<$. (Here we abuse notation by omitting~${\boldsymbol j}_{\<\<U}^{}$ in front of ${\mathbf R} Q_{\<\<U}^{}$, see beginning of~\S\ref{SS:Dvc-and-Dqc}). \penalty-1000 Let ${\mathscr U}\!:=\kappa^{-1}U$, and $\hat A\!:=\Gamma({\mathscr U},{\mathcal O}_{\mathscr X})$, so that $\kappa|_{\mathscr U}$ factors naturally as $$ {\mathscr U}={\mathrm {Spf}}(\hat A)\xrightarrow{\kappa_1\,}U_1\!:= {\mathrm {Spec}}(\hat A)\xrightarrow{k\,}{\mathrm {Spec}}(A)=U. $$ The functors ${\mathbf R} Q_{\lower.3ex\hbox{$\scriptstyle\<\< U$}}k_*$ and\vadjust{\kern.8pt} $k_*{\mathbf R} Q_{ U_1}^{}$, both right-adjoint to the natural composition \smash{$\D_{\mkern-1.5mu\mathrm {qc}}(U)\xrightarrow{\vbox to0pt{\vskip-.8ex\hbox{$\scriptstyle k^*$}\vss}} \D_{\mkern-1.5mu\mathrm {qc}}(U_1)\hookrightarrow{\mathbf D}(U_1)$,} are isomorphic; so there are natural isomorphisms $$ {\mathbf R} Q_{\lower.3ex\hbox{$\scriptstyle\<\<U$}} (\kappa_*\kappa^*{\mathcal G})|_{\lower.3ex\hbox{$\scriptstyle U$}} = {\mathbf R} Q_{\lower.3ex\hbox{$\scriptstyle\<\<U$}} k_{*}\kappa_{1*}^{}\kappa_1^*k^* ({\mathcal G}|_{\lower.3ex\hbox{$\scriptstyle U$}}) \<\iso\< k_{*}{\mathbf R} Q_{U_1}^{} \kappa_{1*}^{}\kappa_1^*k^*({\mathcal G}|_{\lower.3ex\hbox{$\scriptstyle U$}}) \<\smash{\underset{\ref{c-erator}}{\iso}}\< k_{*}k^*({\mathcal G}|_{\lower.3ex\hbox{$\scriptstyle U$}}) $$ and the presheaf becomes $ U\mapsto {\mathrm {Hom}}_{{\mathbf D}(U)}\<\bigl({\mathcal F}|_{\lower.3ex\hbox{$\scriptstyle U$}}[-j], k_{*}k^*({\mathcal G}|_{\lower.3ex\hbox{$\scriptstyle U$}})\bigr). $ The equivalence of categories $\D_{\mkern-1.5mu\mathrm {qc}}(U)\cong {\mathbf D}(\A_{\qc}(U))={\mathbf D}(A)$ indicated at the beginning of~\S\ref{SS:Dvc-and-Dqc} yields an isomorphism $$ {\mathrm {Hom}}_{{\mathbf D}(U)}\<\bigl({\mathcal F}|_{\lower.3ex\hbox{$\scriptstyle U$}}[-j], k_{*}k^*({\mathcal G}|_{\lower.3ex\hbox{$\scriptstyle U$}})\bigr)\iso {\mathrm {Hom}}_{{\mathbf D}(A)}\<\bigl(F[-j], G\otimes_A\hat A\bigr) $$ where $F$ is a complex of $A$-modules\vadjust{\penalty-1000} with ${\mathrm H}^iF =0$ for $i<i_0\>$, and both $G$ and $G\otimes_A\hat A$ are complexes of injective $A$-modules vanishing in all degrees $>n$\vspace{-1pt} (the latter since $\hat A$ is $A$-flat). Hence the presheaf vanishes, and the conclusion follows. \end{proof} \end{parag} \penalty-1500 \begin{parag}\label{bf (e)} (Dualizing complexes.)\index{dualizing complexes|(} Let ${\mathscr X}$ be a noetherian formal scheme, and write ${\mathbf D}$ for ${\mathbf D}({\mathscr X})$, etc. The derived functor\index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!a@$\boldsymbol{\varGamma}\!:={\mathbf R}\iGp{\mathscr X}$ (cohomology colocalization)} $\boldsymbol{\varGamma}\!:={\mathbf R}\iGp{\mathscr X}\colon{\mathbf D}\to{\mathbf D}$ (see \Sref{Gamma'1}) has a right adjoint% \index{ $\mathbf {La}$@${\boldsymbol\Lambda}$ (homology localization)} ${\boldsymbol\Lambda}={\boldsymbol\Lambda}_{\mathscr X}\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}, -)$. This adjunction is given by \eqref{adj}, a natural isomorphism of which we'll need the sheafified form, proved similarly: \begin{equation}\label{adj0} {\mathbf R}\cH{om}^{\bullet}({\mathcal M},{\boldsymbol\Lambda}{\mathcal R}) \cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}\<{\mathcal M},{\mathcal R}). \end{equation} There are natural maps $\boldsymbol{\varGamma}\to\mathbf1\to{\boldsymbol\Lambda}$ inducing isomorphisms ${\boldsymbol\Lambda}\boldsymbol{\varGamma}\iso{\boldsymbol\Lambda}\iso{\boldsymbol\Lambda}\BL$, $\boldsymbol{\varGamma}\BG\iso\boldsymbol{\varGamma}\iso\boldsymbol{\varGamma}{\boldsymbol\Lambda}$ (\Rref{R:Gamma-Lambda}\,(1)). \Pref{formal-GM}, a form of Greenlees-May duality, shows that ${\boldsymbol\Lambda}(\D_{\mkern-1.5mu\mathrm c})\subset\D_{\mkern-1.5mu\mathrm c}$. (Recall that the objects of the $\Delta$-subcategory $\D_{\mkern-1.5mu\mathrm c}\subset{\mathbf D}$ are the complexes whose homology sheaves are all coherent.) Let $\D_{\mkern-1.5mu\mathrm c}^*$ be the essential image of $\boldsymbol{\varGamma}|_{\D_{\mkern-1.5mu\mathrm c}}$, i.e., the full subcategory of~${\mathbf D}$ such that ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^*\Leftrightarrow{\mathcal E}\cong\boldsymbol{\varGamma}{\mathcal F}$ with ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}$. \Pref{Gamma'(qc)} shows that $\D_{\mkern-1.5mu\mathrm c}^*\subset\D_{\mkern-1.5mu\mathrm{qct}}$. It follows from the preceding paragraph that \begin{alignat*}{2} {\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^*&\iff \boldsymbol{\varGamma}{\mathcal E}\iso{\mathcal E}&&\text{ and }\,{\boldsymbol\Lambda}\>{\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}\>,\\ {\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}&\iff {\mathcal F}\iso{\boldsymbol\Lambda}{\mathcal F}\,&&\text{ and }\,\boldsymbol{\varGamma}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^*. \end{alignat*} (In particular, $\D_{\mkern-1.5mu\mathrm c}^*$ is a $\Delta$-subcategory of~${\mathbf D}$.) Moreover $\boldsymbol{\varGamma}$ and ${\boldsymbol\Lambda}$ are quasi-inverse equivalences between the categories $\D_{\mkern-1.5mu\mathrm c}$ and $\D_{\mkern-1.5mu\mathrm c}^*$. \end{parag} \begin{sdef}\label{D:dualizing} A complex ${\mathcal R}$ is a \emph{c-dualizing complex on} ${\mathscr X}\>$ if \begin{enumerate} \item[(i)] ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X}).$ \item[(ii)] The natural map is an isomorphism $\,{\mathcal O}_{\mathscr X}\iso{\mathbf R}\cH{om}^{\bullet}({\mathcal R},{\mathcal R})$. \item[(iii)] There is an integer $b$ such that for every coherent torsion sheaf~${\mathcal M}$ and for every~$i>b$, $\>\E{xt}^i({\mathcal M},{\mathcal R})\!:= H^i\>{\mathbf R}\cH{om}^{\bullet}({\mathcal M},{\mathcal R})=0$. \end{enumerate} A complex ${\mathcal R}$ is a \emph{t-dualizing complex on} ${\mathscr X}\>$ if \begin{enumerate} \item[(i)] ${\mathcal R}\in{\mathbf D}_{\mathrm t}^+({\mathscr X}).$ \item[(ii)] The natural map is an isomorphism $ \,{\mathcal O}_{\mathscr X}\iso{\mathbf R}\cH{om}^{\bullet}({\mathcal R},{\mathcal R}). $ \item[(iii)] There is an integer $b$ such that for every coherent torsion sheaf~${\mathcal M}$ and for every $i>b$, $\>\E{xt}^i({\mathcal M},{\mathcal R})\!:= H^i\>{\mathbf R}\cH{om}^{\bullet}({\mathcal M},{\mathcal R})=0$. \item[(iv)] For some ideal of definition~${\mathscr J}$ of~${\mathscr X}$, ${\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}, {\mathcal R})\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X}).$ \item[{}](Equivalently---by simple arguments---${\mathbf R}\cH{om}^{\bullet}({\mathcal M}, {\mathcal R})\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$ for every coherent torsion sheaf ${\mathcal M}$.) \end{enumerate} \end{sdef} \pagebreak[3] \emph{Remarks.} (1) On an ordinary scheme, (iii) signifies \emph{finite injective dimension} \cite[p.\,83, Definition, and p.\,134, (iii)${}_{\textup c}$]{H1}, so both c-dualizing and t-dualizing mean the same as what is called ``dualizing" in \cite[p.\,258, Definition]{H1}. (For the extension to arbitrary noetherian formal schemes, see (4) below.) (2) By (i) and (iv), \Pref{Gamma'(qc)}(a), and \Cref{qct=plump}, any t-dualizing complex ${\mathcal R}$ is in $\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$; and then (iii) implies that \emph{${\mathcal R}$~is isomorphic in ${\mathbf D}$ to a bounded complex of $\A_{\mathrm {qct}}\<$-injectives.} To see this, begin by imitating the proof of~\cite[p.\,80, (iii)$\Rightarrow$(i)]{H1}, using \cite[Theorem~4.8]{Ye} and \Lref{L:Hom=RHom} below, to reduce to showing that \emph{if\/ ${\mathcal N}\in\A_{\mathrm {qct}}\<({\mathscr X})$ is such that\/ $\E{xt}^1({\mathcal M},{\mathcal N}\>)=0\>$ for every coherent torsion sheaf\/ ${\mathcal M}$ then\/ ${\mathcal N}$ is\/ $\A_{\mathrm {qct}}\<$-injective.} For the last assertion, suppose first that ${\mathscr X}$ is affine. \Lref{Gamma'+qc} implies that \mbox{$\cH{om}({\mathcal M},{\mathcal N}\>)\in\A_{\vec {\mathrm c}}({\mathscr X})$;} and then $\textup{Ext}^1({\mathcal M},{\mathcal N}\>)=0$, by the natural exact sequence $$ 0\underset{\textup{(3.1.8)}}=\textup H^1\bigl({\mathscr X}, \cH{om}({\mathcal M},{\mathcal N}\>)\bigr) \to\textup{Ext}^1({\mathcal M},{\mathcal N}\>)\to \textup H^0\bigl({\mathscr X}, \E{xt}^1({\mathcal M},{\mathcal N}\>)\bigr). $$ Since coherent torsion sheaves generate $\A_{\mathrm {qct}}\<({\mathscr X})$ (\Cref{qct=plump}, \Lref{Gamma'+qc}), a standard argument using Zorn's Lemma shows that ${\mathcal N}$~is indeed $\A_{\mathrm {qct}}\<$-injective. \penalty-1500 In the general case, let ${\mathscr U}\subset{\mathscr X}$ be any affine open subset. For any coherent torsion ${\mathcal O}_{\mathscr U}$-module~${\mathcal M}_0$, \Pref{f-*(qct)} and \Lref{Gamma'+qc} imply there is a coherent torsion ${\mathcal O}_{\mathscr X}$-module~${\mathcal M}$ restricting on~${\mathscr U}$ to~${\mathcal M}_0$, whence $\>\E{xt}^1_{\mathscr U}({\mathcal M}_0, {\mathcal N}\>|_{{\mathscr U}})=0$. By the affine case, then, ${\mathcal N}\>|_{{\mathscr U}}$ is $\A_{\mathrm {qct}}\<({\mathscr U})$-injective, hence $\A_{\mathrm t}\<({\mathscr U})$-injective \cite[Proposition~4.2]{Ye}. Finally, as in \cite[p.\,131, Lemma 7.16]{H1}, using \cite[Lemma 4.1]{Ye},% \footnote{where one may assume that $X$ and ${\mathscr X}$ have the same underlying space} one concludes that ${\mathcal N}$ is $\A_{\mathrm t}\<({\mathscr X})$-injective, hence $\A_{\mathrm {qct}}\<({\mathscr X})$-injective. \smallskip (3) With (2) in mind, one finds that what is called here ``t-dualizing complex" is what Yekutieli\index{Yekutieli, Amnon} calls in \cite[\S5]{Ye} ``dualizing complex." \smallskip (4) \emph{A c-dualizing complex~${\mathcal R}$ has finite injective dimension}: there is an integer~$n_0$ such that for any $i>n_0$ and any ${\mathcal O}_{\mathscr X}$-module~${\mathcal E}$, ${\mathrm {Hom}}_{{\mathbf D}}({\mathcal E}\<, {\mathcal R}[i])=0$. To see this, note first that $$ {\mathrm {Hom}}_{{\mathbf D}}({\mathcal E}\<,{\mathcal R}[i])\cong {\mathrm {Hom}}_{{\mathbf D}}({\mathcal E}\<,{\boldsymbol\Lambda}\boldsymbol{\varGamma}{\mathcal R}[i])\cong {\mathrm {Hom}}_{{\mathbf D}}(\boldsymbol{\varGamma}{\mathcal E}\<,\boldsymbol{\varGamma}{\mathcal R}[i]). $$ \Lref{L:interchange}(b) below and (2) above show that $\boldsymbol{\varGamma}{\mathcal R}$ is isomorphic to a bounded complex of $\A_{\mathrm {qct}}\<$-injectives. The complex $\boldsymbol{\varGamma}{\mathcal E}$---obtained by applying the functor $\iGp{\mathscr X}$ to an injective resolution of~${\mathcal E}$---consists of torsion ${\mathcal O}_{\mathscr X}$-modules, and so as in \cite[Corollary 4.3]{Ye} (see also the proof of \Lref{L:Hom=RHom} below, with \Pref{iso-qct} in place of \Pref{(3.2.1)}), the natural map $$ \textup H^i\bigl({\mathrm {Hom}}^{\bullet}(\boldsymbol{\varGamma}{\mathcal E}\<,\boldsymbol{\varGamma}{\mathcal R})\bigr)\to \textup H^i\bigl({\mathbf R}{\mathrm {Hom}}^{\bullet}(\boldsymbol{\varGamma}{\mathcal E}\<,\boldsymbol{\varGamma}{\mathcal R})\bigr) ={\mathrm {Hom}}_{{\mathbf D}}(\boldsymbol{\varGamma}{\mathcal E}\<, \boldsymbol{\varGamma}{\mathcal R}[i]) $$ is an \emph{isomorphism.} Since $\boldsymbol{\varGamma}{\mathcal E}$ vanishes in degrees $<0$, the asserted result holds for any $n_0$ such that $H^i(\boldsymbol{\varGamma}{\mathcal R})=0$ for $i>n_0\>$. \smallskip (5) For a complex ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}^+\cap\mspace{1.5mu}\D_{\mkern-1.5mu\mathrm c}^-\<$, conditions (ii) and (iii) in \Dref{D:dualizing} hold iff they hold stalkwise for $x\in {\mathscr X}$, with an integer $b$ \emph{independent of\/~$x$.} (The idea is that such an ${\mathcal R}$ is locally resolvable by a bounded-above complex~${\mathcal F}$ of finite-rank locally free ${\mathcal O}_{\mathscr X}$-modules, as is ${\mathcal M}$ in~(iii), and $\cH{om}^{\bullet}({\mathcal F}\<,{\mathcal R})\cong{\mathbf R}\cH{om}^{\bullet}({\mathcal F}\<,{\mathcal R})$\dots.) Proceeding as in the proofs of~\cite{H1}, Proposition~8.2, p.\,288, and Corollary~7.2, p.\,283, one concludes that ${\mathcal R}$ is c-dualizing iff $\>{\mathscr X}$ has finite Krull dimension and ${\mathcal R}_x$~is a dualizing complex for the category of ${\mathcal O}_{{\mathscr X}\<,\>x}$-modules for every $x\in{\mathscr X}$. (It is enough that the latter hold for all \emph{closed} points~$x\in{\mathscr X}$.) \begin{exams}\label{regular} (1) If $\,{\mathcal R}$ is c-(or t-)dualizing then so is ${\mathcal R}\otimes{\mathcal L}[n]$ for any invertible ${\mathcal O}_{\mathscr X}$-module and $n\in\mathbb Z$. The converse also holds, see \Pref{P:uniqueness}. \smallskip (2) (Cf.~\cite[Example 5.12]{Ye}.) If $X$ is an ordinary scheme and $\kappa\colon{\mathscr X}\to X$ is its completion along some closed subscheme~$Z$, then for any dualizing ${\mathcal O}_{\<\<X}$-complex~${\mathcal R}$, $\kappa^*{\mathcal R}$ is c-dualizing on ~${\mathscr X}$, and $\boldsymbol{\varGamma}\kappa^*{\mathcal R}\cong\kappa^*{\mathbf R}\iG Z{\mathcal R}$ (see \Pref{Gammas'+kappas}(c)) is a t-dualizing complex lying in $\D_{\mkern-1.5mu\mathrm c}^*({\mathscr X})$. \emph{Proof.} For $\kappa^*{\mathcal R}$, conditions (i) and (ii) in the definition of c-dualizing follow easily from the same for ${\mathcal R}$ (because of \Lref{L:kappa*Ext}). So does~(iii), after we reduce to the case $X$ affine, where \Pref{(3.2.1)} allows us to write ${\mathcal M}=\kappa^*{\mathcal M}_0$ with ${\mathcal M}_0\in{\mathcal A}(X)$. (Recall from remark (1) above that ${\mathcal R}$ has finite injective dimension.) The last assertion is given by \Lref{L:interchange}(b). \smallskip (3) If ${\mathscr X}={\mathrm {Spf}}(A)$ where $A$ is a complete local noetherian ring topologized by its maximal ideal~$m$---so that ${\mathcal A}({\mathscr X})$ is just the category of $A$-modules---then a \mbox{c-dualizing} ${\mathcal O}_{\mathscr X}$-complex is an $A$-dualizing complex in the usual sense; and by~(2) (via~\cite[p.\,276, 6.1]{H1}), or directly from \Dref{D:dualizing}, the injective hull of $A/m$ is a t-dualizing complex lying in $\D_{\mkern-1.5mu\mathrm c}^*({\mathscr X})$. \smallskip (4) It is clear from \Dref{D:dualizing} and remark~(4) above that ${\mathcal O}_{\mathscr X}$ is c-dualizing iff ${\mathcal O}_{\mathscr X}$ has finite injective dimension over itself. By remark (5), ${\mathcal O}_{\mathscr X}$ is c-dualizing iff ${\mathscr X}$ is finite dimensional and ${\mathcal O}_{{\mathscr X}\<,\>x}$ is \emph{Gorenstein} for all $x\in{\mathscr X}$ \cite[p.\,295, Definition]{H1}. \smallskip (5) For instance, if the finite-dimensional noetherian formal scheme $\>{\mathscr Y}$ is \emph{regular} (i.e., the local rings ${\mathcal O}_{{\mathscr Y}\<,\>y}\ (y\in{\mathscr Y})$ are all regular), and ${\mathcal I}$ is a coherent \mbox{${\mathcal O}_{\mathscr Y}$-ideal}, defining a closed formal subscheme~$i\colon{\mathscr X}\hookrightarrow{\mathscr Y}$ \cite[p.\,441,(10.14.2)]{GD}, then by~ remark~(3), ${\mathbf R}\cH{om}^{\bullet}(i_*{\mathcal O}_{\mathscr X}^{}, \>{\mathcal O}_{\mathscr Y}^{})$ is c-dualizing on~${\mathscr X}$. So \Lref{L:interchange} gives that\looseness=-1 $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}^{}/{\mathcal I}, \>{\mathbf R}\iGp{\mathscr Y}{\mathcal O}_{\mathscr Y}^{}) \underset{\textup{\ref{R:Dtilde}(4)}}\cong {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}(i_*{\mathcal O}_{\mathscr X}^{}, \>{\mathcal O}_{\mathscr Y}^{})\in\D_{\mkern-1.5mu\mathrm c}^*({\mathscr X}) $$ is t-dualizing on~${\mathscr X}$. (This is also shown in \cite[Proposition 5.11, Theorem 5.14]{Ye}.) \end{exams} \pagebreak[3] \begin{slem}\label{L:interchange} \textup{(a)} If\/ ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}^*$ is t-dualizing then\/ ${\boldsymbol\Lambda}{\mathcal R}$ is c-dualizing. \vspace{1pt} \textup{(b)} If\/ ${\mathcal R}$ is c-dualizing then\/ $\boldsymbol{\varGamma}{\mathcal R}$ is t-dualizing, and lies in\/~$\D_{\mkern-1.5mu\mathrm c}^*$. \end{slem} \begin{proof} (a) If $\;{\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}^*$ then of course ${\boldsymbol\Lambda}{\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}\>$. Also, ${\boldsymbol\Lambda}({\mathbf D}^+)\subset{\mathbf D}^+$ because ${\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}$ is given locally by a finite complex ${\mathcal K}_\infty^\bullet\>$, see proof of \Pref{Gamma'(qc)}(a). For condition (ii), note that if ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$ then $\boldsymbol{\varGamma}{\mathcal R}\cong{\mathcal R}$ (\Pref{Gamma'(qc)}), then use the natural isomorphisms (see \eqref{adj0}: $$ {\mathbf R}\cH{om}^{\bullet}({\boldsymbol\Lambda}{\mathcal R}\>,{\boldsymbol\Lambda}{\mathcal R}) \cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\boldsymbol\Lambda}{\mathcal R}\>,{\mathcal R})\cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal R}\>,{\mathcal R})\cong {\mathbf R}\cH{om}^{\bullet}({\mathcal R}\>,{\mathcal R}). \\ $$ For (iii) note that $\boldsymbol{\varGamma}{\mathcal M}\cong{\mathcal M}$ (\Pref{Gamma'(qc)}), then use \eqref{adj0}. \smallskip (b) \Pref{Gamma'(qc)} makes clear that if ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}^+$ then $\boldsymbol{\varGamma}{\mathcal R}\in\D_{\mkern-1.5mu\mathrm{qct}}^+\cap\D_{\mkern-1.5mu\mathrm c}^*$. For (ii) use the isomorphisms (the second holding because ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}$): $$ {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal R},\boldsymbol{\varGamma}{\mathcal R}) \underset{\textup{\ref{C:Hom-Rgamma}}}\cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal R},{\mathcal R}) \underset{\textup{\ref{formal-GM}}}\cong {\mathbf R}\cH{om}^{\bullet}({\mathcal R},{\mathcal R}). $$ For (iii) use the isomorphism ${\mathbf R}\cH{om}^{\bullet}({\mathcal M},\boldsymbol{\varGamma}{\mathcal R}) \underset{\textup{\ref{C:Hom-Rgamma}}}\cong {\mathbf R}\cH{om}^{\bullet}({\mathcal M},{\mathcal R}).$\vspace{2pt} For (iv), note that when ${\mathcal M}={\mathcal O}_{\mathscr X}/{\mathscr J}\,$ (${\mathscr J}$ any ideal of definition) this isomorphism gives $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J},\boldsymbol{\varGamma}{\mathcal R})\cong{\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J},{\mathcal R}) \underset{\textup{\ref{P:Rhom}}}\in\D_{\mkern-1.5mu\mathrm c}\>, $$ \vskip-4.3ex \end{proof} \smallskip \pagebreak[3] The essential \emph{uniqueness} of t-(resp.~c-)dualizing complexes is expressed by: \begin{sprop}\label{P:uniqueness} \textup{(a) (Yekutieli)}\index{Yekutieli, Amnon} If\/ ${\mathcal R}$ is t-dualizing then a complex\/ ${\mathcal R}'$ is t-dualizing iff there is an invertible sheaf\/~${\mathcal L}$ and an integer\/~$n$ such that\/ ${\mathcal R}'\cong{\mathcal R}\otimes{\mathcal L}[n]$. \smallskip \textup{(b)} If\/ ${\mathcal R}$ is c-dualizing then a complex\/ ${\mathcal R}'$ is c-dualizing iff there is an invertible sheaf\/~${\mathcal L}$ and an integer\/~$n$ such that\/ ${\mathcal R}'\cong{\mathcal R}\otimes{\mathcal L}[n]$. \end{sprop} \begin{proof} Part (a) is proved in \cite[Theorem 5.6]{Ye}. \smallskip Now for a fixed invertible sheaf ${\mathcal L}$ there is a natural isomorphism of functors \begin{equation}\label{iso} {\boldsymbol\Lambda}({\mathcal F}\otimes{\mathcal L})\iso{\boldsymbol\Lambda}{\mathcal F}\otimes{\mathcal L}\qquad({\mathcal F}\in{\mathbf D}), \end{equation} as one deduces, e.g., from a readily-established natural isomorphism between the respective right adjoints $$ \boldsymbol{\varGamma}{\mathcal E}\otimes{\mathcal L}^{-1}\osi\boldsymbol{\varGamma}({\mathcal E}\otimes{\mathcal L}^{-1})\qquad({\mathcal E}\in{\mathbf D}). $$ Part (b) results, because $\boldsymbol{\varGamma}{\mathcal R}'$ and~$\boldsymbol{\varGamma}{\mathcal R}$ are t-dualizing (\Lref{L:interchange}), so that by (a) (and taking ${\mathcal F}\!:=\boldsymbol{\varGamma}{\mathcal R}[n]$ in \eqref{iso}) we have isomorphisms \begin{flalign*} \hskip61pt{\mathcal R}'\cong{\boldsymbol\Lambda}(\boldsymbol{\varGamma}{\mathcal R}') &\cong{\boldsymbol\Lambda}(\boldsymbol{\varGamma}{\mathcal R}\otimes{\mathcal L}[n])\qquad &&\quad(\text{${\mathcal L}$ invertible, $n\in\mathbb Z$})\\ &\cong({\boldsymbol\Lambda}\boldsymbol{\varGamma}{\mathcal R})\otimes{\mathcal L}[n] \cong {\mathcal R}\otimes{\mathcal L}[n]. \mkern-3mu \end{flalign*} \vskip-3.8ex \end{proof} \begin{scor}\label{C:Dc*} If\/ ${\mathscr X}$ is locally embeddable in a regular finite-dimensional formal scheme then any t-dualizing complex on\/~${\mathscr X}$ lies in\/~$\D_{\mkern-1.5mu\mathrm c}^*$. \end{scor} \begin{proof} Whether a t-dualizing complex~${\mathcal R}$ satisfies ${\boldsymbol\Lambda}{\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}$ is a local question, so we may assume that ${\mathscr X}$ is a closed subscheme of a finite-dimensional regular formal scheme, and then \Eref{regular}(5) shows that \emph{some}---hence by \Pref{P:uniqueness}, \emph{any}---t-dualizing complex lies in~$\D_{\mkern-1.5mu\mathrm c}^*$. \end{proof} \begin{slem}\label{L:Hom=RHom} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme, let ${\mathcal I}$ be a bounded complex of $\A_{\mathrm {qct}}\<({\mathscr X})$-injectives, say\/ ${\mathcal I}\>^i=0$ for all\/~$i>n,$ and let\/ ${\mathcal F}\in\mathbf D^+({\mathscr X}),$ say \mbox{$H^\ell({\mathcal F})=0$} for all\/~$\ell<\!-m.$ Suppose there exists an open cover\/ $({\mathscr X}_\alpha)$ of\/ ${\mathscr X}$ by completions of ordinary noetherian schemes\/~$X_\alpha$ along closed subsets, with completion maps\/ $\kappa_\alpha\colon{\mathscr X}_\alpha\to X_\alpha\>,$ such that for each\/ $\alpha$ the restriction of\/ ${\mathcal F}$ to ${\mathscr X}_\alpha$ is\/ $\mathbf D$-isomorphic to\/~$\kappa_{\<\alpha}^*F^{}_{\<\alpha}$ for some $F_\alpha\in\mathbf D(X_\alpha).$ Then $$ \E{xt}^i({\mathcal F}\<,\>{\mathcal I}\>)\!:= H^i{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathcal F}\<,\>{\mathcal I}\>)=0\quad\textup{for all\/~$i>m+n$.} $$ Moreover, if\/ ${\mathscr X}$ has finite Krull dimension\/~$d$ then $$ \textup{Ext}^i({\mathcal F}\<,\>{\mathcal I}\>)\!:= H^i{\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr X}({\mathcal F}\<,\>{\mathcal I}\>)=0\quad\textup{for all\/~$i>m+n+d$.} $$ \end{slem} \emph{Remarks.} In the published version of this paper (Contemporary Math.~244) \Lref{L:Hom=RHom} stated: \emph{Let\/ ${\mathcal F}\in\D_{\<\vc}$ and let\/ ${\mathcal I}$ be a bounded-below complex of\/ $\A_{\mathrm {qct}}\<$-injectives. Then the canonical map is a\/ ${\mathbf D}$-isomorphism} $$ \cH{om}^{\bullet}({\mathcal F}\<,{\mathcal I}\>)\iso{\mathbf R}\cH{om}^{\bullet}({\mathcal F}\<,{\mathcal I}\>). $$ Suresh Nayak pointed out that the proof given applies only to $\A_{\vec {\mathrm c}}$-complexes, not, as asserted, to arbitrary ${\mathcal F}\in\D_{\<\vc}$. (Cf.~\cite[Corollary 4.3]{Ye}.) \Lref{L:Hom=RHom} is used four times in \S 2.5, so these four places need to be revisited. (There are no other references to Lemma~2.5.6 in the paper.) \smallskip First, in Remark (2) on p.\,24, the reference to Lemma~2.5.6 is not necessary: the cited theorem 4.8 in \cite{Ye} (see also \Pref{1!} below) shows that the t-dualizing complex~ ${\mathcal R}$ is $\mathbf D$-isomorphic to a bounded-below complex~$\mathcal X'{}^\bullet$ of $\A_{\mathrm {qct}}\<$-injectives; and then one can proceed as indicated to show that for some~$n$ the (bounded) truncation~$\sigma_{\le n} \mathcal X'{}^\bullet$ is $\A_{\mathrm {qct}}\<$-injective and $\mathbf D$-isomorphic to~$\mathcal X'{}^\bullet$. (To follow the details, it helps to keep in mind 5.1.3 and~ 5.1.4 below.) Since, by Remark (2), any t-dualizing complex is $\mathbf D$-isomorphic to a bounded complex of $\A_{\mathrm {qct}}\<$-injectives, in view of Propositions 3.3.1 and~ 5.1.2 one finds that the remaining three references to Lemma 2.5.6 can be replaced by references to the present \Lref{L:Hom=RHom}. For the reference in the proof of 2.5.7(b) this is clear. The same is true for Remark ~(4) on p.\,25, but $i>n_0$ at the end should be~$i>n_0 + d$, where, by Remark~(5), the Krull dimension~$d$ of $\>{\mathscr X}$ is finite. Finally, for the reference in the proof of 2.5.12, one can note, via 5.1.4 and~5.1.2, that $\D_{\mkern-1.5mu\mathrm c}^*\subset\D_{\mkern-1.5mu\mathrm{qct}}\subset\D_{\<\vc}\>$.\vspace{1pt} {\sc Proof of \ref{L:Hom=RHom}}. By the proof of \cite[Proposition 4.2]{Ye}, $\A_{\mathrm {qct}}\<$-injectives are just direct sums of sheaves of the form ${\mathcal J}(x)\ (x\in{\mathscr X})$, where for any open ${\mathscr U}\subset{\mathscr X}$, $\Gamma({\mathscr U},{\mathcal J}(x))$~is a fixed injective hull of the residue field of ${\mathcal O}_{{\mathscr X}\<,x}$ if $x\in{\mathscr U}$, and vanishes otherwise. Hence the restriction of an $\A_{\mathrm {qct}}\<({\mathscr X})$-injective to an open ${\mathscr V}\subset{\mathscr X}$ is $\A_{\mathrm {qct}}\<({\mathscr V})$-injective; and so the first assertion is local. Thus to prove it one may assume that ${\mathscr X}$ itself is a completion, with completion map $\kappa\colon{\mathscr X}\to X\<$, and that in ${\mathbf D}({\mathscr X})$, ${\mathcal F}\cong\kappa^*\<F$ for some $F\in{\mathbf D}(X)$. As $\kappa^*\<$, being exact, commutes with the truncation functor~ $\sigma_{{\scriptscriptstyle\ge} -m}\>$, there are \mbox{$\mathbf D$-isomorphisms} (the first as in \cite[p.\,70]{H1}): \looseness=-1 $$ {\mathcal F}\cong\sigma_{{\scriptscriptstyle\ge} -m}\>{\mathcal F} \cong\sigma_{{\scriptscriptstyle\ge} -m}\kappa^*\<F \cong \kappa^*\sigma_{{\scriptscriptstyle\ge} -m}\>F\>; $$ so one can replace~$F$ by~$\sigma_{{\scriptscriptstyle\ge} -m}\>F$ and assume further that $F^\ell=0$ for all\/~$\>\ell<-m\>$.\vspace{1pt} From the above description of $\A_{\mathrm {qct}}\<$-injectives, one sees that $\kappa_*{\mathcal I}$ is a bounded complex of ${\mathcal O}_{\<\<X}$-injectives, vanishing in degree $>n\>$. Since $\kappa_*$ is exact, therefore for all $i>m+n$, \begin{align*} \kappa_*H^i{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathcal F},\>{\mathcal I}\>)&\cong H^i\kappa_*{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}(\kappa^*\< F,\>{\mathcal I}\>)\\ &\cong H^i{\mathbf R}\cH{om}^{\bullet}_X(F,\>\kappa_*\>{\mathcal I}\>) \quad\qquad\qquad\textup{\cite[p.\,147, 6.7(2)]{Sp}}\\ &\cong H^i\cH{om}^{\bullet}_X(F,\>\kappa_*\>{\mathcal I}\>)=0\>, \end{align*} and hence $H^i{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathcal F},\>{\mathcal I}\>)=0$. \vspace{1pt} If ${\mathscr X}$ has Krull dimension~ $d$, and $\Gamma\!:=\Gamma({\mathscr X},-)$ is the global-section functor, then by a well-known theorem of Grothendieck the restriction of the derived \mbox{functor~${\mathbf R}\Gamma$} to the category of abelian sheaves has cohomological dimension $\le d\>$; and so since ${\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr X}\cong {\mathbf R}\Gamma\>{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}$ \cite[Exercise 2.5.10(b)]{Derived categories}, the second assertion\vspace{.6pt} follows from \cite[Remark 1.11.2(iv)]{Derived categories}. \vspace{2pt}\hfill{$\square$} \Pref{P:dualizing} below brings out the basic property of the \emph{dualizing functors} associated with dualizing complexes. (For illustration, one might keep in mind the special case of \Eref{regular}(3).) \begin{slem}\label{L:dualizing} Let\/ ${\mathcal R}$ be a c-dualizing\/ complex on\/~${\mathscr X},$ let\/ ${\mathcal R_{\>\mathrm t}}$ be the t-dualizing complex ${\mathcal R_{\>\mathrm t}}\!:=\boldsymbol{\varGamma}{\mathcal R},$ and for any\/ ${\mathcal E}\in{\mathbf D}$ set $$ {\mathcal D}{\mathcal E}\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal R}),\qquad \cD_{\<\mathrm t}\>{\mathcal E}\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal R_{\>\mathrm t}}). $$ \textup{(a)} There are functorial isomorphisms $$ {\boldsymbol\Lambda}\cD_{\<\mathrm t}\cong{\boldsymbol\Lambda}{\mathcal D}\cong{\mathcal D}{\boldsymbol\Lambda}\cong{\mathcal D}\cong{\mathcal D}\>\boldsymbol{\varGamma}\cong\cD_{\<\mathrm t}\>\boldsymbol{\varGamma}\<. $$ \textup{(b)} For all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}\>,$ ${\mathcal D}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}$ and there is a natural isomorphism\/ $ \cD_{\<\mathrm t}\>{\mathcal F}\cong\boldsymbol{\varGamma}{\mathcal D}{\mathcal F}$. \end{slem} \begin{proof} (a) For any ${\mathcal E}\in{\mathbf D}$, \Pref{formal-GM} gives the isomorphism $$ {\mathcal D}{\mathcal E}={\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,{\mathcal R}) \cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal E}\<,{\mathcal R}) ={\mathcal D}\boldsymbol{\varGamma}{\mathcal E}. $$ In particular, ${\mathcal D}{\boldsymbol\Lambda}\>{\mathcal E}\cong{\mathcal D}\boldsymbol{\varGamma}\<{\boldsymbol\Lambda}\>{\mathcal E}\cong{\mathcal D}\boldsymbol{\varGamma}{\mathcal E}$. Thus ${\mathcal D}\cong{\mathcal D}\>\boldsymbol{\varGamma}\cong{\mathcal D}{\boldsymbol\Lambda}$. \enlargethispage*{1.5\baselineskip} Furthermore, using that the natural map $\smash{\boldsymbol{\varGamma}{\mathcal O}_{\mathscr X}\Otimes{\mathcal E}}\to\boldsymbol{\varGamma}{\mathcal E}$ is an \emph{isomorphism} (localize, and see \cite[p.\,20, Corollary (3.1.2)]{AJL}) we get natural isomorphisms \vspace{-1pt} \begin{multline*} {\mathbf R}\cH{om}^{\bullet}\bigl(\boldsymbol{\varGamma}{\mathcal O}_{\mathscr X}\>,\>\cH{om}^{\bullet}({\mathcal E},{\mathcal R})\bigr)\iso\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal E}\<,\>{\mathcal R}) \underset{\textup{\ref{C:Hom-Rgamma}}}\cong {\mathbf R}\cH{om}^{\bullet}(\boldsymbol{\varGamma}{\mathcal E},\boldsymbol{\varGamma}{\mathcal R})\\ \cong {\mathbf R}\cH{om}^{\bullet}\bigl(\boldsymbol{\varGamma}{\mathcal O}_{\mathscr X}\>,\>\cH{om}^{\bullet}({\mathcal E},\boldsymbol{\varGamma}{\mathcal R})\bigr), \end{multline*} giving ${\boldsymbol\Lambda}{\mathcal D}\cong{\mathcal D}\boldsymbol{\varGamma}\cong\cD_{\<\mathrm t}\boldsymbol{\varGamma}\cong{\boldsymbol\Lambda}\cD_{\<\mathrm t}$. \smallskip (b) Given remark (2) following \Dref{D:dualizing}, \Lref{L:Hom=RHom} implies that the functor $\cD_{\<\mathrm t}\!:={\mathbf R}\cH{om}^{\bullet}(-, {\mathcal R_{\>\mathrm t}})$ is bounded on~$\D_{\<\vc}\>$. The same holds for~${\mathcal D}=\cD_{\<\mathrm t}\boldsymbol{\varGamma}$ (see (a)), because $\boldsymbol{\varGamma}(\D_{\<\vc})\subset\D_{\mkern-1.5mu\mathrm{qct}}\subset\D_{\<\vc}$ (\Lref{Gamma'+qc}), and $\boldsymbol{\varGamma}$ is bounded. ($\boldsymbol{\varGamma}$~is given locally by tensoring with a bounded flat complex ${\mathcal K}_\infty^\bullet\>$, see proof of \Pref{Gamma'(qc)}(a)). Arguing as in \Pref{P:Rhom}, we see that \mbox{$\cD_{\<\mathrm t}\>{\mathcal F}\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal F}\<,\>{\mathcal R_{\>\mathrm t}})\in\D_{\mkern-1.5mu\mathrm{qct}}$}, so that $\boldsymbol{\varGamma}\cD_{\<\mathrm t}\>{\mathcal F}\iso\cD_{\<\mathrm t}\>{\mathcal F}$ (\Pref{Gamma'(qc)}(a)); and similarly, ${\mathcal D}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}\>$. Furthermore, the argument in \Rref{R:Dtilde}(4) gives an isomorphism $\boldsymbol{\varGamma}\cD_{\<\mathrm t}\>{\mathcal F}\>\cong\boldsymbol{\varGamma}{\mathcal D}{\mathcal F}\<.$ \end{proof} \begin{sprop}\label{P:dualizing} With notation as in \Lref{L:dualizing} we have, for\/ ${\mathcal E},{\mathcal F}\in {\mathbf D}$$:$ \smallskip \noindent\textup{(a)} ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^*\!\iff\!\cD_{\<\mathrm t}\>{\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}$ and the natural map is an isomorphism $\,{\mathcal E}\!\iso\<\cD_{\<\mathrm t}\cDt\>{\mathcal E}$. \smallskip \noindent\textup{(b)} ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}\!\iff\!{\mathcal D}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}$ and the natural map is an isomorphism $\,{\mathcal F}\!\iso\<{\mathcal D}\cD{\mathcal F}$. \smallskip \noindent\textup{(c)} $\,{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}\!\iff\!\cD_{\<\mathrm t}\>{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^*$ and the natural map is an isomorphism $\,{\mathcal F}\!\iso\<\cD_{\<\mathrm t}\cDt\>{\mathcal F}$. \end{sprop} \begin{small} \emph{Remark.} The isomorphism ${\mathcal F}\iso\cD_{\<\mathrm t}\cDt\>{\mathcal F}\>$ is a formal version of ``Affine Duality, " see \cite[\S5.2]{AJL}.\index{Duality!Affine} \end{small} \smallskip \begin{proof} For ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}\>$, \Lref{L:dualizing}(b) gives ${\mathcal D}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}$, so \mbox{$\cD_{\<\mathrm t}{\mathcal F}\cong\boldsymbol{\varGamma}{\mathcal D}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^*$.} Moreover, from the isomorphism $\cD_{\<\mathrm t}\boldsymbol{\varGamma}{\mathcal F}\cong{\mathcal D}{\mathcal F}$ of \Lref{L:dualizing}(a) it follows that $\cD_{\<\mathrm t}(\D_{\mkern-1.5mu\mathrm c}^*)\subset\D_{\mkern-1.5mu\mathrm c}$. The $\Longleftarrow$ implications in (a), (b) and (c) result, as do the first parts of the $\implies$ implications. Establishing the isomorphisms ${\mathcal D}\cD{\mathcal F}\osi{\mathcal F}\iso\cD_{\<\mathrm t}\cDt\>{\mathcal F}$ is a local problem, so we may assume ${\mathscr X}$ affine. Since the functors ${\mathcal D}$ and $\cD_{\<\mathrm t}$ are bounded on~ $\D_{\<\vc}$ (see proof of \Lref{L:dualizing}(b)), and both take $\D_{\mkern-1.5mu\mathrm c}$ into $\D_{\<\vc}\>$, therefore the functors ${\mathcal D}\cD$ and $\cD_{\<\mathrm t}\cDt$ are bounded on~$\D_{\mkern-1.5mu\mathrm c}\>$, and so \cite[p.\,68, 7.1]{H1} (dualized) reduces the problem to the tautological case ${\mathcal F}={\mathcal O}_{\mathscr X}$ (cf.~\cite[p.\,258, Proposition~2.1]{H1}.) \smallskip For assertion (a) one may assume that ${\mathcal E}=\boldsymbol{\varGamma}{\mathcal F}\ ({\mathcal F}\in\D_{\mkern-1.5mu\mathrm c})$, so that there is a composed isomorphism (which one checks to be the natural map): $$ {\mathcal E}=\boldsymbol{\varGamma}{\mathcal F} \cong \boldsymbol{\varGamma}{\mathcal D}\cD{\mathcal F} \underset{\textup{\ref{L:dualizing}(b)}}\cong \cD_{\<\mathrm t}{\mathcal D}{\mathcal F} \underset{\textup{\ref{L:dualizing}(a)}}\cong \cD_{\<\mathrm t}\cDt\boldsymbol{\varGamma}{\mathcal F} = \cD_{\<\mathrm t}\cDt\>{\mathcal E}. \vspace{-5ex} $$ \end{proof} \smallskip \begin{scor}\label{P:ducomp} With the preceding notation, \smallskip \textup{(a)} The functor\/ ${\mathcal D}$ induces an involutive anti-equivalence of\/~$\D_{\mkern-1.5mu\mathrm c}$ with itself. \smallskip \textup{(b)} The functor\/ $\cD_{\<\mathrm t}$ induces quasi-inverse anti-equivalences between\/ $\D_{\mkern-1.5mu\mathrm c}$ and\/~$\D_{\mkern-1.5mu\mathrm c}^*$. \end{scor} \begin{slem}\label{L:building} Let\/ ${\mathscr J}$ be an ideal of definition of\/~${\mathscr X}$. Then a complex\/ ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}$ $(\<$resp.~${\mathcal R}\in\D_{\mkern-1.5mu\mathrm{qct}})$ is c-dualizing $($resp.~t-dualizing$)$ iff for every\/ $n>0$ the complex\/ ${\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n,\>{\mathcal R})$ is dualizing on the scheme\/ $X_n\!:=({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J}^n)$. \end{slem} \begin{proof} Remark (1) after \Dref{D:dualizing} makes it straightforward to see that if ${\mathcal R}$ is either c- or t-dualizing on~${\mathscr X}$ then ${\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n,\>{\mathcal R})$ is dualizing on $X_n$. For the converse, to begin with, \Cref{C:Hom-Rgamma} gives $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n\<,\>{\mathcal R})={\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n\<,\>\boldsymbol{\varGamma}{\mathcal R}), $$ and it follows from \Lref{L:interchange} that it suffices to consider the t-dualizing case. So suppose that ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm{qct}}$ and that for all~$n$,\vspace{.5pt} ${\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n,\>{\mathcal R})$ is dualizing on $X_n$. Taking \smash{$\tilde{\mathcal R}={\mathcal R}$} in the proof of \cite[Theorem 5.6]{Ye},\vspace{.5pt} one gets ${\mathcal O}_{\mathscr X}\iso{\mathbf R}\cH{om}^{\bullet}({\mathcal R}, {\mathcal R})$. \goodbreak It remains to check condition (iii) in \Dref{D:dualizing}. We may assume ${\mathcal R}$ to be K-injective, so that ${\mathcal R}_n\!:=\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J}^n\<,\>{\mathcal R})$ is K-injective on~$X_n$ for all~$n$. Then, since $\iGp{\mathscr X}{\mathcal R}\cong{\mathbf R}\iGp{\mathscr X}{\mathcal R}\cong{\mathcal R}$ (\Pref{Gamma'(qc)}(a)), $$ H^i\>{\mathcal R}\cong H^i\<\iGp{\mathscr X}{\mathcal R}\cong H^i\>\>\smash{\dirlm{n}}\!{\mathcal R}_n\cong \smash{\dirlm{n}}\!H^i\>{\mathcal R}_n\qquad(i\in\mathbb Z). $$ \smallskip\noindent For each $n$, ${\mathcal R}_n$ is quasi-isomorphic to a \emph{residual complex,} which is an injective ${\mathcal O}_{\!X_n}\<$-complex vanishing in degrees outside a certain finite interval $I\!:=[a,b]$ (\cite[pp.\:304--306]{H1}). If $m\le n$, the same holds---with the same $I$---for the complex \mbox{${\mathcal R}_m\cong\cH{om}_{X_n}({\mathcal O}_{\mathscr X}/{\mathscr J}^m\<, {\mathcal R}_n)$.} It follows that $H^i\>{\mathcal R}=0$ for $i\notin I$. In particular, ${\mathcal R}\in\D_{\mkern-1.5mu\mathrm{qct}}^+\>$. So now we we may assume that ${\mathcal R}$ is a bounded-below complex of $\A_{\mathrm {qct}}\<$-injectives \cite[Theorem 4.8]{Ye}. Then for any coherent torsion sheaf~${\mathcal M}$, the homology of $$ {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathcal M},\>{\mathcal R}) \underset{\textup{\ref{L:Hom=RHom}}}\cong \cH{om}^{\bullet}_{\mathscr X}({\mathcal M},\>{\mathcal R})\cong\cH{om}^{\bullet}_{\mathscr X}({\mathcal M},\>\dirlm{n}\!{\mathcal R}_n) \cong\dirlm{n}\cH{om}^{\bullet}_{\mathscr X}({\mathcal M},\>{\mathcal R}_n)\vspace{-5pt}\\ $$ vanishes in all degrees $>b$, as required by (iii). \end{proof} \begin{sprop}\label{P:twisted inverse} Let $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map of noetherian formal schemes. \vspace{1pt} \textup{(a)} If\/ ${\mathcal R}$ is a t\kern.6pt-dualizing complex on\/ ${\mathscr Y},$ then\/ $f_{\mathrm t}^\times{\mathcal R}$ is t\kern.6pt-dualizing on\/~${\mathscr X}$. \smallskip \textup{(b)} If\/ ${\mathcal R}$ is a c-dualizing complex on\/ ${\mathscr Y},$ then\/ $\ush f{\mathcal R}$ is c-dualizing on\/~${\mathscr X}$. \end{sprop} \begin{proof} (a) Let ${\mathscr J}$ be a defining ideal of~${\mathscr X}$, and let ${\mathscr I}$ be\vspace{2pt} a defining ideal of ${\mathscr Y}$ such that ${\mathscr I}{\mathcal O}_{\mathscr X}\subset{\mathscr J}$. Let $X_{\mathscr J}\!:=({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J})\overset{{\vbox to 0pt{\vskip-5pt\hbox{$\scriptstyle \,j$}\vss}}}\hookrightarrow{\mathscr X}$ and $Y_{\mathscr I}\!:= ({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I})\overset{{\vbox to 0pt{\vskip-3.5pt\hbox{$\scriptstyle i$}\vss}}}\hookrightarrow{\mathscr Y}$ be the resulting closed immersions. \Eref{ft-example}(4) shows that $i_{\mathrm t}^{\<\times}{\mathcal R} \cong {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal R}),$ which is a dualizing complex on~$Y_{\mathscr I}$. Pseudo\kern.6pt-properness of~$f$ means the map $f_{{\mathscr I}{\mathscr J}}\colon X_{\mathscr J}\to Y_{\mathscr I}$ induced by $f$ is proper, so as in \cite[p.\,396, Corollary 3]{f!} (hypotheses about finite Krull dimension being unnecessary here for the existence of $f_{\<\mathrm t}^{\mkern-1.5mu\times}\!\<$, etc.), $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J},\>f_{\mathrm t}^\times{\mathcal R})\cong j_{\mathrm t}^{\<\times}\<f_{\mathrm t}^\times{\mathcal R}\cong (f_{{\mathscr I}{\mathscr J}})_{\mathrm t}^{\<\times}i_{\mathrm t}^{\<\times}{\mathcal R} $$ is a dualizing complex on $X_{\mathscr J}$. The assertion is given then by \Lref{L:building}. \smallskip (b) By \Pref{P:coherence}, $\ush f{\mathcal R}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$. By \Cref{C:identities}, \Lref{L:interchange}(b), and the just-proved assertion (a), $$ {\mathbf R}\iGp{\mathscr X}\ush f{\mathcal R}\congf_{\mathrm t}^\times{\mathcal R}\congf_{\mathrm t}^\times\<{\mathbf R}\iGp{\mathscr Y}{\mathcal R}, $$ is t-dualizing on~${\mathscr X}$. So by \Lref{L:interchange}(a), $\ush f{\mathcal R}\cong{\boldsymbol\Lambda}_{\mathscr X}{\mathbf R}\iGp{\mathscr X}\ush f{\mathcal R}$ is c-dualizing. \end{proof} \pagebreak[3] The following proposition generalizes \cite[p.\,291, 8.5]{H1} (see also \cite[middle of p.\,384]{H1} and \cite[p.\,396, Corollary 3]{f!}). \begin{sprop}\label{P:Dual!} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map of noetherian formal schemes. Suppose that\/ ${\mathscr Y}$ has a c-dualizing complex\/~${\mathcal R_{\mathrm c}},$ or equivalently \textup(by \Lref {L:interchange}\,\textup{),} a t\kern.6pt-dualizing complex\/~${\mathcal R_{\>\mathrm t}}\in\D_{\mkern-1.5mu\mathrm c}^*({\mathscr Y}),$ so that $\ush f{\mathcal R_{\mathrm c}}$ is c-dualizing $($resp.~$f_{\mathrm t}^\times{\mathcal R_{\>\mathrm t}}$ is t\kern.6pt-dualizing\/$)$ on\/~${\mathscr X}$ $($\Pref{P:twisted inverse}$\mkern1.5mu)$. Define dualizing\index{dualizing functors} functors\looseness=-1 \begin{align*} \cD_{\<\mathrm t}^{\mathscr Y}(-)&\!:={\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}(-,{\mathcal R_{\>\mathrm t}}), & \qquad \cD_{\<\<\mathrm c}^{\mathscr Y}(-)&\!:={\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}(-,{\mathcal R_{\mathrm c}}), \\ \cD_{\<\mathrm t}^{\mathscr X}(-)&\!:={\mathbf R}\cH{om}^{\bullet}_{\mathscr X}(-,f_{\mathrm t}^\times{\mathcal R_{\>\mathrm t}}), & \qquad \cD_{\<\<\mathrm c}^{\mathscr X}(-)&\!:={\mathbf R}\cH{om}^{\bullet}_{\mathscr X}(-,\ush f{\mathcal R_{\mathrm c}}).\mathstrut \end{align*} Then there are natural isomorphisms \begin{alignat*}{2} f_{\mathrm t}^\times\<{\mathcal E} &\cong\cD_{\<\mathrm t}^{\mathscr X}{\mathbf L} f^*\cD_{\<\mathrm t}^{\mathscr Y}{\mathcal E},\qquad&&\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^*({\mathscr Y})\cap{\mathbf D}^+({\mathscr Y})\bigr),\\ \ush f\<{\mathcal E} &\cong\cD_{\<\<\mathrm c}^{\mathscr X}{\mathbf L} f^*\cD_{\<\<\mathrm c}^{\mathscr Y}{\mathcal E}\qquad&&\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^+\<({\mathscr Y})\bigr). \end{alignat*} \end{sprop} \begin{proof} When ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^*({\mathscr Y})\>\cap\>\>{\mathbf D}^+\<({\mathscr Y})$ (resp.\ ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y}))$ set ${\mathcal F}\!:=\cD_{\<\mathrm t}^{\mathscr Y}{\mathcal E}$ (resp.\ ${\mathcal F}\!:=\cD_{\<\<\mathrm c}^{\mathscr Y}{\mathcal E}$). In either case, ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr Y})$ (\Pref{P:dualizing}), and also ${\mathcal F}\in{\mathbf D}^-({\mathscr Y})$---in the first case by remark (2) following \Dref{D:dualizing} and \Lref{L:Hom=RHom}, in the second case by remark (1) following \Dref{D:dualizing}. So, by \Pref{P:dualizing}, we need to find natural isomorphisms \begin{align*} f_{\mathrm t}^\times\cD_{\<\mathrm t}^{\mathscr Y}{\mathcal F}&\cong\cD_{\<\mathrm t}^{\mathscr X}{\mathbf L} f^*\<{\mathcal F}, \\ \ush f\cD_{\<\<\mathrm c}^{\mathscr Y}{\mathcal F} &\cong \cD_{\<\<\mathrm c}^{\mathscr X}{\mathbf L} f^*\<{\mathcal F}\<. \end{align*} Such isomorphisms are given by the next result---a generalization of~\cite[p.\,194, 8.8(7)]{H1}---for ${\mathcal G}\!:={\mathcal R_{\>\mathrm t}}$ (resp.~${\mathcal R_{\mathrm c}}$).% \end{proof} \pagebreak[3] \begin{sprop}\label{P:Hom!} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of noetherian formal schemes. Then for\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^-({\mathscr Y})$ and\/ ${\mathcal G}\in{\mathbf D}^+({\mathscr Y})$ there are natural isomorphisms \begin{gather*} {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*\<{\mathcal F}\<, f_{\mathrm t}^\times{\mathcal G})\isof_{\mathrm t}^\times{\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}({\mathcal F}\<,{\mathcal G}),\\ {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*\<{\mathcal F}\<, \ush f{\mathcal G})\iso\ush f{\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}({\mathcal F}\<,{\mathcal G}). \end{gather*} \end{sprop} \begin{proof} The second isomorphism follows from the first, since $\ush f=\BLf_{\mathrm t}^\times$ and since there are natural isomorphisms \begin{align*} {\boldsymbol\Lambda}{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*\<{\mathcal F}\<, \>f_{\mathrm t}^\times{\mathcal G}) &={\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\bigl({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>, \> {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*\<{\mathcal F}\<,\>f_{\mathrm t}^\times{\mathcal G})\bigl)\\ &\cong\smash{{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\bigl({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\Otimes {\mathbf L} f^*\<{\mathcal F}\<,\>f_{\mathrm t}^\times{\mathcal G})\bigl)}\\ &\cong{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\bigl({\mathbf L} f^*\<{\mathcal F}\<, {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{},\>f_{\mathrm t}^\times{\mathcal G})\bigl)\\ &={\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf L} f^*\<{\mathcal F}\<, \>\ush f{\mathcal G}). \end{align*} For fixed ${\mathcal F}$ the source and target of the first isomorphism in \Pref{P:Hom!} are functors from ${\mathbf D}^+({\mathscr Y})$ to $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (see \Pref{P:Rhom}), right adjoint, respectively,\vspace{.7pt} to the functors \smash{${\mathbf R f_{\!*}}({\mathcal E}\Otimes{\mathbf L} f^*\<{\mathcal F})$} and \smash{${\mathbf R f_{\!*}}{\mathcal E}\Otimes{\mathcal F}$} $({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}))$. The functorial ``projection" map $$ \smash{{\mathbf R f_{\!*}}{\mathcal E}\Otimes{\mathcal F}\to{\mathbf R f_{\!*}}({\mathcal E}\Otimes{\mathbf L} f^*\<{\mathcal F}),} $$ is, by definition, adjoint to the natural composition $$ \smash{{\mathbf L} f^*({\mathbf R f_{\!*}}{\mathcal E}\Otimes{\mathcal F}\>)\to{\mathbf L} f^*{\mathbf R f_{\!*}}{\mathcal E}\Otimes{\mathbf L} f^*\<{\mathcal F} \to{\mathcal E}\Otimes{\mathbf L} f^*\<{\mathcal F};} $$ and it will suffice to show that this projection map is an isomorphism. For this, the standard strategy is to localize to where ${\mathscr Y}$ is affine, then use boundedness of some functors, and compatibilities with direct sums, to reduce to the trivial case ${\mathcal F}={\mathcal O}_{\mathscr Y}$. Details appear, e.g., in \cite[pp.\,123--125,~Proposition 3.9.4]{Derived categories}, modulo the following substitutions: use $\D_{\<\vc}$ in place of $\D_{\mkern-1.5mu\mathrm {qc}}$, and for boundedness and direct sums use \Lref{Gamma'+qc} and Propositions \ref{Rf_*bounded}(b) and~\ref{P:coprod} below. \end{proof}\index{dualizing complexes|)} \section{Direct limits of coherent sheaves on formal schemes.} \label{properly} In this section we establish, for a locally noetherian formal scheme~${\mathscr X}$, properties\- of $\A_{\vec {\mathrm c}}({\mathscr X})$ needed in \S\ref{sec-th-duality} to adapt Deligne's\index{Deligne, Pierre} proof of global Grothendieck Duality\- to the formal context. The basic result, \Pref{(3.2.2)}, is that $\A_{\vec {\mathrm c}}({\mathscr X})$ is~\emph{plump} (see opening remarks in \S\ref{S:prelim}), hence abelian, and so\ (being closed under~\smash{$\dirlm{}\!$})\vadjust{\kern.7pt} cocomplete, i.e., it has arbitrary small colimits. This enables us to speak about~${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$, and to apply standard adjoint functor theorems to colimit\kern.5pt-preserving functors on~$\A_{\vec {\mathrm c}}({\mathscr X})$. (See e.g., \Pref{A(vec-c)-A}, Grothendieck Duality for the identity map of~${\mathscr X}$). The preliminary paragraph~\ref{SS:vc-and-qc} sets up an equivalence of categories which allows us to reduce local questions about the (globally defined) category $\A_{\vec {\mathrm c}}({\mathscr X})$ to corresponding questions about quasi-coherent sheaves on ordinary noetherian schemes. Paragraph~\ref{SS:Dvc-and-Dqc} extends this equivalence to derived categories. As one immediate application, \Cref{corollary} asserts that the natural functor ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$ is an equivalence of categories when ${\mathscr X}$ is \emph{properly algebraic,}\index{properly algebraic} i.e., the $J$-adic completion of a proper $B$-scheme with $B$ a noetherian ring and $J$ a $B$-ideal. This will yield a stronger version of Grothendieck Duality on such formal schemes---for $\D_{\<\vc}({\mathscr X})$ rather than ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$, see \Cref{cor-prop-duality}. We do not know whether such global results hold over arbitrary noetherian formal schemes. \pagebreak[3] Paragraph~\ref{SS:bounded} establishes boundedness for some derived functors, a condition which allows us to apply them freely to unbounded complexes, as illustrated, e.g., in Paragraph~\ref{3.5}. \begin{parag}\label{SS:vc-and-qc} For $X$ a noetherian ordinary scheme, $\A_{\vec {\mathrm c}}(X)=\A_{\qc}\<(X)$ \cite[p.\,319,~6.9.9]{GD}. The inclusion $j_{\lower.2ex\hbox{$\scriptstyle\<\< X$}}\colon\A_{\qc}\<(X) \to {\mathcal A}(X)$ has a right adjoint $Q_{\<\<X}\colon{\mathcal A}(X) \to \A_{\qc}\<(X)$, the ``quasi-coherator,'' \index{ ${\mathbf R}$@ {}$Q_{\mathscr X}$ (quasi-coherator)\vadjust{\penalty 10000}} necessarily left exact \cite[p.\,187, Lemme 3.2]{I}. (See \Pref{A(vec-c)-A} and~\Cref{C:Qt} for generalizations to formal schemes.) \begin{sprop} \label{(3.2.1)} Let\/ $A$ be a noetherian adic ring with ideal of definition\/~$I,$\ let\/ $f_0\colon X\to{\mathrm {Spec}}(A)$ be a proper map, set $Z:=f_0^{-1}{\mathrm {Spec}}(A/I),$\ and let $$ \kappa\colon{\mathscr X}= X_{/Z}\to X $$ be the formal completion of\/~$X$ along\/~$Z$. Let\/ $Q\!:=Q_{\<\<X}$ be as above. Then\/ $\kappa^*$ induces equivalences of categories from\/ $\A_{\qc}\<(X)$ t\/o $\A_{\vec {\mathrm c}}({\mathscr X})$ and from\/ $\A_{\mathrm c}(X)$ to\/ $\A_{\mathrm c}({\mathscr X}),$\ both with quasi-inverse\/~$Q\kappa_*$. \end{sprop} \begin{proof} For any quasi-coherent ${\mathcal O}_{\<\<X}$-module~${\mathcal G}$ the canonical maps are \emph{isomorphisms} \stepcounter{sth} \renewcommand{\theequation}{\thesth} \begin{equation}\label{3.2.1.1} {\rm H}^i(X\<,{\mathcal G})\iso {\rm H}^i(X\<,\kappa_*\kappa^*{\mathcal G}) = {\rm H}^i({\mathscr X}, \kappa^*{\mathcal G})\qquad(i\ge0). \end{equation}\stepcounter{sth (The equality holds because $\kappa_*$ transforms any flasque resolution of~$\kappa^*{\mathcal G}$ into one of~$\kappa_*\kappa^*{\mathcal G}$.) For, if $({\mathcal G}_\lambda)$ is the family of coherent submodules of~${\mathcal G}$, ordered by inclusion, then $X$ and ${\mathscr X}$ being noetherian, one checks that (\ref{3.2.1.1}) is the composition of the sequence of natural isomorphisms \begin{alignat*}{2} {\rm H}^i(X\<,{\mathcal G})&\iso {\rm H}^i(X, \>\dirlm{\lambda}{\mathcal G}_\lambda)& &\hskip-60pt\mbox{\cite[p.~319,~(6.9.9)]{GD}}\\ &\iso \dirlm{\lambda} {\rm H}^i(X\<,{\mathcal G}_\lambda) &&\\ &\iso \dirlm{\lambda} {\rm H}^i({\mathscr X},\kappa^*{\mathcal G}_\lambda)& &\hskip-60pt\mbox{\cite[p.~125,~(4.1.7)]{EGA}}\\ &\iso {\rm H}^i({\mathscr X},\>\dirlm{\lambda}\kappa^*{\mathcal G}_\lambda)\\ &\iso{\rm H}^i({\mathscr X},\kappa^*\dirlm{\lambda} {\mathcal G}_\lambda) \iso {\rm H}^i({\mathscr X},\kappa^*{\mathcal G}). \end{alignat*} Next, for any ${\mathcal G}$ and ${\mathcal H}$ in $\A_{\qc}\<(X)$ the natural map is an \emph{isomorphism} \begin{equation} {\mathrm {Hom}}_X({\mathcal G},{\mathcal H})\iso {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal G},\>\kappa^*{\mathcal H}) \label{3.2.1.2} \end{equation} For, with ${\mathcal G}_\lambda$ as above, (\ref{3.2.1.2}) factors as the sequence of natural isomorphisms \begin{align*} {\mathrm {Hom}}_{X}({\mathcal G},{\mathcal H})&\iso \inlm{\lambda}{\mathrm {Hom}}_{X}({\mathcal G}_\lambda\>,{\mathcal H}) \\ &\iso \inlm{\lambda} {\rm H}^0\<\bigl({X\<, \cH{om}_{X}({\mathcal G}_\lambda\>,{\mathcal H})}\bigr) \\ &\iso \inlm{\lambda} {\rm H}^0\<\bigl({{\mathscr X},\kappa^* \cH{om}_{X}({\mathcal G}_\lambda\>,{\mathcal H}})\bigr) \qquad\bigl({\mbox{see }(\ref{3.2.1.1})}\bigr)\\ &\iso \inlm{\lambda} {\rm H}^0\< \bigl({{\mathscr X}, \cH{om}_{{\mathscr X}}(\kappa^*{\mathcal G}_\lambda\>,\kappa^*{\mathcal H})}\bigr) \\ &\iso \inlm{\lambda}{\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal G}_\lambda\>,\kappa^*{\mathcal H})\\ &\iso{\mathrm {Hom}}_{\mathscr X}(\dirlm{\lambda} \kappa^*{\mathcal G}_\lambda\>,\kappa^*{\mathcal H}) \iso {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal G},\>\kappa^*{\mathcal H}). \end{align*} Finally, we show the equivalence of the following conditions, for ${\mathcal F}\in{\mathcal A}({\mathscr X})$: \begin{list}% {(\arabic{t})} {\usecounter{t} \setlength{\rightmargin}{\leftmargin}} \item \label{1} The functorial map $\alpha({\mathcal F}\>)\colon\kappa^*Q\kappa_*{\mathcal F}\to{\mathcal F}$ (adjoint to the canonical map $Q\kappa_*{\mathcal F}\to \kappa_*{\mathcal F}\>)$ is an isomorphism. \item \label{2} There exists an isomorphism $\kappa^*{\mathcal G}\iso{\mathcal F}$ with ${\mathcal G}\in\A_{\qc}\<(X)$. \item \label{3} ${\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})$. \end{list} Clearly $(\ref{1}) \Rightarrow (\ref{2})$; and $(\ref{2}) \Rightarrow (\ref{3})$ because \smash{$\dirlm{}_{\!\!{}_\lambda}\,\kappa^*{\mathcal G}_\lambda\iso \kappa^*{\mathcal G}$} (${\mathcal G}_\lambda$ as before). Since $\kappa^*$ commutes with $\smash{\dirlm{}}$\vspace{1pt} and induces an equivalence of categories from $\A_{\mathrm c}(X)$ to $\A_{\mathrm c}({\mathscr X})$ \cite[p.~150,~(5.1.6)]{EGA}, we see that $(\ref{3}) \Rightarrow (\ref{2})$.\vspace{1pt} For ${\mathcal G}\in\A_{\qc}(X)$, let $\beta({\mathcal G})\colon{\mathcal G}\to Q\kappa_*\kappa^*{\mathcal G}$ be the canonical map (the unique one whose composition with $Q\kappa_*\kappa^*{\mathcal G}\to \kappa_*\kappa^*{\mathcal G}$ is the canonical map ${\mathcal G}\to\kappa_*\kappa^*{\mathcal G}$). Then for any ${\mathcal H}\in\A_{\qc}\<(X)$ we have the natural commutative diagram $$ \begin{CD} {\mathrm {Hom}}({\mathcal H},{\mathcal G}) @>\text{via }\beta>> {\mathrm {Hom}}({\mathcal H},Q\kappa_*\kappa^*{\mathcal G})\\ @V\simeq VV @VV\simeq V \\ {\mathrm {Hom}}(\kappa^*{\mathcal H},\kappa^*{\mathcal G}) @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>>{\mathrm {Hom}}({\mathcal H},\kappa_*\kappa^*{\mathcal G}) \end{CD} $$ \smallskip\noindent where the left vertical arrow is an isomorphism by (\ref{3.2.1.2}), the right one is an isomorphism because $Q$ is right-adjoint to $\A_{\qc}\<(X)\hookrightarrow{\mathcal A}(X)$, and the bottom arrow is an isomorphism because $\kappa_*$ is right-adjoint to $\kappa^*$; so ``via $\beta\>$'' is an isomorphism for all~${\mathcal H}$, whence \emph{$\beta({\mathcal G})$ is an isomorphism.} The implication $(\ref{2}) \Rightarrow (\ref{1})$ follows now from the easily checked fact that $\alpha(\kappa^*{\mathcal G}){\<\smcirc\<}\kappa^*\<\beta({\mathcal G})$ is the identity map of $\kappa^*{\mathcal G}$. We see also that $Q\kappa_*(\A_{\mathrm c}({\mathscr X}))\subset\A_{\mathrm c}(X)$, since by \cite[p.~150,~(5.1.6)]{EGA} every \mbox{${\mathcal F}\in\A_{\mathrm c}({\mathscr X})$} is isomorphic to $\kappa^*{\mathcal G}$ for some ${\mathcal G}\in\A_{\mathrm c}(X)$, and $\beta({\mathcal G})$ is an isomorphism. Thus we have the functors $\kappa^*\colon\A_{\qc}\<(X)\to\A_{\vec {\mathrm c}}({\mathscr X})$ and $Q\kappa_*\colon\A_{\vec {\mathrm c}}({\mathscr X})\to\A_{\qc}\<(X)$, both of which preserve coherence, and the functorial isomorphisms $$ \alpha({\mathcal F}\>)\colon\kappa^*Q\kappa_*{\mathcal F} \iso {\mathcal F} \ \ \bigl({{\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})}\bigr); \qquad \beta({\mathcal G})\colon{\mathcal G}\iso Q\kappa_*\kappa^*{\mathcal G} \ \ \bigl({{\mathcal G}\in\A_{\qc}\<(X)}\bigr)\<\<. $$ \Pref{(3.2.1)} results. \end{proof} \smallskip Since $\kappa^*$ is right-exact, we deduce: \begin{scor} \label{coker} For any affine noetherian formal scheme\/~${\mathscr X},$\ ${\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})$ iff\/ ${\mathcal F}$~is a cokernel of a map of free\/ ${\mathcal O}_{\mathscr X}$-modules\/ $($i.e., direct sums of copies of\/ ${\mathcal O}_{\mathscr X})$. \end{scor} \begin{scor}\label{C:vec-c is qc} For a locally noetherian formal scheme\/ ${\mathscr X},$\ $\A_{\vec {\mathrm c}}({\mathscr X})\subset\A_{\qc}\<({\mathscr X}),$\ i.e., any\/ \smash{$\dirlm{}\!\!$} of coherent\/ ${\mathcal O}_{\mathscr X}$-modules is quasi-coherent. \end{scor} \begin{proof} Being local, the assertion follows from \Cref{coker}. \end{proof} \pagebreak[3] \begin{scor}[cf.~{\cite[3.4, 3.5]{Ye}}] \label{C:images} For a locally noetherian formal scheme\/~${\mathscr X}$ let\/ ${\mathcal F}$ and\/ ${\mathcal G}$ be quasi-coherent ${\mathcal O}_{\mathscr X}$-modules. Then\/$:$ \textup{(a)} The kernel, cokernel, and image of any\/ ${\mathcal O}_{\mathscr X}$-homomorphism\/ ${\mathcal F}\to{\mathcal G}$ are quasi-coherent. \textup{(b)} ${\mathcal F}$ is coherent iff\/ ${\mathcal F}$ is locally finitely generated. \textup{(c)} If\/ ${\mathcal F}$ is coherent and\/ ${\mathcal G}$ is a sub- or quotient module of\/~${\mathcal F}$ then\/ ${\mathcal G}$ is coherent. \textup{(d)} If\/ ${\mathcal F}$ is coherent then\/ $\cH{om}({\mathcal F}\<,{\mathcal G})$ is quasi-coherent; and if also\/ ${\mathcal G}$ is coherent then\/ $\cH{om}({\mathcal F}\<,{\mathcal G})$ is coherent. \textup(For a generalization, see \Pref{P:Rhom}.\textup) \end{scor} \begin{proof} The questions being local, we may assume ${\mathscr X}={\mathrm {Spf}}(A)$ ($A$~noetherian adic), and, by \Cref{coker}, that ${\mathcal F}$ and ${\mathcal G}$ are in $\A_{\vec {\mathrm c}}({\mathscr X})$. Then, $\kappa^*$~being exact, \Pref{(3.2.1)} with $X\!:={\mathrm {Spec}}(A)$ and $f_0\!:=\text{identity}$ reduces the problem to noting that the corresponding statements about coherent and quasi-coherent sheaves on~$X$ are true. (These statements are in \cite[p.\,217, Cor.\:(2.2.2) and~p.\,228, \S(2.7.1)]{GD}. Observe also that if $F$ and $G$ are ${\mathcal O}_{\<\<X}$-modules with $F$ coherent then $\cH{om}_{\mathscr X}(\kappa^*\<F,\kappa^* G)\cong \kappa^*\cH{om}_X(F,G)$.) \end{proof} \begin{scor}\label{C:limsub} For a locally noetherian formal scheme\/ ${\mathscr X},$ any\/ \mbox{${\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})$} is the\/ \smash{$\dirlm{}\!\<$} of its coherent\/ ${\mathcal O}_{\mathscr X}$-submodules. \end{scor} \begin{proof} Note that by \Cref{C:images}(a) and~(b) the sum of any two coherent submodules of~${\mathcal F}$ is again coherent. By definition, ${\mathcal F}=\smash{\dirlm{}_{\!\!{}_\mu}\>\>{\mathcal F}_\mu}$ with ${\mathcal F}_\mu$ coherent, and from~ \Cref{C:images}(a) and~(b) it follows that the canonical image of ${\mathcal F}_\mu$ is a coherent submodule of~${\mathcal F}\<$, whence the conclusion. \end{proof} \begin{scor} \label{limit(vec-c)=qc} For any affine noetherian formal scheme\/~${\mathscr X},$\ any\/ ${\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})$ and any\/ $i>0,$ $$ {\rm H}^i({\mathscr X},{\mathcal F}\>)=0. $$ \end{scor} \begin{proof} Taking $f_0$ in \Pref{(3.2.1)} to be the identity map, we have \mbox{${\mathcal F}\cong\kappa^*{\mathcal G}$} with ${\mathcal G}$ quasi-coherent; and so by \eqref{3.2.1.1}, $\mathrm H^i({\mathscr X},{\mathcal F}\>)\cong \mathrm H^i({\mathrm {Spec}}(A),{\mathcal G})=0$. \end{proof} \end{parag} \smallskip \begin{parag} \Pref{(3.2.1)} will now be used to show, for locally noetherian formal schemes~${\mathscr X}$, that $\A_{\vec {\mathrm c}}({\mathscr X})\subset{\mathcal A}({\mathscr X})$ is plump, and that this inclusion has a right adjoint, extending to derived categories. \begin{slem} \label{L:Ext+lim} Let\/ ${\mathscr X}$ be a noetherian formal scheme, let\/ ${\mathcal F}\in\A_{\mathrm c}({\mathscr X}),$\ and let\/ $({\mathcal G}_\alpha\>,\gamma_{\alpha\beta}\colon{\mathcal G}_\beta\to {\mathcal G}_\alpha)_{\alpha,\>\beta\in \Omega}$ be a directed system in\/~$\A_{\mathrm c}({\mathscr X})$. Then for every\/~$q\ge0$ the natural map is an isomorphism $$ \dirlm{\alpha}\mathrm {Ext}^q({\mathcal F}\<,\>{\mathcal G}_\alpha)\iso \mathrm {Ext}^q({\mathcal F}\<,\>\dirlm{\alpha}{\mathcal G}_\alpha). $$ \end{slem} \begin{proof} For an ${\mathcal O}_{\mathscr X}$-module ${\mathcal M}$, let $\mathrm E({\mathcal M})$ denote the usual spectral sequence\looseness=-1 $$ \mathrm E\mspace{.5mu}_2^{pq}({\mathcal M})\!:=\mathrm H^p\bigl({\mathscr X},\>\E{xt}^{\>q}({\mathcal F}\<,{\mathcal M})\bigr) \Rightarrow \mathrm {Ext}^{p+q}({\mathcal F}\<,{\mathcal M}). $$ It suffices that the natural map of spectral sequences be an isomorphism $$ \smash{\dirlm{} \mathrm E({\mathcal G}_\alpha)\iso \mathrm E(\>\dirlm{} {\mathcal G}_\alpha)\qquad(\dirlm{}\!\<:=\dirlm{\alpha}\!),} $$ \smallskip\noindent and for that we need only check out the $\mathrm E\mspace{.5mu}_2^{pq}$ terms, i.e., show that the natural maps $$ \postdisplaypenalty 10000 \smash{\dirlm{}\mathrm H^p\bigl({\mathscr X},\>\E{xt}^{\>q}({\mathcal F}\<, {\mathcal G}_\alpha)\bigr)\to \mathrm H^p\bigl({\mathscr X},\>\dirlm{}\E{xt}^{\>q}({\mathcal F}\<, {\mathcal G}_\alpha)\bigr)\to \mathrm H^p\bigl({\mathscr X},\>\E{xt}^{\>q}({\mathcal F}\<, \dirlm{}{\mathcal G}_\alpha)\bigr)} $$ \smallskip\noindent are isomorphisms. The first one is, because ${\mathscr X}$ is noetherian. So we need only show that the natural map is an isomorphism $$ \dirlm{}\E{xt}^{\>q}({\mathcal F}\<,\>{\mathcal G}_\alpha)\iso \E{xt}^{\>q}({\mathcal F}\<,\>\dirlm{}{\mathcal G}_\alpha). $$ For this localized question we may assume that ${\mathscr X}={\mathrm {Spf}}(A)$ with $A$ a noetherian adic ring. By \Pref{(3.2.1)} (with $f_0$ the identity map of $X\!:={\mathrm {Spec}}(A)$) there is a coherent ${\mathcal O}_{\<\<X}$-module~$F$ and a directed system $(G_\alpha\>,g_{\alpha\beta}\colon G_\beta\to G_\alpha)_{\alpha,\>\beta\in\Omega}$ of coherent ${\mathcal O}_{\<\<X}$-modules such that ${\mathcal F}=\kappa^*F\<$, ${\mathcal G}_\alpha=\kappa^*G_\alpha\>$, and $\gamma_{\alpha,\>\beta}=\kappa^*\<g_{\alpha,\>\beta}$. Then the well-known natural isomorphisms (see \cite[(Chapter 0), p.\,61, Prop.\,(12.3.5)]{EGA}---or the proof of \Cref{(3.2.3)} below) \begin{multline*} \dirlm{}\E{xt}_{\mathscr X}^q({\mathcal F}\<,\>{\mathcal G}_\alpha) \iso\dirlm{}\kappa^*\E{xt}_{\<\<X}^{\>q}(F\<,G_\alpha)\iso \kappa^*\dirlm{}\E{xt}_{\<\<X}^{\>q}(F\<,G_\alpha) \\ \iso\kappa^*\E{xt}_{\<\<X}^{\>q}(F\<,\>\dirlm{}G_\alpha)\iso \E{xt}_{\mathscr X}^q(\kappa^*F\<, \>\kappa^*\dirlm{} \<G_\alpha)\iso \E{xt}_{\<\<X}^{\>q}({\mathcal F}\<, \>\dirlm{} {\mathcal G}_\alpha) \end{multline*} \penalty10000 give the desired conclusion. \end{proof} \begin{sprop} \label{(3.2.2)} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme. If\/ $$ {\mathcal F}_1\to{\mathcal F}_2\to{\mathcal F}\to{\mathcal F}_3\to{\mathcal F}_4 $$ is~an exact sequence of\/ ${\mathcal O}_{\mathscr X}$-modules and if\/ ${\mathcal F}_1\>,$ ${\mathcal F}_2\>,$ ${\mathcal F}_3$ and\/ ${\mathcal F}_4$ are all in\/ $\A_{\qc}\<({\mathscr X})$ $($resp.~$\A_{\vec {\mathrm c}}({\mathscr X}))$ then\/ ${\mathcal F} \in \A_{\qc}\<({\mathscr X})$ $($resp.~$\A_{\vec {\mathrm c}}({\mathscr X}))$. Thus\/ $\A_{\qc}\<({\mathscr X})$ and $\A_{\vec {\mathrm c}}({\mathscr X})$ are plump---hence abelian---subcategories of\/~${\mathcal A}({\mathscr X}),$\ and both $\D_{\mkern-1.5mu\mathrm {qc}}\<({\mathscr X})$ and its subcategory\/~$\D_{\<\vc}({\mathscr X})$ are triangulated subcategories of\/ ${\mathbf D}({\mathscr X})$. Furthermore, $\A_{\vec {\mathrm c}}({\mathscr X})$ is closed under arbitrary small\/ ${\mathcal A}({\mathscr X})$-colimits. \end{sprop} \begin{proof} Part of the $\A_{\qc}$ case is covered by \Cref{C:images}(a), and all of it by \cite[Proposition 3.5]{Ye}. At any rate, since every quasi-coherent ${\mathcal O}_{\mathscr X}$-module is locally in~$\A_{\vec {\mathrm c}}\subset\A_{\qc}$ (see Corollaries~\ref{coker} and~\ref{C:vec-c is qc}), it suffices to treat the $\A_{\vec {\mathrm c}}$ case. Let us first show that the kernel~$\mathcal K$ of an $\A_{\vec {\mathrm c}}$ map $$ \mspace{160mu}\psi\colon\smash{\dirlm{}_{\<\<\!\beta}}\>\>\H_\beta=\H\to{\mathcal G} =\smash{\dirlm{}_{\!\!\alpha}}\>\>{\mathcal G}_\alpha\qquad ({\mathcal G}_\alpha\>,\H_\beta\in\A_{\mathrm c}({\mathscr X})) $$ is itself in $\A_{\vec {\mathrm c}}({\mathscr X})$. It will suffice\vadjust{\kern1.5pt} to do so for the kernel~$\mathcal K_\beta$ of the composition $$ \smash{\psi_\beta\colon\H_\beta\xrightarrow{\text {natural}\,}\H\xrightarrow{\psi\>}{\mathcal G},} $$ since $\mathcal K=\smash{\dirlm{}_{\<\<\!\beta}\>\>\mathcal K_\beta\>}$. \pagebreak[2] By the case\vadjust{\kern.75pt} $q=0$ of \Cref{L:Ext+lim}, there is an $\alpha$ such that $\psi_\beta$ factors as\vadjust{\kern-3pt} $$ \H_\beta\xrightarrow{\psi_{\beta\alpha}\>}{\mathcal G}_\alpha \xrightarrow{\text {natural}\,}{\mathcal G}\>; $$ and then with $\mathcal K_{\beta\alpha'}\ (\alpha'>\alpha)$ the (coherent) kernel of the composed map $$ \H_\beta\xrightarrow{\psi_{\beta\alpha}\>}{\mathcal G}_\alpha \xrightarrow{\text {natural}\,}{\mathcal G}_{\alpha'} $$ we have $\mathcal K_\beta =\dirlm{}_{\<\<\!\alpha'}\>\>\mathcal K_{\beta\alpha'}\in\A_{\vec {\mathrm c}}({\mathscr X})$. Similarly, we find that $\text{coker}(\psi)\in\A_{\vec {\mathrm c}}({\mathscr X})$. Being closed under small direct sums, then, $\A_{\vec {\mathrm c}}({\mathscr X})$ is closed under arbitrary small ${\mathcal A}({\mathscr X})$-colimits \cite[Corollary\,2, p.\,109]{currante}. Consideration of the exact sequence $$ 0\longrightarrow\text{coker}({\mathcal F}_1\to{\mathcal F}_2)\longrightarrow{\mathcal F}\longrightarrow \ker({\mathcal F}_3\to{\mathcal F}_4)\longrightarrow 0 $$ now reduces the original question to where ${\mathcal F}_1={\mathcal F}_4=0$. Since ${\mathcal F}_3$ is the $\smash{\dirlm{}}$ of its coherent submodules (\Cref{C:limsub}) and ${\mathcal F}$ is the $\smash{\dirlm{}}$ of the inverse images of those submodules, we need only show that each such inverse image is in $\A_{\vec {\mathrm c}}({\mathscr X})$. Thus we may assume ${\mathcal F}_3$ coherent (and ${\mathcal F}_2=\smash{\dirlm{}_{\!\!\alpha}}\>{\mathcal G}_\alpha$ with ${\mathcal G}_\alpha$ coherent).\vspace{1pt} \penalty-1000 The exact sequence $0\to{\mathcal F}_2 \to {\mathcal F}\to {\mathcal F}_3 \to 0$ represents an element $$ \eta\in\mathrm{Ext}^1({\mathcal F}_3,\>{\mathcal F}_2)= \mathrm{Ext}^1({\mathcal F}_3,\>\smash{\dirlm{}_{\!\!\alpha}}\>{\mathcal G}_\alpha); $$ and by \Cref{L:Ext+lim}, there is an~$\alpha$ such that $\eta$~is the natural image of an element $\eta_\alpha\in\mathrm{Ext}^1({\mathcal F}_3,\>{\mathcal G}_\alpha)$, represented by an exact sequence~$0\to{\mathcal G}_\alpha \to {\mathcal F}_\alpha\to {\mathcal F}_3 \to 0$. Then ${\mathcal F}_\alpha$~is coherent, and by \cite[p.\,66, Lemma 1.4]{sM75}, we have an isomorphism $$ \postdisplaypenalty 10000 {\mathcal F}\iso {\mathcal F}_2\oplus_{{\mathcal G}_\alpha} \<{\mathcal F}_\alpha\>. $$ Thus ${\mathcal F}$ is the cokernel of a map in~$\A_{\vec {\mathrm c}}({\mathscr X})$, and so as above, ${\mathcal F}\in \A_{\vec {\mathrm c}}({\mathscr X})$. \end{proof} \begin{sprop} \label{A(vec-c)-A}\index{ ${\mathbf R}$@ {}$Q_{\mathscr X}$ (quasi-coherator)\vadjust{\penalty 10000}} On a locally noetherian formal scheme\/~${\mathscr X},$\ the inclusion functor\/ $j_{\lower.2ex\hbox{$\scriptstyle{\mathscr X}$}}\colon \A_{\vec {\mathrm c}}({\mathscr X}) \to {\mathcal A}({\mathscr X})$ has a right adjoint\/ $Q_{{\mathscr X}}\colon{\mathcal A}({\mathscr X}) \to \A_{\vec {\mathrm c}}({\mathscr X});$ and\/ ${\mathbf R} Q_{\mathscr X}^{} $ is right-adjoint to the natural functor ${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to {\mathbf D}({\mathscr X})$. In particular, if\/ $\kappa\colon{\mathscr X}\to X$ is as in \Pref{(3.2.1)} then\/ $Q_{{\mathscr X}}\cong\kappa^*Q_{\<\<X}\kappa_*$ and\/ ${\mathbf R} Q_{{\mathscr X}}\cong\kappa^*{\mathbf R} Q_{\<\<X}\kappa_*\>$. \end{sprop} \begin{proof} Since $\A_{\vec {\mathrm c}}({\mathscr X})$ has a small family of (coherent) generators, and is closed under arbitrary small ${\mathcal A}({\mathscr X})$-colimits, the existence of~$Q_{\mathscr X}^{}$ follows from the Special Adjoint Functor Theorem\index{Special Adjoint Functor Theorem} (\cite[p.\,90]{pF1964} or \cite[p.\,126, Corollary]{currante}).% \footnote{It follows that $\A_{\vec {\mathrm c}}({\mathscr X})$ is closed under \emph{all} ${\mathcal A}({\mathscr X})$-colimits (not necessarily small): if $F$ is any functor into $\A_{\vec {\mathrm c}}(X)$ and ${\mathcal F}\in{\mathcal A}(X)$ is a colimit of $j_{\<\lower.2ex\hbox{$\scriptscriptstyle{\mathscr X}$}}\smcirc F\<$, then $Q_{{\mathscr X}}{\mathcal F}$ is a colimit of $F\<$, and the natural map is an isomorphism ${\mathcal F}\iso j_{\<\lower.2ex\hbox{$\scriptscriptstyle{\mathscr X}$}}Q_{{\mathscr X}}{\mathcal F}$. (Proof: exercise, given in dual form in \cite[p.\,80]{pF1964}.) \looseness=-1} In an abelian category~${\mathcal A}$, a complex~$J$ is, by definition, K-injective if for each exact ${\mathcal A}$-complex $G$, the complex ${\mathrm {Hom}}^{\bullet}_{{\mathcal A}}(G, J)$ is exact too. Since $j_{\mathscr X}^{}$ is exact, it follows that its right adjoint $Q_{\mathscr X}^{}$ transforms K-injective ${\mathcal A}({\mathscr X})$-complexes into \mbox{K-injective} $\A_{\vec {\mathrm c}}({\mathscr X})$-complexes, whence the derived functor ${\mathbf R} Q_{\mathscr X}^{}$ is right-adjoint to the natural functor ${\mathbf D}(\A_{\vec {\mathrm c}}\<({\mathscr X})) \to {\mathbf D}({\mathscr X})$ (see \cite[p.\,129, Proposition~1.5(b)]{Sp}). The next assertion is a corollary of~\Pref{(3.2.1)}: any ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$ is isomorphic to $\kappa^*{\mathcal G}$ for some ${\mathcal G}\in\A_{\qc}\<(X)$, and then for any ${\mathcal N}\in{\mathcal A}({\mathscr X})$ there are natural isomorphisms\looseness=1 \begin{align*} {\mathrm {Hom}}_{\mathscr X}(j_{\lower.2ex\hbox{$\scriptstyle{\mathscr X}$}}{\mathcal M},\>{\mathcal N}\>)&\cong {\mathrm {Hom}}_{\mathscr X}(j_{\lower.2ex\hbox{$\scriptstyle{\mathscr X}$}}\kappa^*{\mathcal G},\>{\mathcal N}\>)\\ &\cong {\mathrm {Hom}}_X(j_{\lower.2ex\hbox{$\scriptstyle\<\<X$}}{\mathcal G},\> \kappa_*\>{\mathcal N}\>)\cong {\mathrm {Hom}}_{\A_{\qc}\<(X)}({\mathcal G},\>Q_{\<\<X}\kappa_*\>{\mathcal N}\>)\\ &\cong {\mathrm {Hom}}_{\A_{\vec {\mathrm c}}({\mathscr X})}(\kappa^*{\mathcal G},\>\kappa^*Q_{\<\<X}\kappa_*\>{\mathcal N}\>)\cong {\mathrm {Hom}}_{\A_{\vec {\mathrm c}}({\mathscr X})}({\mathcal M},\>\kappa^*Q_{\<\<X}\kappa_*\>{\mathcal N}\>). \end{align*} Moreover, since $\kappa_*$ has an exact left adjoint (viz.~$\kappa^*$), therefore, as above, $\kappa_*$~transforms K-injective ${\mathcal A}({\mathscr X})$-complexes into K-injective ${\mathcal A}(X)$-complexes, and it follows at once that ${\mathbf R} Q_{\mathscr X}^{}\cong\kappa^*{\mathbf R} Q_{\<\<X}\kappa_*$. \end{proof} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme. A property~$\mathbf P$ of sheaves of modules is \emph{local} if it is defined on~${\mathcal A}({\mathscr U})$ for arbitrary open subsets\/ ${\mathscr U}$ of\/~${\mathscr X}$, and is such that\/ for any ${\mathcal E}\in{\mathcal A}({\mathscr U})$ and any open covering\/ $({\mathscr U}_\alpha)$ of\/~${\mathscr U},$\ $\mathbf P({\mathcal E})$~holds iff\/ $\mathbf P({\mathcal E}|_{{\mathscr U}_\alpha}\<)$ holds for all~$\alpha$. For example, coherence and quasi-coherence are both local properties---to which by \Pref{(3.2.2)}, the following Proposition applies. \begin{sprop}\label{P:Rhom} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme, and let\/~$\mathbf P$ be a local property of\/ sheaves of modules. Suppose further that for all open\/~${\mathscr U}\subset{\mathscr X}$ the full subcategory\/ ${\mathcal A}_{\mathbf P}\<({\mathscr U})$ of\/~${\mathcal A}({\mathscr U})$ whose objects are all the ${\mathcal E}\in{\mathcal A}({\mathscr U})$ for which\/ $\mathbf P({\mathcal E})$ holds is a \emph{plump} subcategory of~$\>{\mathcal A}({\mathscr U})$. Then for all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^-\<({\mathscr X})$ and\/ ${\mathcal G}\in{\mathbf D}_{\mathbf P}^+\<({\mathscr X}),$\ it holds that ~${\mathbf R}\cH{om}^{\bullet}({\mathcal F},{\mathcal G})\in{\mathbf D}_{\mathbf P}^+\<({\mathscr X})$. \end{sprop} \begin{proof} Plumpness implies that ${\mathbf D}_{\mathbf P}\<({\mathscr X})$ is a triangulated subcategory of~${\mathbf D}({\mathscr X})$, as is $\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$, so \cite[p.\,68, Prop.\,7.1]{H1} gives a ``way-out'' reduction to where ${\mathcal F}$ and~${\mathcal G}$ are ${\mathcal O}_{\mathscr X}$-modules. The question being local on ${\mathscr X}$, we may assume ${\mathscr X}$ affine and replace ${\mathcal F}$ by a quasi-isomorphic bounded-above complex ${\mathcal F}\>^\bullet$ of finite-rank free ${\mathcal O}_{\mathscr X}$-modules, see \cite[p.\,427, (10.10.2)]{GD}. Then ${\mathbf R}\cH{om}^{\bullet}({\mathcal F}\>^\bullet\<,\>{\mathcal G})=\cH{om}^{\bullet}({\mathcal F}\>^\bullet\<,\>{\mathcal G})$, and the conclusion follows easily. \end{proof} \end{parag} \smallbreak \begin{parag} \label{SS:Dvc-and-Dqc} \Pref{A(vec-c)-A} applies in particular to any noetherian scheme~$X\<$. When $X$ is separated, $j_{\lower.2ex\hbox{$\scriptstyle\<\<X$}}$ induces an \emph{equivalence of categories} ${\boldsymbol j}_{\!X}\colon{\mathbf D} (\A_{\qc}\<(X)) \cong \D_{\mkern-1.5mu\mathrm {qc}}\<(X)$,% \index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} with quasi-inverse ${\mathbf R} Q_{\<\<X}|_{\D_{\mkern-1.5mu\mathrm {qc}}\<(X)}$. (See \cite[p.\,133, Corollary 7.19]{H1} for bounded-below complexes, and \cite[p.\,230, Corollary~5.5]{BN} or \cite[p.\,12, Proposition~(1.3)]{AJL} for the general case.) We do not know if such an equivalence, with ``$\vec{\mathrm c}\,$" in place of~``qc," always holds for separated noetherian formal schemes. The next result will at least take care of the ``properly algebraic" case, see \Cref{corollary}. \begin{sprop} \label{c-erator} In \Pref{(3.2.1)}\textup{,} the functor\/ $\kappa^*\colon{\mathbf D}(X)\to {\mathbf D}({\mathscr X})$ induces equivalences from\/ $\D_{\mkern-1.5mu\mathrm {qc}}\<(X)$ to\/ $\D_{\<\vc}({\mathscr X})$ and from\/ $\D_{\mkern-1.5mu\mathrm c}(X)$ to\/ $\D_{\mkern-1.5mu\mathrm c}({\mathscr X}),$\ both with quasi-inverse\/~${\mathbf R} Q\kappa_*$ $($where\/~${\mathbf R} Q$ stands for\/~${\boldsymbol j}_{\!X}\mspace{-1.5mu}\smcirc\<{\mathbf R} Q_{\<\<X}^{})$. \end{sprop} \begin{proof} Since $\kappa^*$ is exact, \Pref{(3.2.1)} implies that $\kappa^*(\D_{\mkern-1.5mu\mathrm {qc}}\<(X))\subset\D_{\<\vc}({\mathscr X})$ and $\kappa^*(\D_{\mkern-1.5mu\mathrm c}(X))\subset \D_{\mkern-1.5mu\mathrm c}({\mathscr X})$. So it will be enough to show that: \smallskip (1) If ${\mathcal F}\in\D_{\<\vc}({\mathscr X})$ then the functorial ${\mathbf D}({\mathscr X})$-map $\kappa^*{\mathbf R} Q\kappa_*{\mathcal F}\to {\mathcal F}$ adjoint to the natural map ${\mathbf R} Q\kappa_*{\mathcal F}\to \kappa_*{\mathcal F}$ is an isomorphism. (2) If ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}\<(X)$ then the natural map ${\mathcal G}\iso {\mathbf R} Q\kappa_*\kappa^*{\mathcal G}$ is an isomorphism. (3) If ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$ then ${\mathbf R} Q\kappa_*{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}(X)$. \smallskip Since $\D_{\<\vc}({\mathscr X})$ is triangulated (\Pref{(3.2.2)}), we can use way-out reasoning \cite[p.~68, Proposition~7.1 and p.~73, Proposition~7.3]{H1} to reduce to where ${\mathcal F}$ or ${\mathcal G}$ is a single sheaf. (For bounded-below complexes we just need the obvious facts that $\kappa^*$ and the restriction of ${\mathbf R} Q\kappa_*$ to~$\D_{\<\vc}({\mathscr X})$ are both bounded-below ($=$~way-out right) functors. For unbounded complexes, we need those functors to be bounded-above as well, which is clear for the exact functor $\kappa^*\<$, and will be shown for ${\mathbf R} Q\kappa_*|_{\D_{\<\vc}({\mathscr X})}$ in \Pref{(3.2.7.1)} below.) Any ${\mathcal F}\in\A_{\vec {\mathrm c}}({\mathscr X})$ is isomorphic to $\kappa^*{\mathcal G}$ for some ${\mathcal G}\in\A_{\qc}\<(X)$; and one checks that the natural composed map $\kappa^*{\mathcal G}\to \kappa^*{\mathbf R} Q\kappa_*\kappa^*{\mathcal G}\to\kappa^*{\mathcal G}$ is the identity, whence $(2)\Rightarrow(1)$. Moreover, if ${\mathcal F}\in\A_{\mathrm c}({\mathscr X}\>)$ then ${\mathcal G}\cong Q\kappa_*{\mathcal F}\in\A_{\mathrm c}(X)$, whence $(2)\Rightarrow(3)$. Now a map $\varphi:{\mathcal G}_1\to{\mathcal G}_2$ in $\D_{\mkern-1.5mu\mathrm {qc}}^+(X)$ is an isomorphism iff \begin{quote} \hskip-2.25em$(*)\colon\!$ the induced map ${\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n], \>{\mathcal G}_1)\to{\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n], \>{\mathcal G}_2)$ is an\newline isomorphism for every ${\mathcal E}\in\A_{\mathrm c}(X)$ and every $n\in\mathbb Z$. \end{quote} (For, if ${\mathcal V}$ is the vertex of a triangle with base~$\varphi$, then $(*)$ says that for all~\mbox{${\mathcal E}$, $n$,} ${\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n], {\mathscr V})=0$; but if $\varphi$ is not an isomorphism, i.e., ${\mathscr V}$ has non-vanishing homology, say $H^n({\mathcal V}) \neq 0$ and $H^i({\mathcal V})= 0$ for all $i<n$, then the inclusion~into~ $H^n({\mathcal V})$ of any coherent non-zero submodule~ ${\mathcal E}$ gives a non-zero map ${\mathcal E}[-n]\to{\mathscr V}$.) So for~(2) it's enough to check that the natural composition\looseness=-2 \begin{align*} {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n], \>{\mathcal G}) &\longrightarrow {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n], {\mathbf R} Q\kappa_*\kappa^*{\mathcal G})\\ &\!\iso {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}[-n],\kappa_*\kappa^*{\mathcal G}) \iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}(\kappa^*{\mathcal E}[-n], \kappa^*{\mathcal G}) \end{align*} is the \emph{isomorphism} ${\rm Ext}_X^n({\mathcal E}\<,{\mathcal G})\<\iso \< {\rm Ext}_{{\mathscr X}}^n(\kappa^*{\mathcal E}\<,\kappa^*{\mathcal G})$ in the following consequence of ~\eqref{3.2.1.1}: \begin{scor} \label{(3.2.3)} With $\kappa \colon{\mathscr X} \to X$ as in \Pref{(3.2.1)} and\/ \mbox{${\mathcal L}\in \D_{\mkern-1.5mu\mathrm {qc}}(X),$} the natural map\/ ${\mathbf R} \Gamma(X,{\mathcal L}) \to {\mathbf R}\Gamma({\mathscr X},\kappa^* {\mathcal L})$ is an isomorphism. In particular, for\/ ${\mathcal E} \in \D_{\mkern-1.5mu\mathrm c}^- (X)$ and\/ ${\mathcal G} \in \D_{\mkern-1.5mu\mathrm {qc}}^+(X)$ the natural map\/ ${\rm Ext}_X^n({\mathcal E}\<,{\mathcal G}) \to {\rm Ext}_{{\mathscr X}}^n(\kappa^*{\mathcal E}\<,\kappa^*{\mathcal G})$ is an isomorphism. \end{scor} {\it Proof.} After ``way-out'' reduction to the case where ${\mathcal L} \in \A_{\qc}\<(X)$ (the ${\mathbf R}\Gamma$'s are bounded, by \Cref{Rf_*bounded}(a) below), the first assertion is given by~\eqref{3.2.1.1}. To get the second assertion, take ${\mathcal L} \!:={\mathbf R}\cH{om}^{\bullet}_{\<\<X}({\mathcal E}\<,{\mathcal G})$ (which is in $\D_{\mkern-1.5mu\mathrm {qc}}^{\raise.2ex\hbox{$\scriptscriptstyle+$}}(X)$, \cite[p.~92,~Proposition~3.3]{H1}), so that $\kappa^*{\mathcal L} \cong {\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}}(\kappa^*{\mathcal E}\<,\kappa^*{\mathcal G})$ (as one sees easily after way-out reduction to where ${\mathcal E}$ and ${\mathcal G}$ are ${\mathcal O}_{\<\<X}$-modules, and further reduction to where $X$ is affine, so that ${\mathcal E}$ has a resolution by finite-rank free modules\dots\!). \end{proof} \begin{sdef} \label{D:propalg}\index{properly algebraic} A formal scheme ${\mathscr X}$ is said to be \emph{properly algebraic} if there exist a noetherian ring~$B$, a $B$-ideal~$J\<$, a proper $B$-scheme $X\<$, and an isomorphism from~${\mathscr X}$ to the $J$-adic completion of~$X\<$. \end{sdef} \begin{scor} \label{corollary} On a properly algebraic formal scheme\/~${\mathscr X}$ the natural functor\/ ${\boldsymbol j}_{\!{\mathscr X}}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} is an equivalence of categories with quasi-inverse\/~${\mathbf R} Q_{{\mathscr X}}^{}\>;$\ and therefore\/ ${\boldsymbol j}_{\!{\mathscr X}}\mspace{-1.5mu}\smcirc\<{\mathbf R} Q_{{\mathscr X}}^{}$ is right-adjoint to the inclusion\/ $\D_{\<\vc}({\mathscr X})\hookrightarrow{\mathbf D}({\mathscr X})$. \end{scor} \begin{proof} \smallskip If ${\mathscr X}$ is properly algebraic, then with $A\!:=J$-adic completion of~$B$ and $I\!:=JA$, it holds that ${\mathscr X}$ is the $I\<$-adic completion of~$X\otimes_{B}A$, and so we may assume the hypotheses and conclusions of \Pref{(3.2.1)}. We have also, as above, the equivalence of categories ${\boldsymbol j}_{\!X}\colon{\mathbf D}(\A_{\qc}(X))\to\D_{\mkern-1.5mu\mathrm {qc}}(X)$; and so the assertion follows from Propositions~\ref{c-erator} and~\ref{A(vec-c)-A}. \end{proof} \begin{sprop}\label{P:Lf*-vc} For a map\/ $g\colon{\mathscr Z}\to{\mathscr X}$ of locally noetherian formal schemes, $$ {\mathbf L} g^*\<(\D_{\<\vc}({\mathscr X}))\subset\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr Z}). $$ If\/ ${\mathscr X}$ is properly algebraic, then $$ {\mathbf L} g^*\<(\D_{\<\vc}({\mathscr X}))\subset\D_{\<\vc}({\mathscr Z}). $$ \end{sprop} \begin{proof} The first assertion, being local on~${\mathscr X}$, follows from the second. Assuming ${\mathscr X}$ properly algebraic we may, as in the proof of \Cref{corollary}, place ourselves in the situation of \Pref{(3.2.1)}, so that any ${\mathcal G}\in\D_{\<\vc}({\mathscr X})$ is, by \Cref{corollary} and \Pref{(3.2.1)}, isomorphic to $\kappa^*{\mathcal E}$ for some ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$. By \cite[p.\,10, Proposition~(1.1)]{AJL}), ${\mathcal E}$ is isomorphic to a \smash{$\dirlm{}\!\<$}\vspace{1pt} of bounded-above quasi-coherent flat complexes (see the very end of the proof of~\emph{ibid.}); and therefore ${\mathcal G}\cong\kappa^*{\mathcal E}$ is isomorphic to a K-flat complex of $\A_{\vec {\mathrm c}}({\mathscr X})$-objects. Since ${\mathbf L} g^*$~agrees with~$g^*$ on K-flat complexes, and $g^*(\A_{\vec {\mathrm c}}({\mathscr X}))\subset\A_{\vec {\mathrm c}}({\mathscr Z})$, we are done. \end{proof} \begin{srems} \label{(3.2.4.1)} (1) Let ${\mathscr X}$ be a properly algebraic formal scheme (necessarily noetherian) with ideal of definition~${\mathscr I}$, and set $I\!:={\rm H}^0({\mathscr X},{\mathscr I})\subset A\!:= {\rm H}^0({\mathscr X},{\mathcal O}_{\mathscr X})$. Then {\it $A$ is a noetherian $I\<$-adic ring, and ${\mathscr X}$ is ${\mathrm {Spf}}(A)$-isomorphic to the $I\<$-adic completion of a proper $A$-scheme}. Hence ${\mathscr X}$ is proper over~${\mathrm {Spf}}(A)$, via the canonical map given by \cite[p.~407,~(10.4.6)]{GD}. \penalty -1000 Indeed, with $B$, $J$ and $X$ as in \Dref{D:propalg}, \cite[p.\,125, Theorem~(4.1.7)]{EGA} implies that the topological ring $$ A=\inlm{\,n>0} {\rm H}^0({\mathscr X}, {\mathcal O}_{\mathscr X}/{\mathscr I}^n{\mathcal O}_{\mathscr X}) = \inlm{\,n>0} {\rm H}^0(X, {\mathcal O}_{\<\<X}/I^n{\mathcal O}_{\<\<X}) $$ is the $J$-adic completion of the noetherian $B$-algebra $A_0:= H^0(X,{\mathcal O}_{X})$, and that the $J$-adic and $I\<$-adic topologies on~$A$ are the same; and then ${\mathscr X}$ is the $I\<$-adic completion of~$X\otimes_{A_0}A$. (2) It follows that a quasi-compact formal scheme ${\mathscr X}$ is properly algebraic iff so~is each of its connected components. (3) While (1) provides a less relaxed characterization of properly algebraic formal schemes than \Dref{D:propalg}, \Cref{(3.5.2)} below provides a more relaxed one. \end{srems} \begin{slem} \label{(3.5.1)} Let\/ $X$ be a locally noetherian scheme, ${\mathcal I}_1\subset{\mathcal I}_2$ quasi-coherent\/ ${\mathcal O}_{\<\<X}$-ideals, $Z_i$ the support of\/ ${\mathcal O}_{\<\<X}/{\mathcal I}_i\>,$\ and\/ ${\mathscr X}_i$ the completion\/ $X_{\!/Z_i}\ (i=1,2)$. Suppose that\/ ${\mathcal I}_1{\mathcal O}_{{\mathscr X}_2}$ is an ideal of definition of\/ ${\mathscr X}_2$. Then\/ ${\mathscr X}_2$ is a union of connected components of\/ ${\mathscr X}_1$ $($with the induced formal-subscheme structure$)$. \end{slem} \begin{proof} We need only show that $Z_2$ is open in~$Z_1$. Locally we have a noetherian ring $A$ and $A$-ideals $I\subset J$ equal to their own radicals such that with ${\hat A}$ the $J$-adic completion, $J^n{\hat A}\subset I{\hat A}$ for some $n>0$; and we want the natural map $A/I\twoheadrightarrow A/J$ to be {\it flat}. (For then with $L\!:= J/I$, $L/L^2={\rm Tor}_1^{A/I}(A/J,A/J)=0$, whence \mbox{$(1-\ell)L = (0)$} for some $\ell\in L$, whence $\ell=\ell^2$ and $L=\ell( A/I)$, so that \mbox{$A/I\cong L\times (A/J)$} and ${\mathrm {Spec}}(A/J) \hookrightarrow{\mathrm {Spec}}(A/I)$ is open.) \begin{comment} \footnote{% If $A$ is a ring, and $L$ a finitely generated $A$-ideal such that the natural surjection $A \to A/L$ is flat, then $L/L^2={\rm Tor}_1^A(A/L,A/L)=0$, whence $(1-\ell)L = (0)$ for some $\ell\in L$, whence $\ell=\ell^2$ and $L=\ell A$, so that $A\cong L\times A/L$ and ${\mathrm {Spec}}(A/L) \hookrightarrow{\mathrm {Spec}}(A)$ is open.% } \end{comment} So it suffices that the localization $(A/I\>)_{1+J}\to (A/J\>)_{1+J} = A/J$ by the multiplicatively closed set~$1+J\>$ be an isomorphism, i.e., that its kernel $J (A/I\>)_{1+J}$ be nilpotent (hence (0), since $A/I$ is reduced.) But this is so because the natural map $A_{1+J}\to {\hat A}$ is faithfully flat, and therefore $J^n\< A_{1+J}\subset IA_{1+J}$. \end{proof} \vspace{-5pt} \begin{scor} \label{(3.5.2)} Let\/ $A$ be a noetherian ring, let\/ $I$ be an\/ $A$-ideal, and let\/ ${\hat A}$ be the\/ $I\<$-adic completion of\/ $A$. Let\/ $f_0\colon X \to{\mathrm {Spec}}(A)$ be a separated finite-type scheme-map, let\/ $Z$ be a closed subscheme of\/ $f_0^{-1}({\mathrm {Spec}}(A/I)),$\ let\/ ${\mathscr X}=X_{\</Z}$ be the completion of\/ $X$ along\/~$Z,$\ and let\/ $f\colon {\mathscr X} \to{\mathrm {Spf}}({\hat A})$ be the formal-scheme map induced by\/ $f_0\>$\textup{:} $$ \begin{CD} {\mathscr X}\!:=X_{\</Z} @>>> X \\ \vspace{-22pt}\\ @VfVV @VVf_0 V \\ \vspace{-22pt}\\ {\mathrm {Spf}}(\hat A) @>>> {\mathrm {Spec}}(A) \end{CD} $$ \smallskip\noindent If\/ $f$ is proper \textup(see\/ \textup{\S\ref{maptypes})} then\/ ${\mathscr X}$ is properly algebraic. \end{scor} \begin{proof} Consider a compactification of $f_0$ (see \cite[Theorem~3.2]{Lu}): $$ X {\begin{array}[t]{c} \hookrightarrow \\[-2.5 mm] \mbox{\tiny open} \end{array}} {\>\>\overline {\<\<X}} {\begin{array}[c]{c} \mbox{\scriptsize $\bar f_{\scriptscriptstyle\<0}^{}$}\\[-2 mm] \longrightarrow \\[-2.5 mm] \mbox{\tiny proper} \end{array}} {\mathrm {Spec}}(A). $$ Since $f$ is proper, therefore $Z$ is proper over ${\mathrm {Spec}}(A)$, hence closed in $\>\>{\overline {\<\<X}}\<$. Thus we may replace\- $f_0$~by~${\bar f_0}\>$, i.e., we may assume $f_0$ proper. Since $f\<$, being proper, is adic, \Lref{(3.5.1)}, with \hbox{$Z_2\!:= Z$} and $Z_1\!:= f_0^{-1}({\mathrm {Spec}}(A/I))$, shows that ${\mathscr X}$ is a union of connected components\- of the properly algebraic formal scheme~$X_{\<\</Z_1}$. Conclude by \Rref{(3.2.4.1)}(2). \end{proof} \end{parag} \pagebreak[3] \begin{parag} \label{SS:bounded} To deal with unbounded complexes we need the following boundedness results on certain derived functors. (See, e.g., Propositions~\ref{P:proper f*} and~\ref{P:kappa-f*} below.) \begin{sparag}\label{note1} Refer to \S\ref{maptypes} for the definitions of separated, resp.~affine, maps. A formal scheme~${\mathscr X}$ is \emph{separated}\index{formal scheme!separated} if the natural map \hbox{$f_{\mathscr X}\colon {\mathscr X}\to\text{Spec}({\mathbb Z})$} is separated, i.e., for some---hence any---ideal of definition~${\mathscr J}$, the scheme $({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J})$ is separated. For example, any locally noetherian affine formal scheme is separated. A locally noetherian formal scheme ${\mathscr X}$ is affine if and only if the map~$f_{\mathscr X}$ is affine, i.e., for some---hence any---ideal of definition~${\mathscr J}$, the scheme $({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J})$ is affine. Hence the intersection ${\mathscr V}\cap{\mathscr V}'$ of any two affine open subsets of a separated locally noetherian formal scheme~${\mathscr Y}$ is again affine. In other words, the inclusion ${\mathscr V}\hookrightarrow{\mathscr Y}$ is an affine map. More generally, if $f\colon{\mathscr X}\to {\mathscr Y}$ is a map of locally noetherian formal schemes, if ${\mathscr Y}$~is separated, and if ${\mathscr V}$ and ${\mathscr V}'$ are affine open subsets of~${\mathscr Y}$ and~${\mathscr X}$ respectively, then $f^{-1}{\mathscr V}\>\cap{\mathscr V}'$ is affine \cite[p.\,282, (5.8.10)]{GD}. \end{sparag} \begin{slem} \label{affine-maps} If\/ $g\colon {\mathscr X} \to {\mathscr Y}$ is an affine map\/ of locally noetherian formal schemes, then every\/ ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$ is $g_*\<$-acyclic, i.e., $R^ig_*{\mathcal M}=0$ for all\/ $i>0$. More generally, if\/ ${\mathcal G}\in\D_{\<\vc}({\mathscr X})$ and\/ $e\in\mathbb Z$ are such that\/ $H^i({\mathcal G})=0$ for all\/ $i\ge e$, then\/ $H^i({\mathbf R} g_*{\mathcal G})=0$ for all\/ $i\ge e$. \end{slem} \begin{proof} $R^ig_*{\mathcal M}$ is the sheaf associated to the presheaf ${\mathscr U} \mapsto {\rm H}^i({g^{-1}({\mathscr U}),{\mathcal M}})$, (${\mathscr U}$~open in~${\mathscr Y}$) \cite[Chap.\,0, (12.2.1)]{EGA}. If $\>{\mathscr U} $ is affine then so is $g^{-1}({\mathscr U})\subset {\mathscr X}$, and \Cref{limit(vec-c)=qc} gives ${\rm H}^i({g^{-1}({\mathscr U}),{\mathcal M}})=0$ for all $i>0$. Now consider in ${\mathbf K}({\mathscr X})$ a quasi-isomorphism ${\mathcal G}\to I$ where $I$ is a ``special" inverse limit of injective resolutions~$I_{-e}$ of the truncations ${\mathcal G}^{{\scriptscriptstyle\ge}e}$ (see \eqref{trunc}), so that $H^i({\mathbf R} g_*{\mathcal G})$ is the sheaf associated to the presheaf $\>{\mathscr U}\mapsto \mathrm H^i(\Gamma(g^{-1}{\mathscr U},I))$, see \cite[p.\,134, 3.13]{Sp}. If $C_{\<-e}$ is the kernel of the split surjection $I_{-e}\to I_{\>1-e}$ then $C_{\<-e}[e]$ is an injective resolution of $H^e({\mathcal G})\in\A_{\vec {\mathrm c}}({\mathscr X})$, and so for any affine open ${\mathscr U}\subset{\mathscr Y}$ and~$i>e$, \mbox{$\mathrm H^i(\Gamma(g^{-1}{\mathscr U},C_{\<-e}))=0$.} Applying \cite[p.\,126, Lemma]{Sp}, one finds then that for $i\ge e$ the natural map $\mathrm H^i(\Gamma(g^{-1}{\mathscr U},I))\to \mathrm H^i(\Gamma(g^{-1}{\mathscr U},I_{-e}))$ is an isomorphism. Consequently if $H^i({\mathcal G})=0$ for all~$i\ge e$ (whence $I_{-e}\cong {\mathcal G}^{{\scriptscriptstyle\ge}e}=0$ in~${\mathbf D}({\mathscr X})$) then $\mathrm H^i(\Gamma(g^{-1}{\mathscr U},I))=0$. \end{proof} \begin{sprop} \label{Rf_*bounded} Let\/ ${\mathscr X}$ be a noetherian formal scheme. Then: \textup{(a)} The functor\/ ${\mathbf R}\Gamma({\mathscr X},-)$ is bounded-above on\/~$\D_{\<\vc}({\mathscr X})$. In other words, there is an integer\/ $e\ge 0$ such that if\/ ${\mathcal G}\in\D_{\<\vc}({\mathscr X})$ and\/ $H^i({\mathcal G}\>)=0$ for all $i\ge i_0$ then\/ $\mathrm H^i({\mathbf R}\Gamma({\mathscr X},-))=0$ for all\/ $i\ge i_0+e$. \textup{(b)} For any formal-scheme map\/ $f\colon{\mathscr X}\to{\mathscr Y}$ with\/ ${\mathscr Y}$ quasi-compact, the functor\/~${\mathbf R f_{\!*}}$ is bounded-above on\/~$\D_{\<\vc}({\mathscr X}),$\ i.e., there is an integer\/ $e\ge 0$ such that if\/~${\mathcal G}\in\D_{\<\vc}({\mathscr X})$ and\/ $H^i({\mathcal G}\>)=0$ for all $i\ge i_0$ then\/ $H^i({\mathbf R f_{\!*}}{\mathcal G}\>)=0$ for all\/ $i\ge i_0+e$. \end{sprop} \begin{proof} Let us prove (b). (The proof of (a) is the same, \emph{mutatis mutandis.}) Suppose first that ${\mathscr X}$ is separated, see~\S\ref{note1}. Since ${\mathscr Y}$ has a finite affine open cover and ${\mathbf R f_{\!*}}$ commutes with open base change, we may assume that ${\mathscr Y}$ itself is affine. Let $n({\mathscr X})$ be the least positive integer~$n$ such that there exists a finite affine open cover ${\mathscr X}=\cup_{i=1}^n {\mathscr X}_i\>$, and let us show by induction on~$n({\mathscr X})$ that $e\!:=n({\mathscr X})-1$ \emph{will do.} \enlargethispage*{.6\baselineskip} The case $n({\mathscr X})=1$ is covered by \Lref{affine-maps}. So assume that $n\!:=n({\mathscr X})\ge 2$, let ${\mathscr X}=\cup_{i=1}^n {\mathscr X}_i$ be an affine open cover, and let $u_{1}\colon\<{\mathscr X}_1\hookrightarrow {\mathscr X}$,\,\ $u_2\colon\!\cup_{i=2}^n {\mathscr X}_i \hookrightarrow {\mathscr X}$, \,\ $u_3\colon\!\cup_{i=2}^n ({\mathscr X}_1 \cap {\mathscr X}_i) \hookrightarrow {\mathscr X}$ be the respective inclusion maps. Note that ${\mathscr X}_1 \cap\> {\mathscr X}_i$ is affine because ${\mathscr X}$ is separated. So by the inductive\vadjust{\penalty-500} hypothesis, the assertion holds for the~maps $f_i\!:=f\smcirc u_i\ (i=1,2,3)$. \pagebreak[3] Now apply the $\Delta$-functor~${\mathbf R f_{\!*}}$ to the ``Mayer\kern.5pt-Vietoris" triangle% \index{Mayer-Vietoris triangle} $$ {\mathcal G} \longrightarrow {\mathbf R} u_{1*}^{}u_1^*{\mathcal G} \oplus {\mathbf R} u_{2*}^{}u_{2}^*{\mathcal G}\longrightarrow {\mathbf R} u_{3*}^{}u_{3}^*{\mathcal G} \stackrel{+1\>}{\longrightarrow} $$ (derived from the standard exact sequence $$ 0\to{\mathcal E}\to u_{1*}^{}u_1^*{\mathcal E} \oplus u_{2*}^{}u_2^*{\mathcal E}\to u_{3*}^{}u_3^*{\mathcal E}\to 0 $$ where \hbox{${\mathcal G}\to{\mathcal E}$} is a K-injective resolution) to get the ${\mathbf D}({\mathscr Y})$-triangle $$ {\mathbf R f_{\!*}}{\mathcal G} \longrightarrow {\mathbf R} f_{1*}^{}u_1^*{{\mathcal G}} \oplus {\mathbf R} f_{2*}^{}u_{2}^*{\mathcal G}\longrightarrow {\mathbf R} f_{3*}^{}u_{3}^*{\mathcal G} \stackrel{+1\>}{\longrightarrow} $$ whose associated long exact homology sequence yields the assertion for~$f$. The general case can now be disposed of with a similar Mayer\kern.5pt-Vietoris induction on the least number of \emph{separated} open subsets needed to cover~${\mathscr X}$. \end{proof} \begin{sprop} \label{(3.2.7.1)} Let\/ $X$ be a separated noetherian scheme, let\/ $Z\subset X$ be a closed subscheme, and let\/ $\kappa_{\mathscr X}^{\phantom{.}}\colon{\mathscr X}=X_{/Z} \to X$ be the completion map. Then the~functor\/~${\mathbf R} Q_{\<\<X} \kappa_*$ is bounded-above on\/ $\D_{\<\vc}({\mathscr X})$. \looseness =1 \end{sprop} \begin{proof} Set $\kappa\!:=\kappa_{\mathscr X}^{}$. Let $n(X)$ be the least number of affine open subschemes needed to cover~$X\<$. When $X$ is affine, $Q_{\<\<X}$ is the sheafification of the global section functor, and since $\kappa_*$ is exact and, being right adjoint to the \emph{exact} functor~$\kappa^*\<$, preserves K-injectivity, we find that for any ${\mathcal F}\in {\mathbf D}({\mathscr X})$, ${\mathbf R} Q_{\<\<X}\kappa_*{\mathcal F}$ is the sheafification of the complex ${\mathbf R} \Gamma(X, \kappa_*{\mathcal F}\>)={\mathbf R} \Gamma({\mathscr X}\<, {\mathcal F}\>)$. Thus \Pref{Rf_*bounded}(a) yields the desired result for $n(X)=1$. Proceed by induction when $n(X)>1$, using a ``Mayer\kern.5pt-Vietoris'' argument as in the proof of \Pref{Rf_*bounded}. The enabling points are that if $v\colon V\hookrightarrow X$ is an open immersion with $n(V)<n(X)$, giving rise to the natural commutative diagram $$ \CD V_{\<\</Z\cap V}=:\:@.{\mathscr V} @>\kappa_{\mathscr V}^{\phantom{.}}>> V \\ @.@V\hat v VV @VVvV \\ @.{\mathscr X}@>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle\kappa_{\mathscr X}^{\phantom{.}}$}\vss}> X \endCD $$ then there are natural isomorphisms, for ${\mathcal F}\in\D_{\<\vc}({\mathscr X})$ and $v_*^{\rm qc}\colon\A_{\qc}\<(V)\to \A_{\qc}\<(X)$ the restriction of $v_*\>$: $$ {\mathbf R} Q_{\<\<X} \kappa_{{\mathscr X}*}^{\phantom{.}}{\mathbf R} {\hat v}_*{\hat v}^*{\mathcal F} \cong {\mathbf R} Q_{\<\<X} {\mathbf R} v_*\kappa_{{\mathscr V}*}^{\phantom{.}}{\hat v}^*{\mathcal F} \cong {\mathbf R} v_*^{\rm qc} {\mathbf R} Q_V \kappa_{{\mathscr V}*}^{\phantom{.}}{\hat v}^*{\mathcal F}, $$ and the functor ${\mathbf R} Q_V\kappa_{{\mathscr V}*}^{\phantom{.}}{\hat v}^*$ is bounded-above, by the inductive hypothesis on $n(V)<n(X)$, as is ${\mathbf R} v_*^{\rm qc}$, by the proof of \cite[p.\,12, Proposition (1.3)]{AJL}. \end{proof} \end{parag} \medskip \pagebreak[3] \begin{parag}\label{3.5} Here are some examples of how boundedness is used. \begin{sprop}\label{P:proper f*} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a proper map of noetherian formal schemes. Then $$ {\mathbf R} f_{\!*}\D_{\mkern-1.5mu\mathrm c}({\mathscr X})\subset\D_{\mkern-1.5mu\mathrm c}({\mathscr Y})\quad\textup{and}\quad {\mathbf R} f_{\!*}\D_{\<\vc}({\mathscr X})\subset\D_{\<\vc}({\mathscr Y}). $$ \end{sprop} \begin{proof} For a coherent\vspace{.4pt} ${\mathcal O}_{\mathscr X}$-module ${\mathcal M}$, ${\mathbf R} f_{\!*}{\mathcal M}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr Y})$ \cite[p.\,119, (3.4.2)]{EGA}. Since ${\mathscr X}$ is noetherian, the homology functors $H^i{\mathbf R} f_{\!*}$ commute with\vspace{.9pt} \smash{$\dirlm{}\!\!$} on ${\mathcal O}_{\mathscr X}$-modules, whence\vspace{.6pt} ${\mathbf R} f_{\!*}{\mathcal N}\in\D_{\<\vc}({\mathscr Y})$ for all ${\mathcal N}\in\A_{\vec {\mathrm c}}({\mathscr X})$. ${\mathbf R} f_{\!*}$ being bounded on~$\D_{\<\vc}({\mathscr X})$ (\Pref{Rf_*bounded}(b)), way-out reasoning \cite[p.\,74, (iii)]{H1} completes the proof.\looseness=-1 \end{proof} \begin{sprop}\label{P:coprod} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of quasi-compact formal schemes, with\/ ${\mathscr X}$ noetherian. Then the functor\/ ${\mathbf R f_{\!*}}|_{\D_{\<\vc}({\mathscr X})}$ commutes with small direct sums, i.e., for any small family\/ $({\mathcal E}_\alpha)$ in\/~$\D_{\<\vc}({\mathscr X})$ the natural map $$ \oplus_{\<\alpha}( {\mathbf R f_{\!*}}{\mathcal E}_\alpha) \to{\mathbf R f_{\!*}}(\oplus_{\<\alpha}\>{\mathcal E}_\alpha) $$ is a\/ ${\mathbf D}({\mathscr Y})$-isomorphism. \end{sprop} \begin{proof} It suffices to look at the induced homology maps in each degree, i.e., setting $R^{\>i}\<\<f_{\!*}\!:= H^i{\mathbf R f_{\!*}}\ (i\in\mathbb Z)$, we need to show that \emph{the natural map} $$ \oplus_{\<\alpha}( R^{\>i}\<\<f_{\!*}{\mathcal E}_\alpha)\iso R^{\>i}\<\<f_{\!*}(\oplus_{\<\alpha}\>{\mathcal E}_\alpha). $$ \emph{is an isomorphism.} For any ${\mathcal F}\in\D_{\<\vc}({\mathscr X})$ and any integer $e\ge0$, the vertex~${\mathcal G}$ of a triangle based on the natural map~$t_{i-e}$ from ${\mathcal F}\>$ to the truncation ${\mathcal F}^{\>{\scriptscriptstyle\ge}i-e}$ (see~\eqref{trunc}) satisfies $H^j({\mathcal G})=0$ for all $j\ge i-e-1$; so if $e$ is the integer in~\Pref{Rf_*bounded}(b), then $R^{\>i-1}\<\<f_{\!*}{\mathcal G}=R^{\>i}\<f_{\!*}{\mathcal G}=0$, and the map induced by $t_{i-e}$ is an \emph{isomorphism}\looseness=-1 $$ R^{\>i}\<\<f_{\!*}{\mathcal F}\iso R^{\>i}\<\<f_{\!*}{\mathcal F}^{\>{\scriptscriptstyle\ge}i-e}\<. $$ We can therefore replace each ${\mathcal E}_\alpha$ by ${\mathcal E}_\alpha^{{\scriptscriptstyle\ge}i-e}$, i.e., we may assume that the ${\mathcal E}_\alpha$ are uniformly bounded below. We may assume further that each complex~${\mathcal E}_\alpha$ is injective, hence $f_{\!*}$-acyclic (i.e., the canonical map is an \emph{isomorphism} $f_{\!*}{\mathcal E}_\alpha\iso {\mathbf R f_{\!*}}{\mathcal E}_\alpha$). Since ${\mathscr X}$ is noetherian, $R^{\>i}\<\<f_{\!*}$ commutes with direct sums; and so each component of $\oplus_{\<\alpha}\>{\mathcal E}_\alpha$ is an $f_{\!*}$-acyclic ${\mathcal O}_{\mathscr X}$-module. This implies that the bounded-below complex $\oplus_{\<\alpha}\>{\mathcal E}_\alpha$ is itself $f_{\!*}$-acyclic. Thus in the natural commutative diagram $$ \begin{CD} \oplus_{\<\alpha}( f_{\!*}{\mathcal E}_\alpha)@>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>>f_{\!*}(\oplus_{\<\alpha}\>{\mathcal E}_\alpha)\\ @V\simeq VV @VV\simeq V\\ \oplus_{\<\alpha}( {\mathbf R f_{\!*}}{\mathcal E}_\alpha)@>>>{\mathbf R f_{\!*}}(\oplus_{\<\alpha}\>{\mathcal E}_\alpha) \end{CD} $$ the top and both sides are isomorphisms, whence so is the bottom. \end{proof} The following Proposition generalizes \cite[p.\,92, Theorem (4.1.5)]{EGA}. \begin{sprop}\label{P:kappa-f*} Let\/ $f_0\colon\! X\to Y$ be a proper map of locally noetherian schemes, let\/ $W\subset Y$ be a closed subset, let\/ $Z\!:=f_0^{-1}W,$\ let\/ $\kappa_{\mathscr Y}^{\phantom{.}}\colon{\mathscr Y}=Y_{/W}\to Y$ and\/~ $\kappa_{\mathscr X}^{\phantom{.}}\colon{\mathscr X}=X_{/Z}\to X$ be the respective \(flat\) completion maps, and let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be the map induced by~$f_0\>$. Then for ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$ the map\/ $\theta_{\<{\mathcal E}}$ adjoint to the natural composition $$ {\mathbf R} f_{\<0*}^{}{\mathcal E}\longrightarrow{\mathbf R} f_{\<0*}^{}\kappa_{{\mathscr X}*}^{\phantom{.}}\kappa_{\mathscr X}^*{\mathcal E}\iso \kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E} $$ is an \emph{isomorphism} $$ \theta_{\<{\mathcal E}}\colon \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathcal E}\iso {\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}\<. $$ \end{sprop} \begin{proof} We may assume $Y$ affine, say $Y={\mathrm {Spec}}(A)$, and then $W={\mathrm {Spec}}(A/I)$ for some $A$-ideal~$I\<$. Let $\hat A$ be the $I\<$-adic completion of~$A$, so that there is a natural cartesian diagram $$ \begin{CD} X\otimes_A\hat A=:\>@. X_1 @>k_{\<X}^{}>> X \\ @.@V f_1 VV @VV f_0 V \\ {\mathrm {Spec}}(\hat A)=:\>@. Y_1 @>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle k_Y^{}$}\vss}> Y \end{CD} $$ Here $k_Y^{}$ is flat, and the natural map is an isomorphism $k_Y^*{\mathbf R} f_{\<0*}^{}{\mathcal E}\iso {\mathbf R} {f_{\<1*}^{}}k_{\<X}^*{\mathcal E}\colon$ since ${\mathbf R} f_{\<0*}^{}$ (resp.~${\mathbf R} {f_{\<1*}^{}})$ is bounded-above on~$\D_{\mkern-1.5mu\mathrm {qc}}(X)$ (resp.~$\D_{\mkern-1.5mu\mathrm {qc}}(X_1)$), see \Pref{Rf_*bounded}(b), way-out reasoning reduces this assertion to the well-known case where ${\mathcal E}$ is a single quasi-coherent ${\mathcal O}_X$-module. Simple considerations show then that we can replace $f_0$ by $f_1$ and ${\mathcal E}$ by $k_{\<X}^*{\mathcal E}$; in other words, we can assume \mbox{$A=\hat A$}. From \Pref{P:proper f*} it follows that ${\mathbf R} f_{\<0*}^{}{\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(Y)$ and ${\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}\in \D_{\<\vc}({\mathscr Y})$. Recalling the equivalences in~\Pref{c-erator}, we see that any ${\mathcal F}\in\D_{\<\vc}({\mathscr Y})$ is isomorphic to~$\kappa_{\mathscr Y}^*{\mathcal F}_0$ for some ${\mathcal F}_0\in\D_{\mkern-1.5mu\mathrm {qc}}(Y)$ (so that ${\mathbf L} f_{\<0}^*{\mathcal F}_0\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$), and that there is a sequence of natural isomorphisms \begin{align*} {\mathrm {Hom}}_{\mathscr Y}({\mathcal F}\<, \>\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathcal E}) &\iso {\mathrm {Hom}}_Y({\mathcal F}_0\>, \>{\mathbf R} f_{\<0*}^{}{\mathcal E}) \\ &\iso {\mathrm {Hom}}_X({\mathbf L} f_{\<0}^*{\mathcal F}_0\>, \>{\mathcal E}) \\ &\iso {\mathrm {Hom}}_{\mathscr X}(\kappa_{\mathscr X}^*{\mathbf L} f_{\<0}^*{\mathcal F}_0\>, \>\kappa_{\mathscr X}^*{\mathcal E}) \\ &\iso {\mathrm {Hom}}_{\mathscr X}({\mathbf L} f^*\<\kappa_{\mathscr Y}^*{\mathcal F}_0\>, \>\kappa_{\mathscr X}^*{\mathcal E}) \iso {\mathrm {Hom}}_{\mathscr Y}({\mathcal F}\<, \>{\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}). \end{align*} The conclusion follows. \end{proof} \end{parag} \section{Global Grothendieck Duality.} \label{sec-th-duality} \index{Grothendieck Duality!global} \begin{thm} \label{prop-duality} Let\/ $f\colon\<{\mathscr X} \to {\mathscr Y}$ be a map of quasi-compact formal schemes, with ${\mathscr X}$~noetherian, and let\/~${\boldsymbol j}\colon\<{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} \to{\mathbf D}({\mathscr X})$ be the natural functor. Then the\/ \hbox{$\Delta$-functor\/}\vadjust{\kern.4pt} ${\mathbf R f_{\!*}} \<\smcirc\>{\boldsymbol j}\>$ has a right\/ $\Delta$-adjoint. In fact there is a bounded-below\/ \hbox{$\Delta$-functor\/} $f^{\times}\<\colon{\mathbf D}({\mathscr Y}) \to{\mathbf D}\left({\A_{\vec {\mathrm c}}({\mathscr X})}\right)\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$f^\times\<\<$}$\vadjust{\kern.3pt} and a map of\/ $\Delta$-functors $\tau\colon{\mathbf R f_{\!*}} \>{\boldsymbol j} f^{\times}\to {\bf 1}$\index{ {}$\tau$ (trace map)} such that for all\/ ${\mathcal G}\in{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ and\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y}),$\ the composed map $($in\/ the derived category of abelian groups\/$)$ \begin{align*} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\A_{\vec {\mathrm c}}({\mathscr X})\<}({\mathcal G},\>f^\times\<\<{\mathcal F}\>) &\xrightarrow{\mathrm{natural}\,} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{{\mathcal A}({\mathscr Y})\<}({\mathbf R f_{\!*}}\>{\boldsymbol j}\>{\mathcal G}, \>{\mathbf R f_{\!*}} \>{\boldsymbol j} f^{\times}\<{\mathcal F}\>)\\ &\xrightarrow{\;\>\mathrm{via}\ \tau\ } {\mathbf R}{\mathrm {Hom}}^{\bullet}_{{\mathcal A}({\mathscr Y})\<}({\mathbf R f_{\!*}} \>{\boldsymbol j}\>{\mathcal G},\>{\mathcal F}\>) \end{align*} is an isomorphism. \end{thm} With \Cref{corollary} this gives: \begin{scor}\label{cor-prop-duality} If\/ ${\mathscr X}$ is properly algebraic, the restriction of\/~${\mathbf R f_{\!*}}$ to\/~$\D_{\<\vc}({\mathscr X})$ has a right\/ $\Delta$-adjoint \textup(also to be denoted $f^\times$ when no confusion results\/\textup). \end{scor} \noindent\emph{Remarks.} 1.~Recall that over any abelian category~${\mathcal A}$ in which each complex~${\mathcal F}$ has a K-injective resolution~$\rho({\mathcal F}\>)$, we can set $$ {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathcal A}({\mathcal G},{\mathcal F}\>)\!:={\mathrm {Hom}}^{\bullet}_{\mathcal A}\bigl({\mathcal G},\rho({\mathcal F}\>)\bigr) \qquad \bigl({\mathcal G},{\mathcal F}\in{\mathbf D}({\mathcal A})\bigr);\index{ ${\mathbf R}$ (right-derived functor)!${\mathbf R}{\mathrm {Hom}}^{\bullet}$} $$ and there are natural isomorphisms $$ \mathrm H^i{\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathcal A}({\mathcal G},{\mathcal F}\>)\cong \mathrm{Hom}_{{\mathbf D}({\mathcal A})}\bigl({\mathcal G},{\mathcal F}[i]\bigr) \qquad(i\in\mathbb Z). $$ 2.~Application of homology to the second assertion in the Theorem reveals that it~is equivalent to the first one. 3.~We do not know in general (when ${\mathscr X}$ is not properly algebraic) that the functor~${\boldsymbol j}$ is fully faithful---${\boldsymbol j}$\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} has a right adjoint $(\text{identity})^{\<\times}\cong{\mathbf R} Q_{\mathscr X}^{}$ (see \Pref{A(vec-c)-A}), but it may be that for some ${\mathcal E}\in\A_{\vec {\mathrm c}}({\mathscr X})$ the natural map $\>{\mathcal E}\to{\mathbf R} Q_{\mathscr X}^{}\>{\boldsymbol j}\>{\mathcal E}$ is not an isomorphism. 4. For a \emph{proper} map $f_0\colon X\to Y$ of \emph{ordinary} schemes it is customary to write~$f_0^!$\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\mathstrut f^!\<$} instead of~$f_0^\times\<$. (Our extension of this notation to maps of formal schemes---introduced immediately after \Dref{D:basechange}---is not what would be expected here.) 5.~\Tref{prop-duality} includes the case when ${\mathscr X}$ and~${\mathscr Y}$ are ordinary noetherian schemes. (In fact the proof below applies with minor changes to arbitrary maps of quasi-compact, quasi-separated schemes, cf.~\cite[Chapter~4]{Derived categories}.) The next Corollary relates the formal situation to the ordinary one. \begin{scor}\label{C:kappa-f^times} Let\/ $A$ be a noetherian adic ring with ideal of definition\/~$I,$\ set\/ $Y\!:={\mathrm {Spec}}(A)$ and $W\!:={\mathrm {Spec}}(A/I)\subset Y\<$. Let\/ $f_0\colon X\to Y$ be a proper map and set~ $Z\!:=f_0^{-1}W,$ so that there is a commutative diagram $$ \begin{CD} {\mathscr X}\!:[email protected]_{/Z} @>\kappa_{\mathscr X}^{\phantom{.}}>> X \\ @.@V f VV @VV f_0 V \\ {\mathscr Y}\!:=\:@.{\mathrm {Spf}}(A)@>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle\kappa_{\mathscr Y}^{\phantom{.}}$}\vss}> Y \end{CD} $$ with\/ $\kappa_{\mathscr X}^{\phantom{.}}$ and\/ $\kappa_{\mathscr Y}^{\phantom{.}}$ the respective \textup(flat\textup) completion maps, and $f$ the \textup(proper\textup) map induced by\/~$f_0\>$. \pagebreak[3] Then the map adjoint to the natural composition $$ {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}} \xrightarrow{\ref{P:kappa-f*}\>} \kappa_{\mathscr Y}^*{\mathbf R} f_{0*} f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}} \longrightarrow \kappa_{\mathscr Y}^*\kappa_{{\mathscr Y}*}^{\phantom{.}}\longrightarrow\mathbf 1 $$ is an isomorphism of functors---from\/ ${\mathbf D}({\mathscr Y})$ to\/ $\D_{\<\vc}({\mathscr X}),$\ see \Cref{cor-prop-duality}--- $$ \kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}\iso f^\times\<. $$ \end{scor} \begin{proof} For any ${\mathcal E}\in\D_{\<\vc}({\mathscr X})$ set ${\mathcal E}_0\!:={\boldsymbol j}_{\!X}{\mathbf R} Q_{\<\<X}^{}\kappa_{{\mathscr X}*}^{\phantom{.}}{\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$ (see \Sref{SS:Dvc-and-Dqc}). Using \Pref{c-erator} we have then for any ${\mathcal F}\in{\mathbf D}({\mathscr Y})$ the natural isomorphisms \begin{align*} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}({\mathcal E},\>\kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) &\iso {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal E}_0\>,\>f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) \\ &\iso {\mathrm {Hom}}_{{\mathbf D}(Y)}({\mathbf R} f_{\<0*}^{}{\mathcal E}_0\>,\>\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) \\ &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}(\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathcal E}_0\>,\>{\mathcal F}\>) \\ &\underset{\ref{P:kappa-f*}}{\iso} {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}({\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}_0\>,\>{\mathcal F}\>) \iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}({\mathbf R} f_{\!*}{\mathcal E},\>{\mathcal F}\>). \end{align*} Thus $\kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}$ is right-adjoint to~${\mathbf R f_{\!*}}|_{\D_{\<\vc}({\mathscr X})}\>$, whence the conclusion. \end{proof} \smallskip \begin{proof}[Proof of \Tref{prop-duality}.] 1. Following Deligne\index{Deligne, Pierre} \cite[p.\,417, top]{H1}, we begin by considering for ${\mathcal M}\in{\mathcal A}({\mathscr X})$ the functorial flasque \emph{Godement resolution} $$ 0\to{\mathcal M}\to G^0({\mathcal M})\to G^1({\mathcal M})\to\cdots\,. $$ Here, with $G^{-2}({\mathcal M})\!:=0$, $G^{-1}({\mathcal M})\!:={\mathcal M}$, and for $i\ge 0$, $K^i({\mathcal M})$ the cokernel of $G^{i-2}({\mathcal M})\to G^{i-1}({\mathcal M})$, the sheaf $G^i({\mathcal M})$ is specified inductively by $$ G^i({\mathcal M})\bigl({\mathscr U}\bigr)\!:=\prod_{x\in{\mathscr U}}\,K^i({\mathcal M})_x \qquad({\mathscr U}\text{ open in }{\mathscr X}). $$ One shows by induction on~$i$ that all the functors $G^i$ and $K^i$ (from ${\mathcal A}({\mathscr X})$ to itself) are \emph{exact.} Moreover, for $i\ge 0$, $G^i({\mathcal M})$, being flasque, is $f_{\!*}$-\emph{acyclic,} i.e., $$ R^j\<\<f_{\!*}G^i({\mathcal M})=0\quad\text{for all }j>0. $$ The category $\A_{\vec {\mathrm c}}({\mathscr X})$ has small colimits (\Pref{(3.2.2)}), and is generated~by its coherent members, of which there exists a small set containing representatives of every isomorphism class. The Special Adjoint Functor Theorem\index{Special Adjoint Functor Theorem} \mbox{(\cite[p.\,90]{pF1964}} or~\cite[p.\,126, Corollary]{currante}) guarantees then that a right-exact functor $F$ from~$\A_{\vec {\mathrm c}}$ into~an abelian category~${\mathcal A}'$ has a right adjoint iff $F$ is \emph{continuous}\index{continuous functor} in the sense that it commutes with filtered direct limits, i.e., for any small directed system \mbox{$({\mathcal M}_\alpha\>,\,\varphi_{\alpha\beta}\colon {\mathcal M}_\beta\to {\mathcal M}_\alpha)$} in~$\A_{\vec {\mathrm c}}\>$, with $\dirlm{\alpha}{\mathcal M}_\alpha=({\mathcal M},\,\varphi_\alpha\colon {\mathcal M}_\alpha\to {\mathcal M})$ it holds that $$ \bigl(F({\mathcal M}), F(\varphi_\alpha)\bigr)=\dirlm{\alpha}\bigl(F({\mathcal M}_\alpha), F(\varphi_{\alpha\beta})\bigr). $$ Accordingly, for constructing right adjoints we need to replace the restrictions of~$G^i$ and~$K^i$ to $\A_{\vec {\mathrm c}}({\mathscr X})$ by continuous functors. \begin{slem}\label{L:vc-functor} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme and let\/ $G$ be a functor from $\A_{\mathrm c}({\mathscr X})$ to a category\/~${\mathcal A}'$ in which direct limits exist for all small directed systems. Let\/ $j\colon\A_{\mathrm c}({\mathscr X})\hookrightarrow\A_{\vec {\mathrm c}}({\mathscr X})$ be the inclusion functor. Then: \smallskip \textup{(a)} There exists a continuous functor\/~$G_{\vec{\mathrm{c}}}\>\colon\A_{\vec {\mathrm c}}({\mathscr X})\to{\mathcal A}'$ and an isomorphism of functors\/ $\varepsilon\colon G\iso G_{\vec{\mathrm{c}}}\smcirc\< j$ such that for any map of functors\/ $\psi\colon G\to F\<\smcirc\< j$ with\/ $F$~ continuous, there is a unique map of functors\/ $\psi_{\vec{\mathrm{c}}}\>\colon G_{\vec{\mathrm{c}}}\to F$ such that\/ $\psi$ factors as $$ G \xrightarrow{\,\varepsilon\,} G_{\vec{\mathrm{c}}}\smcirc\< j \xrightarrow{\textup{via }\psi_{\vec{\mathrm{c}}}\,} F\<\smcirc\< j\>. $$ \smallskip \textup{(b)} Assume that ${\mathcal A}'$ is abelian, and has exact filtered direct limits $($i.e., satisfies Grothendieck's axiom\/ \textup{AB5)}. Then if\/ $G$ is exact, so is $G_{\vec{\mathrm{c}}}\>$. \end{slem} \begin{proof} (a) For ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$, let $({\mathcal M}_\alpha)$ be the directed system of coherent ${\mathcal O}_{\mathscr X}$-submodules of\/~${\mathcal M},$\ and set $$ G_{\vec{\mathrm{c}}}({\mathcal M})\!:=\smash{\dirlm{\alpha}}G({\mathcal M}_\alpha). $$ \smallskip\noindent For any $\A_{\vec {\mathrm c}}({\mathscr X})$-map $\nu\colon{\mathcal M}\to{\mathcal N}$ and any~$\alpha$, there exists a coherent submodule~${\mathcal N}_\beta\subset{\mathcal N}$ such that $\nu|_{{\mathcal M}_\alpha}$ factors as ${\mathcal M}_\alpha\to{\mathcal N}_\beta\hookrightarrow{\mathcal N}$ (\Cref{C:limsub} and \Lref{L:Ext+lim}, with $q=0$); and the resulting composition $$ \nu_\alpha'\colon G({\mathcal M}_\alpha)\to G({\mathcal N}_\beta)\to G_{\vec{\mathrm{c}}}({\mathcal N}\>) $$ does not depend on the choice of~${\mathcal N}_\beta\>$. We define the map $$ G_{\vec{\mathrm{c}}}(\nu)\colon G_{\vec{\mathrm{c}}}({\mathcal M})=\smash{\dirlm{\alpha}G({\mathcal M}_\alpha)}\to G_{\vec{\mathrm{c}}}({\mathcal N}\>) $$ \smallskip\noindent to be the unique one whose composition with $G({\mathcal M}_\alpha)\to G_{\vec{\mathrm{c}}}({\mathcal M})$ is $\nu_\alpha'$ for all~$\alpha$. Verification of the rest of assertion (a) is straightforward. (b) Let $0\to{\mathcal M}\to{\mathcal N}\xrightarrow{\,\pi\,}\mathcal{Q}\to 0$ be an exact sequence in $\A_{\vec {\mathrm c}}({\mathscr X})$. Let $({\mathcal N}_\beta)$ be the filtered system of coherent submodules of~${\mathcal N}\<$, so that ${\mathcal N}=\smash{\dirlm{}\<{\mathcal N}_\beta}$ (\Cref{C:limsub}). Then $({\mathcal M}\cap{\mathcal N}_\beta)$ is a filtered system of coherent ${\mathcal O}_{\mathscr X}$-modules whose $\smash{\dirlm{}}$ is~${\mathcal M}$, and $(\pi{\mathcal N}_\beta)$ is a filtered system of coherent ${\mathcal O}_{\mathscr X}$-modules whose $\smash{\dirlm{}}\vspace{.7pt}$ is~$\mathcal Q$ (see \Cref{C:images}). The exactness of~$G_{\vec{\mathrm{c}}}$ is then made apparent by application of~\smash{$\dirlm{}_{\<\<\!\beta}$} to the system of exact sequences $$ 0\to G({\mathcal M}\cap{\mathcal N}_\beta)\to G({\mathcal N}_\beta)\to G(\pi{\mathcal N}_\beta)\to 0. $$ \vskip-3.8ex \end{proof} \smallskip Now for ${\mathcal M} \in\A_{\vec {\mathrm c}}({\mathscr X})$, the $\smash{\dirlm{}}$\vspace{1pt} of the system of Godement resolutions of all the coherent submodules~${\mathcal M}_\alpha\subset{\mathcal M}$ is a functorial resolution $$ 0\to{\mathcal M}\to G_{\vec{\mathrm{c}}}^0({\mathcal M})\to G_{\vec{\mathrm{c}}}^1({\mathcal M})\to\cdots; $$ and the cokernel of $G_{\vec{\mathrm{c}}}^{i-2}({\mathcal M})\to G_{\vec{\mathrm{c}}}^{i-1}({\mathcal M})$ is\vspace{1pt} $K_{\vec{\mathrm{c}}}^i({\mathcal M})\!:=\smash{\dirlm{}\!}K^i({\mathcal M}_\alpha)$. By~(b) above (applied to the exact functors $G^i$ and $K^i$),\vspace{.8pt} the continuous functors $G_{\vec{\mathrm{c}}}^i$ and $K_{\vec{\mathrm{c}}}^i$ are exact; and $G_{\vec{\mathrm{c}}}^i({\mathcal M})=\smash{\dirlm{}}\>G^i({\mathcal M}_\alpha)$\vspace{1.5pt} is $f_{\!*}$-acyclic since $G^i({\mathcal M}_\alpha)$ is, and---${\mathscr X}$ being noetherian---the functors $R^j\<f_{\!*}$\vspace{1pt} commute with~$\smash{\dirlm{}\!}$.\vspace{1.2pt} \Pref{Rf_*bounded}(b) implies\- then that there is an integer~$e\ge 0$ such that for all ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$, $K_{\vec{\mathrm{c}}}^e({\mathcal M}\>)$ is $f_{\!*}$-acyclic. \vspace{1pt} So if we define the exact functors~${\mathcal D}^i\colon \A_{\vec {\mathrm c}}({\mathscr X})\to{\mathcal A}({\mathscr X})$ by $$ {\mathcal D}^i({\mathcal M}\>)\!= \begin{cases} G_{\vec{\mathrm{c}}}^i({\mathcal M}\>)\qquad &(0\le i< e) \\ K_{\vec{\mathrm{c}}}^e({\mathcal M}\>) \qquad&(i=e) \\ 0\qquad&(i>e) \end{cases} $$ then for ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$, each ${\mathcal D}^i({\mathcal M})$ is $f_{\!*}$-acyclic and the natural sequence $$ 0\longrightarrow{\mathcal M}\xrightarrow{\delta({\mathcal M})\>}{\mathcal D}^0({\mathcal M}\>) \xrightarrow{\delta^0({\mathcal M})\>}{\mathcal D}^1({\mathcal M}\>) \xrightarrow{\delta^1({\mathcal M})\>}{\mathcal D}^2({\mathcal M}\>) \longrightarrow\cdots\longrightarrow{\mathcal D}^e({\mathcal M}\>) \longrightarrow 0 $$ is exact. In short, the sequence ${\mathcal D}^0\to{\mathcal D}^1\to{\mathcal D}^2\to\cdots\to{\mathcal D}^e\to 0$ is an \emph{exact, continuous, $f_{\!*}\<$-acyclic, finite resolution of the inclusion functor $\A_{\vec {\mathrm c}}({\mathscr X})\hookrightarrow{\mathcal A}({\mathscr X})$.} \smallskip \pagebreak[3] 2. We have then a $\Delta$-functor $({{\mathcal D}}^{\bullet}\<,\text{Id})\colon{\mathbf K}(\A_{\vec {\mathrm c}}({\mathscr X}) )\to {\mathbf K}({\mathscr X})$ which assigns an \mbox{$f_{\!*}$-acyclic} resolution to each $\A_{\vec {\mathrm c}}({\mathscr X})$-complex~${\mathcal G} =({\mathcal G}^{\>p})_{p\in\lower.1ex\hbox{$\scriptstyle\mathbb Z$}}\,$: $$ ({\mathcal D}^\bullet\<{\mathcal G}\>)^m\!:=\bigoplus_{p+q=m}{\mathcal D}^q({\mathcal G}^{\>p})\qquad (m\in\mathbb Z,\ 0\le q\le e), $$ the differential $({\mathcal D}^\bullet {\mathcal G}\>)^m\to({\mathcal D}^\bullet {\mathcal G}\>)^{m+1}$ being defined on ${\mathcal D}^q({\mathcal G}^{\>p})$ $(p+q=m)$ to be $d'+(-1)^pd''$ where $d'\colon{\mathcal D}^q({\mathcal G}^{\>p})\to{\mathcal D}^q({\mathcal G}^{\>p+1})$ comes from the differential in ${\mathcal G}$ and $d''=\delta^q({\mathcal G}^{\>p})\colon{\mathcal D}^q({\mathcal G}^{\>p})\to{\mathcal D}^{q+1}({\mathcal G}^{\>p})$. It is elementary to check that the natural map $\delta({\mathcal G}\>)\colon{\mathcal G}\to {{\mathcal D}}^{\bullet}{\mathcal G}$ is a \emph{quasi-isomorphism}. The canonical maps are \emph{${\mathbf D}({\mathscr Y})$-isomorphisms} \begin{equation}\label{f*D} f_{\!*}{{\mathcal D}}^{\bullet}({\mathcal G}) \iso {\mathbf R f_{\!*}}{{\mathcal D}}^{\bullet}({\mathcal G}) \;\underset{{\mathbf R f_{\!*}}\delta({\mathcal G})}{\osi}\;{\mathbf R f_{\!*}}{\mathcal G}, \end{equation} i.e., the natural map $\alpha^i\colon H^i\bigl(f_{\!*}{\mathcal D}^\bullet({\mathcal G})\bigr)\to H^i\bigl({\mathbf R f_{\!*}}{\mathcal D}^\bullet({\mathcal G})\bigr)$ is an isomorphism for all~$i\in\mathbb Z\>\>$: this holds for bounded-below ${\mathcal G}$ because ${{\mathcal D}}^{\bullet}({\mathcal G})$ is a complex of $f_{\!*}$-acyclic objects; and for arbitrary~${\mathcal G}$ since for any $n\in\mathbb Z$, with ${\mathcal G}^{{\scriptscriptstyle\ge}n}$ denoting the truncation\looseness=-1 \stepcounter{numb} \begin{equation}\label{trunc} \cdots \to 0\to 0 \to \textup{coker}({\mathcal G}^{n-1}\to{\mathcal G}^n)\to{\mathcal G}^{n+1}\to{\mathcal G}^{n+2}\to\cdots \end{equation} there is a natural commutative diagram $$ \CD H^i\bigl(f_{\!*}{\mathcal D}^\bullet({\mathcal G})\bigr)@>\alpha^i>>H^i\bigl({\mathbf R f_{\!*}}{\mathcal D}^\bullet({\mathcal G})\bigr) \\ @V\beta_n^i VV @VV\gamma_n^i V \\ H^i\bigl(f_{\!*}{\mathcal D}^\bullet({\mathcal G}^{{\scriptscriptstyle\ge}n})\bigr)@>>\alpha_n^i> H^i\bigl({\mathbf R f_{\!*}}{\mathcal D}^\bullet({\mathcal G}^{{\scriptscriptstyle\ge}n})\bigr) \endCD $$ in which, when $n\ll i$, $\beta_n^i$ is an isomorphism (since ${\mathcal G}$ and ${\mathcal G}^{{\scriptscriptstyle\ge}n}$ are identical in all degrees $>n$), $\gamma_n^i$~is an isomorphism (by \Pref{Rf_*bounded}(b) applied to the mapping cone of the natural composition ${\mathcal D}^\bullet({\mathcal G})\iso{\mathcal G}\longrightarrow{\mathcal G}^{{\scriptscriptstyle\ge}n} \iso{\mathcal D}^\bullet({\mathcal G}^{{\scriptscriptstyle\ge}n})$), and $\alpha_n^i$ is an isomorphism (since ${\mathcal G}^{{\scriptscriptstyle\ge}n}$ is bounded below). Thus we have realized ${\mathbf R f_{\!*}}\smcirc\>\>{\boldsymbol j}$ at the homotopy level, via the functor ${\mathcal C}^\bullet\!:=f_{\!*}{\mathcal D}^\bullet\>$; and our task is now to find a right adjoint at this level. \smallskip 3. Each functor ${{\mathcal C}}^p=f_{\!*}{{\mathcal D}}^p\colon\A_{\vec {\mathrm c}}({\mathscr X})\to{\mathcal A}({\mathscr Y})$ is exact,\vspace{.3pt} since $R^1\<\<f_{\!*}({\mathcal D}^p({\mathcal M}\>))=0$ for all ${\mathcal M}\in\A_{\vec {\mathrm c}}({\mathscr X})$. ${{\mathcal C}}^p$ is continuous, since ${\mathcal D}^p$ is\vspace{.6pt} and, ${\mathscr X}$ being noetherian, $f_{\!*}$ commutes with $\smash{\dirlm{}}\!$. As before, the Special Adjoint Functor Theorem\index{Special Adjoint Functor Theorem}\vspace{1.2 pt} yields that \emph{${\mathcal C}^p$~has a right adjoint ${\mathcal C}_p\colon{\mathcal A}({\mathscr Y}) \to\A_{\vec {\mathrm c}}({\mathscr X})$.}\vadjust{\kern.4 pt} \penalty-1000 For each ${\mathcal A}({\mathscr Y})$-complex ${\mathcal F}=({\mathcal F}^p)_{p\in\lower.1ex\hbox{$\scriptstyle\mathbb Z$}}\>$ let ${\mathcal C}_\bullet \>{\mathcal F}$ be the $\A_{\vec {\mathrm c}}({\mathscr X})$-complex with $$ ({\mathcal C}_\bullet \>{\mathcal F}\>)^m\!:=\prod_{p-q\,=\,m}{\mathcal C}_q{\mathcal F}^p\qquad (m\in\mathbb Z, 0\le q\le e), $$ and with differential $({\mathcal C}_\bullet \>{\mathcal F}\>)^m\to({\mathcal C}_\bullet \>{\mathcal F}\>)^{m+1}$ the unique map making the following diagram commute for all $r, s$ with $r\mspace{-1.5mu}-\mspace{-1.5mu}s\>=\>m\mspace{-1.5mu}+\!1\>$: $$ \CD \underset{p-q\,=\,m}{\prod} {\mathcal C}_q{\mathcal F}\>^p @>\phantom{d_\prime\>+\>(-1)^rd_{\prime\prime}}>> \underset{p-q\,=\,m+1}{\prod} {\mathcal C}_q{\mathcal F}\>^p\\ @VVV @VVV\\ {\mathcal C}_s{\mathcal F}\>^{r-1}\oplus{\mathcal C}_{s+1}{\mathcal F}\>^r @>>d_\prime\>+\>(-1)^rd_{\prime\prime}> {\mathcal C}_s{\mathcal F}\>^r \endCD $$ where: (i) the vertical arrows come from projections, (ii) $d_\prime\colon{\mathcal C}_s{\mathcal F}\>^{r-1}\to{\mathcal C}_s{\mathcal F}\>^r$ corresponds to the differential in ${\mathcal F}$, and (iii) with $\delta_s\colon{\mathcal C}_{s+1}\to {\mathcal C}_s$ corresponding by adjunction to $f_{\!*}(\delta^s)\colon {\mathcal C}^s\to{\mathcal C}^{s+1}$, $$ d_{\prime\prime}\!:=(-1)^s\delta_s({\mathcal F}\>^r)\colon{\mathcal C}_{s+1}{\mathcal F}\>^r\to{\mathcal C}_s{\mathcal F}\>^r. $$ This construction leads naturally to a $\Delta$-functor $({{\mathcal C}}_{\bullet}\>,\text{Id})\colon{\mathbf K}({\mathscr Y})\to {\mathbf K}(\A_{\vec {\mathrm c}}({\mathscr X}))$. The adjunction isomorphism $$ {\mathrm {Hom}}_{\A_{\vec {\mathrm c}}({\mathscr X})}({\mathcal M}\<,{{\mathcal C}}_p\>{\mathcal N}\>) \iso {\mathrm {Hom}}_{{\mathcal A}({\mathscr Y})}({\mathcal C}^p\<{\mathcal M}\<,\>{\mathcal N}\>) \qquad \bigl({\mathcal M}\in \A_{\vec {\mathrm c}}({\mathscr X}),\ {\mathcal N}\in {\mathcal A}({\mathscr Y})\bigr) $$ applied componentwise produces an isomorphism of complexes of abelian groups \stepcounter{sth} \begin{equation}\label{Deligne} {\mathrm {Hom}}^{\bullet}_{\A_{\vec {\mathrm c}}({\mathscr X})}({\mathcal G},\>{\mathcal C}_{\bullet}{\mathcal F}\>) \iso {\mathrm {Hom}}^{\bullet}_{{\mathcal A}({\mathscr Y})}({{\mathcal C}}^{\bullet}{\mathcal G},\>{\mathcal F}\>) \end{equation} for all $\A_{\vec {\mathrm c}}({\mathscr X})$-complexes ${\mathcal G}$ and ${\mathcal A}({\mathscr Y})$-complexes~${\mathcal F}\<$. \smallskip 4. The isomorphism \eqref{Deligne} suggests that we use $\>{\mathcal C}_\bullet$ to construct $f^\times\<$, as follows. Recall~that a complex ${\mathscr J}\in{\mathbf K}(\A_{\vec {\mathrm c}}({\mathscr X}))$ is K-injective iff for each exact complex \hbox{${\mathcal G}\in{\mathbf K}(\A_{\vec {\mathrm c}}({\mathscr X}))$}, the complex ${\mathrm {Hom}}^{\bullet}_{\A_{\vec {\mathrm c}}({\mathscr X})}({\mathcal G},{\mathscr J})$ is exact too. By~\eqref{f*D}, ${\mathcal C}^\bullet{\mathcal G}$ is exact if ${\mathcal G}$ is; so it follows from~\eqref{Deligne} that \emph{if ${\mathcal F}$ is K-injective in~${\mathbf K}({\mathscr Y})\mspace{-.6mu}$ then ${\mathcal C}_\bullet{\mathcal F}$~is \hbox{K-injective} in~${\mathbf K}(\A_{\vec {\mathrm c}}({\mathscr X}))$.} Thus if ${\mathbf K}_{\text{\textbf I}}(-)\subset{\mathbf K}(-)$% \index{ ${\mathbf K}$ (homotopy category)!a@${\mathbf K}_{\text{\textbf I}}$} is the full subcategory of all \mbox{K-injective} complexes, then we have a $\Delta$-functor \hbox{$({\mathcal C}_\bullet\>,\text{Id})\colon {\mathbf K}_{\text{\textbf I}}({\mathscr Y})\to{\mathbf K}_{\text{\textbf I}}(\A_{\vec {\mathrm c}}({\mathscr X}))$.} Associating a K-injective resolution to each complex in~${\mathcal A}({\mathscr Y})$ leads to a $\Delta$-functor \hbox{$(\rho, \Theta)\colon {\mathbf D}({\mathscr Y})\to{\mathbf K}_{\text{\textbf I}}({\mathscr Y})$}.% \footnote {In fact $(\rho, \Theta)$ is an equivalence of $\Delta$-categories, see \cite[\S1.7]{Derived categories}. But note that $\Theta$ need not be the identity morphism, i.e., one may not be able to find a complete family of K-injective resolutions commuting with translation. For example, we do not know that every periodic complex has a periodic K-injective resolution } This $\rho$ is bounded below: an ${\mathcal A}({\mathscr Y})$-complex~${\mathcal E}$ such that $H^i({\mathcal E})=0$ for all $i<n$ is quasi-isomorphic to its truncation ${\mathcal E}^{{\scriptscriptstyle\ge}n}\<$ (see~\eqref{trunc}), which is quasi-isomorphic to an injective complex~${\mathcal F}$ which vanishes in all degrees below~$n$. (Such an~${\mathcal F}$ is K-injective.) Finally, one can define~$f^\times$ to be the composition of the functors $$ {\mathbf D}({\mathscr Y})\xrightarrow{\,\rho\,} {\mathbf K}_{\text{\textbf I}}({\mathscr Y})\xrightarrow{{\mathcal C}_\bullet\>} {\mathbf K}_{\text{\textbf I}}(\A_{\vec {\mathrm c}}({\mathscr X})) \xrightarrow{\text{natural}\,} {\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X})), $$ and check, via \eqref{f*D} and \eqref{Deligne} that \Tref{prop-duality} is satisfied. (This involves some tedium with respect to $\Delta$-details.) \end{proof} \section{Torsion sheaves.} Refer to \S\ref{Gamma'} for notation and first sorites regarding torsion sheaves. Paragraphs~\ref{tors-sheaves} and~\ref{tors-D} develop properties of quasi-coherent torsion sheaves and their derived categories on locally noetherian formal schemes---see e.g., Propositions~\ref{Gamma'(qc)}, \ref{Gammas'+kappas}, \ref{Rf-*(qct)}, and \Cref{C:f* and Gamma}. (There is some overlap here with \S4 in \cite{Ye}.) Such properties will be needed throughout the rest of the paper. For instance, Paragraph~\ref{tors-eqvce} establishes for a noetherian formal scheme~${\mathscr X}$, either separated or finite-dimensional, an \emph{equivalence of categories} ${\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X})){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\approx\mkern17mu}}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$, thereby enabling the use of~ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$---rather than ${\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X}))$---in~\Tref{T:qct-duality} ($\:\cong\:$\Tref{Th2} of \Sref{S:prelim}). Also, \Lref{Gam as holim}, identifying the derived functor ${\mathbf R}\iG{\mathcal J}(-)$ (for any ${\mathcal O}_{\<\<X}$-ideal~${\mathcal J}$, where $X$ is a ringed space) with the homotopy colimit\- of the functors ${\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\<\<X}/\<{\mathcal J}^n\<,-)$, plays a key role in the proof of the Base Change \Tref{T:basechange} ($\:\cong\:$\Tref{Th3}). \pagebreak[3] \begin{parag}\label{tors-sheaves} This paragraph deals with categories of quasi-coherent torsion sheaves on locally noetherian formal schemes. \end{parag} \begin{sprop} \label{f-*(qct)} Let $f\colon {\mathscr X}\to {\mathscr Y}$ be a map of noetherian formal schemes, and let\/ ${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$. Then\/ $f_{\!*}\>{\mathcal M} \in \A_{\mathrm {qct}}\<({\mathscr Y})$. Moreover, if\/ $f$ is pseudo\kern.6pt-proper\/ \(see~\textup{\S\ref{maptypes}}\) and\/ ${\mathcal M}$ is coherent then $f_{\!*}\>{\mathcal M}$ is coherent. \end{sprop} \begin{proof} Let ${\mathscr J}\subset{\mathcal O}_{\mathscr X}$ and ${\mathscr I}\subset{\mathcal O}_{\mathscr Y}$ be ideals of definition such that ${\mathscr I}{\mathcal O}_{\mathscr X}\subset{\mathscr J}$, and let\vspace{-1pt} $$ X_{n}\!:=({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J}^n) \xrightarrow{f_{\<n}^{}\>}({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I}^n)=:Y_{n}\qquad(n>0) $$ be the scheme-maps induced by $f\<$, so that if $j_n$ and $i_n$ are the canonical closed immersions then $fj_n=i_nf_{\<n}^{}$. Let ${\mathcal M}_{n}\!:=\cH{om}({\mathcal O}_{{\mathscr X}}/{\mathscr J}^n\<,\>{\mathcal M})$, so that $$ {\mathcal M}=\iGp{{\mathscr X}}{\mathcal M}= \dirlm{n}\<{\mathcal M}_n=\dirlm{n}j_{n*}j_n^*\>{\mathcal M}_n\>. $$ Since ${\mathscr J}^{n}$ is a coherent ${\mathcal O}_{\mathscr X}$-ideal \cite[p.\,427]{GD}, therefore ${\mathcal M}_n$ is quasi-coherent (\Cref{C:images}(d)), and it is straightforward to check that $i_{n*}f_{\<n*}^{}j_n^*\>{\mathcal M}_n\in\A_{\mathrm {qct}}\<({\mathscr Y})$. Thus, ${\mathscr X}$~being noetherian, and by \Cref{qct=plump} below, $$ f_{\!*}\>{\mathcal M} = f_{\!*}\>\> \dirlm{n}\<{\mathcal M}_n\cong \dirlm{n} f_{\!*}j_{n*}j_n^*\>{\mathcal M}_n = \dirlm{n} i_{n*}f_{\<n*}^{}j_n^*\>{\mathcal M}_n \in\A_{\mathrm {qct}}\<({\mathscr Y}). $$ When $f$ is pseudo\kern.6pt-proper every $f_{\<n}^{}$ is proper; and if ${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$ is coherent then so is $f_{\!*}\>{\mathcal M}$, because for some $n$, $f_{\!*}\>{\mathcal M}=f_{\!*}j_{n*}j_{n}^{*}\>{\mathcal M}_n=i_{n*}f_{\<n*}^{}j_n^*\>{\mathcal M}_n$. \end{proof} \begin{sprop} \label{iso-qct} Let\/ $Z$ be a closed subset of a locally noetherian scheme\/~$X\<,$ and let\/ $\kappa\colon{\mathscr X} \to X$ be the completion of\/ $X$ along $Z$. Then the functors\/ $\kappa^*$ and~$\kappa_*$ restrict to inverse isomorphisms between the categories\/ ${\mathcal A}_Z(X)$ and\/ $\A_{\mathrm t}\<({\mathscr X}),$\ and between the categories\/ $\A_{{\qc}Z}(X)$ and\/ $\A_{\mathrm {qct}}\<({\mathscr X});\>$\ and if\/ ${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$ is coherent, then so is\/ $\kappa_*\>{\mathcal M}$. \end{sprop} \begin{proof} Let ${\mathcal J}$ be a quasi-coherent ${\mathcal O}_X$-ideal such that the support of~${\mathcal O}_X/{\mathcal J}$ is~$Z$. Applying $\dirlm{n}\!\!$ to the natural isomorphisms\vadjust{\kern-2pt} $$ \postdisplaypenalty 1000 \quad\kappa^*\cH{om}_X({\mathcal O}_X/{\mathcal J}^n\<\<,\>\>{\mathcal N}\>)\iso \cH{om}_{\mathscr X}({\mathcal O}_{\mathscr X}/{\mathcal J}^n{\mathcal O}_{\mathscr X}\>,\>\kappa^*{\mathcal N}\>) \qquad ({\mathcal N}\in{\mathcal A}(X),\ n>0) $$ we get a functorial isomorphism $\kappa^*\<\iGp{Z}\iso\iGp{{\mathscr X}}\kappa^*\<$, and hence $\kappa^*({\mathcal A}_Z(X))\subset{\mathcal A}_t({\mathscr X})$. Applying $\dirlm{n}\!\!$ to the natural isomorphisms\vadjust{\kern-1pt} $$ \cH{om}_X({\mathcal O}_X/{\mathcal J}^n\<\<,\>\kappa_*{\mathcal M})\iso \kappa_*\cH{om}_{\mathscr X}({\mathcal O}_{\mathscr X}/{\mathcal J}^n{\mathcal O}_{\mathscr X}\>,\>{\mathcal M})\qquad({\mathcal M}\in{\mathcal A}({\mathscr X}),\ n>0) $$ we get a functorial isomorphism $\iGp{Z}\kappa_*\iso\kappa_*\iGp{{\mathscr X}}\>$, and hence $\kappa_*({\mathcal A}_t({\mathscr X}))\subset{\mathcal A}_Z(X)$. \goodbreak As $\kappa$ is a pseudo\kern.6pt-proper map of locally noetherian formal schemes ((0) being an ideal of definition of~$X$), we see as in the proof of \Pref{f-*(qct)} that for~\mbox{${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$,} $\kappa_*\>{\mathcal M}$ is a \smash{$\dirlm{}\!\!$} of quasi-coherent ${\mathcal O}_X$-modules, so is itself quasi-coherent, and $\kappa_*\>{\mathcal M}$ is coherent whenever ${\mathcal M}$ is.% \footnote{The noetherian assumption in \Lref{f-*(qct)} is needed only for commutativity of $f_{\!*}$ with $\smash{\subdirlm{}}\!$, a condition clearly satisfied by $f=\kappa$ in the present situation.\par} Finally, examining stalks (see \S\ref{Gamma'}) we find that the natural transformations ${\rm 1} \to \kappa_* \kappa^*$ and $\kappa^* \kappa_* \to 1$ induce isomorphisms \begin{align*} \iGp{Z}{\mathcal N} \iso \kappa_*\kappa^*\iGp{Z}{\mathcal N} \qquad & \bigl({\mathcal N} \in {\mathcal A}(X)\bigr)\<, \\ \kappa^* \kappa_*\iGp{{\mathscr X}}{\mathcal M}\iso \iGp{{\mathscr X}}{\mathcal M} \qquad & \bigl({\mathcal M} \in{\mathcal A}({\mathscr X})\bigr)\<. \end{align*} \vspace{-6.2ex} \phantom{xxx} \end{proof} \begin{scor}\label{qct=plump} If\/ ${\mathscr X}$ is a locally noetherian formal scheme then\/~$\A_{\mathrm {qct}}\<({\mathscr X})$ is plump in\/~${\mathcal A}({\mathscr X})$ and closed under small\/ ${\mathcal A}({\mathscr X})$-colimits.% \begin{comment} \footnote {Actually, $\A_{\mathrm {qct}}\<({\mathscr X})$ is closed under \emph{all} ${\mathcal A}({\mathscr X})$-colimits---see footnote under \Pref{A(vec-c)-A}.% } \end{comment} \end{scor} \begin{proof} The assertions are local, and so, since $\A_{\mathrm t}\<({\mathscr X})$ is plump (\S\ref{Gamma'1}), \Pref{iso-qct} (where $\kappa^*$ commutes with~\smash{$\dirlm{}\!$}) enables reduction to well-known facts about $\A_{{\qc}Z}(X)\subset{\mathcal A}(X)$ with $X$ an affine noetherian (ordinary) scheme. \end{proof} \begin{slem} \label{Gamma'+qc} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme. If\/ ${\mathcal M}$ is a quasi-coherent\/ ${\mathcal O}_{\mathscr X}$-module then\/ $\iGp{{\mathscr X}}{\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$ is the\/ \smash{$\dirlm{}\!\!$} of its coherent submodules. In particular, $\A_{\mathrm {qct}}\<({\mathscr X})\subset\A_{\vec {\mathrm c}}({\mathscr X})$. \end{slem} \begin{proof} Let ${\mathscr J}$ be an ideal of definition of ${\mathscr X}\>$. For any positive integer~$n$, let $X_{n}$ be the scheme~$({\mathscr X}, {\mathcal O}_{{\mathscr X}}/{\mathscr J}^n)$, let $j_n\colon X_{n}\to{\mathscr X}$ be the canonical closed immersion, and let ${\mathcal M}_n\!:=\cH{om}({\mathcal O}_{{\mathscr X}}/{\mathscr J}^n\<,\>{\mathcal M})\subset\iGp {\mathscr X}({\mathcal M})$, so that ${\mathcal M}_n\in\A_{\mathrm {qct}}\<({\mathscr X})$ (\Cref{C:images}(d)). Then the quasi-coherent ${\mathcal O}_{\!X_{n}}\<$-module $j_n^*{\mathcal M}_n$ is the $\smash{\dirlm{}}\!\!$\vspace{1pt} of its coherent submodules \cite[p.\,319, (6.9.9)]{GD}, hence so is $\>{\mathcal M}_n=j_{n*}j_n^*{\mathcal M}_n$\vspace{.6pt} (since $j_n^*$ and $j_{n*}$ preserve both $\smash{\dirlm{}}\!\!$ and coherence\vadjust{\kern1pt} \cite[p.\,115, (5.3.13) and~(5.3.15)]{GD}), and therefore so is \mbox{$\iGp{{\mathscr X}}{\mathcal M}=\smash{\dirlm{n}\!{\mathcal M}_n\>}$.} That $\smash{\dirlm{n}\<\<{\mathcal M}_n\>}\in\A_{\mathrm {qct}}\<({\mathscr X})$\vspace{1.5pt} results from \Cref{qct=plump}. \end{proof} \begin{scor}\label{C:Qt} For a locally noetherian formal scheme\/~${\mathscr X},$ the inclusion functor\/ $j^{\mathrm t}_{\<{\mathscr X}}\colon\A_{\mathrm {qct}}\<({\mathscr X})\hookrightarrow {\mathcal A}({\mathscr X})$ has a right adjoint\/~$Q^{\mathrm t}_{\<{\mathscr X}}$.\index{ ${\mathbf R}$@ {}$Q_{\mathscr X}$ (quasi-coherator)\vadjust{\penalty 10000}!$Q^{\mathrm t}_{\<{\mathscr X}}$|(} If\/ moreover\/ ${\mathscr X}$ is noetherian then\/ $Q^{\mathrm t}_{\<{\mathscr X}}$ commutes with\/ \smash{$\dirlm{}\!\<.$} \end{scor} \begin{proof} To show that $j^{\mathrm t}_{\<{\mathscr X}}$ has a right adjoint one can, in view of \Cref{qct=plump} and \Lref{Gamma'+qc}, simply apply the Special Adjoint Functor theorem. More specifically, since $\iGp{{\mathscr X}}$ is right-adjoint to the inclusion $\A_{\mathrm t}\<({\mathscr X})\hookrightarrow{\mathcal A}({\mathscr X})$, and $\A_{\vec {\mathrm c}}({\mathscr X})\subset\A_{\qc}({\mathscr X})$ (\Cref{C:vec-c is qc}), it follows from \Lref{Gamma'+qc} that the restriction of~$\iGp{{\mathscr X}}$ to $\A_{\vec {\mathrm c}}({\mathscr X})$ is right-adjoint to $\A_{\mathrm {qct}}\<({\mathscr X})\hookrightarrow\A_{\vec {\mathrm c}}({\mathscr X})$; and by \Pref{A(vec-c)-A}, $\A_{\vec {\mathrm c}}({\mathscr X})\hookrightarrow{\mathcal A}({\mathscr X})$ has a right adjoint~$Q_{\mathscr X}^{}\>$; so $Q^{\mathrm t}_{\<{\mathscr X}}\!:=\iGp{{\mathscr X}}\smcirc Q_{\mathscr X}^{}$ is right-adjoint to~$j^{\mathrm t}_{\<{\mathscr X}}\>$. (Similarly, $Q_{\mathscr X}^{}\smcirc \iGp {\mathscr X}$ is right-adjoint to~$j^{\mathrm t}_{\<{\mathscr X}}\>$.) Commutativity with \smash{$\dirlm{}\!\!$}\vspace{1pt} means that for any small directed system~$({\mathcal G}_\alpha)$ in~${\mathcal A}({\mathscr X})$ and any ${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$, the natural map $$ \phi\colon{\mathrm {Hom}}({\mathcal M},\>\dirlm\alpha\<Q^{\mathrm t}_{\<{\mathscr X}}\>{\mathcal G}_\alpha)\to {\mathrm {Hom}}({\mathcal M},Q^{\mathrm t}_{\<{\mathscr X}}\>\dirlm\alpha\<{\mathcal G}_\alpha) $$ is an \emph{isomorphism}. This follows from \Lref{Gamma'+qc}, which allows us to assume that ${\mathcal M}$ is coherent, in which case $\phi$ is isomorphic to the natural composed isomorphism $$ \dirlm\alpha{\mathrm {Hom}}({\mathcal M},\>Q^{\mathrm t}_{\<{\mathscr X}}\>{\mathcal G}_\alpha)\iso \dirlm\alpha{\mathrm {Hom}}({\mathcal M},\>{\mathcal G}_\alpha)\iso {\mathrm {Hom}}({\mathcal M},\>\dirlm\alpha\<{\mathcal G}_\alpha). $$ \vskip-4ex \end{proof} \smallskip \emph{Remark.} For an ordinary noetherian scheme~$X$ we have $Q^{\mathrm t}_{\<\<X}=Q_{\<\<X}^{}$\index{ ${\mathbf R}$@ {}$Q_{\mathscr X}$ (quasi-coherator)\vadjust{\penalty 10000}!$Q^{\mathrm t}_{\<{\mathscr X}}$|)} (see \S\ref{SS:vc-and-qc}). More generally, if $\kappa\colon{\mathscr X}\to X$ is as in \Pref{iso-qct}, then $Q^{\mathrm t}_{{\mathscr X}}=\kappa^*\<\iG {Z\>} Q_{\<\<X}^{}\kappa_*$. Hence \Pref{f-*(qct)} (applied to open immersions ${\mathscr X}\hookrightarrow{\mathscr Y}$ with ${\mathscr X}$ affine) lets us construct the functor $Q^{\mathrm t}_{{\mathscr Y}}$ for any noetherian formal scheme~${\mathscr Y}$ by mimicking the construction for ordinary schemes (cf.~ \cite[p.\, 187, Lemme 3.2]{I}.) \begin{parag}\label{tors-D} The preceding results carry over to derived categories. From \Cref{qct=plump} it follows that on a locally noetherian formal scheme~${\mathscr X}$, $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ is a triangulated subcategory of ${\mathbf D}({\mathscr X})$, closed under direct sums. \penalty-1000 \begin{sprop}\label{Gamma'(qc)} \hskip-1pt For a locally noetherian formal scheme\/ ${\mathscr X},$ set\/ $\A_{\mathrm t}\<\!:=\A_{\mathrm t}\<({\mathscr X}),$\ the category of torsion\/ ${\mathcal O}_{\mathscr X}$-modules, and let\/ ${\boldsymbol i}\colon{\mathbf D}(\A_{\mathrm t}\<)\to {\mathbf D}({\mathscr X})$ be the natural \mbox{functor.} Then$\>:$ \textup{(a)} An\/ ${\mathcal O}_{\mathscr X}$-complex\/~${\mathcal E}$ is in\/~$\D_{\mathrm t}\<({\mathscr X})$ iff the natural map\/ ${\boldsymbol i}{\mathbf R}\iGp{\mathscr X}{\mathcal E}\to{\mathcal E}$ is a\/ ${\mathbf D}({\mathscr X})$-isomorphism. \textup{(b)} If\/ ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})$ then\/ ${\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(\A_{\mathrm t}\<)$. \textup{(c)} The functor ${\boldsymbol i}$ and its right adjoint ${\mathbf R}\iGp{{\mathscr X}}$ induce quasi-inverse equivalences between\/ ${\mathbf D}(\A_{\mathrm t}\<)$ and\/~$\D_{\mathrm t}\<({\mathscr X})$ and between $\D_{\mkern-1.5mu\mathrm {qc}}(\A_{\mathrm t}\<)$ and\/~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$.% \footnote{We may therefore sometimes abuse notation and write ${\mathbf R}\iGp{\mathscr X}$ instead of $\>{\boldsymbol i}{\mathbf R}\iGp{\mathscr X}$; but the meaning should be clear from the context.} \end{sprop} \begin{proof} (a) For ${\mathcal F}\in{\mathbf D}(\A_{\mathrm t}\<)$ (e.g., ${\mathcal F}\!:={\mathbf R}\iGp{\mathscr X}{\mathcal E}$), any complex isomorphic to ${\boldsymbol i}{\mathcal F}$ is clearly in~$\D_{\mathrm t}\<({\mathscr X})$. Suppose conversely that ${\mathcal E}\in\D_{\mathrm t}\<({\mathscr X})$. The assertion that ${\boldsymbol i}{\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\cong{\mathcal E}$ is local, so we may assume that ${\mathscr X}={\mathrm {Spf}}(A)$ where $A=\Gamma({\mathscr X},\>{\mathcal O}_{\mathscr X})$ is a noetherian adic ring, so that any defining ideal~${\mathscr J}$ of~${\mathscr X}$ is generated by a finite sequence in~$A$. Then ${\boldsymbol i}{\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\cong {\mathcal K}_\infty^\bullet\otimes\,{\mathcal E}$, where ${\mathcal K}_\infty^\bullet$% \index{ ${\mathbf K}$ (homotopy category)@${\mathcal K}_\infty^\bullet$ (limit of Koszul complexes)} is a bounded flat complex---a \smash{$\>\dirlm{}\!\!$}\vadjust{\kern1pt} of Koszul complexes on powers of the generators of~${\mathscr J}$---see \cite[p.\,18, Lemma 3.1.1]{AJL}. So ${\boldsymbol i}{\mathbf R}\iGp{{\mathscr X}}$ is a bounded functor, and the usual way-out argument reduces the question to where ${\mathcal E}$ is a single torsion sheaf. But then it is immediate from the construction of ${\mathcal K}_\infty^\bullet$ that ${\mathcal K}_\infty^\bullet\<\otimes{\mathcal E}={\mathcal E}$. \smallskip (b) Again, we can assume that ${\mathscr X}={\mathrm {Spf}}(A)$ and ${\mathbf R}\iGp{{\mathscr X}}$ is bounded, and since $\A_{\qc}({\mathscr X})$ is plump in~${\mathcal A}({\mathscr X})$ (\Pref{(3.2.2)}) we can reduce to where ${\mathcal E}$ is a single quasi-coherent ${\mathcal O}_{\mathscr X}$-module, though it is better to assume only that ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}^+({\mathscr X})$, for then we may also assume ${\mathcal E}$ injective, so that $$ {\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\cong\iGp{{\mathscr X}}{\mathcal E}=\dirlm{n>0\,\,\>}\cH{om}({\mathcal O}/{\mathscr J}^n\<,\>{\mathcal E}). $$ From \Cref{C:images}(d) it follows that $\cH{om}({\mathcal O}/{\mathscr J}^n\<,\>{\mathcal E})\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$---for this assertion another way-out argument reduces us again to where ${\mathcal E}$ is a single quasi-coherent ${\mathcal O}_{\mathscr X}$-module---and since homology commutes with \smash{$\dirlm{}\!\!$} and $\A_{\mathrm {qct}}\<$ is closed under \smash{$\dirlm{}\!\!$} (\Cref{qct=plump}), therefore ${\mathbf R}\iGp{{\mathscr X}}{\mathcal E}$ has quasi-coherent homology. \smallskip Assertion (c) results now from the following simple lemma. \end{proof} \enlargethispage*{3pt} \begin{slem}\label{L:j-gamma-eqvce} Let\/ ${\mathcal A}$ be an abelian category, let\/ $j\colon{\mathcal A}_\flat\to{\mathcal A}$ be the inclusion of a plump subcategory such that $j$ has a right adjoint\/~$\varGamma\<\<,$\ and let\/ ${\boldsymbol j}\colon{\mathbf D}({\mathcal A}_\flat)\to{\mathbf D}({\mathcal A})$\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} be the derived-category extension of\/ $j$. Suppose that every\/ ${\mathcal A}$-complex has a K-injective resolution, so that the derived functor\/ ${\mathbf R}\varGamma\colon{\mathbf D}({\mathcal A})\to{\mathbf D}({\mathcal A}_\flat)$ exists. Then ${\mathbf R} \varGamma$ is right-adjoint to~${\boldsymbol j}$. Furthermore, the following conditions are equivalent. \begin{enumerate} \item[(1)] ${\boldsymbol j}$ induces an equivalence of categories from\/ ${\mathbf D}({\mathcal A}_\flat)$ to\/ ${\mathbf D}_\flat({\mathcal A}),$ with quasi-inverse\/~ ${\mathbf R}_\flat\varGamma\!:={\mathbf R}\varGamma|_{{\mathbf D}_\flat({\mathcal A})}$.\vadjust{\kern1pt} \item[(2)] For every\/ ${\mathcal E}\in {\mathbf D}_\flat({\mathcal A})$ the natural map\/ ${\boldsymbol j}{\mathbf R} \varGamma{\mathcal E}\to{\mathcal E}$ is an isomorphism.\vadjust{\kern1pt} \item[$(3)$] The functor\/~${\mathbf R}_\flat\varGamma$ is bounded, and for\/ ${\mathcal E}_0\in {\mathcal A}_\flat$ the natural map\/ ${\boldsymbol j}{\mathbf R} \varGamma{\mathcal E}_0\to{\mathcal E}_0$ is a\/ ${\mathbf D}({\mathcal A})$-isomorphism. \end{enumerate} When these conditions hold, every \/ ${\mathcal A}_\flat$-complex has a K-injective resolution. \end{slem} \begin{proof} Since $\varGamma$ has an exact left adjoint, it takes K-injective ${\mathcal A}$-complexes to K-injective ${\mathcal A}_\flat$-complexes, whence there is a bifunctorial isomorphism in the derived category of abelian groups $$ {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathcal A}({\boldsymbol j}{\mathcal G},\>{\mathcal E})\iso{\mathbf R}{\mathrm {Hom}}^{\bullet}_{{\mathcal A}_\flat}({\mathcal G},\>{\mathbf R}\varGamma{\mathcal E}) \qquad\bigr({\mathcal G}\in{\mathbf D}({\mathcal A}_\flat),\ {\mathcal E}\in{\mathbf D}({\mathcal A})\bigl). $$ (To see this, one can assume ${\mathcal E}$ to be K-injective, and then drop the ${\mathbf R}$'s\dots\!\!) \:Apply homology $\mathrm H^0$ to this isomorphism to get adjointness of ${\boldsymbol j}$ and~${\mathbf R}\varGamma\<$. The implications $(1)\!\Rightarrow\!(3)\!\Rightarrow\!(2)$ are straightforward. For $(2)\!\Rightarrow\!(1)$, one needs that for ${\mathcal G}\in{\mathbf D}({\mathcal A}_\flat)$ the natural map ${\mathcal G}\to{\mathbf R}\varGamma{\boldsymbol j}{\mathcal G}$ is an isomorphism, or equivalently (look at homology), that the corresponding map ${\boldsymbol j}{\mathcal G}\to{\boldsymbol j}{\mathbf R}\varGamma{\boldsymbol j}{\mathcal G}$ is an isomorphism. But the composition of this last map with the isomorphism ${\boldsymbol j}{\mathbf R}\varGamma{\boldsymbol j}{\mathcal G}\iso{\boldsymbol j}{\mathcal G}$ (given by~(2)) is the identity, whence the conclusion. Finally, if ${\mathcal G}$ is an ${\mathcal A}_\flat$-complex and $j{\mathcal G}\to {\mathcal J}$ is a K-injective ${\mathcal A}$-resolution, then as before $\varGamma\<{\mathcal J}$ is a K-injective ${\mathcal A}_\flat$-complex; and (1) implies that the natural composition\looseness=-1 $$ {\mathcal G}\to\varGamma\< j{\mathcal G}\to\varGamma\< {\mathcal J}\ (\:\cong {\mathbf R}\varGamma{\boldsymbol j} {\mathcal G}) $$ is a ${\mathbf D}({\mathcal A}_\flat)$-isomorphism, hence an ${\mathcal A}_\flat$-K-injective resolution. \end{proof} \begin{scor}\label{C:Hom-Rgamma} For any complexes\/ ${\mathcal E}\in\D_{\mathrm t}\<({\mathscr X})$ and\/ ${\mathcal F}\in{\mathbf D}({\mathscr X})$ the natural map\/ ${\mathbf R}\iGp{\mathscr X}{\mathcal F}\to{\mathcal F}$ induces an isomorphism $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal E}, {\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)\iso{\mathbf R}\cH{om}^{\bullet}({\mathcal E}, {\mathcal F}\>). $$ \end{scor} \begin{proof} Consideration of homology presheaves shows it sufficient that for each affine open ${\mathscr U}\subset{\mathscr X}$, the natural map $$ {\mathrm {Hom}}_{{\mathbf D}({\mathscr U})}\bigl({\mathcal E}|_{\mathscr U}\>,\>({\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)|_{\mathscr U}\bigr) \to{\mathrm {Hom}}_{{\mathbf D}({\mathscr U})}\bigl({\mathcal E}|_{\mathscr U}\>,\>{\mathcal F}|_{\mathscr U}\bigr) $$ be an isomorphism. But since ${\mathbf R}\iGp{{\mathscr X}}$ commutes with restriction to~${\mathscr U}$, that is a direct consequence of \Pref{Gamma'(qc)}(c) (with ${\mathscr X}$ replaced by~${\mathscr U}$). \end{proof} Parts (b) and (c) of the following Proposition will be generalized in parts (d) and~(b), respectively, of \Pref{P:f* and Gamma}. \begin{sprop}\label{Gammas'+kappas} Let\/ $Z$ be a closed subset of a locally noetherian scheme\/~$X\<,$ and let\/ $\kappa\colon{\mathscr X} \to X$ be the completion of\/ $X$ along $Z$. Then$\>:$ \vadjust{\kern1pt} \textup{(a)} The exact functors\/ $\kappa^*$ and~$\kappa_*$ restrict to inverse isomorphisms between the categories\/ ${\mathbf D}_{\<Z}(X)$ and\/ $\D_{\mathrm t}\<({\mathscr X}),$\ and between the categories\/ $\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$ and\/ $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X});\>$\ and if\/ ${\mathcal M}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ has coherent homology, then so does\/ $\kappa_*\>{\mathcal M}$. \vadjust{\kern1pt} \textup{(b)} There is a unique derived-category isomorphism $$ {\mathbf R}\iGp Z\kappa_*{\mathcal E}\iso \kappa_*{\mathbf R}\iGp {\mathscr X}{\mathcal E}\qquad\:\bigl({\mathcal E}\in{\mathbf D}({\mathscr X})\bigr) $$ whose composition with the natural map $\kappa_*{\mathbf R}\iGp{\mathscr X}{\mathcal E}\to\kappa_*{\mathcal E}$ is just the natural map \mbox{${\mathbf R}\iGp Z\kappa_*{\mathcal E}\to\kappa_*{\mathcal E}$.} \vadjust{\kern1pt} \textup{(c)} There is a unique derived-category isomorphism $$ \kappa^*{\mathbf R}\iGp Z{\mathcal F} \iso{\mathbf R}\iGp {\mathscr X}\kappa^*\<{\mathcal F}\qquad\bigl({\mathcal F}\in{\mathbf D}(X)\bigr) $$ whose composition with the natural map ${\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F}\to\kappa^*\<{\mathcal F}$ is just the natural map \mbox{$\kappa^*{\mathbf R}\iGp Z{\mathcal F}\to\kappa^*\<{\mathcal F}$.} \end{sprop} \pagebreak[3] \begin{proof} The assertions in (a) follow at once from \Pref{iso-qct}. \vadjust{\kern1pt} (b) Since $\kappa_*$ has an exact left adjoint (namely~$\kappa^*$), therefore $\kappa_*$ transforms \mbox{K-injective} ${\mathcal A}({\mathscr X})$-complexes into K-injective ${\mathcal A}(X)$-complexes, and consequently the isomorphism in~(b) results from the isomorphism $\iGp Z\kappa_*\iso \kappa_*\iGp{\mathscr X}$ in the proof of \Pref{iso-qct}. That the composition in~(b) is as asserted comes down then to the elementary fact that the natural composition $$ \postdisplaypenalty5000 \cH{om}_X({\mathcal O}_X/{\mathcal J}^n\<\<,\>\kappa_*{\mathcal M})\iso \kappa_*\cH{om}_{\>{\mathscr X}}({\mathcal O}_{\mathscr X}/{\mathcal J}^n{\mathcal O}_{\mathscr X}\>,\>{\mathcal M})\longrightarrow \kappa_*{\mathcal M} $$ (see proof of \Pref{iso-qct}) is just the obvious map. Since $\kappa_*{\mathbf R}\iGp{\mathscr X}{\mathcal E}\in{\mathbf D}_{\<Z}(X)$ (by~(a) and \Pref{Gamma'(qc)}(a)), the uniqueness assertion (for the inverse isomorphism) results from adjointness of~${\mathbf R}\iGp Z$ and the inclusion ${\mathbf D}_{\<Z}(X)\hookrightarrow {\mathbf D}(X)$. (The proof is similar to that of \Pref{Gamma'(qc)}(c)). \vadjust{\kern1pt} (c) Using (b), we have the natural composed map $$ \kappa^*{\mathbf R}\iGp Z{\mathcal F} \to \kappa^*{\mathbf R}\iGp Z\kappa_*\kappa^*\<{\mathcal F}\iso \kappa^*\kappa_*{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F}\to{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F}. $$ Showing this to be an isomorphism is a local problem, so assume $X={\mathrm {Spec}}(A)$ with $A$ a noetherian adic ring. Let $K_\infty^\bullet$ be the usual \smash{$\dirlm{}\!\!$}\vspace{.8pt} of Koszul complexes on powers of a finite system of generators\vadjust{\kern.4pt} of an ideal of definition of~$A$ (\cite[\S3.1]{AJL}); and let \smash{$\widetilde K_\infty^\bullet$} be the corresponding quasi-coherent complex on ${\mathrm {Spec}}(A)$, so that the complex ${\mathcal K}_\infty^\bullet$% \index{ ${\mathbf K}$ (homotopy category)@${\mathcal K}_\infty^\bullet$ (limit of Koszul complexes)} in the proof of \Pref{Gamma'(qc)}(a) is just~\smash{$\kappa^*\widetilde K_\infty^\bullet$.} Then one checks via~ \cite[p.\,18, Lemma~(3.1.1)]{AJL} that the map in question is isomorphic to the natural isomorphism of complexes $$ \kappa^*(\widetilde K_\infty^\bullet\otimes_{{\mathcal O}_{\<\<X}}\<{\mathcal F}\>)\iso \kappa^*\widetilde K_\infty^\bullet\otimes_{{\mathcal O}_{{\mathscr X}}}\<\kappa^*\<{\mathcal F}. $$ That the composition in (c) is as asserted results from the following natural commutative diagram, whose bottom row composes to the identity: $$ \minCDarrowwidth=22pt \begin{CD} \kappa^*{\mathbf R}\iGp Z{\mathcal F} @>>> \kappa^*{\mathbf R}\iGp Z\kappa_*\kappa^*\<{\mathcal F} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>>\kappa^*\kappa_*{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F} @>>>{\mathbf R}\iGp{\mathscr X}\kappa^*\<{\mathcal F} \\ @VVV @VV\hskip3.5em\textup{(b)}V @VVV @VVV \\ \kappa^*\<{\mathcal F} @>>>\kappa^*\kappa_*\kappa^*\<{\mathcal F} @= \kappa^*\kappa_*\kappa^*\<{\mathcal F} @>>>\kappa^*\<{\mathcal F} \end{CD} $$ Uniqueness is shown as in (b). \end{proof} \smallskip \begin{scor} \label{C:Gammas'+kappas} The natural maps are isomorphisms \begin{alignat*}{2} {\mathrm {Hom}}_{\<X\<}({\mathcal E},\>{\mathcal F}\>) &\cong{\mathrm {Hom}}_{\<X\<}({\mathcal E},\kappa_*\kappa^*\<{\mathcal F}\>)\cong {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal E},\kappa^*\<{\mathcal F}\>) \quad&&\bigl({\mathcal E}\in{\mathbf D}_{\<Z}(X),\, {\mathcal F}\in{\mathbf D}(X)\bigr), \\ {\mathrm {Hom}}_{\<X\<}({\mathcal E},\>{\mathcal F}\>) &\cong{\mathrm {Hom}}_{\<X\<}({\mathcal E},\kappa_*\kappa^*\<{\mathcal F}\>)\cong {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal E},\kappa^*\<{\mathcal F}\>) \quad&&\bigl({\mathcal E}\in{\mathbf D}(X),\, {\mathcal F}\in{\mathbf D}_{\<Z}(X)\bigr), \\ {\mathrm {Hom}}_{\mathscr X}({\mathcal G},\>\H\>) &\cong{\mathrm {Hom}}_{\mathscr X}(\kappa^*\kappa_*{\mathcal G},\>\H\>)\cong {\mathrm {Hom}}_{\<X\<}(\kappa_*{\mathcal G},\kappa_*\H\>) \quad&&\bigl({\mathcal G}\in\D_{\mathrm t}\<({\mathscr X}),\, \H\in{\mathbf D}({\mathscr X})\bigr). \end{alignat*} \end{scor} \begin{proof} For the first line, use \Pref{Gamma'(qc)} and its analogue for ${\mathbf D}_{\<Z}(X)$, \Lref{L:j-gamma-eqvce}, and \Pref{Gammas'+kappas} to get the equivalent sequence of natural isomorphisms \begin{align*} {\mathrm {Hom}}_{\<X\<}({\mathcal E},{\mathcal F}\>)&\cong {\mathrm {Hom}}_{\<X\<}({\mathcal E},{\mathbf R}\iGp Z{\mathcal F}\>)\\ &\cong {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal E},\kappa^*{\mathbf R}\iGp Z{\mathcal F}\>)\\ &\cong {\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal E},{\mathbf R}\iGp {\mathscr X}\kappa^*\<{\mathcal F}\>)\\ &\cong{\mathrm {Hom}}_{\mathscr X}(\kappa^*{\mathcal E}, \kappa^*\<{\mathcal F}\>)\\ &\cong{\mathrm {Hom}}_{\<X\<}({\mathcal E},\kappa_*\kappa^*\<{\mathcal F}\>). \end{align*} The rest is immediate from \Pref{Gammas'+kappas}(a). \end{proof} \medskip \pagebreak[3] The next series of results concerns the behavior of $\D_{\mkern-1.5mu\mathrm{qct}}$ with respect to maps of formal schemes. \begin{sprop} \label{Rf-*(qct)} Let $f\colon {\mathscr X} \to {\mathscr Y}$ be a map of noetherian formal schemes. Then\/ ${\mathbf R f_{\!*}}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$ is bounded, and $$ {\mathbf R} f_{\!*} \bigl(\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr) \subset \D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}). $$ Moreover, if\/ $f$ is pseudo\kern.6pt-proper and\/ ${\mathcal F}\in\D_{\mathrm t}\<({\mathscr X})$ has coherent homology, then\/ so does ${\mathbf R} f_{\!*}\>{\mathcal F}\in\D_{\mathrm t}\<({\mathscr Y})$. \end{sprop} \begin{proof} Since $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\subset\D_{\<\vc}({\mathscr X})$ (\Lref{Gamma'+qc}), the boundedness assertion is given by \Pref{Rf_*bounded}(b). (Clearly, ${\mathbf R f_{\!*}}$ is bounded-below.) It suffices then for the next assertion (by the usual way-out arguments \cite[p.\,73, Proposition 7.3]{H1}) to show for any ${\mathcal M}\in\A_{\mathrm {qct}}\<({\mathscr X})$ that ${\mathbf R} f_{\!*}{\mathcal M}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})$. Let ${\mathcal E}$ be an injective resolution of~${\mathcal M}$, let ${\mathscr J}$ be an ideal of definition of~${\mathscr X}$, and let ${\mathcal E}_n$ be the flasque complex ${\mathcal E}_n\!:= \cH{om}({\mathcal O}/{\mathscr J}^n\<,\>{\mathcal E})$. Then by \Pref{Gamma'(qc)}(a), \mbox{${\mathcal M}\cong{\mathbf R}\iGp {\mathscr X}{\mathcal M}\cong \smash{\dirlm{}\!\<_n\>\>{\mathcal E}_n}\>$.} Since ${\mathscr X}$ is noetherian, \smash{$\dirlm{}\!\!$}'s of flasque sheaves are \mbox{$f_{\!*}$-acyclic} and \smash{$\dirlm{}\!\!$} commutes with~$f_{\!*}\>$; so with notation as in the proof of \Pref{f-*(qct)}, $$ {\mathbf R f_{\!*}} {\mathcal M}\cong{\mathbf R} f_{\!*}\>{\mathbf R}\iGp {\mathscr X}{\mathcal M} \cong f_{\!*}\>\dirlm{n}\<{\mathcal E}_n \cong \dirlm{n}\<f_{\!*}j_{n*}j_n^*{\mathcal E}_n \cong \dirlm{n} i_{n*}f_{\<n*}^{}j_n^*{\mathcal E}_n\>. $$ Since ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}^+({\mathscr X})$, therefore $$ j_{n*}j_n^*{\mathcal E}_n=\cH{om}({\mathcal O}/{\mathscr J}^n\<,\>{\mathcal E})\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X}), $$ as we see by way-out reduction to where ${\mathcal E}$ is a single quasi-coherent sheaf and then by \Cref{C:images}(d); and hence $j_n^*{\mathcal E}_n\in\D_{\mkern-1.5mu\mathrm {qc}}(X_{n})$ (see \cite[p.\,115, (5.3.15)]{GD}). Now $j_n^*{\mathcal E}_n$~is a flasque bounded-below ${\mathcal O}_{\!X_{\<n}}$-complex, so by way-out reduction to (for example) \cite[p.\,643, corollary~11]{Ke}, $$ f_{\<n*}^{}j_n^*{\mathcal E}_n\cong {\mathbf R} f_{\<n*}^{}j_n^*{\mathcal E}_n\in\D_{\mkern-1.5mu\mathrm {qc}}(Y_{n}); $$ and finally, in view of \Cref{qct=plump}, $$ {\mathbf R} f_{\!*}{\mathcal M}\cong i_{n*}\>\dirlm{n}\< f_{\<n*}^{}j_n^*{\mathcal E}_n \in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}). $$ For the last assertion, we reduce as before to showing for each coherent torsion ${\mathcal O}_{\mathscr X}$-module~${\mathcal M}$ and each $p\ge0$ that ${R}^p\!f_{\!*}{\mathcal M}\!:= H^p{\mathbf R f_{\!*}}{\mathcal M}$ is a coherent ${\mathcal O}_{\mathscr Y}$-module. With notation remaining as in \Pref{f-*(qct)}, the maps $i_n$ and $j_n$ are exact, and for some $n$, \mbox{${\mathcal M}=j_{n*}j_n^*{\mathcal M}_n$.} So $$ R^p\!f_{\!*}\>{\mathcal M}=R^p\!f_{\!*}\>j_{n*}j_n^*{\mathcal M}_n= i_{n*}R^p\!f_{n*}\>j_n^*{\mathcal M}_n, $$ which is coherent since $j_n^*{\mathcal M}_n$ is a coherent ${\mathcal O}_{\!X_n}$-module and $f_n\colon X_n\to Y_n$ is a proper scheme-map. \end{proof} \begin{scor}[cf.~\Cref{P:kappa-f*}]\label{C:kappa-f*t} Let\/ $f_0\colon X\to Y$ be a map of locally noetherian schemes, let\/ $W\subset Y$ and\/ $Z\subset f_0^{-1}W$ be closed subsets, with associated \(\kern.5pt flat\) completion maps\/ $\kappa_{\mathscr Y}^{{\phantom{.}}}\colon{\mathscr Y}=Y_{\</W}\to Y\<, \,$ $\kappa_{\mathscr X}^{{\phantom{.}}}\colon{\mathscr X}=X_{\</Z}\to X\<,$ and let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be the map induced by~$f_0\>$. For\/ ${\mathcal E}\in{\mathbf D}(X)$ let $$ \theta_{\<{\mathcal E}}\colon\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathcal E}\to {\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E} $$ be the map adjoint to the natural composition $$ {\mathbf R} f_{\!0*}{\mathcal E}\longrightarrow{\mathbf R} f_{\<0*}^{}\kappa_{{\mathscr X}*}^{\phantom*}\kappa_{\mathscr X}^*{\mathcal E}\iso \kappa_{{\mathscr Y}*}^{\phantom*}{\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}. $$ Then\/ $\theta_{\<{\mathcal E}}$ is an isomorphism for all\/ ${\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$ . \end{scor} \begin{proof} $\theta_{\<{\mathcal E}}$ is the composition of the natural maps $$ \kappa_{\mathscr Y}^*{\mathbf R} f_{\!0*}{\mathcal E}\to \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}\kappa_{{\mathscr X}*}^{\phantom*}\kappa_{\mathscr X}^*{\mathcal E}\iso \kappa_{\mathscr Y}^*\kappa_{{\mathscr Y}*}^{\phantom*}{\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E} \to {\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}. $$ By \Pref{Gammas'+kappas}, the first map and (in view of \Pref{Rf-*(qct)}) the third~map are both isomorphisms. \end{proof} \begin{sprop}\label{P:f* and Gamma} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of locally noetherian formal schemes. Let\/ ${\mathscr I}$ be a coherent\/ ${\mathcal O}_{\mathscr Y}$-ideal, and let ${\mathbf D}_{\mathscr I}({\mathscr Y})$ be the triangulated subcategory of\/~${\mathbf D}({\mathscr Y})$ whose objects are the complexes\/~${\mathcal F}$ with\/ ${\mathscr I}$-torsion homology\/ \textup(i.e., $\iG{\mathscr I}\<\<H^i{\mathcal F}=H^i{\mathcal F}$ for all\/ $i\in\mathbb Z$---see \S\S\textup{\ref{S:prelim}} and\/~\textup{\ref{Gamma'1}).} Then$\>:$ \smallskip \textup{(a)} ${\mathbf L} f^*\<({\mathbf D}_{\mathscr I}({\mathscr Y}))\subset{\mathbf D}_{{\mathscr I}{\mathcal O}_{\mathscr X}}\<({\mathscr X})$. \smallskip \textup{(b)} There is a unique functorial isomorphism $$ \hskip100pt \xi({\mathcal E})\colon{\mathbf L} f^*{\mathbf R}\iG{\mathscr I}{\mathcal E}\iso {\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathbf L} f^*{\mathcal E} \qquad \ \bigl({\mathcal E}\in{\mathbf D}({\mathscr Y})\bigr) $$ whose composition with the natural map\/ ${\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathbf L} f^*{\mathcal E}\to{\mathbf L} f^*{\mathcal E}$ is the natural map\/ ${\mathbf L} f^*{\mathbf R}\iG{\mathscr I}{\mathcal E}\to{\mathbf L} f^*{\mathcal E}\<$. \smallskip \textup{(c)} The natural map is an isomorphism $$ {\mathbf R}\iGp{\mathscr X}\> {\mathbf L} f^*{\mathbf R}\iGp{\mathscr Y}{\mathcal E}\iso {\mathbf R}\iGp{\mathscr X}\> {\mathbf L} f^*{\mathcal E} \qquad \ \bigl({\mathcal E}\in{\mathbf D}({\mathscr Y})\bigr). $$ \smallskip \textup{(d)} If\/ ${\mathscr X}$ is noetherian, there is a unique functorial isomorphism $$ \hskip100pt{\mathbf R}\iG{\mathscr I}\<{\mathbf R f_{\!*}}\>{\mathcal G} \iso {\mathbf R f_{\!*}}{\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}\qquad \ \bigl({\mathcal G}\in{\mathbf D}^+({\mathscr X})\bigr) $$ whose composition with the natural map\/ ${\mathbf R f_{\!*}}{\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}\to{\mathbf R f_{\!*}}\>{\mathcal G}$ is the natural map\/ ${\mathbf R}\iG{\mathscr I}\<{\mathbf R f_{\!*}}\>{\mathcal G}\to{\mathbf R f_{\!*}}\>{\mathcal G}$. \end{sprop} \begin{proof} (a) Let ${\mathcal F}\in{\mathbf D}_{\mathscr I}({\mathscr Y})$. To show that ${\mathbf L} f^*\<{\mathcal F}\in{\mathbf D}_{{\mathscr I}{\mathcal O}_{\mathscr X}}\<({\mathscr X})$ we may assume that ${\mathcal F}$~is K-injective. Let $x\in{\mathscr X}$, set $y\!:= f(x)$, and let $P_{\<x}^\bullet$ be a flat resolution of the ${\mathcal O}_{{\mathscr Y}\<,y}$-module~${\mathcal O}_{{\mathscr X}\<,x}\>$. Then, as in the proof of \Pref{Gamma'(qc)}(a), there is a canonical ${\mathbf D}({\mathscr Y})$-isomorphism $$ \dirlm{n} \cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}/{\mathscr I}^n\<,\>{\mathcal F}\>)=\iG{\mathscr I}\<{\mathcal F}={\mathbf R}\iG{\mathscr I}\<{\mathcal F}\iso{\mathcal F}, $$ and it follows that for any~$i$ the stalk at~$x$ of the homology~$H^i{\mathbf L} f^*\<{\mathcal F}$ is $$ \textup H^i\bigl(P_{\<x}^\bullet\otimes_{{\mathcal O}_{{\mathscr Y}\<,y}} {\mathcal F}^{\phantom{.}}_{\!y} \bigr) =\dirlm{n}\textup H^i\bigl(P_{\<x}^\bullet\otimes_{{\mathcal O}_{{\mathscr Y}\<,y}} {\mathrm {Hom}}^{\bullet}_{{\mathcal O}_{{\mathscr Y}\<,y}}({\mathcal O}^{\phantom{.}}_{{\mathscr Y}\<,y}/{\mathscr I}^n_{\!y}\>, \>{\mathcal F}^{\phantom{.}}_{\!y})\bigr). $$ Hence each element of the stalk is annihilated by a power of~${\mathscr I}{\mathcal O}_{{\mathscr X}\<,x}\>$, and (a) results. \smallskip (b) The existence and uniqueness of a functorial map $\xi({\mathcal E})$ satisfying everything except the isomorphism property result from (a) and the fact that ${\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}\<\<}$ is right-adjoint to the inclusion ${\mathbf D}_{{\mathscr I}{\mathcal O}_{\mathscr X}}\<\<({\mathscr X})\hookrightarrow{\mathbf D}({\mathscr X})$. To show that $\xi({\mathcal E})$ is an isomorphism we may assume that ${\mathscr Y}$ is affine and that ${\mathcal E}$ is K-flat, and then proceed as in the proof of (the special case) \Pref{Gammas'+kappas}(c), via the bounded flat complex $K_\infty^\bullet\>$. \smallskip (c) Let ${\mathscr I}$, ${\mathscr J}$ be defining ideals of~${\mathscr Y}$ and~${\mathscr X}$ respectively, so that ${\mathcal K}\!:={\mathscr I}{\mathcal O}_{\mathscr X}\subset{\mathscr J}$. The natural map $ {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\iG{\mathcal K}:={\mathbf R}\iG{\mathscr J}{\mathbf R}\iG{\mathcal K}\to{\mathbf R}\iG{\mathscr J}=:{\mathbf R}\iGp{\mathscr X} $ is an \emph{isomorphism,} as one checks locally via ~\cite[p.\,20, Corollary~(3.1.3)]{AJL}. So for any ${\mathcal E}\in{\mathbf D}({\mathscr Y})$, (b) gives $$ {\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*\<{\mathcal E} \cong {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\iG{\mathcal K}{\mathbf L} f^*\<{\mathcal E} \cong {\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*\>{\mathbf R}\iGp{\mathscr Y}\>{\mathcal E}. $$ \smallskip (d) ${\mathcal G}$ may be assumed bounded-below and injective, so that $$ {\mathcal G}_n\!:=\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr I}^n{\mathcal O}_{\mathscr X}\>,\>{\mathcal G}) $$ is flasque. Then, since ${\mathscr X}$ is noetherian, $\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}=\smash{\dirlm{}\!\mkern-1.5mu _n\>\>{\mathcal G}_n}\>$ is flasque too, and $$ {\mathbf R f_{\!*}}\>{\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}\cong {\mathbf R f_{\!*}}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}\cong f_{\!*}\mkern1.5mu{\dirlm{n}{\mathcal G}_n} \cong {\dirlm{n}\<f_{\!*}\>{\mathcal G}_n}\in{\mathbf D}_{\mathscr I}({\mathscr Y}). $$ By \Lref{L:j-gamma-eqvce}, ${\mathbf R}\iG{\mathscr I}$ (resp.~${\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}\<\<}$) is right-adjoint to the inclusion \mbox{${\mathbf D}_{\mathscr I}({\mathscr Y})\hookrightarrow{\mathbf D}({\mathscr Y})$} (resp.~${\mathbf D}_{{\mathscr I}{\mathcal O}_{\mathscr X}}({\mathscr X})\hookrightarrow{\mathbf D}({\mathscr X})$), whence, in particular, the uniqueness in~(d). Moreover, in view of~(a), for any ${\mathcal E}\in{\mathbf D}_{\mathscr I}({\mathscr Y})$ the natural maps are isomorphisms \begin{multline*} {\mathrm {Hom}}_{\mathscr Y}({\mathcal E},\>{\mathbf R}\iG{\mathscr I}\<{\mathbf R f_{\!*}}\>{\mathcal G}) \iso {\mathrm {Hom}}_{\mathscr Y}({\mathcal E}\<,\>{\mathbf R f_{\!*}}\>{\mathcal G}) \iso {\mathrm {Hom}}_{\mathscr X}({\mathbf L} f^*\<{\mathcal E}\<,\>{\mathcal G}) \\ \iso {\mathrm {Hom}}_{\mathscr X}({\mathbf L} f^*\<{\mathcal E}\<,\>{\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}) \iso {\mathrm {Hom}}_{\mathscr Y}({\mathcal E}\<,\>{\mathbf R f_{\!*}}\>{\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}). \end{multline*} It follows formally that the image under this composed isomorphism of the identity map of~${\mathbf R}\iG{\mathscr I}\<{\mathbf R f_{\!*}}\>{\mathcal G}$ is an isomorphism as asserted. (In fact this isomorphism is adjoint to the composition ${\mathbf L} f^*{\mathbf R}\iG{\mathscr I}\<\<{\mathbf R f_{\!*}}{\mathcal G} \xrightarrow[\xi({\mathbf R f_{\!*}}{\mathcal G})]{} {\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathbf L} f^*{\mathbf R f_{\!*}}{\mathcal G} \xrightarrow[\textup{nat'l}]{} {\mathbf R}\iG{{\mathscr I}{\mathcal O}_{\mathscr X}}{\mathcal G}.\>)$ \end{proof} \begin{sdef}\label{D:Dtilde}\index{ ${\mathbf D}$ (derived category)!z@${ \widetilde {\vbox to5pt{\vss\hbox{$\mathbf D$}}}_{\mkern-1.5mu\mathrm {qc}} }$} For a locally noetherian formal scheme~${\mathscr X}$, $$ \wDqc({\mathscr X})\!:={\mathbf R}\iGp{\mathscr X}{}^{-1}(\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})) $$ is the $\Delta$-subcategory of ${\mathbf D}({\mathscr X})$ whose objects are those complexes~${\mathcal F}$ such that ${\mathbf R}\iGp{\mathscr X}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})$---or equivalently, ${\mathbf R}\iGp{\mathscr X}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. \end{sdef} \begin{srems}\label{R:Dtilde} (1) By \Pref{Gamma'(qc)}(b), $\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})\subset\wDqc({\mathscr X})$. Hence $$ {\mathbf R}\iGp{\mathscr X}\bigl(\>\wDqc({\mathscr X})\bigr)\subset\wDqc({\mathscr X}). $$ (2) Since ${\mathbf R}\iGp{\mathscr X}$ is idempotent (see \Pref{Gamma'(qc)}), the vertex of any triangle based on the canonical map ${\mathbf R}\iGp{\mathscr X}{\mathcal E}\to{\mathcal E}\ ({\mathcal E}\in{\mathbf D}({\mathscr X}))$ is annihilated by ${\mathbf R}\iGp{\mathscr X}$. It follows that $\wDqc({\mathscr X})$ is the smallest $\Delta$-subcategory of~${\mathbf D}({\mathscr X})$ containing $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ and all complexes~${\mathcal F}$ such that ${\mathbf R}\iGp{\mathscr X}{\mathcal F}=0$.\vspace{1.5pt} \smallskip (3) The functor ${\mathbf R}\iGp{\mathscr X}\colon{\mathbf D}({\mathscr X})\to {\mathbf D}({\mathscr X})$ has a right adjoint\index{ $\mathbf {La}$@${\boldsymbol\Lambda}$ (homology localization)} $$ {\boldsymbol\Lambda}_{\mathscr X}(-)\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,-). $$ Indeed, there are natural functorial isomorphisms for ${\mathcal E},{\mathcal F}\in{\mathbf D}({\mathscr X})$, \begin{equation}\label{adj} \begin{aligned} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})\<}({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,\>{\mathcal F}\>) &\iso{\mathrm {Hom}}_{{\mathbf D}({\mathscr X})\<}({\mathcal E}\Otimes{\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,\>{\mathcal F}\>)\\ &\iso{\mathrm {Hom}}_{{\mathbf D}({\mathscr X})\<}\bigl({\mathcal E},\>{\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,\>{\mathcal F}\>)\bigr). \end{aligned} \end{equation} (Whether the natural map \smash{${\mathcal E}\Otimes{\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\iso{\mathbf R}\iGp{\mathscr X}{\mathcal E}$} is an isomorphism\vspace{1.3pt} is a local question, dealt with e.g., in \cite[p.\,20, Corollary~(3.1.2)]{AJL}. \vspace{.8pt} The second isomorphism is given, e.g., by \cite[p.\,147, Proposition 6.6\,(1)]{Sp}.)\vspace{.5pt} \pagebreak[3] There is a natural isomorphism ${\mathbf R}\iGp{\mathscr X}\iso{\mathbf R}\iGp{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}^{}$ (see (d) in \Rref{R:Gamma-Lambda} below), and consequently $$ {\boldsymbol\Lambda}_{\mathscr X}^{}\bigl(\>\wDqc({\mathscr X})\bigr)\subset\wDqc({\mathscr X}). $$ \smallskip (4) \emph{If\/ ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^-({\mathscr X})$ and\/ ${\mathcal F}\in\wDqc({\mathscr X})$ then\/ ${\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>)\in\wDqc({\mathscr X})$, and hence\/ ${\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,\>{\mathcal F}\>)\in\wDqc({\mathscr X})$.} Indeed, the natural map $$\postdisplaypenalty10000 {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,{\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)\to{\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>) $$ is an \emph{isomorphism,} since for any ${\mathcal G}$ in $\D_{\mathrm t}\<({\mathscr X})$, $\>\smash{{\mathcal G}\Otimes{\mathcal E}}\in\D_{\mathrm t}\<(X)$ (an assertion which can be checked locally, using \Pref{Gamma'(qc)}(a) and the complex~${\mathcal K}_\infty^\bullet$ in its proof), so that there is a sequence of natural isomorphisms (see \Pref{Gamma'(qc)}(c)): \begin{align*} {\mathrm {Hom}}\bigl({\mathcal G}, {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,{\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)\bigr) &\iso {\mathrm {Hom}}\bigl({\mathcal G}, {\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,{\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)\bigr) \\ &\iso {\mathrm {Hom}}\bigl(\smash{\smash{{\mathcal G}\Otimes{\mathcal E}}}\<, {\mathbf R}\iGp{\mathscr X}{\mathcal F}\bigr) \\ &\iso {\mathrm {Hom}}\bigl(\smash{{\mathcal G}\Otimes{\mathcal E}}\<, {\mathcal F}\bigr) \\ &\iso {\mathrm {Hom}}\bigl({\mathcal G}, {\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>)\bigr)\\ &\iso {\mathrm {Hom}}\bigl({\mathcal G}, {\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>)\bigr). \end{align*} Since $\A_{\mathrm {qct}}\<({\mathscr X})$ is plump in~${\mathcal A}({\mathscr X})$ (\Cref{qct=plump}), \Pref{P:Rhom} shows that \mbox{${\mathbf R}\iGp{\mathscr X}\>{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,{\mathbf R}\iGp{\mathscr X}{\mathcal F}\>)\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$,} whence ${\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>)\in\wDqc({\mathscr X})$. From (3) and the natural isomorphisms $$ {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,\>{\mathcal F}\>)\cong {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\Otimes{\mathcal E}\<,\>{\mathcal F}\>)\cong {\boldsymbol\Lambda}_{\mathscr X}^{}{\mathbf R}\cH{om}^{\bullet}({\mathcal E}\<,\>{\mathcal F}\>) $$ we see then that $$ {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,\>{\mathcal F}\>)\in\wDqc({\mathscr X}). $$ \smallskip \pagebreak[3] (5) For ${\mathcal F}\in{\mathbf D}({\mathscr X})$ it holds that $$ {\mathcal F}\in\wDqc({\mathscr X})\iff{\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/{\mathscr J},\>{\mathcal F}\>)\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}) \text{ for all defining ideals~${\mathscr J}$ of~${\mathscr X}$.} $$ The implication $\implies$ is given, in view of \Cref{C:Hom-Rgamma}, by \Pref{P:Rhom}; and the converse is given by \Lref{Gam as holim}, since \Cref{qct=plump} implies that $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ is a $\Delta$-subcategory of ${\mathbf D}({\mathscr X})$ closed under direct sums. \smallskip (6)\vspace{.7pt} Let $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of locally noetherian formal schemes. For any ${\mathcal F}\in\wDqc({\mathscr Y})$, \Lref{Gamma'+qc} and \Pref{P:Lf*-vc} give $$ {\mathbf L} f^*\>{\mathbf R}\iGp{\mathscr Y}\>{\mathcal F}\in {\mathbf L} f^*(\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}))\subset{\mathbf L} f^*(\D_{\<\vc}({\mathscr Y}))\subset\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})\subset\wDqc({\mathscr X}), $$ and so ${\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*\<{\mathcal F} \underset{\textup{\ref{P:f* and Gamma}(c)}}\cong {\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*{\mathbf R}\iGp{\mathscr Y}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}).$ Thus\vspace{-5pt} $$ {\mathbf L} f^*\bigl(\>\wDqc({\mathscr Y})\bigr)\subset\wDqc({\mathscr X}). $$ \end{srems} \smallskip \begin{scor}\label{C:f* and Gamma} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be an adic map of locally noetherian formal schemes. Then$\>:$ \smallskip \textup{(a)} ${\mathbf L} f^*\<(\D_{\mathrm t}\<({\mathscr Y}))\subset\D_{\mathrm t}\<({\mathscr X})$. \smallskip \textup{(b)} ${\mathbf L} f^*\<(\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}))\subset\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. \smallskip \pagebreak[3] \textup{(c)} There is a unique functorial isomorphism $$ {\mathbf L} f^*{\mathbf R}\iGp{\mathscr Y}{\mathcal E}\iso {\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*{\mathcal E} \qquad \ \bigl({\mathcal E}\in{\mathbf D}({\mathscr Y})\bigr) $$ whose composition with the natural map\/ ${\mathbf R}\iGp{\mathscr X}\>{\mathbf L} f^*{\mathcal E}\to{\mathbf L} f^*{\mathcal E}$ is the natural map\/ ${\mathbf L} f^*{\mathbf R}\iGp{\mathscr Y}{\mathcal E}\to{\mathbf L} f^*{\mathcal E}$. There results a conjugate isomorphism of right-adjoint functors $$ {\mathbf R f_{\!*}}{\boldsymbol\Lambda}_{\mathscr X}\>{\mathcal G}\iso {\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R f_{\!*}}{\mathcal G}\qquad\bigl({\mathcal G}\in{\mathbf D}({\mathscr X})\bigr). $$ whose composition with the natural map\/ ${\mathbf R f_{\!*}}{\mathcal G}\to{\mathbf R f_{\!*}}{\boldsymbol\Lambda}_{\mathscr X}\>{\mathcal G}$ is the natural map\/ ${\mathbf R f_{\!*}}{\mathcal G}\to{\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R f_{\!*}}{\mathcal G}$. \smallskip \goodbreak \textup{(d)} If\/ ${\mathscr X}$ is noetherian then there is a unique functorial isomorphism $$ \hskip100pt{\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}\>{\mathcal G} \iso {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\qquad \ \bigl({\mathcal G}\in{\mathbf D}^+({\mathscr X})\textup{ or }\>{\mathcal G}\in\wDqc({\mathscr X})\bigr) $$ whose composition with the natural map\/ ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\to{\mathbf R f_{\!*}}\>{\mathcal G}$ is the natural map\/ ${\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}\>{\mathcal G}\to{\mathbf R f_{\!*}}\>{\mathcal G}$. \smallskip \textup{(e)} If\/ ${\mathscr X}$ is noetherian then\/ ${\mathbf R f_{\!*}}\<\bigl(\>\wDqc({\mathscr X})\bigr)\subset\wDqc({\mathscr Y}).$ \end{scor} \begin{proof} To get (a) and (c) take ${\mathscr I}$ in \Pref{P:f* and Gamma} to be an ideal of definition of~${\mathscr Y}$. (The second assertion in (c) is left to the reader.) As $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\<=\>\D_{\<\vc}({\mathscr Y})\cap\>\D_{\mathrm t}\<({\mathscr Y})$ (\Cref{C:vec-c is qc} and \Lref{Gamma'+qc}), (b) follows from (a) and \Pref{P:Lf*-vc}. The same choice of~$\>{\mathscr I}$ gives (d) for ${\mathcal G}\in{\mathbf D}^+({\mathscr X})$---and the argument also works for \smash{${\mathcal G}\in\wDqc({\mathscr X})$} once one notes that $$ {\mathbf R f_{\!*}}\>{\mathbf R}\iGp{\mathscr X}\<\bigl(\>\wDqc({\mathscr X})\bigr) \subset{\mathbf R f_{\!*}}\<\bigl(\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr) \underset{\mathstrut\text{\ref{Rf-*(qct)}}}\subset \D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y})\subset\D_{\mathrm t}\<({\mathscr Y}). $$ The isomorphism in (d) gives (e) via \Pref{Rf-*(qct)}. \end{proof} \begin{scor} \label{C:kappa-f*t'} In \Cref{C:kappa-f*t}\textup{,} if\/ ${\mathscr X}$ is noetherian and\/ $Z= f_0^{-1}W$ then for all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$ the map\/ $\theta_{\<{\mathcal F}}'\!:={\mathbf R}\iGp{\mathscr Y}(\theta_{\<{\mathcal F}}\<)$ is an isomorphism $$ \theta_{\<{\mathcal F}}'\colon{\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}\>{\mathcal F}\iso {\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}\kappa_{\mathscr X}^*\>{\mathcal F}. $$ \end{scor} \begin{proof} Arguing as in \Pref{Gamma'(qc)}, we find that ${\mathbf R}\iG Z\>{\mathcal F}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$, so that we have the isomorphism $\theta_{{\mathbf R}\iG Z{\mathcal F}}$ of \Cref{C:kappa-f*t}.\vspace{1pt} Imitating the proof of~ \Cref{C:f* and Gamma}, we get an isomorphism $$ \alpha_{\<{\mathcal F}}^{}\colon{\mathbf R} f_{\<0*}^{}{\mathbf R}\iG Z\>{\mathcal F}\iso{\mathbf R}\iG W{\mathbf R} f_{\<0*}^{}\>{\mathcal F} $$ whose composition with the natural map ${\mathbf R}\iG W{\mathbf R} f_{\<0*}^{}\>{\mathcal F}\to{\mathbf R} f_{\<0*}^{}\>{\mathcal F}$ is the natural map ${\mathbf R} f_{\<0*}^{}{\mathbf R}\iG Z\>{\mathcal F}\to{\mathbf R} f_{\<0*}^{}\>{\mathcal F}$. Consider then the diagram $$ \begin{CD} \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathbf R}\iG Z\>{\mathcal F} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}> \smash{\kappa_{\mathscr Y}^*(\<\alpha_{\<\<{\mathcal F}}^{}\<)}> \kappa_{\mathscr Y}^*{\mathbf R}\iG W{\mathbf R} f_{\<0*}^{}\>{\mathcal F} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\smash{\textup{\ref{Gammas'+kappas}(c)}}> {\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}\>{\mathcal F}@>\textup{nat'l}>> \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}\>{\mathcal F}\\ @V\theta_{{\mathbf R}\iG Z{\mathcal F}} V \simeq V @. @V(1)\hskip8.8em V\theta_{\!{\mathcal F}}'V @VV\theta_{\!{\mathcal F}}^{} V\\ {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG Z\>{\mathcal F} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\textup{\ref{Gammas'+kappas}(c)}> {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*\>{\mathcal F} @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\textup{\ref{C:f* and Gamma}(d)}> {\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}\kappa_{\mathscr X}^*\>{\mathcal F} @>>\textup{nat'l}> {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*\>{\mathcal F}\\ \end{CD} $$ It suffices to show that subdiagram (1) commutes; and since ${\mathbf R}\iGp{\mathscr Y}$ is right-adjoint to the inclusion $\D_{\mathrm t}\<({\mathscr Y})\hookrightarrow{\mathbf D}({\mathscr Y})$ it follows that it's enough to show that the outer border of the diagram commutes. But it is straightforward to check that the top and bottom rows compose to the maps induced by the natural map ${\mathbf R}\iG Z\to\bf 1$, whence the conclusion. \end{proof} \end{parag} \begin{parag}\label{tors-eqvce} From the following key \Pref{1!}---generalizing the noetherian case of \cite[p.\,12, Proposition~(1.3)]{AJL}---there will result, for complexes with quasi-coherent torsion homology, a stronger version of the Duality \Tref{prop-duality}, see \Sref{S:t-duality}. Recall what it means for a noetherian formal scheme~${\mathscr X}$ to be \emph{separated} (\S\ref{note1}). Recall also from \Cref{C:Qt} that the inclusion functor\/ $j^{\mathrm t}_{\<{\mathscr X}}\colon\A_{\mathrm {qct}}\<({\mathscr X})\hookrightarrow {\mathcal A}({\mathscr X})$ has a right adjoint\/~$Q^{\mathrm t}_{\<{\mathscr X}}$. \pagebreak[3] \begin{sprop} \label{1!}Let\/ ${\mathscr X}$ be a noetherian formal scheme. \textup{(a)} The extension of\/~$j^{\mathrm t}_{\<{\mathscr X}}$ induces an \emph{equivalence of categories}\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$!${\boldsymbol j}^{\mathrm t}$} $$ {\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}\colon{\mathbf D}^+(\A_{\mathrm {qct}}\<({\mathscr X})){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\approx\mkern17mu}}\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X}), $$ with bounded quasi-inverse\/ $\RQ^{\mathrm t}_{{\mathscr X}}|_{\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})}$.\vspace{1pt} \textup{(b)} If\/ ${\mathscr X}$ is separated, or of finite Krull dimension, then the extension of\/~$j^{\mathrm t}_{\<{\mathscr X}}$ induces an \emph{equivalence of categories} $$ {\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}\colon{\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X})){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\approx\mkern17mu}}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}), $$ with bounded quasi-inverse\/ $\RQ^{\mathrm t}_{{\mathscr X}}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$. \end{sprop} \begin{proof} (a) The asserted equivalence is given by \cite[Theorem 4.8]{Ye}. The idea is that $\A_{\mathrm {qct}}\<({\mathscr X})$ contains enough $\A_{\mathrm t}\<({\mathscr X})$-injectives \cite[Proposition 4.2]{Ye}, so by \cite[p.\,47, Proposition 4.8]{H1}, ${\mathbf D}^+(\A_{\mathrm {qct}}\<({\mathscr X}))$ is equivalent to $\D_{\mkern-1.5mu\mathrm {qc}}^+(\A_{\mathrm t}\<({\mathscr X}))$, which is equivalent to~$\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$ (\Pref{Gamma'(qc)}(c)). Since $\RQ^{\mathrm t}_{{\mathscr X}}$\vspace{.4pt} is right-adjoint to ${\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}$ (\Lref{L:j-gamma-eqvce}), its restriction to $\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$ is quasi-inverse to ${\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}|_{{\mathbf D}^+(\A_{\mathrm {qct}}\<({\mathscr X}))}$. From the resulting isomorphism $$ \iota^{}_{\mathcal E}\colon{\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}\RQ^{\mathrm t}_{{\mathscr X}}{\mathcal E}\iso{\mathcal E}\qquad ({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})) $$ we see that if $H^i{\mathcal E}=0$ then $H^i\> \RQ^{\mathrm t}_{{\mathscr X}}{\mathcal E}=0$, so that $\RQ^{\mathrm t}_{{\mathscr X}}|_{\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})}$ is bounded. (b) By \Lref{L:j-gamma-eqvce}, and having the isomorphism~$\iota^{}_{\mathcal E}$, we need only show that $\RQ^{\mathrm t}_{\mathscr X}$~is bounded on~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. Suppose that ${\mathscr X}$ is the completion of a separated ordinary noetherian scheme~$X$ along some closed subscheme, and let $\kappa\colon{\mathscr X}\to X$ be the completion map, so that $Q^{\mathrm t}_{{\mathscr X}}=\kappa^*\<\iG Z Q_{\<\<X}^{}\kappa_*^{}$ (see remark following \Cref{C:Qt}). The exact functor~$\kappa_*^{}$ preserves K-injectivity, since it has an exact left adjoint, namely~$\kappa^*\<$. Similarly $Q_{\<\<X}$~transforms K-injective ${\mathcal A}(X)$-complexes into K-injective $\A_{\qc}(X)$-complexes. Hence $\RQ^{\mathrm t}_{\mathscr X}\cong \kappa^*{\mathbf R}\iG Z^{\!\!\textup{qc}}\>{\mathbf R} Q_{\<\<X}^{}\kappa_*^{}$, where $\iG Z^{\!\!\textup{qc}}\colon\A_{\qc}(X)\to\A_{{\qc}Z}(X)$ is the restriction of~$\iG Z\>$. Now by the proof of \cite[p.\,12, Proposition~(1.3)]{AJL}, ${\mathbf R} Q_{\<\<X}^{}$ is bounded on $\D_{\mkern-1.5mu\mathrm {qc}}(X)\supset\kappa_*{}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (\Pref{Gammas'+kappas}). Also, by \cite[p.\,24, Lemma (3.2.3)]{AJL}, ${\mathbf R}\iG Z$ is bounded; and hence by \cite[p.\,26, Proposition (3.2.6)]{AJL}, so is $\iG Z^{\!\!\textup{qc}}\<$. Thus $\RQ^{\mathrm t}_{\mathscr X}$ is bounded on~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. In the general separated case, one proceeds by induction on the least number\- of affine open subsets covering ${\mathscr X}$, as in the proof of \cite[p.\,12, Proposition~(1.3)]{AJL} (which is \Pref{1!} for ${\mathscr X}$ an ordinary scheme), \emph{mutatis mutandis}---namely, substitute ``${\mathscr X}\>$" for ``$\<X\<$,\!" ``qct" for ``qc," ``$Q^{\mathrm t}$" for ``$Q$,\!" and recall for a map~\mbox{$v\colon{\mathscr V}\to{\mathscr X}$} of noetherian formal schemes that \mbox{$v_*(\A_{\mathrm {qct}}\<({\mathscr V}))\subset\A_{\mathrm {qct}}\<({\mathscr X})$} (\Pref{f-*(qct)}), and furthermore that if $v$ is affine then $v_*|_{\A_{\mathrm {qct}}\<({\mathscr V})}$ is \emph{exact} (Lemmas~\ref{Gamma'+qc} and~\ref{affine-maps}). A similar procedure works when the Krull dimension $\dim{\mathscr X}$ is finite, but now the induction is on $n({\mathscr X})\!:={}$least $n$ such that ${\mathscr X}$ has an open covering ${\mathscr X}=\cup_{i=1}^n{\mathscr U}_i$ where for each~$i$ there is a separated ordinary noetherian scheme $U_i$ such that ${\mathscr U}_i$~is isomorphic to the completion of $U_i$ along one of its closed subschemes. (This property of ${\mathscr U}_i$ is inherited by any of its open subsets). \newcommand{v_*^{\mathrm{qct}}}{v_*^{\mathrm{qct}}} The case $n({\mathscr X})=1$ has just been done. Consider, for any open immersion $v\colon{\mathscr V}\hookrightarrow {\mathscr X}$, the functor $v_*^{\mathrm{qct}}\!:= v_*|_{\A_{\mathrm {qct}}\<({\mathscr V})}$. To complete the induction as in the proof of \cite[p.\,12, Proposition~(1.3)]{AJL}, one needs to show that \emph{the derived functor ${\mathbf R} v_*^{\mathrm{qct}}\colon{\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr V}))\to{\mathbf D}({\mathscr X})$ is bounded above.} For $\>{\mathcal N}\in\A_{\mathrm {qct}}\<({\mathscr V})$, let $\>{\mathcal N}\to{\mathcal J}^\bullet\>$ be an $\A_{\mathrm {qct}}\<$-injective---hence flasque---resolution \cite[Proposition 4.2]{Ye}. Now $H^i{\mathbf R}v_*^{\mathrm{qct}}({\mathcal N})$ is the sheafification of the presheaf sending~ an open ${\mathscr W}\subset{\mathscr X}$ to $\textup{H}^i\Gamma({\mathscr W}\cap{\mathscr V},\> {\mathcal J}^\bullet)=\textup H^i({\mathscr W}\cap{\mathscr V},\>{\mathcal N})$, which vanishes when $i>\dim{\mathscr X}$, whence the conclusion (\cite[Proposition (2.7.5)]{Derived categories}). \end{proof} \end{parag} \begin{parag} Let $(X, {\mathcal O}_{\<\<X})$ be a ringed space, and let ${\mathcal J}$ be an ${\mathcal O}_{\<\<X}$-ideal. The next Lemma, expressing ${\mathbf R}\iG{\<{\mathcal J}\>}$ as a ``homotopy colimit," \index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!${\mathbf R}\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ as homotopy colimit|(} lifts back to ${\mathbf D}(X)$ the well-known relation $$ H^i{\mathbf R}\iG{\<{\mathcal J}\>}{\mathcal G}= \dirlm{n}\text{\emph{${\mathcal E}\<$xt}}^i_{{\mathcal O}_X}({\mathcal O}_X/\<{\mathcal J}^n\<,\>{\mathcal G}) \qquad\bigl({\mathcal G}\in {\mathbf D}(X)\bigr). $$ Define ${\boldsymbol h}_n\colon{\mathbf D}(X)\to{\mathbf D}(X)$~by $$ {\boldsymbol h}_n({\mathcal G})\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\<\<X}/\<{\mathcal J}^n\<,\>{\mathcal G})\qquad\bigl(n\ge1,\ {\mathcal G}\in{\mathbf D}(X)\bigr). $$ There are natural functorial maps $s_n\colon {\boldsymbol h}_n\to {\boldsymbol h}_{n+1}$ and $\varepsilon_n\colon {\boldsymbol h}_n\to{\mathbf R} \iG{\<{\mathcal J}\>}$, satisfying \mbox{$\varepsilon_{n+1}s_n=\varepsilon_n$.} The family $$ (1,-s_m)\colon {\boldsymbol h}_m\to {\boldsymbol h}_m\oplus {\boldsymbol h}_{m+1} \subset \oplus_{n\ge1}\>{\boldsymbol h}_n \qquad(m\ge 1) $$ defines a natural map $s\colon {\oplus_{n\ge1}\>{\boldsymbol h}_n}\to\oplus_{n\ge1}\>{\boldsymbol h}_n$. There results, for each ${\mathcal G}\in{\mathbf D}(X)$, a map of triangles $$ \begin{CD} \oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G} @>s>> \oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G} @>>> \textup{??} @>+>> \\ @VVV @VV \<\sum\!\varepsilon_n V @VV\overline\varepsilon V \\ 0 @>>> {\mathbf R} \iG{\<{\mathcal J}\>}{\mathcal G} @= {\mathbf R} \iG{\<{\mathcal J}\>}{\mathcal G} @>+>> \end{CD} $$ \begin{slem} \label{Gam as holim} The map\/ $\overline{\varepsilon}$ is a\/ ${\mathbf D}(X)$-isomorphism, and so we have a triangle $$ \begin{CD} \oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G} @>s>> \oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G} @>\sum\!\varepsilon_n>> {\mathbf R}\iG{\<{\mathcal J}\>}{\mathcal G} @>+>> \end{CD} $$ \end{slem} \begin{proof} In the exact homology sequence $$ \cdots\to H^i\bigl(\!\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G}\bigr) \overset{\sigma^i}{\longrightarrow} H^i\bigl(\!\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G}\bigr) \longrightarrow H^i\bigl(\textup{??}\bigr) \longrightarrow H^{i+1}\bigl(\!\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal G}\bigr) \to\cdots $$ the map $\sigma^i$ is injective, as can be verified stalkwise at each $x\in X$. Assuming, as one may, that ${\mathcal G}$ is K-injective, one deduces that $$ H^i(\textup{??}) = \dirlm{n}\! H^i({\boldsymbol h}_n\>{\mathcal G})=H^i\>\>\dirlm{n}\! ({\boldsymbol h}_n\>{\mathcal G}) = H^i\>\>\dirlm{n}\!\cH{om}^{\bullet}({\mathcal O}_{\<\<X}/\<{\mathcal J}^n\<,\>{\mathcal G})=H^i({\mathbf R}\iG{\<{\mathcal J}\>}{\mathcal G}), $$ whence the assertion. \end{proof} \index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!${\mathbf R}\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ as homotopy colimit|)} \end{parag} \section{Duality for torsion sheaves.} \label{S:t-duality} Paragraph~\ref{T:qct-duality} contains the proof of \Tref{Th2} (section~\ref{S:prelim}), that is, of two essentially equivalent forms of Torsion Duality% \index{Grothendieck Duality!Torsion (global)} on formal schemes---\Tref{T:qct-duality} and~ \Cref{C:f*gam-duality}. The rest of the paragraph deals with numerous relations among the functors which have been introduced, and with compatibilities among dualizing functors occurring before and after completion of maps of ordinary schemes. More can be said for complexes with coherent homology, thanks to Greenlees-May duality.\index{Greenlees-May Duality} This is done in paragraph~\ref{coherent}. \pagebreak[3] Paragraph~\ref{SS:Gam-Lam} discusses additional relations involving ${\mathbf R}\iGp{\mathscr X}\colon{\mathbf D}({\mathscr X})\to{\mathbf D}({\mathscr X})$ and its right adjoint ${\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,-)$ on a locally noetherian formal scheme~${\mathscr X}$. \begin{thm}\label{T:qct-duality} \textup{(a)} Let\/ $f\colon\<{\mathscr X} \to {\mathscr Y}$ be a map of noetherian formal schemes. Assume that\/ $f$ is separated, or\/ ${\mathscr X}$ has finite Krull dimension, or else restrict~to bounded-below complexes. Then the\/ \hbox{$\Delta$-functor\/} \mbox{$\>{\mathbf R f_{\!*}}\colon\<\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\<\xrightarrow{\!\ref{Rf-*(qct)}\>}\<\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}) \hookrightarrow{\mathbf D}({\mathscr Y})$} has a right\/ $\Delta$-adjoint. In~fact there is a\/ bounded-below $\Delta$-functor $f_{\mathrm t}^\times\colon{\mathbf D}({\mathscr Y})\to\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$\vadjust{\kern.3pt}% \index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$f_{\mathrm t}^\times\<\<$} and a map of\/ $\Delta$-functors $\tau_{\<\mathrm t}^{\phantom{.}}\colon{\mathbf R} f_{\!*} f_{\mathrm t}^\times\to {\bf 1}$% \index{ {}$\tau$ (trace map)!$\tau_{\<\mathrm t}$} such that for all\/ ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ and\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y}),$\ the composed map $($in\/ the derived category of abelian groups\/$)$ \begin{align*} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr X}({\mathcal G},\>f_{\mathrm t}^\times\<{\mathcal F}\>) &\xrightarrow{\mathrm{natural}\,} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr Y}({\mathbf R} f_{\!*}{\mathcal G}, \>{\mathbf R} f_{\!*}f_{\mathrm t}^\times\<{\mathcal F}\>) \\ &\xrightarrow{\:\mathrm{via}\ \tau_{\<\mathrm t}^{\phantom{.}}\;\mkern1.5mu}{\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr Y}({\mathbf R} f_{\!*}{\mathcal G},{\mathcal F}\>) \end{align*} is an isomorphism. \smallskip \textup{(b)} If\/ $g\colon {\mathscr Y}\to{\mathscr Z}$ is another such map then there is a natural isomorphism\/ $$ (gf)_{\mathrm t}^{\<\times} \isof_{\mathrm t}^\times\< g_{\mathrm t}^\times\<. $$ \end{thm} \noindent\emph{Proof.} Assertion (b) follows from (a), which easily implies that $(gf)_{\mathrm t}^\times$ and $f_{\mathrm t}^\times\< g_{\mathrm t}^\times$ are both right-adjoint to the restriction of~${\mathbf R}(gf)_*={\mathbf R} g_*{\mathbf R f_{\!*}}$ to~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. As for (a), assuming first that ${\mathscr X}$ is separated or finite-dimensional, or that only bounded-below complexes are considered, we can replace~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ by the \emph{equivalent} category ${\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X}))$ (\Pref{1!}). The inclusion\mbox{ $k\colon\A_{\mathrm {qct}}\<({\mathscr X})\hookrightarrow\A_{\vec {\mathrm c}}({\mathscr X})$} has the right adjoint $\iGp {\mathscr X}$. (\mbox{$\iGp{\mathscr X}(\A_{\vec {\mathrm c}}({\mathscr X}))\subset\A_{\mathrm {qct}}\<({\mathscr X})$,} by \Lref{Gamma'+qc} and~\Cref{C:vec-c is qc}.) So for all $\A_{\mathrm {qct}}\<({\mathscr X})$-complexes~${\mathcal G}'$ and $\A_{\vec {\mathrm c}}({\mathscr X})$-complexes~${\mathcal F}^{\>\prime}$ there~is a natural isomorphism of abelian-group complexes $$ {\mathrm {Hom}}^{\bullet}_{\A_{\mathrm {qct}}\<}({\mathcal G}'\<,\>\iGp{\mathscr X} {\mathcal F}^{\>\prime}\>) \iso {\mathrm {Hom}}^{\bullet}_{\A_{\vec {\mathrm c}}}(k{\mathcal G}'\<,\>{\mathcal F}^{\>\prime}\>). $$ Note that if ${\mathcal F}^{\>\prime}$ is K-injective over $\A_{\vec {\mathrm c}}({\mathscr X})$ then $\iGp{\mathscr X}{\mathcal F}^{\>\prime}$ is K-injective over~$\A_{\mathrm {qct}}\<({\mathscr X})$, because $\iGp{\mathscr X}$ has an exact left adjoint. Combining this isomorphism with the isomorphism (\ref{Deligne}) in the proof of \Tref{prop-duality}, we can conclude just as in part~4 at the end of that proof, with the functor~$f_{\mathrm t}^\times$ defined to be the composition $$ {\mathbf D}({\mathscr Y})\xrightarrow{\,\rho\,} {\mathbf K}_{\text{\textbf I}}({\mathscr Y})\xrightarrow{{\mathcal C}_\bullet\>} {\mathbf K}_{\text{\textbf I}}(\A_{\vec {\mathrm c}}({\mathscr X}))\xrightarrow{\iGp{\mathscr X}\>} {\mathbf K}_{\text{\textbf I}}(\A_{\mathrm {qct}}\<({\mathscr X})) \xrightarrow{\text{natural}\,} {\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X})). $$ (We have in mind here simply that the natural functor ${\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X}))\to{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ has a right adjoint. That is easily seen to be true once one knows the existence of K-injective resolutions in~${\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$; but we don't know how to prove the latter other than by quoting the generalization to arbitrary Grothendieck categories \cite[Theorem~2]{BR}, \cite[Theorem 5.4]{AJS}. The preceding argument avoids this issue. One could also apply Brown Representability\index{Brown Representability} directly, as in the proof of \Tref{Th1} described in the Introduction.) \smallskip Now suppose only that the map $f$ is separated. If ${\mathscr Y}$ is separated then so is~${\mathscr X}$, and the preceding argument holds. For arbitrary noetherian~${\mathscr Y}$ the existence of a bounded-below right adjoint for ${\mathbf R} f_{\!*}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\to{\mathbf D}({\mathscr Y})$ results then from the following Mayer\kern.5pt-Vietoris pasting argument, by induction on the least number of separated open subsets needed to cover~${\mathscr Y}$. Finally, to dispose of the assertion about the ${\mathbf R}{\mathrm {Hom}}^{\bullet}\>$'s apply homology to reduce it to $f_{\mathrm t}^\times$ being a right adjoint. To reduce clutter, we will abuse notation---but only in the rest of the proof of \Tref{T:qct-duality}---by writing ``$f^\times\>$" in place of~``$f_{\mathrm t}^\times\<$." \begin{slem}\label{L:pasting} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}={\mathscr Y}_1\cup{\mathscr Y}_2$ $({\mathscr Y}_i$ open in\/~${\mathscr Y})$ be a map of formal schemes, with\/ ${\mathscr X}$ noetherian. Consider the commutative~diagrams $$ \begin{CD} {\mathscr X}_{12}\!:=@.\;{\mathscr X}_1\cap {\mathscr X}_2 @>q_i>> {\mathscr X}_i @>x_i>> {\mathscr X} \\ @.@Vf_{12}VV @Vf_iVV @VV f\hbox to 0pt{\quad \qquad$(i=1,2)$\hss}V \\ {\mathscr Y}_{12}\!:=@.{\mathscr Y}_1\cap {\mathscr Y}_2 @>>p_i> {\mathscr Y}_i @>>y_i> {\mathscr Y} \end{CD} $$ where\/ ${\mathscr X}_i\!:=f^{-1}{\mathscr Y}_i$\ and all the horizontal arrows represent inclusions. Suppose that for\/ $i=1,2,12,$ the functor\/ ${\mathbf R} f_{i*}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}_i)\to{\mathbf D}({\mathscr Y}_i)$ has a right adjoint\/~$f_i^\times\<$. Then ${\mathbf R} f_{\!*}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\to{\mathbf D}({\mathscr Y})$ has a right adjoint $f^\times;$\ and with the inclusions\/ $y_{12}\!:=y_i\smcirc p_i\>,$\ $x_{12}\!:=x_i\smcirc q_i\>,$\ there is for each\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y})$ a natural\/ ${\mathbf D}({\mathscr X}\>)$-triangle $$ f^\times\<{\mathcal F}\to {\mathbf R}\>x_{1*}^{}f_1^\times y_1^*{\mathcal F} \oplus {\mathbf R}\>x_{2*}^{}f_2^\times y_2^*{\mathcal F}\xrightarrow{\lambda_{\mathcal F}} {\mathbf R}\>x_{12*}^{}f_{12}^\times y_{12}^*{\mathcal F} \to (f^\times\<{\mathcal F}\>)[1]\,. $$ \end{slem} \emph{Remark.} If we expect $f^\times$ to exist, and the natural maps $x_i^*f^\times \to f_i^\times y_i^*$ to be isomorphisms, then there should be such a triangle---the Mayer\kern.5pt-Vietoris triangle \index{Mayer-Vietoris triangle} of~$f^\times\<{\mathcal F}$. This suggests we first define $\lambda_{\mathcal F}\>$, then let $f^\times\<{\mathcal F}$ be the vertex of a triangle based on $\lambda_{\mathcal F}\>$, and verify~\dots \smallskip \begin{proof} There are natural maps $$ \tau_1\colon{\mathbf R} f_{1*}^{}f_1^\times\to \mathbf1,\qquad \tau_2\colon{\mathbf R} f_{2*}^{}f_2^\times\to \mathbf1,\qquad \tau_{12}\colon{\mathbf R} f_{12*}^{}f_{12}^\times\to \mathbf1. $$ For $i=1,2$, define the ``base-change" map $\beta_i\colon q_i^*\<f_i^\times\to f_{12}^\times\> p_i^*$ to be adjoint under \Tref{T:qct-duality} to the map of functors $$ {\mathbf R} f_{12*}^{}q_i^*f_i^{\times}\underset{\>\text{natural}\>}{\xrightarrow{\ \vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}\ }} p_i^*{\mathbf R} f_{i*}f_i^{\times} \xrightarrow{\;\tau_i\;} p_i^*. $$ This $\beta_i$ corresponds to a functorial map $\beta_i'\colon f_i^\times\to {\mathbf R} q_{i*}f_{12}^\times\> p_i^*$, from which we obtain a functorial map $$ \nopagebreak {\mathbf R}\>x_{i*}f_i^\times y_i^* \longrightarrow{\mathbf R}\>x_{i*} {\mathbf R} q_{i*}f_{12}^\times\> p_i^*y_i^* \iso{\mathbf R}\>x_{12*}^{}f_{12}^\times\> y_{12}^*\>\>, $$ and hence a natural map, for any ${\mathcal F}\in{\mathbf D}({\mathscr Y})$: $$ \check D^0({\mathcal F}\>)\!:= {\mathbf R}\>x_{1*}^{}f_1^\times y_1^*{\mathcal F} \oplus {\mathbf R}\>x_{2*}^{}f_2^\times y_2^*{\mathcal F} \xrightarrow{\lambda_{\mathcal F}\>} {\mathbf R}\>x_{12*}^{}f_{12}^\times\> y_{12}^*{\mathcal F}=: \check D^1({\mathcal F}\>)\>. $$ Embed this map in a triangle $\check D({\mathcal F}\>)$, and denote the third vertex by $f^\times({\mathcal F}\>)$: $$ \check D({\mathcal F}\>)\colon\ f^\times\<{\mathcal F}\to\check D^0({\mathcal F}\>)\xrightarrow{\lambda_{\mathcal F}\>} \check D^1({\mathcal F}\>)\to (f^\times\<{\mathcal F}\>)[1]\>. $$ Since $\check D^0({\mathcal F} )$ and $\check D^1({\mathcal F} )$ are in $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (see \Pref{Rf-*(qct)}), therefore so is $f^{\times} {\mathcal F}$ (\Cref{qct=plump}). This is the triangle in \Lref{L:pasting}. Of course we must still show that this~$f^\times$ is functorial, and right-adjoint to~ ${\mathbf R} f_{\!*}$. (Then by uniqueness of adjoints such a triangle will exist no matter which right adjoint~$f^\times$ is used.) \goodbreak Let us next construct a map $ \tau_{\<{\mathcal F}}^{}\colon {\mathbf R} f_{\!*}f^\times\<{\mathcal F}\to{\mathcal F}\ ({\mathcal F}\in{\mathbf D}({\mathscr Y})). $ Set $$ \check C^0({\mathcal F}\>)\!:={\mathbf R} y_{1*}^{} y_1^*{\mathcal F}\oplus {\mathbf R} y_{2*}^{} y_2^*{\mathcal F} ,\qquad \check C^1({\mathcal F}\>)\!:= {\mathbf R} y_{12*}^{} y_{12}^*{\mathcal F}. $$ We have then the Mayer\kern.5pt-Vietoris ${\mathbf D}({\mathscr Y})$-triangle $$ \check C({\mathcal F}\>)\colon\ {\mathcal F} \to \check C^0({\mathcal F}\>) \xrightarrow{\mu_{\mathcal F}\>} \check C^1({\mathcal F}\>) \to {\mathcal F}\>[1], $$ arising from the usual exact sequence (\v Cech resolution) $$ 0\to {\mathcal F} \to y_{1*}^{}y_1^*{\mathcal F}\oplus y_{2*}^{}y_2^*{\mathcal F} \to y_{12*}^{}y_{12}^*{\mathcal F} \to 0, $$ where ${\mathcal F}$ may be taken to be K-injective. Checking commutativity of the following natural diagram is a purely category-theoretic exercise (cf.~\cite[Lemma (4.8.1.2)]{Derived categories} : $$ \CD \quad{\mathbf R} f_{\!*} \check D^0({\mathcal F}\>)\quad \rlap{$\overset{{\mathbf R} f_{\!*}\lambda_{\mathcal F}}{\hbox to 94pt{\rightarrowfill}}$} @.@. {\mathbf R} f_{\!*} \check D^1({\mathcal F}\>)\\ @| @. @| \\ {\mathbf R} f_{\!*}({\mathbf R}\>x_{1*}^{}f_1^\times y_1^*{\mathcal F} \oplus {\mathbf R}\>x_{2*}^{}f_2^\times y_2^*{\mathcal F}\>) @.@. {\mathbf R} f_{\!*}\>{\mathbf R}\>x_{12*}^{}f_{12}^\times\> y_{12}^*{\mathcal F}\\ @V\simeq VV @. @VV\simeq V \\ {\mathbf R} y_{1*}^{}{\mathbf R} f_{1*}^{}f_1^\times y_1^*{\mathcal F} \oplus {\mathbf R} y_{2*}^{}{\mathbf R} f_{2*}^{}f_2^\times y_2^*{\mathcal F} @.\hbox to36pt{}@. {\mathbf R} y_{12*}^{}{\mathbf R} f_{12*}^{}f_{12}^\times\> y_{12}^*{\mathcal F}\\ @V \tau_{\<1}^{}\oplus\tau_2^{} VV @.@VV \tau_{\<12}^{} V \\ {\mathbf R} y_{1*}^{} y_1^*{\mathcal F} \oplus {\mathbf R} y_{2*}^{} y_2^*{\mathcal F} @.@. {\mathbf R} y_{12*}^{} y_{12}^*{\mathcal F} \\ @| @. @| \\ \quad\ \ \,\check C^0({\mathcal F}\>)\quad\ \ \, \rlap{$\underset{\mu_{\mathcal F}}{\hbox to 94pt{\rightarrowfill}}$} @.@. \check C^1({\mathcal F}\>) \endCD $$ This commutative diagram extends to a map $\check\tau_{\<{\mathcal F}}^{}$ of triangles: $$ \begin{CD} {\mathbf R} f_{\!*}f^\times\<{\mathcal F}@>>>{\mathbf R} f_{\!*}\check D^0({\mathcal F}\>) @>>>{\mathbf R} f_{\!*} \check D^1({\mathcal F}\>) @>>>{\mathbf R} f_{\!*}f^\times\<{\mathcal F}[1] \\ @V\tau_{\<\<{\mathcal F}}^{} VV @VVV @VVV @VV\tau_{\<\<{\mathcal F}}^{}[1] V \\ {\mathcal F}@>>> \check C^0({\mathcal F}\>) @>>> \check C^1({\mathcal F}\>) @>>>{\mathcal F}\>[1] \end{CD} $$ The map $\tau_{\<{\mathcal F}}^{}$ is not necessarily unique. But the next Lemma will show, for fixed~${\mathcal F}\<$, that \emph{the pair $(f^{\times}\<{\mathcal F}, \>\tau_{\<{\mathcal F}}^{})$ represents the functor} $$ {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})} ({\mathbf R} f_{\!*} {\mathcal E}, {\mathcal F}\>)\qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr). $$ It follows formally that one can make $f^{\times}$ into a functor and $\tau\colon {\mathbf R} f_{\!*}f^{\times}\to \mathbf 1$ into a morphism of functors in such a way that the pair $(f^\times\<, \tau)$ is a right adjoint for ${\mathbf R} f_{\!*}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}) \to {\mathbf D}({\mathscr Y})$ (cf.~\cite[p. 83, Corollary~2]{currante}); and that there is a unique isomorphism of functors $\Theta\colon f^\times T_2\iso T_1f^\times$ (where $T_1$ and~$T_2$ are the respective translations on $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ and~${\mathbf D}({\mathscr Y})$) such that $(f^\times\<,\Theta)$ is a $\Delta$-functor $\Delta$-adjoint to~${\mathbf R} f_{\!*}$ (cf.~\cite[Proposition (3.3.8)]{Derived categories}). That will complete the proof of \Lref{L:pasting}. \end{proof} \begin{slem}\label{L:f^times} For\/ ${\mathcal E}\in \D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),$ and with\/ $f^\times\<{\mathcal F},$ $\tau_{\<{\mathcal F}}^{}$ as above, the composition \begin{comment} $$ \begin{CD} {\mathrm {Hom}}_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}\<({\mathcal E},\,f^\times\<{\mathcal F}\>) @>\textup{natural\,}>> {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}{\mathcal E},\>{\mathbf R} f_{\!*}f^\times\< {\mathcal F}\>) \\ @>\,\,\textup{via\;}\tau_{\lower.3ex\hbox{$\scriptscriptstyle\<\<{\mathcal F}$}}\,\> >> \hbox spread 1.15cm{$\<{\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}{\mathcal E},\>{\mathcal F}\>) $\hss} \end{CD} $$ \end{comment} $$ {\mathrm {Hom}}_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}\<({\mathcal E},\,f^\times\<{\mathcal F}\>) \xrightarrow{\!\textup{natural}\,} {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}{\mathcal E},\>{\mathbf R} f_{\!*}f^\times\< {\mathcal F}\>) \xrightarrow{\textup{via}\;\tau_{\<\<{\mathcal F}}^{}} {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}{\mathcal E},\>{\mathcal F}\>) $$ is an isomorphism. \end{slem} \begin{proof} In the following diagram, to save space we write $H_{\mathscr X}$ for ${\mathrm {Hom}}_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$, $H_{\mathscr Y}$ for ${\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}$, and $f_{\!*}$ for ${\mathbf R} f_{\!*}\>$: $$ \nopagebreak \def\H#1,{H_{\mathscr X}\bigl({\mathcal E},\>#1\bigr)} \def\h#1,{H_{\mathscr Y}\bigl(f_{\!*}{\mathcal E},\>#1\bigr)} \CD \H (\check D^0{\mathcal F}\>)[-1], @>>> \h f_{\!*}\bigl((\check D^0{\mathcal F}\>)[-1]\bigr), @>>> \h (\check C^0{\mathcal F}\>)[-1], \\ @VVV @VVV @VVV\\ \H (\check D^1{\mathcal F}\>)[-1], @>>> \h f_{\!*}\bigl((\check D^1{\mathcal F}\>)[-1]\bigr), @>>> \h (\check C^1{\mathcal F}\>)[-1], \\ @VVV @VVV @VVV\\ \H f^\times {\mathcal F}, @>>> \h f_{\!*}f^\times {\mathcal F}, @>>> \h {\mathcal F},\\ @VVV @VVV @VVV\\ \H \check D^0{\mathcal F}, @>>> \h f_{\!*}\check D^0{\mathcal F}, @>>> \h \check C^0{\mathcal F}, \\ @VVV @VVV @VVV\\ \H \check D^1{\mathcal F}, @>>> \h f_{\!*}\check D^1{\mathcal F}, @>>> \h \check C^1{\mathcal F}, \endCD $$ The first column maps to the second via functoriality of $f_{\!*}\>$, and the second to the third via the above triangle map $\check \tau_{\<{\mathcal F}}^{}\>$; so the diagram commutes. The columns are exact \cite[p.\,23, Prop.\,1.1\,b)]{H1}, and thus if each of the first two and last two rows is shown to compose to an isomorphism, then the same holds for the middle row, proving \Lref{L:f^times}. Let's look at the fourth row. With notation as in \Lref{L:pasting} (and again, with all the appropriate ${\mathbf R}$'s omitted), we want the left column in the following natural diagram to compose to an isomorphism: $$ \def\H#1,#2,#3,{H_{#1}(#2,\>#3\>)} \begin{CD} \H {\mathscr X},{\mathcal E},x_{i*}f_i^\times y_i^*{\mathcal F}, @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>> \H {\mathscr X}_i,x_i^*{\mathcal E},f_i^\times y_i^*{\mathcal F}, \\ @VVV @VVV \\ \H {\mathscr Y}, f_{\!*}{\mathcal E},f_{\!*}x_{i*}f_i^\times y_i^*{\mathcal F}, @. \H {\mathscr Y}_i,f_{i*}x_i^*{\mathcal E}, f_{i*}f_i^\times y_i^*{\mathcal F}, \\ @V\simeq VV @VV \simeq V \\ \H {\mathscr Y}, f_{\!*}{\mathcal E},y_{i*}f_{i*}f_i^\times y_i^*{\mathcal F}, @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>> \H {\mathscr Y}_i,y_i^*f_{\!*}{\mathcal E}, f_{i*}f_i^\times y_i^*{\mathcal F}, \\ @V\text{via } \tau_i VV @VV\text{via } \tau_i V \\ \H {\mathscr Y}, f_{\!*}{\mathcal E},y_{i*}y_i^*{\mathcal F}, @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>> \H {\mathscr Y}_i,y_i^*f_{\!*}{\mathcal E}, y_i^*{\mathcal F}, \end{CD} $$ Here the horizontal arrows represent adjunction isomorphisms. Checking that the diagram commutes is a straightforward category-theoretic exercise. By hypothesis, the right column composes to an isomorphism. Hence so does the left one. \enlargethispage{-.2\baselineskip} The argument for the fifth row is similar. Using the (easily checked) fact that the morphism $f_{\!*}\check D^0\to \check C^0$ is $\Delta$-functorial, we find that the first row is, up to isomorphism, the same as the fourth row with ${\mathcal F}[-1]$ in place of ${\mathcal F}$, so it too composes to an isomorphism. Similarly, isomorphism for the second row follows from that for the fifth. \end{proof} \penalty-2000 \begin{exams}\label{ft-example} (1) Let $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of quasi-compact formal schemes with ${\mathscr X}$ \emph{properly algebraic,} and let $f^\times$ be the right adjoint given by \Cref{cor-prop-duality}. Using \Pref{Gamma'(qc)} we find then that $f_{\mathrm t}^\times:={\mathbf R}\iGp{\mathscr X}\smcirc \<f^\times$ is a right adjoint for the restriction of $\>{\mathbf R} f_{\!*}$ to~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$.\vadjust{\kern1.5pt} (2) For a noetherian formal scheme~${\mathscr X}$, \Tref{T:qct-duality} gives a right adjoint~\mbox{$\mathbf1^{\<!}\!:=\mathbf1^{\!\times}_{\mathrm t}$} to the inclusion $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\hookrightarrow{\mathbf D}({\mathscr X})$. If ${\mathcal G}\in\wDqc({\mathscr X})$ (i.e., ${\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$, see \Dref{D:Dtilde}), then the natural $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$-map ${\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\to \mathbf1^{\<!}{\mathcal G}$ (corresponding to the natural ${\mathbf D}({\mathscr X})$-map ${\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\to {\mathcal G}$) is an \emph{isomorphism,} see \Pref{Gamma'(qc)}.\vadjust{\kern1.5pt} (3) If ${\mathscr X}$ is \emph{separated} or if ${\mathscr X}$ is \emph{finite-dimensional,} then we have the equivalence ${\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}}\colon{\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X})){{\mkern8mu\longrightarrow \mkern-25.5mu{}^\approx\mkern17mu}}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ of \Pref{1!}, and we can take $\mathbf1^{\<!}\!:={\boldsymbol j}^{\mathrm t}_{\<\<{\mathscr X}} \smcirc\RQ^{\mathrm t}_{\<{\mathscr X}}$, see \Cref{C:Qt} and \Lref{L:j-gamma-eqvce}.\vadjust{\kern1.5pt} (4) Let $f\colon{\mathscr X}\to{\mathscr Y}$ be a closed immersion of noetherian formal schemes (see \cite[p.\,442]{GD}). The functor~$f_{\!*}\colon{\mathcal A}({\mathscr X})\to{\mathcal A}({\mathscr Y})$ is exact, so ${\mathbf R f_{\!*}}=f_{\!*}$. Let ${\mathscr I}$ be the kernel of the surjective map\vadjust{\kern.5pt} ${\mathcal O}_{\mathscr Y}\twoheadrightarrow f_{\!*}{\mathcal O}_{\mathscr X}$\vspace{1pt} and let $\overline{{\mathscr Y}\<}\>$ be the ringed space~$({\mathscr Y}, {\mathcal O}_{\mathscr Y}/{\mathscr I})$, so that $f$~factors naturally as ${\mathscr X}\stackrel {\vbox to 0pt{\vskip-4.5pt\hbox{$\scriptscriptstyle\bar{\! f}\>$}\vss}} \to\overline{{\mathscr Y}\<}\>\stackrel {\vbox to 0pt{\vskip-3pt\hbox{$\scriptscriptstyle i\,$}\vss}}\to{\mathscr Y}$,\vadjust{\kern1.2pt} the map~$\,\>\bar{\<\!f}\>$ being flat. The\vspace{2.5pt} inverse isomorphisms $ {\mathcal A}({\mathscr X})\, \begin{minipage}[b][10pt][c]{22pt} $$ \begin{CD} _{_{\bar{\!f}_{\!\<*}^{}}}\\ \vspace{-31pt}\\ \xrightarrow{\ \ \ }\\ \vspace{-35.5pt}\\ \xleftarrow{\ \ \,}\\ \vspace{-30pt}\\ ^{^{\,\bar {\!f}^{\mkern-.5mu*}}}\\ \vspace{-23.5pt} \end{CD} $$ \end{minipage} \<{\mathcal A}({\mathscr Y}) $ extend to inverse isomorphisms $ {\mathbf D}({\mathscr X})\, \begin{minipage}[b][10pt][c]{22pt} $$ \begin{CD} _{_{\bar{\!f}_{\!\<*}^{}}}\\ \vspace{-31pt}\\ \xrightarrow{\ \ \ }\\ \vspace{-35.5pt}\\ \xleftarrow{\ \ \,}\\ \vspace{-30pt}\\ ^{^{\,\bar {\!f}^{\mkern-.5mu*}}}\\ \vspace{-23.5pt} \end{CD} $$ \end{minipage} \<{\mathbf D}({\mathscr Y}) $ \medskip The functor ${\mathcal H}_{\mathscr I}\colon {\mathcal A}({\mathscr Y})\to{\mathcal A}(\>\overline{\<{\mathscr Y}\<\<}\>\>)$ defined by\vspace{.6pt} $ {\mathcal H}_{\mathscr I}(F\>)\!:=\cH{om}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>F\>) $ has an exact left adjoint, namely $i_*\colon{\mathcal A}(\>\overline{\<{\mathscr Y}\<\<}\>\>)\to{\mathcal A}({\mathscr Y})$, so ${\mathcal H}_{\mathscr I}$ preserves K-injectivity and ${\mathbf R}{\mathcal H}_{\mathscr I}$ is right-adjoint to $i_*\colon{\mathbf D}(\>\overline{\<{\mathscr Y}\<\<}\>\>)\to{\mathbf D}({\mathscr Y})$ (see proof of~\Lref{L:j-gamma-eqvce}). Hence the functor $f^\natural\colon{\mathbf D}({\mathscr Y})\to{\mathbf D}(\>\overline{\<{\mathscr Y}\<\<}\>\>)$ defined by \begin{equation}\label{f^natl} f^\natural({\mathcal F}\>)\!:=\bar{f}^*{\mathbf R}{\mathcal H}_{\mathscr I}({\mathcal F}\>)= \bar{f}^* {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)\qquad\bigl({\mathcal F}\in{\mathbf D}({\mathscr Y})\bigr) \end{equation}\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!{}$f^\natural\<$} is right-adjoint to $f_{\!*}=i_*\bar f_{\!*}$, and $f_{\!*}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\to{\mathbf D}({\mathscr Y})$ has the right adjoint% \index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\mathstrut f^!\<$} $$ f_{\mathrm t}^\times\!:= f^!\!:= \mathbf1^{\<!}\smcirc f^\natural. $$ We recall that ${\mathcal G}\in{\mathcal A}({\mathscr X})$ is quasi-coherent iff $\bar f_{\!*}{\mathcal G}\in\A_{\qc}(\>\overline{\<{\mathscr Y}\<\<}\>)$ iff $f_{\!*}{\mathcal G}\in\A_{\qc}({\mathscr Y})$, see \cite[p.\,115, (5.3.15), (5.3.13)]{GD}. Also, by looking at stalks (see \S\ref{Gamma'1}) we find that \mbox{$f_{\!*}{\mathcal G}\in\A_{\mathrm t}\<({\mathscr Y})\Rightarrow {\mathcal G}\in\A_{\mathrm t}\<({\mathscr X})$.} Hence \Rref{R:Dtilde}(4) together with the isomorphism ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\cong{\mathbf R}\iGp{\mathscr Y}{\mathbf R f_{\!*}}$ of \Cref{C:f* and Gamma}(d) yields that $f^\natural\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\subset\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr X})$; and given \Cref{qct=plump}, \Pref{P:Rhom} yields $f^\natural\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr Y})\subset\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$. Thus if \mbox{${\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})$} then by~(2) above, $f^!\<{\mathcal F}\cong {\mathbf R}\iGp {\mathscr X} f^\natural{\mathcal F}\>$; and if ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr Y})$ then\vspace{1.5pt} $f^!\<{\mathcal F}\cong f^\natural{\mathcal F}$. (5) Let $f\colon{\mathscr X}\to{\mathscr Y}$ be any map satisfying the hypotheses of~ \Tref{T:qct-duality}. Let ${\mathscr J}\subset{\mathcal O}_{\mathscr X}$ and ${\mathscr I}\subset{\mathcal O}_{\mathscr Y}$ be ideals of definition such that ${\mathscr I}{\mathcal O}_{\mathscr X}\subset{\mathscr J}$, and let $$ X_n\!:=({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J}^n) \xrightarrow{f_n^{}\>}({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I}^n)=:Y_n\qquad(n>0) $$ be the scheme-maps induced by $f\<$, so that each $f_n$ also satisfies the hypotheses of~\kern.5pt\Tref{T:qct-duality}. As the target of the functor $(f_n)_{\mathrm t}^{\<\times}$ is $\D_{\mkern-1.5mu\mathrm{qct}}(X_n)=\D_{\mkern-1.5mu\mathrm {qc}}(X_n)$, we write $f_n^\times$ for $(f_n)_{\mathrm t}^{\<\times}$ (see~(1) above). If $\>j_n\colon X_n\hookrightarrow{\mathscr X}$ and $i_n\colon Y_n\hookrightarrow{\mathscr Y}$ are the canonical closed immersions then $fj_n=i_nf_n$, and so $\>j_n^!f_{\mathrm t}^\times\<=f_n^\times i_n^!$. \vspace{.8pt} The functor $j_n^\natural\colon{\mathbf D}({\mathscr X})\to{\mathbf D}(X_{n})$ being as in~\eqref{f^natl}, we have, using~(4), $$ {\boldsymbol h}_n{\mathcal G}\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\<\<X}/\<{\mathscr J}^n\<,\>{\mathcal G}) =j_{n*}j_n^\natural{\mathcal G}\cong j_{n*}j_n^!{\mathcal G} \qquad\bigl({\mathcal G}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr X})\bigr). $$ Hence for ${\mathcal G}\setf_{\mathrm t}^\times\<{\mathcal F}\ ({\mathcal F}\in{\mathbf D}^+({\mathscr Y}))$, \Lref{Gam as holim} gives a% \index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$f_{\mathrm t}^\times\<\<$!as homotopy colimit} ``homotopy colimit" triangle\looseness=-1 $$ \oplus_{n\ge1}\, j_{n*}f_n^\times i_n^!\>{\mathcal F} \longrightarrow \oplus_{n\ge1}\,j_{n*}f_n^\times i_n^!\>{\mathcal F} \longrightarrow f_{\mathrm t}^\times\<{\mathcal F} \overset{+}\longrightarrow $$ \end{exams} \goodbreak Once again, $\wDqc({\mathscr X})\!:=({\mathbf R}\iGp{\mathscr X})^{-1}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ (\Dref{D:Dtilde}). \begin{scor}\label{C:f*gam-duality} \textup{(a)} Let\/ $f\colon\<{\mathscr X} \to {\mathscr Y}$ be a map of noetherian formal schemes. Suppose that\/ $f$ is separated or that\/ ${\mathscr X}$ has finite Krull dimension, or else restrict~to bounded-below complexes. Let\/ ${\boldsymbol\Lambda}_{\mathscr X}\colon{\mathbf D}({\mathscr X})\to{\mathbf D}({\mathscr X})$ be the bounded-below\/ $\Delta$-functor% \index{ $\mathbf {La}$@${\boldsymbol\Lambda}$ (homology localization)} $$ {\boldsymbol\Lambda}_{\mathscr X}(-)\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,-), $$ and let\/ $\ush f\colon{\mathbf D}({\mathscr Y})\to\wDqc({\mathscr X})$ be the\/ $\Delta$-functor $\ush f\!:={\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times$\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\ush f$} \textup(see Example~\textup{\ref{R:Dtilde}(3)).} The functor\/ $\ush f$ is bounded-below, and is right-adjoint to $$ {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\colon\wDqc({\mathscr X})\xrightarrow{\ref{Rf-*(qct)}\,}\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr Y}) \hookrightarrow{\mathbf D}({\mathscr Y}). $$ \textup(In particular with\/ ${\boldsymbol j}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to{\mathbf D}({\mathscr X})$\index{ $\iG{\<{\mathcal J}\>}$@${\boldsymbol j}$} the natural functor, the~functor\/ $$ {\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to{\mathbf D}({\mathscr Y}) $$ has the bounded-below right adjoint\/~${\mathbf R} Q_{\mathscr X}^{} \ush f\<\<$---see Proposition~\textup{\ref{A(vec-c)-A}.)} In fact there is a map of\/ $\Delta$-functors% \index{ {}$\tau$ (trace map)!$\ush\tau$} $$ \ush\tau\colon{\mathbf R} f_{\!*}{\mathbf R}\iGp{\mathscr X} \ush f\to {\bf 1} $$ such that for all\/ ${\mathcal G}\in\wDqc({\mathscr X})$ and\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y}),$\ the~composed map \begin{align*} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr X}({\mathcal G},\>\ush f{\mathcal F}\>) &\xrightarrow{\mathrm{natural}\,} {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr Y}({\mathbf R} f_{\!*}{\mathbf R}\iGp{\mathscr X} {\mathcal G}, \>{\mathbf R} f_{\!*}{\mathbf R}\iGp{\mathscr X} \ush f{\mathcal F}\>) \\ &\xrightarrow{\:\mathrm{via}\ \ush \tau}{\mathbf R}{\mathrm {Hom}}^{\bullet}_{\mathscr Y}({\mathbf R} f_{\!*}{\mathbf R}\iGp{\mathscr X} {\mathcal G},{\mathcal F}\>) \end{align*} is an \emph{isomorphism.} \smallskip \textup{(b)} If\/ $g\colon {\mathscr Y}\to{\mathscr Z}$ is another such map then there is a natural isomorphism\/ $$ \ush{(gf)}\iso\ush f\< \ush g. $$ \end{scor} \begin{proof} (a) The functor ${\boldsymbol\Lambda}_{\mathscr X}$ is bounded below because ${\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}$ is locally isomorphic to the bounded complex~${\mathcal K}_\infty^\bullet$ in the proof of \Pref{Gamma'(qc)}(a), hence homologically bounded-above. Since ${\boldsymbol\Lambda}_{\mathscr X}$ is right-adjoint to ${\mathbf R}\iGp{\mathscr X}$ (see \eqref{adj}), (a) follows directly from \Tref{T:qct-duality}. \smallskip (b) Propositions~\ref{Rf-*(qct)} and~\ref{Gamma'(qc)}(a) show that for any ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ we have \mbox{${\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}{\mathcal G}\cong {\mathbf R f_{\!*}}{\mathcal G}$,} and hence the functors $f_{\mathrm t}^\times\<\<{\boldsymbol\Lambda}_{\mathscr Y}$ and~$f_{\mathrm t}^\times$ are both right-adjoint to~${\mathbf R f_{\!*}}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$, so they are isomorphic. Then \Tref{T:qct-duality}(b) yields functorial isomorphisms $$ \ush{(gf)}={\boldsymbol\Lambda}_{\mathscr X}(gf)_{\mathrm t}^{\<\times}\iso {\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\<g_{\mathrm t}^\times\iso {\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\<{\boldsymbol\Lambda}_{\mathscr Y} g_{\mathrm t}^\times= \ush f\<\ush g. $$ \end{proof} \medskip Here are some ``identities" involving the dualizing functors $f^\times$ (\Tref{prop-duality}), $f_{\mathrm t}^\times$~(\Tref{T:qct-duality}), and $\ush f\!:={\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times$ (\Cref{C:f*gam-duality}). Note that ${\boldsymbol\Lambda}_{\mathscr X}$ is right-adjoint to~${\mathbf R}\iGp{\mathscr X}$, see~\eqref{adj}. Simple arguments show that the natural maps are isomorphisms ${\boldsymbol\Lambda}_{\mathscr X}\iso{\boldsymbol\Lambda}_{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}\>$, ${\mathbf R}\iGp{\mathscr X}\iso{\mathbf R}\iGp{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}\>$, see~(b) and~(d) in \Rref{R:Gamma-Lambda}(1). \pagebreak[3] \begin{scor}\label{C:identities} With the notation of \Cref{C:f*gam-duality}\kern.5pt\textup{,} \smallskip \textup{(a)} There are natural isomorphisms \begin{alignat*}{2} {\mathbf R}\iGp {\mathscr X}\ush f&\isof_{\mathrm t}^\times\<,\qquad & \ush f&\iso {\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\<,\\ {\mathbf R}\iGp{\mathscr X}f_{\mathrm t}^\times&\isof_{\mathrm t}^\times\<,\qquad & \ush f&\iso{\boldsymbol\Lambda}_{\mathscr X}\ush f\<. \end{alignat*} \smallskip \textup{(b)} The natural functorial maps\/ ${\mathbf R}\iGp{\mathscr Y}\to\mathbf 1\to{\boldsymbol\Lambda}_{\mathscr Y}$ induce isomorphisms \begin{gather*} f_{\mathrm t}^\times{\mathbf R}\iGp{\mathscr Y}\isof_{\mathrm t}^\times\isof_{\mathrm t}^\times\<\<{\boldsymbol\Lambda}_{\mathscr Y}, \\ \ush f{\mathbf R}\iGp{\mathscr Y} \iso\ush f \iso \ush f\<\<{\boldsymbol\Lambda}_{\mathscr Y}. \end{gather*} \smallskip \textup{(c)} There are natural pairs of maps \begin{gather*} f_{\mathrm t}^\times\xrightarrow{\alpha_1\>} {\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j} f^\times \xrightarrow{\alpha_2\>}f_{\mathrm t}^\times\<, \\ \ush f\xrightarrow{\beta_1\>} {\boldsymbol\Lambda}_{\mathscr X}\>{\boldsymbol j} f^\times \xrightarrow{\beta_2\>}\ush f\<, \end{gather*} each of which composes to an identity map. If\/ ${\mathscr X}$ is properly algebraic then all of these maps are isomorphisms. \smallskip \textup{(d)} If\/ $f$ is\/ \emph{adic} then the isomorphism\/\vspace{.5pt} ${\mathbf R f_{\!*}}\>{\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j}\osi{\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}\>{\boldsymbol j}$ in\/~\textup{\ref{C:f* and Gamma}(d)} indu\-ces an isomorphism of the right adjoints $(\<$see \Tref{prop-duality}\textup{,} \Pref{A(vec-c)-A}$)$ $$ f^\times\<\<{\boldsymbol\Lambda}_{\mathscr Y}\iso{\mathbf R} Q_{\mathscr X}^{}\ush f\<. $$ \end{scor} \begin{proof} (a) The second isomorphism (first row) is the identity map. \Pref{Gamma'(qc)} yields the third. The first is the composition $$ {\mathbf R}\iGp {\mathscr X}\ush f={\mathbf R}\iGp {\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\iso{\mathbf R}\iGp {\mathscr X}f_{\mathrm t}^\times\isof_{\mathrm t}^\times\<. $$ The fourth is the composition $$ \hskip21pt\ush f={\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\iso{\boldsymbol\Lambda}_{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\iso{\boldsymbol\Lambda}_{\mathscr X}\ush f\<. $$ (b) The first isomorphism results from ${\mathbf R}\iGp{\mathscr Y}$ being right adjoint to the inclusion $\D_{\mathrm t}\<({\mathscr Y})\hookrightarrow{\mathbf D}({\mathscr Y})$ (see \Pref{Gamma'(qc)}(c)). For the second, check that $f_{\mathrm t}^\times$ and~$f_{\mathrm t}^\times\<\<{\boldsymbol\Lambda}_{\mathscr Y}$ are both right-adjoint to ${\mathbf R} f_{\!*}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}\>\dots$ (Or, consider the composition $f_{\mathrm t}^\times\isof_{\mathrm t}^\times{\mathbf R}\iGp{\mathscr Y}\isof_{\mathrm t}^\times{\mathbf R}\iGp{\mathscr Y}{\boldsymbol\Lambda}_{\mathscr Y}\isof_{\mathrm t}^\times\<{\boldsymbol\Lambda}_{\mathscr Y}$.) Then apply ${\boldsymbol\Lambda}_{\mathscr X}$ to the first row to get the second row. \smallskip (c) With $\boldsymbol k\colon{\mathbf D}(\A_{\mathrm {qct}}\<({\mathscr X}))\to{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))$ the natural functor, let $$\alpha\colon\boldsymbol k\RQ^{\mathrm t}_{\mathscr X}f_{\mathrm t}^\times\to f^\times$$ be adjoint to ${\mathbf R} f_{\!*}{\boldsymbol j}\boldsymbol k\RQ^{\mathrm t}_{\mathscr X}f_{\mathrm t}^\times \stackrel{\ref{1!}}{=}{\mathbf R} f_{\!*}f_{\mathrm t}^\times \xrightarrow{\tau_{\textup t}^{\phantom{.}}} \mathbf 1.$ By \Cref{C:Hom-Rgamma}, ${\boldsymbol j}(\alpha)\colonf_{\mathrm t}^\times\to{\boldsymbol j} \< f^\times$ factors naturally as $$ f_{\mathrm t}^\times\xrightarrow{\alpha_{\<1}^{}\>}{\mathbf R}\iGp{\mathscr X}{\boldsymbol j} \< f^\times\to{\boldsymbol j} \< f^\times. $$ Let $\alpha_2$ be the map adjoint to the natural composition ${\mathbf R} f_{\!*}\>{\mathbf R}\iGp{\mathscr X}{\boldsymbol j} \< f^\times\to{\mathbf R} f_{\!*}{\boldsymbol j} \< f^\times\to\mathbf 1$. One checks that $\tau_{\textup t}\smcirc{\mathbf R} f_{\!*}(\alpha_2\alpha_1)=\tau_{\textup t}$ ($\tau_{\textup t}$ as in \Tref{T:qct-duality}), whence $\alpha_2\alpha_1=\text{identity}$. The pair $\beta_1\>,\>\beta_2$ is obtained from $\alpha_1\>,\>\alpha_2$ by application of the functor~${\boldsymbol\Lambda}_{\mathscr X}$---see \Cref{C:Hom-Rgamma}. (Symmetrically, the pair $\alpha_1\>,\>\alpha_2$ is obtained from $\beta_1\>,\>\beta_2$ by application of the functor~${\mathbf R}\iGp{\mathscr X}$.) When ${\mathscr X}$ is properly algebraic, the functor~${\boldsymbol j}$ is fully faithful (\Cref{corollary}); and it follows that ${\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j} \< f^\times$ and~$f_{\mathrm t}^\times$ are both right-adjoint to ${\mathbf R} f_{\!*}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$. \smallskip (d) Straightforward. \end{proof} \pagebreak[3] The next three corollaries deal with compatibilities between formal (local) and ordinary (global) Grothendieck duality. \begin{scor}\label{C:kappa-f^times-tors} Let\/ $f_0\colon X\to Y$ be a map of noetherian ordinary schemes. Suppose either that\/ $f_0$ is separated or that\/ $X$ is finite-dimensional, or else restrict to bounded-below complexes. Let\/ $W\subset Y$ and\/~ $Z\subset f_0^{-1}W$ be closed subsets, $\kappa_{\mathscr Y}^{\phantom{.}}\colon{\mathscr Y}=Y_{/W}\to Y$ and\/ $\kappa_{\mathscr X}^{\phantom{.}}\colon{\mathscr X}=X_{/Z}\to X$ the respective completion maps, and\/ $f\colon{\mathscr X}\to{\mathscr Y}$ the map induced by~$f_0.$ \vadjust{\penalty-750} $$ \begin{CD} {\mathscr X}@.:=X_{\</Z} @>\kappa_{\mathscr X}^{\phantom{.}}>> X \\ @V f VV @. @VV f_0^{} V \\ {\mathscr Y}@.:=Y_{/W}@>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle\kappa_{\mathscr Y}^{\phantom{.}}$}\vss}> Y \end{CD} $$ With $f_{\<0}^{\<\times}\!:=(f_0^{})_{\textup t}^{\<\<\times}$ right-adjoint to ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm {qc}}(X)\to{\mathbf D}(Y),$ let $\tau_{\<\mathrm t}'$ be the composition $$ {\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} \underset{\ref{C:kappa-f*t}\>}{\iso} \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} \longrightarrow \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} \longrightarrow \kappa_{\mathscr Y}^*\kappa_{{\mathscr Y}*}^{\phantom{.}}\longrightarrow\mathbf 1. $$ Then for all ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ and ${\mathcal F}\in{\mathbf D}({\mathscr Y}),$ the composed map \begin{align*} \alpha({\mathcal E}\<,{\mathcal F}\>)\colon{\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<({\mathcal E}\<, \kappa_{\mathscr X}^*{\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} {\mathcal F}\>) &\longrightarrow {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R f_{\!*}}{\mathcal E}\<, \>{\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) \\ &\,\<\underset{\textup{via }\tau_{\<\mathrm t}'}{\longrightarrow} {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R f_{\!*}}{\mathcal E}\<,\>{\mathcal F}\>) \end{align*} \noindent is an isomorphism. Hence the map adjoint to~$\tau_{\<\mathrm t}'$ is an isomorphism of functors $$ \kappa_{\mathscr X}^*{\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} \iso f_{\mathrm t}^\times\<. $$ \end{scor} \begin{proof} For any ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$, set ${\mathcal E}_0\!:=\kappa_{{\mathscr X}*}^{\phantom{.}}{\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$ (\Pref{Gammas'+kappas}). \Pref{Gammas'+kappas} and \cite[p.\,7, Lemma (0.4.2)]{AJL} give natural isomorphisms\vspace{-3pt} \begin{multline*} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<({\mathcal E},\>\kappa_{\mathscr X}^*{\mathbf R}\iG Z{\mathcal G}) \iso {\mathrm {Hom}}_{{\mathbf D}(X)}\<({\mathcal E}_0\>,\>{\mathbf R}\iG Z{\mathcal G}) \iso {\mathrm {Hom}}_{{\mathbf D}(X)}\<({\mathcal E}_0\>,\>{\mathcal G})\\ \bigr({\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)\bigl). \end{multline*} \vspace{-15pt} \noindent (In other words, $\kappa_{\mathscr X}^*{\mathbf R}\iG Z{\mathcal G}=(\kappa_{\mathscr X}^{\phantom{.}})_{\mathrm t}^{\<\times}{\mathcal G}$.) One checks then that the map $\alpha({\mathcal E}\<,{\mathcal F}\>)$ factors as the sequence of natural isomorphisms \begin{align*} {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}\<({\mathcal E},\>\kappa_{\mathscr X}^*{\mathbf R}\iG{Z}f_{\<0}^{\<\times} \kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) &\iso {\mathrm {Hom}}_{{\mathbf D}(X)}\<({\mathcal E}_0\>,\>f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) \\ &\iso {\mathrm {Hom}}_{{\mathbf D}(Y)}\<({\mathbf R} f_{\<0*}^{}{\mathcal E}_0\>,\>\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) \\ &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<(\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathcal E}_0\>,\>{\mathcal F}\>) \\ &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}\kappa_{\mathscr X}^*{\mathcal E}_0\>,\>{\mathcal F}\>) \qquad\textup{(\Cref{C:kappa-f*t})}\\ &\iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr Y})}\<({\mathbf R} f_{\!*}{\mathcal E},\>{\mathcal F}\>). \end{align*} \vskip-3.8ex \end{proof} \vskip1pt \begin{scor}\label{C:kappa+duality} With hypotheses as in \Cref{C:kappa-f^times-tors}\kern.5pt\textup{:} \textup{(a)} There are natural isomorphisms \begin{align*} {\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} &= (\kappa_{\mathscr X}^{\phantom{.}})_{\mathrm t}^{\<\times}\<f_{\<0}^{\<\times} \kappa_{{\mathscr Y}*}^{\phantom{.}} \iso f_{\mathrm t}^\times\<, \\ {\boldsymbol\Lambda}_{\mathscr X}\kappa_{\mathscr X}^*f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} &=\kappa_{\mathscr X}^{\textup{\texttt\#}}f_{\<0}^{\<\times} \kappa_{{\mathscr Y}*}^{\phantom{.}} \iso \ush f; \end{align*} and if $f_0$ is proper, $Y={\mathrm {Spec}}(A)$ \($A$ adic\)$,$ $Z=f_0^{-1}W,$ then with $f^{\<\times}\!$ as in \Cref{cor-prop-duality}\kern.5pt\textup{:}\vspace{-1.4ex} $$ \kappa_{\mathscr X}^*f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}}\iso f^\times\<.\vspace{4pt} $$ \enlargethispage*{\baselineskip} \textup{(b)} The functor $ f_{\<0,Z}^\times\!:={\mathbf R}\iG Z f_{\<0}^{\<\times}\colon{\mathbf D}(Y)\to\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X) $ is right-adjoint to the functor\/ ${\mathbf R f_{\!*}}|_{\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)}\>;$\ and there is an isomorphism\vspace{-1.5pt} $$ \kappa_{\mathscr X}^*f_{\<0,Z}^\times\kappa_{{\mathscr Y}*}^{\phantom{.}}\iso f_{\mathrm t}^\times\<. $$ \pagebreak[3] \textup{(c)} If\/ $X$ is separated then, with notation as in \Sref{SS:Dvc-and-Dqc}\textup{,} the functor $$ \ush{f_{\<0,Z}}\!:= {\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}{\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}\>,\>f_{\<0}^{\<\times}-) \colon{\mathbf D}(Y)\to\D_{\mkern-1.5mu\mathrm {qc}}(X) $$ is right-adjoint to\/~${\mathbf R} f_{\<0*}^{}\>{\mathbf R}\iG Z|_{\D_{\mkern-1.5mu\mathrm {qc}}(X)}\>;$\ \vspace{.6pt}and if\/ ${\mathscr X}$ is properly algebraic, so that we have the equivalence\/~ ${\boldsymbol j}_{\!{\mathscr X}}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$ \(\kern-1pt \Cref{corollary}\kern.7pt\), there\vspace{.6pt} is an isomorphism\vspace{-2pt} $$ \kappa_{\mathscr X}^*\ush{f_{\<0,Z}}\kappa_{{\mathscr Y}*}^{\phantom{.}}\iso {\boldsymbol j}_{\!{\mathscr X}}{\mathbf R} Q_{\mathscr X}^{}\ush f\<. $$ \end{scor} \pagebreak[3] \begin{proof} (a) The first isomorphism combines \Cref{C:kappa-f^times-tors} (in proving which we noted that $\kappa_{\mathscr X}^*{\mathbf R}\iG Z{\mathcal G} =(\kappa_{\mathscr X}^{\phantom{.}})_{\mathrm t}^{\<\times}{\mathcal G}$ for ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)$) and \Pref{Gammas'+kappas}. The second follows from $\ush f={\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times\<$. The third is \Cref{C:kappa-f^times}. (b) The first assertion is easily checked; and the isomorphism is given by \Cref{C:kappa-f^times-tors}. (c) When $X$ is separated, ${\boldsymbol j}_{\!X}^{}$ is an equivalence \cite[p.\,12, Proposition (1.3)]{AJL}, and then the first assertion is easily checked. From \Cref{C:kappa-f^times-tors} and \Pref{Gammas'+kappas} we get an isomorphism $$ {\mathbf R}\iG{Z}f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}}\iso \kappa_{{\mathscr X}*}^\pdf_{\mathrm t}^\times\<. $$ As in \Cref{C:Hom-Rgamma}, the natural map is an isomorphism $$ {\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}\>,{\mathcal G}) \iso {\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}\>, {\mathbf R}\iG Z{\mathcal G})\qquad\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}(X)\bigr). $$ When ${\mathscr X}$ is properly algebraic, ${\boldsymbol j}_{\!{\mathscr X}}^{}{\mathbf R} Q_{\mathscr X}^{}\cong \kappa_{\mathscr X}^*\>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}\kappa_{{\mathscr X}*}^{}$ (\Pref{A(vec-c)-A}). So then we have a sequence of natural isomorphisms \begin{align*} \kappa_{\mathscr X}^*\ush{f_{\<0,Z}}\kappa_{{\mathscr Y}*}^{\phantom{.}} & \ \>\raise.2ex\hbox{\EQAL{17}}\;\> \kappa_{\mathscr X}^*\>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}{\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}, \>f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}} -) \\ &\iso \kappa_{\mathscr X}^*\>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}{\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}, \>{\mathbf R}\iG Z f_{\<0}^{\<\times}\kappa_{{\mathscr Y}*}^{\phantom{.}}- ) \\ &\iso \kappa_{\mathscr X}^*\>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}{\mathbf R}\cH{om}^{\bullet}_X({\mathbf R}\iG Z{\mathcal O}_{\<\<X}, \>\kappa_{{\mathscr X}*}^\pdf_{\mathrm t}^\times -) \\ &\iso \kappa_{\mathscr X}^*\>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}\kappa_{{\mathscr X}*}^{\phantom{.}} {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}(\kappa_{\mathscr X}^*{\mathbf R}\iG Z{\mathcal O}_{\<\<X},\>f_{\mathrm t}^\times -) \\ &\iso {\boldsymbol j}_{\!{\mathscr X}}^{}{\mathbf R} Q_{\mathscr X}^{}{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}({\mathbf R}\iGp {\mathscr X}{\mathcal O}_{{\mathscr X}}\>,\>f_{\mathrm t}^\times-)\\ & \ \>\raise.2ex\hbox{\EQAL{17}}\;\>{\boldsymbol j}_{\!{\mathscr X}}^{}{\mathbf R} Q_{\mathscr X}^{}\ush f. \end{align*} \vskip-3.8ex \end{proof} The following instance of ``flat base change" will be needed in the proof of the general base-change \Tref{Th3}. \begin{scor}\label{C:compln+basechange} In \Cref{C:kappa-f^times-tors}\textup{,} assume further that $Z=f_{\<0}^{-1}W\<$. Then the natural map is an isomorphism $$ {\mathbf R}\iG Zf_{\<0}^{\<\times}\<{\mathcal F}\iso{\mathbf R}\iG Zf_{\<0}^{\<\times}\<\kappa_{{\mathscr Y}*}\kappa_{\mathscr Y}^*{\mathcal F} \qquad\bigl({\mathcal F}\in{\mathbf D}(Y)\bigr), $$ and so there is a composed isomorphism $$ \zeta\colon{\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*f_{\<0}^{\<\times}\<{\mathcal F} \underset{\textup{\ref{Gammas'+kappas}(c)}}\iso \kappa_{\mathscr X}^*{\mathbf R}\iG Z f_{\<0}^{\<\times}\<{\mathcal F} \iso \kappa_{\mathscr X}^*{\mathbf R}\iG Z f_{\<0}^{\<\times}\<\kappa_{{\mathscr Y}*}\kappa_{\mathscr Y}^*{\mathcal F} \underset{\textup{\ref{C:kappa+duality}(b)}}\iso f_{\mathrm t}^\times\<\kappa_{\mathscr Y}^*{\mathcal F}. $$ \end{scor} \begin{proof} First, ${\mathbf R} f_{\<0*}^{}(\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X))\subset\D_{\mkern-1.5mu\mathrm {qc}}{}_W(Y)$. For, by \cite[Proposition~(3.9.2)]{Derived categories}, ${\mathbf R} f_{\<0*}^{}(\D_{\mkern-1.5mu\mathrm {qc}}(X))\subset\D_{\mkern-1.5mu\mathrm {qc}}(Y)$; and then the assertion follows from the natural isomorphism of functors (from $\D_{\mkern-1.5mu\mathrm {qc}}(X)$ to $\D_{\mkern-1.5mu\mathrm {qc}}(Y)$) ${\mathbf R}\iG W{\mathbf R} f_{\<0*}^{}\cong{\mathbf R} f_{\<0*}^{}{\mathbf R}\iG {f^{-1}W}\>,$ because ${\mathcal G}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)$ (resp.~$\H\in\D_{\mkern-1.5mu\mathrm {qc}}{}_W(Y)$) iff ${\mathbf R}\iG Z{\mathcal G}\cong{\mathcal G}$ (resp.~${\mathbf R}\iG W\H\cong\H$), cf.~\Pref{Gamma'(qc)}(a) and its proof. (The said functorial isomorphism arises from the corresponding one without the ${\mathbf R}$'s, since ${\mathbf R} f_{\<0*}^{}$ preserves K-flabbiness, see \cite[5.12, 5.15(b), 6.4, 6.7]{Sp}. Now \Cref{C:Gammas'+kappas} gives that the natural map is an isomorphism $$ {\mathrm {Hom}}_{{\mathbf D}(Y)}\<({\mathbf R} f_{\<0*}^{}{\mathcal E}, {\mathcal F}\>)\iso {\mathrm {Hom}}_{{\mathbf D}(Y)}\<({\mathbf R} f_{\<0*}^{}{\mathcal E},\kappa_{{\mathscr Y}*}\kappa_{\mathscr Y}^*{\mathcal F}\>) \qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu{\mathrm {qc}}Z}(X)\bigr), $$ and the conclusion follows from the adjunction in \Cref{C:kappa+duality}(b). \end{proof} \medskip \begin{parag}\label{coherent} The next Proposition is a special case of Greenlees-May Duality for formal schemes\index{Greenlees-May Duality} (see \cite[Proposition 0.3.1]{AJL$'$}). It is the key to many statements in this paper concerning complexes with coherent homology. \begin{sprop} \label{formal-GM} Let\/ ${\mathscr X}$ be a locally noetherian formal scheme, ${\mathcal E}\in{\mathbf D}({\mathscr X})$. Then for all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr X})$ the natural map\/ ${\mathbf R}\iGp{{\mathscr X}}{\mathcal E}\to {\mathcal E}$ induces an isomorphism $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal E}, \>{\mathcal F}\>) \iso {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{{\mathscr X}}{\mathcal E},\>{\mathcal F}\>). $$ \end{sprop} \begin{proof} The canonical isomorphism (cf.~\eqref{adj}) $$ {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal E}\<,\>{\mathcal F}\>) \iso{\mathbf R}\cH{om}^{\bullet}\bigl({\mathcal E},\>{\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,\>{\mathcal F}\>)\bigr) $$ reduces the question to where ${\mathcal E}={\mathcal O}_{{\mathscr X}}\>$. It suffices then---as in the proof of \Cref{C:Hom-Rgamma}---that for affine~${\mathscr X}={\mathrm {Spf}}(A)$, the natural map be an isomorphism $$ {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}({\mathcal O}_{{\mathscr X}}\>, \>{\mathcal F}\>) \iso {\mathrm {Hom}}_{{\mathbf D}({\mathscr X})}({\mathbf R}\iGp{{\mathscr X}}{\mathcal O}_{{\mathscr X}}\>, \>{\mathcal F}\>) \qquad \bigl({\mathcal F} \in \D_{\mkern-1.5mu\mathrm c}({\mathscr X})\bigr). $$ Let $I$ be an ideal of definition of the adic ring $A$, set $Z\!:={\mathrm {Supp}}(A/I)$, and let $\kappa\colon{\mathscr X}\to X\!:={\mathrm {Spec}}(A)$ be the completion map. The categorical equivalences in \Pref{c-erator} and the isomorphism $\kappa^*{\mathbf R}\iGp Z{\mathcal O}_X\!\iso\!{\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}$ in \Pref{Gammas'+kappas} make the problem whether for all $F\in\D_{\mkern-1.5mu\mathrm c}(X)$ (e.g., $F={\mathbf R} Q\>\kappa_*^{}{\mathcal F}\!:={\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}\kappa_*^{}{\mathcal F}\>$) the~natural map is an isomorphism $$ {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal O}_{\<\<X}, \>F) \iso {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathbf R}\iG Z{\mathcal O}_{\<\<X}, \>F). $$ Now, the canonical functor ${\boldsymbol j}_{\!X}^{}\colon{\mathbf D}(\A_{\qc}(X))\to {\mathbf D}(X)$ induces an equivalence of categories \mbox{${\mathbf D}(\A_{\qc}(X)){{\mkern6mu\longrightarrow \mkern-25.5mu{}^\approx\mkern15mu}}\D_{\mkern-1.5mu\mathrm {qc}}(X)$} (see beginning of \S\ref{SS:Dvc-and-Dqc}), and so we may assume that $F$~is a K-flat quasi-coherent complex. \Lref{L:j-gamma-eqvce} shows that ${\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}$ is right-adjoint to the inclusion $\D_{\mkern-1.5mu\mathrm {qc}}(X)\hookrightarrow{\mathbf D}(X)$. The natural map $$ {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_X, F)\to{\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iG Z{\mathcal O}_X, F)\vspace{-5pt} $$ factors then as\vspace{3pt} \begin{equation}\label{nat} \begin{aligned} {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_X, F)=F &\underset{\ref{c-erator}}\iso {\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}^{}\kappa_*^{}\kappa^*F \\ &\,\longrightarrow\,\mathstrut\kappa_*^{}\kappa^*F\\ &\underset{\vbox to 0pt{\vss\hbox{$\scriptstyle\lambda$}\vskip1pt}}\iso \,\inlm{n}F/(I{\mathcal O}_X)^nF \underset{\vbox to 0pt{\vss\hbox{$\scriptstyle\Phi$}\vskip1pt}}{\iso} {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iG Z{\mathcal O}_X, F), \end{aligned} \end{equation} where the map $\lambda$, obtained by applying $\kappa_*^{}$ to the natural map from $\kappa^*F$ to the completion~$F_{/Z}$, is a ${\mathbf D}(X)$-isomorphism by \cite[p.\,6, Proposition~(0.4.1)]{AJL}; and $\Phi$~is the isomorphism $\Phi(F,{\mathcal O}_X)$ of \cite[\S2]{AJL}. (The fact that $\Phi$ is an isomorphism is essentially the main result in \cite{AJL}.) Also, by adjointness, the natural map is an isomorphism $$ {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal O}_{\<\<X}, \>{\boldsymbol j}_{\!X}^{}{\mathbf R} Q_{\<\<X}\kappa_*^{}\kappa^*F) \iso {\mathrm {Hom}}_{{\mathbf D}(X)}({\mathcal O}_{\<\<X},\>\kappa_*^{}\kappa^*F). $$ Conclude now by applying the functor $\textup H^0{\mathbf R}\Gamma(X,-)$ to~\eqref{nat}. \end{proof} \begin{scor}\label{C:coh-dual} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be as in \Cref{C:f*gam-duality}\textup{,} and assume further that\/ $f$~is adic. Then for all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}({\mathscr Y})$ the map corresponding to the natural composition\/ ${\mathbf R} f_{\!*} {\mathbf R}\iGp{\mathscr X}\>{\boldsymbol j} \< f^\times\<{\mathcal F} \to {\mathbf R} f_{\!*} {\boldsymbol j} \< f^\times\<{\mathcal F} \to {\mathcal F} $ \(see \Tref{prop-duality}\/\) is an isomorphism $$ f^\times\<{\mathcal F}\iso{\mathbf R} Q_{\mathscr X}^{}\ush f\<{\mathcal F}. $$ \end{scor} \begin{proof} By ~\Pref{formal-GM}, ${\mathcal F}\cong{\boldsymbol\Lambda}_{\mathscr Y}\>{\mathcal F}\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr Y}{\mathcal O}_{\mathscr Y}\>,\>{\mathcal F}\>)$; so this Corollary is a special case of \Cref{C:identities}(d). \end{proof} \begin{scor}\label{C:completion-proper} In \Cref{C:kappa-f^times-tors}\textup{,} suppose\/ $Y={\mathrm {Spec}}(A)$ \($A$ adic\/\) and\vadjust{\kern.5pt} that the the map\/~$f_0$ is \emph{proper.} Then with the customary\vadjust{\kern.5pt} notation\/~$f_{\<0}^!\>$ for\/ $f_{\<0}^{\<\times}$ we have, for any\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+\<({\mathscr Y}),$ a natural isomorphism $$ \kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\iso \ush f{\mathcal F} \in\D_{\mkern-1.5mu\mathrm c}^+\<({\mathscr X}). $$ \end{scor} \begin{proof} The natural map $f_{\<0}^!\mkern1.5mu{\boldsymbol j}_{\<Y}\<{\mathbf R} Q_{\<Y}\kappa_{{\mathscr Y}*}^{\phantom{.}}\to f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}$ is an isomorphism of functors from ${\mathbf D}({\mathscr Y})$ to~$\D_{\mkern-1.5mu\mathrm {qc}}(X)$, both being right-adjoint to~$\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}$. \Pref{c-erator} gives ${\boldsymbol j}_{\<Y}{\mathbf R} Q_{\<Y}\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+\<(Y)$; so by \cite[p.\,396, Lemma~1]{f!}, $f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+\<(X)$.% \footnote{% For ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm c}^{\scriptscriptstyle+}(Y)$ one has $f_0^!{\mathcal G}\in\D_{\mkern-1.5mu\mathrm c}^{\scriptscriptstyle+}(X)$: The question being local on $X$ one reduces to where\vadjust{\kern.6pt} \emph{either} $X$ is a projective space ${\mathbf P}^n_Y$ and $f_0$ is projection, so that~ {$f_0^!{\mathcal G}=f_0^*{\mathcal G}\otimes\Omega^n_{X/Y}[n]\in \D_{\mkern-1.5mu\mathrm c}^{\scriptscriptstyle+}(X)$}, \emph{or} $f_0$ is a closed immersion and $f_{0*}f_0^!{\mathcal G}={\mathbf R}\cH{om}^{\bullet}_Y(f_{0*}{\mathcal O}_X,{\mathcal F}\>)\in\D_{\mkern-1.5mu\mathrm c}^{\scriptscriptstyle+}(Y)$ \cite[p.\,92, Proposition~3.3]{H1} whence, again, $f_0^!{\mathcal G}\in\D_{\mkern-1.5mu\mathrm c}^{\scriptscriptstyle+}(X)$ \cite[p.~115, (5.3.13)]{GD}.% \vadjust{\kern 3pt}% } Hence \Pref{formal-GM} and Corollary ~\ref{C:kappa+duality}(a) yield isomorphisms\looseness=-1 $$ \kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F} \iso{\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{}\>,\>\kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F}\>) =:\<\<{\boldsymbol\Lambda}_{\mathscr X}\kappa_{\mathscr X}^*f_{\<0}^!\kappa_{{\mathscr Y}*}^{\phantom{.}}{\mathcal F} \iso\ush f{\mathcal F}\<. $$ \vskip-3.8ex \end{proof} \smallskip \end{parag} \begin{parag} \label{SS:Gam-Lam} More relations, involving the functors ${\mathbf R}\iGp{\mathscr X}$ and~ ${\boldsymbol\Lambda}_{\mathscr X}\!:={\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr X}{\mathcal O}_{\mathscr X}^{},-)$\index{ $\mathbf {La}$@${\boldsymbol\Lambda}$ (homology localization)} on a locally noetherian formal scheme~${\mathscr X}$, will now be summarized. \begin{small} \begin{srems}\label{R:Gamma-Lambda} Let ${\mathscr X}$ be a locally noetherian formal scheme. (1) The functor\index{ $\iG{\raise.3ex\hbox{$\scriptscriptstyle{\ldots}$}}$ (torsion functor)!a@$\boldsymbol{\varGamma}\!:={\mathbf R}\iGp{\mathscr X}$ (cohomology colocalization)} $\boldsymbol{\varGamma}\!:={\mathbf R}\iGp{\mathscr X}\colon{\mathbf D}({\mathscr X})\to{\mathbf D}({\mathscr X})$ admits a natural map \mbox{$\boldsymbol{\varGamma}\xrightarrow{\gamma\>}\mathbf 1$}, which induces a functorial isomorphism \begin{equation} {\mathrm {Hom}}(\boldsymbol{\varGamma}{\mathcal E}, \boldsymbol{\varGamma}{\mathcal F}\>)\iso {\mathrm {Hom}}(\boldsymbol{\varGamma}{\mathcal E},{\mathcal F}\>)\qquad\bigl({\mathcal E},{\mathcal F}\in{\mathbf D}({\mathscr X})), \tag{A} \end{equation} see \Pref{Gamma'(qc)}(c). Moreover $\boldsymbol{\varGamma}$ has a right adjoint, viz.~ ${\boldsymbol\Lambda}\!:={\boldsymbol\Lambda}_{\mathscr X}$ (see~\eqref{adj}). The rest of (1) consists of (well-known) formal consequences of these properties. \smallskip Since $\gamma$ is functorial, it holds that $\gamma({\mathcal F}\>)\smcirc\gamma(\boldsymbol{\varGamma}{\mathcal F}\>) =\gamma({\mathcal F}\>)\smcirc\boldsymbol{\varGamma}(\gamma({\mathcal F}\>))\colon\boldsymbol{\varGamma}\BG{\mathcal F}\to{\mathcal F}\<$, so injectivity of the map in (A) (with ${\mathcal E}=\boldsymbol{\varGamma}{\mathcal F}\>$) yields $ \gamma(\boldsymbol{\varGamma}{\mathcal F}\>)=\boldsymbol{\varGamma}(\gamma({\mathcal F}\>))\colon\boldsymbol{\varGamma}\BG{\mathcal F}\to\boldsymbol{\varGamma}{\mathcal F}; $ and one finds after setting ${\mathcal F}=\boldsymbol{\varGamma}{\mathcal G}$ in (A) that this functorial map is an \emph{isomorphism} \begin{equation} \gamma(\boldsymbol{\varGamma})=\boldsymbol{\varGamma}(\gamma)\colon\boldsymbol{\varGamma}\BG\iso \boldsymbol{\varGamma}. \tag{a} \end{equation} Conversely, given (a) one can deduce that the map in (A) is an isomorphism, whose inverse takes $\alpha\colon\boldsymbol{\varGamma}{\mathcal E}\to{\mathcal F}\>$ to the composition $\boldsymbol{\varGamma}{\mathcal E}\iso\boldsymbol{\varGamma}\BG{\mathcal E}\xrightarrow{\boldsymbol{\varGamma}\alpha\,}\boldsymbol{\varGamma}{\mathcal F}\>$.% \footnote {The \emph{idempotence} of~$\boldsymbol{\varGamma}\<$, expressed by (a) or (A),\vadjust{\kern1.3pt} can be interpreted as follows. Set ${\mathbf D}\!:={\mathbf D}({\mathscr X})$, $\mathbf S\!:=\{\,{\mathcal E}\in{\mathbf D}\mid \boldsymbol{\varGamma}({\mathcal E})=0\,\}\<$,\vspace{.6pt} so that $\boldsymbol{\varGamma}$ factors uniquely as \smash{${\mathbf D} \overset{\vbox to0pt{\vss \hbox{$\scriptstyle q\>$} \vskip-.35ex} }\to {\mathbf D}/\mathbf S \overset{\vbox to0pt{\vss \hbox{$\scriptstyle\,\overline{\!\boldsymbol{\varGamma}\<}\>\>$} \vskip-.35ex} }\to{\mathbf D}$} where $q$ is the ``Verdier quotient" functor. Then \emph{$\,\overline{\!\boldsymbol{\varGamma}}$ is left-adjoint to~$q$}, so that $\mathbf S\subset{\mathbf D}$ admits a ``Bousfield colocalization."\index{Bousfield colocalization} It follows from (c) and (d) below that $\mathbf S=\{\,{\mathcal E}\in{\mathbf D}\mid {\boldsymbol\Lambda}({\mathcal E})=0\,\}$, and (b) below means that \emph{the functor $\bar{\boldsymbol\Lambda}\colon{\mathbf D}/\mathbf S\to{\mathbf D}$ defined by ${\boldsymbol\Lambda}=\bar{\boldsymbol\Lambda}\smcirc q$ is right-adjoint to~$q\>$}; thus $\mathbf S\subset{\mathbf D}$ also admits a ``Bousfield localization." And ${\mathbf D}/\mathbf S$ is equivalent, via $\,\overline{\!\boldsymbol{\varGamma}}$ and $\bar{\boldsymbol\Lambda}$ respectively, to the categories $\D_{\mathrm t}\<\subset{\mathbf D}$ and~${\mathbf D}\>\>\hat{}\>\subset{\mathbf D}$ introduced below---categories denoted by $\mathbf S^\perp$ and ${}^\perp\mathbf S$ in~\mbox{\cite[Chapter 8]{TC}. } The composed functorial map $\lambda\colon\mathbf1\to{\boldsymbol\Lambda}\boldsymbol{\varGamma}\xrightarrow{{\boldsymbol\Lambda}(\gamma)}{\boldsymbol\Lambda}$ induces an isomorphism \begin{equation} {\mathrm {Hom}}({\boldsymbol\Lambda}{\mathcal E},{\boldsymbol\Lambda}{\mathcal F}\>)\iso {\mathrm {Hom}}({\mathcal E}, {\boldsymbol\Lambda}{\mathcal F}\>) \qquad\bigl({\mathcal E},{\mathcal F}\in{\mathbf D}({\mathscr X})), \tag{B} \end{equation} or equivalently (as above), $\lambda$ induces an isomorphism \begin{equation} \lambda({\boldsymbol\Lambda})={\boldsymbol\Lambda}(\lambda)\colon{\boldsymbol\Lambda}\iso{\boldsymbol\Lambda}\BL. \tag{b} \end{equation} Moreover, the isomorphism (A) transforms via adjointness to an isomorphism $$ {\mathrm {Hom}}({\mathcal E}, {\boldsymbol\Lambda}\boldsymbol{\varGamma}{\mathcal F}\>)\iso {\mathrm {Hom}}({\mathcal E},{\boldsymbol\Lambda}{\mathcal F}\>)\qquad\bigl({\mathcal E},{\mathcal F}\in{\mathbf D}({\mathscr X})), $$ whose meaning is that $\gamma$ induces an isomorphism \begin{equation} {\boldsymbol\Lambda}\boldsymbol{\varGamma}\iso {\boldsymbol\Lambda}. \tag{c} \end{equation} Similarly, (B) means that $\lambda$ induces the conjugate isomorphism \begin{equation} \boldsymbol{\varGamma}{\boldsymbol\Lambda}\osi \boldsymbol{\varGamma}. \tag{d} \end{equation} Similarly, that ${\boldsymbol\Lambda}(\lambda({\mathcal F}\>))$---or $\gamma(\boldsymbol{\varGamma}({\mathcal E}))$---is an isomorphism (respectively that $\lambda({\boldsymbol\Lambda}({\mathcal F}\>))$---or~$\boldsymbol{\varGamma}(\gamma({\mathcal E}))$---is an isomorphism) is equivalent to the first (respectively the second) of the following maps (induced by $\lambda$ and $\gamma$ respectively) being an isomorphism: \begin{equation} {\mathrm {Hom}}(\boldsymbol{\varGamma}{\mathcal E}\<,{\mathcal F}\>) \iso{\mathrm {Hom}}(\boldsymbol{\varGamma}{\mathcal E}\<,{\boldsymbol\Lambda}\>{\mathcal F}\>) \osi{\mathrm {Hom}}({\mathcal E}\<,{\boldsymbol\Lambda}\>{\mathcal F}\>). \tag{AB} \end{equation} That (c) is an isomorphism also means that the functor~${\boldsymbol\Lambda}$ factors, via $\boldsymbol{\varGamma}\<$, through the essential image~$\D_{\mathrm t}\<({\mathscr X})$ of~$\boldsymbol{\varGamma}\<$ (i.e., the full subcategory~$\D_{\mathrm t}\<({\mathscr X})$ whose objects are isomorphic to $\boldsymbol{\varGamma}{\mathcal E}$ for some~${\mathcal E}$); and similarly (d) being an isomorphism means that $\boldsymbol{\varGamma}$ factors, via~${\boldsymbol\Lambda}$, through the essential image ${\mathbf D}\>\>\hat{}\>({\mathscr X})$ of~${\boldsymbol\Lambda}$; and the isomorphisms \mbox{$\boldsymbol{\varGamma}{\boldsymbol\Lambda}\boldsymbol{\varGamma}\cong\boldsymbol{\varGamma}$} and ${\boldsymbol\Lambda}\boldsymbol{\varGamma}{\boldsymbol\Lambda}\cong{\boldsymbol\Lambda}$ deduced from~(a)--(d) signify that ${\boldsymbol\Lambda}$ and $\boldsymbol{\varGamma}$ induce quasi-inverse equivalences between the categories $\D_{\mathrm t}\<({\mathscr X})$ and~${\mathbf D}\>\>\hat{}\>({\mathscr X})$. \smallskip (2) If ${\mathscr X}$ is properly algebraic, the natural functor ${\boldsymbol j}\colon{\mathbf D}(\A_{\vec {\mathrm c}}({\mathscr X}))\to\D_{\<\vc}({\mathscr X})$ is an \emph{equivalence,} and the inclusion $\D_{\<\vc}({\mathscr X})\hookrightarrow{\mathbf D}({\mathscr X})$ has a right adjoint~$\mathbf Q:={\boldsymbol j}{\mathbf R} Q_{\mathscr X}^{}$ (\Cref{corollary}.) Then (easily checked, given \Cref{C:vec-c is qc} and \Pref{Gamma'(qc)}) all of~(1) holds with ${\mathbf D}$,~$\D_{\mathrm t}\<\>$, and~${\boldsymbol\Lambda}$ replaced by $\D_{\<\vc}\>$, $\D_{\mkern-1.5mu\mathrm{qct}}\>$, and~${\boldsymbol\Lambda{\!^\vc}}\!:=\mathbf Q{\boldsymbol\Lambda}$, respectively. \smallskip (3) As in (1), ${\boldsymbol\Lambda}$ induces an equivalence from $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ to $\D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}({\mathscr X})$, the essential image of ${\boldsymbol\Lambda}|_{\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})}$---or, since ${\boldsymbol\Lambda}\cong{\boldsymbol\Lambda}\boldsymbol{\varGamma}\<$, of ${\boldsymbol\Lambda}|_{\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})}$ (\Pref{Gamma'(qc)}). So for\vadjust{\kern.7pt} any $f\colon{\mathscr X}\to{\mathscr Y}$ as in \Cref{C:identities}, the functor $$ {\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R} f_{\<\<*}\>{\mathbf R}\iGp{\mathscr X}\colon \D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}({\mathscr X}) \to \D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}({\mathscr Y}) $$ has the right adjoint ${\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times{\mathbf R}\iGp{\mathscr Y}={\boldsymbol\Lambda}_{\mathscr X}f_{\mathrm t}^\times=\ush f\<$. There result two ``parallel" adjoint pseudofunctors \cite[(3.6.7)(d)]{Derived categories} (where ``3.6.6" should be ``3.6.2"): $$ ({\mathbf R} f_{\<\<*}\>,\>f_{\mathrm t}^\times)\text{ (on $\D_{\mkern-1.5mu\mathrm{qct}}$)\quad and\quad } ({\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R} f_{\<\<*}\>{\mathbf R}\iGp{{\mathscr X}},\>\ush f\>) \text{ (on $\D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}\>$)}. $$ Both of these correspond to the same adjoint pseudofunctor on the quotient~$\D_{\mkern-1.5mu\mathrm {qc}}/(\mathbf S\cap\D_{\mkern-1.5mu\mathrm {qc}})$, see footnote under (1). If $f$ is \emph{adic} then ${\mathbf R f_{\!*}}{\boldsymbol\Lambda}_{\mathscr X}\cong{\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R f_{\!*}}$ (\Cref{C:f* and Gamma}(c)), and so \Pref{Rf-*(qct)} gives that ${\mathbf R f_{\!*}}(\D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}\>({\mathscr X}))\subset \D_{\mkern-1.5mu\mathrm {qc}}\mspace{-11mu}\hat{}\mspace{10mu}\>({\mathscr Y})$. Moreover, there are functorial isomorphisms $$ {\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R} f_{\<\<*}\>{\mathbf R}\iGp{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}\cong {\mathbf R} f_{\<\<*}\>{\boldsymbol\Lambda}_{\mathscr X}{\mathbf R}\iGp{\mathscr X}{\boldsymbol\Lambda}_{\mathscr X}\cong {\mathbf R} f_{\<\<*}\>{\boldsymbol\Lambda}_{\mathscr X}\>. $$ Thus for adic $f\<$, ${\boldsymbol\Lambda}_{\mathscr Y}{\mathbf R} f_{\<\<*}\>{\mathbf R}\iGp{\mathscr X}$ can be replaced above by~${\mathbf R f_{\!*}}$.\vspace{1pt} When $f$ is \emph{proper} more can be said, see \Tref{T:properdual}. \end{srems} \end{small} \end{parag} \section{Flat base change.} \label{sec-basechange} \renewcommand{\theequation}{\thesth} A \emph{fiber square}\index{fiber square} of adic formal schemes is a commutative diagram $$ \begin{CD} {\mathscr V}@>v>>{\mathscr X} \\ @VgVV @VVfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ such that the natural map is an \emph{isomorphism} ${\mathscr V}\iso{\mathscr X}\times_{\mathscr Y}{\mathscr U}$. If ${\mathscr I}$, ${\mathscr J}$, $\mathscr K$ are ideals of definition of ${\mathscr Y}$, ${\mathscr X}$, ${\mathscr U}$ respectively, then ${\mathcal L}\!:={\mathscr J}{\mathcal O}_{\mathscr V}+\mathscr K{\mathcal O}_{\mathscr V}$ is an ideal of definition of~$\>{\mathscr V}$, and the scheme $V\!:=({\mathscr V}, {\mathcal O}_{\mathscr V}/{\mathcal L})$ is the fiber product of the $({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I})$-schemes $({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J})$ and $({\mathscr U},{\mathcal O}_{\mathscr U}/\mathscr K)$, see \cite[p.\,417, Proposition~(10.7.3)]{GD}. By \cite[p.\,414, Corollaire~(10.6.4)]{GD}, if $V$ is locally noetherian and the ${\mathcal O}_V$-module~${\mathcal L}/{\mathcal L}^2$ is of finite type then ${\mathscr V}$ is locally noetherian. That happens whenever ${\mathscr X}$, ${\mathscr Y}$ and ${\mathscr U}$ are locally noetherian and either $u$ or $f$ is of pseudo\kern.6pt-finite type.\looseness=-1 Our goal is to prove \Tref{T:basechange} (=\:\Tref{Th3} of the Introduction). That is, given a fiber square as above, with ${\mathscr X}$, ${\mathscr Y}$, ${\mathscr U}$ and ${\mathscr V}$ noetherian, $f$ \emph{pseudo\kern.6pt-proper,} and $u$ \emph{flat,} we want to establish a functorial isomorphism $$ \beta_{\mathcal F}\colon{\mathbf R}\iGp{\mathscr V}\>v^*\<f_{\mathrm t}^\times\<{\mathcal F} \iso g_{\mathrm t}^{\<\times}\>{\mathbf R}\iGp{\mathscr U} u^*{\mathcal F}\ (\cong g_{\mathrm t}^{\<\times}\<u^*\<{\mathcal F}\>) \qquad \bigl({\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr). $$ Some consequences of this theorem will be given in \Sref{Consequences}. In order to define $\beta_{\mathcal F}$ (\Dref{D:basechange}) we first need to set up a canonical isomorphism ${\mathbf R}\iGp{\mathscr U} u^*{\mathbf R f_{\!*}}\iso{\mathbf R}\iGp{\mathscr U} {\mathbf R} g_*v^*\<$. This is done in \Pref{uf=gv}. (When $u$~is \emph{adic} as well as flat, ${\mathbf R}\iGp{\mathscr U}$ can be omitted.) Our proof of \Tref{T:basechange} has the weakness that it \emph{assumes} the case when $f$ is a proper map of noetherian ordinary schemes. As far as we know, the published proofs of this latter result make use of finite-dimensionality hypotheses on the schemes involved (see \cite[p.\, 392, Thm.\,2]{f!}, \cite[p.\,383, Cor.\,3.4]{H1}), or projectivity hypotheses on~$f$ \cite[p.\,191, 5]{H1}). There is however an outline of a proof for the general case, even without noetherian hypotheses, in \cite{Non noetherian}---see Corollary 4.3 there \footnote{ Details may eventually appear in \cite{Derived categories}. It is quite possible that the argument can be adapted to give a direct proof for formal schemes too.} \medskip To begin with, here are several properties of formal-scheme maps (see \S\ref{maptypes}) which propagate across fiber squares. \begin{prop} \label{P:basechange} \textup{(a)} Let\/ $f\colon {\mathscr X}\to{\mathscr Y}$ and\/ $u:{\mathscr U}\to{\mathscr Y}$ be maps of locally noetherian formal schemes, such that the fiber product\/ ${\mathscr X}\times_{{\mathscr Y}} {\mathscr U}$ is locally noetherian \textup(a condition which holds, e.g., if either \/ $f$ or\/ $u$ is of pseudo-finite type, see \cite[p.\,414, Corollaire (10.6.4)]{GD}\textup). If\/ $f$ is\/ \emph{separated} \textup(resp.~\emph{affine,} resp.~\emph{pseudo\kern.6pt-proper,} resp.~\emph{pseudo\kern.6pt-finite,} resp.~\emph{of pseudo\kern.6pt-finite type,} resp.~\emph{adic}\textup) then so is the projection\/ ${\mathscr X}\times_{{\mathscr Y}}{\mathscr U}\to{\mathscr U}$. \vspace{1pt} \textup{(b)} With\/ $f\colon {\mathscr X}\to{\mathscr Y}$ and $u:{\mathscr U}\to{\mathscr Y}$ as in\/ \textup{(a),} assume either that\/ $u$ is adic or that\/ $f$ is of pseudo\kern.6pt-finite type. If\/ $u$ is flat then so is the projection\/ ${\mathscr X}\times_{{\mathscr Y}}{\mathscr U}\to{\mathscr X}$.\vadjust{\kern1pt} \textup{(c)} Let\/ $f\colon {\mathscr X}\to{\mathscr Y}$, $u:{\mathscr U}\to{\mathscr Y}$ be maps of locally noetherian formal schemes, with\/ $u$ flat and locally over\/~${\mathscr Y}$ the completion of a finite-type map of ordinary schemes. Then\/ ${\mathscr X}\times_{{\mathscr Y}}{\mathscr U}$ is locally noetherian, and the projection\/ ${\mathscr X}\times_{{\mathscr Y}}{\mathscr U}\to{\mathscr X}$ is flat. \end{prop} \begin{proof} (a) The adicity assertion is obvious, and the rest follows from corresponding assertions for the ordinary schemes obtained by factoring out defining ideals.\vadjust{\kern1pt} (b) It's enough to treat the case when ${\mathscr Y}$, ${\mathscr X}$, and ${\mathscr U}$ are the formal spectra, respectively, of noetherian adic rings $(A,I)$, $(B,J)$ and $(C,K)$ such that $B$ and~$C$ are $A$-algebras with \mbox{$J\supset IB$} and $K\supset IC$, and such that $B\, \widehat{\otimes}_{\<A} \>\>C$ is noetherian (since ${\mathscr X}\times_{{\mathscr Y}} {\mathscr U}$ is locally noetherian, see \cite[p.\,414, Corollaire (10.6.5)]{GD}). By the following \Lref{(4.1.2)}, the problem is to show that if $C$ is $A$-flat and \emph{either} $K=IC$ ($u$ adic), \emph{or} $B/\<J$ is a finitely-generated $A$-algebra ($f$ of pseudo-finite type), then $B\, {\widehat \otimes}_{\<A}\>\>C$ is $B$-flat. The local criterion of flatness \cite[p.\,98, \S5.2, Thm.\,1 and p.\,101, \S5.4, Prop.\,2]{Bou} reduces the problem further to showing that for all $n>0$, $ (B\, {\widehat\otimes}_{\<A} \>\>C)/J^n(B\, {\widehat\otimes}_{\<A} \>\>C)$ is $(B/\<J^n)$-flat, i.e., that $(B/\<J^n)\, {\widehat\otimes}_{\<A} \>\>C$ is $(B/\<J^n)$-flat. But, $C$ being $A$-flat, if $K=IC$ then $(B/\<J^n)\, {\widehat\otimes}_{\<A} \>\>C =(B/\<J^n)\otimes_{A/I^n} (C/I^nC) $ is clearly $B/\<J^n$-flat; while if $B/J$ is a finitely-generated $A$-algebra, then $(B/\<J^n)\>\otimes_A \>C$ is noetherian and $(B/\<J^n)$-flat, whence so is its $K$-adic completion $(B/\<J^n)\, {\widehat\otimes}_{\<A} \>\>C$. \vadjust{\kern1pt} (c) Proceeding as in the proof of (b), we may assume~$C$ to be the $K'$-adic completion of a finite\kern.5pt-type $A$-algebra~$C'$ ($K'$ a $C'$-ideal). If $C$ is $A$-flat then by \cite[\S 5.4, Proposition 4]{Bou}, the localization $C'{}'\!:= C'[(1+K')^{-1}]$ is $A$-flat, so the noetherian $B$-algebra $ B\otimes_{\<A} C'{}' $ is $B$-flat, as is its (noetherian) completion $B\,\widehat {\otimes}_{\<A}\, C$. \end{proof} \begin{slem} \label{(4.1.2)} Let $\varphi:A \to C$ be a continuous homomorphism of noetherian adic rings. Then $C$ is $A$-flat iff the corresponding map ${\mathrm {Spf}}(\varphi)\colon {\mathrm {Spf}}(\<C)\to{\mathrm {Spf}}(A)$ is~flat, i.e., iff for each open prime~$q\subset C,$ $C_{\{q\}}$ is $A_{\{\varphi^{-1}q\}}\<$-\kern.5pt flat. \end{slem} \begin{proof} Recall that if $K$ is an ideal of definition of~$C$ and $q\supset K$ is an open prime ideal in~$C$, then with $C\setminus q$ ordered by divisibility, $$ C_{\{q\}}\!:={\mathcal O}_{{\mathrm {Spf}}(C)\<,\>q}= \dirlm{{\displaystyle\mathstrut}\hbox to 0pt {\hss$\scriptstyle f\in \>C\setminus q\;$\hss}}C_{\{f\}}\vspace{3pt} $$ \vspace{1pt}% where $C_{\{f\}}$ is the $K$-adic completion of the localization~$C_{\<f}\>$. Now for each $f\notin q$ and $n>0$ the canonical map\vspace{1pt} $C_{\<f}/\<K^n\<C_{\<f} \to C_{\{f\}}/\<K^n\<C_{\{f\}}$~is bijective,\vadjust{\kern.7pt} so the \smash{$\dirlm{}\!\!$} of these maps is an isomorphism\vadjust{\kern.8pt} $C_q/\<K^n\<C_q\iso C_{\{q\}}/\<K^n\<C_{\{q\}}$, whence so is the $K$-adic completion \smash{$\widehat{C_q} \iso \widehat{C_{\{q\}}}$} of the canonical map $C_q\to C_{\{q\}}$. We can therefore apply \cite[\S5.4, Proposition 4]{Bou} twice to get that $C_q$ is $A_{\varphi^{-1}q}$-flat iff $C_{\{q\}}$ is $A_{\{\varphi^{-1}q\}}$-flat. So if $C$ is $A$-flat then ${\mathrm {Spf}}(\varphi)$ is flat; and the converse holds because $C$ is $A$-flat iff $C_m$ is $A_{\varphi^{-1}m}$-flat for every maximal ideal~$m$ in~$C$, and every such $m$ is open since $C$ is complete. \end{proof} \begin{prop} \label{uf=gv} \textup{(a)} Consider a fiber square of noetherian formal schemes $$ \begin{CD} {\mathscr V}@>v>>{\mathscr X} \\ @VgVV @VVfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ with $u$ and $v$ flat. Let $$ \psi_{\<{\mathcal G}}^{}\colon {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal G}\to {\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*\<{\mathcal G} \qquad\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr X})\bigr) $$ be the unique map whose composition with the natural map ${\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*\<{\mathcal G}\to{\mathbf R} g_*v^*\<{\mathcal G}$ is the natural map ${\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal G}\to{\mathbf R} g_* v^*\<{\mathcal G}$. \textup(The existence of\/~$\psi_{\<{\mathcal G}}^{}$ is given by Propositions~\ref{Gamma'(qc)} and ~\ref{Rf-*(qct)}.\textup) Then for all\/ ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),$ $\psi_{\mathcal E}$ is an \emph{isomorphism}. In particular, if\/ $u$ \textup(hence $v)$ is \emph{adic} then\/ $\psi_{{\mathcal E}}$ can be identified with the identity map of\/ $ {\mathbf R} g_*v^*\<{\mathcal E}$. \vspace{2pt} \textup{(b)} Let\/ ${\mathscr X},$\ ${\mathscr Y},$\ ${\mathscr U}$ be noetherian formal schemes, let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ and\/ $u\colon{\mathscr U}\to{\mathscr Y}$ be maps, with\/ $u$ flat, and assume further that one of the following holds:\vadjust{\kern1pt} \item[\hspace{2.87em}(i)] $u$ is adic, and\/ ${\mathscr V}\!:={\mathscr X}\times_{\mathscr Y}{\mathscr U}$ is noetherian, \item[\hspace{2.6em}(ii)] $f$ is of pseudo\kern.6pt-finite type, \item[\hspace{2.3em}(iii)] $u$ is locally the completion of a finite-type map of ordinary schemes;\vadjust{\kern1.5pt} \noindent so that by \Pref{P:basechange} we have a fiber square as in\/ \textup{(a)}. Let $$ \theta_{\mathcal G}\colon u^*\>{\mathbf R f_{\!*}}{\mathcal G}\to {\mathbf R} g_*v^*{\mathcal G}\qquad\bigl ({\mathcal G}\in{\mathbf D}({\mathscr X})\bigr) $$ be adjoint\vadjust{\kern.5pt} to the canonical map\/ ${\mathbf R f_{\!*}} {\mathcal G}\to{\mathbf R f_{\!*}} {\mathbf R} v_* v^*{\mathcal G}={\mathbf R} u_*{\mathbf R} g_*v^*{\mathcal G}$.\vadjust{\kern1pt} Then for all\/ ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),$\ the map $\theta_{\!{\mathcal E}}'\!:={\mathbf R}\iGp{\mathscr U}(\theta_{\<{\mathcal E}})$\index{ {}$\theta'\<$} is an \emph{isomorphism} $$\postdisplaypenalty10000 \theta_{\!{\mathcal E}}'\colon{\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}{\mathcal E}\iso{\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*\<{\mathcal E}. $$ In particular, if\/ $u$ \textup(hence $v)$ is \emph{adic} then\/ $\theta_{\<{\mathcal E}}$ itself is an isomorphism. \smallskip \textup{(c)} Under the hypotheses of\/ \textup{(a)}\vspace{.6pt} resp.~\textup{(b),} if\/ $f$ \textup(hence $g)$ is adic then\/ $\psi_{\mathcal E}$ resp.~$\theta_{\!{\mathcal E}}'$ is an isomorphism for all\/ ${\mathcal E}\in\wDqc({\mathscr X})$ \textup(see \Dref{D:Dtilde}\kern.5pt\textup{).} \end{prop} \begin{proof} (a) Let ${\mathscr J}$ be an ideal of definition of~${\mathscr X}$, and ${\mathcal K}$ of~${\mathscr U}$, so that ${\mathscr J}{\mathcal O}_{\mathscr V}+{\mathcal K}{\mathcal O}_{\mathscr V}$ is an ideal of definition of~${\mathscr V}$. The obvious equality $\iG{{\mathscr J}{\mathcal O}_{\mathscr V}+{\mathcal K}{\mathcal O}_{\mathscr V}}=\iG{{\mathcal K}{\mathcal O}_{\mathscr V}}\iG{{\mathscr J}{\mathcal O}_{\mathscr V}}$, applied to K-injective ${\mathcal O}_{\mathscr V}$-complexes, leads to a natural functorial map $$ {\mathbf R}\iGp{\mathscr V} \underset{\ref{Gamma'1}}{\overset{\textup{def}}=} {\mathbf R}\iG{{\mathscr J}{\mathcal O}_{\mathscr V}+{\mathcal K}{\mathcal O}_{\mathscr V}} \longrightarrow{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}}{\mathbf R}\iG{{\mathscr J} {\mathcal O}_{\mathscr V}} $$ which is an \emph{isomorphism,} as one checks locally via ~\cite[p.\,20, Corollary~(3.1.3)]{AJL}. Also, there are natural isomorphisms $$ {\mathbf R}\iG{{\mathscr J}{\mathcal O}_{\mathscr V}}v^*\<{\mathcal E} \underset{\textup{\ref{P:f* and Gamma}(b)}}{\iso} v^*\>{\mathbf R}\iGp{\mathscr X}\>{\mathcal E}=v^*\>{\mathbf R}\iG{\mathscr J}{\mathcal E} \underset{\textup{\ref{Gamma'(qc)}(a)}}{\iso} v^*\<{\mathcal E} \qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr). $$ Thus the natural map ${\mathbf R}\iGp{\mathscr V}\to{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}}$ induces an \emph{isomorphism}---the composition $$ {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\mkern.6mu v^*\<{\mathcal E} \iso{\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}}{\mathbf R}\iG {{\mathscr J} {\mathcal O}_{\mathscr V}}v^*\<{\mathcal E} \iso{\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}} v^*\<{\mathcal E}. $$ Since $(*)\<\colon$\! $\<{\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}} v^*\<{\mathcal E}\cong{\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\mkern.6mu v^*\<{\mathcal E} \<\in\<\D_{\mathrm t}\<({\mathscr U})$ (Propositions~\ref{Gamma'(qc)} and ~\ref{Rf-*(qct)}) therefore we can imitate the proof of \Pref{P:f* and Gamma}(d)---\emph{without} the boundedness imposed there on~${\mathcal G}$, since that would be needed only to get $(*)$---to see that the map ${\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}} v^*\<{\mathcal E}\to{\mathbf R} g_*v^*\<{\mathcal E}$ induced by ${\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}}\to \mathbf1$ factors uniquely as $$ {\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}} v^*\<{\mathcal E}\iso{\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*\<{\mathcal E}\longrightarrow{\mathbf R} g_*v^*\<{\mathcal E}, $$ with the first map an isomorphism. It follows that $\psi_{\mathcal E}$ is the composed isomorphism $$ {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\mkern.6mu v^*\<{\mathcal E} \iso{\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}}{\mathbf R}\iG {{\mathscr J} {\mathcal O}_{\mathscr V}}v^*\<{\mathcal E} \iso{\mathbf R} g_*{\mathbf R}\iG{{\mathcal K} {\mathcal O}_{\mathscr V}} v^*\<{\mathcal E} \iso{\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*\<{\mathcal E}. $$ The last statement in (a) (for adic~$u$) results then from \Cref{C:f* and Gamma}(b) and Propositions~\ref{Rf-*(qct)} and \ref{Gamma'(qc)}(a). \smallskip (b) Once $\theta_{\!{\mathcal E}}'$ is shown to be an isomorphism, the last statement in (b) (for adic~$u$) follows from \Cref{C:f* and Gamma}(b), and Propositions~\ref{Rf-*(qct)} and \ref{Gamma'(qc)}(a). To show that $\theta_{\!{\mathcal E}}'$ is an isomorphism, it suffices to show that the composition $$ \psi_{\mathcal E}^{-1}\theta_{\!{\mathcal E}}'\colon{\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}{\mathcal E} \to {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal E} \qquad\bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\bigr). $$ is an isomorphism. We use \Lref{Gam as holim} to reduce the problem, as follows. First, the functors $u^*\<$, $v^*\<$, ${\mathbf R}\iGp{\mathscr U}$ and ${\mathbf R}\iGp{\mathscr V}$ are bounded, and commute with direct\- sums: for $u^*\<$ and $v^*\<$ that is clear, and for ${\mathbf R}\iGp{\mathscr U}$ and ${\mathbf R}\iGp{\mathscr V}$ it holds because they can be realized locally by tensoring with a bounded flat complex (see proof of \Pref{Gamma'(qc)}). Furthermore, \Lref{Gamma'+qc}, \Pref{Gamma'(qc)}, and~\Pref{P:Lf*-vc} show that ${\mathbf R}\iGp{\mathscr V}\>v^*\>\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\subset\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr V})$; and the functor ${\mathbf R} g_*$ (resp.~${\mathbf R f_{\!*}}$) is bounded on, and commutes with direct sums in, $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr V})$ (resp.~$\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$), see Propositions~\ref{Gamma'+qc}, ~\ref{Rf_*bounded} and~\ref{P:coprod}. Hence, standard way-out reasoning allows us to assume that ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$. Next, let ${\mathscr J}$ be an ideal of definition of~${\mathscr X}$, $X_{n}\ (n>0)$ the scheme $({\mathscr X},{\mathcal O}_{\mathscr X}/\<{\mathscr J}^n)$, and $j_n\colon X_{n}\hookrightarrow {\mathscr X}$ the associated closed immersion. The functor $j_{n*}\colon{\mathcal A}(X_{n})\to{\mathcal A}({\mathscr X})$ is exact, so it extends to a functor ${\mathbf D}(X_{n})\to{\mathbf D}({\mathscr X})$. The functor $j_n^\natural\colon{\mathbf D}({\mathscr X})\to{\mathbf D}(X_{n})$ being defined as in~\eqref{f^natl}, we have $$ {\boldsymbol h}_n({\mathcal G})\!:={\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr X}/\<{\mathscr J}^n\<,\>{\mathcal G}) =j_{n*}j_n^\natural{\mathcal G}\qquad\bigl({\mathcal G}\in{\mathbf D}({\mathscr X})\bigr). $$ If ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$ then ${\mathcal E}={\mathbf R}\iG{\<{\mathscr J}}\<{\mathcal E}$ (\Pref{Gamma'(qc)}(a)), and, as noted just after~\eqref{f^natl}, $j_n^\natural{\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(X_{n})$. Hence, from the triangle in \Lref{Gam as holim} (with ${\mathcal G}$~replaced by an ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$) we derive a diagram of triangles $$ \minCDarrowwidth=18pt \begin{CD} {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}(\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal E}) @>>> {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}(\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal E}) @>>> {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}({\mathbf R}\iG{\<{\mathscr J}}\<{\mathcal E}) @>+>> \\ @V\simeq VV @V\simeq VV @VV\simeq V \\ \oplus_{n\ge1}\>{\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}} {\boldsymbol h}_n\>{\mathcal E} @>>> \oplus_{n\ge1}\>{\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}} {\boldsymbol h}_n\>{\mathcal E} @>>> {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}{\mathcal E} @>+>> \\ @V\oplus V\psi_{{\boldsymbol h}_{\<n}{\mathcal E}}^{-1}\>\theta_{\< {\boldsymbol h}_{\<n}{\mathcal E}}'V @V\oplus V\psi_{{\boldsymbol h}_{\<n}{\mathcal E}}^{-1}\>\theta_{\<{\boldsymbol h}_{\<n}{\mathcal E}}'V @VV\psi_{\mathcal E}^{-1}\theta_{\!{\mathcal E}}'V \\ \oplus_{n\ge1}{\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\boldsymbol h}_n\>{\mathcal E} @>>> \oplus_{n\ge1}{\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\boldsymbol h}_n\>{\mathcal E} @>>> {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal E} @>+>> \\ @V\simeq VV @V\simeq VV @VV\simeq V \\ {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*(\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal E}) @>>> {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V} v^*(\oplus_{n\ge1}\>{\boldsymbol h}_n\>{\mathcal E}) @>>> {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*({\mathbf R}\iG{\<{\mathscr J}}\<{\mathcal E}) @>+>> \end{CD} $$ From this diagram we see that if each $\psi_{{\boldsymbol h}_n{\mathcal E}}^{-1}\>\theta_{\<{\boldsymbol h}_n{\mathcal E}}'$ is an isomorphism, then so is~$\psi_{\mathcal E}^{-1}\>\theta_{\mathcal E}'$. So we need only prove (b) when ${\mathcal E}=j_{n*}\>{\mathcal F}$ with ${\mathcal F}\!:= j_n^\natural{\mathcal E}\in\D_{\mkern-1.5mu\mathrm {qc}}(X_{n})$.\vspace{1pt} Let us show that in fact \emph{for any $n>0$ and any ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(X_{n})\<,$ $\theta_{\!j_{\<n\<*}{\mathcal F}}'$ is an isomorphism.} \smallskip The assertion~(b) is local both on~${\mathscr Y}$ and on ${\mathscr U}$. Indeed, for (b) to hold it suffices, for every diagram of fiber squares $$ \CD {\mathscr V}@<j'<<{\mathscr V}'@>v'>>{\mathscr X}' @>j>>{\mathscr X} \\ @VgVV@Vg'VV @VVf'V @VVfV\\ {\mathscr U}@<<i'<{\mathscr U}'@>>u'>{\mathscr Y}'@>>i>{\mathscr Y} \endCD $$ where ${\mathscr Y}'$ ranges over a base of open subsets of~${\mathscr Y}$, ${\mathscr U}'$ ranges over a base of open subsets of~$u^{-1}{\mathscr Y}'\<$, $u'$ is induced by~$u$, and $i$, $i'$ are the inclusions, that $i'{}^{\<*}\<\theta_{\!{\mathcal E}}'$ \mbox{$(=\theta_{\!{\mathcal E}}'|_{{\mathscr U}'})$} be an isomorphism. Now when $u$ is an open immersion, $\theta_{\mathcal G}$ is an isomorphism for all ${\mathcal G}\in{\mathbf D}({\mathscr X}\>)$. (One may assume ${\mathcal G}$ to be K-injective and note that $v^*\<$, having the exact left adjoint ``extension by zero," preserves K-injectivity, so that $\theta_{\mathcal G}$ becomes the usual isomorphism $u^*\<\<f_{\!*}\>{\mathcal G}\iso g_*v^*{\mathcal G}$). Thus there are functorial isomorphisms $i'{}^*\>{\mathbf R} g_*\iso{\mathbf R} g_*'\>j'{}^*$ and $i^*\>{\mathbf R f_{\!*}}\iso{\mathbf R} f_{\<\<*}'\>j^*$; and similarly there is an isomorphism $i'{}^*\>{\mathbf R}\iGp{\mathscr U}\iso{\mathbf R}\iGp{{\mathscr U}'} i'{}^*\<$. So it suffices that the composition $$ i'{}^*\>{\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}} {\mathcal E}\xrightarrow{i'{}^{\<\<*}\<\<\theta_{\!{\mathcal E}}'\>} i'{}^*\>{\mathbf R}\iGp{\mathscr U}\>{\mathbf R} g_*v^*\<{\mathcal E}\iso {\mathbf R}\iGp{{\mathscr U}'} i'{}^*\>{\mathbf R} g_*v^*\<{\mathcal E}\iso {\mathbf R}\iGp{{\mathscr U}'} {\mathbf R} g_*'\>j'{}^*v^*\<{\mathcal E} $$ be an isomorphism; and with a bit of patience one identifies this composition with $$ {\mathbf R}\iGp{{\mathscr U}'}u'{}^*i^*\>{\mathbf R f_{\!*}} {\mathcal E}\iso {\mathbf R}\iGp{{\mathscr U}'} u'{}^*\>{\mathbf R} f_{\<\<*}'\>j^*\<{\mathcal E} \xrightarrow{\theta_{\<j^{\<\<*}\!{\mathcal E}}'\>\>} {\mathbf R}\iGp{{\mathscr U}'}{\mathbf R} g_*'v'{}^*\<j^*\<{\mathcal E}, $$ thereby reducing to showing that $\theta_{\<j^{\<\<*}\<\<{\mathcal E}}'$ is an isomorphism. Thus one may assume that both ${\mathscr Y}$ and ${\mathscr U}$ are affine, say ${\mathscr Y}={\mathrm {Spf}}(A)$ and ${\mathscr U}={\mathrm {Spf}}(C)$ with $C$ a flat $A$-algebra (\Lref{(4.1.2)}). Suppose next that ${\mathscr X}$ and ${\mathscr Y}$ are ordinary schemes, so that ${\mathscr Y}={\mathrm {Spec}}(A)$. In cases~(i) and (ii) of (b), set $C'{}'\!:= C$, and in case (iii) let $C'{}'$ be as in the proof of part~(c) of \Pref{P:basechange}. In any case, $C'{}'$ is $A$-flat, $C$ is the $K$-adic completion of~$\>C'{}'$ for some $C'{}'$-ideal~$K\<$, $\>{\mathscr X}\times_{\mathscr Y} {\mathrm {Spf}}(C)$ is the $K$-adic completion of~${\mathscr X}\times_{\mathscr Y} {\mathrm {Spec}}(C'{}')$, and we have a natural commutative diagram $$ \begin{CD} {\mathscr X}\times_{\mathscr Y} {\mathrm {Spf}}(C)@>v_2>>{\mathscr X}\times_{\mathscr Y} {\mathrm {Spec}}(C'{}')@>v_1>>{\mathscr X} \\ @VgVV @Vg_1VV @VVfV \\ {\mathrm {Spf}}(C) @>>u_2> {\mathrm {Spec}}(C'{}') @>>u_1> {\mathscr Y} \end{CD} $$ With $\iGp{}$ denoting $\iGp{{\mathrm {Spf}}(C)}$, $\theta_{\!{\mathcal E}}'=:\theta'({\mathcal E},f,u)$ factors naturally as the composition $$ {\mathbf R}\iGp{}u_2^*u_1^*\>{\mathbf R f_{\!*}}{\mathcal E} \xrightarrow{{\mathbf R}\iGp{}u_2^*(\theta({\mathcal E}\<\<,\>f\<,\>u_1))} {\mathbf R}\iGp{}u_2^*\>{\mathbf R} g_{1\<*}^{}v_1^*{\mathcal E} \xrightarrow{\theta'(v_1^*\<{\mathcal E}\<\<,\>\>g_1\<,\>\>u_2)} {\mathbf R}\iGp{}v_2^*\>{\mathbf R} g_*v_2^*v_1^*{\mathcal E}. $$ Here $\theta({\mathcal E},f,u_1)$ is an isomorphism because all the schemes involved are ordinary schemes. (One argues as in \cite[p.\,111, Prop.\,5.12]{H1}, using \cite[p.\,35, (6.7)]{AHK}; for a fussier treatment see \cite[Prop.\,(3.9.5)]{Derived categories}.) Also, $\theta'(v_1^*{\mathcal E},g_1,u_2)$ is an isomorphism, in case (i) of (b) since then $u_2$ and $v_2$ are identity maps, and in cases (ii) and (iii) by \Cref{C:kappa-f*t'} since then ${\mathscr X}\times_{\mathscr Y}{\mathrm {Spec}}(C'{}')$ is noetherian. Thus: \begin{slem}\label{L:ordinary} \Pref{uf=gv} holds when\/ ${\mathscr X}$ and\/ ${\mathscr Y}$ are both ordinary schemes. \end{slem} We will also need the following special case of \Pref{uf=gv}: \begin{slem}\label{L:closed} Let\/ ${\mathscr I}$ be an ideal of definition of\/ ${\mathscr Y},$\ $Y_n$ the scheme $({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I}^n),$\ and $i_n\colon Y_n\hookrightarrow{\mathscr Y}$ the canonical closed immersion. Let $u_n\colon Y_n\times_{\mathscr Y}{\mathscr U}\to Y_n$ and $p_n\colon Y_n\times_{\mathscr Y}{\mathscr U}\to{\mathscr U}$ be the projections \textup(so that\/ $u_n$ is flat and\/ $p_n$ is a closed immersion, see\/ \textup{\cite[p.\,442, (10.14.5)(ii)]{GD})}. Then the natural map is an isomorphism $$ u^*i_{n*}{\mathcal G}\iso p_{n*}u_n^*{\mathcal G}\qquad\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}(Y_n)\bigr). $$ \end{slem} \begin{proof} Since the functors $u^*\<$, $i_{n*\>}$, $ p_{n*\>}$, and $u_n^*$ are all exact,\vspace{.5pt} we may assume that ${\mathcal G}$ is a quasi-coherent ${\mathcal O}_{Y_n}$-module; and since those functors commute with\vspace{1pt} \smash{$\dirlm{}\!\!$} we may further assume ${\mathcal G}$ coherent, and then refer to \cite[p.\,443, (10.14.6)]{GD}. \end{proof} Finally, for general noetherian formal schemes ${\mathscr X}$ and ${\mathscr Y}$, and ${\mathscr I}$ and $Y_n$ as above, let ${\mathscr J}\supset{\mathscr I}{\mathcal O}_{\mathscr X}$ be an ideal of definition of\/ ${\mathscr X}$, let $X_n$ be the scheme $({\mathscr X},{\mathcal O}_{\mathscr X}/\<{\mathscr J}^n),$\ and let $f_n\colon X_n\to Y_n$ be the map induced by $f$. Then for any ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(X_n)$, it holds that ${\mathbf R} f_{n*}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(Y_n)$. (See \Pref{Rf-*(qct)}---though the simpler case ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}^+(X_n)$ would do for proving \Pref{uf=gv}.) Associated to the natural diagram\looseness=-1 \bigskip \xdef\RestoreCatCode{\catcode`\noexpand\@=\the\catcode`\@} \catcode`@=11 \expandafter\ifx\csname graph\endcsname\relax \alloc@4\box\chardef\insc@unt\graph\fi \expandafter\ifx\csname graphtemp\endcsname\relax \alloc@1\dimen\dimendef\insc@unt\graphtemp\fi \RestoreCatCode \setbox\graph=\vtop{% \vbox to0pt{\hbox{% \special{pn 8}% \special{pn 2}% \special{ia 275 1889 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 1.889in \rlap{\kern 0.275in\lower\graphtemp\hbox to 0pt{\hss ${\mathscr U}$\hss}}% \special{ia 1740 1889 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 1.889in \rlap{\kern 1.74in\lower\graphtemp\hbox to 0pt{\hss ${\mathscr Y}$\hss}}% \special{ia 975 1389 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 1.389in \rlap{\kern 0.975in\lower\graphtemp \hbox to 0pt% {\hss $Y_n\!\<\times_{\<{\mathscr Y}}\!{\mathscr U}$\hss}}% \special{ia 2440 1389 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 1.389in \rlap{\kern 2.44in\lower\graphtemp\hbox to 0pt{\hss $Y_n$\hss}}% \special{pa 0 780}% \special{pa 0 500}% \special{pa 550 500}% \special{pa 550 780}% \special{pa 0 780}% \special{ip}% \graphtemp=.6ex \advance\graphtemp by 0.64in \rlap{\kern 0.275in\lower\graphtemp\hbox to 0pt{\hss ${\mathscr V}$\hss}}% \special{ia 1740 640 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 0.64in \rlap{\kern 1.74in\lower\graphtemp\hbox to 0pt{\hss ${\mathscr X}$\hss}}% \special{pa 700 260}% \special{pa 700 0}% \special{pa 1250 0}% \special{pa 1250 280}% \special{pa 700 280}% \special{ip}% \graphtemp=.6ex \advance\graphtemp by 0.14in \rlap{\kern 0.975in\lower\graphtemp\hbox to 0pt {\hss$X_n\!\<\times_{\<{\mathscr Y}}\! {\mathscr U}\ $\hss}}% \special{ia 2440 140 140 140 0 6.28319}% \graphtemp=.6ex \advance\graphtemp by 0.14in \rlap{\kern 2.44in\lower\graphtemp\hbox to 0pt{\hss $X_n$\hss}}% \graphtemp=0.3\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 1.889in \rlap{\kern 1.007in\lower\graphtemp\hbox to 0pt{\hss $\scriptscriptstyle u$\hss}}% \special{pa 415 1889}% \special{pa 1600 1889}% \special{fp}% \special{pa 1600 1889}% \special{pa 1575 1914}% \special{fp}% \special{pa 1575 1864}% \special{pa 1600 1889}% \special{fp}% \graphtemp=-0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 1.55in \rlap{\kern 1.405in\lower\graphtemp\hbox to 0pt{\hss $\scriptscriptstyle\ \quad u_{\<n}$\hss}}% \special{pa 1250 1389}% \special{pa 1695 1389}% \special{fp}% \special{pa 1785 1389}% \special{pa 2300 1389}% \special{fp}% \special{pa 2300 1389}% \special{pa 2275 1414}% \special{fp}% \special{pa 2275 1364}% \special{pa 2300 1389}% \special{fp}% \graphtemp=-0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 1.639in \rlap{\kern 0.6in\lower\graphtemp\hbox to 0pt {\hss {$\scriptscriptstyle p_{n}\ $}\hss}}% \special{pa 389 1808}% \special{pa 837 1514}% \special{fp}% \special{pa 389 1808}% \special{pa 396 1773}% \special{fp}% \special{pa 389 1808}% \special{pa 424 1815}% \special{fp}% \graphtemp=0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 1.639in \rlap{\kern 2.09in\lower\graphtemp\hbox to 0pt{\hss \raise1ex\hbox{$\scriptscriptstyle\,\ \quad i_{\<n}$}\hss}}% \special{pa 1854 1808}% \special{pa 2326 1471}% \special{fp}% \special{pa 1854 1808}% \special{pa 1861 1773}% \special{fp}% \special{pa 1854 1808}% \special{pa 1889 1815}% \special{fp}% \graphtemp=-0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 0.64in \rlap{\kern 1.075in\lower\graphtemp\hbox to 0pt{\hss $\scriptscriptstyle\qquad v$\hss}}% \special{pa 415 640}% \special{pa 1600 640}% \special{fp}% \special{pa 1600 640}% \special{pa 1575 665}% \special{fp}% \special{pa 1575 615}% \special{pa 1600 640}% \special{fp}% \graphtemp=-0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 0.401in \rlap{\kern 0.564in\lower\graphtemp\hbox to 0pt{\hss {$\scriptscriptstyle\quad\! q_{n}\quad $}\hss}}% \special{pa 389 560}% \special{pa 861 254}% \special{fp}% \special{pa 389 560}% \special{pa 396 525}% \special{fp}% \special{pa 389 560}% \special{pa 424 567}% \special{fp}% \graphtemp=-0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 0.14in \rlap{\kern 1.775in\lower\graphtemp\hbox to 0pt{\hss $\scriptscriptstyle v_{\<n}$\hss}}% \special{pa 1250 140}% \special{pa 2300 140}% \special{fp}% \special{pa 2300 140}% \special{pa 2275 165}% \special{fp}% \special{pa 2275 115}% \special{pa 2300 140}% \special{fp}% \graphtemp=0.5\baselineskip \advance\graphtemp by .6ex \advance\graphtemp by 0.39in \rlap{\kern 2.09in\lower\graphtemp\hbox to 0pt{\hss \raise1ex\hbox{$\scriptscriptstyle\,\ \quad j_{\<n}$}\hss}}% \special{pa 1854 558}% \special{pa 2326 221}% \special{fp}% \special{pa 1854 558}% \special{pa 1861 523}% \special{fp}% \special{pa 1854 558}% \special{pa 1889 565}% \special{fp}% \graphtemp=.6ex \advance\graphtemp by 1.264in \rlap{\kern 1.74in\lower\graphtemp\hbox to 0pt{\raise1.5ex\hbox{$\scriptscriptstyle \ f$}\hss}}% \special{pa 1740 780}% \special{pa 1740 1749}% \special{fp}% \special{pa 1740 1749}% \special{pa 1715 1724}% \special{fp}% \special{pa 1765 1724}% \special{pa 1740 1749}% \special{fp}% \graphtemp=.6ex \advance\graphtemp by 0.665in \rlap{\kern 2.44in\lower\graphtemp% \hbox to 0pt{$\scriptscriptstyle \ f_{\<n}$\hss}}% \special{pa 2440 260}% \special{pa 2440 1249}% \special{fp}% \special{pa 2440 1249}% \special{pa 2415 1224}% \special{fp}% \special{pa 2465 1224}% \special{pa 2440 1249}% \special{fp}% \graphtemp=.6ex \advance\graphtemp by 1.264in \rlap{\kern 0.275in\lower\graphtem \hbox to 0pt{\hss $\scriptscriptstyle g\ $}}% \special{pa 275 780}% \special{pa 275 1749}% \special{fp}% \special{pa 275 1749}% \special{pa 250 1724}% \special{fp}% \special{pa 300 1724}% \special{pa 275 1749}% \special{fp}% \special{pa 975 280}% \special{pa 975 590}% \special{fp}% \graphtemp=.6ex \advance\graphtemp by 0.97in \rlap{\kern 0.975in\lower\graphtemp\hbox to 0pt{\hss \raise2.5ex\hbox{$\scriptscriptstyle g_{\<n}^{\phantom{.}}\; $}}}% \special{pa 975 690}% \special{pa 975 1249}% \special{fp}% \special{pa 975 1249}% \special{pa 950 1224}% \special{fp}% \special{pa 1000 1224}% \special{pa 975 1249}% \special{fp}% \kern 2.58in }\vss}% \kern 2.088in } \stepcounter{sth} \renewcommand{\theequation}{\thesparag} \vspace{45pt} \begin{equation}\label{cube} \end{equation} \vspace{-95pt} \centerline{\quad\ \box\graph} \bigskip \noindent there is a composed isomorphism \begin{align*} {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}} j_{n*}\>{\mathcal F} &\iso {\mathbf R}\iGp{\mathscr U} u^*i_{n*}{\mathbf R} f_{\<\<n*}\>{\mathcal F} &&\bigl ({\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}(X_n)\bigr) \\ &\iso {\mathbf R}\iGp{\mathscr U} \>\>p_{n*}u_n^*\>{\mathbf R} f_{\<\<n*}\>{\mathcal F} &&(\textup{\Lref{L:closed})} \\ &\iso {\mathbf R}\iGp{\mathscr U}\>\> p_{n*}{\mathbf R} g_{n*}v_n^*{\mathcal F} && (\textup{\Lref{L:ordinary})}\\ &\iso {\mathbf R}\iGp{\mathscr U}\>{\mathbf R} g_{*}q_{n*}v_n^*{\mathcal F} \\ &\iso {\mathbf R}\iGp{\mathscr U}\>{\mathbf R} g_{*}v^*\<j_{n*}\>{\mathcal F} &&(\textup{\Lref{L:closed}),} \end{align*} which---the conscientious reader will verify---is just $\theta_{\!j_{\<n\<*}{\mathcal F}}'\>$. Thus $\theta_{\!j_{\<n\<*}{\mathcal F}}'\>$ is indeed an isomorphism. \smallskip (c) By definition ${\mathbf R}\iGp{\mathscr X}(\>\wDqc({\mathscr X}))\subset \D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$, and so by (a) and~(b) it's enough to see, as follows, that the natural map ${\mathbf R}\iGp{\mathscr X}{\mathcal E}\to{\mathcal E}$ induces isomorphisms of the source and target of both $\psi_{\mathcal E}$ and $\theta_{\!{\mathcal E}}'\>$. \Pref{P:f* and Gamma}(c) gives the isomorphism ${\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*{\mathbf R}\iGp{\mathscr X}{\mathcal E} \iso {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal E}$, as well as the second of the following isomorphisms, the first and third of which follow from \Cref{C:f* and Gamma}(d): $$ {\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*{\mathbf R}\iGp{\mathscr X}\>{\mathcal E}\cong {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*{\mathbf R}\iGp{\mathscr X}\>{\mathcal E}\cong {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*{\mathcal E}\cong {\mathbf R}\iGp{\mathscr U}\> {\mathbf R} g_*v^*{\mathcal E}. $$ Likewise, there are natural isomorphisms $$ {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}{\mathcal E}\cong {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}{\mathcal E}\cong {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}{\mathcal E}.\vspace{-3.7ex} $$ \end{proof} \goodbreak \smallskip Notation and assumptions stay as in \Pref{uf=gv}(a). Assume that $f$ and~$g$ satisfy the hypotheses of \Tref{T:qct-duality}, so that the functor \mbox{${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})\to{\mathbf D}({\mathscr Y})$} has a right adjoint~$f_{\mathrm t}^\times\<$, and similarly for~$g$. Recall from \Cref{C:identities}(b) that there is a natural isomorphism $g_{\mathrm t}^{\<\times} {\mathbf R}\iGp{\mathscr U}\isog_{\mathrm t}^{\<\times}$. \begin{defi}\label{D:basechange} \hskip-1pt With conditions as in \Pref{uf=gv}(b), the \kern-.5pt\emph{base-change~map} $$\index{base-change map} \beta_{\mathcal F}\colon{\mathbf R}\iGp{\mathscr V}\> v^*\<f_{\mathrm t}^\times\<\<{\mathcal F} \tog_{\mathrm t}^{\<\times}\>{\mathbf R}\iGp{\mathscr U} u^*\<{\mathcal F} \qquad\bigl({\mathcal F}\in{\mathbf D}({\mathscr Y})\bigr) $$ is defined to be the map adjoint to the natural composition $$ {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\>\>v^*\<\<f_{\mathrm t}^\times\<\<{\mathcal F} \underset{\psi}{\iso} {\mathbf R}\iGp{\mathscr U}\>{\mathbf R} g_* v^*\<\<f_{\mathrm t}^\times\<\<{\mathcal F} \underset{\theta'{}^{-1}}{\iso} {\mathbf R}\iGp{\mathscr U} u^*\>{\mathbf R f_{\!*}}f_{\mathrm t}^\times\<\<{\mathcal F} \to {\mathbf R}\iGp{\mathscr U} u^*\<{\mathcal F} $$ where $\psi\!:=\psi_{\<f_{\mathrm t}^\times\!{\mathcal F}}$ and $\theta'\!:= \theta'_{\!f_{\mathrm t}^\times\!{\mathcal F}}\>\>$. In particular, if $u$ (hence $v$) is \emph{adic} then $$ \beta_{\mathcal F}\colon v^*\<\<f_{\mathrm t}^\times\<\<{\mathcal F}\to g_{\mathrm t}^{\<\times} u^*\<{\mathcal F} $$ is the map adjoint to the natural composition $$ {\mathbf R} g_* v^*\<\<f_{\mathrm t}^\times\<\<{\mathcal F} \underset{\theta^{-1}}{\iso} u^*\>{\mathbf R f_{\!*}}f_{\mathrm t}^\times\<\<{\mathcal F}\to u^*\<{\mathcal F} $$ where $\theta\!:= \theta_{\!f_{\mathrm t}^\times\!{\mathcal F}}\>\>$. \end{defi} \emph{Notation.} For a pseudo\kern.6pt-proper (hence separated) map~$f$ (see \S\ref{maptypes}), we write~$f^!$\index{ $\iG$@$f^{{}^{\>\ldots}}$ (right adjoint of ${\mathbf R} f_{\<\<*}\cdots$)!$\mathstrut f^!\<$} instead of $f_{\mathrm t}^\times\<$. \pagebreak[3] \smallskip \begin{thm}\label{T:basechange}\index{base-change isomorphism} Let\/ ${\mathscr X},$ ${\mathscr Y}$ and\/ ${\mathscr U}$ be noetherian formal schemes, let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map, and let\/ $u\colon {\mathscr U}\to{\mathscr Y}$ be flat, so that in any fiber square $$ \begin{CD} {\mathscr V}@>v>>{\mathscr X} \\ @VgVV @VVfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ the formal scheme\/ ${\mathscr V}$ is noetherian, $g$ is pseudo\kern.6pt-proper, and $v$ is flat \textup(\Pref{P:basechange}\kern.5pt\textup). Then for all\/ ${\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\!:=\wDqc({\mathscr Y})\cap{\mathbf D}^+({\mathscr Y})$ the base-change map\/ $\beta_{\mathcal F}$ is an \emph{isomorphism} $$ \beta_{\mathcal F}\colon{\mathbf R}\iGp{\mathscr V}\>v^*f^!{\mathcal F} \iso g^!\>\>{\mathbf R}\iGp{\mathscr U} u^*{\mathcal F}\ (\cong g^!\<u^*\<{\mathcal F}\>). $$ \end{thm} \begin{small} \emph{Remark.} In \cite[p.\,233, Example 6.5]{N1} Neeman\index{Neeman, Amnon} gives an example where $f$ is a finite\- map of ordinary schemes, $u$ is an open immersion, ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^-({\mathscr Y})$, and $\beta_{\mathcal F}$ is \emph{not} an isomorphism. \end{small} \begin{proof} Recall diagram \eqref{cube}, in which, ${\mathscr I}$ and ${\mathscr J}\supset{\mathscr I}{\mathcal O}_{\mathscr X}$ being defining ideals of ${\mathscr Y}$ and ${\mathscr X}$ respectively, $Y_n$ is the scheme $({\mathscr Y},{\mathcal O}_{\mathscr Y}/{\mathscr I}^n)$ and $X_n$ is the scheme $({\mathscr X},{\mathcal O}_{\mathscr X}/{\mathscr J}^n)$. Let ${\mathcal K}\supset{\mathscr I}{\mathcal O}_{\mathscr U}$ be a defining ideal of~${\mathscr U}$, let ${\mathcal L}\!:={\mathscr J}{\mathcal O}_{\mathscr V}+{\mathcal K}{\mathcal O}_{\mathscr V}\>$, a defining ideal of~${\mathscr V}$, let $V_n\ (n>0)$ be the scheme $({\mathscr V},{\mathcal O}_{\mathscr V}/{\mathcal L}^n)$, and let $ l_n\colon V_n\hookrightarrow{\mathscr V} $ be the canonical closed immersion. Then by \Eref{ft-example}(4), $$ l_{n*}l_n^!{\mathcal G}=l_{n*}l_n^\natural{\mathcal G}={\mathbf R}\cH{om}({\mathcal O}_{\mathscr V}/{\mathcal L}^n\<,{\mathcal G}\>)=:\,{\boldsymbol h}_n({\mathcal G}\>) \qquad \bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr V})\bigr). $$ So in view of the natural isomorphism ${\mathbf R}\iGp{\mathscr V}g^! u^*{\mathcal F}\<\iso\<g^! u^*{\mathcal F}$ (\Pref{Gamma'(qc)}(a)), \Lref{Gam as holim} shows it sufficient to prove that the maps $$ {\boldsymbol h}_n(\beta_{\mathcal F}\>)\colon l_{n*}l_n^!{\mathbf R}\iGp{\mathscr V} v^*\<\<f^!{\mathcal F}\to l_{n*}l_n^!g^!u^*{\mathcal F} \qquad(n>0) $$ are all isomorphisms. \pagebreak[3] Moreover, the closed immersion $l_n$ factors uniquely as $$ V_n\xrightarrow{r_n\>\>} X_n\times_{\mathscr Y}{\mathscr U}\xrightarrow{q_n\>\>}{\mathscr V}, $$so we can replace $l_n^!$ by~$r_n^!q_n^!$ (\Tref{T:qct-duality}(b)). Thus \emph{it will suffice to prove that the maps $$ q_n^!(\beta_{\mathcal F}\>)\colon q_n^!{\mathbf R}\iGp{\mathscr V} v^*\<\<f^!{\mathcal F}\to q_n^!g^!u^*{\mathcal F} \qquad(n>0) $$ are all isomorphisms.} \smallskip In the cube \eqref{cube}, the front, top, rear, and bottom faces are fiber squares, denoted, respectively, by $\square$, $\square_{\textup t}\>$, $\square_{\textup r}\>$ and $\square_{\textup b}\>$; and we have the ``composed" fiber square $\square_{\textup c}\>$: $$ \begin{CD} X_n\times_{\mathscr Y}{\mathscr U} @>v_n>> X_n \\ @Vp_ng_nV \!=\,gq_n V @Vi_nf_nV \!=\,fj_n V \\ {\mathscr U} @>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}> {\mathscr Y} \end{CD} $$ The proper map $f_n$ and the closed immersions~$i_n$ and~$j_n$ are all of pseudo-finite type. Also, it follows from \Pref{P:basechange}(b) that in addition to~$u$, the maps $u$, $u_n$, $v$ and~$v_n$ are all flat. So corresponding to the fibre squares~$\square_\bullet$ we have base-change maps~$\beta_\bullet\>$. \goodbreak Consider the following diagram of functorial maps where, to save space, we set $\blacktriangle\!:= X_n\!\times_{\mathscr Y}\<{\mathscr U}$ and $\blacktriangledown\!:= Y_n\!\times_{\mathscr Y}\<{\mathscr U}$. $$ \minCDarrowwidth=15pt \begin{CD} q_n^!{\mathbf R}\iGp{\mathscr V} v^*\<\<f^!\< @<\;\beta_{\textup t}\; << \<{\mathbf R}\iGp\blacktriangle v_n^*j_n^!f^! @<\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{n}}$}\vskip-8.5pt}<< {\mathbf R}\iGp\blacktriangle v_n^*(fj_n)^! @= {\mathbf R}\iGp\blacktriangle v_n^*(i_nf_n)^!\< @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{n}}$}\vskip-8.5pt}>> \<{\mathbf R}\iGp\blacktriangle v_n^*f_n^!i_n^! \\ @V q_n^!(\beta) VV @. @V\beta_{\textup c} VV @. @VV\beta_{\textup r} V \\ q_n^!g^!u^* @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{n}}$}\vskip-8.5pt}>> (gq_n)^!u^* @= (p_ng_n)^!u^* @<\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{n}}$}\vskip-8.5pt}<< g_n^!p_n^!u^* @<<\hbox to0pt{\hss$\scriptstyle g_n^!(\beta_{\textup b})$\hss}< g_n^!{\mathbf R}\iGp{\blacktriangledown} u_n^*i_n^! \end{CD} $$ As above, we want to see that $q_n^!(\beta)$ is an isomorphism (in the category of functors from $\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})$ to ${\mathbf D}(X_n\!\<\times_{\mathscr Y}\<\<{\mathscr U})$). For that the following assertions clearly suffice:\vspace{2pt} (a) The preceding diagram commutes. (b) The base-change maps $\beta_{\textup t}$ and $\beta_{\textup b}$ are isomorphisms. (c) The base-change map $\beta_{\textup r}$ is an isomorphism. \smallskip\noindent Assertion (a) results from part (b) of the transitivity lemma~\ref{L:trans} below. Since $i_n$ and $j_n$ are closed immersions, assertion~(b) results from \Lref{L:immbc}, which is just \Tref{T:basechange} for the case when $f$ is a closed immersion. Since $f$~is pseudo\kern.6pt-proper therefore $f_n$ is proper, and assertion~(c) is essentially the case of \Tref{T:basechange}---established in \Lref{L:ordbc}---when ${\mathscr X}$ and ${\mathscr Y}$ are ordinary schemes. Thus these three Lemmas will complete the proof of \Tref{T:basechange}. \end{proof} \begin{parag}\label{S:trans} We will need some ``transitivity" properties\index{transitivity} of the maps $\theta_{\!{\mathcal E}}'$ and $\beta_{\mathcal F}$ relative to horizontal and vertical composition of fiber squares of noetherian formal schemes, i.e., diagrams of the form \begin{subequations}\label{E:trans} \begin{equation}\label{E:transh} \begin{CD} {\mathscr V}@>v_2>> {\mathscr V}_1 @>v_1>>{\mathscr X} \\ @VgVV @Vg_1VV @VVfV \\ {\mathscr U} @>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle u_2$}\vss}> {\mathscr U}_1 @>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle u_1$}\vss}> {\mathscr Y} \end{CD} \end{equation} \vspace{10pt} \begin{equation}\label{E:transv} \begin{CD} {\mathscr V} @>v>> {\mathscr X} \\ @Vg_2 VV @VVf_2V \\ {\mathscr W} @>w>>{\mathscr Z} \\ @Vg_1 VV @VVf_1V \\ {\mathscr U} @>>\vbox to 0pt{\vskip-1.3ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} \end{equation} \end{subequations} where all squares are fiber squares, and the maps $u$, $u_i\>$, $v$, $v_i\>$, and~$w$ are all flat. As we will be dealing with several fiber squares simultaneously we will indicate the square with which, for instance, the map $\theta_{\mathcal G}$ in \Pref{uf=gv} is associated, by writing $\theta_{\<\<f\<,u}({\mathcal G})$ instead. The transitivity properties\index{transitivity} begin with: \begin{slem}\label{L:transtheta} Coming out of the fiber square diagrams\/~\eqref{E:transh} and\/ ~\eqref{E:transv}\textup, the following natural diagrams commute for all ${\mathcal G}\in{\mathbf D}({\mathscr X})\colon$ $$ \begin{CD} (u_1u_2)^*\>{\mathbf R f_{\!*}}{\mathcal G} @.\!\!\overset{\theta_{\<\<f\<,\>u_{\mkern-1.5mu 1}^{}\!\<u_{\mkern-1.5mu 2}^{}}({\mathcal G})}{\Rarrow{158pt}}@. {\mathbf R} g_*(v_1v_2)^*{\mathcal G}\\ @V\simeq VV @. @VV\simeq V \\ u_2^*u_1^*\>{\mathbf R f_{\!*}}{\mathcal G} @>>u_2^*(\theta_{\<\<f\<,\>u_{\<1}^{}}({\mathcal G}))> u_{2}^*\>{\mathbf R} g_{1\<*}^{}v_1^*{\mathcal G} @>>\theta_{\<g_{\<1}^{}\<,\>u_{\<2}^{}}(v_1^*{\mathcal G})> {\mathbf R} g_*v_2^*v_1^*{\mathcal G} \end{CD} $$ \vspace{10pt} $$ \begin{CD} u^*\>{\mathbf R} (f_1f_2)_*{\mathcal G} @.\quad\,\overset{\theta_{\<\<f_1\<f_2,\>u}({\mathcal G})}{\Rarrow{189pt}}\ @. {\mathbf R} (g_1g_2)_{\<*}\>v^*{\mathcal G}\\ @V\simeq VV @. @VV\simeq V \\ u^*\>{\mathbf R} f_{1\<*}\>{\mathbf R} f_{\<2*}^{}{\mathcal G} @>>\theta_{\<\<f_{1}\<,\>u}({\mathbf R} f_{\<2*}^{}{\mathcal G})> {\mathbf R} g_{1\<*}^{}\>w^*\>{\mathbf R} f_{\<2*}^{}{\mathcal G} @>>{\mathbf R} g_{1\<*}^{}(\theta_{\<\<f_2,\>w}({\mathcal G}))> {\mathbf R} g_{1\<*}^{}\>{\mathbf R} g_{2*}^{}v^*{\mathcal G} \end{CD} $$ \end{slem} \begin{proof} This is a formal exercise, based on adjointness of $u^*$ and ${\mathbf R} u_*\>$, etc. Details are left to the reader. \end{proof} \begin{slem}\label{L:trans}\index{transitivity} \textup{(a)} In the fiber square diagram\/~\eqref{E:transh}\vspace{.6pt} \(with\/ $u_1\>,$ $v_1\>,$ $u_2$~and $v_2$ flat\), let\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y})$ be such that the maps\/ $\theta_{\<\<1}'\!:=\theta'_{\<\<f\<,\>u_1^{}}(f_{\mathrm t}^\times\<{\mathcal F}\>),$ \mbox{$\theta_{\<2}'\!:= \theta'_{\<g_1,u_2^{}}((g_1^{}\<)_{\textup t}^{\!\times}\<u_1^*{\mathcal F}\>)$} and\/ $\theta_{\<2}'{}\!'\!:= \theta'_{\<g_1,u_2^{}}({\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*f_{\mathrm t}^\times\<{\mathcal F}\>)$ of \Pref{uf=gv} are isomorphisms. Then the map\/ $\theta'\!:=\theta'_{\<\<f\<,\>u_1^{}\<u_2^{}}(f_{\mathrm t}^\times\<{\mathcal F}\>)$ is an isomorphism, so the base-change maps\/ $\beta_1\!:=\beta_{\<\<f\<,\>u_1^{}}({\mathcal F}\>),$ $\beta_2\!:=\beta_{g_1^{},\>u_2^{}}(u_1^*{\mathcal F}\>)$ and\/ $\beta\!:=\beta_{\<\<f\<,\>u_1^{}\<u_2^{}}({\mathcal F}\>)$ can all be defined as in \Dref{D:basechange};\vspace{1pt} and the following natural diagram, all of whose uparrows are isomorphisms, commutes$\>:$ $$ \begin{CD} {\mathbf R} \iGp{\mathscr V}(v_1v_2)^*\<f_{\mathrm t}^\times\<{\mathcal F} @.\hskip-15pt\overset{\beta}{\Rarrow{180pt}} @. g_{\mathrm t}^{\<\times}\<{\mathbf R}\iGp{\mathscr U}(u_1u_2)^*{\mathcal F}\\ @A\simeq AA @. @AA\simeq A \\ {\mathbf R} \iGp{\mathscr V} v_2^*v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} @. {\mathbf R}\iGp{\mathscr V} \>v_2^*(g_1^{}\<)_{\textup t}^{\!\times} u_1^*{\mathcal F} @>\beta_2>> g_{\mathrm t}^{\<\times}\<{\mathbf R}\iGp{\mathscr U} u_2^*u_1^*{\mathcal F}\\ @A\simeq A\textup{\ref{P:f* and Gamma}(c)} A @A\simeq A\textup{\ref{C:identities}(b)}A @A\textup{\ref{P:f* and Gamma}(c)}A\simeq A \\ {\mathbf R}\iGp{\mathscr V} \>v_2^*\>{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*f_{\mathrm t}^\times\<{\mathcal F} @>> {\mathbf R}\iGp{\mathscr V} \>v_2^*(\beta_1)> {\mathbf R}\iGp{\mathscr V} \>v_2^*(g_1^{}\<)_{\textup t}^{\!\times} {\mathbf R}\iGp{{\mathscr U}_1}u_1^*{\mathcal F} @>>\beta_2> g_{\mathrm t}^{\<\times}\<{\mathbf R}\iGp{\mathscr U} u_2^*{\mathbf R}\iGp{{\mathscr U}_1}u_1^*{\mathcal F} \end{CD} $$ \textup{(b)} In the fiber square diagram\/~\eqref{E:transv}---where\/ $u,$ $v$ and~$w$ are assumed flat---set\/ \mbox{$f\!:= f_{\<1}f_2$} and\/ $g\!:= g_1g_2$. Let\/ ${\mathcal F}\in{\mathbf D}({\mathscr Y})$ be such that the maps\/ $\theta_{\<\<1}'\!:=\theta'_{\<\<f_{\<1}\<,\>u} ((f_{\<1}^{}\<)_{\textup t}^{\!\times}{\mathcal F}\>),$ \mbox{$\theta_{\<2}'\!:= \theta'_{\<\<f_2,w}(f_{\mathrm t}^\times\<\<{\mathcal F}\>)$} and\/ $\theta'\!:=\theta'_{\<\<f\<,\>u}(f_{\mathrm t}^\times\<\<{\mathcal F}\>)$ of \Pref{uf=gv} are isomorphisms, so that the base-change maps\/ $\beta_1\!:=\beta_{\<\<f_{\<1},\>u}({\mathcal F}\>),$ $\beta_2\!:=\beta_{\<\<f_2,w}((f_{\<1}^{}\<)_{\textup t}^{\!\times}{\mathcal F}\>)$ and\/ $\beta\!:=\beta_{\<\<f,\>u}({\mathcal F}\>)$ are all defined. Then the following diagram, whose two uparrows are isomorphisms, commutes\/\textup{:} $$ \begin{CD} {\mathbf R}\iGp{\mathscr V} v^*\< f_{\mathrm t}^\times\<{\mathcal F} @. \overset{\beta}{\hskip9.8pt\Rarrow{167pt}} @. g_{\mathrm t}^{\<\times}{\mathbf R}\iGp{\mathscr U} u^*{\mathcal F} \\ @A\simeq AA @. @AA\simeq A \\ {\mathbf R}\iGp{\mathscr V} v^* (f_2^{})\<_{\textup t}^{\!\times}\<(f_{\<1}^{}\<)\<_{\textup t}^{\<\times}{\mathcal F} @>>\beta_2> (g_2^{})\<_{\textup t}^{\<\times}{\mathbf R}\iGp{\mathscr W} \>w^*\<(f_{\<1}^{}\<)\<_{\textup t}^{\<\times}{\mathcal F} @>> (g_2^{})\<_{\textup t}^{\<\times}\<(\beta_1) > (g_2^{})\<_{\textup t}^{\<\times}\<(g_1^{})\<_{\textup t}^{\<\times} {\mathbf R}\iGp{\mathscr U} u^*\<{\mathcal F} \end{CD} $$ \end{slem} \pagebreak[3] \deff^{\<\times}{f^{\<\times}} \begin{proof} (a) The map $$ \gamma\!:= {\mathbf R}\iGp{\mathscr U} u_2^*(\theta_{\<f,u_{\<1}^{}}\!(f_{\mathrm t}^\times\<{\mathcal F}\>))\colon {\mathbf R}\iGp{\mathscr U} u_2^*u_1^*{\mathbf R f_{\!*}} f_{\mathrm t}^\times\<{\mathcal F}\longrightarrow {\mathbf R}\iGp{\mathscr U} u_2^*{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} $$ is isomorphic, by \Pref{P:f* and Gamma}(c), to $$ {\mathbf R}\iGp{\mathscr U} u_2^*(\theta_{\<\<1}')\colon {\mathbf R}\iGp{\mathscr U} u_2^*{\mathbf R}\iGp{{\mathscr U}_1}\<u_1^*{\mathbf R f_{\!*}} f_{\mathrm t}^\times\<{\mathcal F}\longrightarrow {\mathbf R}\iGp{\mathscr U} u_2^*{\mathbf R}\iGp{{\mathscr U}_1}\<{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}\<, $$ and so is an isomorphism (since $\theta_{\<\<1}'$ is). The map $$ \theta_{\<g_1^{}\<\<,u_2^{}}'(v_1^*f_{\mathrm t}^\times\<{\mathcal F}\>)\colon {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \to {\mathbf R} \iGp{\mathscr U} \>{\mathbf R} g_* v_2^*v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} $$ is also an isomorphism, as it is isomorphic to $$ \theta_{\<2}'{}\!'\colon {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \to {\mathbf R} \iGp{\mathscr U} \>{\mathbf R} g_* v_2^*{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}\<, $$ because the natural map ${\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \to {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}$ is the composed isomorphism \begin{multline*} \smash{{\mathbf R} \iGp {\mathscr U}\< u_2^*\>{\mathbf R} g_{1\<*}^{}{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \xrightarrow[\!\!{\mathbf R} \iGp {\mathscr U} u_2^* \psi_{\! f_{\<\<\mathrm t}^{\!\times}\!{\mathcal F}}\!\!]{\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}} {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R}\iGp{{\mathscr U}_1}\<{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}}{}_{\displaystyle\mathstrut} \\ \xrightarrow[\!\textup{\ref{P:f* and Gamma}(c)}\!]{\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}} {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}{} \end{multline*} \smallskip \noindent(see \Pref{uf=gv}(a)); and because ${\mathbf R} \iGp{\mathscr U} \>{\mathbf R} g_*v_2^*{\mathbf R}\iGp{{\mathscr V}_1}\<v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \to {\mathbf R} \iGp{\mathscr U} \>{\mathbf R} g_* v_2^*v_1^*\<f_{\mathrm t}^\times\<{\mathcal F}$ is one of the maps in the commutative diagram (B) below, all of whose other maps are isomorphisms. Thus in the next diagram, whose commutativity results easily from that of the first diagram in \Lref{L:transtheta}, all the maps other than $\theta'$ are isomorphisms, whence so is $\theta'$. $$ \begin{CD} {\mathbf R}\iGp{\mathscr U} \>{\mathbf R} g_* (v_1v_2)^{\<*}\<f_{\mathrm t}^\times\<{\mathcal F} @<\mkern48mu\theta'\mkern48mu<< {\mathbf R} \iGp {\mathscr U} u_2^*u_1^*{\mathbf R f_{\!*}} f_{\mathrm t}^\times\<{\mathcal F} \\ @V\simeq V\hbox{\hskip67.5pt{\footnotesize(A)}}V @V\simeq V\gamma V \\ {\mathbf R} \iGp{\mathscr U} \>{\mathbf R} g_* v_2^*v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} @<\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}<\theta_{\<\<g_1^{}\<\<,u_2^{}}'\<\<(\<v_1^*f_{\mathrm t}^\times\!{\mathcal F} \>)< {\mathbf R} \iGp {\mathscr U} u_2^*\>{\mathbf R} g_{1\<*}^{}v_1^*\<f_{\mathrm t}^\times\<{\mathcal F} \end{CD} $$ Now it suffices to show that the diagram which is \emph{adjoint} to the diagram in~(a) without its southeast (bottom right) corner, commutes. That adjoint diagram is the outer border of the following one, where, to reduce clutter, we omit all occurrences of the symbols~${\mathbf R}$ and ${\mathcal F}$, write $f^{\<\times}$ for $f_{\mathrm t}^\times\<$, etc., and leave some obvious maps unlabeled: $$ \minCDarrowwidth=21.5pt \begin{CD} g_*\iGp{\mathscr V}\> (v_1v_2)^{\<*}\<\<f^{\<\times} @>\psi>> \iGp{\mathscr U} \>g_* (v_1v_2)^*\<\<f^{\<\times} @>\mkern33mu\theta'{}^{-1}\mkern33mu>> \iGp {\mathscr U} u_2^*u_1^*f_{\<\<*} f^{\<\times} @>>> { \iGp {\mathscr U} u_2^*u_1^* } \\ @A AA @A A\hbox{\hskip54.5pt{\footnotesize(A)}}A @A \hbox to 0pt{$\scriptstyle\hskip7pt \gamma^{-1} \hss$} A \hskip76pt\raise4.5ex \UnderElement{\hbox{\footnotesize(C)\hskip22pt}}{\|}{50pt}{} A \\ g_*\iGp{\mathscr V}\> v_2^*v_1^*\<f^{\<\times}\hbox to0pt{\hskip17pt {\footnotesize(B)}\hss} @. \iGp{\mathscr U} \>g_* v_2^*v_1^*\<f^{\<\times} @<<\theta_{\<g_1^{}\<\<,u_2^{}}'(v_1^*f^{\<\times}\<)< \iGp {\mathscr U} u_2^*\>g_{1\<*}^{\phantom{.}}v_1^*\<f^{\<\times} \\ \vspace{-22pt} \\ @A\simeq A\textup{\ref{P:f* and Gamma}(c)}A @AAA @AAA \\ g_*\iGp{\mathscr V}\> v_2^*\iGp{{\mathscr V}_1}\<\<v_1^*\<f^{\<\times} @>>\psi> \iGp{\mathscr U} \>g_* v_2^*\iGp{{\mathscr V}_1}\<\<v_1^*\<f^{\<\times} @>>\mkern31mu\theta_{\<2}'\!{\mkern-1.5mu}_{\phantom A}'\!\!^{-1} \mkern31mu > \iGp {\mathscr U} u_2^*\>g_{1\<*}^{\phantom{.}}\iGp{{\mathscr V}_1}\<\<v_1^*\<f^{\<\times} \\ \vspace{-22pt} \\ @V\beta_1VV @V\beta_1VV @VV\beta_1V \\ g_*\iGp{\mathscr V}\> v_2^*g_1^{\<\times}\< u_1^* @>> \psi > \iGp{\mathscr U} \>g_* v_2^*g_1^{\<\times}\< u_1^* @>>\mkern32mu \theta'_{\<2}{}^{-1} \mkern32mu > \iGp {\mathscr U} u_2^*\>g_{1\<*}^{\phantom{.}}\>g_1^{\<\times}\< u_1^* @>>> \iGp {\mathscr U} u_2^*u_1^* \end{CD} $$ It suffices then that each one of the subrectangles commute. \pagebreak[2] For the three unlabeled subrectangles commutativity is clear. As before, commutativity of subrectangle~(A) follows from that of the first diagram in \Lref{L:transtheta}. Commutativity of~(B) is easily checked after composition with the natural map $\iGp{\mathscr U} \>g_* (v_1v_2)^*\<\<f^{\<\times}\to g_* (v_1v_2)^*\<\<f^{\<\times}\<\<$. (See the characterization of~$\psi$ in~\Pref{uf=gv}(a).) Commutativity of~(C) results from that of the following diagram: $$ \begin{CD} g_{1\<*}^{}\<v_1^*\<f^{\<\times} @= g_{1\<*}^{}\<v_1^*\<f^{\<\times} @< \theta_{\<\<f\<,\>u_1^{}} << u_1^*f_{\<\<*}f^{\<\times} @>>> u_1^* \\ @A A \hbox{\hskip33pt{\footnotesize(D)}} A @AAA @AAA @AAA \\ g_{1\<*}^{}\iGp{{\mathscr V}_1}\<\<v_1^*\<f^{\<\times} @> > \psi > \iGp{{\mathscr U}_1} \>g_{1\<*}^{}v_1^*\<f^{\<\times} @>> \theta_{\<1}'{}^{-1} > \iGp{{\mathscr U}_1}\<\<u_1^*f_{\<\<*} f^{\<\times} @>>> \iGp{{\mathscr U}_1}\<\<u_1^* \\ @V \beta_1 VV @. @. @V \hbox{\footnotesize (E)\hskip109pt} VV \\ g_{1\<*}g_1^{\<\times} u_1^* \hbox to 0pt{\hskip113.5pt\Rarrow{205pt}\hss} @. @. @. u_1^* \end{CD} $$ Here subrectangle~(D) commutes by the characterization of~$\psi$ in~\Pref{uf=gv}(a); and (E) commutes by the very definition of the base-change map~$\beta_1$. \smallskip (b) As in (a), we consider the \emph{adjoint} diagram, essentially the outer border of the following diagram (\ref{L:trans}.1). (Note: The map $\psi\colon\< g_{1\<*}^{}\iGp{\mathscr W} \>w^*\<\<f_{\<2*}^{}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times}\< \to\iGp{\mathscr U}\> g_{1\<*}^{} w^*\<\<f_{\<2*}^{}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} $\vspace{1pt} in the middle of diagram \ref{L:trans}.1 is defined because $f_{\<2*}^{}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} \!:= {\mathbf R} f_{2*}^{}(f_2^{})\<_{\textup t}^{\<\times} \<(f_1^{})\<_{\textup t}^{\<\times} \<{\mathcal F}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr Z})$, by \mbox{\Pref{Rf-*(qct)}.}) For diagram \ref{L:trans}.1, commutativity of subrectangle (B) (resp.~(D)) is given by the definition of~$\beta_2$ (resp.~$\beta_1$.) Commutativity of~(C) follows from that of the second diagram in \Lref{L:transtheta}. Commutativity of~(A) is left as an exercise. (It is helpful to compose with the natural map $\iGp{\mathscr U}\>g_{1\<*}^{}g_{2*}^{}v^*\< \<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} \to g_{1\<*}^{}g_{2*}^{}v^*\<\<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} $ and to use the characterization of $\psi$ in~\Pref{uf=gv}(a).) The rest is straightforward. \end{proof} $$ \minCDarrowwidth=18pt \begin{CD} g_*\iGp{\mathscr V}\> v^*\<\<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} @>\mkern30mu\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}\mkern30mu>> g_{1\<*}^{}g_{2*}^{}\iGp{\mathscr V} v^*\<\<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} @. \kern-17pt \overset{\text{via }\beta_2}{\Rarrow{77pt}} @. \underset{\UnderElement{}{\downarrow}{7.3ex}{}} {\hz{\hss\hskip-10pt$g_{1\<*}^{}g_{2*}^{}g_2^{\<\times}\! \iGp{\mathscr W}\> w^*\!f_{\<\<1}^{\<\<\times} \!$\hss}} \\ \vspace{-20pt} \\ @V\simeq VV @VV\psi V \\ g_*\iGp{\mathscr V}\> v^*\<\<f^{\<\times} @. g_{1\<*}^{}\iGp{\mathscr W} g_{2*}^{}v^*\<\<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} \hbox to0pt{\hskip46pt{\footnotesize(B)}\hss} \\ @V\psi V \hbox{\hskip54pt{\footnotesize{(A)}}} V @V \simeq V g_{1\<*}^{}(\theta_{\<2}'{}^{-1}) V \\ \iGp{\mathscr U} g_*v^*\<\<f^{\<\times} @. g_{1\<*}^{}\iGp{\mathscr W} \>w^*\<f_{\<2*^{}}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} @.\Rarrow{93.5pt} @. g_{1\<*}^{}\iGp{\mathscr W} \>w^*\<\<f_{\<\<1}^{\<\times} \\ @V\simeq VV @VV\psi V @. @| \\ \vspace{-21pt} \\ \underset{\UnderElement{\simeq}{\downarrow}{7.3ex} {\hbox{\hskip51.7pt{\footnotesize{(C)}}}}} {\iGp{\mathscr U}\>g_{1\<*}^{}g_{2*}^{}v^*\<\<f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} } @< \!\!\!\text{via }\theta_{\<\<f_2,w}(f^{\<\times})\!\!\! << \iGp{\mathscr U}\> g_{1\<*}^{} w^*\<\<f_{\<2*}^{}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} @>>> \iGp{\mathscr U}\> g_{1\<*}^{} w^*\<\<f_{\<\<1}^{\<\times} @<\psi<< \underset{\UnderElement{\hbox{\footnotesize{(D)}\hskip21pt}} {\downarrow}{7.3ex}{\!\<\<g_{1\<*}^{}\<(\beta_1\<)}} {g_{1\<*}^{}\iGp{\mathscr W} \>w^*\<\<f_{\<\<1}^{\<\times} } \\ \vspace{-20pt} \\ @. @VV\hbox to0pt{$\scriptstyle \theta'_{f_1\<,u}(f_{\<2*}^{}\<f^{\<\times}\<)^{-1}$\hss} V @VV \theta_{\<\<1}'{}^{\<-1} V \\ @. \iGp{\mathscr U}\> u^*\<\<f_{\<1*}^{}f_{\<2*}^{}f_{\<\<2}^{\<\times}\! f_{\<\<1}^{\<\times} @>>> \iGp{\mathscr U}\> u^*\<\<f_{\<1*}^{}f_{\<\<1}^{\<\times} \\ @. @VV \simeq V @VVV \\ \iGp{\mathscr U}\>g_*v^*\<\<f^{\<\times} @>> \mkern27mu\theta'{}^{-1}\mkern27mu> \iGp{\mathscr U}\> u^*\<\<f_{\!*}f^{\<\times} @>>> \iGp{\mathscr U} u^* @<<< g_{1\<*}^{}g_1^{\<\times}\<\< \iGp{\mathscr U} u^* \end{CD} $$ \bigskip \centerline{\bf(\ref{L:trans}.1)} \end{parag} \pagebreak[2] \begin{parag} This subsection, proving \Lref{L:immbc}, is independent of the preceding one. \begin{slem} \label{L:immbc} \Tref{T:basechange} holds when\/ $f$ is a closed immersion. \end{slem} \begin{proof} The natural isomorphisms ${\mathbf R}\iGp{\mathscr V}\>v^*\<\<f^!{\mathbf R}\iGp{\mathscr Y}{\mathcal F}\iso{\mathbf R}\iGp{\mathscr V}\>v^*\<\<f^!{\mathcal F}$ and $$ g^!u^*\>{\mathbf R}\iGp{\mathscr Y}{\mathcal F} \iso g^!{\mathbf R}\iGp{\mathscr U}\> u^*\>{\mathbf R}\iGp{\mathscr Y}{\mathcal F} \underset{\textup{\ref{P:f* and Gamma}(c)}}\iso g^!{\mathbf R}\iGp{\mathscr U} u^*{\mathcal F} \iso g^! u^*{\mathcal F} $$ (see \Cref{C:identities}(b)) let us replace~${\mathcal F}$ by ${\mathbf R}\iGp{\mathscr Y}{\mathcal F}\<$, i.e., we may assume ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr Y})$. Recall from \Eref{ft-example}(4) that ${\mathbf R f_{\!*}}=f_{\!*}\colon{\mathbf D}({\mathscr X})\to{\mathbf D}({\mathscr Y})$ has a right adjoint~$f^\natural$ such that $f^\natural(\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr Y}))\subset\D_{\mkern-1.5mu\mathrm{qct}}^+({\mathscr X})$; and that there is a natural isomorphism $$ j^{\>{\mathscr X}}_{\mathcal G}\colon {\mathbf R}\iGp{\mathscr X} f^\natural{\mathcal G}\iso\mathbf1^{\<!}\<f^\natural{\mathcal G}\cong f^!{\mathcal G} \qquad\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm {qc}}^+({\mathscr Y})\bigr). $$ The canonical map $f_{\!*} f^!\to\mathbf 1$ is the natural composition $$ f_{\!*} f^!\underset{(j^{\mathscr X})^{-1}}\iso f_{\!*}{\mathbf R}\iGp{\mathscr X} f^\natural\to f_{\!*} f^\natural \to \mathbf 1. $$ Similar remarks hold for $g$---also a closed immersion \cite[p.\,442, (10.14.5)(ii)]{GD}. As in the proof of \Lref{L:closed}, the map $ \theta_{\<{\mathcal E}}\colon u^*\<\<f_{\!*}\>{\mathcal E}\iso g_*v^*{\mathcal E} $ of \Pref{uf=gv} is an isomorphism for all ${\mathcal E}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$. (Recall \Lref{C:vec-c is qc}.) This being so, the base-change map~ $\beta_{\mathcal F}$ is easily seen to factor naturally as $$ {\mathbf R}\iGp{\mathscr V} v^*\<\<f^!{\mathcal F} \to g^!g_*{\mathbf R}\iGp{\mathscr V} v^*\<\< f^!\<{\mathcal F} \to g^! g_* v^*\<\<f^!\<{\mathcal F} \underset{\theta^{-1}}\iso g^! u^*\<f_{\!*}f^!\<{\mathcal F} \to g^! u^*\<{\mathcal F}. $$ Also, we can define the functorial map $\beta_{\<{\mathcal C}}^\natural$ to be the natural composition $$ v^*\!f^\natural{\mathcal C}\to g^\natural g_*v^*\!f^\natural{\mathcal C}\underset{\theta^{-1}}\iso g^\natural u^*\<f_{\!*}f^\natural{\mathcal C}\to g^\natural u^*\<{\mathcal C} \qquad\bigl({\mathcal C}\in\D_{\mkern-1.5mu\mathrm {qc}}({\mathscr Y})\bigr). $$ The maps $\beta_{\<{\mathcal F}}^\natural$ and~$\beta_{\mathcal F}$ are related by commutativity of the following diagram, in which ${\mathscr J}$ is an ideal of definition of ${\mathscr Y}$ (so that ${\mathscr J}{\mathcal O}_{\mathscr X}$ is an ideal of definition of~${\mathscr X}$): $$ \begin{CD} {\mathbf R}\iGp{\mathscr V} v^*{\mathbf R}\iGp{\mathscr X} f^\natural @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>{\textup{\ref{P:f* and Gamma}(b)}}> {\mathbf R}\iGp{\mathscr V}\>{\mathbf R}\iG{{\mathscr J}{\mathcal O}_{\mathscr V}} v^*\<\< f^\natural @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>> {\mathbf R}\iGp{\mathscr V} v^*\<\<f^\natural @>{\mathbf R}\iGp{\mathscr V}(\beta^\natural)>> {\mathbf R}\iGp{\mathscr V} \>g^\natural\> u^* \\ \vspace{-23pt}\\ @V{\mathbf R}\iGp{\mathscr V} v^*(j^{\>{\mathscr X}}) V\simeq V @. @. @V\simeq V j^{\mathscr V} V \\ {\mathbf R}\iGp{\mathscr V} v^*\<\< f^! @.{} @. \mkern-189mu\underset{\beta}{\Rarrow{226pt}} @. g^!u^* \end{CD} $$ (For the unlabeled isomorphism, see the beginning of the proof of \Pref{uf=gv}.) Since ${\mathbf R}\iGp{\mathscr V}$ is right-adjoint to the inclusion $\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr V})\hookrightarrow{\mathbf D}({\mathscr V})$ (\Pref{Gamma'(qc)}), we can verify this commutativity after composing with $g^!u^*\iso {\mathbf R}\iGp{\mathscr V} \>g^\natural u^*\to g^\natural u^*\<\<$, at which point the verification is straightforward. Thus to prove \Lref{L:immbc} we need only show that $\beta_{\mathcal F}^\natural$ is an isomorphism,\vspace{-.6pt} i.e.\ (since $g$ is a closed immersion), that $g_*(\beta_{\mathcal F}^\natural)$ is an isomorphism.\vspace{1.6pt} For that purpose, consider the unique functorial map $$ \sigma=\sigma({\mathcal E},\>{\mathcal G})\colon u^*{\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}({\mathcal E},\>{\mathcal G})\to {\mathbf R}\cH{om}^{\bullet}_{\mathscr U}(u^*{\mathcal E},u^*\<{\mathcal G}) \quad\ \: \bigl({\mathcal E}\in\D_{\mkern-1.5mu\mathrm c}^-({\mathscr Y}),\ {\mathcal G}\in{\mathbf D}^+({\mathscr Y})\bigr) $$ which for bounded-below injective complexes~${\mathcal G}$ is the natural composition $$ u^*{\mathbf R}\cH{om}^{\bullet}_{\mathscr Y}({\mathcal E},\>{\mathcal G})\cong u^*\cH{om}^{\bullet}_{\mathscr Y}({\mathcal E},\>{\mathcal G})\to \cH{om}^{\bullet}_{\mathscr U}(u^*{\mathcal E},u^*{\mathcal G})\to{\mathbf R}\cH{om}^{\bullet}_{\mathscr U}(u^*{\mathcal E},u^*{\mathcal G}). $$ This map is an \emph{isomorphism}. Indeed, it commutes with localization, so we need only check for affine~${\mathscr Y}$, and then, since every coherent ${\mathcal O}_{\mathscr Y}$-module is a homomorphic image of a finite-rank free one (\cite[p.\,427, (10.10.2)]{GD}), a standard way-out argument reduces the problem to the trivial case ${\mathcal E}={\mathcal O}_{\mathscr Y}$. Take ${\mathcal E}\!:= f_{\!*}{\mathcal O}_{\mathscr X}={}$(say)\:${\mathcal O}_{\mathscr Y}/{\mathscr I}$. The source and target of~$\sigma({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)$ are \begin{gather*} u^*{\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)= u^*\<\<f_{\!*}f^\natural{\mathcal F} \cong g_*v^*\!f^\natural\<{\mathcal F}, \\ {\mathbf R}\cH{om}^{\bullet}(u^*({\mathcal O}_{\mathscr Y}/{\mathscr I}),\>u^*{\mathcal F}\>)= g_*g^\natural u^*\<\<{\mathcal F} . \end{gather*} Let ${\mathcal K}$ be a K-injective ${\mathcal O}_{\mathscr U}$-complex quasi-isomorphic to $u^*\<{\mathcal F}\<$. Since the complexes $u^*\cH{om}^{\bullet}_{\mathscr Y}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)$ and $\cH{om}^{\bullet}_{\mathscr U}(u^*{\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal K}\>)\cong{\mathbf R}\cH{om}^{\bullet}_{\mathscr U}(u^*{\mathcal O}_{\mathscr Y}/{\mathscr I},u^*{\mathcal F}\>)$ are both annihilated\- by~${\mathscr I}{\mathcal O}_{\mathscr U}$, we see that the isomorphism $\sigma({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)$ is isomorphic to a map of the form $g_*(\varsigma)$ where $ \varsigma\colon v^*\<\<f^\natural{\mathcal F}\to g^\natural u^*\<\<{\mathcal F} $ is a map in~${\mathbf D}({\mathscr V})$. It suffices then to show that $\varsigma=\beta_{\mathcal F}^\natural\>$, i.e.~(by definition of~$\beta_{\mathcal F}^\natural$), that the natural composition $$ u^*\<\<f_{\!*}f^\natural{\mathcal F}\iso g_*v^*\!f^\natural{\mathcal F}\xrightarrow{g_*(\varsigma)} g_*g^\natural u^*\<\<{\mathcal F} \xrightarrow{\tau_{\<\<u^{\mkern-1.5mu*}\!{\mathcal F}}^\natural} u^*\<{\mathcal F} $$ is induced by the natural map\index{ {}$\tau$ (trace map)!$\tau^\natural$} $$ \tau_{\<{\mathcal F}}^\natural\colon f_{\!*}f^\natural{\mathcal F}={\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)\to {\mathbf R}\cH{om}^{\bullet}({\mathcal O}_{\mathscr Y},\>{\mathcal F}\>)={\mathcal F}. $$ From \Eref{ft-example}(4) one sees, for injective~${\mathcal F}\<$,\vspace{-1pt} that $\tau_{\<{\mathcal F}}^\natural$ takes any homomorphism $\varphi\colon {\mathcal O}_{\mathscr Y}/{\mathscr I}\to{\mathcal F}$ over an open subset of~${\mathscr Y}$ to $\varphi(1)$; and similarly for $\tau_{\<\<u^{\mkern-1.5mu*}\!{\mathcal F}}^\natural\>$. The conclusion follows from the above definition of $\sigma({\mathcal O}_{\mathscr Y}/{\mathscr I},\>{\mathcal F}\>)=g_*(\varsigma)$. \end{proof} \end{parag} \begin{parag}\label{reduction} In this subsection we prove \Tref{T:basechange} in case $f\colon{\mathscr X}\to{\mathscr Y}$ is a proper map of ordinary noetherian schemes, by reduction to the case where ${\mathscr X}$, ${\mathscr Y}$, ${\mathscr U}$ and ${\mathscr V}$ are \emph{all} ordinary schemes---a case which we take for granted (see the introductory remarks for section~\ref{sec-basechange}). Of course when $u$ is \emph{adic} then ${\mathscr U}$ is already an ordinary scheme, and no reduction is needed at all. \begin{slem}\label{L:ordbc} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a proper map of ordinary noetherian schemes. For \Tref{T:basechange} to hold with this\/~$f$ it suffices that it hold whenever\/ ${\mathscr U}$ and\/ ${\mathscr V}$ are ordinary schemes as well. \end{slem} \begin{proof} Without yet assuming that ${\mathscr X}$ and ${\mathscr Y}$ are ordinary schemes, we can reduce \Tref{T:basechange} to the special case where the formal scheme~${\mathscr U}$ is \emph{affine} and $u({\mathscr U})$ is contained in an affine open subset of~${\mathscr Y}$. Indeed, for the base-change map $\beta_{\mathcal F}=\beta_{\<\<f,\>u}({\mathcal F}\>)$ of \Tref{T:basechange} to be an isomorphism, it clearly suffices that for any composition of fiber squares $$ \begin{CD} {\mathscr V}_0@>v_0^{}>>{\mathscr V}@>v>>{\mathscr X} \\ @Vg_0^{}VV @VgVV @VVfV \\ {\mathscr U}_0@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u_0^{}$}\vss}>{\mathscr U} @>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ with $u_0$ the inclusion of an affine open ${\mathscr U}_0\subset{\mathscr U}$ such that $u({\mathscr U}_0)$ is contained in an affine open subset of~${\mathscr Y}$, the map $$ v_0^*(\beta_{\mathcal F}\>)\colon v_0^*{\mathbf R}\iGp{\mathscr V} v^*\<\<f^!\<{\mathcal F}\to v_0^*g^!u^*\<{\mathcal F} $$ be an isomorphism. \Rref{R:Dtilde}(6) yields that ${\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\Rightarrow u^*{\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr U})$.\vspace{.5pt} So if we assume the above-specified special case, then $\beta_{\<\<f\<,\>uu_0^{}\<}({\mathcal F}\>)$\vspace{.5pt} and~$\beta_{\<g,\>u_0^{}\<}(u^*\<{\mathcal F}\>)$ are both isomorphisms. From \Pref{Gamma'(qc)}(a) we have a natural isomorphism $$ v_0^*(\beta_{\mathcal F}\>)\cong {\mathbf R}\iGp{{\mathscr V}_0}\< v_0^*(\beta_{\<\<f,\>u}({\mathcal F}\>)), $$ so \Lref{L:trans}(a) shows that $v_0^*(\beta_{\mathcal F}\>)$ is in fact an isomorphism. With reference to the remarks just preceding \Sref{S:trans}, (a) and (b) having already been proved, only (c) remains, i.e., we need only prove \Tref{T:basechange} for the rear face of diagram~\eqref{cube}. In other words, with the notation of diagram~\eqref{cube}, we may assume in proving \Tref{T:basechange} that $f=f_n$ (a proper map of ordinary schemes), and that $u=u_n$. Moreover $Y_n$ is a closed subscheme of ${\mathscr Y}$, and so if ${\mathscr U}$ is affine and $u({\mathscr U})$ is contained in an affine open subset of~${\mathscr Y}$, then $Y_n\times_{\mathscr Y}{\mathscr U}$ is affine and $u_n(Y_n\times_{\mathscr Y}{\mathscr U})$ is contained in an affine open subset of~$Y_n$. It follows that $Y_n\times_{\mathscr Y}{\mathscr U}$ is the completion of an ordinary affine $Y_n$-scheme. (That can be seen via the one-one correspondence from maps between affine formal schemes to continuous homomorphisms between their associated rings \cite[p.\,407, (10.4.6)]{GD}). \Tref{T:basechange} is thus reduced to the case depicted in the following diagram, where $f\colon{\mathscr X}\to {\mathscr Y}$ is now a proper map of ordinary noetherian schemes, $U$ is an ordinary affine ${\mathscr Y}$-scheme, $\kappa\colon{\mathscr U}\to U$ is a completion map, and $u\colon{\mathscr U}\to{\mathscr Y}$ factors as shown. $$ \begin{CD} {\mathscr X}\times_{\mathscr Y}{\mathscr U}@>>>{\mathscr X}\times_{\mathscr Y} U@>>>{\mathscr X} \\ @VgV\mkern63mu\text{\footnotesize(1)}V @VVV@V\text{\footnotesize(2)}\mkern45mu VfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1.1ex\hbox{$\scriptstyle \kappa$}\vss}>U@>>>{\mathscr Y} \end{CD} $$ We will show that \Tref{T:basechange} holds for subdiagram (1) by identifying the base-change map associated to $\kappa$ with the \emph{isomorphism}~$\zeta$ in~\Cref{C:compln+basechange}. As subdiagram (2) is a fiber square of ordinary schemes, \Lref{L:ordbc} will then result from the preceding reduction and the transitivity \Lref{L:trans}(a). \smallskip \deff_{\mkern-1.5mu0}^{\<\times}{f_{\mkern-1.5mu0}^{\<\times}} It is convenient to re-represent subdiagram~(1) in the notation of~\Cref{C:compln+basechange}. Consider then a diagram $$ \begin{CD} {\mathscr X}@.:=X_{\</Z} @>\kappa_{\mathscr X}^{\phantom{.}}>> X \\ @V f VV @. @VV f_0^{} V \\ {\mathscr Y}@.:=Y_{/W}@>>\vbox to 0pt{\vskip-1.1ex\hbox{$\scriptstyle\kappa_{\mathscr Y}^{\phantom{.}}$}\vss}> Y \end{CD} $$ as in \Cref{C:kappa-f^times-tors}, with $Z=f_0^{-1}W\<$. That $\zeta$ \emph{is} the base-change map means that $\zeta$ is adjoint to the natural composition $$ {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}\underset{\psi}\iso {\mathbf R}\iGp{\mathscr Y}{\mathbf R f_{\!*}}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}\underset{\theta'{}^{-1}}\iso {\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}f_{\mkern-1.5mu0}^{\<\times}\longrightarrow {\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*\longrightarrow \kappa_{\mathscr Y}^*. $$ But by definition, $\zeta$ is adjoint to the natural composition $$ {\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}\underset{\textup{~\ref{Gammas'+kappas}(c)}}\iso {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG Zf_{\mkern-1.5mu0}^{\<\times}\longrightarrow {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG Zf_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*\\ \underset{\tau_{\textup t}'(\kappa_{\mathscr Y}^*)}\longrightarrow\kappa_{\mathscr Y}^* $$ with $\tau_{\textup t}'$ as in~\Cref{C:kappa-f^times-tors}---so that $\tau_{\textup t}'(\kappa_{\mathscr Y}^*)$ factors naturally as \begin{align*} {\mathbf R f_{\!*}}\kappa_{\mathscr X}^*{\mathbf R}\iG Zf_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^* &\underset{\textup{~\ref{C:kappa-f*t}}}\iso \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}{\mathbf R}\iG Zf_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*\\ &\longrightarrow \kappa_{\mathscr Y}^*{\mathbf R} f_{\<0*}^{}f_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*\\ &\longrightarrow \kappa_{\mathscr Y}^*\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*\\ &\xrightarrow{\,\>\,\pi\,\,\>} \kappa_{\mathscr Y}^*\<. \end{align*} It will suffice then to verify that the following natural diagram commutes (where, again, we omit all occurrences of ${\mathbf R}$): $$ \minCDarrowwidth=25pt \defg_*\iGp\V\>v^*\<\<\ush f{f_{\!*}\iGp{\mathscr X}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}} \defg_*g^!u^*{\iGp{\mathscr Y}\< f_{\!*}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}} \def\iGp\Y\kappa_\Y^* f_{\<0*}^{}{\iGp{\mathscr Y}\kappa_{\mathscr Y}^* f_{\<0*}^{}f_{\mkern-1.5mu0}^{\<\times}} \def\kappa_\Y^* \iG W\< f_{\<0*}^{}{\iGp{\mathscr Y}\kappa_{\mathscr Y}^*} \defg_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f{\kappa_{\mathscr Y}^* f_{\<0*}^{}f_{\mkern-1.5mu0}^{\<\times}} \defg_*\iGp\V\BL_{\<\V}\iGp\V\>v^*\<\<\ush f{\kappa_{\mathscr Y}^* f_{\<0*}^{}\iG Zf_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*} \defg_*\iGp\V\BL_{\<\V} g^!u^*{\kappa_{\mathscr Y}^* f_{\<0*}^{}f_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*} \def\kappa_\Y^*\kappa_{\Y^*}^{}\kappa_\Y^*{\kappa_{\mathscr Y}^*\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*} \def\kappa_\Y^*{\kappa_{\mathscr Y}^*} \deff_{\!*}\kappa_\X^*\iG Z\fot\<\kappa_{\Y^*}^{}\kappa_\Y^*{f_{\!*}\kappa_{\mathscr X}^*\iG Zf_{\mkern-1.5mu0}^{\<\times}\<\kappa_{{\mathscr Y}^*}^{}\kappa_{\mathscr Y}^*} \deff_{\!*}\kappa_\X^*\iG Z{f_{\!*}\kappa_{\mathscr X}^*\iG Zf_{\mkern-1.5mu0}^{\<\times}} \def\kappa_\Y^* f_{\<0*}^{}\iG Z{\kappa_{\mathscr Y}^* f_{\<0*}^{}\iG Zf_{\mkern-1.5mu0}^{\<\times}} \deff_{\!*}\kappa_\X^*{f_{\!*}\kappa_{\mathscr X}^*f_{\mkern-1.5mu0}^{\<\times}} \begin{CD} g_*\iGp\V\>v^*\<\<\ush f @>\psi>> g_*g^!u^* @>\theta'{}^{-1}>> \iGp\Y\kappa_\Y^* f_{\<0*}^{} @>>>\kappa_\Y^* \iG W\< f_{\<0*}^{} \\ \vspace{-19pt}\\ @V\textup{\ref{Gammas'+kappas}(c)}VV \text{\footnotesize(A)} @. @VVV @VVV \\ \vspace{-22pt}\\ f_{\!*}\kappa_\X^*\iG Z @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\textup{~\ref{C:kappa-f*t}}> \kappa_\Y^* f_{\<0*}^{}\iG Z @>>> g_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f @>>> \underset{\UnderElement{}{\downarrow}{2.8ex} {\!\!\<\raisebox{.4ex}{$\scriptstyle\iota$}}}{\mkern1mu\strut} \mkern-20mu\hz{$\mkern-8mu\kappa_\Y^*$\hss} \\ \vspace{-18.7pt}\\ @VVV @VVV @VVV @A\pi AA\\ \vspace{-20pt}\\ f_{\!*}\kappa_\X^*\iG Z\fot\<\kappa_{\Y^*}^{}\kappa_\Y^* @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}>\textup{~\ref{C:kappa-f*t}}> g_*\iGp\V\BL_{\<\V}\iGp\V\>v^*\<\<\ush f @>>>g_*\iGp\V\BL_{\<\V} g^!u^* @>>>\kappa_\Y^*\kappa_{\Y^*}^{}\kappa_\Y^*\\ \vspace{-12pt} \end{CD} $$ Given that $\pi\iota=1$, the verification of commutativity is straightforward, except for subrectangle (A). \enlargethispage{-.7\baselineskip} Now there is a functorial isomorphism $\alpha\colon {\mathbf R} f_{0*}{\mathbf R}\iG Z\iso {\mathbf R}\iG W {\mathbf R} f_{0*}$ which arises in the obvious way, via ``K-flabby'' resolutions, from the equality $f_{0*}\iG Z=\iG W f_{0*}$ (see the last paragraph in the Remark following (3.2.5) in~\cite[p.\,25]{AJL}), and whose composition with the natural map ${\mathbf R}\iG W {\mathbf R} f_{0*}\to{\mathbf R} f_{0*}$ is the natural map ${\mathbf R} f_{0*}{\mathbf R}\iG Z\to{\mathbf R} f_{0*}$. And, again, we have the isomorphism ${\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*\iso\kappa_{\mathscr Y}^*{\mathbf R}\iG W$ of \Pref{Gammas'+kappas}(c), whose composition with the natural map $\kappa_{\mathscr Y}^*{\mathbf R}\iG W\to\kappa_{\mathscr Y}^*$ is the natural map ${\mathbf R}\iGp{\mathscr Y}\kappa_{\mathscr Y}^*\to\kappa_{\mathscr Y}^*$. Hence commutativity of (A) follows from that of the outer border---consisting entirely of isomorphisms---of the following diagram: $$ \defg_*\iGp\V\>v^*\<\<\ush f{f_{\!*}\iGp{\mathscr X}\kappa_{\mathscr X}^*} \defg_*g^!u^*{\iGp{\mathscr Y} \<f_{\!*}\kappa_{\mathscr X}^*} \def\iGp\Y\kappa_\Y^* f_{\<0*}^{}{\iGp{\mathscr Y}\kappa_{\mathscr Y}^* f_{\<0*}^{}} \def\kappa_\Y^* \iG W\< f_{\<0*}^{}{\kappa_{\mathscr Y}^* \iG W\< f_{\<0*}^{}} \defg_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f{\kappa_{\mathscr Y}^* f_{\<0*}^{}} \deff_{\!*}\kappa_\X^*\iG Z{f_{\!*}\kappa_{\mathscr X}^*\iG Z} \def\kappa_\Y^* f_{\<0*}^{}\iG Z{\kappa_{\mathscr Y}^* f_{\<0*}^{}\iG Z} \deff_{\!*}\kappa_\X^*{f_{\!*}\kappa_{\mathscr X}^*} \def\rotatebox{-30}{\hbox to 41pt{\rightarrowfill}}{\rotatebox{-30}{\hbox to 41pt{\rightarrowfill}}} \def\rotatebox{30}{\hbox to 41pt{\rightarrowfill}}{\rotatebox{30}{\hbox to 41pt{\rightarrowfill}}} \def\rotatebox{30}{\hbox to 41pt{\leftarrowfill}}{\rotatebox{30}{\hbox to 41pt{\leftarrowfill}}} \def\rotatebox{-30}{\hbox to 41pt{\leftarrowfill}}{\rotatebox{-30}{\hbox to 41pt{\leftarrowfill}}} \begin{CD} \underset{\UnderElement{\textup{~\ref{Gammas'+kappas}(c)}}{\uparrow}{7.2ex} {\<\<\simeq}}g_*\iGp\V\>v^*\<\<\ush f @>\psi>\vbox to0pt{\vskip2pt\hz{\hss\rotatebox{-30}{\hbox to 41pt{\rightarrowfill}}\hss}\vskip9pt \hz{\hss\rotatebox{30}{\hbox to 41pt{\rightarrowfill}}\hss}\vss}> g_*g^!u^* @.\overset{\theta'}{\Larrow{100pt}} @. \underset{\UnderElement{\simeq}{\uparrow}{7.2ex} {\!\!\!\textup{~\ref{Gammas'+kappas}(c)}}} \iGp\Y\kappa_\Y^* f_{\<0*}^{} \\ \vspace{-19pt}\\ @. @VVV \\ \vspace{-22pt}\\ @. f_{\!*}\kappa_\X^* @<<\theta< g_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f\\ \vspace{-21pt}\\ @. @. @AAA\\ f_{\!*}\kappa_\X^*\iG Z @. \overset{\vbox to 0pt{\vss \hbox to 0pt{\hss$\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}$\hss}\vskip-6.5pt \hbox to 0pt{\hss$\scriptstyle\textup{~\ref{C:kappa-f*t}}$\hss} \vskip-12pt\vss} } {\Rarrow{100pt}} @. \kappa_\Y^* f_{\<0*}^{}\iG Z @> \vbox to0pt{\vss\hz{\hss\rotatebox{30}{\hbox to 41pt{\leftarrowfill}}\hss}\vskip-6pt \hz{\hss\rotatebox{-30}{\hbox to 41pt{\leftarrowfill}}\hss}\vskip 2pt\hbox to 0pt{\hss$\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}$\hss}} >\alpha> \kappa_\Y^* \iG W\< f_{\<0*}^{}\text{\large\strut} \end{CD} $$ \medskip Since ${\mathbf R}\iGp{\mathscr Y}$ is right-adjoint to the inclusion $\D_{\mathrm t}\<({\mathscr Y})\hookrightarrow {\mathbf D}({\mathscr Y})$ (\Pref{Gamma'(qc)}), we can check commutativity \emph{after} composing the outer border with the natural map ${\mathbf R}\iGp{\mathscr Y} f_{\!*}\kappa_{\mathscr X}^*\to f_{\!*}\kappa_{\mathscr X}^*\>$, so that it suffices to check commutativity of all the subdiagrams of the preceding one. This is easy to do, as, with ${\mathcal E}\!:=f_{\mkern-1.5mu0}^{\<\times}\<{\mathcal F}$, the maps denoted by~ $\theta_{\<{\mathcal E}}\; (=\theta_{\!f_{\<\halfsize{\footnotesize0}}^{}, \mkern1.5mu\kappa_{\mathscr Y}^{}\!\<}({\mathcal E})$) in \Cref{C:kappa-f*t} and in \Pref{uf=gv} are the same.\vspace{1pt} This completes the proof of \Lref{L:ordbc}, and of \Tref{T:basechange}. \end{proof} \end{parag} \section{Consequences of the flat base change isomorphism.} \label{Consequences}\index{base-change isomorphism} We begin with a flat-base-change theorem for the functor $\ush f={\boldsymbol\Lambda}_{\mathscr X} f^!$ associated to a pseudo\kern.6pt-proper map~$f\colon{\mathscr X}\to Y$ of noetherian formal schemes. (As before,\vspace{.3pt} $f^!\setf_{\mathrm t}^\times$, and $\ush f$ is right-adjoint to the functor ${\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\colon\wDqc({\mathscr X})\to{\mathbf D}({\mathscr Y})$,\vspace{.5pt} where $\wDqc({\mathscr X})$ is the (full) $\Delta$-subcategory of~${\mathbf D}({\mathscr X})$ such that\vspace{.25pt} $$ {\mathcal F}\in\wDqc({\mathscr X})\Leftrightarrow{\mathbf R}\iGp{\mathscr X}{\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}), $$ see \Cref{C:f*gam-duality}.) We deduce a sheafified version \Tref{T:sheafify} of \Tref{Th2} of the Introduction (=\:\Tref{T:qct-duality} + \Cref{C:f*gam-duality}). This is readily seen equivalent to the case of flat base change where $u\colon{\mathscr U}\to{\mathscr Y}$ is an open immersion; in other words, it expresses the local nature, over~${\mathscr Y}$, of $f^!$ and $\ush f\<$. \Sref{S:coherent} establishes the local nature of $f^!$ and $\ush f$ over~${\mathscr X}$. From this we obtain that $\ush f\bigl(\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\bigr)\subset \D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ (\Pref{P:coherence}). This leads further to an improved base-change theorem for bounded-below complexes with coherent homology, and to \Tref{T:properdual}, a duality theorem for such complexes under proper maps. \medskip We consider as in \Tref{T:basechange} a fiber square of noetherian formal schemes $$ \begin{CD} {\mathscr V}@>v>>{\mathscr X} \\ @VgVV @VVfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y} \end{CD} $$ with $f$ and $g$ pseudo\kern.6pt-proper, $u$ and $v$ flat. For any ${\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})$ we have the composed isomorphism $$ \vartheta\colon{\mathbf R}\iGp{\mathscr V}\>v^*\<\<\ush f\<{\mathcal F} \underset{\textup{\ref{P:f* and Gamma}(c)}}\iso {\mathbf R}\iGp{\mathscr V}\>v^*\>{\mathbf R}\iGp{\mathscr X}\ush f\<{\mathcal F} \underset{\textup{\ref{C:identities}(a)}}\iso {\mathbf R}\iGp{\mathscr V}\>v^*\<\<f^!{\mathcal F} \underset{\textup{\ref{T:basechange}}}\iso g^!u^*\<{\mathcal F}. $$ In particular, $v^*\<\<\ush f\<{\mathcal F}\in\wDqc({\mathscr V})$. \pagebreak[3] \stepcounter{numb} \renewcommand{\theequation}{\theparag.\arabic{numb}} \begin{thm}\label{T:sharp-basechange} Under the preceding conditions, let $$ \ush{\beta_{\<\<{\mathcal F}}}\colon v^*\<\<\ush f\<{\mathcal F}\to\ush g u^*\<{\mathcal F}\qquad\bigl({\mathcal F}\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr) $$ be the map adjoint to the natural composition \begin{equation}\label{adjointto} {\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\>v^*\<\<\ush f\<{\mathcal F} \underset{{\mathbf R} g_*\vartheta}\iso{\mathbf R} g_*g^!u^*\<{\mathcal F}\to u^*{\mathcal F}. \end{equation} Then the map ${\boldsymbol\Lambda}_{\<{\mathscr V}}(\ush{\beta_{\<\<{\mathcal F}}})$ is an \emph{isomorphism} $$ {\boldsymbol\Lambda}_{\<{\mathscr V}}(\ush{\beta_{\<\<{\mathcal F}}})\colon{\boldsymbol\Lambda}_{\<{\mathscr V}}\> v^*\<\<\ush f\<{\mathcal F}\iso {\boldsymbol\Lambda}_{\<{\mathscr V}}\>\ush g u^*\<{\mathcal F} \underset{\textup{\ref{C:identities}(a)}}\cong \ush g u^*\<{\mathcal F}. $$ Moreover, if\/ $u$ is an open immersion then\/ $\ush{\beta_{\<\<{\mathcal F}}}$ itself is an isomorphism. \end{thm} \stepcounter{numb} \renewcommand{\theequation}{\theparag.\arabic{numb}} \begin{proof} The map $\ush{\beta}$ factors naturally as \begin{equation}\label{factors} v^*\<\<\ush f \to {\boldsymbol\Lambda}_{\<{\mathscr V}}\>v^*\<\<\ush f \underset{\textup{\ref{R:Gamma-Lambda}(c)}}\iso {\boldsymbol\Lambda}_{\<{\mathscr V}}{\mathbf R}\iGp{\mathscr V}\>v^*\<\<\ush f \underset{{\boldsymbol\Lambda}_{\<{\mathscr V}}\vartheta}\iso {\boldsymbol\Lambda}_{\<{\mathscr V}} g^!u^*=\ush g u^*\<. \end{equation} To see this, one needs to check that \eqref{factors} is adjoint to \eqref{adjointto}. The natural map $\mathbf 1\to{\boldsymbol\Lambda}_{\mathscr V}$ factors naturally as $\mathbf1\to{\boldsymbol\Lambda}_{\mathscr V}{\mathbf R}\iGp{\mathscr V}\to{\boldsymbol\Lambda}_{\mathscr V}$ (easy check), and hence the adjointness in question amounts to the readily-verified commutativity of the outer border of the following diagram (with all occurrences of ${\mathbf R}$ left out): $$ \defg_*\iGp\V\>v^*\<\<\ush f{g_*\iGp{\mathscr V}\>v^*\<\<\ush f} \defg_*g^!u^*{g_*g^!u^*} \defg_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f{g_*\iGp{\mathscr V}{\boldsymbol\Lambda}_{\<{\mathscr V}}\>v^*\<\<\ush f} \defg_*\iGp\V\BL_{\<\V}\iGp\V\>v^*\<\<\ush f{g_*\iGp{\mathscr V}{\boldsymbol\Lambda}_{\<{\mathscr V}}\iGp{\mathscr V}\>v^*\<\<\ush f} \defg_*\iGp\V\BL_{\<\V} g^!u^*{g_*\iGp{\mathscr V}{\boldsymbol\Lambda}_{\<{\mathscr V}} g^!u^*} \begin{CD} g_*\iGp\V\BL_{\<\V}\iGp\V\>v^*\<\<\ush f@<<<g_*\iGp\V\>v^*\<\<\ush f@>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt} >\text{via\;}\vartheta>g_*g^!u^* \\ \vspace{-22pt}\\ @V\simeq VV @AAA @AAA \\ g_*\iGp\V\BL_{\<\V}\>v^*\<\<\ush f @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt} >> g_*\iGp\V\BL_{\<\V}\iGp\V\>v^*\<\<\ush f @>\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt} >\text{via\;}\vartheta> g_*\iGp\V\BL_{\<\V} g^!u^* \end{CD} $$ That ${\boldsymbol\Lambda}_{\<{\mathscr V}}(\ush{\beta_{\<\<{\mathcal F}}})$ is an isomorphism results then from the idempotence of~${\boldsymbol\Lambda}_{\<{\mathscr V}}$ \mbox{(\Rref{R:Gamma-Lambda}(b)).} When $u$---hence $v$---is an open immersion, we have isomorphisms (the first of which is obvious): $$ {\boldsymbol\Lambda}_{\<{\mathscr V}}\>v^*\<\<\ush f\iso v^*\<{\boldsymbol\Lambda}_{\<{\mathscr X}}\ush f \underset{\textup{\ref{C:identities}(a)}}\iso v^*\<\<\ush f, $$ and the last assertion follows. \end{proof} \medskip Next comes the sheafification of \Tref{Th2}. Let $f\colon{\mathscr X}\to{\mathscr Y}$ be a map of locally noetherian formal schemes. For ${\mathcal G}$ and ${\mathcal E}\in{\mathbf D}({\mathscr X})$ we have natural compositions $$ {\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\<({\mathcal G}\<,{\mathcal E}) \to {\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\<({\mathbf L} f^*{\mathbf R f_{\!*}}{\mathcal G}\<,{\mathcal E})\, \xrightarrow[\mkern-15mu\textup{\cite[\kern-1pt p.\kern1pt147, \kern-1pt 6.7]{Sp}}\mkern-15mu]{\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}} \,{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathcal G}\<,{\mathbf R f_{\!*}}{\mathcal E}) $$ and $$ {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\<({\mathcal G}\<,{\mathcal E}) \to {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\<({\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,{\mathcal E}) \xrightarrow[\textup{\ref{C:Hom-Rgamma}}]{\vbox to 0pt{\vss\hbox{$\widetilde{\phantom{nn}}$}\vskip-7pt}} {\mathbf R}\cH{om}^{\bullet}_{\mathscr X}\<({\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,{\mathbf R}\iGp{\mathscr X}{\mathcal E}). $$ \begin{thm}\label{T:sheafify}\index{Grothendieck Duality!Torsion (sheafified)} Let\/ ${\mathscr X}$ and\/ ${\mathscr Y}$ be noetherian formal schemes and let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a pseudo\kern.6pt-proper map. Then the following natural compositions are \emph{isomorphisms:} \begin{align*} \ \ush\delta\<\<\colon\<{\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>\ush f\<{\mathcal F}\>) &\to\< {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\ush f\<{\mathcal F}\>) \\ &\to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathcal F}\>)\vspace{-4pt} \quad\ \bigl({\mathcal G}\in\wDqc({\mathscr X}),\;{\mathcal F}\>\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr); \end{align*} \vspace{-6pt} \noindent $ \ \delta^!\<\<\colon{\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>f^!{\mathcal F}\>) \to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathbf R f_{\!*}} f^!{\mathcal F}\>) \to {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathcal F}\>)\vspace{3pt} $ \rightline{$\bigl({\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X}),\;{\mathcal F}\>\in\wDqc^{\lower.5ex\hbox{$\scriptstyle+$}}({\mathscr Y})\bigr).$} \end{thm} \begin{proof} The map $\ush\delta$ is an isomorphism iff the same is true of ${\mathbf R}\Gamma({\mathscr U},\ush\delta)$ for all open ${\mathscr U}\subset{\mathscr Y}$. (For if ${\mathcal E}$---which may be assumed K-injective---is the vertex of a triangle based on~$\ush\delta\<$, then $\ush\delta$ is an isomorphism $\Leftrightarrow{\mathcal E}\cong 0\Leftrightarrow H^i({\mathcal E})=0$ for all $i\in\mathbb Z\Leftrightarrow$ the sheaf associated to the presheaf ${\mathscr U}\mapsto \textup H^i\Gamma({\mathscr U},{\mathcal E})=\textup H^i{\mathbf R}\Gamma({\mathscr U},{\mathcal E})$ vanishes for all~$i$.) Set ${\mathscr V}\!:= f^{-1}{\mathscr U}$, and let $u\colon{\mathscr U}\hookrightarrow{\mathscr Y}$ and $v\colon{\mathscr V}\hookrightarrow{\mathscr X}$ be the respective inclusions. We have then the fiber square $$ \begin{CD} {\mathscr V}@>v>>{\mathscr X} \\ @VgVV @VVfV \\ {\mathscr U}@>>\vbox to 0pt{\vskip-1ex\hbox{$\scriptstyle u$}\vss}>{\mathscr Y}\hbox to 0pt{,\hss} \end{CD} $$ and need only verify that ${\mathbf R}\Gamma({\mathscr U},\ush\delta)$ is the composition of the following sequence of isomorphisms: \begin{flalign*} {\mathbf R}\Gamma\bigl({\mathscr U},\>{\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>\ush f\<{\mathcal F}\>)\bigr) &\<\iso\< {\mathbf R}\Gamma\bigl({\mathscr V},\>{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>\ush f\<{\mathcal F}\>)\bigr) \quad&&\!\!\!\! \textup{\cite[\!6.4,\:6.7,\:5.15]{Sp}}\\ &\<\iso\< {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\>{\mathscr V}}(v^*\<{\mathcal G}\<,v^*\<\<\ush f\<{\mathcal F}\>) \quad&&\!\!\!\! \textup{\cite[\!5.14,\:5.12,\:6.4]{Sp}}\\ &\<\iso\< {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\>{\mathscr V}}(v^*\<{\mathcal G}\<,\>\ush g u^*{\mathcal F}\>) \quad&&\!\!\!\!(\textup{\Tref{T:sharp-basechange}})\\ &\<\iso\< {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\>{\mathscr U}}\<({\mathbf R} g_*{\mathbf R}\iGp{\mathscr V}\> v^*\<{\mathcal G}\<,\>u^*{\mathcal F}\>) \ &&\ \ \!\textup{\bigl(\kern-1pt\ref{C:f*gam-duality},\! \ref{R:Dtilde}(6)\kern-1pt\bigr)}\\ &\<\iso\< {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\>{\mathscr U}}\<({\mathbf R} g_*v^*{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>u^*{\mathcal F}\>) \ &&\quad\ \ \textup{(elementary)}\\ &\<\iso\< {\mathbf R}{\mathrm {Hom}}^{\bullet}_{\>{\mathscr U}}\<(u^*{\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>u^*{\mathcal F}\>) \ &&\quad\ \ \textup{(elementary)}\\ &\<\iso\< {\mathbf R}\Gamma\bigl({\mathscr U},\>{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr Y}}\<({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathcal F}\>)\bigr) \hbox to0pt{\qquad\;\textup{(as above).}\hss} \hskip-14pt \end{flalign*} This somewhat tedious verification is left to the reader (who may e.g., refer to the proof of $(4.3)^{\textup o}\Rightarrow (4.2)$ near the end of \cite{Non noetherian}). That $\delta^!$ is an isomorphism can be shown similarly---or be deduced via the natural map $f^!\cong{\mathbf R}\iGp{\mathscr X} \ush f\to \ush f$ (\Cref{C:identities}), which for ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr X})$ induces an isomorphism ${\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>f^!{\mathcal F})\iso {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr X}}\<({\mathcal G}\<,\>\ush f{\mathcal F})$ (\Cref{C:Hom-Rgamma}). \end{proof} \pagebreak \begin{parag}\label{S:coherent} For pseudo\kern.6pt-proper~$f\colon{\mathscr X}\to{\mathscr Y}$, the functors $f^!\!:= f_{\mathrm t}^\times$ and $\ush f$ are \emph{local on~${\mathscr X}$}, in the following sense. \begin{sprop}\label{P:local} Let there be given a commutative diagram $$ \begin{CD} {\mathscr U}@>i_1>>{\mathscr X}_1 \\ @V i_2 VV @VV f_1 V \\ {\mathscr X}_2 @>> f_2 > {\mathscr Y} \end{CD} $$ of noetherian formal schemes, with\/ $f_1$ and\/~$f_2$ pseudo\kern.6pt-proper and $i_1$ and\/~$i_2$ open immersions. Then there are functorial isomorphisms $$ i_1^*f_1^! \iso i_2^*f_2^!\>,\qquad\quad i_1^*\ush{f_1} \iso i_2^*\ush{f_2}. $$ \end{sprop} \begin{proof} The second isomorphism results from the first, since for any ${\mathcal F}\>\in{\mathbf D}({\mathscr Y})$ and for $j=1,2$, \begin{align*} i_j^*\ush{f_j}{\mathcal F}\>\overset{\text{\ref{C:f*gam-duality}}}= i_j^*{\mathbf R}\cH{om}^{\bullet}_{{\mathscr X}_j}\<\<({\mathbf R}\iGp{{\mathscr X}_j}\<{\mathcal O}_{{\mathscr X}_j}, f_j^!{\mathcal F}\>) &\cong{\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr U}}\<(i_j^*{\mathbf R}\iGp{{\mathscr X}_j}\<{\mathcal O}_{{\mathscr X}_j}, i_j^*f_j^!{\mathcal F}\>)\\ &\cong {\mathbf R}\cH{om}^{\bullet}_{\>{\mathscr U}}\<({\mathbf R}\iGp{{\mathscr U}}{\mathcal O}_{\mathscr U}, i_j^*f_j^!{\mathcal F}\>). \end{align*} For the first isomorphism, Verdier's\index{Verdier, Jean-Louis} proof of \cite[p.\,395, Corollary 1]{f!}---a special case of \Pref{P:local}---applies verbatim, modulo the following extensions (a), (b) and~(c) of some elementary properties of schemes to formal schemes.\looseness=-1 (a) Since pseudo\kern.6pt-proper maps are separated, the graph of~$i_j$ is a \emph{closed immersion} $\gamma\colon {\mathscr U}\hookrightarrow {\mathscr X}_j\<\<\times_{\<{\mathscr Y}}\<{\mathscr U}$ (see \cite[p.\,445, (10.15.4)]{GD}, where the ``finite-type'' hypothesis is used only to ensure that ${\mathscr X}_j\<\times_{\<{\mathscr Y}}{\mathscr U}$ is locally noetherian, a condition which holds here by the first paragraph in \Sref{sec-basechange}. And if $\>{\mathscr U}\to{\mathscr Y}$ is an open immersion, then so is~$\gamma$ (since then both $\pi_j\colon{\mathscr X}_j\!\times_{\<{\mathscr Y}}\<\<{\mathscr U}\to{\mathscr X}_j$ and $i_j=\pi_j\gamma$ are open immersions). (b) If $s\colon{\mathscr U}\to{\mathscr V}$ is an open and closed immersion, then the exact functors~$s_*$ and~$s^*$ are adjoint, and by \Eref{ft-example}(4) there is a functorial isomorphism $$ s^!{\mathcal F}\cong s^\natural{\mathcal F}\cong s^*{\mathcal F}\qquad\bigl({\mathcal F}\in\D_{\mkern-1.5mu\mathrm{qct}}({\mathscr V})\bigr). $$ (c) (Formal extension of \cite[p.\,325, (6.10.6)]{GD}.) Let ${\mathscr U}\overset{\gamma}\hookrightarrow {\mathscr W} \overset{w}\hookrightarrow{\mathscr Z}$ be maps of locally noetherian formal schemes such that $\gamma$~is a closed immersion and $w$ is an open immersion. (We are interested specifically in the case ${\mathscr W}\!:={\mathscr X}_2\<\times_{\<{\mathscr Y}}\<{\mathscr U}$ and ${\mathscr Z}\!:={\mathscr X}_2\<\times_{\<{\mathscr Y}}\<{\mathscr X}_1$, see (a).) Set $u\!:= w\gamma$. Then \emph{the closure\/~$\overline{\mathscr U}$ of\/~$u({\mathscr U})$ is a formal subscheme of\/~${\mathscr Z}$, and the map\/ ${\mathscr U}\to\overline{\mathscr U}$ induced by\/~$u$ is an open immersion.}\vspace{1pt} Indeed, $\overline{\mathscr U}$ is the support of ${\mathcal O}_{\mathscr Z}/{\mathscr I}$ where ${\mathscr I}$ is the kernel of the natural map ${\mathcal O}_{\mathscr Z}\to u_*{\mathcal O}_{\mathscr U}\>$; and it follows from \cite[p.\,441, (10.14.1)]{GD} that we need only show that ${\mathscr I}$ is \emph{coherent}. The question being local, we may assume that ${\mathscr Z}$ is affine, say ${\mathscr Z}={\mathrm {Spf}}(A)$. Cover ${\mathscr U}$ by a finite number of affine open subschemes~${\mathscr U}_i\ (1\le i\le n)$, with inclusions $u_i\colon{\mathscr U}_i\hookrightarrow{\mathscr U}$. Then there is a natural injection $$ u_*{\mathcal O}_{\mathscr U}\hookrightarrow u_*\<\bigl(\!\oplus_{i=1}^nu_{i*}{\mathcal O}_{{\mathscr U}_i}\bigr)\cong \oplus_{i=1}^n(uu_i)_*{\mathcal O}_{{\mathscr U}_i}\>, $$ so that ${\mathscr I}$ is the intersection of the kernels of the natural maps ${\mathcal O}_{\mathscr Z}\to(uu_i)_*{\mathcal O}_{{\mathscr U}_i}$, giving us a reduction to the case where ${\mathscr U}$ itself is affine, say ${\mathscr U}={\mathrm {Spf}}(B)$. Now if $I$ is the kernel of the ring-homomorphism $\rho\colon A\to B$ corresponding to~$u$, then for any $f\in A$ the kernel of the induced map $\rho_{\{f\}}\colon A_{\{f\}} \to B_{\{f\}} $ is $I_{\{f\}}$;\vspace{1pt} and one deduces that ${\mathscr I}$ is the coherent ${\mathcal O}_{\mathscr Z}$-module denoted by $I^\Delta$ in \cite[p.\,427, (10.10.2)]{GD}. \end{proof} \begin{sprop}\label{P:coherence} If\/ $f\colon{\mathscr X}\to{\mathscr Y}$ is a pseudo\kern.6pt-proper map of noetherian formal schemes then $$ \ush f\bigl(\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\bigr)\subset \D_{\mkern-1.5mu\mathrm c}^+({\mathscr X}). $$ \end{sprop} \begin{proof} Since $\ush f$ commutes with open base change (\Tref{T:sharp-basechange}) we may assume ${\mathscr Y}$ to be affine, say ${\mathscr Y}={\mathrm {Spf}}(A)$. Since $f$ is of pseudo-finite type, every point of ${\mathscr X}$ has an open neighborhood~${\mathscr U}$ such that $f|_{\mathscr U}$ factors as $$ {\mathscr U}\overset{i}\hookrightarrow{\mathrm {Spf}}(B)\xrightarrow{\,p}{\mathrm {Spf}}(A)={\mathscr Y} $$ where $B$ is the completion of a polynomial ring $P\!:= A[T_0, T_1,\dots,T_n]$ with respect\- to an ideal~$I$ whose intersection with $A$ is open, $i$~is a closed immersion, and $p$~corresponds to the obvious continuous ring homomorphism $A\to B$ (see footnote in \Sref{maptypes}). This ${\mathrm {Spf}}(B)$ is an open subscheme of the completion~$\mathscr P$ of the projective space ~$\mathbf P_{\!\!\!A}^n$ along the closure of its subscheme ${\mathrm {Spec}}(P/I)$. Thus by \Pref{P:local} and item~(c) in its proof, we can replace ${\mathscr X}$ by a closed formal subscheme of~ $\mathscr P$ having ${\mathscr U}$ as an open subscheme. In other words, we may assume that $f$ factors as $ {\mathscr X}\overset{i_1^{}\>}\hookrightarrow\mathscr P\xrightarrow{p_1^{}\>}{\mathrm {Spf}}(A)={\mathscr Y} $ with $i_1$ a closed immersion and $p_1$ the natural map. Then $\ush f=\ush{i_1}\ush{p_1}$, and we need only consider the two cases (a) $f=p_1$ and (b) $f=i_1$. \penalty-1000 Case (a) is given by \Cref{C:completion-proper}. In case (b) we see as in example~\ref{ft-example}(4) that for ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$ we have $f^\natural{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ and $$ \ush f\<{\mathcal F}={\boldsymbol\Lambda}_{\mathscr X}{\mathbf R}\iGp{\mathscr X} f^\natural\<{\mathcal F}\underset {\text{or }\ref{C:Hom-Rgamma}}{\overset{\ref{R:Gamma-Lambda}\text{(c)}}{=\!=\!=}}{\boldsymbol\Lambda}_{\mathscr X} f^\natural\<{\mathcal F}\overset{\ref{formal-GM}}{=\!=}f^\natural\<{\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X}). \vspace{-4.3ex} $$ \end{proof} \begin{scor}\label{C:coh-basechange}\index{base-change isomorphism} \smallskip For all\/ ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$ the base-change map\/ $\ush{\beta_{\<\<{\mathcal F}}}$ of \Tref{T:sharp-basechange} is an \emph{isomorphism} $$ \ush{\beta_{\<\<{\mathcal F}}}\colon v^*\<\<\ush f\<{\mathcal F} \iso \ush g u^*\<{\mathcal F}. $$ \end{scor} \begin{proof} \Pref{formal-GM} gives an isomorphism $v^*\<\<\ush f\<{\mathcal F}\iso{\boldsymbol\Lambda}_{\<{\mathscr V}}\> v^*\<\<\ush f\<{\mathcal F}$ . \end{proof} \end{parag} \medskip We have now the following duality theorem for proper maps and bounded-below complexes with coherent homology. \begin{thm}\label{T:properdual}\index{Grothendieck Duality!coherent} Let\/ $f\colon{\mathscr X}\to{\mathscr Y}$ be a proper map of noetherian formal schemes, so that\/ ${\mathbf R f_{\!*}}\(\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X}))\subset\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$ and\/ $\ush f\bigl(\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\bigr)\subset \D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ $($see Propositions~\textup{\ref{P:proper f*}} and~\textup{\ref{P:coherence}).} Then for\/ ${\mathcal G}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ and ${\mathcal F}\in\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$ there are functorial \emph{isomorphisms} \begin{align*} {\mathbf R f_{\!*}}{\mathbf R}\cH{om}^{\bullet}({\mathcal G}\<,\ush f\<{\mathcal F}\>) &\underset{\textup{\ref{T:sheafify}}}\iso {\mathbf R}\cH{om}^{\bullet}({\mathbf R f_{\!*}}{\mathbf R}\iGp{\mathscr X}\>{\mathcal G}\<,\>{\mathcal F}\>) \\ &\underset{\textup{\ref{C:f* and Gamma}(d)}}\iso {\mathbf R}\cH{om}^{\bullet}({\mathbf R}\iGp{\mathscr Y}\>{\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathcal F}\>) \underset{\textup{\ref{formal-GM}}}\iso {\mathbf R}\cH{om}^{\bullet}({\mathbf R f_{\!*}}{\mathcal G}\<,\>{\mathcal F}\>). \end{align*} In particular, $\ush f\colon\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})\to \D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})$ is right-adjoint to\/ ${\mathbf R f_{\!*}}\colon\D_{\mkern-1.5mu\mathrm c}^+({\mathscr X})\to\D_{\mkern-1.5mu\mathrm c}^+({\mathscr Y})$.\vspace{1.5pt} If\/ ${\mathscr X}$ is properly algebraic we can replace\/ $\ush f$ by the functor\/ $f^{\<\times}$ of \Cref{cor-prop-duality}. \end{thm} {\sc Proof} Left to reader. (For the last assertion see Corol\-laries~\ref{C:coh-dual} and~\ref{corollary}.) \enlargethispage*{\baselineskip}

9708

alg-geom/9708016

en

https://arxiv.org/abs/alg-geom/9708016

alg-geom/9708016

Nhadhule

Klaus Hulek

Nef Divisors on Moduli Spaces of Abelian Varieties

LaTeX2e, 23 pages. The proof of the main result has been shortened. In particular, the former technical propositions 4.3 and 4.4 were replaced by a simpler argument

We determine the cone of nef divisors on the Voronoi compactification A_g^* of the moduli space A_g of principally polarized abelian varieties of dimension g for genus g=2,3. As a corollary we obtain that the spaces A_g^*(n) with level-n structure are a minimal, resp. canonical, model for g=2, n>=4, resp. n>=5 and g=3, n>=3, resp. n>=4. We give two proofs: The easy and quick one reduces the problem to \bar M_g where we can use a result of Faber. This approach cannot be generalized to higher genus g. The main point of the paper is, therefore, to give a second proof using theta functions and a result of Weissauer. This technique can be at least partially generalized to higher genus. We formulate a conjecture for the nef cone of A_g^* for all g.

alg-geom

\section{Introduction} Let ${\cal A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$. Over the complex numbers ${\cal A}_g={\mathbb{H}}_g/\Gamma_g$ where ${\mathbb{H}}_g$ is the Siegel space of genus $g$ and $\Gamma_g=\on{Sp}(2g,{\mathbb{Z}})$. We denote the torodial compactification given by the second Voronoi decomposition by ${\cal A}_g^*$ and call it the \emph{Voronoi compactification}. It was shown by Alexeev and Nakamura \cite{A} that ${\cal A}_g^*$ coarsely represents the stack of principally polarized stable quasiabelian varieties. The variety ${\cal A}_g^*$ is projective \cite{A} and it is known that the Picard group of ${\cal A}_g^*, g\ge 2$ is generated (modulo torsion) by two elements $L$ and $D$, where $L$ denotes the (${\mathbb{Q}}$-)line bundle given by modular forms of weight $1$ and $D$ is the boundary (see \cite{Mu2}, \cite{Fa} and \cite{Mu1} for $g=2, 3$ and $\geq4$). In this paper we want to discuss the following \begin{theorem}\label{theo0.1} Let $g=2$ or $3$. A divisor $aL-bD$ on ${\cal A}_g^*$ is nef if and only if $b\geq0$ and $a-12b\geq0$. \end{theorem} The varieties ${\cal A}_g$ have finite quotient singularities. Adding a level-$n$ structure one obtains spaces ${\cal A}_g(n)={\mathbb{H}}_g/\Gamma_g(n)$ where $\Gamma_g(n)$ is the principal congruence subgroup of level $n$. For $n\geq3$ these spaces are smooth. However, the Voronoi compactification ${\cal A}_g^*(n)$ acquires singularities on the boundary for $g\geq5$ due to bad behaviour of the second Voronoi decomposition. There is a natural quotient map ${\cal A}_g^*(n)\to{\cal A}_g^*$. Note that this map is branched of order $n$ along the boundary. Hence Theorem~(\ref{theo0.1}) is equivalent to \begin{theorem}\label{theo0.2} Let $g=2$ or $3$. A divisor $aL-bD$ on ${\cal A}_g^*(n)$ is nef if and only if $b\geq0$ and $a-12\frac{b}{n}\geq0$. \end{theorem} This theorem easily gives the following two corollaries. \begin{corollary}\label{cor0.3} If $g=2$ then $K$ is nef but not ample for ${\cal A}_2^*(4)$ and $K$ is ample for ${\cal A}_2^*(n)$, $n\geq5$; in particular ${\cal A}_2^*(n)$ is a minimal model for $n\geq4$ and a canonical model for $n\geq5$. \end{corollary} This was first proved by Borisov \cite{Bo}. \begin{corollary}\label{cor0.4} If $g=3$ then $K$ is nef but not ample for ${\cal A}_3^*(3)$ and $K$ is ample for ${\cal A}_3^*(n)$, $n\geq4$; in particular ${\cal A}_3^*(n)$ is a minimal model for $n\geq3$ and a canonical model for $n\geq4$. \end{corollary} In this paper we shall give two proofs of Theorem~(\ref{theo0.1}). The first and quick one reduces the problem via the Torelli map to the analogous question for ${\overline{M}}_2$, resp.~${\overline{M}}_3$. Since the Torelli map is not surjective for $g\ge 4$ this proof cannot possibly be generalized to higher genus. This is the main reason why we want to give a second proof which uses theta functions. This proof makes essential use of a result of Weissauer \cite{We}. The method has the advantage that it extends in principle to other polarizations as well as to higher $g$. We will also give some partial results supporting the \begin{conjecture} For any $g\geq2$ the nef cone on ${\cal A}_g^*$ is given by the divisors $aL-bD$ where $b\geq0$ and $a-12b\geq0$. \end{conjecture} \begin{acknowledgement} It is a pleasure for me to thank RIMS and Kyoto University for their hospitality during the autumn of 1996. I am grateful to V.~Alexeev and R.~Salvati Manni for useful discussions. It was Salvati Manni who drew my attention to Weissauer's paper. I would also like to thank R.~Weissauer for additional information on \cite{We}. The author is partially supported by TMR grant ERBCHRXCT 940557. \end{acknowledgement} \section{Curves meeting the interior} We start by recalling some results about the Kodaira dimension of ${\cal A}_g^*(n)$. It was proved by Freitag, Tai and Mumford that ${\cal A}_g^*$ is of general type for $g\geq7$. The following more general result is probably well known to some specialists. \begin{theorem}\label{theo1.1} ${\cal A}_g^*(n)$ is of general type for the following values of $g$ and $n\geq n_0$: \begin{center} \begin{tabular}{c|cccccc} $g$ & $2$ & $3$ & $4$ & $5$ & $6$ & $\geq7$ \\\hline $n_0$ & $4$ & $3$ & $2$ & $2$ & $2$ & $1$ \end{tabular}. \end{center} \end{theorem} \begin{Proof} One can use Mumford's method from \cite{Mu1}. First recall that away from the singularities and the closure of the branch locus of the map ${\mathbb{H}}_g\to{\cal A}_g(n)$ the canonical bundle equals \begin{equation}\label{formula1.1} K\equiv (g+1)L-D. \end{equation} This equality holds in particular also on an open part of the boundary. If $g\leq4$ and $n\geq3$ the spaces ${\cal A}_g^*(n)$ are smooth and hence (\ref{formula1.1}) holds everywhere. If $g\geq5$ then Tai \cite{T} showed that there is a suitable toroidal compactification ${\tilde{{\cal A}}}_g(n)$ such that all singularities are canonical quotient singularities. By Mumford's results from \cite{Mu1} one can use the theta-null locus to eliminate $D$ from formula (\ref{formula1.1}) and obtains \begin{equation}\label{formula1.2} K\equiv \left((g+1)-\frac{2^{g-2}(2^g+1)}{n2^{2g-5}}\right)L+ \frac{1}{n2^{2g-5}}[\Theta_{\on{null}}]. \end{equation} We then have general type if all singularities are canonical and if the factor in front of $L$ is positive. This gives immediately all values in the above table with the exception of $(g,n)=(4,2)$ and $(7,1)$. In the latter case the factor in front of $L$ is negative. The proof that ${\cal A}_7$ is nevertheless of general type is the main result of \cite{Mu1}. The difficulty in the first case is that one can possibly have non-canonical singularities. One can, however, use the following argument which I have learnt from Salvati Manni: An immediate calculation shows that for every element $\sigma\in\Gamma_g(2)$ the square $\sigma^2\in\Gamma_g(4)$. Hence if $\sigma$ has a fixed point then $\sigma^2=1$ since $\Gamma_g(4)$ acts freely. But for elements of order $2$ one can again use Tai's extension theorem (see \cite[Remark after Lemma~4.5]{T} and \cite[Remark after Lemma~5.2]{T}). \end{Proof} \begin{rem}\label{rem1.2} The Kodaira dimension of ${\cal A}_6$ is still unknown. All other varieties ${\cal A}_g(n)$ which do not appear in the above list are either rational or unirational: Unirationality of ${\cal A}_g$ for $g=5$ was proved by Donagi \cite{D} and by Mori and Mukai \cite{MM}. For $g=4$ the same result was shown by Clemens \cite{C}. Unirationality is easy for $g\leq3$. Igusa \cite{I2} showed that ${\cal A}_2$ is rational. Recently Katsylo \cite{Ka} proved rationality of ${\cal M}_3$ and hence also of ${\cal A}_3$. The space ${\cal A}_3(2)$ is rational by work of van Geemen \cite{vG} and Dolgachev and Ortlang \cite {DO}. ${\cal A}_2(3)$ is the Burkhardt quartic and hence rational. This was first proved by Todd (1936) and Baker (1942). See also the thesis of Finkelnberg \cite{Fi}. The variety ${\cal A}_2(2)$ has the Segre cubic as a projective model \cite{vdG1} and is hence also rational. Yamazaki \cite{Ya} first showed general type for ${\cal A}_2(n)$, $n\geq4$. \end{rem} We denote the Satake compactification of ${\cal A}_g$ by $\overline{{\cal A}}_g$. There is a natural map $\pi:{\cal A}_g^*\to\overline{{\cal A}}_g$ which is an isomorphism on ${\cal A}_g$. The line bundle $L$ is the pullback of an ample line bundle on $\overline{{\cal A}}_g$ which, by abuse of notation, we again denote by $L$. In fact the Satake compactification is defined as the closure of the image of ${\cal A}_g$ under the embedding given by a suitable power of $L$ on ${\cal A}_g$. In particular we notice that $L.C\geq0$ for every curve $C$ on ${\cal A}_g^*$ and that $L.C>0$ if $C$ is not contracted to a point under the map $\pi$. Let $F$ be a modular form with respect to the full modular group $\on{Sp}(2g,{\mathbb{Z}})$. Then the \emph{order} $o(F)$ of $F$ is defined as the quotient of the vanishing order of $F$ divided by the weight of $F$. \begin{theorem}[Weissauer]\label{theo1.3} For every point $\tau\in{\mathbb{H}}_g$ and every $\varepsilon>0$ there exists a modular form $F$ of order $o(F)\geq\frac{1}{12+\varepsilon}$ which does not vanish at $\tau$. \end{theorem} \begin{Proof} See \cite{We}. \end{Proof} \begin{proposition}\label{prop1.4} Let $C\subset{\cal A}_g^*$ be a curve which is not contained in the boundary. Then $(aL-bD).C\geq0$ if $b\geq0$ and $a-12b\geq0$. \end{proposition} \begin{Proof} First note that $L.C>0$ since $\pi(C)$ is a curve in the Satake compactification. It is enough to prove that $(aL-bD).C>0$ if $a-12b>0$ and $a,b\geq0$. This is clear for $b=0$ and hence we can assume that $b\neq0$. We can now choose some $\varepsilon>0$ with $a/b>12+\varepsilon$. By Weissauer's theorem there exists a modular form $F$ of say weight $k$ and vanishing order $m$ with $F(\tau)\neq0$ for some point $[\tau]\in C$ and $m/k\geq1/(12+\varepsilon)$. In terms of divisors this gives us that $$ kL=mD+D_F, \quad C\not\subset D_F $$ where $D_F$ is the zero-divisor of $F$. Hence $$ \left(\frac km L-D\right)=\frac1mD_F.C\geq0. $$ Since $a/b>12+\varepsilon\geq k/m$ and $L.C>0$ we can now conclude that $$ \left(\frac abL-D\right).C>\left(\frac kmL-D\right).C\geq0. $$ \end{Proof} \begin{rem}\label{rem1.5} Weissauer's result is optimal, since the modular forms of order $>1/12$ have a common base locus. To see this consider curves $C$ in ${\cal A}_g^*$ of the form $X(1)\times\{A\}$ where $X(1)$ is the modular curve of level $1$ parametrizing elliptic curves and $A$ is a fixed abelian variety of dimension $g-1$. The degree of $L$ on $X(1)$ is $1/12$ (recall that $L$ is a ${\mathbb{Q}}$-bundle) whereas it has one cusp, i.e.~the degree of $D$ on this curve is $1$. Hence every modular form of order $>1/12$ will vanish on $C$. This also shows that the condition $a-12b\geq0$ is necessary for a divisor to be nef. \end{rem} \section{Geometry of the boundary (I)} \setcounter{equation}{0} We first have to collect some properties of the structure of the boundary of ${\cal A}_g^*(n)$. Recall that the Satake compactification is set-theoretically the union of ${\cal A}_g(n)$ and of moduli spaces ${\cal A}_k(n)$, $k<g$ of lower dimension, i.e. $$ \overline{{\cal A}}_g(n)={\cal A}_g(n)\amalg\left(\underset{i_1}{\amalg} {\cal A}_{g-1}^{i_1}(n)\right)\amalg \left(\underset{i_2}{\amalg} {\cal A}_{g-2}^{i_2}(n)\right)\ldots\amalg \left(\underset{i_g}{\amalg} {\cal A}_0^{i_g}(n)\right). $$ Via the map $\pi:{\cal A}_g^*(n)\to\overline{{\cal A}}_g(n)$ this also defines a stratification of ${\cal A}_g^*(n)$: $$ {\cal A}^*_g(n)={\cal A}_g(n)\amalg\left(\underset{i_1}{\amalg} D_{g-1}^{i_1}(n)\right)\amalg \left(\underset{i_2}{\amalg} D_{g-2}^{i_2}(n)\right)\ldots\amalg \left(\underset{i_g}{\amalg} D_0^{i_g}(n)\right). $$ The irreducible components of the boundary $D$ are the closures $\overline{D}_{g-1}^{i_1}(n)$ of the codimension $1$ strata $D_{g-1}^{i_1}(n)$. Whenever we talk about a \emph{boundary component} we mean one of the divisors $\overline{D}_{g-1}^{i_1}(n)$. Then the boundary $D$ is given by $$ D=\sum_{i_1}\overline{D}_{g-1}^{i_1}(n). $$ The fibration $\pi:D_{g-1}^{i_1}(n)\to{\cal A}_{g-1}^{i_1}(n)={\cal A}_{g-1}(n)$ is the universal family of abelian varieties of dimension $g-1$ with a level-$n$ structure if $n\geq3$ resp.~the universal family of Kummer surfaces for $n=1$ or $2$ (see \cite{Mu1}). We shall also explain this in more detail later on. To be more precise we associate to a point $\tau\in{\mathbb{H}}_g$ the lattice $L_{\tau,1}=(\tau,\mathbf{1}){\mathbb{Z}}^{2g}$, resp.~the principally polarized abelian variety $A_{\tau,1}={\mathbb{C}}^g/L_{\tau,1}$. Given an integer $n\geq1$ we set $L_{n\tau,n}=(n\tau,n\mathbf{1}_g){\mathbb{Z}}^{2g}$, resp.~$A_{n\tau,n}= {\mathbb{C}}^g/L_{n\tau,n}$. By $K_{n\tau,n}$ we denote the Kummer variety $A_{n,\tau n}/\{\pm1\}$. \begin{lemma}\label{lemma2.1} Let $n\geq3$. Then for any point $[\tau]\in{\cal A}_{g-1}^{i_1}(n)$ the fibre of $\pi$ equals $\pi^{-1}([\tau])=A_{n,\tau n}$. \end{lemma} \begin{Proof} Compare \cite{Mu1}. We shall also give an independent proof below. \end{Proof} This result remains true for $n=1$ or $2$, at least for points $\tau$ whose stabilizer subgroup in $\Gamma_g(n)$ is $\{\pm1\}$, if we replace $A_{n,\tau n}$ by its associate Kummer variety $K_{n,\tau n}$. \begin{lemma}\label{lemma2.2} Let $n\geq3$. Then for $[\tau]\in{\cal A}_{g-1}^{i_1}(n)$ the restriction of ${D}_{g-1}^{i_1}(n)$ to the fibre $\pi^{-1}([\tau])$ is negative. More precisely $$ D_{g-1}^{i_1}(n)|_{\pi^{-1}([\tau])}\equiv -\frac2nH $$ where $H$ is the polarization on $A_{n\tau,n}$ given by the pull-back of the principal polarization on $A_{\tau,1}$ via the covering $A_{n,\tau n}\to A_{\tau,1}$. \end{lemma} \begin{Proof} Compare \cite[Proposition~1.8]{Mu1}, resp.~see the discussion below. \end{Proof} Again the statement remains true for $n=1$ or $2$ if we replace the abelian variety by its Kummer variety. \begin{Proof*}{First proof of Theorem (\ref{theo0.1})} We have already seen (see Remark~\ref{rem1.5}) that for every nef divisor $aL-bD$ the inequality $a-12b\geq0$ holds. If $C$ is a curve in a fibre of the map ${\cal A}_g^*(n)\to\overline{{\cal A}}_g(n)$, then $L.C=0$. Lemma~(\ref{lemma2.2}) immediately implies that $b\geq0$ for any nef divisor. It remains to show that the conditions of Theorem~(\ref{theo0.1}) are sufficient to imply nefness. For any genus the Torelli map $t:{\cal M}_g\to{\cal A}_g$ extends to a morphism $\overline{t}:\overline{{\cal M}}_g\to {\cal A}_g^*$ (see \cite{Nam}). Here $\overline{{\cal M}}_g$ denotes the compactification of ${\cal M}_g$ by stable curves. For $g=2$ and $3$ the map $\overline{t}$ is surjective. It follows that for every curve $C$ in ${\cal A}_g^*$ there is a curve $C'$ in $\overline{{\cal M}}_g$ which is finite over $C$. Hence a divisor on ${\cal A}_g^*$, $g=2,3$ is nef if and only if this holds for its pull-back to $\overline{{\cal M}}_g$. In the notation of Faber's paper \cite{Fa} $\overline{t}^*L=\lambda$ where $\lambda$ is the Hodge bundle and $\overline{t}^*D=\delta_0$ where $\delta_0$ is the boundary ($g=2$), resp. the closure of the locus of genus $2$ curves with one node ($g=3$) (cf also \cite{vdG2}). The result follows since $a\lambda-b\delta_0$ is nef on $\overline{{\cal M}}_g$, $g=2,3$ for $a-12b\geq0$ and $b\geq0$ (see \cite{Fa}). \end{Proof*} As we have already pointed out the Torelli map is not surjective for $g\geq4$ and hence this proof cannot possibly be generalized to higher genus. The main purpose of this paper is, therefore, to give a proof of Theorem~(\ref{theo0.1}) which does not use the reduction to the curve case. This will also allow us to prove some results for general $g$. At the same time we obtain an independent proof of nefness of $a\lambda-b\delta_0$ for $a-12b\geq0$ and $b\geq0$ on $\overline{{\cal M}}_g$ for $g=2$ and $3$. We now want to investigate the open parts $D_{g-1}^{i_1}(n)$ of the boundary components $\overline{D}_{g-1}^{i_1}(n)$ and their fibration over ${\cal A}_{g-1}(n)$ more closely. At the same time this gives us another argument for Lemmas~(\ref{lemma2.1}) and (\ref{lemma2.2}). At this stage we have to make first use of the toroidal construction. Recall that the boundary components $D_{g-1}^{i_1}(n)$ are in $1:1$ correspondence with the maximal dimensional cusps, and these in turn are in $1:1$ correspondence with the lines $l\subset{\mathbb{Q}}^g$ modulo $\Gamma_g(n)$. Since all cusps are equivalent under the action of $\Gamma_g/\Gamma_g(n)$ we can restrict our attention to one of these cusps, namely the one given by $l_0=(0,\ldots,0,1)$. This corresponds to $\tau_{gg}\to i\infty$. To simplify notation we shall denote the corresponding boundary stratum simply by ${D}_{g-1}^{1}(n)=D_{g-1}(n)$. The stabilizer $P(l_0)$ of $l_0$ in $\Gamma_g$ is generated by elements of the following form (cf.~\cite[Proposition~I.3.87]{HKW}): \begin{align*} g_1&=\begin{pmatrix} A&0&B&0 \\ 0&1&0&0 \\ C&0&D&0 \\ 0&0&0&1 \end{pmatrix},\ \begin{pmatrix} A&B \\ C&D \end{pmatrix} \in \Gamma_{g-1},\\ g_2&=\begin{pmatrix} \mathbf{1}_{g-1}&0&0&0 \\ 0&\pm1&0&0 \\ 0&0&\mathbf{1}_{g-1}&0 \\ 0&0&0&\pm1 \end{pmatrix},\\ g_3&=\begin{pmatrix} \mathbf{1}_{g-1}&0&0&\sideset{^t}{}{\on{\mathit{N}}} \\ M&1&N&0 \\ 0&0&\mathbf{1}_{g-1}&-\sideset{^t}{}{\on{\mathit{M}}} \\ 0&0&0&1 \end{pmatrix},\ M,N \in{\mathbb{Z}}^{g-1},\\ g_4&=\begin{pmatrix} \mathbf{1}_{g-1}&0&0&0 \\ 0&1&0&S \\ 0&0&\mathbf{1}_{g-1}&0 \\ 0&0&0&1 \end{pmatrix},\ S\in{\mathbb{Z}}. \end{align*} \noindent We write $\tau=(\tau_{ij})_{1\leq i,j\leq g}$ in the form $$ \left( \begin{array}{ccc|c} \tau_{11} & \cdots & \tau_{1,g-1} & \tau_{1g}\\ \vdots & & \vdots & \vdots\\ \tau_{1,g-1} & \cdots & \tau_{g-1,g-1} & \tau_{g-1,g}\\\hline \tau_{1,g} & \cdots & \tau_{g-1,g} & \tau_{gg} \end{array} \right) = \left( \begin{array}{c|c} \tau_1 & \sideset{^t}{}{\on{\tau_2}}\\\hline \tau_2 & \tau_3 \end{array} \right). $$ \noindent Then the action of $P(l_0)$ on ${\mathbb{H}}_g$ is given by (cf.~\cite[I.3.91]{HKW}): \begin{align*} g_1(\tau)&=\begin{pmatrix} (A\tau_1+B)(C\tau_1+D)^{-1} & * \\ \tau_2(C\tau_1+D)^{-1} & \tau_3-\tau_2(C\tau_1+D)^{-1}C \sideset{^t}{}{\on{\tau_2}} \end{pmatrix},\\ g_2(\tau)&=\begin{pmatrix} \tau_1 & * \\ \pm\tau_2 & \tau_3 \end{pmatrix},\\ g_3(\tau)&=\begin{pmatrix} \tau_1 & * \\ \tau_2+M\tau_1+N & \tau_3' \end{pmatrix}\\ \intertext{where $\tau_3'=\tau_3+M\tau_1\sideset{^t}{}{\on{\mathit{M}}} + M\sideset{^t}{}{\on{\tau_2}}+ \sideset{^t}{}{\on{(}} M \sideset{^t}{}{\on{\tau_2}})+N\sideset{^t}{}{\on{\mathit{M}}},$} g_4(\tau)&=\begin{pmatrix} \tau_1 & \tau_2 \\ \tau_2 & \tau_3+S \end{pmatrix}. \end{align*} The parabolic subgroup $P(l_0)$ is an extension $$ 1\longrightarrow P'(l_0)\longrightarrow P(l_0)\longrightarrow P''(l_0) \longrightarrow 1 $$ where $P'(l_0)$ is the rank $1$ lattice generated by $g_4$. To obtain the same result for $\Gamma_g(n)$ we just have to intersect $P(l_0)$ with $\Gamma_g(n)$. Note that $g_2$ is in $\Gamma_g(n)$ only for $n=1$ or $2$. The first step in the construction of the toroidal compactification of ${\cal A}_g^*(n)$ is to divide ${\mathbb{H}}_g$ by $P'(l_0)\cap\Gamma(n)$ which gives a map $$ \begin{array}{ccl} {\mathbb{H}}_g &\longrightarrow& {\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}\times{\mathbb{C}}^*\\ \begin{pmatrix} \tau_1 & \sideset{^t}{}{\on{\tau_2}} \\ \tau_2 & \tau_3 \end{pmatrix} &\longmapsto& (\tau_1,\tau_2,e^{2\pi i\tau_3/n}). \end{array} $$ \noindent Partial compactification in the direction of $l_0$ then consists of adding the set ${\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}\times\{0\}$. It now follows immediately from the above formulae for the action of $P(l_0)$ on ${\mathbb{H}}_g$ that the action of the quotient group $P''(l_0)$ on ${\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}\times{\mathbb{C}}^*$ extends to ${\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}\times\{0\}$. Then $D_{g-1}(n)=({\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1})/P''(l_0)$ and the map to ${\cal A}_{g-1}(n)$ is induced by the projection from ${\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}$ to ${\mathbb{H}}_{g-1}$. This also shows that $D_{g-1}(n)\to{\cal A}_{g-1}(n)$ is the universal family for $n\geq3$ and that the general fibre is a Kummer variety for $n=1$ and $2$. Whenever $n_1|n_2$ we have a Galois covering $$ \pi(n_1,n_2):{\cal A}_g^*(n_2)\longrightarrow{\cal A}_g^*(n_1) $$ whose Galois group is $\Gamma_g(n_1)/\Gamma_g(n_2)$. This induces coverings $\overline{D}_{g-1}(n_2)\to\overline{D}_{g-1}(n_1)$, resp.~$D_{g-1}(n_2)\to D_{g-1}(n_1)$. In order to avoid technical difficulties we assume for the moment that ${\cal A}_g^*(n)$ is smooth (this is the case if $g\leq4$ and $n\geq3$). In what follows we will always be able to assume that we are in this situation. Then we denote the normal bundle of $\overline{D}_{g-1}(n)$ in ${\cal A}_g^*(n)$ by $N_{\overline{D}_{g-1}(n)}$, resp.~its restriction to $D_{g-1}(n)$ by $N_{D_{g-1}(n)}$. Since the covering map $\pi(n_1,n_2)$ is branched of order $n_2/n_1$ along the boundary, it follows that $$ \pi^*(n_1,n_2)n_1N_{\overline{D}_{g-1}}(n_1)=n_2N_{\overline{D}_{g-1}}(n_2). $$ We now define the bundle $$ \overline{M}(n):=-nN_{\overline{D}_{g-1}(n)}+L. $$ This is a line bundle on the boundary component $\overline{D}_{g-1}(n)$. We denote the restriction of $\overline{M}(n)$ to $D_{g-1}(n)$ by $M(n)$. We find immediately that $$ \pi^*(n_1,n_2)\overline{M}(n_1)=\overline{M}(n_2). $$ The advantage of working with the bundle $\overline{M}(n)$ is that we can explicitly describe sections of this bundle. For this purpose it is useful to review some basic facts about theta functions. For every element $m=(m',m'')$ of ${\mathbb{R}}^{2g}$ one can define the theta-function $$ \Theta_{m'm''}(\tau,z)=\sum_{q\in{\mathbb{Z}}^g} e^{2\pi i[{(q+m')\tau} {^t(q+m')}/2+{(q+m')}{^t(z+m'')}]}. $$ The transformation behaviour of $\Theta_{m'm''}(\tau,z)$ with respect to $z\mapsto z+u\tau+u'$ is described by the formulae ($\Theta1$)--($\Theta5$) of \cite[pp.~49,~50]{I}. The behaviour of $\Theta_{m'm''}(\tau,z)$ with respect to the action of $\Gamma_g(1)$ on ${\mathbb{H}}_g\times{\mathbb{C}}^g$ is given by the theta transformation formula \cite[Theorem~II.5.6]{I} resp.~the corollary following this theorem \cite[p.~85]{I}. \begin{proposition}\label{prop2.3} Let $n\equiv0\on{mod}4p^2$. If $m',m'',\overline{m}',\overline{m}''\in \frac{1}{2p} {\mathbb{Z}}^{g-1}$, then the functions $\Theta_{m'm''}(\tau,z) \Theta_{\overline{m}'\overline{m}''}(\tau,z)$ define sections of the line bundle $M(n)$ on $D_{g-1}(n)$. \end{proposition} \begin{Proof} It follows from ($\Theta3$) and ($\Theta1$) that for $k,k'\in n{\mathbb{Z}}^{g-1}$ the following holds: $$ \Theta_{m',m''}(\tau,z+k\tau+k')= e^{2\pi i[-\frac12k{\tau} {^tk}-{k}{^t(z+k')}]} \Theta_{m',m''}(\tau,z). $$ Similarly, of course, for $\Theta_{\overline{m}',\overline{m}''}(\tau,z)$. Moreover the theta transformation formula together with formula ($\Theta2$) gives $$ \Theta_{m',m''}(\tau^\#,z^\#) = e^{2\pi i[\frac12z(C\tau+D)^{-1}{C}{^tz}]} \det(C\tau+D)^{1/2} u \Theta_{m',m''}(\tau,z) $$ for every element $\gamma=\begin{pmatrix} A&B\\ C&D \end{pmatrix}\in \Gamma_{g-1}(n)$ and $$ \tau^\#=\gamma(\tau),\quad z^\#=z(C\tau+D)^{-1}. $$ Here $u^2$ is a character of $\Gamma_{g-1}(1,2)$ with $u^2|_{\Gamma_{g-1}(4)}\equiv1$. On the other hand the boundary component $D_{g-1}(n)$ is defined by $t_3=0$ with $t_3=e^{2\pi i\tau_3/n}$. We have already described the action of $P''(l_0)$ on ${\mathbb{H}}_{g-1}\times{\mathbb{C}}^{g-1}$. The result then follows by comparing the transformation behaviour of $(t_3/t_3^2)^n$ with respect to $g_1$ and $g_3$ with the above formulae together with the fact that the line bundle $L$ is defined by the automorphy factor $\det(C\tau+D)$. \end{Proof} This also gives an independent proof of Lemma~(\ref{lemma2.2}). \section{Geometry of the boundary (II)} So far we have described the stratum $D_{g-1}(n)$ of the boundary component $\overline{D}_{g-1}(n)$ and we have seen that there is a natural map $D_{g-1}(n)\to{\cal A}_{g-1}(n)$ which identifies $D_{g-1}(n)$ with the universal family over ${\cal A}_{g-1}(n)$ if $n\geq3$. We now want to describe the closure $\overline{D}_{g-1}(n)$ in some detail. In order to do this we have to restrict ourselves to $g=2$ and $3$. First assume $g=2$. Then the projection $D_1(n)\to{\cal A}_1(n)=X^0(n)$ extends to a projection $\overline{D}_1(n)\to X(n)$ onto the modular curve of level $n$ and in this way $\overline{D}_1(n)$ is identified with Shioda's modular surface $S(n)\to X(n)$. The fibres are either elliptic curves or $n$-gons of rational curves (if $n\ge 3$). Similarly the fibration $D_2(n)\to{\cal A}_2(n)$ extends to a fibration $\overline{D}_2(n)\to{\cal A}_2^*$ whose fibres over the boundary of ${\cal A}_2^*(n)$ are degenerate abelian surfaces. This was first observed by Nakamura \cite{Na} and was described in detail by Tsushima \cite{Ts} whose paper is essential for what follows. We shall now explain the toroidal construction which allows us to describe the fibration $\overline{D}_2(n)\to{\cal A}_2^*(n)$ explicitly. Here we shall concentrate on a description of this map in the most difficult situation, namely in the neighbourhood of a cusp of maximal corank. The toroidal compactification ${\cal A}_g^*(n)$ is given by the second Voronoi decomposition $\Sigma_g$. This is a rational polyhedral decomposition of the convex hull in $\on{Sym}_g^{\geq0}({\mathbb{R}})$ of the set $\on{Sym}_g^{\geq0}({\mathbb{Z}})$ of integer semi-positive $(g\times g)$-matrices. For $g=2$ and $3$ it can be described as follows. First note that $\on{Gl}(g,{\mathbb{Z}})$ acts on $\on{Sym}_g^{\geq0}({\mathbb{R}})$ by $\gamma\mapsto \sideset{^t}{}{\on{\mathit{M}}}\gamma M$. For $g=2$ we define the standard cone $$ \sigma_2={\mathbb{R}}_{\geq0}\gamma_1+{\mathbb{R}}_{\geq0}\gamma_2+{\mathbb{R}}_{\geq0}\gamma_3 $$ with $$ \gamma_1=\begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix},\quad \gamma_2=\begin{pmatrix} 0&0 \\ 0&1 \end{pmatrix},\quad \gamma_3=\begin{pmatrix} 1&-1 \\ -1&1 \end{pmatrix}. $$ Then $$ \Sigma_2=\{M(\sigma_2);\ M\in\on{Gl}(2,{\mathbb{Z}})\}. $$ Similarly for $g=3$ we consider the standard cone $$ \sigma_3={\mathbb{R}}_{\geq0}\alpha_1+{\mathbb{R}}_{\geq0}\alpha_2+{\mathbb{R}}_{\geq0}\alpha_3+ {\mathbb{R}}_{\geq0}\beta_1+{\mathbb{R}}_{\geq0}\beta_2+{\mathbb{R}}_{\geq0}\beta_3 $$ with \begin{gather*} \alpha_1=\begin{pmatrix} 1&0&0 \\ 0&0&0 \\ 0&0&0 \end{pmatrix},\quad \alpha_2=\begin{pmatrix} 0&0&0 \\ 0&1&0 \\ 0&0&0 \end{pmatrix},\quad \alpha_3=\begin{pmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&1 \end{pmatrix},\\ \beta_1=\begin{pmatrix} 0&0&0 \\ 0&1&-1 \\ 0&-1&1 \end{pmatrix},\quad \beta_2=\begin{pmatrix} 1&0&-1 \\ 0&0&0 \\ -1&0&1 \end{pmatrix},\quad \beta_3=\begin{pmatrix} 1&-1&0 \\ -1&1&0 \\ 0&0&0 \end{pmatrix}. \end{gather*} Then $$ \Sigma_3=\{M(\sigma_3);\ M\in\on{Gl}(3,{\mathbb{Z}})\}. $$ We consider the lattices $$ N_3 = {\mathbb{Z}}\gamma_1+{\mathbb{Z}}\gamma_2+{\mathbb{Z}}\gamma_3\\ $$ $$ N_6 = {\mathbb{Z}}\alpha_1+{\mathbb{Z}}\alpha_2+{\mathbb{Z}}\alpha_3+{\mathbb{Z}}\beta_1+{\mathbb{Z}}\beta_2+{\mathbb{Z}}\beta_3. $$ The fans $\Sigma_2$ resp.~$\Sigma_3$ define torus embeddings $T^3\subset X(\Sigma_2)$ and $T^6\subset X(\Sigma_3)$. We denote the divisors of $X(\Sigma_3)$ which correspond to the $1$-dimensional simplices of $\Sigma_3$ by ${\cal D}^i$. Let ${\cal D}={\cal D}^1$ be the divisor corresponding to ${\mathbb{R}}_{\geq0}\alpha_3$. An open part of ${\cal D}$ (in the ${\mathbb{C}}$-topology) is mapped to the boundary component $\overline{D}_2(n)$. In order to understand the structure of ${\cal D}$ we also consider the rank $5$ lattice $$ N_5={\mathbb{Z}}\alpha_1+{\mathbb{Z}}\alpha_2+{\mathbb{Z}}\beta_1+{\mathbb{Z}}\beta_2+{\mathbb{Z}}\beta_3\cong N_6/{\mathbb{Z}}\alpha_3. $$ The natural projection $\rho:N_{6,{\mathbb{R}}}\to N_{5,{\mathbb{R}}}$ maps the cones of the fan $\Sigma_3$ to the cones of a fan $\Sigma_3'\subset N_{5,{\mathbb{R}}}$. This fan defines a torus embedding $T^5=({\cal D}\setminus \bigcup\limits_{i\neq1}{\cal D}^i)\subset X(\Sigma_3')={\cal D}$. The projection $$ \begin{array}{rccl} \lambda:& N_{6,{\mathbb{R}}}\cong\on{Sym}_3({\mathbb{R}}) &\longrightarrow& N_{3,{\mathbb{R}}}\cong\on{Sym}_2({\mathbb{R}})\\ &\begin{pmatrix} a&b&d \\ b&c&e \\ d&e&f \end{pmatrix} &\longmapsto& \begin{pmatrix} a&b \\ b&c \end{pmatrix} \end{array}. $$ maps $\Sigma_3$ to $\Sigma_2$ and factors through $N_{5,{\mathbb{R}}}$. In this way we obtain an induced map $$ \begin{array}{ccc} {\cal D}=X(\Sigma_3') &\longrightarrow& X(\Sigma_2)\\ \cup & & \cup\\ T^5 &\longrightarrow& T^3. \end{array} $$ In order to describe this map we first consider the standard simplices $\sigma_3\subset N_{6,{\mathbb{R}}}$ and $\sigma_2\subset N_{3,{\mathbb{R}}}$, resp.~$\sigma_3'=\rho(\sigma_3)\subset N_{5,{\mathbb{R}}}$. On the torus $T^6$ (and similarly on $T^5$ and $T^3$) we introduce coordinates by $$ t_{ij}=e^{2\pi i\tau_{ij}/n}\qquad (1\leq i,j\leq 3). $$ These coordinates correspond to the dual basis of the basis $U_{ij}^*$ of $\on{Sym}(3,{\mathbb{Z}})$ where the entries of $U_{ij}^*$ are $1$ in positions $(i,j)$ and $(j,i)$ and $0$ otherwise. One easily checks that $T_{\sigma_3}\cong{\mathbb{C}}^6\subset X(\Sigma_3)$ and as coordinates on $T_{\sigma_3}$ one can take the coordinates which correspond to the dual basis of the generators $\alpha_1,\ldots,\beta_3$. Let us denote these coordinates by $T_1,\ldots,T_6$. A straightforward calculation shows that the inclusion $T^6\subset T_{\sigma_3}$ is given by \begin{equation}\label{formula2.1} \begin {array}{l@{\qquad}l@{\qquad}l} T_1 = t_{11}t_{13}t_{12}, & T_2 = t_{22}t_{23}t_{12}, & T_3 = t_{33}t_{13}t_{23},\\ T_4 = t_{23}^{-1}, & T_5 = t_{13}^{-1}, & T_6 = t_{12}^{-1}. \end {array} \end{equation} Then ${\cal D}\cap T_{\sigma_3}=\{T_3=0\}$. For genus $2$ the corresponding embedding $T^3\subset T_{\sigma_2}$ is given by $$ T_1=t_{11}t_{12},\qquad T_2=t_{22}t_{12},\qquad T_3=t_{12}^{-1}. $$ Finally we consider $T_{\sigma_3'}\cong{\mathbb{C}}^5\subset X(\Sigma_3')$. The projection ${\cal D}=X(\Sigma_3')\to X(\Sigma_2)$ map $T_{\sigma_3'}$ to $T_{\sigma_2}$. We can use $T_1,T_2,T_4,T_5,T_6$ as coordinates on $T_{\sigma_3'}$. Since $\lambda(\alpha_1)=\lambda(\beta_2)=\gamma_1$, $\lambda(\alpha_2)=\lambda(\beta_1)=\gamma_2$ and $\lambda(\alpha_3)=\gamma_3$ we find that \begin{equation}\label{formula2.2} \begin{array}{ccl} T_{\sigma_3'}\cong{\mathbb{C}}^5 &\longrightarrow& T_{\sigma_2}\cong{\mathbb{C}}^3\\ (T_1,T_2,T_4,T_5,T_6) &\longmapsto& (T_1T_5,T_2T_4,T_6). \end{array} \end{equation} Given any (maximal dimensional) cone $\sigma'=\rho(\sigma)$ in $\Sigma_3'$ we can describe the map $T_{\sigma'}\to T_{\lambda(\sigma)}$ in terms of coordinates by the method described above. In this way we obtain a complete description of the map ${\cal D}\to X(\Sigma_2)$. Let us now return to the toroidal compactification ${\cal A}_3^*(n)$ of ${\cal A}_3(n)$. Let $u_0\subset{\mathbb{Q}}^6$ be a maximal isotropic subspace. Then we obtain the compactification of ${\cal A}_3(n)$ in the direction of the cusp corresponding to $u_0$ as follows: The parabolic subgroup $P(u_0)\subset\Gamma_3(n)$ is an extension $$ 1\longrightarrow P'(u_0)\longrightarrow P(u_0)\longrightarrow P''(u_0)\longrightarrow 1 $$ where $P'(u_0)$ is a lattice of rank $6$. We have an inclusion ${\mathbb{H}}_g/P'(u_0)\subset T^6\subset X(\Sigma_3)$ and we denote the interior of the closure of ${\mathbb{H}}_g/P'(u_0)$ in $X(\Sigma_3)$ by $X(u_0)$. Then $P''(u_0)$ acts on $X(u_0)$ and we obtain a neighbourhood of the cusp corresponding to $u_0$ by $X(u_0)/P''(u_0)$. We have already described the partial compactification in the direction of a line (in our case $l_0$). Similarly we can define a partial compactification in the direction of an isotropic plane $h_0$. The space ${\cal A}_3^*(n)$ is then obtained by glueing all these partial compactifications. The result of Nakamura and Tsushima can then be stated as follows: The restriction of the map $\pi:{\cal A}_3^*(n)\to\overline{{\cal A}}_3(n)$ to the boundary component $\overline{D}_2(n)$ admits a factorisation $$ \diagram \overline{D}_2(n) \rto^{\pi'} \drto_{\pi} & {\cal A}_2^*(n) \dto^{\pi''}\\ & \overline{{\cal A}}_2(n) \enddiagram $$ where $\pi'':{\cal A}_2^*(n)\to\overline{{\cal A}}_2(n)$ is the natural map of the Voronoi compactification ${\cal A}_2^*(n)$ of ${\cal A}_2(n)$ to the Satake compactification $\overline{{\cal A}}_2(n)$. The map $\pi':\overline{D}_2(n)\to{\cal A}_2^*(n)$ is a flat family of surfaces extending the universal family over ${\cal A}_2(n)$. In order to describe the fibres over the boundary points of ${\cal A}_2^*(n)$ recall that every boundary component of ${\cal A}_2^*(n)$ is isomorphic to the Shioda modular surface $S(n)$. We explain the \emph{type} of a point $P$ in ${\cal A}_2^*(n)$ as follows: $$ \begin{array}{lcl} P\text{ has type I} &\Longleftrightarrow& P\in{\cal A}_2(n)\\ P\text{ has type II} &\Longleftrightarrow& P\text{ lies on a smooth fibre of}\\ && \text{a boundary component }S(n)\\ P\text{ has type IIIa} &\Longleftrightarrow& P\text{ is a smooth point on a singular}\\ && \text{fibre of }S(n)\\ P\text{ has type IIIb} &\Longleftrightarrow& P\text{ is a singular point of an $n$-gon}\\ && \text{in }S(n). \end{array} $$ Points of type IIIb are also often called \emph{deepest points}. \begin{proposition}[Nakamura, Tsushima]\label{prop2.4} Assume $n\geq3$. Let $P$ be a point in ${\cal A}_2^*(n)$ and denote the fibre of the map $\pi':\overline{D}_2(n)\to{\cal A}_2^*(n)$ over $P$ by $A_P$. Then the following holds: \noindent $\on{(i)}$ If $P=[\tau]\in{\cal A}_2(n)$ is of type $\on{I}$ then $A_P$ is a smooth abelian surface, more precisely $A_P\cong A_{n,\tau n}$. \noindent $\on{(ii)}$ if $P$ is of type $\on{II}$, then $A_P$ is a cycle of $n$ elliptic ruled surfaces. \noindent $\on{(iii)}$ If $P$ is of type $\on{IIIa}$, then $A_P$ consists on $n^2$ copies of ${\mathbb{P}}^1\times{\mathbb{P}}^1$. \noindent $\on{(iv)}$ If $P$ is of type $\on{IIIb}$, then $A_P$ consists of $3n^2$ components. These are $2n^2$ copies of the projective plane ${\mathbb{P}}^2$ and $n^2$ copies of ${\tilde{{\mathbb{P}}}}^2$, i.e.~${\mathbb{P}}^2$ blown up in $3$ points in general position. \end{proposition} \begin{Proof} The proof consists of a careful analysis of the map $\overline{D}_2(n)\to{\cal A}_2^*(n)$ using the description of the map ${\cal D}\to X(\Sigma_2)$. For details see \cite[section 4]{Ts}. \end{Proof} \begin{urems} (i) The degenerations of type IIIa and IIIb are usually depicted by the diagrams \begin{gather}\tag{IIIa} \begin{minipage}[c]{3.4cm} \unitlength1cm \begin{picture}(5.5,3.4) \multiput(1,0)(1,0){4}{\line(0,1){3.4}} \multiput(0,0.2)(0,1){4}{\line(1,0){5}} \end{picture} \end{minipage} \end{gather} where each square stands for a ${\mathbb{P}}^1\times{\mathbb{P}}^1$, resp. \begin{gather}\tag{IIIb} \begin{minipage}[c]{9.6cm} \unitlength1cm \begin{picture}(9.6,7.5) \put(0.4,0.2){\line(1,0){8.8}} \put(0.2,2.4){\line(1,0){9.2}} \put(0.2,4.6){\line(1,0){9.2}} \put(0.4,6.8){\line(1,0){8.8}} \put(0,4.1){\line(2,-3){2.735}} \put(0.9,7){\line(2,-3){4.67}} \put(3.9,7){\line(2,-3){4.67}} \put(6.9,7){\line(2,-3){2.705}} \put(2.7,7){\line(-2,-3){2.705}} \put(5.7,7){\line(-2,-3){4.67}} \put(8.7,7){\line(-2,-3){4.67}} \put(9.6,4.1){\line(-2,-3){2.735}} \end{picture} \end{minipage} \end{gather} where the triangles stand for projective planes ${\mathbb{P}}^2$ and the hexagons for blown-up planes ${\tilde{{\mathbb{P}}}}^2$. \noindent (ii) The singular fibres are degenerate abelian surfaces (cf.~\cite{Na}, \cite{HKW}). \noindent (iii) This description must be modified for $n=1$ or $2$. Then the general fibre is a Kummer surface $K_{n,\tau n}$ and the fibres of type (IIIb) consist of $8$ ($n=2$), resp.~$2$ copies of ${\mathbb{P}}^2$. \end{urems} The following is a crucial technical step: \begin{proposition}\label{prop2.5} Let $n\equiv 0\on{mod} 8p^2$. If $m',m'',\overline{m}',\overline{m}''\in \frac{1}{2p}{\mathbb{Z}}^2$ then the sections $\Theta_{m'm''}(\tau,z)\Theta_{\overline{m}'\overline{m}''}(\tau,z)$ of the line bundle $M(n)$ on $D_2(n)$ extend to sections of the line bundle $\overline{M}(n)$ on $\overline{D}_2(n)$. \end{proposition} \begin{Proof} We have to prove that the sections in question extend to the part of $D_2(n)$ which lies over the boundary of ${\cal A}_2^*(n)$. This is a local statement. Moreover it is enough to prove extension in codimension $1$. Due to symmetry considerations we can restrict ourselves to one boundary component in ${\cal A}_2^*(n)$. We shall use the above description of the toroidal compactifications ${\cal A}_2^*(n)$ and ${\cal A}_3^*(n)$ and of the map $\overline{D}_2(n)\to{\cal A}_2^*(n)$. We consider the boundary component of ${\cal A}_2^*(n)$ given by $\{T_2=0\}\subset T_{\sigma_2}\subset X(\Sigma_2)$. Recall the theta functions $$ \Theta_{m'm''}(\tau,z)=\sum_{q\in{\mathbb{Z}}^2} e^{2\pi i[\frac12(q+m'){\tau} {^t(q+m')}+{(q+m')}{^t(z+m'')}]} $$ In our situation $$ \tau=\begin{pmatrix} \tau_{11}&\tau_{12} \\ \tau_{12} & \tau_{22} \end{pmatrix},\quad z=(z_1,z_2)=(\tau_{13},\tau_{23}). $$ In level $n$ we have the coordinates $$ t_{ij}=e^{2\pi i\tau_{ij}/n} $$ and $\Theta_{m'm''}(\tau,z)$ becomes \begin{multline*} \Theta_{m'm''}(\tau,z)=\sum_{q=(q_1,q_2)\in{\mathbb{Z}}^2} t_{11}^{\frac12(q_1+m_1')^2n} t_{12}^{(q_1+m_1')(q_2+m_2')n} t_{22}^{\frac12(q_2+m_2')^2n}\\ t_{13}^{(q_1+m_1')n} t_{23}^{(q_2+m_2')n} e^{2\pi {i (q+m')} {^tm}''}. \end{multline*} We use the coordinates $T_1,T_2,T_4,T_5,T_6$ on $T_{\sigma_3'}$. It follows from (\ref{formula2.1}) that \begin{alignat}{2} t_{11} &= T_1T_5T_6, & \quad t_{22} &= T_2T_4T_6, \notag \\ t_{23} &= T_4^{-1}, & \quad t_{13} &= T_5^{-1},\label{formula2.3}\\ t_{12} &= T_6^{-1}.\notag \end{alignat} This leads to the following expression for the theta-functions \begin{multline*} \Theta_{m'm''}(\tau,z)=\sum_{q\in{\mathbb{Z}}^2} T_1^{\frac12(q_1+m_1')^2n} T_2^{\frac12(q_2+m_2')^2n} T_4^{\frac12(q_2+m_2')(q_2+m_2'-2)n}\\ T_5^{\frac12(q_1+m_1')(q_1+m_1'-2)n} T_6^{\frac12((q_1+m_1')-(q_2+m_2'))^2n} e^{2\pi {i(q+m')} {^tm}''}. \end{multline*} By (\ref{formula2.2}) the locus over $T_2=0\subset T_{\sigma_2}$ in $T_{\sigma_3'}$ is given by $T_2T_4=0$. The equation for the boundary component $\overline{D}_2(n)$ is given by $t_{33}=0$. Since by (\ref{formula2.1}) we have $t_{33}=T_3T_4T_5$ we can assume that the normal bundle and hence $\overline{M}(n)$ (more precisely its pullback to $X(\Sigma_3')$) is trivial outside $T_4T_5=0$. Since the exponent of $T_2$ is a non-negative integer (here we use $n\equiv 0\on{mod} 8p^2$) this shows that the sections extend over $T_2=0$, $T_4\neq0$. To deal with the other components of $T_{\Sigma_3'}$ which lie over $\{T_2=0\}$ in $T_{\sigma_2}$ we use the matrices $$ \nu_{nm}=\begin{pmatrix} 1&0&m \\ 0&1&n \\ 0&0&1 \end{pmatrix} \qquad (n,m\in{\mathbb{Z}}) $$ (cf.~\cite{Ts}) which act on $\on{Sym}_3^{\geq0}({\mathbb{Z}})$ by $$ \gamma\longmapsto\sideset{^t}{_{nm}}{\on{\nu}}\gamma\nu_{nm}. $$ Via $\lambda$ this action lies over the trivial action on $\on{Sym}_2^{\geq0}({\mathbb{Z}})$. This action also factors through $\rho$. Let $(\sigma_3')_{nm}=\rho(\sideset{^t}{_{nm}}{\on{\nu}}\sigma_3\nu_{nm})$. We can then either argue with the symmetries induced by this operation or repeat directly the above calculation for $T_{(\sigma_3')_{nm}}$. Acting with $\nu_{0m}$, $m\in{\mathbb{Z}}$, we can thus treat all components in $X(\Sigma_3')$ lying over $\{T_2=0\}$ in $X(\Sigma_2)$. \end{Proof} \section{Curves in the boundary } We can now treat curves contained in a boundary component. The following technical lemma will be crucial. Its proof uses the ideas of \cite[Abschnitt~4]{We} in an essential way and it can be generalized in a suitable form to arbitrary $g$. We consider the boundary component $\overline{D}_2(n)$ which belongs to the line $l_0=(0,\ldots,0,1)\subset{\mathbb{Q}}^6$. Recall that the open part $D_2(n)$ of $\overline{D}_2(n)$ is of the form $D_2(n)={\mathbb{C}}^2\times{\mathbb{H}}_2/(P''(l_0)\cap \Gamma(n))$ and that the group $P''(l_0)/(P''(l_0)\cap\Gamma(n)$) acts on $\overline{D}_2(n)$. Recall also the fibration $\pi':\overline{D}_2(n)\to {\cal A}_2^*(n)$. We shall denote the boundary of ${\cal A}_2^*(n)$ by $B$. \begin{proposition}\label{prop2.6} Let $(z,\tau)\in{\mathbb{C}}^2\times{\mathbb{H}}_2$. For every $\varepsilon>0$ there exist integers $n,k$ and a section $s\in H^0(\overline{M}(n)^k)$ such that \noindent $\on{(i)}$ $s([z,\tau])\neq0$ where $[z,\tau]\in D_2(n)={\mathbb{C}}^2\times {\mathbb{H}}_2/(P''(l_0)\cap\Gamma(n))$, \noindent $\on{(ii)}$ $s$ vanishes on $\pi^*B$ of order $\lambda$ with $\frac{\lambda}{k}\geq\frac{n}{12+\varepsilon}$. \end{proposition} \begin{Proof} Let $p \ge 3$ be a prime number (which will be chosen later). For $l=2p$ we consider the set of characteristics ${\cal M}$ in $(\frac1l{\mathbb{Z}}/{\mathbb{Z}})^6$ of the form $m=(m_p,m_2)$ in $(\frac1p{\mathbb{Z}}/{\mathbb{Z}})^6\oplus (\frac12{\mathbb{Z}}/{\mathbb{Z}})^6$ with $m_p\not\in{{\mathbb{Z}}}^6$. The group $\Gamma_3(1)$ acts on ${\cal M}$ with $2$ orbits. Assume $\varepsilon>0$ is given and that $\widetilde{\cal M}$ is a subset of ${\cal M}$ with $$ \#\widetilde{\cal M}<\varepsilon\#{\cal M}. $$ Then set $$ \Theta_{{\cal M},\widetilde{\cal M}}(\tau,z)= \prod_{m\in{\cal M}\setminus\widetilde{\cal M}} \Theta_m^l(\tau,z). $$ Let $n=8p^2$. By Proposition~(\ref{prop2.5}) the functions $\Theta_m^l(\tau,z)$ define sections in $\overline{M}(n)^p$. Let $M_1,\ldots,M_N\in \Gamma_2(1)$ be a set of generators of $\Gamma_2(1)/\Gamma_2(n)\cong \on{Sp}(4,{\mathbb{Z}}/n{\mathbb{Z}})$. Then $M_1,\ldots,M_N$, considered as elements in $P(l_0)$, act on the line bundle $\overline{M}(n)$. We set $$ F_r(\tau,z)=\sum_{i=1}^NM_i^*\Theta_{{\cal M},\widetilde{\cal M}}^r. $$ This is a $\Gamma_2/\Gamma_2(n)$-invariant section of $\overline{M}(n)^{pr}$. Now consider the abelian surface $A=A_{\tau,1}={\mathbb{C}}^2/({\mathbb{Z}}^2\tau+{\mathbb{Z}}^2)$. Then $A_{n\tau,n}={\mathbb{C}}^2/((n{\mathbb{Z}})^2\tau+(n{\mathbb{Z}})^2)$ is the fibre of $\pi$ over the point $[\tau]\in{\cal A}_2(n)$. Let $$ \widetilde{\cal M}=\{m\in{\cal M};\ \Theta_m(\tau,z)=0\}. $$ The argument of Weissauer shows that $$ \#\widetilde{\cal M}<\varepsilon\#{\cal M} $$ for $p$ sufficiently large. For some $r$ the section $F_r(\tau,z)$ does not vanish at $[z,\tau]\in D_2(n)$. Let $B'$ be a boundary boundary component of ${\cal A}_2^*(n)$. The inverse image $D'$ of $B'$ under $\pi'$ consists of several components. Using the matrices $\nu_{nm}$ which were introduced in the proof of Proposition~(\ref{prop2.5}) one can, however, show that the vanishing order of the sections $\Theta_m^l(\tau,z)$ on the components of $D'$ only depends on $B'$. Hence one can argue as in \cite{We} and finds that the vanishing order along $\pi^*B$ goes to $\frac{prn}{12}$ as $p$ goes to infinity. Setting $k=pr$ this gives (ii). \end{Proof} We can now start giving the proof of Theorem~(\ref{theo0.1}). Let $$ H=aL-bD\qquad b>0,\ 12a-\frac bn>0 $$ be a divisor on ${\cal A}_g^*(n)$. In view of Proposition~(\ref{prop1.4}) it remains to consider curves $C$ which are contained in the boundary. To simplify notation we write the decomposition of the boundary $D$ as $$ D=\sum_{i=1}^N\overline{D}_{g-1}^i(n) $$ where $N=N(n,g)$ can be computed explicitly. Then \begin{equation}\label{formula2.4} H|_{\overline{D}_{g-1}^1(n)}=\left. \left(aL-b\sum_{i\neq1}\overline{D}_{g-1}^i(n)\right) \right|_{\overline{D}_{g-1}^1(n)}- b\overline{D}_{g-1}^{1}(n)|_{\overline{D}_{g-1}^{1}(n)}. \end{equation} Now let $g=2$ or $3$ where we have the fibration $$ \pi':\overline{D}_{g-1}^1(n)\longrightarrow{\cal A}_{g-1}^*(n). $$ We shall denote the boundary of ${\cal A}_{g-1}^*(n)$ by $B$. Also note that the restriction of $L$ to the boundary equals ${\pi'}^* L_{{\cal A}_{g-1}^*(n)}$ where we use the notation $L$ for both the line bundle on ${\cal A}_g^*(n)$ and ${\cal A}_{g-1}^*(n)$. Thus we find that \begin{equation}\label{formula2.4a} H|_{\overline{D}_{g-1}^1(n)}= {\pi'}^*(aL-bB)-b\overline{D}_{g-1}^{1}(n)|_{\overline{D}_{g-1}^{1}(n)}. \end{equation} In view of the definition of the line bundle $\overline{M}(n)$ this gives \begin{equation}\label{formula2.5} H|_{\overline{D}_{g-1}^1(n)}={\pi'}^*\left(\left(a-\frac bn\right)L-bB\right)+\frac bn\overline{M}(n). \end{equation} \begin{Proof*}{Proof of Theorem~(\ref{theo0.1}) for $g=2$} In this case the boundary components $\overline{D}_{1}^i(n)$ are isomorphic to Shioda's modular surface $S(n)$ and the projection $\pi'$ is just projection to the modular curve $X(n)$. The degree of $L$ on $X(1)$ is $\frac{1}{12}$ and we have one cusp. Hence $$ \deg_{X(n)}(aL-bB)=\mu(n)\left(\frac{a}{12}-\frac bn\right) $$ where $\mu(n)$ is the degree of the Galois covering $X(n)\to X(1)$, i.e.~$\mu(n)=|\on{PSL}(2,{\mathbb{Z}}/n{\mathbb{Z}})|$. This is non-negative if and only if $a-12\frac bn\geq0$. The normal bundle of $\overline{D}_{1}^i(n)$ can also be computed explicitly. This can be done as follows: Using the degree $10$ cusp form which vanishes on the reducible locus one finds the equality $10L=2H_1+D$ on ${\cal A}_2^*$ where $H_1$ is the Humbert surface parametrizing polarized abelian surfaces which are products. Hence we conclude for the canonical bundle on ${\cal A}_2^*(n)$ that $K=(3-\frac{10}{n})L+\frac{2}{n}H_1$. The restriction of the divisor $H_1$ to a boundary component $\overline{D}_{1}^i(n) \cong S(n)$ is the sum of the $n^2$ sections $L_{ij}$ of $S(n)$. The canonical bundle of the surfaces $S(n)$ is equal to the pull-back via ${\pi'}$ of $3L$ minus the divisor of the cusps on the modular curve $X(n)$ (see also \cite{BH}). Hence adjunction together with an easy calculation gives $$ -n\overline{D}_{1}^i(n)|_{\overline{D}_{1}^i(n)} =2{\pi'}^* L_{X(n)}+2\sum {L_{ij}} $$ Since $L_{ij}|_{L_{ij}}=-L_{X(n)}$ one sees immediately that this line bundle is nef and positive on the fibres of $\pi':S(n)\to X(n)$. The result now follows directly from (\ref{formula2.4a}). \end{Proof*} We shall now turn to the case $g=3$. As we have remarked before it remains to consider curves which are contained in the boundary of ${\cal A}_3^*(n)$. Among those curves we shall first deal with curves whose image under the map $\pi'$ meets the interior of ${\cal A}_2(n)$. \begin{proposition}\label{prop2.8} Let $H=aL-bD$ be a divisor on ${\cal A}_3^*(n)$ with $a-12\frac bn>0$, $b>0$. For every curve $C$ in a boundary component $\overline{D}_2(n)$ with $\pi'(C)\cap{\cal A}_2(n)\neq\emptyset$ the intersection number $H.C>0$. \end{proposition} \begin{Proof} We shall use (\ref{formula2.5}) and Proposition~(\ref{prop2.6}). If we replace $n$ by some multiple and consider the pull-back of $H$ the coefficient $b/n$ is not changed. The inverse image of $C$ may have several components. All of these are, however, equivalent under some finite sympectic group and it is sufficient to prove that the degree of $H$ is positive on one (and hence on every) component lying over $C$. After this reduction we can again assume that $C$ is irreducible and by Proposition~(\ref{prop2.6}) we can find for every $\varepsilon>0$ a divisor ${\cal C}$ not containing $C$ with $$ \overline{M}(n)={\cal C} +\frac{\lambda}{k} \pi^*B,\qquad \frac{\lambda}{k}\geq \frac{n}{12+\varepsilon}. $$ By (\ref{formula2.5}) $$ H|_{\overline{D}_2(n)}=\pi^*\left(\left(a-\frac bn\right)L-b\left(1- \frac{\lambda}{nk}\right)B\right)+\frac bn {\cal C}. $$ The assertion follows from the corresponding result for $g=2$ provided $$ \left(a-\frac bn\right)-12\frac bn\left(1-\frac{\lambda}{nk}\right)\ge \left(a-12\frac bn\right)-\frac bn \left(1-\frac{12}{12+\varepsilon}\right)>0. $$ Since $a-12b/n>0$ this is certainly the case for $\varepsilon$ sufficiently small. \end{Proof} We are now left with curves in the boundary of ${\cal A}_3^*(n)$ whose image under $\pi'$ is contained in the boundary of ${\cal A}_2^*(n)$. These are exactly the curves which are contained in more than $1$ boundary component of ${\cal A}_3^*(n)$. Before we conclude the proof, we have to analyze the situation once more. First of all we can assume by symmetry arguments that $C$ is contained in $\overline{D}_2(n)$=$\overline{D}_{2}^1(n)$. Let $B'$ be a component of the boundary $B$ of ${\cal A}_2^*(n)$ which contains $\pi'(C)$. Let $D'=({\pi'})^{-1}(B')$. Then $D'$ consists of $n$ irreducible components and we have the following commutative diagram $(n\ge 3)$: $$ \diagram \overline{D}_2^1(n) \rto^{\pi'} \morphism{\dottedwith{}}\notip\notip[d]|{\displaystyle\cup} &{\cal A}_2^*(n) \morphism{\dottedwith{}}\notip\notip[d]|{\displaystyle\cup} \\ D' \rto^{\pi'} \drto_{\pi} & B'\cong S(n) \dto^{\pi''} \\ & X(n). \enddiagram $$ \noindent Altogether there are three possibilities:\\ (1) $\pi'(C)=pt$, i.e. $C'$ is contained in a fibre of $\pi'$.\\ (2) $\pi(C)=pt, \pi'(C) \neq pt$. Then $\pi'(C)$ is either a smooth fibre of $S(n)$ or a component of a singular $n$--gon.\\ (3) $\pi(C)=X(n)$.\\ The final step in the proof of Theorem (\ref{theo0.1}) is the following: \begin{proposition}\label{prop2.10} Let $C\subset \overline{D}_2(n)$ be a curve whose image $\pi'(C)$ is contained in the boundary of ${\cal A}_2^{*}(n)$. If $H=aL-bD$ is a divisor with $b>0, a-12\frac bn>0$ then $H.C>0$. \end{proposition} \begin{Proof} By induction on $g$ and formula (\ref{formula2.4a}) it is enough to prove that there is some $\overline{D}_{2}^{j}(n)$ with $C.\overline{D}_{2}^{j}(n)\le 0$. Consider the inverse image $D'$ of $B'$ under $\pi'$. Then $D'$ consists of $n$ irreducible components each of which is of the form $\overline{D}_2^i(n)\cap\overline{D}_2^1(n)$ for some $i\neq 1$. We already know that $-B'|_{B'}$ is nef. Hence $$ \left(\sum\limits_{i\in I} \overline{D}_2^i(n)\cap\overline{D}_2^1(n)\right).C\le 0 $$ where $I$ is a suitable set of indices consisting of $n$ elements. In particular $\overline{D}_{2}^{j}(n).C\le 0$ for some index $j$. \end{Proof} \begin{urems} (i) If $\pi'(C)=pt$, then one can give an alternative proof of $\overline{D}_2(n).C>0$ by computing the normal bundle of $\overline{D}_2(n)$ restricted to the singular fibres of $\pi'$. The conormal bundle is ample as in the smooth case (cf. Lemma (\ref{lemma2.2})). \\ (ii) If $\pi'(C)\neq pt$ one can also use the theta functions $\Theta_{m'm''}$ with $m', m''\in\frac 12 {\mathbb{Z}}^2$ to construct sections of $\overline{M}(n)$ which, after subtracting suitable components of the form $\overline{D}_{2}^{i}(n)\cap\overline{D}_2^1(n)$, do not vanish identically on $C$. In this way one can compute similarly to the proof of Proposition (\ref{prop2.8}) that $H.C> 0$. \end{urems} \noindent{\em Proof of Theorem } (0.1)(g=3). This follows now immediately from Proposition (\ref{prop1.4}), Proposition (\ref{prop2.8}) and Proposition (\ref{prop2.10}). \hfill $\Box$\\ \noindent {\em Proof of the corollaries}. These follow immediately from Theorem (\ref{theo0.1}) since the moduli spaces are smooth and since $$ K\equiv(g+1)L-D. $$ Obviously $$ (g+1)-\frac{12}n \ge 0\Leftrightarrow\left\{ \begin{array}{ccl} n \ge 4 &\mbox{ if }& g=2\\ n \ge 3 &\mbox{ if }& g=3. \end{array} \right. $$ \noindent Hence $K$ is nef if $g=2, n\ge 4$ and $g=3, n\ge 3$, resp. numerically positive if $g=2, n\ge 5$ and $g=3, n\ge 4$. It follows from general results of classification theory that $K$ is ample in the latter case.\hfill$\Box$ \bibliographystyle{alpha}

9708

alg-geom/9708021

en

https://arxiv.org/abs/alg-geom/9708021

alg-geom/9708021

Uwe Nagel

M. Kreuzer, J. C. Migliore, U. Nagel, C. Peterson

Determinantal schemes and Buchsbaum-Rim sheaves

20 pages, LaTeX

Let $\phi$ be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $\phi$ has the expected codimension. We call $B_{\phi} = \ker \phi$ a (first) Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing $\phi$ as a matrix (after choosing bases), we say that $X$ is good if one can delete a generalized row from $\phi$ and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown that being a good determinantal scheme of codimension $r+1$ is equivalent to being the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank $r+1$. It is also equivalent to being standard determinantal and locally a complete intersection outside a subscheme $Y \subset X$ of codimension $r+2$. Furthermore, for any good determinantal subscheme $X$ of codimension $r+1$ there is a good determinantal subscheme $S$ codimension $r$ such that $X$ sits in $S$ in a nice way. This leads to several generalizations of a theorem of Kreuzer. For example, we show that for a zeroscheme $X$ in $\proj{3}$, being good determinantal is equivalent to the existence of an arithmetically Cohen-Macaulay curve $S$, which is a local complete intersection, such that $X$ is a subcanonical Cartier divisor on $S$.

alg-geom

\section{Introduction} A natural and efficient method for producing numerous examples of interesting schemes is to consider the vanishing locus of the minors of a homogeneous polynomial matrix. If the matrix satisfies certain genericity conditions then the resulting schemes have a number of well described properties. These objects have been studied in both a classical context and a modern context and go by the name of determinantal schemes. Some of the classical schemes that can be constructed in this manner are the Segre varieties, the rational normal scrolls, and the Veronese varieties. In fact, it can be shown (cf.\ \cite{harris}) that any projective variety is isomorphic to a determinantal variety arising from a matrix with linear entries! Due to their important role in algebraic geometry and commutative algebra, determinantal schemes and their associated rings have both merited and received considerable attention in the literature. Groundbreaking work has been carried out by a number of different authors; we direct the reader to the two excellent sources \cite{bruns-vetter} and \cite {eisenbud} for background, history, and a list of important papers. A homogeneous polynomial matrix can be viewed as defining a map between free modules defined over the underlying polynomial ring. Associated to such a map are a number of complexes. The most important of these are the Eagon-Northcott and Buchsbaum-Rim complexes. Under appropriate genericity conditions, these complexes are exact and it is in this special situation where we will focus our attention. Buchsbaum-Rim sheaves are a family of sheaves associated to the sheafified Buchsbaum-Rim complex. In particular, a first Buchsbaum-Rim sheaf is the kernel of a generically surjective map between two direct sums of line bundles, whose cokernel is supported in the correct codimension. This family of sheaves is described and studied in the two papers \cite{mig-pet}, \cite{MNP}. A certain aspect of these sheaves was found to bear an interesting relationship to earlier work of the first author. In \cite{kreuzer}, Kreuzer obtained the following characterization of 0-dimensional complete intersections in $\proj{3}$: \bigskip \noindent {\bf Theorem} (\cite{kreuzer} Theorem 1.3) \ \ {\em A 0-dimensional subscheme $Y \subset \proj{3}$ is a complete intersection if and only if $Y$ is arithmetically Gorenstein and there exists an arithmetically Cohen-Macaulay, l.c.i.\ curve $C$ such that $Y$ is the associated subscheme of an effective Cartier divisor on $C$ and ${\cal O}_C (Y) \cong \omega_C (-a_Y )$ is globally generated.} \bigskip Complete intersections form a very important subset of the more general class of standard determinantal schemes (i.e the determinantal subschemes of $\proj{n}$ arising from the maximal minors of a homogeneous matrix of the ``right size"). One immediately observes that to every standard determinantal scheme is associated a number of Buchsbaum-Rim sheaves and to every Buchsbaum-Rim sheaf is associated a standard determinantal ideal. We say a standard determinantal scheme is ``good" if one can delete a generalized row from its corresponding matrix and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In particular, complete intersections are good, as are most standard determinantal schemes. The paper is organized as follows. In Section 2 we provide the necessary background information. The next section is the heart of the paper. Here we give several characterizations of standard and good determinantal subschemes. Some of these results are summarized in the following: \bigskip \noindent {\bf Theorem}{\em \ \ Let $X$ be a subscheme of $\proj{n}$ with $codim \ X \geq 2$. The following are equivalent. \newcounter{tempA} \begin{list} {(\alph{tempA})}{\usecounter{tempA}} \setlength{\rightmargin}{\leftmargin} \item $X$ is a good determinantal scheme of codimension $r+1$. \item $X$ is the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank $r+1$. \item $X$ is standard determinantal and locally a complete intersection outside a subscheme $Y \subset X$ of codimension $r+2$ in $\proj{n}$. \end{list} } \noindent Several of our results in Section~\ref{char-good} involve the cokernel of the map of free modules mentioned above. We do not quote these results here since we need some notation from Section~\ref{prelim-sect}. These results are important in Section~\ref{corollaries}, though, where we give our main generalizations of Kreuzer's theorem. We mention two of these. \bigskip \noindent {\bf Corollary}{\em \ \ Let $X \subset \proj{n}$ be a subscheme of codimension $r+1 \geq 3$. Then $X$ is a complete intersection if and only if $X$ is arithmetically Gorenstein and there is a good determinantal subscheme $S \subset \proj{n}$ of codimension $r$ and a canonically defined sheaf ${\cal M}_S$ on $S$ (in codimension two, ${\cal M}_S \cong \omega_S$ up to twist) such that $X \subset S$ is the zero-locus of a regular section $t \in H^0_* (S, {\cal M}_S )$. Furthermore, $S$ and ${\cal M}_S$ can be chosen so that ${\cal M}_S$ is globally generated. } \bigskip \noindent {\bf Corollary}{\em \ \ Suppose $X \subset \proj{3}$ is zero-dimensional. Then the following are equivalent: \newcounter{tempB} \begin{list} {(\alph{tempB})}{\usecounter{tempB}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item $X$ is standard determinantal and a local complete intersection; \item There is an arithmetically Cohen-Macaulay curve $S$, which is a local complete intersection, such that $X$ is a subcanonical Cartier divisor on $S$. \end{list} Furthermore, $X$ is defined by a $t \times (t+r)$ matrix if and only if the Cohen-Macaulay type of $X$ is ${{r+t-1} \choose r}$ and that of $S$ is ${{r+t-1} \choose {r-1}}$. } \bigskip \noindent The last sentence of this corollary gives the connection to Kreuzer's theorem: recall that the only standard determinantal subschemes with Cohen-Macaulay type 1 (i.e.\ arithmetically Gorenstein) are complete intersections. In a similar way we characterize good determinantal subschemes of $\proj{n}$ of any codimension, with special, stronger, results in the case of zeroschemes and the case of codimension two subschemes. We close with a number of examples. \section{Preliminaries}\label{prelim-sect} Let $R=k[x_0,x_1,\dots,x_n]$ be a polynomial ring with the standard grading, where $k$ is an infinite field and $n\geq 2$. For any sheaf $\cal F$ on $\proj{n}$, we define $H^i_*(\proj{n},{\cal F})=\bigoplus_{t\in {\Bbb Z} }H^i(\proj{n},{\cal F}(t))$. For any scheme $V \subset \proj{n}$, $I_V$ denotes the saturated homogeneous ideal of $V$ and ${\cal I}_V$ denotes the ideal sheaf of $V$ (hence $I_V= H^0_* (\proj{n},{\cal I}_V)$). \begin{definition} If $A$ is a homogeneous matrix, we denote by $I(A)$ the ideal of maximal minors of $A$. A codimension $r+1$ scheme, $X$, in $\proj{n}=Proj(R)$ will be called a {\em standard determinantal scheme} if $I_X = I(A)$ for some homogeneous $t \times (t+r)$ matrix, $A$. $X$ will be called a {\em good determinantal scheme} if additionally, $A$ contains a $(t-1) \times (t+r)$ submatrix (allowing a change of basis if necessary-- see Example~\ref{good-ex}) whose ideal of maximal minors defines a scheme of codimension $r+2$. In a similar way we define standard and good determinantal ideals. \end{definition} \begin{example} The ideal defined by the maximal minors of the matrix \[ \left [ \begin{array}{cccccccccc} x_1 & x_2 & x_3 & 0 \\ 0 & x_1 & x_2 & x_3 \end{array} \right ] \] is an example of a standard determinantal ideal which is not good. Note that this ideal is the square of the ideal of a point in $\proj{3}$, and is not a local complete intersection (see Proposition~\ref{good-det}). \end{example} Note that standard determinantal schemes form an important subclass of the more general notion of determinantal schemes, where smaller minors are allowed (among other generalizations). See for instance \cite{bruns-vetter}, \cite{eisenbud}, \cite{harris}. \begin{remark}\label{one-minor} In the next section we will make a deeper study of good determinantal schemes. For now, though, we observe the following. Let $X$ be a standard determinantal scheme coming from a $t \times (t+r)$ matrix $A$. Then $X$ is good if and only if there is a $(t-1) \times (t-1)$ minor of $A$ which does not vanish on any component of $X$ (possibly after making a change of basis). In particular, we formally include the possibility that $t=1$, and we include the complete intersections among the good determinantal schemes. \end{remark} \begin{fact} \label{EN-and-BR} Let ${\cal F}$ and ${\cal G}$ be locally free sheaves of ranks $f$ and $g$ respectively on a smooth variety $Y$. Let $\phi :{\cal F}\rightarrow {\cal G}$ be a generically surjective homomorphism. We can associate to $\phi$ an Eagon-Northcott complex \begin{equation}\label{ENSeq} \begin{array}{cccc} 0 \rightarrow \wedge^f{\cal F} \otimes (S^{f-g} {\cal G})^{\vee} \otimes \wedge^g {\cal G}^{\vee} \rightarrow \wedge^{f-1} {\cal F} \otimes (S^{f-g-1} {\cal G})^{\vee} \otimes \wedge^g {\cal G}^{\vee} \rightarrow\dots \\ \hskip1.5cm \rightarrow \wedge^{g+1} {\cal F} \otimes {\cal G}^{\vee} \otimes \wedge^g {\cal G}^{\vee} \rightarrow \wedge^g {\cal F} \otimes \wedge^g {\cal G}^{\vee} \buildrel \wedge^g \phi \over \rightarrow {\cal O}_Y \rightarrow 0\\ \end{array} \end{equation} and a Buchsbaum-Rim complex \begin{equation}\label{BRSeq} \begin{array}{cccc} 0 \rightarrow \wedge^f {\cal F} \otimes S^{f-g-1} {\cal G}^\vee \otimes \wedge^g {\cal G}^\vee \rightarrow \wedge^{f-1} {\cal F} \otimes S^{f-g-2} {\cal G}^\vee \otimes \wedge^g {\cal G}^\vee \rightarrow \dots \\ \hskip 1cm \rightarrow \wedge^{g+2} {\cal F} \otimes {\cal G}^\vee \otimes \wedge^g {\cal G}^\vee \rightarrow \wedge^{g+1} {\cal F} \otimes \wedge^g {\cal G}^\vee \rightarrow {\cal F} \buildrel \phi \over \rightarrow {\cal G} \rightarrow 0\\ \end{array} \end{equation} (see \cite{GLP}, \cite{eisenbud}, \cite{BR}, \cite{EN}, \cite{buchs64}). If the support of the cokernel of $\phi$ has the expected codimension $f-g+1$ then these complexes are acyclic. \end{fact} The consequences of this fact will play a crucial role throughout the paper and they lead us to the following definition. \begin{definition}\label{def-BR} Let ${\cal F}$ and ${\cal G}$ be two locally free sheaves which split as the sum of line bundles and let $\phi :{\cal F}\rightarrow {\cal G}$ be a generically surjective homomorphism whose cokernel is supported on a scheme with the ``expected'' codimension $f-g+1$. As mentioned in the fact above, the Buchsbaum-Rim complex will be exact and provides a free resolution of the cokernel of the map $\phi$. The kernel of the map $\phi$ will be called a {\it first Buchsbaum-Rim sheaf}. We use the symbol ${\cal B}_{\phi}$ to represent such a sheaf. \end{definition} More generally, the $i^{th}$ Buchsbaum-Rim sheaf associated to $\phi$ is the $(i+1)^{st}$ syzygy sheaf in the Buchsbaum-Rim complex. However, in this paper we will use only the first Buchsbaum-Rim sheaves. \begin{remark} \label{free-is-BR} In Fact~\ref{EN-and-BR} and Definition~\ref{def-BR}, we will allow the rank of $\cal G$ to be zero, and use the convention that even in this case, $\wedge^0 {\cal G}^\vee = {\cal O}_Y$. Moreover, the Buchsbaum-Rim complex becomes $0 \rightarrow {\cal F} \rightarrow {\cal F} \buildrel \phi \over \rightarrow 0$, and it follows that the sheafification of any free module is a first Buchsbaum-Rim sheaf. In Fact~\ref{EN-and-BR} and Definition~\ref{def-BR}, we can also start with free modules $F$ and $G$, and we get Eagon-Northcott and Buchsbaum-Rim complexes of free modules. The corresponding kernel of the map $\phi$ will then be called a {\em first Buchsbaum-Rim module}. Note that in this context $\phi$ can be represented by a homogeneous matrix $\Phi$, and the image of $\wedge^g \phi$ is precisely $I(\Phi )$. Note also that since first Buchsbaum-Rim sheaves (resp.\ modules) are second syzygy sheaves (resp.\ modules), they are reflexive. \end{remark} \begin{fact} (\cite{eisenbud} exer.\ 20.6 or \cite{BE77})\label{annihilator} Let $\Phi$ be a matrix whose ideal $I(\Phi )$ of maximal minors vanishes in the expected codimension, and so $coker \ \Phi$ has a corresponding Buchsbaum-Rim resolution. Then the annihilator of $coker \ \Phi$ is precisely $I(\Phi )$. \end{fact} In this paper, we will often be interested in going in the opposite direction, starting with a standard determinantal ideal $J$ and considering the possible associated matrices and cokernels. With this in mind, we make the following definition. \begin{defn}\label{Mv} \begin{rm} Let $X$ be a standard determinantal scheme of codimension $r+1$ with corresponding ideal $I_X$. Then we set \[ {\cal M}_X := \left \{ \begin{array}{c|c} M & \begin{array}{l} \hbox{\rm $M$ is a f.g.\ graded $R$-module with $Ann_R M = I_X$ and a minimal} \\ \hbox{\rm presentation of the form $\displaystyle R^{r+\mu} \rightarrow R^\mu \rightarrow M \rightarrow 0$} \end{array} \end{array} \right \} \] \end{rm} \end{defn} \bigskip \noindent ${\cal M}_X$ is the set of possible cokernels of homogeneous matrices whose ideals of maximal minors are precisely $I_X$. In some situations, ${\cal M}_X$ consists of just one element (up to isomorphism and twisting). For example, it can be shown that this happens if $r=1$ (i.e.\ codimension 2, using Hilbert-Burch theory-- see Corollary~\ref{codim2}). ${\cal M}_X$ also consists of just one element if $X$ is a complete intersection. We do not know the precise conditions which guarantee that all the elements of ${\cal M}_X$ are isomorphic up to twisting. In any case, we can at least show that the elements of ${\cal M}_X$ look very much alike: \begin{lemma} The elements of ${\cal M}_X$ all have the same graded Betti numbers, up to twisting, and in particular come from matrices of the same size. \end{lemma} \noindent {\em Proof:} Let $M_1, M_2 \in {\cal M}_X$ and assume that $M_i$ has $t_i$ minimal generators, $i = 1,2$. We may also assume that $M_i$ is the cokernel of a $t_i \times (t_i +r)$ matrix $\Phi_i$. By \cite{eisenbud} p.\ 494, $Rad(I(\Phi )) = Rad (Ann_R M_i) = Rad (I_X )$. Hence $I(\Phi)$ is a homogeneous matrix defining a subscheme of $\proj{n}$ of codimension $r+1$, the expected codimension, and we may apply the Eagon-Northcott complex to get a minimal free resolution for $I(\Phi ) = I_X$. Hence $I_X$ has ${{r+t_1} \choose r} = {{r+t_2} \choose r}$ minimal generators, and $t_1 = t_2$. Now let $M \in {\cal M}_X$ and assume that it has $t$ minimal generators. There is a minimal free resolution \[ \cdots \rightarrow F \buildrel \Phi \over \rightarrow G \rightarrow M \rightarrow 0 \] where $rk \ F = t+r$ and $rk \ G = t$. As above, $I(\Phi)$ defines a subscheme of codimension $r+1$, and so the Buchsbaum-Rim complex resolves $M$ and we are done. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{prop}\label{entries-of-syz} Let ${\cal F}$ and ${\cal G}$ be locally free sheaves of ranks $f$ and $g$ respectively on $\proj{n}$. Let $\phi :{\cal F}\rightarrow {\cal G}$ be a generically surjective homomorphism. Assume the cokernel of $\phi$ is supported on a scheme of codimension $f-g+1$. Let $I_{\phi}$ denote the homogeneous ideal of the scheme determined by the cokernel of $\wedge^g\phi$. Let $I_s$ denote the homogeneous ideal of the zero-locus of a section, $s\in H^0(\proj{n},{\cal B}_{\phi})$ (where ${\cal B}_{\phi}$ denotes the local first Buchsbaum-Rim sheaf of $\phi$). Let $I_t$ denote the homogeneous ideal of the zero-locus of a section, $t\in H^0(\proj{n},{\cal B}_{\phi}^*)$ (where ${\cal B}_{\phi}^*$ denotes the dual of ${\cal B}_{\phi}$). Then for any such section, $I_s\subset I_{\phi}$ and $I_t\subset I_{\phi}$ \end{prop} \noindent {\em Proof:} Locally, we can represent the map $\phi$ by an $f\times g$ matrix, $A$. In the same local coordinates, the map from $\wedge^{g+1}{\cal F}\otimes \wedge^g{\cal G}^{\vee}$ to ${\cal F}$ (in the Buchsbaum-Rim complex associated to $\phi$) can be expressed by a matrix, $M$. The entries of $M$ can be written in terms of $A$ as follows. Let $I_A$ denote the ideal of maximal minors of the matrix $A$. $I_A$ locally describes the scheme defined by $I_{\phi}$. Each column in the matrix, $M$, arises from choosing $t+1$ columns of the matrix $A$ and considering all $t\times t$ minors of this submatrix of $A$. Thus, each entry in the matrix $M$ is an element of $I_A$. Locally, sections of the first Buchsbaum-Rim sheaf of $\phi$ are determined by an element of the column space of $M$ (considered as a module). An immediate consequence of this fact is that the vanishing locus of any section of the first Buchsbaum-Rim sheaf of $\phi$ or the dual of the first Buchsbaum-Rim sheaf of $\phi$ will contain the scheme defined by $I_{\phi}$. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{remark} For clarity, and because of its importance, we restrict ourselves to determinantal subschemes of projective space in the body of this paper. However, the reader will observe that many of our arguments hold true for subschemes of a smooth projective variety and some even for determinantal ideals of an arbitrary commutative ring. \end{remark} \section{Characterizations of Good Determinantal Schemes}\label{char-good} In \cite{mig-pet} and \cite{MNP}, regular sections of first Buchsbaum-Rim sheaves were considered, and it was shown that they possess many interesting properties. For example, a regular section of a first Buchsbaum-Rim sheaf of odd rank has a zero-locus whose top dimensional part is arithmetically Gorenstein. In this paper we are primarily concerned with regular sections of the {\em dual} of a first Buchsbaum-Rim sheaf. Our first result gives a property which is analogous to the ones mentioned above for the first Buchsbaum-Rim sheaves. \begin{thm}\label{good-iff-sect} Let $X$ be a subscheme of $\proj{n}$ with $codim \ X \geq 2$. The following are equivalent. \newcounter{temp} \begin{list} {(\alph{temp})}{\usecounter{temp}} \setlength{\rightmargin}{\leftmargin} \item $X$ is a good determinantal scheme of codimension $r+1$. \item $X$ is the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank $r+1$. \end{list} \end{thm} \noindent {\em Proof:} We first prove (a) $\Rightarrow$ (b). By assumption there is a homomorphism $\Phi$ such that $I_X = I(\Phi )$, and we have an exact sequence \begin{equation} 0 \rightarrow B \rightarrow F \buildrel \Phi \over \rightarrow G \rightarrow coker \ \Phi \rightarrow 0 \end{equation} where $rk \ G = t$, $rk \ F = t+r$ and $B$ is a first Buchsbaum-Rim module. If $t=1$ then $I(\Phi )$ is a complete intersection of height $r+1$, which can be viewed as a section of (the dual of) a free module of rank $r+1$. By Remark~\ref{free-is-BR}, a free module is a first Buchsbaum-Rim module. Hence we can assume from now on that $t \geq 2$. Since $X$ is a good determinantal scheme, there is a projection $\pi : G \rightarrow G'$, where $G'$ has rank $t-1$, $G'$ is obtained from $G$ by removing one free summand $R(a)$, and such that $ht ( I(\pi \circ \Phi )) = r+2$. We get a commutative diagram \begin{equation}\label{usual-diag} \begin{array}{ccccccccccc} &&&&&& 0 \\ &&&&&& \downarrow \\ &&&& 0 & \rightarrow & R(a) & \rightarrow & R(a) & \rightarrow & 0 \\ &&&& \downarrow && \downarrow \\ 0 & \rightarrow & B & \rightarrow & F & \buildrel \Phi \over \rightarrow & G & \rightarrow & coker \ \Phi & \rightarrow & 0 \\ &&&& || && \phantom \pi \downarrow \pi \\ 0 & \rightarrow & B' & \rightarrow & F & \buildrel {\Phi '} \over \rightarrow & G' & \rightarrow & coker \ \Phi ' & \rightarrow & 0 \\ &&&& \downarrow && \downarrow \\ &&&& 0 && 0 \\ \end{array} \end{equation} Let $\alpha$ be the induced injection from $B$ to $B'$. Twist everything in (\ref{usual-diag}) by $-a$ and relabel, so that the Snake Lemma gives that $I = coker \ \alpha$ is an ideal and we have an exact sequence \begin{equation}\label{short-exact} 0 \rightarrow R/I \rightarrow coker \ \Phi \rightarrow coker \ \Phi ' \rightarrow 0 \end{equation} It follows that $I_X = I(\Phi ) = Ann ( coker \ \Phi) \subset I$ (see Fact~\ref{annihilator}), where $I_X$ is the saturated ideal of $X$. On the other hand, it follows from the same exact sequence that \[ Ann ( coker \ \Phi' ) \cdot I \subset Ann(coker \ \Phi ) = I(\Phi ) = I_X. \] But since $X$ is good determinantal, it follows that $I(\Phi' ) = Ann ( coker \ \Phi' )$ and $ht ( I(\Phi' )) > ht (I(\Phi ))$. Hence $I \subset I(\Phi )$ and so we conclude $I = I(\Phi ) = I_X$. But then we have a short exact sequence \[ 0 \rightarrow B \rightarrow B' \rightarrow I_X \rightarrow 0 \] and so by sheafifying, it follows that $X$ is the zero-locus of a regular section of the dual of the first Buchsbaum-Rim sheaf ${\cal B}'$ as claimed. (Note that $B'$ is reflexive-- see Remark~\ref{free-is-BR}.) We now prove (b) $\Rightarrow$ (a). Assume that $X$ is the zero-locus of a regular section of a sheaf $({\cal B}')^*$, where ${\cal B'}$ is the sheafification of a first Buchsbaum-Rim module $B'$ of rank $r+1$. We are thus given exact sequences (after possibly replacing $B'$ by a suitable twist) \begin{equation}\label{BRseq} 0 \rightarrow B' \rightarrow F \buildrel {\Phi'} \over \longrightarrow G \rightarrow coker \ \Phi' \rightarrow 0 \end{equation} and \begin{equation}\label{sect} 0 \rightarrow R \rightarrow (B')^* \rightarrow Q \rightarrow 0 \end{equation} such that $rk \ F = t+r$, $rk \ G = t-1$, $Ann ( coker \ \Phi' ) = I(\Phi ')$ (which has height $r+2$) and \[ 0 \rightarrow Q^* \rightarrow B' \rightarrow I \rightarrow 0 \] is exact (again, $B'$ is reflexive), where $I$ is an ideal whose saturation is $I_X$. One can check that dualizing (\ref{BRseq}) provides \[ 0 \rightarrow G^* \rightarrow F^* \rightarrow (B')^* \rightarrow 0. \] The mapping cone procedure applied to (\ref{sect}) then gives \[ 0 \rightarrow R \oplus G^* \rightarrow F^* \rightarrow Q \rightarrow 0. \] Dualizing this, we obtain the following commutative diagram: \[ \begin{array}{cccccccccccc} &&&&&& 0 \\ &&&&&& \downarrow \\ &&0 && 0 && R \\ && \downarrow && \downarrow && \downarrow \\ 0 & \rightarrow & Q^* & \rightarrow & F & \buildrel \Phi \over \rightarrow & R \oplus G & \rightarrow & coker \ \Phi & \rightarrow & 0 \\ && \downarrow && || && \downarrow && \downarrow \\ 0 & \rightarrow & B' & \rightarrow & F & \buildrel {\Phi '} \over \rightarrow & G & \rightarrow & coker \ \Phi ' & \rightarrow & 0 \\ && \downarrow && \downarrow && \downarrow \\ && I && 0 && 0 \\ && \downarrow \\ && 0 \end{array} \] The Snake Lemma then gives \[ 0 \rightarrow R/I \rightarrow coker \ \Phi \rightarrow coker \ \Phi' \rightarrow 0. \] It follows that \[ I \cdot Ann (coker \ \Phi' ) = I \cdot I(\Phi' ) \subset Ann(coker \ \Phi ). \] Thus $ht (Ann ( coker \ \Phi )) \geq r+1$. Note that the maximal possible height of $Ann ( coker \ \Phi )$ is $r+1$, hence we get $ht (Ann ( coker \ \Phi )) = r+1$ and $Q^*$ is a first Buchsbaum-Rim module. From the Buchsbaum-Rim complex one can then check that $H^1_* (\proj{n} , {\cal Q}^* ) = 0$, and hence $I = I_X$ is saturated. Then as in the first part we get $I_X = I(\Phi )$, as desired. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \medskip We now give a result which characterizes the good determinantal schemes among the standard determinantal schemes. We use the set ${\cal M}_X$ introduced in Definition~\ref{Mv}. \begin{prop}\label{good-det} Suppose that $X$ is a standard determinantal scheme of codimension $r+1$. Then the following are equivalent. \newcounter{temp2} \begin{list} {(\alph{temp2})}{\usecounter{temp2}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item There is an $M_X \in {\cal M}_X$ and an embedding $R/I_X \hookrightarrow M_X$ whose image is a minimal generator of $M_X$ as an $R$-module, and whose cokernel is supported on a subscheme of codimension $\geq r+2$. \item There is an element $M_X \in {\cal M}_X$ which is an ideal in $R/I_X$ of positive height. \end{list} Furthermore, if any of the above conditions hold then $X$ is a local complete intersection outside a subscheme $Y \subset \proj{n}$ of codimension $r+2$. \end{prop} \begin{remark} The first two parts of the above proposition do not even require that the field be infinite. \end{remark} \noindent {\em Proof of \ref{good-det}} We begin with (a) $\Rightarrow$ (b). Assume that $X$ is a good determinantal scheme arising from a homogeneous matrix $\Phi$. As in the proof of Theorem~\ref{good-iff-sect} (see the diagram (\ref{usual-diag})), we have (after possibly twisting) a commutative diagram \begin{equation} \begin{array}{ccccccccccc} &&&&&& && 0 \\ &&&&&& && \downarrow \\ && 0 && & & R & \rightarrow & R/I_X & \\ && \downarrow && && \downarrow && \downarrow \\ 0 & \rightarrow & B & \rightarrow & F & \buildrel \Phi \over \rightarrow & G & \rightarrow & M_X & \rightarrow & 0 \\ && \downarrow && || && \phantom \pi \downarrow \pi && \downarrow \\ 0 & \rightarrow & B' & \rightarrow & F & \buildrel {\Phi '} \over \rightarrow & G' & \rightarrow & M_Y & \rightarrow & 0 \\ &&\downarrow && && \downarrow && \downarrow \\ && I_X &&&& 0 && 0 \\ && \downarrow \\ && 0 \end{array} \end{equation} where $rk \ F = t+r$, $rk \ G = t$, $rk \ G' = t-1$, $\Phi'$ is obtained by deleting a suitable row of $\Phi$, $Y$ is the codimension $r+2$ scheme defined by the maximal minors of $\Phi'$, $B$ and $B'$ are the kernels of $\Phi$ and $\Phi'$, respectively, and $M_X$ and $M_Y$ are the respective cokernels. Then all parts of (b) follow immediately. This diagram also proves the last part of the Proposition, since by Theorem~\ref{good-iff-sect} $X$ is the zero-locus of a section of ${\cal B}'$, the sheafification of $B'$, which is locally free of rank $r+1$ outside $Y$. We now prove (b) $\Rightarrow$ (a). The assumptions in (b) imply a commutative diagram \[ \begin{array}{cccccccccc} && 0 && 0 \\ && \downarrow && \downarrow \\ && R & \rightarrow & R/I_X & \rightarrow & 0\\ && \downarrow && \phantom s \downarrow s \\ F & \buildrel \Phi \over \rightarrow & G & \rightarrow & M_X & \rightarrow & 0\\ && \phantom \alpha \downarrow \alpha && \downarrow \\ && G' & \buildrel \beta \over \rightarrow & coker \ s & \rightarrow & 0 \\ && \downarrow && \downarrow \\ && 0 && 0 \end{array} \] with $rk \ F = t+r$, $rk\ G = t$, $rk \ G' = t-1$. Define $\Phi' = \alpha \circ \Phi$. One can then show that \[ F \buildrel {\Phi '} \over \rightarrow G' \buildrel \beta \over \rightarrow coker \ s \rightarrow 0 \] is exact. (Either use a mapping cone argument, splitting off $R$, or else use a somewhat tedious diagram chase.) The assumption on the support of the cokernel of $s$ implies $height (I(\Phi')) = r+2$, so $X$ is good, proving (a). Now we prove (a) $\Rightarrow $ (c). The assumption that $X$ is good implies, in particular, that the ideal of $(t-1) \times (t-1)$ minors of $\Phi$ has height $\geq r+2$. Hence after possibly making a change of basis, we can apply Remark~\ref{one-minor} and \cite{eisenbud} Theorem~A2.14 (p.\ 600) to obtain $M_X = coker \ \Phi \cong J/I_X$, where $J \subset R$ is an ideal of height $\geq r+2$, proving (c). Finally we prove (c) $\Rightarrow$ (b). Since $M_X$ is an ideal of positive height in $R/I_X$, we can find $f \in R$ with $\bar f = f \ mod \ I_X \in M_X$ such that the map $R/I_X \buildrel s \over \rightarrow M_X , \ 1 \mapsto \bar f$ is injective. We can even choose $f$ so that $\bar f$ is a minimal generator of $M_X$, considered as an $R$-module. Then $coker \ s \cong M_X / (\bar f \cdot R/I_X )$ shows that $I_X + (f) \subset Ann_R (coker \ s)$, so $coker \ s$ is supported on a subscheme of height $\geq r+2$. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \bigskip Next, we want to give an intrinsic characterization of good determinantal subschemes. \begin{thm}\label{good-iff-glci} Suppose that $codim \ X = r+1$. Then the following are equivalent: \newcounter{temp5} \begin{list} {(\alph{temp5})}{\usecounter{temp5}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item $X$ is standard determinantal and locally a complete intersection outside a subscheme $Y \subset X$ of codimension $r+2$ in $\proj{n}$. \end{list} \end{thm} \noindent {\em Proof:} In view of Proposition~\ref{good-det}, we only have to prove (b) $\Rightarrow$ (a). We again start with the exact sequence \[ 0 \rightarrow B \rightarrow F \buildrel \Phi \over \rightarrow G \rightarrow M_X \rightarrow 0 \] where $F$ and $G$ are free of rank $t+r$ and $t$ respectively. Now let $P$ be a point of $X$ outside $Y$, with ideal $\wp \subset R$. By assumption, $X$ is a complete intersection at $P$. We first claim that $(M_X)_\wp \cong (R/I_X )_\wp$. To see this, we first note that localizing $\Phi$ at $\wp$, we can split off, say, $s$ direct summands until the resulting map is minimal. Then the ideal of maximal minors of this matrix has precisely ${{r+t-s} \choose {t-s}}$ minimal generators (Eagon-Northcott complex). On the other hand it is a complete intersection, hence $t-s = 1$ and the cokernel $(M_X )_\wp$ of $\Phi_\wp$ is as claimed. Using the above isomorphism, we note that $(M_X )_\wp$ has exactly one minimal generator as an $R_\wp$-module. Then by \cite{bruns-vetter}, Proposition~16.3, it follows that the ideal of submaximal minors of $\Phi$ is not contained in $\wp$. Since $P$ was chosen to be any point outside of $Y$ and $codim \ Y = r+2$, it follows that no component of $X$ lies in the ideal of submaximal minors. That is, the ideal of submaximal minors has height greater than that of $I_X$. Hence by \cite{eisenbud} p.\ 600, Theorem A2.14, we can conclude that $M_X$ is an ideal in $R/I_X$ of positive height. Therefore $X$ is good determinantal, by Proposition~\ref{good-det}, (c). \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{rmk}\label{gci}\begin{rm} Recall that a subscheme of $\proj{n}$ is said to be a {\em generic complete intersection} if it is locally a complete intersection at all its components. In particular, every integral subscheme is a generic complete intersection. This notion occurs naturally in the Serre correspondence which relates reflexive sheaves and generic complete intersections of codimension two (cf., for example, \cite{H2}). Since the locus of points at which a subscheme fails to be locally a complete intersection is closed, for a subscheme $X$ of codimension $r+1$ the conditions being a generic complete intersection and being locally a complete intersection outside a subscheme $Y \subset X$ of codimension $r+2$ in $\proj{n}$ are equivalent. Thus we can reformulate the last result as follows: \begin{quote} A subscheme is good determinantal if and only if it is standard determinantal and a generic complete intersection. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \end{quote} \end{rm} \end{rmk} \begin{lemma}\label{alg-lemma} Let $A$ be a ring and let ${\goth a} \subset A$ be an ideal containing an $A$-regular element $f$. Let ${\goth b} := fA :_A {\goth a} = Ann_A ({\goth a} /fA)$. Then $Hom_A ({\goth a}, A) \cong {\goth b}$. \end{lemma} \noindent {\em Proof:} If $grade \ {\goth a} \geq 2$ then it is well-known that $Hom_A ({\goth a}, A) \cong A$ (up to shift in the graded case). The interesting case is $grade \ {\goth a} = 1$. However, we prove it in the general case. Our main application is to the graded case, where we assume that $\goth a$ and $f$ are homogeneous; then we obtain an isomorphism of graded modules $Hom_A ({\goth a},A) \cong {\goth b}(deg \ f)$. Consider the exact sequence \[ \begin{array}{ccccccccc} 0 & \rightarrow & A & \rightarrow & {\goth a} & \rightarrow & {\goth a} /fA & \rightarrow & 0 \\ &&1 & \mapsto & f \end{array} \] Since $f$ is $A$-regular, dualizing provides \[ \begin{array}{ccccccccccc} 0 & \rightarrow & Hom_A ({\goth a} /fA ,A) & \rightarrow & Hom_A ({\goth a} ,A) & \buildrel \beta \over \rightarrow & Hom_A (A,A) \\ && || &&&& \phantom{\wr} || \wr \\ && 0 &&&& A \end{array} \] We first prove that, up to the isomorphism $Hom_A (A,A) \cong A$, we get $Hom_A ({\goth a},A) \subset {\goth b}$. Let $\phi \in Hom_A ({\goth a},A)$ and let $\psi = \beta (\phi)$. Let $b := \psi (1) = \phi(f)$. Then for any $a \in A$ we have \[ \psi (a) = \phi(f\cdot a) = a \cdot b. \] For any $a \in {\goth a}$ we have \[ f \cdot \phi(a) = \phi (f\cdot a) = \psi (a) = a \cdot b. \] Hence $b \cdot {\goth a} \subset f \cdot A$, i.e.\ $b \in fA :_A {\goth a} = {\goth b}$. It follows that $Hom_A ({\goth a},A) \cong im \ \beta \subset {\goth b}$. For the reverse inclusion we can define for any $b \in {\goth b}$ a homomorphism $\phi \in Hom_A ({\goth a},A)$ as the composition of \[ \begin{array}{ccccccc} {\goth a} & \rightarrow & fA & \hbox{ and } & fA & \buildrel \sim \over \rightarrow & A \\ a & \mapsto & ab \end{array} \] Then $\phi (f) = b$. We conclude that ${\goth b} = im \ \beta \cong Hom_A ({\goth a},A)$. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{thm} \label{std-and-good} Suppose that $r+1 \geq 3$. Then \newcounter{temp3} \begin{list} {(\alph{temp3})}{\usecounter{temp3}} \setlength{\rightmargin}{\leftmargin} \item $X$ is standard determinantal of codimension $r+1$ if and only if there is a good determinantal subscheme $S \subset \proj{n}$ of codimension $r$ such that $X \subset S$ is the zero-locus of a regular section $t \in H^0_* (S, \widetilde M_S ) = M_S$ for some $M_S \in {\cal M}_S$. \item $X$ is good determinantal of codimension $r+1$ if and only if there is a good determinantal subscheme $S \subset \proj{n}$ of codimension $r$, such that $X \subset S$ is the zero-locus of a regular section $t \in H^0_* (S, \widetilde M_S ) = M_S$ for some $M_S \in {\cal M}_S$, and the cokernel of this section is isomorphic to an ideal sheaf in ${\cal O}_X$ of positive height. \end{list} \end{thm} \noindent {\em Proof:} We first assume that $X$ is standard determinantal and we let $\Phi$ be a $t \times (t+r)$ homogeneous matrix with $I(\Phi ) = I_X$. Adding a general row to $\Phi$ gives a homogeneous $(t+1) \times (t+r)$ matrix $\Psi$ whose ideal of maximal minors defines a good determinantal scheme $S \supset X$ of codimension $r$. We have the commutative diagram \[ \begin{array}{cccccccccccc} 0 & \rightarrow & ker \ \Psi & \rightarrow & F & \buildrel \Psi \over \rightarrow & G & \rightarrow & M_S & \rightarrow & 0 \\ &&&& || && \downarrow \\ 0 & \rightarrow & ker \ \Phi & \rightarrow & F & \buildrel \Phi \over \rightarrow & G' & \rightarrow & M_X & \rightarrow & 0 \\ &&&&&&\downarrow \\ &&&&&& 0 \end{array} \] where $rk \ F = t+r$, $rk \ G' = t$ and $rk \ G = t+1$. As in Theorem~\ref{good-iff-sect}, after possibly twisting we get the exact sequence \begin{equation}\label{sect-coker} 0 \rightarrow R/I_S (- \hbox{deg } t) \buildrel t \over \rightarrow M_S \rightarrow M_X \rightarrow 0. \end{equation} Since $S$ is good by construction, Proposition~\ref{good-det} shows that Lemma~\ref{alg-lemma} applies, setting $A := R/I_S$ and ${\goth a} = M_S$. This gives \[ Hom_A (M_S ,A)(-\hbox{deg } t) \cong Ann_A (M_X ) \cong I_X /I_S . \] Now, dualizing (\ref{sect-coker}) we get \[ \begin{array}{cccccccc} 0 & \rightarrow & Hom_A (M_X ,A) & \rightarrow & Hom_A (M_S ,A) & \buildrel {t^*} \over \rightarrow A (\hbox{deg } t) \\ && || \\ && 0 \end{array} \] It follows that $X$ is the zero-locus of $t$, proving the direction $\Rightarrow$ for case (a). In case (b), we are done by applying Proposition~\ref{good-det}. We now consider the direction $\Leftarrow$. Again let $A = R/I_S$, where $I_S = I(\Psi)$ for some homogeneous $(t+1) \times (t+r)$ matrix $\Psi$, and apply the mapping cone construction to the diagram \[ \begin{array}{cccccccc} &&&& 0 \\ &&&& \downarrow \\ && R & \rightarrow & A & \rightarrow & 0 \\ && \downarrow && \phantom{t} \downarrow t \\ F & \buildrel \Psi \over \rightarrow & G & \rightarrow & M_S \\ &&&& \downarrow \\ &&&& coker \ t \\ &&&& \downarrow \\ &&&& 0 \end{array} \] where $rk \ G = t+1$. This gives the exact sequence \[ \cdots \rightarrow F \oplus R \buildrel \Phi \over \rightarrow G \rightarrow coker \ t \rightarrow 0 \] Since $S$ is good, Proposition~\ref{good-det} gives us that $coker \ t \cong M_S /f\cdot A$ for some $A$-regular element $f \in A$ (see the proof of (c) $\Rightarrow$ (b)). It follows that $Ann_R (coker \ t)$ has grade $\geq 1 + grade \ I_S = r+1$, thus $grade \ I(\Phi) = r+1$. Let $Y$ be the subscheme defined by $I(\Phi)$. Then we get as above that $Y$ is the zero-locus of $t$, and so $X = Y$, and we are done in case (a). For case (b), again an application of Proposition~\ref{good-det} completes the argument since $coker \ t \in {\cal M}_X$. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \bigskip Note that Theorem~\ref{std-and-good} does not mention global generation, while Kreuzer's theorem mentioned in the introduction does. Conjecture~\ref{glob-gen-conj} and Remark~\ref{on-conj} address this. \begin{conj}\label{glob-gen-conj} Given $X$ a standard determinantal scheme as in Theorem~\ref{std-and-good}, one can choose $S$ and $M_S \in {\cal M}_S$ such that $X \subset S$ is the zero-locus of a regular section $t \in H^0 (S, \widetilde M_S )$ and such that $\widetilde M_S$ is globally generated. \end{conj} \begin{remark}\label{on-conj} Consider a free presentation of $M_X$ as in the proof of Theorem~\ref{std-and-good}: $$ 0 \rightarrow B \rightarrow F \buildrel \Phi \over \rightarrow G \rightarrow M_X \rightarrow 0. $$ Suppose that $\widetilde G$ is globally generated and furthermore that $\widetilde B^*$ has a regular section $s$. Then we can write $$ 0 \rightarrow {\cal O} \buildrel s \over \rightarrow \widetilde B^* \rightarrow {\cal Q} \rightarrow 0. $$ A mapping cone gives a free resolution $$ 0 \rightarrow {\cal O} \oplus \widetilde G^* \rightarrow \widetilde F^* \rightarrow {\cal Q} \rightarrow 0. $$ Dualizing this sequence gives $$ 0 \rightarrow {\cal Q}^* \rightarrow \widetilde F \buildrel \Psi \over \rightarrow {\cal O} \oplus \widetilde G \rightarrow {\cal E}xt^1({\cal Q}, {\cal O}) \rightarrow 0. $$ Since $s$ is a regular section, ${\cal E}xt^1({\cal Q}, {\cal O})$ is supported on a scheme of codimension one less than the codimension of $X$. We conclude that $\Psi$ is a Buchsbaum-Rim matrix, and hence $\widetilde M_S ={\cal E}xt^1({\cal Q}, {\cal O})$ for the scheme $S$ defined by the maximal minors of $\Psi$. As in the proof of Theorem~\ref{std-and-good}, we obtain the exact sequence $$ 0 \rightarrow R/I_S \rightarrow M_S \rightarrow M_X \rightarrow 0. $$ Since ${\cal O} \oplus \widetilde G$ is globally generated, we see that $\widetilde M_S$ is globally generated as an ${\cal O}$-module (and hence as an ${\cal O}_S$-module). We have just shown that Conjecture~\ref{glob-gen-conj} is true whenever we can simultaneously guarantee that $\widetilde M_X$ is globally generated and $\widetilde B^*$ has a regular section. Note in particular that $\widetilde B^*$ will have a regular section if $\widetilde F^*$ is globally generated. The latter holds true, for example, if $X$ is a complete intersection and we choose $M_X = R/I_X$. \end{remark} \begin{remark}\label{CM-type} Analyzing the proof of Theorem~\ref{std-and-good} and noting that $X$ and $S$ are defined by the maximal minors of a $t \times (t+r)$ matrix and a $(t+1) \times (t+r)$ matrix, respectively, one observes that there is the following relation between the Cohen-Macaulay types of $X$ and $S$, respectively: \begin{quote} \begin{em} $X$ has Cohen-Macaulay type ${r + t - 1 \choose r}$ $\Leftrightarrow$ $S$ has Cohen-Macaulay type ${r + t -1 \choose r - 1}$. \end{em} \end{quote} This follows from the corresponding Eagon-Northcott resolutions. \end{remark} \section{Applications and Examples} \label{corollaries} In this section we draw some consequences of the results we have shown. We begin with a characterization of complete intersections. It is well-known that every complete intersection is arithmetically Gorenstein but the converse fails in general unless the subscheme has codimension two. For subschemes of higher codimension we have: \begin{cor} \label{complete_intersection} Let $X \subset \proj{n}$ be a subscheme of codimension $r+1 \geq 3$. Then $X$ is a complete intersection if and only if $X$ is arithmetically Gorenstein and there is a good determinantal subscheme $S \subset \proj{n}$ of codimension $r$ such that $X \subset S$ is the zero-locus of a regular section $t \in H^0_* (S, \widetilde M_S ) = M_S$ for some $M_S \in {\cal M}_S$. Furthermore, $S$ and $M_S$ can be chosen so that $\widetilde M_S$ is globally generated. \end{cor} \noindent {\em Proof:} The result follows immediately from Theorem~\ref{std-and-good}, Remark~\ref{on-conj}, and Remark~\ref{CM-type}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \bigskip Next, we consider subschemes of low codimension. As remarked after Definition~\ref{Mv}, in the case of codimension two we know that ${\cal M}_X$ consists of precisely one element (up to isomorphism). \begin{cor}\label{codim2} Suppose $X \subset \proj{n}$ ($n \geq 2$) has codimension two. Then \newcounter{temp4} \begin{list} {(\alph{temp4})}{\usecounter{temp4}} \setlength{\rightmargin}{\leftmargin} \item $X$ is standard determinantal if and only if $X$ is arithmetically Cohen-Macaulay. \item The following are equivalent: \newcounter{temp8} \begin{list} {(\roman{temp8})}{\usecounter{temp8}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item $X$ is arithmetically Cohen-Macaulay and there are an integer $e \in {\Bbb Z}$ and a section $s \in H^0 (X, \omega_X (e))$ generating $\omega_X (e)$ outside a subscheme of codimension 3 as an ${\cal O}_X$-module and such that $s$ is a minimal generator of $H^0_* (\omega_X )$; \item $X$ is arithmetically Cohen-Macaulay and a generic complete intersection . \end{list} \end{list} \end{cor} \noindent {\em Proof:} Part (a) is just the Hilbert-Burch theorem. For (b), the fact that the codimension of $X$ is 2 implies that $\widetilde M_X \cong \omega_X (e)$ for some $e \in {\Bbb Z}$. Then (b) is just a corollary of \linebreak Proposition~\ref{good-det} and Theorem~\ref{good-iff-glci}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{cor} Suppose that $X \subset \proj{n}$ has codimension 3. Then $X$ is good determinantal if and only if there is a good determinantal subscheme $S \subset \proj{n}$ of codimension 2 such that $X \subset S$ is the zero-locus of a regular section $t \in H^0 (S, \omega_S (e))$ (for suitable $e \in {\Bbb Z}$) whose cokernel is supported on a subscheme of codimension $\geq 4$ and isomorphic to an ideal sheaf of ${\cal O}_X$. \end{cor} \noindent {\em Proof:} This is immediate from Theorem~\ref{std-and-good}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{rmk} \begin{rm} In general, if $X$ is a good determinantal subscheme of codimension $r+1$ in $\proj{n}$ then there is a flag of {\em good} determinantal subschemes $X_i$ of codimension $i$: \[ X = X_{r+1} \subset X_r \subset \cdots \subset X_2 \subset X_1 \subset \proj{n}. \] In the next corollary we will show that we can choose the various $X_i$ in such a way that they have even better properties than guaranteed by the results of the previous section. \ \ \ \ \ \ \hbox{$\rlap{$\sqcap$}\sqcup$} \end{rm} \end{rmk} \begin{cor}\label{exists-glci-scheme} If $X \subset \proj{n}$ has codimension $ r + 1 \geq 2$ then the following are equivalent: \newcounter{temp9} \begin{list} {(\alph{temp9})}{\usecounter{temp9}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item There is a good determinantal subscheme $S$ of codimension $r$ which is a local complete intersection outside a subscheme of codimension $r + 2$, and a section $t \in H^0_* (S, \widetilde M_S )$ inducing an exact sequence \[ 0 \rightarrow {\cal O}_S(e) \buildrel t \over \rightarrow \widetilde M_S \rightarrow \widetilde M_X \rightarrow 0 \] for suitable $M_S \in {\cal M}_S$ and $M_X \in {\cal M}_X$. \end{list} \end{cor} \noindent {\em Proof:} We first prove (a) $\Rightarrow $ (b). The existence of a good determinantal subscheme $S$ and a section $t$ as in the statement follows from Theorem~\ref{std-and-good} and the exact sequence (\ref{sect-coker}) in particular. The only thing remaining to prove is that $S$ can be chosen to be a local complete intersection outside a subscheme of codimension $r + 2$ (rather than codimension $r+1$, as guaranteed by Proposition~\ref{good-det}). Assume that the matrix $\Phi$, whose maximal minors define $X$, is a homogeneous $t \times (t+r)$ matrix. The scheme $S$ is constructed in Theorem~\ref{std-and-good} by adding a ``general row'' to $\Phi$, producing a $(t+1) \times (t+r)$ matrix, $\Psi$. One of the points of the proof of Theorem~\ref{good-iff-glci} is that the locus $Y$ where $S$ fails to be a local complete intersection is a subscheme of the scheme defined by the ideal of submaximal minors of $\Psi$. In particular, $Y$ is a subscheme of $X$. The fact that $S$ can be chosen to be a local complete intersection outside a subscheme of codimension $r+2$ will then follow once we show that, given a general point $P$ in any component of $X$, there is at least one submaximal minor of $\Psi$ that does not vanish at $P$. Since $X$ is good, after a change of basis if necessary we may assume that there is a $(t-1) \times (t+r)$ submatrix $\Phi'$ whose ideal of maximal minors defines a scheme of codimension $r+2$ which is disjoint from $P$. Hence there is a maximal minor $A$ of $\Phi'$ which does not vanish at $P$. (We make our change of basis, if necessary, before adding a row to construct $\Psi$. Note that we formally include the possibility that $t=1$, i.e.\ that $X$ is a complete intersection-- see Remark~\ref{one-minor}, Remark~\ref{free-is-BR} and Theorem~\ref{good-iff-sect}.) Concatenate another column of $\Phi'$ to $A$ (by abuse we denote by $A$ both the submatrix and its determinant), forming a $(t-1) \times t$ submatrix of $\Phi'$. Now concatenate the corresponding elements of the ``general row'' to this matrix, forming a $t \times t$ matrix, $B$, whose determinant is a submaximal minor of $\Psi$. Expanding along this latter row and using the fact that its elements were chosen generally and that $A$ does not vanish at $P$, we get that the determinant of $B$ does not vanish at $P$, as desired. This completes the proof that (a) $\Rightarrow$ (b). The converse follows exactly as in the proof of Theorem~\ref{std-and-good} (b). Note that the condition of being a local complete intersection away from a subscheme of codimension $r+2$ is irrelevant in this direction. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{rmk} \begin{rm} (i) Using the notation of the previous proof we have seen that given a good determinantal subscheme $X$ we can find subschemes $Y, S$ such that $Y \subset X \subset S$ have decreasing codimensions, $X$ is the zero-locus of a section of $H^0_* (S, \widetilde M_S )$ and $X, S$ are local complete intersections outside $Y$. In this situation we want to call $X$ a {\em Cartier divisor on $S$ outside $Y$}. If $Y$ is empty then $X$ is a Cartier divisor on $S$ in the usual sense. (ii) Let $X$ be a good determinantal subscheme of codimension $r+1$ in $\proj{n}$ and let $X_{r+2} \subset X$ be a subscheme of codimension $r+2$ such that $X$ is a local complete intersection outside $X_{r+2}$. Then Corollary \ref{exists-glci-scheme} implies that there is a flag of {\em good} determinantal subschemes $X_i$ of codimension $i$: \[ X = X_{r+1} \subset X_r \subset \cdots \subset X_2 \subset X_1 \subset X_0 = \proj{n} \] such that $X_{i+1}$ is a Cartier divisor on $X_i$ outside $X_{i+2}$ for all $i = 0,\dots,r$. \ \ \ \ \ \ \hbox{$\rlap{$\sqcap$}\sqcup$} \end{rm} \end{rmk} \begin{cor}\label{exists-lci-curve} If $X \subset \proj{n}$ is zero-dimensional then the following are equivalent: \newcounter{temp6} \begin{list} {(\alph{temp6})}{\usecounter{temp6}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item There is a good determinantal curve $S$ which is a local complete intersection such that $X$ is a Cartier divisor on $S$ associated to a section $t \in H^0_* (S, \widetilde M_S )$ inducing an exact sequence \[ 0 \rightarrow {\cal O}_S (e) \buildrel t \over \rightarrow \widetilde M_S \rightarrow {\cal O}_X (f) \rightarrow 0. \] \end{list} \end{cor} \noindent {\em Proof:} Note that under the hypotheses that $X$ is zero-dimensional and good, we get in the commutative diagram (\ref{usual-diag}) that $coker \ \Phi'$ has finite length, and hence its sheafification is zero. Hence by the exact sequence (\ref{short-exact}), we get that the sheafification of $coker \ \Phi$ is just ${\cal O}_X$. Then the result follows from Corollary~\ref{exists-glci-scheme}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \bigskip \begin{cor} \label{points_in_three_space} Suppose $X \subset \proj{3}$ is zero-dimensional. Then the following are equivalent: \newcounter{temp7} \begin{list} {(\alph{temp7})}{\usecounter{temp7}} \setlength{\rightmargin}{\leftmargin} \item $X$ is good determinantal; \item There is an arithmetically Cohen-Macaulay curve $S$, which is a local complete intersection, such that $X$ is a subcanonical Cartier divisor on $S$. \end{list} Furthermore, $X$ is defined by a $t \times (t+r)$ matrix if and only if the Cohen-Macaulay type of $X$ is ${{r+t-1} \choose r}$ and that of $S$ is ${{r+t-1} \choose {r-1}}$. \end{cor} \noindent {\em Proof:} Since $S$ has codimension two the exact sequence in the previous result specializes to the sequence \[ 0 \rightarrow {\cal O}_S (e) \buildrel t \over \rightarrow \omega_S \rightarrow {\cal O}_X (f) \rightarrow 0 \] by Corollary~\ref{codim2}. Since $S$ is a local complete intersection it implies that $X$ is subcanonical. The statement about the Cohen-Macaulay types is just Remark~\ref{CM-type}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \begin{rmk} \begin{rm} In view of Remark~\ref{on-conj} and Remark~\ref{CM-type}, Corollaries \ref{complete_intersection}, \ref{exists-lci-curve} and \ref{points_in_three_space} are generalizations of Theorem~1.3 of \cite{kreuzer}. \hskip 1cm $\rlap{$\sqcap$}\sqcup$ \end{rm} \end{rmk} \begin{example}\label{good-ex} In view of Theorem~\ref{good-iff-glci}, we give examples of curves in $\proj{3}$ (both of degree 3) to show that a good determinantal scheme need not be either reduced or a local complete intersection. For the first, consider the curve defined by the matrix \[ \left [ \begin{array}{ccccc} x_0 & x_1 & x_2 \\ 0 & x_0 & x_3 \end{array} \right ] \] For the second, consider the curve defined by the matrix \[ \left [ \begin{array}{cccc} -x_3 & x_2 & 0 \\ 0 & -x_2 & x_1 \end{array} \right ] \] This is the defining matrix for the ``coordinate axes,'' which fail to be a complete intersection precisely at the ``origin.'' (Recall that in the definition of a good determinantal scheme we allowed for the removal of a {\em generalized} row.) \end{example} \begin{example} The point of Corollary~\ref{exists-lci-curve} is that given a zero-scheme $X$, there is so much ``room'' to choose the curve $S$ containing it, that $S$ can be assumed to be a local complete intersection even at $X$, where one would normally expect it to have problems. One naturally can ask if there is so much room that $S$ can even be taken to be smooth. The answer is no: for example, the zeroscheme in $\proj{3}$ defined by the complete intersection $(X_1^2 ,X_2^2 ,X_3^2 )$ lies on no smooth curve. One can ask, though, if there is any matrix condition analogous to the main result of \cite{chiantini-orecchia} which guarantees that a ``general'' choice of $S$ will be smooth. \end{example} \begin{example} Any regular section of any twist of the tangent bundle of $\proj{n}$ defines a good determinantal zero-scheme in $\proj{n}$, by Theorem~\ref{good-iff-sect}. In fact, it can be shown that if $\cal E$ is any rank $n$ vector bundle on $\proj{n}$ with $H^i_* (\proj{n} ,{\cal E}) = 0$ for $1 \leq i \leq n-2$, then any regular section of $\cal E$ defines a good determinantal zero-scheme in $\proj{n}$. \end{example}

9708

alg-geom/9708026

en

https://arxiv.org/abs/alg-geom/9708026

alg-geom/9708026

Frank Sottile

Frank Sottile (University of Toronto)

Pieri-type formulas for maximal isotropic Grassmannians via triple intersections

LaTeX 2e, 24 pages (9 pages is an appendix detailing the proof in the symplectic case). Expanded version of MSRI preprint 1997-062

Colloquium Mathematicum, Vol. 82 (1999), 49--63.

We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an explicit description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are deduced.

alg-geom

\section*{Introduction} The goal of this paper is to give an elementary geometric proof of Pieri-type formulas in the cohomology of Grassmannians of maximal isotropic subspaces of odd orthogonal or symplectic vector spaces. For this, we explicitly compute a triple intersection of Schubert varieties, where one is a special Schubert variety. Previously, Sert\"oz~\cite{Sertoz} had studied such triple intersections in orthogonal Grassmannians, but was unable to determine the intersection multiplicities and obtain a formula. These multiplicities are either 0 or powers of 2. Our proof explains them as the intersection multiplicity of a linear subspace (defining the special Schubert variety) with a collection of quadrics and linear subspaces (determined by the other two Schubert varieties). This is similar to triple intersection proofs of the classical Pieri formula ({\em cf.}~\cite{Hodge_intersection}\cite[p. 203]{Griffiths_Harris}% \cite[\S 9.4]{Fulton_tableaux}) where the multiplicities (0 or 1) count the number of points in the intersection of linear subspaces. A proof of the Pieri-type formula for classical flag varieties~\cite{Sottile_pieri_schubert} was based upon those ideas. Similarly, the ideas here provide a basis for a proof of Pieri-type formulas in the cohomology of symplectic flag varieties~\cite{Bergeron_Sottile_Lagrangian_Pieri}. These Pieri-type formulas are due to Hiller and Boe~\cite{Hiller_Boe}, whose proof used the Chevalley formula~\cite{Chevalley91}. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski~\cite{Pragacz_Ratajski_Operator}. These formulas also arise in the theory of projective representations of symmetric groups~\cite{Schur,Hoffman_Humphreys} as product formulas for Schur $P$- and $Q$-functions, and were first proven in this context by Morris~\cite{Morris}. The connection of Schur $P$- and $Q$-functions to geometry was noticed by Pragacz~\cite{Pragacz_88} (see also~\cite{Jozefiak} and~\cite{Pragacz_S-Q}). \smallskip In Section 1, we give the basic definitions and state the Pieri-type formulas in both the orthogonal and symplectic cases, and conclude with an outline of the proof in the orthogonal case. Since there is little difference between the proofs in each case, we only do the orthogonal case in full. In Section 2, we describe the intersection of two Schubert varieties, which we use in Section 3 to complete the proof. \section{The Grassmannian of maximal isotropic subspaces} Let $V$ be a ($2n+1$)-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form $\beta$ and $W$ a $2n$-dimensional complex vector space equipped with a non-degenerate alternating bilinear form, also denoted $\beta$. A subspace $H$ of $V$ or of $W$ is {\em isotropic} if the restriction of $\beta$ to $H$ is identically zero. Isotropic subspaces have dimension at most $n$. The {\em Grassmannian of maximal isotropic subspaces} $B_n$ or $B(V)$ (respectively $C_n$ or $C(W)$) is the collection of all isotropic $n$-dimensional subspaces of $V$ (respectively of $W$). The group $\mbox{\em So}_{2n+1}{\mathbb C} = {\rm Aut}(V,\beta)$ acts transitively on $B_n$ with the stabilizer $P_0$ of a point a maximal parabolic subgroup associated to the short root, hence $B_n = \mbox{\em So}_{2n+1}{\mathbb C}/P_0$. Similarly, $C_n= \mbox{\em Sp}_{2n}{\mathbb C}/P_0$, where $P_0$ is a maximal parabolic associated to the long root. Both $B_n$ and $C_n$ are smooth complex manifolds of dimension $\binom{n+1}{2}$. While they are not isomorphic if $n>1$, they have identical Schubert decompositions and Bruhat orders. Another interesting connection is discussed in Remark~\ref{rem:similarities}. We describe the Schubert decomposition. For an integer $j$, let $\overline{\jmath}$ denote $-j$. Choose bases $\{e_{\overline{n}},\ldots,e_n\}$ of $V$ and $\{f_{\overline{n}},\ldots,f_n\}$ of $W$ for which $$ \beta(e_i,e_j)\ =\ \left\{ \begin{array}{ll} 1&\mbox{ \ if } i=\overline{\jmath}\\ 0&\mbox{ \ otherwise}\end{array}\right. \quad\mbox{and}\quad \beta(f_i,f_j)\ =\ \left\{ \begin{array}{ll} j/|j|&\mbox{ \ if } i=\overline{\jmath}\\ 0 &\mbox{ \ otherwise}\end{array}\right. . $$ For example, $\beta(e_1,e_0)=\beta(f_{\overline{2}},f_1)=0$ and $\beta(e_0,e_0)= \beta(f_{\overline{1}},f_1)=-\beta(f_1,f_{\overline{1}})= 1$. Schubert varieties are determined by sequences $$ \lambda:\ n\geq \lambda_1>\lambda_2>\cdots>\lambda_n\geq \overline{n} $$ whose set of absolute values $\{|\lambda_1|,\ldots,|\lambda_n|\}$ equals $\{1,2,\ldots,n\}$. Let ${\mathbb{SY}}_n$ denote this set of sequences. The Schubert variety $X_\lambda$ of $B_n$ is $$ \{H\in B_n\mid\dim H\cap\Span{e_{\lambda_j},\ldots,e_n}\geq j \mbox{\ for } 1\leq j\leq n\} $$ and the Schubert variety $Y_\lambda$ of $C_n$ $$ \{H\in C_n\mid\dim H\cap\Span{f_{\lambda_j},\ldots,f_n}\geq j \mbox{\ for } 1\leq j\leq n\}. $$ Both $X_\lambda$ and $Y_\lambda$ have codimension $|\lambda|:=\lambda_1+\cdots+\lambda_k$, where $\lambda_k>0>\lambda_{k+1}$. Given $\lambda,\mu\in{\mathbb{SY}}_n$, we see that $$ X_\mu\supset X_\lambda\ \Longleftrightarrow\ Y_\mu\supset Y_\lambda\ \Longleftrightarrow\ \mu_j\leq \lambda_j \mbox{ for } 1\leq j\leq n. $$ Define the {\em Bruhat order} by $\mu\leq \lambda$ if $\mu_j\leq \lambda_j$ for $1\leq j\leq n$. Note that $\mu\leq \lambda$ if and only if $\mu_j\leq \lambda_j$ for those $j$ with $0<\mu_j$. \begin{ex}{\rm Suppose $n=4$. Then $X_{3\,2\,\overline{1}\,\overline{4}}$ consists of those $H\in B_4$ such that $$ \dim H\cap\Span{e_3,e_4}\geq 1,\ \dim H\cap\Span{e_2,e_3,e_4}\geq 2,\ \mbox{and}\ \dim H\cap\Span{e_{\overline{1}},\ldots,e_4}\geq 3. $$ } \end{ex} Define $P_\lambda:=[X_\lambda]$, the cohomology class Poincar{\'e} dual to the fundamental cycle of $X_\lambda$ in the homology of $B_n$.\ Likewise set $Q_\lambda:=[Y_\lambda]$. Since Schubert varieties are closures of cells from a decomposition into (real) even-dimensional cells, these {\em Schubert classes} $\{P_\lambda\}$, $\{Q_\lambda\}$ form bases for integral cohomology: $$ H^*B_n\ =\ \bigoplus_{\lambda} P_\lambda \cdot {\mathbb Z} \qquad \mbox{and}\qquad H^*C_n\ =\ \bigoplus_{\lambda} Q_\lambda \cdot {\mathbb Z}. $$ Each $\lambda\in{\mathbb{SY}}_n$ determines and is determined by its diagram, also denoted $\lambda$. The diagram of $\lambda$ is a left-justified array of $|\lambda|$ boxes with $\lambda_j$ boxes in the $j$th row, for $\lambda_j>0$. Thus $$ 3\,2\,\overline{1}\,\overline{4} \ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(30,20)(0,7) \put( 0, 0){\line(0,1){20}}\put(0, 0){\line(1,0){20}} \put(10, 0){\line(0,1){20}}\put(0,10){\line(1,0){30}} \put(20, 0){\line(0,1){20}}\put(0,20){\line(1,0){30}} \put(30,10){\line(0,1){10}}\end{picture}\, \qquad\mbox{ and }\qquad 4\,2\,1\,\overline{3} \ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(40,20)(0,12) \put( 0, 0){\line(0,1){30}}\put(0,30){\line(1,0){40}} \put(10, 0){\line(0,1){30}}\put(0,20){\line(1,0){40}} \put(20,10){\line(0,1){20}}\put(0,10){\line(1,0){20}} \put(30,20){\line(0,1){10}}\put(0, 0){\line(1,0){10}} \put(40,20){\line(0,1){10}}\end{picture}\, . \raisebox{-15pt}{\rule{0pt}{5pt}} $$ The Bruhat order corresponds to inclusion of diagrams. Given $\mu\leq \lambda$, let $\lambda/\mu$ be their set-theoretic difference. For instance, $$ 4\,2\,1\,\overline{3}/3\,2\,\overline{1}\,\overline{4}\ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(40,20)(0,12) \put(30,20){\line(0,1){10}}\put(30,30){\line(1,0){10}} \put(40,20){\line(0,1){10}}\put(30,20){\line(1,0){10}} \put( 0, 0){\line(0,1){10}}\put( 0,10){\line(1,0){10}} \put(10, 0){\line(0,1){10}}\put( 0, 0){\line(1,0){10}} \put( 0,20){\dashbox{2}(10,10)[t]{}}\put( 0,10){\dashbox{2}(10,10)[t]{}} \put(10,20){\dashbox{2}(10,10)[t]{}}\put(10,10){\dashbox{2}(10,10)[t]{}} \put(20,20){\dashbox{2}(10,10)[t]{}} \end{picture} \qquad\mbox{ and }\qquad 3\,2\,\overline{1}\,\overline{4}/ 1\,\overline{2}\,\overline{3}\,\overline{4}\ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(30,20)(0,7) \put( 0, 0){\line(0,1){10}}\put( 0, 0){\line(1,0){20}} \put(10, 0){\line(0,1){20}}\put( 0,10){\line(1,0){30}} \put(20, 0){\line(0,1){20}}\put(10,20){\line(1,0){20}} \put(30,10){\line(0,1){10}}\put( 0,10){\dashbox{2}(10,10)[t]{}} \end{picture}\, . \raisebox{-15pt}{\rule{0pt}{5pt}} $$ Two boxes are connected if they share a vertex or an edge; this defines {\em components} of $\lambda/\mu$. We say $\lambda/\mu$ is a {\em skew row} if $\lambda_1\geq\mu_1\geq\lambda_2\geq\cdots\geq\mu_n$ equivalently, if $\lambda/\mu$ has at most one box in each column. Thus $4\,2\,1\,\overline{3}/3\,2\,\overline{1}\,\overline{4}$ is a skew row, but $3\,2\,\overline{1}\,\overline{4}/ 1\,\overline{2}\,\overline{3}\,\overline{4}$ is not. The {\em special Schubert class} $p_m\in H^*B_n$ ($q_m\in H^*C_n$) is the class whose diagram consists of a single row of length $m$. Hence, $p_2 = P_{2\,\overline{1}\,\overline{3}\,\overline{4}}$. A {\em special Schubert variety} $X_K$ ($Y_K$) is the collection of all maximal isotropic subspaces which meet a fixed isotropic subspace $K$ nontrivially. If $\dim K=n+1-m$, then $[X_K]=p_m$ and $[Y_K]=q_m$. \begin{thm}[Pieri-type Formula]\label{thm:pieri} For any $\mu\in{\mathbb{SY}}_n$ and $1\leq m\leq n$, \begin{enumerate} \item ${\displaystyle P_\mu\cdot p_m\ =\ \sum_{\lambda/\mu\ \mbox{\scriptsize skew row}} 2^{\delta(\lambda/\mu)-1}\, P_\lambda}$ \qquad and \item ${\displaystyle Q_\mu\cdot q_m\ =\ \sum_{\lambda/\mu\ \mbox{\scriptsize skew row}}^{\rule{0pt}{5pt}} 2^{\varepsilon(\lambda/\mu)}\, Q_\lambda}$, \end{enumerate} where $\delta(\lambda/\mu)$ counts the components of the diagram $\lambda/\mu$ and $\varepsilon(\lambda/\mu)$ counts the components of $\lambda/\mu$ which do not contain a box in the first column. \end{thm} \begin{ex}\label{ex:prod} {\em For example, \begin{eqnarray*} P_{3\,2\,\overline{1}\,\overline{4}}\cdot p_2 &=& 2\cdot P_{4\,2\,1\,\overline{3}}\ \,+ \ \ \, P_{4\,3\,\overline{2}\,\overline{1}}\qquad\mbox{ and}\\ Q_{3\,2\,\overline{1}\,\overline{4}}\cdot q_2 &=& 2\cdot Q_{4\,2\,1\,\overline{3}}\ +\ 2\cdot Q_{4\,3\,\overline{2}\,\overline{1}}. \end{eqnarray*} } \end{ex} For $\lambda,\mu,\nu\in{\mathbb{SY}}_n$, there exist integral constants $g^\lambda_{\mu,\nu}$ and $h^\lambda_{\mu,\nu}$ defined by the identities $$ P_\mu\cdot P_\nu\ =\ \sum_\lambda g^\lambda_{\mu,\nu}\,P_\lambda \qquad \mbox{and}\qquad Q_\mu\cdot Q_\nu\ =\ \sum_\lambda h^\lambda_{\mu,\nu}\,Q_\lambda. $$ These constants were first given a combinatorial formula by Stembridge~\cite{Stembridge_shifted}.\smallskip Define $\lambda^c$ by $\lambda^c_j:=\overline{\lambda_{n+1-j}}$. Let $[\mbox{pt}]$ be the class dual to a point. The Schubert basis is self-dual with respect to the intersection pairing: If $|\lambda|=|\mu|$, then \begin{equation}\label{eq:poincare} P_\mu\cdot P_{\lambda^c} \ =\ Q_\mu\cdot Q_{\lambda^c} \ =\ \left\{\begin{array}{ll} [\mbox{pt}]&\ \ \mbox{if}\ \lambda=\mu\\ 0 &\ \ \mbox{otherwise}\end{array}\right.. \end{equation} Define the Schubert variety $X'_{\lambda^c}$ to be $$ \{H\in B_n\mid \dim H\cap\Span{e_{\overline{n}},\ldots,e_{\lambda_j}}\geq n+1-j \ \mbox{for}\ 1\leq j \leq n\}. $$ This is a translate of $X_{\lambda^c}$ by an element of $\mbox{\em So}_{2n+1}{\mathbb C}$. In a similar fashion, define $Y'_{\lambda^c}$, a translate of $Y_{\lambda^c}$ by an element of $\mbox{\em Sp}_{2n}{\mathbb C}$. For any $\lambda,\mu$, $X_\mu\bigcap X'_{\lambda^c}$ is generically transverse~\cite{Kleiman}. This is because if $X_\mu$ and $X'_{\lambda^c}$ are {\em any} Schubert varieties in general position, then there is a basis for $V$ such that these varieties and the quadratic form $\beta$ are as given. The analogous facts hold for the varieties $Y'_{\lambda^c}$. We see that to establish the Pieri-type formula, it suffices to compute the degree of the 0-dimensional schemes $$ X_\mu\bigcap X'_{\lambda^c}\bigcap X_K \quad\mbox{and}\quad Y_\mu\bigcap Y'_{\lambda^c}\bigcap Y_K $$ where $K$ is a general isotropic ($n+1-m$)-plane and $|\lambda|=|\mu|+m$.\smallskip We only do the (more difficult) orthogonal case of Theorem~\ref{thm:pieri} in full, and indicate the differences for the symplectic case. We first determine when $X_\mu\bigcap X'_{\lambda^c}$ is non-empty. Let $\mu,\lambda\in{\mathbb{SY}}_n$. Then, by the definition of Schubert varieties, $H\in X_\mu\bigcap X'_{\lambda^c}$ implies $\dim H\bigcap \Span{e_{\mu_j},\ldots,e_{\lambda_j}}\geq 1$, for every $1\leq j\leq n$. Hence $\mu\leq\lambda$ is necessary for $X_\mu\bigcap X'_{\lambda^c}$ to be nonempty. In fact, $$ X_\mu \bigcap X'_{\lambda^c}\ =\ \left\{ \begin{array}{ll} \Span{e_{\lambda_1},\ldots,e_{\lambda_n}}&\mbox{ if } \lambda=\mu\\ \emptyset &\mbox{ otherwise}, \end{array}\right. $$ which establishes~(\ref{eq:poincare}). Suppose $\mu\leq \lambda$ in ${\mathbb{SY}}_n$. For each component $d$ of $\lambda/\mu$, let col$(d)$ index the columns of $d$ together with the column just to the left of $d$, which is $0$ if $d$ meets the first column, in that it has a box in the first column. For each component $d$ of $\lambda/\mu$, define a quadratic form $\beta_d$: $$ \beta_d\ :=\ \sum_{\stackrel{\mbox{\scriptsize $\overline{n}\leq j\leq n$}}% {\overline{\jmath}\ {\rm or}\ j\in \mbox{\scriptsize col}(d)}} x_j x_{\overline{\jmath}}, $$ where $x_{\overline{n}},\ldots,x_n$ are coordinates for $V$ dual to the basis $e_{\overline{n}},\ldots,e_n$. For each {\em fixed point} of $\lambda/\mu$ ($j$ such that $\lambda_j=\mu_j$), define the linear form $\alpha_j:=x_{\overline{\lambda_j}}$. If there is no component meeting the first column, then we say that $0$ is a fixed point of $\lambda/\mu$ and define $\alpha_0:=x_0$. Let $Z_{\lambda/\mu}$ be the common zero locus of these forms $\alpha_j$ and $\beta_d$. \begin{lemma}\label{lemma:vanish} Suppose $\mu\leq\lambda$ and $H\in X_\mu\bigcap X'_{\lambda^c}$. Then $H\subset Z_{\lambda/\mu}$. \end{lemma} Let ${\mathcal Q}$ be the isotropic points in $V$, the zero locus of $\beta$. For each $0\leq i\leq n$, there is a unique form among the $\alpha_j$, $\beta_d$ in which one (or both) of the coordinates $x_{i},x_{\overline{\imath}}$ appears. Thus $\beta$ is in the ideal generated by these forms $\alpha_j$, $\beta_d$ and we see that they are dependent on ${\mathcal Q}$. However, if $\delta=\delta(\lambda/\mu)$ counts the components of $\lambda/\mu$ and $\varphi$ the number of fixed points, then the collection of $\varphi$ forms $\alpha_j$ and $\delta-1$ of the forms $\beta_d$ {\em are} independent on ${\mathcal Q}$. Moreover, Lemma~\ref{lem:columns} shows that $$ n +1 \ =\ \varphi+\delta+\#\mbox{\rm columns of $\lambda/\mu$}. $$ Thus, if $m=|\lambda|-|\mu|$, then $\varphi+\delta-1\leq n-m$, with equality only when $\lambda/\mu$ is a skew row. Since ${\mathcal Q}$ has dimension $2n$, it follows that a general isotropic ($n+1-m$)-plane $K$ meets $Z_{\lambda/\mu}$ only if $\lambda/\mu$ is a skew row. We deduce \begin{thm}\label{thm:geom} Let $\mu,\lambda\in{\mathbb{SY}}_n$ and suppose $K$ is a general isotropic $(n+1-m)$-plane with $|\mu|+m = |\lambda|$. Then $$ X_\mu \bigcap X'_{\lambda^c}\bigcap X_K $$ is non-empty only if $\mu\leq \lambda$ and $\lambda/\mu$ is a skew row. \end{thm} \noindent{\bf Proof of Theorem~\ref{thm:pieri}. } Suppose $\lambda,\mu\in{\mathbb{SY}}_n$ with $|\lambda|-|\mu|=m>0$. Let $K$ be a general isotropic $(n+1-m)$-plane in $V$. We compute the degree of \begin{equation}\label{eq:zero-scheme} X_\mu \bigcap X'_{\lambda^c}\bigcap X_K. \end{equation} By Theorem~\ref{thm:geom}, this is non-empty only if $\mu\subset\lambda$ and $\lambda/\mu$ is a skew row. Suppose that is the case. Then the forms $\alpha_j$ and $\beta_d$ (which define $Z_{\lambda/\mu}$) determine $2^{\delta(\lambda/\mu)-1}$ isotropic lines in $K$. Theorem~\ref{thm:unique} asserts that a general isotropic line in $Z_{\lambda/\mu}$ is contained in a unique $H\in X_\mu \bigcap X'_{\lambda^c}$, which shows that (\ref{eq:zero-scheme}) has degree $2^{\delta(\lambda/\mu)-1}$. This completes the proof of Theorem~\ref{thm:pieri}. \QED \begin{ex} {\em Let $n=4$ and $m=2$, so that $n+1-m=3$. We show that if $K\subset {\mathcal Q}$ is a general 3-plane, then $$ \#\, X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c} \bigcap X_K\ =\ 2. $$ Note that 2 is the coefficient of $P_{4\,2\,1\,\overline{3}}$ in the product $P_{3\,2\,\overline{1}\,\overline{4}}\cdot p_2$ of Example~\ref{ex:prod}. First, the local coordinates for $X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$ described in Lemma~\ref{lem:loc_coords} show that, for any $x,z\in{\mathbb{C}}$, the row span $H$ of the matrix with rows $g_i$ and columns $e_j$ $$ \begin{array}{l|cccc|c|cccc} {} &e_{\overline{4}}&e_{\overline{3}} &e_{\overline{2}}&e_{\overline{1}}&e_0&e_1&e_2&e_3&e_4\\ \hline g_1&0&0&0&0&0&0&0&-x&1\\ g_2&0&0&0&0&0&0&1&0&0\\ g_3&0&0&0&1&2z&-2z^2&0&0&0\\ g_4&x&1&0&0&0&0&0&0&0 \end{array} $$ is in $X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$. Suppose $K$ is the row span of the matrix with rows $v_i$ $$ \begin{array}{l|cccc|c|cccc} {} &e_{\overline{4}}&e_{\overline{3}} &e_{\overline{2}}&e_{\overline{1}}&e_0&e_1&e_2&e_3&e_4\\ \hline v_1&0&1&0&1&0& 0&1&0&1\\ v_2&1&1&0&1&2&-2&1&1&-1\\ v_3&0&0&1&0&0&-1&0&0&0 \end{array} $$ Then $K$ is an isotropic 3-plane, and the forms \begin{eqnarray*} \beta_0&=& 2x_{\overline{1}}x_1 + x_0^2\\ \beta_d&=&x_{\overline{4}}x_4 + x_{\overline{3}}x_3\\ \alpha_2&=& x_{\overline{2}} \end{eqnarray*} define the 2 isotropic lines $\Span{v_1}$ and $\Span{v_2}$ in $K$. Lastly, for $i=1,2$, there is a unique $H_i\in X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$ with $v_i\in H_i$. In these coordinates, $$ H_1\ :\ x=z=0\quad\mbox{and}\quad H_2\ :\ x=z=1. $$ } \end{ex} In the symplectic case, isotropic $K$ are not contained in a quadric ${\mathcal Q}$, the form $\alpha_0=x_0$ does not arise, only components which do not meet the first column give quadratic forms $\beta_d$, and the analysis of Lemma~\ref{lem:components}~(2) in Section~\ref{sec:triple} is (slightly) different. \section{The intersection of two Schubert varieties} We study the intersection $X_\mu\bigcap X'_{\lambda^c}$ of two Schubert varieties. Our main result, Theorem~\ref{thm:main}, expresses $X_\mu\bigcap X'_{\lambda^c}$ as a product whose factors correspond to components of $\lambda/\mu$, and each factor is itself an intersection of two Schubert varieties. These factors are described in Lemmas~\ref{lemma:0comp} and~\ref{lem:offdiag}, and in Corollary~\ref{cor:classical}. These are crucial to the proof of the Pieri-type formula that we complete in Section 3. Also needed is Lemma~\ref{lem:subspace}, which identifies a particular subspace of $H\cap \Span{e_1,\ldots,e_n}$ for $H\in X_\mu\bigcap X'_{\lambda^c}$. For Lemma~\ref{lem:subspace}, we work in the (classical) Grassmannian $G_k(V^+)$ of $k$-planes in $V^+:=\Span{e_1,\ldots,e_n}$. For basic definitions and results see any of ~\cite{Hodge_Pedoe,Griffiths_Harris,Fulton_tableaux}. Schubert subvarieties $\Omega_\sigma,\Omega'_{\sigma^c}$ of $G_k(V^+)$ are indexed by partitions $\sigma\in{\mathbb{Y}}_k$, that is, integer sequences $\sigma=(\sigma_1,\ldots,\sigma_k)$ with $n - k\geq \sigma_1\geq\cdots\geq\sigma_k\geq0$. For $\sigma\in{\mathbb{Y}}_k$ define $\sigma^c\in{\mathbb{Y}}_k$ by $\sigma^c_j=n-k-\sigma_{k+1-j}$. For $\sigma,\tau\in{\mathbb{Y}}_k$, define \begin{eqnarray*} \Omega_\tau&:=& \{H\in G_k(V^+)\mid \dim H\cap\Span{e_{k+1-j+\tau_j},\ldots,e_n}\geq j,\ 1\leq j\leq k\}\\ \Omega'_{\sigma^c}&:=& \{H\in G_k(V^+)\mid \dim H\cap\Span{e_1,\ldots,e_{j+\sigma_{k+1-j}}}\geq j,\ 1\leq j\leq k\}. \end{eqnarray*} Let $\lambda,\mu\in\mathbb{SY}_n$ with $\mu\leq \lambda$, and suppose $\mu_k>0>\mu_{k+1}$. Define partitions $\sigma$ and $\tau$ in ${\mathbb{Y}}_k$ (which depend upon $\lambda$ and $\mu$) by \begin{eqnarray*} \tau&:=& \mu_1-k\geq \cdots\geq \mu_k-1\geq 0\\ \sigma&:=& \lambda_1-k\geq \cdots\geq \lambda_k-1\geq0 \end{eqnarray*} \begin{lemma}\label{lem:subspace} Let $\mu\leq \lambda\in{\mathbb{SY}}_n$, and define $\sigma,\tau\in{\mathbb{Y}}_k$, and $k$ as above. If $H\in X_\mu\bigcap X'_{\lambda^c}$, then $H\cap V^+$ contains a $k$-plane $L\in\Omega_\tau\bigcap\Omega'_{\sigma^c}$. \end{lemma} \noindent{\bf Proof. } Suppose first that $H\in X_\mu$ with $\dim H\cap \Span{e_{\mu_j},\ldots,e_n}=j$ for $j=k$ and $k+1$. Since $\mu_k>0>\mu_{k+1}$, we see that $L:= H\cap V^+$ has dimension $k$. If $H\in X'_{\lambda^c}$ in addition, it is an exercise in the definitions to verify that $L\in\Omega_\tau\bigcap\Omega'_{\sigma^c}$. The lemma follows as such $H$ are dense in $X_\mu$. \QED The first step towards Theorem~\ref{thm:main} is the following combinatorial lemma. \begin{lemma}\label{lem:columns} Let $\varphi$ count the fixed points and $\delta$ the components of $\lambda/\mu$. Then $$ n +1\ =\ \varphi+\delta+\#\mbox{\rm columns of $\lambda/\mu$}, $$ and $\mu_j>\lambda_{j+1}$ precisely when $|\mu_j|$ is an empty column of $\lambda/\mu$. \end{lemma} \noindent{\bf Proof. } Suppose $k$ is a column not meeting $\lambda/\mu$. Thus, there is no $i$ for which $\mu_i<k\leq\lambda_i$. Let $j$ be the index such that $|\mu_j|=k$. If $\mu_j=k$, then we must also have $\lambda_{j+1}<k$, as $\mu_{j+1}<k$. Either $\mu_j=\lambda_j$ is a fixed point of $\lambda/\mu$ or else $\mu_j<\lambda_j$, so that $k$ is the column immediately to the left of a component $d$ which does not meet the first column. If $\mu_j=-k$, then $\lambda_j=-k$, for otherwise $k=\lambda_i$ for some $i$, and for this $i$ we must necessarily have $\mu_i<k$, contradicting $k$ being an empty column. This proves the lemma, as $0$ is either a fixed point of $\lambda/\mu$ or else $\lambda/\mu$ has a component meeting the first column, but not both.\QED Let $d_0$ be the component of $\lambda/\mu$ meeting the first column (if any). Define mutually orthogonal subspaces $V_\varphi,V_0$, and $V_d$, for each component $d$ of $\lambda/\mu$ not meeting the first column ($0\not\in\mbox{col}(d)$) as follows: \begin{eqnarray*} V_\varphi\: &:=& \Span{e_{\mu_j},e_{\overline{\mu_j}}\mid \mu_j=\lambda_j},\\ V_0\:\, &:=& \Span{e_0,e_k,e_{\overline{k}}\mid k\in\mbox{col}(d_0)},\\ V^-_d &:=& \Span{e_k\mid k\in\mbox{col}(d)},\\ V^+_d &:=& \Span{e_{\overline{k}}\mid k\in\mbox{col}(d)}, \end{eqnarray*} and set $V_d:= V^-_d\oplus V^+_d$. Then $$ V\ =\ V_\varphi\oplus V_0\oplus \bigoplus_{0\not\in\mbox{\scriptsize col}(d)} V_d. $$ For each fixed point $\mu_j=\lambda_j$ of $\lambda/\mu$, define the linear form $\alpha_j:=x_{\overline{\mu_j}}$. For each component $d$ of $\lambda/\mu$, let the quadratic form $\beta_d$ be the restriction of the form $\beta$ to $V_d$. Composing with the projection of $V$ to $V_d$ gives a quadratic form (also written $\beta_d$) on $V$. If there is no component meeting the first column, define $\alpha_0:=x_0$ and call $0$ a fixed point of $\lambda/\mu$. If $0\not\in\mbox{col}(d)$, then the form $\beta_d$ identifies $V^+_d$ and $V^-_d$ as dual vector spaces. \begin{lemma}\label{lemma:intersection} Let $H\in X_\mu\bigcap X'_{\lambda^c}$. Then \begin{enumerate} \item $H\bigcap V_\varphi = \Span{e_{\mu_j}\mid\mu_j=\lambda_j}$. \item $\dim H\bigcap V_0 = \#\mbox{col}(d_0)$. \item For all components $d$ of $\lambda/\mu$ which do not meet the first column, \begin{eqnarray*} \dim H\bigcap V^+_d&=& \#\mbox{rows of }d,\\ \dim H\bigcap V^-_d&=& \#\mbox{col}(d) - \#\mbox{rows of }d, \end{eqnarray*} and $\left(H\bigcap V^-_d\right)^\perp = H\bigcap V^+_d$. \end{enumerate} \end{lemma} \noindent{\bf Proof of Lemma~\ref{lemma:intersection}. } Let $H\in X_\mu\bigcap X'_{\lambda^c}$. Suppose $\mu_j>\lambda_{j+1}$ so that $|\mu_j|$ is an empty column of $\lambda/\mu$. Then the definitions of Schubert varieties imply $$ H\ =\ H\cap\Span{e_{\overline{n}},\ldots,e_{\lambda_{j+1}}}\oplus H\cap\Span{e_{\mu_j},\ldots,e_n}. $$ Suppose $d$ is a component not meeting the first column. If the rows of $d$ are $j,\ldots,k$, then \begin{eqnarray*} H\cap V^+_d&=& H\cap\Span{e_{\mu_k},\ldots,e_{\lambda_j}}\\ &=& H \cap\Span{e_{\overline{n}},\ldots,e_{\lambda_j}} \cap\Span{e_{\mu_k},\ldots,e_n}, \end{eqnarray*} and so has dimension at least $k-j+1$. Similarly, if $l,\ldots,m$ are the indices $i$ with $\overline{\lambda_{j}}\leq \mu_i,\lambda_i\leq \overline{\mu_k}$, then $H\bigcap V^-_d$ has dimension at least $l-m+1$. Hence $\dim V_d/2 = \#\mbox{col}(d)=k+m-l-j+2$, as $\lambda_j,\ldots,\lambda_k,\overline{\lambda_l},\ldots,\overline{\lambda_m}$ are the columns of $d$. Since $H$ is isotropic, $\dim H^+_d + \dim H^-_d \leq \#\mbox{col}(d)$, which proves the first part of (3). Moreover, $H\bigcap V^+_d=\left(H\bigcap V^-_d\right)^\perp$: Since $H$ is isotropic, we have $\subset$, and equality follows by counting dimensions. Similar arguments prove the other statements. \QED For $H\in X_\mu\bigcap X'_{\lambda^c}$, define $H_\varphi:= H\bigcap V_\varphi$, $H_0:= H\bigcap V_0$, $H^+_d := H\bigcap V^+_d$, and $H^-_d := H\bigcap V^-_d$. Then $H_\varphi\subset V_\varphi$ is the zero locus of the linear forms $\alpha_j$, $H_0$ is isotropic in $V_0$, and, for each component $d$ of $\lambda/\mu$ not meeting the first column, $H_d:= H^+_d\oplus H^-_d$ is isotropic in $V_d$, which proves Lemma~\ref{lemma:vanish}. Moreover, $H$ is the orthogonal direct sum of $H_\varphi$, $H_0$, and the $H_d$. \begin{thm}\label{thm:main} The map $$ \{H_0\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \times \prod_{0\not\in\mbox{\scriptsize col}(d)} \{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\} \longrightarrow\ X_\mu\bigcap X'_{\lambda^c} $$ defined by $$ (H_0,\ldots,H_d,\ldots) \longmapsto \Span{H_\varphi,H_0,\ldots,H_d,\ldots} $$ is an isomorphism. \end{thm} \noindent{\bf Proof. } By the previous discussion, it is an injection. For surjectivity, note that both sides have the same dimension. Indeed, $\dim X_\mu\bigcap X'_{\lambda^c}=|\lambda|-|\mu|$, the number of boxes in $\lambda/\mu$. Lemmas~\ref{lemma:0comp} and \ref{lem:offdiag} show that the factors of the domain each have dimension equal to the number of boxes in the corresponding components. \QED Suppose there is a component, $d_0$, meeting the first column. Let $l$ be the largest column in $d_0$, and define $\lambda(0),\mu(0)\in{\mathbb{SY}}_l$ as follows: Let $j$ be the first row of $d_0$ so that $l=\lambda_j$. Then, since $d_0$ is a component, for each $j\leq i<j+l-1$, we have $\lambda_{i+1}\geq \mu_i$ and $l=\overline{\mu_{j+l-1}}$. Set \begin{eqnarray*} \mu(0)&:=& \mu_j>\cdots >\mu_{j+l-1}\\ \lambda(0)&:=& \lambda_j>\cdots >\lambda_{j+l-1} \end{eqnarray*} Define $\lambda(0)^c$ by $\lambda(0)^c_p:= \overline{\lambda(0)_{l+1-p}} = \overline{\lambda_{j+l-p}}$. The following lemma is a straightforward consequence of these definitions. \begin{lemma}\label{lemma:0comp} With the above definitions, $$ \{H_0\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ =\ X_{\mu(0)}\bigcap X'_{\lambda(0)^c} $$ as subvarieties of $B_k\simeq B(V_0)$, and $\lambda(0)/\mu(0)$ has a unique component meeting the first column and no fixed points. \end{lemma} We similarly identify $\{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}$ as an intersection $X_{\mu(d)}\cap X'_{\lambda(d)^c}$ of Schubert varieties in $B_{\#\mbox{\scriptsize col}(d)}\simeq B(\Span{e_0,V_d})$. Let $j,\ldots,k$ be the rows of $d$ and $l,\ldots,m$ be the indices $i$ with $\overline{\lambda_j}\leq \mu_i,\lambda_i\leq \overline{\mu_k}$, as in the proof of Lemma~\ref{lemma:intersection}. Let $p=\#\mbox{col}(d)$ and define $\lambda(d),\mu(d)\in {\mathbb{SY}}_p$ as follows. Set $a = \mu_k$, and define \begin{eqnarray*} \mu(d)&:=& \mu_j-a+1>\cdots> \hspace{24pt}1\hspace{24pt} >\mu_l+a-1>\cdots>\mu_m+a-1\\ \lambda(d)&:=& \lambda_j-a+1>\cdots>\lambda_k-a+1> \lambda_l+a-1>\cdots>\lambda_m+a-1 \end{eqnarray*} As with Lemma~\ref{lemma:0comp}, the following lemma is straightforward. \begin{lemma}\label{lem:offdiag} With these definitions, $$ \{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \simeq\ X_{\mu(d)}\bigcap X'_{\lambda(d)^c} $$ as subvarieties of $B_p\simeq B(\Span{e_0,V_d})$ and $\lambda(d)/\mu(d)$ has a unique component not meeting the first column with only 0 as a fixed point. \end{lemma} Suppose now that $\mu,\lambda\in{\mathbb{SY}}_n$ where $\lambda/\mu$ has a unique component $d$ not meeting the first column and no fixed points. Suppose $\lambda$ has $k$ rows. A consequence of Lemma~\ref{lemma:intersection} is that the map $H^+_d \mapsto \Span{H^+_d,\left(H^+_d\right)^\perp}$ gives an isomorphism \begin{equation}\label{isomorphism} \{ H^+_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \stackrel{\sim}{\longrightarrow}\ X_\mu\bigcap X'_{\lambda^c}. \end{equation} The following corollary of Lemma~\ref{lem:subspace} identifies the domain. \begin{cor}\label{cor:classical} With $\mu,\lambda$ as above and $\sigma,\tau$, and $k$ as defined in the paragraph preceding Lemma~\ref{lem:subspace}, we have: $$ \{H^+_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ =\ \Omega_{\tau}\bigcap\Omega_{\sigma^c}, $$ as subvarieties of $G_k(V^+)$. \end{cor} \begin{rem}\label{rem:similarities} {\em The symplectic analogs of Lemma~\ref{lem:offdiag} and Corollary~\ref{cor:classical}, which are identical save for the necessary replacement of $Y$ for $X$ and $C_p$ for $B_p$, show an unexpected connection between the geometry of the symplectic and orthogonal Grassmannians. Namely, suppose $\lambda/\mu$ has no component meeting the first column. Then the projection map $V \twoheadrightarrow W$ defined by $$ e_i\ \longmapsto\ \left\{\begin{array}{ll}0&\ \mbox{if}\ i=0\\ f_i&\ \mbox{otherwise} \end{array}\right. $$ and its left inverse $W\hookrightarrow V$ defined by $f_j\mapsto e_j$ induce isomorphisms $$ X_\mu\bigcap X'_{\lambda^c}\ \stackrel{\sim}{\longleftrightarrow}\ Y_\mu\bigcap Y'_{\lambda^c}. $$ } \end{rem} \section{Pieri-type intersections of Schubert varieties}\label{sec:triple} Fix $\lambda/\mu$ to be a skew row with $|\lambda|-|\mu|=m$. Let $Z_{\lambda/\mu}\subset{\mathcal Q}$ be the zero locus of the forms $\alpha_j$ and $\beta_d$ of \S 2. If $\lambda/\mu$ has $\delta$ components, then as a subvariety of ${\mathcal Q}$, $Z_{\lambda/\mu}$ is the generically transverse intersection of the zero loci of the forms $\alpha_j$ and any $\delta-1$ of the forms $\beta_d$. It follows that a general $(n+1-m)$-plane $K\subset{\mathcal Q}$ meets $Z_{\lambda/\mu}$ in $2^{\delta-1}$ lines. Thus if $\Span{v}\subset Z_{\lambda/\mu}$ is a general line, then $$ \# X_\mu\bigcap X'_{\lambda^c}\bigcap X_K \ =\ 2^\delta\cdot \# X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}. $$ Theorem~\ref{thm:pieri} is a consequence of this observation and the following: \begin{thm}\label{thm:unique} Let $\lambda/\mu$ be a skew row, $Z_{\lambda/\mu}$ be as above, and $\Span{v}$ a general line in $Z_{\lambda/\mu}$. Then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \end{thm} \noindent{\bf Proof. } Let ${\mathcal Q}_0$ be the cone of isotropic points in $V_0$ and ${\mathcal Q}_d$ the cone of isotropic points in $V_d$ for $d\neq d_0$. Since $$ Z_{\lambda/\mu}\ =\ H_\varphi \oplus {\mathcal Q}_0\oplus \bigoplus_{0\not\in\mbox{\scriptsize col}(d)} {\mathcal Q}_d, $$ we see that a general non-zero vector $v$ in $Z_{\lambda/\mu}$ has the form $$ v\ =\ \sum_{\mu_j=\lambda_j} a_je_{\mu_j}\ +\ v_0\ + \sum_{0\not\in\mbox{\scriptsize col}(d)} v_d, $$ where $a_j\in {\mathbb C}^\times$ and $v_0\in {\mathcal Q}_0$, $v_d\in {\mathcal Q}_d$ are general vectors. Thus, if $H\in X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$, we see that $v_0\in H_0$ and $v_d\in H_d$. By Theorem~\ref{thm:main}, $H$ is determined by $H_0$ and the $H_d$, thus it suffices to prove that $H_0$ and the $H_d$ are uniquely determined. The identifications of Lemmas~\ref{lemma:0comp} and~\ref{lem:offdiag} show that this is just the case of the theorem when $\lambda/\mu$ has a single component, which is Lemma~\ref{lem:components} below. \QED \begin{lemma}\label{lem:components} Suppose $\lambda,\mu\in{\mathbb{SY}}_n$ where $\lambda/\mu$ is a skew row with a unique component and no non-zero fixed points. \begin{enumerate} \item If $\lambda/\mu$ does not meet the first column and $v\in {\mathcal Q}_d$ is a general vector, then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \item If $\lambda/\mu$ meets the first column and $v\in {\mathcal Q}$ is general, then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \end{enumerate} \end{lemma} \noindent{\bf Proof of (1). } Let $v\in{\mathcal Q}_d$ be a general vector. Since ${\mathcal Q}_d\subset V^+\oplus V^-$, $v=v^+\oplus v^-$ with $v^+ \in V^+$ and $v^-\in V^-$. Suppose $\mu_k>0>\mu_{k+1}$. Consider the set $$ \{H^+\in G_k(V^+)\mid v\in H^+\oplus\left(H^+\right)^\perp\} \ =\ \{H^+\mid v^+\in H^+\subset (v^-)^\perp\}. $$ This is a Schubert variety $\Omega''_{h(n-k,k)}$ of $G_kV^+$, where $h(n-k,k)$ is the partition of hook shape with a single row of length $n-k$ and a single column of length $k$. Under the isomorphisms of (\ref{isomorphism}) and Lemma~\ref{lem:offdiag}, and with the identification of Corollary~\ref{cor:classical}, we see that $$ X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}\ \simeq\ \Omega_\tau\bigcap\Omega'_{\sigma^c}\bigcap\Omega''_{h(n-k,k)}, $$ where $\sigma,\tau$ are as defined in the paragraph preceding Corollary~\ref{cor:classical}. For $\rho\in{\mathbb{Y}}_k$, let $S_\rho:=[\Omega_\rho]$ be the cohomology class Poincar\'e dual to the fundamental cycle of $\Omega_\rho$ in $H^*G_kV^+$. The multiplicity we wish to compute is \begin{equation}\label{Schur:calc} \deg (S_\tau\cdot S_{\sigma^c}\cdot S_{h(n-k,k)}). \end{equation} By a double application of the classical Pieri's formula (as $S_{h(n-k,k)}=S_{n-k}\cdot S_{1^{k-1}}$), we see that (\ref{Schur:calc}) is either 1 or 0, depending upon whether or not $\sigma/\tau$ has exactly one box in each diagonal. But this is the case, as the transformation $\mu,\lambda \longmapsto \tau,\sigma$ takes columns to diagonals. \QED Our proof of Lemma~\ref{lem:components} (2) uses a system of local coordinates for $X_\mu\bigcap X'_{\lambda^c}$. Let $\lambda/\mu$ be as in Lemma~\ref{lem:components} (2), and suppose $\lambda_{k+1}=1$. For $y_2,\ldots,y_n,x_0,\ldots,x_{n-1}\in{\mathbb C}$, define vectors $g_j\in V$ as follows: \begin{equation}\label{loc_coords} g_j\ := \ \left\{\begin{array}{ll} {\displaystyle e_{\lambda_j} + \sum_{i=\mu_j}^{\lambda_j-1}x_i\,e_i}&\ j\leq k\\ {\displaystyle -2x_0^2e_1 + 2x_0e_0+ e_{\overline{1}} + \sum_{i=\mu_{k+1}}^{\overline{2}}y_i\,e_i}&\ j= k+1\\ {\displaystyle e_{\lambda_j} + \sum_{i=\mu_j}^{\lambda_j-1}y_i\,e_i}&\ j>k+1\\ \end{array}\right.. \end{equation} \begin{lemma}\label{lem:loc_coords} Let $\lambda,\mu\in{\mathbb{SY}}_n$ where $\lambda/\mu$ is a skew row meeting the first column with no fixed points, and define $\tau,\sigma\in{\mathbb{Y}}_k$, and $k$ as for Lemma~\ref{lem:subspace}. Then \begin{enumerate} \item For any $x_1,\ldots,x_{n-1}\in{\mathbb C}$, we have $\Span{g_1,\ldots,g_k}\ \in\ \Omega_\tau\bigcap\Omega'_{\sigma^c}$. \item For and $x_0,\ldots,x_{n-1}\in{\mathbb C}$ with $x_{\overline{\mu_{k+1}}},\ldots,x_{\overline{\mu_{n-1}}}\neq 0$, the condition that $H:= \Span{g_1,\ldots,g_n}$ is isotropic determines a unique $H\in X_\mu\bigcap X'_{\lambda^c}$. \end{enumerate} Moreover, these coordinates parameterize dense subsets of the intersections. \end{lemma} \noindent{\bf Proof. } The first statement is immediate from the definitions. For the second, note that each $g_j\in{\mathcal Q}$. The conditions that $\Span{g_1,\ldots,g_n}$ is isotropic are $$ \beta(g_i,g_j)\ =\ 0\quad \mbox{for}\quad i\leq k< j. $$ Only $n-1$ of these are not identically zero. Indeed, for $i\leq k<j$, $$ \beta(g_i,g_j)\ \not\equiv\ 0\ \ \Longleftrightarrow\ \ \left\{\begin{array}{ll} \mbox{either}\ & \overline{\lambda_j}<\mu_i<\overline{\mu_j},\\ \mbox{or}& \mu_i<\overline{\mu_j}<\lambda_i. \end{array}\right. $$ Moreover, if we order the variables $y_2<\cdots<y_n<x_0<\cdots<x_{n-1}$, then, in the lexicographic term order, the leading term of $\beta(g_i,g_j)$ for $i\leq k<j$ is $$ \begin{array}{cl} y_{\lambda_i}&\ \mbox{if}\ \overline{\lambda_j}<\mu_i<\overline{\mu_j},\\ y_{\overline{\mu_j}}x_{\overline{\mu_j}} &\ \mbox{if}\ \mu_i<\overline{\mu_j}<\lambda_i,\quad \mbox{or}\\ y_n=y_{\overline{\mu_n}}&\ \mbox{if}\ i=1,\ j=n.\end{array} $$ Since $\{2,\ldots,n\}=\{\lambda_2,\ldots,\lambda_{k-1}, \overline{\mu_k},\ldots,\overline{\mu_n}\}$, each $y_l$ appears as the leading term of a unique $\beta(g_i,g_j)$ with $i<k\leq j$, thus these $n-1$ non-trivial equations $\beta(g_i,g_j)=0$ determine $y_2,\ldots,y_n$ uniquely. These coordinates parameterize an $n$-dimensional subset of $X_\mu\bigcap X'_{\lambda^c}$. Since $\dim(X_\mu\bigcap X'_{\lambda^c})=n$ and $X_\mu\bigcap X'_{\lambda^c}$ is irreducible~\cite{Deodhar}, this subset is dense, which completes the proof. \QED \begin{ex} {\em Let $\lambda=6\,5\,3\,1\,\overline{2}\,\overline{4}$ and $\mu = 5\,3\,1\,\overline{2}\,\overline{4}\,\overline{6}$ so $k=3$. We display the components of the vectors $g_i$ in a matrix $$ \begin{array}{l|cccccc|c|cccccc} {} &e_{\overline{6}}&e_{\overline{5}}&e_{\overline{4}}&e_{\overline{3}} &e_{\overline{2}}&e_{\overline{1}}&e_0&e_1&e_2&e_3&e_4&e_5&e_6\\ \hline g_1&0&0&0&0&0&0&0&0&0&0&0&x_5&1\\ g_2&0&0&0&0&0&0&0&0&0&x_3&x_4&1&0\\ g_3&0&0&0&0&0&0&0&x_1&x_2&1&0&0&0\\ g_4&0&0&0&0&y_2&1&2x_0&-2x_0^2&0&0&0&0&0\\ g_5&0&0&y_4&y_3&1&0&0&0&0&0&0&0&0\\ g_6&y_6&y_5&1&0&0&0&0&0&0&0&0&0&0 \end{array} $$ Then there are 5 non-zero equations $\beta(g_i,g_j)=0$ with $i\leq 3<j$: \begin{eqnarray*} 0\ =\ \beta(g_3,g_4) &=& y_2 x_2 + x_1\\ 0\ =\ \beta(g_3,g_5) &=& y_3+x_2\\ 0\ =\ \beta(g_2,g_5) &=& y_4x_4 + y_3x_3\\ 0\ =\ \beta(g_2,g_6) &=& y_5+x_4\\ 0\ =\ \beta(g_1,g_6) &=& y_6+x_5y_5 \end{eqnarray*} Solving, we obtain: $$ y_2=-x_1/x_2,\ y_3=-x_2,\ y_4=-y_3x_3/x_4,\ y_5=-x_4,\ \mbox{and}\ y_6=-x_5y_5. $$ }\end{ex} \noindent{\bf Proof of Lemma~\ref{lem:components} (2). } Suppose $\lambda,\mu\in{\mathbb{SY}}_n$ where $\lambda/\mu$ is a skew row with a single component meeting the first column and no fixed points. Let $v\in{\mathcal Q}$ be a general vector and consider the condition that $v\in H$ for $H\in X_\mu\bigcap X'_{\lambda^c}$. Let $\sigma,\tau\in{\mathbb Y}_k$ be defined as in the paragraph preceding Lemma~\ref{lem:subspace}. We first show that there is a unique $L\in\Omega_\tau\bigcap\Omega_{\sigma^c}$ with $L\subset H$, and then argue that $H$ is unique. The conditions on $\mu$ and $\lambda$ imply that $\mu_n=\overline{n}$ and $\mu_j=\lambda_{j+1}$ for $j<n$. We further suppose that $\lambda_{k+1}=1$, so that the last row of $\lambda/\mu$ has length 1. This is no restriction, as the isomorphism of $V$ defined by $e_j\mapsto e_{\overline{\jmath}}$ sends $X_{\mu}\bigcap X'_{\lambda^c}$ to $X_{\lambda^c}\bigcap X'_{(\mu^c)^c}$ and one of $\lambda/\mu$ or $\mu^c/\lambda^c$ has last row of length 1.\smallskip Let $v\in {\mathcal Q}$ be general. If necessary, scale $v$ so that its $e_{\overline{1}}$-component is 1. Let $2z$ be its $e_0$-component, then necessarily its $e_1$-component is $-2z^2$. Let $v^-\in V^-$ be the projection of $v$ to $V^-$. Similarly define $v^+\in V^+$. Set $v':= v^+ + 2z^2e_1$, so that $\beta(v^-,v')=0$ and $$ v\ =\ v^- + 2z(e_0 - z e_1) + v'. $$ Let $H\in X_\mu\bigcap X'_{\lambda^c}$, and suppose that $v\in H$. In the notation of Lemma~\ref{lem:subspace}, let $L\in \Omega_\tau\bigcap\Omega_{\sigma^c}$ be a $k$-plane in $H\bigcap V^+$. If $H$ is general, in that $$ \dim H\bigcap\Span{e_{\overline{n}},\ldots,e_{\lambda_{k+2}}}\ =\ \dim H\bigcap\Span{e_{\overline{n}},\ldots,e_0}\ =\ n-k-1, $$ then $\Span{L,e_1}$ is the projection of $H$ to $V^+$. As $v\in H$, we have $v^+\in \Span{L,e_1}$. Since $L\subset v^\perp\bigcap V^+=(v^-)^\perp$, we see that $v'\in L$, and hence $$ v'\ \in\ L\ \subset\ (v^-)^\perp. $$ As in the proof of part (1), there is a (necessarily unique) such $L\in\Omega_\tau\bigcap\Omega_{\sigma^c}$ if and only if $\sigma/\tau$ has a unique box in each diagonal. But this is the case, as the transformation $\mu,\lambda \longrightarrow \tau,\sigma$ takes columns (greater than 1) to diagonals. \smallskip To complete the proof, we use the local coordinates for $X_\mu\bigcap X'_{\lambda^c}$ and $\Omega_\tau\bigcap\Omega_{\sigma^c}$ of Lemma~\ref{lem:loc_coords}. Since $v$ is general, we may assume that the $k$-plane $L\in\Omega_\tau\bigcap\Omega_{\sigma^c}$ determined by $v'\in L\subset(v^-)^\perp$ has non-vanishing coordinates $x_{\overline{\mu_{k+1}}},\ldots,x_{\overline{\mu_{n-1}}}$, so that there is an $H\in X_\mu\bigcap X'_{\lambda^c}$ in this system of coordinates with $L=H\cap V^+$. Such an $H$ is determined up to a choice of coordinate $x_0$. The requirement that $v\in H$ forces the projection $\Span{e_{\overline{1}}+ 2x_0e_0}$ of $H$ to $\Span{e_{\overline{1}},e_0}$ to contain $e_{\overline{1}}+ 2ze_0$, the projection of $v$ to $\Span{e_{\overline{1}},e_0}$. Hence $x_0=z$, and it follows that there is at most one $H\in X_\mu\bigcap X'_{\lambda^c}$ with $v\in H$. Let $g_1,\ldots,g_n$ be the vectors~(\ref{loc_coords}) determined by the coordinates $x_1,\ldots,x_{n-1}$ for $L$ with $x_0=z$. We claim $v\in H:=\Span{g_1,\ldots,g_n}$. Indeed, since $v'\in L$ and $v^-\in L^\perp=\Span{g_{k+1}-2z(e_0-ze_1),g_{k+2},\ldots,g_n}$, there exists $\alpha_1,\ldots,\alpha_n\in{\mathbb C}$ with $$ v^-+v'\ =\ \alpha_1 g_1+\cdots + \alpha_{k+1}(g_{k+1}-2z(e_0-ze_1)) +\cdots+\alpha_ng_n. $$ We must have $\alpha_{k+1}=1$, since the $e_{\overline{1}}$-component of both $v$ and $g_{k+1}$ is 1. It follows that $$ v \ =\ \sum_{i=1}^n \alpha_i g_i\quad \in \ H.\qquad\QED $$ \noindent{\bf Remarks. } \begin{enumerate} \item Our desire to give elementary proofs led us to restrict ourselves to the complex numbers. With the appropriate modifications, these arguments give the same results for Chow rings of these varieties over any field of characteristic $\neq 2$. For example, the appropriate intersection-theoretic constructions and the properness of a general translate provide a substitute for our use of transversality. Then one could argue for the multiplicity of $2^{\delta-1}$ as follows: If $\lambda,\mu\in{\mathbb{SY}}_n$ and $K$ is a linear subspace in ${\mathcal{Q}}$, then the scheme-theoretic intersection $X_\mu\bigcap X'_{\lambda^c}\bigcap X_K$ is $pr_* \pi^*(K)$, where $$ \begin{picture}(200,52) \put(0,0){$K\ \hookrightarrow\ {\mathcal{Q}}$} \put(80,0){$C_n$} \put(60,40){$\Xi\ =\ \{(p,H)\mid p\in H\in X_\mu\bigcap X'_{\lambda^c}\}$} \put(60,35){\vector(-1,-2){11}} \put(68,35){\vector(1,-2){11}} \put(46,24){\scriptsize$\pi$} \put(78,24){\scriptsize$pr$} \end{picture} $$ Then intersection theory on the quadric ${\mathcal{Q}}$ (a homogeneous space) and Kleiman's Theorem that the intersection with a general translate is proper~\cite{Kleiman} gives a factor of $2^{\delta-1}$ from the intersection multiplicity of $K$ and the subvariety $Z_{\lambda/\mu}$ of ${\mathcal Q}$ consisting of the image of $\pi$. The arguments of Section 3 show that $\pi$ has degree 1 onto its image. \item Conversely, similar to the proof of Lemma~\ref{lem:loc_coords}, we could give local coordinates for any intersection $X_\mu\bigcap X'_{\lambda^c}$. Such a description would enable us to establish transversality directly, and to dispense with the intersection theory of the classical Grassmannian. This would work over any field whose characteristic is not 2, but would complicate the arguments we gave. \item We have not investigated to what extent these methods would work in characteristic 2. \end{enumerate}

9708

alg-geom/9708004

en

https://arxiv.org/abs/alg-geom/9708004

alg-geom/9708004

Mark De Cataldo

Mark Andrea A. de Cataldo

Effective nonvanishing, effective global generation

LaTex (article) 13 pages; revised: one section added; to appear in Ann. Inst. Fourier

We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.

alg-geom

\section{Introduction} \label{intr} Koll\'ar's nonvanishing theorem \ci{koebpf}, 3.2 is an instrument to make Kawamata-Shokurov base-point-freeness assertion into an effective one. His result can be applied to a variety of other situations; see \ci{koebpf}, \S4, \ci{koshafinv}, \S8 and \ci{koshaf}, \S14. The basic set-up is as follows. Let $g: X \to S$ be a surjective morphism of proper varieties, where $X$ in nonsingular and complete, $M$ be a nef and $g$-big line bundle on $X$, $L$ be a nef and big line bundle on $S$ and $N=K_X + M + mg^*L$ be a line bundle varying with the positive integer $m$. Koll\'ar proves, under the necessary assumption that the first direct image sheaf $g_*N\not= 0$, that $h^0(X, N)=h^0(S,g_*N)>0$ and the sections of $g_*N$ generate this sheaf at a general point of $S$ for {\em every} $\, m> (1/2)(\dim{S}^2 + \dim{S})$ (this is what makes the result into an ``effective" tool). The proof starts with the choice of a very general point $x$ on $S$ and ends with producing sections of $g_*N$ which generate at $x$ and therefore at a general point. \medskip The purpose of this note is to observe that we can obtain more precise statements by considering the local Seshadri constants of $L$ on $S$ and we can also simplify considerably the proof. See the discussion at the beginning of \S\ref{nonvan} and Remark \ref{why}. Our main result is the Effective Nonvanishing Theorem \ref{effnonvan}, a ``multiple-points higher-jets" version of \ci{koebpf}, Theorem 3.2. The proof hinges on Demailly's observation that given a nef line bundle $\cal L$ on $X$, a big enough local Seshadri constant for $\cal L$ at a point $x$ can be used together with Kawamata-Viehweg Vanishing Theorem to produce sections of the adjoint line bundle $K_X + {\cal L}$ with nice generating properties at $x$ (cf. \ci{dem94}, Proposition 7.10). An effective way to force a big enough local Seshadri constant is Theorem \ref{ekltm}, which is due Ein-K\"uchle-Lazarsfeld. \smallskip As applications we offer some generalizations (to the case of nef vector bundles) of the results concerning line bundles in \ci{ekl} and \ci{koshafinv}: effective construction of rational and birational maps, and nonvanishing on varieties with big enough algebraic fundamental group. These results in the case of one point follow easily from results of Koll\'ar by considering the tautological line bundle of the projectivization of the vector bundles in question. We present them as an exemplification of the unifying character of Theroem \ref{effnonvan} and also because they are new in the case of multiple-points and higher-jets. \noindent We also show how the global generation results for line bundles of Anghern-Siu, Demailly, Tsuji and Siu (see \ci{dem94} and \ci{deshm} for a bibliography) generalize to vector bundles of the form $K_X^{\otimes a} \otimes E\otimes \det{E} \otimes L^{\otimes m}$, where $a$ and $m$ are appropriate positive integers, $E$ is a nef vector bundle and $L$ is and ample line bundle. We give explicit upper bounds on $m$ which depend only on the dimension of the variety, and not also on the Chern classes of the variety and the bundles in question. However, we do not expect these bounds to be otpimal since they do not match with the line bundle case ({\em i.e.} assuming that the vector bundle $E$ is the trivial line bundle). In our paper \ci{deshm} we prove upper bounds as above for vector bundles $E$ subject to curvature conditions which seem to be the natural differential-geometric analogue of nefness and indeed imply nefness. These bounds match exactly the results in the line bundle case. The methods employed there are analytical. In the final section we prove, using the language and techniques of algebraic Nadel ideals, a global generating statement for nef vector bundles which indeed matches the result of Anghern and Siu in the line bundle case. \medskip The paper is organized as follows. \S1 is preliminary and consists of easy and mostly known facts about local Seshadri constants, and of more elaborate ones, such as Theorem \ref{ekltm}, which makes Theorem \ref{effnonvan} into an effective statement. \S2 is devoted to the main result, Theorem \ref{effnonvan}. \S3 is devoted to the applications discussed above. \S4 is devoted to the proof of Corollary 4.6, a major improvment of the results of Theorem 3.7.1 and 3.7.3. \bigskip \S4 has been written while the author enjoyed the hospitality of the Max-Planck-Institut f\'ur Mathematik of Bonn. \section{Notation and preliminaries} \label{not} We mostly employ the notation of \ci{k-m-m}. A {\em variety} is an integral separated scheme of finite type over an uncountable algebraically closed field of characteristic zero $k$. We say that {\em a property holds at a very general point on $X$} if it holds for every point in the intersection, $\frak U$, of some at most countable family of Zariski-open dense subsets of $X$. Any such set $\frak U$ meets any Zariski-open dense subset of $X$. The term ``point" refers to a closed one. Vector bundles and associated locally free sheaves are identified. Cartier divisors are at times identified with the associated invertible sheaves and the additive and multiplicative notation are both used, at times simultaneously. The symbol $B(a,b)$ denotes the usual binomial coefficient. Let $X$ be a variety, $n$ be its dimension and $Div(X)$ be the group of Cartier divisors on $X$. A {\em $\rat$-Cartier divisor} is an element of $Div(X)\otimes {\rat}$. The linear and numerical equivalence of $\rat$-divisors are denoted by ``$\approx$" and ``$\equiv$," respectively. A {\em $\rat$-divisor} is an element in $Z_{n-1}(X) \otimes {\rat}$, where $Z_{n-1}(X)$ is the free group of Weil divisors on $X$. The symbols $\lfloor a\rfloor$ and $\langle a \rangle$ denote the biggest integer less than or equal to $a$, and $a- \lfloor a\rfloor$, respectively. These symbols are used in conjunction with $\rat$-divisors when these divisors are written as a $\rat$-combination of {\em distinct} prime divisors. Given any proper morphism of varieties $\pi: X \to S$, we have the notions of ($\pi$-)ample, ($\pi$-)nef, ($\pi$-)big and ($\pi$-)nef and ($\pi$-)big for (numerical equivalence classes of) $\rat$-Cartier divisors on $X$. Let ${\cal A} \in (Div(X) \otimes {\rat})/\equiv$ be a numerical class and $D$ be a $\rat$-Cartier divisor on $X$. By abuse of notation, ${\cal A} \equiv D$ means that $A \equiv D$ for one, and thus all, the elements in ${\cal A}$. This remark plays a role when we use the canonical divisor class together with $\rat$-Cartier divisors. In what above, the field $\real$ can replace $\rat$ with minor changes. \medskip The following two vanishing-injectivity theorems are needed for Theorem \ref{effnonvan}. \begin{tm} \label{van} {\rm (Cf. \ci{k-m-m}, 1.2.3)} Let $X$ be a nonsingular variety and $\pi: X \to S$ be a proper morphism onto a variety $S$. Assume that $N$ is a Cartier divisor on $X$ and that $M$ and $\Delta$ are $\rat$-Cartier divisors on $X$ with the following properties: $(1)$ $M$ is $\pi$-nef and $\pi$-big, $(2)$ the support of $\Delta$ is a divisor with normal crossings, and $\lfloor \Delta \rfloor = 0$, and $(3)$ $N\equiv M+ \Delta$. \smallskip \noindent Then $R^i\pi_*\odix{X} (K_X + N) =0$ for $i>0$. \end{tm} \begin{tm} \label{inj} {\rm (Cf. \ci{koshaf}, 10.13 and 9.17, and \ci{e-v}, 5.12.b)} Let $\pi:X \to S$ and $\Delta$ be as above with $X$ projective, $D$ be an effective Cartier divisor on $X$ such that it does not dominate $S$ via $\pi$, and $L$ be a nef and big $\rat $-Cartier divisor on $S$. Let $N$ be a Cartier divisor such that $N\equiv \Delta + \pi^*L$. \noindent Then the following natural homomorphisms are injective for every $i\geq 0$: $$ H^i (X, K_X + N) \longrightarrow H^i(X, K_X + N + D). $$ \end{tm} \medskip \noindent {\bf Local Seshadri constants.} Good references for what follows are \ci{dem94}, \S7 and \ci{ekl}. \begin{defi} \label{seshco} {\rm Let $X$ be a complete variety, $L$ be a {\em nef} $\,$ $\rat$-Cartier divisor on $X$, and $x$ be a point on $X$. The following nonnegative real number is called the {\em Shesadri constant of $L$ at $x$}: $$ \e(L,x)= \inf \{ \frac{L\cdot C}{mult_x C} \}, $$ where the infimum is taken over all integral curves passing through $x$ and $mult_x C$ is the multiplicity of $C$ at $x$. } \end{defi} Let $x$ be a point in $X_{reg}$, $b_x: X' \to X$ be the blowing-up of $X$ at $x$ and $E$ be the corresponding exceptional divisor on $X'$. The $\rat$-Cartier divisor $b_x^* L$ on $X'$ is nef as well. In particular, there is a well-defined nonnegative real number: $$ \e'(L,x) : = \sup \{\e' \in \rat | \, b_x^*L - \e' E \,\,\, is \,\, nef \,\}. $$ It is clear that the $\real$-Cartier divisor $ b_x^*L - \e'(L,x) E$ is nef, and that the $\rat$-Cartier divisor $ b_x^*L - \e' E$ is nef for every rational number $\e'$ with the property that $0\leq \e' \leq \e' (L,x)$. \noindent \begin{fact} \label{same} We have that $\e(L,x)=\e'(L,x)$ for every $x\in X_{reg}$. {\rm This follows from the formula: $(b_x^*L - \e E)\cdot \tilde{C}= L\cdot C - \e \, mult_x C$, where $\e$ is any real number, $\tilde{C}$ is any integral curve in $X'$ not contained in $E$ and $C:=b_x(\tilde{C})$. } \end{fact} \smallskip We collect the simple properties of $\e(L,x)$ which, together with Theorem \ref{ekltm}, are essential in the sequel of the paper. \begin{lm} \label{basic} Let $L$ be a nef $\,\rat$-Cartier divisor on a complete variety $X$ and $x$ be a point in $X_{reg}$. Then \noindent {\rm (\ref{basic}.1)} $L^n \geq \e(L,x)^n$; \noindent {\rm (\ref{basic}.2)} for every $t\in \rat^+$, $\e(tL,x)=t\,\e(L,x)$; \noindent {\rm (\ref{basic}.3)} Let $f: X' \to X$ be a proper and birational morphism and $x$ be a point on $X$ over which $f$ is an isomorphism; then $\e(L,x)=\e(f^*L, f^{-1}\{x\})$; \noindent {\rm (\ref{basic}.4)} if $L$ is Cartier, ample and generated by its global sections on $X$, then $\e(L,x) \geq 1$; \noindent {\rm (\ref{basic}.5)} if $L$ is Cartier and the global sections of $L$ generate jets of order $s$ at $x$, {\em i.e.} the natural evaluation map $H^0(X,L) \to \odixl{X}{L}/{\frak m}_x^{s+1} \odixl{X}{L}$ is surjective, then $\e(L,x) \geq s$. \end{lm} \noindent {\em Proof.} The first property follows from the fact that since $b_x^*L - \e(L,x)E$ is nef, then its top self-intersection is nonnegative. The second one is an obvious consequence of the bilinearity of the intersection product. \noindent The third property follows from the fact that there is a natural bijection, given by taking strict transforms, between the sets of integral curves on $X$ through $x$ and on $X'$ through $x':=f^{-1}\{x\}$. If $C$ and $C'$ correspond to each other in this bijection, then $L \cdot C= b_x^*L \cdot C'$ and $mult_xC=mult_{x'}C'$ so that the two local Seshadri constants are the same. \noindent If $L$ is ample, Cartier and generated by its global sections on $X$, then the rational map $\varphi$ defined by $|L|$ is a finite morphism. Let $C$ be any integral curve on $X$ passing through $x$. Since $C$ is not contracted by $\varphi$, there is an effective divisor $D$ in $|L|$ passing through $x$ but not containing $C$. It follows that $L\cdot C=D\cdot C \geq mult_x C$. This implies the third property. \noindent Finally, if $s=0$, then there is nothing left to prove. Assume that $s\geq 1$. Then the global sections of $L\otimes {\frak m}_x^{s}$ generate $L\otimes {\frak m}_x^{s}$ at $x$. Given any integral curve $\tilde{C}$ on $X'$ not contained in $E$, we find a divisor $D \in |L\otimes {\frak m}_x^s|$ not containing the curve $b_x (\tilde{C})$. It follows that the effective divisor $b_x^* (D) \in |b_x^*L -sE|$ does not contain $\tilde{C}$. In particular, $(b_x^*L -sE)\cdot \tilde{C} \geq 0$. This is enough to establish the last property. \blacksquare \medskip If $X$ is complete and $L$ is a nef $\rat$-Cartier divisor on $X$, then Shesadri's criterion of ampleness asserts that $L$ is ample iff $\e(L): = \inf \{ \e(L,x) | \, x \in X \} >0$. An example of R. Miranda's (cf. \ci{dem94}, 7.14) shows that given any positive real number $\e$, there exists a nonsingular rational surface $X$, a point $x \in X$ and an ample line bundle $L$ on $X$, such that $\e(L,x) \leq \e$. \noindent In particular, we {\em cannot} expect to have a statement of the form: {\em let $X$ be a nonsingular projective variety of dimension $n$ and $L$ be an ample line bundle on $X$, then $\e(L) \geq C_n$, for some positive constant depending only on $n$}. \smallskip What is known is the following result of Ein, K\"uchle and Lazarsfeld. The authors prove it for projective varieties, but by Chow's Lemma and Lemma \ref{basic}.3 the statement is true for every complete variety. \begin{tm} \label{ekltm} {\rm (Cf. \ci{ekl}, Theorem 1)} Let $L$ be a nef and big Cartier divisor on a complete variety $X$ of dimension $n$. Then at a very general point $x$ on $X$ we have: $$ \e (L,x) \geq \frac{1}{n}. $$ \end{tm} The example that follows shows that Theorem \ref{ekltm} cannot hold as stated for an ample and effective integral $\rat$-Cartier $\rat$-divisor on a normal projective variety. As is pointed out in \ci{ekl}, if $m$ is the smallest positive integer such that $mL$ is Cartier, then Theorem \ref{ekltm} holds if we replace ``$\e(L,x) \geq \frac{1}{n}$" by ``$\e(L,x) \geq \frac{1}{nm}$." \begin{ex} \label{exindex} {\rm Let $S_m\subset \pn{m+1}$ be the surface which is a cone of vertex $v$ over the rational normal curve of degree $m$ in $\pn{m}$ and $\frak l$ be any line belonging to the ruling of $S_m$. The Weil divisor $\frak l$ is an integral $\rat$-Cartier $\rat$-divisor; to be precise it is $m$-Cartier. The Cartier divisor $m{\frak l}$ is very ample so that, for every $x \in S_m \setminus \{v \}$, we have that $\e({\frak l},x) \geq \frac{1}{m}$. On the other hand, fix $x \in S_m \setminus \{v \}$ and let $ C$ be the line on $S_m$ passing through $x$. We have ${\frak l} \cdot C=\frac{1}{m}$, so that $\e( {\frak l},x ) \leq \frac{1}{m}$. It follows that $\e({\frak l},x) = \frac{1}{m}$, for every $x \in S_m \setminus \{v \}$. } \end{ex} \section{An effective nonvanishing theorem} \label{nonvan} In this section we prove a nonvanishing theorem very similar to \ci{koebpf}, Theorem 3.2. While the statement is clearly inspired by \ci{koebpf}, Theorem 3.2, its simpler proof is inspired by \ci{dps}, Lemma 3.21. The basic nonvanishing and global generation at a generic point follow easily from \ci{koebpf}, Theorem 3.2 (and in fact are slightly weaker than this latter result). The ``multiple-points higher-jets" statements do not follow directly from the results in the literature. \noindent Let us point out that Koll\'ar's result implies a version of Theorem \ref{effnonvan} with $x$ general, p=1 and s=0. However, one can use this result in place of Koll\'ar's in proving the effective base-point-freeness result 1.1 of \ci{koebpf}. The advantages of Theorem \ref{effnonvan} are at least two. \noindent The former is the simplicity of its proof which consists of basic yoga and one blowing-up procedure. However, we should stress that more often than not this becomes an effective result if used in conjunction with the non-trivial result Theorem \ref{ekltm}. \noindent The latter is that it is a ``multiple-point higher-jets" effective result which, at least in principle, can be applied to prescribed points and can give more than mere nonvanishing. For example, one can use this result to obtain increased lower bounds of log-plurigenera (cf. \ci{koebpf}, \S4). We shall see some other applications in the following sections. \begin{rmk} \label{logsmooth} {\rm Let $g:Y \to S$ be a proper morphism of varieties with $Y$ nonsingular and $\Delta$ a divisor on $Y$ such that $Supp(\Delta)$ has simple normal crossings. \noindent By virtue of generic smoothness, there exists a largest Zariski-open dense subset $U$ of $S$ such that (i) $g_{|g^{-1}(U)}: g^{-1}(U) \to U$ is smooth, (ii) for every point $x \in S$, any irreducible component $F$ of the fiber $F_x$ of $g$ is not contained in $Supp(\Delta)$ and (iii) $Supp(\Delta)$ has simple normal crossings on $F$. } \end{rmk} \begin{tm} \label{effnonvan} {\rm ({\bf Effective Nonvanishing})} Let the following data be given. \noindent {\rm (\ref{effnonvan}.1)} $(Y,\Delta)$: a log-pair, where $Y$ is nonsingular and complete, $\lfloor\Delta \rfloor = 0$ and $Supp(\Delta)$ has simple normal crossings. \noindent {\rm (\ref{effnonvan}.2)} $N$: a Cartier divisor on $Y$. \noindent {\rm (\ref{effnonvan}.3)} \noindent - $g: Y \to S$: a proper and surjective morphism onto a complete variety $S$ of positive dimension, \noindent - $U=U(g,\Delta)$: the Zariski-open dense set of $S$ defined in {\rm Remark \ref{logsmooth}}, \noindent - $V$: the Zariski-open dense subset of $S$ over which the formation of $g_*$ for $N$ commutes with base extensions. \noindent {\rm (\ref{effnonvan}.4)} \noindent - $p$: a positive integer, \noindent - $\{s_1, \ldots, s_p\}$: a $p$-tuple of non-negative integers, \noindent - $\{x_1, \ldots, x_p\}$: $p$ distinct points in $U\cap V$. \noindent {\rm (\ref{effnonvan}.5)} $M$: a $\rat$-Cartier divisor on $Y$ such that either it is nef and $g$-big, or $X$ is projective and $M\equiv 0$. \noindent {\rm (\ref{effnonvan}.6)} $L_1, \ldots, L_p$: $p$ $\rat$-Cartier divisors on $S$ such that all $L_j$ are nef and big and either \noindent (a) $\e(L_j, x_j) > \dim S + s_j$, $\forall j=1, \ldots, p$, or \noindent (b) $\e(L_j, x_j) \geq \dim S + s_j$, $\forall j=1, \ldots, p$ and $L_{j_0}^{\dim{S}} > \e(L_{j_0})^{\dim{S}}$ for at least one index $j_0$, $1 \leq j_0 \leq p$. \smallskip \noindent Assume that $$ N \equiv K_Y + \Delta + M + g^*\sum_{j=1}^p L_j. $$ \smallskip \noindent Then the following natural map is surjective $$ H^0(X,N) \simeq H^0(S, g_*N) \surj \bigoplus_{j=1}^p \, \, \frac{g_*N}{{\frak m}_{x_j}^{s_j+1} \cdot g_*N}. $$ In particular, if $g_*N$, which is torsion-free, is not the zero sheaf, then $H^0(X,N)\not= \{0\}$. \end{tm} \begin{rmk} \label{why} {\rm The reason for calling this theorem ``Effective Nonvanishing" is the last assertion of the theorem and the fact that, for example, if all the $L_j$ were Cartier, then we could make sure, by virtue of Theorem \ref{ekltm}, that condition (\ref{effnonvan}.6) is fulfilled at very general points by taking sufficiently high multiples of the $L_j$. \noindent Note also that the conclusion of the theorem holds trivially also for $\dim{S}=0$, but that in this case (2.1.6) is not meaningful. } \end{rmk} \noindent {\em Proof.} The proof is divided into two cases. The former deals with $M$ nef and $g$-big. The latter with $X$ projective and $M\equiv 0$. Each case is divided into two sub-cases corresponding to the two numerical assumptions (a) and (b) in (\ref{effnonvan}.6). \smallskip \noindent CASE I: {\em $M$ is nef and $g$-big}. \noindent First we show that in this case $U=U\cap V$. By virtue of \ref{van}, we know that $R^ig_* N=0$ for $i >0$. By the smoothness of $g$ over $U$, $N$ is flat over $U$. By well-known results of Grothendieck (see \ci{gro}, III.7.7.10) $g_*N$ is locally free on $U$ and the formation of $g_*$ commutes with base extension over $U$. \smallskip \noindent In particular, if $Y_{x_j}^{\s}:= Y\times_S Spec \, (\odix{S,x_j}/{\frak m}_{x_j}^{\s})$ is the ``$\s$-thickened fiber" of $g$ at $x_j$ and $N_{x_j}^{\s}$ is the pull-back of $N$ to $Y_{x_j}^{\s}$ via the natural projection, then the following natural maps are isomorphisms: $$ \frac{g_*N}{{\frak m}_{x_j}^{s_j+1} \cdot g_*N}= g_*N \otimes ( \odix{S,x_j}/{\frak m}_{x_j}^{s_j+1} ) \longrightarrow H^0(Y_{x_j}^{s_j+1}, N_{x_j}^{s_j+1}). $$ \smallskip \noindent To prove CASE I it is enough to show that the natural map \begin{equation} \label{1} H^0(Y,N) \longrightarrow \bigoplus_{j=1}^p \, \, H^0(Y_{x_j}^{s_j+1}, N_{x_j}^{s_j+1}), \end{equation} which factors through $g_*N \otimes \odix{S,x_j}/{\frak m}_{x_j}^{s_j+1}$, is surjective. \smallskip \noindent Consider the following cartesian diagram: $$ \begin{array}{lll} \hspace{1cm} Y' & \stackrel{B}\longrightarrow & Y \\ \hspace{1cm} \downarrow {g'} & \ & \downarrow g \\ \hspace{1cm} S' & \stackrel{b}\longrightarrow & S \end{array} $$ where $b$ is the blowing-up of $S$ at all the simple points $x_j$. Let $F:=\coprod F_j$ be the scheme-theoretic-fiber of $g$ corresponding to the union of the points $x_j$, $j=1,\ldots, p$. Since $g$ is smooth over $U$ and all the $x_j$ are in $U$, we see that $B$ coincides with the blowing-up of $Y$ along $F$. In particular, $Y'$ is a nonsingular variety. Let $E=\sum E_j$ be the exceptional divisor of $b$ and $D=\sum D_j$ the one of $B$; we have that $D_j={g'}^*E_j$, for every $j=1, \ldots, p$. \smallskip \noindent The map (\ref{1}) is surjective iff the natural map $H^1(Y',B^*N-\sum (s_j+1)D_j) \to H^1(Y',B^*N)$ is injective. It is this injectivity that we are going to establish using Theorem \ref{van}. \smallskip \noindent Note that $K_{Y'} \approx B^*K_Y + (\dim S -1)\sum{D_j}$ and that since no irreducible component of any $F_j$ is contained in any $\Delta_i$ and if any such component meets any $\Delta_i$ it does so transversally, we have that a) $\Delta':=B^*\Delta=B^{-1}_*\Delta$, {\em i.e.} the pull-back is the strict transform, b) $\lfloor \Delta' \rfloor=0$ and c) the support of $\Delta'$ has simple normal crossings. The following numerical equality is easily checked: \begin{equation} \label{2} B^*N - \sum{(s_j+1)D_j} \equiv K_{Y'} + \Delta' + B^*M + B^*g^*\sum{L_j} - \sum{(\dim{S} + s_j)D_j}. \end{equation} \smallskip \noindent SUB-CASE I.A: {\em Assume that $\e(L_j,x_j)> \dim{S} + s_j$, for every index $j$, $1 \leq j \leq p$.} \noindent Since for every index $j$ we have that $\e(L_j,x_j)> \dim{S} + s_j$, there exists a positive rational number $0<t<1$ such that $\e((1-t)L_j,x_j)> \dim{S} + s_j$ for every $j$, $1\leq j \leq p$. Using the fact that $B^*g^*={g'}^*b^*$ we can re-write the r.h.s. of equation (\ref{2}) as \begin{equation} \label{3} K_{Y'} + \Delta' + B^*(M + tg^*\sum{L_j}) + {g'}^*\sum {\left[ b^*(1-t)L_j - (\dim{S}+ s_j)E_j \right]}. \end{equation} The last summand is nef by the very definition of $\e((1-t)L_j,x_j)$. \noindent Since $M$ is nef and $g$-big and $t>0$, the $\rat$-divisor $M + tg^*\sum{L_j}$ is nef and big. In particular, $B^*(M + tg^*\sum{L_j})$ is nef and big. It follows that the l.h.s. of (\ref{2}) is a Cartier divisor satisfying the assumptions of Kawamata-Viehweg Vanishing Theorem so that $H^1(Y', B^*N-\sum (s_j+1)D_j)=\{0\}$ and (\ref{1}) is surjective. \medskip \noindent SUB-CASE I.B: {\em Assume that $\e(L_j, x_j) \geq \dim S + s_j$, $\forall j, \, 1\leq j \leq p$ and that $L_{j_0}^{\dim{S}} > \e(L_{j_0})^{\dim{S}}$ for at least one index $j_0$, $1 \leq j_0 \leq p$.} \noindent Using the fact that $B^*g^*={g'}^*b^*$ and isolating the index $j_0$ we write the r.h.s. of (\ref{2}) as \begin{equation} \label{4} K_{Y'} + \Delta' + B^*M + \sum_{j\not= j_0}{{g'}}^* \left[ b^*{L_j}- (\dim{S}+ s_j)E_j\right] + {{g'}}^* \left[ b^*{L_{j_0}}- (\dim{S}+ s_{j_0})E_{j_0} \right]. \end{equation} Since $M$ is nef and $g$-big and $\sum_{j\not= j_0}{{g'}}^* (b^*{L_j}- (\dim{S}+ s_j)E_j)$ is nef we see that $B^*M + \sum_{j\not= j_0}{{g'}}^* (b^*{L_j}- (\dim{S}+ s_j)E_j)$ is nef and ${g'}$-big. Since $L_{j_0}^{\dim{S}} > \e(L_{j_0},x_{j_0})^{\dim{S}}$, we see, as in the proof of Lemma \ref{basic}.1, that $ (b^*{L_{j_0}}- (\dim{S}+ s_{j_0})E_{j_0})$ is nef and big. It follows that $ B^*M + \sum_{j\not= j_0}{{g'}}^* (b^*{L_j}- (\dim{S}+ s_j)E_j) + {{g'}}^* (b^*{L_{j_0}}- (\dim{S}+ s_{j_0})E_{j_0})$ is nef and big and we conclude as in SUB-CASE I.A. \medskip \noindent CASE II: {\em $X$ is projective, $M\equiv0$ and the points $x_j$ are in $U\cap V$.} \noindent We by-pass the first paragraph in the proof of CASE I. We proceed {\em verbatim} as in that case until we hit again (\ref{2}). We delete $M$. We can again divide the analysis into two separate sub-cases. We do so and obtain that in the two distinct sub-cases the l.h.s. of (\ref{2}) is numerically equivalent to the r.h.s. of (\ref{3}) and (\ref{4}), respectively and, in both cases, we are in the position to apply Theorem \ref{inj} to the morphism ${g'}:Y' \to S'$ and infer the desired injectivity statement. \blacksquare \section{Applications} \label{firstapp} The local Seshadri constant can be linked, via Kawamata-Viehweg Vanishing Theorem to the production of sections for the adjoint to nef and big line bundles. This observation is due to Demailly; see \ci{dem94}, Proposition 7.10 and \ci{ekl}, Proposition 1.3. In this section we apply Theorem \ref{effnonvan} to nef vector bundles. Actually, a factor $\det{E}$ appears and is necessary in our proof. We ignore if it is necessary for the truth of the various statements that follow. First we fix some notation. \bigskip Let $E$ be a rank $r$ vector bundle on a nonsingular complete variety $X$. We denote by ${\Bbb P}_X(E)$ the projectivized bundle of hyperplanes, by $\pi: {\Bbb P}_X(E) \to X$ the natural morphism and by $\xi$ or $\xi_E$ the tautological line bundle $\odixl{{\Bbb P}_X(E)}{1}$. We say that $E$ is nef if $\xi$ is nef. Let $p$ be any positive integer. We say that {\em the global sections of $E$ generate jets of order $s_1,\ldots, s_p \in {\Bbb N}$ at $p$ distinct points $\{ x_1, \ldots ,x_p \} $ of $X$} if the following natural map is surjective: $$ H^0(X,E) \longrightarrow \bigoplus_{\i=1}^p E_{x_{i}} \otimes \odix{X}/{\frak m}^{s_{i}+1}_{x_{i}}. $$ \noindent We say that {\em the global sections of $E$ separate $p$ distinct points $\{x_1, \ldots, x_p \}$ of $X$ } if the above holds with all $s_{i}=0$. The case $p=1$ is equivalent to $E$ being generated by its global sections ({\em generated}, for short) at the point in question. \smallskip \noindent {\bf Rational maps to Grassmannians.} Let $V:= H^0(X,E)$ and $h^0:=$ $h^0(X,E):=$ $\dim_k H^0(X,E)$. Consider the Grassmannian $G:= G(r,h^0)$ of $r$-dimensional quotients of $V$, the universal quotient bundle $\QQ$ of $G$ and the determinant of $\QQ$, $\q$. \noindent As soon as $E$ is generated at some point of $X$, we get a rational map $\varphi:X --> G$ assigning to each point $y \in X$ where $E$ is generated the quotient $E_y \otimes k(y)$. \noindent If $E$ is generated at every point of $X$, then $f:=\varphi$ is a morphism and $E \simeq f^* \QQ$. \noindent It is clear that: - $V$ separates arbitrary pairs of points of $X$ iff $f$ is bijective birational onto its image; - If $V$ separates every pair of points of $X$ and generates jets of order $1$ at every point of $X$, then $f$ is a closed embedding (the converse maybe false if $r>1$). \medskip In the three propositions that follow we generalize to the case of higher rank results in \ci{ekl}. The analogues to these facts involving arbitrary $p$ and $\{s_1, \ldots, s_p\}$ are clear, and left to the reader. We give the reference to the analogous results for line bundles, but we prove only the first of the three propositions to illustrate the method. \noindent \begin{pr} \label{sect-s} {\rm (Cf. \ci{ekl}, 1.3 and 4.4)} Let $X$ be a nonsingular complete variety of dimension $n$. Let $E$ be a rank $r$ nef vector bundle on $X$, $L$ be a nef and big $\rat$-Cartier divisor on $X$, $\Delta'$ be a $\rat$-Cartier divisor on $X$ such that $\lfloor \Delta' \rfloor =0$ and $Supp(\Delta')$ has simple normal crossings, and $N'$ be a Cartier divisor on $X$ such that $N'\equiv L + \Delta'$. \noindent Let $s$ be a nonnegative integer and $x$ be a point of $X\setminus Supp(\Delta)$. \noindent Assume that either $\e(L,x) > n+s$, or $\e(L,x)\geq n+s$ and $L^n > \e(L,x)^n$. \noindent Then $H^0(X, K_X \otimes E \otimes \det{E} \otimes N')$ generates $s$-jets at $x$ and the rational map $\varphi$ as above is defined. Moreover, $$ h^0(X, K_X \otimes E \otimes \det{E} \otimes N') \geq r B(n+s,s). $$ In particular, if $\cal L$ is a nef and big Cartier divisor on $X$, then $$ H^0(X, K_X \otimes E \otimes \det{E} \otimes {\cal L}^{\otimes m}) \geq r B(n+s,s), \quad \forall \, m \geq n^2 +ns. $$ \end{pr} \noindent {\em Proof.} Set $Y:={\Bbb P}_X(E)$, $S:=X$, $g:=\pi$, $\Delta:=g^*\Delta'$, $M:=(r+1)\xi$, $N:=K_Y + (r+1)\xi + g^*N'$, $p=1$, $s_1=s$. Note that $M$ is nef and $g$-big and that $g_*N=K_X \otimes E \otimes \det{E} \otimes N'$. \noindent Apply Theorem \ref{effnonvan}. The only issue is whether $x\in U$; this is why the point $x$ is assumed to be outside of $Supp(\Delta)$. \noindent The lower bound on $h^0$ stems from the surjection given by Theorem \ref{effnonvan} and the fact that $$\dim_k \odix{X,x}/{\frak m}_x^{s+1}=B(n+s,n).$$ \noindent The statement about $\cal L$ is a special case after Theorem \ref{ekltm}: there exists $x\in X$ such that $\e({\cal L},x)\geq 1/n$. If $m\geq n^2 +ns$, then $\e(m{\cal L}, x) \geq n+s$ and equality holds iff $\e({\cal L},x)=1/n$ and $m=n^2+ns$; in this case the inequality ${\cal L}^n \geq 1 > \e({\cal L},x)^n$ is automatic. \blacksquare \begin{pr} \label{genbirat} {\rm (Cf. \ci{ekl}, 4.5)} Let $X$, $n$, $E$, $L$, $\Delta'$ and $N'$ be as above. Assume that either $n\geq 2$ and $\e(L,x)\geq 2n$ for every $x$ very general, or that $n=1$ and $\deg{N'}\geq 3$. \noindent Then the rational map $\varphi$ associated with $H^0(X, K_X \otimes E \otimes \det{E} \otimes N' )$ is defined and is birational onto its image. \noindent In particular, if ${\cal L}$ is a nef and big Cartier divisor on $X$, then the rational map $\varphi$ associated with $H^0(X, K_X \otimes E \otimes \det{E} \otimes {\cal L}^{\otimes m} )$ is defined and birational onto its image for every $m \geq 2n^2$. \end{pr} \begin{pr} \label{lt} {\rm (Cf. \ci{ekl}, 4.6)} Let $X$ be a complete variety of dimension $n$ with only terminal singularities and of global index $i$ such that $K_X$ is nef and big, ({\em i.e.} $X$ is normal, $\rat$-Gorenstein and a minimal variety of general type, and $i$ is the smallest positive integer such that the Weil divisor class $iK_X$ is a Cartier divisor class), and $E$ be a nef vector bundle on $X$. \noindent Then the rational map associated with $H^0(X, {\cal O}_X(miK_X) \otimes E \otimes \det{E} )$ is defined and is birational onto its image for every $m\geq 2n^2 +1$. \end{pr} \medskip The following follows from results in \ci{koshafinv}, \S8. As is already pointed out in \ci{ekl}, a generically large algebraic fundamental group on the base variety $S$ can be used to produce section by increasing the local Seshadri constants on finite \'etale covers of $S$. The reader can consult \ci{koshafinv} for the relevant definitions. \begin{pr} \label{nonvanfundgrp} {\rm (Cf. \ci{koshafinv}, 8.4)} Let $X$ be a normal and complete variety, $N'$ be an integral big $\rat$-Cartier $\rat$-divisor on $X$, and $E$ be a nef vector bundle on $X$. \noindent Assume that $X$ has generically large algebraic fundamental group. \noindent Then $h^0(X, \odixl{X}{K_X +N'} \otimes E \otimes \det{E} ) > 0$. \end{pr} \noindent {\em Sketch of proof.} By the proof of \ci{koshafinv}, Corollary 8.4 and by the first part of the proof of \ci{koshafinv}, Theorem 8.3 we are reduced to the case in which $X$ is nonsingular and $N'\equiv L + \Delta$, where $L$ and $\Delta'$ are $\rat$-Cartier divisors, $L$ is nef and big, $\lfloor \Delta' \rfloor =0$ and $Supp(\Delta')$ has simple normal crossings. \noindent Pick a point $x\in X$ such that $\e(L,x) >0$. By \ci{koshafinv}, Lemma 8.2 there is a finite \'etale map of varieties $m:X'' \to X$ and a point $x''\in X''$ such that $\e(m^*L,x'') \geq n$. \noindent Denote $\deg{m}$ by $d$, $m^*L$ by $L''$, $m^*\Delta'$ by $\Delta''$, $m^*N'$ by $N''$ and $m^*E$ by $E''$. \noindent Apply Proposition \ref{sect-s} to $X''$, $L''$, $\Delta''$, $N''$, $E''$ and $s=0$. We get $h^0(K_{X''}\otimes E'' \otimes \det{E''} \otimes N'')>0$. \noindent Kawamata-Vieheweg Vanishing Theorem applied to the nef and big $\rat$-divisor $(r+1)\xi_{E''} + {\pi''}^*L''$ gives, via Leray spectral sequence, $h^i(X'', K_{X''}\otimes E'' \otimes \det{E''} \otimes N'')=0$, for every $i>0$. The analogous statement holds on $X$. \noindent The above vanishing and the multiplicative behavior of Euler-Poincar\'e characteristics of coherent sheaves under finite \'etale maps of nonsingular proper varieties gives: $$ h^0(X, K_{X}\otimes E \otimes \det{E} \otimes N')= \chi (X,-)= \frac{1}{d}\chi (X'',-'')=\frac{1}{d} h^0(X'',K_{X''}\otimes E'' \otimes \det{E''} \otimes N'') >0. $$ \blacksquare Let us point out a consequence of $\ci{koshafinv}, 8.3$ as a corollary to the result above. Recall that the integers $I_{lm}^i: =h^i(X,S^l (\Omega_X^1) \otimes K_X^{\otimes m})$ are birational invariants of a nonsingular and complete variety $X$ for every $m,l\geq 0$ and that they are independent of the more standard invariants like the plurigenera or the cohomology groups of the sheaves $\Omega^p_X$; for some facts about these invariants and some references see \ci{rack}. The assumptions of the ``sample" corollary that follows are fulfilled, for example, by projective varieties whose universal covering space is the unit ball in $\comp^n$. \begin{cor} \label{eximple} {\rm (Cf. \ci{koshafinv}, 8.5)} Let $X$ be a nonsingular complete variety with $K_X$ nef and big, $\Omega^1_X$ nef and generically large algebraic fundamental group. \noindent Then $I_{1m}^0\geq 0$ for every $l \geq 0$ and $m\geq 3$. \end{cor} \medskip We now observe that the global generation results of Anghern-Siu, Demailly, Tsuji and Siu can be used to deduce analogous statements for vector bundles of the form $K_X^{\otimes a} \otimes E \otimes \det{E} \otimes L^{\otimes m}$, where $E$ and $L$ are a nef vector bundle and an ample line bundle on $X$, respectively. The idea is simple: once the sections of a line bundle of the form ${\cal L}:=K_X+mL$ generate the $s$ jets at every point, the local Seshadri constant is at least $s$ at every point by virtue of Lemma \ref{basic}.5. We then use Proposition \ref{sect-s}. However, this idea is applied here in a rather primitive way; we expect these results to be far from otpimal. \noindent We shall give statements concerning $p=1,2$ and low values for the jets. In the same way one can prove statements concerning more points and higher jets. We omit the details. For ease of reference we collect the line bundle results in the literature in the following result. First some additional notation. Let $n$ and $p$ be positive integers and $\{ s_1, \ldots , s_p \}$ be a $p$-tuple of nonnegative integers. Let us define the following integers: $$ m_1 (n,p) : = \frac{1}{2}(n^2 +2pn -n +2 ), $$ $$ m_2(n,p;s_1, \ldots , s_p)= 2n \sum_{i=1}^p B(3n + 2s_i -3, n) + 2pn +1. $$ \begin{tm} \label{effres} Let $X$ be a nonsingular projective variety of dimension $n$, and $L$ be an ample Cartier divisor on $X$. \medskip \noindent { \rm (\ref{effres}.1) (Cf. \ci{siu94b})} if $m\geq m_2(n,p; s_1, \ldots, s_p)$, then the global sections of $2K_X + mL$ generate simultaneous jets of order $s_1,\ldots, s_p \in {\Bbb N}$ at arbitrary $p$ distinct points of $X$; \medskip \noindent { \rm (\ref{effres}.2) (Cf. \ci{an-siu})} If $m\geq m_1(n,p)$, then the global sections of $K_X + mL$ separate arbitrary $p$ distinct points of $X$; \end{tm} \begin{tm} \label{effresvb} Let $X$, $n$ and $L$ be as above. Let $E$ be a nef vector bundle on $X$. Then the vector bundles $K_X^{\otimes a} \otimes E \otimes \det{E} \otimes L^{\otimes m}$: \noindent {\rm (\ref{effresvb}.1)} are generated by their global sections and the associated morphism to a Grassmannian $f:X \to G$ is finite, for $a=2$ and for every $m\geq (1/2)(m_2(n,1;2n)+1)$; \noindent {\rm (\ref{effresvb}.2)} have global sections which separate arbitrary pairs of points, $1$-jets at an arbitrary point, and $f$ is a closed embedding, for $a=2$ and for every $m\geq (1/2) ( m_2(n,1;4n) + 1)$; \noindent {\rm (\ref{effresvb}.3)} are generated by their global sections and $f$ is finite, for $a=n+1$ and for every $m\geq nm_1(n,1)$; \noindent {\rm (\ref{effresvb}.4)} have global sections which separate arbitrary pairs of points, $1$-jets at an arbitrary point, and $f$ is a closed embedding, for $a=2n+1$ and for every $m\geq 2nm_1(n,1)$. \end{tm} \noindent {\em Proof.} Let us observe that all the vector bundles in question are ample. One sees this easily by observing that $K_X + (n+1)L$ is always nef (Fujita) and that ``nef $\otimes$ ample $=$ ample." As soon as $f$ is defined, these bundles are pull-backs under $f$ so that they can be ample only if $f$ is finite. \smallskip \noindent Let $L':= K_X + (1/2) (m_2(n,1;2n) + 1)L$. By virtue of Theorem \ref{effres}.1, the global sections of $2L'$ generate $2n$ jets at every point $x\in X$. By virtue of Lemma \ref{basic}.5, $\e(L',x)\geq n$ for every $x\in X$. We can apply Proposition \ref{sect-s} which assumptions are readily verified. This proves (\ref{effresvb}.1). \noindent The proof of (\ref{effresvb}.2) is similar. We observe that we need $\e(L',x) \geq 2n$ to separate points and $\e(L',x)\geq n+1$ to separate $1$-jets. We then use Proposition \ref{genbirat} in the former case and Proposition \ref{sect-s} with $s=1$ in the latter. \noindent (\ref{effresvb}.3) and (\ref{effresvb}.4) are proved similarly using Theorem \ref{effres}.2 and Lemma \ref{basic}.4. \blacksquare \section{Better bounds for global generation} \label{einz} In this section we greatly improve upon Theorem \ref{effresvb}.1 and \ref{effresvb}.3. The method is similar to the one of the previous section. However, it does not use Seshadri constants. It needs a similar, local positivity result which allows one to apply the same techniques used before, and based on Theorem \ref{effnonvan}, to produce sections. Once the local positivity at one point has been established, the technique employed in \ref{effnonvan} emerges in all its simplicity. Let us recall, for the readers's convenience few basic facts about the algebraic counterparts to Nadel Ideals. The reference is \ci{ein}. \smallskip Let $X$ be a nonsingular projective variety and $D$ be an effective $\rat$-divisor. Let $f: X'\to X$ be an embedded resolution for $(X,D)$. The integral divisor $K_{X'/X} - f^* \lfloor D \rfloor$ can be written as $P-N$, where $P$ and $N$ are integral divisors without common components and $P$ is $f$-exceptional. The {\em multiplier ideal} ${\cal I} (D)$ associated with $(X,D)$ is, by definition, $$ {\cal I} (D) := f_* \odixl{X'}{P-N} = f_* \odixl{X'}{-N} \subseteq \odix{X}. $$ One checks that this ideal sheaf is independent of the resolution chosen and that $\odixl{X'}{P-N}$ has trivial higher direct images. As a consequence, we get the following vanishing result. \begin{pr} \label{evr} {\rm (Cf. \ci{ein}, 1.4)} Let $X$ be a nonsingular projective variety, $L$ be a line bundle on $X$ and $D$ be an effective $\rat$-divisor on $X$. Assume that $L-D$ is nef and big. \noindent Then $H^j(X, K_X \otimes L \otimes {\cal I} (D)) = \{0 \}$, for every $j>0$. \end{pr} The following lemma is an easy consequence of the definitions and is a functorial property of these ideals. \begin{lm} \label{funct} Let $\pi : P \to X$ be a smooth morphism of nonsingular projective varieties and $D$ be an effective $\rat$-divisor on $X$. Then $\pi^* {\cal I} (D) = {\cal I} (\pi^* D)$. \end{lm} {\em Proof.} Consider the following cartesian diagram $$ \begin{array}{ccc} P' & \stackrel{f'}\longrightarrow & P \\ \downarrow{\pi'} & {\Box} &\downarrow{\pi} \\ X'& \stackrel{f}\longrightarrow & X \end{array} $$ where $f:X' \to X$ is an embedded resolution of singularities of the log-pair $(X,D)$. Since $\pi$ is smooth, $f': P' \to P$ is a resolution of $(P, \pi^*D)$. \noindent We have ${\cal I}(\pi^* D) = f'_* (K_{P'/P} - \lfloor {f'}^* (\pi^* D)\rfloor )=$ $ {f'}_* ({\pi '}^* K_{X'/X} - \lfloor {\pi '}^* f^*D\rfloor )=$ ${f'}_* ( {\pi '}^* K_{X'/X} - {\pi '}^* \lfloor f^*D\rfloor )=$ ${f'}_* ( {\pi '}^* ( K_{X'/X} - \lfloor f^*D\rfloor )) = $ $\pi^* ( f_* (K_{X'/X} - \lfloor f^*D\rfloor )) = $ $\pi^* {\cal I} (D)$, where: the second equality holds because the formation of the sheaf of relative differentials $\Omega^1_{*/*}$ commutes with base change and the relative cononical sheaf is, in the simple case under scrutiny, the determinant of $\Omega^1_{*/*}$; the third equality holds because $\pi '$ is smooth; the fifth stems from the fact that cohomology commutes with the flat base extension $\pi$. \blacksquare The following result is a $\rat$-divisors reformulation of the result of Anghern-Siu. The result is due to Koll\'ar \ci{koslp}. The formulation given below in terms of algebraic multiplier ideals is due to Ein \ci{ein}. \begin{tm} \label{ein} Let $X$ be a nonsingular projective variety of dimension $n$ and $L$ be an ample line bundle on $X$ such that $$ L^d \cdot Z > B(n+1,2)^d $$ for every $d$-dimensional integral cycle $Z$ on $X$. Then, for every point $x\in X$ there exists an effective $\rat$-divisor $D$ such that $D\equiv \lambda L$ for some positive rational number $0< \lambda <1$ and $x$ is in the support of an isolated component of $V( {\cal I} (D) )$. \end{tm} \begin{rmk} \label{sevpts} {\rm A similar statement holds if we consider several distinct points. } \end{rmk} \begin{tm} \label{asvb} Let $\pi : P \to X$ be a smooth morphism with connected fibers of nonsingular projective varieties, $n$ be the dimension of $X$, $M$ be a nef and $\pi$-big line bundle on $P$, $\cal L$ be an ample line bundle on $X$ such that $$ {\cal L}^d \cdot Z > B(n+1,2)^d $$ for every $d$-dimensional integral cycle $Z$ on $X$. \noindent Then, the vector bundle $\pi_* (K_P + M)\otimes {\cal L}$ is generated by its global sections. \noindent In particular, if $L$ is any ample line bundle on $X$, then we can choose, ${\cal L}:= B(n+1,2) \, L$. \end{tm} {\em Proof.} Let $x \in X$ be an arbitrary point. Let $D$ be a $\rat$-divisor as in Theorem \ref{ein}. Since ${\cal L} - D$ is ample and $M$ is nef and $\pi$-big, the $\rat$-divisor $M+ \pi^* ({\cal L} -D)$ is nef and big on $P$. The smoothness of $\pi$ implies, by virtue of Lemma \ref{funct}, that $\pi^* {\cal I} (D) = {\cal I} (\pi^* D)$. It follows that $H^1(P, (K_P + M + \pi^* {\cal L}) \otimes \pi^* {\cal I} (D))=$ $ H^1(P, (K_P + M + \pi^*{\cal L}) \otimes {\cal I} (\pi^* D) ) =\{0\}$, the second equality stemming from Ein's version of Nadel Vanishing Theorem Proposition \ref{evr}. Since $V({\cal I} (D))$ has isolated support at $x$, we conclude that, if we denote by $F_x$ the fiber of $\pi$ over $x$: $$ H^0(P, K_P + M + \pi^* {\cal L} ) \surj H^0 (F_x, (K_P + M + \pi^* {\cal L})\otimes \odix{F_x} ). $$ The result follows by the natural identification between the map given above and the map $$ H^0 (X, \pi_*(K_P + M) \otimes {\cal L}) \to \pi_* (K_P+M) \otimes {\cal L} \otimes \odix{X}/{\frak m}_x, $$ which holds because $R^1\pi_* (K_P+M)=0$ is the zero sheaf by relative vanishing. \blacksquare \begin{cor} \label{good} Let $X$ be a nonsingular projective variety of dimension $n$, $E$ be a nef vector bundle on $X$, and $\cal L$ be an ample line bundle on $X$. Assume that $$ {\cal L}^d \cdot Z > B(n+1,2)^d $$ for every $d$-dimensional integral cycle $Z$ on $X$. \noindent Then $K_X \otimes E \otimes \det{E} \otimes {\cal L}$ is generated by its global sections at every point of $X$. \noindent In particular, if $L$ is any ample line bundle on $X$, then we can choose ${\cal L} = B(n+1, 2) \, L$. \end{cor} {\em Proof.} Set $P:= {\Bbb P }(E)$, $\pi:=$ the natural projection onto $X$, $M: = (r+1)\xi_E$, where $r$ is the rank of $E$, and apply Theorem \ref{asvb}. \blacksquare \begin{rmk} {\rm A similar statement hold for the simultaneous generation at several points; see Remark \ref{sevpts}. The same is true for Theorem \ref{asvb}. } \end{rmk} \begin{rmk} {\rm The paper \ci{deshm} contains similar results without the factor $\det{E}$. However, the assumption $E$ nef is there replaced by a stronger curvature condition on $E$, and the methods are purely analytic } \end{rmk} \begin{rmk} {\rm We do not know if similar statements hold without the factor $\det {E}$. } \end{rmk}

9708

alg-geom/9708025

en

https://arxiv.org/abs/alg-geom/9708025

alg-geom/9708025

Georg Hein

Georg Hein

Duality Construction of Moduli Spaces

12 pages LaTeX using pb-diagram.sty

We show for the moduli space of rank-2 coherent sheaves on an algebraic surface that there exists a 'dual' moduli space. This dual space allows a construction of the first one without using the GIT construction. Furthermore, we obtain a Barth-morphism, generalizing the concept of jumping lines. This morphism is by construction a finite morphism.

alg-geom

\section*{Introduction} In \S 1 of Faltings' article \cite{Fal} a ``GIT-free'' construction is given for the moduli spaces of vector bundles on curves using generalized theta functions. Incidentally, this construction is implicitly described in Le Potier's article \cite{LP2}. The aim of this paper is to generalize the {\em duality construction} to projective surfaces. For a rank two vector bundle $E$ on the projective plane $I \!\! P^2$, the divisor $D_E$ of its jumping lines is a certain generalization of the Chow divisor of a projective scheme. We give a generalization of this divisor for coherent sheaves on surfaces. Using this duality we construct the moduli space of coherent sheaves on a surface that does not use Mumford's geometric invariant theory. Furthermore, we obtain a finite morphism from this moduli space to a linear system, which generalizes the divisors of jumping lines. Applying this construction to curves, we get exactly Faltings' construction. The moduli space we construct here can also be obtained by using GIT. This construction is carried out in \S8.2 of the book \cite{HL} of Huybrechts and Lehn. Le Potier obtained this moduli space in \cite{LP1} for surfaces with ``many lines'' (see \S \ref{SBAR} for an exact definition). However, it is the modest hope of the author that the construction presented here provides new insight into the geometry of moduli spaces. First we outline this concept, which generalizes the famous {\em strange duality} to moduli of coherent sheaves on surfaces. To do so we define duality between schemes in part \ref{SDUAL}, giving three examples of ``natural dualities''. In section \ref{SDCON} the duality construction is given. In order to avoid a too technic presentation of the construction itself, we defer the proofs to the following section. The last section is dedicated to the Barth morphism. In order to simplify the discussion we restrict ourselves to moduli spaces of sheaves of rank two with trivial determinant. The interested reader will be able to extend this to arbitrary rank and determinant. The author is thankful to his thesis advisor, H.~Kurke, for many fruitful discussions. \section{Duality of schemes}\label{SDUAL} \subsection{Definitions} Let $(X,{\cal O}_X(D_X))$ and $(Y, {\cal O}_Y(D_Y))$ be two schemes with line bundles. A duality between these two pairs is given by a nontrivial section $s \in H^0({\cal O}_X(D_X)) \otimes H^0({\cal O}_Y(D_Y))$. We will identify $s$ with its vanishing divisor $D=V(s) \subset X \times Y$. \vspace{1em} \hspace{12em} \setlength{\unitlength}{0.0012cm} \begin{picture}(4512,4548)(2389,-6097) \thicklines \put(4801,-1561){\line( 0,-1){2400}} \put(4801,-3961){\line(-1, 0){2400}} \put(2401,-3961){\line( 0, 1){2400}} \put(2401,-1561){\line( 1, 0){2400}} \put(4801,-1561){\line(-1, 0){2400}} \put(2401,-3961){\line( 1, 1){2400}} \put(6601,-1561){\line( 0,-1){2400}} \put(2401,-5761){\line( 1, 0){2400}} \put(5251,-2761){\vector( 1, 0){900}} \put(3526,-4336){\vector( 0,-1){1125}} \put(3100,-2986){\makebox(0,0)[cc]{$D$}} \put(5476,-2611){\makebox(0,0)[cc]{$q$}} \put(6999,-2911){\makebox(0,0)[cc]{$Y$}} \put(3976,-6061){\makebox(0,0)[cc]{$X$}} \put(3888,-4936){\makebox(0,0)[cc]{$p$}} \put(3851,-3686){\makebox(0,0)[cc]{$X \times Y$}} \end{picture}\\ We obtain a rational morphism $$\begin{array}{cccc} s_X: & X & - - - \rightarrow & |D_Y| \\ & x & \mapsto & q(D \cap p^{-1}(x)) \, . \\ \end{array}$$ The base locus of this morphism $s_X$ consists of all points $x$ of $X$ such that the vertical component $x \times Y$ is contained in $D$. This motivates the following {\vspace{0.5em} } {\bf Definition: } The duality $D$ between $(X,D_X)$ and $(Y,D_Y)$ is called\\ {\em generated,} if $s_X$ is a morphism;\\ {\em generated ample,} if $s_X$ is a finite morphism;\\ {\em very ample,} if $s_X$ is an embedding.\\ \subsection{Examples} The first example demonstrates that the above definitions are something with which we are familiar. {\vspace{0.5em} } {\bf The duality of a linear system} Let $X$ be a given scheme with an effective divisor $D_X$, and let $Y \subset |D_X|$ be a linear system. We take $D$ to be the incident divisor, i.e. $D= \{ (x, H) \, | x \in H \}$. Then the notions for $D$ given in the above definition correspond to those for the linear system. {\vspace{0.5em} } {\bf Strange duality} Let $C$ be a smooth projective curve of genus $g$ over the complex numbers. We fix two positive integers $m$ and $n$ and a theta characteristic $A$, i.e. $A \in {\rm Pic}^{g-1}(C)$ and $A^{\otimes 2} \cong \omega_C$. We consider the following moduli schemes. $$\begin{array}{ccl} X & = & {\rm U}_C(n,n(g-1)) \\ & = & \left\{ E \, \left| \, \begin{array}{l} E \mbox{ semistable } C \mbox{-vector bundles with } \\ {\rm rk}(E)=n \mbox{ and } \deg(E)=n(g-1) \\ \end{array} \right. \right\} \\ D_X & = & \{ E \in X \, | h^0(E)=h^1(E)>0 \} \\ Y & = & {\rm SU}_C(m) \\ & = & \left\{ F \, \left| \, \begin{array}{l} F \mbox{ semistable } C \mbox{-bundles with } \\ {\rm rk}(E)=m \mbox{ and } \det(E) \cong {\cal O}_C \\ \end{array} \right. \right\} \\ D_Y & = & \{ F \in Y \, | h^0(F \otimes A)=h^1(F \otimes A)>0 \} \\ D & = & \{ (E,F) \, | \, h^0(E\otimes F) = h^1(E \otimes F) >0 \} \\ \end{array}$$ The line bundle ${\cal O}_{X \times Y}(D)$ is isomorphic to $p^*{\cal O}_X(D_X)^{\otimes m} \otimes q^*{\cal O}_Y(D_Y)^{\otimes n}$. By the duality we obtain a linear map from $H^0(X,{\cal O}_X(D_X)^{\otimes m})^\lor \rarpa{s} H^0(Y,q^*{\cal O}_Y(D_Y)^{\otimes n})$. According to the Verlinde formula, both spaces have the same dimension. The natural conjecture that $s$ is an isomorphism is called the {\em Strange Duality Conjecture}. For more details on this topic, see Beauville's survey article \cite{Bea}. {\vspace{0.5em} } {\bf Duality between moduli spaces on polarized surfaces} The next example is our main example, the notions of which will be used for the remainder of the article. We describe the moduli spaces here only set theoretically. Their construction uses the duality, thus giving a rough idea of the construction which is the object of the following section. The concept of semistability used here is the Mumford (slope) semistability. Let $(S,{\cal O}_S(1) = {\cal O}_S(H))$ be a projective polarized surface. Fix a class $c_2 \in H^4(S,Z \!\!\! Z)$. We will consider a duality between the following two coarse moduli spaces: $$\begin{array}{rcl} X & = & M_S(2,0,c_2) \\ & = & \left\{ E \left| \, \begin{array}{l} E \mbox{ semistable torsion free sheaf on } S, \mbox{ with} \\ {\rm rk}(E)=2 \quad \det(E) \cong {\cal O}_S \quad c_2(E)=c_2 \\ \end{array} \right. \right\} \\ \\ Y & = & M_{|H|}(2,\omega_{|H|}) \\ & = & \left\{ F \left| \, \begin{array}{l} F \mbox{ semistable torsion sheaf on } S, \mbox{ with} \\ Z={\rm supp}(F) \in |H| \quad {\rm rk}_Z(F)=2 \quad \det_Z(F) \cong \omega_Z \\ \end{array} \right. \right\} \\ \end{array}$$ The duality will be given by the $X \times Y$ divisor $$D= \{ (E,F) | H^*(E \otimes F) \not= 0 \}\, .$$ \section{The duality construction}\label{SDCON} Using the notation introduced in the last example, we give a construction of the coarse moduli scheme $X$ using the duality morphism $s$. $X$ will be obtained together with a polarization and the {\em Barth morphism}, which will be finite by construction. The steps for this construction are listed below, and proofs for all pertinent theorems are provided in the next section. \begin{description} \item [Boundedness of $X$] There exists a projective scheme $Q$ and a torsion free sheaf ${\cal E}$ on $Q \times S$ flat over $Q$ which (over)parameterizes the moduli problem. More precisely denote by $p$, $q$ the two projections $$Q \larpa{p} Q \times S \rarpa{q} S \, .$$ For any sheaf $E$ of $X$ let $Q_E$ be the subscheme $$Q_E = \{ q\in Q \, | \, {\cal E}_q \cong E \}$$ of $Q$. All $Q_E$ are required to be connected and nonempty. Since this is the same starting point like in the GIT construction, it is obvious that $Q$ can be taken to be a suitable Quot scheme (see \cite{Gro}). \item [Elements of $Y$ give sections in a $Q$-line bundle ${\cal L}$] (see \S \ref{P1}) We will show that there exists a $Q$-line bundle ${\cal L}$ and a global section $s_F \in H^0(Q, {\cal L})$, for any $F \in Y$. The vanishing locus of $s_F$ is given by $$V(s_F) = \{ q \in Q \, | \, H^*(S, {\cal E}_q \otimes F) \not= 0 \} \, .$$ \item [Base points correspond to unstable objects] (see \S \ref{P2}) The base locus $B({\cal L})$ with respect to the sections given by $Y$ is the scheme theoretic intersection of all $V(s_F)$ for all $F \in Y$. Since $Q$ is noetherian we can write $$B({\cal L}) = \bigcap_{i=0}^N V(s_{F_i}) \, .$$ It will be shown that $B({\cal L})$ consists exactly of those points $q \in Q$ for which the sheaf ${\cal E}_q$ is not semistable. \item [Properness of $X$] (see \S \ref{P3}) We have to show that semistable limits of semistable families exist. \item [The line bundle ${\cal L}$ is $X$-positive] (see \S \ref{P4}) It will be shown that the degree of ${\cal L}$ on a curve $C$ parameterizing semistable objects is zero only if the curve parameterizes Jordan-H\"older equivalent sheaves. \item [The duality construction] The rational morphism $s=(s_{F_0}: \ldots:s_{F_N})$ from $Q$ to $I \!\! P^N$ leads to a morphism $\bar Q \rarpa{\varphi} I \!\! P^N$ after a blow up of $Q$. We consider the following diagram $$\begin{diagram} \node{\bar Q} \arrow{s,r}{\pi} \arrow{se,t}{\varphi} \arrow{e,t}{\varphi_0} \node{X} \arrow{s,r}{\varphi_1} \\ \node{Q} \arrow{e,t,..}{s} \node{I \!\! P^N} \end{diagram} \quad.$$ Here $\varphi = \varphi_1 \circ \varphi_0$ is the Stein factorization of $\varphi$. Hence the Barth morphism $\varphi_1$ is finite. By the above, any point of $X$ corresponds to exactly one Jordan-H\"older equivalence class of semistable bundles. \end{description} \section{Details and proofs} \subsection{The $Q$-line bundle ${\cal L}$ and its invariant sections}\label{P1} ${\cal L}$ is defined to be the determinant bundle ${\cal L} = det(p_!({\cal E} \otimes q^*F)^{-1}$. The definition does not depend on the choice of $F \in Y$, because these elements coincide in the Grothendieck group $K_0(S)$. {\vspace{0.5em} } {\bf Framed elements of $Y$ give sections in $\Gamma({\cal L})$}\\ Let $F$ be a semistable element in $Y$ with support $Z \in |H|$. Then there exists a short exact sequence $$0 \rightarrow F \rightarrow {\cal O}_Z(M)^{\oplus 3} \rarpa{\alpha} \omega_Z^\lor(3M) \rightarrow 0 \, .$$ We remark that $M>>0$ can be chosen for all $F \in Y$. Define $s_F$ to be the section $\det(Rp_*({\cal E} \otimes q^* \alpha))$. It is clear from the construction that the vanishing divisor $V(s_F)$ is supported on those $q\in Q$, for which $H^*({\cal E}_q \otimes q^*F)$ is not zero. {\bf Remark: } By abuse of notation we simply write $s_F$ and do not explicitly refer to the framing. {\vspace{0.5em} } {\bf Global sections in ${\cal L}^{\otimes k}$} Let $\tilde F$ be a rank two vector bundle on a curve $\tilde Z$ from the linear system $|kH|$. We require the determinant of $\tilde G$ to be isomorphic to $\omega_{\tilde Z}$. Using adjunction to express $\omega_{\tilde F}$ and $\omega_F$, the following computation in the Grothendieck group $K_0(S)$ shows that $[\tilde F] =k[F]$ for any $F \in Y$: $$\begin{array}{ccl} [\tilde F] & = & ([{\cal O}_S]-[{\cal O}_S(-kH)])([{\cal O}_S] + [K_S(kH)]) \\ \\ & = & ([{\cal O}_S] -[{\cal O}_S(-H)]) \left(\sum\limits_{i=0}^{k-1}[{\cal O}_S(-iH)]) ([{\cal O}_S] + [K_S(kH)] \right) \\ \\ & = & ([{\cal O}_S] -[{\cal O}_S(-H)]) k([{\cal O}_S] + [K_S(H)]) \, = \, k[F] \, \, .\\ \end{array}$$ Hence $\tilde F$ (together with a framing) defines a global section in ${\cal L}^{\otimes k}$. \subsection{Semistability}\label{P2} Assume that $H$ is big enough, which means that ${\cal O}_S(H)$ is globally generated, and the following two conditions hold: (i) $H^2>4c_2$, and (ii) The positive generator $a$ of the $Z \!\!\! Z$ ideal $\{ D.H \, | \, D \in {\rm NS}(S) \}$ satisfies $a>c_2$. Under these assumptions we have the following \begin{satzdef}\label{SEMISTABLE} For a torsion free rank two $S$-sheaf $E$ with $\det(E) \cong {\cal O}_S$ and $c_2(E)=c_2$ the following four conditions are equivalent: \begin{enumerate} \item There exists an $F \in Y$ such that $H^*(S,E\otimes F)=0$; \item For all rank one subsheaves $M \subset E$, the inequality $c_1(M).H \leq 0$ holds; \item The restriction of $E$ to a general divisor $Z \in |H|$ is semistable, i.e. all $Z$-line bundles contained in $E_Z$ have nonpositive degree; \item For $Z \in |H|$ general, there exists a $Z$-line bundle $A$ such that $H^*(Z, L \otimes E|_Z)=0$. \end{enumerate} If one of these conditions is satisfied we call $E$ a semistable $S$-sheaf. \end{satzdef} {\bf Proof: } {\bf (1) $\Rightarrow$ (2) } Suppose there exists an $M \subset E$ such that $c_1(M).H>0$. Then $M$ restricted to $Z = {\rm supp}(F)$ is of positive degree. Therefore the Euler characteristic of $M \otimes F$ is positive. Since the sheaf is one-dimensional there are global sections. Hence there are global sections in $H^0(E \otimes F)$, which contradicts the assumption of (1). {\bf (2) $\Rightarrow$ (3) } This is a restriction theorem that follows from Bogomolov's inequality (see \cite{Bog}). For a complete proof of this implication see \cite{HL} theorem 7.3.5. {\bf (3) $\Rightarrow$ (4)} This result goes back to Raynaud (\cite{Ray}). For a shorter proof see \cite{He1}. {\bf (4) $\Rightarrow$ (1) } Denote the genus of $Z$ by $g$. It follows by Riemann-Roch that $A \in {\rm Pic}^{g-1}(Z)$. The condition $H^*(E|_Z \otimes A) =0$ is satisfied on a nonempty open subset of the Jacobian ${\rm Pic}^{g-1}(Z)$. Hence the condition $H^*(E|_Z \otimes \omega_Z \otimes A^{-1})=0$ is again open and not empty. Consequently we can choose $F$ to be a direct sum $A \oplus (\omega_Z\otimes A^{-1})$. { \hfill $\Box$} \subsection{Properness}\label{P3} Although the properness of the moduli functor $X$ is well known, a new proof is given here which is shorter than Langton's original proof in \cite{Lan}. The main idea to get ``more and more'' semistable extensions of a generic semistable family by elementary transformations comes from Langton's proof. However, using the invariant functions, we can control the maximal number of elementary transformations required. Therefore the proof fits into the concept of the duality construction. \begin{theorem} (\cite{Lan}) Let $R$ be a discrete valuation ring with ${\rm Spec}(R)=\{ 0, \eta \}$. Let ${\cal E}_\eta$ be a semistable torsion free sheaf on $\eta \times S$. Then there exists an extension ${\cal E}_R$ of ${\cal E}_\eta$ which is semistable in the special fiber as well. \end{theorem} {\bf Proof: } We consider the following morphisms: $$ {\rm Spec}(R) \stackrel{p}{\leftarrow} {\rm Spec}(R) \times S \stackrel{q}{\rightarrow} S \quad .$$ Since the Quot scheme is projective, there exist torsion free extensions of ${\cal E}_\eta$. For an extension ${\cal E}$, we define its badness\footnote{We use the word badness because $b$ measures how far ${\cal E}$ is from being a semistable extension. So badness zero implies semistability.} $b({\cal E},F)$ with respect to an $F \in Y$ as $$b({\cal E},F)= \left\{ \begin{array}{ll} \infty & \mbox{ if } {\rm supp} R^1p_*({\cal E} \otimes q^*F)={\rm Spec}(R) \, , \\ {\rm length}(R^1p_*({\cal E} \otimes q^*F)) & \mbox{ otherwise.} \\ \end{array} \right.$$ The absolute badness $b({\cal E})$ of ${\cal E}$ is defined to be the minimum of all these numbers: $$ b({\cal E}) = \min_{F \in Y} \{ b({\cal E},F) \} \, .$$ Since ${\cal E}_\eta$ is semistable the badness $b({\cal E})$ has to be finite. We suppose that ${\cal E}$ is an extension with minimal possible badness. If the badness is zero, the special fiber ${\cal E}_0$ is semistable by \ref{SEMISTABLE}. Hence we may assume that $b({\cal E})>0$. Since ${\cal E}_0$ is not semistable, there is a surjection ${\cal E}_0 \rightarrow L \otimes {\cal J}_Z$ with $L.H< 0$, ${\cal J}_Z$ being the ideal sheaf of a codimension two subscheme of $S$. We choose an element $F \in Y$ subject to the following three open conditions: {\vspace{0.5em} } (i) $H^0(L \otimes F) =0$, (ii) $b({\cal E},F) = b({\cal E})$, (iii) ${\rm supp}(F) \cap Z = \emptyset $. {\vspace{0.5em} } Define the elementary transformation ${\cal E}'$ of ${\cal E}$ by the exact sequence $$0 \rightarrow {\cal E}' \rightarrow {\cal E} \rightarrow L \otimes {\cal J}_Z \rightarrow 0 \, .$$ Applying the functor $p_*(- \otimes q^*F)$ to that sequence, we obtain $$\begin{array}{cccc} $$p_*(L \otimes q^*F) & \rightarrow R^1p_*({\cal E}' \otimes q^*F) \rightarrow R^1p_*({\cal E} \otimes q^*F) \rightarrow & R^1p_*(L \otimes q^*F) & \rightarrow 0\, .\\ || && \parallel \!\!\!\!\!\! - \\ 0 && 0 \\ \end{array}$$ This contradicts the minimality assumption on the badness of ${\cal E}$ by the very definition of this number. { \hfill $\Box$} \subsection{$X$-positivity of the line bundle ${\cal L}$}\label{P4} We have to consider the equivalence classes of semistable sheaves parameterized by our moduli space $X$ using the following equivalence relation. {\vspace{0.5em} } {\bf Definition (trivially connected equivalence)} Two semistable $X$-sheaves $E$ and $E'$ on $S$ are called trivially connected if there exists a connected projective curve $B$ and a family ${\cal E}$ on $B \times S$ such that\\ - the determinant line bundle ${\cal L}_B$ on $B$ is trivial and\\ - there are points $b$ and $b'$ in $B$ with $E \cong {\cal E}_b$ and $E' \cong {\cal E}_{b'}$. {\vspace{0.5em} } There is a second equivalence relation that reflects the geometry of the sheaves. We start with some preparations. If $\tau$ is a coherent sheaf of dimension zero, then we define its trivialisation ${\rm triv}(\tau)$ by $$ {\rm triv}(\tau) := \bigoplus_{P \in X} k(P)^{\oplus {\rm length}_P(\tau)} \, .$$ For a torsion free sheaf $G$, let $G^{\lor \lor}$ be its double dual and $\tau(G)$ be the cokernel of the injection $G \hookrightarrow G^{\lor \lor}$. Define by $${\rm triv}(G) = G^{\lor \lor} \oplus {\rm triv}(\tau(G))$$ the trivialisation\footnote{This definition is good enough for our purposes. However it should be replaced by ${\rm triv}(G) = G^{\lor \lor} \ominus {\rm triv}(\tau(G))$.} of $G$. {\vspace{0.5em} } Define the graded object of a stable sheaf $E$ to be $E$ itself: ${\rm gr}_H(E) =E$. If $E$ is a semistable but not stable sheaf, then there exists a short exact sequence $0 \rightarrow A' \rightarrow E \rightarrow A'' \rightarrow 0$ with $A'$ a saturated subsheaf of $E$ and $c_1(A').H=0$. In this case we define the graduated object ${\rm gr}_H(E)$ of $E$ to be the direct sum $A' \oplus A''$. {\vspace{0.5em} } {\bf Definition (Jordan-H\"older equivalence)} Two $X$-sheaves $E$ and $E'$ on $S$ are called Jordan-H\"older equivalent if and only if ${\rm triv}({\rm gr}_H(E)) \cong {\rm triv}({\rm gr}_H(E'))$. {\vspace{0.5em} } This definition implies, in particular, that the equivalence class of a stable vector bundle consists of one element up to isomorphism. In the course of the next result, it will be shown that Jordan-H\"older equivalence coincides with trivially connected equivalence. \begin{theorem}\label{POSITIVE} Let $B$ be a smooth projective connected curve and ${\cal E}_B$ be a family of sheaves on $B \times S$. Assume that the sheaf parameterized by the generic point of $B$ is semistable. Then the $B$-line bundle ${\cal L}_B$ is trivial or ample. If ${\cal L}_B$ is trivial, then all $S$-sheaves parameterized by $B$ are Jordan-H\"older equivalent. \end{theorem} First we note that there are nontrivial sections in ${\cal L}_B$, because there are semistable objects parameterized by points of $B$. Therefore this line bundle is either trivial or ample. Consequently the proof reduces to showing that ${\cal O}_B \cong {\cal L}_B$ (by definition, the trivially connected equivalence) induces the Jordan-H\"older equivalence. To prove the above theorem some preparations will be needed. We retain the above notations. \begin{theorem}\label{POSCURVE} (\cite{Fal} theorem I.4) Let $C$ be smooth projective curve and ${\cal E}$ a vector bundle on $B \times C$ with $deg_C({\cal E}_b) = 0$ for all $b\in B$. Denote the projections of $B \times C$ to the components by $p$ and $q$. Suppose there exists a $C$-line bundle $M$ such that $R^*p_*({\cal E} \otimes q^*M) =0$ holds. Then the $C$-objects parameterized by $B$ are all $S$-equivalent. \end{theorem} {\bf Proof: } We proceed in steps. \step{1} There exists a $B$-bundle $G$ such that ${\cal E}_P \cong G$ for all points $P \in C$. Let $P$ and $Q$ be two arbitrary points of $C$. The set of all line bundles $M$ in ${\rm Pic}^{g_C-1}(C)$ with $R^*p_*({\cal E} \otimes q^*M) =0$ is open. Hence there is a line bundle $\tilde M \in {\rm Pic}^g(C)$ such that $\tilde M(-P)$ and $\tilde M(-Q)$ are in this open set. From the exact sequence $0 \rightarrow \tilde M(-P) \rightarrow \tilde M \rightarrow k(P) \rightarrow 0$ we obtain $$p_*({\cal E} \otimes q^*M) \cong p_*({\cal E} \otimes q^*k(P)) \cong p_*({\cal E}|_{ B \times \{ P \} }) \cong {\cal E}|_{ B\times \{ P \} }\, .$$ Analogously, $p_*({\cal E} \otimes q^*M) \cong {\cal E}|_{ B\times \{ Q \} }$ which proves the assertion of the first step. \step{2} Set ${\cal G} = p^*G$. There are three distinct cases to be considered. \case{1} $G$ is stable Since $G$ is simple it follows that $N = q_*({\cal G}^\lor \otimes {\cal E})$ is a $C$-line bundle. But ${\cal G} \otimes q^*N$ is isomorphic to ${\cal E}$, therefore all objects parameterized by $B$ are isomorphic to $N \oplus N$. \case{2} $G$ is semistable but not stable After a twist with a line bundle we may assume $G$ to be of degree zero. By theorem \ref{SEMISTABLE} there exists a $B$-line bundle $A$ such that $G \otimes B$ has no cohomology. This implies $R^*q_*({\cal E} \otimes p^*A) =0$, and, as in the first step, all $C$-objects parameterized by $B$ are isomorphic. \case{3} $G$ is not semistable Let $0 \subset A \subset G$ be the Harder-Narasimhan filtration of $G$, i.e. $A$ is the subline bundle of $G$ of maximal degree. We denote the quotient $G/A$ by $A'$. By the uniqueness of $A$, $N = q_*(p^*A^\lor \otimes {\cal E})$ is a line bundle on $C$. We find that $p^*A \otimes q^*N$ is a subbundle of ${\cal E}$ with cokernel isomorphic to $p^*A' \otimes q^*N'$ for a $C$-line bundle $N'$. Using the short exact sequence $$ 0 \rightarrow p^*A \otimes q^*(N \otimes M) \rightarrow {\cal E} \otimes q^*M \rightarrow p^*A' \otimes q^*(N' \otimes M) \rightarrow 0$$ to compute the degree of $R^*p_*({\cal E} \otimes q^*M)$, we have $$0 = \deg(A)\deg(N)+\deg(A')\deg(N') \, .$$ Since ${\cal E}$ is a family of degree zero sheaves on $C$, it follows that $\deg(N') = - \deg(N)$. Hence we obtain the equality $$0 = (\deg(A) - \deg(A'))\deg(N) \, .$$ By assumption the first factor is strictly positive, thus $\deg(N) =\deg(N')=0$. But this means that all objects parameterized by $B$ are extensions of two line bundles of the same degree. { \hfill $\Box$} \begin{lemma}\label{BOUNDLINE} The set of all line bundles $L$ on $S$ such that $L.H=0$ and for which there exists a nontrivial homomorphism in ${\rm Hom}(E,L)$ for some $E \in X$ is bounded. Subsequently, these line bundles can be parameterized by an noetherian scheme. \end{lemma} {\bf Proof: } It is enough to show that the set of Hilbert polynomials of these line bundles $L$ is finite. For any such line bundle $L$, there is an exact sequence $$0 \rightarrow L^{-1} \otimes {\cal J}_{Z_1} \rightarrow E \rightarrow L \otimes {\cal J}_{Z_2} \rightarrow 0 \, .$$ Since $E$ is semistable, ${\cal J}_{Z_2}$ is the ideal sheaf of some zero dimensional scheme. Using the above sequence, the second Chern class can be computed, and indeed, $c_2=-L^2+{\rm length}(Z_1) +{\rm length}(Z_2)$. Hence we conclude by the Hodge index theorem that $L^2$ is in the interval $[-c_2,0]$ and that $(H.(K_S \pm L))^2 \leq H^2 (K_S \pm L)^2$, which gives lower and upper bounds for $K_S.L$. The Hilbert polynomial of $L$ with respect to $H$ is determined completely by the numbers $L^2$, $L.K_S$ and $L.H$. { \hfill $\Box$} \begin{lemma}\label{BIGK} There exists a positive number $k$ such that, for all $X$ sheaves $E$ and $E'$ on $S$ and all line bundles on $S$ with $H.L=0$ and ${\rm Hom}(E,L) \not=0$, the groups ${\rm Ext}^1(E, L(-kH))$ and ${\rm Ext}^1(E,E'(-kH)^{\lor \lor})$ vanish. \end{lemma} {\bf Proof: } By Serre duality ${\rm Ext}^1(E,L(-kH)) \cong H^1(E(K_S+kH) \otimes L^{-1})^\lor$. For any pair $(E,L)$ there exists a number $k$ such that the cohomology group vanishes. By lemma \ref{BOUNDLINE} the set of all these pairs is bounded. Hence there exists a global $k$. The same argument shows the vanishing of ${\rm Ext}^1(E,E'(-kH)^{\lor \lor})$ for a given $k$. { \hfill $\Box$} \begin{lemma}\label{BIGGERK} There exists an integer $k$ such that for all semistable $X$ sheaves $E$ on $S$ the following holds: Let $Z$ be a smooth curve in the linear system $|kH|$ such that $E|_Z$ is a vector bundle on $Z$. If $E|_Z \rarpa{\bar \alpha} \bar M$ is a surjection onto a $Z$-line bundle $\bar M$ of degree zero, then $\bar \alpha$ is the restriction of a morphism $E \rarpa{\alpha} M$ to $Z$, where $M$ is a $S$-line bundle with $M.H=0$. \end{lemma} {\bf Proof: } This lemma follows from Bogomolov's inequality, as in the proof of (2) $\Rightarrow$ (3) of \ref{SEMISTABLE}. For details see \cite{Bog2} theorem 2.3 or \cite{HL} theorem 7.3.5. {\vspace{0.5em} } {\bf Proof of the positivity theorem \ref{POSITIVE}}\\ Since ${\cal L}_B$ is assumed to be the trivial line bundle, we may pass to a power ${\cal L}_B^{\otimes k}$. Choose $k$ such that lemmas \ref{BIGK} and \ref{BIGGERK} apply. Now let $F$ be a torsion sheaf supported on a smooth divisor $Z \in |kH|$ such that the global section of ${\cal L}_B^{\otimes k}$ defined by $F$ is nontrivial. By construction it is clear that ${\cal E}$ restricted to $Z$ is a vector bundle. Now by theorem \ref{POSCURVE} this yields the $S$-equivalence of the restriction ${\cal E}|_Z$, and from the proof given here, it follows that we are in one of the following two cases. \case{1} All $Z$-vector bundles parameterized by $B$ are stable and isomorphic. Let $P$ and $Q$ be two geometric points of the curve $B$. We consider the following long exact sequence $$0 \rightarrow {\rm Hom}({\cal E}_P, {\cal E}_Q(-kH)^{\lor \lor}) \rightarrow {\rm Hom}({\cal E}_P, {\cal E}_Q^{\lor \lor}) \rarpa{\alpha} $$ $$\rarpa{\alpha} {\rm Hom}({\cal E}_P, {\cal E}_Q^{\lor \lor}|_Z) \rightarrow {\rm Ext}^1({\cal E}_P, {\cal E}_Q(-kH)^{\lor \lor}) \rightarrow \, .$$ Since $E_P$ and $E_Q$ are semistable the group ${\rm Hom}({\cal E}_P, {\cal E}_Q(-kH)^{\lor \lor})$ vanishes. Hence by lemma \ref{BIGK} the morphism $\alpha$ is an isomorphism. The support of the cokernels of the nontrivial morphisms ${\cal E}_P$ to ${\cal E}_Q^{\lor \lor}$ can not change because they never meet the ample divisor $Z$. This shows the Jordan-H\"older equivalence of ${\cal E}_Q$ and ${\cal E}_P$. \case{2} All $Z$ vector bundles parameterized by $B$ have a surjection to a $Z$-line bundle $\bar M$ of degree zero. First we remark that $\bar M$ is the restriction of an $S$-line bundle $M$ to $Z$ by lemma \ref{BIGGERK}. As in the first case, all sheaves parameterized by $B$ are Jordan-H\"older equivalent. { \hfill $\Box$} \section{The Barth morphism}\label{SBAR} Our construction gives us the moduli space $X$ together with a finite morphism $X \rarpa{\varphi_1} I \!\! P^N$, which we call the Barth morphism. In this section we show that this morphism for ``surfaces with many lines'' assigns a sheaf $E$ its divisor of jumping curves. We say that a polarized surface $(S,{\cal O}_S(H))$ has many lines if the linear system $|H|$ is globally generated and the generic curve of this linear system is rational. By adjunction we have $H.(H+K_S) = -2$. Hence any given rank two sheaf $E$ can be normalised such that $H.c_1(E) \in \{-3, -2 , -1, 0 \}$ by twisting ${\cal O}_S(H+K_S)^{\otimes k}$. In the even case the assumptions made in \ref{SEMISTABLE} are not needed because of the following theorem. \begin{theorem} {\bf (Grauert-M\"ulich theorem) } If $(S,H)$ is a surface with many lines and $E$ a torsion free rank two sheaf on $S$ with $c_1(E).H$ even, then $E$ is semistable if and only if the restriction of $E$ to the general curve $l$ in the linear system $|H|$ is isomorphic to the direct sum of two isomorphic $l$-line bundles. \end{theorem} {\bf Proof: } See \cite{OSS} II theorem, 2.1.4. or \cite{He2} Lemma 3.6. {\vspace{0.5em} } This theorem provides us with a very explicit description of the dual moduli space $Y$ as we will see in a moment. We have to distinguish the following two cases: {\bf Case 1: $H.c_1(E) =-2$} By the Grauert-M\"ulich theorem, $H^*(E \otimes {\cal O}_l)=0$ for $E$ semistable and $l$ general in $|H|$. Hence we may take $|H|$ to be the dual moduli space. Recalling our construction we work here with a square root of the line bundle ${\cal L}$. {\bf Case 2: $H.c_1(E) =0$ } In this case we again identify $Y$ with the complete linear system $|H|$. Note that for any $\l \in |H|$ the dimension of the ${\rm Ext}^1_{{\cal O}_l}({\cal O}_l, {\cal O}_l(K_l+H))$ is one. Hence there is a unique nontrivial extension $\xi_l$. By assigning to $l$ this torsion sheaf $\xi_l$, $|H|$ is identified with the dual moduli space $Y$. {\vspace{0.5em} } Any semistable sheaf $D_E$ defines now a divisor $D_E$ in the dual moduli space $Y$ consisting of all curves $l \in |H|$ where $ \otimes {\cal O}_l$ is not of the expected type. Therefore this divisor $D_E$ is called the {\em divisor of jumping curves}. By straightforward calculations we find the \begin{proposition} The degree $d_E$ of the divisor $D_E$ of jumping curves equals $$\begin{array}{ll} d_E=c_2(E)-2+\frac{(K_S-c_1(E)).c_1(E)}{2} & \mbox{ if } H.c_1(E)=-2\, ;\\ \\ d_E=2c_2(E)-2-c_1(E)^2 & \mbox{ if } H.c_1(E)=0\, .\\ \end{array}$$ \end{proposition} If now we choose $N+1 = { (H^2 +1 ) + d_E \choose d_E }$ curves $\{ l_i \}_{i+0}^N$ in $|H|$ such that no divisor of degree $d_E$ contains all the $l_i$, then the duality construction gives us a finite morphism $X \rarpa{\varphi_1} I \!\! P^N$. The name Barth morphism is used because in \cite{Bar} Barth studied rank-2 vector bundles on the projective plane via their divisors of jumping lines. This morphism assigns every semistable rank two sheaf $E$ its jumping divisor $D_E$. As a corollary of this construction we have the \begin{theorem} The Barth morphism $X \rarpa{\varphi_1} I \!\! P^N$ is finite. \end{theorem}

9708

alg-geom/9708003

en

https://arxiv.org/abs/alg-geom/9708003

alg-geom/9708003

Mark De Cataldo

Mark Andrea A. de Cataldo

Singular hermitian metrics on vector bundles

LaTex (article) 25 pages; revised: minor changes; to appear in Crelle's J; dedicated to Michael Schneider

We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of $L^2$-estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain $d''$-complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, $t$-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form $E=F \otimes L$, where $F$ is a $t$-nef vector bundle and $L$ is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.

alg-geom

\section{Introduction} In this study I introduce a notion of singular hermitian metrics ({\em s.h.m.}) on holomorphic vector bundles over complex manifolds. The original motivation was to explore the possibility of employing, in the setting of vector bundles, the new transcendental techniques developed by Demailly and Siu in order to study global generation problems for (adjoint) line bundles. The notes \ci{dem94} are an excellent introduction to these techniques and to the results in the literature. One can consult the lucid notes \ci{eln} for an algebraic counterpart to these techniques. \bigskip Let me discuss the case of line bundles. Let $X$ be a non-singular projective manifold of dimension $n$, $ L$ and $E$ be an ample and a nef line bundle on $X$, respectively, $a$ be a non-negative integer and $m$ be a positive one. \noindent {\em Under which conditions on $a$ and $m$ will the line bundle $$ {\frak P}: =K_X^{\otimes a }\otimes { L}^{\otimes m} \otimes {E} $$ be generated by its global sections (free)?} \noindent More generally, we can ask for conditions on $a$ and $m$ under which the simultaneous generation of the higher jets of $\frak P$ at a prescribed number of points on $X$ is ensured. It is clear that $m\gg 0$ answers the question. However, how big $m$ should be could depend, {\em a priori}, on $X$. For example, Matsusaka Big Theorem asserts that $L^{\otimes m}$ is very ample for every $ m \geq M:=M(n, L^n, K_X\cdot L^{n-1})$. An effective value for $M$ has been recently determined in \ci{siumat} and \ci{dem96}; see also \ci{fdb} for the case of surfaces. The presence of the canonical line bundle, i.e. $a>0$, changes dramatically the shape of the lower bound on $m$. Fujita's Conjecture speculates that $K_X\otimes {L}^{\otimes m}$ should be free as soon as $ m\geq n+1$. This conjecture is true for $n\leq 4$ by the work of Reider, Ein-Lazarsfeld and Kawamata. In the papers \ci{an-siu} and \ci{tsu} it is proved that $m\geq \frac{1}{2}(n^2 + n +2)$ gives freeness. Effective results depending only on $n$ are proved for $a\geq 1$ by several authors. The seminal paper is \ci{dem93b} where it is proved, by (differential-geometric-){\em analytic methods}, that $K_X^{\otimes 2}\otimes L^{\otimes m}$ is very ample for all $ m \geq 12n^n$. Then followed the paper \ci{ko}, where a similar result is proved using {\em algebraic-geometric methods}. Since then, several papers have appeared on the subject. The reader may consult the following references to compare the various results and techniques: \ci{dem94} (an account of the analytic approach with a rather complete bibliography), \ci{eln} (an account of the algebraic approach and of how many of the analytic instruments may be re-tooled and made algebraic), \ci{an-siu} and \ci{tsu} (freeness; written in the analytic language, but apt to be completely translated into the algebraic language after observations by Koll\'ar \ci{kos}, \S5; see also \ci{siu-tak}), \ci{dem96}, \ci{siu-va} and \ci{siu94b} (very ampleness; analytic), \ci{siumat} and \ci{dem96} (an effective version of Matsusaka Big Theorem; analytic). \bigskip An extra nef factor $E$ plays a minor role and all of the results quoted above hold in its presence. This simple fact was the starting point of my investigation. \begin{??} \label{qk} Can we obtain effective results on $a$ and $m$ for the global generation of the vector bundles $\frak P$ by assuming that $E$ is a suitably semi-positive vector bundle of rank $r$? More generally, can we obtain similar results about the simultaneous generation of the higher jets of $\frak P$ at a prescribed number of points on $X$\,? \end{??} \medskip I expected that the statements in the aforementioned literature concerning the line bundles $\frak P$ with the nef line bundle $E$ should carry over, {\em unchanged}, to the case in which $E$ is a nef vector bundle. \noindent On a projective manifold a nef line bundle can be endowed with hermitian metrics whose curvature forms can be made to have arbitrarily small negative parts (cf. Definition \ref{defneflb}). In the analytic context this fact can be used to make the presence of a nef line bundle $E$ harmless. The same is true in the algebraic context because of the the numerical properties of nefness. \noindent A natural algebraic approach to the case of higher rank is to consider an analogous question for the tautological line bundle $\xi$ of the projectivized bundle $\pi: {\Bbb P}({E}) \to X$. The results I obtain with the algebraic approach are for ${\frak P}\otimes \det E$; compare Remark \ref{algnef} with the sample effective global generation result presented below; see \ci{deeff}. \noindent On the analytic side, the problem is that the nefness of a vector bundle $E$ does not seem to be linked to a curvature condition on $E$ itself. \smallskip As far as Question \ref{qk} is concerned, nefness does not seem to give enough room to work analytically with higher rank vector bundles. \noindent Instead I introduce, for every vector bundle $E$ and every positive integer $t$, the notion of {\em $t$-nefness} which is a new semi-positivity concept for vector bundles. In some sense it is in between the algebraic notion of nefness and the differential-geometric notion of $t$-semipositivity. It is a natural higher rank curvature analogue of the aforementioned characterization of nef line bundles. The property of $t$-nefness is checked by considering tensors in $T_X\otimes {E}$ of rank at most $t$; such tensors have ranks never bigger than $N:=\min (\dim X, {\rm rank} \,{E})$. Incidentally, $(t+1)$-nefness implies $t$-nefness for all positive integers $t$, $1$-nefness implies nefness and I do not know whether nefness implies $1$-nefness. \smallskip Though, as I show in Theorem \ref{umemura}, on curves $1$-nefness is equivalent to nefness, the notion of $t$-nefness is rather difficult to check in an algebraic context. However, see Example \ref{listnef} for a list of nef bundles which I know to be $N$-nef or from which it is easy to obtain $N$-nef bundles (e.g. nef bundles on curves, nef line bundles, flat bundles, nef bundles on toric or abelian varieties, the tangent bundles of low-dimensional K\"ahler manifolds with nef tangent bundles, pull-backs, etc.). \smallskip Assuming that $E$ is $N$-nef, I prove for the vector bundles $\frak P$ the same statements as the ones in the literature for the line bundle case; see Theorem \ref{effres}. Moreover, if $\,{E}$ is $1$-nef, then the same results hold replacing $E$ by $E \otimes \det E$. The scheme of the proofs is the same as in the rank one case (see Proposition \ref{cucu}, \S\ref{vecbl}, and of course \ci{dem94}, \S5 and \S8). However, at each and every step we need higher rank analogues of the analytic package developed for the line bundle case by Demailly and Nadel: regularization, $L^2$-estimates, coherence of relevant sheaves and vanishing theorems. For the purpose of proving these effective results for the vector bundles $\frak P$, one would have to make precise the notion of singular hermitian metrics with positivity and prove their relevant properties in a special case: the one of a hermitian vector bundle twisted by a line bundle endowed with a singular metric. Then one would have to prove the relevant vanishing theorems. All this can be done by building on \ci{dem82}, \S5 and \S9. \noindent However, I felt that it should be worthwhile to develop a general theory of {\em singular hermitian metrics} on vector bundles with special regards towards positivity. \medskip Inspired by the case of line bundles, in this paper I develop such a theory and obtain as an application the effective results mentioned above. To get a flavor of the results let me state (\ref{effres}.$1'$), which constitutes an answer to Question \ref{qk} (see Remark \ref{geoint} for a geometric interpretation of these kind of results): \medskip \noindent {\bf Effective global generation.} {\em Let $E$ be $N$-nef. Then $K_X \otimes L^{\otimes m} \otimes {E}$ is globally generated by its global sections for all $m\geq \frac{1}{2}(n^2 + n +2)$. Moreover, if $E$ is $1$-nef, then the previous statement is true if we replace $E$ by ${E} \otimes \det E$. } \bigskip The paper is organized as follows. \noindent \S{\bf 1} fixes the notation. \S{\bf 2} is devoted to s.h.m. which are defined in \S2.1. The case of line bundles is discussed in \S\ref{exlbshm}. In \S2.3 we introduce the sheaf $\E (h)$ which generalizes Nadel multiplier ideal sheaf. In \S2.4 we define positivity for s.h.m. and study some of its properties. \S{\bf 3} revolves about the notion of $t$-nefness. The definition and the basic properties are to be found in \S3.1 and \S3.2, respectively. \S3.3 is devoted to the proof of Theorem \ref{umemura} which ensures that on curves the algebraic-geometric notion of nefness can be characterized by the differential-geometric notion of $1$-nefness. \S3.4 consists of a footnote to \ci{d-p-s}, Theorem 1.12: ampleness for a vector bundle $E$ can be characterized by a curvature condition on a system of metrics on {\em all} symmetric powers $S^pE$ of $E$, though positivity may occur only for $p\gg 0$. \S{\bf 4} is devoted to vanishing theorems. The basic one is Theorem \ref{vanish}, a generalization of Nadel Vanishing Theorem; Proposition \ref{coherent} asserts that $\E(h)$ is coherent in the presence of suitable positivity. \S4.2 links $t$-nefness and positivity via vanishing; see Theorem \ref{myvan}. Theorem \ref{kv} is a generalization of Kawamata-Viehweg Vanishing Theorem. \S{\bf 5} contains the effective results concerning the vector bundles $\frak P$. \S5.1 contains, for the reader's convenience, a summary of the results of Anghern-Siu and Siu concerning special s.h.m. on line bundles which, transplanted to $N$-nef vector bundles, will provide the global generation of jets. We also offer the simple Lemma \ref{freetojet}, which constructs metrics with similar properties starting from free line bundles. \S\ref{vecbl} contains our effective results concerning the vector bundles $\frak P$; see Theorem \ref{effres}. \medskip \noindent {\bf Acknowledgments}. I heartily thank J.-P. Demailly for reading a preliminary and rough version of this paper and for suggesting some improvements. I am indebted to J. Koll\'ar for posing a question similar to Question \ref{qk}. I thank L. Ein and R. Lazarsfeld for convincing me to think about an algebraic proof of the results of Theorem \ref{effres}; this has lead me to the statements of Remark \ref{algnef}; see \ci{deeff}. It is a pleasure to thank the participants of the lively algebraic geometry seminar at Washington University in St. Louis for their encouragment and useful criticisms: V. Masek, T. Nguyen, P. Rao and D. Wright. I would like to thank N.M. Kumar for many pleasant and useful conversations. \section{Notation and preliminaries} Our basic reference for the language of complex differential geometry is \ci{g-h}. Sufficient and more self-contained references are \ci{dem82}, \S 2 and \ci{dem94}, \S 3. All manifolds are second countable, connected and complex; the dimension is the complex one. All vector bundles are holomorphic. The term {\em hermitian metric} always refers to a hermitian metric of class ${\cal C}^2$. A {\em hermitian bundle} $(E,h)$ is the assignment of a vector bundle $E$ together with a hermitian metric $h$ on it. Duality for vector bundles is denoted by the symbol $`` \, ^ * \, "$ and ${\rm End} (E)$ is the vector bundle of endomorphisms of $E$. We often do not distinguish between vector bundles and associated sheaves of holomorphic sections; at times, we employ simultaneously the additive and multiplicative notation for line bundles. \smallskip - $d=d' + d''$ denotes the natural decomposition of the exterior derivative $d$ into its $(1,0)$ and $(0,1)$ parts; $d''$ denotes also the usual operator associated with a vector bundle $E$. \smallskip If $(E,h)$ is a rank $r$ hermitian vector bundle on a manifold $X$ of dimension $n$, then we denote by: - $D_h(E)$ the associated {\em hermitian connection} which is also called the {\em Chern connection}; - $\T{h}{E}=iD^2_h(E)$ the associated {\em curvature tensor}; \noindent in particular, if $L$ is a line bundle with a metric $h$, represented locally on some open set $U$ by $e^{-2\varphi}$, then we have $ \T{h}{L}_{|U}= 2i \D{\varphi}$; - $\tilde{\Theta}_h (E)$ the associated hermitian form on $T_X \otimes E$. \smallskip If $\theta$ is a hermitian form on a complex vector space $V$, we denote $\theta (v,v)$ by $\theta (v)$; if in addition, $\theta$ is positive definite, then we denote $\theta(v)$ by $|v|^2_{\theta}$. - ${\rm Herm}_h (V)$ is the set of endomorphisms $\alpha$ of a hermitian vector space $(V,h)$ such that $h(\alpha(v),w)=h(v,\alpha (w))$, $\forall v,\, w \in V$. \noindent Given $\Theta$, a real $(1,1)$-form with values in ${\rm Herm}_h (E)$, we denote the associated hermitian form on $T_X \otimes E$ by $\Theta_h$, or by $\Theta$, if no confusion is likely to arise. The hermitian form $\TT{h}{E}$ will be denoted from now on by $\T{h}{E}$. If $\omega$ is a real $(1,1)$-form, e.g. the one associated with a hermitian metric on $X$, then $\omega \otimes {\rm Id}_E$ has values in ${\rm Herm}_h(E)$ and we denote the associated hermitian form by $\omega \otimes {\rm Id}_{E_h}$ so that $\omega \otimes {\rm Id}_{E_h}(t\otimes e)=$ $\omega (t,it) |e|^2_h$, $\forall x \in X$, $\forall t \in T_{X,x}$ and $\forall e \in E_x$. \medskip \noindent {\bf The rank of a tensor.} Let $V$ and $W$ be complex vector spaces of finite dimensions $r$ and $s$, respectively, $v=\{v_{i}\}_{i=1}^r$ and $w=\{w_{\alpha}\}_{\alpha=1}^s$ be bases for $V$ and $W$, respectively; tensor products are taken over $\comp$. \noindent Every tensor $\tau \in V \otimes W$ defines two linear maps $\alpha_{\tau}: W^* \to V$ and $\beta_{\tau}: V^* \to W$; moreover, we can write $\tau=\sum_{i\alpha} \tau_{i\alpha}v_i\otimes w_{\alpha}$ and associate with $\tau$ the $r\times s$ matrix $||\tau_{i\alpha}||$. The integer $\rho (\tau):= {\rm rank} (\alpha_{\tau})=$ ${\rm rank} (\beta_{\tau})=$ ${\rm rank} ||\tau_{i\alpha}||$ is called the {\em rank} of the tensor $\tau$. \noindent Tensors of rank zero or one are called {\em decomposable}; they have the form $\tau=v\otimes w$, for some $v \in V$ and $w\in W$. For any non-zero tensor $\tau\in V \otimes W$ we have that $1 \leq \rho (\tau) \leq \min (r, s)$. In particular, if either $r=1$, $s =1$, or both, then every tensor $\tau\in V \otimes W$ is decomposable. \medskip \noindent {\bf Inequalities associated with the rank.} Given two hermitian forms $\theta_1$ and $\theta_2$ on $V\otimes W$, we can compare them on tensors of various rank. Let $t$ be any positive integer. We write $\theta_1 \geq_t \theta_2$ if the hermitian form $\theta_1 - \theta_2$ is semi-positive definite on all tensors in $V \otimes W$ of rank $\rho \leq t$. If $\theta \geq_t 0$, then $\theta \geq_{t'} 0$ for every $ t' \leq t$. If $\theta_1 \geq_{\min (r,s) } \theta_2$, then $\theta_1 \geq _t \theta_2$ for every $t$. The symbol $>_t$ can be defined analogously and it enjoys similar properties. \medskip These considerations and this language are easily transferred to vector bundles. \section{Singular hermitian metrics on vector bundles} \label{shm} In this section we define singular hermitian metrics on vector bundles, discuss the case of line bundles, introduce the sheaf $\E (h)$ and define positivity. \subsection{The definition of singular hermitian metrics} \label{dshm} Let $X$ be a manifold of dimension $n$, $E$ be a rank $r$ vector bundle over $X$ and $\bar{E}$ the conjugate of $E$. Let $h$ be a section of the smooth vector bundle $E^*\otimes \bar{E}^*$ with measurable coefficients, such that $h$ is an almost everywhere (a.e.) positive definite hermitian form on $E$; we call such an $h$ a {\em measurable metric} on $E$. A measurable metric $h$ on $E$ induces naturally measurable metrics on $E^*$, on any tensor representation of $E$, e.g. $T^{\alpha} E$, $S^{\beta} E$, $\wedge^{\gamma} E$ etc., on any quotient bundle of $E$, etc. In practice these metrics $h$ occur as {\em degenerate metrics} of some sorts, e.g. $h$ is a hermitian metric outside a proper analytic subset $\Sigma$ of $X$, so that the curvature tensor is well-defined outside $\Sigma$. \smallskip We are interested in those $h$ for which the curvature tensor has a global meaning. We propose the following simple-minded definition. \begin{defi} \label{defshm} {\rm(s.h.m.)} Let $X$, $E$ and $h$ be as above and $\Sigma \subseteq X$ be a closed set of measure zero. Assume that there exists a sequence of hermitian metrics $h_s$ such that: $$ \lim_{s \to \infty} h_s = h \qquad in \,\, the \, \, {\cal C}^{2}-topology \,\, on\,\, X \setminus \Sigma. $$ We call the collection of data $(X,E,\Sigma, h,h_s)$ a singular hermitian metric (s.h.m.) on $E$. We call $\T{h}{E_{|X\setminus \Sigma}}$ the curvature tensor of $(X,E,\Sigma,h,h_s)$ and we denote it by $\T{h}{E}$. $\T{h}{E}$ has continuous coefficients and values in ${\rm Herm}_h(E)$ away from $\Sigma$; we denote the a.e.-defined associated hermitian form on $T_X \otimes E$ by the same symbol $\T{h}{E}$. \end{defi} If no confusion is likely to arise, we indicate a s.h.m by $(E,h)$ or simply by $h$. \medskip The guiding principle which subtends this definition can be formulated as follows. \smallskip \noindent {\em Assume that we would like to prove a property $P$ for $h$ which is true for all metrics $h'$ of class ${\cal C}^2$ in the presence of a certain curvature condition $C$ on $h'$; if $h$ has the required property $C$ and we can find hermitian metrics $h_s$ which regularize $h$ ``maintaining" $C$, then $P$ holds for all $h_s$ and we can try to prove, using limiting arguments, that $P$ holds for $h$.} \smallskip \noindent This principle has been successfully exploited in \ci{dem82}, \S5; see \S\ref{exlbshm} for a brief discussion. We will take this principle as the definition of positivity; see Definition \ref{ipo} and Proposition \ref{l2}, where $P$ is the solution to the $d''$-problem with $L^2$-estimates and $C$ is ``positivity." \medskip Because of the convergence in the ${\cal C}^2$-topology, the notion of s.h.m. is well behaved under the operations of taking quotients, dualizing, forming direct sums, taking tensor products, forming tensor representations, etc. \subsection{Discussion of the line bundle case: curvature current, positivity, Nadel Ideal, Nadel Vanishing Theorem, and the production of sections} \label{exlbshm} We now remark that the singular metrics on line bundles to be found in the literature are s.h.m. We also discuss some of the relevant features of these metrics in the presence of positivity. Basic references for what follows are \ci{dem94}, \S5, \ci{dem82}, \S9 and \S5. A technical remark: for the mere purpose of being consistent with Definition \ref{defshm}, in what follows we assume that plurisubharmonic (psh) functions are ${\cal C}^2$ outside a closed set of measure zero. In all the applications one uses {\em algebraic singular metrics} as in \ci{dem96}, so that this condition is automatically satisfied. However, all the theory described below and its applications work without this restriction; see also \ci{dem92}, \S3. \smallskip Note that in what follows we can replace the hermitian line bundle $(L,h_0)$ by a hermitian vector bundle $(E,h_0)$ by operating minor changes. \medskip A {\em singular metric} on a line bundle $L$ over a manifold $X$ is, by definition, a metric of the form $h=h_0 e^{-2\varphi}$, where $h_0$ is a hermitian metric on $L$ and $\varphi$ is a locally integrable function on $X$. We shall always assume that $X$ is K\"ahler and that $\varphi$ is {\em almost psh}, i.e. it can be written, locally on $X$, as the sum $\varphi=\alpha + \psi$, where $\alpha$ is a local function of class ${\cal C}^2$ and $\psi$ is a local psh function. By taking $d'd''$ in the sense of distributions, we can define the associated {\em curvature $(1,1)$-current}: $$ T:= \T{h_0}{L} + 2i d'd'' \varphi_{ac} + 2i d'd'' \varphi_{sing}, $$ where $2id'd'' \varphi_{\rm ac}$ and $2i d'd'' \varphi_{sing}$ are the absolutely continuous and singular part of $2i d'd'' \varphi$, respectively; $2id'd'' \varphi_{\rm ac}$ has locally integrable coefficients and $2id'd''\varphi_{\rm sing}$ is supported on some closed set $\Sigma$ of measure zero. A regularizing-approximating result of Demailly's exhibits these singular metrics on line bundles as s.h.m. by constructing the necessary regularizing hermitian metrics $\{h_s \}_{s=1}^{\infty}$. We have $\T{h}{L}=\T{h_0}{L} + id'd''\varphi_{ac}$. Similar considerations hold for metrics dual to metrics as above. \begin{ex} \label{ef} {\rm (Cf. \ci{dem94}, Example 3.11 and \ci{dem96}, page 246) Let $D=\sum m_iD_i$ be a divisor with coefficients $m_i \in\zed$. The associated line bundle carries a singular metric with curvature current $T=2\pi \sum m_i[D_i] $ where the $[D_i]$ are the currents of integration over the subvarieties $D_i$. These currents are positive if and only if all $m_i\geq 0$. More generally, given a finite number of non-trivial holomorphic sections of a multiple of a line bundle $L$, we can construct a s.h.m. on $L$. This metric will be singular only at the common zeroes of the sections in question. } \end{ex} \smallskip The {\em Nadel ideal} $\id (h)$ (see \S \ref{mi}) {\em is coherent}. This is an essential feature in view of the use of this ideal in conjunction with Riemann-Roch Theorem. \medskip Let $\omega$ be a K\"ahler metric on a weakly pseudoconvex manifold $X$. Assume that $\T{h}{L} \geq \e\omega$ as a $(1,1)$-current, for some positive and continuous function $\e$ on $X$. Then we have {\em Nadel Vanishing Theorem}: $ H^q(X,K_X\otimes L \otimes \id (h))=0, \quad \forall q >0;$ see \ci{na}. This can be seen as a consequence of the solution to the $d''$-problem for $(L,h)$ with $L^2$-estimates; see \ci{dem94}, \S5. \medskip As an easy consequence of Nadel Vanishing Theorem we have the following result which lays the basis for the effective results for the global generation of adjoint line bundles etc. See \ci{dem94}, Corollary 5.12. \begin{pr} \label{cucu} Let $(X,\omega)$ be as above and $\cal L$ be a line bundle over $X$ equipped with a s.h.m. $h$ such that $\T{h}{L} \geq \e \omega$ for some continuous and positive function $\e$ on $X$. Assume that $p$ is a positive integer and that $s_1, \ldots, s_p$ are non-negative ones. Let $x_1, \ldots, x_p$ be distinct isolated points of the complex space $V(\id (h))$ such that $\id (h) \subseteq {\frak m}_{x_i}^{s_i +1}$. Then there is a surjective map $$ H^0 (X, K_X + {\cal L} ) \surj \bigoplus_{i=1}^p {\cal O}(K_X + {\cal L} )\otimes {\cal O}_{X,x_i}/ {\frak m}_{x_i}^{s_i +1}. $$ \end{pr} Once the analytic package (definition of s.h.m., regularization-approximation, solution of $d''$ with $L^2$-estimates, coherence of Nadel ideal and Nadel Vanishing Theorem) has been developed, in order to solve the global generation problem one needs s.h.m. as in Proposition \ref{cucu}. This requires hard work and it has been done by Anghern-Siu, Demailly, Siu and Tsuji. The coherence and the vanishing theorem are utilized together with a clever use of Noetherian Induction. \medskip We are about to provide a similar analytic package for the case of vector bundles. \subsection{The subsheaf $\E (h)$ associated with a measurable metric $(E,h)$} \label{mi} If $h$ is a measurable metric on $E$ and $e$ is a measurable section of $E$, then the function $|e|_h$ is measurable. \begin{defi} \label{iande} Let $h$ be a measurable metric on $E$. \noindent Let $\id (h)$ be the analytic sheaf of germs of holomorphic functions on $X$ defined as follows: $$ \id(h)_x:= \{ f_x \in \odix{X,x} \!: \, \, |f_x e_x|^2_{h}\, \mbox{is integrable in some neighborhood of \,} \, x, \, \forall \, e_x \in E_x\}. $$ Analogously, we define an analytic sheaf $\E (h)$ by setting: $$ \E(h)_x := \{ e_x \in E_x \, : \, \, |e_x|^2_{h}\, \mbox{ is integrable in some neighborhood of } \, x\,\}. $$ \end{defi} \begin{rmk} \label{sub} {\rm It is easy to show, using the triangle inequality, that $\id (h)\otimes E \subseteq \E(h)$. } \end{rmk} We call $\id(h)$ the {\em multiplier ideal} of $(E,h)$. Note that if $E$ is a line bundle together with a measurable metric $h$, then $\E(h)=\id (h) \otimes E$. \medskip There are other subsheaves of $E$, associated with a measurable metric $h$. \smallskip Given any measurable metric $h$ on a vector bundle $E$, the tautological line bundle $\xi := \odixl{{\Bbb P}(E)}{1}$ inherits a natural measurable metric ${\frak h}$, the quotient metric of the surjection $\pi^* E \to \xi$; here $\pi: {\Bbb P}(E) \to X$ is the structural morphism of the projectivized bundle and we are using Grothendieck's notation. We thus get two sheaves $\id ({\frak h})$ and $\xi \otimes \id (\frak h)$. If we apply $\pi_*$, then we get two other subsheaves of $E$. \medskip In summary, associated with $(E,h)$ there are four subsheaves of $E$: \smallskip \centerline{ $\id (h)\otimes E \subseteq \E(h)$, $ \quad \pi_* \id ({\frak h}) \otimes E \quad $ and $\quad \pi_* \,\, \xi \otimes \id ({\frak h})$.} \begin{rmk} {\rm The inclusion above may be strict. In fact, consider the vector bundle $\Delta \times \comp^2$, where $(\Delta,z)$ is the unit disk in $\comp^1$; define a s.h.m. by setting $h={\rm diag} (e^{-2\log{|z|}}, e^{-4\log{|z|}})$. Then one checks that $\id (h) =z^2 \cdot \odix{\Delta}$ and that $\E (h)= z\cdot \odix{\Delta} \oplus z^2\cdot \odix{\Delta}$. The same example shows that $\E (h)$ is not in general equal to neither $\pi_* \,\, \xi \otimes \id ({\frak h})$, nor $\pi_* \id ({\frak h}) \otimes E$. In fact, a direct computation shows that: $\id ({\frak h})=\pi^*(z)$. We have $ \id(h) \otimes E$ $\subset$ $\E(h)$ $\subset$ $\pi_* \, \id ({\frak h}) \otimes \xi=$ $\pi_* \, \id ({\frak h}) \otimes E$. } \end{rmk} What is, among the four sheaves above, the ``right" object to look at? To answer this question we consider: \medskip \noindent {\bf the complex $({\frak L}^{\bullet}, d'')$}. Let $h$ be a measurable metric on a vector bundle $E$ and $\omega $ be a hermitian metric on $X$. By following the standard conventions in \ci{we}, we obtain a metric with measurable coefficients for the fibers of ${T_X^{p,q}}^* \otimes E$; we denote this metric again by $h$. We define a complex $({\frak L}^{\bullet}, d'')$ of sheaves on $X$ as follows. This complex is independent of the choice of $\omega$. \noindent Let ${\frak L}^q$ be the sheaf of germs of $(n,q)$-forms $u$ with values in $E$ and square-integrable coefficients such that $|u|_h^2$ is locally integrable, $d''u$ is defined in the sense of distributions with square-integrable coefficients and $|d''u|_h^2$ is locally integrable. \medskip The kernel of $d''$ in degree zero is $K_X \otimes \E(h)$ (cf. \ci{g-h}, page 380). A solution to the $d''$-problem with $L^2$-estimates for $(E,h)$ would imply the vanishing of the higher cohomology of $K_X \otimes \E(h)$. See Theorem \ref{vanish}. \medskip If we are aiming at vanishing theorems as in the line bundle case, the sheaf $\E(h)$ seems to be the right object to look at. \subsection{Positivity} \label{p} As is well-known, the curvature tensor $\T{h}{L}$ of a hermitian line bundle $(L,h)$ is decomposable and can be identified with a real $(1,1)$-form on $X$. This latter is a positive $(1,1)$-form if and only if the hermitian form $\T{h}{L}$ is positive on $T_X \otimes L$. It is therefore natural to define positivity for singular metrics on line bundles using the notion of {\em positive currents} according to Lelong; see \ci{le}, \S 2. \medskip In the higher rank case the curvature tensor is not, in general, decomposable. We introduce a notion of positivity which incorporates what is needed to obtain $L^2$-estimates-type results. \medskip Let $\omega $ be a hermitian metric on $X$, $\theta $ be a hermitian form on $T_X$ with continuous coefficients and $(X,E,\Sigma, h,h_s)$ be a s.h.m.; in particular, the curvature tensor and the curvature form $\T{h}{E}$ are defined a.e. (i.e. outside of $\Sigma$) and have measurable coefficients. \begin{defi} \label{ipo} {\rm ($\geq_t^{\mu}$; compare with \ci{dem82}, \S 5.)} Let things be as above and $t$ be a positive integer. \noindent We write: $$ \T{h}{E} \geq_t^{\mu} \theta \otimes {\rm Id}_{E_h} $$ if the following requirements are met. \noindent There exist a sequence of hermitian forms $\theta_s$ on $T_X\otimes E$ with continuous coefficients, a sequence of continuous functions $\lambda_s$ on $X$ and a continuous function $\lambda$ on $X$ subject to the following requirements: \noindent {\rm (\ref{ipo}.1)} $\forall x \in X$: $|e_x|_{h_s}\leq |e_x|_{h_{s+1}}$, $\forall s \in \nat$ and $\forall e_x \in E_x$; \noindent {\rm (\ref{ipo}.2)} $\theta_s\geq_t \theta \otimes {\rm Id}_{E_{h_s}}$; \noindent {\rm (\ref{ipo}.3)} $\T{h_s}{E}\geq_t \theta_s - \lambda_s \omega \otimes {\rm Id}_{E_{h_s}}$; \noindent {\rm (\ref{ipo}.4)} $\theta_s \to \T{h}{E}$ a.e. on $X$; \noindent {\rm (\ref{ipo}.5)} $\lambda_s \to 0$ a.e. on $X$; \noindent {\rm (\ref{ipo}.6)} $0\leq \lambda_s \leq \lambda$, $\forall s$. \end{defi} Conditions {\rm (\ref{ipo}.1)} and {\rm (\ref{ipo}.6)} are needed to apply Lebesgue's theorems on monotonic and dominated convergence. In order to obtain $L^2$-estimates-type results, we also need the remaining four conditions to make precise the sought-for control of the curvature by the regularizing and approximating metrics $h_s$. \begin{rmk} \label{refcoherent} {\rm As an application of the $L^2$-estimates, we will see that if $\T{h}{E} \geq_N^{\mu} \theta \otimes {\rm Id}_{E_h}$, for some continuous $\theta$, then the sheaf $\E(h)$ is coherent; see Proposition \ref{coherent}. } \end{rmk} \begin{ex} {\rm If $(E,h)$ is a hermitian bundle with $\T{h}{E} \geq_t \theta \otimes {\rm Id}_{E_h}$, then it is easy to exhibit $h$ as a s.h.m. such that $\T{h}{E} \geq_t^{\mu} \theta \otimes {\rm Id}_{E_h}$; just set $h_s:=h$ $\forall s$, etc. } \end{ex} \begin{ex} \label{regappr} {\rm Let $(E,h)$ be a vector bundle together with a {\em continuous } s.h.m metric. Under certain positivity conditions on the current $id'd'' h^*$which is defined on the total space of $E^*$ (see \ci{cm}, \S7.1) we can exhibit $h$ as a s.h.m. with positivity in the sense of Definition \ref{ipo}. This is achieved in two steps. In the first one $h^*$ is regularized by using Riemannian convolution coupled with the parallel transport associated with an arbitrary hermitian metric on $E^*$ (see \ci{cm}, Lemme 7.2). In the second one the resulting metrics are modified so that they have the prescribed properties; this technical modification follows ideas in \ci{dem82}, \S8. Details will appear elsewhere. } \end{ex} \begin{ex} {\rm Let $h=h_0e^{-2 \varphi}$ be a singular metric on a line bundle $L$ with $T \geq \theta$ as currents where $\theta $ is a continuous and real $(1,1)$-form. \ci{dem82}, Th\'eor\`eme 9.1, exhibits these data as a s.h.m. $h$ with $\T{h}{L}\geq_1^{\mu} \theta \otimes {\rm Id}_{L_{h}}$. \noindent Conversely, if we have a s.h.m. $h$ with $\T{h}{L}\geq_1^{\mu} \theta \otimes {\rm Id}_{L_h}$, then we have $\T{h_s}{L}\geq \theta_s - \lambda_s\omega$ and $T\geq T_{ac} \geq \theta$. } \end{ex} \begin{rmk} {\rm The existence of a s.h.m. $h$ on a line bundle $L$ for which $\T{h}{L} \geq_1^{\mu} 0$ does not imply that $L$ is nef. See {\rm \ci{d-p-s}}, {\rm Remark 1.6}. \noindent What is true is that if $L$ is nef, then $L$ will admit a metric $h=h_0e^{-2\varphi}$ with $h_0$ a hermitian metric on $L$ and $\varphi$ almost psh such that $\T{h}{L} \geq_1^{\mu} 0$. This can be seen by using \ci{d-p-s}, Proposition 1.4, \ci{dem92}, Proposition 3.7 and {\rm \ci{dem82}}, Th\'eor\`eme $9.1$. \noindent Similar remarks hold for big line bundles on projective manifolds (cf. \ci{dem94}, Proposition 6.6). } \end{rmk} The following lemma is elementary. \begin{lm} \label{fund} Let $(E,\Sigma_E, h,h_s)$ and $(F,\Sigma_F, g,g_s)$ be s.h.m. on two vector bundles $E$ and $F$ over $X$, $\s_1$ and $\s_2$ be two real $(1,1)$-forms with continuous coefficients such that $\T{h}{E} \geq_{t_1}^{\mu} \s_1 \otimes {\rm Id}_{E_h}$ and $\T{g}{F} \geq_{t_2}^{\mu} \s_2 \otimes {\rm Id}_{F_g}$. \noindent Then $H:=h\otimes g$ on $E\otimes F$ can be seen as a s.h.m. by setting $H_s:=h_s \otimes g_s$ and $$ \T{H}{E\otimes F} \geq_{\min (t_1, t_2)}^{\mu} (\s_1 + \s_2) \otimes {\rm Id}_{{(E\otimes F)}_H} \, . $$ \end{lm} Note that if the rank of $F$ is one, then $\min (t_1,t_2)= t_1$. \medskip We now prove that positivity is inherited by quotient metrics. \begin{lm} \label{posquot} Let $(X,E,\Sigma, h,h_s)$ be a s.h.m such that $\T{h}{E} \geq_t^{\mu} \theta \otimes {\rm Id}_{E_h}$, $\phi:E\to Q$ be a surjection of vector bundles with kernel $K$. Then $Q$ admits a s.h.m. $(Q, \Sigma' \subseteq \Sigma, q_s, q)$ such that $\T{q}{Q}\geq_1^{\mu} \theta \otimes {\rm Id}_{Q_q}$. \end{lm} \noindent {\em Proof.} Consider the dual exact sequence $ 0 \to Q^* \to E^* \to K^* \to 0. $ Each hermitian metric $h_s^*$ defines by restriction a hermitian metric $q_s^*$ on $Q^*$; analogously we get $q^*:=h^*_{|Q^*}$. Clearly $(Q^*,\Sigma', q,q_s)$ is a s.h.m for an appropriate $\Sigma'\subseteq \Sigma$. \noindent For every $s$ we have that $\T{q_s^*}{Q^*}=\T{h_s^*}{E^*}_{|{Q^*}} + i\beta_s^* \wedge \beta_s$, where $\beta_s$ is a $(1,0)$-form with values in ${\rm Hom} (Q^*, K^*)$, ${\cal C}^1$ coefficients and $\beta^*$ is its adjoint. Moreover $i\beta_s^* \wedge \beta_s \leq_1 0$; see \ci{dem82}, Lemme 6.6. The statement follows easily by dualizing again, which has the effect of transposing and changing the signs. \blacksquare \medskip More generally, a s.h.m. on $E$ ``with positivity" will induce s.h.m. ``with positivity" on $T^{\alpha}E $, $S^{\beta}E$ and $\wedge^{\gamma} E$. We leave the various formulations and elementary proofs to the reader. \section{t-nef vector bundles} \label{tnef} \subsection{The definition of $t$-nefness} \label{cn} Let $X$ be a compact manifold of dimension $n$, $\omega$ be a hermitian metric on $X$, $E$ be a vector bundle of rank $r$ on $X$, $N:=\min (n,r)$ and $L$ be a line bundle on $X$. Every tensor in $T_X \otimes E$ has rank $\rho \leq N$. \medskip There are notions of semi-positivity associated with every positive integer $t$. The standard one is the following. \begin{defi} {\rm (t-semi-positive vector bundle)} We say that a vector bundle $E$ is $t$-semi-positive, if $E$ admits a hermitian metric $h$ such that $\T{h}{E} \geq_t 0$. \end{defi} Note that $E$ is $1$-semi-positive if and only if it is Griffiths-semi-positive, and that $E$ is $N$-semi-positive if and only if it is Nakano-semi-positive. A similar remark holds for strict inequalities. \medskip In algebraic geometry the most natural semi-positivity concept is {\em nefness}. A differential-geometric characterization of this concept can be given as follows. \begin{defi} \label{defneflb} {\rm (Nef line bundle and nef vector bundle)} We say that $L$ is nef if for every $ \e >0$ there exists a hermitian metric $h_{\e}$ on $L$ such that $\T{h_{\e}}{L}\geq -\e \omega$ as $(1,1)$-forms or, equivalently, if $\T{h_{\e}}{L}\geq_1 -\e \omega \otimes {\rm Id}_{L_{h_{\e}}}$ as hermitian forms on $T_X\otimes L$. \noindent We say that $E$ is nef if the tautological line bundle $\xi:=\odixl{{\Bbb P}(E)}{1}$ is nef. \end{defi} Note that the compactness of $X$implies that the definitions given above are independent of the choice of $\omega$. The same holds true for all the other definitions given below which involve a choice of $\omega$. \medskip If $X$ is projective, then Definition \ref{defneflb} is equivalent to the usual one: {\em $L$ is nef if $L\cdot C \geq 0$, for every integral curve $C$ in $X$}; see \ci{dem94}, Proposition 6.2. \medskip Unfortunately a nef line bundle is not necessarily $1$-semi-positive ($\geq_1 0$). See {\rm \ci{d-p-s}, Example 1.7}, where an example is given of a nef rank two vector bundle $E$ on an elliptic curve such that the nef tautological line bundle $\xi$ on ${\Bbb P} (E)$ is not $\geq_1 0$. Moreover, $E$ is not $\geq_1 0$ (otherwise $\xi$ would be $\geq_1 0$); this shows that even on curves nefness and Griffiths-semi-positivity do not coincide. Recall Theorem 1.12, \ci{d-p-s}, which states that nefness of a vector bundle $E$ can be characterized by the presence of a system of hermitian metrics on all bundles $S^{\alpha}E$ such that they are suitably semi-positive for $\alpha \gg 0$. I do not know if nefness can be characterized in terms of hermitian metrics on the vector bundle itself. The two facts above and the need to express semi-positivity in terms of curvature have motivated my introducing the notion of $t$-nefness. \begin{defi} {\rm (t-nef vector bundle)} We say that a vector bundle $E$ is $t$-nef if for every $ \e>0$ there exists a hermitian metric $h_{\e}$ on $E$ such that $\T{h_{\e}}{E} \geq_t -\e\omega \otimes {\rm Id}_{E_{h_\e}}$. \end{defi} \smallskip Every flat vector bundle is $N$-nef. \noindent If $E$ is $t$-semi-positive, then $E$ is $t$-nef. As pointed out above, the converse is not true in general; see {\rm \ci{d-p-s}, Example 1.7}. \noindent If $E$ is $t$-semi-positive {\rm (}$t$-nef, respectively{\rm )}, then $E$ is $t'$-semi-positive {\rm (}$t'$-nef, respectively{\rm )}, for every $ t'$ such that $1\leq t' \leq t$. \noindent By definition, a line bundle is nef if and only if it is $1$-nef. A $1$-nef vector bundle is nef as we will see in {\rm Proposition \ref{list}}. The converse is true on curves as we will see in {\rm Theorem \ref{umemura}}. We do not know whether the converse is true or false when $\dim X \geq 2$. This problem is the analogue of Griffiths' question: does ampleness imply Griffiths-positivity? \noindent We have checked that if $E$ is nef and is the tangent bundle of a compact complex surface or of a compact K\"ahler threefold, then $E$ is $1$-nef. This is done by using the classification results contained in {\rm \ci{d-p-s}} and {\rm Proposition \ref{list}}. According to conjectures in {\rm \ci{d-p-s}}, the same should be true for compact K\"ahler manifolds of arbitrary dimension. \noindent ({\bf From nefness to $1$-nefness}) On special manifolds, such as toric and abelian varieties, we have that if $E$ is nef, then $E \otimes \det E$ is $1$-nef; see {\rm \ci{cm}}, {\rm \S7.2.1}. By the following paragraph, if $E$ is a rank $r$ nef vector bundle on such a variety, then $E \otimes (\det \, E)^{\otimes r+2}$ is $N$-nef. \noindent ({\bf From $1$-nefness to $N$-nefness}) On any compact manifold, if $E$ is $1$-nef, then $E\otimes \det E$ is $N$-nef. See {\rm \ci{dem-sk}}. \begin{ex} \label{listnef} {\rm ({\bf Some $N$-nef vector bundles}) The results mentioned above and the ones of sections \S\ref{bbpp} and \S\ref{nefoncurves} give us the following list of examples. \smallskip \noindent 1) A nef vector bundle over a curve is $N$-nef. A nef line bundle is $t$-nef for every $t$. \noindent 2) A flat vector bundle is $N$-nef. \noindent 3) If $X$ is a special manifold such as a toric or an abelian variety and $E$ is nef of rank $r$, then ${E} \otimes \det {E}$ is $1$-nef and ${E} \otimes (\det {E})^{\otimes r+2}$ is $N$-nef. \noindent 4) If $X$ is a K\"ahler manifold of dimension $n\leq 3$ with nef tangent bundle $T_X$, then $T_X$ is $1$-nef and $K_X^{\otimes -1}\otimes T_X$ is $N$-nef. \noindent 5) Every Nakano-semipositive vector bundle is $N$-nef (the converse is not true). If $E$ is a Griffiths-semipositive vector bundle, then $E\otimes \det E$ is $N$-nef. \noindent 6) The extension of two $t$-nef vector bundles is $t$-nef. Positive tensor representations of a $t$-nef vector bundle are $t$-nef. If $E_1$ and $E_2$ are $t$-nef, then $E_1\otimes E_2$ is $t$-nef. If $E$ is $t$-nef and $L$ is a nef line bundle, then $E \otimes L$ is $t$-nef. \noindent 7) If $E$ is $1$-nef, then $E \otimes \det E$ is $N$-nef. \noindent 8) If $f: X \to Y$ is a morphism and $E$ is a $t$-nef vector bundle on $Y$, then $f^* E$ is $t$-nef. If $t=1$ and, either $f$ is finite and surjective, or $X$ is projective and $f$ has equidimensional fibers, then the converse is true. } \end{ex} At this point the following question is only natural. \begin{??} Is every nef vector bundle $1$-nef? \end{??} \subsection{Basic properties of $t$-nefness} \label{bbpp} Let us list and prove some basic properties of $t$-nef vector bundles. We start with functorial ones. \begin{pr} \label{fctr} Let $f:X \to Y$ be a holomorphic map, where $X$ and $Y$ are compact manifolds and $E$ is a vector bundle on $Y$. \medskip {\rm (1)} If $E$ if $t$-nef, then $f^*E$ is $t$-nef. \medskip {\rm (2)} Assume that $f$ is surjective and that the rank of $E$ is one. \noindent Then $f^*E$ is $1$-nef {\rm (}=\,nef {\rm )} if and only if $E$ is $1$-nef {\rm (}=\,nef {\rm )}. \medskip {\rm (3)} Assume that $f$ is finite and surjective, that $Y$ (and thus $X$) is K\"ahler and let $E$ be of any rank. Then $f^*E$ is $1$-nef if and only if $E$ is $1$-nef. \end{pr} \noindent {\em Proof.} \medskip \noindent (1). Let $\omega$ and $\omega'$ be two hermitian metrics on $X$ and $Y$, respectively. Let $A$ be a positive constant such that $A \omega \geq f^* \omega'$. Fix $\e >0$ and let $\e':=\frac{\e}{A}$. Let $h'$ be a hermitian metric on $E$ such that $\T{h'}{E} \geq_t -\e' \omega' \otimes {\rm Id}_{E_{h'}}$. Endow $f^*E$ with the pull-back metric $h:=f^*h'$. The claim follows from the formula $\T{h}{f^*E}= f^* \T{h'}{E}$. \medskip \noindent {\rm (2).} See \ci{d-p-s}, Proposition 1.8.ii for the case of equidimensional fibers and \ci{pm} for the general statement. \medskip \noindent (3) It follows easily from \ci{cm}, \S7.1: assign to $E$ the appropriate trace metrics and regularize. \blacksquare \begin{rmk} {\rm As pointed out in {\rm Example \ref{regappr}}, the regularizing metrics in (3) can be chosen to satisfy favorable conditions towards $L^2$-estimates. } \end{rmk} \begin{??} {\rm Can we drop the assumption of finiteness from (3). A. J. Sommese has pointed that the answer is positive when $X$ is projective and the fibers are equidimensional: slice $X$ with sufficiently ample general divisors to reduce to the case in which the morphism is finite. Is (3) true if we replace $1$-nef by $t$-nef, with $t>1$? } \end{??} \begin{pr} \label{list} Let $X$, $E$ and $r$ be as above. Then: \medskip {\rm (1)} Let $E \to Q$ be a surjection of vector bundles. If $E$ is $1$-nef, then $Q$ is $1$-nef. \medskip {\rm (2)} If $E$ is $1$-nef, then $E$ is nef. \medskip {\rm (3)} If $S^m E $ is $1$-nef, then $E$ is nef. \medskip {\rm (4)} Let $0 \to K \to E \to Q\to 0$ be an exact sequence of vector bundles. If $K$ and $Q$ are $t$-nef, then $E$ is $t$-nef. \medskip {\rm (5)} Let $E=E_1 \oplus E_2$. The vector bundle $E$ is $t$-nef if and only if $E_1$ and $E_2$ are $t$-nef. \medskip {\rm (6)} Let $F$ be another vector bundle. Assume that $E$ and $F$ are $t$-nef and $t'$-nef respectively; then $E \otimes F$ is $\min (t, t' )$-nef. \medskip {\rm (7)} Assume that $E$ is $t$-nef. \noindent Then $S^m E$ and $\wedge^l E $ are $t$-nef for all $ m\geq 0$ and for $ 0\leq l \leq r$. \noindent Moreover, $\Gamma^a E$ is $t$-nef, where $\Gamma^a E$ is the irreducible tensor representation of $Gl(E)$ of highest weight $a=(a_1, \ldots, a_r) \in {\zed}^r$, with $a_1\geq \ldots a_r \geq 0$. \medskip {\rm (8)} Let $0 \to E \to E' \to \tau \to 0$ be an exact sequence with $E'$ a vector bundle on $X$ and $\tau$ a sheaf, quotient of a $1$-nef vector bundle $E''$. If $E$ is $1$-nef, then so is $E'$. \medskip {\rm (9)} Let $0 \to K \to E \to Q\to 0$ be an exact sequence of vector bundles. If $E$ and $\det Q^*$ are $1$-nef, then $K$ is $1$-nef. \medskip {\rm (10)} Assume that $\det E$ is hermitian flat; the vector bundle $E$ is $t$-nef if and only if $E^*$ is $t$-nef. \medskip {\rm (11)} Let $E$ be $1$-nef and $s\in \Gamma (E^*)$. Then $s$ has no zeroes. \end{pr} \noindent {\em Proof.} Fix, once and for all, $\omega$ a hermitian metric on $X$. \medskip \noindent (1). Let $\e >0$ and $h_{\e}$ be a hermitian metric on $E$ with $\T{h_{\e}}{E} \geq_1 -\e \omega \otimes {\rm Id}_{E_{h_{\e}}}$. Endow $Q$ with the quotient metric $h'_{\e}$; $Q$ can be seen as a smooth sub-bundle of $E$ via the ${\cal C}^{\infty}$ orthogonal splitting of $E\to Q$ determined by $h_{\e}$, so that ${h_{\e}}_{|Q}=h'_{\e}$. It is well-known (e.g. \ci{dem82}, Lemme 6.6) that $\T{h'_{\e}}{Q} \geq_1 {{\T{h_{\e}}{E}}_{|Q}}_{h'_{\e}}$ and it is clear that ${ {\rm Id}_{ {E_{ h_{\e} } }}}_{|Q} ={\rm Id}_{Q_{h'_{\e}}}$. The claim follows. \medskip \noindent (2). Let $\pi : {\Bbb P}(E) \to X$ be the canonical projection. By virtue of \ref{fctr}.1 we have that $\pi^* E$ is $1$-nef; (1) and the canonical surjection $\pi^*E \to \odixl{{\Bbb P}(E)}{1}$ imply that this latter line bundle is $1$-nef. \medskip \noindent (3) $\pi^* S^m E$ is $1$-nef by \ref{fctr}.1, so that $\odixl{{\Bbb P}(E)}{m}$, being a quotient of $\pi^* S^m E$, is $1$-nef, by (1). It follows that $\odixl{{\Bbb P}(E)}{1}$ is $1$-nef and thus nef. \medskip \noindent (4). Fix a ${\cal C}^{\infty}$ vector bundle isomorphism $\Phi: E \to K \oplus Q$. Let and $\e >0$. By assumption, there are metrics $h_{K,\e}$ and $h_{Q,\e}$ such that $\T{h_{K,\e}}{K} \geq_t -\frac{\e}{3} \omega \otimes {\rm Id}_{K_{h_{K,\e}}}$ and $\T{h_{Q,\e}}{Q} \geq_t -\frac{\e}{3} \omega \otimes {\rm Id}_{Q_{h_{Q,\e}}}$. \noindent Fix an arbitrary positive real number $\rho >0$ and consider the automorphism $\phi_{\rho}: Q \to Q$ defined by multiplication by the factor $\rho^{-1}$. \noindent Let $\Phi_{\rho}:=({\rm Id}_K\oplus \phi_{\rho})\circ \Phi: E \to K \oplus Q$; denote the first component of $\Phi_{\rho}$ by $\Phi_{K,\rho}$ and the second one by $\Phi_{Q,\rho}$ \noindent Define a hermitian metric on $E$ by setting $h_{\e, \rho}:=$ $\Phi_{K,\rho}^*h_{K, \e} \oplus \Phi_{Q,\rho}^*h_{Q, \e}$. Its associated Chern connection has the form: \[ D_{h_{\e, \rho}} = \left( \begin{array}{cc} D_{h_{K,\e}} & - \beta^*_{\rho} \\ \beta_{\rho} & D_{h_{Q,\e}} \end{array} \right ), \] \noindent where $\beta_{\rho}=\rho \beta_1$ is a $(1,0)$-form with values in ${\rm Hom}(K,Q)$. By calculating $D^2$ we see that $$ \T{h_{\e, \rho}}{E} \geq_t -\frac{2}{3}\e \omega \otimes {\rm Id}_{E_{h_{\e, \rho}}} + O(\rho)\omega \otimes {\rm Id}_{E_{h_{\e, \rho}}}. $$ The claim follows by recalling that $X$ is compact and by taking $\rho$ sufficiently small. \medskip \noindent (5). The ``if" part follows from (4). The converse follows by observing that if $E$ has a metric $h$, then each $E_i$ inherits a metric $h_i$ for which $D_{h_i}={D_{h}}_{|E_i}$. The same holds for the curvature tensors. \medskip \noindent (6). The proof is immediate once one recalls the formula for the curvature of the tensor product of two hermitian metrics: $\T{h_1 \otimes h_2}{E_1\otimes E_2}=$ $\T{h_1}{E_1}\otimes {\rm Id}_{E_2}+$ ${\rm Id}_{E_1} \otimes \T{h_2}{E_2}$. \medskip \noindent (7). The tensor powers $T^n(E)$ are $t$-nef by virtue of (6). $S^n(E)$ and $\wedge^n(E)$ are both direct summands of $T^n(E)$ so that they are $t$-nef by (5). Recall that $\Gamma^a E$ is a direct summand of the vector bundle $\otimes_{i=1}^r S^{a_i}(\wedge^i E)$ which is $t$-nef by what above, (5) and (6). \medskip \noindent {\rm (8).} The ``pull-back" construction gives the following commutative diagram of coherent sheaves: $$ \begin{array}{lllllllll} \hspace{1cm} 0 & \to & E & \to & E' & \to & \tau & \to & 0 \\ \hspace{1cm} \, & \ & \uparrow {\rm Id}_E& \ & \uparrow q & \ & \uparrow & \ & \, \\ \hspace{1cm} 0 & \to & E & \to & E''' & \to & E'' &\to & 0, \end{array} $$ \noindent where $q$ is surjective. Since $E$ and $E''$ are locally free and $1$-nef, so is $E'''$ by $(4)$. Since $q$ is surjective, it follows that $E'$ is $1$-nef by $(1)$. \smallskip \noindent The proofs of (9) and (10) are the same as in the nef case; the proof of (11) is in fact easier. The reader can consult \ci{d-p-s}. \blacksquare \begin{rmk} \label{etl} {\rm It is easy to show, using $(6)$, that if $E$ is $t$-nef and $L$ is a positive line bundle, then $E\otimes L$ admits a hermitian metric $h$ with curvature $\Theta_h (E\otimes L) >_t 0$. In particular, if $E$ is $N$-nef, then $E \otimes L$ is Nakano-positive. A similar remark holds for the symbol $\geq_t^{\mu}$; see {\rm Lemma \ref{fundamental}}. } \end{rmk} \begin{rmk} {\rm As far as $(1)$ above is concerned, it is not true that if $E$ is $t$-nef, then $Q$ is $t$-nef. In fact, consider the canonical surjection $\odix{\pn{2}}^3 \to T_{\pn{2}}(-1)$: $\odix{\pn{2}}^3$ is $2$-nef, but if $T_{\pn{2}}(-1)$ were $2$-nef, then $T_{\pn{2}}=$ $T_{\pn{2}}(-1) \otimes \odixl{\pn{2}}{1}$ would be $>_2 0$, i.e. Nakano-positive and this is a contradiction. This example also shows that $1$-nefness is strictly weaker than $2$-nefness. We do not know whether $(8)$ is false when we replace $1$ by $t$. } \end{rmk} \subsection{Nefness and $t$-nefness on curves} \label{nefoncurves} It is an outstanding problem in Hermitian differential geometry to determine whether an ample vector bundle is Griffiths-positive. In \ci{ume}, Umemura proves that on curves ampleness and Griffiths-positivity coincide. As is was pointed out to me by N.M. Kumar, the part of the argument that needs a result analogue to Proposition \ref{list}.8 is omitted in \ci{ume}. \medskip We now prove that on curves nefness and $1$-nefness coincide: the algebraic notion of nefness can be characterized in differential-geometric terms. Recall that Example 1.7, \ci{d-p-s}, implies that even over a curve, a nef vector bundle is not necessarily Griffiths-semi-positive. \begin{tm} \label{umemura} Let $X$ be a nonsingular projective curve and $E$ be a vector bundle of rank $r$ on $X$. The following are equivalent. \smallskip {\rm ($i$)} $E$ is $1$-nef; \smallskip {\rm ($ii$)} $E$ is nef; \smallskip {\rm ($iii$)} every quotient bundle of $E$, and in particular $E$, has non-negative degree. \end{tm} \noindent {\em Proof.} ($i$)$ \Rightarrow$($ii$). This is Proposition \ref{list}.2. \smallskip ($ii$)$ \Rightarrow$($iii$). In fact they are equivalent by \ci{c-p}, Proposition 1.2.7. \medskip ($iii$)$ \Rightarrow$($i$). We divide the proof in three cases, according to whether $g=0$, $g=1$ or $g\geq 2$. Let $d$ be the degree of $E$. By assumption $d\geq 0$. \medskip If the genus $g(X)=0$, then $E$ splits into a direct sum of line bundles and the statement follows easily. \medskip Let $g(X)=1$. It is enough to consider the case when $E$ is indecomposable. Let us first assume that $d\geq r$. By \ci{at}, Lemma 11, $E$ admits a maximal splitting $(L_1,\ldots , L_r)$ with $L_i$ ample line bundles on $X$. It follows that $E$ could then be constructed inductively from (ample =) positive line bundles by means of extensions. A repeated use of Proposition \ref{list}.4 would allow us to conclude. We may thus assume, without loss of generality, that $0\leq d <r$. If $r=1$, then $E$ is either ample or hermitian flat; in both cases we are done. We now proceed by induction on the rank of $E$. Assume that we have proved our contention for every vector bundle of rank strictly less than $r$. By \ci{at}, Lemma 15 and Theorem 5, $E$ sits in the middle of an exact sequence: $$ 0 \to A \to E \to B \to 0, $$ where $B$, being a quotient of $E$, enjoys property ({\em iii}) and $A$ is either a trivial vector bundle (if $d>0$) or a hermitian flat line bundle (if $d=0$). In any case $A$ is clearly $1$-nef and $B$ is $1$-nef by the induction hypothesis. We can apply \ref{list}.4 and conclude that $E$ is $1$-nef. This proves the case $g(X)=1$. \medskip We now assume that $g(X)\geq 2$. The proof will be by induction on $r$. If $r=1$, then we are done since $\deg{E}\geq 0$ implies that either $E$ is ample or it is hermitian flat. Assume that we have proved our assertion for every vector bundle of rank strictly less than $r$. \smallskip \noindent There are two cases. \noindent In the first one we suppose that $E$ contains a non-trivial vector sub-bundle $K$ which is $1$-nef. Consider the exact sequence of coherent shaves: $$ 0 \to K \to E \to Q:=E/K \to 0. $$ There are two sub-cases. In the first one we assume that $Q$ is locally free. By assumption every quotient vector bundle of $Q$, being in turn a quotient bundle of $E$, has positive degree. The induction hypothesis forces $Q$ to be $1$-nef. Proposition \ref{list}.4 allows us to conclude that $E$ is $1$-nef as well. \noindent In the second sub-case $Q\simeq F \oplus \tau$, with $F$ locally free and $\tau$ has zero-dimensional support; in particular there is a surjection $\odix{X}^m \to \tau$. If $K'$ is the kernel of the surjection $E\to F$, then we have the exact sequence $$ 0 \to K \to K' \to \tau \to 0, $$ so that, by \ref{list}.$8$, $K'$ is $1$-nef and we are reduced to the first sub-case. \noindent In the second case we are allowed to assume that $E$ does not contain properly any non-trivial vector bundle $K$ which is $1$-nef. \smallskip {CLAIM.} $E$ is stable. \noindent To prove this we start a new proof by induction. Let $K$ be a vector bundle contained in $E$, neither trivial nor equal to $E$. Since (iii) implies that $\deg{(E)}\geq 0$, to prove that $E$ is stable it is enough to show that $\deg {K} <0$. Seeking a contradiction, let us assume that $\deg {K} \geq 0$. Let $s$ be the rank of $K$. If $s=1$, since $K$ is not $1$-nef by the working assumption of this second case, we see that $\deg{K}<0$ (otherwise $K$ would be either ample or hermitian flat) and we have reached a contradiction if $s=1$. Assume that, for every non-trivial $K''\subseteq E $ with rank strictly less than $s$, $\deg{K''} <0$. Each sub-bundle of $K$ has, by this second inductive hypothesis, negative degree. Since we are assuming that $\deg {K} \geq 0$, it follows that every quotient of $K$, including $K$ itself has non-negative degree, so that, by the first induction hypothesis, $K$ is $1$-nef and we have reached a contradiction for every $s$: the degree of $K$ must be negative, $E$ is stable and the claim is proved. \smallskip \noindent Since $\det E$ has non-negative degree by assumption, $\det E$ is $1$-nef. Since $E$ is stable, for any hermitian metric $h$ on $\det E$ of curvature $\T{h}{\det E}$, a standard calculation (see \ci{ume}, Lemma 2.3) yields a hermitian metric $H$ on $E$ of curvature $\frac{1}{r}\T{h}{\det E} \otimes {\rm Id}_E$. This proves that $E$ is $1$-nef also in the second case. \blacksquare \subsection{A differential-geometric characterization of ampleness for vector bundles} \label{dg} We now prove a characterization of ampleness by means of curvature properties which is a simple consequence of \ci{d-p-s}, Theorem 1.12. \begin{pr} \label{1.12} Let $X$ be a compact manifold equipped with a hermitian metric $\omega$ and $E$ be a vector bundle on $X$. Then $E$ is ample if and only if there exists a sequence of hermitian metrics $h_m$ on $S^m E$ such that \smallskip {\rm (i)} the sequence of metrics on $\odixl{{\Bbb P}(E)}{1}$ induced by the surjective morphisms $$\pi^* S^m E \to \odixl{{\Bbb P}(E)}{m}$$ converges uniformly to a hermitian metric $h$ of positive curvature on $ \odixl{{\Bbb P}(E)}{1}$ and \smallskip {\rm (ii)} there exist $\eta >0$ and $m_0 \in \nat$ such that $\forall m \geq m_0$: $$ \label{F&A} \T{h_m}{S^m(E)} \geq_1 m\eta \omega \otimes {\rm Id}_{{S^m E}_{h_m}}. $$ \end{pr} \begin{rmk} \label{known} {\rm If $E$ is ample, then the fact that some metrics $h_m$ with property (ii) exist for all $ m \gg 0$ is a well known consequence of {\rm \ci{griff}}, theorems F and A. The point made by the statement above is that the metrics $h_m$ are constructed on {\em all} symmetric powers $S^m(E)$, and that they are all built starting from a suitable metric on $\odixl{{\Bbb P}(E)}{1}$; see {\rm \ci{dem92}, Theorem 4.1}. } \end{rmk} \noindent {\em Proof.} The proof of the implication ``$\Leftarrow$" follows easily from (i): $ \odixl{{\Bbb P}(E)}{1}$ is positive by the existence of $h$ so that $E$ is ample. \noindent For the reverse implication ``$\Rightarrow$" we argue as follows. Fix a hermitian metric $\omega'$ on ${\Bbb P}(E)$. The ampleness of $E$ implies the ampleness of $\odixl{{\Bbb P}(E)}{1}$, which then admits a hermitian metric $h$ of positive curvature; the compactness of $X$ ensures us that there exist $\alpha >0$ and $A>0$ such that $$ \T{h}{\odixl{{\Bbb P}(E)}{1}} \geq \alpha \omega' \geq \alpha A \pi^* \omega. $$ Define $\eta:=\frac{2}{3} \alpha A$. We are now in the position of using \ci{dem92}, Theorem 4.1 with $v:=\frac{3}{2}\eta \omega$ and $\e:=\frac{1}{2}\eta$. \blacksquare \section{Vanishing theorems} \label{vt} In this section we link the positivity of $h$ to the vanishing of the cohomology of $K_X \otimes \E (h)$. \subsection{The basic $L^2$-estimate and vanishing theorem, and the coherence of $\E (h)$} Following Demailly, \ci{dem82}, \S 5, we say that a s.h.m. $(X,E,\Sigma,h,h_s)$ is {\em $t$-approximable} if $\T{h}{E} \geq_t^{\mu} 0$ (cf. Definition \ref{ipo}). We denote the space of $(p,q)$-forms with values in $E$ and coefficients which are locally square-integrable by $L_{p,q}^2(X,E,{\rm loc})$. As usual, $n=\dim{X}$, $r$ is the rank of $E$ and $N:={\rm min} (n,r)$. \begin{pr} \label{l2} {\rm (See \ci{dem82}, Th\'eor\`eme 5.1)} Let $(X,\omega)$ be K\"ahler, where either $\omega$ is complete or $X$ is weakly pseudoconvex. Assume that $(E,h)$ is a s.h.m. with the property that $\T{h}{E}\geq_{n-q+1}^{\mu} \e \omega \otimes {\rm Id}_{E_h}$, where $\e$ is a non-negative and continuous function on $X$ and $q>0$ is a positive integer. \noindent Let $g\in L^2_{n,q}(X,E,{\rm loc})$ be such that $$ d''g=0, \qquad \qquad \int_X{ |g|^2_h \, dV_{\omega}} < + \infty \qquad and \qquad \int_X{ \frac{1}{\e} |g|^2_h \, dV_{\omega}} < + \infty . $$ \noindent Then there exists $f \in L^2_{n,q-1}(X,E,{\rm loc})$ such that $$d''f=g \qquad and \qquad \int_X |f|^2_h dV \leq \frac{1}{q} \int_X \frac{1}{\e} |g|^2_h \, dV_{\omega}. $$ \end{pr} \noindent {\em Sketch of proof.} Th\'eor\`eme 5.1 states something slightly different but it is immediate to recover the statement of the proposition. We merely point out, for the reader's convenience, the minor changes to be implemented to obtain the above statement. The notation is from \ci{dem82}. \noindent The assumption $\T{h}{E}\geq_{n-q+1}^{\mu} \e \omega \otimes {\rm Id}_{E_h}$ has two consequences. The former is that $h$ is $n-q+1$-approximable. The latter is that, by virtue of \ci{dem82}, Lemme 3.2 (3.4): $$ |g|^2_{\T{h}{E}} \leq \frac{1}{q\e} |g|^2_h \quad a.e. $$ We can apply the aforementioned theorem and conclude. \blacksquare \medskip The following generalizes Nadel Vanishing Theorem. It is an easy consequence of the proposition above. \medskip \begin{tm} \label{vanish} Let $(X, \omega)$ be K\"ahler with $X$ weakly pseudoconvex. Assume that $(E,h)$ is a s.h.m. such that $\T{h}{E} \geq_{N}^{\mu} \epsilon \omega \otimes {\rm Id}_{E_h}$ for some positive and continuous function $\epsilon$. Then, $H^q(X,K_X\otimes \E (h))=0$, $\forall q>0$. \end{tm} \noindent {\em Proof.} The complex $({\frak L}^{\bullet}, d'')$ of \S\ref{mi} is exact by Proposition \ref{l2} applied to small balls. This complex is therefore an acyclic resolution of $K_X \otimes \E(h)$ whose cohomology is isomorphic to the cohomology of the complex of global sections of $({\frak L}^{\bullet}, d'')$. This latter cohomology is trivial for every positive value of $q$ by Proposition \ref{l2} (modify the metric as in \ci{dem94}, Proposition 5.11). \blacksquare \medskip \medskip We now prove that if $h$ is suitably positive, then $\E (h)$ is coherent. The line bundle case is due to Nadel. \begin{pr} \label{coherent} Let $X$ be a complex manifold, $(X,E,\Sigma,h,h_s)$ be a s.h.m., and $\theta$ be a continuous real $(1,1)$-form on $X$ such that $\T{h}{E} \geq_N^{\mu} \theta \otimes {\rm Id}_{E_h}$. Then $\E (h)$ is coherent. \end{pr} \noindent {\em Proof.} We make the necessary changes from the line bundle case (cf. \ci{dem94}, Proposition 5.7). \noindent Note that the condition $\T{h}{E} \geq_N^{\mu} \theta \otimes {\rm Id}_{E_h}$ implies that $h\geq h_1$ a.e. \noindent The statement being local, we may assume that $X$ is a ball centered about the origin in $\comp^n$ with holomorphic coordinates $(z)$, that $E$ is trivial and that $\theta$ has bounded coefficients. Let $\omega$ be the $(1,1)$-form associated with the euclidean metric on $X$. Let $\frak S$ be the vector space of holomorphic sections $f$ of $E$ such that $\int_X{|f|^2_h \, d\lambda} < \infty$, where $d\lambda$ is the Lebesgue measure on $\comp^n$. Consider the natural evaluation map $ev:{\frak S} \otimes_{\comp} \odix{X} \to E$. The sheaf ${\frak E}:= Im(ev)$ is coherent by Noether Lemma (cf. \ci{gr-re}, page 111) and it is contained in $\E (h)$. \noindent We want to prove that $\E (h)_x = {\frak E}_x$ for all $x \in X$. In view of Nakayama's Lemma, \ci{at-mac}, Corollary 2.7, it is enough to show that $ {\frak E}_x + {{\frak m}^{\gamma}_x} \cdot \E (h)_x = \E (h)_x$ for some $\gamma \geq 1$. \medskip \noindent STEP I. Assume that we could prove that: \smallskip \noindent $(\bullet) \qquad \qquad \qquad \qquad $ $ {\frak E}_x + \E (h)_x \cap {\frak m}^l_x \cdot E_x =\E (h)_x$ for every positive integer $l$. \smallskip \noindent By the Artin-Rees Lemma, \ci{at-mac}, Corollary 10.10, there would be a positive integer $k=k(x)$ such that $$ \E(h)_x = {\frak E}_x + \E(h)_x\cap {\frak m}_x^l \cdot E_x \subseteq {\frak E}_x + {\frak m}_x^{l-k}\cdot \E(h)_x \subseteq {\frak E}_x + {\frak m}_x \cdot \E(h)_x \subseteq \E(h)_x $$ for all $l\geq k$. All symbols ``$\subseteq$" could be replaced by equalities and we could conclude that ${\frak E}_x = \E (h)_x$ by Nakayama's Lemma as above. \medskip \noindent STEP II. We now prove $(\bullet)$. \noindent Let $f$ be a germ in $\E (h)_x$ and $\s$ be a smooth cut-off function such that is identically $1$ around $x$ and that has compact support small enough so that $\s f$ is smooth on $X$. \noindent For every positive integer $l$ define a strictly psh function $\varphi_{l}:= (n+l)\ln |z-x| +C|z|^2$ where $C$ is a positive constant chosen so that $2id'd'' (C|z|^2) + \theta \geq \e \omega$, for some positive constant $\e$. \noindent Define a metric on $E$ by setting $H_l:=$ $h e^{-2\varphi_{l}}$. Since both $\ln |z-x|$ and $|z|^2$ are psh, we can apply the results of \ci{dem82}, \S9 to $\varphi$ and deduce, with the aid of Lemma \ref{fund}, that $H_{l}$ is a s.h.m. on $E$ with $\T{H_{l}}{E}=\T{h}{E} + 2id'd''\varphi_{l} \otimes {\rm Id}_E$ and such that $\T{H_{l}}{E} \geq_N^{\mu} \e \omega \otimes {\rm Id}_{ E_{H_{l}} }$. \noindent Consider the smooth $(0,1)$- form $g:=d''(\s f)$ which has compact support and is identically zero around $x$. The function $|z-x|^{-2n -2l}$ is continuous outside $x$. It follows that: $$ \int_X{|g|_{H_{l}}^2\, d \lambda}= \int_X{ |g|_h^2 \, |z-x|^{-2n- 2l}\, e^{-2C|z|^2} \, d \lambda < \infty. } $$ We solve, for every index $l$, the equation $d''u=g$ with $L^2$-estimates relative to $H_{l}$ using Proposition \ref{l2}. We obtain a set of solutions $u_{l}$ such that $$ \int_X{|u_{l}|^2_{H_{l}}} \, d\lambda= \int_X{|u_{l}|^2_{h} \, |z-x|^{-2n-2l} \, e^{-2C |z|^2} \, d\lambda < \infty}. $$ Since the factor $e^{-2C |z|^2}$ does not affect integrability we get that $$ \int_X{|u_{l}|^2_{h} \, |z-x|^{-2n-2l} \, d\lambda < \infty}. $$ Since $d'' (\s f - u_{l}) =0$ and $h \leq H_{l}$, we see that $ \s f- u_{l}=: F_{l} \in {\frak E}$ (cf. \ci{g-h} page 380). The germ $u_{l,x}=f - F_{l,x}$ is holomorphic. Since $h\geq h_1$ and $h_1$ is continuous, there is a positive constant $B$ such that: $$ \int_X{ B \, |u_{l}|^2 \, |z-x|^{-2n-2l} } \, d\lambda \leq \int_X{|u_{l}|^2_h \, |z-x|^{-2n-2l} } \, d\lambda < \infty. $$ Let $u_{l}^{\{j \}}$ be the $j$-th coordinate function of $u_{l}$, $j=1, \ldots ,r$. By a use of Parseval's formula (cf. \ci{dem94}, 5.6.b) we see that $u_{l}^{\{j \}} \in {\frak m}^l_x$ for every index $j$. It follows that $(\bullet)$ holds and we are done. \blacksquare \subsection{$t$-nefness and vanishing} We now show how to use Theorem \ref{vanish} to infer the vanishing of cohomology in the case of a $N$-nef vector bundle twisted by a line bundle which can be endowed with a positive s.h.m. \smallskip The following is an elementary consequence of Lemma \ref{fund}: \begin{lm} \label{fundamental} Let $E$ be a $t$-nef vector bundle on a compact manifold $X$, $\omega$ be a hermitian metric on $X$, $\theta$ be a real $(1,1)$-form with continuous coefficients and $(F,g,g_s)$ be a vector bundle endowed with a s.h.m. such that $\T{g}{F} \geq_t^{\mu} \theta \otimes {\rm Id}_{F_g}$. \noindent Then for every constant $\eta >0$ there is a s.h.m. $H_{\eta}$ on $E\otimes F$ for which: $$ \T{H_{\eta}}{E\otimes F} \geq_t^{\mu} (\theta - \eta \omega) \otimes {\rm Id}_{(E \otimes F)_{H_{\eta}}}. $$ Moreover, if $F$ is a line bundle and $\T{g}{F} \geq \theta$ as $(1,1)$-forms, then the same conclusion holds. \end{lm} \begin{lm} \label{ELco} Let $(F,h_F)$ be a hermitian vector bundle on a manifold $X$ and $(L,h_L)$ be a line bundle on $X$ endowed with a singular metric $h_L$ as in {\rm \S\ref{exlbshm}}. Consider the vector bundle $E:=F\otimes L$ endowed with the measurable metric $h:=h_F\otimes h_L$. \noindent Then $ \E (h) =\id (h_L) \otimes E$ and $\E (h)$ is coherent. \end{lm} \noindent {\em Proof.} The statement $\E (h) = \id (h_L)\otimes E$ is local on $X$ so that we may assume that $X$ is a ball in $\comp^n$, that $F$ and $L$ are trivial, that $h_L=e^{-2\varphi}$ with $\varphi$ almost psh and that $h_F$ has bounded coefficients. \noindent Let us first prove that $ \id (h_L)_x \otimes E_x \subseteq \E(h)_x$. Let $e_x\in E_x$ and $f_x \in \id (h_L)_x$. Since $h_F$ is continuous, we have that $|f_x e_x|^2_h= |f_x|^2 |e_x|^2_{h_F} e^{-2\varphi}$ is locally integrable. \noindent Let us prove the reverse inclusion $ \E (h)_x \subseteq \id (h_L)_x\otimes E_x $ for every $x$ in $ X$. There exists a constant $\tilde\epsilon >0$ such that $h_F \geq \tilde\epsilon \Delta$, where $\Delta$ is the standard euclidean metric on the fibers of $E$. Fix $x \in X$. Assume that $\E (h)_x \ni e_x=<f_1,\ldots, f_r>$. Then $|e_x|^2_h=|e_x|^2_{h_F} e^{-2\phi} \geq \tilde \epsilon \sum |f_i|^2 e^{-2\varphi}$. As the left hand side of the inequality is integrable around $x$, so is each summand on the right. This proves the reverse inclusion. To conclude recall that $\id (h_L)$ is coherent (or apply Proposition \ref{coherent}). \blacksquare \medskip The following result is the key to the proofs of the effective statements to be found in \S\ref{effective}. See Ex. \ref{listnef} for examples of $N$-nef vector bundles. \begin{tm} \label{myvan} Let $(X,\omega)$ be as in {\rm Theorem \ref{vanish}}, and $(F,h_F)$, $(L,h_L)$ and $(E,h)$ be as in {\rm Lemma \ref{ELco}}. \noindent If $\T{h}{E}\geq_N^{\mu} \e \omega \otimes {\rm Id}_E$ for some positive and continuous function $\e$, then $$ H^q(X, K_X \otimes F \otimes L \otimes \id (h_L) )= H^q(X, K_X \otimes \E (h) ) =0, \qquad \forall \, q>0. $$ Moreover, if $X$ is compact, $F$ is $N$-nef and $(L,h)$ is such that $\T{h}L\geq \e \omega$, for some positive constant $\e$, then the same conclusion holds. \end{tm} \noindent {\em Proof.} By Lemma \ref{ELco}, we have that $\E(h) =\id (h_L) \otimes E$. We conclude in view of Theorem \ref{vanish}. \noindent The case of $X$ compact is a special case after Lemma \ref{fundamental}. \blacksquare \medskip The following is not needed in the sequel. We include it since it is a generalization of Kawamata-Vieheweg Vanishing Theorem (K-V) and it can be proved along the lines of \ci{dem94}, 6.12 by using Theorem \ref{myvan} instead of Nadel Vanishing Theorem. The ``$1$-nef" case follows easily from K-V Theorem and the Leray spectral sequence by looking at the projectivization of $E$. The statement in the ``$N$-nef" case seems new for $0<q< {\rm rank} \, E$ and the vanishing in the complementary range follows from K-V and Le-Potier spectral sequence. See \ci{dem94} for the particular language employed in the statement below. \begin{tm} \label{kv} Let $(X,E,F)$ be the datum of: $X$ a projective manifold, $E$ a $N$-nef vector bundle on $X$, $F$ a line bundle on $X$ such that some positive multiple $mF$ can be written as $mF=L+D$, where $L$ is a nef line bundle and $D$ is an effective divisor. Then $$ H^q(X, K_X \otimes E \otimes F \otimes \id(\frac{1}{m}D))=0 \qquad for\, \, q > \dim X - \nu (L), $$ where $\nu(L)$ is the numerical dimension of $L$ and $\id(\frac{1}{m}D)$ is the multiplier ideal of the singular local weights associated with the $m-$roots of the absolute values of local equations for $D$. \noindent As a special case, we have that if $F$ is a nef line bundle, then $$ H^q(X, K_X \otimes E \otimes F )=0 \qquad for \, \, q > \dim X - \nu (F); $$ in particular, if $F$ is nef and big, then $$ H^q(X, K_X \otimes E \otimes F )=0 \qquad for \, \, q > 0. $$ \end{tm} \section{Effective results} \label{effective} \subsection{Special s.h.m. on line bundles after Anghern-Siu, Demailly, Siu and Tsuji} \label{demsiu} The following proposition is at the heart of the effective base-point-freeness, point-separation and jet-separation results in \ci{an-siu}, \ci{tsu}, \ci{siu-va}, \ci{siu94b} and \ci{dem96}; it provides us with the necessary s.h.m. which we transplant to the vector bundle case and use in connection with Theorem \ref{myvan}. \medskip First we need to fix some notation. \bigskip Let $F$ be a rank $r$ vector bundle on a complex manifold $X$ and $p$ be any positive integer. We say that {\em the global sections of $F$ generate simultaneous jets of order $s_1,\ldots, s_p \in {\Bbb N}$ at arbitrary $p$ distinct points of $X$} if the natural maps $$ H^0(X,F) \to \bigoplus_{\i=1}^p {\cal O}(F)_{x_{i}} \otimes \odix{X}/{\frak m}^{s_{i}+1}_{x_{i}} $$ are surjective for every choice of $p$ distinct points $x_1, \ldots , x_p$ in $X$. \noindent We say that {\em the global sections of $F$ separate arbitrary $p$ distinct points of $X$ } if the above holds with all $s_{i}=0$. \smallskip Assume that $X$ is compact. Let $V:= H^0(X,F)$ and $h^0:=h^0(X,F):= \dim_{\comp} H^0(X,F)$. Consider $G:= G(r,h^0)$ the Grassmannian of $r$-dimensional quotients of $V$, $\QQ$ the universal quotient bundle of $G$ and $\q$ the determinant of $\QQ$. \noindent As soon as $F$ is generated by its global sections (which corresponds to the above conditions being met for $p=1$ and $s_1=0$), we get a morphism $f:X \to G$ assigning to each $x \in X$ the quotient $F_x \otimes k(x)$ and such that $F \simeq f^* \QQ$. The Pl\"ucker embedding defined by $\q$ gives a closed embedding into the appropriate projective space $\iota: G \to {\Bbb P}$. We obtain a closed embedding $\hat f := \iota \circ f: X \to {\Bbb P}$. It is clear that: \noindent - $V$ separates arbitrary $2$ points of $X$ iff $f$ is bijective birational onto its image; \noindent - If $V$ separates arbitrary pairs of points of $X$ and generates jets of order $1$ at an arbitrary point of $X$, then $f$ is a closed embedding. \smallskip Given $n$, $p$ and $\{ s_1, \ldots , s_p \}$ as above let us define the following integers: $$ m_1 (n,p) : = \frac{1}{2}(n^2 +2pn -n +2 ), $$ $$ m_2(n,p;s_1, \ldots , s_p)= 2n \sum_{i=1}^p B(3n + 2s_i -3, n) + 2pn +1, $$ where $B(a,b)$ denotes the usual binomial coefficient, $$ m_3(n,p;s_1, \ldots , s_p) = (pn + \sum_{i=1}^p s_i) \, m_1(n,1) $$ and $$ m_4 (n)= (n+1) \, m_1(n,1). $$ \begin{pr} \label{heart} Let $X$ be a projective manifold of dimension $n$ and $L$ be an ample line bundle on $X$. Fix a K\"ahler form $\omega$ on $X$. \medskip \noindent {\rm (\ref{heart}.1) (Cf. \ci{an-siu} and \ci{tsu}.)} Let $p$ be a positive integer. Assume that $m \geq m_1(n,p)$. \noindent Then for any set of $p$ distinct points $\{x_1, \ldots x_p\}$ of $X$, there exists a nonempty subset $J_0\subset \{1,\ldots , p\}$ with the following property: \noindent there exist $\e>0$, a s.h.m. $h$ for $mL$ with $\T{h}{mL} \geq^{\mu}_1 \epsilon \omega \otimes {\rm Id}_{L_h}$ and with the property that the multiplier ideal $\id (h)$ of $h$ is such that the closed subscheme given by $\id (h)$ has the points $x_{i}$ as isolated points $\forall i \in J_0$ and contains all the points $\{ x_{i}\}$. \medskip \noindent {\rm (\ref{heart}.2) (Cf. \ci{siu94b}; see also \ci{dem96}.)} Fix a positive integer $p$ and a sequence of non-negative integers $\{s_1, \ldots , s_p\}$. Assume that $ m\geq m_2(n,p; s_1, \ldots , s_p). $ \noindent Then for any set of $p$ distinct points $\{x_1, \ldots x_p\}$ of $X$ there exist $\e>0$, a s.h.m. $h$ for $K_X+mL$ with $\T{h}{K_X + mL} \geq^{\mu}_1 \epsilon \omega \otimes {\rm Id}_{L_h}$ and with the property that the multiplier ideal $\id (h)$ satisfies $\id(h)_{x_{i }} \subseteq {\frak m}_{x_i}^{s_i+1}$, for every $1\leq i \leq p$, and is such that the closed subscheme given by $\id (h)$ has all the points $x_{i}$ as isolated points. \end{pr} The easy lemma that follows is probably well-known and makes precise a well-understood principle: {\em it is easy to go from global generation to the generation of higher jets}. Though the presence of the nef line bundle $M$ is redundant in the statement, we use it because of the application of this lemma to the case of higher rank. \begin{lm} \label{freetojet} {\rm (From freeness to the generation of jets)} Let $X$, $n$, $p$ and $\{ s_1, \ldots , s_p \}$ be as above, $F$, $A$ and $M$ be line bundles on $X$ such that $F$ is ample and generated by its global sections, $A$ is ample and $M$ is nef. Then the global sections of $K_X + (pn + \sum_{i=1}^p s_i)F +A + M$ generate simultaneous jets of order $s_1, \ldots, s_p$ at arbitrary distinct points $x_1, \ldots , x_p$ of $X$. \noindent Moreover, $ K_X + (n + 1)F +A + M$ is very ample. \end{lm} \noindent {\em Proof.} Fix $\omega$ a hermitian metric on $X$ and $g$ a hermitian metric on $A$ with positive curvature $\T{h}{A} \geq \frac{3}{2}\e \omega$ for some $\e>0$. \noindent Since $F$ is ample and the linear system $|F|$ is free of base-points, for every index $i$ there are $n$ sections $\{\s_{ij}\}_{j=1}^n$ of $F$ such that their common zero locus is zero-dimensional at $x_i$. \noindent Define a s.h.m. $h$ on $(pn+\sum_i{s_i}) F $ by first defining metrics $h_i$ on $(n+ s_i)F$: $$ h^{-1}_i:= \left[ \sum_{j=1}^n{|\s_{ij}^2|} \right]^{n+s_i} $$ and then by multiplying them together $$ h: = \prod_{i=1}^p h_i. $$ \noindent Since $M$ is nef, one can choose a hermitian metric $l$ on it such that $\T{l}{M} \geq - \frac{1}{2} \e \omega$. \noindent Define a metric $H$ on $(pn + \sum_{i=1}^p s_i)F +A + M$ by setting $$ H:=h\otimes g \otimes l. $$ We have that $\T{h_i}{(n+s_i)F}\geq 0$, $\forall i$ so that $\T{h}{(np + \sum s_i)F} \geq 0$. It follows that $\Theta_H \geq \e \omega$. \noindent Since $g$ and $l$ are continuous, $\id(H)=\id (h)$. \noindent By virtue of \ci{dem94}, Lemma 5.6.b, we have that $\id (H)_{x_i}=\id (h)_{x_i} \subseteq {\frak m}_{x_i}^{s_i +1}$, $\forall i$ and that the scheme associated with $\id (H)$ is zero-dimensional at all the points $x_i$. We conclude by Proposition \ref{cucu}. \noindent The second part of the statement is \ci{an-siu}, Lemma 11.1 (the proof of which contains minor inaccuracies but it is correct). \blacksquare \subsection{Effective results on vector bundles} \label{vecbl} We now see how to ``transplant" the metrics of Proposition \ref{heart} to vector bundles and how to use the results of \S\ref{vt} to prove effective results for the vector bundles of the form $\frak P$ as in the Introduction. \medskip Let us remark that the lower bounds on $m$ given in the various statements of the theorem that follows are only indicative. Any improvement of these bounds in the line bundle case that can be obtained using strictly positive singular metrics would give an analogous improvement in the vector bundle case; see \ci{siu94b}, Proposition 5.1 for example. \medskip Let $n$, $p$, $\{s_1, \ldots , s_p \}$ and the various $m_i$ be as in section \S\ref{demsiu}. Assume that $E$ is a rank $r$ vector bundle on $X$ and let $N:=\min \{n, r \}$. \begin{rmk} \label{more} {\rm See Ex. \ref{listnef} for examples of $N$-nef vector bundles. } \end{rmk} \begin{tm} \label{effres} Let $X$ be a projective manifold of dimension $n$, $E$ be $N$-nef, $A$ and $L$ be ample line bundles on $X$. \medskip \noindent { \rm (\ref{effres}.1)} If $m\geq m_1(n,p)$, then the global sections of $K_X\otimes E \otimes (mL)$ separate arbitrary $p$ distinct points of $X$; \medskip \noindent { \rm (\ref{effres}.$1'$)} if $m\geq \frac{1}{2}(n^2 +n +2)$, then $K_X\otimes E \otimes (mL)$ is generated by its global sections; \medskip \noindent { \rm (\ref{effres}.2)} if $m\geq m_2(n,p; s_1, \ldots, s_p)$, then the global sections of $2K_X \otimes E \otimes (mL)$ generate simultaneous jets of order $s_1,\ldots, s_p \in {\Bbb N}$ at arbitrary $p$ distinct points of $X$; \medskip \noindent { \rm (\ref{effres}.$2'$)} if $m\geq m_2(n,1;1)$, then the global sections of $2K_X \otimes E \otimes (mL)$ separate arbitrary pair of points of $X$ and generate jets of order $1$ at an arbitrary point of $X$. \medskip \noindent { \rm (\ref{effres}.3)} if $m\geq m_3(n,p; s_1, \ldots , s_p)$, then the global sections of $ (pn+\sum s_i +1)K_X \otimes E \otimes (mL) \otimes A $ generate simultaneous jets of order $s_1,\ldots, s_p \in {\Bbb N}$ at arbitrary $p$ distinct points of $X$; \medskip \noindent { \rm (\ref{effres}.4)} if $m\geq m_4(n)$, then the global sections of $(n+2)K_X \otimes E \otimes (mL) \otimes A$ separate arbitrary pair of points of $X$ and generate jets of order $1$ at an arbitrary point of $X$; \medskip \noindent { \rm (\ref{effres}.5)} the global sections of $E\otimes (mL)$ separate arbitrary pair of points of $X$ and generate jets of order $1$ at an arbitrary point of $X$ as soon as $$ m\geq C_n (L^n)^{3^{(n-2)}} ( n+2 + \frac{L^{n-1} \cdot K_X}{L^n})^{3^{(n-2)}( \frac{n}{2} + \frac{3}{4})+ \frac{1}{4}}, $$ where $C_n=(2n)^{\frac{3^{(n-1)} - 1}{2}}(n^3-n^2 -n -1)^{ 3^{(n-2)}( \frac{n}{2} + \frac{3}{4})+ \frac{1}{4}}$. \end{tm} \begin{rmk} \label{geoint} {\rm Let us give a geometric interpretation to, say, (\ref{effres}.$2'$). We employ the notation of \S\ref{demsiu}. Let $(X,E,L,m)$ be as in (\ref{effres}.$2'$). Let $E':= E \otimes (K_X + mL)$ and $L'=(r+1)(2K_X+ mL) + \det E$; note that $h^0:=h^0(X,E')=\chi (X,E')$ and that $L'$ is very ample. Then there is a closed embedding $$ \phi := f \times g : X \longrightarrow G(r,h^0) \times \pn{n} $$ such that $E \simeq f^*(\QQ \otimes \q) \otimes g^*\odixl{\pn{n}}{-1}$, $\deg \hat{f}(X)= (\det E')^n$ and $g$ is finite surjective with $\deg g =L'^n$. \noindent Let $\{X_i,E_i,L_i\}_{i\in I}$ be a set of triplets as above. If we can bound from above $h^0_i$, $\deg \hat{f_i}(X_i)$ and ${L'_i}^n$, then we can find embeddings $\phi_i : X_i \to G \times \pn{n}$ with $G=G(r,\max_I (h^0_i))$ such that the relevant invariants are bounded from above. This applies, for example, to the set of flat vector bundles of fixed rank over a (family of) projective manifold(s), to the set of all nef vector bundles of fixed rank over curves of fixed genus, to the set of projective surfaces with nef tangent bundles, etc. By virtue of Remark \ref{algnef}, a similar remark holds, more generally, for nef vector bundles. } \end{rmk} \medskip \noindent {\em Proof of Theorem \ref{effres}.} Note that $(5.2.2.1')$ and $(5.2.2.2')$ are special cases of $(5.2.2.1)$ and $(5.2.2.2)$, respectively. We shall prove $(5.2.2.1)$ and $(5.2.2.2)$ in detail to illustrate the method. The remaining three assertions are left to the reader and can be proved using the same method with the aid of Lemma \ref{freetojet} for the second and third to last, and with the guideline of \ci{dem96}, 4.7 for the last one. \medskip \noindent Proof of (\ref{effres}.1). We follow closely \ci{an-siu}. The proof is by induction on $p$. Let $p=1$. Let $x\in X$ be arbitrary. By (\ref{heart}.1) we have a strictly positive s.h.m. $h$ on $mL$ such that $x$ is an isolated point of the scheme associated with $\id (h)$. By virtue of Theorem \ref{myvan}, $H^1(X, K_X \otimes E \otimes (mL) \otimes \id (h))=0$ and the following surjections imply the case $p=1$: $$ H^0(X, K_X \otimes E \otimes (mL)) \surj H^0(X, K_X \otimes E \otimes (mL) \otimes \odix{X}/\id (h)) $$ $$ \surj H^0(X, K_X \otimes E \otimes (mL) \otimes \odix{X}/ {\frak m}_x). $$ Let us assume that (\ref{effres}.1) is true for all integers $\rho \leq p-1$ and prove the case $\rho=p$. Let $h$ be as in (\ref{heart}.1) and $\id (h)$ be its multiplier ideal. By virtue of Theorem \ref{myvan}, we have that $H^1(K_X\otimes E \otimes (mL) \otimes \id (h))=0$. Let $\cal J$ be the ideal sheaf on $X$ which agrees with $\id (h)$ on $X\setminus J_0$ and which agrees with $\odix{X}$ on $J_0$. Relabel the points so that $J_0=\{ 1, \ldots , l\}$. By tensoring the exact sequence $$ 0 \to \id (h) \to {\cal J} \to {\cal J}/\id(h) \to 0 $$ with $K_X \otimes E \otimes mL$ we get the surjection: $$ H^0(X, K_X\otimes E \otimes (mL) \otimes {\cal J}) \surj \bigoplus_{i=1}^{l} {\cal O}( K_X\otimes E \otimes (mL))_{x_i} \otimes \odix{X,x_{i}}/{\frak m}_{x_{i}} $$ which implies that we can choose sections ${a_{1,j}} \in H^0(X, K_X \otimes E \otimes (mL))$ vanishing at $x_2, \ldots , x_p$, but generating the stalk $ (K_X \otimes E \otimes (mL))_{x_1}$. We now apply the induction hypothesis to the set of $p-1$ points $\{x_2, \ldots , x_p\}$. \ By repeating the above procedure, and keeping in mind that at each stage we may have to relabel the points, we obtain sections $\{a_{i,j_{i}}\}\in H^0(X, K_X \otimes E \otimes (mL))$, $\forall \, 1\leq i \leq p$ vanishing at $\{x_{i +1}, \ldots , x_p\}$ but generating the stalk $ (K_X \otimes E \otimes (mL))_{x_{i}}$. Given any point $x_{i}$, with $1\leq i \leq r$, and any vector $w \in (K_X \otimes E \otimes (mL) )_{x_{i}}\otimes \odix{X, x_{i}}/ m_{x_{i}}$ it is now easy to find a linear combination of the sections $a_{i,j_i}$ which is $w$ at $x_{i}$ and zero at all the other $p-1$ points. This proves (\ref{effres}.1). \medskip \noindent Proof of (\ref{effres}.2). We fix the integers $p,$ $s_1, \ldots, s_p$ and $p$ arbitrary distinct points on $X$. We take a singular metric $h$ on $K_X +mL$ with $m\geq m_1$ for which the associated multiplier ideal $\id (h)$ has the properties ensured by (\ref{heart}.2). Theorem \ref{myvan} gives us the vanishing of $H^1(K_X \otimes K_X \otimes E \otimes (mL) \otimes \id(h))$ which, in turn, gives the wanted surjection in view of the obvious surjections $$ \odix{X,x_{i}}/\id (h)_{x_{i}} \to \odix{X,x_{i}}/{\frak m}_{x_{i}}^{s_{i} +1}, \quad \forall \, \, 1 \leq i \leq p. $$ \blacksquare \begin{rmk} {\rm Both statements in {\rm Proposition \ref{heart}} have counterparts entailing not powers $mL$ of an ample line bundle $L$, but directly an ample line bundle $\frak{L}$ which has ``intersection theory" large enough. See {\rm \ci{an-siu}}, {\rm Theorem 0.3}, {\rm \ci{dem96} Theorem 2.4.b} and {\rm \ci{siu94b}}. \noindent As a consequence one has statements similar to the ones of {\rm Theorem \ref{effres}} with $mL$ substituted by an ample line bundle $\frak{L}$ with intersection theory large enough; we omit the details. } \end{rmk} \begin{rmk} \label{algnef} {\rm Let $X$, $n$, $E$, $L$ be as in this section except that $E$ is only assumed to be nef. Using algebraic techniques we can see that $K_X \otimes E \otimes \det \, E \otimes L^{m}$ is globally generated for $m\geq \frac{1}{2}(n^2 +n+2) $. Statements involving higher jets can be proved as well. Details will appear in \ci{deeff}. } \end{rmk} \begin{??} {\rm Let $X$, $E$ and $L$ be as above. Is the vector bundle $K_X \otimes E \otimes L^{\otimes m}$ generated by global sections for every $m\geq \frac{1}{2}(n^2 +n+2)$? } \end{??}

9708

alg-geom/9708001

en

https://arxiv.org/abs/alg-geom/9708001

alg-geom/9708001

Rahul Pandharipande

T. Graber and R. Pandharipande

Localization of virtual classes

29 pages, LaTeX2e, General revision including error corrections

We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.

alg-geom

\section{\bf{Introduction}} We prove a localization formula for the virtual fundamental class in the general context of $\mathbb{C}^*$-equivariant perfect obstruction theories. Let $X$ be an algebraic scheme with a $\mathbb{C}^*$-action and a $\mathbb{C}^*$-equivariant perfect obstruction theory. The virtual fundamental class $[X]^{\it{vir}}$ in the expected equivariant Chow group $A_*^{\mathbb{C}^*}(X)$ may be constructed by the methods of Li-Tian [LT] and Behrend-Fantechi [B], [BF]. Each connected component $X_i$ of the fixed point scheme carries an associated $\mathbb{C}^*$-fixed perfect obstruction theory. A virtual fundamental class in $A_*(X_i)$ is thus determined. The virtual normal bundle to $X_i$ is obtained from the moving part of the virtual tangent space determined by the obstruction theory. The localization formula is then: \begin{equation} \label{exloc} [X]^{\it{vir}} = \iota_* \sum \frac{[X_i]^{\it{vir}}}{e(N^{\it{vir}}_i)} \end{equation} in $A_*^{\mathbb{C}^*}(X) \otimes \mathbb{Q}[t,\frac{1}{t}]$ where $t$ is the generator of the equivariant ring of $\mathbb{C}^*$. This localization formula is the main result of the paper. The proof requires an additional hypothesis on $X$: the existence of a $\mathbb{C}^*$-equivariant embedding in a nonsingular variety $Y$. In case $X$ is nonsingular with the trivial perfect obstruction theory, equation (\ref{exloc}) reduces immediately to the standard localization formula [Bo], [AB]. Originally, this localization was proven in equivariant cohomology. Algebraic localization in equivariant Chow theory has recently been established in [EG2]. The point of view of our paper is entirely algebraic. The definitions and constructions related to the virtual localization formula (\ref{exloc}) are discussed in Section \ref{locfor}. The simplest example of a $\mathbb{C}^*$-equivariant perfect obstruction theory is given by the following situation. Let $Y$ be a nonsingular algebraic variety with a $\mathbb{C}^*$-action. Let $V$ be a $\mathbb{C}^*$-equivariant bundle on $Y$. Let $v\in H^0(Y, V)^{\mathbb{C}^*}$ be a $\mathbb{C}^*$-fixed section. Let $X$ be the scheme-theoretic zero locus of $v$. $X$ is naturally endowed with an equivariant perfect obstruction theory which yields the refined Euler class (top Chern class) of $V$ as the virtual fundamental class. The localization formula in this basic setting is proven in Section \ref{localcase}. The method is to deduce (\ref{exloc}) for $X$ from the known localization formula for the nonsingular variety $Y$. The proof of (\ref{exloc}) for general $\mathbb{C}^*$-equivariant perfect obstruction theories on an algebraic scheme $X$ proceeds in a similar manner. Again, formula (\ref{exloc}) is deduced from the ambient localization formula for $Y$. The argument here is more subtle: explicit manipulation of cones and a rational equivalence due to Vistoli [V] are necessary. This proof is given in Section \ref{gencase}. There are two immediate applications of the virtual localization formula. First, a local complete intersection scheme is endowed with a canonical perfect obstruction theory obtained from the cotangent complex. A localization formula is thus obtained for these singular schemes (at least when equivariant embeddings in nonsingular varieties exist). Second, the proper Deligne-Mumford moduli stack $\overline{M}_{g,n}(V, \beta)$ of stable maps to a nonsingular projective variety $V$ is equipped with a canonical perfect obstruction theory. If $V$ has a $\mathbb{C}^*$-action, then a natural $\mathbb{C}^*$-action on $\overline{M}_{g,n}(V, \beta)$ is defined by translation of the map. An equivariant perfect obstruction theory on $\overline{M}_{g,n}(V, \beta)$ can be obtained. Moreover, $\overline{M}_{g,n}(V, \beta)$ admits an equivariant embedding in a nonsingular Deligne-Mumford stack. As a result, the virtual localization formula holds for $\overline{M}_{g,n}(V, \beta)$. In the last two sections of the paper, consequences of the localization formula in Gromov-Witten theory are explored. In Section \ref{projj}, an explicit graph summation formula for the Gromov-Witten invariants (for all genera) of $\mathbf P^r$ is presented via localization on the moduli space of maps $\overline{M}_{g,n}(\mathbf P^r,d)$. The invariants are expressed as a sum over graphs corresponding to the fixed point loci. For each graph, the summand is a product over vertex terms. The vertex terms are integrals over associated spaces $\overline{M}_{g',n'}$ of the Chern classes of the cotangent line bundles and the Hodge bundle. All these integrals may be calculated from Witten's conjectures (Kontsevich's theorem) by a method due to Faber [Fa]. Similar graph sum formulas exist for Gromov-Witten invariants (and their descendents) of all the compact algebraic homogeneous spaces $\mathbf{G}/\mathbf{P}$ via the action of the maximal torus $\mathbf{T}\subset \mathbf{G}$. The positive degree Gromov-Witten invariants of $\mathbf P^2$ coincide with enumerative geometry: they count the number $N^{g}_d$ of genus $g$, degree $d$, nodal plane curves passing through $3d+g-1$ points in $\mathbf P^2$. Localization presents a solution of this enumerative geometry problem via integrals of tautological classes over the moduli space of pointed curves. The numbers $N^{g}_d$ have been computed via more classical degeneration methods in [R1], [CH]. The character of the solutions in [R1], [CH] is markedly different: it is by recursion over wider classes of enumerative questions. In Section \ref{ellip}, localization is applied to a question suggested to us by S. Katz: the calculation of excess integrals on the moduli spaces $\overline{M}_{g,0}(\mathbf P^1,d)$ that arise in the study of Calabi-Yau 3-folds. Under suitable conditions, the integral \begin{equation} \label{exxx} \int _{[ \overline{M}_{g,0}(\mathbf P^1,d)]^{\it{vir}}} c_{\rm{top}} (R^1 \pi_* \mu^* N) \end{equation} is the contribution to the genus $g$ Gromov-Witten invariant of a Calabi-Yau 3-fold of multiple covers of a fixed rational curve (with normal bundle $N={\mathcal{O}}(-1)\oplus {\mathcal{O}}(-1)$). In [M], the integral (\ref{exxx}) is explicitly evaluated to be $1/d^3$ in the genus $g=0$ case via localization on the nonsingular stack $\overline{M}_{0,0}(\mathbf P^1,d)$. A trick of setting one of the $\mathbb{C}^*$-weights on $\mathbf P^1$ to be 0 is used. We evaluate the excess integral in the genus $g=1$ case in Section \ref{ellip} via virtual localization on $\overline{M}_{1,0}(\mathbf P^1,d)$. Manin's trick [M] and formulas for cotangent line integrals on $\overline{M}_{1,n}$ are used to handle the graph sum. The answer obtained, $1/12d$, agrees with the physics result of [BCOV]. The higher genus integrals may be explicitly evaluated in any given case by virtual localization and the algorithm implemented by Faber [Fa] to calculate the vertex integrals. The conjecture obtained from these calculations is: for $g\geq 2$, \begin{equation} \label{conjj} \int _{[ \overline{M}_{g,0}(\mathbf P^1,d)]^{\it{vir}}} c_{\rm{top}} (R^1 \pi_* \mu^* N) = \frac{|B_{2g}| \cdot d^{2g-3}}{2g\cdot (2g-2)!} = |\chi(M_g)| \frac{d^{2g-3}}{(2g-3)!} \end{equation} where $B_{2g}$ is the $2g^{th}$ Bernoulli number and $\chi(M_g)= B_{2g}/2g(2g-2)$ is the orbifold Euler characteristic of $M_g$. This conjecture was made jointly with C. Faber. We have not yet been able to evaluate the graph sums uniformly to establish (\ref{conjj}). The localization formula and graph sum formulas were first introduced in the context of stable maps by Kontsevich in [K] following related work of Ellingsrud and Str\o mme [ES]. Kontsevich studied the convex genus 0 case where the moduli spaces are nonsingular Deligne-Mumford stacks. Many ideas about the virtual fundamental class and localization described here are implicit in [K]. In particular, the higher genus formulas of Section \ref{projj} are identical to the genus 0 formulas of [K] except for the new Hodge bundle terms. However, the higher genus map spaces are in general nonreduced, reducible, and singular, so the virtual localization formula (\ref{exloc}) is essential. Givental has stated a localization axiom for genus 0 Gromov-Witten invariants of toric varieties in [G] which follows from (\ref{exloc}). Localization formulas are used in [G] to prove predictions of mirror symmetry in the case of Calabi-Yau complete intersections in toric varieties. The authors thank P. Aluffi, K. Behrend, D. Edidin, W. Fulton, E. Getzler, L. G\"ottsche, S. Katz, A. Kresch, J. Li, B. Seibert, M. Thaddeus, and A. Vistoli for many valuable conversations. A special thanks is due to C. Faber for his computations of the vertex integrals in (\ref{conjj}) and to B. Fantechi for her tireless explanations of obstruction theories and virtual classes. The first author was supported by an NSF graduate fellowship. The second author was partially supported by an NSF post-doctoral fellowship. The authors also thank the Mittag-Leffler Institute for support. \section{\bf{The virtual localization formula}} \label{locfor} Let $X$ be an algebraic scheme over $\mathbb{C}$. A perfect obstruction theory consists of the following data: \begin{enumerate} \item[(i)] A two term complex of vector bundles $E^\bullet = [E^{-1} \rightarrow E^0]$ on X. \item[(ii)] A morphism $\phi$ in the derived category (of quasi-coherent sheaf complexes bounded from above) from $E^\bullet$ to the cotangent complex $L^\bullet X$ of $X$ satisfying two properties. \begin{enumerate} \item[(a)] $\phi$ induces an isomorphism in cohomology in degree 0. \item[(b)] $\phi$ induces a surjection in cohomology in degree -1. \end{enumerate} \end{enumerate} The constructions of [LT], [BF] give rise to a virtual fundamental class, $[X]^{\it{vir}}$ in $A_d(X)$ where $d= \mbox{rk}(E^0)-\mbox{rk}(E^{-1})$ is the expected dimension. Let $E_\bullet=[ E_0 \rightarrow E_1]$ denote the dual complex of $E^\bullet$. If $X$ admits a global closed embedding in a nonsingular scheme (or Deligne-Mumford stack) $Y$, one can give a relatively straightforward construction of the virtual class as follows. In this situation, the two term cut-off of the cotangent complex can be taken to be: $$L^\bullet X = [I/I^2 \rightarrow \Omega_Y]$$ where $I$ is the ideal sheaf of $X$ in $Y$. Since only this cut-off will be used, the cotangent complex will be identified with its cut-off throughout this section. We assume for simplicity that \begin{equation} \label{assum} \phi: E^\bullet \rightarrow [I/I^2 \rightarrow \Omega_Y] \end{equation} is an actual map of complexes. This hypothesis is not required for the constructions of [LT], [BF]. However, if $X$ has enough locally frees, such a representative $(E^\bullet, \phi)$ may always be chosen in the derived category. The mapping cone associated to the morphism $\phi$ of complexes yields an exact sequence of sheaves: \begin{equation} \label{mppp} E^{-1} \rightarrow E^0 \oplus I/I^2 \stackrel{\gamma}{\rightarrow} \Omega_Y \rightarrow 0. \end{equation} In fact, $\phi$ satisfies (a) and (b) if and only if (\ref{mppp}) is exact. We consider the associated exact sequence of abelian cones \begin{equation} \label{fff} 0 \rightarrow TY \rightarrow C(I/I^2) \times_X E_0 \rightarrow C(Q)\rightarrow 0 \end{equation} where $C(Q)$ is the cone associated to the kernel $Q$ of $\gamma$. As $Q$ is a quotient of $E^{-1}$, $C(Q)$ embeds in $E_1$. The normal cone of $X$ in $Y$, $C_{X/Y}$, is naturally a closed subscheme of $C(I/I^2)$. If we define $D=C_{X/Y} \times_X E_0$, then $D$ is a $TY$-cone (see [BF]), and the quotient of $D$ by $TY$ is a subcone of $C(Q)$ which we will denote by $D^{\it{vir}}$. The virtual fundamental class of $X$ associated to this obstruction theory is then the refined intersection of $D^{\it{vir}}$ with the zero section of the vector bundle $E_1$. Suppose $X\subset Y$ is equipped with an equivariant $\mathbf{G}$-action together with a lifting to the complex $E^\bullet$ such that $\phi$ is a morphism in the derived category of $\mathbf{G}$-equivariant sheaves (with respect to the natural $\mathbf{G}$-action on $L^\bullet X$). The above construction then yields an equivariant virtual fundamental class in the equivariant Chow group $A^\mathbf{G}_d(X)$ since the cones used are invariant. In fact, to define the equivariant virtual class, global equivariant embeddings are not necessary. We now assume the group $\mathbf{G}$ is the torus $\mathbb{C}^*$. We expect to be able to reduce integrals over $[X]^{\it{vir}}$ to integrals over the fixed point set. Let $X^f$ be the maximal $\mathbb{C}^*$-fixed closed subscheme of $X$. $X^f$ is the natural scheme theoretic fixed point locus. If $X={\rm Spec}(A)$, then the ideal of $X^f$ is generated by the $\mathbb{C}^*$-eigenfunctions with nontrivial characters. For nonsingular $Y$, $Y^f$ is the nonsingular set theoretic fixed point locus [I]. For $X\subset Y$, the relation $X^f= X\cap Y^f$ holds. We let $Y_f= \bigcup Y_i$ be the decomposition into irreducible components. Let $X_i= X\cap Y_i$. $X_i$ is possibly reducible. Let $S$ be a coherent sheaf on a fixed component $X_i$ with a $\mathbb{C}^*$-action. $S$ decomposes as direct sum, $$S = \bigoplus_{k\in \mathbb{Z}} S^{k},$$ of $\mathbb{C}^*$-eigensheaves of ${\mathcal{O}}_{X_i}$-modules. If $S$ is locally free, each summand is also locally free. We denote the fixed subsheaf $S^{0}$ by $S^f$ and the moving subsheaf $\oplus_{k\neq 0} S^k$ by $S^{m}$. There is a natural isomorphism $\Omega_Y |_{Y_i}^f= \Omega_{Y_i}$ [I]. It is easy to then deduce: $$\Omega_X |_{X_i}^f = \Omega_{X_i}$$ from the equality $X_i = X \cap Y_i$ or the universal property of K\"ahler differentials. Let $E^{\bullet}_i$ denote the restriction of $E^\bullet$ to $X_i$. Let $E^{\bullet,f}_i$ denote the fixed part of the complex $E^{\bullet}_i$. $E^{\bullet,f}_i$ is a two term complex of bundles. There exists a canonical map, $$\psi_i: E^{\bullet,f}_i \rightarrow L^\bullet X_i,$$ determined by the following construction. Let $\phi_i: E^\bullet_i \rightarrow L^\bullet X |_{X_i}$ be the pull-back of $\phi$, and let $\phi_i^f: E_i^{\bullet,f} \rightarrow L^\bullet X|_{X_i}^f$ be the associated fixed map. Similarly let $\delta_i: L^\bullet X|_{X_i} \rightarrow L^\bullet X_i$ be the canonical morphism, and let $\delta_i^f$ be the associated fixed map. Then, $\psi_i= \delta_i^f \circ \phi_i^f$. \begin{pr} The map $\psi_i: E^{\bullet,f}_i \rightarrow L^\bullet X_i$ is a canonical perfect obstruction theory on $X_i$. \end{pr} \noindent {\em Proof.} To show $\psi_i$ satisfies properties (a) and (b), it suffices to show both maps $\phi_i^f$ and $\delta_i^f$ satisfy these properties. A map of complexes $\nu: A^\bullet \rightarrow B^\bullet$ satisfies (a) and (b) if and only if the sequence $$ A^{-1} \oplus B^{-2} \rightarrow A^0 \oplus B^{-1} \rightarrow B^0 \rightarrow 0$$ is exact. Since tensor product is right exact, the joint validity of (a) and (b) is preserved under pull-back. As $\phi$ is a perfect obstruction theory, $\phi_i$ satisfies (a) and (b). The fixed map $\phi_i^f$ also satisfies properties (a) and (b) since taking invariants is exact. The cotangent complex of $X_i$ can be represented by the embedding $X_i \subset Y_i$: $$L^\bullet X_i =[ I_{X_i/Y_i}/ I^2_{X_i/Y_i} \rightarrow \Omega_{Y_i} |_{X_i}].$$ The zeroth cohomology of $L^\bullet X|_{X_i}^f$ is $\Omega_{X}|_{X_i}^f = \Omega_{X_i}.$ Thus, $\delta_i^f$ satisfies property (a). Property (b) for $\delta^f_i$ will now be established. The map $\delta^f_i$ is represented by the natural diagram: \begin{equation*} \begin{CD} I_{X/Y}/I^2_{X/Y} | _{X_i}^f @>>> \Omega_{Y} |_{X_i}^f \\ @V{d^{-1}}VV @V{d^0}VV \\ I_{X_i/Y_i}/I^2_{X_i/Y_i} @>>> \Omega_{Y_i}|_{X_i} \end{CD} \end{equation*} Since $X_i= X\cap Y_i$, the map $$I_{X/Y}/I^2_{X/Y}|_{X_i} \rightarrow I_{X_i/Y_i}/I^2_{X_i/Y_i}$$ is surjective. Hence, $d^{-1}$ is surjective. As $d^0$ is an isomorphism, $\delta_i^f$ is surjective on cohomology in degree $-1$. \qed \vspace{+10pt} \noindent The virtual structure on $X_i$ is defined to be the one induced by the perfect obstruction theory $\psi_i: E_i^{\bullet,f} \rightarrow L^\bullet X_i$. We define the virtual normal bundle, $N^{\it{vir}}_i$ to $X_i$ to be the moving part of $E_{\bullet,i}$. Note that $E_{\bullet,i}$ is a complex, and not a single bundle. Also note that in the non-virtual case, when the complex has just one term, this coincides with the usual normal bundle. Define the Euler class (top Chern class) of a two term complex $[B_0 \rightarrow B_1]$ to be the ratio of the Euler classes of the two bundles: $e(B_0)/e(B_1)$. We arrive at the following natural formulation of the virtual Bott residue formula for the Euler class of a bundle $A$ of rank equal to the virtual dimension of $X$: \begin{equation} \label{boott} \int_{[X]^{\it{vir}}} e(A) = \sum \int_{[X_i]^{\it{vir}}} \frac{e(A_i)}{e(N^{\it{vir}}_i)} \end{equation} where the Euler classes on the right hand side are equivariant classes. Since $N^{\it{vir}}_i$ is a complex of bundles with nonzero $\mathbb{C}^*$-weights, the Euler class $e(N^{\it{vir}}_i)$ is invertible in the localized ring $$A^{\mathbb{C}^*}_*(X)_{{t}}= A^{\mathbb{C}^*}_*(X) \otimes_{\mathbb{Q}[t]}\mathbb{Q}[t, \frac{1}{t}].$$ Chow groups will always be taken with $\mathbb{Q}$-coefficients. As in the case of the standard Bott residue formula, equation (\ref{boott}) should be a consequence of a localization formula in equivariant Chow groups. On a nonsingular variety $Y$, the fundamental result which immediately implies the residue formula is: $$[Y]= \iota_* \sum \frac{[Y_i]} {e(N_i)}$$ in $A^{\mathbb{C}^*}_* (Y) _t.$ The obvious generalization to the virtual setting which would just as readily imply the virtual residue formula is: \begin{equation} \label{vbott} [X]^{\it{vir}} = \iota_* \sum \frac{[X_i]^{\it{vir}}} {e(N^{\it{vir}}_i)}. \end{equation} It is worth remarking that in the case of most interest to us, the moduli space of maps to projective space, the right side of (\ref{vbott}) is directly accessible. In this case, three special properties hold. First, the fixed loci $X_i$ for a general $\mathbb{C}^*$-action have been identified by Kontsevich in [K]: they are indexed by graphs and are essentially products of Deligne-Mumford moduli spaces of pointed curves. Second, $[X_i]^{\it{vir}}=[X_i]$. Finally, $e(N^{\it{vir}}_i)$ is expressible in terms of tautological classes on $X_i$ via the deformation theory of curves and maps. Thus (\ref{vbott}) provides a concrete way to calculate virtual integrals on moduli spaces where it seems quite difficult to directly evaluate the virtual fundamental class. \section{\bf{Proof in the basic case}} \label{localcase} As a first motivational step, we prove the virtual localization formula (\ref{exloc}) in the following situation. Let $Y$ be a nonsingular variety equipped with a $\mathbb{C}^*$-action, a $\mathbb{C}^*$-equivariant bundle $V$, and an invariant section $v$ of $V$. The zero scheme $X$ of $v$ carries a natural equivariant perfect obstruction theory. The two term complex of bundles on $X$, $$E^\bullet =[V^\vee \rightarrow \Omega_Y],$$ is obtained from the the section $v$. The required morphism to the cotangent complex $L^\bullet X =[ I/I^2 \rightarrow \Omega_Y]$ is obtained from the natural map $V^\vee \rightarrow I/I^2$ on $X$. The definitions show the virtual fundamental class in this case is just the refined Euler class of $V$. That is, if we consider the graph of the section, and take its refined intersection product with the zero section, we get a Chow homology class supported on the zero locus $X$. The definitions of the virtual fundamental class for general spaces are specifically designed to recover this refined Euler class from the local data of the two term complex, and to generalize this class in cases where such a geometric realization of the deformation complex does not necessarily exist. In this basic situation, we can express all of the virtual objects in the localization formula in terms of familiar data on $Y$. As in Section \ref{locfor}, we denote the components of the fixed locus of $Y$ by $Y_i$. $V$ splits into eigenbundles on $Y_i$. Since $v$ is a $\mathbb{C}^*$-invariant section, it is necessarily a section of the weight 0 bundle $V_i^f$. $Y_i$ is a smooth manifold with a vector bundle and a section which vanishes exactly on $X_i=X\cap Y_i$. The associated $\mathbb{C}^*$-fixed obstruction theory defined in Section \ref{locfor}, $$[(V_i^{f}) ^\vee \rightarrow \Omega_{Y_i}],$$ is exactly the perfect obstruction theory obtained from the pair $V_i^f$ and $v\in H^0(Y_i, V_i^f)$. Note the maps to the cotangent complex must be checked to agree. It follows that the virtual fundamental class of $X_i$ is the same as the refined Euler class of $V_i^f$ on $Y_i$. The virtual normal bundle is by definition the moving part of the complex $[TY \rightarrow V]$. The moving part of $TY$ is just the normal bundle to $Y_i$. Hence $N^{\it{vir}}_i$ is the complex $[N_{Y_i/Y} \rightarrow V_i^m]$. By the definition of Euler class of a complex, we obtain: $$e(N^{\it{vir}}_i)=\frac{e(N_{Y_i/Y})} { e(V_i^m)}.$$ After substituting this expression for $e(N^{\it{vir}}_i)$ into the virtual localization formula (\ref{exloc}), we see the equality we want to prove in $A_*^{\mathbb{C}^*}(X)_t$ is: \begin{equation} \label{goall} e_{\rm{ref}}(V) = \iota_* \sum \frac{e_{\rm{ref}}(V_i^f) \cap e(V_i^m)} { e(N_{Y_i/Y})} \end{equation} where $e_{\rm{ref}}(V)$ is the refined Euler class of $V$ as a Chow homology class on $X$. We know by the localization formula on $Y$: $$[Y] = \iota_* \sum \frac{[Y_i]}{e(N_{Y_i/Y})}. $$ Intersecting both sides with $e_{\rm{ref}}(V)$ yields: $$e_{\rm{ref}}(V) = \iota_* \sum \frac{e_{\rm{ref}}(V)\cap [Y_i]}{ e(N_{Y_i/Y})}.$$ Since taking refined Euler class commutes with pullback, the numerators on the right hand side are just the refined Euler classes of $V_i$. On each component, we have the splitting $V_i=V_i^f \oplus V_i^m$. Since the section lives entirely in $V_i^f$, it follows that $e_{\rm{ref}}(V_i)= e_{\rm{ref}}(V_i^f) \cap e(V_i^m)$. Formula (\ref{goall}) is thus obtained. The proof of (\ref{exloc}) in the basic case is complete. \section{\bf{Proof in the general case}} \label{gencase} In this section, we prove the virtual localization formula for an arbitrary scheme $X$ which admits an equivariant embedding in a nonsingular scheme $Y$. Recall from the construction of the virtual class in Section \ref{locfor}, the two cones $D$ and $D^{\it{vir}}$ satisfy: \begin{equation} \label{seq1} 0 \rightarrow TY \rightarrow D \rightarrow D^{\it{vir}} \rightarrow 0 \end{equation} \begin{equation} \label{seq2} D= C_{X/Y} \times E_0. \end{equation} $D^{\it{vir}}$ is a embedded as a closed subcone of $E_1$. The virtual class is defined by $[X]^{\it{vir}} = s_{E_1}^* [D^{\it{vir}}]$. Alternatively, there is a fiber square: \begin{equation} \label{sqq} \begin{CD} TY @>>> D \\ @VVV @VVV \\ X @>{0_{E_1}}>> E_1 \end{CD} \end{equation} where the bottom map is the zero section. Then, $[X]^{\it{vir}}= s_{TY}^* 0^{!}_{E_1} [D].$ Let $X_i= X\cap Y_i$ be defined as in Section \ref{localcase}. $X_i$ is a union of connected components. $\mathbb{C}^*$-fixed analogues of (\ref{seq1}) and (\ref{seq2}) hold for the embeddings $X_i \subset Y_i$: $$0 \rightarrow TY_i \rightarrow D_i \rightarrow D^{\it{vir}}_i \rightarrow 0,$$ $$D_i= C_{X_i/Y_i} \times E_0^f.$$ $D^{\it{vir}}_i$ is a embedded as a closed subcone of $E^f_1$, and $[X_i]^{\it{vir}}= s_{E_1^f}^*(D^{\it{vir}}_i)$. Since $X_i$ is possibly disconnected, it should be noted that the ranks of the bundles $E^f_{0,i}$ and $E^f_{1,i}$ may vary on the connected components. The Euler classes of these bundles on $X_i$ are taken with respect to their ranks on each component. For notational convenience, the restriction subscript $i$ will be dropped. Similarly, the pull-backs of $TY$ and $TY_i$ to $X_i$ will be denoted by $TY$ and $TY_i$. The virtual localization formula for $X$ will be deduced from localization for $Y$. We start with the equality $$[Y] = \iota_* \sum \frac{[Y_i]}{e(TY^m)}$$ in $A^{\mathbb{C}^*}_*(Y)_t$. The refined intersection product with $[X]^{\it{vir}}$ yields: $$[X]^{\it{vir}} = \iota_* \sum \frac{[X]^{\it{vir}} \cdot [Y_i]}{e(TY^m)}$$ in $A^{\mathbb{C}^*}_*(X)_t$. Comparing this equation with our desired virtual localization formula, we see that it suffices to establish: \begin{equation} \label{point} \frac{[X]^{\it{vir}} \cdot [Y_i]} { e(TY^m)} = \frac{ [X_i]^{\it{vir}} \cap e(E_1^m)} { e(E_0^m)} \end{equation} in $A^{\mathbb{C}^*}(X_i) _t$. The refined intersection of a basic linear equivalence due to Vistoli [V] with the zero section of a bundle will yield equation (\ref{point}). The method follows similar arguments in [BF]. We first review Vistoli's rational equivalence. Consider the following Cartesian diagram: \begin{equation} \label{carty} \begin{CD} \iota^* C_{X/Y} @>>> C_{X/Y} \\ @VVV @ VVV\\ X_i @>>> X \\ @VVV @VVV \\ Y_i @>{\iota}>> Y \end{CD} \end{equation} The cone $C_{X_i/Y_i}$ naturally embeds in $\iota^* C_{X/Y}$. Vistoli [V] has constructed a rational equivalence in $N_{Y_i/Y} \times \iota^* C_{X/Y}$ which implies \begin{equation} \label{dwdw} \iota^![C_{X/Y}]= [C_{X_i/Y_i}] \end{equation} in $A_*(\iota^* C_{X/Y})$ (see [BF]). Applying Vistoli's equivalence to the $\mathbb{C}^*$-homotopy quotients yields equation (\ref{dwdw}) in $A^{\mathbb{C}^*}_*(\iota^* C_{X/Y})$. We will consider the pull-back of this relation to $\iota^*D= \iota^*C_{X/Y} \times E_0$: \begin{equation} \label{vist} \iota^![D] = [D_i \times E_0^m] \end{equation} in $A_*^{\mathbb{C}^*}(\iota^* D)$. Consider the pull-back of the exact sequence of abelian cones (\ref{fff}) to $X_i$: $$0 \rightarrow TY \rightarrow \iota^*C(I/I^2) \times E_0 \rightarrow \iota^* C(Q) \rightarrow 0.$$ There is an inclusion $\iota^* C(Q) \subset E_1$. The natural inclusion $\iota^* D \subset \iota^* C(I/I^2) \times E_0$ is $TY$-invariant. Hence, the quotient cones $$\iota^*D/ TY_i \rightarrow \iota^*D/ TY \subset \iota^* C(Q)$$ exist. We obtain a three level Cartesian diagram: \begin{equation} \label{cartone} \begin{CD} TY @>>> \iota^*D \\ @VVV @VVV \\ TY/TY_i @>>> \iota^*D/TY_i \\ @VVV @VVV \\ X_i @>{0_{E_1}}>> E_1 \end{CD} \end{equation} Note that $TY/TY_i= TY^m$. We now start the derivation of equation (\ref{point}). The first steps are: \begin{eqnarray*} [X]^{\it{vir}} \cdot [Y_i] & = & \iota^! s_{TY}^* 0^!_{E_1} [D] \\ & = & s_{TY}^* 0^!_{E_1} \iota^! [D] \\ & = & s_{TY}^* 0^!_{E_1} [D_i \times E_0^m] \end{eqnarray*} in $A_*^{\mathbb{C}^*}(X_i)$. The first equality is by the definition of $[X]^{\it{vir}}$. The second is obtained from the commutativity of the intersection product. The third follows from equation (\ref{vist}). The $TY_i$-action on $\iota^* D$ leaves the cycle $D_i \times E_0^m$ invariant (since $TY_i$ acts naturally on $D_i$ and trivially on $E_0^m$). By definition, $$D_i/TY_i= D^{\it{vir}}_i.$$ The class $[D_i\times E_0^m]\in A_*^{\mathbb{C}^*}(\iota^* D)$ is thus the pull-back of $[D^{\it{vir}}_i \times E_0^m] \in A_*^{\mathbb{C}^*}(\iota^*D/TY_i)$. Hence, $$s_{TY}^* 0^!_{E_1} [D_i \times E_0^m] = s_{TY^m}^* 0^!_{E_1} [D^{\it{vir}}_i \times E_0^m].$$ The scheme-theoretic intersection $0^{-1}_{E_1} (D^{\it{vir}}_i \times E_0^m)$ certainly lies in $TY^m$. The map $$D^{\it{vir}}_i \times E_0^m \rightarrow E_1$$ is the product of the inclusion $D^{\it{vir}}_i \subset E_1^f$ and the natural map from the obstruction theory $E_0^m \rightarrow E_1^m$. We thus observe $0^{-1}_{E_1}(D^{\it{vir}}_i \times E_0^m)$ also lies in $E_0^m$. We conclude the existence of the following diagram: \begin{equation} \label{ccccc} \begin{CD} 0^{-1}_{E_1}(D^{\it{vir}}_i \times E_0^m) @>>> E_0^m \\ @VVV @VVV \\ TY^m @>>> X_i \\ \end{CD} \end{equation} To proceed, we need a relation among Gysin maps. \begin{lm} \label{tww} Let $B_0$ and $B_1$ be $\mathbb{C}^*$-equivariant bundles on $X_i$. Let $Z$ be a scheme equipped with two equivariant inclusions $j_0$, $j_1$ over $X_i$: \begin{equation} \begin{CD} Z @>>> B_1 \\ @VVV @VVV \\ B_0 @>>> X_i \\ \end{CD} \end{equation} Let $\zeta \in A^{\mathbb{C}^*}_*(Z)$. Then, $$ s^*_{B_0} j_{0*}(\zeta) \cap e(B_1) = s^*_{B_1} j_{1*}(\zeta) \cap e(B_0) \ \ \in A^{\mathbb{C}^*}_* (X_i).$$ \end{lm} \noindent {\em Proof.} Consider the family of inclusions $j_t:Z \hookrightarrow B_0 \times B_1$ defined for $t \in \mathbb{C}$ by: $$j_t= (1-t)\cdot j_0 + t\cdot j_1.$$ The existence of this family implies: $$s_{B_0 \times B_1}^*j_{0*} (\zeta)= s_{B_0 \times B_1}^*j_{1*}(\zeta).$$ This yields the Lemma by the excess intersection formula. \qed \vspace{+10pt} Applying Lemma \ref{tww} to diagram (\ref{ccccc}) and the class $\zeta= 0^!_{E_1}[D^{\it{vir}}_i \times E_0^m]$ yields: \begin{equation} \label{kkkk} [X]^{\it{vir}} \cdot [Y_i] = s_{E_0^m}^* \Big( 0^!_{E_1} [D^{\it{vir}}_i \times E_0^m] \Big) \cdot \frac{e(TY^m)}{e(E_0^m)}. \end{equation} The class $0^!_{E_1} [D^{\it{vir}}_i \times E_0^m]$ is now considered to lie in $A_*^{\mathbb{C}^*}(E_0^m)$. As this class does not depend on the bundle map \begin{equation} \label{aaaa} E_0^m \rightarrow E_1^m, \end{equation} we are free to assume (\ref{aaaa}) is trivial. Then, the equality \begin{equation} \label{kdf} s_{E_0^m}^* \Big( 0^!_{E_1} [D^{\it{vir}}_i \times E_0^m] \Big) = [X_i]^{\it{vir}} \cap e(E_1^m) \end{equation} follows easily from the definition of $[X_i]^{\it{vir}}$ and the excess intersection formula. Equation (\ref{point}) is a consequence of (\ref{kkkk}) and (\ref{kdf}). The proof of the virtual localization formula is complete. \section{\bf{The formula for $\mathbf P^r$}} \label{projj} We can use the virtual localization formula (\ref{exloc}) to derive an expression for the higher genus Gromov-Witten invariants of projective space analogous to the one given for genus 0 invariants in [K]. The additional arguments needed to justify formula (\ref{exloc}) in the category of Deligne-Mumford stacks for the moduli space of maps are given in the Appendices. We first establish our conventions about the torus action on projective space. Let $V=\mathbb{C}^{r+1}$. Let $p_i\in \mathbf P(V)$ be the points determined by the basis vectors. Let $\mathbb{C}^*$ act on $V$ with generic weights $-\lambda_0, \ldots ,-\lambda_r$. Then, the induced action on the tangent space to $\mathbf P(V)$ at $p_i$ has weights $\lambda_i - \lambda_j$ for $j \neq i$. Let $\mathbf{T}$ be the full diagonal torus acting on $\mathbf P^r$. Following [K], we can identify the components of the fixed point locus of the $\mathbf{T}$-action on $\Mgn (\proj^r,d)$ with certain types of marked graphs. Let $f:C \rightarrow \mathbf P^r$ be a $\mathbf{T}$-fixed stable map. The image of $C$ is a $\mathbf{T}$-invariant curve in $\mathbf P(V)$, and the images of all marked points, nodes, contracted components, and ramification points are $\mathbf{T}$-fixed points. The $\mathbf{T}$-fixed points on $\mathbf P^r$ are $p_0 , \ldots, p_r$, and the only invariant curves are the lines joining the points $p_i$. It follows that each non-contracted component of $C$ must map onto one of these lines, and be ramified only over the two fixed points. This forces such a component to be rational, and the map restricted to this component is completely determined by its degree. We are led to identify the components of the fixed locus with marked graphs. To an invariant stable map $f$, we associate a marked graph $\Gamma$ as follows. $\Gamma$ has one edge for each non-contracted component. The edge $e$ is marked with the degree $d_e$ of the map from that component to its image line. $\Gamma$ has one vertex for each connected component of $f^{-1}(\{p_0, \ldots, p_r\})$. Define the labeling map $$i: \text{Vertices} \rightarrow \{0, \ldots, r\}$$ by $f(v)= p_{i(v)}$. The vertices have an additional labeling $g(v)$ by the arithmetic genus of the associated component. (Note the component may be a single point, in which case its genus is 0.) Finally, $\Gamma$ has $n$ numbered legs coming from the $n$ marked points. These legs are attached to the appropriate vertex. An edge is incident to a vertex if the two associated subschemes of $C$ are incident. The set of all invariant stable maps whose associated graph is $\Gamma$ is naturally identified with a finite quotient of a product of moduli spaces of pointed curves. Define: $${\barr M}_\Gamma = \prod_{{\rm vertices}} {\barr M}_{g(v),{\rm val} (v)}.$$ ${\barr M}_{0,1}$ and ${\barr M}_{0,2}$ are interpreted as points in this product. Over the Deligne-Mumford stack ${\barr M}_\Gamma$, there is a canonical family $$\pi: \mathcal{C} \rightarrow {\barr M}_\Gamma$$ of $\mathbb{C}^*$-fixed stable maps to $\mathbf P^r$ yielding a morphism $$\gamma: {\barr M}_\Gamma \rightarrow \Mgn (\proj^r,d).$$ There is a natural automorphism group $\mathbf{A}$ acting $\pi$-equivariantly on $\mathcal{C}$ and ${\barr M}_\Gamma$. $\mathbf{A}$ is filtered by an exact sequence of groups: $$ 1 \rightarrow \prod_{\rm edges} {\mathbb{Z}}/{d_e} \rightarrow \mathbf{A} \rightarrow {\rm Aut}(\Gamma) \rightarrow 1$$ where ${\rm Aut} (\Gamma)$ is the automorphism group of $\Gamma$ (as a marked graph). ${\rm Aut}(\Gamma)$ naturally acts on $\prod_{\rm edges} \mathbb{Z}/ d_e$ and $\mathbf{A}$ is the semidirect product. The induced map: $$\gamma/ \mathbf{A} : {\barr M}_\Gamma / \mathbf{A} \rightarrow \Mgn (\proj^r,d)$$ is a closed immersion of Deligne-Mumford stacks. It should be noted that the subgroup $\prod_{\rm edges} \mathbb{Z}/ d_e$ acts trivially on ${\barr M}_\Gamma$ and that ${\barr M}_\Gamma / \mathbf{A}$ is nonsingular. A component of the $\mathbb{C}^*$-fixed stack of $\Mgn (\proj^r,d)$ is supported on ${\barr M}_\Gamma/ \mathbf{A}$. The fixed stack will be shown to be nonsingular by analysis of the $\mathbb{C}^*$-fixed perfect obstruction theory which yields the Zariski tangent space. This nonsingularity is surprising since the moduli stack $\Mgn (\proj^r,d)$ is singular. A generic $\mathbb{C}^*\subset \mathbf{T}$ will have the same fixed point loci in $\Mgn (\proj^r,d)$. Via this fixed point identification, the virtual localization formula will relate the Gromov-Witten invariants of $\mathbf P^r$ to integrals over moduli spaces of pointed curves. Following [K], we define a flag $F$ of the graph $\Gamma$ to be an incident edge-vertex pair $(e,v)$. Define $i(F)=i(v)$. The edge $e$ is incident to one other vertex $v'$. Define $j(F)=i(v')$. Define: $$\omega_F=\frac{\lambda_{i(F)}-\lambda_{j(F)} }{d_e}.$$ This is the weight of the induced action of $\mathbb{C}^*$ on the tangent space to the rational component $C_e$ of $C$ corresponding to $F$ at its preimage over $p_{i(F)}$. This fact follows from the corresponding calculation on the weight of the action on the tangent space to the image line, with a factor of $\frac{1}{d_e}$ coming from the $d_e$-fold ramification of the map at the fixed point. We describe the obstruction theory of $\Mgn (\proj^r,d)$ restricted to ${\barr M}_\Gamma/ \mathbf{A}$. Define sheaves $\mathcal{T}^1$ and $\mathcal{T}^2$ on ${\barr M}_\Gamma/ \mathbf{A}$ via the cohomology of the restriction of the canonical (dual) perfect obstruction theory $E_\bullet$ on $\Mgn (\proj^r,d)$: \begin{equation} \label{eggg} 0 \rightarrow \mathcal{T}^1 \rightarrow E_{0,\Gamma} \rightarrow E_{1, \Gamma} \rightarrow \mathcal{T}^2 \rightarrow 0. \end{equation} There is a tangent-obstruction exact sequence of sheaves on the substack ${\barr M}_\Gamma/\mathbf{A}$: \begin{equation} \label{tanob} 0 \rightarrow {\rm Ext}^0(\Omega_C(D), {\mathcal{O}}_C) \rightarrow H^0(C,f^*TX) \rightarrow \mathcal{T}^1 \rightarrow \end{equation} $$ \rightarrow {\rm Ext}^1(\Omega_C(D), {\mathcal{O}}_C) \rightarrow H^1(C,f^*TX) \rightarrow \mathcal{T}^2 \rightarrow 0.$$ The marked point divisor on $C$ is denoted by $D$. The 4 terms other than the sheaves $\mathcal{T}^i$ are vector bundles and are labeled by their fibers. This sequence can be viewed as filtering the deformations of the maps by those which preserve the domain curves. It arises via the pull-back to ${\barr M}_\Gamma/\mathbf{A}$ of a distinguished triangle of complexes on $\Mgn (\proj^r,d)$ (see Appendix B). These results may be found in [LT], [R2], [B]. In the remainder of this section, the fixed and moving parts of the 4 bundles in the tangent-obstruction complex are explicitly identified following [K]. It is simpler to carry out the bundle analysis on the prequotient $\overline{M}_\Gamma$ to avoid monodromy in the nodes. In fact, the final integrals over the fixed locus will be evaluated on ${\barr M}_\Gamma$ and corrected by the order of $\mathbf{A}$. It will be seen that there are exactly 3 fixed pieces in the 4 bundles. They occur in the $1^{\rm st}$, $2^{\rm nd}$, and $4^{\rm th}$ terms of the complex. The fixed piece in the $1^{\rm st}$ term maps isomorphically to the fixed piece of the $2^{\rm nd}$. $\mathcal{T}^{1,f}$ is thus isomorphic to the fixed piece in the $4^{\rm th}$ term. The latter is canonically the tangent bundle to ${\barr M}_\Gamma$. Also, $\mathcal{T}^{2,f}=0$. We can conclude that the fixed stack is nonsingular and equal to ${\barr M}_\Gamma/\mathbf{A}$. The two exact sequences (\ref{eggg}) and (\ref{tanob}) imply: $$e(N^{\it{vir}})= \frac{e(B_2^{m}) e(B_4^{m})}{e(B_1^m)e(B_5^m)}$$ where, for example, $B_2^m$ denotes the moving part of the $2^{\rm nd}$ term in (\ref{tanob}). We first calculate the contribution coming from the bundle $${\rm Aut}(C)={\rm Ext}(\Omega_C(D), {\mathcal{O}}_C)$$ parameterizing infinitesimal automorphisms of the pointed domain. For each non-contracted component of $C$, there is a weight zero piece coming from the infinitesimal automorphism of that component fixing the two special points. This term will cancel with a similar term in $H^0(f^*T\mathbf P^r)$. Also, since there is no moving part, $e(B_1^{m})=1$. If it is the case that the special points are not marked or nodes, that is the associated vertex of the graph has genus 0 and valence one, there would be an extra automorphism with nontrivial weight. We will leave this case and the case of a genus 0 valence 2 vertex to the reader. No extra trivial weight pieces arise in these two cases. As in [K], the (integrated) final formulas for the genus 0 vertex contributions will still be correct. Next, we consider the bundle ${\rm Def}(C)={\rm Ext}^1(\Omega_C(D), {\mathcal{O}}_C)$ parameterizing deformations of the pointed domain. A deformation of the contracted components (as marked curves) is a weight zero deformation of the map which yields the tangent space of ${\barr M}_\Gamma/ \mathbf{A}$ as a summand in the weight zero piece of ${\rm Def}(C)$. The other deformations of $C$ come from smoothing nodes of $C$ which join contracted components to non-contracted components. This space splits into a product of spaces corresponding to deformations which smooth each node individually. The one dimensional space associated to each node is identified as a bundle with the tensor product of the tangent spaces of the two components at the node. We see that the tangent space to the non-contracted curve forms a trivial bundle with weight $\omega_F$ while the tangent space to the contracted curve varies but has trivial weight. Let $e_F$ denote the line bundle on ${\barr M}_\Gamma$ whose fiber over a point is the cotangent space to the component associated to $F$ at the corresponding node. Therefore, $$e(B_4^{m})= \prod_{\rm flags}(\omega_F - e_F).$$ To compute the contribution coming from $H^\bullet(f^*T\mathbf P^r)$, we consider the normalization sequence resolving all of the nodes of $C$ which are forced by the graph type $\Gamma$. $$0 \rightarrow {\mathcal{O}}_C \rightarrow \bigoplus_{\rm vertices} {\mathcal{O}}_{C_v} \oplus \bigoplus_{\rm edges} {\mathcal{O}}_{C_e} \rightarrow \bigoplus_{\rm flags} {\mathcal{O}}_{x_F} \rightarrow 0$$ Twisting by $f^*(T\mathbf P^r)$ and taking cohomology yields: $$0 \rightarrow H^0(f^*T\mathbf P^r) \rightarrow \bigoplus_{\rm vertices} H^0(C_v, f^*T\mathbf P^r) \oplus \bigoplus_{\rm edges}H^0(C_e, f^*T\mathbf P^r) \rightarrow$$ $$\rightarrow \bigoplus_{\rm flags} T_{p_{i(F)}}\mathbf P^r \rightarrow H^1(f^*T\mathbf P^r) \rightarrow \bigoplus_{\rm vertices} H^1(C_v, f^*T\mathbf P^r) \rightarrow 0$$ where we have used the fact that there will be no higher cohomology on the non-contracted components since they are rational. Also note that $H^0(C_v, f^*(T\mathbf P^r) = T_{p_{i(v)}}\mathbf P^r$ since $C_v$ is connected and $f$ is constant on it. Thus, we obtain: $$ H^0 - H^1 = \begin{array}{cccc} + & {\displaystyle \bigoplus_{\rm vertices} T_{p_{i(v)}} \mathbf P^r}& +&{\displaystyle \bigoplus_{\rm edges} H^0(C_e, f^*T\mathbf P^r)} \\ - & {\displaystyle \bigoplus_{\rm flags} T_{p_{i(F)}}\mathbf P^r}& -& {\displaystyle \bigoplus_{\rm vertices} H^1(C_v, f^*T\mathbf P^r)} \end{array} $$ As non-contracted components are rigid, we see that $H^0(C_e,f^*T\mathbf P^r)$ is trivial as a bundle, but we need to determine its weights. We do this via the Euler sequence. On $\mathbf P^r$ we have: $$0 \rightarrow {\mathcal{O}} \rightarrow {\mathcal{O}}(1) \otimes V \rightarrow T\mathbf P^r \rightarrow 0. $$ Pulling back to $C_e$ and taking cohomology gives us: $$ 0 \rightarrow \mathbb{C} \rightarrow H^0({\mathcal{O}}(d_e))\otimes V \rightarrow H^0(f^*T\mathbf P^r) \rightarrow 0 $$ Here the weight on $\mathbb{C}$ is trivial, and the weights on $H^0({\mathcal{O}}(d_e))$ are given by $\frac{a}{d_e}\lambda_i + \frac{b}{d_e}\lambda_j$ for $a+b=d_e$. The weights on $V$ are $-\lambda_0, \ldots ,-\lambda_r$. So the weights of the middle term are just the pairwise sums of these, $\frac{a}{d_e}\lambda_i + \frac{b}{d_e}\lambda_j -\lambda_k$. There are exactly 2 zero weight terms here coming from $a=0, k=j$ and $b=0, k=i$. These cancel the zero weight term from the $\mathbb{C}$ on the left, and the zero weight term occurring in ${\rm Aut}(C)$. Breaking up the remaining terms into two groups corresponding to $k=i,j$ and $k \neq i,j$, we obtain the contribution of $H^0(C_e,f^*T\mathbf P^r)$ to the Euler class ratio $e(B_2^{m})/e(B_5^{m})$: $$(-1)^{d_e} \frac{{d_e!}^2 }{ d_e^{2d_e}} (\lambda_i - \lambda_j)^{2d_e} \cdot \prod_{\stackrel{a+b=d_e}{ k\neq i,j}} (\frac{a} {d_e}\lambda_i + \frac{b}{d_e}\lambda_j -\lambda_k).$$ Finally, we evaluate the contribution of $H^1(C_v, f^* T\mathbf P^r)$. This is simply $H^1(C_v, {\mathcal{O}}_{C_v})\otimes T_{p_{i(v)}}\mathbf P^r$. As a bundle, $H^1(C_v, {\mathcal{O}}_{C_v})$ is the dual of the Hodge bundle $E=\pi_* \omega$ on the moduli space ${\barr M}_{g(v), {\rm val}(v)}$. The bundle $H^1(C_v, {\mathcal{O}}_{C_v})\otimes T_{p_{i(v)}}\mathbf P^r$ splits into $r$ copies of $E^\vee$ twisted respectively by the $r$ weights $\lambda_i - \lambda_j$ for $j \neq i$. Taking the equivariant top Chern class of this bundle yields: $$\prod_{j \neq i} c_{(\lambda_i - \lambda_j)^{-1}}(E^\vee)\cdot (\lambda_i - \lambda_j)^{g(v)}$$ where for a bundle $Q$ of rank $q$: $$c_t(Q)= 1+ t c_1(Q) + \ldots t^q c_{q}(Q).$$ We arrive at the following form of the inverse Euler class of the virtual normal bundle to the fixed point locus corresponding to the graph $\Gamma$. \begin{eqnarray*} & &\prod_{\rm flags} \frac{1}{\omega_F - e_F} \prod_{j \neq i(F)} (\lambda_{i(F)} -\lambda_j) \\ \frac{1}{e(N^{\it{vir}})}& =&\prod_{\rm vertices} \prod_{j \neq i(v)} c_{(\lambda_{i(v)} - \lambda_j)^{-1}}(E^\vee) \cdot (\lambda_{i(v)} -\lambda_j)^{g(v)-1} \\ & &\prod_{\rm edges} \frac{(-1)^{d_e} d_e^{2d_e}} { (d_e!)^2 (\lambda_i -\lambda_j)^{2d_e}} \prod_{\stackrel{a+b=d_e} {k\neq i,j}} \frac{1}{ \frac{a}{d_e}\lambda_i + \frac{b}{d_e}\lambda_j -\lambda_k} \end{eqnarray*} In addition, the virtual fundamental class of the fixed locus must be identified. We have already seen $\mathcal{T}^{1,f}$ is the tangent bundle of ${\barr M}_\Gamma$. and $\mathcal{T}^{2,f}=0$. It then follows from (\ref{eggg}) that the $\mathbb{C}^*$-fixed (dual) perfect obstruction theory is equivalent on the fixed stack to the trivial perfect obstruction theory. The virtual fundamental class of the fixed stack is simply the ordinary fundamental class. The above expression for $\frac{1}{e(N^{\it{vir}})}$ can be used in the virtual localization formula to deduce formulas expressing Gromov-Witten invariants of projective space in terms of integrals on moduli spaces of pointed curves. The numerator terms, coming from the cohomology classes of $\mathbf P^r$ are identical in this higher genus case to the terms appearing in [K]. In particular, they contribute only additional weights, and no cohomological terms. Let $[n]=\{1,\ldots,n\}$ be the marking set of an $n$-pointed graph $\Gamma$. Let $i:[n] \rightarrow \{0, \ldots, r\}$ be defined by $f(m)= p_{i(m)}$. The final expression for the Gromov-Witten invariants of $\mathbf P^r$ is: $$ I_{g, d}^{\mathbf P^r}(H^{l_1}, \ldots, H^{l_n})= \sum_{\Gamma} \frac{1}{|\mathbf{A}_\Gamma|} \int_{{\barr M}_\Gamma} \frac{\prod_{[n]} \lambda_{i(m)}^{l_m}} {e(N^{\it{vir}}_\Gamma)}.$$ The sum is over all graphs $\Gamma$ indexing fixed loci of $\Mgn (\proj^r,d)$. To evaluate the integral, one expands the terms of the form $\frac{1}{\omega - e}$ as formal power series, and then integrates all terms of the appropriate degree. Each integral that is encountered will naturally split as a product of integrals over the different moduli spaces of pointed curves. We remark that the integrals over genus 0 spaces are identical to the ones which are dealt with in [K]. In particular, while the formula given above is incorrect for graphs with vertices of genus 0 and valence 1 or 2, the formulas obtained in [K] after integrating over ${\barr M}_{0,n}$ hold for these degenerate cases as well. In higher genera, we know of no closed formulas for the integrals which occur in these calculations. However, C. Faber has constructed an algorithm in [Fa] which determines all such integrals. Thus, this formula, together with Faber's algorithm, gives a method in principle to determine arbitrary Gromov-Witten invariants of projective space. \section{\bf{Multiple cover calculations}} \label{ellip} Let $C\subset X$ be a nonsingular rational curve with balanced normal bundle $N\stackrel{\sim}{=}{\mathcal{O}}(-1) \oplus {\mathcal{O}}(-1)$ in a nonsingular Calabi-Yau 3-fold $X$. Let $[C]\in H_2(X, \mathbb{Z})$ be the homology class of $C$. The space of stable elliptic maps to $X$ representing the curve class $d[C]$ contains a component $Y_d$ consisting of maps which factor through a $d$-fold cover of $C$. $Y_d$ is naturally isomorphic to $\overline{M}_{1,0}(C,d)$, the space of unpointed, genus 1 stable maps. The contribution of $Y_d$ to the elliptic Gromov-Witten invariant $I_{1,d[C]}^X$ has been computed in physics [BCOV]. The answer obtained is $\frac{1}{12d}$ (accounting for the differing treatment of the elliptic involution). Mathematically, the excess contribution of $Y_d$ is expressed as an integral over $\overline{M}_{1,0}(C,d)$. The integral is computed here for all $d$ via localization. Localization reduces the contribution to a graph sum which can be explicitly evaluated by Manin's trick [M] and a formula for intersections of cotangent lines on $\overline{M}_{1,n}$. Let $\pi: U \rightarrow \overline{M}_{1,0}(C,d)$ be the universal family over the moduli space. Let $\mu: U \rightarrow C$ be the universal evaluation map. The expected dimension of $\overline{M}_{1,0}(C,d)$ is $2d$. By the cohomology and base change theorems, $R^1 \pi_* \mu^* N$ is a vector bundle of rank $2d$ on $\overline{M}_{1,0}(\mathbf P^1,d)$. The contribution of $Y_d$ to the elliptic Gromov-Witten invariant of curve class $d[C]$ is: \begin{equation} \label{exx} \int _{[ \overline{M}_{1,0}(C,d)]^{\it vir}} c_{2d} (R^1 \pi_* \mu^* N). \end{equation} Natural lifts of $\mathbb{C}^*$-actions on $C$ to $\overline{M}_{1,0}(C,d)$, $N$, and $R^1 \pi_* \mu^* N$ exist. The localization formula can therefore be applied to compute (\ref{exx}). The answer obtained agrees with the physics calculation. \begin{pr} \label{mule} $$\int _{[ \overline{M}_{1,0}(C,d)]^{\it vir}} c_{2d} (R^1 \pi_* \mu^* N) = \frac{1}{12 d}.$$ \end{pr} Let $V\stackrel{\sim}{=} \mathbb{C}^2$. Let $C= \mathbf P(V)$. Let $\mathbb{C}^*$ act by weights $0$ and $-1$ on $V$. Let $x_0$ and $x_{-1}$ be the respective fixed points in $C$. The $\mathbb{C}^*$-action lifts naturally to the tautological line ${\mathcal{O}}(-1)$ and thus to $N$. Consider the graph sum obtained by the localization formula for the integral (\ref{exx}). The 0 weight leads to a drastic collapse of the sum. This was observed by Manin in [M] for an analogous excess integral over a space of genus 0 maps. In fact, the only graphs which contribute are comb graphs where the backbone is an elliptic curve contracted over $x_{-1}$ and the teeth are rational curves multiple covering $\mathbf P(V)$. The degree $d$ is distributed over the teeth by $\sum_{1}^{k} m_i =d$. The denominator terms in the localization formula are determined by the results of Section \ref{projj}. The numerator is given by the bundle $R^1\pi_* \mu^*N$ which is decomposed on each fixed point locus via the natural normalization sequence. The formula \begin{equation} \label{mast} \sum_{m\vdash d} \frac{(-1)^{d-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int_{\overline{M}_{1,L(m)}} \frac{1+\lambda}{\Pi_{1}^{L(m)} (1-m_ie_i)} \end{equation} is obtained for the degree $d$ contribution. The sum is over all positive partitions: $$m=(m_1, \ldots, m_k), \ \ m_i>0, \ \ \sum_{1}^{k} m_i =d.$$ $L(m)$ denotes the length of $m$. ${\rm Aut}(m)$ is the order of the stabilizer of the symmetric group $S_k$-action on the string $(m_1, \ldots, m_k)$. The class $\lambda$ in the numerator is the first Chern class of the Hodge bundle on $\overline{M}_{1,n}$. As before, $e_i$ is the $i^{th}$ cotangent line bundle on $\overline{M}_{1,n}$. The integral (\ref{mast}) is calculated in two parts to prove Proposition \ref{mule}. \begin{lm} \label{AAA} $$ \sum_{m\vdash d} \frac{(-1)^{d-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int_{\overline{M}_{1,L(m)}} \frac{\lambda}{\Pi_{1}^{L(m)} (1-m_ie_i)}= \frac{d}{24}$$ \end{lm} \begin{lm} \label{BBB} $$ \sum_{m\vdash d} \frac{(-1)^{d-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int_{\overline{M}_{1,L(m)}} \frac{1}{\Pi_{1}^{L(m)} (1-m_ie_i)}= \frac{d}{24}$$ \end{lm} We start with Lemma \ref{AAA}. The first step is to use the boundary expression for $\lambda$ to reduce to an integral over genus 0 pointed moduli spaces. On $\overline{M}_{1,1}$, the equation: \begin{equation} \label{lamb} \lambda = \frac{\Delta_0}{12} \end{equation} holds where $\Delta_0$ is the irreducible boundary divisor. Since $\lambda$ on $\overline{M}_{1,n}$ is a pull-back from a one pointed space, (\ref{lamb}) is valid on $\overline{M}_{1,n}$. Using the standard identification of $\Delta_0$ with the $\mathbb{Z}/2\mathbb{Z}$-quotient of $\overline{M}_{0,n+2}$, the equality: $$\int_{\overline{M}_{1, L(m)}} \frac{\lambda}{\Pi_{1}^{L(m)} (1-m_ie_i)} = \frac{1}{24} \int_{\overline{M}_{0, L(m)+2}} \frac{1}{\Pi_{1}^{L(m)} (1-m_ie_i)}$$ is obtained. Next, using the well-known formula for intersection numbers on the genus 0 spaces, we see: $$\int_{\overline{M}_{0, L(m)+2}} \frac{1}{\Pi_{1}^{L(m)} (1-m_ie_i)} = (\sum_{1}^{L(m)} m_i)^{L(m)-1}= d^{L(m)-1}.$$ After substituting these equalities, the sum of Lemma \ref{AAA} is transformed to: \begin{equation} \label{manin} \frac{(-1)^d}{24d} \sum_{m\vdash d} \frac{(-1)^{-L(m)}}{{\rm Aut}(m)\ \Pi_1^{L(m)}m_i} {d}^{L(m)}. \end{equation} The summation term in (\ref{manin}) was encountered by Manin in [M]. It evaluates explicitly to $(-1)^d$ via a generating function argument (see [M] p.416). The value of (\ref{manin}) is thus $\frac{1}{24d}$. Lemma \ref{AAA} is established. We now prove Lemma \ref{BBB}. A generating function approach is taken. For $d\geq 1$, let $$g_d= \sum_{m\vdash d} \frac{(-1)^{d-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int_{\overline{M}_{1,L(m)}} \frac{1}{\Pi_{1}^{L(m)} (1-m_ie_i)}.$$ Define $\gamma(t)$ by: $$\gamma(t)= \sum _{\alpha \geq 1} (-1)^{\alpha} g_\alpha t^\alpha.$$ An important observation is that $\gamma(t)$ can be rewritten in the following form: \begin{equation} \label{oboe} \gamma(t)= < \exp \ (-\sum_{\alpha\geq 1} \sum_{i\geq 0}\alpha^{i-1} t^\alpha \sigma_i) >_1. \end{equation} Here, Witten's notation, \begin{equation} \label{jenny} <\sigma_{0}^{r_0} \sigma_{1}^{r_1} \cdots \sigma_{k}^{r_k}>_1, \end{equation} is used to denote the integral: $$\int_{\overline{M}_{1,r}} \underbrace {e_{r_0+1} \ldots e_{r_0+r_1}}_{r_1} \cdot \underbrace {e^2_{r_0+r_1+1} \ldots e^2_{r_0+r_1+r_2}}_{r_2} \ldots \underbrace {e^k_{r-r_k+1} \ldots e^k_{r}} _{r_k}$$ where $r=\sum_1^k r_i$. Equality (\ref{oboe}) is a simply a rewriting of terms. The genus 1 integrals (\ref{jenny}) are determined from genus 0 integrals by a beautiful formula in the formal variables $\{z_i\} _{i \geq 0}$: \begin{equation} \label{dike} < \exp \sum_{i\geq 0} z_i \sigma_i >_1 = \frac{1}{24} \log < \sigma_0^3 \exp \sum_{i\geq 0} z_i \sigma_i >_0 . \end{equation} Formula (\ref{dike}) can be found, for example, in [D]. Let $z_i=-\sum_{\alpha\geq 1} \alpha^{i-1}t^\alpha$. Using (\ref{oboe}) and (\ref{dike}), $\gamma(t)$ may be expressed as: \begin{equation} \label{gam} \gamma(t)=\frac{1}{24} \log < \sigma_0^3 \exp (-\sum_{\alpha\geq 1} \sum_{i\geq 0}\alpha^{i-1} t^\alpha \sigma_i) >_0 \end{equation} Equation (\ref{gam}) will be used to determine $\gamma(t)$. First, define another generating function $\psi(t)$ by: $$\psi(t)= 1 + \sum_{\beta} s_\beta t^\beta$$ where the coefficients $s_\beta$ are: \begin{equation} \label{quail} s_\beta= \sum_{m\vdash \beta} \frac{ (-1)^{-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int _{\overline{M}_{0, L(m)+3}} \frac{1} {\Pi_{1}^{L(m)} (1-m_ie_i)}. \end{equation} As before, the equality: $$\psi(t)= < \sigma_0^3 \exp(-\sum_{\alpha\geq 1} \sum_{i\geq 0}\alpha^{i-1} t^\alpha \sigma_i) >_0$$ is a rewriting of terms. However, the expression (\ref{quail}) may be explicitly evaluated by the genus 0 intersection formulas and Manin's summation argument to yield: $$ s_\beta= (-1)^\beta.$$ Hence, $\psi(t)$ is simply $1/(1+t)$, and $$\gamma(t)= -\frac{\log(1+t)}{24}= \frac{1}{24}(-t + \frac{t^2}{2}- \frac{t^3}{3} + \ldots).$$ Thus, $g_d= \frac{1}{24d}$. Lemma \ref{BBB} is proven. Proposition \ref{mule} follows from (\ref{mast}) and the two Lemmas. Localization may be applied to the analogous excess integrals for arbitrary genus $g$. The resulting formula is: \begin{equation} \label{mast2} \int _{[ \overline{M}_{g,0}(\mathbf P^1,d)]^{\it vir}} c_{\rm {top}} (R^1 \pi_* \mu^* N) = \end{equation} \vspace{+15pt} $$ \sum_{m\vdash d} \frac{(-1)^{d-L(m)}}{{\rm Aut}(m) \ \Pi_{1}^{L(m)} m_i} \int_{\overline{M}_{g,L(m)}} \frac{1+c_1(E)+\ldots + c_g(E)}{\Pi_{1}^{L(m)} (1-m_ie_i)}$$ \vspace{+15pt} \noindent where $E$ is the Hodge bundle. For $g\geq 2$, we have conjectured with C. Faber the above integral sum is equal to: \begin{equation*} \frac{|B_{2g}| \cdot d^{2g-3}}{2g\cdot (2g-2)!} = \frac{|\chi(M_g)|\cdot d^{2g-3}}{(2g-3)!}. \end{equation*} This equality has been verified in case $g+d \leq 7$.

9702

alg-geom/9702015

en

https://arxiv.org/abs/alg-geom/9702015

alg-geom/9702015

Rick Miranda

C. Ciliberto (U. of Rome II), R. Miranda (Colorado State U.)

Degenerations of Planar Linear Systems

material is streamlined and some is moved to a forthcoming paper

Fixing $n$ general points $p_i$ in the plane, what is the dimension of the space of plane curves of degree $d$ having multiplicity $m_i$ at $p_i$ for each $i$? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all $n$ and for $m_i$ constant, at most 3. This application, while previously known (see \cite{hirschowitz1}), demonstrates the utility of our approach, which is based on an analysis of the corresponding linear system on a degeneration of the plane itself, leading to a simple recursion for these dimensions. We also obtain results in the ``quasi-homogeneous'' case when all the multiplicities are equal except one; this is the natural family to consider in the recursion.

alg-geom

\section*{Introduction} Fix the projective plane ${\mathbb{P}}^2$ and $n+1$ general points $p_0, p_1, \ldots, p_n$ in it. Let $H$ denote the line class of the plane. Consider the linear system consisting of plane curves of degree $d$ (that is, divisors in $|dH|$) with multiplicity $m_0$ at $p_0$ and multiplicity $m_i$ at $p_i$ for $i \geq 1$. If all $m_i$ for $i \geq 1$ are equal, to $m$ say, we denote this system by $\L = \L(d,m_0,n,m)$ and call the system \emph{quasi-homogeneous}. Define its {\em virtual dimension} \[ \v= \v(d,m_0,n,m) = d(d+3)/2 - m_0(m_0+1)/2 - nm(m+1)/2); \] the linear system of all plane curves of degree $d$ has dimension $d(d+3)/2$ and a point of multiplicity $k$ imposes $k(k+1)/2$ conditions. Of course the actual dimension of the linear system cannot be less than $-1$ (projectively, dimension $-1$ means an empty system); hence we define the {\em expected dimension} to be \[ e = e(d,m_0,n,m) = \max\{-1, \v(d,m_0,n,m)\}. \] As the points $p_i$ vary on ${\mathbb{P}}^2$, the dimension of this linear system is upper semi-continuous; therefore on a Zariski open set in the parameter space of $(n+1)$-tuples of points, the dimension achieves its minimum value, which we (abusing notation slightly) call the {\em dimension} of $\L$, and denote by $\l = \l(d,m_0,n,m)$. We always have that \[ \l(d,m_0,n,m) \geq e(d,m_0,n,m); \] equality implies (when the numbers are at least $-1$) that the conditions imposed by the multiple points are independent. We will say that the generic system $\L$ is {\em non-special} if equality holds, i.e., that either the system is empty or that the conditions imposed by the multiple points are independent. If $\l > e$ then we say the system is {\em special}. In this article we discuss the speciality of such linear systems of plane curves, and classify the special linear systems with low multiplicity $m$. There is a long history for this problem; we will not make an attempt here to review it. The ``homogeneous'' cases with $m_0=0$, and $m \leq 2$, are discussed in \cite{arbarello-cornalba}, and in \cite{hirschowitz1}; this last reference also has results on the homogeneous $m=3$ case. In \cite{alexander}, \cite{alex-hirsch1}, \cite{alex-hirsch2}, \cite{evain}, \cite{gergimhar}, \cite{gergimpit}, \cite{harbourne1}, \cite{harbourne2}, \cite{harbourne3}, \cite{hirschowitz3}, \cite{mignon}, and \cite{segre} one may also find related conjectures and results on the general case. The reader may consult \cite{gimigliano} for a survey. In Section 1 we lay out some basic notation and elementary observations. In Section 2 we describe in detail our approach, which is based on a degeneration of the plane and of the corresponding linear system. This leads to a recursion for the sought-after dimension, which relies on a transversality theorem for the pair of linear systems to which the recursion is reduced; this is described in Section 3. The failure of systems to be non-special is always due (in our examples) to the presence of multiple $(-1)$-curves in the base locus; we formalize this in Section 4 (calling such systems ``$(-1)$-special'') and give a classification of quasi-homogeneous $(-1)$-curves in Section 5. In Section 6 we present a computation of the dimension of the linear system $\L(d,m_0,n,m)$ with large $m_0$, which is particularly useful in our inductive approach; this uses Cremona transformations in an essential way. Using this, we present a list of the $(-1)$-special systems in Section 7, for $m \leq 3$. We then prove in Section 8 that all quasi-homogeneous special systems with $m \leq 3$ are $(-1)$-special. In the paper \cite{cm2} we turn our attention to the homogeneous case. We classify all $(-1)$-special homogeneous systems, and prove every special homogeneous system with $m \leq 12$ is $(-1)$-special. The main technique which is used for this is the one described in the present paper. The authors would like to thank L. Caporaso, A. Geramita, B. Harbourne, J. Harris, and A. Tjurin for some useful discussions. \section{Basic Facts} We keep the notation of the Introduction. If there is no danger of confusion, we will omit in what follows the indication of the data $(d,m_0,n,m)$. Also, if $n=0$, we omit $n$ and $m$ from the notation and speak of the linear system $\L(d,m_0)$. We note that the speciality of $\L$ is equivalent to a statement about linear systems on the blowup ${\mathbb{F}}_1$ of ${\mathbb{P}}^2$ at the point $p_0$. If $H$ denotes the class of the pullback of a line and $E$ denotes the class of the exceptional divisor, then the linear system on ${\mathbb{P}}^2$ transforms to the linear subsystem of $|dH-m_0E|$ consisting of those curves having multiplicity $m$ at each of the $n$ transformed points $p_i, i \geq 1$ (none of which lie on $E$). If we further blow up the points $p_i$ to $E_i$, we obtain the rational surface ${\mathbb{P}}^\prime$; the linear system of proper transforms is the complete linear system $|dH - m_0E-\sum_i m_iE_i|$ and we denote by $\L^\prime = \L^\prime(d,m_0,n,m)$ the corresponding line bundle. Then the original system $\L$ is non-special if and only if \[ h^1(\L^\prime) = \max\{0, -1-\v \}. \] In particular if the system is non-empty (which means that $H^0(\L^\prime)$ is non-zero), it is non-special if and only if the $H^1$ is zero. More precisely if the virtual dimension $\v \geq -1$ then non-speciality means that the $H^1$ is zero, or, equivalently, that the conditions imposed by the multiple base points are linearly independent. There is one situation in which a completely general statement can be made, namely the case of simple base points. Suppose that $\mathcal{M}$ is a linear system on a variety $X$, and $p_1,\ldots,p_n$ are general points of $X$. Let $\mathcal{M}(-\sum_i p_i)$ be the linear subsystem consisting of those divisors in $\mathcal{M}$ which pass through the $n$ given points. \begin{multonelemma \label{multone} $\dim \mathcal{M}(-\sum_{i=1}^n p_i) = \max\{-1, \dim\mathcal{M} - n\}$. \end{multonelemma} \begin{pf} The proof is obtained by induction on $n$; one simply chooses the next point not to be a base point of the previous linear system, if this is non-empty. \end{pf} We can speak of the \emph{self-intersection} ${\L}^2$ and the \emph{genus} $g_{\L}$ of the curves of the system ${\L}(d,m_0,n,m)$, which will be the self-intersection and the arithmetic genus of $\L^\prime$ on the blowup ${\mathbb{P}}^\prime$. We have: \[ {\L}^2=d^2-m^2_0-nm^2 \;\;\;\mathrm{ and }\;\;\; 2g_{\L}-2=d(d-3)-m_0(m_0-1)-nm(m-1). \] Notice the basic identity: \begin{equation} \label{RR} {\v}={\L}^2-g_{\L}+1. \end{equation} We can also speak of the \emph{intersection number} ${\L}(d,m_0,n,m)\cdot{\L}(d',m_0',n',m')$, $n'\leq n$, which is of course given by: \[ {\L}(d,m_0,n,m)\cdot{\L}(d',m'_0,n',m) := dd' - m_0m_0' - n'mm' \] We collect here certain initial observations. \begin{lemma}\label{basic1}\mbox{} \begin{itemize} \item[a.] $\L(d,0,1,m)$ is non-special for all $d$, $m$. \item[b.] If $e(d,m_0,n,m)\geq -1$ and $\L(d,m_0,n,m)$ is non-special then $\L(d',m_0',n',m')$ is non-special whenever $d' \geq d$, $m_0' \leq m_0$, $n' \leq n$, and $m' \leq m$. \item[c.] If $e(d,m_0,n,m) = -1$ and $\L(d,m_0,n,m)$ is non-special (i.e., the system $\L$ is empty) then $\L(d',m_0',n',m')$ is non-special (and therefore empty) whenever $d' \leq d$, $m_0' \geq m_0$, $n' \geq n$, and $m' \geq m$. \item[d.] If $d \geq 2$ then $\L(d,0,2,d)$ and $\L(2d,0,5,d)$ are special. \end{itemize} \end{lemma} \begin{pf} Statement (a) is elementary. For statement (b), we note that if $d'=d$, then the inequalities on $m_0'$, $n'$, and $m'$ imply that the conditions imposed on the curves of degree $d$ for the system $\L(d,m_0',n',m')$ are a subset of the conditions for the system $\L(d,m_0,n,m)$; since those are independent by the assumptions that $\L$ is non-special and non-empty, so are the conditions in the subset. and therefore $\L(d,m_0',n',m')$ is non-special. Therefore to prove (b) we may assume that $m_0'=m_0$, $n'=n$, and $m'=m$, and using induction that $d'=d+1$. If we now pass to the blow-up ${{\mathbb{P}}}'$ of ${{\mathbb{P}}}^2$, we have the following exact sequence: \[ 0 \to {\L}^\prime(d,m_0,n,m) \to {\L}^\prime(d+1,m_0,n,m) \to \O_H(d+1) \to 0. \] Since $H^1(H,\O_H(d+1))=0$, and by assumption $H^1(({{\mathbb{P}}}^\prime,{\L}^\prime(d,m_0,n,m))=0$, we have the assertion. Statement (c) is similarly proved, using the same argument to reduce to $m_0'=m_0$, $n'=n$, and $m'=m$; then the same exact sequence, at the $H^0$ level, proves the result. For (d) we note that $\v(d,0,2,d) = \v(2d,0,5,d) = d(1-d)/2 < 0$ but through $2$ general points there is always the line counted with multiplicity $d$, and through $5$ general points the conic counted with multiplicity $d$. \end{pf} Also, the case of large $m_0$ is easy to understand. Denote by $T_i$ the line joining $p_0$ to $p_i$. \begin{lemma}\label{basic2}\mbox{} \begin{itemize} \item[a.] $\L(d,m_0,n,m)$ is non-special if $m_0 > d$ ($\l = e = -1$ and the system is empty). \item[b.] $\l(d,d,n,m) = \max\{-1,d-nm\}$ and $\L(d,d,n,m)$ is special if and only if $n\geq 1$, $m \geq 2$, and $d \geq nm$. \item[c.] If $m+m_0 \geq d+1$ then \[ \l(d,m_0,n,m) = \l(d-n(m+m_0-d),m_0-n(m+m_0-d),n,d-m_0). \] \item[d.] If $m \geq 2$ then $\l(d,d-1,n,m) = \max\{-1,2d-2nm+n\}$. \end{itemize} \end{lemma} \begin{pf} Statement (a) is obvious. To see (c), note that $m+m_0 \geq d+1$ implies that each line $T_i$ must be a component of any divisor in the linear system; therefore factoring out these $n$ lines implies that $\l(d,m_0,n,m) = \l(d-n,m_0-n,n,m-1)$. This is iterated $m+m_0-d$ times, leading to $\l(d,m_0,n,m) = \l(d-n(m+m_0-d),m_0-n(m+m_0-d),n,m-(m+m_0-d))$ as claimed. Now (b) follows from (c), since if $m \geq 1$ then (c) gives $\l(d,d,n,m) = \l(d-nm,d-nm,n,0)$. The final statement is also proved by applying (c) and using the Multiplicity One Lemma \ref{multone}. \end{pf} Later we will find it useful to apply Cremona transformations to the plane to relate different linear systems and to compute their dimensions. Let $p_0,p_1,\ldots,p_n$ be general points in the plane, and $C$ be a curve of degree $d$ having multiplicity $m_\ell$ at $p_\ell$ for each $\ell \geq 0$. Fix three of the points $p_i$, $p_j$, and $p_k$; note that $2d \geq m_i+m_j+m_k$ since otherwise the linear system of such curves would have negative intersection number with every conic through these three points, and therefore would be empty. Then the effect of performing a quadratic Cremona transformation based at these three points $p_i$, $p_j$, and $p_k$ is to transform $C$ to a curve of degree $2d-m_i-m_j-m_k$, and having multiplicities at least $d-m_j-m_k$ at $q_i$, $d-m_i-m_k$ at $q_j$, $d-m_i-m_j$ at $q_k$, and $m_\ell$ at $p_\ell$ for $\ell \neq i,j,k$ (where $q_i$, $q_j$, and $q_k$ are the images of the three lines joining the three points). If we consider the entire linear system of such curves, then it is clear that the dimension of the linear system does not change upon performing the Cremona transformation. Moreover, if the linear system contains irreducible curves before applying the Cremona transformation, it will contain irreducible curves after applying the Cremona transformation, except possibly the inclusion of one or more of the three lines joining the three points as base locus. In addition, the virtual dimension does not change upon applying the Cremona transformation, if all of the numbers involved are nonnegative. Therefore Cremona transformation are a useful tool for analysing the speciality of systems in certain situations. \section{The Degeneration of the Plane} In this section we describe the degeneration of the plane which we use in the analysis. It is related to that used by Ran \cite{ran} in several enumerative applications. Let $\Delta$ be a complex disc around the origin. We consider the product $V = {\mathbb{P}}^2 \times \Delta$, with the two projections $p_1:V \to \Delta$ and $p_2:V \to {\mathbb{P}}^2$. We let $V_t = {\mathbb{P}}^2\times \{t\}$. Consider a line $L$ in the plane $V_0$ and blow it up to obtain a new three-fold $X$ with maps $f:X\to V$, $\pi_1 = p_1 \circ f: X \to \Delta$, and $\pi_2 = p_2 \circ f: X \to {\mathbb{P}}^2$. The map $\pi_1:X \to \Delta$ is a flat family of surfaces over $\Delta$. We denote by $X_t$ the fibre of $\pi_1$ over $t \in \Delta$. If $t \neq 0$, then $X_t = V_t$ is a plane ${\mathbb{P}}^2$. By contrast $X_0$ is the union of the proper transform ${\mathbb{P}}$ of $V_0$ and of the exceptional divisor ${\mathbb{F}}$ of the blow-up. It is clear that ${\mathbb{P}}$ is a plane ${\mathbb{P}}^2$ and ${\mathbb{F}}$ is a Hirzebruch surface ${\mathbb{F}}_1$, abstractly isomorphic to the plane blown up at one point. They are joined transversally along a curve $R$ which is a line $L$ in ${\mathbb{P}}$ and is the exceptional divisor $E$ on ${\mathbb{F}}$. Notice that the Picard group of $X_0$ is the fibered product of $\mathrm{Pic}({\mathbb{P}})$ and $\mathrm{Pic}({\mathbb{F}})$ over $\mathrm{Pic}(R)$. In other words, giving a line bundle $\mathcal{X}$ on $X_0$ is equivalent to giving a line bundle $\mathcal{X}_{{\mathbb{P}}}$ on ${{\mathbb{P}}}$ and a line bundle $\mathcal{X}_{{\mathbb{F}}}$ on ${\mathbb{F}}$ whose restrictions to $R$ agree. The Picard group $\mathrm{Pic}({\mathbb{P}})$ is generated by $\O(1)$, while the Picard group of $\mathrm{Pic}({\mathbb{F}})$ is generated by the class $H$ of a line and the class $E$ of the exceptional divisor. Since $H\cdot R = 0$ and $E\cdot R = -1$, we have $\O_{\mathbb{F}}(H)|_R \cong \O_R$ and $\O_{\mathbb{F}}(E)|_R \cong \O_R(-1)$. Hence in order that the restrictions to $R$ agree, one must have $\mathcal{X}_{{\mathbb{P}}}\cong{\O}_{{\mathbb{P}}}(d)$ and $\mathcal{X}_{{\mathbb{F}}}\cong{\O}_{{\mathbb{F}}}(cH-dE)$ for some $c$ and $d$. We will denote this line bundle on $X_0$ by $\mathcal{X}(c,c-d)$. The normal bundle of ${\mathbb{P}}$ in the $3$-fold $X$ is $-L$; the normal bundle of ${\mathbb{F}}$ in $X$ is $-E$. Hence for example the bundle $\O_X({\mathbb{P}})$ restricts to ${\mathbb{P}}$ as ${\O}_{{\mathbb{P}}}(-1)$ and restricts to ${\mathbb{F}}$ as ${\O}_{{\mathbb{F}}}(E)$. Let ${\O}_X(d)$ be the line bundle $\pi_2^*({\O}_{{\mathbb{P}}^2}(d))$. If $t \not=0$, then the restriction of $\O_X(d)$ to $X_t\cong {\mathbb{P}}^2$ is isomorphic to ${\O}_{{\mathbb{P}}^2}(d)$ whereas the restriction of ${\O}_X(d)$ to $X_0$ is the bundle $\mathcal{X}(d,0)$, (whose restriction to ${\mathbb{P}}$ is the bundle ${\O}_{{\mathbb{P}}}(d)$ and whose restriction to ${\mathbb{F}}$ is the bundle ${\O}_{{\mathbb{F}}}(dH-dE)$). Let us denote by ${\O}_X(d,k)$ the line bundle ${\O}_X(d)\otimes {\O}_X(k{\mathbb{P}})$. The restriction of ${\O}_X(d,k)$ to $X_t$, $t\not=0$, is still the same, i.e. it is isomorphic to ${\O}_{{\mathbb{P}}^2}(d)$, but the restriction to $X_0$ is now different: it is isomorphic to $\mathcal{X}(d,k)$ (whose restriction to ${\mathbb{P}}$ is the bundle ${\O}_{{\mathbb{P}}}(d-k)$ and whose restriction to ${\mathbb{F}}$ is the bundle ${\O}_{{\mathbb{F}}}(dH-(d-k)E)$). We therefore see that all of the bundles $\mathcal{X}(d,k)$ on $X_0$ are flat limits of the bundles $\O_{{\mathbb{P}}^2}(d)$ on the general fiber $X_t$ of this degeneration. Fix a positive integer $n$ and another non-negative integer $b\leq n$. Let us consider $n-b+1$ general points $p_0, p_1, \ldots, p_{n-b}$ in ${\mathbb{P}}$ and $b$ general points $p_{n-b+1},...,p_n$ in ${\mathbb{F}}$. We can consider these points as limits of $n$ general points $p_{0,t}, p_{1,t}, \ldots, p_{n,t}$ in $X_t$. Consider then the linear system ${\L}_t(d,m_0,n,m)$ which is the system ${\L}(d,m_0,n,m)$ in $X_t \cong {{\mathbb{P}}}^2$ based at the points $p_{0,t}, p_{1,t},...,p_{n,t}$. We now also consider the linear system ${\mathcal{L}_0}:={\mathcal{L}_0}(d,k,m_0,n,b,m)$ on $X_0$ which is formed by the divisors in $|\mathcal{X}(d,k)|$ having a point of multiplicity $m_0$ at $p_0$ and points of multiplicity $m$ at $p_1,...,p_n$. According to the above considerations, any one of the systems ${\mathcal{L}_0}(d,k,m_0,n,b,m)$ can be considered as a flat limit on $X_0$ of the system ${\L}_t(d,m_0,n,m)={\L}(d,m_0,n,m)$. We will say that ${\mathcal{L}_0}$ is obtained from $\L$ by a \emph{$(k,b)$-degeneration}. We note that the system $\mathcal{L}_0$ restricts to ${\mathbb{P}}$ as a system $\mathcal{L}_\bP$ of the form $\L(d-k,m_0,n-b,m)$ and $\mathcal{L}_0$ restricts to ${\mathbb{F}}$ as a system $\mathcal{L}_\bF$ of the form $\L(d,d-k,b,m)$. Indeed, at the level of vector spaces, the system $\mathcal{L}_0$ is the fibered product of $\mathcal{L}_\bP$ and $\mathcal{L}_\bF$ over the restricted system on $R$, which is $\O_R(d-k)$. Specifically, if $\mathcal{L}_\bP$ is the projectivization of the vector space $W_{\mathbb{P}}$, and $\mathcal{L}_\bF$ is the projectivization of the vector space $W_{\mathbb{F}}$, then by restriction to the double curve $R$ we have maps $W_{\mathbb{P}} \to H^0(R,\O(d-k))$ and $W_{\mathbb{F}} \to H^0(R,\O(d-k))$, and the fibered product $W = W_{\mathbb{P}} \times_{H^0(R,\O(d-k))}W_{\mathbb{F}}$ gives the linear system $\mathcal{L}_0 = {\mathbb{P}}(W)$ as its projectivization. Since the linear system $\mathcal{L}_0$ is a linear system on a reducible scheme, its elements come in three types. The first type of element of $\mathcal{L}_0$ consists of a divisor $C_{\mathbb{P}}$ on ${\mathbb{P}}$ in the system $|(d-k)H|$ and a divisor $C_{\mathbb{F}}$ on ${\mathbb{F}}$ in the system $|dH-(d-k)E|$ (both of which satisfying the multiple point conditions) which restrict to the same divisor on the double curve $R$. We will then say that $C_{\mathbb{P}}$ and $C_{\mathbb{F}}$ \emph{match} to give a divisor in $\mathcal{L}_0$. The second type is a divisor corresponding to a section of the bundle which is identically zero on ${\mathbb{P}}$, and gives a general divisor in the system $\L(d,d-k,b,m)$ on ${\mathbb{F}}$ which contains the double curve $E$ as a component; that is, an element of the system $E + \L(d,d-k+1,b,m)$. The third type is the opposite, corresponding to a section of the bundle which is identically zero on ${\mathbb{F}}$, and gives a general divisor in the system $\L(d-k,m_0,n-b,m)$ on ${\mathbb{P}}$ which contains the double curve $L$ as a component; that is, an element of the system $L + \L(d-k-1,m_0,n-b,m)$. We denote by $\ell_0$ the dimension of the linear system $\mathcal{L}_0$ on $X_0$. By semicontinuity, this dimension $\ell_0$ is at least that of the linear system on the general fiber, i.e., \[ \ell_0 = \dim(\mathcal{L}_0) \geq \l(d,m_0,n,m). \] Therefore we have the following: \begin{lemma} \label{lo=Ethenreg} If $\ell_0 = e(d,m_0,n,m)$ then the system $\L(d,m_0,n,m)$ is non-special. \end{lemma} The basis of our method is to compute $\ell_0$ by a recursion. The easy case is to compute this dimension when all divisors in the linear system are of the second or third type, that is, come from sections which are identically zero on one of the components ${\mathbb{P}}$ or ${\mathbb{F}}$. In this case one simply obtains the dimension of the linear system on the other component, which gives us the following. \begin{lemma} \label{dimLo1} Fix $d$, $k$, $m_0$, $n$, $b$, and $m$. \begin{itemize} \item[a.] If $\l(d-k,m_0,n-b,m) < 0$ then $\ell_0 = \l(d,d-k+1,b,m)$. \item[b.] If $\l(d,d-k,b,m) < 0$ then $\ell_0 = \l(d-k-1,m_0,n-b,m)$. \end{itemize} \end{lemma} We will define $\hat{\ell}_0$ to be the dimension of the linear system $\hat{\mathcal{L}}_0$ of divisors in $\mathcal{L}_0$ which have the double curve $R$ as a component. We need to extend Lemma \ref{dimLo1} to handle the cases when there are divisors in $\mathcal{L}_0$ which are not identically zero on either component. Fix $d$, $k$, $m_0$, $n$, $b$, and $m$. We will refer to the system $\L = \L(d,m_0,n,m)$ as the \emph{general system}. The system on ${\mathbb{P}}$ restricts to a system $\mathcal{R}_\mathbb{P}$ on the double curve $R=L$, and the kernel is, at the level of linear systems, the system $\hat{\mathcal{L}}_\mathbb{P} = \L(d-k-1,m_0,a,m)$. Similarly, the system on ${\mathbb{F}}$ restricts to a system $\mathcal{R}_\mathbb{F}$ on the double curve $R=E$, and the kernel is, at the level of linear systems, the system $\hat{\mathcal{L}}_\mathbb{F} = \L(d,d-k+1,b,m)$. We denote by \[ \begin{array}{ll} \v = \v(d,m_0,n,m) & \text{ the virtual dimension of the general system,} \\ v_\mathbb{P} = \v(d-k,m_0,n-b,m) & \text{ the virtual dimension of the system on } {\mathbb{P}}, \\ v_\mathbb{F} = \v(d,d-k,b,m) & \text{ the virtual dimension of the system on } {\mathbb{F}}, \\ \hat{v}_\mathbb{P} = \v(d-k-1,m_0,n-b,m) & \text{ the virtual dimension of the kernel system on } {\mathbb{P}}, \\ \hat{v}_\mathbb{F} = \v(d,d-k+1,b,m) & \text{ the virtual dimension of the kernel system on } {\mathbb{F}}, \\ \l = \l(d,m_0,n,m) & \text{ the dimension of the general system}, \\ \ell_\mathbb{P} = \l(d-k,m_0,n-b,m) & \text{ the dimension of the system on } {\mathbb{P}}, \\ \ell_\mathbb{F} = \l(d,d-k,b,m) & \text{ the dimension of the system on } {\mathbb{F}}, \\ \hat{\ell}_\mathbb{P} = \l(d-k-1,m_0,n-b,m) & \text{ the dimension of the kernel system on } {\mathbb{P}}, \\ \hat{\ell}_\mathbb{F} = \l(d,d-k+1,b,m) & \text{ the dimension of the kernel system on } {\mathbb{F}}, \\ r_\mathbb{P} = \ell_\mathbb{P}-\hat{\ell}_\mathbb{P}-1 & \text{ the dimension of the restricted system $\mathcal{R}_\mathbb{P}$}\\ &\text{ on $R=L$ from ${\mathbb{P}}$, and} \\ r_\mathbb{F} = \ell_\mathbb{F}-\hat{\ell}_\mathbb{F}-1 & \text{ the dimension of the restricted system $\mathcal{R}_\mathbb{F}$}\\ &\text{ on $R=E$ from ${\mathbb{F}}$}. \\ \end{array} \] One has the following lemma, whose immediate proof can be left to the reader: \begin{lemma} \label{identities} The following identities hold: \begin{itemize} \item[a.] ${v_\mathbb{P}}+{v_\mathbb{F}}=\v + d-k$. \item[b.] ${\hat{v}_\mathbb{P}}+{v_\mathbb{F}}={\v}-1$. \item[c.] ${v_\mathbb{P}}+{\hat{v}_\mathbb{F}}={\v}-1$. \item[d.] ${\hat{\ell}_0}={\hat{\ell}_\mathbb{P}}+{\hat{\ell}_\mathbb{F}}+1$. \end{itemize} \end{lemma} The restricted systems $\mathcal{R}_\mathbb{P}$ and $\mathcal{R}_\mathbb{F}$ on the double curve may intersect in various dimensions a priori. The dimension $\ell_0$ of the linear system $\mathcal{L}_0$ on $X_0$ depends on the dimension of this intersection, since $\mathcal{L}_0$ is obtained as a fibered product. Returning to the notation above, we have that $\mathcal{L}_0 = {\mathbb{P}}(W)$ where $W$ is the fibered product $W_{\mathbb{P}} \times_{H^0(R,\O(d-k))} W_{\mathbb{F}}$. Hence at the level of vector spaces \[ W = \{(\alpha,\beta) \in W_{\mathbb{P}} \times W_{\mathbb{F}} \;|\; \alpha|_R = \beta|_R\}. \] Let $W_R$ be the vector space corresponding to the intersection of the restricted systems $\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}$, so that $\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F} = {\mathbb{P}}(W_R)$. Then an element of $W$ is determined by first choosing an element $\gamma \in W_R$, then choosing pre-images $\alpha \in W_{\mathbb{P}}$ and $\beta \in W_{\mathbb{F}}$ of $\gamma$. Using vector space dimensions, the choice of $\gamma$ depends on $1+\dim(\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F})$ parameters, and then once $\gamma$ is chosen the choice of $\alpha$ depends on the vector space dimension of the kernel system, which is $1+\hat{\ell}_\mathbb{P}$, and similarly the choice of $\beta$ depends on $1+\hat{\ell}_\mathbb{F}$ parameters. Therefore $\dim(W) = 1+\dim(\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}) + 1+\hat{\ell}_\mathbb{P} + 1+\hat{\ell}_\mathbb{F}$. Projectivizing gives the dimension of the linear system $\mathcal{L}_0$, and we have proved the following: \begin{lemma} \label{dimLo2} With the above notation, \[ \ell_0 = \dim(\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}) + \hat{\ell}_\mathbb{P} + \hat{\ell}_\mathbb{F} + 2. \] \end{lemma} This is the extension of Lemma \ref{dimLo1} which we were seeking. \section{The Transversality of the Restricted Systems} It is clear from the previous Lemma that the computation of $\ell_0$ depends on the knowledge of the dimension of the intersection $\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}$ of the restricted linear systems. The easiest case to handle would be if these two systems were transverse (as linear subspaces of the projective space of divisors of degree $d-k$ on the double curve $R$); then a formula for the dimension of the intersection is immediate. It turns out that this is always the case, which is a consequence of the following Proposition, first proved to our knowledge by Hirschowitz in \cite{hirschowitz2}, using the Borel fixed point theorem. Our proof is a variation on the theme using the finiteness of inflection points of linear systems. \begin{proposition} \label{SL2prop} Let $G = PGL(2,\mathbb{C})$ be the automorphism group of $\mathbb{P}^1$. Let $X$ be the linear system of divisors of degree $d$ on $\mathbb{P}^1$. Note that $G$ acts naturally on $X$, and on linear subspaces of $X$ of any dimension. Then for any two nontrivial linear subspaces $V$ and $W$ of $X$, there is an element $g \in G$ such that $V$ meets $gW$ properly. \end{proposition} \begin{pf} It suffices to prove the assertion when $V$ and $W$ have complementary dimensions $k$ and $d-k-1$ respectively. We argue in this case by contradiction: suppose that for every $g \in G$, the intersection $V\cap gW$ is nonempty. Fix a general coordinate system $[x:y]$ on $\mathbb{P}^1$, and consider the element $g_t \in G$ given by $g_t[x:y] = [tx:t^{-1}y]$. Suppose that in this coordinate system we have a basis $\{v_0,\dots,v_k\}$ for $V$ and a basis $\{w_{k+1},\ldots,w_d\}$ for $W$. Write each $v_i$ as a polynomial in $x$ and $y$ as $v_i=\sum_j a_{ij}x^jy^{d-j}$ and similarly write each $w_i$ as $w_i=\sum_j b_{ij}x^jy^{d-j}$. Note that with this notation we have that the subspace $g_tW$ has as its basis the polynomials $g_tw_i = \sum_j t^{2j-d}b_{ij}x^jy^{d-j}$. Let $A$ be the $(k+1) \times (d+1)$ matrix of the $a_{ij}$ coefficients, and let $B_t$ be the $(d-k)\times (d+1)$ matrix of the $t^{2j-d}b_{ij}$ coefficients. Notice that $B:=B_1$ is the matrix of coefficients for the original subspace $W$. Let $C_t$ be the square matrix whose with $A$ as its first $k+1$ rows and $B_t$ as its last $d-k$ rows. Since the subspaces $V$ and $g_tW$ intersect nontrivially, they cannot span the whole space $X$; hence the matrix of coefficients $C_t$ must have trivial determinant. Note that this determinant is a Laurent polynomial in $t$, and hence each coefficient of $t$ in this polynomial must vanish. By expressing this determinant in the Laplace expansion using the minors of the first $k+1$ rows against the minors of the last $d-k$ rows, we see that the top coefficient of the determinant is the product of the minor of $A$ using the first $k+1$ columns with the minor of $B$ using the last $d-k$ columns. Since this coefficient is zero, we have that either the first minor of $A$ is zero or the last minor of $B$ is zero. If the first minor of $A$ is zero, then there exists in $V$ a polynomial whose first $k+1$ coefficients are zero, and hence vanishes at $[0:1]$ to order at least $k+1$. Hence $[0:1]$ would be an inflection point for the system $V$. Similarly, if the last minor of $B$ is zero, we conclude in the same way that the point $[1:0]$ is an inflection point for the system $W$. Since the coordinate system was chosen to be general, we see that there are infinitely many inflection points for at least one of the two systems. This is a contradiction, finishing the proof. \end{pf} Note that the given any automorphism $g$ of a line in the plane, there is a lift of $g$ to an automorphism of the plane fixing the line. By using this and the previous Proposition, one immediately deduces the following. \begin{corollary} The restricted systems $\mathcal{R}_\mathbb{P}$ and $\mathcal{R}_\mathbb{F}$ on the double line intersect properly. \end{corollary} The previous Corollary, combined with Lemma \ref{dimLo2}, gives the formula for $\ell_0$: \begin{proposition} \label{dimLo3} \mbox{} \begin{itemize} \item[(a)] If $r_\mathbb{P}+r_\mathbb{F} \leq d-k-1$, then \[ \ell_0 = \hat{\ell}_\mathbb{P} + \hat{\ell}_\mathbb{F} + 1. \] \item[(b)] If $r_\mathbb{P}+r_\mathbb{F} \geq d-k-1$, then \[ \ell_0 = \ell_\mathbb{P}+ \ell_\mathbb{F} - d + k. \] \end{itemize} \end{proposition} \begin{pf} If $r_\mathbb{P}+r_\mathbb{F} \leq d-k-1$, then the transversality of $\mathcal{R}_\mathbb{P}$ and $\mathcal{R}_\mathbb{F}$ implies that $\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}$ is empty, of dimension $-1$. This gives (a), using Lemma \ref{dimLo2}. If $r_\mathbb{P}+r_\mathbb{F} \geq d-k-1$, then $\dim(\mathcal{R}_\mathbb{P}\cap\mathcal{R}_\mathbb{F}) = r_\mathbb{P}+r_\mathbb{F}-d+k$ (again using the transversality) and by Lemma \ref{dimLo2}, we have \begin{align*} \ell_0 &= r_\mathbb{P}+r_\mathbb{F}-d+k + \hat{\ell}_\mathbb{P} + \hat{\ell}_\mathbb{F} + 2\\ &= \ell_\mathbb{P} +\ell_\mathbb{F} - d + k \end{align*} using the definition of $r_\mathbb{P}=\ell_\mathbb{P}-\hat{\ell}_\mathbb{P}-1$ and $r_\mathbb{F}=\ell_\mathbb{F}-\hat{\ell}_\mathbb{F}-1$. \end{pf} Our method for proving that the system $\L$ is non-special is to find appropriate integers $k$, $a$, and $b$ with $n = a+b$ such that $\ell_0 = e(d,m_0,a+b,m)$, and to invoke Lemma \ref{lo=Ethenreg}. The computation of $\ell_0$ is done by recursively using Proposition \ref{dimLo3}. The dimension computed in part (b) of the Proposition is, miraculously, the virtual dimension of the system on the general fiber, if each of the systems involved in (b) has the virtual dimension, using Lemma \ref{identities}(a). Therefore (b) is useful for proving that $\L$ has the correct minimal dimension and is therefore non-special. Statement (a) is more useful for proving that $\L$ is empty. The following makes these two strategies explicit. \begin{corollary} \label{cordimLk} Fix $d$, $m_0$, $n$, and $m$. \begin{itemize} \item[a.] Suppose that positive integers $k$ and $b$ exist, with $0 < k < d$ and $0 < b < n$, such that \begin{enumerate} \item $\L(d-k-1,m_0,n-b,m)$ is empty, \item $\L(d,d-k+1,b,m)$ is empty, and \item $\dim \L(d-k,m_0,n-b,m) + \dim \L(d,d-k,b,m) \leq d-k-1$. (This is automatic if $\v(d,m_0,n,m) \leq -1$ and both these systems are non-special with virtual dimension at least $-1$.) \end{enumerate} Then $\L(d,m_0,n,m)$ is empty (and therefore non-special). \item[b.] Suppose that positive integers $k$ and $b$ exist, with $0 < k < d$ and $0 < b < n$, such that \begin{enumerate} \item $\v(d-k,m_0,n-b,m) \geq -1$ and $\L(d-k,m_0,n-b,m)$ is non-special, \item $\v(d,d-k,b,m) \geq -1$ and $\L(d,d-k,b,m)$ is non-special, and \item $\dim \L(d-k-1,m_0,n-b,m) + \dim \L(d,d-k+1,b,m) \leq \v(d,m_0,n,m)-1$. (This is automatic if both these systems are non-special with virtual dimension at least $-1$.) \end{enumerate} Then $\L(d,m_0,n,m)$ is non-special, with virtual dimension $\v(d,m_0,n,m)$ at least $-1$. \end{itemize} \end{corollary} \begin{pf} In case (a), the first two hypotheses say that $\hat{\ell}_\mathbb{P} = \hat{\ell}_\mathbb{F} = -1$, so that $r_\mathbb{P} = \ell_\mathbb{P}$ and $r_\mathbb{F} = \ell_\mathbb{F}$. Hence by the third hypothesis \[ r_\mathbb{P}+r_\mathbb{F} = \ell_\mathbb{P} + \ell_\mathbb{F} \leq d-k-1 \] so that using Proposition \ref{dimLo3}(a) we have $\ell_0 = \hat{\ell}_\mathbb{P} + \hat{\ell}_\mathbb{F} + 1 = -1$, proving that $\L$ is empty by semicontinuity. The parenthetical statement in the third hypothesis follows from Lemma \ref{identities}(a). For (b), the first two hypotheses say that $\ell_\mathbb{P}=v_\mathbb{P}$ and $\ell_\mathbb{F}=v_\mathbb{F}$; then \begin{align*} r_\mathbb{P}+r_\mathbb{F} &= \ell_\mathbb{P} - \hat{\ell}_\mathbb{P} + \ell_\mathbb{F} - \hat{\ell}_\mathbb{F} - 2 \\ &= v_\mathbb{P} - \hat{\ell}_\mathbb{P} + v_\mathbb{F} - \hat{\ell}_\mathbb{F} - 2 \\ &\geq v_\mathbb{P}+v_\mathbb{F} - 1 - \v \;\;\;\text{ using the third hypothesis}\\ &= d-k-1 \end{align*} using Lemma \ref{identities}(a). Hence Proposition \ref{dimLo3}(b) implies that $\ell_0 = \ell_\mathbb{P}+\ell_\mathbb{F}-d+k = v_\mathbb{P} + v_\mathbb{F} - d + k = \v(d,m_0,n,m)$ again using Lemma \ref{identities}(a). The parenthetical statement in the third hypothesis follows from Lemma \ref{identities}(a), (b), and (c). \end{pf} \section{$(-1)$-Special Systems and the Main Conjecture} A linear system $\L(d,m_0,n,m)$ with $\L^2 = -1$ and $g_\L = 0$ will be called a \emph{quasi-homogeneous $(-1)$-class}. By (\ref{RR}), we see that $\v = 0$, so that every quasi-homogeneous $(-1)$-class is effective. Suppose that $A$ is an irreducible rational curve and is a member of a linear system $\L=\L(d,m_0,n,m)$, and suppose that on the blowup ${\mathbb{P}}^\prime$ of the plane the proper transform of $A$ is smooth, of self-intersection $-1$. We say then that $A$ is a \emph{$(-1)$-curve}. In this case $\L$ is a quasi-homogeneous $(-1)$-class. Indeed, if this happens, then $\L = \{A\}$: if $D \in \L$, then $D\cdot A < 0$ on ${\mathbb{P}}^\prime$, so that $D$ must contain $A$ as a component, and then be equal to $A$ since they have the same divisor class. Therefore such a linear system $\L$ is non-special, of dimension $0$. A quasi-homogeneous $(-1)$-class containing a $(-1)$-curve will be called an \emph{irreducible $(-1)$-class}. Let $A$ be a $(-1)$-curve and suppose that $2A$ is a member of a linear system $\L$. Then $\L^2 = -4$ and $g_\L = -2$ so by (\ref{RR}) $\v = -1$ and the system is expected to be empty; however it clearly contains the divisor $2A$ (and is equal in fact to $\{2A\}$). Therefore such a linear system is special. More generally we have the following. \begin{lemma} \label{minusonespecial} Let $\L$ be a nonempty linear system, Suppose that $A_1,\ldots,A_r$ are $(-1)$-curves which meet $\L$ negatively; write $\L \cdot A_j = - N_j$ with $N_j \geq 1$ for each $j$. Then: \begin{itemize} \item[(a)] $\L$ contains $\sum_j N_jA_j$ as a fixed divisor. \item[(b)] If $i \neq j$ then $A_i\cdot A_j = 0$. \item[(c)] If $\mathcal{M} = \L - \sum_j N_j A_j$ is the residual system, then \[ \v(\mathcal{M}) - \v(\L) = \sum_j N_j(N_j-1)/2. \] \item[(d)] If $N_j \geq 2$ for some $j$ then $\L$ is special. \end{itemize} \end{lemma} \begin{pf} We work on the blowup ${\mathbb{P}}^\prime$, and we note that since $\L\cdot A_j = -N_j$, $N_jA_j$ is certainly a fixed divisor; hence we have $\L = \sum_j N_jA_j + \mathcal{M}$ for some linear system $\mathcal{M}$, with $\dim(\L) = \dim(\mathcal{M})$, which proves (a). If a linear system $\L$ meets two $(-1)$-curves $A$ and $B$ negatively, then both $A$ and $B$ must be part of the fixed divisor of $\L$. If $A$ and $B$ meet then since $\v(A+B)=A\cdot B$, we would have that $A+B$ moves, and could not be part of the fixed divisor of $\L$. This proves (b). Note also that $\mathcal{M}^2 = \L^2 + \sum_j N_j^2$, and $\mathcal{M}\cdot K = \L \cdot K + \sum_j N_j$. Then \begin{align*} \v(\mathcal{M}) - v(\L) &= (\mathcal{M}^2-\mathcal{M}\cdot K)/2 - (\L^2-\L\cdot K)/2 \\ &= (\L^2+\sum_j N_j^2-\L\cdot K-\sum_j N_j)/2 - (\L^2- \L \cdot K)/2 \\ &= \sum_j N_j(N_j-1)/2 > 0 \end{align*} which proves (c). If any $N_j \geq 2$ then we see that $v(\mathcal{M}) > \v(L)$ and hence \[ \dim(\L) = \dim(\mathcal{M}) \geq \v(\mathcal{M}) > \v(\L) \] proving the speciality of $\L$. \end{pf} The above Lemma suggests the following. \begin{definition} A linear system $\L$ is \emph{$(-1)$-special} if there are $(-1)$-curves $A_1,\ldots,A_r$ such that $\L\cdot A_j = -N_j$ with $N_j \geq 1$ for every $j$ and $N_j \geq 2$ for some $j$, with the residual system $\mathcal{M} = \L - \sum_j N_j A_j$ having non-negative virtual dimension $\v(\mathcal{M}) \geq 0$, and having non-negative intersection with every $(-1)$-curve. \end{definition} We remark that if $\L$ is a linear system satisfying all of the hypotheses of the above definition except the last one, then in fact it is $(-1)$-special; the residual system can meet only finitely many additional $(-1)$-curves negatively, and after adding these to the fixed part we get a residual system satisfying the final condition. By the Lemma, every $(-1)$-special system is special; the condition that $\v(\mathcal{M}) \geq 0$ implies that $\mathcal{M}$ is non-empty, and hence that $\L$ is non-empty. The following conjecture is a restatement of a conjecture of Hirschowitz (see \cite{hirschowitz3}), also made by Harbourne (see \cite{harbourne1} and \cite{harbourne2}). \begin{mainconj} Every special system is $(-1)$-special. \end{mainconj} \section{Quasi-homogeneous $(-1)$-classes} It is clear from the previous section that a classification of $(-1)$-classes is important in understanding speciality of linear systems. Fortunately it is not hard to classify all quasi-homogeneous $(-1)$-classes, as we now do. Suppose that $\L(d,m_0,n,m)$ is a quasi-homogeneous $(-1)$-class. Then \begin{equation} \label{QH1} d^2-m_0^2-nm^2 = -1 \end{equation} (since $\L^2 = -1$) and \begin{equation} \label{QH2} 3d-m_0-nm=1 \end{equation} (which is equivalent to the genus condition, and is specifically the condition that $\L\cdot K = -1$ on the blowup surface). Solving (\ref{QH2}) for $m_0$ gives $m_0=3d-nm-1$ and plugging this into (\ref{QH1}) yields \begin{equation*} 8d^2 - 6dnm + n^2m^2 - 6d + 2nm + nm^2 = 0 \end{equation*} which can be rewritten as \begin{equation} \label{QH4} (4d-nm)(2d-nm) + (m-1)(4d-nm) - (2m+1)(2d-nm) = 0. \end{equation} This suggests the change of variables \[ u = 4d - nm, \;\;\; v = 2d - nm \] so that (\ref{QH4}) now becomes \begin{equation} \label{QH5} uv + (m-1)u - (2m+1)v = 0. \end{equation} Reversing this change of coordinates gives \begin{equation} \label{QH7} d = \frac{u-v}{2},\;\;\; nm = u-2v,\;\;\; m_0 = \frac{u+v}{2} - 1. \end{equation} Hence we seek integral solutions $(u,v)$ to the equation (\ref{QH5}) with $u\equiv v \mod{2}$ (so that $d$ and $m_0$ are integers) and $m|(u-2v)$ (so that $n$ is an integer), and all quantities $u-v$, $u-2v$, and $u+v$ positive. If $m=1$, the curve (\ref{QH5}) is $uv-3v=0$, so either $u=3$ or $v=0$. If $u=3$ then the positivity conditions are that $3-v>0$, $3-2v>0$, and $3+v>0$, so $-3 < v < 3/2$ and must be odd; only $v = \pm 1$ are possibilities. The solution $(3,-1)$ gives $(d,m_0,n,m)=(2,0,5,1)$, and the solution $(3,1)$ gives $(d,m_0,n,m)=(1,1,1,1)$. If $v=0$ we only must have $u>0$ and even, say $u=2e$. This gives $(d,m_0,n,m)=(e,e-1,2e,1)$, for any $e \geq 1$. These are all the solutions with $m=1$. {}From now on we assume that $m \geq 2$. In this case the hyperbola (\ref{QH5}) in the $(u,v)$ plane has the horizontal asymptote $v = 1-m$, the vertical asymptote $u = 2m+1$, and passes through the origin with slope $(m-1)/(2m+1) < 1/2$. This slope condition implies that in the third quadrant the hyperbola lies entirely above the line $v = u/2$. Now $nm = u-2v > 0$ so $v < u/2$. Moreover $m_0 \leq d-1$ hence $(u+v)/2 -1 \leq (u-v)/2 -1$, implying that $v \leq 0$. These two inequalities imply that the only integral points of interest lie on the branch of the hyperbola in the fourth quadrant, with $v < 1-m$ and $u > 2m+1$. Hence we may assume that \begin{equation} \label{QH8} u \geq 2m+2 \;\;\;\mathrm{and}\;\;\; v \leq -m. \end{equation} Finally make the change of coordinates \begin{equation} x = u-2m-1, \;\;\; y = 1-m-v, \;\;\; u=x+2m+1,\;\;\; v = 1-m-y \end{equation} which transforms the hyperbola (\ref{QH5}) into \begin{equation} \label{QH9} xy = (m-1)(2m+1). \end{equation} The branch of (\ref{QH9}) corresponding to the branch of (\ref{QH5}) in the fourth quadrant is the one in the first quadrant, with $x\geq 1$ and $y\geq 1$. Clearly the integral points on (\ref{QH9}) come simply from the possible factorizations of $(m-1)(2m+1)$. This gives the following classification. \begin{proposition} \label{QH-1general} The quasi-homogeneous $(-1)$-classes are the classes $\L(d,m_0,n,m)$ with $(d,m_0,n,m)$ on the following list: \begin{itemize} \item[(a)] $(2,0,5,1)$ and $(1,1,1,1)$. \item[(b)] $(e,e-1,2e,1)$ with $e \geq 1$. \item[(c)] For any $m \geq 2$, and any $x\geq 1$, $y\geq 1$ with \begin{itemize} \item[(i)] $xy = (m-1)(2m+1)$, \item[(ii)] $x+m \geq y$, \item[(iii)] $x-y \equiv m \mod{2}$, and \item[(iv)] $m | x+2y-1$, \end{itemize} the four-tuple \[ (\frac{x+y+3m}{2}, \frac{x-y+m}{2}, \frac{x+2y-1}{m}+4, m). \] \end{itemize} \end{proposition} In part (c), condition (ii) is the non-negativity of $m_0$, while conditions (iii) and (iv) are needed to insure that $d$, $m_0$, and $n$ are integral. It is easy to classify all homogeneous $(-1)$-classes from the above Proposition. \begin{corollary} The classes $\L(2,0,5,1)=\L(2,1,4,1)$ and $\L(1,0,2,1)=\L(1,1,1,1)$ are the only homogeneous $(-1)$-classes. \end{corollary} \begin{pf} Clearly these are the only ones with $m=1$. For $m \geq 2$, we must have the factors $x$ and $y$ satisfying $y=x+m$, and so $x(x+m)=(m-1)(2m+1)$. If $x \leq m-1$ then the other factor $x+m$ must be at least $2m+1$, a contradiction. If $x \geq m+1$ then $x(x+m) \geq (m+1)(2m+1)$, which is too big. Hence only $x=m$ is a possibility, which in fact does not work. \end{pf} \begin{example} \label{QH-1extremal} For any $m\geq 2$, set $x = (m-1)(2m+1)$ and $y = 1$. Conditions (i), (ii), and (iii) clearly hold, and $x+2y-1 = (2m^2 -m - 1) +2-1 = m(2m-1)$ so that also (iv) holds. This gives \[ d=m^2+m, \;\;\; m_0 = m^2-1,\;\;\; n=2m+3. \] \end{example} \begin{example} Fix an integer $z\neq 0$, let $m=4z^2+3z$, $x = 8z^2+2z-1$ and $y = 4z^2+5z+1$. This gives the solution \[ d=12z^2+8z, \;\;\; m_0 = 4z^2-1,\;\;\; n=8,\;\;\; m=4z^2+3z. \] \end{example} It is an exercise to check that the previous two examples produce all of the quasi-homogeneous $(-1)$-classes with $n=8$. \begin{example} \label{QH-1list} The following is a complete list of all quasi-homogeneous $(-1)$-classes with $m \leq 7$: \begin{center} \begin{tabular}{cccccc} $d$ & $m_0$ & $n$ & $m$ & ($x$ & $y$) \\ \hline 1 & 1 & 1 & 1 & - & - \\ 2 & 0 & 5 & 1 & - & - \\ $e\geq 1$ & $e-1$ & $2e$ & $1$ & - & - \\ 6 & 3 & 7 & 2 & (5 & 1)\\ 12 & 8 & 9 & 3 & (14 & 1)\\ 20 & 15 & 11 & 4 & (27 & 1) \\ 30 & 24 & 13 & 5 & (44 & 1) \\ 42 & 35 & 15 & 6 & (65 & 1) \\ 20 & 3 & 8 & 7 & (9 & 10) \\ 27 & 17 & 9 & 7 & (30 & 3) \end{tabular} \end{center} \end{example} We note that Cremona transformations may be used to give a numerical criterion for deciding when a quasi-homogeneous $(-1)$-class is an irreducible class: if it can be transformed, by a series of quadratic Cremona transformations, to the class of a line through two points $\L(1,0,2,1)$. \begin{proposition}\mbox{} \begin{itemize} \item[(a)] All quasi-homogeneous $(-1)$-classes having $m =1$ are irreducible. \item[(b)] All quasi-homogeneous $(-1)$-classes with $m \geq 2$ of the form $\L(d=m^2+m, m_0=m^2-1, n=2m+3, m)$ (obtained by the factorization $x = (m-1)(2m+1)$, $y=1$, as in Example \ref{QH-1extremal}) are irreducible. \item[(c)] All quasi-homogeneous $(-1)$-classes having $m \leq 6$ are irreducible. \item[(d)] The $(-1)$-class $\L(27,17,9,7)$ is not irreducible. \end{itemize} \end{proposition} \begin{pf} It is obvious that $\L(1,1,1,1)$ and $\L(2,0,5,1)$, corresponding to a line through two points and a conic through $5$, are irreducible. To see that $\L(e,e-1,2e,1)$ is irreducible, note that it is irreducible for $e=1$: again this is a line through $2$ points. For $e \geq 2$, applying a Cremona transformation to $\L(e,e-1,2e,1)$ at the points $p_0$, $p_1$, $p_2$ transforms the system to $\L(e-1,e-2,2e-2,1)$, and so by induction all these systems are irreducible. This proves (a). To prove (b), apply the quadratic Cremona transformation to $\L(m^2+m, m^2-1, 2m+3, m)$ exactly $m+1$ times, at $p_0, p_{2i-1}, p_{2i}$, for $i=1,\ldots m+1$. It is easy to see that this transforms the system to $\L(m+1,m,2m+2,1)$, which is irreducible by (a). Part (c) now is a consequence of (a) and (b), given the list of Example \ref{QH-1list}. To prove (d), note that $\L(12,8,9,3)$ is irreducible, and $\L(27,17,9,7)\cdot \L(12,8,9,3) = 12\cdot 27 - 8\cdot 17 - 9\cdot 3 \cdot 7 = -1$, so that if $A$ is the $(-1)$-curve in $\L(12,8,9,3)$, then $A$ is a fixed curve of $\L(27,17,9,7)$. The residual system is $\L(15,9,9,4)$, which has virtual dimension $0$ and is therefore non-empty. \end{pf} Recall that we are interested in $(-1)$-curves because they are useful in constructing special linear systems. Suppose a quasi-homogeneous system $\L(d,m_0,n,m)$ meets negatively a $(-1)$-curve $A$ of degree $\delta$, having multiplicities $\mu_0,\mu_1,\ldots,\mu_n$ at the points $p_0,\ldots,p_n$. Since the points are general, a monodromy argument implies that for any permutation $\sigma \in \Sigma_n$, $\L$ also meets negatively the $(-1)$-curve $A_\sigma$ of degree $\delta$, having multiplicity $\mu_0$ at $p_0$, and having multiplicities $\mu_{\sigma(i)}$ at $p_i$ for each $i \geq 1$. There may of course be repetitions among the $A_\sigma$'s. If this happens, no two of the $(-1)$-curves $A_\sigma$ can meet, by Lemma \ref{minusonespecial}. Hence, if $m+0\neq0$, the Picard group of the blowup surface ${\mathbb{P}}^\prime$ has rank $n+2$, and there can be at most $n+1$ of these disjoint $(-1)$-curves. In the homogeneous case where $m_0=0$, we do not blow up $p_0$, and the rank of the Picard group is only $n+1$; therefore in this case there can be at most $n$ of these disjoint $(-1)$-curves. The sum of all of these $A_\sigma$'s must also be quasi-homogeneous, and if there are $k$ of them, is therefore of the form $\L(k\delta,k\mu_0,n,\mu^\prime)$ for some $\mu^\prime$. An elementary counting argument shows that if the $\mu_i$'s (for $i\geq 1$) occur in subsets of size $k_1\leq k_2\leq\dots\leq k_s$, constant in each subset, then the number of distinct $A_\sigma$'s is \[ \frac{n!}{k_1!k_2!\cdots k_s!}. \] The only way this can be less than or equal to $n+1$ is if $s=1$ (and $A$ is then quasi-homogeneous) or if $s=2$ and $k_1=1$, $k_2=n-1$. The classification in case $s=1$ we have discussed above. In the case $s=2$, there are exactly $n$ $A_\sigma$'s, and the sum of the $A_\sigma$'s is quasi-homogeneous, and is of the form $\L(n\delta,n\mu_0,n,\mu_1+(n-1)\mu_2)$ if $\mu_i = \mu_2$ for $i \geq 2$. The condition that $A$ and $A_\sigma$ do not meet is that \[ \delta^2 - \mu_0^2 - 2\mu_1\mu_2 - (n-2)\mu_2^2 = 0 \] while that fact that $A$ is a $(-1)$-curve implies that \[ \delta^2 - \mu_0^2 - \mu_1^2 - (n-1)\mu_2^2 = -1. \] Subtracting these two equations gives \[ {(\mu_1-\mu_2)}^2 = 1 \] so that $\mu_1 = \mu_2 \pm 1$. We call such a system a \emph{quasi-homogeneous $(-1)$-configuration}. We say that the configuration is \emph{compound} if it consists of more than one $(-1)$-curve. It is not hard to completely classify these classes for low $m$: \begin{example} \label{compoundQH-1list} The following is a complete list of all of the quasi-homogeneous $(-1)$-configurations $\L(d,m_0,n,m)$ with $m \leq 10$: \begin{center} \begin{tabular}{cccccccl} $d$ & $m_0$ & $n$ & $m$ & ($\delta$ & $\mu_0$ & $\mu_1$ & $\mu_2=\cdots = \mu_n$) \\ \hline $e \geq 2$ & $e$ & $e$ & 1 & (1 & 1 & 1 & 0) \\ 3 & 0 & 3 & 2 & (1 & 0 & 0 & 1) \\ 10 & 5 & 5 & 4 & (2 & 1 & 0 & 1) \\ 12 & 0 & 6 & 5 & (2 & 0 & 0 & 1) \\ 21 & 14 & 7 & 6 & (3 & 2 & 0 & 1) \\ 18 & 6 & 6 & 7 & (3 & 1 & 2 & 1) \\ 21 & 0 & 7 & 8 & (3 & 0 & 2 & 1) \\ 36 & 27 & 9 & 8 & (4 & 3 & 0 & 1) \\ 55 & 44 & 11 & 10 & (5 & 4 & 0 & 1) \\ \end{tabular} \end{center} \end{example} Using the classification of quasi-homogeneous $(-1)$-classes, one can easily give a complete classification of homogeneous $(-1)$-configurations, i.e., those with $\mu_0=0$. \begin{proposition} \label{homog-1} The following is a complete list of homogeneous $(-1)$-configurations: \[ \begin{array}{cl} \L(1,0,2,1) & \mathrm{\;\;\;not\;\;compound} \\ \L(2,0,5,1) & \mathrm{\;\;\;not\;\;compound} \\ \L(3,0,3,2) & \mathrm{\;\;\;compound\;\;with\;\;\;} \delta=1, n=3, \mu_1=0, \mu_2=1\\ \L(12,0,6,5) & \mathrm{\;\;\;compound\;\;with\;\;\;} \delta=2, n=6, \mu_1=0, \mu_2=1\\ \L(21,0,7,8) & \mathrm{\;\;\;compound\;\;with\;\;\;} \delta=3, n=7, \mu_1=2, \mu_2=1\\ \L(48,0,8,17) & \mathrm{\;\;\;compound\;\;with\;\;\;} \delta=6, n=8, \mu_1=3, \mu_2=2. \end{array} \] \end{proposition} \begin{pf} The non-compound $(-1)$-curves $\L(1,0,2,1)$ and $\L(2,0,5,1)$ we have seen before. If $A$ is a $(-1)$-curve producing a homogeneous $(-1)$-configuration, then, with the notation above, $A$ has degree $\delta$, one point of multiplicity $\mu_1$, and $n-1$ points of multiplicity $\mu_2 = \mu_1 \pm 1$. Therefore, shifting the indices of the points, we see that $A$ is quasi-homogeneous, in the class $\L(\delta,\mu_1,n-1,\mu_2)$. Hence we may appeal to Proposition \ref{QH-1general}. If $\mu_2 = 1$, then either $\mu_1=0$ (giving the two possibilities $\delta = 1, n-1=2$, and the compound configuration $\L(3,0,3,2)$, or $\delta=2$, $n-1=5$, and the compound configuration $\L(12,0,6,5)$) or $\mu_1=2$ (giving $\delta=3$, $n-1=6$, and the compound configuration $\L(21,0,7,8)$). If $\mu_2 \geq 2$, then the class $A$ comes from a factorization $xy = (\mu_2-1)(2\mu_2+1)$, and then $\mu_1 = (x-y+\mu_2)/2$. This is $\mu_2\pm 1$ if and only if $x-y = \mu_2 \pm 2$. Now $x = 2\mu_2+1$, $y = \mu_2-1$ is a factorization with $x-y = \mu_2+2$, but now we look at the requirement that $\mu_2$ divides $x+2y-1 = 4\mu_2-2$. This forces $\mu_2 = 2$, giving $x=5$, $y=1$, and $\delta = 6, n-1=7$, leading to the compound configuration $\L(48,0,8,17)$. This is the only solution with $x-y=\mu_2+2$. Hence what is left is to discuss the possible cases with $x-y = \mu_2-2$. To obtain such a factorization $x$, $y$, we must have $x < 2\mu_2+1$ and $y > \mu_2-1$. Neither $x = 2\mu_2$ nor $y = \mu_2$ are possible factors, so in fact we must have $x \leq 2\mu_2-1$ and $y \geq \mu_2+1$. However the difference $x-y$ being $\mu_2-2$ then forces $x = 2\mu_2-1$ and $y = \mu_2+1$, whose product never equals $(\mu_2-1)(2\mu_2+1)$. Thus there are no more homogeneous $(-1)$-configurations. \end{pf} \section{The Dimension for Large $m_0$} \label{sec10} In our application we will usually make a $(k,b)$-degeneration with $k$ near $m$. This leads to linear systems on ${\mathbb{F}}$ which have the form $\L(d,m_0,b,m)$ with $m_0$ near $d-m$. These linear systems may usually be effectively analyzed by applying Cremona transformations, since one of the multiplicities is so large with respect to the degree. Let us first consider quasi-homogeneous systems of the form $\L(d,d-m,n,m)$. \begin{lemma} \label{m0=d-m_algorithm} Fix $d \geq m\geq 0$. Consider the general linear system $\L=\L(d,d-m,n,m)$, whose dimension is $\l$. \begin{itemize} \item[(a)] If $m=0$ then $\L$ is non-special and $\l=\v(d,d)=d$. \item[(b)] If $m=1$ then $\L$ is non-special and $\l = e(d,d-1,n,1)=\max\{-1,2d-n\}$. \item[(c)] If $n=0$ then $\L$ is non-special and $\l = \v(d,d-m) = d+dm-m^2/2+m/2$. \item[(d)] If $n=1$ then $\L$ is non-special and $\l = \v(d,d-m,1,m) = d+m(d-m)$. \item[(e)] If $n = 2$ and $m \leq d \leq 2m$ then $\l = (d-m)(d-m+3)/2$. In this case $\L$ is special if $d \leq 2m-2$ and $\L$ is non-special if $d=2m-1$ or $d=2m$. \item[(f)] If $n = 2$ and $d \geq 2m+1$ then $\L$ is non-special and $\l = dm+d-3m^2/2-m/2$. \item[(g)] If $n \geq 2$ and $d \geq 2m$ then $\l=\l(d-m,d-2m,n-2,m)$. \item[(h)] If $2 \leq m \leq d \leq 2m-1$ and $n \geq 3$ then $\L$ is empty (and therefore non-special), so that $\l = -1$. \end{itemize} \end{lemma} \begin{pf} Statements (a) and (b), where we have either no base points or simple base points, are trivial. Statements (c), (d), (e), and (f), where we have $3$ or fewer multiple points, are handled easily by putting the $3$ points at the coordinate points of the plane and counting homogeneous monomials. Statement (g) is obtained by making a quadratic Cremona transformation at the point $p_0$ of multiplicity $d-m$ and two of the $n$ points of multiplicity $m$; we note that the resulting linear system is of the same form, namely that it is quasi-homogeneous with $m_0=d-m$. Finally we turn to statement (h). If $n \geq 3$, $d < 2m$ and $\L$ is nonempty then the line $L_{ij}$ through any two of the $n$ points $p_i$ and $p_j$ must split off the linear system, since $\L\cdot\L(1,0,2,1) = d-2m$. If in fact $n \geq 4$, then the two lines $L_{12}$ and $L_{34}$ become $(-1)$-curves on the blowup of the plane, which meet at one point; hence the sum $L_{12}+L_{34}$ moves in a pencil, and so cannot be part of the fixed part of the system $\L$. This contradiction shows that $\L$ must be empty. To finish we may therefore assume that $n=3$. Again the three lines through the three points split off the system, (in fact each splits off $2m-d$ times) leaving the residual system $\L(4d-6m,d-m,3,2d-3m)$. Therefore $\L$ is clearly empty if $2d<3m$. If $2d \geq 3m$, then the $3$ lines through $p_0$ and the $3$ points $p_i$ split off the residual system, since the intersection number $\L(4d-6m,d-m,3,2d-3m)\cdot\L(1,1,1,1) = (4d-6m)-(d-m)-(2d-3m) = d-2m<0$. Indeed, we see that these three lines each must split off $2m-d$ times, leaving as the further residual system $\L(7d-12m,4d-7m,3,3d-5m)$ if all of these numbers are non-negative. If $7d < 12m$ we see then that the residual system, and hence $\L$, is empty, and we are done. Note that if $7d \geq 12m$ then $3d \geq 5m$ (since $5/3<12/7$). If $4d\leq 7m$ then the residual system is $\L(7d-12m,0,3,3d-5m)$; again the $3$ lines through the $3$ points split off this system, each $2m-d$ times, leaving the system $\L(10d-18m,0,3,5d-9m)$; but if $4d\leq 7m$ then (since $7/4<9/5$) we have that $5d<9m$, so that this further residual system is empty, and we are done. If on the other hand $4d > 7m$ then the residual system is actually $\L(7d-12m,4d-7m,3,3d-5m)$, and all these numbers are strictly positive. Define the ratio $r = d/m$. We have shown that if $r\leq 7/4$ then $\L(d,d-m,3,m)$ is empty, while if $r > 7/4$ then $\l(d,d-m,3,m)=\l(7d-12m,4d-7m,3,3d-5m)$. This residual system is of the same form ($m_0=d-m$) and has as its ratio $s = (7d-12m)/(3d-5m) = (7r-12)/(3r-5)$. Therefore if $s < 7/4$, which happens for $r<13/7$, the system is again empty. We claim that by iterating this procedure enough times we will be done, i.e., for any $r < 2$ there is an iterate $s^{(n)}(r)$ which is less than $7/4$. The function $s(r)$ maps the interval $r\in(7/4,2)$ onto the interval $s \in (1,2)$, and moreover $s(r) < r$ for each such $r$. Hence the iterates $s^{(n)}(r)$ form a decreasing sequence, and if they never go below $7/4$, they must converge to a fixed point of the function $s(r)$. This is impossible, since the only fixed point is at $r=2$. \end{pf} The above Lemma allows the construction of an algorithm to compute $\l(d,d-m,n,m)$. If $n \geq 2$ and $d \geq 2m$ one uses (g) to reduce the numbers, and hence one may assume that either $d < 2m$ or $n<2$. Each of these cases is covered by the Lemma. One can turn this algorithm into a formula, and a criterion for speciality, without too much difficulty. \begin{proposition} \label{m0=d-m} Let $\L=\L(d,d-m,n,m)$ with $2 \leq m \leq d$. Write $d = qm+\mu$ with $0\leq \mu\leq m-1$, and $n = 2h+\epsilon$, with $\epsilon \in\{0,1\}$. Then the system $\L$ is special if and only if $q = h$, $\epsilon = 0$, and $\mu \leq m-2$. More precisely: \begin{itemize} \item[(a)] If $q \geq h+1$ then $\L$ is nonempty and non-special. In this case \[ \dim\L=d(m+1)-\binom{m}{2}-n\binom{m+1}{2}. \] \item[(b)] If $q=h$ and $\epsilon = 1$ the system $\L$ is empty and non-special. \item[(c)] If $q=h$, $\epsilon = 0$, and $\mu = m-1$, the system $\L$ is nonempty and non-special; in this case \[ \dim\L=(m-1)(m+2)/2. \] \item[(d)] If $q=h$, $\epsilon = 0$, and $\mu \leq m-2$, the system $\L$ is special; in this case \[ \dim \L = \mu(\mu+3)/2. \] \item[(e)] If $q\leq h-1$ the system $\L$ is empty and non-special. \end{itemize} \end{proposition} \begin{pf} If $q \geq h+1$, we may apply quadratic Cremona transformations $h$ times, arriving at the system $\L(d-hm,d-(h+1)m,\epsilon,m)$, which is nonempty and non-special. If $q \leq h$, we may apply quadratic Cremona transformations $q-1$ times, arriving at the system $\L(\mu+m,\mu,2(h-q)+\epsilon+2,m)$. If either $q < h$, or $\epsilon = 1$, then we apply Lemma \ref{m0=d-m_algorithm}(h) and conclude that $\L$ is empty, and therefore non-special. We are left with the case $q=h$ and $\epsilon = 0$, for which we have the system $\L(\mu+m,\mu,2,m)$; we then apply Lemma \ref{m0=d-m_algorithm}(e) to conclude the proof. \end{pf} This analysis of the $m_0=d-m$ case applies immediately when $m_0 > d-m$ also. Consider the system $\L(d,d-m+k,n,m)$ with $k \geq 1$. We note that the $n$ lines through $p_0$ and $p_i$ split off, each $k$ times, leaving as the residual system the system $\L(d-kn,d-kn-m+k,n,m-k)$ (which is of the type discussed above). The speciality of $\L$ is then deduced from this residual system: \begin{corollary} \label{m0=d-m+k} Let $\L = \L(d,d-m+k,n,m)$ with $k \geq 1$, and let \[ \L' = \L(d-kn,d-kn-m+k,n,m-k). \] Then $\dim \L = \dim \L'$ and $\L$ is non-special unless either \begin{itemize} \item[(a)] $k \geq 2$ and $\L'$ is nonempty and non-special, or \item[(b)] $\L'$ is special. \end{itemize} \end{corollary} Finally we turn to the case when $m_0 = d-m-1$. \begin{proposition} \label{m0=d-m-1} Let $\L=\L(d,d-m-1,n,m)$ with $2 \leq m \leq d-1$. Write $d = q(m-1)+\mu$ with $0\leq \mu \leq m-2$, and $n = 2h+\epsilon$, with $\epsilon \in\{0,1\}$. Then the system $\L$ is non-special of dimension $d(m+2)-(n+1)m(m+1)/2$ unless \begin{itemize} \item[(a)] $q=h+1$, $\mu=\epsilon=0$, and $(m-1)(m+2) \geq 4h$, in which case \[ \dim \L = (m-1)(m+2)/2 - 2h, \] or \item[(b)] $q = h$, $\epsilon = 0$, and $4q \leq \mu(\mu+3)$, in which case $\dim \L = \mu(\mu+3)/2 - 2q$. \end{itemize} \end{proposition} \begin{pf} First we note that performing a quadratic Cremona transformation to the system $\L(d,d-m-1,n,m)$ gives the subsystem of $\L(d-m+1,d-2m,n-2,m)$ (which is of the same type, namely ``$m_0=d-m-1$'') with two general simple base points. Therefore by induction if we perform $k$ such transformations, we obtain the subsystem of $\L(d-k(m-1),d-(k+1)(m-1)-2,n-2k,m)$ with $2k$ general simple base points, if $d-(k+1)(m-1)-2 \geq 0$ and $n \geq 2k$. If $q \geq h+2$, or if $q = h+1$ and $\mu \geq 2$, then we may perform $h$ quadratic Cremona transformations and arrive at the subsystem of $\L(d-h(m-1),d-(h+1)m-2,\epsilon,m)$ with $2h$ general simple base points. This system is non-special. We now analyze the case with $q=h+1$ and $\mu \leq 1$. If $m=2$ the system is easily seen to be empty, and therefore non-special; hence we assume that $m \geq 3$. We perform $h-1$ transformations, leading to the subsystem of $\L(d-(h-1)(m-1),d-h(m-1)-2,2+\epsilon,m) =\L(2m-2+\mu,m-3+\mu,2+\epsilon,m)$ with $2h-2$ general base points. We note that each line joining $2$ of the $2+\epsilon$ points of multiplicity $m$ in this system splits off, with multiplicity $2-\mu$; hence if $\mu = 0$ and the system is not empty, it is certainly special. If $\mu = 0$ and $\epsilon = 1$, when $m=3$ the original system is $\L(4,0,3,3)$, which is empty; if $m \geq 4$ the residual system is $\L(2m-8,m-3,3,m-4)$. Performing a Cremona transformation on this gives the obviously empty system $\L(m-4,m-3)$. If $\mu = 1$ and $\epsilon = 0$ the residual system is $\L(2m-2,m-2,2,m-1)$ which transforms to the non-special system $\L(m,0,2,1)$; hence $\L$ is non-special in this case. If $\mu=\epsilon = 1$, then the residual system is $\L(2m-4,m-2,3,m-2)$ which transforms to $\L(m-2,0,1,m-2)$ which is again non-special. If $\mu = \epsilon = 0$, then the residual system is $\L(2m-4,m-3,2,m-2)$ which transforms to $\L(m-1,0,2,1)$ and is therefore non-special. The original system is therefore nonempty if and only if $(m-1)(m+2) \geq 4h$, and this is the only special case. If $q \leq h$, we perform $q-1$ transformations, arriving at the system $\L(\mu+m-1,\mu-2,2(h-q)+\epsilon+2,m)$, with in addition $2q-2$ simple base points. (If $\mu = 0$ the system is empty, and therefore non-special; if $\mu = 1$, the system is also empty unless $q=h=1$ and $\epsilon=0$, but this implies $d=m$ which is impossible.) If $\mu \geq 2$, and $h > q$, then we argue as in the proof of Lemma \ref{m0=d-m_algorithm}(h) and conclude that the system is empty. We are left to analyze the case $\mu \geq 2$ and $h=q$. If $\epsilon=1$, then the three lines through the three points split off, each with multiplicity $m-\mu+1$; therefore if $2m \geq 4\mu-3$, the system is empty (the residual has negative degree). Otherwise the residual system is $\L(4\mu-2m-4,\mu-2,3,2\mu-m-2)$ and which transforms to $\L(2\mu-m-2,\mu-2)$. Since $\mu \leq m-2$, this system is empty. Finally we take up the case where $\mu \geq 2$, $h=q$ and $\epsilon=0$, in which case the line through the two remaining points splits off $m+1-\mu$ times, leaving the residual system $\L(2\mu-2,\mu-2,2,\mu-1)$ (with $2q-2$ simple base points). We perform one more transformation, giving the system of plane curves of degree $\mu$ with $2q$ simple base points, leading to the last exception. \end{pf} \section{$(-1)$-Special Systems with $m \leq 3$} Suppose $\L(d,m_0,n,m)$ is a $(-1)$-special system with $m \leq 3$. Then $\L$ must be of the form $\L = \mathcal{M} + NC$ for some $N = 2$ or $3$, where $C\in\L(\delta,\mu_0,n,1)$ is either a quasi-homogeneous $(-1)$-curve or a (compound) quasi-homogeneous $(-1)$-configuration, and $\v(\mathcal{M}) \geq 0$ and $\mathcal{M} \cdot C = 0$. This implies that $C$ is either: \begin{center} \begin{tabular}{cc} $\L(2,0,5,1)$ & \\ $\L(e,e-1,2e,1)$ & with $e \geq 1$, or \\ $\L(e,e,e,1)$ & with $e \geq 1$. \end{tabular} \end{center} The last one on the list is compound when $e \geq 2$; all others are irreducible $(-1)$-curves. These observations are sufficient to classify such systems. \begin{lemma} \label{obirreg23} The quasi-homogeneous $(-1)$-special systems with $m \leq 3$ are the systems $\L(d,m_0,n,m)$ on the following list: \begin{center} \begin{tabular}{cccc} $\L(4,0,5,2)$ & & $\v=-1$ & $\l = 0$ \\ $\L(2e,2e-2,2e,2)$, & $e \geq 1$ & $v = -1$ & $\l = 0$\\ $\L(d,d,e,2)$, & $d \geq 2e\geq 2$ & $\v = d-3e$ & $\l = d-2e$ \\ $\L(4,0,2,3)$ & & $\v=2$ & $\l = 3$ \\ $\L(6,0,5,3)$ & & $\v=-3$ & $\l=0$ \\ $\L(6,2,4,3)$ & & $\v=0$ & $\l=1$ \\ $\L(3e,3e-3,2e,3)$, & $e\geq 1$ & $\v = -3$ & $\l=0$ \\ $\L(3e+1,3e-2,2e,3)$, & $e\geq 1$ & $\v=1$ & $\l=2$ \\ $\L(4e,4e-2,2e,3)$, & $e\geq 1$ & $\v = -1$ & $\l=0$ \\ $\L(d,d-1,e,3)$, & $2d\geq 5e \geq 5$ & $\v=2d-6e$ & $\l = 2d-5e$ \\ $\L(d,d,e,3)$, & $d \geq 3e \geq 3$ & $\v = d-6e$ & $\l = d-3e$ . \end{tabular} \end{center} \end{lemma} \begin{pf} We'll only present the analysis for $m = 3$; the $m=2$ case is similar and easier, and we leave it to the reader. Using the notation above, with $\L = \mathcal{M} + N C$, in this case $N$ may be either $2$ or $3$. We first discuss when $N=2$. Suppose that $\L(2,0,5,1)$ splits twice off $\L(d,m_0,5,3)$. Then $\mathcal{M}=\L(d-4,m_0,5,1)$, and $\mathcal{M}\cdot\L(2,0,5,1) = 2d-13$, which can never be zero; hence this case cannot occur. Suppose that $A=\L(e,e-1,2e,1)$ splits twice off $\L(d,m_0,2e,3)$. Then $\mathcal{M}=\L(d-2e,m_0-2e+2,2e,1)$, and \begin{align*} 0 = \mathcal{M}\cdot A &= (d-2e)e-(m_0-2e+2)(e-1)-2e \\ &= de-m_0(e-1)-6e+2 &= e(d-m_0-6)+m_0+2 \end{align*} so that certainly $m_0\geq d-5$. Clearly for $\v(\mathcal{M}) \geq 0$ we must have $m_0\leq d-2$, but in this case \begin{align*} 0 &= de -m_0(e-1) -6e +2 \\ &\geq de -(d-2)(e-1) -6e +2 \\ &= d-4e \end{align*} so that $d \leq 4e$. We take up in turn the various possibilities for $m_0$. If $m_0=d-5$, then $e=m_0+2$ for $\mathcal{M}\cdot A = 0$, which gives that $m_0 = e-2$, a contradiction since $m_0 \geq 2e-2$ and $e \geq 1$. If $m_0=d-4$, then $\v(\mathcal{M}) = 3d-8e-1\geq 0$, but then $0=\mathcal{M}\cdot A = m_0+2-2e=d-2-2e$, so that $d=2e+2$ and hence $3(2e+2)-8e-1\geq 0$, forcing $e\leq 2$. When $e=1$ we have $\L(4,0,2,3)$, which has $\v = 2$ but $\l = 3$. When $e=2$ we have $\L(6,2,4,3)$, which has $\v=0$ but $\l=1$. If $m_0=d-3$, then $\v(\mathcal{M}) = 2d-6e$, so $d \geq 3e$; then $0=\mathcal{M}\cdot A = m_0+2-3e = d-3e-1$, so that $d=3e+1$. This gives the system $\L(3e+1,3e-2,2e,3)$, which has $\v=1$ but after splitting off $A$ twice leaves the system $\mathcal{M}=\L(e+1,e,2e,1)$, which has $\l=2$. If $m_0=d-2$, then $\v(\mathcal{M}) = d-4e$, so $d\geq 4e$, and now $0 = \mathcal{M}\cdot A = m_0+2-4e$, so that $m_0=4e-2$, forcing $d=4e$. This gives the special system $\L(4e,4e-2,2e,3)$, which has $\v = -1$ but consists of the fixed curve $2A$ and the residual system $\mathcal{M} = \L(2e,2e,2e,1)$ which is non-empty (it is a quasi-homogeneous $(-1)$-configuration, consisting of the $2e$ lines through $p_0$ and $p_i$). Finally suppose that the compound class $\L(e,e,e,1)$ splits twice off $\L(d,m_0,e,3)$. Here $A = \L(1,1,1,1)$, and $\mathcal{M}=\L(d-2e,m_0-2e,e,1)$, so that $0 = \mathcal{M}\cdot A = d-m_0-1$, forcing $m_0=d-1$. Then $\v(\mathcal{M}) = 2d-5e$, leading to the special systems $\L(d,d-1,e,3)$ with $2d\geq 5e$. This completes the analysis for $m=3$ and $N=2$. We now turn to the $m=N=3$ case. Suppose first that $A=\L(2,0,5,1)$ splits three times off $\L(d,m_0,5,3)$. Then $\mathcal{M}=\L(d-6,m_0)$, which has $\v\geq 0$ if $m_0\leq d-6$ and $\mathcal{M}\cdot A = 2d-12$, forcing $d = 6$, and $m_0=0$, leading to the system $\L(6,0,5,3)$. Suppose that $A=\L(e,e-1,2e,1)$ splits three times off $\L(d,m_0,2e,3)$. Then $\mathcal{M} = \L(d-3e,m_0-3e+3)$ so that $d\geq m_0+3$ for $\v(\mathcal{M})\geq 0$, and $0=A\cdot\mathcal{M} = de-m_0(e-1)-6e+3 = e(d-m_0-6)+m_0+3$, so that certainly $m_0\geq d-5$. If $m_0=d-5$, then $e=m_0+3=d-2$; but $e+2 = d\geq 3e$ forces $e=1$ and $m_0= -2$, a contradiction. If $m_0=d-4$, then $m_0=2e-3$, and so $d=2e+1$. Again since $d \geq 3e$ forces $e=1$ but then $m_0 = -1$, a contradiction. If $m_0 = d-3$, then $m_0 = 3e-3$, and so $d = 3e$, so that $\mathcal{M}=0$ and we have the system $\L(3e,3e-3,2e,3)$. Finally suppose that the compound class $\L(e,e,e,1)$ splits three times off $\L(d,m_0,e,3)$. Here $A = \L(1,1,1,1)$, and $\mathcal{M}=\L(d-3e,m_0-3e)$, so that $0 = \mathcal{M}\cdot A = d-m_0$, forcing $m_0=d$. Then $\v(\mathcal{M}) = d-3e$, leading to the special systems $\L(d,d,e,3)$ with $d\geq 3e$. This completes the $m=N=3$ analysis. \end{pf} Since we have discussed in some detail the speciality of systems $\L(d,m_0,n,m)$ with $m_0 \geq d-m-1$ in Section \ref{sec10}, we take the opportunity to make the following observation: \begin{corollary} Let $\L = \L(d,m_0,n,m)$ with $m_0 \geq d-m-1$ and $m \in \{2,3\}$. Then $\L$ is special if and only if $\L$ is $(-1)$-special. \end{corollary} \begin{pf} Let us first discuss the $m=2$ case. Suppose that $m_0 = d-3$. Then Proposition \ref{m0=d-m-1} gives no special systems. Suppose next that $m_0 = d-2$. The Proposition \ref{m0=d-m} gives that the only such special system is $\L(2e,2e-2,2e,2)$, which is $(-1)$-special. If $m_0 = d-1$, Corollary \ref{m0=d-m+k} gives no special systems. Finally if $m_0 = d$, Corollary \ref{m0=d-m+k} gives the assertion. Now we turn to the $m=3$ case, and first suppose that $m_0 = d-4$. Then Proposition \ref{m0=d-m-1}(a) leads only to the systems $\L(4,0,2,3)$ and $\L(6,2,4,3)$ which are $(-1)$-special. Proposition \ref{m0=d-m-1}(b) gives no special systems with $m=3$. Next suppose that $m_0=d-3$. Then Proposition \ref{m0=d-m} gives that the only such special systems are $\L(3e,3e-3,2e,3)$ and $\L(3e+1,3e-2,2e,3)$ for $e \geq 1$. If $m_0 = d-2$, Corollary \ref{m0=d-m+k} gives only the special systems $\L(4e,4e-2,2e,3)$, for $e\geq 1$. The other cases of $m_0 \geq d-1$ are trivial. \end{pf} \section{The Classification of Special Systems with $m \leq 3$} \begin{theorem} \label{thmreg2} A system ${\L}(d,m_0,n,m)$ with $m \leq 3$ is special if and only if it is a $(-1)$-special system, i.e., it is one of the systems listed in Lemma \ref{obirreg23}. \end{theorem} \begin{pf} We will outline the proof in the case $m=3$; the $m=2$ case is analogous in every way. We will assume that $m_0 \neq 1,3$. We may also assume $\L$ is not empty, otherwise it is certainly non-special. We will prove the theorem by induction on $n$; the assertion is true for $n\leq 2$. So we may assume $n\geq 3$. The theorem is easily seen to be true for $d\leq 5$. So we will assume $d\geq 6$. Furthermore the cases $d<m_0$ are trivial and the cases $d\leq m_0+4$ are taken care of by the results of Section \ref{sec10}. Hence we may assume that $d \geq m_0+5$, or that $m_0 \leq d-5$. The general approach is to assume $\L$ is not $(-1)$-special and prove that it is non-special. We proceed by induction and we assume the theorem holds for lower values of $n$. We start with the case ${\v}\leq -1$. We perform a $(2,b)$-degeneration and we require that both kernel linear systems $\hat{\mathcal{L}}_\mathbb{P}$, $\hat{\mathcal{L}}_\mathbb{F}$ are empty. The requirement that $\hat{\mathcal{L}}_\mathbb{F}={\L}(d,d-1,b,3)$ is empty translates into the inequality $5b>2d$. As for requirement that $\hat{\mathcal{L}}_\mathbb{P}={\L}(d-3,m_0,n-b,3)$ is empty, we can use induction and impose that its virtual dimension ${\hat{v}_\mathbb{P}}$ is negative, unless $\hat{\mathcal{L}}_\mathbb{P}$ is a $(-1)$-special system. This might only happen (when $m_0 \leq d-5$) if: (i) $m_0=d-5$, $d-3=4e$, $n-b=2e$; (ii) $m_0=d-6$, $d-3=3e,3e+1$, $n-b=2e$; (iii) $d=9, m_0=0, n-b=5$; (iv) $d=9, m_0=2, n-b=4$; (v) $d=7, m_0=0, n-b=2$. Since ${\v}(d,m_0,n,3)\leq -1$, the condition ${\v}(d-3,m_0,n-b,3)<0$ is implied by ${\v}(d-3,m_0,n-b,3)\leq {\v}(d,m_0,n,3)$, which reads $2b\leq d$. In conclusion we need to choose a $b$ such that $\frac{2d}{5} < b \leq \frac{d}{2}$. For later purposes we will need $b < \frac{d}{2}$ if $d$ is divisible by $4$. Since $\frac{d}{2} - \frac{2d}{5} = \frac{d}{10}$ we see that we can choose a suitable $b$ as soon as $d\geq 11$, but in fact one sees that for $d$ between $6$ and $10$ and unequal to $8$, the maximum $b$ such that $2b \leq d$ works. On the other hand one proves directly the theorem for $d=8$, so we can dispense with this case. With this choice, only the $(-1)$-special systems with negative virtual dimension could occur, i.e. (i), (ii) with $d=3e+3$, (iii). We also remark that, with this choice of $b$, one has $b<n$. Suppose we are in case (i). Then $d=4e+3$ and if $d \geq 19$, then we may choose $b$ in at least two ways, and avoid the $(-1)$-special systems. We are left with the cases $d=7,11,15$. Then $b=3,5,7$ respectively, and, if $\hat{\mathcal{L}}_\mathbb{P}$ is a $(-1)$-special system, then $n=5,9,13$ respectively. But in each one of these cases the virtual dimension of $\L$ is $2$, contrary to the hypothesis. Suppose we are in case (ii), with $d=3e+3$. Again we can choose $b$ in at least two ways, and avoid the $(-1)$-special systems, as soon as $d\geq 18$. The remaining cases $d=6,9,12,15$, in which the values of $b$ are $b=3,4,5,7$ respectively and therefore $n=5,8,11,15$, also have non-negative virtual dimension for $d \geq 9$; the case $d=6$ is a $(-1)$-special system. In case (iii), $b=4$ and therefore $n=9$, in which case $\L$ has virtual dimension $0$, a contradiction. We have therefore arranged to choose a $b$ in such a way that both kernel linear systems $\hat{\mathcal{L}}_\mathbb{F}$ and $\hat{\mathcal{L}}_\mathbb{P}$ are empty, unless $\L$ is $(-1)$-special. Now we claim that, with the choices we made, ${\mathcal{L}_0}$ is empty, which implies that also $\L$ is empty, hence non-special. The assertion is clear if either one of the two systems $\mathcal{L}_\bP$, $\mathcal{L}_\bF$ is empty. So we may assume that both systems $\mathcal{L}_\bP$ and $\mathcal{L}_\bF$ are not empty. The system $\mathcal{L}_\bF$ is ${\L}(d,d-2,b,3)$ and ${v_\mathbb{F}}=3d-1-6b$. The $b$ lines through $p_0$ split off this system, and the residual system is $\L(d-b,d-b-2,b,2)$, which by Theorem \ref{thmreg2} and Lemma \ref{obirreg23} could only be special if $d=4e$ and $b=2e$. Because of our choice of $b$, this does not occur; hence $\mathcal{L}_\bF$ is non-special and therefore its dimension is ${\ell_\mathbb{F}}={v_\mathbb{F}}=3d-1-6b > -1$. Let us examine the possibility that $\mathcal{L}_\bP=\L(d-2,m_0,n-b,3)$ is a $(-1)$-special system. For $d \geq 9$, or $d=6$ or $7$, this can only be the case if: (i) $m_0=d-5$, $d=3e+2,3e+3$, $n=b+2e$; (ii) $d=6, m_0=0, n=b+2$. In case (ii) $\L$ is the $(-1)$-special system $\L(6,0,5,3)$. In cases (i) with $d=3e+2$, then ${r_\mathbb{P}} = \ell_\mathbb{P} =0$; since $b>0$, the linear series $\mathcal{R}_\mathbb{F}$ contains $b$ general fixed points, and hence $\mathcal{R}_\mathbb{F} \cap \mathcal{R}_\mathbb{P}$ is empty. Therefore by Lemma \ref{dimLo2} ${\mathcal{L}_0} = -1$. Similarly in case (i), if $d=3e+3$, then ${\ell_\mathbb{P}}=2$ but $b>2$, which again forces $\mathcal{R}_\mathbb{F} \cap \mathcal{R}_\mathbb{P}$ to be empty. Hence we may now assume that also $\mathcal{L}_\bP$ is non-special. Now we conclude that $\L$ is empty by Corollary \ref{cordimLk}(a). Now we consider the case ${\v}\geq 0$. Again we may assume $m_0 \leq d-5$. We observe that we may assume $n>\frac{5}{6}d-1$, otherwise ${\v}(d,d-4,n,3)\geq 1$ and therefore ${\L}(d,d-4,n,3)$ is non-special, hence also ${\L}(d,m_0,n,3)$ is non-special. We propose to perform a $(3,h)$-degeneration, where we write $d = 2h-\epsilon$ with $\epsilon \in \{0,1\}$, With this choice $\hat{\mathcal{L}}_\mathbb{F}=\L(d,d-2,h,3)$ is empty if $d$ is not divisible by $4$, and has dimension $0$ if $4$ divides $d$, since $\l(d,d-2,h,3)=\l(d-h,d-h-2,h,2)$ and $\v(d-h,d-h-2,h,2) = 3d-6h-1 = -3\epsilon-1 < 0$, and is non-special unless it is $(-1)$-special, which can only happen if $h=2e$ and $d=2h$. Now $\mathcal{L}_\bF = \L(d,d-3,h,3)$ which is never $(-1)$-special; hence $\ell_\mathbb{F} = \v(d,d-3,h,3) = 4d-3-6h = d-3(\epsilon+1)$. Therefore $r_\mathbb{F} = d-3(\epsilon+1) - \eta$ where $\eta = 1$ if $4|d$ and is zero otherwise. Now note that $v_\mathbb{P}-\v = \v(d-3,m_0,n-h,3)-\v(d,m_0,n,3) = 6h-3d = 3\epsilon \geq 0$, so that $\mathcal{L}_\bP$ is nonempty. Suppose first that $\mathcal{L}_\bP$ is non-special, and $\hat{\mathcal{L}}_\mathbb{P}$ is empty. Then we have $r_\mathbb{P} = v_\mathbb{P} = \v+3\epsilon$, and so $r_\mathbb{F}+r_\mathbb{P} = d-3 - \eta + \v \geq d-4$. Hence we apply Proposition \ref{dimLo3}(b) and finish. Suppose next that $\mathcal{L}_\bP$ is non-special, and $\hat{\mathcal{L}}_\mathbb{P}$ is nonempty and non-special. Then $r_\mathbb{P}=v_\mathbb{P}-\hat{v}_\mathbb{P}-1 = d-3$ so that Proposition \ref{dimLo3}(b) applies and we finish. Next we suppose that $\mathcal{L}_\bP$ is non-special but $\hat{\mathcal{L}}_\mathbb{P}$ is nonempty and special. In all the possibilities for special $\hat{\mathcal{L}}_\mathbb{P}$ with $d \geq 9$, except one, we have $\hat{\ell}_\mathbb{P} \leq 2$; hence $r_\mathbb{P} \geq d-6$, which forces $r_\mathbb{P}+r_\mathbb{F} \geq d-4$ easily, and we again finish using Proposition \ref{dimLo3}(b). The one exception to this is when $m_0 = d-5$ and $2d-8 \geq 5(n-h)$. However in this case $\hat{\ell}_\mathbb{P} = 2(d-4)-5(n-h)$, and since $\v \geq 0$, we see easily that $r_\mathbb{P} +r_\mathbb{F} \geq d-4$ again. Finally we must discuss the case that ${\mathcal{L}_\bP}$ is $(-1)$-special. Since we have already shown that $v_\mathbb{P} \geq 0$, this can only be the case (given that $d\geq 9$) when: (i) $d=9$, $m_0=2$, $h=5$, $n = 9$; (ii) $d=3e+4$, $m_0=3e-2$, $n=h+2e$. In the first case we see that $\v = \v(9,2,9,3) = -3$, a contradiction. As to the second case, we suggest instead a $(3,h+1)$-degeneration. In this case $\hat{\mathcal{L}}_\mathbb{F}$, which had dimension at most $0$ before, is now empty. Moreover $\mathcal{L}_\bF = \L(d,d-3,h+1,3)=\L(3e+4,3e+1,(3e+6+\epsilon)/2,3)$ is again never $(-1)$-special. Hence $r_\mathbb{F} = \ell_\mathbb{F} = v_\mathbb{F} = 3e-5-3\epsilon$. Now $\hat{\mathcal{L}}_\mathbb{P} = \L(3e,3e-2,2e-1,3)$ which is never $(-1)$-special; since $\hat{v}_\mathbb{P} = 5-3e < 0$, we see that $\hat{\mathcal{L}}_\mathbb{P}$ is empty. The system $\mathcal{L}_\bP = \L(3e+1,3e-2,2e-1,3)$ is also never $(-1)$-special; moreover $v_\mathbb{P} = 7$, so that $\mathcal{L}_\bP$ is not empty and also $r_\mathbb{P} = 7$. Therefore $r_\mathbb{F}+r_\mathbb{P} = 3e+2-3\epsilon = d-2-3\epsilon$; hence if $\epsilon = 0$, we conclude the proof using Proposition \ref{dimLo3}(b). However if $\epsilon = 1$, then $e$ is odd, and $h = (3e+5)/2$, so that $n = (7e+5)/2$; computing the virtual dimension of the system $\L$ we find $\v = -2$, a contradiction. This completes the proof in the $m=3$ case. As noted above, we leave the details of the $m=2$ case to the reader. The outline of the proof is to make a $(1,b)$ degeneration, and again use induction on $n$. We may assume that $d \geq m_0+4$. When ${\v}\leq -1$, we need to prove that if $\L$ is not $(-1)$-special, then it is empty. We perform a $(1,b)$-degeneration, where $b$ is the minimum integer with $2b>d$; the relevant linear systems are $\mathcal{L}_\bP = \L(d-1,m_0,n-b,2)$, $\mathcal{L}_\bF = \L(d,d-1,b,2)$, $\hat{\mathcal{L}}_\mathbb{P} = \L(d-2,m_0,n-b,2)$, and $\hat{\mathcal{L}}_\mathbb{F} = \L(d,d,b,2)$. The reader can check that with this choice of $b$, both $\hat{\mathcal{L}}_\mathbb{F}$ and $\hat{\mathcal{L}}_\mathbb{P}$ are empty. Hence if we can show that $\ell_\mathbb{P}+\ell_\mathbb{F} \leq d-2$, then we can apply Corollary \ref{cordimLk}(a) and conclude that $\L$ will be empty. This is automatic if either $\mathcal{L}_\bP$ or $\mathcal{L}_\bF$ is empty, so one may assume that both systems are not empty. If so, one checks as in the $m=3$ case that we do have $\ell_\mathbb{P}+\ell_\mathbb{F} \leq d-2$ as needed. In case ${\v}\geq 0$, still assuming that $d \geq 4$ and $m_0 \leq d-3$. We notice we can also assume $n\geq d$; otherwise, as we proved already, ${\L}(d,d-2,n,2)$ is non-special and not empty, and therefore also ${\L}(d,m_0,n,2)$ is non-special for all $m_0\leq d-3$ by Lemma (0.1)\ref{basic1}(c). Again we perfom a $(1,b)$-degeneration, and we take $b$ to be the maximum such that $2b \leq d+1$. Then $\hat{\ell}_\mathbb{F} = \l(d,d,b,2) = d-2b$, and $\mathcal{L}_\bF$ has dimension $\ell_\mathbb{F} = 2d-3b$. Therefore the system $\mathcal{L}_\bF$ is non-special, and $r_\mathbb{F} = \ell_\mathbb{F}-\hat{\ell}_\mathbb{F}-1 = (2d-3b)-(d-2b)-1 = d-b-1$. One checks that $\mathcal{L}_\bP$ is non-special and non-empty. Since we already know that $\mathcal{L}_\bF$ is non-special and non-empty, we will be done if we show that $r_\mathbb{P}+r_\mathbb{F} \geq d-2$, by applying Proposition \ref{dimLo3}(b), and noting that in this case $\ell_0=\ell_\mathbb{P}+\ell_\mathbb{F}-d+1=v_\mathbb{P}+v_\mathbb{F}-d+1=\v$. Since we have seen above that $r_\mathbb{F}=d-b-1$, we need only to show that $r_\mathbb{P} \geq b-1$ to finish the proof. This we leave to the reader. \end{pf}

9702

alg-geom/9702013

en

https://arxiv.org/abs/alg-geom/9702013

alg-geom/9702013

Gian Mario Besana

Alberto Alzati, Marina Bertolini, Gian Mario Besana

Numerical Criteria for vey Ampleness of Divisors on Projective Bundles over an elliptic curve

AMS-Latex, 18 pages, Canadian Journal of Math, Dec 1996

In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a sufficient condition for a line bundle associated with a divisor D to be normally generated on $X=P(E)$ where E is a vector bundle over a smooth curve C. A line bundle which is ample and normally generated is automatically very ample. Therefore the condition found in Butler's work, together with Miyaoka's well known ampleness criterion, give a sufficient condition for the very ampleness of D on X. This work is devoted to the study of numerical criteria for very ampleness of divisors D which do not satisfy the above criterion, in the case of C elliptic. Numerical conditions for the very ampleness of D are proved,improving existing results. In some cases a complete numerical characterization is found.

alg-geom

\section{Introduction} Ampleness of divisors on algebraic varieties is a numerical property. On the other hand it is in general very difficult to give numerical necessary and sufficient conditions for the very ampleness of divisors. In \cite{bu} the author gives a sufficient condition for a line bundle associated with a divisor $D$ to be normally generated on $X =\Bbb{P}(E)$ where $E$ is a vector bundle over a smooth curve $C.$ A line bundle which is ample and normally generated is automatically very ample. Therefore the condition found in \cite{bu}, together with Miyaoka's well known ampleness criterion, give a sufficient condition for the very ampleness of $D$ on $X.$ This work is devoted to the study of numerical criteria for very ampleness of divisors $D$ which do not satisfy the above criterion, in the case of $C$ elliptic. With this assumption Biancofiore and Livorni \cite{bi-li3} (see also \cite[Prop.8.5.8]{BESO} for a generalization) gave a necessary and sufficient condition when $E$ is indecomposable, rk$E$ = 2 and deg$E$ = 1. Gushel \cite{gu} also gave a complete characterization of the very ampleness of $D$ assuming that $E$ is indecomposable and $|D|$ embeds $X$ as a scroll. This work deals with the general situation and addresses the cases still open. The main technique used here is a very classical one. A suitable divisor $A$ on $X$ is chosen such that there exists a smooth $S \in |A|$ containing every pair of points, possibly infinitely near. Appropriate vanishing conditions are established to assure that the natural restriction map $H^0(X, \cal{O}_X(D)) \to H^0(S, \cal{O}_X(D)_{|S})$ is surjective. In this way we get that a divisor $D$ of $X$ is very ample if and only if $D_{|S}$ is very ample. In this context $S$ is chosen as $S = \Bbb{P}(E')$ where $E'$ is a quotient of $E,$ thus with rank smaller than rank $E.$ Therefore an inductive process on the rank can be set up. This process is not always easy to carry on. For example if $E$ is assumed to be indecomposable there is no guarantee that $E'$ will still be indecomposable. Since ampleness is inherited by quotients, we will require at some stage that $E$ be ample. The paper is organized as follows. Section 2 contains notation, known and preliminary results used in the sequel. In section 3 the case of rank $E = 2$ is fully treated. We recover Biancofiore and Livorni's results and deal with the case of $E$ decomposable. Section 4 deals with the case of rank $E = 3$ while section 5 contains the study of case rank $E \ge 4.$ In particular in the case of rank $E = 3$ we get the following result (see section \ref{notation} for notation): \begin{introteo} Let $E$ be a rank $3$ vector bundle on an elliptic curve $C$ and let $D\equiv aT+bf$ be a line bundle on $ X =\Bbb{P}(E).$ \begin{itemize} \item[(a)] If $E$ is indecomposable then \begin{itemize} \item[(a1)] if $d=0$ (mod $3$) , $D$ is very ample if and only if $b+a\mu^-(E) \geq 3$ \item[(a2)]if $d=1$ (mod $3$) , $D$ is very ample if $b+a\mu^-(E) > 1$ \item[(a3)]if $d=2$ (mod $3$) , $D$ is very ample if $b+a\mu^-(E) >\frac{4}{3}$ \end{itemize} \item[(b)] if $E$ is decomposable, then $D$ is very ample if and only if $b+a\mu^-(E) \geq 3$ except when $E=E_1\oplus E_2$, with rk$E_1=1$, rk$E_2=2$, deg$E_2$ odd and deg$E_1> \frac {degE_2}2.$ In the latter case the condition is only sufficient. \end{itemize} \end{introteo} Notice that the above theorem shows the existence, among others, of a smooth threefold of degree 20 embedded in $\Bbb{P}^9$ as a fibration of Veronese surfaces over an elliptic curve, choosing $a = 2, b= -1$ and $ d= 4.$ The authors would like to thank Enrique Arrondo and Antonio Lanteri for their friendly advice. The third author would like to thank the Department of Mathematics of Oklahoma State University for the kind hospitality and warm support during the final stages of this work. \section{General Results and Preliminaries} \label{prelimsec} \subsection{Notation} \label{notation} The notation used in this work is mostly standard from Algebraic Geometry. Good references are \cite{H} and \cite{gh}. The ground field is always the field $\Bbb{ C}$ of complex numbers. Unless otherwise stated all varieties are supposed to be projective. $\Bbb{P}^{n}$ denotes the n-dimensional complex projective space and $\Bbb{C}^*$ the multiplicative group of non zero complex numbers. Given a projective n-dimensional variety $X$, ${\cal O}_X$ denotes its structure sheaf and $Pic(X)$ denotes the group of line bundles over $X.$ Line bundles, vector bundles and Cartier divisors are denoted by capital letters as $L, M, \dots.$ Locally free sheaves of rank one, line bundles and Cartier divisors are used interchangeably as customary. Let $L, M \in Pic(X)$, let $E$ be a vector bundle of rank $r$ on $X$, let $\cal{F}$ be a coherent sheaf on $X$ and let $Y\subset X$ be a subvariety of $X.$ Then the following notation is used: \begin{enumerate} \item[ ] $L C$ the intersection number of $L$ with a curve $C,$ \item[ ] $L^{n}$ the degree of $L,$ \item[ ] $|L|$ the complete linear system of effective divisors associated with $L$, \item[ ]$L_{|Y}$ the restriction of $L$ to $Y,$ \item[ ] $L \sim M$ the linear equivalence of divisors \item[ ] $L \equiv M$ the numerical equivalence of divisors \item[ ] Num$(X)$ the group of line bundles on $X$ modulo the numerical equivalence \item[ ] $ E^*$ the dual of $E.$ \item[ ] $\Bbb{P}(E)$ the projectivized bundle of $E$ \item[ ] $H^i(X, \cal{F})$ the $i^{th}$ cohomology vector space with coefficient in ${\cal F},$ \item[ ] $h^i(X,\cal{F})$ the dimension of $H^i(X, \cal{F}).$ \end{enumerate} If $C$ denotes a smooth projective curve of genus $ g$, and $E$ a vector bundle over $C$ of deg $E= c_1(E)= $d and rk $E=r$, we need the following standard definitions: \begin{enumerate} \item[ ] $E$ is $\it normalized$ if $h^0(E)\ne 0$ and $h^0(E \otimes L)=0$ for any invertible sheaf $L$ over $C$ with deg$L<0$. \item[ ] $E$ has slope $\mu(E) = \frac{d}{r}$. \item[ ] $E$ is $\it semistable$ if and only if for every proper subbundle $S$, $\mu(S) \leq \mu(E)$. It is $\it stable$ if and only if the equality is strict. \item[ ] The Harder-Narasimhan filtration of $E$ is the unique filtration: $$0=E_0\subset E_1\subset ....\subset E_s=E$$ such that $\frac{E_i}{E_{i-1}}$ is semistable for all $i$, and $\mu_i(E)=\mu (\frac{E_i}{E_{i-1}})$ is a strictly decreasing function of $\it i$. \end{enumerate} We recall now some definitions from \cite{bu} which we will use in the following: let $0=E_0 \subset E_1 \subset ....\subset E_s=E$ be the Harder-Narasiman filtration of a vector bundle $E$ over $C$. Then \begin{enumerate} \item[]$\mu^-(E)=\mu_s(E)=\mu (\frac{E_s}{E_{s-1}})$ \item[]$\mu^+(E)=\mu_1(E)=\mu (E_1)$ \item[]or alternatively \item[]$\mu^+(E)= $max $\{\mu(S) |0 \to S \to E \}$ \item[]$\mu^-(E)= $min $\{\mu(Q) |E \to Q \to 0 \}$. \end{enumerate} We have also $\mu^+(E) \geq \mu(E) \geq \mu^-(E)$ with equality if and only if $E$ is semistable. In particular if $C$ is an elliptic curve, an indecomposable vector bundle $E$ on $C$ is semistable and hence $\mu(E) = \mu^-(E)$. Moreover if $F,G$ are indecomposable and hence semistable vector bundles on an elliptic curve $C$ and $F \to G$ is a non zero map, it follows that $\mu(F)\leq\mu(G)$. \medskip \subsection{General Results} \medskip Let $C$ be a smooth projective curve of genus $g$, $E$ a vector bundle of rank $r$, with $r \ge 2$, over $C$ and $\pi : X =\Bbb{P}(E) \to C $ the projective bundle associated to $E$ with the natural projection $\pi$. With standard notations denote with $\cal {T} = \cal {O}_{\Bbb{P}(E)} (1) $ the tautological sheaf and with $\cal {F}_P= \pi^*\cal {O}_{C}(P) $ the line bundle associated with the fiber over $P\in C.$ Let $T$ and $f$ denote the numerical classes respectively of $\cal T$ and $\cal {F}_P$. Let $D\sim a\cal{T} + \pi^*B$, with $ a\in \Bbb{Z}$, $B\in Pic(C)$ and deg$B = b$, then $ D \equiv aT+bf .$ Moreover $\pi_*D = S^{a}(E) \otimes \cal {O}_{C}(B) $ and hence $\mu^-(\pi_*D)=a\mu^-(E) +b$ (see \cite{bu}). Regarding the ampleness, the global generation, and the normal generation of $D$, the following criteria are known: \begin{Thm}[Miyaoka \cite{Miyao3}] \label{miyaoteo} Let $E$ be a vector bundle over a smooth projective curve $C$ of genus $g$, and $X =\Bbb{P}(E)$ . If $D\equiv aT+bf$ is a line bundle over $X$, then $D$ is ample if and only if $a>0$ and $b+a \mu^-(E) >0$. \end {Thm} \begin{Prop}[Gushel \cite{gu2}, proposition 3.3] \label{guongg} Let $D \sim a\cal{T}+\pi^*B$ where $a > 0$ and $B \in$ Pic($C$), be a divisor on a projective bundle $\pi : X =\Bbb{P}(E) \to C $ . Then: \begin{itemize} \item[i)]if $a=1$, the bundle $\pi_* (D)$ is generated by global sections if and only if the divisor $D$ is \item[ii)]if $a \geq 2$, and the vector bundle $\pi_*(D)$ is generated by global sections, then also the divisor $D$ is. \end {itemize} \end {Prop} \begin {Lem}[Gushel \cite{gu}, Proposition 3.2] \label{ggforindec} Let $E$ be an indecomposable vector bundle over an elliptic curve $C$. $E$ is globally generated if and only if deg$E > $rank$E$. \end{Lem} \begin{Lem} \label{buongg} (see e.g. \cite{bu}, lemma 1.12) Let $E$ be a vector bundle over $C$ of genus $g$. \begin{itemize} \item[i)]if $\mu^-(E) > 2g-2$ then $h^1(C,E)=0$ \item[ii)]if $\mu^-(E) > 2g-1$ then $E$ is generated by global sections. \end{itemize} \end{Lem} For the following theorem we need a definition: \begin{Def}[Butler, \cite{bu}] Let $E$ be a vector bundle over a variety $Y$, and let $\pi: X =\Bbb{P}(E) \to Y$ be the natural projection. A coherent sheaf $\cal F$ over $X$ is said to be $t \pi-regular$ if, for all $i>0$, $$ \cal {R}^{i}\pi_{*}(\cal {F}(t-i)) =0. $$ \end{Def} \begin{Thm}[Butler,\cite{bu}] Let $E$ be a vector bundle over a smooth projective curve $C$ of genus $g$, and $X =\Bbb{P}(E)$ . If $D$ is a $(-1){\pi}-$ regular line bundle over X, with $\mu^-(\pi_*D) >2g $, then $ D$ is normally generated. \label{butlerteo} \end{Thm} \begin{rem} \label{criteriodelbutler} Let $D$ be a divisor of $X =\Bbb{P}(E)$, with $E$ vector bundle on a smooth projective curve of genus $g.$ As $h^{i}(\cal {F}_p,D_ {|\cal{F}_p}(-1-i))=0$ for $i\ge1$ , the $(-1)\pi $- regularity of $D$ is satisfied, hence the condition $a\mu^-(E) +b >2g$ implies that $D$ is normally generated. If a line bundle $D$ on a projective variety $X$ is ample and normally generated it is very ample. Hence from Theorem \ref{miyaoteo} and \ref{butlerteo} we get that $D$ is very ample on $ X =\Bbb{P}(E) $ if \begin{equation} \label{condizionedelbutler} b+a\mu^-(E) > 2g. \end{equation} Hence, if $g=1$, the very ampleness of $D \equiv aT+bf$ is an open problem only in the range \begin{equation} \label{range} 0<b+a\mu^-(E) \le2. \end{equation} \end{rem} \medskip \subsection{Preliminaries} \medskip The following result is standard from the theory of vector bundles (see \cite{H}): \begin {Lem} \label{degofnormed} Let $E$ be an indecomposable vector bundle of rank $r$ on an elliptic curve. If $E$ is normalized then $0 \leq deg E \leq r-1$. \end {Lem} \begin{Lem} Let $E=\bigoplus_{i=1}^{n} E_i$ be a decomposable vector bundle over an elliptic curve $C$, with $E_i$ indecomposable vector bundles. Then $\mu^-(E)=$ min$ \,\mu(E_i)$. \label{mimenodec} \end{Lem} \begin{pf} For the proof we need the following three claims. \begin{claim} Let $E=\bigoplus_{i} E_i$ be as above, then $\mu(E) \ge$ min$\,\mu(E_i)$. \label{claim1} \end{claim} \begin{pf} Let us denote by $r=rk(E)$ $r_i=rk(E_i)$ $d=deg(E)$, $d_i=deg(E_i)$. Let us consider the vectors $\underline{v}_i$ in $\Bbb{R}^2$ whose coordinates are $(r_i,d_i)$ and the vector $\underline{v} = \sum_{i} \underline{v}_i$. Let $\alpha_i$ be the angle between the $r-$axis and $\underline {v}_i$. Let $\alpha$ be the angle between the $r-$axis and $\underline {v}$. It is $\mu(E) = \frac{d}{r}=$ tg$(\alpha) \geq $ min$_i$ tg$(\alpha_i) = $ min$_i$ $(\frac{d_i}{r_i}) = $ min$_i$ $ \mu(E_i)$. \end{pf} \begin{claim} Let $E=\bigoplus_{i} E_i$ be as above, and $\mu(E_i)=\frac{d_i}{r_i}=h \in \Bbb {Q},$ for all $i$. Then $\, \mu^-(E) = h$. \label{claim2} \end{claim} \begin{pf} Notice that under this hypothesis $\mu(E)=h$. Moreover, by definition, it is $\mu^-(E) =$ min $\{\mu(Q)\, | E \to Q \to 0 \}$. If $Q$ is decomposable in the direct sum of indecomposable vector bundles $ Q_k$, the existence of a surjective map $ E \to Q \to0 $ implies the existence of surjective maps $ E \to Q_k \to 0 $ for all $k$ and consequently from Claim \ref{claim1}, $\mu^-(E) =$ min $\{\mu(Q)\, | E \to Q \to 0,$ and $ Q$ indecomposable \}. Now let $Q_o$ be an indecomposable vector bundle which realizes the minimum, i.e. $\mu(Q_o)= \mu^-(E).$ From $\oplus_i E_i \to Q_o \to 0 $ it follows that there exists at least an index ${i_0}$ such that the map $E_{i_0} \to Q$ is not zero and $\mu(E_{i_0})\leq\mu(Q_o).$ Therefore it is $h\leq\mu^-(E)$. As $h\geq\mu^-(E)$, the Claim is proved. \end{pf} \begin{claim} Let $E=\bigoplus_{i}E_i$ be as in Claim (2). Then E is semistable. \label{claim3} \end{claim} \begin{pf} It is enough to prove that for any $ S$ vector bundle on $C$ such that there exists a map $0 \to S \to E$ then $\mu(S) \leq \mu(E) =h$. If we consider the dual map $E^* \to S^* \to 0$ we have $\mu(S^*) \geq \mu^-(E^*) =\mu^-(\bigoplus_{i} E{_i}{^*})$ and, as $\mu(E^*{_i}) =- \frac{d_i}{r_i} = -h$, from Claim \ref{claim2} applied to $E^*$ we have $\mu^-(E^*)= -h$. Hence $\mu(S^*)=-\mu(S) \geq -h$ and $\mu(S) \leq h$. \end{pf} The Lemma can now be proved. Let $E=\bigoplus_{i} E_i$ be as in the hypothesis of Lemma, and denote by $\mu_i =\mu(E_i)$. We can choose an ordering such that $E=E_1 \oplus E_2 \oplus E_3 .....$ and $\mu_1\geq \mu_2 \geq \mu_3 ...$. Let $E=\bigoplus_{k=1}^{s} A_k$ be a new decomposition of $E$ such that each $A_k$ is an indecomposable vector bundle or a sum of indecomposable vector bundles $E_i$ with the same $\mu_i$. In this way we get a strictly decreasing sequence $\mu(A_1)>\mu(A_2)>...>\mu(A_s)$, and by claim (3) each $A_k$ is semistable. Moreover the sequence $ 0\subset F_1 \subset F_2 \subset ... \subset F_s =E$ with $F_i = A_1 \oplus A_2 \oplus ...\oplus A_i$ with $ 1\leq i \leq s$, is the Harder-Narasimhan filtration of $E$ because the sequence of the slopes $\mu(\frac {F_i}{F_{i-1}}) = \mu(A_i)$ is strictly decreasing and each $\frac {F_i}{F_{i-1}}=A_i$ is semistable for all $i=1...s$. Hence we get $ \mu^-(E) = \mu(\frac {E_s}{E_{s-1}}) = \mu(A_s) =$ min $ \mu(E_i).$ \end {pf} \begin{Lem} Let $D\sim a\cal{T} + \pi^*B$ be a line bundle in $X =\Bbb{P}(E)$ over a curve $C$ of genus $g=1$, with $B\in Pic(C)$ , $a \geq 1$ and deg$B = b $ . \begin{itemize} \item[i)]If $a=1$ $D$ is globally generated if and only if $b+ \mu^-(E) > 1$ \item[ii)]If $a\geq2$ $D$ is globally generated if $b+a\mu^-(E) > 1$ \end{itemize} \end{Lem} \begin{pf} To prove ii) it is sufficient to apply Proposition \ref{guongg} and Lemma \ref{buongg}. To prove i) notice that if $E$ is indecomposable it is enough to apply Proposition \ref{guongg} and Lemma \ref{ggforindec} , observing that an indecomposable vector bundle $E$ over an elliptic curve is semistable and hence $\mu^-(E)=\mu(E).$ Let now $E$ be decomposable and hence $E \otimes B$ decomposable over $C$. In particular let $ E \otimes B= \bigoplus_{q=1}^{s} A_q $ be a decomposition of $ E \otimes B$ in indecomposable vector bundles $A_q$ over $C$. By Lemma \ref{ggforindec} every $A_q$, for $q=1...s$ is globally generated if and only if deg $A_q >$rk $A_q$, i.e. if and only if $\mu(A_q)>1,$ for all $q$. From Lemma \ref{ggforindec}, Lemma\ref{mimenodec} and Proposition \ref{guongg} we get the following chain of equivalences which conclude the proof: $\mu^- (E)+b > 1\Leftrightarrow \mu^-( E \otimes B)=$ min$_q \mu(A_q) >1 \Leftrightarrow \mu(A_q) >1$ for all $q$ $\Leftrightarrow A_q$ is globally generated for all $q$ $ \Leftrightarrow \pi_*D$ is globally generated on $C \Leftrightarrow D $ is globally generated on $X$ \end{pf} The above Lemma is partially contained in \cite[Prop. 3.3]{gu}. Unfortunately the proof presented there is based on \cite[Prop.1.1 (iv)]{gu}, which is not correct, as the following counterexample shows. Let $E$ be an indecomposable vector bundle over an elliptic curve with deg $E = 1$ and rank $E = 2.$ Then $2 \cal{T} = \cal{O}_{\Bbb{P}(E)}(2)$ is generated by global sections, according to \cite[Prop. 8.5.8]{BESO}. On the other hand let $\pi_*(2\cal{T}) = S^2 E = \bigoplus_q A_q,$ where $A_q$ is indecomposable for all $q.$ Then $S^2E$ is generated by global sections if and only if $A_q$ is such, for all $q.$ >From Lemma \ref{ggforindec} it follows that $S^2E$ is globally generated if and only if $\mu(A_q) > 1$ for all $q,$ i.e. if and only if $\mu^-(S^2E) > 1,$ i.e. if and only if $ 2 \mu^-(E) = 2 \mu(E) > 1$ which is false. \medskip If we consider an indecomposable vector bundle of degree $d=0$, we have the following proposition. It is contained in \cite[Theorem 3.9]{gu2}, but we prefer to give here a simpler proof. \begin{Prop} \label{vadeg0modr} Let $E$ be an indecomposable rank r vector bundle over an elliptic curve $C$ with deg$E=0$ (mod $r$), and let $D\equiv aT+bf$ be a line bundle on $ X =\Bbb{P}(E) $. Then $D$ is very ample on $X$ if and only if $b+a\mu^-(E) = b+a\mu(E) \ge 3$. \end {Prop} \begin{pf} It is enough to consider the case in which $E$ is normalized , as if $E$ is not normalized we can consider its normalization $\bar {E}=E \otimes L$ with deg$L=l$. If $D \equiv aT+bf$ in Num$\Bbb{P}(E)$ then in Num$\Bbb{P}(\bar{E})$ we get \begin{equation} \label{contonorm} D \equiv a{\bar{T}}+(b-al)\bar{f}\thinspace,\ \ \ \ \bar{d}= deg\bar{E}=d+rl,\ \ \ \ \mu(\bar{E}) = \mu(E) + al. \end{equation} Let $E$ be normalized, hence $d=0$ and $E = F_r$ in the notation of \cite {At} (recall that $F_1 = \cal{O}_C$). According to (\ref{range}) the only cases to be considered are $b=1$ and $b=2$ and hence $D\equiv aT+f$ or $D\equiv aT+2f$. We want to show that in both these cases $D$ is not very ample. Assume the contrary and proceed by induction on $r$. Let $r=2.$ As $D T = 1$ or $2$, the smooth elliptic curve $\Gamma$, which is the only element of $|\cal T|$, is embedded by $\phi_{|D|}$ as a line or a conic which is a contradiction. Assume now the proposition true for $F_{r-1}$ and recall that there is a short exact sequence (see \cite{At} pag 432) \begin{equation} \label{succdegliFr} 0 \to \cal{O}_C \to F_r \to F_{r-1} \to 0.\end{equation} Let $T'= T_{|Y}$ and $f' = f_{|Y}$ the generators of Num$(Y)$ where $ Y =\Bbb{P}(F_{r-1}) \subset X =\Bbb{P}(E) $. If $D$ is very ample, $D_{|Y}$ is very ample too; but $D_{|Y} \equiv aT'+bf'$ and it is not very ample by induction hypothesis. Hence $D$ is very ample if and only if $b \geq 3$. \end {pf} The following Lemma, which gives a sufficient condition for the very ampleness of a divisor $D$ on $X =\Bbb{P}(E),$ will be needed later on. \begin {Lem} \label{lemmadiEnrique} Let $E$ be a rank $r$ vector bundle over a curve $C$ and let $D\equiv aT+bf$ be a line bundle on $ X =\Bbb{P}(E) $,with $a \geq 1$. If $ \pi_*D$ is a very ample vector bundle on the curve $C$, then $D$ is very ample on $\Bbb{P}(E)$. Moreover if $a=1$, $D$ is very ample on $X$ if and only if $\pi_*D$ is very ample on $C.$ \end {Lem} \begin {pf} We give only a sketch of the proof. A divisor $D\equiv aT+bf$ on $X$ defines a map $\varphi_{|D|}$ in a suitable projective space such that $X'= \varphi_{|D|}(X)$ is a bundle on $C$ whose fibers are the Veronese embedding of the fibers of $X =\Bbb{P}(E)$. Moreover each fiber of $X'$ is embedded in a fiber of the projective bundle $\Bbb{P}(S^{a}(E) \otimes \cal {O}_{C}(B))$.It follows that the very ampleness of $S^{a}(E) \otimes \cal {O}_{C}(B)$ and hence of its tautological bundle implies that the map $\varphi_{|D|}$ gives an embedding and hence that $D$ is very ample. The case $a=1$ follows immediatly from the above considerations. \end {pf} \medskip \subsection{The case $a=1$} \medskip We want to investigate the very ampleness of $D \equiv aT+bf$ in dependence of $a$ and $b$. As we have remarked at the end of section 2.2, the problem is open only when $0 < b+a\mu^-(E) \leq 2$. Let us begin with the case $a=1$. In this case we have the following theorem: \begin{Thm}[Gushel,\cite{gu} theorem 4.3] Let $D \sim \cal{T}+ \pi^*B$ be a divisor on $\Bbb{P}(E) $, where $E$ is an indecomposable and normalized vector bundle of rank $r$ over an elliptic curve $C$. If $b= $deg$B$, the divisor $D$ is very ample if and only if: \begin{enumerate} \item[i)] $b \geq 3$ if deg$E=0$ \item[ii)] $b \geq 2$ if $0< $deg$E < r$. \end{enumerate} \end{Thm} Now it is easy to prove the following (see (\ref{contonorm})): \begin{Prop} \label{vaindeca=1} In the above assumptions and notations, if $E$ is indecomposable but not normalized, it follows that $D$ is very ample if and only if the following conditions hold : \begin{enumerate} \item[i)] $b + \mu(E) \geq 3$ if $d=0$ (mod $r$) . \item[ii)] $b + \mu(E) \geq 2$ otherwise. \end{enumerate} \end{Prop} \medskip \begin{rem} The previous results consider the case in which $E$ is indecomposable. If $E$ is decomposable, by Lemma 2.12, we can argue as follows: firstly in this case, as $a=1$, $D$ is very ample if and only if $\pi_*(D)$ is very ample. Secondly we have $D \sim \cal T + \pi^{*}B$, $\pi_*(D) \simeq E \otimes \cal {O}_{C}(B) = \bigoplus E_j \otimes \cal {O}_{C}(B)$, with $E_j$ indecomposable vector bundles . Moreover $E \otimes \cal {O}_{C}(B) $ is very ample if and only if every $E_j \otimes \cal {O}_{C}(B) $ is very ample. Let deg$E_j = d_j$ and rk$E_j = r_j$, and assume that $d_j= 0$ (mod $r_j$), possibly only for $j = 1...t.$ Then $D$ is very ample if and only if $ b+ \frac{d_j}{r_j} \geq 3$ for $j=1...t$, and $ b+ \frac{d_j}{r_j} \geq 2$ for the remaining $j$'s, by Proposition \ref{vaindeca=1}. \end{rem} \medskip Having dealt above with the case $a=1$, from now on the blanket assumption $a\ge 2$ will be in effect. \section{Rank 2} Let $E$ be a rank $2$ vector bundle on an elliptic curve $C$ and let $D\equiv aT+bf$ be a line bundle on $ X =\Bbb{P}(E)$. Assume that $E$ is indecomposable. If $E$ is normalized then deg$E = 0,1$ by Lemma \ref{degofnormed}. If deg$E=0,$ from Proposition \ref{vadeg0modr} it follows that $D$ is very ample if and only if $b \geq 3$. If deg$E=1$, necessary and sufficient conditions for the very ampleness of $D$ are given by the following Theorem, reformulated under our assumption that $a\ge 2.$ \begin{Thm}[Biancofiore - Livorni, \cite{bi-li3},Theo 6.3] Let $D \sim a\cal{T}+ \pi^*B$ be a divisor on $\Bbb{P}(E) $, where $E$ is an indecomposable normalized vector bundle of rank $2$ and degree $1$ over an elliptic curve $C$. If $b= $deg$B$, the divisor $D$ is very ample if and only if $b + \frac{a}{2} > 1$. \end{Thm} The following Proposition can now be easily proved (see (\ref{contonorm})). \begin{Prop} \label{rangodue} In the above hypothesis, if $E$ is indecomposable but not normalized, $D$ is very ample if and only if the following conditions hold : \begin{enumerate} \item[ii)] $b + a\mu^{-}(E) \geq 3$ if $d=0$ (mod $2$) . \item[i)] $b + a\mu^{-}(E) > 1$ if $d=1$ (mod $2$). \end{enumerate} \end{Prop} The case $E$ decomposable is treated by the following Theorem. \begin{Thm} Let $D \sim a\cal{T}+ \pi^*B$ be a divisor on $\Bbb{P}(E) $, where $E$ is a decomposable vector bundle of rank $2$ over an elliptic curve $C$, $b =$ deg $B$. The divisor $D$ is very ample if and only if $b + a\mu^{-}(E) \geq 3.$ \end{Thm} \begin{pf} To prove the sufficient condition let $E$ be decomposable as $H\bigoplus G$ where $H$ and $G$ are line bundles on $C$ with deg$H=h \geq$ deg $G=g$. By Lemma \ref{mimenodec} it is $\mu^{-}(E) = g$. By Lemma \ref{lemmadiEnrique}, a sufficient condition for the very ampleness of $D$ on $X$ is that $\pi_*(D) $ is very ample as a vector bundle on $C$. In our hypothesis $$\pi_*(D) = S^a(E) \otimes \cal {O}_C(B) = \bigoplus_{q=0}^{a}H^{\otimes q} \otimes G^{\otimes a-q} \otimes B.$$ Now $\pi_{*}(D) $ is very ample if each element of its decomposition has degree $\geq 3$, i.e. if $qh + (a-q)g +b \geq 3$, for all $q=0,...a$. As the minimum of $qh + (a-q)g +b$ is realized for $q=0$, $\pi_{*}(D) $ is very ample if and only if $ag +b = b+a \mu^{-}(E) \geq 3$. This condition is also necessary for the very ampleness of $D$. Indeed the projective bundle $\Bbb{P}(G) $, by the exact sequence $$0 \to H \to E \to G \to 0$$ \cite{gu}, Proposition 1.1, gives an elliptic curve $\Gamma$ on $X,$ $\Gamma \in | \cal{T}+ \pi^{*}(H^{*})| .$ Notice that $h^0(X, \cal{T}+ \pi^{*}(H^{*}))>0.$ If $D$ is very ample it must be $D \Gamma = (aT+bf)(T-hf)= ag+b= b+a \mu^{-}(E) \geq 3$. \end{pf} \medskip \section{Rank 3} \label{rango3sec} In this section and in the next one, we will prove the very ampleness of a divisor on a smooth variety following a classical method, based on the following lemmata. \begin{Lem} \label{reductiontoS} Let $X$ be a smooth variety, $D$ a divisor in $Pic(X)$ and let $A$ be another element of $Pic(X)$, such that $h^1(X, \cal{O}_X(D-A))=0$. If, for each pair of points $R,Q \in X$ (possibly infinitely near) it is possible to find a smooth element $S \in |A|$ containing $R$ and $Q$, then $D$ is very ample on $X$ if and only if $D_{|S}$ is very ample on $S$. \end{Lem} \begin{pf} If $D$ is very ample then obviously $D_{|S}$ is very ample. On the other hand, pick any two points $R,Q \in X$ (distinct or infinitely near) and choose $S \in |A|$ such that $R,Q \in S$. As $D_{|S}$ is very ample there exist sections of $D_{|S}$ separating $R$ and $Q$. Now look at the following exact sequence $$0 \to \cal{O}_X(D-A) \to \cal{O}_X(D) \to \cal{O}_S(D_{|S}) \to 0.$$ From the assumptions above we get that the map $H^0(X,\cal{O}_X(D)) \to H^0(S,\cal{O}_S(D_{|S}))$ is surjective and hence $D$ is very ample on $X$ if and only if $D_{|S}$ is very ample on $S$. \end{pf} \begin{Lem} \label{T+F} Let $E$ be an ample vector bundle over an elliptic curve $C$ such that deg~$E< $~rk~$E$. Let $ X =\Bbb{P}(E) $, let $P$ be a fixed point of $C$, $D\sim a\cal{T} +\pi^*B$ and $A = \cal{T}+\cal{F_P}$ be line bundles on $X$, with $b =$ deg $B$ , $b-1+(a-1)\mu^-(E) >0$ and $h^0(X,\cal{O}_X(A)) \geq $ deg $E +3$. Then the hypothesis of Lemma \ref{reductiontoS} are satisfied for $A$. \end{Lem} \begin{pf} If $ (a-1)\mu^-(E)+b-1>0$ then by Lemma \ref{buongg} it is $h^1(X, \cal{O}_X(D-A))=0.$ Moreover being $h^0(X, \cal{O}_X(A)) \geq $ deg $E +3,$ for each pair of points $R, Q \in X$ there exists a linear subsystem $\cal L \subset |A|$, with dim$\cal L \ge$ deg $E$ , all the elements of which contain $R$ and $Q$. Moreover in $\cal L$ there is at least one smooth element $S.$ In fact, assume that all the elements of $\cal L$ are singular. Note that any singular element of $\cal L$ must be reducible as $\Gamma \cup \cal{F}_P$, with $\Gamma \in |\cal{T}|$, $\Gamma$ smooth, because we have that any divisor numerically equivalent to $T-f$ is not effective as deg $E< $ rk $E$. As $h^0(X, \cal{O}_X(\cal{F}_p)) =1,$ for all $P \in C$, by Bertini's theorem all the elements of $\cal L$ are singular only if $\cal{F_P}$ is fixed and $\Gamma$ varies in a subsystem of $|\cal{T}|$ of dimension deg $E$. This is impossible as $h^0(X,\cal{O}_X(\cal{T}))= $ deg $E$. \end{pf} \begin{Lem} Let $E$ be an ample vector bundle over an elliptic curve $C$ such that deg$E<$rk$E$. Let $ X =\Bbb{P}(E) $, and let $D \sim a\cal{T}+\pi^*B$ be a line bundle on $X$, with $b= $ deg $B$ , $b+(a-1)\mu^-(E) >0$ and $h^0(X,\cal{O}_X(\cal{T})) \geq 3.$ Then the hypothesis of Lemma \ref{reductiontoS} are satisfied, with $A = \cal T$. \label{T} \end{Lem} \begin{pf} If $ (a-1)\mu^-(E)+b>0$ then Lemma \ref{buongg} gives $h^1(X, \cal{O}_X(D-A))=0.$ Moreover as each element of $|\cal{T}|$ is smooth, because we have that any divisor numerically equivalent to $T-f$ is not effective as deg$E<$rk$E$, the condition $h^0(X,\cal{O}_X(\cal{T})) \geq 3$ shows that it is possible to find a smooth element $S \in |\cal{T}|$ containing each fixed pair of points $R,Q \in X$. \end{pf} The following Lemma will be very useful to obtain the vanishing condition required by Lemma \ref{reductiontoS} in many borderline cases. The notation used here is the classical notation used by Atiyah in \cite{At}. \begin{Lem} \label{AA} Let $E$ be an indecomposable vector bundle over an elliptic curve $C$ with rank $E = r$ and deg $E = d.$ Let $X=\Bbb{P}(E)$ and let $\pi : X \to C$ be the natural projection. Let $D =a\cal{T}+ \pi^*(B)$ for a line bundle $B$ with deg $B = b$ and let $A=\cal{T} +\pi^*(\cal{O}_C(P))$ where $P$ is a point in $C.$ If $\frac{(a-1) d}{r} + b -1=0,$ it is possible to choose $P\in C$ such that $h^1(X, D - A) = 0.$ \end{Lem} \begin{pf} It is enough to show that $h^1(C, S^{a-1} E \otimes B \otimes \cal{O}_C(-P)) = 0.$ Since deg $S^{a-1} E \otimes B \otimes \cal{O}_C(-P) = 0$ by Riemann Roch it is enough to show that $h^0(S^{a-1} E \otimes B \otimes \cal{O}_C(-P)) = 0.$ Because $S^{a-1}(E)$ is a direct summand of $E^{\otimes (a-1)}$ it is enough to show that $h^0( E^{\otimes (a-1)} \otimes B~\otimes \cal{O}_C(-P)) = 0.$ Let $h =$ gcd $(d, r).$ Then by \cite{At} Lemma 24 and 26 it is \begin{equation} \label{EconFh} E = E' \otimes F_h \end{equation} where $d'=$ deg $E' = \frac{d}{h}$ and $r' = $ rank $E' = \frac{r}{h}$ so that gcd$(d',r') = 1,$ $F_h$ is as in \cite{At} Theorem 5 and $E'$ is indecomposable . Being $r'$ and $d'$ relatively prime, the condition $\frac{(a-1) d'}{r'} + b -1=0$ shows that $r'$ divides $(a-1).$ Therefore following \cite{H2} Proposition 1.4 it follows that \begin{equation} \label{EprimoconFri} E^{' \otimes (a-1)} = \bigoplus_i(F_{r_i} \otimes L_i) \end{equation} Therefore putting (\ref{EconFh}) and (\ref{EprimoconFri}) together we get $$ E^{\otimes(a-1)} =( E' \otimes F_h)^{\otimes (a-1)} = \bigoplus_i(F_{r_i} \otimes L_i) \otimes F_h^{\otimes (a-1)}.$$ Theorem 8 in \cite{At} shows that tensor powers of $F_l$ 's are direct sums of $F_k$'s so we conclude that $$ E^{(a-1)} = \bigoplus_j (F_{r_j} \otimes L_j).$$ It is then enough to show that for all $j$ it is $h^0(F_{r_j} \otimes L_j \otimes B \otimes \cal{O}_C(-P)) = 0.$ Let $\cal{L}_{j,P} = L_j \otimes B\otimes \cal{O}_C(-P) .$ Recall that the $F_{r_j}$ are obtained as successive extensions of each other by $\cal{O}_C,$ i.e. for every $r$ we have the sequence (\ref{succdegliFr}) (see proof of Prop .2.11). This shows that it is $h^0(F_{r_j} \otimes \cal{L}_{j,P}) = 0$ unless $\cal{L}_{j,P} = \cal{O}_C.$ It is then enough to choose a point $P$ such that $L_j \otimes B \otimes \cal{O}_C(-P) \neq \cal{O}_C$ for all $j.$ Since $B$ is a fixed line bundle and $j$ runs over a finite set, a $P$ that works for all $j$ can certainly be found. \end{pf} The following Theorem collects our results for the case rk $E = 3.$ \begin{Thm} \label{Thmrango3} Let $E$ be a rank $3$ vector bundle on an elliptic curve $C$ and let $D\equiv aT+bf$ be a line bundle on $ X =\Bbb{P}(E).$ \begin{itemize} \item[(a)] If $E$ is indecomposable then \begin{itemize} \item[(a1)] if $d=0$ (mod $3$) , $D$ is very ample if and only if $b+a\mu^-(E) \geq 3$ \item[(a2)]if $d=1$ (mod $3$) , $D$ is very ample if $b+a\mu^-(E) > 1$ \item[(a3)]if $d=2$ (mod $3$) , $D$ is very ample if $b+a\mu^-(E) >\frac{4}{3}$ \end{itemize} \item[(b)] if $E$ is decomposable, then $D$ is very ample if and only if $b+a\mu^-(E) \geq 3$ except when $E=E_1\oplus E_2$, with rk$E_1=1$, rk$E_2=2$, deg$E_2$ odd and deg$E_1> \frac {degE_2}2.$ In the latter case the condition is only sufficient. \end{itemize} \end{Thm} \begin{pf} Firstly we consider the case $E$ indecomposable and normalized. By Lemma \ref{degofnormed} and Proposition \ref{vadeg0modr} only the cases $d=1$ and $d=2$ need to be considered. \begin{case} $d=1$. \end{case} Let $A$ be as in Lemma \ref{T+F} and notice that $h^0(A) =$ deg $E + 3.$ By Remark \ref{criteriodelbutler}. we can assume $b+\frac{a}{3}= 1+\frac {\varepsilon}{3},$ where $\varepsilon = 1,2,3.$ If $\varepsilon =2,3$ then $ (a-1)\mu^-(E)+b-1 = b+ \frac {a}{3} - \frac {4}{3} >0.$ If $\varepsilon = 1$ Lemma \ref{AA} still allows the use of Lemma \ref{reductiontoS}. Therefore Lemma \ref{reductiontoS} and \ref{T+F} can be applied. Let $S$ be a smooth element in $|\cal{T}+\cal{F_P}|.$ It is enough to check when $D_{|S}$ is very ample. By \cite{gu}, prop 1.1, $S=\Bbb{P}(E')$ where $E'$ is a rank $2$ vector bundle on the curve $C$, with deg$E'=2$, defined by the sequence \begin{equation} 0 \to \cal {O_C}(-P) \to E \to E' \to 0. \label{secondasucc} \end{equation} As Num($S$) is generated by $T'$ and $f'$, with $T'=T_{|S}$ and $f'=f_{|S}$, $D_{|S} \equiv aT'+bf'$ and moreover by section $3$, $D_{|S}$ is very ample in our range if and only if $b+ a\mu^-(E') \geq 3$. If $E'$ is indecomposable then $\mu^-(E')=\mu(E')=1$. If $E'$ is decomposable, i.e. $E'= H \oplus G $, both $\mu(H) \geq \mu(E) = \frac{1}{3}$ and $\mu(G) \geq \mu(E) = \frac{1}{3}$ because from (\ref{secondasucc}) there exist non zero surjective morphisms $E \to H$ and $E \to G.$ Hence deg$H$= deg$G$ =$1$, $\mu(H)=\mu(G)= 1$ and $\mu^-(E')=1.$ In every case the condition $b+ a\mu^-(E') \geq 3$ is satisfied in the range under consideration. \begin{case} $d=2$. \end{case} Let $A$ and $S$ be as in Lemma \ref{reductiontoS} and \ref{T+F}. In this case it is $h^0(X,A) = 5 \geq $ deg $E + 3$. By Remark \ref{criteriodelbutler} we can assume $b+a\frac{2}{3}=1+\frac{\varepsilon}{3}$ with $\varepsilon=1,2,3$. If $\varepsilon=3$ the hypothesis of Lemma \ref{T+F} are satisfied. If $\varepsilon=2$ Lemma \ref{AA} still allows the use of Lemma \ref{reductiontoS}. Therefore it suffices to investigate the very ampleness of $D_{|S}.$ Let $S=\Bbb{P}(E')$ with $E'$ a rank $2$ vector bundle on $C$, with deg$E'=3$ defined again by (\ref{secondasucc}). If $E'$ is indecomposable, $D_{|S} \equiv aT'+bf'$ is very ample if and only if $b+a\mu^-(E') > 1$, i.e. $b+\frac{3}{2} a >1$ i.e. for all $D$ in the range under consideration. If $E'$ is decomposable then $E'= H \oplus G $, with deg$H$ and deg$G$ $\geq \mu(E)=\frac{2}{3}.$ Hence we can assume deg$H=1$ and deg$G=2.$ By Lemma \ref{mimenodec}, $ \mu^-(E)=1$ and the very ampleness condition is $b+a \geq 3$ which is satisfied by every $D$ in the range under consideration. Notice that in the case $b+\frac{2}{3} a = \frac{4}{3}$, i.e. $\varepsilon=1$, a very ampleness result for all $D$ in our range cannot be expected. For example, $D\sim 2\cal{T}$ is not very ample as $D_{|Y}$ is not very ample for each smooth surface $Y\in |\cal{T}|$ by section $3$. If $E$ is indecomposable but not normalized, the result is obtained by similar computations (see (\ref{contonorm})). To prove (b), let now $E$ be decomposable. Firstly we prove the sufficient condition. By Proposition \ref{guongg} it suffices to prove that $\pi_{*}(D)$ is very ample. Let us consider $\pi_{*}(D) = S^{a}(E) \otimes \cal {O}_C(B) = \bigoplus_{q}A_q$ where $A_q$ is an indecomposable vector bundle on $C$ for all $q.$ $ S^{a}(E) \otimes \cal {O}_C(B)$ is very ample if and only if $A_q$ is very ample for all $ q$ i.e. if $\mu(A_q)\geq 3 $ for all $q.$ This condition is satisfied if min$_q \mu(A_q) = \mu^-( S^{a}(E) \otimes \cal {O}_C(B))=b+a\mu^-(E) \geq 3$ which is what we wanted to show. To prove the necessary condition, two cases will be considered: i) $E$ is sum of three line bundles, $E = W \oplus G \oplus H$ respectively of degrees $w\leq g \leq h$. By lemma \ref{mimenodec} $\mu^-(E) = w$, and $d= $deg$E=w+g+h.$ From \cite{gu}, Prop 1.1 it follows that $\Bbb{P}(G \oplus H)$ is a subvariety of $X$ corresponding to a line bundle numerically equivalent to $T-wf$ while $\Bbb{P}(W)$ is an elliptic curve $\Gamma$ on $X$, isomorphic to $C$, which is numerically equivalent to $T^2+xT f$ , for some $x \in \Bbb{Z}$. As the cycles corresponding to $\Bbb{P}(W)$ and $\Bbb{P}(G \oplus H)$ do not intersect, from $(T^2+xTf) (T-wf)=0$ we get $x=-(g+h)$. If $D$ is a very ample line bundle on $X$, $D_{|\Gamma}$ is very ample, hence $D \Gamma = b+a\mu^-(E) \geq 3.$ ii) $E=H \oplus G$ where rk$H=1$, rk$G=2$, $h=$ deg$H$, $g= $ deg$G$. As in i), we get that $Z= \Bbb{P}(H)$ is numerically equivalent to $T^2-gTf$ and the very ampleness of $D_{|Z}$ implies $b+ah \geq 3$. If $h \leq \frac{g}{2}$ this concludes the proof. Otherwise let us denote by $Y$ the smooth surface $\Bbb{P}(G)$. As usual Num($Y$) is generated by $T'=T_{|Y}$ and $f'=f_{|Y}$. The very ampleness of $D$ implies the one of $D_{|Y}$ and by section 3, $D_{|Y}$ is very ample if and only if \begin{itemize} \item[]$b+a \frac {g}{2} \geq 3$ if $g$ is even and \item[]$b+a \frac {g}{2} > 1$ if $g$ is odd. \end{itemize} If $g$ is even, a necessary condition for the very ampleness of $D$ is $b+a\frac{g}{2} \geq 3$ i.e. $b+a\mu^-(E) \geq 3$ which is the desired condition. If $g$ is odd, necessary conditions for the very ampleness of $D$ are both $b+ah \geq 3$ and $b+a\frac {g}{2} > 1.$ Hence only the sufficient condition $b+a\mu^-(E) \geq 3$ is obtained in this case. \end{pf} \section {rank $r$} To deal with the case of $E$ vector bundle on an elliptic curve $C$, of rank $r>3$ we need to recall first the following \begin{Thm}[\cite{H2}] Let $E$ be a vector bundle of rank $r$ on an elliptic curve $C.$ $E$ is ample if and only if every indecomposable direct summand $E_i$ of $E$ has deg$E_i >0.$ \end{Thm} Our method based on Lemma \ref{reductiontoS}, \ref {T+F} and \ref{T} forces us to investigate first the case deg$E =3$, then deg$E=1, 2$ and finally deg$E\geq 4$. \subsection{ deg$E=3$} \begin{Thm} \label{r4d3teo} Let $E$ be a vector bundle over an elliptic curve $C$, with deg$E=3$ and rank$E=4$, and let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E)$. Then the following conditions hold: \begin{enumerate} \item[i)] If $E$ is indecomposable, and $b+\frac{3}{4} a >\frac{3}{4}$, $D$ is very ample if and only if $b+a \geq 3$ \item[ii)] If $E$ is decomposable and ample, and $b+ \frac{a}{3} >\frac{1}{3}$ $D$ is very ample if $b+ \frac{a}{2} >2$. \end{enumerate} \end{Thm} \begin{pf} i) If deg$E=h^0(\cal{T})=3$ then we can apply Lemma 4.3 and 4.1 with $A= \cal T$, as $b+(a-1)\mu^-(E) = b+\frac{3}{4} a- \frac{3}{4} >0$ by hypothesis. Hence $D$ is very ample if and only if $D_{|S} $ is very ample, where $S$ is a suitable element of $|\cal{T}|$. There exists a vector bundle $E'$ with rk$E'=3$, deg$E'=3$ , given by \begin{equation} \label{succdelT} 0\to \cal{O}_C \to E \to E' \to 0 , \end{equation} such that $S=\Bbb{P}(E')$ and Num(S) is generated by $T'$ and $f'$, where $T'= T_{|S}$ and $f'= f_{|S}$ so that $D_{|S}\equiv aT'+bf'$. By section 4, $D_{|S}$ is very ample if and only if $b+a \geq 3$. Indeed if $E'$ is indecomposable the necessary and sufficient condition for the very ampleness is $b+a \geq 3$. If $E'=\bigoplus_i E'_i$ then $\mu(E'_i) \geq \frac{3}{4}$ for all $i.$ Hence the only possibilities for a decomposition of $E'$ are: \begin{enumerate} \item[1)] $E'=F \oplus G \oplus H$, with $F,G,H$ line bundles all of degree $1.$ \item[2)] $E'= H \oplus G$, with rank $G=2$, rank $H=1$, deg $G=2$, deg $H=1.$, \end{enumerate} Again by section 4 in both the above cases, the necessary and sufficient condition for the very ampleness of $D_{|S}$ is $b+a \geq 3$. ii) Let us suppose that $E$ is decomposable and ample. Then the possible decompositions for $E$ give $\mu^-(E) = \frac{2}{3},\frac{1}{3},\frac{1}{2}.$ Note that the condition $b+ \frac{a}{3} > \frac{1}{3}$ allows us to apply lemma 4.3 with $A= \cal T$ in any case. If $S$ is the usual element of $|\cal{T}|$,we get that $D$ is very ample on $X$ if and only if $D_{|S}$ is. If $S=\Bbb{P}(E')$ , $E'$ could be decomposable. In this case $\mu^-(E') \ge \frac{1}{2}.$ Therefore the condition $b+ \frac{a}{2} > 2$ guarantees the very ampleness of $D_{|S}$ by section \ref{rango3sec}. \end{pf} Because $a\geq 2,$ Theorem \ref{r4d3teo} immediately gives the following: \begin{Cor} \label{corond3r4} Let $E$ be an ample vector bundle over an elliptic curve $C$, with deg$E=3$ and rank$E=4$, and let $D \equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E)$ with $b+ \frac{a}{3} > \frac{1}{3}$. If $b+ \frac{a}{2} >2$ then $D$ is very ample. \end{Cor} \begin{Thm} Let $E$ be an ample vector bundle over an elliptic curve $C$, with deg $E=3$ and $r=$ rank$E\geq 4$, and let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E)$ with $b+a\mu^-(E) >\frac{3}{5}$. Then $D$ is very ample if $b+\frac{a}{2} >2$ and $b+\frac{a}{3} > \frac{1}{3}$. \label{deg3anyr} \end{Thm} \begin{pf} Proceed by induction on $r=$ rank $E$. If $r=4$ the inductive hypothesis is verified by Corollary \ref{corond3r4}. Let $r \geq 5.$ It is $h^0(X, \cal T)=h^0(C, E)=3.$ Notice that $\mu^-(E) \le \mu(E) \le\frac{3}{r}\leq\frac{3}{5}.$ Therefore $b + (a-1) \mu^-(E) > 0$ and Lemma \ref{T} can be applied, with $A = \cal{T}.$ Let $S=\Bbb{P}(E')$ be as in (\ref{succdelT}) with deg $(E')=3$ and rk $(E')=r-1\geq 4$. Num$(S)$ is generated by $ T'=T_{|S}$ and $ f'=f_{|S},$ so that $ D_{|S} \equiv aT'+bf'$. Notice that $E'$ is ample being a quotient of $E.$ Notice that $\mu^-(E) \leq \mu^-(E')$. Indeed there exists a map from at least one direct summand $E_k$ of $E$ and the summand $E'_j$ of $E'$ which realizes $\mu^-(E')$ and so we get $\mu^-(E) \leq \mu(E_k) \leq \mu(E_j) = \mu^-(E')$. As $b+a\mu^-(E') \geq b+a\mu^-(E) >\frac{3}{5}$, by induction $D_{|S} $ is very ample if $b+\frac{a}{2} >2$, $b+\frac{a}{3}>\frac{1}{3}$. \end{pf} \subsection{deg $E =2$} \begin{Thm} \label{d2anyrindecteo} Let $E$ be an indecomposable vector bundle over an elliptic curve $C$, with deg$E=2$, rk $E= r \ge4.$ Let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E).$ Then the following conditions hold: \begin{enumerate} \item[i)] If $r=4$ , $D$ is very ample if $b+\frac{a}{2} >2$ \item[ii)] If $r \ge 5 $, $D$ is very ample if $b+\frac{a}{2} >2$ , $b+\frac{a}{3} >\frac{1}{3}$ and $b+a\frac{2}{r} > 1+ \frac{1}{r}$. \end{enumerate} \end{Thm} \begin{pf} Let $A$ be as in Lemma \ref{T+F} and notice that $h^0(A) = 2 + r > $ deg $E + 3.$ By Remark \ref{criteriodelbutler} and the assumptions in i) and ii) we can assume \begin{equation} \label{rangeconepsilon} b + a\frac{2}{r} = 1+\frac{\varepsilon}{r}, \end{equation} where $\varepsilon = 1,2....r.$ If $b+a\frac{2}{r} >1+\frac{2}{r}$ the assumptions of Lemma \ref{reductiontoS} and \ref{T+F} are satisfied. If $b+a\frac{2}{r} = 1+\frac{2}{r}$ the line bundle $D-A$ has degree $0$ and Lemma \ref{AA} shows that Lemma \ref{reductiontoS} can be used if \begin{equation} \label{1+2sur} b+a\frac{2}{r} \geq 1+\frac{2}{r}. \end{equation} Therefore condition (\ref{1+2sur}) can be rewritten using (\ref{rangeconepsilon}) as \begin{equation} \label{1+1sur} b+a\frac{2}{r} > 1+\frac{1}{r}. \end{equation} Let $S=\Bbb{P}(E')$, where $E'$ is as in (\ref{secondasucc}), where deg $ E'=3$ , rk $ E'=r-1\geq 3$, Num$(S)$ is generated by $ T'=T_{|S}$ and $ f'=f_{|S},$ and $D_{|S} \equiv aT'+bf'$. Notice that $E'$ is indecomposable or decomposable and ample because $ \mu^-(E') \geq \mu^-(E)=\frac{2}{r} > 0$. To prove i) assume $r=4.$ In this case rk $E'=3$ and deg $E'=3$. If $E'$ is indecomposable, as $b+a \mu^-(E') >b+a\mu^-(E)=b+\frac{a}{2}> 1,$ by Theorem \ref{Thmrango3} $D_{|S}$ is very ample if $b+a \geq 3$ . By the same theorem if $E'$ is decomposable and ample then $D_{|S}$ is very ample if $b+a \mu^-(E') > 2.$ The possible values for $\mu^-(E')$ are $1$ or $\frac{1}{2}.$ Therefore by Theorem \ref{Thmrango3} it follows that $D_{|S}$ is very ample if \begin{equation} b+\frac{a}{2} >2. \label{b+asu2} \end{equation} Putting (\ref{1+1sur}) and (\ref{b+asu2}) together it follows that in the case $r=4$ $D$ is very ample if $b+\frac{a}{2} > \frac {5}{4}$ and $b+\frac{a}{2} >2$ i.e. $b+\frac{a}{2} >2$. To prove ii) assume $r-1\geq 4.$ Since $b+a\mu^-(E') \geq b+a\mu^-(E) = b+a\frac{2}{r} >1+\frac{1}{r}>\frac{3}{5}$ from Theorem \ref{deg3anyr} it follows that $D_{|S}$ is very ample if $b+\frac{a}{2} >2$ and $b+\frac{a}{3} >\frac{1}{3}.$ Therefore if $r\geq 5$, $D$ is very ample if $b+\frac{a}{2} >2$ , $b+\frac{a}{3} >\frac{1}{3}$ and $b+a\frac{2}{r} > 1+ \frac{1}{r}$. \end{pf} \begin{Thm} \label{d2anyrdecteo} Let $E$ be a decomposable and ample vector bundle of rank $ r\ge 4$ over an elliptic curve $C$, with deg $E=2.$ Let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E).$ Then the following conditions hold: \begin{enumerate} \item[i)] If $r =4$ , $D$ is very ample if $b+\frac{a}{2} >2$ and $b+a\mu^-(E) > \frac{3}{2}$ \item[ii)] If $r \geq 5$, $D$ is very ample if $b+a\mu^-(E) > 1+\frac{2}{r} $, $b+\frac{a}{3}>\frac{1}{3}$ and $b+ \frac{a}{2} > 2$. \end{enumerate} \end{Thm} \begin{pf} Let $A$ be as in Lemma \ref{T+F} and notice that $h^0(A) = r+2 \geq$ deg $E + 3.$ Also notice that $\mu^-(E) \leq \mu(E) = \frac{2}{r}$. In our hypothesis we have $b+(a-1)\mu^-(E)-1\geq b+a\mu^-(E) -\frac{2}{r}-1 > 0$. Therefore Lemma \ref{reductiontoS} and \ref{T+F} can be applied. Let $S=\Bbb{P}(E')$, where $E'$ is as in (\ref{secondasucc}) where deg $E'=3$ and $rk(E')=r-1 \geq 3$. If $r \geq 5$ we can apply Theorem \ref{deg3anyr} to $D_{|S}. $ Indeed $b+a\mu^-(E') \geq b+a\mu^-(E) > 1+ \frac{2}{r} > \frac{3}{5}$. Hence $D_{|S}$ is very ample if $b+\frac{a}{3}>\frac{1}{3}$ and $b+ \frac{a}{2} > 2.$ If $r =4$ we can apply Theorem \ref{Thmrango3}. If $E'$ is indecomposable then $D_{|S}$ is very ample if $b+a \geq 3.$ If $E'$ is decomposable, noticing that $E'$ is ample, the condition is $ b+a\mu^-(E') > 2.$ In this case $\mu^-(E')$ can be $1$ or $\frac {1}{2}$ and hence the worst sufficient condition becomes $b+ \frac{a}{2} > 2$. \end{pf} Theorem \ref{d2anyrindecteo} and \ref{d2anyrdecteo} give the following Corollary. \begin{Cor} \label{corond2} Let $E$ be an ample vector bundle over an elliptic curve $C$, with deg $E=2$ and let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E).$ Then the following conditions hold: \begin{enumerate} \item[i)] If rank $E \ge 5$ , then $D$ is very ample if $b+a\mu^-(E) >1+\frac{2}{r}$, $b+ \frac{a}{2} > 2$ and $b+ \frac{a}{3} > \frac{1}{3}.$ \item[ii)] If rank $E=4$, $D$ is very ample if $b+a\mu^-(E) > \frac{3}{2}$ and $b+ \frac{a}{2} > 2$. \end{enumerate} \end{Cor} \subsection{deg $E =1$} Note that if deg$E=1$, $E$ is ample if and only if it is indecomposable. \begin{Thm} Let $E$ be an indecomposable vector bundle over an elliptic curve $C$, with deg $E=1$, rk $E = r \geq 4$, and let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E)$ with $b+ \frac{a}{r} >1$. Then the following conditions hold: \begin{enumerate} \item[i)] If $r= 4,$ $D$ is very ample if $b+\frac{a}{2} > 2$ \item[ii)] If $r = 5,$ $D$ is very ample if $b+\frac{a}{3}>\frac{3}{2}$ and $b+\frac{a}{2} > 2$. \item[iii)] If $r \geq 6$, $D$ is very ample if $b+\frac{a}{r-2} > 1+ \frac{2}{r - 1}$ and $b+\frac{a}{2} >2$. \end{enumerate} \end{Thm} \begin{pf} In our hypothesis it is $(a-1)\frac{1}{r} +b-1 > 0.$ Lemma \ref{AA} allows us to apply Lemma \ref{reductiontoS} and \ref{T+F} when $(a-1)\frac{1}{r} +b-1 \ge 0,$ noticing that if $A = \cal{T}+\cal{F_P}$ it is $h^0(A)=1+r \geq 5$. If $S=\Bbb{P}(E')$, where $E'$ is as in (\ref{secondasucc}), notice that $E'$ is ample because $E$ is. It is rk $E'=r-1\geq 3$ and deg $E'=2.$ Let us now distinguish the three cases according to the values of $r.$ Theorem \ref{deg3anyr} and Corollary \ref{corond2} will be used. \begin{enumerate} \item[i)] If rank$E=4$, rank$E'=3$, and $D_{|S}$ is very ample if $b+\frac{2}{3}a >\frac{4}{3} $ when $E'$ is indecomposable while if $E'$ is decomposable and ample the condition is $b+a\mu^-(E') >2$ In the worst case $ \mu^-(E')= \frac{1}{2}$ so we have $b+\frac{a}{2} >2$. \item[ii)] If rank $E=5$ , rank $E'=4$, and $D_{|S}$ is very ample if $b+a\mu^-(E') > \frac{3}{2}$, and $b+\frac{a}{2} >2$. As $E'$ is indecomposable or decomposable and ample, in the worst case $ \mu^-(E')= \frac{1}{3}$ and so it is enough to ask $b+\frac{a}{3} > \frac{3}{2}$ and $b + \frac{a}{2} > 2.$ \item[iii)] If rank $E \geq 6$ then rank $E' \geq 5$, hence by Corollary \ref{corond2} $D_{|S}$ is very ample if $b+a\mu^-(E') > 1+\frac{2}{r-1} $, $b+\frac{a}{3}>\frac{1}{3}$ and $b+ \frac{a}{2} > 2$. If $E'$ is indecomposable then $\mu^-(E') = \frac{2}{r-1}$ while if $E'$ is decomposable and ample, then $\mu^-(E') = min (\frac{1}{s}, \frac{1}{r-1-s})$ with $ s= 1..r-2$. So the condition $b+a\mu^-(E') > 1+\frac{2}{r-1} $ can be substituted by $b+\frac {a}{r-2} >1+\frac{2}{r-1} $ which implies the condition $b+\frac{a}{3}>\frac{1}{3}.$ \end{enumerate} \end{pf} \subsection{deg $E \geq 4$} \begin{Thm} Let $E$ be an ample vector bundle over an elliptic curve $C$, with deg$E=d \geq 4$, rk $E = r \geq 4$ and $d < r.$ Let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E).$ If $b+ \frac{a}{d-1} > 2$ and $b + (a-1) \mu^-(E) > 0$ then $D$ is very ample. \end{Thm} \begin{pf} The proof proceeds by induction on $r.$ If $r=4$ the smallest possible value of $\mu^-(E)$ is $\frac{1}{3}$. Since $d\ge 4$ it is $\frac{1}{d-1} \le \frac{1}{3}.$ Therefore $b + a\mu^-(E) \ge b + \frac{a}{3} \ge b + \frac{a}{d-1} > 2.$ Lemma \ref{lemmadiEnrique} and Theorem \ref{butlerteo} conclude the proof of the initial inductive step. Let us suppose the statement true for $r-1$ and prove it for $r$. Let $A$ be as in Lemma \ref{T} and notice that $h^0(A) = d \geq 4.$ Therefore Lemma \ref{T} and \ref{reductiontoS} can be applied. By considering $S=\Bbb{P}(E')$ where $E'$ is as in (\ref{succdelT}), we get that $E'$ is ample with deg $E'=d \ge 4$, rk $E'= r' =r-1 \ge 4.$ Because $\mu^-(E') \ge \mu^-(E)$ it is $b+ (a-1) \mu^-(E') \ge b + (a-1)\mu^-(E) >0$ and again $b + \frac{a}{d-1} > 2.$ Hence we can conclude by induction hypothesis. \end{pf} \begin{rem} Note that in the above theorem, if $E$ is indecomposable, by normalizing $E$ we can always assume $d < r.$ \end{rem} In a very particular case we can say a little more: \begin{Prop} Let $E$ be an indecomposable vector bundle over an elliptic curve $C$, with deg $E=d \geq 4$, rk $E = d+1$, and let $D\equiv aT+bf$ be a line bundle on $X=\Bbb{P}(E).$ If $b+(a-1)\frac{d}{d+1} >0$ and $b+ a > 2$ then $D$ is very ample. \end{Prop} \begin{pf} It is $h^0(X, \cal{T}) = d \ge 4.$ Our hypothesis $b+(a-1)\frac{d}{d+1} >0$ shows that Lemma \ref{reductiontoS} and \ref{T} can be applied. By considering $S=\Bbb{P}(E')$ , where $E'$ is as in (\ref{succdelT}) it is deg $E'$ = rk $E'$= $d$. A sufficient condition for the very ampleness of $D_{|S}$ is $b+ a \mu^-(E') > 2$ by Lemma \ref{lemmadiEnrique} and Theorem \ref{butlerteo}. We claim that in this case $\mu^-(E')=1$. If $E'$ is indecomposable it is $\mu^-(E') = \mu(E') =1$. If $E'$ is decomposable we can suppose that $E'= \oplus G_j$, with $r_j = $ rk $G_j \geq 1$ , $d_j=$ deg $G_j \geq 1$ (as $\mu(G_j) \geq \mu(E)=\frac {d}{d+1}$) and $\sum d_j =\sum r_j =d$. Hence we have $\frac{d_j}{r_j} \geq \frac {d}{d+1},$ for all $j,$ which implies $r_j-d_j \leq \frac{d_j}{d} < 1,$ for all $ j.$ Hence $r_j - d_j \leq 0 ,$ for all $j$ i.e. $d_j=r_j+s_j$ with $s_j \geq 0,$ for all $j.$ But $d= \sum d_j = \sum (d_j+s_j )= d+ \sum s_j$ and $\sum s_j =0$ i.e. $s_j=0,$ for all $ j$. Hence $\mu^-(E') =1$. \end{pf}

9702

alg-geom/9702010

en

https://arxiv.org/abs/alg-geom/9702010

alg-geom/9702010

Michael Finkelberg

Michael Finkelberg and Alexander Kuznetsov (Independent University of Moscow)

Global Intersection Cohomology of Quasimaps' Spaces

21 pages, AmsLatex 1.1

Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $\pi: \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincar\'e polynomial'') of the direct sum $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ and construct a natural action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.

alg-geom

\section{Introduction} \subsection{} Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space is noncompact. Some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ were constructed by Drinfeld, Laumon and Kontsevich respectively. In ~\cite{k} it was proved that the natural map $\pi:\ \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps' spaces. \subsection{} It appears that $\CQ^L_\alpha$ admits a cell decomposition, whence its cohomology has a pure Tate Hodge structure. It was essentially computed by G.Laumon (see ~\cite{la}, Theorem 3.3.2). For the reader's convenience we reproduce the computation in section 2. We calculate the generating function $P_G(t)$ (``Poincar\'e polynomial'') of the direct sum $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ as a formal cocharacter of $G=SL_n$ with coefficients in the Laurent polynomials in $t$ (a formal variable of degree 2). It is given by the following formula: $$P_G(t)=\frac{e^{2\rho}t^{-\frac{1}{2}\dim\CB}\sum_{w\in W}t^{\ell(w)}} {\prod_{\theta\in R^+}(1-te^\theta)(1-t^{-1}e^\theta)}$$ where $W=S_n$ is the Weyl group of $G$ with its standard length function, $R^+$ is the set of positive coroots of $G$, and $2\rho$ stands for $\sum_{\theta\in R^+}\theta$. \subsection{} For any $\alpha,\gamma\in\BN[I]$ there is a closed subvariety of middle dimension $\fE^\alpha_\gamma\subset\CQ^L_\alpha\times\CQ^L_{\alpha+\gamma}$. It is formed by pairs of quasiflags such that the second one is a subflag of the first one. The top-dimensional irreducible components of $\fE^\alpha_\gamma$ are naturally numbered by the Kostant partitions $\bc\in\fK(\gamma)$ of $\gamma$, independently of $\alpha$. For $\bc\in\fK(\gamma)$ the corresponding irreducible component $\fE^\alpha_\bc$, viewed as a correspondence between $\CQ^L_\alpha$ and $\CQ^L_{\alpha+\gamma}$, defines two operators: $$e_\bc:\ H^\bullet(\CQ^L_\alpha)\rightleftharpoons H^\bullet(\CQ^L_{\alpha+\gamma})\ :f_\bc$$ adjoint to each other with respect to Poincar\'e duality. \subsubsection{} Let $\fn$ denote the Lie subalgebra of upper-triangular matrices in the Lie algebra ${\frak{sl}}_n$. Let $U(\fn)$ denote the Kostant integral form (with divided powers) of the universal enveloping algebra of $\fn$. It turns out that the linear span of operators $e_\bc$ is closed under composition; the algebra they form is naturally isomorphic to $U(\fn)$, and the isomorphism takes $e_\bc$ to the corresponding element of Poincar\'e-Birkhoff-Witt-Kostant basis of $U(\fn)$. \subsubsection{} Moreover, all the operators $e_\bc,f_\bc$ together generate an action of the universal enveloping algebra of ${\frak{sl}}_n$ on $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$. The character of this module is given by $\dfrac{|W|e^{2\rho}}{\prod_{\theta\in R^+}(1-e^\theta)^2}$. \subsection{} We conjecture that the ${\frak{sl}}_n$-module $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$ is isomorphic to $H^\nu_\fn(\CN,\CO)$. Here $\CN$ stands for the nilpotent cone of ${\frak{sl}}_n$, and $H^\nu_\fn(\CN,\CO)$ is the cohomology of the structure sheaf with supports in $\fn$ of degree $\nu=\dim\fn=\frac{n(n-1)}{2}$. To verify this conjecture it would be enough to check that $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$ is free as a $U(\fn)$-module (see section 6). \subsection{} The above conjecture is motivated by B.Feigin's conjecture about the semiinfinite cohomology $H_\fu^{\frac{\infty}{2}+\bullet}$ of small quantum group $\fu$ of type $A_{n-1}$ (see ~\cite{ar} and section 6). Let us add a few more words about motivation. We believe that the Drinfeld's spaces $\CQ^D_\alpha$ are the basic building blocks of the would-be {\em Semiinfinite Flag Space} (cf. ~\cite{fm}). On the other hand, it was conjectured by G.Lusztig and B.Feigin that an appropriate category of perverse sheaves on Semiinfinite Flags is equivalent to a regular block of the category $\CC$ of graded $\fu$-modules. In this equivalence, the algebraic counterpart of the global Intersection Cohomology $H^\bullet(\CQ^D_\alpha,IC)$ is exactly $_{(\alpha+2\rho)}H_\fu^{\frac{\infty}{2}+\bullet}$ --- the (co)weight $(\alpha+2\rho)$ space of the ${\frak{sl}}_n$-module of semiinfinite cohomology $H_\fu^{\frac{\infty}{2}+\bullet}$. In fact, another geometric realization of the category $\CC$ was constructed in ~\cite{fs}. One of the main theorems of {\em loc. cit.} canonically identifies $_{(\alpha+2\rho)}H^{\frac{\infty}{2}+\bullet}_\fu$ with the Intersection Cohomology of a certain one-dimensional local system on a certain configuration space of $C$. Combining all the established equalities of characters (see section 6) we see that the above Intersection Cohomology has the same graded dimension as $H^\bullet(\CQ^D_\alpha,IC)$. We believe that it would be extremely interesting and important to find a direct explanation of this coincidence. In fact, this (conjectural) coincidence was the main impetus for the present work. The desired explanation might be not that easy since $H^\bullet(\CQ^D_\alpha,IC)$ has a Tate Hodge structure while the Intersection Cohomology of the above local system has quite a nontrivial Hodge structure (e.g. of elliptic curves or K3-surfaces) already in the simplest examples. \subsection{} The idea to realize the algebra $U(\fn)$ in correspondences (or in the $K$-groups of constructible sheaves on certain spaces) is not new: see e.g. the remarkable works ~\cite{lu}, ~\cite{blm}, ~\cite{gi}, ~\cite{naa}. What seems to be new compared to {\em loc. cit.} is the reason behind the relations in $U(\fn)$ (or $U({\frak{sl}}_n)$). Say, the divided powers of simple generators appear in {\em loc. cit.}, roughly, due to the fact that the flag manifold of $SL_d$ has Euler characteristic $d!$; while in the present work the divided powers appear, roughly, due to the fact that the Cartesian power $C^d$ is a $d!$-fold cover of the symmetric power $C^{(d)}$. Thus, the present construction may be viewed as a sort of globalization of {\em loc. cit.} in the particular case of Dynkin diagram of type $A_{n-1}$ (one might say that ~\cite{blm} and ~\cite{gi} lived in the formal neighbourhood of a point $0\in C$, while we work over the whole $C$). Note that the constructions of ~\cite{lu} and ~\cite{naa} can be (and are) generalized to arbitrary Dynkin graphs and quantized. It would be extremely interesting to generalize the results of the present note to an arbitrary Dynking graph (or even quantize them). On the other hand, the idea to realize Lie algebras' representations via ``global'' correspondences is not new either: see e.g. the remarkable works ~\cite{gr}, ~\cite{na}. These works realize some irreducible representations of infinite dimensional Lie algebras (Heisenberg and Clifford) in the cohomology of Hilbert schemes of {\em surfaces}. Thus the present work may be viewed as a baby version of {\em loc. cit.} Note though that in all the previous cases the representations of Lie algebras realized geometrically turned out to be irreducible, while we expect our modules to be {\em nonsemisimple}. In fact, we conjecture that they are {\em tilting} (see ~\cite{ap} and section 6). Also, in the global context, the appearence of Serre relations seems to be new. \subsection{} It is clear from the above explanations how much we were influenced by all the above cited works. It is a pleasure to thank B.Feigin, S.Arkhipov, and V.Ostrik for the numerous illuminating discussions and suggestions. Above all we are obliged to I.Mirkovic who spent half a year teaching one of us (M.F.) the beautiful geometry of affine and semiinfinite flag spaces, of which the present results are but a superficial manifestation. \subsection{} The present note is a sequel to ~\cite{k}. We will freely refer the reader to {\em loc. cit.} \section{Cohomology of $\CQ^L_\alpha$} \subsection{Notations} \subsubsection{} \label{not} We choose a basis $\{v_1,\ldots,v_n\}$ in $V$. This choice defines a Cartan subgroup $H\subset G$ of matrices diagonal with respect to this basis, and a Borel subgroup $B\subset G$ of matrices upper triangular with respect to this basis. We have $\CB=G/B$. Let $I=\{1,\ldots,n-1\}$ be the set of simple coroots of $G=SL_n$. Let $R^+$ denote the set of positive coroots, and let $2\rho=\sum_{\theta\in R^+}\theta$. For $\alpha=\sum a_ii\in\BN[I]$ we set $|\alpha|:=\sum a_i$. Recall the notations of ~\cite{k} concerning Kostant's partition function. For $\gamma\in\BN[I]$ a {\em Kostant partition} of $\gamma$ is a decomposition of $\gamma$ into a sum of positive coroots with multiplicities. The set of Kostant partitions of $\gamma$ is denoted by $\fK(\gamma)$. For $\kappa\in\fK(\gamma)$ let $|\kappa|=\gamma$, $||\kappa||=|\gamma|$ and let $K(\kappa)$ be the number of summands in $\kappa$. There is a natural bijection between the set of pairs $1\leq q\leq p\leq n-1$ and $R^+$, namely, $(p,q)$ corresponds to $i_q+i_{q+1}+\ldots+i_p$. Thus a Kostant partition $\kappa$ is given by a collection of nonnegative integers $(\kappa_{p,q}), 1\leq q\leq p\leq n-1$. Following {\em loc. cit.} (9) we define a collection $\mu(\kappa)$ as follows: $\mu_{p,q}=\sum_{r\leq q\leq p\leq s}\kappa_{s,r}$. \subsubsection{} For the definition of Laumon's Quasiflags' space $\CQ^L_\alpha$ the reader may consult ~\cite{la} 4.2, or ~\cite{k} 1.4. It is the space of complete flags of locally free subsheaves $$0\subset E_1\subset\dots\subset E_{n-1}\subset V\otimes\CO_C$$ such that rank$(E_k)=k$, and $\deg(E_k)=-a_k$. It is known to be a smooth projective variety of dimension $2|\alpha|+\dim\CB$. \subsubsection{} For the definition of Drinfeld's Quasimaps' space $\CQ^D_\alpha$ the reader may consult ~\cite{k} 1.2. It is the space of collections of invertible subsheaves $\CL_\lambda\subset V_\lambda\otimes\CO_C$ for each dominant weight $\lambda\in X^+$ satisfying Pl\"ucker relations, and such that $\deg\CL_\lambda=-\langle\lambda,\alpha\rangle$. It is known to be a (singular, in general) projective variety of dimension $2|\alpha|+\dim\CB$. \subsection{} Given a quasiflag $E_\bullet\in\CQ^L_\alpha$ and a point $x\in C$ the {\em type $\kappa(E_\bullet),\mu(E_\bullet)$ (of defect) of $E$ at $x$} was defined in {\em loc. cit.} (6)--(11). {\bf Definition.} For $\gamma\leq\alpha,\ \kappa\in\fK(\gamma)$ we denote by $\CZ^\kappa_\alpha\subset\CQ^L_\alpha$ the locally closed subspace formed by the quasiflags with defect of type $\kappa$ at $\infty\in C$. In particular, $\CZ^0_\alpha$ is an open subset of $\CQ^L_\alpha$. {\em Normalization} at $\infty\in C$ (see {\em loc. cit.} 1.5.1) defines a map $$\varpi^\kappa_\alpha:\ \CZ^\kappa_\alpha\lra\CZ^0_{\alpha-|\ka|}$$ Evaluation at $\infty\in C$ defines a map $$\Upsilon_\alpha:\ \CZ^0_\alpha\lra\CB$$ Evidently, $\Upsilon_\alpha$ is a locally trivial fibration. We will denote the fiber of $\Upsilon_\alpha$ over the point $B\in\CB=G/B$ by $\CY_\alpha$. \subsection{} \label{retract} The point $B\in\CB$ is represented by a flag $0\subset V_1\subset\ldots \subset V_{n-1}\subset V$. Let $E^0_\bullet$ denote the corresponding trivial flag of subbundles: $E^0_i=V_i\otimes\CO_C$. For the point $0\in C$ the {\em simple fiber} $F(E^0_\bullet,\alpha0)\subset \CQ^L_\alpha$ was introduced in {\em loc. cit.} 2.1.2. Its cohomology was computed in {\em loc. cit.} 2.4.4: its Poincar\'e polynomial equals $\CK_\alpha(t)=t^{|\alpha|}\sum_{\kappa\in\fK(\alpha)}t^{-K(\kappa)}$ --- the Lusztig-Kostant polynomial ($t$ has degree 2). {\bf Lemma.} The closed embedding $F(E^0_\bullet,\alpha0)\hookrightarrow \CY_\alpha$ induces an isomorphism of cohomology: $$H^\bullet(\CY_\alpha,\BQ)\iso H^\bullet(F(E^0_\bullet,\alpha0),\BQ)$$ {\em Proof.} We restrict the natural map $\pi:\ \CQ^L_\alpha\lra\CQ^D_\alpha$ to the locally closed subvariety $\CY_\alpha\subset\CQ^L_\alpha$. The image $\pi(\CY_\alpha)\subset\CQ^D_\alpha$ is denoted by $\CZ_\alpha$. It consists of quasimaps regular at $\infty\in C$ and taking there the value $B\in\CB$. We will preserve the same name for the restriction of $\pi$ to $\CY_\alpha\lra\CZ_\alpha$. It follows from the main Theorem of {\em loc. cit.} that $\pi$ is a small resolution of singularities. Hence $H^\bullet(\CY_\alpha,\BQ)=H^\bullet(\CZ_\alpha,IC)$. Now $\BC^*$ acts on $C$ by dilations preserving $0,\infty$, and thus it acts on both $\CY_\alpha$ and $\CZ_\alpha$, and the map $\pi$ is equivariant with respect to this action. The space $\CZ_\alpha$ has the only point $Z$ fixed by $\BC^*$: it is the point where all the defect is concentrated at $0\in C$. The {\em simple fiber} $F(E^0_\bullet,\alpha0)$ is nothing else than the fiber $\pi^{-1}(Z)$. So the stalk $IC_{(Z)}$ of $IC$-sheaf at the point $Z$ equals $H^\bullet(F(E^0_\bullet,\alpha0),\BQ)$. On the other hand, since the $\BC^*$-action contracts $\CZ_\alpha$ to $Z$, we have $H^\bullet(\CZ_\alpha,IC)=IC_{(Z)}$. The proposition is proved. Alternatively, instead of using Intersection Cohomology, we could argue that according to ~\cite{sl}, 4.3, $F(E^0_\bullet,\alpha0)=\pi^{-1}(Z)$ is a deformation retract of $\CY_\alpha$. $\Box$ \subsubsection{Remark} The space $\CZ_\alpha$ plays a central role and is extensively studied in ~\cite{fm}. \subsubsection{Corollary} \label{van} The odd-dimensional cohomology of $\CY_\alpha$ vanishes. {\em Proof.} Follows immediately from ~\cite{k}, 2.4.4. $\Box$ \subsection{} \label{fib} We consider the locally trivial fibration $\Upsilon_\alpha:\ \CZ^0_\alpha \lra\CB$ with the fiber $\CY_\alpha$. Since the odd-dimensional cohomology of both the fiber and the base vanishes, the Leray spectral sequence of this fibration degenerates, and we arrive at the following Lemma: {\bf Lemma.} The odd-dimensional cohomology of $\CZ^0_\alpha$ vanishes. The Poincar\'e polynomial $P(H^\bullet(\CZ^0_\alpha),t)$ equals $\CK_\alpha(t)\sum_{w\in W}t^{\ell(w)}$. $\Box$ \subsection{Lemma} \label{fig} The Poincar\'e polynomial of the cohomology with compact support $P(H_c^\bullet(\CZ^0_\alpha),t)$ equals $t^{\dim\CB+2|\alpha|} \CK_\alpha(t^{-1})\sum_{w\in W}t^{-\ell(w)}$. {\em Proof.} The space $\CZ^0_\alpha$ is smooth of dimension $\dim\CB+2|\alpha|$. Now apply the Poincar\'e duality and the Lemma ~\ref{fib}. $\Box$ \subsection{Lemma} \label{mig} The odd-dimensional cohomology with compact support of $\CZ^0_\alpha$ vanishes. The Poincar\'e polynomial of the cohomology with compact support $P(H_c^\bullet(\CZ^\kappa_\alpha),t)$ equals $$ t^{\dim\CB+2|\alpha|-||\kappa||-K(\kappa)} \CK_{\alpha-|\kappa|}(t^{-1})\sum_{w\in W}t^{-\ell(w)}. $$ {\em Proof.} The normalization map $\varpi^\kappa_\alpha:\ \CZ^\kappa_\alpha\lra\CZ^0_{\alpha-|\kappa|}$ is a locally trivial fibration with a fiber isomorphic to a pseudoaffine space $\fS_{\mu(\kappa)}$ (see ~\cite{k}~(16)) of dimension $||\kappa||-K(\kappa)$ (see {\em loc. cit.}~(21)). Now apply the Lemma ~\ref{fig}. $\Box$ \subsection{} \label{character} We consider the stratification $$\CQ^L_\alpha=\bigsqcup_{|\kappa|\leq\alpha}\CZ^\kappa_\alpha$$ and the corresponding Cousin spectral sequence converging to the compactly supported cohomology of $\CQ^L_\alpha$ (equal to $H^\bullet(\CQ^L_\alpha,\BQ)$ by Poincar\'e duality). Since the odd-dimensional compactly supported cohomology of every stratum vanishes, the Cousin spectral sequence degenerates, and we are able to compute the Poincar\'e polynomial of the space $\CQ^L_\alpha$. To write it down in a neat form we will need some minor preparations. First of all, we shift the cohomological degree so that the cohomology becomes symmetric around zero degree: we consider $H^\bullet(\CQ^L_\alpha,\BQ[\dim\CQ^L_\alpha])$. Recall that $\dim\CQ^L_\alpha= 2|\alpha|+\dim\CB=2|\alpha|+\frac{n(n-1)}{2}$. Second, we will consider the generating function for $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ[\dim\CQ^L_\alpha])$. To record the information on $\alpha$ we will consider this generating function as a formal cocharacter of $H$ with coefficients in the Laurent polynomials in $t$. Formal cocharacters will be written multiplicatively, so that the cocharacter corresponding to $\alpha$ will be denoted by $e^\alpha$. Finally, for the reasons which will become clear later (see Proposition ~\ref{h}), we will make the following rescaling. We will attach to $H^\bullet(\CQ^L_\alpha,\BQ[\dim\CQ^L_\alpha])$ the cocharacter $e^{\alpha+2\rho}$. With all this in mind, the Poincar\'e polynomial $P_G(t)$ of $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ[\dim\CQ^L_\alpha])$ is calculated as follows: {\bf Theorem.} $$P_G(t)=\frac{e^{2\rho}t^{-\frac{1}{2}\dim\CB} \sum_{w\in W}t^{\ell(w)}} {\prod_{\theta\in R^+}(1-te^\theta)(1-t^{-1}e^\theta)}$$ $\Box$ \subsection{} \label{IC} The main Theorem of ~\cite{k} asserts that the natural map $\pi:\ \CQ^L_\alpha\lra\CQ^D_\alpha$ is a small resolution of singularities. Hence the Intersection Cohomology complex on $\CQ^D_\alpha$ is the direct image of the constant sheaf on $\CQ^L_\alpha:\ IC=\pi_*\underline{\BQ}[\dim\CQ^L_\alpha]$. This implies that the global Intersection Cohomology $H^\bullet(\CQ^D_\alpha,IC)$ coincides with the cohomology $H^\bullet(\CQ^L_\alpha,\BQ[\dim\CQ^L_\alpha])$. Thus we obtain the following theorem. {\bf Theorem.} The generating function for the global Intersection Cohomology of Drinfeld's Quasimaps' spaces $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ is given by $$P_G(t)=\frac{e^{2\rho}t^{-\frac{1}{2}\dim\CB}\sum_{w\in W}t^{\ell(w)}} {\prod_{\theta\in R^+}(1-te^\theta)(1-t^{-1}e^\theta)}$$ $\Box$ \subsection{} \label{main} {\bf Theorem.} There is a cell decomposition $$\CQ^L_\alpha=\bigsqcup\Delta(w,\kappa^0,\kappa^\infty)$$ into cells numbered by the following data: $w\in W$; partition $\kappa^0$ (resp. $\kappa^\infty$) of $\gamma^0\in\BN[I]$ (resp. $\gamma^\infty\in\BN[I]$) such that $\gamma^0+\gamma^\infty=\alpha$. \subsubsection{} The proof of the Theorem will occupy the rest of the section. \subsection{} We will consider a torus action on $\CQ^L_\alpha$ with finitely many fixed points, and the Bialynicki-Birula decomposition defined by this action will give the desired cell decomposition. \subsubsection{} \label{zvezdochka} The Cartan group $H$ acts on $V$ and hence on $\CQ^L_\alpha$. The group $\BC^*$ of dilations of $C={\Bbb P}^1$ preserving $0$ and $\infty$ also acts on $\CQ^L_\alpha$ commuting with the action of $H$. Hence we obtain the action of a torus $\BT:=H\times\BC^*$ on $\CQ^L_\alpha$. \subsubsection{} We fix a coordinate $z$ on $C={\Bbb P}^1$ such that $z(0)=0,\ z(\infty)=\infty$. \subsection{} \label{fixpoint} Let us describe the fixed point set $(\CQ^L_\alpha)^\BT$. Given a triple $(w,\kappa^0,\kappa^\infty)$ as in the Theorem ~\ref{main}, we define the point $\delta(w,\kappa^0,\kappa^\infty)\in(\CQ^L_\alpha)^\BT$ as follows. It is a quasiflag $(E_1,\ldots,E_{n-1})$ such that its normalization $(\tilde{E}_1,\ldots,\tilde{E}_{n-1})$ (see ~\cite{k}, Definition ~1.5.1) is a constant flag with $\tilde{E}_1$ spanned by $v_{w(1)}$; $\tilde{E}_2$ spanned by $v_{w(1)}$ and $v_{w(2)};\ \ldots;\ \tilde{E}_{n-1}$ spanned by $v_{w(1)},v_{w(2)},\ldots,v_{w(n-1)}$. Its defect is a collection of torsion sheaves (see {\em loc. cit.}) on $C$ supported at $0$ and $\infty$. In a neighbourhood of $0\in C$ the quasiflag $(E_1,\ldots,E_{n-1})$ is defined as follows: $$ {\arraycolsep=1pt \begin{array}{llrlcrlcccrlc} E_1 & =\langle & z^{d^0_{1,1}} & v_{w(1)}&\rangle \\ E_2 & =\langle & z^{d^0_{2,1}} & v_{w(1)}&,& z^{d^0_{2,2}} & v_{w(2)} &\rangle\\ \ \vdots && \vdots &&& \vdots \\ E_{n-1} & =\langle & z^{d^0_{n-1,1}} & v_{w(1)} &,& z^{d^0_{n-1,2}} & v_{w(2)} & , & \dots & , & z^{d^0_{n-1,n-1}} & v_{w(n-1)}&\rangle \\ \end{array} } $$ where the collection $(d^0_{p,q})_{1\leq q\leq p\leq n-1}$ (resp. $(d^\infty_{p,q})_{1\leq q\leq p\leq n-1}$) is defined via $\kappa^0$ (resp. $\kappa^\infty$) as follows: $$ d^\bullet_{p,q}=\sum_{r=p}^{n-1}\kappa^\bullet_{r,q}. $$ Finally, in a neighbourhood of $\infty\in C$ the quasiflag $(E_1,\ldots,E_{n-1})$ is defined exactly as around $0$, with the replacement of $d^0$ by $d^\infty$ and $z$ by $z^{-1}$. \subsection{Proposition} The fixed point set $(\CQ^L_\alpha)^\BT$ coincides with the collection of points $\delta(w,\kappa^0,\kappa^\infty)$ numbered by the triples as in the Theorem ~\ref{main}. {\em Proof.} Easy. $\Box$ \subsection{} \label{cells} It is well known that for a generic choice of one-parametric subgroup $\fG\subset \BT$ we have $(\CQ^L_\alpha)^\fG=(\CQ^L_\alpha)^\BT$. Hence we can apply the main Theorem 4.4 of ~\cite{bb} to obtain the desired decomposition $$\CQ^L_\alpha=\bigsqcup\Delta(w,\kappa^0,\kappa^\infty)$$ into locally closed subchemes. Each one of them is an affine space containing exactly one fixed point: namely, $\Delta(w,\kappa^0,\kappa^\infty)\ni \delta(w,\kappa^0,\kappa^\infty)$. This completes the proof of the Theorem ~\ref{main}. $\Box$ \subsubsection{Remark} It would be desirable to make a wise canonical choice of $\fG$ producing some canonical cell decomposition (or, moreover, a stratification) of $\CQ^L_\alpha$. For instance, we expect that for such a wise choice the dimension of $\Delta(w,\kappa^0,\kappa^\infty)$ would be given by $d(w,\kappa^0,\kappa^\infty)=\ell(w)+||\kappa^0||+||\kappa^\infty||+ K(\kappa^0)-K(\kappa^\infty)$. This would give a more natural proof of the Theorem ~\ref{character}. Furthermore, such decomposition would produce a canonical basis in $H^\bullet(\CQ^L_\alpha,\BQ)$ (Poincar\'e duals of fundamental classes of cells) which in turn might prove useful in checking the freeness of $U(\fn)$-action on $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$ defined in the next section (see the Conjecture ~\ref{conjecture} and the Remark ~\ref{freeness}). Unfortunately, we were not able to make such a wise choice of $\fG$. \section{Simple correspondences} \subsection{}\label{def} For any $i\in I$ and $\alpha\in\BN[I]$ we introduce the following closed subvariety $\fE_i^\alpha\subset\CQ^L_\alpha\times\CQ^L_{\alpha+i}$. {\bf Definition.} $\fE_i^\alpha:= \{((E_1,\ldots,E_{n-1}),(E'_1,\ldots,E'_{n-1}))$ such that for $j\not=i$ we have $E_j=E'_j$, while $E_i\supset E'_i$, and $E_i/E'_i$ is a torsion sheaf of length one$\}$. There are natural maps $$ \bp:\fE_i^\al\to\QL\al,\qquad \barq:\fE_i^\al\to\QL{\al+i},\quad\text{and}\quad \br:\fE_i^\al\to C. $$ The first and second maps are induced by the projections of $\QL\al\times\QL{\al+i}$ onto the first and second factor and the third is defined as $$ \br((E_\bullet,E'_\bullet))=\supp{E_i/E'_i}. $$ The following Lemma describes the fibers of the map $$ \bp\times \br:\fE_i^\al\to\QL\al. $$ \subsection{Lemma} \label{fibersofpr} Let $E_\bullet\in\QL\al$, $x\in C$. The fiber $(\bp\times \br)^{-1}(E_\bullet)$ is naturally isomorphic to the projective space $\PP(\Hom(E_i/E_{i-1},\CO_x))$. The map $(\bp\times \br)$ is an isomorphism over the space of pairs $(E_\bullet,x)$ such that $E_\bullet\in\CQ_\al$ is a flag of subbundles. {\em Proof.} The fiber $(\bp\times \br)^{-1}(E_\bullet,x)$ is evidently isomorphic to the space of all subsheaves $E'_i\subset E_i$ such that $E_i/E'_i=\CO_x$ and embedding $E_{i-1}\hookrightarrow E_i$ factors through $E'_i$. In other words it is the space of all diagrams $$ \begin{CD} E_{i-1} @>>> E'_i @>>> E'_i/E_{i-1} \\ @| @VVV @VVV \\ E_{i-1} @>>> E_i @>>> E_i/E_{i-1} \\ @. @VVV @VVV \\ @. \CO_x @= \CO_x \end{CD} $$ with exact rows and columns. But such a diagram can be uniquely reconstructed from the map $E_i/E_{i-1}\to\CO_x$. The first part of the Lemma follows. If $E_\bullet\in\CQ_\al$ then the quotient $E_i/E_{i-1}$ is locally free, hence $\Hom(E_i/E_{i-1},\CO_x)=\CCC$ for all $x$. This means that $\bp\times \br$ is an isomorphism over the space $\CQ_\al\times C\subset\QL\al\times C$. $\Box$ \subsubsection{} Let $\fE_{\lbr i\}\}}^\al$ be the closure of the space $(\bp\times \br)^{-1}(\CQ_\al\times C)$. Recall (see ~\cite{k}, 1.4.1) that $\dim\CQ^L_\alpha=\dim\CB+2|\alpha|$. The map $(\bp\times \br)_{|\fE_{\lbr i\}\}}^\al}: \fE_{\lbr i\}\}}^\al\to\QL\al\times C$ is birational, hence $\fE_{\lbr i\}\}}^\al$ is a $(\dim\CB+2|\al|+1)$-dimensional irreducible variety. \begin{lem}{irred} The space $\fE_{\lbr i\rbr }^\al$ is a unique $(\dim\CB+2|\al|+1)$-dimensional irreducible component of $\fE_i^\al$. \end{lem} {\em Proof.} It is a particular case of the Proposition ~\ref{dims}. Alternatively, a direct proof goes as follows. Consider the following stratification of $\QL\al\times C$ $$ \QL\al\times C=\bigsqcup\begin{Sb}\gamma\le\alpha\\\ka\in\fK(\gamma)\end{Sb} Z^\al_\ka, $$ where $Z^\al_\ka\subset\QL\al\times C$ is the subspace of pairs $(E_\bullet,x)$ such that $\ka\in\fK(\ga)$ is the type of the defect of $E_\bullet\in\QL\al$ at the point $x\in C$ (see the section 2 of ~\cite{k}). Considering the map $$ \Pi:Z^\al_\ka\to\QL{\al-|\ka|}\times C,\qquad (E_\bullet,x)\mapsto (\ti E_\bullet,x), $$ where $\ti E_\bullet$ is the normalization at $x$ of $E_\bullet$ and applying the Lemma~2.4.3 of {\em loc.\ cit.} we see that $$ \dim Z^\al_\ka=\dim\CB+2|\al|-||\ka||-K(\ka)+1. $$ On the other hand, it is easy to see that over the stratum $Z^\al_\ka$ we have $$ \length(T_x)=\sum_{p=1}^{i-1}\ka_{i-1,p}, $$ where $T_x$ is the part of the torsion $T$ in the quotient sheaf $E_i/E_{i-1}$ with support at $x$. Using obvious inequality $$ \dim\Hom(T,\CO_x)\le\length(T_x) $$ we see that the dimension of the fiber of $\fE_i^\al$ over the point $(E_\bullet,x)\in Z^\al_\ka$ is not greater than $\sum_{p=1}^{i-1}\ka_{i-1,p}$. But $$ ||\ka||+K(\ka)\ge\sum_{p=1}^{i-1}\ka_{i-1,p} $$ and equality is possible only for $\gamma=0$ (it follows easily from {\em loc.\ cit.}~(9)). This means that for any $0<\gamma\le\alpha$ and $\ka\in\fK(\gamma)$ $$ \dim \bp^{-1}(Z^\al_\ka)<\dim\CB+2|\al|+1 $$ and the Lemma follows. $\Box$ \subsection{} \label{ef} According to Lemma ~\ref{irred}, we may consider the Poincar\'e dual of the fundamental class $[\fE_{\lbr i\rbr }^\alpha]\in H^\bullet(\CQ^L_\alpha\times \CQ^L_{\alpha+i},\BQ)$. Viewed as a correspondence, it defines two operators: $$e_i:\ H^\bullet(\CQ^L_\alpha,\BQ)\rightleftharpoons H^\bullet(\CQ^L_{\alpha+i},\BQ)\ :f_i$$ adjoint to each other with respect to Poincar\'e duality. The operator $e_i$ increases the cohomological degree by 2, and the operator $f_i$ decreases it by 2. \subsubsection{Remark}\label{rem1} We may also consider the operators $$ \hat e_i:\ H^\bullet(\CQ^L_\alpha,\BQ)\rightleftharpoons H^\bullet(\CQ^L_{\alpha+i},\BQ)\ :\hat f_i $$ defined by the fundamental class $[\fE_i^\al]\in H^\bullet(\CQ^L_\alpha\times\CQ^L_{\alpha+i},\BQ)$. Consider the decompositions of the operators $\hat e_i$ and $\hat f_i$ into the sum of operators shifting the cohomological degree by $k$. $$ \hat e_i=\sum\hat e_i^k,\qquad \hat f_i=\sum\hat f_i^k $$ The Lemma~\ref{irred} implies that $$ e_i=\hat e_i^2,\qquad f_i=\hat f_i^{-2}, $$ and $$ \hat e_i^k=0\quad\text{for }k>2\qquad\text{and}\qquad \hat f_i^k=0\quad\text{for }k<-2. $$ \subsubsection{} We fix an orientation $\Omega=(1\lra2\lra\ldots\lra n-1)$ of the Dynkin graph with the set of vertices $I$. Note that any flag of subsheaves (subbundles) in the trivial bundle $V\otimes\CO_C$ can be considered as a representation of the quiver $\Omega$ in the category of subsheaves (subbundles) of $V\otimes\CO_C$. Therefore, given a pair of flags $E'_\bullet\subset E_\bullet$ we have the quotient representation $T_\bullet=E_\bullet/E'_\bullet$. This is a representation of the quiver $\Omega$ in the category of torsion sheaves on $C$. Let us denote the category of such representations by $\RT$. Define the {\em dimension} and {\em local dimension at $x\in C$} of $T=(T_1,\dots,T_{n-1})\in\Ob(\RT)$ as the coroots $$ \dim T=\sum_{i\in I}\length(T_i)i\in\NNN[I],\qquad \dim_x T=\sum_{i\in I}\length_x(T_i)i\in\NNN[I]. $$ Let $T\in\Ob(\RT)$ and $\dim T=\gamma$. Given a filtration $0=F_0\subset F_1\dots\subset F_m=T$ of $T$ by subrepresentations we say that it is a filtration of the type $(\gamma_1,\dots,\gamma_m)$ if $\dim F_k/F_{k-1}=\gamma_k$. Let us denote by $\CO_x[i]$ a simple $i$-dimensional object in the category $\RT$, consisting of the sheaf $\CO_x$ which lives over the $i$-th vertex of $\Omega$. \subsection{Proposition} \label{triv} Given $i,j\in I$ such that $|i-j|>1$ we have a) $e_ie_j=e_je_i$; b) $f_if_j=f_jf_i$. {\em Proof.} Instead of $e_ie_j$ and $e_je_i$ we will consider the components of $\hat e_i\hat e_j$ and $\hat e_j\hat e_i$ shifting the cohomological degree by 4 (it suffices by the Remark~\ref{rem1}). To this end, consider the spaces $$ \fE^\al_{i,j}=\bp_{12}^{-1}(\fE^\al_i)\cap \bp_{23}^{-1}(\fE^{\al+i}_j)\subset \QL\al\times\QL{\al+i}\times\QL{\al+i+j} $$ and $$ \fE^\al_{j,i}=\bp_{12}^{-1}(\fE^\al_j)\cap \bp_{23}^{-1}(\fE^{\al+j}_i)\subset \QL\al\times\QL{\al+j}\times\QL{\al+i+j}, $$ where $\bp_{ab}$ denotes the projection of the product $\QL\al\times\QL{\al+i}\times\QL{\al+i+j}$ (resp.\ $\QL\al\times\QL{\al+j}\times\QL{\al+i+j}$) onto the product of the $a$-th and $b$-th factors. The definition of $\fE^\al_i$ implies that $\fE^\al_{i,j}=\{(E_\bullet,E'_\bullet,E'''_\bullet)$ such that $E_\bullet\supset E'_\bullet\supset E'''_\bullet$, $E_\bullet/E'_\bullet=\CO_x[i]$ and $E'_\bullet/E'''_\bullet=\CO_y[j]$ for some $x,y\in C\,\}$. This means that $E_\bullet/E'''_\bullet$ is an extension of $\CO_y[j]$ by $\CO_x[i]$. But it is easy to see that $|i-j|>1$ implies $\Ext_\RT(\CO_y[j],\CO_x[i])=0$, hence $E_\bullet/E'''_\bullet=\CO_x[i]\oplus\CO_y[j]$. Let $E''$ be the kernel of the composition $E_\bullet\to\CO_x[i]\oplus\CO_y[j]\to\CO_y[j]$. Then we have $E_\bullet\supset E''_\bullet\supset E'''_\bullet$, $E_\bullet/E''_\bullet=\CO_y[j]$ and $E''_\bullet/E'''_\bullet=\CO_x[i]$. This means that $(E,E',E''')\to(E,E'',E''')$ is a map from $\fE^\al_{i,j}$ to $\fE^\al_{j,i}$. This map is certainly an isomorphism. Now the composition of correspondences $[\fE^\al_i]\circ[\fE^\al_j]$ is the correspondence given by the cycle $$ {\bp_{13}}_*(\bp_{12}^*[\fE^\al_i]\cap \bp_{23}^*[\fE^{\al+i}_j])= {\bp_{13}}_*[\fE^\al_{i,j}]={\bp_{13}}_*[\fE^\al_{j,i}]= {\bp_{13}}_*(\bp_{12}^*[\fE^\al_j]\cap \bp_{23}^*[\fE^{\al+j}_i]), $$ i.e.\ $$ \hat e_i\hat e_j=\hat e_j\hat e_i\qquad\hat f_i\hat f_j=\hat f_j\hat f_i $$ and Proposition follows. $\Box$ \subsection{Proposition} \label{Serre} Given $i,j\in I$ such that $|i-j|=1$ we have a) $e_i^2e_j-2e_ie_je_i+e_je_i^2=0$; b) $f_i^2f_j-2f_if_jf_i+f_jf_i^2=0$. {\em Proof.} Let $j=i-1$. Consider the space $\fE^\al_{2i+j}\subset\QL\al\times\QL{\al+2i+j}$ of all pairs $(E,E')$ such that $E'\subset E$ and $\dim(E/E')=2i+j$. Let $T=(0,\dots,0,T_j,T_i,0,\dots,0)=E/E'$ be the quotient representation. Let $\bp:\fE^\al_{2i+j}\to\QL\al$ denote the map induced by the projection of $\QL\al\times\QL{\al+2i+j}$ onto the first factor, and let $\br:\fE^\al_{2i+j}\to C^{2i+j}$ denote the map sending a pair $(E,E')$ to $\sum\limits_{x\in C}\dim_x(T)x=j\,\supp T_j+i\,\supp T_i$. Recall the diagonal stratification $$C^{2i+j}=C^{2i+j}_{\lbr i,i,j\rbr }\sqcup C^{2i+j}_{\lbr 2i,j\rbr }\sqcup C^{2i+j}_{\lbr i,i+j\rbr }\sqcup C^{2i+j}_{\lbr 2i+j\rbr }$$ introduced e.g. in ~\cite{k}, 1.3. Consider the map $\bp\times \br:\fE^\al_{2i+j}\to\QL\al\times C^{2i+j}$. This map is an isomorphism over the open set $\CQ_\al\times C^{2i+j}_{\lbr i,i,j\rbr }\subset\QL\al\times C^{2i+j}$. This can be proved by the same arguments as the Lemma~\ref{fibersofpr} (the fiber over the point $(E,jx\!+\!iy\!+\!iz)$ is isomorphic to $\PP(\Hom(E_{i-1}/E_{i-2},\CO_x))\times \Big(\PP(\Hom(E_i/E_{i-1},\CO_y)) \times\PP(\Hom(E_i/E_{i-1},\CO_z))/\ZZZ_2\Big)$ which is a single point in our case). On the other hand, over the subset $\CQ_\al\times C^{2i+j}_{\lbr i,i+j\rbr }\subset\QL\al\times C^{2i+j}$ the fibers of the map $\bp\times \br$ are one-dimensional (the fiber over a point $(E,(i\!+\!j)x\!+\!iy)$ is naturally isomorphic to $\PP(\Hom(E_i/E_{i-2},\CO_x))\times\PP(\Hom(E_i/E_{i-1},\CO_y))$ which is $\PP^1$ in our case). Note that for the generic element $(E,E')$ of the fiber the map $T_{i-1}\to T_i$ in the quotient representation is non-zero. Let $\fE^\al_{\lbr i,i,j\rbr }$ denote the closure of $(\bp\times \br)^{-1}(\CQ_\al\times C^{2i+j}_{\lbr i,i,j\rbr })$, and let $\fE^\al_{\lbr i,i+j\rbr }$ denote the closure of $(\bp\times \br)^{-1}(\CQ_\al\times C^{2i+j}_{\lbr i,i+j\rbr })$. The spaces $\fE^\al_{\lbr i,i,j\rbr }$ and $\fE^\al_{\lbr i,i+j\rbr }$ are irreducible $(\dim\CB+2|\al|+3)$-dimensional components of $\fE^\al_{2i+j}$. \subsubsection{Claim}\label{dim2} All other irreducible components of $\fE^\al_{2i+j}$ have smaller dimension. {\em Proof.} It is just a particular case of the Proposition ~\ref{dims}. $\Box$ Now we can finish the proof of the proposition. To this end consider the spaces $$ {\arraycolsep=3pt \begin{array}{lcccccccccccccc} \fE^\al_{i,i,j} & = & \bp_{12}^{-1}(\fE^\al_i) & \cap & \bp_{23}^{-1}(\fE^{\al+i}_i) & \cap & \bp_{34}^{-1}(\fE^{\al+2i}_j) & \subset & \QL\al & \times & \QL{\al+i} & \times & \QL{\al+2i} & \times & \QL{\al+2i+j}\\ \fE^\al_{i,j,i} & = & \bp_{12}^{-1}(\fE^\al_i) & \cap & \bp_{23}^{-1}(\fE^{\al+i}_j) & \cap & \bp_{34}^{-1}(\fE^{\al+i+j}_i) & \subset & \QL\al & \times & \QL{\al+i} & \times & \QL{\al+i+j} & \times & \QL{\al+2i+j}\\ \fE^\al_{j,i,i} & = & \bp_{12}^{-1}(\fE^\al_j) & \cap & \bp_{23}^{-1}(\fE^{\al+j}_i) & \cap & \bp_{34}^{-1}(\fE^{\al+i+j}_i) & \subset & \QL\al & \times & \QL{\al+j} & \times & \QL{\al+i+j} & \times & \QL{\al+2i+j} \end{array} } $$ It is easy to see that the space $\fE^\al_{i,i,j}$ (resp.\ $\fE^\al_{i,j,i}$, $\fE^\al_{j,i,i}$) is isomorphic to the space of triples $(E,E',F)$, where $E\in\QL\al$, $E'\in\QL{\al+2i+j}$ such that $E'\subset E$ and $\dim(E/E')=2i+j$, and $F$ is a filtration (by subrepresentations) in the quotient representation $0\subset F_1\subset F_2\subset F_3=E/E'$ of the type $(i,i,j)$ (resp.\ $(i,j,i)$, $(j,i,i)$). Consider the projection $\bp_{14}:\fE^\al_{i,i,j}\to\QL\al\times\QL{\al+2i+j}$ (and two others). It is clear that the images of $\fE^\al_{i,i,j}$ (resp.\ $\fE^\al_{i,j,i}$, $\fE^\al_{j,i,i}$) lie in $\fE^\al_{2i+j}$ and the fibers of these projections over the point $(E,E')$ can be identified with the set of all filtrations $F$ in the quotient representation $T=E/E'$ of the corresponding type. \subsubsection{Lemma}\label{filt} a) If $\ jx\!+\!iy\!+\!iz\in C^{2i+j}_{\lbr i,i,j\rbr }$ and $T$ is a $(2i+j)$-dimensional representation of the quiver $\Omega$ with $\supp T_{i-1}=x$, $\supp T_i=\{y,z\}$ then $T$ admits two filtrations of type $(i,i,j)$, two filtrations of the type $(i,j,i)$ and two filtrations of the type $(j,i,i)$. b) If $\ (i\!+\!j)x\!+\!iy\in C^{2i+j}_{\lbr i,i+j\rbr }$ and $T$ is $(2i+j)$-dimensional representation of the quiver $\Omega$ with $\supp T_{i-1}=x$, $\supp T_i=\{x,y\}$ and non-zero map $T_{i-1}\to T_i$ then $T$ admits two filtrations of the type $(i,i,j)$, one filtration of the type $(i,j,i)$ and no filtrations of the type $(j,i,i)$. {\em Proof.} Trivial. $\Box$ Now we are ready to compute the compositions of the correspondences. \begin{multline*} [\fE^\al_i]\circ[\fE^{\al+i}_i]\circ[\fE^{\al+2i}_j]= (\bp_{14})_* (\bp_{12}^*[\fE^\al_i]\cap \bp_{23}^*[\fE^{\al+i}_i]\cap \bp_{34}^*[\fE^{\al+2i}_j])=\\ =(\bp_{14})_*[\fE^\al_{i,i,j}]=2[\fE^\al_{\lbr i,i,j\rbr }]+ 2[\fE^\al_{\lbr i,i+j\rbr }]+ \text{terms of smaller dimension.} \end{multline*} Similarly, $$ [\fE^\al_i]\circ[\fE^{\al+i}_j]\circ[\fE^{\al+i+j}_i]= 2[\fE^\al_{\lbr i,i,j\rbr }]+ [\fE^\al_{\lbr i,i+j\rbr }]+\text{terms of smaller dimension,} $$ and $$ [\fE^\al_j]\circ[\fE^{\al+j}_i]\circ[\fE^{\al+i+j}_i]= 2[\fE^\al_{\lbr i,i,j\rbr }]+\text{terms of smaller dimension.} $$ The Proposition in the case $j=i-1$ follows (recall Remark~\ref{rem1}). The case $j=i+1$ can be treated similarly. $\Box$ \subsection{Proposition} \label{h} a) For $i\not=j$ we have $e_if_j=f_je_i$; b) On $H^\bullet(\CQ^L_\alpha,\BQ)$ we have $e_if_i-f_ie_i=\langle i',\alpha+2\rho\rangle$ (multiplication by a constant). Here $i'\in X$ stands for the simple root dual to $i$, and $\langle\;,\,\rangle:\ X\times Y\lra\BZ$ stands for the nondegenerate pairing between cocharacters and weights. Finally, $2\rho\in\BN[I]$ is the sum of all positive coroots. {\em Proof.} Consider the following spaces: $$ {\arraycolsep=2pt \begin{array}{lcccccccccc} \fEF^\al_{i,j} & = & \bp_{12}^{-1}(\fE^\al_i) & \cap & \bp_{23}^{-1}\left((\fE^{\al+i-j}_j)^T\right) & \subset & \QL\al & \times & \QL{\al+i} & \times & \QL{\al+i-j} \\ \fFE^\al_{i,j} & = & \bp_{12}^{-1}\left((\fE^{\al-j}_j)^T\right) & \cap & \bp_{23}^{-1}(\fE^{\al-j}_i) & \subset & \QL\al & \times & \QL{\al-j} & \times & \QL{\al+i-j} \end{array} } $$ Here $(\fE^\al_i)^T$ denotes the subvariety in $\QL{\al+i}\times\QL\al$ transposed to $\fE^\al_i\subset\QL\al\times\QL{\al+i}$. It is easy to see that $\fEF^\al_{i,j}$ is the space of triples $(E,E',E''')\in\QL\al\times\QL{\al+i}\times\QL{\al+i-j}$ such that $E\supset E'\subset E'''$ and $\fFE^\al_{i,j}$ is the space of triples $(E,E'',E''')\in\QL\al\times\QL{\al-j}\times\QL{\al+i-j}$ such that $E\subset E''\supset E'''$. Consider the projections $\bp_{13}:\fEF^\al_{i,j}\to\QL\al\times\QL{\al+i-j}$ and $\bp_{13}:\fFE^\al_{i,j}\to\QL\al\times\QL{\al+i-j}$. Over the set $U=\{(E,E''')\ |\ E\ne E'''\}\subset\QL\al\times\QL{\al+i-j}$ (in the case $i\ne j$ we have $U=\QL\al\times\QL{\al+i-j}$) the spaces $\fEF^\al_{i,j}$ and $\fFE^\al_{i,j}$ are isomorphic. The isomorphisms are given by formulas $$ (E,E',E''')\mapsto(E,E+E''',E''')\qquad\text{and}\qquad (E,E'',E''')\mapsto(E,E\cap E''',E'''). $$ Let $\wti \fEF^\al_{i,j}$ (resp.\ $\wti \fFE^\al_{i,j}$) denote the closure of $\bp_{13}^{-1}(U)$ in $\fEF^\al_{i,j}$ (resp. in $\fFE^\al_{i,j}$). In the case $i\ne j$ we have $\wti \fEF^\al_{i,j}=\fEF^\al_{i,j}$ (resp.\ $\wti \fFE^\al_{i,j}=\fFE^\al_{i,j}$). We have $(\bp_{13})_*[\wti \fEF^\al_{i,j}]=(\bp_{13})_*[\wti \fFE^\al_{i,j}]$. Since $$ [\fE^\al_i]\circ[(\fE^{\al+i-j}_j)^T]=(\bp_{13})_*[\fEF^\al_{i,j}],\qquad [(\fE^{\al-j}_j)^T]\circ[\fE^{\al-j}_i]=(\bp_{13})_*[\fFE^\al_{i,j}], $$ the case $i\ne j$ follows. In the case $i=j$ it remains to compare the contribution of components of $\fEF^\al_{i,i}$ and $\fFE^\al_{i,i}$ over the diagonal $\QL\al @>\Delta>> \QL\al\times\QL\al$. Let $\fEF^\al_i$ (resp.\ $\fFE^\al_i$) be the preimage of the diagonal $\QL\al\subset\QL\al\times\QL\al$, and let $\bp:\fEF^\al_i\to\QL\al$ (resp.\ $\barq:\fFE^\al_i\to\QL\al$) be the corresponding projection. It is easy to see that $\fEF^\al_i$ is isomorphic to $\fE^\al_i$ and $\fFE^\al_i$ is isomorphic to $\fE^{\al-i}_i$ (and the maps $\bp$, $\barq$ are the same as in \ref{def}). Hence the dimension of $\fFE^\al_i$ is equal to $\dim\CB+2|\al|-1$ which is less than the expected dimension of $[(\fE^{\al-i}_i)^T]\circ[\fE^{\al-i}_i]$ equal to $\dim\CB+2|\al|$. Thus the contribution of $\fFE^\al_i$ in $[(\fE^{\al-i}_i)^T]\circ[\fE^{\al-i}_i]$ lives in the dimension smaller than $\dim\CB+2|\al|$. On the other hand the dimension of $\fEF^\al_i$ is equal to $\dim\CB+|\al|+1$ which is greater than the expected dimension. According to the Intersection Theory (see ~\cite{fu}) in this case we have $$ [\fE^\al_i]\circ[(\fE^\al_i)^T]-[(\fE^{\al-i}_i)^T]\circ[\fE^{\al-i}_i]= (\bp_{13})_*(c_1(\CL)), $$ where $\CL$ is a certain line bundle on $\fEF^\al_i$ defined in ~\ref{L} below. We know that only one component of $\fE^\al_i$ (namely $\fE^\al_{\lbr i\rbr}$) dominates $\QL\al$. This means that $$ [\fE^\al_i]\circ[(\fE^\al_i)^T]-[(\fE^{\al-i}_i)^T]\circ[\fE^{\al-i}_i]= (\bp_{13})_*(c_1(\CL_{|\fE^\al_{\lbr i\rbr}}))= \Delta_*\bp_*c_1(\CL_{|\fE^\al_{\lbr i\rbr}})+\text{terms of smaller dimension}. $$ Since over the generic point of $\QL\al$ the fiber of $\fE^\al_{\lbr i\rbr}$ is one-dimensional we have $$ [\fE^\al_i]\circ[(\fE^\al_i)^T]-[(\fE^{\al-i}_i)^T]\circ[\fE^{\al-i}_i]= b^i_\al[\Delta]+\text{terms of smaller dimension,} $$ where $b^i_\al$ is the degree of the restriction of $\CL$ to the generic fiber of $\fE^\al_{\lbr i\rbr}$ over $\QL\al$. Thus to prove the Proposition it suffices to compute the integers $b^i_\al$. The calculation of $b^i_\al$ will be given in the next section. $\Box$ \subsubsection{Definition of $\CL$ and $b^i_\al$} \label{L} Since we are ultimately interested in the degree of $\CL$ restricted to the generic fiber which belongs to the smooth locus of $\fE^\al_{\lbr i\rbr}$, below we will restrict ourselves to this smooth locus. We have the following diagram $$ \begin{CD} \fE^\al_{\lbr i\rbr} @>\id\times \bp>> \fE^\al_i\times\QL\al \\ @V \bp\times\id VV @V\id\times\id VV \\ \QL\al\times(\fE^\al_i)^T @>\id\times\id^T>> \QL\al\times\QL{\al+i} \times\QL\al \end{CD} $$ where $\id$ denotes either identity map or natural embedding and $T$ denotes the transposition. According to the Intersection Theory, $\CL$ is the cokernel of the natural map of normal bundles $$ \CN_{\fE^\al_{\lbr i\rbr}/(\QL\al\times(\fE^\al_i)^T)}\lra (\id\times \bp)^*\CN_{(\fE^\al_i\times\QL\al)/(\QL\al\times\QL{\al+i} \times\QL\al)} \lra\CL\lra0. $$ The first term is evidently isomorphic to $\bp^*\CT_{\QL\al}$ and the second term is isomorphic to $\CN_{\fE^\al_{\lbr i\rbr}/(\QL\al\times\QL{\al+i})}$. Consider the following commutative diagram $$ \begin{CD} @. \CT_{\fE^\al_{\lbr i\rbr}} @= \CT_{\fE^\al_{\lbr i\rbr}}\\ @. @VVV @VVV \\ \bp^*\CT_{\QL\al} @>>>(\CT_{\QL\al\times\QL{\al+i}})_{|\fE^\al_{\lbr i\rbr}} @>>> \barq^*\CT_{\QL{\al+i}} \\ @| @VVV \\ \bp^*\CT_{\QL\al} @>>> \CN_{\fE^\al_{\lbr i\rbr}/(\QL\al\times\QL{\al+i})} \end{CD} $$ with exact middle row and exact middle column. This diagram implies that we have the following exact sequence: $$ \CT_{\fE^\al_{\lbr i\rbr}} \lra \barq^*\CT_{\QL{\al+i}} \lra \CL \lra 0. $$ Let $D^\al_i=\barq(\fE^\al_{\lbr i\rbr})$. This is a divisor in $\QL{\al+i}$. Let $\vphi\in\CQ_\al$, hence $\bp^{-1}(\vphi)\cong C$. Since the restriction of $\barq$ to the open subset $\bp^{-1}(\CQ_\al)$ is an embedding we have $$ \CL_{|\bp^{-1}(\vphi)}\cong \barq^*\CN_{D^\al_i/\QL{\al+i}} $$ Thus we have proved the following. \begin{lem}{balpha} $b^i_\al=\deg \barq^*\CN_{D^\al_i/\QL{\al+i}}$ where $\vphi\in\CQ_\al$, and $\barq:C=\bp^{-1}(\vphi)\to\QL{\al+i}$ is the map, induced by the projection $\barq:\fE^\al_i\to\QL{\al+i}$. \end{lem} The calculation of these integers will be given in the next section (see the Proposition~\ref{degN}) with the help of Kontsevich's compactification $\CQ^K_\al$ of the space $\CQ_\al$. \subsection{} \label{sl} Let us define an operator $h_i:\ H^\bullet(\CQ^L_\alpha,\BQ)\lra H^\bullet(\CQ^L_\alpha,\BQ)$ as a scalar multiplication by $\langle i',\alpha+2\rho\rangle$. Combining the Propositions ~\ref{triv}, ~\ref{Serre}, ~\ref{h} together with the Theorem ~\ref{character} we arrive at the following Theorem: {\bf Theorem.} The operators $e_i,f_i,h_i(i\in I)$ extend to the action of Lie algebra $\frak{sl}_n$ on $\bigoplus\limits_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$. The character of this $\frak{sl}_n$-module is equal to $\dfrac{|W|e^{2\rho}}{\prod_{\theta\in R^+}(1-e^\theta)^2}$. $\Box$ \subsubsection{Remark} We would like to emphasize that the Lie algebra $\frak{sl}_n$ acting by correspondences in the cohomology of Laumon's spaces should be viewed as the {\em Langlands dual} of the original group $G=SL_n$. In effect, the character of the $\frak{sl}_n$-module is naturally a formal cocharacter of $G$ (cf. ~\ref{character}). \section{Kontsevich's compactification} \subsection{} Recall the notion of a {\em stable map} (see \cite{kt} for details). \begin{defn}{st} A stable map is a datum $(\CC;x_1,\dots,x_m;f)$ consisting of a connected compact reduced curve $\CC$ with $m\ge 0$ pairwise distinct marked non-singular points and at most ordinary double singular points, and a map $f:\CC @>>> \CX$ having no non-trivial first order infinitesimal automorphisms, identical on $\CX$ and $x_1,\dots,x_m$ (stability). \end{defn} \subsubsection{} \begin{defn}{qk} $\QK\al=\overline\CM_{0,0}(C\times\CB,1\oplus\al)$ is the moduli space of stable maps to $C\times\CB$ of curves of arithmetic genus 0 with no marked points such that $f_*([\CC])=1\oplus\al\in H_2(C,\ZZZ)\oplus H_2(\CB,\ZZZ)=\ZZZ\oplus\ZZZ[I]$. \end{defn} Given a stable map $(\CC;f)\in\QK\al$ we denote by $f':\CC\to C$ and $f'':\CC\to\CB$ the induced maps; we denote by $\CC_0$ the irreducible component of $\CC$ such that $f'_*[\CC_0]=[C]$; we denote by $\CC_1,\dots,\CC_m$ the connected components of $\CC\setminus\CC_0$; we denote by $f_r$ (resp.\ $f'_r$, $f''_r$) the restriction of $f$ (resp.\ $f'$, $f''$) to $\CC_r$. Finally, let $\beta$ be the degree of $f''_0$ and $\gamma_r$ be the degree of $f_r''$ ($r=1,\dots,m$). The space of maps $\CQ_\al$ is naturally embedded into $\QK\al$ (to every map $\vphi:C\to\CB$ we associate its graph $\Gamma_\vphi\subset C\times\CB$) and can be identified with the space of all stable maps $(\CC,f)$ such that $\CC$ is irreducible. Hence we can consider $\QK\al$ as a compactification of $\CQ_\al$. \subsection{The birational correspondence between $\QK\al$ and $\QL\al$} Let $0\subset\CF_1\subset\dots\subset\CF_{n-1}\subset V\otimes\CO_\CB$ be the universal flag of vector bundles over the flag variety $\CB$. A stable map $(\CC,f)\in\QK\al$ gives rise to the following flag of vector bundles over $C$: $$ 0\subset f'_*{f''}^*\CF_1\subset\dots\subset f'_*{f''}^*\CF_{n-1}\subset V\otimes f'_*{f''}^*\CO_\CB=V\otimes\CO_C. $$ Note that the above inclusions are no longer inclusions of vector subbundles, but only of coherent sheaves. Let us denote this flag by $\Phi(\CC,f)$. Let $U^K_\al\subset\QK\al$ denote the open subspace consisting of all stable maps $(\CC,f)$ such that $|\ga_1+\dots+\ga_m|<2$. \begin{lem}{Phi} Given $(\CC,f)\in U^K_\al$ we have $\Phi(\CC,f)\in\QL\al$; hence $\Phi$ is a map $\Phi:U^K_\al\to\QL\al$. \end{lem} {\em Proof.} Easy. $\Box$ \begin{rem}{remPhi} In general, the degree of the quasiflag $\Phi(\CC,f)$ is smaller than $\alpha$, hence the map $\Phi$ is not defined on the whole $\QK\al$. \end{rem} \subsubsection{} Given a quasiflag $E_\bullet\in\QL\al$ define its graph $\Gamma_E$ in $C\times\CB$ as follows: \begin{multline*} \Gamma_E=\{ (x,F_\bullet)\ |\ \text{the composition } E_i \lra V\otimes\CO_C \lra V/F_i\otimes\CO_C\\ \text{ vanishes at the point }x\text{ for all $i=1,\dots,n-1.$}\} \end{multline*} Let $U^L_\al\subset\QL\al$ denote the open subspace consisting of all quasiflags $E_\bullet$ such that $|\deff E|<2$, where $\deff E$ stands for the defect of $E_\bullet$. \begin{lem}{Gamma} The graph of $E_\bullet\in U^L_\al$ is a stable curve. Its natural embedding in $C\times\CB$ is a stable map of degree $1\oplus\alpha$; hence the correspondence $E_\bullet\mapsto\Gamma_E$ defines a map $\Gamma: U^L_\al\to\QK\al$. \end{lem} {\em Proof.} Evident. $\Box$ \subsubsection{} \begin{prop}{iso} The maps $\Phi$ and $\Gamma$ define the mutually inverse isomorphisms $\Phi:U^K_\al\rightleftarrows U^L_\al:\Gamma$, which are identical on the subspace $U^K_\al\supset\CQ_\al\subset U^L_\al$. \end{prop} {\em Proof.} Clear. $\Box$ \subsubsection{} Let $\tilde D^\al_i$ be closed subspace in $U^K_{\al+i}$ consisting of all stable maps $(\CC,f)$ with $\beta=\al$, $\ga_1=i$. \begin{lem}{D} The maps $\Phi$ and $\Gamma$ induce the isomorphisms $\Phi:D^\al_i\cap U^L_{\al+i}\rightleftarrows\tilde D^\al_i:\Gamma$. Hence $\barq^*\CN_{D^\al_i/\QL{\al+i}}\cong \tilde \barq^*\CN_{\tilde D^\al_i/\QK{\al+i}}$ where $\vphi\in\CQ_\al,\ \barq:C=\bp^{-1}(\vphi)\to\QL{\al+i}$ is the map induced by the projection $\barq:\fE^\al_i\to\QL{\al+i}$, and $\tilde \barq$ is the composition $C\stackrel{\barq}{\lra}\QL{\al+i}\stackrel{\Gamma}{\dasharrow}\QK{\al+i}$. \end{lem} {\em Proof.} Easy. $\Box$ \subsection{}\label{CBi} Let $\CP_i\subset G$ be the minimal parabolic subgroup of type $i$ containing the Borel subgroup $B$. Let $\CB_i=G/\CP_i$ be the corresponding homogenuous space, and let $\sigma_i$ stand for the natural projection $\sigma_i:\CB=G/B\to\CB_i$. The map $\sigma_i:\CB\to\CB_i$ is a $\PP^1$-fibration, and its relative tangent bundle $\CT_{\CB/\CB_i}$ is canonically isomorphic to the line bundle $\CL_{i'}$ corresponding to the simple root $i'$, considered as a character of $B$. \subsubsection{Lemma} Let $\vphi\in\CQ_\al$ be a map from $C$ to $\CB$ of degree $\al$. The map $\tilde \barq:C\to\QK{\al+i}$ can be described as follows: $$ x\in C\mapsto \Gamma_\vphi\cup\{x\}\times\sigma_i^{-1}(\sigma_i(\vphi(x))) \subset C\times\CB. $$ Here the RHS is a stable curve and $\tilde \barq(x)$ is this curve with its natural embedding into $C\times\CB$. {\em Proof.} Apply the definitions of $\fE^\al_i$ and of the map $\Gamma$. $\Box$ \subsection{Proposition}\label{degN} Given $\vphi\in\CQ_\al$ we have $$ \deg \barq^*\CN_{D^\al_i/\QL{\al+i}}=\langle i',\al+2\rho\rangle. $$ {\em Proof.} Recall that the fiber of the normal bundle to the divisor $\tilde D^\al_i$ in the space of stable maps $\QK{\al+i}$ at the point $(\CC=\CC_0\cup\CC_1,f)$ is canonically isomorphic to $(\CT_{\CC_0})_P\otimes(\CT_{\CC_1})_P$ where $P$ is the point of intersection $P=\CC_0\cap\CC_1$. {\sloppy The canonical isomorphisms $(\CT_{\CC_0})_P\cong{f'_0}^*(\CT_C)_{f'(P)}$, $(\CT_{\CC_1})_P\cong{f''_1}^*(\CT_{\CB/\CB_i})_{f''(P)}$ imply that $\tilde \barq^*\CN_{\tilde D^\al_i/\QK{\al+i}}= \CT_C\otimes\vphi^*\CL_{i'}$. Hence its degree equals $$ \deg(\CT_C)+\deg\vphi^*\CL_{i'}=2+\langle i',\al\rangle= \langle i',2\rho\rangle+\langle i',\al\rangle=\langle i',\al+2\rho\rangle. $$ Now the Proposition follows from the Lemma~\ref{D}. $\Box$ } \section{More on correspondences} In this section we will follow the notations of ~\cite{lu} in the particular case of Dynkin graph of type $A_{n-1}$. \subsection{} We fix an orientation $\Omega=(1\lra2\lra\ldots\lra n-1)$ on the Dynkin graph with the set of vertices $I$. We fix a sequence $\mbox{\bf{i}}\in\CX$ adapted to $\Omega$; let $\theta^1,\ldots,\theta^\nu$ be the corresponding total order on $R^+$ (see {\em loc. cit.}, \S4); here $\nu=\frac{n(n-1)}{2}$. G.Lusztig has introduced in {\em loc. cit.} a bijection $\bc\mapsto\Vc$ between $\BN^\nu$ and the set of isomorphism classes of representations of the quiver $\Omega$. For $\bc\in\BN^\nu$ we will denote by $d(\bc)\in\BN[I]$ the dimension of $\Vc$. In the notations of ~\ref{not} we have $\bc\in\fK(\gamma)\Leftrightarrow \gamma=d(\bc)=|\bc|$. \subsection{} For $\gamma\in\BN[I]$ we introduce the closed subvariety $\fE^\al_\gamma\subset\QL\al\times\QL{\al+\gamma}$ as follows. {\bf Definition.} $\fE^\al_\gamma=\{(E_\bullet,E'_\bullet)$ such that $E'_\bullet\subset E_\bullet\}$. There are natural maps $$ \bp:\fE^\al_\gamma\to\QL\al,\qquad \barq:\fE^\al_\gamma\to\QL{\al+\gamma} \quad\text{and}\quad \br:\fE^\al_\gamma\to C^\gamma, $$ where $C^\gamma$ is the configuration space (see e.g. \cite{k}, 1.3). The first and second maps are induced by the projections of $\QL\al\times\QL{\al+\gamma}$ onto the first and second factors and the third one is defined as $$ \br((E_\bullet,E'_\bullet))=\sum_{x\in C}\dim_x(E'_\bullet/E_\bullet)\cdot x. $$ We will be interested in irreducible components of $\fE^\al_\gamma$ of the middle dimension $\dim\CB+2|\al|+|\gamma|$. \subsubsection{} Recall that for $\gamma\in\BN[I]$ we denote by $\Gamma(\gamma)$ the set of all partitions of $\gamma$, i.e. multisubsets (subsets with multiplicities) $\Gamma=\lbr \gamma_1,\ldots,\gamma_m\rbr $ of $\BN[I]$ with $\sum_{r=1}^m\gamma_r=\gamma,\ \gamma_r>0$ (see e.g. ~\cite{k}, 1.3). The diagonal stratification $C^\gamma=\sqcup_{\Gamma\in\Gamma(\gamma)} C^\gamma_\Gamma$ was introduced e.g. in {\em loc. cit.} Recall that for $\Gamma=\lbr \gamma_1,\ldots,\gamma_m\rbr $ we have $\dim C^\gamma_\Gamma=m$. Given a partition $\Gamma=\lbr \gamma_1,\dots,\gamma_m\rbr \in\Gamma(\gamma)$ consider the following closed subspace of $\fE^\al_\gamma$: $$ \fE^\al_\Gamma=\overline{(\bp\times \br)^{-1}(\CQ_\al\times C^\gamma_\Gamma)}. $$ \begin{lem}{EG} If $\gamma_r\in R^+$ for any $r=1,\ldots,m$ (i.e. $\Gamma\in\fK(\gamma)$ is a Kostant partition of $\gamma$), then $\fE^\al_\Gamma$ is irreducible of dimension $\dim\CB+2|\al|+|\gamma|$. \end{lem} {\em Proof.} Since $\CQ_\al\times C^\gamma_\Gamma$ is irreducible of dimension $\dim\CB+2|\al|+m$ ($m$ is the number of elements in the partition), we need to check that the fibers of the projection $\pr:(\bp\times \br)^{-1}(\CQ_\al\times C^\gamma_\Gamma)\to \CQ_\al\times C^\gamma_\Gamma$ are irreducible of dimension $|\gamma|-m=\sum_{r=1}^m(|\gamma_r|-1)$. Let $\ga_r=i_{q_r}+i_{q_r+1}+\dots+i_{p_r}$. Then the fiber over a point $(E_\bullet,\sum\ga_rx_r)\in\CQ_\al\times C^\ga_\Gamma$ is naturally isomorphic to $$ \prod_{r=1}^m\PP(\Hom(E_{p_r}/E_{q_r-1},\CO_{x_r})). $$ Since $E_{p_r}/E_{q_r-1}$ is locally free of rank $p_r-q_r+1=|\gamma_r|$ the Lemma follows. $\Box$ \subsubsection{Remark}\label{generic} Denote by $\EGo\subset(\pr)^{-1}(\CQ_\al\times C^\ga_\Gamma)$ the open subspace of $E^\al_\Gamma$ with the fiber over the point $(E_\bullet,\sum\ga_rx_r)\in\CQ_\al\times C^\ga_\Gamma$ equal to $$ \prod_{r=1}^m\big(\PP(\Hom(E_{p_r}/E_{q_r-1},\CO_{x_r}))\setminus \PP(\Hom(E_{p_r}/E_{q_r},\CO_{x_r}))\big). $$ Then for a point $(E_\bullet,E_\bullet')\in\EGo$ we have the following decomposition of the quotient $E_\bullet/E_\bullet'$ (in the category $\RT$) $$ E_\bullet/E_\bullet'=\bigoplus_{r=1}^m\bM_{\gamma_r}\otimes\CO_{x_r}, $$ where $\bM_{\gamma_r}$ is the indecomposable representation of $\Omega$ (in the category of representations in vector spaces), corresponding to coroot $\gamma_r\in R^+$. \subsection{Proposition}\label{dims} Dimension of any irreducible component of $\fE^\al_\ga$ is not greater than $\dim\CB+2|\al|+|\ga|$. Any component of this dimension coincides with $\fE^\alpha_\Gamma$ for some $\Gamma\in\fK(\gamma)$ (see~\ref{EG}). {\em Proof.} Consider the stratification of $\QL\al\times C^\ga$ via the defect of $E_\bullet$ at the support of $\sum\ga_rx_r\in C^\ga$, namely $$ \QL\al\times C^\ga=\bigsqcup \begin{Sb} \Gamma\in\Gamma(\ga)\\ |\ka'_1|+\dots+|\ka'_m|=\ga'\le\al \end{Sb} \CZ^\Gamma_{\ka'_1,\dots,\ka'_m}. $$ Here $\CZ^\Gamma_{\ka'_1,\dots,\ka'_m}\subset\QL\al\times C^\ga$ is the subspace of all pairs $(E_\bullet,\sum\ga_rx_r)$ such that $\lbr \ga_1,\dots,\ga_m\rbr =\Gamma$ and the defect of $E_\bullet$ at the point $x_r$ is of type $\ka'_r$ ($r=1,\dots,m$). We evidently have \begin{multline*} \dim\CZ^\Gamma_{\ka'_1,\dots,\ka'_m}= \dim\CB+2\left|(\al-\sum|\ka'_r|)\right|+\sum(||\ka'_r||-K(\ka'_r))+m=\\= \dim\CB+2|\al|-\sum||\ka'_r||-\sum K(\ka'_r)+m= \dim\CB+2|\al|+|\ga|+\sum(1-|\ga_r+\ga'_r|-K(\ka'_r)). \end{multline*} \subsubsection{} Given $(E_\bullet,\sum\ga_rx_r)\in\CZ^\Gamma_{\ka'_1,\dots,\ka'_m}$ we define $\CF(E_\bullet,\sum\ga_rx_r)$ as $(\pr)^{-1}((E_\bullet,\sum\ga_rx_r))$. \begin{lem}{product} $\CF(E_\bullet,\sum\ga_rx_r)=\prod\limits_{r=1}^m\CF(E_\bullet,\ga_rx_r)$. \end{lem} {\em Proof.} Absolutely similar to the proof of Proposition 2.1.2 in \cite{k}. $\Box$ \subsubsection{}\label{hnu} Fix $x\in C$ and $\CE_\bullet\in\QL\al$ such that the defect of $\CE_\bullet$ at the point $x$ is of type $\ka'\in\fK(\ga')$. Here we will study the variety $\CF(\CE_\bullet,\ga x)$. To this end we may (and will) replace $C$ by the formal neighbourhood of $x$. Let $\CE_q^p$ be the normalization of $\CE_q$ in $\CE_p$ and let $\ti\CE_q$ be the normalization of $\CE_q$ in $V\otimes\CO_C$. Then we evidently have $\CE_q=\CE_q^q\subset\dots\subset\CE_q^{n-1}\subset\ti\CE_q$. Given $E_\bullet\in\CF(\CE_\bullet,\ga x)$ we define $$ \hnu_{pq}(E_\bullet)= \length\left(\frac{\CE_q^p\cap E_{p+1}}{\CE_q^p\cap E_p}\right) \quad(1\le q\le p\le n-1), $$ $$ \tnu_{pq}(E_\bullet)= \length\left(\frac{\ti\CE_q\cap E_{p+1}}{\ti\CE_q\cap E_p}\right) \quad(1\le q\le p\le n-1), $$ $$ \tka_{pq}(E_\bullet)=\tnu_{pq}-\tnu_{p.q-1}\quad(1\le q\le p\le n-1) $$ (cf. ~\cite{k} (10), (8)). Note that $\tka$ is nothing else then the type of the defect of $E_\bullet$, hence $\tka\in\fK(\ga'+\ga)$. \subsubsection{Lemma}\label{less} For all $1\le q\le p\le n-1$ we have $\hnu_{pq}\le\tnu_{pq}$. {\em Proof.} Since $\CE_q^p\cap E_p=\left(\CE_q^p\cap E_{p+1}\right)\bigcap \left(\ti\CE_q\cap E_p\right)$ the Lemma follows from {\em loc.\ cit.}, 2.2.1. $\Box$ \subsubsection{} Let $\fS_\hnu\subset\CF(\CE_\bullet,\ga x)$ be the subspace of all $E_\bullet$ such that $\hnu(E_\bullet)=\hnu$. \begin{lem}{pseu} $\fS_\hnu$ is a pseudoaffine space of dimension $\sum\limits_{1\le q<p\le n-1}\hnu_{pq}$. \end{lem} {\em Proof.} Same as the proof of {\em loc.\ cit.}, Theorem 2.3.3. $\Box$ \subsubsection{} Since $$ \sum_{1\le q<p\le n-1}\hnu_{pq}\le\sum_{1\le q<p\le n-1}\tnu_{pq}= ||\tka||-K(\tka) $$ and recalling \ref{hnu}, \ref{pseu} we get the following estimate: $$ \dim\CF(\CE_\bullet,\ga x)\le\max_{\tka\in\fK(\ga'+\ga)}(||\tka||-K(\tka))= |\ga+\ga'|-\min_{\tka\in\fK(\ga'+\ga)}K(\tka). $$ Comparing it with the formula for dimension of $\CZ^\Gamma_{\ka'_1,\dots,\ka'_m}$ we get $$ \dim(\pr)^{-1}(\CZ^\Gamma_{\ka'_1,\dots,\ka'_m})\le \dim\CB+2|\al|+|\ga|+ \sum_{r=1}^m\left(1-K(\ka'_r)- \min_{\tka_r\in\fK(\ga'_r+\ga_r)}K(\tka_r)\right). $$ Since $\ga_r\ne0$ we have $K(\tka_r)\ge1$, therefore the last term is allways non-positive and the first part of the Proposition follows. Furthermore, the last term is equal to zero only if for all $r$ we have $K(\ka'_r)=0$ (hence $\ga'_r=0$) and $\ga_r\in R^+$ for any $r$. But this is exactly the case of Lemma~\ref{EG}. $\Box$ \subsection{} Recall that the set $\fK(\gamma)\subset\Gamma(\gamma)$ of Kostant partitions consists of all partitions $\lbr \ga_1,\dots,\ga_m\rbr $ of $\ga$ such that $\gamma_r\in R^+$ for any $r=1,\ldots,m$. We have an obvious bijection between $\fK(\gamma)$ and the set of all $\bc\in\BN^\nu$ with $c_1\theta_1+\dots+c_\nu\theta_\nu=\ga$. For $\bc\in\BN^\nu$ we introduce the closed subvariety $\fE_\bc^\alpha\subset \CQ^L_\alpha\times\CQ^L_{\alpha+d(\bc)}$ as follows. {\bf Definition.} $\fE_\bc^\alpha:=\fE^\al_\Gamma$ (see ~\ref{EG}), where $\Gamma$ is the partition corresponding to $\bc$. \subsection{} \label{pbw} Consider the Poincar\'e dual of the fundamental class in the middle cohomology $[\fE_\bc^\alpha]\in H^\bullet(\CQ^L_\alpha\times\CQ^L_{\alpha+d(\bc)},\BQ)$. Viewed as a correspondence, it defines two operators: $$e_\bc:\ H^\bullet(\CQ^L_\alpha,\BQ)\rightleftharpoons H^\bullet(\CQ^L_{\alpha+d(\bc)},\BQ)\ :f_\bc$$ adjoint to each other with respect to Poincar\'e duality. \subsection{Proposition} \label{divided} (cf. ~\cite{lu} 5.4.c) For $\bc=(c_1,\ldots,c_\nu)$ we have a) $e_\bc=e_{\theta_1}^{(c_1)}\cdots e_{\theta_\nu}^{(c_\nu)}$; b) $f_\bc=f_{\theta_1}^{(c_1)}\cdots f_{\theta_\nu}^{(c_\nu)}$ where $f^{(c)}$ stands for the divided power $\dfrac{f^c}{c!}$. {\em Proof.} Let $c_1\theta_1+\dots+c_\nu\theta_\nu=\ga$. Let $$ \eac=\bp_{12}^{-1}(\fE^\al_{\theta_1})\cap\dots\cap \bp_{N-1,N}^{-1}(\fE^{\al+\ga-\theta_\nu}_{\theta_\nu})\subset \QL\al\times\QL{\al+\theta_1}\times\dots \times\QL{\al+\ga-\theta_\nu}\times\QL{\al+\ga}, $$ where $N=c_1+\dots+c_\nu+1$ and $\bp_{ab}$ stands for the projection onto the product of $a$-th and $b$-th factors. Obviously $$ \eac=\{(E_\bullet\supset E_\bullet'\supset\dots\supset E_\bullet^{(N)})\}\subset \QL\al\times\QL{\al+\theta_1}\times\dots \times\QL{\al+\ga-\theta_\nu}\times\QL{\al+\ga}, $$ hence $\bp_{1N}(\eac)\subset \fE^\al_\ga$. This implies that \begin{multline*} \underbrace{[\fE^\al_{\theta_1}]\circ\dots\circ [\fE^{\al+(c_1-1)\theta_1}_{\theta_1}]}_{c_1} \circ\dots\circ\underbrace{[\fE^{\al+\ga-c_\nu\theta_nu}_{\theta_\nu}] \circ\dots\circ[\fE^{\al+\ga-\theta_\nu}_{\theta_\nu}]}_{c_\nu}= {\bp_{1N}}_*\left[\eac\right]=\\= \sum_\Gamma a_\Gamma[\fE^\al_\Gamma]+\text{terms of smaller dimension}, \end{multline*} where $\Gamma=\lbr \ga_1,\dots,\ga_m\rbr \in\fK(\ga)$ is a Kostant partition of $\ga$ and $a_\Gamma$ is the number of points in the generic fiber of $\eac$ over $\fE^\al_\Gamma$. The space $\eac$ is naturally isomorphic to the space of triples $(E_\bullet,E_\bullet^{(N)},F)$, where $E_\bullet\in\QL\al$, $E_\bullet^{(N)}\in\QL{\al+\ga}$ such that $E_\bullet^{(N)}\subset E_\bullet$ and $F$ is a filtration in the quotient representation $0=F_N\subset\dots\subset F_1= E_\bullet/E_\bullet^{(N)}$ of the type $(\underbrace{\theta_1,\dots,\theta_1}_{c_1},\dots, \underbrace{\theta_\nu,\dots,\theta_\nu}_{c_\nu})$ (``of type $\bc$'' for short). Hence $a_\Gamma$ is the number of filtrations of the type $\bc$ in the quotient $E_\bullet/E_\bullet'$ for generic $(E_\bullet,E_\bullet')\in \fE^\al_\Gamma$. Let $(E_\bullet,E_\bullet')\in\EGo$ (see Remark~\ref{generic}). Then $E_\bullet/E_\bullet'=\oplus\bM_{\ga_r}\otimes\CO_{x_r}$. Assume that $F$ is a filtration of type $\bc$ on $E_\bullet/E_\bullet'$ and let $\ti F_k=F_{c_1+\dots+c_k+1}$. Then $0=\ti F_\nu\subset\dots\subset\ti F_1\subset\ti F_0=E_\bullet/E_\bullet'$ (resp. $0=H^0(\ti F_\nu)\subset\dots\subset H^0(\ti F_1)\subset H^0(\ti F_0)= H^0(E_\bullet/E_\bullet')=\bM:=\oplus\bM_{\ga_r}$) is a filtration of type $(c_1\theta_1,\dots,c_\nu\theta_\nu)$ in the category $\RT$ (resp. in the category of representations of $\Omega$ in vector spaces). But existence of the latter filtration means that the isomorphism class of $\bM$ is $\bc$ (see \cite{lu}), hence integer $a_\Gamma$ is not zero only for $\Gamma$, corresponding to $\bc$. Therefore, in order to prove the Proposition it remains to check that for $\Gamma$ corresponding to $\bc$ we have $a_\Gamma=c_1!\cdot\dots\cdot c_\nu!$. By the Proposition 4.9 of {\em loc.\ cit.} the filtration $H^0(\ti F)$ in $\bM$ is unique, hence the filtration $\ti F$ on $E_\bullet/E_\bullet'$ is unique, hence we need to compute the number of refinements of the filtration $\ti F$ to a filtration $F$ of type $\bc$. But $\ti F_{k-1}/\ti F_k= \bigoplus\limits_{t=1}^{c_k}\bM_{\theta_k}\otimes\CO_{x_{r^k_t}}$, where $\{r^k_1,\dots,r^k_{c_k}\}=\{r\in\{1,\dots,m\}\ |\ \ga_r=\theta_k\}$. Hence the set of these refinements is isomorphic to the set of all orderings of subsets $\{r^k_1,\dots,r^k_{c_k}\}$ and the Proposition follows. $\Box$ \section{Conjectures} \subsection{} The formal character appearing in the Theorem ~\ref{sl} is not new in the representation theory of $\frak{sl}_n$. Let us recall its previous appearences. First of all, let $\fn\subset\frak{sl}_n$ be the nilpotent subalgebra generated by the simple generators $e_1,\ldots,e_{n-1}$. Let $\CN\subset\frak{sl}_n$ be the nilpotent cone. The Lie algebra $\frak{sl}_n$ acts on the cohomology $H^\nu_\fn(\CN,\CO)$ of $\CN$ with supports in $\fn$. The character of this module is exactly $\dfrac{|W|e^{2\rho}}{\prod_{\theta\in R^+}(1-e^\theta)^2}$ (see e.g. ~\cite{ar}, Appendix A). \subsection{} Let $\zeta\in\BC$ be a root of unity of degree $p>2n$ and let $\fu$ be a small quantum group defined by G.Lusztig for the root datum $(X,Y,\ldots)$ of type $G$ and $\zeta$ (see e.g. ~\cite{luu}). S.Arkhipov has introduced in ~\cite{ar} the graded vector space of semiinfinite cohomology $H^{\frac{\infty}{2}+\bullet}_\fu$ along with the action of $\frak{sl}_n$ on it. The (graded) character of this module is given by $$P_G(t):=\frac{e^{2\rho}t^{-\frac{1}{2}\dim\CB}\sum_{w\in W}t^{\ell(w)}} {\prod_{\theta\in R^+}(1-te^\theta)(1-t^{-1}e^\theta)}$$ (~\cite{ar}, Theorem 4.5). \subsection{} B.Feigin has conjectured (1993, unpublished) that the $\frak{sl}_n$-modules $H^{\frac{\infty}{2}+\bullet}_\fu$ and $H^\nu_\fn(\CN,\CO)$ are isomorphic. S.Arkhipov has checked this conjecture at the level of characters in ~\cite{ar}. \subsection{} \label{conjecture} We propose the following conjecture. {\bf Conjecture.} $\frak{sl}_n$-module $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$ is isomorphic to $H^\nu_\fn(\CN,\CO)$. \subsubsection{Remark} \label{freeness} Note that $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^L_\alpha,\BQ)$ is evidently selfdual (by Poincar\'e duality) while $H^\nu_\fn(\CN,\CO)$ can be easily seen to be $\fn$-free, i.e. to posess a Verma filtration. Thus the Conjecture admits a funny corollary that both of modules in question are {\em tilting} (see ~\cite{ap}, chapter 1). The conjecture would follow in turn from this funny corollary since a tilting module is defined up to isomorphism by its character. \subsubsection{Remark} (V.Ostrik) Here is a sketch of a nondegenerate $\frak{sl}_n$-invariant contragredient self-pairing on $H^\nu_\fn(\CN,\CO)$. We will prove that $H^\nu_\fn(\CN,\CO)$ is self-dual with respect to the standard contragredient duality in the BGG category $\CO$. We prefer to use another duality, without Chevalley involution on $\frak{sl}_n$, exchanging highest and lowest weight modules. To this end it suffices to construct a nondegenerate $\frak{sl}_n$-invariant pairing between $H^\nu_\fn(\CN,\CO)$ and $H^\nu_{\fn_-}(\CN,\CO)$ where $\fn_-$ denotes the nilpotent subalgebra of $\frak{sl}_n$ generated by the simple generators $f_1,\ldots,f_{n-1}$. The desired pairing is the composition of the cup-product $\cup:\ H^\nu_\fn(\CN,\CO)\times H^\nu_{\fn_-}(\CN,\CO)\lra H^{2\nu}_0(\CN,\CO)$ and the trace (residue) morphism $Res_0:\ H^{2\nu}_0(\CN,\CO)\lra H^0(0,\CO)=\BC$. Note that the dualizing complex of $\CN$ is isomorphic to its structure sheaf $\CO$, whence the trace morphism above.

9702

alg-geom/9702016

en

https://arxiv.org/abs/alg-geom/9702016

alg-geom/9702016

Miles Reid

Miles Reid (Nagoya and Warwick)

McKay correspondence

V2 cured 2 misguided crossreferences and some errors of punctuation. This v3 gives references sent in by listeners to this network, and centres the graphics, a triumph of mind over computer manual!

Proc of Kinosaki conference (Nov 1996), and Warwick preprint 1997

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G <--> cohomology of resolution (2) conjugacy classes of G <--> homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.

alg-geom

\section{Introduction}\label{sec:intro} \begin{conjecture}[since 1992]\label{conj:1992} $G\subset\SL(n,\C)$ is a finite subgroup. Assume that the quotient $X=\C^n/G$ has a crepant resolution $f\colon Y\to X$ (this just means that $K_Y=0$, so that $Y$ is a ``noncompact Calabi--Yau manifold''). Then there exist ``natural'' bijections \begin{align} \{\text{\em irreducible representations of $G$}\} &\to \text{\em basis of $H^*(Y,\Z)$} \\ \{\text{\em conjugacy classes of $G$}\} &\to \text{\em basis of $H_*(Y,\Z)$} \end{align} As a slogan ``$\text{representation theory of $G$}=\text{homology theory of $Y$}$''. Moreover, these bijections satisfy ``certain compatibilities'' \begin{equation} \left. \begin{array}{rr} \text{\em character table of $G$}\\ \text{\em McKay quiver} \end{array} \right\} \bij \left\{ \begin{array}{ll} \text{\em duality}\\ \text{\em cup product} \end{array} \right. \notag \end{equation} \end{conjecture} As you can see, the statement is still too vague because I don't say what ``natural'' means, and what ``compatibilities'' to expect. At present it seems most useful to think of this statement as pointer towards the truth, rather than the truth itself (compare Main Conjecture~\ref{conj:K}). The conjecture is known for $n=2$ (Kleinian quotient singularities, Du Val singularities). McKay's original treatment was mainly combinatorics \cite{McK}. The other important proof is that of Gonzales-Sprinberg and Verdier \cite{GSp-V}, who introduced the GSp--V or tautological sheaves, also my main hope for the correspondence (1). For $n=3$ a weak version of the correspondence (2) is proved in \cite{IR}. We hope that a modification of this idea will work in general for (2); for details, see \S\ref{sec:IR}. \paragraph{Contents} This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how McKay correspondence relates to mirror symmetry. The main aim is to give numerical examples of how the McKay correspondences (1) and (2) must work, and to restate Conjecture~\ref{conj:1992} as a {\em tautology}, like the cohomology or K-theory of projective space $\proj^n$ (see Main Conjecture~\ref{conj:K}). Introduction to Nakamura's results on the Hilbert scheme of $G$-clusters. \paragraph{Credits} Very recent results of I. Nakamura on $G$-Hilb, who sent me a first draft of \cite{N3} and many helpful explanations. Joint work with Y.~Ito. Moral support and invaluable suggestions of S. Mukai. Support Sep--Nov 1996 by the British Council--Japanese Ministry of Education exchange scheme, and from Dec 1996 by Nagoya Univ., Graduate School of Polymathematics. \subsection{History} Around 1986 Vafa and others defined the {\em stringy Euler number} for a finite group $G$ acting on a manifold $M$: \begin{equation} \begin{aligned} e_{\text{string}}(M,G)&=\text{crazy formula (you'd better forget it!)}\\ &=\sum_{H\subset G} e(X_H) \times \card\{\text{conjugacy classes in $H$}\}. \end{aligned} \tag{$*$} \end{equation} Here $X=M/G$, and $X$ is stratified by stabiliser subgroups: for a subgroup $H\subset G$, set \begin{align*} M_H &=\{Q\in M | \Stab_G Q=H\},\\ X_H &=\pi(M_H)\\ &=\{P\in X | \text{for $Q\in\pi\1(P)$, $\Stab_G Q$ is conjugate to $H$}\}. \end{align*} The sum in ($*$) runs over all subgroups $H$, and $e(X_H)$ is the ordinary Euler number. The mathematical formulation ($*$) is due to Hirzebruch--H\"ofer \cite{HH} and Roan \cite{Roan}. If $M=\C^n$ and $G\subset\GL(n,\C)$ only fixes the origin, then the closure of each $X_H$ is contractible, so that only the origin $\{0\}=X_G$ contributes to the sum in ($*$), and \begin{equation} e_{\text{string}}(\C^n,G)=\card\{\text{conjugacy classes in $G$}\}. \notag \end{equation} At the same time, Vafa and others conjectured the following: \begin{conjecture}[``physicists' Euler number conjecture'']\label{conj:vaf} In appropriate circumstances, \begin{equation} e_{\text{\em string}}(M,G)=\text{\em Euler number of minimal resolution of $M/G$.} \notag \end{equation} \end{conjecture} The context is string theory of $M=\text{CY 3-fold}$, and the $G$ action on $M$ is Gorenstein, meaning that it fixes a global basis $s\in\om_M=\Oh(K_M)\iso\Oh_M$ (dualising sheaf $\om_M=\Om^3_M$). In particular, for any point $Q\in M$, the stabiliser subgroup is in $\SL(T_QM)$. At that time, the physicists possibly didn't know that there was a generation of algebraic geometers working on minimal models of 3-folds, and possibly naively assumed that in their cases, there exists a unique minimal resolution $Y\to X=M/G$, so that $e_{\text{string}}(M,G)=e(Y)$. A number of smart-alec \hbox{3-folders} raised various instinctive objections, that a minimal model may not exist, is usually not unique etc. However, it turns out that the physicists were actually nearer the mark. One of the points of these lectures is that, in flat contradiction to the official 3-fold ideology of the last 15 years, in many cases of interest, there {\em is} a distinguished crepant resolution, namely Nakamura's $G$-Hilbert scheme. My guess of the McKay correspondences follow on naturally from Vafa's conjecture, by the following logic. If $M=\C^n$, then one sees easily that for any reasonable resolution of singularities $Y\to X=\C^n/G$, the cohomology is spanned by algebraic cycles, so that \begin{equation} e(Y)=\sum H^{p,p}=\card\{\text{algebraic cycles of $Y$}\}. \notag \end{equation} It seems unlikely that we could prove the numerical concidence \begin{equation} e(Y)=\card\{\text{conjugacy classes of $G$}\} \notag \end{equation} without setting up some kind of bijection between the two sets. \cite{IR} does so for $G\subset\SL(3,\C)$. \subsection{Relation with mirror symmetry, applications} Consider: \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item the search for mirror pairs; \item Vafa's conjecture; \item conjectural McKay correspondence; \item speculative theory of equivariant mirror symmetry ($G$-mirror symmetry). \end{enumerate} Historically, (a) led to (b), (b) led to (c), and logically (c) implies (b). I have long speculated that (c) is connected to (d), and maybe even that it would eventually be proved in terms of (d). The point is that up to now, the known proofs of the McKay correspondence (even in 2 dimensions) rely on the explicit classification of the groups, plus quite detailed calculations, and it would be very interesting to get more direct relations. I suggest below in \S\ref{sec:taut} that the McKay correspondence can be derived in tautological terms. If this works, it will have applications to (d). Some trivial aspects of this are already contained in Candelas and others' example of the mirror of the quintic 3-fold \cite{C}, where you could take intermediate quotients in the $(\Z/5)^3$ Galois tower. My suggestion is that $G$-mirror symmetry should relate pairs of CYs with group actions, and include the character theory of finite groups as the zero dimensional case. I guess you're supposed to add an analog of ``complexified K\"ahler parameters'' to the conjugacy classes, and ``complex moduli'' to the irreducible representations. Another application (more speculative, this one) might be to wake up a few algebraists. \subsection{Conjecture~\ref{conj:1992}, (1) or (2), which is better?} I initially proposed Conjecture~\ref{conj:1992} in 1992 in terms of irreducible representations, an analog of the formulations of McKay and of \cite{GSp-V}. I was persuaded by social pressure around the Trento conference and by my coauthor Yukari Ito to switch to (2); its advantage is that the two sides are naturally graded, and we could prove a theorem \cite{IR}. Batyrev and Kontsevich and others have argued more recently that (2) is the more fundamental statement. However, the version of correspondence (2) in cohomology stated in \cite{IR} gives a $\Q$-basis only: the crepant divisors do not base $H^2(Y,\Z)$ in general: fractional combinations of them turn up as $c_1(\sL)$ for line bundles on $Y$ that are eigensheaves of the group action, that is, GSp-V sheaves for 1-dimensional representations of $G$. These lectures return to (1), passing via K-theory; in this context, the natural structure on the right hand side of (1) is not the {\em grading} of $H^*$, but the {\em filtration} of $K_0Y$. In fact, my thoughts on (2) in general are, to be honest, in a bit of a mess at present (see \S\ref{sec:IR} and \S\ref{sec:Kexs} below). \section{First examples} These preliminary examples illustrate the following points: \begin{enumerate} \item To construct a resolution of a quotient singularity $\C^n/G$, and a very ample linear system on it, rather than {\em invariant rational functions}, it is more efficient to use {\em ratios of covariants}, that is, ratios of functions in the same character space. This leads directly to the Hilbert scheme as a natural candidate for a resolution. \item Functions in a given character space $\rho$ define a tautological sheaf $\sF_\rho$ on the resolution $Y\to X$, and in simple examples, you easily cook up combinations of Chern classes of the $\sF_\rho$ to base the cohomology of $Y$. \end{enumerate} I fix the following notation: $G\subset\GL(n,\C)$ is a finite subgroup, $X=\C^n/G$ the quotient, and $Y\to X$ a crepant resolution (if it exists). For a given cyclic (or Abelian) group, I choose eigencoordinates $x_1,\dots,x_n$ or $x,y,z,\dots$ on $\C^n$. I write $\frac1{r}(a_1,\dots,a_n)$ for the cyclic group $\Z/r$ action given by $x_i\mapsto \ep^{a_i}x_i$, where $\ep=\exp(2\pi i/r)=\text{fixed primitive $r$th root of 1}$. Other notation, for example the lattice $L=\Z^n+\Z\cdot\frac1{r}(a_1,\dots,a_n)$ of weights, and the junior simplex $\De_{\text{junior}}\subset L_\R$ are as in \cite{IR}. \begin{example}\label{ex:A_n} The \begin{figure}[thb] \centering\mbox{\epsfbox{An.ps}} \caption{$E_0$ and $E_r$ are the image of the $x$ and $y$ axes} \label{fig:A_n} \end{figure} quotient singularity $\frac1{r}(1,-1)$. The notation means the cyclic group $G=\Z/r$ acting on $\C^2$ by $(x,y)\mapsto(\ep x,\ep^{r-1}y)$. Everyone knows the invariant monomials $u=x^r,v=xy,w=y^r$, the quotient map \begin{equation} \C^2\to X=\C^2/G=\text{Du Val singularity $A_{r-1}$}:(uw=v^r)\subset\C^3, \notag \end{equation} and the successive blowups that give the resolution $Y\to X$ and its chain of $-2$-curves $E_1,\dots,E_{r-1}$ (Figure~\ref{fig:A_n}). However, the new point to note is that each $E_i$ is naturally parametrised by the ratio $x^i:y^{r-i}$. More precisely, an affine piece $Y_i\subset Y$ of the resolution is given by $\C^2$ with parameters $\la,\mu$, and the equations \begin{equation} x^i=\la y^{r-i},\quad y^{r-i+1}=\mu x^{i-1}\quad\text{and}\quad xy=\la\mu \label{eq:Ac} \end{equation} define the $G$-invariant rational map $\C^2\mathrel{{\relbar\kern-.2pt\rightarrow}} Y_i$ (quotient map and resolution at one go). The ratio $x^i:y^{r-i}$ defines a linear system $|L(i)|$ on $Y$, with intersection numbers \begin{equation} L(i)\cdot E_j=\de_{ij}\quad\text{(Kronecker $\de$).} \notag \end{equation} Thus, writing $\sL(i)$ for the corresponding sheaf or line bundle gives a natural one-to-one correspondence from nontrivial characters of $G$ to line bundles on $Y$ whose first Chern classes $c_1(\sL(i))\in H^2(Y,\Z)$ give the dual basis to the natural basis $[E_i]$ of $H_2(Y,\Z)$. \end{example} \begin{example}\label{ex:max} One way of generalising Example~\ref{ex:A_n} to dimension 3. Let \begin{equation} G=\Span{\frac1{r}(1,-1,0),\frac1{r}(0,1,-1),\frac1{r}(-1,0,1,)} =(\Z/r)^2\subset\SL(3,\C) \notag \end{equation} be the maximal diagonal Abelian group of exponent $r$. Then the first quadrant of $L_\R$ has an obvious triangulation \begin{figure}[ht] \centering\mbox{\epsfbox{fig2.ps}} \caption{Triangulation of $\protect\De_{\protect\text{junior}}$ in Example~\ref{ex:max}} \label{fig:max1} \end{figure} by regular simplicial cones that are basic for $L$ and have vertexes in the junior simplex $\De_{\text{junior}}$. By toric geometry and the standard discrepancy calculation \cite{YPG}, this triangulation defines a crepant resolution $Y\to X=\C^3/G$. From now on, restrict for simplicity to the case $r=5$ (featured on the mirror of the quintic \cite{C}), whose triangulation is illustrated in Figure~\ref{fig:max1}. $X=\C^3/G$ has lines of Du Val singularities $A_4=\frac15(1,-1)$ along the 3 coordinate axes, the fixed locuses of the 3 generating subgroups $\frac15(1,-1,0)$ etc., of $G$. \begin{figure}[ht] \centering\mbox{\epsfbox{fig3.ps}} \caption{The resolution corresponding to the triangulation of Figure~\ref{fig:max1}} \label{fig:max2} \end{figure} As illustrated in Figure~\ref{fig:max2}, the resolution $Y$ has 3 chains of 4 ruled surfaces over the coordinate axes of $X$, and 6 del Pezzo surfaces of degree 6 (``regular hexagons'') over the origin. Every exceptional curve stratum in the resolution is a $(-1,-1)$ curve. Functions on the quotient $X=\C^3/G$ are given by $G$-invariant polynomials, $k[X]=\C[x,y,z]^G$. To get more functions on $Y$ (and a projective embedding of $Y$), consider the following ratios of monomials in the same eigenspace of the $G$ action: \begin{equation} x^i:(yz)^{5-i} \quad\text{for $i=1,\dots,4$, and permutations of $x,y,z$.} \label{eq:ratio} \end{equation} Each ratio (\ref{eq:ratio}) defines a free linear system on $Y$, and all together, they define a relative embedding of $Y$ into a product of many copies of $\proj^1$. \begin{figure}[hbt] \centering\mbox{\epsfbox{fig4.ps}} \caption{Two affine pieces near the hexagon at (3,1,1)} \label{fig:max3} \end{figure} For example, as shown in Figure~\ref{fig:max3}, the toric stratum at $(2,2,1)$ is a del Pezzo surface of degree 6 embedded by the 3 ratios $x^3:y^2z^2$, $y^3:x^2z^2$ and $z^4:xy$ (having product the trivial ratio $1:1$). Figure~\ref{fig:max3} shows two affine pieces of $Y$, of which the right-hand one is $\C^3$ with coordinates $\la,\mu,\nu$ related to $x,y,z$ by a set of equations generalising (\ref{eq:Ac}): \begin{equation} \begin{matrix} x^3&=&\la y^2z^2\\ y^4&=&\mu xz\\ z^4&=&\nu xy \end{matrix} \qquad \begin{matrix} y^3z^3&=&\mu \nu x^2\\ x^2z^2&=&\la \nu y^3\\ x^2y^2&=&\la \mu z^3 \end{matrix} \quad\text{and}\quad xyz=\la\mu\nu. \label{eq:maxr} \end{equation} Denote the linear system $|x^i:(yz)^{5-i}|$ by $|L(x^i)|$, and similarly for permutations of $x,y,z$. The sum of all the $|L(x^i)|$ is very ample on $Y$, but their first Chern classes do not span $H^2(Y,\Z)$. To see this, recall the del Pezzo surface $S_6$ of degree 6, the 3 point blowup of $\proj^2$ familiar from Cremona and Max Noether's elementary quadratic transformation. It has 3 maps to $\proj^1$ and 2 maps to $\proj^2$; write $e_1,e_2,e_3$ for the divisor classes of the maps to $\proj^1$, and $f_1,f_2$ for the maps to $\proj^2$. Then clearly, \begin{equation} \begin{gathered} e_1,e_2,e_3,f_1,f_2\quad \text{span} \quad H^2(S_6,\Z),\\ \text{with the single relation}\quad e_1+e_2+e_3=f_1+f_2. \end{gathered} \label{eq:reln} \end{equation} For $S_6$ one of the hexagons of Figure~\ref{fig:max2}, the 3 maps to $\proj^1$ are provided by certain of the linear systems $|L(x^i)|$. The two maps to $\proj^2$ are provided by other character spaces: for example, for the $(2,2,1)$ hexagon of Figure~\ref{fig:max3}, $f_1$ and $f_2$ are given by the linear systems $|L(x^3y)|$ and $|L(xy^3)|$ corresponding respectively to the ratios $$ \left(x^2z^4:x^3y:y^3z^2\right) \quad\text{and}\quad \left(xy^3:y^2z^4:x^3z^2\right)= \left({1\over x^2z^4}:{1\over x^3y}:{1\over y^3z^2} \right). $$ For each surface $S_6$, the generators $e_1,e_2,e_3,f_1,f_2$ correspond to certain characters of $G$. For example, if I choose the 3 generators $\frac15(1,-1,0)$, $\frac15(0,1,-1)$ and $\frac15(-1,0,1)$ of $G$, the characters of $x,y,z$ are $$ \renewcommand{\arraystretch}{1.2} \begin{array}{rrr} x & y & z \\ \hline 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{array} \quad\text{and my $(2,2,1)$ hexagon has} \quad \begin{array}{ccc|cc} e_1 & e_2 & e_3 & f_1 & f_2\\ x^3 & y^3 & z^4 & x^3y & xy^3\\ \hline 3 & 2 & 0 & 2 & 3\\ 0 & 3 & 1 & 1 & 3\\ 2 & 0 & 4 & 2 & 4 \end{array} $$ Moreover, you see easily that the relations (\ref{eq:reln}) actually hold in $H^2(Y,\Z)$, not just in $H^2(S_6,\Z)$. Represent each character of $G$ by a monomial $x^m$ (such as $x^i$ or $x^3y$); this corresponds to a free linear system $|L(x^m)|$ on $Y$, in much the same way as the $L(x^i:(yz)^{r-i})$ or $L(x^2z^4:x^3y:y^3z^2)$ just described. Now the McKay correspondence (1) of Conjecture~\ref{conj:1992} is the following recipe: \begin{equation} \text{monomial $x^m$} \mapsto \text{line bundle $\sL(x^m)$} \mapsto c_1(\sL(x^m))\in H^2(Y,\Z). \notag \end{equation} These elements generate $H^2(Y,\Z)$, with one relation of the form (\ref{eq:reln}) for every regular hexagon $S_6$ of the picture. Moreover, each relation (\ref{eq:reln}) gives an element \begin{equation} c_2(L(e_1)\oplus L(e_2)\oplus L(e_3)) -c_2(L(f_1)\oplus L(f_2))\in H^4(Y,\Z), \label{eq:reln2} \end{equation} which is the dual element to $[S_6]\in H_4(Y,\Z)$. Indeed, \begin{align*} c_2(L(e_1)\oplus L(e_2)\oplus L(e_3))\cdot S_6&=e_1e_2+e_1e_3+e_2e_3=3,\\ \text{and}\quad c_2(L(f_1)\oplus L(f_2))\cdot S_6&=f_1f_2=2. \end{align*} I draw the McKay correspondence resulting from this cookery in Figure~\ref{fig:max4}: each edge $E\iso\proj^1$ is labelled by the linear system $L(x^m)$ with $L(x^m)\cdot E=1$, and each hexagon $S_6$ by 2 characters corresponding to the two extra generators $f_1,f_2$ of $H^2(S_6,\Z)$ with the relation which gives the dual element of $H^4(Y,\Z)$. \begin{figure}[ht] \centering\mbox{\epsfbox{fig5.ps}} \caption{McKay correspondence for Example~\ref{ex:max}} \label{fig:max4} \end{figure} \end{example} One of the morals of this example is that we get a basis of cohomology in terms of Chern classes of virtual sums of tautological bundles; this suggests using the tautological bundles to base the K-theory of $Y$, and passing from K-theory to cohomology by Chern classes or Chern characters. In fact, the combinations used in (\ref{eq:reln2}) were fixed up to have zero first Chern class, exactly what you must do if you want the second Chern character to come out an integral class. \section{Ito--Reid, and the direct correspondence (2)}\label{sec:IR} A group $G\subset\SL(n,\C)$ has a natural filtration by {\em age}. Namely, any element $g\in G$ can be put in diagonal form by choosing $x_1,\dots,x_n$ to be eigencoordinates of $g$. We write $g=\frac1{r}(a_1,\dots,a_n)$ to mean that \begin{equation} g: (x_1,x_2,\dots,x_n) \mapsto (\ep^{a_1}x_1,\ep^{a_2}x_2,\dots,\ep^{a_n}x_n), \notag \end{equation} where $\ep=\exp(2\pi i/r)=\text{fixed primitive $r$th root of 1}$, and $a_i\in[0,1,\dots,n-1]$. Toric geometry tells us to consider the lattice \begin{equation} L=\Z^n+\Z \frac1{r}(a_1,\dots,a_n) \notag \end{equation} (more generally for $A\subset G$ an Abelian group, we would add in lots of vectors $\frac1{r}(a_1,\dots,a_n)$ for each $g\in A$). This consists of weightings on the $x_i$, so that the invariant monomials have integral weights. Then for any element $b=\frac1{r}(b_1,\dots,b_n)\in L$ with all $b_i\ge0$ (that is, $b$ in the positive quadrant), define \begin{equation} \age(b)=\frac1{r}\sum b_i. \notag \end{equation} In particular, for $g=\frac1{r}(a_1,\dots,a_n)$ in the unit cube, \begin{equation} \age(g)=\frac1{r} \sum a_i; \notag \end{equation} this is obviously an integer (because $g\in\SL(n,\C))$ in the range $[0,n-1)$, and this defines the age filtration. Now any primitive vector $b=\frac1{r}(b_1,\dots,b_n)\in L$ and in the positive quadrant defines a {\em monomial valuation} $v_b$ on the function field $k(X)$ of $X$. Furthermore, the standard discrepancy calculation (see \cite{YPG}) says that \begin{equation} \disc(v_b)=\age(b) - 1. \notag \end{equation} \paragraph{Reminder:} The {\em discrepancy} $\disc v_b$ means that if I make a blowup $W_b\to X$ so that $v_b$ is the valuation at a prime divisor $F_b\subset W_b$, then $K_{W_b}=K_X+\disc(v_b) F_b$. Note also that {\em junior} means $\age=1$, and {\em crepant} means $\text{discrepancy}=0$. Any other questions? The valuation $b$ defines a locus $E_b=\centre(v_b)\subset Y$. Consider only weightings $b$ such that $v_b$ is the valuation of $E_b\subset Y$; this means that if I blow up $Y$ along $E_b$, and $F_b$ is the exceptional divisor, then $v_b$ is the valuation associated with the prime divisor $F_b\subset\widetilde Y$. Since $Y$ is crepant, the adjunction formula for a blowup gives \begin{equation} \disc(v_b)=\codim E_b - 1,\quad\text{that is,}\quad\codim E_b=\age(b). \notag \end{equation} In \cite{IR}, we uses this idea to give a bijection \begin{equation} \{\text{junior conjugacy classes of $G$}\} \to \{\text{crepant valuations of $X$}\} \notag \end{equation} which gave us a basis of $H^2(Y,\Q)$, and we dealt with $H^4(Y,\Q)$ by Poincar\'e duality. Thus \cite{IR} only used the valuation theoretic construction \begin{equation} b \mapsto v_b \mapsto E_b \notag \end{equation} for $b$ in the junior simplex $\De_{\text{junior}}$. However, the same idea obviously extends to give a correspondence from certain ``good'' elements $b$ to a set of locuses in $Y$ which generate $H_*(Y,\Z)$. Thus the idea for the direct correspondence (2) is \begin{equation} \begin{aligned} G\ni g & \mapsto \text{collection of suitable $b$}\\ & \mapsto \text{collection of locuses $E_b\subset Y$}. \end{aligned} \notag \end{equation} The first step is by a mysterious cookery, which I only indicate by the labelling in the two examples of \S\ref{sec:Kexs} below (it should be possible to extract a good conjectural statement from this data). \section{Tautological sheaves and the main conjecture}\label{sec:taut} These lectures are mainly concerned with providing experimental data for a suitably rephrased Conjecture~\ref{conj:1992}, (1). In this section, I speculate on a framework to explain what is going on, that might eventually lead to a proof. The following is the main idea of \cite{GSp-V}. Given $G\subset\SL(n,\C)$, we choose once and for all a complete set of irreducible representations $\rho\colon G\to\GL(V_\rho)$. I use $\pi_*$ to view sheaves on $\C^n$ such as the structure sheaf $\Oh_{\C^n}$ as sheaves on the quotient $\pi\colon\C^n\to X$. Since $X$ is affine, these are really simply modules over $k[X]=k[\C^n]^G$, so I usually omit $\pi_*$. Note that $k(\C^n)$ is a Galois extension of $k(X)$, so that, by the cyclic element theorem of Galois theory, it is the regular representation of $G$, that is, $k(\C^n)=k(X)[G]$; thus $\pi_*\Oh_{\C^n}$ is generically isomorphic to the regular representation $\Oh_X[G]$. For each $\rho$, set \begin{equation} \sF'_\rho:=\Hom(V_\rho,\Oh_{\C^n})^G \notag \end{equation} Then $\sF'_\rho\tensor V_\rho\subset\Oh_{\C^n}$ is the character subsheaf corresponding to $V_\rho$; by the usual decomposition of the regular representation, $\sF'_\rho$ is a sheaf of $\Oh_X$-modules of rank $\deg\rho$. And there is a canonical decomposition \begin{equation} \Oh_{\C^n}=\sum_\rho \sF'_\rho\tensor V_\rho\quad\text{as $\Oh_X[G]$ modules.} \notag \end{equation} Now let $f\colon Y\to X$ be a given resolution. Each $\sF'_\rho$ has a {\em birational transform} $\sF_\rho$ on $Y$. This means that $\sF_\rho$ is the torsion free sheaf of $\Oh_Y$ modules generated by $\sF'_\rho$, or if you prefer, $\sF_\rho=f^*\sF'_\rho/(\text{torsion})$. The sheaves $\sF_\rho$ are the {\em GSp-V sheaves}, or the {\em tautological sheaves} of $Y$. Note that by definition, the $\sF_\rho$ are generated by their $H^0$. \begin{conjecture}[Main conjecture]\label{conj:K} Under appropriate circumstances, the\linebreak tautological sheaves $\sF_\rho$ form a $\Z$-basis of the Grothendieck group $K_0(\Coh Y)$, and a certain cookery with their Chern classes leads to a $\Z$-basis of $H^*(Y,\Z)$. A slightly stronger conjecture is that the $\sF_\rho$ form a $\Z$-basis of the derived category $D^b(\Coh Y)$. \end{conjecture} \begin{remark} ``Appropriate circumstances'' in the conjecture include all cases when $G\subset\SL(n,\C)$ and $Y=\GHilb$ is a crepant resolution. In this case, these tautological sheaves $\sF_\rho$ have lots of good properties (see \S\ref{sec:hilb}). But flops should not make too much difference to the statement -- one expects a flopped variety $Y'$ to have more or less the same homology and cohomology as $Y$, at least additively. \end{remark} \begin{example}\label{ex:Ver} $\frac1n(1,\dots,1)$ (with $n$ factors). The quotient $X$ is the cone on the $n$th Veronese embedding of $\proj^{n-1}$, and the resolution $Y$ is the anticanonical bundle of $\proj^{n-1}$, containing the exceptional divisor $\proj^{n-1}$ with normal bundle $\Oh(-n)=\om_{\proj^n}$. The tautological sheaves are \begin{equation} \Oh, \Oh(1),\dots,\Oh(n-1). \notag \end{equation} That is, these are sheaves on $Y$ restricting down to the first $n$ multiples of $\Oh(1)$ on $\proj^{n-1}$. It is well known that these sheaves form a $\Z$-basis of the Grothendieck group $K_0(\proj^{n-1})$. It is a standard (not quite trivial) bit of cookery with Chern classes and Chern characters to go from this to a $\Z$-basis of $H^*(\proj^{n-1},\Z)$. \end{example} \begin{remark} Recall the original (1977) {\em Beilinson diagonal trick}: the diagonal $\De_{\proj^{n-1}}\subset\proj^{n-1}\times\proj^{n-1}$ is defined by the section \begin{equation} s_\De=\sum x_i'{\partial\ \over\partial x_i}\in p_1^*\Oh_{\proj^{n-1}}(1)\otimes p_2^*T_{\proj^{n-1}}(-1). \notag \end{equation} Therefore, it follows (tautologically) that the derived category $D^b(\Coh\proj^{n-1})$ (hence also the K theory $K_0$) has two ``dual'' bases \begin{equation} \Oh, \Om^1(1),\dots,\Om^{n-1}(n-1) \quad\text{vs.}\quad \Oh, \Oh(-1),\dots,\Oh(-(n-1)). \notag \end{equation} \end{remark} \subsection*{Lame attempt to prove Conjecture~\ref{conj:K}} \paragraph{Step~I} The resolution $Y\to X$ is the quotient $A/H$ of an open set $A\subset\C^N$ by a connected algebraic group $H$. In other words, by adding extra variables in a suitable way, we can arrange to make the finite quotient $X=\C^n/G$ equal to the quotient $\C^N/H$ of a bigger space by the action of a connected group $H$ (the quotient singularities arise from jumps in the stabiliser group of the $H$-action); moreover, we can arrange to obtain the resolution $Y\to X$ by first deleting a set of ``unstable'' points of $\C^N$ and then taking the new quotient $A/H$. For example, the Veronese cone singularity of Example~\ref{ex:Ver} is $\C^{n+1}$ divided by \begin{equation} \C^*\ni\la\colon(x_1,\dots,x_n;z)\mapsto(\la x_1,\dots,\la x_n;\la^{-n}z). \notag \end{equation} (Obvious if you think about the ring of invariants). The finite group $\Z/n$ is the stabiliser group of a point of the $z$-axis. The blowup is the quotient $A/\C^*$, where $A=\C^{n+1}\setminus\text{$z$-axis}$. (Because at every point of $A$, at least one of the $x_i\ne0$, so the invariant ratios $x_j/x_i$ are defined locally as functions on the quotient.) \paragraph{Step~II} Most optimistic form: the Beilinson diagonal trick may apply to a quotient of the form obtained in Step~I. That is, the diagonal $\De_Y\subset Y\times Y$ has ideal sheaf $\sI_{\De_Y}$ resolved by an exact sequence in which all the other sheaves are of the form $\sF_i\boxtimes\sG_i=p_1^*\sF_i\tensor p_2^*\sG_i$, where the $\sF_i$ and $\sG_i$ are combinations of the tautological bundles. It's easy enough to get an expression for the tangent sheaf of $Y$, in terms of an Euler sequence arising by pushdown and taking invariants from the exact sequence of vector bundles over $A$ \begin{equation} \Lie(H)\to T_A\to f^*(T_Y)\to 0, \label{eq:Eu} \end{equation} where $\im\Lie(H)$ is the foliation by $H$-orbits. Maybe one can define a filtration of this sequence corresponding to characters, and write the equations of $\De_Y$ in terms of successive sections of twists of the graded pieces. For example, the resolution $Y$ in Example~\ref{ex:Ver} is an affine bundle over $\proj^{n-1}$, and the diagonal in $Y$ is defined by first taking the pullback of the diagonal of $\proj^{n-1}$ (defined by the section $\sum x_i'\partial/\partial x_i\in \Oh_{\proj^{n-1}}(1)\otimes T_{\proj^{n-1}}(-1)$, the classic case of the Beilinson trick), then taking the relative diagonal of the line bundle $\Oh(-n)$ over $\proj^{n-1}$. \paragraph{Step~III} The sheaves $\sF_i$ or $\sG_i$ appearing in a Beilinson resolution form two sets of generators of the derived category $D^b(\Coh Y)$. Indeed, for a sheaf on $Y$, taking $p_1^*$, tensoring with the diagonal $\Oh_{\De_Y}$, then taking $p_{2*}$ is the identity operation. However, a Beilinson resolution means that $\Oh_{\De_Y}$ is equal in the appropriate derived category to a complex of sheaves of the form $\sF_i\boxtimes\sG_i$. (This is a tautology, like saying that if $V$ is a vector space, and $f_i\in V$, $g_i\in V^*$ elements such that $\id_V=\sum f_ig_i$, then $f_i$ and $g_i$ span $V$ and $V^*$.) It should be possible to go from this to a basis of $D^b(\Coh Y)$ by an argument involving Serre duality and the assumption $K_Y=0$. In this context, it is relevant to note that the Beilinson trick leads to line bundles in the range $K<\sF_i\le\Oh$ as one of the dual bases (for $\proj^{n-1}$, I believe also in all the other known cases). \section{Generalities on $\protect\GHilb$}\label{sec:hilb} The next sections follow Nakamura's ideas and results, to the effect that the Hilbert scheme of $G$-orbits often provides a preferred resolution of quotient singularities (see \cite{N1}--\cite{N3}, \cite{IN1}--\cite{IN3}, compare also \cite{N}, Theorem~4.1 and \cite{GK}); the results here are mostly due to Nakamura. I write $M=\C^n$, and let $G\subset\GL(n,\C)$ be a finite subgroup. \begin{definition} $\GHilb $ is the fine moduli space of $G$-clusters $Z\subset M$. Here a $G$-{\em cluster} means a subscheme $Z$ with defining ideal $\sI_Z\subset\Oh_M$ and structure sheaf $\Oh_Z=\Oh_M/\sI_Z$, having the properties: \begin{enumerate} \item $Z$ is a {\em cluster} (that is, a 0-dimensional subscheme). (Request to 90\% of the audience: please suggest a reasonable translation of cluster into Chinese characters (how about {\em tendan}, cf.\ {\em seidan} = constellation, as in the Pleiades cluster?) \item $Z$ is $G$-invariant. \item $\deg Z=N=|G|$. \item $\Oh_Z\iso k[G]$ (the regular representation of $G$). For example, $Z$ could be a general orbit of $G$ consisting of $N$ distinct points. \end{enumerate} \end{definition} \begin{remark} \begin{enumerate} \item A quotient set $M/G$ is traditionally called an {\em orbit space}, and that's exactly what $\GHilb M$ is -- the space of clusters of $M$ which are scheme theoretic orbits of $G$. \item There is a canonical morphism $\GHilb M\to M/G$, part of the general nonsense of Hilbert and Chow schemes: $\GHilb $ parametrises $Z$ by considering the ideal $\sI_Z\subset\Oh_M$ as a point of the Grassmannian, whereas the corresponding point of $M/G$ is constructed from the set of hyperplanes (in some embedding $M\into\proj^{\text{large}}$) that intersect $Z$. \item If $\pi\colon M\to M/G$ is the quotient morphism, and $P\in M/G$ a ramification point, the scheme theoretic fibre $\pi^*P$ is always much too fat; over such a point, a point of $\GHilb M$ adds the data of a subscheme $Z$ of the right length. \item I hope we don't need to know anything at all about $\Hilb^N M$ (all clusters of degree $N=|G|$), which is pathological if $N,n\ge3$. Morally, $\GHilb$ is a moduli space of points of $X=M/G$, and the right way to think about it should be as a {\em birational change of GIT quotient} of $M/G$. \end{enumerate} \end{remark} \begin{conjecture}[Nakamura]\label{conj:N} \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $Hilb^G M$ is irreducible. \item For $G\subset\SL(3,\C)$, $Y=\GHilb \C^3\to X=\C^3/G$ is a crepant resolution of singularities. (This is mostly proved, see \cite{N3} and below.) \item For $G\subset\SL(n,\C)$, if a crepant resolution of\/ $\C^n/G$ exists, then $\GHilb\C^n$ is a crepant resolution. \item If $N$ is normal in $G$ and $T=G/N$ then $\Hilb^T\Hilb^N=\GHilb $. \end{enumerate} \end{conjecture} \begin{remark} For $n\ge4$, a crepant resolution $Y\to X$ usually does not exist, but the cases when it does seem to be rather important. As Mukai remarks, a famous theorem of Chevalley, Shephard and Todd says that for $G\subset\GL(n,\C)$, the quotient $\C^n/G$ is nonsingular if and only if $G$ is generated by quasi\-reflections. Since we want to view $\GHilb \C^n$ as a different way of constructing the quotient, the question of characterising $G$ for which $\GHilb\C^n$ is nonsingular (or crepant over $\C^n/G$) is a natural generalisation. We know that the answer is yes for groups $G\subset\SL(2,\C)$, probably also $\SL(3,\C)$, so by analogy with Shephard--Todd, I conjecture that it is also yes for groups generated by subgroups in $G\subset\SL(2,\C)$ or $\SL(3,\C)$. For cyclic coprime groups $\frac1{r}(a,b,c,d)$, based on not much evidence, I guess there is a crepant resolution iff there are $\frac13(r-1)$ junior elements, that is, exactly one third of the internal points of $\Box$ lie on the junior simplex (see \cite{IR}); this is very rare -- by volume, you expect approx 4 middle-aged elements for each junior one (as in most math departments). An easy example to play with is $\frac1{r}(1,1,1,-3)$, which obviously has a crepant resolution \begin{align*} \iff\enspace &\text{the simplex $\Span{([\frac r3],[\frac r3],[\frac r3], r-3[\frac r3]), (1000),(0100),(0010)}$ is basic}\\ \iff\enspace &r\equiv1\mod3. \end{align*} For more examples, see also \cite{DHZ}. \end{remark} \begin{proposition}[Properties of $\protect\GHilb $]\label{prop:GHilb} Assume Conjecture~\ref{conj:N}, (1). (In most cases of present interest, one proves that $\GHilb$ is a nonsingular variety by direct calculation; alternatively, if Conjecture~\ref{conj:N}, (1) fails, replace $Hilb^G M$ by the irreducible component birational to $M/G$.) \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item The tautological sheaves $\sF_\rho$ on $Y$ are generated by their $H^0$. \item They are vector bundles. \item Their first Chern classes or determinant line bundles \begin{equation} \sL_\rho=\det \sF_\rho=c_1(\sF_\rho) \notag \end{equation} define free linear systems $|L_\rho|$ according to (1), and are therefore nef. \item Any strictly positive combination $\sum a_\rho L_\rho$ of the $L_\rho$ is ample on Y. \item These properties characterise $\GHilb$ among varieties birational to $X$ (or the irreducible component). \end{enumerate} \end{proposition} \begin{remark} If $G\subset\SL(n,\C)$ and $M=\C^n$, and $Y=\GHilb M$ is nonsingular, the McKay correspondence says in particular that the $L_\rho$ span $\Pic Y=H^2(Y,Z)$ (this much is proved). In the 3-fold case, when $Y$ is a crepant resolution, (3--4) resolve the contradiction with the expectation of 3-folders, because they show how $\GHilb $ is distinguished among all crepant resolutions of $X$. For if we flip $Y$ in some curve $C\subset Y$, then by (4) we know that $LC>0$ for some $L=L_\rho$, and it follows that the flipped curve $C'\subset Y'$ has $L'_\rho C'<0$. Thus (1--3) do not hold on $Y'$. \end{remark} \paragraph{Proof} Write $Y=\GHilb M$. By definition of the Hilbert scheme, there exists a universal cluster $\sZ\subset Y\times M$, whose first projection $p\colon\sZ\to Y$ is finite, with every fibre a $G$-cluster $Z$. Now from the defining properties of clusters $p_*\Oh_Z$ is locally isomorphic to $\Oh_Y[G]$, the regular representation of $G$ over $\Oh_Y$. In particular, it is locally free, and therefore so are its irreducible factors $\sF_\rho\otimes V_\rho$. Since $Z\subset M=\C^n$, the polynomial ring $k[M]$ maps surjectively to every $\Oh_Z$, so that $p_*\Oh_Z$ is generated by its $H^0$. This proves (1--3). For any $G$-cluster $Z\in\GHilb M$, the defining exact sequence \begin{equation} 0\to\sI_Z\to\Oh_{C^n}\to\Oh_Z\to0 \label{eq:defg} \end{equation} splits as a direct sum of exact sequences (I omit $\pi_*$, remember): \begin{equation} 0\to\sI_{Z,\rho}\to\sF'_\rho\tensor V_\rho\to F_{Z,\rho}\tensor V_\rho\to0 \notag \end{equation} Therefore $Z$ is uniquely determined by the set of surjective maps $\sF_\rho\to F_{Z,\rho}$. This proves (4). I now explain (5). The linear systems $|L_\rho|$ are birational in nature, coming from linear systems of Weil divisors $|L_\rho|_X$ on the quotient $X=M/G$, and their birational transforms on any partial resolution $Y'\to X$. Now (5) says there is a unique model $Y$ on which these linear systems are all free and their sum is very ample: namely, for a single linear system, the blowup, and for several, the birational component of the fibre product of the blowups. This also gives a plausibility argument for Conjecture~\ref{conj:N}, (iii): if we believe in the existence of one crepant resolution $Y'$, and we admit the doctrine of flops from Mori theory, we should be able to flop our way from $Y'$ to another model $Y$ on which the $|L_\rho|_Y$ are all free linear systems. (This is not a proof: a priori, if the $L_\rho$ are dependent in $\Pic Y$, a flop that makes one nef might mess up the nefdom of another. However, it seems that the dependences are quite restricted; compare the discussion at the end of Example~\ref{ex:II}.) \QED I go through these properties again in the Abelian case, which is fun in its own right, and useful for the examples in \S\ref{sec:Kexs}. Then an irreducible representation $\rho$ is an element of the dual group \begin{equation} \widehat G=\bigl\{\text{homomorphisms $a\colon G\to r$th roots of 1 in $\C^*$}\bigr\}, \notag \end{equation} where $r$ is the exponent of $G$. I write $\Oh_X(a)$ for the eigensheaf, and $\sL_Y(a)$ for the tautological line bundle on $Y$ (previously $\sF'_\rho$ and $\sF_\rho$ respectively). For any $Z$, the sequence (\ref{eq:defg}) splits as \begin{equation} 0\to\bigoplus m_a \to\bigoplus \Oh_X(a) \to\bigoplus k_a \to 0 \quad\text{(sum over $a\in\widehat G$)}, \notag \end{equation} where $k_a$ is the 1-dimensional representation corresponding to $a$ (because of the assumption $\Oh_Z=k[G]$). Thus a $G$-cluster is exactly the same thing as a set of maximal subsheaves \begin{equation} m_a\subset\Oh_X(a),\quad\text{one for every $a\in\widehat G$,} \notag \end{equation} subject to the condition that $\sum m_a$ is an ideal in $\Oh_{C^n}$, that is, that $m_a\Oh_X(b)\subset\Oh_X(a+b)$ for every $a,b\in\widehat G$. Now it is an easy exercise to see that the Hilbert scheme parametrising maximal subsheaves of $\Oh_X(a)$ is the blowup of $X$ in $\Oh_X(a)$, which I write $\Bl_a X\to X$, and in particular, it is birational. It follows that $\GHilb $ is contained in the product of these blowups: \begin{equation} \GHilb \subset\prod \Bl_a X \tag{$*$} \end{equation} (where the product is the fibre product over $X$ of all the $\Bl_a X$ for $a\in\widehat G$), and is the locus defined in this product by the ideal condition: \begin{equation} m_a\Oh_X(b) \subset \Oh_X(a+b) \quad\text{for every $a,b\in\widehat G$} \tag{$**$} \end{equation} (this obviously defines an ideal of $\Bl_a\times_X\Bl_b$). By contruction of a blowup, each $\Bl_a$ has a tautological sheaf $\Oh_a(1)$, which is relatively ample on $\Bl_a$. The tautological sheaves on $\GHilb $ are simply the restrictions of the $\Oh_a(1)$ to the subvariety ($*$). This proves (1--4) again. \QED \begin{remark} The fibre product in ($*$) is usually reducible, with big components over the origin (the product of the exceptional locuses of the $\Bl_a$). However, it is fairly plausible that the relations ($**$) define an irreducible subvariety. This is the reason for Conjecture~\ref{conj:N}, (1). \end{remark} \section{Examples of Hilbert schemes}\label{sec:Kexs} More experimental data, to support the following conclusions: \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $Y=\GHilb$ can be calculated directly from the definition; for 3-fold Gorenstein quotients, it gives a crepant resolution, distinguished from other models as embedded in projective space by ratios of functions in the same character spaces. \item Conjecture~\ref{conj:1992} can be verified in detail in numerically complicated cases. It amounts to a funny labelling by $a\in\widehat G$ of curves and surfaces on the resolution. \item The relations in $\Pic Y$ between the tautological line bundles, whose $c_2$ give higher dimensional cohomology classes, come from equalities between products of monomial ideals. \end{enumerate} \begin{example}\label{ex:hirz} Examples~\ref{ex:A_n}--\ref{ex:max} are $G$-Hilbert schemes. In fact the equations (\ref{eq:Ac}) and (\ref{eq:maxr}) were written out to define $G$-clusters. Next, it is a pleasant surprise to note that the famous Jung--Hirzebruch continued fraction resolution of the surface cyclic quotient singularity $\frac1{r}(1,q)$ is the $G$-Hilbert scheme $(\Z/r)\text{-}\Hilb\C^2$. To save notation, and to leave the reader a delightful exercise, I only do the example $\frac15(1,2)$, where $5/2=[3,2]=3-1/2$; the invariant monomials and weightings are as in Figure~\ref{fig:hirz}. As usual, $X=\C^2/G$ and $Y\to X$ is the minimal resolution, with two exceptional curves $E_1$ and $E_2$ with $E_1^2=-2$, $E_2^2=-3$. \begin{figure}[ht] $$ \renewcommand{\arraycolsep}{2pt} \begin{array}{lllll} x^5\\[6pt] & x^3y \\[6pt] && xy^2 \\ &&&\kern1cm y^5 \\[10pt] &&\text{(a)} \end{array} \kern2cm \renewcommand{\arraycolsep}{0pt} \begin{array}{lllll} (0,5)\\[18pt] && (1,2) \\[-3pt] &&& \kern4pt(3,1) \\[-3pt] &&&& \kern4pt(5,0) \\[10pt] &&\text{(b)} \end{array} $$ \caption{Newton polygons (a) of invariant monomials and (b) of weights} \label{fig:hirz} \end{figure} In toric geometry, $E_1$ corresponds to $(3,1)$ (as a vertex of the Newton polygon (b) in the lattice of weights, or a ray of the fan defining the resolution $Y$); the parameter along $E_1\iso\proj^1$ is $x:y^3$. Similarly, $E_2$ corresponds to $(1,2)$ and has parameter $x^2:y$. Exactly as in Figure~\ref{fig:A_n} and (\ref{eq:Ac}), a neighbourhood $Y_1$ of the point $E_1\cap E_2$ is $\C^2$ with parameters $\la,\mu$, and the rational map $\C^2\mathrel{{\relbar\kern-.2pt\rightarrow}} Y_1$ is determined by equations analogous to (\ref{eq:Ac}): \begin{equation} x^2=\la y,\quad y^3=\mu x,\quad\text{and}\quad xy^2=\la\mu. \label{eq:52clus} \end{equation} These equations define a $G$-cluster $Z$: for a basis of $\Oh_Z=k[x,y]/((\ref{eq:52clus}))$ is given by $1,y,y^2,x,xy$. Every $G$-cluster is given by these equations, or by one of the following other two types: $x^5=\la',y=\mu'x^2$ or $x=\la''y^3,y^5=\mu''$; the 3 cases correspond to the 3 affine pieces with coordinates $\la,\mu$, etc. covering $Y$. The generic $G$-cluster is $G\cdot(a,b)$ with $a,b\ne0$; all the equations \begin{equation} x^5=a^5,\enspace x^3y=a^3b,\enspace xy^2=ab^2,\enspace y^5=b^5,\enspace bx^2=a^2y,\enspace ay^3=b^3x \notag \end{equation} vanish on $G\cdot(a,b)$, and since $a,b\ne0$, generators of its ideal can be chosen in lots of different ways from among these, including the 3 stated forms. The ratio $x:y^3$ along $E_1$ and $x^2:y$ along $E_2$ define free linear systems $|L(1)|$, $|L(2)|$ on $Y$ corresponding to the two characters $1,2$ of $G=\Z/5$, with \begin{equation} \begin{aligned} &L(1)\cdot E_1=1\\ &L(1)\cdot E_2=0 \end{aligned} \quad\text{and}\quad \begin{aligned} &L(2)\cdot E_1=0\\ &L(2)\cdot E_2=1 \end{aligned} \notag \end{equation} These two give a dual basis of $H^2(Y,\Z)$, a truncated McKay correspondence. \paragraph{Exercise--Problem} The case of general $\frac1{r}(1,q)$ can be done likewise; see for example \cite{R}, p.~220 for the notation, and compare also \cite{IN2}. Problem: I believe that the minimum resolution of the other surface quotient singularities is also a $G$-Hilbert scheme. The best way of proving this may not be to compute $\GHilb$ exhaustively. In the $\SL(2,\C)$ case, Ito and Nakamura get the result $K_Y=0$ automatically, because the moduli space $\GHilb$ carries a symplectic form. \end{example} \subsection*{The toric treatment of $\GHilb$} From now on, I deal mainly with isolated Gorenstein cyclic quotient 3-fold singularities $\frac1{r}(a,b,c)$, where $a,b,c$ are coprime to $r$ and $a+b+c=r$. If $G$ is Abelian diagonal, then $X$ is obviously toric; however, it turns out that so is the $G$-Hilbert scheme. There are two proofs; the better proof is that due to Nakamura, described in \S\ref{sec:nak}. I now give a garbled sketch of the first proof: I claim that the $G$-Hilbert scheme $\GHilb\C^n=Y(\Si)$ is the toric variety given by the fan $\Si$, the ``simultaneous dual Newton polygon'' of the eigensheaves $\Oh_X(a)$, defined thus: \begin{quote} for every character $a\in\widehat G$, write $\Oh_X(a)$ for the eigenspace of $a$, $L(a)$ for the set of monomial minimal generators of $\Oh_X(a)$, and construct the Newton polyhedron $\operatorname{Newton}(L(a))$ in the space of monomials. Then $\Si$ is the fan in the space of weights consisting of the cones $\Span{A_1,\dots,A_k}$ where the $A_i$ are weights having a common minimum in every $L(a)$. This means that the 1-skeleton $\Si^1$ consists of weights $A$ which either support a wall (= $(n-1)$-dimensional face) of $\operatorname{Newton}(L(a))$ for some $a$, or which support positive dimensional faces of a number of $L(a_j)$ whose product is $n-1$ dimensional (in other words, ratios between monomials in the various $L(a_j)$ which are minima for $A$ generate a function field of dimension $n-1$). Then $\Span{A_1,\dots,A_k}$ is a cone of $\Si$ if and only if $\{A_i\}$ is a complete set of weights in $\Si^1$ having a common minimum in every $L(a)$; and $\Span{A_1,\dots,A_k}$ has dimension $d$ if and only if the ratio between these minima span an $(n-d)$ dimensional space. \end{quote} This definition is algorithmic, but quite awkward to use in calculations: you have to list the minimal generators in each character space, and figure out where each weight $A_i$ takes its least values; when $n=3$, you soon note that the key point is the ratios like $x^3y:z^5$ between two monomials on an edge of the Newton boundary. \begin{figure}[th] \centering\mbox{\epsfbox{fig7.ps}} \caption{$\protect\GHilb$ for $\protect\frac1{r}(1,2,-3)$. $B_i$ is joined to $A_{2i-2},A_{2i-1},A_{2i}$} \label{fig:II} \end{figure} \paragraph{Sketch proof} Because $\Oh_Z=k[G]$ for $Z\in\GHilb$, for every character $a$ of $G$, the generators of $L(a)$ map surjectively to the 1-dimensional character space $k_a$, so there is a well defined ratio between the generators of $\sI_Z(a)$. This means that for fixed $Z$ and every $L(a)$, we mark one monomial $s_a=x^{m(Z,a)}\in L(a)$ as the minimum of all the valuations $A_1,\dots,A_k$ spanning a cone, and, using it as a generator, we get the invariant ratios $x^{m'}/s_a$ as regular functions on $\GHilb$ near $Z$. \begin{example}\label{ex:II} Consider $\frac1{r}(1,2,-3)$ where $r=6k+1$. The quotient $X=\C^3/(\Z/r)$ is toric, and the $G$-Hilbert scheme is given by the triangulation of the first quadrant of Figure~\ref{fig:II}. \begin{figure}[ht] \centering\mbox{\epsfbox{fig8.ps}} \caption{$\protect\GHilb$ for $\protect\frac1{13}(1,2,10)$: why join $(8,3,2)$---$(2,4,7)$?} \label{fig:13a} \end{figure} This can be proved by carrying out the above proof explicitly. I omit the laborious details, concentrating on one point: how does the Hilbert scheme construction choose one triangulation in preference to another? For simplicity, consider only $r=13$, so the triangulation simplifies to Figure~\ref{fig:13a}. How do I know to join $(8,3,2)$---$(2,4,7)$ by a cone $\si$, rather than $(7,1,5)$---$(3,6,4)$? By calculating $2\times2$ minors of $\left(\begin{smallmatrix}8&3&2\\2&4&7\end{smallmatrix}\right)$, we see that the parameter on the corresponding line $E_\si\in Y$ should be the ratio $xz^2:y^4$, where $xz^2,y^4\in L(8)$. The Newton polygon of $L(8)$ is \begin{figure}[bht] \centering\mbox{\epsfbox{fig9.ps}} \caption{The McKay correspondence for $\frac1{13}(1,2,10)$} \label{fig:13b} \end{figure} \end{example} $$ \renewcommand{\arraystretch}{1.4} \begin{matrix} x^8&x^6y&x^4y^2&x^2y^3&y^4\\ &&\kern-1cm(2,4,7)\kern-.6cm\\ && xz^2&&y^2z^3\\ &&&\kern-1cm(8,3,2)\kern-.6cm\\ &&&& z^6 \end{matrix} $$ (The figure is not planar: $xz^2$ and $y^4$ are ``lower''.) Here $(2,4,7)$ and $(8,3,2)$ have minima on the two planes as indicated, with common minima on $xz^2$ and $y^4$, so that the linear system $|xz^2:y^4|$ can be free on $L_\si$. But $(7,1,5)$ and $(3,6,4)$ don't have a common minimimum here: $(7,1,5)$ prefers $y^4$ only, and $(3,6,4)$ prefers $xz^2$ only. If I join $(7,1,5)$---$(3,6,4)$, the linear system $|xz^2:y^4|$ would have that line as base locus. The resolution is as in Figure~\ref{fig:13b}. The McKay correspondence marks each exceptional stratum: a line $L$ parametrised by a ratio $x^{m_1}:x^{m_2}$ is marked by the common character space of $x^{m_1},x^{m_2}$. In other words, a linear system such as $xz^2:y^4$ corresponds to a tautological line bundle $\sL(xz^2:y^4)=\sL(8)$ with $c_1(\sL(8))\cdot L=1$. The surfaces are marked by relations between the $c_1(\sL(i))$. In this case, because there are no hexagons, these all arise from surjective maps $\Oh_X(i)\otimes\Oh_X(j)\onto\Oh_X(i+j)$. For example, generators of the character spaces $1,2,3$ are given by monomials (written out as Newton polygons) \begin{equation} \renewcommand{\arraycolsep}{3pt} L(1):\enspace \begin{array}{lll} x & & y^7\\ & y^2z\\ z^4 \end{array}, \quad L(2):\enspace \begin{array}{llll} x^2 & & y\\ xz^4\\ z^8 \end{array} \quad\text{and}\quad L(3):\enspace \begin{array}{llll} x^3 & xy & y^8\\ x^2z^4 & y^3z\\ xz^8\\ z^{12} \end{array} \quad \notag \end{equation} \begin{figure}[t!hb] \centering\mbox{\epsfbox{fig10.ps}} \caption{The McKay correspondence for $\frac1{37}(1,5,31)$} \label{fig:37} \end{figure} Clearly, $L(1)\otimes L(2)\onto L(3)$. (Thus this guy $\sL(3)$ is not active in the resolution; in fact he's completely useless, so deserves to be senior.) This means that on the resolution \begin{equation} c_1(\sL(3)-\sL(1)-\sL(2))=0, \notag \end{equation} and $c_2(\sL(3)-\sL(1)-\sL(2))$ is the dual class to the top left surface in Figure~\ref{fig:13b}. \begin{example}\label{ex:III} The $G$-Hilbert scheme for $\frac1{37}(1,5,31)$ is given by the triangulation in Figure~\ref{fig:37}, which also indicates the labelling by characters of the McKay correspondence. I confine myself to a few comments: on the right-hand side, \begin{align*} (1,5,31)\text{--}(9,8,20) \quad &\text{are joined by the ratio}\quad x^4z:y^7 \\ (8,3,26)\text{--}(23,4,10) \quad &\text{are joined by the ratio}\quad x^2z:y^{14} \end{align*} for reasons similar to those explained in Example~\ref{ex:II}. The resolution has 3 regular hexagons (del Pezzo surfaces $S_6$), coming from the regular triangular pattern on the left-hand side of Figure~\ref{fig:37}. Tilings by regular hexagons appear quite often among the exceptional surfaces of the Hilbert scheme resolution $Y$, as we saw in Figure~\ref{fig:max2}. The reason for this is taken up again at the end of \S\ref{sec:nak}, see Figure~\ref{fig:37b}. The cohomology classes dual to these 3 surfaces are given as in (\ref{eq:reln2}) by taking $c_2$ of the relation $e_1+e_2+e_3-f_1-f_2$, where the $f_1,f_2$ are the characters written in each little hexagonal box of Figure~\ref{fig:37}, and $e_1,e_2,e_3$ are the characters marking the 3 lines through the box. The relation $e_1+e_2+e_3=f_1+f_2$ can also be expressed as equality between two products of monomial ideals. \end{example} \section{Nakamura's proof that $\protect\GHilb$ is a crepant resolution}\label{sec:nak} \begin{theorem}[Nakamura, very recent]\label{th:N} For $G$ a finite diagonal subgroup of $\SL(3,\C)$, $Y=\GHilb\to X=\C^3/G$ is a crepant resolution. \end{theorem} \paragraph{Proof} I start from the {\em McKay quiver} of $G$ with the 3 given characters $a,b,c$, corresponding to the eigencoordinates $x,y,z$, satisfying $a+b+c=0$; to get the full symmetry, draw this as a doubly periodic tesselation of the plane by regular hexagons, labelled by characters in $\widehat G$: \begin{equation} \renewcommand{\arraystretch}{1.3} \renewcommand{\arraycolsep}{7pt} \setcounter{MaxMatrixCols}{15} \begin{matrix} & & & & & & & & & & & & & &\\[-18pt] &&& \kern-1cm \cdots \kern-1cm \\ &&&& \kern-1cm 2b \kern-1cm \\ &&&&& \kern-1cm b \kern-1cm && \kern-1cm a+b \kern-1cm && \kern-1cm 2a+b \kern-1cm \\ \kern-1cm \cdots \kern-1cm && \kern-1cm 2b+2c \kern-1cm && \kern-1cm b+c \kern-1cm && \kern-1cm 0 \kern-1cm && \kern-1cm a \kern-1cm && \kern-1cm 2a \kern-1cm && \kern-1cm 3a \kern-1cm && \kern-1cm \cdots \kern-1cm \\ &&&&& \kern-1cm c \kern-1cm && \kern-1cm a+c \kern-1cm \\ &&&& \kern-1cm \cdots \kern-1cm \end{matrix} \label{fig:honey} \end{equation} corresponding to the monomials \begin{equation} \renewcommand{\arraystretch}{1.4} \renewcommand{\arraycolsep}{8pt} \setcounter{MaxMatrixCols}{13} \begin{matrix} & & & & & & & & & & & &\\[-18pt] & \kern-1cm \cdots \kern-1cm \\ && \kern-1cm y^2 \kern-1cm \\ &&& \kern-1cm y \kern-1cm && \kern-1cm xy \kern-1cm && \kern-1cm x^2y \kern-1cm \\ \kern-1cm y^2z^2 \kern-1cm && \kern-1cm yz \kern-1cm && \kern-1cm 1 \kern-1cm && \kern-1cm x \kern-1cm && \kern-1cm x^2 \kern-1cm && \kern-1cm x^3 \kern-1cm && \kern-1cm \cdots \kern-1cm \\ &&& \kern-1cm z \kern-1cm && \kern-1cm xz \kern-1cm \\ && \kern-1cm \cdots \kern-1cm \end{matrix} \notag \end{equation} For $\frac1{37}(1,5,31)$, we get Figure~\ref{fig:honey37}; it is a quiver, with arrows in the 3 principal directions ``add 1, 5 or 31''. Or you can view it as the lattice of monomials modulo $xyz$, labelled with their characters in the $\frac1{37}(1,5,31)$ action; then the arrows are multiplication by $x,y,z$. \begin{figure}[h!b] \centering\mbox{\epsfbox{fig11.ps}} \caption{The McKay quiver for $\frac1{37}(1,5,31)$.} \label{fig:honey37} \end{figure} The whole of this business is contained one way or another in the hexagonal figure (\ref{fig:honey}), together with its period lattice $\Pi$, and the many different possible ways of choosing nice fundamental domains for the periodicities; that is, we are doing Escher periodic jigsaw patterns on a fixed honeycomb background. First of all, note that the periodicity of (\ref{fig:honey}) is exactly the lattice of invariant Laurent monomials modulo $xyz$. Call this $\Pi$. The proof of Nakamura's theorem follows from the following proposition: \begin{proposition}\label{prop:N} For every $G$-cluster $Z$, the defining equations (that is, the generators of $\sI_Z$) can be written as 7 equations in one of the two following forms: either \begin{equation} \renewcommand{\arraycolsep}{1.5pt} \begin{matrix} x^{a+d+1}&=&\la y^bz^f\\ y^{b+e+1}&=&\mu z^cx^d\\ z^{c+f+1}&=&\nu x^ay^e \end{matrix} \qquad \begin{matrix} y^{b+1} z^{f+1}&=&\mu \nu x^{a+d}\\ z^{c+1} x^{d+1}&=&\la \nu y^{b+e}\\ x^{a+1} y^{e+1}&=&\la \mu z^{c+f} \end{matrix} \quad\text{and}\quad xyz=\la\mu\nu,\tag{$\uparrow$} \notag \end{equation} for some $a,b,c,d,e,f\ge0$; or \begin{equation} \renewcommand{\arraycolsep}{1.5pt} \begin{matrix} x^{a+d}&=&\be\ga y^{b-1}z^{f-1}\\ y^{b+e}&=&\al\ga z^{c-1}x^{d-1}\\ z^{c+f}&=&\al\be x^{a-1}y^{e-1} \end{matrix} \qquad \begin{matrix} y^b z^f&=&\al x^{a+d-1}\\ z^c x^d&=&\be y^{b+e-1}\\ x^a y^e&=&\ga z^{c+f-1} \end{matrix} \quad\text{and}\quad xyz=\al\be\ga,\tag{$\downarrow$} \notag \end{equation} for some $a,b,c,d,e,f\ge1$. \end{proposition} \paragraph{Proof of Theorem~\ref{th:N}, assuming the proposition} Nakamura's theorem follows easily, because $\GHilb$ is a union of copies of $\C^3$ with coordinates $\la,\mu,\nu$ (or $\al,\be,\ga$), therefore nonsingular. Every affine chart is birational to $X$, because it contains points with none of $\la,\mu,\nu=0$ (or none of $\al,\be,\ga=0$). Moreover, an easy linear algebra calculation shows that the equations ($\uparrow$) or ($\downarrow$) correspond to basic triangles of the junior simplex, so that each affine chart of $\GHilb $ is crepant over $X$. In more detail: \paragraph{Case ($\uparrow$)} Write out the 3 x 3 matrix of exponents of the first three equations of ($\uparrow$): $$ \begin{matrix} a+d+1&-b&-f\\ -d&b+e+1&-c\\ -a&-e&c+f+1 \end{matrix} $$ (note that each of the 3 columns add to 1, more less equivalent to the junior condition). The 2 x 2 minors of this give the 3 vertexes \begin{align*} P&=(bc+bf+ef+b+c+e+f+1,\quad ac+cd+df+d,\quad ab+ae+de+a),\\ Q&=(bc+bf+ef+b,\quad ac+cd+df+a+d+c+f+1,\quad ab+ae+de+e),\\ R&=(bc+bf+ef+f,\quad ac+cd+df+b,\quad ab+ae+de+a+b+d+e+1). \end{align*} The triangle PQR ``points upwards'', in the sense that \begin{align*} &\text{$P$ is closest to $(1,0,0)$,}\\ &\text{$Q$ is closest to $(0,1,0)$,}\\ &\text{$R$ is closest to $(0,0,1)$.} \end{align*} The 3 given ratios $x^{a+d+1}:y^bz^f$, etc.\ correspond to the 3 sides of triangle $PQR$. In any case, all the vertexes belong to the junior simplex, so that this piece of $\GHilb $ is crepant over $X$. \paragraph{Case ($\downarrow$)} Write out the exponents of the second set of three equations: $$ \begin{matrix} -(a+d)+1 & b & f\\ d & -(b+e)+1 & c\\ a & e & -(c+f)+1 \end{matrix} $$ again, each of the 3 columns add to 1, and the $2\times2$ minors of this give the 3 vertexes \begin{align*} P&=(bc+bf+ef-b-c-e-f+1, ac+cd+df-d, ab+ae+de-a ),\\ Q&=( bc+bf+ef-b, ac+cd+df-a-d-c-f+1, ab+ae+de-e ),\\ R&=( bc+bf+ef-f, ac+cd+df-b, ab+ae+de-a-b-d-e+1), \end{align*} all of which again belong to the junior simplex, so this affine chart is also crepant over $X$. This time the triangle $PQR$ ``points downwards'', in the sense that \begin{align*} &\text{$P$ is furthest from $(1,0,0)$,}\\ &\text{$Q$ is furthest from $(0,1,0)$,}\\ &\text{$R$ is furthest from $(0,0,1)$.} \end{align*} The 3 given ratios $x^{a+d-1}:y^bz^f$, etc.\ again correspond to the 3 sides. Q.E.D. for the theorem, assuming the proposition. \paragraph{Proof of Proposition~\ref{prop:N}} Most of this is very geometric: any reasonable choice of monomials in $x,y,z$ whose classes in $\Oh_Z$ form a basis is given by a polygonal region $M$ of the honeycomb figure (\ref{fig:honey}) satisfying 2 conditions: \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item in each of the 3 triants (triangular sector) it is concave, that is, a downwards staircase: because it is a Newton polygon for an ideal; \item it is a fundamental domain of the periodicity lattice $\Pi$: because we assume that $\Oh_Z=k[G]$, therefore every character appears exactly once. \end{enumerate} The condition (ii) means that $M$ and its translates by $\Pi$ tesselate the plane, so they form a kind of jigsaw pattern like the Escher periodic patterns. However, in each of the 3 principal directions corresponding to the $a$, $b$, and $c$-axes, there is only one acute angle, namely the summit at the end of the $a$-axis (etc.). Therefore $M$ can only have one valley (concave angle) in the $b,c$ triant. As a result, there is only one geometric shape for the polygon $M$, the {\em tripod} or {\em mitsuya} (3 valleys, or 3 arrows) of Figure~I. I introduce some terminology: the {\em tripod} $M$ has 3 {\em summits} at the end of the axis of monomials $x^i$, and 3 {\em triants} or sectors of $120^\circ$ containing monomials $x^iy^j$. Each triant has one {\em valley} and two {\em shoulders} (incidentally, the 6 shoulders give the {\em socle} of $\Oh_Z$). \begin{remark}\label{rem:deg} There are degenerate cases when some of the valleys or summits are trivial (for example, $a=0$ in $\uparrow$). The most degenerate case is a straight lines, when $\Oh_Z$ is based by powers of $x$ (say), and the equations boil down to $y=x^i,z=x^j$ (the $x$-{\em corner} of the resolution). I omit discussion of these cases, since the equations of the cluster $Z$ are always a lot simpler. \end{remark} \begin{verbatim} I o o o o I o o o o o I o o o o o I o o o o o o o o I o o o o o o o o o I I I I I I I (Figure I) o o o I o o o o o o o o I o o o I o o I o o \end{verbatim} Thus there is only one ``geometric'' solution to the Escher jigsaw puzzle, namely \begin{verbatim} ... I I I I I I I I I I u u u I u u u u u u u u u u u u u I u u I o o o u I u u o I o o o I u u I v v v o o I o o o v I v v v o o I o o o o o o v v I v v v o o I o o o o o o v v I v v v v v v ... (Figure II) o o o I I I I I I I v v I v o o o I o o o o o o v v ... o o I o o v v o I o o I o o \end{verbatim} In particular, the external sides (going out to the 3 summits) are equal plus-or-minus 1 to the opposite internal sides (going in to the 3 valleys). \begin{figure}[th] \centering\mbox{\epsfbox{dmz.ps}} \caption{Two different cocked hats} \label{fig:dmz} \end{figure} However, the geometric statement of Figure~II is only exact for closed polygons, whereas our tripods are Newton polygons spanned by integer points, and are separated by a thin ``demilitarised zone'' between the integer points. When you consider the tripods together with the integer lattices, there are two completely different ways in which the three shoulders of neighbouring tripods can fit together (corresponding to the two cyclic orders, or the two cocked hats of Figure~\ref{fig:dmz}), namely either ($\uparrow$) \begin{verbatim} y y y y z y y y z x x x z z x x x z z \end{verbatim} where the last $\tt y$ is just after the last $\tt x$, and the shoulder of the $\tt z$ is level with the top row of $\tt x$ or ($\downarrow$) \begin{verbatim} y y y z z y y z z x x x z z x x x z z \end{verbatim} where the last $\tt y$ is just before the last $\tt x$ and the top row of $\tt x$ is just below the shoulder of the $\tt z$. The two different forms ($\uparrow$) and ($\downarrow$) come from this patching. \begin{remark}[Algorithm for $\GHilb$] Nakamura \cite{N3} gives an algorithm to compute $\GHilb $ in this case as a toric variety. This can be viewed as a way of classifying all the possible tripods in terms of elementary operations, which correspond to the 0-strata and the 1-strata of the toric variety $\GHilb $. You pass from an $\uparrow$ tripod to a $\downarrow$ one by shaving off a layer of integer points one thick around one valley (assumed to have thickness $\ge1$), and glueing it back around the opposite summit. And vice versa to go from $\downarrow$ to $\uparrow$. You can start from anywhere you like, for example from the $x$-corner (see Remark~\ref{rem:deg}). Nakamura's algorithm applied to the statement in Proposition~\ref{prop:N} expressed in terms of the fan triangulating the junior simplex, gives that if $\uparrow$ and $a,b,c,d,\allowbreak e,f\ge k\ge2$ (say) then you can cross any wall of the ``upwards'' triangle of the fan to get a new $\downarrow$ coordinate patch with $a',b',c',d',e',f'\ge k-1$, which corresponds to a ``downwards'' triangle, and vice-versa. It follows that the first triangle is surrounded by a patch of width $k-1$ which is triangulated by the regular triangular lattice, so that the resolution has a corresponding patch of regular hexagons (that is, del Pezzo surfaces of degree 6). Figure~\ref{fig:honey} shows the McKay quiver of $\frac1{37}(1,5,31)$ and Figure~\ref{fig:37b} its fundamental domain giving the equations of $G$-clusters \begin{equation} x^4=\la y^2z,\quad y^4=\mu xz^3,\quad z^5=\nu x^2y,\quad \text{etc.} \notag \end{equation} on the coordinate chart of the resolution of $\frac1{37}(1,5,31)$, corresponding to the starred triangle of Figure~\ref{fig:37}. \begin{figure}[ht] \centering\mbox{\epsfbox{fig12.ps}} \caption{A fundamental domain of the McKay quiver for $\frac1{37}(1,5,31)$} \label{fig:37b} \end{figure} \end{remark}

9702

alg-geom/9702003

en

https://arxiv.org/abs/alg-geom/9702003

alg-geom/9702003

Tohsuke Urabe

Tohsuke Urabe (Department of Mathematics Tokyo Metropolitan University, Hachioji-shi, Tokyo, Japan)

Dual varieties and the duality of the second fundamental form

LaTeX2e+AmsLaTeX. 3 pages. This manuscript was submitted to Proceedings of Symposium Real Analytic and Algebraic Singularities(IS(J held at Nagoya University in September - October, 1996. Adobe PDF version is available also at http://urabe-lab.math.metro-u.ac.jp/

First, we consider a compact real-analytic irreducible subvariety $M$ in a sphere and its dual variety $M^\vee$. We explain that two matrices of the second fundamental forms for both varieties $M$ and $M^\vee$ can be regarded as the inverse matrices of each other. Also generalization in hyperbolic space is explained.

alg-geom

\section{Spherical case} \label{sphere} In this article I would like to explain main ideas in my recent results on duality of the second fundamental form. (Urabe\cite{{urabe;dual}}.) Theory of dual varieties in the complex algebraic geometry is very interesting. (Griffiths and Harris~\cite{griffiths-harris;geo}, Kleiman~\cite{kleiman;enume}, Piene~\cite{piene;polar}, Urabe~\cite{urabe;polar}, Wallace~\cite{wallace;tangency}.) Let $\mathbf P$ be a complex projective space of dimension $N$, and $X\subset \mathbf P$ be a complex algebraic subvariety. The set of all hyperplanes in $\mathbf P$ forms another projective space $\mathbf P^\vee$ of dimension $N$, which is called the \emph{dual projective space} of $\mathbf P$. The dual projective space $(\mathbf P^\vee)^\vee$ of $\mathbf P^\vee$ is identified with $\mathbf P$. The closure in $\mathbf P^\vee$ of the set of tangent hyperplanes to $X$ is called the \emph{dual variety} of $X$, and is denoted by $X^\vee$. We say that a hyperplane $H$ in $\mathbf P$ is tangent to $X$, if we have a smooth point $p\in X$ such that $H$ contains the embedded tangent space of $X$ at $p$. It is known that the dual variety $X^\vee$ is again a complex algebraic variety, and the dual variety $(X^\vee)^\vee$ of $X^\vee$ coincides with $X$. We would like to develop similar theory in the real-analytic category. (Obata~\cite{obata;gauss}.) First, we fix the notations. Let $N$ be a positive integer, and $L$ be a vector space of dimension $N+1$ over the real field $\mathbf R$. A fixed positive-definite inner product on $L$ is denoted by $(\ \ ,\ \ \ )$. By $S=\{a\in L|(a,a)=1\}$ we denote the unit sphere in $L$. The sphere $S$ has dimension $N$. We consider a compact real-analytic irreducible subvariety $M$ in $S$. We assume moreover that $M$ has only ordinary singularities as singularities. We have to explain the phrase of ``ordinary singularity'' here. Let $X\subset L$ be a real-analytic subset. For every point $p\in X$ we can consider the germ $(X,p)$ of $X$ around $p$. The germ $(X,p)$ is decomposed into into irreducible components. By $\dim (X,p)$ we denote the dimension of the germ $(X,p)$. The germ $(X,p)$ is said to be \emph{smooth}, if $(X,p)$ is real-analytically isomorphic to $(\mathbf R^n,0)$ where $n=\dim (X,p)$ and $0$ is a point of $\mathbf R^n$. A point $p$ of $X$ is said to be smooth, if the germ $(X,p)$ is smooth. We say that $X$ has an \emph{ordinary singularity} at $p\in X$, if every irreducible component of $(X,p)$ is smooth. Let $M_{smooth}\subset M$ be the set of smooth points $p\in M$ with $\dim (M,p)=\dim M$. Under our assumption $M_{smooth}$ is dense in $M$. For every point $p\in M_{smooth}$ the tangent space $T_p(M)$ of $M$ at $p$ is defined. Note in particular that $T_p(M)$ is not an affine subspace but a vector subspace in $L$ passing through the origin. The tangent space $T_p(M)$ has dimension equal to $\dim M$. A point $q\in S$ is a normal vector of $M$ in $S$ at a point $p\in M$, if $q$ is orthogonal to $p$ and $T_p(M)$. We say that a point $q\in S$ is a normal vector of $M$ in $S$, if $q$ is a normal vector of $M$ in $S$ at some point $p\in M$. By $M^\vee$ we denote the closure in $S$ of the set of normal vectors $a$ of $M$ in $S$ with $(a,a)=1$, and we call $M^\vee\subset S$ the \emph{dual variety} of $M\subset S$. The dual variety $M^\vee$ has a lot of interesting properties. However, $M^\vee$ is not a real-analytic subset in general. \begin{prop} \label{dense} Under our assumption the dual variety $M^\vee$ contains a dense smooth real-analytic subset whose connected components have the same dimension. \end{prop} Let $X\subset S$ be a subset containing a dense smooth real-analytic subset whose connected components have the same dimension. Obviously we can define the dual variety $X^\vee$ of $X$ by the essentially same definition as above. \begin{thm} \label{duality} Under our assumption $(M^\vee)^\vee=M\cup\tau(M)$, where $\tau:S\rightarrow S$ denotes the antipodal map $\tau(q) =-q$. \end{thm} \begin{rem} Note that $M\cup\tau(M)$ is a compact real-analytic subset only with ordinary singularities as singularities. For any compact real-analytic subset in $L$ only with ordinary singularities as singularities, the irreducible decomposition is possible. Therefore, $M$ is an irreducible component of $M\cup\tau(M)$, and we can recover $M$ from $M\cup\tau(M)$. \end{rem} There exists an open dense smooth real-analytic subset $V$ of $M^\vee$ such that for every point $q\in V$ there exists a point $p\in M$ such that \begin{enumerate} \item $q$ is a normal vector of $M$ in $S$ at $p$, and \item $p$ is a normal vector of $M^\vee$ in $S$ at $q$. \end{enumerate} Moreover, there exists an open dense smooth real-analytic subset $U$ of $M$ such that for every point $p\in U$ there exists a point $q\in V$ satisfying the same conditions 1 and 2 above. Choose arbitrarily a pair $(q,p)$ of a smooth point $q\in M^\vee$ and a smooth point $p\in M$ satisfying conditions 1 and 2, and fix it. The second fundamental form of $M$ at $p$ in the normal direction $q$ $$\widetilde{II}:\: T_p(M)\times T_p(M)\longrightarrow \mathbf R$$ and the second fundamental form of $M^\vee$ at $q$ in the normal direction $p$ $$\widetilde{II}^\vee:\: T_q(M^\vee)\times T_q(M^\vee) \longrightarrow \mathbf R$$ are defined. We set \begin{eqnarray*} \mathrm{rad}\,\widetilde{II}&=&\{X\in T_p(M)|\mbox{For every }Y\in T_p(M),\: \widetilde{II}(X,Y)=0\}\\ \mathrm{rad}\,\widetilde{II}^\vee&=&\{X\in T_q(M^\vee)|\mbox{For every } Y\in T_q(M^\vee),\: \widetilde{II}^\vee(X,Y)=0\}. \end{eqnarray*} \begin{thm}[Duality of the second fundamental form] \label{second} \ \newline\vspace*{-10pt} \begin{enumerate} \item $T_p(M)=\mathrm{rad}\,\widetilde{II}+(T_p(M)\cap T_q(M^\vee))$ (orthogonal direct sum) \item $T_q(M^\vee)=\mathrm{rad}\,\widetilde{II}^\vee+(T_p(M)\cap T_q(M^\vee))$ (orthogonal direct sum) \item $L=\mathbf R p+\mathrm{rad}\,\widetilde{II}+(T_p(M)\cap T_q(M^\vee)) +\mathrm{rad}\,\widetilde{II}^\vee+\mathbf R q$ (orthogonal direct sum) \item Let $X_1, X_2,\ldots, X_r$ be an orthogonal normal basis of $T_p(M)\cap T_q(M^\vee)$. The matrix $(\widetilde{II}(X_i, X_j))$ is the inverse matrix of $(\widetilde{II}^\vee(X_i, X_j))$. \end{enumerate} \end{thm} Proposition~\ref{dense} is the most difficult part to show in our theory. Once we obtain Proposition~\ref{dense}, it is not difficult to deduce Theorem~\ref{duality} applying analogous arguments in complex projective algebraic geometry. Theorem~\ref{duality} and Theorem~\ref{second} can be shown through computation on Maurer-Cartan forms. Theorem~\ref{second} seems to have a lot of applications in theory of subvarieties in a sphere. You can download my preprint~\cite{urabe;dual} containing verification at \begin{center} \begin{tabular}{ll} http://urabe-lab.math.metro-u.ac.jp/ & (Japanese)\\ http://urabe-lab.math.metro-u.ac.jp/DefaultE.html & (English). \end{tabular} \end{center} \section{Hyperbolic case} \label{hyperbolic} We can consider similar situations in hyperbolic case. (Obata~\cite{obata;gauss}.) Let $L$ be a vector space of dimension $N+1$ over the real field $\mathbf R$ as in Section~\ref{sphere}. Now, we consider a non-degenerate inner product $(\ \ ,\ \ \ )$ on $L$ with signarutre $(N, 1)$. By $S$ we denote one of the two connected components of the set $\{a\in L|(a, a)=-1\}$ in $L$. The hyperbolic space $S$ has dimension $N$. Also in this case we consider a compact real-analytic irreducible subvariety $M$ in $S$ only with ordinary singularities as singularities. Let $S^\vee=\{a\in L|(a, a)=1\}$. Note that also $S^\vee$ is a smooth real-analytic connected variety with dimension $N$. However, $S\cap S^\vee=\emptyset$, and the metric on $S^\vee$ is not definite. We can define the dual variety $M^\vee$ of $M$ as a subset of $S^\vee$ by the essentially same definition as above. The dual variety $(M^\vee)^\vee$ of $M^\vee$ can be defined as a subset of $S$. Proposition~\ref{dense} and Theorem~\ref{second} hold also in this case without any modification. Theorem~\ref{duality} is replaced by the following brief theorem: \begin{thm} \label{duality2} In hyperbolic case under our assumption $(M^\vee)^\vee=M$. \end{thm} \begin{prob} Give genelarization of theory of dual varieties in $C^\infty$-category. \end{prob}

9702

alg-geom/9702019

en

https://arxiv.org/abs/alg-geom/9702019

alg-geom/9702019

Alan Durfee

Alan H. Durfee

Five Definitions of Critical Point at Infinity

20 pages, Latex, 4 figures

This survey paper discusses five equivalent ways of defining a ``critical point at infinity'' for a complex polynomial of two variables.

alg-geom

\section{#1}} \newcounter{mycounter}[section] \renewcommand{\themycounter}{\arabic{section}.\arabic{mycounter}} \newenvironment{theorem}% {\medskip \refstepcounter{mycounter} {\bf \noindent Theorem \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{proposition}% {\medskip \refstepcounter{mycounter} {\bf \noindent Proposition \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{lemma}% {\medskip \refstepcounter{mycounter} {\bf \noindent Lemma \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{corollary}% {\medskip \refstepcounter{mycounter} {\bf \noindent Corollary \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{remark}% {\medskip \refstepcounter{mycounter} {\bf \noindent Remark \themycounter. \ }}% {\medskip } \newenvironment{definition}% {\medskip \refstepcounter{mycounter} {\bf \noindent Definition \themycounter. \ }}% {\medskip } \newenvironment{example}% {\medskip \refstepcounter{mycounter} {\bf \noindent Example \themycounter. \ }}% {\medskip } \newenvironment{problem}% {\medskip \refstepcounter{mycounter} {\bf \noindent Problem \themycounter. \ }}% {\medskip } \newenvironment{xproof}% {\medskip \noindent {\bf Proof. \ }}% {$\square$ \medskip } \newcommand{}{} \begin{document} \maketitle \begin{abstract} This survey paper discusses five equivalent ways of defining a ``critical point at infinity'' for a complex polynomial of two variables. \end{abstract} \section{Introduction} A proper smooth map without critical points from one manifold to another is a locally trivial fibration by a well-known theorem of Ehresmann. On the other hand, a nonproper map without critical points may not be a fibration. This phenomenon occurs for complex polynomials. A simple example is provided by $f: \bf C^2 \to \bf C$ defined by the polynomial $f(x,y) = y(xy-1)$. This map has no critical points, but the fiber over the origin is different from the other fibers. (In fact, the fiber over the origin is two rational curves, one punctured at two points and the other at one point, whereas the general fiber is a cubic curve, punctured at two points.) One would like to identify these ``critical values'' where the topology changes and their corresponding ``critical points at infinity''. We first review the history of this subject. Let $f: \bf C^{n} \to \bf C$ be a complex polynomial. There is a finite set $\Sigma \in \bf C$ such that $$ f: \bf C^n - f^{-1}(\Sigma) \to \bf C - \Sigma $$ is a fibration. This is a form of Sard's theorem for polynomials; the set $\Sigma$ is finite because it is algebraic. For a proof, see \cite[Proposition 1]{Broughton-83} (based on work of Verdier), \cite[Appendix A1]{Pham-83}, \cite[Theorem 1]{Ha-Le-84} or \cite{Ha-89-2}. We let $$\Sigma = \Sigma_{fin} \cup \Sigma_{\infty}$$ where $\Sigma_{fin}$ is the set of critical values coming from critical points in $\bf C^n$, and $\Sigma_\infty$ is the set of critical values ``coming from infinity''. Of course these two sets may have nonempty intersection. Broughton in \cite{Broughton-83,Broughton-88} calls the polynomial $f$ {\em tame} if there is a $\delta > 0$ such that the set $\{ x: | grad \, f(x) | \leq \delta \}$ is compact. He proved that if $f$ is tame, then $\Sigma_\infty$ is empty. Thus if the gradient of a polynomial goes to zero along some path going to infinity, then something bad may happen. Topics surrounding the gradient of the polynomial are treated in Section 4 of this paper. The speed at which the gradient of $f$ goes to zero is measured by the Lojasiewicz number at infinity; see \cite{Ha-90, Ha-P91, Ha-94, Cassou-Ha-P92, Cassou-Ha-95}. There followed many efforts in the case $n = 2$ to identify the set $\Sigma_\infty$ more precisely. Suzuki \cite[Corollary 1]{Suzuki-74} provides an estimate on the number of points in $\Sigma$. In \cite{Ha-Le-84} it is shown that $c \in \Sigma$ if and only if $\chi (f^{-1} (c) ) \neq \chi (f^{-1} (t) )$, where $f^{-1} (t)$ is a generic fiber of $f$ and $\chi$ denotes Euler characteristic. Further work on identifying $\Sigma_\infty$ ca be found in \cite{Ha-Nguyen-89, Ha-89-2, Nemethi-Zaharia-90, Nemethi-Zaharia-92, Le-Oka-P93}. The homology and homotopy of the fibers of the polynomial $f$ were also computed, leading to various numerical invariants which will be discussed in the Section 2 of this paper. Suzuki \cite[Proposition 2]{Suzuki-74} shows that $$rank \, H_1(f^{-1}(t)) = \mu + \lambda$$ where $f^{-1}(t)$ is a generic fiber, $\mu$ is the sum of the Milnor numbers at the critical points of $f$ in $\bf C^2$, and $\lambda$ is the sum of all the ``jumps'' in the Milnor numbers at infinity. (In the terminology of Section 2, $\lambda = \sum \nu_{p,c}$, where the sum is over $c \in \bf C$ and $ p \in {\Bbb L}_\infty$, and $\nu_{p,c}$ is the jump in the Milnor number at the point $p \in {\Bbb L}_\infty$ and value $c \in \bf C$.) More on the topology of the fiber can be found in \cite{Bartolo-Cassou-Dimca-P96}. The polynomial $f$ extends to a function on projective space ${\Bbb P}^2$ which is well defined except at a finite number of points. The points of indeterminacy can be easily resolved, and the structure of the resolution contains information about these points \cite{Le-Weber-95, Le-Weber-P96}. These topics are discussed in Section 3. Other topics investigated (but not discussed in this paper) include Newton diagrams \cite[Proposition 3.4]{Broughton-88}, \cite{Nemethi-Zaharia-90, Cassou-P96}, knots \cite{Neumann-89, Ha-91}, and the Jacobian conjecture \cite{Le-Weber-95}. Papers in higher dimensions (that is, $n > 2$) include \cite{Parusinski-P95, Siersma-Tibar-95, Tibar-P96}. Broughton \cite[Proposition 3.2]{Broughton-88} shows that the tame polynomials form a dense constructible set in the set of polynomials of a given degree; Cassou-Nogues \cite[Example V]{Cassou-P96} gives an example to show that this set is not open in dimension $n=3$. Although the scene in higher dimensions is not yet settled, the situation in dimension two is now clear. The purpose of this paper is to collect together five definitions of ``critical point at infinity'' in this low-dimensional case and prove that they are equivalent. These definitions have appeared in the literature in some form or other, usually in a global affine context; the purpose of this paper is to give these definitions and prove their equivalence in a purely local setting near a point on the line at infinity. Many examples are also given. It should be noted that this material can be tricky, despite its apparent simplicity, and one should take care to make precise statements and proofs as well as to check examples. If $f(x,y)$ is a polynomial, $p \in {\Bbb L}_\infty$ is a point through which the level curves of $f$ pass, and $c \in \bf C$, we say that the pair $(p,c)$ is a {\em regular point at infinity} for $f(x,y)$ if it satisfies any one of the following equivalent conditions. (Otherwise it is a {\em critical point at infinity}.) \begin{itemize} \item Condition M (\ref{condition-m}): There is no jump in the Milnor number (2.1): $\nu_{p,c} = 0$. \item Condition E (\ref{condition-e}): The family of germs $f(x,y) = tz^d$ at $p$ is equisingular at $t = c$. \item Condition F (\ref{condition-f}): The map $f$ is a smooth fiber bundle near $p$ and the value $c$. \item Condition R (\ref{condition-r}): There is a resolution $\tilde{f}: M \to {\Bbb P}^1$ with $\pi: M \to {\Bbb P}^2$ and a neighborhood $U$ of $p \in {\Bbb P}^2$ such that $\{ \tilde{f} = c \} \cap \pi^{-1}(U)$ is smooth and intersects the exceptional set $ \pi^{-1}(p)$ transversally. \item Condition G (\ref{condition-g}): There does not exist a sequence of points $ \{ p_k \} \in \bf C^2 $ with $ p_k \to p$, $grad \, f(p_k) \to 0$ and $f(p_k) \to c$ as $k \to \infty $. \end{itemize} Most of these equivalences are well known; we give either proofs or references for proofs in the pages that follow. There are several new results in this paper. First, we define (\ref{nu-infinity}) an invariant $\nu_{p, \infty}$ which measures the number of vanishing cycles at a point $p$ on the line at infinity for the critical value infinity, and show that this invariant has many of the same properties that $\nu_{p,c}$ does for $c \in \bf C$. Secondly, we define $g_{p,c}$ to be the number of isotopy classes of paths $\alpha : \bf R \to \bf C^2$ such that $\alpha (t) \to p$, $grad \, f(\alpha (t) ) \to 0$ and $f(\alpha(t)) \to c$ as $t \to + \infty$. We use this to give a new proof that Condition M implies Condition G. In fact, we will show (Proposition \ref{nu-geq-g}) that $\nu_{p,c} \geq g_{p,c}$. The work described in this paper started in 1989 when the author supervised a group of undergraduates in the Mount Holyoke Summer Research Institute in Mathematics who were working on corresponding problems for real polynomials. These results are described in \cite{REU}, with further results in \cite{Durfee-P95}. The work for this paper was carried out at the Tata Institute, Bombay, Martin-Luther University, Halle (with support from IREX, the International Research and Exchanges Board), the University of Nijmegen, Warwick University, the Massachusetts Institute of Technology and the University of Bordeaux. The author would like to thank all of them for their hospitality. Earlier versions of this paper included results on deformations of critical points at infinity; these will appear elsewhere. \section{Numerical Invariants} We will use coordinates $(x,y)$ for the complex plane $\bf C^2$, and coordinates $[x,y,z]$ for the projective plane ${\Bbb P} ^2$. We let $${\Bbb L}_\infty = \{ [x,y,z] \in {\Bbb P}^2: z=0 \}$$ be the line at infinity. We let $d$ be the degree of the polynomial $f(x,y)$. We let $f_d$ denote the homogeneous term of degree $d$ in $f$. If $p = [a,b,0] \in {\Bbb L}_\infty$, we let $d_p$ be the multiplicity of the factor $(bx-ay)$ in $f_d$. Suppose that the level sets of $f$ intersect ${\Bbb L}_\infty$ at $p$. Let $$F_t(x,y,z) = z^df(x/z, y/z) - tz^d$$ be the homogenization of the polynomial $f(x,y)-t$, where $t \in \bf C$, and let $g_{p,t}$ be the local equation of $F_t$ at $p$. If $p = [1,0,0]$, then $g_{p.t}$ is given in local coordinates $$(u,v) = (y/x, 1/x)$$ by $$g_{p,t}(u,v) = F_t(1,u,v)= v^df(1/v,u/v) - tv^d$$ Note that the multiplicity of $g_{p,t}$ at $(0,0)$ is at most $d_p$. \begin{definition} \label{Milnor-number} The {\em Milnor number} $\mu_{p,t}$ of $f(x,y)$ at $(p,t) \in {\Bbb L}_\infty \times \bf C$ is the Milnor number of the germ $g_{p,t}$ at $(0,0)$ in the usual sense. The {\em generic Milnor number} $\mu_{p,gen}$ is the Milnor number $\mu_{p,t}$ for generic $t$. The number of {\em vanishing cycles} at $(p,t)$ is $$\nu_{p,t} = \mu_{p,t} - \mu_{p,gen}$$ \end{definition} \begin{example} Let $f(x,y) = y(xy-1)$ and $p = [1,0,0]$. Then $g_{p,t}(u,v) = u^2-uv^2-tv^3$. We have $\mu_{p,gen}=2$, $\nu_{p,0}=1$, and all other $\nu_{p,t} = 0$. In fact, for $t \neq 0$, the singularity is of type $A_2$, and for $t=0$, the singularity is of type $A_3$. This well-known example is the simplest ``critical point at infinity". More generally, if $f(x,y) = y(x^ay-1)$ then for $t \neq 0$, $\mu_{p,t} = a+1$ and there is a singularity of type $A_{a+1}$. For $t=0$, $\mu_{p,0} = 2a+1$ and there is a singularity of type $A_{2a+1}$. \end{example} \begin{example} Let $f(x,y)=x(y^2-1)$ and $p = [1,0,0]$. Then $g_{p,t}(u,v) = u^2-v^2-tv^3$. For all $t$, $\mu_{p,t}=1$; the family is equisingular, and there is no ``critical point at infinity''. This is another basic example. \end{example} \begin{example} \label{two-max-ex} Here is a more complicated example (see \cite{REU,Durfee-P95}): Let $f(x,y) = (xy^2-y-1)^2 + (y^2-1)^2$. At $p = [1,0,0]$ we have $\mu_{gen} = 15$, $\nu_{p,1}=2$, $\nu_{p,2} = 1$ and $\nu_{p,c}=0$ for all other $c$. \end{example} Next we relate $\nu_{p,t}$ to homological vanishing cycles. Fix $p \in {\Bbb L}_\infty$ and $c \in \bf C \cup \{\infty \}$. Let $U \subset \bf C^2$ be an open set such that the closure in projective space of the set $$ \{ (x,y) \in \bf C^2 : (x,y) \in U \mbox{\ and \ } f(x,y)= t \} $$ is $p$ for $t$ near $c$. Choose $C > 0$ large. We define the {\em Milnor fiber} of $f$ at $(p,c)$ to be $$\tilde{F}_{p,c} = \overline{ \{ (x,y) \in \bf C^2 : (x,y) \in U \mbox{\ and \ } |(x,y)| \geq C \mbox{\ and \ } f(x,y)= t \} } $$ where the overbar indicates closure in projective space, and, if $c \in \bf C$, then $t$ is near, but not equal to, $c$, and if $c = \infty$, then $t$ is large. \begin{proposition} \label{betti} For $p \in {\Bbb L}_\infty$ and $c \in \bf C$, $$\nu_{p,c} = rank \, H_1 (\tilde{F}_{p,c})$$ \end{proposition} \begin{xproof} Without loss of generality, we may assume that $p = [1,0,0]$. The number $\nu_{p,c}$ is the difference of the Milnor number $\mu_{p,c}$ and the generic Milnor number $\mu_{p,gen}$. The number $\mu_{p,gen}$ is the Milnor number of $g_{p,t}$ for $t$ near, but not equal to, $c$. By the usual argument, this difference is $rank \, H_1 (\{g_{p,c}= 0 \} \cap B_0)$ where $B_0$ is the small ball for the Milnor number of $g_{p,t}$. We may replace $\{g_{p,c}= 0 \} \cap B_0$ by $$F'_{p,c} = \{ (u,v) \in \bf C^2 : |v| \leq \epsilon' \mbox{\ and \ } g_{p,t}(u,v) = 0 \} $$ We may replace $\tilde{F}_{p,c}$ by $$\tilde{F}'_{p,c} = \overline{ \{ (x,y) \in \bf C^2 : (x,y) \in U \mbox{\ and \ } |x| \geq C \mbox{\ and \ } f(x,y)= t \} } $$ The change of coordinates $x = 1/v$ and $y = u/v$ takes $\tilde{F}_{p,c}$ to $F'_{p,c}$. \end{xproof} To define ``vanishing cycles'' for the critical value $c = \infty$, we take the above proposition to be a definition: \begin{definition} \label{nu-infinity} For $p \in {\Bbb L}_\infty$ we let $$\nu_{p,\infty} = rank \, H_1 (\tilde{F}_{p,\infty})$$ \end{definition} \begin{remark} Here is a topological interpretation of the number of vanishing cycles at infinity: Suppose the level curves of the polynomial $f$ of degree $d$ intersect ${\Bbb L}_\infty$ at $k$ points (counted without multiplicities). Then $\nu_{p,\infty} = 0$ for all $p \in {\Bbb L}_\infty$ if and only if $\overline{ \{ f(x,y) = t \} }$ for $t$ large is homeomorphic to a $d$-fold cover of ${\Bbb L}_\infty$ branched at $k$ points. For example, $y(xy-1) = t$ (where $\nu_{p,\infty} = 0$ for all $p$) is a three-fold cover of ${\Bbb P}^1$ branched at two points, but $y^2-x = t$ (where $\nu_{[1,0,0],\infty} = 1$) is not a two-fold cover of ${\Bbb P}^1$ branched at one point. \end{remark} Next we describe three ways of computing the number of vanishing cycles $\nu_{p,c}$. The first is to compute (perhaps with a computer algebra progam) $\mu_{p,c}$ and $\mu_{p,gen}$ and subtract. The second is by counting nondegerate critical points, as described in the proposition below. This is similar to computing the usual Milnor number by counting the number of nondegerate critical points in a Morsification (see \cite[vol II, p. 31]{AGV}), and the proof is similar. \begin{proposition} \label{nu-equals-critical-values} Let $p \in {\Bbb L}_\infty$ and $c \in \bf C \cup \{ \infty \}$. The number $\nu_{p,c}$ is equal to the number of critical points (assumed nondegenerate) $q \neq (0,0)$ of the function $g_{p,t}$ such that $q \to (0,0)$ as $t \to c$. \end{proposition} \begin{example} Let $f(x,y) = y(xy-1)$ and $p = [1,0,0]$. Then $g_{p,t}(u,v) = u^2-uv^2-tv^3$. For $t \neq 0$ the function $g_{p,t}$ has a (degenerate) critical point at $(0,0)$ with critical value 0, and a nondegenerate critical point at $((9/2) t^2, -3t)$ with critical value $(27/4)t^4$. As $t \to 0$ the second critical point approaches $(0,0)$. Thus $\nu_{p,0} = 1$. \end{example} \begin{example} Let $f(x,y) = x-y^2$ and $p = [1,0,0]$. Then $g_{p,t}(u,v) = v - u^2 - tu^2$. The function $g_{p,t}$ has a single nondegenerate critical point at $(0,1/(2t))$ with critical value $1/(4t)$. As $t \to \infty$ this critical point approaches $(0,0)$, so $\nu_{p,\infty} = 1$. \end{example} The next proposition describes the result of computing $\nu_{p,\infty}$ by similar methods. \begin{proposition} For $p \in {\Bbb L}_\infty$, $$\nu_{p,\infty} = (d_p-1)(d-1) - \mu_{p,gen}$$ \end{proposition} \begin{xproof} Without loss of generality $p = [1,0,0]$. The intersection multiplicity of the curves $(g_{p,t})_u$ and $(g_{p,t})_v$ at $(0,0)$ for $t = \infty$, where $(u,v)$ are local coordinates at $(0,0)$, can be computed using the algorithm in \cite{Fulton}, and is found to be $(d_p-1)(d-1)$. (To compute the intersection multiplicity at $t = \infty$, we let $s = 1/t$ and compute it at $s = 0$.) For large $t \neq \infty$, the intersections split into those at $(0,0)$, the number of which is $\mu_{p,gen}$, and those not at $(0,0)$, the number of which is $\nu_{p, \infty}$. \end{xproof} \begin{example} The polynomial $f(x,y) = y^a + x^{a-2}y+x$ has $\nu_{p,\infty} = a^2-2a$ at the point $p = [1,0,0]$, and all other $\nu_{p,c} = 0$. \end{example} Finally, $\nu_{p,t}$ can computed a third way by using polar curves, as described below. (See \cite[1.6, 1.8]{Ha-Nguyen-89}.) This method also shows that some vanishing cycles are easy to ``see'' from a contour plot, since they are where the level curves of the polynomial have a vertical tangent. \begin{proposition} \label{polar-curves} Suppose $p = [1,0,0]$, $c \in \bf C \cup \{\infty\}$ and the level sets of $f$ pass through $p$. Then $\nu_{p,c}$ is the number of points of intersection $q \in \bf C^2$ (assumed transverse) of the curves $f = t$ and $f_y = 0$ in $\bf C^2$ such that $q \to p$ as $t \to c$. \end{proposition} \begin{xproof} The set $F'_{p,c}$ from the proof of Proposition \ref{betti} is a connected branched cover of the disk $|v| \leq \epsilon'$ in the $uv$-plane. Two sheets come together at each branch point, and all the sheets come together over $p$. The result follows from Hurwitz's formula. \end{xproof} \begin{example} If $f(x,y) = x(y^2-1)$, the curves $f=t$ and $f_y =0$ intersect at $(-t,0)$. As $t \to \infty$, the intersection point $(-t,0) \to [1,0,0]$ and $f(t,0) \to \infty$. Thus $\nu_{[1,0,0],\infty} = 1$. All other $\nu_{[1,0,0],c} = 0$. \end{example} Next we give three definitions of ``critical point at infinity''. \begin{definition} \label{condition-m} The polynomial $f(x,y)$ satisfies Condition M at the point $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$ if $\nu_{p,c} = 0$. \end{definition} \begin{definition} \label{condition-e} The polynomial $f(x,y)$ satisfies Condition E at the point $(p,c) \in {\Bbb L}_\infty \times \bf C$ if the family of germs $g_{p,t}$ at (0,0) is equisingular at $t = c$. \end{definition} A proof that Condition M for $c \in \bf C$ is equivalent to Condition E may be found at the end of \cite{Le-Ramanujam}). There are various equivalent ways of specifying equisingularity; see for instance the papers by Zariski in volume IV of \cite{Zariski-works}. One that will be useful for us is the following: The family of germs $g_{p,t}$ is equisingular if the germs $g_{p,t}= 0$ at (0,0) form a fiber bundle near $t = c$. \begin{definition} \label{condition-f} The polynomial $f(x,y)$ satisfies Condition F at a point $(p,c) \in {\Bbb L}_\infty \times \bf C$ if the map $f$ is a smooth fiber bundle near $p$ and the value $c$. (More precisely, a polynomial satisfies Condition F if there is a $U \subset \bf C^2$ with $p$ in the closure of $U$ in projective space and $C > 0$ and $\beta > 0$ such that, letting $$B = \{ t \in \bf C : |t - c | \leq \beta \}$$ and $$N = \{ (x,y) \in \bf C^2 : (x,y) \in U \mbox{\ and \ } |(x,y)| \geq C \mbox{\ and \ } f(x,y) \in B \} $$ then $$f: N \to B $$ is a smooth fiber bundle.) \end{definition} \begin{proposition} \label{E-equivalent-F} A polynomial $f(x,y)$ satisfies Condition E at a point $(p,c) \in {\Bbb L}_\infty \times \bf C$ if and only if it satisfies Condition F at that point. \end{proposition} \begin{xproof} The proof is straight-forward, and just involves replacing the ``spherical'' Milnor fiber by one in a ``box'': Without loss of generality, we may assume that $p = [1,0,0]$. We may replace Condition E by the following: There is an $\epsilon' > 0$ and a $\delta' > 0$ such that, letting $$D' = \{ t \in \bf C : |t - c | < \delta' \} $$ and $$M' = \{ (u,v,t) \in \bf C^2 \times \bf C : |v| \leq \epsilon' \mbox{\ and \ } t \in D' \mbox{\ and \ } g_{p,t}(u,v) = 0 \} $$ then the restriction of the projection to the third coordinate $$\pi: M' \to D' $$ is a fiber bundle. We may do this since the germs $g_{p,t}(u,v) = 0$ never have $v = 0$ as a component. We may also replace Condition F by the following: There is a $U' \subset \bf C^2$ with $p$ in the closure of $U'$ in projective space and $C' > 0$ and $\beta' > 0$ such that, letting $$B' = \{ t \in \bf C : |t - c | \leq \beta' \}$$ and $$N' = \{ (x,y) \in \bf C^2 : (x,y) \in U' \mbox{\ and \ } |x| \geq C' \mbox{\ and \ } f(x,y) \in B' \} $$ then $$f: N' \to B' $$ is a smooth fiber bundle. The change of coordinates $x = 1/v$ and $y = u/v$ takes $f(x,y) = t$ to $g_{p,t}(u,v)= 0$ and $N'$ to $M'$. \end{xproof} \section{Resolutions} \label{sec-resolutions} The polynomial $$f: \bf C^2 \to \bf C $$ extends to a map $$\hat{f} : {\Bbb P}^2 \to {\Bbb P} $$ which is undefined at a finite number of points on the line at infinity ${\Bbb L}_\infty$. By blowing up these points one gets a manifold $M$ and a map $$ \pi : M \to {\Bbb P}^2 $$ such that the map $$ \tilde{f} : M \to {\Bbb P} $$ which is the lift of $\hat{f}$ is everywhere defined. We call the map $ \tilde{f}$ a {\em resolution of $f$}. Some interesting results on the structure of resolutions are announced in \cite[Theorems 2, 3, 4]{Le-Weber-95}. For example, a resolution (the minimal resolution) of $y(xy-1)$ is given in Figure \ref{std-crpt-res-g}. \begin{figure} \postscript{std-crpt-res-g.eps}{0.7} \caption{Resolution of $y(xy-1)$} \label{std-crpt-res-g} \end{figure} The symbol $c^{m}$ next to a divisor means that at each smooth point of the divisor there are local coordinates $(z,w)$ in a neighborhood of the point such that the divisor is $z=0$ and $\tilde{f}(z,w) =(z-c)^m$. The proper transform of level curves of $f$ have arrowheads on them; the exceptional sets do not. Resolution are easy compute. For example, starting with $f(x,y) = y(xy-1)$ which we wish to resolve near $[1,0,0]$, the function in local coordinates at $[1,0,0]$ is $u(u-v^2)/v^3$, and we blow up in the standard fashion until it is everywhere defined. More examples are shown in Figures \ref{std-nocrpt-res} and \ref{two-max-res}. \begin{figure} \postscript{std-nocrpt-res.eps}{0.7} \caption{Resolution of $x(y^2-1)$ at $[1,0,0]$} \label{std-nocrpt-res} \end{figure} \begin{figure} \postscript{two-max-res.eps}{0.7} \caption{Resolution of $(xy^2-y-1)^2 + (y^2-1)^2$ at $[1,0,0]$} \label{two-max-res} \end{figure} Next we give a condition for ``regular point at infinity'' in terms of a resolution. (See also \cite[Theorem 5]{Le-Weber-95}.) \begin{definition} \label{condition-r} The polynomial $f(x,y)$ satisfies Condition R at a point $(p,c) \in {\Bbb L}_\infty \times \bf C$ if there is a resolution $\tilde{f}: M \to {\Bbb P}^1$ with $\pi: M \to {\Bbb P}^2$ and a neighborhood $U$ of $p \in {\Bbb P}^2$ such that $\{ \tilde{f} = c \} \cap \pi^{-1}(U)$ is smooth and intersects the exceptional set $ \pi^{-1}(p)$ transversally. \end{definition} \begin{example} \label{Krasinski} Let $f(x,y) =y-(xy-1)^2$ near $[1,0,0]$ (See \cite{Krasinski}). In this example the level curve of the function $\tilde{f} = 0$ is smooth, but it does not intersect the exceptional divisor transversally; see Figure \ref{kras-res}. Hence $(p,c) = ([1,0,0],0)$ does not satisfy Condition R. (Here $\nu_{[1,0,0],0} = 1$.) \begin{figure} \postscript{kras-res.eps}{0.7} \caption{Resolution of $y-(xy-1)^2$ at $[1,0,0]$} \label{kras-res} \end{figure} \end{example} \begin{proposition} \label{E-equivalent-R} A point $p \in {\Bbb L}_\infty$ and a value $c \in \bf C$ for a polynomial $f(x,y)$ satisfies Condition E if and only if it satisfies Condition R. \end{proposition} \begin{xproof} Suppose $(p,c)$ satisfies Condition E. Let $U$ be a neighborhood of $p$ in ${\Bbb P} ^2$ containing no critical points of $f$ in $\bf C ^2$ or points on ${\Bbb L}_\infty$ though which the level curves of $f$ pass. Find a resolution $\tilde{f}$ of $f$. By further blowing up (if necessary), we may assume that $\tilde{f}^{-1}(c)$ is a divisor with normal crossings transversally intersecting the exceptional set where $\tilde{f}$ is not constant. Equisingularity in the form of Zariski's (b)-equivalence \cite[p. 513]{Zariski-65} implies that the functions $g_{p,t}$ for $t$ near $c$ have the same resolution as the function $g_{p,c}$. This can only happen if $\{\tilde{f}^{-1}(t) \cap \pi^{-1}(U) \}$ is smooth and transversally intersects the exceptional set $\pi ^{-1}(p)$. Thus $p$ and $c$ satisfy Condition R. Conversely, if $p$ and $c$ satisfy Condition R, then the resolutions of $ \{ g_{p,t} = 0 \}$ for $t$ near $c$ are (b)-equivalent and hence equisingular. \end{xproof} \section{The Gradient} If $f$ is a complex polynomial, we define $grad \, f$ as in \cite{Milnor} to be the complex conjugate of the vector of partial derivatives. Of course $p \in \bf C^2$ is a regular point for a function $f$ with regular value $c \in \bf C$ if $f(p) = c$ and $grad \, f(p) \neq 0$. An equivalent definition would be to say that there is no sequence of points $ \{ p_k \} $ with $ p_k \to p$, $grad \, f(p_k) \to 0$ and $f(p_k) \to c$ as $k \to \infty $. We can now imitate this definition for $p \in {\Bbb L}_\infty$ as follows: \begin{definition} \label{condition-g} The polynomial $f(x,y)$ satisfies Condition G at a point $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$ if there does not exist a sequence of points $ \{ p_k \} \in \bf C^2 $ with $ p_k \to p$, $grad \, f(p_k) \to 0$ and $f(p_k) \to c$ as $k \to \infty $. \end{definition} If $(p,c)$ does not satisfy Condition G, then a version of Milnor's curve selection lemma (see for instance \cite[Lemma 3.1]{Ha-P91} or \cite[Lemma 2]{Nemethi-Zaharia-92}) implies that the sequence of points can be replaced by a curve: \begin{lemma} \label{curveselectionlemma} If $(p,c)$ does not satisfy Condition G, then there is a smooth real algebraic curve $\alpha : \bf R^+ \to \bf C^2$ such that $\alpha(t) \to p$, \ $grad \, f(\alpha(t)) \to 0$ and $f(\alpha(t)) \to c$ as $t \to + \infty$. \end{lemma} By ``real algebraic curve'' we mean that the image of $\alpha$ in $\bf C^2$ is contained in an irreducible component of the zero locus of a real polynomial. \begin{example} Let $f(x,y) = y(xy-1)$. Let $\alpha(t) = (t, 1/(2t))$. As $t \to + \infty$, $\alpha(t) \to [1,0,0]$, the gradient of $f$ goes to $0$ and the value of the function approaches $0$. \end{example} \begin{example} (c.f. Example \ref{two-max-ex}.) Let $f(x,y) = (xy^2-y-1)^2 + (y^2-1)^2$. Let $\alpha(t) = (t+t^2, \pm 1/t)$. As $t \to + \infty$, $\alpha(t) \to [1,0,0]$, the gradient of $f$ goes to $0$ and the function approaches the value $1$. If $\beta(t) = (t/2, 1/t)$, then as $t \to + \infty$, $\beta(t) \to [1,0,0]$, the gradient of $f$ goes to $0$ and the function approaches the value $2$. (These paths were found by Ian Robertson in the Mount Holyoke REU program in the summer of 1992.) \end{example} \begin{example} If $f(x,y) = x^2y + xy^2 + x^5y^3 + x^3y^5$ and $q \to [1,0,0]$ along the curve $y^2x^3 = -1/3$, then $grad \, f(q) \to 0$ and $f(q) \to \infty$. Here $v_{p,\infty} = 1$. This polynomial is ``quasi-tame'' but not ``tame'' \cite{Nemethi-Zaharia-92}. It would be interesting to find more examples like this. \end{example} The following proposition is well-known. It was first proved in the global case by Broughton \cite{Broughton-88}; see also \cite{Nemethi-Zaharia-90}, proof of Theorem 1, and \cite{Siersma-Tibar-95}, proof of Proposition 5.5. The idea of the proof is to use integral curves of the vector field $grad \, f / |grad \, f |^2$ to identify the fibers. \begin{proposition} \label{G-implies-F} If a polynomial $f(x,y)$ satisfies Condition G at $(p,c) \in {\Bbb L}_\infty \times \bf C$, then it satisfies Condition F at this point. \end{proposition} Next we will show that Condition M implies Condition G; this has been shown in \cite{Ha-90, Ha-P91, Siersma-Tibar-95, Parusinski-P95}. Here we will prove a stronger result by different methods. \begin{definition} \label{def-gpc1} For $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$, let $g_{p,c}$ be the number of isotopy classes of smooth real algebraic curves $\alpha : \bf R \to \bf C^2$ such that $\alpha (t) \to p$, $grad \, f(\alpha (t) ) \to 0$ and $f(\alpha(t)) \to c$ as $t \to + \infty$. \end{definition} \begin{example} If $f(x,y) = y^5+x^2y^3-y$ and $p = [1,0,0]$, then $\nu_{p,0} = 2$. There are two isotopy classes of curves approaching $p$ along which $grad \, f$ goes to zero, namely the ones containing the two branches of the curve $f_y = 0$ at $p$. Hence $g_{p,0} =2$. (This example is from \cite{REU}.) \end{example} Clearly $(p,c) \in {\Bbb L}_\infty \times (\complex \cup \{\infinity\})$ satisfies Condition G if and only if $g_{p,c} = 0$. Now let $\pi : M \to {\Bbb P}^2$ be a resolution of $f$, $f_x$, and $f_y$ (so that $\tilde{f}$, $\widetilde{(f_x)}$ and $\widetilde{(f_y)}$ are defined on $M$), and let $$G_{p,c} = \{ q \in M : \pi(q) = p,\ \tilde{f}(q) = c,\ \widetilde{(f_x)}(q) = 0 \mbox{ \ and \ } \widetilde{(f_y)}(q) = 0 \} $$ \begin{definition} \label{def-gpc2} For $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$, let $\tilde{g}_{p,c}$ be the number of connected components of $G_{p,c}$. \end{definition} The number $\tilde{g}_{p,c}$ is independent of the resolution by the usual argument. \begin{example} In the minimal resolution of $f(x,y) = y(xy-1)$ at $p = [1,0,0]$ (Figure \ref{std-crpt-res-g}), the functions $f_x$ and $f_y$ are defined. The zero locus of the lift of $f_x$ contains the exceptional set where the lift of $f$ is zero, and the zero locus of the lift of $f_y$ intersects this set transversally. Thus $G_{p,0}$ consists of a single point, and $\tilde{g}_{p,0} = 1$. If $f(x,y) = y^5+x^2y^3-y$ and $p = [1,0,0]$, one finds similarly that $G_{p,0}$ consists of two points. \end{example} The two definitions are equivalent by the following proposition. \begin{proposition} For $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$, $g_{p,c} = \tilde{g}_{p,c}$. \end{proposition} \begin{xproof} Let $\pi: M \to {\Bbb P}^2$ be a resolution of $f$, $f_x$ and $f_y$. We will show that there is a one-one correspondence between isotopy classes of curves satisfying (\ref{def-gpc1}) and connected components of $G_{p,c}$. Suppose that $\alpha : \bf R^+ \to \bf C^2$ is a smooth real algebraic curve satisfying the conditions of (\ref{def-gpc1}). Since $\alpha$ is real algebraic, it lifts to to a map $\tilde{\alpha}: \bf R^+ \cup \{ \infty \} \to M$ with $\tilde{\alpha}(\infty) \in \pi^{-1}({\Bbb L}_\infty)$. Let $q = \tilde{\alpha}(\infty)$. Then $\tilde{f}(q) = c$ and $\widetilde{(f_x)}(q) = 0$ and $\widetilde{(f_y)}(q) = 0$. Thus $q \in G_{p,c}$. If $\alpha_0$ is isotopic to $\alpha_1$ through curves $\alpha_t$ satisfying (\ref{def-gpc1}), then the curves $\alpha_t$ lift to $M$ and are isotopic. In particular, $\tilde{\alpha_0}(\infty)$ and $\tilde{\alpha_1}(\infty)$ are in the same connected component of $G_{p,c}$. For each $q \in G_{p,c}$ there is an algebraic curve $\tilde{\alpha}: \bf R^+ \cup \{ \infty \} \to M$ with $\tilde{\alpha}(\infty) = q$ and $\tilde{\alpha}(R^+) \subset \pi^{-1}(\bf C^2)$. Let $\alpha = \pi \circ \tilde{\alpha}: \bf R^+ \to \bf C^2$. Then $\alpha$ satisfies the conditions of (\ref{def-gpc1}). If $q_0, q_1 \in G_{p,c}$, then there are two such curves $\tilde{\alpha}_0, \tilde{\alpha}_1$. If $q_0$ and $q_1$ are in the same connected component of $G_{p,c}$, then $\tilde{\alpha}_0$ is isotopic to $\tilde{\alpha}_1$ through a family of such curves $\tilde{\alpha}_t$. Hence $\alpha_0$ is isotopic to $\alpha_1$ through curves satisfying (\ref{def-gpc1}). If $q_0$ and $q_1$ are in different connected components, then $\alpha_0$ is not isotopic to $\alpha_1$ through curves satisfying (\ref{def-gpc1}). Thus $g_{p,c} = \tilde{g}_{p,c}$. \end{xproof} \begin{proposition} \label{nu-geq-g} For $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$, $$\nu_{p,c} \geq g_{p,c}$$ \end{proposition} \begin{xproof} We will show that $\nu_{p,c} \geq \tilde{g}_{p,c}$ and will use Proposition \ref{polar-curves} to compute $\nu_{p,c}$. We may assume without loss of generality that $p = [1,0,0]$. Pick a connected component $G'$ of $G_{p,c}$. Let $t$ be near $c$. We will show that $f = t$ intersects $f_y = 0$ in $\bf C^2$ near $G'$. There is a $q \in G'$ and a component $C$ of $f_y = 0 $ in $\bf C ^2$ such that $q$ is in the closure of $C$ in $M$: We have that $\tilde{f_y} = 0$ on $G'$. Blow down $G'$ to a point $q'$, and let $E'$ be the image of $\pi^{-1}(p)$. Then the lift of $f_y$ is not constant on $E'$ near $q'$, so there is a component of $f_y= 0$ passing through $q'$. Lift this component back to $M$. Next, $f$ is not constant on $C$: If it were, then the gradient vector of $f$ would be horizontal, so $C$ would be of the form $x = const$, and $p$ would not be in the closure of $C$. Thus $f = t$ intersects $C$ near $q$ for small $\epsilon \neq 0$, and the intersection points are in $\bf C^2$. \end{xproof} \begin{remark} The inequality of the proposition can be strict, as it is for the polynomial $y(x^2y-1)$ at $p = [1,0,0]$ and $c = 0$, where $\nu_{p,c} = 2$ and $g_{p,c} = 1$. \end{remark} \begin{corollary} If $p \in {\Bbb L}_\infty$ and $c \in \complex \cup \{\infinity\}$ satisfy Condition M, then they satisfy Condition G. \end{corollary} \begin{remark} The converse to this corollary is not true for $c = \infty$: For example, the polynomial $x(y^2-1)$ has a gradient whose magnitude is bounded below for large $x$ and hence satisfies Condition G at $p = [1,0,0]$, yet $\nu_{p,\infty} = 1$. \end{remark} \bibliographystyle{alpha} \newcommand{\etalchar}[1]{$^{#1}$}

9702

alg-geom/9702018

en

https://arxiv.org/abs/alg-geom/9702018

alg-geom/9702018

Furuya Masako

Masako Furuya

On $\delta_m$ constant locus of versal deformations of nondegenerate hypersurface simple K3 singularities

AMS-LaTeX v1.2, 35 pages with 5 figures

Hypersurface simple K3 singularities defined by nondegenerate quasi-homogeneous polynomials are classified into ninety five classes in term of the weight of the polynomial by T. Yonemura. We consider versal deformations of them. It has been conjectured that the stratum $\mu$ =const of the versal deformation of any nondegenerate hypersurface simple K3 singularity is equivalent to the $\delta_m$ constant locus. It holds true for the case deformations are also nondegenerate by K. Watanabe. On the other hand, it follows from Reid that the $\delta_m$ constant locus includes the $\mu$ constant locus generally. We show the conjecture holds true in general for No.10-14, 46-51 and 83 in the table of Yonemura.

alg-geom

\section*{Introduction} Simple $K3$ singularities are regarded as natural generalizations in three-dimensional case of simple elliptic singularities. The notion of a simple $K3$ singularity was defined by S. Ishii and K. Watanabe [IW] as a three-dimensional Gorenstein purely elliptic singularity of (0,2)-type, whereas a simple elliptic singularity is a two-dimensional purely elliptic singularity of (0,1)-type. It is also pointed out in [IW] that a simple $K3$ singularity is characterized as a quasi-Gorenstein singularity such that the exceptional set of any minimal resolution is a normal $K3$ surface. Let $ f \in {\bold C}[x,y,z,w] $ be a polynomial which is nondegenerate with respect to its Newton boundary $\Gamma(f)$ in the sense of [V1], and whose zero locus $ X=\{f=0\} $ in ${\bold C}^4$ has an isolated singularity at the origin $ 0 \in {\bold C}^4 $. Then the condition for $(X,0)$ to be a simple $K3$ singularity is given by a property of the Newton boundary $\Gamma(f)$ of $f$ (cf. Theorem 1.6). Hypersurface simple $K3$ singularities defined by nondegenerate quasi-homogeneous polynomials are classified into ninety five classes in term of the weight of the polynomial by Yonemura [Yo]. We consider versal deformations of them. It has been conjectured that the stratum $\mu$ =const of the versal deformation of any nondegenerate hypersurface simple $K3$ singularity is equivalent to the $\delta_m$ constant locus by Ishii. It holds true for the case deformations are also nondegenerate by 1.7 (1) [W]. On the other hand, it follows from 2.2 ([R1], [R2]) that the $\delta_m$ constant locus includes the $\mu$ constant locus generally. We show the conjecture holds true in general for No.10-14, 46-51 and 83 in the table of [Yo]. I would like to express my sincere gratitude to Professor Shihoko Ishii for telling me about the conjecture and giving valuable advice. I also express my gratitude to \\ Professors Masataka Tomari and Kei-ichi Watanabe for their useful comments \\ concerning Theorem 2.4, and to Professor Takao Fujita for his helpful remark on \\ Section 1 and 2. I also thank Mr. Hironobu Ishihara who pointed out grammatical mistakes throughout this paper. \section{Preliminary} In this section, we recall some definitions and facts from [I1], [IW], [W] and [Yo]. First we define the plurigenera $ \delta_m, \; m \in \bold N $, for normal isolated singularities and define purely elliptic singularities. Let $(X,x)$ be a normal isolated singularity in an $n$-dimensional analytic space $X$, and $ \pi \; : \; (\tilde{X},E) \longrightarrow (X,x) $ a good resolution. In the following, we assume that $X$ is a sufficiently small Stein neighbourhood of $x$. \begin{defn}[Watanabe {\rm [W]-Def. 1.2}] Let $(X,x)$ be a normal isolated singularity. For any positive integer $m$, $$ \delta_m(X,x):=\dim_{\bold C}\Gamma(X-\{x\},{\cal O}(mK))/L^{2/m}(X-\{x\}) , $$ where $K$ is the canonical line bundle on $ X-\{x\} $, and $ L^{2/m}(X-\{x\}) $ is the set of all $L^{2/m}$-integrable (at $x$) holomorphic $m$-ple $n$-forms on $ X-\{x\} $. \end{defn} Then $ \delta_m $ is finite and does not depend on the choice of a Stein neighbourhood $X$. \begin{defn}[Watanabe {\rm [W]-Def. 3.1}] A singularity $(X,x)$ is said to be purely elliptic if $ \delta_m(X,x)=1 $ for every $ m \in \bold N $. \end{defn} In the following, we assume that $(X,x)$ is quasi-Gorenstein, i.e., there exists a non-vanishing holomorphic 3-form on $ X-\{x\} $. Let $ E=\bigcup E_i $ be the decomposition of the exceptional set $E$ into irreducible components, and write $ K_{\tilde{X}} = \pi^{\ast}K_X+\sum_{i \in I}m_i E_i-\sum_{j \in J}m_j E_j $ with $ m_i \geq 0, \; m_j > 0 $. Ishii [I1] defined the essential part of the exceptional set $E$ as $ E_J = \sum_{j \in J}m_j E_j $, and showed that if $(X,x)$ is purely elliptic, then $ m_j=1 $ for all $ j \in J $. \begin{defn}[Ishii {\rm [I1]-Def. 4.1}] A quasi-Gorenstein purely elliptic singularity $(X,x)$ is of $(0,i)$-type if $ H^{n-1}(E_J, {\cal O}_{E_J}) $ consists of the $(0,i)$-Hodge component $ H_{n-1}^{0,i}(E_J) $, where $$ {\bold C} \cong H^{n-1}(E_J, {\cal O}_{E_J}) = Gr_F^0 H^{n-1}(E_J) = \bigoplus_{i=0}^{n-1} H_{n-1}^{0,i}(E_J). $$ \end{defn} \begin{defn-prop}[Ishii-Watanabe {\rm [IW]-Def. 4}] A three-dimensional \\ singularity $(X,x)$ is a simple $K3$ singularity if the following two equivalent conditions are satisfied\rom: \rom{(1)} \ \ $(X,x)$ is Gorenstein purely elliptic of (0,2)-type. \rom{(2)} \ \ The exceptional divisor $E$ is a normal $K3$ surface for any minimal resolution $ \pi : (\tilde{X},E) \longrightarrow (X,x) $. \end{defn-prop} \begin{rec} A minimal resolution $ \pi : (\tilde{X},E) \longrightarrow (X,x) $ is a proper morphism with $ \tilde{X}-E \cong X-\{x\} $, where $\tilde{X}$ has only terminal $ \bold Q $-factorial singularities and $K_{\tilde{X}}$ is numerically effective with respect to $\pi$. \end{rec} Next we consider the case where $(X,x)$ is a hypersurface singularity defined by a nondegenerate polynomial $f= \sum a_{\nu} z^{\nu} \in {\bold C}[z_0,z_1,\dotsb,z_n]$, and $x=0 \in {\bold C}^{n+1}$. \begin{rec} The Newton boundary $\Gamma(f)$ of $f$ is the union of the compact faces of $\Gamma_+(f)$, where $\Gamma_+(f)$ is the convex hull of $ \bigcup_{a_{\nu} \ne 0}(\nu+{\bold R}_{\geq 0}^{n+1}) $ in ${\bold R}^{n+1}$. For any face $\Delta$ of $\Gamma_+(f)$, set $ f_{\Delta}:=\sum_{\nu \in \Delta}a_{\nu}z^{\nu} $. We say $f$ to be nondegenerate, if $$ \displaystyle \frac{\partial f_{\Delta}}{\partial z_0} = \frac{\partial f_{\Delta}}{\partial z_1} = \dotsb = \frac{\partial f_{\Delta}}{\partial z_n} = 0 $$ has no solution in $ ({\bold C}-\{0\})^{n+1} $ for any face $\Delta$. \end{rec} When $f$ is nondegenerate, the condition for $(X,x)$ to be a purely elliptic singularity is given as follows: \begin{thm}[Watanabe {\rm [W]-Prop. 2.9, Cor. 3.14}] Let $f$ be a nondegenerate \\ polynomial and suppose $X=\{f=0\}$ has an isolated singularity at $x=0 \in {\bold C}^{n+1}$. \rom{(1)} $(X,x)$ is purely elliptic if and only if $(1,1,\dotsb,1) \in \Gamma(f)$. \rom{(2)} Let n=3 and let $\Delta_0$ be the face of $\Gamma (f)$ containing $(1,1,1,1)$ in the relative interior of $\Delta_0$. Then $(X,x)$ is a simple $K3$ singularity if and only if $\dim_{\bold R} \Delta_0=3$. \end{thm} Thus if $f$ is nondegenerate and defines a simple $K3$ singularity, then $f_{\Delta_0}:=\sum_{\nu \in \Delta_0} a_{\nu} z^{\nu}$ is a quasi-homogeneous polynomial of a uniquely determined weight $\alpha$ called the weight of $f$ and denoted $\alpha(f)$. Namely, $\alpha =(\alpha_1,\dotsb,\alpha_4) \in {\bold Q_{>0}}^4$ and $\deg_{\alpha}(\nu):=\sum_{i=1}^4 \alpha_i \nu_i =1$ for any $\nu \in \Delta_0$. In particular, $\sum_{i=1}^4 \alpha_i=1$, since $(1,1,1,1)$ is always contained in $\Delta_0$. \begin{thm}[Yonemura {\rm [Yo]-Prop. 2.1}] The cardinality of \\ $ \{ \alpha (f) \; ; f \text{ is nondegenerate and defines a simple } K3 \text{ singularity}, \; \alpha_1 \geq \alpha_2 \geq \alpha_3 \geq \alpha_4 \} $ \\ is 95. \end{thm} In Table 2.2 of [Yo] can be found the complete list of weights $\alpha=\alpha(f)$ and examples of $f=\sum \alpha_{\nu} z^{\nu}$ such that $f$ is quasi-homogeneous and that $\{f=0\} \subset {\bold C}^4$ has a simple $K3$ singularity at the origin $0 \in {\bold C}^4$. We describe a weight $\alpha=\alpha(f)$ as $\alpha=(p_1/p,p_2/p,p_3/p, p_4/p)$, where $p, \; p_i$ are \\ positive integers with $ {\bold {gcd}}(p_1,p_2,p_3,p_4)=1.$ \\ Next we consider the versal deformation of simple $K3$ singularity $(X=\{f=0\}, \; 0)$ such that $f$ is a nondegenerate quasi-homogeneous polynomial. \begin{defn} A deformation of an isolated singularity $(X,x)$ is a flat family of singularities: $$ {\frak X} =\{(X_t,x_t) \; ; \; t \in U \subset {\bold C}^N \} \overset{flat}{\longrightarrow} U $$ such that $(X_0,x_0) \cong (X,x)$ as germs of holomorphic functions. \\ Where $U$ is a sufficiently small open neighbourhood of 0 in $ {\bold C}^N $. \end{defn} \begin{defn} A deformation $\frak X$ of an isolated singularity $(X,x)$ is versal if for any deformation ${\frak X'} \longrightarrow U'$ of $(X,x)$, the following is satisfied: \begin{equation*} \begin{CD} {\frak X}{\times}_U U' \overset{isom}{\cong} {\frak X}' @>>> U' \\ @. @VV{\exists \; holomorphic}V \\ {\hspace{2.1cm} \frak X} @>>> U \\ \end{CD} \end{equation*} \end{defn} \begin{thm}[{\rm [KS], [Tj]}] The versal deformation $\frak X$ of an isolated singularity \\ $(\{f=0\}, \; 0)$ is described by $$ {\frak X} =\{(\{f+\sum \lambda_i g_i=0\}, \; 0) \; ; \; (\lambda_i) \in U \subset {\bold C}^N \}, $$ where the $g_i$ determine a $\bold C$-basis of the vector space ${\bold C}\{z_0,\dotsb,z_n\}/(f, f_{z_0}, \dotsb, f_{z_n})$. \end{thm} The purpose of this paper is to show that the following conjecture holds true for No.10-14, 46-51 and 83 in Table 2.2 of [Yo]. \begin{prob}[Ishii] Let $(X=\{f=0\}, \; 0)$ be a hypersurface simple $K3$ singularity defined by a nondegenerate quasi-homogeneous polynomial $f$, and let $\frak X$ be the versal deformation of $(X,0)$. Then, \\ \hspace*{1.5cm} $ \{ \lambda \in U \subset {\bold C}^N \; ; \; \mu(X,0)=\mu(X_{\lambda} ,0), \; (X_{\lambda} ,0) \in {\frak X} \} $ \\ \hspace*{1cm} $ = \{ \lambda \in U \subset {\bold C}^N \; ; \; 1=\delta_m (X,0)=\delta_m (X_{\lambda} ,0) \; \text{for all } m \geq 1, \; (X_{\lambda} ,0) \in {\frak X} \}. \qquad \square $ \end{prob} Since $(X,0)$ and $(X_{\lambda} ,0)$ are hypersurface isolated singularities, they are normal Gorenstein, and so $$ P_g=0 \; \Longleftrightarrow \; \delta_m=0 \text{\; for all \ } m \geq 1.$$ On the other hand, $(X_{\lambda} ,0)$ is a deformation of a purely elliptic singularity $(X,0)$, so it is either rational or purely elliptic. Therefore, this problem is equivalent to: $$ \mu(X,0)=\mu(X_{\lambda} ,0) \quad \Longleftrightarrow \quad 1=P_g(X,0)=P_g(X_{\lambda} ,0). $$ Furthermore, since $\mu$ and $P_g$ are upper semi-continuous in respect of deformation \\([M], [Te], [E]-Thm. 1, [Ya]-Thm. 2.6), it is equivalent to: $$ \mu (X,0) > \mu (X_{\lambda} ,0) \quad \Longleftrightarrow \quad 1 = P_g (X,0) > P_g (X_{\lambda} ,0) = 0. $$ \section{Reduction of the problem} Considering some facts, we can reduce the problem posed in section 1 to one about the weight of the defining polynomial of a hypersurface singularity. \begin{thm}[Varchenko {\rm [V2]-Thm. 2}] Let $f \in {\bold C}[z_0,\dotsb,z_n]$ be quasi-homogeneous of weight $ \alpha $ with $ \alpha_0+\dotsb+\alpha_n=1 $, and $\{f=0\}$ has an isolated singularity at $0 \in {\bold C}^{n+1}$. Let $$ \mu := \mu (f,0)<+\infty, \qquad f_{\lambda} := f+\sum_{i=1}^{\mu} \lambda_i g_i, $$ where $g_i \in {\bold C}\{z_0,\dotsb,z_n\}/(f_{z_0},\dotsb,f_{z_n})$ are generators of the Jacobi ring, which are monomials. \rom(Since $f$ is quasi-homogeneous, $g_i$ can be taken as monomials.\rom) Then, \\ \hspace*{1.5cm} $ \{ \lambda =(\lambda_1,\dotsb,\lambda_{\mu} ) \in U \subset {\bold C}^{\mu} \; ; \; \mu (f,0)=\mu (f_{\lambda},0)\} $ \\ \hspace*{1cm} $ = \{ \lambda =(\lambda_1,\dotsb,\lambda_{\mu} ) \in U \subset {\bold C}^{\mu} \; ; \; \lambda_i=0 \text{\ for all i satisfying that } \deg_{\alpha} (g_i)<1 \}, $ \\ where $ \deg_{\alpha}(z_0^{i_0} \cdot \dotsb \cdot z_n^{i_n}) :=\alpha_0 i_0+ \dotsb +\alpha_n i_n. \qquad \square $ \end{thm} From 1.7 (1) and 2.1, it follows that $$ \mu(f,0)= \mu(f_{\lambda},0) \quad \Longleftrightarrow \quad P_g(f,0)=P_g(f_{\lambda},0) $$ holds true if $f_{\lambda}$ is also nondegenerate. Though $f_{\lambda}$ is not always nondegenerate, the \\ following theorem is useful as well. \begin{thm}[Reid {\rm [R1]-Thm. 4.1, [R2]-Thm. 4.6}] Let $(X=\{f=0\}, \; 0) \subset ({\bold C}^{n+1},0)$ be a hypersurface singularity and let $\alpha=(\alpha_0,\dotsb,\alpha_n)=(p_0/p,\dotsb,p_n/p) \in {{\bold Q}_{>0}}^{n+1}$ such that $(p_0,\dotsb,p_n) \in {\bold N}^{n+1}$ is a primitive vector. Then, \\ \hspace*{2cm} $ (X,0): \; \text{canonical} \quad \Longrightarrow \quad \deg_{\alpha}(z_0\cdot\dotsb\cdot z_n)>\deg_{\alpha}(f), $ \\ where $ \deg_{\alpha}(f):=\min\{\deg_{\alpha}(z^{\nu}) \; ; \; z^{\nu} \in f \}. \qquad \square $ \end{thm} A hypersurface singularity is canonical if and only if it is rational, and so it follows that $$ \mu(f,0)=\mu(f_{\lambda},0) \Longrightarrow P_g(f,0)=P_g(f_{\lambda},0) $$ holds always true from 2.2. Hence we should show the converse proposition. \begin{rem} Let $(X,x)$ be a n-dimensional normal Gorenstein singularity and $$ (\Tilde{\Tilde X},\tilde{E}) \overset{\pi'}{\longrightarrow} (\tilde{X},E) \overset{\pi}{\longrightarrow} (X,x) , $$ where $\tilde{X}$ has at most rational singularities and $\pi'':=\pi \circ \pi'$ is a resolution. Let $ E=\bigcup_{i \in I}E_i $ be the decomposition of the exceptional set $E$ into irreducible components, and write $ K_{\tilde{X}}=\pi^{\ast} K_X+\sum_{i \in I} m_i E_i $. Then, it follows that $ \pi'_{\ast}({\cal O}(K_{\Tilde{\Tilde X}}))={\cal O}(K_{\tilde{X}}) $ since $\tilde{X}$ has at most rational singularities ([KKMS]-p. 50), and so {\allowdisplaybreaks \begin{align*} P_g(X,x) & :=\dim_{\bold C} (R^{n-1}{\pi''}_{\ast} {\cal O}_{\Tilde{\Tilde X}})_x \\ & =\dim_{\bold C}\Gamma(X-\{x\},{\cal O} (K_X))/L^2(X-\{x\}) \\ & =\dim_{\bold C}\Gamma(\Tilde{\Tilde X}-\tilde{E}, {\cal O} (K_{\Tilde{\Tilde X}}))/ \Gamma(\Tilde{\Tilde X}, {\cal O} (K_{\Tilde{\Tilde X}})) \\ & =\dim_{\bold C} \Gamma(\Tilde{\Tilde X}, {\cal O} ({\pi''}^{\ast} K_X))/ \Gamma(\Tilde{\Tilde X}, {\cal O} (K_{\Tilde{\Tilde X}})) \\ & =\dim_{\bold C}\Gamma(\tilde{X}, {\cal O} (\pi^{\ast} K_X))/ \Gamma(\tilde{X}, \pi'_{\ast}({\cal O}(K_{\Tilde{\Tilde X}}))) \\ & =\dim_{\bold C}\Gamma(\tilde{X}, {\cal O} (\pi^{\ast} K_X))/ \Gamma(\tilde{X}, {\cal O} (K_{\tilde{X}})). \end{align*}} Therefore, $$ \exists \; i \in I \; ; \; m_i<0 \; \Longleftrightarrow \; \Gamma(\tilde{X},{\cal O} (\pi^{\ast}K_X)) \supsetneq \Gamma(\tilde{X},{\cal O} (K_{\tilde{X}})) \; \Longleftrightarrow \; P_g(X,x) > 0, $$ namely, $$ \forall \; i \in I , \; m_i \geq 0 \; \Longleftrightarrow \; P_g(X,x)=0. $$ \end{rem} \begin{thm}[Tomari-Watanabe {\rm [TW]-Thm. 5.6}] Let $(X=\{f=0\}, \; x)$ be a \\ n-dimensional hypersurface isolated singularity and \\ \hspace*{1cm} $ f=f_0+f_1+f_2+\dotsb, $ \\ \hspace*{2cm} $ f_i : $ quasi-homogeneous polynomial of weight \; $ \alpha=(\alpha_0,\dotsb,\alpha_n) \; ; $ \\ \hspace*{2cm} $ 1=\deg_{\alpha}(f_0)<\deg_{\alpha}(f_1)<\deg_{\alpha}(f_2) <\dotsb, $ \\ and \\ \begin{figure}[h] \setlength{\unitlength}{1mm} \begin{picture}(117,41)(-35,0) \put(0,0){\makebox(45,7){$ E={\pi}^{-1}(x) $}} \put(45,0){\makebox(25,7){$ \longrightarrow $}} \put(70,0){\makebox(12,7){$ x $.}} \put(0,7){\makebox(45,9){$ \bigcup $}} \put(70,7){\makebox(12,9){$ \cup $}} \put(76,10.5){\line(0,1){2.5}} \put(0,16){\makebox(45,7) {$ \tilde{X}=\overline{{\Pi}^{-1}(X)-{\Pi}^{-1}(x)} $}} \put(45,16){\makebox(25,9) {$ @>{\pi=\Pi {\vert}_{\tilde{X}}}>> $}} \put(70,16){\makebox(12,7){$ X $}} \put(0,23){\makebox(45,9){$ \bigcup $}} \put(70,23){\makebox(12,9){$ \bigcup $}} \put(0,32){\makebox(45,7){$ V $}} \put(45,32){\makebox(25,9) {$ @>{\Pi \; : \; \alpha-blow-up}>> $}} \put(70,32){\makebox(12,7){$ {\bold C}^{n+1} $}} \end{picture} \end{figure} \\ Assume that $f_0$ is irreducible and both $X-\{x\}$ and $\{f_0=0\}-\{x\}$ have at most rational singularities around $x$. Then $\tilde{X}$ has at most rational singularities. \qquad $\square$ \end{thm} Thus we expect a partial resolution $\pi$ in 2.3 is given by a weighted blow-up. \begin{rem}[Ishii {\rm [I2]-Prop. 1.3, 1.6}] Under the notation in 2.4, $ \Pi^{\ast} X \subset V $ and $ K_{{\bold C}^4} $ are principal divisors, hence, {\allowdisplaybreaks \begin{align*} \Pi^{\ast} X & = \tilde{X} + p F, \\ K_V & = \Pi^{\ast} K_{{\bold C}^4} + (p_1+p_2+p_3+p_4-1)F, \\ K_X & = (K_{{\bold C}^4}+X)|_X, \\ \text{thus,} \qquad K_{\tilde{X}} & = (K_V+\tilde{X})|_{\tilde{X}} \\ & = (\Pi^{\ast}(K_{{\bold C}^4}+X) + (p_1+p_2+p_3+p_4-1-p) F)|_ {\tilde{X}} \\ & = \pi^{\ast}K_X+(p_1+p_2+p_3+p_4-1-p)\sum_i k_i E_i, \end{align*}} where \ $ \alpha = (\alpha_1, \; \alpha_2, \; \alpha_3, \; \alpha_4) = (p_1/p, \; p_2/p, \; p_3/p, \; p_4/p) $ with $ {\bold {gcd}}(p_1, \; p_2, \; p_3, \; p_4)=1 $,\\ \hspace*{1.3cm} $ F={\Pi}^{-1}(0), \quad F|_{\tilde{X}}=E=\sum_i k_i E_i, \quad k_i>0. $ \\ Thus, \ \ $ p<p_1+\dotsb+p_4 $ if and only if $ R^2 \pi_{\ast} {\cal O}_{\tilde{X}}=0 $. \end{rem} \begin{prob} Let $f$ be a nondegenerate quasi-homogeneous polynomial which \\ defines a simple $K3$ singularity at $0$, and $(X_{\lambda}=\{f_{\lambda}=0\}, \; 0)$ a versal deformation of \\ $(X=\{f=0\}, \; 0)$ such that $ \mu(X_{\lambda},0) < \mu(X,0) $. Then find a weight $\alpha'=(\alpha'_1,\dotsb,\alpha'_4) =(p'_1/p',\dotsb,p'_4/p')$ of $f_{\lambda}$ with $ 1<\alpha'_1+\dotsb+\alpha'_4 $ such that: \\ \hspace*{1cm} $ f_{\lambda}=f_0+f_1+f_2+\dotsb, $ \\ \hspace*{2cm} $ f_i $ : quasi-homogeneous polynomial of weight $ \alpha', $ \\ \hspace*{2cm} $ 1=\deg_{\alpha'}(f_0)<\deg_{\alpha'}(f_1)<\deg_{\alpha'}(f_2)< \dotsb, $ \\ \hspace*{2cm} $ f_0 $ is irreducible, \\ \hspace*{1cm} and $ \{f_0=0\}-\{0\} $ has at most rational singularities around $ 0 $. \end{prob} If there exists such weight $\alpha'$ then $P_g(X_{\lambda},0)=0$ by 2.4 and 2.5. We show there exists such weight $\alpha'$ as in 2.6 for $ f=x^2+y^3+\dotsb, $ of No.10-14, 46-51 and 83 in Table 2.2 of [Yo] to obtain our main result as follows: \begin{thm} Let $(X=\{f=0\}, \; 0)$ be a hypersurface simple $K3$ singularity defined by a nondegenerate quasi-homogeneous polynomial $ f=x^2+y^3+\dotsb, $ which is one of No.10-14, 46-51 and 83 in Table 2.2 of [Yo]\rom: \\ \hspace*{1.5cm} $($No. 10$)$ \ \ $ f = x^2 + y^3 + z^{12} + w^{12} $,\\ \hspace*{1.5cm} $($No. 11$)$ \ \ $ f = x^2 + y^3 + z^{10} + w^{15} $,\\ \hspace*{1.5cm} $($No. 12$)$ \ \ $ f = x^2 + y^3 + z^9 + w^{18} $,\\ \hspace*{1.5cm} $($No. 13$)$ \ \ $ f = x^2 + y^3 + z^8 + w^{24} $,\\ \hspace*{1.5cm} $($No. 14$)$ \ \ $ f = x^2 + y^3 + z^7 + w^{42} $,\\ \hspace*{1.5cm} $($No. 46$)$ \ \ $ f = x^2 + y^3 + z^{11} + z w^{12} $,\\ \hspace*{1.5cm} $($No. 47$)$ \ \ $ f = x^2 + y^3 + y z^7 + z^9 w^2+w^{14}$,\\ \hspace*{1.5cm} $($No. 48$)$ \ \ $ f = x^2 + y^3 + z^9 w + w^{16} $,\\ \hspace*{1.5cm} $($No. 49$)$ \ \ $ f = x^2 + y^3 + z^8 w + w^{21} $,\\ \hspace*{1.5cm} $($No. 50$)$ \ \ $ f = x^2 + y^3 + y z^5 + z^7 w^2+w^{30}$,\\ \hspace*{1.5cm} $($No. 51$)$ \ \ $ f = x^2 + y^3 + z^7 w + w^{36} $,\\ \hspace*{1.5cm} $($No. 83$)$ \ \ $ f = x^2 + y^3 + y w^9+z^{10} w+z^2w^{11}$,\\ and let $\frak X$ be the versal deformation of $(X,0)$. Then, \\ \hspace*{1.5cm} $ \{ \lambda \in U \subset {\bold C}^N \; ; \; \mu(X,0)=\mu(X_{\lambda} ,0), \; (X_{\lambda} ,0) \in {\frak X} \} $ \\ \hspace*{1cm} $ = \{ \lambda \in U \subset {\bold C}^N \; ; \; 1=\delta_m (X,0)=\delta_m (X_{\lambda} ,0) \; \text{for all } m \geq 1, \; (X_{\lambda} ,0) \in {\frak X} \}. $ \end{thm} \section{Proof of Theorem 2.7} We prepare the following Lemma prior to the proof of Theorem 2.7. \begin{lem} Let $f$ be a nondegenerate quasi-homogeneous polynomial listed by \\ Yonemura [Yo] which defines a simple $K3$ singularity, and $f_{\lambda}$ a versal deformation of $f$ such that $ \mu(f_{\lambda},0) < \mu(f,0) $. Under this original local coordinate system, if $ \alpha'=(\alpha'_1,\dotsb,\alpha'_4) \in {{\bold Q}_{>0}}^4 $ satisfies $ \deg_{\alpha'}(f_{\lambda})=1 $ then $ 1<\alpha'_1+\dotsb+\alpha'_4. $ \end{lem} \begin{pf} For any i=1,2,3,4, one of the following is satisfied: (a) \ $ p_i | p $, (b) \ $ p_i | (p-p_j) $ for some $ j \neq i $,\\ namely, (a) \ $ z_i^I \in f \qquad (\exists I \geq 2) $, (b) \ $ z_i^I z_j \in f \qquad (\exists I \geq 2) $ for some $ j \neq i $.\\ By assumptions, $ p \alpha_1+q \alpha_2+r \alpha_3+s \alpha_4<1 $ for some $ {z_1}^p {z_2}^q {z_3}^r {z_4}^s \in f_{\lambda} $,\\ \hspace*{2.2cm} and $ p \alpha'_1+q \alpha'_2+r \alpha'_3+s \alpha_4 \geq 1 $ for all $ {z_1}^p {z_2}^q {z_3}^r {z_4}^s \in f_{\lambda} $,\\ so there exists $ i \in \{1,2,3,4\} $ such that $ \alpha_i<\alpha'_i \qquad \dotsb \circledast.$ \\ So the defining polynomials $f$ can be classified as below.\\ Case $\bold 1.$ \ \ $ z_i^I \in f \quad (\exists I \geq 2) $.\\ \hspace*{1cm} Then $ I \alpha_i=1 \leq I \alpha'_i $, so we have $ \alpha_i \leq \alpha'_i. $ \\ \hspace*{0.2cm} ($\bold 1$-I) \ \ $ z_i^I, \; z_j^J \in f \quad (\exists I, \; \exists J \geq 2) $.\\ \hspace*{1.2cm} Since $ J \alpha_j=1 \leq J \alpha'_j $, we have $ \alpha_j \leq \alpha'_j. $ \\ \hspace*{0.4cm} ($\bold 1$-I-i) \ \ $ z_i^I, \; z_j^J, \; z_k^K \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} From $ K \alpha_k=1 \leq K \alpha'_k $, we have $ \alpha_k \leq \alpha'_k. $ \\ \hspace*{0.6cm} ($\bold 1$-I-i-a) (No.1-14) \ \ $ z_i^I, \; z_j^J, \; z_k^K, \; z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} From $ L \alpha_l=1 \leq L \alpha'_l $, we have $ \alpha_l \leq \alpha'_l $, so $ \alpha_1+\dotsb+\alpha_4 <\alpha'_1+\dotsb+\alpha'_4 $ by $\circledast$. \\ \hspace*{0.6cm} ($\bold 1$-I-i-b) (No.15-51) \ \ $ z_i^I, \; z_j^J, \; z_k^K, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} From $ \alpha_k+L \alpha_l=1 \leq \alpha'_k+L \alpha'_l $, we have $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l. $\\ \hspace*{1.6cm} If $ \alpha_l \leq \alpha'_l $ then $ \alpha_1+\dotsb+\alpha_4 <\alpha'_1+\dotsb+\alpha'_4 $ by $\circledast$. \\ \hspace*{1.6cm} If $ \alpha_l>\alpha'_l $ then \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ \hspace*{0.4cm} ($\bold 1$-I-ii) \ \ $ z_i^I, \; z_j^J, \; z_k^K z_l \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2)$.\\ \hspace*{1.4cm} $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_k+\alpha_l+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{0.6cm} ($\bold 1$-I-ii-a) (No.78) \ \ $ z_i^I, \; z_j^J, \; z_k^K z_l, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_k \leq \alpha'_k $ and $ \alpha_l \leq \alpha'_l $ then $ \alpha_1+\dotsb+\alpha_4 < \alpha'_1+\dotsb+\alpha'_4 $ by $\circledast$. \\ \hspace*{1.6cm} If $ \alpha_k \leq \alpha'_k $ and $ \alpha_l > \alpha'_l $ then \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ \hspace*{1.6cm} If $ \alpha_k > \alpha'_k $ and $ \alpha_l \leq \alpha'_l $ then \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ \hspace*{0.6cm} ($\bold 1$-I-ii-b) (No.52, 54-74, 76, 77, 79-83) \\ \hspace*{1.6cm} $ z_i^I, \; z_j^J, \; z_k^K z_l, \; z_j z_l^L \in f $ \quad $(\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_j+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_j+\alpha'_l $. \ \ The rest is similar to $\bold 1$-I-ii-a .\\ \hspace*{0.4cm} ($\bold 1$-I-iii) \ \ $ z_i^I, \; z_j^J, \; z_j z_k^K \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2)$.\\ \hspace*{1.4cm} Similarly, $ \alpha_j+\alpha_k+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_j+\alpha'_k $.\\ \hspace*{0.6cm} ($\bold 1$-I-iii-a) (No.66, 67, 72, 75, 81, 82) \\ \hspace*{1.6cm} $ z_i^I, \; z_j^J, \; z_j z_k^K, \; z_j z_l^L \in f $ \quad $(\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} Since there exists $ z_k^r z_l^s \in f $, we have $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} Moreover $ \alpha_j+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_j+\alpha'_l $ by $ z_j z_l^L \in f $.\\ \hspace*{1.6cm} So we have $ \alpha_1+\dotsb+\alpha_4 < \alpha'_1+\dotsb+\alpha'_4 $ similarly as in $\bold 1$-I-ii-a.\\ \hspace*{0.6cm} ($\bold 1$-I-iii-b) (No.53, 57, 58, 62-64, 66, 67, 69-72) \\ \hspace*{1.6cm} $ z_i^I, \; z_j^J, \; z_j z_k^K, \; z_i z_l^L \in f $ \quad $(\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $ then the rest is similar to $\bold 1$-I-ii-a. \\ \hspace*{1.6cm} If $ \alpha_k > \alpha'_k $ and $ \alpha_l > \alpha'_l $ then\\ \hspace*{1.6cm} $ \alpha_i+\alpha_j+\alpha_k+\alpha_l < \alpha_i+\alpha_j+\alpha_k+\alpha_l+(K-1)(\alpha_k-\alpha'_k) +(L-1)(\alpha_l-\alpha'_l) $ \\ \hspace*{4.8cm} $ \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l .$ \\ \hspace*{0.2cm} ($\bold 1$-II) \ \ $ z_i^I, \; z_j^J z_k \in f \quad (\exists I, \; \exists J \geq 2) $.\\ \hspace*{1.2cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_k \leq \alpha'_k $, \ and $ \alpha_j+\alpha_k+(J-1)(\alpha_j-\alpha'_j) \leq \alpha'_j+\alpha'_k $.\\ \hspace*{0.4cm} ($\bold 1$-II-i) \ \ $ z_i^I, \; z_j^J z_k, \; z_k^K z_l \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_k+\alpha_l+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_k+\alpha'_l.$\\ \hspace*{0.6cm} ($\bold 1$-II-i-a) (No.88, 90-93) \ \ $ z_i^I, \; z_j^J z_k, \; z_k^K z_l, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} Since there exists $ z_j^q z_l^s \in f $, we have $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} Moreover $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l $ by $ z_k z_l^L \in f $.\\ \hspace*{1.6cm} If $ \alpha_j \leq \alpha'_j $, $ \alpha_k \leq \alpha'_k $ and $ \alpha_l \leq \alpha'_l $, then the assertion holds true. \\ \hspace*{1.6cm} If $ \alpha_j \leq \alpha'_j $, $ \alpha_k \leq \alpha'_k $ and $ \alpha_l > \alpha'_l $, then \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ \hspace*{1.6cm} The assertion holds true similarly for both the case of $ \alpha_j \leq \alpha'_j $, $ \alpha_k > \alpha'_k $, \\ \hspace*{1.6cm} $ \alpha_l \leq \alpha'_l $, and the case of $ \alpha_j > \alpha'_j $, $ \alpha_k \leq \alpha'_k $, $ \alpha_l \leq \alpha'_l $. \\ \hspace*{0.6cm} ($\bold 1$-II-i-b) (No.86-88, 92) \ \ $ z_i^I, \; z_j^J z_k, \; z_k^K z_l, \; z_j z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_j+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_j+\alpha'_l.$\\ \hspace*{1.6cm} The rest is similar to $\bold 1$-II-i-a. \\ \hspace*{0.6cm} ($\bold 1$-II-i-c) (No.84-89) \ \ $ z_i^I, \; z_j^J z_k, \; z_k^K z_l, \; z_i z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l.$ \\ \hspace*{1.6cm} When $ \alpha_k \leq \alpha'_k $, the rest is similar to $\bold 1$-I-iii-b. \\ \hspace*{1.6cm} If $ \alpha_k > \alpha'_k $ then $ \alpha_j \leq \alpha'_j $ and $ \alpha_l \leq \alpha'_l $, so \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ \hspace*{0.4cm} ($\bold 1$-II-ii) \ \ $ z_i^I, \; z_j^J z_k, \; z_j z_k^K \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} $ \alpha_j+\alpha_k+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_j+\alpha'_k.$\\ \hspace*{0.6cm} ($\bold 1$-II-ii-a) (No.89) \ \ $ z_i^I, \; z_j^J z_k, \; z_j z_k^K, \; z_i z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l.$\\ \hspace*{1.6cm} For the case $ \alpha_l \leq \alpha'_l $, the assertion holds true. \\ \hspace*{1.6cm} If $ \alpha_l > \alpha'_l $ then \\ \hspace*{1.6cm} $ \alpha_i+\alpha_j+\alpha_k+\alpha_l < \alpha_i+\alpha_j+\alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) $ \\ \hspace*{1.6cm} $ < \begin{cases} & \alpha_i+\alpha_j+\alpha_k+\alpha_l +(J-1)(\alpha_j-\alpha'_j)+(L-1)(\alpha_l-\alpha'_l) \quad ( \text{if} \; \alpha_j>\alpha'_j ) \\ & \alpha_i+\alpha_j+\alpha_k+\alpha_l +(K-1)(\alpha_k-\alpha'_k)+(L-1)(\alpha_l-\alpha'_l) \quad ( \text{if} \; \alpha_k>\alpha'_k ) \end{cases} $ \\ \hspace*{1.6cm} $ \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l $.\\ \hspace*{0.4cm} ($\bold 1$-II-iii) \ \ $ z_i^I, \; z_j^J z_k, \; z_i z_k^K \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2)$.\\ \hspace*{1.4cm} $ \alpha_i+\alpha_k+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_i+\alpha'_k.$\\ \hspace*{0.6cm} ($\bold 1$-II-iii-a) (No.89) \ \ $ z_i^I, \; z_j^J z_k, \; z_i z_k^K, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{1.6cm} Moreover $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $ since there exists $ z_j^q z_l^s \in f $.\\ \hspace*{1.6cm} So $ \alpha_1+\dotsb+\alpha_4 < \alpha'_1+\dotsb+\alpha'_4 $ similarly as $\bold 1$-II-i-a.\\ \hspace*{0.6cm} ($\bold 1$-II-iii-b) (No.85, 87, 89) \ \ $ z_i^I, \; z_j^J z_k, \; z_i z_k^K, \; z_i z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} Since there exists $ z_k^r z_l^s \in f $, we have $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} Moreover $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l $ by $ z_i z_l^L \in f $.\\ \hspace*{1.6cm} When $ \alpha_k \leq \alpha'_k $, the rest is similar to $\bold 1$-I-iii-b. \\ \hspace*{1.6cm} If $ \alpha_k > \alpha'_k $ then $ \alpha_j \leq \alpha'_j $ and $ \alpha_l \leq \alpha'_l $, so \\ \hspace*{1.6cm} $ \alpha _i+\alpha _j+\alpha _k+\alpha _l < \alpha _i+\alpha _j+\alpha _k+\alpha _l +(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l . $ \\ Case $\bold 2.$ \ \ $ z_i^I z_j \in f \quad (\exists I \geq 2) $.\\ \hspace*{1cm} $ \alpha_i \leq \alpha'_i $ or $ \alpha_j \leq \alpha'_j $, \ and $ \alpha_i+\alpha_j+(I-1)(\alpha_i-\alpha'_i) \leq \alpha'_i+\alpha'_j $.\\ \hspace*{0.2cm} ($\bold 2$-I) \ \ $ z_i^I z_j, \; z_j^J z_k \in f \quad (\exists I, \; \exists J \geq 2) $.\\ \hspace*{1.2cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_k \leq \alpha'_k $, \ and $ \alpha_j+\alpha_k+(J-1)(\alpha_j-\alpha'_j) \leq \alpha'_j+\alpha'_k $.\\ \hspace*{0.4cm} ($\bold 2$-I-i) \ \ $ z_i^I z_j, \; z_j^J z_k, \; z_k^K z_l \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_k+\alpha_l+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{0.6cm} ($\bold 2$-I-i-a) (No.94, 95) \ \ $ z_i^I z_j, \; z_j^J z_k, \; z_k^K z_l, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{1.6cm} Since there exists $ z_j^q z_l^s \in f $, we have $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_i \leq \alpha'_i $ then the rest is similar to $\bold 1$-II-i-a. \\ \hspace*{1.6cm} If $ \alpha_i > \alpha'_i $ then $ \alpha_j \leq \alpha'_j $ and so \\ \hspace*{1.6cm} $ \alpha_i+\alpha_j+\alpha_k+\alpha_l < \alpha_i+\alpha_j+\alpha_k+\alpha_l+(I-1)(\alpha_i-\alpha'_i) $ \\ \hspace*{1.6cm} $ < \begin{cases} & \alpha_i+\alpha_j+\alpha_k+\alpha_l +(I-1)(\alpha_i-\alpha'_i)+(K-1)(\alpha_k-\alpha'_k) \quad ( \text{if} \; \alpha_k>\alpha'_k ) \\ & \alpha_i+\alpha_j+\alpha_k+\alpha_l +(I-1)(\alpha_i-\alpha'_i)+(L-1)(\alpha_l-\alpha'_l) \quad ( \text{if} \; \alpha_l>\alpha'_l ) \end{cases} $ \\ \hspace*{1.6cm} $ \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l.$ \\ \hspace*{0.6cm} ($\bold 2$-I-i-b) (No.94, 95) \ \ $ z_i^I z_j, \; z_j^J z_k, \; z_k^K z_l, \; z_j z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_j+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_j+\alpha'_l $.\\ \hspace*{1.6cm} Since there exists $ z_i^p z_l^s \in f $, we have $ \alpha_i \leq \alpha'_i $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_i \leq \alpha'_i $ then the rest is similar to $\bold 1$-II-i-a. \\ \hspace*{1.6cm} If $ \alpha_i > \alpha'_i $ then $ \alpha_j \leq \alpha'_j $ and $ \alpha_l \leq \alpha'_l $, \ and so \\ \hspace*{1.6cm} $ \alpha_i+\alpha_j+\alpha_k+\alpha_l < \alpha_i+\alpha_j+\alpha_k+\alpha_l+(I-1)(\alpha_i-\alpha'_i) +(K-1)(\alpha_k-\alpha'_k) $ \\ \hspace*{4.8cm} $ \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l $ \qquad for the case of $ \alpha_k > \alpha'_k $.\\ \hspace*{0.6cm} ($\bold 2$-I-i-c) (No.94, 95) \ \ $ z_i^I z_j, \; z_j^J z_k, \; z_k^K z_l, \; z_i z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_i \leq \alpha'_i $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_i \leq \alpha'_i $ and $ \alpha_j \leq \alpha'_j $ then the assertion holds true. \\ \hspace*{1.6cm} If $ \alpha_i \leq \alpha'_i $ and $ \alpha_j > \alpha'_j $ then $ \alpha_k \leq \alpha'_k $, so the rest is similar to $\bold 1$-I-iii-b. \\ \hspace*{1.6cm} If $ \alpha_i > \alpha'_i $ and $ \alpha_j \leq \alpha'_j $ then $ \alpha_l \leq \alpha'_l $, so the rest is similar to $\bold 1$-I-iii-b. \\ \hspace*{0.2cm} ($\bold 2$-II) \ \ $ z_i^I z_j, \; z_i z_j^J \in f \quad (\exists I, \; \exists J \geq 2) $.\\ \hspace*{1.2cm} $ \alpha_i+\alpha_j+(J-1)(\alpha_j-\alpha'_j) \leq \alpha'_i+\alpha'_j $.\\ \hspace*{0.4cm} ($\bold 2$-II-i) \ \ $ z_i^I z_j, \; z_i z_j^J, \; z_k^K z_l \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} $ \alpha_k \leq \alpha'_k $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_k+\alpha_l+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{0.6cm} ($\bold 2$-II-i-a) \ \ $ z_i^I z_j, \; z_i z_j^J, \; z_k^K z_l, \; z_k z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2)$.\\ \hspace*{1.6cm} $ \alpha_k+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_k+\alpha'_l $.\\ \hspace*{1.6cm} $ \alpha_i+\alpha_j+\alpha_k+\alpha_l $ \\ \hspace*{1.6cm} $ < \begin{cases} & \alpha_i+\dotsb+\alpha_l+(I-1)(\alpha_i-\alpha'_i) \qquad ( \text{if} \; \alpha_i>\alpha'_i ) \\ & \dotsb\dotsb\dotsb\dotsb \hspace{4.5cm} \dotsb\dotsb \\ & \alpha_i+\dotsb+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \qquad ( \text{if} \; \alpha_l>\alpha'_l ) \end{cases} $ \\ \hspace*{1.6cm} $ < \begin{cases} & \alpha_i+\dotsb+\alpha_l +(I-1)(\alpha_i-\alpha'_i)+(K-1)(\alpha_k-\alpha'_k) \quad ( \alpha_i>\alpha'_i, \; \alpha_k>\alpha'_k ) \\ & \alpha_i+\dotsb+\alpha_l +(I-1)(\alpha_i-\alpha'_i)+(L-1)(\alpha_l-\alpha'_l) \quad ( \alpha_i>\alpha'_i, \; \alpha_l>\alpha'_l ) \\ & \alpha_i+\dotsb+\alpha_l +(J-1)(\alpha_j-\alpha'_j)+(K-1)(\alpha_k-\alpha'_k) \quad ( \alpha_j>\alpha'_j, \; \alpha_k>\alpha'_k ) \\ & \alpha_i+\dotsb+\alpha_l +(J-1)(\alpha_j-\alpha'_j)+(L-1)(\alpha_l-\alpha'_l) \quad ( \alpha_j>\alpha'_j, \; \alpha_l>\alpha'_l ) \end{cases} $ \\ \hspace*{1.6cm} $ \leq \alpha'_i+\alpha'_j+\alpha'_k+\alpha'_l.$ \\ \hspace*{0.4cm} ($\bold 2$-II-ii) \ \ $ z_i^I z_j, \; z_i z_j^J, \; z_j z_k^K \in f \quad (\exists I, \; \exists J, \; \exists K \geq 2) $.\\ \hspace*{1.4cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_k \leq \alpha'_k $, \ and $ \alpha_j+\alpha_k+(K-1)(\alpha_k-\alpha'_k) \leq \alpha'_j+\alpha'_k $.\\ \hspace*{0.6cm} ($\bold 2$-II-ii-a) \ \ $ z_i^I z_j, \; z_i z_j^J, \; z_j z_k^K, \; z_j z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_j+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_j+\alpha'_l $.\\ \hspace*{0.6cm} ($\bold 2$-II-ii-b) (No.94, 95) \ \ $ z_i^I z_j, \; z_i z_j^J, \; z_j z_k^K, \; z_i z_l^L \in f \quad (\exists I, \; \exists J, \; \exists K, \; \exists L \geq 2) $.\\ \hspace*{1.6cm} $ \alpha_i \leq \alpha'_i $ or $ \alpha_l \leq \alpha'_l $, \ and $ \alpha_i+\alpha_l+(L-1)(\alpha_l-\alpha'_l) \leq \alpha'_i+\alpha'_l $.\\ \hspace*{1.6cm} Since there exists $ z_i^p z_k^r \in f $, we have $ \alpha_i \leq \alpha'_i $ or $ \alpha_k \leq \alpha'_k $.\\ \hspace*{1.6cm} Since there exists $ z_j^q z_l^s \in f $, we have $ \alpha_j \leq \alpha'_j $ or $ \alpha_l \leq \alpha'_l $.\\ \hspace*{1.6cm} If $ \alpha_l \leq \alpha'_l $ then the rest is similar to $\bold 1$-II-i-a. \\ \hspace*{1.6cm} If $ \alpha_l > \alpha'_l $ then $ \alpha_i \leq \alpha'_i $ and $ \alpha_j \leq \alpha'_j $, so the rest is similar to $\bold 2$-I-i-b. \\ \end{pf} \begin{lem} Let $ f=\alpha(x,y,z)x^2+\beta(y,z)y^3+\phi(z)y^2+\varphi(z)y+\psi(z) \in {\bold C}[x,y,z] $ define an isolated singularity at the origin $ 0 \in {\bold C}^3 $, which satisfies one of the following\rom: $(1)$ \ $ \beta(y,z)=\beta_0+\text{higher terms}, \; 0 \ne \beta_0 \in {\bold C}, \; \text{ord}\phi=1 $, and \\ \hspace*{1.1cm} $ \beta_0 y^3+\phi_0(z)y^2+\varphi_0(z)y+\psi_0(z) $ has no triple factor, $(1')$ \ $ \beta \equiv 0 $ or $ \text{ord}\beta \geq 1 $, \ and $ \text{ord}\phi=1 $, $(2)$ \ $ \beta(y,z)=\beta_0+\text{higher terms}, \; 0 \ne \beta_0 \in {\bold C}, \; \; \text{ord}\varphi \leq 3 $ or $ \text{ord}\psi \leq 5 $, \ and \\ \hspace*{1.1cm} $ \beta_0 y^3+\phi_0(z)y^2+\varphi_0(z)y+\psi_0(z) $ has no triple factor, \\ where $ \text{ord}\alpha=0 $, and $ \phi_0, \; \varphi_0, \; \psi_0 $ are the initial parts of $ \phi, \; \varphi, \; \psi $, respectively. Then $(f,0)$ is rational. \end{lem} \begin{pf} \hspace*{0.2cm} We may assume that $ \alpha(x,y,z)=1, \; \beta_0=1 $. \\ (1) \ If $ \text{ord}\varphi \leq 1 $ or $ \text{ord}\psi \leq 2 $ then the assertion holds true. So we may assume $ \text{ord}\varphi \geq 2 $ and $ \text{ord}\psi \geq 3 $. Let $f_0$ be the initial part of $f$ with respect to the weight $ \alpha = (1/2, \; p, \; q) $, where $\alpha$ satisfies the conitions $$ \frac13 \leq p = \frac{1-q}{2} < \frac12 < p+q \left( = \frac{1+q}{2} \right), $$ $$ \deg_{\alpha}(\varphi(z)y), \; \deg_{\alpha}\psi(z) \geq 1, \quad \text{and} \quad \deg_{\alpha}(\varphi(z)y) \; \text{or} \; \deg_{\alpha}\psi(z)=1. $$ (There exists such weight $\alpha$ because $$ \frac{1-q}{2} \geq \frac13, \; \frac{1-q}{2}+Nq \geq 1 \quad \text{for } N \geq 2, \; \frac13 \geq q \geq \frac{1}{2N-1}.) $$ If $ f_0-x^2 \in {\bold C}[y,z] $ has no double factor, then $(f_0, 0)$ is an isolated singularity, so $(f_0, 0)$ is rational. So $ \{f_0=0\} \subset {\bold C}^3 $ has only rational singularities around the origin $ 0 \in {\bold C}^3 $. Therefore $(f,0)$ is also rational from 2.4 and 2.5, because $ \displaystyle \frac12+p+q>1 $.\\ If $ f_0-x^2 $ has a double factor, namely, \\ \hspace*{1cm} $ f=x^2+(y+\gamma_1 z)^2 (y+\gamma_2 z)+\text{higher terms}, \; \gamma_1 \ne \gamma_2 $, \\ then taking the coordinate changes $ Y:=y+\gamma_1 z $ and $ z':=(\gamma_2-\gamma_1)z $, thus \\ \hspace*{1cm} $ f=x^2+\beta'(Y,z')Y^3+\phi'(z')Y^2+\varphi'(z')Y+\psi'(z')$,\\ for some $ \beta' \in {\bold C}[Y,z'], \; \phi', \; \varphi', \; \psi' \in {\bold C}[z'] $ with $ \beta'=1+\text{higher terms}, \; \phi'=z'+\text{higher terms}, \; 2=\text{ord}\varphi<\text{ord}\varphi', \; 3=\text{ord}\psi<\text{ord}\psi' $. Let $ \varphi'_0, \; \psi'_0 $ be the initial parts of $ \varphi', \; \psi' $, respectively. We replace the weight $ \alpha = (1/2, \; p, \; q) $ with $ \alpha' = (1/2, \; p', \; q') $, which satisfies: $$ \frac13 = p < p' = \frac{1-q'}{2} < \frac12 < p'+q' \left( = \frac{1+q'}{2} \right), $$ $$ \deg_{\alpha'}(\varphi'(z')Y), \; \deg_{\alpha'}\psi'(z') \geq 1, \quad \text{and} \quad \deg_{\alpha'}(\varphi'(z')Y) \; \text{or} \; \deg_{\alpha'}\psi'(z')=1. $$ (There exists such weight $\alpha'$ because $$ \frac{1-q'}{2} > \frac13, \; \frac{1-q'}{2}+Nq' \geq 1 \quad \text{for } N > 2, \; \frac13 > q' \geq \frac{1}{2N-1}.) $$ If \ $ Y^2 z'+Y\varphi'_0(z')+\psi'_0(z')=(Y+g(z'))^2 z' $ \; for some $ g \in {\bold C}[z'] \; \; (\text{ord}(g) > 1) $,\\ then taking the coordinate change $ Y':=Y+g(z') $, thus \\ \hspace*{1cm} $ f=x^2+\beta''(Y',z'){Y'}^3+\phi''(z'){Y'}^2 +\varphi''(z')Y'+\psi''(z') $, \\ for some $ \beta'' \in {\bold C}[Y',z'], \; \phi'', \; \varphi'', \; \psi'' \in {\bold C}[z'] $ with $ \beta''=1+\text{higher terms}, \; \phi''=z'+\text{higher terms}, \; \text{ord} \varphi' < \text{ord} \varphi'', \; \text{ord} \psi' < \text{ord} \psi'' $. Let $ \varphi''_0, \; \psi''_0 $ be the initial parts of $ \varphi'', \; \psi'' $, respectively. If $$ {Y'}^2 z'+Y'\varphi''_0(z')+\psi''_0(z')=(Y'+g'(z'))^2 z' $$ for some $ g' \in {\bold C}[z'] \; (1 < \text{ord}(g) < \text{ord}(g')) $, then taking the coordinate change \\ $ Y'':=Y'+g'(z') $, $\dotsb\dotsb$. If this procedure continues infinitely, then \\ \hspace*{2.5cm} $ \displaystyle \frac13=p<p'<\dotsb<p^{(n)}=\frac{1-q^{(n)}}{2}<\dotsb <\frac12<p^{(n)}+q^{(n)} $,\\ \hspace*{2.5cm} $ \displaystyle \frac13=q>q'>\dotsb>q^{(n)}>\dotsb>0 $, $$ \dim_{\bold C}{\bold C}\{z'\} \left/ \left(\varphi^{(n)}, \frac{\partial \psi^{(n)}} {\partial z'} \right) \right. \leq \mu(f,0) \overset{n \to +\infty}{\longrightarrow} +\infty, $$ a contradiction. Therefore $(f,0)$ is rational by 2.4 and 2.5. \\ (1$'$) \ We may assume $ \phi=z+\text{higher terms}, \; \text{ord}\varphi \geq 2 $ and $ \text{ord}\psi \geq 3 $. Let $f_0$ be the initial part of $f$ with respect to the weight $ \alpha = (1/2, \; p, \; q) $, where $\alpha$ satisfies: $$ \frac13 \leq p = \frac{1-q}{2} < \frac12 < p+q \left( = \frac{1+q}{2} \right), $$ $$ \deg_{\alpha}(\varphi(z)y), \; \deg_{\alpha}\psi(z) \geq 1, \quad \text{and} \quad \deg_{\alpha}(\varphi(z)y) \; \text{or} \; \deg_{\alpha}\psi(z)=1. $$ If $ f_0-x^2 \in {\bold C}[y,z] $ has no double factor, then $(f_0, 0)$ is rational similarly as in (1). \\ If $ f_0-x^2 $ has a double factor, namely, \\ \hspace*{1cm} $ y^2 z+y\varphi_0(z)+\psi_0(z)=(y+g(z))^2 z $ \; for some $ g \in {\bold C}[z] \; \; (\text{ord}(g) \geq 1) $,\\ (where $ \varphi_0, \; \psi_0 $ are the initial parts of $ \varphi, \; \psi $, respectively,) then taking the coordinate change $ Y:=y+g(z) $, thus \\ \hspace*{1cm} $ f= \begin{cases} x^2+\phi(z)Y^2+\varphi'(z)Y+\psi'(z), \qquad & (\beta \equiv 0) \\ x^2+\beta'(Y,z)Y^3+\phi'(z)Y^2+\varphi'(z)Y+\psi'(z), \qquad & (\text{ord}\beta \geq 1) \end{cases} $ \\ for some $ \beta' \in {\bold C}[Y,z], \; \phi', \; \varphi', \; \psi' \in {\bold C}[z] $ with $ 1 \leq \text{ord}\beta', \; \phi'=z+\text{higher terms}, \; 2 \leq \text{ord} \varphi < \text{ord} \varphi', \; 3 \leq \text{ord} \psi < \text{ord} \psi' $. We replace the weight $ \alpha = (1/2, \; p, \; q) $ with $ \alpha' = (1/2, \; p', \; q') $, which satisfies the conitions $$ \frac13 \leq p < p' = \frac{1-q'}{2} < \frac12 < p'+q' \left( = \frac{1+q'}{2} \right), $$ $$ \deg_{\alpha'}(\varphi'(z)Y), \; \deg_{\alpha'}\psi'(z) \geq 1, \quad \text{and} \quad \deg_{\alpha'}(\varphi'(z)Y) \; \text{or} \; \deg_{\alpha'}\psi'(z)=1. $$ If this procedure continues infinitely, then $ \mu(f,0) \overset{n \to +\infty}{\longrightarrow} +\infty, $ a contradiction. \\ (2) \ Let $f_0$ be the initial part of $f$ with respect to the weight $ \alpha = (1/2, \; 1/3, \; q) $.\\ If $ f_0-x^2 \in {\bold C}[y,z] $ has no double factor, then $(f_0, 0)$ is an isolated singularity, and so $(f_0, 0)$ is rational for the case of $ \text{ord}\varphi \leq 3 $ or $ \text{ord}\psi \leq 5 $. Therefore $(f,0)$ is also rational from 2.4 and 2.5, because $ \displaystyle \frac12+\frac13+q>1 $ for $ \displaystyle q>\frac16 $.\\ So consider the case of $ f=x^2+(y+\gamma_1 z)^2 (y+\gamma_2 z)+\text{higher terms}, \; \gamma_1 \ne \gamma_2 $. \\ Then the situation is similar to (1). \end{pf} \begin{exmp} \hspace*{0.3cm} (See Theorem 2.7.) \\ \hspace*{1cm} $ f = x^2 + y^3 + z^9 + w^{18} $, \ \ \ (No.12) \\ \hspace*{1cm} $ f_{\lambda} = x^2 + y^3 + (- \frac{27}{4}c_{3 0}^2 (z+\gamma_1 w)^4 (z-2\gamma_1 w)^2 + \varphi(z,w))y $ \\ \hspace*{2cm} $ + (z+\gamma_1 w)^6 (z-2\gamma_1 w)^3 + \psi(z,w) $,\\ where $ - \frac{27}{4}c_{3 0}^3 = 1, \; \text{ord}\varphi \geq 7, \; \text{ord}\psi \geq 10, \; \psi(z,w) \ni w^{18} $ \ (Case 10-b). \\ Taking the coordinate change $ Y:= y + \frac32 c_{3 0} (z+\gamma_1 w)^2 (z-2\gamma_1 w) $, we have \\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 c_{3 0} (z+\gamma_1 w)^2 (z-2\gamma_1 w)Y^2 + \varphi(z,w)Y $ \\ \hspace*{2cm} $ - \frac32 c_{3 0} (z+\gamma_1 w)^2 (z-2\gamma_1 w) \varphi(z,w) + \psi(z,w) $,\\ and then $ z':= z+\gamma_1 w $,\\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 c_{3 0} ({z'}^3 - 3\gamma_1 {z'}^2 w)Y^2 + \varphi'(z',w)Y + \psi'(z',w) $ \\ for some $ \varphi', \; \psi' \in {\bold C}[z',w], \; \text{ord}\varphi' \geq 7, \; \text{ord}\psi' \geq 10 $.\\ Let $ \alpha':=(1/2, 1/3, 1/8, 1/12), \; \alpha'':=(1/2, 17/50, 3/25, 2/25), \; c'_{2 1}:=-3\gamma_1 c_{3 0} $, and \\ \hspace*{1cm} $ \varphi'=-3(3c'_{2 1}{z'}^2 w+c'_{0 4}w^4)c'_{0 4}w^4+\Phi', \qquad \quad \; \Phi'_0=c'_{1 7}z'w^7 $,\\ \hspace*{1cm} $ \psi'=(-\frac92c'_{2 1}{z'}^2 w-2c'_{0 4}w^4)(c'_{0 4}w^4)^2 +\Psi', \qquad \Psi'_0=c'_{5 5}{z'}^5 w^5+c'_{1 \; 11}z' w^{11} $,\\ where $ \Phi'_0, \; \Psi'_0 $ are the initial parts of $ \Phi', \; \Psi' $ with respect to the weight $ \alpha' $. Then \\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 (c_{3 0}{z'}^3+c'_{2 1}{z'}^2 w)Y^2 + (-3(3c'_{2 1}{z'}^2 w+c'_{0 4}w^4)c'_{0 4}w^4+\Phi')(z',w)Y $\\ \hspace*{2cm} $ +((-\frac92c'_{2 1}{z'}^2 w-2c'_{0 4}w^4)(c'_{0 4}w^4)^2+\Psi') (z',w) $ \\ \hspace*{1.5cm} $ = x^2+(Y+c'_{0 4}w^4)^2 (Y-\frac92c'_{2 1}{z'}^2 w-2c'_{0 4}w^4) $ \\ \hspace*{2cm} $ -\frac92c_{3 0}{z'}^3 Y^2+\Phi'(z',w)Y+\Psi'(z',w) $ \qquad \qquad (Case II-A-II$'$-ii-i). \\ Taking the coordinate change $ Y':=Y+c'_{0 4}w^4 $, we have \\ \hspace*{1cm} $ f_{\lambda} = x^2 + {Y'}^3 - (\frac92c_{3 0}{z'}^3 +\frac92c'_{2 1}{z'}^2 w+3c'_{0 4}w^4){Y'}^2 $ \\ \hspace*{2cm} $ + (9c_{3 0}c'_{0 4}{z'}^3 w^4+\Phi'(z',w))Y' - \frac92c_{3 0}{c'}_{0 4}^2 {z'}^3 w^8 - c'_{0 4}w^4\Phi'(z',w) + \Psi'(z',w) $,\\ \hspace*{1cm} $ f_0 = x^2 - (\frac92c'_{2 1}{z'}^2 w+3c'_{0 4}w^4){Y'}^2 - \frac92c_{3 0}{c'}_{0 4}^2 {z'}^3 w^8 - c'_{0 4}w^4\Phi'_0(z',w) + \Psi'_0(z',w) $.\\ Then $(f_0, 0)$ is an isolated singularity or Case II-A-I, and $ 1/2+17/50+3/25+2/25>1 $, so $ \{f_0=0\}-\{0\} \subset {\bold C}^4 $ has at most rational singularities around the origin $ 0 \in {\bold C}^4 $. \end{exmp} \begin{rem} We will take the following arguments to show Theorem 2.7. Taking suitable local coordinate changes finitely, if necessary, we have a three-dimensional face $\Delta$ of $\Gamma(f_{\lambda})$ such that $(1,1,1,1)$ lies strictly above the hyperplane including $\Delta$, that $ \operatorname{Sing}(f_0) := \operatorname{Sing}(\{f_0=0\}) = \bigcup_j C_j $ with $ \dim_{\bold C} C_j \leq 1 $, and that $ (f_0, P):=(\{f_0=0\}, P) $ is rational for any $ P \in C_j \subset \operatorname{Sing}(f_0) $ with $ P \ne 0 $ and $ 0 \in C_j $, where $ f_0 := f_{\lambda \; \Delta} = \sum_{\nu \in \Delta} a_{\nu} z^{\nu} \subset f_{\lambda} $. Therefore $ \{f_0=0\}-\{0\} $ has at most rational singularities around $0$, and so $(f_{\lambda},0)$ is rational. More precisely, any three-dimensional face $ \Delta = \{ (X,Y,Z,W) \in {{\bold R}_{\geq 0}}^4 \; ; \; \frac12 X + \beta Y + \gamma Z + \delta W = 1, \; 0 \leq X \leq 2, \; j \leq Y \leq j', \; k \leq Z \leq k', \; l \leq W \leq l' \} $ of $\Gamma(f_{\lambda})$ satisfies $ 1 < \frac12 + \beta + \gamma + \delta $ under the original local coordinates from Lemma 3.1. Choose a three-dimensional face $\Delta$ of $\Gamma(f_{\lambda})$ suitably and let $ f_0 := f_{\lambda \; \Delta} = \sum_{\nu \in \Delta} a_{\nu} z^{\nu} \subset f_{\lambda} := \sum_{\nu} a_{\nu} z^{\nu} $.\\ \\ Case (0). \ $ (f_0,0) $ is an isolated singularity. Then $ (f_0,0) $ is rational and so $ \{ f_0=0 \} \subset {\bold C}^4 $ has only rational singularities around the origin $ 0 \in {\bold C}^4 $. \\ \\ Case (I). \ $ \operatorname{Sing}(f_0) = \bigcup_j C_j, \qquad \dim_{\bold C} C_j \leq 1 $ for all j. Since $f_0$ is quasi-homogeneous, we have $ \operatorname{Sing}(C_j)=\{0\} $. If an arbitrary point $ P \in C_j-\{0\} $ is rational on $\{ f_0=0 \}$ for each irreducible curve $C_j$ with $ 0 \in C_j $, then $ \{ f_0=0 \}-\{ 0 \} $ has at most rational singularities around 0. Rationality of $P$ is shown by using the following fact: \\ \hspace*{1cm} if there exists an hyperplane cut $ (H,P) \subset ( \{ f_0=0 \}, \; P ) $ which is rational \\ \hspace*{1cm} and Gorenstein, then $ ( \{ f_0=0 \}, \; P ) $ is also rational and Gorenstein. When we can not tell whether $P$ is rational or not, take suitable local coordinate change and repeat the same procedure as above. The condition $ 1 < \frac12 + \beta^{(n)} + \gamma^{(n)} + \delta^{(n)} $ is still satisfied for a certain $\Delta^{(n)}$ after each coordinate change. This procedure must finish in finite times from the assumption $ \mu(f_{\lambda},0)<\mu(f,0) $.\\ \\ Case (II). \ $ \operatorname{Sing}(f_0) = \bigcup_j C_j, \qquad \dim_{\bold C} C_j = 2 $ for some j. After suitable local coordinate change, choose another three-dimensional face $\Delta$ of $\Gamma(f_{\lambda})$ properly, and let $ f_0 := f_{\lambda \; \Delta} = \sum_{\nu \in \Delta} a_{\nu} z^{\nu} $. (II-I$^{(\prime)}$). \ $ \operatorname{Sing}(f_0) = \bigcup_j C_j, \qquad \dim_{\bold C} C_j \leq 1 $ for all j. \\ \hspace*{0.5cm} Then the proof is completed similarly as in (I). (II-II$^{(\prime)}$). \ $ \operatorname{Sing}(f_0) = \bigcup_j C_j, \qquad \dim_{\bold C} C_j = 2 $ for some j. \\ \hspace*{0.5cm} After suitable local coordinates change, the condition $ 1 < \frac12 + \beta' + \gamma' + \delta' $ is still satisfied for a certain face $ \Delta'= \{ (X,Y,Z,W) \; ; \; \frac12 X + \beta' Y + \gamma' Z + \delta' W = 1 \} $ of $\Gamma(f_{\lambda})$. Choose such three-dimensional face $ \Delta'$ properly; if Case (I$^{(\prime)}$) then the assertion is \\ concluded. If Case (II$^{(\prime)}$) again, take suitable coordinate change once more. This \\ procedure must finish in finite times from the assumption $ \mu(f_{\lambda},0)<\mu(f,0) $. \end{rem} {\it Proof of Theorem 2.7}. If there exists $ y^j z^k w^l \in f_{\lambda} $ such that $ j+k+l \leq 2 $ then $(f_{\lambda},0)$ is at most rational. So we may assume $ j+k+l \geq 3 $ for all $ y^j z^k w^l \in f_{\lambda} $. Let $ \Lambda :=\{(k,l) \; ; z^k w^l \in f_{\lambda}\} \bigcup\{\frac32(k,l) \; ; y z^k w^l \in f_{\lambda}\} $ and let $\Gamma$ be the union of the compact faces of the convex hull of $ \bigcup_{\nu \in \Lambda} (\nu + {{\bold R}_{\geq 0}}^2) $ in $ {\bold R}^2 $. For any one-dimensional face $ \Delta = \{ (Z,W) \; ; \; \gamma Z + \delta W = 1 \; , k_1 \leq Z \leq k_0, \; l_0 \leq W \leq l_1 \} $ of $\Gamma$, we get $ \gamma+\delta>1/6 $ from Lemma 3.1. Choose such $\Delta$ satisfies $ k_1<6 $ and $ l_0<6 $. (See {\sc Figure} 1.) Then, \\ \hspace*{1cm} $ f_{\lambda} = x^2+y^3+ \sum_{\gamma k+\delta l \geq 2/3} b_{kl} y z^k w^l + \sum_{\gamma k+\delta l \geq 1} c_{kl} z^k w^l $,\\ \hspace*{1cm} $ f_0 = x^2+ y^3+ \sum_{\gamma k+\delta l = 2/3} b_{kl} y z^k w^l + \sum_{\gamma k+\delta l = 1} c_{kl} z^k w^l $ \\ with respect to the weight $ \alpha:=(1/2, 1/3, \gamma, \delta) $.\\ \begin{figure}[h] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(85,85)(-15,-15) \put(0,0){\vector(1,0){60}} \put(0,0){\vector(0,1){60}} \put(-5,-5){0} \put(63,-2){$Z$} \put(-2,63){$W$} \put(55,5){\line(-2,1){20}} \put(35,15){\thicklines\line(-1,1){20}} \put(15,35){\line(-1,4){5}} \put(30,30){\circle*{1}} \put(30,30){\makebox(15,6){(6,6)}} \put(35,15){\makebox(17,6){($k_0,l_0$)}} \put(15,35){\makebox(17,6){($k_1,l_1$)}} \put(15,50){$\Gamma$} \multiput(50,0)(-1,1){15}{\circle*{0.2}} \multiput(15,35)(-1,1){15}{\circle*{0.2}} \put(50,-10){$ \displaystyle \frac{1}{\gamma} $} \put(-5,50){$ \displaystyle \frac{1}{\delta} $} \put(20,20){$\Delta$} \end{picture} \caption{ } \end{center} \end{figure} \\ Case I. \ $ h:=f_0-x^2 \in {\bold C} [y,z,w] $ \ has no double factor. \\ Since $ h $ and $ \displaystyle \frac{\partial h}{\partial y} $ have no common factor, $ \dim_{\bold C} \operatorname{Sing}(f_0) \leq 1 $. Let $ C_j \subset \operatorname{Sing}(f_0) $ be an irreducible curve with $ 0 \in C_j $. If $ C_j \ni P = (0,a,b,c) \ne (0,0,0,0) $ then $ b \ne 0 $ or $ c \ne 0 $.\\ Let $ P \ne 0 $ be an arbitrary point on $C_j$. \\ \\ (I-A). \ $ C_j \ni P = (0,a(t),b(t),t) \; ; \; t \ne 0 $.\\ Since $f_0$ is quasi-homogeneous, we have $ a(t)=a't^{1/3\delta}, \; b(t)=b't^{\gamma/\delta} $ for some $ a', \; b' \in {\bold C} $.\\ Let $ \eta:=y-a, \; \zeta:=z-b $, \ and $ f_0(w=t):=f_0(x, \eta+a, \zeta+b, t) $.\\ Then $ f_0(w=t)-x^2 = {\eta}^3+3a{\eta}^2+\dotsb \in {\bold C} [\eta,\zeta] $ has no double factor. \\ (In fact, \ if $ f_0(w=t)-x^2=(\eta+\varphi(\zeta))^2 (\eta+\psi(\zeta)) $ then {\allowdisplaybreaks \begin{align*} f_0(w=t) - x^2 & = {\eta}^3 + 3a{\eta}^2 - 3 \varphi(\varphi-2a)\eta - {\varphi}^2 (2\varphi-3a) \\ & = y^3 - 3(\varphi(z-b(t))-a(t))^2 y - 2(\varphi(z-b(t))-a(t))^3 ,\\ f_0 - x^2 & = (y+\varphi(z-b(w))-a(w))^2 (y-2\varphi(z-b(w))+2a(w)) \\ & = y^3 - 3(\varphi(z-b(w))-a(w))^2 y - 2(\varphi(z-b(w))-a(w))^3 . \end{align*}} It follows that $ \varphi(z-b(w))-a(w) \in {\bold C} [z,w] $ from $ \varphi(z-b(w))-a(w) \in {\bold C} [z,w^{\gamma/\delta}] $,\\ $(\varphi(z-b(w))-a(w))^2, \; (\varphi(z-b(w))-a(w))^3 \in {\bold C} [z,w]$.)\\ Therefore $ (f_0(w=t), \; (0,0,0)) $ is an isolated singularity under the local coordinate system $ (x,\eta,\zeta) $.\\ If $ a \ne 0 $ then $(f_0(x,y,z,t), \; (0,a,b))$ is rational, so $(f_0, \; (0,a,b,t))$ is rational.\\ If $ a=0 $ then $ \eta=y $, $ f_0(w=t)=x^2+y^3+\sum b_{k l} y (\zeta+b)^k t^l +\sum c_{k l} (\zeta+b)^k t^l $.\\ If $ b=0 $ then $ \zeta=z $, $ f_0(w=t)=x^2+y^3+\sum b_{k l} y z^k t^l +\sum c_{k l} z^k t^l $. By Lemma 3.2, $ (f_0(x,y,z,t), \; (0,0,0)) $ is rational, so $ (f_0, \; (0,0,0,t)) $ is rational. \\ So we consider the case $ b \ne 0 $.\\ If there exists $ i \leq 5 $ such that $ {\zeta}^i \in f_0(w=t) $ or $ j \leq 3 $ such that $ y {\zeta}^j \in f_0(w=t), $ then $(f_0(w=t),(0,0,0))$ is rational under the local coordinate system $(x,y,\zeta)$. So we assume the coefficient of $ {\zeta}^i $ is $0$ for all $ i \leq 5 $ and the coefficient of $ y {\zeta}^j $ is $0$ for all $ j \leq 3 $. Furthermore we may assume $ \gamma \geq \delta $. ( Indeed, if $ \gamma < \delta $ we consider $ f_0(x,y,s,w) $ instead of $ f_0(x,y,z,t) $.) Thus $ b=b(t) $ is written as $ b(t)=b't^m, \; m := q/p = \gamma/\delta \geq 1, \; p, q \in {\bold N}, \; (p,q)=1 $. Since the coefficient of $ {\zeta}^i $ is $ \displaystyle \frac{1}{i!}\frac{{\partial}^i f_0}{(\partial z)^i} (0,0,b,t) $ and the coefficient of $ y {\zeta}^j $ is $ \displaystyle \frac{1}{j!}\frac{{\partial}^{j+1} f_0}{\partial y (\partial z)^j} (0,0,b,t) $, it follows that: \\ \hspace*{1.5cm} the coefficient of \ $ {\zeta}^i $ in $ f_0 (w=t) $ is $0$ \\ \hspace*{1cm} $ \Longleftrightarrow f_0 (0,0,z,t)=(z-b)^{i+1} \varphi(z,t) $, for some $ \varphi(z,t) \in {\bold C} [z,t^m] $ \\ \hspace*{1cm} $ \Longleftrightarrow f_0 (0,0,z,w)=(z-b'w^m)^{i+1} \varphi(z,w) $, for some $ \varphi(z,w) \in {\bold C} [z,w^m] $,\\ and \\ \hspace*{1.5cm} the coefficient of \ $ y {\zeta}^j $ in $ f_0 (w=t) $ is $0$ \\ \hspace*{1cm} $ \displaystyle \Longleftrightarrow \frac{\partial f_0}{\partial y}(0,0,z,t)=(z-b)^{j+1} \psi(z,t) $, for some $ \psi(z,t) \in {\bold C} [z,t^m] $ \\ \hspace*{1cm} $ \displaystyle \Longleftrightarrow \frac{\partial f_0}{\partial y}(0,0,z,w) =(z-b'w^m)^{j+1} \psi(z,w) $, for some $ \psi(z,w) \in {\bold C} [z,w^m] $.\\ The number $ m = \gamma/\delta $ is a integer by the assumptions. \\ ( In fact, if $ p \ne 1 $ then $ f_0(0,0,z,w) $ and $ \displaystyle \frac{\partial f_0}{\partial y}(0,0,z,w) $ are written as \\ $ f_0(0,0,z,w)=(z-b'w^m)^6 \varphi(z,w)=(z^p-b''w^q)^6 \varphi'(z,w) , \, b'' \in {\bold C}, \; \varphi'(z,w) \in {\bold C} [z,w] $,\\ $ \displaystyle \frac{\partial f_0}{\partial y}(0,0,z,w)=(z-b'w^m)^4 \psi(z,w) =(z^p-b'''w^q)^4 \psi'(z,w) , $ \\ \hspace*{10cm} $ b''' \in {\bold C}, \; \psi'(z,w) \in {\bold C} [z,w] $,\\ respectively. Since $ 2 \leq p \leq q $, we have $ \displaystyle \frac{1}{6p}+\frac{1}{6q} \leq \frac16 \quad \text{and \ } \frac{1}{4p}+\frac{1}{4q} \leq \frac14 $. This is a contradiction to the condition $ \gamma + \delta > 1/6 $.) Let $ z':=z-b'w^m $, \ $ \Lambda':=\{(k,l) \; ; {z'}^k w^l \in f_{\lambda}\} \bigcup\{\frac32(k,l) \; ; y {z'}^k w^l \in f_{\lambda}\} $, and let $\Gamma'$ be the union of the compact faces of the convex hull of $ \bigcup_{\nu \in \Lambda'} (\nu + {{\bold R}_{\geq 0}}^2) $ in $ {\bold R}^2 $. Then: \begin{claim} For any one-dimensional face $ \Delta' =\{ \gamma' Z + \delta' W = 1 , \; k'_1 \leq Z \leq k'_0, \; l'_0 \leq W \leq l'_1 \} $ of $\Gamma'$, the condition $ \gamma'+\delta'>1/6 $ is satisfied. \end{claim} A proof of this claim is found at the end of this paper. \\ There exists $ k_1 < i \leq k_0 $ such that $ i = \max \{ i' \in {\bold N} \; ; \; (z-b'w^m)^{i'} | f_0(0,0,z,w) \} $ or $ \frac23 k_1 < j \leq \frac23 k_0 $ such that $ \displaystyle j = \max \left\{ j' \in {\bold N} \; ; \; (z-b'w^m)^{j'} \left| \frac{\partial f_0}{\partial y}(0,0,z,w) \right. \right\} $. Choose $ \Delta' $ such that $ k'_0 \leq \min \{i, \; \frac32 j \} $, $ k'_1 < 6 $ and $ l'_0 < 6 $. Then $ k'_0 \leq k_0 $, \ $ l_0 \leq l'_0 $, and $ \displaystyle 1 \leq \frac{\gamma}{\delta} < \frac{\gamma'}{\delta'} $. Let $ f_0 $ be the initial part of $ f \in {\bold C} [x,y,z',w] $ with respect to the weight $(1/2, 1/3, \gamma', \delta')$. If $ h:=f_0-x^2 \in {\bold C} [y,z',w] $ has a double factor, then Case (II-A). Now we assume $ h $ \ has no double factor. If $ k'_0 < 6 $ then our assertion is concluded. So we assume $ k'_0 \geq 6 $. Repeating same argument as above, \\ \hspace*{1.5cm} $ z'':=z'-b''w^{m'}, \dotsb , z^{(n)}:=z^{(n-1)}-b^{(n)}w^{m^{(n-1)}}, \dotsb , $ \\ \hspace*{1.5cm} $ k_0 \geq k'_0 \geq k''_0 \geq \dotsb \geq k_0^{(n)} \geq \dotsb , $ \\ \hspace*{1.5cm} $ \displaystyle 1 \leq \frac{\gamma}{\delta} < \frac{\gamma'}{\delta'} < \frac{\gamma''}{\delta''} < \dotsb < \frac{{\gamma}^{(n)}}{{\delta}^{(n)}} < \dotsb , $ \\ \hspace*{1.5cm} $ \frac16 < {\gamma}^{(n)} + {\delta}^{(n)} $ \ for all $ n \in {\bold N} $.\\ (See {\sc Figure} 2.) If there exists $ n \in {\bold N} $ such that $ k_0^{(n)} < 6 $, then this proof is completed. So consider the case $ k_0^{(n)} \geq 6 $ for all $ n \in {\bold N} $. Then $ \displaystyle \frac{{\gamma}^{(n)}}{{\delta}^{(n)}} \in {\bold N} $ for all $ n \in {\bold N} $. If $ k_0^{(n)} > 6 $ for all $ n \in {\bold N} $ then $ \displaystyle \frac{{\gamma}^{(n)}}{{\delta}^{(n)}} < - \frac{l_0^{(n)}-6}{k_0^{(n)}-6} $, and so this procedure must finish in finite times (See {\sc Figure} 3). Thus assume there exists $ n \in {\bold N} $ such that $ 6=k_0^{(n)}= k_0^{(n+1)}=k_0^{(n+2)}=\dotsb . $ If this procedure continues infinitely then $ \mu(f_{\lambda}, 0) \gg 1 $ , a contradiction. This completes Case (I-A). \\ \begin{figure}[h] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(90,80)(0,0) \put(70,10){\line(-1,1){40}} \put(60,20){\line(-2,3){24}} \put(50,35){\line(-1,4){10}} \put(70,5){\makebox(18,7){($k_0,l_0$)}} \put(12,45){\makebox(18,7){($k_1,l_1$)}} \put(60,20){\makebox(18,6){($k'_0,l'_0$)}} \put(17,55){\makebox(18,7){($k'_1,l'_1$)}} \put(50,33){\makebox(18,7){($k''_0,l''_0$)}} \put(22,75){\makebox(18,7){($k''_1,l''_1$)}} \put(45,25){$\Delta$} \put(50,55){$\Delta''$} \end{picture} \caption{ } \end{center} \end{figure} \begin{figure}[h] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(85,85)(-10,-10) \put(0,0){\vector(1,0){60}} \put(0,0){\vector(0,1){65}} \put(63,-2){$Z$} \put(-2,68){$W$} \put(-5,-5){0} \put(39,-5){6} \put(-5,39){6} \put(0,0){\dashbox(40,40)} \put(50,10){\line(-1,1){40}} \put(50,10){\line(-1,2){25}} \put(50,10){\line(-1,3){15}} \put(50,10){\makebox(20,7){($k_0^{(n)},l_0^{(n)}$)}} \end{picture} \caption{ } \end{center} \end{figure} \\ (I-B). \ $ C_j \ni P = (0,a(s),s,c(s)) \; ; \; s \ne 0 $.\\ Let $ \eta:=y-a, \; \omega:=w-c $, \; and $ f_0(z=s):=f_0(x, \eta+a, s, \omega+c) $. Then $ f_0(z=s)-x^2 \in {\bold C} [\eta,\omega] $ has no double factor. If $ a \ne 0 $ or $ a=c=0 $ then $(f_0(x,y,s,w), \; (0,a,c))$ is rational, so $(f_0, \; (0,a,s,c))$ is rational similarly as in Case (I-A). Thus we assume $ a=0, \; c \ne 0 $. If $ \gamma \geq \delta $ then our argument can be reduced to the case $ a=0, \; b \ne 0 $ and $ \gamma \geq \delta $ of Case (I-A). So it is sufficient to consider the case $ \gamma < \delta $. Furthermore we may assume that the coefficient of $ {\omega}^i $ in $ f_0(z=s) $ is $0$ for all $ i \leq 5 $ and the coefficient of $ y {\omega}^j $ in $ f_0(z=s) $ is $0$ for all $ j \leq 3 $. Thus $ c=c(s) $ is written as $ c(s)=c's^m, \; c' \in {\bold C}, \; m = \delta/\gamma > 1, \; m \in {\bold N} $. Therefore $ l_1 < 6 $ for No.11-14, 46-51 and $ l_1 \leq 6 $ for No.10, 83. (Because $ z^{12} \in f $ for No.10, $ z^{10} \in f $ for No.11, $ z^9 \in f $ for No.12, $ z^8 \in f $ for No.13, $ z^7 \in f $ for No.14, $ z^{11} \in f $ for No.46, $ y z^7 \in f $ for No.47, $ z^9 w \in f $ for No.48, $ z^8 w \in f $ for No.49, $ z^7 w^2 \in f $ for No.50, $ z^7 w \in f $ for No.51, $ z^{10} w \in f $ for No.83.) Thus it is sufficient to consider the case $ (k_1,l_1)=(0,6) $ of No.10 and 83. Let $ w':=w-c'z^m, \; \Lambda':=\{(k,l) \; ; z^k {w'}^l \in f_{\lambda}\} \bigcup \{\frac32(k,l) \; ; y z^k {w'}^l \in f_{\lambda}\} $, and let $\Gamma'$ be the union of the compact faces of the convex hull of $ \bigcup_{\nu \in \Lambda'} (\nu + {{\bold R}_{\geq 0}}^2) $ in ${\bold R}^2$. Then for any one-dimensional face $ \Delta' =\{ \gamma' Z + \delta' W = 1 , \; k'_1 \leq Z \leq k'_0, \; l'_0 \leq W \leq l'_1 \} $ of $\Gamma'$, $ \gamma'+\delta'>1/6 $ and $ \displaystyle 1 < \frac{\delta}{\gamma} < \frac{\delta'}{\gamma'} $ are satisfied. Repeating same argument as in Case (I-A), \\ \hspace*{1.5cm} $ w'':=w'-c''z^{m'}, \dotsb , w^{(n)}:=w^{(n-1)}-c^{(n)}z^{m^{(n-1)}}, \dotsb , $ \\ \hspace*{1.5cm} $ l_1 \geq l'_1 \geq l''_1 \geq \dotsb \geq l_1^{(n)} \geq \dotsb , $ \\ \hspace*{1.5cm} $ \displaystyle 1 < \frac{\delta}{\gamma} < \frac{\delta'}{\gamma'} < \frac{\delta''}{\gamma''} < \dotsb < \frac{{\delta}^{(n)}}{{\gamma}^{(n)}} < \dotsb , $ \\ \hspace*{1.5cm} $ \frac16 < {\gamma}^{(n)} + {\delta}^{(n)} $ \ for all $ n \in {\bold N} $.\\ If there exists $ n \in {\bold N} $ such that $ l_1^{(n)} < 6 $ then the assertion holds true. If $ l_1^{(n)} = 6 $ for all \\ $ n \in {\bold N} $ then this procedure continues infinitely, therefore $ \mu(f_{\lambda}, 0) \gg 1 $, a contradiction. This completes Case (I).\\ \\ For convenience, we write $ z, \gamma, \delta, $ instead of $ z^{(n)}, {\gamma}^{(n)}, {\delta}^{(n)}, $ etc.\\ \\ Case II. \ $ h:=f_0-x^2 \in {\bold C} [y,z,w] $ \ has a double factor. \\ \hspace*{1cm} $ f_0 = x^2+y^3-\frac{27}{4}g(z,w)^2 y-\frac{27}{4}g(z,w)^3 = x^2+(y+\frac32 g(z,w))^2 (y-3g(z,w)) $.\\ Let $ Y:= y+\frac32 g(z,w) $, then $f_0$ and $f_{\lambda}$ are written as:\\ \hspace*{1cm} $ f_0 = x^2+Y^3-\frac92 g(z,w) Y^2 $,\\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 g(z,w) Y^2 + \varphi(z,w) Y + \psi(z,w) $, \\ for some $ \varphi(z,w), \; \psi(z,w) \in {\bold C} [z,w] $ with $ \varphi \not\equiv 0 $ or $ \psi \not\equiv 0 $, \ $ \varphi \equiv 0 $ or $ \deg_{\alpha}\varphi>2/3 $, \ $ \psi \equiv 0 $ or $ \deg_{\alpha}\psi>1 $.\\ \\ (II-A). \ When $ \gamma \geq \delta $, $g(z,w)$ can be classified into ten cases as below: \\ \hspace*{1cm} (1) \ \ $ g = c_{1 0} z + c_{0 L} w^L $, \\ \hspace*{1cm} (2) \ \ $ g = c_{1 1} z w + c_{0 (L+1)} w^{L+1} $, \\ \hspace*{1cm} (3) \ \ $ g = c_{2 0} z^2 + c_{0 L} w^L , \qquad 3 \leq L $, \quad L is odd, \\ \hspace*{1cm} (4-a) \ $ g = c_{2 0} (z + \gamma_1 w^L)(z + \gamma_2 w^L) , \qquad \gamma_1 \ne \gamma_2 $,\\ \hspace*{1cm} (4-b) \ $ g = c_{2 0} (z + \gamma_1 w^L)^2 $, \\ \hspace*{1cm} (5) \ \ $ g = c_{2 1} z^2 w + c_{0 (L+1)} w^{L+1} , \qquad 3 \leq L $, \quad L is odd, \\ \hspace*{1cm} (6-a) \ $ g = c_{2 1} (z + \gamma_1 w^L)(z + \gamma_2 w^L) w, \qquad \gamma_1 \ne \gamma_2 $,\\ \hspace*{1cm} (6-b) \ $ g = c_{2 1} (z + \gamma_1 w^L)^2 w $, \\ \hspace*{1cm} (7) \ \ $ g = c_{3 0} z^3 + c_{0 4} w^4 $, \\ \hspace*{1cm} (8) \ \ $ g = c_{3 0} z^3 + c_{1 3} z w^3 $, \\ \hspace*{1cm} (9) \ \ $ g = c_{3 0} z^3 + c_{0 5} w^5 $, \\ \hspace*{1cm} (10-a) \ $ g = c_{3 0} (z + \gamma_1 w)(z + \gamma_2 w) (z + \gamma_3 w) , \qquad \gamma_i \ne \gamma_j $ for $ i \ne j $,\\ \hspace*{1cm} (10-b) \ $ g = c_{3 0} (z + \gamma_1 w)^2 (z + \gamma_2 w) , \qquad \gamma_1 \ne \gamma_2 $,\\ \hspace*{1cm} (10-c) \ $ g = c_{3 0} (z + \gamma_1 w)^3 $. \\ After the local coordinate change $ z':=z+\gamma_1 w^L \; ( L=1 $ for (10)) around $0$, \\ \hspace*{1cm} (4-a) \ $ g = c_{2 0} {z'}^2 + c'_{1 L} z' w^L $, \\ \hspace*{1cm} (4-b) \ $ g = c_{2 0} {z'}^2 $, \\ \hspace*{1cm} (6-a) \ $ g = c_{2 1} {z'}^2 w + c'_{1 (L+1)} z' w^{L+1} $,\\ \hspace*{1cm} (6-b) \ $ g = c_{2 1} {z'}^2 w $, \\ \hspace*{1cm} (10-a) \ $ g = c_{3 0} {z'}^3 + c'_{2 1} {z'}^2 w + c'_{1 2} z' w^2 $, \\ \hspace*{1cm} (10-b) \ $ g = c_{3 0} {z'}^3 + c'_{2 1} {z'}^2 w $, \\ \hspace*{1cm} (10-c) \ $ g = c_{3 0} {z'}^3 $. \\ (See {\sc Figure} 4.)\\ \begin{figure}[h] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(70,90)(-10,-10) \put(0,0){\vector(1,0){50}} \put(0,0){\vector(0,1){70}} \put(-5,-5){0} \put(9,-5){1} \put(19,-5){2} \put(29,-5){3} \put(39,-5){4} \put(-5,9){1} \put(-5,19){2} \put(-5,29){3} \put(-5,39){4} \put(-5,49){5} \put(-5,59){6} \put(53,-2){$Z$} \put(-2,73){$W$} \put(20,20){\circle*{1}} \put(20,20){\makebox(12,6){(2,2)}} \put(10,0){\line(-1,3){10}} \put(10,10){\line(-1,5){10}} \put(20,0){\line(-1,1){10}} \put(20,10){\line(-1,3){10}} \put(30,0){\line(-3,4){30}} \put(30,0){\line(-2,3){20}} \put(30,0){\line(-3,5){30}} \multiput(30,0)(-1,1){20}{\circle*{0.2}} \put(20,0){\line(-2,3){20}} \put(10,0){\circle*{1}} \put(20,0){\circle*{1}} \put(30,0){\circle*{1}} \put(10,10){\circle*{1}} \put(20,10){\circle*{1}} \put(0,30){\circle*{1}} \put(10,30){\circle*{1}} \put(0,40){\circle*{1}} \put(10,40){\circle*{1}} \put(0,50){\circle*{1}} \put(0,60){\circle*{1}} \put(10,20){\circle{1}} \end{picture} \caption{ } \end{center} \end{figure} Let $ \Lambda':=\{(k,l) \; ; {z'}^k w^l \in f_{\lambda}\} \bigcup\{\frac32(k,l) \; ; Y {z'}^k w^l \in f_{\lambda}\} \bigcup\{3(k,l) \; ; Y^2 {z'}^k w^l \in f_{\lambda}\} $, and let $\Gamma'$ be the union of the compact faces of the convex hull of $ \bigcup_{\nu \in \Lambda'} (\nu + {{\bold R}_{\geq 0}}^2) $ in $ {\bold R}^2 $. Then for any one-dimensional face $ \Delta' =\{ \gamma' Z + \delta' W = 1 \; , \; k'_1 \leq Z \leq k'_0, \; l'_0 \leq W \leq l'_1 \} $ of $\Gamma'$, the condition $ \gamma'+\delta'>1/6 $ is satisfied. (In fact, No.11, 13, 14 and 49-51 can not become Case (10). About Case (10-c) which comes from No.10, 12, 46-48 or 83, the condition $ \gamma'+\delta'>1/6 $ is satisfied as in Claim 3.5.) \\ \\ First we consider Case (1), (2), (3), (4-a), (5), (6-a), (7), (8), (9) and (10-a). \\ We replace the weight $ \alpha=(1/2,1/3,\gamma,\delta) $ with $ \alpha'=(1/2,\beta',\gamma',\delta') $ which satisfies the \\ conditions \\ \hspace*{1cm} $ \frac13 = \beta < \beta' < \frac12 < \beta' + \gamma' + \delta', \quad \gamma'/\delta' = \gamma/\delta, $ \\ \hspace*{1cm} $ \deg_{\alpha'}(g(z,w)Y^2) = 1, $ \; $ \deg_{\alpha'}(\varphi(z,w)Y), \; \deg_{\alpha'}\psi(z,w) \geq 1 $, \; and \\ \hspace*{1cm} $ \deg_{\alpha'}(\varphi(z,w)Y) $ or $ \deg_{\alpha'}\psi(z,w) = 1 $,\\ and let $ f_0 $ be the initial part of $ f_{\lambda} $. (There exists such weight $\alpha'$ because \\ \hspace*{1cm} $ \beta' + \gamma' + \delta' = \beta' + (\gamma+\delta)(1-2\beta')/(1-2\beta) > \frac12 $ and \\ \hspace*{1cm} $ \deg_{\alpha'}(g(z,w)Y^2) = 2\beta'+\deg_{\alpha}g(z,w)(1-2\beta')/(1-2\beta) = 1 $ \\ for all $ \beta, \gamma, \delta, \beta', \gamma', \delta' $ satisfying \\ \hspace*{1cm} $ \beta, \; \beta' < \frac12 < \beta+\gamma+\delta $ and $ \gamma'/\gamma=\delta'/\delta=(1-2\beta')/(1-2\beta) $.) \\ Then \\ \hspace*{1cm} $ f_0 = x^2 - \frac92 g(z,w) Y^2 + \varphi_0 (z,w) Y + \psi_0 (z,w) $,\\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 g(z,w) Y^2 + \varphi (z,w) Y + \psi (z,w) $,\\ where $ \varphi_0 \; (\text{resp. } \psi_0) $ is either 0 or the initial part of $ \varphi \; (\text{resp. }\psi) $. \\ \\ (II-A-I). \ $ h:=f_0-x^2 \in {\bold C} [Y,z,w] $ \ has no double factor. \\ Let $ C_j \subset \operatorname{Sing}(f_0) $ be an irreducible curve with $ 0 \in C_j $, and $ P = (0,a,b,c) \ne 0 $ an arbitrary point on $C_j$. \\ \\ (II-A-I-i). \ $ C_j \ni P = (0,a(t),b(t),t) \; ; \; t \ne 0 $.\\ Then, $ a(t)=a't^{\beta'/\delta'}, \; b(t)=b't^{\gamma'/\delta'} $ for some $ a', \; b' \in {\bold C} $.\\ Let $ \eta:=Y-a, \; \zeta:=z-b $, \ and $ f_0(w=t):=f_0(x, \eta+a, \zeta+b, t) $, then \\ \hspace*{1cm} $ f_0(w=t) = x^2 - \frac92 g(\zeta+b,t) (\eta+a)^2 + \varphi_0 (\zeta+b,t) (\eta+a) + \psi_0 (\zeta+b,t) $ \\ \hspace*{2.8cm} $ = x^2 - \frac92 g(\zeta+b,t) \eta^2 + (-9a g(\zeta+b,t) + \varphi_0 (\zeta+b,t)) \eta $ \\ \hspace*{3.2cm} $ - \frac92 a^2 g(\zeta+b,t) + a \varphi_0 (\zeta+b,t) + \psi_0 (\zeta+b,t) $.\\ Then both $ g(\zeta+b,t) \in {\bold C} [\zeta] $ and $ f_0(w=t)-x^2 \in {\bold C} [\eta,\zeta] $ have no double factor.\\ Therefore $ (f_0(w=t), \; (0,0,0)) $ is an isolated singularity under the coordinates $ (x, \eta, \zeta) $. \\ If $ g(b,t) \ne 0 $ then $ {\eta}^2 \in f_0(w=t) $. Otherwise, $ \zeta \in g(\zeta+b,t) $, so $ {\eta}^2 \zeta \in f_0(w=t) $. \\ Hence $ (f_0(w=t), \; (0,0,0)) $ is rational under the coordinates $ (x, \eta, \zeta) $ by Lemma 3.2. \\ \\ (II-A-I-ii). \ $ C_j \ni P = (0,a(s),s,0) \; ; \; s \ne 0 $.\\ Let $ \eta:=Y-a $, then \\ \hspace*{1cm} $ f_0(z=s) = x^2 - \frac92 g(s,w) (\eta+a)^2 + \varphi_0 (s,w) (\eta+a) + \psi_0 (s,w) $ \\ \hspace*{2.8cm} $ = x^2 - \frac92 g(s,w) {\eta}^2 + ( -9 a g(s,w) + \varphi_0 (s,w)) \eta $ \\ \hspace*{3.3cm} $ - \frac92 a^2 g(s,w) + a \varphi_0 (s,w) + \psi_0 (s,w). $ \\ $ (f_0(z=s), \; (0,0,0)) $ is an isolated singularity under the coordinates $ (x, \eta, w) $ because $ h:=f_0(z=s)-x^2 \in {\bold C}[\eta,w] $ has no double factor. We have $ {\eta}^2 \in f_0(z=s) $ for (1), (3), (4-a), (7), (8), (9), (10-a), and $ {\eta}^2 w \in f_0(z=s) $ for (2), (5), (6-a), since $ s \ne 0 $. Therefore $ (f_0(z=s), \; (0,0,0)) $ is rational. \\ \\ (II-A-I-iii). \ $ C_j \ni P = (0,a,0,0) \; ; \; a \ne 0 $.\\ Since $ f_0(Y=a) = x^2 - \frac92g(z,w) a^2 + \varphi_0(z,w) a + \psi_0(z,w) $, it follows that \\ $ (f_0(Y=a), \; (0,0,0)) $ is rational under the coordinates $ (x,z,w) $ by Lemma 3.2.\\ \\ (II-A-II). \ $ h:=f_0-x^2 \in {\bold C}[Y,z,w] $ has a double factor. \\ \hspace*{1cm} $ f_{\lambda} = x^2 + Y^3 - \frac92 g(z,w) Y^2 + \varphi (z,w) Y + \psi (z,w) $,\\ \hspace*{1cm} $ f_0 = x^2 - \frac92 g(z,w) Y^2 + \varphi_0 (z,w) Y + \psi_0 (z,w) $ \\ \hspace*{1.5cm} $ = x^2 - \frac92 g(z,w) (Y+\phi(z,w))^2 $.\\ After the coordinate change $ Y':=Y+\phi(z,w) $, \\ \hspace*{1cm} $ f_{\lambda} = x^2 + {Y'}^3 + (-\frac92 g - 3\phi)(z,w){Y'}^2 + \varphi'(z,w) Y'+ \psi'(z,w) $ \\ for some $ \varphi', \; \psi' \in {\bold C}[z,w] $ with $ \deg_{\alpha'} \varphi < \deg_{\alpha'} \varphi', \; \deg_{\alpha'} \psi < \deg_{\alpha'} \psi' $. We replace the weight $ \alpha'=(1/2,\beta',\gamma',\delta') $ with $ \alpha'' =(1/2,\beta'',\gamma'',\delta'') $ which satisfies the conditions \\ \hspace*{1cm} $ \frac13 < \beta'' < \frac12 < \beta''+\gamma''+\delta'', \quad \gamma''/\delta'' = \gamma'/\delta', $ \\ \hspace*{1cm} $ \deg_{\alpha''}(g(z,w){Y'}^2) = 1, \quad \deg_{\alpha''}(\varphi'(z,w) Y'), \; \deg_{\alpha''}\psi'(z,w) \geq 1 $, \; and \\ \hspace*{1cm} $ \deg_{\alpha''}(\varphi'(z,w) Y') \; \text{or} \; \deg_{\alpha''}\psi'(z,w) = 1 $,\\ and let $ f_0 $ be the initial part of $ f_{\lambda} $. Then \\ \hspace*{1cm} $ f_0 = x^2 - \frac92 g(z,w) {Y'}^2 + \varphi'_0 (z,w) Y' + \psi'_0 (z,w) $, \\ \hspace*{1cm} $ \beta' < \beta'', \quad \gamma' > \gamma'', \quad \delta' > \delta'', $ \\ \hspace*{1cm} $ \deg_{\alpha'}g(z,w) < \deg_{\alpha'}\phi(z,w), $ \\ \hspace*{1cm} $ \deg_{\alpha'}\varphi_0(z,w) < \deg_{\alpha'}\varphi'_0(z,w) $ or $ \varphi'_0(z,w) \equiv 0, $ \\ \hspace*{1cm} $ \deg_{\alpha'} \psi_0(z,w) < \deg_{\alpha'} \psi'_0(z,w) $ or $ \psi'_0(z,w) \equiv 0. $ \\ If $ h:=f_0-x^2 \in {\bold C}[Y',z,w] $ has no double factor, then Case (II-A-I). \\ If $h$ has a double factor, namely, \\ \hspace*{1.5cm} $ f_0 = x^2 - \frac92 g(z,w) (Y'+\phi'(z,w))^2 $,\\ then after the coordinate change $ Y'':=Y'+\phi'(z,w) $, \\ \hspace*{1cm} $ f_{\lambda} = x^2 + {Y''}^3 + (-\frac92 g - 3(\phi+\phi'))(z,w){Y''}^2 + \varphi''(z,w) Y''+ \psi''(z,w) $ \\ \hspace*{1.5cm} $ =: x^2 + {Y''}^3 + (-\frac92 g - 3\phi)(z,w){Y''}^2 + \varphi''(z,w) Y''+ \psi''(z,w) $ \\ for some $ \varphi'', \; \psi'' \in {\bold C}[z,w] $ with $ \deg_{\alpha''} \varphi' < \deg_{\alpha''} \varphi'', \; \deg_{\alpha''} \psi' < \deg_{\alpha''} \psi'' $.\\ If this procedure continues infinitely, then \\ \hspace*{1cm} $ f_{\lambda} = x^2 + {Y^{(n)}}^3 + (-\frac92 g - 3\phi)(z,w){Y^{(n)}}^2 + \varphi^{(n)}(z,w)Y^{(n)}+\psi^{(n)}(z,w) $,\\ \hspace*{1cm} $ \frac13 = \beta < \beta' < \beta'' < \dotsb < \beta^{(n)} < \dotsb < \frac12, $ \\ \hspace*{1cm} $ \gamma > \gamma' > \gamma'' > \dotsb > \gamma^{(n)} > \dotsb, $ \\ \hspace*{1cm} $ \delta > \delta' > \delta'' > \dotsb > \delta^{(n)} > \dotsb, $ \\ \hspace*{1cm} $ \gamma/\delta = \gamma'/\delta' = \gamma''/\delta'' = \dotsb = \gamma^{(n)}/\delta^{(n)} = \dotsb, $ \\ \hspace*{1cm} $ \frac12 < \beta^{(n)} + \gamma^{(n)} + \delta^{(n)} \quad \text{for all } n \in {\bold Z}_{\geq 0}, $ \\ \hspace*{1cm} $ \deg_{\alpha} g(z,w) < \deg_{\alpha} \phi(z,w), $ \\ \hspace*{1cm} $ \deg_{\alpha} \varphi_0(z,w) < \deg_{\alpha} \varphi'_0(z,w) < \deg_{\alpha} \varphi''_0(z,w) < \dotsb < \deg_{\alpha} \varphi_0^{(n)}(z,w) < \dotsb, $ \\ \hspace*{1cm} $ \deg_{\alpha} \psi_0(z,w) < \deg_{\alpha} \psi'_0(z,w) < \deg_{\alpha} \psi''_0(z,w) < \dotsb < \deg_{\alpha} \psi_0^{(n)}(z,w) < \dotsb, $ \\ so $ \displaystyle \mu(f_{\lambda},0) \geq \dim_{\bold C} {\bold C} \{ w \} \left/ \left( \varphi^{(n)}(0,w), \; \frac{\partial \psi^{(n)}}{\partial z}(0,w), \; \frac{\partial \psi^{(n)}}{\partial w}(0,w) \right) \right. \gg 1 $,\\ a contradiction. \\ \\ Next we consider Case (4-b), (6-b), (10-b), (10-c). Choose a $\Delta'$ such that: \\ \hspace*{1cm} $ (k'_0, l'_0) $ = (6, 0) for (4-b), \ (6, 3) for (6-b) and (10-b), \\ \hspace*{2.8cm} $ k'_1<6 $ and $ l'_0<6 $ for (10-c), \\ respectively. Then $ \displaystyle 1 \leq \frac{\gamma}{\delta} < \frac{\gamma'}{\delta'} $. Let $f_0$ be the initial part of $ f_{\lambda} \in {\bold C} [x,Y,z',w] $ with respect to the weight $ \alpha':=(1/2, 1/3, \gamma', \delta') $, and $ m':=\gamma'/\delta' $.\\ For the case ${z'}^3 Y^2$ is not contained in $f_0$ of (10-c), the situation is similar to (I-A), but $ h:=f_0-x^2 \in {\bold C}[Y,z^{(n)},w] $ can not have a double factor for $ n \in {\bold N} $. So we consider other cases, i.e., \begin{align*} \hspace*{1cm} f_0 &= x^2 + Y^3 - \frac92c_{kl}^{(\prime)}{z'}^k w^l Y^2 + \varphi'_0(z',w)Y + \psi'_0(z',w) \\ &= \begin{cases} x^2 + Y^3 - \frac92c_{20}{z'}^2 Y^2 + \varphi'_0(z',w)Y + \psi'_0(z',w), \qquad & \text{(4-b)} \\ x^2 + Y^3 - \frac92c_{21}{z'}^2 w Y^2 + \varphi'_0(z',w)Y + \psi'_0(z',w), \qquad & \text{(6-b)} \\ x^2 + Y^3 - \frac92c'_{21}{z'}^2 w Y^2 + \varphi'_0(z',w)Y + \psi'_0(z',w), \qquad & \text{(10-b)} \\ x^2 + Y^3 - \frac92c_{30}{z'}^3 Y^2 + \varphi'_0(z',w)Y + \psi'_0(z',w), \qquad & \text{(10-c)} \end{cases} \end{align*} where $ \deg\varphi'_0(z',t)<2k, \; \deg\psi'_0(z',t)<3k \quad (0 \ne t \in {\bold C}). $ \\ \\ (II-A-I$'$). \ $ h:=f_0-x^2 \in {\bold C} [Y,z',w] $ \ has no double factor. \\ (II-A-I$'$-i). \ $ \text{Sing}(f_0) \supset C_j \ni P=(0,a(t),b(t),t) \; ; \; t \ne 0 $.\\ Let $ \eta:=Y-a, \; \zeta:=z'-b $ and $ f_0(w=t):=f_0(x, \eta+a, \zeta+b, t) $, then $ f_0(w=t)-x^2 $ has no double factor. \\ (Indeed, assume $ f_0(w=t)-x^2=(\eta+A(\zeta))^2(\eta+B(\zeta)) $ for some $ A, \; B \in {\bold C}[\zeta] $, then \\ \hspace*{1cm} $ f_0(w=t)-x^2=(Y-a(t)+A(z'-b(t)))^2(Y-a(t)+B(z'-b(t))) $,\\ \hspace*{1cm} $ f_0-x^2=(Y-a(w)+A(z'-b(w)))^2(Y-a(w)+B(z'-b(w))) $ \\ \hspace*{2.4cm} $ =(Y+G(z',w))^2(Y+H(z',w)) $ \\ \hspace*{2.4cm} $ (G:=-a(w)+A(z'-b(w)), \; H:=-a(w)+B(z'-b(w)) \in {\bold C}[z',w^{m'}].) $ \\ \hspace*{2.4cm} $ =Y^3+(2G+H)Y^2+G(G+2H)Y+G^2H $ \\ \hspace*{2.4cm} $ =Y^3-\frac92g'Y^2-G(9g'+3G)Y-G^2(\frac92g'+2G) $,\\ where $ g'(z',w):=c_{k l}^{(\prime)}{z'}^k w^l $. It follows that $ \deg G(z',t) < k $ from $ \deg \varphi'_0(z',t) < 2k, \; \deg \psi'_0(z',t) < 3k $, and that $ G \in {\bold C}[z',w] $ from $ g', \; G(9g'+3G), \; G^2(\frac92g'+2G) \in {\bold C}[z',w], \; G \in {\bold C}[z',w^{m'}] $.)\\ If $ a=b=0 $ then \\ \hspace*{2cm} $ f_0(w=t) = x^2 + Y^3 - \frac92c_{k l}^{(\prime)}{z'}^k t^l Y^2 + \varphi'_0(z',t)Y + \psi'_0(z',t), $ \\ with $ \text{ord}\varphi'_0(z',t) < 4 \leq 2k $ or $ \text{ord}\psi'_0(z',t) < 6 \leq 3k $.\\ If $ a=0, \; b \ne 0 $, then \\ \hspace*{1cm} $ f_0(w=t) = x^2 + Y^3 - \frac92c_{k l}^{(\prime)} ({\zeta}^k+kb{\zeta}^{k-1} +\dotsb+kb^{k-1}\zeta+b^k)t^l Y^2 $ \\ \hspace*{3.3cm} $ + \varphi'_0(\zeta+b,t)Y + \psi'_0(\zeta+b,t) \ni Y^2. $ \\ If $ a \ne 0, \; b=0 $, then \\ \hspace*{1cm} $ f_0(w=t) = x^2 + {\eta}^3 + 3a{\eta}^2 + (3a^2 + \varphi'_0(z',t))\eta - \frac92c_{k l}^{(\prime)}{z'}^k t^l (\eta + a)^2 + \dotsb \ni {\eta}^2. $ \\ If $ a \ne 0, \; b \ne 0 $, then $ {\eta}^2 \in f_0(w=t) $. (Indeed, the coefficient of ${\eta}^2$ in $f_0(w=t)$ is $ \displaystyle \frac12 \frac{\partial^2 f_0}{(\partial Y)^2}(0,a,b,t) $, so it follows that:\\ \hspace*{2.5cm} the coefficient of ${\eta}^2$ in $f_0(w=t)$ is 0 \\ \hspace*{2cm} $ \Longleftrightarrow f_0(0,Y,b(t),t)=(Y-a(t))^3 $ \\ \hspace*{2cm} $ \Longleftrightarrow f_0(0,Y,b(w),w)=(Y-a(w))^3 $ \\ \hspace*{2cm} $ \Longleftrightarrow f_0(0,Y,z',w)=(Y-\frac32c_{k l}^{(\prime)}{z'}^k w^l)^3 $.)\\ Hence $(f_0(w=t), 0)$ is rational. \\ \\ (II-A-I$'$-ii). \ $ \text{Sing}(f_0) \supset C_j \ni P=(0,a(s),s,0) \; ; \; s \ne 0 $.\\ Then $ a=0 $ because $ \displaystyle 0=f_0(0,a,s,0)=\frac{\partial f_0}{\partial Y}(0,a,s,0) =\varphi'_0(s,0)=\psi'_0(s,0) $.\\ Furthermore, \\ \hspace*{1cm} $ f_0(z'=s) - x^2 :=f_0(x, Y, s, w) - x^2 = Y^3 - \frac92c_{k l}^{(\prime)}s^k w^l Y^2 + \varphi'_0(s,w)Y + \psi'_0(s,w) $ \\ has no double factor. So it follows that $(f_0(z'=s),0)$ is rational from \\ \hspace*{1cm} $ Y^2 \in f_0(z'=s) $ for (4-b) and (10-c), \\ \hspace*{1cm} $ Y^2 w \in f_0(z'=s) $ and $ w^i, \; Y w^j $ are not contained in $f_0(z'=s)$ for $ i \leq 3, \; j \leq 2 $ \\ \hspace*{1cm} for (6-b) and (10-b).\\ \\ (II-A-II$'$). \ $ h:=f_0-x^2 \in {\bold C} [Y,z',w] $ \ has a double factor. \\ $f_0$ and $f_{\lambda}$ are written as: \\ \hspace*{1cm} $ f_0=x^2+(Y+G(z',w))^2 (Y+(-\frac92g'-2G)(z',w)) $ \\ \hspace*{1.5cm} $ =:x^2+{Y'}^3-(\frac92g'+3G)(z',w){Y'}^2, \qquad (Y':=Y+G(z',w).) $ \\ \hspace*{1cm} $ f_{\lambda}=: \begin{cases} x^2+{Y'}^3-\frac92g''(z',w){Y'}^2+\varphi''(z',w)Y'+\psi''(z',w), \qquad \; \text{(4-b, 6-b, 10-c)} \\ x^2+{Y'}^3-\frac92(c_{3 0}{z'}^3+g''(z',w)){Y'}^2+\varphi''(z',w)Y' +\psi''(z',w), \qquad \text{(10-b)} \end{cases} $ \\ \\ for some $ g'':=g'+\frac23G, \; \varphi'', \; \psi'' \in {\bold C}[z',w] $ with $ \deg G(z',t) < k \quad (0 \ne t \in {\bold C}), \\ \deg_{\alpha'}\varphi''>2/3=2(k\gamma'+l\delta'), \; \deg_{\alpha'}\psi''>1 $.\\ Then $g''$ is one of the following:\\ \\ $ \hspace*{1cm} \left. \begin{gathered} \text{(3) \ } g'' = c_{2 0} {z'}^2 + c'_{0 L'} w^{L'}, \qquad 3 \leq L', \quad L' \; \text{is odd}, \hspace{1.2cm} \\ \text{(4-a) \ } g''= c_{2 0}(z'+ \gamma'_1 w^{L'})(z'+ \gamma'_2 w^{L'}), \quad 2 \leq L', \; \gamma'_1 \ne \gamma'_2, \\ \text{(4-b) \ } g'' = c_{2 0} (z' + \gamma'_1 w^{L'})^2, \qquad 2 \leq L', \hspace{3.2cm} \end{gathered} \right\} \quad \text{(become from 4-b)} \\ \\ \hspace*{1cm} \left. \begin{gathered} \text{(5) \ } g'' = c_{2 1}^{(\prime)}{z'}^2 w + c'_{0 (L'+1)}w^{L'+1}, \qquad 3 \leq L', \quad L' \; \text{is odd}, \; \\ \text{(6-a) \ } g'' = c_{2 1}^{(\prime)} (z' + \gamma'_1 w^{L'}) (z' + \gamma'_2 w^{L'}) w, \quad 2 \leq L', \; \gamma'_1 \ne \gamma'_2, \\ \text{(6-b) \ } g'' = c_{2 1}^{(\prime)} (z' + \gamma'_1 w^{L'})^2 w, \qquad 2 \leq L', \hspace{3.2cm} \end{gathered} \right\} \quad \begin{gathered} \text{(become from} \\ \text{6-b or 10-b)} \end{gathered} \\ \\ \hspace*{1cm} \left. \begin{gathered} \text{(7) \ } g'' = c_{3 0} {z'}^3 + c'_{0 4} w^4, \quad \\ \text{(8) \ } g'' = c_{3 0} {z'}^3 + c'_{1 3} z' w^3, \\ \text{(9) \ } g'' = c_{3 0} {z'}^3 + c'_{0 5} w^5, \quad \end{gathered} \right\} \quad \text{(become from 10-c).} $ \\ \\ For 3, 4-a, 5, 6-a, 7, 8 and 9, we replace the weight $ \alpha'=(1/2, 1/3, \gamma', \delta') $ with $ \alpha''=(1/2, \beta'', \gamma'', \delta'') $ which satisfies the conditions \\ \hspace*{1cm} $ \frac13 < \beta'' < \frac12 < \beta''+\gamma''+\delta'', \quad \gamma''/\delta'' = \gamma'/\delta', $ \\ \hspace*{1cm} $ \deg_{\alpha''}(g''(z',w){Y'}^2) = 1, \quad \deg_{\alpha''}(\varphi''(z',w) Y'), \; \deg_{\alpha''}\psi''(z',w) \geq 1 $, \; and \\ \hspace*{1cm} $ \deg_{\alpha''}(\varphi''(z',w) Y') \; \text{or} \; \deg_{\alpha''}\psi''(z',w) = 1 $,\\ and let $ f_0 $ be the initial part of $ f_{\lambda} $.\\ If $ h:=f_0-x^2 \in {\bold C}[Y',z',w] $ has no double factor then Case (II-A-I). Otherwise, Case (II-A-II), and this procedure must finish in finite times from the assumption. \\ For 4-b, 6-b, after the coordinate change $z'':=z'+\gamma'_1 w^{L'}$, we define $\Lambda''$, $\Gamma''$ under the coordinates $(x, Y', z'', w)$ similarly as before and choose a $\Delta''$ corresponding to a weight $ \alpha''=(1/2, 1/3, \gamma'', \delta'') $ which satisfies $ (k''_0, l''_0) =(6, 0) $ for (4-b), \ (6, 3) for (6-b). Then $ \displaystyle \frac16 < \gamma''+\delta''$ and $ \displaystyle 1 \leq \frac{\gamma}{\delta} < \frac{\gamma'}{\delta'} < \frac{\gamma''}{\delta''} $. Let $f_0$ be the initial part of $f_{\lambda}$. Then, \\ \\ \hspace*{1cm} $ f_0 = \begin{cases} x^2+{Y'}^3-\frac92 c_{2 0}{z''}^2{Y'}^2+\varphi'''_0(z'',w)Y' +\psi'''_0(z'',w), \qquad & \text{(4-b)} \\ x^2+{Y'}^3-\frac92 c_{2 1}^{(\prime)}{z''}^2 w{Y'}^2+\varphi'''_0(z'',w)Y' +\psi'''_0(z'',w), \qquad & \text{(6-b)} \end{cases} $ \\ \\ where $ \deg\varphi'''_0(z'',t) < 4, \; \deg\psi'''_0(z'',t) < 6 \quad (0 \ne t \in {\bold C}) $. \\ If $ h:=f_0-x^2 \in {\bold C}[Y',z'',w] $ has no double factor then Case (II-A-I$'$). Otherwise, Case (II-A-II$'$), and this procedure must finish in finite times from the assumption. \\ \\ (II-B). \ When $ \gamma < \delta $, $ g(z,w) $ can be classified as below: \\ \hspace*{1cm} (1$'$) \ \ $ g = c_{0 1} w + c_{K 0} z^K, \qquad 2 \leq K $,\\ \hspace*{1cm} (2$'$) \ \ $ g = c_{1 1} z w + c_{(K+1) 0} z^{K+1}, \qquad 2 \leq K $, \\ \hspace*{1cm} (3$'$) \ \ $ g = c_{0 2} w^2 + c_{K 0} z^K, \qquad 3 \leq K $, \ K is odd, \\ \hspace*{1cm} (4-a$'$) \ $ g = c_{0 2} (w + \gamma_1 z^K)(w + \gamma_2 z^K), \quad 2 \leq K, \ \gamma_1 \ne \gamma_2 $, \ ( for No.10, 83 only ), \\ \hspace*{1cm} (4-b$'$) \ $ g = c_{0 2} (w + \gamma_1 z^K)^2, \qquad 2 \leq K $, \qquad ( for No.10, 83 only ), \\ \hspace*{1cm} (5$'$) \ \ $ g = c_{1 2} z w^2 + c_{(K+1) 0} z^{K+1}, \quad 3 \leq K $, \ K is odd, \qquad ( for No.10 only ),\\ \hspace*{1cm} (7$'$) \ \ $ g = c_{0 3} w^3 + c_{4 0} z^4 $, \qquad ( for No.10 only ). \\ For Cases (1$'$), (2$'$), (3$'$), (4-a$'$), (5$'$) and (7$'$), the situation is similar to (II-A). For Case (4-b$'$) of No.10 and 83, we define $ \Lambda', \Gamma', \Delta', f_0 $ and $h$ similarly as in (II-A), and repeat the same argument as in (II-A). Namely, if $ h:=f_0-x^2 \in {\bold C}[Y,z,w'] $ has no double factor then (II-B-I$'$), and if $h$ has a double factor then (II-B-II$'$). Then the condition $ \beta^{(n)}+\gamma^{(n)}+\delta^{(n)}>\frac12 $ must be always satisfied for No.10, because \\ $ \mu(f_{\lambda},0)<\mu(f,0)=242 $. For the case of $(k_1,l_1)=(k'_1,l'_1)=(0,6)$ of No.83, if it becomes (II-B-II$'$), then $ z^k \in f_{\lambda} $ for some $ k \geq 15 $. And so, if $ \beta^{(n)}+ \gamma^{(n)}+\delta^{(n)} \leq 1/2 $ for some $ n \in {\bold N} $ then $ \mu(f_{\lambda},0) \geq (3-1)(10-1)(15-1)=252 $, which is a contradiction to the condition $ \mu(f_{\lambda},0)<\mu(f,0)=245 $. And this argument must finish finitely since $ \mu(f_{\lambda},0)<\mu(f,0) $.\\ For arrangement, let us illustrate the above argument as follows:\\ \\ {\small $ \text{I} \begin{cases} \text{I-A} \begin{cases} a \ne 0 \quad \; \; : \text{O.K.} \dotsb \; (*) \\ a=b=0 : \text{O.K.} \dotsb \; (**) \\ a=0, \; b \ne 0 \begin{cases} \zeta^i \; (\exists \; i \leq 5) \; \text{or} \; y\zeta^j \; (\exists \; j \leq 3) \in f_0(w=t) : \text{O.K.} \dotsb \; (\text{***}) \\ \text{otherwise} : z':=z-b'w^m \; \longrightarrow \begin{cases} \text{I-A} \\ \text{II-A} \end{cases} \end{cases} \end{cases}\\ \\ \text{I-B} \begin{cases} a \ne 0 \quad \; \; : \text{O.K.} \dotsb \; (*') \\ a=c=0 : \text{O.K.} \dotsb \; (**') \\ a=0, \; c \ne 0 \begin{cases} \omega^i \; (\exists \; i \leq 5) \; \text{or} \; y\omega^j \; (\exists \; j \leq 3) \in f_0(z=s) : \text{O.K.} \dotsb \; (\text{***}') \\ \text{otherwise} : w':=w-c'z^m \; \longrightarrow \begin{cases} \text{I-B} \\ \text{II-B} \end{cases} \end{cases} \end{cases} \end{cases} \\ \\ \text{II} \begin{cases} \text{II-A} \begin{cases} \begin{gathered} \text{1, 2, 3, 4-a, 5, } \\ \text{6-a, 7, 8, 9, 10-a } \end{gathered} \begin{cases} \text{II-A-I : O.K.} \dotsb \; (*'') \\ \text{II-A-II} : Y':=Y+\phi \; \to \begin{cases} \text{II-A-I : O.K.} \dotsb \; (*'') \\ \text{II-A-II} : Y'':=Y'+\phi' \; \to \; \dotsb \end{cases} \end{cases} \\ \\ \begin{gathered} \text{4-b} \\ \text{6-b} \\ \text{10-b} \end{gathered} : z':=z+\gamma_1 w^L \; \to \begin{cases} \text{II-A-I$'$ : O.K.} \dotsb \; (**'') \\ \text{II-A-II$'$} : Y':=Y+G \to \begin{cases} \begin{gathered} \text{3, 4-a, } \\ \text{5, 6-a} \end{gathered} \begin{cases} \text{II-A-I : O.K.} \dotsb \; (*'') \\ \text{II-A-II} : Y'':=Y'+\phi' \to \end{cases} \\ \\ \begin{gathered} \text{4-b} \\ \text{6-b} \end{gathered} : z'':=z'+\gamma'_1 w^{L'} \\ \hspace*{0.7cm} \to \begin{cases} \text{II-A-I$'$ : O.K.} \dotsb (**'') \\ \text{II-A-II$'$} : Y'':=Y'+G' \to \end{cases} \end{cases} \end{cases}\\ \\ \text{10-c} : z':=z+\gamma_1 w \to \begin{cases} {z'}^3 Y^2 \in f_0 \begin{cases} \text{II-A-I$'$ : O.K.} \dotsb \; (**'') \\ \text{II-A-II$'$} : Y':=Y+G \to \begin{cases} \text{II-A-I : O.K.} \dotsb \; (*'') \\ \text{II-A-II} : Y'':=Y'+\phi' \to \end{cases} \end{cases}\\ \text{otherwise} : \text{I-A} \end{cases} \end{cases}\\ \\ \text{II-B} \begin{cases} \begin{gathered} \text{1$'$, 2$'$, 3$'$, } \\ \text{4-a$'$, 5$'$, 7$'$ } \end{gathered} \begin{cases} \text{II-B-I : O.K.} \dotsb \; (*''') \\ \text{II-B-II} : Y':=Y+\phi \; \to \begin{cases} \text{II-B-I : O.K.} \dotsb \; (*''') \\ \text{II-B-II} : Y'':=Y'+\phi' \; \to \; \dotsb \end{cases} \end{cases} \\ \\ \text{4-b$'$} : w':=w+\gamma_1 z^K \; \to \begin{cases} \text{II-B-I$'$ : O.K.} \dotsb \; (**''') \\ \text{II-B-II$'$} : Y':=Y+G \to \begin{cases} \text{3$'$, 4-a$'$} \begin{cases} \text{II-B-I : O.K.} \dotsb \; (*''') \\ \text{II-B-II} : Y'':=Y'+\phi' \to \end{cases} \\ \\ \text{4-b}' : w'':=w'+\gamma'_1 z^{K'} \\ \hspace*{0.5cm} \to \begin{cases} \text{II-B-I$'$ : O.K.} \dotsb (**''') \\ \text{II-B-II$'$} : Y'':=Y'+G' \to \end{cases} \end{cases} \end{cases} \end{cases} \end{cases} $} \\ \\ The procedures of combinations of ``a coordinate change $\longrightarrow$" must finish in finite times. Namely, taking suitable coordinate changes in finite times if necessary, the situation can be reduced to the case that $(\{f_0=0\} \cap H, P)$ is rational i.e. the case \\ $(*\dotsb*^{(m)})$ of (I) or (II), after all. Thus, there exist a local coordinate system and a weight such that $(f_0, P)$ is rational. Furthermore, the condition $ \frac12 + \beta^{(n)} + \gamma^{(n)} + \delta^{(n)} > 1 $ is satisfied under each coordinate system appearing in the above procedures. \\ Thus it is enough to show Claim 3.5 for the proof of Theorem 2.7. \\ \\ {\it Proof of Claim 3.5}.\\ Step 1. Since $(f_{\lambda},0)$ is an isolated singularity, it follows that: \\ \hspace*{2cm} $ z^k $ or $ y z^k $ or $ z^k w \in f_{\lambda}, $ \; and \; $ w^l $ or $ y w^l $ or $ z w^l \in f_{\lambda}. $ \\ Taking suitable coordinate change \\ \hspace*{1cm} $ y' := y + b_1 z^{K_1} + c_1 w^{L_1}, \quad z' := z + c_2 w^{L_2}, \quad w' := w + b_2 z^{K_2} $ \\ for some sufficiently large $ K_i, \; L_j \in {\bold N} $ and some $ b_i, \; c_j \in {\bold C} $, we have $ {z'}^{k'}, \; {w'}^{l'} \in f_{\lambda} $ and $ \Gamma(f_{\lambda}) $ is the same as before except adding compact faces touching some coordinate planes.\\ \\ Step 2 ([A]-Thm. XXII, [AGV]-12.7, Kouchnirenko [K]-Thm. I). Let $ y^J, z^K, w^L \in F(y,z,w) \in {\bold C}[y,z,w] $ and $ V=\bigcup_{i=0} ^3 V_i $ be a decomposition of the three-dimensional region of the positive orthant below the Newton boundary $ \Gamma(F) \subset {{\bold R}_{\geq 0}}^3 = \{ (Y,Z,W) \in {{\bold R}_{\geq 0}}^3 \} $ which satisfies: \\ \hspace*{1cm} $V_0$ is the three-dimensional simplicial cone which has vertex $ (0,0,0), \; (j,0,0), $ \\ \hspace*{1cm} $ (0,k,0), \; (0,0,l) $ with $ j, k, l \in {\bold Q}_{>0} \; ; \; j \leq J, \; k \leq K, \; l \leq L $,\\ \hspace*{1cm} $V_1$ has a vertex $ (J,0,0) $, \ $V_2$ has a vertex $ (0,K,0) $, \ $V_3$ has a vertex $ (0,0,L) $, \\ \hspace*{1cm} $ \dim_{\bold R}(V_i \cap V_j) \leq 2 $ for $ i \ne j $, \; and $ S_{1 2} = S_{2 3} = S_{3 1} = \emptyset $ for \\ \hspace*{1cm} $ S_{i 1} := V_i \cap (YZ \text{-plane}), \; S_{i 2} := V_i \cap (ZW \text{-plane}), \; S_{i 3} := V_i \cap (WY \text{-plane}) $.\\ (See {\sc Figure} 5.) \begin{figure}[h] \setlength{\unitlength}{1mm} \begin{picture}(155,155)(-60,-60) \multiput(0,0)(1.5,0){50}{\circle*{0.2}} \multiput(0,0)(0,1.5){50}{\circle*{0.2}} \multiput(0,0)(-1,-1){40}{\circle*{0.2}} \put(75,0){\vector(1,0){10}} \put(0,75){\vector(0,1){10}} \put(-40,-40){\vector(-1,-1){10}} \put(-2,87){$Y$} \put(-55,-55){$Z$} \put(87,-2){$W$} \put(25,-10){\thicklines\line(1,3){10}} \put(35,20){\thicklines\line(3,-2){15}} \put(50,10){\thicklines\line(2,-1){10}} \put(60,5){\thicklines\line(3,-1){15}} \put(75,0){\thicklines\line(-5,-1){50}} \put(-10,25){\thicklines\line(3,2){15}} \put(5,35){\thicklines\line(1,0){15}} \put(20,35){\thicklines\line(-3,4){15}} \put(5,55){\thicklines\line(-1,4){5}} \put(0,75){\thicklines\line(-1,-5){10}} \put(-40,-40){\thicklines\line(2,1){50}} \put(10,-15){\thicklines\line(-2,1){10}} \put(0,-10){\thicklines\line(-1,1){10}} \put(-10,0){\thicklines\line(-1,2){5}} \put(-15,10){\thicklines\line(-1,-2){25}} \put(-15,10){\thicklines\line(1,3){5}} \put(20,35){\thicklines\line(1,-1){15}} \put(10,-15){\thicklines\line(3,1){15}} \multiput(10,-15)(-1.5,-0.5){25}{\circle*{0.2}} \multiput(-15,10)(-0.5,-1.5){25}{\circle*{0.2}} \multiput(25,-10)(1.5,0.5){20}{\circle*{0.2}} \multiput(55,0)(-1,1){20}{\circle*{0.2}} \multiput(20,35)(-1,1){20}{\circle*{0.2}} \multiput(0,55)(-0.5,-1.5){20}{\circle*{0.2}} \put(-5,75){$J$} \put(-40,-45){$K$} \put(75,-5){$L$} \put(-15,60){$j$} \put(-20,-40){$k$} \put(60,-15){$l$} \put(10,10){$V_0$} \put(10,55){$V_1$} \put(-40,-25){$V_2$} \put(55,10){$V_3$} \put(-10,60){\line(2,-1){9}} \put(60,-10){\line(-1,2){4.5}} \put(-20,-35){\line(-1,1){7}} \end{picture} \caption{ } \end{figure} \\ Suppose that $(F,0)$ is an isolated singularity, then {\allowdisplaybreaks \begin{align*} \mu(F,0) & \geq 3! \left( \sum_{i=0}^3 V_i \right) - 2! \left( \sum_{i=0}^3 \sum_{j=1}^3 S_{i j} \right) + 1!(J+K+L) - 1 \\ & = jkl-(jk+kl+lj)+j+k+l-1 \\ & \quad + \sum_{i=1}^3 \left( 6V_i-2\sum_{j=1}^3 S_{i j} \right) + (J-j) + (K-k) + (L-l) \\ & \geq (j-1)(k-1)(l-1). \end{align*}} Step 3. If $ 0 < \delta' < \delta \leq \gamma < \gamma' < 1 $ and $ \gamma + \delta = \gamma' + \delta' $ then $$ \left( \frac{1}{\gamma}-1 \right) \left( \frac{1}{\delta}-1 \right) < \left( \frac{1}{\gamma'}-1 \right) \left( \frac{1}{\delta'}-1 \right). $$ In fact, let $ \gamma + \delta = \gamma' + \delta' =: 1/c, \; c \in {\bold R} $, \ then {\allowdisplaybreaks \begin{align*} & \quad \frac{1}{\gamma' \delta'}-\frac{1}{\gamma'}-\frac{1}{\delta'}- \left( \frac{1}{\gamma \delta}-\frac{1}{\gamma}-\frac{1}{\delta} \right) \\ & = (c-1)(\gamma'-\gamma)(c(\gamma+\gamma')-1)/ \gamma\gamma'(1-c\gamma)(1-c\gamma') > 0. \end{align*}} Step 4. From Step 1 - Step 3, if $ \gamma'+\delta'= 1/c \leq 1/6 $ then {\allowdisplaybreaks \begin{align*} \mu(f_{\lambda},0) & \geq (3-1) \left( \frac{1}{\gamma'}-1 \right) \left( \frac{1}{\delta'}-1 \right) \\ & \geq (3-1) \left( \frac{6}{c\gamma'}-1 \right) \left( \frac{6}{c\delta'}-1 \right) \\ & > (3-1) \left( \frac{1}{\alpha_3}-1 \right) \left( \frac{1}{\alpha_4}-1 \right) = \mu(f,0), \end{align*}} a contradiction. This completes the proof of Claim 3.5 and Theorem 2.7. \ \ \ Q.E.D.\\ \vspace{1cm}

9702

alg-geom/9702004

en

https://arxiv.org/abs/alg-geom/9702004

alg-geom/9702004

Alice Silverberg

A. Silverberg and Yu. G. Zarhin

Semistable reduction of abelian varieties over extensions of small degree

LaTeX2e

We obtain necessary and sufficient conditions for abelian varieties to acquire semistable reduction over fields of low degree. Our criteria are expressed in terms of torsion points of small order defined over unramified extensions.

alg-geom

\section{Introduction} In this paper we obtain criteria for abelian varieties to acquire semistable reduction over fields of certain given (small) degrees. Our criteria are expressed in terms of unramified torsion points. Suppose that $X$ is an abelian variety defined over a field $F$, and $n$ is a positive integer not divisible by the characteristic of $F$. Let $X^\ast$ denote the dual abelian variety of $X$, let $X_n$ denote the kernel of multiplication by $n$ in $X(F^s)$, where $F^s$ denotes a separable closure of $F$, let $X_n^\ast$ denote the kernel of multiplication by $n$ in $X^\ast(F^s)$, and let $\boldsymbol \mu _n$ denote the $\mathrm{Gal} (F^s/F)$-module of $n$-th roots of unity in $F^s$. The Weil pairing $e_n : X_n \times X_n^\ast \to {\boldsymbol \mu}_n$ is a $\mathrm{Gal}(F^s/F)$-equivariant nondegenerate pairing. If $S$ is a subgroup of $X_n$, let $$S^{\perp_n} = \{ y \in X_n^\ast : e_n(x,y) = 1 \text{ for every } x \in S \} \subseteq X_n^\ast.$$ For example, if $n = m^2$ and $S = X_m$, then $S^{\perp_n} = X_m^\ast$. If $X$ is an elliptic curve and $S$ is a cyclic subgroup of order $n$, then $S^{\perp_n} = S$. Suppose that $v$ is a discrete valuation on $F$ whose residue characteristic does not divide $n$. Previously we showed that if $n \ge 5$ then $X$ has semistable reduction at $v$ if and only if there exists a subgroup $S$ of $X_n$ such that all the points on $S$ and on $S^{\perp_n}$ are defined over an extension of $F$ unramified over $v$ (see Theorem 4.5 of \cite{dpp}; see also Theorem 6.2 of \cite{semistab}). In the current paper we show that if there exists a subgroup $S$ of $X_n$, for $n = 2$, $3$, or $4$ (respectively), such that all the points on $S$ and on $S^{\perp_n}$ are defined over an extension of $F$ unramified over $v$, then $X$ acquires semistable reduction over every degree $4$, $3$, or $2$ (respectively) extension of $F$ totally ramified above $v$. We also give necessary and sufficient conditions for semistable reduction over quartic, cubic, and quadratic extensions. Namely, if $L$ is a totally ramified extension of $F$ of degree $4$, $3$, or $2$, respectively, then $X$ has semistable reduction over $L$ if and only if there exist a finite unramified extension $K$ of $F$, an abelian variety $Y$ over $K$ which is $K$-isogenous to $X$, and a subgroup $S$ of $Y_n$, for $n = 2$, $3$, or $4$, respectively, such that all the points of $S$ and of $S^{\perp_n}$ are defined over an unramified extension of $K$. If $X$ is an elliptic curve one may take $Y = X$. This is not true already for abelian surfaces. However, one may take $Y = X$ in the special case where $X$ has purely additive and potentially good reduction, with no restriction on the dimension. The study of torsion subgroups of abelian varieties with purely additive reduction was initiated in \cite{LenstraOort} and pursued in \cite{Lorenzini} (see \cite{Frey} and \cite{Flexor-Oesterle} for the case of elliptic curves). See \cite{Kraus} for a study of the smallest extension over which an elliptic curve with additive and potentially good reduction acquires good reduction. We state and prove Theorem \ref{oneway} in the generality $n \ge 2$ (rather than just $2 \le n \le 4$) since doing so requires no extra work and affords us the opportunity to give a slightly different exposition from that in \cite{dpp} for $n \ge 5$, which highlights the method. See \S\ref{ssredsect} for our major results, see \S\ref{appsect} for applications and refinements, and see \S\ref{exssect} for examples which demonstrate that our results are sharp. \section{Notation and definitions} Define $$R(n) = 1 \text{ if } n \ge 5, \quad R(4) = 2, \quad R(3) = 3, \quad R(2) = 4.$$ If $X$ is an abelian variety over a field $F$, and $\ell$ is a prime not equal to the characteristic of $F$, let $$\rho_{\ell,X} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(T_\ell(X))$$ denote the $\ell$-adic representation on the Tate module $T_\ell(X)$ of $X$. We will write $\rho_\ell$ when there is no ambiguity. Let $V_\ell(X) = T_\ell(X)\otimes_{{\mathbf Z}_\ell}{\mathbf Q}_\ell$. If $L$ is a Galois extension of $F$ and $w$ is an extension of $v$ to $L$, let ${\mathcal I}(w/v)$ denote the inertia subgroup at $w$ of $\mathrm{Gal}(L/F)$. Throughout this paper we will let ${\mathcal I}$ denote ${\mathcal I}({\bar v}/v)$, where ${\bar v}$ is a fixed extension of $v$ to $F^s$, and we will let ${\mathcal J}$ denote the first ramification group (i.e., the wild inertia group). We also write ${\mathcal I}_{w}$ for ${\mathcal I}({\bar v}/w)$. \begin{defn} Suppose $L/F$ is an extension of fields, $w$ is a discrete valuation on $L$, and $v$ is the restriction of $w$ to $F$. Let $e(w/v) = [w(L^\times):v(F^\times)]$. We say that $w/v$ is {\em unramified} if $e(w/v) = 1$ and the residue field extension is separable. We say that $w/v$ is {\em totally ramified} if $w$ is the unique extension of $v$ to $L$ and the residue field extension is purely inseparable. We say that $w/v$ is {\em tamely ramified} if the residue field extension is separable and $e(w/v)$ is not divisible by the residue characteristic. \end{defn} \section{Preliminaries} \begin{thm} \label{quasithm} Suppose $n$ is an integer, $n \ge 2$, ${\mathcal O}$ is an integral domain of characteristic zero such that no rational prime which divides $n$ is a unit in ${\mathcal O}$, $\alpha \in {\mathcal O}$, $\alpha$ has finite multiplicative order, and $(\alpha-1)^2 \in n{\mathcal O}$. Then $\alpha^{R(n)} = 1$. \end{thm} \begin{proof} See Corollary 3.3 of \cite{serrelem}. \end{proof} \begin{lem}[Lemma 5.2 of \cite{semistab}] \label{localglobal} Suppose that $d$ and $n$ are positive integers, and for each prime $\ell$ which divides $n$ we have a matrix $A_\ell \in M_{2d}({\mathbf Z}_\ell)$ such that the characteristic polynomials of the $A_\ell$ have integral coefficients independent of $\ell$, and such that $(A_\ell-1)^2 \in nM_{2d}({\mathbf Z}_\ell)$. Then for every eigenvalue $\alpha$ of $A_\ell$, $(\alpha-1)/\sqrt{n}$ satisfies a monic polynomial with integer coefficients. \end{lem} \begin{thm}[Galois Criterion for Semistable Reduction] \label{galcrit} Suppose $X$ is an abel\-ian variety over a field $F$, $v$ is a discrete valuation on $F$, and $\ell$ is a prime not equal to the residue characteristic of $v$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has semistable reduction at $v$, \item[(ii)] ${\mathcal I}$ acts unipotently on $T_\ell(X)$; i.e., all the eigenvalues of $\rho_{\ell}(\sigma)$ are $1$, for every $\sigma \in {\mathcal I}$, \item[(iii)] for every $\sigma \in {\mathcal I}$, $(\rho_{\ell}(\sigma)-1)^2 = 0$. \end{enumerate} \end{thm} \begin{proof} See Proposition 3.5 and Corollaire 3.8 of \cite{SGA} and Theorem 6 on p.~184 of \cite{BLR}. \end{proof} \begin{lem} \label{primrtsof1} Suppose $\ell$ is a prime number and $\zeta$ is a primitive $\ell^s$-th root of unity. Then $$\frac{(\zeta-1)^{\varphi(\ell^s)}}{\ell}$$ is a unit in ${\mathbf Z}[\zeta]$. \end{lem} \begin{proof} See for example the last two lines on p.~9 of \cite{Wash}. \end{proof} \section{Lemmas} \label{ssredlemsect} \begin{rem} \label{cyclicrem} Suppose $w$ is a discrete valuation on a field $L$, $L$ is a finite extension of a field $F$, $v$ is the restriction of $w$ to $F$, and $w/v$ is totally and tamely ramified. Then the maximal unramified extension $L_{nr}$ of $L$ is the compositum of $L$ with the maximal unramified extension $F_{nr}$ of $F$. Further, $L_{nr}/F_{nr}$ is a cyclic extension whose degree is $[L:F]$ (see \S8 of \cite{Frohlich}, especially Corollary 3 on p.~31). Since passing to the maximal unramified extensions does not change the inertia groups, it follows that ${\mathcal I}_{w}$ is a normal subgroup of ${\mathcal I}$, and ${\mathcal I}/{\mathcal I}_{w}$ is cyclic of order $[L:F]$. \end{rem} \begin{lem} \label{rootsofone} Suppose $v$ is a discrete valuation on a field $F$ with residue characteristic $p \ge 0$, $R$ is a positive integer, $\ell$ is a prime, $p$ does not divide $R\ell$, and $L$ is a degree $R$ extension of $F$ which is totally ramified above $v$. Suppose that $X$ is an abelian variety over $F$, and for every $\sigma \in {\mathcal I}$, all the eigenvalues of $\rho_\ell(\sigma)$ are $R$-th roots of unity. Then $X$ has semistable reduction at the extension of $v$ to $L$. \end{lem} \begin{proof} This was proved in Lemma 5.5 of \cite{semistab} in the case where $L$ is Galois over $F$. However, the same proof also works in general. This follows from the fact that in the proof we replaced $F$ by its maximal unramified extension. For fields which have no non-trivial unramified extensions, every totally and tamely ramified extension is cyclic (and therefore Galois), and for each degree prime to the residue characteristic, there is a unique totally ramified extension of that degree. See \S8 of \cite{Frohlich}, especially Corollary 3 on p.~31. \end{proof} The following result yields a converse of Theorem 5.1 of \cite{serrelem}. \begin{lem} \label{algprop} Suppose ${\mathcal O}$ is an integral domain of characteristic zero, and $\ell$ is a prime number. Suppose $k$, $r$, and $m$ are positive integers such that $k \ge m\varphi(\ell^r)$. Suppose $\alpha \in {\mathcal O}$ and $\alpha^{\ell^r} = 1$. Then $(\alpha-1)^{k} \in \ell^m{\mathbf Z}[\alpha]$. \end{lem} \begin{proof} Let $s$ be the smallest positive integer such that $\alpha^{\ell^s} = 1$. Then $$(\alpha-1)^{k} \in (\alpha-1)^{m\varphi(\ell^s)}{\mathbf Z}[\alpha] \subseteq \ell^m{\mathbf Z}[\alpha],$$ by Lemma \ref{primrtsof1}. \end{proof} \begin{lem} \label{ssprelem} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ and $m$ are integers, and $n$ is not divisible by the residue characteristic of $v$. Suppose $\sigma \in {\mathcal I}$. If there exists a subgroup $S$ of $X_n$ such that $(\sigma^m-1)S = 0$ and $(\sigma^m-1)S^{\perp_n} = 0$, then $(\sigma^m-1)^2 X_n = 0$. \end{lem} \begin{proof} The map $x \mapsto (y \mapsto e_n(x,y))$ induces a $\mathrm{Gal}(F^s/F)$-equivariant isomorphism from $X_n/S$ onto $\mathrm{Hom}(S^{\perp_n},\boldsymbol \mu_n)$. Since $\sigma = 1$ on $\boldsymbol \mu_n$, and $\sigma^m = 1$ on $S^{\perp_n}$, it follows that $\sigma^m = 1$ on $X_n/S$. Therefore, $(\sigma^m-1)^2X_n \subseteq (\sigma^m-1)S = 0$. \end{proof} \begin{lem} \label{sslem} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ is an integer not divisible by the residue characteristic of $v$, and $S = X_n^{\mathcal I}$. Then ${\mathcal I}$ acts as the identity on $S^{\perp_n}$ if and only if $(\sigma-1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$. \end{lem} \begin{proof} Applying Lemma \ref{ssprelem} with $m = 1$, we obtain the forward implication. Conversely, suppose that $(\sigma-1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$. Writing $\sigma^n = ((\sigma-1)+1)^n$, it is easy to see that $\sigma^n = 1$ on $X_n$ for every $\sigma \in {\mathcal I}$. Since $n$ is not divisible by the residue characteristic of $v$, $X_n$ and $X_n^\ast$ are tamely ramified at $v$. Then the action of ${\mathcal I}$ on $X_n$ and on $X_n^\ast$ factors through the tame inertia group ${\mathcal I}/{\mathcal J}$. Let $\tau$ denote a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$. Since $$e_n((\tau-1)X_n,(X_n^\ast)^{\mathcal I}) = 1,$$ we have $$\#((X_n^\ast)^{\mathcal I})\#((\tau-1)X_n) \le \#X_n^\ast.$$ The map from $X_n$ to $(\tau-1)X_n$ defined by $y \mapsto (\tau-1)y$ defines a short exact sequence $$0 \to S \to X_n \to (\tau-1)X_n \to 0.$$ Therefore, $$\#S\#((\tau-1)X_n) = \#X_n = \#S\#S^{\perp_n}.$$ Similarly, $$\#((X_n^\ast)^{\mathcal I})\#((\tau-1)X_n^\ast) = \#X_n^\ast.$$ Therefore, $$\#S^{\perp_n} = \#((\tau-1)X_n) \le \#((\tau-1)X_n^\ast).$$ Since $(\tau-1)X_n^\ast \subseteq S^{\perp_n}$, we conclude that $$S^{\perp_n} = (\tau-1)X_n^\ast.$$ From the natural $\mathrm{Gal}(F^s/F)$-equivariant isomorphism $X_n^\ast \cong \mathrm{Hom}(X_n,\boldsymbol \mu_n)$ it follows that $(\tau-1)^2X_n^\ast = 0$. Therefore, ${\mathcal I}$ acts as the identity on $S^{\perp_n}$. \end{proof} \begin{lem} \label{sslem2} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, and $n$ is an integer not divisible by the residue characteristic of $v$. If $X$ has semistable reduction at $v$, then \begin{enumerate} \item[(i)] $(\sigma-1)^2X_n = 0$ for every $\sigma \in {\mathcal I}$, \item[(ii)] ${\mathcal I}$ acts as the identity on $(X_n^{\mathcal I})^{\perp_n}$, \item[(iii)] $(\sigma^n-1)X_n = 0$ for every $\sigma \in {\mathcal I}$; in particular, $X_n$ is tamely ramified at $v$. \end{enumerate} \end{lem} \begin{proof} By Theorem \ref{galcrit}, we have (i). By Lemma \ref{sslem}, we have (ii). In the proof of Lemma \ref{sslem}, we showed that (i) implies (iii). \end{proof} \begin{lem} \label{tame} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$, and $\ell$ is a prime number not equal to $p$. If $X_\ell$ is tamely ramified at $v$, then $T_\ell(X)$ is tamely ramified at $v$. \end{lem} \begin{proof} If $p = 0$ then the wild inertia group ${\mathcal J}$ is trivial and we are done. Suppose $p > 0$ and $\sigma \in {\mathcal J}$. Since $p \ne \ell$, $\rho_{\ell}({\mathcal J})$ is a finite $p$-group. Therefore, $\rho_{\ell}(\sigma)$ has order a power of $p$. Since $X_\ell$ is tamely ramified, $\rho_{\ell}(\sigma)-1 \in \ell\mathrm{End}(T_\ell(X))$. It follows that $\rho_{\ell}(\sigma) = 1$ if $\ell \ge 3$, and $\rho_{\ell}(\sigma)^2 = 1$ if $\ell = 2$. Since $p$ and $\ell$ are relatively prime, $\rho_{\ell}(\sigma) = 1$. \end{proof} \begin{lem} \label{lemT} Suppose $X$ is an abelian variety over a field $F$, $n = 2$, $3$, or $4$, $\ell$ is the prime divisor of $n$, $v$ is a discrete valuation on $F$ whose residue characteristic is not $\ell$, $t$ is a non-negative integer, $L$ is an extension of $F$ of degree $R(n)^{t+1}$ which is totally ramified above $v$, and $X$ has semistable reduction over $L$ above $v$. Let $\tau$ denote a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$. Let $\gamma = \rho_{\ell}(\tau)^{R(n)^t}$, let $\lambda = (\gamma - 1)^2/n$, and let $$T = T_\ell(X) + {\lambda}T_\ell(X) + {\lambda^2}T_\ell(X) + \cdots + {\lambda^{R(n)-1}}T_\ell(X).$$ Then: \begin{enumerate} \item[(a)] $T$ is the smallest $\lambda$-stable ${\mathbf Z}_\ell$-lattice in $V_\ell(X)$ which contains $T_{\ell}(X)$, \item[(b)] $(\gamma^{R(n)} - 1)^{2} = 0$, \item[(c)] $n^{R(n)-1}T \subseteq T_{\ell}(X) \subseteq T$, \item[(d)] $(\gamma - 1)^{2R(n)} \subseteq nT_{\ell}(X)$, \item[(e)] if $n = 2$ or $3$, then $nT \subseteq T_{\ell}(X)$ if and only if $(\gamma - 1)^{4}T_{\ell}(X) \subseteq nT_{\ell}(X)$, \item[(f)] if $n = 2$, then $4T \subseteq T_{2}(X)$ if and only if $(\gamma - 1)^{6}T_{2}(X) \subseteq 2T_{2}(X)$, \item[(g)] if $n = 4$, then $2T \subseteq T_{2}(X)$ if and only if $(\gamma - 1)^{2}T_{2}(X) \subseteq 2T_{2}(X)$. \end{enumerate} \end{lem} \begin{proof} Let $w$ denote the restriction of ${\bar v}$ to $L$. By Remark \ref{cyclicrem}, ${\mathcal I}/{\mathcal I}_{w}$ is cyclic of order $R(n)^{t+1}$. By Theorem \ref{galcrit}, we have (b). It follows that $(\lambda + \gamma)^2(\lambda + \gamma - 1)^2 = 0$ if $n = 2$, $\lambda(\lambda + \gamma)^2 = 0$ if $n = 3$, and $\lambda(\lambda + \gamma) = 0$ if $n = 4$. Therefore, $\lambda$ satisfies a polynomial over ${\mathbf Z}[\gamma]$ of degree $R(n)$, and we have (a) and (c). From the definition of $T$ we easily deduce (e), (f), and (g). Further, (d) follows from (b). \end{proof} We will apply the following result only in Corollary \ref{4326cor}e. \begin{thm} \label{divby23prop} Suppose $L/F$ is a finite separable field extension, $w$ is a discrete valuation on $L$, and $v$ is the restriction of $w$ to $F$. Suppose $X$ is a $d$-dimensional abelian variety over $F$ which has semistable reduction at $w$ but not at $v$. Then $[{\mathcal I}_{v}:{\mathcal I}_{w}]$ has a prime divisor $q$ such that $q \le 2d + 1$. \end{thm} \begin{proof} Let $\ell$ be a prime not equal to the residue characteristic $p$, and let $${{\mathcal I}_{v,X}} = \{\sigma \in {\mathcal I}_{v} : \sigma \text{ acts unipotently on } V_{\ell}(X) \}.$$ We have ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}} \subsetneqq {\mathcal I}_{v}$ by Theorem \ref{galcrit}, since $X$ has semistable reduction at $w$ but not at $v$. Let $F_{v}$ be the completion of $F$ at $v$ and let $F_{v}^{nr}$ be the maximal unramified extension of $F_{v}$. Then ${{\mathcal I}_{v,X}}$ is an open normal subgroup of ${\mathcal I}_{v}$, is independent of $\ell$, and cuts out the smallest Galois extension $F'$ of $F_{v}^{nr}$ over which $X$ has semistable reduction (see pp.~354--355 of \cite{SGA}). We have $\mathrm{Gal}(F'/F_{v}^{nr}) \cong {\mathcal I}_{v}/{{\mathcal I}_{v,X}}$. By a theorem of Raynaud (see Proposition 4.7 of \cite{SGA}), $X$ has semistable reduction over $F_{v}^{nr}(X_{n})$, for every integer $n$ not divisible by $p$ and greater than $2$. The intersection $M$ of these fields therefore contains $F'$. As on the top of p.~498 of \cite{SerreTate}, every prime divisor of $[M:F_{v}^{nr}]$ is at most $2d+1$ (see Theorem 4.1 and Formula 3.1 of \cite{JPAA} for an explicit integer that $[M:F_{v}^{nr}]$ divides). Thus, if $q$ is a prime divisor of $[{\mathcal I}_{v}:{{\mathcal I}_{v,X}}]$ then $q \le 2d+1$. Since ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}} \subsetneqq {\mathcal I}_{v}$, we obtain the desired result. \end{proof} \begin{rem} \label{divby23} With hypotheses and notation as in Theorem \ref{divby23prop}, let $k_{w}$ and $k_{v}$ denote the residue fields. Then $[{\mathcal I}_{v}:{\mathcal I}_{w}] = e(w/v)[k_{w}:k_{v}]_{i}$, where the subscript $i$ denotes the inseparable degree (see Proposition 21 on p.~32 of \cite{Corps} for the case where $L/F$ is Galois. In the non-Galois case, take a Galois extension $L'$ of $F$ which contains $L$, and apply the result to $L'/L$ and $L'/F$, to obtain the result for $L/F$). Taking completions, then $[L_{w}:F_{v}] = e(w/v)[k_{w}:k_{v}] = [{\mathcal I}_{v}:{\mathcal I}_{w}][k_{w}:k_{v}]_{s}$, where the subscript $s$ denotes the separable degree. Therefore, the prime $q$ from Theorem \ref{divby23prop} divides $[L_{w}:F_{v}]$. \end{rem} \section{Semistable reduction} \label{ssredsect} The results in this section extend the results of \cite{dpp} to the cases $n = 2, 3, 4$. Theorem \ref{oneway} is also a generalization of Corollary 7.1 of \cite{semistab}. \begin{rem} \label{lemrem} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, and $n$ is an integer greater than $1$ which is not divisible by the residue characteristic of $v$. By Lemma \ref{sslem}, the following two statements are equivalent: \begin{enumerate} \item[(a)] there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$, \item[(b)] $(\sigma - 1)^2X_n = 0$ for every $\sigma \in {\mathcal I}$. \end{enumerate} \end{rem} \begin{thm} \label{oneway} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, and $n$ is an integer greater than $1$ which is not divisible by the residue characteristic of $v$. Suppose there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. Then $X$ has semistable reduction over every degree $R(n)$ extension of $F$ totally ramified above $v$. \end{thm} \begin{proof} Suppose $\sigma \in {\mathcal I}$. By Lemma \ref{sslem}, $(\sigma-1)^2 X_n = 0$. Let ${\mathcal I}' \subseteq {\mathcal I}$ be the inertia group for the prime below ${\bar v}$ in a finite Galois extension of $F$ over which $X$ has semistable reduction. Then $\sigma^r \in {\mathcal I}'$ for some $r$. Let $\ell$ be a prime divisor of $n$. Theorem \ref{galcrit} implies that $(\rho_{\ell}(\sigma)^r-1)^2 = 0$. Let $\alpha$ be an eigenvalue of $\rho_{\ell}(\sigma)$. Then $(\alpha^r-1)^2 = 0$. Therefore, $\alpha^r = 1$. By our hypothesis, $$(\rho_{\ell}(\sigma)-1)^2 \in n\mathrm{M}_{2d}({\mathbf Z}_\ell),$$ where $d = \mathrm{dim}(X)$. By Th\'eor\`eme 4.3 of \cite{SGA}, the characteristic polynomial of $\rho_{\ell}(\sigma)$ has integer coefficients which are independent of $\ell$. By Lemma \ref{localglobal}, $(\alpha-1)^2 \in n{\bar {\mathbf Z}}$, where ${\bar {\mathbf Z}}$ denotes the ring of algebraic integers. By Theorem \ref{quasithm} we have $\alpha^{R(n)} = 1$. The result now follows from Lemma \ref{rootsofone}. \end{proof} \begin{cor}[Theorem 4.5 of \cite{dpp}] \label{fromdpp} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$, $n$ is an integer not divisible by the residue characteristic of $v$, and $n \ge 5$. Then $X$ has semistable reduction at $v$ if and only if there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. \end{cor} \begin{proof} If $X$ has semistable reduction at $v$, then by Theorem \ref{galcrit}, $(\sigma - 1)^2X_n = 0$ for every $\sigma \in {\mathcal I}$. Apply Lemma \ref{sslem}. For the converse, apply Theorem \ref{oneway} with $n \ge 5$. \end{proof} \begin{rem} \label{otherway} It follows immediately from Theorem \ref{galcrit} and Lemma \ref{sslem} that if $X$ has semistable reduction above $v$ over a degree $m$ extension of $F$ totally ramified above $v$, then there exists a subgroup $S$ of $X_n$ such that ${\mathcal I}$ acts via a cyclic quotient of order $m$ on $S$ and on $S^{\perp_n}$. (If $L$ is the extension of $F$, let $w$ be the restriction of ${\bar v}$ to $L$ and let $S = X_n^{{\mathcal I}_w}$.) Theorem \ref{bothways} below gives a different result in the direction converse to Theorem \ref{oneway}, and, further, gives conditions for semistable reduction which are both necessary and sufficient, thereby giving a generalization of Corollary \ref{fromdpp} to the cases $n = 2, 3, 4$. Note that in the case $n \ge 5$, the equivalence of (i) and (ii) in Theorem \ref{bothways} is just a restatement of Corollary \ref{fromdpp} (since $R(n) = 1$ if $n \ge 5$). We remark that in that case, one can take (in the notation of Theorem \ref{bothways}) $Y = X$ and $\varphi$ the identity map. \end{rem} \begin{thm} \label{bothways} Suppose $n = 2$, $3$, or $4$, respectively. Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete valuation on $F$ whose residue characteristic does not divide $n$. Suppose $t$ is a non-negative integer and $L$ is an extension of $F$ of degree $R(n)^{t+1}$ which is totally ramified above $v$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has semistable reduction over $L$ above $v$, \item[(ii)] there exist an abelian variety $Y$ over a finite extension $K$ of $F$ unramified above $v$, a separable $K$-isogeny $\varphi : X \to Y$, and a subgroup $S$ of $Y_n$ such that ${\mathcal I}$ acts via a cyclic quotient of order $R(n)^t$ on $S$ and on $S^{\perp_n}$. \end{enumerate} One can take $\varphi$ so that its kernel is killed by $8$, $9$, or $4$, respectively. If $X$ has potentially good reduction at $v$, then one can take $\varphi$ so that its kernel is killed by $2$, $3$, or $2$, respectively. \end{thm} \begin{proof} Let $\ell$ denote the prime divisor of $n$. Suppose $K$ is a finite extension of $F$ unramified above $v$, $Y$ is an abelian variety over $K$, $X$ and $Y$ are $K$-isogenous, and $S$ is a subgroup of $Y_n$ such that ${\mathcal I}$ acts via a cyclic quotient of order $R(n)^t$ on $S$ and on $S^{\perp_n}$. Suppose $\sigma \in {\mathcal I}$. By Lemma \ref{ssprelem}, $(\sigma^{R(n)^t}-1)^2Y_n = 0$, i.e., $$(\rho_{\ell,Y}(\sigma^{R(n)^t})-1)^2 \in nM_{2d}({\mathbf Z}_\ell).$$ Let $\alpha$ be an eigenvalue of $\rho_{\ell,Y}(\sigma)$. Since $Y$ has potentially semistable reduction, $\alpha$ is a root of unity. By Theorem \ref{quasithm}, $(\alpha^{R(n)^t})^{R(n)} = 1$. Therefore, all eigenvalues of $\rho_{\ell,Y}(\sigma)$ are ${R(n)}^{t+1}$-th roots of unity. By Lemma \ref{rootsofone}, $Y$ has semistable reduction over $LK$ above $v$. Since $X$ and $Y$ are $K$-isogenous and $K/F$ is unramified above $v$, $X$ has semistable reduction over $L$ above $v$. Conversely, suppose $X$ has semistable reduction over $L$ above $v$. By Lemma \ref{sslem2}iii, for every $\sigma \in {\mathcal I}$ we have $(\sigma^{nR(n)^{t+1}}-1)X_n = 0$. Since $nR(n)^{t+1}$ is not divisible by the residue characteristic, $X_n$ is tamely ramified at $v$. Then the action of ${\mathcal I}$ on $X_n$ factors through ${\mathcal I}/{\mathcal J}$. Let $\tau$ denote a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$. Let $T$ denote the ${\mathbf Z}_\ell$-lattice obtained from Lemma \ref{lemT}. By Lemma \ref{tame}, $T$ is stable under ${\mathcal I}$. Note that $n^{R(n)-1} = 8$, $9$, or $4$ when $n = 2$, $3$, or $4$, respectively. Let $C = T/T_\ell(X)$, and view $C$ as a subgroup of $X_8$, $X_9$, or $X_4$, respectively. Let $Y = X/C$. Then the projection map $X \to Y$ is a separable isogeny defined over a finite separable extension $K$ of $F$ which is unramified over $v$, $$T_\ell(Y) = T, \qquad \text{ and } \qquad (\rho_{\ell,Y}(\tau)^{R(n)^t}-1)^2Y_n = 0.$$ Let $K'$ (respectively, $L'$) be the maximal unramified extension of $K$ (respectively, $L$) in $F^{s}$, let $M$ be the degree ${R(n)^t}$ extension of $K'$ in $K'L'$ cut out by $\tau^{R(n)^t}$, let $w$ be the restriction of ${\bar v}$ to $M$, and let $S = Y_n^{{\mathcal I}_w}$. Then $\tau^{R(n)^t}$ is a lift to ${\mathcal I}_w$ of a topological generator of the pro-cyclic group ${\mathcal I}_w/{\mathcal J}_w$, where ${\mathcal J}_w$ is the first ramification group of ${\mathcal I}_w$. By Lemma \ref{sslem}, $\tau^{R(n)^t}$ acts as the identity on $S$ and on $S^{\perp_n}$. Therefore, ${\mathcal I}$ acts on $S$ and on $S^{\perp_n}$ via the cyclic group ${\mathcal I}/{\mathcal I}_w \cong \mathrm{Gal}(M/K')$. As in Lemma \ref{lemT}, let $\gamma = \rho_{\ell,X}(\tau)^{R(n)^t}$ and let $\lambda = (\gamma - 1)^2/n$. If $X$ has potentially good reduction at $v$, then $\gamma^{R(n)} = 1$. Let $\mu = \lambda + \gamma$. Then $\mu^2 = \mu$ and $T = T_\ell(X) + \mu T_\ell(X)$. Since $\mu = (\gamma^2 + 1)/2$ if $n = 2$, $\mu = (\gamma^2+\gamma+1)/3$ if $n = 3$, and $\mu = (\gamma + 1)/2$ if $n = 4$, it follows that $C$ is a subgroup of $X_2$, $X_3$, or $X_2$, respectively. \end{proof} Since the most interesting case of Theorem \ref{bothways} is the case $t = 0$, we explicitly state that case. \begin{cor} \label{bothcor} Suppose $n = 2$, $3$, or $4$, respectively. Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete valuation on $F$ whose residue characteristic does not divide $n$. Suppose $L$ is an extension of $F$ of degree $4$, $3$, or $2$, respectively, which is totally ramified above $v$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has semistable reduction over $L$ above $v$, \item[(ii)] there exist an abelian variety $Y$ over a finite extension $K$ of $F$ unramified above $v$, a separable $K$-isogeny $\varphi : X \to Y$, and a subgroup $S$ of $Y_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$. \end{enumerate} Further, $\varphi$ can be taken so that its kernel is killed by $8$, $9$, or $4$, respectively. If $X$ has potentially good reduction at $v$, then $\varphi$ can be taken so that its kernel is killed by $2$, $3$, or $2$, respectively. \end{cor} \section{Applications and refinements} \label{appsect} In the next result we show that the numbers in Theorem \ref{bothways} and Corollary \ref{bothcor} can be improved for abelian varieties of dimension $1$, $2$ (if $n = 2$ or $3$), and $3$ (if $n = 2$). In \S\ref{exssect} we show that the numbers in Theorem \ref{ellcor} are sharp. See also \cite{Katz}, which deals with other problems concerned with finding a ``good'' abelian variety in an isogeny class, with an answer depending on the dimension. \begin{thm} \label{ellcor} In Theorem \ref{bothways} and Corollary \ref{bothcor}, with $d = \mathrm{dim}(X)$, $\varphi$ can be taken so that its kernel is killed by $4$ if $d = 3$ and $n = 2$, by $3$ if $d =2$ and $n = 3$, and by $2$ if $d = n = 2$. If $d = 1$, then we can take $Y = X$ and $\varphi$ the identity map. \end{thm} \begin{proof} We use the notation from Lemma \ref{lemT} and from the proof of Theorem \ref{bothways}. Suppose $n = 2$ or $3$. By Lemma \ref{lemT}d, $\gamma$ acts unipotently on the ${\mathbf F}_{\ell}$-vector space $X_{\ell} \cong \frac{1}{\ell}T_{\ell}(X)/T_{\ell}(X)$. Therefore, $(\gamma - 1)^{2d}X_{\ell} = 0$. By Lemma \ref{lemT}e, if $d = 2$ then $C$ is killed by $n$. By Lemma \ref{lemT}f, if $n = 2$ and $d = 3$, then $C$ is killed by $4$. If $d = 1$, then $\lambda$ is an endomorphism of $T_{\ell}(X)$, so $T = T_{\ell}(X)$ and $Y = X$. Suppose $d = 1$ and $n = 4$. Since $\tau \in {\mathcal I}$, we have $\gamma \in \mathrm{SL}_{2}({\mathbf Z}_{2})$. Therefore, the eigenvalues of $\gamma$ are either both $1$ or both $-1$. Therefore either $(\gamma - 1)^{2} = 0$ or $(\gamma + 1)^{2} = 0$. In both cases, $(\gamma - 1)^{2}X_{4} = 0$. Therefore, $\lambda$ is an endomorphism of $T_{2}(X)$ and $Y = X$. \end{proof} We can therefore take $Y = X$ in Theorem \ref{bothways} and Corollary \ref{bothcor} when $X$ is an elliptic curve. This is not the case in general for abelian varieties of higher dimension, as shown by the examples in the next section. However, in Corollary \ref{paddcor} below we will show that a result of this sort does hold for abelian varieties with purely additive potentially good reduction. Next, we will give criteria for an elliptic curve to acquire semistable reduction over extensions of degree $2$, $3$, $4$, and either $6$ or $12$. \begin{cor} \label{4326cor} Suppose $X$ is an elliptic curve over a field $F$, and $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$. \begin{enumerate} \item[(a)] If $p \ne 2$, then $X$ has semistable reduction above $v$ over a totally ramified quartic extension of $F$ if and only if $X$ has an ${\mathcal I}$-invariant point of order $2$. \item[(b)] If $p \ne 3$, then $X$ has semistable reduction above $v$ over a totally ramified cubic extension of $F$ if and only if $X$ has an ${\mathcal I}$-invariant point of order $3$. \item[(c)] If $p \ne 2$, then $X$ has semistable reduction above $v$ over a quadratic extension of $F$ if and only if either $X$ has an ${\mathcal I}$-invariant point of order $4$, or all the points of order $2$ on $X$ are ${\mathcal I}$-invariant. \item[(d)] If $p \ne 2$ and $X$ has bad but potentially good reduction at $v$, then $X$ has good reduction above $v$ over a quadratic extension of $F$ if and only if $X$ has no ${\mathcal I}$-invariant point of order $4$ and all its points of order $2$ are ${\mathcal I}$-invariant. \item[(e)] Suppose $p$ is not $2$ or $3$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $2$ or $3$, \item[(ii)] there does not exist a finite separable extension $L$ of $F$ of degree less than $6$ such that $X$ has semistable reduction at the restriction of ${\bar v}$ to $L$. \end{enumerate} \item[(f)] Suppose $p$ is not $2$ or $3$. Then the following are equivalent: \begin{enumerate} \item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $4$ or $3$ and not all the points of order $2$ are ${\mathcal I}$-invariant, \item[(ii)] there does not exist a finite separable extension $L$ of $F$ of degree less than $4$ such that $X$ has semistable reduction at the restriction of ${\bar v}$ to $L$. \end{enumerate} \end{enumerate} \end{cor} \begin{proof} Theorem \ref{ellcor} implies that, for $n = 2$, $3$, or $4$, if $L$ is an extension of $F$ of degree $R(n)$ which is totally ramified above $v$, then $X$ has semistable reduction over $L$ above $v$ if and only if there exists a subgroup ${\mathfrak S}$ of $X_n$ such that ${\mathcal I}$ acts as the identity on ${\mathfrak S}$ and on ${\mathfrak S}^{\perp_n}$. Parts (a), (b), and (c) are a reformulation of this. For (d), note that by Theorem 7.4 of \cite{semistab}, if $X$ has an ${\mathcal I}$-invariant point of order $4$ then $X$ has good reduction at $v$. In case (e), if $X$ has an ${\mathcal I}$-invariant point of order $2$ (respectively, $3$), then $X$ has semistable reduction above $v$ over a totally ramified extension of degree $4$ (respectively, $3$), by part (a) (respectively, (b)). Conversely, suppose $L/F$ is a finite separable extension of degree less than $6$, and suppose $X$ has semistable reduction at the restriction $w$ of ${\bar v}$ to $L$. If $X$ has semistable reduction at $v$, then we are done by Corollary \ref{fromdpp} with $n = 6$. Otherwise, taking completions we have $[L_{w}:F_{v}] = 2$, $3$, or $4$ by Remark \ref{divby23}. There exists an intermediate unramified extension $M/F_{v}$ such that $L_{w}/M$ is totally ramified. By parts (a), (b), and (c) applied to $M$ in place of $F$, then $X$ has an ${\mathcal I}$-invariant point of order $2$ or $3$. Case (f) proceeds the same way as case (e). \end{proof} \begin{rem} Note that if the elliptic curve $X$ has additive reduction at $v$, but has multiplicative reduction over an extension $L$ of $F$ which is totally and tamely ramified above $v$, then $X$ has multiplicative reduction over a quadratic extension of $F$, but not over any non-trivial totally and tamely ramified extension of $F$ of odd degree (since $(x+1)^2$ is the only possibility for the characteristic polynomial of $\rho_\ell(\tau)$, where $\tau$ is as before). Therefore in case (b) of Corollary \ref{4326cor}, either $X$ already has semistable reduction at $v$, or else $X$ has good (i.e., does not have multiplicative) reduction above $v$ over a cubic extension of $F$. In case (e), $X$ has good reduction over an extension of degree $6$ or $12$ (see Proposition 1 of \cite{Kraus}). \end{rem} \begin{cor} \label{paddcor} Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$, and $X$ has purely additive and potentially good reduction at $v$. \begin{enumerate} \item[(a)] If $p \ne 2$, then $X$ has good reduction above $v$ over a quadratic extension of $F$ if and only if there exists a subgroup $S$ of $X_4$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_4}$. \item[(b)] If $p \ne 3$, then $X$ has good reduction above $v$ over a totally ramified cubic extension of $F$ if and only if there exists a subgroup $S$ of $X_3$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_3}$. \item[(c)] Suppose $p \ne 2$, and $L/F$ is a degree $4$ extension, totally ramified above $v$, which has a quadratic subextension over which $X$ has purely additive reduction. Then $X$ has good reduction above $v$ over $L$ if and only if there exists a subgroup $S$ of $X_2$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_2}$. \end{enumerate} \end{cor} \begin{proof} The backwards implications follow immediately from Corollary \ref{bothcor}. Let $n = 4$, $3$, and $2$ and $\ell = 2$, $3$, and $2$, in cases (a), (b), and (c), respectively. Let $\tau$ be a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$, and let $\gamma = \rho_{\ell}(\tau)$. If $X$ acquires good reduction over a totally ramified degree $R(n)$ extension, then $\gamma^{R(n)} = 1$, by Remark \ref{cyclicrem}. Since $X$ has purely additive reduction at $v$, $1$ is not an eigenvalue of $\gamma$ (see \cite{LenstraOort}). In case (c), $-1$ is not an eigenvalue of $\gamma$, since $X$ has purely additive reduction over a ramified quadratic extension. It follows that in cases (a), (b), and (c), respectively, we have $$\gamma+1 = 0, \qquad \gamma^2+\gamma+1=0, \qquad {\text{ and }}\qquad \gamma^2+1=0$$ in $\mathrm{End}(V_\ell(X))$. We deduce that $(\gamma-1)^2T_{\ell}(X) \subseteq nT_{\ell}(X)$, i.e., $(\tau-1)^2X_n = 0$. The result now follows from Lemma \ref{sslem}. \end{proof} \section{Examples} \label{exssect} We will show that the numbers in Corollary \ref{bothcor} and Theorem \ref{ellcor} are sharp. First, we will show that Corollary \ref{bothcor} is sharp in the case of potentially good reduction. This will show that we cannot take $Y = X$ in general. In the next 3 examples, we have $n = 2$, $3$, or $4$, respectively. Let $\ell$ denote the prime divisor of $n$. Suppose that $F$ is a field with a discrete valuation $v$ of residue characteristic not equal to $\ell$. Suppose $E$ and $E'$ are elliptic curves over $F$, $E$ has good reduction at $v$, and $E'$ has additive reduction at $v$ but acquires good reduction over an extension $L$ of $F$ of degree $R(n)$. Let $Y = E \times E'$. As shown in the proof of Theorem \ref{bothways}, the action of ${\mathcal I}$ on $Y_n$ factors through ${\mathcal I}/{\mathcal J}$. Let $\tau$ be a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group ${\mathcal I}/{\mathcal J}$, and let $g = \rho_{\ell,Y}(\tau)$. Note that $g^{R(n)} = 1$. Let $G$ denote the cyclic group generated by $g$. In each example we will construct a certain ${\mathbf Z}_{\ell}[G]$-module $T$ such that $T \subset T_\ell(Y) \subset \frac{1}{\ell}T$. Let $C' = \frac{1}{\ell}T/T_\ell(Y)$, view $C'$ as a subgroup of $Y_\ell$, and let $X = Y/C'$. Then $T_\ell(X) \cong T$. Viewing $T_{\ell}(Y)/T$ as a subgroup $C$ of $X_{\ell}$, we have $Y = X/C$. In our 3 examples, $C$ is stable under ${\mathcal I}$, $(\tau - 1)^2X_n \ne 0$, and $(\tau - 1)^2Y_n = 0$. By Remark \ref{lemrem}, there is a subgroup $S \subseteq Y_n$ such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$, but there does not exist a subgroup ${\mathfrak S} \subseteq X_n$ such that ${\mathcal I}$ acts as the identity on ${\mathfrak S}$ and on ${\mathfrak S}^{\perp_n}$. We see that $X$ and $Y$ satisfy (ii) of Corollary \ref{bothcor}. \begin{ex} \label{exfor2} Let $n = 2$. Suppose that $E'$ does not acquire good reduction over a quadratic subextension of $L/F$. As ${\mathbf Z}_{2}[G]$-modules, we have $$T_2(Y) \cong ({\mathbf Z}_2[x]/(x-1))^2 \oplus {\mathbf Z}_2[x]/(x^2+1),$$ where $g$ acts via multiplication by $x$. Let $$T = {\mathbf Z}_2[x]/(x-1) \oplus {\mathbf Z}_2[x]/(x-1)(x^2+1),$$ and view $T$ as a submodule of $T_2(Y)$ via the natural injection. For example, one could take $F = {\mathbf Q}$, $v = 3$, and $E$ and $E'$, respectively, the elliptic curves 11A3 and 36A1 from the tables in \cite{Cremona}. \end{ex} \begin{ex} \label{exfor3} Let $n = 3$. As ${\mathbf Z}_{3}[G]$-modules, we have $$T_3(Y) \cong ({\mathbf Z}_3[x]/(x-1))^2 \oplus {\mathbf Z}_3[x]/(x^2+x+1),$$ where $g$ acts via multiplication by $x$. Let $$T = {\mathbf Z}_3[x]/(x-1) \oplus {\mathbf Z}_3[x]/(x^3-1),$$ and view $T$ as a submodule of $T_3(Y)$ via the natural injection. For example, one could take $F = {\mathbf Q}$, $v = 2$, and $E$ and $E'$, respectively, the elliptic curves 11A3 and 20A2 from the tables in \cite{Cremona}. \end{ex} \begin{ex} Let $n = 4$. As ${\mathbf Z}_{2}[G]$-modules, we have $$T_2(Y) \cong ({\mathbf Z}_2[x]/(x-1))^2 \oplus ({\mathbf Z}_2[x]/(x+1))^2 \cong ({\mathbf Z}_2[G])^2,$$ where $g$ acts via multiplication by $x$. Let $$T = {\mathbf Z}_2[x]/(x-1) \oplus {\mathbf Z}_2[x]/(x^2-1) \oplus {\mathbf Z}_2[x]/(x+1),$$ and view $T$ as a submodule of $T_2(Y)$ via the natural injection. One could take $F = {\mathbf Q}$, $v = 3$, and $E$ and $E'$, respectively, the elliptic curves 11A3 and 99D1 from the tables in \cite{Cremona}. \end{ex} Next, we will show that the numbers $8$, $9$ and $4$ (respectively) in Corollary \ref{bothcor} are sharp. \begin{ex} Let $n = 2$, $3$, or $4$. For ease of notation, let $R = R(n)$. Let $\ell$ be the prime divisor of $n$. Let $F$ be a field with a discrete valuation $v$ of residue characteristic not equal to $\ell$, and suppose $E$ is an elliptic curve over $F$ with multiplicative reduction at $v$. Suppose that $M$ is a degree $R$ Galois extension of $F$ which is totally ramified above $v$. Let $\chi$ be the composition $$\mathrm{Gal}(F^{s}/F) \to \mathrm{Gal}(M/F) \cong {\mathbf Z}/R{\mathbf Z} \hookrightarrow \mathrm{Aut}_{F}(E^{R}),$$ where the image of the last map is generated by a cyclic permutation of the factors of $E^{R}$, and $E^{R}$ is the $R$-fold product of $E$ with itself. Let $A$ denote the twist of $E^{R}$ by $\chi$. Let $\tau$ denote a lift to ${\mathcal I}$ of a generator of ${\mathcal I}/{\mathcal J}$. As ${\mathbf Q}_{\ell}[\tau]$-modules, $V_{\ell}(A) \cong {\mathbf Q}_{\ell}[\tau]/(\tau^{R}-1)^{2}$. Let ${\tilde T}$ be the inverse image of ${\mathbf Z}_{\ell}[\tau]/(\tau^{R}-1)^{2}$ in $V_{\ell}(A)$. Then for some integer $k$, we have $T_{\ell}(A) \subseteq \ell^{k}{\tilde T}$. View $\ell^{k}{\tilde T}/T_{\ell}(A)$ as a finite subgroup of $A$ and let $X$ be the quotient of $A$ by this subgroup. Then $X$ is defined over an extension $K$ of $F$ unramified above $v$, and $X$ acquires semistable reduction over $KM$ above $v$. We have ${\tilde T} = T_{\ell}(X)$, and the minimal polynomial of $\tau$ on $X_{\ell}$ is $(x^{R}-1)^{2} \equiv (x-1)^{2R} \pmod{\ell}$. Therefore, $$(\tau - 1)^{6}X_{2} \ne 0 \text{ if } n = 2, \,\, (\tau - 1)^{4}X_{3} \ne 0 \text{ if } n = 3, \text{ and } (\tau - 1)^{2}X_{2} \ne 0 \text{ if } n = 4.$$ From Lemma \ref{lemT} (with $t = 0$, $F = K$, and $L = KM$) we obtain a lattice $T$ such that $$8T \subseteq T_{2}(X) \subseteq T \,\, \text{ if } \,\, n = 2, \qquad 9T \subseteq T_{3}(X) \subseteq T \,\, \text{ if }\,\, n = 3,$$ $$\text{ and } \quad 4T \subseteq T_{2}(X) \subseteq T \,\, \text{ if } \,\, n = 4.$$ Let $C = T/T_{\ell}(X)$, view $C$ as a subgroup of $X_{\ell}$, and let $Y = X/C$. As we saw in the proof of Theorem \ref{bothways}, $(\tau - 1)^{2}Y_{n} = 0$, and $C$ is killed by $8$, $9$, or $4$ if $n = 2$, $3$, or $4$ respectively. By Lemma \ref{lemT}efg, the group $C$ is not killed by $4$, $3$, or $2$, respectively. Suppose $K'$ is a finite extension of $K$ unramified above $v$, $Y'$ is an abelian variety over $K'$, $\varphi : X \to Y'$ is a separable $K'$-isogeny, and $(\tau - 1)^{2}Y'_{n} = 0$. Suppose that the kernel of $\varphi$ is killed by some positive integer $s$. Then we can suppose $sT_{\ell}(Y') \subseteq T_{\ell}(X) \subseteq T_{\ell}(Y')$. Let $\lambda = (\tau^{2} - 1)/n$. Since $T_{\ell}(Y')$ is a $\lambda$-stable ${\mathbf Z}_{\ell}$-lattice in $V_{\ell}(X)$ which contains $T_{\ell}(X)$, we have $T \subseteq T_{\ell}(Y')$ by Lemma \ref{lemT}a. Therefore, $sT \subseteq T_{\ell}(X)$. Then $C$ is killed by $s$, and therefore $s$ cannot be $4$, $3$, or $2$, respectively. This shows that the numbers $8$, $9$, and $4$ are sharp in Corollary \ref{bothcor}. Note that $\mathrm{dim}(X) = 4$, $3$, or $2$, respectively. By Theorem \ref{ellcor}, these are the smallest dimensions for which such examples exist. \end{ex} \begin{ex} Let $F$ be a field with a discrete valuation $v$ of residue characteristic not equal to $2$, and suppose $E$ is an elliptic curve over $F$ with multiplicative reduction at $v$. Suppose that $M$ is a degree $4$ Galois extension of $F$ which is totally ramified above $v$. Let $\chi$ be the composition $$\mathrm{Gal}(F^{s}/F) \to \mathrm{Gal}(M/F) \cong {\mathbf Z}/4{\mathbf Z} \hookrightarrow \mathrm{Aut}_{F}(E^{4}),$$ where the image of the last map is generated by a cyclic permutation of the factors of $E^{4}$. Let $$B = \{(e_{1},e_{2},e_{3},e_{4}) \in E^{4} : e_{1} + e_{2} + e_{3} + e_{4} = 0\} \cong E^{3},$$ and let $A$ be the twist of $B$ by $\chi$. Let $\tau$ denote a lift to ${\mathcal I}$ of a generator of ${\mathcal I}/{\mathcal J}$, and let $f(x) = (x^{3}+x^{2}+x+1)^{2}$. As ${\mathbf Q}_{2}[\tau]$-modules, $V_{2}(A) \cong {\mathbf Q}_{2}[\tau]/f(\tau)$. Let ${\tilde T}$ be the inverse image of ${\mathbf Z}_{2}[\tau]/f(\tau)$ in $V_{2}(A)$. As in the previous example, we obtain an abelian variety $X$ such that ${\tilde T} = T_{2}(X)$, and such that the minimal polynomial of $\tau$ on $X_{2}$ is $f(x) \equiv (x-1)^{6}$ (mod $2$). Therefore, $(\tau - 1)^{4}X_{2} \ne 0$. As above, we see that $X$ is isogenous over an unramified extension to an abelian variety $Y$ such that $(\tau-1)^{2}Y_{2} = 0$ and such that the kernel of the isogeny is killed by $4$. Using Lemma \ref{lemT}e, we see that there does not exist such a $Y$ where the kernel is killed by $2$. This shows that the result in Theorem \ref{ellcor} for $d=3$ and $n=2$ is sharp. The sharpness of the other numbers in Theorem \ref{ellcor} follows from Examples \ref{exfor3} and \ref{exfor2}. \end{ex}

9702

alg-geom/9702008

en

https://arxiv.org/abs/alg-geom/9702008

alg-geom/9702008

Eleny-Nicoleta Ionel

Eleny-Nicoleta Ionel, Thomas H. Parker

The Gromov Invariants of Ruan-Tian and Taubes

AMS-LaTeX, 11 pages

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants. This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants $Gr_{\delta}$, $\delta=0,1,2,...$. The first of these, $Gr_0$, generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants $Gr_{\delta}$ count immersed curves with $\delta$ double points. In particular, these results give an independent proof that Taubes' invariants are well-defined. They also show that some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants.

alg-geom

\section{Gromov Invariants} Fix a closed symplectic four-manifold $(X,\omega)$. Following the ideas of Gromov and Donaldson, one can define symplectic invariants by introducing an almost complex structure $J$ and counting (with orientation) the number of $J$-holomorphic curves on $X$ satisfying certain constraints. Unfortunately, technical difficulties make it necessary to modify the straightforward count in order to obtain an invariant. In this section we review the general construction and describe how the technicalities have led to two types of Gromov invariants. Given $(X,\omega)$, one can always choose an almost complex structure $J$ tamed by $\omega$, i.e. with $\omega(Z,JZ)>0$ for all tangent vectors $Z$. A map $f:\Sigma \to X$ from a topological surface $\Sigma$ is called $J$-holomorphic if there is a complex structure $j$ on $\Sigma$ such that \begin{equation} \overline{\partial}_Jf=0 \label{1.holomorphicmapeq} \end{equation} where $\overline{\partial}_Jf =\frac12(df \circ j - J \circ df)$. The image of such a map is a $J$-holomorphic curve. Conversely, each immersed $J$-holomorphic curve is uniquely specified by the equivalence class of a $J$-holomorphic pair $(f,j)$ under the action of the group of diffeomorphisms of $\Sigma$. These equivalence classes $[(f,j)]$ form moduli spaces $$ {\cal M}_{{A,g}} $$ labeled by the genus $g$ of $\Sigma$ and the class $A\in H_2(X)$ of the image (and implicitly depending on $J$). The formal tangent space to ${\cal M}_{{A,g}}$ at $[(f,j)]$ can be identified with the kernel of the operator \begin{equation} D_{f,j}:\Gamma(f^*N)\to \Omega^{0,1}(f^*N) \label{1.kernelD} \end{equation} obtained by linearizing (\ref{1.holomorphicmapeq}) and restricting to the normal bundle $N$ along the image of $f$. The Riemann-Roch Theorem shows that $$ \mbox{dim}\ {\cal M}_{{A,g}} =2[g-1-\kappa\cdot A] \label{1.dimM} $$ where $\kappa$ is the canonical class of $(X,J)$. We can elaborate on this construction by marking $d$ points $x_i$ on $\Sigma$. The equivalence classes $[(f,j,x_1,\dots ,x_d)]$ of marked $J$-holomorphic curves then form a moduli space ${\cal M}_{{A,g,d}}$ of dimension $2[g-1-\kappa\cdot A+d]$, and the evaluations $x_i\mapsto f(x_i)$ define a map $$ ev:{\cal M}_{{A,g,d}} \to X^d=X\times\dots\times X $$ The marked points enable us to restrict attention to curves satisfying constraints. For our purposes it is almost always enough to consider point constraints. Thus we pick $$ d=d_{A,g}=g-1-\kappa\cdot A $$ generic points $p_i$ in $X$ and consider the constrained moduli space $$ {\cal M}'_{{A,g}}=ev^{-1}(p_1,\dots, p_d) $$ of all $J$-holomorphic curves that pass through the (ordered) points $p_i$. For generic $J$ and $\{p_i\}$, this constrained moduli space is zero-dimensional and its formal tangent space at $C=[(f,j,x_1,\dots ,x_d)]$ is the kernel of the restriction $D_C$ of (\ref{1.kernelD}) to the subspace of $\Gamma(f^*N)$ that vanishes at each marked point. Each curve $C\in {\cal M}'_{{A,g}}$ therefore has a sign given by $(-1)^{\em SF}$ where SF denotes the spectral flow from $D_C$ to any complex operator $\overline{\partial}_C$ which is a compact perturbation of $D_C$. Counting the points in ${\cal M}'_{{A,g}}$ with sign gives a ``Gromov invariant'' \begin{equation} Gr_{A,g}(p^d)=\sum_{C\in{\cal M}'_{{A,g}}}{\mbox{sgn}\ C} = \sum_{C\in{\cal M}'_{{A,g}}}\ (-1)^{\em{SF}\,(D_C)}. \label{1.SF} \end{equation} One then tries to mimic Donaldson's cobordism arguments to show that $Gr_{A,g}$ is independent of $J$ and $\{p_i\}$, and hence defines a symplectic invariant. This involves considerable analysis, and along the way one encounters a major technical difficulty --- ${\cal M}_{A,g}$ may not be a manifold at the multiply-covered maps. There currently exist two distinct ways of overcoming this difficulty. \begin{enumerate} \item Taubes restricts $g$ to be the genus expected for embedded curves and counts embedded, not necessarily connected, $J$-holomorphic curves, dealing with the complications associated with multiply-covered curves. In the end he obtains `Gromov-Taubes' invariants that we will denote by $GT_0(A)$. \item Ruan-Tian [RT] observed that the difficulties with multiply-covered maps can be overcome by replacing (\ref{1.holomorphicmapeq}) by the inhomogeneous equation $$ \overline{\partial}_Jf=\nu \label{1.pertholomorphicmapeq} $$ where $\nu$ is an appropriate perturbation term. We will denote the resulting symplectic invariants by $RT(A,d)$. \end{enumerate} The next two sections give some details about these two sets of invariants and describe generating functions involving them. \medskip \medskip \section{The Taubes Series} The details of Taubes' construction are interesting and surprisingly subtle. Given $A\in H_2(X,{ \Bbb Z})$, Taubes fixes the genus to be $$ g_A=1+\frac12(A\cdot A +\kappa\cdot A). $$ The moduli space of such curves has $\mbox{dim}\, {\cal M}_A =A\cdot A-\kappa\cdot A$, so we constrain by $d_A=\frac12 (A\cdot A-\kappa\cdot A)$ points. The adjunction formula implies that each constrained curve is embedded unless $A$ lies in the set $$ T=\{\; A\in H_2(X,{ \Bbb Z})\;|\; A^2=0 \ \ \mbox{and}\ \ \kappa\cdot A=0\;\}, $$ in which case the curve is a multiple cover of an embedded torus and $d_A=0$. Similarly, each constrained curve in $$ {\cal E}=\{\; A\in H_2(X,{ \Bbb Z})\;|\; A^2=-1\;\}, $$ is an embedded ``exceptional'' sphere. More generally, for each class $A$ and $d\geq 0$ we get a count of {\it connected} curves through $d$ generic points \begin{eqnarray*} Gr(A,d) \label{2.AnotinS} \end{eqnarray*} defined by (\ref{1.SF}) with $g=d+1+\kappa\cdot A$. Note that by the adjunction formula \begin{equation} d_A-d=g_A-g=\delta \geq 0, \label{2.defDelta} \end{equation} so $0\le d\le d_A$ with $d_A=0$ for $A\in {\cal E}\cup T$. Geometrically, $\delta$ is the number of double points on a generic immersed $A$-curve. Taubes observed that for $A\in T$, $Gr(A,0)$ depends on $J$, as follows. For an embedded torus $C$, let $L_i,\ i=1,2,3$ be the three non-trivial real line bundles over $C$. Twisting the linearization $D_C$ by $L_i$ gives operators $$ D_i:\Gamma(f^*N\otimes L_i)\to \Omega^{0,1}(f^*N\otimes L_i). $$ The space of almost complex structures is divided into chambers by the codimension one ``walls'' consisting of those $J$ for which there is a $J$-holomorphic curve with either $D_C$ or one of the $D_i$ not invertible. The value of $Gr(A,0)$ changes as $J$ crosses a wall. Within a chamber, there are four types of $J$-holomorphic tori, labeled by the number $k=0,1,2,3$ of the $D_i$ whose sign (determined by the spectral flow) is negative. Thus for generic $J$, the moduli space of $J$-holomorphic $A$-curves is the disjoint union of four zero-dimensional moduli spaces ${\cal M}_{A,k}$. Counting with sign gives four ``Taubes numbers'' \begin{equation} \tau(A,k)= \sum_{C\in{\cal M}_{A,k}}\ \mbox{sgn}\ C. \label{2.AinS} \end{equation} Taubes derived wall-crossing formulas and showed that a certain combination of the $\tau(A,k)$ is independent of $J$. The right combination is best described by assembling the counts (\ref{2.AnotinS}) and (\ref{2.AinS}) into a single quantity associated with $X$. For that purpose, we introduce formal symbols $t_A$ for $A\in H_2(X;{ \Bbb Z})$ with relations $t_{A+B}=t_At_B$ and specify three ``generating functions'' $e(t),f(t)$ and $g(t)$. From $f$ we construct functions $f_k$ corresponding to the four types of curves by setting \begin{eqnarray}\label{T gen fc} f_0=f,\qquad f_1(t)={f(t)\over f(t^2)},\qquad f_2(t)={f(t)f(t^4)\over f^2(t^2)},\qquad f_3(t)={f(t)f(t^4)\over f^3(t^2)}. \end{eqnarray} in accordance to the wall crossing formulas in [T2]. We will also use another variable $s$ to keep track of the number of double points. \begin{defn} The {\em Taubes Series} of $(X,\omega)$ with generating functions $e, f$ and $g$ is the formal power series in the variables $t_A$ and $s$ defined by \begin{eqnarray} GT_X(t,s)=\prod_{E\in {\cal E}}e(t_E)^{Gr(A,0)} \cdot \prod_{A\notin T\cup {\cal E}}\prod_{d=0}^{d_A} g\l(t_A{s^{d}\over d! }\r)^{Gr(A,d)} \cdot \prod_{A\in T}\ma\prod_{ k=0}^3 f_k(t_A)^{\tau(A,k)} \label{egfgeneratingfnc} \end{eqnarray} with the $f_k$ given by (\ref{T gen fc}). \end{defn} We then get a sequence of maps $GT_{\delta}: H_2(X;{ \Bbb Z})\to { \Bbb Z}$ by expanding (\ref{egfgeneratingfnc}) as a power series in $s$: \begin{eqnarray} GT(t,s)\ =\ \sum_{A} \sum_{\delta=d_A-d}GT_{\delta}(A)\ t_A\, \frac{s^d}{d!} \label{egfgeneratingfnc2} \end{eqnarray} where we have labeled the coefficients by $\delta=d_A-d$ rather than $d$. \begin{prop} With the choice \begin{eqnarray} e(t)=1+t,\qquad f(t)=\frac{1}{1-t},\quad \mbox{and}\qquad g(t)=e^t, \label{2.taubesgeneratingfncs} \end{eqnarray} the degree zero component $GT_{0}$ in (\ref{egfgeneratingfnc2}) is the Gromov invariant defined by Taubes in [T2]. \end{prop} \noindent {\bf Proof. } The coefficient $GT_{\delta}(A)$ of $t_A s^d/d!$ in (\ref{egfgeneratingfnc}) is a sum of coefficients, one for each product of monomials $(t_{A_i}s^{d_i})^{n_i}$ with $d=\sum n_id_i$ and $A=\sum n_iA_i$, where the $A_i$ are distinct homology classes, $n_i\geq 0$, and $n_i=1$ for all $A_i\in {\cal E}$ (because the generating function is $e(t)=1+t$). Given such a decomposition, we can expand $\delta=d_A-d =d_A-\sum n_id_i$ by writing $\delta_i=d_{A_i}-d_i\ge 0$ as in (\ref{2.defDelta}) and using the definition of $d_A$. This gives \begin{eqnarray*} \delta & = & \frac12\left[(\ma\sum n_i A_i)^2-\ma\sum n_i A_i^2\right] +\sum n_i \delta_i \\ & = & \ma\sum \frac12 n_i(n_i-1)\ A_i^2 +\sum_{i<j} n_i n_j A_i A_j +\sum n_i\delta_i \end{eqnarray*} Each of the terms in this sum are nonnegative since (a) $A_i^2\ge 0$ for $A_i\notin {\cal E}$ and $n_i=1$ for $A_i\in {\cal E}$, and (b) $A_i\cdot A_j\ge 0$ for $i\ne j$ because the $A_i$ are distinct. Consequently, the only monomials that contribute to the $\delta=0$ term are those corresponding to decompositions of $A$ and $d$ with \smallskip \ \hskip1in (a) $n_i=1$ unless $A_i^2=0$, \ \hskip1in (b) $A_i\cdot A_j=0$ for all $i\ne j$, \ \hskip1in (c) $d_i=d_{A_i}$. \smallskip \noindent Let ${\cal S}={\cal S}(A)$ be the set of such decompositions. For each $y=\{(n_i,A_i)\}$ in ${\cal S}$, let $y'$ be the set of those $(n_i,A_i)\in y$ with $A_i\notin T$, let $y''$ be the set of those $(n_i,A_i)\in y$ with $A_i$ primitive and $A_i\in T$, and let $t_{y'}$ and $t_{y''}$ be the corresponding monomials. Putting the functions (\ref{2.taubesgeneratingfncs}) into (\ref{egfgeneratingfnc}), one sees that the coefficient of $t_A s^{d_A}/d_A!$ has the form \begin{eqnarray} \label{2.T1} Gr_0(A)\ =\ \sum_{y\in{\cal S}} R(y') Q(y''). \end{eqnarray} Here $R(y')$ is the coefficient of $t_{y'}s^d/d!$ in $$ \prod_{A_i\notin T} \left[\mbox{exp}\left(t_{A_i}\frac{s^{d_i}}{d_i!}\right)\right]^{Gr(A_i,d _i)} $$ (after noting that $t_{y'}$ is at most linear in $t_{A_i}$ for each $A_i\in {\cal E}$ and $\mbox{exp}\,t=1+t+O(t^2)$), so \begin{eqnarray} \label{2.T2} R(y')\ =\ d!\,\prod_{(n_i,A_i)\in y'} \frac{Gr(A_i,d_{i})^{n_i}} {n_i!\,(d_i!)^{n_i}}. \end{eqnarray} Similarly, $Q(y'')$ is the coefficient of $t_{y''}$ in $$ \prod_{A_i\in T}\ \prod_{ k=0}^3 f_k(t_{A_i})^{\tau(A_i,k)}= \prod_{A_i\in T \atop primitive}\ \prod_{q=1}^{\infty}\; \ma\prod_{ k=0}^3 f_k(t_{A_i}^q)^{\tau(qA_i,k)}. $$ Then \begin{eqnarray} \label{2.T3} Q(y'')\ =\ \prod_{(n_i,A_i)\in y''}\ Q(n_i,A_i) \end{eqnarray} where $Q(n,A)$ is the coefficient of $t_{A}^n$ in $$ \ma\prod_{q=1}^{\infty}\ma\prod_{ k=0}^3 f_k(t_{qA})^{\tau(qA,k)}. $$ For each embedded, holomorphic torus $C$, let $f_C$ denote the function $f_k$ (resp. $1/f_k$) when $C$ is of type $k$ and has positive (resp. negative) sign. Expanding $f_{C}(t)=\sum_{m} r(C,m) t^m$, we have \begin{eqnarray} \label{2.T4} Q(n,A)\ =\ \sum_{{\cal D}}\prod r(C_j,m_j), \end{eqnarray} where ${\cal D}$ is the set of all pairs $(m_j,C_j)$ of $J$-holomorphic curves $C_j$ and multiplicities $m_j$ with $[C_j]=q_jA$ and $\sum m_j q_j=n$. Together, (\ref{2.T1}) -- (\ref{2.T4}) exactly agree with the invariant defined by Taubes ([T2] section 5d). \qed \bigskip \begin{rem} Taubes chooses the functions (\ref{2.taubesgeneratingfncs}) to make his invariants agree with the Seiberg-Witten invariants. \end{rem} \medskip The numbers $Gr_{\delta}(A)$ defined by (\ref{egfgeneratingfnc}) and (\ref{egfgeneratingfnc2}) count the $J$-holomorphic $A$-curves (of any genus and any number of components) with $\delta$ double points, and thus generalize Taubes' count of embedded curves. Below, we will verify that the $Gr_{\delta}(A)$ are symplectic invariants by relating the Taubes Series to Ruan-Tian invariants. \medskip \medskip \setcounter{equation}{0} \section{The RT Series} Ruan and Tian [RT] define symplectic invariants $RT_{A,g,d}(\alpha_1,\dots,\alpha_d)$ by taking the moduli space ${\cal M}_{A,g,d}$ of {\em connected, perturbed} holomorphic $A$-curves with genus $g$ and $d$ marked points, restricting to the subset ${\cal M}'_{A,g,d}$ where the marked points lie on fixed constraint surfaces representing the $\alpha_i\in H_*(X)$, and counting with orientation (assuming ${\cal M}'_{A,g,d}$ is zero-dimensional). In particular, when the $\alpha_i$ are all points and $g=d+1+\kappa\cdot A$ we get invariants \begin{equation} RT(A,d)\ =\ RT_{A,d+1+\kappa\cdot A,d}(p^d). \label{3.1} \end{equation} This section describes how to assemble these invariants into a series analogous to (\ref{egfgeneratingfnc}). First we must deal with a technical problem. In [RT], the invariants $RT_{A,g,d}$ are defined only for the ``stable range'' $2g+d\geq 3$. This leaves $RT(A,d)$ undefined for two types of curves: tori with no marked points, which occur when $d=\kappa\cdot A=0$, and spheres with fewer than three marked points, which occur when $d=0,1,2$ and $d+1=-\kappa\cdot A$. But we can extend definition (\ref{3.1}) to these cases by imposing additional ``constraints'' which are automatically satisfied. For this, choose a class $\beta\in H_{2}(X)$ with $A\cdot \beta \neq 0$ and set \begin{equation} RT(A,0) \ =\ \frac{1}{A\cdot \beta}\, RT_{A,1,1}(\beta) \qquad \mbox{if}\ \kappa\cdot A =0 \label{3.2} \end{equation} and $$ RT(A,d) \ =\ \frac{1}{(A\cdot \beta)^{3-d}}\,RT_{A,0,3}(p^d\beta^{3-d})\qquad \mbox{if }\ d=\kappa\cdot A-1=0,1,2. $$ Thus defined, these invariants count perturbed holomorphic curves. For example, when $\kappa\cdot A =0$ each genus one curve $C$ (without marked points) representing $A$ is a map $f:T^2\to X$, well-defined up the automorphisms of $T^2$ with the induced complex structure. Fix a point $p\in T^2$ and represent $\beta$ by a cycle in general position. Then $C\cap \beta$ consists of $A\cdot \beta$ distinct points. Hence $C$ is the image of exactly $A\cdot \beta$ maps $f:T^2\to X$ with $f(p)\in\beta$ and these are counted by $RT_{A,1,1}(\beta)$. \medskip Now fix a generating function $F_A$ for each class $A$ and assign a factor $F_A(t_A)$ to each curve that contributes $+1$ to the count $RT(A,d)$, and a factor $1/F_A(t_A)$ to each curve that contributes $-1$. Taking the product gives a series in the variables $t_A$ $$ \ma\prod_{A\in H_2(X)}F_A(t_A)^{RT(A,d)} $$ which is an invariant of the deformation class of the symplectic structure of $(X,\omega)$. As with the Taubes Series, different choices of the $F_A$ give different series, but all encode the same data. We will choose three generating functions and form a series resembling (\ref{egfgeneratingfnc}). \begin{defn} The {\em Ruan-Tian Series} of $(X,\omega)$ defined by $e(t)$, $F(t)$ and $g(t)$ is \begin{eqnarray}\label{3.defgr} RT_X(t,s)\ =\ \prod_{E\in {\cal E}}e(t_E)^{RT(A,0)} \cdot \prod_{A\notin T\cup {\cal E}} g\l(t_A{s^{d}\over d! }\r)^{RT(A,d)}\ \cdot\ \ma\prod_{A\in T}F(t_A)^{RT(A,0)} \end{eqnarray} Expanding in power series as in (\ref{egfgeneratingfnc2}) gives invariants $RT_{\delta}:H_2(X;{ \Bbb Z})\to{ \Bbb Z}$. \end{defn} To make this more concrete, we could take $e(t)$, $F(t)$ and $g(t)$ to be the specific functions given in (\ref{2.taubesgeneratingfncs}). That choice, however, overcounts tori with self-intersection zero. It turns out that the formulas are simpler if $F$ satisfies \begin{eqnarray}\label{prodF=t} \prod_{k=1}^\infty F(t^k)=e^t. \end{eqnarray} Thus it is appropriate to make the more awkward-looking choice \begin{eqnarray}\label{MoebiusF} e(t)=1+t,\qquad F(t)=\exp\l(\ma\sum_{m= 1}^\infty \mu(m)t^m \r) ,\qquad \mbox{and}\qquad g(t)=e^t, \end{eqnarray} where $\mu$ is the M\"{o}bius function. (The M\"{o}bius function is defined by $\mu(1)=1$, $\mu(m)=(-1)^k$ if $m$ is a product of $k$ distinct primes, and $\mu=0$ otherwise.) One can then verify (\ref{prodF=t}) by writing $\ell=mk$ and using the basic fact that $$ \ma\sum_{m|\ell}\mu(m)\ =\ \left\{ \begin{array}{ll} 1\qquad & \mbox{if }\ell=1,\\ 0 & \mbox{otherwise.} \end{array} \right. $$ We will see next how the generating functions (\ref{MoebiusF}) lead back to the Taubes Series and the Seiberg-Witten invariants. \medskip \medskip \setcounter{equation}{0} \section{Equivalence of the Invariants} In this section we will prove that the Taubes and Ruan-Tian Series are equal for any closed symplectic four-manifold. The proof is straightforward for classes $A\notin T$, but for the toroidal classes $A\in T$ it requires some combinatorics. \smallskip For classes $A\notin T$, the moduli space of $J$-holomorphic curves of genus $g_A$ passing through $d$ points contains no multiply covered curves for generic $J$ (cf. [R], [T2]). Consequently, the moduli space of such curves is smooth and the linearized operator has no cokernel. The Implicit Function Theorem then implies that each of these curves (but none of their multiple covers) can be uniquely perturbed to a solution of the equation $\overline \partial_{j} f=\nu$ for small $\nu$. Thus \begin{eqnarray} Gr(A,d)=RT(A,d)\qquad \mbox{ for } \quad A\notin T, \label{4.1} \end{eqnarray} so the first two factors in the products (\ref{egfgeneratingfnc}) and (\ref{3.defgr}) are equal. The computations for $A\in T$ are more complicated because multiple covers {\em do} contribute. In this case, the moduli space ${\cal M}_A$ of $J$-holomorphic, connected, embedded $A$-curves is finite for generic $J$, and each curve $C\in {\cal M}_A$ is a torus. The last part of the Gromov series (\ref{3.defgr}) has the form $$ Gr^T\ =\ \prod_{A\in T} \ \prod_{C\in {\cal M}_A} \phi_C(t_A) $$ for some function $\phi_C$ that we must determine. To do that, we fix one torus $C\in{\cal M}_A$ defined by an embedding $(T^2,x_0,j_0)\rightarrow X$ and regard the domain $(T^2,x_0,j_0)$ as the quotient of the complex plane by the lattice $$ \Lambda_0={ \Bbb Z}\oplus \tau{ \Bbb Z}. $$ Curves $C'$ which are $m$-fold covers of $C$ are given by pairs $(\psi,j)$ where $\psi:(T^2,x_0,j)\rightarrow (T^2,x_0,j_0)$ is an $m$-fold cover map; these are classified (up to diffeomorphisms of the domain) by index $m$ sublattices $\Lambda \subset \Lambda_0$. Let ${\cal L}_m$ be the set of all such lattices. For generic $J$ the linearized operator has zero cokernel (it is invertible with index zero). Hence each $m$-fold cover can be uniquely perturbed to a solution of $\overline \partial_{j} f=\nu$, which contributes to $RT_{mA,1,1}$. The total contribution of the multiple covers of $C$ to $RT_{mA,1,1}$ is $$ \ma\sum_{\Lambda\in{\cal L}_m} \mbox{sgn }\Lambda \label{sumofsgns} $$ where $\mbox{sgn }\Lambda$ is the sign of the multiple cover $C'$ described by $\Lambda$. Thus, after stabilizing as in (\ref{3.2}), \begin{eqnarray} \phi_C(t_A)\ =\ \prod_{m=1}^{\infty}F(t_A^m)^{\frac{1}{m} \ma\sum_{\Lambda\in{\cal L}_m} \mbox{sgn }\Lambda} \label{phiC} \end{eqnarray} To proceed, we must determine $\mbox{sgn }\Lambda$ using the orientation prescribed by Ruan-Tian. As in Section 1, this is given by the the spectral flow of the linearization $D_C$ (the exposition in [RT] is obscure, but this is clearly the orientation that the authors intended to specify). This sign is independent of $\nu$ for small $\nu$, so we can assume that $\nu=0$ in the subsequent calculations. \begin{lemma} The sign of a curve $C'={ \Bbb C}/\Lambda$ is \begin{equation} \mbox{\em sgn} \, \Lambda\ =\ \mbox{\em sgn} \,D_0\,\ma\prod \mbox{\em sgn}\,D_i \label{sgnLambda} \end{equation} where the product is over all $i=1,2,3$ such that $\Lambda_0$ is a sublattice of $\Lambda_i$ with $\Lambda_i$ defined by (\ref{lambdai}). \end{lemma} \noindent {\bf Proof. } Looking at the explicit formula for $D_{C'}$ [T2], one sees that $D_{C'}$ is the pullback of $D_{C}$ (it depends only on the 1-jet of $J$ along $C$). Fix a complex operator $\ov\partial$ on $C$, choose a path from $\ov\partial$ to $D_{C}$, and let $D_t$ be the lifted path of operators on $C'$; each $D_t$ is invariant under deck transformations. As in [T2], we can assume that $\mbox{ker}\, D_t=\{0\}$ except at finitely many values of $t=t_k$, where $\mbox{ker}\, D_t$ is one-dimensional. The translations of ${ \Bbb C}$ by 1 and $\tau$ respectively induce deck transformations $\tau_1$ and $\tau_2$ of $C'\to C$; these generate the abelian group $G=\Lambda_0/\Lambda$ of all deck transformations. At each $t=t_k$, $\mbox{ker}\, D_t$ is a one-dimensional representation $\rho_i$ of $G$, so is one of four possibilities: $$ \left\{\begin{array}{l}\rho_0\tau_1(\xi)=\xi\\ \rho_0\tau_2(\xi)=\xi\end{array}\right.\qquad \left\{\begin{array}{l}\rho_1\tau_1(\xi)=-\xi\\ \rho_1\tau_2(\xi)=\xi\end{array}\right.\qquad \left\{\begin{array}{l}\rho_2\tau_1(\xi)=\xi\\ \rho_2\tau_2(\xi)=-\xi\end{array}\right.\qquad \left\{\begin{array}{l}\rho_3\tau_1(\xi)=-\xi\\ \rho_3\tau_2(\xi)=-\xi\end{array}\right. $$ where $\xi$ is a generator of the kernel. Call these kernels of type 0,\,1,\,2 and 3 respectively. Then $$ SF=\sum_{i=0}^3 SF_i $$ where $\mbox{SF}_i$ is the number of $t_k$ of type $i$ (counted with orientation), and \begin{equation} \mbox{sgn } \Lambda\ =\ (-1)^{\mbox{SF}}\ =\ \prod (-1)^{\mbox{SF}_i}. \label{sgnLambda2} \end{equation} Note that each $\xi$ of type 0 descends to a section of $\mbox{ker }\,D_t$ on ${ \Bbb C}/\Lambda_0$. In fact, this is a one-to-one correspondence, so $SF_0$ is the spectral flow of the path $D_t$ on the base curve $C$ and $(-1)^{\mbox{SF}}$ is the sign of $D_0$. The remaining representations determine three index two sublattices \begin{eqnarray} \Lambda_i=\mbox{ker}\, \rho_i \label{lambdai} \end{eqnarray} of $\Lambda$. Thinking of $\xi$ as a $\Lambda$-invariant section on ${ \Bbb C}$, one sees that (a) a type $i$ kernel cannot appear unless $\Lambda\subset\Lambda_i$, and (b) if $\Lambda\subset\Lambda_i$ then $\xi$ descends to an element of $\mbox{ker}\, D_t$ over the double cover ${ \Bbb C}/\Lambda_i$. Thus $\mbox{SF}_i$ vanishes if $\Lambda$ is not a subset of $\Lambda_i$, and when $\Lambda\subset\Lambda_i$ $\mbox{SF}_i$ coincides with the spectral flow of Taubes' operator $D_i$. Then (\ref{sgnLambda2}) is the same as (\ref{sgnLambda}). \qed \bigskip \addtocounter{theorem}{1 A set of representatives of the lattices in ${\cal L}_m$ is \begin{equation} \Lambda=a{ \Bbb Z}+(b\tau+p){ \Bbb Z} \qquad\mbox{where}\ \ m=ab,\ \ p=0,\dots,a-1. \label{lattices} \end{equation} The group of deck transformations is $G\cong { \Bbb Z}_{a}\times{ \Bbb Z}_{b}$, and the three the lattices (\ref{lambdai}) are $$ \Lambda_1={ \Bbb Z}+2\tau{ \Bbb Z}, \qquad \Lambda_2=2{ \Bbb Z}+\tau{ \Bbb Z}, \qquad \Lambda_3=2{ \Bbb Z}+(1+\tau){ \Bbb Z}. $$ \medskip Now fix $m$ and separate the set of lattices ${\cal L}$ into: \begin{eqnarray*} {\cal L}^0\ &=&\{ \Lambda\in{\cal L}\;|\; \Lambda \mbox{ is contained in none of the lattices }\Lambda_1,\Lambda_2,\Lambda_3\;\}\\ {\cal L}^i\ &=&\{ \Lambda\in{\cal L}\;|\; \Lambda \mbox{ is contained in $\Lambda_k$ only for $k=i$}\;\}\\ {\cal L}^{123}&=&\{ \Lambda\in{\cal L}\;|\; \Lambda \mbox{ is contained in }\Lambda_1,\Lambda_2\ \mbox{and}\ \Lambda_3\;\}. \end{eqnarray*} Note that if $\Lambda$ is contained in two of the $\Lambda_i$ then it is contained in the third. Thus the above sets constitute a partition $$ {\cal L}= {\cal L}^0 \cup {\cal L}^1 \cup {\cal L}^2 \cup {\cal L}^3 \cup {\cal L}^{123}. $$ Furthermore, there are automorphisms of $\Lambda_0$ that interchange the lattices $\Lambda_i, i=1,2,3$, so the sets ${\cal L}^1$, ${\cal L}^2$, and ${\cal L}^3$ have the same cardinality. Hence from (\ref{sgnLambda}) we have \begin{eqnarray} \sum_{\Lambda\in{\cal L}_m} \mbox{sgn }\Lambda\ =\ \mbox{sgn}\; D_0\left\{ A+B\;\sum_{i=1}^3 \mbox{sgn}\; D_i+C\prod_{i=1}^3 \mbox{sgn}\; D_i \right\} \label{RTm3} \end{eqnarray} where $A=|{\cal L}^0|$ is the number of elements of ${\cal L}^0$, $B=|{\cal L}^1|$, and $C=|{\cal L}^{123}|$. \medskip \begin{lemma} \label{abclemma} Set $\sigma(m)=\ma\sum_{a|m}a$ if $m$ is a positive integer, and $\sigma=0$ otherwise. Then $$ A+3B+C=\sigma(m), \qquad B+C=\sigma(m/2), \qquad C=\sigma(m/4). $$ \end{lemma} \noindent {\bf Proof. } Using the representatives (\ref{lattices}) of ${\cal L}$, we have \begin{eqnarray*} A+3B+C=|{\cal L}|=\sum_{m=ab}\;\sum_{p=0}^{a-1} 1=\ma\sum_{a|m}a=\sigma(m). \end{eqnarray*} Next, $B+C$ is the number of lattices $\Lambda\in{\cal L}$ which contain $\Lambda_1$. These are the lattices (\ref{lattices}) with $b=2\beta$ even, so \begin{eqnarray*} B+C=\sum_{m=a\cdot2\beta}\;\sum_{p=0}^{a-1}1= \sum_{a|{m\over 2}} a=\sigma\l({m\over 2}\r). \end{eqnarray*} Finally, ${\cal L}^{123}$ is the set of all lattices $\Lambda$ such that $a, b$ and $p$ are all even. Writing $a=2\alpha$, $b=2\beta$, and $p=2q$, we obtain \begin{eqnarray*} C=\left|{\cal L}^{123}\right|= \sum_{m=4\alpha \beta}\; \sum_{0\leq 2q\leq 2\alpha-1} 1=\sum_{\alpha|{m\over 4}} \alpha =\sigma\l({m\over 4}\r).\qquad \Box \end{eqnarray*} \medskip \begin{prop} \label{rel RT-Ta} The generating function $\phi_C$ of an embedded torus $C$ is \begin{eqnarray} \phi_C(t)=\l[f(t)f(t^2)^{s_1/ 2}f(t^4)^{s_2/4}\r]^{\mbox{{\em sgn} C}} \label{proprelRT-Ta} \end{eqnarray} where \begin{eqnarray}\label{s_i's} s_1=\ma\sum_{i=1}^3 \mbox{\em sgn}\,D_i-3, \qquad s_2=\ma\prod_{i=1}^3 \mbox{\em sgn}\,D_i-\ma\sum_{i=1}^3 \mbox{\em sgn}\,D_i+2, \end{eqnarray} and \begin{eqnarray} f(t)=\prod_{m\ge 1} F(t^m)^{\sigma(m)/m}. \label{deff(t)} \end{eqnarray} \end{prop} \noindent {\bf Proof. } From equations (\ref{phiC}) and (\ref{RTm3}) and Lemma \ref{abclemma}, we obtain \begin{eqnarray*} \log \phi_C(t_A)= \mbox{sgn}\; D_0 \ma \sum_{m= 1}^\infty \frac1m\left[\sigma(m)+s_1\cdot\sigma\l({m\over 2}\r)+ s_2\cdot\sigma\l({m\over 4}\r)\right]\ \log F(t_A^m) \end{eqnarray*} After substituting in (\ref{deff(t)}), this gives (\ref{proprelRT-Ta}). \qed \bigskip When $C$ has Taubes' type 0 all three $D_i$ have positive sign, so $s_1=s_2=0$ in (\ref{s_i's}). Similarly, $(s_1,s_2)$ is $(-2,0)$ for type 1, $(-4,4)$ for type 2, and $(-6,4)$ for type 3. Thus (\ref{proprelRT-Ta}) gives $$ \prod_{A\in T} \ \ma \prod_{C\in{\cal M}_A} \phi_C(t_A)\ =\ \prod_{A\in T} \ \ma\prod_{ k=0}^3 f_k(t_A)^{\tau(A,k)} $$ where $f_0=f$ and $$ f_1(t) ={f(t)\over f(t^2)},\qquad f_2(t)={f(t)f(t^4)\over f^2(t^2)},\qquad f_3(t)={f(t)f(t^4)\over f^3(t^2)} $$ --- exactly as in (\ref{T gen fc})! Since the first factors in (\ref{egfgeneratingfnc}) and (\ref{3.defgr}) are equal by (\ref{4.1}), we have the following equivalence. \begin{theorem} For any closed symplectic four-manifold $(X,\omega)$, the Taubes and Ruan-Tian Series (\ref{egfgeneratingfnc}) and (\ref{3.defgr}) coincide when $f$ and $F$ are related by (\ref{deff(t)}): $$ GT_X(t,s)\ =\ RT_X(t,s). $$ Hence Taubes' Gromov invariants $GT_{\delta}(A)$ depend only of the deformation class of $\omega$ and are computable from the Ruan-Tian invariants. \label{4.mainthm} \end{theorem} \medskip If we use the particular form of $F$ satisfying (\ref{prodF=t}) and make the change of variable $m=ab$, we obtain \begin{eqnarray*} \log f(t)&=&\sum_{m= 1}^\infty \;\sum_{a|m}{a\over m}\log F(t^m) = \; \sum_{b= 1}^\infty {1\over b}\;\sum_{a=1}^\infty \log F(t^{ab})= \ma \sum_{b= 1}^\infty {1\over b}\; t^b=\;\log {1\over 1-t}. \end{eqnarray*} Thus the Ruan-Tian Series with the generating functions defined in (\ref{MoebiusF}) exactly reproduces the Taubes Series with his choice of generating functions (\ref{2.taubesgeneratingfncs}). \bigskip \medskip

9312

alg-geom/9312007

en

https://arxiv.org/abs/alg-geom/9312007

alg-geom/9312007

Gerd Dethloff, Georg Schumacher, Pit-Mann Wong

Hyperbolicity of the complement of plane algebraic curves

LaTeX

Amer. J. Math. 117, 573-599 (1995)

The paper is a contribution of the conjecture of Kobayashi that the complement of a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadrics is hyperbolic and hyperbolically embedded as well as the complement of two quadrics and a line.

alg-geom

\section{Introduction} Hyperbolic manifolds have been studied in complex analysis as the generalizations of hyperbolic Riemann surfaces to higher dimensions. Moreover, the theory of hyperbolic manifolds is closely related to other areas (cf.\ eg. \cite{LA1}). However, only very few quasi-projective (non closed) hyperbolic manifolds are known. But one still believes that e.g.\ the complements of `most' hypersurfaces in $\Bbb P_n$ are hyperbolic, if only their degree is at least 2n+1, more precisely: \begin{conj} Let ${\cal C}(d_1,\ldots ,d_k)$ be the space of $k$ tupels of hypersurfaces $\,\Gamma = (\Gamma_1 , \ldots , \Gamma_k )\,$ in $\Bbb P_n$, where ${\rm deg}(\Gamma_i)=d_i$. Then for all $(d_1,\ldots ,d_k)$ with $\, \sum_{i=1}^k d_i =:d \geq 2n+1\,$ the set $\,{\cal H}(d_1,\ldots ,d_k)= \{ \Gamma \in {\cal C}(d_1,\ldots ,d_k) : \Bbb P_n \setminus \bigcup_{i=1}^k \Gamma_i\, $ {\rm is complete hyperbolic and hyperbolically embedded}$\}\,$ contains the complement of a proper algebraic subset of ${\cal C}(d_1,\ldots ,d_k)$. \end{conj} For complements of hypersurfaces in $\Bbb P_n$ this was posed by Kobayashi as `Problem 3' in his book \cite{KO}, and later by Zaidenberg in his paper \cite{ZA}. In this paper, we shall deal with the complements of plane curves i.e. the case n=2. Other than in the case of 5 lines $({\cal C}(1,1,1,1,1))$, the conjecture was previously proved by M.~Green in \cite{GRE2} in the case of a curve $\Gamma$ consisting of one quadric and three lines (${\cal C}(2,1,1,1)$). Furthermore, it was shown for ${\cal C}(d_1,\ldots,d_k)$, whenever $k\geq5$, by Babets in \cite{BA}. A closely related result by Green in \cite{GRE1} is that for any four non-redundant hypersurfaces $\Gamma_j$, $j=1,\ldots 4$ in $\Bbb P_2$ any entire curve $f:\Bbb C \to \Bbb P_2 \setminus \bigcup_{j=1}^4 \Gamma_j$ is algebraically degenerate. (The degeneracy locus of the Kobayashi pseudometric was studied by Adachi and Suzuki in \cite{A--S1}, \cite{A--S2}). In fact, for generic configurations, any such algebraically degenerate map is constant, hence the conjecture is true for any family ${\cal C}(d_1,\ldots,\linebreak d_k)$ with $k \geq 4$ (cf.\ cf.\ Theorem \ref{4c}). This includes the case of a curve $\Gamma$ consisting of 2 quadrics and 2 lines. We also give another proof of Green's result, which yields a slightly stronger result related to the statement of a second main theorem of value distribution theory in this situation. It seems that the conjecture is the more difficult the smaller k is. Already the case k=3 seems to be very hard: In 1989 H.~Grauert worked on the case of a curve $\Gamma$ consisting of 3 quadrics, i.e. ${\cal C}(2,2,2)$, in \cite{GR}, using sophisticated differential geometric methods including Jet-metrics. We believe that the methods developed there might be suited for proving major parts of the conjecture. For the time being, however, certain technical problems still exist with these methods including the case ${\cal C}(2,2,2)$. The main result of this paper (Theorem \ref{mt}) is a proof of the conjecture for 3 quadrics. Our methods are completely different from those used in \cite{GR} --- instead of differential geometry we use value distribution theory: For any pair of quadrics which intersect transversally, there are 6 lines through the intersection points, out of which 4 are in general position. We first show that we can assign a set of 12 lines in general position to any generic system $C$ of 3 quadrics. Let $f: \Bbb C \rightarrow \Bbb P_2 \setminus C$ be an entire holomorphic curve. Our method now essentially consists of showing that the defect of $f$ with respect to the above 12 lines had to be at least equal to 4 unless $f$ is algebraically degenerate. (For technical reasons our exposition is based on the Second Main Theorem rather than the defect relation). The last step is to show that this fact is actually sufficient for generic complements of 3 quadrics to be complete hyperbolic and hyperbolically embedded. For ${\cal C}(2,2,1)$, i.e. two quadrics and a line, our result states the existence of an open set, which contains a quasi-projective set of codimension one, of configurations, where the conjecture is true (Theorem $\ref{thm221}$). The somewhat lengthy proof is based on a generalized Borel lemma. With the same methods we prove that also the complement of three generic Fermat quadrics is hyperbolic. The paper is organized as follows: In section~2 we collect, for the convenience of the reader, some basics from value distribution theory, and, in section~3, some consequences from Brody's techniques for later reference. In section~4 we prove some `algebraic' hyperbolicity of generic complements of certain curves. Next, in section~5 we prove Theorem \ref{4c}. In section~6 we study linear systems of lines associated to systems of 3 quadrics. Section~7 contains the proof of Theorem~\ref{mt}. In section~8 we treat complements of two quadrics and a line and complements of three Fermat quadrics. The first named author would like to thank S.~Frankel (Nantes), H.~Grauert (G"ottingen), S.~Kosarew (Grenoble) and M.~Zaidenberg (Grenoble) for valuable discussions, the Department of Mathematics at Notre Dame for its hospitality, and the DFG, especially the `Schwer\-punkt Kom\-ple\-xe Man\-nig\-fal\-tig\-kei\-ten' in Bochum for support. The second named author would like to thank H.~Grauert, W.Stoll (Notre Dame) and M.Zaidenberg for valuable discussions, and the Department of Mathematics at Notre Dame and the SFB~170 in G\"{o}ttingen for its hospitality and the Schwerpunkt `Komplexe Mannigfaltig\-kei\-ten' for support. The third named author would like to thank the SFB 170 and the NSF for partial support. \section{Some tools from Value Distribution Theory} In this section we fix some notations and quote some facts from Value Distribution Theory. We give references but do not trace these facts back to the original papers. We define the characteristic function and the counting function, and give some formulas for these. Let $\,||z||^2= \sum_{j=0}^n |z_j|^2$, where $(z_0,\ldots ,z_n) \in \Bbb C^{n+1}$, let $\Delta_t = \{\xi \in \Bbb C : |\xi| < t \}$, and let $d^c = (i/4 \pi) (\overline{\partial} - \partial)$. Let $r_0$ be a fixed positive number and let $\,r \geq r_0$. Let $\,f:\Bbb C \rightarrow \Bbb P_n\,$ be entire, i.e. $f$ can be written as $\, f=[f_0:\ldots :f_n]\,$ with holomorphic functions $\, f_j : \Bbb C \rightarrow \Bbb C\, , j=0,\ldots ,n\,$ without common zeroes. Then the {\it characteristic function} $T(f,r)$ is defined as $$ T(f,r) = \int_{r_0}^r \frac{dt}{t} \int_{\Delta_t} dd^c \log ||f||^2$$ Let furthermore $\, D=\{ P=0\}\,$ be a divisor in $\Bbb P_n$, given by a homogeneous polynomial $P$. Assume $\, f(\Bbb C) \not\subset \hbox{ {\rm support}}(D)$. Let $\,n_f(D,t)\,$ denote the number of zeroes of $\, P \circ f\,$ inside $\, \Delta_t\,$ (counted with multiplicities). Then we define the {\it counting function} as $$ N_f(D,r) = \int_{r_0}^r n_f(D,t) \frac{dt}{t} $$ Stokes Theorem and transformation to polar coordinates imply (cf.\ \cite{WO}): \begin{equation} \label{1} T(f,r) = \frac{1}{4 \pi} \int_0^{2 \pi} \log ||f||^2 (re^{i \vartheta})d \vartheta + O(1). \end{equation} The characteristic function as defined by Nevanlinna for a holomorphic function $\,f: \Bbb C \rightarrow \Bbb C$ is $$ T_0(f,r) = \frac{1}{2 \pi} \int_0^{2 \pi} \log ^+ |f(re^{i \vartheta})| d \vartheta . $$ For the associated map $\, [f:1]: \Bbb C \rightarrow \Bbb P_1$ one has \begin{equation} \label{2} T_0(f,r) = T([1:f],r) + O(1) \end{equation} (cf.\ \cite{HA}). By abuse of notation we will, from now on, for a function $\, f: \Bbb C \rightarrow \Bbb C$, write $T(f,r)$ instead of $T_0(f,r)$. Furthermore we sometimes use $N(f,r)$ instead of $N_f([z_0=0],r)$. The concept of finite order is essential for later applications. \begin{defi} Let $s(r)$ be a positive, monotonically increasing function defined for $\,r \geq r_0$. If $$ \overline{\lim_{r \rightarrow \infty} } \frac{\log s(r)}{\log r} = \lambda$$ then $s(r)$ is said to be of order $\lambda$. For entire $\,f:\Bbb C \rightarrow \Bbb P_n\,$ or $\, f: \Bbb C \rightarrow \Bbb C\,$ we say that $f$ is of order $\lambda$, if $T(f,r)$ is. \end{defi} \begin{rem}\label{remfo} Let $f=[f_0:\ldots:f_n]:\Bbb C \to \Bbb P_n$ be a holomorphic map of finite order $\lambda$. Then $\log T(f,r)= O(\log r)$. \end{rem} We need the following: \begin{lem} \label{e} Assume that $\,f: \Bbb C \rightarrow \Bbb P_n\,$ is an entire map and misses the divisors $\,\{ z_j = 0\}\,$ for $j=0,\ldots,n$ (i.e. the coordinate hyperplanes of $\Bbb P_n$). Assume that $f$ has order at most $\lambda$. Then $f$ can be written as $\,f = [1:f_1:\ldots :f_n]\,$ with $\, f_j(\xi) = e^{P_j(\xi)}$, where the $P_j(\xi)$ are polynomials in $\xi$ of degree $d_j\leq \lambda$. \end{lem} {\it Proof:} We write $\, f=[1:f_1:\ldots :f_n]\,$ with holomorphic $\,f_j: \Bbb C \rightarrow \Bbb C \setminus \{0\}$. Now we get with equations (\ref{1}) and (\ref{2}) for $j=1,\ldots ,n$: $$ T(f_j,r) = T([1:f_j],r) + O(1) \leq T(f,r) + O(1), $$ hence the $f_j$ are nonvanishing holomorphic functions of order at most $\lambda$. This means that $$ \lim\/{\rm sup}_{r \rightarrow \infty} \frac{T(f_j,r)}{r^{\lambda + \epsilon}} =0 $$ for any $\, \epsilon > 0$. From this equation our assertion follows with the Weierstra\char\ss theorem as it is stated in \cite{HA}. \qed The previous Lemma is helpful because we can use it to `calculate' $T(f,r)$ by the Ahlfors-Lemma (cf.\ \cite{ST}) \begin{lem} \label{A} Let $\, P_0,\ldots ,P_n\,$ be polynomials of degree at most $\,\lambda \in \Bbb N\,$. Let $\, \alpha_j \in \Bbb C$ be the coefficients of $\,x^{\lambda}\,$ in $P_j$ (possibly equal to zero). Let $\, L(\alpha_0,\ldots ,\alpha_n)\,$ be the length of the polygon defined by the convex hull of the $\, \alpha_0,\ldots ,\alpha_n$. If $$f=[e^{P_0}:\ldots :e^{P_n}]:\Bbb C \rightarrow \Bbb P_n$$ then $$ \lim_{r \rightarrow \infty} \frac{T(f,r)}{r^{\lambda}} = \frac{L(\alpha_0,\ldots ,\alpha_n)}{2 \pi}$$ \end{lem} We state the First and the Second Main Theorem of Value Distribution Theory which relate the characteristic function and the counting function (cf.\ \cite{SH}): Let $\,f:\Bbb C \rightarrow \Bbb P_n\,$ be entire, and let $D$ be a divisor in $\Bbb P_n$ of degree $d$, such that $\,f(\Bbb C) \not\subset \hbox{ {\rm support}}(D)$. Then: \medskip {\bf First Main Theorem} $$ N_f(D,r) \leq d \cdot T(f,r) + O(1)$$ Another way of stating this theorem is the following: The quantity $$ \delta_f(D)=\liminf_{r\to \infty}\left(1- {N_f(D,r)\over d \cdot T(f,r)}\right) $$ is called {\it defect} of $D$ with respect to $f$. Then $$ \delta_f(D)\geq 0. $$ Assume now that $\, f(\Bbb C)\,$ is not contained in any hyperplane in $\Bbb P_n$, and let $\, H_1,\ldots ,H_q\,$ be distinct hyperplanes in general position. Then \medskip {\bf Second Main Theorem} $$ (q-n-1)T(f,r) \leq \sum_{j=0}^q N_f(H_j,r) + S(r) $$ where $\: S(r) \leq O(\log (rT(f,r)))\,$ for all $\,r \geq r_0\,$ except for a set of finite Lebesque measure. If $f$ is of finite order, then $\, S(r) \leq O(\log r)\,$ for all $\,r \geq r_0$. We examine how the characteristic function behaves under morphisms of the projective space: \begin{lem} \label{m} Let $$ R=[R_0:\ldots :R_N]: \Bbb P_n \rightarrow \Bbb P_N$$ be a morphism with components of degree $p$, and let $\, f:\Bbb C \rightarrow \Bbb P_n\,$ be entire. Then $$ T(R \circ f,r) = p \cdot T(f,r) + O(1)$$ \end{lem} {\it Proof:} Define $$\mu([z_0:\ldots :z_n]) = \frac{|R_0|^2+\ldots + |R_N|^2}{(|z_0|^2+\ldots +|z_n|^2)^p} $$ Since $R$ is a morphism the $\,R_j, j=0,\ldots ,N\,$ have no common zeroes, hence there exist constants $\,A,B >0\,$ with $\:0 < A \leq \mu \leq B\:$ on $\Bbb P_n$. From that and equation (\ref{1}) we get: $$T(R \circ f,r) - p \cdot T(f,r) = \frac{1}{4 \pi} \int_0^{2 \pi} (\log ||R \circ f||^2(re^{i\vartheta}) - p \cdot \log ||f||^2(re^{i\vartheta}))d \vartheta +O(1) $$ $$= \frac{1}{4 \pi} \int_0^{2 \pi} \log (\mu \circ f)(re^{i\vartheta}) d\vartheta +O(1)$$ In the last term the integral is bounded by $\, \frac{1}{2}\log A\,$ and $\,\frac{1}{2} \log B\,$ independently of $r$. \qed \section{Some consequences of Brody's techniques} In this section we list briefly some consequences of Brody's techniques for later application. The first is a corollary of a well known theorem of M. Green. It shows how to use entire curves $\,f:\Bbb C \rightarrow \Bbb P_2\,$ of finite order to prove hyperbolicity of quasiprojective varieties. The second follows from of a theorem of M.Zaidenberg. \medskip a) The main theorem of \cite{GRE2} implies: \begin{cor} \label{c} Let $D$ be a union of curves $\, D_1,\ldots ,D_m\,$ in $\Bbb P_2$ such that for all $\, i=1,\ldots ,m\,$ the number of intersection points of $\,D_i\,$ with $\:\bigcup_{j=1,\ldots ,m;j \neq i} D_j\:$ is at least three. Then $\,\Bbb P_2 \setminus D\,$ is complete hyperbolic and hyperbolically embedded, if there does not exist a non-constant entire curve $\, f:\Bbb C \rightarrow \Bbb P_2\,$ of order at most two which misses $D$. \end{cor} b) The following proposition shows that the property of a union of curves having hyperbolic complement is essentially a (classically) open condition. \begin{prop} \label{z} Let $\,H_1,\ldots ,H_m\,$ be hypersurfaces in $\, \Bbb P_2 \times (\Delta_t)^n\,$ for some $\,t>0$, $n \in \Bbb N$. Let $\: \pi : \Bbb P_2 \times (\Delta_t)^n \rightarrow (\Delta_t)^n\:$ be the projection. Assume that 1) for all $\, z \in (\Delta_t)^n\,$ and all $i=1,\ldots ,m$ the fibers $\: \pi^{-1}(z) \cap H_i\:$ are curves in $\Bbb P_2$ 2) for all $i=1,\ldots ,m$ the number of intersection points of $\: \pi^{-1}(0) \cap H_i\:$ and \\ $\: \bigcup_{j=1,\ldots ,m;j \neq i}(\pi^{-1}(0) \cap H_j))\:$ is at least three. 3) $\: \Bbb P_2 \setminus \bigcup_{j=1,\ldots ,m} (\pi^{-1}(0) \cap H_j)\:$ is hyperbolically embedded in $\Bbb P_2$. Then $\: \Bbb P_2 \setminus \bigcup_{j=1,\ldots ,m}( \pi^{-1}(z) \cap H_j)\:$ is complete hyperbolic and hyperbolically embedded for all $\, z \in (\Delta_s)^n\,$ for some $\,s \leq t \,$. \end{prop} {\it Proof:} In the terminology of \cite{ZA}, the $\: \pi^{-1}(0) \cap H_i\:$ form an absorbing $H$-stratification (cf.\ \cite{ZA}, p. 354 f.), for which we can apply Theorem~2.1 of \cite{ZA}. Complete hyperbolicity follows from \cite{LA2}, p.36. \qed \section{Nonexistence of algebraic entire curves in generic complements} In this section we prove that the complement of 3 generic quadrics, or of any 4 generic curves other than 4 lines, does not contain non-constant entire curves contained in an algebraic curve. Because of Corollary \ref{c} this can be regarded as a statement of `algebraic' hyperbolicity. Let us first make precise what we mean by generic. The space of curves $ \Gamma_i $ of degree $ d_i $ in $\Bbb P_2$, which we define as the projectivized space of homogeneous polynomials of degree $d_i$, is a projective space of dimension $\: n_i = \frac{1}{2}(d_i+2)(d_i+1) -1 $. Hence $ {\cal C}(d_1,\ldots ,d_k) = \prod_{i=1}^k \Bbb P_{n_i} $ is projective algebraic. In order to simplify notations we denote this space by $S$ in all what follows, and its elements by $ s \in S$, and by $ \Gamma_i(s) $ the curve given by the i-th component of $ s \in S$. \begin{prop} \label{a} Let $ S={\cal C}(2,2,2) $ or $ S= {\cal C}(d_1,\ldots ,d_k) $ with $ k \geq 4 $ and $ d= \sum_{i=1}^k d_i \geq 5 $. Then there exists a proper algebraic variety $ V \subset S $ st. for $ s \in S \setminus V $ the following holds:\\ For any irreducible plane algebraic curve $A\subset \Bbb P_2$ the punctured Riemann surface $A\setminus \bigcup_{i=1}^k \Gamma_i(s)$ is hyperbolic, in particular any holomorphic map $f:\Bbb C \to \Bbb P_2\setminus\bigcup_{i=1}^k \Gamma_i(s)$ with $f(\Bbb C)\subset A$ (which may also be reducible) is constant. \end{prop} {\it Proof:} In order to define $ V \subset S $ we list 5 conditions: (1) All $ \Gamma_i (s) $ are smooth (and of multiplicity one). (2) The $ \Gamma_i (s),\: i=1,\ldots ,k $ intersect transversally, in particular no 3 of these intersect in one point. (3) In the case of $ {\cal C}(2,2,2) $: For any common tangent line of two of the quadrics $\Gamma_j(s)$ which is tangential to these in points $P$ and $Q$ resp. the third quadric does not intersects the tangent in both points $P$ and $Q$. (4) In the case of $ {\cal C}(d_1,d_2,1,1),\:d_1,d_2 \geq 2 $: There does not exist a common tangent $L$ to $ \Gamma_1(s) $ and $ \Gamma_2(s) $ such that $L\cap\Gamma_1(s)=\{P\}$ and $L\cap\Gamma_2(s)=\{Q\}$ such that the lines $ \Gamma_3(s) $ and $\Gamma_4(s) $ contain $P$ and $Q$ resp.. (5) In the case of $ {\cal C}(d_1,1,1,1) $: There does not exist a tangent line $L$ at $ \Gamma_1 (s) $ with $L\cap \Gamma_1 (s)= \{P\}$ such that $P$ is contained in one of the lines $ \Gamma_i (s),\:i=2,3,4 $ and $L$ contains the intersection points of the other two lines. Define $ V \subset S $ to be the set of those points $s \in S$ such that the $\Gamma_i(s)$ violates one of the above conditions. This set is clearly algebraic and not dense in $S$. For intersections of at least five curves (2) implies that any irreducible algebraic curve $A$ intersects $\bigcup_{i=1}^k \Gamma_i(s)$ in at least three different points, which proves the claim. Assume that there exists an irreducible algebraic curve $A \subset \Bbb P_2 $ and $s\in S$ such that $A\setminus \bigcup_{i=1}^k \Gamma_i(s)$ is not hyperbolic. By condition (2) we know that $ A \cap \bigcup_{i=1}^k \Gamma_i(s) $ consists of at least 2 points $P$ and $Q$. Moreover, $A$ cannot have a singularity at $P$ or $Q$ with different tangents, because $A$ had to be reducible in such a point, and $A\setminus \bigcup_{i=1}^k \Gamma_i(s)$ could be identified with an irreducible curve with at least three punctures. (This follows from blowing up such a point or considering the normalization). So $ A \cap \bigcup_{i=1}^k \Gamma_i(s) $ consists of exactly 2 points $P$ and $Q$ with simple tangents. We denote the multiplicities of $A$ in $P$ and $Q$ by $ m_P $ and $ m_Q $. Let $d_0= \deg(A)$. Then the inequality (cf.\ \cite{FU}, p.117) $$ m_P(m_P-1)+m_Q(m_Q-1) \leq (d_0-1)(d_0-2) $$ implies \begin{equation} \label{*} m_P , m_Q < d_0 \hbox{ {\rm or} }d_0=m_P=m_Q=1. \end{equation} Let us now first treat the case $ k=4 $: Each $ \Gamma_i(s) $ contains exactly one of the points $P$ and $Q$. Let $\Gamma_j(s)$ and $\Gamma_k(s)$ resp. intersect $A$ in $P$ and $Q$ resp. not tangential, i.e. with tangents different from those of $A$ in these points. Let $d_j$ and $d_k$ be the degrees of these components. We compute intersection multiplicities according to \cite{FU}, p.75 $$ m_P = I(P, A \cap \Gamma_j (s)) = d_j d_0\quad {\rm and} \quad m_Q= I(Q, A \cap \Gamma_k (s)) = d_k d_0.$$ Hence $$ d_0=d_j=m_P=1\:\: {\rm and} \:\: d_0=d_k=m_Q=1.$$ In particular $A, \Gamma_j$ and $\Gamma_k$ are lines. These situations are excluded by (4) and (5). Now let us treat the case of 3 quadrics. After a suitable enumeration of its components we may assume that $P \in \Gamma_1 (s) \cap \Gamma_2 (s) $ and $ Q \in \Gamma_3 (s)$. If $Q \not\in \Gamma_2 (s) \cup \Gamma_1 (s) $ we are done, since then we may assume that $A$ is not tangential to $ \Gamma_2 (s)$, and again $$ m_P = I(P,A \cap \Gamma_2 (s)) = 2d_0. $$ contradicts equation (\ref{*}). So we may assume that $\, Q \in \Gamma_2(s) \cap \Gamma_3(s)$. Now $A$ has to be tangential to $\, \Gamma_1(s)$ in $P$ and to $\, \Gamma_3(s)$ in $Q$, otherwise we again get $\, m_P=2d_0\,$ or $\,m_Q =2d_0\,$ what contradicts equation (\ref{*}). But then $\, \Gamma_2(s)$ is not tangential to $A$ in $P$ or $Q$, so we have $$m_P + m_Q = I(P,A \cap \Gamma_2(s)) + I(Q, A \cap \Gamma_2(s)) = 2d_0$$ Again by equation (\ref{*}) this is only possible if $\, m_P=m_Q=d_0=1$, but then we are in a situation which we excluded in condition (3), which is a contradiction. \qed \section{Hyperbolicity of generic complements of at least four curves} In this section we prove a result in the direction towards a generalized second main theorem. As a corollary we get a new proof of the fact that for any generic collection of four hypersurfaces $\Gamma_j$, $j=1,\ldots 4$ in $\Bbb P_2$ any entire curve $f:\Bbb C \to \Bbb P_2 \setminus \bigcup_{j=1}^4 \Gamma_j$ has to be algebraically degenerate. This fact, combined with our result in the previous section implies the hyperbolicity of the complement of such a configuration. \begin{theo} \label{4c} Let $ S= {\cal C} (d_1,\ldots ,d_k) $ with $ k \geq 4$, $d= \sum_{i=1}^k d_i \geq 5$. Then there exists an algebraic variety $ V \subset S $ such that for $ s \in S \setminus V $ the following holds: Assume that $ f: \Bbb C \rightarrow \Bbb P_2 \setminus \bigcup_{j=1}^3 \Gamma_j (s) $ is a non-constant holomorphic map. Then $ \delta_f (\Gamma_l (s))=0 $ for $l=4,\ldots ,k$. In particular, $f$ cannot miss any $\Gamma_l(s)$, $l=4,\ldots ,k$. \end{theo} {\it Proof:} Let $ V \subset S $ be defined like in the proof of Proposition \ref{a}, i.e. $ s \in S \setminus V $, iff the conditions (1) to (5) given there are satisfied. Let $ \Gamma_i(s) = \{P_i(s) = 0 \} $ for $i=1,\ldots ,k$. For suitable powers $a_j$ we have because of condition (2) a morphism \begin{equation} \label{a1} \Phi : \Bbb P_2 \rightarrow \Bbb P_2; [z_0:z_1:z_2] \rightarrow [P_1^{a_1}(s):P_2^{a_2}(s):P_3^{a_3}(s)] \end{equation} Since there exists no non-constant morphism on projective spaces, whose image is of lower dimension, for all $ s \in S \setminus V $ the image $ \Phi (\Bbb P_2) $ is not contained in an algebraic curve. From now on, we keep some $s \in S\setminus V $ fixed and drop the parameter $s$ for the rest of the proof. Furthermore let $ \Phi (\Gamma_4) =\{ Q=0 \} $, where $$Q(w_0,w_1,w_2) = \sum_{i_0 +i_1+ i_2 =e} a_{i_0i_1i_2} w_0^{i_0}w_1^{i_1}w_2^{i_2},$$ so $\deg Q=e$. Finally, let $\: \Phi^{-1}(\Phi (\Gamma_4)) = \Gamma_4 \cdot R \:$ be the decomposition of the inverse image curve of the curve $ \Phi(\Gamma_4) $ in $\Gamma_4$ and the other components (which possibly may contain $\Gamma_4$ as well). Now the proof consists of 3 steps:\\ a) We have $ a_{e00} \not= 0,\: a_{0e0} \not= 0,\:a_{00e} \not= 0 $, i.e. the polynomial $Q$ contains the $e$-th powers of the coordinates:\\ We prove that indirectly, so without loss of generality we may assume that $ a_{e00}=0$. Then we have $ Q([1:0:0])=0$, i.e. $ [1:0:0] \in \Phi (\Gamma_4)$. So there exists a point $ z \in \Gamma_4 $ with $ P_1(z) \not= 0,\:P_2(z)=0,\:P_3(z)=0 $. But that means that the 3 curves $\Gamma_2$, $\Gamma_3$ and $\Gamma_4$ have a common point which contradicts our condition (2).\\ b) We show by using the Second Main Theorem that $ \delta_{\Phi \circ f}( \Phi (\Gamma_4))=0$:\\ Let $ J=\{ (\underline{i} = (i_0,i_1,i_2) : a_{i_0i_1i_2} \not= 0 \}$ and $ \kappa : J \rightarrow \{0,1,\ldots ,p \} $ be an enu\-me\-ra\-tion of $J$. Let $ Q_j = w_0^{i_0}w_1^{i_1}w_2^{i_2} $ if $ \kappa ((i_0,i_1,i_2)) =j$. Then by part a) the map $$ \Psi : \Bbb P_2 \rightarrow \Bbb P_p; [w_0:w_1:w_2] \rightarrow [Q_0:\ldots :Q_p] $$ is a morphism with components of degree $ e=\deg(Q)$. The $p+2$ lines $ L_i= \{ \xi_i=0 \},\:i=0,\ldots ,p $ and $ L= \{\sum_{\underline{i} \in J} a_{\underline{i}} \xi^{\kappa (\underline{i})} =0 \} $ are in general position. Furthermore the map $\: \Psi \circ \Phi \circ f : \Bbb C \rightarrow \Bbb P_p $ is linearly non degenerate: By Proposition \ref{a}, $ f(\Bbb C) $ is not contained in an algebraic curve, so especially not in an algebraic curve of the form $ \sum_{\underline{i} \in J} b_{\underline{i}}(P_1^{a_1})^{i_0} (P_2^{a_2})^{i_1}(P_3^{a_3})^{i_2} $, resulting from such a line in $ \Bbb P_p$, unless the latter is identically zero. But this is impossible, since the map $\Phi$ is surjective. So we have by the Second Main Theorem: $$T(\Psi \circ \Phi \circ f,r) \leq N_{\Psi \circ \Phi \circ f}(L,r) + \sum_{i=0}^p N_{\Psi \circ \Phi \circ f}(L_i,r) + S(r) $$ and by the First Main Theorem $$N_{\Psi \circ \Phi \circ f}(L,r) \leq T(\Psi \circ \Phi \circ f,r) + O(1)$$ Observe that all $N_{\Psi \circ \Phi \circ f}(L_i,r)$ vanish. Together with Lemma \ref{m} this yields, since $\deg Q=e$ \begin{equation} \label{a2} \delta_{\Phi \circ f} (\Phi (\Gamma_4)) = \liminf_{r \rightarrow \infty} (1- \frac{N_{\Phi \circ f}(\Phi(\Gamma_4),r)}{\deg(Q)T(\Phi \circ f,r)}) = \liminf_{r \rightarrow \infty} (1- \frac{N_{\Psi \circ \phi \circ f} (L,r)}{ T(\Psi \circ \Phi \circ f,r)}) =0 \end{equation} c) We finally show that $ \delta_f(\Gamma_4)=0$:\\ By equation (\ref{a2}) and Lemma \ref{m} we have: $$1 = \limsup_{r \rightarrow \infty} \frac{N_{\Phi \circ f}( \Phi (\Gamma_4),r)}{ \deg(Q)T(\Phi \circ f,r)} = \limsup_{r \rightarrow \infty} \frac{N_f(\Phi^{-1} \Phi (\Gamma_4),r)}{\deg(Q \circ \Phi) T(f,r)}$$ $$= \limsup_{r \rightarrow \infty} \frac{N_f(\Gamma_4,r) + N_f(R,r)}{\deg(Q \circ \Phi) T(f,r)} = \limsup_{r \rightarrow \infty} \frac{N_f(\Gamma_4,r) + N_f(R,r)}{(\deg(\Gamma_4) + \deg(R)) T(f,r)}$$ or short: \begin{equation} \label{a3} \limsup_{r \rightarrow \infty} \frac{N_f(\Gamma_4,r)}{T(f,r)} + \limsup_{r \rightarrow \infty} \frac{N_f(R,r)}{T(f,r)} = \deg(\Gamma_4) + \deg(R) \end{equation} By the First Main Theorem we have: $$ \limsup_{r \rightarrow \infty} \frac{N_f(R,r)}{\deg(R)T(f,r)} \leq 1 ,\:\: \limsup_{r \rightarrow \infty} \frac{N_f (\Gamma_4,r)}{\deg(\Gamma_4)T(f,r)} \leq 1$$ and hence with equation (\ref{a3}): $$\limsup_{r \rightarrow \infty} \frac{N_f(\Gamma_4,r)}{T(f,r)} = \deg (\Gamma_4) ,\:{\rm i.e.}\: \delta_f(\Gamma_4)=0 .$$ \qed \section{Line systems through intersection points of three quadrics} In this section, we study certain configurations of 18 lines associated to three smooth quadrics. These lines are needed in order to apply Value Distribution Theory to prove our main theorem in the next section. Let $V' \subset S={\cal C}(2,2,2)$ be the algebraic variety defined by the conditions (1), (2), and (3) given in the Proof of Proposition \ref{a}, namely $s\in S \setminus V'$, iff \begin{description} \item[(1)] All $\Gamma_i(s)$ are smooth quadrics. \item[(2)] The $\Gamma_i(s),\:i=1,2,3$ intersect transversally (in particular not all 3 intersect in one point) \item[(3)] For any common tangent line of two of the quadrics $\Gamma_j(s)$ which is tangential to these in points $P$ and $Q$ resp. the third quadric does not intersects the tangent in $P$ and $Q$. \end{description} In order to prove our main theorem we will need one further condition of `genericity' related to those 18 lines already mentioned above. For this condition it is quite not so obvious any more that it yields a quasiprojective set. We shall give an argument for this in Proposition \ref{ls}. Let us first state the extra condition: Because of (2) any two of the three quadrics $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ intersect in 4 distinct points $A_1, A_2, A_3, A_4$ which give rise to six lines \begin{equation} \label{!} \overline{A_1A_2},\,\overline{A_3A_4} \hbox{ \rm and }\overline{A_1A_3},\,\overline{A_2A_4} \hbox{ \rm and } \overline{ A_1A_4},\,\overline{A_2A_3}. \end{equation} So all three pairs of quadrics give rise to three sets $L_{12}(s)$, $L_{13}(s)$ and $L_{23}(s)$ of six lines each, i.e. a collection $L(s)$ of 18 lines. We will show in the proof of Proposition \ref{ls} that as a consequence of (1) and (2) they are pairwise distinct. Now our condition (4) reads: \begin{description} \item[(4)] The 18 lines $L(s)$ intersect as follows: At any point of $\Gamma_i(s)\cap \Gamma_j(s)$, $i \not= j$ there intersect exactly 3 of the 18 lines, and in every other point of $\Bbb P_2$ there intersect at most 2 of the 18 lines. \end{description} Now we have: \begin{prop} \label{ls} Define $V \subset S$ to be the set of all $s \in S$ such that one of the conditions (1) to (4) is not satisfied. Then $V \subset S$ is a proper algebraic subset. \end{prop} {\it Proof:}\/ In order to prove the Proposition we use an argument which involves an elementary case of a Chow scheme. We denote by $\Bbb P_2^\vee$ the space of all lines in $\Bbb P_2$. Look at the following rational map $$ \psi: (\Bbb P_2)^4 \to (\Bbb P_2^\vee)^6 $$ $$ (A_1,A_2,A_3,A_4) \mapsto (A_j \wedge A_k)_{j<k} $$ where the wedge product of two points is considered as an element of the dual projective space. This map descends to a rational map of symmetric spaces: $$ \Psi: S^4(\Bbb P_2) \to S^6(\Bbb P_2^\vee). $$ Over the complement of a proper algebraic subset it assigns to a set of four distinct points the configuration of six lines through these points. Now we assign to any $s \in S\setminus V'$ the tripel of sets $(\Gamma_1(s)\cap\Gamma_2(s),\Gamma_1(s)\cap\Gamma_3(s),\Gamma_2(s)\cap\Gamma_3(s))$, which amounts to a morphism $$ \rho:S\setminus V' \to (S^4(\Bbb P_2))^3. $$ Observe that $\Xi:=(\Psi)^3\circ \rho : S\setminus V' \to (S^6(\Bbb P_2^\vee))^3$ is a morphism. Now we can rephrase condition (4): Let $U\simeq \Bbb C^3$ and $W\simeq (U)^6$. Then we consider $W^3 = \{ (a_{jk})|a_{jk} \in U ; j=1,\ldots,6; k=1,2,3 \}$ and look at the linear subspace $B\subset W^3$ defined by the condition that at least three components $a_{j_1 k_1}$, $a_{j_2 k_2}$ and $a_{j_3 k_3}$ are {\it linearly dependent} where {\it not all $k_j$ are the same}. (We needn't care about the system of the six lines given by the four intersection points of two fixed quadrics, since they automatically have the desired intersection properties, because no three of the four intersection points of the two quadrics can be collinear.) Obviously $B$ descends to an algebraic set $\tilde B \in (S^6(\Bbb P_2^\vee))^3$. Now (4) means for $s\in S\setminus V'$ that $\Xi(s) \not \in \tilde B$. The construction immediately implies that $\,V \setminus V' \subset S \setminus V'\,$ is algebraic, and since $\, V' \subset S$ is algebraic, we have that $\, V= \overline{V \setminus V'} \cup V'\,$ is algebraic in $S$, where the closure here means the Zariski closure. We have to show that $V \neq S$. The existence of an $s \in S\setminus V$ is proved by a deformation argument: We start with any $s \in S \setminus V'$ (then the $\Gamma_i(s)$ are smooth and we have 12 different intersection points of two of the 3 quadrics each). It is easy to see that then we really have 18 different lines, otherwise 4 of the 12 intersection points of the 3 quadrics had to be contained in a line (because no three quadrics pass through a line). It follows from the construction, that this line would intersect one of the quadrics in 4 points, which is impossible. Let $k=k(s)$ be the largest number of lines among the 18 lines (determined by the parameter $s$) which run through some point. Let $\nu_k=\nu_k(s)$ be the number of points in $\Bbb P_2$ which are contained in $k(s)$ of the lines. We will proceed now as follows: We observe that $k$ lines running through a point is a closed condition with respect to the classical topology of $S$. That means that in a neighborhood $U$ of a point $s_0 \in S$ we have $k(s)\leq k(s_0)$, and at least $\nu_k(s)\leq\nu_k(s_0)$, if $k(s)= k(s_0)$. We will show that for some $s\in U$ actually $k(s)<k(s_0)$ or at least $\nu_k(s)<\nu_k(s_0)$, if $k(s)= k(s_0)$, as long as $k(s_0)>3$ or $k(s_0)=3$ but $\nu_k(s_0)> 12 $. Iterating this procedure we are done if we can show: Consider the 18 lines in $L(s_0)$. If $k \geq 4$ take any of these intersection points where $k$ lines intersect (call it $T$), if $k=3$ take such an intersection point $T$ which is not intersection point of two of the quadrics. Then we can find $s \in S$ arbitrarily near to $s_0$ st. over $s$ the point $T$ `breaks up' into intersection points of strictly less then $k(s_0)$ lines. But then $k(s)<k(s_0)$, or at least $k(s)=k(s_0)$ and $\nu_k(s)<\nu_k(s_0)$. Let us now prove that: Take 3 of the lines running through $T$ over $s_0$ and denote them by $L_1,L_2,L_3$. Each of them is defined by construction by two of the intersection points of two of the 3 quadrics. Let $L_1$ be defined by such points $T_1,T_2$, let $L_2$ be defined by $T_3,T_4$ and let $L_3$ be defined by $T_5,T_6$. We may assume that no 3 of the 6 points $T_1,\ldots ,T_6$ are equal to $T$ (this could only occur if $T$ is an intersection point of 2 of our 3 quadrics, but then $k \geq 4$ and we just have taken the 3 lines defined by $T$ and one other intersection point each, so we can choose a different line). So without loss of generality we may assume that $T_1 \not= T \not= T_2$, and we have the following 3 possibilities for $L_1,L_2,L_3,T,T_1,\ldots ,T_6$: \moveleft1cm \hbox{ \beginpicture \setcoordinatesystem units <0.2em,0.2em> \unitlength0.2em \setplotarea x from -30 to 30, y from -50 to 50 \put {\line(1,-2){20}} [Bl] at 0 0 \put {\line(-1,-2){20}} [Bl] at 0 0 \put {\line(1,0){20}} [Bl] at 0 0 \put {\line(1,2){20}} [Bl] at 0 0 \put {\line(-1,2){20}} [Bl] at 0 0 \put {\line(-1,0){20}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at -10 20 \put {\circle*{1.1}} [Bl] at 10 -20 \put {\circle*{1.1}} [Bl] at 10 20 \put {\circle*{1.1}} [Bl] at -10 -20 \put {\circle*{1.1}} [Bl] at 10 0 \put {\circle*{1.1}} [Bl] at -10 0 \put {$L_1$} [Bl] at -18 40 \put {$L_2$} [Bl] at 22 40 \put {$L_3$} [Bl] at 22 -2 \put{$T_1$} [Bl] at -10 22 \put{$T_2$} [Bl] at 12 -18 \put{$T_3$} [Bl] at 12 22 \put{$T_4$} [Bl] at -8 -22 \put{$T_5$} [Bl] at 12 2 \put{$T_6$} [Bl] at -8 2 \put{$T$} [Bl] at 3 2 \endpicture \beginpicture \setcoordinatesystem units <0.2em,0.2em> \unitlength0.2em \setplotarea x from -30 to 30, y from -50 to 50 \put {\line(1,-2){20}} [Bl] at 0 0 \put {\line(-1,-2){20}} [Bl] at 0 0 \put {\line(1,0){20}} [Bl] at 0 0 \put {\line(1,2){20}} [Bl] at 0 0 \put {\line(-1,2){20}} [Bl] at 0 0 \put {\line(-1,0){20}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at -10 20 \put {\circle*{1.1}} [Bl] at 10 -20 \put {\circle*{1.1}} [Bl] at 10 20 \put {\circle*{1.1}} [Bl] at -10 -20 \put {\circle*{1.1}} [Bl] at -10 0 \put {$L_1$} [Bl] at -18 40 \put {$L_2$} [Bl] at 22 40 \put {$L_3$} [Bl] at 22 -2 \put{$T_1$} [Bl] at -10 22 \put{$T_2$} [Bl] at 12 -18 \put{$T_3$} [Bl] at 12 22 \put{$T_4$} [Bl] at -8 -22 \put{$T_6$} [Bl] at -8 2 \put{$T=T_5$} [Bl] at 3 2 \endpicture \beginpicture \setcoordinatesystem units <0.2em,0.2em> \unitlength0.2em \setplotarea x from -30 to 30, y from -50 to 50 \put {\line(1,-2){20}} [Bl] at 0 0 \put {\line(-1,-2){20}} [Bl] at 0 0 \put {\line(1,0){20}} [Bl] at 0 0 \put {\line(1,2){20}} [Bl] at 0 0 \put {\line(-1,2){20}} [Bl] at 0 0 \put {\line(-1,0){20}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at 0 0 \put {\circle*{1.1}} [Bl] at -10 20 \put {\circle*{1.1}} [Bl] at 10 -20 \put {\circle*{1.1}} [Bl] at -10 -20 \put {\circle*{1.1}} [Bl] at -10 0 \put {$L_1$} [Bl] at -18 40 \put {$L_2$} [Bl] at 22 40 \put {$L_3$} [Bl] at 22 -2 \put{$T_1$} [Bl] at -10 22 \put{$T_2$} [Bl] at 12 -18 \put{$T_4$} [Bl] at -8 -22 \put{$T_6$} [Bl] at -8 2 \put{$T=T_3=T_5$} [Bl] at 3 2 \endpicture } \noindent The point $T_1$ lies on 2 of the $\Gamma_i$, assume on $\Gamma_1 \cap \Gamma_2$. Then at most 3 of the 4 or 5 different points in $\{T_2,\ldots,T_6\}$ can also be in $\Gamma_1 \cap \Gamma_2$. So there exists one of them, call it $T_0$, which does lie on $\Gamma_1 \cap \Gamma_3$ or $\Gamma_2 \cap \Gamma_3$, assume on $ \Gamma_2 \cap \Gamma_3$. So at most 4 of the points $T_2,\ldots ,T_6$ are contained in $\Gamma_1$. So we can `move' $\Gamma_1$ while keeping these 4 points fixed and keeping $\Gamma_2$ and $\Gamma_3$ fixed. But that means that there is a non-constant variation of $s$ where we keep all of the points $T_2,\ldots,T_6$ fixed. Hence the lines $L_2$ and $L_3$ and their intersection point $T$ are kept fixed. We claim that for some small such variation the line $L_1$ does not pass any longer through $T$. If it would, it had to be fixed, since $T_2$ is kept fixed. By definition we have $T_1 \in L_1 \cap \Gamma_1 \cap \Gamma_2$ and $ L_1 \cap \Gamma_2$ is a discrete set. Hence $T_1$ remains fixed. But that would mean that any quadric $\Gamma_1$ through the at most 4 of the fixed points $T_2,\ldots,T_6$ contained in $\Gamma_1$ must contain a fifth fixed point $T_1$. This is certainly a contradiction, since the space of plane quadrics is of dimension five. \qed \pagebreak {}From any of the configurations of 18 lines in Proposition \ref{ls} we can pick 12 in general position: \begin{cor} \label{12l} There exists an algebraic variety $V \subset S$ st. for all $ s \in S \setminus V$ we have subsets of 12 of the 18 lines of Proposition \ref{ls} which are in general position. \end{cor} {\it Proof:} For each pair $\Gamma_i(s),\Gamma_j(s),\: i \not=j$ we have constructed 3 pairs of lines (defined by equation (\ref{!})). Choose, for fixed $s \in S$, for each pair of quadrics two of these pairs of lines. \qed At last we prove the simple fact that the pairs of lines as defined in equation (\ref{!}) are contained in the linear system spanned by the two quadrics. \begin{prop} \label{sy} Let $\Gamma_1,\Gamma_2$ be two smooth quadrics intersecting in 4 different points $ A_1,A_2,A_3,A_4$, and let the lines $L_1$ resp. $L_2$ be given by $A_1,A_2$ and $A_3,A_4$ resp. Then $L_1L_2$ is a degenerate quadric contained in the linear system spanned by $\Gamma_1$ and $\Gamma_2$, i.e. $\: L_1L_2 = a \Gamma_1 + b \Gamma_2$. \end{prop} {\it Proof:} Look at the set ${\cal L}$ of all quadrics (possibly singular) which run through the 4 points $A_1,A_2,A_3,A_4$. Then ${\cal L}$ is a one dimensional linear system containing $L_1L_2$. Since it is one dimensional, it is spanned by any 2 of its elements, e.g.\ by $\Gamma_1$ and $\Gamma_2$. \qed \section{Hyperbolicity of generic complements of three quadrics} We will prove: \begin{theo} \label{mt} Let $V \subset S$ be the variety defined in Proposition \ref{ls}. Let $s \in S \setminus V$. Then the quasiprojective variety $\: \Bbb P_2 \setminus \bigcup_{i=1}^3 \Gamma_i(s)\:$ is complete hyperbolic and hyperbolically embedded. \end{theo} \begin{rem} {\it The variety $S\setminus V$ is certainly not contained in an open subset of the space of all divisors of degree $6$ whose complement in $\Bbb P_2$ is hyperbolic (cf.\ also \cite{ZA}): Take any quadratic polynomials $P_1,P_2, P_3$ corresponding to some $s\in S\setminus V$. Then with respect to suitable coordinates we have $P_1=z_0^2-z_1z_2$. Set $Q=(z_1^6 + P_1\cdot F)$, where $F$ is an arbitrary polynomial of degree $4$, $P=P_1\cdot P_2 \cdot P_3$, and $R_t= P + t\cdot Q$, $t\in \Bbb C$. Then the zero set of $R_0$ is just $\bigcup_{i=1}^3 \Gamma_i(s)$. However, for $t\not=0$ the intersection of $V(R_t)$ with the rational curve $V(P_1)$ consists only of the point $[0:0:1]\in \Bbb P_2$.} \end{rem} \qed {\it Proof of the Theorem:} By Corollary \ref{c} it is sufficient to show that there doesn't exist a non-constant entire curve $\:f: \Bbb C \rightarrow \Bbb P_2 \setminus \bigcup_{i=1}^3 \Gamma_i(s)$ of order at most 2. Assume there exists such a non-constant entire curve $f$. From Proposition \ref{a} we know that $f$ is not algebraically degenerate. For simplicity of notation we drop the $s$ in the rest of the proof. Furthermore we enumerate the 12 lines which we constructed in Proposition \ref{ls} and Corollary \ref{12l} as follows: $L_1L_2$ and $L_3L_4$ are in the linear system of $\Gamma_1$ and $\Gamma_2$ $L_5L_6$ and $L_7L_8$ are in the linear system of $\Gamma_1$ and $\Gamma_3$ $L_9L_{10}$ and $L_{11}L_{12}$ are in the linear system of $\Gamma_2$ and $\Gamma_3$. Let $\Gamma_i = \{P_i = 0\}$ with a homogeneous polynomial $P_i$ of degree 2. The map $\: \Phi = [P_1:P_2:P_3]: \Bbb P_2 \rightarrow \Bbb P_2$ is a morphism (because $\Gamma_1 \cap \Gamma_2 \cap \Gamma_3 = \emptyset$). Furthermore the map $\: \Phi \circ f : \Bbb C \rightarrow \Bbb P_2$ again is an entire curve and the map $\Phi \circ f$ is again of finite order at most 2, because by Lemma \ref{m} we have \begin{equation} \label{*1} T( \Phi \circ f,r) = 2 \cdot T(f,r) +O(1) \end{equation} Since $f$ misses the divisor $\Gamma_1\Gamma_2\Gamma_3$ the map $\Phi \circ f$ misses the divisors $\{ z_i = 0 \}, \:i=1,2,3$ and hence by Lemma \ref{e} we can write \begin{equation} \label{*2} \Phi \circ f = [g_0:g_1:g_2] \end{equation} with $$ g_i = e^{\alpha_i \xi^2 + \beta_i \xi + \gamma_i};\: \alpha_i,\:\beta_i,\:\gamma_i \in \Bbb C $$ where $ \:g_i = (P_i \circ f) \cdot h$; $h: \Bbb C \rightarrow \Bbb C^*$ are entire functions. We may assume that not all three $\alpha_j$ are equal: Assume $\alpha_1 = \alpha_2 = \alpha_3$, then we can divide out the function $e^{\alpha_1 \xi^2}$ and then compose the resulting functions with $\xi\mapsto \xi^2$, i.e. we may consider the function $ \Phi \circ f (\xi^2)$. This map is again of order at most 2 and we have $\: g_i = e^{\beta_i \xi^2 + \gamma_i}$. If now $\beta_1 = \beta_2 = \beta_3$, the map $ \Phi \circ f$ would be constant, which is impossible, since $f$ is algebraically non degenerate. So we exclude the case $\alpha_1=\alpha_2 = \alpha_3$ without loss of generality. The Ahlfors Lemma \ref{A} allows the computation of some limits of characteristic functions: For $1 \leq i < j \leq 3$: \begin{equation} \label{*3} \lim_{r \rightarrow \infty} \frac{T([P_i \circ f: P_j \circ f],r)}{r^2} = \lim_{r \rightarrow \infty} \frac{T([g_i:g_j],r)}{r^2} = \frac{2|\alpha_i - \alpha_j|}{2 \pi} \end{equation} and \begin{equation} \label{*4} \lim_{r \rightarrow \infty} \frac{T([P_1 \circ f: P_2 \circ f: P_3 \circ f],r)}{ r^2} = \lim_{r \rightarrow \infty} \frac{T([g_1:g_2:g_3])}{r^2} \end{equation} $$ = \frac{|\alpha_1 - \alpha_2| + |\alpha_1 - \alpha_3| + |\alpha_2 - \alpha_3|}{2 \pi} $$ hold. Now we want to relate the counting functions of the 12 lines to the characteristic functions used in equations (\ref{*3}) and (\ref{*4}): We know that $L_1L_2$ is in the linear system of $\Gamma_1$ and $\Gamma_2$, i.e. $\: L_1L_2 = a \Gamma_1 + b \Gamma_2\:$ with $a,b \not= 0$ since $\Gamma_1$ and $\Gamma_2$ are smooth quadrics. We consider the map $$[P_1 \circ f: P_2 \circ f] : \Bbb C \rightarrow \Bbb P_1.$$ Its image is not contained in a hyperplane in $\Bbb P_1$, i.e. a point, since $f$ is algebraically non degenerate. Furthermore the 3 divisors $$\: [z_0 = 0], [z_1 = 0], [az_0 + bz_1 = 0]$$ are in general position in $\Bbb P_1$, i.e. distinct. The Second Main Theorem yields $$T([P_1 \circ f: P_2 \circ f],r) \leq N_{[P_1 \circ f: P_2 \circ f]} ([z_0 = 0],r)$$ $$ + N_{[P_1 \circ f: P_2 \circ f]} ([z_1 = 0],r) + N_{[P_1 \circ f: P_2\circ f]} (az_0 + bz_1 =0],r) + O(\log r) =$$ $$ N_f ([P_1 = 0],r) + N_f([P_2=0],r) + N_f([aP_1 + bP_2 =0],r) + O(\log r)=$$ $$0+0+N_f([L_1L_2=0],r) +O(\log r) = N_f([L_1=0],r) + N_f([L_2=0],r) + O(\log r)$$ where $N_f([P_i=0],r)=0$ because $f$ misses $ \Gamma_i = [P_i=0]$, and where we identify the line $L_i$ with its defining equation, so that $[L_i = 0]$ makes sense. On the other hand we have by the First Main Theorem $$N_{[P_1 \circ f: P_2 \circ f]}([az_0 + bz_1 =0],r) \leq T([P_1 \circ f: P_2 \circ f],r) + O(1)$$ and hence $$T([P_1 \circ f: P_2 \circ f],r) = N_f([L_1=0],r) + N_f([L_2=0],r) + O(\log r) $$ The corresponding equations hold for all other lines as well, i.e. we have: $$T([P_1 \circ f:P_2 \circ f],r) = N_f([L_1=0],r) + N_f([L_2=0],r)+ O(\log r)$$ $$ = N_f([L_3=0],r) + N_f([L_4=0],r)+ O(\log r) $$ \begin{equation} \label{*5} T([P_1 \circ f:P_3 \circ f],r) = N_f([L_5=0],r) + N_f([L_6=0],r)+ O(\log r) \end{equation} $$ = N_f([L_7=0],r) + N_f([L_8=0],r)+ O(\log r)$$ $$T([P_2 \circ f: P_3 \circ f],r)= N_f([L_9=0],r) + N_f([L_{10}=0],r) +O(\log r)$$ $$ =N_f([L_{11}=0],r) + N_f([L_{12}=0],r) +O(\log r). $$ Since $ f: \Bbb C \rightarrow \Bbb P_2$ is not linearly degenerate and the 12 lines $L_1,\ldots ,L_{12}$ are in general position, we can again apply the Second Main Theorem and get \begin{equation} \label{*6} 9 \cdot T(f,r) \leq \sum_{i=1}^{12} N_f([L_i=0],r) + O(\log r). \end{equation} The equations (\ref{*1}), (\ref{*5}) and (\ref{*6}) imply $$ \frac{9}{2} \cdot T(\Phi \circ f,r) = 9 \cdot T(f,r) +O(1) \leq \sum_{i=1}^{12} N_f([L_i=0],r) + O(\log r)$$ $$= 2 \cdot (T([P_1 \circ f:P_2 \circ f],r) + T([P_1 \circ f:P_3 \circ f],r) + T([P_2 \circ f: P_3 \circ f],r) + O(\log r)$$ Hence together we have \begin{equation} \label{*7} 9 \cdot T(\Phi \circ f,r) \leq 4 \cdot ( \sum_{1 \leq i < j \leq 3} T([P_i \circ f: P_j \circ f],r)) + O(\log r). \end{equation} We now divide equation (\ref{*7}) by $r^2$ and take $\lim_{r \rightarrow \infty}$. Using the equations (\ref{*3}) and (\ref{*4}) we obtain: $$ 9 \cdot \frac{|\alpha_1 - \alpha_2| + |\alpha_1 - \alpha_3| + |\alpha_2 - \alpha_3|}{2 \pi} \leq 4 \cdot 2 \cdot \frac{|\alpha_1 - \alpha_2| + |\alpha_1 - \alpha_3|+|\alpha_2 - \alpha_3|}{2 \pi}. $$ This can only hold if $\alpha_1 = \alpha_2 =\alpha_3 $, which is a contradiction. \qed \def\hbox{\rlap{$\sqcap$}$\sqcup$}{\hbox{\rlap{$\sqcap$}$\sqcup$}} \def\qed{\ifmmode\hbox{\rlap{$\sqcap$}$\sqcup$}\else{\unskip\nobreak\hfil \penalty50\hskip1em\null\nobreak\hfil\hbox{\rlap{$\sqcap$}$\sqcup$} \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi} \def\bbbr{{\rm I\!R}} \def\bbbn{{\rm I\!N}} \def{\rm I\!M}{{\rm I\!M}} \def{\rm I\!H}{{\rm I\!H}} \def{\rm I\!K}{{\rm I\!K}} \def{\rm I\!P}{{\rm I\!P}} \def\Bbb C{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} \def{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}{{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\scriptstyle Z\kern-0.3em Z$}} {\hbox{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}} \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise .15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} \def\hbox{ \vrule height 7pt width 4pt depth 0pt} {\hbox{ \vrule height 7pt width 4pt depth 0pt} } \newtheorem{expl}[defi]{Example} \section{Complements of two quadrics and a line} In this section we need the following theorem of M.~Green \cite{GRE3} (in degree $d=2$) which generalizes in a sense the classical Borel lemma. \begin{theo}\label{grebor} a) Let $g_0$, $g_1$, $g_2$ be entire holomorphic functions of finite order, $g_1$ and $g_2$ both nowhere vanishing. Assume that $$ g_0^2+ g_1^2+g_2^2 = 1. \eqno{(A)}$$ Then the set $$\{1,g_0,g_1,g_2\}$$ of holomorphic functions has to be linearly dependent.\\ b) Let $g_0$ and $g_1$ be entire holomorphic functions of finite order, $g_1$ nowhere vanishing. Assume that $$ g_0^2+ g_1^2 = 1. $$ Then $g_0$ and $g_1$ must be constant. \end{theo} We consider the complement of three quadrics. We allow one of these to be also a double line. (The case, where one of the three quadrics degenerates to two distinct lines, i.e. two quadrics and two lines, has already been treated above). Since in this section we work also with double lines we will distinguish between $\Gamma$ and $P$, where $\Gamma=V(P)$ (for simplicity reasons we didn't always do this in the previous sections). Before we state the main result of this section, we observe that also some singular configurations of two quadrics in the projective plane can be treated by means of the generalized Borel lemma. \begin{prop}\label{propQ} Let $\Gamma_j=\{Q_j=0\}\subset {\rm I\!P}_2$, $j=1,2$ be two smooth distinct quadrics, whose intersection consists of exactly one point. Then any holomorphic map $f:\Bbb C \to {\rm I\!P}_2\setminus (\Gamma_1\cup \Gamma_2)$ of finite order has values in a quadric (which may degenerate to a double line) from the linear system spanned by $Q_1, Q_2$. \end{prop} {\it Proof.}\/ Let the common tangent line to $\Gamma_1$ and $\Gamma_2$ through the intersection point be defined by the linear equation $L=0$. One verifies immediately that \begin{equation} L^2=aQ_1+bQ_2, \end{equation} $a,b \not = 0$. Let $q_j$ be entire non-vanishing holomorphic functions, $j=1,2$ such that $q_j^2 = Q_j \circ f$, and $q_0=L\circ f$. Then Theorem~8.1 b) implies that $Q_1 \circ f = c \cdot Q_2 \circ f$. \qed Another case is the following. \begin{prop} Let $\Gamma_j=\{Q_j=0\}\subset {\rm I\!P}_2$, $j=1,2$ be two smooth distinct quadrics, which intersect exactly at two points tangentially. Then any holomorphic map $f:\Bbb C \to {\rm I\!P}_2\setminus (\Gamma_1\cup \Gamma_2)$ of finite order has values in a quadric contained in the linear system spanned by $Q_1, Q_2$. \end{prop} {\it Proof.}\/ The linear system spanned by $Q_1$ and $Q_2$ contains $L^2$, where $L$ is the line through the two points of intersection. Using this the statement follows as above. \qed \begin{theo}\label{borhyp} Let $0\not=Q_j \in \Bbb C[z_0,z_1,z_2]$, $j=1,2,3$ be quadratic polynomials, where either all $Q_j$ are irreducible or all but one which may be a square of a linear function. Let $\Gamma_j\subset {\rm I\!P}_2$ be the zero-sets. Assume \begin{description} \item[(1)] no more than two of these intersect at one point, \item[(2)] no tangent to a smooth quadric $\Gamma_j$ at a point of intersection with some other $\Gamma_k$ contains a further intersection point of the curves $\Gamma_l$, \item[(3)] there exists a linear combination of the $Q_j$ which is a square: \begin{equation}\label{Sum} \sum_{j=1}^3 a_j Q_j = P^2,\quad P\in \Bbb C[z_0,z_1,z_2], \end{equation} where at least two coefficients $a_j$ are different from zero. \end{description} Then any holomorphic map $$ f:\Bbb C \to {\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j $$ has values in a quadric (which may be degenerate to a double line). \end{theo} We call a holomorphic map $\Bbb C \to {\rm I\!P}_2$ {\it linearly or quadratically degenerate}\/, if its values are contained in a line or a (possibly degenerate) quadric resp.. \begin{cor}\label{corhyp} \phantom{abc} Let $\Gamma_j=V(Q_j)\subset {\rm I\!P}_2$, $j=2,3$ be smooth quadrics and $\Gamma_1=L_1=V(Q_1)\subset {\rm I\!P}_2$ a line, where $Q_1$ is the square of a linear polynomial, and let the assumptions of \ref{borhyp} be satisfied. \begin{description} \item[(1)] The quasiprojective variety ${\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j$ is Brody-hyperbolic, unless there exists a smooth quadric or a line $\Gamma$ such that after choosing the notation accordingly ($p$, $q$ distinct points): \begin{description} \item[(a)] $\Gamma \cap \Gamma_2=\{p,q\}$, $\Gamma \cap\Gamma _3= \{p\}$, $\Gamma \cap L_1=\{q\}$ \item[(b)] $\Gamma \cap \Gamma_2=\{p\}$, $\Gamma \cap\Gamma _3= \{p\}$, $\Gamma \cap L_1=\{q\}$ \item[(c)] $\Gamma \cap \Gamma_2=\{p\}$, $\Gamma \cap\Gamma _3= \{q\}$, $\Gamma \cap L_1=\{p\}$ \item[(d)] $\Gamma \cap \Gamma_2=\{p\}$, $\Gamma \cap\Gamma _3= \{q\}$, $\Gamma \cap L_1=\{p,q\}$ \end{description} \item[(2)] The quasiprojective variety ${\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j$ is complete hyperbolic and hyperbolically embedded, unless \begin{description} \item[(e)] at least two of the $\Gamma_j$ are tangent to each other at some point, \item[(f)] there exists a smooth quadric, which has only one point of intersection with each of $\Gamma_2$ and $\Gamma_3$ with both of these points contained in $\Gamma_1$, \item[(g)] There exists a tangent to one of $\Gamma_2$ and $\Gamma_3$ at a point of intersection with $\Gamma_1$ which is tangent to the other smooth quadric. \end{description} \end{description} \end{cor} \qed We introduce the following polynomials which will take care of a necessary elimination process in the proof of \ref{borhyp}. \begin{defi}\label{defR} Let the homogeneous polynomial $R_j(y_0,\ldots,y_j)\in \Bbb C[y_0,\ldots,y_j]$ of degree $2^{j-1}$ be defined by the equation $$R_j(x_0^2,\ldots,x_j^2)= \prod_{(\epsilon_1,\ldots,\epsilon_j)\in \{1,-1\}^j}(x_0+ \epsilon_1 x_1 + \ldots + \epsilon_j x_j).$$ \end{defi} For later applications we need some properties of the $R_j$: \begin{lem}\label{lemR} a) $\; R_2 (x,y,z)= x^2 +y^2 +z^2 -2xy -2xz -2yz$\\ b) Let $a,b,c \in \Bbb C$. Then $$ S(x,y,z):=R_3(ax+by+cz,x,y,z) $$ has the following properties: \begin{description} \item[1)] The coefficient of $x^4$ equals $(a-1)^4$. \item[2)] The coefficient of $x^2y^2$ equals $2(3a^2(b-1)^2-2a(b-1)(3b+1)+3b^2+2b+3)$. In particular, if the coefficient of $y^4$ vanishes, the coefficient of $x^2y^2$ equals $16$. \item[3)] Assume that all coefficients of forth powers in $S$ vanish. Then $$ S(x,y,z)= 16(x^2y^2+x^2z^2+y^2z^2)-32(x^2yz+xy^2z+xyz^2). $$ \end{description} \end{lem} We omit the computational proof. \qed {\it Proof of Theorem \ref{borhyp}.} Since the map $f$ has no values in the given quadrics $\Gamma_j$, there exist entire holomorphic functions $q_j$, $j=1,2,3$ such that $q_j^2= Q_j\circ f$. If we put then $q_0=P\circ f$ and $g_j=q_j/q_3$ for $j=0,1,2 $. We apply the generalized Borel lemma (Theorem 8.1): If one of the $a_j$ vanishes, we get immediately quadratic degeneracy from part b) of this theorem. So from now on we assume that all $a_j$ are non-zero. Thereom~8.1~a) implies that the set of functions $\{1,g_0,g_1,g_2\}$ is linearly dependent, i.e. $\{q_0,\ldots,q_3\}$ has this property. Let \begin{equation}\label{Sum1} \sum_{j=0}^3 \alpha_j q_j =0, \quad {\hbox{not all }} \alpha_j=0, \end{equation} and let $R=R_3$ be the polynomial of \ref{defR}. It has been chosen in a way such that $R(\alpha_0^2q_0^2,\ldots,\alpha_3^2q_3^2)=0$. The assumption (\ref{Sum}) means that $q_0^2=a_1q_1^2+a_2q_2^2+a_3q_3^2$. Now the curve defined by the equation \begin{equation}\label{RGl} \tilde R(z_0,z_1,z_2)= R(\alpha_0^2(a_1Q_1+a_2Q_2+a_3Q_3),\alpha_1^2Q_1,\alpha_2^2Q_2, \alpha_3^2Q_3)=0 \end{equation} contains the image of $f$ and is of degree at most eight. We have to show that $\tilde R$ is not identically zero. Otherwise, since $(Q_1,Q_2,Q_3)$ defines a morphism, i.e. an epimorphism $Q:{\rm I\!P}_2\to {\rm I\!P}_2$, the polynomial $R(\alpha_0^2(a_1 y_1^2 + a_2 y_2^2 + a_3 y_3^2), \alpha_1^2 y_1^2, \alpha_2^2 y_2^2,\alpha_3^2 y_3^2) \in \Bbb C[y_1,y_2,y_3]$ would be the zero polynomial. The definition of $R$ would imply that $\alpha_0^2(a_1 y_1^2 + a_2 y_2^2 + a_3 y_3^2) = (\sum_1^3 \delta_j \alpha_j y_j)^2$ for certain $\delta_j=\pm1$. Thus at least two of $\alpha_1, \alpha_2, \alpha_3$ must vanish. However, by assumption, the $a_j$ are different from zero. From this fact it follows immediately that all $\alpha_j=0$, which is a contradiction. We have shown that $f(\Bbb C)$ is contained in an algebraic curve of degree at most eight which is defined by a polynomial of degree four in $Q_1,Q_2,Q_3$. \qed Before we proceed with the proof of Theorem~\ref{borhyp}, we give an application of the classical Borel Lemma. Let ${\rm I\!P}_2=\{z_0,z_1,z_2\}$, and $H_j=\{z_j=0\}$ be the coordinate hyperplanes. \begin{rem}\label{Rem1} Let $f:\Bbb C \to {\rm I\!P}_2\setminus (H_0\cup H_1\cup H_2)$ be a holomorphic map. Assume that $f$ is algebraically degenerate, i.e. its values are contained in an algebraic curve $C$. Then $f(\Bbb C)\subset C'$, where $C'$ is the zero-set of a polynomial of the form $z_0^k-\beta z_1^lz_2^m$, $\beta\not =0$, $k,l+m \leq deg(C)$ (after a suitable reordering of indices). \end{rem} {\it Proof.}\/ Let $C$ be the zero-set of some polynomial $P(z_0,z_1,z_2)$. Denote by $f_j$ the components of $f$. The classical lemma of Borel, applied to the monomials in the expansion of $P$, implies that there exist at least two such monomials, which are proportional after composing with $f$. Thus $f_0^r f_1^s f_2^t = \beta f_0^u f_1^v f_2^w$ for some $\beta\neq 0$. \qed (The statement of the Lemma has an obvious generalization to ${\rm I\!P}_n$.) An immediate consequence of Theorem \ref{borhyp}, as far as we have proved is yet, and of Remark \ref{Rem1} is, since all $Q_j\circ f$ have no zeroes, \begin{lem}\label{lem1} Given the assumptions of {\rm\ref{borhyp}}, the image $f(\Bbb C)$ is contained in a curve of the form $$Q_u^k-\alpha Q_v^l Q_w^m = 0, \quad \alpha\not = 0,$$ where $\{u,v,w\} =\{1,2,3\}$, and $k, l+m \leq 4$. \end{lem} \qed We note the following fact: \begin{lem}\label{lem2} Let $f:\Bbb C \to {\rm I\!P}_2$ be as above. \begin{description} \item[(1)] Let $f(\Bbb C)$ be contained in the zero-set \begin{equation}\label{equiv} V(Q_1^{k_1}Q_2^{k_2}Q_3^{k_3}- \alpha Q_1^{l_1}Q_2^{l_2}Q_3^{l_3}), \end{equation} with $\sum k_j = \sum l_j = 4$, and $k_j=l_j$ for at least one $j$, or \item[(2)] let $f(\Bbb C)$ be contained in both zero-sets \begin{equation}\label{Gl1} V(Q_1^{k_1}Q_2^{k_2}Q_3^{k_3}- \alpha Q_1^{l_1}Q_2^{l_2}Q_3^{l_3}) \end{equation} and \begin{equation}\label{Gl2} V(Q_1^{m_1}Q_2^{m_2}Q_3^{m_3}- \beta Q_1^{n_1}Q_2^{n_2}Q_3^{n_3}) \end{equation} with $\sum k_\nu =\sum l_\nu=\sum m_\nu=\sum n_\nu=4$ such that the vector $$ (k_1-l_1,k_2-l_2,k_3-l_3) $$ is not a rational multiple of the vector $$ (m_1-n_1,m_2-n_2,m_3-n_3). $$ \end{description} Then $f(\Bbb C)$ is contained in a quadric curve, which is a member of the linear system generated by two of the quadrics $Q_j$. \end{lem} We call the monomials $Q_1^{k_1}Q_2^{k_2}Q_3^{k_3}$ and $Q_1^{l_1}Q_2^{l_2}Q_3^{l_3}$ satisfying (\ref{equiv}) {\it equivalent with respect to $f$}. {\it Proof.}\/ We can eliminate one of the $Q_j$, say $Q_1$, and obtain that $f(\Bbb C)$ is contained in $V(Q_2^r - \gamma Q_3^r)$ for some integer $r$, and $\gamma \in \Bbb C$, because the $Q_j\circ f$ have no zeroes. Taking roots we find that $f(\Bbb C)\subset V(\sigma Q_2 - \tau Q_3)$, $\sigma,\tau\in \Bbb C$ not both equal to zero. \qed We return to the proof of Theorem~\ref{borhyp}, and consider under which conditions the above lemma can be applied. Let the situation of Theorem~\ref{borhyp} be given. Denote by $$ T(Q_1,Q_2,Q_3)= R(\alpha_0^2(a_1Q_1+a_2Q_2+a_3Q_3),\alpha_1^2Q_1,\alpha_2^2Q_2, \alpha_3^2Q_3) $$ the polynomial of (\ref{RGl}). We know that $T$ is not the zero-polynomial. We already reduced the proof of Theorem~\ref{borhyp} to the case, where all $a_j$ in (\ref{Sum}) are different from zero. {\it First part:}\/ Let all $\alpha_j \neq 0$. Thus we can (after normalizing these constants to $1$) apply Lemma~\ref{lem2}. {\it First case:}\/ We claim that the conditions of Lemma \ref{lem2} (2) are satisfied, if at least two of the coefficients of $Q_j^4$ in $T$, say those of $Q_1^4$ and $Q_2^4$, are different from zero. If $Q_1^4$ and $Q_2^4$ are equivalent, $Q_1\circ f$ is a constant multiple of $Q_2\circ f$, and $f$ is quadratically degenerate. Otherwise there exist exponents $(r_1,r_2,r_3)$, $(s_1,s_2,s_3)$ of $Q_j$ that match $(4,0,0)$ and $(0,4,0)$ resp. in the sense of (\ref{Gl1}) and (\ref{Gl2}) resp. because of the classical Borel Lemma. Assume that the assumptions of Lemma~\ref{lem2}~(2) are not fulfilled, so there exists a rational number $c$ such that $$ (4,0,0)-(r_1,r_2,r_3)= c((0,4,0)-(s_1,s_2,s_3)). $$ Since $(r_1,r_2,r_3)\not=(4,0,0)$ we have $r_1<4$, thus $s_1>0$ and $c<0$. Now $0 \geq -r_3=c(-s_3)\geq 0$ implies $r_3=s_3=0$, so we can apply Lemma \ref{lem2} (1). {\it Second case:}\/ The next case to consider is, where exactly one forth power occurs, say $Q_1^4$. According to Lemma~\ref{lemR}~2), the coefficient of $Q_1^2Q_2^2$ in $T$ must be different from zero. Assume first that $Q_1^4$ is equivalent to $Q_1^2Q_2^2$ with respect to $f$. Then Lemma~\ref{lem2}~(1) is applicable. If these monomials are not equivalent, we have some (non trivial) relations $Q_1^4\sim Q_1^{r_1}Q_2^{r_2}Q_3^{r_3}$ and $Q_1^2Q_2^2\sim Q_1^{s_1}Q_2^{s_2}Q_3^{s_3}$. If the assumptions of Lemma~\ref{lem2}~(2) would not hold, we had $$ (4,0,0)-(r_1,r_2,r_3)= c((2,2,0)-(s_1,s_2,s_3)) $$ for some $0\neq c \in \bbbq$ and $0\leq r_j,s_j \leq 4$, $\sum r_j=\sum s_j=4$. For $r_3=0$ Lemma~\ref{lem2}~(1) could be applied. Only $0<r_3\leq 4$ is left; in particular $-r_3=c(-s_3)$ implies $c>0$, $s_3 > 0$. Now $r_1\neq 4$. Thus $4-r_1=c(2-s_1)$ gives $s_1=0$ or $s_1=1$. Furthermore $-r_2= c(2-s_2)$ holds. Again $r_2=0$ makes \ref{lem2}~(1) applicable so that we are left with $s_2=3$ or $s_2=4$. Thus $(s_1,s_2,s_3)=(0,3,1)$. Hence $f$ has values in the quartic curve $Q_1^2-\gamma Q_2 Q_3=0$ for some $\gamma\in \Bbb C $. Let $C$ be the curve $V(Q_1^2- \gamma Q_2 Q_3)$. We note first that $$ C\cap (\Gamma_1\cup\Gamma_2\cup\Gamma_3)= \Gamma_1 \cap (\Gamma_2\cup\Gamma_3) $$ The case, where two smooth quadrics $\Gamma_j$ intersect in exactly one point, yields immediately quadratic degeneracy by Proposition~\ref{propQ}, and we are done. If one of the $\Gamma_j$ is a line, it cannot be tangent to both of the further given smooth quadrics --- this is also excluded by assumption (2). Thus $C\cap (\Gamma_1\cup\Gamma_2\cup\Gamma_3)$ consists of at least three points. As $f(\Bbb C)$ is contained in $C\setminus(\Gamma_1\cup\Gamma_2\cup\Gamma_3)$, the curve $C$ cannot be irreducible unless $f$ is constant. We are left with the case where $C$ decomposes into a line $l$ and a cubic. We have $C\cap\Gamma_1=\Gamma_1\cap(\Gamma_2\cup\Gamma_3)= C\cap(\Gamma_1\cap(\Gamma_2\cup\Gamma_3))$, which implies $l\cap \Gamma_1=(l\cap \Gamma_2)\cup(l\cap \Gamma_3)$. This equality means that $l\neq \Gamma_1$ and that $l \cap \Gamma_1$ consists of two distinct points $p'$ and $p''$, (since no more than two of the $\Gamma_j$ pass through a point). Let $l\cap\Gamma_2=\{p'\}$ and $l\cap\Gamma_3=\{p''\}$. This means that $l$ is at least tangent to one of the smooth quadrics and passes through one further intersection point of the $\Gamma_j$. This was excluded by assumption (2). {\it Third case:}\/ Assume finally that all coefficients of $Q_j^4$ in $T$ vanish. According to Lemma~\ref{lemR}~3) the non-zero monomials in $T$ are $Q_j^2Q_k^2$, $j\neq k$ and $Q_j^2Q_kQ_l$, where $(j,k,l)$ run through all cyclic permutations of $(1,2,3)$. We pick $Q_1^2Q_2^2$ and check to which of the monomials it can be equivalent with respect to $f$. Lemma~\ref{lem2}~(1) is directly applicable to all possible cases but $Q_1^2Q_2^2 \sim Q_1Q_2Q_3^2$ which implies $Q_1Q_2 \sim Q_3^2$. This case was already treated. The claim is now shown under the assumption that all $\alpha_j$ are different from zero. If two or more of the $\alpha_j$ vanish, the claim is already clear from (\ref{Sum1}): We then get the equation $\:\alpha_j q_j = -\alpha_k q_k\,$, which, after squaring both sides, yields us quadratic degeneracy immediately, or if $q_0$ is involved, by using that at least two of the $a_i$ are not zero. The remaining case is, where exactly one $\alpha_j=0$. Here we use $R_2$ from \ref{defR} and arrive at a polynomial $U(y_1,y_2,y_3)$ of degree two, such that $f(\Bbb C)$ is contained in the zero-set of $U(Q_1,Q_2,Q_3)$. Again Borel's lemma is applied to its monomials. A non-empty subset of $\{Q_1^2,Q_2^2,Q_3^2,Q_1Q_2,Q_1Q_3,Q_2Q_3\}$ has to be divided into sets of $f$-equivalent polynomials. In the view of \ref{lem2}~(1) the only case to remain is $Q_j^2\sim Q_kQ_l$ where $(j,k,l)$ is a cyclic permutation of $(1,2,3)$. This case was treated above. \qed In the sequel we treat the case of the complement of two plane quadrics and a line and the case of three Fermat quadrics. We show that \begin{theo}\label{thm221} There exist \begin{enumerate} \item[(a)] a quasiprojective set $V\subset {\cal C}(2,2,1)$ of codimension one and \item[(b)] an open, non-empty subset $U\subset {\cal C}(2,2,1)$ containing $V$ \end{enumerate} such that for all $s\in U$ the space ${\rm I\!P}_2\setminus \Gamma(s)$ is complete hyperbolic and hyperbolically embedded. \end{theo} {\it Proof.} The set $V$ will be constructed in a such a way that the configurations $\Gamma(s)$ for $s\in V$ satisfy the conditions of Proposition~3.2 so that $(b)$ will follow from the first statement. Let ${\rm I\!P}_2=\{[z_0:z_1:z_2]\}$ and \begin{eqnarray} l &=& c_0 z_0 + c_1 z_1 + c_2 z_2 \label{B0}\\ Q_0&=&l^2 \label{B1}\\ Q_j&=&\sum_{k=0}^2 a_{jk}z_k^2 + b_{j0} z_0z_1 + b_{j1} z_0z_2+ b_{j2} z_1z_2\label{B2} \end{eqnarray} for $j=1,2$. We include $Q_0$ in this notation and compute $a_{0k}, b_{0k}$ in terms of $c_l$. We shall discuss, when (\ref{Sum}) holds for these. Let $A=(a_{jk})$ and $B=(b_{jk})$. Let $\hat{A}$ be the adjoint matrix of $A$, i.e. $\hat{A} \cdot A = \hbox{det}(A) E$. For $\kappa^2,\lambda^2,\mu^2 \in \Bbb C$ we consider the following linear combination \begin{equation}\label{B3} (\kappa^2,\lambda^2,\mu^2)\cdot \left( \hbox{det}(A) \left( \begin{array}{c} z_0^2\\ z_1^2\\ z_2^2 \end{array} \right) + \hat{A} B \cdot \left( \begin{array}{c} z_0z_1\\ z_0z_2\\ z_1z_2 \end{array} \right) \right) = (\kappa^2,\lambda^2,\mu^2)\cdot \hat{A}\cdot \left( \begin{array}{c} Q_0\\ Q_1\\ Q_2 \end{array} \right) \end{equation} Looking at the left hand side one verifies that this expression is a square of a linear polynomial, if and only if the following equation holds: \begin{equation}\label{B4} (\kappa^2,\lambda^2,\mu^2)\cdot \hat{A} B = 2\hbox{det}(A) (\kappa \lambda, \kappa \mu, \lambda\mu) \end{equation} We set $a=(a_{jk})_{j>0} \in \Bbb C^6$, $b=(b_{jk})_{j>0} \in \Bbb C^6$, and $c=(c_l)\in \Bbb C^3$. So $A$, $B$ and $\hat{A}$ are now given in terms of $a,b,c$. We define $M\subset \Bbb P_2 \times \Bbb C^3 \times \Bbb C^6 \times \Bbb C^6$ to be the set of all points $([\kappa:\lambda:\mu],c,a,b)$ for which (\ref{B4}) holds. For all $m\in M$ the inequality ${\rm dim}_m M \geq 14$ holds, since (\ref{B4}) consists of three equations in $\kappa, \lambda, \mu, A,\hat{A},B$ and hence in $\kappa, \lambda, \mu, a, b, c$. Consider the canonical projection ${\rm pr}: {\rm I\!P}_2 \times \Bbb C^3 \times \Bbb C^6 \times \Bbb C^6 \to \Bbb C^3 \times \Bbb C^6 \times \Bbb C^6$. Let $c_0=(1,0,0)$, $a_0= \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right)$, and $b\in \Bbb C^6$ arbitrary. Then we calculate that $ ({\rm I\!P}_2\times \{(c_0,b,a_0)\})\cap M $ is zero-dimensional. In Example \ref{expl} we shall give an explicit example of a point $m_0= ([\kappa_0:\lambda_0:\mu_0],c_0,b_0,a_0)$ which is contained in such a zero dimensional set, where $\kappa_0,\lambda_0, \mu_0 \neq 0$. Denote by $M_0\subset M$ an irreducible component of $M$ containing $m_0$. Now ${\rm pr}(M_0)\subset \Bbb C^3 \times \Bbb C^6 \times \Bbb C^6$ is algebraic and at least of dimension 14, because the fiber is zero dimensional. (One can check easily that ${\rm pr}(M_0) \neq \Bbb C^3 \times\Bbb C^6 \times\Bbb C^6$). Let $$ (\kappa^2,\lambda^2,\mu^2)\cdot \hat{A}=(\phi,\psi,\chi), $$ and $N=V(\phi \cdot \psi \cdot \chi \cdot {\rm det}(A))\subset M$. Observe $M_0\setminus N \neq \emptyset$, since $m_0 \not\in N$ (what can be checked easily). Let $V'\subset \Bbb C^3\times\Bbb C^6\times\Bbb C^6$ be the quasi projective hypersurface $V'={\rm pr}(M_0)\setminus {\rm pr}(N) \subset {\rm pr}(M_0\setminus N)$, which is not empty: The fiber of ${\rm pr}|M_0$ at $m_0$ is of dimension zero, hence ${\rm dim (pr} (N)) \leq {\rm dim} N < {\rm dim} M_0 = {\rm dim (pr}(M_0))$. By means of the assignment $\Bbb C^3\times\Bbb C^6\times\Bbb C^6 \ni (c,b,a)\mapsto (l,Q_1,Q_2) \in \Bbb C^3\times\Bbb C^6\times\Bbb C^6 $ we associate to any point of ${\rm pr}(M_0)$ a triple consisting of one linear and two quadratical polynomials. Now ${\rm pr}(M_0)$ as well as ${\rm pr}(N)$ are invariant under the canonical action of $(\Bbb C^*)^3$, given by multiplication of $l$, $Q_1$, $Q_2$ by elements of $\Bbb C^*$. This follows from the original definition of $M$ and $N$ (the existence of a linear combination of the $Q_0, Q_1, Q_2$ to a square and the number of coefficients which are zero is independent of the $\Bbb C^*$ action on $l$, $Q_1$, $Q_2$) and the fact that under this action $(\Bbb C^*)^3\times M_0$ has values in some irreducible component of $M$, which has to be $M_0$. Now ${\rm pr}(M) \setminus {\rm pr}(N)$ defines a quasi projective subvariety $V' \subset {\cal C}(1,2,2)$ of codimension one. Our aim is to construct a quasi projective variety $V \subset {\cal C}(1,2,2)$ of codimension one, which is contained in $V'$ satisfying the further conditions of Corollary~\ref{corhyp}(2), and hence proving the Theorem. We already chose $M$ and $N$ in a way that $V'$ satisfies condition (3) of \ref{borhyp}. All of the configurations which had to be excluded because of the further conditions in \ref{borhyp} and \ref{corhyp} define a proper algebraic subset $W\subset {\cal C}(1,2,2)$. All we need is to see that $V:= V'\setminus W$ is not empty. But we have ${\rm pr}(m_0) \in V' \setminus W$ for our point $m_0$ coming from the example below. \begin{expl}\label{expl} The following set of quadratic polynomials defines an element of $V$. In particular the complement of its zero-sets in ${\rm I\!P}_2$ is complete hyperbolic and hyperbolically embedded. \begin{eqnarray} Q_0&=& z_0^2\\ Q_1&=&z_1^2+ z_0z_1 + z_0z_2 + (1/25) z_1z_2 \\ Q_2&=&z_2^2 + 50 z_0z_1 - 10 z_0z_2 + 9z_1z_2 \end{eqnarray} \end{expl} One checks immediately that $225 Q_0 + 100 Q_1 + 4 Q_2$ is a square. Set $\Gamma_j=V(Q_j)$. Furthermore: 1) No more than two $\Gamma_j$ intersect in one point. 2) None of the $\Gamma_j$ are tangent to any other $\Gamma_k$. 3) No tangent to one of $\Gamma_2$ and $\Gamma_3$ at a point of intersection with any $\Gamma_j$ contains a further point of intersection of the $\Gamma_j$. 4) No tangent to one of $\Gamma_2$ and $\Gamma_3$ at a point of intersection with $\Gamma_1$ is tangent to the other smooth quadric. 5) There exists no smooth quadric $\Gamma$ with $\, \Gamma_2 \cap \Gamma = \{p'\}$, $\Gamma_3 \cap \Gamma = \{p''\}$ and $\{p',p''\} \subset \Gamma_1$.\\ How to check 1) to 4) is obvious. If $T'$ resp. $T''$ are the linear polynomials which give the tangents at $\Gamma_2$ in $p'$ resp. at $\Gamma_3$ in $p''$ we have $\:Q = a Q_1 + b(T')^2\,$, $\: Q = c Q_3 + d (T'')^2\,$, where $\Gamma = V(Q)$, $\Gamma_i = V(Q_i)$. Now solve for $a,b,c,d$, and show that only the trivial solution exists. \qed For intersections of three smooth quadrics Theorem~\ref{borhyp} is not quite superseeded by the more general statement of Theorem~7.1. as the application to intersections to Fermat quadrics shows. We first note a further corollary to Theorem~\ref{borhyp}. \begin{cor} Let $\Gamma_j=V(Q_j)\subset {\rm I\!P}_2$, $j=1,2,3$ be smooth quadrics, and let the assumptions of \ref{borhyp} be satisfied. \begin{description} \item[(1)] The quasiprojective variety ${\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j$ is Brody-hyperbolic, unless there exists a smooth quadric or a line $\Gamma$ such that after choosing the notation accordingly ($p$, $q$ distinct points): \begin{description} \item[(a)] $\Gamma \cap \Gamma_1=\{p,q\}$, $\Gamma \cap\Gamma _2= \{p\}$, $\Gamma \cap \Gamma_3=\{q\}$ \item[(b)] $\Gamma \cap \Gamma_1=\{p\}$, $\Gamma \cap\Gamma _2= \{p\}$, $\Gamma \cap \Gamma_3=\{q\}$ \end{description} \item[(2)] The above conditions (a) and (b) can be replaced by the following (somewhat stronger) condition: \begin{description} \item[(c)] all of the $\Gamma_j$ intersect transversally. \end{description} In this case ${\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j$ is complete hyperbolic and hyperbolically embedded. \end{description} \end{cor} \qed We apply the Corollary to the following \begin{prop} Let $$ Q_j=a_j x^2 + b_j y^2 + c_j z^2 ; \quad j=1,2,3 $$ be linearly independent polynomials, whose zero-sets $\Gamma_j$ are smooth. Assume \begin{description} \item[(1)] no more than two of the $\Gamma_j$ intersect at one point, \item[(2)] no tangent to a quadric $\Gamma_j$ at a point of intersection with some other $\Gamma_k$ contains a further intersection point of the curves $\Gamma_l$, \item[(3)] none of the $\Gamma_j$ are tangent to each other at any point. \end{description} Then ${\rm I\!P}_2\setminus \bigcup_{j=1}^3 \Gamma_j$ is complete hyperbolic and hyperbolically embedded. \end{prop} \qed

9312

alg-geom/9312011

en

https://arxiv.org/abs/alg-geom/9312011

alg-geom/9312011

Charles Walter

Charles H. Walter

Components of the Stack of Torsion-Free Sheaves of Rank 2 on Ruled Surfaces

16 pages, LATeX 2.09

Math. Ann. 301 (1995), 699-715

Let S be a ruled surface without sections of negative self-intersection. We classify the irreducible components of the moduli stack of torsion-free sheaves of rank 2 sheaves on S. We also classify the irreducible components of the Brill-Noether loci in Hilb^N(P1xP1) given by W_N^0(D)={[X] | h^1(I_X(D)) >= 1 } for D an effective divisor class. Our methods are also applicable to P2 giving new proofs of theorems of Stromme (slightly extended) and Coppo.

alg-geom

\section{\@startsection{section}{1}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf}} \def\subsection{\@startsection {subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subsubsection{\@startsection {subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\paragraph{\@startsection {paragraph}{3}{\z@}{2ex plus 0.6ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subparagraph{\@startsection {subparagraph}{3}{\parindent}{2ex plus 0.6ex minus .2ex}{-1pt}{\normalsize\sl}} \def\RIfM@{\relax\ifmmode} \def0{0} \newif\iffirstchoice@ \firstchoice@true \def\the\scriptscriptfont\@ne{\the\textfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi{\RIfM@\expandafter\RIfM@\expandafter\text@\else\expandafter\text@@\fi@\else\expandafter\RIfM@\expandafter\text@\else\expandafter\text@@\fi@@\fi} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi@@#1{\leavevmode\hbox{#1}} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi@#1{\mathchoice {\hbox{\everymath{\displaystyle}\def\the\scriptscriptfont\@ne{\the\textfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@}\textdef@@ T#1}} {\hbox{\firstchoice@false \everymath{\textstyle}\def\the\scriptscriptfont\@ne{\the\textfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@}\textdef@@ T#1}} {\hbox{\firstchoice@false \everymath{\scriptstyle}\def\the\scriptscriptfont\@ne{\the\scriptfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\scriptfont\tw@}\textdef@@ S\rm#1}} {\hbox{\firstchoice@false \everymath{\scriptscriptstyle}\def\the\scriptscriptfont\@ne {\the\scriptscriptfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}} \def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl\textdef@#1\it} \def\DN@{\def\next@} \def\eat@#1{} \def\textdef@#1#2{% \DN@{\csname\expandafter\eat@\string#2fam\endcsname}% \if S#1\edef#2{\the\scriptfont\next@\relax}% \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}% \else\edef#2{\the\textfont\next@\relax}\fi\fi} \catcode`\@=12 \begin{document} \maketitle \begin{abstract} \noindent Let $S$ be a ruled surface without sections of negative self-intersection. We classify the irreducible components of the moduli stack of torsion-free sheaves of rank $2$ sheaves on $S$. We also classify the irreducible components of the Brill-Noether loci in ${\rm Hilb}^N({\bf P}% ^1\times {\bf P}^1)$ given by $W_N^0(D)=\{[X]\mid h^1({\cal I}_X(D))\geq 1\}$ for $D$ an effective divisor class. Our methods are also applicable to ${\bf % P}^2$ giving new proofs of theorems of Str\o mme (slightly extended) and Coppo. \bigskip\ \end{abstract} Let $\pi {:}~S={\bf P}({\cal A})\rightarrow C$ be a ruled surface with tautological line bundle ${\cal O}(1):={\cal O}_{{\bf P}({\cal A})}(1)$. The current classification of isomorphism classes of rank $2$ vector bundles $% {\cal E}$ on $S$ (\cite{BS} \cite{B} \cite{HS} \cite{Ho}) proceeds by stratifying the moduli functor (or stack) and then classifying the sheaves in each stratum independently. The numerical data used to distinguish the strata are usually (i) the splitting type ${\cal O}_{{\bf P}^1}(a)\oplus {\cal O}_{{\bf P}^1}(b)$ of the generic fiber of $\pi $ (with $a\geq b$), and (ii) the degree of the locally free sheaf $\pi _{*}({\cal E}(-a))$ on $C$% . On each stratum, ${\cal U}:=\pi ^{*}(\pi _{*}({\cal E}(-a)))(a)$ is naturally a subsheaf of ${\cal E}$, and the possible quotient sheaves ${\cal % E}/{\cal U}$ and extension classes ${\rm Ext}^1({\cal E}/{\cal U},{\cal U})$ have been classified. To the author's knowledge, rank $2$ torsion-free sheaves on $S$ have not been given a similar classification, but one could clearly adapt the ideas used for vector bundles. What this approach has usually not described is the relationship between the strata particularly for the strata parametrizing only unstable sheaves. In this paper we give a first result along these lines by describing which strata are generic, i.e.\ which are open in the (reduced) moduli stack. Thus we are really classifying the irreducible components of the moduli stack of rank $2$ torsion-free sheaves on $S$. We use a method developed by Str\o mme \cite{St} for rank $2$ vector bundles on ${\bf P}^2$ modified by deformation theory techniques which originate in \cite{DLP}. We will divide our irreducible components into two types. The first type we call prioritary because the general member of a component of this type is a prioritary sheaf in the sense that we used in \cite{W}. That is, if for each $p\in C$ we write $f_p:=\pi ^{-1}(p)$ for the corresponding fiber, then a coherent sheaf ${\cal E}$ on $S$ is {\em prioritary} if it is torsion-free and satisfies ${\rm Ext}^2({\cal E},{\cal E}(-f_p))$ for all $p$. We showed in \cite{W} Lemma 7, that if one polarizes $S$ by an ample divisor $H$ such that $H\cdot (K_S+f_p)<0$, then $H$-semistable sheaves are prioritary. Thus the prioritary components should be viewed as playing a role one might otherwise assign to semistable components. But the condition of priority is simpler to use than semistability because it does not depend on the choice of a polarization, and moreover the existence problem has a simpler solution (particularly in higher rank). The second type of components are nonprioritary ones. Our main result is the following. We use the convention that if $D\in {\rm NS% }(S)$, then ${\cal O}_S(D)$ is the line bundle corresponding to the generic point of the corresponding component ${\rm Pic}^D(S)$ of the Picard scheme. This is well-defined on all surfaces for which numerical and algebraic equivalence coincide, including all of ours. \begin{theorem} \label{ruled}Let $S$ be a ruled surface without curves of negative self-intersection, and let $f$ be the numerical class of a fiber of $S$. Let $c_1\in {\rm NS}(S)$ and $c_2\in {\bf Z}$. The irreducible components of the stack ${\rm TF}_S(2,c_1,c_2)$ of torsion-free sheaves on $S$ of rank $2$ and Chern classes $c_1$ and $c_2$ are the following: (i)\quad A unique prioritary component if $c_1f$ is even and $c_2\geq \frac 14c_1^2$, or if $c_1f$ is odd and $c_2\in {\bf Z}$. This component is generically smooth of dimension $-\chi ({\cal E},{\cal E})$, and the general sheaf in the component is locally free. (ii)\quad For every pair $(D,n)\in {\rm NS}(S)\times {\bf Z}$ such that $% Df\leq -1+\frac 12c_1f$ and $0\leq n\leq c_2+D(D-c_1)\leq \chi ({\cal O}% _S(-c_1))+D(2D-2c_1-K_S)$ a unique nonprioritary component whose general member is a general extension% $$ 0\rightarrow {\cal I}_{Z_1}(c_1-D)\rightarrow {\cal E}\rightarrow {\cal I}% _{Z_2}(D)\rightarrow 0 $$ where $Z_1$ (resp.\ $Z_2)$ is a general set of $n$ (resp.\ $n^{\prime }:=c_2+D(D-c_1)-n$) points on $S$. These components have dimensions $-\chi (% {\cal E},{\cal E})+\chi ({\cal O}_S(-c_1))+D(D-c_1-K_S)-c_2$ but have generic embedding codimension $n^{\prime }+h^1({\cal O}_S(2D-c_1)$. The general sheaf in the component is locally free except at $Z_1$. \end{theorem} For ${\bf P}^2$ the components of ${\rm TF}_{{\bf P}^2}(2,c_1,c_2)$ containing locally free sheaves were classified by Str\o mme using a similar method (\cite{St} Theorem 3.9). We wish to add to his classification the components of ${\rm TF}_{{\bf P}^2}(2,c_1,c_2)$ whose general member is not locally free. We recall from \cite{HL} that a prioritary sheaf ${\cal E}$ on ${\bf P}^2$ is one that is torsion-free and satisfies ${\rm Ext}^2({\cal E},% {\cal E}(-1))=0$. \begin{theorem} \label{P2}Let $S$ be ${\bf P}^2$ and let $f\in {\rm NS}(S)$ be the class of a line. Let $(c_1,c_2)\in {\bf Z}^2$. Then the irreducible components of $% {\rm TF}_{{\bf P}^2}(2,c_1,c_2)$ have the same classification as in Theorem \ref{ruled} except that the prioritary component exists if and only if $% c_2\geq \frac 14c_1^2-\frac 14$. \end{theorem} The uniqueness of the prioritary components was proven for ruled surfaces (resp.\ for ${\bf P}^2$) in \cite{W} (resp.\ \cite{HL}) although of course there were many earlier results by many authors concerning semistable components on ${\bf P}^2$ and on various ruled surfaces. The classification of the irreducible components of the stacks of torsion-free sheaves has an interesting application to Brill-Noether problems. Let $S$ be a smooth projective algebraic surface, $E$ an effective divisor class on $S$, and $N$ a positive integer such that $N\leq h^0({\cal O% }_S(E))$. For simplicity we will assume that $H^1({\cal O}_S)=H^1({\cal O}% _S(E))=0$. We consider the Brill-Noether loci in ${\rm Hilb}^NS$ defined as% $$ W_N^i(E)=\{[X]\in {\rm Hilb}^NS\mid h^1({\cal I}_X(E))\geq i+1\}. $$ Thus $W_N^i(E)$ parametrizes those $0$-schemes of length $N$ which impose at least $i+1$ redundant conditions on divisors in $|E|$. What we wish to consider is: \paragraph{The Brill-Noether Problem.} Classify the irreducible components of the $W_N^i(E)$ and compute their dimensions.\medskip\ It is known from general principles that each component has codimension at most $(\chi +i+1)(i+1)$ in ${\rm Hilb}^NS$ where $\chi =h^0({\cal O}% _S(E))-N\geq 0$, but there can be many components of various smaller codimensions. The Brill-Noether problem is related to the problem of classifying irreducible components of the stack of torsion-free sheaves on $S$ as follows. By an elementary argument (cf.\ \cite{C} p.\ 732) the general $X$ in any component of $W_N^i(E)$ has $h^1({\cal I}_X(E))=i+1$. One then uses Serre duality $H^1({\cal I}_X(E))^{*}\cong {\rm Ext}^1({\cal I}_X(E-K_S),% {\cal O}_S)$ to get an extension% $$ 0\rightarrow {\cal O}_S^{\oplus i+1}\rightarrow {\cal E\rightarrow I}% _X(E-K_S)\rightarrow 0. $$ So we get a Serre correspondence between $X\in W_N^i(E)-W_N^{i-1}(E)$ and pairs $({\cal E},V)$ where ${\cal E}$ is torsion-free of rank $i+2$ with $% c_1({\cal E})=E-K_S$, $c_2({\cal E})=N$, $H^1({\cal E}(K_S))=H^2({\cal E}% (K_S))=0$, and for which there exists an $(i+1)$-dimensional subspace $% V\subset H^0({\cal E})$ such that the natural map $V\otimes {\cal O}% _S\rightarrow {\cal E}$ is injective with torsion-free quotient. Note that these properties are all open conditions on ${\cal E}$ within the stack of torsion-free sheaves on $S$. So Theorems \ref{ruled} and \ref{P2} yields a classification of the irreducible components of the $W_N^0(E)$ for ${\bf P}% ^1\times {\bf P}^1$ and ${\bf P}^2$. This classification has been previously obtained by Coppo for ${\bf P}^2$ by a different method (\cite{C} Th\'eor\`eme 3.2.1) but seems new for ${\bf P}^1\times {\bf P}^1$. \begin{theorem} \label{BN}Let $S$ be ${\bf P}^2$ (resp.\ ${\bf P}^1\times {\bf P}^1$), let $% E $ be an effective divisor of degree $e$ (resp.\ of bidegree $(e_1,e_2)$), and let $N$ be an integer such that $0<N\leq \chi ({\cal O}_S(E))$. Then the irreducible components of the Brill-Noether locus $W_N^0(E)$ are the following: (i)\quad For every pair $(D,n)\in {\rm NS}(S)\times {\bf Z}$ such that $D$ is an effective and irreducible divisor class of degree $d$ on ${\bf P}^2$ (resp.~of bidegree $(d_1,d_2)$ on ${\bf P}^1\times {\bf P}^1$) such that $% d\leq \frac 12(e+1)$\ (resp.\ $d_2\leq \frac 12e_2$), $D(E-D)\leq \chi (% {\cal O}_S(E))-N$, $n\geq 0$, and $0\leq N-D(E-D-K_S)-n\leq \chi ({\cal O}% _S(D+K_S))$, there exists a unique irreducible component of codimension $% D(E-D)+1$ in ${\rm Hilb}^N(S)$ whose general member is the union of $n$ general points of $S$ and $N-n$ points on a curve in $\left| D\right| $. (ii)\quad If $S$ is ${\bf P}^2$ (resp.\ if $S$ is ${\bf P}^1\times {\bf P}^1$ and $e_2$ is even, resp.\ if $S$ is ${\bf P}^1\times {\bf P}^1$ and $e_2$ is odd), then there exists one additional component of codimension $\chi ({\cal % O}_S(E))$$-N+1$ in ${\rm Hilb}^N(S)$ if $N\geq \frac 14(e+2)(e+4)$ (resp.~$% N\geq \frac 12(e_1+2)(e_2+2)$, resp.\ $N\geq \frac 12(e_1+2)(e_2+1)+1$). If $% S={\bf P}^1\times {\bf P}^1$ and $(e_1,e_2,N)=(e_1,1,e_1+2)$ there is also one additional component of codimension $\chi ({\cal O}_S(E))$$-N+1$. \end{theorem} In part (i) the $N-n$ points on the curve $C\in |D|$ have the property that their union is a divisor on $C$ belonging to a linear system of the form $% \left| \Gamma +E{\mid }_C-K_C\right| $ with $\Gamma $ an effective divisor satisfying $h^0({\cal O}_C(\Gamma ))=1$. The necessary condition $0\leq \deg (\Gamma )\leq g(C)$ is exactly the condition $0\leq N-D(E-D-K_S)-n\leq \chi (% {\cal O}_S(D+K_S))$. The main tool which we use to obtain our results is interesting in its own right. We use the notation $\chi ({\cal F},{\cal G})=\sum (-1)^i\dim {\rm Ext% }^i({\cal F},{\cal G})$. \begin{proposition} \label{unobst}Let $S$ be a projective surface, and ${\cal E}$ a coherent sheaf on $S$ with a filtration $0=F_0({\cal E})\subset F_1({\cal E})\subset \cdots \subset F_r({\cal E})={\cal E}$. Suppose that the graded pieces ${\rm % gr}_i({\cal E}):=F_i({\cal E})/F_{i-1}({\cal E})$ satisfy ${\rm Ext}^2({\rm % gr}_i({\cal E}),{\rm gr}_j({\cal E}))$ for $i\geq j$. Then (i)\quad the deformations of ${\cal E}$ as a filtered sheaf are unobstructed, (ii)\quad if ${\cal E}$ is a generic filtered sheaf, then the ${\rm gr}_i(% {\cal E})$ are generic, and (iii)\quad if ${\cal E}$ is generic as an unfiltered sheaf, then also $\chi (% {\rm gr}_i({\cal E}),{\rm gr}_{i+1}({\cal E}))\geq 0$ for $i=1,\ldots ,r-1$. \end{proposition} The outline of the paper is as follows. In the first section we review some necessary facts about algebraic stacks and their dimensions. In the second section we prove our technical tool Proposition \ref{unobst} and describe some situations where it applies. It the third section we classify the prioritary components of the ${\rm TF}_S(2,c_1,c_2)$ and the $W_N^0(E)$. In the fourth section we classify the nonprioritary components. In the short final section we complete the proofs of the main theorems. This paper was written in the context of the group on vector bundles on surfaces of Europroj. The author would like to thank A.\ Hirschowitz and M.-A.\ Coppo for some useful conversations. \section{Algebraic Stacks} In this paper we use stacks because in that context there exist natural universal families of coherent (or torsion-free) sheaves. The paper should be manageable even to the reader unfamiliar with algebraic stacks if he accepts them as some sort of generalization of schemes where there are decent moduli for unstable sheaves. For the reader who wishes to learn about algebraic stacks we suggest \cite{LMB}. Alternative universal families of coherent sheaves which stay within the category of schemes would be certain standard open subschemes of Quot schemes. This is the approach taken in \cite {St}. But the language of algebraic stacks is the natural one for problems which involve moduli of unstable sheaves. Stacks differ from schemes is in the way their {\em dimensions} are calculated. For the general definition of the dimension of an algebraic stack at one of its points the reader should consult \cite{LMB} \S 5. But the dimension of the algebraic stack ${\rm Coh}_S$ of coherent sheaves on $S$ (or of any open substack of ${\rm Coh}_S$ such as a ${\rm TF}_S(r,c_1,c_2)$) at a point corresponding to a sheaf ${\cal E}$ is the dimension of the Kuranishi formal moduli for deformations of ${\cal E}$ (i.e.\ the fiber of the obstruction map $({\rm Ext}^1({\cal E},{\cal E}),0)^{\wedge }\rightarrow ({\rm Ext}^2({\cal E},{\cal E}),0)^{\wedge }$) minus the dimension of the automorphism group of ${\cal E}$. Thus if we write $e_i=\dim {\rm Ext}^i(% {\cal E},{\cal E})$, then $-e_0+e_1-e_2\leq \dim {}_{[{\cal E}]}{\rm Coh}% _S\leq -e_0+e_1$. If $S$ is a surface, this means \begin{equation} \label{dimcoh}-\chi ({\cal E},{\cal E})\leq \dim {}_{[{\cal E}]}{\rm Coh}% _S\leq -\chi ({\cal E},{\cal E})+e_2. \end{equation} If ${\cal E}$ is a stable sheaf, then $\dim {}_{[{\cal E}]}{\rm Coh}_S$ is one less than the dimension of the moduli scheme at $\left[ {\cal E}\right] $ because ${\cal E}$ has a one-dimensional family of automorphisms, the homotheties. Generally speaking, the dimension of an algebraic stack are well-behaved. It is constant on an irreducible component away from its intersection with other components; the dimension of a locally closed substack is smaller than the dimension of the stack; etc. But stacks can have negative dimensions. \section{When is a Filtered Sheaf Generic?} In this section we prove our main technical tool Proposition \ref{unobst} and then give two corollaries applying the proposition to birationally ruled surfaces. \paragraph{Proof of Proposition \ref{unobst}.} We begin by recalling some of the deformation theory of \cite{DLP}. We consider the abelian category of sheaves with filtrations of a fixed length $% r$:% $$ 0=F_0({\cal E})\subset F_1({\cal E})\subset \cdots \subset F_r({\cal E})=% {\cal E.} $$ On this category we can define functors% $$ \begin{array}{rcl} {\rm Hom}_{-}({\cal E},{\cal F}) & = & \{\phi \in {\rm Hom}({\cal E},{\cal F})\mid \phi (F_i({\cal E}))\subseteq F_i({\cal F}) \RIfM@\expandafter\text@\else\expandafter\text@@\fi{ for all }i\}, \\ {\rm Hom}_{neg}({\cal E},{\cal F}) & = & \{\phi \in {\rm Hom}({\cal E},{\cal F})\mid \phi (F_i({\cal E}))\subseteq F_{i-1}({\cal % F})\RIfM@\expandafter\text@\else\expandafter\text@@\fi{ for all }i\}. \end{array} $$ These have right-derived functors denoted ${\rm Ext}_{-}^p$ and ${\rm Ext}% _{neg}^p$ which may be computed by the spectral sequences (\cite{DLP} Proposition 1.3)% \begin{eqnarray} E_1^{pq} \ = & \left\{ \begin{array}{ll} \prod_i{\rm Ext}^{p+q}({\rm gr}_i({\cal E}),{\rm gr}_{i-p}({\cal E})) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }p\geq 0 \\ 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }p\le -1 \end{array} \right\} & \Rightarrow \ {\rm Ext}_{-}^{p+q} ({\cal E},{\cal E}), \label{specf} \\ E_1^{pq} \ = & \left\{ \begin{array}{ll} \prod_i{\rm Ext}^{p+q}({\rm gr}_i({\cal E}),{\rm gr}_{i-p}({\cal E})) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }p\geq 1 \\ 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }p\leq 0 \end{array} \right\} & \Rightarrow \ {\rm Ext}_{neg}^{p+q}({\cal E},{\cal E}). \label{specminus} \end{eqnarray} There is also a long exact sequence \begin{equation} \label{extneg}\cdots \rightarrow {\rm Ext}_{neg}^p({\cal E},{\cal E}% )\rightarrow {\rm Ext}_{-}^p({\cal E},{\cal E})\rightarrow \prod_i{\rm Ext}% ^p({\rm gr}_i({\cal E}),{\rm gr}_i({\cal E}))\rightarrow {\rm Ext}% _{neg}^{p+1}({\cal E},{\cal E})\rightarrow \cdots . \end{equation} (i) The tangent space for the deformations of ${\cal E}$ as a filtered sheaf is ${\rm Ext}_{-}^1({\cal E},{\cal E})$ and the obstruction space is ${\rm % Ext}_{-}^2({\cal E},{\cal E})$. The latter vanishes because of the spectral sequence (\ref{specf}). (ii) From (\ref{specminus}) and (\ref{extneg}) we see that the map ${\rm Ext}% _{-}^1({\cal E},{\cal E})\rightarrow \prod_i{\rm Ext}^1({\rm gr}_i({\cal E}),% {\rm gr}_i({\cal E}))$ is surjective. Thus any first-order infinitesimal deformation of the ${\rm gr}_i({\cal E})$ can be induced from a first-order infinitesimal deformation of ${\cal E}$ as a filtered sheaf. But because of (i) any first-order infinitesimal deformation of the filtered sheaf ${\cal E} $ is induced by a noninfinitesimal deformation of ${\cal E}$. So if ${\cal E} $ is generic, then the ${\rm gr}_i({\cal E})$ must also be generic in their respective stacks. (iii) We consider ${\cal E}$ with two filtrations: the original filtration and its subfiltration obtained by suppressing the term $F_i({\cal E})$. We write ${\rm Ext}_{-}^p$ (resp.\ ${\rm Ext}_{-,sub}^p$) for the ${\rm Ext}% _{-}^p$ associated to these two filtrations. We have a long exact sequence $$ \cdots \rightarrow {\rm Ext}_{-}^p({\cal E},{\cal E})\rightarrow {\rm Ext}% _{-,sub}^p({\cal E},{\cal E})\rightarrow {\rm Ext}^p({\rm gr}_i({\cal E}),% {\rm gr}_{i+1}({\cal E}))\rightarrow {\rm Ext}_{-}^{p+1}({\cal E},{\cal E}% )\rightarrow \cdots . $$ Also ${\rm Ext}_{-}^2({\cal E},{\cal E})=0$ by (i). So the formal moduli for the deformations of ${\cal E}$ as a filtered sheaf for the full filtration is of dimension $\dim {\rm Ext}_{-}^1({\cal E},{\cal E})$. The formal moduli for the deformations of ${\cal E}$ as a filtered sheaf for the subfiltration is by general principles of dimension at least% $$ \dim {\rm Ext}_{-,sub}^1({\cal E},{\cal E})-\dim {\rm Ext}_{-,sub}^2({\cal E}% ,{\cal E})\geq \dim {\rm Ext}_{-}^1({\cal E},{\cal E})-\chi ({\rm gr}_i(% {\cal E}),{\rm gr}_{i+1}({\cal E})). $$ So if $\chi $$({\rm gr}_i({\cal E}),{\rm gr}_{i+1}({\cal E}))<0$, then the natural morphism from the formal moduli for the deformations of ${\cal E}$ with the full filtration to the formal moduli for the deformations of ${\cal % E}$ with the subfiltration could not be surjective. So there would be finite deformations of ${\cal E}$ which preserve the subfiltration but not the full filtration. This would contradict the genericity of ${\cal E}$ as an unfiltered sheaf.\TeXButton{qed}{\hfill $\Box$ \medskip} There are several situations in which there are filtrations to which Proposition \ref{unobst} applies. For the first situation, let $S$ be a smooth projective surface and $H$ an ample divisor on $S$. Recall that the $% H $-slope of a torsion-free sheaf ${\cal F}$ on $S$ is $\mu _H({\cal F}% ):=(Hc_1({\cal F}))/{\rm rk}({\cal F})$. We write $\mu _{H,\max }({\cal F})$ is the maximum $H$-slope of a nonzero subsheaf of ${\cal F}$, and $\mu _{H,\min }({\cal F})$ is the minimum slope of a nonzero torsion-free quotient sheaf of ${\cal F}$. \begin{lemma} \label{mu}Let $S$ be a smooth projective surface and $H$ an ample divisor on $S$ such that $HK_S<0$. Let ${\cal F}$ and ${\cal G}$ be torsion-free sheaves on $S$ such that $\mu _{H,\max }({\cal F})+HK_S<\mu _{H,\min }({\cal % G})$. Then ${\rm Ext}^2({\cal F},{\cal G})=0$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}By Serre duality we have ${\rm Ext}^2({\cal F},% {\cal G})\cong {\rm Hom}({\cal G},{\cal F}(K_S))^{*}$. If there were a nonzero $\phi \in {\rm Hom}({\cal G},{\cal F}(K_S))$, then we would have $$ \mu _{H,\max }({\cal F})+HK_S=\mu _{H,\max }({\cal F}(K_S))\geq \mu ({\rm im}% (\phi ))\geq \mu _{H,\min }({\cal G}), $$ a contradiction.\TeXButton{qed}{\hfill $\Box$ \medskip} It follows that if $(S,{\cal O}_S(H))$ is a polarized surface such that $% HK_S<0$, then Proposition \ref{unobst} applies to the Harder-Narasimhan filtration of a torsion-free sheaf ${\cal E}$ on $S$. It also applies to the weak Harder-Narasimhan filtration for torsion-free sheaves on ${\bf P}^2$ described in \cite{W2}. The other situation in which Proposition \ref{unobst} applies is the relative Harder-Narasimhan filtration of a torsion-free sheaf ${\cal E}$ on a ruled surface $\pi {:}~S\rightarrow C$. To describe this let $f_\eta $ be the generic fiber of $\pi $. Write ${\cal E}{\mid }_{f_\eta }\cong \bigoplus_{i=1}^s{\cal O}_{f_\eta }(e_i)^{n_i}$ with $e_1>e_2>\cdots >e_s$ and the $n_i>0$. There exists a unique filtration $0=F_0({\cal E})\subset F_1({\cal E})\subset \cdots \subset F_s({\cal E})={\cal E}$ such that the graded pieces ${\rm gr}_i({\cal E})$ are torsion-free and satisfy ${\rm gr}% _i({\cal E}){\mid }_{f_\eta }\cong {\cal O}_{f_\eta }(e_i)^{n_i}$. The $F_i(% {\cal E})$ may be obtained as the inverse image in ${\cal E}$ of the torsion subsheaf of ${\cal E}/{\cal E}_i$ where ${\cal E}_i$ is the image of the natural map $\pi ^{*}($$\pi _{*}({\cal E}(-e_i)))(e_i)\rightarrow {\cal E}$. Proposition \ref{unobst} applies to this relative Harder-Narasimhan filtration because of \begin{lemma} \label{fiber}Let $\pi {:}~S\rightarrow C$ be a ruled surface, and let ${\cal % E}$ and ${\cal G}$ be torsion-free sheaves on $S$. Suppose that the restrictions of ${\cal E}$ and ${\cal G}$ to a general fiber $F$ of $\pi $ are of the forms ${\cal E}{\mid }_F\cong \bigoplus_i{\cal O}_F(e_i)$ and $% {\cal G}{\mid }_F\cong \bigoplus_j{\cal O}_F(g_j)$ with $\max \{e_i\}-2<\min \{g_j\}.$ Then ${\rm Ext}^2({\cal E},{\cal G})=0$. In particular, if $\max \{e_i\}-\min \{e_j\}<2$, then ${\cal E}$ is prioritary. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}Again Serre duality gives ${\rm Ext}^2({\cal E},% {\cal G})\cong {\rm Hom}({\cal G},{\cal E}(K_S))^{*}$. If there were a nonzero $\phi \in {\rm Hom}({\cal G},{\cal E}(K_S))$, then there would be a nonzero $\phi {\mid }_F\in \bigoplus_{i,j}H^0({\cal O}_F(e_i-2-g_j))$. This is impossible since $f_i-2-g_j<0$ for all $i$ and $j$. If $\max \{e_i\}-\min \{e_i\}<2$, then we may set ${\cal G}={\cal E}(-f_p)$ for any fiber $f_p=\pi ^{-1}(p)$ to get ${\rm Ext}^2({\cal E},{\cal E}% (-f_p))=0$ for all $p\in C$. Thus ${\cal E}$ is prioritary.\TeXButton{qed} {\hfill $\Box$ \medskip} \section{Prioritary Components} In this section we prove the necessary lemmas for classifying the principal components of the ${\rm TF}_S(2,c_1,c_2)$ and $W_N^0(E)$. We use \cite{W} and \cite{HL} as our basic sources for existence and uniqueness results because these use our preferred language of prioritary sheaves. But existence and uniqueness results for only marginally different classes of sheaves on ${\bf P}^2$ and of rank $2$ sheaves on ruled surfaces had already been proven in \cite{Ba} \cite{BS} \cite{B} \cite{DLP} \cite{E} \cite{ES} \cite{HS} \cite{Ho} \cite{Hu1} \cite{Hu2} (and perhaps elsewhere). \begin{proposition} \label{discquad}Let $\pi {:}~S\rightarrow C$ be a ruled surface, and let $% f\in {\rm NS}(S)$ be the numerical class of a fiber of $\pi $. Then ${\rm TF}% _S(r,c_1,c_2)$ has a unique prioritary component if $r$ divides $c_1f$ and $% 2rc_2\geq (r-1)c_1^2$, or if $r$ does not divide $c_1f$. Otherwise it has no prioritary components. The prioritary component is smooth of dimension $% -\chi ({\cal E},{\cal E})$. \end{proposition} \TeXButton{Proof}{\paragraph{Proof. }}The uniqueness and smoothness of the prioritary component was proven in \cite{W} Proposition 2. Since by definition a prioritary sheaf ${\cal E}$ satisfies ${\rm Ext}^2({\cal E},{\cal E}% (-f_p))=0 $ for all $p\in C$, it also satisfies ${\rm Ext}^2({\cal E},{\cal E% })=0$. So the prioritary component has dimension $-\chi ({\cal E},{\cal E})$ according to (\ref{dimcoh}). For existence of a prioritary sheaf ${\cal E}$, note that $2rc_2-(r-1)c_1^2$ is invariant under twist as is the residue of $c_1f$ modulo $r$. Then by replacing ${\cal E}$ by a twist ${\cal E}(n)$ if necessary, we may assume that $d:=-c_1f$ satisfies $0\leq d<r$. In the proof of \cite{W} Proposition 2 it was shown that a general prioritary sheaf with such a $c_1$ fits into an exact sequence \begin{equation} \label{Beil}0\rightarrow \pi ^{*}({\cal K})\rightarrow {\cal E}\rightarrow \pi ^{*}({\cal L})\otimes \Omega _{S/C}(1)\rightarrow 0 \end{equation} where ${\cal K}$ is a vector bundle on $C$ of rank $r-d$ and ${\cal L}$ a coherent sheaf on $C$ of rank $d$. Let $k=\deg ({\cal K})$ and $l=\deg (% {\cal L})$. Write $h=c_1({\cal O}(1))$ so that $\{h,f\}$ is a basis of ${\rm % NS}(S)$. Then ${\cal E}$ has rank $r$ and Chern classes $c_1=(k+l)f-dh$ and $% c_2=\frac 12d(d-1)h^2-(k+l)d+l$. So to finish the proof of the lemma we need to show that if $0<d<r$, then there exist prioritary sheaves of the form (% \ref{Beil}) for all $k$ and $l$, while if $d=0$, then there exist prioritary sheaves of that form if and only if $(k,l)$ satisfies $l\geq 0$. If $0<d<r$, then for any $k$ and $l$ and any locally free sheaves ${\cal K}$ (resp.\ ${\cal L})$ on $C$ of rank $r-d$ and degree $k$ (resp.\ rank $d$ and degree $l$), the sheaf ${\cal F}:=\pi ^{*}({\cal K})\oplus \left[ \pi ^{*}(% {\cal L})\otimes \Omega _{S/C}(1)\right] $ has splitting type ${\cal O}_{% {\bf P}^1}^{r-d}\oplus {\cal O}_{{\bf P}^1}(-1)^d$ on all fibers and hence is prioritary by Lemma \ref{fiber}. If $d=0$, then ${\cal L}$ has rank $0$. So its degree $l$ must be nonnegative. Conversely if $k$ is any integer and $l\geq 0$, then an ${\cal E% }$ as in (\ref{Beil}) can be constructed for any locally free sheaf ${\cal K} $ of rank $r$ and degree\thinspace $k$ on $C$ as an elementary transform of $% \pi ^{*}({\cal K})$ along $l$ fibers of $\pi $. Such an ${\cal E}$ is prioritary by Lemma \ref{fiber} because its restriction to the general fiber of $\pi $ is trivial. Thus for $d=0$ there exists prioritary sheaf ${\cal E}$ of the form (\ref{Beil}) for and only for those $(k,l)$ satisfying $l\geq 0$% . This completes the proof of the lemma. \TeXButton{qed}{\hfill $\Box$ \medskip} \begin{proposition} \label{discp2}{\rm (Hirschowitz-Laszlo)} The stack ${\rm TF}_{{\bf P}% ^2}(r,c_1,c_2)$ has a unique prioritary component if $2rc_2-(r-1)c_1^2\geq -d(r-d)$ where $c_1\equiv -d\pmod{r}$ and $0\leq d<r$. Otherwise it has no prioritary components. The prioritary component is smooth of dimension $% -\chi ({\cal E},{\cal E})$. \end{proposition} \TeXButton{Proof}{\paragraph{Proof. }}Let $c_1=mr-d$. Let $\mu =c_1/r$ be the slope and $% \Delta =(2rc_2-(r-1)c_1^2)/2r^2$ the discriminant of ${\cal E}$. Then in \cite{HL} Chap.~I, Propositions 1.3 and 1.5 and Th\'eor\`eme 3.1, it is shown that ${\rm TF}_{{\bf P}^2}(r,c_1,c_2)$ has a priority component if and only if the Hilbert polynomial $P(n)=r\left( \frac 12(\mu +n+2)(\mu +n+1)-\Delta \right) $ is nonpositive for some integer $n$, and that in that case the prioritary component is unique and smooth. The dimension of such a component is again $-\chi ({\cal E},{\cal E})$ by (\ref{dimcoh}) because the prioritary condition implies ${\rm Ext}^2({\cal E},{\cal E})=0$. We show that the Hilbert polynomial criterion of \cite{HL} is equivalent to the criterion asserted by the lemma. But $P(n)-P(n-1)=\mu +n+1=m-\frac dr+n+1 $ is nonnegative if and only if $n\geq -m$. So $\min _{n\in {\bf Z}% }P(n)=P(-m-1)=r\left( \frac 12(1-\frac dr)(-\frac dr)-\Delta \right) $, and this is nonpositive if and only if $2r^2\Delta \geq -d(r-d)$.\thinspace \TeXButton{qed}{\hfill $\Box$ \medskip} We recall the Riemann-Roch formula for a coherent sheaf ${\cal E}$ of rank $% r $ and Chern classes $c_1$ and $c_2$ on a surfaces $S$: \begin{equation} \label{RR}\chi ({\cal E})=r\chi ({\cal O}_S)+\frac 12c_1\left( c_1-K_S\right) -c_2. \end{equation} \begin{lemma} \label{diminution}Let $\pi {:}~S\rightarrow C$ be a ruled surface or let $S$ be ${\bf P}^2$. Suppose ${\cal E}$ is a prioritary sheaf on $S$ of rank $% r\geq 2$ such that $H^1({\cal E})=H^2({\cal E})=0$. Let $H$ be a very ample divisor on $S$. (i)\quad If ${\cal F}$ is a general prioritary sheaf of rank $r$ and Chern classes $c_1=c_1({\cal E})$ and $c_2\geq c_2({\cal E})$ such that $\chi (% {\cal F})\geq 0$, then $H^1({\cal F})=H^2({\cal F})=0$. (ii)\quad If in addition $H^1({\cal E}(H))=H^2({\cal E}(H))=0$ and $\chi (% {\cal E}(H))\geq \chi ({\cal E})$, then for all $n\geq 2$ the sheaf ${\cal F}% (nH)$ is generated by global sections and its general section has degeneracy locus of codimension $2$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) By semicontinuity it is enough to exhibit one such ${\cal F}$. We go by induction on $c_2$. If $c_2=c_2({\cal E})$, we may take ${\cal F}={\cal E}$. If $c_2>c_2({\cal E})$, let ${\cal G}$ be a prioritary sheaf of rank $r$ and Chern classes $c_1$ and $c_2-1$ with $H^1(% {\cal G})=H^2({\cal G})=0$. By (\ref{RR}) we have $\chi ({\cal G})=\chi (% {\cal F})+1>0$. So ${\cal G}$ must have a nonzero global section $s$. If $% x\in S$ is a general point of $S$ and ${\cal G}\otimes k(x)\TeXButton{-->>} {\twoheadrightarrow}k(x)$ a general one-dimensional quotient of the fiber of ${\cal G}$ at $x$, then the image of $s$ in $k(x)$ is nonzero. So if ${\cal F% }$ is the kernel% $$ 0\rightarrow {\cal F}\rightarrow {\cal G}\rightarrow k(x)\rightarrow 0, $$ then $h^0({\cal F})=h^0({\cal G})-1$ and $H^1({\cal F})=H^2({\cal F})=0$. (ii) Under the added hypotheses the general ${\cal F}$ also satisfies $H^1(% {\cal F}(H))=H^2({\cal F}(H))=0$. But $H^1({\cal F}(H))=H^2({\cal F})=0$ implies that ${\cal F}(nH)$ is generated by global sections for all $n\geq 2$ by the Castelnuovo-Mumford lemma. The general section of ${\cal F}(nH)$ will drop rank in codimension $2$ by Bertini's theorem. \begin{lemma} \label{goodp2}Let ${\cal F}$ be a generic prioritary sheaf of rank $2$ and Chern classes $c_1\geq -4$ and $c_2$ on ${\bf P}^2$. The two conditions (a) $% H^1({\cal F})=H^2({\cal F})=0$ and (b) ${\cal F}(3)$ has a section with degeneracy locus of codimension $2$ hold if and only if $\chi ({\cal F})\geq 0$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}If (a) and (b) hold, then clearly $\chi ({\cal F}% )=h^0({\cal F})\geq 0$. Conversely, suppose ${\cal F}$ is generic prioritary of rank $2$ with $c_1\geq -4$ and $\chi ({\cal F})\geq 0$. If $c_1=2a$ is even, then let ${\cal E}={\cal O}_{{\bf P}^2}(a)^2$. Then $c_1=c_1({\cal E})$ and $c_2\geq \frac 14c_1^2=a^2=c_2({\cal E})$ by Proposition \ref{discp2}. Moreover, $H^1({\cal E})=H^2({\cal E})=H^1({\cal E}(1))=H^2({\cal E}(1))=0$ and $\chi ({\cal E}(1))=\chi ({\cal E})+2a+4\geq \chi ({\cal E})$. Hence conditions (a) and (b) follow from Lemma \ref{diminution}. If $c_1=2a+1$ is odd, we may apply Lemma \ref{diminution} with ${\cal E}=% {\cal O}_{{\bf P}^2}(a)\oplus {\cal O}_{{\bf P}^2}(a+1)$.\TeXButton{qed} {\hfill $\Box$ \medskip} \begin{lemma} \label{goodquad}Let ${\cal F}$ be a generic prioritary sheaf of rank $2$ and Chern classes $c_1=(a_1,a_2)$ and $c_2$ on ${\bf P}^1\times {\bf P}^1$. Suppose the $a_i\geq -2$. (i)\quad If $a_2$ is even, then the two conditions (a) $H^1({\cal F})=H^2(% {\cal F})=0$ and (b) ${\cal F}(2,2)$ has a section with degeneracy locus of codimension $2$ hold if and only if $\chi ({\cal F})\geq 0$. (ii)\quad If $a_2$ is odd, then (a) and (b) hold if and only if one has both $\chi ({\cal F})\geq 0$ and either $c_2\geq \frac 12a_1(a_2-1)-1$ or $% (a_1,a_2,c_2)=(a_1,-1,-a_1-2)$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) We apply Lemma \ref{diminution} with ${\cal E}$ either ${\cal O}(\frac{a_1}2,\frac{a_2}2)^2$ or ${\cal O}(\frac{a_1-1}2, \frac{a_2}2)\oplus {\cal O}(\frac{a_1+1}2,\frac{a_2}2)$. (ii) Before beginning, recall that if $a_2=2b-1$ is odd, then by (\ref{Beil}% ) the general prioritary sheaf of the given rank and Chern classes is of the form% $$ 0\rightarrow {\cal O}(a_1-p,b)\rightarrow {\cal F\rightarrow O}% (p,b-1)\rightarrow 0 $$ with $p$ determined by $c_2=(a_1-p)(b-1)+pb=\frac 12a_1(a_2-1)+p$. Now suppose that (a) and (b) hold. Then clearly $\chi ({\cal F})=h^0({\cal F}% )\geq 0$. If $p\geq -1$, then $c_2\geq \frac 12a_1(a_2-1)-1$ as desired. If on the other hand $p\leq -2$, then the sequence splits. So if (a) and (b) hold, then $H^1({\cal O}(p,b-1))=0$ while ${\cal O}(p+2,b+1)$ is generated by global sections. These are possible simultaneously only if $p=-2$ and $% b=0 $. Conversely, if $\chi ({\cal F})\geq 0$ and $c_2\geq \frac 12a_1(a_2-1)-1$, then (a) and (b) hold by Lemma \ref{diminution} using ${\cal E}={\cal O}% (a_1+1,b)\oplus {\cal O}(-1,b-1)$. If $(a_1,a_2,c_2)=(a_1,-1,-a_1-2)$, then one may pick ${\cal F}={\cal O}(a_1+2,0)\oplus {\cal O}(-2,-1)$. \TeXButton{qed}{\hfill $\Box$ \medskip} \section{Nonprioritary Components} In this section we study nonprioritary components of ${\rm TF}_S(r,c_1,c_2)$ and of $W_N^0(E)$. According to Proposition \ref{unobst} on a ruled surface $% \pi {:}~S\rightarrow C$ or on ${\bf P}^2$ with $f$ denoting the numerical class either of a fiber of $\pi $ or of a line in ${\bf P}^2$, the general member of any nonprioritary component of ${\rm TF}_S(r,c_1,c_2)$ a {\em % nonprioritary generic extension of twisted ideal sheaves,} i.e.\ an extension \begin{equation} \label{HN}0\rightarrow {\cal I}_{Z_1}(L_1)\rightarrow {\cal E}\rightarrow {\cal I}_{Z_2}(L_2)\rightarrow 0 \end{equation} such that the ${\cal O}_S(L_i)$ are generic line bundles having $L_1f>L_2f+1$ and the $Z_i$ are generic sets of $n_i$ points in $S$. In addition, the proposition says that \begin{equation} \label{chi}\chi ({\cal I}_{Z_1}(L_1),{\cal I}_{Z_2}(L_2))=\chi ({\cal O}% (L_2-L_1))-n_1-n_2\geq 0. \end{equation} Moreover, the extension is uniquely determined by ${\cal E}$ since it defines the Harder-Narasimhan filtration of ${\cal E}$ with respect to a suitable polarization of the surface. The next two lemmas show that if $S$ is ${\bf P}^2$ or a semistable ruled surface, then a generic extension of twisted ideal sheaves satisfying (\ref {chi}) is the generic sheaf of an irreducible component of the stack of torsion-free rank $2$ sheaves on $S$. \begin{lemma} \label{divisor}Suppose either that $\pi {:}~S\rightarrow C$ is a birationally ruled surface or $S$ is ${\bf P}^2$. If a nonprioritary generic extension of twisted ideal sheaves ${\cal E}$ as in (\ref{HN}) specializes to another nonprioritary generic extension of twisted ideal sheaves $% 0\rightarrow {\cal I}_{Z_1^{\prime }}(L_1^{\prime })\rightarrow {\cal E}% ^{\prime }\rightarrow {\cal I}_{Z_2^{\prime }}(L_2^{\prime })\rightarrow 0$, then (i)\quad $\chi ({\cal I}_{Z_1^{\prime }}(L_1^{\prime }),{\cal I}% _{Z_2^{\prime }}(L_2^{\prime }))<\chi ({\cal I}_{Z_1}(L_1),{\cal I}% _{Z_2}(L_2))$ and (ii)\quad there exists an effective divisor $\Gamma $ on $S$ such that $% -\Gamma \cdot \Gamma >(L_1-L_2+K_S)\cdot \Gamma $. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) Let ${\rm FiltCoh}_S$ be the stack parametrizing filtered coherent sheaves ${\cal F}_1\subset {\cal F}.$ Because the tangent space for automorphisms of ${\cal F}_1\subset {\cal F}$ (resp.\ the tangent space for deformations of ${\cal F}_1\subset {\cal F}$, resp.\ the obstruction space for deformations of ${\cal F}_1\subset {\cal F}$% ) is ${\rm Ext}_{-}^i({\cal F},{\cal F})$ for $i=0$ (resp.$\ i=1$, resp.\ $% i=2)$, one has% $$ -\chi _{-}({\cal F},{\cal F})\leq \dim {}_{[{\cal F}_1\subset {\cal F}]}{\rm % FiltCoh}_S\leq -\chi _{-}({\cal F},{\cal F})+\dim {\rm Ext}_{-}^2({\cal F},% {\cal F}) $$ where $\chi _{-}({\cal F},{\cal F})=\sum (-1)^i\dim {\rm Ext}_{-}^i({\cal F},% {\cal F})$. The forgetful functor ${\rm FiltCoh}_S\rightarrow {\rm Coh}_S$ defined by $[{\cal F}_1\subset {\cal F}]\mapsto [{\cal F}]$ induces maps on infinitesimal automorphism, tangent, and obstruction spaces ${\rm Ext}_{-}^i(% {\cal F},{\cal F})\rightarrow {\rm Ext}^i({\cal F},{\cal F})$. So if ${\rm % Hom}_{+}({\cal F},{\cal F}):={\rm Hom}({\cal F}_1,{\cal F}/{\cal F}_1)=0$, then the morphism ${\rm FiltCoh}_S\rightarrow {\rm Coh}_S$ is unramified at $% [{\cal F}_1\subset {\cal F}]$, and ${\rm FiltCoh}_S$ can be viewed as more or less a locally closed substack of ${\rm Coh}_S$ in a neighborhood of $[% {\cal F}]$. In our case the subsheaf ${\cal F}_1={\cal I}_{Z_1}(L_1)$ is unique, so ${\rm FiltCoh}_S$ is a locally closed substack of ${\rm Coh}_S$ in a neighborhood of $[{\cal F}]$. Thus the dimension of the locally closed substack of torsion-free sheaves numerically equivalent to ${\cal E}$ which admit a filtration with the subsheaf numerically equivalent to ${\cal I}_{Z_1}(L_1)$ (resp.\ to ${\cal I}% _{Z_1^{\prime }}(L_1^{\prime })$) and with ${\rm Ext}_{-}^2({\cal E},{\cal E}% )=0$ is% $$ -\chi _{-}({\cal E},{\cal E)=}-\chi ({\cal E},{\cal E})+\chi ({\cal I}% _{Z_1}(L_1),{\cal I}_{Z_2}(L_2)) $$ (resp.\ $-\chi ({\cal E},{\cal E})+\chi ({\cal I}_{Z_1^{\prime }}(L_1^{\prime }),{\cal I}_{Z_2^{\prime }}(L_2^{\prime }))$). If the former substack contains the latter in its closure, its dimension must be larger. (ii) As ${\cal E}$ specializes to ${\cal E}^{\prime }$, its subsheaf ${\cal I% }_{Z_1}(L_1)$ specializes to a subsheaf of ${\cal E}^{\prime }$. Because this subsheaf destabilizes ${\cal E}^{\prime }$, it must be contained in $% {\cal I}_{Z_1^{\prime }}(L_1^{\prime })$. Hence ${\cal O}_S(L_1)$ specializes to a line bundle of the form ${\cal O}_S(L_1^{\prime }-\Gamma )$ with $\Gamma $ an effective divisor, and\ ${\cal O}_S(L_2)$ specializes to $% {\cal O}_S(L_2^{\prime }+\Gamma )$. Since $L_2^{\prime }-L_1^{\prime }\equiv L_2-L_1-2\Gamma $, the Riemann-Roch formula leads to% $$ \chi ({\cal O}(L_2^{\prime }-L_1^{\prime }))=\chi ({\cal O}(L_2-L_1))+\left( 2(L_1-L_2+\Gamma )+K_S\right) \cdot \Gamma \RIfM@\expandafter\text@\else\expandafter\text@@\fi{.} $$ We also have $$ n_1+n_2+\left( L_1\cdot L_2\right) =c_2({\cal E})=c_2({\cal E}^{\prime })=n_1^{\prime }+n_2^{\prime }+\left( L_1^{\prime }\cdot L_2^{\prime }\right) , $$ from which we see that% $$ n_1^{\prime }+n_2^{\prime }=n_1+n_2+(L_1-L_2+\Gamma )\cdot \Gamma . $$ Thus% $$ \chi ({\cal I}_{Z_1^{\prime }}(L_1^{\prime }),{\cal I}_{Z_2^{\prime }}(L_2^{\prime }))=\chi ({\cal I}_{Z_1}(L_1),{\cal I}_{Z_2}(L_2))+(L_1-L_2+% \Gamma +K_S)\cdot \Gamma . $$ Because of (i) this now implies the lemma.\TeXButton{qed}{\hfill $\Box$ \medskip} \begin{lemma} \label{tilt}Suppose either that $\pi {:}~S\rightarrow C$ is a ruled surface without curves of negative self-intersection or that $S$ is ${\bf P}^2$. If a nonprioritary generic extension of twisted ideal sheaves ${\cal E}$ as in (% \ref{HN}) specializes to another generic extension of twisted ideal sheaves $% {\cal E^{\prime }}$, then $\chi ({\cal O}(L_2-L_1))\leq 0$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}Because ${\cal E}$ is not prioritary, the restriction of ${\cal O}(L_2-L_1)$ to a general fiber of $\pi $ or a general line of ${\bf P}^2$ is of negative degree. So $H^0({\cal O}(L_2-L_1))=0$. Since $S$ contains no curves of negative self-intersection, Lemma \ref {divisor}(ii) says that there is an effective divisor $\Gamma $ on $S$ such that $(L_1-L_2+K_S)\cdot \Gamma <0$. Since $S$ contains no curves of negative self-intersection, ${\cal O}(L_1-L_2+K_S)$ cannot be effective. Thus $H^0({\cal O}(L_1-L_2+K_S))=0$ and by Serre duality $H^2({\cal O}% (L_2-L_1))=0$. It follows that $\chi ({\cal O}(L_2-L_1))=-h^1({\cal O}% (L_2-L_1))\leq 0$ as asserted. \TeXButton{qed}{\hfill $\Box$ \medskip} \begin{lemma} \label{nonprior}Suppose either that $\pi {:}~S\rightarrow C$ be a ruled surface without curves of negative self-intersection and $f\in {\rm NS}(S)$ is the class of a fiber of $\pi $, or that $S$ is ${\bf P}^2$ and $f$ is the class of a line. Let $c_1\in {\rm NS}(S)$ and $c_2\in {\bf Z}$. Let $% (D,n_1,n_2)\in {\rm NS}(S)\times {\bf Z}^2$. Then ${\rm TF}_S(2,c_1,c_2)$ has a unique component whose general member ${\cal E}$ is a nonprioritary generic extension of twisted ideal sheaves \begin{equation} \label{again}0\rightarrow {\cal I}_{Z_1}(c_1-D)\rightarrow {\cal E}% \rightarrow {\cal I}_{Z_2}(D)\rightarrow 0 \end{equation} with $\deg (Z_i)=n_i$ if and only if $Df\leq -1+\frac 12c_1f$, and the $n_i$ are nonnegative and satisfy $n_1+n_2=c_2-D(D-c_1)\leq \chi ({\cal O}% _S(2D-c_1))$. Such a component of ${\rm TF}_S(2,c_1,c_2)$ has dimension $% -\chi ({\cal E},{\cal E})+\chi ({\cal O}_S(2D-c_1))-n_1-n_2$ and generic embedding codimension $n_2+h^1({\cal O}_S(2D-c_1))$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}If ${\cal E}$ is a generic nonprioritary sheaf, then its restriction to a general fiber $F$ of $\pi $ (resp.\ to a generic line of ${\bf P}^2$) must be of the form ${\cal E}{\mid }_F\cong {\cal O}% _F(a)\oplus {\cal O}_F(b)$ with $a\geq b+2$ by Lemma \ref{fiber} (resp. by \cite{HL} Chap. I, Proposition 1.2). Hence the relative Harder-Narasimhan filtration of ${\cal E}$ which was described before Lemma \ref{fiber} (resp.~the Harder-Narasimhan filtration of ${\cal E}$ on ${\bf P}^2$) must be of the form $0\subset {\cal I}_{Z_1}(c_1-D)\subset {\cal E}$ with ${\cal E% }/{\cal I}_{Z_1}(c_1-D)\cong {\cal I}_{Z_2}(D)$ for some divisor $D$ on $S$ and some $0$-dimensional subschemes $Z_i\subset S$ such that $(c_1-D)f\geq Df+2$, or $Df\leq -1+\frac 12c_1f$. Clearly one has $n_i:=\deg (Z_i)\geq 0$ and $c_2=D(c_1-D)+n_1+n_2$. Lemma \ref{fiber} shows that Proposition \ref {unobst} is applicable to the filtered sheaf $0\subset {\cal I}% _{Z_1}(c_1-D)\subset {\cal E}$. So $\chi ({\cal I}_{Z_1}(c_1-D),{\cal I}% _{Z_2}(D))=\chi ({\cal O}_S(2D-c_1))-n_1-n_2\geq 0$. Thus to any nonprioritary irreducible component of ${\rm TF}_S(2,c_1,c_2)$ there is an associated triple $(D,n_1,n_2)$ satisfying the asserted numerical conditions. Conversely suppose $(D,n_1,n_2)$ satisfy all the numerical conditions. Let $% Z_i$ be a general set of $n_i$ points on $S$ and let ${\cal E}$ be a generic extension as in (\ref{again}). Then ${\cal E}$ cannot be a specialization of another nonprioritary generic extension $0\rightarrow {\cal I}_{Z_1^{\prime }}(c_1-D^{\prime })\rightarrow {\cal E^{\prime }}\rightarrow {\cal I}% _{Z_2^{\prime }}(D^{\prime })\rightarrow 0$ because in that case Lemma \ref {divisor}(i) would imply% $$ \chi ({\cal O}_S(2D^{\prime }-c_1))-n_1^{\prime }-n_2^{\prime }>\chi ({\cal O% }_S(2D-c_1))-n_1-n_2\geq 0 $$ contradicting Lemma \ref{tilt}. Nor can ${\cal E}$ be a specialization of a generic prioritary sheaf because it is the sheaf corresponding to a generic point of a locally closed substack of ${\rm TF}_S(2,c_1,c_2)$ whose dimension was calculated in the proof of Lemma \ref{divisor}(i) as $-\chi (% {\cal E},{\cal E})+\chi ({\cal O}_S(2D-c_1))-n_1-n_2$. This is at least $% -\chi ({\cal E},{\cal E})$, the dimension of the prioritary component. So $% {\cal E}$ is the generic sheaf of an irreducible component of ${\rm TF}% _S(2,c_1,c_2)$ of dimension $-\chi ({\cal E},{\cal E})+\chi ({\cal O}% _S(2D-c_1))-n_1-n_2$. The embedding codimension is the dimension of the cokernel of the map $% \alpha $ between the tangent spaces of the stack of filtered sheaves and the stack of unfiltered sheaves which is given by% $$ {\rm Ext}_{-}^1({\cal E},{\cal E})\stackrel{\alpha }{\rightarrow }{\rm Ext}% ^1({\cal E},{\cal E})\rightarrow {\rm Ext}^1({\cal I}_{Z_1}(c_1-D),{\cal I}% _{Z_2}(D))\rightarrow 0. $$ But since ${\rm Hom}({\cal I}_{Z_1}(c_1-D),{\cal I}_{Z_2}(D))=0$, the dimension of ${\rm cok}(\alpha )$ is the difference between the two numbers \begin{eqnarray*} & \dim {\rm Ext}^2({\cal I}_{Z_1}(c_1-D),{\cal I}_{Z_2}(D))=h^0({\cal I}% _{Z_1}(c_1-2D+K_S))=\left[ h^2({\cal O}(2D-c_1))-n_1\right] _{+}, & \\ & \chi ({\cal I}_{Z_1}(c_1-D),{\cal I}_{Z_2}(D))=\left[ h^2({\cal O}% (2D-c_1))-n_1\right] -\left[ h^1({\cal O}(2D-c_1))+n_2\right] . & \end{eqnarray*} Because the $\chi $ is nonnegative, we see that $h^2({\cal O}% (2D-c_1))-n_1\geq 0$, and that therefore the difference between the two numbers is $n_2+h^1({\cal O}(2D-c_1))$. \TeXButton{qed}{\hfill $\Box$ \medskip} \begin{remark} The components of ${\rm TF}_{{\bf P}^2}(2,c_1,c_2)$ containing locally free sheaves were already classified by Str\o mme in \cite{St} Theorem 3.9, but he made one minor error with the embedding codimensions. The prioritary components of ${\rm TF}_{{\bf P}^2}(2,c_1,\frac 14c_1^2+1)$ are generically smooth like all prioritary components. But they appear in Str\o mme's classification in \cite{St} Theorem 3.9 as the component with $% (d,c_1,c_2)=(0,0,1)$ which was said to be nonreduced with generic embedding codimension $1$. The computation of the $h^i({\cal E}nd({\cal E}))$ in \cite {St} Proposition 1.4 is wrong in that single case. \end{remark} We now consider what the classification of generic rank $2$ sheaves entails for Brill-Noether loci. For the sake of simplicity, we will restrict ourselves to those surfaces covered by Lemma \ref{tilt} which also have vanishing irregularity, thus ${\bf P}^2$ and ${\bf P}^1\times {\bf P}^1$, so that we do not need to analyze nongeneric line bundles which might have more cohomology than the corresponding generic line bundles. \begin{lemma} \label{final}Let $S$ be ${\bf P}^1\times {\bf P}^1$ (resp.\ ${\bf P}^2$) and let $f\in {\rm NS}(S)$ be the class of a fiber of ${\rm pr}_1$ (resp.\ a line). Let ${\cal F}$ be a nonprioritary generic extension of twisted ideal sheaves \begin{equation} \label{third}0\rightarrow {\cal I}_{Z_1}(c_1-D)\rightarrow {\cal F}% \rightarrow {\cal I}_{Z_2}(D)\rightarrow 0 \end{equation} with $c_1-K_S$ effective, $Df\leq -1+\frac 12c_1f$, and $\chi ({\cal O}% _S(2D-c_1))\geq 0$. Let $n_i:=\deg (Z_i)$. Then the two conditions (a) $H^1(% {\cal F})=H^2({\cal F})=0$ and (b) ${\cal F}(-K_S)$ has a section with degeneracy locus of codimension $2$ hold if and only if the three conditions hold: (i) $\chi ({\cal F})\geq 0$, (ii) $D-K_S$ is an effective and irreducible divisor class, and (iii) $n_2\leq h^0({\cal O}_S(D))$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}We will prove the lemma for ${\bf P}^1\times {\bf P}% ^1$ only. The proof for ${\bf P}^2$ is similar and actually simpler. Let $(a_1,a_2)$ be the bidegree of $c_1-D$ and $(b_1,b_2)$ the bidegree of $% D $. We claim that the hypotheses of the lemma imply that $a_1\geq 0$ and $% a_2\geq 0$. To see this first note that the effectiveness of $c_1-K_S$ is equivalent to $a_1+b_1\geq -2$ and $a_2+b_2\geq -2$. The condition $Df\leq -1+\frac 12c_1f$ is equivalent to $b_2\leq -1+\frac 12(a_2+b_2)$, or $% a_2-b_2\geq 2$. Adding gives $a_2\geq 0$ as claimed. Also we have% $$ 0\leq \chi ({\cal O}_S(2D-c_1))=(b_1-a_1+1)(b_2-a_2+1). $$ Since $b_2-a_2+1<0$, this gives $b_1-a_1+1\leq 0$. Thus $a_1>b_1$. Adding this to $a_1+b_1\geq -2$ now gives $a_1\geq 0$ as claimed. The fact that $c_1-D$ has bidegree $(a_1,a_2)$ with $a_1\geq 0$ and $a_2\geq 0$ implies that $H^i({\cal O}_S(c_1-D))=0$ for $i=1,2$ and that ${\cal O}% _S(c_1-D+K_S)\TeXButton{ncong}{\ncong}{\cal O}_S$. Now suppose that (a) and (b) hold. Then $\chi ({\cal F})=h^0({\cal F})\geq 0$% , whence (i). To prove conditions (ii) and (iii), note first that $H^1({\cal % F})$ and $H^2({\cal F})$ vanish because of (a) while $H^2({\cal I}% _{Z_2}(c_1-D))\cong H^2({\cal O}_S(c_1-D))$ vanishes because of the previous paragraph. So $H^1({\cal I}_{Z_2}(D))=H^2({\cal I}_{Z_2}(D))=0$ by (\ref {third}). This implies that $H^1({\cal O}_S(D))=H^2({\cal O}_S(D))=0$ and $% n_2\leq h^0({\cal O}_S(D))$. Thus we have (iii) plus $H^1({\cal O}% _S(b_1,b_2))=H^2({\cal O}_S(b_1,b_2))=0$. These vanishings imply either that both $b_i\geq -1$ and hence that the divisor $D-K_S$ of bidegree $% (b_1+2,b_2+2)$ is very ample, or that $(b_1,b_2)$ is $(-1,d)$ or $(d,-1)$ with $d\leq -2$. But if $(b_1,b_2)$ had of one of these last two forms, and if also $d\leq -3$, then ${\cal O}_S(D-K_S)={\cal O}_S(b_1+2,b_2+2)$ would not have any global sections. Hence all sections of ${\cal F}(-K_S)$ would lie in ${\cal I}_{Z_1}(c_1-D-K_S)$. But we have shown that the line bundle $% {\cal O}_S(c_1-D-K_S)$ is always nontrivial. So all global sections of $% {\cal F}(-K_S)$ would degenerate along a nontrivial curve, contradicting (b). Hence the only possible cases where (a) and (b) hold with $D-K_S$ not very ample are the cases where $D-K_S$ is of bidegree $(0,1)$ or $(1,0)$, whence (ii). Thus (a) and (b) imply (i), (ii) and (iii). Conversely, suppose (i), (ii) and (iii) hold for ${\cal F}$. We begin by proving (a) in the special case where $n_1=0$. Condition (ii) implies that either both $b_i\geq -1$ or one $b_i=-1$. Therefore $H^1({\cal O}_S(D))=H^2(% {\cal O}_S(D))=H^2({\cal I}_{Z_2}(D))=0$. Because $Z_2$ consists of $n_2\leq h^0({\cal O}_S(D))$ generic points of $S$ by condition (iii), we have $H^1(% {\cal I}_{Z_2}(D))=0$ also. And we have already shown that $H^i({\cal O}% _S(c_1-D))=0$ for $i=1,2$. It now follows by (\ref{third}) that $H^1({\cal F}% )=H^2({\cal F})=0$. Thus (a) holds in the special case where $n_1=0$. If $n_1>0$, then we may prove (a) by induction on $n_1$ using the same method as in the proof of Lemma \ref{diminution}(i). For (b) let $H$ be a divisor of bidegree $(1,1)$. Then% $$ \chi ({\cal F}(H))=\chi ({\cal F})+(c_1+2H)H+2>\chi ({\cal F})\geq 0 $$ since $c_1+2H=c_1-K_S$ is effective. So (i) holds for ${\cal F}(H)$. Since (ii) holds for ${\cal F}$, the divisor $D-K_S$ is base-point-free, so $% D+H-K_S$ is very ample. Hence (ii) holds for ${\cal F}(H)$. And $$ h^0({\cal O}_S(D+H))=h^0({\cal O}_S(D))+(D+H)H+1\geq h^0({\cal O}_S(D))\geq n_2 $$ since according to (ii) $D+H$ is either effective or of bidegree $(0,-1)$ or $(-1,0)$. So (iii) also holds for ${\cal F}(H)$. By what we have already verified, the fact that (i), (ii) and (iii) all hold for ${\cal F}$ and $% {\cal F}(H)$ implies that (a) also holds for ${\cal F}$ and ${\cal F}(H)$. Hence $H^1({\cal F}(H))=H^2({\cal F})=0$. So ${\cal F}(2H)={\cal F}(-K_S)$ is generated by global sections by the Castelnuovo-Mumford lemma. Condition (b) now follows from Bertini's theorem. \TeXButton{qed}{\hfill $\Box$ \medskip} \section{Proofs of the Theorems} \paragraph{Proof of Theorems \ref{ruled} and \ref{P2}.} Theorem \ref{ruled} follows from the classification of the prioritary components of ${\rm TF}_S(2,c_1,c_2)$ in Lemma \ref{discquad} and the classification of the nonprioritary components of ${\rm TF}_S(2,c_1,c_2)$ in Lemma \ref{nonprior}. Note that the expression $\chi ({\cal O}% _S(-c_1))+D(2D-2c_1-K_S)$ appearing the Theorem \ref{ruled} is equal to the expression $\chi ({\cal O}_S(2D-c_1))$ appearing in Lemma \ref{nonprior} by a simple application of the Riemann-Roch formula for a line bundle on $S$. Theorem \ref{P2} follows from Lemmas \ref{discp2} and \ref{nonprior} in the same manner. \paragraph{Proof of Theorem \ref{BN}.} According to the argument given before the statement of Theorem \ref{BN} there is a correspondence between irreducible components of $W_N^0(E)$ correspond to the irreducible components of ${\rm TF}_S(2,E-K_S,N)$ whose general member ${\cal E}$ satisfies $H^1({\cal E}(K_S))=H^2({\cal E}(K_S))=0$ and has a section with zero locus of codimension $2$. These irreducible components of ${\rm TF}_S(2,E-K_S,N)$ may either be nonprioritary or prioritary. According to Theorems \ref{ruled} and \ref{P2} the nonprioritary components of ${\rm TF}_S(2,E-K_S,N)$ correspond to pairs $(D,n)\in {\rm NS}% (S)\times {\bf Z}$ such that $Df\leq -1+\frac 12(E-K_S)f$ and $0\leq n\leq N+D(D-E+K_S)$ and $N\leq \chi ({\cal O}_S(-E+K_S))+D(D-E)=\chi ({\cal O}% _S(E))-D(E-D)$. According to Lemma \ref{final} the general member of such an irreducible component has $H^1({\cal E}(K_S))=H^2({\cal E}(K_S))=0$ and a section with zero locus of codimension $2$ if and only if (i) $\chi ({\cal E}% (K_S))\geq 0$, (ii) $D$ is an effective and irreducible divisor class, and (iii) $n_2=N+D(D-E+K_S)-n\leq \chi ({\cal O}_S(D+K_S))$. Since $\chi ({\cal E% }(K_S))=\chi ({\cal I}_X(E))+1=\chi ({\cal O}_S(E))-N+1>0$, we see that the irreducible components of $W_N^0(E)$ with nonprioritary ${\cal E}$ are precisely the components described in part (i) of Theorem \ref{BN}. Moreover, the geometry of $X$ can be recovered from ${\cal E}$, $D$ and $n$ via the diagram% $$ \begin{array}{ccccccc} & & & 0 & & 0 & \\ & & & \downarrow & & \downarrow & \\ & & & {\cal O}_S & = & {\cal O}_S & \\ & & & \downarrow & & \downarrow & \\ 0\rightarrow & {\cal I}_{Z_1}(E-K_S-D) & \rightarrow & {\cal E} & \rightarrow & {\cal I}_{Z_2}(D) & \rightarrow 0 \\ & \parallel & & \downarrow & & \downarrow & \\ 0\rightarrow & {\cal I}_{Z_1}(E-K_S-D) & \rightarrow & {\cal I}_X(E-K_S) & \rightarrow & {\cal K} & \rightarrow 0 \\ & & & \downarrow & & \downarrow & \\ & & & 0 & & 0 & \end{array} $$ The bottom row must be a twist of the residual exact sequence for ${\cal I}% _X(E-K_S)$ with respect to a curve $C\in |D|$. So ${\cal K}={\cal I}_{X\cap C/C}(E-K_S)$. Thus $X=Z_1\cup (X\cap C)$ with $Z_1$ a generic set of $n$ points of $S$ and $X\cap C$ a set of $N-n$ points on $C$. Thus the irreducible components of $W_N^0(E)$ such that ${\cal E}$ is nonprioritary are exactly those described in part (i) of Theorem \ref{BN}. In addition ${\rm TF}_S(2,E-K_S,N)$ may have a unique prioritary component. For ${\bf P}^2$ this component exists if and only if $N\geq \frac 14(e+4)(e+2)$ by Theorem \ref{P2}. Its general member ${\cal E}$ always satisfies $H^1({\cal E}(K_S))=H^2({\cal E}(K_S))=0$ and has a section with zero locus of codimension $2$ according to Lemma \ref{goodp2} because $\chi (% {\cal E}(K_S))=\chi ({\cal O}_S(E))-N+1>0$. If $S$ is ${\bf P}^1\times {\bf P% }^1$ and $e_2$ is even, the prioritary component exists if and only if $% N\geq \frac 12(e_1+2)(e_2+2)$ according to Theorem \ref{ruled}, and its general member always satisfies $H^1({\cal E}(K_S))=H^2({\cal E}(K_S))=0$ and has a section with zero locus of codimension $2$ according to Lemma \ref {goodquad} because $\chi ({\cal E}(K_S))>0$. If $S$ is ${\bf P}^1\times {\bf % P}^1$ and $e_2$ is odd, then the prioritary component exists for all $N$ according to Theorem \ref{ruled}. But according to Lemma \ref{goodquad} its general member ${\cal E}$ satisfies $H^1({\cal E}(K_S))=H^2({\cal E}(K_S))=0$ and has a section with zero locus of codimension $2$ only if either $N\geq \frac 12(e_1+2)(e_2+1)+1$ or $(e_1,e_2,N)=(e_1,1,e_1+2)$. This gives all the components described in part (ii) of Theorem \ref{BN}. The dimension of a component of $W_N^0(E)$ is the dimension of the corresponding component of ${\rm TF}_S(2,E-K_S,N)$ plus $h^0({\cal E})-1$ (for the choice of a section of ${\cal E}$ modulo $k^{\times }$) plus $1$ (to cancel the negative contribution of $\dim {\rm Aut}({\cal I}_X(E))$ in the stack computations). Hence the prioritary components have dimension $% -\chi ({\cal E},{\cal E})+\chi ({\cal E})$ which a straightforward Riemann-Roch computation shows is $2N-(\chi ({\cal O}_S(E)-N+1)$. Since $% \dim {\rm Hilb}^N(S)=2N$, this is the asserted codimension $(\chi ({\cal O}% _S(E)-N+1)$. The nonprioritary components of ${\rm TF}_S(2,E-K_S,N)$ have dimensions greater by $\chi ({\cal O}_S(E))+D(D-E)-N$. So the nonprioritary components of $W_N^0(E)$ have codimensions $D(E-D)+1$. \TeXButton{qed} {\hfill $\Box$ \medskip}

9312

alg-geom/9312004

en

https://arxiv.org/abs/alg-geom/9312004

alg-geom/9312004

Alexander Polischuk

A. Polishchuk

On Koszul property of the homogeneous coordinate ring of a curve

17 pages, Latex

The following corollary has been added: for general tetragonal curve $C$ of genus $g\ge 9$ the homogeneous coordinate ring of $C$ defined by the line bundle $K(-T)$, where $K$ is the canonical class, $T$ is the tetragonal series, is Koszul. Also some misprints are corrected.

alg-geom

\section{Introduction} This paper is devoted to Koszul property of the homogeneous coordinate algebra of a smooth complex algebraic curve in the projective space (the notion of a Koszul algebra is some homological refinement of the notion of a quadratic algebra, for precise definition see next section). It grew out from the attempt to understand methods of M. Finkelberg and A. Vishik in their paper \cite{FV} proving this property for the canonical algebra of a curve in the case it is quadratic. The basic ingredient of their proof is the following lemma on special divisors. \begin{lem}{\rm (\cite{GL})} Let $C$ be a non-hyperelliptic non-trigonal curve which is not a plane quintic. Then there exists a divisor $D$ of degree $g-1$ on $C$ such that $|D|$ and $|K(-D)|$ are base-point free linear systems of dimension 1 where $K$ is the canonical class. \end{lem} In the cited paper of M. Green and R. Lazarsfeld it is used for the vector bundle proof of Petri's theorem which asserts that if $C$ satisfies the conditions of this lemma then the canonical algebra of $C$ is quadratic. In this note we will show that Koszul property can be derived from this lemma by purely homological technique combined with a simple statement concerning Koszul property of the homogeneous coordinate algebra of a finite set of points in the projective space --- in particular we obtain a new proof of Petri's theorem. It turns out that the same technique works for the proof of Koszul property of the homogeneous coordinate ring of a curve of genus $g$ embedded by complete linear system of degree $\ge 2g+2$ (this result is due to D. Butler \cite{B}) and also for embeddings defined by the complement to some very special linear systems (the simplest case being that of tetragonal systems) in the canonical class. In particular, we prove Koszul property for a general tetragonal curve of genus $g\ge 9$ embedded by $K(-T)$ where $T$ is the tetragonal series. It is worthy to mention here that if $C$ is a non-hyperelliptic curve and $L$ is a linear bundle of degree $2g+1$ such that $H^0(L\otimes K^{-1})=0$ then the coordinate algebra of $C$ in the embedding defined by $L$ is quadratic (see e.g. \cite{L}). It is still unknown (at least to me) whether it is Koszul or not. It seems that the following general question is also open: whether the homogeneous coordinate algebra of a projectively normal smooth connected complex curve is Koszul provided it is quadratic? To illustrate our technique we present here the proof of Petri's theorem based on the lemma mentioned above. \begin{thm} Assume that there exists a divisor $D$ on $C$ of degree $g-1$ such that $|D|$ and $|K(-D)|$ are base-point-free linear series of dimension 1. Then the canonical algebra $R$ is quadratic. \end{thm} \noindent {\it Proof} . Choose the divisor $P_1+\ldots+P_{g-1}$ in the linear system $|D|$ such that these $g-1$ points are distinct (then they are in general linear position in the $(g-3)$-dimensional space they span). Let $V=H^0(K(-D))\subset H^0(K)=R_1$ be the corresponding 2-dimensional subspace. Denote the symmetric algebra $Sym(R_1)$ by $S$ and let $J\subset S$ be the homogeneous ideal of $C$ so that $R=S/J$ by Noether's theorem (obviously $C$ is non-hyperelliptic ). Consider the following exact sequence $$0\rightarrow VS\cap J\rightarrow J\rightarrow J/(VS\cap J)\rightarrow 0$$ where $VS$ is the ideal in $S$ generated by $V$. It is easy to see that it is enough to check the following two statements: \begin{enumerate} \item $J/(VS\cap J)$ is generated over $S$ by elements of degree 2; \item $VS\cap J$ is generated over $S$ by elements of degree 2 modulo $VJ$. \end{enumerate} We will use the following lemma. \begin{lem} Let $|D|$ be a base-point-free linear system of dimension 1. Then the natural homomorphism $H^0(D)\otimes H^0(K^n)\rightarrow H^0(K^n(D))$ is surjective for $n\ge 0$. \end{lem} The proof is left to reader. Now for the proof of 1) we claim that $VS+J$ is exactly the homogeneous ideal of the set of points $P_1,\ldots, P_{g-1}$ so the statement follows from the quadratic property of an algebra of $g-1$ points in general position in $\P^{g-3}$ (see sect. 3 for more general result in this direction). Indeed let $A$ be the coordinate algebra of these points. Then $A$ is the factoralgebra of $R$ by the ideal $I=\oplus_{n\ge1}H^0(K^n(-D))$. Now it follows from the above lemma that $I$ is generated by $I_1=V$ over $R$ so $A=R/(VR)=S/(VS+J)$ and our assertion follows. It remains to check 2). For this consider the multiplication map \\ $\mu_S:V\otimes S(-1)\rightarrow S$. It is enough to verify that $\mu_S^{-1}(J)/(V\otimes J(-1))$ is generated over $S$ by elements of degree 2. We can rewrite this as follows: let $\mu_R:V\otimes R(-1)\rightarrow R$ be another multiplication map then ker$\mu_R$ is generated by elements of degree 2 over $R$. But this kernel is equal to $\oplus_{n\ge2}H^0(K^{n-2}(D))$ (by base-point-pencil trick applied to $K(-D)$) so we are done by the lemma above. \ \vrule width2mm height3mm \vspace{3mm} {\it Acknowledgments.} I am grateful to A. Bondal, L. Positselsky and A. Vishik for stimulating discussions. Also I thank J. Kollar and University of Utah for their hospitality while carrying out this research. \section{Some homological algebra} In this section we prove a general criterion for Koszul property of a graded algebra given its Koszul factoralgebra and some information about this factoralgabra as a module over original algebra. Recall that graded (associative but not necessary commutative) algebra $A=A_0\oplus A_1\oplus\ldots$ over the field $k$ is called Koszul if $A_0=k$ and ${\rm Ext}^n(k,k(-m))=0$ for $n\neq m$. Here functor Ext is taken in the category of graded (left) $A$-modules, $k$ is trivial $A$-module concentrated in degree 0, for graded $A$-module $M=\oplus M_i$ we define the shifted module as $M(l)=\oplus M_{i+l}$. Note that the above condition on Ext's for $n=1$ means that $A$ is generated by elements of degree 1, for $n=1$ and $n=2$ - that $A$ is quadratic (i.e. in addition it has defining relations of degree 2). For equivalent definitions via exactness of Koszul complex and distributivity of some lattices see \cite{P},\cite{Ba1},\cite{BGS}. The following theorem is the generalization of lemma 7.5 in \cite{BP} where it was proved in the particular case when $A=k$. \begin{thm} Let $R$ be graded algebra over $k$ with $R_0=k$ and $A$ -- its Koszul (graded) factoralgebra. Assume that there exists a complex $K^{\cdot}$ of free right $R$-modules of the form $$\ldots\rightarrow V_2\otimes R(-2)\rightarrow V_1\otimes R(-1)\rightarrow R$$ (so $K^i=V^i\otimes R(-i)$ where $V^i$ are finite-dimensional $k$-linear spaces, $V^0=k$) such that $H_0(K^.)=A$, $H_p(K^.)_j=0$ for $p\ge 1$, $j>p+1$. Then $R$ is Koszul and $A$ has a linear free resolution as $R$-module that is a resolution of the form $$\ldots\rightarrow U_2\otimes R(-2)\rightarrow U_1\otimes R(-1)\rightarrow R\rightarrow A\rightarrow 0$$ \end{thm} \noindent {\it Proof} . Note that the conditions on the homology $H_p(K^{\cdot})$ for $p\ge 1$ in formulation of the theorem mean that $H_p(K^{\cdot})$ as $R$-module is an extension of some multiples of trivial modules $k(-p)$ and $k(-p-1)$. Indeed this follows directly from the fact that $H_p(K^{\cdot})_j=0$ for $j<p$ which is evident. Now we prove by induction on $n\ge 0$ that $Ext_R^n(k,k(-m))=0$ if $m\neq n$. Case $n=0$ is trivial. Assume that $n>0$, ${\rm Ext}_R^i(k,k(-m))=0$ for $i<n$, $m\neq i$ and let us prove the assertion for $n$. Considering the spectral sequence $$E_2^{p,q}={\rm Ext}_A^q(Tor_p^R(A,k),k(-m))\Rightarrow{\rm Ext}_R^{p+q}(k,k(-m))$$ it is easy to see that it is enough to prove that ${\rm Tor}_j^R(A,k)_i=0$ if $i\neq j$, $j\le n$ (here we use Koszul property of $A$). The latter is equivalent to ${\rm Ext}_R^j(A,k(-i))=0$ for $i\neq j$, $j\le n$ (here ${\rm Ext}$ is taken in the category of right $R$-modules). Consider the spectral sequence associated with complex $K^{\cdot}$ and cohomological functor ${\rm Hom}_{D^-(R)}(.,k(-m))$ where $D^-(R)$ is the derived category of complexes bounded from the right: $$E_2^{p,q}={\rm Ext}_R^q(H_p(K^{\cdot}),k(-m))\Rightarrow {\rm Hom}^{p+q}_{D^-(R)}(K^{\cdot},k(-m))$$ The limit on the right can be computed easily with help of another spectral sequence with $E_1$ obtained by applying the same functor to the terms of the complex $K^{\cdot}$. Due to the form of this complex it degenerates at $E_1$ which gives an equality ${\rm Hom}^n(K^{\cdot},k(-m))=0$ if $n\neq m$. On the other side for $p\ge1$ the group $H_p(K^{\cdot})$ is an extension of the direct sums of several copies of $k(-p)$ and $k(-p-1)$ and we obtain by assumption that $E_2^{p,q}={\rm Ext}_R^q(H_p(K^{\cdot}),k(-m))=0$ if $p\ge1$, $q<n$, $m\neq p+q$, $m\neq p+q+1$. This implies that the terms $E_2^{0,q}$ must survive and contribute to $E_{\infty}$ if $q\le n$, $q\le m-1$. Indeed all the differentials $d_r: E_r^{r-1,q-r}\rightarrow E_r^{0,q}$ are zero for $r\ge2$, $q\le {\rm min}(n,m-1)$ because $E_r^{r-1,q-r}$ is zero for these values of $r$ and $q$. Therefore our computation of the limit implies an equality $E_2^{0,q}={\rm Ext}_R^q(A,k(-m))=0$ for $q\le{\rm min}(n,m-1)$. On the other side obviously ${\rm Ext}_R^q(A,k(-m))=0$ for $q>m$ so we are done. \ \vrule width2mm height3mm \vspace{3mm} \section{Application to the coordinate ring of a curve} Now we are going to apply the theorem of the previous section to the homogeneous coordinate algebra of a curve. So let $C$ be a curve, $L$ be a very ample linear bundle on $C$. We are interested in the algebra $R=R_L=\\ =\oplus_{n\ge0}H^0(L^n)$. Our approach is as follows: we consider the factoralgebra $A$ of $R$ associated with some effective divisor $D$ on $C$ namely $A=A_D=R/J_D$ where $J_D=\oplus_{n\ge1}H^0(L^n(-D))$ is an ideal in $R$. To apply the above theorem to this situation we have to construct a complex of free $R$-modules which is an "almost resolution" of $A$ and has the required form, and to check Koszul property of the algebra $A$. The latter is simple provided that points of the divisor $D$ are "sufficiently linear independent" in the embedding defined by $L$. To construct the desired complex we assume that $\O(D)$ and $L(-D)$ are base-point-free so that we have the following exact triples: $$0\rightarrow \O(-D)\rightarrow V\otimes \O\rightarrow \O(D)\ra0$$ $$0\rightarrow L^{-1}(D)\rightarrow U\otimes\O\rightarrow L(-D)\ra0$$ where $V$ and $U$ are some vector space of dimension 2. Now the complex $K$ has the following form: $$\ldots \rightarrow U\otimes R(-3)\stackrel{d_3}{\rightarrow} V\otimes R(-2)\stackrel{d_2}{\rightarrow} U\otimes R(-1)\stackrel{d_1}{\rightarrow} R$$ where $d_1$ is induced by the composition of maps $$U\rightarrow H^0(L(-D))\rightarrow H^0(L)=R_1,$$ the $n$-th component of $d_{2k}$ is the composition $$V\otimes H^0(L^{n-2k})\rightarrow H^0(L^{n-2k}(D))\rightarrow U\otimes H^0(L^{n-2k+1}),$$ and that of $d_{2k+1}$ is the composition $$U\otimes H^0(L^{n-2k-1})\rightarrow H^0(L^{n-2k}(-D))\rightarrow V\otimes H^0(L^{n-2k})$$ Using the exact triples above one can compute easily the homology of $K$. Indeed $${\rm ker}(d_{2k+1})_n={\rm ker}(U\otimes H^0(L^{n-2k-1})\rightarrow H^0(L^{n-2k}(-D)))\simeq H^0(L^{n-2k-2}(D))$$ Therefore $$H_{2k+1}(K)_n\simeq{\rm coker}(V\otimes H^0(L^{n-2k-2})\rightarrow H^0(L^{n-2k-2}(D)))$$ $$\simeq{\rm ker}(H^1(L^{n-2k-2}(-D))\rightarrow V\otimes H^1(L^{n-2k-2})$$ It follows that if $H^1(L^2(-D))=0$ then $H_{2k+1}(K)_n=0$ for $n\ge 2k+4$. Futhermore if the map $\alpha:H^1(L(-D))\rightarrow V\otimes H^1(L)$ is injective then $H_{2k+1}(K)_{2k+3}=0$ as well. In analogous way we obtain that if $H^1(L(D))=0$ and $\beta:H^1(D)\rightarrow U\otimes H^1(L)$ is injective then $H_{2k}(K)_{\ge 2k+2}=0$. Note that the condition $H^1(L(D))=0$ implies surjectivity of $\alpha$ so in this case injectivity is equivalent to the equality of dimensions: $h^1(L(-D))=2h^1(L)$. Analogously if $H^1(L^2(-D))=0$ then injectivity of $\beta$ is equivalent to equality $h^1(D)=2h^1(L)$. It easy to see that if all these conditions hold and in addition $h^0(L(-D))=2$ that is $U\simeq H^0(L(-D))$, then image of $d_1$ is equal to the ideal $J_D$; thus all the homological conditions of Theorem 1 are satisfied except for Koszul property of an algebra $A$. At this step we use the following result of G. Kempf. \begin{thm}{\rm (\cite{K})} Homogeneous coordinate algebra of the finite set of $d$ distinct points in general linear position in $\P^{d-p}$ is Koszul provided that $p\le d/2$. \end{thm} \begin{rem} In the case $p\le 3$ this is particularly easy: the points lie on a rational normal curve (see \cite{GH}) so the statement can be deduced easily from the main theorem of \cite{Ba2} (see also \cite{ERT}). Note also that if $p=1$ (resp. $p=2$) then the set of points in question is a hyperplane section of a rational (resp. an elliptic) normal curve so the Koszul property in these two cases follows from the same property of the coordinate algebra of a rational (resp. an elliptic) normal curve. So we can avoid the reference to the results above if we are interested in the case $h^1(L)\le 1$ only. \end{rem} Now we are ready to prove our main theorem. \begin{thm} Let $L$ be a very ample linear bundle on $C$ of degree \\ ${\rm deg}L\ge g+3$ such that corresponding embedding of $C$ into $\P(H^0(L)^*)$ is projectively normal. Assume that there exists a divisor $D=P_1+\ldots+P_d$ of the degree $d={\rm deg}L-g-1+2h^1(L)$ such that the linear series $|D|$ and $|L(-D)|$ are base-point free of the dimensions ${\rm deg}L-2g+4h^1(L)-1$ and 1 correspondingly. Assume also that $h^1(L(D))=h^1(L^2(-D))=0$ and that any $h^1(L)$ points of $D$ impose independent conditions on $K\otimes L^{-1}(D)$. Then algebra $R_L$ is Koszul. \end{thm} \noindent {\it Proof} . By Riemann-Roch we obtain that $h^1(D)=h^1(L(-D))=2h^1(L)$ so it follows from the discussion above that we only have to check Koszul property of the set of points ${P_1,\ldots,P_d}$ embedded by $L$. The dimension of the linear subspace spanned by these points is equal to $h^0(L)-3=d-h^1(L)-1$. The condition $p=h^1(L)+1\le d/2$ is satisfied by assumption so by Theorem 5 it is sufficient to verify that any $d-h^1(L)$ of $P_1, \ldots, P_d$ impose independent conditions on $|L|$. But this is equivalent to the property we have assumed that any $h^1(L)$ of them impose independent conditions on $K\otimes L^{-1}(D)$. \ \vrule width2mm height3mm \vspace{3mm} \begin{rem} The condition of the theorem concerning independence of any $h^1(L)$ points is satisfied automatically if $h^1(L)\le 1$. If $h^1(L)=2$ and ${\rm deg}L=2g-6$ so that the dimension of $|D|$ is 1 then the sufficient condition is that $K\otimes L^{-1}(2D)$ is very ample. \end{rem} \begin{cor} If $C$ is a non-hyperelliptic, non-trigonal curve which is not a plane quintic then the canonical algebra $R_K$ is Koszul. \end{cor} \noindent {\it Proof} . It follows from the Green-Lazarsfeld's lemma and the theorem above. \ \vrule width2mm height3mm \vspace{3mm} \begin{cor} For any curve of genus $g$ and any linear bundle $L$ of degree $\ge 2g+2$ algebra $R_L$ is Koszul. \end{cor} \noindent {\it Proof} . In this case $h^1(L)=0$ and we can choose $D$ as above in the linear system $L(-P_1-\ldots -P_{g+1})$ for general $g+1$ points $P_1,\ldots ,P_{g+1}$. Note also that Koszul property in this case follows trivially from Kempf's Theorem applied to a general hyperplane section of $C$. \ \vrule width2mm height3mm \vspace{3mm} \begin{cor} Under the assumptions of the theorem the following natural map is surjective: $$H^0(L)\otimes H^0(D)\rightarrow H^0(L(D))$$ Also if we define a vector bundle $M_D$ from the exact triple $$0\rightarrow M_D\rightarrow H^0(D)\otimes \O\rightarrow \O(D)\rightarrow 0$$ then $M_D\otimes L$ is generated by global sections. \end{cor} \noindent {\it Proof} . Both statements follow from the second part of Theorem 4. \ \vrule width2mm height3mm \vspace{3mm} It remains to analyze the case $h^1(L)\ge 2$ of our theorem. Put $L=K(-A)$, dim$|A|=r$. Then the condition $h^0(D)\ge 2$ implies the following inequality: deg$A\le 4r$. On the other hand considering the natural map $$|A|\times |D|\rightarrow |K\otimes L^{-1}(D)|$$ we obtain another inequality: $$r+h^0(D)-1\leq h^1(L(-D))-1$$ which is equivalent to deg$A\ge 3r$. Also considering the map $$|D|\times|L(-D)|\rightarrow |L|$$ we obtain the restriction $r\le g/3-1$. In the case $r=1$ we obtain from the above inequalities that $|A|$ is either trigonal or tetragonal system. Furthermore it is easy to see that the former case is impossible so $A=T$ is a tetragonal system. One can check that $T$ should be base-point-free otherwise $C$ fails to be cut out (even set-theoretical) by quadrics in the embedding defined by $|L|$. Also the fact that $L$ is very ample implies that $C$ is not hyperelliptic. To satisfy the conditions of the theorem we should have a decomposition of $L$ into the sum of two divisors of degree $g-3$ each defining the base-point free linear system of dimension 1. In the next section we will give some examples when this situation occurs. \begin{rem} So far I don't know much about the case $h^0(L)>2$. I hope that there should be examples when the above technique applies to this case too. \end{rem} \section{Tetragonal curves} In this section we study the case of the embedding of a tetragonal curve $C$ by the complete linear system $|K(-T)|$ where $T$ is a (base-point-free) tetragonal series. As a tetragonal series is not unique in general it is natural to consider the moduli space ${\cal M}_g^t$ of pairs $(C, T)$ where $T$ is such a series on a non-hyperelliptic curve $C$ of genus $g$. By the well-known construction (see \cite{S}) $T$ gives an embedding of $C$ into 3-dimensional rational normal scroll $X$. Furthermore it is easy to see that $C$ is a complete intersection of two divisors on $X$. Thus there is a stratification of ${\cal M}_g^t$ by the type of scroll $X$ and the type of complete intersection, and all the strata are irreducible {}. Note that the Hilbert series of $R_{K(-T)}$ is constant over ${\cal M}_g^t$ so by the well-known result the set of pairs $(C,T)$ for which $R_{K(-T)}$ is Koszul is an intersection of countably many open subsets in ${\cal M}_g^t$. The problem to be solved is to find all the strata ${\cal N}$ such that for general pair $(C,T)\in {\cal N}$ the algebra $R_{K(-T)}$ is Koszul (we say that such a stratum is Koszul). Here "general" means "in the complement of countably many proper subvarieties" so a stratum satisfies this property if it contains at least one such pair. Note that if a stratum ${\cal N}_1$ is Koszul and it is contained in the closure of a stratum ${\cal N}_2$ then ${\cal N}_2$ is also Koszul. The analogous problem for quadratic algebras is easy and we will see that the most degenerate (with respect to complete intersection type) quadratic strata are Koszul, so it is natural to expect that this is true in general. Also we construct examples of pairs $(C,T)$, for which algebra $R_{K(-T)}$ is Koszul, on some other strata cosidering ramified double coverings of hyperelliptic curves and applying the method of the previous section. As an evident consequence we obtain that for general tetragonal curve of genus $g\ge 9$ algebra $R_{K(-T)}$ is Koszul (note that for general tetragonal curve $T$ is unique). We begin with recalling the construction of a 3-dimensional rational normal scroll containing tetragonal curve $C$. Let $T$ be a base-point free tetragonal system on $C$. It defines a 4-sheeted covering $\pi:C\rightarrow\P^1$. Applying the relative duality to the canonical non-vanishing section $\O\rightarrow \pi_*\O$ we obtain the surjective homomorphism $\pi_*K_C\rightarrow\O(-2)$. Let $V$ be its kernel so that we have the following exact triple on $\P^1$: $$0\rightarrow V\rightarrow \pi_*K_C\rightarrow \O(-2)\rightarrow 0$$ It follows that the homomorphism $V\rightarrow \pi_*K_C$ induces an isomorphism of global sections and therefore the corresponding homomorphism $\pi^*V\rightarrow K_C$ is surjective. Thus we obtain a morphism $\phi:C\rightarrow X$ where $X=\P(V^{\vee})$ such that $\phi^*\O_X(1)\simeq K_C$. Note that it follows from the exact triple above that $h^0(V(-i))=h^0(K_C(-iT))$. In particular $h^0(V)=g$, $h^0(V(-1))=g-3$ so that $V\ge0$ in the sense that all linear direct summands in $V$ has form $\O(l)$ with $l\ge0$. Therefore $\O_X(1)$ is base-point free and defines a morphism from $X$ to $\P^{g-1}$ inducing the canonical morphism by composition with $\phi$. It follows that $\phi$ is an embedding (we have assumed that $C$ is non-hyperelliptic). Furthermore $h^0(V(-2))=g-6$ if and only if $h^0(2T)=3$ and assuming that we obtain that $V(-1)\ge0$ and hence $\O_X(1)$ is very ample. Let $p:X\rightarrow \P^1$ be the projection. The push forward by $p$ of the natural homomorphism $\O(2)\rightarrow \O(2)|_C$ gives rise to a homomorphism $f:S^2V\rightarrow\\ \rightarrow \pi_*K_C^2$ where $S^2V\simeq p_*\O(2)$ is the second symmetric power of $V$. We claim that $f$ is surjective. Indeed it suffices to check this pointwise so we have to verify that for each $q\in \P^1$ the four points of $\pi^{-1}(q)$ impose independent conditions on quadrics in the corresponding projective plane $p^{-1}(q)$. But this is evidently true because these points span that plane by the geometric form of the Riemann-Roch Theorem. Let us denote the kernel of $f$ by $E$. This is a rank-2 vector bundle on $\P^1$ of degree $g-5$ and it fits in the following exact sequence $$0\rightarrow E\rightarrow S^2V\rightarrow \pi_*K^2\ra0$$ Now we claim that the corresponding homomorphism $p^*E\rightarrow \O_X(2)$ vanishes exactly along $C\in X$. Indeed as any divisor in $|T|$ contains four points no three of which lie on a line they are cut out by two quadrics in the corresponding plane. More than that, comparing arithmetical genera we conclude that $C$ as a subscheme of $X$ coincides with the zero-locus of the corresponding regular section of $p^*E^{\vee}(2)$. Now $E\simeq \O(a)\oplus\O(b)$ where $a+b=g-5$ so $C$ is in fact a complete intersection of two divisors $S_1\in|\O_X(2)(-aH)|$ and $S_2\in|\O_X(2)(-bH)|$ where $H=p^*\O(1)$. Assume now that $K_C(-T)$ is projectively normal. Then the homomorphism $S^2H^0(K_C(-T))\rightarrow H^0(K_C^2)$ is surjective. As $V(-1)\ge0$ it follows that the natural homomorphism $S^2H^0(V(-1))\rightarrow H^0(S^2V(-2))$ is surjective too. Hence the homomorphism $S^2V(-2)\rightarrow \pi_*(K_C^2)(-2)$ induces a surjection on global sections and consequently $H^1(E(-2))=0$ that is $a,b\ge1$. Conversely it easy to see that if $C$ is a complete intersection as above with $a,b\ge1$ then $K_C(-T)=(\O_X(1)(-H))|_C$ is projectively normal. We fix these results in the following proposition. \begin{prop} Let $C$ be a non-hyperelliptic curve of genus $g$ with a base-point free tetragonal system $T$ on it. Then we can present $C$ as a complete intersections of two divisors from the linear systems $|\O_X(2)(-aH))|$ and $|\O_X(2)(-bH)|$ on a 3-dimensional rational normal scroll $X=\P(V^{\vee})$, where $V\ge 0$ is rk-3 vector bundle of degree $g-3$ on $\P^1$, in such a way that $T=H|_C$ (here $H$ is the pull-back of $\O(1)$ from $\P^1$, $a+b=g-5$). Furthermore $h^0(2T)=3$ if and only if $V(-1)\ge0$ and if this condition is satisfied then $K_C(-T)$ is projectively normal if and only if $a\ge1, b\ge1$. \end{prop} \begin{rem} Note that if $K_C(-T)$ is very ample then $a,b\ge0$. In any case $a,b\ge -1$. \end{rem} Our next remark is that under notations of the previous proposition $R_{K_C(-T)}$ is quadratic if and only if $a,b\ge 2$ (this is very easy to verify using exact sequences of the restriction to a divisor). So it makes reasonable the following conjecture. \begin{conj} Under assumptions and notations above if $a,b\ge 2$ then the algebra $R_{K_C(-T)}$ is Koszul . \end{conj} \begin{rem} It is easy to see that this is true at least when $a=2$ or $b=2$. Indeed first we can prove that the coordinate algebra of any divisor $S\in |\O(2)(-aH)|$ under the embedding defined by $\O(1)(-H)$ is Koszul provided that $a\ge 2$. The reason is that its hyperplane section is a curve of genus $g-5-a$ embedded by the complete linear system of degree $2g-10-a$ so we can apply the corollary 2 above. Then our curve $C$ is an intersection of $S$ with a quadric so its coordinate algebra is Koszul by the result of \cite{BF}. This proves the conjecture for $g=9,10$. To prove the Koszul property for general pair $(C,T)$ of some stratum with $a,b\ge 2$ it would be sufficient to prove that this stratum can be degenerated into one with $a=2$ or $b=2$. \end{rem} Our method of proving Koszul property suggests the following conjecture which implies the previous one. \begin{conj} Under the same assumptions there exists a decomposition of $K_C(-T)$ into the sum of two base-point free pencils of degree $g-3$. \end{conj} Now we consider the specific case when our tetragonal curve $C$ is a double covering of a hyperelliptic curve. We are going to construct some examples when the required decomposition of $K_C(-T)$ into the sum of two pencils exists. For this we will use the well-known connection between linear bundles over the double covering and rk-2 bundles with a Higgs field (which is a twisted endomorphism of a bundle) over the base of the covering (see for example \cite{Hi}). So let $C_h$ be the hyperelliptic curve of genus $g_h$, $\pi_h:C_h\rightarrow\P^1$ be the corresponding double covering, ${\Gamma}=\pi_h^*\O(1)$ be the hyperelliptic system. Then $K_h=\O((g_h-1){\Gamma})$ is the canonical class of $C_h$. Now we consider a Higgs bundle $(F,\phi)$ where $F=\O(D_1)\oplus \O(D_2)$, $D_i$ being the divisors of degree $g_h-2$ and $\phi:F\rightarrow F(M)$ is a homomorphism defined by the sections $s_1\in H^0(M(D_2-D_1))$ and $s_2\in H^0(M(D_1-D_2))$ (other enties of $\phi$ being zero) -- here M is some linear bundle which has sufficiently large degree to be estimate later. These data define the line bundle $\O(D)$ over the double covering $\pi:C\rightarrow C_h$ such that $\pi_*\O(D)\simeq F$. Simple computation shows that deg$D=g-3$ where $g=2g_h-1+{\rm deg}M$ is the genus of $C$. Now we look under what conditions $|D|$ is a base-point-free linear system of dimension 1. First we should have that $F$ is globally generated outside the ramification divisor $s=s_1s_2\in H^0(M^2)$. Hence $h^0(D_1)=h^0(D_2)=1$ and the unique divisor of $|D_i|$ is contained in the zero divizor of $s$. Furthermore at the point $x$ of ramification global sections of $F$ should generate the unique $\phi$-invariant 1-dimensional factorspace of the stalk $F_x$. It is easy to see that these conditions are satisfied if we put $s_i=u_it_i$ where $u_i\in H^0(D_i)$ are non-zero sections, $(i=1,2)$, $t_1\in H^0(M(D_2-2D_1))$, $t_2\in H^0(M(D_1-2D_2))$ provided that zero divisors of $u_1, u_2, t_1, t_2$ are all disjoint. Note that the change of $D$ by $K_C(-T-D)$ where $T=\pi^*{\Gamma}$ leads to the change of $D_i$ by $(g_h-2){\Gamma}-D_i$ with essentially the same Higgs field. In particular if we choose $D_2=(g_h-2){\Gamma}-D_1$ then we will have $\O(2D)\simeq K_C(-T)$. Now it is clear that if $M$ is a general linear bundle of degree at least $2g_h-1$ then we can find $(F,\phi)$ as above such that corresponding divisor $D$ on $C$ satisfies the conditions of Theorem 2. Indeed then deg$M(D_2-2D_1)\ge g_h+1$ and we can use the fact that general bundle of degree $\ge g_h+1$ is base-point free. Also as we have mentioned above under suitable choices we'll have $\O(2D+T)\simeq K_C$ which implies the last condition of Theorem 2 (see remark after it). Now in order to verify projective normality of $K_C(-T)$ we have to compute the discrete invariants described above (namely the bundles $V$ and $E$ on $\P^1$) of the pair $(C,T)$ obtained in this way. First we have the canonically splitting exact triple $$0\rightarrow K_h\otimes M\rightarrow \pi_*K_C\rightarrow K_h\rightarrow 0$$ Now $V$ is the kernel of composition $$(\pi_h)_*(\pi_*K_C)\rightarrow(\pi_h)_*K_h\rightarrow \O(-2)$$ It follows that $V$ fits into the splitting exact sequence $$0\rightarrow (\pi_h)_*M(g_h-1)\rightarrow V\rightarrow \O(g_h-1)\rightarrow 0$$ Now it is easy to check that $(\pi_h)_*\pi_*(K_C^2)\simeq((\pi_h)_*(M^2)\oplus (\pi_h)_*M)(2g_h-2)$ and the natural map $$ S^2((\pi_h)_*M\oplus \O)\simeq S^2V(-2g_h+2)\rightarrow (\pi_h)_*\pi_*(K_C^2)(-2g_h+2)\simeq$$ $$ \simeq (\pi_h)_*(M^2)\oplus(\pi_h)_*M$$ is induced by the natural maps $\psi:S^2((\pi_h)_*M)\rightarrow (\pi_h)_*(M^2)$, $\O\rightarrow (\pi_h)_*(M^2)$ and the identity map of $(\pi_h)_*M$. It follows that there is an exact sequence $$0\rightarrow {\rm ker}\psi \rightarrow E(-2g_h+2)\rightarrow \O\rightarrow 0$$ provided that $\psi$ is surjective. Assume that deg$M\ge 2g_h+1$ then $\psi$ is evidently surjective and ker$\psi\simeq\O({\rm deg}M-2g_h-2)$ hence we obtain that $E(-2g_h+2)\simeq \O\oplus \O({\rm deg}M-2g_h-2)$. At last note that as $M$ is general the splitting type of $(\pi_h)_* M$ is either $(i,i)$ or $(i,i+1)$ depending on the parity of deg$M$. Summarizing the discussion above we obtain the following statement. \begin{thm} Let $\cal N$ be the stratum of the moduli space ${\cal M}_g^t$ $(g\ge 9)$ with one of the following splitting types of $V$ and $E$: \begin{enumerate} \item $(g_h-1,g_h-1+i,g_h-1+i)$ and $(2g_h-2,g_h-3+2i)$ where $g=3g_h+2i$, $i\ge g_h/2$, $g_h\ge2$; \item $(g_h-1,g_h-2+i,g_h-1+i)$ and $(2g_h-2,g_h-4+2i)$ where $g=3g_h+2i-1$, $i\ge(g_h+1)/2$, $g_h\ge2$. \end{enumerate} Then for general $(C,T)\in{\cal N}$ algebra $R_{K(-T)}$ is Koszul. \end{thm} \begin{rem} The condition $g\ge9$ excludes the case $g_h=2, i=1$ in 1). \end{rem} \begin{cor} For general tetragonal curve of genus $g\ge 9$ algebra $R_{K(-T)}$ is Koszul. \end{cor} \noindent {\it Proof} . It is sufficient to note that the pairs $(C,T)$ constructed above (for which this algebra is Koszul) lie in the open subset $U$ of ${\cal M}_g^t$ for which $a,b\ge 1$ (notation as above), and it is known (see \cite{S}) that for such curves $C$ the series $T$ is unique. Hence $U$ embeds as an open subset in the locus of tetragonal curves inside ${\cal M}_g$ which is irreducible, so $U$ itself is irreducible. \ \vrule width2mm height3mm \vspace{3mm} \section{On regularity of modules over a commutative Koszul algebra} In this section we give a geometric bound for the regularity of a module over a commutative Koszul algebra. The result is not new but our proof seems to be very simple so we present it here. Let $X$ be a projective scheme, $L=\O_X(1)$ be a very ample line bundle on $X$ such that the corresponding algebra $R=R_L$ is Koszul. For a sheaf $F$ on $X$ we denote tensor product of $F$ with the $n$-th power of $L$ by $F(n)$. With these assumptions we have the following. \begin{thm} For any coherent sheaf $F$ on $X$ if $H^i(F(-i))=0$ for any $i>0$ then the corresponding $R$-module $M=\oplus_{i\ge 0}H^0(F(i))$ has a linear free resolution that is a resolution of the form $$\ldots \rightarrow V_2 \otimes R(-2) \rightarrow V_1 \otimes R(-1) \rightarrow V_0 \otimes R \rightarrow M \rightarrow 0$$ where $V_i$ are some vector spaces. \end{thm} \noindent {\it Proof} . Koszul property of $R$ means that there is a resolution of the trivial module (of degree zero) which has form $$\ldots \rightarrow Q_2 \otimes R(-2) \rightarrow Q_1 \otimes R(-1) \rightarrow Q_0\otimes R \rightarrow k \rightarrow 0$$ where $Q_0=k$. It induces the following exact sequence of sheaves on $X$ $$\ldots \rightarrow Q_2 \otimes \O(-2) \rightarrow Q_1 \otimes \O(-1) \rightarrow \O \rightarrow 0$$ Now we can tensor it by $F(n)$ and consider the corresponding spectral sequence computing hypercohomology $$E_1^{p,q}=Q_{-p}\otimes H^q(F(i+p))\Rightarrow 0$$ with differentials $d_r$ of bidegree $(r,-r+1)$. Now $F$ is $0$-regular in the sense of Castelnuovo-Mumford (see \cite{M}) so we have $E_1^{p,q}=0$ if $p+q\ge -i,\ q\ge 1$. It follows that the complex $E_1^{\cdot,0}$ is exact in terms $p>-i$. But this complex computes syzigies of $M$, namely its cohomology in $p$-th term is Tor$^R_p(k,M)_(i+2p)$. So we have proved that Tor$_p(k,M)_j=0$ for $j>p$ which is equivalent to the existence of linear free resolution in question. \ \vrule width2mm height3mm \vspace{3mm} \begin{rem} The statement of this theorem is essentially equivalent to the statement of the main theorem of \cite{AE} for the case of Koszul algebras $R$ which has form $R_L$. Indeed that theorem asserts that the regularity of a module $M$ over $R$ is bounded by its regularity as a module over a symmetric algebra $S$ which surjects onto $R$. We have proved that for a modules which come from coherent sheaves the regularity is bounded by its geometric counterpart --- the regularity in the sense of Castelnuovo-Mumford. Note that the regularity of an arbitrary module (not necessary coming from coherent sheaf) can be bounded easily using the bound for that special type of modules. Now the point is that for the case of symmetric algebra these two regularities are equal so we arrive to the formulation of \cite{AE}. \end{rem}

9312

alg-geom/9312010

en

https://arxiv.org/abs/alg-geom/9312010

alg-geom/9312010

Charles Walter

Charles H. Walter

On the Harder-Narasimhan Filtration for Coherent Sheaves on P2: I

14 pages, LATeX 2.09

Let E be a torsion-free sheaf on P2. We give an effective method which uses the Hilbert function of E to construct a weak version of the Harder-Narasimhan filtration of a torsion-free sheaf on P2 subject only to the condition that E be sufficiently general among sheaves with that Hilbert function. This algorithm uses on a generalization of Davis' decomposition lemma to higher rank.

alg-geom

\section{\@startsection{section}{1}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf}} \def\subsection{\@startsection {subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subsubsection{\@startsection {subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\paragraph{\@startsection {paragraph}{3}{\z@}{2ex plus 0.6ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subparagraph{\@startsection {subparagraph}{3}{\parindent}{2ex plus 0.6ex minus .2ex}{-1pt}{\normalsize\sl}} \renewcommand{\theequation}{\thesubsection.\arabic{equation}} \@addtoreset{equation}{subsection} \def\RIfM@{\relax\ifmmode} \def0{0} \newif\iffirstchoice@ \firstchoice@true \def\the\textfont\@ne{\the\textfont\@ne} \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi{\RIfM@\expandafter\RIfM@\expandafter\text@\else\expandafter\text@@\fi@\else\expandafter\RIfM@\expandafter\text@\else\expandafter\text@@\fi@@\fi} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi@@#1{\leavevmode\hbox{#1}} \def\RIfM@\expandafter\text@\else\expandafter\text@@\fi@#1{\mathchoice {\hbox{\everymath{\displaystyle}\def\the\textfont\@ne{\the\textfont\@ne}% \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@}\textdef@@ T#1}} {\hbox{\firstchoice@false \everymath{\textstyle}\def\the\textfont\@ne{\the\textfont\@ne}% \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\textfont\tw@}\textdef@@ T#1}} {\hbox{\firstchoice@false \everymath{\scriptstyle}\def\the\textfont\@ne{\the\scriptfont\@ne}% \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\scriptfont\tw@}\textdef@@ S\rm#1}} {\hbox{\firstchoice@false \everymath{\scriptscriptstyle}\def\the\textfont\@ne {\the\scriptscriptfont\@ne}% \def\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}} \def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl\textdef@#1\it} \def\DN@{\def\next@} \def\eat@#1{} \def\textdef@#1#2{% \DN@{\csname\expandafter\eat@\string#2fam\endcsname}% \if S#1\edef#2{\the\scriptfont\next@\relax}% \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}% \else\edef#2{\the\textfont\next@\relax}\fi\fi} \def\mathop{\textstyle \sum }{\mathop{\textstyle \sum }} \def\binom#1#2{{#1 \choose #2}} \catcode`\@=12 \begin{document} \maketitle \begin{abstract} \noindent Let ${\cal E}$ be a torsion-free sheaf on ${\bf P}^2$. We give an effective method which uses the Hilbert function of ${\cal E}$ to construct a weak version of the Harder-Narasimhan filtration of a torsion-free sheaf on ${\bf P}^2$ subject only to the condition that ${\cal E}$ be sufficiently general among sheaves with that Hilbert function. This algorithm uses on a generalization of Davis' decomposition lemma to higher rank. \bigskip\ \end{abstract} \TeXButton{subsection1}{\refstepcounter{subsection}}Consider the following problem. Let ${\cal E}$ be an explicit torsion-free sheaf on ${\bf P}^2$ given by a presentation \begin{equation} \label{pres}0\rightarrow \bigoplus_{n\in {\bf Z}}{\cal O}_{{\bf P}% ^2}(-n)^{b(n)}\stackrel{\phi }{\rightarrow }\bigoplus_{n\in {\bf Z}}{\cal O}% _{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}\rightarrow 0. \end{equation} How does one go about effectively computing the Harder-Narasimhan filtration of ${\cal E}$, i.e.\ the unique filtration% $$ 0=F_0({\cal E})\subset F_1({\cal E})\subset \cdots \subset F_s({\cal E})=% {\cal E} $$ such that the graded pieces ${\rm gr}_i({\cal E}):=F_i({\cal E})/F_{i-1}(% {\cal E})$ are semistable in the sense of Gieseker-Maruyama and their reduced Hilbert polynomials $P_i(n)=\chi ({\rm gr}_i({\cal E})(n))/{\rm rk}(% {\rm gr}_i({\cal E}))$ satisfy $P_1(n)>P_2(n)>\cdots >P_s(n)$ for $n\gg 0?$ In this paper and its planned sequel we consider the problem under the simplifying assumption that the matrix $\phi $ of homogeneous polynomials is general, i.e.\ that ${\cal E}$ is general among torsion-free sheaves with the same Hilbert function as ${\cal E}$. Our solution to the problem then divides into two parts. In this first part we construct a filtration of $% {\cal E}$ of the type \begin{equation} \label{genfilt}0\subset {\cal E}_{\leq \tau _1}\subset \cdots \subset {\cal E% }_{\leq \tau _s}\subset {\cal E} \end{equation} where ${\cal E}_{\leq n}$ denotes the subsheaf of ${\cal E}$ which is the image of the natural evaluation map $H^0({\cal E}(n))\otimes {\cal O}_{{\bf P% }^2}(-n)\rightarrow {\cal E}$. We give an algorithm for picking the $\tau _i$ so that the filtration approximates the true Harder-Narasimhan filtration but groups together all pieces of the Harder-Narasimhan filtration with slopes between two consecutive integers. The associated graded sheaves ${\rm % gr}_i({\cal E}):={\cal E}_{\leq \tau _i}/{\cal E}_{\leq \tau _{i-1}}$ are not always semistable, but they do share a number of properties with semistable sheaves which suffice for a number of applications. For instance they are of {\em rigid splitting type}, i.e.\ their restrictions to a general line $L$ of ${\bf P}^2$ are of the form ${\cal O}_L(n_i)^{\alpha _i}\oplus {\cal O}_L(n_i+1)^{\beta _i}$ for some $n_i$, $\alpha _i$ and $% \beta _i$. They also satisfy ${\rm Hom}({\rm gr}_i({\cal E}),{\rm gr}_i(% {\cal E})(-1))=0$. We call our filtration the Weak Harder-Narasimhan (or WHN) filtration of $% {\cal E}$. It is fine enough to give graded pieces of rigid splitting type but is otherwise deliberately as coarse as possible in order to keep the algorithm for picking the $\tau _i$ as simple as possible (and also because further refinement can actually be counterproductive in such applications as the classification of irreducible components of the moduli stack of torsion-free sheaves on ${\bf P}^2$). Thus in some cases the WHN filtration may not even be the finest filtration of ${\cal E}$ by subsheaves of the form ${\cal E}_{\leq n}$ which is compatible with the Harder-Narasimhan filtration, although such a refinement could certainly be computed by the methods of this paper by adding an extra step to the end of the algorithm of paragraph (\ref{effective}). The true Harder-Narasimhan filtration is not always given by subsheaves of the form ${\cal E}_{\leq n}$ and can therefore be much harder to compute. For example the sheaf ${\cal E=O}_{{\bf P}% ^2}\oplus \Omega _{{\bf P}^2}(2)$ has Harder-Narasimhan filtration $0\subset \Omega _{{\bf P}^2}(2)\subset {\cal E}$ but is unfilterable by subsheaves of the form ${\cal E}_{\leq n}$ since ${\cal E}_{\leq n}=0$ for $n\leq -1$ and $% {\cal E}_{\leq n}={\cal E}$ for $n\geq 0$. In the planned part II we will show how to refine the WHN filtration of a sufficiently general sheaf to the true Harder-Narasimhan filtration using exceptional objects and mutations. The precise formulation of the WHN filtration requires a certain number of numerical definitions. We consider a general sheaf ${\cal E}$ with a presentation of the form (\ref{pres}) for given functions $a(n)$ and $b(n)$ of finite support. We define $r(n)$ and $h(n)$ as the first and second integrals of $a(n)-b(n)$, i.e.% \begin{eqnarray} r(n) & := & \sum_{m\leq n}\left\{ a(m)-b(m)\right\} , \label{r(n)} \\ h(n) & := & \sum_{m\leq n}r(m)=\sum_{m\leq n}(n-m+1)\left\{ a(m)-b(m)\right\} .\label{h(n)} \end{eqnarray}The function $h$, $r$, and $a-b$ are respectively the first, second, and third differences of the Hilbert function of ${\cal E}$ defined by $n\mapsto h^0({\cal E}(n))$. We will assume that the $a(n)$ and $b(n)$ are such that $r(n)\geq 0$ for all $n$. The general $\phi {:}~{\cal % \bigoplus O}(-n)^{b(n)}\rightarrow \bigoplus {\cal O}(-n)^{a(n)}$ is injective if and only if this is the case (see \cite{Ch} or Theorem \ref {Chang} below). Depth considerations show that the cokernel ${\cal E}$ of such an injective $\phi $ will have no subsheaves supported at isolated points, but ${\cal E}$ is permitted to have torsion supported along a curve. We now define further auxiliary functions by% \begin{eqnarray} \tilde h(n) & := & \max \, \{h(m)+(n-m)r(m) \mid m\geq n \}, \label{htilde(n)} \\ t(n) & := & \max \,\{m\geq n\mid \tilde h(n)=h(m)-(m-n)r(m)\}\in {\bf Z}\cup \{+\infty \}.\label{t(n)} \end{eqnarray}We will show in Lemma \ref{combin} that if $a(n)$ and $b(n)$ are such that $r(n)\geq 0$ for all $n$, then the function $t$ is nondecreasing and takes only finitely many values $\tau _0<\tau _1<\cdots <\tau _s<\tau _{s+1}=+\infty $. These $\tau _i$ may be effectively computed by an algorithm we will give in paragraph (\ref{effective}). We set also $% \tau _{-1}=-\infty $. Then we define the {\em WHN filtration} of ${\cal E}$ as the filtration \begin{equation} \label{wfilt}0={\cal E}_{\leq \tau _{-1}}\subset {\cal E}_{\leq \tau _0}\subset \cdots \subset {\cal E}_{\leq \tau _s}\subset {\cal E}_{\leq \tau _{s+1}}={\cal E} \end{equation} with graded pieces ${\rm gr}_i({\cal E}):={\cal E}_{\leq \tau _i}/{\cal E}% _{\leq \tau _{i-1}}$ for $0\leq i\leq s+1$. Our main result is: \begin{theorem} \label{main}Let $a$, $b{:}~{\bf Z}\rightarrow {\bf Z}_{\geq 0}$ be functions of finite support such that the function $r(n)$ of (\ref{r(n)}) is nonnegative. Let ${\cal E}$ be the cokernel of an injection $\phi % {:}~\bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_n% {\cal O}_{ {\bf P}^2}(-n)^{a(n)}$. If $\phi $ is sufficiently general, then the WHN filtration of ${\cal E}$ defined in (\ref{wfilt}) has the following properties: (i)\quad For all $0\leq i\leq s+1$ the sheaf ${\rm gr}_i({\cal E})$ has resolution% $$ 0\rightarrow \bigoplus_{\tau _{i-1}<n\leq \tau _i}{\cal O}_{{\bf P}% ^2}(-n)^{b(n)}\rightarrow \bigoplus_{\tau _{i-1}<n\leq \tau _i}{\cal O}_{% {\bf P}^2}(-n)^{a(n)}\rightarrow {\rm gr}_i({\cal E})\rightarrow 0. $$ (ii)\quad The subsheaf ${\cal E}_{\leq \tau _0}={\rm gr}_0({\cal E})$ is the torsion subsheaf of ${\cal E}$. (iii)\quad For $1\leq i\leq s+1$ the sheaf ${\rm gr}_i({\cal E})$ is torsion-free and of rigid splitting type, i.e.\ if $L$ is a general line of $% {\bf P}^2$, then ${\rm gr}_i({\cal E}){\mid }_L\cong {\cal O}_L(-\nu _i)^{\beta _i}\oplus {\cal O}_L(-\nu _i-1)^{\rho _i-\beta _i}$ for some integers $\nu _i$, $\beta _i$ and $\rho _i$. Moreover $\nu _1<\nu _2<\cdots <\nu _{s+1}$ and ${\cal E}{\mid }_L\cong \bigoplus_{i=0}^{s+1}{\rm gr}_i(% {\cal E}){\mid }_L$. (iv)\quad For $i\leq j$ we have ${\rm Hom}({\rm gr}_i({\cal E}),{\rm gr}_j(% {\cal E})(-1))=0$. (v)\quad The Harder-Narasimhan filtration of ${\cal E}/{\cal E}_{\leq \tau _0}$ for Gieseker-Maruyama stability is a refinement of the filtration (\ref {wfilt}) of ${\cal E}/{\cal E}_{\leq \tau _0}$. Indeed ${\rm gr}_i({\cal E})$ collects all pieces of the Harder-Narasimhan filtration with slopes $\mu $ satisfying $-\nu _i-1<\mu <-\nu _i$ as well as some of those of slopes $-\nu _i-1$ and $-\nu _i$. \end{theorem} The outline of the paper is as follows. In the first section we prove a number of numerical lemmas leading to a method of filtering the Hilbert function of ${\cal E}$. The key definition is that the Hilbert function of $% {\cal E}$ (or its differences $h(n)$, $r(n)$ or $a(n)-b(n)$ as defined above) is {\em filterable} at $m$ if the function $r(n)$ of (\ref{r(n)}) satisfies $r(n)\geq r(m)$ for all $n\geq m$. The $a(n)$ and $b(n)$ then split into $$ a_m^{{\rm sub}}(n):=\left\{ \begin{array}{ll} a(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n\leq m, \\ 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n\geq m, \end{array} \right. $$ and analogous functions $a_m^{{\rm quot}}(n)$, $b_m^{{\rm sub}}(n)$ and $% b_m^{{\rm quot}}(n)$. The $r_m^{{\rm sub}}$, $r_m^{{\rm quot}}$, $h_m^{{\rm % sub}}$ and $h_m^{{\rm quot}}$ are defined by integrating. If a Hilbert function is filterable at several integers $m_i$, it may be split into several graded pieces this way. The lemmas of the section show that the Hilbert function of ${\cal E}$ is filterable at the $\tau _i$ and that its graded pieces satisfy conditions analogous to the conditions of parts (ii)-(v) of Theorem \ref{main}. In the second section we show that such filtrations of Hilbert functions correspond to filtrations of ${\cal E}$ by subsheaves of the form ${\cal E}% _{\leq m}$ if ${\cal E}$ is sufficiently general among coherent sheaves with the same Hilbert function. The key lemma is the following which may be regarded as a generalization of Davis' decomposition lemma \cite{D} to higher rank. \TeXButton{Davis.lemma} {\begin{trivlist} \item [\hskip \labelsep{\bf Lemma \ref{gener}.}]{\sl Suppose that ${\cal E}$ is a coherent sheaf on ${\bf P}^2$ without zero-dimensional associated points such that the Hilbert function of ${\cal E}$ is filterable at an integer $m$. Write ${\cal E}$ as the cokernel of an injection $\phi {:}\ \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{a(n)}$. If the matrix $\phi $ is sufficiently general, then ${\cal E}_{\leq m}$ and ${\cal E}/{\cal E}_{\leq m}$ have resolutions}\begin{eqnarray*} & 0\rightarrow \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}_{\leq m}\rightarrow 0 , & \\ & 0\rightarrow \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}_{>m}\rightarrow 0 . & \end{eqnarray*} \end{trivlist} } The rest of the section is devoted to showing that Theorem \ref{main} follows from this lemma and from the numerical lemmas proved in the first section. This paper was written in the context of the group on vector bundles on surfaces of Europroj. The author would like to thank A.\ Hirschowitz for some useful conversations. \section{Filtering Hilbert Functions} This section contains the purely combinatorial part of the proof of the Theorem \ref{main}. It consists of a number of numerical lemmas on Hilbert functions of coherent sheaves on ${\bf P}^2$. We begin by fixing some terminology. We use the notation $(x)_{+}=\max (x,0)$. Our fundamental invariant is the difference $a(n)-b(n)$ between the functions of (\ref{pres}). We assume that $a(n)-b(n)$ is an integer for all $% n$ and vanishes for all but finitely many $n$ and that the associated function $r(n)=\sum_{m\leq n}\left\{ a(n)-b(n)\right\} \geq 0$ for all $n$. We call the associated function $h(n)=\sum_{m\leq n}r(n)$ of (\ref{h(n)}) the {\em FDH function} (or first difference of a Hilbert function). It is the FDH functions which will play the major role in our computations. An intrinsic definition is: \begin{definition} An FDH function is a function $h{:}~{\bf Z}\rightarrow {\bf Z}_{\geq 0}$ such that $r(n)=\Delta h(n)\geq 0$ for all $n$, $h(n)=0$ for $n\ll 0$, and $% h(n)$ is linear of the form $\rho n+\sigma $ for $n\gg 0$. We call $\rho $ the {\em rank} of $h$, $\sigma -\rho $ its {\em degree}, and $\sum_{n\in {\bf Z}}\left\{ (\rho n+\sigma )_{+}-h(n)\right\} $ its {\em deficiency}. An FDH function is {\em torsion} if its rank is $0$. An FDH function $h$ is {\em torsion-free} if $r(m)\geq 1$ implies that $r(n)\geq 1$ for all $n\geq m $. An FDH function $h$ is {\em locally free} if $h$ is torsion-free and additionally $r(m)\geq 2$ implies that $r(n)\ge 2$ for all $n\geq m$. (This terminology will be justified by Theorem \ref{Chang}.) \end{definition} These functions have the following basic properties: \begin{lemma} \label{combin}Suppose $h$ is an FDH function of rank $\rho $ and degree $% \sigma -\rho $. Let $r=\Delta h$, and let $\tilde h$ and $t$ be as in (\ref {htilde(n)}) and (\ref{t(n)}). Then (i)\quad For all $n$ one has $0\leq h(n)\leq \tilde h(n)$ and $t(n)>n$. (ii)\quad If $m>t(n)$, then $r(m)>r(t(n))$. (iii)\quad The function $t$ is nondecreasing and takes only finitely many distinct values. If we write these as $\tau _0<\tau _1<\cdots <\tau _s<\tau _{s+1}=\infty $, then $0=r(\tau _0)<r(\tau _1)<\cdots <r(\tau _s)<\rho $. (iv)\quad Let $\nu _i=\min \{n\mid t(n)=\tau _i\}$. If $\nu _i\leq n<\nu _{i+1}$, then $t(n)=\tau _i$ and $\tilde h(n)=nr(\tau _i)+\left( h(\tau _i)-\tau _ir(\tau _i)\right) $. Moreover $n<\nu _i$ if and only if \begin{equation} \label{nu(i)}nr(\tau _{i-1})+\left( h(\tau _{i-1})-\tau _{i-1}r(\tau _{i-1})\right) >nr(\tau _i)+\left( h(\tau _i)-\tau _ir(\tau _i)\right) \end{equation} (v)\quad If $\nu _i\leq n\leq m\leq \tau _i$, then $h(\tau _i)+(n-\tau _i)r(\tau _i)\geq h(m)+(n-m)r(m)$. (vi)\quad If $n<\nu _1$, then $t(n)=\tau _0=\max \{n\mid r(n)=0\}$ and $% \tilde h(n)=h(\tau _0)$. In particular, if $h$ is torsion-free then $\tilde h(n)=0$ and $t(n)=\max \{m\mid h(m)=0\}$ for $n<\nu _1$. (vii)\quad For $n\geq \nu _{s+1}$ one has $\tilde h(n)=\rho n+\sigma $ and $% t(n)=\tau _{s+1}=+\infty $. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) From the definitions we see that $% h(n)=h(n+1)-r(n+1)\leq \tilde h(n)$ and that this implies that $t(n)\geq n+1$% {}. (ii) We go by induction on $m$. Thus we assume that $r(i)>r(t(n))$ for $% t(n)<i<m$, and we will show that $r(m)>r(t(n))$ as well. But the definitions of $\tilde h(n)$ and $t(n)$ yield immediately% $$ h(t(n))+(n-t(n))r(t(n))=\tilde h(n)>h(m)-(m-n)r(m). $$ Hence% \begin{eqnarray*} (m-n)r(m) & > & (t(n)-n)r(t(n))+h(m)-h(t(n)) \\ & = & (t(n)-n)r(t(n))+\mathop{\textstyle \sum }_{i=t(n)+1}^mr(i) \\ & \geq & (t(n)-n)r(t(n))+(m-t(n)-1)r(t(n))+r(m) \end{eqnarray*}whence (ii). (iii) Since $t(n-1)\geq n$ by (i), we see from the definitions that $$ \tilde h(n)\geq h(t(n-1))+(n-t(n-1))r(t(n-1))=\tilde h(n-1)+r(t(n-1)). $$ Thus $r(t(n-1))\leq \Delta \tilde h(n)$. Similarly% $$ \tilde h(n-1)\geq h(t(n))+(n-1-t(n))r(t(n))=\tilde h(n)-r(t(n)) $$ and $\Delta \tilde h(n)\leq r(t(n))$. Hence $r\circ t$ is nondecreasing. Because of (ii) this implies that $t$ is nondecreasing. The function $t$ can only take finitely many values since by (ii) $r$ takes a different value at each value of $t$, and the values of $r$ are bounded since $r(n)$ is constant for $n\ll 0$ and for $n\gg 0$. If $\tau _i=$$t(n)<+\infty $, then for $m\gg \tau _i$ (ii) yields $\rho =r(m)>r(\tau _i)$. Finally for $n\ll 0$ one has $0\leq \tilde h(n)=h(\tau _0)+(n-\tau _0)r(\tau _0)$ which implies that $r(\tau _0)\leq 0$. But since $h$ is an FDH function, $r(\tau _0)\geq 0$% . So $r(\tau _0)=0$. This finishes (iii). (iv) If $\nu _i\leq n<\nu _{i+1}$, then $t(n)=\tau _i$. According to the definitions, this implies $$ \tilde h(n)=h(\tau _i)-(\tau _i-n)r(\tau _i)=nr(\tau _i)+\left( h(\tau _i)-\tau _ir(\tau _i)\right) . $$ As for the inequality (\ref{nu(i)}), because both its sides are linear and the slope on the left side is less than that on the right side, it is enough to show that the inequality holds for $n=\nu _i-1$ but fails for $n=\nu _i$. But because of the definition of $t$, this follows immediately from $t(\nu _i-1)=\tau _{i-1}<$$\tau _i=t(\nu _i)$. (v) The proof is divided into several cases. First if $n<\nu _{i+1}$, then by (iv) the inequality becomes $\tilde h(n)\geq h(m)-(m-n)r(m)$ which follows from the definition of $\tilde h(n)$. If $n\ge \nu _{i+1}$ but $% r(m)\leq r(\tau _i)$, then the inequality follows from the case $n=\nu _i$ by% \begin{eqnarray*} h(\tau _i)+(n-\tau _i)r(\tau _i) & = & \tilde h(\nu _i)+(n-\nu _i)r(\tau _i) \\ & \geq & h(m)+(\nu _i-m)r(m)+(n-\nu _i)r(\tau _i) \\ & \geq & h(m)+(n-m)r(m). \end{eqnarray*}Finally if $n\ge \nu _{i+1}$ but $r(m)>r(\tau _i)$, then let $% m^{\prime }:=\min \{M>m\mid r(M)\leq r(\tau _i)\}\leq \tau _i$. Then using the previous case applied with $m^{\prime }-1$ substituted for $n$ and $% m^{\prime }$ substituted for $m$ we see that% \begin{eqnarray*} h(m)+(n-m)r(m) & = & \left[ h(m^{\prime })-r(m^{\prime })\right] - \sum_{i=m+1}^{m^{\prime }-1}r(i)-(m-n)r(m) \\ & \leq & \left[ h(\tau _i)+(m^{\prime }-1-\tau _i)r(\tau _i)\right] - (m^{\prime }-1-n)r(\tau _i) \\ & = & h(\tau _i)+(n-\tau _i)r(\tau _i) . \end{eqnarray*} (vi) If $n<\nu _1$, then $t(n)=\tau _0$ by (iv), $\tau _0=$$\max \{n\mid r(n)=0\}$ by (iii) and (ii), and $\tilde h(n)=h(\tau _0)$ by (iv). (vii) For $n\geq \nu _{s+1}$ we have $t(n)=+\infty $. This means that there exists a sequence of integers $m_i\rightarrow +\infty $ such that% $$ \tilde h(n)=h(m_i)-(m_i-n)r(m_i)=(\rho m_i+\sigma )-(m_i-n)\rho =\rho n+\sigma . $$ \subsection{Filtering FDH Functions.\label{filter}} Let $h$ be an FDH function. We will say that $h$ is {\em filterable at }$m$ if the associated function $r$ satisfies $r(n)\geq r(m)$ for all $n\geq m$. A {\em filtration} of $h$ is a sequence of integers $m_0<m_1<\cdots <m_s$ at which $h$ is filterable. Given such a filtration we decompose $h$ into a sum of $s+2$ function $h_0,\ldots ,h_{s+1}$ defined as follows. We set $% m_{-1}=-\infty $ and $m_{s+1}=+\infty $. Then the second difference $\Delta ^2h(n)=a(n)-b(n)$ may be decomposed by% $$ a_i(n)-b_i(n):=\left\{ \begin{array}{ll} a(n)-b(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }m_{i-1}<n\leq m_i, \\ 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{otherwise,} \end{array} \right. $$ with $r_i(n)=\sum_{m\leq n}\left\{ a_i(m)-b_i(m)\right\} $ and $% h_i(n)=\sum_{m\leq n}r_i(n)$ defined as in (\ref{r(n)}) and (\ref{h(n)}). If we write $H_i(n):=h(m_i)+(n-m_i)r(m_i)$, then the $r_i(n)$ and $h_i(n)$ satisfy% \begin{eqnarray*} & r_i(n) = \left\{ \begin{array}{ll} 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n\leq m_{i-1} \\ r(n)-r(m_{i-1}) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }m_{i-1}<n\leq m_i, \\ r(m_i)-r(m_{i-1}) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n>m_i. \end{array} \right. \label{ri(n)} & \\ & h_i(n) = \left\{ \begin{array}{ll} 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n\leq m_{i-1} \\ h(n)-H_{i-1}(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }m_{i-1}<n\leq m_i, \\ H_i(n)-H_{i-1}(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{if }n>m_i. \end{array} \right. \label{h_i(n)} & \end{eqnarray*}The filterability of $h$ at the $m_i$ implies that $% r_i(n)\geq 0$ for all $n$ and $i$. So the $h_i(n)$ are all FDH functions. We call the functions $h_i(n)$ the {\em graded pieces} of the filtration. We will say that a filtration is {\em trivial} if all but one of its graded pieces vanish. Now let us consider the associated function $t$ of (\ref{t(n)}). By Lemma \ref{combin}(iii) the sequence $\tau _0<\tau _1<\cdots <\tau _s$ of all distinct finite values of $t$ form a filtration of $h$ which we call the {\em WHN filtration} (or weak Harder-Narasimhan filtration). Some of the properties of this filtration are \begin{lemma} \label{WHN}Let $h$ be an FDH function, let $\tau _0<\tau _1<\cdots <\tau _s$ be the WHN filtration of $h$, and let $h_0,h_1,\ldots ,h_{s+1}$ be the graded pieces of the filtration. For each $i$ let $\nu _i=\min \{n\mid t(n)=\tau _i\}$. Then (i)\quad The FDH function $h_0$ is torsion. It vanishes if $h$ is torsion-free. (ii)\quad The FDH functions $h_1,h_2,\ldots ,h_{s+1}$ are torsion-free. (iii)\quad For $i=1,\ldots ,s$ the function $t_i$ associated to $h_i$ by (% \ref{t(n)}) satisfies $t_i(n)=\tau _{i-1}$ for $n<\nu _i$, and $t(n)=+\infty $ for $n\geq \nu _i$. Thus the $h_i$ are FDH functions with trivial WHN filtrations. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) This is a direct translation of Lemma \ref {combin}(vi). (ii) This follows directly from Lemma \ref{combin}(ii) and the formula for $% r_i(n)$. (iii) We first suppose $n<\nu _i$. We will compute $\widetilde{h_i}(n)$ and $% t_i(n)$ according to the definitions (\ref{htilde(n)}) and (\ref{t(n)}). This means first computing $h_i(m)+(n-m)r_i(m)$ for all $m\geq n$. If $m\leq \tau _{i-1}$, then $h_i(m)=r_i(m)=0$ and so $h_i(m)+(n-m)r_i(m)=0$. If $\tau _{i-1}<m\leq \tau _i$, then% $$ h_i(m)+(n-m)r_i(m)=\left\{ h(m)+(n-m)r(m)\right\} -\left\{ h(\tau _{i-1})+(n-\tau _{i-1})r(\tau _{i-1})\right\} . $$ But since $m>\tau _{i-1}$, the definitions of $\tilde h(n)$ and $t(n)$ imply that the right side of this equation is negative for all $n$ such that $% t(n)=\tau _{i-1}$, including $n=\nu _i-1$. The right hand side is also linear in $n$ with slope $r(m)-r(\tau _{i-1})$ which is positive by Lemma \ref{combin}(ii), so it must be negative for all $n<\nu _i$. Thus $% h_i(m)+(n-m)r_i(m)<0$ if $\tau _{i-1}<m\leq \tau _i$. If $m\geq \tau _i$, then $h_i(m)+(n-m)r_i(m)=h_i(\tau _i)+(n-\tau _i)r_i(\tau _i)<0$ because $h_i$ is linear in this range. So by the definitions we have $\widetilde{h_i}(n)=0$ and $\nu _i(n)=\tau _{i-1}$ for $n<\nu _i$. Now we suppose that $\nu _i\leq n\leq \tau _i$. Then after subtracting $% h(\tau _{i-1})+(n-\tau _{i-1})r(\tau _{i-1})$ from both sides of the inequality of Lemma \ref{combin}(v), we see that for all $n\leq m\leq \tau _i $ we have \begin{equation} \label{tau(i)}h_i(\tau _i)+(n-\tau _i)r_i(\tau _i)\geq h_i(m)+(n-m)r_i(m). \end{equation} And for all $m\geq \tau _i$ we have equality in (\ref{tau(i)}) because $h_i$ is linear in this range. So by the definitions, $\tilde h_i(n)=h_i(m)+(n-m)r_i(m)$ for all $m\geq \tau _i$, and $t_i(n)=+\infty $. Finally if $n\geq \tau _i$, then for all $m\geq n$ we have equality in (\ref {tau(i)}), so again we have $t_i(n)=+\infty $. \TeXButton{qed}{\hfill $\Box$ \medskip} \subsection{Torsion-free FDH functions $h$ with trivial WHN-filtrations.} We wish to decompose an $h$ of this type in a certain way. For $n\gg 0$ the function $h(n)$ is linear, so we may write it in the form $\rho (n-\nu )+\beta $ with $\rho $, $\nu $, and $\beta $ integers such that $0\leq \beta <\rho $. But then if $n<\nu _1$ we have by Lemma \ref{combin}(vi) that $% h(n)=\tilde h(n)=0$ and $t(n)=\max \{n\mid h(n)=0\}$. If $n\geq \nu _1$ then by Lemma \ref{combin}(vii) we have $t(n)=+\infty $ and $0\leq h(n)\leq $$% \tilde h(n)=\rho (n-\nu )+\beta $. Moreover, $\nu _1=\nu $ by Lemma \ref {combin}(iv). We now define% \begin{eqnarray} \gamma^i(n) & = & \left\{ \begin{array}{ll} (n-\nu +1)_{+} & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=1,\ldots ,\beta , \\ (n-\nu )_{+} & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=\beta +1,\ldots ,\rho , \end{array} \right. \\ h^i(n) & = & \min \left\{ \gamma^i(n),\left[ h(n)-\sum_{k=1}^{i-1}\gamma ^k(n)\right] _{+}\right\}. \label{hi(n)} \end{eqnarray}These functions have the following properties: \begin{lemma} \label{hi}Suppose $h$ is a torsion-free FDH function such that the associated function $t$ of (\ref{t(n)}) takes only two distinct values. For $% i=1,\ldots ,\rho $ let $h^i{:}~{\bf Z}\rightarrow {\bf Z}_{\geq 0}$ be the function defined in (\ref{hi(n)}). Then (i)\quad The $h^i$ satisfy $\sum_{i=1}^\rho h^i=h$, (ii)\quad The $h^i$ are torsion-free FDH functions of rank $1$. The degree of $h^i$ is $-\nu $ for $i=1,\ldots ,\beta $, and $-\nu -1$ for $i=\beta +1,\ldots ,\rho $. (iii)\quad The deficiency of $h^i$ is positive for $i=1,\ldots ,\beta $. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}(i) First note that if $n<\nu $, then $\gamma ^i(n)=h^i(n)=0$ for all $i$. But $h(n)=0$ as well. So this case is fine. If $% n\geq \nu $, then we have $0\leq h(n)\leq \tilde h(n)=\sum_{i=1}^\rho \gamma ^i(n)$. So there exists a $k$ such that $\sum_{i=1}^{k-1}\gamma ^i(n)\leq h(n)\leq \sum_{i=1}^k\gamma ^i(n)$. Then \begin{equation} \label{h(i)(n)}h^i(n)=\left\{ \begin{array}{ll} \gamma ^i(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=1,\ldots ,k-1 \\ h(n)-\sum_{i=1}^{k-1}\gamma ^i(n) & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=k \\ 0 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=k+1,\ldots ,\rho \end{array} \right. \end{equation} and the sum is $h(n)$. This completes the proof of (i). For (ii) we first show that the $h^i$ are torsion-free FDH functions, i.e.\ that $\Delta h^i(n)>0$ for all $n$ such that $h^i(n)>0$. To verify this, we may clearly assume that $n\geq \nu $ since otherwise $h^i(n)=0$. Now note that if one has a function of the form $h^i=\min (f,g)$, then in order to show that $h^i(n)>0$ implies $\Delta h^i(n)>0$ it is enough to show that $% f(n)>0$ implies $\Delta f(n)>0$ and that $g(n)>0$ implies $\Delta g(n)>0$. So now consider the case $i=1,\ldots ,\beta $. The function $f(n):=\gamma ^i(n)$ satisfies $\Delta \gamma ^i(n)=1>0$. And if $g(n):=\left[ h(n)-\sum_{k=1}^{i-1}\gamma ^k(n)\right] _{+}>0$, then $h(n)>(i-1)(n-\nu +1)$% . But from the definition of $\tilde h(n)$ we have% $$ 0=\tilde h(\nu -1)\geq h(n)-(n-\nu +1)\Delta h(n). $$ So $(n-\nu +1)\Delta h(n)\geq h(n)$. Thus $\Delta h(n)>i-1$, and $\Delta g(n)>0$. This proves that $h^i$ is a torsion-free FDH function for $% i=1,\ldots ,\beta $. The proof that $h^i$ is a torsion-free FDH function for $i=\beta +1,\ldots ,\rho $ is similar except that one uses $\tilde h(\nu )=\beta $ to obtain $% (n-\nu +1)\Delta h(n)\geq h(n)-\beta $. For the rank and degree of $h^i$ note that for $n\gg 0$ the formula (\ref {hi(n)}) becomes% $$ h^i(n)=\min \left\{ \gamma ^i(n),\sum_{k=i}^\rho \gamma ^k(n)\right\} =\gamma ^i(n)=\left\{ \begin{array}{ll} n-\nu +1 & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=1,\ldots ,\beta , \\ n-\nu & \RIfM@\expandafter\text@\else\expandafter\text@@\fi{for }i=\beta +1,\ldots ,\rho . \end{array} \right. $$ For (iii) note that for $i=1,\ldots ,\beta $ the deficiency of $h^i$ is $% \sum_{n\geq \nu }(n-\nu +1-h^i(n))$. All the terms in this sum are nonnegative, so it is enough to show that $h^i(\nu )=0$. But recall that $% t(\nu -1)=\max \{n\mid h(n)=0\}$. And by Lemma \ref{combin}(i) $t(\nu -1)\geq \nu $. Thus $h(\nu )=0$, from which $h^i(\nu )=0$ for all $i$ by (i). Part (iii) now follows. \TeXButton{qed}{\hfill $\Box$ \medskip} As a final numerical result we wish to compare the decompositions of Lemmas \ref{WHN} and \ref{hi}. To do this we introduce an order on torsion-free FDH functions. Namely if $h$ and $h^{\prime }$ are torsion-free FDH functions of ranks $\rho $ and $\rho ^{\prime }$, degrees $d$ and $d^{\prime }$, and deficiencies $\delta $ and $\delta ^{\prime }$, then $h\succeq h^{\prime }$ (resp.\ $h\succ h^{\prime }$) if $d/\rho >d^{\prime }/\rho ^{\prime }$ or if $d/\rho =d^{\prime }/\rho ^{\prime }$ and $\delta /\rho \leq \delta ^{\prime }/\rho ^{\prime }$ (resp.\ $\delta /\rho <\delta ^{\prime }/\rho ^{\prime }$% ). \begin{lemma} \label{compar}Let $h$ be an FDH function, let $h_0,h_1,\ldots ,h_{s+1}$ be the graded pieces of the WHN filtration of $h$ of Lemma \ref{WHN}. For $% i=1,\ldots ,s+1$, let $\rho _i$ denote the rank of $h_i$, and let $% h_i=\sum_{j=1}^{\rho _i}h_i^j$ be the decomposition of $h_i$ of Lemma \ref {hi}. Then (i)\quad $h_i^1\succeq h_i^2\succeq \cdots \succeq h_i^{\rho _i}$ for $% i=1,\ldots ,s+1$, and (ii)\quad $h_i^{\rho _i}$ $\succ h_{i+1}^1$ for $i=1,\ldots ,s$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}First we introduce some notation. As in Lemma \ref {hi} if $n\gg 0$, then we may write $h_i(n)=\rho _i(n-\nu _i)+\beta _i$ with $0\leq \beta _i<\rho _i$. For each $i$ and $j=1,\ldots ,\rho _i$ we define% $$ \begin{array}{lclcl} \eta _i^j & := & \min \{n\mid h_i^j(n)>0\} & = & \min \{n\mid h_i(n)>\sum_{k=1}^{j-1}\gamma _i^k(n)\}, \\ \zeta _i^j & := & \min \{n\mid h_i^j(n)=\gamma _i^j(n)>0\} & = & \min \{n\mid h_i(n)\geq \sum_{k=1}^j\gamma _i^k(n)>0\}. \end{array} $$ where $\gamma _i^k$ is as in Lemma \ref{hi}. Then if $1\leq j\leq \beta _i$ (resp.\ if $\beta _i+1\leq j\leq \rho _i$), the FDH function $h_i$ has rank $% 1$, degree $d_i^j=-\nu _i$ (resp.\ $d_i^j=-\nu _i-1$), and deficiency $% \delta _i^j$ satisfying $\binom{\eta _i^j+d_i^j+1}2\leq \delta _i^j\leq \binom{\zeta _i^j+d_i^j+1}2$. (i) For all $i$ and $j=1,\ldots ,\rho _i-1$ we have $\zeta _i^j\leq \eta _i^{j+1}$. Hence $h_i^1\succeq h_i^2\succeq \cdots \succeq h_i^{\beta _i}$ for all $i$ because all these functions have the same rank, the same degree $% \alpha _i$, and nondecreasing deficiencies. Similarly $h_i^{\beta _i+1}\succeq h_i^{\beta _i+2}\succeq \cdots \succeq h_i^{\beta _i}$. And $% h_i^{\beta _i}\succ h_i^{\beta _i+1}$ by reason of degree. (ii) Note that by Lemmas \ref{combin} (v) and \ref{WHN}, the FDH function $% h_i^{\rho _i}$ has degree $-\nu _i-1$ and has $$ \zeta _i^{\rho _i}=\min \{n>\nu _i\mid h(n)=h(\tau _i)+(n-\tau _i)r(\tau _i)\}\leq \tau _i-1, $$ while $h_{i+1}^1$ has degree $-\nu _{i+1}$ or $-\nu _{i+1}-1$ and has% $$ \eta _{i+1}^1=\min \{n>\nu _i\mid h(n)>h(\tau _i)+(n-\tau _i)r(\tau _i)\}=\tau _i+1. $$ Since $\nu _i<\nu _{i+1}$, the degree of $h_i^{\rho _i}$ is at least that of $h_{i+1}^1$, and in case of equality the former function has a smaller deficiency than the latter.\TeXButton{qed}{\hfill $\Box$ \medskip} \subsection{Effective computation of the $\tau _i$ and $\nu _i$.\label {effective}} The $\tau _i$ and $\nu _i$ defined in Lemma \ref{combin} and referred to in the statement of Theorem \ref{main} may be effectively computed from $h(n)$ or $r(n)=\Delta h(n)$. Note that $\Delta r(n)$ is $0$ for all but finitely many $n$. For $t=+\infty $ we write $r(t)=\rho $ and $h(t)-tr(t)=\sigma $. According to Lemma \ref{combin}(ii), $$ \{\tau _i\}_{i=0}^{s+1}\subset T:=\left\{ n\mid r(m)>r(n)\RIfM@\expandafter\text@\else\expandafter\text@@\fi{ for all }% m>n\right\} \cup \{+\infty \}. $$ The set $T$ may be computed by passing through the finite set% $$ T^{\prime }:=\left\{ n\mid \Delta r(n+1)>0\right\} \cup \{+\infty \} $$ in descending order and purging those $n\in T^{\prime }$ such that $r(n)\geq r(m)$ where $m$ is the smallest unpurged element of $T^{\prime }$ larger than $n$. The minimal element of $T$ is $\tau _0$ since it is the unique $% t\in T$ such that $r(t)=0$. The other $\tau _i$ and $\nu _i$ may be computed recursively as follows. Suppose we have computed $\tau _0,\ldots ,\tau _{i-1}$ and $\nu _1$,$\ldots ,\nu _{i-1}$. We now need to find for which $x>\nu _{i-1}$ there is a $t\in T $ with $t>\tau _{i-1}$ such that% $$ h(t)+(x-t)r(t)\geq h(\tau _{i-1})+(x-\tau _{i-1})r(\tau _{i-1}). $$ For $t=+\infty $, the left side should be read as $\rho x+\sigma $ according to our above conventions. Each of the inequalities is equivalent to% $$ x\geq x_t:=\frac{\left( h(t)-tr(t)\right) -\left( h(\tau _{i-1})-\tau _{i-1}r(\tau _{i-1})\right) }{r(t)-r(\tau _{i-1})} $$ So $\nu _i=\min \left\{ \left\lceil x_t\right\rceil \mid t\in T\RIfM@\expandafter\text@\else\expandafter\text@@\fi{ and }% t>\tau _{i-1}\right\} $ where the notation $\left\lceil x_t\right\rceil $ means the smallest integer greater than or equal to $x_t$. We then look at those $t$ such that $\left\lceil x_t\right\rceil =\nu _i$, and pick out those among them for which $h(t)+(\nu _i-t)r(t)$ is maximal. The largest of these $t$ is $\tau _i$. We continue until some $\tau _i=+\infty $. The other invariants in the statement of Theorem \ref{main} and the proof of Lemma \ref{compar} may be computed as% \begin{eqnarray*} \rho _i & = & r(\tau _i)-r(\tau _{i-1}), \\ \beta _i & = & \left( h(\tau _i)+(\nu _i-\tau _i)r(\tau _i)\right) -\left( h(\tau _{i-1})+(\nu _i-\tau _{i-1})r(\tau _{i-1})\right) . \end{eqnarray*} \section{A Special Filtration on ${\bf P}^2$\label{sec2}} In this section we proceed to give sheaf-theoretic significance to the numerical computations of the previous section. We do this by introducing the WHN filtration on the general torsion-free sheaf ${\cal E}$ with a given Hilbert function. The Hilbert function of the graded pieces ${\rm gr}_i(% {\cal E})$ of the filtration are those given by Lemma \ref{WHN}. Lemma \ref {hi} is then used to show that the graded pieces satisfy ${\rm Hom}({\rm gr}% _i({\cal E}),{\rm gr}_j({\cal E})(-1))=0$ for all $i\leq j$. Lemma \ref {compar} is used to show that the WHN filtration is compatible with the Harder-Narasimhan filtration for Gieseker-Maruyama stability. \subsection{Hilbert Functions.} Recall that any coherent sheaf ${\cal E}$ on ${\bf P}^2$ without zero-dimensional torsion has a free resolution of the form \begin{equation} \label{phi}0\rightarrow \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{b(n)}\stackrel{% \phi }{\rightarrow }\bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}\rightarrow 0. \end{equation} The $a(n)$ and $b(n)$ are related to the Hilbert function $n\mapsto h^0(% {\cal E}(n))$ of ${\cal E}$ via \begin{equation} \label{Hilbert}\sum_nh^0({\cal E}(n))\,t^n=(1-t)^{-3}\sum_n\{a(n)-b(n)\}t^n \end{equation} or $a(n)-b(n)=\Delta ^3h^0({\cal E}(n))$. So the $a(n)$ and $b(n)$ determine the Hilbert function of ${\cal E}$ which conversely determines the differences $a(n)-b(n)$. If ${\cal E}$ is sufficiently general, then for all $n$ either $a(n)=0$ or $b(n)=0$ according to the sign of $a(n)-b(n)$, so the Hilbert function then actually determines the $a(n)$ and $b(n)$. The next theorem is the filtered Bertini theorem as applied to the special case of ${\bf P}^2$: \begin{theorem} {\rm \label{Chang}(Chang \cite{Ch})} A general map $\phi {:}\ \bigoplus_n% {\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_n{\cal O}_{{\bf P}% ^2}(-n)^{a(n)}$ is injective if and only if the function $h$ whose Poincar\'e series is% $$ \sum_nh(n)t^n=(1-t)^{-2}\sum_n\{a(n)-b(n)\}t^n $$ is an FDH function. Moreover, the cokernel ${\cal E}$ of a general $\phi $ is $\bullet $ torsion-free with at worst singular points of multiplicity $1$ if $h$ is torsion-free, $\bullet $ locally free if $h$ is locally free, $\bullet $ a line bundle on a curve with normal crossings if $h$ is torsion, $\bullet $ a line bundle on a smooth curve if $h$ is torsion and unfilterable. \end{theorem} Here the {\em multiplicity} of a singular point $P$ of a torsion-free sheaf $% {\cal E}$ on a smooth surface is the length of ${\cal E}_P^{\vee \vee }/% {\cal E}_P$. Theorem \ref{Chang} and formula (\ref{Hilbert}) allow us to define the {\em % FDH function of a coherent sheaf }${\cal E}$ without zero-dimensional torsion as $h:=\Delta h^0({\cal E}(n))$. The rank of the FDH function $h$ is then ${\rm rk}({\cal E})$, and the degree of $h$ is $c_1({\cal E)}$. If this rank is $\rho $, and the degree is written as $\rho \alpha +\beta $ with $% \alpha $ and $\beta $ integers such that $0\leq \beta <\rho $, then the deficiency of $h$ is $c_2({\cal E})-c_2({\cal F})$ where ${\cal F}={\cal O}_{% {\bf P}^2}(\alpha +1)^\beta \oplus {\cal O}_{{\bf P}^2}(\alpha )^{\rho -\beta }$. We may speak of a generic sheaf with FDH function $h$ because of the following fact, which is well known and which we therefore state without proof: \begin{lemma} The coherent sheaves without zero-dimensional torsion on ${\bf P}^2$ with a fixed Hilbert function or FDH function form an irreducible and smooth locally closed substack of the stack of coherent sheaves on ${\bf P}^2$. \end{lemma} Our next lemma relates the filterability of the FDH function $h$ to sheaf theory. It is essentially a generalization of Davis' decomposition lemma \cite{D} to higher rank. \begin{lemma} \label{gener}Suppose $h$ is an FDH function which is filterable at an integer $m$. Suppose ${\cal E}$ is a general coherent sheaf without zero-dimensional torsion with FDH function $h$. Let ${\cal E}_{\leq m}$ be the subsheaf of ${\cal E}$ generated by $H^0({\cal E}(m))$. Then ${\cal E}% _{\leq m}$ has FDH function $h_m^{{\rm sub}}$, and ${\cal E}/{\cal E}_{\leq m}$ has FDH function $h_m^{{\rm quot}}$. Moreover, if ${\cal E}$ is generic, then so are ${\cal E}_{\leq m}$ and ${\cal E}/{\cal E}_{\leq m}$. If ${\cal E% }$ is the cokernel of an injection $\phi {:}\ \bigoplus_n{\cal O}_{{\bf P}% ^2}(-n)^{b(n)}\rightarrow \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{a(n)}$, then $% {\cal E}_{\leq m}$ and ${\cal E}/{\cal E}_{\leq m}$ have resolutions% \begin{eqnarray*} & 0\rightarrow \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}_{\leq m}\rightarrow 0 & \\ & 0\rightarrow \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}% _{>m}\rightarrow 0 & \end{eqnarray*} \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}Consider the morphism of exact sequences% $$ \begin{array}{ccccccl} 0\rightarrow & \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{b(n)} & \rightarrow & \bigoplus {\cal O}_{{\bf P}^2}(-n)^{b(n)} & \rightarrow & \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{b(n)} & \rightarrow 0 \\ & \downarrow \phi ^{\prime \prime } & & \downarrow \phi & & \downarrow \phi ^{\prime } & \\ 0\rightarrow & \bigoplus_{n\leq m}{\cal O}_{{\bf P}^2}(-n)^{a(n)} & \rightarrow & \bigoplus {\cal O}_{{\bf P}^2}(-n)^{a(n)} & \rightarrow & \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{a(n)} & \rightarrow 0. \end{array} $$ The Poincar\'e series associated to $\phi ^{\prime }$ and $\phi ^{\prime \prime }$ are, respectively,% \begin{eqnarray*} (1-t)^{-2}\sum_{n>m}\{a(n)-b(n)\}t^n & = & \sum_nh_m^{\rm quot}(n)t^n, \\ (1-t)^{-2}\sum_{n\leq m}\{a(n)-b(n)\}t^n & = & \sum_nh_m^{\rm sub}(n)t^n. \end{eqnarray*}Since $h$ is filterable at $m$, $h_m^{{\rm quot}}$ and $h_m^{% {\rm sub}}$ are both FDH functions by (\ref{filter}). If $\phi $ is general, then $\phi ^{\prime }$ is general and so injective by the filtered Bertini Theorem \ref{Chang}. In any case the snake lemma yields an exact sequence% $$ 0\rightarrow \ker (\phi ^{\prime })\rightarrow {\rm cok(\phi ^{\prime \prime })}\stackrel{\psi }{\rightarrow }{\cal E}\rightarrow {\rm cok(\phi ^{\prime })}\rightarrow 0 $$ such that ${\rm im}(\psi )={\cal E}_{\leq m}$. So if $\phi $ is general, then ${\rm cok(\phi ^{\prime \prime })}={\cal E}_{\leq m}$ and ${\rm % cok(\phi ^{\prime })}={\cal E}/{\cal E}_{\leq m}$, and they have FDH functions $h_m^{{\rm sub}}$ and $h_m^{{\rm quot}}$, respectively. Finally, if ${\cal E}$ is generic, then $\phi $ is generic, which implies the genericity of $\phi ^{\prime }$ and $\phi ^{\prime \prime }$ and thus of $% {\cal E}_{\leq m}$ and ${\cal E}/{\cal E}_{\leq m}$. \TeXButton{qed} {\hfill $\Box$ \medskip} \subsection{Davis' Decomposition Lemma.} If ${\cal E}$ is torsion-free and $r(m)=1=\min _{n\geq m}r(n)$, then the lemma holds without any condition that ${\cal E}$ be general. This is because the vanishing of $\ker (\phi ^{\prime })$ may be shown without invoking the filtered Bertini theorem. For the condition $r(m)=1$ implies that ${\rm cok(\phi ^{\prime \prime })}$ is of rank $1$. Since it has nonzero image in ${\cal E}$ which is torsion-free, it follows that $\ker (\phi ^{\prime })$ is of rank $0$. But since $\ker (\phi ^{\prime })\subset \bigoplus_{n>m}{\cal O}_{{\bf P}^2}(-n)^{b(n)}$, it is also torsion-free. So it vanishes. In the case of an ${\cal E}$ of rank one, this is more or less Davis' Decomposition Lemma \cite{D}. \subsection{The WHN Filtration of a General Sheaf.\label{WHNfilt}} Because of the last lemma, if $h$ is an FDH function with a filtration $% m_0<m_1<\cdots <m_s$, then the general sheaf ${\cal E}$ with FDH function $h$ will have a filtration \begin{equation} \label{filt.sheaf}0\subset {\cal E}_{\leq m_0}\subset {\cal E}_{\leq m_1}\subset \cdots \subset {\cal E}_{\leq m_s}\subset {\cal E.} \end{equation} If we write ${\rm gr}_i({\cal E})={\cal E}_{\leq m_i}/{\cal E}_{\leq m_{i-1}} $ for $i=1,\ldots ,s$, and ${\rm gr}_0({\cal E})={\cal E}_{\leq m_0} $ and ${\rm gr}_{s+1}({\cal E})={\cal E}/{\cal E}_{\leq m_s}$, then the FDH function of ${\rm gr}_i({\cal E})$ is the function $h_i$ of (\ref{filter}% ). If ${\cal E}$ has resolution% $$ 0\rightarrow \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{b(n)}\rightarrow \bigoplus_n{\cal O}_{{\bf P}^2}(-n)^{a(n)}\rightarrow {\cal E}\rightarrow 0, $$ then each ${\rm gr}_i({\cal E})$ has resolution% $$ 0\rightarrow \bigoplus_{m_{i-1}<n\leq m_i}{\cal O}_{{\bf P}% ^2}(-n)^{b(n)}\rightarrow \bigoplus_{m_{i-1}<n\leq m_i}{\cal O}_{{\bf P}% ^2}(-n)^{a(n)}\rightarrow {\rm gr}_i({\cal E})\rightarrow 0. $$ If we apply this with the WHN filtration of $h$ of Lemma \ref{WHN}, then we call the resulting filtration of ${\cal E}$ the {\em WHN filtration }of the sheaf ${\cal E}$. This filtration only exists for a general sheaf with FDH function $h$ because the construction of the filtration of the sheaf depended ultimately on the filtered Bertini theorem.\bigskip\ We now recall some terminology. If ${\cal E}$ is a coherent sheaf of rank $% \rho >0$ on ${\bf P}^2$, then its reduced Hilbert polynomial is $P_{{\cal E}% }(n):=\chi ({\cal E}(n))/\rho $. Such polynomials may be ordered by $P_{% {\cal E}}\succ P_{{\cal F}}$ (resp.$\ P_{{\cal E}}\succeq P_{{\cal F}}$) if $% P_{{\cal E}}(n)>P_{{\cal F}}(n)$ (resp.\ $P_{{\cal E}}(n)\geq P_{{\cal F}% }(n) $) for $n\gg 0$. This order is compatible with the order on FDH functions of Lemma \ref{compar} in the sense that if ${\cal E}$ has FDH function $h_{{\cal E}}$ and ${\cal F}$ has FDH function $h_{{\cal F}}$, then $P_{{\cal E}}\succ P_{{\cal F}}$ if and only if $h_{{\cal E}}\succ h_{{\cal F% }}$, and $P_{{\cal E}}\succeq P_{{\cal F}}$ if and only if $h_{{\cal E}% }\succeq h_{{\cal F}}$. \begin{lemma} \label{ideals}Let $h$ be an FDH function, let the $h_i$, $\rho _i$, $\nu _i$% , $\beta _i$, and $h_i^j$ be as in Lemma \ref{compar}. Then there exists a coherent sheaf ${\cal F}$ with FDH function $h$ of the form ${\cal F}% =\bigoplus_{i=0}^{s+1}{\cal F}_i$ such that (i)$\quad {\cal F}$ admits the WHN filtration with graded pieces ${\rm gr}_i(% {\cal F})\cong {\cal F}_i$. (ii)\quad ${\cal F}_0$ is the torsion subsheaf of ${\cal F}$. (iii)\quad For $i=1,\ldots ,s+1$ we have ${\cal F}_i=\bigoplus_{j=1}^{\beta _i}{\cal I}_{Z_i^j}(-\nu _i)\oplus \bigoplus_{j=\beta _i+1}^{\rho _i}{\cal I}% _{Z_i^j}(-\nu _i-1)$ where the $Z_i^j$ are disjoint sets of distinct points. Moreover, $Z_i^j\neq \emptyset $ for $1\leq j\leq \beta _i$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}Let ${\cal F}_0$ be a general sheaf with FDH function $h_0$, and let ${\cal F}_i^j$ be a general sheaf with FDH function $% h_i^j$. For $i=1,\ldots ,s+1$, let ${\cal F}_i=\bigoplus_{j=1}^{\rho _i}% {\cal F}_i^j$ and ${\cal F}=\bigoplus_{i=0}^{s+1}{\cal F}_i$. Then ${\cal F}$ has FDH function $h_0+\sum_{i=1}^{s+1}\sum_{j=1}^{\rho _i}h_i^j=h$. We now verify the three asserted conditions in reverse order. (iii) By Lemma \ref{hi} (ii), the $h_i^j$ are torsion-free sheaves of rank $% 1 $. So by Theorem \ref{Chang} ${\cal F}_i^j$ is a twist of an ideal sheaf of a set $Z_i^j$ of distinct points. The twist is given by the degree of $% h_i^j$, which is $-\nu _i$ if $1\leq j\leq \beta _i$ (resp.\ $-\nu _i-1$ if $% \beta _i+1\leq j\leq \rho _i$). Replacing the $Z_i^j$ by projectively equivalent sets of points if necessary, we may assume that the $Z_i^j$ are disjoint. Finally, the cardinality of $Z_i^j$ is the deficiency of $h_i^j$, which is positive if $1\leq j\leq \beta _i$ by Lemma \ref{hi} (iii). (ii) The sheaf ${\cal F}_0$ is torsion because the function $h_0$ is torsion by Lemma \ref{WHN} (i). The other factors ${\cal F}_i$ in the direct sum $% {\cal F}$ are torsion-free by part (iii) which we just proved. So ${\cal F}% _0 $ is exactly the torsion subsheaf of ${\cal F}$. (i) We need to show that for $i=0,\ldots ,s$, the subsheaf ${\cal F}_{\leq \tau _i}\subset {\cal F}$ is $\bigoplus_{k=0}^i{\cal F}_k$. So we show that $% {\cal F}_{i,\leq \tau _i}={\cal F}_i$ and ${\cal F}_{i+1,\leq \tau _i}=0$. First suppose that $g$ is an FDH function of rank $\rho $ and degree $d$, and if $\tau $ is an integer such that $g(n)=\rho (n+1)+d$ for all $n\geq \tau -1$, then $g$ is filterable at $\tau $, and $g_\tau ^{{\rm sub}}=g$ and $g_\tau ^{{\rm quot}}=0$. So if ${\cal G}$ is a general sheaf with FDH function $g$, then ${\cal G}_{\leq \tau }={\cal G}$. If we apply this with $% g=h_0$ and $\tau =\tau _0$, we see that ${\cal F}_{0,\leq \tau _0}={\cal F}% _0 $. We may also apply it with $g=h_i^j$ and $\tau =\tau _i$ because $\tau _i\geq \zeta _i^j+1$ where $\zeta _i^j$ is as in the proof of Lemma \ref {compar}. Thus for $i=1,\ldots ,s$, we have ${\cal F}_{i,\leq \tau _i}=\bigoplus_{j=1}^{\rho _i}{\cal F}_{i,\leq \tau _i}^j=\bigoplus {\cal F}% _i^j={\cal F}_i$. To show that ${\cal F}_{i+1,\leq \tau _i}=0$ we need to show that all $% h_{i+1}^j(\tau _i)=0$. But if $\eta _{i+1}^j$ is as in the proof of Lemma \ref{compar}, then $h_{i+1}^j(n)$ for all $n<\eta _{i+1}^j$. But as we showed there $\eta _{i+1}^j\geq \eta _{i+1}^1\geq \tau _i+1$. This completes the proof of the lemma.\TeXButton{qed}{\hfill $\Box$ \medskip} \paragraph{Proof of Theorem \ref{main}.} Part (i) was shown in (\ref{WHNfilt}). Parts (ii), (iv), and (v) then describe open properties, so it is enough to verify them for the sheaf $% {\cal F}$ of Lemma \ref{ideals}. Part (ii) then follows from Lemma \ref {ideals}(ii). To derive (iv) from \ref{ideals}(iii), note that the fact that the $Z_i^k$ are all disjoint implies that ${\cal H}om({\cal I}_{Z_i^k},{\cal % I}_{Z_j^l})\cong {\cal I}_{Z_j^l}$. So ${\rm Hom}({\rm gr}_i({\cal E}),{\rm % gr}_j({\cal E})(-1))$ is a sum of terms of the forms $H^0({\cal I}% _{Z_j^l}(\nu _i-\nu _j))$, $H^0({\cal I}_{Z_j^l}(\nu _i-\nu _j-1))$ and $H^0(% {\cal I}_{Z_j^l}(\nu _i+1-\nu _j))$. In the first two forms the cohomology vanishes because the twists $\nu _i-\nu _j$ or $\nu _i-\nu _j-1$ are negative. For the third form the twist $\nu _i+1-\nu _j$ is nonpositive, but even if it is zero $H^0({\cal I}_{Z_j^l})$ vanishes because this form only occurs with $1\leq l\leq \beta _j$ and in that case $Z_j^l\neq \emptyset $. Before beginning on (v) note that since the ${\rm gr}_i({\cal F}% )=\bigoplus_{j=1}^{\rho _i}{\cal F}_i^j$ is a direct sum of semistable sheaves, the graded pieces of the Harder-Narasimhan of ${\rm gr}_i({\cal F})$ are direct sums of ${\cal F}_i^j$'s with proportional Hilbert polynomials. Hence any non-torsion quotient sheaf ${\cal G}$ of ${\rm gr}_i({\cal F})$ has $P_{{\cal G}}\succeq \min _j\{P_{{\cal F}_i^j}\}$, which is $P_{{\cal F}% _i^{\beta _i}}$ by Lemma \ref{compar}(i), and any nonzero subsheaf ${\cal H}$ has $P_{{\cal H}}\preceq $ $\max _j\{P_{{\cal F}_i^j}\}=P_{{\cal F}_i^1}$. Now to show that the Harder-Narasimhan filtration of ${\cal F}$ is a refinement of the WHN filtration, we need to show that for $1\leq i\leq s$, if ${\cal G}$ is a nonzero torsion-free quotient of ${\rm gr}_i({\cal F})$ and ${\cal H}$ a nonzero subsheaf of ${\rm gr}_{i+1}({\cal F})$, then $P_{% {\cal G}}\succ P_{{\cal H}}$. But by the previous paragraph and Lemma \ref {compar}(ii) we have $P_{{\cal G}}\succeq P_{{\cal F}_i^{\beta _i}}\succ P_{% {\cal F}_{i+1}^1}\succeq P_{{\cal H}}$. To show the second assertion of (v) we now need to show that every nonzero subsheaf of ${\rm gr}_i({\cal F})$ has slope at most $-\nu _i$ and every non-torsion quotient sheaf has slope at least $-\nu _i-1$. But this is now clear. In part (iii) the isomorphisms ${\rm gr}_i({\cal F}){\mid }_L\cong {\cal O}% _L(-\nu _i)^{\beta _i}\oplus {\cal O}_L(-\nu _i-1)^{\rho _i-\beta _i}$ follow from Lemma \ref{ideals}(iii). Because these latter sheaves are rigid (i.e.\ generic in the stack of coherent sheaves on $L$), a general ${\cal E}$ must have ${\rm gr}_i({\cal E}){\mid }_L$ isomorphic to ${\cal O}_L(-\nu _i)^{\beta _i}\oplus {\cal O}_L(-\nu _i-1)^{\rho _i-\beta _i}$. We now claim that any filtered sheaf ${\cal H}$ such that ${\rm Ext}^1({\rm gr}_i({\cal H}% ),{\rm gr}_j({\cal H}))=0$ for all $i>j$ has ${\cal H}\cong \bigoplus_i{\rm % gr}_i({\cal H})$. This claim can easily be verified by induction on the length of the filtration. To apply this to ${\cal E}{\mid }_L$, we need to verify that if $i>j$, then ${\rm Ext}^1({\rm gr}_i({\cal E}){\mid }_L,{\rm gr% }_j({\cal E}){\mid }_L)=0$. But ${\rm Ext}^1({\rm gr}_i({\cal E}){\mid }_L,% {\rm gr}_j({\cal E}){\mid }_L)$ is a direct sum of terms of the form $H^1(% {\cal O}_L(\nu _i-\nu _j+\epsilon ))$ with $\epsilon \in \{-1,0,1\}$. Since the $\nu _i$ form a strictly increasing sequence of integers, the twists $% \nu _i-\nu _j+\epsilon $ are all nonnegative$.$ So the $H^1$ vanish. Therefore ${\cal E}{\mid }_L\cong \bigoplus_i{\rm gr}_i({\cal E}){\mid }_L$, completing the proof of (iii). \TeXButton{qed}{\hfill $\Box$ \medskip}

9312

alg-geom/9312008

en

https://arxiv.org/abs/alg-geom/9312008

alg-geom/9312008

Gerd Dethloff, Georg Schumacher, Pit-Mann Wong

On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case

26 pages, LaTeX

Duke Math. J. 78, 193-212 (1995)

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a line and three quadrics. The main results are Let C be the union of three curves in P_2 whose degrees are at least two, one of which is at least three. Then for generic such configurations the complement of C is hyperbolic and hyperbolically embedded. The same statement holds for complements of curves in generic hypersurfaces X of degree at least five and curves which are intersections of X with hypersurfaces of degree at least five. Furthermore results are shown for curves on surfaces with picard number one.

alg-geom

\section{Introduction} In complex analysis hyperbolic manifolds have been studied extensively, with close relationships to other areas (cf.\ eg. \cite{LA1}). Hyperbolic manifolds are generalizations of hyperbolic Riemann surfaces to higher dimensions. Despite the fact that the general theory of hyperbolic manifolds is well-developed, only very few classes of hyperbolic manifolds are known. But one could hope that `most' of the pseudoconvex quasi-projective varieties are in fact hyperbolic. In particular it is believed that e.g. the complements of most hypersurfaces in $\Bbb P_n$ are hyperbolic, if only their degree is at least 2n+1. More precisely according to Kobayashi \cite{KO}, and later Zaidenberg \cite{ZA} one has the following: {\bf Conjecture:} {\it Let ${\cal C}(d_1, \ldots , d_k)$ be the space of $k$ tupels of hypersurfaces $\, C = (C_1 , \ldots , C_k )\, $ in $\Bbb P_n$, where ${\rm deg}(C_i)=d_i$. Then for all $(d_1, \ldots , d_k)$ with $\, \sum_{i=1}^k d_i =:d \geq 2n+1\, $ the set $\, {\cal H}(d_1, \ldots , d_k)= \{ C \in {\cal C}(d_1, \ldots , d_k) : \Bbb P_n \setminus \bigcup_{i=1}^k C_i\, $ {\rm is complete hyperbolic and hyperbolically embedded}$\}\, $ contains the complement of a proper algebraic subset of ${\cal C}(d_1, \ldots , d_k)$.} In this paper we shall restrict ourselves to the two dimensional case. However we consider also more general quasi-projective complex surfaces than the complements of curves in the projective plane. Concerning the above conjecture, the following was known: It seems that the conjecture is the more difficult the smaller $k$ is. Other than in the case of 5 lines $({\cal C}(1, 1, 1, 1, 1))$, the conjecture was previously proved by M.~Green in \cite{GRE2} in the case of a curve $C$ consisting of one quadric and three lines (${\cal C}(2, 1, 1, 1)$). Furthermore, it was shown for ${\cal C}(d_1, \ldots, d_k)$, whenever $k\geq5$, by Babets in \cite{BA}. A result which went much further was given by Eremenko and Sodin in \cite{E-S}, where they proved a Second Main Theorem of value distribution theory in the situation $k \geq 5$. Green proved in \cite{GRE1} that for any hypersurface $C$ consisting of at least four components in $\Bbb P_2$ any entire curve $f:\Bbb C \to \Bbb P_2 \setminus C$ is algebraically degenerate. Knowing this, it follows immediately that for generic configurations, any such algebraically degenerate map is constant, hence the conjecture is true for any family ${\cal C}(d_1, \ldots, d_k)$ with $k \geq 4$ (cf. \cite{DSW}). (The degeneracy locus of the Kobayashi pseudometric was studied by Adachi and Suzuki in \cite{A--S1}, \cite{A--S2}). In our paper \cite{DSW} we gave a proof of the conjecture for 3 quadrics (${\cal C}(2, 2, 2)$), based on methods from value distribution theory. The three quadric case had been previously studied by Grauert in \cite{GR} who used differential geometry. However, certain technical problems still exist with this approach. For ${\cal C}(2, 2, 1)$, i.e. two quadrics and a line, we proved with similar methods the existence of an open set in the space of all such configuarions, which contains a quasi-projective set of codimension one, where the conjecture is true. The paper contains two main results. The first is Theorem~\ref{MT}. It states that the conjecture is true for almost all three component cases, namely for ${\cal C}(d_1, d_2, d_3)$ with $d_1, d_2, d_3 \geq 2$ and at least one $d_i \geq 3$. Together with our result for three quadrics (which, by the way, occur on the borderline of the method used in this paper) this means that the conjecture is true for three components whenever none of them is a line. We finally remark that we get a weaker conclusion also for ${\cal C}(d_1, d_2, d_3)$ where, up to enumeration, $d_1=1$, $d_2 \geq 3$, $d_3 \geq 4$: Namely we show that any holomorphic map $f:\Bbb C \to X$ is algebraically degenerated, i.e.\ $f(\Bbb C)$ is contained in a proper algebraic subset of $X$. The other main result is Theorem~\ref{MT1}. We consider a smooth surface $\bar X$ in $\Bbb P_3$ of degree at least five for which every curve on $\bar X$ is the complete intersection with another hypersurface. Surfaces of this kind are much more general than $\Bbb P_2$ -- by the Noether-Lefschetz theorem (cf. \cite{N-L}) the 'generic' surface in $\Bbb P_3$ of any given degree at least four has this property (`generic' here indicates the complement of a countable union of proper varieties). Let $C$ be a curve on $\bar X$ consisting of three smooth components intersecting transversally. From our assumptions we know that $C$ is a complete intersection of $\bar X$ and a hypersurface $B$. We assume that the degree of $B$ is at least five. Now Theorem~\ref{MT1} states the hyperbolicity of any such $X=\bar X \setminus C$. Moreover $X$ is complete hyperbolic and hyperbolically embedded. Our method of proof is the following: We heavily use a theorem due to S.~Lu (cf. \cite{Lu}). It states that for a certain class of differentials $\sigma$, which may have logarithmic poles along the curve $C$, and any holomorphic map $f: \Bbb C \rightarrow \Bbb P_2 \setminus C$ the pull-back $f^*(\sigma)$ vanishes identically. This can be interpreted as algebraic degeneracy of the tangential map corresponding to $f: \Bbb C \rightarrow X$. Our aim is to show algebraic degeneracy of the map $f$ itself. The paper is organized as follows: In section~2 we collect, for the convenience of the reader, some basics from value distribution theory. (Readers who are familar with these may skip this section). In section~3 we fix the notation and quote some theorems which are needed in the following proof, especially Lu's theorem. Furthermore we examine more closely the spaces of sections which are used in Lu's theorem and get sections with special zero sets. The essential step of our paper is the proof of Theorem~\ref{deg} in section~4. It states the algebraic degeneracy of holomorphic maps $f:\Bbb C \to X$, if ${\rm Pic}(\bar X)=\Bbb Z$, and under assumptions on the determinant bundle and the Chern numbers of the logarithmic cotangent bundle on $\bar X$ with respect to $C$. The proof uses value distribution theory and the existence of the special sections which were constructed in section~3. In section~5 we compute the Chern numbers and the determinant bundle in the situation where $\bar X$ is a complete intersection (Theorem~\ref{main}). We apply this to $\bar X = \Bbb P_2$, and to hypersurfaces in $\Bbb P_3$ using the Noether Lefschetz theorem (Theorem~\ref{main2}). Finally in section~6 we apply Theorem~\ref{main2} and get Theorem~\ref{MT}, using an argument like in our paper \cite{DSW} to prove the nonexistence of algebraic entire curves in generic complements. Furthermore we apply Theorem~\ref{main2} using results of Xu \cite{Xu} and Clemens \cite{Cl} to get Theorem~\ref{MT1}. The first named author would like to thank S.~Kosarew (Grenoble) for valuable discussions. The second named author would like to thank the SFB 170 at G"ottingen, and the third named author would like to thank the SFB 170 and the NSF for partial support. \section{Some tools from Value Distribution Theory} In this section we fix some notations and quote some facts from Value Distribution Theory. We give references but do not trace these facts back to the original papers. We define the characteristic function and the counting function, and give some formulas for these. Let $\,||z||^2= \sum_{j=0}^n |z_j|^2$, where $(z_0,\ldots ,z_n) \in \Bbb C^{n+1}$, let $\Delta_t = \{\xi \in \Bbb C : |\xi| < t \}$, and let $d^c = (i/4 \pi) (\overline{\partial} - \partial)$. Let $r_0$ be a fixed positive number and let $\,r \geq r_0$. Let $\,f:\Bbb C \rightarrow \Bbb P_n\,$ be entire, i.e. $f$ can be written as $\, f=[f_0:\ldots :f_n]\,$ with holomorphic functions $\, f_j : \Bbb C \rightarrow \Bbb C\, , j=0,\ldots ,n\,$ without common zeroes. Then the {\it characteristic function} $T(f,r)$ is defined as $$ T(f,r) = \int_{r_0}^r \frac{dt}{t} \int_{\Delta_t} dd^c \log ||f||^2$$ Let furthermore $\, D=V(P)$ be a divisor in $\Bbb P_n$, given by a homogeneous polynomial $P$. Assume $\, f(\Bbb C) \not\subset \hbox{ {\rm support}}(D)$. Let $\,n_f(D,t)\,$ denote the number of zeroes of $\, P \circ f\,$ inside $\, \Delta_t\,$ (counted with multiplicities). Then we define the {\it counting function} as $$ N_f(D,r) = \int_{r_0}^r n_f(D,t) \frac{dt}{t} $$ Stokes Theorem and transformation to polar coordinates imply (cf. \cite{WO}): \begin{equation} \label{1} T(f,r) = \frac{1}{4 \pi} \int_0^{2 \pi} \log ||f||^2 (re^{i \vartheta})d \vartheta + O(1). \end{equation} The characteristic function as defined by Nevanlinna for a holomorphic function $\,f: \Bbb C \rightarrow \Bbb C$ is $$ T_0(f,r) = \frac{1}{2 \pi} \int_0^{2 \pi} \log ^+ |f(re^{i \vartheta})| d \vartheta . $$ For the associated map $\, [f:1]: \Bbb C \rightarrow \Bbb P_1$ one has \begin{equation} \label{2} T_0(f,r) = T([1:f],r) + O(1) \end{equation} (cf. \cite{HA}). By abuse of notation we will, from now on, for a function $\, f: \Bbb C \rightarrow \Bbb C$, write $T(f,r)$ instead of $T_0(f,r)$. Furthermore we sometimes use $N(f,r)$ instead of $N_f({z_0=0},r)$. We state some elementary properties of the characteristic function: \begin{lem}\label{calc} Let $f,g,f_j: \Bbb C \to \Bbb C$ be entire holomorphic functions for $j=0,\ldots,n$. Then \begin{description} \item[a)] $$ T(f\cdot g, r) \leq T(f,r) + T(g,r) + O(1) $$ \item[b)] $$ T([f_0:\ldots:f_n],r) \leq \sum_{j=0}^n T(f_j,r) + O(1) $$ \item[c)] $$ T(f+g,r) \leq T(f,r)+ T(g,r) + O(1) $$ \end{description} \end{lem} {\it Proof:}\/ Propterty a) is obvious for $T_0$ and generalizes to $T$ because of (\ref{2}). Property b) is a consequence of $$ \log \sum_{j=0}^n|f_j|^2 \leq \sum_{j=0}^n \log (1+|f_j|^2). $$ Property c) is a consequence of $$ \log^+|f+g| \leq \log (1+|f|+|g|) \leq \log(1+|f|) +\log(1+|g|) \leq \log^+|f| + \log^+|g| +2 $$ \qed Later on we will use the concept of finite order. \begin{defi} Let $s(r)$ be a positive, monotonically increasing function defined for $\,r \geq r_0$. If $$ \overline{\lim_{r \rightarrow \infty} } \frac{\log s(r)}{\log r} = \lambda$$ then $s(r)$ is said to be of order $\lambda$. For entire $\,f:\Bbb C \rightarrow \Bbb P_n\,$ or $\, f: \Bbb C \rightarrow \Bbb C\,$ we say that $f$ is of order $\lambda$, if $T(f,r)$ is. \end{defi} \begin{rem}\label{remfo} Let $f=[f_0:\ldots:f_n]:\Bbb C \to \Bbb P_n$ be a holomorphic map of finite order $\lambda$. Then $\log T(f,r)= O(\log r)$. \end{rem} For holomorphic maps to $\Bbb P^1$ whose characteristic function only grows like $\log r$ we have the following characterization (cf. \cite{HA}): \begin{lem}\label{logr} Let $f=[f_0:f_1]: \Bbb C \rightarrow \Bbb P^1$ be entire. Then $T(f,r) =O(\log r)$ if and only if the meromorphic function $f_0/f_1$ is equal to a quotient of two polynomials. \end{lem} We need the following: \begin{lem} \label{e} Assume that $\,f: \Bbb C \rightarrow \Bbb P_n\,$ is an entire map and misses the divisors $\,\{ z_j = 0\}\,$ for $j=0,\ldots,n$ (i.e. the coordinate hyperplanes of $\Bbb P_n$). Assume that $f$ has order at most $\lambda$. Then $f$ can be written as $\,f = [1:f_1:\ldots :f_n]\,$ with $\, f_j(\xi) = e^{P_j(\xi)}$, where the $P_j(\xi)$ are polynomials in $\xi$ of degree $d_j\leq \lambda$. \end{lem} {\it Proof:} We write $\, f=[1:f_1:\ldots :f_n]\,$ with holomorphic $\,f_j: \Bbb C \rightarrow \Bbb C \setminus \{0\}$. Now we get with equations (\ref{1}) and (\ref{2}) for $j=1,\ldots ,n$: $$ T(f_j,r) = T([1:f_j],r) + O(1) \leq T(f,r) + O(1), $$ hence the $f_j$ are nonvanishing holomorphic functions of order at most $\lambda$. This means that $$ {\rm lim sup}_{r \rightarrow \infty} \frac{T(f_j,r)}{r^{\lambda + \epsilon}} =0 $$ for any $\, \epsilon > 0$. From this equation our assertion follows with the Weierstra\char\ss theorem as it is stated in \cite{HA}. \qed We state the First and the Second Main Theorem of Value Distribution Theory which relate the characteristic function and the counting function (cf. \cite{SH}): Let $\,f:\Bbb C \rightarrow \Bbb P_n\,$ be entire, and let $D$ be a divisor in $\Bbb P_n$ of degree $d$, such that $\,f(\Bbb C) \not\subset \hbox{ {\rm support}}(D)$. Then: \medskip {\bf First Main Theorem} $$ N_f(D,r) \leq d \cdot T(f,r) + O(1)$$ Assume now that $\, f(\Bbb C)\,$ is not contained in any hyperplane in $\Bbb P_n$, and let $\, H_1,\ldots ,H_q\,$ be distinct hyperplanes in general position. Then \medskip {\bf Second Main Theorem} $$ (q-n-1)T(f,r) \leq \sum_{j=0}^q N_f(H_j,r) + S(r) $$ where $\: S(r) \leq O(\log (rT(f,r)))\,$ for all $\,r \geq r_0\,$ except for a set of finite Lebesque measure. If $f$ is of finite order, then $\, S(r) \leq O(\log r)\,$ for all $\,r \geq r_0$. \section{Setup and Basic Methods} We denote by $\bar X$ a non-singular projective surface and by $C$ a curve in $\bar X$ whose irreducible components are smooth and intersect each other only in normal crossings. Let $X=\bar X \setminus C$. We denote by $E$ the dual of the bundle $\Omega^1_{\bar X}(\log C)$ of holomorphic one forms of $\bar X$ with logarithmic poles along $C$. Then we define the projectivized logarithmic tangent bundle $p:\Bbb P (E)\to \bar X$ over $\bar X$ to be the projectivized bundle whose fibers correspond to the one dimensional subspaces of the fibers of $E$. Furthermore let ${\cal O}_{\Bbb P (E)}(-1)$ be the sheaf associated to the tautological line bundle on $\Bbb P (E)$, for which we have the canonical isomorphism between the total space of ${\cal O}_{\Bbb P (E)}(-1) \setminus \{{\rm zero section}\}$ and the total space of $E \setminus \{ {\rm zero section}\}$. Let $D$ be a divisor on $\bar X$. According to a Theorem of Kobayashi-Ochiai (cf. \cite{K-O}), the cohomology, in particular the holomorphic sections, of a symmetric power of $E^*$ tensorized with the bundle $[-D]$, corresponds to the cohomology of the m-th power of the dual of the tautological line bundle on $\Bbb P(E)$, tensorized with the pull-back of $[-D]$: $$ H^0(\bar X, S^m(E^*)\otimes [-D]) \simeq H^0(\Bbb P(E),{\cal O}_{\Bbb P(E)}(m)\otimes p^*[-D]), $$ and in particular $$ H^0(\bar X, S^m(E^*)) \simeq H^0(\Bbb P(E),{\cal O}_{\Bbb P(E)}(m)). $$ Let $f:\Bbb C \to X$ be a holomorphic map. Denote by $$(f, f'): T(\Bbb C) \rightarrow T(X)$$ the induced map from $T(\Bbb C)$ to the holomorphic tangent bundle $T(X)$, which gives rise to a meromorphic map $$F: \Bbb C \rightarrow \Bbb P (T(X))$$ from $\Bbb C$ to $\Bbb P(T(X))$. Since the domain is of dimension one, points of indeterminacy can be eliminated, more precisely the map $F$ extends holomorphically into the points $\xi \in \Bbb C$ where $f' (\xi)=0$. We denote the extended holomorphic map on $\Bbb C$ again by $F$. Since the restriction of $E$ to $X$ is isomorphic to the holomorphic tangent bundle of $X$, any map $$f: \Bbb C \rightarrow X$$ has a unique holomorphic lift $F:\Bbb C \to \Bbb P(E)$. The following theorem, which is a special case of Theorem~2 of Lu in \cite{Lu} (it actually follows already from Proposition~4.1 there) imposes restrictions to such lifts $F$. \begin{theo}[Lu]\label{Lutheo} Assume that the divisor $D$ is ample and that there exist a non-trivial holomorphic section $$ 0\neq \sigma \in H^0(\Bbb P(E),{\cal O}_{\Bbb P(E)}(m)\otimes p^*[-D]). $$ Then for any non-constant holomorphic map $f:\Bbb C \to X$ the holomorphic lift $F:\Bbb C \to \Bbb P(E)$ has values in the zero-set of $\sigma$. \end{theo} In order to apply Lu's theorem in a given situation it is important to guarantee the existence of suitable sections. Let $(\bar X, C)$ be given as above. The logarithmic Chern classes $\bar c_j(X)$ are by definition the Chern classes of the logarithmic tangent bundle $E$: $$ \bar c_j(X)= c_j(E) = c_j(X,(\Omega^1_{\bar X}(\log C))^*). $$ Now the existence of suitable sections is guaranteed by the following theorem of Bogomolov (cf. \cite{Bo} and also Lu \cite{Lu} ( Proof of Proposition~3.1 and localization to the divisor $D$). \begin{theo}[Bogomolov]\label{Bogo} Let $D$ be a divisor on $\bar X$, $D$ effective (i.e.\ \\ $D \geq 0$). Assume that $$ \bar c_1^2(X) - \bar c_2(X) > 0, $$ and that $$ {\rm det}(E^*) $$ is effective. Then there exist positive constants $A, B$ and $m_0, n_0 \in \Bbb N$, such that $$ A\cdot m^3 \leq h^o(\bar X, S^{mn_0}(E^*)\otimes [-D]) \leq B\cdot m^3 $$ for all $m\geq m_0$. \end{theo} We have the following nonexistence statement, which is a consequence of the logarithmic version of the Bogomolov's lemma due to Sakai (cf. \cite{Sak}). It will also become important for the following proofs. \begin{lem}\label{lem2} Assume that the divisor $D$ is ample. Then the following group vanishes: $$ H^0(\Bbb P(E), {\cal O}_{\Bbb P(E)}(1)\otimes p^*[-D])=\{0\}. $$ In particular, there is no logarithmic $1$-form on $\bar X$ which vanishes on $D$. \end{lem} {\it Proof:}\/ The existence of a non-trivial section $s \in H^0( \bar X, \Omega^1_{\bar X}(\log C)\otimes [-D])$ implies that the invertible sheaf ${\cal L}:=[D]$ can be realized as a subsheaf of $\Omega^1_{\bar X}(\log C)$. According to a result of Sakai \cite{Sak}, (7.5), this implies that the $\cal L$-dimension of $ \bar X$ equals one, which is clearly impossible since $[D]$ is ample. \qed Next we deal with divisors in $\Bbb P (E)$ which project down to all of $\bar X$: \begin{defi} Consider the projection $p:\Bbb P(E) \to \bar X$. We call a divisor $Z\subset \Bbb P(E)$ horizontal, if $p(Z)=\bar X$. \end{defi} Those horizontal divisors which occur as parts of the zero sets $V(\sigma)$ of sections $0\neq \sigma \in H^0(\Bbb P(E), {\cal O}_{\Bbb P(E)}(m)\otimes p^*[-D])$ will play an important role in the sequel. We study this relationship somewhat closer. \begin{lem}\label{horcomp} Given $$ 0\neq \sigma \in H^0(\Bbb P (E), {\cal O}(m)\otimes p^*[-D]) $$ there exist divisors $E_j$ $ j=1, \ldots, l$ on $\bar X$, and numbers $a_j, n_j\in \Bbb N$ such that $[\sum a_j\cdot E_j - D] \geq 0$ and sections $s_j\in H^0(\Bbb P(E), {\cal O}(n_j)\otimes p^*[-E_j])$, $\tau \in H^0(\Bbb P(E), p^*[\sum a_j\cdot E_j - D])$ such that $\sigma = \tau \otimes_{1\leq j \leq l} s_j^{a_j}$ with the following property: The zero-sets of $s_j$ are precisely the irreducible horizontal components of $V(\sigma)$. \end{lem} {\it Proof:}\/ Let $0\neq \sigma \in H^0(\Bbb P(E), {\cal O}(m)\otimes p^*[-D])$ be a non-trivial section and $V(\sigma)$ its zero divisor. We denote by $S_j$; $j=1, \ldots l$ the irreducible horizontal components of $V(\sigma)$. Since ${\rm Pic }(\Bbb P(E))={\rm Pic }(X) \oplus \Bbb Z$, we get $[S_j]= {\cal O}_{\Bbb P(E)}(n_j)\otimes p^*[-E_j]$ for certain divisors $E_j\subset \bar X$ with $n_j \geq 1$. This fact follows by restricting the bundles $[S_j]$ to a generic fiber of $p$. Let $a_j$ be the multiplicities of $\sigma$ with respect to $S_j$, then in particular $a_1n_1+\ldots a_ln_l =m$. (This fact follows again by restricting bundles and sections to a generic fiber of $p$.) Canonical sections of $[S_j]$ give rise to non-trivial sections $s_j\in H^0(\Bbb P(E), {\cal O}(n_j)\otimes p^*[-E_j])$ which vanish exactly on $S_j$. Thus $\tau:= \sigma/( s_1^{a_1}\cdot\ldots\cdot s_l^{a_l})$ is a (holomorphic) section of $H^0(\Bbb P(E), p^*[\sum a_j\cdot E_j - D])$. In particular $[\sum a_j\cdot E_j - D] \geq 0$. \qed In order to control the horizontal divisors of a section of $H^0(\Bbb P(E), {\cal O}(m)\otimes p^*[-D])$, the number $m$ will be chosen minimal in the following sense. For any $k\in \Bbb N$ we set $$ \mu_k:=\inf\{m;h^0(\Bbb P(E), {\cal O}(m)\otimes p^*[-kD])\neq 0 \}. $$ and $$ \mu := \inf_{k \in \Bbb N} \{\mu_k\}. $$ \begin{lem}\label{onecomp} Assume that ${\rm Pic}(\bar X)= \Bbb Z$ and that $[D]$ is the ample generator of ${\rm Pic}( \bar X)$.\\ Then we have $$ 2 \leq \mu < \infty $$ and if $k_0 ={\rm min}\{k \in \Bbb N :\mu_k = \mu\}$, there exists a non-trival section $$ 0\neq \sigma \in H^0(\Bbb P (E), {\cal O}(\mu)\otimes p^*[-k_0D]) $$ such that exactly one horizontal component of $V(\sigma)$ exists and has multiplicity one. \end{lem} {\it Proof:}\/ Since some multiple of $D$ is a very ample and hence linear equivalent to an effective divisor, we have $\mu < \infty$ from Theorem~\ref{Bogo}. From Lemma~\ref{lem2} we then get $\mu \geq 2$.\\ Now take any section $0\neq \sigma \in H^0(\Bbb P (E), {\cal O}(\mu)\otimes p^*[-k_0D])$. We use Lemma~\ref{horcomp}. Since $[D]$ is a generator of ${\rm Pic }(\bar X)$, there exist $b_j \in \Bbb Z$ such that $[E_j]=b_j\cdot [D]$. Since $[\sum a_j\cdot E_j - k_0 D] \geq 0$ we have $\sum a_jb_j \geq k_0$. Since all $a_j\geq 0$, there must be at least one $b_j>0$, say $b_1> 0$. Now $s_1\in H^0(\Bbb P(E), {\cal O}(n_1)\otimes p^*[-b_1\cdot D])$ is a non-trivial section. By definition of $\mu$ we have $n_1\geq \mu$ which means $n_1=\mu$, since $\sum a_jn_j = \mu$. So in terms of the notion of Lemma~\ref{horcomp} $\sigma=\tau \cdot s_1$, i.e.\ $S$ contains only one horizontal component. This component has multiplicity one. \qed \section{Algebraic Degeneracy Of Entire Curves} Let $\bar X $ be a non-singular (connected) projective surface. \begin{defi} Let $f:\Bbb C \to \bar X$ be a holomorphic map. We call $f$ {\em algebraically degenerate}, if there exists an algebraic curve $A \subset \bar X$ such that $f(\Bbb C)$ is contained in $A$. \end{defi} Our main result on algebraic degeneration is: \begin{theo}\label{deg} Let $C \subset \bar X$ be a curve consisting of three smooth components with normal crossings. Assume that:\\ i) ${\rm Pic}(\bar X) = \Bbb Z$\\ ii) The logarithmic Chern numbers of $X = \bar X \setminus C$ satisfy the inequality $$\bar c_1^2(X) - \bar c_2(X) > 0$$ iii) The line bundle ${\rm det}(E^*)$ is effective, where $E^* = \Omega_X^1(\log C)$ is the logarithmic cotangent bundle.\\ Then any holomorphic map $f:\Bbb C \to \bar X\setminus C$ of order at most two is algebraically degenerate. \end{theo} Remark: The theorem also holds without the assumption on the order of the map $f$, but since we are mostly interested in the hyperbolicity of the complement, we include this assumption, because it slightly simplifies the proof.\\ The rest of this section is devoted to the proof of this Theorem. Let again $[D]$ be an ample generator of ${\rm Pic}(\bar X)$. Let $k \in \Bbb N$ be a natural number such that $[kD]$ is very ample. Then by Theorem~\ref{Bogo} there exists a symmetric differential $\omega \in H^0(\bar X, S^m(E^*) \otimes [-kD])$ which is not identically zero. By Theorem~\ref{Lutheo} we know that $f^*\omega \equiv 0$.\\ The proof now will work as follows: The three components of the curve $C$ give rise to a morphism $\Phi:\bar X \to \Bbb P_2$ which maps $C$ to the union of the three coordinate axis. In the first step of the proof we show that we can `push down' the symmetric differential $\omega$ by this morphism to some symmetric rational differential $\Omega$ on $\Bbb P_2$ and that we still have $(\Phi \circ f)^*(\Omega) \equiv 0$. Since $\Phi \circ f$ maps the complex plane to the complement of the three coordinate hyperplanes in $\Bbb P_2$, we will be able to interpret this, in the second step of the proof, as an equation for nonvanishing functions with coefficients which may have zeroes, but which grow of smaller order, only. In such a situation we then can apply Value Distribution Theory. {\bf First step:} We first remark that the intersection number of any two curves $D_1$ and $D_2$ is positive (including self intersection numbers). Let $[D]$ be the ample generator of ${\rm Pic}(\bar X) = \Bbb Z$. Now $[D_j] = a_j [D]$; $a_j \in \Bbb Z$, and $0< D_j \cdot D = a_j D^2$ (cf.\ the easy implication of the Nakai criterion). Hence all $a_j$ are positive, and $$ D_1 \cdot D_2 = a_1 a_2 D^2 >0. $$ We can find $a_j \in \Bbb N$; $ j=1, 2, 3$ such that $[a_1C_1]=[a_2C_2]=[a_3C_3]$, since the divisors $C_j$ $ j=1, 2, 3$ are effective. Let $\sigma_j \in H^0(\bar X, L)$ be holomorphic sections which vanish exactly on $C_j$. Then $$ \Phi = [\sigma_1:\sigma_2:\sigma_3]: \bar X \rightarrow \Bbb P_2 $$ defines a rational map, which is a morphism, since the three components do not pass through any point of $\bar X$. \begin{lem} \label{dom} The morphism $\Phi$ is a branched covering. \end{lem} {\it Proof:}\/ Since $C_2\cdot C_3 >0$, the fiber $\Phi^{-1}(1:0:0) = C_2 \cap C_3$ is non-empty. By assumption $C_2 \cap C_3$ consists of at most finitely many points. Hence $\Phi$ is surjective and has discrete generic fibers. Finally $\Phi$ has no positive dimensional fibers at all: Applying Stein factorization we would get a bimeromorphic map. Since there are no curves of negative self-intersection, no exceptional curves exist on $\bar X$(cf. \cite{BPV}). Hence there exist no positive dimensional fibers of $\Phi$. \qed Hence the morphism $\Phi$ is a finite branched covering of $\bar X$ over $\Bbb P_2$ with, let us say $N$ sheets. Let $R$ be the ramification divisor of $\Phi$, $B=\Phi(R)$ the branching locus and $R'=\Phi^{-1}(B)$. Then $$\Phi : \bar X \setminus R' \rightarrow \Bbb P_2 \setminus B$$ is an unbranched covering with $N$ sheets. We now want to construct a meromorphic symmetric $mN$-form $\Omega$ defined on $\Bbb P_2 \setminus B$ from the meromorphic symmetric $m$-form $\omega$ on $\bar X$: For any point $w^0 \in \Bbb P_2 \setminus B$, there exists a neighborhood $U=U(w^0)$ of $w^0$ and $N$ holomorphic maps $a_i(w)$, $a_i: U\to \bar X \setminus R'$; $ i=1, \ldots, N$ such that $\Phi \circ a_i= {\rm id}_U$. By pulling back the symmetric $m$-form $\omega$ by means of these maps we get $N$ meromorphic symmetric $m$-forms $ (a_i)^*(\omega)(w)$ on $U$. Taking now the symmetric product of these $m$-forms, we get the symmetric $(Nm)$-form $\Omega$ on $U$: $$\Omega (w) = \prod_{i=1}^N a_i^*\omega(w). $$ Let $M=Nm$. Defining $g=\Phi \circ f$, we then have: \begin{lem} \label{meromext} The form $\Omega$ extends to a rational symmetric $M$-form on $\Bbb P_2$, which we again denote by $\Omega$. We have $\Omega \not\equiv 0$, but $g^*\Omega =0$. \end{lem} The first statement of this lemma is probably well known, and the second statement is considered to be obvious. But since we did not find a reference, we will include a proof of this Lemma at the end of this section. We proceed with the proof of Theorem~\ref{deg}. Denote the homogeneous coordinates of $\Bbb P_2$ by $w_0,w_1,w_2$. On $\Bbb P_2 \setminus V(w_0)$ we have inhomogeneous coordinates $\xi_1=w_1/w_0, \xi_2 = w_2/w_0$. Hence on $\Bbb P_2 \setminus V(w_0)$ the symmetric $M$-form $\Omega$ can be written as \begin{equation}\label{om} \Omega = \sum_{i=1}^M R_i(\xi_1,\xi_2) (d\xi_1)^i (d\xi_2)^{M-i} \end{equation} where multiplication means the symmetric tensor product here, and the coefficients $R_i(\xi_1, \xi_2)$ are rational functions in $\xi_1$ and $\xi_2$. Now $g:\Bbb C \rightarrow \Bbb P_2$ has values in the complement $\Bbb P_2 \setminus V(w_0w_1w_2)$ of the three coordinate axis, hence the functions $g_j =\xi_j \circ g$ are holomorphic and without zeroes. Since $g^*\omega \equiv 0$ on $\Bbb C$, equation (\ref{om}) implies \begin{equation}\label{omg} \sum_{i=1}^M R_i(g_1(\eta),g_2(\eta)) (g_1'(\eta))^i(g_2'(\eta))^{M-i} \equiv 0 \end{equation} for all $\eta \in \Bbb C$. This equation still holds if we clear the denominators of the $R_i(\xi_1, \xi_2)$ simultaneously, so without loss of generality we may assume from now on that in equation (\ref{omg}) the $R_i(g_1(\eta), g_2(\eta))$ are polynomials in $g_1(\eta)$ and $g_2(\eta)$, i.e. we have \begin{equation} \label{terms} R_i(g_1(\eta),g_2(\eta))= \sum_{j,k} a_{ijk} (g_1(\eta))^j (g_2(\eta))^k \end{equation} Under our assumptions we are able to say more about the functions $g_i$; $i=1, 2$. Since the holomorphic map $f:\Bbb C \rightarrow \bar X \setminus C$ was of finite order at most two, this is also true for $g=\Phi \circ f$ by Lemma~\ref{calc}, since the components of $g$ are polynomials in the components of $f$. Hence by Lemma~\ref{e}, we have \begin{equation} \label{gi} g_i (\eta) = \exp (p_i(\eta)) \end{equation} where the $p_i(\eta)$; $ i=1,2$ are polynomials in $\eta$ of degree at most two. Furthermore we may assume that both polynomials are non-constant, otherwise $g$ would be linearly degenerate and so $f$ would be algebraically degenerate, and we were done. Replacing equation (\ref{gi}) and equation (\ref{terms}) in equation (\ref{omg}) we get \begin{equation}\label{new} \sum_{i=1}^M \sum_{j,k} a_{ijk} \exp\{(i+j)p_1(\eta))+(M-i+k)p_2(\eta)\} (p_1'(\eta))^i(p_2'(\eta))^{M-i} \equiv 0. \end{equation} If we still allow linear combinations of the above summands with constant coefficients $c_{ijk}$ in equation (\ref{new}) we can pass to a subset $S$ of indices which occur in this equation and get a relation \begin{equation}\label{news} \sum_{(i, j, k) \in S} c_{ijk} a_{ijk} \exp\{(i+j)p_1(\eta))+(M-i+k)p_2(\eta)\} (p_1'(\eta))^i(p_2'(\eta))^{M-i} \equiv 0 \end{equation} but now with the additional property that $S$ is minimal with equation (\ref{news}). Let $S$ have $L$ elements. Since we may assume that the polynomials $p_i(\eta)$ are nonconstant, we know that the $p_i' (\eta)$ are not identically zero and hence that $L \geq 2$. For the rest of this proof we will distinguish between two cases:\\ {\bf Case 1:}\/ There exist two summands in equation (\ref{news}) the quotient of which is not a rational function in the variable $\eta$.\\ {\bf Case 2:}\/ The quotient of any two summands in equation (\ref{news}) is a rational function in the variable $\eta$.\\ We shall show that the first case is impossible whereas in the second case algebraic degeneracy is shown. {\bf Case 1:} We could immediately finish up the proof under the assumptions of case 1 by using a Second Main Theorem for moving targets to equation~(\ref{news}), as to be found e.g. in the paper of Ru and Stoll \cite{R-S}. Another approach is to treat equation~(\ref{news}) directly with a generalized Borel's theorem (we can regard this equation as a sum of nonvanishing holomorphic functions with coefficients which may vanish, but which grow of a smaller order than the nonvanishing functions, only). We present here a more elementary argument based on the Second Main Theorem which might also be considered somewhat simpler. First, it is easy to see that $L \geq 3$, since for $L=2$ we would get, by dividing in equation~(\ref{news}) through one of the exponential terms, that the exponential of a nonconstant polynomial is equal to a quotient of two other polynomials, which is absurd. Let $\psi_1,\ldots, \psi_L$ be some enumeration of the summands which occur in equation (\ref{news}). Then, after factoring out possible common zeroes of the entire holomorphic functions $\psi_1, \ldots, \psi_L$ we get an entire holomorphic curve $$ \Psi : \Bbb C \rightarrow \Bbb P^{L-1}; \eta \rightarrow [\psi_1(\eta):\ldots:\psi_L(\eta]. $$ If we denote the homogenous coordinates of this $\Bbb P^{L-1}$ by $[z_1:\ldots:z_L]$, the image of $\Psi$ is contained in the hyperplane $H=\{z_1+\ldots+z_L=0\}$ and does not hit any of the coordinate hyperplanes $H_i=\{z_i=0\}$. So we can regard $\Psi$ also as an entire holomorphic mapping with values in the hyperplane $H$ (which is isomorphic to $\Bbb P^{L-2}$) which does not intersect the $L$ different hyperplanes $H \cap H_i$ in $H$. It is now an important fact that these hyperplanes are in general position in $H$, and that the entire curve $\Psi$ is not mapping $\Bbb C$ entirely into any hyperplane in $H$ (the latter follows from the minimality condition in equation (\ref{news})), because under these conditions we can apply the Second Main Theorem (cf. section~2), which yields: \begin{equation} \label{smt} (L-(L-2)-1) T(\Psi, r) \leq \sum_{i=1}^L N_{\Psi}(H \cap H_i, r) + O(\log r) \end{equation} because the entire curve $\Psi$ is of finite order at most two by Lemma~\ref{calc}. Now we have $$ N_{\Psi}(H \cap H_i, r) = N(\{\psi_i=0\}, r) \leq M (N(\{p_1' =0\}, r)+N(\{p_2' =0\}, r)). $$ The First Main Theorem (cf. section~2) and Lemma~\ref{logr} imply that the right hand side grows at most of order $O(\log r)$ only, so equation (\ref{smt}) yields that \begin{equation} \label{log} T(\Psi, r) =O(\log r). \end{equation} We know by the assumption of case 1 that there exist indices $i, j$ such that $\psi_i(\eta)/\psi_j(\eta)$ is not a rational function in $\eta$. Then $[\psi_i(\eta):\psi_j(\eta)]:\Bbb C \rightarrow \Bbb P^1$ is an entire curve for which by Lemma~\ref{logr} the characteristic function $T([\psi_i:\psi_j])$ grows faster than $\log r$, so (by the formula for the characteristic function given in equation~(\ref{1})) this is also true for $T(\Psi, r)$ contradicting equation~(\ref{log}). So we have shown that under the assumptions of case 1 we get a contradiction. {\bf Case 2:} We want to show first that there exist nonvanishing complex numbers $\gamma$ and $\lambda$ such that \begin{equation} \label{deriv} \lambda p_1'(\eta) = \gamma p_2'(\eta). \end{equation} We only need to show that $p_1'(\eta)$ and $p_2'(\eta)$ are linearly dependent, because if one of them is the zero polynomial, we have algebraic degeneracy of $g$ and hence of $f$. So assume that $p_1'(\eta)$ and $p_2'(\eta)$ are linearly independent. Then no linear combination of $p_1(\eta)$ and $p_2(\eta)$ is a constant polynomial. So under the assumptions of case 2 get that for all $(i, j, k) \in S$ the terms $i+j$ in the summands $$ c_{ijk} a_{ijk} \exp((i+j)p_1(\eta))+(M-i+k)p_2(\eta)) (p_1'(\eta))^i(p_2'(\eta))^{M-i} $$ are equal, and also the terms $k+(M-i)$ are the same as well. But then for a given $i_0$ there can be at most one $(i_0, j, k) \in S$. So by factoring out the exponential function in equation (\ref{news}) we get a nontrivial homogenous equation of degree $M$ in $p_1'(\eta)$ and $p_2'(\eta)$, which then can be factored in linear factors. Since then one of the linear factors has to vanish identically we get the linear dependency of $p_1'(\eta)$ and $p_2' (\eta)$ again, so the assumption of linear independency was wrong. We now want to construct a special symmetric form with at most logarithmic poles as singularities along the curve $C$ which is annihilated by $f$. Let us simply state equation~(\ref{deriv}) in terms of the original entire curve $f$. We have \begin{equation} \label{ww} p_i'(\eta) = \frac{dg_i(\eta)}{g_i(\eta)} = (\Phi \circ f)^* \frac{d\xi_i}{\xi_i}= f^* \omega_i \end{equation} where $\omega_i$; $i=1,2$ is a differential one form on $\bar X$ with at most logarithmic poles along $C$. Define $\omega_0 = \lambda \omega_1 - \gamma \omega_2$. Then $\omega_0 \in H^0(\bar X, E^*)$, and since $$\omega_0 =\Phi^* (\lambda \frac{d\xi_1}{\xi_1} - \gamma \frac{d\xi_2}{\xi_2}) $$ and the map $\Phi$ is a local isomorphism outside the branching, we have $$\omega_0 \not= 0$$ Furthermore by equations (\ref{ww}) and (\ref{deriv}) we have $$ f^*\omega_0 \equiv 0$$ Now the proof of the fact that $f$ is algebraically degenerate is almost finished: Let $\sigma \in H^0(\Bbb P(E), {\cal O}(\mu) \otimes p^*[-k_0D])$ be the section constructed in Lem\-ma~\ref{onecomp} and $\tilde\sigma \in H^0(\Bbb P(E), {\cal O} (1))$ the section which corresponds to $\omega_0$. We recall that both sections are nontrivial, that $\mu \geq 2$, and that $V(\sigma)$ contains only one horizontal component, which we will denote by $S_{\sigma}$, with multiplicity one. We also recall that by Theorem~\ref{Lutheo}, the lift of $f$ to $\Bbb P(E)$, which we denoted by $F$, maps entirely into $V(\sigma)$. We may assume that it maps into $S_{\sigma}$, otherwise by projecting down to $\bar X$ we get that $f$ is algebraically degenerate and we are done. If $\tilde\sigma$ does not vanish identically on $S_{\sigma}$, $F$ maps into the zero set of $\tilde\sigma$ in $S_{\sigma}$, which has codimension at least two. So projecting down to $\bar X$ again yields algebraic degeneracy of $f$. Hence we now may assume that $\tilde\sigma$ vanishes identically on $S_{\sigma}$. Since $\tilde\sigma \in H^0(\Bbb P(E), {\cal O}(1))$ the degree of $V(\tilde\sigma)$ with respect to a generic fiber of the map $p:\Bbb P(E) \rightarrow \bar X$ is one (cf. the argument in the proof of Lemma~\ref{horcomp}). However since $S_{\sigma}$ is the only horizontal component of the zero set of $\sigma \in H^0(\Bbb P(E), {\cal O}(\mu) \otimes p^*[-k_0D])$ with $\mu \geq 2$ and has multiplicity one, and since $\tilde\sigma$ vanishes on $S_{\sigma}$, the degree of $V(\tilde\sigma)$ with respect to such a generic fiber must be at least two, which is a contradiction. So this case cannot occur and the proof of Theorem~\ref{deg} is complete. \qed \begin{small} {\it Proof of Lemma~\ref{meromext}:}\/ The assertion $g^*\Omega \equiv 0$ is clear from $f^*\omega \equiv 0$ and the construction of $\Omega$. In order to prove the assertion $\Omega \not\equiv 0$, we choose a point $\xi^0 \in \Bbb P_2 \setminus (B \cup \{w_0=0\})$. In a small neighborhood $U(\xi^0)$ we have the $N$ biholomorphic functions $a_i(\xi), i=1,...,N$ which invert the map $\Phi$ on $U(\xi^0)$. Then we have \begin{equation} \label{express} ((a_i)^*(\omega))(\xi) = \sum_{j=0}^m b_{ij}(\xi) (d\xi_1)^j(d\xi_2)^{m-j} \end{equation} After possibly moving the point $\xi^0$ in $U(\xi^0)$ we may assume that the meromorphic functions $b_{ij}(\xi)$ either vanish identically on $U(\xi^0)$ or have no zero or singularity in $\xi^0$. Let now for each $i=1,...,N$ the index $j(i)$ be the maximal $j \in \{0,...,m\}$ such that $b_{ij}(\xi^0) \not= 0$. Let $k=\sum_{i=1}^N j(i)$. Then the $(d\xi_1)^k(d\xi_2)^{M-k}$-monomial of $\Omega$ in the point $\xi_0$ is equal to $\prod_{i=1}^N b_{ij(i)}(\xi^0)$, which is not equal to zero by construction. Last we have to show that $\Omega$ extends to a rational symmetric $M$-form on $\Bbb P_2$. We only have to show how $\Omega$ can be extended over smooth points of the branching locus, because by Levi's extension theorem (cf. \cite{G--R}) we then can extend it over the singular locus which is of codimension two. Then, by Chow's Theorem it is rational. So assume $P \in B$ is a smooth point of $B$. Then (cf. \cite{G-R}) there exists a neighborhood of $P$ over which $\Phi$ is an analytically branched covering of a very special form: For every point $Q$ over $P$ one can introduce local coordinates $\xi_1,\xi_2$ around $P$ and $z_1,z_2$ around $Q$ such that $\xi_1(P)=\xi_2(P)=z_1(Q)=z_2(Q)=0$, and neighborhoods $U=\{ |\xi_1|<1, |\xi_2|<1\}$, $V= \{|z_1|<1, |z_2|<1 \}$ such that, for some $b \in \{1,...,N\}$, we have \begin{equation} \label{abc} \Phi :V \rightarrow U; (z_1,z_2) \rightarrow (z_1^b,z_2) \end{equation} In order to prove our assertion in a neighborhood of $P \in B$, it is sufficient to prove it for the analytically branched covering in equation (\ref{abc}). For $k=0,...,b-1$ let $$g_k: V \rightarrow V; (z_1,z_2) \rightarrow (\exp(\frac{2\pi i k}{b}) z_1,z_2).$$ Then $G= \{g_0,...,g_{b-1} \}$ is just the group of deck transformations, i.e.\ automorphisms which respect $\Phi$. For the meromorphic symmetric $m$-form $\omega$ on $V$, let $\tilde\Omega$ be the symmetric product of the $b$ meromorphic symmetric $m$-forms $(g_i)^*(\omega)$ on $V$. We are done, if we show that by projecting down with $\Phi$ this form gives rise to a meromorphic symmetric $M$-form on $U$, because in $U \setminus B$ this is just the form $\Omega$. The symmetric $M$-form $\tilde\Omega$ can be uniquely written in the form \begin{equation} \label{uni} \tilde\Omega (z_1,z_2) = \sum_{i=0}^M r_{ij}(z_1, z_2)(\frac{dz_1}{z_1})^i (dz_2)^{M-i} \end{equation} with meromorphic functions $r_{ij}$ in the variables $(z_1,z_2)$. Now $\tilde\Omega(z_1, z_2)$ is invariant under the action of $G$, $(\frac{dz_1}{z_1})^i$ and $(dz_2)^{M-i}$ are also $G$-invariant. Moreover $b \frac{dz_1}{z_1}=\frac{d\xi_1}{\xi_1}$ and $dz_2=d\xi_2$. Hence the $r_{ij}$ are $G$-invariant functions, i.e.\ these are pull-backs of meromorphic functions on $U$. \qed \end{small} \section{Application to the Projective Plane and Complete Intersections} We shall apply Theorem~\ref{deg}. Throughout this section, we make the following assumptions: Let the smooth complex surface $\bar X$ be a complete intersection $$\bar X = V_2^{(a_1, \ldots , a_r)} \subset \Bbb P_{r+2}\, , \: r \geq 0 $$ of hypersurfaces of degrees $a_j$, $j=1, \ldots , r$ in $\Bbb P_{r+2}$. Set $A=\prod_{i=1}^r a_i$ and $a=\sum_{i=1}^r a_i$. Let smooth curves $C_j$, $j=1, 2, 3$ be given in $\bar X$ which intersect in normal crossings. We assume that these curves are transversal intersections of $\bar X$ with hypersurfaces of degrees $b_j$. We set $b=b_1+b_2+b_3$. \begin{lem}\label{riro} The Euler numbers of $\bar X$ and $C_j$ are: $$e(\bar X)=A(2+(a-r-1)^2)$$ and $$e(C_j)=Ab_j(3+r-a-b_j).$$ \end{lem} The {\it Proof}\/ is a direct consequence of the Riemann-Roch Theorem. According to \cite{HI}, Theorem~22.1.1, the $\chi_y$-characteristic of a complete intersection can be computed from a generating function. Its value at $y=-1$ yields the Euler number. \qed In order to determine, when the assumptions of Theorem~\ref{deg} are satisfied, we first compute $\bar c_1^2(X) - \bar c_2(X)$. \begin{prop}\label{num} In the above situation $$ \bar c_1^2(X) - \bar c_2(X) = A((a-r-3)(b-4) -6 + \sum_{i<j}b_i\cdot b_j), $$ and $${\rm det}(E^*)= (a+b-3-r)\tilde{H},$$ where $\tilde{H}$ is a hyperplane section. \end{prop} {\it Proof:}\/ According to a result of Sakai \cite{Sak} we have ${\rm det}(E^*)=[\Gamma]$, where $\Gamma= K_{\bar X} + C$. Then the second claim follows from the Adjunction Formula. Furthermore (cf. \cite{Sak}), $$ c_1^2(E)-c_2(E)= c_1^2(E^*)-c_2(E^*)= \Gamma^2 - e(\bar X \setminus C) = \Gamma^2 - e(\bar X) + e(C), $$ where $\Gamma^2$ denotes the self intersection. It equals $$ \Gamma^2= A(a+b-r-3)^2. $$ For the Euler number of $\bar X$ we use Proposition~\ref{riro}. The Euler number of $C$ is evaluated in terms of the Euler numbers $e(C_j)$ of the components and the respective intersection numbers $$ C_i\cdot C_j= A b_i b_j $$ to be $$ e(C) = \sum_{j=1}^3 e(C_j) - \sum_{i<j} C_i\cdot C_j. $$ {}From these equalities we get immediately the above formula for $c_1^2-c_2$.\qed Now Theorem \ref{deg} yields \pagebreak \begin{theo}\label{main} Let $X=\bar X \setminus C$ as above. Then any entire holomorphic curve $f:\Bbb C \to X$ of order at most two is algebraically degenerate, if \begin{itemize} \item[i)] ${\rm Pic}(\bar X)= \Bbb Z$ \item[ii)] $(a-r-3)(b-4)+ \sum_{i<j}b_ib_j > 6$ \item[iii)] $a+b \geq r+3$ \end{itemize} \end{theo} The {\it Proof}\/ follows from Theorem~\ref{deg}, and Proposition~\ref{num}. \qed \begin{theo} \label{main2} Let $X=\bar X \setminus C$ as above. Then any entire holomorphic curve of order at most two $f:\Bbb C \to X$ is algebraically degenerate in any of the following cases: \begin{itemize} \item[a)] ${\rm Pic}(\bar X)= \Bbb Z$, and $a \geq r+3$, $b \geq 5$. \item[b)] $\bar X \subset \Bbb P_3$ is a `generic' hypersurface of degree at least four, and $b \geq 5$. \item[c)] Let $\bar X =\Bbb P_2$ (i.e.\ $a_1= \ldots a_r =1$, $r \geq 0$). Let $b_1, b_2, b_3 \geq 2$ and at least one $b_j \geq 3$, or up to enumeration $b_1=1, b_2\geq 3, b_3 \geq 4$. \end{itemize} \end{theo} Remark: `Generic' indicates the complement of a countable union of proper varieties in space of all hypersurfaces. {\it Proof:}\/ Case a) is obvious. Case b) is an application of the Noether-Lefschetz theorem \cite{N-L} and case a). For case c) we set e.g.\ $r=a_1=1$. Then $$ \bar c_1^2(X) - \bar c_2(X) = -3(b-4) -6 + \sum_{i<j}b_i\cdot b_j $$ is equal to $$ (b_1-2)(b_2-2)+(b_1-2)(b_3-2)+(b_2-2)(b_3-2)+b-6$$ or to $$ (b_1-1)(b_2-1)+(b_1-1)(b_3-2)+(b_2-3)(b_3-4)+(2b_2+b_3)-9$$ From these facts the assertion of case c) follows immediately.\qed \section{Algebraic Degeneracy of Entire Curves Versus Hyperbolicity} \begin{theo} \label{MT} Let $C$ be the union of three smooth curves $C_j$ $ j=1,2,3$ in $\Bbb P_2$ of degree $d_j$ with $$ d_1,d_2,d_3\geq 2 \hbox{ and at least one } d_j\geq 3. $$ Then for generic such configurations $ \Bbb P^2 \setminus C$ is complete hyperbolic and hyperbolically embedded in $\Bbb P_2$. More precisely this is the case, if the curves intersect only in normal crossings, and if one curve is a quadric there must not exist a line which intersects the two other curves only in one point each and which intersects the quadric just in these two points.\\ \end{theo} {\it Proof:}\/ In order to prove that $\Bbb P_2 \setminus C$ is hyperbolic and hyperbolically embedded in $\Bbb P_2$, we only will have to prove, by an easy Corollary of a Theorem of M.Green (cf. \cite{GRE2}), that there does not exist a non-constant entire curve $\, f:\Bbb C \rightarrow \Bbb P_2 \setminus C$ of order at most two. We know from Theorem~\ref{main2} that the entire curve $f:\Bbb C \to \Bbb P_2\setminus C$ of order at most two is contained in an algebraic curve $A\subset \Bbb P_2$ of degree $d_0$ say. Now the proof is almost the same as in \cite{DSW}. Assume that there exists an irreducible algebraic curve $A \subset \Bbb P_2 $ such that $A\setminus C$ is not hyperbolic. We know that $ A \cap C$ consists of at least 2 points $P$ and $Q$. Moreover, $A$ cannot have a singularity at $P$ or $Q$ with different tangents, because $A$ had to be reducible in such a point, and $A\setminus C$ could be identified with an irreducible curve with at least three punctures. (This follows from blowing up such a point or considering the normalization). So $ A \cap C$ consists of exactly 2 points $P$ and $Q$ with simple tangents. We denote the multiplicities of $A$ in $P$ and $Q$ by $ m_P $ and $ m_Q $. Then the inequality (cf. \cite{FU}) $$ m_P(m_P-1)+m_Q(m_Q-1) \leq (d_0-1)(d_0-2) $$ implies \begin{equation} \label{*} m_P , m_Q < d_0 \hbox{ {\rm or} }d_0=m_P=m_Q=1. \end{equation} After a suitable enumeration of its components we may assume that $P \in C_1 \cap C_2 $ and $ Q \in C_3$. If $Q \not\in C_2 \cup C_1 $ we are done, since then we may assume that $A$ is not tangential to $ C_2$, and then, computing intersection multiplicities according to \cite{HA}, we have $$ m_P = I(P,A \cap C_2) = d_2d_0 $$ which contradicts equation (\ref{*}). So we may assume that $\, Q \in C_2 \cap C_3$. Now $A$ has to be tangential to $C_1$ in $P$ and to $\, C_3$ in $Q$, otherwise we again get $\, m_P=d_1d_0\,$ or $\,m_Q =d_3d_0\,$ what contradicts equation (\ref{*}). But then $\,C_2$ is not tangential to $A$ in $P$ or $Q$, so we have $$m_P + m_Q = I(P,A \cap C_2) + I(Q, A \cap C_2) = d_2d_0$$ Again by equation (\ref{*}) this is only possible if $d_2=2$ and $\, m_P=m_Q=d_0=1$, but then we are in a situation which we excluded in Theorem~\ref{MT}, which is a contradiction. \qed We make the same assumptions as in section~5. \begin{theo} \label{MT1} Let $\bar X \subset \Bbb P_3$ be a `generic' smooth hypersurface of degree $d \geq 5$ and $b \geq 5$. Then $X=\bar X \setminus C$ is hyperbolic and hyperbolically embedded in $\bar X$. \end{theo} {\it Proof:}\/ According to Xu \cite{Xu} and Clemens \cite{Cl} $\bar X$ does not contain any rational or elliptic curves. Hence Theorem~\ref{main2} yields the claim. \qed \pagebreak

9312

alg-geom/9312006

en

https://arxiv.org/abs/alg-geom/9312006

alg-geom/9312006

Fabrizio Broglia

F. Acquistapace, F.Broglia, M.Pilar Velez

An algorithmic criterion for basicness in dimension 2

23 pages, amslatex (+bezier.sty) report: 1.89.(766) october 1993

We give a constructive procedure to check basicness of open (or closed) semialgebraic sets in a compact, non singular, real algebraic surface $X$. It is rather clear that if a semialgebraic set $S$ can be separated from each connected component of $X\setminus(S\cup\frz S)$ (when $\frz S$ stands for the Zariski closure of $(\ol S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$), then $S$ is basic. This leads to associate to $S$ a finite family of sign distributions on $X\setminus\frz S$; we prove the equivalence between basicness and two properties of these distributions, which can be tested by an algorithm. There is a close relation between these two properties and the behaviour of fans in the algebraic functions field of $X$ associated to a real prime divisor, which gives an easy proof, for a general surface $X$, of the well known 4-elements fan's criterion for basicness (Brocker, Andradas-Ruiz). Furthermore, if the criterion fails, using the description of fans in dimension 2, we find an algorithmic method to exhibit the failure. Finally, exploiting this thecnics of sign distribution we give one improvement of the 4-elements fan's criterion of Brocker to check if a semialgebraic set is principal.

alg-geom

\section*{Introduction.} In this paper we give a constructive procedure to check basicness of open (or closed) semialgebraic sets in a compact, non singular, real algebraic surface $X$. It is rather clear that if a semialgebraic set $S$ can be separated from each connected component of $X\setminus(S\cup\partial _{\rm z} S)$ (when $\partial _{\rm z} S$ stands for the Zariski closure of $(\overline S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$), then $S$ is basic. This leads to associate to $S$ a finite family of sign distributions on $X\setminus\partial _{\rm z} S$; we prove the equivalence between basicness and two properties of these distributions, which can be tested by an algorithm described in [ABF]. By this method we find for surfaces a general result of [AR2] about the ``ubiquity of Lojasiewicz's example" of non basic semialgebraic sets (2.9). There is a close relation between these two properties and the behaviour of fans in the algebraic functions field of $X$ associated to a real prime divisor (Lemmas 3.5 and 3.6). We use this fact to get an easy proof (Theorem 3.8), for a general surface $X$, of the well known 4-elements fan's criterion for basicness (see [Br] and [AR1]). Furthermore, if the criterion fails, using the description of fans in dimension 2 [Vz], we find an algorithmic method to exhibit the failure. Finally, exploiting this thecnics of sign distribution we give one improvement of the 4-elements fan's criterion in [Br] to check if a semialgebraic set is principal. In fact, one goal of this paper is to give purely geometric proofs, in the case of surfaces, of the theory of fans currently used in Semialgebraic Geometry ([Br]). In particular, we only need the definitions of that theory. It is at least remarkable that while the notion of fan is highly geometric in nature all known proofs of the main results are pure quadratic forms theory. \section{Geometric review of basicness} Let $X\subset {\Bbb R}^n$ be an algebraic surface. Denote by ${\cal R}(X)$ the ring of regular functions on $X$. Let $S\subset X$ be a {\em semialgebraic set}, that is $$S=\bigcup_{i=1}^p\{x\in X:f_{i1}(x)>0,\dots,f_{ir_i}(x)>0,g_i(x)=0\}$$ with $f_{i1},\dots,f_{ir_i},g\in {\cal R}(X)$, for $i=1,\dots,p$.\\ We will simply write: $S=\bigcup\{f_{i1}>0,\dots,f_{ir_i}>0,g_i=0\}$. \begin{defn} A semialgebraic set $S$ is {\em basic open} (resp. {\em basic closed}) if there exist $f_1,\dots,f_r\in {\cal R}(X)$ such that $$S=\{x\in X:f_1(x)>0,\dots,f_r(x)>0\}$$ $$({\it resp.}\; S=\{x\in X:f_1(x)\geq 0,\dots,f_r(x)\geq 0\})$$ \end{defn} \begin{defn} A semialgebraic set $S\subset X$ is {\em generically basic} if there exist a Zariski closed set $C\subset X$, with ${\rm dim}(C)\leq 1$, such that $S\setminus C$ is basic open. \end{defn} Denote by $\partial _{\rm z} S$ the Zariski closure of the set $\partial(S)=(\overline S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$. It is known that in dimension 2 {\em basic} and {\em generically basic} have almost the same meaning (see [Br]). We give here a direct proof of this fact. \begin{lem} Let $S$ be an open semialgebraic set in $X$, if $S$ is generically basic then $S\cap \partial _{\rm z} S$ is a finite set. \end{lem} \begin{pf} Suppose there exist $f_1,\dots,f_s\in {\cal R}(X)$ and an algebraic set $C\subset X$ such that ${\rm dim}(C)\leq 1$ and $$S\setminus C=\{f_1>0,\dots,f_s>0\}.$$ Suppose also that there is an irreducible component $H$ of $\partial _{\rm z} S$ such that ${\rm dim}(H\cap S)=1$. Let ${\frak p}$ be the ideal of $H$ in ${\cal R}(X)$ and pick any $x_0\in {\rm Reg}(H)$. Then ${\cal R}(X)_{x_0}$ (the localization of ${\cal R}(X)$ at the maximal ideal ${\frak m}$ of $x_0$, so ${\frak p}\subset{\frak m}$) is a factorial ring and ${\rm ht}({\frak p}{\cal R}(X)_{x_0})=1$, hence ${\frak p}{\cal R}(X)_{x_0}=h{\cal R}(X)_{x_0}$ for some $h\in {\frak p}$. Take $g_1,\dots,g_r\in{\cal R}(X)$ such that ${\frak p}{\cal R}(X)=(g_1,\dots,g_r)$, then there exist $\lambda_i,s_i\in{\cal R}(X)$ with $s_i(x_0)\not= 0$ (in particular, $s_i\not\in{\frak p}$) such that $s_ig_i=\lambda_ih$, for $i=1,\dots,r$.\\ \indent Consider $U=X\setminus\{s_1\cdots s_r=0\}$ which is Zariski open in $X$. Then we have ${\frak p}{\cal R}(X)_x=(h){\cal R}(X)_x$ for all $x\in U_1=U\cap{\rm Reg}(H)$.\\ \indent Take now $x_0\in U_1$, we have $f_j=\rho_jh^{\alpha_j}$ with $\alpha_j\geq 0$ and $\rho_j\in {\cal R}(X)_{x_0}$ such that $h$ does not divide $\rho_j$ (in particular, $\rho_j\not\in {\frak p}{\cal R}(X)_{x_0}$). Then $\rho_j=p_j/q_j$, $j=1,\dots,s$, where $p_j,q_j\in{\cal R}(X)$ and $p_j(x_0)\not=0,q_j(x_0)\not=0$. Let $U_2$ be the Zariski open set $U_1\setminus\{q_1\cdots q_s=0\}\subset H$, then for all $x\in U_2$, $q_jf_j=p_jh^{\alpha_j}$, $p_j,q_j$ do not change sign in a neighbourhood of $x$, for $j=1,\dots,s$ and $h$ changes sign in a neighbourhood of $x$, since locally $h$ is a parameter of $H$ in $x$. Hence for any $j=1,...,s$, $f_j$ changes sign in a neighbourhood of $x$ if $\alpha_j$ is odd and does not change sign if $\alpha_j$ is even.\\ \indent Using the fact that ${\rm dim}(S\cap H)=1$, ${\rm dim}(C)\leq 1$ and $s_i,p_j,q_j\not\in{\frak p}$, there exist a Zariski dense open set ${\mit\Omega}$ in $U_2$ such that $f_j$ does not change sign through ${\mit\Omega}$, for all $j=1,\dots,s$, then $\alpha_j$ is even for all $j=1,\dots,s$. But also there is a Zariki dense open set ${\mit\Omega}'$ in $U_2$ such that ${\mit\Omega}'\subset \overline S\setminus S$, then there is $l\in \{1,\dots,s\}$ such that $f_l$ changes sign through ${\mit\Omega}'$, and $\alpha_l$ is odd, which is impossible. \end{pf} \begin{prop} Let $S$ be a semialgebraic set in $X$.\\ \indent (1) Let $S$ be open. Then $S$ is generically basic if and only if there exist $p_1,\dots,p_l\in \partial _{\rm z} S$ such that $S\setminus\{p_1,\dots,p_l\}$ is basic open.\\ \indent (2) $S$ is basic open if and only if $S$ is generically basic and $S\cap\partial _{\rm z} S=\emptyset$.\\ \indent (3) Let $S$ be closed. Then $S$ is generically basic if and only if $S$ is basic closed. \end{prop} \begin{pf} First we prove {\it (1)}. The ``if part'' is trivial. Suppose $S$ to be open and generically basic, that is there are $f_1,...,f_r\in{\cal R}(X)$ and an algebraic set $C\subset X$ with ${\rm dim}(C)\leq 1$ such that $$S\setminus C=\{f_1>0,\dots,f_r>0\}.$$ \noindent We suppose first that $C$ is a curve and we can also suppose that $\{f_1\cdots f_r=0\}\subset C$.\\ \indent $C\cap S$ is an open semialgebraic set in $C$, then using [Rz, 2.2], there exist $g_1\in {\cal R}(C)$ such that $$C\cap S=\{x\in C:g_1(x)>0\}$$ $$\overline{C\cap S}=\{x\in C:g_1(x)\geq 0\}$$ \noindent choose $g_1$ to be the restriction of a regular function $g\in {\cal R}(X)$.\\ \indent For each $i=1,\dots,r$ consider the open sets $B_i=\{x\in X:f_i(x)<0\}$ \noindent and the closed sets $T_i=(\overline S\cap\{g\leq 0\})\cup(\overline{B_i}\cap\{g\geq 0\}).$\\ \indent Applying [BCR, 7.7.10] to $T_i$, $f_i$ and $g$ for $i=1,...,r$, we can find $p_i,q_i\in {\cal R}(X)$, with $p_i>0$, $q_i\geq 0$, such that\\ \indent {\em (i)} $F_i=p_if_i+q_ig$ has the same signs as $f_i$ on $T_i$;\\ \indent {\em (ii)} The zero set $Z(q_i)$ of $q_i$ verifies, $Z(q_i)={\rm Adh}_{\rm z}(Z(f_i)\cap T_i)$.\\ We remark the following:\\ \indent {\em a)} $F_i(S\setminus C)>0$ for $i=1,\dots,r$, since $F_i$ has the same signs as $f_i$ on $T_i\cap S$ and outside is the sum of a strictly positive function and a nonnegative one.\\ \indent {\em b)} $F_i(B_i)<0$ for $i=1,\dots,r$, by the same reasons.\\ \indent {\em c)} $Z(q_i)\subset \partial _{\rm z} S$. In fact, denote $$Z_1^i=Z(f_i)\cap \overline S\cap\{g\leq 0\}$$ $$Z_2^i=Z(f_i)\cap \overline{B_i}\cap\{g\geq 0\}$$ \noindent then we have $Z(q_i)={\rm Adh}_{\rm z}(Z_1^i)\cup{\rm Adh}_{\rm z}(Z_2^i)\subset Z(f_i)\subset C.$ Since $g$ is positive on $C\cap S$ and $Z_1^i\subset C\cap \{g\leq 0\}$, we have $Z_1^i\cap S=\emptyset$, hence $Z_1^i\subset \partial(S)$. Indeed, since $B_i\cap S=\emptyset$ and $S$ is open we have $\overline{B_i}\cap S=\emptyset$. Moreover $C\cap \overline{B_i}\subset\{g\leq 0\}$, hence $Z_2^i\subset\{g=0\}\cap C\subset \overline S$, then $Z_2^i\subset\partial(S)$.\\ \indent From these remarks, denoting $Z=\bigcup_{i=1}^rZ(q_i)$, we have $$S\setminus Z=\{F_1>0,\dots,F_r>0\}.$$ \noindent In fact, if $x\in S\setminus C$ then $F_i(x)>0$ for $i=1,...,r$, if $x\in (C\cap S)\setminus Z$ then $f_i(x)\geq 0$, $q_i(x)\geq 0$ and $g(x)>0$, hence $F_i(x)>0$ for $i=1,...,r$ and $x\in S\setminus Z$. Otherwise, suppose $x\not\in S\setminus Z$, then $x\in (X\setminus S)\cup(S\cap Z)$: if $x\in X\setminus S$ there is $l\in \{1,\dots,r\}$ such that $f_l(x)\leq 0$, and we can have $x\not\in C$ or $x\in C$, in the first case $f_i(x)\not= 0$ for all $i$, so $x\in B_l$ and $F_l(x)<0$, in the second case $g(x)\leq 0$, $q_l(x)\geq 0$, so $F_l(x)\leq 0$; if $x\in S\cap Z$ there is $l\in \{1,\dots,r\}$ such that $q_l(x)=0$, then $f_l(x)=0$ and $F_l(x)=0$. In any way, there is $l$ such that $F_l(X)\leq 0$ if $x\not\in S\setminus Z$.\\ \indent By 1.3 and remark {\em c)} above we have that there exist $p_1,\dots,p_l\in\partial _{\rm z} S$ such that $\bigcup_{i=1}^rZ(q_i)\cap S=\{p_1,\dots,p_l\}$, hence $$S\setminus\{p_1,\dots,p_l\}=\{F_1>0,\dots,F_r>0\}.$$ \indent If $C=\{a_1,\dots,a_m\}$ is a finite set we have to check that we can throw out from $C$ all the $a_i$ which do not lie in $\partial _{\rm z} S$. This can be done as before by taking the function 1 at the place of $g$ and putting $T_i=\overline{B_i}$.\\ \indent From {\it (1)} we have immediately {\it (2)}, because if $S$ is generically basic and $\partial _{\rm z} S\cap S=\emptyset$, $S$ is basic open, since following the proof above we have $Z(q_i)\cap S=\emptyset$ for $i=1,...,r$. On the countrary, if $S$ is basic open then it is generically basic and $\partial _{\rm z} S\cap S=\emptyset$ (because $\partial _{\rm z} S\subset\{f_1\cdots f_r=0\}$ if $S=\{f_1>0,\dots,f_r>0\}$).\\ \indent Finally we prove {\it (3)}. The ``if part" is trivial. Then suppose $S$ to be closed and generically basic, i. e. $$S\setminus C=\{f_1>0,\dots,f_r>0\}$$ \noindent with $f_1,\dots,f_r\in {\cal R}(X)$ and ${\rm dim}(C)<2$. We can suppose $\{f_1\cdots f_r=0\}\subset C$.\\ \indent $C\cap S$ is a closed semialgebraic in $C$, by [Rz, 2.2], there is $g_1\in {\cal R}(C)$ such that $$C\cap S=\{x\in C:g_1(x)\geq 0\}.$$ \noindent Take $g\in {\cal R}(X)$ as above, $f\in {\cal R}(X)$ a positive equation of $C$ and $T=S\cap \{g\leq 0\}$. Applying again [BCR, 7.7.10] to $T$, $f$ and $g$, we find $p,q\in{\cal R}(X)$ with $p>0$, $q\geq 0$ such that\\ \indent {\em (i)} $h=pf+qg$ has the same sign as $f$ on $T$;\\ \indent {\em (ii)} $Z(q)={\rm Adh}_{\rm z}(Z(f)\cap T)$.\\ Notice that $h(S)\geq 0$ and $Z(f)\cap T=C\cap S\cap\{g\leq 0\}\subset\{g=0\}\cap C$, because $C\cap S\subset\{g\geq 0\}$; but $\{g=0\}\cap C$ is a finite set contained in $S$, then $Z(q)$ is a finite set contained in $S$.\\ \indent We will prove that $$S=\{f_1\geq 0,\dots,f_r\geq 0,h\geq 0\}.$$ \noindent In fact, if $x\in S\setminus C$ then $f_i(x)>0$ for all $i$ and $h(x)>0$, if $x\in S\cap C$ then $f_i(x)\geq 0$ for all $i$ (because ${\rm dim}(C)<2$) and $h(x)\geq 0$, hence $S\subset\{f_1\geq 0,\dots,f_r\geq 0,h\geq 0\}$. Otherwise suppose $x\not\in S\supset S\setminus C$, then there is $l\in \{1,...,r\}$ such that $f_l(x)\leq 0$, if $x\not\in C$, $f_i(x)\not= 0$ for all $i=1,...,r$, then $f_l(x)<0$; if $x\in C\setminus (C\cap S)$, $f_l(x)\leq 0$, $q(x)\not= 0$ and $g(x)<0$, then $h(x)<0$. \end{pf} \begin{rmks} {\em (1)} {\rm Let $S$ be a closed semialgebraic set, then $S$ is basic closed if and only if $S\setminus\partial _{\rm z} S$ is basic open.\\ \indent In fact, if $S$ is basic closed then it is generically basic, hence $S\setminus\partial _{\rm z} S$ is generically basic and as $(S\setminus\partial _{\rm z} S)\cap\partial _{\rm z}(S\setminus\partial _{\rm z} S)=\emptyset$, $S\setminus\partial _{\rm z} S$ is basic; on the contrary, if $S\setminus\partial _{\rm z} S$ is basic open then $S$ is basic closed, since $S$ is closed.} \indent {\em (2)} {\rm Let $S$ be a semialgebraic set, and let $S^\ast$ denote the set ${\rm Int}(\overline S)$. Then, $S$ is basic open if and only if $S^\ast$ is generically basic and $S\cap\partial _{\rm z} S=\emptyset$.\\ \indent In fact, $S$ and $S^\ast$ are generically equal.} \end{rmks} \section{Basicness and sign distributions} We recall some definitions and results from [ABF]. Let $X$ be a compact, non singular, real algebraic surface and $Y\subset X$ an algebraic curve.\\ \indent Consider a {\em partial sign distribution} $\sigma$ on $X\setminus Y$, which gives the sign $1$ to some connected components of $X\setminus Y$ (whose union is denoted by $\sigma^{-1}(1)$) and the sign $-1$ to some others (whose union is denoted by $\sigma^{-1}(-1)$). So $\sigma^{-1}(1)$ and $\sigma^{-1}(-1)$ are disjoint open semialgebraic sets in $X$. \begin{defns} {\em (1)} A sign distribution $\sigma$ is {\em completable} if $\sigma^{-1}(1)$ and $\sigma^{-1}(-1)$ can be separated by a regular function, i.e. there is $f\in{\cal R}(X)$ such that $f(\sigma^{-1}(1))>0$, $f(\sigma^{-1}(-1))<0$ and $f^{-1}(0)\supset Y$. Briefly, we say that {\em $f$ induces $\sigma$}. {\em (2)} A sign distribution $\sigma$ is {\em locally completable} at a point $p\in Y$ if there is $f\in{\cal R}(X)$ such that $f$ induces $\sigma$ on a neighbourhood of $p$. {\em (3)} An irreducible component $Z$ of $Y$ is a {\em type changing component} with respect to $\sigma$ if there exist two nonempty open sets ${\mit\Omega}_1,{\mit\Omega}_2\subset Z\cap{\rm Reg}(Y)$ such that\\ \hspace{.2in} (a) ${\mit\Omega}_1\subset\overline{\sigma^{-1}(1)}\cap\overline{\sigma^{-1}(-1)}$,\\ \hspace{.2in} ($b_+$) ${\mit\Omega}_2\subset {\rm Int}(\overline{\sigma^{-1}(1)})$ \hspace{.2in} or \hspace{.2in} ($b_-$) ${\mit\Omega}_2\subset {\rm Int}(\overline{\sigma^{-1}(-1)})$.\\ If ($b_+$) (resp. ($b_-$)) holds we say that $Z$ is {\em positive type changing} (resp. {\em negative type changing}) with respect to $\sigma$. {\em (4)} An irreducible component $Z$ of $Y$ is a {\em change component} if there exist a nonempty open set ${\mit\Omega}\subset Z\cap{\rm Reg}(Y)$ verifying (a). \end{defns} Completable sign distributions are characterized by the following theorem. \begin{thm} {\rm (See [ABF, 1.4 and 1.7])} Denote by $Y^c$ the union of the change components of $Y$ with respect to $\sigma$. Then $\sigma$ is completable if and only if\\ \indent {\em (1)} $Y$ does not have type changing component with respect to $\sigma$;\\ \indent {\em (2)} $\sigma$ is locally completable at any point $p\in{\rm Sing}(Y)$;\\ \indent {\em (3)} There exist an algebraic curve $Z\subset X$ such that $Z\cap(\sigma^{-1}(1)\cup\sigma^{-1}(-1))=\emptyset$ and $[Z]=[Y^c]$ in ${\rm H}_1(X,{\Bbb Z}_2)$.\\ \indent Moreover condition {\em (2)} becomes condition {\em (1)} after the blowings-up of the canonical desingularization of $Y$, namely each point where $\sigma$ is not locally completable corresponds to at least one type changing component of the exceptional divisor with respect to the lifted sign distribution. \end{thm} \begin{prop} {(\rm See [ABF, ``the procedure"])} Condition {\em (2)} of theorem 2.2 can be tested, without performing the blowings-up, by an algorithm that only uses the Puiseux expansions of the branches of $Y$ at its singular points. \end{prop} In fact, there are two algorithms: \noindent \fbox{A1} (see [ABF, 2.4.19]) Given a branch $C$ of an algebraic curve through a point $p_0$ and an integer $\rho>0$, it is possible to find explicitely, in terms of the Puiseux expansion of $C$, the irreducible Puiseux parametrizations of all analytic arcs $\gamma$ through $p_0$ with the following properties:\\ \indent a) Denoting respectively by $\gamma_{\rho}$ and by $C_{\rho}$ the strict transform of $\gamma$ and $C$ after $\rho$ blowings-up in the standard resolution of $C$ and by $D_{\rho}$ the exceptional divisor arising at the last blowing-up, then $\gamma_{\rho-1}$ is parametrized by \[ \left\{ \begin{array}{l} x=t\\ y=at+\cdots\end{array}\right. \] \noindent and $C_{\rho-1}\cap\gamma_{\rho-1}=0\in D_{\rho-1}$.\\ \indent b) $\gamma_{\rho-1}$ and $C_{\rho-1}$ have distinct tangents at 0. \noindent \fbox{A2} (see [ABF, solution to problem 2]) Given an analytic arc $\gamma$ and a region of ${\Bbb R}^2$ bounded by two analytic arcs $\gamma_1,\gamma_2$, with $\gamma,\gamma_1,\gamma_2$ through $0\in {\Bbb R}^2$, it is possible to decide, looking at the Puiseux parametrizations, whether $\gamma$ crosses the region or not. Using \fbox{A1} and \fbox{A2} one can decide whether $D_{\rho}$ is positive or negative type changing with respect to the lifted sign distribution $\sigma_{\rho}$ without performing the blowings up, because for the family of arcs given by \fbox{A1}, whose strict transform are transversal to $D_{\rho}$, we can decide by \fbox{A2} whether or not $\sigma$ changes sign along some elements of the family and whether or not $\sigma$ has constant positive or constant negative sign along some other ones.\\ Now let $S$ be an open semialgebraic set. \begin{nott} {\rm Let $A_1,\dots,A_t$ be the connected component of $X\setminus(S\cup\partial _{\rm z} S)$. For each $i=1,\dots,t$ we denote by $\sigma_i^S$ (or simply $\sigma_i$ when there is not risk of confusion) the following sign distribution on $X\setminus\partial _{\rm z} S$: \begin{eqnarray*} (\sigma_i^S)^{-1}(1)&=&S\setminus\partial _{\rm z} S\\ (\sigma_i^S)^{-1}(-1)&=&A_i \end{eqnarray*} } \end{nott} \begin{lem} Let $S$ be a semialgebraic set and $S^\ast={\rm Int}(\overline S)$. Then, ${\rm dim}(S^\ast\cap\partial _{\rm z} S^\ast)=1$ if and only if there exists $i\in\{1,\dots,t\}$ such that $\partial _{\rm z} S$ has a positive type changing component with respect to $\sigma_i^S$. \end{lem} \begin{pf} Suppose ${\rm dim}(S^\ast\cap\partial _{\rm z} S^\ast)=1$, then there is an irreducible component $H$ of $\partial _{\rm z} S^\ast$ such that ${\rm dim}(H\cap S^\ast)=1$. So we can find an open 1-dimensional set ${\mit\Omega}\subset H\cap S^\ast$, hence $${\mit\Omega}\subset{\rm Int}(\overline S)={\rm Int}(\overline{S\setminus\partial _{\rm z} S})={\rm Int}(\overline{\sigma_i^{-1}(1)})$$ \noindent for each $i=1,\dots,t$. And we can find another open 1-dimensional set ${\mit\Omega}'\subset\partial(S^\ast)\cap H$ such that $${\mit\Omega}'\subset\overline{S^\ast}=\overline S=\overline{S\setminus\partial _{\rm z} S}=\overline{\sigma_i^{-1}(1)}$$ \noindent for each $i=1,\dots,t$ and $${\mit\Omega}'\subset X\setminus S^\ast=\bigcup_{i=1}^t\overline{A_i}\; .$$ \noindent Then ${\mit\Omega}'\subset\bigcup_{i=1}^t(\overline{A_i}\setminus A_i)$, since $A_i\cap S=\emptyset$ and $S$ is open. Hence, there exist a 1-dimensional open set ${\mit\Omega}''\subset{\mit\Omega}'$ and $i_0\in\{1,...,t\}$ such that ${\mit\Omega}''\subset\overline{A_{i_0}}\setminus A_{i_0}$, because ${\rm dim}({\mit\Omega}')=1$ and we have a finite number of $A_i$. Hence ${\mit\Omega}''\subset\overline{\sigma_{i_0}^{-1}(1)}\cap\overline{\sigma_{i_0}^{-1}(-1)}$. But since $H$ is a 1-dimensional component of $\partial _{\rm z} S^\ast$ and $\partial _{\rm z} S^\ast \subset\partial _{\rm z} S$, $H$ is an irreducible component of $\partial _{\rm z} S$ of dimension 1; then if we take ${\mit\Omega}_1={\mit\Omega}''\cap{\rm Reg}(\partial _{\rm z} S)$ and ${\mit\Omega}_2={\mit\Omega}\cap{\rm Reg}(\partial _{\rm z} S)$, $H$ is a positive type changing component with respect to $\sigma_{i_o}$.\\ \indent On the contrary suppose that $H$ is an irreducible component of $\partial _{\rm z} S$ which is a positive type changing component with respect to some $\sigma_l$ ($l=1,...,t$). So there exist open sets ${\mit\Omega}_1,{\mit\Omega}_2\subset H$ of ${\rm Reg}(\partial _{\rm z} S)$ such that ${\mit\Omega}_1\subset\overline{\sigma_l^{-1}(1)}\cap\overline{\sigma_l^{-1}(-1)}$ and ${\mit\Omega}_2\subset{\rm Int}(\overline{\sigma_l^{-1}(1)})$. Then ${\mit\Omega}_2\subset S^\ast$ and ${\rm dim}(S^\ast\cap H)=1$. But ${\mit\Omega}_1\subset\overline{S\setminus\partial _{\rm z} S}=\overline {S^\ast}$ and ${\mit\Omega}_1\subset\overline{A_i}$, then ${\mit\Omega}_1\subset\overline{S^\ast}\setminus S^\ast$ because $X\setminus S^\ast=\bigcup\overline{A_i}$. So $H$ is an irreducible component of $\partial _{\rm z} S^\ast$, hence ${\rm dim}(S^\ast\cap\partial _{\rm z} S^\ast)=1$. \end{pf} \begin{prop} Let $S$ be a semialgebraic set in the sphere ${\Bbb S}^2$ such that $\partial _{\rm z} S\cap S=\emptyset$ (resp. a closed semialgebraic set in ${\Bbb S}^2$). Then $S$ is basic open (resp. closed) if and only if for each $i=1,\dots,t$, $\sigma_i^S$ is completable. \end{prop} \begin{pf} It suffices to prove the result for a semialgebraic $S$ such that $\partial _{\rm z} S\cap S=\emptyset$, because if $S$ is closed, applying 1.5 we have done.\\ \indent Suppose $S$ to be basic open, then $S^\ast$ is generically basic and ${\rm dim}(S^\ast \cap\partial _{\rm z} S^\ast)<1$. By lemma 2.5 no irreducible component of $\partial _{\rm z} S$ is positive type changing with respect to $\sigma_i^S$ for each $i=1,...,t$. But also they cannot be negative type changing, because any curve in ${\Bbb S}^2$ divides ${\Bbb S}^2$ into connected components in such a way that none of them lies on both sides of a branch of the curve (because the curves in ${\Bbb S}^2$ have orientable neighbourdhoods).\\ \indent If for some $i\in\{1,...,t\}$, $\sigma_i^S$ were not locally completable at some point $p\in\partial _{\rm z} S$, we could find (Theorem 2.2) a non singular surface $V$ together with a contraction $\pi:V\to{\Bbb S}^2$ of an algebraic curve $E\subset V$ to the point $p$, with $\pi^{-1}(\partial _{\rm z} S)$ normal crossing in $V$, such that an irreducible component $D$ of $E$ would be type changing with respect to $\sigma_i'=\sigma_i^S\cdot \pi$. But $p\not\in S\cup A_i$, so $\sigma_i'$ is defined on $V\setminus\pi^{-1}(\partial _{\rm z} S)$ by \begin{eqnarray*} (\sigma_i')^{-1}(1)=&\pi^{-1}(S)=&T\\ (\sigma_i')^{-1}(-1)=&\pi^{-1}(A_i)&; \end{eqnarray*} \noindent and as $\pi:V\setminus(\pi^{-1}(\partial _{\rm z} S))\to {\Bbb S}^2\setminus\partial _{\rm z} S$ is a biregular isomorphism, $\sigma_i'=\sigma_i^T$. So $T$ and $T^\ast$ are generically basic, being biregularly isomorphic to $S$, hence ${\rm dim}(\partial _{\rm z} T^\ast\cap T^\ast)<1$; so $D$ cannot be positive type changing by lemma 2.5. We have to exclude that $D$ is negative type changing. Suppose it is so; then we can find two open sets ${\mit\Omega},{\mit\Omega}'\subset D$ such that $\pi^{-1}(A_i)$ lies on both sides of ${\mit\Omega}$ and ${\mit\Omega}'$ divides $T$ from $\pi^{-1}(A_i)$. Then there exists an irreducible component $Z$ of $\partial _{\rm z} S$ such that its strict transform $Z'$ crosses $D$ between ${\mit\Omega}$ and ${\mit\Omega}'$. Then $Z'$ must cross $\pi^{-1}(A_i)$, because this open set is connected. This is impossible since $A_i$ does not lie on both sides of $Z$. Then no irreducible component of $\partial _{\rm z} S$ is type changing with respect to $\sigma_i^S$ and $\sigma_i^S$ is locally completable at any $p\in\partial _{\rm z} S$ for all $i$; so $\sigma_i^S$ is completable, since ${\rm H}_1({\Bbb S}^2,{\Bbb Z}_2)=0$ (2.2).\\ \indent Suppose now that $\sigma_i^S$ is completable for each $i=1,...,t$ and let $f_i\in{\cal R}({\Bbb S}^2)$ be a regular function inducing $\sigma_i^S$. Clearly $S\subset\{f_1>0,\dots,f_t>0\}.$ But if $x\not\in S$ then $x\in A_l$ for some $l=1,...,t$ or $x\in\partial _{\rm z} S$. If $x\in A_l$, then $f_l<0$; if $x\in\partial _{\rm z} S$, then $f_i(x)=0$ for all $i=1,...,t$, since $f_i$ vanishes on $\partial _{\rm z} S$ by the very definition of completability. So $S=\{f_1>0,\dots,f_t>0\}$ and it is basic. \end{pf} Before proving an analogous result for a general surface we need a lemma. \begin{lem} Let $S\subset X$ be an open semialgebraic set such that $S=S^\ast$, $\partial _{\rm z} S\cap S=\emptyset$ and $\partial _{\rm z} S$ is normal crossing. Let $H$ be an irreducible component of $\partial _{\rm z} S$. Then there exist a non singular algebraic set $H'\subset X$ such that\\ \indent (a) $[H]=[H']$ in ${\rm H}_1(X,{\Bbb Z}_2)$,\\ \indent (b) $H$ and $H'$ are transversals,\\ \indent (c) $H'\cap S=\emptyset$. \end{lem} \begin{pf} $H$ is a smooth compact algebraic curve, composed by several ovals and locally disconnects its neighbourhood; moreover $S$ is locally only on one side of $H$, because $S=S^\ast$ and $H\cap S^\ast=\emptyset$, at least outside a neighbourhood of the singular points lying in $H$, namely the points where $H$ crosses an other component of $\partial _{\rm z} S$. At each of these points we have only two possible situations (see figures 2.7). From this it is clear how to construct a smooth differentiable curve $C_H$ with $[C_H]=[H]$ in ${\rm H}_1(X,{\Bbb Z}_2)$ such that $C_H$ is transversal to each irreducible component of $\partial _{\rm z} S$ and $C_H\cap S=\emptyset$, using a suitable tubular neighbourhood of $H$. \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(100,40) \put(0,20){\line(1,0){40}} \put(60,20){\line(1,0){40}} \put(20,0){\line(0,1){40}} \put(80,0){\line(0,1){40}} \put(78,4){\makebox(0,0){$ _H$}} \put(18,4){\makebox(0,0){$ _H$}} \put(5,35){\makebox(0,0){$S$}} \put(65,35){\makebox(0,0){$S$}} \multiput(0,22)(0,2){5}{\line(1,0){19.5}} \multiput(10,32)(0,2){4}{\line(1,0){9.5}} \multiput(60,22)(0,2){5}{\line(1,0){19.5}} \multiput(70,32)(0,2){4}{\line(1,0){9.5}} \multiput(80.5,2)(0,2){9}{\line(1,0){19.5}} \put(23,2){\line(0,1){36}} \put(26,35){\makebox(0,0){$ _{C_H}$}} \put(89,35){\makebox(0,0){$ _{C_H}$}} \bezier{100}(85,39)(85,25)(80,20) \bezier{100}(80,20)(75,15)(75,1) \put(20,-4){\makebox(0,0){{\small Figure 2.7.a}}} \put(80,-4){\makebox(0,0){{\small Figure 2.7.b}}} \end{picture} \end{center} \vskip .2cm \indent Now we want to approximate $C_H$ by a nonsingular algebraic curve $H'$ with the same properties. To do this we use the fact that $[H]$ gives a strongly algebraic line bundle $\pi:E\rightarrow X$ on $X$ (see [BCR, 12.2.5]). For this line bundle, $H$ is the zero set of an algebraic section $h$ and $C_H$ is the zero set of a ${\cal C}^\infty$ section $c$. Let $Q_1,\dots,Q_k$ be the points in the set $H\cap C_H$. We can take a finite open covering $V_1,\dots,V_l$ of $X$ with the following properties:\\ \indent 1) $Q_i\in V_i$ for $i=1,...,k$ and $V_1,\dots,V_k$ are pairwise disjoint.\\ \indent 2) For each $j=1,...,l$ there exist an algebraic section $s_j$ of $E$ such that $s_j(x)\not=0$ for each $x\in V_j$ ($s_j$ generates $E_x$ for each $x\in V_j$).\\ \indent Take a ${\cal C}^\infty$ partition of the unity $\{\varphi_1,\dots,\varphi_l\}$ associated to the covering, with the property that for $i=1,...,k$, $\varphi_i^{-1}(1)$ is a closed neighbourhood of $Q_i$ in $V_i$, $\varphi_i(Q_j)=0$ for $j\not= i$. For $j=1,...,l$ we can write $c\,\vline_{V_j}=\alpha_j s_j,$ \noindent with $\alpha_j\in {\cal C}^\infty(V_j)$ and $\alpha_j(Q_j)=0$ if $j=1,...,k$, because $s_j$ generates the fiber. Then we have $$c=\sum_{j=1}^l\varphi_jc=\sum_{j=1}^l(\varphi_j\alpha_j)s_j$$ \noindent and the smooth functions $\beta_j=\varphi_j\alpha_j\in{\cal C}^\infty(X)$ vanishes at $Q_i$ for $i=1,...,k$.\\ \indent By a classical relative approximation theorem (see [BCR, 12.5.5]) we can approximate $\beta_j$ on $X$ by a regular function $f_j$ on $X$ vanishing at $\{Q_1,\dots,Q_k\}$. Then the algebraic section $$s=\sum_{j=1}^lf_js_j$$ \noindent has an algebraic zero set $H'$ passing through $Q_1,\dots,Q_k$. Moreover $H'\cap S=\emptyset$, because $H'$ is very close to $C_H$, and $[H']=[H]$ in ${\rm H}_1(X,{\Bbb Z}_2)$. \end{pf} Finally we have. \begin{thm} Let $X$ be a compact, non-singular, real algebraic surface and $S\subset X$ be a semialgebraic set with $\partial _{\rm z} S\cap S=\emptyset$ (resp. $S\subset X$ be a closed semialgebraic set).\\ \indent Then $S$ is basic if and only if for each $i=1,\dots,t$ the sign distribution $\sigma_i^S$ verifies the following two properties:\\ \indent (a) No irreducible component of $\partial _{\rm z} S$ is positive type changing with respect to $\sigma_i^S$.\\ \indent (b) No irreducible component of the exceptional divisor of a standard resolution \linebreak $\pi:V\to X$ of $\partial _{\rm z} S$ is positive type changing with respect to $\sigma_i'=\sigma_i^S\cdot\pi$. \end{thm} \begin{pf} It suffices to prove for $S$ with $S\cap\partial _{\rm z} S=\emptyset$, because this implies that $S$ is open and by remark 1.5.1 we have done for $S$ closed.\\ \indent The ``only if part" is the same as for ${\Bbb S}^2$, without the argument proving there are not negative type changing component. For the ``if part" we can reason as follows.\\ \indent Let $\pi:V\to X$ be the standard resolution of $\partial _{\rm z} S$. Denote by $Y$ the curve $\pi^{-1}(\partial _{\rm z} S)$ (see [EC] or [BK]), $Y$ is normal crossing in $V$. Each irreducible component of $Y$ is a non-singular curve consisting possibly of several ovals. Each of this ovals can have an orientable neighbourhood (in which case it is homologically trivial) or a non orientable neighbourhood isomorphic to the M\" obius band.\\ \indent Denote as it is usual $\sigma_i'$ the sign distribution $\sigma_i^S\cdot \pi$ and consider the sets $T=\pi^{-1}(S)$ and $T^\ast={\rm Int}(\overline T)$. Conditions {\em (a)} and {\em (b)} for $\sigma_i^S$ imply that $Y$ has not positive type changing components with respect to $\sigma_i'$. So by 2.5, $\partial _{\rm z} T^\ast\cap T^\ast$ is a finite set, and as $\partial _{\rm z} T^\ast\subset Y$ has not isolated points and $T^\ast$ is open, we have $T^\ast\cap \partial _{\rm z} T^\ast=\emptyset$.\\ \indent Consider now the sign distributions $\sigma_j^{T^\ast}$ on $V\setminus \partial _{\rm z} T^\ast$ defined as before for $j=1,\dots,l$, where $l$ is the number of connected component of $V\setminus(T^\ast\cup\partial _{\rm z} T^\ast)$; clearly they do not have positive type changing components. Apply 2.7 to each irreducible component $H$ of $\partial _{\rm z} T^\ast$ being a change component for some $\sigma_j^{T^\ast}$. Then, we find a non-singular algebraic set $H'$ such that $H'\cap T^\ast=\emptyset$ and $[H]=[H']$ in ${\rm H}_1(V,{\Bbb Z}_2)$. The union of all $H'$ gives an algebraic set $Z\subset V$.\\ \indent Remark that if $H$ is a negative type changing component with respect some $\sigma_j^{T^\ast}$ then this phenomenon occurs along an oval of $H$ whose neighbourhood is a M\" obius band, because in the other case the two sides of the oval whould be in different connected components of $V\setminus\partial _{\rm z} T^\ast$. Take the sign distributions $\tau_K$ on $V\setminus(\partial _{\rm z} T^\ast\cup Z)$, $k=1,\dots,m$, defined by \begin{eqnarray*} (\tau_k)^{-1}(1)&=&T^\ast\\ (\tau_k)^{-1}(-1)&=&B_k, \end{eqnarray*} \noindent where $B_1,\dots,B_m$ are the connected components of $V\setminus (T^\ast\cup\partial _{\rm z} T^\ast\cup Z)$.\\ \indent We claim that $\tau_k$ is completable for each $k=1,...,m$. In fact, we prove that conditions (1), (2) and (3) of 2.2 are verified.\\ \indent (1) No irreducible component of $\partial _{\rm z} T^\ast\cup Z$ is neither positive nor negative type changing with respect $\tau_k$: this is true because now the sign $-1$ can occur at most on one side of each irreducible component $H$ of $\partial _{\rm z} T^\ast\cup Z$ (a M\" obius band is divided by two transversal generators of its not vanishing homological class into two connected components), then there are not negative type changing components with respect $\tau_k$.\\ \indent (2) $\tau_k$ is completable at each point $p\in \partial _{\rm z} T^\ast\cup Z$. In fact, if $\partial _{\rm z} T^\ast\cup Z$ is normal crossing in $p$, since there are not type changing components, $\tau_k$ is locally completable at $p$. But, in general $\partial _{\rm z} T^\ast\cup Z$ is not normal crossing. If $p_0$ is not normal crossing we have, by construction 2.7, two irreducible components $H,H_1$ of $\partial _{\rm z} T^\ast$ and one irreducible component $H'$ of $Z$, with $[H']=[H]$, meeting pairwise transversally at $p_0$. \begin{center}\setlength{\unitlength}{1mm}\begin{picture}(40,40) \put(0,20){\line(1,0){37}} \put(20,2){\line(0,1){35}} \put(35,5){\line(-1,1){30}} \put(40,20){\makebox(0,0){$ _{H_1}$}} \put(20,40){\makebox(0,0){$ _H$}} \put(38,2){\makebox(0,0){$ _{H'}$}} \put(10,10){\makebox(0,0){$+$}} \put(30,30){\makebox(0,0){$+$}} \put(30,30){\makebox(0,0){$+$}} \put(30,15){\makebox(0,0){$-$}} \put(10,25){\makebox(0,0){$-$}} \put(20,-3){\makebox(0,0){\small Figure 2.8}} \end{picture} \end{center} \vskip .2cm \noindent Then by construction we have two signs $+1$ between $H$ and $H_1$ near $p_0$ (if not $H'$ would not cross $H$) and at most two signs $-1$ between $H$ and $H'$ or between $H'$ and $H_1$ (see figure 2.8). So $\tau_k$ is locally completable at $p_0$.\\ \indent (3) If $H$ is a change component for $\tau_k$, then $H\subset \partial _{\rm z} T^\ast$ and if $[H]\not= 0$ then for some irreducible component $Z_H$ of $Z$, $[H\cup Z_H]=0$ and $Z_H\cap T^\ast=\emptyset$, $Z_H\cap B_k=\emptyset$, by construction.\\ \indent So $\tau_k$ is completable for $k=1,...,m$. Let $P_k$ be a regular function inducing $\tau_k$. Then $$T^\ast\subset\{P_1>0,\dots,P_m>0\},$$ \noindent but if $x\not\in T^\ast$, $x\in (\bigcup_{k=1}^mB_k)\cup\partial _{\rm z} T^\ast\cup Z$, hence at least one among $P_k$ verifies $P_k(x)\leq 0$. So $$T^\ast=\{P_1>0,\dots,P_m>0\}$$ \noindent then it is basic, hence $S$ is generically basic, but $S\cap\partial _{\rm z} S=\emptyset$ so, by 1.4, $S$ is basic. \end{pf} From 2.8 we find for surfaces a geometric proof of a general basicness characterization in [AR2]. We call {\em birational model} of a semialgebraic set $S$ any semialgebraic set obtained from $S$ by a birational morphism on $X$. \begin{cor} Let $X$ be a surface and $S\subset X$ a semialgebraic set. Then, $S$ is basic open if and only if $\partial _{\rm z} S\cap S=\emptyset$ and for each birational model $T$ of $S$ we have $\partial _{\rm z} T^\ast\cap T^\ast$ is a finite set. \end{cor} \begin{pf} By 1.4 we are done the {\em only if part}. Suppose now that $\partial _{\rm z} S\cap S=\emptyset$ and for each birational model $T$ of $S$ we have that $\partial _{\rm z} T^\ast\cap T^\ast$ is a finite set. Then, $S$ is open and we will prove that it is basic open.\\ \indent Take a compactification of $X$ (for instance its closure in a projective space) and then take a non-singular birational model $X_1$ of $X$, obtained by a finite sequence of blowings-up along smooth centers. The strict transform $S_1$ of $S$ is a birational model of $S$.\\ \indent Consider now the standard resolution $\pi:X_2\to X_1$ of $\partial _{\rm z} S_1$ and take $S_2$ the strict transform of $S_1$ by $\pi$. Then $\partial _{\rm z} S_2$ is normal crossing and $\partial _{\rm z} S_2^\ast\cap S_2^\ast$ is a finite set, because $S_2$ is a birational model of $S$. But, $\partial _{\rm z} S_2^\ast\subset \partial _{\rm z} S_2$ has not isolated points (it is normal crossing) and $S_2^\ast$ is open, then $\partial _{\rm z} S_2^\ast\cap S_2^\ast=\emptyset$. So by 2.5 and 2.8, $S_2^\ast$ is basic open. Hence $S$ is generically basic and, as $\partial _{\rm z} S\cap S=\emptyset$, $S$ is basic open. \end{pf} \section{Geometric review of fans} For all notions of real algebra, real spectra, specialization, real valuation rings, etc., we refer to [BCR]. Only for tilde operation we use a slightly different definition: for a semialgebraic set $S$ in an algebraic set $X$, $\tilde S$ is the constructible set of ${\rm Spec}_r ({\cal R}(X))$ (instead of ${\cal P}(X)$) defined by the same formula which defines $S$. The properties of this tilde operation are the the same as the usual ones (see [BCR, chap.7]). Let $K$ be a real field, a subset $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ of ${\rm Spec}_r K$ is a {\em 4-element fan} (or simply a {\em fan}) if each $\alpha_i$ is the product of the other three, that is for each $f\in K$ we have $$\alpha_i\alpha_j\alpha_k(f)=\alpha_l(f)$$ \noindent for all $\{i,j,k,l\}=\{1,2,3,4\}$, where $\alpha(f)$ denotes the sign ($1$ or $-1$) of $f$ in the ordering $\alpha\in {\rm Spec}_r(K)$.\\ \indent Given a fan $F$ we can find a valuation ring $V$ of $K$ such that\\ \indent a) Each $\alpha_i\in F$ is compatible with $V$; that is, the maximal ideal ${\frak m}_V$ of $V$ is $\alpha_i$-convex.\\ \indent b) $F$ induces at most two orderings in the residue field $k_V$ of $V$.\\ In this situation we say that $F$ trivializes along $V$ (see [BCR, chap.10] and [Br]). Let $X$ be a real algebraic set, and ${\cal K}(X)$ be the function field of $X$, that is a finitely generated real extension of ${\Bbb R}$. Denote $K={\cal K}(X)$. \begin{defn} A fan $F$ of $K$ is {\em associated to a real prime divisor $V$} if\\ \indent {\em (a)} $V$ is a discrete valuation ring such that $F$ trivializes along $V$.\\ \indent {\em (b)} The residue field $k_V$ of $V$ is a finitely generated real extension of ${\Bbb R}$ such that ${\rm dg.tr.}[K:{\Bbb R}]={\rm dg.tr.}[k_V:{\Bbb R}]+1$. \end{defn} \begin{rmk} {\rm Let $F$ be a not trivial fan (i.e. the $\alpha_i$'s are distincts) associated to a real prime divisor $V$, then it induces two distinct orderings $\tau_1,\tau_2$ in $k_V$ ([BCR, 10.1.10]). If $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ we suppose that $\alpha_1,\alpha_3$ (resp. $\alpha_2,\alpha_4$) induce $\tau_1$ (resp. $\tau_2$) and we write this \[\begin{array}{ccccccccc} V& &\alpha_1& &\alpha_3& &\alpha_2& &\alpha_4\\ \downarrow& & &\searrow\swarrow& & & &\searrow\swarrow& \\ k_V& & &\tau_1& & & &\tau_2& \end{array}\] Conversely, let $\tau_1,\tau_2\in{\rm Spec}_r(k_V)$ be distinct, and let $t\in V$ be a uniformizer for $V$. Each $f\in V$ can be written as $f=t^nu$, where $n$ is the valuation of $f$ and $u$ is a unit in $V$. Denote by $\overline u$ the class of $u$ in $k_V$ and consider the orderings in $K$ defined as follows: \[\begin{array}{ccc} \alpha_1(f)=\tau_1(\overline u)&;&\alpha_3(f)=(-1)^n\tau_1(\overline u)\\ \alpha_2(f)=\tau_2(\overline u)&;&\alpha_4(f)=(-1)^n\tau_2(\overline u)\\ \end{array}\] \noindent They form a fan $F$ of $K$ associated to the real prime divisor $V$.\\ \indent We may consider $\tau_1,\tau_2\in{\rm Spec}_r(k_V)$ as elements of ${\rm Spec}_r(V)$ with ${\frak m}_V$ as support. Then we have that $\alpha_1,\alpha_3$ (resp. $\alpha_2,\alpha_4$) specialize to $\tau_1$ (resp. $\tau_2$) in ${\rm Spec}_r(V)$.\\ \indent When $\alpha$ specializes to $\tau$, we write $\alpha\to \tau$.} \end{rmk} From now we consider the field of rational functions ${\cal K}(X)$ of a compact non-singular real algebraic surface $X$, which is a finitely generated real extension of ${\Bbb R}$ with transcendence degree over ${\Bbb R}$ equal to 2. \begin{rmk} {\rm Let $F$ be a fan in ${\cal K}(X)$ associated to a real prime divisor V of ${\cal K}(X)$. Then, ${\cal R}(X)\subset V$ (because $X$ is compact); consider the real prime ideal ${\frak p}={\cal R}(X)\cap{\frak m}_V$. We have that $V$ dominates ${\cal R}(X)_{\frak p}$ and there are two possibilities:\\ \indent 1) If the height of ${\frak p}$ is 1, it is the ideal of an irreducible algebraic curve $H\subset X$. Since $X$ is non-singular, ${\cal R}(X)_{\frak p}$ is a discrete valuation ring, which is dominated by $V$. Hence $V={\cal R}(X)_{\frak p}$ and $k_V$ is the function field ${\cal K}(H)$ of $H$; so $\tau_1,\tau_2\in{\rm Spec}_r(H)$.\\ \indent 2) If ${\frak p}$ is a maximal ideal, it is the ideal of a point $p\in X$, because $X$ is compact.} \end{rmk} \begin{defn} Let $F$ be a fan of ${\cal K}(X)$ associated to a real prime divisor $V$ and let ${\frak p}={\cal R}(X)\cap{\frak m}_V$. The {\em center of $F$} is the zero set $Z({\frak p})$ of ${\frak p}$.\\ \indent We say that {\em $F$ is centered at a curve (resp. a point)} if ${\frak p}$ has height 1 (resp. is maximal). \end{defn} \begin{lem} Let $S\subset X$ be an open semialgebraic set. Then the following facts are equivalent:\\ \indent {\em (i)} For each fan $F$ of ${\cal K}(X)$ centered at a curve, $\#(F\cap\tilde S)\not= 3$\\ \indent {\em (ii)} $\partial _{\rm z} S$ has not positive type changing components with respect to the sign distributions $\sigma_i^S$ for $i=1,...,t$, defined in 2.4. \end{lem} \begin{pf} Suppose to have a fan $F$ centered at a curve $H\subset X$, such that $\#(F\cap\tilde S)=3$. Then by remarks 3.2 and 3.3 we have:\\ \indent a) $F$ is associated to a real prime divisor $V={\cal R}(X)_{\frak p}$, where ${\frak p}$ is the ideal of $H$.\\ \indent b) If $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$, then $\alpha_1,\alpha_3\to\tau_1\; ,\;\alpha_2,\alpha_4\to\tau_2$ in ${\rm Spec}_r(V)$, with $\tau_1\not=\tau_2$ and $\tau_1,\tau_2\in{\rm Spec}_r({\cal K}(H))$.\\ \indent Suppose $\alpha_1,\alpha_2,\alpha_3\in \tilde S$ and $\alpha_4\not\in\tilde S$. Remark that an element of ${\rm Spec}_r({\cal R}(X)_{\frak p})$ is a prime cone of ${\rm Spec}_r({\cal R}(X))$ which support is contained in ${\frak p}$. So we can consider $\alpha_i,\tau_j\in{\rm Spec}_r({\cal R}(X))$ ($i=1,2,3,4$, $j=1,2$) with $\tau_1,\tau_2\in\tilde H$ and $\alpha_1,\alpha_3\to\tau_1\; ,\;\alpha_2,\alpha_4\to\tau_2$ in ${\rm Spec}_r({\cal R}(X))$.\\ \indent We have $\tau_1\in\overline{\tilde S}=\tilde{\overline S}$, because $\alpha_1,\alpha_3\in \tilde S$. But by [BCR, 10.2.8] there are precisely two prime cone different from $\tau_1$ in ${\rm Spec}_r({\cal R}(X))$ specializing to $\tau_1$, so they are $\alpha_1,\alpha_3$. And as $\alpha_1,\alpha_3,\tau_1\in\tilde{\overline S}$, we get that $\tau_1$ is an interior point of $\tilde{\overline S}$, so $\tau_1\in\tilde{S^\ast}$. This means $\tau_1\in\tilde H\cap\tilde{S^\ast}$, so ${\rm dim}(H\cap S^\ast)= 1$.\\ \indent Now $\tau_2\in\tilde{\overline S}$, because $\alpha_2\in\tilde S$ and $\alpha_2\to\tau_2$. Again [BCR, 10.2.8] the prime cones specializing to $\tau_2$ and different from it are precisely $\alpha_2,\alpha_4$. Since $\alpha_4\not\in\tilde S$ and $\tilde S\cap{\rm Spec}_r({\cal K}(X))=\tilde{\overline S}\cap{\rm Spec}_r({\cal K}(X))$, we have that $\alpha_4\not\in\tilde{\overline S}$, so $\tau_2$ is not interior to $\tilde{\overline S}$, that is $\tau_2\not\in\tilde{S^\ast}$. But $\overline S=\overline{S^\ast}$, then $$\tau_2\in\widetilde{\overline{S^\ast}\setminus S^\ast}\subset\widetilde{\partial _{\rm z} S^\ast} \,.$$ \noindent This implies that ${\rm dim}(H\cap\partial _{\rm z} S^\ast)=1$, then $H$ is an irreducible component of $\partial _{\rm z} S^\ast$. So ${\rm dim}(S^\ast\cap\partial _{\rm z} S^\ast)=1$ and by 2.5 $\partial _{\rm z} S$ has a positive type changing component with respect $\sigma_i^S$ for some $i=1,...,t$.\\ \indent Conversely, let $H$ be a irreducible component of $\partial _{\rm z} S$ which is positive type changing with respect $\sigma_i^S$ for some $i$. Then we can find open sets ${\mit\Omega}_1,{\mit\Omega}_2\in H\cap{\rm Reg}(\partial _{\rm z} S)$ such that\\ \indent a) ${\mit\Omega}_1\subset\overline{\sigma_i^{-1}(1)}\cap\overline{\sigma_i^{-1}(-1)}=\overline S\cap\overline A_i$\\ \indent b) ${\mit\Omega}_2\subset{\rm Int}(\overline{\sigma_i^{-1}(1)})=S^\ast$\\ Let ${\frak p}$ be the ideal of $H$ in ${\cal R}(X)$ and $V$ be the discrete valuation ring ${\cal R}(X)_{\frak p}$. Consider two orderings $\tau_1,\tau_2\in{\rm Spec}_r({\cal K}(H))$, with $\tau_1\in \tilde{\mit\Omega}_1$, $\tau_2\in\tilde{\mit\Omega}_2$, and let $F$ be the fan defined by $\tau_1,\tau_2$ as in 3.2. So $\alpha_1,\alpha_3\to\tau_1\; ,\;\alpha_2,\alpha_4\to\tau_2$ in ${\rm Spec}_r({\cal R}(X))$, as before.\\ \indent We have $\alpha_2,\alpha_4\in\tilde S^\ast$, because $\tau_2\in\tilde S^\ast$ and $S^\ast$ is open. But $\alpha_2,\alpha_4\in{\rm Spec}_r({\cal K}(X))$ and $\tilde S\cap{\rm Spec}_r({\cal K}(X))=\tilde S^\ast\cap{\rm Spec}_r({\cal K}(X))$, then $\alpha_2,\alpha_4\in\tilde S$.\\ \indent On the other hand, $\tau_1\in \tilde{\overline S}\cap\tilde{\overline A_i}$. So there exist $\alpha\in\tilde S$ and $\beta\in\tilde A_i$ with $\alpha,\beta\to\tau_1$. Again by [BCR, 10.2.8] we must have $\alpha=\alpha_1$ and $\beta=\alpha_3$. So $\#(F\cap\tilde S)=3$. \end{pf} \begin{lem} Let $S$ be a open semialgebraic set in $X$ such that $\partial _{\rm z} S^\ast\cap S^\ast$ is a finite set. Fix $p\in\partial S$. Then the following facts are equivalent:\\ \indent {\em (i)} For each fan $F$ centered at $p$, $\#(F\cap\tilde S)\not= 3$.\\ \indent {\em (ii)} For each contraction $\pi:X'\to X$ of a curve $E$ to the point $p$, no irreducible component of $E$ is positive type changing with respect to $\sigma_i'=\sigma_i^S\cdot \pi$, for $i=1,...,t$. \end{lem} \begin{pf} Suppose that there exist a contraction $\pi:X'\to X$ of a curve $E$ to the point $p$ and $i=1,...,t$ such that an irreducible component $H$ of $E$ is positive type changing with respect to $\sigma_i'$. But if $T=\pi^{-1}(S)$, then $(\sigma_i')^{-1}(1)=T$, $(\sigma_i')^{-1}(-1)=\pi^{-1}(A_i)$; moreover, as $p\not\in S$, the set $\{\pi^{-1}(A_i):i=1,...,t\}$ is precisely the set of connected component of $X'\setminus(T\cup\partial _{\rm z} T)$, then $\sigma_i'=\sigma_i^T$. Now by 3.5 there exists a fan $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ of ${\cal K}(X')$ with center the curve $H$ such that $\#(F\cap\tilde T)=3$. \\ \indent The contraction $\pi$ induces a field isomorphism $$\pi_*:{\cal K}(X)\to{\cal K}(X')$$ \noindent and an injective ring homomorphism $\pi_*\,\vline_{{\cal R}(X)}:{\cal R}(X)\to{\cal R}(X').$ Let $G$ be the fan of ${\cal K}(X')$ inverse image of $F$ by $\pi_*$, namely $$G=\{\pi_*^{-1}(\alpha_1),\pi_*^{-1}(\alpha_2),\pi_*^{-1}(\alpha_3),\pi_*^{-1}(\alpha_4)\}\subset{\rm Spec}_r({\cal K}(X))$$ Then $\#(G\cap\tilde S)=3$ and we have to prove that $p$ is the center of $G$. Let $V$ be the real prime divisor associated to $F$, then $\pi_*^{-1}(V)=W$ is the real prime divisor associated of $G$, so $${\frak m}_W\cap{\cal R}(X)=\pi_*^{-1}({\cal J}(H))$$ \noindent where ${\cal J}(H)$ denotes the ideal of $H$ in ${\cal R}(X')$. Hence, $${\frak m}_W\cap{\cal R}(X)={\cal J}(\pi(H))={\frak m}_p$$ \noindent with ${\frak m}$ the maximal ideal of $p$.\\ \indent On the contrary, we suppose that no irreducible component of $E$ is positive type changing with respect to $\sigma_i'=\sigma_i\cdot\pi$, for each contraction $\pi$ of a curve to $p$. Take a neighbourhood $U$ of $p$, homeomorphic to a disk in ${\Bbb R}^2$ ($X$ is non-singular), such that $U$ does not meet any irreducible component of $\partial _{\rm z} S$ unless it contains $p$. Consider the sign distributions $\delta_j$ in $X\setminus (\partial _{\rm z} S\cup \partial U)$ for $j=1,...,l$, defined by \begin{eqnarray*} \delta_j^{-1}(1)&=&U\cap S\\ \delta_j^{-1}(-1)&=&B_j \end{eqnarray*} \noindent where $B_1,\dots,B_l$ are the connected components of $U\setminus(S\cup\partial _{\rm z} S)$. As $\partial _{\rm z} S^\ast\cap S^\ast$ is a finite set, by 2.5 $\partial _{\rm z} S$ has not positive type changing components with respect to $\sigma_i^S$ for all $i=1,...,t$. We claim that $\partial _{\rm z}(U\cap S)$ has no type changing components at all. In fact, as $B_j\subset A_i$ for some $i$, there are not positive ones; $\partial U$ cannot be type changing because the signs may lie only on one side of it and no other component can be negative type changing, for the same reasons as in the proof of 2.6.\\ \indent Now if $\partial _{\rm z} S$ is normal crossing at $p$, by 2.2 $\delta_j$ is locally completable at $p$ for $i=1,...,l$. If not, consider the standard singularity resolution $\pi:X'\to X$ of $\partial _{\rm z} S$ at $p$ and the sign distributions \[\begin{array}{c} \sigma_i'=\sigma_i^S\cdot\pi,\;{\it for}\;i=1,...,t\\ \delta_j'=\delta_j\cdot\pi,\;{\it for}\;i=1,...,l \end{array} \] \noindent As no irreducible component of $\pi^{-1}(p)$ is positive type changing with respect to $\sigma_i'$ for all $i$, the same is true with respect to $\delta_j'$ ($j=1,...,l$). More over each such component has a M\"obius neighbourhood in $\pi^{-1}(U)$ where the sign minus can occur locally only on one side of the curve, so it cannot be negative type changing. Then, by 2.2 $\delta_j$ is locally completable at $p$ for each $j=1,..,l$.\\ \indent Hence, for each $j=1,...,l$, take $f_j\in{\cal R}(X)$ and an open set $U_j\ni p$, $U_j\subset U$, such that $f_j$ induces $\delta_j$ on $U_j$. Consider $A=\bigcap_{j=1}^lU_j$, then $$S\cap A=\{f_1>0,\dots,f_l>0\}\cap A.$$ \noindent In fact, by definition of locally completable we have $$S\cap A\subset \{f_1>0,\dots,f_l>0\};$$ \noindent but if $x\in A\setminus S$, then $x\in(\bigcup B_j)\cup\partial _{\rm z} S$, so there is a $j_0$ such that $f_{j_0}(x)\leq 0$.\\ \indent Let $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ a fan centered at $p$. Clearly $F\subset\tilde A$, because each $\alpha_i$ specializes in ${\rm Spec}_r({\cal R}(X))$ to the prime cone having ${\frak m}_p$ as support, i.e the unique prime cone giving to $f\in{\cal R}(X)$ the sign of $f(p)$ ([BCR, 10.2.3]). Suppose that $\alpha_1,\alpha_2,\alpha_3\in\tilde S$ and $\alpha_4\not\in\tilde S$; then $\alpha_1,\alpha_2,\alpha_3\in\tilde A\cap\tilde S$ and $\alpha_4\not\in\tilde A\cap\tilde S$. So for all $j=1,...,l$, $\alpha_i(f_j)>0$ for $i=1,2,3$ and there is $j_0$ such that $\alpha_4(f_{j_0})<0$. But this is imposible because $F$ is a fan and $\alpha_1\alpha_2\alpha_3(f_{j_0})\not=\alpha_4(f_{j_0})$. \end {pf} \begin{rmk} {\rm Let $S$ be an open semialgebraic set such that $\partial _{\rm z} S^\ast\cap S^\ast$ is a finite set. Let $p\in\partial _{\rm z} S$ such that $p\not\in\partial S$ or $\partial _{\rm z} S$ is normal crossing at $p$. Then for each fan centered at $p$, $\#(F\cap\tilde S)\not= 3$.} \end{rmk} \begin{thm} {\rm (See [Br] and [AR1])} Let $X$ be a real irreducible algebraic surface. Let $S$ be a semialgebraic set such that $\partial _{\rm z} S\cap S=\emptyset$ (resp. a closed semialgebraic set). Then, $S$ is basic open (resp. basic closed) if and only if for each fan $F$ of ${\cal K}(X)$ which is associated to a real prime divisor, $\#(F\cap\tilde S)\not=3$. \end{thm} \begin{pf} Suppose $S$ to be basic open, then $$S=\{f_1>0,\dots,f_r>0\},\;{\rm with}\; f_1,\dots,f_r\in{\cal R}(X).$$ \noindent Let $F=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ be a fan in ${\cal K}(X)$ and suppose that $\alpha_1,\alpha_2,\alpha_3\in S$ and $\alpha_4\not\in S$. Then for all $i=1,...,r$, $f_i(\alpha_j)>0$ ($j=1,2,3$) and there exists $i_0\in\{1,...,r\}$ such that $\alpha_4(f_{i_0})>0$; which is impossible because $F$ is a fan.\\ \indent Conversely, suppose $S$ to be not basic open. If $X$ is compact and non singular, by 2.8, 3.5 and 3.6 we have done. If not, take a birational model $X_1$ of $X$ obtained compactifying and desingularizing $X$. Let $S_1$ be the strict transform of $S$ in $X_1$. Then $S_1$ is not basic open and $\partial _{\rm z}(S_1)\cap S_1=\emptyset$, because $S$ verifies these properties. So, by 2.8, 3.5 and 3.6, we can find a fan $F$ of ${\cal K}(X_1)$ associated to a real prime divisor such that $\#(F\cap\tilde S_1)=3$. Since ${\cal K}(X)$ and ${\cal K}(X_1)$ are isomorphic, $F$ gives a fan $G$ of ${\cal K}(X)$ such that $G$ is associated to a real prime divisor and $\#(G\cap\tilde S)=3$. \end{pf} \section{The algorithms} By 2.3 (see also [ABF]) there is an algorithmic method for checking properties {\em (a)} and {\em (b)} of 2.8; so we can decide algorithmically if a semialgebraic $S$ with $\partial _{\rm z} S\cap S=\emptyset$ is open basic. This method works as follows:\\ \indent 1) It calculates ${\rm dim}(S^\ast\cap\partial _{\rm z} S^\ast)$ by tecniques of cilindrical algebraic descomposition (C.A.D.), for instance (see [BCR]). If it is equal to 1, we know that $S$ is not basic. If not, we continue with 2).\\ \indent 2) It decides if some irreducible component of the exceptional divisor of the standard resolution of $\partial _{\rm z} S$ is positive type changing using \framebox{A1} and \framebox{A2} in the points of $\partial S$ which are not normal crossing (remark that \framebox{A2} decides if the non-local completability at a point is due to a positive or a negative type changing component after some blowing-up). Moreover, from [Vz] we have a complete description for fans in ${\Bbb R}(x,y)$ associated to a real prime divisor.\\ \indent Consider the field ${\Bbb R}((u,v))$ of formal series in two variables over ${\Bbb R}$, with the ordering which extends $0^+$ in ${\Bbb R}((u))$ by $v>0$. Any ordering in ${\Bbb R}(x,y)$ is defined by an ordered ${\Bbb R}$-homomorphism $\psi:{\Bbb R}(x,y)\to{\Bbb R}((u,v))$ (see [AGR]). So a non-trivial fan $F$ is given by 4 homomorphisms $\psi_1,\psi_2,\psi_3,\psi_4$. More precisely: \begin{thm} {\rm (See [Vz])} Let $F$ be a fan in ${\Bbb R}(x,y)$. Then $F$ is described as follows:\\ \indent {\em 1)} If $F$ has as center an irreducible curve $H\subset{\Bbb S}^2$ and if $P(x,y)\in{\Bbb R}[x,y]$ is a polynomial generating the ideal ${\cal J}(H)\subset{\Bbb R}[x,y]$ of the image of $H$ by a suitable stereographic projection, then $$\psi_i:{\Bbb R}(x,y)\to {\Bbb R}((t,z)),\; for\; i=1,2,3,4$$ \noindent are defined (possibly interchanging $x$ and $y$) as follows: \[\left\{ \begin{array}{l} \psi_1(x)=a_1+\delta t^N\\ \psi_1(y)=a_2+\sum_{i\geq 1}c_it^{n_i}+z \end{array}\right. \qquad \left\{\begin{array}{l} \psi_2(x)=b_1+\delta't^M\\ \psi_2(y)=b_2+\sum_{i\geq 1}d_it^{m_i}+z \end{array}\right.\] \[\left\{ \begin{array}{l} \psi_3(x)=a_1+\delta t^N\\ \psi_3(y)=a_2+\sum_{i\geq 1}c_it^{n_i}-z \end{array}\right. \qquad \left\{\begin{array}{l} \psi_4(x)=b_1+\delta't^M\\ \psi_4(y)=b_2+\sum_{i\geq 1}d_it^{m_i}-z \end{array}\right.\] \noindent where $(a_1+\delta t^N,\, a_2+\sum_{i\geq 1}c_it^{n_i})$ and $(b_1+\delta't^M,\, b_2+\sum_{i\geq 1}d_it^{m_i})$ are irreducible Puiseux parametrizations of two half-branches of $H$, centered respectively at $(a_1,a_2)$, $(b_1,b_2)$. {\em 2)} If $F$ is centered at a point $p\in{\Bbb S}^2$, we may suppose $p=(0,0)$ in a suitable stereographic projection, then $$\psi_i:{\Bbb R}(x,y)\to {\Bbb R}((z,t)),\; for\; i=1,2,3,4$$ \noindent are given by one of the following expressions:\\ \indent {\em a)} \[\left\{ \begin{array}{l} \psi_1(x)=t\\ \psi_1(y)=tw \phantom{-} \end{array}\right. \qquad \left\{\begin{array}{l} \psi_2(x)=tw'\phantom{-}\\ \psi_2(y)=t \end{array}\right.\] \[\left\{ \begin{array}{l} \psi_3(x)=-t\\ \psi_3(y)=-tw \end{array}\right. \qquad \left\{\begin{array}{l} \psi_4(x)=-tw'\\ \psi_4(y)=-t \end{array}\right.\] \noindent with $w\in\{z+a,-z+a:a\in{\Bbb R}\}$, $w'\in\{z,-z\}$.\\ \indent {\em b)} Up to interchanging $x$ and $y$, \[\left\{ \begin{array}{l} \psi_1(x)=\delta t^N\\ \psi_1(y)=\sum_{i=1}^sc_it^{n_i}+t^mw\phantom{(-1)^{n_i}(-1)^m} \end{array}\right. \qquad \left\{\begin{array}{l} \psi_2(x)=\delta t^N\\ \psi_2(y)=\sum_{i=1}^sc_it^{n_i}+t^mw'\phantom{(-1)^{n_i}(-1)^m} \end{array}\right.\] \[\left\{ \begin{array}{l} \psi_3(x)=(-1)^N\delta t^N\\ \psi_3(y)=\sum_{i=1}^s(-1)^{n_i}c_it^{n_i}+(-1)^mt^mw \end{array}\right. \qquad \left\{\begin{array}{l} \psi_4(x)=(-1)^N\delta t^N\\ \psi_4(y)=\sum_{i=1}^s(-1)^{n_i}c_it^{n_i}+(-1)^mt^mw' \end{array}\right.\] \noindent with $\delta\in\{1,-1\}$; $c_i\in{\Bbb R}$ for $i=1,...,s$; $N\leq n_1<n_2<\dots <n_s$, ${\rm g.c.d.}(N,n_1,...,n_s)=d$ and ${\rm g.c.d}(d,m)=1$; and $w,w'\in\{z+a,-z+a,1/z,-1/z:a\in {\Bbb R}\}$, with $w\not= w'$ if $d$ is odd and $w\not= w',w\not= -w'$ if $d$ is even. If $N=1$, $c_1=\dots=c_s=0$, then $w,w'\not\in\{1/z,-1/z\}$.\\ \indent {\em c)} Up to interchanging $x$ and $y$, \[\left\{ \begin{array}{l} \psi_1(x)=\delta t^N\\ \psi_1(y)=\sum_{i=1}^sc_it^{n_i}+t^mw \end{array}\right. \qquad \left\{\begin{array}{l} \psi_2(x)=(-1)^{N/d}\delta t^N\\ \psi_2(y)=\sum_{i=1}^s(-1)^{n_i/d}c_it^{n_i}+t^mw' \end{array}\right.\] \[\left\{ \begin{array}{l} \psi_3(x)=\delta t^N\\ \psi_3(y)=\sum_{i=1}^sc_it^{n_i}-t^mw \end{array}\right. \qquad \left\{\begin{array}{l} \psi_4(x)=(-1)^{N/d}\delta t^N\\ \psi_4(y)=\sum_{i=1}^s(-1)^{n_i/d}c_it^{n_i}-t^mw' \end{array}\right.\] \noindent with $\delta,c_i,N,n_i,w,w'$ as in {\em b)}, but $d$ always even and without supplementary conditions on $w,w'$. \end{thm} \begin{rmk} {\rm To each fan $F$ in ${\Bbb R}(x,y)$ centered at $(0,0)$ we can associate two families of arcs through $(0,0)$ which are parametrized by $z\in (0,\epsilon)$. In fact, for each fixed $z\in(0,\epsilon)$, $\psi_1$ and $\psi_3$ (resp. $\psi_2$ and $\psi_4$) define the two half-branches of the same curve germ $\gamma_1^z$ (resp. $\gamma_2^z$). This curves, $\gamma_1^z$ and $\gamma_2^z$, verify one with respect to the other conditions a) and b) of 2.3 (see [ABF, 2.19]).} \end{rmk} We want to know the relations between these two families of arcs associated to $F$ and the families of arcs of \framebox{A1} and \framebox{A2} (2.3). Let $C$ be a curve germ through $(0,0)\in{\Bbb R}^2$; and consider the standard resolution $\pi=\pi_N\cdots\pi_1$ of $C$ at $(0,0)$. For each $i=1,\dots,N$, denote by $C_i$ the curve $(\pi_i\cdots\pi_1)^{-1}(C)$, by $D_i$ the exceptional divisor arising during the $i^{th}$ blowing-up, and by $E_i$ the exceptional curve after $i$ blowings-up (i.e. $E_1=(\pi_i\cdots\pi_1)^{-1}(0,0)$). $D_i$ is an irreducible component of $E_i$. \begin{defn} Let $F$ be a fan of ${\Bbb R}(x,y)$ with center $(0,0)\in{\Bbb R}^2$, for $i=1,...,N$ denote by $F_i$ the fan obtained from $F$ after $i$ blowings-up by lifting the orderings of $F$. We say that $F$ has the {\em property $\star(\rho)$} with respect to $C$ if it verifies:\\ \indent {\em a)} $F_{\rho-1}$ is centered at the point $0=C_{\rho-1}\cap D_{\rho-1}$.\\ \indent{\em b)} $F_{\rho-1}$ is described in the sense of 4.1 as in {\em 2-a)} or {\em 2-b)} with $N=1$, $c_1=\dots=c_s=0$.\\ \indent {\em c)} $C_{\rho-1}$ is not tangent to any curve of the two families associates to $F_{\rho-1}$. \end{defn} \begin{rmk} {\rm A fan $F$ verifies $\star(\rho)$ with respect to a curve $C$ if and only if for each $z\in(0,\epsilon)$ the curves $\gamma_1^z$ and $\gamma_2^z$ verify a) and b) of 2.3 with respect to $C$ (see also [ABF, 2.9]).} \end{rmk} Finally we have: \begin{thm} Let $S$ be an open semialgebraic set in $X$, such that $\partial _{\rm z} S\cap S=\emptyset$. Suppose that $\partial _{\rm z} S^\ast\cap S^\ast$ is a finite set and $S$ is not basic. Then, there exists an algorithmic method for finding a fan $F$ of ${\cal K}(X)$ with $\#(F\cap\tilde S)=3$. \end{thm} \begin{pf} By 2.5 $\partial _{\rm z} S$ has not type changing components with respect to $\sigma_i^S$; then by 2.8 there is at least one irreducible component $D_\rho$ of the exceptional divisor of a standard resolution $\pi$ of $\partial _{\rm z} S$ at a point $O=(0,0)$ which is positive type changing with respect to some $\sigma_i'=\sigma_i^S\cdot\pi$. By 2.3 we can find algorithmically two arcs $\gamma_1,\gamma_2$ with the properties a) and b) of 2.3 with respect to an irreducible component of $\partial _{\rm z} S$ through $p=\pi(D)$, for $\rho>0$, such that $\gamma_1$ joins two regions in $(\sigma_i^S)^{-1}(1)$, while $\gamma_2$ joins a region in $(\sigma_i^S)^{-1}(1)$ to a region in $(\sigma_i^S)^{-1}(-1)$. Moreover, each $\gamma_i$ ($i=1,2$) is defined by open conditions.\\ \indent Then $(\gamma_1)_\rho$ and $(\gamma_2)_\rho$ are smooth arcs wich meet $D_\rho$ transversally in different points $p,q\in D_\rho$ and $\rho$ is the first level in the resolution process at wich $\gamma_1$ and $\gamma_2$ are separated. By [ABF, 2.19], $\gamma_1$ and $\gamma_2$ are parametrized by \[ \gamma_1:\left\{ \begin{array}{l} x=\delta t^N\\ y=\sum_{i=1}^sc_it^{n_i}+f(t) \end{array}\right. \qquad \gamma_2:\left\{\begin{array}{l} x=\delta' t^N\\ y=\sum_{i=1}^sd_it^{n_i}+g(t) \end{array}\right.\] \noindent where $\delta,\delta'\in\{1,-1\}$, $c_i,d_i\in{\Bbb R}$ are determined by [ABF, 2.19] as follows: $\delta'=\delta$ and $d_i=c_i$ or $\delta'=(-1)^{N/d}\delta$ and $d_i=(-1)^{n_i/d}c_i$, for $i=1,...,s$, with $d={\rm g.c.d.}(N,n_1,\dots,n_s)$; and $f(t)=at^m+...$, $g(t)=bt^m+...$ with $a\not= b$.\\ \indent We can construct four fans with the property $\star(\rho)$ with respect to $\gamma_1$ and $\gamma_2$ as follows:\\ \indent At the level $\rho$ of the resolution process we have four fans centered at $D_\rho$ in half-branches at $p$ and $q$ (Fig. 4.5), obtained by taking respectively a half-branch of $D_\rho$ at $p$ and an other at $q$. \begin{center}\setlength{\unitlength}{1mm}\begin{picture}(100,30) \multiput(20,2)(20,0){4}{\line(0,1){26}} \multiput(20,30)(20,0){4}{\makebox(0,0){$_{D_\rho}$}} \multiput(16,7)(0,1){3}{\line(4,1){4}} \multiput(56,7)(0,1){3}{\line(4,1){4}} \multiput(20,8)(0,1){3}{\line(4,-1){4}} \multiput(60,8)(0,1){3}{\line(4,-1){4}} \multiput(16,21)(0,1){3}{\line(4,-1){4}} \multiput(36,21)(0,1){3}{\line(4,-1){4}} \multiput(20,20)(0,1){3}{\line(4,1){4}} \multiput(40,20)(0,1){3}{\line(4,1){4}} \multiput(36,11)(0,1){3}{\line(4,-1){4}} \multiput(76,11)(0,1){3}{\line(4,-1){4}} \multiput(40,10)(0,1){3}{\line(4,1){4}} \multiput(80,10)(0,1){3}{\line(4,1){4}} \multiput(56,17)(0,1){3}{\line(4,1){4}} \multiput(76,17)(0,1){3}{\line(4,1){4}} \multiput(60,18)(0,1){3}{\line(4,-1){4}} \multiput(80,18)(0,1){3}{\line(4,-1){4}} \multiput(20,10)(20,0){4}{\circle*{1.5}} \multiput(20,20)(20,0){4}{\circle*{1.5}} \multiput(22,12)(40,0){2}{\makebox(0,0){$_p$}} \multiput(22,18)(20,0){2}{\makebox(0,0){$_q$}} \multiput(42,8)(40,0){2}{\makebox(0,0){$_p$}} \multiput(62,22)(20,0){2}{\makebox(0,0){$_q$}} \put(50,-1){\makebox(0,0){Figure 4.5}} \end{picture} \end{center} \indent Going back by the birational morphism $\pi_\rho\cdots\pi_1$ we obtain four fans centered at $O=(0,0)$ such that all they have the property $\star(\rho)$ with respect to $\gamma_1$ and $\gamma_2$. More precisely applying again [ABF, 2.19] we can describe them in terms of 4.1: for every pair $\eta,\eta'\in\{1,-1\}$ we have one of this fans $F_{\eta,\eta'}$ and his associated arcs are \[ \gamma_1^z:\left\{ \begin{array}{l} x=\delta t^N\\ y=\sum_{i=1}^sc_it^{n_i}+(\eta z+a)t^m \end{array}\right. \qquad \gamma_2^z:\left\{\begin{array}{l} x=\delta' t^N\\ y=\sum_{i=1}^sd_it^{n_i}+(\eta'z+b)t^m \end{array}\right.\] \indent By the construction of $\gamma_1,\gamma_2$ it is easy to check that $\#(F_{\eta,\eta'}\cap\tilde S)=3$. \end{pf} \begin{rmk} {\rm In the hypotesis of 3.13 there are in fact infinite fans $F$ of ${\cal K}(X)$ verifying $\#(F\cap\tilde S)=3$, because there are infinite pairs of arcs joining respectively two region with sign $1$ and a region with sign $1$ with a region with sign $-1$.\\ \indent So we find only fans with $w,w'\in\{z+a,-z+a:a\in{\Bbb R}\}$, according to description 4.1.2. In other case, $(\gamma_1^z)_{\rho}$ or $(\gamma_2^z)_{\rho}$ for some $\rho$, would be tangent to $D_{\rho}$, but applying \framebox{A1} and \framebox{A2} as in [ABF] we take $\gamma_1$, $\gamma_2$ without this property. This means that if it exists a fan $F$ with $w\;{\rm or}\; w'\in\{1/z,-1/z\}$ and $\#(F\cap\tilde S)=3$, there is another fan $F'$ with $w,w'\not\in\{1/z,-1/z\}$ and $\#(F'\cap\tilde S)=3$.} \end{rmk} \section{Principal sets} Using the results of the previous sections, we obtain a simple characterization of principal open (resp. closed) sets. In order to conserve the unity of this paper we give all results about principal sets in dimension 2, but in fact they can be extended to arbitrary dimension following similar proofs (Remarks 5.9). Details can be found in [Vz].\\ \indent Let $X$ be a compact, non singular, real algebraic surface. \begin{defn} A semialgebraic set $S\subset X$ is {\em principal open} (resp. {\em principal closed}) if there exists $f\in{\cal R}(X)$ such that $$S=\{x\in X:f(x)>0\}$$ $$(resp.\; S=\{x\in X:f(x)\geq 0\})$$ \end{defn} \begin{defn} A semialgebraic set $S\in X$ is {\em generically principal} if there exists a Zariski closed set $C\in X$ with ${\rm dim} (C)\leq 1$ such that $S\setminus C$ is principal open. \end{defn} \begin{rmk} {\rm A semialgebraic set $S$ is principal closed if and only if $X\setminus S$ is principal open.\\ \indent Then it suffices to work with principal open sets.} \end{rmk} \begin{nott} {\rm Let $S$ be an open semialgebraic set and $Y=\partial _{\rm z} S$. We denote by $S^c$ the open semialgebraic set $X\setminus(S\cup Y)$ and by $A_1,\dots,A_t$ (resp. $B_1,\dots,B_l$) the connected component of $S^c$ (resp. $S\setminus Y$).\\ \indent Let $\sigma_i$ be the sign distributions $\sigma_i^S$ for $i=1,\dots,t$ and $\sigma_j^c$ be the sign distributions $\sigma_j^{S^c}$ for $j=1,\dots,l$ defined as in 2.4. And denote by $\delta$ the total sign distribution defined by \begin{eqnarray*} \delta^{-1}(1)&=&S\setminus Y\\ \delta^{-1}(-1)&=&S^c \end{eqnarray*} } \end{nott} \begin{rmk} {\rm A semialgebraic set $S$ such that $\partial _{\rm z} S\cap S=\emptyset$ is principal open if and only if the sign distribution $\delta$ is admissible (that is, there exists $f\in{\cal R}(X)$ such that $f$ induces $\delta$ on $X\setminus\partial _{\rm z} S$).} \end{rmk} \begin{thm} Let $S$ be a semialgebraic set such that $\partial _{\rm z} S\cap S=\emptyset$. Then $S$ is principal open if and only if $S^*\cap\partial _{\rm z} S^*$ and $(S^c)^*\cap\partial _{\rm z}(S^c)^*$ are finite sets. \end{thm} \begin{pf} Suppose that $S$ is principal then $S$ and $S^c$ are basic and by 2.8 no irreducible component of $\partial _{\rm z} S$ is positive type changing with respect to $\sigma_i$ and $\sigma_j^c$ for each $i=1,...,t$, $j=1,...,l$. Applying now 2.5 we have that $S^*\cap\partial _{\rm z} S^*$ and $(S^c)^*\cap\partial _{\rm z}(S^c)^*$ are finite sets.\\ \indent Conversely suppose that $S^*\cap\partial _{\rm z} S^*$ and $(S^c)^*\cap\partial _{\rm z}(S^c)^*$ are finite sets, then by 2.5 again no irreducible component of $\partial _{\rm z} S$ is positive type changing with respect to $\sigma_i$ and $\sigma_j^c$ for $i=1,...,t$, $j=1,...,l$.\\ \indent Remark that an irreducible component $H$ of $\partial _{\rm z} S$ is positive (resp. negative) type changing with respect to $\delta$ if and only if $H$ is positive type changing with respect to $\sigma_i$ for some $i$ (resp. $\sigma_j^c$ for some $j$).\\ \indent Hence no irreducible component of $\partial _{\rm z} S$ is type changing with respect to $\delta$. Denote by $Z^c$ the union of all the change components of $\partial _{\rm z} S$ and by $Z$ the set of points where $Z^c$ has dimension 1. So $[Z]=[Z^c]$ and $[Z]=0$ in ${\rm H}_1(M,{\Bbb Z}_2)$, because it bounds the open sets ${\rm Int}(\overline{\sigma^{-1}(1)})$ and ${\rm Int}(\overline{\sigma^{-1}(-1)})$. Then by [BCR, 12.4.6] the ideal ${\cal J}(Z^c)$ of $Z^c$ is principal. Let $f$ be a generator of ${\cal J}(Z^c)$. Again by [BCR, 12.4.6] for each irreducible component $H_k$ of $\partial _{\rm z} S$ not lying in $Z^c$ we can choose a generator $h_k$ of ${\cal J}(H_k)^2$ (wich exists because $2[H_k]=0$). Then the regular function $f\cdot\prod h_k$ induces $\delta$ or $-\delta$. So $\delta$ is admisible and $S$ principal. \end{pf} Remark that this proof is almost the same as the proof of [AB, Proposition 2]. \begin{thm} Let $S$ be a semialgebraic set in $X$.\\ \indent (1) $S$ is principal open if and only if $\partial _{\rm z} S\cap S=\emptyset$ and for each fan $F$ centered at a curve, $\#(F\cap\tilde S)\not= 1,3$.\\ \indent (2) $S$ is principal closed if and only if $\partial _{\rm z} S\cap(X\setminus S)=\emptyset$ and for each fan $F$ centered at a curve, $\#(F\cap\tilde S)\not= 1,3$. \end{thm} \begin{pf} It is immediately using 5.6 and 3.5. \end{pf} \begin{rmks} {\rm (1) Results 5.6 and 5.7 can be generalized to an arbitrary surface compactifying and desingularizing as in 2.9.\\ \indent (2) All this section can be generalized to a compact, non singular, real algebraic set $X$, because the results of the previus sections used here (specifically 1.3 and 2.5) can be generalized to arbitrary dimension. Moreover, defining fan centered at a hypersurface $H$ of $X$ as a fan $F$ associated to a real prime divisor $V$ such that the prime ideal ${\frak p}={\frak m}_V\cap{\cal R}(X)$ has height $1$ and ${\cal Z}({\frak p})=H$, we find an improvement of 4-elements fans criterion [Br, 5.3].\\ \indent (3) For a compact, non singular, real algebraic set we obtain:\\ {\em A semialgebraic set $S$ is principal open (resp. closed) if and only if $\partial _{\rm z} S\cap S=\emptyset$ (resp. $\partial _{\rm z} S\cap(X\setminus S)=\emptyset$) and $S$ is generically principal.}\\ } \end{rmks}

9312

alg-geom/9312012

en

https://arxiv.org/abs/alg-geom/9312012

alg-geom/9312012

Israel Vainsencher

Israel Vainsencher

Enumeration of $n$-fold tangent hyperplanes to a surface

34 pages, Latex (Corrects Latex errors of previous version, minor changes)

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.

alg-geom

\section{Introduction} \normalsize The purpose of this article is to present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface for $1\leq n\leq6$. The method also yields formulas for the number of multi--tangent planes to a hypersurface. In particular, it enables us to find the number {\bf17,601,000} of rational ({\em singular}) plane {quintic} curves in a generic {quintic} threefold. We give several examples and discuss the difficulties involved for $n\geq7$. Our motivation was in response to a question asked by A. Lopez and C. Ciliberto regarding the number of rational curves in the system of hyperplane sections in a generic $K3-$surface imbedded in \p{4} (resp. \p{5}) as a $(2,3)$ (resp. $(2,2,2)$)--complete intersection. In \cite{clm} (joint with Miranda) they study degenerations $K3\mbox{${\rightarrow}$}$ union of 2 scrolls. According to A. Lopez (priv. comm.), the consideration of limit curves in the scrolls suggests a formula for the number of rational curves in the $K3-$surface. However, the numbers they have found are so far in disagreement with those obtained by the formulas presented here for $n=4,5,6$ (cf. \ref{k3}). A similar question communicated by S. Katz concerns the number of {\em plane} rational curves of degree 5 contained in a generic quintic $3-$fold in \p{4}. The subject was raised by Clemens \cite{c} and has received striking contributions from physicists (cf. Morrison \cite{mor}, Piene \cite{ragni}, Bershadski {\it et al.} \cite{vafa}). The total number of rational curves of degree $\geq4$ has not been verified so far {\it au go\^ut du jour des math\'ematiciens}. The cases of degrees 1 and 2 were treated by Harris \cite{har} and Katz \cite{katz}. As for degree 3, it required a thorough investigation of the Chow ring of the variety of twisted cubics (cf. Ellingsrud and Str{\o}mme \cite{twc},\cite{twc2} (see also \cite{twc3} for a simpler approach). A pleasant byproduct was the development of the computer package {\sc schubert}\ by Katz and Str{\o}mme \cite{schub}. The work of Coray \cite{co} reduces certain enumerative questions concerning rational curves in \p3 to the question of finding the numbers $\Delta_{\mu,\nu}$ of irreducible rational curves of bidegree $(\mu,\nu)$ passing through $2(\mu+\nu)-1$ general points on a quadric surface. He computes $\Delta_{2,3}$ and $\Delta_{2,4}$ (in addition to a few trivial cases). We also obtain here $\Delta_{3,3}$ (\ref{d33}), $\Delta_{2,5}$ (\ref{d25}), and $\Delta_{3,4}$ (\ref{d34}). Counting hyperplanes multi--tangent to a curve is well known as a particular case of the classical formula of De Jonqui\`eres \cite{harris}, \cite{i}. For surfaces, the cases $n\leq 3$ are classical and have been checked with currently standard tools of intersection theory, cf. Kleiman \cite{kleimansing},\cite{i}. The degrees of the ``Severi varieties'' of nodal curves in the plane were computed (in principle) by Ran in \cite{ranb},\cite{ranb1}. Although we have at our disposal multiple point formulas (Kleiman \cite{kleimanmultpts}, Ran \cite{ran}), they do not give the correct answer for multi--tangencies already for $n=$2 or 3 due to the presence of cusps. There are also formulas taking into account stationary multiple-points (Colley \cite{susan}). However, for $n\geq4$ the relevant map does {\em not} satisfy a required curvilinearity hypothesis. This is due to the existence of curves with a triple point in virtually any linear system of dimension $\geq4$ on a surface. Our approach is based on the iteration procedure presented in \cite{i0},\cite{i} (also explored in a broader context in \cite{kleimanmultpts},\cite{ran},\cite{susan}). We obtain, for each $n=1,\dots,6$ a formula for the degree of a zero cycle supported on the set of sequences $(C,y_1,\dots,y_n)$ such that $C$ is a member of a (sufficiently general) linear system of dimension $n$ and $y_1$ is a singular point of $C$, $y_{2}$ is a singular point of the blowup of $C$ at $y_1$, and so on (roughly speaking, cf. \S2 and (\ref{step}),(\ref{dejonq}) for the precise statement). The main novelty here is, essentially, detecting the contribution to that zero cycle due to singularities worse than nodes (cf.\ref{formulas}). We also sharpen the scope of validity of the formulas, now requiring only that the relevant loci be finite (\ref{step}). Thanks are due to the MSRI for the stimulating environment and to P. Aluffi, E. Arrondo, S. Katz, A. Lopez and M. Pedreira for many pleasant conversations and to C. Schoen for the comments following Example 4.6. I'm also indebted to S.L. Kleiman for reading a preliminary version and helping to clarify the proof of the Lemma \ref{reduced}. We also thank {\sc schubert} \cite{schub}, for patiently allowing us to verify many examples. \section{Notation and basic definitions} We recall, for the reader's benefit, some definitions from \cite{i}. Let $Y$ be a smooth variety. For each sequence of integers ${\underline{m}} =(m_1, \dots,$ $ m_r)$ we say an effective divisor $D$ has a singularity of ({\em weak}) type ${\underline{m}} $ if the following holds: \begin{itemize}}\def\ei{\end{itemize} \item there is a point $y_1$ of multiplicity$\geq m_1$ in $D$; next \item blowup $Y$ at $y_1$, let $E_1$ denote the exceptional divisor and let $D_1$ denote the {\em total} transform of $D$; then \item require that the effective divisor $D_1-m_1E_1$ have a point $y_2$ of multiplicity $\geq m_2$, and so on. \ei The sequence $(y_1,y_2,\dots)$ thus constructed is called a {\em singularity of type} ${\underline{m}}$~ of $D$. We further say the type is {\em strict} if all inequalities are equalities and each $y_i$ lies off the exceptional divisor. One may also consider $nested$ sequences $(\dots, m_i$ $(m_{i+1},$ $\dots), \dots)$ and say a singularity is of such type if $y_i$ is of multiplicity$ \geq m_i$ and $y_{i+1}$ is infinitely near to $y_i,\ i.e.,$ lies on the exceptional divisor besides being of multiplicity $\geq m_{i+1}$, etc. We write $ m^{[k]}$ to indicate $k$ repetitions of $m$. \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx} \label{triple} Let $Y$ be a surface and $y_1$ a triple point on the curve $C$. Then of course $C$ has a singularity of strict type $(3)$. However, if the 3 tangents are distinct, $C$ also has a singularity of weak type $(2^{[4]})$ due to the intersections of the strict transform of $C$ and the exceptional line $E_1$: \ex \vskip5pt $$\begin{picture}(0,0)(0,0) \put(-54,-4){\footnotesize o}\put(-47,-3){$y_1$}\put(-52,-2){\line( 0,-1){20}} \put(-52,-2){\line( 0,1){20}}\put(-52,-2){\line( 1,1){20}} \put(-52,-2) {\line( -1,-1){20}} \put(-52,-2){\line( 1,1){20}}\put(-52,-2){\line( 1,-1){20}} \put(-52,-2) {\line( -1,1){20}} \put(0,0){\vector(-1,0){15}} \put(52,-2){\line( 1,0){25}}\put(52,-2) {\line(-1,0){25}} \put(37,-2){\line(0,1){20}}\put(37,-2){\line(0,-1){20}} \put(52,-2) {\line( 0,1){20}} \put(52,-2){\line(0,-1){20}}\put(67,-2){\line(0,1){20}}\put(67,-2) {\line(0,-1){20}} \put(34,-4){{\footnotesize o}}\put(26,-9){$y_2$}\put(49,-4){{\footnotesize o}} \put(41,-9){$y_3$}\put(64,-4){{\footnotesize o}}\put(56,-9){$y_4$} \put(83,-4){$E_1$} \end{picture}$$ \vskip10pt\begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx} \label{tritwo} On the other hand, if $y_1$ is of type $(3(2))$, it follows that $C$ has a singularity of type $(2^{[6]})$!\ex\vskip2pt $$\begin{picture}(40,0)(0,0) \put(-105,6){\oval(25,18)[br]} \put(-101,7){\oval(25,20)[bl]} \put(-105,-12){\oval(25,18)[tr]} \put(-101,-13){\oval(25,20)[tl]} \put(-104,-4){\footnotesize o}\put(-92,-3){$y_1$} \put(-102,-2){\line( 0,-1){12}} \put(-102,-2){\line( 0,1){12}} \put(-55,0){\vector(-1,0){10}} \put(-3,-2){\line( 1,0){30}}\put(-3,-2){\line(-1,0){30}} \put(-23,-2){\line(0,1){12}}\put(-23,-2){\line(0,-1){12}} \put(-26,-4){{\footnotesize o}}\put(-34,-11){$y_3$} \put(6,-2){\line( 1,1){12}}\put(6,-2){\line(-1,-1){12}} \put(6,-2){\line(-1,1){12}}\put(6,-2){\line(1,-1){12}} \put(5,-4){{\footnotesize o}}\put(2,-12){$y_2$} \put(32,-4){$E_1$} \put(65,0){\vector(-1,0){10}} \put(113,-2){\line( 1,0){30}}\put(113,-2){\line(-1,0){30}} \put(91,-2){\line(0,1){12}}\put(91,-2){\line(0,-1){12}} \put(90,-4){{\footnotesize o}}\put(81,-10){$y_3$} \put(104,-2){\line(0,1){12}}\put(104,-2){\line(0,-1){12}} \put(103,-4){{\footnotesize o}}\put(101,14){{\footnotesize$E_1'$}} \put(117,-2){\line(0,1){12}}\put(117,-2){\line(0,-1){12}} \put(116,-4){{\footnotesize o}}\put(118,-10){$y_{2_1}$} \put(134,-2){\line(0,1){12}}\put(134,-2){\line(0,-1){12}} \put(133,-4){{\footnotesize o}} \put(135,-10){$y_{2_2}$} \put(150,-4){$E_2$} \end{picture}$$\vskip10pt\noindent Indeed, let $y_2$ be the double point infinitely near to the triple point $y_1$, and let $C_1$ denote the total transform of $C$; then $C_1-3E_1$ is effective and intersects $E_1$ twice at $y_2$ and once at the (smooth) branch $y_3$. Thus, the divisor $C':=C_1-2E_1$ has multiplicity $3$ at the point $y_2$. Blowing it up, let $C_2$ be the total transform of $C'$; now $C_2-2E_2$ still contains the exceptional line $E_2$ once and therefore has 4 double points: one for the intersection of $E_2$ and the strict transform of $E_1$, two for the branches over $y_2$ and finally one over $y_3$. \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}\label{nonred} Let $Y$ be a surface and $y_1$ a fourfold point on the curve $C$. Then $C_1-2E_1$ is nonreduced, hence $C$ has a singularity $(y_1,\dots, y_r)$ of type $(2^{[r]})$ for any $r$. \ex This ilustrates a main difficulty in our approach to enumeration of singularities. Formulas for a given type are usually not hard to obtain, at least in principle (cf. (\ref{dejonq}) below), but the exact contribution of each strict type actually occurring seems less evident. For the case we're interested in, we have the following description of the possible singularity types. \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx} \label{tipos} Let $Y$ be a smooth surface; fix $n\!\in\!\{1,\dots,6\}$. Let $D$ be an ample divisor on $Y$. Then there exists $r_0$ such that for all $r\geq r_0$ and any sufficiently general linear subsystem $S$ of $|rD|$ of dimension $n$, there are at most finitely many members $C\!\in\! S$ with a singularity of type $(2^{[n]})$. Moreover, we have the following list of possible strict types actually occurring in type $(2^{[n]})$:\vskip-5pt $$\ba{c} n\leq3\Rightarrow (2^{[n]})\ only;\\ n=4\Rightarrow (2^{[4]})\ or\ (3);\\ n=5\Rightarrow (2^{[5]})\ or\ (3,2)\ or\ (2,3);\\ n=6\Rightarrow (2^{[6]})\ or\ ( 3(2))\ or\ any\ of\ (3,2,2),(2,3,2),(2,2,3). \end{array}$$ \ep \vskip-10pt \vskip10pt\n{\bf Proof.\hskip10pt} Set $\L=\O(D)$ and let ${\cal M}_y$\ be the ideal sheaf of a point $y\!\in\! Y$. The members of $|D|$ with an $m-$fold point at $y$ come from $H^0(Y,{\cal M}^m_y\otimes \L)$. Let ${Y}_n^{\hbox{o}}$ denote the complement of the diagonals in $Y^{\times{n}}$. Given a sequence of positive integers, $(m_1,\dots,m_n)$, replacing $\L$ by a sufficiently high power, we may assume $H^1(Y,{\cal M}^{m_1}_{y_1}\dots {\cal M}^{m_n}_{y_n} \otimes \L)=0$ for all $(y_1,\dots,y_n)\!\in\! {Y}_n^{\hbox{o}}$. It follows that the set $$\{(C,y_1, \dots, y_n) \!\in\!|D|\times {Y}_n^{\hbox{o}}\ |\ \hbox{mult}_{y_i}C\geq{m_i}\}$$ is a projective bundle over ${Y}_n^{\hbox{o}}$ with fibre dimension $=\hbox{dim} |D| - \Sigma m_i(m_i+1)/2$. Its image in $|D|$ is of codimension $\Sigma m_i(m_i+1)/2-2$. Therefore no sufficiently general subsystem of dimension$\leq3$ (resp. $\leq7$) has a member with a triple (resp. $4-$fold) point. It can be easily checked that a singularity of type $(2(2))$ ($i.e.$, a double point with another infinitely near) (resp. $(2(2),2)$ or $(2,2(2)))$ imposes 3 (resp. 4) independent conditions. Let $(y_1,\dots,y_6)$ be a singularity of weak type $(2^{[6]})$ occuring in a general $\infty^6$ linear system. As explained just above, a $4-$fold point imposes 8 conditions, so each $y_i$ is at worst a triple point. Moreover, it can be checked that 2 triple points (infinitely near or not) impose at least 8 conditions, thus at most one of the $y_i$ is triple. We claim that $y_i$ cannot be a triple point unless $i\leq3$. Indeed, the imposition of 3 double points $(y_1,y_2,y_3)$ costs at least 3 parameters, leaving less than the 4 required for the acquisition of an additional triple point. A similar argument rules out other sequences of double points (with some possibly infinitely near) different from those listed. \hfill\mbox{$\Box$} \section{Basic setup} \label{setup} Let $f:X\mbox{${\rightarrow}$} S$ be proper and smooth. Let $\L$ be an invertible $\O_X-$module and let $D\subset X$ be the scheme of zeros of a section of $\L$. As in \cite{i}, we construct a scheme $\Sigma({\underline{m}};D)$ whose fibre over each $s\!\in\! S$ consists of the sequences of singularities of type ${\underline{m}}$~ of the fibre $D_s$. Set $X_0=S,\ X_1=X,\ f_{1}=f:X_1\mbox{${\rightarrow}$} X_0.$ For $r\geq1$ denote by $$b_{r+1}:X_{r+1}\mbox{${\rightarrow}$} X_{r}\times_{f_{r}}X_r \quad\hbox{ and }\quad p_{r+1,i}:X_{r}\times_{f_{r}}X_r\mbox{${\rightarrow}$}{}X_r$$ respectively the blowup of the diagonal and the projection. Set $f_{r+1,i}=p_{r+1,i}\circ{} b_{r+1}$. We think of each $X_r$ as a scheme over $X_{r-1}$ with structure map $f_r=f_{r,1}$. Write $E_{1,r}$ for the exceptional divisor of $b_{r}$. For $2\leq j<r$ set $E_{r-j+1,j}=f_{r,2}^*\cdots f_{j+1,2}^*E_{1,j}$. By abuse, still denote by the same symbol pullbacks of $E_{r-j+1,j}$ via compositions of the structure maps $f_3,f_4,\dots$. Notice the 2\up{nd} index in $E_{r-j+1,j}$ indicates where the divisor first appears in the sequence of blowups, whereas $r-j$ keeps track of the number of pullbacks via the $f_{k,2}$. For each sequence of nonnegative integers ${\underline{m}} =(m_1,\dots, m_r)$ we define the divisor on $X_{r+1}$, $$ {\underline{m}} E=m_rE_{1,r+1}+\cdots +m_2E_{r-1,3}+m_1E_{r,2}. $$ Let $y_1\!\in\! X_1$ lie over $s\!\in\! X_0$. Notice that, by construction, the fibre $X_{2{y_1}}$ of $f_2$ over $y_1$ is equal to the blowup of the fibre $f_1^{-1}(s)$ at $y_1$. By the same token, a point in $X_r$ lying over $s$ should be thought of as a sequence $(y_1,\dots,y_r)$ of points in $f_1^{-1}(s)$ each possibly infinitely near to a previous one. Also, the fibre of ${\underline{m}} E$ over a point $(y_1,\dots,y_r)\!\in\! X_r$ is equal to $m_rE_{y_r}+\cdots +m_1E_{y_1}$, where $E_{y_i}\subset X_{i+1y_i}$ denotes (for $i<r$, the total transform of) the exceptional divisor of the blowup of $X_{iy_{i-1}}$ at ${y_i}$. We set $$ \L({\underline{m}})=f_{r+1,2}^*\cdots f_{2,2}^*\L\otimes\O_{X_{r+1}}(-{\underline{m}} E). $$ Pulling back the section of $\L$ defining $D$, we get the diagram of maps of $\O_{X_{r+1}}-$modules, \begin{equation}}\def\ee{\end{equation}\label{s} \ba{l} \O_{X_{r+1}}\\ \downarrow\put(20,10){\vector(3,-1){40}} \put(40,10){$\sigma_{{\underline{m}}}^D$}\\ \hskip-2cm f_{r+1,2}^*\cdots f_{2,2}^*\L\hskip10pt\mbox{${\longrightarrow}$}\hskip10pt f_{r+1,2}^*\cdots f_{2,2}^*\L\otimes\O_{{\underline{m}} E}. \end{array} \ee \noindent By construction, $\sigma_{{\underline{m}}}^D$ vanishes on a fibre $f_{r+1}^{-1}(y_1, \dots,y_r)$ iff $y_1,\dots,y_r$ is a singularity of type ${\underline{m}}$~ of $D_{s}$, where $s= f_1(y_1)$. We define the ${\underline{m}}-${\em contact sheaf} as the $\O_{X_r}-$module, $$ \eml{{\underline{m}}}{\L}=f_{r+1*}(\O_{{\underline{m}} E}\otimes f_{r+1,2}^*\cdots f_{2,2}^*\L). $$ \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} \label{sh} Notation as above, we have: \begin{enumerate}}\def\en{\end{enumerate} \item \eml{{\underline{m}}}{\L} is a locally free \O$_{X_r}-$ module of rank $\Sigma \ (^{dim f+m_i-1}_{\ \ \ \ dim f})$ and its formation commutes with base change; \item there are exact sequences, $$0\mbox{${\longrightarrow}$} \eml{m_r}{\L({\underline{m}}')}\mbox{${\longrightarrow}$} \eml{{\underline{m}}}{\L} \mbox{${\longrightarrow}$} f_{r}^*\eml{{\underline{m}}'}{\L}\lar0,$$ where ${\underline{m}}'$ denotes the truncated sequence $(m_1,\dots,m_{r-1});$ \item we have $\eml{1}\L=\L$ and for $\mu\geq2$ we have an exact sequence, $$0\mbox{${\longrightarrow}$} \L\otimes{}Sym^{\mu-1}\Omega^1_{X/S}\mbox{${\longrightarrow}$} \eml{\mu}{\L} \mbox{${\longrightarrow}$} \eml{\mu-1}{\L} \lar0.$$ \en \el \vskip10pt\n{\bf Proof.\hskip10pt} The inclusion $f_{r+1,2}^*{\underline{m}}'E\subset {\underline{m}} E$ yields the exact sequence \begin{equation}}\def\ee{\end{equation}\label{exs} \ba{ccc} 0\hskip3pt\mbox{${\longrightarrow}$} &\hskip-15pt\O_{{\underline{m}} E}(-f_{r+1,2}^*{\underline{m}}'E)&\hskip-15pt\mbox{${\longrightarrow}$} \hskip3pt\O_{{\underline{m}} E} \hskip3pt\mbox{${\longrightarrow}$}\hskip3pt \O_{f_{r+1,2}^*{\underline{m}}'E}\hskip3pt \lar0\\ &||&\\ &\hskip-20pt\O_{m_rE_{1,r+1}}(-f_{r+1,2}^*{\underline{m}}'E)& \end{array} \ee Notice $f_{r+1,2}^*{\underline{m}}'E$ and $m_rE_{1,r+1}$ are $f_{r+1,1}-$flat. Indeed, for a divisor such as $E_{2,r}:=f_{r+1,2}^*E_{1,r}$ which intersects the blowup center $\Delta(X_r)$ properly (along $\Delta(E_{1,r})$), the total and strict transforms are one and the same. Thus, to show $f_{r+1,1}-$ flat\-ness of $E_{2,r}$ it suffices to verify that each power of the ideal sheaf of $\Delta(E_{1,r})$ in $p_{r+1,2}^*E_{1,r}$ is $p_{r+1,1}-$flat. This is a consequence of the following. \vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} \label{flat} Let $p:X\mbox{${\rightarrow}$} Y$ be a smooth map of smooth varieties. Let $Z\subset X$ be a smooth subvariety of $X$ such that the restriction of $p$ induces an isomorphism $Z\stackrel{\sim}{\mbox{${\rightarrow}$}} p(Z)$ onto a hypersurface of $Y$. Let ${\cal I}$ denote the ideal of $Z$ in $X$. Then each power ${\cal I} ^m$ is $p-$flat. \el\vskip10pt\n{\bf Proof.\hskip10pt} We assume for simplicity dim $p$=1 (hence codim($Z,X$) $=2$). There is a local representation of $p$ by a ring homomorphism ${\cal A\mbox{${\rightarrow}$} B}$ fitting into a commutative diagram, $$\ba{ccc} {\cal A} &\mbox{${\longrightarrow}$} & {\cal B}\\ \uparrow&&\uparrow\\ {\cal C}:={\cal R}[u]&\mbox{${\longrightarrow}$}& {\cal D}:={\cal R}[u,v] \end{array} $$ such that the vertical maps are \'etale, ${\cal R}$ is regular, $u,v$ denote indeterminates and the image of $u$ (resp. $u,v$) generates the ideal of $p(Z)$ (resp. $Z$) (cf. \cite{ak0}, p. 128--130). Under these circumstances, let ${\cal M}$ be a ${\cal D}-$module flat/${\cal C}$. Then ${\cal B}\otimes_{{\cal D}} {\cal M}$ is flat/${\cal A}$. Indeed, put ${\cal A}':={\cal A}\otimes_{{\cal C}}{\cal D}$; clearly ${\cal M}_{{\cal A}}:={\cal A}\otimes_{{\cal C}}{\cal M}$ is an ${\cal A}'-$module flat/${\cal A}$. Notice ${\cal A}\mbox{${\rightarrow}$} B$ factors as ${\cal A}\mbox{${\rightarrow}$} {\cal A}' \mbox{${\rightarrow}$} {\cal B}$ and ${\cal B}$ is \'etale, hence flat/${\cal A}'$ . Let ${{\cal J}}\subset {\cal A}$ be an ideal. We have $0\mbox{${\rightarrow}$} {\cal J}\otimes_{{\cal A}}{\cal M}_{{\cal A}}\mbox{${\rightarrow}$} {\cal M}_{{\cal A}}$ exact. Hence $$\ba{cccc} 0\mbox{${\rightarrow}$} & {\cal B}\otimes_{{\cal A}'}{\cal J}\otimes_{{\cal A}}{\cal M}_{{\cal A}}&\mbox{${\rightarrow}$} & {\cal B}\otimes_{{\cal A}'}{\cal M}_{A}\\ &|| & &|| \\ 0\mbox{${\rightarrow}$} &{\cal J}\otimes_{{\cal A}}{\cal B}\otimes_{{\cal D}}{\cal M}&\mbox{${\rightarrow}$} & {\cal B}\otimes_{{\cal D}}{\cal M} \end{array} $$ is exact by flatness of ${\cal B}$/${\cal A}'$. Apply this to the ideal ${\cal M}=(u,v)^m{\cal D}$, which is a flat, in fact free ${\cal C}-$module with basis $\{u^m,\dots,uv^{m-1},v^m,\dots\}$. \hfill\mbox{$\Box$(for \ref{flat})}\vskip10pt \noindent The same argument applies to all $E_{j,r-j+1}$. Since a sum of flat divisors is flat, we've proved that ${\underline{m}} E$ is $f_{r+1,1}-$flat. \noindent Tensoring (\ref{exs}) with $f_{r+1,2}^*\cdots f_{2,2}^*\L$ and pushing forward by $f_{r+1}=p_{r+1,1}b_{r+1}$, the assertions follow by a standard base change argument (cf. \cite{i}, p. 411).\vskip1pt\hfill\mbox{$\Box$(for \ref{sh})} \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx} \label{step} Let $\Sigma({\underline{m}};D)\subset X_r$ be the scheme of zeros of the section $\sigma_{{\underline{m}}}^D:\O_{X_r}\mbox{${\rightarrow}$} \eml{{\underline{m}}}{\L}$ defined in {\rm(\ref{s})}. Then: \begin{enumerate}}\def\en{\end{enumerate} \item $\Sigma({\underline{m}};D)$ is equal to the scheme of zeros of $ \sigma_{{\underline{m}}}^D$ along the fibres of $f_{r+1}$, thus parametrizing the singularities of type ${\underline{m}}$~ of the fibres of $D$; \item with notation as in Lemma \ref{sh}, setting $D'=$ $f_{r+1,2}^*(f_{r,2}^*\dots f_{2,2}^*D-{\underline{m}}'E)$ restricted over $\Sigma({\underline{m}}';D)$, we have $$\Sigma({\underline{m}};D)=\Sigma((m_r);D');$$ \item each component of $\Sigma({\underline{m}};D)$ is of codimension$\leq \rho= \Sigma\ (^{dim f+m_i-1}_{\ \ \ \ dim f});$ \item \label{mreg} if $\Sigma({\underline{m}};D)$ is empty or of the right codimension $\rho$ then its class in the Chow group of $X_r$ is given by the formula, \en \begin{equation}}\def\ee{\end{equation} \label{dejonq} [\Sigma({\underline{m}};D)]=c_\rho(\eml{{\underline{m}}}{\L})\cap[X_r]. \ee\ep \vskip10pt\n{\bf Proof.\hskip10pt} The 1\up{st} assertion follows from \cite{ak}, Prop.(2.3). The 2\up{nd} one derives from the exact sequence in Lemma \ref{sh}(2). The remaining are well known facts (cf. Fulton\cite{f}).\hfill\mbox{$\Box$} \bs{Remark.}\label{defmreg} In practice, the formula (\ref{dejonq}) may be computed using the exact sequences in \ref{sh}. However, it is only useful to the extent the conditions of (\ref{step})\ref{mreg} are met; we then say $D$ is ${\underline{m}}-${\em regular}. We refer to \cite{i} for sufficient conditions for ${\underline{m}}-$regularity.\end{sex}\rm \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx}\label{iter} Let $D\subset X\mbox{${\rightarrow}$} S$ be as in the beginning of \S\ref{setup}. Set $S'=\Sigma(2,S)$. Fix $P\!\in\! D$. Assume that \begin{enumerate}}\def\en{\end{enumerate} \item $S$ is regular at the image of $P$; \item the ``total space'' $D$ is smooth at $P$ and \item the fibre of $D$ through $P$ has an ordinary double point ({\em odp}) there. \en Then we have that $S'$ is smooth at $P$. Moreover, $D':=f_{2,2}^*D_{|S'}-2E_{1,2|S'}$ is regular along the inverse image of $P$. \ep\vskip10pt\n{\bf Proof.\hskip10pt} We assume for simplicity dim$X/S=2$ and dim$S\geq1$. The question is local analytic. Let $A$ be a regular local ring and ${\cal M}$ its maximal ideal, let $ h\!\in\! B=A[|x_1,x_2|]$ and set ${\cal N}={\cal M} B + (x_1,x_2)B$. Assume that $B/(h)$ is regular and $h=x_1x_2\ mod (x_1, x_2)^3 +{{\cal M}}B$. Then $\bar{B}:=B/(h,h_{x_1},h_{x_2})$ is regular. Indeed, we may write $h=t+m_1x_1+m_2x_2 $ $+x_1x_2+\cdots$, with $t,m_1,$ $m_2\!\in\!{{\cal M}}$. Notice that, since $h\!\in\!{\cal N}-{\cal N} ^2$, we have in fact $t\!\in\!{{\cal M}}-{{\cal M}}^2$. From $h_{x_i}=m_i + x_j + \cdots (\{i,j\} = \{1,2\})$, it follows that $h, h_{x_1}, h_{x_2}$ are linearly independent $mod\ {\cal N}^2$, as desired for the regularity of $\bar{B}$. Let $t_1=t,\dots,t_n$ generate ${\cal M}$ minimally. We may replace $S$ by the germ of curve defined by $t_2,\dots,t_n$. Thus $t$ is a uniformizing parameter of $A$. Since the map germ of $D\mbox{${\rightarrow}$} S$ has an ordinary quadratic singularity at $P$, there are regular parameters $\bar{x}_1$,$\bar{x}_2$ of $D$ around $P$ such that $t\mapsto\bar{x}_1\bar{x}_2$. So now we have reduced to the following. The completion of the local ring of $S$ at the image of $P$ may be writen as $A[|t|]$ for some power series ring $A$. The completion of the local ring of $X$ (resp. $D$) at $P$ is of the form $B=A[|t,x_1,x_2|]$ (resp. $B/(t-x_1x_2))$. Hence $S'$ is represented by the ideal $(t,x_1,x_2)\subset B$. The diagonal and the fibre product $X\times_S{X}$ are represented by $(x_1- x_1',x_2-x_2') \subset{A}[|t,x_1,x_2,x_1',x_2'|]$. The blowup $X_2\mbox{${\rightarrow}$}{X}\times_S{X}$ is given by the inclusion ${A}[|t,x_1,x_2,x_1',x_2'|]\subset{A}[|t,x_1,x_2,x_1',u|]$ defined by $x_2'=x_2+u(x_1'-x_1)$. Restriction over $S'$ therefore takes on the form, $A\mbox{${\rightarrow}$}{A}[|x_1',x_2'|]\subset{A}[|x_1',u|]$, with $D'$ defined by $u$. \hfill\mbox{$\Box$} \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx} \label{reduced} Let $Y$ be a smooth, projective surface and let $D$ be an ample divisor on $Y$. Fix $n\!\in\!\{1,\dots,6\}$. Then there exists an integer $r_0$ such that, for all $r\geq r_0$, for all linear subsystems $S$ of $|rD|$ of dimension $n$ in an open dense subset of the appropriate grassmannian, the following holds: \begin{itemize}}\def\ei{\end{itemize} \item[] $\Sigma((2^{[n]});S)$ is finite, reduced, and for $(C,y_1,\dots,y_n)\!\in\! \Sigma((2^{[n]});S)$ we have that $(y_1,\dots, y_n)$ is a singularity of one of the strict types described in Prop.\ref{tipos}. \ei\ep \vskip10pt\n{\bf Proof.\hskip10pt} As observed in the proof of Prop.\ref{tipos}, ampleness ensures that for any fixed sequence ${\underline{m}}=(m_1,\dots, m_n)$ of positive integers there exists $r_0$ such that, for all $r\geq r_0$, and for any sequence $(y_1,\dots,y_n)$ of distinct points in $Y$, the sheaf ${\cal M}_{y_1}^{m_1}\cdots{\cal M}_{y_n}^{m_n} \otimes \O(rD)$ is generated by global sections. It follows that distinct $y_i$'s impose independent conditions to be a singularity of strict type ${\underline{m}}$ on the system $|rD|$ and in fact, $\Sigma({\underline{m}},rD)$ restricted to the complement of the union of the exceptional divisors in $Y_n$ is a projective bundle. In \cite{i} ((9.1),p. 417) it is shown the same is true over all of $Y_n$ provided ${\underline{m}}$ satisfies the relaxed proximity inequalities $m_{i}\geq- 1+m_{i+1}+\cdots+m_n$ for $i=1,\dots,n-1$. As this is no longer the case for ${\underline{m}}=(2^{[n]}),\ n\geq3$, a direct verification of smoothness is required. At any rate, $\Sigma((2);S)$ and $\Sigma((2,2);S)$ are smooth for all sufficiently ample complete system $S$ and remain smooth upon replacing $S$ by a general subsystem by transversality of a general translate \cite{kltransv}. For $n\geq3$ we proceed by the following iteration argument. Recall from Prop.\ref{step} that for any $D\subset X\mbox{${\rightarrow}$} S$ as in \S\ref{setup}, we have $$\Sigma((2^{[3]});D)=\Sigma((2);D'),$$ where $D'=(f_{3,2}^*(f_{2,2}^*D-2E_{1,2})-2E_{1,3})_{|\Sigma((2,2);D)}$. If $S$ is a sufficiently ample complete system, one checks that $D'$ is regular. In fact, it is the total space of a family of basepoint--free divisors in the fibres of $Y_3\mbox{${\rightarrow}$}{Y_2}$. Indeed, let $Y'\mbox{${\rightarrow}$}{Y}$ be the blowup at $y_1\!\in\!{Y}$ and let $Y''\mbox{${\rightarrow}$}{Y'}$ be the blowup at $y_2\!\in\!{Y'}$. Let $y_3\!\in\!{Y''}$. Let $\L$ be an ample line bundle over $Y$. Then $$H^1(Y'',\L^{\otimes{r}}\otimes{\O_{Y''}(-2E_{y_2})}\otimes {\O_{Y'}( 2E_{y_1})}\otimes{{\cal M}_{y_3}})=0$$ for $r>>0$ because the sequence $(2,2,1)$ satisfies the relaxed proximity inequalities. Hence Prop.\ref{iter} implies that $\Sigma((2^{[3]},D)$ is regular at any $(C,y_1,$ $y_2,$ $y_3)$ such that $y_3$ is an {\em odp} of $C-2E_{y_1}-2E_{y_2}$. Now, if $y_3$ were a triple point (allowed if $n=6$), then we would certainly have $y_3$ not infinitely near ${y_2}$. Let $\pi$ be the involution of $X_2 \times_{X} X_2$ (so that $p_{3,2}\pi=p_{3,1}$). It lifts to an involution of $X_3$ that leaves $\Sigma((2^{[3]},D)$ invariant. Since $\pi$ maps $(C,y_1,y_2,y_3)$ to $(C,y_1,y_3,y_2)$, we get regularity at the latter as well. The same argument yields regularity of $\Sigma((2^{[n]},D)$ for $n=4,5,6$ and $S$ generic, $\infty^n$. For instance, to show $\Sigma((2^{[6]},D)$ is regular at $(C,y_1,\dots,y_6)$ such that $y_1$ is of strict type $(3(2))$ and $y_2$ is the double point infinitely near (cf.\ref{tritwo}), we argue by regularity of $\Sigma((2^{[2]},D)$ at $(y_1,y_2)$ and apply iteration, observing that $(y_3,\dots,y_6)$ are all ordinary quadratic singularities. If $y_2$ were the intersection of the exceptional line and the smooth branch, then $y_3$ must be the double point infinitely near to $y_1$. In this case apply first a permutation and argue as before. \hfill\mbox{$\Box$} \section{Applications} Here are two situations we may apply the above constructions to. \n1:\underline{Linear systems}. Let $Y$ be a smooth projective surface, let ${\cal M}$ be an invertible \O$_Y-$module and let $V\subset H^0(Y,{\cal M})$ be a subspace. Set $S=\p{}(V^*), X=S\times Y$ and let $f:X\mbox{${\rightarrow}$} S$ be the projection. Then $\L={\cal M}\otimes \O(1)$ has a section defining the universal divisor $D$ of the linear system parametrized by $S$. We also write in this case, $\Sigma({\underline{m}};S):=\Sigma({\underline{m}};D)$. \vskip10pt \n2:\underline{Hypersurfaces}. Let $S=Gr(2,N)$ be the Grassmann variety of planes in \p{N}, with tautological quotient sheaf $\O^{N+1}\surj \mbox{${\cal Q}$}$, where rank$\mbox{${\cal Q}$}$=3. Let $X=\p{}(\mbox{${\cal Q}$})\subset S\times\p N$ be the universal plane in \p N. Set $\L=\O\!_{\pp{}({\cal Q})}(d)$ and let $D\subset X$ be defined by a form of degree $d$. Thus the fibre of $D$ over $s\!\in\! S$ is the intersection of a fixed hypersurface with the plane $s$ represents. Using Prop. \ref{reduced} we get the following formulas for the number $tg_n$ of $n-$nodal curves in an $\infty^n$ family of curves, for $n\!\in\!\{1,\dots,6\}$. \vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx} \label{formulas} Fix $n\!\in\!\{1,\dots,6\}$. Let $D\subset X\mbox{${\rightarrow}$} S$ be a family of curves in a smooth family of surfaces of dimension $n$. Assume $\Sigma((2^{[n]};D)$ is reduced and receives contributions only from the strict types described in Prop.\ref{tipos}. Then we have: \begin{itemize}}\def\ei{\end{itemize} \item[] $tg_n=(\#\Sigma((2^{[n]});S)/n!\hbox{\quad for }n\!\in\!\{1,2,3\};$ \item[] $tg_4=(\#\Sigma((2^{[4]});S)-6\#\Sigma((3);S))/4!;$ \item[] $tg_5:=(\#\Sigma((2^{[5]});S)-30\#\Sigma((3,2);S))/5!;$ \item[] $tg_6:=(\#\Sigma((2^{[6]});S)-30\#\Sigma((3(2));S)- 90 \#\Sigma((3,2,2);S))/6!.$ \ei\ep \vskip10pt\n{\bf Proof.\hskip10pt} Let us explain for instance the coefficient 90 appearing in the formula for $tg_6$. Pick $(C,z_1,z_2,z_3)$ in $\Sigma((3,2,2);S)$. Here $C$ is a curve in the system $S$ and $(z_1,z_2,z_3)$ is a singularity of strict type $(3,2,2)$. Let $z_{11},z_{12},z_{13}$ be the branches over $z_1$. It gives rise to the following list of singularities $(y_1,\dots,y_6)$ of weak type $2^{[6]}$ on $C$: \begin{enumerate}}\def\en{\end{enumerate} \item[] $y_1=z_1$ and $(y_2,\dots,y_6)=$ any permutation of $\{z_2,z_3,z_{11},z_{12},z_{13}\}$\vskip1pt\hfill SUBTOTAL:~120. \item[] $y_1=z_i ,~y_2=z_1$ and $(y_3,\dots,y_6)=$ any permutation of $z_j,z_{11},z_{12},z_{13}$ with $\{i,j\}=\{2,3\}$\hfill SUBTOTAL:~48. \item[] $y_1\!=\!z_i,y_2\!=\!z_j,y_3\!=\!z_1$ and $(y_4,y_5,y_6)=$ any permutation of $z_{11},z_{12},z_{13}$ with $\{i,j\}=\{2,3\}$ \hfill SUBTOTAL:~12. \en The factor 180/2 comes from the fact that $(C,z_1,z_2,z_3)$ and $(C,z_1,z_3,z_2)$ yield the same contributions to $\Sigma((2^{[6]});S)$ \hfill\mbox{$\Box$} \vskip10pt Using the formula (\ref{dejonq}) in Prop.\ref{step} the $rhs$ can be computed in terms of Chern classes for each of the two situtions envisaged above. We've made extensive use of {\sc maple\cite{maple}\ \& schubert\cite{schub}}). See the appendix for the computations. \section{Surfaces} For the case of linear systems on a surface $Y$, setting for short, $$\ba{ll} c_2=degree(c_2\Omega^1_Y),\ &k_1=degree(c_1\Omega^1_Yc_1\L),\\ k_2=degree((c_1\Omega^1_Y)^2),\ &d=degree((c_1\L)^2). \end{array}$$ we get from (\ref{sh}), (\ref{formulas}) and (\ref{dejonq}), \vskip8pt \footnotesize \noindent$tg_1:=3 d\!+\!2 k_1\!+\!c_2; $\vskip8pt\noindent$ tg_2:=(tg_1 (\!-\!7\!+\!3 d\!+\!2 k_1\!+\!c_2)\!-\!6 k_2\!-\!25 k_1\!- \!21 d)/2 ;$\vskip8pt \noindent$tg_3:=(2tg_2 (\!-\!14\!+\!3 d\!+\!2 k_1\!+\!c_2)\!+\!tg_1 (\!- \!6 k_2\!-\!25 k_1\!-\!21 d\!+\!40)\!+\! (\!-\!6 k_2\!-\!25 k_1\!-\!21 d) c_2\!-\!63 d^2\!+\!(\!-\!18 k_2\!-\!117 k_1\!+\!672) d\!-\!50 k_1^2\!+\! (\!-\!12 k_2\!+\!950) k_1\!+\!292 k_2)/6;$\vskip8pt \noindent$tg_4= (81d^4\!+\!(216k_1 \!+\! 108c_2 \!-\! 2268)d^3\!+\! (54c_2^2 \!+\! (216k_1 \!-\! 1890)c_2 \!-\! 324k_2 \!+\! 21852 \!-\! 5130k_1 \!+\! 216k_1^2)d^2 \!+\! (12c_2^3 \!+\! ( \!-\! 504 \!+\! 72k_1)c_2^2 \!+\! ( \!-\! 216k_2 \!+\! 8940 \!+\! 144k_1^2 \!-\! 2916k_1)c_2 \!-\! 3816k_1^2 \!+\! 39780k_1 \!+\! 96k_1^3 \!+\! 6024k_2 \!-\! 72360 \!-\! 432k_1k_2)d \!+\! c_2^4 \!+\! ( \!-\! 42 \!+\! 8k_1)c_2^3 \!+\! ( \!-\! 402k_1 \!-\! 36k_2 \!+\! 24k_1^2 \!+\! 699)c_2^2 \!+\! ( \!-\! 3888 \!-\! 144k_1k_2 \!+\! 1756k_2 \!+\! 9046k_1 \!-\! 1104k_1^2 \!+\! 32k_1^3)c_2 \!-\! 144k_1^2k_2 \!+\! 16k_1^4 \!+\! 108k_2^2 \!+\! 4412k_1k_2 \!-\! 936k_1^3 \!+\! 17171k_1^2 \!-\! 28842k_2 \!-\! 95670k_1)/24$; \vskip8pt\noindent$ tg_5 = 81/40 d^5\!+\!(27/8 c_2\!+\!27/4 k_1\!-\!189/2) d^4\!+\!(9/4 c_2^2\!+\!(\!-\!441/4\!+\!9 k_1)c_2\!+\! 9 k_1^2\!- \!27/2 k_2\!-\!1107/4 k_1\!+\!3393/2) d^3\!+\!(3/4 c_2^3\!+\!(\!- \!189/4\!+\!9/2 k_1) c_2^2\!+\! (9 k_1^2\!-\!981/4 k_1\!+\!2469/2\!-\!27/2 k_2) c_2\!-\!27 k_1 k_2\!+\!6 k_1^3\!- \!603/2 k_1^2\!-\! 13875\!+\!471k_2\!+\!8463/2 k_1) d^2\!+\!(1/8 c_2^4\!+\!(\!-\!35/4\!+\! k_1) c_2^3\!+\!(3 k_1^2\!-\! 285/4 k_1\!+\!2207/8\!-\!9/2 k_2) c_2^2\!+\!(4 k_1^3\!-\!4789\!-\!18 k_1 k_2\!-\!180 k_1^2\!+\! 565/2 k_2\!+\!8589/4 k_1) c_2\!-\!145 k_1^3\!-\!22445/4 k_2\!+\!27403/8 k_1^2\!+\!2 k_1 ^4\!+\!27/2 k_2^2\!+\!1355/2 k_1 k_2\!-\!111959/4 k_1\!+\!217728/5\!-\!18 k_1^2 k_2) d\!+\! 1/120 c_2^5\!+\!(1/12 k_1\!-\!7/12) c_2^4\!+\!(141/8\!+\!1/3 k_1^2\!-\!1/2 k_2\!-\!27/4 k_1) c_2^3\!+\!(251/6 k_2\!-\!53/2 k_1^2\!-\!3 k_1 k_2\!+\!2/3 k_1^3\!-\!485/2\!+\!1547/6 k_1) c_2^2 \!+\!(\!-\!17881/12 k_2\!+\!3516/5\!+\!1229/6 k_1 k_2\!-\!68137/12 k_1\!-\!131/3 k_1^3\!+\!9/2 k_2^2\!+\!21551/24 k_1^2\!+\!2/3 k_1^4\!-\!6 k_1^2 k_2) c_2\!+\!727/3 k_1^2 k_2\!-\! 188k_2^2\!-\!8827/2 k_1 k_2\!+\!321882/5 k_1\!+\!9 k_2^2 k_1\!+\!22695 k_2\!+\! 10867/12 k_1^3\!-\! 26189/2 k_1^2\!-\!4 k_1^3 k_2\!+\!4/15 k_1^5\!-\!26 k_1^4;$ \vskip8pt\noindent$ tg_6= ( 81/80 ) d^6\!+\!( 81/40 c_2\!-\!567/8\!+\!81/20 k_1 ) d^5\!+\!( 27/16 c_2^2\!+\!(27/4 k_1\!-\!1701/16) c_2\!-\!81/8 k_2\!+\!8109/4\!+\!27/4 k_1^2\!-\!4077/16 k_1 ) d^4\!+\! ( 3/4 c_2^3\!+\!(9/2 k_1\!-\!63) c_2^2\!+\!(8523/4\!-\!27/2 k_2\!+\!9 k_1^2\!-\!1233/4 k_1) c_2\!+\! 1131/2 k_2\!+\!6 k_1^3\!- \!29601\!-\!27 k_1 k_2\!-\!729/2 k_1^2\!+\!25671/4 k_1 ) d^3\!+\! ( 3/16 c_2^4\!+\!(3/2 k_1\!-\!147/8) c_2^3\!+\!(12909/16\!-\!27/4 k_2\!+\!9/2 k_1^2\!-\!1107/8 k_1) c_2^2\!+\!(2073/4 k_2\!- \!76959/4\!-\!27 k_1 k_2\!+\!41493/8 k_1\!+\!6 k_1^3\!-\!333 k_1^2) c_2\!+\! 3 k_1^4\!+\!81/4 k_2^2\!-\!27 k_1^2 k_2\!- \!96699/8 k_2\!-\!519/2 k_1^3\!+\!1102009/5\!+\! 119961/16 k_1^2\!-\!639927/8 k_1\!+\!4821/4 k_1 k_2 ) d^2\!+\! ( 1/40 c_2^5\!+\!(1/4 k_1\!-\!21/8) c_2^4\!+\!(\!-\!3/2 k_2\!+\!3071/24\!-\!109/4 k_1\!+\!k_1^2) c_2^3 \!+\!(\!-\!201/2 k_1^2\!+\!157 k_2\!+\!2 k_1^3\!-\!29213/8\!-\!9 k_1 k_2\!+\!5421/4 k_1) c_2^2\!+\! (\!-\!26787/4 k_2\!+\!648997/10\!- \!74149/2 k_1\!-\!159 k_1^3\!+\!1481/2 k_1 k_2\!+\!27/2 k_2^2\!+\! 32959/8 k_1^2\!+\!2 k_1^4\!-\!18 k_1^2 k_2) c_2\!+\!853 k_1^2 k_2\!-\!18481 k_1 k_2\!-\! 1317/2 k_2^2\!+\!27 k_2^2 k_1\!+\!1401361/12 k_2\!+\!28988249/60 k_1\!+\!46109/12 k_1^3\!-\! 12 k_1^3 k_2\!+\!4/5 k_1^5\!-\!668388\!-\!554465/8 k_1^2\!-\!92 k_1^4 ) d\!+\!1/720 c_2^6\!+\!(\!-\!7/48\!+\! 1/60 k_1) c_2^5\!+\!(\!-\!1/8 k_2\!+\!1/12 k_1^2\!-\!95/48 k_1\!+\!331/48) c_2^4 \!+\!(\!-\!k_1 k_2\!-\!10 k_1^2\!+\!8147/72 k_1\!-\!8095/48\!+\!565/36 k_2\!+\!2/9 k_1^3) c_2^3\!+\! (\!- \!145/6 k_1^3\!+\!15347/10\!+\!1355/12 k_1 k_2\!-\!3 k_1^2 k_2\!+\!1/3 k_1^4\!+\!9/4 k_2^2\!-\! 190339/48 k_1\!+\!26519/48 k_1^2\!-\!10891/12 k_2) c_2^2\!+\!(\!-\!4 k_1^3 k_2\!-\!85/3 k_1^4\!+\! 4291/4 k_1^3\!+\!9 k_2^2 k_1\!+\!10998\!-\!815/4 k_2^2\!-\!807341/48 k_1^2\!+\!790/3 k_1^2 k_2\!+\! 4/15 k_1^5\!- \!62339/12 k_1 k_2\!+\!691883/24 k_2\!+\!$ $10672201/120 k_1) c_2\!-\! 311237/16 k_1^3\!-\!9/2 k_2^3\!+\!4/45 k_1^6\!+\!7001519/72 k_1 k_2\!-\!2 k_1^4 k_2\!-\! 1855/4 k_2^2 k_1\!+\!9 k_1^2 k_2^2\!+\!1805/9 k_1^3 k_2\!- \!1080646 k_1\!+\! 86753363/360 k_1^2\!+\!200477/36 k_2^2\!+\!26297/36 k_1^4\!-\!13 k_1^5\!-\! 55951/8 k_1^2 k_2\!- \!2567321/6 k_2.$ \normalsize \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf $Y=\p{2}$.} \label{p2} We make the substitutions,\ex \centerline{$c_2=3,d=m^2,k_1=-3 m,k_2=9.$} \bs{ $n=4.$} The expression for $tg_4$ above reduces to\end{sex}\rm \footnotesize $$tg_{4,\p{2}}(m)=\!-\! 8865\!+\!18057/4 m\!+\!37881/8 m^2\!-\!2529 m^3\!-\!642 m^4\!+\!1809/4 m^5 \!-\!27m^7\!+\!27/8 m^8.$$ \noindent\normalsize Setting $m=4$ we get $666=126+540$ for the number of 4--nodal quartics through 10 general points. Indeed, a plane quartic with 4 nodes splits as a union of 2 conics, 126 of which pass through 10 points, or of a singular cubic and a line through $10$ points. \bs{$n=5.$} We find,\end{sex}\rm \noindent\footnotesize $tg_{5,\p{2}}(m)= 81/40 m^{10}\!-\!81/4 m^9\!-\!27/8 m^8\!+\! 2349/4 m^7\!-\!1044 m^6\!-\!127071/20 m^5\!+\!128859/8 m^4\!+\!59097/2 m^3-3528381/40 m^2-946929/20 m\!+\!153513$. \normalsize \vskip8pt\noindent Setting $m=4$ and picking a system of quartics through 9 general points, we do get the right answer, 378=\bin{9}{5}$\times$3. Indeed, a plane quartic with 5 nodes can only be a union of a conic and line pair: hence \bin{9}{5} for the choice of 5 points determining a conic, times the number 3 of line pairs through the 4 remaining points... \bs{ $n=6.$} We have,\par \noindent\footnotesize $tg_{6,\p{2}}(m)=$ $81/80 m^{12}$ - $243/20 m^{11}$ $-$ $81/20 m^{10}$ + $8667/16 m^9$ $-$ $9297/8 m^8 $ $-$ $47727/5 m^7$ + $2458629/ 80 m^6$ + $3243249/40 m^5$ $-$ $6577679/20 m^4$ $-$ $ 25387481/80 m^3$ + $6352577/4 m^2$ + $8290623/20 m $ $-$ $2699706.$ \normalsize \vskip8pt\noindent Again setting $m=4$, we find 105 for the number of 6--nodal quartics through 8 general points: the configurations must consist of 4 lines. \begin{ssex}\em}\def\ess{\end{ssex}\rm Setting $m=5$, we can find the number of {\em irreducible} rational plane quintic curves through 14 general points. This is $tg_{6,\p{2}}(5)-\bin{14}{2}tg_{2,\p{2}}(4)-\bin{14}{5}$ $= 109781-20475-2002=87304$. The corrections are due to the reducible 6--nodal quintics: either line+binodal quartic or conic+cubic. \ess\end{sex}\rm \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf$Y=\mbox{$\p1\!\times\!\p1$}$.} For a system of curves of type $(m_1,m_2)$, we set\newline \centerline{$c_2=4,k_2=8,k_1=-2(m_1+m_2),d=2 m_1m_2$.}\ex \bs{$n=4.$} We get, \vskip8pt\noindent\footnotesize $tg_{4,\mbox{\footnotesize$\pp1\!\!\times\!\pp1$}}(m_1,m_2) = (32/3\!-\!64 m_2\!+\!144 m_2^2\!-\!144 m_2^3\!+\!54 m_2^4) m_1^4\!+\!( 808/3\!-\!3112/3 m_2\!+\!1230 m_2^2\!-\!324 m_2^3\!- \! 144 m_2^4) m_1^3\!+\! (11987/6\!-\!3494 m_2\!-\!2 m_2^2\!+\! 1230 m_2^3 \!+\! 144 m_2^4 ) m_1^2 \!+\!(17359/6\!+\!11333/3 m_2\!-\!3494 m_2^2\!-\!3112/3 m_2^3 \!-\!64 m_2^4) m_1 \!- \!7460\!+\!17359/6 m_2\!+\!11987/6 m_2^2\!+\!808/3 m_2^3 \!+\! 32/3 m_2^4. $\vskip10pt \normalsize \begin{ssex}\em}\def\ess{\end{ssex}\rm If $m_1=m_2=2$, it checks with the number $6$ of configurations of $4$ lines in the system $(2,2)$ through 4 general points on a quadric. Indeed, since $p_a=1$, the curve splits in one of the types: $(1,1)+(1,1)$, $(2,0)+(0,2)$, $(2,1)+(0,1)$ or $(1,2)+(1,0)$. The latter two cases consist of the union of a twisted cubic and a bi-secant line, hence get for free two nodes due to the intersections. In order to present $4$ nodes, the twisted cubic must split further. One easily sees that the only possibility is indeed a configuration $(2,0)+(0,2)$ of $4$ lines. We may assume no $2$ of the $4$ points are on a ruling. Label the points $1,2$ so that the lines composing the curve $(2,2)$ through them are both of system $(1,0)$; this forces the other $2$ lines to be of the opposite system $(0,1)$. Thus, the choice of $1,2$ completely determines the solution, hence \bin{4}{2}. \ess \begin{ssex}\em}\def\ess{\end{ssex}\rm For $(m_1,m_2)=$ $(2,3)$, we find $tg_{4,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2,3) =133$. As $p_a=2$, we obtain again reducible configurations. Notice the system $|(2,3)|$ is $\infty^{11}$. Let the $\infty^4$ subsystem be defined by imposing 7 points. Possible splitting types? (i)$(2,0)+(0,3)$ is $\infty^{5}$, too small. (ii)$(2,1)+(0,2)$ is $\infty^{7}$; 4 nodes due to intersection, $\bin{7}{2}=21$ choices for configuration consisting of twisted cubic$\in\!|(2,1)|$ through $5$ points and line pairs$\in\!| (0,2)|$ through $2$ points. SUBTOTAL:~ 21. (iii)$(1,1)+(1,2)$ is $\infty^{3+5}$; $3$ nodes due to intersection, hence need additional node for either $(1,1)$ or $(1,2)$ component. If the new node is on $(1,1)$, this curve must be a line pair; make it pass through 2 of the points ($\bin{7}{2}$ choices for these) $\times$ 2 (=number of such line pairs for each choice of 2 points), total 42. One takes the $(1,2)-$component through the remaining 5 points, unique choice. SUBTOTAL:~ 42. If the new node is on a $(1,2)-$curve, this must split as $(0,1) +(1,1)$, so the actual solutions are of the form $(1,1)+(0,1)+(1,1)$; if the $7\up{th}$ point is on the line, the remaining 6 will be on $\bin{6}{3}/2$ conic pairs. SUBTOTAL:~ 70. (iv)$(2,2)+(0,1)$ has 2 nodes due to intersection, hence need two additional nodes for $(2,2)-$component; now if a $(2,2)-$curve acquires 2 double points, it splits as $(2,1)+(0,1)$ or $(1,2)+(1,0)$ or $(1,1)+(1,1)$; these have already been accounted for! Thus it all happily adds up to the right TOTAL:~ 133. \ess \begin{ssex}\em}\def\ess{\end{ssex}\rm For $(m_1,m_2)=$ $(2,4)$, we find $tg_{4,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2,4) = $ 1261. The system $|(2,4)|$ is $\infty^{14}$. We impose 10 points to select an $\infty^{4}$ subsystem. Possible splitting types? (i) $(2,3)+(0,1)$ is $\infty^{12}$; $2$ nodes due to intersection. Impose $2$ new nodes for $(2,3)-$component; there are $tg_{2,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2, 3)=105$ through each of the 10 choices of 9 points. Notice that among these 1050 curves there are 90 in $|(2,2)+(0,2)|$. These will be accounted for separately below. SUBTOTAL:~ 960. (ii) $(2,2)+(0,2)$ is $\infty^{10}$; 4 nodes due to intersection; $\bin{10}{2}=45$ choices for 2 points determining a line pair in the system $|(0,2)|$, the remaining 8 points singling out a member in $|(2,2)|$. SUBTOTAL:~ 45. (iii) $(2,1)+(0,3)$ is $\infty^{8}$: too small! Similarly for $(2,0)+(0,4)$. (iv) $(1,4)+(1,0)$ is $\infty^{10}$; 4 nodes due to intersection; 10 choices for the point determining the component $(1,0)$. SUBTOTAL:~ 10. (v) $(1,3)+(1,1)$ is $\infty^{10}$; 4 nodes due to intersection; $\bin{10}{3}$ choices for 3 points determining a conic while the 7 other points determine the component $(1,3)$. SUBTOTAL:~ 120. (vi) $(1,2)+(1,2)$ is $\infty^{10}$; 4 nodes due to intersection; $\bin{10}{5}$ choices for 5 points determining a twisted cubic $(1,2)$ SUBTOTAL:~ 126. It gives the expected TOTAL:~ 1261. \ess \begin{ssex}\em}\def\ess{\end{ssex}\rm{\bf Irreducible rational curves with $p_a=4$ on \mbox{$\p1\!\times\!\p1$}.}\label{d33} We may compute the number {\bf3510} of irreducible rational curves of type $|(3,3)|$ passing through $11=15-4$ general points. We subtract from $tg_{4,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(3,3) =4115$, the contributions given by: (i) \big(nodal $(3,2)$ through 10 points + $(0,1)$ through the 11\up{th}\big): $20\times11=220$; (ii) \big(nodal $(2,3)$ through 10 points + $(1,0)$\big): 220; (iii) \big($(2,2)$ through 8 points + $(1,1)$ through 3 others\big): \bin{11}{8}=165. (Note that $(3,1)+(0,2)$ is $\infty^{9}-$ too small.) \ess\begin{ssex}\em}\def\ess{\end{ssex}\rm\label{d25} Reasoning as above, we also find the number {\bf3684} of irreducible rational curves in the system $(2,5)$ passing through $13=17-4$ general points. This is \big($tg_{4,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2,5) =7038$\big) minus \big($(tg_{2,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2,4) =252)\times13$ due to binodal $(2,4)+(0,1)$\big) minus \big(\bin{13}{11}=78 due to curves $(2,3)$ through 11 points + $(0,2)$ through 2 others\big). \ess\end{sex}\rm \bs{$n=5.$} The first interesting check is provided by the system $|(3,3)|$ on $\mbox{$\p1\!\times\!\p1$}$. We find $tg_{5,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(3,3) =3702$. Here we have $p_a=4$, hence imposing 5 nodes will force again reducible curves. Fix 10 points in general position to define an $\infty^5$ subsystem of $|(3,3)|$. Possible splitting types? (i)$(3,1)+2(0,1)$ is $\infty^{7+1} $ and $(3,1)+(0,2)$ is $\infty^{7+2}$, both too small. (ii)$(3,2)+(0,1)$: $\infty^{12}$; there are 3 nodes due to intersection. Look at members of$|(3,2)|$ through 9 points and with 2 additional nodes: we find $tg_{2,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(3,2)=105$. Among these, 9 split further as $(2,2)+(1,0)$ and will be accounted for separately in (iv). Since there are 10 choices for the 9 points, we have the SUBTOTAL:~ 960. (iii)$(2,3)+(1,0)$: just as in (ii), SUBTOTAL:~960. (iv)$(2,2)+(1,0)+(0,1)$: there are $\bin{10}{2}=45$ times 2 for choices of points and system of line through them. SUBTOTAL:~90. (v)$(2,2)+(1,1)$: we have 4 nodes due to intersection. When the aditional node is on the $(2,2)$ component which passes through 7 points, we find $tg_{1,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(2,2)=12$, times $\bin{10}{3}$ obtaining the SUBTOTAL:~1440. If the additional node be on $(1,1)$, the type becomes $(2,2)+(1,0)+(0,1)$, already accounted for in (iv) above. (vi)$(2,1)+(1,2)$:there are 5 nodes due to intersection; contributes $\bin{10}{5}$, SUBTOTAL:~ 252, fortunately totaling 3702. \begin{ssex}\em}\def\ess{\end{ssex}\rm{How about the irreducible rational curves with $p_a=5$ on \mbox{$\p1\!\times\!\p1$}?} The possible bidegrees are $(2,6),(6,2)$. One expects finitely many of these passing through 15 points. However we notice that {\em any} subsystem $S\subset|(2,6)|$ of codimension 15 meets the family of curves of type $(2,4)+2(0,1)$. Since these present a nonreduced component, therefore $\Sigma((2^{[5]});S)$ contains components of wrong dimension (cf.\ref{nonred}), so that the formula is {\em not} applicable to the present case. It would be nice to compute the equivalence of these bad components. \ess\end{sex}\rm \bs{$n=6.$} \begin{ssex}\em}\def\ess{\end{ssex}\rm We look again at the system $|(3,3)|$ on $\mbox{$\p1\!\times\!\p1$}$. We find $tg_{6,\mbox{\footnotesize$\pp1\!\!\times\!\pp1$}}$ $(3,3)=$ 2224. Fix 9 points in general position to define an $\infty^6$ subsystem. Possible splitting types? (i)$(3,1)+2(0,1):\ \infty^{7+1}$, too small. (ii)\big(nodal$(2,2)$\big) $+(1,0)+(0,1)$: $ 12\times\bin{9}{7}\times2$. SUBTOTAL:~864. (iii)$(3,1)+(0,2):\infty^{7+2}$; contributes $\bin{9}{7}.$ SUBTOTAL:~ 36. (iv)$(1,3)+(2,0)$: SUBTOTAL:~ 36. (iv) $(2,1)+(1,1)+(0,1):\infty^{5+3+1}$; contributes $\bin{9}{5}\times\bin{4}{3}.$ SUBTOTAL:~ 504. (v)$ (1,2)+(1,1)+(1,0)$. SUBTOTAL:~ 504. For several days, we had found only these 1944. The 280 then missing were pointed out to me (after a lunch break at the MSRI) by Enrique Arrondo: $\bin{9}{3}\times\bin{6}{3}/6=280$ curves of the form $(1,1)+(1,1)+(1,1)$!!! \ess \begin{ssex}\em}\def\ess{\end{ssex}\rm{\bf Irreducible rational curves of bidegree $(3,4)$}\label{d34} passing through $13=19-6$ general points: {\bf90508}. We subtract from $tg_{6,\mbox{\footnotesize\mbox{$\p1\!\times\!\p1$}}}(3,4) =122865 $, the contributions given by: (i)\big(trinodal $(3,3)$ through 12 points\big) + \big($(0,1)$ through the 13\up{th}\big): $1944\times13=25272$; (ii)\big(nodal $(2,3)$ through 10 points\big) + \big($(1,1) $ through 3 others\big): 20$\times$\bin{13}{3}=5720; (iii) \big($(2,2)$ through 8 points\big) + \big($(1,2)$ through 5 others\big): \bin{13}{8}= 1287; (iv)\big($(3,2)$ through 11 points\big) + \big($(0,2)$ through 2 others\big): \bin{13}{2}=78. \ess\end{sex}\rm \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf Del-Pezzo surface:} $Y=\p{2}$ blown up at 5 points, imbedded in \p{4} as a $(2,2)$ intersection by the system of plane cubics through the 5 points. There are \bf40 \rm fourfold tangent hyperplanes. Indeed, label the points $\{1,\dots,5\}$; draw the lines $\overline{12},\ \overline{15},\ \overline{34}$; let $a=\overline{12} \cap \overline{34},\ b= \overline{15} \cap \overline{34}.$ Note $1$ is double on $\overline{12} + \overline{15} + \overline{34}$. After blowing up, the hyperplane system $|3L-e_1-\cdots-e_5|$ will contain the curve $e_1 + \overline{12}' + \overline{15}' + \overline{34}'$ (the $'$ denoting strict transform). It presents the 2 double points $a',b'$ and two others on $e_1$. The number of such configurations can be counted as 5 choices for the point labeled 1, times \bin{4}{2} choices for $\overline{12}, \overline{15}$, totaling 30. In addition to these configurations of lines, we may also take the conic $c$ and a line through a pair of the points, say $\overline{12}$; then we get the hyperplane section $c' + \overline{12}' +e_1 +e_2$. This gives 10 more, totaling 40, as predicted by the formula. \ex \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf Surfaces of degree $9$ in \p{4}.} Substituting $$[d=9,k_1=2p_a-11,k_2=6\chi-5p_a+23, c_2=12\chi-k_2]$$ in $tg_4$ with the list of possible pairs (cf. \cite{aure}) $[p_a=\hbox{sectional genus};\chi=(c_2+k_2)/12]$ yields the table, {\footnotesize $$\ba{cccccccc} [6;1]& [7;1]& [7;2]& [8;2]& [8;3]& [9;4]&[10;5]& [12;9]\\ \bf15645 &\bf 57162 &\bf 107646 &\bf 248671 &\bf 388846 & \bf 1022595 &\bf 2222868 &\bf 10957224\rm. \end{array}$$}\ex \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf$K3-$surfaces.}\label{k3} Let $Y$ be embedded by a complete system $|C|$ of curves of genus $n\!\in\!\{3,4,5,6\}$. We have $2n-2=C\cdot(C+K_Y)$. Substituting $[d=2n-2,k_1=0,k_2=0, c_2=24]$ in $tg_n$ we find $$\ba{rcccc} n: &3&4&5&6\\ tg_n: &\bf3200&\bf 25650 &\bf 176256 &\bf 1073720\rm \end{array}$$ For $n\!\in\!\{4,5,6\}$, the values given above for $tg_n$ are smaller then those predicted by a formula Ciliberto and Lopez (priv. communication) obtained by a degeneration argument. A related development is the work of Manoil \cite{m}, where he addresses the question of existence of rational points on $K3-$surfaces defined over a number field. He proves the existence of curves of geometric genus $\leq1$ for a certain class of surfaces by counting singular curves. \ex \begin{exx}{\bf{Example. }}\em}\def\ex{\rm\end{exx}{\bf Abelian surfaces } $Y\subset \p{4}$. Here we find the number \bf150 \rm of $4-$fold tangent hyperplanes. It might be more than just a coincidence the fact that the contribution from $\#\Sigma(3;S)$ is also $=150$, suspiciously a factor of the number $15,000$ of symmetries of the Horrocks-Mumford bundle, a generic section of which is known to vanish precisely on $Y\dots$ The following comments were kindly communicated by Chad Schoen. Let $Y$ be an Abelian surface with a polarization of type $(1,5)$. Any Horrocks-Mumford Abelian surface is of this type. The converse is almost true. I believe that any simple Abelian surface with a $(1,5)$ polarization is a Horrocks-Mumford Abelian surface. Let $N$ be an invertible sheaf giving the $(1,5)$ polarization. A curve in $|N|$ has self-intersection $10$. This is the degree of the normal sheaf which is also the dualizing sheaf. Thus the arithmetic genus is $6$. If the curve is irreducible and has $4$ nodes it's normalization has genus 2. If $Y$ is ``general'' its Picard number is $1$ and any hyperplane section must be irreducible. Let $C$ be such a $4-$nodal curve and $\tilde{C}$ its normalization. There is an isogeny $Jac(\tilde{C}) \mbox{${\rightarrow}$} Y$ taking $\tilde{C}$ to $C$. Again if $Y$ has Picard number $1$, there is no choice but for this map to have degree 5. Now the degree 5 unramified covers of $Y$ are parametrized by the subgroups of order 5 in the fundamental group of $Y$. Write $L$ for this lattice and $L'\subset L$ for the index 5 subgroup. Assuming that $Y$ has Picard number $1$, the 5 fold cover $f:J\mbox{${\rightarrow}$} Y$ will be the Jacobian of a genus 2 curve if and only if $J$ is principally polarized. This will occur if and only if the pull back of the $(1,5)$ polarization on $Y$ is 5 times a polarization on $J$. In terms of lattices and the Riemann form associated to the polarization we have: $$A:(1/5L)/L \times L/5L \mbox{${\rightarrow}$}(1/5\mbox{$Z$})/\mbox{$Z$}=\mbox{$Z$}/5.$$ This alternating form on the 5 torsion of $Y$ has a two dimensional radical--call it $K$. ($K$=vectors in $(1/5)L/L$ which are orthogonal to the whole space). Now the pull back to $J$ is divisible by 5 if and only if the restriction of $A$ to $(1/5)L'/L \times L'/5L \mbox{${\rightarrow}$} (1/5)\mbox{$Z$}/\mbox{$Z$}$ is identically zero. This occurs exactly when $K$ lies in $(1/5)L'/L$. We can count all such $L'$. They are hyperplanes in \p{3} containing a fixed \p{1} all over the field $\mbox{$Z$}/5$. Thus the $L'$ 's are parametrized by \p{1}$(\mbox{$Z$}/5)$. There are 6 possible $L'$ 's. Thus 6 possible $J$'s. Finally we note that translation by elements of $K=\mbox{$Z$}/5\times \mbox{$Z$}/5$ give automorphisms of $Y$ preserving the $(1,5)$ polarization. This gives $6 \times25 =150 $ four-nodal hyperplane sections. There are only 6 different isomorphism classes of genus 2 curve which occur as normalizations. {\bf Question:} Inversion in the Abelian variety should also preserve the polarization (I {\it (C. Schoen)} think). How does this permute the 4--nodal hyperplane sections? \ex \section{Threefolds} The same method yields the formula, \vskip10pt\footnotesize \noindent$ tg_{6,m}=( m^{18}-12m^{17}+24m^{16}+155m^{15}-405m^{14} +1082m^{13}- 18469m^{12}+66446m^{11} - 192307m^{10}+1242535m^{9} -4049006m^{8}+11129818m^{7}- 53664614m^{6}+166756120m^{5} -415820104m^{4}+ 1293514896m^{3}- 2517392160m^{2}+1781049600m)/6! $ \vskip10pt\noindent\normalsize for the number of planes in \p{4} that are 6-fold tangent to a hypersurface of degree $m$. \subsection{Quartics.} For $m=4$, the formula above gives \bf5600\rm. This can be verified by the following direct calculation via the Fano variety $F$ (cf.\cite{ak}) of $\infty^1$ lines contained in a 4\up{ic} threefold {\bf T}. Presently the counting refers to the set $$\{(\ell_1,\dots,\ell_4)\in F^{\times4}|\exists\hbox{ plane }\pi\ s.t.\ \ell_1+\cdots+\ell_4= \pi\cap {\bf T}\}$$ of 4--tuples of coplanar lines in that family. Let $\mbox{${\cal S}$}_i\mbox{\raise-.03cm\hbox{\mbox{\tiny\bf$^\succ$} \O^{\oplus5}\surj\mbox{${\cal Q}$}_i$ (rank $\mbox{${\cal Q}$}_i= i+1$) denote the tautological sequence over the Grassmann variety $G_i:=Gr(i,4)$ of $i-$dimensional subspaces of \p{4}. Go to the incidence variety ${\bf I}}\def\l{{\bf L}:=\{(\ell,\pi)\!\in\! G_1\times{}G_2|\ell\subset\pi\}.$ It carries the diagram of locally free sheaves, (omitting pullbacks) \begin{equation}}\def\ee{\end{equation}\label{taut}\ba{ccccc} \mbox{${\cal S}$}_2 & \mbox{\raise-.03cm\hbox{\mbox{\tiny\bf$^\succ$} & \mbox{${\cal S}$}_1 & \surj &{\cal M}\\ || & &\injdown& &\injdown\\ \mbox{${\cal S}$}_2 & \mbox{\raise-.03cm\hbox{\mbox{\tiny\bf$^\succ$} &\O^{\oplus5}&\surj &\mbox{${\cal Q}$}_2\\ & & \mbox{\hskip.01cm\raise-.37cm\hbox{\mbox{\large$\check{}$}& &\mbox{\hskip.01cm\raise-.37cm\hbox{\mbox{\large$\check{}$}\\ & & \mbox{${\cal Q}$}_1 & = & \mbox{${\cal Q}$}_1 \end{array}\ee The universal plane \p{}$(\mbox{${\cal Q}$}_2)$ contains the total space $D$ of the family of intersections with the fixed 4\up{ic} hypersurface. Our goal is to compute the intersection class supported by $${\bf I}}\def\l{{\bf L}_3:=\{(\ell_1,\ell_2,\ell_3,\pi)\!\in\!{}{\bf I}}\def\l{{\bf L}\times_{G_2}{\bf I}}\def\l{{\bf L} \times_{G_2}{\bf I}}\def\l{{\bf L} | D_\pi\geq\ell_1+\ell_2+\ell_3\}.$$ Set ${\bf I}}\def\l{{\bf L}_1=\{(\ell,\pi) | \ell\subset \pi\cap D_\pi\}.$ This is expressible as zeros of a section of a suitable bundle. Indeed, up on $\p{}(\mbox{${\cal Q}$}_2)_{|{\bf I}}\def\l{{\bf L}}$, we have the Cartier divisors $D_{|{\bf I}}\def\l{{\bf L}}$ and $\l_1:=\p{}(\mbox{${\cal Q}$}_1)_{|{\bf I}}\def\l{{\bf L}}$. One checks that ${\bf I}}\def\l{{\bf L}_1$ is exactly the locus in ${\bf I}}\def\l{{\bf L}$ where ``$\l_1 \subset D$'' holds along fibers. Studying the natural diagram of $\O_{\pp{}(\mbox{${\cal Q}$}_2)_{|{\bf I}}\def\l{{\bf L}}}-$modules, $$\ba{l} \O\\ \downarrow\hskip.7cm\sear{s}\\ \O(D)\mbox{${\rightarrow}$}{}\O_{\l_1}(D) \end{array}$$ one sees that the slant arrow $s$ vanishes on the fiber over $(\ell,\pi)\!\in\! {\bf I}}\def\l{{\bf L}$ iff $\ell\subset\pi\cap D_{\pi}$. Let $p:{\p{}(\mbox{${\cal Q}$}_2)_{|{\bf I}}\def\l{{\bf L}}}\mbox{${\rightarrow}$} {{\bf I}}\def\l{{\bf L}}$ denote the structure map; it follows that ${\bf I}}\def\l{{\bf L}_1$ is the scheme of zeros of the section $p_*s$ of the direct image $sym_4Q_1$ of $\O_{\l_1}(D)=\O_{\l_1}(4)$. We obtain $[{\bf I}}\def\l{{\bf L}_1]=c_5sym_4Q_1$. Pulling back $D$ to ${\bf I}}\def\l{{\bf L}_1$ (and abusing notation), it splits as $D=D_1+\l_1$, thus defining $D_1$. Moreover, since $\p{}(\mbox{${\cal Q}$}_1)$ is the divisor of zeros of a section of $\O_{Q_2}(1)\otimes{}{\cal M}^*$, we have $\O(D_1)=\O_{Q_2}(4)\otimes\O_{Q_2}(-1)\otimes{}{\cal M}$. We may ask when does $D_1$ split further. Go to ${\bf I}}\def\l{{\bf L}_1\times_{G_2} {\bf I}}\def\l{{\bf L}$. Set $\l_2={\bf I}}\def\l{{\bf L}_1 \times_{G_2} \l_1$ and define ${\bf I}}\def\l{{\bf L}_2$ by imposing the fibers of $D_1$ to contain a 2\up{nd} line. As before, ${\bf I}}\def\l{{\bf L}_2$ is given by the vanishing of a section of the pushforward of $\O_{\l_2}(D_1)$. Denoting by $_{(i)}$ the pullback to ${\bf I}}\def\l{{\bf L}\times_{G_2}{\bf I}}\def\l{{\bf L}\cdots$ via $i$\up{th} projection, we find $[{\bf I}}\def\l{{\bf L}_2]=c_4({\cal M}_{(2)}\otimes{}sym_3\mbox{${\cal Q}$}_{1(2)}).$ Similarly, pulling back $D_1$ over ${\bf I}}\def\l{{\bf L}_2$ yields $D_1=D_2+\l_2$ and we get $[{\bf I}}\def\l{{\bf L}_3]=c_3({\cal M}_{(3)}\otimes{}sym_2Q_{1(3)})$. See in the Appendix a script for the actual computation using {\sc schubert}\cite{schub}. Observing that a 6--fold tangent plane $\pi$ to a 4\up{ic} hypersurface cuts 4 lines, the computation gives 134400/24=\bf5600 \rm as asserted. \subsection{Quintics} Recall that a general 5\up{ic}threefold {\bf T}$\subset\!\p{4}$ contains 2,875 lines and 609,250 conics (cf. \cite{har}, \cite{katz}). The plane through a conic counts as a 6--fold tangent since its intersection wih {\bf T} splits as a $conic+cubic$, thereby presenting 6 nodes. Through each line, there are $\infty^2$ planes in \p{4}. The intersection of any such plane with {\bf T} splits as $line+quartic$ thereby counting as a 4--fold tangent. The plane is a 6--fold tangent iff the residual plane quartic is binodal. Fix a line $\ell\subset{\bf T}$. Let us find, among these $\infty^2$ residual plane quartic curves the number of those with 2 double points. This requires the computation of $\Sigma((2,2);D)$ for the family $D\!\subset{}X\mbox{${\rightarrow}$}{}S$ of residual plane quartic we now describe. Notation as in the previous example, let $\mbox{${\cal S}$}_2\mbox{\raise-.03cm\hbox{\mbox{\tiny\bf$^\succ$} \O^{\oplus5}\surj\mbox{${\cal Q}$}_2$ (rank $\mbox{${\cal Q}$}_2= 3$) denote the tautological sequence over the Grassmann variety $G_2$ of planes in \p {4}. Let $G_{2,\ell}$ be the Schubert subvariety of all 2-planes through a fixed line $\ell$. Let $X=\p{}(\mbox{${\cal Q}$}_2)_{|G_{2,\ell}}{\subset} G_{2,\ell}\times\p 4$ be the restriction over $G_{2,\ell}$ of the universal plane in \p 4. Restricting the sequence over $G_{2,\ell} $ yields an exact sequence, (cf. top sequence in (\ref{taut})) $\mbox{${\cal S}$}_2\mbox{\raise-.03cm\hbox{\mbox{\tiny\bf$^\succ$} \O^{\oplus3}=\mbox{${\cal S}$}_{1|\ell}\surj {\cal M}$, where ${\cal M}$ is a line subbundle of $\mbox{${\cal Q}$}_2$ with Chern class $x:=$ $c_1{\cal M}=-c_1\mbox{${\cal S}$}=c_1\mbox{${\cal Q}$}_2$. Over $X$, we have the natural commutative diagram of maps of locally free sheaves, $$\ba{ccc} {\cal M}&&\\ \injdown&\sear{}&\\ \mbox{${\cal Q}$}_2&\surj{}&\O\!_{{\cal Q}_2}(1) \end{array}$$ where the bottom line is the tautological 1--quotient on the projective bundle\break {\bf Proj}$(Sym(\mbox{${\cal Q}$}_2))$. One checks that $\ell':=G_{2,\ell}\times\ell$ is the divisor in $\p{}(\mbox{${\cal Q}$}_2)_\ell$ of zeros of the slant arrow ${\cal M}\mbox{${\rightarrow}$}\O(1)$. Therefore, setting $y=c_1\O(1)$ we have $\O(\ell')=\O(y-x)$. Now let $ D_{\mbox{\footnotesize\bf T}} \subset\p{}({\mbox{${\cal Q}$}_2})$ be the divisor defined by intersection with {\bf T}, so that $\O(D_{\mbox{\footnotesize\bf T}} =\O(5\cdot y)$. Restriction over $G_{2,\ell}$ splits $D_{\mbox{\footnotesize\bf T}} =D+\ell'$. By construction, $D$ is the total space of the family of plane quartic curves residual to $\ell$. Finally, we have $\L:=\O(D)=\O(5\cdot y-(y-x))=\O(4y+x)$. Using {\sc schubert}\cite{schub} we may compute $ \int_{G_{2,\ell}}(c_6\eml{(2,2)}{\L}/2= 1,185$ (see the appendix) and find the number $$\bf17\!,\!601\!,\!000\rm=tg_{6,5} - 609250 - 1185\times2875$$ of $irreducible$ plane rational quintic curves contained in a generic 5\up{ic} threefold. The 1\up{st} correction is due to $conic+cubic$ and the 2\up{nd} to $line+binodal$ $quartic$. \section{Final comments} An additional difficulty appears for the case of $7-$fold tangent hyperplanes. Indeed, for a general $7-$dimensional linear system, we'd expect $\Sigma(2^{[7]};S)$ to receive contributions from $\Sigma(3(2),2;S),\ \Sigma(3,2^{[3]};S),\ \Sigma(3(2)';S)$, so that a na\"{\i}ve count would predict $$\mbox{\footnotesize$tg_7:=(\#\Sigma(2^{[7]};S)- 210\#\Sigma(3(2),2;S)-1260\#\Sigma(3,2^{[3]};S)/6)- 30\#\Sigma(3(2)';S))/7!,$}$$ \noindent\normalsize where $\Sigma(3(2)';S)$ denotes a cycle supported on the set of $(C,y_1,\dots,y_7)$ such that $C\!\in\! S$ has a triple point $y_1$ with the infinitely near double point $y_2$ presenting a branch tangent to the exceptional line over $y_1$. However, barring some computational error, in fact the rhs did not yield an integer for any of the examples we've experimented with. This seems to indicate that $\Sigma(2^{[7]};S)$ may not be reduced at some of the points involving singularities worse than nodes. In fact, the argument of Prop.\ref{reduced} does not apply. This would imply that the coefficients $210$, $1260$ and $30$, postulated by the na\"{\i}ve count of permutations, must be modified. For $n\geq8$, we face the intrusion of a component of wrong dimension in $\Sigma(2^{[n]};S)$ due to 4--fold points. In this case, the technique of residual intersections might shed some light. \section{Appendix: computations} \baselineskip6p \footnotesize \begin{verbatim} ###CUT HERE FOR MAPLE with(schubert):with(SF): #PRINCIPAL PARTS of order n, # f =cotg,d=linebundle princ:= proc(n,f,d)local i:d&*sum('symm(i,f)',i=0..n):end: whichmon:=proc(f,vars)local i,v,z: z:=expand(f): if type(z,`*`)or type(z,`^`)or type(z,`name`) then v:=[seq(vars[i]=1,i=1..nops(vars))]: RETURN(f/subs(v,f)): else ERROR(`invalid arg`) fi:end: #SUBS EXACT MONOMIAL RELATIONS submonpol:=proc(f,vars,rels)local z,i,j,term,mono,temp: z:=expand(f):temp:=0: if type(z,`+`)then for i to nops(z)do term:=op(i,z): mono:=whichmon(",vars): for j to nops(rels) while mono<>lhs(rels[j])do od: if j<=nops(rels)then temp:=temp+term/mono*rhs(rels[j]) else temp:=temp+term: fi: od: elif type(z,`*`)or type(z,`^`)or type(z,`name`)then term:=z: mono:=whichmon(term,vars): for j to nops(rels)while mono<>lhs(rels[j])do od: if j<=nops(rels)then temp:=temp+term/mono*rhs(rels[j]) else temp:=temp+term fi: fi: RETURN(temp)end: #KILL TERMS IN VARS OF TOTDEG>DIM dimsimpl:= proc(x,vars,degs,dim)local i,j,temp,par,n: temp:=expand(x): if type(temp,`+`)then par:=0:n:=nops(temp): for i to n do op(i,temp): degree(collect(subs([seq(vars[j]=t_^degs[j]*vars[j], j=1..nops(vars))],"),t_),t_): if "<=dim then par:=par+"":fi: od: temp:=par: else degree(collect(subs([seq(vars[j]=t_^degs[j]*vars[j], j=1..nops(vars))],"),t_),t_): if ">dim then temp:=0 fi: fi: RETURN(temp):end: simplification:=proc () local i, j, n, z, zz: n:= args[1]: z:=args[2]: if nargs=3 and type(args[3],set) then zz:=args[3] else zz:={n} fi: for i from n by -1 to 2 do for j to n+1-i do if 2 < degree(collect(z,e[j,i]),e[j,i]) then z:=rem(collect(z,e[j,i]),relexc.i.j,e[j,i]): zz:=zz union{i+j-1}: fi: od:od: if opt_=5 then for i in zz do if 2 < degree(collect(z,y.i),y.i) then z:=rem(z,rely.i,y.i): fi: od: else for i in zz do z:=dimsimpl(z,var0.(i),deg1,2):od: fi : RETURN(z)end: #MAIN PROCEDURE FOR PUSHFORWARD {n}->{n-1} push:= proc(n,f) local z,z0,z2,zz,mons,i,j,i1,j1,temp,varn,var0,degn,dd: option remember: if opt_=5 and type(relpush_5,set)=false then relpush_5:={}:fi: if opt_<>5 and type(relpush_,set)=false then relpush_:={}:fi: if n=1 then if opt_=5 then rem(f,rely1,y1):z:=coeff(collect(",y1),y1,2): else subs([seq(var1[i]=0,i=1..nops(var1))],f): f-":dimsimpl(",var0.n,deg1,2): submonpol(",var1,{c[1,2]=chi,c[1,1]^2=k2,h[1]^2=d, h[1]*c[1,1]=hk}): z:=submonpol(",var1,{c[1,1]=0,h[1]=0}) fi: RETURN(z): else convert(var.(n-1),set) minus convert(var.(n-2),set): var0:=[op(")]: degn:=[seq(1,i=1..nops(var0))]: if opt_<>5 then var0:=[op(var0),c[n-1,2]]:degn:=[op(degn),2] fi: varn:=[seq(p.n.2 &^* var0[i],i=1..nops(var0))]: subs([e[1,n]=0,seq(varn[i]=0,i=1..nops(varn))],f): z:=collect(f-",e[1,n]): if 2<degree(z,e[1,n]) then z:=rem(z,relexc.n.1,e[1,n]): fi: z:=collect(z,e[1,n]): z:=collect(z-e[1,n]*coeff(z,e[1,n],1),e[1,n]): if z<>0 then z0:=coeff(z,e[1,n],0): if z0<>0 then simplification(n,z0):z0:=collect(",e[2,n-1]): z0:=z0-e[2,n-1]*coeff(z0,e[2,n-1],1) fi: zz:=collect(subs([seq(varn[i]=t_*varn[i], i=1..nops(varn))],z0),t_): dd:=degree(zz,t_): temp:=0: for i from dd by -1 to 1 do z0:=expand(coeff(zz,t_,i)): if z0 <> 0 then if (opt_=5 and type(relpush_5.n.i,list)=false or (opt_<>5 and type(relpush.n.i,list)=false)) then if opt_=5 then print(`BUILD RELPUSH_5`.n.i): elif opt_<>5 then print(`BUILD RELPUSH`.n.i): fi: mons :=monomials(i,var0,degn): z2:={}: for j to nops(mons) do dimsimpl(mons[j],var0.(n-1),deg1,2): if degree(collect(",e[1,n-1]),e[1,n-1])<>1 and member(true,{seq(type("/e[j1,n-j1]^3, polynom),j1= 1..n-2)})=false then z2:=z2 union {mons[j]} fi: od: mons:=[seq(p.n.2&^* z2[j]=push(n-1,z2[j]),j= 1..nops(z2))]: if opt_=5 then relpush_5.n.i:=mons: else relpush.n.i:=mons: fi: elif opt_=5 and member([n,i],relpush_5)=false then print(`USING RELPUSH_5`.n.i.` BUILT BEFORE`): relpush_5:=relpush_5 union{[n,i]}: elif opt_<>5 and member([n,i],relpush_)=false then print(`USING RELPUSH`.n.i.` BUILT BEFORE`): relpush_:=relpush_ union{[n,i]} fi: if opt_=5 then mons:=relpush_5.n.i: else mons:=relpush.n.i: fi: z0:=submonpol(z0,varn,mons) fi: temp:=temp+z0: od: z0:=temp: z2:=-coeff(z,e[1,n],2): for i to nops(varn) while z2 <> 0 do z2:=collect(z2,varn[i]): if degree(z2,varn[i]) <> 0 then z2:=rem(z2,varn[i]-var0[i],varn[i]): fi: od: if z2<>0 then z2:=simplification(n-1,z2) fi: z:=z0+z2: fi: RETURN(z) fi: end: #of push # CALCULATIONS for opt_ in[5,0]do if opt_=5 then grass(3,5,x,all): Grass(g,1,Qx,y,all): omega1:=dual(g[tangentbundle_]): rely1:=chern(3,"):var1:=[y1]:deg1:=[1]: variety(S1,dim=8,vars=var1,degs=deg1): else var1:=[c[1,1],h[1],c[1,2]]:deg1:=[1,1,2]: variety(S1,dim=2,vars=var1,degs=deg1): fi: var01:=var1: for n to 6 do if n=1 then if opt_=5 then DIM:=3: #ONLY FOR THE SAKE OF RANKS... L:=o(4*y1+x1): #FOR BINODAL 4ICS princ(1,omega1,L):chern(3,"):FB1:=rem(",rely1,y1): print(`done FB1`): DIM:=6:M:=o(m*y1):DIM:=3: else opt_:=0:goto(S1):bundle(2,c): omega1:=subs([c1=c[1,1],c2=c[1,2]],"): #COTANGENT BUNDLE M:=o(h[1]): fi: princ(1,omega1,M): if opt_=5 then chern(3,"):else chern("): fi: F1:=simplification(n,"):print(`done F1`): if opt_=5 then DIM:=6: #ONLY FOR THE SAKE OF RANKS... fi: princ(2,bundle(2,c),M): subs([seq(c.i=chern(i,omega1),i=1..2)],"): if opt_=5 then chern(6,"):else chern("): fi: E_31:=simplification(1,"):print(`done E_31`): elif n>=2 then var0.n:=[y.n]: var.n:=[seq(y.j,j=1..n), seq(seq(e[j,k],j=1..n-k+1),k=2..n-1),e[1,n]]: deg.n:=[seq(1,j=1..n), seq(seq(1,j=1..n-k+1),k=2..n-1),1]: if opt_<>5 then var0.n:=[c[n,1],h[n],c[n,2]]: var.n:=[seq(c[j,1],j=1..n), seq(c[j,2],j=1..n),op(subs([seq(y.j=h[j],j=1..n)],var.n))]: deg.n:=[seq(1,j=1..n),seq(2,j=1..n),op(deg.n)] fi: if opt_=5 then rely.n:=subs(y1=y.n,rely1): fi: variety(S.n,dim=6/5*opt_+2*n,vars=var.n,degs=deg.n): morphism(p.(n).2,S.n,S.(n-1),subs([seq(var0.(n-1)[k]=var0.(n)[k], k=1..nops(var.01)),seq(e[n-k,k]=e[n-k+1,k],k=2..n-1)], var.(n-1))):print(`built S`.n): DIM:=3: #OK since ranks<=3 omega.n:=((p.n.2)&^*(omega.(n-1)))&*o(e[1,n])+o(-e[1,n])-1: chern(3,omega.n): #Will set=0 since rk.omega=2 print(`DONE OMEGA`.n): relexc.n.1:=rem(",rely.n,y.n): for i from n-1 by -1 to 2 do relexc.i.(n+1-i):=(p.n.2)&^*(relexc.i.(n-i)): if degree(collect(relexc.n.1,e[n+1-i,i]),e[n+1-i,i])>2 then relexc.n.1:=rem(relexc.n.1,",e[n+1-i,i]) fi: od: M:=collect((p.n.2)&^*M&*o(-2*e[1,n]),t): #Adjust M princ(1,omega.n,M): if opt_=5 then chern(3,"):else chern("): fi: F.n:=simplification(n,"):print(`done F`.n): if n=2 then #E_32 collect(M&*o(-e[1,2]),t):princ(1,omega2,"): if opt_=5 then E_32:=chern(3,"): L:=collect((p.n.2)&^*L&*o(-2*e[1,n]),t): #Adjust L princ(1,omega.n,L):chern(3,"): FB.n:=simplification(n,"):print(`done FB`.n): else E_32:=chern("): fi: E_32:=simplification(n,E_32):print(`done E_32`): E_3_2:=rem(collect("*e[1,2],e[1,2]),relexc21,e[1,2]): print(`done E_3_2`): fi:#n=2 if n=3 then #E_33 collect(M&*o(-e[2,2]),t):princ(1,omega3,"): if opt_=5 then chern(3,"):else chern("): fi: E_33:=simplification(n,"):print(`done E_33`): fi: #E_33 fi: od: if opt_=5 then 1: for i from 6 by -1 to 1 do F.i:=push(i,"*F.i):print(i) od: E_33:=push(3,E_33):E_32:=push(2,E_32*E_33): E_3_2:=push(2,E_3_2):e_322:=push(1,E_31*E_32): e_3_2:=push(1,E_31*E_3_2): tg.6:=integral(Gx,F1-30*e_3_2-90 *e_322)/6!: lprint(`#4-coplanar lines in 4ic 3fld: `,subs(m=4,tg.6)): lprint(`#6-nodal plane sections of 5ic 3fold: `,subs(m=5,tg.6)): lprint(`#binodal plane 4ic residul to line in 5ic 3fld: `, 1/2*integral(Gx,(x1^2-x2)^2*push(1,FB1*push(2,FB2)))): lprint(`#6-nodal IRREDUCIBLE plane sections of 5ic 3fold: `, subs(m=5,tg.6) - 609250 - 1185*2875): #4-COPLANAR LINES VIA FANO DIM:=3: for i to 3 do relm.i.1:=chern(3,5-bundle(2,z.i)-o(m.i.1)): for j to 2 do z.i.j:=chern(j,bundle(3,x)-o(m.i.1)): od od: chern(3,symm(2,bundle(2,z3))&*o(2*x1-z11-z21)): rem(",relm31,m31): I_3:=coeff(",m31,2): DIM:=4: chern(4,symm(3,bundle(2,z2))&*o(x1-z11))*I_3: rem(",relm21,m21): I_2:=coeff(",m21,2): DIM:=5: I_1:=chern(5,symm(4,bundle(2,z1)))*I_2: I_1:=rem(I_1,relm11,m11): I_1:=coeff(I_1,m11,2): integral(Gx,")/4!: else for j to 6 do for i from 0 to 3 do F.j.i:=coeff(collect(F.j,t),t,i): if j<=3 then E_3.j.i:=coeff(collect(E_3.j,t),t,i):fi: if j=2 then E_3_2.(i):=coeff(collect(E_3.2,t),t,i)*e[1,2]:fi: od:od: for i from 1 to 3 do F6.i:=push(6,F6.i):od: ftg6:=ftg5*F62: for j5 from 1 to 3 do print(`j5=`.j5): a5:=push(5,`F`.5.j5*F63): #dim 10-j5-1=9-j5<=8 ok for j4 from 3-j5 to 3 do a4:=push(4,`F`.4.j4*a5): # 6>=dim 9-j5-j4 >=0 for j3 from max(0,5-j5-j4) to 3 do a3:=push(3,`F`.3.j3*a4): #4>=dim 9-j5-j4-j3 >=0 for j2 from max(0,7-j5-j4-j3) to min(3,9-j5-j4-j3) do a2:=push(2,`F`.2.j2*a3): #2>=dim 9-j5-j4-j3-j2 j1 :=9-j5-j4-j3-j2 : lprint(`j5=`.j5,` j4=`.j4,` j3=`.j3,` j2=`.j2,` j1=`.j1): ftg6:=ftg6+push(1,`F`.1.j1*a2): od:od:od:od: ########### for i from 1 to 3 do F5.i:=push(5,F5.i):od: ftg5:=ftg4*F52: for j4 from 1 to 3 do a4:=push(4,`F`.4.j4*F53): #dim 8-j4-1=7-j4<=6 ok for j3 from 3-j4 to 3 do a3:=push(3,`F`.3.j3*a4): #4>=dim 7-j4-j3 >=0 for j2 from max(0,5-j4-j3 ) to min(3,7-j4-j3) do a2:=push(2,`F`.2.j2*a3): #2>=dim 7-j4-j3-j2>=0 j1 :=7-j4-j3-j2 : lprint(`j4=`.j4,` j3=`.j3,` j2=`.j2,` j1=`.j1): ftg5:=ftg5+push(1,`F`.1.j1*a2): od:od:od: ########### for i from 1 to 3 do F4.i:=push(4,F4.i):od: ftg4:=ftg3*F42: for j3 from 1 to 3 do a3:=push(3,`F`.3.j3*F43): #4>=dim 5-j3 >=0 for j2 from max(0,3-j3) to min(3,5-j3) do a2:=push(2,`F`.2.j2*a3): #2>=dim 5-j3-j2>=0 j1 :=5-j3-j2 : lprint(`j3=`.j3,` j2=`.j2,` j1=`.j1): ftg4:=ftg4+push(1,`F`.1.j1*a2): od:od: ########## for i from 1 to 3 do F3.i:=push(3,F3.i):E_33.i:=push(3,E_33.i): od: ftg3:=ftg2*F32: for j2 from 1 to 3 do a2:=push(2,`F`.2.j2*F33): #2>=dim 3-j2 >=0 j1 :=3-j2 : lprint(` j2=`.j2,` j1=`.j1): ftg3:=ftg3+push(1,`F`.1.j1*a2): od: e_322:=e_32*E_332: for j2 from 1 to 3 do a2:=push(2,`E_3`.2.j2*E_333): #2>=dim 3-j2 >=0 j1 :=3-j2: lprint(`j2=`.j2,` j1=`.j1): e_322:=e_322+push(1,`E_3`.1.j1*a2): od: ########## for i from 1 to 3 do F2.i:=push(2,F2.i): E_32.i:=push(2,E_32.i): E_3_2.(i):=push(2,E_3_2.(i)): od: ftg2:=ftg1*F22+push(1,`F`.1.1*F23): e_32:=e_3*E_322+push(1,`E_3`.1.1*E_323): e_3_2:=e_3*E_3_21+push(1,`E_3`.1.1*E_3_22)+ push(1,`E_3`.1.0*E_3_23): ftg1:=push(1,F12):e_3:=push(1,E_312): for n to 3 do tg.n:=ftg.n /n! :od: tg.4:=(ftg4-6*e_3)/4!: tg.5:=(ftg5-30*e_32)/5!: tg.6:=(ftg6-30*e_3_2-90*e_322)/6!: p2:=proc(m,tg)subs([chi=3,d=m^2,hk=-3*m,k2=9], tg):end: p1xp1:=proc(m1,m2,tg)subs([chi=4,k2=8,hk=-2*m2-2*m1, d=m1*m2*2],tg):end: K_3:=proc(g)subs([chi=24,k2=0,hk=0,d=2*g-2],`tg`.g):end: for i from 3 to 6 do lprint(`# `.i.`-nodal hyperplane sections of K3-sfce in P`. i.`: `,K_3(i)): od: for i from 4 to 6 do lprint(`# `.i.`-nodal plane quartics through `. (14-i).` general points: `,p2(4,tg.i)): od: fi: od: #CUT HERE FOR MAPLE \end{verbatim} \vfill\eject

9606

alg-geom/9606001

en

https://arxiv.org/abs/alg-geom/9606001

alg-geom/9606001

Peter Schenzel

Peter Schenzel

Descent from the form ring and Buchsbaum rings

To appear in Comm. Algebra Latex2e

There is a spectral sequence technique in order to estimate the local cohomology of a ring by the local cohomology of a certain form ring. As applications there are information on the descent of homological properties (Cohen-Macaulay, Buchsbaum etc.) from the form ring to the ring itself. In the case of Buchsbaum ring there is a discussion of the descent of the surjectivity of a natural map into the local cohomology.

alg-geom

\section{Introduction and Main Results} One of the major problems in commutative algebra is to recover information about a commutative ring $A$ from known properties of the form ring $G := G_A(\mathfrak q) = \oplus_{n\geq 0} {\mathfrak q}^n/{\mathfrak q}^{n+1}$ with respect to some ideal $\mathfrak q$ of $A$. There are Krull's classical results saying that $A$ is an integral domain resp. a normal domain if $G$ is an integral domain resp. a normal domain. It follows from the work \cite{AA}, \cite{CN}, \cite{HR} that several other properties of a homological nature, like regularity, Cohen-Macaulayness, Gorensteinness etc., descend from $G$ to $A$. In this note we want to pursue this point of view further. To this end let $Q$ denote the homogeneous ideal of $G$ generated by all the inital forms of element of $\mathfrak q$. For our purposes here we investigate the local cohomology modules $H^{\bullet}_Q(G)$ and $H^{\bullet}_{\mathfrak q}(A)$ of $G$ with respect to $Q$ and of $A$ with respect to $\mathfrak q$ resp. For their definition and basic properties see \cite{aG67}. The first result concerns the descent of the finiteness from $H^i_Q(G)$ to $H^i_{\mathfrak q}(A)$. \begin{theorem} \label{1.1} Let $\mathfrak q$ be an ideal of a commutative Noetherian ring $A$. Assume $H^i_Q(G)$ is a finitely generated graded $G$-module for all $i<t$. Then $H^i_{\mathfrak q}(A), \enspace i<t$, is a finitely generated $A$-module. \end{theorem} Furthermore, let $H^i(Q;G)$ and $H^i({\mathfrak q};A)$ denote the Koszul cohomology of $G$ with respect to $Q$ and of $A$ with respect to $\mathfrak q$ resp. Note that changing the basis yields isomorphic cohomology modules. Hence, it is not necessary to fix a basis in our notation. It is well-known, see e.g. \cite{aG67}, that there are canonical homomorphisms $$ \begin{array}{lclcl} f^i_G & : & H^i(Q;G) & \to & H^i_Q(G) \quad \text{and} \\\vspace*{.5pt} f^i_A & : & H^i({\mathfrak q};A) & \to & H^i_{\mathfrak q}(A). \end{array} $$ Our next result concerns the descent of the surjectivity from $f^i_G$ to the surjectivity of $f^i_A$. Note that the surjectivity of $f^i_A$ is the crucial point in the investigation of local Buchsbaum rings, see \cite{jS80}. For a graded $G$-module $M$ let $[M]_k, \enspace k \in \mathbb Z$, denote its $k$-th graded piece. \begin{theorem} \label{1.2} Assume there are integers $k$ and $t$ such that $$ \begin{array}{lcl} [H^i_Q(G)]_{n-i} & = & 0 \enspace \text{for} \enspace n \not= k-1, k \enspace \text{and} \enspace 0 \leq i < t \enspace \text{ and } \\\vspace*{.2pt} [H^t_Q(G)]_{n-t} & = & 0 \enspace \text{for} \enspace n> k. \end{array} $$ Then $f^i_A$ is surjective for all $i<t$ if and only if $f^i_G$ is surjective for all $i<t$. \end{theorem} If $(A,{\mathfrak m})$ is a local Noetherian ring, then the Buchsbaum property does not descend from $G=G_A(\mathfrak m)$ to $A$. Counterexamples are given by Steurich (University of Essen), cf. \ref{4.3}. So our result yields as a corollary, cf. \ref{4.1}, that under certain additional assumptions on $G$ the basic ring $A$ is a Buchsbaum ring if $G$ is a Buchsbaum ring. The third result has supplementary character. It gives a sufficient condition for $f^i_G, \enspace i<t$, to be a surjective homomorphism. As shown in Example \ref{4.4} it is not necessary. \begin{theorem} \label{1.3} Let $t$ be an integer. Assume for each $0 \leq i<j<t$ and all integers $p$ and $q$ such that \begin{displaymath} [H^i_Q(G)]_{p-i} \not= 0 \enspace\text{and} \enspace [H^j_Q(G)]_{q-j} \not= 0 \end{displaymath} we have $p-q \not=1$. Then the canonical homomorphism \begin{displaymath} f^i_G : H^i(Q;G) \to H^i_Q(G) \end{displaymath} is surjective for $i<t$, provided $QH^i_Q(G)=0$ for $i<t$. \end{theorem} The proof of \ref{1.1} and \ref{1.2} is based on a spectral sequence technique of Serre \cite{jpS}, devised for passing from the form ring to the underlying ring, see 2. Another spectral sequence argument yields the proof of \ref{1.3}. The above results apply to Buchsbaum rings, for this see 4. We refer to \cite[5.6]{cW} for an introduction and the baic results concerning spectral sequences. The author is grateful to R\"udiger Achilles for stimulating discussions during the preliminary draft of this note several years ago. \section{Auxiliary Spectral Sequences} A filtration of a ring $A$ is a decreasing sequence of ideals $({\mathfrak a}_n)_{n \in\mathbb Z}$ of $A$ such that ${\mathfrak a}_m {\mathfrak a}_n \subseteq {\mathfrak a}_{m+n}$ a nd ${\mathfrak a}_0=A$. A filtered $A$-module is an $A$-module $M$ with a decreasing sequence $(M_n)_{n \in\mathbb Z}$ of $A$-submodules of $M$ such that ${\mathfrak a}_m M_n \subseteq M_{m+n}$. Let $\mathfrak a$ be an ideal of $A$. A filtration $(M_n)_{n \in\mathbb Z}$ of $M$ is called essentially $\mathfrak a$-adic if ${\mathfrak a}M_n \subseteq M_{n+1}$ for all $n$ and ${\mathfrak a}M_n = M_{n+1}$ for all suffciently large $n.$, By virtue of \cite[(0.11.1.3)]{aG61} a filtration is called exhaustive (resp. co-discrete) provided $\cap _{n\in \mathbb Z} M_n = M$ (resp. there is an integer $m$ such that $M_m = M$). \begin{proposition} \label{2.1} Let $\mathfrak q$ be an ideal of a commutative Noetherian ring $A$. Then there exists a spectral sequence \begin{displaymath} E^{pq}_1 = [H^{p+q}_Q(G)]_p \Longrightarrow_p E^{p+q} = H^{p+q}_{\mathfrak q}(A) \end{displaymath} whose $E^{pq}_{\infty}$-term is the $p$-th component of the graded module associated to an essentially $\mathfrak q$-adic filtration of $H^{p+q}_{\mathfrak q}(A)$. \end{proposition} \begin{proof} Based on Serre's technique \cite{jpS} Achilles and Avramov, see \cite{AA}, constructed a spectral sequence \begin{displaymath} E^{pq}_1 = [\Ext^{p+q}_G(G(M),G(N))]_p \Longrightarrow_p E^{p+q} = \Ext^{p+q}_A(M,N), \end{displaymath} where $M,N$ are finitely generated $A$-modules with $G(M), \enspace G(N)$ their form modules with respect to $\mathfrak q$. Set $N=A$ and $M=A/{\mathfrak q}^n$. Because $G(A/{\mathfrak q}^n) = G/Q^n, \enspace n \geq 0$, the $E_1$-term becomes \begin{displaymath} E^{pq}_1 = [\Ext^{p+q}_G(G/Q^n,G)]_p. \end{displaymath} By virtue of the canonical homomorphism induced by \begin{displaymath} A/{\mathfrak q}^{n+1} \to A/{\mathfrak q}^n \enspace\text{and}\enspace G/Q^{n+1} \to G/Q^n \end{displaymath} both sides from a direct system. According to \cite[(0.11.1)]{aG61} we may form its direct limit, which yields the desired spectral sequence. Note that the filtration induced by the direct limit of the spectral sequences is in general not co-discrete. But it is always exhaustive. \end{proof} The above proposition is the crucial point of our investigation. It is based on Serre's technique \cite{jpS}, devised for passing from the tangent cone of a variety to the variety itself. For further investigations see \cite{AA}. The next proposition concerns the Koszul cohomology. Its proof follows from \cite{jpS}, see also \cite{AA}. \begin{proposition} \label{2.2} Let ${\mathfrak q}, A,Q$, and $G$ as before. Then, there is a convergent spectral sequence \begin{displaymath} E^{pq}_1 = [H^{p+q}(Q;G)]_p \Longrightarrow_p E^{p+q} = H^{p+q}({\mathfrak q};A) \end{displaymath} whose $E^{pq}_{\infty}$-term is the $p$-th component of the graded module associated to an essentially $\mathfrak q$-adic filtration of $H^{p+q}({\mathfrak q};A)$. \end{proposition} \noindent {\bf Proof of \ref{1.1}.} Because $H^i_Q(G), \enspace i<t$, is a finitely generated graded $G$-module it follows that $[H^i_Q(G)]_n =0$ for all $|n| \gg 0$ and $0 \leq i < t$. That is, the filtration of $H^i_{\mathfrak q} (A), \enspace i<t $, given in \ref{2.1}, is finite. Since all the $E^{pq}_{\infty}$-terms, $p+q<t$, are annihilated by $\mathfrak q$ we see that $H^i_{\mathfrak q}(A), \enspace i < t$, is annihilated by a power of $\mathfrak q$. According to \cite[Lemma 3]{gF}, this completes the proof. \hfill $\Box$ \vspace*{5pt} A litte bit more is true, if we assume $H^i_Q(G), \enspace i<t$, a $G$-module of finite length. \begin{corollary} \label{2.4} Suppose $H^i_Q(G), \enspace i<t$, is a $G$-module of finite length. Then $H^i_{\mathfrak q}(A), \enspace i<t$, is an $A$-module of finite length and $$ L_G(H^i_Q(G)) \geq L_A(H^i_{\mathfrak q}(A)) \text{ for } i<t. $$ \end{corollary} The proof of \ref{2.4} is similarly to the proof of \ref{1.1}. Hence we omit it. The particular case of \ref{2.4} for a local ring $A$ and an $\mathfrak m$-primary ideal has been shown in \cite[(4.2)]{pS84}, by a completely different technique. As a further auxiliary result we shall use the following well-known spectral sequence, see e.g. \cite[Section 2]{pS95}. \begin{proposition} \label{2.5} There is a convergent spectral sequence \begin{displaymath} E^{pq}_2 = H^p(Q;H^q_Q(G)) \Longrightarrow E^{p+q} = H^{p+q}(Q;G) \end{displaymath} for computing the Koszul cohomology. \end{proposition} \section{Proof of (1.2) and (1.3)} \noindent{\bf Proof of \ref{1.2}.} Firstly, we consider the spectral sequence given in \ref{2.1}. We claim that $E^{pq}_1 = E^{pq}_{\infty}$ for $p+q<t$. In order to show this we consider the subsequent stages, i.e. \begin{displaymath} E^{p-r,q+r-1}_r \to E^{pq}_r \to E^{p+r,q-r+1}_r. \end{displaymath} Assume $E^{pq}_r \not= 0$ for some $p+q=i<t$. Then $E^{pq}_1 \not= 0$, i.e. $p=k-i-1$ or $p=k-i$ by virtue of the assumption. Now $E^{p+r,q-r+1}_r$ is a subquotient of $E^{p+r,q-r+1}_1 = [H^{i+1}_Q(G)]_{p+r} =0$ for $p=k-i-1$ or $p=k-i$. Therefore, if $E^{pq}_r \not= 0$ we have $E^{p+r,q-r+1}_r =0$. The same arguments yield $E^{p-r,q+r-1}_r =0$ if $E^{pq}_r \not= 0$. That is, $E^{pq}_1 = E^{pq}_{\infty}$ for all $p+q = i<t$. Thus, $H^i_{\mathfrak q}(A), \enspace i<t$, possesses a filtration of two terms. Hence, there is a short exact sequence \begin{displaymath} 0 \to [H^i_Q(G)]_{k-i} \to H^i_{\mathfrak q}(A) \to [H^i_Q(G)]_{k-i-1} \to 0. \end{displaymath} Secondly, we use the spectral sequence given in \ref{2.5} in order to show that \begin{eqnarray*} [H^i(Q;G)]_{n-i} & = & 0 \enspace\text{for} \enspace n \not= k-1, k \enspace\text{and} \enspace 0 \leq i<t \enspace\text{and} \\\vspace*{.2pt} [H^t(Q;G)]_{n-t} & = & 0 \enspace\text{for} \enspace n>k. \end{eqnarray*} To this end we note that $E^{pq}_2 = H^p(Q;H^q_Q(G))$ is by definition a subquotient of $H^q_Q(G)(p)^{\binom{m}{p}}$, where $m$ denotes the number of generators of $Q$. Therefore, $E^{pq}_2$ vanishes for $p+q=i<t$ (resp. $p+q=t$) in all graded pieces $\not= k-i-1$, $k-i$ (resp. $>k-t$). Hence, the spectral sequence proves the claim. Thirdly, we apply this result in order to investigate the spectral sequence given in \ref{2.2}. Similarly as in the first part of the proof it yields a short exact sequence \begin{displaymath} 0 \to [H^i(Q;G)]_{k-i} \to H^i({\mathfrak q};A) \to [H^i(Q;G)]_{k-i-1} \to 0 \end{displaymath} for $i<t$. Therefore, the canonical homomorphisms $f^i_G$ and $f^i_A$ induce a commutative diagram with exact rows $$ \begin{array}{ccccccccc} 0 & \to & [H^i(Q;G)]_{k-i} & \to & H^i({\mathfrak q};A) & \to & [H^i(Q;G)]_{k-i-1} & \to & 0 \\ & & \downarrow [f^i_G]_{k-i} & &\downarrow f^i_A & & \downarrow [f^i_G]_{k-i-1} & & \\ 0 & \to & [H^i_Q(G)]_{k-i} & \to & H^i_{\mathfrak q}(A) & \to & [H^i_G(G)]_{k-i-1} & \to & 0 \end{array} $$ for $i<t$, where $[f^i_G]_n$ denotes the $n$-th graded piece of $f^i_G.$ According to \ref{3.2} $[f^i_G]_{k-i}$ is always surjective. Hence, the snake lemma yields that $f^i_A$, $i<t$, is surjective if and only if $f^i_G$, $i<t$, is ssurjective. \hfill $\Box$ \begin{proposition} \label{3.2} Suppose there exists an integer $k$ such that $$ [H^i_Q(G)]_{k+1-i} =0 \text{ and all } i \in \mathbb Z. $$ Then the canonical homomorphism \begin{displaymath} [f^i_G]_{k-i} : [H^i(Q;G)]_{k-i} \to [H^i_Q(G)]_{k-i} \end{displaymath} is surjective for all $i$. \end{proposition} \begin{proof} For this we have to investigate the spectral sequence given in \ref{2.5}. We claim $[E^{0i}_2]_{k-i} = [E^{0i}_{\infty}]_{k-i}$ for all $i$. To this end we are looking at the subsequent stages. Incoming $d$'s come from subquotients of $E^{-r,i+r-1}_2 = 0$. Outgoing $d$'s land in subquotients of $$ [E^{r,i-r+1}_2]_{k-i} = [H^r(Q;H^{i-r+1}_Q (G))]_{k-i} $$ which is a subquotient of $[H^{i-r+1}_Q (G)(r)^{\binom{m}{r}}]_{k-i} = 0$, by virtue of the assumption. Here $m$ denotes the number of generators of $Q$. Hence the claim is proved. Next we note that \begin{displaymath} [E^{0i}_2]_{k-i} = [H^0(Q;H^i_Q(G))]_{k-i} \simeq [H^i_Q(G)]_{k-i} \end{displaymath} because $[H^i_Q(G)]_{n-i} = 0$ for $n>k$ by the assumption. But now the module $[H^i(Q;G)]_{k-i}$ possesses a filtration whose associated $0$-th graded piece is $[E^{0i}_{\infty}]_{k-i},$ i.e. there is a canonical surjective mapping \begin{displaymath} [H^i(Q;G)]_{k-i} \to [H^i_Q(G)]_{k-i} \end{displaymath} for all $i$. By virtue of the functoriality of the spectral sequence it is noting else but $[f^i_G]_{k-i}$. \end{proof} \noindent {\bf Proof of \ref{1.3}.} We use the spectral sequence \begin{displaymath} E^{pq}_2 = H^p(Q;H^q_Q(G)) \Longrightarrow E^{p+q}(Q;G) \end{displaymath} given in \ref{2.5}. Because of $QH^i_Q(G) = 0, i<t$, it follows that $$ E^{pq}_2 = H^q_Q(G)(p)^{\binom{m}{p}} \text{ for } p+q<t. $$ Here $m$ denotes the number of generators of $Q$. In the subsequent stages we have \begin{displaymath} E^{p-r,q+r-1}_r \to E^{pq}_r \to E^{p+r,q-r+1}_r. \end{displaymath} Suppose that \begin{displaymath} [E^{pq}_r]_n \not= 0 \text{ for some } p+q<t, \end{displaymath} then $[E^{pq}_2]_n \not= 0$. Because $E^{p+r,q-r+1}_r$ resp. $E^{p-r,q+r-1}_r$ is derived from $H^{p+r}(Q;H^{q-r+1}_Q(G))$ resp. $H^{p-r}(Q;H^{q+r-1}_Q (G))$ it follows that \begin{displaymath} [E^{p+r,q-r+1}_r]_n = 0 \text{ resp.} [E^{p-r,q+r-1}_r]_n =0 \end{displaymath} by virtue of the assumption. That is, $E^{pq}_2 = E^{pq}_{\infty}$ for $p+q<t$. Because $H^i(Q;G)$ possesses a filtration whose associated $i$-th graded piece is $E^{0i}_{\infty}$ there is a canonical surjective homomorphism \begin{displaymath} H^i(Q;G) \to H^i_Q(G), \enspace i<t. \end{displaymath} By virtue of the functoriality of the considered spectral sequence it is nothing else but $f^i_G$, i.e. $f^i_G$, $i<t$, is surjective, as required. \hfill $\Box$ \section{Applications to Buchsbaum Rings} In this section $(A,\mathfrak m)$ denotes a local Noetherian ring of dimension $d$ with $\mathfrak m$ its maximal ideal. Then $A$ is called a Buchsbaum ring if the difference $C(A) := L(A/\mathfrak q) - e({\mathfrak q};A)$ is an invariant of $A$ not depending on the choice of a parameter ideal $\mathfrak q$ of $A.$ Here $L(A/\mathfrak q)$ and $e({\mathfrak q};A)$ denote resp. the length of $A/\mathfrak q$ and the multiplicity of $A$ with respect to $\mathfrak q,$ see \cite{SV} for further details. In his crucial paper \cite{jS80} St\"uckrad showed that $A$ is a Buchsbaum ring if and only if the canonical homomorphism $f^i_A : H^i({\mathfrak m};A) \to H^i_{\mathfrak m} (A)$, $i<d$, is surjective. Let $G=G_A(\mathfrak m)$ denote the associated form ring and $M=G_+$ the irrelevant maximal ideal of $G$. The graded ring $G$ is called a Buchsbaum ring if $G_M$ is a local Buchsbaum ring. Hence, our main results \ref{1.2} and \ref{1.3} apply to the situation of Buchsbaum rings. \begin{corollary} \label{4.1} Let $G=G_A(\mathfrak m)$ denote the associated graded form ring. Assume there is an integer $k$ such that \begin{eqnarray*} [H^i_M(G)]_{n-i} = 0 & \text{for} & n \not= k-1, k \enspace\text{and}\enspace 0 \leq i < d \enspace\text{ and } \\\vspace*{.1pt} [H^d_M(G)]_{n-d} = 0 & \text{ for } & n>k. \end{eqnarray*} Then $A$ is a Buchsbaum ring if and only if $G$ is a Buchsbaum ring. In this case $L(H^i_M(G)) = L(H^i_{\mathfrak m}(A))$ for all $0 \leq i<d$. \end{corollary} \begin{proof} Readily it follows from \ref{1.2} using the characterization of Buchsbaum rings in terms of the surjectivity of $f^i_A$ and $f^i_G$. The statement on the length of the local cohomology modules is clear by virtue of the short exact sequence given in the proof of \ref{1.2}. \end{proof} The `only if' part of \ref{4.1} is one of the main results of Goto's paper \cite[Theorem (1.1)]{sG82}. His proof is completely different. It does not use Serre's spectral sequence technique for passing from the tangent cone to the ring. \begin{corollary} \label{4.2} Let $G$ be as in \ref{4.1}. Assume for each $0 \leq i<j<d$ and all integers $p$ and $q$ with \begin{displaymath} [H^i_M(G)]_{p-i} \not= 0 \enspace\text{and} \enspace [H^j_M(G)]_{q-j} \not= 0 \end{displaymath} we have that $p-q \not= 1$. Then $G$ is a Buchsbaum ring, provided $MH^i_M(G) =0$ for $i<d$. \end{corollary} The proof follows by virtue of \ref{1.3} accordingly to the fact that $G$ is a Buchsbaum ring if $f^i_G$, $i<d$, is surjective. Note that \ref{4.2} was shown independently by St\"uckrad \cite[Prop. 3.10]{SV} not using a spectral sequence technique. Particular cases of it were obtained in \cite{pS82}, resp. by Goto and others. We conclude this section with two examples concerning the assumptions in \ref{1.2} and \ref{1.3}. The following example was given by Steurich, see also \cite[(4.10)]{sG82}. \begin{example} \label{4.3} The condition in \ref{1.2} is the best possible. Set \begin{displaymath} A=k[|x,y,z|]/(x^2,xy,xz-y^r,y^{r+1},xz^2), \, r \geq 3 \text{ an integer, } \end{displaymath} where $k[|x,y,z|]$ denotes the formal power series ring over a field $k$. Note that $\dim A=1$. Then \begin{displaymath} G := G_A(\mathfrak m) = k[x,y,z]/(x^2,xy,,xz,y^{r+1},y^r z) \end{displaymath} is a Buchsbaum ring, i.e. $f^0_G$ is surjective. Furthermore, \begin{displaymath} \begin{array}{ccl} H^0_M(G) & = & G/M(-1) \oplus G/M(-r), \\\vspace*{.2pt} [H^1_M(G)]_n & = & 0 \text{ for } n>r-2 \text{ and } [H^1_M(G)]_{r-2} \not= 0. \end{array} \end{displaymath} On the other hand, $A$ is not a Buchsbaum ring, i.e. $f^0_A$ is not surjective. \end{example} While the finite length of $H^i_M(G), i \not= \dim G,$ is inherited to $H^i_{\mathfrak m}(A), i \not= \dim A,$ for the form ring $G = G_{\mathfrak m}(A)$ of a local ring $(A, \mathfrak m),$ see \ref{1.1}, this is not true in the Buchsbaum case. It does not hold even in the quasi-Buchsbaum case, where quasi-Buchsbaum means that for $i \not= \dim A$ the local cohomology is a vector space over $A/\mathfrak m.$ This follows because the local ring $(A, \mathfrak m)$ in \ref{4.3} is not quasi-Buchsbaum. So one might continue in order to improve the result in \ref{1.1} by taking into acount the more subtle behaviour of $k$-Buchsbaum rings. The next example shows that the assumption in \ref{1.3} is not necessary for $f_G$, $i<t$, surjective. \begin{example} \label{4.4} Let $R=k[x_1,\ldots,x_6]$ denote the polynomial ring over an infinite field $k$. Using a technique of Griffith and Evans in \cite{sG81} Goto constructed examples of homogeneous prime ideals $P \subset R$ such that $R_1 := R/P$ is a 4-dimensional graded domain with $$ \begin{array}{ccl} H^i_M(R_1) & = & 0, \enspace i \not=1,4, \quad H^1_M(R_1) \simeq k(-2), \enspace\text{ and } \\\vspace*{.2pt} [H^4_M(R_1)]_n & = & 0 \enspace\text{ for all } n \geq 0. \end{array} $$ Let $R_2 =k[y_1,y_2,y_3]/F$, $F$ a homogeneous form of degree 3, be the coordinate ring of a plane cubic. Note that $R_2$ is a two-dimensional Cohen-Macaulay ring with $[H^2_M(R_2)]_n = 0$ for all $n \geq 1$ and $[H^2_M(R_2)]_0 = k$. Let $S$ denote the Segre product of $R_1$ and $R_2$ over $k$. Using the K\"unneth formula, as done in \cite[Section 5]{pS82}, we obtain \begin{eqnarray*} H^i_M(S) & = & 0, \enspace i \not= 1,2,5, \quad H^1_M(S) \simeq k^{10}(-2), \enspace\text{ and } \\ H^2_M(S) & \simeq & k(0). \end{eqnarray*} Therefore, the assumptions of \ref{1.3} are not fulfilled because in this case $(1+2) - (2+0) = 1$. We show that $f^i_S$, $i<5$, is surjective. For $i \not= 0,3,4$ this is trivially true. Using \ref{2.5} it follows readily for $i=1$. Therefore, it is enough to show the surjectivity of $[f^2_S]_0$. The K\"unneth relations induce a commutative diagram \begin{eqnarray*} [R_1]_0 \otimes [H^2(M;R_2)]_0 & \to & [H^2(M;S)]_0 \\ \downarrow [\text{id}]_0 \otimes [f^2_{R_2}]_0 & & \downarrow [f^2_S]_0 \\\vspace*{.1pt} [R_1]_0 \otimes [H^2_M (R_2)]_0 &\simeq & [H^2_M(S)]_0. \end{eqnarray*} Because $R_2$ is a Cohen-Macaulay ring with $[H^2_M(R_2)]_n = 0$ for all $n \geq 1$ it follows easily that $[f^2_{R_2}]_0 : [H^2(M;R_2)]_0 \to [H^2_M(R_2)]_0$ is an isomorphism. Hence, $[f^2_S]_0$ is surjective as required. \end{example}

9606

alg-geom/9606014

en

https://arxiv.org/abs/alg-geom/9606014

alg-geom/9606014

Sheldon Katz, Zhenbo Qin, and Yongbin Ruan

Composition law and Nodal genus-2 curves in P^2

13 pages, AMS-TeX

OSU Math 1996-18

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree-$d$ genus-$2$ plane curves whose normalization has a fixed complex structure and which pass through $3d - 2$ general points in $\Bbb P^2$.

alg-geom

\section{1. Introduction} Enumerative algebraic geometry is an old field of algebraic geometry. There are many fascinating problems going back more than a hundred years to the Italian school. The most famous one is perhaps the counting problem for the number of holomorphic curves in $\Pee^2$. There are in fact two different counting problems. Let $\tilde N_{g,d}$ be the number of irreducible, reduced, nodal, degree-$d$ genus-$g$ curves which pass through $3d + (g-1)$ general points in $\Pee^2$. The integer $\tilde N_{g,d}$ is often referred to as the Severi number. These numbers have recently been computed in \cite{C-H}. A companion number is the number $N_{g,d}$ of irreducible, reduced, nodal, degree-$d$ genus-$g$ curves whose normalization has a fixed complex structure and which pass through $m(d)$ general points in $\Bbb P^2$. Here $m(d)$ stands for $3d -1$ when $g = 0, 1$ and $3d - 2(g-1)$ when $g \ge 2$. Clearly, $\tilde{N}_{0,d}=N_{0,d}$. The goal of this paper is to compute $N_{2, d}$ for $d \ge 4$ (see (1.1)). The lower degree cases for $N_{g,d}$ were known classically for many years. Not much progress has been made until the introduction of quantum cohomology theory. Based on ideas from physics, a recursion formula for $N_{0,d}$ was proved by \cite{R-T, K-M}. The influence of physical ideas opens up entirely new directions in enumerative geometry. Roughly speaking, $N_{0,d}$ can be interpreted as a correlation function in a certain topological quantum field theory (topological sigma model \cite{Wit}). All topological quantum field theories have a composition law, which in this instance gives the beautiful recursion formula for $N_{0,d}$. Furthermore, the composition law naively suggests a recursion formula for $N_{g,d}$ in terms of $N_{0,d}$. Unfortunately, a simple calculation of lower degree elliptic curves showed that the formula from physics always gives a wrong answer for higher genus case $g>0$. It was showed in \cite{R-T} that the correlation function $\Psi_{g,d}$ (Gromov-Witten invariants) of the topological sigma model counts the number of perturbed pseudo-holomorphic maps. Moreover, it satisfies the composition law physicists predicted and hence can be computed by $N_{0,d}$. However, $\Psi_{g,d} \neq N_{g, d}$ for $g>0$. The original problem of computing $N_{g,d}$ remains to be solved. One obvious approach is to compute the error term $\Psi_{g,d}-N_{g,d}$ and then to use the formula of $\Psi_{g,d}$ to compute $N_{g,d}$. Such an approach involves some delicate obstruction analysis and was carried out in \cite{Ion} for $N_{1,d}$. At the same time, a direct argument for $N_{1,d}$ was given independently in \cite{Pa1}. To explain our approach, we have to explain the composition law of the topological sigma model. Recall that we fix a complex structure on a genus-$g$ curve $\Sigma_g$ to define $N_{g, d}$ (and $\Psi_{g,d}$). Roughly speaking, the composition law gives an explicit formula of $\Psi_{g,d}$ in terms of $\Psi_{g',d'}$ with $g'<g, d'\leq d$ when we degenerate $\Sigma_g$ to a stable curve. As we mentioned, such a composition law fails for $N_{g,d}$. Our observation is that if we degenerate $\Sigma_g$ to a stable curve $C_0$ with only rational components each of which contains exactly three nodal points, then an analogue of the composition law might still hold. In fact, we shall prove that for $d \ge 4$, $$N_{2, d} = {(d-1)(d-2)(d-3) \over 2d} N_d$$ $$+ \sum \limits_{d_1+d_2=d} {d_1d_2(d_1d_2d-6d+18)-4d \over 12d} {3d-2 \choose 3d_1 -1} d_1d_2 N_{d_1} N_{d_2} \eqno (1.1)$$ where for simplicity, we have used $N_d$ to stand for both $N_{0, d}$ and $\tilde N_{0, d}$. Our arguments are parallel to those of Pandharipande \cite{Pa1}. We expect that the same method works for any $g$. The difficulty is a technical one which becomes harder as $g$ gets larger. However, we believe that $N_{g, d}$ is closely related to the number of irreducible, reduced, degree-$d$ plane rational curves which pass through certain general points in $\Pee^2$ and have certain types of singularities. Indeed, it is not difficult to see that one term involved in $N_{g, d}$ with $g > 2$ is $$(3d - 2(g-1))! \cdot \sum \limits_{d_1 + \ldots + d_{2(g-1)} = d} \, \prod \limits_{i=1}^{2(g-1)} {d_i^3 \cdot N_{d_i} \over (3d_i -1)!}. \eqno (1.2)$$ It is interesting to note that this term appears explicitly in $\Psi_{g,d}$. It would be very interesting to figure out the other terms in $N_{g, d}$. This paper is organized as follows. In section 2, the formula (1.1) is proved. In the proof, we need to know the number of constraints on stable maps that are degenerations of maps on irreducible and smooth curves, and the number of irreducible, reduced, degree-$d$ rational plane curves that pass through $(3d-2)$ general points in $\Pee^2$ and have exactly one triple point with all other singularities being nodes. These two numbers are studied in section 3. \section{2. Proof of (1.1)} First of all, we recall some definitions and notations for $g \ge 2$. Let $\overline{\frak M}_g$ be the Deligne-Mumford moduli space of stable genus-$g$ curves, and let $$\overline{\frak M}_{g}(d) \quad {\overset \text{def} \to =} \quad \overline{\frak M}_{g, 3d-2(g-1)}(\Bbb P^2, d) \eqno (2.1)$$ be the moduli space of stable maps from $(3d - 2(g-1))$-pointed genus-$g$ curves to $\Bbb P^2$ such that the homology class of the images is $d[\ell]$ where $[\ell]$ stands for the homology class of a line $\ell$ in $\Bbb P^2$. Then, there is a natural map $\pi: \overline{\frak M}_{g}(d) \to \overline{\frak M}_g$ obtained by forgetting the stable maps and all the marked points, and then contracting any unstable components. For a stable genus-$g$ curve $C$, let $\overline{\frak M}_C(d)$ be the moduli space of stable maps to $\Bbb P^2$ from curves $D$ stably equivalent to $C$ with $(3d - 2(g-1))$ marked points such that the homology class of the images is $d[\ell]$. In the above, we say that $D$ is {\it stably equivalent\/} to $C$ if contracting the unstable components of $D$ yields $C$. By the universal properties of moduli spaces, there is a canonical bijection $$\overline{\frak M}_C(d) \to \pi^{-1}([C]) \eqno (2.2)$$ which is an isomorphism when $[C] \in \overline{\frak M}_g$ is general. Let $W_g(d) \subset \overline{\frak M}_{g}(d)$ be the locus of stable maps whose domains are irreducible, and let $\overline{W}_g(d)$ be the closure of $W_g(d)$ in $\overline{\frak M}_{g}(d)$. Then $W_g(d)$ is a reduced and irreducible open subset of dimension $6d - (g -1)$. For $i = 1, \ldots, 3d - 2(g-1)$, define the evaluation map $e_i: \overline{W}_g(d) \to \Bbb P^2$ by $[\mu: (D, p_1, \ldots, p_{3d - 2(g-1)})] \mapsto \mu(p_i)$, and $\Cal L_i = e_i^*(\Cal O_{\Bbb P^2}(1))$. Put $Z = c_1(\Cal L_1)^2 \cap \ldots \cap c_1(\Cal L_{3d-2(g-1)})^2\cap [\overline{W}_g(d)]$. In the sequel, we will usually think of $Z$ as the cycle determined by the condition that $\mu(p_i)$ is a fixed general point of $\Pee^2$. Now for a general curve $[C] \in \overline{\frak M}_g$, the intersection $$\pi^{-1}([C]) \cap [\overline{W}_g(d)-{W}_g(d)]$$ has codimension at least one in the irreducible and reduced subvariety $\pi^{-1}([C]) \cap \overline{W}_g(d)$. Since the linear series $e_i^*|\ell|$ are base-point-free, $$\pi^{-1}([C]) \cap Z = [\pi^{-1}([C]) \cap \overline{W}_g(d)] \cap Z = [\pi^{-1}([C]) \cap {W}_g(d)] \cap Z.$$ Moreover, by Bertini's Theorem, the intersection cycle $[\pi^{-1}([C]) \cap {W}_g(d)] \cap Z$ consists of finitely many reduced points in $\pi^{-1}([C]) \cap {W}_g(d)$. The number of these points is precisely $N_{g, d}$. Thus for a general curve $[C] \in \overline{\frak M}_g$, $$N_{g, d} = [\pi^{-1}([C]) \cap {W}_g(d)] \cap Z = \pi^{-1}([C]) \cap Z = \pi^{-1}([C]) \cdot Z.$$ It follows that for every stable curve $[C] \in \overline{\frak M}_g$, we have $$N_{g, d} = \pi^{-1}([C]) \cdot Z. \eqno (2.3)$$ Next, we construct a special stable genus-$g$ curve $C_{0, g}$ in $\overline{\frak M}_g$ by induction on $g$. First of all, $C_{0, 2}$ consists of two smooth rational curves intersecting transversely at three points. To get $C_{0, 3}$, we blow-up two nodal points in $C_{0, 2}$ by adding two smooth rational curves which intersect transversely at one point. In general, to obtain $C_{0, g}$ from $C_{0, g-1}$, we blow-up two nodal points in $C_{0, g-1}$ by adding two smooth rational curves which intersect transversely at one point. Thus, $C_{0, g}$ consists of $2(g-1)$ smooth rational curves which are denoted by $R_1, \ldots, R_{2(g-1)}$, and has $3(g-1)$ nodal points. Moreover, if $g > 2$, then for each smooth rational curve $R_i$ in $C_{0, g}$, there exist three other smooth rational curves $R_{k_1}, R_{k_2}, R_{k_3}$ in $C_{0, g}$ such that $R_i$ and each $R_{k_j}$ ($j = 1, 2, 3$) intersect transversely at one point. For simplicity, we denote the curve $C_{0, g}$ by $C_0$. Let $\omega_{C_0}$ be the dualizing sheaf of $C_0$. Then the restriction of $\omega_{C_0}$ to each smooth rational curve $R_i$ in $C_0$ has degree $1$. Thus if $L$ is a line bundle on $C_0$ such that $\text{deg}(L|_{R_i}) \ge 0$ for all $i$ with $1 \le i \le 2(g-1)$, $\text{deg}(L|_{R_{i_1}}) > 1$ for at least one $i_1$, and $\text{deg}(L|_{R_{i_2}}) = 0$ for at most one $i_2$, then $$H^1(C_0, L) \cong H^0(C_0, L^{-1} \otimes \omega_{C_0}) = 0. \eqno (2.4)$$ Let $[C] \in \overline{\frak M}_g$ be generic, and $|\text{Aut}(C)|$ be the order of the automorphism group of $C$. Then $|\text{Aut}(C)| = 2$ when $g = 2$, and $|\text{Aut}(C)| = 1$ when $g > 2$. By (2.3), $$N_{g, d} = \pi^{-1}([C]) \cdot Z = \pi^{-1}([C_0]) \cdot Z = {|\text{Aut}(C_0)| \over |\text{Aut}(C)|} \cdot \big (\overline{\frak M}_{C_0}(d) \cdot Z \big ).$$ Here $\overline{\frak M}_{C_0}(d)$ is identified with $\pi^{-1}([C_0])$ but with the reduced scheme structure. So to prove (1.1), it suffices to show that for $g = 2$ and $d \ge 4$, \smallskip $$\overline{\frak M}_{C_0}(d) \cdot Z = {1 \over |\text{Aut}(C_0)|} \cdot \bigg[ {(d-1)(d-2)(d-3) \over d} N_d$$ $$+ \sum \limits_{d_1+d_2=d} {d_1d_2(d_1d_2d-6d+18)-4d \over 6d} {3d-2 \choose 3d_1 -1} d_1d_2 N_{d_1} N_{d_2} \bigg]. \tag 2.5$$ \smallskip\noindent Note that $\overline{\frak M}_{C_0}(d) \cap Z \subset \overline{\frak M}_{C_0}(d) \cap \overline{W}_g(d)$. Let $[\mu: (D, p_1, \ldots, p_{3d - 2(g-1)})]$ be a point in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_g(d)$. Then $D$ consists of $(k+2(g-1))$ smooth rational curves with $k \ge 0$. For simplicity, we also use $R_1, \ldots, R_{2(g-1)}$ to stand for the $2(g-1)$ smooth rational curves in $D$ which are identified with the $2(g-1)$ smooth rational curves $R_1, \ldots, R_{2(g-1)}$ in $C_0$ after $D$ is contracted to $C_0$. Dropping $R_1, \ldots, R_{2(g-1)}$ from $D$ results in a disjoint union $T_1 \coprod \ldots \coprod T_s$ of trees of smooth rational curves. \lemma{2.6} Assume $[\mu: (D, p_1, \ldots, p_{3d - 2(g-1)})] \in \overline{\frak M}_{C_0}(d) \cap Z$. Let $D_1, \ldots, D_m$ be all the irreducible components of $D$ such that $\mu|_{D_i}$ are not constant, and let $b_i = \text{deg}(\mu|_{D_i})$ for $1 \le i \le m$. Then $m \le 2(g - 1)$; moreover, when $m = 2(g - 1)$, $\mu(D_1), \ldots, \mu(D_m)$ have at most nodal singularities, intersect each other transversally at nonsingular points, and have degrees $b_1, \ldots, b_m$ respectively. \endproclaim \noindent {\it Proof.} Note that $\sum_{i=1}^m b_i = d$. For $1 \le i \le m$, let $\tilde b_i$ be the degree of $\mu(D_i)$ in $\Pee^2$. Then, $\tilde b_i \le b_i$. On the one hand, since $[\mu: (D, p_1, \ldots, p_{3d - 2(g-1)})] \in \overline{\frak M}_{C_0}(d) \cap Z$, $\mu(D)$ has to pass $3d - 2(g-1)$ general points in $\Pee^2$. On the other hand, the degree-$\tilde b_i$ irreducible rational curve $\mu(D_i)$ can pass through at most $(3 \tilde b_i - 1)$ general points in $\Pee^2$. So $\mu(D)$ can pass through at most $\sum_{i=1}^m (3 \tilde b_i - 1) \le (3d -m)$ general points in $\Pee^2$. Thus $m \le 2(g - 1)$. Moreover, if $m = 2(g - 1)$, then $\tilde b_i = b_i$ and $\mu(D_1), \ldots, \mu(D_m)$ have at most nodal singularities and intersect each other transversally. \qed \bigskip\noindent {\it Proof of} (1.1): We shall now prove formula (1.1) by verifying (2.5). So let $g = 2$ and $d \ge 4$. Put $d_i = \text{deg}(\mu|_{R_i})$ for $i = 1, 2$. Then, there are three cases: \roster \item"{(i)}" both $d_1$ and $d_2$ are positive; \item"{(ii)}" exactly one of $d_1$ and $d_2$ is positive (so the other is zero); \item"{(iii)}" $d_1 = d_2 = 0$. \endroster Our strategy is the following. Fix the nonnegative integer $k$. In each of the above three cases, we shall estimate the number $n(k)$ of moduli of various points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$. Since the linear systems $e_i^*|\ell|$ are base-point-free, it follows that if $n(k) < 6d - 4$, then the case will not contribute to the intersection number $\overline{\frak M}_{C_0}(d) \cap Z$. We shall show that only cases (i) and (ii) with $k= 0$ may contribute. Furthermore, all cases (i) and (ii) with $k = 0$ actually contribute, i.e. any such map is actually in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$. So formula (2.5) will be the sum of the contributions of cases (i) and (ii) with $k = 0$. Notice that by Lemma 2.6, we may assume that $m \le 2$ and that if $m = 2$, then $\mu(D_1)$ and $\mu(D_2)$ have at most nodal singularities and intersect transversally. First of all, we consider case (i), that is, both $d_1$ and $d_2$ are positive. By Lemma~2.6, $\mu$ is constant on the trees $T_i$. If $k=0$, then the number of moduli of these points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d)$ is $$n(0) \le (3d - 2) + (3d - 2) = 6d - 4$$ where the first $(3d-2)$ is the number of moduli of the stable map $\mu$, and the second $(3d-2)$ is the number of moduli of the $(3d-2)$ marked points. We claim that for arbitrary $k$, the number of moduli of these types of maps satisfies $n(k)\le 6d-4-k$. To see this, consider the effect of adding an additional rational component $D'$ to $D$ on which $\mu$ is constant. If $D'$ meets the other components in one point, it must contain two marked points for stability, and these points do not have moduli. Since $D'$ can replace at most one point that has no moduli (the point $D'\cap D$), we have $n(k+1)\le n(k)-1$ for such types of maps. The other possibility is for $D'$ to meet the other components in two points. Then $D'$ must contain at least one point without moduli, but no marked points have been replaced, so we obtain $n(k+1)\le n(k)-1$ in this case as well. This proves the claim. Thus only the cases with $k=0$ can occur, as claimed in the preceding paragraph. Furthermore, all cases (i) with $k = 0$ actually contribute. Indeed, suppose that we have a map $\mu: C_0 (= D) \to \Pee^2$ with $d_1>0$ and $d_2>0$. Consider a general flat family of curves $\eta:\Cal E\to\Delta_t$ with $\eta^{-1}(0)=C_0$ and $\eta^{-1}(t)$ smooth for $t\ne 0$. Consider the moduli functor of relative line bundles on families of curves. The obstruction space for this functor is 0, since it lies in an appropriate $\text{Ext}^2$ on a curve. So the functor is smooth. Since $C_0$ is stable, we can find a line bundle ${\Cal L}$ on $\Cal E$ which restricts to $\mu^*{\Cal O}_{\Pee^2}(1)$ on $C_0$ (possibly after a finite base change). Since $d>2$, either $d_1 \ge 2$ or $d_2 \ge 2$. By (2.4), $H^1(C_0, \mu^*{\Cal O}_{\Pee^2}(1)) = 0$. It follows that $R^1\eta_*{\Cal L}=0$ and that $\eta_*{\Cal L}$ is locally free of rank $(d-1)$. Thus there is no obstruction to extending the three sections of $\mu^*{\Cal O}_{\Pee^2}(1)$ which determine the map $\mu$ in the standard coordinates of $\Bbb P^2$. This shows that $\mu$ is in $\overline{W}_2(d)$. So $\mu \in \overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$. Since $H^1(C_0, \mu^*T_{\Pee^2}) = 0$ for all such stable maps $\mu$ and the linear series $e_i^*|\ell|$ are base-point-free, the contribution of case (i) to $\overline{\frak M}_{C_0}(d) \cap Z$ consists of finitely many reduced points. Notice that for a fixed pair of integers $d_1$ and $d_2$ with $d_1 > 0$ and $d_2 = d - d_1 > 0$, the number of unordered pairs $\{C_1, C_2\}$ of irreducible, reduced, nodal, degree-$d_1$ and degree-$d_2$ rational plane curves $C_1$ and $C_2$ whose union $C_1 \cup C_2$ pass through $(3d - 2)$ general points in $\Pee^2$ is ${3d - 2 \choose 3d_1 -1} N_{d_1} N_{d_2}$. Thus the contribution of case (i) to $\overline{\frak M}_{C_0}(d) \cdot Z$ is $${1 \over |\text{Aut}(C_0)|} \cdot \sum \limits_{d_1+d_2=d} {d_1 d_2 \choose 3} {3d - 2 \choose 3d_1 -1} N_{d_1} N_{d_2}. \eqno (2.7)$$ For case (ii), we assume without loss of generality that $d_1 > 0$ and $d_2 = 0$. We start with $m = 2$. Let $\Pee$ be the unique smooth rational component of $D$ (aside from $R_1$) with nonconstant $\mu|_{\Pee}$. Let $T$ be the tree containing $\Pee$. Then $T \cap R_1$ either is empty or consists of precisely one point. If $T \cap R_1$ is empty, or if $T \cap R_1$ is nonempty but the unique point in $T \cap R_1$ is not identified with one of the three singular points of $C_0$, then $\mu(R_1)$ has at least a triple point. If the unique point in $T \cap R_1$ is one of the singular points, then the other two singular points of $C_0$, when identified with points of $R_1$, are mapped by $\mu$ to a double point of $\mu(R_1)$, through which $\mu(\Pee)$ must pass. By Lemma 2.6, the points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ can not be contained in $\overline{\frak M}_{C_0}(d) \cap Z$. So the case $m=2$ does not contribute. Next, let $m = 1$. Then $\mu$ is constant on the closure $\overline{D \backslash R_1}$ of ${D \backslash R_1}$ in $D$. So there exist at least three points $q_1, q_2, q_3$ in $R_1$ such that the images $\mu(q_1), \mu(q_2), \mu(q_3)$ are the same, i.e. every divisor in $(\mu|_{R_1})^*|\ell|$ which contains $p_1$ must contain both $p_2$ and $p_3$. This imposes $4$ independent conditions in choosing the linear series $(\mu|_{R_1})^*|\ell|$ from the complete linear system $|(\mu|_{R_1})^*\ell|$. If $k=0$, the number of moduli of these points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$ is at most $$n(0) = [(3d -1) + (3-4)] + (3d-2) = 6d - 4.$$ Here, $[(3d -1) + (3-4)]$ is an upper bound for the number of moduli of $\mu(R_1)$, where the integer $3$ in $(3-4)$ is the number of moduli of the three points $q_1, q_2, q_3$ varying in $R_1$. As in case (i) above, we see that for general $k$, the number of moduli satisfies $n(k)\le 6d-4-k$. It follows that only the case $k=0$ can occur. Again by arguments similar to those in case (i), we see that all cases (ii) with $k = 0$ actually contribute and that the contribution to $\overline{\frak M}_{C_0}(d) \cap Z$ consists of finitely many reduced points. Furthermore, if $[\mu: (C_0, p_1, \ldots, p_{3d - 2})]$ is such a reduced point in $\overline{\frak M}_{C_0}(d) \cap Z$, then $\mu(C_0) = \mu(R_1)$ is an irreducible and degree-$d$ rational plane curve that passes through $(3d - 2)$ general points in $\Pee^2$ and has at least one triple point. In fact, such a curve in $\Pee^2$ must have exactly one triple point with all other singularities being nodes. Now there are only finitely many irreducible, reduced, degree-$d$ rational plane curves that pass through $(3d - 2)$ general points in $\Pee^2$ and have exactly one triple point with all other singularities being nodes. The number $\tilde N_d$ of such curves is given by Lemma 3.2 which will be proved in the next section. Taking into account of the automorphism group of $C_0$ and the symmetry between $R_1$ and $R_2$, we see that the contribution of case (ii) to $\overline{\frak M}_{C_0}(d) \cdot Z$ is $${1 \over |\text{Aut}(C_0)|} \cdot 2 \tilde N_d. \eqno (2.8)$$ Next we consider case (iii), that is, both $\mu|_{R_1}$ and $\mu|_{R_2}$ are constant. We start with $m = 1$. Let $\Pee$ be the unique smooth rational component of $D$ with nonconstant $\mu|_{\Pee}$, and let $T$ be the unique tree in $T_1, \ldots, T_s$ such that $\Pee \subset T$. Here and in other subcases of case (iii), we will be able to assume without loss of generality that the tree $T$ is actually a particularly simple chain. The recurring theme will be that if $W$ is a subvariety of $\overline{\frak M}_{C_0}(d)$ such that $T$ is a certain type of chain and $\dim(W\cap\overline{W}_g(d))<6d-4$, then the same conclusion will hold for subvarieties $W'$ associated to more complicated trees, since in these situations $W'$ will always be a subvariety of the closure $\overline{W}$ obtained from contracting suitable $\mu$-constant curves. Returning to the subcase at hand, we may for the above reason assume that $\Pee=T$ has one component. If $T$ intersects $R_1$ or $R_2$ but not both, then by Lemma 3.7 and Lemma 3.13, there exist $4$ independent conditions in choosing the linear series $(\mu|_T)^*|\ell|$ from $|(\mu|_T)^*\ell|$. So the number of moduli of the points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$ is at most $$n(k) \le [(3d - k) + (2-4)] + (3d-2) = 6d - 4 - k \le 6d - 5$$ where the integer $2$ in $(2-4)$ is the number of the moduli of the point $T \cap (R_1 \cup R_2)$ varying in both $T$ and $R_1 \cup R_2$. Thus this case makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. If $T$ intersects both $R_1$ and $R_2$, then by Remark 3.12 (ii) and Lemma 3.7, there still exist $4$ independent conditions in choosing $(\mu|_T)^*|\ell|$ from $|(\mu|_T)^*\ell|$. So the number of moduli of the points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$ is at most $$n(k) \le [(3d - k) + (2-4)] + (3d-2) = 6d - 4 - k \le 6d - 5$$ where the integer $2$ in $(2-4)$ is the number of the moduli of the two points $T \cap R_1$ and $T \cap R_2$ varying in $T$. It follows that this case makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. We are left with case (iii) and $m = 2$. Then $k \ge 2$. Let $D_1, D_2$ be the two rational components in $D$ such that $\mu|_{D_1}, \mu|_{D_2}$ are nonconstant. Then $D_i \ne R_j$ for $i, j = 1, 2$. First, assume that $D_1$ and $D_2$ are contained in the same tree $T$ from $T_1, \ldots, T_s$. If $T$ intersects $R_1$ or $R_2$ but not both, then by Lemma 3.7, there exist $4$ independent conditions in choosing the linear series $(\mu|_{D_1})^*|\ell|$ and $(\mu|_{D_2})^*|\ell|$ from the complete linear systems $|(\mu|_{D_1})^*\ell|$ and $|(\mu|_{D_2})^*\ell|$. So the number of moduli of the points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$ is at most $$n(k) \le [(3d - k) + (2-4)] + (3d-2) = 6d - 4 - k \le 6d - 6$$ where the integer $2$ in $(2-4)$ is the number of the moduli of the point $T \cap (R_1 \cup R_2)$ varying in both $T$ and $R_1 \cup R_2$, and this case makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. Similarly, by Remark 3.12 (ii) and Lemma 3.7, the case when $T$ intersects both $R_1$ and $R_2$ makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. Next, assume that $D_1$ and $D_2$ are contained in two different trees, say $T_1$ and $T_2$, from $T_1, \ldots, T_s$. If $T_1$ intersects both $R_1$ and $R_2$ and if $D_1$ intersects $\overline{D \backslash D_1}$ at least twice, then $\mu(D_2)$ passes through a singular point of $\mu(D_1)$. By Lemma 2.6, this case makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. So we may assume that if $T_i$ intersects both $R_1$ and $R_2$, then $D_i$ intersects $\overline{D \backslash D_i}$ once. By Lemma 3.7 and Remark 3.12 (ii), the number of moduli of the points $[\mu: (D, p_1, \ldots, p_{3d - 2})]$ in $\overline{\frak M}_{C_0}(d) \cap \overline{W}_2(d)$ is at most $$n(k) \le [(3d - k) + (4-4)] + (3d-2) = 6d - 2 - k \le 6d - 4.$$ If $n(k) = 6d - 4$, then $k = s = 2$ and $T_i = D_i$ ($i = 1, 2$) intersects $R_1$ and $R_2$ but not both. Since the number of the moduli for the two points $D_1 \cap (R_1 \cup R_2)$ and $D_2 \cap (R_1 \cup R_2)$ varying in $R_1 \cup R_2$ is $2$, the pairs $(\mu(D_1), \mu(D_2))$ of curves form a codimension-$2$ subset in the $(3d-2)$-dimensional variety $S(0, d_1) \times S(0, d_2)$ where $S(0, d_i)$ stands for the moduli space of irreducible, reduced, nodal, degree-$d_i$ rational plane curves. It follows that this case makes no contribution to $\overline{\frak M}_{C_0}(d) \cap Z$. Summing up (2.7) and (2.8) and using (3.3), we obtain (2.5) and hence (1.1). \qed \bigskip Finally, some remarks about the number $N_{g, d}$ with $g > 2$ follow. We expect that only the cases with $k = 0$ (that is, stable maps $[\mu: (D, p_1, \ldots, p_{3d - 2(g-1)})]$ with $D = C_0 = C_{0, g}$) contribute to $N_{g, d} = |\text{Aut}(C_0)| \cdot \big (\overline{\frak M}_{C_0}(d) \cdot Z \big )$. For instance, as in the proof of the above case (i), the case when $k = 0$ and $\text{deg}(\mu|_{R_i}) > 0$ for all $1 \le i \le 2(g-1)$ contributes to $N_{g, d}$, and its contribution is $$(3d - 2(g-1))! \cdot \sum \limits_{d_1 + \ldots + d_{2(g-1)} = d} \, \prod \limits_{i=1}^{2(g-1)} {d_i^3 \cdot N_{d_i} \over (3d_i -1)!}. \eqno (2.9)$$ It is interesting to notice that for $g>2$, the term~(2.9) is precisely a term that arises in the calculation of $\Psi_{g,d}$. In general, let $S$ be any nonempty subset of $\{1, \ldots, 2(g-1) \}$. We believe that the contribution of the cases with $k = 0$ and $\text{deg}(\mu|_{R_i}) > 0$ if and only if $i \in S$ is closely related to the number of irreducible, reduced, degree-$d$ plane rational curves which pass through certain general points in $\Pee^2$ and have certain types of singularities. \bigskip\noindent {\bf 3. Rational plane curves with triple points and constraint on stable maps} In this section, we prove the lemmas quoted in the previous section. We shall compute the number $\tilde N_d$ of irreducible, reduced, degree-$d$ rational plane curves that pass through $(3d-2)$ general points in $\Pee^2$ and have exactly one triple point with all other singularities being nodes. Then we analyze the constraints on stable maps that are degenerations of maps whose domains are irreducible and smooth curves. \ssection{3.1. Rational plane curves with triple points} Fix $d \ge 3$. Intersections of $\Bbb Q$-divisors in the moduli space $\overline{\frak M}_{0, 0}(\Bbb P^2, d)$ has been studied by Pandharipande. We recall some results from \cite{Pa2, Pa3}. The boundary $\Delta$ of $\overline{\frak M}_{0, 0}(\Bbb P^2, d)$ is the locus corresponding to stable maps whose domains are reducible, and is of pure codimension-$1$. For $1 \le i \le \left [ {d \over 2} \right ]$, let $K^i$ be the irreducible component of $\Delta$ whose general elements are the form $\mu: D_1 \cup D_2 \to \Pee^2$ such that $\text{deg}(\mu|_{D_1}) = i$, and $D_1$ and $D_2$ are smooth rational curves meeting transversely at one point. Let $\Cal H$ be the locus of $\overline{\frak M}_{0, 0}(\Bbb P^2, d)$ corresponding to maps whose images pass through a fixed point in $\Pee^2$. Then $\Cal H$ is a Cartier divisor in $\overline{\frak M}_{0, 0}(\Bbb P^2, d)$, and $\text{Pic}(\overline{\frak M}_{0, 0}(\Bbb P^2, d)) \otimes \Bbb Q$ is generated by $\Cal H$ and $K^i$ with $1 \le i \le \left [ {d \over 2} \right ]$; moreover, $$\Cal H^{3d-1} = N_d, \quad K^i \cdot \Cal H^{3d-2} = \cases {3d-2 \choose 3i -1} i(d-i) N_i N_{d-i}, &\text{if $i \ne {d \over 2}$}\\ {1 \over 2} {3d-2 \choose 3{d \over 2} -1} \left ({d \over 2} \right )^2 N_{d \over 2}^2, &\text{if $i = {d \over 2}$}. \endcases \tag 3.1$$ Let $Z \subset \overline{\frak M}_{0, 0}(\Bbb P^2, d)$ be the subvariety consisting of degree-$d$ maps $\mu: \Pee^1 \to \Pee^2$ such that $\mu(\Pee^1)$ has exactly one triple point with all other singularities being nodes, and let ${\overline Z}$ be the closure of $Z$ in $\overline{\frak M}_{0, 0}(\Bbb P^2, d)$. Then $Z$ is reduced and of pure codimension-$1$, and ${\overline Z}$ is a Weil divisor. Similar arguments as in section (3.4) of \cite{Pa2} show that the intersection ${\overline Z} \cap \Cal H^{3d-2}$ determined by $(3d-2)$ general points in $\Pee^2$ consists of finitely many reduced points in $Z$. Thus $\tilde N_d = {\overline Z} \cdot \Cal H^{3d-2}$. \lemma{3.2} For $d \ge 3$, $\tilde N_d$ can be expressed in terms of the $N_d$: $${(d-1)(d-2)(d-3) \over 2d} N_d - \sum_{d_1+d_2=d} {d_1 d_2 (d-6)+2d \over 4d} {3d-2 \choose 3d_1 -1} d_1 d_2 N_{d_1} N_{d_2}. \eqno (3.3)$$ \endproclaim \noindent {\it Proof.} Let ${\overline Z} = a \Cal H + \sum_{i = 1}^{\left [ {d \over 2} \right ]} a_i K^i$ where $a, a_i \in \Bbb Q$. By (3.1), it suffices to determine $a$ and $a_i$. Fix a nonsingular curve $C$. As in section (4.5) of \cite{Pa2}, we compute the intersection number $C \cdot \lambda^*{\overline Z}$ for some morphism $\lambda: C \to \overline{\frak M}_{0, 0}(\Bbb P^2, d)$ which is constructed as follows. Let $\pi_2: S = \Pee^1 \times C \to C$ be the second projection, and let $\Cal N$ be a line bundle on $S$ of degree type $(d, k)$ where $k$ is a very large integer. Let $z_0, z_1, z_2 \in H^0(S, \Cal N)$ determine a rational map $\phi: S \dashrightarrow \Pee^2$ such that \roster \item"{(i)}" every base point of $\phi$ is simple. Here a base point $s$ of $\phi$ is {\it simple of degree $i$ with $1 \le i \le d$} if the blowing-up of $S$ at $s$ resolves $\phi$ locally at $s$ and the resulting map is of degree $i$ on the exceptional divisor; \item"{(ii)}" there exist only finitely many points $c_1, \ldots, c_n \in C$ such that for each $i$, $c_i$ is not the projection of base points of $\phi$ and $\overline \phi(\overline \pi^{-1}(c_i))$ contains one and exactly one triple point. Here $\overline S$ is the blow-up of $S$ at the base points, $\overline \pi: \overline S \to C$ is the projection to $C$, and $\overline \phi: \overline S \to \Pee^2$ is the resolution of $\phi$. \endroster Now $\overline \phi: \overline S \to \Pee^2$ and $\overline \pi: \overline S \to C$ induce a morphism $\lambda: C \to \overline{\frak M}_{0, 0}(\Bbb P^2, d)$. A point of the intersection $C \cdot \lambda^*K^i$ can arise in two cases, that is, a simple base point of degree $i$ or $(d-i)$ can be blown-up. Let $C \cdot \lambda^*K^i = x_i + y_i$ where $x_i$ and $y_i$ are the number of instances of the first and second case respectively. On the one hand, $C \cdot \lambda^*\Cal H = 2dk - \sum_{i= 1}^{\left [ {d \over 2} \right ]} [i^2x_i + (d-i)^2y_i]$ according to \cite{Pa2}. Thus, $$C \cdot \lambda^*{\overline Z} = 2adk - \sum_{i= 1}^{\left [ {d \over 2} \right ]} a[i^2x_i + (d-i)^2y_i] + \sum_{i = 1}^{\left [ {d \over 2} \right ]} a_i (x_i + y_i). \eqno (3.4)$$ On the other hand, the triple-point formula of \cite{Kle} can be applied to the map $(\overline{\phi},\overline{\pi}):\overline{S}\to \Pee^2\times C$ to yield $$C \cdot \lambda^*{\overline Z} = (d-1)(d-2)(d-3)k + \sum_{i= 1}^{\left [ {d \over 2} \right ]} (-{1\over 2} i^2 d^2 +3i^2d -{1\over 2} id-1+3i-5i^2) x_i$$ $$+\sum_{i= 1}^{\left [ {d \over 2} \right ]} (-{1\over 2} (d-i)^2 d^2 +3(d-i)^2d -{1\over 2} (d-i)d-1+3(d-i)-5(d-i)^2) y_i. \eqno (3.5)$$ Parts of this computation were performed using Schubert \cite{K-S}. Comparing the coefficients of $k, x_i, y_i$ in (3.4) and (3.5) leads to $$a = {(d-1)(d-2)(d-3) \over 2d} \qquad \text{and} \qquad a_i = -{i(d-i)(d-6)+2d \over 2d}. \eqno (3.6)$$ Therefore the formula for $\tilde N_d$ follows from (3.1), (3.6) and $$\tilde N_d = {\overline Z} \cdot \Cal H^{3d-2} = a \Cal H^{3d-1} + \sum_{i = 1}^{\left [ {d \over 2} \right ]} a_i K^i \cdot \Cal H^{3d-2}. \qed$$ \ssection{3.2. Constraint on stable maps} Let $\eta: \Cal E \to \Delta_t$ be a flat family of stable genus-$g$ curves in $\overline{\frak M}_g$ satisfying \roster \item"{(a)}" $\eta^{-1}(t)$ is irreducible and smooth for all $t \ne 0$. \item"{(b)}" for every smooth point $p \in C_0 {\overset \text{def} \to =} \eta^{-1}(0)$, there exists a basis $\Lambda_1, \ldots, \Lambda_g$ for $H^0(C_0, \omega_{C_0})$ such that locally near $p$, we have $\Lambda_j = v^{j-1} f_j(v) \cdot \text{d}v$ for some holomorphic functions $f_j(v)$ satisfying $f_j(0) \ne 0$ for $1 \le j \le g$ where $v$ is a local coordinate of $C_0$ centered at $p$. \endroster Let $\hat \eta: \hat \Cal E \to \Delta_t$ be the family obtained by blowing-up $\Cal E$ at points in $C_0$ (possibly infinitely near) and by adding $(3d-2(g-1))$ markings. Let $D = \hat \eta^{-1}(0)$, and $T_1, \ldots, T_s$ be all the connected components of the closure $\overline{D \backslash C_0}$ of $D \backslash C_0$ in $D$. Then each $T_i$ is a tree of smooth rational curves, and $D = C_0 \cup (\coprod_{i=1}^s T_i)$. Assume that $\mu: \hat \Cal E \to \Pee^2$ is a morphism such that $\mu, \hat \eta$, and the $(3d-2(g-1))$ markings determine a family of stable maps in $\overline{W}_g(d)$. Furthermore, suppose that there exist smooth rational components $D_1, \ldots, D_m$ contained in the trees $T_1, \ldots, T_s$ with $\text{deg}(\mu|_{D_i}) > 0$ for each $i$ and $\sum_{i = 1}^m \text{deg}(\mu|_{D_i}) = d$. \lemma{3.7} Let $g \ge 2$ and $d > 2(g-1)$. Assume that $\hat \Cal E$ is obtained from $\Cal E$ by a chain of blowups at smooth points of $C_0$ (possibly infinitely near). If $m \le 2$, then there exist $2g$ independent conditions in choosing the $m$ linear series $(\mu|_{D_i})^*|\ell|$, $1 \le i \le m$ from the $m$ complete linear systems $|(\mu|_{D_i})^*\ell|$, $1 \le i \le m$. \endproclaim \proof There are three separate cases: (i) $m = 1$; (ii) $m = 2$, and $D_1, D_2$ are contained in the same tree in $T_1, \ldots, T_s$; (iii) $m = 2$, and $D_1, D_2$ are contained in the two different trees in $T_1, \ldots, T_s$. The proofs of (ii) and (iii) are very similar to the proof of (i) but need some extra preparation. {\bf Case (i)}: We follow the approach in \cite{Pa1}. For simplicity, we first assume that $\hat \Cal E$ is the blow-up of $\Cal E$ at a smooth point $p \in C_0$. Then $s=1= i$, and $\Pee = T_1 = \Pee^1$ is the exceptional divisor. For each $j$ with $1 \le j \le d$, let $\Cal G_j = \Cal H_j \subset \Cal E$ be the open subset of $\Cal E$ on which the morphism $\eta$ is smooth. Put $$X = \Cal G_1 \times_{\Delta_t} \ldots \times_{\Delta_t} \Cal G_d \times \Cal H_1 \times_{\Delta_t} \ldots \times_{\Delta_t} \Cal H_d.$$ Then $X$ is a smooth open subset of the $2d$-fold fiber product of $\Cal E$ over $\Delta_t$. Let $Y \subset X$ be the subset of points $y = (g_1, \ldots, g_d, h_1, \ldots, h_d)$ such that the two divisors $\sum_{j} g_j$ and $\sum_{j} h_j$ are linearly equivalent on the curve $\eta^{-1}(\eta(y))$. Let $\gamma: \Delta_t \to \Cal E$ be a local holomorphic section of $\eta$ such that $\gamma(0) = p$. Let $V$ be a nonvanishing local holomorphic field of vertical tangent vectors to $\Cal E$ on an open subset containing $p$. The section $\gamma$ and the vertical vector field $V$ together determine local holomorphic coordinates $(t, v)$ on $\Cal E$ at $p$. Let $\phi_V: \Cal E \times \Cee \to \Cal E$ be the holomorphic flow of $V$ defined locally near $(p, 0) \in \Cal E \times \Cee$. Then the coordinate map $\psi: \Cee^2 \to \Cal E$ is defined by $\psi(t, v) = \phi_V(\gamma(t),v)$. Since $p \in C_0$ is a smooth point in $C_0$, $p \in \Cal G_j$ and $p \in \Cal H_j$ for $1 \le j \le d$. Put $x_p = (p, \ldots, p, p, \ldots, p) \in X$. Then the local coordinates on $X$ near $x_p$ are given by $(t, v_1, \ldots, v_d, w_1, \ldots, w_d)$, and the coordinate map $\psi_X$ is defined by putting $\psi_X(t, v_1, \ldots, v_d, w_1, \ldots, w_d)$ to be $$(\psi(t, v_1), \ldots, \psi(t, v_d), \psi(t, w_1), \ldots, \psi(t, w_d)) \in X.$$ By our assumption, there exists a basis $\Lambda_1, \ldots, \Lambda_g$ of $H^0(C_0, \omega_{C_0})$ such that near $p$, we have $\Lambda_k(v) = v^{k-1} f_k(v) \cdot \text{d}v$ and $f_k(0) \ne 0$ for $1 \le k \le g$. Let $\Lambda_k(t, v)$ ($1 \le k \le g$) be a local holomorphic extension of $\Lambda_k(v)$ at $0 \in \Delta_t$ such that for a fixed $t$, $\Lambda_1(t, v), \ldots, \Lambda_g(t, v)$ form a basis of $H^0(\eta^{-1}(t), \omega_{\eta^{-1}(t)})$ using the coordinate map $\psi$. This basis may be chosen so that $$\Lambda_k(t, v) = v^{k-1} f_k(t, v) \cdot \text{d}v$$ with $f_k(0, 0) = f_k(0) \ne 0$. Now the local equations of $Y \subset X$ at the point $x_p$ are $$\sum_{j=1}^d \left ( \int_0^{v_j} \Lambda_k(t, v) - \int_0^{w_j} \Lambda_k(t, v) \right ) = 0, \qquad 1 \le k \le g. \eqno (3.8)$$ Let $L_1$ and $L_2$ be general divisors in $\mu^*|\ell|$ such that each intersects $\Pee$ at $d$ distinct points. For $1 \le \alpha \le 2$, $L_\alpha$ determines local holomorphic sections $s_{\alpha, 1} + \ldots + s_{\alpha, d}$ of $\hat \eta$ at $0 \in \Delta_t$. These sections $s_{\alpha, j}$ with $1 \le \alpha \le 2$ and $1 \le j \le d$ determine a map $\lambda: \Delta_t \to Y$ locally at $0 \in \Delta_t$. Let an affine coordinate on $\Pee^1$ be given by $\xi$ corresponding to the normal direction $${\text{d}\gamma \over \text{d}t}|_{t = 0} + \xi \cdot V(p).$$ Let $s_{1, j}(0) = \nu_j \in \Cee^1 \subset \Pee^1$ and $s_{2, j}(0) = \omega_j \in \Cee^1 \subset \Pee^1$ be given in terms of the affine coordinates $\xi$. Then the map $\lambda$ has the form $$\lambda(t) = (t, \nu_1(t), \ldots, \nu_d(t), \omega_1(t), \ldots, \omega_d(t)) \eqno (3.9)$$ where $$\nu_j(t) = \nu_j t + O(t^2), \ \omega_j(t) = \omega_j t + O(t^2),\qquad 1 \le j \le d. \eqno (3.10)$$ Since $Y$ is defined by the equations (3.8), we obtain $$\sum_{j=1}^d \left ( \int_0^{\nu_j(t)} v^{k-1} f_k(t, v) \cdot \text{d}v - \int_0^{\omega_j(t)} v^{k-1} f_k(t, v) \cdot \text{d}v \right ) = 0, \quad 1 \le k \le g. \eqno (3.11)$$ Differentiating (3.11) $k$-times with respect to $t$ and evaluating at $t = 0$ results in $$\sum_{j=1}^d \left ( (k-1)! f_k(0,0) \cdot \nu_j^k - (k-1)! f_k(0,0) \cdot \omega_j^k \right ) = 0, \qquad 1 \le k \le g.$$ Since $f_k(0, 0) = f_k(0) \ne 0$, we have $\sum_{j=1}^d \nu_j^k = \sum_{j=1}^d \omega_j^k$ where $1 \le k \le g$. Let $\beta_k$ be the $k$-th elementary symmetric function in $d$ variables. Then, $\beta_k(\nu_1, \ldots, \nu_d) = \beta_k(\omega_1, \ldots, \omega_d)$ for $1 \le k \le g$. Put $\beta_k' = (-1)^k \cdot \beta_k(\nu_1, \ldots, \nu_d)$ for $1 \le k \le g$. Then the divisors in $(\mu|_{\Pee})^*|\ell|$ correspond to degree-$d$ polynomials of the form $$K(\xi^d + \beta_1' \cdot \xi^{d-1} + \ldots + \beta_g' \cdot \xi^{d-g} + \ldots)$$ where $K$ stands for constants. It follows that the linear series $(\mu|_{\Pee})^*|\ell|$ has vanishing sequence $\{0, \ge (g+1), *\}$ at the point $\xi = \infty$ which is the intersection $C_0 \cap \Bbb P$. Since the complete linear system $|(\mu|_{\Pee})^*\ell|$ is base-point-free, the existence of a vanishing sequence of the form $\{0, \ge (g+1), *\}$ for the linear series $(\mu|_{\Pee})^*|\ell|$ imposes $2g$ independent conditions in choosing $(\mu|_{\Pee})^*|\ell|$ from $|(\mu|_{\Pee})^*\ell|$. The general case arises when $n$ blowups are needed to obtain $\Pee$. In this situation, using automorphisms of the rational components in $T_i$, we may assume that the form of $\lambda$ is again given by~(3.9) with~(3.10) replaced by $$\nu_j(t) = \nu_j t^n + O(t^{n+1}), \, \omega_j(t) = \omega_j t^n + O(t^{n+1}), \qquad 1 \le j \le d.$$ Then the calculation concludes as before. (Compare with \cite{Pa1}.) {\bf Case (ii)}: For simplicity, we assume that $\hat \Cal E$ is the $2$-fold blow-up of $\Cal E$ at a smooth point $p \in C_0$. Then $s=1= i$, and $T_1 = D_1 \cup D_2$ is the union of the two exceptional divisors. Let $d_i = \text{deg}(\mu|_{D_i})$ for $i = 1, 2$. Then, $d_1 > 0$, $d_2 > 0$, and $d_1 + d_2 = d$. Let $L_1$ and $L_2$ be general divisors in $\mu^*|\ell|$ such that each intersects $D_i$ at $d_i$ distinct points. Let other notations be as in Case (i). Then locally at $0 \in \Delta_t$, $L_1$ and $L_2$ induce a map $\lambda: \Delta_t \to Y$ sending $t \in \Delta_t$ to the following point in $Y$: $$\lambda(t) = (t, \nu_{1,1}(t), \ldots, \nu_{1, d_1}(t), \nu_{2,1}(t), \ldots, \nu_{2, d_2}(t),$$ $$\quad \omega_{1,1}(t), \ldots, \omega_{1, d_1}(t), \omega_{2,1}(t), \ldots, \omega_{2, d_1}(t))$$ where for $1 \le i \le 2$ and $1 \le j \le d_i$, we may assume that $\nu_{i, j}(t) = \nu_{i, j} t^i + O(t^{i+1})$ and $\omega_{i, j}(t) = \omega_{i, j} t^i + O(t^{i+1})$ for some constants $\nu_{i, j}$ and $\omega_{i, j}$. A similar argument as in Case (i) shows that the linear series $(\mu|_{D_1})^*|\ell|$ has vanishing sequence $\{0, \ge (g+1), *\}$ at the point $C_0 \cap D_1$. So there exist $2g$ independent conditions in choosing $(\mu|_{D_1})^*|\ell|$ from the complete linear system $|(\mu|_{D_1})^*\ell|$. {\bf Case (iii)}: Again for simplicity, we assume that $\hat \Cal E$ is the $2$-fold blow-up of $\Cal E$ at two smooth points $p_1, p_2 \in C_0$. Then $s=2$, and $\{ T_1, T_2 \} = \{D_1, D_2\}$ is the set of the two exceptional divisors. Let $d_i = \text{deg}(\mu|_{D_i})$ for $i = 1, 2$. Then, $d_1 > 0$, $d_2 > 0$, and $d_1 + d_2 = d$. Since $d > 2(g-1)$, we may assume that $d_1 \ge g$. As in Case (i), we construct local coordinates $(t, v_i)$ and coordinate map $\psi_i$ on $\Cal E$ at each point $p_i$. Let $X$ and $Y$ be as in Case (i). Define a point $x_{p_1,p_2} \in X$ by $$x_{p_1,p_2} = (\underbrace{p_1, \ldots, p_1}_{\text {$d_1$ times}}, \underbrace{p_2, \ldots, p_2}_{\text {$d_2$ times}}, \underbrace{p_1, \ldots, p_1}_{\text {$d_1$ times}}, \underbrace{p_2, \ldots, p_2}_{\text {$d_2$ times}}).$$ The local coordinates on $X$ near $x_{p_1,p_2}$ are given by $(t, v_1, \ldots, v_d, w_1, \ldots, w_d)$, and the coordinate map $\psi_X$ is defined by putting $\psi_X(t, v_1, \ldots, v_d, w_1, \ldots, w_d)$ to be $$(\psi_1(t, v_1), \ldots, \psi_1(t, v_{d_1}), \psi_2(t, v_{d_1 +1}), \ldots, \psi_2(t, v_d),$$ $$\psi_1(t, w_1), \ldots, \psi_1(t, w_{d_1}), \psi_2(t, w_{d_1 +1}), \ldots, \psi_2(t, w_d)).$$ Note that $x_{p_1,p_2} \in Y$ and the local equations of $Y \subset X$ at $x_{p_1,p_2}$ are given by (3.8) where $\Lambda_1(t, v), \ldots, \Lambda_g(t, v)$ are chosen so that $\Lambda_k(t, v_1) = v_1^{k-1} f_k(t, v_1) \cdot \text{d}v_1$ with $f_k(0, 0) \ne 0$. Let $L_1$ and $L_2$ be general divisors in $\mu^*|\ell|$ such that each intersects $D_i$ at $d_i$ distinct points. Then locally at $0 \in \Delta_t$, $L_1$ and $L_2$ induce $\lambda: \Delta_t \to Y$ by $$\lambda(t) = (t, \nu_{1,1}(t), \ldots, \nu_{1, d_1}(t), \nu_{2,1}(t), \ldots, \nu_{2, d_2}(t),$$ $$\quad \omega_{1,1}(t), \ldots, \omega_{1, d_1}(t), \omega_{2,1}(t), \ldots, \omega_{2, d_1}(t))$$ where for $1 \le i \le 2$ and $1 \le j \le d_i$, we have $\nu_{i, j}(t) = \nu_{i, j} t + O(t^2)$ and $\omega_{i, j}(t) = \omega_{i, j} t + O(t^2)$ for some constants $\nu_{i, j}$ and $\omega_{i, j}$. Now a similar argument as in Case (i) shows that for $1 \le k \le g$, $\sum_{j=1}^{d_1} \nu_{1,j}^k - \sum_{j=1}^{d_1} \omega_{1, j}^k$ is a homogeneous polynomial in $\nu_{2,1}, \ldots, \nu_{2,d_2}$ and in $\omega_{2,1}, \ldots, \omega_{2,d_2}$. Therefore for a fixed linear series $(\mu|_{D_2})^*|\ell|$ in $|(\mu|_{D_2})^*\ell|$, these exist $2g$ independent conditions in choosing $(\mu|_{D_1})^*|\ell|$ from $|(\mu|_{D_1})^*\ell|$, i.e. there exist $2g$ independent conditions in choosing the linear series $(\mu|_{D_1})^*|\ell|$ and $(\mu|_{D_2})^*|\ell|$ from the complete linear systems $|(\mu|_{D_1})^*\ell|$ and $|(\mu|_{D_2})^*\ell|$. This completes the proof of Case (iii) and hence the proof of the lemma. \endproof \noindent {\it Remark 3.12.} (i) It is reasonable to expect that Lemma 3.7 holds for any $m$. \par (ii) In Lemma 3.7, we have assumed that $\hat \Cal E$ is the blow-up of $\Cal E$ at smooth points of $C_0$. However, a slight modification of its proof shows that the conclusion is still true if $g = 2$ and the blowing-up $\hat \Cal E \to \Cal E$ also takes place at nodal points of $C_0$. Finally, we show that the stable genus-$2$ curve $C_0$ constructed in Section~2 satisfies hypothesis (b) leading up to the statement of Lemma~3.7. \lemma{3.13} For every smooth point $p \in C_0$, there exists a basis $\Lambda_1, \Lambda_2$ for $H^0(C_0, \omega_{C_0})$ such that $\Lambda_j = v^{j-1} f_j(v) \cdot \text{d}v$ and some holomorphic functions $f_j(v)$ such that $f_j(0) \ne 0$ for $1 \le j \le 2$ where $v$ is a local coordinate of $C_0$ centered at $p$. \endproclaim \noindent {\it Proof.} Assume that $p \in R_1 = \Pee^1$. Choose an affine coordinate $z$ for $R_1$ such that the three nodal points in $C_0$ are identified with $0, 1, \infty$ in $R_1$. Then a basis for $H^0(C_0, \omega_{C_0})$ can be identified with $\Lambda_1' = {1 \over z} \cdot \text{d}z, \Lambda_2' = {1 \over z-1} \cdot \text{d}z.$ Let $z_0$ be the coordinate of $p \in R_1$, and let $v = z-z_0$. Then the desired basis consists of $$\Lambda_1 = {1 \over z} \cdot \text{d}z = {1 \over v+z_0} \cdot \text{d}v, \quad \Lambda_2 = {(z-z_0) \over z(z-1)} \cdot \text{d}z = {v \over (v+z_0)(v+z_0-1)} \cdot \text{d}v. \qed$$ \Refs \widestnumber\key{MMM} \ref\key C-H \by L.~Caporaso and J.~Harris \paper Counting plane curves of any genus \jour Preprint \endref \ref\key Ion \by E.-M. Ionel \jour Michigan State University Ph.D. Thesis \yr 1996 \endref \ref\key K-S \by S. Katz and S.A.~Str\o mme \book Schubert: A Maple Package for Intersection Theory in Algebraic Geometry \bookinfo Available by anonymous ftp from ftp.math.okstate.edu or linus.mi.uib.no, cd pub/schubert \endref \ref\key Kle \by S.~Kleiman \paper Multiple point formulas I: Iteration \jour Acta Math. \vol 147 \pages 13-49 \yr 1981 \endref \ref\key K-M \by M. Kontsevich, Y. Manin \paper Gromov-Witten classes, quantum cohomology, and enumerative geometry \jour Commun. Math. Phys. \vol 164 \pages 525-562 \yr 1994 \endref \ref\key Pa1 \by R. Pandharipande \paper A note on elliptic plane curves with fixed $j$-invariant \jour Preprint \endref \ref\key Pa2 \bysame \paper Intersection of $\Bbb Q$-divisors on Kontsevich's moduli space $\overline{M}_{0, n}(\Pee^r, d)$ and enumerative geometry \jour Preprint \endref \ref\key Pa3 \bysame \paper Notes on Kontsevich's compactification of the space of maps \jour Preprint \endref \ref\key R-T \by Y. Ruan, G. Tian \paper A mathematical theory of quantum cohomology \jour J. Diffeo. Geom. \vol 42 \pages 259-367 \yr 1995 \endref \ref\key Wit \by E. Witten \paper Topological sigma models \jour Commun. Math. Phys. \vol 118 \pages 411-449 \yr 1988 \endref \endRefs \enddocument

9606

alg-geom/9606010

en

https://arxiv.org/abs/alg-geom/9606010

alg-geom/9606010

Vladimir Hinich

Vladimir Hinich

Descent of Deligne groupoids

Minor corrections made AMSLaTeX v 1.2 (Compatibility mode)

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of Kan simplicial sets. There is a natural homotopy equivalence between $\Sigma_g$ and the Deligne groupoid corresponding to $g$. The main result of the paper claims that the functor $\Sigma$ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman which implies that if a deformation problem is described ``locally'' by a sheaf of dg Lie algebras $g$ on a topological space $X$ then the global deformation problem is described by the homotopy Lie algebra $R\Gamma(X,g)$.

alg-geom

\section{Introduction} \subsection{} \label{i1} Let $\fg$ be a dg Lie algebra over a field $k$ of characteristic zero concentrated in non-negative degrees. The algebra $\fg$ defines a functor $$ \CC_{\fg}:\art/k\to\Grp$$ from the category of local artinian $k$-algebras with the residue field $k$ to the category of groupoids --- see~\ref{del-gr} or~\cite{gm1}, sect.~2. A common belief is that any "reasonable" formal deformation problem can be described by the functor $\CC_{\fg}$ where $\fg$ is an appropriate "Lie algebra of infinitesimal automorphisms". This would imply, for instance, that if $H^0(\fg)=0$ (i.e. if the automorphism group of the deformed object is discrete) then the completion of the local ring of a moduli space at a given point is isomorphic to the completion of the $0$-th cohomology group of $\fg$. If we are dealing with the deformations of algebraic structures (associative, commutative or Lie algebras or so), the Lie algebra $\fg$ is just the standard complex calculating the cohomology of the appropriate type. In this paper we prove the following claim conjectured by V.~Schechtman in ~\cite{s,s1,hs}. It allows one to construct a dg Lie algebra which governs various formal deformations in the non-affine case. {\em Let $X$ be a topological space and let $\fg$ be a sheaf of dg Lie $k$-algebras. Let $\CU=\{U_i\}$ be an locally finite open covering of $X$, $\CG_i=\CC_{\Gamma(U_i,\fg)}$ and let $\CG$ be the groupoid of "descent data" for the collection $\CG_i$ (see~\ref{tot:simpl},{\em2}). Then $\CG$ is naturally equivalent to the groupoid $\CC_L$ where $L$ is a dg Lie algebra representing the Cech complex $\Cech(\CU,\fg)$.} This result implies, in particular, \Cor{generalization} which claims that Theorem 8.3 of~\cite{hdtc} remains valid without the assumption of formal smoothness. \subsection{Structure of the Sections} In Section~\ref{sigma} we define the {\em contents} $\Sigma(\fg)$ of a nilpotent dg Lie algebra $\fg$. It is a Kan simplicial set homotopically equivalent to the Deligne groupoid $\CC(\fg)$. In Section~\ref{tot} we recall the definition of the total space functor in different categories. We also present~\Prop{criterion} giving a sufficient condition for a map of bisimplicial dg algebras to be an acyclic fibration in the sense of~\ref{bisim-alg}. Now the claim~\ref{i1} can be interpreted as the commutativity of $\Sigma$ with the functors $\Tot$. The main \Thm{main} is proven in Section~\ref{thm}. In the last Section~\ref{app} we deduce from~\Thm{main} an application to formal deformation theory. The idea that the generalization of the result of~\cite{hdtc} to the non-smooth case should follow from a descent property for Deligne groupoids belongs to V.~Schechtman. I am very grateful to him for helpful discussions on the subject. \Prop{kan} claiming that the content $\Sigma(\fg)$ and the Deligne groupoid $\CC(\fg)$ are homotopy equivalent is consonant to the Main Homotopy Theorem of Schlessinger--Stasheff, cf.~ \cite{ss}. I am greatly indebted to J.~Stasheff who read the first draft of the manuscript and made some important remarks. \subsection{Notations} Throughout this paper $k$ is a fixed field of characteristic zero. $\art/k$ denotes the category of commutative local artinian $k$-algebras having the residue field $k$. $\dgl(k)$ (resp., $\dgc(k)$) is the category of non-negatively graded dg Lie (resp., commutative) algebras over $k$. $\Delta$ is the category of ordered sets $[n]=\{0,\ldots,n\},\ n\geq 0$ and monotone maps; $\simpl$ is the category of simplicial sets; $\Delta^n\in\simpl$ are the standard $n$-simplices. $\Kan\subseteq\simpl$ is the full subcategory of Kan simplicial sets. $\Ab$ is the category of abelian groups. $C(\CA)$ (sometimes $C(R)$) is the category of complexes over an abelian category $\CA$ (over the category of $R$-modules). $C^{\geq 0}$ denotes the full subcategory of non-negatively graded complexes. \section{Contents of a nilpotent Lie dg algebra} \label{sigma} \subsection{} For any $n\geq 0$ denote by $\Omega_n$ the $k$-algebra of polynomial differential forms on the standard $n$-simplex $\Delta^n$ --- see~\cite{bg}. One has $$\Omega_n=k[t_0,\ldots,t_n,dt_0,\ldots,dt_n]/(\sum t_i-1,\sum dt_i).$$ The algebras $\Omega_n$ form a simplicial commutative dg algebra: a map $u:[p]\to[q]$ induces the map $\Omega(u):\Omega_q\to\Omega_p$ defined by the formula $\Omega(u)(t_i)=\sum_{u(j)=i}t_j$. If $\fg$ is a dg Lie $k$-algebra and $A$ is a commutative dg $k$-algebra then the tensor product $A\otimes\fg$ is also a dg Lie $k$-algebra. Thus, any dg Lie algebra $\fg$ gives rise to a simplicial dg Lie algebra $$\fg_{\bullet}=\{\fg_n=\Omega_n\otimes\fg\}_{n\geq 0}.$$ For any dg Lie algebra $\fg$ denote by $\MC(\fg)$ the set of elements $x\in\fg^1$ satisfying the Maurer-Cartan equation: $$ dx+\frac{1}{2}[x,x]=0.$$ \subsubsection{} \begin{defn}{contents} Let $\fg\in\dgl(k)$ be nilpotent. Its {\em contents} $\Sigma(\fg)\in\simpl$ is defined as $$\Sigma(\fg)=\MC(\fg_{\bullet}).$$ \end{defn} \subsubsection{} Recall (see~\cite{bg}) that the collection of commutative dg algebras $\Omega_n$ defines a contravariant functor $$ \Omega:\simpl\to\dgc(k)$$ so that $\Omega(\Delta^n)=\Omega_n$ and $\Omega$ carries direct limits in $\simpl$ to inverse limits. \begin{lem}{repr} Let $S\in\simpl$. There is a natural map $$\MC(\Omega(S)\otimes\fg)\to\Hom(S,\Sigma(\fg))$$ which is bijective provided $S$ is finite (i.e., has a finite number of non-degenerate simplices). \end{lem} \begin{pf} This is because tensoring by $\fg$ commutes with finite limits --- compare to~\cite{bg}, 5.2. \end{pf} \subsubsection{} \begin{defn}{AF} A map $f:\fg\to\fh$ of nilpotent algebras in $\dgl(k)$ will be called an {\em acyclic fibration} if it is surjective and induces a quasi-isomorphism of the corresponding lower central series. \end{defn} \subsubsection{} \begin{lem}{AFsur} Let $f:\fg\to\fh$ be an acyclic fibration of nilpotent dg Lie algebras in $\dgl(k)$. Then the induced map $\MC(f):\MC(\fg)\to\MC(\fh)$ is surjective. \end{lem} \begin{pf} Induction by the nilpotence degree of $\fg$ --- similarly to~\cite{gm1},~Th. 2.4. \end{pf} \subsubsection{} \begin{lem}{A+L}Let $f:A\to B$ be a surjective map in $\dgc(k)$ and $g:\fg\to\fh$ be a surjective map in $\dgl(k)$. Then the map $$ A\otimes\fg\to (A\otimes\fh)\times_{(B\otimes\fh)}(B\otimes\fg)$$ is an acyclic fibration provided either (a) $f$ is quasi-isomorphism or (b) $g$ is acyclic fibration. \end{lem} \begin{pf} Since for any commutative dg algebra $A$ (with 1) the functor $A\otimes\_$ transforms the lower central series of $\fg$ into the lower central series of $A\otimes\fg$, it suffices to check that the above map is a surjective quasi-isomorphism. This is fairly standard. \end{pf} \subsubsection{} \begin{prop}{AFaf} Let $f$ be as above. Then $\Sigma(f):\Sigma(\fg)\to\Sigma(\fh)$ is an acyclic fibration of simplicial sets. \end{prop} \begin{pf} Lemmas~\ref{repr} and~\ref{AFsur} reduce the question to the following. Let $K\to L$ be an injective map of simplicial sets. Then one has to show that the induced map of nilpotent Lie algebras $$ \Omega(L)\otimes\fg\to\Omega(L)\otimes\fh\times_{\Omega(K)\otimes\fh} \Omega(K)\otimes\fg$$ is an acyclic fibration. This follows from~\ref{A+L}(b). \end{pf} \subsection{Deligne groupoid} \label{del-gr} Recall (cf.~\cite{gm1}) that for a nilpotent dg Lie algebra $\fg\in\dgl(k)$ defines the {\em Deligne groupoid} $\CC(\fg)$ as follows. The Lie algebra $\fg^0$ acts on $\MC(\fg)$ by vector fields: $$ \rho(y)(x)=dy+[x,y]\text{ for }y\in\fg^0, x\in\fg^1.$$ This defines the action of the nilpotent group $G=\exp(\fg^0)$ on the set $\MC(\fg)$. Then the groupoid $\CC=\CC(\fg)$ is defined by the formulas $$ \Ob\CC=\MC(\fg)$$ $$ \Hom_{\CC}(x,x')=\{g\in G|x'=g(x)\}.$$ \subsubsection{} \begin{lem}{exp-1} Let $\fg\in\dgl(k)$ be nilpotent. The natural map $\fg\to\fg_n=\Omega_n\otimes\fg$ induces an equivalence of groupoids $\CC(\fg)\to\CC(\fg_n)$. \end{lem} \begin{pf} It suffices to check the claim when $n=1$. In this case an element $z=x+dt\cdot y\in\fg_{1}=\fg[t,dt]$ with $x\in\fg^1[t],y\in\fg^0[t]$ satisfies MC iff $x(0)\in\MC(\fg)$ and $x$ satisfies the differential equation $$ \dot{x}=dy+[x,y].$$ This means precisely that $x=g(x(0))$ where $g\in G_1=\exp(\fg_1)$ is given by the differential equation $$ \dot{g}=g(y)$$ with the initial condition $g(0)=1$. \end{pf} \subsubsection{Explicit description of $\Sigma(\fg)$} The notations are as in~\ref{del-gr}. For any $n\geq 0$ let $G_n=\exp(\fg^0_n)$ be the group of polynomial maps from the standard $n$-simplex $\Delta_n$ to the group $G$. The collection $G_{\bullet}=\{G_n\}$ forms a simplicial group. Right multiplication defines on $G_{\bullet}$ a right $G$-action. \begin{prop}{explicit} There is a natural bijection $$G_{\bullet}\times^G\MC(\fg)\to\Sigma(\fg)$$ of simplicial sets. Here, as usual, $X\times^GY$ is the quotient of the cartesian product $X\times Y$ by the relation $$(xg,y)\sim (x,gy)\text{ for } x\in X,\ y\in Y,\ g\in G.$$ \end{prop} \begin{pf} This immediately follows from~\Lem{exp-1}. \end{pf} Let $N\CC(\fg)\in\simpl$ be the nerve of $\CC(\fg)$. Define the map $\tau:\Sigma(\fg)\to N\CC(\fg)$ by the formula $$\tau(g,x)=(g_0(x),g_1g_0^{-1},\ldots,g_ng_{n-1}^{-1})$$ where $g\in G_n,x\in\MC(\fg), g_i=v_i(g)$ with $v_i$ being the $i$-th vertex. \Prop{explicit} implies the following \subsubsection{} \begin{prop}{kan} $\Sigma(\fg)$ is a Kan simplicial set. The map $\tau$ is an acyclic fibration identifying $\CC(\fg)$ with the Poincar\'{e} groupoid of $\Sigma(\fg)$. More generally, if $f:\fg\to\fh$ is a surjective map of nilpotent dg Lie algebras, then the induced map $$ \Sigma(\fg)\to\CC(\fg)\times_{\CC(\fh)}\Sigma(\fh)$$ is an acyclic fibration. \end{prop} \begin{pf} The question reduces to the following. Given a pair of polynomial maps $\alpha:\partial\Delta^n\to\fg^0,\ \beta:\Delta^n\to\fh^0$ satisfying $f\alpha=\beta|_{\partial\Delta^n}$ find a map $\gamma:\Delta^n\to\fg^0$ such that $\alpha=\gamma|_{\partial\Delta^n},\ \beta=f\gamma$. This is always possible sincec the canonical map $\Omega_n\to\Omega(\partial\Delta^n)$ is surjective. \end{pf} \subsubsection{} \begin{rem}{} If one does not require $\fg$ to be non-negatively graded, its contents is still a Kan simplicial set. In this case it can probably be considered as a generalization of the notion of Deligne groupoid. \end{rem} \section{"Total space" functor and $\CM$-simplicial sets} \label{tot} \subsection{Generalities} \subsubsection{The category $\CM$} \label{catM} Here and below $\CM$ denotes the following category of morphisms of $\Delta$: The objects of $\CM$ are morphisms $[p]\to [q]$ in $\Delta$. A morphism from $[p]\to [q]$ to $[p']\to [q']$ is given by a commutative diagram $$ \begin{array}{ccc} [p] & {\lra} & [q] \\ {{\scriptstyle\alpha}\uparrow} & & {{\scriptstyle\beta}\downarrow} \\ {[p']} & {\lra} & {[q']} \\ \end{array} $$ The morphism in $\CM$ corresponding to $\alpha=\id,\beta=\sigma^i$, is denoted by $\sigma^i$; the one corresponding to $\alpha=\id,\beta=\partial^i$, is denoted by $\partial^i$. "Dually", the morphism corresponding to $\alpha=\partial^i,\beta=\id$, is denoted by $d_i$ and the one with $\alpha=\sigma^i,\beta=\id$, is denoted by $s_i$. \subsubsection{Total space} Let $\CC$ be a simplicial category having inverse limits and functorial {\em function objects} $\uhom(S,X)\in\CC$ for $X\in\CC,\ S\in\simpl$ --- see~\cite{bk}, IX.4.5 and the examples below. The total space $\Tot(X)$ of a cosimplicial object $X\in\Delta\CC$ is defined by the formula $$ \Tot(X)=\invlim_{\phi\in\CM}\uhom(\Delta^p,X^q)$$ where $\phi:[p]\to [q]$. \subsection{Examples} We will use three instances of the described construction. \subsubsection{} \label{tot:simpl} Let $\CC=\simpl$. Then the above definition coincides with the standard one given in~{\em loc. cit.}, XI.3. Let $G\in\Delta\Grp$ be a (strict) cosimplicial groupoid. We will consider $\Grp$ as a full subcategory of $\simpl$, so a simplicial set $\Tot(G)$ is defined. \begin{lem}{tot-gr} $T=\Tot(G)$ is a groupoid. The objects of $T$ are collections $\{a\in\Ob G^0, \theta:\partial^1(a)\overset{\sim}{\ra}\partial^0(a)\}$ with $\theta$ satisfying the cocycle condition: $$ \sigma^0(\theta)=\id(a); \ \partial^1\theta=\partial^0\theta\circ\partial^2\theta.$$ A morphism in $T$ from $\{a,\theta\}$ to $\{b,\theta'\}$ is a morphism $a\to b$ compatible with $\theta,\theta'$. Thus, $\Tot(G)$ is ``the groupoid of descent data'' for $G$. \end{lem} \subsubsection{} \label{tot:compl} Let $\CC=C(k)$ be the category of complexes over $k$. For $S\in\simpl$ and $X\in C(k)$ the complex $\uhom(S,X)$ is defined to be $\Hom(C_{\bullet}(S),X)$ where $C_{\bullet}$ is the complex of normailized chains of $S$ with coefficients in $k$. The above definition of the functor $\Tot$ coincides with the standard one. \subsubsection{} \label{tot:lie} Let $\CC=\dgl(k)$. For $S\in\simpl$ and $\fg\in\dgl(k)$ define $\uhom(S,\fg)=\Omega(S)\otimes\fg$. Then the functor $\Tot:\Delta\dgl(k)\to \dgl(k)$ coincides with the Thom-Sullivan functor described in~\cite{hdtc}, 5.2.4. The De Rham theorem (see, e.g., {\em loc. cit.}, 5.2.8) shows that the functor $\Tot$ commutes up to homotopy with the forgetful functor $ \#:\dgl(k)\to C(k).$ \subsection{$\CM$-simplicial sets} In the sequel functors $X:\CM\to\simpl$ will be called $\CM$-simplicial sets. We now wish to find a sufficient condition for a map $f:X\to Y$ of $\CM$-simplicial sets to induce an acyclic fibration $\invlim f$ of the inverse limits. This will be an important technical tool to prove the main theorem~\ref{main}. \subsubsection{Matching spaces} Fix $n\in\Bbb N$. Let $\partial^i:[n-1]\to[n]$, $i=0,\ldots,n$, be the standard face maps and let $\sigma^i:[n]\to[n-1]$, $i=0,\ldots,n-1$, be the standard degeneracies. The $n$-th matching space of a $\CM$-simplicial set $X$ is a simplicial subset $\mu_n(X)$ of the product $$\prod_{i=0}^nX(\partial^i)\times\prod_{i=0}^{n-1}X(\sigma^i)$$ consisting of the collections $\left(x_i\in X(\partial^i),y^i\in X(\sigma^i)\right) $ satisfying the following three conditions: ($d$): $d_ix_j=d_{j-1}x_i\text{ for } i<j$. ($\sigma$): $\sigma^jy^i=\sigma^iy^{j+1}\text{ for } i\leq j$. ($d\sigma$): $\sigma^jx_i=d_iy^j\text{ for all } i,j$. One has a canonical map $X(\id_n)\to\mu_n(X)$ which sends an element $x\in X(\id_n)$ to the collection $(d_0x,\ldots,d_nx,\sigma^0x,\ldots,\sigma^{n-1}x)$. \subsubsection{} \begin{defn}{Maf} A map $f:X\to Y$ of $\CM$-simplicial sets is called an {\em acyclic fibration} if for any $n\in\Bbb N$ the commutative square $$ \begin{array}{ccc} X(\id_n) & {\lra} & Y(\id_n) \\ {\downarrow} & & {\downarrow} \\ \mu_n(X) & {\lra} & \mu_n(Y) \\ \end{array} $$ defines an acyclic fibration $$X(\id_n)\to Y(\id_n)\times_{\mu_n(Y)}\mu_n(X).$$ \end{defn} \subsubsection{} \begin{lem}{afM->ss} Let $f:X\to Y$ be an acyclic fibration of $\CM$-simplicial sets. Then the induced map of the corresponding inverse limits, $\invlim f$, is an acyclic fibration. \end{lem} \begin{pf} Let $\CM_{\leq n}$ be the full subcategory of $\CM$ consisting of morphisms $\alpha:[p]\to[q]$ with $p\leq n,q\leq n$. Put $X(n)=\invlim X|_{\CM_{\leq n}}$. Then it is easy to see that $X(n)=X(\id_n)\times_{\mu_n(X)}X(n-1)$. This immediately proves the lemma. \end{pf} \subsection{Bisimplicial algebras} \label{bisim-alg} Now we will present a source of various $\CM$-simplicial sets in this paper. \subsubsection{} Fix a cosimplicial nilpotent dg Lie algebra $\fg$. Any bisimplicial commutative dg algebra $A\in(\Delta^0)^2\dgc$ defines a $\CM$-simplicial set $\Sigma(A,\fg)$ as follows: For $a:[p]\to[q], n\in{\Bbb N}$ one has $$\Sigma(A,\fg)(a)_n=\MC(A_{np}\otimes\fg^q).$$ Define also a simplicial set $\sigma(A,\fg)$ to be the inverse limit of $\Sigma(A,\fg)$ as a functor from $\CM$ to $\simpl$. We provide now a sufficient condition for a map $f:A\to B$ of bisimplicial commutative dg algebras to induce an acyclic fibration $\Sigma(f,\fg)$ for any cosimplicial dg Lie algebra. According to~\Lem{afM->ss} this implies that $\sigma(f,\fg)$ is also an acyclic fibration. \subsubsection{} Any bisimplicial abelian group $A$ gives rise to a functor $$A:\simpl\times\simpl\lra\Ab$$ which is uniquely described by the following properties: --- $A(\Delta^m,\Delta^n)=A_{mn}$ --- $A$ carries direct limits over each one of the arguments to inverse limit. We will identify bisimplicial abelian groups with the functors they define. \subsubsection{} \begin{defn}{bi-match} The matching space $M_{mn}(A)$ of a bisimplicial abelian group $A$ is defined to be $$ M_{mn}(A)=A(\partial\Delta^m,\Delta^n)\times _{A(\partial\Delta^m,\partial\Delta^n)} A(\Delta^m,\partial\Delta^n)$$ where $\partial\Delta^n$ is the boundary of the $n$-simplex. One has a canonical map $A_{mn}\to M_{mn}(A).$ \end{defn} \subsubsection{} \begin{defn}{bi-af} A map $f:A\to B$ in $(\Delta^0)^2C(\Bbb Z)$ is called an {\em acyclic fibration} if for any $m,n$ the canonical map $$ A_{mn}\to B_{mn}\times_{M_{mn}(B)} M_{mn}(A)$$ is a surjective quasi-isomorphism. \end{defn} \subsubsection{} \begin{prop}{bi-af->af}Let $f:A\to B$ in $(\Delta^0)^2\dgc(k)$ be an acyclic fibration. Let $\fg$ be a cosimplicial nilpotent dg Lie algebra. Then the induced map $\Sigma(f,\fg)$ is an acyclic fibration of $\CM$-simplicial sets. \end{prop} \begin{pf} This is a direct calculation using~\ref{AFsur},~\ref{A+L}(a). Here we use that the natural map from $\fg^{n+1}$ to the $n$-th matching space $M^n(\fg)$ (see~\cite{bk}, X.5) is surjective. \end{pf} Now we wish to formulate a sufficient condition for $f:A\to B$ to be an acyclic fibration of bisimplicial dg algebras. The following trivial lemma will be useful. \subsubsection{} \begin{lem}{trivial} Let in a commutative square $$ \begin{array}{ccc} A & \overset{f}{\lra} & B \\ {{\scriptstyle g}\downarrow} & & {{\scriptstyle h}\downarrow} \\ C & {\lra} & D \\ \end{array} $$ of abelian groups the map $g$ and the map $\Ker(g)\to\Ker(h)$ be surjective. Then the induced map $A\to B\times_DC$ is also surjective. \end{lem} We start with a simplicial case. Recall that a simplicial abelian group $A\in\Delta^0\Ab$ defines a functor $A:\simpl\to\Ab$ by the formula $$ A(S)=\Hom(S,A).$$ \subsubsection{} \begin{lem}{simpl-af}Let $f:A\to B$ be a map in $\Delta^0C^{\geq 0}(\Bbb Z)=C^{\geq 0}(\Delta^0\Ab)$. Suppose that (a) $f_n:A_n\to B_n$ is a quasi-isomorphism in $C(\Bbb Z)$ (b) for any $S\in\simpl$ the map $f(S):A(S)\to B(S)$ is surjective (c) for any $d\in\Bbb Z$ the $d$-components $A^d$ and $B^d$ are contractible simplicial abelian groups. Then for any injective map $\alpha:S\to T$ the induced map $$ A(T)\to A(S)\times_{B(S)}B(T)$$ is a surjective quasi-isomorphism. \end{lem} \begin{pf} For any $S\in\simpl$ the map $f(S):A(S)\to B(S)$ is quasi-isomorphism --- this follows from~\cite{le}, remark at the end of III.2, applied to~\cite{ha}, Thm.~1.5.1. This immediately implies that for any $\alpha:S\to T$ the induced map $$ A(T)\to A(S)\times_{B(S)}B(T)$$ is a quasi-isomorphism. Let us prove that it is surjective if $\alpha$ is injective. Since $A^d$ are contractible (and Kan), the map $A(\alpha):A(T)\to A(S)$ is surjective. Thus, by~\Lem{trivial}, it suffices to check that the map $$ \Ker A(\alpha)\to\Ker B(\alpha)$$ is surjective. Since the functors $A,B:\simpl\to C(\Bbb Z)$ carry colimits to limits, the map $\Ker A(\alpha)\to\Ker B(\alpha)$ is a direct summand of the map $f(T/S): A(T/S)\to B(T/S)$ which is surjective by (b). \end{pf} Now we are able to prove the following criterion for a map of bisimplicial complexes of abelian groups to be an acyclic fibration. \subsubsection{} \begin{prop}{criterion} Let $f:A\to B$ be a map in $(\Delta^0)^2C^{\geq 0}(\Bbb Z)$ satisfying: (a) for any $S,T\in\simpl$ the map $f(S,T)$ is surjective (b) for any $S\in\simpl,\ p\in\Bbb N$ the map $f(S,\Delta^p): A(S,\Delta^p)\to B(S,\Delta^p)$ is a quasi-isomorphism in $C(\Bbb Z)$. (c) for any $S\in\simpl$ and any $d$ the simplicial abelian groups $A^d(S,\_)$, $B^d(S,\_)$ are contractible. (d) for any $p,d$ the simplicial abelian group $A^d(\_,\Delta^p)$ is contractible. Then $f$ is an acyclic fibration. \end{prop} \begin{pf}Apply~\Lem{simpl-af} to the map $f(S,\_):A(S,\_)\to B(S\_)$. We immediately get that for any $S\in\simpl$ and any injective map $\alpha:T\to T'$ the induced map $$ A(S,T')\to A(S,T)\times_{B(S,T)}B(S,T')$$ is a surjective quasi-isomorphism. Put $T'=\Delta^p,\ T=\partial\Delta^p.$ The map $A_{np}\to B_{np}\times_{M_{np}(B)}M_{np}(A)$ is then automatically quasi-isomorphism, an we have only to check it is surjective. According to (d) the map $ A(S',T')\to A(S,T')$ is surjective for $T'=\Delta^p$. Define $$X(T)=\Ker(A(S',T)\to A(S,T)),\ Y(T)=\Ker(B(S',T)\to B(S,T)).$$ The groups $X(T)$ and $Y(T)$ are direct summands of $A(S'/S,T)$ and of $B(S'/S,T)$ respectively. Hence the map $X\to Y$ satisfies the hypotheses of~\Lem{simpl-af}. Therefore the map $$X(T')\to X(T)\times_{Y(T)}Y(T')$$ is surjective by~\Lem{trivial}. Proposition is proven. \end{pf} \section{The main theorem} \label{thm} Now we came to the main result of the paper. Let $\fg\in\Delta\dgl(k)$ be a cosimplicial nilpotent dg Lie $k$-algebra. Suppose that $\fg$ is {\em finitely dimensional in the cosimplicial sense}, i.e. that the normalization $$N^n(\fg)=\{x\in\fg^n|\sigma^i(x)=0\text{ for all } i\}$$ vanishes for sufficiently big $n$. \subsection{} \begin{thm}{main} There is a natural homotopy equivalence $$\Sigma(\Tot(\fg))\lra \Tot(\Sigma({\fg}))$$ in $\Kan$. \end{thm} Taking into account \ref{kan}, we easily get \subsubsection{} \begin{cor}{cor(main)} Let $\fg\in\Delta\dgl(k)$ be a nilpotent cosimplicial dg Lie $k$-algebra. Suppose that $\fg$ is finitely dimensional in the cosimplicial sense. Then there is a natural equivalence of groupoids $$ \CC(\Tot(\fg))\ra\Tot(\CC(\fg)).$$ \end{cor} \begin{pf} One has to check that the functor $\Tot$ carries the map $\tau:\Sigma(\fg)\to\CC(\fg)$ to a homotopy equivalence. By~\cite{bk},~X.5, it suffices to check $\tau$ is a fibration in sense of {\em loc. cit.} This follows from (the second claim of)~\Prop{kan} since for any $n$ the natural map from $\fg^{n+1}$ to the $n$-th matching space $M^n(\fg)$ (in notations of {\em loc. cit}) is surjective. \end{pf} \subsection{Proof of the theorem} The set of $n$-simplices of the left-hand side is $$ \MC(\Omega_n\otimes\invlim \Omega_p\otimes\fg^q)$$ and for the right-hand side: $$ \invlim\MC(\Omega(\Delta^n\times\Delta^p)\otimes\fg^q).$$ Here the inverse limits are taken over the category $\CM$ defined in~\ref{catM} and~\Lem{repr} is used to get the second formula. Taking into account that the functor $\MC$ commutes with the inverse limits, a canonical map $\Sigma\circ{\Tot(\fg)}\to\Tot\circ\Sigma({\fg})$ is defined by the composition $$\Omega_n\otimes\invlim \Omega_p\otimes\fg^q\to \invlim \Omega_n\otimes\Omega_p\otimes\fg^q\to \invlim\Omega(\Delta^n\times\Delta^p)\otimes\fg^q$$ the latter arrow being induced by the canonical projections of $\Delta^n\times\Delta^p$ to $\Delta^n$ and to $\Delta^p$. We wish to prove that the map described induces a homotopy equivalence. Since $\fg$ is finitely dimensional, the first map in the composition is bijective --- see~\cite{hdtc}, Thm. 6.11. In order to prove that the second map is a homotopy equivalence, let us fix $n$ and $p$ and present the map $$\alpha:\Omega_n\otimes\Omega_p\to\Omega(\Delta^n\times\Delta^p)$$ as the composition $$\Omega_n\otimes\Omega_p\overset{\beta}{\lra} \Omega(\Delta^n\times\Delta^{p+1})\otimes\Omega_p\overset{\pi}{\lra} \Omega(\Delta^n\times\Delta^p)$$ where $\beta$ is induced by the projection $\Delta^n\times\Delta^{p+1}\to\Delta^n$ and $\pi$ by the pair of maps $\id\times\partial^{p+1}:\Delta^n\times \Delta^p\to\Delta^n\times\Delta^{p+1}$, $\pr_2:\Delta^n\times\Delta^p\to\Delta^p$. We will check below that the maps $\beta$ and $\pi$ induce homotopy equivalences for different reasons: $\beta$ induces a strong deformation retract and $\pi$ induces an acyclic (Kan) fibration. This will prove the theorem. \subsubsection{Notations} Define $\Delta^{+1}$ to be the cosimplicial simplicial set with $(\Delta^{+1})^n=\Delta^{n+1}$ whose cofaces and codegeneracies are the standard maps between the standard simplices (they all preserve the final vertex). Put $A_{np}=\Omega_n\otimes\Omega_p$, $B_{np}=\Omega(\Delta^n\times \Delta^p)$, $C_{np}=\Omega(\Delta^n\times\Delta^{p+1})\otimes\Omega_p$. $A$ and $B$ are bisimplicial commutative dg algebras by an obvious reason. Bisimplicial structure on $C$ in defined by the cosimplicial structure on $\Delta^{+1}$. Our aim is to prove that the maps $\beta:A\to C$ and $\pi:C\to B$ induce homotopy equivalences $\sigma(\beta,\fg)$ and $\sigma(\pi,\fg)$. We will check immediately that $\sigma(\beta,\fg)$ is a strong deformation retract. Afterwards, using the criterion~\ref{criterion} we will get that $\sigma(\pi,\fg)$ is an acyclic fibration. \subsubsection{Checking $\beta$} For $S\in\simpl$ and nilpotent $\fg\in\dgl(k)$ denote by $\Sigma^S(\fg)$ (or just $\Sigma^S$ when $\fg$ is one and the same) the simplicial set whose set of $n$-simplicies is $\MC(\Omega(\Delta^n\times S)\otimes\fg)$. Any map $f:K\times S\to T$ in $\simpl$ induces a map $$\Sigma^f:K\times \Sigma^T\to \Sigma^S$$ as follows. Let $k\in K_n,\ x\in\Sigma^T_n=\MC(\Omega(\Delta^n\times T)\otimes\fg)$. Denote by $F_k$ the composition $$ \Delta^n\times S\overset{\diag\times 1}{\lra} \Delta^n\times\Delta^n\times S\overset{1\times k\times 1}{\lra} \Delta^n\times K\times S\overset{1\times f}{\lra} \Delta^n\times T.$$ Then $\Sigma^f(k,x)$ is defined to be $(\Omega(F_k)\otimes\id_{\fg})(x)$. Define the map $\Phi_p:\Delta^1\times \Delta^{p+1}\to\Delta^{p+1}$ as the one given on the level of posets by the formula $$ \Phi_p(i,j)=\begin{cases} j \text{ if } i=0\\ p+1\text{ if } i=1 \end{cases} $$ The maps $\Phi_p$ induce the maps $\Sigma^{\Phi_p}(\Omega_p\otimes\fg^q)$ which define for any $a:[p]\to [q]$ in $\CM$ the simplicial set $\Sigma(A,\fg)(a)=\MC(A_{\bullet p}\otimes\fg^q)$ as a strong deformation retract of $\Sigma(C,\fg)(a)$. The retractions $\Sigma^{\Phi_p}(\Omega_p\otimes\fg^q)$ are functorial in $a\in\CM$, therefore the inverse image map $\sigma(\beta,\fg)$ is also strong deformation retract. \subsubsection{Checking $\pi$} Now we will prove that the map $\pi:C\to B$ of bisimplicial dg algebras defined above induces an acyclic fibration $\sigma(\pi,\fg)$. Let us check the hypotheses of~\ref{criterion}. First of all, let us check that $\pi(S,T)$ is surjective. For this consider $D_{np}=\Omega(\Delta^n,\Delta^{p+1})$ and the map $\rho_{np}:D_{np}\to B_{np}$ which is the composition of $\pi$ with the natural embedding $D_{np}\to C_{np}=D_{np}\otimes\Omega_p$. Surely, it suffices to prove that $\rho(S,T)$ is surjective. For this we note that $D(S,T)=\Omega(S\times\cone{T})$ where $\cone{\ }:\simpl\to\simpl$ is the functor satisfying the condition $\cone{\ }|_{\Delta}=\Delta^{+1}$ and preserving colimits (see~\cite{gz},II.1.3). Then, since the map $S\times T\to S\times\cone{T}$ is injective, the map $$\Omega(S\times\cone{T})\to \Omega(S\times T)$$ is surjective. Next, one has $$C(S,\Delta^p)=\Omega(S\times\Delta^{p+1})\otimes\Omega_p,$$ $$B(S,\Delta^p)=\Omega (S\times\Delta^p)$$ so the condition~\ref{criterion} (b) is fulfilled. The simplicial abelian groups $C^d(S,\_)$ are contractible by the K\"unneth formula. The abelian groups $B^d(S,\_)=\Omega^d(S\times\_)$ are contractible since any injective map $T\to T'$ induces a surjection $\Omega(S\times T')\to\Omega(S\times T)$. The same reason proves the condition (d). Therefore, the map $\pi:C\to B$ of bisimplicial commutative dg algebras is an acyclic fibration by~\Prop{criterion} and then by~\Prop{bi-af->af} the map $\sigma(\pi,\fg)$ is an acyclic fibration in $\simpl$. The Theorem is proven. \section{Application to Deformation theory} \label{app} In this Section we describe how to deduce from~\Cor{cor(main)} the description of universal formal deformations for some typical deformation problems. Consider, for example, three deformation problems which have been studied in~\cite{hdtc}. Let $X$ be a smooth separated scheme $X$ over a field $k$ of characteristic $0$, $G$ an algebraic group over $k$ and $p: P\lra X$ a $G$-torsor over $X$. Consider the following deformation problems. {\bf Problem 1.} Flat deformations of $X$. {\bf Problem 2.} Flat deformations of the pair $(X,P)$. {\bf Problem 3.} Deformations of $P$ ($X$ being fixed). To each problem one can assign a sheaf of $k$-Lie algebras $\fg_i$ on $X$ (these are the sheaves ${\cal A}_i,\ i=1,2,3$ from {\em loc. cit.}, Section 8). According to Grothendieck, to each problem corresponds a (2-)functor of infinitesimal deformations $$ F_i:\art/k\lra\Grp $$ from the category of local artinian $k$-algebras with the residue field $k$ to the (2-)category of groupoids. In each case, $\fg_i$ is "a sheaf of infinitesimal automorphisms" corresponding to $F_i$ (in the sense of ~\cite{sga1}, Exp.III, 5, especially Cor. 5.2 for Problem 1; for the other problems the meaning is analogous). If $X$ is affine, the functor $F_i$ is equivalent to the Deligne groupoid $\CC_L,\ L=\Gamma(X,\fg_i)$ defined as the functor $$ \CC_L:\art/k\to\Grp$$ which is given by the formula $$\CC_L(A)=\CC(\fm\otimes L)$$ where $\fm$ is the maximal ideal of $A$. The deformation functor $F_i$ defines a stack $\CF_i$ in the Zariski topology of $X$. The Deligne functor defines a fibered category $\CC_{\fg_i}$ which assigns a groupoid $\CC_{\Gamma(U,\fg_i)}(A)$ to each Zariski open set $U$ and to each $A\in\art/k$. We have a canonical map of fibered categories $ \CC_{\fg_i}\to\CF_i$ so that $\CF_i$ is equivalent to the stack associated to the fibered category $\CC_{\fg_i}$. Using~\Cor{cor(main)} we immediately get \subsection{} \begin{cor}{} The deformation groupoid $F_i$ is naturally equivalent to the Deligne groupoid associated with the dg Lie algebra $\Right\Gamma^{\Lie}(X,\fg_i)$. \end{cor} In particular, the following generalization of~\cite{hdtc}, Thm.~8.3 takes place. \subsection{} \begin{cor}{generalization} Suppose that $H^0(X,\fg_i)=0$; let $\fS=\Spf(R)$ be the base of the universal formal deformation for Problem $i$. Then we have a canonical isomorphism $$ R^*=H^{Lie}_0(\Right\Gamma^{Lie}(X,\fg_i)) $$ where $R^*$ denotes the space of continuous $k$-linear maps $R\lra k$ ($k$ considered in the discrete topology) and $H^{\Lie}_0$ denotes the $0$-th Lie homology. \end{cor} Recall that this result has been proven in~{\em loc. cit.} for $\fS$ formally smooth.

9606

alg-geom/9606006

en

https://arxiv.org/abs/alg-geom/9606006

alg-geom/9606006

Dmitri O. Orlov

Dmitri Orlov

Equivalences of derived categories and K3 surfaces

28 pages, LaTeX file

J. Math. Sci. (New York) 84 (1997), no. 5, 1361--1381

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.

alg-geom

\section*{Introduction} Let $\db{X}$ be the bounded derived category of coherent sheaves on a smooth projective variety $X.$ The category $\db{X}$ has the structure of a triangulated category (see \cite{Ver}, \cite{GM}). We shall consider $\db{X}$ as a triangulated category. In this paper we are concerned with the problem of description for varieties, which have equivalent derived categories of coherent sheaves. In the paper \cite{Mu1}, Mukai showed that for an abelian variety $A$ and its dual $\hat{A}$ the derived categories $\db{A}$ and $\db{\hat{A}}$ are equivalent . Equivalences of another type appeared in \cite{BO}. They are induced by certain birational transformations which are called flops. Further, it was proved in the paper \cite{BOr} that if $X$ is a smooth projective variety with either ample canonical or ample anticanonical sheaf, then any other algebraic variety $X'$ such that $\db{X'}\simeq\db{X}$ is biregularly isomorphic to $X.$ The aim of this paper is to give some description for equivalences between derived categories of coherent sheaves. The main result is Theorem \ref{main} of $\S 2.$ It says that any full and faithful exact functor $F: \db{M}\longrightarrow \db{X}$ having left (or right) adjoint functor can be represented by an object $E\in\db{M\times X},$ i.e. $F(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})\cong R^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}\pi_*(E\stackrel{L}{\otimes}p^*(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})),$ where $\pi$ and $p$ are the projections on $M$ and $X$ respectively. In $\S 3,$ basing on the Mukai's results \cite{Mu}, we show that two K3 surfaces $S_1$ and $S_2$ over field $\Bbb{C}$ have equivalent derived categories of coherent sheaves iff the lattices of transcendental cycles $T_{S_1}$ and $T_{S_2}$ are Hodge isometric. I would like to thank A.~Polishchuk for useful notices. \section{Preliminaries} \refstepcounter{THNO}\par\vspace{1.5ex We collect here some facts relating to triangulated categories. Recall that a triangulated category is an additive category with additional structures: a) {\it an additive autoequivalence $T : {\mathcal D}\longrightarrow {\mathcal D},$ which is called a translation functor} (we usually write $X[n]$ instead of $T^n(X)$ and $f[n]$ instead of $T^n(f)$), b) {\it a class of distinguished triangles:} $$ X\stackrel{u}{\to}Y\stackrel{v}{\to}Z\stackrel{w}{\to}X[1]. $$ And these structures must satisfy the usual set of axioms (see \cite{Ver}). If $X,$ $ Y$ are objects of a triangulated category ${\mathcal D},$ then ${\H i,{\mathcal D}, X, Y}$ means ${\H{},{\mathcal D}, X, {Y[i]}}.$ An additive functor $F : {\mathcal D}\longrightarrow{\mathcal D}'$ between two triangulated categories ${\mathcal D}$ and ${\mathcal D}'$ is called {\sf exact} if a) {\it it commutes with the translation functor, i.e there is fixed an isomorphism of functors:} $$ t_F : F\circ T\stackrel{\sim}{\longrightarrow}T'\circ F, $$ b) {\it it takes every distinguished triangle to a distinguished triangle} (using the isomorphism $t_F,$ we replace $F(X[1])$ by $F(X)[1]$). The following lemma will be needed for the sequel. \th{Lemma}\cite{BK} If a functor $G : {\mathcal D}'\longrightarrow{\mathcal D}$ is a left (or right) adjoint to an exact functor $F : {\mathcal D}\longrightarrow{\mathcal D}'$ then functor $G$ is also exact . \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Since $G$ is the left adjoint functor to $F,$ there exist canonical morphisms of functors $id_{\mathcal D'}\to F\circ G,\; G\circ F\longrightarrow id_{\mathcal D}.$ Let us consider the following sequence of natural morphisms: $$ G\circ T'\longrightarrow G\circ T'\circ F\circ G\stackrel{\sim}{\longrightarrow} G\circ F\circ T\circ G\longrightarrow T\circ G $$ We obtain the natural morphism $G\circ T'\longrightarrow T\circ G.$ This morphism is an isomorphism. Indeed, for any two objects $A\in {\mathcal D}$ and $B\in {\mathcal D}'$ we have isomorphisms : $$ \begin{array}{l} {\h G(B[1]), A}\cong{\h B[1], {F(A)}}\cong{\h B, {F(A)[-1]}}\cong\\\\ {\h B, {F(A[-1])}}\cong{\h G(B), {A[-1]}}\cong{\h G(B)[1], A}\\ \end{array} $$ This implies that the natural morphism $G\circ T'\longrightarrow T\circ G$ is an isomorphism. Let now $A\stackrel{\alpha}{\longrightarrow}B\longrightarrow C\longrightarrow A[1]$ be a distinguished triangle in ${\mathcal D}'.$ We have to show that $G$ takes this triangle to a distinguished one. Let us include the morphism $G(\alpha) : G(A)\to G(B)$ into a distinguished triangle: $$ G(A)\longrightarrow G(B)\longrightarrow Z\longrightarrow G(A)[1]. $$ Applying functor $F$ to it, we obtain a distinguished triangle: $$ FG(A)\longrightarrow FG(B)\longrightarrow F(Z)\longrightarrow FG(A)[1] $$ (we use the commutation isomorphisms like $T'\circ F\stackrel{\sim}{\to} F\circ T$ with no mention). Using morphism $id\to F\circ G,$ we get a commutative diagram: $$ \begin{array}{ccccccc} A&\stackrel{\alpha}{\longrightarrow}&B&\longrightarrow& C&\longrightarrow& A[1]\\ \big\downarrow&&\big\downarrow&&&&\big\downarrow\\ FG(A)&\stackrel{FG(\alpha)}{\longrightarrow}& FG(B)&\longrightarrow& F(Z)&\longrightarrow& FG(A)[1] \end{array} $$ By axioms of triangulated categories there exists a morphism $\mu : C\to F(Z)$ that completes this commutative diagram. Since $G$ is left adjoint to $F,$ the morphism $\mu$ defines $\nu : G(C)\to Z.$ It is clear that $\nu$ makes the following diagram commutative: $$ \begin{array}{ccccccc} G(A)&{\longrightarrow}&G(B)&\longrightarrow&G(C)&\longrightarrow&G(A)[1]\\ \wr\big\downarrow&&\wr\big\downarrow&&\big\downarrow\rlap{\ss{\nu}}&&\wr\Big\downarrow\\ G(A)&\longrightarrow&G(B)&\longrightarrow& Z&\longrightarrow&G(A)[1]\\ \end{array} $$ To prove the lemma, it suffices to show that $\nu$ is an isomorphism. For any object $Y\in{\mathcal D}$ let us consider the diagram for $\mbox{Hom}$: $$ \begin{array}{cccccccc} \to{\h G(A)[1], Y}&\to&{\h Z, Y}&\to&{\h G(B), Y}\to\\ \big\downarrow\wr&&\big\downarrow\rlap{\ss{\mr{H}_Y (\nu)}}&& \big\downarrow\wr\\ \to{\h G(A)[1], Y}&{\to}&{\h G(C), Y}&\to&{\h G(B), Y} \to\\ \big\downarrow\wr&&\big\downarrow\wr&&\big\downarrow\wr\\ \to{\h A[1], {F(Y)}}&\to&{\h C, {F(Y)}}&\to&{\h B, {F(Y)}} \to\\ \end{array} $$ Since the lower sequence is exact, the middle sequence is exact also. By the lemma about five homomorphisms, for any $Y$ the morphism $\mr{H}(\nu)$ is an isomorphism . Thus $\nu : G(C)\to Z$ is an isomorphism too. This concludes the proof. $\Box$ \refstepcounter{THNO}\par\vspace{1.5ex Let $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}} = \{ X^{c}\stackrel{d^{c}}{\to}X^{c+1}\stackrel{d^{c+1}}{\to} \cdots\to X^0\}$ be a bounded complex over a triangulated category ${\mathcal D},$ i.e. all compositions $d^{i+1}\circ d^i$ are equal to $0$ ($c< 0$). A left Postnikov system, attached to $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}},$ is, by definition, a diagram \begin{picture}(400,100) \put(24,75){\vector(1,-2){30}} \put(64,15){\vector(1,2){30}} \put(104,75){\vector(1,-2){30}} \put(144,15){\vector(1,2){30}} \put(184,75){\vector(1,-2){30}} \put(304,15){\vector(1,2){30}} \put(344,75){\vector(1,-2){30}} \put(40,82){\vector(1,0){40}} \put(120,82){\vector(1,0){40}} \put(120,2){\vector(-1,0){40}} \put(200,2){\vector(-1,0){40}} \put(250,2){\vector(-1,0){10}} \put(285,2){\vector(-1,0){10}} \put(360,2){\vector(-1,0){40}} \put(10,80){$X^{c}$} \put(90,80){$X^{c+1}$} \put(170,80){$X^{c+2}$} \put(32,0){$Y^{c}=X^{c}$} \put(130,0){$Y^{c+1}$} \put(210,0){$Y^{c+2}$} \put(330,80){$X^0$} \put(370,0){$Y^0$} \put(290,0){$Y^{-1}$} \put(252,0){$\cdots$} \put(3,32){$i_c=id$} \put(62,43){$ j_c$} \put(100,32){$ i_{c+1}$} \put(134,43){$ j_{c+1}$} \put(178,32){$ i_{c+2}$} \put(302,43){$ j_{-1}$} \put(342,32){$ i_0$} \put(55,85){$ d^{c}$} \put(135,85){$ d^{c+1}$} \put(50,60){$ \circlearrowleft$} \put(90,26){$ \star$} \put(130,60){$ \circlearrowleft$} \put(170,26){$ \star$} \put(330,26){$ \star$} \put(90,7){$ [1]$} \put(170,7){$ [1]$} \put(330,7){$ [1]$} \end{picture} \bigskip \noindent in which all triangles marked with $\star$ are distinguished and triangles marked with $\circlearrowleft$ are commutative (i.e. $j_k\circ i_k = d^{k}$). An object $E\in\mbox{Ob}{\mathcal D}$ is called a left convolution of $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}},$ if there exists a left Postnikov system, attached to $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ such that $E=Y^0.$ The class of all convolutions of $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ will be denoted by $\mbox{Tot} (X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}).$ Clearly the Postnikov systems and convolutions are stable under exact functors between triangulated categories. The class $\mbox{Tot}(X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}})$ may contain many non-isomorphic elements and may be empty. Further we shall give a sufficient condition for $\mbox{Tot}(X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}})$ to be non-empty and for its objects to be isomorphic. The following lemma is needed for the sequel(see \cite{BBD}). \th{Lemma}\label{tr} Let $g$ be a morphism between two objects $Y$ and $Y',$ which are included into two distinguished triangles: $$ \begin{array}{ccccccc} X&\stackrel{u}{\longrightarrow}&Y&\stackrel{v}{\longrightarrow}&Z&\stackrel{w}{\longrightarrow}&X[1]\\ \begin{picture}(4,15)\multiput(2,9)(0,-2){5}{\circle*{1}\rlap{\ss{f}}&&\big\downarrow\rlap{\ss{g}}&&\begin{picture}(4,15)\multiput(2,9)(0,-2){5}{\circle*{1}\rlap{\ss{h}}&&\begin{picture}(4,15)\multiput(2,9)(0,-2){5}{\circle*{1}\rlap{\ss{f[1]}}\\ X'&\stackrel{u'}{\longrightarrow}&Y'&\stackrel{v'}{\longrightarrow}&Z'&\stackrel{w'}{\longrightarrow}&X'[1] \end{array} $$ If $v'gu=0,$ then there exist morphisms $f : X\to X'$ and $h : Z\to Z'$ such that the triple $(f, g, h)$ is a morphism of triangles. If, in addition, ${\h X[1], {Z'}}=0$ then this triple is uniquely determined by $g.$ \par\vspace{1.0ex}\endgroup Now we prove two lemmas which are generalizations of the previous one for Postnikov diagrams. \th{Lemma}\label{pd1} Let $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}} = \{ X^{c}\stackrel{d^{c}}{\to}X^{c+1}\stackrel{d^{c+1}}{\to} \cdots\to X^0\}$ be a bounded complex over a triangulated category ${\mathcal D}.$ Suppose it satisfies the following condition: \begin{equation}\label{ex} {\H i, {}, X^a, {X^b}}=0\; \mbox{ for }\; i<0 \;\mbox{ and }\; a<b. \end{equation} Then there exists a convolution for this complex and all convolutions are isomorphic (noncanonically). If, in addition, \begin{equation}\label{une} {\H i, {}, X^a, {Y^0}}=0 \;\mbox{ for }\; i<0\; \mbox{ and for all }\; a \end{equation} for some convolution $Y^0$ (and, consequently, for any one), then all convolutions are canonically isomorphic. \par\vspace{1.0ex}\endgroup \th{Lemma}\label{pd2} Let $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_1$ and $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_2$ be bounded complexes that satisfy (\ref{ex}), and let $(f_c,...,f_0)$ be a morphism of these complexes: $$ \begin{array}{ccccccc} X^c_1&\stackrel{d^c_1}{\longrightarrow}&X^{c+1}_1&\longrightarrow&\cdots&\longrightarrow&X^0_1\\ \big\downarrow\rlap{\ss{f_c}}&&\big\downarrow\rlap{\ss{f_{c+1}}}&&&&\big\downarrow\rlap{\ss{f_0}}\\ X^c_2&\stackrel{d^c_2}{\longrightarrow}&X^{c+1}_2&\longrightarrow&\cdots&\longrightarrow&X^0_2\\ \end{array} $$ Suppose that \begin{equation}\label{exm} {\H i, {}, X^a_1, {X^b_2}}=0 \;\mbox{ for }\; i<0 \;\mbox{ and }\; a<b. \end{equation} Then for any convolution $Y^0_1$ of $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_1$ and for any convolution $Y^0_2$ of $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_2$ there exists a morphism $f : Y^0_1\to Y^0_2$ that commutes with the morphism $f_0.$ If, in addition, \begin{equation} \label{unm} {\H i, {}, X^a_1, {Y^0_2}}=0 \;\mbox{ for }\; i<0\; \mbox{ and for all }\; a \end{equation} then this morphism is unique. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } We shall prove both lemmas together. Let $Y^{c+1}$ be a cone of the morphism $d^c$: $$ X^c\stackrel{d^c}{\longrightarrow}X^{c+1}\stackrel{\alpha}{\longrightarrow} Y^{c+1}\longrightarrow X^c [1] $$ By assumption $d^{c+1}\circ d^c =0$ and ${\h X^c [1], {X^{c+2}}}=0,$ hence there exists a unique morphism $\bar{d}^{c+1} : Y^{c+1}\to X^{c+2}$ such that $\bar{d}^{c+1}\circ \alpha = d^{c+1}.$ Let us consider a composition $d^{c+2}\circ \bar{d}^{c+1} : Y^{c+1}\to X^{c+3}.$ We know that $d^{c+2}\circ \bar{d}^{c+1}\circ\alpha = d^{c+2}\circ d^{c+1}=0 ,$ and at the same time we have ${\h X^c [1], {X^{c+3}}}=0.$ This implies that the composition $d^{c+2}\circ \bar{d}^{c+1}$ is equal to $0.$ Moreover, consider the distinguished triangle for $Y^{c+1}.$ It can easily be checked that ${\H i, {}, Y^{c+1},{ X^b}}=0$ for $i<0$ and $b>c+1.$ Hence the complex $Y^{c+1}\longrightarrow X^{c+2}\longrightarrow\cdots\longrightarrow X^0$ satisfies the condition (\ref{ex}). By induction, we can suppose that it has a convolution. This implies that the complex $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ has a convolution too. Thus, the class $\mbox{Tot}(X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}})$ is non-empty. Now we shall show that under the conditions (\ref{exm}) any morphism of complexes can be extended to a morphism of Postnikov systems. Let us consider cones $Y^{c+1}_1$ and $Y^{c+1}_2$ of the morphisms $d^c_1$ and $d^c_2.$ There exists a morphism $g_{c+1} : Y^{c+1}_1\to Y^{c+1}_2$ such that one has the morphism of distinguished triangles: $$ \begin{array}{ccccccc} X^c_1&\stackrel{d^c_1}{\longrightarrow}&X^{c+1}_1&\stackrel{\alpha}{\longrightarrow}& Y^{c+1}_1&\longrightarrow&X^c_1 [1]\\ \big\downarrow\rlap{\ss{f_c}}&&\big\downarrow\rlap{\ss{f_{c+1}}}&&\big\downarrow\rlap{\ss{g_{c+1}}}& &\big\downarrow\rlap{\ss{f_c [1]}}\\ X^c_2&\stackrel{d^c_2}{\longrightarrow}&X^{c+1}_2&\stackrel{\beta}{\longrightarrow}& Y^{c+1}_2&\longrightarrow&X^c_2 [1]\\ \end{array} $$ As above, there exist uniquely determined morphisms $ \bar{d}^{c+1}_i : Y^{c+1}_i\to X^{c+2}_i$ for $i=1,2.$ Consider the following diagram: $$ \begin{array}{ccc} Y^{c+1}_1&\stackrel{\bar{d}^{c+1}_1}{\longrightarrow}&X^{c+2}_1\\ \big\downarrow\rlap{\ss{g_{c+1}}}&&\big\downarrow\rlap{\ss{f_{c+2}}}\\ Y^{c+1}_2&\stackrel{\bar{d}^{c+1}_2}{\longrightarrow}&X^{c+2}_2 \end{array} $$ Let us show that this square is commutative. Denote by $h$ the difference $f_{c+2}\circ \bar{d}^{c+1}_1 - \bar{d}^{c+1}_2\circ g_{c+1}.$ We have $h\circ \alpha = f_{c+2}\circ {d}^{c+1}_1 - {d}^{c+1}_2\circ f_{c+1} = 0$ and, by assumption, ${\h X^c_1 [1], {X^{c+2}_2}}=0.$ It follows that $h=0.$ Therefore, we obtain the morphism of new complexes: $$ \begin{array}{ccccccc} Y^{c+1}_1&\stackrel{\bar{d}^{c+1}_1}{\longrightarrow}&X^{c+2}_1&\longrightarrow&\cdots&\longrightarrow&X^0_1\\ \big\downarrow\rlap{\ss{g_{c+1}}}&&\big\downarrow\rlap{\ss{f_{c+2}}}&&&&\big\downarrow\rlap{\ss{f_0}}\\ Y^{c+1}_2&\stackrel{\bar{d}^{c+1}_2}{\longrightarrow}&X^{c+2}_2&\longrightarrow&\cdots&\longrightarrow&X^0_2\\ \end{array} $$ It can easily be checked that these complexes satisfy the conditions (\ref{ex}) and (\ref{exm}) of the lemmas. By the induction hypothesis, this morphism can be extended to a morphism of Postnikov systems, attached to these complexes. Hence there exists a morphism of Postnikov systems, attached to $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_1$ and $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_2.$ Moreover, we see that if all morphisms $f_i$ are isomorphisms, then a morphism of Postnikov systems is an isomorphism too. Therefore, under the condition (\ref{ex}) all objects from the class $\mbox{Tot}(X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}})$ are isomorphic. Now let us consider a morphism of the rightmost distinguished triangles of Postnikov systems: $$ \begin{array}{ccccccc} Y^{-1}_1&\stackrel{j_{1, -1}}{\longrightarrow}&X^0_1&\stackrel{i_{1, 0}}{\longrightarrow}& Y^0_1&\longrightarrow&Y^{-1}_1 [1]\\ \big\downarrow\rlap{\ss{g_{-1}}}&&\big\downarrow\rlap{\ss{f_0}}&&\big\downarrow\rlap{\ss{g_0}}& &\big\downarrow\rlap{\ss{g_{-1} [1]}}\\ Y^{-1}_2&\stackrel{j_{2, -1}}{\longrightarrow}&X^0_2&\stackrel{i_{2, 0}}{\longrightarrow}& Y^0_2&\longrightarrow&Y^{-1}_2 [1]\\ \end{array} $$ If the complexes $X^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_i$ satisfy the condition (\ref{unm}) ( i.e.${\H i, {}, X^a_1, {Y^0_2}}=0$ for $ i<0 $ and all $ a$), then we get ${\h Y^{-1}_1 [1], {Y^0_2}}=0.$ It follows from Lemma \ref{tr} that $g_0$ is uniquely determined. This concludes the proof of both lemmas. $\Box$ \section{Equivalences of derived categories} \refstepcounter{THNO}\par\vspace{1.5ex Let $X$ and $M$ be smooth projective varieties over field $k.$ Denote by $\db{X}$ and $\db{M}$ the bounded derived categories of coherent sheaves on $X$ and $M$ respectively. Recall that a derived category has the structure of a triangulated category. For every object $E\in \db{M\times X}$ we can define an exact functor $\Phi_E$ from $\db{M}$ to $\db{X}.$ Denote by $p$ and $\pi$ the projections of ${M\times X}$ onto $M$ and $X$ respectively: $$ \begin{array}{ccc} M\times X&\stackrel{\pi}{\longrightarrow}&X\\ \llap{\ss{p}}\big\downarrow&&\\ M&& \end{array} $$ Then $\Phi_E$ is defined by the following formula: \begin{equation}\label{dfun} \Phi_E(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}):= \pi_*(E\otimes p^*(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})) \end{equation} (we always shall write shortly $f_* , f^*, \otimes$ and etc. instead of $R^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}f_*, L^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}f^*, \stackrel{L}{\otimes},$ because we consider only derived functors). The functor $\Phi_E$ has the left and the right adjoint functors $\Phi_E^*$ and $\Phi_E^!$ respectively, defined by the following formulas: $$ \begin{array}{l} \Phi_E^* (\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}) = p_*(E^{\vee}\otimes \pi^*(\omega_X [dimX]\otimes (\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}))),\\\\ \Phi_E^! (\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}) =\omega_M [dimM]\otimes p_*(E^{\vee}\otimes (\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})), \end{array} $$ where $\omega_X$ and $\omega_M$ are the canonical sheaves on $X$ and $M,$ and $E^{\vee}:= {\pmb R}^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}{\mathcal H}om( E, {\mathcal O}_{M\times X} ).$ Let $F$ be an exact functor from the derived category $\db{M}$ to the derived category $\db{X}.$ Denote by $F^*$ and $F^!$ the left and the right adjoint functors for $F$ respectively, when they exist. Note that if there exists the left adjoint functor $F^*,$ then the right adjoint functor $F^!$ also exists and $$ F^! = S_M\circ F^*\circ S_X^{-1}, $$ where $S_X$ and $S_M$ are Serre functors on $\db{X}$ and $\db{M}.$ They are equal to $(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})\otimes\omega_X [dimX]$ and $(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture})\otimes\omega_M [dimM]$ (see \cite{BK}). What can we say about the category of all exact functors between $\db{M}$ and $\db{X}$? It seems to be true that any functor can be represented by an object on the product $M\times X$ for smooth projective varieties $M$ and $X.$ But we are unable prove it. However, when $F$ is full and faithfull, it can be represented. The main result of this chapter is the following theorem. \th{Theorem}\label{main} Let $F$ be an exact functor from $\db{M}$ to $\db{X},$ where $M$ and $X$ are smooth projective varieties. Suppose $F$ is full and faithful and has the right (and,consequently, the left) adjoint functor. Then there exists an object $E\in\db{M\times X}$ such that $F$ is isomorphic to the functor $\Phi_E$ defined by the rule (\ref{dfun}), and this object is unique up to isomorphism. \par\vspace{1.0ex}\endgroup \refstepcounter{THNO}\par\vspace{1.5ex Let $F$ be an exact functor from a derived category $\db{\mathcal A}$ to a derived category $\db{\mathcal B}.$ We say that $F$ is {\sf bounded} if there exist $z\in {\Bbb{Z}}, n\in \Bbb{N}$ such that for any $A\in{\mathcal A}$ the cohomology objects ${H}^i (F(A))$ are equal to $0$ for $i\not\in [z, z+n].$ \th{Lemma} Let $M$ and $X$ be smooth projective varieties. If an exact functor $F : \db{M}\longrightarrow\db{X}$ has a left adjoint functor then it is bounded. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Let $G : \db{X}\longrightarrow\db{M}$ be a left adjoint functor to $F.$ Take a very ample invertible sheaf ${\mathcal L}$ on $X.$ It gives the embedding $i : X \hookrightarrow {\Bbb{P}}^N.$ For any $i<0$ we have right resolution of the sheaf ${\mathcal O}(i)$ on ${\Bbb{P}}^N$ in terms of the sheaves ${\mathcal O}(j),$ where $j=0, 1,.., N$(see \cite{Be}). It is easily seen that this resolution is of the form $$ {\mathcal O}(i)\stackrel{\sim}{\longrightarrow}\Bigl\{ V_0\otimes{\mathcal O}\longrightarrow V_1\otimes{\mathcal O}(1)\longrightarrow\cdots\longrightarrow V_N\otimes{\mathcal O}(N)\longrightarrow 0 \Bigl\} $$ where all $V_k$ are vector spaces. The restriction of this resolution to $X$ gives us the resolution of the sheaf ${\mathcal L}^{ i}$ in terms of the sheaves ${\mathcal L}^{ j},$ where $j=0, 1,..., N.$ Since the functor $G$ is exact that there exist $z'$ and $n'$ such that ${H}^{k}(G({\mathcal L}^{ i}))$ are equal $0$ for $k\not\in [z', z'+n'].$ This follows from the existence of the spectral sequence $$ E^{p,q}_1 = V_p\otimes {H}^q(G({\mathcal L}^{p})) \Rightarrow {H}^{p+q}(G({\mathcal L}^{i})). $$ As all nonzero terms of this spectral sequence are concentrated in some rectangle, so it follows that for all $i$ cohomologies ${H}^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}(G({\mathcal L}^{i}))$ are concentrated in some segment. Now, notice that if ${\H j, {}, {\mathcal L}^{i}, {F(A)}}= 0$ for all $i\ll0,$ then ${H}^j (F(A))$ is equal to $0.$ Further, by assumption, the functor $G$ is left adjoint to $F,$ hence $$ {\H j, {}, {\mathcal L}^{i}, {F(A)}}\cong {\H j, {}, G({\mathcal L}^{i}), A}.\\ $$ If now $A$ is a sheaf on $M,$ then ${\H j, {}, G({\mathcal L}^{i}), A}=0$ for all $i<0$ and $j\not\in [-z'-n', -z'+ dimM],$ and thus ${H}^j (F(A))=0$ for the same $j.$ $\Box$ \th{Remark}\label{boun} We shall henceforth assume that for any sheaf ${\mathcal F}$ on $M$ the cohomology objects ${H}^i (F({\mathcal F}))$ are nonzero only if $i\in [-a, 0].$ \par\vspace{1.0ex}\endgroup \refstepcounter{THNO}\par\vspace{1.5ex \label{cons} Now we begin constructing an object $E\in\db{M\times X}.$ Firstly, we shall consider a closed embedding $j : M\hookrightarrow{\Bbb{P}}^N$ and shall construct an object $E'\in \db{{\Bbb{P}}^N \times X}.$ Secondly, we shall show that there exists an object $E\in\db{M\times X}$ such that $E'=(j\times id)_* E.$ After that we shall prove that functors $F$ and $\Phi_E$ are isomorphic. Let ${\mathcal L}$ be a very ample invertible sheaf on $M$ such that ${\mr H}^i ({\mathcal L}^{k})=0$ for any $k>0,$ when $i\not=0.$ By $j$ denote the closed embedding $j : M\hookrightarrow{\Bbb{P}}^N$ with respect to ${\mathcal L}.$ Recall that there exists a resolution of the diagonal on the product ${\Bbb{P}}^N \times {\Bbb{P}}^N$ (see\cite{Be}). Let us consider the following complex of sheaves on the product: \begin{equation}\label{di} 0\to {\mathcal O}(-N)\boxtimes\Omega^N (N)\stackrel{d_{-N}}{\to}{\mathcal O}(-N+1)\boxtimes \Omega^{N-1} (N-1)\to\cdots\to{\mathcal O}(-1)\boxtimes\Omega^{1} (1) \stackrel{d_{-1}}{\to}{\mathcal O}\boxtimes{\mathcal O} \end{equation} This complex is a resolution of the structure sheaf ${\mathcal O}_{\Delta}$ of the diagonal $\Delta.$ Now by $F'$ denote the functor from $\db{{\Bbb{P}}^N}$ to $\db{X},$ which is the composition $F\circ j^*.$ Consider the product $$ \begin{array}{ccc} {\Bbb{P}}^N \times X&\stackrel{\pi^{'}}{\longrightarrow}&X\\ \llap{\ss{q}}\big\downarrow&&\\ {\Bbb{P}}^N \end{array} $$ Denote by $$ d'_{-i}\in{\H {}, {{\Bbb{P}}^N \times X}, {\mathcal O}(-i)\boxtimes F'(\Omega^i (i)), {{\mathcal O}(-i+1)\boxtimes F'(\Omega^{i-1} (i-1))}} $$ the image $d_{-i}$ under the following through map. $$ \begin{array}{l} {\h {\mathcal O}(-i)\boxtimes \Omega^i (i), {{\mathcal O}(-i+1)\boxtimes \Omega^{i-1} (i-1)}}\stackrel{\sim}{\longrightarrow}\\\\ {\h {\mathcal O}\boxtimes \Omega^i (i), {{\mathcal O}(1)\boxtimes \Omega^{i-1} (i-1)}}\stackrel{\sim}{\longrightarrow}\\\\ {\h \Omega^i (i), {{\mr H}^0 ({\mathcal O}(1))\otimes \Omega^{i-1} (i-1)}}\longrightarrow\\\\ {\h F'(\Omega^i (i)), {{\mr H}^0 ({\mathcal O}(1))\otimes F'(\Omega^{i-1} (i-1))}}\stackrel{\sim}{\longrightarrow}\\\\ {\h {\mathcal O}\boxtimes F'(\Omega^i (i)), {{\mathcal O}(1)\boxtimes F'(\Omega^{i-1} (i-1))}}\stackrel{\sim}{\longrightarrow}\\\\ {\h {\mathcal O}(-i)\boxtimes F'(\Omega^i (i)), {{\mathcal O}(-i+1)\boxtimes F'(\Omega^{i-1} (i-1))}} \end{array} $$ It can easily be checked that the composition $d_{-i+1}\circ d_{-i}$ is equal to $0.$ We omit the check. Consider the following complex $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ $$ C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}} :=\{ {\mathcal O}(-N)\boxtimes F'(\Omega^N (N))\stackrel{d'_{-N}}{\longrightarrow} \cdots\longrightarrow {\mathcal O}(-1)\boxtimes F'(\Omega^{1} (1))\stackrel{d'_{-1}}{\longrightarrow} {\mathcal O}\boxtimes F'({\mathcal O}) \} $$ over the derived category $\db{{\Bbb{P}}^N \times X}.$ For $l<0$ we have $$ \begin{array}{l} {\H l, {}, {\mathcal O}(-i)\boxtimes F'(\Omega^i (i)), {{\mathcal O}(-k)\boxtimes F'(\Omega^{k} (k))}}\cong\\\\ {\H l, {}, {\mathcal O}\boxtimes F'(\Omega^i (i)), {{\mr H}^0 ({\mathcal O}(i-k))\otimes F'(\Omega^{k} (k))}}\cong\\\\ {\H l, {}, j^* (\Omega^i (i)), {{\mr H}^0 ({\mathcal O}(i-k))\otimes j^* (\Omega^{k} (k))}}=0 \end{array} $$ Hence, by Lemma \ref{pd1}, there exists a convolution of the complex $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}},$ and all convolutions are isomorphic. By $E'$ denote some convolution of $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ and by $\gamma_0$ denote the morphism ${\mathcal O}\boxtimes F'({\mathcal O})\stackrel{\gamma_0}{\longrightarrow}E'.$ (Further we shall see that all convolutions of $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ are canonically isomorphic). Now let $\Phi_{E'}$ be the functor from $\db{{\Bbb{P}}^N}$ to $\db{X},$ defined by (\ref{dfun}). \th{Lemma}\label{tran} There exist canonically defined isomorphisms $f_k : F'({\mathcal O}(k)) \stackrel{\sim}{\longrightarrow}\Phi_{E'}({\mathcal O}(k))$ for all $k\in{\Bbb{Z}},$ and these isomorphisms are functorial, i.e. for any $\alpha : {\mathcal O}(k)\to {\mathcal O}(l)$ the following diagram commutes $$ \begin{array}{ccc} F'({\mathcal O}(k))&\stackrel{F'(\alpha)}{\longrightarrow}&F'({\mathcal O}(l))\\ \llap{\ss{f_k}}\big\downarrow&&\big\downarrow\rlap{\ss{f_l}}\\ \Phi_{E'}({\mathcal O}(k))&\stackrel{\Phi_{E'}(\alpha)}{\longrightarrow}&\Phi_{E'}({\mathcal O}(l)) \end{array} $$ \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } At first, assume that $k\ge0.$ Consider the resolution (\ref{di}) of the diagonal $\Delta\subset{\Bbb{P}}^N \times{\Bbb{P}}^N$ and, after tensoring it with ${\mathcal O}(k)\boxtimes{\mathcal O},$ push forward onto the second component. We get the following resolution of ${\mathcal O}(k)$ on ${\Bbb{P}}^N$ $$ \{ {\mr H}^0 ({\mathcal O}(k-N))\otimes\Omega^N (N){\longrightarrow}\cdots\longrightarrow{\mr H}^0 ({\mathcal O}(k-1)) \otimes\Omega^{1}(1){\longrightarrow}{\mr H}^0 ({\mathcal O}(k))\otimes{\mathcal O} \}\stackrel{\delta_k}{\longrightarrow}{\mathcal O}(k) $$ Consequently $F'({\mathcal O}(k))$ is a convolution of the complex $D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k$: $$ {\mr H}^0 ({\mathcal O}(k-N))\otimes F'(\Omega^N (N)){\longrightarrow}\cdots\longrightarrow{\mr H}^0 ({\mathcal O}(k-1)) \otimes F'(\Omega^{1}(1)){\longrightarrow}{\mr H}^0 ({\mathcal O}(k))\otimes F'({\mathcal O}) $$ over $\db{X}.$ On the other hand, let us consider the complex $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k := q^* {\mathcal O}(k)\otimes C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ on ${\Bbb{P}}^N\times X$ with the morphism $\gamma_k : {\mathcal O}(k)\boxtimes F'({\mathcal O})\longrightarrow q^*{\mathcal O}(k)\otimes E',$ and push it forward onto the second component. It follows from the construction of the complex $C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}$ that $\pi'_* (C^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k)=D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k.$ So we see that $F'({\mathcal O}(k))$ and $\Phi_{E'}({\mathcal O}(k))$ both are convolutions of the same complex $D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k.$ By assumption the functor $F$ is full and faithful, hence, if ${\mathcal G}$ and ${\mathcal H}$ are locally free sheaves on ${\Bbb{P}}^N$ then we have $$ {\H i, {}, F'({\mathcal G}), {F'({\mathcal H})}}= {\H i, {}, j^* ({\mathcal G}), {j^*({\mathcal H})}}=0 $$ for $i<0.$ Therefore the complex $D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k$ satisfies the conditions (\ref{ex}) and (\ref{une}) of Lemma \ref{pd1} Hence there exists a uniquely defined isomorphism $f_k : F'({\mathcal O}(k))\stackrel{\sim}{\longrightarrow} \Phi_{E'}({\mathcal O}(k)),$ completing the following commutative diagram $$ \begin{array}{ccc} {\mr H}^0 ({\mathcal O}(k))\otimes F'({\mathcal O})&\stackrel{F'(\delta_k)}{\longrightarrow} &F'({\mathcal O}(k))\\ \llap{\ss{id}}\big\downarrow&&\big\downarrow\rlap{\ss{f_k}}\\ {\mr H}^0 ({\mathcal O}(k))\otimes F'({\mathcal O})&\stackrel{\pi'_* (\gamma_k)}{\longrightarrow}&\Phi_{E'}({\mathcal O}(k)) \end{array} $$ Now we have to show that these morphisms are functorial. For any $\alpha : {\mathcal O}(k)\to{\mathcal O}(l)$ we have the commutative squares $$ \begin{array}{ccc} {\mr H}^0 ({\mathcal O}(k))\otimes F'({\mathcal O})&\stackrel{F'(\delta_k)}{\longrightarrow} &F'({\mathcal O}(k))\\ \llap{\ss{{\mr H}^0 (\alpha)\otimes id}}\big\downarrow&&\big\downarrow\rlap{\ss{F'(\alpha)}}\\ {\mr H}^0 ({\mathcal O}(l))\otimes F'({\mathcal O})&\stackrel{F'(\delta_l)}{\longrightarrow}& F'({\mathcal O}(l)) \end{array} $$ and $$ \begin{array}{ccc} {\mr H}^0 ({\mathcal O}(k))\otimes F'({\mathcal O})& \stackrel{\pi'_* (\gamma_k)}{\longrightarrow} &\Phi_{E'}({\mathcal O}(k))\\ \llap{\ss{{\mr H}^0 (\alpha)\otimes id}}\big\downarrow&& \big\downarrow\rlap{\ss{\Phi_{E'}(\alpha)}}\\ {\mr H}^0 ({\mathcal O}(l))\otimes F'({\mathcal O})& \stackrel{\pi'_* (\gamma_l)}{\longrightarrow}&\Phi_{E'}({\mathcal O}(l)) \end{array} $$ Therefore we have the equalities: $$ \begin{array}{l} f_l\circ F'(\alpha)\circ F'(\delta_k)=f_l\circ F'(\delta_l)\circ ({\mr H}^0 (\alpha)\otimes id)= \pi'_* (\gamma_l)\circ ({\mr H}^0 (\alpha)\otimes id)= \Phi_{E'}(\alpha)\circ \pi'_* (\gamma_k)= \\ \Phi_{E'}(\alpha)\circ f_k \circ F'(\delta_k) \end{array} $$ Since the complexes $D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_k$ and $D^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_l$ satisfy the conditions of Lemma \ref{pd2} there exists only one morphism $h: F'({\mathcal O}(k)) \to \Phi_{E'}({\mathcal O}(l))$ such that $$ h\circ F'(\delta_k) = \pi'_* (\gamma_l)\circ ({\mr H}^0 (\alpha)\otimes id) $$ Hence $f_l\circ F'(\alpha)$ coincides with $\Phi_{E'}(\alpha)\circ f_k.$ Now, consider the case $k<0.$ Let us take the following right resolution for ${\mathcal O}(k)$ on ${\Bbb{P}}^N.$ $$ {\mathcal O}(k)\stackrel{\sim}{\longrightarrow}\bigl\{ V^k_0 \otimes{\mathcal O}\longrightarrow\cdots\longrightarrow V^k_N \otimes {\mathcal O}(N)\bigr\} $$ By Lemma \ref{pd2}, the morphism of the complexes over $\db{X}$ $$ \begin{array}{ccccc} V^k_0 \otimes F'({\mathcal O})&\longrightarrow&\cdots&\longrightarrow& V^k_N \otimes F'({\mathcal O}(N))\\ \llap{\ss{id\otimes f_0}}\big\downarrow\wr&&&&\llap{\ss{id\otimes f_N}}\big\downarrow\wr\\ V^k_0 \otimes \Phi_{E'}({\mathcal O})&\longrightarrow&\cdots&\longrightarrow& V^k_N \otimes \Phi_{E'}({\mathcal O}(N))\\ \end{array} $$ gives us the uniquely determined morphism $f_k : F'({\mathcal O}(k))\longrightarrow \Phi_{E'}({\mathcal O}(k)).$ It is not hard to prove that these morphisms are functorial. The proof is left to a reader. $\Box$ \refstepcounter{THNO}\par\vspace{1.5ex Now we must prove that there exists an object $E\in\db{M\times X}$ such that $j_* E\cong E'.$ Let ${\mathcal L}$ be a very ample invertible sheaf on $M$ and let $j: M\hookrightarrow {\Bbb{P}}^N$ be an embedding with respect to ${\mathcal L}.$ By $A$ denote the graded algebra $\bigoplus\limits^{\infty}_{i=0} {\mr H}^0 (M, {\mathcal L}^{i}).$ Let $B_0=k,$ and $B_1=A_1.$ For $m\ge 2,$ we define $B_m$ as \begin{equation} \label{dual} B_m = Ker( B_{m-1}\otimes A_1 \longrightarrow B_{m-2}\otimes A_2 ) \end{equation} \th{Definition} $A$ is said to be $n$-Koszul if the following sequence is exact $$ B_n \otimes_k A\longrightarrow B_{n-1}\otimes_k A\longrightarrow \cdots\longrightarrow B_1\otimes_k A\longrightarrow A\longrightarrow k\longrightarrow 0 $$ \par\vspace{1.0ex}\endgroup Assume that $A$ is n-Koszul. Let $R_0 = {\mathcal O}_M.$ For $m\ge 1,$ denote by $R_m$ the kernel of the morphism $B_m\otimes{\mathcal O}_M \longrightarrow B_{m-1}\otimes{\mathcal L}.$ Using (\ref{dual}), we obtain the canonical morphism $R_m\longrightarrow A_1\otimes R_{m-1}.$ (actually, ${\h R_m, {R_{m-1}}}\cong A_1^*$). Since $A$ is $n$-Koszul, we have the exact sequences $$ 0\longrightarrow R_m \longrightarrow B_m\otimes {\mathcal O}_M \longrightarrow B_{m-1}\otimes{\mathcal L}\longrightarrow\cdots \longrightarrow B_1 \otimes {\mathcal L}^{{m-1}}\longrightarrow {\mathcal L}^{m}\longrightarrow 0 $$ for $m\le n.$ We have the canonical morphisms $f_m : j^* \Omega^m (m)\longrightarrow R_m,$ because $\Lambda^i A_1 \subset B_i$ and there exist the exact sequences on ${\Bbb{P}}^N$ $$ 0\longrightarrow \Omega^m (m) \longrightarrow \Lambda^m A_1 \otimes {\mathcal O}\longrightarrow \Lambda^{m-1} A_1 \otimes {\mathcal O}(1)\longrightarrow \cdots \longrightarrow {\mathcal O}(m)\longrightarrow 0 $$ It is known that for any $n$ there exists $l$ such that the Veronese algebra $A^l = \bigoplus\limits^{\infty}_{i=0} {\mr H}^0 (M, {\mathcal L}^{il})$ is $n$-Koszul.( Moreover, it was proved in \cite{Bec} that $A^l$ is Koszul for $l\gg 0$). Using the technique of \cite{IM} and substituting ${\mathcal L}$ with ${\mathcal L}^j,$ when $j$ is sufficiently large , we can choose for any $n$ a very ample ${\mathcal L}$ such that 1) algebra $A$ is $n$-Koszul, 2) the complex $$ {\mathcal L}^{ -n}\boxtimes R_n\longrightarrow\cdots\longrightarrow {\mathcal L}^{ -1}\boxtimes R_1\longrightarrow {\mathcal O}_M \boxtimes R_0 \longrightarrow {\mathcal O}_{\Delta} $$ on $M\times M$ is exact, 3) the following sequences on $M.$ $$ A_{k-n}\otimes R_n \longrightarrow A_{k-n+1}\otimes R_{n-1}\longrightarrow\cdots\longrightarrow A_{k-1}\otimes R_1 \longrightarrow A_k \otimes R_0 \longrightarrow {\mathcal L}^{k}\longrightarrow 0 $$ are exact for any $k\ge 0.$ Here, by definition, if $ k-i<0,$ then $A_{k-i}=0.$ (see Appendix for proof). Let us denote by $T_k$ the kernel of the morphism $A_{k-n}\otimes R_n \longrightarrow A_{k-n+1}\otimes R_{n-1}.$ Consider the following complex over $\db{M\times X}$ \begin{equation}\label{nob} {\mathcal L}^{ -n}\boxtimes F(R_n)\longrightarrow\cdots\longrightarrow {\mathcal L}^{ -1}\boxtimes F(R_1)\longrightarrow {\mathcal O}_M \boxtimes F(R_0) \end{equation} Here the morphism ${\mathcal L}^{-k}\boxtimes F(R_k)\longrightarrow {\mathcal L}^{-k+1}\boxtimes F(R_{k-1})$ is induced by the canonical morphism $R_k\longrightarrow A_1 \otimes R_{k-1}$ with respect to the following sequence of isomorphisms $$ {\h {\mathcal L}^{-k}\boxtimes F(R_k), {{\mathcal L}^{-k+1}\boxtimes F(R_{k-1})}}\cong {\h F(R_k), {{\mr H}^0 ({\mathcal L})\otimes F(R_{k-1})}}\cong $$ $$ \cong{\h R_k, {A_1 \otimes R_{k-1}}} $$ By Lemma \ref{pd1}, there is a convolution of the complex (\ref{nob}) and all convolutions are isomorphic. Let $G\in \db{M\times X}$ be a convolution of this complex. For any $k\ge 0,$ object $\pi_* (G\otimes p^* ({\mathcal L}^k))$ is a convolution of the complex $$ A_{k-n}\otimes F(R_n)\longrightarrow A_{k-n+1}\otimes F(R_{n-1})\longrightarrow\cdots\longrightarrow A_k \otimes F(R_0). $$ On the other side, we know that $T_k[n]\oplus{\mathcal L}^k$ is a convolution of the complex $$ A_{k-n}\otimes R_n\longrightarrow A_{k-n+1}\otimes R_{n-1}\longrightarrow\cdots\longrightarrow A_k \otimes R_0, $$ because ${\E n+1 , {}, {\mathcal L}^k , {T_k}}=0$ for $n\gg 0.$ Therefore, by Lemma \ref{pd1}, we have $\pi_* (G\otimes p^* ({\mathcal L}^k))\cong F(T_k[n]\oplus {\mathcal L}^k).$ It follows immediately from Remark \ref{boun} that the cohomology sheaves ${H}^i (\pi_* (G\otimes p^* ({\mathcal L}^k)))= {H}^i (F(T_k)[n])\oplus{H}^i (F( {\mathcal L}^k))$ concentrate on the union $[-n-a, -n]\cup [-a, 0]$ for any $k>0$ ($a$ was defined in \ref{boun}). Therefore the cohomology sheaves ${H}^i (G)$ also concentrate on $[-n-a, -n]\cup [-a, 0].$ We can assume that $n> dimM + dimX + a.$ This implies that $G\cong C\oplus E,$ where $E, C$ are objects of $\db{M\times X}$ such that ${H}^i (E)=0$ for $i\not\in [-a, 0]$ and ${H}^i (C)=0$ for $i\not\in [-n-a, -n].$ Moreover, we have $\pi_* (E\otimes p^* ({\mathcal L}^k))\cong F({\mathcal L}^k).$ Now we show that $j_* (E)\cong E'.$ Let us consider the morphism of the complexes over $\db{{\Bbb{P}}^N \times X}.$ $$ \begin{array}{ccccc} {\mathcal O}(-n)\boxtimes F' (\Omega^n (n))&\longrightarrow&\cdots&\longrightarrow&{\mathcal O}\boxtimes F' ({\mathcal O})\\ \big\downarrow\rlap{\ss{can\boxtimes F(f_n)}}&&&&\big\downarrow\rlap{\ss{can\boxtimes F(f_0)}}\\ j_* ({\mathcal L}^{-n}) \boxtimes F(R_n)&\longrightarrow&\cdots&\longrightarrow& j_* ({\mathcal O}_M)\boxtimes F(R_0) \end{array} $$ By Lemma \ref{pd2}, there exists a morphism of convolutions $\phi : K\longrightarrow j_* (G).$ If $N>n,$ then $K$ is not isomorphic to $E',$ but there is a distinguished triangle $$ S\longrightarrow K\longrightarrow E'\longrightarrow S[1] $$ and the cohomology sheaves ${H}^i (S)\ne 0$ only if $i\in [-n-a, -n].$ Now, since ${\h S, {j_* (E)}}=0$ and ${\h S[1], {j_* (E)}}=0,$ we have a uniquely determined morphism $\psi : E' \longrightarrow j_* (E)$ such that the following diagram commutes $$ \begin{array}{ccc} K&\stackrel{\phi}{\longrightarrow}& j_* (G)\\ \big\downarrow&&\big\downarrow\\ E'&\stackrel{\psi}{\longrightarrow}& j_* (E) \end{array} $$ We know that $\pi'_*( E'\otimes q^*({\mathcal O}(k)))\cong F({\mathcal L}^k)\cong \pi_*( E\otimes p^*({\mathcal L}^k)).$ Let $\psi_k$ be the morphism $ \pi'_*( E'\otimes q^*({\mathcal O}(k)))\longrightarrow \pi_*( E\otimes p^*({\mathcal L}^k))$ induced by $\psi.$ The morphism $\psi_k$ can be included in the following commutative diagram: $$ \begin{array}{ccccc} S^k A_1 \otimes F({\mathcal O})&\stackrel{can}{\longrightarrow}& F({\mathcal L}^k)&\stackrel{\sim}{\longrightarrow} &\pi'_*( E'\otimes q^*({\mathcal O}(k)))\\ \llap{\ss{can}}\big\downarrow&&&&\big\downarrow\rlap{\ss{\psi_k}}\\ A_k \otimes F({\mathcal O})&\stackrel{can}{\longrightarrow}& F({\mathcal L}^k)&\stackrel{\sim}{\longrightarrow} &\pi_*( E\otimes p^*({\mathcal L}^k)) \end{array} $$ Thus we see that $\psi_k$ is an isomorphism for any $k\ge 0.$ Hence $\psi$ is an isomorphism too. This proves the following: \th{Lemma}\label{obj} There exists an object $E\in\db{M\times X}$ such that $j_* (E)\cong E',$ where $E'$ is the object from $\db{{\Bbb{P}}^N \times X},$ constructed in \ref{cons}. \par\vspace{1.0ex}\endgroup \refstepcounter{THNO}\par\vspace{1.5ex Now, we prove some statements relating to abelian categories. they are needed for the sequel. Let ${\mathcal A}$ be a $k$-linear abelian category (henceforth we shall consider only $k$-linear abelian categories). Let $\{ P_i \}_{i\in\Bbb Z}$ be a sequence of objects from ${\mathcal A}.$ \th{Definition} We say that this sequence is {\sf ample} if for every object $X\in {\mathcal A}$ there exists $N$ such that for all $i<N$ the following conditions hold: a) the canonical morphism ${\h P_i, X}\otimes P_i \longrightarrow X$ is surjective, b) ${\E j, {}, P_i, X}=0$ for any $j\not=0,$ c) ${\h X, {P_i}}=0.$ \par\vspace{1.0ex}\endgroup It is clear that if ${\mathcal L}$ is an ample invertible sheaf on a projective variety in usual sense, then the sequence $\{ {\mathcal L}^i \}_{i\in{\Bbb{Z}}}$ in the abelian category of coherent sheaves is ample. \th{Lemma}\label{zer1} Let $\{ P_i \}$ be an ample sequence in an abelian category ${\mathcal A}.$ If $X$ is an object in $\db{\mathcal A}$ such that ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, P_i, X}=0$ for all $i\ll 0,$ then $X$ is the zero object. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } If $i\ll 0$ then $${\h P_i, {{H}^k (X)}}\cong{\H k, {}, P_i, X}=0$$ The morphism ${\h P_i, {{H}^k (X)}}\otimes P_i\longrightarrow {H}^k (X)$ must be surjective for $i\ll 0,$ hence ${H}^k (X)=0$ for all $k.$ Thus $X$ is the zero object. $\Box$ \th{Lemma}\label{zer2} Let $\{ P_i \}$ be an ample sequence in an abelian category ${\mathcal A}$ of finite homological dimension. If $X$ is an object in $\db{\mathcal A}$ such that ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, X, {P_i}}=0$ for all $i\ll 0.$ Then $X$ is the zero object. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Assume that the cohomology objects of $X$ are concentrated in a segment $[a, 0].$ There exists the canonical morphism $X\longrightarrow {H}^0 (X).$ Consider a surjective morphism $P_{i_1}^{\oplus k_1}\longrightarrow {H}^0 (X).$ By $Y_1$ denote the kernel of this morphism. Since ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, X, {P_{i_1}}}=0$ we have ${\H 1, {}, X, {Y_1}}\not=0.$ Further take a surjective morphism $P_{i_2}^{\oplus k_2}\longrightarrow Y_1.$ By $Y_2$ denote the kernel of this morphism. Again, since ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, X, {P_{i_2}}}=0,$ we obtain ${\H 2, {}, X, {Y_2}}\not=0.$ Iterating this procedure as needed, we get contradiction with the assumption that ${\mathcal A}$ is of finite homological dimension. $\Box$ \th{Lemma}\label{f&f} Let ${\mathcal B}$ be an abelian category, ${\mathcal A}$ an abelian category of finite homological dimension, and $\{ P_i \}$ an ample sequence in ${\mathcal A}.$ Suppose $F$ is an exact functor from $\db{\mathcal A}$ to $\db{\mathcal B}$ such that it has right and left adjoint functors $F^!$ and $F^*$ respectively. If the maps $$ {\H k, {}, P_i, {P_j}}\stackrel{\sim}{\longrightarrow}{\H k, {}, F(P_i), {F(P_j)}} $$ are isomorphisms for $i<j$ and all $k.$ Then $F$ is full and faithful. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Let us take the canonical morphism $f_i : P_i\longrightarrow F^! F(P_i)$ and consider a distinguished triangle $$ P_i\stackrel{f_i}{\longrightarrow} F^! F(P_i)\longrightarrow C_i\longrightarrow P_i [1]. $$ Since for $j\ll 0$ we have isomorphisms: $$ {\H k, {}, P_j, {P_i}}\stackrel{\sim}{\longrightarrow}{\H k, {}, F(P_j), {F(P_i)}} \cong{\H k, {}, P_j, {F^! F(P_i)}}. $$ We see that ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, P_j, {C_i}}=0$ for $j\ll 0.$ It follows from Lemma \ref{zer1} that $C_i =0.$ Hence $f_i$ is an isomorphism. Now, take the canonical morphism $g_X : F^* F(X)\longrightarrow X$ and consider a distinguished triangle $$ F^* F(X)\stackrel{g_X}{\longrightarrow} X\longrightarrow C_X\longrightarrow F^* F(X)[1] $$ We have the following sequence of isomorphisms $$ {\H k, {}, X, {P_i}}\stackrel{\sim}{\longrightarrow}{\H k, {}, X, {F^! F(P_i)}} \cong{\H k, {}, F^* F(X), {P_i}} $$ This implies that ${\H {\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}, {}, C_X, {P_i}}=0$ for all $i.$ By Lemma \ref{zer2}, we obtain $C_X=0.$ Hence $g_X$ is an isomorphism. It follows that $F$ is full and faithful. $\Box$ Let ${\mathcal A}$ be an abelian category possessing an ample sequence $\{ P_i\}.$ Denote by $\db{\mathcal A}$ the bounded derived category of ${\mathcal A}.$ Let us consider the full subcategory $j: {\mathcal C}\hookrightarrow\db{\mathcal A}$ such that $\mbox{Ob} {\mathcal C}:=\{ P_i\; |\; i\in{\Bbb{Z}} \}.$ Now we would like to show that if there exists an isomorphism of a functor $F : \db{{\mathcal A}}\longrightarrow\db{{\mathcal A}}$ to identity functor on the subcategory ${\mathcal C},$ then it can be extended to the whole $\db{\mathcal A}.$ \th{Proposition}\label{ext} Let $F :\db{{\mathcal A}}\longrightarrow\db{{\mathcal A}}$ be an autoequivalence. Suppose there exists an isomorphism $f : j\stackrel{\sim}{\longrightarrow}F\mid_{\mathcal C} $ ( where $j : {\mathcal C}\hookrightarrow \db{{\mathcal A}}$ is a natural embedding). Then it can be extended to an isomorphism $id\stackrel{\sim}{\longrightarrow}F$ on the whole $\db{\mathcal A}.$ \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } First, we can extend the transformation $f$ to all direct sums of objects ${\mathcal C}$ componentwise , because $F$ takes direct sums to direct sums. Note that $X\in\db{\mathcal A}$ is isomorphic to an object in ${\mathcal A}$ iff ${\H j, {}, P_i, X}=0$ for $j\not=0$ and $i\ll 0.$ It follows that $F(X)$ is isomorphic to an object in ${\mathcal A},$ because $$ {\H j, {}, P_i, {F(X)}}\cong{\H j, {}, F(P_i), {F(X)}}\cong {\H j, {}, P_i, X}=0 $$ for $j\not=0$ and $i\ll 0.$ \refstepcounter{SNO}\par\vspace{1ex At first, let $X$ be an object from ${\mathcal A}.$ Take a surjective morphism $v : P^{\oplus k}_i\longrightarrow X.$ We have the morphism $f_i : P^{\oplus k}_i\longrightarrow F(P^{\oplus k}_i)$ and two distinguished triangles: $$ \begin{array}{ccccccc} Y&\stackrel{u}{\longrightarrow}&P^{\oplus k}_i&\stackrel{v}{\longrightarrow}&X&\longrightarrow&Y [1]\\ &&\big\downarrow\rlap{$f_i$}&&&& \\ F(Y)&\stackrel{F(u)}{\longrightarrow}&F(P^{\oplus k}_i)&\stackrel{F(v)}{\longrightarrow}&F(X)&\longrightarrow&F(Y)[1]\\ \end{array} $$ Now we show that $F(v)\circ f_i\circ u=0.$ Consider any surjective morphism $ w : P^{\oplus l}_j\longrightarrow Y.$ It is sufficient to check that $F(v)\circ f_i\circ u\circ w=0.$ Let $f_j : P^{\oplus l}_j\longrightarrow F(P^{\oplus l}_j)$ be the canonical morphism. Using the commutation relations for $f_i$ and $f_j,$ we obtain $$ F(v)\circ f_i\circ u\circ w = F(v)\circ F(u\circ w)\circ f_j= F(v\circ u\circ w)\circ f_j =0 $$ because $v\circ u=0.$ Since ${\h Y[1], {F(X)}}=0,$ by Lemma \ref{tr}, there exists a unique morphism $f_X : X\longrightarrow F(X)$ that commutes with $f_i.$ \refstepcounter{SNO}\par\vspace{1ex Let us show that $f_X$ does not depend from morphism $v : P^{\oplus k}_i \longrightarrow X.$ Consider two surjective morphisms $v_1 : P^{\oplus k_1}_{i_1}\longrightarrow X$ and $v_2 : P^{\oplus k_2}_{i_2}\longrightarrow X.$ We can take two surjective morphisms $w_1 : P^{\oplus l}_{j}\longrightarrow P^{\oplus k_1}_{i_1}$ and $w_2 : P^{\oplus l}_{j}\longrightarrow P^{\oplus k_2}_{i_2}$ such that the following diagram is commutative: $$ \begin{array}{ccc} P^{\oplus l}_{j}&\stackrel{w_2}{\longrightarrow}& P^{\oplus k_2}_{i_2}\\ \big\downarrow\rlap{\ss{w_1}}&& \big\downarrow\rlap{\ss{v_2}}\\ P^{\oplus k_1}_{i_1}&\stackrel{v_1}{\longrightarrow}& X\\ \end{array} $$ Clearly, it is sufficient to check the coincidence of the morphisms, constructed by $v_1$ and $v_1\circ w_1.$ Now, let us consider the following commutative diagram: $$ \begin{array}{ccccc} P^{\oplus l}_{j}&\stackrel{w_1}{\longrightarrow}& P^{\oplus k_1}_{i_1}& \stackrel{v_1}{\longrightarrow}&X\\ \big\downarrow\rlap{\ss{f_j}}&&\big\downarrow\rlap{\ss{v_2}}&&\big\downarrow\rlap{\ss{f_X}}\\ F(P^{\oplus l}_{j})&\stackrel{F(w_1)}{\longrightarrow}&F(P^{\oplus k_1}_{i_1})& \stackrel{F(v_1)}{\longrightarrow}&F(X)\\ \end{array} $$ Here the morphism $f_X$ is constructed by $v_1.$ Both squares of this diagram are commutative. Since there exists only one morphism from $X$ to $F(X)$ that commutes with $f_j,$ we see that the $f_X,$ constructed by $v_1,$ coincides with the morphism, constructed by $v_1\circ w_1.$ \refstepcounter{SNO}\par\vspace{1ex Now we must show that for any morphism $X\stackrel{\phi}{\longrightarrow}Y$ we have equality: $$ f_Y \circ \phi = F(\phi )\circ f_X $$ Take a surjective morphism $P^{\oplus l}_j\stackrel{v}{\longrightarrow} Y.$ Choose a surjective morphism $P^{\oplus k}_i\stackrel{u}{\longrightarrow}X$ such that the composition $\phi\circ u$ lifts to a morphism $\psi : P^{\oplus k}_i{\longrightarrow} P^{\oplus l}_j .$ We can do it, because for $i\ll 0$ the map ${\h P^{\oplus k}_i, {P^{\oplus l}_j}}\to {\h P^{\oplus k}_i, Y}$ is surjective. We get the commutative square: $$ \begin{array}{ccc} P^{\oplus k}_{i}&\stackrel{u}{\longrightarrow}&X\\ \big\downarrow\rlap{\ss{\psi}}&&\big\downarrow\rlap{\ss{\phi}}\\ P^{\oplus l}_{j}&\stackrel{v}{\longrightarrow}&Y\\ \end{array} $$ By $h_1$ and $h_2$ denote $f_Y\circ\phi$ and $F(\phi)\circ f_X$ respectively. We have the following sequence of equalities: $$ h_1\circ u=f_Y\circ\phi\circ u=f_Y\circ v\circ\psi=F(v)\circ f_j\circ\psi= F(v)\circ F(\psi)\circ f_i $$ and $$ h_2\circ u=F(\phi)\circ f_X\circ u =F(\phi)\circ F(u)\circ f_i = F(\phi\circ u)\circ f_i = F(v\circ \psi)\circ f_i = F(v)\circ F(\psi)\circ f_i $$ Consequently, the following square is commutative for $t=1,2.$ $$ \begin{array}{ccccccc} Z&\longrightarrow&P^{\oplus k}_{i}&\stackrel{u}{\longrightarrow}&X&\longrightarrow&Z[1]\\ &&\llap{\ss{F(\psi)\circ f_i}}\big\downarrow&&\big\downarrow\rlap{\ss{h_t}}&&\\ F(W)&\longrightarrow&F(P^{\oplus l}_{j})&\stackrel{F(v)}{\longrightarrow}&F(Y)&\longrightarrow&F(W)[1]\\ \end{array} $$ By Lemma \ref{tr}, as ${\h Z[1], {F(Y)}}=0,$ we obtain $h_1=h_2.$ Thus, $f_Y \circ \phi = F(\phi )\circ f_X.$ Now take a cone $C_X$ of the morphism $f_X.$ Using the following isomorphisms $$ {\h P_i, X}\cong{\h F(P_i), {F(X)}}\cong{\h P_i, {F(X)}}, $$ we obtain ${\H j, {}, P_i, {C_X}}=0$ for all $j,$ when $i\ll 0.$ Hence, by Lemma \ref{zer1}, $C_X=0$ and $f_X$ is an isomorphism. \refstepcounter{SNO}\par\vspace{1ex Let us define $f_{X[n]} : X[n]\longrightarrow F(X[n])\cong F(X)[n]$ for any $X\in{\mathcal A}$ by $$ f_{X[n]}=f_X [n]. $$ It is easily shown that these transformations commute with any $u\in{\E k, {}, X, Y}.$ Indeed, since any element $u\in{\E k, {}, X, Y}$ can be represented as a composition $ u = u_0 u_1 \cdots u_k $ of some elements $u_i\in{\E 1, {}, Z_i, {Z_{i+1}}}$ and $Z_0=X, Z_k=Y,$ we have only to check it for $u\in{\E 1, {}, X, Y}$ . Consider the following diagram: $$ \begin{array}{ccccccc} Y&\longrightarrow&Z&\longrightarrow&X&\stackrel{u}{\longrightarrow}&Y[1]\\ \llap{\ss{f_Y}}\big\downarrow&&\big\downarrow\rlap{\ss{f_Z}}&&&&\big\downarrow\rlap{\ss{f_Y [1]}}\\ F(Y)&\longrightarrow&F(Z)&\longrightarrow&F(X)&\stackrel{F(u)}{\longrightarrow}&F(Y)[1]\\ \end{array} $$ By an axiom of triangulated categories there exists a morphism $h : X\to F(X)$ such that $(f_Y, f_Z, h)$ is a morphism of triangles. On the other hand, since ${\h Y[1], {F(X)}}=0,$ by Lemma \ref{tr}, $h$ is a unique morphism that commutes with $f_Z.$ But $f_X$ also commutes with $f_Z.$ Hence we have $h=f_X.$ This implies that $$ f_Y [1] \circ u = F(u) \circ f_X $$ \refstepcounter{SNO}\par\vspace{1ex The rest of the proof is by induction over the length of a segment, in which the cohomology objects of $X$ are concentrated. Let $X$ be an object from $ \db{\mathcal A}$ and suppose that its cohomology objects ${H}^p (X)$ are concentrated in a segment $[a, 0].$ Take $v: P^{\oplus k}_i\longrightarrow X$ such that \begin{eqnarray}\label{tt} a)& {\H j, {}, P_i, {{H}^p (X)}}=0& \mbox{ for all }\; p\; \mbox{ and for } \; j\not=0,\nonumber\\ b)& u: P_i^{\oplus k}\longrightarrow {H}^0 (X)& \mbox{is the surjective morphism},\\ c)&{\h {H}^0 (X), {P_i}}=0.&\nonumber \end{eqnarray} Here $u$ is the composition $v$ with the canonical morphism $X\longrightarrow {H}^0 (X).$ Consider a distinguished triangle: $$ Y[-1]\longrightarrow P^{\oplus k}_i\stackrel{v}{\longrightarrow}X\longrightarrow Y $$ By the induction hypothesis, there exists the isomorphism $f_Y$ and it commutes with $f_i.$ So we have the commutative diagram: $$ \begin{array}{ccccccc} Y[-1]&\longrightarrow&P^{\oplus k}_i&\stackrel{v}{\longrightarrow}&X&\longrightarrow&Y\\ \llap{\ss{f_Y [-1]}}\big\downarrow&&\big\downarrow\rlap{\ss{f_i}}&&&&\big\downarrow\rlap{\ss{f_Y }}\\ F(Y)[-1]&\longrightarrow&F(P^{\oplus k}_i)&\stackrel{F(v)}{\longrightarrow}&F(X)&\longrightarrow&F(Y)\\ \end{array} $$ Moreover we have the following sequence of equalities $$ {\h X, {F(P^{\oplus k}_i)}}\cong{\h X, {P^{\oplus k}_i}}\cong {\h {H}^0 (X), {P^{\oplus k}_i}}=0 $$ Hence, by Lemma \ref{tr}, there exists a unique morphism $f_X : X\longrightarrow F(X)$ that commutes with $f_Y.$ \refstepcounter{SNO}\par\vspace{1ex We must first show that $f_X$ is correctly defined. Suppose we have two morphisms $v_1 : P_{i_1}^{\oplus k_1}\longrightarrow X$ and $v_2 : P_{i_2}^{\oplus k_2}\longrightarrow X.$ As above, we can find $P_j$ and surjective morphisms $w_1, w_2$ such that the following diagram is commutative $$ \begin{array}{ccc} P^{\oplus l}_{j}&\stackrel{w_2}{\longrightarrow}& P^{\oplus k_2}_{i_2}\\ \big\downarrow\rlap{\ss{w_1}}&& \big\downarrow\rlap{\ss{u_2}}\\ P^{\oplus k_1}_{i_1}&\stackrel{u_1}{\longrightarrow}&{H}^0 (X)\\ \end{array} $$ We can find a morphism $\phi : Y_j\longrightarrow Y_{i_1}$ such that the triple $(w_1, id, \phi)$ is a morphism of distinguished triangles. $$ \begin{array}{ccccccc} P^{\oplus l}_j&\stackrel{v_1\circ w_1}{\longrightarrow}&X&\longrightarrow&Y_j&\longrightarrow&P^{\oplus l}_j [1]\\ \llap{\ss{w_1}}\big\downarrow&&\big\downarrow\rlap{\ss{id}}&&\big\downarrow\rlap{\ss{\phi}}&&\big\downarrow\rlap{\ss{w_1 [1]}}\\ P^{\oplus k_1}_{i_1}&\stackrel{v_1}{\longrightarrow}&X&\longrightarrow&Y_{i_1}&\longrightarrow& P^{\oplus k_1}_{i_1} [1]\\ \end{array} $$ By the induction hypothesis, the following square is commutative. $$ \begin{array}{ccc} Y_j&\stackrel{\phi}{\longrightarrow}&Y_{i_1}\\ \llap{\ss{f_{Y_j}}}\big\downarrow&&\big\downarrow\rlap{\ss{f_{Y_i}}}\\ F(Y_j)&\stackrel{F(\phi)}{\longrightarrow}&F(Y_{i_1})\\ \end{array} $$ Hence, we see that the $f_X,$ constructed by $v_1\circ w_1,$ commutes with $f_{Y_{i_1}}$ and, consequently, coincides with the $f_X,$ constructed by $v_1$; because such morphism is unique by Lemma \ref{tr}. Therefore morphism $f_X$ does not depend on a choice of morphism $v: P^{\oplus k}_i\longrightarrow X.$ \refstepcounter{SNO}\par\vspace{1ex Finally, let us prove that for any morphism $\phi : X\longrightarrow Y$ the following diagram commutes \begin{equation}\label{comd} \begin{array}{ccc} X&\stackrel{\phi}{\longrightarrow}&Y\\ \llap{\ss{f_X}}\big\downarrow&&\big\downarrow\rlap{\ss{f_Y }}\\ F(X)&\stackrel{F(\phi)}{\longrightarrow}&F(Y) \end{array} \end{equation} Suppose the cohomology objects of $X$ are concentrated on a segment $[a, 0]$ and the cohomology objects of $Y$ are concentrated on $[b, c].$ \noindent{\it Case 1.} If $c<0,$ we take a morphism $v: P^{\oplus k}_i \longrightarrow X$ that satisfies conditions (\ref{tt}) and ${\h P^{\oplus k}_i, Y}=0.$ We have a distinguished triangle: $$ \begin{array}{ccccccc} P^{\oplus k}_i&\stackrel{v_1}{\longrightarrow}&X&\stackrel{\alpha}{\longrightarrow}&Z&\longrightarrow& P^{\oplus k}_i [1]\\ \end{array} $$ Applying the functor ${\h -, Y}$ to this triangle we found that there exists a morphism $\psi : Z \longrightarrow Y$ such that $\phi = \psi \circ \alpha.$ We know that $f_X,$ defined above, satisfy $$ F(\alpha )\circ f_X = f_Z \circ \alpha $$ If we assume that the diagram $$ \begin{array}{ccc} Z&\stackrel{\psi}{\longrightarrow}&Y\\ \llap{\ss{f_Z}}\big\downarrow&&\big\downarrow\rlap{\ss{f_Y }}\\ F(Z)&\stackrel{F(\psi)}{\longrightarrow}&F(Y)\\ \end{array} $$ commutes, then diagram (\ref{comd}) commutes too. This means that for verifying the commutativity of (\ref{comd}) we can substitute $X$ by an object $Z.$ And the cohomology objects of $Z$ are concentrated on the segment $[a, -1].$ \noindent{\it Case 2.} If $c\ge0,$ we take a surjective morphism $v: P^{\oplus k}_i\longrightarrow Y[c]$ that satisfies conditions (\ref{tt}) and ${\h {H}^c (X), {P^{\oplus k}_i}}=0.$ Consider a distinguished triangle $$ \begin{array}{ccccccc} P^{\oplus k}_i [-c]&\stackrel{v[-c]}{\longrightarrow}&Y&\stackrel{\beta}{\longrightarrow}&W&\longrightarrow& P^{\oplus k}_i [-c+1]\\ \end{array} $$ Note that the cohomology objects of $W$ are concentrated on $[b, c-1].$ By $\psi$ denote the composition $\beta\circ\phi.$ If we assume that the following square $$ \begin{array}{ccc} X&\stackrel{\psi}{\longrightarrow}&W\\ \llap{\ss{f_X}}\big\downarrow&&\big\downarrow\rlap{\ss{f_W }}\\ F(X)&\stackrel{F(\psi)}{\longrightarrow}&F(W)\\ \end{array} $$ commutes, then, since $F(\beta)\circ f_Y = f_W \circ \beta,$ $$ F(\beta)\circ(f_Y\circ\phi - F(\phi)\circ f_X)= f_W \circ \psi - F(\psi ) \circ f_X =0. $$ We chose $P_i$ such that ${\h X, {P^{\oplus k}_i [-c]}}=0.$ As $F( P_i^{\oplus k})$ is isomorphic to $P_i^{\oplus k},$ then ${\h X, {F(P^{\oplus k}_i [-c])}}=0.$ Applying the functor ${\h X, {F(-)}}$ to the above triangle we found that the composition with $F(\beta )$ gives an inclusion of ${\h X, {F(Y)}}$ into ${\h X, {F(W)}}.$ This follows that $f_Y\circ\phi = F(\phi)\circ f_X,$ i.e. our diagram (\ref{comd}) commutes. Combining case 1 and case 2, we can reduce the checking of commutativity of diagram (\ref{comd}) to the case when $X$ and $Y$ are objects from the abelian category ${\mathcal A}.$ But for those it has already been done. Thus the proposition is proved. $\Box$ \refstepcounter{THNO}\par\vspace{1.5ex {\bf Proof of theorem}. 1) {\sc Existence}. Using Lemma \ref{obj} and Lemma \ref{tran}, we can construct an object $E\in\db{M\times X}$ such that there exists an isomorphism of the functors $\bar{f} : F\bigl|_{\mathcal C}\stackrel{\sim}{\longrightarrow}\Phi_E \bigr|_{\mathcal C}$ on full subcategory ${\mathcal C}\subset\db{M},$ where ${\mr O}{\mr b}{\mathcal C}= \{ {\mathcal L}^{i} \mid i\in{\Bbb{Z}} \}$ and ${\mathcal L}$ is a very ample invertible sheaf on $M$ such that for any $k>0$ ${\rm H}^i ( M, {\mathcal L}^k )=0$ , when $i\ne 0.$ By Lemma \ref{f&f} the functor $\Phi_E$ is full and faithfull. Moreover, the functors $F^!\circ \Phi_E$ and $\Phi_E^* \circ F$ are full and faithful too, because we have the isomorphisms: $$ F^! (\bar{f}) : F^!\circ F\bigl|_{\mathcal C}\cong id_{\mathcal C}\stackrel{\sim}{\longrightarrow}F^! \circ \Phi_E\bigl|_{\mathcal C} $$ $$ \Phi_E^* (\bar{f}) : \Phi_E^*\circ F\bigl|_{\mathcal C}\stackrel{\sim}{\longrightarrow} \Phi_E^*\circ \Phi_E\bigl|_{\mathcal C}\cong id_{\mathcal C} $$ and conditions of Lemma \ref{f&f} is fulfilled. Further, the functors $F^!\circ \Phi_E$ and $\Phi_E^* \circ F$ are equivalences, because they are adjoint each other. Consider the isomorphism $F^! (\bar{f}) : F^!\circ F\bigl|_{\mathcal C}\cong id_{\mathcal C}\stackrel{\sim}{\longrightarrow}F^! \circ \Phi_E\bigl|_{\mathcal C}$ on the subcategory ${\mathcal C}.$ By Proposition \ref{ext} we can extend it onto the whole $\db{M},$ so $id\stackrel{\sim}{\longrightarrow}F^!\circ\Phi_E.$ Since $F^!$ is the right adjoint to $F,$ we get the morphism of the functors $f : F\longrightarrow \Phi_E$ such that $f|_{\mathcal C} =\bar{f}.$ It can easily be checked that $f$ is an isomorphism. Indeed, let $C_Z$ be a cone of the morphism $f_Z : F(Z)\longrightarrow \Phi_E (Z).$ Since $F^! (f_Z)$ is an isomorphism, we obtain $F^! (Z)=0.$ Therefore, this implies that ${\h F(Y), {C_Z}}=0$ for any object $Y.$ Further, there are isomorphisms $F({\mathcal L}^k)\cong \Phi_E ({\mathcal L}^k)$ for any $k.$ Hence, we have $$ {\H i, {},{\mathcal L}^k, {\Phi^!_E (C_Z)}}={\H i, {},\Phi_E ({\mathcal L}^k ), {C_Z)}}={\H i, {}, F({\mathcal L}^k ), {C_Z)}}=0 $$ for all $k$ and $i.$ Thus, we obtain $\Phi^!_E (C_Z)=0.$ This implies that ${\h \Phi_E (Z), {C_Z}}=0.$ Finally, we get $F(Z)=C_Z [-1]\oplus \Phi_E (Z).$ But we know that ${\h F(Z)[1], {C_Z}}=0.$ Thus, $C_Z=0$ and $f$ is an isomorphism. 2) {\sc Uniqueness}. Suppose there exist two objects $E$ and $E_1$ of $D^{b}(M\times X)$ such that $\Phi_{E_1}\cong F\cong\Phi_{E_2}.$ Let us consider the complex (\ref{nob}) over $D^{b}(M\times X)$(see the proof Lemma \ref{obj}). $$ {\mathcal L}^{ -n}\boxtimes F(R_n)\longrightarrow\cdots\longrightarrow {\mathcal L}^{ -1}\boxtimes F(R_1)\longrightarrow {\mathcal O}_M \boxtimes F(R_0) $$ By Lemma \ref{pd1}, there exists a convolution of this complex and all convolutions are isomorphic. Let $G\in \db{M\times X}$ be a convolution of the complex (\ref{nob}). Now consider the following complexes $$ {\mathcal L}^{ -n}\boxtimes F(R_n)\longrightarrow\cdots\longrightarrow {\mathcal L}^{ -1}\boxtimes F(R_1)\longrightarrow {\mathcal O}_M \boxtimes F(R_0)\longrightarrow E_k $$ Again by Lemma \ref{pd1}, there exists a unique up to isomorphism convolutions of these complexes. Hence we have the canonical morphisms $G\longrightarrow E_1$ and $G\longrightarrow E_2.$ Moreover, it has been proved above (see the proof of Lemma \ref{obj}) that $ C_1\oplus E_1\cong G\cong C_2 \oplus E_2$ for large $n,$ where $E_k, C_k$ are objects of $\db{M\times X}$ such that ${H}^i (E_k)=0$ for $i\not\in [-a, 0]$ and ${H}^i (C_k)=0$ for $i\not\in [-n-a, -n]$ ($a$ was defined in \ref{boun}). Thus $E_1$ and $E_2$ are isomorphic. This completes the proof of Theorem \ref{main} $\Box$ \th{Theorem}\label{eqo} Let $M$ and $X$ be smooth projective varieties. Suppose $F : \db{M}\longrightarrow\db{X}$ is an equivalence. Then there exists a unique up to isomorphism object $E\in \db{M\times X}$ such that the functors $F$ and $\Phi_E$ are isomorphic. \par\vspace{1.0ex}\endgroup It follows immediately from Theorem \ref{main} \section{Derived categories of K3 surfaces} \refstepcounter{THNO}\par\vspace{1.5ex In this chapter we are trying to answer the following question: When are derived categories of coherent sheaves on two different K3 surfaces over field $\Bbb{C}$ equivalent? This question is interesting, because there exists a procedure for recovering a variety from its derived category of coherent sheaves if the canonical (or anticanonical) sheaf is ample. Besides, if $\db{X}\simeq\db{Y}$ and $X$ is a smooth projective K3 surface, then $Y$ is also a smooth projective K3 surface. This is true, because the dimension of a variety and Serre functor are invariants of a derived category. The following theorem is proved in \cite{BOr}. \th{Theorem}(see \cite{BOr})\label{rec} Let $X$ be smooth irreducible projective variety with either ample canonical or ample anticanonical sheaf. If $D=\db{X}$ is equivalent to $\db{X'}$ for some other smooth algebraic variety, then $X$ is isomorphic to $X'.$ \par\vspace{1.0ex}\endgroup However, there exist examples of varieties that have equivalent derived categories, if the canonical sheaf is not ample. Here we give an explicit description for K3 surfaces with equivalent derived categories. \th{Theorem}\label{K3} Let $S_1$ and $S_2$ be smooth projective K3 surfaces over field $\Bbb{C}.$ Then the derived categories $\db{S_1}$ and $\db{S_2}$ are equivalent as triangulated categories iff there exists a Hodge isometry $f_{\tau} : T_{S_1}\stackrel{\sim}{\longrightarrow}T_{S_2}$ between the lattices of transcendental cycles of $S_1$ and $S_2.$ \par\vspace{1.0ex}\endgroup Recall that the lattice of transcendental cycles $T_S$ is the orthogonal complement to Neron-Severi lattice $N_S$ in $H^2 ( S, {\Bbb{Z}} ).$ {\sf Hodge} isometry means that the one-dimensional subspace $H^{2,0} (S_1)\subset T_{S_1}\otimes\Bbb{C}$ goes to $H^{2,0} (S_2)\subset T_{S_2}\otimes\Bbb{C}.$ Now we need some basic facts about K3 surfaces (see \cite{Mu}). If $S$ is a K3 surface, then the Todd class $td_S$ of $S$ is equal to $1+2w,$ where $1\in H^0 ( S, {\Bbb{Z}} )$ is the unit element of the cohomology ring $H^* ( S, {\Bbb{Z}} )$ and $w\in H^4 ( S, {\Bbb{Z}} )$ is the fundamental cocycle of $S.$ The positive square root $\sqrt{td_S}= 1+w$ lies in $H^* ( S, {\Bbb{Z}} )$ too. Let $E$ be an object of $\db{S}$ then the Chern character $$ ch(E) = r(E) + c_1 (E) + \frac{1}{2} (c^2_1 - 2c_2) $$ belongs to integral cohomology $H^* ( S, {\Bbb{Z}} ).$ For an object $E,$ we put $v(E) = ch(E)\sqrt{td_S}\in H^* ( S, {\Bbb{Z}} )$ and call it the vector associated to $E$ (or Mukai vector). We can define a symmetric integral bilinear form $(,)$ on $H^* ( S, {\Bbb{Z}} )$ by the rule $$ (u, u') = rs' + sr' - \alpha\alpha' \in H^4 ( S, {\Bbb{Z}} )\cong{\Bbb{Z}} $$ for every pair $u=(r, \alpha, s), u'=(r', \alpha', s')\in H^0 ( S, {{\Bbb{Z}}} ) \oplus H^2 ( S, {{\Bbb{Z}}} )\oplus H^4 ( S, {{\Bbb{Z}}} ).$ By $\widetilde{H} ( S, {{\Bbb{Z}}} )$ denote $ H^* ( S, {\Bbb{Z}} )$ with this inner product $(,)$ and call it Mukai lattice. For any objects $E$ and $F,$ inner product $(v(E), v(F))$ is equal to the $H^4$ component of $ch(E)^{\vee} \cdot ch(F)\cdot td_S.$ Hence, by Riemann-Roch- Grothendieck theorem, we have $$ (v(E), v(F)) = \chi (E, F):=\sum_i (-1)^i dim{\E i, {}, E, F} $$ Let us suppose that $\db{S_1}$ and $\db{S_2}$ are equivalent. By Theorem \ref{main} there exists an object $E\in\db{S_1\times S_2}$ such that the functor $\Phi_E$ gives this equivalence. Now consider the algebraic cycle $Z:=p^*\sqrt{td_{S_1}}\cdot ch(E)\cdot \pi^*\sqrt{td_{S_2}}$ on the product $S_1\times S_2,$ where $p$ and $\pi$ are the projections $$ \begin{array}{ccc} S_1\times S_2&\stackrel{\pi}{\longrightarrow}&S_2\\ \llap{\ss{p}}\big\downarrow&&\\ S_1&& \end{array} $$ It follows from the following lemma that the cycle $Z$ belongs to integral cohomology $H^* (S_1\times S_2, {\Bbb{Z}}).$ \th{Lemma}\cite{Mu} For any object $E\in\db{S_1\times S_2}$ the Chern character $ch(E)$ is integral, which means that it belongs to $H^* (S_1\times S_2, {\Bbb{Z}})$ \par\vspace{1.0ex}\endgroup The cycle $Z$ defines a homomorphism from integral cohomology of $S_1$ to integral cohomology of $S_2$: $$ \begin{array}{cccc} f:&H^* (S_1, {\Bbb{Z}})&\longrightarrow&H^* (S_2, {\Bbb{Z}})\\ &\cup&&\cup\\ &\alpha&\mapsto&\pi_* (Z\cdot p^* (\alpha)) \end{array} $$ The following proposition is similar to Theorem 4.9 from \cite{Mu}. \th{Proposition}\label{mu} If $\Phi_E$ is full and faithful functor from $\db{S_1}$ to $\db{S_2}$ then: 1) $f$ is an isometry between $\widetilde{H} (S_1, {\Bbb{Z}})$ and $\widetilde{H} (S_2, {\Bbb{Z}}),$ 2) the inverse of $f$ is equal to the homomorphism $$ \begin{array}{cccc} f':&H^* (S_2, {\Bbb{Z}})&\longrightarrow&H^* (S_1, {\Bbb{Z}})\\ &\cup&&\cup\\ &\beta&\mapsto&p_* (Z^{\vee}\cdot \pi^* (\beta)) \end{array} $$ defined by $Z^{\vee}= p^* \sqrt{td_{S_1}}\cdot ch(E^{\vee})\cdot \pi^* \sqrt{td_{S_2}},$ where $E^{\vee}:={\pmb R^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}}{\mathcal H}om( E, {\mathcal O}_{S_1\times S_2}).$ \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } The left and right adjoint functors to $\Phi_E$ are: $$ \Phi_E^* =\Phi_E^! = p_* (E^{\vee}\otimes \pi^*(\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}))[2] $$ Since $\Phi_E$ is full and faithful, the composition $\Phi_E^*\circ \Phi_E$ is isomorphic to $id_{\db{S_1}}.$ Functor $id_{\db{S_1}}$ is given by the structure sheaf ${\mathcal O}_{\Delta}$ of the diagonal $\Delta\subset S_1\times S_1.$ Using the projection formula and Grothendieck-Riemann-Roch theorem, it can easily be shown that the composition $f'\circ f$ is given by the cycle $p_1^* \sqrt{td_{S_1}}\cdot ch({\mathcal O}_{\Delta})\cdot p_2^* \sqrt{td_{S_1}},$ where $p_1, p_2$ are the projections of $S_1\times S_1$ to the summands. But this cycle is equal to $\Delta.$ Therefore, $f'\circ f$ is the identity, and, hence, $f$ is an isomorphism of the lattices, because these lattices are free abelian groups of the same rank. Let $\nu_S : S\longrightarrow Spec\Bbb{C}$ be the structure morphism of $S.$ Then our inner product $(\alpha, \alpha')$ on $\widetilde{H} (S, {\Bbb{Z}})$ is equal to $\nu_*(\alpha^{\vee} \cdot\alpha').$ Hence, by the projection formula, we have $$ \begin{array}{rcl} (\alpha, f(\beta))&=&\nu_{S_2, *}(\alpha^{\vee}\cdot\pi_* (\pi^* \sqrt{td_{S_2}} \cdot ch(E)\cdot p^* \sqrt{td_{S_1}}\cdot p^* (\beta)))=\\ &=&\nu_{S_2, *}\pi_* (\pi^*(\alpha^{\vee})\cdot p^* (\beta)\cdot ch(E)\cdot \sqrt{td_{S_1\times S_2}})=\\ &=&\nu_{S_1\times S_2, *} (\pi^* (\alpha^{\vee})\cdot p^* (\beta)\cdot ch(E) \cdot \sqrt{td_{S_1\times S_2}}) \end{array} $$ for every $\alpha\in H^* ( S_2, {\Bbb{Z}} ), \beta\in H^* ( S_1, {\Bbb{Z}} ).$ In a similar way, we have $$ (\beta, f'(\alpha)) =\nu_{S_1\times S_2, *} ( p^* (\beta^{\vee})\cdot \pi^* (\alpha)\cdot ch(E)^{\vee} \cdot \sqrt{td_{S_1\times S_2}}) $$ Therefore, $(\alpha, f(\beta))=(f'(\alpha), \beta).$ Since $f'\circ f$ is the identity, we obtain $$ (f(\alpha), f(\alpha'))=(f'f(\alpha), \alpha')=(\alpha, \alpha') $$ Thus, $f$ is an isometry. $\Box$ \refstepcounter{THNO}\par\vspace{1.5ex Consider the isometry $f.$ Since the cycle $Z$ is algebraic, we obtain two isometries $f_{alg} : -N_{S_1}\bot U\stackrel{\sim}{\longrightarrow} -N_{S_2}\bot U$ and $f_{\tau} : T_{S_1}\stackrel{\sim}{\longrightarrow} T_{S_2},$ where $N_{S_1}, N_{S_2}$ are Neron-Severi lattices, and $T_{S_1}, T_{S_2}$ are the lattices of transcendental cycles. It is clear $f_{\tau}$ is a Hodge isometry. Thus we have proved that if the derived categories of two K3 surfaces are equivalent, then there exists a Hodge isometry between the lattices of transcendental cycles. \refstepcounter{THNO}\par\vspace{1.5ex Let us begin to prove the converse. Suppose we have a Hodge isometry $$ f_{\tau} : T_{S_2}\stackrel{\sim}{\longrightarrow}T_{S_1} $$ It implies from the following proposition that we can extend this isometry to Mukai lattices. \th{Proposition}\cite{Ni}\label{Ni} Let $\phi_1 , \phi_2 : T\longrightarrow H$ be two primitive embedding of a lattice $T$ in an even unimodular lattice $H.$ Assume that the orthogonal complement $N:=\phi_1 (T)^{\perp}$ in $H$ contains the hyperbolic lattice $U=\left(\begin{array}{cc}0&1\\1&0\\\end{array}\right)$ as a sublattice. Then $\phi_1$ and $\phi_2$ are equivalent, that means there exists an isometry $\gamma$ of $H$ such that $\phi_1 = \gamma\phi_2.$ \par\vspace{1.0ex}\endgroup We know that the orthogonal complement of $T_S$ in Mukai lattice $\widetilde{H}( S, {\Bbb{Z}} )$ is isomorphic to $N_S \perp U.$ By Proposition \ref{Ni}, there exists an isometry $$ f : \widetilde{H}( S_2 , {\Bbb{Z}} )\stackrel{\sim}{\longrightarrow} \widetilde{H}( S_1 , {\Bbb{Z}} ) $$ such that $f\bigl|_{T_{S_2}} = f_{\tau}.$ Put $v=f(0,0,1)=(r,l,s)$ and $u=f(1,0,0)=(p,k,q).$ We may assume that $r>1.$ One may do this, because there are two types of isometries on Mukai lattice that are identity on the lattice of transcendental cycles. First type is multiplication by Chern character $e^m$ of a line bundle: $$ \phi_m (r,l,s)=(r,l+rm,s+(m,l)+\frac{r}{2}m^2 ) $$ Second type is the change $r$ and $s$(see \cite{Mu}). So we can change $f$ in such a way that $r>1$ and $f\bigl|_{T_{S_2}}=f_{\tau}.$ First, note that vector $v\in U\perp -N_{S_1}$ is isotropic, i.e $(v,v)=0.$ It was proved by Mukai in his brilliant paper \cite{Mu} that there exists a polarization $A$ on $S_1$ such that the moduli space ${\mathcal M}_A (v)$ of stable bundles with respect to $A,$ for which vector Mukai is equal to $v,$ is projective smooth K3 surface. Moreover, this moduli space is fine, because there exists the vector $u\in U\perp -N_{S_1}$ such that $(v,u)=1.$ Therefore we have a universal vector bundle ${\mathcal E}$ on the product $S_1 \times {\mathcal M}_A (v).$ The universal bundle ${\mathcal E}$ gives the functor $\Phi_{\mathcal E} : \db{{\mathcal M}_A (v)}\longrightarrow\db{S_1}.$ Let us assume that $\Phi_{\mathcal E}$ is an equivalence of derived categories. In this case, the cycle $Z=\pi^*_{S_1}\sqrt{td_{S_1}}\cdot ch({\mathcal E})\cdot p^*\sqrt{td_{\mathcal M}}$ induces the Hodge isometry $$ g : \widetilde{H} ( {\mathcal M}_A (v), {\Bbb{Z}} ) \longrightarrow\widetilde{H} ( S_1, {\Bbb{Z}} ), $$ such that $g(0,0,1)=v=(r,l,s).$ Hence, $f^{-1}\circ g$ is an isometry too, and it sends $(0,0,1)$ to $(0,0,1).$ Therefore $f^{-1}\cdot g$ gives the Hodge isometry between the second cohomologies, because for a K3 surface $S$ $$ \begin{array}{c} H^2 ( S, {\Bbb{Z}} )=(0,0,1)^{\perp}\Big/{\Bbb{Z}}(0,0,1). \end{array} $$ Consequently, by the strong Torelli theorem (see \cite{Lo}), the surfaces $S_2$ and ${\mathcal M}_A (v)$ are isomorphic. Hence the derived categories of $S_1$ and $S_2$ are equivalent. \refstepcounter{THNO}\par\vspace{1.5ex Thus, to conclude the proof of Theorem \ref{K3}, it remains to show that the functor $\Phi_{\mathcal E}$ is an equivalence. First, we show that the functor $\Phi_{\mathcal E}$ is full and faithful. This is a special case of the following more general statement, proved in \cite{BO}. \th{Theorem}\cite{BO}\label{mai} Let $M$ and $X$ be smooth algebraic varieties and\hfill\\ $E\in{\db {M\times X}}.$ Then $\Phi_{E}$ is fully faithful functor, iff the following orthogonality conditions are verified: $$ \begin{array}{lll} i) & {\H i, X, \Phi_E({\mathcal O}_{t_1}), {\Phi_{E}({\mathcal O}_{t_2})}} = 0 & \qquad \mbox{for every }\: i\;\mbox{ and } t_1\ne t_2.\\ &&\\ ii) & {\H 0, X, \Phi_E({\mathcal O}_t), {\Phi_E({\mathcal O}_t)}} = k,&\\ &&\\ & {\H i, X, \Phi_E({\mathcal O}_t), {\Phi_E({\mathcal O}_t)}} = 0 , & \qquad \mbox{ for }i\notin [0, dim M]. \end{array} $$ Here $t,$ $t_{1},$ $t_{2}$ are points of $M,$ ${\mathcal O}_{t_{i}}$ are corresponding skyscraper sheaves. \par\vspace{1.0ex}\endgroup In our case, $\Phi_{\mathcal E} ({\mathcal O}_t)=E_t,$ where $E_t$ is stable sheaf with respect to the polarization $A$ on $S_1$ for which $v(E_t)=v.$ All these sheaves are simple and ${\E i, {}, E_t , {E_t}}=0$ for $i\not\in [0,2].$ This implies that condition 2) of Theorem \ref{mai} is fulfilled. All $E_t$ are stable sheaves, hence ${\h E_{t_1}, {E_{t_2}}}=0.$ Further, by Serre duality ${\E 2, {}, E_{t_1}, {E_{t_2}}}=0.$ Finally, since the vector $v$ is isotropic, we obtain ${\E 1, {}, E_{t_1}, {E_{t_2}}}=0.$ This yields that $\Phi_{\mathcal E}$ is full and faithful. As our situation is not symmetric (a priori), it is not clear whether the adjoint functor to $\Phi_{\mathcal E}$ is also full and faithful. Some additional reasoning is needed. \th{Theorem}\label{equi} In the above notations, the functor $\Phi_{\mathcal E}: \db{{\mathcal M}_A (v)}\longrightarrow\db{S_1}$ is an equivalence. \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Assume the converse, i.e. $\Phi_{\mathcal E}$ is not an equivalence, then, since the functor $\Phi_{\mathcal E}$ is full and faithful, there exists an object $C\in \db{S_1}$ such that $\Phi^*_{\mathcal E} (C)=0.$ By Proposition \ref{mu}, the functor $\Phi_{\mathcal E}$ induces the isometry $f$ on the Mukai lattices, hence the Mukai vector $v(C)$ is equal to $0.$ Object $C$ satisfies the conditions ${\H i, {}, C, {E_t}}=0$ for every $i$ and all $t\in {\mathcal M}_A (v),$ where $E_t$ are stable bundles on $S_1$ with the Mukai vector $v.$ Denote by $H^i (C)$ the cohomology sheaves of the object $C.$ There is a spectral sequence which converges to ${\H i, {}, C, {E_t}}$ \begin{equation}\label{seq} E^{p,q}_2 = {\E p, {}, H^{-q} (C), {E_t}} \Longrightarrow {\H {p+q}, {}, C, {E_t}} \end{equation} It is depicted in the following diagram \begin{picture}(400,160) \put(200,10){\vector(0,1){140}} \put(130,80){\vector(1,0){160}} \put(201,145){$q$} \put(275,85){$p$} \put(200,120){\circle*{3}} \put(200,100){\circle*{3}} \put(200,60){\circle*{3}} \put(200,80){\circle*{3}} \put(240,120){\circle*{3}} \put(240,100){\circle*{3}} \put(240,80){\circle*{3}} \put(240,60){\circle*{3}} \put(220,120){\circle*{3}} \put(220,100){\circle*{3}} \put(220,80){\circle*{3}} \put(220,60){\circle*{3}} \put(212,135){$\vdots$} \put(212,40){$\vdots$} \put(203,78){\vector(2,-1){33}} \put(203,98){\vector(2,-1){33}} \put(203,118){\vector(2,-1){33}} \put(219,72){\scriptsize{$d_2$}} \put(219,92){\scriptsize{$d_2$}} \put(219,112){\scriptsize{$d_2$}} \end{picture} We can see that ${\E 1, {}, H^q (C), {E_t}}=0$ for every $q$ and all $t,$ and every morphism $d_2$ is an isomorphism. To prove the theorem, we need the following lemma. \th{Lemma}\label{GN} Let $G$ be a sheaf on K3 surface $S_1$ such that ${\E 1, {}, G, {E_t}}=0$ for all $t.$ Then there exists an exact sequence $$ 0\longrightarrow G_1 \longrightarrow G\longrightarrow G_2 \longrightarrow 0 $$ that satisfies the following conditions: $$ \begin{array}{llll} 1)\; {\E i, {}, G_1, {E_t}}=0 \quad \mbox{ for every }\; i\ne 2,\;\mbox{and} & {\E 2, {}, G_1, {E_t}}\cong{\E 2, {}, G, {E_t}} \\ 2)\; {\E i, {}, G_2, {E_t}}=0 \quad \mbox{ for every } \; i\ne 0,\;\mbox{and} & {\H {}, {}, G_2, {E_t}}\cong{\H {}, {}, G, {E_t}} \end{array} $$ and $p_A (G_2) < p_A (G) < p_A (G_1)$ (where $p_A(F)$ is a Gieseker slope, i.e., a polynomial such that $p_A(F)(n)=\chi(F(nA))/r(F).$) \par\vspace{1.0ex}\endgroup \par\noindent{\bf\ Proof. } Firstly, there is a short exact sequence $$ 0\longrightarrow T\longrightarrow G\longrightarrow \widetilde{G}\longrightarrow 0, $$ where $T$ is a torsion sheaf, and $\widetilde{G}$ is torsion free. Secondly, there is a Harder-Narasimhan filtration $0=I_0\subset ...\subset I_n =\widetilde{G}$ for $\widetilde{G}$ such that the successive quotients $I_i / I_{i-1}$ are $A$-semistable, and $p_A (I_i / I_{i-1})>p_A( I_j / I_{j-1} )$ for $i<j.$ Now, combining $T$ and the members of the filtration for which $p_A (I_i / I_{i-1})>p_A (E_t)$ (resp. $=,$ $<$) to one, we obtain the 3-member filtration on $G$ $$ 0=J_0 \subset J_1 \subset J_2 \subset J_3 =G. $$ Let $K_i$ be the quotients sheaves $J_i / J_{i-1}.$ We have $$ p_A (K_1) > p_A (K_2)=p_A (E_t) > p_A (K_3) $$ (we suppose, if needed, $p_A (T)= +\infty$). Moreover, it follows from stability of $E_t$ that $$ {\h K_1, {E_t}}=0 \qquad \mbox{ and }\qquad {\E 2, {}, K_3, {E_t}}=0 $$ Combining this with the assumption that ${\E 1, {}, G, {E_t}}=0,$ we get ${\E 1, {}, K_2, {E_t}}=0.$ To prove the lemma it remains to show that $K_2 =0.$ Note that $K_2$ is $A$-semistable. Hence there is a Jordan-H\"older filtration for $K_2$ such that the successive quotients are $A$-stable. The number of the quotients is finite. Therefore we can take $t_0$ such that $$ {\h K_2, {E_{t_0}}}=0 \qquad \mbox{ and }\qquad {\E 2, {}, K_2, {E_{t_0}}}=0 $$ Consequently, $\chi ( v(K_2), v(E_t) )=0.$ Thus, as ${\E 1, {}, K_2, {E_t}}=0$ for all $t,$ we obtain ${\E i, {}, K_2, {E_t}}=0$ for every $i$ and all $t.$ Further, let us consider $\Phi^*_{\mathcal E} (K_2).$ We have $$ {\H \begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture} , {}, \Phi^*_{\mathcal E}(K_2), {{\mathcal O}_t}}\cong {\H \begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture} , {}, K_2, {E_t}}=0, $$ This implies $\Phi^*_{\mathcal E} (K_2)=0.$ Hence $v(K_2)=0,$ because $f$ is an isometry. And, finally, $K_2=0.$ The lemma is proved. $\Box$ Let us return to the theorem. The object $C$ possesses at least two non-zero consequent cohomology sheaves $H^p (C)$ and $H^{p+1} (C)$ . They satisfy the condition of Lemma \ref{GN} Hence there exist decompositions with conditions 1),2): $$ 0 \longrightarrow H^p_1 \longrightarrow H^p (C) \longrightarrow H^p_2 \longrightarrow 0 \quad\mbox{and}\quad 0 \longrightarrow H^{p+1}_1 \longrightarrow H^{p+1} (C) \longrightarrow H^{p+1}_2 \longrightarrow 0 $$ Now consider the canonical morphism $H^{p+1} (C) \longrightarrow H^p (C)[2].$ It induces the morphism $ s : H^{p+1}_1 \longrightarrow H^p_2 [2].$ By $Z$ denote a cone of $s.$ Since $d_2$ of the spectral sequence (\ref{seq}) is an isomorphism, we obtain $$ {\H \begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}, {}, Z, {E_t}}=0 \qquad \mbox{ for all } t. $$ Consequently, we have $\Phi^*_{\mathcal E} (Z)=0.$ On the other hand, we know that $p_A (H^{p+1}_1)>p_A (E_t)>p_A (H^p_2).$ Therefore $v(Z)\ne 0.$ This contradiction proves the theorem. $\Box$ There exists the another version of Theorem \ref{K3} \th{Theorem} Let $S_1$ and $S_2$ be smooth projective K3 surfaces over field $\Bbb{C}.$ Then the derived categories $\db{S_1}$ and $\db{S_2}$ are equivalent as triangulated categories iff there exists a Hodge isometry $f: \widetilde{H} ( S_1, {\Bbb{Z}} )\stackrel{\sim}\longrightarrow \widetilde{H} ( S_2, {\Bbb{Z}} )$ between the Mukai lattices of $S_1$ and $S_2.$ \par\vspace{1.0ex}\endgroup Here the `{\sf Hodge} isometry' means that the one-dimensional subspace $H^{2,0} (S_1)\subset \widetilde{H} ( S_1, {\Bbb{Z}} )\otimes\Bbb{C}$ goes to $H^{2,0} (S_2)\subset \widetilde{H} ( S_2, {\Bbb{Z}} )\otimes\Bbb{C}.$ \sec{Appendix.} The facts, collected in this appendix, are not new; they are known. However, not having a good reference, we regard it necessary to give a proof for the statement, which is used in the main text. We exploit the technique from \cite{IM}. Let $X$ be a smooth projective variety and $L$ be a very ample invertible sheaf on $X$ such that ${\rm H}^i ( X , L^k ) =0$ for any $k>0$ , when $i\ne 0.$ Denote by $A$ the coordinate algebra for $X$ with respect to $L,$ i.e. $A = \bigoplus\limits^{\infty}_{k=0} {\rm H}^0 ( X , L^k ).$ Now consider the variety $X^n.$ First, we introduce some notations. Define subvarieties $\Delta^{(n)}_{(i_1 , ..., i_k )(i_{k+1}, ..., i_m )} \subset X^n$ by the following rule: $$ \Delta^{(n)}_{(i_1 , ..., i_k )(i_{k+1}, ..., i_m )} :=\{ (x_1 , ..., x_n ) | x_{i_1}=\cdots = x_{i_k} ; x_{i_{k+1}}=\cdots = x_{m} \} $$ By $S^{(n)}_i$ denote $\Delta^{(n)}_{(n,..., i)}.$ It is clear that $S^{(n)}_i \cong X^i.$ Further, let $T^{(n)}_i := \bigcup\limits^{i-1}_{k=1} \Delta^{(n)}_{(n,..., i)(k, k-1)}$ (note that $T^{(n)}_1$ and $T^{(n)}_2$ are empty) and let $\Sigma^{(n)} := \bigcup\limits^{n}_{k=1} \Delta^{(n)}_{(k, k-1)}.$ We see that $T^{(n)}_i \subset S^{(n)}_i.$ Denote by ${\mathcal I}^{(n)}_i$ the kernel of the restriction map ${\mathcal O}_{S^{(n)}_i}\longrightarrow {\mathcal O}_{T^{(n)}_i}\longrightarrow 0.$ Using induction by $n,$ it can easily be checked that the following complex on $X^n$ $$ P^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_n : 0\longrightarrow J_{\Sigma^{(n)}} \longrightarrow {\mathcal I}^{(n)}_n \longrightarrow {\mathcal I}^{(n)}_{n-1} \longrightarrow \cdots \longrightarrow {\mathcal I}^{(n)}_2 \longrightarrow {\mathcal I}^{(n)}_1 \longrightarrow 0 $$ is exact. (Note that ${\mathcal I}^{(n)}_1 = {\mathcal O}_{\Delta^{(n)}_{n,...,1}}$ and ${\mathcal I}^{(n)}_2 = {\mathcal O}_{\Delta^{(n)}_{n,...,2}}$). For example, for $n=2$ this complex is a short exact sequence on $X\times X$: $$ P^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_2 : 0\longrightarrow J_{\Delta}\longrightarrow {\mathcal O}_{X\times X}\longrightarrow {\mathcal O}_{\Delta}\longrightarrow 0 $$ Denote by $\pi^{(n)}_i$ the projection of $X^n$ onto $i^{th}$ component, and by $\pi^{(n)}_{ij}$ denote the projection of $X^n$ onto the product of $i^{th}$ and $j^{th}$ components. Let $B_n := {\rm H}^0 ( X^n , J_{\Sigma^{(n)}}\otimes (L\boxtimes \cdots \boxtimes L))$ and let $R_{n-1} := R^{0} \pi^{(n)}_{1 *} (J_{\Sigma^{(n)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L)).$ \ap{Proposition} Let $L$ be a very ample invertible sheaf on $X$ as above. Suppose that for any $m$ such that $1< m \le n+dim X+2$ the following conditions hold: $$ \begin{array}{lll} a)& {\rm H}^i ( X^m , J_{\Sigma^{(m)}}\otimes (L\boxtimes \cdots \boxtimes L))=0&\; \mbox{for} \quad i\ne 0\\ b)& R^{i} \pi^{(m)}_{1 *} (J_{\Sigma^{(m)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L))=0&\; \mbox{for}\quad i\ne 0\\ c)& R^{i} \pi^{(m)}_{1m *} (J_{\Sigma^{(m)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L\boxtimes {\mathcal O}))=0&\; \mbox{for} \quad i\ne 0 \end{array} $$ Then we have: 1) algebra $A$ is n-Koszul, i.e the sequence $$ B_n \otimes_k A\longrightarrow B_{n-1}\otimes_k A\longrightarrow \cdots\longrightarrow B_1\otimes_k A\longrightarrow A\longrightarrow k\longrightarrow 0 $$ is exact; 2) the following complexes on $X$: $$ A_{k-n}\otimes R_n \longrightarrow A_{k-n+1}\otimes R_{n-1}\longrightarrow\cdots\longrightarrow A_{k-1}\otimes R_1 \longrightarrow A_k \otimes R_0 \longrightarrow {\mathcal L}^{k}\longrightarrow 0 $$ are exact for any $k\ge 0$ (if $ k-i<0,$ then $A_{k-i}=0$ by definition); 3) the complex $$ { L}^{ -n}\boxtimes R_n\longrightarrow\cdots\longrightarrow {L}^{ -1}\boxtimes R_1\longrightarrow {\mathcal O}_M \boxtimes R_0 \longrightarrow {\mathcal O}_{\Delta} $$ gives n-resolution of the diagonal on $X\times X,$ i.e. it is exact. \par\endgroup \par\noindent{\bf\ Proof. } 1) First, note that $$ {\rm H}^i ( X^m , {\mathcal I}^{(m)}_k \otimes (L\boxtimes \cdots \boxtimes L))={\rm H}^i ( X^{k-1} , J_{\Sigma^{(k-1)}}\otimes (L\boxtimes \cdots \boxtimes L))\otimes A_{m-k+1} $$ By condition a), they are trivial for $i\ne 0.$ Consider the complexes $P^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_m \otimes (L\boxtimes\cdots \boxtimes L)$ for $m\le n+dim X +1.$ Applying the functor ${\rm H}^0$ to these complexes and using condition a), we get the exact sequences: $$ 0\longrightarrow B_m \longrightarrow B_{m-1}\otimes_k A_1 \longrightarrow \cdots\longrightarrow B_1\otimes_k A_{m-1} \longrightarrow A_m \longrightarrow 0 $$ for $m\le n+dimX+1.$ Now put $m=n+dimX+1.$ Denote by $W^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_m$ the complex $$ {\mathcal I}^{(m)}_m \longrightarrow {\mathcal I}^{(m)}_{m-1} \longrightarrow \cdots \longrightarrow {\mathcal I}^{(m)}_2 \longrightarrow {\mathcal I}^{(m)}_1 \longrightarrow 0 $$ Take the complex $W^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_m \otimes (L\boxtimes\cdots\boxtimes L\boxtimes L^i )$ and apply functor ${\rm H}^0$ to it. We obtain the following sequence: $$ B_{m-1}\otimes_k A_i \longrightarrow B_{m-2}\otimes_k A_{i+1} \longrightarrow\cdots\longrightarrow B_1\otimes_k A_{m-1} \longrightarrow A_m \longrightarrow 0 $$ Its cohomologies are ${\rm H}^j ( X^m , J_{\Sigma^{(m)}}\otimes (L\boxtimes \cdots \boxtimes L\boxtimes L^i)).$ It follows from condition b) that $$ {\rm H}^j ( X^m , J_{\Sigma^{(m)}}\otimes (L\boxtimes \cdots \boxtimes L\boxtimes L^i))={\rm H}^j ( X, R^0 \pi^{(m)}_{m*}(J_{\Sigma^{(m)}}\otimes (L\boxtimes \cdots \boxtimes L\boxtimes {\mathcal O}))\otimes L^i ). $$ Hence they are trivial for $j> dimX.$ Consequently, we have the exact sequences: $$ B_n \otimes_k A_{m-n+i-1}\longrightarrow B_{n-1}\otimes_k A_{m-n+i} \longrightarrow \cdots\longrightarrow B_1\otimes_k A_{m+i-2}\longrightarrow A_{m+i-1} $$ for $i\ge 1.$ And for $i\le 1$ the exactness was proved above. Thus, algebra $A$ is n-Koszul. 2) The proof is the same as for 1). We have isomorphisms $$ R^i \pi^{(m)}_{1*}({\mathcal I}^{(m)}_k \otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L))\cong R^i \pi^{(k-1)}_{1*}( J_{\Sigma^{(k-1)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L))\otimes A_{m-k+1} $$ Applying functor $R^0 \pi^{(m)}_{1*}$ to the complexes $P^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_{m} \otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L))$ for $m\le n+dimX+2,$ we obtain the exact complexes on $X$ $$ 0\longrightarrow R_{m-1} \longrightarrow A_{1}\otimes R_{m-2}\longrightarrow\cdots\longrightarrow A_{m-2}\otimes R_1 \longrightarrow A_{m-1} \otimes R_0 \longrightarrow {\mathcal L}^{m-1}\longrightarrow 0 $$ for $m\le n+dimX+2.$ Put $m=n+dimX+2.$ Applying functor $R^0 \pi^{(m)}_{1*}$ to the complex $W^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_m \otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L \boxtimes L^i)),$ we get the complex $$ A_{i}\otimes R_{m-2}\longrightarrow\cdots\longrightarrow A_{m+i-3}\otimes R_1 \longrightarrow A_{m+i-2} \otimes R_0 \longrightarrow {\mathcal L}^{m+i-2}\longrightarrow 0 $$ The cohomologies of this complex are $$ R^{j} \pi^{(m)}_{1 *} (J_{\Sigma^{(m)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L \boxtimes L^i))\cong R^{j}p_{1*}(R^{0} \pi^{(m)}_{1m *} (J_{\Sigma^{(m)}}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L\boxtimes {\mathcal O}))\otimes ({\mathcal O}\boxtimes L^i)) $$ They are trivial for $j> dimX.$ Thus, the sequences $$ A_{k-n}\otimes R_n \longrightarrow A_{k-n+1}\otimes R_{n-1}\longrightarrow\cdots\longrightarrow A_{k-1}\otimes R_1 \longrightarrow A_k \otimes R_0 \longrightarrow {\mathcal L}^{k}\longrightarrow 0 $$ are exact for all $k\ge 0.$ 3) Consider the complex $W^{\begin{picture}(4,4)\put(2,3){\circle*{1.5}}\end{picture}}_{n+2}\otimes ({\mathcal O}\boxtimes L\boxtimes \cdots \boxtimes L \boxtimes L^{-i}).$ Applying the functor $R^0 \pi^{(n+2)}_{1(n+2)*}$ to it, we obtain the following complex on $X\times X$: $$ {L}^{-n}\boxtimes R_n\longrightarrow\cdots\longrightarrow {L}^{-1}\boxtimes R_1\longrightarrow {\mathcal O}_M \boxtimes R_0 \longrightarrow {\mathcal O}_{\Delta} $$ By condition c), it is exact. This finishes the proof. Note that for any ample invertible sheaf $L$ we can find $j$ such that for the sheaf $L^j$ the conditions a),b),c) are fulfilled.

9606

alg-geom/9606003

en

https://arxiv.org/abs/alg-geom/9606003

alg-geom/9606003

V. Batyrev

Victor V. Batyrev, Yuri Tschinkel

Height Zeta Functions of Toric Varieties

27 pages, AMS-LaTeX

LMENS-96-9

We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors

alg-geom

\section{Introduction} \bigskip Let $X$ be a $d$-dimensional algebraic variety defined over a number field $F$. Denote by ${\cal L}=(L,\{\|\cdot\|_v\})$ a metrized line bundle on $X$ , i.e. a line bundle $L$ together with a family of $v$-adic metrics, where $v$ runs over the set ${\operatorname{Val} }(F)$ of all valuations of $F$. For any locally closed algebraic subset $Y\subset X$ we denote by $Y(F)$ the set of $F$-rational points in $Y$. A metrized line bundle ${\cal L}$ defines a height function $$ H_{\cal L} \, :\, X(F) \rightarrow {\bold R }_{>0}. $$ Assume that a subset $Y \subset X$ and a bundle $L$ are choosen in such a way that $$ N(Y,{\cal L},B) := \# \{ x\in Y(F) \mid H_{\cal L}(x)\le B \} < \infty $$ for all $B \in {\bold R }_{>0}$ (e.g., this holds for any $Y\subset X$ if $L$ is ample). Then the asymptotic behavior of the counting function $N(Y,{\cal L},B)$ as $B \rightarrow \infty$ is determined by analytic properties of the {\em height zeta function} $Z(Y,{\cal L},s)$ defined by the series $$ Z(Y,{\cal L},s) : =\sum_{x\in Y(F)} H_{\cal L}(x)^{-s} $$ which converges for ${ \operatorname{Re} }(s)\gg 0$. More precisely, one has the following Tauberian statement: \begin{theo} {\rm \cite{BaTschi1}} Assume that the series $Z(Y,{\cal L},s)$ is absolutely convergent for ${ \operatorname{Re} }(s) > a> 0$ and that there exists some positive integer $b$ such that $$ Z(Y,{\cal L},s)=\frac{g(s)}{(s-a)^b} + h(s) $$ where $g(s)$ and $h(s)$ are functions holomorphic in the domain ${ \operatorname{Re} }(s)\ge a$ and $g(a)\neq 0$. Then the following asymptotic formula holds: $$ N(Y,{\cal L},B)=\frac{g(a)}{a(b-1)!} B^{a}(\log B)^{b-1}(1+o(1))\;{ for} \;B\rightarrow \infty. $$ \end{theo} \begin{dfn} {\rm Denote by $NS(X)$ be the Neron-Severi group of $X$ and by $NS(X)_{{\bold R }} = NS(X)\otimes {\bold R }$. The {\em cone of effective divisors of $X$} is the closed cone $ \Lambda _{\rm eff}(X)\subset NS(X)_{{\bold R }}$ generated by the classes of effective divisors. } \end{dfn} \noindent Let $[L] \in NS(X)$ be the first Chern class of $L$. Denote by ${\cal K}_X = (K_X,\{\|\cdot\|_v\})$ the metrized canonical line bundle on $X$. \begin{dfn} {\rm Let $L$ be any line bundle on $X$. Define $$ a(L) : =\inf \{ a\in {\bold R } \,|\, a[L]+[K_X]\in \Lambda _{\rm eff}(X)\}. $$} \end{dfn} One of our main results in this paper is the following theorem: \begin{theo} Let $T$ be a $d$-dimensional algebraic torus over a number field $F$, $X$ a smooth projective toric variety containing $T$ as a Zariski open subset, and ${\cal L}$ a metrized line bundle on $X$ $($with the metrization introduced in {\rm \cite{BaTschi1}}$)$. Assume that the class $[L]$ is contained in the interior of the cone of effective divisors. Then the height zeta function has the following representation: $$ Z(T,{\cal L},s)=\frac{g(s)}{(s-a(L))^{b(L)}} + h(s) $$ where $g(s)$ and $h(s)$ are functions holomorphic in the domain ${ \operatorname{Re} }(s)\ge a(L)$, $g(a(L))\neq 0$, and $b(L)$ is the codimension of the minimal face of $ \Lambda _{\rm eff}(X)$ which contains the class $a(L)[L]+[K_X]$. \label{t1} \end{theo} \begin{coro} We have the following asymptotic formula $$ N(T,{\cal L},B)= \frac{g(a(L))}{a(L)(b(L)-1)!}B^{a(L)}(\log B)^{b(L)-1}(1+o(1))\;{ for} \;B\rightarrow \infty. $$ \end{coro} \noindent The paper is organized as follows: \medskip The technical heart of the paper is contained in Section 2, where we investigate analytic properties of some complex valued functions related to convex cones. In Section 3, we review basic facts from harmonic analysis on the adele group of an algebraic torus. In Section 4, we recall the terminology from the theory of toric varieties as well as the definition and main properties of heights on toric varieties. In Section 5, we give the proof of \ref{t1}. We remark that the most subtle part in the statement of \ref{t1} is the nonvanishing of the asymptotic constant $g(a(L)) \neq 0$. \bigskip \section{Technical theorems} Let $I$ and $J$ be two positive integers, ${{\bold R }}\lbrack {\bold s},{\bold t} \rbrack$ (resp. ${{\bold C }}\lbrack {\bold s}, {\bold t} \rbrack$) the ring of polynomials in $I +J$ variables $s_1, \ldots, s_I, t_1, \ldots, t_J$ with coefficients in ${{\bold R }}$ (resp. in ${{\bold C }}$) and ${{\bold C }}\lbrack \lbrack {\bold s}, {\bold t} \rbrack \rbrack$ the ring of formal power series in $s_1, \ldots, s_I,t_1, \ldots, t_J$ with complex coefficients. \begin{dfn} {\rm Two elements $f({\bold s},{\bold t}),\, g({\bold s},{\bold t}) \in {{\bold C }}\lbrack \lbrack {\bold s}, {\bold t} \rbrack \rbrack$ will be called {\em coprime}, if $g.c.d.(f({\bold s},{\bold t}),\, g({\bold s},{\bold t})) =1$. } \end{dfn} \begin{dfn} {\rm Let $f({\bold s},{\bold t})$ be an element of ${{\bold C }}\lbrack \lbrack {\bold s}, {\bold t} \rbrack \rbrack$. By the {\em order} of a monomial $s_1^{ \alpha _1} \cdots s_I^{ \alpha _I}t_1^{ \beta _1} \cdots t_J^{ \beta _J}$ we mean the sum of the exponents $$ \alpha _1+ \cdots + \alpha _I + \beta _1 + \cdots + \beta _J.$$ By the {\em multiplicity $\mu(f({\bold s},{\bold t}))$ of $f({\bold s},{\bold t})$ at ${\bold 0} = (0, \ldots, 0)$} we always mean the minimal order of non-zero monomials appearing in the Taylor expansion of $f({\bold s},{\bold t})$ at ${\bold 0}$ . } \label{mult1} \end{dfn} \begin{dfn} {\rm Let $f({\bold s},{\bold t})$ be a meromorphic at ${\bold 0}$ function. Define the {\em multiplicity $\mu(f({\bold s},{\bold t}))$ } of $f({\bold s},{\bold t})$ at ${\bold 0}$ as \[ \mu(f({\bold s},{\bold t})) = \mu(g_1({\bold s},{\bold t})) - \mu(g_2({\bold s},{\bold t})) \] where $g_1({\bold s},{\bold t})$ and $g_2({\bold s},{\bold t})$ are two coprime elements in ${{\bold C }}\lbrack \lbrack {\bold s},{\bold t} \rbrack \rbrack$ such that $f = g_1/g_2$. } \label{mult2} \end{dfn} \begin{rem} {\rm It is easy to show that for any two meromorphic at ${\bold 0}$ functions $f_1({\bold s},{\bold t})$ and $f_2({\bold s},{\bold t})$, one has (i) $\mu(f_1 \cdot f_2) = \mu(f_1) + \mu(f_2)$ (in particular, one can omit "coprime" in Definition \ref{mult2}); (ii) $\mu(f_1 + f_2) \geq \min \{ \mu(f_1), \mu(f_2) \}$; (iii) $\mu(f_1 + f_2) = \mu(f_1)$ if $\mu(f_2) > \mu(f_1)$. } \label{mult3} \end{rem} Using the properties \ref{mult3}(i)-(ii), one immediately obtains from Definition \ref{mult1} the following: \begin{prop} Let $f_1({\bold s},{\bold t}) \in {{\bold C }}\lbrack \lbrack {\bold s}, {\bold t} \rbrack \rbrack$ and $f_2({\bold s}) \in {{\bold C }}\lbrack \lbrack {\bold s}\rbrack \rbrack$ be two analytic at ${\bold 0}$ functions, $l({\bold s}) \in {\bold R}[{\bold s}]$ a homogeneous linear function in the variables $s_1, \ldots, s_I$, $${\bold \gamma} = ({\bold \gamma}^I, {\bold \gamma}^J) = ( \gamma_1, \ldots, \gamma_I, \gamma_1', \ldots, \gamma_J') \in {{\bold C }}^{I +J}$$ an arbitrary complex vector with $l({\bold \gamma}^I) \neq 0$, and $g({\bold s},{\bold t}) := f_1({\bold s},{\bold t})/f_2({\bold s})$. Then the multiplicity of the function $$ \tilde{g}({\bold s},{\bold t}): = \left(\frac{\partial}{\partial z}\right)^k g({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z\cdot {\bold \gamma}^J ) |_{z = - l({\bold s})/l({\bold \gamma}^I)} $$ at ${\bold 0}$ is at least $\mu(g) - k$, if $$ f_2({\bold s} + z \cdot \gamma^I) |_{z = - l({\bold s})/l({\bold \gamma}^I)} $$ is not identically zero. \label{mult4} \end{prop} Let $\Gamma \subset {{\bold Z }}^{I+J}$ be a sublattice, $\Gamma_{{\bold R }} \subset {{\bold R }}^{I + J}$ (resp. $\Gamma_{{\bold C }} \subset {{\bold C }}^{I +J}$) the scalar extension of $\Gamma$ to a ${{\bold R }}$-subspace (resp. to a ${{\bold C }}$-subspace). We always assume that $\Gamma_{{\bold R }} \cap {{\bold R }}_{\geq 0}^{I +J} = 0$ and $ \Gamma _{{\bold R }} \cap {\bold R }^J = 0$. We set $P_{{\bold R }}: = {{\bold R }}^{I +J}/\Gamma_{{\bold R }}$ and $P_{{\bold C }}: = {{\bold C }}^{I+J}/\Gamma_{{\bold C }}$. Let $\pi^I$ be the natural projection ${\bold C }^{I+J} \rightarrow {\bold C }^I$. Denote by $\psi$ (resp. by $\psi^I$) the canonical surjective mapping ${\bold C }^{I+J} \rightarrow P_{{\bold C }}$ (resp. ${\bold C}^{I} \rightarrow {\bold C }^I/\pi^I( \Gamma _{{\bold C }})$). \begin{dfn} {\rm A complex analytic function $$ f({\bold s}, {\bold t})= f(s_1, \ldots, s_I, t_1, \ldots, t_J): U \rightarrow {\bold C} $$ defined on an open subset $U \subset {\bold C}^{I +J}$ is said to {\em descend to $P_{{\bold C }}$} if $ f({\bold u} ) = f({\bold u}')$ for all ${\bold u}, {\bold u}' \in U$ with ${\bold u} - {\bold u}' \in \Gamma _{{\bold C }}$. } \end{dfn} \begin{rem} {\rm By definition, if $f({\bold s}, {\bold t})$ descends to $P_{{\bold C }}$, then there exists an analytic function $g$ on $\psi(U) \subset P_{{\bold C }}$ such that $f = g \circ \psi$. Using Cauchy-Riemann equations, one immediatelly obtains that $f$ descends to $P_{{\bold C }}$ if and only if for any vector ${\bold \alpha } \in \Gamma_{{\bold R }}$ and any ${\bold u}= (u_1, \ldots, u_{I+J}) \in U$ such that ${\bold u} + i{\bold \alpha } \in U$, one has \[ f({\bold u}+ i{\bold \alpha }) = f({\bold u}). \]} \label{desc} \end{rem} \begin{dfn} {\rm An analytic function $f({\bold s}, {\bold t})$ in the domain ${ \operatorname{Re} }({\bold s}) \in {{\bold R }}_{>0}^I$, ${ \operatorname{Re} }({\bold t}) \in {{\bold R }}_{> - \delta _0}^J$ $(${for some} $ \delta _0 >0)$ is called {\em good with respect to $\Gamma$ and the set of variables $\{s_1,...,s_I\}$} if it satisfies the following conditions: {(i)} $f({\bold s},{\bold t})$ descends to $P_{{\bold C }}$; {(ii)} There exist pairwise coprime linear homogeneous polynomials $$ l_1({\bold s}), \ldots, l_p({\bold s}) \in {{\bold R }}\lbrack s_1, \ldots, s_I \rbrack$$ and positive integers $k_1, \ldots, k_p$ such that for every $j \in \{1, \ldots, p \}$ the linear form $l_j({\bold s})$ descends to $P_{{\bold C }}$, $l_j({\bold s})$ does not vanish for ${ \operatorname{Re} }({\bold s}) \in {{\bold R }}_{>0}^{I}$, and $$ q({\bold s},{\bold t}) = f({\bold s}, {\bold t}) \cdot \prod_{j =1}^p l_j^{k_j}({\bold s}) $$ is analytic at ${\bold 0}$. (iii) There exists a nonzero constant $C(f)$ and a homogeneous polynomial $q_0({\bold s})$ of degree $\mu(q)$ in variables $s_1, \ldots, s_I$ such that $$ q({\bold s}, {\bold t}) = q_0({\bold s}) + q_1({\bold s}, {\bold t}) $$ and $$ \frac{q_0({\bold s})}{\prod_{j =1}^p l_j^{k_j}({\bold s})} = C(f) \cdot {\cal X}_{ \Lambda (I)}(\psi^I({\bold s})), $$ where $q_1({\bold s}, {\bold t})$ is an analytic function at ${\bold 0}$ with $\mu(q_1) > \mu(q_0)$, both functions $q_0$, $q_1$ descend to $P_{{\bold C }}$, and ${\cal X}_{ \Lambda (I)}$ is the ${\cal X}$-function of the cone $ \Lambda (I) = \psi^I({{\bold R }}^I_{\geq 0}) \subset \psi^I({{\bold R }}^I)$ (see Definition \ref{c.func}). } \label{def.good} \end{dfn} \begin{rem} {\rm Let $q({\bold s}, {\bold t})$ be an arbitrary analytic at ${\bold 0}$ function. Collecting terms in the Taylor expansion of $q$, we see that there exists a unique homogeneous polynomial $q_0({\bold s}, {\bold t})$ and an analytic at ${\bold 0}$ function $q_1({\bold s}, {\bold t})$ such that $$ q({\bold s}, {\bold t}) = q_0({\bold s}, {\bold t}) + q_1 ({\bold s}, {\bold t}) $$ with $\mu(q) = \mu(q_0) < \mu(q_1)$. In particular, the polynomial $q_0$ and the function $q_1$ in \ref{def.good} are uniquely defined.} \label{unique} \end{rem} \begin{dfn} {\rm If $f({\bold s},{\bold t})$ is good with respect to $\Gamma$ and the set of variables $\{s_1,...,s_I\}$ as above, then the meromorphic function $$ \frac{q_0({\bold s})}{\prod_{j =1}^p l_j^{k_j}({\bold s})} $$ will be called the {\em principal part of $f({\bold s}, {\bold t})$ at ${\bold 0}$} and the constant $C(f)$ the {\em principal coefficient of $f({\bold s}, {\bold t})$ at ${\bold 0}$}. } \end{dfn} Suppose that ${\rm dim }\, \psi^I ({{\bold R }}^I) \geq 2$. Let ${\bold \gamma} = ({\bold \gamma}^I, {\bold \gamma}^J) \in {{\bold Z }}^{I +J}$ be an element which is not contained in $\Gamma$, $\tilde{\Gamma}: = \Gamma \oplus {\bold Z } < {\bold \gamma} >$, $\tilde{\Gamma}_{{\bold R }} := \Gamma_{{\bold R }} \oplus {\bold R } < {\bold \gamma} >$, $\tilde{P}_{{\bold R }} := {{\bold R }}^{I +J} /\tilde{\Gamma}_{{\bold R }}$ and $\tilde{P}_{{\bold C }} := {{\bold C }}^{I+J} /\tilde{\Gamma}_{{\bold C }}$. We assume that $\tilde{ \Gamma } \cap {\bold R }^J = {\bold 0}$ and $\tilde{\Gamma}_{{\bold R }} \cap {{\bold R }}_{\geq 0}^{I+J} = 0$. We denote by $\tilde{\psi}$ (resp. by $\tilde{\psi}^I$) the natural projection ${{\bold C }}^{I+J} \rightarrow \tilde{P}_{{\bold C }}$ (resp. ${{\bold C }}^{I} \rightarrow {{\bold C }}^I/\pi^I(\tilde{ \Gamma }_{{\bold C }})$). The following easy statement will be helpful in the sequel: \begin{prop} Let $f({\bold s}, {\bold t})$ be an analytic at ${\bold 0}$ function, $l({\bold s}) \in {\bold R }[{\bold s}]$ a homogeneous linear function such that $l({\bold \gamma}^I) \neq 0$. Assume that $f({\bold s}, {\bold t})$ and $l({\bold s})$ descend to $P_{{\bold C }}$. Then $$ \tilde{f}({\bold s}, {\bold t}) : = f \left({\bold s} - \frac{l({\bold s})}{l({\bold \gamma}^I)} \cdot {\bold \gamma}^I, {\bold t} - \frac{l({\bold s})}{l({\bold \gamma}^I)} \cdot {\bold \gamma}^J\right) $$ descends to $\tilde{P}_{{\bold C }}$. \label{desc2} \end{prop} \begin{theo} Let $f({\bold s},{\bold t})$ be a good function with respect to $\Gamma$ and the set of variables $\{s_1,...,s_I\}$ as above, $$ \Phi({\bold s}) = \prod_{j\;:\; l_j({\bold \gamma}^I)=0} l_j^{k_j}({\bold s}) $$ the product of those linear forms $l_j({\bold s})$ $(j \in \{ 1, \ldots, p\})$ which vanish on ${\bold \gamma}^I$. Assume that the following statements hold: {\rm (i)} The integral $$ \tilde{f}({\bold s}, {\bold t}) : = \frac{1}{2\pi i} \int_{{ \operatorname{Re} }(z) = 0} f({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J) dz , \;\; z \in {\bold C } $$ converges absolutely and uniformly to a holomorphic function on any compact in the domain ${ \operatorname{Re} }({\bold s}) \in {{\bold R }}_{>0}^{I}$, ${ \operatorname{Re} }({\bold t}) \in {{\bold R }}_{> - \delta _0}^{J}$; {\rm (ii)} There exists $ \delta > 0$ such that the integral $$ \frac{1}{2\pi i} \int_{{ \operatorname{Re} }(z) = \delta } \Phi({\bold s}) \cdot f({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J) dz $$ converges absolutely and uniformly in an open neighborhood of ${\bold 0}$. Moreover, the multiplicity of the meromorphic function $$ \tilde{f}_{ \delta }({\bold s}, {\bold t}): = \frac{1}{2\pi i} \int_{{ \operatorname{Re} }(z) = \delta } f({\bold s} + z \cdot \gamma) dz $$ at ${\bold 0}$ is at least $1 - {\operatorname{dim} }\, \tilde{\psi}^I ({{\bold R }}^I)$; {\rm (iii)} For any ${ \operatorname{Re} }({\bold s}) \in {{\bold R }}_{>0}^I$ and ${ \operatorname{Re} }({\bold t}) \in {{\bold R }}_{> - \delta _0}^{J}$, one has $$ \lim_{ \lambda \rightarrow + \infty} \left( \sup_{0 \leq { \operatorname{Re} }(z) \leq \delta , \, |{ \operatorname{Im} }(z)| = \lambda } |f({\bold s}+ z\cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J)| \right) = 0. $$ Then $\tilde{f}({\bold s})$ is a good function with respect to $\tilde{\Gamma}$ and $\{s_1,...,s_I\}$, and $C({\tilde{f}}) = C(f)$. \label{desc3} \end{theo} \noindent {\em Proof.} By our assumption on $\tilde{ \Gamma }$, $ \gamma^I \neq {\bold 0}$. We can assume that $l_j({\bold \gamma}^I) < 0$ for $j=1, \ldots, p_1$, $l_j({\bold \gamma}^I) = 0$ for $j=p_1 +1, \ldots, p_2$, and $l_j({\bold \gamma}^I) > 0$ for $j=p_2 +1, \ldots, p$. In particular, one has \[ \Phi({\bold s}) = \prod_{j = p_1 + 1}^{p_2} l_j^{k_j}({\bold s}), \] where $k_j$ $(j =p_1 +1 , \ldots, p_2)$ are some positive integers. Denote by $z_j$ the solution of the following linear equation in $z$: \[ l_j({\bold s}) + z l_j({\bold \gamma}^I) = 0,\;\;j =1, \ldots, p_1. \] Let $U$ be the intersection of ${{\bold R }}^{I+J}_{>0}$ with an open neighborhood of ${\bold 0}$ such that $ \Phi({\bold s}) \cdot \tilde{f}_{ \delta }({\bold s}, {\bold t})$ is analytic for all $({\bold s}, {\bold t}) \in U$. By the property (i), both functions $\tilde{f}_{ \delta }({\bold s}, {\bold t})$ and $\tilde{f}({\bold s}, {\bold t})$ are analytic in $U$. Moreover, the integral formulas for $\tilde{f}_{ \delta }({\bold s},{\bold t})$ and $\tilde{f}({\bold s}, {\bold t})$ show that the equalities $\tilde{f}_{ \delta }({\bold u}+ iy \cdot {\bold \gamma}) =\tilde{f}_{ \delta }({\bold u})$ and $\tilde{f}({\bold u}+ iy \cdot {\bold \gamma}) =\tilde{f}({\bold u})$ hold for any $y \in {\bold R }$ and ${\bold u},{\bold u}+ iy \cdot {\bold \gamma} \in U$. Therefore, both functions $\tilde{f}_{ \delta }({\bold s}, {\bold t})$ and $\tilde{f}({\bold s}, {\bold t})$ descend to $\tilde{P}_{\bold C}$ (see Remark \ref{desc}). Using the properties (i)-(iii), we can apply the residue theorem and obtain \[ \tilde{f}({\bold s},{\bold t}) - \tilde{f}_{ \delta }({\bold s},{\bold t}) = \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} f({\bold s} + z \cdot { \gamma}^I,{\bold t} + z \cdot {\bold \gamma}^J)\] for ${\bold s},{\bold t} \in U$. We denote by $U({\bold \gamma})$ the open subset of $U$ which is defined by the inequalities $$ \frac{l_j({\bold s})}{l_j({\bold \gamma}^I)} \neq \frac{l_{m}({\bold s})}{l_{m}({\bold \gamma}^I)}\;\; \mbox{\rm for all $j \neq m$, $\;\;j,m \in \{ 1, \ldots, p\}$.} $$ The open set $U({\bold \gamma})$ is non-empty, since we assume that $g.c.d.(l_j, l_{m})=1$ for $j \neq m$. Moreover, for $({\bold s},{\bold t}) \in U({\bold \gamma})$, we have \\ $ {\rm Res}_{z = z_j} f({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J) = $ $$ = \frac{1}{(k_j-1)!} \left( \frac{\partial}{\partial z} \right)^{k_j-1} \frac{l_{j}({\bold s} + z \cdot {\bold \gamma}^I)^{k_j} q({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J)}{l_j^{k_j} ({\bold \gamma}^I) \cdot \prod_{m =1}^p l_{m}^{k_m}({\bold s} + z \cdot {\bold \gamma}^I) }|_{z = z_j}, $$ where $$ z_j = - \frac{l_j({\bold s})}{l_j({\bold \gamma}^I)}\, \;(j=1, \ldots, p_1) . $$ Let $$ f({\bold s},{\bold t}) \cdot \prod_{j =1}^p l_j^{k_j}({\bold s}) = {q}({\bold s},{\bold t}) = {q}_0({\bold s}) + {q}_1({\bold s},{\bold t}) $$ and $$ \frac{q_0({\bold s})}{\prod_{j =1}^p l_j^{k_j}({\bold s})} = C(f) \cdot {\cal X}_{ \Lambda (I)}({\psi}^I({\bold s})), $$ where ${q}_0({\bold s})$ is a uniquely determined homogeneous polynomial (see Remark \ref{unique}), ${q}_0({\bold s},{\bold t})$ is an analytic at ${\bold 0}$ function with $\mu({q}) = \mu({q}_0) < \mu({q}_1)$ and ${\cal X}_{ \Lambda (I)}({\psi}^I({\bold s}))$ is the ${\cal X}$-function of the cone $ \Lambda (I) = \psi^I({{\bold R }}^{I}_{\geq 0})$. We set $$ R_0({\bold s}) : = \frac{q_0({\bold s})}{\prod_{j =1}^p {l}^{k_j}_j({\bold s})}, \;\; R_1({\bold s},{\bold t}) : = \frac{q_1({\bold s},{\bold t})}{\prod_{j =1}^p {l}_j^{k_j}({\bold s})}. $$ Then $\mu(f)= \mu (R_0) < \mu (R_1)$. Moreover, $\mu(R_0) = - {\operatorname{dim} }\, \psi^I({\bold R }^I)$ (see Prop. \ref{merom}). Define $$ \tilde{R}_0({\bold s}):= \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_0({\bold s}+ z\cdot {\bold \gamma}^I) $$ and $$ \tilde{R}_1({\bold s},{\bold t}):= \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_1({\bold s}+ z\cdot {\bold \gamma}^I,{\bold t} + z \cdot {\bold \gamma}^J). $$ We claim that $$ \tilde{R}_0({\bold s}) = C(f) \cdot {\cal X}_{\tilde{ \Lambda }(I)} (\tilde{\psi}^I({\bold s})), $$ where ${\cal X}_{\tilde{ \Lambda }(I)}(\tilde{\psi}^I({\bold s}))$ is the ${\cal X}$-function of the cone $\tilde{ \Lambda }(I) = \tilde{\psi}^I({{\bold R }}^{I}_{\geq 0})$. Indeed, repeating for ${\cal X}_{ \Lambda (I)}(\psi({\bold s}))$ the same arguments as for $f({\bold s},{\bold t})$, we obtain $$ \frac{1}{2\pi i} \int_{{ \operatorname{Re} }(z) = 0} {\cal X}_{ \Lambda (I)}(\psi({\bold s} + z \cdot {\bold \gamma}^I)) dz - \frac{1}{2\pi i} \int_{{ \operatorname{Re} }(z) = \delta } {\cal X}_{ \Lambda (I)}(\psi({\bold s} + z \cdot {\bold \gamma}^I)) dz $$ $$ = \sum_{j=1}^{k_1} {\rm Res}_{z = z_j} {\cal X}_{ \Lambda (I)}(\psi({\bold s} + z_j \cdot {\bold \gamma}^I)). $$ Moving the contour of integration ${ \operatorname{Re} }(z) = \delta $ $( \delta \rightarrow + \infty)$, by residue theorem, we obtain $$ \int_{{ \operatorname{Re} }(z) = \delta } {\cal X}_{ \Lambda (I)}(\psi({\bold s} + z \cdot {\bold \gamma}^I)) dz =0. $$ On the other hand, $$ {\cal X}_{\tilde{ \Lambda }(I)}(\tilde{\psi}({\bold s})) = \frac{1}{2\pi i}\int_{{ \operatorname{Re} }(z) = 0} {\cal X}_{ \Lambda (I)}(\psi({\bold s} + z \cdot {\bold \gamma}^I)) dz $$ (see Theorem \ref{char0}). Consider the decomposition of $\tilde{f}$ into the sum: $$ \tilde{f}({\bold s},{\bold t}) = \tilde{f}_{ \delta }({\bold s},{\bold t}) + \tilde{R}_0({\bold s}) + \tilde{R}_1({\bold s},{\bold t}). $$ By our assumption in (ii), $\mu(\tilde{f}_{ \delta }) \geq 1 -{\operatorname{dim} }\, \tilde{\psi}^I({\bold R }^I)$. By Proposition \ref{mult4}, we have $\mu (\tilde{R}_1) \geq 1+ \mu(R_1) \geq 2 + \mu(R_0)= 1- {\rm dim}\, \tilde{\psi}^I({\bold R }^I)$. Using \ref{mult3}(iii), we obtain that $\mu (\tilde{f}) = \mu (\tilde{R}_0) = - {\rm dim}\, \tilde{\psi}^I({\bold R }^I)$ and $ \mu (\tilde{f}_{\delta} + \tilde{R}_1) > \mu(\tilde{f})$. By \ref{desc2}, the linear forms \[ h_{m,j}({\bold s}):= l_{m}({\bold s} + z_j \cdot {\bold \gamma}^I) = l_{m}({\bold s}) - \frac{l_j({\bold s})}{l_{j}({\bold \gamma}^I)} l_{m}({\bold \gamma}^I), \; \; (j =1, \ldots, p_1,\; m \neq j) \] and the analytic in the domain $U({\bold \gamma})$ functions \[ {\rm Res}_{z = z_j} f({\bold s} + z \cdot {\bold \gamma}^I,{\bold t} + z \cdot {\bold \gamma}^J),\;\; j =1, \ldots, p_1 \] and \[ {\rm Res}_{z = z_j} R_0({\bold s} + z \cdot {\bold \gamma}^I), \;\; j =1, \ldots, p_1 \] descend to $\tilde{P}_{{\bold C }}$. For any $j \in \{ 1, \ldots, p_1\}$, let us denote $$ Q_j ({\bold s}) = \prod_{m \neq j, m=1}^p h_{m,j}^{k_m}({\bold s}). $$ It is clear that $$ Q_j^{k_j}({\bold s}) \cdot {\rm Res}_{z = z_j} f({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J)\;$$ and $$Q_j^{k_j}({\bold s}) \cdot {\rm Res}_{z = z_j} R_0({\bold s} + z \cdot {\bold \gamma}^I) $$ are analytic at ${\bold 0}$ and $\Phi({\bold s})$ divides each $Q_j ({\bold s})$. Hence, we obtain that \\ $ \tilde{f}({\bold s},{\bold t}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bold s}) = $ $$ = \left( \tilde{f}_{ \delta }({\bold s},{\bold t}) + \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} f({\bold s} + z \cdot {\bold \gamma}^I, {\bold t} + z \cdot {\bold \gamma}^J) \right) \prod_{j=1}^{p_1} Q_j^{k_j}({\bold s}) $$ and $$ \tilde{R}_0({\bold s}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bold s}) = \left( \sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_0({\bold s} + z \cdot {\bold \gamma}^I) \right) \prod_{j=1}^{p_1} Q_j^{k_j}({\bold s}) $$ are analytic at ${\bold 0}$. Let us define the set $\{ \tilde{l}_1({\bold s}), \ldots, \tilde{l}_{\tilde{p}}({\bold s}) \}$ as a subset of pairwise coprime elements in the set of homogeneous linear forms $\{ h_{m,j}({\bold s}) \}$ $(m \in \{1, \ldots, p\}, \; j \in \{1, \ldots, p_1\})$ such that there exist positive integers $n_1, \ldots, n_{\tilde{p}}$ and a representation of the meromorphic functions $\tilde{f}({\bold s},{\bold t})$ and $\tilde{R}_0({\bold s})$ as quotients \[ \tilde{f}({\bold s},{\bold t}) = \frac{\tilde{q}({\bold s},{\bold t})}{\prod_{j =1}^{\tilde{p}} \tilde{l}^{n_j}_j({\bold s})},\;\; \tilde{R}_0({\bold s}) = \frac{\tilde{q}_0({\bold s})}{\prod_{j =1}^{\tilde{p}} \tilde{l}^{n_j}_j({\bold s})},\] where $\tilde{q}({\bold s},{\bold t})$ is analytic at ${\bold 0}$, $\tilde{q}_0({\bold s})$ is a homogeneous polynomial, and none of the forms $\tilde{l}_1({\bold s}), \ldots, \tilde{l}_q({\bold s})$ vanishes for $({\bold s},{\bold t}) \in {{\bold R }}_{>0}^{I +J}$ (the last property can be achieved, because both functions $\tilde{f}({\bold s},{\bold t})$ and $\tilde{R}_0({\bold s},{\bold t})$ are analytic in $U$ and the closure of $U$ is equal to ${{\bold R }}_{\geq 0}^{I +J}$). \noindent Define $$ \tilde{q}_1({\bold s},{\bold t}) = \left( \tilde{f}_{ \delta }({\bold s},{\bold t}) + \tilde{R}_1({\bold s},{\bold t}) \right) \cdot \prod_{j =1}^{\tilde{p}} \tilde{l}^{n_j}_j({\bold s}). $$ Then $$ \tilde{q}({\bold s},{\bold t}) = \tilde{q}_0({\bold s}) + \tilde{q}_1({\bold s},{\bold t}) $$ where $\tilde{q}_0({\bold s},{\bold t})$ is a homogeneous polynomial and $\tilde{q}_1({\bold s},{\bold t})$ is an analytic at ${\bold 0}$ function such that $\mu(\tilde{q}) = \mu(\tilde{q}_0) < \mu(\tilde{q}_1)$. Moreover, $$ \frac{\tilde{q}_0({\bold s})}{\prod_{j =1}^{\tilde{p}} \tilde{l}_j^{n_j}({\bold s})} = C(f) \cdot {\cal X}_{\tilde{ \Lambda }(I)}(\tilde{\psi}^I({\bold s})), $$ i.e., $\tilde{f}$ is good. \hfill $\Box $ \begin{dfn} {\rm For any finite dimensional Banach space $V$ over ${\bold R }$ we denote by $\| \cdot\|$ a representative in the class of equivalent norms on $V$. For ${\bold y}=(y_1,...,y_r)\in {\bold R }^r$ we will set $$ \|{\bold y}\|: =\sum_{j=1}^r |y_j|. $$} \end{dfn} \noindent The following lemma is elementary: \begin{lem} Let $V=V_1\oplus V_2$ be a direct sum of finite dimensional vector spaces over ${\bold R }$, $r_2$ is the dimension of $V_2$, and $r_2>0$. Let $f({\bold x})$ be a complex valued function on $V$ satisfying the inequality $$ |f({\bold x})|\le \frac{c}{(1+\|{\bold x}\|)^{r_1+r_2+2 \varepsilon }}, $$ for any ${\bold x}=({\bold x}_1,{\bold x}_2)\in V$ and some constants $c, \varepsilon >0$. Let $W\subset V_2$ be a locally closed subgroup such that $V_2/W$ is compact. Choose any Haar measure ${\bold dw}$ on $W$. Then there exists a constant $c'>0$ such that we have the estimate $$ \int_W |f({\bold x}_1+{\bold w})|{\bold dw} \le \frac{c'}{(1+\|{\bold x}_1\|)^{r_1+ \varepsilon }} $$ for any ${\bold x}_1 \in V_1$. \label{trivial} \end{lem} \begin{theo} Let $f({\bold s}, {\bold t})$ be an analytic function for ${ \operatorname{Re} }({\bold s})\in {\bold R }^{I}_{>0}$, ${ \operatorname{Re} }({\bold t}) \in {\bold R }^{J}_{> - \delta _0}$ $($for some $ \delta _0>0)$, $ \Gamma \subset {\bold Z }^{I+J}$ a sublattice of rank $t<I$ with $ \Gamma _{{\bold R }} \cap {\bold R }^J=0$ and $ \Gamma _{{\bold R }} \cap {\bold R }^{I+J}_{\geq 0} = 0$. Assume that there exist constants $ \varepsilon , \varepsilon _0>0$ such that the following holds: {\rm (i)} The function $$ g({\bold s}, {\bold t})=s_1\cdots s_If({\bold s}, {\bold t}) $$ is holomorphic in the domain ${ \operatorname{Re} }({\bold s}) \in {\bold R }^{I}_{> - \varepsilon }$, ${ \operatorname{Re} }({\bold t}) \in {\bold R }^{J}_{> - \delta _0}$ and $C(f):=g({\bold 0})\neq 0$; {\rm (ii)} For all $ \varepsilon _1$ $($with $0 < \varepsilon _1< \varepsilon $$)$ there exist a constant $C( \varepsilon _1)>0$ and an estimate $$ |f({\bold s}+i{\bold y}_I, {\bold t} + i{\bold y}_J) | \leq \frac{C( \varepsilon _1)}{(1 + \|{\bold y }\|)^{t + \varepsilon _0}}, $$ $$\; {\bold y} =( {\bold y}_I, {\bold y}_J), \;\; \|{\bold y}\| = \|{\bold y}_I \| + \| {\bold y}_J \|, $$ which holds for all ${\bold s}$ such that one of the two inequalities $- \varepsilon < { \operatorname{Re} }(s_j) < \varepsilon _1$ or ${ \operatorname{Re} }(s_j) > \varepsilon _1$ is satisfied for every $j=1,...,I$. Then the integral $$ \frac{1}{(2\pi )^t} \int_{ \Gamma _{{\bold R }}}f({\bold s}+i{\bold y}_I, {\bold t} + i {\bold y}_J){\bold dy} $$ is a good function with respect to $ \Gamma $ and the set of variables $\{s_1,...,s_I\}$, and $C(f)$ is its principal coefficient. \label{integral} \end{theo} \noindent {\em Proof.} Without loss of generality we can assume that $ \Gamma $ is not contained in any of $I$ coordinate hyperplanes $s_j =0$ $(j =1,...,I)$, otherwise we reduce the problem to a smaller value of $I$. Therefore, we can choose a basis ${ \gamma}^1,...,{ \gamma}^t$ of $ \Gamma $ such that all first $I$ coordinates of ${ \gamma}^u= ({ \gamma}_I^u, { \gamma}_J^u) \in {\bold Z }^{I+J}$ are not equal to $0$ for every $u=1,...,t$. For any non-negative integer $u\le t$ we define a subgroup $ \Gamma ^{(u)} \subset \Gamma $ of rank $u$ as follows: $$ \Gamma ^{(0)}= 0;\;\; \Gamma ^{(u)}:= \bigoplus_{j=1}^u {\bold Z }<{ \gamma}^u>,\;u=1,...,t. $$ We introduce some auxiliary functions $$ f^{(0)}({\bold s}, {\bold t})=f({\bold s}, {\bold t}); $$ $$ f^{(u)}({\bold s}, {\bold t})= \frac{1}{(2\pi )^u} \int_{ \Gamma ^{(u)}_{{\bold R }}} f({\bold s}+i{\bold y}^{(u)}_I, {\bold t} +i {\bold y}^{(u)}_J) {\bold dy}^{(u)},\;\;u=1,...,t, $$ where ${\bold dy}^{(u)}$ is the Lebesgue measure on $ \Gamma ^{(u)}_{{\bold R }}$ normalised by the lattice $ \Gamma ^{(u)}$. Denote by $P_{{\bold C }}^{(u)}= {\bold C }^r/ \Gamma ^{(u)}_{{\bold C }}$. By the estimate in (ii), $f^{(u)}({\bold s}, {\bold t})$ is a holomorphic function in the domain $({\rm Re}({\bold s}),{\rm Re}({\bold t}) ) \in {\bold R }^{I +J}_{>0}$ and descends to $P^{(u)}_{{\bold C }}$. We prove by induction that $f^{(u)}({\bold s}, {\bold t})$ is good with respect to $ \Gamma ^{(u)}\subset {\bold Z }^{I+J}$ and $\{s_1,...,s_I\}$. By (i), $f^{(0)}({\bold s}, {\bold t})$ is good. By induction assumption, we know that $f^{(u-1)}({\bold s}, {\bold t})$ is good with respect to $ \Gamma ^{(u-1)}$ and $\{s_1,...,s_I\}$. Moreover, we have $$ f^{(u)}({\bold s}, {\bold t})=\frac{1}{(2\pi i )}\int_{{ \operatorname{Re} }(z)=0} f^{(u-1)}({\bold s} + z \cdot { \gamma}^u_I, {\bold t} + z \cdot { \gamma}^u_J ) dz. $$ Choose $ \delta _u>0$ in such a way that for every $j=1,...,I$ one of the following two inequalities is satisfied: $$ - \varepsilon < \delta _u \gamma_j^u \ <- \varepsilon _1, \;\; \mbox{\rm or}\;\; \delta _u \gamma_j^u> \varepsilon _1 $$ for some $0< \varepsilon _1< \varepsilon $. By (ii), the integral $$ f_{ \delta }^{(u)}({\bold s}, {\bold t})= \frac{1}{(2\pi i )} \int_{{ \operatorname{Re} }(z) = \delta _u} f^{(u-1)}({\bold s} + z \cdot { \gamma}^u_I, {\bold t} + z \cdot { \gamma}^u_J) dz $$ $$ =\frac{1}{(2\pi)^u} \int_{ \Gamma _{{\bold R }}^{(u)}} f({\bold s} + \delta _u \gamma^{u}_I + i{\bold y}^{(u)}_I,{\bold t} + \delta _u \gamma^{u}_J + i{\bold y}^{(u)}_J ) {\bold dy}^{(u)} $$ converges absolutely and uniformly in an open neighborhood of ${\bold 0}$, i.e. the multiplicity of $f_{ \delta }^{(u)}({\bold s}, {\bold t})$ is at least $0\ge 1+ {\rm rk}\, \Gamma ^{(u)} -I$. Hence it is holomorphic at ${\bold 0}$ and satisfies assumption (ii) of \ref{desc3}. By lemma \ref{trivial}, the property \ref{desc3} (iii) holds. Applying theorem \ref{desc3}, we conclude that $f^{(u)}({\bold s}, {\bold t})$ is a good function with the principal coefficient $g({\bold 0})$. \hfill $\Box$ \section{Fourier analysis on algebraic tori} Let $X_F$ be an algebraic variety over a number field $F$ and $E/F$ a finite extension of number fields. We shall denote by $X_E$ the $E$-variety obtained by base change from $X_F$ and by $X(E)$ the set of $E$-rational points of $X_F$. Sometimes we omit the subscript in $X_E$ if the field is clear from the context. Let ${\bold G}_m= {\rm Spec}(F[x,x^{-1}])$ be the multiplicative group scheme over $F$. A $d$-dimensional algebraic torus $T$ is a group scheme over $F$ such that over some finite field extension $E/F$ we have $T_E \cong ({\bold G}_{m})^d$. We call the minimal $E$ with this property the splitting field of $T$. Denote by $G={\rm Gal }(E/F)$ the Galois group of $E$ over $F$. For every $G$-module $A$, $A^G$ stands for the submodule of elements fixed by $G$. For any field $E$ we denote by $\hat{T}_E$ the $G$-module ${\rm Hom}(T_E,{\bold G}_{m})$ of $E$-rational characters of $T$. If $E$ is the splitting field of $T$, we put $M: =\hat{T}_E$ and $N: ={\rm Hom} (M,{\bold Z })$ the dual $G$-module. We denote by $t$ the rank of the lattice $M^G$. Let $T$ be an algebraic torus over a number field $F$. Denote by ${\operatorname{Val} }(F)$ the set of valuations of $F$ and by ${\operatorname{Val} }_{\infty}(F)$ the set of archimedian valuations. Let $F_v$ be the completion of $F$ with respect to $v\in {\operatorname{Val} }(F)$, ${\cal V}$ an extension of $v$ to $E$, $$ G_v\; :\;= {\rm Gal}(E_{\cal V}/F_v)\subset {\rm Gal }(E/F) $$ the decomposition group at $v$, $T(F_v)$ the group of $F_v$-rational points of $T$ and $T({\cal O}_v)$ its maximal compact subgroup. We have the canonical embeddings $$ \pi_v\; :\;T(F_v)/T({\cal O}_v)\hookrightarrow N^{G_v} $$ for all non-archimedian $v \in {\operatorname{Val} }(F)$ and $$ \pi_v\; :\;T(F_v)/T({\cal O}_v)\hookrightarrow N^{G_v}_{{\bold R }} $$ for all $v \in {\operatorname{Val} }_{\infty}(F)$. Denote by $\overline{x}_v$ the image of $x_v\in T(F_v)$ in $N^{G_v}$ (resp. $N_{{\bold R }}^{G_v}$) under $\pi_v$. \begin{dfn} {\rm We call a valuation $v \in {\operatorname{Val} }(F)$ {\em good}, if the mapping $\pi_v$ is an isomorphism. We denote by $S$ a finite subset in ${\operatorname{Val} }(F)$ containing ${\operatorname{Val} }_{\infty}(F)$ and all valuations $v \in {\operatorname{Val} }(F)$ which are not good. } \end{dfn} Let us recall some basic arithmetic properties of algebraic tori over the ring of adeles ${\bold A}_F$. Define $$ T^1({\bold A}_F)=\{ {\bold x}\in T({\bold A}_F) \, \mid \, \prod_{v\in {\operatorname{Val} }(F)} |m(x_v)|_v =1, \, \; {\rm for}\,\; {\rm all}\,\; m\in M^G\}. $$ Let ${\bold K}_T=\prod_{v\in {\operatorname{Val} }(F)} T({\cal O}_v) $ be the maximal compact subgroup of $T({\bold A}_F)$. \begin{prop} The groups $T({\bold A}_F), T^1({\bold A}_F), T(F), {\bold K}_T$ have the following properties: {\rm (i)} $T({\bold A}_F)/T^1({\bold A}_F) \cong N_{{\bold R }}^G \cong {\bold R }^t$; {\rm (ii)} $T^1({\bold A}_F)/T(F)$ is compact; {\rm (iii)} $T^1({\bold A}_F)/T(F){\bold K}_T$ is isomorphic to the product a finite group ${\bold cl}(T)$, and a connected compact abelian group; {\rm (iv)} $w(T)={\bold K}_T\cap T(F)$ is a finite abelian group of torsion elements in $T(F)$. \label{tori.adelic} \end{prop} Let $T({\cal O}) \subset T(F)$ be the subgroup of ${\cal O}_F$-integral points. Then $T({\cal O})$ contains $w(T)$, and ${\cal E}_T : = T({\cal O}_F)/w(T)$ has a canonical embedding, as a discrete subgroup, into the archimedian logarithmic space $$ N_{{\bold R },\infty}= \bigoplus_{v\in {\operatorname{Val} }_{\infty}(F)}N_{{\bold R }}^{G_v}= \bigoplus_{v\in {\operatorname{Val} }_{\infty}(F)}T(F_v)/T({\cal O}_v). $$ Moreover, the image of ${\cal E}_T$ in $N_{{\bold R },\infty}$ is contained in the ${{\bold R }}$-subspace $N_{{\bold R },\infty}^1$ defined as $$ N_{{\bold R },\infty}^1 : = \{ \overline{x} \in N_{{\bold R },\infty} | \sum_{v \in {\operatorname{Val} }_{\infty}(F)} m({\overline{x}}_v) = 0\;\; \mbox{\rm for all $m \in M^G$} \}, $$ and the quotient $N_{{\bold R },\infty}^1/ {\cal E}_T$ is compact. \begin{dfn} {\rm Let $T$ be an algebraic torus over a number field $F$. We define $$ {\cal H}_T:=(T({\bold A}_F)/T(F))^* $$ as the group of topological characters of $T({\bold A}_F)$ which are trivial on $T(F)$. Define the group ${\cal D}_T$ as $$ {\cal D}_T:=(T^1({\bold A}_F)/T(F))^*. $$ Define the group ${\cal U}_T$ as: $$ {\cal U}_T := (T^1({\bold A}_F)/T(F){\bold K}_T)^*. $$ We call the characters $\chi \in {\cal D}_T$ {\em discrete} and $\chi \in {\cal U}_T$ {\em unramified}.} \end{dfn} \noindent Using \ref{tori.adelic} (i), we see that a choice of a splitting of the exact sequence $$ 1 \rightarrow T^1({\bold A}_F) \rightarrow T({\bold A}_F) \rightarrow T({\bold A}_F)/T^1({\bold A}_F) \rightarrow 1 $$ defines isomorphisms $$ {\cal H}_T \cong M^G_{{\bold R }} \oplus {\cal D}_T, $$ $$ N_{{\bold R },\infty} = N_{{\bold R }}^G \oplus N_{{\bold R },\infty}^1, $$ and $$ M_{{\bold R },\infty} = M_{{\bold R }}^G \oplus M_{{\bold R },\infty}^1, $$ where $$ M_{{\bold R },\infty} = \bigoplus_{v \in {\operatorname{Val} }_{\infty}(F)} M_{{\bold R }}^{G_v}. $$ and $M_{{\bold R },\infty}^1$ is the minimal ${{\bold R }}$-subspace in $M_{{\bold R },\infty}$ containing the image of ${\cal U}_T$ under the canonical mapping $$ {\cal U}_T \rightarrow M_{{\bold R },\infty}. $$ >From now on we fix such a non-canonical splitting. This allows to consider ${\cal U}_T$ as a subgroup of ${\cal H}_T$. By \ref{tori.adelic}, we have: \begin{prop} There is an exact sequence $$ 0 \rightarrow {\bold cl}^*(T) \rightarrow {\cal U}_T \rightarrow {\cal M}_T \rightarrow 0, $$ where ${\cal M}_T$ is the image of the canonical projection of ${\cal U}_T$ to $M_{{\bold R },\infty}^1$ and ${\bold cl}^*(T)$ is a finite abelian group dual to ${\bold cl}(T)$. \label{ex.seq} \end{prop} We see from \ref{ex.seq} that a character $\chi \in M_{{\bold R }}^G \oplus {\cal U}_T$ is determined by its archimedian component which is an element in $M_{{\bold R },\infty}$ up to a finite choice. Denote by $y(\chi ) \in M_{{\bold R }}^G \oplus {\cal M}_T$ the image of $\chi \in M_{{\bold R }}^G \oplus {\cal U}_T$ in $M_{{\bold R },\infty}$. \noindent For all valuations $v$ we choose Haar measures $d\mu_v$ on $T(F_v)$ normalized by $$ \int_{T({\cal O}_v)} d\mu_v =1. $$ We define the canonical measure on the group $T({\bold A}_F)$ $$ \omega = \prod_{v\in {\operatorname{Val} }(F)}d\mu_v. $$ For archimedian valuations the Haar measure $d\mu_v$ is the pullback of the Lebesgue measure on $ N_{{\bold R }}^{G_v}$ under the logarithmic map $$ T(F_v)/T({\cal O}_v)\rightarrow N_{{\bold R }}^{G_v}. $$ Let ${\bold dx}$ be the Lebesgue measure on $T({\bold A}_F)/T^1({\bold A}_F)$. There exists a unique Haar measure $\omega^1$ on $T^1({\bold A}_F)$ such that $\omega=\omega^1{\bold dx}$. We define $$ b(T)=\int_{T^1({\bold A}_F)/T(F)}\omega^1. $$ For any $L^1$-function $f$ on $T({\bold A}_F)$ and any topological character $\chi $ we denote by $\hat{f}(\chi ) $ its global Fourier transform with respect to $\omega$ and by $\hat{f}_v(\chi_v ) $ the local Fourier transforms. We will use the following version of the Poisson formula: \begin{theo} Let ${\cal G}$ be a locally compact abelian group with Haar measure $dg, {\cal G}_0\subset {\cal G} $ a closed subgroup with Haar measure $dg_0$. The factor group ${\cal G}/{\cal G}_0$ has a unique Haar measure $dx$ normalized by the condition $dg=dx\cdot dg_0$. Let $f\,:\, {\cal G} \rightarrow {\bold C } $ be an ${L}^1$-function on ${\cal G}$ and $\hat{f}$ its Fourier transform with respect to $dg$. Suppose that $\hat{f}$ is also an ${L}^1$-function on ${\cal G}_0^{\perp}$, where ${\cal G}_0^{\perp}$ is the group of topological characters $\chi $ which are trivial on ${\cal G}_0$. Then $$ \int_{{\cal G}_0} f(x)dg_0=\int_{{\cal G}_0^{\perp}}\hat{f}(\chi) d\chi, $$ where $d\chi$ is the orthogonal Haar measure on ${\cal G}_0^{\perp}$ with respect to the Haar measure $dx$ on ${\cal G}/{\cal G}_0$. \label{poi} \end{theo} \noindent We will apply this formula with ${\cal G}= T({\bold A}_F)$, ${\cal G}_0= T(F)$, $dg=\omega $ and $dg_0 $ is the discrete measure on $T(F)$. The Haar measure $d\chi$ induces the Lebesgue measure on $M_{{\bold R }}^G$ normalized by the lattice $M^G\subset M_{{\bold R }}^G$ and the discrete measure on ${\cal D}_T$. \begin{dfn}{\rm Let $T$ be an algebraic torus over $F$ and $\overline{T(F)}$ the closure of $T(F)$ in $T({\bold A}_F)$ in the {\em direct product topology}. Define the obstruction group to weak approximation as $$ A(T)= T({\bold A}_F)/\overline{T(F)}. $$ } \label{WA} \end{dfn} \section{Geometry of toric varieties} \begin{dfn} {\rm A complete regular $d$-dimensional fan $G$-invariant $ \Sigma $ is a finite set of convex rational polyhedral cones in $N_{{\bold R }}$ satisfying the following conditions: (i) every cone $ \sigma \in \Sigma $ contains $0\in N_{{\bold R }}$; (ii) every face $ \sigma '$ of a cone $ \sigma \in \Sigma $ belongs to $ \Sigma $; (iii) the intersection of any two cones in $ \Sigma $ is a face of both cones; (iv) $N_{{\bold R }}$ is the union of cones from $ \Sigma $; (v) every cone $ \sigma \in \Sigma $ is generated by a part of a ${\bold Z }$-basis of $N$; (vi) For any $g\in G$ and any $ \sigma \in \Sigma $, one has $g( \sigma )\in \Sigma $. } \end{dfn} A complete regular $d$-dimensional fan $ \Sigma $ defines a smooth toric variety $X_{ \Sigma ,E}$ as follows: $$ X_{ \Sigma ,E}=\bigcup_{ \sigma \in \Sigma } U_{ \sigma }=\bigcup_{ \sigma \in \Sigma } {\rm Spec }(E[M\cap\check{ \sigma }]), $$ where $\check{ \sigma }\subset M_{{\bold R }}$ is the dual to $ \sigma $ cone. We can see that $T_E\subset U_{ \sigma }$ for all $ \sigma \in \Sigma $ and that $U_0=T$. \begin{theo}\cite{vosk1} Let $ \Sigma $ be a complete regular $G$-invariant fan in $N_{{\bold R }}$. Assume that the complete toric variety $X_{ \Sigma ,E}$ defined over the splitting field $E$ by $ \Sigma $ is projective. Then there exists a unique complete algebraic variety $X_{ \Sigma ,F}$ over $F$ such that its base extension to $E$ is isomorphic to $X_{ \Sigma ,E}$. \end{theo} Denote by $ \Sigma (j)$ the subset of $j$-dimensional cones in $ \Sigma $ and by $N_{ \sigma ,{\bold R }}\subset N_{{\bold R }}$ the minimal linear subspace containing $ \sigma $. Let $\{e_1,...,e_n\}$ be the set of $1$-dimensional generators of $ \Sigma $. Denote by $PL( \Sigma )$ the lattice of piecewise linear integral functions on $N$. By definition, a function $ \varphi \in PL( \Sigma )$ iff $ \varphi (N)\subset {\bold Z }$ and the restriction of $ \varphi $ to every cone $ \sigma \in \Sigma $ is a linear function; equivalently, there exist elements $m_{ \sigma }\in M$ such that the restriction of $ \varphi $ to $ \sigma $ is given by $< \cdot ,m_{ \sigma }> $ where $< \cdot ,\cdot> $ is induced from the pairing between $N$ and $M$. The $G$-action on $M$ (and $N$) induces a $G$-action on the free abelian group $PL( \Sigma )$. Let $$ \Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1)$$ be the decomposition of $ \Sigma (1)$ into a union of $G$-orbits. A $G$-invariant piecewise linear function $ \varphi \in PL( \Sigma )^G $ is determined by the vector ${\bold u}=(u_1,...,u_r)$, where $u_i$ is the value of $ \varphi $ on the generator of some $1$-dimensional cone in the $G$-orbit $ \Sigma _i(1), (i=1,...,r)$. It will be convenient for us to consider complex valued piecewise linear functions and to identify $ \varphi = \varphi _{\bold u}\in PL( \Sigma )_{{\bold C }}^G$ with its complex coordinates ${\bold u}=(u_1,...,u_r)\in PL( \Sigma )_{{\bold C }}^G$. \begin{theo} The toric variety $X_{ \Sigma }$ has the following properties: (i) There is a representation of $X_{ \Sigma ,E}$ as a disjoint union of split algebraic tori $T_{ \sigma ,E}$ of dimension $\operatorname{dim} T_{ \sigma ,E}= d-\operatorname{dim} \sigma $. For each $j$-dimensional cone $ \sigma \in \Sigma (j)$ we denote by $T_{ \sigma ,E}$ the kernel of a homomorphism $T_E\rightarrow {\bold G}_{m,E}^j$ defined by a ${\bold Z }$-basis of the sublattice $N\cap N_{ \sigma ,{\bold R }}$. (ii) The closures of $(d-1)$-dimensional tori corresponding to the $1$-dimensional cones ${\bold R }_{\ge 0}e_1,...,{\bold R }_{\ge 0}e_n\in \Sigma (1)$ define divisors $\overline{T}_1,...,\overline{T}_n$. We can identify the lattices $PL( \Sigma )=\oplus_{j=1}^n {\bold Z } [\overline{T}_j]$. (iii) There is an exact sequence of $G$-modules $$ 0\rightarrow M\rightarrow PL( \Sigma )\rightarrow {\rm Pic}(X_{ \Sigma ,E})\rightarrow 0, $$ moreover, we have ${\rm Pic}(X_{ \Sigma ,F})={\rm Pic}(X_{ \Sigma ,E})^G$; (iv) The cone of effective divisors $ \Lambda _{\rm eff}(X_{ \Sigma ,F})\subset {\rm Pic}(X_{ \Sigma ,F})_{{\bold R }}$ is generated by the classes of $G$-invariant divisors $$ D_j=\sum_{{\bold R }_{\ge 0}e_i\in \Sigma _j(1)} \overline{T}_i. $$ (v) The class of the anticanonical divisor $-[K_{ \Sigma }]$ is given by the class of the $G$-invariant divisor $$ -[K_{ \Sigma }]=[D_1+...+D_r]. $$ \label{toric.geom} \end{theo} \begin{rem} We note that for toric varieties we have ${\rm Pic}(X_{ \Sigma ,F})=NS(X_{ \Sigma ,F})$. \end{rem} \begin{dfn}{\rm Let $ \varphi \in PL( \Sigma )_{{\bold C }}^G$ be a complex valued piecewise linear function. Let $v\in {\operatorname{Val} }(F)$ be a non-archimedian valuation. Denote by $q_v$ the order of the residue field of $F_v$. For $x_v\in T(F_v)$ we define the complex local height function $$ H_{ \Sigma ,v}(x_v, \varphi ) = e^{ \varphi (\overline{x}_v)\log q_v}. $$ Let $v$ be an archimedian valuation. The complex local height function is defined as $$ H_{ \Sigma ,v}(x_v, \varphi ) = e^{ \varphi (\overline{x}_v)}. $$ } \end{dfn} \begin{rem}{\rm This provides a {\em piecewise smooth} metrization of line bundles on the toric variety $X_{ \Sigma }$. One can show that this metrization is, in a sense, "canonical". Namely, an algebraic torus admits a morphism to itself (n-th power morphism), which extends to a compactification. Using the construction of Tate one can obtain a metrization on a line bundle by a limiting process. This metrization coincides with ours. } \end{rem} \begin{dfn} {\rm Let $x\in T(F)\subset X_{ \Sigma }(F)$ be a rational point. The global height function is defined by $$ H_{ \Sigma }(x, \varphi )=\prod_v H_{ \Sigma ,v}(x_v, \varphi ). $$ \label{height} } \end{dfn} By the product formula, the function $H_{ \Sigma }(x, \varphi )$ as a function on $T(F)$ descends to the complexified Picard group ${\rm Pic}(X_{ \Sigma })_{{\bold C }}$. Moreover, we have the following \begin{prop} {\rm \cite{BaTschi1}} Let $X_{ \Sigma } $ be an smooth projective toric variety. For all $x\in T(F)\subset X_{ \Sigma }(F)$ the function $H_{ \Sigma }(t, \varphi )$ coincides with a classical height corresponding to some metrization of the line bundle $L$ represented by a piecewise linear function $ \varphi \in PL( \Sigma )^G$. \end{prop} Let $X_{ \Sigma }$ be a toric variety and ${H}_{ \Sigma }$ the height pairing. Clearly, it extends to a pairing $T({\bold A}_F)\times PL( \Sigma )_{{\bold C }}^G\rightarrow {\bold C }$. Moreover, it is invariant under the maximal compact subgroup ${\bold K}_T=\prod_{v\in {\operatorname{Val} }(F)} T({\cal O}_v)$. Therefore, its Fourier transform $\hat{H}_{ \Sigma }(\chi,-{\bold s})$ equals zero for characters $\chi\in {\cal H}_T$ which are non-trivial on ${\bold K}_T$. By \ref{toric.geom}, we have an exact sequence of ${\bold Z }$-modules $$ 0\rightarrow M^G\rightarrow PL( \Sigma )^G\rightarrow {\rm Pic}(X_{ \Sigma ,F})\rightarrow H^1(G,M)\rightarrow 0. $$ It induces a surjective map of tori $ a\,:\, \prod_{j=1}^r R_{F_j/F}({\bold G}_{m}) \rightarrow T$ and a surjective homomorphism $$ a\,:\,\prod_{hj=1}^r {\bold G}_m({\bold A}_{F_j})/{\bold G}_m(F_j)\rightarrow T({\bold A}_F)/T(F) $$ Every character $\chi\in {\cal H}_T$ defines $r$ Hecke characters $\chi_1,...,\chi_r$ of the groups ${\bold G}_m({\bold A}_{F,j})/{\bold G}_m(F_j)$ by $\chi \circ a$. It is known \cite{draxl}, that ${\rm Coker}(a)$ is isomorphic to the obstruction group to weak approximation $A(T)$ (see \ref{WA}). Similarly, every local character $\chi_v$ defines local characters $\chi_{1,v},...\chi_{r,v}$. If $\chi$ is trivial on ${\bold K}_T$ then all $\chi_j$ are trivial on the maximal compact subgroups in in ${\bold G}_m({\bold A}_{F_j})$, in other words, all $\chi_j$ are unramified. Their local components for all valuations are given by $$ \chi_{j,v}\,:\, {\bold G}_m(F_{j,v})/{\bold G}_m({\cal O}_{j,v})\rightarrow {\bold C }^* $$ $$ \chi_{j,v}(x_v)=|x_v|_v^{iy_{j,v}} $$ for some real $y_{j,v}$. \medskip In the remaining part of this section we recall some estimates which will be used it the study of analytic properties of the height zeta function (see {\rm \cite{BaTschi1}}). Let us consider a Hecke character $\chi\in ({\bold G}_m({\bold A}_F)/{\bold G}_m(F))^*$ and the corresponding Hecke $L$-function $L(\chi,u)$. The following estimate can be proved using the Phragmen-Lindel\"of principle \cite{rademacher}. \begin{prop} For all $ \varepsilon >0$ there exists a constant $c_1( \varepsilon )$ such that for all unramified $\chi $ and all $u$ with ${ \operatorname{Re} }(u)>1+ \varepsilon $ $$ |L(\chi,u)|<c_1( \varepsilon ). $$ For all $ \varepsilon >0$ there exists a $ \delta>0$ such that for all unramified $\chi$ and all $u$ with ${ \operatorname{Re} }(u)$ contained in any compact ${\bold K}$ in the domain $ 0<| { \operatorname{Re} }\,(u) - 1| <\delta$ there exists a constant $c({\bold K}, \varepsilon )$ depending only on ${\bold K}$ and $ \varepsilon $ such that \[ | L(\chi,u) | \leq c({\bold K}, \varepsilon ) (1 + |{ \operatorname{Im} }(u)|+ \|y(\chi)\|)^{ \varepsilon }. \] \label{m.estim} \end{prop} Let $T$ be an algebraic torus and $\chi\in {\cal U}_T$ an unramified character. Denote by $\chi_v$ its local components and by $\chi_1,...,\chi_r$ the induced unramified Hecke characters of ${\bold G}_m({\bold A}_{F_j})$. \begin{dfn} {\rm Define $$ \zeta_{fin}(\chi,-{\bold u}):= \frac{\prod_{v\not\in {\operatorname{Val} }_{\infty}(F)}\hat{H}_{ \Sigma ,v}(\chi_v,-{\bold u})}{ \prod_{j=1}^rL_{F_j}(\chi_j,u_j)}, $$ where for every field $F_j$ we denoted by $L_{F_j}(\chi_j,u) $ the standard Hecke $L$-function of $F_j$. For any $\chi\in {\cal D}_T$ we define $$ \zeta_{\infty}(\chi,{\bold u}):=\zeta_{fin}(\chi,-{\bold u})\cdot \prod_{v\in {\operatorname{Val} }_{\infty}(F)} \hat{H}_{ \Sigma ,v}(\chi,-{\bold u}). $$ } \label{defin} \end{dfn} \begin{prop} {\rm \cite{BaTschi1} } For every $ \delta _0>0$ there exist constants $0<c_1<c_2$ such that for any ${\bold u}$ with ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1/2+ \delta _0}$ and any $\chi\in {\cal U}_T$ we have $$ c_1<|\zeta_{fin}(\chi,-{\bold u})|<c_2. $$ \end{prop} \begin{prop}{\rm \cite{BaTschi1}} Let $\chi\in {\cal U}_T$ be an unramified character and $y(\chi)$ its image in $M_{{\bold R },\infty}$. For all $ \delta _0 >0$ there exists a constant $c( \delta _0 )$ such that for any ${\bold u}$ in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1/2+ \delta _0}$ we have the following estimate $$ |\prod_{v\in {\operatorname{Val} }_{\infty}(F)} \hat{H}_{ \Sigma ,v}(\chi,-{\bold u})| \le \frac{c( \delta _0)}{(1+\|y(\chi)\|+\|{ \operatorname{Im} }({\bold u})\|)^{\rho+t+1}}, $$ where $\rho+t$ is the dimension of the real vector space $M_{{\bold R },\infty}$. \label{estimates-inf} \end{prop} \begin{coro} For any $ \delta _0>0$, there exists a constant $c( \delta _0)$ such that for any $\chi\in {\cal U}_T$ and any ${\bold u}$ in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1/2+ \delta _0}$ we have the following estimate: $$ |\zeta_{\infty}(\chi,{\bold u})|\le \frac{c( \delta _0)}{(1+\|y(\chi)\|+\|{ \operatorname{Im} }({\bold u})\|)^{\rho+t+1}}. $$ \label{est-inf} \end{coro} \section{Analytic properties of height zeta functions} \begin{dfn}{\rm Let $X_{ \Sigma }$ be a smooth projective toric variety. Let $ \varphi = \varphi _{\bold u}\in PL( \Sigma )_{{\bold C }}^G$ be a complexified piecewise linear function. Let $Y\subset X_{ \Sigma }$ be a locally closed subset. The height zeta function with respect to $Y$ is defined as $$ Z_{ \Sigma }(Y, {\bold u})=\sum_{x\in Y(F)} H_{ \Sigma }(x,-{\bold u}). $$ } \end{dfn} \noindent Let us formulate the first main result. \begin{theo}{\rm \cite{BaTschi1}} The height zeta function $Z_{ \Sigma }(T,{\bold u})$ as a function on $PL( \Sigma )_{{\bold C }}^G$ is holomorphic for ${ \operatorname{Re} }({\bold u})\in {\bold R }_{>1}^r$. Moreover, it descends to ${\rm Pic}(X_{ \Sigma })_{{\bold C }}$ and is holomorphic for ${ \operatorname{Re} }({\bold u})$ contained in the open cone $ \Lambda _{\rm eff}^{\circ}(X_{ \Sigma })+[K_{ \Sigma }]$. \label{convergence-cone} \end{theo} \begin{theo} (Poisson formula) {\rm \cite{BaTschi1,BaTschi2}} For all ${\bold u}$ with ${ \operatorname{Re} }({\bold s})\in {\bold R }^r_{>1}$ we have the following formula: $$ Z_{\Sigma}(T,{\bold u})=\frac{1}{(2\pi )^t b(T)} \int_{{\cal H}_T} \hat{H}_{ \Sigma }(\chi,-{\bold u}) d\chi, $$ The integral converges absolutely and uniformly to a holomorphic function in ${\bold u}$ in any compact in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1}$. \label{poiss} \end{theo} Let ${\cal L}$ be a line bundle on $X_{ \Sigma }$ metrized as above, such that its class $[L]$ is contained in the interior of the cone of effective divisors $ \Lambda _{\rm eff}(X_{ \Sigma })\subset {\rm Pic}(X_{ \Sigma })$. We have defined $a(L)$ as $$ a(L):=\inf \{a\in {\bold R } \mid a[L]+ [K_{ \Sigma }] \in \Lambda _{\rm eff}(X_{ \Sigma }) \}. $$ By our assumptions, we have $a(L)>0$, since $-[K_{ \Sigma }]\in \Lambda ^{\circ}_{\rm eff}(X_{ \Sigma })$. Denote by $ \Lambda (L)$ the face of maximal codimension of the cone $ \Lambda _{\rm eff}(X_{ \Sigma }) $ which contains $a(L)[L]+ [K_{ \Sigma }] $. Let $J(L)\subset [1,...,r]$ be the set of indices such that $[D_j]\in \Lambda (L)$ for $j\in J(L)$ and $I(L)=[1,...,r]\backslash J(L)$. We set $I=|I(L)|$ and $J=|J(L)| = r - I$. Without loss of generality, we assume that $I(L) = \{ 1, \ldots, I \}$ and $J(L) = \{ I+1, \ldots, r \}$. Since $a(L)[L]+ [K_{ \Sigma }]$ is an interior point of $ \Lambda (L)$ it follows that there exists a representation $$ a(L)[L]+ [K_{ \Sigma }]=\sum_{j\in J(L)} \lambda _j [D_j], $$ where $ \lambda _j \in {\bold Q }_{>0}$. Therefore, $$ [L]= \sum_{j\in J(L)} \frac{ \lambda _j +1}{a(L)}[D_j] +\sum_{i\in I(L)}\frac{1}{a(L)}[D_i]. $$ Fix these $ \lambda _j$ and choose $ \varepsilon >0$ such that $2 \varepsilon < \min_{j\in J(L)} \lambda _j$. We denote by $\varphi_L$ the piecewise linear function from $PL( \Sigma )^G_{{\bold R }}$ such that $a(L) \varphi_L(e_i) =1 $ for $i =1, \ldots, I$ and $a(L) \varphi_L(e_j) = 1 + \lambda _j$ for $j \in J(L)$. Here $e_i$ are generators of one-dimensional cones ${\bold R }_{\ge 0}e_i$ in the $G$-orbits $ \Sigma _i(1)$. We introduce the lattice $$ M_J=\{m\in M\,| \, <e,m>=0\, \;{\it for}\,\; {\bold R }_{\ge 0}e\in \cup_{i = 1}^I \Sigma _i(1) \}. $$ Define $M_I \cong M/M_J$. The following diagram is commutative $$ \begin{array}{ccccc} 0\rightarrow & M_J &\rightarrow & \bigoplus_{j\in J(L)}{\bold Z }[G_j] \\ & \downarrow & & \downarrow \\ 0\rightarrow & M &\rightarrow & \bigoplus_{i=1}^r{\bold Z }[G_i] \\ & \downarrow & & \downarrow \\ 0\rightarrow & M_I &\rightarrow & \bigoplus_{i\in I(L)}{\bold Z }[G_i]. \end{array} $$ The exact sequence of $G$-modules $$ 0\rightarrow M_J\rightarrow M\rightarrow M_I\rightarrow 0 $$ induces the exact sequence of algebraic tori $$ 1\rightarrow T_I\rightarrow T\rightarrow T_J\rightarrow 1. $$ It will be convenient to introduce new coordinates ${\bold s}=(s_i)_{i\in I(L)}, {\bold t}=(t_j)_{j\in J(L)}$ on $PL({ \Sigma })_{{\bold C }}^G$, where $s_i = u_i -1$ $( i = 1, \ldots, I)$, $t_j = u_{I +j} - 1 + \varepsilon $ $(j =1, \ldots, J)$. We shall write $({\bold s},{\bold t})=(s_1,...,s_I, t_1,...,t_J)$. \begin{theo} The height zeta function $Z_{ \Sigma }(T,{\bold s},{\bold t})$ is good with respect to the lattice $M^G_I$ and variables $\{s_1,...,s_I\}$ in the domain ${ \operatorname{Re} }({\bold s}) \in {\bold R }_{>0}^I$, ${ \operatorname{Re} }({\bold t}) \in {\bold R }_{> - \delta _0}^J$ for some positive $ \delta _0 < \varepsilon $. \label{Z-analytic} \end{theo} {\it Proof.} Recall that $Z_{ \Sigma }(T,{\bold u}) $ has the following integral representation in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1}$ (\ref{poiss}): $$ Z_{\Sigma}(T,{\bold u})=\frac{1}{(2\pi )^t b(T)} \int_{{\cal H}_T}\hat{H}_{ \Sigma }(\chi, -{\bold u})d\chi. $$ \noindent Using the explicit computation of the Fourier transform of local height functions and the absolute convergence of the integral in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1}$, we have $$ Z_{\Sigma}(T,{\bold u})=\frac{1}{(2\pi )^t b(T)} \int_{M_{{\bold R }}^G}{\bold dy} \left( \sum_{\chi\in{\cal U}_T} \hat{H}_{ \Sigma }(\chi, -{\bold u}+i{\bold y}) \right), $$ because the local height functions are invariant under the maximal compact subgroups $T({\cal O}_v) \subset T(F_v)$ and $\hat{H}_{ \Sigma }(\chi, -{\bold u}) = 0$ for all $\chi$ which are not trivial on the maximal compact subgroup ${\bold K}_T$. By \ref{defin}, we have: $$ \hat{H}_{ \Sigma }(\chi,-{\bold u}) =\prod_{j=1}^{r} L(\chi_j,u_j)\times \zeta_{\infty}(\chi,{\bold u}), $$ where $\chi_1,...,\chi_r$ are unramified Hecke characters of ${\bold G}_m^1({\bold A}_{F_j})$ induced from a character $\chi\in {\cal U}_T$, and $\zeta_{\infty}(\chi,{\bold u})$ is a function in ${\bold u}$ which is holomorphic in the domain ${ \operatorname{Re} }({\bold u})\in {\bold R }^r_{>1/2+ \delta _0}$ (for all $ \delta _0>0$). \noindent We have $$ {Z}_{ \Sigma }(T,{\bold s}, {\bold t})=\frac{1}{(2\pi )^t b(T)} \int_{M_{I,{\bold R }}^G} f_{ \Sigma }({\bold s}-i{\bold y}_I,{\bold t}-i{\bold y}_J){\bold dy}^I, $$ where $$ f_{ \Sigma }({\bold s},{\bold t}):= \sum_{\chi\in {\cal U}_T} \prod_{i\in I(L)}L(\chi_i,s_i+1)\times b_{ \Sigma }(\chi,{\bold s},{\bold t}) $$ $$ b_{ \Sigma }(\chi,{\bold s},{\bold t})= $$ $$ =\int_{M_{J,{\bold R }}^G}\prod_{j\in J(L)} L_{F_j}(\chi_j,t_j+1+ \varepsilon -iy_j) \times \zeta_{\infty}(\chi,{\bold s}-i{\bold y}_I,{\bold t}-i{\bold y}_J)){\bold dy}^J, $$ ${\bold dy}^I$ is the Lebesgue measure on $M_{I,{\bold R }}^G$ and ${\bold dy}^J $ the Lebesgue measure on $M_{J,{\bold R }}^G$. Using the estimates \ref{trivial}, \ref{estimates-inf}, \ref{m.estim}, \ref{est-inf}, we see that the sums and integrals above converge absolutely and uniformly to an analytic function in any compact in the domain ${ \operatorname{Re} }({\bold s})\in {\bold R }^I_{>0}$ and ${ \operatorname{Re} }({\bold t})\in {\bold R }^J_{> - \delta _0}$ for some $ \delta _0>0$ $( \delta _0< \varepsilon )$. Now the fact that the function $Z_{ \Sigma }(T,{\bold s},{\bold t}) $ is good with respect to the lattice $M_I^G\subset {\bold Z }^I$ and the variables $(s_1,...,s_I)$ follows from \ref{integral} and the following statement: \begin{theo} $$ \lim_{{\bold s} \rightarrow {\bold 0}} s_1\cdots s_I f_{ \Sigma }({\bold s},{\bold 0}) $$ exists and is not equal to zero. \label{nonzero} \end{theo} We divide the proof of Theorem \ref{nonzero} into a sequence of lemmas: \begin{lem} Let ${\cal U}_T(I)$ be the subgroup of ${\cal U}_T$ consisting of characters $\chi \in {\cal U}_{T}$ such that the corresponding Hecke characters $\chi_i \; \,(i =1, \ldots, I) $ are trivial. Denote \[ f_{ \Sigma }^I({\bold s},{\bold t})=\sum_{\chi\in{\cal U}_T(I)} \prod_{i\in I(L)} L_{F_i}(\chi_i,s_i+1) b_{ \Sigma }(\chi,{\bold s}, {\bold t}).\] Then \[ \lim_{{\bold s} \rightarrow {\bold 0}} s_1\cdots s_I f_{ \Sigma }({\bold s},{\bold 0}) = \lim_{{\bold s} \rightarrow {\bold 0}} s_1\cdots s_I f_{ \Sigma }^I({\bold s},{\bold 0}). \] \end{lem} \begin{lem} (Poisson formula) For ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^I_{>0} \times {\bold R }^J_{> - \delta _0}$, one has: \[ f_{ \Sigma }^I({\bold s},{\bold t}) = \int_{{\cal A}} H_{ \Sigma }(x, (-{\bold s},- {\bold t})) d{ \alpha }, \] where the subgroup ${\cal A} \subset T({\bold A}_F)$ is defined as $${\cal A}:= T(F) \overline{T_I(F)}. $$ \end{lem} {\em Proof. } By definition of $f_{ \Sigma }^I({\bold s},{\bold t})$, we conclude that this function equals to the integral of the Fourier transform of the adelic height function over the subgroup of characters $\chi$ of $T({\bold A}_F)$ which are trivial on $T(F)$ and such that the induced Hecke characters $\chi_i$ are trivial for $i \in I(L)$. It follows from the diagram $$ \begin{array}{cccccc} \prod_{i\in I(L)} {\bold G}_m({\bold A}_{F_i})/ {\bold G}_m(F_i) & \rightarrow & T_I({\bold A}_F)/T_I(F) & \rightarrow & A(T_I) &\rightarrow 0 \\ \downarrow & & \downarrow & & \downarrow & \\ \prod_{i=1}^r {\bold G}_m({\bold A}_{F_i})/ {\bold G}_m(F_j) & \rightarrow & T({\bold A}_F)/T(F) & \rightarrow & A(T) &\rightarrow 0 \\ \downarrow & & \downarrow & & \downarrow & \\ \prod_{j\in J(L)} {\bold G}_m({\bold A}_{F_j})/ {\bold G}_m(F_j) & \rightarrow & T_J({\bold A}_F)/T_J(F) & \rightarrow & A(T_J) &\rightarrow 0. \end{array} $$ that the common kernel of all such characters is $T(F) \overline{T_I(F)}$ (here we used the isomorphism $A(T_I) = T_I({\bold A}_F)/\overline{T_I(F)}$). The proof of the absolute convergence of the integral over ${\cal A}$ in the domain ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^I_{>0} \times {\bold R }^J_{> - \delta _0}$ is analogous to the proof of theorem 4.2 in \cite{BaTschi2}. \hfill $\Box$ \begin{lem} The function \[ s_1\cdots s_If_{ \Sigma }^I({\bold s},{\bold t}) \] extends to an analytic function in the domain ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^r_{>- \delta _0}$. \end{lem} {\em Proof. } The proof is similar to the proof of theorem 4.2 in \cite{BaTschi2}. The integral \[ \int_{{\cal A}} H_{ \Sigma }(x, (-{\bold s},- {\bold t})) d{ \alpha } \] can be estimated from above by an Euler product which is absolutely convergent in the domain ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^r_{>- \delta _0}$ times a product of zeta functions $\prod_{i =1}^I \zeta_{F_i}(s_i+1)$. \hfill $\Box$ \smallskip \noindent For $({\bold s},{\bold t})\in {\bold R }^r$ the function $H_{ \Sigma }(x, (-{\bold s},- {\bold t}))$ has values in positive real numbers. Therefore, to prove the non-vanishing of the constant, it suffices to show the following: \begin{lem} The value of \[ s_1\cdots s_I \int_{\overline{T_I(F)}} H_{ \Sigma }(x, (-{\bold s},- {\bold t})) d{ \alpha }_I \] at $({\bold 0}, {\bold 0})$ is positive. Here $d{ \alpha }_I$ is the induced Haar measure on $\overline{T_I(F)}$ \end{lem} {\em Proof.} For some finite subset $S \subset {\operatorname{Val} }(F)$, we can split the group $\overline{T_I(F)}$ into the direct product \[ \overline{T_I(F)}_S \times T_I({\bold A}_{F,S}),\] where $\overline{T_I(F)}_S$ is the image of $\overline{T_I(F)}$ in the finite product $\prod_{v \in S} T_I(F_v)$ and $$T_I({\bold A}_{F,S})= T_I({\bold A}_F) \cap \prod_{v \not\in S} T_I(F_v). $$ Hence, \[ \int_{\overline{T_I(F)}} H_{ \Sigma }(x, (-{\bold s},- {\bold t})) d{ \alpha }_I = \] \[ =\int_{\overline{T_I(F)}_S} H_{ \Sigma }(x, (-{\bold s},- {\bold t})) d{ \alpha _S} \times \prod_{v \not\in S} \int_{{T_I(F_v)}} H_{ \Sigma ,v}(x, (-{\bold s},- {\bold t})) d{ \alpha _v}. \] Here we denoted by $d{ \alpha _S}$ and $d{ \alpha _v}$ the Haar measures induced from $d{ \alpha }_I$. We claim that \[ \prod_{i =1}^I \zeta^{-1}_{F_i}(s_i +1) \prod_{v \not\in S} \int_{{T_I(F_v)}} H_{ \Sigma ,v}(x, (-{\bold s},- {\bold t})) d{ \alpha _v}. \] is an absolutely convergent Euler product for ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^r_{>- \delta _0}$. This statement follows from the explicit calculation of the local integrals (see \ref{localint}). \begin{lem} For all good valuations $v \not\in S$, the local integral \[ \int_{{T_I(F_v)}} H_{ \Sigma ,v}(x, (-{\bold s},- {\bold t})) d{ \alpha _v} = \prod_{i =1}^I \prod_{{\cal V}|v} \zeta_{F_i,{\cal V}}(s_i +1) \left( 1 + o(q_v^{-1 - \varepsilon _0}) \right) \] for some $ \varepsilon _0 > 0$ and all ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^r_{>- \delta _0}$ \label{localint} \end{lem} {\em Proof.} Denote by $N_{{\bold R }}(I)$ the minimal ${{\bold R }}$-subspace of $N_{{\bold R }}$ spanned by all $e$ with ${\bold R }_{\ge 0}e$ contained in the set of $1$-dimensional cones in $\cup_{i\in I(L)} \Sigma _i(1)$. Let $ \Sigma (L)$ be the complete $G$-invariant fan of cones in $N_{{\bold R }}(I)$ which consists of intersections of cones in $ \Sigma \subset N_{{\bold R }}$ with the subspace $N_{{\bold R }}(I)$. Since $ \Sigma (L)$ is not necessary a regular fan, we construct a new $G$-invariant fan $\tilde{ \Sigma }(L)$ by subdivision of cones in $ \Sigma (L)$ into regular ones using the method of Brylinski \cite{bryl}. This reduces the computation of the local intergral to the one made for local height functions on smooth toric varieties in \cite{BaTschi1}, theorem 2.2.6. Let $ \sigma _1,..., \sigma _{\tilde{n}}$ be the set of representatives of $G_v$-orbits in the set of $1$-dimensional cones in ${\tilde{ \Sigma }(L)}\subset N_{{\bold R }}(I)$. We obtain $$ \int_{{T_I(F_v)}} H_{ \Sigma ,v}(x_v, (-{\bold s},- {\bold t})) d{ \alpha _v} = $$ $$ = Q_{\tilde{ \Sigma }(L)}(q_v^{-l_{ \sigma _1}({\bold s}, {\bold t})},..., q_v^{-l_{ \sigma _{\tilde{n}}}({\bold s}, {\bold t})} )\prod_{j=1}^{\tilde{n}} \left( 1 - \frac{1}{q_v^{-l_{ \sigma _j}({\bold s}, {\bold t})}} \right), $$ where $l_{ \sigma }({\bold s}, {\bold t})$ are linear forms which are $\geq 1 - \varepsilon _0$ in the domain ${ \operatorname{Re} }({\bold s}, {\bold t}) \in {\bold R }^r_{>- \delta _0}$, and $ Q_{\tilde{ \Sigma }(L)}({\bold z}) $ is a polynomial in the variables ${\bold z}=(z_1,...,z_{\tilde{n}})$ such that all monomials in $Q_{\tilde{ \Sigma }(L)}({\bold z}) -1$ have degree $\geq 2$. Now we notice that $l_{ \sigma }({\bold 0}, {\bold 0}) =1$ iff $ \sigma $ is a $1$-dimensional cone in $ \Sigma $ and therefore, the cone ${\bold R }_{\ge 0}e_i$ for some $i\in I(L)$ is contained in the $G_v$-orbit of $ \sigma $ (see \ref{sigma}). \hfill $\Box$ \begin{lem} The set of lattice vectors $e \in N$ such that $a(L)\varphi_L (e) =1$ coincides with the set of lattice vectors $e_i\in N_{{\bold R }}(I)$ with ${\bold R }_{\geq 0}e_i \in \Sigma (1)$ and $a(L)\varphi_L (e_i) =1$. \label{sigma} \end{lem} {\em Proof. } Let $e$ be a lattice point in $N$. Since $ \Sigma $ is complete, there exists a $d$-dimensional cone $\sigma \in \Sigma $ such that $e \in \sigma$. We claim that the property $a(L)\varphi_L (e) =1$ implies that $e$ is a generator of a $1$-dimensional face of $\sigma$. Indeed, we have $a(L)\varphi_L(x) \geq \varphi_{ \Sigma }(x)$ for all $x \in N_{{\bold R }}$. On the other hand, $\sigma$ is generated by a basis of $N$ and $\varphi_{ \Sigma }$ has value $1$ on these generators. Hence, $e$ must be one of the generators of $\sigma$. It remains to show that the property $a(L)\varphi_L (e_i) =1$ for some generator $e_i$ of a $1$-dimensional cone ${\bold R}_{\geq 0}e_i \in \Sigma $ implies that $e_i \in N_{{\bold R }}(L)$. But this follows from the definition of $N_{{\bold R }}(L)$ as the subspace in $N_{{\bold R }}$ generated by all elements $e_i \in N$ such that ${\bold R}_{\geq 0}e_i \in \Sigma $ and $a(L)\varphi_L (e_i) =1$. \hfill $\Box$ \begin{theo} There exists a $ \delta >0$ such that the zeta function $Z_{ \Sigma }(T,{\cal L},s)$ obtained by restriction of the zeta function $Z_{ \Sigma }(T,{\bold s})$ to the complex line $s[L]\in {\rm Pic}(X_{ \Sigma })_{{\bold C }}$ has a representation of the form $$ Z_{ \Sigma }(T,{\cal L},s)= \frac{\Theta_{\cal L}( \Sigma )}{(s-a(L))^{b(L)}} +\frac{h(s)}{(s-a(L))^{b(L)-1}} $$ with some function $h(s)$ which is holomorphic in the domain ${ \operatorname{Re} }(s)>a(L)- \delta $ and a nonzero constant $\Theta_{\cal L}( \Sigma )$. \end{theo} \section{Appendix: ${\cal X}$-functions of polyhedral cones} Let $(A, A_{\bold R}, \Lambda ) $ be a triple consisting of a free abelian group $A$ of rank $k$, a $k$-dimensional real vector space $A_{\bold R} = A \otimes {\bold R}$ containing $A$ as a sublattice of maximal rank, and a convex $k$-dimensional finitely generated polyhedral cone $\Lambda \subset A_{{\bold R }}$ such that $\Lambda \cap - \Lambda = 0 \in A_{{\bold R }}$. Denote by $ \Lambda ^{\circ}$ the interior of $ \Lambda $ and by ${ \Lambda }_{\bold C}^{\circ} = { \Lambda }^{\circ} + iA_{{\bold R }}$ the complex tube domain over ${ \Lambda }^{\circ}$. Let $( A^*, A^*_{{\bold R }}, \Lambda ^*) $ be the triple consisting of the dual abelian group $A^* = {\rm Hom}(A, {\bold Z })$, the dual real vector space $A^*_{{\bold R }} = {\rm Hom}(A_{{\bold R }}, {\bold R })$, and the dual cone $ \Lambda ^* \subset A^*_{{\bold R }}$. We normalize the Haar measure $ {\bold d}{\bold y}$ on $A_{{\bold R }}^*$ by the condition: ${\rm vol}(A^*_{{\bold R }}/A^*)=1$. \begin{dfn}{\rm The {\em ${\cal X}$-function of} ${ \Lambda }$ is defined as the integral \[ {\cal X}_{ \Lambda }({\bold s}) = \int_{{ \Lambda }^*} e^{- \langle {\bold s}, {\bold y} \rangle} {\bold d}{\bold y}, \] where ${\bold s} \in { \Lambda }_{\bold C}^{\circ}$. } \label{c.func} \end{dfn} \begin{prop} One has ${\cal X}_{ \Lambda }({\bold s})$ is a rational function $$ {\cal X}_{ \Lambda }({\bold s}) = \frac{P({\bold s})}{Q({\bold s})}, $$ where $P$ is a homogeneous polynomial, $Q$ is a product of all linear homogeneous forms defining the codimension $1$ faces of $ \Lambda $, and ${\rm deg}\, P - {\rm deg}\, Q = -k$. In particular, if $(A, A_{{\bold R }}, \Lambda )=({\bold Z }^k,{\bold R }^k,{\bold R }^k_{\ge 0})$, then $$ {\cal X}_{ \Lambda }({\bold s}) = \frac{1}{s_1\cdots s_k}. $$ \label{merom} \end{prop} \begin{prop} {\rm \cite{BaTschi2} } Let $(A, A_{{\bold R }}, \Lambda )$ and $(\tilde{A}, \tilde{A}_{{\bold R }}, \tilde{ \Lambda })$ be two triples as above, $k = {\rm rk}\, A$ and $\tilde{k} = {\rm rk}\, \tilde{A}$, and $\psi\;:\; A \rightarrow \tilde{A}$ a homomorphism of free abelian groups with a finite cokernel ${\rm Coker} (\psi )$ (i.e., the corresponding linear mapping of real vector spaces $\psi \;:\; A_{{\bold R }} \rightarrow \tilde{A}_{{\bold R }}$ is surjective), and $\psi( \Lambda ) = \tilde{ \Lambda }$. Let $ \Gamma = {\rm Ker}\, \psi \subset A$, ${\bold d}{\bold y}$ the Haar measure on $ \Gamma _{{\bold R }} = \Gamma \otimes {{\bold R }}$ normalized by the condition ${\rm vol}( \Gamma _{{\bold R }}/ \Gamma )=1$. Then for all ${\bold s}$ with ${ \operatorname{Re} }({\bold s}) \in \Lambda^{\circ}$ the following formula holds: $$ {\cal X}_{\tilde{ \Lambda }}(\psi({\bold s})) = \frac{1}{(2\pi)^{k-\tilde{k}}|{\rm Coker}(\psi )|} \int_{ \Gamma _{{\bold R }}} {\cal X}_{{ \Lambda }} ({\bold s} + i {\bold y}) {\bold dy}, $$ where $|{\rm Coker} (\psi )|$ is the order of the finite abelian group ${\rm Coker} (\psi )$. \label{char0} \end{prop} Assume that a $\tilde{k}$-dimensional rational finite polyhedral cone $\tilde{\Lambda} \subset \tilde{A}_{{\bold R }}$ contains exactly $r$ one-dimensional faces with primitive lattice generators $a_1, \ldots, a_r \in \tilde{A}$. We set $k := r$, $A := {{\bold Z }}^r$ and denote by $\psi$ the natural homomorphism of lattices ${\bold Z}^r \rightarrow \tilde{A}$ which sends the standard basis of ${{\bold Z }}^r$ into $a_1, \ldots, a_r \in \tilde{A}$, so that $\tilde{ \Lambda }$ is the image of the simplicial cone ${\bold R }^r_{\ge 0}\subset {\bold R }^r$ under the surjective map of vector spaces $\psi\; : \; {\bold R}^r \rightarrow A_{{\bold R }}$. Denote by $ \Gamma $ the kernel of $\psi$. By \ref{char0} we obtain the following: \begin{coro}{\rm Let ${\bold s}=(s_1,...,s_r)$ be the standard coordinates in ${\bold C }^r$. Then $$ {\cal X}_{ \Lambda }(\psi({\bold s}))= \frac{1}{(2\pi )^{r-k}|{\rm Coker}(\psi )|} \int_{ \Gamma _{{\bold R }}}\frac{1}{\prod_{j=1,n}(s_j+iy_j)} {\bold d}{\bold y}, $$ where ${\bold dy}$ is the Haar measure on the additive group $ \Gamma _{{\bold R }}$ normalized by the lattice $ \Gamma $, $y_j$ are the coordinates of ${\bold y}$ in ${\bold R }^r$, and $|{\rm Coker} (\psi )|$ is the index of the sublattice in $\tilde{A}$ generated by $a_1, \ldots, a_r$. \label{int.formula} } \end{coro}

9606

alg-geom/9606019

en

https://arxiv.org/abs/alg-geom/9606019

alg-geom/9606019

Misha Verbitsky

Dmitry Kaledin, Misha Verbitsky

Non-Hermitian Yang-Mills connections

48 pages, LaTeX 2e

Selecta Math. 4 (1998) 279-320

We study Yang-Mills connections on holomorphic bundles over complex K\"ahler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk\"ahler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K\"ahler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the space of rational curves in the twistor space of the ``Mukai dual'' hyperk\"ahler manifold.

alg-geom

\section{Introduction.} \subsection{An overview} In this paper we study non-Hermitian Yang-Mills (NHYM) connections on a complex vector bundle ${\cal B}$ over a K\"ahler manifold. By definition, a connection $\nabla$ in ${\cal B}$ is Yang-Mills if its curvature $\Theta$ satisfies \begin{equation} \label{intro-Yang-Mills_Equation_} \begin{cases} \Lambda(\Theta)&=\text{const}{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}\operatorname{id}\\ \Theta \phantom{\Lambda()}&\in \Lambda^{1,1}(M, \operatorname{{\cal E}\!{\it nd}}({\cal B})), \end{cases} \end{equation} where $\Lambda$ is the standard Hodge operator, and $\Lambda^{1,1}(M, \operatorname{{\cal E}\!{\it nd}}({\cal B})$ is the space of $(1,1)$-forms with coefficients in $\operatorname{{\cal E}\!{\it nd}}({\cal B})$ (see Definition \ref{_NHYM_Definition_} for details). This definition is standard \cite{UY}, \cite{Donaldson:surfa}. However, usually $\nabla$ is assumed to be compatible with some Hermitian metric in $\cal B$. This is why we use the term ``non-Hermitian Yang-Mills'' to denote Yang-Mills connections which are not necessarily Hermitian. An important analogy for our construction is the one with flat connections on a complex vector bundle ${\cal B}$. Recall that when $c_1({\cal B}) = c_2({\cal B}) = 0$, Hermitian Yang-Mills connections are flat (L\"ubcke's principle; see \cite{S}). The moduli of flat, but not necessary unitary bundles is a beautiful subject, well studied in literature (see, e. g. \cite{S2}). This space has dimension twice the dimension of the moduli space of unitary flat bundles and has a natural holomorphic symplectic form. Also, generic part of the moduli of non-unitary flat bundles is equipped with a holomorphic Lagrangian\footnote{Lagrangian with respect to the holomorphic symplectic form.} fibration over the space of unitary flat connections. When $c_1({\cal B}) = c_2({\cal B}) = 0$, the flat connections are in one-to-one correspondence with those holomorphic structures on ${\cal B}$ which make it a polystable\footnote{By {\em polystable} we will always mean ``a direct sum of stable''. Throughout the paper, stability is understood in the sense of Mumford-Takemoto.} holomorphic bundle \cite{UY}, \cite{S}. For arbitrary bundle ${\cal B}$, a similar statement holds if we replace ``flat unitary'' by ``Hermitian Yang-Mills''. Thus, it is natural to weaken the flatness assumption and consider instead all Hermitian Yang-Mills connections. The non-Hermitian Yang-Mills connections that we define correspond then to connections that are flat but not necessarily unitary. The basic properties listed above for non-unitary flat bundles hold here as well. We show that the moduli space of NHYM connections has dimension twice the dimension of the moduli of Hermitian Yang-Mills connections and is naturally equipped with a holomorphic symplectic form. As in the case of flat bundles, generic part of the moduli of NHYM connections has a holomorphic Lagrangian fibration over the space of Hermitian Yang-Mills connections. Let us give a brief outline of the paper. Fix a compact K\"ahler manifold $M$ with a complex vector bundle ${\cal B}$. Let ${\cal M}^s$ be the set of equivalence classes of NHYM connections on ${\cal B}$, and let ${\cal M}^s_0 \subset {\cal M}^s$ be the subset of connections admitting a compatible Hermitian metric. Both sets turn out to have natural structures of complex analytic varieties. Recall that ${\cal M}^s_0$ is a moduli space of stable holomorphic bundles (\cite{UY}; see also \ref{_Uhle-Yau_Theorem_}). After giving the relevant definitions, in Section 1 we study the structure of ${\cal M}^s$ in the neighborhood of ${\cal M}^s_0$. We prove that $\dim{\cal M}^s = 2 \dim {\cal M}^s_0$ in a neighborhood of ${\cal M}^s_0$. Moreover, we identify the ring of germs of holomorphic functions on ${\cal M}^s$ near ${\cal M}^s_0$ with the ring of real-analytic complex-valued functions on ${\cal M}^s_0$. Thus an open neighborhood $U \supset {\cal M}^s$ is a complexification of ${\cal M}^s_0$ in the sense of Grauert. We also construct a holomorphic $2$-form on ${\cal M}^s$ which is symplectic in a neighborhood of ${\cal M}^s_0$. This picture is completely analogous to that for the space of flat connections, studied by Hitchin, Simpson and others (\cite{H},\cite{S2}). For concrete examples and applications of our theory, we consider the case of NHYM-connections over a hyperk\"ahler manifold (Definition \ref{defn.hyperkahler}). In this case, it is natural to modify the NHYM condition. Every hyperk\"ahler manifold is equipped with a quaternion action in its tangent space. Since the group of unitary quaternions is isomorphic to $SU(2)$, a hyperk\"ahler manifold has the group $SU(2)$ acting on its tangent bundle. Consider the corresponding action of $SU(2)$ on the space $\Lambda^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M)$ of differential forms over a hyperk\"ahler manifold $M$. Then all $SU(2)$-invariant $2$-forms satisfy \eqref{intro-Yang-Mills_Equation_} (Lemma \ref{primitive}). Thus, if the curvature $\Theta$ of a bundle $({\cal B}, \nabla)$ is $SU(2)$-invariant, ${\cal B}$ is NHYM. Converse is {\it a priori} non-true when $\dim_{\Bbb R} M >4$: there are $2$-forms satisfying \eqref{intro-Yang-Mills_Equation_} which are not $SU(2)$-invariant. However, as the discussion at the end of Section \ref{_autodu_Section_} shows, for bundles over compact manifolds $SU(2)$-invariance of the curvature is a good enough approximation of the NHYM property. A connection in ${\cal B}$ is called {\bf autodual} if its curvature is $SU(2)$-invariant (Definition \ref{_autodual_Definition_}). For $\dim_{\Bbb R} M=4$, the autoduality, in the sense of our definition, is equivalent to the anti-autoduality in the sense of 4-dimensional Yang-Mills theory. Hermitian autodual bundles were studied at great length in \cite{Vb}.\footnote{In \cite{Vb}, the term {\bf hyperholomorphic} was used for ``Hermitian autodual''.} Most of this paper is dedicated to the study of non-Hermitian autodual bundles over compact hyperk\"ahler manifolds. Consider the natural action of $SU(2)$ in the cohomology of a compact hyperk\"ahler manifold (see the beginning of Section \ref{_autodu_Section_}). Let ${\cal B}$ be a bundle with the first two Chern classes $c_1({\cal B})$, $c_2({\cal B})$ $SU(2)$-invariant. In \cite{Vb} we proved that every Hermitian Yang-Mills connection in ${\cal B}$ is autodual. It is natural to conjecture that in such a bundle every NHYM connection is autodual. In Theorem~\ref{_NHYM-are-autodu_Theorem_}, we prove a weaker form of this statement: namely, in a neighbourhood of the space of Hermitian Yang-Mills connections, every NHYM connection is autodual, assuming the first two Chern classes are $SU(2)$-invariant. This is done by constructing an explicit parametrization of this neighbourhood (Proposition~\ref{series}). Throughout the rest of this paper (starting from Section \ref{_twistors_Section_}) we study algebro-geometrical aspect of autodual connections. Two interdependent algebro-geometric interpretations of autoduality arise. Both of these itterpretations are related to the twistor formalism, which harks back to the works of Penrose and Salamon \cite{Sal}. Twistor contruction is explained in detail in Section \ref{_twistors_Section_}; here we give a brief outline of this formalism. Every hyperk\"ahler manifold $M$ has a whole 2-dimensional sphere of integrable complex structures, called {\bf induced complex structures}; these complex structures correspond bijectively to ${\Bbb R}$-algebra embeddings from complex numbers to quaternions (see Definition \ref{_induced_co_str_Definition_}). We identify this 2-dimensional sphere with ${\Bbb C} P^1$. Gluing all induced complex structures together with the complex structure in ${\Bbb C} P^1$, we obtain an almost complex structure on the product $M \times {\Bbb C} P^1$ (Definition \ref{twistor}). As proven by Salamon \cite{Sal}, this almost complex structure is integrable. The complex manifold obtained in this way is called {\bf the twistor space for $M$}, denoted by $\operatorname{Tw}(M)$. Consider the natural projections \[ \sigma:\; \operatorname{Tw}(M)= M\times {\Bbb C} P^1 \longrightarrow M, \ \ \ \pi:\; \operatorname{Tw}(M)= M\times {\Bbb C} P^1 \longrightarrow {\Bbb C} P^1;\] the latter map is holomorphic. The key statement to the twistor transform is the following lemma. \begin{lemma}\label{_autodua_(1,1)-on-twi-intro_Lemma_} Let $({\cal B}, \nabla)$ be a bundle with a connection over a hyperk\"ahler manifold $M$, and \[ (\sigma^*{\cal B}, \sigma^* \nabla)\] be the pullback of $({\cal B}, \nabla)$ to the twistor space. Then $(\sigma^*{\cal B}, \sigma ^* \nabla)$ is holomorphic if and only if $({\cal B}, \nabla)$ is autodual. \end{lemma} \par\noindent{\bf Proof.}\ This is a restatement of Lemma \ref{_autodua_(1,1)-on-twi_Lemma_}. \endproof This gives a natural map from the space of autodual connections on $M$ to the space of holomorphic bundles on $\operatorname{Tw}(M)$. We prove that this map is injective, and describe its image explicitly. For every points $x\in M$, the set $\sigma^{-1}(x)$ is a complex analytic submanifold of the twistor space. The projection $\pi\restrict{\sigma^{-1}(x)}:\; \sigma^{-1}(x) \longrightarrow {\Bbb C} P^1$ gives a canonical identification of $\sigma^{-1}(x)$ with ${\Bbb C} P^1$. The rational curve $\sigma^{-1}(x) \subset \operatorname{Tw}(M)$ is called {\bf a horisontal twistor line in $\operatorname{Tw}(M)$} (Definition \ref{twistor}). The following proposition provides an inverse to the map given by Lemma \ref{_autodua_(1,1)-on-twi-intro_Lemma_}. \begin{prop} Let $M$ be a hyperk\"ahler manifold, $\operatorname{Tw}(M)$ its twistor space and ${\cal E}$ a holomorphic bundle over $\operatorname{Tw}(M)$. Then ${\cal E}$ comes as a pullback of an autodual bundle $({\cal B}, \nabla)$ if and only if restriction of ${\cal E}$ to all horisontal twistor lines is trivial as a holomorphic vector bundle. Moreover, this autodual bundle is unique, up to equivalence. \end{prop} \par\noindent{\bf Proof.}\ This is a restatement of Theorem \ref{_twisto_transfo_equiva_Theorem_}. \endproof We obtained an identification of the set of equivalence classes of autodual bundles with a subset of the set of equivalence classes of bundles over the twistor space. We would like to interpret this identification geometrically, as an identification of certain moduli spaces. The autodual bundles are NHYM. The set of equivalence classes of stable NHYM bundles is equipped with a natural complex structure and is finite-dimensional, as we prove in Section \ref{_NHYM_Section_}. This general construction is used to build the moduli space of autodual bundles. It remains to define the notion of stability for holomorphic bundles over the twistor space and to construct the corresponding moduli space. The usual (Mumford-Takemoto) notion of stability does not work, because twistor spaces are not K\"ahler.\footnote{Moreover, as can be easily shown, the twistor space of a compact hyperk\"ahler manifold admits no K\"ahler metric.} We apply results of Li and Yau \cite{yl}, who define a notion of stability for bundles over complex manifolds equipped with a Hermitian metric satisfying a certain condition (see \eqref{yl}). The twistor space $\operatorname{Tw}(M)$ is isomorphic as a smooth manifold to ${\Bbb C} P^1 \times M$ and as such is equipped with the product metric. This metric is obviously Hermitian. We check the condition of Li and Yau for twistor spaces by computing the terms of \eqref{yl} explicitly. This enables us to speak of stable and semistable bundles over twistor spaces. Let ${\cal E}$ be a holomorphic bundle over $\operatorname{Tw}(M)$ obtained as a pullback of an autodual bundle on $M$. We prove that ${\cal E}$ is semistable. This gives a holomorphic interpretation of the moduli of autodual bundles on $M$. This is the first of our algebro-geometric interpretations. The second interpretation involves significantly more geometry, but yields a more explicit moduli space. Let $M$ be a compact hyperk\"ahler manifold, and ${\cal B}$ a complex vector bundle with first two Chern classes invariant under the natural $SU(2)$-action. Let $\widehat M$ be the moduli space for the Hermitian Yang-Mills connections on ${\cal B}$.\footnote{Such connections are always autodual, \cite{Vb}.} Then $\widehat M$ is equipped with a natural hyperk\"ahler structure (\cite{Vb}). The first result of this type was obtained by Mukai \cite{_Mukai:K3_} in the context of his duality between K3-surfaces; we use the term ``Mukai dual'' for $\widehat M$ in this more general situation. Let $X\stackrel{\pi}{\longrightarrow} {\Bbb C} P^1$, $\widehat X\stackrel{\hat \pi}{\longrightarrow} {\Bbb C} P^1$ be the twistor spaces for $M$, $\widehat M$, equipped with the natural holomorphic projections to ${\Bbb C} P^1$. For an induced complex structure $L$ on the hyperk\"ahler manifold $M$, we denote by $(M, L)$ the space $M$ considered as a complex K\"ahler manifold with $L$ as a complex structure. Identifying the set of induced complex structures with ${\Bbb C} P^1$, we consider $L$ as a point in ${\Bbb C} P^1$. Then, the complex manifold $(M, L)$ is canonically isomorphic to the pre-image $\pi^{-1}(L) \subset X$. By $i_L$ we denote the natural embedding $(M, L) = \pi^{-1}(L) \stackrel{i_L}{\hookrightarrow} X$. Let $B$ be a stable holomorphic bundle over $(M, L)$, with the complex vector bundle ${\cal B}$ as underlying complex vector space. In \cite{Vb}, we produce a canonical identification between the moduli space of such stable bundles and the space $\widehat M$ of autodual connections in ${\cal B}$. Let ${\cal F}_{B} = {i_L}_* {B}$ be the coherent sheaf direct image of ${B}$ under $i_L$. The moduli space of such sheaves ${\cal F}_{B}$ is naturally identified with $\widehat X$ (Section \ref{_lines_Section_}; see also \cite{Vb}). Consider a holomorphic section $s$ of the map $\hat \pi:\; \widehat X \longrightarrow {\Bbb C} P^1$, that is, a holomorphic embedding $s:\; {\Bbb C} P^1 \longrightarrow \widehat X$ such that $s\circ \hat \pi = \operatorname{id}$. The image of such embedding is called {\bf a twistor line in $\widehat X$} (Section \ref{_twistors_Section_}). Let ${\cal E}$ be a vector bundle over $X$ such that the pullback ${i_L}^* {\cal E}$ is stable for all induced complex structures $L\in {\Bbb C} P^1$. Such a bundle ${\cal E}$ is called {\bf fiberwise stable} (Definition \ref{fib.st}). {}From Lemma \ref{gen.st} it follows that fiberwise stable bundles are also stable, in the sense of Li--Yau. We restrict our attention to those bundles ${\cal E}$ which are, as $C^\infty$-vector bundles, isomorphic to $\sigma^*({\cal B})$, where ${\cal B}$ is our original complex vector bundle on $M$. Every fiberwise stable bundle ${\cal E}$ on $X$ gives a twistor line $s_{\cal E}:\; {\Bbb C} P^1 \longrightarrow \widehat X$ in $\widehat X$, where $s_{\cal E}$ associates a sheaf ${i_L}_* {i_L}^* {\cal E}$ to $L\in {\Bbb C} P^1$. Since points of $\widehat X$ are identified with isomorphism classes of such sheaves, the sheaf ${i_L}_* {i_L}^* {\cal E}$ can be naturally considered as a point in $\widehat X$. Clearly, the moduli $St_f(X)$ space of fiberwise stable bundles is open in the moduli $St(X)$ of stable bundles on $X$. This gives a complex structure on $St_f(X)$. The set $Sec(\widehat X)$ of twistor lines in $\widehat X$ is equipped with a complex structure as a subset of the Douady space of rational curves in $\widehat X$.\footnote{Douady spaces are analogues of Chow schemes, defined in the complex-analytic (as opposed to algebraic) setting.} We constructed a holomorphic map from $St_f(X)$ to $Sec(\widehat X)$. We prove that this map is in fact an isomorphism of complex varieties (Theorem \ref{iso}). The direct and inverse twistor transform give a canonical identification between the moduli of autodual bundles on $M$ and an open subset of the moduli of semi-stable bundles on $X$. Thus, we obtain an identification of the moduli of autodual bundles on $M$ and the space of twistor lines in $\widehat X$. We must caution the reader that in this introduction we mostly ignore the fact that all our constructions use different notions of stability; thus, all identifications are valid only locally in the subset where all the flavours of stability hold. The precise statements are given in Sections \ref{_twisto-tra_Section_}--\ref{_lines_Section_}. \subsection{Contents} \begin{itemize} \item Here are the contents of our article. \item The Introduction is in two parts: the first part explains the main ideas of this paper, and the second gives an overview of its content, section by section. These two parts of Introduction are independent. The introduction is also formally independent from the main part and vice versa. The reader who prefers rigorous discourse might ignore the introduction and start reading from Section \ref{_NHYM_Section_}. \item Section \ref{_NHYM_Section_} contains the definition of NHYM (non-\-Her\-mi\-tian Yang-\-Mills) connection. We give the definition of $(0,1)$-sta\-bi\-lity for NHYM connections and consider the natural forgetful map \begin{equation} \label{_(0,1_stable_to_holo-Equation_} \pi:\; {\cal M}^s \longrightarrow {\cal M}^s_0 \end{equation} from the space of $(0,1)$-stable NHYM-connections to the moduli space of stable holomorphic bundles. The fiber of this map is described explicitly through a power series and the Green operator (Proposition \ref{series}). This map is also used to show that the moduli space of $(0,1)$-stable NHYM-connections is correctly defined and finite-dimensional (Corollary \ref{_NHYM_finite-dim_Corollary_}). Uhlenbeck--Yau theorem (Theorem \ref{_Uhle-Yau_Theorem_}; see also \cite{UY}) provides a compatible Hermitian Yang-Mils connection for every stable holomorphic bundle. This gives a section ${\cal M}^s_0 \stackrel i \hookrightarrow {\cal M}^s$ of the map \eqref{_(0,1_stable_to_holo-Equation_}. We study the structure of ${\cal M}^s$ in the neighbourhood of $i\left({\cal M}^s_0\right)$, and prove that this neighbourhood is isomorphic to the complexification of ${\cal M}^s_0$ in the sense of Grauert (Proposition \ref{_complexi_Graue_Theorem_}). \item In Section \ref{_autodu_Section_}, we recall the definition of a hyperk\"ahler manifold and consider NHYM bundles over a complex manifold with a hyperk\"ahler metric. We define autodual bundles over hyperk\"ahler manifolds (Definition \ref{_autodual_Definition_}) and show that all autodual bundles are NHYM (Proposition \ref{_autodual_is_NHYM_Proposition_}). We cite the result of \cite{Vb}, which shows that all Hermitian Yang-Mills connections on a bundle ${\cal B}$ are autodual, if the first two Chern classes of ${\cal B}$ satisfy a certain natural assumption ($SU(2)$-invariance; see Theorem \ref{HYM.inv}). We also prove that, for a NHYM connection $\nabla$ sufficiently close to Hermitian, $\nabla$ is autodual (Theorem \ref{_NHYM-are-autodu_Theorem_}). \item Further on, we restrict our attention to autodual connection over hy\-per\-k\"ah\-ler manifolds. \item Section \ref{_twistors_Section_} gives a number of definition and preliminary results from algebraic geometry of the twistor spaces. We define the twistor space for an arbitrary hyperk\"ahler manifold (Definition \ref{twistor}). The twistor space is a complex manifold equipped with a holomorphic projection onto ${\Bbb C} P^1$. For most hyperk\"ahler manifolds (including all compact ones), the twistor space does not admit a K\"ahler metrics. This makes it difficult to define stability for bundles over twistor spaces. We overcome this difficulty by applying results of Li and Yau (\cite{yl}). We consider the differential form which is an imaginary part of the natural Hermitian metric on the twistor space. To apply \cite{yl}, we compute explicitly the de Rham differential of this form (Lemma \ref{_differe_of_Hermi_on_twistors_Lemma_}). \item In Section \ref{_twisto-tra_Section_} we define the direct and inverse twistor transform relating autodual bundles over a hyperk\"ahler manifold $M$ and holomorphic bundles over the corresponding twistor space $X$. There is a map from the set of isomorphism classes of autodual bundles on $M$ to the set of isomorphism classes of holomorphic bundles on $X$ (Lemma \ref{_autodua_(1,1)-on-twi_Lemma_}). We show that this map is an embedding and describe its image explicitly (Theorem \ref{_twisto_transfo_equiva_Theorem_}). \item In Section \ref{_stabi_of_twi_tra_Section_}, we consider holomorphic bundles on the twistor space obtained as a result of a twistor transform. We prove semistability of such bundles. Thus, twistor transform is interpreted as a map between moduli spaces. \item In Section \ref{_lines_Section_} we return to the study the algebro-geometric properties of the twistor space. For a compact hyperk\"ahler manifold $M$ and a stable holomorphic bundle $B$ on $M$, we consider the space $\widehat M$ of deformations of $B$. When the first two Chern classes of $B$ are $SU(2)$-invariant, the space $\widehat M$ has a natural hyperk\"ahler structure; this space is called then {\bf Mukai dual} to $M$. Let $X$, $\widehat X$ be the twistor spaces for $M$ and $\widehat M$. We interpret the space of stable bundles on $X$ in terms of rational curves on $\widehat X$ (Theorem \ref{iso}). \item In Section \ref{_conje_Section_}, we relate a number of conjectures and open questions from the geometry of NHYM and autodual bundles. \end{itemize} \section{The general case.} \label{_NHYM_Section_} \subsection{Definition of NHYM connections} Let $M$ be a K\"ahler manifold of dimension $n$ with the real valued K\"ahler form $\omega$. Consider a complex vector bundle ${\cal B}$ on $M$. Denote by ${\cal A}^n({\cal B})$ the bundle of smooth ${\cal B}$-valued $n$-forms on ${\cal B}$. Let $$ {\cal A}^n({\cal B}) = \bigoplus_{i+j=n} {\cal A}^{i,j}({\cal B}) $$ be the Hodge type decomposition. The bundle $\operatorname{{\cal E}\!{\it nd}}{\cal B}$ of endomorphisms of ${\cal B}$ is also a complex vector bundle. As usual, let $$ L:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ be the operator given by multiplication $\omega$. Let $$ \Lambda:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ be the adjoint operator with respect to the trace form on $\operatorname{{\cal E}\!{\it nd}}{\cal B}$. \begin{defn}\label{_NHYM_Definition_} A connection $\nabla:{\cal B} \to {\cal A}^1({\cal B})$ is called {\bf non-Hermitian Yang-Mills} (NHYM for short) if its curvature $R \in {\cal A}^2(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ is of Hodge type $(1,1)$ and satisfies $$ \Lambda \circ R = c \operatorname{Id} $$ for a certain constant $c \in {\Bbb C}$. \end{defn} \begin{rem} This terminology is perhaps unfortunate, in that a NHYM connection can (but need not) be Hermitian. We use the term for lack of better one. \end{rem} To simplify exposition, we will always consider only NHYM connections with the constant $c=0$. Let $\nabla$ be a NHYM connection on ${\cal B}$. Since the $(0,2)$-component of its curvature vanishes, the $(0,1)$-component $\nabla^{0,1}:{\cal B} \to {\cal A}^{0,1}({\cal B})$ defines a holomorphic structure on ${\cal B}$. We will call this {\bf the holomorphic structure associated to $\nabla$}. Let $\overline{\E}^*$ be the dual to the complex-conjugate to the complex bundle ${\cal B}$. Every NHYM-connection $\nabla$ obviously induces a NHYM connection $\nabla^*$ on the dual bundle ${\cal B}^*$. Let $\overline{\nabla}$ be the connection on $\overline{\E}^*$ complex-conjugate to $\nabla^*$. The connection $\overline{\nabla}$ is also obviously NHYM. We will call it {\bf the adjoint connection to $\nabla$}. The holomorphic structure on $\overline{\E}^*$ associated to $\overline{\nabla}$ will be called {\bf the adjoint holomorphic structure associated to $\nabla$}. Note that the adjoint holomorphic structure depends only on the $(1,0)$-part $\nabla^{1,0}$ of the connection $\nabla$. \subsection{Stability and moduli of NHYM connections.} Fix a compact K\"ahler manifold $M$ and a complex vector bundle ${\cal B}$ on $M$. Consider the space ${\cal A}$ of all connections on ${\cal B}$ and let ${\cal A}_0$ be the subspace of NHYM connnections. The space ${\cal A}$ is a complex-analytic Banach manifold, and ${\cal A}_0 \subset {\cal A}$ is an analytic subspace of ${\cal A}$. Let ${\cal G} = \operatorname{Maps}(M,\operatorname{Aut}{\cal B})$ be the complex Banach-Lie group of automorphisms of ${\cal B}$. The group ${\cal G}$ acts on ${\cal A}$ preserving the subset ${\cal A}_0$. In order to obtain a good moduli space for NHYM connections, we need to impose some stability conditions. \begin{defn} A NHYM connection $\nabla$ is called {\bf $(0,1)$-stable} if the bundle ${\cal B}$ with the associated holomorphic structure is a stable holomorphic bundle. \end{defn} Further on, we sometimes use the term {\it stable} to denote $(0,1)$-stable connections. \begin{rem}\label{too.strong} This definition is sufficient for our present purposes. However, it is unnaturally restrictive. See \ref{_stabili_for_hyper_redu-Definition_} for a more natural definition. \end{rem} Let ${\cal A}^s \subset {\cal A}_0$ be the open subset of $(0,1)$-stable NYHM connections and let $$ {\cal M}^s = {\cal A}_s / {\cal G} $$ be the set of {\em equivalence classes} of $(0,1)$-stable NHYM connections on ${\cal B}$ endowed with the quotient topology. Choose a connection $\nabla \in {\cal A}_0, \nabla:{\cal B} \to {\cal A}^1({\cal B})$. Let $\nabla = \nabla^{1,0} + \nabla^{0,1}$ be the type decomposition and extend both components to differentials \begin{align*} D:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) &\to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})\\ \overline{D}:{\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}} &\to {\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \end{align*} The tangent space $T_\nabla({\cal A})$ equals $T_\nabla({\cal A}) = {\cal A}^1(\operatorname{{\cal E}\!{\it nd}}{\cal B})$. The NHYM equations define a complex-analytic map $$ YM:{\cal A} \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^{0,2} \oplus {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}). $$ It is easy to see that the differential of $YM$ at the point $\nabla$ is given by $$ YM_\nabla = D + \overline{D} + \Lambda \nabla: {\cal A}^1(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^{0,2}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ On the other hand, the differential at $\nabla$ of the ${\cal G}$-action on ${\cal A}$ is given by $$ \nabla:{\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^1(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ \begin{defn} The complex $$ 0 \to {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^1(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^{0,2}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ is called {\bf the deformation complex} of the NHYM connection $\nabla$. \end{defn} \begin{rem} The deformation complex has a natural structure of a differential graded Lie algebra. \end{rem} \begin{prop} The deformation complex is elliptic. \end{prop} \par\noindent{\bf Proof.}\ Indeed, the complex $$ 0 \to {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{0,1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{0,2}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ is the Dolbeault complex for the holomorphic bundle $\operatorname{{\cal E}\!{\it nd}}{\cal B}$ and is therefore elliptic. Hence it is enough to prove that \begin{equation}\label{kernel} 0 \to {\cal A}^{1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \end{equation} is elliptic. By Kodaira identity $\Lambda D = \sqrt{-1} D^*$ on ${\cal A}^{1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$, and this complex is the same as $$ 0 \to D \oplus D^*:{\cal A}^{1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^0(\operatorname{{\cal E}\!{\it nd}}{\cal B}), $$ where $D^*$ is defined by means of the trace form on $\operatorname{{\cal E}\!{\it nd}}{\cal B}$. This complex is obviously elliptic. \endproof \begin{corr} Let $\widetilde{G} \subset {\cal G}$ be the stabilizer of $\nabla \in {\cal A}$. Then \begin{enumerate} \item $\widetilde{G}$ is a finite dimensional complex Lie group. \item There exists a finite dimensional locally closed complex-analytic Stein subspace $\widetilde{{\cal M}^s} \subset {\cal A}$ containing $\nabla$ and invariant under $\widetilde{G}$ such that the natural projection $$ \widetilde{{\cal M}^s} / \widetilde{G} \to {\cal M}^s $$ is an open embedding. \end{enumerate} \end{corr} \par\noindent{\bf Proof.}\ This is the standard application of the Luna's slice theorem, see \cite{Kod}. \endproof \begin{corr} \label{_NHYM_finite-dim_Corollary_} The topological space ${\cal M}^s$ has a natural structure of a comp\-lex\--analytic space. \end{corr} \par\noindent{\bf Proof.}\ Indeed, since $\widetilde{{\cal M}^s}$ is Stein, the quotient $\widetilde{{\cal M}^s} / \widetilde{G}$ is a Stein complex-analytic space. Now by the standard argument (\cite{Kod}) the induced complex analytic charts on ${\cal M}^s$ glue together to give a complex-analytic structure on the whole of ${\cal M}^s$. \endproof \subsection[Hermitian Yang-Mills bundles and the theorem of Uhlenbeck--Yau]{Hermitian Yang-Mills bundles and the theorem of \\ Uhlenbeck--Yau} For every complex bundle ${\cal B}$ on $M$ denote by ${\cal M}^s_0({\cal B})$ the moduli space of stable holomorphic structures on ${\cal B}$. Fix ${\cal B}$ and consider the space ${\cal M}^s$ of $(0,1)$-stable NHYM connections on ${\cal B}$. Taking the associated holomorphic structure defines a map $\pi:{\cal M}^s \to {\cal M}^s_0({\cal B})$. \begin{lemma} The map $\pi$ is holomorphic. \end{lemma} \par\noindent{\bf Proof.}\ Clear. \endproof Since every complex vector bundle admits an Hermitian metric, the complex vector bundles ${\cal B}$ and $\overline{\E}^*$ are isomorphic. Therefore the moduli spaces ${\cal M}^s_0({\cal B})$ and ${\cal M}^s_0(\overline{\E}^*)$ are naturally identified. Denote the space ${\cal M}^s_0({\cal B}) = {\cal M}^s_0(\overline{\E}^*)$ simply by ${\cal M}^s_0$, and let $\overline{\cal M}^s_0$ be the complex-conjugate space. Consider the open subset ${{\cal M}^{gs}} \subset {\cal M}^s$ of $(0,1)$-stable NHYM connections on ${\cal B}$ such that the adjoint connection on $\overline{\E}^*$ is also $(0,1)$-stable. Taking the adjoint holomorphic structure defines a map $\overline{\pi}:{{\cal M}^{gs}} \to \overline{\cal M}^s_0$. This map is also obviously holomorphic. \begin{lemma} \label{_compa_Hermi_pi=barpi_Lemma_} A NHYM connection $\nabla \in {{\cal M}^{gs}}$ satisfies $\pi(\nabla) = \overline{\pi}(\nabla)$ if and only if it admits a compatible Hermitian metric. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, $\pi(\nabla) = \overline{\pi}(\nabla)$ if and only if there exists an isomorphism $h:{\cal B} \to \overline{\E}^*$ sending $\nabla$ to $\overline{\nabla}$. This isomorphism defines an Hermitian metric on ${\cal B}$ compatible with the connection $\nabla$. \endproof To proceed further we need to recall the following fundamental theorem. \begin{theorem}[Uhlenbeck,Yau] \label{_Uhle-Yau_Theorem_} Every stable holomorphic bundle ${\cal B}$ on a K\"ahler manifold $M$ admits a unique Hermitian Yang-Mills connection $\nabla$. Vice versa, every holomorphic bundle admitting such a connection is polystable. \end{theorem} We will call such a metric {\bf a Uhlenbeck-Yau metric} for the holomorphic bundle ${\cal B}$. \begin{corr} Let ${\cal M}^s_u \in {{\cal M}^{gs}}$ be subset of equivalence classes of Hermitian connections. The product map $\pi \times \overline{\pi}:{{\cal M}^{gs}} \to {\cal M}^s_0 \times \overline{\cal M}^s_0$ identifies ${\cal M}^s_u$ with the diagonal in ${\cal M}^s_0 \times \overline{\cal M}^s_0$. \end{corr} \par\noindent{\bf Proof.}\ Clear. \endproof \begin{rem} Note that the subset ${\cal M}^s_u \subset {{\cal M}^{gs}}$ is not complex-analytic, but only real-analytic. \end{rem} \subsection{Moduli of NHYM connections as a complexification of the moduli of stable bundles} Let $\nabla \in {\cal M}^s_u \subset {\cal M}^s$ be an Hermitian Yang-Mills connection, and let ${\cal F}_\nabla = \pi^{-1}(\pi(\nabla)) \subset {\cal M}^s$ be the fiber of $\pi$ over $\nabla$. In order to study the map $\pi \times \overline{\pi}:{{\cal M}^{gs}} \to {\cal M}^s_0 \times \overline{\cal M}^s_0$ in a neighborhood of $\nabla$, we first study the restriction of the map $\overline{\pi}$ to ${\cal F}_\nabla$. We begin with the following. \begin{theorem}\label{zhopa} Let $\overline{D}:{\cal B} \to {\cal A}^{0,1}({\cal B})$ be a representative in the equivalence class $\pi(\nabla) \in {\cal M}^s_0$ of holomorphic structures on ${\cal B}$ and let $D:{\cal B} \to {\cal A}^{1,0}({\cal B})$ be the operator adjoint to $\overline{D}$ with respect to the Uhlenbeck-Yau metric. The fiber ${\cal F}_\nabla$ is isomorphic to the set of all $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $(1,0)$-forms $\theta$ satisfying \begin{equation}\label{formula} \begin{cases} D\theta + \theta \wedge \theta &= 0,\\ D^*\theta &= 0. \end{cases} \end{equation} \end{theorem} \par\noindent{\bf Proof.}\ To define the desired isomorphism, choose for any equivalence class $\nabla_1 \in \pi^{-1}(\pi(\nabla))$ a representative $\nabla_1 = \nabla^{1,0}_1 + \overline{D}:{\cal B} \to {\cal A}^1({\cal B})$. Every two representatives must differ by a gauge transformation $g:{\cal B} \to {\cal B}$. The map $g$ must preserve the holomorphic structure $\overline{D}$. However, this holomorphic structure is by assumption stable. Therefore $g = c\operatorname{Id}$ for $c \in {\Bbb C}$, and the operator $\nabla^{1,0}_1:{\cal B} \to {\cal A}^{1,0}({\cal B})$ is defined uniquely by its class in ${\cal F}_\nabla$. Take $\theta = \nabla^{1,0}_1 - D$; the equations~\eqref{formula} follow directly from the definition of NHYM connections. \endproof In order to apply this Theorem, note that by the second of the equations~\eqref{formula} every NHYM connection $\nabla_1 \in {\cal F}_\nabla$ defines a $D^*$-closed $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $(1,0)$-form $\theta$. Complex conjugation with respect to the Uhlenbeck-Yau metric $h_\nabla$ identifies the space of $D^*$-closed $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $(1,0)$-forms with the space of $\overline{D}^*$-closed $(0,1)$-forms, and it also identifies the respective cohomology spaces. But the cohomology spaces of the Dolbeault complex ${\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ with respect to $\overline{D}$ and $\overline{D}^*$ are both equal to the space of harmonic forms, hence naturally isomorphic. Collecting all this together, we define a map $\rho:{\cal F}_\nabla \to \overline{H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})}$ by the rule \begin{equation} \label{_from_NHYM_to_classes_Equation_} \nabla_1 \mapsto \langle \text{ class of } \theta \text{ in } H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) \rangle. \end{equation} \begin{prop}\label{kur} The map $\rho$ is a closed embedding in a neighborhood of $\nabla \in {\cal M}^s_u \in {\cal F}_\nabla$. \end{prop} This Proposition can be deduced directly from Theorem~\ref{zhopa}. However, we prefer to prove a stronger statement. To formulate it, consider the adjoint holomorphic structure $\overline{\pi}(\nabla)$ on the complex bundle $\overline{\E}^* \cong {\cal B}$. Recall the following standard fact from the deformation theory of holomorphic bundles. \begin{theorem} Let $\bar\partial:{\cal B} \to {\cal A}^{0,1}({\cal B})$ be a stable holomorphic structure on a complex Hermitian bundle ${\cal B}$. There exists a neighborhood $U \subset {\cal M}^s_0$ of $\bar\partial$ such that every $\bar\partial_1 \in U$ can be represented uniquely by an operator $\bar\partial + \theta:{\cal B} \to {\cal A}^{0,1}({\cal B})$ satisfying \begin{equation}\label{kuranishi} \begin{cases} \bar\partial\theta = \theta \wedge \theta\\ \bar\partial^*\theta = 0 \end{cases} \end{equation} The {\bf Kuranishi map} $U \to H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ defined by $$ \bar\partial_1 \to \langle \text{ class of } \theta \text{ in } H^1_{\bar\partial^*}({\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})) \cong H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) \rangle $$ is a locally closed embedding. \end{theorem} \begin{corr}\label{compl} The map $\overline{\pi}:{\cal F}_\nabla \to \overline{\cal M}^s_0$ is biholomophic in a neighborhood $V$ of $\nabla \in {\cal M}^s_u \subset {{\cal M}^{gs}}$, and the map $\rho:V \to H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ is the composition of $\overline{\pi}:{\cal F}_\nabla \to \overline{\cal M}^s_0$ and the Kuranishi map. \end{corr} \par\noindent{\bf Proof.}\ Complex conjugation sends equations~\eqref{formula} precisely to \eqref{kuranishi}, and thus establishes a bijection between neighborhoods of $\nabla \in {\cal F}_\nabla$ and $\overline{\pi}(\nabla) \in \overline{\cal M}^s_0$. \endproof This statement, in turn, implies the following. \begin{prop} \label{_complexi_Graue_Theorem_} The product map $\pi \times \overline{\pi}:{{\cal M}^{gs}} \to {\cal M}^s_0 \to \overline{\cal M}^s_0$ is biholomorphic on an open neighborhood $U$ of the subset ${\cal M}^s_u \in {{\cal M}^{gs}}$. \end{prop} \par\noindent{\bf Proof.}\ Consider both ${{\cal M}^{gs}}$ and ${\cal M}^s_0 \times \overline{\cal M}^s_0$ as spaces over ${\cal M}^s_0$. The map $\pi \times \overline{\pi}$ is a map over ${\cal M}^s_0$, and it is locally biholomorphic on every fiber of the natural projections ${{\cal M}^{gs}} \to {\cal M}^s_0$, ${\cal M}^s_0 \times \overline{\cal M}^s_0 \to {\cal M}^s_0$. \endproof Thus $\dim{\cal M}^s = 2\dim{\cal M}^s_0$, and an open neighborhood $U$ of the subspace ${\cal M}^s_u \subset {{\cal M}^{gs}}$ is the complexification of ${\cal M}^s_0$ is the sense of Grauert. \subsection{Holomorphic symplectic form on the moduli of NHYM bundles} In order to construct a holomorphic symplectic $2$-form on ${\cal M}^s$, we need to restrict our attention to a smooth open subset of ${\cal M}^s$. \begin{defn} A NHYM connection $\nabla$ is called {\bf smooth} if both $\nabla \in {\cal M}^s$ and $\pi(\nabla) \in {\cal M}^s_0$ are smooth points. \end{defn} Let $\nabla \in {\cal M}^s$ be a smooth NHYM connection and denote by ${\cal C}_\nabla$ its deformation complex. By construction the holomorphic tangent space $T_\nabla({\cal M}^s)$ is identified with a subspace of the first cohomology space $H^1({\cal C}_\nabla)$. By definition $\pi(\nabla)$ is a smooth point, and the tangent space $T_{\pi(\nabla)}({\cal M}^s_0)$ is a subspace of the first cohomology space $H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ of $M$ with coefficients in ${\cal B}$ equipped with the induced holomorphic structure. Consider the natural projection map ${\cal C}_\nabla \to {\cal A}^{0,i}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ from ${\cal C}_\nabla$ to the Dolbeault complex of the bundle $\operatorname{{\cal E}\!{\it nd}}{\cal B}$. Denote by $$ \operatorname{pr}:H^1({\cal C}_\nabla) \to H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ the induced map on the cohomology spaces and let $$ W = \operatorname{Ker}\operatorname{pr} \subset H^1({\cal C}_\nabla). $$ \begin{prop}\label{diagramma} Let $d\pi:T_\nabla({\cal M}^s) \to T_{\pi(\nabla)}({\cal M}^s_0)$ be the differential of the map $\pi:{\cal M}^s \to {\cal M}^s_0$ in the smooth point $\nabla \in {\cal M}^s_0$. The diagram $$ \begin{CD} 0 @>>> T_\nabla({\cal F}_\nabla) @>>> T_\nabla({\cal M}^s) @>d{\pi}>> T_{\pi(\nabla)} @>>> 0 \\ @VVV @VVV @VVV @VVV @VVV \\ 9 @>>> W @>>> H^1({\cal C}_\nabla) @>{\operatorname{pr}}>> H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) @>>> 0 \end{CD} $$ is commutative. \end{prop} \par\noindent{\bf Proof.}\ Clear. \endproof We first construct a symplectic form on the space $H^1({\cal C}_\nabla)$. To do this, we first identify the space $W \subset H^1({\cal C}_\nabla)$. \begin{lemma} The space $W$ is naturally isomorphic to $H^{n-1}(M, \operatorname{{\cal E}\!{\it nd}}{\cal B} \otimes {\cal K})$, where $K$ is the canonical line bundle on $M$. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, the space $W$ is isomorphic to the space of $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $(1,0)$-forms satisfying \begin{equation}\label{pipupa} \begin{cases} D\theta = 0\\ \Lambda \overline{D} \theta = 0 \end{cases} \end{equation} Consider the map $$ \bullet \wedge \omega^{n-1}:{\cal A}^{1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{n,n-1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}). $$ By Kodaira identities a form $\theta \in {\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ satisfies \eqref{pipupa} if and only if $\theta \wedge \omega^{n-1}$ is harmonic. Hence $\bullet \wedge \omega^{n-1}$ identifies $W$ with $H^{n-1}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B} \otimes {\cal K})$. \endproof Consider now two $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $1$-forms $\theta_0,\theta^1 \in {\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ and let $$ \Omega(\theta_0,\theta_1) = \int_M \operatorname{tr}(\theta_0 \wedge \theta_1 \wedge \omega^{n-1}), $$ where $\operatorname{tr}$ is the trace map, $n = \dim M$ and $\omega$ is the K\"ahler form on $M$. \begin{lemma} If $\theta_0 = \nabla g$ for some section $g \in {\cal A}^0(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$, then $$ \Omega(\theta_0,\theta_1) = 0 $$ for any $\theta_1 \in {\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ satisfying $\Lambda \nabla(\theta_1) = 0$. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, \begin{multline*} \Omega(\theta_0,\theta_1) = \int_M \operatorname{tr}( \nabla(\omega^{n-1}g)) \wedge \theta_1) = \\ = \int_M \operatorname{tr}(\omega^{n-1} \wedge \nabla(g\theta_1)) - \int_M \operatorname{tr}( g \omega^{n-1} \wedge \nabla(\theta_1)) = \int_M \operatorname{tr}(g \Lambda \nabla(\theta_1)) \omega^n = 0 \end{multline*} \endproof \begin{corr}\label{bububuj} \begin{enumerate} \item The form $\Omega$ defines a complex $2$-form on the space $H^1({\cal C}_\nabla)$. \item The subspace $W$ is isotropic, and the induced pairing $$ (W \cong H^{n-1}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}\otimes{\cal K}))\otimes H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\Bbb C} $$ is non-degenerate. \end{enumerate} \end{corr} \par\noindent{\bf Proof.}\ The first statement is clear. It is easy to see that the pairing induced by $\Omega$ is exactly the one defined by the Serre duality, which proves the second statement. \endproof Restricting to the subspace $T_\nabla({\cal M}^s)$, we get a $2$-form $\Omega$ on ${\cal M}^s$. This form is obviously holomorphic. \begin{prop}\label{symplectic} Assume that either $\nabla \in {\cal M}^s$ is Hermitian, or $T_\nabla({\cal M}^s) = H^1({\cal C}_\nabla)$. Then the form $\Omega$ on $T_\nabla({\cal M}^s)$ is non-degenerate. The map ${\cal M}^s \to {\cal M}^s_0$ is a Lagrangian fibration in the neghrborhood of $\nabla$. \end{prop} \par\noindent{\bf Proof.}\ It is easy to see that $T_\nabla({\cal F}_\nabla) \subset T_\nabla({\cal M}^s)$ is isotropic. If the connection $\nabla$ is {\em unobstructed}, that is, the embedding $T_\nabla({\cal M}^s) \hookrightarrow H^1({\cal C}_\nabla)$ is actually an isomorphism, then the statement follows from Corollary~\ref{bububuj}. Suppose now that the inclusion $T_\nabla({\cal M}^s) \subset H^1({\cal C}_\nabla)$ is proper. By assumption the connection $\nabla$ is Hermitian in this case, therefore the complex conjugation map $\overline{\phantom{\E}}:{\cal A}^{0,1}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{1,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ is defined. It is easy to see that this map identifies $\operatorname{Ker}\operatorname{pr}$ with $\overline{H^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})}$. By Corollary~\ref{compl} it also identifies $T_\nabla({\cal F}_\nabla)$ with $\overline{T_{\pi(\nabla)}}$. The form $\Omega(\bullet,\overline{\bullet})$ is a non-degenerate Hermitian form on $\operatorname{Ker}\operatorname{pr}$. Therefore its restriction to $T_\nabla(M_\nabla)$ is also non-degenerate. The last statement now follows directly from Proposition~\ref{diagramma} and Corollary~\ref{bububuj}. \endproof \subsection{Local parametrization of the moduli of NHYM connections} In the last part of this section we give a more explicit description of the embedding ${\cal F}_\nabla \to H^1(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ for an Hermitian Yang-Mills connection $\nabla$ in the spirit of \cite{Vb}. This description is of independent interest, and we will also use it in the next section in the study of NHYM connections on hyperk\"ahler manifolds. Fix an Uhlenbeck-Yau metric on ${\cal B}$ compatible with $\nabla$ and let $\Delta = DD^* + D^*D$ be the associated Laplace operator on ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$. Let $G$ be the Green operator provided by the Hodge theory. Recall that we have the Hodge decomposition $$ {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) = {\cal H}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{ex}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ into the space of harmonic form ${\cal H}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}$ and its orthogonal complement ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{ex}}$. This complement is further decomposed as $$ {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{ex}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) = D{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}-1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus D^*{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ The composition $DD^*G$ is by definition the projection onto $D{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}-1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \subset {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$. Take now a small neighborhood $U \subset {\cal F}_\nabla$ of the Hermitian connection $\nabla \in {\cal F}_\nabla$ and a NHYM connection $\nabla_1 \in U$. Let $\theta = \nabla_1 - \nabla$, $\theta \in {\cal A}^{1,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ and let $K(\theta) \in \overline{H^{1,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})}$ be the associated cohomology class. Shrinking $U$ if neccesary, we see that by Proposition~\ref{kur} the connection $\nabla_1$ is uniquely determined by the class $K(\theta)$. Let $\theta_0 \in {\cal A}^{1,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ be the harmonic form representing the class $K(\theta)$. Define by induction $$ \theta_n = D^*G \sum_{0 \leq k < n} \theta_k \wedge \theta_{n-1-k}. $$ \begin{prop}\label{series} Let $\nabla$ be an Hermitian Yang-Mills connection. There exists a neighborhood $V \subset {\cal F}_\nabla$ of $\nabla \in {\cal F}_\nabla$ such that for every $\nabla_1 \in V \subset {\cal F}_\nabla$ the series \begin{equation}\label{rjad} \sum_{0 \leq k} \theta_k \end{equation} converges to the form $\theta = \nabla_1 - \nabla$. \end{prop} \par\noindent{\bf Proof.}\ The metric on $\operatorname{{\cal E}\!{\it nd}}{\cal B}$ defines a norm $\|\bullet\|$ on ${\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$. We can assume that $\left\| \theta_0 \right\| < \varepsilon$ for any fixed $\varepsilon > 0$. Since the Hodge decomposition is orthogonal, \begin{multline*} \left\|D \theta_n\right\| = \left\|DD^*G\left(\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-k}\right)\right\| \\ \leq \left\|\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-k}\right\| \leq \sum_{0 \leq k < n} \left\|\theta_k\right\| {\:\raisebox{3.5pt}{\text{\circle*{1.5}}}} \left\|\theta_{n-k}\right\|. \end{multline*} Since $D:D^*({\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})) \to {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ is injective and elliptic, there exists a constant $C > 0$ such that $$ \|Df\| > C\|f\| $$ for all $f \in D^*({\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B}))$. Let $a_n = \frac{(2n)!}{(n!)^2}$ be the Catalan numbers. By induction $$ \left\|\theta_n\right\| < a_n \left| \frac{\varepsilon}{C} \right|. $$ Since $A(z) = \sum a_n z^n$ satisfies $A(z) = 1 + z(A(z))^2$, it equals $$ A(z) = \frac{1 - \sqrt{1-4z}}{2} $$ and converges for $z < \frac{1}{4}$. Therefore the series~\eqref{rjad} converges for $4\varepsilon < C$. To prove that it converges to $\theta$, let $\chi_0 = \theta$ and let $$ \chi_n = \theta - \sum_{0 \leq k < n} \theta_k $$ for $n \geq 1$. Since both $\nabla$ and $\nabla + \theta$ are NHYM, we have $D\chi_1 = \chi_0 \wedge \chi_0$. Therefore $\chi_0 \wedge \chi_0 \in {\cal A}^{2,0}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ and $$ D \chi_1 = D D^* G (\chi_0 \wedge \chi_0) = \chi_0 \wedge \chi_0. $$ By induction $$ D \chi_n = \chi_0 \wedge \chi_{n-1} + \sum_{0 \leq k \leq n} \chi_k \wedge \theta_{n - 1 - k}, $$ and $\chi_n \in D^*({\cal A}^{2,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}))$ for all $n > 0$. Again by induction, $$ \left\|\chi_n\right\| < a_{n+1} \left(\frac{\varepsilon}{C}\right)^n. $$ Therefore $\chi_n \to 0$ if $4\varepsilon < C$, which proves the Proposition. \endproof \begin{rem} This Proposition can be strengthened somewhat. Namely, for any harmonic $\operatorname{{\cal E}\!{\it nd}}_{\cal B}$-valued $(1,0)$-form $\theta_0$ in a small neighborhood of $0$ the series~\eqref{rjad} converges to a form $\theta$. As follows from \cite{Vb}, the connection $\nabla + \theta$ is NHYM provided the following holds. \begin{description} \item[*] All the forms $\theta_p \wedge \theta_q \in {\cal A}^{2,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ lie in ${\cal A}^{2,0}_{\operatorname{ex}}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$. \end{description} This condition is also knows as vanishing of all of the so-called Massey products $[\theta_0 \wedge \ldots \wedge \theta_0]$. \end{rem} \section{Autodual and NHYM connections in the hy\-per\-k\"ah\-ler case.} \label{_autodu_Section_} We now turn to the study of NHYM connections on hyperk\"ahler manifolds. First we recall the definitions and some general facts. \begin{defn}[\cite{Cal}] \label{defn.hyperkahler} A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ equipped with two integrable almost complex structures $I,J$ which are parallel with respect to the Levi-Civita connection and satisfy $$ I \circ J = - J \circ I. $$ \end{defn} Let $M$ be a hyperk\"ahler manifold. The operators $I,J$ define an action of the quaternion algebra ${\Bbb H}$ on the tangent bundle $TM$. This action is also parallel. Every imaginary quaternion $a \in {\Bbb H}$ satisfying $a^2 = -1$ defines an almost complex structure on $M$. This almost complex structure is parallel, hence integrable and K\"ahler. \begin{defn} \label{_induced_co_str_Definition_} A complex structure on $M$ corresponding to an imaginary quaternion $a \in {\Bbb H}$ with $a^2 = -1$ is said to be {\bf induced by $a$}. \end{defn} For every such $a \in {\Bbb H}$ we will denote by $\omega_a$ the K\"ahler form in the complex structure induced by $a$. We will always assume fixed a preferred complex K\"ahler structure $I$ on $M$. Recall that every hyperk\"ahler manifold is equipped with a canonical holomorphic symplectic $2$-form $\Omega$. If $J,K \in {\Bbb H}$ satisfy $J^2 = -1$, $IJ=K$ then this form equals $$ \Omega = \omega_J + \sqrt{-1}\omega_K. $$ The group $U({\Bbb H})$ of all unitary quaternions is isomorphic to ${SU(2)}$. Thus every hyperk\"ahler manifold comes equipped with an action of ${SU(2)}$ on its tangent bundle, and, {\it a posteriori}, with an action of its Lie algebra $\frak{su}(2)$. Extend these actions to the bundles $\Lambda^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}$ of differential forms and let $\Lambda^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{inv}} \subset \Lambda^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}$ be the subbundle of ${SU(2)}$-invariant forms. The ${SU(2)}$-action does not commute with the de Rham differential. However, it does commute with the Laplacian (see \cite{Vl}). Therefore it preserves the subspace of harmonic forms. Identifying harmonic forms with coholomogy classes, we get an action of ${SU(2)}$ on the cohomology spaces $H^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M)$. Let $\Lambda:\Lambda^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+2} \to \Lambda^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}$ be the Hodge operator associated to the K\"ahler metric on $M$. \begin{defn} A differential form $\theta$ on $M$ is called {\bf primitive} if $\Lambda\theta = 0$. \end{defn} \begin{lemma}\label{primitive} \begin{enumerate} \item All ${SU(2)}$-invariant forms are primitive. All ${SU(2)}$-in\-va\-ri\-ant $2$-forms are of Hodge type $(1,1)$ for every one of the induced complex structures on $M$. Vice versa, if a form is of type $(1,1)$ for all the induced complex structures, it is ${SU(2)}$-invariant. \item The same statements hold for de Rham cohomology classes instead of forms. \end{enumerate} \end{lemma} \par\noindent{\bf Proof.}\ See \cite{Vb}, Lemma 2.1 \endproof \begin{rem} The converse is true for $\dim_{\Bbb C} M = 2$, but in higher dimensiona there are primitive forms that are not ${SU(2)}$-invariant. \end{rem} Consider a complex bundle ${\cal B}$ on $M$ and let $\nabla:{\cal B} \to {\cal A}^1({\cal B})$ be a connection on ${\cal B}$. \begin{defn}\label{_autodual_Definition_} The connection $\nabla$ is called {\bf autodual} if its curvature $R \in {\cal A}^2(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ is ${SU(2)}$-inavariant. \end{defn} \begin{rem} The terminology comes from the $4$-dimensional topology: autodual connections on hyperk\"ahler surfaces are anti-selfdual in the usual topological sense. \end{rem} We will call an autodual connection $\nabla$ {\bf $(0,1)$-stable} if its $(1,0)$-part defines a stable holomorphic structure on the bundle ${\cal B}$. Denote by ${\cal M}^s_{\operatorname{inv}}$ the set of equivalence classes of $(0,1)$-stable autodual connections on the bundle ${\cal B}$. \begin{rem} Like in Remark~\ref{too.strong}, this $(0,1)$-stability condition may be too restrictive. \end{rem} Let $\nabla$ be an autodual connection on ${\cal B}$. By Lemma~\ref{primitive} for every $J \in {{\Bbb CP}^1}$ the $(0,1)$-component of the connection $\nabla$ with respect to the complex structure induced by $J$ defines a holomorphic structure on ${\cal B}$. We will call it {\bf the holomorphic structure induced by $J$}. \begin{prop} \label{_autodual_is_NHYM_Proposition_} Let $M$ be a hyperk\"ahler manifold and let ${\cal B}$ be a complex bundle on $M$. Every autodual connection $\nabla$ on ${\cal B}$ is NHYM. \end{prop} \par\noindent{\bf Proof.}\ Immediately follows from Lemma~\ref{primitive}. \endproof Therefore there exists a natural embedding ${\cal M}^s_{\operatorname{inv}} \hookrightarrow {\cal M}^s$ from ${\cal M}^s_{\operatorname{inv}}$ to the moduli space ${\cal M}^s$ of NHYM connections on ${\cal B}$. In the rest of this section we give a partial description of the image of this embedding. We will use the following. \begin{theorem}\label{HYM.inv} Assume that the first two Chern classes $$ c_1({\cal B}),c_2({\cal B}) \in H^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M) $$ of the bundle ${\cal B}$ are ${SU(2)}$-invariant. Then every Hermitian Yang-Mills connection $\nabla$ on ${\cal B}$ is autodual. \end{theorem} \par\noindent{\bf Proof.}\ See \cite{Vb}. \endproof Therefore, if the Chern classes $c_1({\cal B}),c_2({\cal B})$ are ${SU(2)}$-invariant, then the closed subset ${\cal M}^s_u \in {\cal M}^s$ of Hermitian Yang-Mills connections lies in ${\cal M}^s_{\operatorname{inv}}$. \begin{theorem} \label{_NHYM-are-autodu_Theorem_} Let $M$ be a hyperk\"ahler manifold and let ${\cal B}$ be a complex vector bundle on $M$ such that the Chern classes $c_1({\cal B}),c_2({\cal B})$ are ${SU(2)}$-invariant. Then subset ${\cal M}^s_{\operatorname{inv}} \subset {\cal M}^s$ of autodual connections contains an open neighborhood of the subset ${\cal M}^s_u \subset {\cal M}^s$ of connections admitting a compatible Hermitian metric. \end{theorem} \par\noindent{\bf Proof.}\ Let $\nabla$ be an Hermitian Yang-Mills connection. It is autodual by Theorem~\ref{HYM.inv}, and it is enough to show that every $\nabla_1 \in {\cal F}_\nabla$ sufficiently close to $\nabla$ is also autodual. Let $\theta = \nabla_1 - \nabla$ and let $\theta_n, n \geq 0$ be as in \eqref{rjad}. By Propostion~\ref{series} we can assume that $$ \theta = \sum_k \theta_k. $$ It is enough to prove that $\overline{D}\theta$ is ${SU(2)}$-invariant. We will prove that $\overline{D}\theta_k$ is ${SU(2)}$-invariant for all $k \geq 0$. To do this, we use results of \cite{Vb}. Consider the operator $$ L^\Omega:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})\to{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+2,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ given by multiplication by the canonical holomorphic $2$-form $\Omega$ on $M$. By \cite{Vl} the operator $$ [L_\Omega,\Lambda]:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}-1}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ coincides with the action of a nilpotent element in the Lie algebra $\frak{su}(2)$. Therefore it is a derivation with respect to the algebra structure on the complex ${\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$. Moreover, a $(1,1)$-form $\alpha$ is ${SU(2)}$-invariant if and only if $[L_\Omega,\Lambda]\alpha = 0$. Let $\partial^J:{\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$ be the commutator $$ \partial^J = \left[\overline{D} , \left[L_\Omega,\Lambda\right]\right]. $$ This map again is a derivation with respect to the algebra structure on ${\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}(\operatorname{{\cal E}\!{\it nd}}{\cal B})$, and it is enough to prove that $\partial^J\theta_n=0$ for all $n \geq 0$. Moreover, $$ \partial^J = \left[\overline{D}, \left[L_\Omega,\Lambda\right]\right] = \left[L_\Omega, \left[\overline{D}, \Lambda\right]\right] + \left[\Lambda, \left[L_\Omega,\overline{D}\right]\right], $$ The second term is zero since $\Omega$ is holomorphic, the first term is $[L_\Omega,\sqrt{-1}D^*]$ by Kodaira identity. Hence $D^*$ and $\partial^J$ anticommute. Finally, the Laplacian $\Delta_J = \partial_J\partial_J^* + \partial_J^*\partial_J$ is proportional by \cite{Vb} to the Laplacian $\Delta = DD^* + D^*D$. Therefore the Laplacian $\Delta$ and the Green operator $G$ also commute with $\partial^J$. Now, the form $\theta_0$ is by definition $\Delta$-harmonic. Therefore it is also $\Delta_J$-harmonic, and $\partial^J\theta_0 = 0$. To prove that $\partial^J\theta_n = 0$, use induction on $n$. By definition \begin{multline*} \partial^J\theta_n = \partial^JD^*G\left(\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-1-k}\right) = \\ = -D^*G\left(\sum_{0 \leq k <n} \partial^J\theta_k \wedge \theta_{n-1-k} + \theta_k \wedge \partial^J\theta_{n-1-k} \right). \end{multline*} The right hand side is zero by the inductive assumption. Thus all $\partial^J\theta_k$ are zero, and all the $\overline{D}\theta_k$ are ${SU(2)}$-invariant, which proves the Theorem. \endproof \section{Stable bundles over twistor spaces.} \label{_twistors_Section_} \subsection{Introduction} To further study autodual connections on a bundle ${\cal B}$ over a hyperk\"ahler manifold $M$, we need to introduce the so-called ``twistor space'' $X$ for $M$. This is a certain non-K\"ahler complex manifold associated to $M$. Autodual connections give rise to holomorphic bundles on $X$ by means of a construction known as ``twistor transform''. This construction turns out to be essentially invertible, thus providing additional information on the moduli space ${\cal M}^s_{\operatorname{inv}}$. We develop the twistor transform machinery in the next section. In this section we give the necessary preliminaries: the definition and some properties of the twistor space $X$, and a discussion of the notion of stability for holomorphic bundles over $X$. \subsection{Twistor spaces} Let $M$ be a hyperk\"ahler manifold. Consider the product manifold $X = M \times S^2$. Embed the sphere $S^2 \subset {\Bbb H}$ into the quaternion algebra ${\Bbb H}$ as the subset of all quaternions $J$ with $J^2 = -1$. For every point $x = m \times J \in X = M \times S^2$ the tangent space $T_xX$ is canonically decomposed $T_xX = T_mM \oplus T_JS^2$. Identify $S^2 = {{\Bbb CP}^1}$ and let $I_J:T_JS^2 \to T_JS^2$ be the complex structure operator. Let $I_m:T_mM \to T_mM$ be the complex structure on $M$ induced by $J \in S^2 \subset {\Bbb H}$. The operator $I_x = I_m \oplus I_J:T_xX \to T_xX$ satisfies $I_x \circ I_x = -1$. It depends smoothly on the point $x$, hence defines an almost complex structure on $X$. This almost complex structure is known to be integrable (see \cite{Sal}). \begin{defn}\label{twistor} The complex manifold $\langle X, I_x \rangle$ is called {\bf the twistor space} for the hyperk\"ahler manifold $M$. \end{defn} By definition the twistor space comes equipped with projections $\sigma:X \to M$, $\pi: X \to {{\Bbb CP}^1}$. The second projection is holomorphic. For any point $m \in M$ the section $\widetilde{m}:{{\Bbb CP}^1} \to X$ with image $m \times {{\Bbb CP}^1} \subset X$ is also holomorphic. We will call this section $\widetilde{m}$ {\bf the horizontal twistor line} corresponding to $m \in {\cal M}^s$. Let $\iota:{{\Bbb CP}^1} \to {{\Bbb CP}^1}$ be the real structure on ${{\Bbb CP}^1}$given by the antipodal involution. Then the product map $$ \iota = \operatorname{id} \times \iota:X \to X $$ defines a real structure on the complex manfiold $X$. The following fundamental property of twistor spaces is proved, e.g., in \cite{HKLR}. \begin{theorem}\label{inv} Let $M$ be a hyperk\"ahler manifold and let $X$ be its twistor space. Then a holomorphic section ${{\Bbb CP}^1} \to X$ of the natural projection $\pi:X \to {{\Bbb CP}^1}$ is a horizontal twistor line if and only if it commutes with natural real structure $\iota:X \to X$. \end{theorem} Let $\operatorname{Sec}$ be the Douady moduli space of holomorphic sections ${{\Bbb CP}^1} \to X$ of the projection $\pi:X \to M$. Then conjugation by $\iota$ defines a real structure on the complex-analytic space $\operatorname{Sec}$. Theorem~\ref{inv} identifies the susbet of real points of $\operatorname{Sec}$ with the hyperk\"ahler manifold $M$. We will call arbitrary holomorphic sections ${{\Bbb CP}^1} \to X$ {\bf twistor lines} in $X$. \subsection{Li--Yau theorem} \label{ss.twistor} The twistor space, in general, does not admit a K\"ahler metric. In order to obtain a good moduli space for holomorphic bundles on $X$ we use a genralization of the notion of stability introduced by Li and Yau in \cite{yl}. We reproduce here some of their results for the convenience of the reader. Let $X$ be an $n$-dimensional Riemannian complex manifold and let $\sqrt{-1}\omega$ be the imaginary part of the metric on $X$. Thus $\omega$ is a real $(1,1)$-form. Assume that the form $\omega$ satisfies the following condition. \begin{equation}\label{yl} \omega^{n-2} \wedge d\omega = 0. \end{equation} For a closed real $2$-form $\eta$ let $$ \deg\eta = \int_X \omega^{n-1} \wedge \eta. $$ The condition \eqref{yl} ensures that $\deg\eta$ depends only on the cohomology class of $\eta$. Thus it defines a degree functional $\deg:H^2(X,{\Bbb R}) \to {\Bbb R}$. This functional allows one to repeat verbatim the Mumford-Takemoto definitions of stable and semistable bundles in this more general situation. Moreover, the Hermitian Yang-Mills equations also carry over word-by-word. Yau and Li proved the following. \begin{theorem}[\cite{yl}] Let $X$ be a complex Riemannian manifold satisfying \eqref{yl}. Then every stable holomorpic bundle ${\cal B}$ on $X$ admits a unique Hermitian Yang-Mills connection $\nabla$. Vice versa, every bundle ${\cal B}$ admitting an Hermitian Yang-Mills connection is polystable. \end{theorem} Just like in the K\"ahler case, this Theorem allows one to construct a good moduli space for holomorphic bundles on $X$. (See \cite{Kod}.) \subsection{Li--Yau condition for twistor space} The twistor space $X = M \times {{\Bbb CP}^1}$ is equipped with a natural Riemannian metric, namely, the product of the metrics on $M$ and on ${{\Bbb CP}^1}$. To apply the Li-Yau theory to $X$, we need to check the condition \eqref{yl}. First, we identify the form $\omega$. Let $\omega = \omega_M + \omega_{{\Bbb CP}^1}$ be the decomposition associated with the product decomposition $X = M \times {{\Bbb CP}^1}$. By the definition of the complex structure on $X$, the form $\omega_{{\Bbb CP}^1}$ is the pullback $\pi^*\omega$ of the usual K\"ahler form on ${{\Bbb CP}^1}$, while $\omega_M$ is a certain linear combination of pullbacks of K\"ahler forms on $M$ associated to different induced complex structures. Let $W \in {\Bbb H}$ be the $3$-dimensional subspace of imaginary quaternions. For every $a \in W$ the metric $\langle {\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}\rangle$ on the hyperk\"ahler manifold $M$ defines a real closed $2$-form $\omega_a$ on $M$ by the rule $$ \omega_a({\:\raisebox{3.5pt}{\text{\circle*{1.5}}}},{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}) = \langle {\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}, a {\:\raisebox{3.5pt}{\text{\circle*{1.5}}}} \rangle. $$ This construction is linear in $a$, hence defines an embedding $W \hookrightarrow {\cal A}^2(M,{\Bbb R})$. Let ${\cal W}$ be the trivial bundle on ${{\Bbb CP}^1}$ with the fiber $W$. The embedding $W \hookrightarrow {\cal A}^2(M,{\Bbb R})$ extends to an embedding $$ {\cal W} \hookrightarrow \pi_*\sigma^*{\cal A}^2(M,{\Bbb R}) \subset \pi_*{\cal A}^2(X). $$ Since $X = M \times {{\Bbb CP}^1}$, the de Rham differential $d = d_X:{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(X) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}(X)$ decomposes into the sum $d = d_M + d_{{\Bbb CP}^1}$. The differential $d_{{\Bbb CP}^1}$ defines a flat connection on the bundle $\pi_*\sigma^*{\cal A}^2(M,{\Bbb R})$. The subbundle ${\cal W} \subset \pi_*\sigma^*{\cal A}^2(M,{\Bbb R})$ is flat with respect to $d_{{\Bbb CP}^1}$. The space $W$ is equipped with an euclidian metric, thus $W = W^*$. Since we have an embedding ${{\Bbb CP}^1} = S^2 \hookrightarrow W$, the bundle ${\cal W} = TW|_{{\Bbb CP}^1} = T^*W|_{{\Bbb CP}^1}$ decomposes orthogonally $$ {\cal W} = {\Bbb R} \oplus {\cal O}(-2) $$ into the sum of the conormal and the cotangent bundles to ${{\Bbb CP}^1} \subset W$. The conormal bundle is the trivial $1$-dimesional real bundle ${\Bbb R}$, and the cotangent bundle is isomorphic to the complex vector bundle ${\cal O}(-2)$ on ${{\Bbb CP}^1}$. The connection $d_x|_{\cal W}$ induces the trivial connection on ${\Bbb R}$ and the usual metric connection on ${\cal O}(2)$. The embedding ${\cal W} \to \pi_*\sigma{\cal A}^2(M,{\Bbb R})$ decomposes then into a real $2$-form $$ \omega \in \sigma^*{\cal A}^2(M,{\Bbb R}) $$ and a complex ${\cal O}(2)$-valued $2$-form $$ \Omega \in \sigma^*{\cal A}^2(M,{\Bbb R}) \otimes \pi^*{\cal O}(2). $$ The form $\Omega$ is holomorphic, while the form $\sqrt{-1}\omega_M$ is precisely the imaginary part of the Hermitian metric on $X$. Let now $\upsilon \in {\cal A}^{2,1}(X)$ be the $(2,1)$-form corresponding to the holomorphic form $$ \Omega \in {\cal A}^{2,0}(X,\pi^*{\cal O}(2)) $$ under the identification ${\cal O}(2) \cong {\cal A}^{0,1}({{\Bbb CP}^1})$ provided by the metric on ${{\Bbb CP}^1}$. \begin{lemma}\label{_differe_of_Hermi_on_twistors_Lemma_} $$ d \omega = \upsilon + \overline{\upsilon}. $$ \end{lemma} \par\noindent{\bf Proof.}\ Indeed, $d\omega_{{\Bbb CP}^1} = 0$ and $d_M\omega_M = 0$, therefore it is enough to compute $d_{{\Bbb CP}^1}\omega_M$. The bundle ${\cal W} \in {\cal A}^2(X)$ is invariant under $d_{{\Bbb CP}^1}$, and $d$ induces the trivial connection $\nabla$ on ${\cal W}$. Let $$ d = \begin{bmatrix} \nabla_{\Bbb R} &\theta_{10}\\\theta_{01} & \nabla_{{\cal O}(2)} \end{bmatrix} $$ be the decomposition of $\nabla$ with respect to ${\cal W} = {\Bbb R} \oplus {\cal O}(2)$. The connection $\nabla_{\Bbb R}$ is trivial, therefore $\nabla_{\Bbb R}\omega = 0$. An easy computation shows that $$ \theta_{10} \in {\cal A}^{1}({{\Bbb CP}^1},{\cal O}(2)) $$ induces the isomorphism ${\cal A}^1({{\Bbb CP}^1}) \cong {\cal O}(2)$. Thus $$ d\omega_M = d_{{\Bbb CP}^1}\omega_M = \theta_{10} \wedge \Omega = \upsilon + \overline{\upsilon}. $$ \endproof Now we can prove that the twistor space $X$ satisfies the condition \eqref{yl}. \begin{prop} The canonical $(1,1)$-form $\omega$ on the twistor space $X$ satisfies $$ \omega^{n-1} \wedge d\omega = 0 $$ for $n = \dim M =\dim X - 1$. \end{prop} \par\noindent{\bf Proof.}\ Let $x = m \times J \in X$ be a point in $X$. Choose a local coordinate $z$ on ${{\Bbb CP}^1}$ near the point $J \in {{\Bbb CP}^1}$. In a neighborhood of $x \in X$ we have $$ \upsilon = f(z)\Omega \wedge d\bar z $$ for some holomorphic function $f(z)$. Therefore $$ \omega^{n-1} \wedge d\omega = f(z)\omega^{n-1} \wedge \Omega \wedge d\bar z + \overline{f(z)}\omega^{n-1} \wedge \overline{\Omega} \wedge dz. $$ But $\omega^{n-1} \wedge \Omega$ and $\omega \wedge \overline{\Omega}$ are both forms on the $n$-dimensional manifold $M$, of Hodge types $(n+1,n-1)$ and $(n-1,n+1)$. Hence both are zero. To prove the Proposition, it remains to see that $\omega^{n-1} - \omega_M^{n-1}$ is divisible by $\omega_{{\Bbb CP}^1}$, and $\omega_{{\Bbb CP}^1} \wedge d z = \omega_{{\Bbb CP}^1} \wedge d \bar z = 0$. \endproof \section{Twistor transform.} \label{_twisto-tra_Section_} \subsection{Twistor transform} We now introduce the direct and inverse twistor transforms which relate autodual bundles on the hyperk\"ahler manifold $M$ and holomorphic bundles on its twistor space $X$. Let ${\cal B}$ be a complex vector bundle on $M$ equipped with a connection $\nabla$. The pullback $\sigma^*{\cal B}$ of ${\cal B}$ to $X$ is then equipped with a connection $\sigma^*\nabla$. \begin{lemma} \label{_autodua_(1,1)-on-twi_Lemma_} The connection $\nabla$ is autodual if and only if the connection $\sigma^*\nabla$ has curvature of Hodge type $(1,1)$. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, the curvature $R_X$ of $\sigma^*\nabla$ is equal to the pullback $\sigma^*R_M$ of the curvature $R_M$ of $\nabla$. Therefore it is of Hodge type $(1,1)$ on $X$ if and only if for every $I \in {{\Bbb CP}^1}$ the form $R_M$ is of type $(1,1)$ in the induced complex structure $I$. By Lemma~\ref{primitive} this happens if and only if $R_M$ is ${SU(2)}$-invariant. \endproof \hspace{-8pt} In particular, for every autodual bundle $\langle {\cal B}, \nabla \rangle$ the $(0,1)$-com\-po\-nent ${\sigma^*\nabla}^{0,1}$ of the connection $\sigma^*\nabla$ satisfies ${\sigma^*\nabla}^{0,1} \circ {\sigma^*\nabla}^{0,1} = 0$ and defines a holomorphic structure on the bundle $\sigma^*{\cal B}$. \begin{defn} The holomorphic bundle $\langle \sigma^*{\cal B}, {\sigma^*\nabla}^{0,1} \rangle$ is called {\bf the twistor transform} of the autodual bundle $\langle {\cal B}, \nabla \rangle$. \end{defn} \subsection{${\Bbb C} P^1$-holomorphic bundles over twistor spaces} The twistor transform is in fact invertible. To construct an inverse transform, we begin with some results on differential forms on the twistor space $X$. The product decomposition $X = M \times {{\Bbb CP}^1}$ induces the decomposition ${\cal A}^1(X) = \sigma^*{\cal A}^1(M) \oplus \pi^*{\cal A}^1({{\Bbb CP}^1})$ of the bundle ${\cal A}^1(M)$ of $1$-forms. By the definition of the complex structure on $X$, the projection onto the subbundle of $(0,1)$-forms commutes with the projection onto the bundle $\pi^*{\cal A}^1({{\Bbb CP}^1})$. Therefore a Dolbeault differential $$ \bar\partial_{{\Bbb CP}^1}:{\cal A}^0(X) \to \pi^*{\cal A}^{0,1}({{\Bbb CP}^1}) $$ is well-defined. \begin{defn} A {\bf ${{\Bbb CP}^1}$-holomorphic bundle} on $X$ is a complex vector bundle ${\cal B}$ on $X$ equipped with an operator $\bar\partial_{{\Bbb CP}^1}:{\cal B} \to {\cal B} \otimes \pi^*{\cal A}^{0,1}({{\Bbb CP}^1})$ satisfying $$ \bar\partial_{{\Bbb CP}^1}(fa) = \bar\partial_{{\Bbb CP}^1}(f)a + f\bar\partial_{{\Bbb CP}^1}(a) $$ for a function $f$ and a local section $a$ of ${\cal B}$. \end{defn} \begin{rem} For any point $m \in M$ the restriction $\widetilde{m}^*{\cal B}$ of a ${{\Bbb CP}^1}$-holomrphic bundle ${\cal B}$ to the horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ is holomorphic in the usual sense. \end{rem} A $\bar\partial_{{\Bbb CP}^1}$-closed smooth section $a \in \Gamma(X,{\cal B})$ of a ${{\Bbb CP}^1}$-holomorphic bundle ${\cal B}$ will be called {\bf ${{\Bbb CP}^1}$-holomorphic}. Tensor products and $\operatorname{{\cal H}\!{\it om}}$-bundles of ${{\Bbb CP}^1}$-holomorphic bundles are again ${{\Bbb CP}^1}$-holomorphic. A differential operator $f:{\cal B}_0 \to {\cal B}_1$ will be called {\bf ${{\Bbb CP}^1}$-holomorphic} if $\bar\partial f(a)=0$ for every local ${{\Bbb CP}^1}$-holomorphic section $a$ of the bundle ${\cal B}$. For every complex vector bundle ${\cal B}$ on $M$ the bundle $\sigma^*{\cal B}$ on $X$ is canonically ${{\Bbb CP}^1}$-holomorphic. For a ${{\Bbb CP}^1}$-holomorphic bundle ${\cal B}$ let $\sigma_*{\cal B}$ be the sheaf on $M$ of ${{\Bbb CP}^1}$-holomorphic sections of ${\cal B}$. The functors $\sigma_*$ and $\sigma^*$ are adjoint. \begin{defn}\label{const} A ${{\Bbb CP}^1}$-holomorphic bundle ${\cal B}$ on $X$ is called {\bf ${{\Bbb CP}^1}$-constant} if it is isomorphic to $\sigma^*{\cal B}_1$ for a complex vector bundle ${\cal B}$ on $M$. \end{defn} Since $\sigma_*\sigma^*{\cal B}_1 \cong {\cal B}_1$ for every complex bundle ${\cal B}_1$ on $M$ , a bundle ${\cal B}$ on $X$ is ${{\Bbb CP}^1}$-constant if and only if the canonical map ${\cal B} \to \sigma^*\sigma_*{\cal B}$ is an isomorphism. The functor $\sigma^*$ is therefore an equivalence between the category of complex vector bundles on $M$ and the category of ${{\Bbb CP}^1}$-constant ${{\Bbb CP}^1}$-holomorphic bundles on $X$. \begin{lemma}\label{constant} A ${{\Bbb CP}^1}$-holomorphic bundle ${\cal B}$ on $X$ is ${{\Bbb CP}^1}$-constant if and only if for every horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ the restriction $\widetilde{m}^*{\cal B}$ is trivial. \end{lemma} \par\noindent{\bf Proof.}\ Clear. \endproof Note that if $\dim\Gamma({{\Bbb CP}^1},\widetilde{m}^*{\cal B})$ is the same for every horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$, then $\sigma_*{\cal B}$ is the sheaf of smooth sections of a complex vector bundle on $M$. \subsection{Differential forms and ${\Bbb C} P^1$-holomorphic bundles} Let ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_M(X) = \sigma^*{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M) \subset {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(X)$ be the subcomplex of relative ${\Bbb C}$-valued forms on $X$ over ${{\Bbb CP}^1}$ and let ${\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M(X)$ be the quotient complex of forms of Hodge type $(0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}})$. Denote by $d_M$ and $\bar\partial_M$ the corresponding differentials and let $P:{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_M \to {\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M$ be the natural projection. By definition all the bundles ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_M$ and ${\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M$ are ${{\Bbb CP}^1}$-holomorphic, and the differentials $d_M$ and $\bar\partial_M$ are ${{\Bbb CP}^1}$-holomorphic. Let ${\cal B}$ be a holomorphic bundle on $X$. Then the complex structure operator $\bar\partial:{\cal B} \to {\cal B} \otimes {\cal A}^{0,1}(X,{\cal B})$ can be decomposed $$ \bar\partial = \bar\partial_M + \bar\partial_{{\Bbb CP}^1} $$ into an operator $\partial_{{\Bbb CP}^1}:{\cal B} \to \otimes {\cal B} \otimes \pi^*{\cal A}^{0,1}({{\Bbb CP}^1})$ and an operator $\bar\partial_M:{\cal B} \to {\cal B} \otimes {\cal A}^{0,1}_M$. The operator $\bar\partial_{{\Bbb CP}^1}$ is a ${{\Bbb CP}^1}$-holomorphic structure on the bundle ${\cal B}$, and the operator $\bar\partial_M$ is ${{\Bbb CP}^1}$-holomorphic. This constrution is in fact invertible. Namely, we have the following. \begin{lemma}\label{rel.holo} The correspondence ${\cal B} \mapsto \langle {\cal B}, \bar\partial_M\rangle$ is an equivalence between the category of holomorphic bundles on $X$ and the category of ${{\Bbb CP}^1}$-ho\-lo\-mor\-phic bundles ${\cal B}$ on $X$ equipped with a ${{\Bbb CP}^1}$-holomorphic operator $\bar\partial_M:{\cal B} \to {\cal B} \otimes {\cal A}^{0,1}_M$ satisfying $0 = \bar\partial_M^2:{\cal B} \to {\cal B} \otimes {\cal A}^{0,2}_M$ and $$ \bar\partial_M(fa) = \bar\partial(f)a + f \bar\partial_M(a) $$ for a function $f$ and a local section $a$ of ${\cal B}$. \end{lemma} \par\noindent{\bf Proof.}\ Clear. \endproof In particlular, every holomorphic bundle ${\cal B}$ on $X$ is canonically ${{\Bbb CP}^1}$-holomorphic. We will call a holomorphic bundle ${\cal B}$ on $X$ {\bf ${{\Bbb CP}^1}$-constant} if the corresponding ${{\Bbb CP}^1}$-holomorphic bundle is ${{\Bbb CP}^1}$-constant in the sense of Definition~\ref{const}. \subsection{The complex ${\cal A}^{\bullet}_{{\mathrm top}}(M)$: definition.} For any horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ the restriction $\widetilde{m}^*{\cal A}^i_M$ is trivial, while $\widetilde{m}^*{\cal A}^{0,i}_M$ is a sum of several copies of the bundle ${\cal O}(i)$ on ${{\Bbb CP}^1}$ (see, e.g., \cite{HKLR}). Therefore $\sigma_*{\cal A}^*{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_M \cong {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M,{\Bbb C})$, and the map $P$ induces a projection $$ P:{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M,{\Bbb C}) \cong \sigma_*{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_M(X) \to \sigma_*{\cal A}^{0,1}_M(X). $$ Let ${\cal A}^{1,0}_M(X)$ be the sheaf of relative forms of type $(1,0)$. By definition we have an exact sequence $$ 0 \to {\cal A}^{1,0}_M(X) \to {\cal A}^1_M(X) \to {\cal A}^{0,1}_M(X) \to 0 $$ of ${{\Bbb CP}^1}$-holomorphic bundles on $X$. Consider the associated long exact sequence for $\sigma_*$. The restriction of ${\cal A}^{1,0}_M$ to any horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ is a sum of several copies of ${\cal O}(-1)$. Therefore this long sequence reduces to the map $$ \sigma_*{\cal A}^1_M(X) \overset{P}{\longrightarrow} \sigma_*{\cal A}^{0,1}_M(X), $$ which is therefore an isomorphism. Recall that the bundle ${\cal A}^i(M)$ carries a representation of the group ${SU(2)}$ for every $i \geq 0$. This representation is completely reducible \footnote{${SU(2)}$ acts along the fibers, which are finite-dimensional}, and contains isotypical components of highest weights $\leq i$. Let ${\cal A}^i_{\operatorname{top}} \subset {\cal A}^i$ be the component of highest weight exactly $i$. \begin{lemma}\label{iso.complex} The map $P:{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M,{\Bbb C}) \to \sigma_*{\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M(X)$ is compatible with the ${SU(2)}$-action on ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}(M,{\Bbb C})$. The restriction $P{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{top}}(M,{\Bbb C}) \to \sigma_*{\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M$ is an isomorphism. \end{lemma} \par\noindent{\bf Proof.}\ We already know the claim for $\sigma_*{\cal A}^{0,1}_M$. Let $i > 1$. Note that $\sigma_*{\cal A}^{0,i}_M$ is equipped with an ${SU(2)}$-action by the Borel-Weyl theory. The corresponding representation is of highest weight $i$. The map $P$ is obviously compatible with this action, which proves the first statement. To prove the second, it is enough to prove that $P$ is invertible on the subbundles of highest vectors. But both these subsundles equal ${\cal A}^{0,i}(M)$. \endproof \subsection{The complex ${{\cal A}}^{\bullet}_{{\mathrm top}}(M)$ and autodual bundles.} The complex ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{top}}(M) \cong \sigma_*{\cal A}^{0,1}_M$ plays the same role for autodual bundles as the Dolbeault resp. de Rham complexes play for holomorphic resp. flat ones. Precisely, let ${\cal B}$ be a complex vector bundle on $M$ equipped with a connection $\nabla:{\cal B} \to {\cal A}^1({\cal B})$. Extend the operator $\nabla$ to a differential operator $D:{\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}_{\operatorname{top}}({\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}_{\operatorname{top}}({\cal B})$ by means of the embedding ${\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}({\cal B}) \hookrightarrow {\cal A}^{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}({\cal B})$ and the natural ${SU(2)}$-invariant projection ${\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}({\cal B}) \to {\cal A}^{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}_{\operatorname{top}}({\cal B})$. \begin{lemma}\label{autodual.complex} The connection $\nabla$ is autodual if and only if its extension $D$ satisfies $D^2 = 0$. \end{lemma} \par\noindent{\bf Proof.}\ The operator $D^2$ is the multiplication by the ${\cal A}^2_{\operatorname{top}}$-part of the curvature $R$ of the bundle ${\cal B}$ with respect to the decomposition $$ {\cal A}^2(\operatorname{{\cal E}\!{\it nd}}{\cal B}) = {\cal A}^2_{\operatorname{top}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^2_{\operatorname{inv}}(\operatorname{{\cal E}\!{\it nd}}{\cal B}). $$ Thus it vanishes if and only if $R$ is ${SU(2)}$-invariant, which by definition means that $\nabla$ is autodual. \endproof Let ${\cal B}$ be a complex vector bundle equipped with an autodual connection $\nabla$. Since ${\cal A}^1({\cal B}) \cong \sigma_*{\cal A}^{0,1}_M(X) \times {\cal B}$, the map $\nabla:{\cal B} \to {\cal B} \otimes \sigma_*{\cal A}^{0,1}_M(X)$ defines a ${{\Bbb CP}^1}$-holomorphic map $$ \bar\partial_M:\sigma^*{\cal B} \to \sigma^*{\cal B} \otimes {\cal A}^{0,1}_M(X) $$ of ${{\Bbb CP}^1}$-holomorphic bundles on $X$. By Lemmas~\ref{iso.complex} and \ref{autodual.complex} the map $\bar\partial_M$ extends to a map ${\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}}_M(X) \otimes \sigma^*{\cal B} \to {\cal A}^{0,{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}+1}_M(X) \otimes {\cal B}$ satisfying $\bar\partial_M^2 = 0$. By Lemma~\ref{rel.holo} this map defines a holomorphic structure on the bundle $\sigma^*{\cal B}$. \begin{lemma} The holomorphic bundle $\langle\sigma^*{\cal B},\bar\partial_M\rangle$ on $X$ is isomorphic to the twistor transform of the autodual bundle ${\cal B}$. \end{lemma} \par\noindent{\bf Proof.}\ Clear. \endproof Let now ${\cal B}$ be an arbitrary ${{\Bbb CP}^1}$-constant holomorphic bundle on $X$. Then the sheaf $\sigma_*{\cal B}$ is the sheaf of sections of a vector bundle. Since ${\cal B}$ is ${{\Bbb CP}^1}$-constant, ${\cal B} \cong \sigma^*\sigma_*{\cal B}$. The operator $$ \partial_M:{\cal B} \cong \sigma^*\sigma_*{\cal B} \to {\cal B} \otimes {\cal A}^{0,1}_M(X) $$ gives by adjuction an operator $$ \nabla:\sigma_*{\cal B} \to \sigma_*\left({\cal B} \otimes {\cal A}^{0,1}_M(X)\right) \cong \sigma_*{\cal B} \otimes {\cal A}^1_{\operatorname{top}}(M). $$ By Lemmas~\ref{iso.complex} and \ref{autodual.complex} the operator $\nabla$ is an autodual connection on $\sigma_*{\cal B}$. \begin{defn} The autodual bundle $\langle \sigma_*,\nabla \rangle$ on $M$ is called {\bf the inverse twistor transform} of the ${{\Bbb CP}^1}$-constant holomorphic bundle ${\cal B}$ on $X$. \end{defn} \begin{theorem} \label{_twisto_transfo_equiva_Theorem_} The direct and inverse twistor transforms are mutually inverse equivalences between the category of autodual bundles on $M$ and the category of ${{\Bbb CP}^1}$-constant holomorphic bundles on $X$. \end{theorem} \par\noindent{\bf Proof.}\ Clear. \endproof \section{Stability of the twistor transform.} \label{_stabi_of_twi_tra_Section_} \subsection{Introduction} Let $M$ be a hyperk\"ahler manifold and let $X$ be its twistor space. Consider a semistable autodual bundle $\langle {\cal B},\nabla \rangle$ on $M$ and let $\sigma^*{\cal B}$ be its twistor transform. The bundle $\sigma^*{\cal B}$ is a holomorphic bundle on $X$. In this section we prove under certain conditions that $\sigma^*{\cal B}$ is semistable in the sense of \ref{ss.twistor}. More precisely, we have the following. \begin{prop} Let $M$ be a hyperk\"ahler manifold. Denote its twistor space by $X$. Let $\langle {\cal B}, \nabla \rangle$ be a semistable autodual bundle on $M$ and let $\sigma^*{\cal B}$ be its twistor transform. Then for every coherent subsheaf ${\cal F} \subset \sigma^*{\cal B}$ we have $$ \frac{\deg c_1({\cal F})}{\operatorname{rank}{\cal F}} \leq \frac{\deg c_1(\sigma^*{\cal B})}{\operatorname{rank}\sigma^*{\cal B}}, $$ where $\operatorname{rank}{\cal F}$ is the rank of the generic fiber of ${\cal F}$ and $\deg$ is understood in the sense of subsection~\ref{ss.twistor}. \end{prop} \subsection{Semistability for ${\Bbb C} P^1$-constant bundles} Before we give a proof of this Proposition, we prove the following. \begin{lemma}\label{constant.ss} Let ${\cal B}$ be a holomorphic bundle on the twistor space $X$. Assume that ${\cal B}$ is ${{\Bbb CP}^1}$-constant (that is, for every horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ the restriction $\widetilde{m}^*{\cal B}$ is trivial). Then $\deg {\cal B} = 0$, and the bundle ${\cal B}$ is semistable. \end{lemma} \par\noindent{\bf Proof.}\ Since $H^1({{\Bbb CP}^1},{\Bbb R}) = 0$, \begin{equation}\label{ff} H^2(X,{\Bbb R}) = H^2(M,{\Bbb R}) \oplus H^2({{\Bbb CP}^1},{\Bbb R}). \end{equation} Since ${\cal B}$ is ${{\Bbb CP}^1}$-constant, $c_1({\cal B}) \in H^2(M,{\Bbb R})$. For every $I \in {{\Bbb CP}^1}$ let $X_I = \pi^{-1}(I) \subset X$ be the fiber over $I$. Since $$ c_1({\cal B})_{X_I} = c_1({\cal B}|_{X_I}) \in H^{1,1}_I(M) $$ is of Hodge type $(1,1)$ for every $I \in {{\Bbb CP}^1}$, the class $c_1({\cal B}) \in H^2(M,{\Bbb R})$ is ${SU(2)}$-invariant by Lemma~\ref{primitive}. Therefore $\deg c_1({\cal B}) = \Lambda c_1({\cal B}) = 0$. To prove semistability, let ${\cal F} \subset {\cal B}$ be a coherent subsheaf. It is enough to prove that $\deg c_1({\cal F}) \leq 0$. Let $c_1({\cal F}) = c_M + c_{{\Bbb CP}^1}$ be the decomposition associated with \eqref{ff}. Again, by Lemma~\ref{primitive} $c_M$ is ${SU(2)}$-invariant, and $\deg c_1({\cal F}) = \deg c_{{\Bbb CP}^1}$. For a generic horizontal twistor line $\widetilde{m}:{{\Bbb CP}^1} \to X$ we have $c_{{\Bbb CP}^1} = c_1(\widetilde{m}^*{\cal F}) \in H^2({{\Bbb CP}^1},{\Bbb R})$. Since $\widetilde{m}^*{\cal B}$ is trivial, it is semistable, and $\deg c_1(\widetilde{m}^*{\cal F}) \leq 0$. \endproof Let ${\cal M}^{ss}_X$ be the moduli space of semistable holomorphic bundles on $X$. Lemma~\ref{constant.ss} implies that the set ${\cal M}^s_{const}$ of equivalence classes of ${{\Bbb CP}^1}$-constant holomorphic bundles on $X$ is a subset of ${\cal M}^{ss}_X$. The subset ${\cal M}^s_{const} \subset {\cal M}^{ss}_X$ is open. \subsection{Conclusion} The Propostion now follows directly from Lemma~\ref{constant.ss}. Moreover, the twistor transform provides an isomorphism ${\cal M}^s_{\operatorname{inv}} \to {\cal M}^s_{const} \subset {\cal M}^{ss}_X$ from the moduli space ${\cal M}^s_{\operatorname{inv}}$ of autodual bundles on $M$ to the open subset of ${{\Bbb CP}^1}$-constant bundles in the moduli space ${\cal M}^{ss}_X$ \section[Stable bundles and projective lines in twistor spaces.] {Stable bundles and projective lines \\ in twistor spaces.} \label{_lines_Section_} \subsection{Hyperk\"ahler structure on the Mukai dual space} \label{_Mukai_dual_Subsection_} Let $M$ be a compact hyperk\"ahler manifold and let ${\cal B}$ be a complex vector bundle on $M$ with ${SU(2)}$-invariant Chern classes $c_1({\cal B})$ and $c_2({\cal B})$. Consider the moduli space ${\cal M}^s_0$ of stable holomorphic structures on ${\cal B}$ and let ${\cal M}^{reg}_0 \subset {\cal M}^s_0$ be the dense open subset of smooth points in ${\cal M}^s_0$. Recall that the subset ${\cal M}^{reg}_0$ is equipped with a natural K\"ahler metric, called {\em the Weil-Peterson metric}. It was proved in \cite{Vb} that the Weil-Peterson metric on ${\cal M}^{reg}_0$ is actually hyperk\"ahler. Moreover, the complex manifold $\left({\cal M}^{reg}_0\right)_J$ with the complex structure induced by a quaternion $J \in {{\Bbb CP}^1} \subset {\Bbb H}$ was naturally identified with the subset of smooth points in the moduli space of stable holomorphic structures on ${\cal B}$ with respect to the complex structure $J$ on $M$. Let ${\cal X}_{reg}$ be the twistor space of the hyperk\"ahler manifold ${\cal M}^{reg}_0$. Consider the topological space ${\cal X} = {\cal M}^s_0 \times {{\Bbb CP}^1}$. We have a natural embedding ${\cal X}_{reg} \subset X$. In \cite{Vb} the complex structure on ${\cal X}_{reg}$ and the real structure $\iota:{\cal X}_{reg} \to {\cal X}_{reg}$ were naturally extended to the whole of ${\cal X}$. The complex-analytic space ${\cal X}$ is in general singular. However, the fundamental Theorem~\ref{inv} still holds for the natural projection $\pi:{\cal X} \to {{\Bbb CP}^1}$. We will call holomorphic sections ${{\Bbb CP}^1} \to {\cal X}$ of the projection $\pi:{\cal X} \to {{\Bbb CP}^1}$ {\bf twistor lines} in ${\cal X}$. The space ${\cal M}^s_0$ is then naturally isomorphic to the subset of real twistor lines in ${\cal X}$. These data define {\bf singular hy\-per\-k\"ah\-ler structure} on ${\cal M}^s_0$ (see \cite{Vb} for details). The space ${\cal M}^s_0$ with this hyperk\"ahler structure is called {\bf Mukai dual} to $M$ (results of \cite{Vb} generalise Mukai's work about duality of K3 surfaces). We must caution the reader that this version of Mukai duality is not involutive, as the term ``dual'' might erroneously imply. \subsection{Fiberwise stable bundles} Let $X$ be the twistor space of the hyperk\"ahler manifold $M$. Let ${\cal M}^s_{const} \subset {\cal M}^{ss}_X$ be the open subset of ${{\Bbb CP}^1}$-constant holomorphic structures in the moduli space ${\cal M}^{ss}_X$ of semistable holomorphic structures on the bundle $\sigma^*{\cal B}$. In the last section we have identified ${\cal M}^s_{const}$ with the space ${\cal M}^s_{\operatorname{inv}}$ of $(0,1)$-stable autodual connections on the bundle ${\cal B}$. In this section we will need still another notion of stability for holomorphic bundles over $X$. \begin{defn}\label{fib.st} Call a stable holomorphic structure $\bar\partial$ on $\sigma^*{\cal B}$ {\bf fiberwise stable} if for any $L \in {{\Bbb CP}^1}$ the restriction of $\langle \sigma^*{\cal B}, \bar\partial \rangle$ to the fiber $X_L = \pi^{-1}(L) \subset X$ is stable. \end{defn} Let ${\cal M}^s_{\operatorname{fib}} \subset {\cal M}^{ss}_X$ be the subset of fiberwise stable holomorphic structures. The intersection $\left({\cal M}^s_{const} \cap {\cal M}^s_{\operatorname{fib}}\right) \subset {\cal M}^{ss}_X$ is, then, isomorphic to the moduli space of autodual connections on ${\cal B}$ inducing a stable holomorphic structure on ${\cal B}$ for every $I \in {{\Bbb CP}^1}$. The goal of this section is to prove the following. \begin{theorem}\label{iso} The space ${\cal M}^s_{\operatorname{fib}}$ is naturally isomorphic to the space $\operatorname{Sec}$ of twistor lines in the manifold ${\cal X}$. \end{theorem} \subsection{Stability of fiberwise-stable bundles} We begin by noting that one of the conditions in Definition~\ref{fib.st} is in fact redundant. \begin{lemma}\label{gen.st} Let ${\cal B}$ be a holomorphic bundle on the twistor space $X$. If the restriction $i^*{\cal B}$ is stable for a generic\footnote{In the sense of \cite{Vsym}} point $I \in {{\Bbb CP}^1}$, then the bundle ${\cal B}$ is stable. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, it was proved in \cite{Vsym} that for a generic point $I \in {{\Bbb CP}^1}$ every rational $(1,1)$-cohomology class for the fiber $X_I$ is of degree zero. Therefore a stable holomorphic bundle on $X_I$ has no proper subsheaves. Hence for a proper subsheaf ${\cal F} \subset {\cal B}$ either ${\cal F}$ or ${\cal B}/{\cal F}$ is supported on non-generic fibers of $\pi:X \to {{\Bbb CP}^1}$. In particular, either ${\cal F}$ or ${\cal B}/{\cal F}$ is a torsion sheaf. This implies that the bundle ${\cal B}$ is stable. \endproof \subsection{Modular interpretation of the Mukai dual twistor space} We now construct a map ${\cal M}^s_{\operatorname{fib}} \to \operatorname{Sec}$. To do this, we give a modular interpretation of the space ${\cal X}$. For any point $I \in {{\Bbb CP}^1}$ let $i:X_I \hookrightarrow X$ be the natural embedding of the fiber $X_I = \pi^{-1}(I) \subset X$. For a stable holomorphic bundle ${\cal B}$ on the fiber $X_I$ call the coherent sheaf $i_*{\cal B}$ on $X$ {\bf a stable sheaf on $X$ supported in $I$}, or simply a {\bf fiber-supported stable sheaf}. More generally, for a complex analytic space $Z$ call a coherent sheaf ${\cal E}$ on $Z \times X$ {\bf a family of fiber-supported stable sheaves on $X$} if there exists a holomorphic map $f_Z:Z \to {{\Bbb CP}^1}$ such that ${\cal B}$ and a holomorphic bundle ${\cal E}_0$ on the subspace $Z \times_{{\Bbb CP}^1} X \subset Z \times X$ such that \begin{enumerate} \item ${\cal E} \cong i_*{\cal E}_0$, where $i:Z \times_{{\Bbb CP}^1} X \to Z \times X$ is the natural embedding. \item For every point $z \in Z$ the restriction of ${\cal E}$ to $z \times \pi^{-1}(f_Z(z)) \subset Z \times X$ is a stable holomorphic bundle. \end{enumerate} The space ${\cal X}$ is obviously the moduli space for families of fiber-supported stable sheaves on $X$. The holomorphic map $f_{\cal X}:{\cal X} \to {{\Bbb CP}^1}$ is the natural projection. Let now ${\cal B}$ be a fiberwise stable holomorphic bundle on $X$. For every $I \in {{\Bbb CP}^1}$ the coherent sheaf $i_*i^*{\cal B}$ on $X$ is a stable sheaf supported in $I$. The correspondence $I \mapsto i_*i^*{\cal B}$ defines a holomorphic map ${{\Bbb CP}^1} \to {\cal X}$. This map is a section of the projection ${\cal X} \to {{\Bbb CP}^1}$, hence defines a point $\psi({\cal B}) \in \operatorname{Sec}$. The correspondence ${\cal B} \mapsto \psi({\cal B})$ comes from a holomorphic map $\psi:{\cal M}^s_{\operatorname{fib}} \to \operatorname{Sec}$ of the corresponding moduli spaces. \subsection{Coarse and fine moduli spaces: a digression} In order to prove that the map $\psi:{\cal M}^s_{\operatorname{fib}} \to \operatorname{Sec}$ is an isomorphism, we need to make a digression about universal objects and coarse moduli spaces. Let ${\frak V}{\frak a}{\frak r}$ be the category of complex-analytic varieties and let ${\cal F}:{\frak V}{\frak a}{\frak r} \to \operatorname{Sets}$ be a functor. Recall that a complex-analytic space $Y$ is said to be a fine moduli space for the functor ${\cal F}$ if ${\cal F} \cong \operatorname{{\cal H}\!{\it om}}(\bullet, Y)$. This implies that there exists an element $C \in {\cal F}(Y)$ such that for every complex-analytic space $U$ and an element $a \in {\cal F}(U)$ there exists a unique map $f:U \to Y$ such that $a ={\cal F}(f)(C)$. Such an element $C$ is called {\em the universal solution} to the moduli problem posed by ${\cal F}$. It is well-known that geometric moduli problems only rarely admit fine moduli spaces. The common way to deal with this is to introduce a weaker notion of a {\em coarse moduli space}. For the purposes of this paper the following notion suffices. \begin{defn} A complex-analytic space $Y$ is called a {\bf coarse moduli space for the problem posed by ${\cal F}$} if for any complex-analytic space $Z$ and an element $a \in {\cal F}(Z)$ there exist a unique map $f:Z \to Y$, an open covering $U_\alpha$ of the space $Y$ and a collection $C_\alpha \in {\cal F}(U_\alpha)$ such that for every index $\alpha$ $$ {\cal F}(f)(C_\alpha) = a|_{f^{-1}(U_\alpha)}. $$ \end{defn} Heuristically, a coarse moduli space $Y$ admits locally a universal solution for the moduli problem ${\cal F}$, but these solutions need not come from a single global solution in ${\cal F}(Y)$. All the moduli spaces constructed as infinite-dimensional quotients by means of the slice theorem are coarse moduli spaces in the sense of this definition. This applies to all the moduli spaces considered in this paper, and to the space ${\cal X}$ in particular. Therefore for every point $x \in {\cal X}$ there exists a neighborhood $U_x \subset X$ and a coherent sheaf ${\cal E}_x$ on $U \times X$ which is a family of fiber-supported stable sheaves on $X$ universal for the moduli problem. Let $U$ be such a neighborhood. Then the universality of the sheaf ${\cal E}$ implies that $$ \operatorname{Aut}{\cal E} = \Gamma(U,{\cal O}^*). $$ \begin{lemma}\label{coarse} Let ${{\Bbb CP}^1} \to {\cal X}$ be a section of the projection $\pi:{\cal X} \to {{\Bbb CP}^1}$. There exists a coherent sheaf\/ ${\cal E}$ on ${{\Bbb CP}^1} \times X$ such that for every $x \in {{\Bbb CP}^1} \subset {\cal X}$ the restriction\/ ${\cal E}|_{U_x \times X}$ is isomorphic to the universal sheaf\/ ${\cal E}_x$. \end{lemma} \par\noindent{\bf Proof.}\ Indeed, cover ${{\Bbb CP}^1}$ by open subsets of the form $U_x$ and choose a finite subcovering $U_\alpha$. In order to define a sheaf ${\cal E}$, it is enough to choose a system of isomorphisms $$ g_{\alpha\beta}:{\cal E}_\alpha|_{(U_\alpha \cap U_\beta) \times X} \to {\cal E}_\beta|_{(U_\alpha \cap U_\beta) \times X} $$ for every intersection $U_\alpha \cap U_\beta$ so that $g_{\alpha\beta} \circ g_{\beta\gamma} = g_{\alpha\gamma}$ for every three indices $\alpha,\beta,\gamma$. Since $\operatorname{Aut}{\cal E}_\alpha = {\cal O}^*_{U_\alpha}$, the obstruction to finding such a system of isomorphisms lies in the second \v{C}ech cohomology group $H^2({{\Bbb CP}^1},{\cal O}^*)$. Consider the long exact sequence $$ H^2({{\Bbb CP}^1} ,{\cal O}) \longrightarrow H^2({{\Bbb CP}^1}, {\cal O}^*) \longrightarrow H^3({{\Bbb CP}^1}, {\Bbb Z}) $$ associated to the exponential exact sequence $$ 0 \longrightarrow {\Bbb Z} \longrightarrow {\cal O} \longrightarrow {\cal O}^* \longrightarrow 0. $$ Since $H^2({{\Bbb CP}^1}, {\cal O}) = H^3({{\Bbb CP}^1}, {\Bbb Z}) = 0$, the group $H^2({{\Bbb CP}^1} ,{\cal O}^*)$ vanishes. \endproof \subsection{Conclusion} We can now finish the proof of Theorem~\ref{iso}. It remains to prove that the map $\psi:{\cal M}^s_{\operatorname{fib}} \to {\cal X}$ is an isomorphism. We will construct an inverse map $\psi^{-1}:{\cal X} \to {\cal M}^s_{\operatorname{fib}}$. Let $x \in \operatorname{Sec}$ be a point and let $\widetilde{x}:{{\Bbb CP}^1} \to {\cal X}$ be the corresponding section. Let ${\cal E}$ be the coherent sheaf on $\widetilde{x}({{\Bbb CP}^1}) \times X$ constructed in Lemma~\ref{coarse}. Let $\Delta = \pi \times \operatorname{id}:X \hookrightarrow {{\Bbb CP}^1} \times X$ be the embedding of $X$ into ${{\Bbb CP}^1} \times X$ as the preimage of the diagonal under the natural projection $\operatorname{id} \times \pi:{{\Bbb CP}^1} \times {\cal X} \to {{\Bbb CP}^1} \times {{\Bbb CP}^1}$. The sheaf ${\cal E}$ is by definition isomorphic to the direct image of a holomorphic vector bundle ${\cal B}$ on $X$: ${\cal E} \cong \Delta_*{\cal B}$. For every point $I \in {{\Bbb CP}^1}$ the coherent sheaf $i_*i^*{\cal B}$ on $X$ is canonically isomorphic to the restriction of ${\cal E}$ to $I \times X \subset {{\Bbb CP}^1} \times X$. Therefore the bundle ${\cal B}$ is stable by Lemma~\ref{gen.st}. Let $\psi^{-1}(x) \in {\cal M}^s_{\operatorname{fib}}$ be the corresponding point in the moduli space ${\cal M}^s_{\operatorname{fib}}$. By construction $\psi(\psi^{-1}(x)) = x$. To prove that $\psi^{-1} \circ \psi = \operatorname{id}$, consider a stable bundle ${\cal B} \in {\cal M}^s_{\operatorname{fib}}$. Let ${\cal E}$ be the coherent sheaf on ${{\Bbb CP}^1} \times X$ constructed in Lemma~\ref{coarse} and let $p:{{\Bbb CP}^1} \times X \to X$ be the projection onto the second factor. By definition $$ {\cal E} \cong \Delta_*\Delta^*(p^*{\cal B}) \cong \Delta_*(\Delta \circ p)^*{\cal B} \cong \Delta_*{\cal B}. $$ Therefore $\psi^{-1}(\psi({\cal B})) = {\cal B} \in {\cal M}^s_{\operatorname{fib}}$. This finishes the proof of Theorem~\ref{iso}. \section{Conjectures and open questions.} \label{_conje_Section_} \subsection{NHYM moduli spaces and hyperk\"ahler reduction} \label{_Hyperkae_redu_Subsection_} \subsubsection{} Let $M$ be a K\"ahler manifold and let ${\cal M}^s$ be the moduli space of NHYM connections on a complex bundle ${\cal B}$ over $M$. We have shown in Section~\ref{_NHYM_Section_} that the space ${\cal M}^s$ is equipped with a natural closed holomorphic $2$-form $\Omega$ which is is symplectic at least in a neighborhood of the subset of Hermitian connections. In fact one could hope for a much stronger statement. \begin{conjecture}\label{hyp.nhymspace} There exists a hyperk\"ahler metric on ${\cal M}^s$ such that $\Omega$ is the associated holomorphic symplectic from. \end{conjecture} Note that the construction of the form $\Omega$ is completely parallel to a construction of a holomorphic symplectic form on the Hitchin-Simpson moduli space ${\cal M}^s_{DR}$ of flat connections on ${\cal B}$ (\cite{S2}). The analog of Conjecture~\ref{hyp.nhymspace} for ${\cal M}^s_{DR}$ is known. \subsubsection{} To provide some evidence for Conjecture~\ref{hyp.nhymspace}, we give an interpretation of the NHYM equation in the context of hyperk\"ahler reduction. Let ${\cal A}$ be the space of all connections on the complex vector bundle ${\cal B}$. The space ${\cal A}$ is an affine space over the complex vector space ${\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$ of $\operatorname{{\cal E}\!{\it nd}}{\cal B}$-valued $1$-forms on $M$. Choose an Hermitian metric $h$ on the bundle ${\cal B}$. The decomposition $$ {\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) = {\cal A}^{1,0}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) \oplus {\cal A}^{0,1}(M,\operatorname{{\cal E}\!{\it nd}}{\cal B}) $$ allows one to define a quaternionic structure on the space ${\cal A}^1(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$. Together with the natural trace metric, this structure makes the space ${\cal A}$ an (infinite-dimesional) hyperk\"ahler manifold. The complex gauge group ${\cal G} = \operatorname{Maps}(M,\operatorname{Aut}{\cal B})$ acts on the space ${\cal A}$. This action is compatible with the hyperk\"ahler structure on ${\cal A}$. Therefore one can apply to the space ${\cal A}$ the machinery of {\bf hyperk\"ahler reduction} (see \cite{HKLR}). It turns out that the complex moment map ${\cal A} \to {\Bbb C}$ is equal to the map $YM:{\cal A} \to \Gamma(M,\operatorname{{\cal E}\!{\it nd}}{\cal B})$, $\nabla \mapsto \Lambda\nabla^2$. Vanishing of this map is precisely the NHYM condition. Let ${\cal A}_0 = YM^{-1}(0) \subset {\cal A}$ be the subset of connections with $\Lambda\nabla^2 = 0$. By the general principles of hyperk\"ahler reduction the quotient ${\cal A}_0 / {\cal G}$ should be hyperk\"ahler. The NHYM moduli space ${\cal M}^s$ is the closed subset ${\cal M}^s \subset {\cal A}_0 / {\cal G}$ of equivalence classes of connections with curvature $R=\nabla^2$ which satisfy $\Lambda R = 0$ and are, in addition, of Hodge type $(1,1)$. We expect that the embedding ${\cal M}^s \hookrightarrow {\cal A}_0 / {\cal G}$ is compatible with the hyperk\"ahler structure on ${\cal A}_0 /\ G$ and gives a hyperk\"ahler structure on ${\cal M}^s$ by restriction. \subsubsection{} The hyperk\"ahler reduction construction of the NHYM moduli space also allows to formulate an analog of the Uhlenbeck-Yau Theorem for NHYM-bundles. We first give a new definition of stability for NHYM bundles, more natural than the $(0,1)$-stability used in the body of the paper. Let $\overline{M}$ be the complex-conjugate complex manifold to $M$. Since $M$ and $\overline{M}$ are the same as smooth manifolds, the bundle ${\cal B}$ can be also considered as a complex vector bundle on $\overline{M}$. For every connection $\nabla$ on ${\cal B}$ the $(1,0)$-part $\nabla^{1,0}$ defines a holomorphic structure on the complex bundle ${\cal B}$ on $\overline{M}$. \begin{defn} Let $\langle{\cal B}, \nabla\rangle$ be a bundle with a $(1,1)$-connection over a complex manifold $X$. Let $U\subset X$ be a Zariski open subset in $X$, and let ${\cal F}\subset {\cal B}\restrict{U}$ be a subbundle which is preserved by $\nabla$. Then ${\cal F}$ is called {\bf a subsheaf of $\langle{\cal B}, \nabla\rangle$} if the following two conditions hold: \begin{description} \item[(i)] Consider ${\cal B}$ as a holomorphic bundle over $M$, with a holomorphic structure defined by the $(0,1)$-part of the connection. Then there exist a coherent subsheaf $\widetilde {\cal F}\subset {\cal B}$ on $M$ such that the restriction $\widetilde {\cal F}\restrict{U}$ is a sub-bundle of ${\cal B}$ which coinsides with ${\cal F}$. \item[(ii)] Consider ${\cal B}$ as a holomorphic bundle over $\overline{M}$, with a holomorphic structure defined by the $(1,0)$-part of the connection. Then there exist a coherent subsheaf $\widetilde {\cal F}\subset {\cal B}$ on $\overline{M}$ such that the restriction $\widetilde {\cal F}\restrict{U}$ is a sub-bundle of ${\cal B}$ which coinsides with ${\cal F}$. \end{description} \end{defn} For a subsheaf ${\cal F}\subset {\cal B}$, it is straightforward to define the Chern classes and the degree. As usually, ${\cal F}$ is called {\bf destabilizing} if \[ \frac{\deg {\cal F}}{\operatorname{rank} {\cal F}} \geq \frac{\deg {{\cal B}}}{\operatorname{rank} {{\cal B}}} \] \begin{defn}\label{_stabili_for_hyper_redu-Definition_} Let $\langle{\cal B}, \nabla\rangle$ be a bundle with $(1,1)$-connection over a compact K\"ahler manifold $M$. Then $\langle{\cal B}, \nabla\rangle$ is called $\nabla$-stable if there are no destabilizing subsheaves ${\cal F}\subset \langle{\cal B}, \nabla\rangle$. \end{defn} This definition generalizes the definition \cite{_Simpson:harmonic_} of stability for flat bundles. \begin{rem} Clearly, for NHYM bundles, $(0,1)$-stability implies the stability in the sense of Definition \ref{_stabili_for_hyper_redu-Definition_} \end{rem} An analogy with the Kempf-Ness Theorem suggests that every stable ${\cal G}$-orbit in ${\cal A}_0$ has non-trivial intersection with the zero set of the real moment map ${\cal A}_0 \to \Gamma(M,\operatorname{{\cal E}\!{\it nd}}_{\Bbb R}{\cal B})$ from ${\cal A}_0$ to the space of anti-Hermitian endomorphisms of the bundle ${\cal B}$. This moment map can be described more explicitly. \begin{defn}[pseudocurvature] Let $\langle{\cal B}, \nabla\rangle$ be a bundle with $(1,1)$-connection and a Hermitian metric $h$, not necessary compatible. Let $\nabla = \nabla' + \nabla''$ be the decomposition of $\nabla$ onto $(1,0)$ and $(0,1)$-parts. Consider the connection in $\overline{{\cal B}}^*$ associated with $\nabla$. Since $h$ identifies ${\cal B}$ and $\overline{{\cal B}}^*$, this gives another connection in ${\cal B}$, denoted by $\nabla_h$. The average $\nabla_h= \frac{\nabla +\nabla_h}{2}$ is again a connection, and is compatible with $h$. Let $\theta$ be the difference $\theta:= \frac{\nabla -\nabla_h}{2}$, which is a tensor. Applying $\nabla_h$ to $\theta$, we obtain a 2-form $\Xi$ with coefficients in $\operatorname{{\cal E}\!{\it nd}}({\cal B})$. This form $\Xi$ is called {\bf the pseudocurvature} of the triple $\langle {\cal B}, \nabla, h\rangle$. \end{defn} It turns out that the real moment map on a NHYM connection $\nabla$ is given by $$ \nabla \mapsto \Lambda(\Xi), $$ where $\Xi$ is the pseudocurvature. \begin{defn} \label{_harmonic_me_Definition_} Let $\langle{\cal B}, \nabla\rangle$ be a bundle with a NHYM $(1,1)$-connection, and let $h$ be an Hermitian metric on ${\cal B}$, not necessarily compatible with $\nabla$ Then $h$ is called {\bf harmonic} if $\Lambda \Xi =0$, where $\Xi$ is the pseudocurvature of $\langle {\cal B}, \nabla, h \rangle$. \end{defn} \begin{conjecture} Let $\langle{\cal B}, \nabla\rangle$ be a bundle with NHYM connection $\nabla$. Then there exists a harmonic metric $h$ on ${\cal B}$ if and only if ${\cal B}$ is a direct sum of $\nabla$-stable bundles. Also, if ${\cal B}$ itself is $\nabla$-stable, then $h$ is unique, up to a constant factor. \end{conjecture} An analogous statement is known for flat connections. See \cite{_Simpson:harmonic_} for a discussion. \subsection{K\"ahler base manifold: open questions.} In this subsection, we relate questions pertaining to the case of base manifold $M$ compact and K\"ahler, but not necessarily hyperk\"ahler. \begin{question} \label{_NHYM_flat?_Question_} Let $(B, \nabla)$ be a NHYM-connection in a bundle with zero Chern classes. Is it true that $\nabla$ is necessarily flat? \end{question} In Hermitian case, the answer is affirmative by L\"ubcke \cite{_Lubcke_} and Simpson \cite{S}. In a neighbourhood of Hermitian Yang-Mills connection, all NHYM connections on a bundle with zero Chern classes are also flat, at one can see, e. g., from Proposition \ref{series}. The hyperk\"ahler analogue of this question is Question \ref{_NHYM-are-autodu_Question_}. \hfill Let $B$ be a stable holomorphic bundle over $M$ and let $St(B)$ be the deformation space of stable holomorphic structures on $B$. In Section \ref{_NHYM_Section_}, we defined a Kuranishi map $\phi:\; U \hookrightarrow H^1(\operatorname{{\cal E}\!{\it nd}}(B))$, where $U$ is a neighbourhood of $[B]$ in $St(B)$. The map $\phi$ is, locally, a closed embedding, and its image in a neighbourhood of zero in $H^1(\operatorname{{\cal E}\!{\it nd}}(B))$ is an algebraic subvariety, defined by the zeroes of so-called Massey products. Let $C$ be the Zariski closure of the image $\phi(U)$ in $H^1(\operatorname{{\cal E}\!{\it nd}}(B))$. Let $\operatorname{NHYM}(B)$ be the space of NHYM connections inducing the same holomorphic structure. In \eqref{_from_NHYM_to_classes_Equation_}, we construct the map $\operatorname{NHYM}(B) \stackrel{\rho}{\longrightarrow} \overline C$, where $\overline C$ is a complex conjugate manifold to $C$, and prove that in a neighbourhood of zero $\rho$ is isomorphism. Two questions arise: \begin{question}\label{_rho_surjec_Question_} Is the map $\rho$ surjective? \end{question} \begin{question} \label{_rho_etale_Question_} Is the map $\rho$ etale? Bijective? \end{question} These two questions might be reformulated in a purely algebraic way. Let $B$ be a stable holomorphic bundle equipped with a Hermitian Yang-Mills metric, and $X$ be the space of all $(1,0)$-forms $\theta\in \Lambda^{1,0}(\operatorname{{\cal E}\!{\it nd}} (B))$ satisfying \begin{equation} \begin{cases} \partial \theta &= \theta\wedge \theta \\ \partial^* \theta& =0, \end{cases} \end{equation} where $\partial$ is the $(1,0)$-part of the connection, and $\partial^*$ the adjoint operator. The middle cohomology space of the complex \[ \Lambda^{2,0}(\operatorname{{\cal E}\!{\it nd}}(B)) \stackrel{\partial^*}{\longrightarrow} \Lambda^{1,0}(\operatorname{{\cal E}\!{\it nd}}(B)) \stackrel{\partial^*}{\longrightarrow} \operatorname{{\cal E}\!{\it nd}}(B) \] is naturally isomorphic to the complex conjugate space to $H^1(\operatorname{{\cal E}\!{\it nd}}(B))$. This gives a map \[ X \stackrel{\rho} {\longrightarrow} \overline{H^1(\operatorname{{\cal E}\!{\it nd}}(B))}, \] associating to $\theta$ its cohomology class. Then, Proposition \ref{kur} implies that the image of $\rho$ lies in $\overline C$ and locally in a neighbourhood of zero, $\rho: \; X \longrightarrow \overline C$ is an isomorphism. Question \ref{_rho_surjec_Question_} asks whether $\rho$ is surjective onto $\overline C$, and Question \ref{_rho_etale_Question_} asks whether $\overline \pi$ is etale, or even invertible. \subsection{Autodual and NHYM connections over a hyperk\"ahler base.} \subsubsection{} The first and foremost question (partially answered in Theorem \ref{_NHYM-are-autodu_Theorem_}; see also Question \ref{_NHYM_flat?_Question_}): \begin{question} \label{_NHYM-are-autodu_Question_} Let $(B, \nabla)$ be a NHYM bundle over a hyperk\"ahler base manifold. Is $(B, \nabla)$ necessarily autodual? \end{question} \subsubsection{} \label{compa_smoo_Mukai_du_Subsubsection_} Let $M$ be a compact hyperk\"ahler manifold, and let $\cal S$ be a connected component of the moduli of autodual connections on a complex vector bundle ${\cal B}$. Assume that $\cal S$ contains a point $B$ which is Hermitian autodual. Consider the ``Mukai dual'' space $\widehat M$, that is, the moduli space of Hermitian autodual connections on ${\cal B}$ (Subsection \ref{_Mukai_dual_Subsection_}). Assume that the connected component of $\widehat M$ containing $B$ is smooth and compact. Clearly, then, all connections from $\cal S$ are fiberwise stable, in the sense of Definition \ref{fib.st}. Thus, Theorem \ref{iso} gives an isomorphism between $\cal S$ and the space $Sec(\widehat M)$ of twistor lines in $\operatorname{Tw}(\widehat M)$. In such situation, we are going to give a conjectural description of the space $\cal S$, assuming that the answer to \ref{_rho_surjec_Question_}---\ref{_rho_etale_Question_} is affirmative. \subsubsection{} \begin{defn}[Twisted cotangent bundle] Let $M$ be a K\"ahler manifold, $\Omega^1M$ its holomorphic cotangent bundle. The K\"ahler class \[ \omega \in H^1(\Omega^1 M)= Ext^1({\cal O} (M), \Omega^1 M) \] gives by Yoneda an exact sequence \[ 0 \longrightarrow \Omega ^1 M \longrightarrow E \stackrel e \longrightarrow {\cal O}(M) \longrightarrow 0, \] where ${\cal O}(M)$ is the trivial one-dimensional bundle. Let $\nu$ be a non-zero section of ${\cal O}(M)$, and $E_\nu$ be the set of all vectors \[ \left\{ v\in E\restrict m \;\; \left|\vphantom{\bigcup\limits_M^M}\right. \;\; e(v) = \nu\restrict m\right\} \] where $m$ runs through all points of $M$. Consider $E_\nu$ as a submanifold in the total space of $E$. Then $E_\nu$ is called {\bf a twisted cotangent bundle of $M$}, denoted by $\Omega_\omega M$. \end{defn} The space $\Omega_\omega M$ has a natural action of $\Omega^1 M$ considered as a group scheme over $M$, and as such is a torsor over $\Omega ^1 M$ \subsubsection{} The affirmative answer to the stronger form of \ref{_rho_etale_Question_} would give the proof of the following conjecture. \begin{conjecture} \label{_sec_to_twi_cota_Conjecture_} Under assumptions of \ref{compa_smoo_Mukai_du_Subsubsection_}, there exists a natural isomorphism of complex manifolds \[ Sec(\widehat M) \cong \Omega_\omega M, \] where $\Omega_\omega M$ is the twisted cotangent bundle. \end{conjecture} In the general situation, there is a natural map from the space of twistor lines $Sec(M)$ of a compact hyperk\"ahler manifold to $\Omega_\omega M$. However, in general there are no approaches to the proof of surjectivity. \begin{example} Let $M$ be a compact complex torus, $\dim_{\Bbb C} M = 2n$, and $B$ a trivial line bundle. Clearly, $M$ is hyperk\"ahler. Then $\widehat M$ is the dual torus, and $\cal S$ is the space of local systems on $M$, which is isomorphic to $({\Bbb C}^*)^{2n}$. The space $C$ of \ref{_rho_etale_Question_} is isomorphic to $H^1({\cal O} M)$, and the answer to \ref{_rho_etale_Question_} is obviously affirmative. Thus, $Sec(M)$ and $\Omega_\omega M$ are also isomorphic to $({\Bbb C}^*)^{2n}$ and are Stein. \end{example} In the following subsection, we shall see that this is indeed a general phenomenon -- the space of twistor lines is equipped with a canonical plurisubharmonic function and is likely to be Stein. However, we don't know a general argument constructing plurisubharmonic functions on the twisted cotangent bundle -- this is one more mystery. \subsection{Plurisubharmonic functions on moduli spaces.} Let $M$ be a hyperk\"ahler manifold $\operatorname{Tw} \stackrel \pi \longrightarrow {\Bbb C} P^1$ its twistor space, and $Sec(M)$ the space of sections $s:\; {\Bbb C} P^1 \longrightarrow \operatorname{Tw}$ of the map $\pi$, also called {\it twistor lines} (Section \ref{_twistors_Section_}). It is easy to equip $Sec(M)$ with a natural plurisubharmonic function. Recall that $\operatorname{Tw}$ is isomorphic as a $C^\infty$-manifold to $M \times {\Bbb C} P^1$. This decomposition gives a natural (non-K\"ahler) Hermitian metric on $\operatorname{Tw}$. \begin{prop} \label{_volume_twi_line_plurisubharmo_Proposition_} Consider the function $v:\; Sec(M) \longrightarrow {\Bbb R} ^+$ which maps a line $s\in \operatorname{Tw}$ to its Hermitian volume, taken with respect to the Hermitian metric on $\operatorname{Tw}$. Then $v$ is strictly plurisubharmonic. \end{prop} \par\noindent{\bf Proof.}\ Let $\omega$ be the differential 2-form which is the symplectic part of the Hermitian metric on $\operatorname{Tw}$. Since the twistor lines are complex subvarieties in $\operatorname{Tw}$, $v(s) = \int_s \omega$ for all twistor lines $s$. Then, for all bivectors $x, \bar x$ in $T_s Sec(M)$, we have \begin{equation}\label{_6bar6omega_Equation_} \partial\bar\partial v(x, \bar x) = \int_s \partial\bar\partial \omega({\bf x},\bar{\bf x}), \end{equation} where ${\bf x},\bar{\bf x}$ are the sections of $T \operatorname{Tw}\restrict{s}$ corresponding to $x, \bar x$. Then, to prove that $v$ is plurisubharmonic it suffices to show that $\partial\bar\partial \omega({\bf x},\bar{\bf x})$ is positive. {}From Lemma \ref{_differe_of_Hermi_on_twistors_Lemma_}, it is easy to see that $\partial\bar\partial \omega = \omega\wedge \pi^* \operatorname{FS}({\Bbb C} P^1)$, where $\operatorname{FS}({\Bbb C} P^1)$ is the Fubini-Study form on ${\Bbb C} P^1$. Clearly, then, $\partial\bar\partial\omega({\bf x},\bar{\bf x})$ is positive, and $v$ is plurisubharmonic. This proves Proposition \ref{_volume_twi_line_plurisubharmo_Proposition_}. \endproof One of the most intriguing questions of hyperk\"ahler geometry is to learn whether the function $v$ is exhausting. \hfill In notation and assumptions of \ref{_sec_to_twi_cota_Conjecture_}, consider the space $Sec(\widehat M)$ which is isomorphic to the space $\cal S$ of autodual connections. There is the canonical Weil-Petersson metric on $\cal S$, coming from results of Subsection \ref{_Hyperkae_redu_Subsection_}. This metric is hyperk\"ahler. This metric is given by a potential, which is equal to the integral of the square of the absolute value of the curvature. \begin{question} Is the Weil--Petersson metric related to the metric given by $v$? \end{question} \subsubsection*{Acknowledgements:} The autors are grateful to S.-T. Yau, who stimulated the interest to the problem, D. Kazhdan and T. Pantev for valuable discussions, S. Arkhipov, M. Finkelberg and L. Positselsky for their attention, and to Soros Foundation which is our source of livelihood.

9606

alg-geom/9606017

fr

https://arxiv.org/abs/alg-geom/9606017

alg-geom/9606017

Emmanuel Ullmo

Emmanuel Ullmo

Positivite et discretion des points algebriques des courbes

We prove the discreteness of algebraic points (with respect to the Neron-Tate height) on a curve of genus greater than one embedded in his jacobian. This result was conjectured by Bogomolov. We also prove the positivity of the self intersection of the admissible dualizing sheaf.

alg-geom

\section{Introduction} Soient $K$ un corps de nombres et $\overline{K}$ sa cl\^oture alg\'ebrique. Soient $X_K$ une courbe propre, lisse, g\'eom\'etriquement connexe de genre $g\ge 2$ sur $K$ et $J$ sa jacobienne. Soit $D_0$ un diviseur de degr\'e 1 sur $X$ et $\phi_{D_0}$ le plongement de $X_K$ dans $J$ d\'efini par $D_0$. On note $h_{NT}(x)$ la hauteur de N\'eron-Tate d'un point $x\in J(\overline{K})$. On montre dans ce texte l'\'enonc\'e suivant qui a \'et\'e conjectur\'e par Bogomolov \cite{Bo}: \begin{teo}\label{teo1} Il existe $\epsilon >0$ tel que $\{P\in X_K(\overline{K}) \vert h_{NT}(\phi_{D_O} (P))\le \epsilon \}$ est fini. \end{teo} Notons que Raynaud \cite{Ra} a prouv\'e que l'ensemble des points $P\in X_K(\overline{K})$ tels que $\phi_{D_0}(P)$ est de torsion dans $J$ est fini. Le th\'eor\`eme \ref{teo1} g\'en\'eralise cet \'enonc\'e car la condition $\phi_{D_0}(P)$ est \'equivalente \`a $h_{NT}(\phi_{D_0}(P))=0$. Le lecteur s'assurera que la d\'emonstration du th\'eor\`eme \ref{teo1} est ind\'ependante de celle de Raynaud. Le th\'eor\`eme \ref{teo1} a \'et\'e obtenu dans de nombreux cas par Szpiro \cite{Sz} et Zhang \cite{Za1}. Soit $\Omega^1_X$ le faisceau des diff\'erentielles holomorphes sur $X_K$. Quand $X_K$ a un mod\`ele propre et lisse sur l'anneau des entiers $O_K$ de $K$, Szpiro \cite{Sz} a montr\'e le th\'eor\`eme \ref{teo1} quand la classe $[\Omega^1_X-(2g-2)D_0]$ n'est pas de torsion dans $J$ ou la self intersection $(\omega_{Ar},\omega_{Ar})_{Ar}$ du dualisant relatif au sens d'Arakelov est non nulle. Il a aussi expliqu\'e comment un \'equivalent du th\'eor\`eme de Nakai et Moishezon en th\'eorie d'Arakelov permet de prouver que la non nullit\'e de $(\omega_{Ar},\omega_{Ar})_{Ar}$ est \'equivalente \`a l'\'enonc\'e du th\'eor\`eme \ref{teo1}. Zhang \cite{Za3} a montr\'e l'analogue du th\'eor\`eme de Nakai et Moishezon (voir aussi les travaux de Kim \cite{Kim} pour une autre approche). Notons enfin que toujours dans le cas o\`u $X_K$ admet un mod\`ele propre et lisse sur $O_K$, Burnol \cite{Bu} et Zhang \cite{Za2} ont donn\'e des conditions suffisantes pour que $(\omega_{Ar},\omega_{Ar})_{Ar}$ soit strictement positif. Pour g\'en\'eraliser les travaux de Szpiro (concernant le cas o\`u le mod\`ele minimal non singulier $\cal X$ de $X_K$ sur l'anneau des entiers $O_K$ de $K$ est lisse), Zhang \cite{Za1} a introduit un accouplement admissible $(\ ,\ )_a$ g\'en\'eralisant celui d'Arakelov $(\ ,\ )_{Ar}$ sur $\cal X$. Il a d\'efini un faisceau dualisant relatif $\omega_a$ qui co\"\i ncide avec le faisceau dualisant relatif $\omega_{Ar}$ dans le cas o\`u $\cal X$ est lisse et qui v\'erifie \begin{equation} (\omega_{Ar},\omega_{Ar})_{Ar}\ge (\omega_a,\omega_a)_a \ge 0. \end{equation} Il a montr\'e le th\'eor\`eme \ref{teo1} quand $(\omega_a,\omega_a)_a>0$ et quand la classe $[\Omega^1_X-(2g-2)D_0]$ n'est pas de torsion dans $J$. Il a enfin montr\'e, que si $(\omega_a,\omega_a)_a = 0$ et $D_0=\frac{\Omega^1_X}{2g-2}$ le th\'eor\`eme \ref{teo1} est en d\'efaut. On obtient ainsi dans ce texte la preuve du r\'esultat suivant : \begin{teo}\label{teo4} Soit $\cal X\longrightarrow \mbox{Spec}(O_K)$ le mod\`ele minimal r\'egulier d'une courbe lisse g\'eom\'etriquement connexe $X_K$ sur $K$ de genre $g\ge 2$. Si ${\cal X}$ a r\'eduction semi--stable alors : \begin{equation} (\omega_{Ar},\omega_{Ar})_{Ar}\ge (\omega_a,\omega_a)_a>0. \end{equation} \end{teo} Notons que l'in\'egalit\'e $(\omega_{Ar},\omega_{Ar})_{Ar}>0$ a \'et\'e d\'emontr\'ee par Zhang \cite{Za1} dans le cas o\`u ${\cal X}$ n'est pas lisse sur $O_K$. Un \'enonc\'e un peu moins g\'en\'eral a \'et\'e obtenu par Burnol \cite{Bu}. Zhang \cite{Za2} a aussi montr\'e l'in\'egalit\'e $(\omega_a,\omega_a)_a>0$ (et donc le th\'eor\`eme \ref{teo1}) quand $\mbox{End}(J)\otimes {\mathbb R}$ n'est pas isomorphe \`a ${\mathbb R}$, ${\mathbb C}$ o\`u ${\mathbb H}$ (alg\`ebre des quaternions). Pour prouver les th\'eor\`emes \ref{teo1} et \ref{teo4}, on suppose que $(\omega_a,\omega_a)_a = 0$ et que $D_0=\frac{\Omega^1_X}{2g-2}$. On dispose alors d'une suite infinie de points $x_n$ de $X_K(\overline{K})$ telle que $h_{NT} (\phi_{D_0}(x_n))$ tend vers $0$ quand $n$ tend vers l'infini. En utilisant des th\'eor\`emes d'\'equidistributions des petits points \` a une suite $y_n$ de $X_K^g$, construite \`a partir de $x_n$, et \`a son image $z_n$ dans $J$ on obtient une contradiction. \section{Pr\'eliminaire en th\'eorie d'Arakelov} \subsection{Vari\'et\'es arithm\'etiques, hauteurs } \bigskip Soit $K$ un corps de nombres, $O_K$ son anneau d'entiers et $S=\mbox{Spec} (O_K)$. On note $S_{\infty, K}$ l'ensemble des places \`a l'infini de $K$. Une vari\'et\'e arithm\'etique sur $S$ est la donn\'ee d'un $S$--sch\`ema plat et projectif $X$ dont la fibre g\'en\'erique $X_K$ est lisse. Pour toute extension $K_1$ de $K$, on note $X_{K_1}=X_K\otimes_K K_1$. Les points de $X({\mathbb C})$ vus comme ${\mathbb Z}$--sch\'ema s'\'ecrivent comme la r\'eunion disjointe $X({\mathbb C})=\displaystyle \cup_{\sigma\in S_{\infty,K}} X_{\sigma}({\mathbb C})$, o\`u $X_{\sigma}({\mathbb C})= X\otimes_{\sigma}{\mathbb C}$. Un fibr\'e inversible $\overline{L}=(L, \Vert\ \Vert_{\sigma})$ sur $X$ est la donn\'ee d'un fibr\'e inversible $L$ sur $X$ et pour tout $\sigma\in S_{\infty,K}$ d'une m\'etrique $C^{\infty}$, invariante par la conjugaison complexe, sur le fibr\'e inversible $L_{\sigma}=L\otimes_{\sigma}{\mathbb C}$ de $X_{\sigma}({\mathbb C})$. On note alors $\overline{L}_{\sigma}$ le fibr\'e inversible hermitien $(L_{\sigma},\Vert\ \Vert_{\sigma})$ de $X_{\sigma}({\mathbb C})$. On note $\overline{\mbox{Pic}}(X)$ la cat\'egorie des fibr\'es inversibles hermitiens sur $X$. On dit que deux \'el\'ements ${\cal \overline{L}}$, ${\cal \overline{L}}'$ de $\overline{\mbox{Pic}}(X)\otimes {\mathbb Q}$ coincident sur la fibre g\'en\'erique s'il existe sur la fibre g\'en\'erique un isomorphisme de ${\cal L}_K$ sur ${\cal L}'_K$ qui est une isom\'etrie en toute place \`a l'infini. Dans cette situation, on se permet d'\'ecrire ${\cal \overline{L}}_K={\cal \overline{L}}'_K$ En particulier, un fibr\'e inversible hermitien $\overline{L}$ sur $S$ est la donn\'ee d'un $O_K$--module projectif de rang 1 et de m\'etriques hermitiennes sur le ${\mathbb C}$--espace vectoriel de dimension 1, $L_{\sigma}$, pour toute place \`a l'infini $\sigma$ de $K$. Le degr\'e d'un fibr\'e inversible hermitien $\overline{L}$ sur $S$ est d\'efini par l'\'egalit\'e : \begin{equation} \deg_{Ar}(\overline{L}) =\log \#(L/O_K .s)-\sum_{\sigma\in S_{\infty,K}} \log \Vert s\Vert_{\sigma} \end{equation} pour une section arbitraire $s$ de $L$. Soit $X$ une vari\'et\'e arithm\'etique sur $S$ de dimension (absolue) $d$ et $\overline{L}$ un fibr\'e inversible hermitien sur $X$. Pour tout $x\in X_K(\overline{K})$, on note $D_x$ la cl\^oture de Zariski de $x$ dans $X$ et $K(x)$ son corps de rationalit\'e . La hauteur $h_{\overline{L}}(x)$ est alors d\'efinie par la formule $\displaystyle h_{\overline{L}}(x)=\frac{\deg_{Ar}(\overline{L}\vert D_x)}{[K(x):K]}$. Si on part d'une vari\'et\'e projective lisse $X_K$ sur $K$, que l'on fixe une extension $K_1$ de $K$, un mod\`ele ${\cal X}_1$ de $X_{K_1}$ sur l'anneau des entiers $O_{K_1}$ de $K_1$ et un fibr\'e inversible hermitien ${\cal \overline{L}}$ sur ${\cal X}_1$, on peut encore d\'efinir une hauteur sur $X_K(\overline{K})$. En effet pour tout point $x\in X_K(\overline{K})$, $\mbox{Spec}(K(x)\otimes_K K_1)$ est une r\'eunion de points $(x_1,\dots,x_r)$ de $X_L(\overline{L})$. On pose alors $$ h_{{\cal \overline{L}}} (x)=\frac{\displaystyle\sum_{i=1}^r h_{{\cal \overline{L}}}(x_i)}{[K_1:K]}. $$ Soit $T$ une vari\'et\'e analytique complexe de dimension $d-1$ et $\overline{L}=(L,\Vert\ \Vert)$ un fibr\'e inversible hermitien sur $T$. Soit $K$ la forme de courbure associ\'ee \`a $\overline{L}$ \cite{GH}. On notera dans la suite $c_1(\overline{L})$ la $(1,1)$--forme ferm\'ee $\frac{i}{2\pi}K$. Par ailleurs on notera $c_1(L)$ la premi\`ere classe de Chern d'un fibr\'e inversible $L$ sur une vari\'et\'e alg\'ebrique $T$. A travers l'application degr\'e, on identifiera $c_1(L)^{d-1}$ \`a un entier naturel. Rappelons que Gillet et Soul\'e \cite{GS1} \cite{GS3} ont d\'efini pour une vari\'et\'e arithm\'etique $X$ de dimension $d$ des groupes de Chow arithm\'etiques $\widehat{CH} ^i(X)$ pour tout entier naturel $i$. Quand $X$ est irr\'eductible, on a $\widehat{CH}^0(X)\simeq {\mathbb Z}$. On dispose d'une application degr\'e $$ \widehat{CH}^d(X) \longrightarrow {\mathbb R}. $$ Pour tout fibr\'e inversible hermitien $\overline{L}$ sur $X$, on dispose d'une premi\`ere classe de Chern arithm\'etique $\mbox{\^c}_1(\overline{L})\in \widehat{CH}^1(X)$. Gr\^ace au produit d'intersection $$ \widehat{CH}^i(X)\times \widehat{CH}^j(X)\longrightarrow \widehat{CH}^{i+j}(X)\otimes_{{\mathbb Z}} {\mathbb Q}, $$ on sait d\'efinir $\mbox{\^c}_1(\overline{L})^i\in \widehat{CH}^i(X)\otimes_{{\mathbb Z}}{\mathbb Q}$ pour tout $i\in {\mathbb N}^*$. On d\'efinit $\mbox{\^c}_0(\overline{L})=1$ et on voit $\mbox{\^c}_1(\overline{L})^d$ comme un nombre r\'eel \`a travers l'application degr\'e. Rappelons aussi que dans le cas des surfaces arithm\'etiques, on dispose d'une th\'eorie due \`a Arakelov \cite{Ar}, Faltings \cite{Fa} et Zhang \cite{Za1} qui pr\'ecise les choix de m\'etriques sur les fibr\'es que l'on est amen\'e \`a \'etudier. Soient donc $X$ une courbe lisse et g\'eom\'etriquement connexe de genre $g$ non nul sur $K$ et ${\cal X}$ son mod\`ele r\'egulier minimal. Quitte \`a \'elargir $K$, on suppose que ${\cal X}$ est semi--stable. Pour tout plongement $\sigma$ de $K$ dans $\Bbb C$, on note $X_{\sigma}$ la surface de Riemann obtenue \`a partir de $X$ par le changement de base d\'efini par $\sigma$. La surface $X_{\sigma}$ est munie d'une $(1,1)$--forme canonique \[ \nu_{\sigma}=\frac{i}{2g}\sum_{i=1}^{g}\omega_i\wedge \overline{\omega}_i, \] pour une base orthonorm\'ee $(\omega_1,\dots ,\omega_g)$ de $H^0(X_{\sigma},\Omega^1)$ o\`u $\Omega^1$ est le faisceau des diff\'erentielles holomorphes sur $X_{\sigma}$ pour le produit scalaire~: \begin{equation}\label{prodscal} \langle \alpha,\beta \rangle=\frac{i}{2}\int_{X_\sigma}\alpha\wedge \overline{\beta}. \end{equation} Arakelov a d\'efini une th\'eorie des intersections $(\ ,\ )_{Ar}$ pour les \'el\'ements de la cat\'egorie $\mbox{Pic}_{Ar}({\cal(X)})$ des fibr\'es inversibles sur ${\cal X}$ munis en chaque place \`a l'infini $\sigma$ de $K$ d'une m\'etrique permise (\`a courbure proportionnelle \`a $\nu_{\sigma}$). Le faisceau $\omega_{{\cal X}/\OK}$, dualisant relatif de ${\cal X}$ sur $\mbox{Spec} (O_K)$ est canoniquement muni de m\'etriques permises \cite{Ar}. On note $\omega_{Ar}=\overline{\omega_{{\cal X}/\OK}}$ l'\'el\'ement de $\mbox{Pic}_{Ar}({\cal(X)})$ ainsi obtenu. Zhang \cite{Za1} a \'etendu et g\'en\'eralis\'e l'intersection d'Arakelov. Il a d\'efini une notion d'admissibilit\'e en toute place de $K$ qui co\"\i ncide avec celle d'Arakelov aux places \`a l'infini. On note $\mbox{Pic}_a({\cal X})$ la cat\'egorie des fibr\'es inversibles admissibles au sens de Zhang \cite{Za1}. On dispose d'une th\'eorie des intersections $(\ ,\ )_a$ sur $\mbox{Pic}_a({\cal X})$. Zhang a aussi d\'efini des m\'etriques admissibles en toute place de $K$ sur le fibr\'e $\omega_{{\cal X}/\OK}$. On note dans ce texte $\omega_a$ l'\'el\'ement de $\mbox{Pic}_a({\cal X})$ ainsi obtenu. On dispose ainsi d'une hauteur $h_{\omega_a}$ sur ${\cal X}$ qui est un repr\'esentant de la classe des hauteurs de Weil associ\'es au fibr\'e $\Omega^1_X$ (voir \cite{Si} pour une introdution au hauteurs). On aura besoin du r\'esultat suivant qui r\'esulte imm\'ediatement de \cite{Za1} (th\'eor\`eme 2-4). \begin{lem}\label{aprox} Soit ${\cal X}\longrightarrow \mbox{Spec}(O_K)$ le mod\`ele minimal non singulier d'une courbe lisse, g\'eom\'etriquement connexe, de genre $g\ge 2$ sur $K$. On suppose que ${\cal X}$ a r\'eduction semi--stable. Il existe une suite d'extension $K_n$ de $K$, telle que si on note ${\cal X}_n$ le mod\`ele minimal non singulier de $X_{K_n}$ sur l'anneau des entiers $O_{K_n}$ de $K_n$, il existe une suite d'\'el\'ements ${\cal \overline{L}}_n$ de ${\rm Pic}_{Ar} ({\cal X}_n)\otimes_{{\mathbb Z}} {\mathbb Q}$ telle que pour tout $n\in {\mathbb N}$, ${\cal \overline{L}}_n$ co\"\i ncide sur la fibre g\'en\'erique avec $\omega_{Ar}$ comme fibr\'e inversible hermitien et telle que $$ \sup_{x\in X_K(\overline{K})} \vert h_{{\cal \overline{L}}_n}(x)-h_{\omega_a}(x)\vert $$ tende vers $O$ quand $n$ tend vers l'infini. \end{lem} Dans la suite de ce texte, on fixe une surface arithm\'etique ${\cal X}\rightarrow \mbox{Spec}(O_K)$ telle que $\omega_a^2=(\omega_a,\omega_a)_a=0$. On choisit un diviseur $D_0$ sur $X_K$ tel que $D_0=\frac{\Omega^1_X}{2g-2}$. Comme $D_0$ est fix\'e jusqu'\`a la fin de ce texte on se permet de noter $j=\phi_{D_0}$ le plongement de $X_K$ dans sa jacobienne d\'efini par $D_0$ et en faisant une extension convenable, on suppose que $D_0$ est rationnel sur $K$. On note encore $D_0$ le diviseur horizontal de $\cal X$ de fibre g\'en\'erique $D_0$. Le lemme suivant r\'esulte imm\'ediatement de \cite{Za1} (preuve du th\'eor\`eme 5-6). \begin{lem}\label{NT-ARAK} Pour tout point $P\in X(\overline{K})$ on a : \begin{equation} h_{NT}(j(P))=\frac{g}{2g-2}h_{\omega_a}(P). \end{equation} \end{lem} \subsection{Th\'eor\`emes d'\'equidistribution} Une suite de points $(u_n)$ d'une vari\'et\'e alg\'ebrique, irr\'eductible, $X$ est dite g\'en\'erique, si $u_n$ converge, au sens de la topologie de Zariski, vers le point g\'en\'erique de $X$ (autrement dit, si pour toute sous-vari\'et\'e stricte $Y$ de $X$, il existe au plus un nombre fini d'indices $i\in {\mathbb N}$ tels que $x_i$ soit un point de $Y$). On a montr\'e dans \cite{SUZ} le th\'eor\`eme d'\'equidistribution des petits points des vari\'et\'es ab\'eliennes suivant: \medskip \begin{teo}\label{teo2} Soit $A$ une vari\'et\'e ab\'elienne sur un corps de nombres $K$. Soit $(x_n)$ une suite g\'en\'erique de points de $A$ telle que $h_{NT}(x_n)$ converge vers $0$. Soit $O(x_n)$ l'orbite de $x_n$ sous l'action du groupe de Galois $G_K=\mbox{Gal}(\overline{K}/K)$. Pour toute place \`a l'infini $\sigma$, la suite $$ \frac{1}{\#O(x_n)}\sum_{x\in \sigma(O(x_n))} \delta_x $$ converge faiblement vers la mesure de Haar de masse totale $1$ $d\mu_{\sigma}$ de $A_{\sigma}({\mathbb C})\simeq A\otimes_{\sigma}{\mathbb C} $. \end{teo} L'\'enonc\'e suivant qui g\'en\'eralise le th\'eor\`eme d'\'equidistribution des petits points des vari\'et\'es arithm\'etiques d\'emontr\'e dans \cite{SUZ} nous sera utile dans la suite de ce texte. \medskip \begin{teo}\label{equi} Soit $X\rightarrow {\rm Spec}(O_K)$ une vari\'et\'e arithm\'etique de dimension $d$. Soit $\overline{L}$ un fibr\'e inversible hermitien sur $X$ tel que $L_K$ soit ample et $c_1(\overline{L}_{\sigma})$ soit positif pour toute place \`a l'infini $\sigma$ de $K$. Soit $h$ une hauteur de Weil sur $X_K$ associ\'ee \`a $L_K$ telle que $h(P)\ge 0$ pour tout $P\in X_K(\overline{K})$. On suppose qu'il existe une suite $K_n$ d'extensions finies de $K$, une suite ${\cal X}_n$ de mod\`eles projectifs de $X_{K_n}$ sur l'anneau des entiers $O_{K_n}$ de $K_n$ et une suite $\overline{L}_n$ d'\'el\'ements de $\overline{{\rm Pic}}({\cal X}_n)\otimes{\mathbb Q}$, telle que pour tout $n\in{\mathbb N}$ et pour toute place \`a l'infini $\sigma$ de $K_n$ on ait: \begin{equation} \overline{L}_n\otimes_{O_{K_n}} K_n=\overline{L}_{K_n}. \end{equation} \begin{equation} \sup_{x\in X_K(\overline{K})} \vert h_{\overline{L}_n}(x)-h(x)\vert \longrightarrow 0\mbox{ quand } n\rightarrow \infty. \end{equation} Soit $(x_n)$ une suite g\'en\'erique de points de $X_K(\overline{K})$ tel que $h(x_i)$ converge vers $0$. Alors pour toute place \`a l'infini $\sigma_0$ de $K$ et toute fonction continue $f$ sur $X_{\sigma_0}({\mathbb C})$ la suite $$ \frac{1}{\# O(x_n)} \sum_{x_n^g\in O(x_n)} f(\sigma_0(x_n^g)) $$ converge vers $\displaystyle \int_{X_{\sigma_0}({\mathbb C})} f(x) d\mu(x)$ o\`u $$ d\mu =\frac{c_1 (\overline{L}_{\sigma_0})^{d-1}}{c_1(L_{\sigma_0})^{d-1} } $$ est vu comme une mesure sur $X_{\sigma_0}({\mathbb C})$ de volume 1. \end{teo} {\it Preuve.} La preuve donn\'ee ici est une simple adaptation de celles des th\'eor\`emes similaires de \cite{SUZ}. Soit $f$ une fonction continue sur $X_{\sigma_0}({\mathbb C})$ telle que $$ u_n=\frac{1}{\# O(x_n)} \sum_{x_n^g\in O(x_n)} f(\sigma_0(x_n^g)) $$ ne converge pas vers $\displaystyle \int_{X_{\sigma_0}({\mathbb C})} f(x) d\mu(x)$. On peut supposer que : \medskip \par a) $\displaystyle \int_{X_{\sigma_0}({\mathbb C})} f(x) d\mu(x)=0$ (changer $f$ en $f-\displaystyle \int_{X_{\sigma_0}({\mathbb C})} f(x) d\mu(x)$). \par b) La suite $\displaystyle\frac{1}{\# O(x_n)} \sum_{x_n^g\in O(x_n)} f(\sigma_0(x_n^g))$ converge vers une constante $C<0$ (extraire une sous--suite convergente et changer si n\'ec\'essaire $f$ en $-f$). \par c) La fonction $f$ est $C^{\infty}$ sur $X_{\sigma_0}({\mathbb C})$. \medskip Pour tout r\'eel positif $\lambda$, on note $\overline{{\cal O}}_n(\lambda f)$ le fibr\'e inversible hermitien ${\cal O}_{{\cal X}_n}$ de ${\cal X}_n$ muni en toute place \`a l'infini ne divisant pas $\sigma_0$ de la m\'etrique triviale et en toute place \`a l'infini $\sigma$ divisant $\sigma_0$ de la m\'etrique v\'erifiant $\Vert 1 \Vert_{\sigma}(x)=\exp(-\lambda f(x))$ (noter que cela a un sens car pour tout $\sigma$ divisant $\sigma_0$ on a $X_{\sigma}\simeq X_{\sigma_0})$. On note $\overline{L}_n(\lambda f)=\overline{L}_n \otimes \overline{{\cal O}}_n(\lambda f)$. Pour $\lambda$ suffisament petit et pour toute place \`a l'infini $\sigma$ de $K_n$, $c_1(\overline{L}_n(\lambda f)_{\sigma})$ est positive (et cela ind\'ependament de $n$.) On remarque que l'on a pour tout $x\in X_K(\overline{K})$ : \begin{equation} h_{\overline{L}_n(\lambda f)}(x) =h_{\overline{L}_n}(x)+\frac{\lambda}{\#(O(x))}\sum_{x^g\in O(x)} f(x^g). \end{equation} Par ailleurs, \begin{equation}\label{eq6} \frac{\mbox{\^c}_1(\overline{\cal L}_n\otimes {\cal O}_n(\lambda f))^d}{[K_n:K]}= \frac{\mbox{\^c}_1(\overline{\cal L}_n)^d}{[K_n:K]}+\mbox{\rm O}(\lambda^2) \end{equation} pour une fonction $\mbox{\rm O}(\lambda^2)$ ind\'ependante de $n$ (utiliser a et le fait que $\overline{\cal L}_{n,\sigma}$ en tant que fibr\'e inversible hermitien sur $X_{\sigma}\simeq X_{\sigma_0}$ est ind\'ependant de $n$). D'autre part en utilisant le th\'eor\`eme 5-2 de \cite{Za} et l'existence de la suite $u_n$, on voit que quand $n$ tend vers l'infini, $\displaystyle\frac{\mbox{\^c}_1(\overline{\cal L}_n)^d}{[K_n:K]}$ converge vers $0$. Soit $\varepsilon$ un nombre r\'eel positif. Par la convergence uniforme de $h_{\overline{\cal L}_n}$ vers $h$ et la discussion pr\'ec\'edente, il existe $N\in {\mathbb N}$ tel que pour tout $n\ge N$ on a: \medskip \begin{equation} \displaystyle\sup_{x\in X_K(\overline{K})}\vert h_{\overline{\cal L}_n}(x)-h(x)\vert \leq \varepsilon. \end{equation} \begin{equation} \displaystyle\vert \frac{\mbox{\^c}_1(\overline{\cal L}_n)^d}{[K_n:{\mathbb Q}]d.c_1(L)^{d-1} } \vert\leq \varepsilon. \end{equation} \medskip En utilisant le th\'eor\`eme 5-2 de \cite{Za}, on voit que \begin{equation} \lim_{\i\rightarrow\infty} (\ h_{\overline{\cal L}_N}(x_i) + \frac{\lambda}{\# O(x_i)} \sum_{x_i^g\in O(x_i)} f(\sigma_0(x_i^g))\ ) \ge \frac{(\mbox{\^c}_1(\overline{\cal L}_N)+\mbox{\^c}_1(\overline{{\cal O}}(\lambda f))^d}{[K_N:K]d.c_1(L_{K_N})^{d-1}}. \end{equation} On en d\'eduit donc que : \begin{equation} \lambda C \ge O(\lambda^2)-2\varepsilon \end{equation} En faisant tendre $\varepsilon$, puis $\lambda$ vers $0$, on montre que $C\ge 0$. Cette contradiction termine la preuve du th\'eor\`eme \ref{equi} \section{Suites g\'en\'eriques de petits points de $X^g$} On rappelle que l'on a fix\'e une surface arithm\'etique semi--stable ${\cal X}\rightarrow \mbox{Spec}(O_K)$, qui est le mod\`ele minimal \r'egulier d'une courbe $X_K$ lisse g\'eom\'etriquement connexe de genre $g\ge 2$ sur $K$, telle que $(\omega_a,\omega_a)_a=0$. De plus $D_0=\frac{\Omega^1_X}{2g-2}$ est suppos\'e \^etre rationnel sur $K$. En utilisant le corrolaire 5-7 de \cite{Za1} et le lemme \ref{NT-ARAK} on peut construire une suite g\'en\'erique $(t_k)$ de points de $X_K(\overline{K})$ telle que $$ h_{\omega_a}(t_k)=\displaystyle \frac{2g-2}{g}h_{NT}(j(t_k)) $$ tend vers $0$ quand $k$ tend vers l'infini. On poursuit cette id\'ee en travaillant sur $X_K^g(\overline{K})$. On note $s$ l'application de $X_K^g$ dans $J$ telle que : $$ s(P_1,\dots,P_g)=j(P_1)+\dots+j(P_g). $$ Pour toute extension $L$ de $K$, on note $G_L$ le groupe de Galois de $\overline{K}$ sur $L$. Si $K_2$ est une extension galoisienne de $K_1$, on note $\mbox{Gal}(K_2/K_1)$ le groupe de Galois de $K_2$ sur $K_1$. Soient $Y$ une vari\'et\'e alg\'ebrique d\'efinie sur $K$ et $x\in Y(\overline{K})$. Pour toute extension $L$ de $K$, telle que $L\subset K(x)$, on note $O_L(x)=G_L.x$ l'orbite sous $G_L$ de $x$. On fait la convention $O_K(x)=O(x)$. Le but de cette partie est de montrer la proposition suivante. \begin{prop}\label{suite} Il existe une suite g\'en\'erique $y_n=(x_{n,1},\dots ,x_{n,g})$ de points de $X_K^g(\overline{K})$ telle que : \par 1) Pour tout $i\in [1,\dots,g]$, $h_{\omega_a}(x_{n,i}) \longrightarrow 0.$ \par 2) La suite $z_n=s(y_n)$ est une suite g\'en\'erique de $J$ et $h_{NT}(z_n)$ converge vers 0 quand $n$ tend vers l'infini.. \par 3) L'application $s$ induit une bijection de $O(y_n)$ sur $O(z_n)$ \end{prop} {\it Preuve.} Dans la suite, on fixe un plongement $\sigma_0=id$ de $\overline{K}$ dans ${\mathbb C}$ et on identifie $X_K(\overline{K})$ \`a un sous--ensemble de $X_{{\mathbb C}}=X_K\otimes_{\sigma}{\mathbb C}$. Pour toute extension $L$ de $K$, on note $L^c$ la plus petite extension galoisienne de $K$ contenant $L$. On choisit une distance $d$ sur $X_{{\mathbb C}}$ d\'efinissant la topologie complexe. Comme cela a \'et\'e indiqu\'e au d\'ebut de cette section, on dispose d'une suite g\'en\'erique $t_k$ de points de $X_K(\overline{K})$ telle que $h_{\omega_a}(t_k)$ tend vers 0. Par le th\'eor\`eme d'\'equidistribution \ref{equi} et le lemme \ref{aprox} on sait que $\{ O(t_k) \ \vert k\in {\mathbb N}\} $ est dense pour la topologie complexe de $X_{\mathbb C} $. Pour tout $n\in {\mathbb N}$, il existe donc un indice $k\in {\mathbb N}$ tel que l'on ait \`a la fois : \begin{equation}\label{x1-1} h_{\omega_a}(t_k)\le \frac{1}{n} \end{equation} \begin{equation}\label{x1-2} \mbox{ pour tout $x\in X_{{\mathbb C}}({\mathbb C})$ il existe $\alpha\in O(t_k)$ tel que $d(x,\alpha) \le \frac{1}{n}$} \end{equation} On choisit un tel indice $k$ et on pose $x_{n,1}=t_k$. \medskip On pose alors $$ \mbox{Gal}({K(x_{n,1})^c}/K)=\{\overline{\sigma}_1,\dots,\overline{\sigma}_r \} $$ et $X_i=X_K\otimes_{\overline{\sigma}_i}K(x_{n,1})^c$. On note encore $\omega_a$ le faisceau dualisant relatif (au sens de Zhang) sur le mod\`ele minimal non singulier ${\cal X}_i$ de $X_{\overline{\sigma}_i}$ sur l'anneau des entiers $O_{K(x_{n,1})^c}$ de $K(x_{n,1})^c$. On dispose par la th\'eorie de Galois \'el\'ementaire d'un morphisme surjectif: $$ \pi_k \ :\ \mbox{Gal}({K(x_{n,1},t_k)^c}/K)\ \longrightarrow \mbox{Gal}({K(x_{n,1})^c}/K). $$ Pour tout $i\in [1,\dots,r]$ et tout $k\in{\mathbb N}$, on choisit $\sigma_{i,k}\in \mbox{Gal}({K(x_{n,1},t_k)^c}/K)$ tel que $$ \pi_k(\sigma_{i,k})=\overline{\sigma}_i. $$ On constate que les $\sigma_{i,k}(t_k)$ pour $i\in [1,\dots,r]$ sont des suites g\'en\'eriques de points de $X_i(\overline{K})$ telles que $h_{\omega_a}(\sigma_{i,k}(t_k))$ tend vers 0. On en d\'eduit que pour tout $k$ assez grand, pour tout $i\in [1,\dots, r] $ et pour tout $x\in X_{{\mathbb C}}$ on a : \begin{equation}\label{x2-1} h_{\omega_a}(\sigma_{i,k}(t_k)) \le \frac{1}{n}. \end{equation} \begin{equation}\label{x2-2} \mbox{ Il existe $ \alpha_i\in O_{K(x_{n,1})^c}(\sigma_{i,k}(t_k))$ tel que } d(x,\alpha_i)\le \frac{1}{n} \end{equation} \begin{equation}\label{x2-3} \mbox{dim }H^0(X_{{\mathbb C}},{\cal O}(x_{n,1}+t_k))=1 \end{equation} (Remarquer pour ce dernier point que la suite $t_k$ est g\'en\'erique et que $\{ P\in X({\mathbb C}) \ \vert \ \mbox{dim}\ H^0(X_{{\mathbb C}},{\cal O}(x_{n,1}+P)) >1 \}$ est fini). On fixe un tel $k$ et on pose $x_{n,2}=t_k$ et $\sigma_{i,k}=\sigma_i$. La suite $(x_{n,1},x_{n,2})$ a la propri\'et\'e suivante : \begin{lem} Pour tout $(x,y)\in X_{{\mathbb C}}\times X_{{\mathbb C}}$, il existe $(\alpha_1,\alpha_2) \in O(x_{n,1},x_{n,2})$ tel que $\max(d(x,\alpha_1),d(y,\alpha_2))\le \frac{1}{n}.$ \end{lem} {\it Preuve}. D'apr\`es (\ref{x1-2}), il existe $\overline{\sigma}_i\in \mbox{Gal}({K(x_{n,1})^c}/K)$ tel que $d(x,\overline{\sigma}_i(x_{n,1}))\le \frac{1}{n}$. Par ailleurs d'apr\`es (\ref{x2-2}) il existe $\gamma \in \mbox{Gal}({K(x_{n,1},x_{n,2})^c}/K(x_{n,1})^c)$ tel que $$ d(y,\gamma \sigma_i(x_{n,2}))\le \frac{1}{n}. $$ On constate que $(\alpha_1,\alpha_2)= \gamma \sigma_i ((x_{n,1},x_{n,2}))$ convient. \medskip En proc\'edant de m\^eme, on construit la suite $y_n=(x_{n,1},\dots,x_{n,g})$ de $X_K^g$ telle que $$ \mbox{dim } H^0(X_{{\mathbb C}},{\cal O}(x_{n,1}+\dots+x_{n,g}))=1 $$ (utiliser le fait que $t_k$ est g\'en\'erique et \cite{Mi} lemme 5-2), $h_{\omega_a}(x_{n,i})$ converge vers 0 pour tout $i$ et telle que pour tout $(x_1,\dots,x_g)\in X_{{\mathbb C}}^g$, il existe $$ (\alpha_1,\dots,\alpha_g)\in O(x_{n,1},\dots,x_{n,g}) $$ v\'erifiant $\displaystyle\max_i( d(x_i,\alpha_i))\le \frac{1}{n}$. Cette derniere propri\'et\'e nous prouve que la suite $y_n$ est g\'en\'erique. La deuxi\`eme partie de la proposition se d\'eduit de la premi\`ere et du fait que la suite $y_n$ est g\'en\'erique en utilisant le lemme \ref{NT-ARAK} et le fait que $s$ est surjective. Comme $D_0$ est rationnel sur $K$, $s$ induit une surjection de $O(y_n)$ sur $O(z_n)$. Soit $(\alpha_1,\dots,\alpha_g)\in O(y_n)$ tel que $$ z_n=x_{n,1}+\dots+x_{n,g}=\alpha_1+\dots+\alpha_g $$ On sait que les fibres du morphisme $s$ correspondent aux syst\`emes lin\'eaires sur $X$ (voir \cite{Mi} chap\^\i tre 5 dans le cas o\`u $D_0$ est un point rationnel sur $K$ et se ramener \`a ce cas apr\`es translation). Comme $$ \mbox{dim } H^0(X_{{\mathbb C}},{\cal O}(x_{n,1}+\dots+x_{n,g}))=1, $$ on voit que $s$ est fini au dessus de $z_n$. Pour $n$ assez grand la construction de la suite prouve que $x_{n,i}$ ne peut pas \^etre dans $O(x_{n,j})$ pour $i\neq j$. On a donc $\alpha_i=x_{n,i}$ pour tout $i$ et $s$ est injective. Ceci termine la preuve de la derni\`ere partie de la proposition. \section{Preuve du th\'eor\`eme \ref{teo1}} Dans la suite, on notera $\pi_i$ la $i$--\`eme projection de ${\cal X}^g= {\cal X}\times_{O_K}\dots\times_{O_K}{\cal X}$ sur ${\cal X}$. On notera aussi $\pi_i$ la $i$--\`eme projection de $X_K^g$ sur $X_K$. On rappelle que $\nu$ d\'enote la $1$-$1$--forme canonique sur $X_{{\mathbb C}}$. \begin{lem}\label{sxg} Soit $y_n$ la suite pr\'ec\'edemment construite. La suite $$ \frac{1}{\#(O(y_n))}\sum_{(\alpha_1,\dots,\alpha_g)\in O(y_n)} \delta_{(\alpha_1,\dots,\alpha_g)} $$ converge faiblement vers la mesure $\pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu$ de $X_{{\mathbb C}}^g$. \end{lem} {\it Preuve}. On note $\overline{L}=\pi_1^*\omega_{{\cal X}/\OK}\otimes\dots\otimes\pi_g^*\omega_{{\cal X}/\OK}$ le fibr\'e inversible m\'etris\'e sur ${\cal X}^g$ obtenu en munissant $L_K=\pi_1^*\Omega^1_X\otimes\dots\otimes\pi_g^*\Omega^1_X$, en toute place \`a l'infini, de la m\'etrique produit des images inverses des m\'etriques canoniques sur $\Omega^1_X$. On note $h$ le repr\'esentant de la classe des hauteurs Weil, sur $X_K^g(\overline{K})$ associ\'e \`a $L_K$ v\'erifiant : $$ h(P_1,\dots,P_g)= \sum_{i=1}^g h_{\omega_a} (P_i) $$ Pour tout $(P_1,\dots,P_g)\in X_K^g(\overline{K})$. On a $h(P_1,\dots,P_g)\ge 0$. On fixe dans la suite une suite d'extension $K_n$ de $K$ et une suite ${\cal \overline{L}}_n$ sur le mod\`ele minimal r\'egulier ${\cal X}_n$ de $X_{K_n}$ donn\'ees par le lemme \ref{aprox}. On pose alors $$ {\cal X}^n_g ={\cal X}_n\otimes_{O_{K_n}} \dots\otimes_{O_{K_n}} {\cal X}_n $$ $$ \overline{L}_n=\pi_1^*{\cal \overline{L}}_n\otimes\dots\otimes\pi_g^*{\cal \overline{L}}_n $$ On constate alors que $\overline{L}_n$ co\"\i ncide g\'en\'eriquement (en tant que fibr\'e inversible hermitien) avec $\overline{L}_{K_n}$ et que l'on a pour tout $(x_1,\dots,x_g)\in X_K(\overline{K})^g$ : $$ h_{\overline{L}_n}(x_1,\dots,x_g)=\sum_{i=1}^g h_{{\cal \overline{L}}n}(x_i). $$ On en d\'eduit alors que $$ \sup_{x\in X_K^g(\overline{K})} \vert h_{\overline{L}_n}(x)-h(x)\vert $$ tend vers $0$ quand $n$ tend vers l'infini. Or $L_K$ est ample sur $X_K$, en toute place \`a l'infini $\sigma$ de $K$, $c_1(\overline{L})_{\sigma}$ est une $1$-$1$--forme positive et on a construit une suite g\'en\'erique $y_n$ telle que $h(y_n)$ tend vers 0. Le th\'eor\`eme d'\'equidistribution \ref{equi} nous permet alors de conclure que la suite $$ \frac{1}{\#(O(y_n))}\sum_{(\alpha_1,\dots,\alpha_g)\in O(y_n)} \delta_{(\alpha_1,\dots,\alpha_g)} $$ converge faiblement vers la mesure $$ \frac{c_1(\overline{L})^g}{c_1(L_K)^g}= \pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu. $$ \begin{lem}\label{Haar} Soit $z_n$ la suite de points de $J$ pr\'ec\'edemment construite. La suite $$ \frac{1}{\#(O(z_n))}\sum_{z\in O(z_n)} \delta_z $$ converge faiblement vers la mesure de Haar normalis\'ee $d\mu_{H}$ de $J$. \end{lem} {\it Preuve}. C'est une cons\'equence imm\'ediate de la deuxi\`eme partie de la proposition \ref{suite}, du th\'eor\`eme d'\'equidistribution \ref{teo2} des petits points des vari\'et\'es ab\'eliennes et du lemme \ref{NT-ARAK}. \begin{lem}\label{eg-mes} Pour toute fonction $f$, continue sur $J$, \`a valeurs dans ${\mathbb R}$, on a: \begin{equation} \int_J f(z) d\mu_H =\int_{X_{{\mathbb C}}^g} f. s(x_1,\dots,x_g) \pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu. \end{equation} \end{lem} {\it Preuve}. En utilisant le lemme \ref{Haar}, on voit que la suite $$ \frac{1}{\#(O(z_n))}\sum_{z\in O(z_n)} f(z) $$ converge vers $\displaystyle\int_J f(z) d\mu_H$. Or d'apr\`es la troisi\`eme partie de la proposition \ref{suite} on a : \begin{equation} \frac{1}{\#(O(z_n))}\sum_{z\in O(z_n)} f(z) =\frac{1}{\#(O(y_n))}\sum_{(\alpha_1,\dots,\alpha_g)\in O(y_n)} f. s(\alpha_1,\dots,\alpha_g). \end{equation} D'apr\`es le lemme \ref{sxg}, cette derni\`ere somme converge vers $$ \int_{X_{{\mathbb C}}^g} f. s(x_1,\dots,x_g) \pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu. $$ \begin{lem}\label{contradiction} On \`a l'\'egalit\'e : \begin{equation} s^*d\mu_H=g!\ \pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu. \end{equation} \end{lem} {\it Preuve}. Cela r\'esulte du lemme \ref{eg-mes} et de la formule de changement de variables quand on a remarqu\'e que $s$ est une application g\'en\'eriquement finie de degr\'e $g!$. {\it Preuve du th\'eor\`eme \ref{teo1}}. On va prouver le th\'eor\`eme \ref{teo1} en montrant que cette derni\`ere \'egalit\'e ne peut \^etre r\'ealis\'ee. Pour cela explicitons ces deux mesures. Soit $(\omega_1,\dots,\omega_g)$ une base orthonorm\'ee de $H^0(X_{{\mathbb C}},\Omega^1_X)$ pour le produit scalaire d\'efini dans l'\'equation \ref{prodscal}. On note $\omega^J_i$ l'unique forme diff\'erentielle de type $1$-$0$ sur $J$, invariante par translation, telle que $j^*\omega^J_i=\omega_i$. On a vu que l'on a : $$ \nu=\frac{i}{2g}\sum_{i=1}^g \omega_i\wedge\overline{\omega_i} $$ On pose alors $$ \mu=\frac{i}{2g}\sum_{i=1}^g \omega^J_i\wedge\overline{\omega_i}^J. $$ On a alors $d\mu_H=\frac{g^g}{g!}\mu^g$ et $\nu=j^*\mu$ et pour tout $i\in \{1,\dots,g \}$ on a : $$ s^*\omega_i^J=\sum_{j=1}^{g} \pi_j^*\omega_i $$ On trouve alors \begin{equation} s^*d\mu_H=\frac{g^g}{g!}(\sum_{i=1}^g \pi_i^*\omega_1)\wedge (\sum_{i=1}^g \pi_i^*\overline{\omega_1}) \wedge\dots\wedge (\sum_{i=1}^g \pi_i^*\omega_g)\wedge (\sum_{i=1}^g \pi_i^*\overline{\omega_g}) \end{equation} On voit donc que pour tout point $P=(P_1,\dots,P_g)\in X^g$ telle que $$ H^0(X_{{\mathbb C}},\Omega^1_X(-P_1-\dots-P_g))>0 $$ (par exemple $P=(P_0,\dots,P_0)$ avec $P_0$ point de Weierstrass de $X_{{\mathbb C}}$) on a $s^*d\mu_H(P)=0$ . Par contre la forme de type $g$-$g$, $$ \pi_1^*\nu\wedge\dots\wedge \pi_g^*\nu $$ est partout strictement positive. Ceci prouve la contradiction du lemme \ref{contradiction} et termine la preuve du th\'eor\`eme.

9606

alg-geom/9606009

en

https://arxiv.org/abs/alg-geom/9606009

alg-geom/9606009

Francisco Jose Plaza Martin

A. \'Alvarez V\'azquez, J. M. Mu\~noz Porras, F. J. Plaza Mart\'in

The algebraic formalism of soliton equations over arbitrary base fields

Minor changes in Section 5 and References

Variedades Abelianas y Funciones Theta, Ap. Mat. Serie Investigaci\'on No. 13, Sociedad Matem\'atica Mexicana, M\'exico 1998

The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassmannian of a $k$-vector space $V$ in such a way that its rational points are precisely the points of the Grassmannian defined by Segal-Wilson, although the points over an arbitrary $k$-scheme $S$ have been not previously considered. This definition of the functor $\fu{\gr}(V,V^+)$ allows us to prove that it is representable by a separated $k$-scheme $\gr(V,V^+)$. Using the theory of determinants of Knudsen and Mumford, the determinant bundle is constructed. This is one of the main results of the paper because it implies that we can define ``infinite determinants'' in a completely algebraic way.

alg-geom

\section{Introduction} The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles. As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. Recently G.~Anderson ([{\bf A}]) has constructed the infinite-dimensional Grassmannians and $\tau$-functions over $p$-adic fields; his constructions are basically the same as in the Segal-Wilson paper ([{\bf SW}]) but he replaces the use of the theory of determinants of Fredholm operators over a Hilbert space by the theory of $p$-adic infinite determinants (Serre [{\bf S}]). Our point of view is completely different and the formalism used is valid for arbitrary base fields; for example, for global number fields or fields of positive characteristic. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassmannian of a $k$-vector space $V$ (with a fixed $k$-vector subspace $V^+\subseteq V$) in such a way that the points $\fu{\gr}(V,V^+)(\spk)$ are precisely the points of the Grassmannian defined by Segal-Wilson or Sato-Sato ([{\bf SW}], [{\bf SS}]) although the points over an arbitrary $k$-scheme $S$ have been not previously considered by other authors. This definition of the functor $\fu{\gr}(V,V^+)$, which is a sheaf in the category of $k$-schemes, allows us to prove that it is representable by a separated $k$-scheme $\gr(V,V^+)$. The universal property of the $k$-scheme $\gr(V,V^+)$ implies, as in finite-dimensional Grassmannians, the existence of a universal submodule, $\L_V$, of $\pi^*V$ ($\pi:\gr(V,V^+)\to\spk$ being the natural projection). These constructions allow us to use the theory of determinants of Knudsen and Mumford ([{\bf KM}]) to construct the determinant bundle over $\gr(V,V^+)$. This is one of the main results of the paper because it implies that we can define ``infinite determinants'' in a completely algebraic way. From this definition of the determinant bundle, we show in $\S 3$ that global sections of the dual determinant bundle can be computed in a very natural form. The construction of $\tau$-functions and Baker functions is based on the algebraic version, given in $\S4$, of the group $\Gamma$ of continuous maps $S^1\to{\Bbb C}^*$ defined by Segal-Wilson ([{\bf SW}]) which acts as a group of automorphisms of the Grassmannians. We replace the group $\Gamma$ by the representant of the following functor over the category of $k$-schemes $$S\rightsquigarrow H^0(S,\o_S)((z))^*= H^0(S,\o_S)[[z]][z^{-1}]^*$$ This is one of the points where our view differs essentially from other known expositions ([{\bf A}], [{\bf AD}], [{\bf SW}], [{\bf SS}]). Usually, the elements of $\Gamma$ are described as developments, of the type $f=\sum_{-\infty}^{+\infty}\lambda_i\,z^i$ ($\lambda_k\in{\Bbb C}$), but in the present formalism the elements of $\Gamma$ with values in a $k$-algebra $A$ are developments $f=\sum_{i\geq -N}^{}\lambda_i\,z^i\in A((z))$ such that $\lambda_{-1},\dots,\lambda_{-N}$ are nilpotent elements of $A$. In future papers we shall apply the formalism offered here to arithmetic problems (Drinfeld moduli schemes and reprocity laws) and shall give an algebraic formalism of the theory of KP-equations related to the characterization of Jacobians and Prym varieties. We also hope that this formalism might clarify the algebro-geometric aspects of conformal field theories over base fields different from $\Bbb R$ or $\Bbb C$ in the spirit of the paper of E.~Witten ([{\bf W}]). \section{Infinite Grassmannians\label{grass-section}} Let $V$ be a vector space over a field $k$. \begin{defn}{(Tate [{\bf T}])} Two vector spaces $A$ and $B$ of $V$ are commensurable if ${A+B}/{A\cap B}$ is a vector space over $k$ of finite dimension. We shall use the symbol $A\sim B$ to denote commensurable vector subspaces. \end{defn} Let us observe that commensurability is an equivalence relation between vector subspaces. The addition and intersection of two vector subspaces commensurable with a vector subspace $A$ is also commensurable with $A$. Let us fix a vector subspace $V^+\subseteq V$. The equivalence class of vector subspaces commensurable with $V^+$ allows one to define on $V$ a topology, which will be called $V^+$-topology: a basis of neighbourhoods of $0$ in this topology is the set of vector subspaces of $V$ commensurable with $V^+$. $V$ is a Haussdorff topological space with respect to the $V^+$-topology. \begin{defn} The completion of $V$ with respect to the $V^+$-topology is defined by: $$ \w V= \underset{A\sim V^+}{\varprojlim} ( V/A)$$ \end{defn} Analogously, given a vector subspace $B\subseteq V$ we define the completions of $B$ and $ V/B$ with respect to $B\cap V^+$ and ${B+V^+}/{B}$, respectively. The homomorphism of completion $V@>i>> \w V$ is injective and $V$ is said to be complete if $V@>i>> \w V$ is an isomorphism. \begin{exam} \begin{itemize} \item $(V, V^+=0)$; $V$ is complete. \item $V=k((t)),$ $V^+=k[[t]]$; $V$ is complete. \item Let $(X, \o_X)$ be a smooth, proper and irreducible curve over the field $k$, and let $V$ be the ring of adeles of the curve and $V^+=\underset p{\prod} \w {\o_p}$ ( $\o_p$ being the $\frak m_p$-adic completion of the local ring of $X$ in the point $p$); $V$ is complete with respect the $V^+$-topology. \end{itemize} \end{exam} \begin{prop} The following conditions are equivalent: \begin{enumerate} \item $V$ is complete. \item $V^+$ is complete. \item Each vector subspace commensurable with $V^+$ is complete. \end{enumerate} \end{prop} \begin{pf} This follows easily from the following commutative diagram for every $A\sim V^+$: $$ \CD 0 @>>> \widehat{A} @>>> \widehat{V} @>>> \widehat{ V/A} @>>> 0 \\ @. @A{i_A}AA @A{i_V}AA @A{\simeq}AA @. \\ 0 @>>> A @>>> V @>>> V/A @>>> 0 \endCD$$ \end{pf} \begin{defn} Given a $k$-scheme $S$ and a vector subspace $B\subseteq V$, we define: \begin{enumerate} \item $\w V_S= \lim ( V / A\underset k{\otimes}{\cal O}_S).$ \item $\widehat B_S=\lim ( {B} /{ A\cap B}) \underset k{\otimes}{\cal O}_S$. \item$\widehat{( V/B)}_S=\lim (( {V}/{A+B})\underset k{\otimes} {\cal O}_S)$. \end{enumerate} \end{defn} \begin{prop} $\w{ V_S}$ is a sheaf of $\o_S$-modules and given $B\sim V^+$, we have: $$\w{(V/B)_S}={\w{ V_S}}/{\w{ B_S}}=(V/B)\underset k{\otimes} \o_S$$ \end{prop} \begin{pf} This is an easy exercise of linear algebra. \end{pf} Let $V$ be a $k$-vector space and $V^+$ a vector subspace determining a class of commensurable vector subspaces. \begin{defn} A discrete vector subspace of $V$ is a vector subspace, $L\subseteq V$, such that $L\cap V^+$ and ${V}/{L+V^+}$ are $k$-vector spaces of finite dimension. \end{defn} We aim to define a Grassmannian scheme $\gr(V,V^+)$, defining its functor of points $\fu{\gr}(V,V^+)$ and proving that it is representable in the category of $k$-schemes. If $V$ is complete, the rational points of our Grassmannian will be precisely the discrete vector spaces of $V$; that is, $\fu{\gr}(V,V^+)(\spk)$ as a set coincides with the usual infinite Grassmannian defined by Pressley and Segal [{\bf PS}] or M. and Y. Sato [{\bf SS}]. \begin{defn} Given a $k$-scheme $S$, a discrete submodule of $\w{V_S}$ is a sheaf of quasi-coherent $\o_S$-submodules $\L \subset \w{V_S}$ such that $\L_{S'}\subset \w{V_{S'}}$ for every morphism $S'\to S$ and for each $s\in S$, $\L\underset{\o_S}\otimes k(s)\subset \w{V_S}\underset{\o_S}\otimes k(s)$ and there exists an open neighbourhood $U_s$ of $s$ and a commensurable $k$-vector subspace $B\sim V^+$ such that: $\L_{U_s}\cap \w{B_{U_s}}$ is free of finite type and ${\w{V_{U_s}}}/{\L_{U_s}+ \w{B_{U_s}}}=0.$ \end{defn} \begin{prop} \label{prop-loc-free} With the notations of the above definition, given another commensurable $k$-vector space, $B'\sim V^+$, such that $B\subseteq B'$, $\L_{U_s}\cap \w{{B'}_{U_s}}$ is locally free of finite type. \end{prop} \begin{pf} This follows easily from the commutative diagram $$ \CD 0 @>>> \w {B_{U_s}} @>>> \w {{B'}_{U_s}} @>>> (( B'/{B}) \underset k{\otimes} {\cal O}_{U_s})= {\w {B'_{U_s}}}/{\w {B_{U_s}}} @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> {\w {V_{U_s}}}/{\L _{U_s}} @>>> {\w {V_{U_s}}}/{\L _{U_s}} @>>> 0 @>>> 0 \endCD$$ using the snake Lemma. \end{pf} \begin{defn} Given a $k$-vector space $V$ and $V^+\subseteq V$, the Grassmannian functor, $\fu{\gr}(V,V^+)$, is the contravariant functor over the category of $k$-schemes defined by $$\fu{\gr}(V,V^+)(S)=\left\{ \begin{gathered} \text{discrete sub-$\o_S$-modules of $\w{V_S}$} \\ \text{with respect the $V^+$-topology} \end{gathered} \right\}$$ \end{defn} \begin{rem} Note that if $V$ is a finite dimensional $k$-vector space and $V^+=(0)$, then $\fu{\gr}(V,(0))$ is the usual Grassmannian functor defined by Grothendieck [{\bf EGA}]. \end{rem} \begin{defn} Given a commensurable vector subspace $A\sim V^+$, the functor $\fu{F_A}$ over the category of $k$-schemes is defined by: $$\fu{F_A}(S)=\{\text{ sub-$\o_S$-modules $\L\subset \w {V_S}$ such that $\L\oplus \w {A_S}=\w {V_S}$ }\}$$ (That is: $ \L \cap \w {A_S}=(0)$ and $ \L + \w {A_S}= \w {V_S}$). \end{defn} \begin{lem}\label{Frepre} For every commensurable subspace $B\sim V^+$, the contravariant functor $\fu{F_B}$ is representable by an affine and integral $k$-scheme $F_B$. \end{lem} \begin{pf} Let $L_0$ be a discrete $k$-subspace of $V$ such that $L_0\oplus B = V$; we then have: $$ F_B(S)=\fu {Hom}_{\o_S}((\L_0)_S,\w B_S) =\lim \fu{Hom}_{\o_S}((\L_0)_S,{ B}/{B\cap A}\underset k{\otimes}\o_S))$$ If we denote by $F_{ B/{{B\cap A}}}(S)$ the set $\fu{ Hom}_{\o_S}((\L_0)_S, B/{B\cap A}{\underset k{\otimes}{\cal O}_S)})$, it is obvious that the functor $F_{ B/{{B\cap A}}}(S)$ is representable by an affine and integral $k$-scheme since $ B/{{B\cap A}}$ is a finite dimensional $k$-vector space. But $\fu{F_B}$ is now a projective limit of functors representable by affine schemes, so we conclude that $\fu{F_B}$ is representable by an affine $k$-scheme. \end{pf} \begin{lem}\label{anterior} Let $\L$ be an element in $\fu{\gr}_{V^+}(V)(S)$ and $A$ and $B$ are two $k$-subspaces of $V$ commensurable with $V^+$. It holds that: \item{a)} if ${\w V_{S}} /{\L+ \w A_{S}} =0$, then $\L\cap \w A_{S}$ is a finite type locally free of $\o_S$-module. \item{b)} ${\w V_{S}} /{\L+ \w B_{S}}$ is an $\o_S$-module locally of finite presentation. \end{lem} \begin{pf} \item{a)} By proposition {\ref{prop-loc-free}}, for each point $s\in S$ there exists an open neighbourhood $U_s$ and a commensurable $k$ subspace $A'\sim V^+$ such that: $A\subseteq A'$, $ { \w V_{U_s}}/{\L_{U_s}+ \w A'_{U_s}} =0$ and $\L_{U_s}\cap \w A'_{U_s}$ is free of finite type. From the exact sequence: $$0\to \L\cap \w A_{S} \to \L \to ({\w V_S}/{\w A_{S}})= ( V/A)_S\to 0$$ one deduces that $\L\cap \w A_{S}$ is quasicoherent and $$0\to (\L\cap \w A_{S})_{U_s} \to \L_{U_s} \to ({\w V_{U_s}}/{\w A_{U_s}})=( V/A)_{U_s} \to 0$$ Let us consider the commutative diagram: $$\CD 0 @>>> \L\oplus \w {A_S} @>>> \L\oplus \w {{A'}_S } @>>> ( {A'}/ A)_S @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> \w{V_S} @>>> \w {V_S} @>>> 0 @>>> 0 \endCD$$ By using the snake lemma we have an exact sequence: $$0\to (\L_{S}\cap \w A_{S}) \to \L_{S} \cap \w A'_{S} \to ( A'/A)_{S} \to {\w{V}_{S}} /{{\L_{S}}+ \w{ A }_{S}}\to {\w{ V}_{S}}/{{\L_{S}}+ \w{ A'}_{S}}\to 0$$ In our conditions for $A$ and $A'$ we have: $$0\to (\L_{U_s}\cap\w A_{U_s}) \to \L_{U_s} \cap \w A'_{U_s} \to ( A'/A)_{U_s} \to 0$$ Then, $(\L\cap \w A_{S})_{U_s}= \L_{U_s}\cap\w A_{U_s}$ is the kernel of a surjective homomorphism between two free $\o_{U_s} $-modules of finite type, and we conclude the proof. \item{b)} For a given $s\in S$, let us take $B\subseteq B'$ such that $B'\sim V^+$ is a commensurable subspace, $\L_{U_s} \cap \w{B'}_{U_s}$ is free of finite type and ${\w{ V}_{U_s}} /{{\L_{U_s}}+ \w{ B'}_{U_s}}=0 $. We then have the exact sequence: $$ \L_{U_s} \cap \w{B'}_{U_s}\to B'/B \underset k\otimes \o_{U_s} \to {\w{ V}_{U_s}} /{{\L_{U_s}}+ \w{ B }_{U_s}} \to 0 $$ and so we conclude. \end{pf} \begin{thm} The functor $\fu{\gr}(V,V^+)$ is representable by a $k$-scheme $\gr(V,V^+)$. \end{thm} \begin{pf} The proof is modeled on the Grothendieck construction of finite Grassmannians [{\bf EGA}]; that is: It is sufficient to prove that $\{\fu{F_A}, A\sim V^+\}$ is a covering of $\fu{\gr}(V,V^+)$ by open subfunctors: \item {1)}{\sl For every $A\sim V^+$, the morphism of functors $\fu{F_A}\to \fu{\gr}(V,V^+)$ is representable by an open immersion}: \newline That is, given morphism of functors $X^{\punt}\to \fu{\gr}(V,V^+)$ (where $X$ is a $k$-scheme), the functor $$X^\punt \underset{\fu{\gr}(V,V^+)}\times \fu{F_A} \hookrightarrow X^\punt $$ is represented by an open subscheme of $X$. This is equivalent to proving that given $\L \in \fu{\gr}(V,V^+)(X)$, the set : $${\cal U}(A,\L)=\{x\in X\quad\text{ such that }\quad\L_x= \L\underset{\o_X}\otimes k(x)\in F_A(\sp(k(x)))\}$$ is open in $X$. \newline If $\L_x \in F_A(\sp(k(x))$, then: $$ {\w V_{k(x)}}/{{\L_x} + \w A_{k(x)}} =0$$ but ${\w V_{X}}/{\L + \w {A_X}}$ is a $\o_X$-module of finite presentation and, applying the lemma of Nakayama, there exists an open neighbourhood $U_x$ of $x$, such that $${\w V_{U_x}}/{\L_{U_x}+\w A_{U_x}}=0$$ \newline By lemma {\ref{anterior}}, $\L_{U_x}\cap \w {A_{U_x}}$ is a $\o_{U_x}$-module coherent. However, bearing in mind that $\L_x \cap \w {A_{k(x)} }=0$, there exists another open neighbourhood of $x$, $U'_x\subseteq U_x$, such that $\L_{U'_x}\cap \w {A_{U'_x}}=0$ and therefore $\L_{U'_x}\in F_A({U'}_x)$. \item{2)} {\sl For every $k$-scheme $X$ and every morphism of functors $$X^\punt\to\fu{\gr}(V,V^+)$$ the open subschemes $\{ {\cal U}(A,\L),\, A\sim V^+\}$ defined above are a covering of $X$.} \newline That is, given $\L \in \fu{\gr}(V,V^+)(X)$ and a point $x\in X$, there exists an open neighbourhood, $U_x$, of $x$ and a commensurable subspace $A\sim V^+$ such that: $$\L_{{U }_x}\in F_A({U }_x)$$ Let $A$ be a commensurable subspace such that: $$ \L_x \cap \w {A_{k(x)}} =0$$ since $ {\w V_{k(x)}}/{\L_x+ \w{ A_{k(x)}}}$ is a $k(x)$-vector space of finite dimension, we can choose a basis $(\overline {e_1 \otimes 1}, \dots,\overline {e_k\otimes 1})$ of $ {\w {V_{k(x)}}}/{ \L _x+ \w{ A_{k(x)}}}$ where $e_i\in V$. Defining $$B=A+\langle e_1,\dots,e_k\rangle$$ obviously $B\sim V^+$. One can easily prove that there exists an open subset ${U'}_x \subseteq U$ such that $\L_{{U'}_x} \in F_B({U'}_x)$ and this completes the proof of the theorem. \end{pf} \begin{lem}\label{anterior2} Let $A$, $B$ be two $k$-vector spaces of $V$ commensurable with $V^+$. A necessary and sufficient condition for the existence of $\L\in\fu{\gr}(V,V^+)(S)$ such that $\L\oplus\w{A_S}=\L\oplus\w{B_S}=\w{V_S}$, is that there should exist an isomorphism of $k$-vector spaces $$\tau: B/{A\cap B}\iso A /{A\cap B}$$ \end{lem} \begin{pf} Let us consider the decomposition: $$ \w {V_S}= \w {(A\cap B)_S}\oplus \left( B /{A\cap B}\right)_S\oplus \left( { A /{A\cap B}}\right)_S\oplus \left( V/ {A+B}\right)_S$$ If the isomorphism $\tau$ exists, we take: $$\L=\left\{(a,b,\tau(b)), \quad a \in \left( V/ {A+B}\right)_S ,\quad b\in \left( B/ {B\cap A}\right)_S \right\}$$ Conversely, assume that $\L\oplus\w{A_S}=\L\oplus\w {B_S}=\w{V_S}$. We then have: $$\L \oplus \left( B /{B\cap A}\right)_S\oplus \left(\w{B\cap A}\right)_S\simeq \L \oplus \w {B _S}\simeq \L\oplus \w{ A _S}\simeq \L \oplus \left( A/ {A\cap B}\right)_S \oplus \left(\w {B\cap A}\right)_S$$ from which we deduce that: $$\left( B /{A\cap B}\right)_S\simeq \left( A/{A\cap B}\right)_S$$ \end{pf} \begin{thm} $\gr(V,V^+)$ is a separated scheme. \end{thm} \begin{pf} Let $F_B=\sp(A_B) =({\cal U}_B) $ be the affine open subschemes of the Grassmannian constructed in lemma {\ref{Frepre}}. It suffices to prove that given two commensurable subspaces $B'$ and $B$ such that $F_B\cap F_{B'}\neq\emptyset$ then $F_B\cap F_{B'}$ is affine. By lemma {\ref{anterior2}} $$ F_B\cap F_{B'} \neq \emptyset$$ implies the existence of $\L_0\in{F_B}(\spk)\cap F_{B'}(\spk)$ and bearing in mind that $$ \fu{F_B} \cap\fu{F_{B'}} = \fu{F_B} \underset{\fu{F_{B+B'}}}\times\fu{F_{B'}} $$ we conclude the proof. \end{pf} \begin{defn} The discrete submodule corresponding to the identity $$Id\in \fu{\gr} (V,V^+)\left(\gr(V,V^+)\right)$$ will be called the universal module and will be denoted by $$\L_V \subset \w V_{\gr(V,V^+) }$$ \end{defn} \begin{rem} In this section we have constructed infinite-dimensional Grassmannian schemes in an abstract way. Since we select particular vector spaces $(V,V^+)$ we obtain different classes of Grassmannians. Two examples are relevant: \begin{enumerate} \item $V=k((t))$, $V^+=k[[t]]$. In this case, $\gr(k((t)),k[[t]])$ is the algebraic version of the Grassmannian constructed by Pressley, Segal, and M. and Y. Sato ([{\bf PS}], [{\bf SS}]) and this Grassmannian is particularly suitable for studying problems related to the moduli of curves (over arbitrary fields) and KP-equations. \item Let $(X,\o_X)$ be a smooth, proper and irreducible curve over the field $k$ and let $V$ be the adeles ring over the curve and $V^+=\underset p\prod \w{\o_p}$ (Example 1.3.3). In this case $\gr(V,V^+)$ is an adelic Grassmannian which will be useful for studying arithmetic problems over the curve $X$ or problems related to the classification of vector bundles over a curve (non abelian theta functions...). \newline Instead of adelic Grassmanians, we could define Grassmanians associated with a fixed divisor on $X$ in an analogous way. \newline These adelic Grassmanians will be also of interest in the study of conformal field theories over Riemann surfaces in the sense of Witten ([{\bf W}]). \end{enumerate} \end{rem} \begin{rem} From the universal properties satisfied by the Grassmannian one easily deduces the well known fact that given a geometric point $W\in \gr(V)(Spec(K))$ ($k\hookrightarrow K$ being an extension of fields), the Zariski tangent space to $\gr(V)$ at the point $W$ is the $K$-vector space: $$T_W \gr(V)=Hom(W, \w{V_K}/W)$$ \end{rem} \section{Determinant Bundles \label{det-bundles}} In this section we construct the determinant bundle over the Grassmannian following the idea of Knudsen and Mumford ([{\bf{KM}}]). This allow us to define determinants algebraically and over arbitrary fields (for example for $k=\Bbb Q$ or $k={\Bbb F}_q$). Let us set a pair of vector spaces, $V^+\subseteq V$. As in section {\ref{grass-section}}, we will denote the Grassmannian $\gr(V,V^+)$ simply by $\grv$. \begin{defn} For each $A\sim V^+$ and each $L\in\grv(S)$ we define a complex, $\c_A(L)$, of $\o_S$-modules by: $$\c_A(L) \equiv \dots\to 0\to L\oplus \hat A_S @>{\delta}>> \hat V_S\to 0\to\dots$$ $\delta$ being the addition homomorphism. \end{defn} \begin{thm} $\c_A(L)$ is a perfect complex of $\o_S$-modules. \end{thm} \begin{pf} We have to prove that the complex of $\c_A(L)$ is locally quasi-isomorphic to a bounded complex of free finitely-generated modules. Let us note that the homomorphism of complexes given by the diagram: $$\CD \dots @>>> 0 @>>> L\oplus \hat A_S @>{\delta}>>\hat V_S @>>> 0 @>>> \dots \\@. @. @V{p_1}VV @VVV @. @. \\ \dots@>>> 0 @>>> L @>{\phantom\delta}>> ( V/A)_S^{\hat{}} @>>> 0 @>>>\dots \endCD$$ ($p_1$ being the natural projection) is a quasi-isomorphism. The problem is local on $S$, and hence for each $s\in S$ we can assume the existence of an open neighbourhood, $U$, and a commensurable subspace $B\sim V^+$ such that $A\subseteq B$ and: $$ \hat V_U/{(L_U,\hat B_U)} = 0\qquad,\qquad L_U\cap \hat B_U \text{ is free and finitely-generated}$$ We then have the exact sequence: $$0 \to L_U\cap\hat A_U \to L_U\cap\hat B_U \to ( B/A)_U \to{\hat V_U}/{(L_U+\hat A_U)} \to 0$$ from which we deduce that the homomorphism of complexes given by the following diagram is a quasi-isomorphism: $$\CD \dots @>>> 0 @>>> L_U\cap\hat B_U @>>>( B/A)^{\hat{}}_U @>>> 0 @>>>\dots \\ @. @. @VVV @VVV @. @.\\ \dots @>>> 0 @>>> L_U @>>> ( V/A)^{\hat{}}_U @>>> 0 @>>>\dots \endCD$$ That is, $\c_A(L)\vert_U$ is quasi-isomorphic to the complex $0 \to L_U\cap\hat B_U \to ( B/A)_U \to 0$, which is a complex of free and finitely-generated modules. \end{pf} \begin{defn} The index of a point $L\in\grv(S)$ is the locally constant function $i_L\colon S\to\Z$ defined by: $$i_L(s)=\text{ Euler-Poincar\'e characteristic of } \c_{V^+}(L)\otimes k(s)$$ $k(s)$ being the residual field of the point $s\in S$. (For the definition of the Euler-Poincar\'e characteristic of a perfect complex see {\rm [{\bf{KM}}]}). \end{defn} \begin{rem} The following properties of the index are easy to verify: \begin{enumerate} \item Let $f:T\to S$ be a morphism of schemes and $L\in\grv(S)$; then: $i_{f^*L}=f^*(i_L)$. \item The function $i$ is constantly zero over the open subset $F_{V^+}$. \item If $B\sim V^+$, $V^+\subseteq B$ and ${\hat V_U}/{L_U+\hat B_U}=0$ over an open subscheme $U\subseteq S$, then $i_L(s)=\dim_{k(s)}(L_s\cap \hat B_s)-\dim_{k(s)}({B_s}/{V^+_s})$. \item If $V$ is a finite-dimensional $k$-vector space and $V^+=V$ and $L\in\grv(S)$, then $i_L=\operatorname{rank}(L)$. \item For any rational point $L\in\grv(\spk)$ one has: $$i_L=\dim_k(L\cap V^+)-\dim_k({\hat V}/{L+\hat V^+})$$ \end{enumerate} \end{rem} \begin{thm} Let $\gr^n(V)$ be the subset over which the index takes values equal to $n\in\Z$. $\gr^n(V)$ are open connected subschemes of $\grv$ and the decomposition of $\grv$ in connected components is: $$\grv=\underset{n\in\Z}\coprod\gr^n(V)$$ \end{thm} \begin{pf} This is obvious from the properties of the index. \end{pf} Given a point $L\in\grv(S)$ and $A\sim V^+$, we denote by $\det\c_A(L)$ the determinant sheaf of the perfect complex $\c_A(L)$ in the sense of [{\bf{KM}}]. \begin{thm} With the above notations the invertible sheaf over $S$, $\det\c_A(L)$, does not depend on $A$ (up to isomorphisms). \end{thm} \begin{pf} Let $A$ and $A'$ be two commen\-surable subspaces. It suffices to prove that: $$\det\c_A(L)\iso\det\c_{A'}(L)$$ in the case $A\subseteq A'$. In this case we have a diagram: $$\CD \dots@>>> 0@>>> \L\oplus\hat A_S @>{\delta}>>\hat V_S @>>>0@>>>\dots \\ @. @. @VVV @V{Id}VV @. @. \\ \dots@>>>0@>>>\L\oplus\hat A'_S @>{\delta}>>\hat V_S @>>>0@>>>\dots \endCD$$ and by the additivity of the functor $\det(-)$ we obtain: $$\det\c_A(L)\otimes\det( {A'}/A)_S\iso \det\c_{A'}(L)$$ However $ {A'}/A$ is free and we conclude the proof. \end{pf} \begin{defn} The determinant bundle over $\grv$, $\det_V$, is the invertible sheaf: $$\det\c_{V^+}(\L_V)$$ $\L_V$ being the universal submodule over $\grv$. \end{defn} \begin{prop}{(Functoriality)} Let $L\in\grv(S)$ be a point given by a morphism $f_L:S\to\grv$. There exists a functorial isomorphism: $$f^*_L\det_V\iso\det\c_A(L)$$ We shall denote this sheaf by $\det_V(L)$. \end{prop} \begin{pf} The functor $\det(-)$ is stable under base changes. \end{pf} \begin{rem} Let $L\in\grv(\sp(K))$ be a rational point and let $A\sim V^+$ such that $L\cap \hat A_K$ and ${\hat V_K}/{L+\hat A_K}$ are $K$-vector spaces of finite dimension. In this case we have an isomorphism: $$\det_V(L)\simeq \wedge^{max}(L\cap\hat A_K)\otimes\wedge^{max}( {\hat V_K}/{(L+\hat A_K)})^\ast$$ That is, our determinant coincides, over the geometric points, with the determinant bundles of Pressley, Segal, Wilson and M. and Y. Sato ([{\bf PS}], [{\bf SW}], [{\bf SS}]). \end{rem} We shall now state with precision the connection between the determinant bundle $\det_V$ and the determinant bundle over the finite Grassmannianns. Let $L,L'\in\grv(\spk)$ such that $L\subseteq L'$. In these conditions, ${L'}/L$ is a $k$-vector space of finite dimension. The natural projection $\pi:L'\to{L'}/L$ induces an injective morphism of functors: $$\fu\gr({L'}/L)\hookrightarrow\fu\gr(V)$$ defined by: $$j(M)=\pi^{-1}(M)\qquad\text{for each }M\in\fu\gr({L'}/L)(S)$$ We then have a morphism of schemes: $$j:\gr({L'}/L)\hookrightarrow\grv$$ It is not difficult to prove that $j$ is a closed immersion. \begin{thm} With the above notations, there exists a natural isomorphism: $$j^*\det_V\iso\det_{{L'}/L}$$ $\det_{{L'}/L}$ being the determinant bundle over the finite Grassmannian $\gr({L'}/L)$. \end{thm} \begin{pf} Let $\L^f$ be the universal submodule over $\gr({L'}/L)$. By definition $\det_{{L'}/L}=\det(\L^f\to{L'}/L)$, which is isomorphic to $\det(\pi^{-1}\L^f\to L')$. By the definition of $j$, one has $j^*\L_V\iso\pi^{-1}\L^f$ and hence: $$j^*\det_V\simeq\det(\pi^{-1}\L^f\oplus\hat V^+\to\hat V)$$ and from the exact sequence of complexes: $$\CD \pi^{-1}\L^f @>>> \pi^{-1}\L^f\oplus \hat V^+ @>>> \hat V^+ \\ @VVV @VVV @VVV \\ L' @>>> \hat V @>>> {\hat V}/{L'} \endCD$$ we deduce that $j^*\det_V\simeq\det_{{L'}/L}$. \end{pf} \begin{cor} Let $i$ be the index function over $\grv$. For each rational point $M\in\gr({L'}/L)$ one has: $$i(j(M))=i(L')+\dim_k({L'}/{M+L})$$ \end{cor} \begin{pf} Obvious. \end{pf} \subsection{Global sections of the determinant bundles and Pl\"ucker morphisms \label{det-section}} It is well known that the determinant bundle have no global sections. We shall therefore explicitly construct global sections of the dual of the determinant bundle over the connected component $\gr^0(V)$ of index zero. We use the following notations: $\wedge^\bullet E$ is the exterior algebra of a $k$-vector space $E$; $\wedge^r E$ its component of degree $r$, and $\wedge E$ is the component of higher degree when $E$ is finite-dimensional. Given a perfect complex $\c$ over $k$-scheme $X$, we shall write $\det^*\c$ to denote the dual of the invertible sheaf $\det\c$. To explain how global sections of the invertible sheaf $\det^*\c$ can be constructed, let us begin with a very simple example: \noindent Let $f:E\to F$ be a homomorphism between finite-dimensional $k$-vector spaces of equal dimension. This homomorphism induces: $$\wedge(f):\wedge E\to\wedge F$$ and $\wedge(f)\ne 0 \iff f$ is an isomorphism. $\wedge(f)$ can be expressed as a homomorphism: $$\wedge(f):k\to \wedge F\otimes (\wedge E)^*$$ Thus, if we consider $E@>{f}>> F$ as a perfect complex, $\c$, over $\spk$, we have defined a {\bf canonical section} $\wedge(f)\in H^0(\spk,\det^*\c)$. Let us now consider a perfect complex $\c\equiv(E@>{f}>>F)$ of sheaves of $\o_X$-modules over a $k$-scheme $X$, with Euler-Poincar\'e characteristic ${\cal X}(\c)=0$. Let $U$ be an open subscheme of $X$ over which $\c$ is quasi-isomorphic to a complex of finitely-generated free modules. By the above argument, we construct a canonical section $det(f\vert_U)\in H^0(U,\det^*\c)$ and for other open subset, $V$, there is a canonical isomorphism $det(f\vert_U)\vert_{U\cap V}\simeq det(f\vert_V)\vert_{U\cap V}$ and we therefore have a canonical section $\det(f)\in H^0(X,\det^*\c)$. If the complex $\c$ is acyclic, one has an isomorphism: $$\aligned \o_X &\iso\det^\ast\c\\ 1 &\mapsto det(f) \endaligned$$ (for details see [{\bf KM}]). Let $0\to H^\bullet\to\c_1\to\c_2\to 0$ be an exact sequence of perfect complexes. There exists a functorial isomorphism $$\det^\ast\c_1\iso \det^\ast H^\bullet\otimes_{\o_X}\det^\ast\c_2$$ If $H^\bullet$ is acyclic, we obtain an isomorphism $$ H^0(X,\det^\ast\c_2)\iso H^0(X,\det^\ast\c_1)$$ In the case $H^\bullet\equiv (E @>{Id}>> E)$, $\c_1\equiv (V@>{f}>>V)$, $\c_2\equiv (F @>{f'}>>F)$, and ${\cal X}(\c_i)=0$ ($i=1,2$), we obtain the following commutative diagram: $$\CD \o_X @>{\sim}>>\det^\ast H^\bullet\otimes\o_X \simeq\o_X \\@VVV @VVV \\ \det^\ast\c_1 @>{\sim}>> \det^\ast H^\bullet\otimes\det^\ast\c_2\simeq\det^\ast\c_2 \endCD$$ from which we deduce that $det(f)=det(f')$. Moreover, if $F$ is locally free of finite rank, this means that computation of $det(f)$ is reduced to computation of $det(f')$, as mentioned above. Let $V$ be a $k$-vector space and $V^+\subseteq V$ and $A\sim V^+$ a commensurable vector subspace. Let us consider the perfect complex $\c_A\equiv(\L\oplus \hat A@>{\delta_A}>>\hat V)$ over $\grv$ defined in {\ref{det-bundles}} ($\L$ being the universal discrete submodule over $\grv$). \begin{lem} $F_A\subseteq\gr^0(V)$ if and only if $$\dim_k( A/{A\cap V^+})-\dim_k({V^+}/{A\cap V^+})=0$$ \end{lem} \begin{pf} Obvious. \end{pf} \begin{cor} The open subschemes $F_A$ with $\dim_k( A/{A\cap V^+})-\dim_k({V^+}/{A\cap V^+})=0$ are a covering of $\gr^0(V)$. Given $A,B\sim V^+$ under the assumption $F_A,F_B\subseteq\gr^0(V)$ one has $\dim_k( A/{A\cap B})-\dim_k( B/{A\cap B})=0$. \end{cor} \begin{pf} Obvious. \end{pf} Given $A\sim V^+$ with $F_A\subseteq \gr^0(V)$, let us note that $\c_A\vert_{F_A}$ is an acyclic complex. One then has an isomorphism: $$\aligned \o_X\vert_{F_A}&@>\sim>> \det^\ast\c_A\vert_{F_A}\\ 1 &\mapsto s_A=det(\delta_A)\vert_{F_A} \endaligned$$ We shall prove that the section $s_A\in H^0(F_A,\det^*\c_A)$ can be extended in a canonical way to a global section of $\det^*\c_A$ over the Grassmannian $\gr^0(V)$. Let $B\sim V^+$ be such that $F_B\subseteq\gr^0(V)$ and let us consider the complex: $$\c_{AB}\equiv(\L\oplus \hat A@>{\delta_{AB}}>> \L\oplus \hat B)$$ where $\delta_{AB}=\delta_B^{-1}\circ\delta_A$. Obviously $\delta_{AB}\vert_{(0,\hat A\cap\hat B)}= Id_{\hat A\cap\hat B}$ and $\delta_{AB}\vert_{(\L,0)}= Id_\L$, we then have an exact sequence of complexes: $$\CD @. @. @. A/{A\cap B} \\ @. @. @. @V{\simeq}VV \\ 0@>>>\L\oplus(\hat A\cap\hat B) @>>>\L\oplus\hat A @>>> {(\L\oplus\hat A)}/{\L\oplus(\hat A\cap\hat B)}@>>>0 \\ @. @V{Id}VV @V{\delta_{AB}}VV @V{\phi_{AB}}VV @. \\ 0 @>>>\L\oplus(\hat A\cap\hat B) @>>>\L\oplus\hat B @>>> {(\L\oplus\hat B)}/{\L\oplus(\hat A\cap\hat B)}@>>> 0 \\ @. @. @. @V{\simeq}VV \\ @. @. @. B/{A\cap B} \\ \endCD$$ and from the discussion at the beginning of this section we have that $det(\phi_{AB})=det(\delta_{AB})\in H^0(F_B,\det^\ast\c_{AB})$ and $det(\delta_{AB})$ satisfies the cocycle condition: $$\aligned &det(\delta_{AA})=1 \\ &det(\delta_{AB})\cdot det(\delta_{BC}) = det(\delta_{AC}) \qquad\text{over $F_B\cap F_C$ for any }C\sim V^+ \endaligned$$ Over $F_A\cap F_B$ we have canonical isomorphisms: $$\aligned \o_{F_A\cap F_B}&\iso \det^\ast\c_A\vert_{F_A\cap F_B} \\ 1&\mapsto s_A \endaligned$$ $$\aligned \o_{F_A\cap F_B}&\iso \det^\ast\c_B\vert_{F_A\cap F_B} \\ 1&\mapsto s_B \endaligned$$ $$\aligned \o_{F_A\cap F_B}&\iso \det^\ast\c_{AB}\vert_{F_A\cap F_B} \\ 1&\mapsto det(\delta_{AB}) \endaligned$$ which are compatible, therefore: $$(s_B\cdot det(\delta_{AB}))\vert_{F_A\cap F_B}= s_A\vert_{F_A\cap F_B}$$ $s_B\cdot det(\delta_{AB})$ being the image of $s_B\otimes det(\delta_{AB})$ by the homomorphism: $$ H^0(F_B,\det^\ast\c_B)\otimes H^0(F_B,\det^\ast\c_{AB})\to\ H^0(F_B,\det^\ast\c_A)$$ defined by the isomorphism of sheaves: $$\det\c_A\simeq \det\c_B\otimes \wedge( A/{A\cap B})\otimes \wedge( B/{A\cap B})^\ast\simeq \det\c_B\otimes\det\c_{AB}$$ \begin{defn} The global section $\omega_A\in H^0(\gr^0(V),\det^*\c_A)$ defined by: $$\{s_B\cdot det(\delta_{AB})\}_{B\sim V^+}$$ will be called the canonical section of $\det^\ast\c_A$. \end{defn} This result allows us to compute many global sections of $\detd_V=\det\c_{V^+}$ over $\gr^0(V)$: \noindent Given $A\sim V^+$ such that $F_A\subseteq\gr^0(V)$ the isomorphism $\det^*\c_A\iso\detd_V$ is not canonical, and in fact we have a canonical isomorphism: $$\det^*\c_A\iso\detd_V\otimes\bigwedge( A/{A\cap V^+})\otimes \bigwedge({V^+}/{A\cap V^+})^*$$ Therefore to give an isomorphism $\det^*\c_A\iso\detd_V$ depends on the choice of bases for the vector spaces $ A/{A\cap V^+}$ and ${V^+}/{A\cap V^+}$. \subsection{Computations for finite-dimensional Grassmannians.}\label{comp-fin-dim} Let $V$ be a d-dimensional $k$-vector space with a basis $\{e_1,\dots,e_d\}$, let $\{e^\ast_1,\dots,e^\ast_d\}$ be its dual basis, and $V^+=<e_{k+1},\dots,e_d>\subseteq V$. In this case $\gr^0(V,V^+)$ is the Grassmannian of $V$ classifying $k$-dimensional vector subspaces of $V$. Given a family of indexes $1\leq i_1<\dots< i_l\leq d$ ($1\le l\le d$), let $A(i_1,\dots,i_l)$ be the vector subspace generated by $\{e_{i_1},\dots,e_{i_l}\}$. One has that $F_A\subseteq\gr^0(V)$ is equivalent to saying that $l=d-k$. Let us set $A=A(i_1,\dots,i_{d-k})$. Now, the canonical section $\omega_A\in H^0(\gr^0(V),\det^\ast\c_A)$ is the section whose value at the point $L=<l_1,\dots,l_k>\in\gr^0(V)$ is given by: $$\omega_A(L)=\pi_A(l_1)\wedge\dots\wedge\pi_A(l_k) \otimes l^\ast_1\wedge\dots\wedge l^\ast_k \in\wedge V/A\otimes\wedge L^\ast=(\det^\ast\c_A)_L$$ $\{l^\ast_1,\dots,l^\ast_k\}$ being the dual basis of $\{l_1,\dots,l_k\}$ and $\pi_A\colon L\to V/A$ the natural projection. Note that $\{e_{j_1},\dots,e_{j_k}\}$ is a basis of $V/A$ where $\{j_1,\dots,j_k\}=\{1,\dots,d\}-\{i_1,\dots,i_{d-k}\}$, and that its dual basis is $\{e^\ast_{j_1},\dots,e^\ast_{j_k}\}$ in $( V/A)^\ast\subset V^\ast$. We have now: $$\omega_A(L)= (e^\ast_{j_1}\wedge\dots\wedge e^\ast_{j_k}) (l_1\wedge \dots\wedge l_k)\cdot e_{j_1}\wedge\dots\wedge e_{j_k}\otimes l^\ast_1\wedge \dots\wedge l^\ast_k$$ Observe that the $k$-vector space $$\wedge( A/{A\cap V^+})^*\otimes\wedge( V^+/{A\cap V^+})$$ is generated by: $$e_A=e^*_{m_1}\wedge\dots\wedge e^*_{m_r}\otimes e_{n_1}\wedge\dots\wedge e_{n_r}$$ where $$\aligned &\{i_1,\dots,i_{d-k}\}-\{k+1,\dots,d\}=\{m_1,\dots,m_r\}\\ &\{k+1,\dots,d\}-\{i_1,\dots,i_{d-k}\}=\{n_1,\dots,n_r\} \endaligned$$ And tensorializing by $e_A$ gives an isomorphism: $$ H^0(\gr^0(V),\det^\ast\c_A) @>{\otimes e_A}>> H^0(\gr^0(V),\detd_V)$$ Let $\Omega_A$ be the image of the canonical section $\omega_A$. The explicit expression of $\Omega_A$ is: $$\Omega_A(L)= (e^*_{j_1}\wedge\dots\wedge e^*_{j_k})(l_1\wedge \dots\wedge l_k) \cdot e_1\wedge\dots\wedge e_{d-k}\otimes l^*_1\wedge \dots\wedge l^*_k \in(\det_V^\ast)_L$$ Let $\L\subseteq V_{\gr^0(V)}$ be the universal submodule. One has a canonical epimorphism: $$\wedge^k V^*_{\gr^0(V)}\to\wedge^k\L^*$$ and bearing in mind the canonical isomorphism $\detd_V\simeq \wedge( V/{V^+})\otimes\wedge^k\L^*$ one obtains a canonical homomorphism: $$\aligned \wedge^k V^\ast= H^0(\gr^0(V),\wedge^k V^\ast)&\to H^0(\gr^0(V),\detd_V)\\ e^*_{j_1}\wedge\dots\wedge e^*_{j_k}&\longmapsto \Omega_{A(i_1,\dots,i_{d-k})} \endaligned$$ (where $\{j_1,\dots,j_k\}\coprod\{i_1,\dots,i_{d-k}\}=\{1,\dots,d\}$). It is well known that this homomorphism is in fact an isomorphism. \subsection{Computations for infinite-dimensional Grassmannians.} In {\ref{comp-fin-dim}} we discussed well known facts about the determinants of finite-dimen\-sional Grassmannians but have stated these results in an intrinsic language, which can easily be generalized to the infinite-dimensional case. Let $V$ be a $k$-vector space. We shall assume that there exists a family of linearly independent vectors $\{e_i,i\in\Z\}$ such that: \begin{enumerate} \item $<\{e_i\}, i\ge 0>$ is dense in $\hat V^+$ (with respect to the $V^+$-topology), \item $<\{e_i\}, i\in\Z>$ is dense in $\hat V$. \end{enumerate} \begin{rem} The above conditions are satisfied for example by $V=k((t))$ and $V^+=k[[t]]$. \end{rem} \begin{defn} Let ${\cal S}$ be the set of sequences $\{s_0,s_1,\dots\}$ of integer numbers satisfying the following conditions: \begin{enumerate} \item the sequence is strictly increasing, \item there exists $s\in\Z$ such that $\{s,s+1,s+2,\dots\}\subseteq\{s_0,s_1,\dots\}$, \item $\#(\{s_0,s_1,\dots\}-\{0,1,\dots\})= \#(\{0,1,\dots\}-\{s_0,s_1,\dots\})$. \end{enumerate} \end{defn} The sequences of ${\cal S}$ are usually called Maya's diagrams or Ferrer's diagrams of virtual cardinal zero (this is condition 3). For each $S\in{\cal S}$, let $A_S$ be the vector subspace of $V$ generated by $\{e_{s_i}, i\ge 0\}$. By the condition 3 one has: $$\dim_k({A_S}/{A_S\cap V^+})= \dim_k({V^+}/{A_S\cap V^+})$$ and hence: $A_S\sim V^+$ and $F_{A_S}\subseteq\gr^0(V)$. Further, $\{F_{A_S},S\in{\cal S}\}$ is a covering of $\gr^0(V)$. Let $\{e^*_i\}$ be a dual basis of $\{e_i\}$; that is, elements of $V^*$ given by $e^*_i(e_j)=\delta_{ij}$. For each finite set of increasing integers, $J=\{j_1,\dots,j_r\}$, let us define $e_J=e_{j_1}\wedge\dots\wedge e_{j_r}$ and $e^*_J=e^*_{j_1}\wedge\dots\wedge e^*_{j_r}$. Given $S\in{\cal S}$, choose $J,K\subseteq\Z$ such that $\{e_j\}_{j\in J}$ is a basis of ${A_S}/{A_S\cap V^+}$ and $\{e^*_k\}_{k\in K}$ of ${V^+}/{A_S\cap V^+}$. We have seen that tensorializing by $e_J\otimes e^*_K$ defines an isomorphism: $$ H^0(\gr^0,\det^*\c_{A_S}) @>{\,\otimes(e_J\otimes e^*_K)\,}>> H^0(\gr^0,\detd_V)$$ \begin{defn}\label{global-section} For each $S\in{\cal S}$, $\Omega_S$ is the global section of $\detd_V$ defined by: $$\Omega_S=\omega_{A_S}\otimes e_J\otimes e^*_K$$ We shall denote by $\Omega_+$ the canonical section of $\detd_V$. \end{defn} Let $\Omega({\cal S})$ be the $k$-vector subspace of $ H^0(\gr^0,\detd_V)$ generated by the global sections $\{\Omega_S,S\in{\cal S}\}$. We define the Pl\"ucker morphism: $$\aligned{\cal P}_V: \gr^0(V) &\to \check\P\Omega(S) \\ L &\mapsto \{\Omega_S(L)\}\endaligned$$ as the morphism of schemes defined by the homomorphism of sheaves: $$\Omega(S)_{\grv}\to\detd_V\to 0$$ (by the universal property of $\check\P$). \begin{rem} Given $L,L'\in\grv(\spk)$ such that $L\subseteq L'$, let $j:\gr^0({L'}/L)\hookrightarrow\gr^0(V)$ be the natural closed immersion. Since $j^*\det_V\simeq\det_{{L'}/L}$, one can easily see that the composition: $$ \gr^0({L'}/L)\overset j\hookrightarrow\gr^0(V) \overset{\quad{\frak p}\quad}\to \check\P\Omega(S)$$ factors through the Pl\"ucker immersion of the finite-dimensional Grassmannian $\gr^0({L'}/L)$: $${\frak p}_{{L'}/L}:\gr^0({L'}/L)\to \proj S^\punt H^0(\gr^0({L'}/L),\det^*_{{L'}/L})$$ \end{rem} \begin{thm} The Pl\"ucker morphism is a closed immersion. \end{thm} \begin{pf} Going on with the analogy with finite grassmannians, we will show that this morphism is locally given as the graph of a suitable morphism. Consider the morphism: $$F_{A_S}\hookrightarrow\gr^0(V) @>{\cal P}>> \check\P\Omega(S)$$ From the universal property of $\check\P$, we deduce a epimorphism: $$f_S:\Omega(S)\underset{k}\otimes B\to B$$ (where $\sp(B)=F_{A_S}$, and $\detd_V\vert_{F_{A_S}}$ is a line bundle). Note that it has a section, since the image of $\Omega_S$ is a everywhere non null function. That is, there exists a subspace $W\subset \Omega(S)$, and an isomorphism of $k$-vector spaces: $$<\Omega_S>\oplus W \iso \Omega(S)$$ such that $f_S$ is the projection onto the first factor. In other words, ${\cal P}\vert_{F_{A_S}}$ is the graph of a morphism. \end{pf} \begin{rem}\label{section-ogr} Note that considering the chain of finite-dimensional Grassmannians $\gr^0({L_i}/{L_{-i}})$ ($L_i$ being the subspaces $<\{e_j\}_{j\leq i}>$), which are closed subschemes of $\gr^0(V)$, one easily deduces that $ H^0(\gr^0(V),\o_{\gr^0(V)})=k$ from the fact that the homomorphism: $$ H^0(\gr^0(V),\o_{\gr^0(V)})\to\limp H^0(\gr^0({L_i}/{L_{-i}}),\o_{\gr^0({L_i}/{L_{-i}})})$$ is injective. \end{rem} \section{Automorphisms of the Grassmannian and the ``formal geometry'' of local curves} \label{aut-grass} Let $(V,V^+)$ be a pair of a $k$-vector space and a vector subspace $V^+\subseteq V$ and let $\grv$ denote the corresponding Grassmannian. We shall define the algebraic analogue of the restricted linear group defined by Pressley, Segal and Wilson ([{\bf PS}], [{\bf SW}]). This group is too large to be representable by a $k$-scheme and we therefore define it as a sheaf of groups in the category of $k$-schemes. For each $k$-scheme $S$, let us denote by $\aut_{\o_S}(\hat V_S)$ the group of automorphisms of the $\o_S$-module $\hat V_S$. \begin{defn} \item{a)} A sub-$\o_S$-module ${\cal B}\subseteq \hat V_S$ is said to be locally commensurable with $V^+$ if for each $s\in S$ there exists an open neighbourhood $U_s$ of $s$ and a commensurable vector subspace $B\sim V^+$ such that ${\cal B}\vert_{U_s}=\hat B_{U_s}$. \item{b)} An automorphism $g\in\aut_{\o_S}(\hat V_S)$ is called bicontinuous with respect to the $V^+$-topo\-logy if $g(\hat V^+_S)$ and $g^{-1}(\hat V^+_S)$ are $\o_S$-modules of $\hat V_S$ locally commensurable with $V^+$. \item{c)} The linear group, $\glv$, of $(V,V^+)$ is the contravariant functor over the category of $k$-schemes defined by: $$S\rightsquigarrow \glv(S)=\{g\in\aut_{\o_S} (\hat V_S)\text{ such that $g$ is bicontinuous }\}$$ \end{defn} \begin{thm} There exists a natural action of $\glv$ over the functor of points of the Grassmannian $\grv$: $$\aligned \glv\times & \fu\grv @>{\mu}>>\fu\grv \\ (g, & L)\phantom{xx}\longmapsto g(L) \endaligned$$ \end{thm} \begin{pf} Let $g\in\glv(S)$ and $L\in\fu\grv(S)$. We have: $${\hat V_S}/{g(L)+\hat V^+_S}\simeq {\hat V_S}/{L+g^{-1}{\hat V^+_S}}$$ and by definition of bicontinuous automorphisms, for each $s\in S$ there exist an open neighbourhood $U_s$ and a commensurable $A\sim V^+$ such that $g^{-1}\hat V^+\vert_{U_s}=\hat A_{U_s}$. Then: $${\hat V_{U_s}}/{g(L)_{U_s}+\hat V^+_{U_s}}\simeq {\hat V_{U_s}}/{L_{U_s}+\hat A^+_{U_s}}$$ from which we deduce that $g(L)\in\fu\grv(S)$. \end{pf} \begin{thm} There exists a canonical central extension of functors of groups over the category of $k$-schemes: $$0\to{\Bbb G}_m\to \glve \to \glv \to 0$$ and a natural action $\bar\mu$ of $\glve$ over the vector bundle ${\Bbb V}(\det_V)$ defined by the determinant bundle, such that the following diagram is commutative: $$\CD \glve\times{\Bbb V}(\detd_V) @>{\bar\mu}>>{\Bbb V}(\detd_V) \\ @VVV @VVV \\ \glv\times\grv @>{\mu}>> \grv \endCD$$ \end{thm} \begin{pf} Let us define $\glve(S)$ as the set of commutative diagrams: $$\CD {\Bbb V}(\detd_V) @>{\bar g}>>{\Bbb V}(\detd_V) \\@VVV @VVV \\ \grv @>g>>\grv \endCD$$ for each $g\in\glv(S)$, and the homomorphism $\glve\to\glv$ given by $\bar g\mapsto g$. The rest of the proof follows immediately from the fact that $ H^0(\gr^0(V),\o_{\gr^0(V)})=k$ (remark {\ref{section-ogr}}) and $g^*\detd_V\simeq\detd_V$ for every $g\in\glv$. \end{pf} \begin{rem}\label{G-extension} Let $G$ be a commutative subgroup of $\glv$ (a subfunctor of commutative groups). The central extension of $\glv$ gives an extension of $G$: $$ 0\to {\Bbb G}_m\to\tilde G @>{\pi}>> G\to 0$$ and the commutator of $\tilde G$: $$\aligned \tilde G\times\tilde G &\to\tilde G\\ (\tilde a,\tilde b)& \mapsto \tilde a\tilde b\tilde a^{-1}\tilde b^{-1} \endaligned$$ induces a pairing: $$\aligned G\times G & @>{[\, , \,]}>> {\Bbb G}_m \\ (g_1,g_2) &\mapsto [g_1,g_2]= \tilde g_1\tilde g_2\tilde g_1^{-1}\tilde g_2^{-1} \qquad \left(\tilde g_i\in\pi^{-1}(g_i)\right) \endaligned$$ When $V$ is a local field or a ring of adeles, this pairing will be of great importance in the study of arithmetic problems because it is connected with the formulation of reprocity laws. The same construction of the extensions $\tilde G$ applied to the Lie algebra of $G$ gives an extension of Lie algebras (taking the points of $G$ with values in $k[x]/x^2$): $$ 0\to {\Bbb G}_a=Lie({\Bbb G}_m)\to Lie(\tilde G) @>{d\pi}>> Lie(G)\to 0$$ and a pairing: $$\aligned Lie(G)\times Lie(G) & @>{R}>> {\Bbb G}_a \\ (D_1,D_2) &\mapsto R((D_1,D_2))=[\tilde D_1,\tilde D_2]= \tilde D_1\tilde D_2 - \tilde D_2\tilde D_1 \endaligned$$ ($\tilde D_i$ being a preimage of $D_i$). \end{rem} The pairing $R$ is an abstract generalization of the definition of Tate [{\bf T}] of the residue pairing. There are several subgroups of special relevance in the application of this theory to the study of moduli problems and soliton equations. Firstly, we are concerned with the algebraic analogue of the group $\Gamma$ ([{\bf SW}] \S 2.3) of continuous maps $S^1\to{\Bbb C}^*$ acting as multiplication operators over the Grassmannian. The main difference between our definition of the group $\Gamma$ and the definitions offered in the literature ([{\bf SW}], [{\bf PS}]) is that in the algebro-geometric setting the elements $\sum_{-\infty}^{+\infty}g_k\,z^k$ with infinite positive and negative coefficients do not make sense as multiplication operators over $k((z))$. Let us now consider the case $V=k((t)), V^+=k[[t]]$. The main idea for defining the algebraic analogue of the group $\Gamma$ is to construct a ``scheme'' whose set of rational points is precisely the multiplicative group $k((z))^*$. \begin{defn} The contravariant functor, $\kz$, over the category of $k$-schemes with values in the category of commutative groups is defined by: $$S\rightsquigarrow \kz(S)= H^0(S,\o_S)((z))^*$$ Where for a $k$-algebra $A$, $A((z))^*$ is the group of invertible elements of the ring $A((z))=A[[z]][z^{-1}]$. \end{defn} \begin{lem}\label{v-loc-const} For each $k$-scheme $S$ and $f\in\kz(S)$, the function: $$\aligned S&\to \Z \\ s &\mapsto v_s(f)= \text{order of $f_s\in k(s)((z))$} \endaligned$$ is locally constant. \end{lem} \begin{pf} We can assume that $S=\sp(A)$, $A$ being a $k$-algebra. Let $f=\sum_{i\ge n}^{}a_i\,z^i$ be an element of $A((z))^*$ ($n\in\Z$). There then exists another element $g=\sum_{i\ge -m}^{}b_i\,z^i$ ($m\in \Z$) such that $f\cdot g=1$. This implies the following relations (from now on we assume $n=0$ to simplify the calculations): \beq \begin{aligned} & 0=b_{-m}\, a_0 \\ & 0=b_{-m}\,a_1+b_{-m+1}\,a_0 \\ & \dots \\ & 0=b_{-m}\,a_{m-1}+\dots+ b_{-1}\,a_0 \\ & 1=b_{-m}\,a_m+\dots+ b_0\,a_0 \end{aligned}\label{relations} \end{equation} Let us distinguish two cases: \item{a)} $b_{-m}$ is not nilpotent in $A$: from {\ref{relations}} we obtain: $$b_{-m}\,a_0=b_{-m}^2\,a_1=\dots b_{-m}^m\,a_{m-1}=0$$ That is, $a_0,\dots,a_{m-1}$ are equal zero in the ring $A_{(b_{-m})}$ and for each $s\in\sp(A)-(b_{-m})_0$ one has $b_{-m}(s)\,a_m(s)=1$ and therefore $v_{s}(f)=m$. We conclude by proving that in this case $(b_{-m})_0$ is also an open subset of $\sp(A)$: \indent From the equations {\ref{relations}} we deduce: $$\gathered (b_{-m},a_{m-1},\dots,a_0)_0= (b_{-m})_0\cap (a_{m-1})_0 \cap \dots\cap(a_0)_0=\emptyset\\ (b_{-m})_0\cup(a_i)_0=\sp(A)\quad,\quad i=0,\dots,n-1 \endgathered$$ \indent and hence: $$(b_{-m})_0\cup\left(\cap_{i=0}^{n-1}(a_i)_0\right)=\sp(A)$$ \item{b)} Let us assume that $b_{-m},\dots,b_{-r-1}$ are nilpotent elements of $A$ and that $b_{-r}$ is not nilpotent. The same argument as in case a) proves that $v_s(f)$ is constant in the closed subscheme $(b_{-r})_0$ and that its complementary in $\sp(A)$ is $\cap_{i=0}^{r-1}(a_i)_0$, from which we conclude the proof. \end{pf} \begin{cor} For an affine irreducible $k$-scheme $S=\sp(A)$ one has that: \begin{enumerate} \item $v_s$ is a constant function over $S$, \item $$\left\{f\in A((z))^* \,\vert\, v(f)=n\right\}= \left\{\gathered \text{series }\,a_{n-r}\,z^r+\dots+a_n\,z^n+\dots\text{ such that}\\ a_{n-r},\dots,a_{n-1}\text{ are nilpotent and }a_n\in A^* \endgathered\right\}$$ \item If $A$ is also a reduced $k$-algebra: $$A((z))^*=\coprod_{n\in \Z}\left\{\sum_{i\ge n} a_i\,z^i \quad a_i\in A\text{ y }a_n\in A^*\right\}$$ \end{enumerate} \end{cor} \begin{pf} This is obvious from lemma {\ref{v-loc-const}}. \end{pf} \begin{thm} The subfunctor $\kz_{red}$ of $\kz$ defined by: $$S\rightsquigarrow \kz_{red}(S)= \coprod_{n\in \Z}\left\{z^n+\sum_{i> n} a_i\,z^i \quad a_i\in H^0(S,\o_S)\right\}$$ is representable by a group $k$-scheme whose connected component of the origin will be denote by $\Gamma_+$. \end{thm} \begin{pf} It suffices to observe that the functor: $$S\rightsquigarrow \left\{z^n+\sum_{i> n} a_i\,z^i \quad a_i\in H^0(S,\o_S)\right\}$$ is representable by the scheme: $$\sp(\limil{l} k[x_1,\dots,x_l])=\limpl{l}\A^l_k$$ and the group law is given by the multiplication of series. \end{pf} \begin{thm} Let $\kz_{nil}$ be the subfunctor of $\kz$ defined by: $$S\rightsquigarrow \kz_{nil}(S)= \coprod_{n>0}\left\{\gathered \text{ finite series }\,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\text{ such that } \\ a_i\in H^0(S,\o_S)\text{ are nilpotent and $n$ arbitrary} \endgathered\right\}$$ There exists a formal $k$-scheme $\Gamma_-$ representing $\kz_{nil}$, that is: $$\hom_{\text{for-sch}}(S,\Gamma_-)=\kz_{nil}(S)$$ for every $k$-scheme $S$. \end{thm} \begin{pf} Let us define the ring of ``infinite'' formal series in infinite variables (which is different from the ring of formal series in infinite variables) by: $$k\{\{x_1,\dots\}\}=\underset n\limp k[[x_1,\dots,x_n]]$$ the morphisms of the projective system being: $$\aligned k[[x_1,\dots,x_{n+1}]]&\to k[[x_1,\dots,x_n]] \\ x_i&\mapsto x_i\qquad\text{for }i=1,\dots,n-1 \\ x_{n+1}&\mapsto 0 \endaligned$$ Note that: $$k\{\{x_1,\dots\}\}=\underset n\limp{k[x_1,\dots,x_n]}/{(x_1,\dots,x_n)^n}$$ It is therefore an admissible linearly topological ring ([{\bf EGA}]~{\bf 0}.7.1) and there therefore exists its formal spectrum $\sf(k\{\{x_1,\dots\}\})$. Let us denote by $J_n$ the kernel of the natural projection $k\{\{x_1,\dots\}\}\to {k[x_1,\dots,x_n]}/{(x_1,\dots,x_n)^n}$ and $J=\limp (x_1,\dots,x_n)$. Let us now prove that $\Gamma_-=\sf(k\{\{x_1,\dots\}\})$: \noindent For every $k$-scheme $S$, considering over $H^0(S,\o_S)$ the discrete topology, we have: $$\aligned\hom_{\text{for-sch}}(S,\Gamma_-)=& \hom_{\text{cont-$k$-alg}}((k\{\{x_1,\dots\}\},H^0(S,\o_S))= \\=&\left\{\gathered f\in\hom_{\text{$k$-alg}}(k\{\{x_1,\dots\}\},H^0(S,\o_S)) \text{ such that} \\ \text{there exists $n\in{\Bbb N}$ satisfying }J_n\subseteq f^{-1}((0))\endgathered\right\} \endaligned$$ However the condition $J_n\subseteq f^{-1}((0))$ is equivalent to saying that $f(x_1),\dots,f(x_n)$ are nilpotent and $f(x_i)=0$ for $i>n$, from which one concludes the proof. \end{pf} \begin{rem} Note that $\Gamma_-$ is the inductive limit in the category of formal schemes ([{\bf EGA}]~{\bf I}.10.6.3) of the schemes which represent the subfunctors: $$S\rightsquigarrow \Gamma^n_-(S)=\left\{\gathered \,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\text{ such that}\\ a_i\in H^0(S,\o_S)\text{ and the ${\text{n}}^{\text{th}}$ power} \\ \text{ of the ideal $(a_1,\dots,a_n)$ is zero} \endgathered\right\}$$ \end{rem} \begin{rem}{\bf Group laws of $\Gamma_+$ and $\Gamma_-$} The group law of $\Gamma_+=\sp(k[x_1,\dots])$ is given by: $$\aligned k[x_1,\dots]&\to k[x_1,\dots]\otimes_k k[x_1,\dots] \\ x_i &\mapsto x_i\otimes 1+\sum_{j+k=i}x_j\otimes x_k+1\otimes x_i \endaligned$$ The group law of $\Gamma_-=\sf k\{\{x_1,\dots\}\}$ is given by: $$\aligned k\{\{x_1,\dots\}\}&\to k\{\{x_1,\dots\}\}\hat\otimes_k k\{\{x_1,\dots\}\} \\ x_i &\mapsto x_i\otimes 1+\sum_{j+k=i}x_j\otimes x_k+1\otimes x_i \endaligned$$ \end{rem} Let be $\kz_0$ be the connected component of the origin in the functor of groups $\kz$. \begin{thm} The natural morphism of functors of groups over the category of $k$-schemes: $$\fu{\Gamma_-}\times\fu{{\Bbb G}_m}\times\fu{\Gamma_+}\to\kz$$ is injective and for $char(k)=0$ gives an isomorphism with $\kz_0$. $\kz_0$ is therefore representable by the (formal) $k$-scheme: $$\Gamma=\Gamma_-\times{\Bbb G}_m\times\Gamma_+$$ \end{thm} \begin{pf} The morphism from $\fu{{\Bbb G}_m}$ to $\kz$ is the one induced by the natural inclusion $H^0(S,\o_S)^*\hookrightarrow H^0(S,\o_S)((z))^*$. The injectivity of $\fu\Gamma\hookrightarrow\kz$ follows from the fact that $\Gamma_-\cap\Gamma_+=\{1\}$. The rest of the proof is trivial from the above results and from the properties of the exponential map we shall see below. \end{pf} \begin{rem} Our group scheme $\Gamma$ is the algebraic analogue of the group $\Gamma$ of Segal-Wilson [{\bf SW}]. Note that the indexes ``-'' and ``+'' do not coincide with the Segal-Wilson notations. Replacing $k((z))$, by $k((z^{-1}))$ we obtain the same notation as in the paper of Segal-Wilson. \end{rem} Let us define the exponential maps for the groups $\Gamma_-$ and $\Gamma_+$. Let $\A_n$ be the $n$ dimensional affine space over $\spk$ with the additive group law, and $\hat\A_n$ the formal group obtained as the completion of $\A_n$ at the origin. We define $\hat\A_\infty$ as the formal group $\limil{n}\hat\A_n$. Obviously $\hat\A_\infty$ is the formal scheme: $$\hat\A_\infty=\sf k\{\{y_1,\dots\}\}$$ with group law: $$\aligned k\{\{y_1,\dots\}\}&\to k\{\{y_1,\dots\}\}\hat\otimes_k k\{\{y_1,\dots\}\} \\ y_i& \longmapsto y_i\otimes 1+1\otimes y_i \endaligned$$ \begin{defn} If the characteristic of $k$ is zero, the exponential map for $\Gamma_-$ is the following isomorphism of formal group schemes: $$\aligned \hat\A_\infty &@>{\exp}>> \Gamma_- \\ \{a_i\}_{i>0} &\mapsto \exp(\sum_{i>0}a_i\,z^{-i}) \endaligned$$ This is the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}& @>{\qquad\exp^*\qquad}>> k\{\{y_1,\dots\}\}\\ x_i &\mapsto \text{ coefficient of $z^{-i}$ in the series } \exp(\sum_{j>0}y_j\,z^{-j})\endaligned$$ \end{defn} \begin{defn}\label{exp-gamma-minus} If the characteristic of $k$ is $p>0$, the exponential map for $\Gamma_-$ is the following isomorphism of formal schemes: $$\aligned \hat\A_\infty &\to \Gamma_- \\ \{a_i\}_{i>0}&\mapsto \prod_{i>0}(1-a_i\,z^{-i})\endaligned$$ which is the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}&@>\exp^*>> k\{\{y_1,\dots\}\}\\ x_i &\mapsto \text{ coefficient of $z^{-i}$ in the series } \prod_{i>0}(1-a_i\,z^{-i})\endaligned$$ \end{defn} Note that this latter exponential map is not a isomorphism of groups. Considering over $\hat \A_\infty$ the law group induced by the isomorphism, $\exp$, of formal schemes, we obtain the Witt formal group law. Analogously, we define the exponential maps for the group $\Gamma_+$: \begin{defn} Let $\A^\infty$ be the group scheme over $k$ defined by $\limpl{n} \A_n$ (where $\A_{n+1}=\sp k[x_1,\dots,x_{n+1}]\to \A_n=\sp k[x_1,\dots,x_n]$ is the morphism defined by forgetting the last coordinate) with its additive group law. The exponential map when $char(k)=0$ is the isomorphism of group schemes: $$\aligned \A^\infty &\to \Gamma_+ \\ \{a_i\}_{i>0}&\mapsto \exp(\sum_{i>0}a_i\,z^i)\endaligned$$ If $char(k)=p\ne 0$, the exponential map is the isomorphism of schemes: $$\aligned \A^\infty &\to \Gamma_+ \\ \{a_i\}_{i>0}&\mapsto \prod_{i>0}(1-a_i\,z^i)\endaligned$$ which is not a morphism of groups. (See {\rm [{\bf B}]} for the connection of these definitions and the Cartier-Dieudonn\'e theory). \end{defn} It should be noted that the formal group scheme $\Gamma_-$ has properties formally analogous to the Jacobians of the algebraic curves: one can define formal Abel maps and prove formal analogues of the Albanese property of the Jacobians of smooth curves (see {\rm [{\bf KSU},{\bf C}]}). Let $\hat C=\sf(k[[t]])$ be a formal curve. We define the Abel morphism of degree $1$ as the morphism of formal schemes: $$\phi_1: \hat C\to \Gamma_-$$ given by $\phi_1(t)=(1-\frac{t}z)^{-1}=1+\sum_{i>0}^{}\frac{t^i}{z^i}$; that is, the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}&\to k[[t]]\\ x_i\,&\mapsto t^i\endaligned$$ Note that the Abel morphism is the algebro-geometric version of the function $q_\xi(z)$ used by Segal and Wilson ([{\bf SW}] page~32) to study the Baker function. Let us explain further why we call $\phi_1$ the ``Abel morphism'' of degree 1. If $char(k)=0$, composing $\phi_1$ with the inverse of the exponential map, we have: $$\bar\phi_1:\hat C@>{\phi_1}>> \Gamma_-@>>{\exp^{-1}}>> \hat\A_\infty$$ and since $(1-\frac{t}z)^{-1}=\exp(\sum_{i>0}^{}\frac{t^i}{i\,z^i})$ (see [{\bf SW}] page~33), $\bar\phi_1$ is the morphism defined by the ring homomorphism: $$\aligned k\{\{y_1,\dots\}\} &\to k[[t]]\\ y_i&\mapsto \frac{t^i}i \endaligned$$ or in terms of the functor of points: $$\aligned\hat C&@>{\bar\phi_1}>>\hat\A_\infty\\ t&\mapsto \{t,\frac{t^2}2,\frac{t^3}3,\dots\} \endaligned$$ Observe that given the basis $\omega_i=t^i\,dt$ of the differentials $\Omega_{\hat C}=k[[t]]dt$, $\bar\phi_1$ can be interpreted as the morphism defined by the ``abelian integrals'' over the formal curve: $$\bar\phi_1(t)=\left( \int\omega_0,\int\omega_1,\dots,\int\omega_i,\dots \right)$$ which coincides precisely with the local equations of the Abel morphism for smooth algebraic curves over the field of complex numbers. In general, for each integer number $n>0$, we define the Abel morphism of degree $n$ as the morphism of formal schemes: $$\bar\phi_n:\hat C\times\overset n\dots\times\hat C=\hat C^n\to\Gamma_-$$ given by $\bar\phi_n(t_1,\dots,t_n)= \prod_{i=1}^n\left(1-\frac{t_i}z\right)^{-1}$; that is, the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}&\to k[[t_1]]\hat\otimes\overset n\dots\hat\otimes k[[t_n]]\\ x_i\,&\mapsto \text{ coefficient of $z^{-i}$ in the series } \prod_{i=1}^n(1-\frac{t_i}{z})^{-1}\endaligned$$ Note that $\bar\phi_n$ factorizes through a morphism, $\phi_n$ from the $n^{\text{th}}$-symmetric product of $\hat C$ to $\Gamma_-$, which is the true Abel morphism; moreover $\phi_n$ is an immersion. \begin{thm} $(\Gamma_-,\phi_1)$ satisfies the Albanese property for $\hat C$; that is, every morphism $\psi:\hat C\to X$ in a commutative group scheme (which sends the unique rational point of $\hat C$ to the $0\in X$) factors through the Abel morphism and a homomorphism of groups $\Gamma_-\to X$. \end{thm} \begin{pf} Let $\psi:\hat C\to X$ be a morphism from the formal scheme $\hat C$ to a group scheme $X$ such that $\psi(\text{rational point})=0$. For each $n>0$, one constructs a morphism: $$\hat C^n @>{\bar\psi_n}>> X$$ which is the composition of $\psi\times\dots\times\psi: \hat C\times\overset n\dots\times\hat C \to X\times\overset n\dots\times X$ and the addition morphism $X\times\overset n\dots\times X\to X$. Observe that $\bar \psi_n$ factors through a morphism: $$S^n\hat C@>{\psi_n}>> X$$ and bearing in mind that $\Gamma_-=\limil{n} S^n\hat C$ (as formal group schemes) we conclude the proof of the existence of a homomorphism of groups $\bar\psi:\Gamma_-\to X$ satisfying the desired condition. \end{pf} \section{$\tau$-functions and Baker functions} This section is devoted to algebraically defining the $\tau$-functions and the Baker functions over an arbitrary base field $k$. Following on with the analogy between the groups $\Gamma$ and $\Gamma_-$ and the Jacobian of the smooth algebraic curves, we shall make the well known constructions for the jacobians of the algebraic curves for the formal curve $\hat C$ and the group $\Gamma$: Poincar\'e bundle over the dual jacobian and the universal line bundle over the jacobian. In the formal case these constructions are essentially equivalent to defining the $\tau$-functions and the Baker functions. Using the notations of section {\ref{aut-grass}}, let us consider the Grassmannian $\grv$ of $V=k((z))$ and the group $$\Gamma=\Gamma_-\times {\Bbb G}_m \times \Gamma_+$$ acting on $\grv$ by homotheties. As we have shown in {\ref{G-extension}}, there exists a central extension of $\Gamma$: $$0\to {\Bbb G}_m \to \widetilde \Gamma \to \Gamma \to 0$$ given by a pairing: \beq\quad \Gamma\times \Gamma \to {\Bbb G}_m \label{pairing}\end{equation} \begin{prop} The extension $\tilde\Gamma_+$ of $\Gamma_+$ is trivial. \end{prop} \begin{pf} We will construct a section $s$ (as groups) of $\tilde\Gamma_+\to\Gamma_+$; that is, for an element $g\in\Gamma_+$ we give $s(g)\in\tilde\Gamma_+$ such that $s$ is a morphism of groups. Denote by $\mu:\Gamma\times\grv\to\grv$ the action of $\Gamma$ on $\grv$ and by $\mu_g$ the automorphism of $\grv$ induced by the homothety $\cdot g:V\to V$ for $g\in\Gamma$. Fix $g\in\Gamma_+$. Observe that there exists a quasi-isomorphism of complexes: $$\CD \L @>>> V/V^+ \\ @V{\cdot g}VV> @VV{\cdot g}V \\ \mu_g^*(\L) @>>> V/V^+ \endCD$$ since $g\cdot V^+\simeq V^+$. We have thus an isomorphism $\detd_V\simeq \mu_g^*\detd_V$ in a canonical way, and hence a well-defined element $s(g)\in\tilde\Gamma_+$. Since this construction is canonical and $\mu_{g'}\circ\mu_g=\mu_{g'\cdot g}$ it follows easily that $s(g')\cdot s(g)=s(g'\cdot g)$. \end{pf} \begin{prop} For a rational point $U\in\grv$, let $\mu_U$ be the morphism $\Gamma\times\{U\}\to\grv$ induced by $\mu$. Then, the line bundle $\mu_U^*\detd_V\vert\Gamma_-$ is trivial, and the extension $\tilde\Gamma_-$ is thus trivial. \end{prop} \begin{pf} Assume $U\in F_{V^+}$ (the general case is anologous). It is no difficult to obtain the following equality for $g\in\Gamma_-$: $$(\mu_U^*\Omega_+)(g)\,=\,\Omega_+(g\cdot U)\,=\, \Omega_+(U)+\sum_S \chi_S(g)\cdot\Omega_S(U)$$ where the sum is taken over the set of Young diagrams and $\chi_S$ is the Schur polynomial (in the coefficients of $g$) corresponding to $S$. Since $\Omega_+(U)\neq0$ and the coefficients of $g$ are nilpotents, it follows that $\mu_U^*\Omega_+$ is a no-where vanishing section of $\mu_U^*\detd_V$, and this bundle is therefore trivial. Observe now that since $\tilde\Gamma_-$ can be thought as the sheaf of automorphisms of $\mu_U^*\detd_V$ one has that $\tilde\Gamma_-$ is a trivial extension. \end{pf} \begin{cor} The restrictions of the pairing {\ref{pairing}} to the subgroups $\Gamma_-$ and $\Gamma_+$ are trivial. \end{cor} We define the Poincar\'e bundle over $\Gamma\times \grv$ as the invertible sheaf: $${\frak P}=\mu^*\det_V^*$$ For each point $U\in \grv$, let us define the Poincar\'e bundle over $\Gamma\times \Gamma$ associated with $U$ by: $${\frak P}_U=(1\times \mu_U)^*{\frak P}=m^*({\mu_U}^*\det_V^*)$$ where $m:\Gamma\times\Gamma\to\Gamma$ is the group law. The sheaf of $\tau$-functions of a point $U\in \grv$, $\widetilde {\L_\tau}(U)$, is the invertible sheaf over $\Gamma\times \{U\}$ defined by: $$\widetilde{\L_\tau}(U)={\frak P}\vert_{\Gamma\times\{U\}}$$ Let us note that the sheaf $\widetilde{\L_\tau}(U)$ is defined for arbitrary points of the Grassmannian and not only for geometric points. The restriction homomorphism induces the following homomorphism between global sections: \beq H^0\left(\Gamma\times \grv, \mu^*\det_V^*\right)\to H^0\left(\Gamma\times \{U\},\widetilde {\L_\tau}(U) \right) \label{restriction}\end{equation} \begin{defn} The $\tau$-function of the point $U$ over $\Gamma$ is defined as the image $\widetilde {\tau }_U$ of the section $\mu^*\Omega_+$ by the homomorphism {\ref{restriction}} ($\Omega_+$ being the global section defined in {\ref{global-section}}). \end{defn} Obviously $\widetilde {\tau }_U$ is not a function over $\Gamma\times \{U\}$ since the invertible sheaf $\widetilde {\L_\tau}(U)$ is not trivial. The algebraic analogue of the $\tau$-function defined by M. and Y. Sato, Segal and Wilson ([{\bf SS}], [{\bf SW}]) is obtained by restricting the invertible sheaf $\widetilde {\L_\tau}(U)$ to the formal subgroup $\Gamma_-\subset\Gamma$. To see this, fix a rational point $U\in\grv$ and define: $$\L_\tau(U)=\widetilde{\L_\tau}(U)\vert_{\Gamma_-\times\{U\}}$$ which is a trivial invertible sheaf over $\Gamma_-$. To obtain a trivialization of ${\L_\tau}(U)$ which will allow us to identify global sections with functions over $\Gamma_-$ we must fix a global section of ${\L_\tau}(U)$ without zeroes in $\Gamma_-$. Recall that $\tilde\Gamma_-$ is a trivial extension of $\Gamma_-$ and it has therefore a section $s$. It follows that the group $\Gamma_-$ acts on $\L_\tau(U)$ (through $s$) and on $\Gamma_-$ by translations. One has easily that the morphism: $${\mathbb V}(\L_\tau(U)^*)\to\Gamma_-$$ is equivariant with respect to these actions. Note now that: $$\hom_{\Gamma_-\text{-equiv}} \left(\Gamma_-,{\mathbb V}(\L_\tau(U)^*)\right)\subseteq \hom_{\Gamma_-\text{-esq}} \left(\Gamma_-,{\mathbb V}(\L_\tau(U)^*)\right)= H^0(\Gamma_-,\L_\tau(U))$$ Let $\delta$ be an non-zero element in the fibre of ${\mathbb V}(\L_\tau(U)^*)$ over the point $1$ of $\Gamma_-$ (1 being the identity of $\Gamma_-$). Let $\sigma_0$ be the unique morphism $\Gamma_-\to{\mathbb V}(\L_\tau(U)^*)$ $\Gamma_-$-equivariant such that $\sigma_0(1)=\delta$, and denote again by $\sigma_0$ the corresponding section of $\L_\tau(U)$. Observe that $\sigma_0$ is a constant section and since it has no zeros it gives a trivialization of $\L_\tau(U)$. Through this trivilization, the global section of ${\L_\tau}(U)$ defined by $\tilde\tau_U$ is identified with the function $\tau_U\in\o(\Gamma_-)= k\{\{x_1,\dots\}\}$ given by Segal-Wilson [{\bf SW}]: $${\tau}_U(g)=\frac{\tilde\tau_U(g)}{\sigma_0(g)}= \frac{\mu^*\Omega_+(g)}{\sigma_0(g)}=\frac{\Omega_+(gU)}{\delta}$$ Finally, if $U\in F_{V^+}$ then one can choose $\delta=\Omega_+(U)$. Observe that the $\tau$-function ${\tau}_U$ is not a series of infinite variables but an element of the ring $k\{\{x_1, \dots\}\}$. The subgroup $\Gamma_+$ of $\Gamma$ acts freely over $\grv$. Accordingly the orbits of the rational points of $\grv$ under the action of $\Gamma_+$ are isomorphic, as schemes, to $\Gamma_+$. Let $X$ be the orbit of $V^-=z^{-1}\cdot k[z^{-1}]\subset V$ under $\Gamma_+$. The restrictions of $\det_V$ and $\detd_V$ to $X$ are trivial invertible sheaves. Bearing in mind that the points of $X$ are $k$-vector subspaces of $V$ whose intersection with $V^+$ is zero, one has that the section $\Omega_+$ of $\det_V^* $ defines a canonical trivialization of $\detd_V$ over $X$. \begin{thm} The restriction homomorphism $\detd_V \to\detd_V\vert_X$ induces a homomorphism between global sections: $$ B: H^0\left(\grv,\detd_V \right)\longrightarrow H^0\left(X,\detd_V\vert_X \right)\simeq \o(\Gamma_+)=k[x_1,\dots]$$ which is an isomorphism between the $k$-vector subspace $\Omega(S)$ defined in {\ref{global-section}} and $\o(\Gamma_+)$. The isomorphism $ H^0(X,\detd_V\vert_X)\overset \sim\rightarrow \o(\Gamma_+)$ is the isomorphism induced by the trivialization defined by $\Omega_+$. In the literature, the isomorphism $B:\Omega(S)\overset \sim \longrightarrow \o(\Gamma_+)$ is usually called the bosonization isomorphism. \end{thm} \begin{pf} All one has to prove is that $B(\Omega_S)=F_S(x)$, $\Omega_S$ being the Pl\"ucker sections of $\detd_V$ defined in {\ref{global-section}} and $F_S(x_1,x_2,\dots)$ being the Schur functions. Proof of the identity $B(\Omega_S)=F_S(x)$ is essentially the same as in the complex analytic case; see [{\bf SW}] and [{\bf PS}]. In some of the literature, the $\tau$ function of a point $U\in\grv$ is defined as the Pl\"ucker coordinates of the point $U$. Let us therefore explain in which sense both definitions are equivalent. The canonical homomorphism: $$H^0(\detd_V)\otimes\o_{\grv}\longrightarrow\detd_V\to 0$$ induces a homomorphism: $$\det_V=\det_V^{**}\overset{\bar\tau}\hookrightarrow H^0(\detd_V)^*\otimes\o_{\grv}$$ \end{pf} \begin{defn} Given a point $\widetilde U\in\det_V$ in the fibre of $U\in\grv$, the $\bar\tau$-function of $\widetilde U$ is defined as the element $\bar \tau(U)\in H^0\left (\detd_V\right)^*\otimes k(U)$ ($k(U)$ being the residual field of $U$). This is essentially the definition of $\tau$-functions given in the papers of M. and Y. Sato, Arbarello and De Concini, and Kawamoto and others ({\rm [{\bf SS}], [{\bf AD}], [{\bf KNTY}]}). \end{defn} \begin{lem} There exists an isomorphism of $k$-vector spaces: $$\o(\Gamma_+)^*\to \o(\Gamma_-)$$ \end{lem} \begin{pf} Recall that $\o(\Gamma_+)=k[x_1,\dots]$ and that $\o(\Gamma_-)=k\{\{x_1,x_2,\dots\}\}$. Now think that $x_i$ is the $i$-symmetric function of other variables, say $t_1,t_2,\dots$. It is known that the Schur polynomials $\{F_S\}$ (where $S$ is a partition) of the $t$'s are polynomials in the $x$'s and are in fact a basis of the $k$-vector space $k[x_1,\dots]$. The isomorphism is the induced by the pairing: $$\aligned \o(\Gamma_+)\times \o(\Gamma_-) &\longrightarrow k \\ (F_S,F_{S'}) & \longmapsto \delta_{S,S'}\endaligned$$ (see [{\bf Mc}]). \end{pf} The composition of the homomorphism $B^*$ (the dual homomorphism of $B$) and the isomorphism of the above lemma gives an homomorphism: $$\tilde B^*:\o(\Gamma_+)^*=k\{\{x_1,x_2,\dots\}\} \longrightarrow H^0\left (\detd_V\right)^*$$ The connection between $\tau_U$ and $\bar \tau (\widetilde U)$ is the following: $$\tilde B^*(\tau_U)=\lambda \cdot(\bar \tau (\widetilde U))$$ $\lambda$ being a non-zero constant. (Of course, if $U$ is not rational but a point with values in a scheme $S$, $\lambda \in H^0\left(S,\o_S\right)^*$). The connection of the $\tau$-functions with autoduality (in the sense of group schemes) properties of the group $\Gamma= \Gamma_-\times {\Bbb G}_m\times \Gamma_+$ implicit in the above discussion, is studied with detail in [{\bf C},{\bf P}]. L. Breen in [{\bf B2}] outlines also some of these properties from another point of view. Once we have algebraically defined the $\tau$-functions, we can define the Baker functions using formula~5.14. of [{\bf SW}]; this is the procedure used by several authors. However, we prefer to continue with the analogy with the classical theory of curves and jacobians and define the Baker functions as a formal analogue of the universal invertible sheaf of the Jacobian. Let us consider the composition of morphisms: $$\tilde\beta:\hat C\times\Gamma\times\grv \overset {\phi\times Id}\longrightarrow \Gamma\times\Gamma\times\grv\overset{m\times Id} \longrightarrow\Gamma\times\grv$$ $\phi:\hat C=\sf k[[z]] \to \Gamma$ being the Abel morphism (taking values in $\Gamma_-\subset \Gamma$) and $m:\Gamma\times \Gamma\to\Gamma$ the group law. \begin{defn} The sheaf of Baker-Akhiezer functions is the invertible sheaf over $\hat C\times \Gamma\times \grv$ defined by: $$\widetilde {\L_B}=(\phi\times Id)^*(m\times Id)^*{\frak P}$$ Let us define the sheaf of Baker functions at a point $U\in \grv$ as the invertible sheaf: $$\widetilde {\L_B}(U)=\widetilde {\L_B}\vert_{\hat C\times \Gamma\times \{U\}}=\tilde {\beta_U}^*\widetilde {\L_\tau}(U)$$ (where $\tilde {\beta_U}^*$ is the following homomorphism between global sections: $$ H^0(\Gamma\times \{U\},\widetilde {\L_\tau}(U)) \overset{\widetilde{\beta_U}^*}\longrightarrow H^0(\hat C\times\Gamma\times \{U\},\widetilde {\L_B}(U))$$ \end{defn} By the definitions, $\widetilde {\L_B}(U)\vert_{\hat C\times \Gamma_-\times \{U\}}={\L_B}(U)$ is a trivial invertible sheaf over $\hat C \times \Gamma_-$. Observe that for each element $u\in\Gamma_-(S)\subseteq \fu{k((z))^*}(S)=H^0(S,\o_S)((z))^*$ we can define a fractionary ideal of the formal curve $\w {C_S}$ by: $$I_u=u\cdot\o_S((z))$$ in such a way that we can interpret the formal group $\Gamma_-$ as a kind of Picard scheme over the formal curve. The universal element of $\Gamma_-$ is the invertible element of $\kz(\Gamma_-)$ given by: $$v=1+\underset {i \geq 1}\sum x_i\,z^{-i}\in k((z))\hat \otimes k\{\{x_{ 1},x_{ 2},\dots\}\}$$ This universal element will be the formal analogue of the universal invertible sheaf for the formal curve $\hat C$. \begin{defn} The Baker function of a point $U\in \grv$ is $\psi_U=v^{-1}\cdot \beta^*_U(\tilde\tau_U)$, where $$\beta_U^*: H^0\left( \Gamma\times \{U\},\widetilde {\L_\tau}(U)\right)\longrightarrow H^0\left(\hat C\times \Gamma\times \{U\},\widetilde {\L_B}(U)\right)$$ is the homomorphism induced by $\widetilde{\beta_U}^*$. \end{defn} Observe that the Baker function of $V^-=z^{-1}\,k[z^{-1}]$ is the universal invertible element $v^{-1}$. Note that, analogously to the case of $\tau$-function, we can choose a trivialization of $\widetilde {\L_B}(U)$ over $\hat C\times\Gamma_-\times \{U\}$ in such a way that the function asociated to the section $v^{-1}\cdot \beta^*_U(\tau_U)$ is: \beq\psi_U(z,g)=v^{-1}\cdot \frac{\tau_U\left(g\cdot \phi_1\right)}{\tau_U(g)} \label{baker-expr}\end{equation} which is the classical expression for the Baker function. When the characteristic of the base field $k$ is zero, we can identify $\Gamma_-$ with the additive group scheme $\hat{\A}_{\infty}$ through the exponential and expression {\ref{baker-expr}} is the classical expression for the Baker functions ([{\bf SW}]~5.16): $$\psi_U(z , t)=\left( \frac{\tau_U(t+[z])}{\tau_U(t) }\right)\cdot \exp(-\sum t_i\,z^{-i})$$ where $[z]=(z,\frac{1}{2}z^2,\frac{1}{3} z^3,\dots)$ and $t=(t_1,t_2,\dots)$ and $v=\exp(\sum t_i\,z^{-i})$ through the exponential map. For the general case, we obtain explicit expressions for $\psi_U$ as a function over $\hat C\times \hat\A_{\infty}$ but considering in $\hat\A_{\infty}$ the group law induced by the exponential {\ref{exp-gamma-minus}}: $$\psi_U(z,g)=v(z,g)^{-1}\cdot \frac{\tau_U\left(t* \phi(z)\right)}{\tau_U(t) }$$ ($*$ being the group law of $\hat\A_\infty$). The classical properties characterizing the Baker functions (for example proposition~5.1 of [{\bf SW}]) can be immediately generalized for the Baker functions over arbitrary fields. \begin{rem} Note that our definitions of $\tau$-functions and Baker functions are valid over arbitrary base fields and that can be generalized for $\Z$. One then has the notion of $\tau$-function and Baker functions for families of elements of $\grv$ and, if we consider the Grassmannian of $\Z((z))$ one then has $\tau$-functions and Baker functions of the rational points of $\gr\left(\Z((z))\right)$ and the geometric properties studied in this paper have a translation into arithmetic properties of the elements of $\gr\left(\Z((z))\right)$. The results stated by Anderson in [{\bf A}] are a particular case of a much more general setting valid not only for $p$-adic fields but also for arbitrary global field numbers. Our future aims are to study the arithmetic properties related to these constructions. \end{rem} \vskip2truecm

9606

alg-geom/9606007

en

https://arxiv.org/abs/alg-geom/9606007

alg-geom/9606007

Joost van Hamel

Fr\'ed\'eric Mangolte and Joost van Hamel

Algebraic cycles and topology of real Enriques surfaces

18 pages AMS-LaTeX v 1.2

For a real Enriques surface Y we prove that every homology class in H_1(Y(R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface.

alg-geom

\subsubsection*{Acknowledgements}}{\par} \begin{document} \title[Real Enriques surfaces]{Algebraic cycles and topology \\ of real Enriques surfaces} \author{Fr\'ed\'eric Mangolte\and Joost van Hamel} \keywords{Algebraic cycles, Real algebraic surfaces, Enriques surfaces, Galois-Maximality} \subjclass{14C25 14P25 14J28} \address{Fr\'ed\'eric Mangolte, D\'epartement de Math\'ematiques, Universit\'e Montpellier II, 34095 Montpellier Cedex 5, France, Tel: (33) 67 14 35 05, Fax: (33) 67 14 35 58} \email{[email protected]} \address{Joost van Hamel, Faculteit der Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands, Tel: (31) 20 444 76 94, Fax: (31) 20 444 76 53} \email{[email protected]} \thanks{The second author was supported in part by EC grant~CHRX-CT94-O506.} \begin{abstract} For a real Enriques surface $Y$ we prove that every homology class in $H_1(Y(\BR), {\BZ/2})$ can be represented by a real algebraic curve if and only if all connected components of $Y(\BR)$ are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or ${\Bf Z}$-Galois-Maximal and we determine the Brauer group of any real Enriques surface. \end{abstract} \maketitle \section{Introduction}\label{sec intro} Let $Y$ be a complex algebraic surface. Let us denote by $Y(\BC)$ the set of closed points of $Y$ endowed with the Euclidean topology and let $H_2^{\mathrm{alg}}(Y(\BC),{\Bf Z})$ be the subgroup of the homology group $H_2(Y(\BC),{\Bf Z})$ generated by the fundamental classes of algebraic curves on $Y$. If $Y$ is an Enriques surface, we have $$ H_2^{\mathrm{alg}}(Y(\BC),{\Bf Z})= H_2(Y(\BC),{\Bf Z}). $$ One of the goals of the present paper is to prove a similar property for real Enriques surfaces with orientable real part. See Theorem~\ref{theo main} below. By an \emph{algebraic variety $Y$ over ${\Bf R}$} we mean a geometrically integral scheme of finite type over the real numbers. The Galois group $G = \{1, \sigma\}$ of ${\Bf C} / {\Bf R}$ acts on $Y(\BC)$, the set of complex points of $Y$, via an antiholomorphic involution, and the real part $Y(\BR)$ is precisely the set of fixed points under this action. An algebraic variety $Y$ over ${\Bf R}$ will be called a \emph{real Enriques surface}, a real K3-surface, etc., if the complexification $Y_{\Bf C} = Y \otimes {\Bf C}$ is a complex Enriques surface, resp.\ a complex K3-surface, etc. Consider the following two classification problems: -- classification of topological types of algebraic varieties $Y$ over ${\Bf R}$ (the manifolds $Y(\BC)$ up to equivariant diffeomorphism), -- classification of topological types of the real parts $Y(\BR)$. For real Enriques surfaces the two classifications have been investigated recently by Nikulin in \cite{Ni}. The topological classification of the real parts was completed by Degtyarev and Kharlamov who give in \cite{DeKha1} a description of all 87 topological types. Let us mention here that the real part of a real Enriques surface $Y$ need not be connected and that a connected component $V$ of $Y(\BR)$ is either a nonorientable surface of genus $\leq 11$ or it is homeomorphic to a sphere or to a torus. The problem of classifying $Y(\BC)$ up to equivariant diffeomorphism still lacks a satisfactory solution. In the attempts to solve this problem, equivariant (co)homology plays an important role (see \cite{Ni}, \cite{N-S}, \cite{DeKha2}). It establishes for any algebraic variety $Y$ over ${\Bf R}$ a link between the action of $G$ on the (co)homology of $Y(\BC)$ and the topology of $Y(\BR)$. For example, the famous inequalities \begin{align} \label{eq GM1} \dim H_*(Y(\BR),{\BZ/2}) & \leq \sum_{r=0}^{2n} \dim H^1(G, H_r(Y(\BC),{\BZ/2})) \\ \label{eq GM2} \dimH_\mathrm{even}(Y(\BR),{\BZ/2}) & \leq \sum_{r=0}^{2n} \dim H^2(G, H_r(Y(\BC),{\Bf Z})) \\ \label{eq GM3} \dimH_\mathrm{odd}(Y(\BR),{\BZ/2}) & \leq \sum_{r=0}^{2n} \dim H^1(G, H_r(Y(\BC),{\Bf Z})) \end{align} (cf.\ \cite{Kr1} or \cite{Si}) can be proven using equivariant homology. We will say that $Y$ is \emph{Galois-Maximal} or a \emph{{\itshape{GM}}-variety} if the first inequality turns into equality, and $Y$ will be called \emph{${\Bf Z}$-Galois-Maximal}, or a \emph{{\itshape{$\BZ$-GM}}-variety} if inequalities \eqref{eq GM2} and \eqref{eq GM3} are equalities. When the homology of $Y(\BC)$ is torsion free, the two notions coincide (see \cite[Prop.~3.6]{Kr1}). A nonsingular projective surface $Y$ over ${\Bf R}$ with $Y(\BR) \neq \emptyset$ is both {\itshape{GM}}\ and {\itshape{$\BZ$-GM}}\ if it is simply connected (see \cite[\S 5.3]{Kr1}). If $H_1(Y(\BC), {\Bf Z}) \neq 0$, as in the case of an Enriques surface, the situation can be much more complicated. The necessary and sufficient conditions for a real Enriques surface $Y$ to be a {\itshape{GM}}-variety were found in \cite{DeKha2}; in the present paper we will give necessary and sufficient conditions for $Y$ to be {\itshape{$\BZ$-GM}}. See Theorem \ref{theo galmax}. As far as we know, this is the first paper on real Enriques surfaces in which equivariant (co)homology with integral coefficients is studied instead of coefficients in ${\BZ/2}$. We expect that the extra information that can be obtained this way (compare for example equations \eqref{eq GM1}--\eqref{eq GM3} ) will be useful in the topological classification of real Enriques surfaces. In Section~\ref{sec brauer} we demonstrate the usefulness of integral coefficients by computing the Brauer group $\operatorname{Br}(Y)$ of any real Enriques surface $Y$. This completes the partial results on the $2$-torsion of $\operatorname{Br}(Y)$ obtained in \cite{N-S} and \cite{N1}. See Theorem~\ref{theo brauer}. \subsection{Main results} Let $Y$ be an algebraic variety over ${\Bf R}$. Denote by $H^{\mathrm{alg}}_{n}(Y(\BR),{\BZ/2})$ the subgroup of the homology group $H_{n}(Y(\BR),{\BZ/2})$ generated by the fundamental classes of $n$-dimensional Zariski-closed subsets of $Y(\BR)$, see \cite{BoHa} or \cite{BCR}. We will say that these classes can be \emph{represented by algebraic cycles.} The problem of determining these groups is still open for most classes of surfaces. For a real rational surface $X$ we always have $H_2^{\mathrm{alg}}(X(\BC),{\Bf Z})=H_2(X(\BC),{\Bf Z})$ and $H_1^{\mathrm{alg}}(X(\BR),{\BZ/2})= H_1(X(\BR),{\BZ/2})$, see \cite{Si}. For real K3-surfaces, the situation is not so rigid. In most connected components of the moduli space of real K3-surfaces the points corresponding to a surface $X$ with $\dim H_1^\mathrm{alg}(X(\BR), {\BZ/2}) \geq k$ form a countable union of real analytic subspaces of codimension $k$ for any $k \leq \dim H_1(X_0({\Bf R}), {\BZ/2})$, where $X_0$ is any K3-surface corresponding to a point from that component. In some components this is only true for $k < \dim H_1(X_0({\Bf R}), {\BZ/2})$; these components do not contain any point corresponding to a surface $X$ with $H_1^\mathrm{alg}(X(\BR), {\BZ/2}) = H_1(X(\BR), {\BZ/2})$, see \cite{Man}. For real Abelian surfaces the situation is similar, see \cite{Huisman:abelian}. \begin{theo}\label{theo main} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. If all connected components of the real part $Y(\BR)$ are orientable, then \[H_1^{\mathrm{alg}}(Y(\BR),{\BZ/2})= H_1(Y(\BR),{\BZ/2}).\] Otherwise, $$ \dim H_1^{\mathrm{alg}}(Y(\BR),{\BZ/2}) = \dim H_1(Y(\BR),{\BZ/2}) - 1. $$ \end{theo} \noindent See Theorem~\ref{theo c alg} for more details. In order to state further results we should mention that the set of connected components of the real part of a real Enriques surface $Y$ has a natural decomposition into two parts $Y(\BR)=Y_1\bigsqcup Y_2$. Following \cite{DeKha1} we will refer to these two parts as the two \emph{halves} of the real Enriques surface. In \cite{N1} it is shown that $Y$ is {\itshape{GM}}\ if both halves of $Y(\BR)$ are nonempty. It follows from \cite[Lem.~6.3.4]{DeKha2} that if precisely one of the halves of $Y(\BR)$ is empty, then $Y$ is {\itshape{GM}}\ if and only if $Y(\BR)$ is nonorientable. This result plays an important role in the proof of many of the main results of that paper (see Section~7 of \emph{loc.\ cit.}). In the present paper we will see in the course of proving Theorem~\ref{theo main} that a real Enriques surface with orientable real part is not a {\itshape{$\BZ$-GM}}-variety. In Section~\ref{sec galmax} we also tackle the nonorientable case and combining our results with the results for coefficients in ${\BZ/2}$ that were already known we obtain the following theorem. \begin{theo}\label{theo galmax} Let $Y$ be a real Enriques surface with nonempty real part. \begin{enumerate} \item Suppose the two halves $Y_1$ and $Y_2$ are nonempty. Then $Y$ is {\itshape{GM}}. Moreover, $Y$ is {\itshape{$\BZ$-GM}}\ if and only if $Y(\BR)$ is nonorientable. \item Suppose one of the halves $Y_1$ or $Y_2$ is empty. Then $Y$ is {\itshape{GM}}\ if and only if $Y(\BR)$ is nonorientable. Moreover, $Y$ is {\itshape{$\BZ$-GM}}\ if and only if $Y(\BR)$ has at least one component of odd Euler characteristic. \end{enumerate} \end{theo} \noindent There are examples of all cases described in the above theorem (see \cite[Fig.~1]{DeKha1}). In Section~\ref{sec brauer} we study the Brauer group $\operatorname{Br}(Y)$ of a real Enriques surface $Y$ using the fact that $\operatorname{Br}(Y)$ is isomorphic to the cohomological Brauer group $\operatorname{Br}'(Y) = H^2_{\text{\textrm{\'et}}}(Y, {\mathbb{G}}_m)$, since $Y$ is a nonsingular surface. In \cite{N-S} Nikulin and Sujatha gave various equalities and inequalities relating the dimension of the 2-torsion of $\operatorname{Br}(Y)$ to other topological invariants of a real Enriques surface $Y$. It was shown in \cite{N1} that $$ \dim_{\BZ/2} \operatorname{Tor}(2, \operatorname{Br}(Y)) \geq 2s -1 $$ where $s$ is the number of connected components of $Y(\BR)$, and that equality holds if $Y$ is {\itshape{GM}}. Using the results in Section~\ref{sec galmax} on equivariant homology with integral coefficients we can compute the whole group $\operatorname{Br}(Y)$. \begin{theo}\label{theo brauer} Let $Y$ be a real Enriques surface. Let $s$ be the number of connected components of $Y(\BR)$. If $Y(\BR) \neq \emptyset$ is nonorientable then $$ \operatorname{Br}(Y) \simeq ({\BZ/2})^{2s-1}. $$ If $Y(\BR) \neq \emptyset$ is orientable then $$ \operatorname{Br}(Y) \simeq \begin{cases} ({\BZ/2})^{2s-2} \oplus {\Bf Z}/4 & \text{if both halves are nonempty},\\ ({\BZ/2})^{2s} & \text{if one half is empty.} \end{cases} $$ If $Y(\BR) = \emptyset$ then $$ Br(Y) \simeq {\BZ/2}. $$ \end{theo} \begin{acknowledgements} Large parts of this paper were written during visits of the first author to the \emph{Vrije Universiteit, Amsterdam} and of the second author to the \emph{Universit\'e Montpellier II}. We want to thank J.~Bochnak and R.~Silhol for the invitations, and the \emph{Thomas Stieltjes Instituut} and the \emph{Universit\'e Montpellier II} for providing the necessary funds. We are grateful to A.~Degtyarev and V.~Kharlamov for giving us preliminary versions of their papers. \end{acknowledgements} \section{Equivariant homology and cohomology }\label{sec equiv} Since the group $G = \operatorname{Gal} ({\Bf C}/{\Bf R})$ acts in a natural way on the complex points of an algebraic variety $Y$ defined over ${\Bf R}$, the best homology and cohomology theories for studying the topology of $Y(\BC)$ are the ones that take this group action into account. In \cite{N-S} \'etale cohomology $H^*_{\text{\textrm{\'et}}} (Y, {\BZ/2})$ is used, and in \cite{N1} the observation is made that this is essentially the same as equivariant cohomology $H^*(Y(\BC); G, {\BZ/2})$. In \cite{DeKha2} Degtyarev and Kharlamov do not use equivariant cohomology as such, but instead a `stabilized' form of the Hochschild-Serre spectral sequence $ E_{p,q}^2(X; G, {\BZ/2}) = H^p(G, H^q(X, {\BZ/2})) $. This construction, due to I.~Kalinin, is based on the fact that if $G = {\BZ/2}$ then $H^{p+2}(G, M) = H^p(G, M)$ for any group $M$ and any $p > 0$, and if $M$ is a ${\BZ/2}$-module then even $H^{p+1}(G, M) = H^p(G, M)$ for any $p > 0$, so it is possible to squeeze the Hochschild-Serre spectral sequence into 1, or at most 2 diagonals. They also use the analogue of this Kalinin spectral sequence in homology. In the present paper we stick to the original equivariant cohomology supplemented with a straightforward dual construction which we call equivariant Borel-Moore homology. First we will recall some properties of equivariant cohomology for a space with an action of $G = {\BZ/2}$. Then we will give the definition of equivariant Borel-Moore homology and list the properties that we are going to need. In Section~\ref{sec fund} we give a short treatment of the fundamental class of $G$-manifolds and formulate Poincar\'e duality in the equivariant context. Let $X$ be a topological space with an action of $G = {\BZ/2}$. We denote the fixed point set of $X$ by $X^G$. In \cite{Grothendieck} the groups $H^*(X; G, \mathcal{F} )$ are defined for a $G$-sheaf $\mathcal{F}$ on $X$, which is a sheaf with a $G$-action compatible with the $G$-action on $X$. Writing $G = \{1, \sigma \}$, this just means that we are given an isomorphism of sheaves $\varsigma:\mathcal{F} \to \sigma^* \mathcal{F}$ satisfying $\sigma^*(\varsigma) \circ \varsigma = \operatorname{id}$. Now define $$ H^p(X; G, - ) = R^p \Gamma(X, -)^G $$ the $p$-th right derived functor of the $G$-invariant global sections functor. We have natural mappings $$ e^p_{\mathcal{F}} \fcolon H^p(X; G, \mathcal{F}) \to H^p(X, \mathcal{F})^G $$ which are the edge morphisms of the \emph{Hochschild-Serre spectral sequence} $$ E_{p,q}^2(X; G, \mathcal{F}) = H^p(G, H^q(X, \mathcal{F})) \Rightarrow H^{p+q}(X; G, \mathcal{F}) $$ For us, the most important $G$-sheaves will be the constant sheaf ${\BZ/2}$ and the constant sheaves constructed from the $G$-modules ${\Bf Z}(k)$ for $k \in {\Bf Z}$. Here we define ${\Bf Z}(k)$, to be the group of integers, equipped with an action of $G$ defined by $\sigma \cdot z = (-1)^k z$. We will use the notation $A(k)$ to denote either ${\BZ/2}$ or ${\Bf Z}(k)$, and we will sometimes use $A$ instead of $A(k)$ if $k$ is even. There is a cup-product $$ H^p(X; G, A(k)) \otimes H^q(X; G, A(l)) \to H^{p+q}(X; G, A(k+l)) $$ and a pull-back $f^*$ for any continuous equivariant mapping $f \fcolon X \to Y$, which both have the usual properties. If $X$ is a point, $H^p(\mathrm{pt}; G, M ) = H^p(G, M), $ which is cohomology of the group $G$ with coefficients in $M$. Recall that as a graded ring, $H^*(G,{\BZ/2})$ is isomorphic to the polynomial ring ${\BZ/2}[\eta]$, where $\eta$ is the nontrivial element in $H^1(G,{\BZ/2})$. By abuse of notation, we will also use the notation $\eta$ for the nontrivial element in $H^1(G, {\Bf Z}(1)) \simeq {\BZ/2}$ and $\eta^2$ for the nontrivial element in $H^2(G, {\Bf Z}) \simeq {\BZ/2}$. This notation is justified by the fact that $\eta \in H^1(G, {\Bf Z}(1))$ maps to $\eta \in H^1(G, {\BZ/2})$ under the reduction modulo 2 mapping and $\eta^2 \in H^2(G, {\Bf Z})$ maps to $\eta^2 \in H^2(G, {\BZ/2})$. The constant mapping $X \to \mathrm{pt}$ induces a mapping $H^*(G, {\BZ/2}) \to H^*(X; G, {\BZ/2})$ and we have a natural injection $H^p(X^G, {\BZ/2}) \hookrightarrow H^p(X^G; G, {\BZ/2})$, so cup-product gives us for any $G$-space $X$ a mapping $$ {H^*(X^G,{\BZ/2})}{\otimes}{H^*(G, {\BZ/2})}\to{H^*(X^G;G,{\BZ/2})} $$ which is well-known to be an isomorphism. Taking the inverse of this isomorphism and sending $\eta$ to the unit element in $H^*(X^G, {\BZ/2})$ we obtain a surjective homomorphism of rings $H^*(X^G;G,{\BZ/2}) \to H^*(X^G, {\BZ/2})$ and we define for $A = {\Bf Z}$ or ${\BZ/2}$ and any $k \in {\Bf Z}$ the homomorphism of rings $$ \beta \fcolon H^*(X; G, A(k)) \to H^*(X^G, {\BZ/2})$$ to be the composite mapping $$ H^*(X; G, A(k)) \labelto{i^*} H^*(X^G; G, A(k)) \labelto{\bmod 2} H^*(X^G; G, {\BZ/2}) \to H^*(X^G, {\BZ/2}), $$ where $i^*$ is induced by the inclusion $i \fcolon X^G \hookrightarrow X$. Note that $\beta$ coincides with the mapping $\beta'$ in \cite{Kr3}. It is clear from the definition that $$ \beta(f^* \omega) = f^* \beta(\omega). $ We use the notation $$ \beta^{n,p}\fcolon H^n(X; G, A(k)) \to H^p(X^G; {\BZ/2}) $$ for the mapping induced by $\beta$. In Section~\ref{sec galmax}, we will need one technical lemma which can easily be proven using the Hochschild-Serre spectral sequence. \begin{lem} \label{lem e2-not-surj} Let $X$ be a $G$-space with $X^G \ne \emptyset$. Then if $e^2_{A(k)}$ is not surjective on $H^2(X,A(k))^G$, there is a class $\omega \in H^1(X; G, A(k-1))$ such that $e^1_{A(k-1)}(\omega) \ne 0$, but $\beta(\omega) = 0$. \end{lem} The homology theory we are going to use is the natural dual to equivariant cohomology. For an extensive treatment of its properties, see \cite{JvH}. Here we will give a short account without proofs. In the rest of this section we assume $X$ to be a locally compact space of finite cohomological dimension with an action of $G = {\BZ/2}$, and $A(k)$ will be as above. We define the \emph{equivariant Borel-Moore homology of $X$ with coefficients in $A(k)$} by $$ H_p(X; G, A(k)) = R^{-p} \operatorname{Hom}_G ( R \Gamma_c (X, {\Bf Z}), A(k)) \text{ for $p \in {\Bf Z}$} $$ where $\operatorname{Hom}_G$ stands for homomorphisms in the category of $G$-modules and $\Gamma_c$ stands for global sections with compact support; this is the natural equivariant generalization of the usual Borel-Moore homology in the context of sheaf theory (see, for example, \cite[Ch.IX]{Iversen}). If $X$ is homeomorphic to an $n$-dimensional locally finite simplicial complex with a (simplicial) action of $G$, the we can determine $H_p(X; G, A(k))$ from a double complex analogous to the double complex~(1-12) in {N1}, which is used for the calculation of equivariant cohomology. Consider the oriented chain complex $ \mathcal{C}^\infty_n \to \mathcal{C}^\infty_{n-1} \to \dots \to \mathcal{C}^\infty_0 $ with closed supports (i.e., the elements of $\mathcal{C}^\infty_p$ are $p$-chains that can be infinite). The chain complex with coefficients in $A(k)$ is defined by $$ \mathcal{C}^\infty_p(A(k)) = \mathcal{C}^\infty_p \tensor A(k), $$ and we give it the diagonal $G$-action. Then $H_p(X; G, A(k))$ is naturally isomorphic to to the $(-p)$th homology group of the total complex associated to the double complex $$\begin{CD} \dots & & \dots & & \dots \\ @AAA @AAA @AAA\\ \mathcal{C}^\infty_{n-1}(A(k)) @>{1-\sigma}>> \mathcal{C}^\infty_{n-1}(A(k)) @>{1+\sigma}>> \mathcal{C}^\infty_{n-1}(A(k)) @>{1-\sigma}>> \cdots \\ @AAA @AAA @AAA\\ \mathcal{C}^\infty_{n}(A(k)) @>{1-\sigma}>> \mathcal{C}^\infty_{n}(A(k)) @>{1+\sigma}>> \mathcal{C}^\infty_{n}(A(k)) @>{1-\sigma}>> \cdots \\ \end{CD}$$ where the lower left hand corner has bidegree $(-n,0)$. Note that by construction $H_p(\mathrm{pt}; G, A(k)) = H^{-p}(G, A(k))$, so Poincar\'e duality holds trivially when $X$ is a point (and the proof of Poincar\'e duality in higher dimensions, as stated in Proposition~\ref{prop poincare}, is no more difficult than in the nonequivariant case). In particular, $H_p(X; G, A(k))$ need not be zero for $p < 0$. The groups $H_p(X; G, A(k))$ are covariantly functorial in $X$ with respect to equivariant proper mappings and the homomorphisms ${\Bf Z}(k) \to {\BZ/2}$ induce homomorphisms $H_p(X; G, {\Bf Z}(k)) \to H_p(X; G, {\BZ/2})$ that fit into a long exact sequence \begin{multline} \label{les coeff} \cdots \labelto{} H_p(X; G, {\Bf Z}(k)) \labelto{\times 2} H_p(X; G, {\Bf Z}(k)) \labelto{} \\ \labelto{} H_p(X; G, {\BZ/2}) \labelto{} H_{p-1}(X; G, {\Bf Z}(k)) \labelto{} \cdots \end{multline} As in the case of cohomology, there are natural homomorphisms $$ e^{A(k)}_p \fcolon H_p(X; G, A(k)) \to H_p(X, A(k))^G $$ which are the edge morphisms of a Hochschild-Serre spectral sequence \begin{equation*} \label{spectral} E^2_{p,q}(X; G, A(k)) = H^{-p}(G, H_q (X, A(k))) \Rightarrow H_{p+q}(X; G, A(k)). \end{equation*} If no confusion is likely, we use $e$ instead of $e^{A(k)}_p$; otherwise we will often write $e^{+}_p = e^{{\Bf Z}(2k)}_p$, $e^{-}_p = e^{{\Bf Z}(2k+1)}_p$, and $e_p = e^{{\BZ/2}}_p$, and we have similar conventions for the edge morphisms $e^p_{A(k)}$ in cohomology. There is a cap-product between homology and cohomology \begin{equation*} \anpairing{H_p(X; G, A(k))}{\otimes}{H^q(X; G, A(l))}->% {H_{p-q}(X; G, A(k-l))},% {\gamma}{\otimes}{\omega}|->{\gamma \cap \omega}, \end{equation*} and of course we have \begin{align} \label{cap-and-cup} \gamma \cap (\omega \cup \omega') &= (\gamma \cap \omega) \cap \omega',\\ \label{cap-and-edge} e(\gamma \cap \omega) &= e(\gamma) \cap e(\omega),\\ \intertext{and for any proper equivariant mapping $f\fcolon X \to Y$} \label{cap-and-f} (f_* \gamma) \cap \omega &= f_* (\gamma \cap f^* \omega). \end{align} Recall that $\eta$ is the nontrivial element in $H^1(G, A(1))$. Cap-product with $\eta$ considered as an element of $H^1(X; G, A(1))$ defines a map \mapping{ s^{A(k)}_p}{H_p(X; G, A(k))}{H_{p-1}(X; G, A(k+1))}% {\gamma}{\gamma \cap \eta} It can be shown, that the $e^{A(k)}_p$ and $s^{A(k)}_p$ fit into a long exact sequence \begin{multline}\label{se edge} \cdots \xrightarrow{s^{A(k-1)}_{p+1}} H_p(X;G,A(k)) \xrightarrow{e^{A(k)}_p} H_p(X,A) \to \\ \to H_p(X ;G,A(k-1))\xrightarrow{s^{A(k-1)}_{p}} H_{p-1}(X ;G, A(k)) \to \cdots \end{multline} For $s^{A(k)}_p$ we adopt the same notational conventions as for $e_p^{A(k)}$. The natural mapping $ H_p(X^G, A) \to H_p(X^G; G, A) $ and the cap-product give us a homomorphism \begin{equation*} H_*(X^G,{\BZ/2}){\otimes}{H^*(G, {\BZ/2})}\to{H_*(X^G;G,{\BZ/2})}, \end{equation*} which is an isomorphism. Taking the inverse of this isomorphism and sending the nontrivial element $\eta \in H^1(G,{\BZ/2})$ to the unit element in $H^*(X^G, {\BZ/2})$ we obtain a surjective homomorphism $$H_*(X^G;G,{\BZ/2}) \to H_*(X^G,{\BZ/2}).$$ Furthermore, the mapping $i_* \fcolon H_n(X^G; G, {\BZ/2}) \to H_n(X; G, {\BZ/2})$ induced by the inclusion $i \fcolon X^G \to X$ is an isomorphism for any $n < 0$, so we can define a homomorphism $$ \rho \fcolon H_*(X;G,A(k)) \to H_*(X^G, {\BZ/2}) $$ by taking the composite mapping \begin{multline*} H_*(X;G,A(k)) \labelto{\bmod 2} H_*(X; G, {\BZ/2}) \labelto{\cap \eta^{N}} H_{< 0} (X;G,{\BZ/2}) \labelto{(i_*)^{-1}} \\ \to H_*(X^G; G, {\BZ/2}) \to H_*(X^G, {\BZ/2}), \end{multline*} where $N$ is any integer greater than the (cohomological) dimension of $X$. We use the notation $\rho_n$ for the restriction of $\rho$ to $H_n(X;G,A(k))$, we write $\rho_{n,p}$ for the composition of $\rho_n$ with the projection $H_*(X^G, {\BZ/2}) \to H_p(X^G, {\BZ/2})$, and similar definitions hold for $\rho_{n, \mathrm{even}}$ and $\rho_{n, \mathrm{odd}}$. It is clear from the above that \begin{equation} \label{s-and-rho} \rho \circ s = \rho, \end{equation} and that the mapping $$ \rho_n \fcolon H_{n}(X;G,{\BZ/2}) \to H_*(X^G,{\BZ/2}) $$ induced by $\rho$ is surjective if $n < 0$. Note that, together with the Hochschild-Serre spectral sequence $E^r_{p,q}(X; G, {\BZ/2})$, this proves equation (\ref{eq GM1}). Equations \eqref{eq GM2} and \eqref{eq GM3} can be derived from the Hochschild-Serre spectral sequence with coefficients in ${\Bf Z}$ and the following proposition. \begin{prop} \label{prop zhomdecompo} Let $X$ be a locally compact space of finite cohomological dimension with an action of $G={\BZ/2}$. Then $$ \rho_{n,\mathrm{even}} \fcolon H_n(X; G, {\Bf Z}(k)) \to H_\mathrm{even}(X^G, {\BZ/2}) $$ is an isomorphism if $n < 0$ and $n + k$ is even, and $$ \rho_{n,\mathrm{odd}} \fcolon H_n(X; G, {\Bf Z}(k)) \to H_\mathrm{odd}(X^G, {\BZ/2}) $$ is an isomorphism if $n < 0$ and $n + k$ is odd. \end{prop} Observe that it is not claimed that $\rho_n \left( H_n(X; G, {\Bf Z}(k)) \right) \subset H_*(X^G,{\BZ/2})$ is contained in $H_\mathrm{even}(X^G, {\BZ/2})$ (resp.\ $H_\mathrm{odd}(X^G, {\BZ/2})$). In fact this is often not the case: for any $\gamma \in H_n(X; G, {\Bf Z}(k))$ there is a $p \equiv n + k \bmod 2$ such that \begin{equation} \label{eq bockstein} \rho(\gamma) = \rho_{n,p} (\gamma) + \delta(\rho_{n,p} (\gamma)) + \rho_{n,p - 2} (\gamma) + \delta(\rho_{n,p - 2} (\gamma)) + \dotsb, \end{equation} where $\delta$ is the Bockstein homomorphism $H_{p+1}(X^G, {\BZ/2}) \to H_{p}(X^G, {\BZ/2})$ associated to the short exact sequence $$0 \to {\Bf Z}/2 \to {\Bf Z}/4 \to {\Bf Z}/2 \to 0$$ (compare \cite[Th.~0.1]{Kr3}). We will also use the symbol $\delta$ for the connecting homomorphism $H_{n+1}(X; G, {\BZ/2}) \to H_{n}(X; G, {\Bf Z}(k))$ of the long exact sequence \eqref{les coeff}, and we have \begin{align} \label{rho-and-delta} \rho_{n, \mathrm{even}}(\delta(\gamma)) &= \rho_{n+1, \mathrm{even}}(\gamma) + \delta(\rho_{n+1, \mathrm{odd}}(\gamma)) && \text{if $n + k$ is even,}\\ \rho_{n, \mathrm{odd}}(\delta(\gamma)) &= \rho_{n+1, \mathrm{odd}}(\gamma) + \delta(\rho_{n+1, \mathrm{even}}(\gamma)) && \text{if $n + k$ is odd.} \end{align} It is clear from the definition and the projection formula \eqref{cap-and-f} that \begin{align} \label{cap-and-rho} \rho(\gamma) \cap \beta(\omega)&= \rho(\gamma \cap \omega), \\ \intertext{and for any proper mapping $f \fcolon X \to Y$ of $G$-spaces} \label{f-and-rho} \rho(f_* \gamma)&= f_* \rho(\gamma). \end{align} There are canonical isomorphisms $H_0(\mathrm{pt}; G, A) \simeq A$ and $H_0(\mathrm{pt}, A) = A$, so the homomorphisms induced by the constant mapping $\varphi \fcolon X \to \mathrm{pt}$ give us for every compact $G$-space $X$ the \emph{degree maps} \begin{align*} \operatorname{deg_G} \fcolon H_0(X;G, A) &{}\to A\\ \intertext{and} \deg \fcolon H_0(X, A) &{}\to A, \end{align*} which satisfy the equality \begin{equation} \label{deg-and-edge1} e \circ \operatorname{deg_G} = \deg{} \circ e. \end{equation} Extending the degree map on $H_0(X^G, {\BZ/2})$ by $0$ to the whole of $H_*(X^G, {\BZ/2})$, we have by equation~(\ref{f-and-rho}) that \begin{equation} \label{deg-and-rho1} \operatorname{deg_G}(\gamma) \equiv \deg( \rho(\gamma)) \bmod2, \end{equation} for any $\gamma \in H_0(X; G, A)$. Finally, define $$ H_*(X^G, A)^0 = \ker \left\{\deg \fcolon H_*(X^G, A) \to A \right\}, $$ and $H_\mathrm{even}(X^G, {\BZ/2})^0 = H_\mathrm{even}(X^G, {\BZ/2}) \cap H_*(X^G, {\BZ/2})^0$. We will record three technical lemmas for use in Section~\ref{sec galmax}. They can be proven by a careful inspection of either the Hochschild-Serre spectral sequence $E_{p,q}(X; G, A(k))$ or the long exact sequence~(\ref{se edge}) with the appropriate coefficients. \begin{lem} \label{lem rho-zz} Let $X$ be a compact connected $G$-space with $X^G \neq \emptyset$. Then $$ \rho_2 \fcolon H_2(X; G, {\BZ/2}) \to H_*(X^G, {\BZ/2})^0 $$ is surjective if and only if the composite mapping $$ H_1 (X; G, {\BZ/2}) \xrightarrow{e_1} H_1(X, {\BZ/2}) ^G \xrightarrow{\cup \eta^2} H^2(G,H_1(X, {\BZ/2})) $$ is zero. \end{lem} \begin{lem} \label{lem even-rho-z} Let $X$ be a compact connected $G$-space. Then $$ \rho_{2,\mathrm{even}} \fcolon H_2(X; G, {\Bf Z}) \to H_\mathrm{even}(X^G, {\BZ/2})^0 $$ is surjective if and only if the composite mapping $$ H_1 (X; G, {\Bf Z}(1)) \xrightarrow{e_1^-} H_1(X, {\Bf Z}(1)) ^G \xrightarrow{\cup \eta^2} H^2(G,H_1(X, {\Bf Z}(1)) $$ is zero. \end{lem} \begin{lem} \label{lem odd-rho} Let $X$ be a locally compact connected $G$-space with $X^G \neq \emptyset$. Then the mapping $$ \rho_{2,\mathrm{odd}} \fcolon H_2(X; G, {\Bf Z}(1)) \to H_\mathrm{odd}(X^G, {\BZ/2}) $$ is surjective if and only if the composite mapping $$ H_1 (X; G, {\Bf Z}) \xrightarrow{e_1^+} H_1(X, {\Bf Z}) ^G \xrightarrow{\cap \eta^2} H^2(G,H_1(X, {\Bf Z})) $$ is zero. \end{lem} \section{The fundamental class of a $G$-manifold} \label{sec fund} Let again $A$ be ${\BZ/2}$ or ${\Bf Z}$. Let $X$ be an $A$-oriented topological manifold of finite dimension $d$ with an action of $G=\{1,\sigma\}$. We will define the fundamental class of $X$ in equivariant homology with coefficients in $A(k)$ for $k$ even or odd. It is well-known, that $H_d(X, A) = A$, and the $A$-orientation determines a generator $\mu_X$ of $H_d(X, A)$. Observe that we do not need to require $X$ to be compact, since we use Borel-Moore homology. If $G$ acts via an $A$-orientation preserving involution, then $\mu_X \in H_d(X, A)^G$, otherwise $\mu_X \in H_d(X, A(1))^G$. By the Hochschild-Serre spectral sequence \eqref{spectral} we have for $k\in{\Bf Z}$ an isomorphism $H_d(X; G, A(k)) \simeq H_d(X, A(k))^G$, given by the edge morphisms $e^{A(k)}_d$, so we a the fundamental class $$ \mu_X \in H_d(X; G, A(k)) $$ where $k$ must have the right parity. \begin{prop}[Poincar\'e duality] \label{prop poincare} Let $X$ be a $G$-manifold with fundamental class $\mu_X \in H_d(X; G, A(k))$. Then the mapping \anmapping{H^i(X; G, A(l))}{H_{d-i}(X; G, A(k-l))}{\omega}{\mu_X\cap\omega} is an isomorphism. \end{prop} Assuming that the action of $G$ is \emph{locally smooth} (i.e., each fixed point has a neighbourhood that is equivariantly homeomorphic to ${\Bf R}^d$ with an orthogonal $G$-action), the fixed point set of $X^G$ is again a topological manifold, but it need not be $A$-orientable and it need not be equi-dimensional. However, if $V$ is a connected component of $X^G$ and $V$ has dimension $d_0$, then it has a fundamental class $\mu_{V} \in H_{d_0}(V, {\BZ/2})$, and we have that the restriction of $\rho_{d,d_0}(\mu_X) \in H_{d_0}(X^G, {\BZ/2})$ to $V$ equals $\mu_V$ (see \cite{JvH}). If $X$ is a closed sub-$G$-manifold of a $G$-manifold $Y$, then the embedding $j \fcolon X \to Y$ is proper, so it induces a mapping in equivariant homology. We define the class in $H_d(Y; G, A(k))$ \emph{represented by $X$} to be $j_* \mu_X$. Now let $X$ be an algebraic variety defined over ${\Bf R}$. We want to define the class in $H_{2d}(X; G, {\Bf Z}(d))$ represented by a subvariety of dimension $d$. As in \cite{Fulton}, we will distinguish two kinds of subvarieties, the \emph{geometrically irreducible subvarieties}, which are varieties over ${\Bf R}$ themselves, and the \emph{geometrically reducible subvarieties}, which are irreducible over ${\Bf R}$, but which split into two components when tensored with ${\Bf C}$. Then the complex conjugation exchanges these two components. Any complex algebraic variety $V$ of dimension $d$ has a fundamental class $\mu_V \in H_{2d}(V({\Bf C}), {\Bf Z})$, and the complex conjugation on ${\Bf C}^d$ preserves orientation if $d$ is even, and reverses orientation if $d$ is odd. This implies that if $j \fcolon Z \hookrightarrow X$ is the inclusion of a subvariety of dimension $d$ defined over ${\Bf R}$, then $\mu_{Z_{\Bf C}}$ is a generator of $H_d(Z({\Bf C}), {\Bf Z}(d))^G$ if $Z_{\Bf C}$ is irreducible, and $H_d(Z({\Bf C}), {\Bf Z}(d))^G$ is generated by $\mu_{Z_1} + \mu_{Z_2}$ if $Z_{\Bf C}$ is the union of two distinct complex varieties $Z_1$ and $Z_2$ of dimension $d$. Hence we define the fundamental class $\mu_Z \in H_{2d}(Z({\Bf C}); G, {\Bf Z}(d))$ of $Z$ to be the inverse image of $\mu_{Z _{\Bf C}}$ (resp.\ of $\mu_{Z_1} + \mu_{Z_2}$) under $e^{{\Bf Z}(d)}_{2d}$. The class $[Z] \in H_{2d}(X({\Bf C}); G, {\Bf Z}(d))$ represented by $Z$ is of course defined to be $j_* \mu_Z$. If we use the notation $[Z({\Bf R})] \in H_d(X(\BR), {\BZ/2})$ for the homology class represented by $Z({\Bf R})$, as defined in \cite{BoHa}, then indeed \begin{equation} \label{rho-and-cycle1} \rho_{2d,d}([Z]) = [Z({\Bf R})]. \end{equation} If $Z, Z'$ are subvarieties of $X$ defined over ${\Bf R}$ which are rationally equivalent over ${\Bf R}$ (see \cite{Fulton} for a definition), then $[Z] = [Z']$, so we get for every $d \leq \dim X$ a well-defined cycle map $${CH}_d(X) \to H_{2d}(X(\BC); G, {\Bf Z}(d))$$ from the Chow group in dimension $d$ to equivariant homology. The image will be denoted by $H_{2d}^\mathrm{alg}(X(\BC); G, {\Bf Z}(d))$, and we see by equation~(\ref{rho-and-cycle1}), that \begin{equation} \label{rho-and-cycle2} \rho_{2d,d}\left(H_{2d}^\mathrm{alg}(X(\BC); G, {\Bf Z}(d))\right) = H_d^\mathrm{alg}(X(\BR), {\BZ/2}). \end{equation} For $X$ nonsingular projective of dimension $n$, this map coincides with the composition of the mapping $${CH}_d(X) \to H^{2(n-d)}(X(\BC); G, {\Bf Z}(n-d))$$ as defined in \cite{Kr2} and the Poincar\'e duality isomorphism. As a consequence we can use the following description of the image of the cycle map in codimension 1, where we use the notation $H^{2}_\mathrm{alg}(X(\BC); G, {\Bf Z}(1))$ for the image of ${CH}_{n-1}(X)$ in cohomology. \begin{prop} \label{prop lefschetz} Let $X$ be a nonsingular projective algebraic variety over ${\Bf R}$. Let $\mathcal{O}_h$ be the sheaf of germs of holomorphic functions on $X(\BC)$. Then $H^2_\mathrm{alg}(X(\BC); G, {\Bf Z}(1))$ is the kernel of the composite mapping $$ H^2(X(\BC); G, {\Bf Z}(1)) \labelto{e^2_-} H^2(X(\BC), {\Bf Z}) \labelto{} H^2(X(\BC), \mathcal{O}_h) $$ \end{prop} \begin{proof} This follows immediately from Proposition~1.3.1 in \cite{Kr2}, which states that $H^2_\mathrm{alg}(X(\BC); G, {\Bf Z}(1))$ is the image of the connecting morphism $$ H^1(X(\BC); G, \mathcal{O}^*_h) \to H^2(X(\BC); G, {\Bf Z}(1)) $$ in the long exact sequence induced by the exponential sequence of $G$-sheaves $$ 0 \to {\Bf Z}(1) \to \mathcal{O}_h \to \mathcal{O}^*_h \to 0 $$ \end{proof} \section{Algebraic cycles}\label{sec enriques} The following facts about real Enriques surfaces can be found in \cite{Ni} or \cite{DeKha1}. Let $Y$ be a real Enriques surface. Let $X \to Y_{\Bf C}$ be the double covering of $Y_{\Bf C}$ by a complex K3-surface $X$. Since $X(\BC)$ is simply connected, $X(\BC)$ is the universal covering space of $Y(\BC)$ and $H_1(Y(\BC), {\Bf Z}) = {\BZ/2}$. The complex conjugation $\sigma$ on $Y(\BC)$ can be lifted to the covering $X(\BC)$ in two different ways. If $Y(\BR) \neq \emptyset$ this is easy to see; if $Y(\BR) = \emptyset$ we need to use the fact that a smooth manifold diffeomorphic to a K3-surface does not admit a free ${\Bf Z}/4$-action, see \cite[p.~439]{Hitchin}. Hence we can give $X$ the structure of a variety over ${\Bf R}$ in two different ways, which we will denote by $X_1$ and $X_2$. The two halves $Y_1$ and $Y_2$ of $Y(\BR)$ mentioned in the introduction consist of the components covered by $X_1({\Bf R})$ and $X_2({\Bf R})$, respectively. All connected components of $X_1({\Bf R})$ and $X_2({\Bf R})$ are orientable, as is the case for the real part of any real K3-surface. If a connected component of a half $Y_i$ is orientable, then it is covered by two components of $X_i({\Bf R})$, which are interchanged by the covering transformation of $X$. A nonorientable component of $Y_i$ is covered by just one component of $X_i({\Bf R})$; this is the orientation covering. Since for an Enriques surface $H^2(Y(\BC), \mathcal{O}_h) = 0$ (see \cite[V.23]{BPV}), we see by Proposition~\ref{prop lefschetz} and Poincar\'e duality that $H_2^\mathrm{alg}(Y(\BC); G, {\Bf Z}(1)) = H_2(Y(\BC); G, {\Bf Z}(1))$, so $H_1^\mathrm{alg}(Y(\BR), {\BZ/2})$ is the image of the mapping $$ \alpha_2 =\rho_{2, 1} \fcolon H_2(Y(\BC); G, {\Bf Z}(1)) \to H_1(Y(\BR), {\BZ/2}). $$ In order to determine the image of $\alpha_2$ we will define $\alpha_n$ for any $n \in {\Bf Z}$ by $$ \alpha_n = \rho_{n,1} \fcolon H_n(Y(\BC); G, {\Bf Z}(n-1)) \to H_1(Y(\BR), {\BZ/2}). $$ Observe, that $\alpha_n = \alpha_{n-1} \circ s^{+/-}_n$. \begin{lem}\label{lem codim} For a real Enriques surface $Y$, the codimension of $\operatorname{Im} \alpha_2$ in $H_1(Y(\BR),{\BZ/2})$ does not exceed $1$. \end{lem} \begin{proof} We may assume that $Y(\BR) \ne \emptyset$. Using the fact that $\alpha_{-1}$ is an isomorphism by Proposition~\ref{prop zhomdecompo}, and both $s^{-}_0$ and $s^{+}_{1}$ are surjective by the long exact sequence~(\ref{se edge}), we see that $\alpha_1$ is surjective. Since $\alpha_1 = \alpha_2 \circ s^-_2$, it suffices to remark that if the cokernel of $s^-_2\fcolon H_2(Y(\BC); G, {\Bf Z}(1)) \to H_1(Y(\BC); G, {\Bf Z})$ is nonzero, it is isomorphic to $H_1(Y(\BC), {\Bf Z}) = {\BZ/2}$. \end{proof} \begin{prop}\label{prop rho} Let $Y$ be a real Enriques surface. A class $\gamma\inH_1(Y(\BR),{\BZ/2})$ is contained in the image of $\alpha_2$ if and only if $$ \deg(\gamma \cap w_1(Y(\BR))) = 0, $$ where $w_1(Y(\BR)) \in H^1(Y(\BR), {\BZ/2})$ is the first Stiefel-Whitney class of $Y(\BR)$. \end{prop} \begin{proof} Again we may assume that $Y(\BR) \ne \emptyset$. Denote by $\Omega$ the subspace of $H_1(Y(\BR),{\BZ/2})$ whose elements $\gamma$ verify $\deg(\gamma\cap w_1(Y(\BR))) = 0$. If $Y(\BR)$ is orientable, $w_1(Y(\BR))=0$ and $\Omega=H_1(Y(\BR),{\BZ/2})$. Furthermore, we have a surjective morphism $$ H_1(X_1({\Bf R}),{\BZ/2}) \oplus H_1(X_2({\Bf R}),{\BZ/2})\toH_1(Y(\BR),{\BZ/2}) $$ where the $X_1$ and $X_2$ are the two real K3-surfaces covering $Y$ (see the beginning of this section). This morphism fits in a commutative diagram $$ \begin{CD} H_2(X_1({\Bf C}); G, {\Bf Z}(1))\oplusH_2(X_2({\Bf C}); G, {\Bf Z}(1)) @>>> H_2(Y(\BC); G,{\Bf Z}(1))\\ @V{\alpha_2^{X_1}\oplus\alpha_2^{X_2}}VV @VV{\alpha_2}V \\ H_1(X_1({\Bf R}),{\BZ/2}) \oplus H_1(X_2({\Bf R}),{\BZ/2})@>>>H_1(Y(\BR),{\BZ/2}) \end{CD} $$ Here the $\alpha_n^{X_i} \fcolon H_n(X_1({\Bf C}); G, {\Bf Z}(n-1)) \to H_1(X_1({\Bf R}),{\BZ/2})$ are defined in the same way as $\alpha_n$. As $H_1(X(\BC),{\Bf Z})=0$ for a real K3-surface $X$, it follows from Lemma~\ref{lem odd-rho}, that $\alpha_2^{X_1}$ and $\alpha_2^{X_2}$ are surjective, which implies the surjectivity of $\alpha_2$. In other words, $\operatorname{Im} \alpha_2=\Omega$. Now assume that $Y(\BR)$ is nonorientable. Then $w_1(Y(\BR))\ne 0$, and by nondegeneracy of the cap-product pairing $\operatorname{codim}\Omega = 1$. First we will prove that $\operatorname{Im} \alpha_2 \subset \Omega$. Let $K=-cw_1(Y(\BC))\inH^2(Y(\BC); G, {\Bf Z}(1))$, where $cw_1(Y(\BC))$ is the first mixed characteristic class of the tangent bundle of $Y(\BC)$ as defined in \cite[3.2]{Kr2}. Then $e(K) \in H^2(Y(\BC),{\Bf Z})$ is the first Chern class of the canonical line bundle of $Y$, so $2 e(K)=0$ (see \cite[V.32]{BPV}). This means that for any $\gamma\inH_2(Y(\BC); G, {\Bf Z}(1))$ we have $\operatorname{deg_G}(\gamma\cap K)=\deg(e(\gamma) \cap e(K)) = 0$, so $\deg(\rho(\gamma) \cap \beta(K)) = 0$ by equations \eqref{deg-and-rho1} and \eqref{cap-and-rho}. The projection $\rho_{2,2}(\gamma)$ of $\rho(\gamma) \in H_*(Y(\BR), {\BZ/2})$ to $H_2(Y(\BR), {\BZ/2})$ is zero by equation \eqref{eq bockstein} and the projection $\beta^{2,0}(K)$ of $\beta(K) \in H^*(Y(\BR), {\BZ/2})$ to $H^0(Y(\BR), {\BZ/2})$ is zero by \cite[Th.~0.1]{Kr3}. This implies $$ \deg(\rho(\gamma) \cap \beta(K)) = \deg (\rho_{2,1}(\gamma) \cap \beta^{2,1}(K) ), $$ but $\beta^{2,1}(K) = w_1(Y(\BR))$ by \cite[Th.~3.2.1]{Kr2}, and $\rho_{2,1}(\gamma)= \alpha_2(\gamma)$ by definition, so $\deg(\alpha_2(\gamma) \cap w_1(Y(\BR)))=0$. In other words, $\operatorname{Im} \alpha_2\subset\Omega$. Lemma~\ref{lem codim} now gives us that $\operatorname{Im} \alpha_2 = \Omega$. \end{proof} \begin{cor} \label{cor alpha} With the above notation, $\alpha_2$ is surjective if and only if $Y({\Bf R})$ is orientable. \end{cor} Theorem~\ref{theo main} in the introduction is an immediate consequence of Proposition~\ref{prop rho}. We can even give an explicit description of $H_1^\mathrm{alg}(Y(\BR), {\BZ/2})$. \begin{theo}\label{theo c alg} Let $Y$ be a real Enriques surface. A class $\gamma\inH_1(Y(\BR),{\BZ/2})$ can be represented by an algebraic cycle if and only if $\deg(\gamma \capw_1(Y(\BR))) = 0$. \end{theo} \section{Galois-Maximality}\label{sec galmax} The aim of this section is to describe which Enriques surfaces are {\itshape{$\BZ$-GM}}-varieties and/or {\itshape{GM}}-varieties in terms of the orientability of the real part and the distribution of the components over the halves. See the introduction for the definition of Galois-Maximality and Section \ref{sec enriques} for the definition of 'halves'. The proof of Theorem \ref{theo galmax} will consist of a collection of technical results and explicit constructions of equivariant homology classes. For completeness we also prove the parts of Theorem \ref{theo galmax} concerning coefficients in ${\BZ/2}$, although these results are not new (see the introduction). \begin{lem}\label{caract gm} Let $Y$ be an algebraic variety over ${\Bf R}$. Then \begin{enumerate} \item $Y$ is {\itshape{$\BZ$-GM}}\ if and only if $e^+_p$ is surjective on $H_p(Y(\BC),{\Bf Z})^G$ and $e^-_p$ is surjective onto $H_p(Y(\BC),{\Bf Z}(1))^G$ for all $p$. \item $Y$ is {\itshape{GM}}\ if and only $e_p$ is surjective onto $H_p(Y(\BC),{\BZ/2})^G$ for all $p$. \end{enumerate} \end{lem} \begin{proof} This follows from the fact that $Y$ is {\itshape{GM}}\ (resp. {\itshape{$\BZ$-GM}}) if and only if the Hochschild-Serre spectral sequence $E^r_{p,q}(Y(\BC); G, A)$ is trivial for $A = {\BZ/2}$ (resp. ${\Bf Z}$), and this can be checked by looking at the edge morphisms, since we have for every $k \geq 0$ and every $G$-module $M$ natural surjections $H^k(G, M) \to H^{k+2}(G, M)$, and $H^k(G, M) \to H^{k+1}(G, M(1))$, which are isomorphisms for $k > 0$. \end{proof} \begin{lem}\label{lem enr gm} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. Then \begin{enumerate} \item for any $p\in\{0,2,3,4\}$, $e^{+/-}_p$ is surjective onto $H_p(Y(\BC),{\Bf Z}(k))^G$, \item for any $p\in\{0,3,4\}$, $e_p$ is surjective onto $H_p(Y(\BC),{\BZ/2})^G$. \end{enumerate} \end{lem} \begin{proof} This can be seen from the Hochschild-Serre spectral sequences (cf. \cite[\S~5]{Kr1}). \end{proof} \begin{cor} \label{cor enr gm} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. Then $Y$ is {\itshape{$\BZ$-GM}}\ if and only if $e_1^{+/-}$ is surjective onto $H_1(Y(\BC), {\Bf Z}(k))^G$ for $k=0, 1$. Moreover, $Y$ is {\itshape{GM}}\ if and only if $e_1$ and $e_2$ are surjective onto $H_1(Y(\BC), {\BZ/2})^G$, resp.\ $H_2(Y(\BC), {\BZ/2})^G$. \end{cor} \begin{lem}\label{e2e1} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. If $e_2$ is not surjective onto $H_2(Y(\BC),{\BZ/2})^G$, then $e_1$ is not surjective onto $H_1(Y(\BC),{\BZ/2})^G$. \end{lem} \begin{proof} By Poincar\'e duality we see that if $e_2$ is not surjective onto $H_2(Y(\BC),{\BZ/2})^G$, then $e^2$ is not surjective onto $H^2(Y(\BC),{\BZ/2})^G$. Let us assume that $e^2$ is not surjective. Then by Lemma~\ref{lem e2-not-surj} there exists an $\omega \in H^1(Y(\BC); G, A(k-1))$ such that $e^1_{A(k-1)}(\omega) \ne 0$, but $\beta(\omega) = 0$. Now suppose $e_1$ is surjective onto $H_1(Y(\BC),{\BZ/2})^G$, then there exists a $\gamma\in H_1(Y(\BC);G,{\BZ/2})$ such that $$ \deg(e_1(\gamma)\cap e^1(\omega))\ne 0. $$ This means that $\operatorname{deg_G}(\gamma \cap \omega) \ne 0$, but this contradicts $$ \operatorname{deg_G}(\gamma \cap \omega) = \deg(\rho(\gamma) \cap \beta(\omega)) = \deg(\rho(\gamma) \cap 0) = 0. $$ Hence $e_1$ is not surjective. \end{proof} \begin{prop}\label{prop zgmzzgm} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. Then \begin{enumerate} \item $Y$ is {\itshape{$\BZ$-GM}}\ if and only if $e^+_1$ and $e^-_1$ are nonzero. \item $Y$ is {\itshape{GM}}\ if and only if $e_1$ is nonzero. \item If $e_1$ is zero then $e^+_1$ and $e^-_1$ are zero. In particular, if $Y$ is {\itshape{$\BZ$-GM}}, then $Y$ is also {\itshape{GM}}. \end{enumerate} \end{prop} \begin{proof} If $Y$ is an Enriques surface, $$ H_1(Y(\BC),{\Bf Z})=H_1(Y(\BC),{\BZ/2})={\BZ/2}, $$ so $e^{+/-}_1$ and $e_1$ are surjective if and only if they are nonzero. By Lemma~\ref{e2e1}, $e_2$ is surjective if $e_1 \ne 0$, so we obtain the first two assertions from Corollary~\ref{cor enr gm}. The last assertion follows from the commutative diagram $$\begin{CD} H_1(Y(\BC); G, {\Bf Z}(k)) @>{e^{+/-}_1}>> H_1(Y(\BC), {\Bf Z}(k))\\ @VVV @VVV \\ H_1(Y(\BC); G, {\BZ/2}) @>{e_1}>> H_1(Y(\BC), {\BZ/2}) \end{CD}$$ \end{proof} \begin{lem}\label{lem e1+} Let $Y$ be a real Enriques surface with $Y(\BR)\ne\emptyset$. Then $e_1^+=0$ if and only if $Y(\BR)$ is orientable. \end{lem} \begin{proof} We know from Corollary~\ref{cor alpha}, that $\alpha_2$ is surjective if and only if $Y(\BR)$ is orientable. Since $H_1(Y(\BC),{\Bf Z})= {\BZ/2}$, the mapping $H_1(Y(\BC), {\Bf Z}) ^G \xrightarrow{\cup \eta^2} H^2(G,H_1(Y(\BC), {\Bf Z}))$ is an isomorphism, so Lemma~\ref{lem odd-rho} gives us that $\alpha_2$ is surjective if and only if $e_1^+=0$. \end{proof} \begin{lem}\label{lem e1-} If the two halves $Y_1$ and $Y_2$ of a real Enriques surface $Y$ are nonempty, then $e^-_1\ne 0$. \end{lem} \begin{proof} Let $X$ be the K3-covering of $Y_{\Bf C}$, let $\tau$ be the deck transformation of this covering and denote by $\sigma_1$ and $\sigma_2$ the two different involutions of $X(\BC)$ lifting the involution $\sigma$ of $Y(\BC)$. Let $X_i({\Bf R})$ be the set of fixed points under $\sigma_i$ and let $p_i$ be a point in $X_i({\Bf R})$ for $i = 1,\;2$. Let $l$ be an arc in $X(\BC)$ connecting $p_1$ and $p_2$ without containing any other point of $X_1({\Bf R})$ or $X_2({\Bf R})$. Then the union $L$ of the four arcs $l$, $\sigma_1(l)$, $\sigma_2(l)$, $\tau(l)$ is homeomorphic to a circle, and we have that $\tau(L)=L$. This implies that the image $\lambda$ of $L$ in $Y(\BC)$ is again homeomorphic to a circle; we choose an orientation on $\lambda$. Now $G$ acts on $\lambda$ via an orientation reversing involution, so $\lambda$ represents a class $[\lambda]$ in $H_1(Y(\BC); G, {\Bf Z}(1))$. Since $X(\BC) \to Y(\BC)$ is the universal covering, and the inverse image of $\lambda$ is precisely $L$, hence homeomorphic to a circle, the class of $\lambda$ is nonzero in $H_1(Y(\BC),{\Bf Z})$, so $e^-_1([\lambda]) \ne 0$. \end{proof} \begin{lem}\label{lem one or} If exactly one of the halves $Y_1$, $Y_2$ of a real Enriques surface $Y$ is empty, then $e_1 = 0$ if and only if $Y(\BR)$ is orientable. \end{lem} \begin{proof} If $e_1=0$, we have $e_1^+=0$ by Proposition~\ref{prop zgmzzgm} and then $Y(\BR)$ is orientable by \ref{lem e1+}. Conversely, if $Y(\BR)$ is orientable and $X_2({\Bf R}) = \emptyset$, then $X_1({\Bf R}) \to Y({\Bf R})$ is the trivial double covering, so it induces a surjection $H_*(X_1({\Bf R}), {\BZ/2})^0 \to H_*(Y(\BR), {\BZ/2})^0$, where $H_*({-}, {\BZ/2})^0$ denotes the kernel of the degree map as defined in Section~\ref{sec equiv}. Since $H_1(X(\BC), {\BZ/2}) = 0$, the mapping $\rho\fcolon H_2(X_1({\Bf C}); G, {\BZ/2}) \to H_*(X_1({\Bf R}), {\BZ/2})^0$ is surjective by Lemma~\ref{lem rho-zz}. Now the functoriality of $\rho$ with respect to proper equivariant mappings (equation (\ref{f-and-rho})) implies $$ \rho_2 \fcolonH_2(Y({\Bf C}); G, {\BZ/2}) \to H_*(Y({\Bf R}), {\BZ/2}) $$ is surjective, and Lemma~\ref{lem rho-zz} then gives that $e_1$ is zero. \end{proof} \begin{lem}\label{lem euler char} If exactly one of the halves $Y_1$, $Y_2$ of a real Enriques surface $Y$ is empty, then $e^-_1 \ne 0$ if and only if $Y(\BR)$ has components of odd Euler characteristic. \end{lem} \begin{proof} Assume $Y_2 = \emptyset$. By Lemma~\ref{lem even-rho-z}, it suffices to show that $$ \rho_{2,\mathrm{even}} \fcolon H_2(Y(\BC); G, {\Bf Z}) \to H_\mathrm{even}(Y(\BR), {\BZ/2})^0 $$ is surjective if and only if $Y(\BR)$ has no components of odd Euler characteristic. Although $Y(\BR)$ need not be orientable, we can apply the K3-covering as in the previous lemma and prove that the image of $\rho_{2,\mathrm{even}}$ contains a basis for the subgroup $H_0(Y(\BR), {\BZ/2}) \cap H_\mathrm{even}(Y(\BR), {\BZ/2})^0$, so $\rho_{2,\mathrm{even}}$ is surjective if and only if $$ \rho_{2,2} \fcolon H_2(Y(\BC); G, {\Bf Z}) \to H_2(Y(\BR), {\BZ/2}) $$ is surjective. We will use that $H_2(Y(\BR), {\BZ/2})$ is generated by the fundamental classes of the connected components of $Y(\BR)$. Pick a component $V$ of $Y(\BR)$. If $V$ is orientable, it gives a class in $H_2(Y(\BC); G, {\Bf Z})$, which maps to the fundamental class of $V$ in $H_2(Y(\BR), {\BZ/2})$. Now assume $V$ is nonorientable. Let $[V]$ be the fundamental class of $V$ in $H_2(Y(\BR), {\BZ/2})$, let $[V]_G$ be the class represented by $V$ in $H_2(Y(\BC); G, {\BZ/2})$, and let $\gamma = \delta([V]_G)$ be the Bockstein image in $H_1(Y(\BC); G, {\Bf Z}(1))$. Then $\rho_{1,2}(\gamma) = \rho_{2,2}([V]_G) = [V]$ by equation~(\ref{rho-and-delta}), so $[V]$ is in the image of $H_2(Y(\BC); G, {\Bf Z})$ under $\rho_{2,2}$ if and only if $e^-_1(\gamma) = 0$. From the construction of $\gamma$ we see that $e^-_1(\gamma) = i_* \delta([V])$, where $i \fcolon V \to Y(\BC)$ is the inclusion and $\delta([V]) \in H_1(V, {\Bf Z})$ is the Bockstein image of $[V]$. Therefore $e^-_1(\gamma)$ can be represented by a circle $\lambda$ embedded in $V$. Since $X(\BC) \to Y(\BC)$ is the universal covering, $e^-_1(\gamma)$ is zero if and only if the inverse image $L$ of $\lambda$ in $X(\BC)$ has two connected components. Let $W$ be the component of $X_1({\Bf R})$ covering $V$. Then $W$ is the orientation covering of $V$ and $L \subset W$. If $V$ has odd Euler characteristic, then it is the connected sum of a real projective plane and an orientable compact surface. We see by elementary geometry that $L$ is connected. If $V$ has even Euler characteristic, it is the connected sum of a Klein bottle and an orientable compact surface, and we see that $L$ has two connected components. \end{proof} \begin{proof}[Proof of Theorem \ref{theo galmax}] By Proposition~\ref{prop zgmzzgm}, the first part of the theorem follows from Lemma~\ref{lem e1+} and Lemma~\ref{lem e1-}, and the second part of the theorem follows from Lemma~\ref{lem one or} and lemma~\ref{lem euler char}. \end{proof} \section{The Brauer group}\label{sec brauer} Let $Y$ be a nonsingular projective algebraic variety defined over ${\Bf R}$. Let $$\operatorname{Br}'(Y) = H^2_{\text{\textrm{\'et}}}(Y, {\mathbb{G}}_m)$$ be the cohomological Brauer group of $Y$, and let $\operatorname{Tor}(n, \operatorname{Br}'(Y))$ be the $n$-torsion of $\operatorname{Br}'(Y)$. We have a canonical isomorphism \begin{multline} \label{eq torbr} \operatorname{Tor}(n, \operatorname{Br}'(Y)) \simeq \\ \simeq \operatorname{Coker} \left \{H^2_\mathrm{alg}(Y(\BC); G, {\Bf Z}(1)) \labelto{\bmod n} H^2(Y(\BC); G, {{\Bf Z}/n}(1)) \right \}, \end{multline} as can be deduced from the Kummer sequence $$ 1 \labelto{} {\boldsymbol{\mu}}_n \labelto{} {\mathbb{G}}_m \labelto{\times n} {\mathbb{G}}_m \labelto{} 1, $$ and the well-known identifications \begin{align*} H^k_{\text{\textrm{\'et}}}(Y, {\boldsymbol{\mu}}_n) & \simeq H^k(Y(\BC); G, {{\Bf Z}/n}(1)) \\ \intertext{and} H^1(Y, {\mathbb{G}}_m) & \simeq \operatorname{Pic}(Y). \end{align*} It can be checked, that the mapping $$\beta^{2,0} \fcolon H^2(Y(\BC); G, {\BZ/2}) \to H^0(Y(\BR), {\BZ/2})$$ induces a well-defined homomorphism \begin{equation} \label{CTP} \operatorname{Tor}(2, \operatorname{Br}'(Y)) \to H^0(Y(\BR), {\BZ/2}). \end{equation} If $\dim Y \leq 2$, in particular if $Y$ is a real Enriques surface, we may identify $\operatorname{Br}'(Y)$ with the classical Brauer group $\operatorname{Br}(Y)$ (see \cite[II, Th.~2.1]{Brauer}). Two of the main problems considered in \cite{N-S} and \cite{N1} are the calculation of $\dim_{\BZ/2} \operatorname{Tor}(2, \operatorname{Br}(Y))$ and the question whether the mapping \eqref{CTP} is surjective for every real Enriques surface $Y$. Both problems were solved for certain classes of real Enriques surfaces. The second problem has been completely solved in \cite{Kr3}, where it is shown that the mapping \eqref{CTP} is surjective for any nonsingular projective surface $Y$ defined over ${\Bf R}$ (see Remark~3.3 in \emph{loc.\ cit.}). The results in Section~\ref{sec galmax} will help us to solve the first problem for every Enriques surface $Y$ by determining the whole group $\operatorname{Br}(Y)$. \begin{lem}\label{lem torbr} Let $Y$ be a nonsingular projective algebraic variety defined over ${\Bf R}$ such that $$ H^2_\mathrm{alg}((Y(\BC); G, {\Bf Z}(1)) = H^2(Y(\BC); G, {\Bf Z}(1)). $$ Then $$ \operatorname{Tor}(\operatorname{Br}'(Y)) \simeq \operatorname{Tor}(H^3(Y(\BC); G, {\Bf Z}(1))). $$ \end{lem} \begin{proof} By the hypothesis and the isomorphism \eqref{eq torbr} there is for every integer $n > 0$ a short exact sequence $$ H^2(Y(\BC); G, {\Bf Z}(1)) \tensor {{\Bf Z}/n} \to H^2(Y(\BC); G, {{\Bf Z}/n}(1)) \to \operatorname{Tor}(n, \operatorname{Br}'(Y)), $$ hence we deduce from the long exact sequence in equivariant cohomology associated to the short exact sequence $$ 0 \to {\Bf Z}(1) \labelto{\times n} {\Bf Z}(1) \to {{\Bf Z}/n}(1) \to 0 $$ that we have for every $n > 0$ a natural isomorphism $$ \operatorname{Tor}(n, \operatorname{Br}'(Y)) \simeq \operatorname{Tor}(n, H^3(Y(\BC); G, {\Bf Z}(1))). $$ \end{proof} \begin{proof}[Proof of Theorem \ref{theo brauer}] By \cite[I.2 and II, Th.~2.1]{Brauer} we have $\operatorname{Br}(Y) = \operatorname{Tor}(\operatorname{Br}(Y)) = \operatorname{Tor}(\operatorname{Br}'(Y))$. On the other hand, $\operatorname{Tor}(H^3(Y(\BC); G, {\Bf Z}(1))) = H^3(Y(\BC); G, {\Bf Z}(1)) $ since $H^3(Y(\BC), {\Bf Z}) = {\BZ/2}$. Hence, by Lemma~\ref{lem torbr} and Poincar\'e duality $$\operatorname{Br}(Y) \simeq H_1(Y(\BC); G, {\Bf Z}(1)). $$ Now consider the long exact sequence~\eqref{se edge} for $A(k) = {\Bf Z}$: $$ \dots \labelto{e_1^+} H_1(Y(\BC), {\Bf Z}) \to H_1(Y(\BC); G, {\Bf Z}(1)) \labelto{s_1^-} H_0(Y(\BC); G, {\Bf Z}) \to \dotsb $$ It follows from Proposition~\ref{prop zhomdecompo} and the long exact sequence~\eqref{se edge} for $A(k) = {\Bf Z}(1)$ that $\rho \fcolon H_*(Y(\BC); G, {\Bf Z}) \to H_*(Y(\BR), {\BZ/2})$ induces an isomorphism $$ \operatorname{Im} s_1^- \overset{\sim}{\to} H_\mathrm{even}(Y(\BR), {\BZ/2})^0.$$ We obtain an exact sequence \begin{equation}\label{se br} \dots \labelto{e_1^+} {\BZ/2} \to H_1(Y(\BC) ; G, {\Bf Z}(1)) \to H_\mathrm{even}(Y(\BR), {\BZ/2})^0\to 0. \end{equation} \noindent If $Y(\BR) \neq \emptyset$ is nonorientable, then $e_1^+ \neq 0$ by Lemma~\ref{lem e1+}, so $H_1(Y(\BC) ; G, {\Bf Z}(1)) \simeq ({\BZ/2})^{2s-1}$, which proves the first part of the theorem. Now assume $Y(\BR) \neq \emptyset$ is orientable. Then $e_1^+ = 0$ by Lemma~\ref{lem e1+}, so we get from~\eqref{se br} an exact sequence \begin{equation*} 0 \to {\BZ/2} \to H_1(Y(\BC); G, {\Bf Z}(1)) \to ({\BZ/2})^{2s -1} \to 0, \end{equation*} hence $H_1(Y(\BC); G, {\Bf Z}(1)) \simeq ({\BZ/2})^{2s}\ \text{or}\ ({\BZ/2})^{2s-2} \oplus ({\Bf Z}/4).$ In order to decide between these two possibilities, consider the following commutative diagram with exact rows. $$\begin{CD} H_2(Y(\BC); G, {\BZ/2}) @>{\delta^-}>> H_1(Y(\BC); G, {\Bf Z}(1)) @>{\times 2}>> H_1(Y(\BC); G, {\Bf Z}(1)) \\ @V{e_2}VV @V{e_1^-}VV @V{e_1^-}VV \\ H_2(Y(\BC), {\BZ/2}) @>{\delta}>> H_1(Y(\BC), {\Bf Z}) @>{\times 2}>> H_1(Y(\BC), {\Bf Z}) \\ @A{e_2}AA @A{e_1^+}AA @A{e_1^+}AA \\ H_2(Y(\BC); G, {\BZ/2}) @>{\delta^+}>> H_1(Y(\BC); G, {\Bf Z}) @>{\times 2}>> H_1(Y(\BC); G, {\Bf Z}) \end{CD}$$ We have that $H_1(Y(\BC); G, {\Bf Z}(1))$ is pure $2$-torsion if and only if $\delta^-$ is surjective. We claim that $\delta^-$ is surjective if and only if $e_1^- = 0$. Together with Lemmas~\ref{lem euler char} and \ref{lem e1-} this would prove the second part of the theorem. Let us prove the claim. Since $e_1^+ = 0$, we have $\delta \circ e_2 =0$. If $e_1^- \neq 0$, an easy diagram chase shows that $\delta^-$ is not surjective. On the other hand the following diagram can be shown to be commutative. \[ \setlength{\unitlength}{0.00083300in} \begin{picture}(4700,1200)(58,-319) \thinlines \put(751,464){\vector( 0,-1){525}} \put(1576,539){\vector( 3,-1){1350}} \put(1576,-211){\vector( 1, 0){1350}} \put(751,614){\makebox(0,0)[b]{\smash{$H_2(Y(\BC); G, {\Bf Z})$}}} \put(751,-286){\makebox(0,0)[b]{\smash{$H_2(Y(\BC) ; G, {\BZ/2})$}}} \put(3226,-286){\makebox(0,0)[lb]{\smash{$H_1(Y(\BC); G, Z(1))$}}} \put(2176,-136){\makebox(0,0)[b]{\smash{$\scriptstyle \delta^-$}}} \put(2176,464){\makebox(0,0)[lb]{\smash{$\scriptstyle s^+_2$}}} \put(676,239){\makebox(0,0)[rb]{\smash{$\scriptstyle \bmod 2$}}} \end{picture} \] In other words, $\operatorname{Im} s_2^+ \subset \operatorname{Im} \delta^-$. Now $\ker e_1^- = \operatorname{Im} s_2^+$, so if $e_1^- = 0$, then $\delta^-$ is surjective. Finally, we will consider the short exact sequence~\eqref{se br} for the case $Y(\BR) = \emptyset$. Then $G$ acts freely on $Y(\BC)$, so we have for all $k$ that $H_k(Y(\BC); G, {\BZ/2}) = H_k(Y(\BC)/G, {\BZ/2})$. By the remarks made in the introduction of Section~\ref{sec enriques}, this means that $H_1(Y(\BC); G, {\BZ/2}) = {\BZ/2} \times {\BZ/2}$, and we can see from the long exact sequence~\eqref{se edge} for $A(k) = {\BZ/2}$ that $e_1 = 0$. This implies that $e^+_1 = 0$ (see Proposition~\ref{prop zgmzzgm}.iii), hence $H_1(Y(\BC); G, {\Bf Z}(1)) = {\BZ/2}$. \end{proof}

9606

alg-geom/9606008

en

https://arxiv.org/abs/alg-geom/9606008

alg-geom/9606008

Michal Kwiecinski

Michal Kwiecinski and Piotr Tworzewski

Finite sets in fibres of holomorphic maps

LaTeX v. 2.09, 16 pages

IMUJ preprint 1996/08, Jagiellonian Univ., Krakow

We consider the maximal number of arbitrary points in a special fibre that can be simultaneously approached by points in one sequence of general fibres. Several results about this topological invariant and their applications describe the structure of holomorphic maps. In particular, we get a lower bound on the number of points in the general fibre of a generically finite map.

alg-geom

\section{Introduction.} From the work of Thom \cite{Thom}, Fukuda \cite{Fukuda} and Nakai \cite{Nakai}, it follows that one cannot stratify arbitrary complex algebraic maps so as to have local topological triviality, such as in the case of Whitney stratified spaces. Indeed, an arbitrary complex map can have a locally infinite number of local topological types at points of the source space. Thus, research on the topology of complex maps was mainly focused on maps satisfying Thom's $a_f$ condition or similar conditions implying some kind of local topological triviality and for example leading the way to vanishing cycles (see e.g. \cite{BMM}, \cite{HMS} and references therein for both classical and recent results on $a_f$ maps). Therefore, the topology of equidimensional maps seems to have received much greater attention than that of non-equidimensional ones. In particular, it seems to have been unknown, that if the generic fibre is discrete but there are special fibres of positive dimension, then there is a lower bound on the number of points in the generic fibre (Theorem \ref{npoints}), which has a simple form if the fibres of positive dimension are isolated (Theorem \ref{isonpoints}). Not having topological constructibility in general, we can still get some insight into the topological structure of holomorphic maps. Using Hironaka's flattening theorem \cite{Hironaka} (and its local version by Hironaka, Lejeune-Jalabert and Teissier \cite{HLT}), Sabbah proves that any map can be made into an $a_f$ map, after a base change by a blowup, thus giving a precise meaning to Thom's ``hidden blowups". A recent result of Parusi\'nski \cite{Parusinski} states that the set of points at which a holomorphic map is not open is analytically constructible. In this paper we shall deal with the following natural problem concerning fibres of holomorphic maps which, to our best knowledge, has not been treated even for complex algebraic maps. Take $i$ points in a fibre of a holomorphic map $f$ and ask whether one can approximate them simultaneously by systems of $i$ points in arbitrarily general neighbouring fibres. Then ask for what maximal $i$ this is always possible and call that number $\app $. Our aim is to prove several theorems about $\app $ and give some applications of it. As will become clear from our results, $\app$ gives some idea of how general fibres converge to special fibres. In particular we shall prove, that for maps to a locally irreducible space, $\app$ is infinite iff the map is open and on the other hand, if $\app$ is finite, then it is smaller than the dimension of the target space (Theorem \ref{mainapp}). We also have similar results for maps to general spaces (Theorems \ref{opennessapp} and \ref{redapp}). For maps to a smooth space, we shall obtain an effective formula for $\app$ in terms of dimensions of the loci where fibres have constant dimension (Theorem \ref{eff}). As a consequence, we obtain a lower bound for the number of points in a generic discrete fibre of a holomorphic map, which also has fibres of small positive dimension (Theorems \ref{npoints} and \ref{isonpoints}). \bigskip \section{Statement of results.} We start by defining $\app$ precisely. For the sake of clarity, as above we break the definition up into two parts. \begin{Def} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Let $x_1,\dots,x_i$ belong to one fibre $f^{-1}(y)$. We say that the sequence of points $x_1,\dots,x_i$ can be approximated by general fibres iff for any boundary set (set with empty interior) $B\subset Y$, there exists a sequence $\{ y_j\}$ in $Y-B$ such that $y_j\to y$ and sequences $\{ x_{1j}\},\dots,\{ x_{ij}\}$ such that for all $k$, $x_{kj}\in f^{-1}(y_j)$, for all $j$ and $x_{kj}\to x_k$, with $j\to\infty$. \end{Def} \begin{Def} For a holomorphic map of analytic spaces $f:X\to Y$ define $\app$ as the supremum of all $i$, such that any sequence of $i$ points in (any) one fibre of $f$ can be approximated by general fibres (and as zero if no such $i$ exists). \end{Def} In this paper {\it analytic space} means reduced complex analytic in the sense of Serre (cf. \cite{Lojasiewicz}). Analytic spaces shall always be considered with their transcendental topology (and not the Zariski topology). While no assumption will be made on the source space $X$, our results will depend on the different assumptions that we shall make on the target space $Y$. In particular, throughout the paper we assume that $Y$ is of finite dimension. Notice that the value of $\app$ will not change if in the definition we demand only that any sequence of pairwise different points in one fibre can be approximated. The arbitrary choice of the boundary set translates the intuitive notion of arbitrarily general fibre. It will follow easily from our results that for proper maps ``boundary" can be replaced by ``nowhere dense analytic", without changing $\app$. For (not necessarily proper) algebraic maps ``boundary" can be replaced by ``nowhere dense algebraic". The following examples illustrate the meaning of the number $\apps$. The value of $\apps$ is infinite for a locally trivial fibration and is zero for a closed (nontrivial) embedding in a complex manifold. For a blowup and more generally for any modification, $\apps$ is equal to 1. The example below, shows that different values of $\apps$ are possible. \begin{Exa} \label{breakpoint} Fix an integer $d\geq 1$. Consider ${\bf C}^{d}\times{\bf C}^{d}$ with variables $(y_1,\dots,y_d,x_1,\dots,x_d)$ and let $X$ be the hypersurface given by the equation $y_1x_1+\cdots+y_dx_d=0$. Let $f:X\to {\bf C}^{d}$ be the restriction of the first projection. Then it is easy to see that $\app=d-1$. \end{Exa} \bigskip The map in the above example is in fact the canonical projection of a spectrum of a symmetric algebra \cite{Vasconcelos} of a ${\bf C}[y_1,\dots,y_d]$ - module to the spectrum of ${\bf C}[y_1,\dots,y_d]$. Using different terminology \cite{Fischer}, one would say that it is the structural projection of a linear space (``vector bundle with singularities") associated to a coherent sheaf on ${\bf C}^{d}$. In this particular case, an equivalent problem to that of bounding $\app$ has been studied in \cite{Michal} and has produced a criterion for projectivity. The main goal of this paper is to prove the following four theorems. \begin{Thm} \label{mainapp} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being of dimension $d$ and locally irreducible. Then the following conditions are equivalent: \begin{enumerate} \item $\app=\infty$, \item $\app\geq d$, \item $f:X\to Y$ is an open map. \end{enumerate} \end{Thm} \vskip 3mm The above theorem says in particular that if $\app<\infty$, then then the number of points in a special fibre that can be approximated by points in a general fibre is small: $\app\leq d-1$. Notice that this bound does not depend on the source space and the map, but only on the dimension of the target space. This theorem can be generalized to the case of a non locally irreducible target in two ways. The first one is yet again a characterization of openness. \begin{Thm} \label{opennessapp} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Let $d=\dim Y$ and let $\pi:\hat Y\to Y$ be the normalization of $Y$. Let $\hat f:\fibprod{\hat Y}YX\to \hat Y$ be the canonical map. Then the following conditions are equivalent: \begin{enumerate} \item $\apps(\hat f)=\infty$, \item $\apps(\hat f)\geq d$, \item $f:X\to Y$ is an open map. \end{enumerate} \end{Thm} \vskip 3mm Remember that we are dealing with the transcendental topology, where the openness of a map does not have to agree with openness in the Zariski topology (consider the normalization of an irreducible curve with an ordinary double point). In fact, in the algebraic case, openness in the transcendental topology is equivalent to universal openness in the Zariski topology (see \cite{Parusinski}). Another generalization of theorem \ref{mainapp} requires us to recall some notions. Recall (cf.\cite{Lojasiewicz} p.295, \cite{Stoll} p.16) that for any holomorphic map $f:Z\to Y$ of analytic spaces, the {\it fibre dimension} and the {\it Remmert Rank} of $f$ at $z\in Z$ are defined by $${\rm fbd}_z f=\dim_z f^{-1}(f(z)),\qquad {\rm\rho}_z f=\dim_z Z-{\rm fbd}_z f.$$ Recall also, that we have the inequality ${\rm\rho}_zf\leq\dim_{f(z)}Y$. As in \cite{Lojasiewicz}, given a map $f:Z\to Y$ and a subset $V\subset Y$, we shall denote the two-sided restriction $f\vert_{f^{-1}V}:f^{-1}V\to V$, by the symbol $f^{V}$. \begin{Thm} \label{redapp} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Suppose that there is an integer $D$ such that the sum of dimensions of irreducible components of any germ of \ $Y$ is at most equal $D$. Then the following conditions are equivalent: \begin{enumerate} \item $\app=\infty$, \item $\app\geq D$, \item for any $y\in Y$, there is a neighbourhood $V$ of $y$ in $Y$, an irreducible component $V_1$ of \ $V$ passing through $y$ and irreducible at $y$, such that for any $x\in f^{-1}(y)$ we have ${\rm\rho}_xf^{V_1}=\dim_y V_1$. \end{enumerate} \end{Thm} As we shall see from further results, the above theorem is a direct generalization of the locally irreducible case. Two things differ: $f$ need not be an open map if $\app$ is infinite and the value of $\app$ can be greater than the dimension of $Y$ even if $\app$ is finite. The examples below illustrate these two phenomena respectively. \begin{Exa} \label{notopen} Consider ${\bf C}^4$ with variables $(y_1,y_2,y_3,y_4)$ and let $Y\subset{\bf C}^4$ be the ``cross", given by the equations $y_1y_3 = y_1y_4 = y_2y_3 = y_2y_4 =0$. Consider $Y\times{\bf C}^2$ with additional variables $(x_1,x_2)$ and let $X\subset Y\times{\bf C}^2$ be given by the equation $y_1x_1+y_2x_2=0$. Let $f:X\to Y$ be the restriction of the first projection. Then $f$ is not open, but $\app=\infty$. \end{Exa} \vskip 5mm \begin{Exa} \label{D} Fix a positive integer $n$ and positive integers $d_1,\dots,d_n$. Let $D=d_1+\dots+d_n$ and consider ${\bf C}^D$ with variables $(y_{jk})$, $j=1,\dots,n$ and $k=1,\dots,d_j$. Let $Y$ be the reduced subspace of ${\bf C}^D$, defined by the monomial equations $$\ \{y_{j_1k_1}\cdots y_{j_nk_n}=0\ \vert\ j_s\neq s\ {\rm and }\ k_s=1,\dots,d_{j_s},\ {\rm for}\ s=1,\dots,n\}\ .$$ Then it is obvious, that the germ of $Y$ at $0$ has $n$ irreducible components, with respective dimensions $d_1,\dots,d_n$. Now fix $j$ and consider $Y\times{\bf C}^{d_j}$, with additional variables $(x_1,\dots,x_{d_j})$ on ${\bf C}^{d_j}$. Let $X_j$ be the subspace of $Y\times{\bf C}^{d_j}$, defined by the equation $$y_{j1}x_1+\cdots+y_{jd_j}x_{d_j}=0.$$ Let $f_j:X_j\to Y$ be the restriction of the natural projection. Now, define the space $X$ as the disjoint sum of the spaces $X_1,\dots,X_{n}$. Consider the map $f:X\to Y$, which coincides with $f_j$ on each $X_j$. One can easily calculate that $\app=D-1$. \end{Exa} Our final results will be an effective formula for $\app$, for maps to a smooth space and its consequences. Our invariant will be read off a partition of the source space which is well behaved with respect to the Remmert Rank. \begin{Def} \label{goodpar} \ \ Let $f:X\to Y$ be a holomorphic map of analytic spaces. A countable partition $\{ X_p \} _{p\in P} $ of $X$ is called {\bf a rank partition} $(for \ f)$ if for each $p\in P$: \begin{enumerate} \item $X_p$ is a nonempty irreducible locally analytic subset of $X$, \item $f|_{X_p}:X_p\to Y$ has constant Remmert Rank. \end{enumerate} \vskip 2mm \end{Def} \bigskip \noindent Standard arguments in stratification theory provide us with the following proposition. \begin{Prop} \label{exgoodpar} For any holomorphic map $f:X\to Y$, there exists a rank partition of $X$. \end{Prop} Actually it is always possible to find a locally finite rank partition which also has the property that the closure of each set $X_p$ is analytic. With a little more work one can prove that any holomorphic map has a rank stratification, i.e. a partition as above, which satisfies the boundary condition: for any $p,q$ if $\bar{X_p}\cap X_q\neq\emptyset$, then $\bar{X_p}\supset X_q$. However, a partition is more than enough for our results. \vskip 1truemm From a rank partition as above, we can read off some numerical data: $r_p$ -- the constant Remmert Rank of $f|_{X_p}$, $k_p=\dim X_p$, $h_p=\min\{\dim_xX:x\in X_p \}$. \bigskip \noindent These data alone allow us to evaluate our invariant. This is done in the theorem below. \begin{Thm} \label{eff} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being smooth of pure dimension $d$. Let $\{ X_p \}_{p\in P}$ be a rank partition of $X$. Then $$ \app=\inf \biggl \{ \biggl [\frac{d-r_p-1\phantom{w}} {(k_p-r_p)-(h_p-d)}\biggr ] \ : \ p\in P, \ k_p-r_p>h_p-d \ \biggr \} , $$ where the square brackets indicate the integer part of a rational number. \end{Thm} \bigskip The geometric meaning of the fraction in the above theorem is roughly the following: the denominator is the difference between the dimension of a special fibre and the dimension of the general fibre and the numerator is the codimension in $Y$ of the locus where that change in dimension occurs minus one. Although, from our proof it will easily follow that $\app$ is smaller or equal to the infimum in the above theorem even if $Y$ is not smooth, equality no longer holds in the general case. This is shown in the following example. \begin{Exa} \label{matrix} Let $Y$ be the space of 2 by 2 complex matrices with vanishing determinant. Let $X$ be the subset of $Y\times{{\bf P}^1\complex}$, consisting of all points $(A,(\lambda:\mu))$ satisfying the equation $A\left({\lambda\atop\mu}\right)=0$. Let $f:X\to Y$ be the restriction of the first projection. Then $\app=1$, but the infimum in Theorem \ref{eff} is equal to $2$. \end{Exa} In fact, a formula for $\app$, in the case when $Y$ is singular, would have to include more data than are used in the formula of theorem \ref{eff}: \begin{Exa} \label{moredata} Embed the space $Y$ of the preceding example as the hypersurface of ${\bf C}^4$ satisfying $xy-zw=0$. Define $X'\subset Y\times{{\bf P}^1\complex}$, by the equation $x\lambda^2+y\lambda\mu+(z-w)\mu^2=0$ and again, let $g:X'\to Y$ be the restriction of the first projection. The numerical data used in the formula in theorem \ref{eff} corresponding to $f$ and $g$ are the same, but $\app=1$ and $\apps(g)=2$. \end{Exa} Theorem \ref{eff} gives us a relationship between the dimensions of different fibres and the way they are attached to each other. From it one can deduce other information concerning the topology of holomorphic maps in more specific cases. For example, when dealing with a map $f$ whose generic fibres are discrete sets, it is obvious that if $f$ also has fibres of positive dimension, then $\app$ is not greater than the number of points in a generic fibre. In this situation, Theorem \ref{eff} provides the following lower bound for such a number (square brackets still denote the integer part). \begin{Thm} \label{npoints} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being smooth and both $X$ and $Y$ being of pure dimension $d$. Let $\{ X_p \}_{p\in P}$ be a rank partition of $X$. Suppose that $f^{-1}(f(x))$ is a discrete set for $x$ belonging to some open dense subset of $X$ and that $f$ has at least one fibre of positive dimension. Then there exists an open dense subset $U$ of $X$, such that for all $x\in U$ $$ \# f^{-1}(f(x)) \geq\inf \biggl \{ \biggl [\frac{d-r_p-1\phantom{.}}{k_p-r_p}\biggr ] \ : \ p\in P, \ k_p>r_p \ \biggr \} \ . $$ \end{Thm} \bigskip \noindent For an isolated fibre of positive dimension this gives: \begin{Thm} \label{isonpoints} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being smooth and both $X$ and $Y$ being of pure dimension $d$. Let $y_0$ be a point in $Y$, such that $\dim f^{-1}(y_0)=w_0>0$ and $\dim f^{-1}(y)=0$, for $y\neq y_0$. Then there exists an open dense set $U$ of $X$, such that for all $x\in U$ $$ \# f^{-1}(f(x)) \geq \biggl [\frac{d-1}{w_0}\biggr ]\ . $$ \end{Thm} The meaning of the above theorems is that if a map has discrete generic fibre, but also special fibres of small positive dimension along small sets, then there must be many points in the discrete generic fibre. An example of this situation is the universal homogeneous polynomial : \begin{Exa} Consider ${\bf C}^d$ with coordinates $x_0,\dots,x_{d-1}$ and ${{\bf P}^1\complex}$ with homogeneous coordinates $(\lambda:\mu)$. Let $X$ be the subspace of ${\bf C}^d\times{{\bf P}^1\complex}$ defined by the equation \newline $x_0\lambda^{d-1}+x_1\lambda^{d-1}\mu+\cdots +x_{d-1}\mu^{d-1}$ and let $f:X\to {\bf C}^d$ be the restriction of the first projection. Then the fibre of $f$ at $0$ is of dimension one; all the other fibres are zero-dimensional and the generic fibre has $d-1$ points (its smallest possible cardinality by theorem \ref{isonpoints}). \end{Exa} Again, if $Y$ is singular, then theorems \ref{npoints} and \ref{isonpoints} fail, as is seen from example \ref{matrix}. Further counterexamples are provided by "small contractions" (see \cite{CKM}), which are the basis of the study of threefolds. The invariant $\app$ is much less well behaved in real geometry. For example, in theorem \ref{mainapp} the only implications that are true are: openness implies $\app=\infty$ implies $\app\geq d$. In particular one can find real algebraic maps to ${\bf R}^2$ with arbitrary value of $\app$ : \begin{Exa} Fix a positive integer $n$. Let $\kappa_n=\tan\left({1\over2} \left({1\over{n+1}}+{1\over n}\right)\pi\right)$. Define the the real algebraic subset $X_n\subset{\bf R}^5$ by the equation $$ x_5^2=(x_1x_4+x_2x_3)(x_1x_4+x_2x_3-\kappa_n(x_1x_3-x_2x_4)) $$ and let $f_n:X_n\to{\bf R}^2$ be the restrction of the orthogonal projection on the $(x_1,x_2)$ plane. It is easy to calculate that $\apps(f_n)=n$. \end{Exa} \section{Fibred powers and quasiopenness.} In the category of analytic spaces fibred products exist are isomorphic to the usual ones (see \cite{Kaups}, p.200) after reduction. \begin{Def} Let $f:X\to Y$ be a holomorphic map of analytic spaces and $i\geq 1$. By the $i$-th {\bf fibred power} of $f$, we mean the pair $(\Xfib i,\ffib i)$ consisiting of the space $\Xfib i=\Xfiblong i$ and the canonical map $\ffib i :\Xfib i\to Y$. \end{Def} We shall use the same definition for fibred powers of continuous maps of topological spaces. The $i$-fold direct product of a space by itself will be denoted $X^i$. By definition, $X^0$ will be a point. Since a point in $\Xfib i$ is nothing else but a sequence of $i$ points in a fibre of $f$, we can easily obtain the following: \begin{Rem} For any $i\geq 1$, $\app \geq i$ iff $\apps(\ffib i)\geq 1$. \end{Rem} Hence, it is natural to determine what maps $f$ have $\app\geq 1$. We introduce the following notion. \begin{Def} \label{qodef} A map of topological spaces $f:Z\to Y$ is called {\bf quasiopen} if for any subset $A\subset Z$ with nonempty interior in $Z$, its image $f(A)$ has nonempty interior in $Y$. \end{Def} Any open map is quasiopen. The blowup ${\bf C}^2$ at the origin is an example of a quasiopen map which is not open. It is immediate that a map is quasiopen if and only if the image of any nonempty open set has nonempty interior. By elementary point-set topology one proves the following for first countable topological spaces. \begin{Rem} \label{basic} For a map of topological spaces $f:Z \to Y$ the following conditions are equivalent. \begin{enumerate} \item $f$ is quasiopen, \item for any boundary set $B\subset Y$ its inverse image $f^{-1}(B)$ is a boundary set in $Z$, \item $\app\geq 1$. \end{enumerate} \end{Rem} Thus, by the third equivalent condition, what we shall be looking at, will be the quasiopenness of fibred powers of holomorphic maps. The above two remarks easily imply the following \begin{Prop} \label{eval} $\app\ =\ \sup\left(\{0\}\cup\{i\geq 1:\ffib i {\rm\ is\ quasiopen}\}\right).$ \end{Prop} We must see more closely what quasiopenness means in the analytic case. The following proposition shows us that. We leave out its proof, which can be done by standard techniques of analytic geometry. \begin{Prop} \label{qoanal} For a holomorphic map of analytic spaces $f:Z\to Y$, the following conditions are equivalent: \begin{enumerate} \item $f$ is quasiopen, \item the restriction of $f$ to each irreducible component of $Z$ is quasiopen, \item the image by $f$ of each irreducible component of $Z$ has nonempty interior in $Y$. \end{enumerate} \end{Prop} \section{The Remmert Rank.} In this section we have gathered some facts about the Remmert Rank which we shall need in the sequel. The usefullness of the Remmert Rank comes from the following well known theorem (see e.g. \cite{Lojasiewicz}, p. 296). \medskip \noindent{\bf Remmert Rank Theorem }{\it Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $X$ being of pure dimension. Suppose that ${\rm\rho}_x f=k$ for all $x\in X$. Then every point of $X$ has an arbitrarily small open neighbourhood, whose image is a locally analytic subset of $Y$, of pure dimension $k$.} \medskip We shall need mainly some results about the sets where the Remmert Rank takes on a different value from its generic value. The first of these are two remarks. \begin{Rem} \label{subqo} Let $f:W\to Y$ be a quasiopen holomorphic map to an analytic space of pure dimension $d$. Let $W_1$ be an irreducible locally analytic subset of $W$ such that ${\rm\rho}_z(f\vert_{W_1})<d$ for all $z\in W_1$. Then $\dim W_1<\min\{\dim_zW\vert z\in W_1\}\ .$ \end{Rem} \proof{Remark \ref{subqo}} If the conclusion of the remark were false, then $W_1$ would contain a nonempty open subset of $W$. By the Remmert Rank Theorem this would contradict quasiopenness. {$ $} \noindent Remark \ref{subqo} immediately implies the next one. \begin{Rem} \label{qofibre} Let $f:Z\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being irreducible of positive dimension. If $f$ is quasiopen, then ${\rm fbd}_zf<\dim_zZ$ for any $z\in Z$. \end{Rem} \medskip In the following sections we shall also make use of a lemma describing the "critical values" with respect to the Remmert Rank. \begin{Lem} \label{sard} {\bf (Sard theorem for the Remmert Rank.)} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being irreducible of dimension $d$. Then the set $C(f)=f(\{x\in X\vert{\rm\rho}_xf<d\})$ is a first category set. \end{Lem} \proof{lemma \ref{sard}} Take a rank partition $\{X_p\}_{p\in P}$ for $f$. The lemma will follow from the the Remmert Rank Theorem if we prove that the set $C(f)$ is contained in the union of images of those sets $X_p$ for which $r_p<d$. (Notice that not all points $x\in X$ with ${\rm\rho}_xf<d$ have to belong to some $X_p$ with $r_p<d$. ) So, take $y\in C(f)$. There exists a point $x$ in the fibre $f^{-1}(y)$ such that ${\rm\rho}_xf<d$. Let $Z$ be a component of $f^{-1}(y)$ passing through $x$, of maximal dimension among such components. Then it is clear that $\min\{\dim_zX:z\in Z\}-\dim Z<d$. Then the family $\{Z\cap X_p\}_{p\in P}$ is an analytic partition of $Z$ and therefore for some $p$, $X_p$ contains a nonempty open subset of $Z$. Then $y\in f(X_p)$ and from the above inequality it follows that $r_p<d$. {$ $} \section{Maps to a locally irreducible space.} Theorem \ref{mainapp} is an immediate corollary of the theorem below and Proposition \ref{eval}. \begin{Thm} \label{main} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being of dimension $d$ and locally irreducible. Then the following conditions are equivalent: \begin{enumerate} \item the maps $\ffib i :\Xfib i\to Y$ are quasiopen for all $i=1,2,\dots$, \item the map $\ffib d :\Xfib d\to Y$ is quasiopen, \item the map $\ffib i :\Xfib i\to Y$ is quasiopen for some $i\geq d$, \item the map $f:X\to Y$ is open. \end{enumerate} \end{Thm} \vskip 3mm The above theorem provides an effective way of checking whether a given holomorphic map is open. Indeed, by condition 2 of theorem \ref{main}, one has to investigate the quasiopenness of the $d$-th fibred power of the map, which by conidtion 3 of proposition \ref{qoanal} can be tested just by looking at images of irreducible components. Thus, combined with primary decomposition algorithms (\cite{Eisenbud}), it provides algorithms for testing the openness of a map. \vskip 5mm \proof{Theorem \ref{main}} The space $Y$ being locally irreducible, its irreducible components are actually its connected components. Their dimensions are bounded from above by $d$. Therefore it is clear that in the proof of theorem \ref{main} we can assume that $Y$ is actually irreducible. Our proof will be structured as follows. First, we observe that condition 1 implies condition 2 and condition 2 implies condition 3 in a trivial way. It is also fairly easy to see that condition 4 implies condition 1, when one notices that each map $\ffib i :\Xfib i\to Y$ is actually open as the restriction of the open map $(f,\dots,f):X\times\dots\times X\to Y\times\dots\times Y$ to the inverse image of the diagonal in $Y\times\dots\times Y$. The hard part of the proof of Theorem \ref{main} lies in showing that condition 3 implies condition 4, which we shall now do. We shall need the following lemma. \begin{Lem} \label{rank} Let $f:X\to Y$ be a holomorphic map of analytic spaces, the space $Y$ being irreducible of dimension $d$. Suppose that $i\geq d$ and $\ffib i :\Xfib i\to Y$ is quasiopen. Then ${\rm\rho}_x f=d$ for every $x\in X$. \end{Lem} \vskip 4mm \noindent Notice that in the above lemma we do not need $Y$ to be locally irreducible. \vskip 5mm \proof{Lemma \ref{rank}} Fix $x_0\in X$ and suppose that $\rho_{x_0} f=d-k, \ 0\leq k \leq d$. Let $m=\dim_{x_0}X$. Without loss of generality we can assume that $\dim X=m$. Let $C(f)$ be as in lemma \ref{sard}. Observe that $\dim f^{-1}(y)\leq m-d$ and hence $\dim (\ffib i)^{-1}(y)\leq i(m-d)$ for $y\notin C(f)$. Now, in $\Xfib i$ consider the subset $A=(\ffib i)^{-1}(C(f))$. Since, by lemma \ref{sard}, $C(f)$ is a boundary set, therefore by remark \ref{basic}, $A$ is a boundary set in $\Xfib i$. Therefore we have $$\dim \Xfib i\ =\ \sup\{\dim_z\Xfib i : z\notin A\}\leq \ i(m-d)+d.$$ \noindent We can restrict our attention to the case $d\geq 1$. Set $z_0=(x_0,\dots ,x_0)\in\Xfib i$ and observe that ${\rm fbd}_{x_0} f=m-d+k$ and so ${\rm fbd}_{z_0}\ffib i=i(m-d+k)$. Since, by remark \ref{qofibre}, ${\rm fbd}_{z_0} \ffib i< \dim \Xfib i$, we get $k<\frac di$ and so $k=0$. This completes the proof of the lemma. {$ $} \vskip 5mm Now we can conclude the proof of theorem \ref{main}. Take $x_0\in X$. By lemma \ref{rank}, ${\rm\rho}_{x_0}f=d$. Let $X_1$ be an irreducible component of maximal dimension passing through $x_0$. Notice that also ${\rm\rho}_x(f\vert_{X_1})=d$, for any $x$ in a small neighbourhood $U$ of $x_0$ in $X_1$. Since $Y$ is locally irreducible, by the Remmert Rank Theorem $f\vert_U$ is open. Therefore, for any neighbourhood $V$ of $x_0$ in $X$, the image $f(V)$ contains $f(V\cap U)$ and hence is a neighbouhood of $f(x_0)$. Since this holds for any $x_0\in X$, the map $f$ is open. {$ $} \vskip 5mm To conclude this section, remark that since condition 4 of theorem \ref{main} implies openness of all maps $\ffib i$, $i=1,2,\dots ,$ as an immediate corollary we obtain that for any $i\geq d$ the map $\ffib i$ is quasiopen iff it is open. \vskip 5mm Notice that Theorem \ref{main} and Lemma \ref{rank} combined provide an easy proof of Remmert's Open Mapping Theorem. \section{Openness in the general case.} \vskip 5mm To prove Theorem \ref{opennessapp}, we first state and prove a purely topological proposition. \vskip 5mm \begin{Prop} \label{basechange} Let $f:X\to Y$ be a map of topological spaces. Let $\pi:\hat Y\to Y$ be a surjective, continuous map of topological spaces with the property that for any point $y\in Y$ and for any open neighbourhood $U$ of $\pi^{-1}(y)$ in $\hat Y$, $\pi (U)$ is a neighbourhood of $y$. Let $\hat f:\fibprod {\hat Y}YX\to\hat Y$ be the canonical (base change) map. Then $f$ is open if and only if $\hat f$ is open. \end{Prop} \vskip 5mm \proof{Proposition \ref{basechange}} If $f$ is open, then $\hat f$ is open just by the continuity of $\pi$: embedding $\fibprod{\hat Y}YX$ in $\hat Y\times X$ one verifies easily that for open sets $U\subset\hat Y$ and $V\subset X$, one has $\hat f((U\times V)\cap(\fibprod{\hat Y}YX))\ =\ U\cap\pi^{-1}(f(V))\ .$ Thus $\hat f$ is indeed open. Now, suppose that $\hat f$ is open. For any point $x\in X$, taking a neighbourhood $V$ of $x$ in $X$, one observes that $f(V)\ =\ \pi (\hat f((\hat Y\times V)\cap(\fibprod{\hat Y}YX)))\ ,$ and thus $f(V)$ is a neighbourhood of $y=f(x)$ by the openness of $\hat f$ and the properties of $\pi$. Hence $f$ is open. This ends the proof of proposition \ref{basechange}. {$ $} \begin{Rem} \label{whichpi} If $\pi:\hat Y\to Y$ is a closed, surjective, continuous map, then the condition imposed on $\pi$ in proposition \ref{basechange} is satisfied. In particular this is the case when $Y$ is Hausdorff and first countable and $\pi$ is proper, surjective and continuous. \end{Rem} \vskip 5mm Proposition \ref{basechange} and remark \ref{whichpi} easily imply the following (cf. also Lemma 1.5 in \cite{Parusinski}). \vskip 5mm \begin{Prop} \label{normopen} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Let $\pi:\hat Y\to Y$ be the normalization of $Y$ and let $\hat f:\fibprod{\hat Y}YX\to \hat Y$ be the canonical map. Then $f$ is open if and only if $\hat f$ is open. \end{Prop} \vskip 5mm Since the normalization of an analytic space is locally irreducible, we can apply theorem \ref{mainapp} to the map $\hat f$. Then, together with proposition \ref{normopen} they imply theorem \ref{opennessapp}. {$ $} \vskip 7mm \section{Quasiopen fibred powers in the general case.} \vskip 6mm This section is devoted to proving theorem \ref{redapp}. As before, it will follow easily from a theorem about the quasiopenness of fibred powers and Proposition \ref{eval}. \begin{Thm} \label{reducible} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Suppose that there is an integer $D$ such that the sum of dimensions of irreducible components of any germ of \ $Y$ is at most equal $D$. Then the following conditions are equivalent: \begin{enumerate} \item the maps $\ffib i :\Xfib i\to Y$ are quasiopen for all $i=1,2,\dots ,$ \item the map $\ffib D :\Xfib D\to Y$ is quasiopen, \item the map $\ffib i :\Xfib i\to Y$ is quasiopen for some $i\geq D$, \item for any $y\in Y$, there is a neighbourhood $V$ of $y$ in $Y$, an irreducible component $V_1$ of \ $V$ passing through $y$ and irreducible at $y$, such that for any $x\in f^{-1}(y)$ we have ${\rm\rho}_xf^{V_1}=\dim_y V_1$. \end{enumerate} \end{Thm} \medskip To prove theorem \ref{reducible}, we shall need the following lemma. \begin{Lem} \label{pointwise} Let $f:X\to Y$ be a holomorphic map of analytic spaces. Fix $y\in Y$ and suppose that $Y$ is locally irreducible at $y$. If $d=\dim_y Y$, then the following conditions are equivalent: \begin{enumerate} \item for all $x\in f^{-1}(y)$ we have ${\rm\rho}_xf=d$, \item for any $i=1,2,\dots$, for any open set $U$ in $\Xfib i$, with $y\in \ffib i(U)$, $\ffib i (U)$ has nonempty interior, \item for any open set $U$ in $\Xfib d$, with $y\in \ffib d(U)$, $\ffib d (U)$ has nonempty interior. \end{enumerate} \end{Lem} \proof{Lemma \ref{pointwise}} To prove that condition 1 implies condition 2, first observe, that by the Remmert Rank Theorem condition 1 implies that for any $x\in f^{-1}(y)$, the image by $f$ of any neighbourhood of $x$ is a neighbourhood of $y$. Now take $U$ as in condition 2. The fact that $y\in\ffib i(U)$, implies that $U$ contains an element $z=(x_1,\dots,x_i)$, with $f(x_1)=\dots=f(x_i)=y$. Thus there are neighbourhoods $U_j$ of each $x_j$ in $X$, such that $U\supset \Xfib i\cap (U_1\times\cdots\times U_i)$ (here $\Xfib i$ is embedded in $X^i$). Hence $\ffib i(U)$ contains the intersection of all the $f(U_j)$, which as we have observed is a neigbourhood of $y$. In particular it has nonempty interior, thus showing that condition 2 is fulfilled. It is trivial that 2 implies 3. To prove that condition 3 implies condition 1, let $Z$ be the sum of those components of $\Xfib d$ on which $\ffib d$ is quasiopen. Remark that condition 3 implies that the fibre $(\ffib d)^{-1}(y)$ is contained in $Z$. Now we can copy the proof of lemma \ref{rank}, taking $i=d$ and replacing $\Xfib i$ by $Z$. We have thus ended the proof of lemma \ref{pointwise}. {$ $} \medskip \proof{Theorem \ref{reducible}} Again, 1 implies 2 implies 3 in a trivial way. Now let us prove that 3 implies 4. Suppose that condition 4 is not fulfilled for a point $y$ in $Y$. Take a neighbourhood $V$ of $y$ in $Y$ such that all the irreducible components $V_1,\dots,V_s$ of $V$ contain $y$ and are locally irreducible at $y$. Let $d_j=\dim_y V_j$. By our assumption, for each $j$, one can choose a point $x_j\in f^{-1}(y)$, such that ${\rm\rho}_{x_j}f^{V_j}<d_j$. Embed canonically $\fib {(f^{-1}V_j)}{d_j}\subset\Xfib {d_j}$. By lemma \ref{pointwise}, for each $j$ there is an open subset $U_j$ of $\Xfib {d_j}$, with $y\in \ffib {d_j} (U_j)$ and such that the set $\fib {(f^{V_j})}{d_j}(U_j\cap \fib{(f^{-1}V_j)} {d_j} )$ has empty interior in $V_j$. In other words, the set $\ffib {d_j}(U_j)\cap V_j$ has empty interior in $V$. Now, fix $i\geq D$ and embedding $\Xfib i\subset \Xfib {d_1}\times\cdots\times\Xfib {d_s}\times X^{i-(d_1+\cdots+d_s)}$, let $U=\Xfib i\cap(U_1\times\dots\times U_s\times X^{i-(d_1+\cdots+d_s)})$. Now we have $y\in\ffib {d_j}(U_j)$ for all $j$ and hence $U$ is a nonempty (open) set in $\Xfib i$. Furthermore, for all $j$, the intersection $\ffib i(U)\cap V_j$ is contained in $\ffib {d_j} (U_j)\cap V_j$ and hence has empty interior. Therefore, $\ffib i(U)$ has empty interior in $V$. We have thus proved that $\ffib i$ is not quasiopen and so ended the proof of this implication. Now we shall prove that 4 implies 1. Fix $i$ and take a nonempty open set $W$ in $\Xfib i$. Choose $z\in W$ and $y=f(z)$. Take $V_1$ from condition 4 and apply lemma \ref{pointwise} to $f^{V_1}$, to find that $\ffib i(W)\cap V_1$ has nonempty interior. Hence $\ffib i(W)$ has nonempty interior. We have shown quasiopenness, ending the proof. {$ $} \bigskip \section{Maps to a smooth space.} \vskip 5mm This section is devoted to the proof of Theorem \ref{eff}. First notice, that if the map $f$ itself is not quasiopen, then there exists $p$, with $h_p=k_p$ and $r_p<d$. Therefore, the formula in Theorem \ref{eff} produces $0$ as it should. Hence, we can suppose that $f=\ffib 1$ is quasiopen in our proof. For convenience, in addition to the numerical data defined after the statement of Proposition \ref{exgoodpar}, we shall denote $w_p=k_p-r_p={\rm fbd}_x(f|_{X_p})$ for all $x\in X_p$. Given $(p_1,\dots,p_i)\in P^i$ we shall denote $X_{p_1}\times_Y\cdots\times_YX_{p_i}$ by $X^{(p_1,\dots,p_i)}$. We shall use the expression of $\app$ given in Proposition $\ref{eval}$. The proof will be carried out in 8 steps. \medskip \proof{Theorem \ref{eff}} \begin{Step} $\dim{X^{(p_1,\dots,p_i)}}\leq r_{p_j}+(w_{p_1}+\dots+w_{p_i})$ for $j=1,\dots,i$. \end{Step} Fix $j$. The fibres of the natural map ${X^{(p_1,\dots,p_i)}}\to Y$ are of dimension $w_{p_1}+\dots+w_{p_i}$. The image of a small neighbourhood of any point $(x_1,\dots,x_i)\in{X^{(p_1,\dots,p_i)}}$ is contained in the image of a small neighbourhood of $x_j\in X_{p_j}$, which is of dimension $r_{p_j}$ by the Remmert Rank Theorem. The inequality follows. \begin{Step} $\dim (X_p)^{\{ i\} }=r_p+iw_p$ and $\dim_z (X_p)^{\{ i\} }=r_p+iw_p$ for each point $z$ on the diagonal in $(X_p)^{\{ i\} }$. \end{Step} By the previous step we have $\dim (X_p)^{\{ i\} }\leq r_p+iw_p$. The converse inequality follows from the second part of the statement, which is a simple consequence of the Remmert Rank Theorem. Notice that $(X_p)^{\{ i\} }$ need not be of pure dimension. \begin{Step} $\dim \Xfib i =\sup\{ r_p+iw_p : p\in P\}$ . \end{Step} Since $\Xfib i$ is the union of all ${X^{(p_1,\dots,p_i)}}$, there exist $(p_1,\dots,p_i)$ such that $\dim \Xfib i =\dim {X^{(p_1,\dots,p_i)}}$. Now take $j$, such that $w_{p_j}=\max\{w_{p_1},\dots,w_{p_i}\}$. By step 1 we obtain $\dim \Xfib i\leq r_{p_j}+iw_{p_j}$ and hence $\dim \Xfib i \leq\sup\{ r_p+iw_p : p\in P\}$. On the other hand $\Xfib i$ contains all the $(X_p)^{\{ i\} }$ and so by step 2 we also have the converse inequality. \begin{Step} If $\ffib i$ is quasiopen and $h_p-d<w_p$ then $r_p+iw_p<\dim \Xfib i$. \end{Step} Let $W_1$ be any irreducible component of maximal dimension of $(X_p)^{\{ i\} }$. By step 2, for any $z\in W_1$ we have in particular ${\rm\rho}_z(f\vert_{W_1})\leq r_p$ and by the assumption on $p$, $r_p<d$. The inequality now follows from Remark \ref{subqo} and Step 2. \begin{Step} If $\ffib i$ is quasiopen then $\dim \Xfib i=d+i(\dim X -d)$ . \end{Step} This follows from the formula in Step 3 in which we can eliminate certain indices $p$, by Step 4. \begin{Step} If $\ffib i$ is quasiopen and $w_p>h_p-d$, then $i\leq \frac{d-r_p-1\phantom{.}} {w_p-(h_p-d)}$. This implies one inequality in Theorem \ref{eff}. \end{Step} Let $W$ be the union of irreducible components of $X$ with dimension not greater than $h_p$ minus the other components. Now we can apply step 4, with $X$ replaced by $W$ and $X_p$ replaced by $X_p\cap W$. We obtain $r_p+iw_p<\dim W^{\{ i\} }$. The previous step gives us a formula for $\dim W^{\{ i\} }$, which implies the inequality. \medskip \begin{Step} Suppose that $\ffib {i+1}$ is not quasiopen. Choose $(p_1,\dots,p_{i+1})$, such that $X^{(p_1,\dots,p_{i+1})}$ contains a nonempty open subset $U$ of $\Xfib {i+1}$, which is irreducible (as a locally analytic set) and whose image by $\ffib {i+1}$ has empty interior in $Y$. Then $\dim U\geq h_{p_1}+\cdots+h_{p_{i+1}}-id\ .$ \end{Step} Embed $X$ in a smooth complex space $M$ (if this can only be done locally, one can carry out a slightly more cumbersome proof using the same idea). For each $j=1,\dots,i+1$, let $Z_j$ be the union of irreducible components of $X$ of dimension not greater than $h_{p_j}$. Now $\Xfib {i+1}$ is isomorphic to the subspace of the smooth space $M^{i+1}\times Y^{i+1}$, defined as the intersection of the graph of the product map $(f\vert_{Z_1},\dots,f\vert_{Z_{i+1}}): Z_1\times\cdots \times Z_{i+1}\to Y^{i+1}$ and the product space $M^{i+1}\times \Delta$, where $\Delta$ is the diagonal subspace of $Y^{i+1}$. The bound then follows directly from the estimate of the codimension of components of an intersection in a smooth space. \begin{Step} If $\ffib {i+1}$ is not quasiopen, then for some $p\in P$, with $w_p>h_p-d$ $$i\geq\left [ \frac{d-r_p-1\phantom{.}} {w_p-(h_p-d)} \right]\ ,$$ where square brackets denote the integer part of a rational number. This proves the remaining inequality in Theorem \ref{eff}. \end{Step} We have $\dim U\leq\dim X^{(p_1,\dots ,p_{i+1})}$ and hence by Step 1, $\dim U= r+(w_{p_1}+\dots +w_{p_{i+1}})\ $, for some $r$ with $r\leq r_j$ for all $j=1,\dots,i+1$. Further, because the Remmert Rank of $f\vert_U$ is strictly smaller than $d$, we have $r<d$. Combining the above expression of $\dim U$ with the inequality from Step 7 and taking $j$ such that $w_{p_j}-h_{p_j}=\max\{w_{p_1}-h_{p_1},\dots, w_{p_{i+1}}-h_{p_{i+1}}\} $ one obtains $d-r\leq (i+1)(d+w_{p_j}-h_{p_j})$. Take $p=p_j$. First, we see that $w_p>h_p-d$. Then, since $r\leq r_p$, we have $d-r_p-1< (i+1)(d+w_p-h_p)$ and so $$i+1> \frac{d-r_p-1\phantom{.}} {w_p-(h_p-d)}\ ,$$ from where the inequality follows automatically. Theorem \ref{eff} follows immediately from steps 6 and 8.{$ $} \vskip 3mm

9606

alg-geom/9606015

en

https://arxiv.org/abs/alg-geom/9606015

alg-geom/9606015

Ines Quandt

Ines Quandt

On a relative version of the Krichever correspondence

59 pages LaTeX with inputs of AMSTeX; In addition to some corrections the main change consists in the extension of the Krichever correspondence to all locally noetherian base schemes

Bayreuther Mathematische Schriften 52 (1997), p.1-74

For a given base scheme, a correspondence is established between a class of sheaves on curves over this base scheme and certain points of infinite Grassmannians. This equivalence extends to a characterization of commutative algebras of ordinary differential operators with coefficients in the ring of formal power series over a given $k$-algebra. Our construction generalizes the approach of M.Mulase, which gives the above connection in the case that the base scheme is one closed point.

alg-geom

\section*{Preface} This PhD thesis is the result of my work in the Graduiertenkolleg "Geometrie und Nichtlineare Analysis" at Humboldt University Berlin and in the DFG project KU 770/1-3. \vspace{0.5cm}\\ It is published in the {\em Bayreuther Mathematische Schriften} {\bf 52} (1997), p.1-74. \vspace{0.5cm}\\ At this point, I would like to express my thanks to all of the people who supported my mathematical development. \vspace{0.5cm}\\ My special thanks go to Doz.~Dr.~sc.~W.~Kleinert, my thesis advisor and my professor since the very beginning of my studies. He also introduced me to the fascinating area of algebraic geometry, turned my attention to the theory of evolution equations, and kindly supported my work on this thesis. \vspace{0.5cm}\\ The subject of the present work has been suggested to me by Prof.~M.~Mulase, whom I would like to express my gratitude for his interest in my work and the inspiration and the encouragement he gave me. \vspace{0.5cm}\\ I gratefully thank Prof.~H.~Kurke for his keen interest in my work and lots of valuable hints. The discussions with him have been a wonderful help during the completion of this paper. \vspace{0.5cm}\\ My special thanks also go to G.~Hein , A.~Matuschke, Dr.~M.~Pflaum and D.~Ro\ss berg for numerous inspiring discussions. \vspace{1.5cm}\\ Berlin, October 1996 \hfill Ines Quandt \newpage \tableofcontents \newpage \addtocounter{section}{-1} \section{Introduction} The aim of this paper is to construct a link ranging from a class of sheaves on curves over some base scheme via infinite Grassmannians to commutative algebras of differential operators and evolution equations. \vspace{0.5cm} The idea of studying relationships between algebraic curves, algebras of differential operators and partial differential equations is not new. This connection has been studied already at the beginning of the 20th century by G.~Wallenberg, I.~Schur, J.~L.~Burchnall and T.~W.~Chaundy. G.~Wallenberg \cite{W} tried to find all commuting pairs of ordinary differential operators. During his classification of commuting operators $P$ and $Q$ of order 2 and 3, respectively, he found a certain relation to a plane cubic curve. However, Wallenberg did not continue to explore this relation. The motivation to study commutative algebras instead of commuting pairs of differential operators was given by I.~Schur \cite{Sc} in 1905, when he proved the following remarkable fact: \vspace{0.5cm}\\ {\em Let $P$ be an ordinary differential operator of order greater than zero, and let $B_{P}$ be the set of all differential operators which commute with $P$. Then $B_{P}$ is a commutative algebra. } \vspace{0.5cm} The work of J.~L.~Burchnall and T.~W.~Chaundy (\cite{BC1}-\cite{BC3}) about the relations between commuting differential operators and affine algebraic curves is extensive. For example, they proved that for commuting $P$ and $Q$ of positive order, the ring $\poly{\Bbb{C}}{P,Q}$ has dimension 1. Furthermore, they analyzed at length certain examples of affine curves and the related differential operators. It is remarkable that a large part of the methods which have been systematically developed more than 50 years later, virtually already exist in these early papers by Burchnall and Chaundy, although mainly in examples. \vspace{0.5cm} After a long period of stagnation, another break-through came with the work of P.~Lax \cite{L} about isospectral deformations of differential operators in the late 60's. It has been realized, already in the early stage of the theory, that a commutative algebra of differential operators carries a lot more information than only its algebraic-geometric spectrum. In the 70's, I.~M.~Krichever analyzed the behaviour of an operator at infinity, i.e., he constructed the line bundle on a complete curve corresponding to a given algebra by special extensions of the trivial bundle to the point at infinity (see \cite{K}, \cite{K1}, \cite{K2}). His approach may be considered as the source of the algebraic-geometric correspondences established later on. Almost simultaneously, and inspired by the work of Krichever \cite{K}, D.~Mumford \cite{Mum1} established a correspondence between pointed curves equipped with a fixed line bundle and certain commutative algebras of differential operators. This article and the one of J.-L.~Verdier \cite{V} also contain the very first constructions in the case of higher rank vector bundles. \vspace{0.5cm} Later on, infinite Grassmannians emerged in the study of evolution equations and their relations to vector bundles on curves. Without claiming to be complete, let us just mention the work of M.~Sato \cite{S}, E.~Previato, G.~Segal and G.~Wilson (\cite{PW}, \cite{SW}) and M.~Mulase \cite{M2}. Among one of the culmination points of this newly established theory was the complete classification of elliptic commutative algebras of ordinary differential operators, and one of the affirmative solutions of the Schottky problem by M.~Mulase in \cite{M1}.\vspace{0.5cm} In the approach of M.~Mulase, some questions arise quite naturally: Can one generalize the correspondence between vector bundles on curves and elements of infinite Grassmannians to the case where the curve is not defined over a field, but for example over a $k$-algebra? In this case, does one also get a correspondence with commutative algebras of differential operators with coefficients in more general rings? The present paper gives an affirmative answer to both questions. These questions are interesting from two points of view. First, the established correspondence enables us to construct certain classes of commutative algebras of partial differential operators. On the other hand, it gives us a powerful tool for the study of degenerations of curves and vector bundles in terms of differential operators and differential equations. \vspace{0.5cm} In the recently published paper \cite{AMP} the authors also give a generalization of infinite Grassmannians to the relative case, which overlaps with ours in the case where the base scheme is defined over a field, and they investigate this Grassmannian from the point of view of representation theory. \vspace{0.5cm} The method presented in our paper serves the purpose of generalizing the techniques developed in \cite{M1} to the relative case. Therefore, this article can be considered as a general reference and often will not be quoted in the sequel. \vspace{0.5cm} A preliminary version of this paper appeared as a preprint \cite{Q}. In addition to some slight corrections, a great extension has been made: In the preprint only integral noetherian base schemes were allowed, whereas now we only need to assume the base scheme to be locally noetherian. \vspace{0.5cm} The paper is organized as follows: The first chapter is devoted to the generalization of the notion of infinite Grassmannians. In the second and third, the link with sheaves on relative curves is established. The fourth chapter aims at illustrating this correspondence. In chapter 5 we give a complete characterization of commutative elliptic algebras of differential operators with coefficients in the ring of formal power series over a $k$-algebra, $k$ being a field of characteristic zero. The appendix is included in order to help those readers who are interested in the details of the computations. \newpage \section{Relative infinite Grassmannians} To begin with, we want to generalize the notion of infinite Grassmannians. This can be done for arbitrary base schemes. To this end, let $S$ be any algebraic scheme. We denote by $\Opower{S}{z}$ the sheaf defined by $$\Opower{S}{z}(U) := \power{{\cal O}_{S}(U)}{z},$$ and $\Onegpower{S}{z}$ is defined as the sheaf $$\Onegpower{S}{z}(U) := \negpower{{\cal O}_{S}(U)}{z},$$ where $\negpower{{\cal O}_{S}(U)}{z}$ stands for the ring of formal Laurent series in $z$ with coefficients in ${\cal O}_{S}(U)$. $\Onegpower{S}{z}$ has a natural filtration by subsheaves of the form $\Opower{S}{z}\cdot z^{n}$. \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} Neither $\Opower{S}{z}$ nor $\Onegpower{S}{z}$ are quasicoherent sheaves of ${\cal O}_{S}$-modules. To see this, take an open affine subset $Spec(R)$ of $S$ and choose an element $f\in R$ which is not a zero divisor. Then there is a natural inclusion $$\power{R}{z}_{f} \hookrightarrow \power{R_{f}}{z}$$ which, in general, is not an isomorphism. Therefore, $\Opower{S}{z}$ is not quasicoherent. The same holds true for $\Onegpower{S}{z}$. However, for each integer $n\in\Bbb{Z}$, the quotient sheaf $$\Onegpower{S}{z}/\Opower{S}{z}\cdot z^{n} \cong \bigoplus_{m<n} {\cal O}_{S}\cdot z^{m}$$ is quasicoherent. \begin{definition} Let $U$ be an open subset of $S$ and $v$ a local section of $\Onegpower{S}{z}^{\oplus r}$ over $U$, for some $r$. Then the {\em order of $v$} is defined to be the minimum integer $n$ such that $v \in \power{{\cal O}_{S}(U)}{z}^{\oplus r} \cdot z^{-n}$. If $\cal V$ is a subsheaf of $\Onegpower{S}{z}^{\oplus r}$, then we define $${\cal V}^{(n)} := {\cal V} \cap \Opower{S}{z}^{\oplus r} \cdot z^{-n}.$$ \end{definition} {\bf Remark} $\Onegpower{S}{z}$ acts on $\Onegpower{S}{z}^{\oplus r}$ by the natural assignment \begin{equation} \begin{array}{cccr} \negpower{{\cal O}_{S}(U)}{z} \times \negpower{{\cal O}_{S}(U)}{z}^{\oplus r} & \longrightarrow & \negpower{{\cal O}_{S}(U)}{z}^{\oplus r} &{}\\ (f,g=(g_{1},\ldots,g_{r})) & \longmapsto & (f\cdot g_{1},\ldots, f\cdot g_{r}) &. \end{array} \end{equation} For the multiplication defined this way we have the estimate: $$ ord(f\cdot g) \leq ord(f) + ord(g).$$ If $S$ is an integral scheme, then both sides are equal. For more properties of power series and Laurent series with coefficients in arbitrary rings, the reader is referred to the appendix. \begin{definition} The {\em Grothendieck group} $K(S)$ is defined to be the quotient of the free abelian group generated by all coherent sheaves on $S$, by the subgroup generated by all expressions $${\cal F} - {\cal F}' -{\cal F}''$$ whenever there is an exact sequence $$0 \rightarrow {\cal F}' \rightarrow {\cal F} \rightarrow {\cal F}''\rightarrow 0$$ of coherent sheaves on $S$. If $\cal F$ is a coherent sheaf on $S$ then we denote by $\gamma({\cal F})$ its image in $K(S)$. \end{definition} \begin{definition} For any natural number $r$, integer $\alpha$ and element $F\in K(S)$, we define {\em the infinite Grassmannian of rank $r$, index $F$ and level $\alpha$ over $S$} to be the set ${\frak G}^{r}_{F,\alpha}(S)$ consisting of all quasicoherent subsheaves of ${\cal O}_{S}$-modules ${\cal W} \subseteq \snegpower{{\cal O}_{S}}{z}{r}$ such that ${\cal W} \cap \spower{{\cal O}_{S}}{z}{r}\cdot z^{-\alpha}$ and the quotient $\snegpower{{\cal O}_{S}}{z}{r}/({\cal W}+ \spower{{\cal O}_{S}}{z}{r}\cdot z^{-\alpha}) $ are coherent and furthermore: $$F = \gamma({\cal W} \cap \spower{{\cal O}_{S}}{z}{r}\cdot z^{-\alpha}) - \gamma(\snegpower{{\cal O}_{S}}{z}{r}/({\cal W}+ \spower{{\cal O}_{S}}{z}{r}\cdot z^{-\alpha})).$$ \end{definition} {\bf Remark}\hspace{0.3cm} The introduction of the level has a merely technical meaning. It will be used only in Chapter \ref{Fam. DO}. \vspace{0.5cm} Now we introduce the concept of a Schur pair. \begin{definition} By a {\em Schur pair of rank $r$ and index $F$ over $S$} we mean a pair $({\cal A},{\cal W})$ consisting of elements ${\cal A} \in {\frak G}^{1}_{G}(S)$, for some $G\in K(S)$, and ${\cal W} \in {\frak G}^{r}_{F}(S)$ such that \begin{itemize} \item ${\cal A}$ is a sheaf of ${\cal O}_{S}$ - subalgebras of $\Onegpower{S}{z}$, \item The natural action of \negpower{{\cal O}_{S}}{z} on \snegpower{{\cal O}_{S}}{z}{r} induces an action of ${\cal A}$ on ${\cal W}$, i.e., ${\cal A}\cdot{\cal W}\subseteq{\cal W}$. \end{itemize} We denote by $\frak{S}^{r}_{F}(S)$ the set of Schur pairs of rank $r$ and index $F$ over $S$. \end{definition} {\bf Remark 1}\hspace{0.3cm} Let us include here a remark on Grothendieck groups. First assume that $S$ is integral. Then there is a surjective group homomorphism $$rk: K(S) \longrightarrow \Bbb{Z}$$ induced by the map $$\gamma({\cal F}) \longmapsto rk({\cal F}),$$ where $rk({\cal F})$ denotes the rank of $\cal F$ at the generic point of $S$. If $S$ equals $Spec(k)$, for some field $k$, then the homomorphism $rk$ is an isomorphism. If $S$ is reduced we still can define a ``multirank'' by taking the rank at every irreducible component. However, if $S$ is not reduced, the rank is no longer well-defined. That is why we use the Grothendieck group to define Grassmannians. \newpage {\bf Remark 2}\label{Grass Mulase}\hspace{0.3cm} How are the notions of infinite Grassmannians and Schur pairs related to those introduced by M.~Mulase? To answer this question, consider the embedding $$ \begin{array}{rcl} \Onegpower{S}{z} &\hookrightarrow &\Onegpower{S}{y}\\ z & \mapsto & y^{r}. \end{array} $$ This leads to the natural identification: $$ \begin{array}{rcl} \Onegpower{S}{z}^{\oplus r} & = & \bigoplus_{i=0}^{r-1} \Onegpower{S}{y^{r}}\cdot y^{i}\\ &=& \Onegpower{S}{y}. \end{array} $$ In particular, let $S=Spec(k)$ for some field $k$. Then ${\frak G}^{r}_{F}(S)$ consists of all subspaces $W\subset \negpower{k}{z}^{\oplus r}$ such that the composition of morphisms $$W\hookrightarrow \negpower{k}{z}^{\oplus r} \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } \negpower{k}{z}^{\oplus r}/ \power{k}{z}^{\oplus r}$$ is a Fredholm operator of index $rk(F)$. In view of the previous identifications, this redefines the notion of an infinite Grassmannian as used by M.~Mulase (cf. \cite{M1}). However, comparing the notions of Schur pairs, we see that our definition is a restricted version of the one given by Mulase. Later we will see that in the case where the ground field is of characteristic zero, this restriction is not substantial. For more details see Section \ref{Fam. DO}. \vspace{0.5cm} \begin{definition} Let $\alpha: S\rightarrow S'$ be a morphism, and $({\cal A}', {\cal W}')$ a Schur pair of rank $r$ over $S'$. We denote by $\alpha^{(*)}{\cal A}'$ (resp. $\alpha^{(*)}{\cal W}'$) the image of ${\cal A}'$ (resp. ${\cal W}'$) under the map $$\alpha^{(*)} : \Onegpower{S'}{z}^{\oplus r} \rightarrow \Onegpower{S}{z}^{\oplus r}$$ which is given by the pull-back of the coefficients. \end{definition} {\bf Remark}\hspace{0.3cm} $(\alpha^{(*)}{\cal A}',\alpha^{(*)}{\cal W}')$ is a Schur pair of rank $r$ over $S$. \vspace{0.5cm} Now we can define homomorphisms of Schur pairs. \begin{definition} \label{xi} Let $({\cal A},{\cal W})$ and $({\cal A}',{\cal W}')$ be Schur pairs over $S$ (resp. $S$') of rank $r$ (resp. $r'$). Then a homomorphism $(\alpha,\xi): ({\cal A}',{\cal W}')\rightarrow ({\cal A},{\cal W})$ consists of \begin{enumerate} \item A morphism $\alpha: S \rightarrow S'$ such that $\alpha^{(*)}{\cal A}' \subseteq {\cal A}$; \item A homomorphism $\xi \in {\cal H}om_{\Opower{S}{z}}(\Opower{S}{z}^{\oplus r'}, \Opower{S}{z}^{\oplus r})$ such that for the induced homomorphism $\xi \in {\cal H}om_{\Onegpower{S}{z}}(\Onegpower{S}{z}^{\oplus r'}, \Onegpower{S}{z}^{\oplus r})$ the inclusion $$\xi(\alpha^{(*)}{\cal W}') \subseteq {\cal W}$$ holds. \end{enumerate} \end{definition} In this way, we get the category ${\frak S}$ of Schur pairs. \begin{definition} We define a full subcategory ${\frak S}'$ of ${\frak S}$ as follows: $({\cal A},{\cal W}) \in \frak{S}'^{r}_{F}(S)$ if and only if $({\cal A},{\cal W}) \in\frak{S}^{r}_{F}(S)$ and ${\cal A} \cap \Opower{S}{z} = {\cal O}_{S}$. \end{definition} The sense of this definition will become clear later on. For the time being, take it simply as a notation. \section{Families of curves and sheaves} In this section, we fix the geometric objects that we want to investigate, and we prove some basic properties of them. The definitions we are going to make might seem a little technical. That is why much space is given to illustrations and examples. Even more examples may be found in Chapter \ref{APPL}. Our first aim is to study sheaves over families of curves. As to that, we need to fix three objects, namely: the base scheme, the total space and a sheaf on the total space. \subsection{Families of curves} As base schemes $S$ we allow all locally noetherian schemes. \begin{definition} \label{total C} By a {\em pointed relative curve over $S$} we understand a scheme $C$ together with a locally projective morphism $\pi:C\rightarrow S$ and a section $P\subset C$ of $\pi$ such that the following holds: \begin{enumerate} \item $P$ is a relatively ample Cartier divisor in $C$. \item For the sheaf ${\cal I}:= {\cal I}_{P}$ defining $P$ in $C$, ${\cal I}/{\cal I}^{2}$ is a free ${\cal O}_{P}$-module of rank 1. \item Let $\widehat{\cal O}_{C}$ denote the formal completion of ${\cal O}_{C}$ with respect to the ideal $\cal I$. Then $\widehat{\cal O}_{C}$ is isomorphic to $\Opower{P}{z}$ as a formal ${\cal O}_{P}$-algebra. \item $\bigcap_{n\geq 0}\pi_{*}{\cal O}_{C}(-nP)=(0)$. \end{enumerate} \end{definition} Let us include here a couple of remarks and examples. \vspace{0.5cm}\\ {\bf Remark} \begin{itemize} \item Since $P$ is a section, $\pi|P : P \rightarrow S$ is an isomorphism. The sheaves ${\cal I}/{\cal I}^{2}$ and $\widehat{\cal O}_{C}$ have their support in $P$. Consequently, Condition 2 is equivalent to $\pi_{*}({\cal I}/{\cal I}^{2})\cong {\cal O}_{S}$, while Condition 3 translates into: $\pi_{*}\widehat{\cal O}_{C}$ is isomorphic to $\Opower{S}{z}$ as a formal ${\cal O}_{S}$-algebra. \item Condition 4 is equivalent to the fact that, for every integer $n\in\Bbb{Z}$, the natural map $$\pi_{*}{\cal O}_{C}(nP) \rightarrow \pi_{*}\widehat{{\cal O}_{C}(nP)}$$ is injective. \end{itemize} {\bf Example 1}\hspace{0.3cm} Let $S=Spec(k)$ for some field $k$. Then $C$ is a complete curve and $P$ corresponds to some smooth, $k$-rational point of $C$. The $k$-rationality is a consequence of the fact that $P$ is a section. Since $\widehat{\cal O}_{C,P}\cong \power{k}{z}$, the ring ${\cal O}_{C,P}$ is regular. The Condition 4 is satisfied if and only if the curve $C$ is reduced and irreducible. \vspace{0.5cm}\\ {\bf Example 2}\hspace{0.3cm} The motivation for studying curves over base schemes which are different from one point comes mainly from the desire to investigate families of curves as considered in Example 1. Let us take, as an example, such a family over an integral $k$-scheme $S$. Since $S$ and the fibres of $\pi$ are irreducible, $C$ is automatically irreducible. Let us assume, in addition, that $C$ is reduced. Then Condition 4 of the previous definition is satisfied. Conditions 2 and 3 amount to saying that our family is constant locally around the section $P$. \vspace{0.5cm} Notice that the condition we impose on $P$ by assuming the triviality of ${\cal I}/{\cal I}^{2}$ is very restrictive in the case where our base scheme $S$ is complete. This is expressed in the following proposition, which can be found in \cite{A1}: \begin{proposition} Let $f:X\rightarrow S$ be a family of nodal curves of genus $g > 0$ over a reduced and irreducible complete curve, and let $\Gamma \subset X$ be a section of $f$ not passing through any of the singular points of the fibres. Suppose that the general fibre of $f$ is smooth. Then $$(\Gamma\cdot\Gamma)\leq 0.$$ Moreover, if $(\Gamma\cdot\Gamma)=0$, then the family $f:X\rightarrow S$, together with the section $\Gamma$, is an isotrivial family of 1-pointed nodal curves. \end{proposition} It is not hard to generalize the whole set-up to the case where ${\cal I}/{\cal I}^{2}$ is locally free, but not free. One simply has to use $\prod_{n\geq 0}({\cal I}/{\cal I}^{2})^{n}$ instead of $\Opower{P}{z}$. However, since we are mainly interested in local considerations, this generalization does not play such an important role that it would justify the technical effort. Now, the first condition needs to be examined. The following lemma shows that it is almost automatically satisfied: \begin{lemma} \label{ample} \begin{itemize} \item If $S$ is irreducible, then $P$ is a Cartier divisor if and only if the conormal sheaf ${\cal I}/{\cal I}^{2}$ is a line bundle on $P$. \item If $P$ is a Cartier divisor on $C$, and if the morphism $\pi$ has irreducible fibres, then $P$ is relatively ample. \end{itemize} \end{lemma} {\bf Remark}\hspace{0.3cm} The assumption on $P$ translates as follows: ${\cal I}={\cal I}_{P}$ is locally generated by one element which is not a zero divisor. \vspace{0.5cm}\\ {\bf Proof of the lemma}\hspace{0.3cm} The first statement is easy. Let us prove the second one. The question is local on $S$. So we are in the following situation:\\ $R$ is a noetherian ring, $\pi:C \rightarrow Spec(R)$ is a projective morphism, and $P\subset C$ is a section of $\pi$ and an effective Cartier divisor. For all $n\in \Bbb{N}$, we have the following exact sequence: $$ 0 \rightarrow {\cal O}_{C}((n-1)P) \rightarrow {\cal O}_{C}(nP) \rightarrow {\cal O}_{P}(nP) \rightarrow 0.$$ Since $P$ is affine, this induces a long exact sequence of cohomology groups: $$ \begin{array}{ccccccccc} 0 & \rightarrow & H^{0}({\cal O}_{C}((n-1)P)) & \rightarrow & H^{0}({\cal O}_{C}(nP)) & \rightarrow & H^{0}({\cal O}_{P}(nP)) & {} & {}\\ {} & \rightarrow & H^{1}({\cal O}_{C}((n-1)P)) & \rightarrow & H^{1}({\cal O}_{C}(nP)) & \rightarrow & 0 . \end{array}$$ Hence we have surjections $ H^{1}({\cal O}_{C}((n-1)P))) \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{1}({\cal O}_{C}(nP))$. Composing these surjections, we obtain, for each $n\in \Bbb{N}$, an epimorphism $$\alpha_{n}:H^{1}({\cal O}_{C}) \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{1}({\cal O}_{C}(nP)).$$ Let $M_{n}$ denote the kernel of the mapping $\alpha_{n}$, which is an $R$ - submodule of $H^{1}({\cal O}_{C})$. From the very definition we get: $M_{n} \subseteq M_{n+1}$. But \cite{H1}, Thm. III.5.2., tells us that $H^{1}({\cal O}_{C})$ is a finitely generated $R$-module, hence noetherian. Therefore, there is an integer $N$ such that $M_{N} = M_{N+1}=\ldots$. This implies that, for all $n > N$, the following sequence is exact: \begin{equation} \label{noether} 0 \rightarrow H^{0}({\cal O}_{C}((n-1)P))) \rightarrow H^{0}({\cal O}_{C}(nP)) \rightarrow H^{0}({\cal O}_{P}(nP)) \rightarrow 0. \end{equation} But $P$ is affine. Therefore ${\cal O}_{P}(nP))$ is globally generated. This implies, using Nakayama's lemma and (\ref{noether}), that the global sections of ${\cal O}_{C}(nP)$ generate this sheaf in some neighborhood of $P$. By definition, base-points of the sheaf ${\cal O}_{C}(nP)$ are contained in $P$. Thus ${\cal O}_{C}(nP)$ itself has no base-points, i.e., it is globally generated. The sections of ${\cal O}_{C}(nP)$ define an $R$ - morphism $$\beta:C \rightarrow \Bbb{P}^{M}_{R}.$$ As $\beta$ is an $R$ - morphism, it is compatible with $\pi$, i.e., $\pi= pr\circ \beta$, where $pr$ denote the natural projection from $\Bbb{P}^{M}_{R}$ onto $Spec(R)$. Since $P$ is a section, $\beta$ restricts to a closed embedding on $P$. In addition, we know that $\beta^{-1}(\beta(P)) = P$. Now we want to prove that $\beta$ has finite fibres. Assume, on the contrary, that there is a point $q\in \beta(C)$ such that $dim \beta^{-1}(q) \geq 1$. Let $X$ be an irreducible component of $\beta^{-1}(q)$ of dimension greater than 0. It is clear that $X$ does not intersect the divisor $P$. In particular, the intersection of $X$ with each fibre of $\pi$ is a closed subset of codimension at least 1. But the fibres of $\pi$ are assumed to be irreducible of dimension 1. Therefore $X$ cannot be contained in any fibre of $\pi$. But this is a contradiction, since $\pi(\beta^{-1}(q)) = pr(q)$. So $\beta$ is a quasi-finite morphism. By \cite{H1}, Cor. II.4.8., $\beta$ is proper. From the Stein-factorization theorem (cf. \cite{G1}, Cor. 4.3.3.) one knows that in this case $\beta$ is also finite. By construction, ${\cal O}_{C}(nP) = \beta^{*}({\cal O}_{\Bbb{P}^{M}_{R}}(1))$. ${\cal O}_{\Bbb{P}^{M}_{R}}(1)$ induces a very ample line bundle on $\beta(C)$. Since $\beta$ is a finite morphism, this implies that ${\cal O}_{C}(nP)$ is ample, too (cf. \cite{H2}, Prop.I.4.4.). \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} We have seen that a family of curves whose fibres satisfy the conditions of Definition \ref{total C} almost matches the definition already. However, the converse is definitely not true. Namely, if $C$ is as in Definition \ref{total C}, then it may occur that some fibres of $\pi$ are not even integral (see Section \ref{base change} for examples). \vspace{0.5cm}\\ After this illustration we return to our general definition. \begin{lemma} \label{affine covering} Let $C$ be as in Definition \ref{total C}. Then: \begin{enumerate} \item $C$ is locally noetherian. \item For each open affine subset $U$ of $S$, $\pi^{-1}(U)\setminus P$ is affine. \item $S$ can be covered by open affine sets, $U_{i} = Spec(R_{i})$, such that for each $i$ there is an open affine subset $V_{i} = Spec(B_{i})$ of $\pi^{-1}(U_{i})$ containing $P\cap \pi^{-1}(U_{i})$. \end{enumerate} \end{lemma} {\bf Proof}\hspace{0.3cm} \begin{enumerate} \item $\pi$ is locally projective, hence it is locally of finite type. Therefore, together with $S$, $C$ is also locally noetherian. \item This is a consequence of the relative ampleness of $P$. \item $\pi$ is locally of finite type, i.e., we can choose an open covering of $S$ by affine sets $U_{k}'= Spec (R_{k}')$ such that, for all $k$, $\pi:\pi^{-1}(U'_{k}) \rightarrow U'_{k}$ is proper, and there are finitely many open affine sets $V_{k,l}' = Spec(B_{k,l}')$ satisfying: \begin{itemize} \item $\pi^{-1}(U_{k}') = \bigcup_{l} V_{k,l}'$, \item $B_{k,l}'$ is a finitely generated $R_{k}'$ - algebra. \end{itemize} Let $U' := U_{k}'$ for some $k$. Choose a point $Q\in P\cap \pi^{-1}(U')$. $Q$ is contained in one of the $V_{k,l}'$ which we denote by $V'$ for short. If $V'$ contains $P\cap \pi^{-1}(U')$, then we are done. Now let us assume that $V'$ does not contain $P\cap \pi^{-1}(U')$. $P\cap \pi^{-1}(U')$ is a closed subset of $\pi^{-1}(U')$. From the closedness of $\pi$ we conclude that $\pi((P\cap \pi^{-1}(U'))\setminus V')$ is closed in $U'$, and that $V'$ contains $P\cap \pi^{-1}(U'\setminus \pi(P\setminus V'))$. The open set $U'\setminus \pi(P\setminus V')$ can be covered by open affine sets of the kind $U'(f) := Spec((R_{k})_{f})$, for certain elements $f\in R_{k}$, and we see that over each of the $U'(f)$'s the affine set $V'(f) = V_{k,l}'(f) := Spec ((B_{k,l}')_{f})$ has exactly the required property. \mbox{\hspace*{\fill}$\Box$} \end{enumerate} Now let us illustrate Condition 3 of Definition \ref{total C}. \begin{lemma} \label{powerseries} Assume that ${\cal I}/{\cal I}^{2}$ is trivial. Then $\widehat{\cal O}_{C}$ is isomorphic to $\Opower{P}{z}$ if and only if each section of ${\cal I}/{\cal I}^{2}\cong {\cal O}_{P}$ lifts to a section of $\widehat{\cal O}_{C}$. In particular, one may interpret $z$ as the lift of a generating section of ${\cal I}/{\cal I}^{2}$. \end{lemma} {\bf Remark}\hspace{0.3cm} Consequently, $\widehat{\cal O}_{C}$ is isomorphic to $\Opower{P}{z}$ if and only if, for any integer $n\in\Bbb{N}$, $n\geq 2$, the map $$H^{0}({\cal I}/{\cal I}^{n}) \rightarrow H^{0}({\cal I}/{\cal I}^{2})$$ is surjective. \vspace{0.5cm}\\ {\bf Proof of the lemma}\hspace{0.3cm} First assume that $\widehat{\cal O}_{C} \cong \Opower{P}{z}$. Then ${\cal I}/{\cal I}^{2} \cong {\cal O}_{P} \cdot z$ and of course $$H^{0}(\Opower{P}{z}) = \power{H^{0}({\cal O}_{P})}{z} \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{0}({\cal O}_{P})\cdot z.$$ Now let $z$ be a global section of $\widehat{\cal O}_{C}$ such that $z(mod ({\cal I}^{2}))$ generates ${\cal I}/{\cal I}^{2}$. The sheaf $\widehat{\cal O}_{C}$ is defined to be the limit taken over the projective system ${\cal O}_{C}/{\cal I}^{n}$. We restrict our consideration to an open affine set $U = Spec(R) \subseteq S$ such that there is an open subset $V = Spec(B)$ of $\pi^{-1}(U)$ which contains $p(U)$ and which is so that $I :={\cal I}(V)$ is free. Then we construct the required isomorphism of sheaves locally on $U$. Choose an element $b\in I$ such that $I=bB$ and $[b] = [z] \in I/I^{2}$. Now take $f \in B/I^{m}$ for some $m$. Identifying $R$ with $\pi^{*}(R) \subset B$, there is a uniquely determined element $f_{1} \in R$ such that $f-f_{1} \in I/I^{m}$. By assumption, $I/I^{2}$ equals $[b]\cdot R$, and thus there is a $f_{2} \in R$ such that $f-f_{1}-f_{2}b(mod I^{m}) \in I^{2}/I^{m}$. Since $I^{n}/I^{n+1}$ is generated by $b^{n}(mod I^{n+1})$, we can continue this process and get well-defined maps $$B/I^{m} \longrightarrow R \oplus R\cdot b \ldots \oplus R\cdot b^{m-1}.$$ These maps give rise to a homomorphism of formal $R$ - algebras: $$ \widehat{B} \longrightarrow \power{R}{b},$$ where $\widehat{B}$ denotes the completion of $B$ with respect to the ideal $I$. By \cite{Mat1}, Thm.8.12., this is an isomorphism. The fixed global section $z$ of $\widehat{\cal O}_{C}$ restricts on $U$ to an element of $\widehat{B}$. Since $[z], [b]\in I/I^{2}$ coincide, the homomorphism constructed above maps $z$ to an element $b\cdot (1+\alpha)$ for some $\alpha\in \power{R}{b}\cdot b$. Remark that all those elements $1+\alpha$ are invertible in $\power{R}{b}$ (cf. the appendix). Therefore the formal power series rings \power{R}{b} and \power{R}{z} are naturally isomorphic. So we finally get a well-defined isomorphism of $R$ - algebras: $$\rho : \widehat{B} \longrightarrow \power{R}{b} \stackrel{\sim}{\longrightarrow} \power{R}{z}.$$ Here, $\rho$ does not depend on the choice of the local lift $b$ of $[z]$. Therefore, these locally defined isomorphisms glue together. It follows from Lemma \ref{affine covering} that $S$ has a covering by sets $U$ as considered above. Thus the construction gives a well-defined isomorphism of sheaves $$\rho : \widehat{\cal O}_{C} \stackrel{\sim}{\longrightarrow} \Opower{P}{z}.$$ \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} \subsection{Families of sheaves} \begin{definition} \label{sheaf F} As sheaves $\cal F$ on $C$ we admit all coherent sheaves such that \begin{enumerate} \item The formal completion $\widehat{\cal F}$ of $\cal F$ along $P$ is a free $\widehat{\cal O}_{C}$-module. \item $\bigcap_{n\geq 0}\pi_{*}{\cal F}(-nP) =(0)$. \end{enumerate} \end{definition} {\bf Remark 1}\hspace{0.3cm} Again, the second condition is equivalent to: $$\pi_{*}{\cal F}(nP) \hookrightarrow \pi_{*}\widehat{{\cal F}(nP)}, \quad \textrm{for all } n\in \Bbb{Z}.$$ {\bf Examples} \begin{itemize} \item If $C$ is an integral scheme, then the second condition is satisfied if and only if $\cal F$ is torsion free. \item If $C$ is a complete curve, then the first condition is also satisfied for torsion free sheaves, since these are free in smooth points. \item For general $S$ and $C$ as in Definition \ref{total C} and for any vector bundle $\cal F$ on $C$, Condition 2 of Definition \ref{sheaf F} is satisfied. \end{itemize} {\bf Remark 2}\hspace{0.3cm} The isomorphism $$\rho : \widehat{\cal O}_{C}\stackrel{\sim}{\longrightarrow} \Opower{P}{z}$$ makes $\widehat{\cal F}$ into an $\Opower{P}{z}$-module. If $\cal F$ satisfies Condition 1, then there is an isomorphism of $\widehat{\cal O}_{C}$-modules $$\Phi : \widehat{\cal F}\stackrel{\sim}{\rightarrow} \widehat{\cal O}_{C}^{\oplus s}$$ or, equivalently, an isomorphism of $\Opower{P}{z}$-modules $$\rho\circ\Phi : \widehat{\cal F} \stackrel{\sim}{\rightarrow} \Opower{P}{z}^{\oplus s}.$$ A natural question is: What do the properties of $\cal F$ imply in general? In order to get a flavor of what is happening, we prove the following lemma. \begin{lemma} \label{loc. free} Let $\cal F$ be a coherent sheaf of rank $r$, as in Definition \ref{sheaf F}. Then the following holds: \begin{enumerate} \item[( i)] The rank of $\widehat{\cal F}$ as an $\widehat{\cal O}_{C}$ - module equals to the rank of $\cal F$. In particular, there is an open, dense subset of $C$ on which $\cal F$ has constant rank. \item[( ii)] ${\cal F}|P$ is free of rank $r$. \item[(iii)] $\cal F$ is locally free in a neighborhood of $P$. \end{enumerate} \end{lemma} {\bf Proof}\hspace{0.3cm} (i) is a consequence of (iii). Let us turn to the proof of (ii). We assume that $\widehat{\cal F}\cong \widehat{\cal O}_{C}^{\oplus s}$, for some $s\in \Bbb{N}$. Thus $$ \begin{array}{rcl} {\cal F}|P & = & {\cal F}\otimes_{{\cal O}_{C}} {\cal O}_{C}/{\cal I}\\ {} & = & \widehat{\cal F}/{\cal I}\widehat{\cal F}\\ {} & = & (\widehat{\cal O}_{C}/{\cal I}\widehat{\cal O}_{C})^{\oplus s}\\ {} & = & ({\cal O}_{C}/{\cal I})^{\oplus s}\\ {} & = & {\cal O}_{P}^{\oplus s}, \end{array} $$ i.e., ${\cal F}|P$ is free of rank $s$. Now let $x$ be a point of $P$. By Nakayama's lemma we get a surjection $${\cal O}_{C,x}^{\oplus s} \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } {\cal F}_{x}.$$ Denote by $K$ the kernel of this morphism. Taking the formal completion along $P$ we get an exact sequence $$0\rightarrow \widehat{K} \rightarrow \widehat{{\cal O}}_{C,x}^{\oplus s} \rightarrow \widehat{{\cal F}}_{x}\rightarrow 0.$$ By \cite{Mat1}, Thm.2.4., $\widehat{K}$ vanishes. On the other hand, we know that $\widehat{{\cal O}}_{C,x}$ is faithfully flat over ${\cal O}_{C,x}$ (cf. \cite{Mat1}, Thm.8.14.). This implies that $K=0$ and we conclude: ${\cal O}_{C,x}^{\oplus s} \cong {\cal F}_{x}$. This completes the proof of the lemma. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} In Section \ref{Eogd} we will see that, under certain conditions, the properties (ii) and (iii) of Lemma \ref{loc. free} are equivalent to Condition 1 in Definition \ref{sheaf F}. This will give us an explicit method to construct examples. A first step in this direction is the following lemma. \begin{lemma} \label{COMP.F} Assume that $C$ is as in Definition \ref{total C}. Assume furthermore that $\cal F$ is locally free of rank $r$ in some neighborhood of the section $P$ and that the restriction of $\cal F$ to $P$ is free. Then $\widehat{\cal F}$ is a free $\widehat{\cal O}_{C}$-module if and only if, for all $n\in\Bbb{N}$, the maps $$H^{0}({\cal F}/{\cal I}^{n}\otimes {\cal F}) \rightarrow H^{0}({\cal F}/{\cal I}\otimes {\cal F})$$ are surjective. \mbox{\hspace*{\fill}$\Box$} \end{lemma} \subsection{Definition of geometric data} To end this chapter, let us precisely define the geometric objects which we want to relate to Schur pairs. Let $S$ be a locally noetherian scheme. \begin{definition} \label{def data} By a {\em geometric datum of rank $r$ and index $F$ over $S$}, we mean a tupel $$(C,\pi,S,P,\rho,{\cal F},\Phi)$$ such that \begin{enumerate} \item $C$ is a scheme. \item $\pi:C\rightarrow S$ is a locally projective morphism. \item $P \subset C$ is a section of $\pi$ such that \begin{itemize} \item $P$ is a relatively ample Cartier divisor in $C$. \item For the sheaf ${\cal I}:= {\cal I}_{P}$ defining $P$ in $C$, ${\cal I}/{\cal I}^{2}$ is a trivial line bundle on $P$. \item Let $\widehat{\cal O}_{C}$ denote the formal completion of ${\cal O}_{S}$ with respect to the ideal $\cal I$. Then $\widehat{\cal O}_{C}$ is isomorphic to $\Opower{P}{z}$ as a formal ${\cal O}_{P}$-algebra. \item $\bigcap_{n\geq 0}\pi_{*}{\cal O}_{C}(-nP)=(0)$. \end{itemize} \item $\rho : \widehat{\cal O}_{C} \stackrel{\sim}{\longrightarrow} \Opower{P}{z}$ is an isomorphism of formal ${\cal O}_{P}$-algebras. \item $\cal F$ is a coherent sheaf of rank $r$ on $C$ such that \begin{itemize} \item The formal completion $\widehat{\cal F}$ of $\cal F$ along $P$ is a free $\widehat{\cal O}_{C}$-module of rank $r$. \item $\bigcap_{n\geq 0}\pi_{*}{\cal F}(-nP) =(0)$. \item $F = \gamma(\pi_{*}{\cal F}) - \gamma(R^{1}\pi_{*}{\cal F})\in K(S)$. \end{itemize} \item $\Phi : \widehat{\cal F} \stackrel{\sim}{\longrightarrow} \widehat{\cal O}_{C}^{\oplus r}$ is an isomorphism of sheaves of $\widehat{\cal O}_{C}$ - modules. \end{enumerate} \end{definition} \begin{definition} \label{ident} Two geometric data $$(C,\pi,S,P,\rho,{\cal F},\Phi) \textrm{ and } (C',\pi',S',P',\rho',{\cal F}',\Phi')$$ are identified if and only if \begin{itemize} \item $S=S'$; \item There is an isomorphism $\beta:C\rightarrow C'$ such that \begin{itemize} \item The diagram $$ \begin{array}{ccccc} P \subset& C & \stackrel{\beta}{\rightarrow} & C'&\supset P'\\ \sim \searrow&\pi\downarrow&&\downarrow\pi'&\swarrow \sim\\ & S &=&S \end{array} $$ is commutative; \item $\rho = \widehat{\beta}^{*}(\rho')$; \end{itemize} \item There is an isomorphism $\Psi: \beta^{*}{\cal F}' \rightarrow {\cal F}$ such that $\widehat{\beta}^{*}(\Phi') = \Phi\circ \widehat{\Psi}$. \end{itemize} In the sequel, we denote by ${\frak D}^{r}_{F}(S)$ the set of equivalence classes of geometric data of rank $r$ and index $F$ over $S$. \end{definition} \begin{definition} \label{homo data} A {\em homomorphism of geometric data} is a collection $$(\alpha,\beta,\Psi):(C,\pi,S,P,\rho,{\cal F},\Phi)\rightarrow (C',\pi',S',P',\rho',{\cal F}',\Phi')$$ consisting of \begin{enumerate} \item A morphism $\alpha:S\rightarrow S'$; \item A morphism $\beta:C\rightarrow C'$ such that \begin{enumerate} \item The following diagram is commutative: $$ \begin{array}{ccccc} P \subset &C & \stackrel{\beta}{\rightarrow} & C'&\supset P' \\ \sim\searrow&\pi \downarrow &&\downarrow \pi'&\swarrow\sim\\ &S & \stackrel{\alpha}{\rightarrow} & S' \end{array} $$ \item $\beta^{*}(P') = P$ as Cartier divisors; \item $\rho=\widehat{\beta}^{*}(\rho')$; \end{enumerate} \item A homomorphism of sheaves $\Psi: \beta^{*}{\cal F}' \rightarrow {\cal F}$. \end{enumerate} Two homomorphisms are identified iff they differ only by an identification isomorphism as defined in the previous definition. \end{definition} This establishes the category $\frak D$ of geometric data. \vspace{0.5cm}\\ Definition \ref{homo data} requires some justification. \begin{lemma} Let $(C,\pi,S,P,\rho,{\cal F},\Phi)$ and $(C',\pi',S',P',\rho',{\cal F}',\Phi')$ be geometric data and $\alpha:S\rightarrow S'$ and $\beta:C\rightarrow C'$ morphisms such that Conditions 2(a) and 2(b) of Definition \ref{homo data} are satisfied. Then \begin{itemize} \item $\widehat{\beta}^{*}\widehat{\cal O}_{C'} \cong \widehat{\cal O}_{C}$ and \item $\widehat{\beta}^{*}{\widehat{{\cal F}}'} \cong \widehat{\beta^{*}{\cal F}'}$, \end{itemize} i.e., Condition 2(c) is well-formulated and for a homomorphism of sheaves $\Psi : \beta^{*}{\cal F}' \rightarrow {\cal F}$, the composition $\pi_{*}((\rho\circ\Phi)\circ\widehat{\Psi}\circ(\widehat{\beta}^{*} (\rho'\circ\Phi'))^{-1})$ belongs to ${\cal H}om_{\power{{\cal O}_{S}}{z}}(\power{{\cal O}_{S}}{z}^{\oplus r'}, \Opower{S}{z}^{\oplus r})$. \end{lemma} {\bf Proof}\hspace{0.3cm} We consider the exact sequence of ${\cal O}_{C'}$-modules \begin{equation} \label{just 1} 0\rightarrow {\cal I}'^{n}\rightarrow {\cal O}_{C'} \rightarrow {\cal O}_{C'}/{\cal I}'^{n} \rightarrow 0. \end{equation} The support of the sheaf ${\cal O}_{C'}/{\cal I}'^{n}$ is contained in $P$, and, as a sheaf on $P$, ${\cal O}_{C'}/{\cal I}'^{n}$ is known to be free. Therefore we can conclude, using \cite{Mat1}, ch.18, Lemma 2, that $${\cal T}or^{{\cal O}_{C'}}_{1}({\cal O}_{C'}/{\cal I}'^{n}, {\cal O}_{C}) \cong {\cal T}or^{{\cal O}_{P'}}_{1}({\cal O}_{C'}/{\cal I}'^{n}, {\cal O}_{P}) =(0).$$ Consequently, the sequence $$ 0\rightarrow \beta^{*}{\cal I}'^{n}\rightarrow \beta^{*}{\cal O}_{C'} \rightarrow \beta^{*}({\cal O}_{C'}/{\cal I}'^{n}) \rightarrow 0 $$ is exact, as well. Using this we start our calculation: $$ \begin{array}{rcl} \widehat{\beta}^{*}\widehat{\cal O}_{C'} & = & lim_{n\to\infty} \beta^{*}({\cal O}_{C'}/{\cal I}'^{n})\\ &\cong& lim_{n\to\infty} \beta^{*}{\cal O}_{C'}/\beta^{*}{\cal I}'^{n}\\ &\cong &lim_{n\to\infty}{\cal O}_{C}/{\cal I}^{n}\\ & = & \widehat{\cal O}_{C}. \end{array} $$ We proceed analogously for the sheaf ${\cal F}'$. The sequence (\ref{just 1}) stays exact after tensoring with ${\cal F}'$, since we assumed ${\cal F}'$ to be locally free near $P'$. With the same conclusions as above we finally get an exact sequence $$ 0\rightarrow \beta^{*}({\cal F}'\otimes{\cal I}'^{n})\rightarrow \beta^{*}({\cal F}') \rightarrow \beta^{*}({\cal F}'\otimes({\cal O}_{C'}/{\cal I}'^{n})) \rightarrow 0 $$ and, of course, ${\cal F}'\otimes({\cal O}_{C'}/{\cal I}'^{n}) \cong {\cal F}'/({\cal F}'\otimes {\cal I}'^{n})$. Now we can calculate again $$ \begin{array}{rcl} \widehat{\beta}^{*}\widehat{{\cal F}'}& = & lim_{n\to\infty} \beta^{*}({\cal F}'/{\cal F}'\otimes {\cal I}'^{n})\\ & \cong & lim_{n\to\infty} \beta^{*}({\cal F}')/\beta^{*}({\cal F}'\otimes {\cal I}'^{n})\\ &\cong & lim_{n\to\infty} \beta^{*}({\cal F}')/(\beta^{*}({\cal F}')\otimes {\cal I}^{n})\\ & = & \widehat{\beta^{*}({\cal F}')}. \end{array} $$ \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Let us include one more definition. \begin{definition} We define a full subcategory ${\frak D}'$ of $\frak D$ as follows:\\ The objects of this category are the equivalence classes of geometric data $(C,\pi,S,P,\rho,{\cal F},\Phi)$ such that $$\pi_{*}{\cal O}_{C} = {\cal O}_{S}.$$ \end{definition} This subcategory will play an important role in Chapter \ref{Fam. DO}. \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} \label{field2} \begin{itemize} \item Assume that $S=Spec(k)$, for some field $k$. Then the geometric datum $(C,\pi,S,P,\rho,{\cal F},\Phi)$ reduces to $(C,P,\rho,{\cal F},\Phi)$. Using the same method as in the remark on page \pageref{Grass Mulase}, we see that this datum corresponds to a {\em quintet} defined by M.~Mulase \cite{M1}, where $$\rho : \widehat{\cal O}_{C} \hookrightarrow \power{k}{y}$$ decomposes into $$\widehat{\cal O}_{C} \stackrel{\sim}{\rightarrow} \power{k}{y^{r}} \hookrightarrow \power{k}{y}.$$ \item Now let $S$ be any scheme, $(C,\pi,S,P,\rho,{\cal F},\Phi)$ a geometric datum and $s\in S$ a closed point. Then we can restrict everything to the fibre $C_{s}$ of $C$ over $s$ and get a collection $$(C_{s},\pi|C_{s},\{s\},P|\{s\},\rho|C_{s},{\cal F}|C_{s},\Phi|C_{s}).$$ Assume that this is also a geometric datum. Then the restriction defines a morphism of geometric data $$(C_{s},\pi|C_{s},\{s\},P|\{s\},\rho|C_{s},{\cal F}|C_{s},\Phi|C_{s}) \rightarrow (C,\pi,S,P,\rho,{\cal F},\Phi).$$ \item Another example of morphisms of geometric data is the following: Let $(C,\pi,S,P,\rho,{\cal F},\Phi)$ be a geometric datum and $\alpha:S'\rightarrow S$ a flat morphism. Then the fibre product gives rise to another geometric datum. This fact will be shown in Section \ref{base change}. \end{itemize} \section{The relative Krichever functor} After having defined both sides we want to relate, let us start with the construction of a bijective contravariant functor between the category of Schur pairs and the category of geometric data. Throughout the chapter let us assume that $S$ is a locally noetherian scheme. \subsection{Constructing Schur pairs} Assume we are given a geometric datum $(C,\pi,S,P,\rho,{\cal F},\Phi)$ of rank $r$ and index $F$. Let us start with \begin{lemma} \label{Cartier} For each integer $n$, the maps $\rho$ and $\Phi$ induce isomorphisms: $$ \begin{array}{*{5}{c}} \rho & : & \widehat{{\cal O}_{C}(n\cdot P)} & \stackrel{\sim}{\longrightarrow} & \Opower{P}{z}\cdot z^{-n},\\ \Phi & : & \widehat{{\cal F}(n\cdot P)} & \stackrel{\sim}{\longrightarrow} & \widehat{{\cal O}_{C}(n\cdot P)}^{\oplus r}. \end{array} $$ \end{lemma} {\bf Proof}\hspace{0.3cm} This is clear from the fact that $${\cal I} \cong {\cal O}_{C}(-P).$$ \mbox{\hspace*{\fill}$\Box$} \begin{lemma} \label{twist up} For any open affine subset $U$ of $S$, the natural maps \begin{equation} \label{loc g} \begin{array}{ccc} H^{0}(\pi^{-1}(U), {\cal O}_{C}(n\cdot P)) & \rightarrow & H^{0}(\pi^{-1}(U)\setminus P, {\cal O}_{C})\\ H^{0}(\pi^{-1}(U), {\cal F}(n\cdot P)) & \rightarrow & H^{0}(\pi^{-1}(U)\setminus P, {\cal F})\\ \end{array} \end{equation} induce isomorphisms $$ \begin{array}{cccr} lim_{n\to\infty}H^{0}(\pi^{-1}(U), {\cal O}_{C}(n\cdot P)) & \stackrel{\sim}{\rightarrow} & H^{0}(\pi^{-1}(U)\setminus P, {\cal O}_{C})&{}\\ lim_{n\to\infty}H^{0}(\pi^{-1}(U), {\cal F}(n\cdot P)) & \stackrel{\sim}{\rightarrow} & H^{0}(\pi^{-1}(U)\setminus P, {\cal F})&.\\ \end{array} $$ \end{lemma} {\bf Proof}\hspace{0.3cm} The maps in (\ref{loc g}) are inclusions, because $P$ is locally given by an element which is neither a zero divisor in ${\cal O}_{C}$ nor in $\cal F$. We know that $P$ is ample relative to $S$. Therefore, for sufficiently large $n\in \Bbb{N}$, ${\cal O}_{\pi^{-1}(U)}(n\cdot P)$ has a nonconstant global section. Now the assertion is a direct consequence of \cite{H1}, Lemma II.5.14. \mbox{\hspace*{\fill}$\Box$} \begin{definition} For a given geometric datum $(C,\pi,S,P,\rho,{\cal F},\Phi)$ of rank $r$ and index $F$, we define $$({\cal A},{\cal W}) := \chi_{r,F}(C,\pi,S,P,\rho,{\cal F},\Phi)$$ as follows: \begin{displaymath} \begin{array}{rcl} {\cal A}(U) & := & \pi_{*}(\rho) (H^{0}(\pi^{-1}(U)\setminus P, {\cal O}_{C}))\\ {} & {}= &\pi_{*}(\rho) (lim_{n\to\infty} H^{0}(\pi^{-1}(U), {\cal O}_{C}(n\cdot P)))\\ {} & {}= & \pi_{*}(\rho)(lim_{n\to\infty} H^{0}(U, \pi_{*}{\cal O}_{C}(n\cdot P)))\\ {} & {}\subset & \negpower{{\cal O}_{S}(U)}{z},\\ {} & {} & {}\\ {\cal W}(U) & := & \pi_{*}(\rho\circ\Phi)(H^{0}(\pi^{-1}(U)\setminus P, {\cal F}))\\ {} & {}= &\pi_{*}(\rho\circ\Phi) (lim_{n\to\infty} H^{0}(\pi^{-1}(U), {\cal F}(n\cdot P)))\\ {} & {}= & \pi_{*}(\rho\circ\Phi)(lim_{n\to\infty} H^{0}(U, \pi_{*}{\cal F}(n\cdot P)))\\ {} & {}\subset & \negpower{{\cal O}_{S}(U)}{z}^{\oplus r}. \end{array} \end{displaymath} \end{definition} {\bf Remark }\hspace{0.3cm} \label{field1} Assume that $S=Spec(k)$ for some field $k$. Then a geometric datum $(C,\pi,S,P,\rho,{\cal F},$ $\Phi)$ gives us a quintet $(C,P,\rho,{\cal F},\Phi)$ as defined in \cite{M1}, and $({\cal A},{\cal W})$ is the corresponding Schur pair defined there. \vspace{0.5cm} In analogy to the case of a curve over a field we now want to identify the (relative) cohomology of ${\cal O}_{C}$ and $\cal F$ via the sheaves $\cal A$ and $\cal W$. At first, we will see that the pole order of a local section along $P$ is exactly the order of the corresponding formal power series: \begin{proposition} \label{order} For all integers $n\in\Bbb{Z}$: $$ \begin{array}{*{5}{c}} {\cal A} & \cap & \power{{\cal O}_{S}}{z}\cdot z^{-n} & = & \pi_{*}(\rho)(\pi_{*}({\cal O}_{C}(n\cdot P))),\\ {\cal W} & \cap & \power{{\cal O}_{S}}{z}^{\oplus r}\cdot z^{-n} & = & \pi_{*}(\rho\circ\Phi)(\pi_{*}({\cal F}(n\cdot P))). \end{array} $$ \end{proposition} {\bf Proof}\hspace{0.3cm} This is obviously a local property. Therefore let $S=Spec(R)$ be affine, $R$ a local ring, $P \subset V=Spec(B)\subset C$, $B$ another local ring, $I:={\cal I}_{P}(V)=b\cdot B$, $b$ a non-zero divisor in $B$, and $M:=H^{0}(V,{\cal F})$. The composition of the maps $$H^{0}(C, {\cal F}(nP)) \rightarrow H^{0}(V,{\cal F}(nP))\rightarrow H^{0}(\widehat{{\cal F}(nP)})$$ is assumed to be injective. Therefore the first map must be injective. The second one is injective, a priori, since $B$ is a local ring. So we only have to show that $$M_{b} \cap \widehat{M} = M.$$ Of course, we are done if we can show: $$M \cap b\widehat{M} = bM,$$ i.e., $$M\cap I\widehat{M} = IM.$$ But this clear: Take $m\in M$ and consider its image in $\widehat{M}$. It can be identified with the sequence $(m(mod I^{n}M))_{n\in\Bbb{N}}$. This sequence belongs to $I\widehat{M}$ if and only if, for all $n$, there are elements $i_{n}\in IM$ such that $m-i_{n}\in I^{n}M$. This implies $m\in IM$ immediately. Of course, the statement concerning $\cal A$ is proved in the same way. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now we want to prove: \begin{proposition} \label{cohomology} There are isomorphisms of sheaves of ${\cal O}_{S}$ - modules $$ \begin{array}{*{4}{c}} {}&\frac{\negpower{{\cal O}_{S}}{z}}{{\displaystyle {\cal A}} + \power{{\cal O}_{S}}{z}} & \cong & R^{1}\pi_{*}{\cal O}_{C}\\ {} & {} & {}\\ \textrm{and }&\frac{\negpower{{\cal O}_{S}}{z}^{\oplus r}}{{\displaystyle {\cal W}} + \power{{\cal O}_{S}}{z}^{\oplus r}} & \cong & R^{1}\pi_{*}{\cal F}. \end{array} $$ \end{proposition} For the proof we need the following lemma. \begin{lemma} \label{iso completion} Assume that $U=Spec(R)$ is an open affine subset of $S$, and $V=Spec(B) \subseteq \pi^{-1}(U)$ is an open affine set containing $P\cap \pi^{-1}(U)$ such that, on $V$, the divisor $P$ is given by a single element $b\in B$ and the restriction of $\cal F$ to $V$ is free. Then the natural maps $$ \begin{array}{ccc} H^{0}(V,{\cal O}_{C}) &\rightarrow &H^{0}(V,\widehat{\cal O}_{C}),\\ H^{0}(V,{\cal F})& \rightarrow &H^{0}(V,\widehat{\cal F}) \end{array} $$ induce isomorphisms of $R$ - modules: {\Large $$ \begin{array}{*{5}{c}} &\frac{H^{0}(V\setminus P,{\cal O}_{C})}{H^{0}(V,{\cal O}_{C})} & \stackrel{\sim}{\longrightarrow} & \frac{H^{0}(V\setminus P,\widehat{\cal O}_{C})}{H^{0} (V,\widehat{\cal O}_{C})}&\\ {\normalsize\textrm{and }}&\frac{H^{0}(V\setminus P,{\cal F})}{H^{0}(V,{\cal F})} & \stackrel{\sim}{\longrightarrow} & \frac{H^{0}(V\setminus P,\widehat{\cal F})}{H^{0} (V,\widehat{\cal F})}&. \end{array} $$} \end{lemma} {\bf Proof}\hspace{0.3cm} Let us start with the investigation of the structure sheaf. The map $$B\rightarrow \widehat{B}$$ must be injective, since all elements of $B$ are locally given by quotients of elements of \linebreak $\bigoplus_{n\geq 0} H^{0}(\pi^{-1}(U), {\cal O}_{C}(nP))$. Let us fix a natural number $n$. We obtain a diagram of embeddings: \begin{equation} \begin{array}{*{8}{c}} B & = & H^{0}(V,{\cal O}_{C}) & \rightarrow & H^{0}(V,\widehat{\cal O}_{C}) & = & \widehat{B} &{}\\ {} & {} & \downarrow & {} & \downarrow&{}\\ \frac{1}{b^{n}}B & = & H^{0}(V,{\cal O}_{C}(n\cdot P)) & \rightarrow & H^{0}(V,\widehat{{\cal O}_{C}(n\cdot P))} & = & \frac{1}{b^{n}} \widehat{B}&. \end{array} \end{equation} Now the proof of Proposition \ref{order} implies that $$ \widehat{B}\cap \frac{1}{b^{n}} B = B. $$ {}From this we obtain a natural inclusion: $$\frac{\frac{1}{b^{n}} B}{B}\hookrightarrow \frac{\frac{1}{b^{n}} \widehat{B}}{\widehat{B}}. $$ Taking the limit over $n$, we get a monomorphism of $R$ - modules: $$ \frac{B_{b}}{B} = \frac{{\displaystyle H^{0}(V\setminus P,{\cal O}_{C})}}{{\displaystyle H^{0}(V,{\cal O}_{C})}} \hookrightarrow lim_{n\to \infty}\frac{{\displaystyle H^{0}(V,\widehat{{\cal O}_{C} (n\cdot P)})}}{{\displaystyle H^{0}(V,\widehat{\cal O}_{C})}} = \frac{(\widehat{B})_{b}}{\widehat{B}}. $$ We claim that this is, in fact, an isomorphism. Of course, we have $\poly{R}{b}\subseteq B$. So we obtain a diagram of inclusions: $$ \begin{array}{ccc} \poly{R}{b} & \rightarrow & B\\ \downarrow&&\downarrow\\ \negpoly{R}{b} & \rightarrow & B_{b}, \end{array} $$ where $\negpoly{R}{b}\cap B = \poly{R}{b}$, since $b$ is a non-zero divisor in $B$. Therefore we end up with a chain of inclusions: $$ \frac{\negpoly{R}{b}}{\poly{R}{b}} \hookrightarrow \frac{{\displaystyle H^{0}(V\setminus P,{\cal O}_{C})}}{{\displaystyle H^{0}(V,{\cal O}_{C})}}\hookrightarrow \frac{{\displaystyle H^{0}(V\setminus P,\widehat{\cal O}_{C})}} {{\displaystyle H^{0}(V,\widehat{\cal O}_{C})}} = \frac{\negpower{R}{b}}{\power{R}{b}}. $$ But the first and the last term are canonically isomorphic $R$ - modules. Therefore all the monomorphisms appearing above are really isomorphisms. Now we turn our attention to $\cal F$. The isomorphism ${\cal F}|V \cong {\cal O}_{V}^{\oplus r}$ extends to an isomorphism $\widehat{\cal F}\cong \widehat{\cal O}_{V}^{\oplus r}$, which also respects the localization by $b$. So we get the claim concerning $\cal F$ by applying the according statement on ${\cal O}_{C}$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Let us proceed to the \vspace{0.5cm}\\ {\bf Proof of the Proposition \ref{cohomology}}\hspace{0.3cm} As the isomorphism constructed in Lemma \ref{iso completion} is natural, it is compatible with intersections of affine sets. Now, once again, it is sufficient to consider an open affine set $U$ of $S$, as in Lemma \ref{iso completion}. Then $\pi^{-1}(U)$ is covered by the affine sets $V$ and $\pi^{-1}(U)\setminus P$. Since $\pi|\pi^{-1}(U)$ is a separated morphism over an affine scheme, we can apply \cite{H1}, Thm. III.4.5., and get, with the aid of Lemmas \ref{twist up} and \ref{iso completion}, $$ \begin{array}{rcl} R^{1}\pi_{*}{\cal O}_{C}(U) & = & H^{1}(\pi^{-1}(U), {\cal O}_{C})\\ {}&{}&{}\\ {} & = & \frac{{\displaystyle H^{0}(V \setminus P, {\cal O}_{C})}}{{ \displaystyle H^{0}(V, {\cal O}_{C}) + H^{0}(\pi^{-1}(U)\setminus P, {\cal O}_{C})}}\\ {}&{}&{}\\ {} & = & \frac{{\displaystyle H^{0}(V\setminus P, \widehat{\cal O}_{C})}}{{\displaystyle H^{0}(V, \widehat{\cal O}_{C}) + H^{0}(\pi^{-1}(U)\setminus P, {\cal O}_{C})}}\\ {}&{}&{}\\ {} & \cong & \frac{{\displaystyle \Onegpower{S}{z}(U)}}{{\displaystyle \Opower{S}{z}(U) + {\cal A}(U)}}. \end{array} $$ The proof of the statement concerning $\cal F$ can be given in the same way. \mbox{\hspace*{\fill}$\Box$} \begin{corollary} If $(C,\pi,S,P,\rho,{\cal F},\Phi)$ is a geometric datum of rank $r$ and index $F$, then $({\cal A},{\cal W}) = \chi_{r,F}(C,\pi,S,P,\rho,{\cal F},\Phi)$ is a Schur pair of rank $r$ and index $F$ over $S$. \end{corollary} {\bf Proof}\hspace{0.3cm} By construction, $\cal A$ is a quasicoherent sheaf of ${\cal O}_{S}$ - subalgebras of $\Onegpower{S}{z}$, $\cal W$ is a quasicoherent sheaf of ${\cal O}_{S}$ - modules, and ${\cal A}\cdot{\cal W}\subseteq{\cal W}$. The fact that $\cal A$ and $\cal W$ are elements of the infinite Grassmannians of rank 1 and $r$, respectively, follows from Propositions \ref{order} and \ref{cohomology}, and the fact that $R^{i}\pi_{*}{\cal O}_{C}$ and $R^{i}\pi_{*}{\cal F}$ are coherent sheaves for locally projective morphisms $\pi$ and for all $i$ (cf. \cite{H1}, Thm.III.8.8). \mbox{\hspace*{\fill}$\Box$} \begin{corollary} \label{ident. Schur} If $(C,\pi,S,P,\rho,{\cal F},\Phi)$ and $(C',\pi',S',P',\rho',{\cal F}',\Phi')$ are identified as geometric data, then we obtain identical corresponding Schur pairs $({\cal A}, {\cal W})$ and $({\cal A}', {\cal W}')$. \end{corollary} {\bf Proof}\hspace{0.3cm} This follows from an easy calculation. We use the isomorphisms $\beta$ and $\Psi$ and get: $$ \begin{array}{rcl} {\cal A} & := & \pi_{*}(\rho)(lim_{n\to\infty} \pi_{*}{\cal O}_{C}(n\cdot P))\\ & {}= & \pi_{*}(\widehat{\beta}^{*}(\rho'))(lim_{n\to\infty} \pi_{*}(\beta^{*}{\cal O}_{C'}(n\cdot P')))\\ & {}= & \pi'_{*}(\rho')(lim_{n\to\infty} \pi'_{*}{\cal O}_{C'}(n\cdot P'))\\ & {}= & {\cal A}'. \end{array} $$ Analogously: $$ \begin{array}{rcl} {\cal W}' & := & \pi'_{*}(\rho'\circ \Phi')(lim_{n\to \infty}\pi'_{*} {\cal F}'(n\cdot P'))\\ & {}= & \pi'_{*}(\rho'\circ \Phi')(lim_{n\to \infty} \pi_{*}\beta^{*} {\cal F}'(n\cdot P'))\\ & {}= & \pi_{*}(\widehat{\beta}^{*}(\rho'\circ \Phi'))(lim_{n\to \infty} \pi_{*}((\beta^{*} {\cal F}')(n\cdot P)))\\ & {}= & \pi_{*}(\rho\circ \Phi\circ\widehat{\Psi})(lim_{n\to \infty} \pi_{*}((\beta^{*} {\cal F}')(n\cdot P)))\\ & {}= & \pi_{*}(\rho\circ \Phi)(lim_{n\to \infty} \pi_{*}( {\cal F}(n\cdot P)))\\ & {}=&{\cal W}. \end{array} $$ \mbox{\hspace*{\fill}$\Box$} \begin{proposition} \label{morph} Homomorphisms of geometric data induce homomorphisms of the corresponding Schur pairs. \end{proposition} {\bf Proof}\hspace{0.3cm} Let $(\alpha,\beta,\Psi):(C,\pi,S,P,\rho,{\cal F},\Phi) \rightarrow (C',\pi',S',P',\rho',{\cal F}',\Phi')$ be a homomorphism of geometric data. Let $({\cal A}, {\cal W})$ and $({\cal A}', {\cal W}')$ be the Schur pairs associated to the given geometric data. We want to construct a homomorphism of Schur pairs $$(\alpha,\xi):({\cal A}', {\cal W}')\rightarrow ({\cal A}, {\cal W}).$$ Of course, the morphism $\alpha:S\rightarrow S'$ is taken directly from $(\alpha,\beta,\Psi)$, whereas $\xi$ is defined as $\pi_{*}((\rho\circ\Phi)\circ\widehat{\Psi}\circ(\widehat{\beta}^{*} (\rho'\circ\Phi'))^{-1})$. Now we apply the properties of the given morphism $\beta:C\rightarrow C'$ and derive: $$ \begin{array}{rcl} \alpha^{(*)}{\cal A}' & = & \alpha^{(*)}(\pi'_{*}(\rho')(lim_{n\to\infty} \pi'_{*}{\cal O}_{C'}(n\cdot P')))\\ {} & \subseteq & \pi_{*}(\widehat{\beta}^{*}(\rho'))(lim_{n\to\infty} \pi_{*} \beta^{*}{\cal O}_{C'}(n\cdot P'))\\ {} & = & \pi_{*}(\rho)( lim_{n\to\infty} \pi_{*} {\cal O}_{C}(n\cdot P))\\ {} & = & {\cal A}. \end{array} $$ As for ${\cal W}$ and ${\cal W}'$, we obtain: $$ \begin{array}{rcl} \xi(\alpha^{(*)}{\cal W}') & = & \xi(\alpha^{(*)}(\pi'_{*}(\rho'\circ\Phi')(lim_{n\to\infty} \pi'_{*}{\cal F}'(n\cdot P'))))\\ &\subseteq& (\xi\circ\pi_{*}(\widehat{\beta}^{*}(\rho'\circ\Phi')))(lim_{n\to\infty} \pi_{*}\beta^{*}{\cal F}'(n\cdot P'))\\ &=& \pi_{*}(\rho\circ\Phi\circ\widehat{\Psi})(lim_{n\to\infty} \pi_{*}\beta^{*}{\cal F}'(n\cdot P'))\\ &\subseteq&\pi_{*}(\rho\circ\Phi)(lim_{n\to\infty} \pi_{*}{\cal F}(n\cdot P'))\\ &=& {\cal W}. \end{array} $$ So we really have constructed a homomorphism of Schur pairs. \mbox{\hspace*{\fill}$\Box$} \begin{corollary} $\chi$ is a contravariant functor from the category of geometric data to the category of Schur pairs. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \begin{definition} The functor $\chi$ is called the {\em Krichever functor}. \end{definition} \subsection{Constructing geometric data} Now assume that we are given a Schur pair $({\cal A},{\cal W})$ of rank $r$ and index $F$ over the scheme $S$. We start with some general observations. \begin{lemma} \label{A0} \begin{enumerate} \item For all $n\in\Bbb{Z}$, ${\cal A}^{(n)}$ and ${\cal W}^{(n)}$ are coherent sheaves. In particular, ${\cal A}^{(0)}$ is a coherent sheaf of ${\cal O}_{S}$-algebras. \item If $U = Spec(R)$ is an open affine subset of $S$, then there is an integer $M\in\Bbb{Z}$ (possibly depending on $U$) such that $${\cal A}^{(-M)}(U) = {\cal W}^{(-M)}(U) = (0).$$ \item All local sections of ${\cal A}^{(-1)}$ are nilpotent. \end{enumerate} \end{lemma} {\bf Proof}\hspace{0.3cm} \begin{enumerate} \item Let $U=Spec(R)$ be an affine open subset of $S$. We have to show that ${\cal A}^{(n)}(U)$ and ${\cal W}^{(n)}(U)$ are finitely generated. ${\cal A}^{(0)}(U)$ and ${\cal W}^{(0)}(U)$ are finitely generated by the definition of Schur pairs. Since $R$ is a noetherian ring, this immediately proves the statement for all $n\leq 0$. Now assume that $n>0$. Then ${\cal A}^{(n)}(U)/{\cal A}^{(0)}(U)$ is isomorphic to a submodule of $\power{R}{z}\cdot z^{-n}/\power{R}{z}$, hence finitely generated. But this already implies that ${\cal A}^{(n)}(U)$ itself is finitely generated. The same method of proof may be used for $\cal W$. \item Without loss of generality we prove the statement for $\cal A$. ${\cal A}^{(0)}$ is generated by finitely many elements $f_{1},\ldots,f_{m}$. Let us write $$f_{i} = \sum_{j\geq 0} \lambda_{i,j} z^{j}.$$ The statement ${\cal A}^{(-M)}(U)\neq (0)$ is equivalent to: There are elements $\mu_{1},\ldots, \mu_{m} \in R$ such that $\sum_{i=1}^{m} \mu_{i}\lambda_{i,j} = 0$ for $j=0,\ldots, M-1$, but $\sum_{i=1}^{m} \mu_{i} f_{i} \neq 0$. We denote by ${\cal N}_{l}$ the submodule of $R^{m}$ generated by the vectors $(\lambda_{\cdot ,0}), \ldots, (\lambda_{\cdot ,l})$. These modules form an ascending chain of submodules of $R^{m}$: $${\cal N}_{0}\subseteq {\cal N}_{1}\subseteq{\cal N}_{2}\subseteq \ldots \subseteq R^{m}.$$ Since $R^m$ is a noetherian module, there is an integer $T$ such that ${\cal N}_{l}={\cal N}_{T}$, for all $l\geq T$. We claim that ${\cal A}^{(-T-1)}(U)=(0)$. If this were false, we could find elements \linebreak[4] $\mu_{1},\ldots, \mu_{m} \in R$ such that $\sum_{i=1}^{m} \mu_{i}\lambda_{i,j} = 0$, for $j=0,\ldots, T$. But by the definition of ${\cal N}_{l}$ this already implies $\sum_{i=1}^{m} \mu_{i} f_{i} = 0$. So we are done. \item The last part of the lemma is easy. Assume that for some $U$, ${\cal A}^{(-1)}(U)$ would contain an element $f$ which is not nilpotent. Then, for all $n\in\Bbb{N}$, $$0\neq f^{n} \in {\cal A}^{(-n)}(U),$$ and this is a contradiction. \end{enumerate} \mbox{\hspace*{\fill}$\Box$} \begin{lemma} \label{a and b} Let $U = Spec(R)$ be an open affine subset of $S$. Then ${\cal A}(U)$ is a finitely generated $R$ - algebra of relative dimension 1. \end{lemma} {\bf Proof}\hspace{0.3cm} By assumption, $\frac{\negpower{R}{z}}{\power{R}{z} + {\displaystyle {\cal A}(U)}}$ is a finitely generated $R$ - module. So we can choose finitely many elements $b_{1},\ldots,b_{m}\in \negpower{R}{z}$ such that $[b_{1}],\ldots,[b_{m}]$ are generators of this module. Denote by $N$ the maximum of the orders of the $b_{j}$'s. We may assume that $N\geq 2$. Now it is straightforward that: $$\negpower{R}{z} = \power{R}{z}\cdot z^{-N} + {\cal A}(U).$$ Therefore, for all $n>N$, there is a monic element of order $n$ in ${\cal A}(U)$. Let us choose monic elements $a, b\in {\cal A}(U)$ of order $2N+2$ and $2N+1$, respectively. We claim that \begin{equation} \label{a and b,2} \negpower{R}{z} = \power{R}{z}\cdot z^{-(2N+1)(2N+2)} + \poly{R}{a,b}. \end{equation} In order to prove this, we choose an integer $n \geq (2N+1)(2N+2)$. Then we can find $m \geq 0$ and $0 \leq l < 2N+2$ such that $$ \begin{array}{*{4}{c}} n & = & (2N+1)(2N+2) + m\cdot (2N+2) + l &{}\\ {} & = & (2N+1)(2N+2) + m\cdot (2N+2) + l\cdot((2N+2) - (2N+1)) &{}\\ {} & = & (m + l)\cdot (2N+2) + (2N+2 - l)\cdot (2N+1) &{}\\ {} & = & ord( a^{m+l}\cdot b^{2N+2-l} ) &. \end{array} $$ Since $a$ and $b$ are monic, this proves (\ref{a and b,2}) and, in particular, the identity \begin{equation} {\cal A}(U) = {\cal A}(U)^{((2N+1)(2N+2))} + \poly{R}{a,b}. \end{equation} ${\cal A}(U)^{((2N+1)(2N+2))}$ is a finitely generated $R$ - module (cf. Lemma \ref{A0}). This implies that ${\cal A}(U)$ is a finitely generated $R$ - algebra of relative dimension at most 2. From the choice of $a$ and $b$, it is also clear that the relative dimension is greater than zero. Now we only need to prove that $a$ and $b$ satisfy a polynomial relation with coefficients in $R$. Obviously, $b$ does not lie in $\poly{R}{a}$. Let $\{u_{1},\ldots,u_{q}\}$ be a set of generators of $\poly{R}{a,b}^{((2N+1)(2N+2)-1)}$ as an $R$ - module, and put $$v_{n} := a^{m+l}\cdot b^{2N+2-l}$$ for $n\geq (N+1)(N+2)$ and $n = (2N+1)(2N+2) + m\cdot (2N+2) + l$ (see above). Then the set $\{u_{1},\ldots,u_{q}\}\cup \{v_{n}\}_{n\geq (2N+1)(2N+2)}$ generates the $R$ - module $\poly{R}{a,b}$. For all $M$, $a^{M}$ can be written as a linear combination of the $u_{j}$'s and $v_{j}$'s. Since none of the $v_{j}$'s is a power of $a$, and since there are only finitely many $u_{j}$'s, the representing linear combination for a sufficiently high power of $a$ is exactly the required polynomial relation between $a$ and $b$. This completes the proof. \mbox{\hspace*{\fill}$\Box$} \begin{lemma} \label{m+n} Let $U=Spec(R)$ be an open affine subset of $S$ and let $N\in\Bbb{N}$ be such that $\negpower{R}{z} = \power{R}{z}\cdot z^{-N} + {\cal A}(U)$ (cf. Lemma \ref{a and b}). Then, for all $n,m \geq 2N+1$: $${\cal A}(U)^{(m)}\cdot{\cal A}(U)^{(n)}={\cal A}(U)^{(m+n)}.$$ \end{lemma} {\bf Proof}\hspace{0.3cm} Of course, ${\cal A}(U)^{(m)}\cdot{\cal A}(U)^{(n)}\subseteq {\cal A}(U)^{(m+n)}$. To see the other inclusion it is sufficient to show that ${\cal A}(U)^{(m)}\cdot{\cal A}(U)^{(n)}$ contains monic elements of the orders $m+1, \ldots, m+n$. First take $1\leq i\leq n-N$. Then we can split $m+i = (m-N) + (N+i)$. Since $m-N$ is greater than $N$, by assumption, ${\cal A}(U)^{(m)}$ contains a monic element of order $m-N$. On the other hand, $N+1\leq N+i \leq n$. Therefore, ${\cal A}(U)^{(n)}$ contains a monic element of order $N+i$. As a second case consider now the terms $m+i$ for $n-N<i\leq n$. In this case, $i$ is greater than $N$, so ${\cal A}(U)^{(n)}$ contains a monic element of order $i$, and, of course, ${\cal A}(U)^{(m)}$ contains a monic element of order $m$. This completes the proof. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now we aim at defining geometric objects from a given Schur pair. First, we define a sheaf of graded ${\cal O}_{S}$ - algebras as follows: $$grd({\cal A}) := {\cal O}_{S}\oplus\bigoplus_{n\geq 1} {\cal A}^{(n)},$$ i.e., $grd({\cal A})_{0} = {\cal O}_{S}$ and $grd({\cal A})_{n} = {\cal A}^{(n)}$, for $n\geq 1$. Now, we define a scheme $C$ by $$C := Proj(grd({\cal A})).$$ This scheme comes equipped with a projection morphism $\pi$ to $S$, and Lemma \ref{a and b} says that $C$ is a curve over $S$. \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} \begin{itemize} \item The morphism $\pi$ factors over the scheme $Spec({\cal A}^{(0)})$. By Lemma \ref{A0}, this scheme is finite over $S$, and $Spec({\cal A}^{(0)})_{red}=S_{red}$. \item It is well-known that, for any $m\in \Bbb{N}$, the scheme $C$ is naturally isomorphic to $$C^{(m)} := Proj({\cal O}_{S}\oplus\bigoplus_{n\geq 1} {\cal A}^{(mn)}).$$ On the other hand, Lemma \ref{m+n} implies that for any affine open subset $U\subseteq S$ there is a number $m$ such that ${\cal O}_{S}(U)\oplus\bigoplus_{n\geq 1} {\cal A}^{(mn)}(U)$ is generated by ${\cal A}^{(m)}(U)$ as an ${\cal O}_{S}(U)$ - algebra. Moreover, we know that, for all $m$, ${\cal A}^{(m)}$ is coherent (see Lemma \ref{A0}). Then we obtain from the general theory developed in \cite{H1}, II.7., that $C$ is locally projective over $S$. \item The localization of $grd({\cal A})$ by the section $1\in grd({\cal A})_{1}$ can be identified with $\cal A$, thus $C$ contains the (relatively) affine subset $Spec({\cal A})$. \end{itemize} \vspace{0.5cm} Analogously, we define $$grd({\cal W}) := \bigoplus_{n\in \Bbb{Z}} {\cal W}^{(n)}.$$ This gives us a locally finitely generated $grd({\cal A})$ - module. The sheaf $\cal F$ on $C$ is defined to be $${\cal F} := (grd({\cal W}))^{\sim}.$$ We would like to see that the curve $C$ just constructed is exactly a curve of the type we started with. \begin{theorem} \label{Schur curve} There is a section $P\subset C$ of $\pi$ such that $C\setminus P$ is precisely $Spec({\cal A})$. $P$ is a relatively ample Cartier divisor, its conormal sheaf is free of rank 1 on $P$, and $\widehat{\cal O}_{C}$ is isomorphic to $\Opower{P}{z}$. Finally, $\bigcap_{n\geq 0} \pi_{*}{\cal O}_{C}(-nP) = (0)$. \end{theorem} {\bf Proof}\hspace{0.3cm} Again, we restrict everything to an affine open subset $U= Spec(R)$ of $S$. Let $a$ and $b$ be the elements we constructed in Lemma \ref{a and b}. We can view $a$ as an element of $(grd({\cal A}))(U)_{2N+2}$. Let us localize $grd({\cal A})$ by $a$: $$ \begin{array}{*{5}{c}} B & := & (grd({\cal A}))(U)_{(a)} & = & \left\{ \frac{g}{a^{n}}/ n\in \Bbb{N}, g \in (grd({\cal A}))(U)_{(2N+2)\cdot n} \right\}\\ &&{} & = & \left\{ \frac{g}{a^{n}}/ n\in \Bbb{N}, g \in {\cal A}(U)^{((2N+2)\cdot n)} \right\}\\ &&{} & \subseteq & \power{R}{z}. \end{array} $$ Especially, $y := \frac{b}{a}$ gives us a monic element of $B$ of order -1. But of course, for this element $y$ we obtain: $$\power{R}{y} \cong \power{R}{z},$$ since all elements of order zero with invertible leading term are invertible in \power{R}{z}. Thus, we have to consider the situation: \begin{equation} \label{ideal in local} \poly{R}{y} \subseteq B \subseteq \power{R}{y} = \power{R}{z} . \end{equation} Let $I := B \cap \power{R}{z}\cdot z$. Obviously, this is an ideal of $B$. Let $P$ be the closed subscheme of $Spec(B)$ defined by $I$. One easily sees from (\ref{ideal in local}) that $B/I$ is naturally isomorphic to $R$. Therefore, $P$ is a section of the projection morphism $\pi$. Observe that the definition of $P$ does not depend on the choice of $a$ and $b$. For the ideal sheaf $\cal I$ of $P$, we get immediately: $${\cal I}/{\cal I}^{2} = [y] \cdot ({\cal O}_{C}/{\cal I}) \cong {\cal O}_{P}\cdot z,$$ i.e., ${\cal I}/{\cal I}^{2}$ is free of rank 1. Now let us prove that $C\setminus P= Spec({\cal A})$. We do this again on the affine open subset $U=Spec(R)$ of $S$. In a first step, we restrict our consideration to the open set $D_{+}(a)=Spec(B)$. $P$ is contained in this set, and $D_{+}(a)$ and $Spec({\cal A}(U))$ cover $\pi^{-1}(U)$. Assume, on the contrary, that there is a graded prime ideal ${\frak p} \subset grd({\cal A}(U))$ such that $1\in {\frak p}_{1}$ and $a\in {\frak p}_{2N+2}$. By the choice of $a$, this already implies that ${\frak p}_{n} = grd({\cal A}(U))_{n}$ for all sufficiently large $n$ (cf. Lemma \ref{m+n}). This is a contradiction. On $D_{+}(a)= Spec(B)$, $Spec({\cal A}(U))$ is given as $D(\frac{1}{a})$. Now the second step is the following: Let ${\frak p}$ be a prime ideal of $B$ containing $\frac{1}{a}$. We want to show that $\frak p$ belongs to $P$. This holds if and only if $\frak p$ contains the ideal $$\left\{ \frac{g}{a^{n}}/ n\in\Bbb{N}, ord(g)<ord(a)n \right\}.$$ Assume that $ord(g)<ord(a)n$ and set $m:=n\cdot ord(a)-ord(g)$. Then: $$\frac{g^{ord(a)}}{a^{n\cdot ord(a)}}=\frac{g^{ord(a)}}{a^{ord(g)+m}} = \frac{g^{ord(a)}}{a^{ord(g)}}\cdot\frac{1}{a^{m}}\in {\frak p}.$$ But $\frak p$ was assumed to be prime, hence $\frac{g}{a^{n}}\in {\frak p}$. So we see that $\frac{1}{a}\cdot B$ defines the same closed subset of $D_{+}(a)$ as $I$. As for the last step, we have to prove that $D(\frac{1}{a})$ does not intersect $P$. Take a prime ideal ${\frak p}\in P$. It is sufficient to show that $$B_{1/a}\cdot {\frak p} = B_{1/a}= {\cal A}_{a}.$$ But this is obvious. Our next aim is to show that the completion of ${\cal O}_{C}$ along $P$ is isomorphic to $\Opower{P}{z}$. To do this, we use the inclusions (\ref{ideal in local}). By \cite{Mat1}, Thm. 8.1., we only need to prove that the $(y)$-adic topology on $\power{R}{y}$ induces the $I$-adic topology on $B$, and this one induces the $(y)$-adic topology on $\poly{R}{y}$. Note that once we have shown the first fact, the second one follows immediately. We saw that $I$ and $\frac{1}{a}B$ define the same closed subset of $Spec(B)$. Therefore, both ideals define the same topology on $B$. Since, obviously, $$(\frac{1}{a}B)^{n} \subseteq \power{R}{y}\cdot y^{n(2N+2)}\cap B,$$ there only remains to show that for each $k\in \Bbb{N}$ there is an integer $N(k)$ such that $$\power{R}{y}\cdot y^{N(k)} \cap B \subseteq (\frac{1}{a}B)^{k}.$$ We claim that this is true for $N(k) = k\cdot (2N+2) = k\cdot (ord(a))$. {}From the definition of $B$ we get: $$\power{R}{y}\cdot y^{k\cdot (ord(a))} \cap B = \{ \frac{g}{a^{\alpha}} / g\in {\cal A}(U), ord(g) \leq ord(a)(\alpha -k) \}.$$ Let us consider such an element $\frac{g}{a^{\alpha}}\in \power{R}{y}\cdot y^{k\cdot (ord(a))} \cap B$. The inequality $ord(g) \leq ord(a)(\alpha -k)$ implies $ord(g\cdot a^{k}) \leq ord(a)\alpha $. Therefore, $\frac{g\cdot a^{k}}{a^{\alpha}}$ is an element of $B$, i.e., $\frac{g}{a^{\alpha}} \in (\frac{1}{a}B)^{k}$. So we have shown that the $I$-adic completion of $B$ is isomorphic to $\power{R}{y} = \power{R}{z}$. This isomorphism obviously does not depend on the choice of $a$ and $b$. Therefore, it extends to an isomorphism $$\rho: \widehat{\cal O}_{C} \stackrel{\sim}{\longrightarrow} \Opower{P}{z}.$$ {}Furthermore, it is now an easy consequence of \cite{Mat1}, Thm. 7.5., that the ideal $I$ is locally free of rank 1. In fact, along $P$, this ideal is generated by the element $y$. This implies that $P$ is a Cartier divisor. The relative ampleness of $P$ is an easy consequence of the fact that for each open affine subset $U$ of $S$, $\pi^{-1}(U)\setminus P = Spec({\cal A}(U))$ is affine. Finally, one easily sees that $\pi_{*}({\cal O}_{C}(nP))$ can be identified with ${\cal A}^{(n)}$. So, the fact that $\bigcap_{n\geq 0} \pi_{*}{\cal O}_{C}(-nP) = (0)$ is a consequence of Lemma \ref{A0}. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ Analogously, one can prove : \begin{theorem} $\widehat{\cal F}$ is a free $\widehat{\cal O}_{C}$-module, and the inclusion of $\cal W$ in \linebreak[4] $\Onegpower{S}{z}^{\oplus r}$ induces an isomorphism of sheaves of $ \widehat{\cal O}_{C}$ - modules: $$\Phi : \widehat{\cal F} \stackrel{\sim}{\longrightarrow} \widehat{\cal O}_{C}^{\oplus r}.$$ The intersection $\bigcap_{n\geq 0} \pi_{*}{\cal F}(-nP)$ vanishes. {}Furthermore, $F = \gamma(\pi_{*}{\cal F}) - \gamma(R^{1}\pi_{*}{\cal F})$. \end{theorem} The proofs of the first statements are pure analogies to Theorem \ref{Schur curve}. The very last assertion is a consequence of Theorem \ref{cohomology}. \mbox{\hspace*{\fill}$\Box$} \begin{definition} {}For a given Schur pair $({\cal A},{\cal W})$ of rank $r$ and index $F$, we define $$\eta_{r,F}({\cal A},{\cal W}) :=(C,\pi,S,P,\rho,{\cal F},\Phi)$$ with the objects described above. This defines a map $$\eta_{r,F}:{\frak S}^{r}_{F}(S) \longrightarrow {\frak D}^{r}_{F}(S).$$ \end{definition} Now we are ready to prove the converse of Theorem \ref{morph}. \begin{theorem} Homomorphisms of Schur pairs induce homomorphisms of the corresponding geometric data. \end{theorem} {\bf Proof}\hspace{0.3cm} Let $(\alpha,\xi):({\cal A}', {\cal W}')\rightarrow ({\cal A}, {\cal W})$ be a homomorphism of Schur pairs. We want to construct a homomorphism $$(\alpha,\beta,\Psi):(C,\pi,S,P,\rho,{\cal F},\Phi)\rightarrow (C',\pi',S',P',\rho',{\cal F}',\Phi')$$ of the associated geometric data. We proceed in two steps: First we assume that $({\cal A}, {\cal W})=(\alpha^{(*)}{\cal A}', \alpha^{(*)}{\cal W}')$. This gives us a morphism $$\beta: C=Proj(grd(\alpha^{(*)}{\cal A}'))\rightarrow C'$$ which makes the following diagram commute $$ \begin{array}{rcl} C & \stackrel{\beta}{\rightarrow} & C'\\ \pi \downarrow &&\downarrow \pi'\\ S & \stackrel{\alpha}{\rightarrow} & S'. \end{array} $$ Note that, in general, $C$ is different from the fibre product $C'\times_{S'} S$ (cf. Section \ref{base change}). Since $C\setminus P=Spec(\alpha^{(*)}{\cal A}')$ maps to $Spec({\cal A}') = C'\setminus P'$, we get $\beta^{-1}(P') = P$. Then, from the construction of $P$ in Theorem \ref{Schur curve}, it is also clear that $\beta^{*}P'=P$. Recall that the local trivializations $\rho$ and $\rho'$ have been defined by the inclusions $\alpha^{(*)}{\cal A}' \subseteq \Onegpower{S}{z}$ and ${\cal A}' \subseteq \Onegpower{S'}{z}$. So it is obvious that $\rho=\widehat{\beta}^{*}(\rho')$. Finally, we consider the sheaves $\cal F$ and ${\cal F}'$. As ${\cal F}= (grd(\alpha^{(*)}{\cal W}'))^{\sim}$ and ${\cal F}'= (grd({\cal W}'))^{\sim}$, there is a natural map $\beta^{*}{\cal F}'\rightarrow {\cal F}$ which is an isomorphism near $P$. \vspace{0.5cm} Now let us return to the general case. As $({\cal A}', {\cal W}')$ is a Schur pair, the induced object $(\alpha^{(*)}{\cal A}', \alpha^{(*)}{\cal W}')$ is also a Schur pair, i.e., the given homomorphism decomposes as follows: $$({\cal A}', {\cal W}') \stackrel{(\alpha,id)}{\longrightarrow} (\alpha^{(*)}{\cal A}', \alpha^{(*)}{\cal W}') \stackrel{(id_{S},\xi)}{\longrightarrow} ({\cal A}, {\cal W}).$$ So it just remains to consider the case where $S=S'$, $\alpha=id_{S}$, ${\cal A}'\subseteq {\cal A}$ and $\xi({\cal W}')\subseteq {\cal W}$. The inclusion ${\cal A}'\subseteq {\cal A}$ induces $grd({\cal A}')\hookrightarrow grd({\cal A})$ and, therefore, a morphism $$\beta: Proj(grd({\cal A})) = C \rightarrow C' = Proj(grd({\cal A}'))$$ which restricts to $$\beta : Spec({\cal A}) \rightarrow Spec({\cal A}')$$ and which, in addition, is an isomorphism near $P$. Therefore, $\beta$ fits into the diagram $$ \begin{array}{ccccc} P \subset & C & \stackrel{\beta}{\rightarrow} & C'&\supset P'\\ \sim\searrow&\pi \downarrow &&\downarrow \pi'&\swarrow\sim\\ &S&=& S \end{array} $$ and $\beta^{*}(P') = P$. The statement $\rho=\widehat{\beta}^{*}(\rho')$ is obvious. Now let us define the homomorphism of sheaves. We know that $\xi({\cal W}') \subseteq {\cal W}$. Since $({\cal A},{\cal W})$ is a Schur pair, this implies: $$\xi({\cal A}\cdot{\cal W}') ={\cal A}\cdot\xi({\cal W}') \subseteq {\cal W}.$$ $\xi$ is determined by the images of the basis elements $e_{1},\ldots,e_{r'}$. By the definition of $\xi$, either $ord(\xi(e_{j}))\leq 0$ or $\xi(e_{j})= 0$. Therefore, $\xi(({\cal A}\cdot{\cal W}')^{(n)})\subseteq {\cal W}^{(n)}$, for all $n\in \Bbb{Z}$, and $\xi$ induces a homomorphism $\xi:grd({\cal A}\cdot {\cal W}') \longrightarrow grd({\cal W})$, i.e., a homomorphism of sheaves $$\Psi : grd({\cal A}\cdot {\cal W}')^{\sim} = \beta^{*}{\cal F}' \longrightarrow {\cal F} = grd({\cal W})^{\sim}.$$ Obviously, for this homomorphism $\Psi$, $\xi$ is recovered by $\xi = \pi_{*}((\rho\circ\Phi)\circ \widehat{\Psi}\circ \widehat{\beta}^{*}(\rho'\circ \Phi')^{-1})$. \mbox{\hspace*{\fill}$\Box$} \begin{corollary} $\eta$ is a contravariant functor from the category of Schur pairs to the category of geometric data. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \begin{theorem} The Krichever functor $\chi$ and the functor $\eta$ are equivalences of the categories $\frak D$ and $\frak S$ and inverse to each other. Under this categorical equivalence, the subcategory ${\frak D}'$ corresponds to ${\frak S}'$. \end{theorem} {\bf Proof}\hspace{0.3cm} This is an easy consequence of Theorem II.5.14 \cite{H1}. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} \section{Applications} \label{APPL} Once the correspondence between geometric data and Schur pairs is established, we are, of course, interested in seeing how this relation works practically. For example, assume that the given geometric objects have additional properties. How do these properties display in the corresponding Schur pair? On the other hand, we had to impose some strong conditions on our family of curves and the sheaf on it (cf. Definitions \ref{total C} and \ref{sheaf F}). How substantial are these conditions? Are there still interesting and significant examples? \subsection{Translation of geometric properties} \label{geom. properties} Let $(C,\pi,S,P,\rho,{\cal F},\Phi)$ be a geometric datum of rank $r$ and index $F$, and $({\cal A},{\cal W})$ the associated Schur pair. A particular question is: What happens if $C$ or $\cal F$ is $S$-flat? \begin{lemma} \label{flat} The sheaf $\cal F$ is flat over $S$ if and only if ${\cal W} \subset \Onegpower{S}{z}^{\oplus r}$ is locally free. $\pi$ is a flat morphism if and only if ${\cal A} \subset \Onegpower{S}{z}$ is locally free. \end{lemma} {\bf Proof}\hspace{0.3cm} Flatness is a local property. So we may assume that $S=Spec(R)$, $R$ a noetherian ring. We know that ${\cal O}_{C}(P)$ is ample on $C$ relative to $S$. Let $N$ be so that ${\cal O}_{C}(N\cdot P)$ is very ample relative to $S$. We know from the proof of \cite{H1}, Thm. III.9.9., that $\cal F$ is $S$ - flat if and only if, for sufficiently large $n$, $\pi_{*}({\cal F} (nN\cdot P))$ is a locally free sheaf of ${\cal O}_{S}$ - modules of finite rank. Remember that $\Phi$ and $\rho$ induce an isomorphism of $\pi_{*} ({\cal F}(nN\cdot P))$ with ${\cal W} \cap \power{{\cal O}_{S}}{z}^{\oplus r}\cdot z^{-nN}$ (cf. Corollary \ref{order}). By assumption, ${\cal W} \cap \spower{{\cal O}_{S}}{z}{r}$ is coherent, hence is of finite rank. As $S$ is assumed to be noetherian, this implies that, for all $m$, ${\cal W} \cap \power{{\cal O}_{S}}{z}^{\oplus r}\cdot z^{-m}$ is also of finite rank. This implies that the $S$ - flatness of $\cal F$ is equivalent to the local freeness of ${\cal W} \cap \power{{\cal O}_{S}}{z}^{\oplus r} \cdot z^{-nN}$ for sufficiently large $n$. But on the other side, we also know that $\snegpower{{\cal O}_{S}}{z}{r}/ ({\cal W}+\spower{{\cal O}_{S}}{z}{r})$ is coherent. Set $W:=H^{0}({\cal W})$. Then $\negpower{R}{z}^{\oplus r}/(W+\power{R}{z}^{\oplus r})$ is a finitely generated $R$-module, hence, for sufficiently large $n$: $$W+\power{R}{z}^{\oplus r}\cdot z^{-nN} = \negpower{R}{z}^{\oplus r}.$$ Denote by $\{e_{1},\ldots,e_{r}\}$ the standard basis of the $\power{R}{z}$ - module $\power{R}{z}^{\oplus r}$. Then, for all $i=1,\ldots,r$ and $j > nN$, there are elements $$w_{i,j} = e_{i}\cdot z^{-j} + \textrm{ terms of lower order } \in W.$$ Let $\bar{W}$ be the free $R$-submodule of $W$ generated by these elements. Then, of course, $$W = \bar{W} \oplus (W\cap \power{R}{z}^{\oplus r}\cdot z^{-nN}),$$ and we see that $W$ is locally free if and only if this holds true for $W\cap \power{R}{z}^{\oplus r}\cdot z^{-nN}$. The proof of the second statement may be completed in the same way. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} At this time, let us outline a result which follows immediately from Lemma \ref{A0}. \begin{lemma} \label{downtwist} If $(C,\pi,S,P,\rho,{\cal F},\Phi)$ is a geometric datum then $\pi_{*}{\cal O}_{C}(-nP)$ and $\pi_{*}{\cal F}(-nP)$ vanish for sufficiently large $n\in\Bbb{N}$. \mbox{\hspace*{\fill}$\Box$} \end{lemma} Now we turn our attention to the stability of sheaves. \begin{definition} We call $\cal F$ {\em strongly semistable with respect to the section $P$} iff there is an integer $N$ such that $$\pi_{*}{\cal F}(N\cdot P) = R^{1}\pi_{*}{\cal F}(N\cdot P) = 0.$$ \end{definition} Later on we will see that this notion of semistability is the most convenient one for the examination of commutative algebras of differential operators corresponding to sheaves over relative curves. Translating the last definition in terms of Schur pairs, we get immediately: \begin{lemma}\label{stab.Schur} $\cal F$ is strongly semistable with respect to $P$ if and only if $${\cal W} \oplus \Opower{S}{z}^{\oplus r}\cdot z^{-N} = \Onegpower{S}{z}^ {\oplus r},$$ for some $N\in \Bbb{Z}$. \mbox{\hspace*{\fill}$\Box$} \end{lemma} \begin{corollary} If $\cal F$ is strongly semistable with respect to $P$ then, in particular, $\cal F$ is flat over $S$. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \vspace{0.5cm} \begin{definition} A coherent sheaf $\cal F$ on $C$ is called {\em simple} if $$ \pi_{*} {\cal E}nd_{{\cal O}_{C}}({\cal F}) = {\cal O}_{S}.$$ \end{definition} In the set-up of Schur pairs it is easier to handle isomorphisms than homomorphisms. That is why we are interested in the following statement. \begin{lemma} \label{simple} Let $S$ be reduced and assume that, for each point $s\in S$, the residue field $k(s)$ is infinite. Then $\cal F$ is simple if and only if $ \pi_{*} {\cal A}ut_{{\cal O}_{C}}({\cal F}) = {\cal O}_{S}^{*}$. \end{lemma} {\bf Proof}\hspace{0.3cm} One implication is obvious. To prove the other one, we can assume, without loss of generality, that $S = Spec(R)$ is affine. We want to prove $$End_{{\cal O}_{C}}({\cal F}) = R$$ under the assumption that $Aut_{{\cal O}_{C}}({\cal F}) = R^{*}$. Obviously, $R$ is contained in $End_{{\cal O}_{C}}({\cal F})$. Now assume that $End_{{\cal O}_{C}}({\cal F})$ contains an element $\phi$ which does not belong to $R$. For $r \in R$, we consider the endomorphism $r+\phi$ and restrict it to the fibres of $\pi$: $(r+\phi)_{s} := (r+\phi)|_{C_{s}}$. Since $S$ is reduced, $r+\phi$ is an isomorphism if and only if, for all $s\in S$, $(r+\phi)_{s}$ is an isomorphism. We define $$S(r) := \{s\in S/ (r+\phi)_{s} \textrm{ is an isomorphism }\}.$$ Obviously, these are open, possibly empty, subsets of $S$. Now we show that, for each $s\in S$, there is an element $r\in R$ such that $s \in S(r)$:\\ Let us fix $s\in S$. We write $${\cal G}_{r} := ker((r+\phi)_{s}) .$$ Then ${\cal G}_{r}$ is a subsheaf of ${\cal F}_{C_{s}}$. We show that the sheaves ${\cal G}_{r}$ and ${\cal G}_{r'}$ intersect only in the zero section whenever $r-r'$ is not contained in the prime ideal defining $s$ in $Spec(R)$. This can be seen locally on $C_{s}$. Let $V$ be an open affine subset of $C_{s}$, and $F := {\cal F}(V)$. Assume there is an element $f\in F$ such that $(r+\phi)_{s}(f) = (r'+\phi)_{s}(f)=0$. Then $(r-r')\cdot f = 0$, which implies that $f=0$. As $k(s)$ is assumed to be infinite, there are infinitely many elements $r\in R$ so that their pairwise differences are not contained in the ideal of $s$. Hence we obtain an infinite chain $${\cal G}_{1}\subseteq\ldots\subseteq {\cal G}_{1} \oplus {\cal G}_{2}\oplus \ldots \subseteq {\cal F}.$$ Since $C_{s}$ is a noetherian scheme and ${\cal F}_{C_{s}}$ is coherent, this chain must become stationary, i.e., there are infinitely many $r\in R$ such that $(r+\phi)_{s}$ is an injective homorphism between two sheaves with the same Hilbert polynomial, hence it must be an isomorphism, and this means $s \in S(r)$ for all those $r$. This proves that the sets $S(r)$ form a covering of $S$. Now, $(r+\phi)|S(r)$ is an isomorphism, hence corresponds to an element of $H^{0}(S(r),{\cal O}_{S_{r}})^{*}$. Therefore, $\phi|S(r) \in H^{0}(S(r),{\cal O}_{S_{r}})$, i.e., $\phi \in H^{0}(S,{\cal O}_{S})=R$. This is a contradiction. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now let us return to Schur pairs. The sheaf of groups $Isom_{\Opower{S}{z}}(\Opower{S}{z}^{\oplus r})$ acts on $\frak{G}^{r}_{F}(S)$ , for every $F\in K(S)$. Using Corollary \ref{ident. Schur} we draw the following two conclusions: \begin{corollary} Let $S$ be as in Lemma \ref{simple}. Then ${\cal W}$ corresponds to a simple sheaf if and only if $$Stab_{{\cal W}} Isom_{\Opower{S}{z}}(\Opower{S}{z}^{\oplus r}) = {\cal O}_{S}^{*}.$$ \mbox{\hspace*{\fill}$\Box$} \end{corollary} \begin{corollary} Again let $S$ be as above. Then a sheaf ${\cal F}$ belonging to a geometric datum $(C,\pi,S,P,\rho,{\cal F},\Phi)$ is simple if and only if for all equivalent geometric data $(C,\pi,S,P,\rho,{\cal F},\Phi')$: $$\Phi' = \lambda \Phi$$ for some $\lambda\in H^{0}(S,{\cal O}_{S})^{*}$. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \vspace{0.5cm} Now let us see to what determinant line bundles correspond. Assume that $S$ is noetherian, regular and separated. ${\cal W}$ is a quasicoherent subsheaf of $\Onegpower{S}{z}^ {\oplus r}$. We know that $\Onegpower{S}{z}^{\oplus r}/({\cal W} + \Opower{S}{z}^{\oplus r})$ is coherent. Together with the fact that the base scheme $S$ is noetherian, this implies that there is an integer $N$ satisfying \begin{equation} \label{determinant} \Onegpower{S}{z}^{\oplus r}={\cal W} + \Opower{S}{z}^{\oplus r}\cdot z^{-N}. \end{equation} Therefore, $$ \begin{array}{rcl} {\cal W}/{\cal W}^{(N)} & = & ({\cal W} + \Opower{S}{z}^{\oplus r}\cdot z^{-N})/ \Opower{S}{z}^{\oplus r}\cdot z^{-N} \\ & = & \Onegpower{S}{z}^{\oplus r}/\Opower{S}{z}^{\oplus r}\cdot z^{-N} \end{array} $$ which is a trivial sheaf of ${\cal O}_{S}$ - modules. Subsequently, $det({\cal W})=det({\cal W}^{(N)})$ is a well-defined line bundle on $S$. Note that this definition does not depend on the choice of the integer $N$ occuring in the condition (\ref{determinant}). Furthermore, additivity holds for exact sequences. On the other hand, for the given sheaf $\cal F$ of ${\cal O}_{C}$-modules we may consider the so-called {\em determinant of the cohomology} (after P.~Deligne) $$\lambda({\cal F}) := det(\pi_{*}{\cal F})\otimes (det (R^{1} \pi_{*}{\cal F}))^{-1}.$$ We can prove the following result. \begin{proposition} $\lambda({\cal F})\cong det({\cal W})$. \end{proposition} {\bf Proof}\hspace{0.3cm} This is an easy consequence of Proposition \ref{cohomology}. Using the fact proven there, we get: $$\lambda({\cal F}) \cong det ({\cal W}^{(0)}) \otimes det (\Onegpower{S}{z}^{\oplus r}/({\cal W} + \Opower{S}{z}^{\oplus r}))^{-1}.$$ Now we consider the exact sequences of quasicoherent sheaves on $S$: $$0\rightarrow {\cal W}/{\cal W}^{(0)}\rightarrow \Onegpower{S}{z}^{\oplus r}/\Opower{S}{z}^{\oplus r} \rightarrow \Onegpower{S}{z}^{\oplus r}/({\cal W} + \Opower{S}{z}^{\oplus r}) \rightarrow 0$$ and $$0\rightarrow {\cal W}^{(0)} \rightarrow {\cal W} \rightarrow {\cal W}/{\cal W}^{(0)} \rightarrow 0.$$ The statement of the lemma follows then from the additivity of $det$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} In the case that $\cal F$ is flat and $S$ is separable, $det({\cal W})$ is again well-defined, even if $S$ is not regular. Consequently this determinant generalizes the determinant of the cohomology. \vspace{0.5cm} \subsection{Examples of geometric data} \label{Eogd} First we prove a criterion which will be highly useful for the construction of examples. \begin{proposition} \label{lifting1} Assume that the base scheme $S$ satisfies: $H^{1}(S,{\cal O}_{S}) = 0$, and that, for the section $P$, ${\cal I}_{P}/{\cal I}_{P}^{2}$ is free of rank 1. Then \begin{enumerate} \item $\widehat{\cal O}_{C} \cong \Opower{P}{z}$. \item If $\cal F$ is a coherent sheaf of ${\cal O}_{C}$ - modules such that \begin{itemize} \item $\cal F$ is locally free in a neighborhood of $P$, \item ${\cal F}|P \cong {\cal O}_{P}^{\oplus r}$ \end{itemize} then $\widehat{\cal F} \cong \widehat{\cal O}_{C}^{\oplus r}$. \end{enumerate} \end{proposition} {\bf Remark}\hspace{0.3cm} For example, the cohomological condition is fulfilled for all affine schemes $S$. \vspace{0.5cm}\\ {\bf Proof of the proposition}\hspace{0.3cm} At first, observe that the condition that \linebreak[4] $H^{1}(S,{\cal O}_{S})$ vanishes is, of course, equivalent to $H^{1}(P,{\cal O}_{P}) = 0$. By Lemma \ref{powerseries} and the remark following it the first claim is equivalent to: $$H^{0}(C,{\cal I}/{\cal I}^{n}) \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{0}(C,{\cal I}/{\cal I}^{2}) \quad \forall n \in \Bbb{N}, n\geq 2.$$ One easily sees that this is the case if and only if for all $n\geq 2$: $$H^{0}(C,{\cal I}/{\cal I}^{n+1}) \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{0}(C,{\cal I}/{\cal I}^{n}).$$ We have the exact sequence of sheaves of ${\cal O}_{P}$-modules \begin{equation} \label{equ.1} 0 \rightarrow {\cal I}^{n}/{\cal I}^{n+1} \rightarrow {\cal I}/{\cal I}^{n} \rightarrow {\cal I}/{\cal I}^{n+1}\rightarrow 0 \end{equation} which induces a long exact sequence of cohomology groups $$0 \rightarrow H^{0}({\cal I}^{n}/{\cal I}^{n+1}) \rightarrow H^{0}({\cal I}/{\cal I}^{n}) \rightarrow H^{0}({\cal I}/{\cal I}^{n+1}) \rightarrow H^{1}({\cal I}^{n}/{\cal I}^{n+1}) \rightarrow \ldots$$ By assumption, ${\cal I}^{n}/{\cal I}^{n+1} = ({\cal I}/{\cal I}^{2})^{n} \cong {\cal O}_{P}$. So our assumption on $S$ implies that $H^{1}({\cal I}^{n}/{\cal I}^{n+1}) = 0$ for all $n\in \Bbb{N}$ and we are done. Now we come to the second part. By Lemma \ref{COMP.F} the claim is equivalent to the following fact: \begin{equation} \label{equ.0} H^{0}({\cal F}/({\cal I}^{n}\otimes {\cal F})) \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } H^{0}({\cal F}/({\cal I}\otimes {\cal F})) \quad, \forall n\in \Bbb{N}. \end{equation} We consider one more exact sequence of coherent sheaves of ${\cal O}_{C}$ - modules: \begin{equation} \label{equ.2} 0 \rightarrow {\cal I}^{n} \rightarrow {\cal O}_{C} \rightarrow {\cal O}_{C}/{\cal I}^{n} \rightarrow 0. \end{equation} Since $\cal F$ is locally free in some neighborhood of $P$, and ${\cal O}_{C}/{\cal I}^{n} = 0$ outside $P$, the sequences (\ref{equ.1}) and (\ref{equ.2}) stay exact when we tensor with $\cal F$. So we get \begin{equation} \label{equ.3} 0 \rightarrow ({\cal I}^{n-1}/{\cal I}^{n})\otimes {\cal F} \rightarrow ({\cal O}_{C}/{\cal I}^{n})\otimes {\cal F} \rightarrow ({\cal O}_{C}/{\cal I}^{n-1})\otimes {\cal F} \rightarrow 0 \end{equation} and (\ref{equ.2}) implies: $({\cal O}_{C}/{\cal I}^{n})\otimes {\cal F} \cong {\cal F}/({\cal I}^{n}\otimes {\cal F})$. Now we write down the long exact sequence of cohomology groups induced by (\ref{equ.3}): \begin{equation} \label{equ.4} \begin{array}{cccccc} 0 & \rightarrow & H^{0}(({\cal I}^{n-1}/{\cal I}^{n})\otimes {\cal F}) & \rightarrow & H^{0}({\cal F}/({\cal I}^{n}\otimes {\cal F})) & \rightarrow \\ &\rightarrow & H^{0}({\cal F}/({\cal I}^{n-1}\otimes {\cal F})) & \rightarrow & H^{1}(({\cal I}^{n-1}/{\cal I}^{n})\otimes {\cal F}). \end{array} \end{equation} Since the restriction of $\cal F$ to $P$ is free, and ${\cal I}^{n-1}/{\cal I}^{n}$ is isomorphic to ${\cal O}_{P}$, $({\cal I}^{n-1}/{\cal I}^{n})\otimes_{{\cal O}_{C}} {\cal F}$ is a free ${\cal O}_{P}$-module. Therefore we finally get: $$H^{1}(({\cal I}^{n-1}/{\cal I}^{n})\otimes {\cal F}) \cong H^{1}(P,{\cal O}_{P})^{\oplus r} = 0,$$ which, together with the sequence (\ref{equ.4}), implies the surjectivity in (\ref{equ.0}). \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now we come to explicit examples. \subsubsection{Trivial families of curves} The easiest case, but which is not without interest, is the one of a trivial family of curves with some sheaf on it. \begin{proposition} \label{CxS} Let $K$ be a complete, integral curve over some field $k$ and $p\in K$ a smooth, $k$-rational point. Let $S$ be a locally noetherian $k$-scheme and set $C:= K\times_{Spec(k)} S$. Denote by $\pi$ the projection from $C$ to $S$, and by $P$ the section $\{ (p,s)/s\in S\} $. Then $\widehat{\cal O}_{C} \cong \Opower{P}{z}$. \end{proposition} {\bf Proof}\hspace{0.3cm} Let ${\cal J} = {\cal J}_{p}$ be the sheaf of ideals defining $p$ in $K$. Since $p$ is a smooth point, $\cal J$ is generated by one element $z$ near $p$, and we get: \begin{itemize} \item ${\cal J}/{\cal J}^{2} = [z]\cdot ({\cal O}_{K}/{\cal J})$; \item $\widehat{\cal O}_{K} \cong \power{k}{z}$. \end{itemize} Since $C=K\times_{Spec(k)} S$ and $P=\{ p \} \times_{Spec(k)} S$, we obtain: \begin{itemize} \item ${\cal I}_{P} = {\cal J}\otimes_{k} {\cal O}_{S}$, hence ${\cal I}_{P}/{\cal I}_{P}^{2} = [z]\cdot({\cal O}_{C}/{\cal I}_{P})=[z]\cdot{\cal O}_{P}$. \item $\widehat{\cal O}_{C} = \widehat{\cal O}_{K} \otimes_{k} {\cal O}_{S}$, i.e., $\widehat{\cal O}_{C} \cong \Opower{P}{z}$. \end{itemize} \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} \subsubsection{Elliptic curves} \label{Elliptic curves} To describe a nontrivial family of integral curves which fits into our set-up, we define a family of elliptic curves over $\Bbb{A}^{2}=\Bbb{A}^{2}_{\Bbb{C}}$ as follows: \begin{equation} \begin{array}{ccc} C\textrm{ }& := & \{(A,B,z_{0}:z_{1}:z_{2}) \in \Bbb{A}^{2}\times \Bbb{P}^{2} / z_{0}z_{2}^{2}= z_{1}^{3} + A z_{0}^{2}z_{1} + B z_{0}^{3} \}\\ \downarrow \pi & {} &{}\\ \Bbb{A}^{2}\textrm{ }&{}&{} \end{array} \end{equation} A section of $\pi$ can be defined by $P:=\{(A,B,0:0:1)/(A,B)\in\Bbb{A}^{2}\}$. One easily sees that $\pi$ is a flat, projective morphism with reduced, irreducible fibres of dimension 1 and that $C$ is reduced. We want to study the conormal sheaf of $P$. Since $P$ does not intersect the hyperplane $\Bbb{A}^{2}\times(z_{2}=0)$, we can restrict our consideration to its affine complement. Let us denote $y_{i} := z_{i}/z_{2}$ for $i=0,1$. Then $C\cap(z_{2}\neq 0)\subset \Bbb{A}^{2}\times \Bbb{A}^{2}$ is given by the equation \begin{equation} \label{elliptic} y_{0} = y_{1}^{3} + A y_{0}^{2}y_{1} + B y_{0}^{3}. \end{equation} Let $R$ be the affine coordinate ring of $C\cap(z_{2}\neq 0)$. The ideal $I$ of $P$ in $R$ is generated by $y_{0}$ and $y_{1}$. But $y_{0} \in I^{2}$, i.e. $I/I^{2} = y_{1}R/I$. This implies that ${\cal I}/{\cal I}^{2}$ is a trivial line bundle. \vspace{0.5cm}\\ Note that $\widehat{\cal O}_{C} \cong \Opower{P}{z}$ since $H^{1}(\Bbb{A}^{2}) =0$. Now we want to find a suitable (formal) local parameter on $C$ along $P$. From the above calculation we know that $y_{1}=z_{1}/z_{2}$ is such a local parameter, i.e., the formal completion $\widehat{R}$ of $R$ with respect to $I$ equals $k\/[\/A,B\/]\/[[\/y_{1}\/]]$. We claim that for $\alpha := \sqrt{z_{0}/z_{1}}$, $\widehat{R} \cong k\/[\/A,B\/]\/[[\/\alpha\/]]$. We use the equation (\ref{elliptic}) and calculate: $$ \begin{array}{rcl} y_{0} & = & y_{1}^{3} + A y_{0}^{2}y_{1} + B y_{0}^{3}\\ y_{0}/y_{1} & = & y_{1}^{2} + A y_{0}^{2} + B y_{0}^{3}/y_{1}\\ -y_{1}^{2} - A y_{0}^{2} & = & (B y_{0}^{2} -1)y_{0}/y_{1}. \end{array} $$ $(B y_{0}^{2} -1)$ is an invertible element of $k\/[\/A,B\/]\/[[\/y_{1}\/]]$. Therefore $$z_{0}/z_{1}= y_{0}/y_{1} = (B y_{0}^{2} -1)^{-1}(-y_{1}^{2} - A y_{0}^{2}) \in k\/[\/A,B\/]\/[[\/y_{1}\/]]$$ is an element of order $-2$ with leading coefficient 1. So, the square root of $z_{0}/z_{1}$ is a well-defined monic element $\alpha$ of $\widehat{R}$ of order $-1$. This implies that $\widehat{R} \cong k\/[\/A,B\/]\/[[\/\alpha\/]]$. Now we construct the corresponding subring $\cal A$ of $k\/[\/A,B\/]\/[[\/\alpha\/]]$. We see that $P$ is exactly the intersection of $C$ with the hyperplane $(z_{0}=0)$. Therefore, the affine ring of coordinates of $C\setminus P$ is $$k\/[\/A,B,z_{1}/z_{0}, z_{2}/z_{0}\/]/((z_{2}/z_{0})^{2}- (z_{1}/z_{0})^{3} -A (z_{1}/z_{0})^{2} - B).$$ We express the generating elements of this $k\/[\/A,B\/]$ - algebra in terms of the above chosen formal parameter $\alpha$: $$ \begin{array}{rcl} z_{1}/z_{0} & = & \alpha^{-2}\\ z_{2}/z_{0} & = & \alpha^{-2}\cdot y_{1}^{-1}. \end{array} $$ $y_{1}^{-1}$ is an element of $k\/[\/A,B\/]\/[[\/\alpha\/]]$. We want to find out its special form. Using the equation (\ref{elliptic}) we get $$ \begin{array}{rcl} 1/y_{1}^{2} & = & y_{1}/y_{0} + A (y_{0}/y_{1})+ B (y_{0}/y_{1})^{2}\\ {} & = & \alpha^{-2} + A \alpha^{2}+ B\alpha^{4}. \end{array}$$ So, finally: $${\cal A } = k\/[\/A,B,\alpha^{-2}, \alpha^{-2}\cdot\sqrt{\alpha^{-2} + A \alpha^{2}+ B\alpha^{4}}\/].$$ The term $\alpha^{-2}\in {\cal A}$ reflects the 2:1 covering $$C\rightarrow \Bbb{A}^{2}\times \Bbb{P}^{1}.$$ \subsubsection{Families of line bundles over a curve} \label{Poincare} Let $C$ be a complete integral complex curve and $p\in C$ a point. We choose a formal local trivialization $\rho$ of $C$ near $p$ and construct the corresponding subring $A\subset \negpower{\Bbb{C}}{z}$. We take the Picard variety $Pic^{n}(C)$ of $C$, for some $n$, as a base scheme. As described in Proposition \ref{CxS}, $\rho$ extends to a local trivialization of $C\times Pic^{n}(C)$ near $\{p\}\times Pic^{n}(C)$ and we get for the corresponding sheaf of ${\cal O}_{Pic^{n}(C)}$- algebras: $${\cal A} = A\otimes_{\Bbb{C}}{\cal O}_{Pic^{n}(C)}\subseteq \Onegpower{Pic^{n}(C)}{z}.$$ Now we consider the Poincar\'{e} bundle ${\cal P}^{n}_{C}$ of degree $n$ on $C$ (normalized with respect to the fixed point $p$). ${\cal P}^{n}_{C}$ is a line bundle on $C\times Pic^{n}(C)$ satisfying: \begin{itemize} \item ${\cal P}^{n}_{C}|C\times \{L\} \cong L$ for every $L\in Pic^{n}(C)$, \item ${\cal P}^{n}_{C}|\{p\}\times Pic^{n}(C)$ is trivial, \item ${\cal P}^{n}_{C}$ is flat over $Pic^{n}(C)$. \end{itemize} For more details see \cite{LB}. Let $U\subset Pic^{n}(C)$ be an open affine subset. We apply Proposition \ref{lifting1} and conclude that ${\cal P}^{n}_{C}|(C\times U)$ satisfies Condition 1 of Definition \ref{sheaf F}. So we can construct (for some local trivialization) the corresponding sheaf ${\cal W}(U)\subseteq \Onegpower{U}{z}$ and we obtain a Schur pair $({\cal A}(U), {\cal W}(U))$. (Keep in mind that $\cal A$ is globally defined whereas $\cal W$ is not!) ${\cal W}(U)$ defines a map from $U$ to ${\frak G}^{1}_{\mu}(Spec(\Bbb{C}))$, $\mu = n+1-g(C)$, and as the image we get $U$ as a subset of the Grassmannian. M.~Mulase proved in \cite{M1} that every finite dimensional integral manifold of the KP-flows on some quotient of the Grassmannian ${\frak G}^{1}_{\mu}(Spec(\Bbb{C}))$ has the linear structure of a Jacobian of a curve. The quotient has been taken in order to eliminate different local trivializations. In particular, the differences arising from the local construction of $\cal W$ cancel out completely. So, the above result implies that the integral manifold also carries the algebraic-geometric structure of the Jacobian. \subsubsection{Base change} \label{base change} Here, we want to investigate the behaviour of geometric data under base changes. In general, the pull-back of a geometric datum over $S$ under a base change $\alpha:S'\rightarrow S$ does not give rise to another geometric datum. However, let us start with a positive example. \begin{lemma} Let $(C,\pi,S,P,\rho,{\cal F},\Phi)$ be a geometric datum of rank $r$ over $S$ and $\alpha:S'\rightarrow S$ a flat morphism. The fibre product construction then defines a collection $(C',\pi',S',P',\rho',{\cal F}',\Phi')$. We claim that this collection forms a geometric datum of rank $r$ over $S'$ and that $(\alpha,\alpha',id)$ is a homomorphism of geometric data $$(\alpha,\alpha',id): (C',\pi',S',P',\rho',{\cal F}',\Phi') \rightarrow (C,\pi,S,P,\rho,{\cal F},\Phi),$$ where $\alpha'$ is defined as the fibre product morphism, $$ \begin{array}{rcl} {}C' & \stackrel{\alpha'}{\rightarrow} & C\\ \pi'\downarrow && \downarrow \pi\\ {}S' & \stackrel{\alpha}{\rightarrow}& S. \end{array} $$ \end{lemma} {\bf Proof}\hspace{0.3cm} Let us check the properties listed in Definition \ref{def data}. Some of them are easy to see. Of course, $C'$ is a scheme, $\pi':C'\rightarrow S'$ is a locally projective morphism, $P'$ is a relatively ample Cartier divisor and ${\cal I}'/{\cal I}'^{2} = \alpha'^{*}({\cal I}/{\cal I}^{2})$ is free of rank 1 on $P'$. In Lemma \ref{powerseries} we saw that the condition $\widehat{\cal O}_{C} \cong \Opower{P}{z}$ is equivalent to the fact that the section $1\in H^{0}(C, {\cal I}/{\cal I}^{2})= H^{0}(P,{\cal O}_{P})$ lifts to a section of ${\cal I}/{\cal I}^{n}$ for all $n\geq 2$. Working through the diagram $$ \begin{array}{ccccccc} H^{0}({\cal I}/{\cal I}^{n}) & \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } & H^{0}({\cal I}/{\cal I}^{2}) & = & H^{0}({\cal O}_{P}) & \ni & 1\\ \alpha^{*} \downarrow&& \alpha^{*}\downarrow\\ H^{0}({\cal I}'/{\cal I}'^{n}) & \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } & H^{0}({\cal I}'/{\cal I}'^{2}) & = & H^{0}({\cal O}_{P'}) & \ni & 1 \end{array} $$ we obtain: $\widehat{\cal O}_{C'} \cong \Opower{P'}{z}$. Of course, $\rho$ and $\alpha$ induce an isomorphism $$\rho': \widehat{\cal O}_{C'} \stackrel{\sim}{\rightarrow} \Opower{P'}{z}.$$ Now we turn our attention to the sheaf ${\cal F}'$. It is easy to see that $${\cal F}'/{\cal I}'\otimes {\cal F}' \cong \alpha'^{*}({\cal F}/{\cal I}\otimes {\cal F}) \cong {\cal O}_{P'}^{\oplus r}$$ and that ${\cal F}'$ is locally free near $P'$. In order to prove that the completion of ${\cal F}'$ along $P'$ is free, we have to show that the generating sections $e_{1},\ldots, e_{r}$ of $H^{0}({\cal O}_{P'}^{\oplus r}) = H^{0}({\cal F}'/{\cal I}'\otimes {\cal F}')$ lift to sections of ${\cal F}'/{\cal I}'^{n}\otimes {\cal F}'$ for all $n\in\Bbb{N}$. This is done as above. Again, $\Phi$ and $\alpha$ induce an isomorphism of $\widehat{\cal O}_{C'}$-modules $$\Phi' : \widehat{{\cal F}'} \stackrel{\sim}{\rightarrow} \widehat{\cal O}_{C'}^{\oplus r}.$$ Finally, note that for sufficiently large $n\in\Bbb{N}$, $$\pi_{*}{\cal O}_{C}(-nP) =0 \textrm{ and } \pi_{*}{\cal F}(-nP) =0$$ (cf. Lemma \ref{downtwist}). Consequently, using \cite{H1}, Thm.II.9.3., $$\pi'_{*}{\cal O}_{C'}(-nP') = \alpha^{*}\pi_{*}{\cal O}_{C}(-nP)=0 \textrm{ and } \pi'_{*}{\cal F}'(-nP') =\alpha^{*}\pi_{*}{\cal F}(-nP)=0$$ for sufficiently large $n$. This completes the list of the properties which we had to check and $(C',\pi',S',P',\rho',{\cal F}',\Phi')$ is really a geometric datum. The fact that $(\alpha,\alpha', id)$ is a morphism of geometric data is straightforward. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} As we mentioned at the beginning, the pull-back of a geometric datum over $S$ via a morphism $\alpha:S'\rightarrow S$ does not always define a new geometric datum. A typical situation for this is the restriction to one point of the base scheme $S$. Let us give an example in terms of Schur pairs. As a base scheme we choose $S=\Bbb{A}^{1}_{k} = Spec (\poly{k}{t})$, and our Schur pair is given by $(A,A)$ for $$A = \poly{k}{t} \oplus t\poly{k}{t}z^{-1} \oplus \poly{\poly{k}{t}}{z^{-1}}\cdot z^{-2}.$$ Now we consider the fibre of the corresponding projective curve $C$ over the point $0\in S$. Its affine part outside the section $P$ is given by the ring $$A_{0} = k \oplus (t\poly{k}{t}/t^{2}\poly{k}{t})z^{-1} \oplus \poly{k}{z^{-1}}\cdot z^{-2}.$$ This is a cuspidal curve with an embedded point. In particular, the fibre is not reduced and therefore cannot be a part of a geometric datum over $k$. However, $(A,A)$ in fact induces a Schur pair over $S'=Spec(k)$, namely (A',A') with $$A' = Im(A\subset \negpower{\poly{k}{t}}{z} \mbox{ $\rightarrow\!\!\!\!\!\!\!\rightarrow$ } \negpower{(\poly{k}{t}/t\cdot\poly{k}{t})}{z}) = k\oplus \poly{k}{z^{-1}} z^{-2}.$$ This ring corresponds to the cuspidal curve, i.e., to the integral component of the fibre passing through the section $P$. \newpage \section{Families of commutative algebras of differential operators} \label{Fam. DO} In \cite{M1} M.~Mulase used the equivalence of the category of Schur pairs and the category of quintets for a complete classification of commutative algebras of ordinary differential operators with coefficients in $\power{k}{x}$. This leads us to the natural question of whether it is possible to extend these results to the relative case, at least in some special situations. It is hard to do this in the set-up of sheaves. Therefore we will restrict our observations to the case of an affine base scheme $S=Spec(R)$, where $R$ is a commutative noetherian $k$-algebra for some field $k$ of characteristic zero. Before beginning let us fix a convention: Whenever in this chapter we speak about Schur pairs we mean elements of ${\frak S}'$ and all geometric data occuring here belong to ${\frak D}'$. Let us start analyzing our objects in this special case. \subsection{Schur pairs over affine base schemes} \label{affine base} \begin{definition} Let $A$ be an $R$-subalgebra of $\negpower{R}{y}$, and $r\in\Bbb{N}$. $A$ is said to be an {\em algebra of pure rank $r$}, if \begin{enumerate} \item $r=gcd(ord(a)/a\in A)$ and \item There are monic elements $a$ and $b$ of positive order in $A$ such that $gcd(ord(a),ord(b))=r$. \end{enumerate} \end{definition} Let us see what these properties imply: \begin{lemma} \label{pure} Let $A\subseteq \negpower{R}{y}$ be an $R$-subalgebra of pure rank $r$. Then there is a monic element $z\in \power{R}{y}$ of order $-r$ such that \begin{itemize} \item $A\subseteq \negpower{R}{z}$, \item $\negpower{R}{z}/(A+\power{R}{z})$ is a finitely generated $R$-module. \end{itemize} \end{lemma} {\bf Proof}\hspace{0.3cm} Choose monic elements $a$ and $b$ of $A$ of positive order such that $gcd(ord(a),ord(b))=r$. Then there are natural numbers $i$ and $j$ such that $$r=i(ord(a))-j(ord(b)).$$ Define $z:= a^{-i}b^{j}$. Since the inverse of a monic element of $\negpower{R}{y}$ is again a well-defined element of $\negpower{R}{y}$, we have constructed a monic element \linebreak[4] $z\in \power{R}{y}$ of order $-r$. Now let us prove that the localization of $A$ by $a$ is contained in $\negpower{R}{z}$: Of course, $z\in A_{a}$. We choose an element $v$ of $A_{a}$ and denote its order by $\alpha$. $v$ has the form $\frac{w}{a^{m}}$ for some elements $w\in A$ and $m\in\Bbb{N}$. $r$ divides the orders of $w$ and $a$, therefore $r$ divides $\alpha$. Since $z$ is monic of order $-r$, there is some $n\in\Bbb{Z}$ and $v_{0}\in R$, such that $v-v_{0}\cdot z^{n} \in A_{a}$ is an element of order less than $\alpha$. Now the assertion is proved inductively. In particular, this shows that $A$ itself is contained in $\negpower{R}{z}$. For the second part see the proof of Lemma \ref{a and b}. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} The converse of this lemma is true, as well:\\ Let $A\subset \negpower{R}{y}$ be an $R$-subalgebra satisfying \begin{itemize} \item $A\subset \negpower{R}{z}$ for some monic element $z$ of order $-r$; \item $\negpower{R}{z}/(A+\power{R}{z})$ is a finitely generated $R$-module \end{itemize} then $A$ is an $R$-algebra of pure rank $r$. \vspace{0.5cm}\\ Now let us define \begin{definition} As an {\em embedded Schur pair of rank $r$, index $F$ and level $\alpha$ over $Spec(R)$} we denote a pair $(A,W)$ consisting of \begin{itemize} \item $A\subseteq \negpower{R}{y}$ an $R$-subalgebra of pure rank $r$ satisfying $A\cap\power{R}{y}=R$ ; \item $W\subseteq \negpower{R}{y}$ with $W\in \frak{G}^{1}_{F,\alpha}(Spec(R))$ \end{itemize} such that $A\cdot W \subseteq W$. We write $\frak{E}_{\alpha}\frak{S}'^{r}_{F}(R)$ for the set of embedded Schur pairs of rank $r$, index $F$ and level $\alpha$ over $Spec(R)$. \end{definition} This is a natural generalization of the notion of Schur pairs introduced by M.~Mulase \cite{M1}. Now we want to see how these objects are related to the Schur pairs on $Spec(R)$ we have defined earlier. \begin{proposition} For all $\alpha\in\Bbb{Z}$ there is a canonical one-to-one correspondence between Schur pairs of rank $r$ and index $F$ and embedded Schur pairs $(A,W)$ of rank $r$, index $F$ and level $\alpha$ with the extra-condition that $$A\subset \negpower{R}{y^{r}}.$$ \end{proposition} {\bf Proof}\hspace{0.3cm} The method of the proof has been outlined already in the remark on page \pageref{Grass Mulase}. Let us start with an embedded Schur pair $(A,W)$ of rank $r$, index $F$ and level $\alpha$ such that $A\subset \negpower{R}{y^{r}}$. Set $z:=y^{r}$. Then, by Lemma \ref{pure}, $A\subset\negpower{R}{z}$ is an element of ${\frak G}^{1}_{G}(Spec(R))$ for some $G\in K(Spec(R))$. Now we identify: $$ \begin{array}{rcl} \power{R}{y}\cdot y^{-\alpha} & = & \bigoplus_{i=-\alpha}^{-\alpha+r-1} \power{R}{y^{r}}\cdot y^{i}\\ &=& \power{R}{z}^{\oplus r}. \end{array} $$ This identification extends to an isomorphism of $\negpower{R}{y}$ with $\negpower{R}{z}^{\oplus r}$ and so we end up with $W\subset \negpower{R}{z}^{\oplus r}$. Since $\power{R}{y}\cdot y^{-\alpha}$ translates into $\power{R}{z}^{\oplus r}$, $W$ gives an element of ${\frak G}^{r}_{F}(Spec(R))$. That also clarifies the inverse construction. We formally set $z:=y^{r}$ and translate the data back using Lemma \ref{pure} and its converse. \mbox{\hspace*{\fill}$\Box$}\vspace{0.5cm} \subsection{Formal pseudo-differential operators} \label{pseudo DO} We saw that embedded Schur pairs are closely related to Schur pairs, while Schur pairs themselves correspond to geometric data via the Krichever functor. Now a natural question is how to identify embedded Schur pairs which lead to ``similar'' geometric data, where similar means that they differ only by a very special change of the local trivializations. This is done with the help of formal pseudo-differential operators. Furthermore, formal pseudo-differential operators will be the main tool for the classification of commutative algebras of differential operators.\vspace{0.5cm} Consider the ring $\power{R}{x}$ of formal power series in one variable with coefficients in $R$, and write $\partial :=\frac{d}{dx}$. $\partial$ acts on $\power{R}{x}$ by derivation: $$\partial(\sum_{i\geq 0}a_{i} x^{i}) = \sum_{i\geq 1} ia_{i}x^{i-1},$$ while, for any $n\in\Bbb{N}$, $\partial^{n}$ acts by repeated derivation: $$\partial^{n} (f) := \partial(\partial^{n-1} f)$$ for $f\in\power{R}{x}$. $\partial^{0}$ is defined to be the identity. For given elements $f,g\in\power{R}{x}$ and $n\in\Bbb{N}$ we define: $$\begin{array}{rcl} (f\partial^{n})(g) &=&f\partial^{n}(g),\\ (\partial^{n}f)(g)&=&\partial^{n}(fg). \end{array}$$ In this way, the ring of ordinary differential operators with coefficients in $\power{R}{x}$, $D := \poly{\power{R}{x}}{\partial}$, turns out to be a subring of the endomorphism ring $End_{R}(\power{R}{x})$. The multiplication of elements of $D$ is determined by the {\em Leibniz rule}: \vspace{0.5cm}\\ For $f,g \in \power{R}{x}$ and $n\in \Bbb{N}$: \begin{equation} \label{Leib} \partial^{n}(fg) = \sum_{i=0}^{\infty} {n \choose i} f^{(i)} \partial^{n-i}(g). \end{equation} It is our aim to make the operator $\partial$ invertible. In fact, we want to introduce $\partial^{-1}$ with $\partial\partial^{-1} = \partial^{-1}\partial = 1$ and define a multiplication on the set $$E:=\{\sum_{n\in \Bbb{Z}} f_{n}\partial^{n}/ f_{n}\in \power{R}{x}, f_{n} = 0 \textrm{ for } n \gg 0\}\supset D,$$ which is compatible with the multiplication on $D$. One can define $\partial^{-1}$ as an endomorphism on $\power{R}{x}$ by formal integration: $$\partial^{-1}(\sum_{i\geq 0}a_{i} x^{i}) = \sum_{i\geq 0} \frac{a_{i}}{i+1}x^{i+1}.$$ But obviously, the so-defined $\partial^{-1}$ is not inverse to $\partial$ as an endomorphism of $\power{R}{x}$. That is why we consider the action of $\partial$ and $\partial^{-1}$ on the quotient $R$-module $\power{R}{x}/\poly{R}{x}$. One easily sees that one may interpret $\poly{\poly{R}{x}}{\partial, \partial^{-1}}$ as a subring of $End_{R}(\power{R}{x}/\poly{R}{x})$ and that for this embedding $\partial^{-1}$ is the inverse of $\partial$. Observe that the Leibniz rule (\ref{Leib}) also holds true for negative $n$, i.e., for formal integration, where, for arbitrary $n\in\Bbb{Z}$ and $i\in\Bbb{N}$, the binomial coefficient ${n \choose i}$ is defined as follows: $${n \choose i} := \frac{n\cdot(n-1)\cdot\ldots\cdot(n-i+1)}{i\cdot (i-1)\cdot\ldots\cdot 1} \in \Bbb{N}.$$ For negative $n\in\Bbb{Z}$, the summation in (\ref{Leib}) is really an infinite one, while, for nonnegative $n$, it is finite. The formula (\ref{Leib}) defines a multiplication rule for elements of $E$ of the form $$\sum_{n=-M}^{N} (\sum_{i=0}^{\alpha_{n}} f_{i,n} x^{i})\partial^{n}.$$ But now it is clear that the so-defined multiplication extends to a multiplication on all of $E$, which restricts to the usual one on $D$ and has the form: $$ \begin{array}{l} (\sum_{m=0}^{\infty}a_{m}\partial^{M-m})\cdot (\sum_{n=0}^{\infty}b_{n}\partial^{N-n})=\\ \hspace{3cm}=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{i=0}^{\infty} {M-m \choose i} a_{m}b_{n}^{(i)}\partial^{M+N-m-n-i}\\ \hspace{3cm}=\sum_{l=0}^{\infty}(\sum_{m=0}^{l}\sum_{i=0}^{l-m} {M-m \choose i} a_{m}b_{l-m-i}^{(i)})\partial^{M+N-l}. \end{array} $$ \begin{definition} $E$ is called the {\em ring of formal pseudo-differential operators with coefficients in $\power{R}{x}$}. \end{definition} {\bf Remark 1}\hspace{0.3cm} In our notation we follow M.~Mulase \cite{M1}. Other authors use the name of {\em micro-differential operators} for the objects which we call formal pseudo-differential operators. \vspace{0.5cm}\\ {\bf Remark 2} \begin{itemize} \item {}From the above construction it is clear that $E$ is an associative, non-commutative ring, which has the additional structure of a left $\power{R}{x}$-module. \item $E$ has a filtration by left $\power{R}{x}$-submodules $$E^{(m)}:=\left\{\sum_{n\in \Bbb{Z}} f_{n}\partial^{n}/ f_{n}\in \power{R}{x}, f_{n} = 0 \textrm{ for } n>m\right\}.$$ The {\em order} of an element $P\in E$ is defined to be the minimum $m\in \Bbb{Z}$ such that $P\in E^{(m)}$. In particular, the order of an element of $D$ coincides with its degree when we consider it as a polynomial in the variable $\partial$. For an operator $P=\sum_{n=0}^{\infty} f_{n}\partial^{N-n}$ of order $N$, $f_{0}$ is called its {\em leading coefficient}. \item An operator $P\in E$ can be written in the {\em right normal form} $P= \sum_{m=0}^{\infty}a_{m}\partial^{M-m}$ or in the {\em left normal form} $P= \sum_{n=0}^{\infty}\partial^{N-n}b_{n}$. It is an easy consequence of the Leibniz rule that the order of an operator is the same in the left and the right normal form and the leading coefficient does not change. So we see that $E$ is also a right $\power{R}{x}$-module and that $E^{(m)}$ gives rise to a filtration of $E$ by right $\power{R}{x}$-submodules. For more properties of formal pseudo-differential operators see the appendix \ref{B}. \end{itemize} $E$ contains the right ideal $xE$ generated by $x$. Denote by $\sigma: E\rightarrow E/xE$ the projection. So we have for formal pseudo-differential operators in the right normal form: $$ \sigma(\sum_{n=0}^{\infty} f_{n} \partial^{N-n}) = \sum_{n=0}^{\infty} f_{n}(0) \partial^{N-n}\in \negpower{R}{\partial^{-1}} = E/xE.$$ Let us set $$y := \partial^{-1}.$$ Obviously we get, for all $n\in\Bbb{Z}$: $$\sigma(E^{(n)}) = \power{R}{y}\cdot y^{-n}.$$ \begin{definition} \label{act PDO} The projection map $\sigma$ defines an action of $E$ on $\negpower{R}{y}$ as follows:\\ Take $P\in E$ and $v\in\negpower{R}{y}$. Then there is an operator $Q\in E$ such that $v=\sigma(Q)$. Define $$P(v) := \sigma(QP).$$ \end{definition} {\bf Remark} \begin{itemize} \item This definition does not depend on the choice of $Q$. Note that for an invertible operator $P\in E$, $P^{-1}:\negpower{R}{y} \rightarrow \negpower{R}{y}$ is inverse to the map $P:\negpower{R}{y} \rightarrow \negpower{R}{y}$. If $P$ is an operator of order 0 with invertible leading coefficient, then $P$ is invertible and the induced map is an automorphism preserving orders, i.e., for all $n\in\Bbb{Z}$, $$ P : \power{R}{y}\cdot y^{n} \stackrel{\sim}{\rightarrow} \power{R}{y}\cdot y^{n}.$$ \item If $P$ is a formal pseudo-differential operator with constant coefficients, $P\in \negpower{R}{\partial^{-1}}$, then we can regard $P$ as an element of $\negpower{R}{y}$ and it is easy to see that in this case the action of $P$ on $\negpower{R}{y}$ coincides with the usual multiplication in $\negpower{R}{y}$: $$\sigma(QP) = \sigma(Q)\cdot P= P\cdot \sigma(Q).$$ \item At this point, our way differs slightly from the one taken by M.~Mulase \cite{M1}. There, the quotient is taken by $Ex$ and operators $P\in E$ act on $\negpower{R}{y}$ from the left. Our approach will find its justification in Section \ref{eigenvalue}. \end{itemize} \begin{proposition} Let $P\in E$ be an operator of order 0 with invertible leading coefficient and $P:\negpower{R}{y} \rightarrow \negpower{R}{y}$ the induced automorphism defined above. Then $P$ induces an automorphism $$P:\frak{G}_{F,\alpha}^{1}(Spec(R)) \rightarrow \frak{G}_{F,\alpha}^{1}(Spec(R))$$ for all integers $\alpha\in \Bbb{Z}$ and all elements $F\in K(Spec(R))$. \end{proposition} {\bf Proof}\hspace{0.3cm} cf. \cite{M1}, Prop. 4.2. \begin{definition} A formal pseudo-differential operator $T\in E$ is called {\em admissible} if it is an operator of order 0 with invertible leading coefficient such that $$T\partial T^{-1} \in \negpower{R}{\partial^{-1}}.$$ The group of admissible operators is denoted by $\Gamma_{a}$. \end{definition} \begin{lemma} \label{admissible} \begin{enumerate} \item An operator $T$ is admissible if and only if it has the form $$T = exp(c_{1}x)\cdot(\sum_{i=0}^{\infty} f_{i}\partial^{-i}),$$ where $c_{1}\in R$, $f_{i}\in\power{R}{x}$ is a polynomial of degree at most $i$, and $f_{0}\in R$ is invertible. \item Let $v\in\negpower{R}{\partial^{-1}}$ be a monic element of order $-r$, $r\neq 0$. Then there is an admissible operator $T\in\Gamma_{a}$ such that $T\partial^{-r}T^{-1} = v$. \end{enumerate} \end{lemma} {\bf Proof}\hspace{0.3cm} This is a direct consequence of Lemma \ref{XLX} and its corollaries. \begin{definition} Two embedded Schur pairs $(A_{1},W_{1})$ and $(A_{2},W_{2})$ of rank $r$, index $F$ and level $\alpha$ are said to be {\em equivalent} if there is an admissible operator T such that $$T^{-1}A_{2} T = A_{1}, \quad TW_{2} = W_{1},$$ where $A_{1}$ and $A_{2}$ are understood to be subalgebras of $\negpower{R}{\partial^{-1}}$, i.e., $T$ acts by conjugation, while $W_{1}$ and $W_{2}$ are understood to be subspaces of $\negpower{R}{y}$ and the action of $T$ on $W_{2}$ is defined by Definition \ref{act PDO}. So we get $\frak{E}_{\alpha}\frak{S}'^{r}_{F}(Spec(R))/\Gamma_{a}$. \end{definition} Now let $(A,W)$ be an arbitrary embedded Schur pair of rank $r$. By Lemma \ref{pure}, $A\subset\negpower{R}{z}$ for some element $z\in\negpower{R}{y}$ of order $-r$. By Lemma \ref{admissible} there is an operator $T\in\Gamma_{a}$ such that $$ T^{-1}zT = y^{r}.$$ Consequently, $A\subset\negpower{R}{z}$ implies that $$T^{-1}AT \subset T^{-1}\negpower{R}{z}T = \negpower{R}{T^{-1}zT}= \negpower{R}{y^{r}}.$$ This proves \begin{lemma} Every equivalence class of embedded Schur pairs contains a representative which corresponds to a Schur pair. \mbox{\hspace*{\fill}$\Box$} \end{lemma} \begin{definition} Let $\alpha\in\Bbb{Z}$ be an integer. Two Schur pairs of rank $r$ and index $F$ are said to be {\em $\alpha$-equivalent} if the associated embedded Schur pairs of level $\alpha$ are equivalent. We call two geometric data {\em $\alpha$-equivalent} if the corresponding Schur pairs are $\alpha$-equivalent. \end{definition} {\bf Remark}\hspace{0.3cm} Observe that the $\alpha$-equivalence depends on the congruence class of $\alpha$ modulo $r$. \vspace{0.5cm} Now let us study the $(-1)$-equivalence in more detail. \begin{lemma} Let $T\in\Gamma_{a}$ be such that for two algebras $A_{1}$ and $A_{2}$ of pure rank $r$ which are both contained in $\negpower{R}{y^{r}}$: $$T A_{1} T^{-1} = A_{2}.$$ Then $T \power{R}{y^{r}} T^{-1} = \power{R}{y^{r}}$. Conversely, for every algebra $A_{1}$ of pure rank $r$ with $A_{1} \subseteq \negpower{R}{y^{r}}$ and every $T\in\Gamma_{a}$ satisfying $T \power{R}{y^{r}} T^{-1} = \power{R}{y^{r}}$, $TA_{1}T^{-1} \subseteq \negpower{R}{y^{r}}$ is also an algebra of pure rank $r$. \end{lemma} {\bf Proof}\hspace{0.3cm} The last part is obvious. As for the first one, remember that there are elements $a,b\in A_{1}$ such that $$a^{-1}b = y^{r} + \sum_{i\geq 2} \alpha_{i}y^{ir}$$ (cf. Lemma \ref{pure}). Then $TA_{1}T^{-1} = A_{2} \subseteq \negpower{R}{y^{r}}$ implies: $$ \begin{array}{rcl} Ta^{-1}bT^{-1} & = & Ta^{-1}T^{-1}TbT^{-1}\\ &=& (TaT^{-1})^{-1} TbT^{-1}\\ & \in & \power{R}{y^{r}} \end{array} $$ hence $T\power{R}{a^{-1}b} T^{-1} \subseteq \power{R}{y^{r}}$. But $\power{R}{y^{r}}$ equals $\power{R}{a^{-1}b}$, and so we obtain $T \power{R}{y^{r}} T^{-1} \subseteq \power{R}{y^{r}}$. The second inclusion we get using the fact that $A_{1}= T^{-1}A_{2}T$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now let us consider the action of $T\in\Gamma_{a}$ on the second component of an embedded Schur pair $(A,W)$ of level $-1$ $$W\subset \negpower{R}{y} = \bigoplus_{i=1}^{r} \negpower{R} {y^{r}}\cdot y^{i}.$$ For $c_{i}\in \negpower{R}{y^{r}}$ we have $$T(\sum_{i=1}^{r} c_{i} y^{i}) = \sum_{i=1}^{r} (T^{-1}c_{i} T)T(y^{i}). $$ In particular, by Corollary \ref{XLX2}, $T$ is determined by $A$ and $T^{-1}AT$ up to an operator with constant coefficients. So we obtain: \begin{proposition} \begin{enumerate} \item If two geometric data $(C,\pi,Spec(R),P,\rho_{1},{\cal F},\Phi_{1})$ and $(C,\pi,Spec(R),$ $P,\rho_{2},{\cal F},\Phi_{2})$ of rank $r$ and index $F$ are $(-1)$-equivalent then there is an automorphism of $R$-algebras $$h: \power{R}{z} \stackrel{\sim}{\rightarrow} \power{R}{z}$$ satisfying $\rho_{2} = h\circ \rho_{1}$. \item Two geometric data $$(C,\pi,Spec(R),P,\rho,{\cal F},\Phi_{1}) \textrm{ and } (C,\pi,Spec(R),P,\rho,{\cal F},\Phi_{2})$$ of rank $r$ and index $F$ are $(-1)$-equivalent if and only if there are elements $d_{1},\ldots,d_{r-1}\in \power{R}{z}$ and $d_{0}\in \power{R}{z}^{*}$ such that for $$ M = \left( \begin{array}{*{5}{c}} d_{0} & d_{1}&\ldots &d_{r-2}&d_{r-1}\\ d_{r-1}y^{r}& d_{0} & \ldots & d_{r-3}& d_{r-2}\\ \multicolumn{5}{c}{\dotfill}\\ d_{1}y^{r}& d_{2}y^{r}&\ldots& d_{r-1}y^{r}& d_{0} \end{array} \right) $$ $$\rho\Phi_{2} = M\circ\rho \Phi_{1}.$$ \end{enumerate} \mbox{\hspace*{\fill}$\Box$} \end{proposition} \begin{corollary} Note that in the case $r=1$, two geometric data $(C,\pi,Spec(R),P,\rho_{1}, {\cal F},\Phi_{1})$ and $(C,\pi,Spec(R),P,\rho_{2},{\cal F},\Phi_{2})$ are always $(-1)$-equivalent. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \subsection{Classification of commutative algebras of differential operators} After these preliminaries we now come to the main object: the classification of commutative algebras of ordinary differential operators with coefficients in $\power{R}{x}$, where $R$ is a commutative noetherian $k$-algebra, for some field $k$ of characteristic zero. \begin{definition} A commutative subalgebra $B$ of $D$ is said to be {\em elliptic of pure rank $r$} if \begin{itemize} \item $r=gcd(ord(P)/P\in B)$; \item There are monic elements $P,Q\in B$ such that $gcd(ord(P),ord(Q))=r$. \end{itemize} The set of all such subalgebras of $D$ is denoted by ${\cal B}_{r}(R)$. \end{definition} Let us start the observations with the following \begin{lemma} \label{XB} If $B$ is an elliptic subalgebra of rank $r$ of $D$ then there is a formal pseudo-differential operator $X$ of order 0 with coefficients in $\power{R}{x}$ and invertible leading coefficient such that $$A:=X^{-1} B X \subseteq \negpower{R}{\partial^{-1}} $$ and $A$ is an algebra of pure rank $r$. \end{lemma} {\bf Proof}\hspace{0.3cm} Choose a monic operator $P\in B$ of order $N$ greater than 0. Then by Lemma \ref{XLX} there exists a formal pseudo-differential operator $X$ of order 0 with invertible leading coefficient such that $$X^{-1} P X = \partial^{N}.$$ Let $Q\in B$ be an arbitrary element of $B$. Since $P$ and $Q$ commute, we get $$ \begin{array}{rcl} 0&=& X^{-1}(PQ-QP)X\\ &=& (X^{-1}PX)(X^{-1}QX) - (X^{-1}QX)(X^{-1}PX)\\ &=& \partial^{N}(X^{-1}QX) - (X^{-1}QX)\partial^{N}. \end{array} $$ {}From the proof of Corollary \ref{XLX2} one gets that then $X^{-1}QX$ must have constant coefficients, i.e., $X^{-1}QX\in \negpower{R}{\partial^{-1}}$. Finally, observe that the rank and the monicity of an operator are preserved under conjugation by $X$. So, $A$ is in fact an algebra of pure rank $r$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Let us look how far the name ``elliptic'' is justified for such an algebra $B$. Let $\frak{m}$ be a maximal ideal of $R$. Then $B/\frak{m}$ is a commutative algebra of differential operators with coefficients in $\power{K}{x}$ for some field $K$ containing $k$. A monic operator $P\in B$ of positive order, which exists by definition, gives us a monic operator $P\in B/\frak{m}$ of positive order. From Lemma \ref{XB}, applied to $B/\frak{m}$, we know that every $Q\in B/\frak{m}$ must have constant leading coefficient. In particular, since $K$ is a field, the leading coefficient of $Q$ is an element of $K^{*}$. This implies that $Q$ is really an elliptic ordinary differential operator, i.e., $B$ is a family of algebras of elliptic operators parametrized by $R$. However, this is not the only way to interpret $B$. Let us take a commutative subalgebra $R\subset \multpower{k}{t}{m}\/[\/\frac{d}{dt_{1}}, \ldots , \frac{d}{dt_{m}}\/]$. Then $Q\in B$ is a partial differential operator, which does {\bf not} need to be elliptic. \vspace{0.5cm} Let us continue with our construction. \begin{lemma}[Sato] \label{Sato1} A formal pseudo-differential operator $P\in E$ is a differential operator if and only if it preserves $\sigma(D)$ in $\negpower{R}{y}$, i.e., $$P\sigma(D) \subseteq \sigma(D).$$ \end{lemma} In the proof one may follow \cite{M1}, Lemma 7.2. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now let us have a look at another definition of infinite Grassmannians \begin{definition} The {\em Sato Grassmannian} is defined to be $$SG^{+} := \{J\subset E \textrm{closed subspace }/ J\oplus E^{(-1)} = E, DJ\subseteq J\}$$ \end{definition} This is a relative version of the Grassmannian originally used by Sato. There is the following connection to the Grassmannians we have considered until now: \begin{theorem}[Sato] \label{Sato2} \begin{enumerate} \item Let $\Gamma_{m}$ be the group of monic formal pseudo-differential operators of order zero and let $SG^{+}$ be the Sato Grassmannian defined as above. Then there is a natural bijection $\alpha:\Gamma_{m} \stackrel{\sim}{\rightarrow} SG^{+}$ given by $$\Gamma_{m} \ni X \stackrel{\alpha}{\mapsto} \alpha(X) = J = DX^{-1} \in SG^{+}.$$ \item Set $\frak{G}^{+}(Spec(R)):=\{W\in \frak{G}^{1}_{0,-1} (Spec(R))/W\oplus \power{R}{y}y =\negpower{R}{y}\}$. Then the projection $\sigma : E\rightarrow \negpower{R}{y}$ induces a bijection $$\sigma : SG^{+} \stackrel{\sim}{\rightarrow} \frak{G}^{+}(Spec(R)).$$ \end{enumerate} \end{theorem} {\bf Proof}\hspace{0.3cm} Lemma \ref{inv pseudo} implies that $\Gamma_{m}$ is a group. Now we can apply the proof of \cite{M1}, Thm. 7.4., and obtain our result. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} $\frak{G}^{+}(Spec(R))$ is the generalization of the {\em big cell of the Grassmannian of rank 1, index 0 and level $-1$} (cf. \cite{M1}). \begin{definition} We call two algebras $B_{1},B_{2}\in {\cal B}_{r}(R)$ {\em equivalent} if there is an invertible element $f\in\power{R}{x}$ such that $$B_{1}= f B_{2} f^{-1}.$$ We denote by $\bar{{\cal B}}_{r}(R)$ the set of these equivalence classes. \end{definition} \begin{theorem} For all $r\geq 1$, there is a canonical bijection $$\mu_{r} : \bar{{\cal B}}_{r}(R) \rightarrow \frak{E}_{-1}\frak{S}'^{r,+}_{0}(Spec(R))/\Gamma_{a},$$ where $\frak{E}_{-1}\frak{S}'^{r,+}_{0} (Spec(R))$ denotes the subset of $\frak{E}_{-1}\frak{S}'^{r}_{0} (Spec(R))$ consisting of embedded Schur pairs $(A,W)$ with $W\in \frak{G}^{+}(Spec(R))$. \end{theorem} {\bf Proof}\hspace{0.3cm} Given $B\in {\cal B}_{r}(R)$. Using Lemma \ref{XB} and Theorem \ref{Sato2}, we construct $A:=X^{-1}BX\subseteq \negpower{R} {\partial^{-1}}$ and $W:= X\sigma(D)\in \frak{G}^{+} (Spec(R))$. $A$ is an algebra of pure rank $r$. Since $B$ is contained in $D$, we get $B\sigma(D)\subseteq \sigma(D)$ (cf. Lemma \ref{Sato1}). This implies $$ A\cdot W = X^{-1} B X (\sigma(DX)) = \sigma(DXX^{-1}BX) = \sigma(DBX) \subseteq X\sigma(D) =W. $$ So we really constructed an embedded Schur pair of the required type. If $X_{1}$ is another operator satisfying $X_{1}^{-1}BX_{1}\subseteq \negpower{R}{\partial^{-1}}$ then $T:= X^{-1}X_{1}$ is an admissible operator and we end up with an equivalent embedded Schur pair. Now, what happens if we take $f B f^{-1}$ instead of $B$ for some $f\in\power{R}{x}^{*}$? Then: $A'= (fX)^{-1} (f B f^{-1}) (fX) = A$ and $W' = (fX)(\sigma(D)) = f_{0}\cdot W = W$, where $f_{0}$ denotes the (invertible) constant coefficient of $f$. \vspace{0.5cm} As for the inverse way, let us take $(A,W)\in \frak{E}_{-1}\frak{S}'^{r,+}_{0} (Spec(R))$. From Theorem \ref{Sato2} we get a unique operator $S\in \Gamma_{m}$ such that $W = S\sigma(D)$. Let us define $B:= SAS^{-1} \subseteq E$. Using Lemma \ref{Sato1} we obtain that $B\subseteq D$. We only have to check that $\sigma(DB)$ is contained in $\sigma(D)$, or, equivalently, that $S\sigma(DB)$ is a subset of $S\sigma(D)$: $$ S\sigma(DB) = \sigma(DBS) = \sigma(DSS^{-1}BS) = A\cdot W \subseteq W = S\sigma(D).$$ If we start with an equivalent embedded Schur pair $T(A,W)$, for some $T\in \Gamma_{a}$, we get an operator $S_{1}\in \Gamma_{m}$ such that $$S_{1} (\sigma(D)) = TW = TS (\sigma(D)).$$ Again Lemma \ref{Sato1} implies $S_{1}^{-1}TS\in D$. But this operator is an invertible formal pseudo-differential operator of order 0. So $S_{1}^{-1}TS$ is in fact an invertible element $f$ of $\power{R}{x}$ and we conclude that $$B'= S_{1} TAT^{-1} S_{1}^{-1} = f S A S^{-1} f^{-1} = fBf^{-1}.$$ So we in fact end up with an equivalent algebra. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} We have established a one-to-one correspondence of elliptic commutative algebras of differential operators with coefficients in $\power{R}{x}$ and certain equivalence classes of embedded Schur pairs. Using the results of the sections \ref{affine base} and \ref{pseudo DO} we can now state: \begin{corollary} There is a bijection between equivalence classes of commutative elliptic subalgebras of $\poly{\power{R}{x}}{\frac{d}{dx}}$ of pure rank $r$ and $(-1)$-equivalence classes of geometric data $$(C,\pi,Spec(R),P,\rho,{\cal F},\Phi)$$ of rank $r$ and index $0$ with the extra-condition that $$H^{0}({\cal F}) = H^{1}({\cal F}) = 0.$$ In particular, every sheaf corresponding to a commutative algebra of differential operators is strongly semistable with respect to $P$ (cf. Section \ref{geom. properties}). \mbox{\hspace*{\fill}$\Box$} \end{corollary} \subsection{Eigenvalue problems} \label{eigenvalue} As a motivation, let us approach the above constructed relation from another side. We take an elliptic commutative algebra $B$ of differential operators with coefficients in $\power{R}{x}$. It is an interesting problem to find out the common eigenfunctions of all operators belonging to $B$. Let $f\in\power{R}{x}$ be such a common eigenfunction, i.e., for each $P\in B$, there is some $\lambda(P)\in R$ such that $$P(f) = \lambda(P)\cdot f.$$ One easily sees that, for a given $f$, the map $$ \begin{array}{ccccc} \lambda & : & B & \rightarrow & R\\ && P & \mapsto & \lambda(P) \end{array} $$ is a homomorphism of $R$-algebras. On the other hand, given a homomorphism $\lambda : B \rightarrow R$, what are the eigenfunctions of $B$ with respect to $\lambda$? Let $(A,W)$ be an embedded Schur pair corresponding to $B$, $W=\sigma(DS)$, $A=S^{-1}BS$, for a formal pseudo-differential operator $S$ of order zero with invertible leading coefficient. For a given function $f\in\power{R}{x}$, we define an $R$-linear map \begin{equation} \label{eigenf} f : W \rightarrow R \end{equation} by $$f(\sigma(QS)) := \sigma(Q(f)), \textrm{ for } Q\in D.$$ This map is well-defined. For, if $\sigma(QS) = \sigma(Q'S)$, then $\sigma(Q) = \sigma(Q')$, i.e., $Q-Q'\in xE$. Consequently, $\sigma((Q-Q')(f)) = 0$. Now we claim \begin{proposition} $f\in\power{R}{x}$ is a common eigenfunction of the elements of $B$ with the eigenvalue $\lambda$ if and only if $\lambda$ makes the map (\ref{eigenf}) $A$-linear, i.e., for $a\in A$, $a=S^{-1}PS$, $P\in B$: $$f(a\cdot w) = \lambda(P)\cdot f(w).$$ \end{proposition} {\bf Proof}\hspace{0.3cm} First assume that $f$ is an eigenfunction as above. Then $P(f)=\lambda(P)\cdot f$ for all $P\in B$. Therefore, for $w= \sigma(QS)\in W$ and $a=S^{-1}PS\in A$ $$ \begin{array}{rcl} f(a\cdot w) & = & f(\sigma(QSS^{-1}PS))\\ &=& f(\sigma(QPS))\\ &=& \sigma((QP)(f))\\ &=& \sigma(Q(\lambda(P)\cdot f))\\ &=& \lambda(P)\cdot \sigma(Q(f))\\ &=& \lambda(P)\cdot f(w). \end{array} $$ On the other hand, assume that the $A$-linearity holds. Then, for all ordinary differential operators $Q$, $$\sigma(Q(P(f))) = \sigma(Q(\lambda(P)\cdot f)).$$ We claim that this implies $P(f) = \lambda(P)\cdot f$. But this is clear, since, for every $g=\sum_{n\geq 0}g_{n}x^{n}\in\power{R}{x}$, $\sigma(\partial^{n}(g)) = n!\cdot g_{n}$, i.e., $g$ is in fact determined by $\sigma(Q(g))$, $Q\in D$. \mbox{\hspace*{\fill}$\Box$} \subsection{Examples} To get a better idea of the formal correspondence constructed in this chapter, let us give some easy examples. Certainly, the results we are going to obtain are not new. They only serve to illustrate the construction. We start with the easiest case: Let $B\in {\cal B}_{1}(R)$ be an algebra containing a monic element of order 1. We prove \begin{lemma} All algebras $B\in {\cal B}_{1}(R)$ containing a monic element of order 1 are equivalent. \end{lemma} {\bf Proof}\hspace{0.3cm} Take $u = \sum_{i\geq 0} v_{i} x^{i} \in \power{R}{x}$. It suffices to prove that there is an invertible formal power series $f = \sum_{i\geq 0} f_{i} x^{i} \in \power{R}{x}$ such that $$f^{-1} \partial f = \partial+u.$$ Let us construct $f$: $$f^{-1} \partial f = f^{-1} f \partial + f^{-1} f' = \partial + f^{-1} f'; $$ therefore we only need to construct $f$ such that $u = f^{-1} f'$, i.e., $fu=f'$. We write this expression as a formal power series $$\sum_{i\geq 0} (\sum_{j=0}^{i} f_{j} u_{i-j}) x^{i} = \sum_{i\geq 0} (i+1) f_{i+1} x^{i},$$ so that we get as a necessary and sufficient condition: $$f_{i+1} = \frac{1}{i+1} \sum_{j=0}^{i} f_{j} u_{i-j},$$ which is solvable for an arbitrarily given $u$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} What does this say in terms of curves and sheaves? The curve associated to such a $B$ is obviously $\Bbb{P}^{1}_{R}$ with some section $P$. One possible line bundle with vanishing cohomologies is ${\cal O}_{\Bbb{P}^{1}_{R}}(-P)$. The different local parametrizations of $\Bbb{P}^{1}_{R}$ and local trivializations of ${\cal O}_{\Bbb{P}^{1}_{R}}(-P)$ along $P$ factor out under the equivalence relation. Now the above lemma reads as: \begin{corollary} For any two sections of $\Bbb{P}^{1}_{R}$ there is an isomorphism of $\Bbb{P}^{1}_{R}$ mapping one section into the other. Furthermore, given a section $P$, ${\cal O}_{\Bbb{P}^{1}_{R}}(-P)$ is the only coherent sheaf on $\Bbb{P}^{1}_{R}$ of rank 1 with vanishing cohomology groups which satisfies the conditions of Definition \ref{def data}. \mbox{\hspace*{\fill}$\Box$} \end{corollary} \vspace{0.5cm} The next interesting case is that of algebras $B\in {\cal B}_{1}(R)$ containing monic elements of order 2 and 3, but without any element of order 1. These correspond to families of reduced and irreducible curves with arithmetic genus 1. First take $R=k$. Then we get a single curve. One may ask how the singularity of the curve is displayed in the associated algebra of differential operators: \begin{proposition} An algebra $B$ as above corresponds to a singular plane cubic with its point at infinity as a section if and only if there is a formal pseudo-differential operator $T$ of order 0 with invertible leading coefficient such that $TBT^{-1}$ is an algebra of {\bf differential} operators with constant coefficients. \end{proposition} {\bf Remark}\hspace{0.3cm} Keep in mind that this does not say that $B$ is {\em equivalent} to an algebra of differential operators with constant coefficients. The above transformation changes the sheaf induced by $B$. \vspace{0.5cm}\\ {\bf Proof}\hspace{0.3cm} Let $B$ be given as above. Then for the corresponding Schur pair $(A,W)$ we get: $$A = k\/[\/y^{-2}+\alpha y^{-1}+\beta, y^{-3} + \gamma y^{-2} + \delta y^{-1} + \epsilon\/] $$ with coefficients $\alpha, \beta, \gamma, \delta, \epsilon \in k$. Note that of course $\beta$ and $\epsilon$ may be changed arbitrarily, and $\gamma$ may be set to 0. Assume that $\alpha$ is different from 0. We show that such an $A$ (understood as an element of ${\cal B}_{1}(k)$ via the identification $\partial^{-1}=y$) is in fact equivalent to an algebra of differential operators containing $\partial^{2}$, i.e., we could have chosen $T$ such that $TBT^{-1}$ already contains $\partial^{2}$. Given $\alpha, \delta \in k$ we want to construct an invertible power series $$f=\sum_{i\geq 0} f_{i} x^{i} \in \power{k}{x}$$ such that: \begin{itemize} \item $f^{-1}\partial^{2}f = \partial^{2}+\alpha \partial+\beta$ for some $\beta\in k$; \item There are numbers $\epsilon, a, b, c \in k$ such that $$f^{-1} (\partial^{3} + a \partial^{2} + b \partial + c ) f = \partial^{3} + \delta \partial + \epsilon.$$ \end{itemize} Let's start the calculation: $$ f^{-1}\partial^{2}f = f^{-1}(f \partial^{2} + 2f' \partial + f'') = \partial^{2} + 2 f^{-1} f' \partial + f^{-1} f''. $$ We have already seen that the equation $f'=\frac{\alpha}{2}f$ is solvable. For $f$ chosen as such we get: $$ \begin{array}{l} f^{-1}\partial^{2}f = \partial^{2} + \alpha \partial + \frac{\alpha^{2}}{4} \textrm{ and}\\ \\ f^{-1} (\partial^{3} + a \partial^{2} + b \partial + c ) f=\\ = f^{-1}(f \partial^{3} + 3 f'\partial^{2} + 3 f''\partial + f'''+ af\partial^{2} + 2af' \partial + af'' + bf\partial + bf' + cf)\\ = \partial^{3} + (3 f^{-1}f' + a)\partial^{2} + (3 f^{-1}f'' + 2a f^{-1} f'+ b)\partial + \\ \hfill + (f^{-1}f''' + af^{-1}f'' + bf^{-1}f' + c)\\ = \partial^{3} + (\frac{3\alpha}{2} + a)\partial^{2} + (\frac{3\alpha^{2}}{4} + a\alpha+ b)\partial + (\frac{\alpha^{3}}{8} + \frac{a\alpha^{2}}{4} + b\frac{\alpha}{2} + c). \end{array} $$ {}From this we see that all conditions can be satisfied. \vspace{0.5cm} We still have to show that the rings $A = k\/[\/y^{-2}, y^{-3} + \delta y^{-1}\/]$ correspond to singular plane cubics and that all such cubics occur. A singular plane cubic has the equation $$z_{0}z_{2}^{2} = z_{1}(z_{1} + \delta z_{0})^{2}$$ with $P=(0:0:1)$ being its point at infinity. Note that it has a cusp if $\lambda = 0$ and a node in the remaining cases. Using the methods of \ref{Elliptic curves} we get $$A = k\/[\/y^{-2}, y^{-3} + \delta y^{-1}\/]$$ as the associated ring, and the proof is complete. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm} Now let us have a look at the partial differential equations which are produced by the algebras $B$ as considered above. $B$ is generated by two elements $$\begin{array}{rcl} L&=&\partial^{2} + u \partial + v\\ P&=&\partial^{3} + \alpha \partial^{2} + \beta \partial + \gamma. \end{array} $$ We have seen before that we can assume that $u=0$ if we are only interested in the equivalence class of $B$. \vspace{0.5cm} What does it mean for $B$ to be commutative? $[P,L]=0$ can be interpreted as $$ \begin{array}{lrcl} (I)& 2\alpha' & = & 0\\ (II)& \alpha'' + 2\beta' -3v'&=&0\\ (III)& \beta'' + 2\gamma' - 3 v'' - 2\alpha v'&=&0\\ (IV)& \gamma'' - v'''-\alpha v'' -\beta v' & = & 0. \end{array} $$ The first equation says that $\alpha$ is a constant. So we can choose another normal form $$\begin{array}{rcl} L&=&\partial^{2} + v\\ P&=&\partial^{3} + \beta \partial + \gamma, \end{array} $$ $v$ and $\gamma$ without constant term. The class of the algebra $B$ uniquely determines $P$ and $L$, and on the other side, these two elements, and by (II) and (III) in fact $\beta$, uniquely determine the equivalence class of $B$. Substituting (I), (II) and (III) in (IV) we get $$\frac{1}{6} \beta''' - \frac{2}{3} \beta\beta' = 0,$$ which is nothing but the stationary Korteveg - de Vries equation. We saw that $B$ corresponds to a singular curve with its point at infinity if it may be transformed into $\bar{B} = k\/[\/\partial^{2},\partial^{3} + \beta \partial \/]$, $\beta$ being a constant. So we get: \begin{corollary} A pointed integral curve $(C,p)$ of arithmetic genus 1 is isomorphic to a singular plane cubic with its point at infinity if and only if there is a torsion free sheaf on $C$ generating a constant solution of the KdV equation. The singularity is a node if this constant if different from zero; it is a cusp if the constant equals to zero. \mbox{\hspace*{\fill}$\Box$} \end{corollary} After this study of special cases we are now interested in the general structure of ${\cal B}_{1}(\Bbb{C})$. {}From the correspondence established in the first part of the section \ref{Fam. DO} it is clear that two algebras of differential operators lead to the same pointed curve if and only if they differ only by conjugation with a formal pseudo-differential operator of order 0 with invertible leading coefficient. So we obtain: \begin{proposition} For any $B\in {\cal B}_{1}(\Bbb{C})$ such that $Spec(B)$ is smooth, the set $$\bar{{\cal B}}_{1}(\Bbb{C})(B):=\left\{ \begin{array}{r} B'\in \bar{{\cal B}}_{1}(\Bbb{C})/ \exists\textrm{ pseudo-diff. op. } X \textrm{ of order 0 with inv.}\\ \textrm{leading coeff. such that } X^{-1}B'X=B \end{array} \right\} $$ has the structure of a reduced, irreducible, affine complex variety of dimension $$g=card\{n\in \Bbb{N}/ \textrm{ there is no differential operator of order $n$ in $B$} \}.$$ \end{proposition} {\bf Proof}\hspace{0.3cm} Let $(C,p)$ be the pointed curve given by $B$. We consider $Pic^{g-1}(C)$. The theta divisor in it is given by the condition $h^{0}\neq 0$. This divisor is known to be ample, so its complement $$U = \{ {\cal L} \textrm{ line bundle of degree $g-1$ on } C/ h^{0} ({\cal L}) = h^{1}({\cal L}) = 0 \}$$ is affine. It is also reduced and irreducible. Now we take the Poincar\'{e} bundle ${\cal P}^{g-1}_{C}$ on $Pic^{g-1}(C) \times C$ (normalized with respect to $p$) and restrict it to $U\times C$. We choose some local parametrization of $U\times C$ near $U\times\{p\}$ and some local trivialization of ${\cal P}^{g-1}_{C}$ along $U\times\{p\}$ (for more details see section \ref{Poincare}). Different parametrizations and trivializations cancel out under the equivalence relation. Now we construct the associated algebra \boldmath$B$\unboldmath {} of ordinary differential operators with coefficients in $\power{R}{x}$ for $U=Spec(R)$. From the universality of the Poincar\'{e} bundle we get that \boldmath$B$\unboldmath {} uniquely parametrizes the equivalence classes of algebras $B'\in {\cal B}_{1}(\Bbb{C})$ corresponding to the given pointed curve $(C,p)$, hence all of $\bar{{\cal B}}_{1}(\Bbb{C})(B)$. \mbox{\hspace*{\fill}$\Box$} \vspace{0.5cm}\\ {\bf Remark}\hspace{0.3cm} In the case that $g=1$ the so-defined \boldmath$B$\unboldmath$\in{\cal B}_{1}(R)$ carries a nontrivial family of solutions of the KdV equation.

9606

alg-geom/9606002

en

https://arxiv.org/abs/alg-geom/9606002

alg-geom/9606002

Rita Pardini

Rita Pardini

On the period map for abelian covers of algebraic varieties

LaTeX, 17 pages

We show that infinitesimal Torelli for $n$-forms holds for abelian covers of algebraic varieties of dimension $n\ge 2$, under some explicit ampleness assumptions on the building data of the cover. Moreover, we prove a variational Torelli result for some families of abelian covers.

alg-geom

\section{Introduction} \setcounter{defn}{0} \setcounter{equation}{0} This paper is devoted to the study of the period map for abelian covers of smooth projective varieties of dimension $n\ge 2$. Our viewpoint is very close to that of Green in \cite{suffampio}, namely we look for results that hold for abelian covers of an arbitrary variety whenever certain ampleness assumptions on the building data defining the cover are satisfied. We focus on two questions: the infinitesimal and the variational Torelli problems. Infinitesimal Torelli for the periods of $k$-forms holds for a smooth projective variety $X$ if the map ${\rm H}^1(X,T_X)\to \oplus_p{\rm Hom}\left ({\rm H}^p(X,\Omega_X^{k-p}),{\rm H}^{p+1}(X, \Omega_X^{k-p-1})\right)$, expressing the differential of the period map for $k$-forms, is injective. This is expected to be true as soon as the canonical bundle of $X$ is ``sufficiently ample''. There are many results in this direction, concerning special classes of varieties, as hypersurfaces (see, for instance, \cite{suffampio}), complete intersections (\cite{konnocompl}) and simple cyclic covers (\cite{konnociclico}, \cite{lwp},\cite{peters}). Here we continue the work on abelian covers of \cite{torelli}, and prove (see \ref{mainthm1}): \begin{thm}\label{introinf} Let $G$ be an abelian group and let $f:X\to Y$ be a $G$-cover, with $X$, $Y$ smooth projective varieties of dimension $n\ge2$. If properties $(A)$ and $(B)$ of \ref{ipotesi} are satisfied, then infinitesimal Torelli for the periods of $n$-forms holds for $X$. \end{thm} Properties $(A)$ and $(B)$ amount to the vanishing of certain cohomology groups and are certainly satisfied if the building data of the cover are sufficiently ample. If $Y$ is a special variety (e.g., $Y={\bf P}^n$), then thm. \ref{introinf} yields an almost sharp statement (see theorem \ref{mainthm1p}). In general, a result of Ein-Lazarsfeld (\cite{el}) and Griffiths vanishing theorem enable us (see prop. \ref{effective}) to give explicit conditions under which $(A)$ and $(B)$ are satisfied, and thus to deduce an effective statement from thm. \ref{introinf} (see thm. \ref{mainthm1e}). In order to extend to the case of arbitrary varieties the infinitesimal Torelli theorem obtained in \cite{torelli} for a special class of surfaces, we introduce a generalized notion of prolongation bundle and give a Jacobi ring construction analogous to those of \cite{suffampio} and \cite{konnovar}. This is also a starting point for attacking the variational Torelli problem, which asks whether, given a flat family ${\cal X}\to B$ of smooth polarized varieties, the map associating to a point $b\in B$ the infinitesimal variation of Hodge structure of the fibre $X_b$ is generically injective, up to isomorphism of polarized varieties. A positive answer to this problem has been given for families of projective hypersurfaces (\cite{cggh},\cite{donagi}), for hypersurfaces of high degree of arbitrary varieties (\cite{suffampio}), for some complete intersections (\cite{konnovar}) and for simple cyclic covers of high degree (\cite{ciclico}). The most effective tool in handling these problems is the symmetrizer, introduced by Donagi, but unfortunately, an analogous construction does not seem feasible in the case of abelian covers. However, exploiting the variational Torelli result of \cite{suffampio}, we are able to obtain, under analogous assumptions, a similar result for a large class of abelian covers. More precisely, we prove (see thm. \ref{mainthm2}): \begin{thm}\label{intro2} (Notation as in section \ref{covers}.) \noindent Let $Y$ be a smooth projective variety of dimension $n\ge 2$, with very ample canonical class. Let $G$ be a finite abelian group and let $f:X\to Y$ be a smooth $G$-cover with sufficiently ample building data $L_{\chi}$, $D_i$, $\chi\in G^*$, $i=1,\ldots r$. Assume that for every $i=1,\ldots r$ the identity is the only automorphism of $Y$ that preserves the linear equivalence class of $D_i$; moreover, assume that for $i=1,\ldots r$ there exist a $\chi\in G^*$ (possibly depending on $i$) such that $\chi(g_i)\ne 1$ and $L_{\chi}(-D_i)$ is ample. Let ${\cal X}\to \tilde{W}$ be the family of the smooth $G$-covers of $Y$ obtained by letting the $D_i$'s vary in their linear equivalence classes: then there is a dense open set $V\subset\tilde{W}$ such that the fibre $X_s$ of ${\cal X}$ over $s\in V$ is determined by its IVHS for $n$-forms plus the natural $G$-action on it. \end{thm} The paper is organized as follows: section $1$ is a brief review of abelian covers, sections $2$ and $3$ contain the technical details about prolongation bundles and the Jacobi ring construction, section $4$ contains the statements and proofs of the results on infinitesimal Torelli, and section $5$ contains some technical lemmas that allow us to prove in section $6$ the variational Torelli theorem \ref{intro2}. \noindent {\em Acknowledgements:} I wish to thank Mark Green for communicating me the proof of lemma \ref{green}. \paragraph{} {\bf Notation and conventions:} All varieties are smooth projective varieties of dimension $n\ge 2$ over the field ${\bf C}$ of complex numbers. We do not distinguish between vector bundles and locally free sheaves; as a rule, we use the additive notation for divisors and the multiplicative notation for line bundles. For a divisor $D$, $c_1(D)$ denotes the first Chern class of $D$ and $|D|$ the complete linear system of $D$. If $L$ is a line bundle, we also denote by $|L|$ the complete linear system of $L$, and we write $L^k$ for $L^{\!\otimes\!^k}$ and $L^{-1}$ for the dual line bundle. As usual, $T_Y$ denotes the tangent sheaf of $Y$, $\Omega^k_Y$ denotes the sheaf of regular $k$-forms on $Y$, $\omega_Y=\Omega^n_Y$ denotes the canonical bundle and ${\rm Pic}(Y)$ the Picard group. If ${\cal F}$ is a locally free sheaf, we denote by ${\cal F}^*$ the dual sheaf, by $S^k{\cal F}$ the $k$-th symmetric power of ${\cal F}$ and by $\det {\cal F}$ the determinant bundle. Consistently, the dual of a vector space $U$ is denoted by $U^*$; the group of linear automorphisms of $U$ is denoted by $GL(U)$. \noindent $[x]$ denotes the integral part of the natural number $x$. \section{Abelian covers and projection formulas}\label{covers} \setcounter{defn}{0} \setcounter{equation}{0} In this section we recall some facts about abelian covers that will be needed later. For more details and proofs, see \cite{abeliani}. Let $G$ be a finite abelian group of order $m$ and let $G^*={\rm Hom}(G,{\bf C}^*)$ be the group of characters of $G$. A {\em $G$-cover} of a smooth $n$-dimensional variety $Y$ is a Galois cover $f:X\to Y$ with Galois group $G$, with $X$ normal. Let ${\cal F}$ be a $G$-linearized locally free sheaf of ${\cal O}_X$-modules: under the action of $G$, the sheaf $f_*{\cal F}$ splits as the direct sum of the eigensheaves corresponding to the characters of $G$. We denote by $(f_*{\cal F})^{(\chi)}$ the eigensheaf corresponding to a character $\chi\in G^*\setminus\{1\}$ and by $(f_*{\cal F})^{inv}$ the invariant subsheaf. In particular, when ${\cal F}={\cal O}_X$ , we have $(f_*{\cal O}_X)^{inv}={\cal O}_Y$ and $(f_*{\cal O}_X)^{(\chi)}=L_{\chi}^{-1}$, with $L_{\chi}$ a line bundle. Let $D_1,\ldots D_r$ be the irreducible components of the branch locus $D$ of $f$. For each index $i$, the subgroup of $G$ consisting of the elements that fix the inverse image of $D_i$ pointwise is a cyclic group $H_i$, the so-called {\em inertia subgroup} of $D_i$. The order $m_i$ of $H_i$ is equal to the order of ramification of $f$ over $D_i$ and the representation of $H_i$ obtained by taking differentials and restricting to the normal space to $D_i$ is a faithful character $\chi_i$. The choice of a primitive $m$-th root $\zeta$ of $1$ defines a map from $\{1,\ldots r\}$ to $G$: the image $g_i$ of $i$ is the generator of $H_i$ that is mapped to $\zeta^{m/m_i}$ by $\chi_i$. The line bundles $L_{\chi}$, $\chi\in G^*\setminus\{1\}$, and the divisors $D_i$, each ``labelled'' with an element $g_i$ of $G$ as explained above, are the {\em building data} of the cover, and determine $f:X\to Y$ up to isomorphism commuting with the covering maps. The building data satisfy the so-called {\em fundamental relations}. In order to write these down, we have to set some notation. For $i=1,\ldots r$ and $\chi\in G^*$, we denote by $a^i_{\chi}$ the smallest positive integer such that $\chi(g_i)=\zeta^{ma^i_{\chi}/m_i}$; for each pair of characters $\chi$,$\phi$ we set $\epsilon_{\chi,\phi}^i=[(a^i_{\chi}+a^i_{\phi})/m_i]$ (notice that $\epsilon_{\chi,\phi}^i=0$ or $1$) and $D_{\chi,\phi}=\sum_{i=1}^r\epsilon_{\chi,\phi}^iD_i$. In particular, $D_{\chi,\chi^{-1}}$ is the sum of the components $D_i$ of $D$ such that $\chi(g_i)\ne 1$. Then, the fundamental relations of the cover are the following: \begin{equation} L_{\chi}+L_{\phi}\equiv L_{\chi\phi}+ D_{\chi,\phi}, \qquad \forall \chi,\phi\in G^* \label{fundrel} \end{equation} When $\phi=\chi^{-1}$, the fundamental relations read: \begin{equation}\label{fundrelbis} L_{\chi}+L_{\chi^{-1}}\equiv D_{\chi,\chi^{-1}}. \end{equation} The cover $f:X\to Y$ can be reconstructed from the building data as follows: if one chooses sections $s_i$ of ${\cal O}_Y(D_i)$ vanishing on $D_i$ for $i=1,\ldots r$, then $X$ is defined inside the vector bundle $V=\oplus_{\chi\ne 1}L_{\chi}$ by the equations: \begin{equation}\label{equazioni} z_{\chi} z_{\phi}=\left(\Pi_{i}s_i^{\epsilon^i_{\chi,\phi}}\right)z_{\chi\phi}, \quad\forall \chi,\phi\in G^*\setminus\{1\} \end{equation} where $z_{\chi}$ denotes the tautological section of the pull-back of $L_{\chi}$ to $V$. Conversely, for every choice of the sections $s_i$, equations (\ref{equazioni}) define a scheme $X$, flat over $Y$, which is smooth iff the zero divisors of the $s_i$'s are smooth, their union has only normal crossings singularities and, whenever $s_{i_1},\ldots s_{i_t}$ all vanish at a point $y$ of $Y$, the group $H_{i_1}\times\cdots\times H_{i_t}$ injects into $G$. So, by letting $s_i$ vary in $\HH{0}(Y,{\cal O}_Y(D_i))$, one obtains a flat family ${\cal X}$ of smooth $G$-covers of $X$, parametrized by an open set $W\subset\oplus_i\HH{0}(Y,{\cal O}_Y(D_i))$. Throughout all the paper we will make the following \begin{assu}\label{ampleness} The $G$-cover $f:X\to Y$ is smooth of dimension $n\ge 2$; the building data $L_{\chi}$, $D_i$ and the adjoint bundles $\omega_Y\!\otimes\! L_{\chi}$, $\omega_Y(D_i)$ are ample for every $\chi\in G^*\setminus\{1\}$ and for every $i=1,\ldots r$. \end{assu} Assumption \ref{ampleness} implies that the cover is {\em totally ramified}, namely that $g_1,\ldots g_r$ generate $G$. Actually, this is equivalent to the fact that the divisor $D_{\chi,\chi^{-1}}$ is nonempty if the character $\chi$ is nontrivial, and also to the fact that none of the line bundles $L_{\chi}$, $\chi\in G^*\setminus\{1\}$, is a torsion point in ${\rm Pic}(Y)$. Since $X$ is smooth, assumption \ref{ampleness} implies in particular that for each subset $\{i_1,\ldots i_t\}\subset\{1,\ldots r\}$, with $t\leq n$, the cyclic subgroups generated by $g_{i_1},\ldots g_{i_t}$ give a direct sum inside $G$. \paragraph{} In principle, all the geometry of $X$ can be recovered from the geometry of $Y$ and from the building data of $f:X\to Y$. The following proposition is an instance of this philosophy. \begin{prop}\label{projform} Let $f:X\to Y$ be a $G$-cover, with $X$, $Y$ smooth of dimension $n$. For $\chi\in G^*$, denote by $\Delta_{\chi}$ the sum of the components $D_i$ of $D$ such that $a_{\chi}^i\neq m_i-1$. Then, for $1\le k\le n$ there are natural isomorphisms: $$(f_*\Omega_X^k)^{(\chi)}=\Omega^k_Y(\log D_{\chi,\chi^{-1}})\!\otimes\! L_{\chi}^{-1}$$ $$(f_*T_X)^{(\chi)}=T_Y(-\log\Delta_{\chi})\!\otimes\! L_{\chi}^{-1}$$ and,in particular: $$(f_*\Omega^k_X)^{inv}=\Omega^k_Y, \quad (f_*T_X)^{inv}=T_Y(-\log D), \quad (f_*\omega_X)^{(\chi)}=\omega_Y\!\otimes\! L_{\chi^{-1}}.$$ \end{prop} {\bf Proof:\,} This is a slight generalization of Proposition $4.1$ of \cite{abeliani}, and it can be proven along the same lines. The identification $(f_*\omega_X)^{(\chi)}=\omega_Y\!\otimes\! L_{\chi^{-1}}$ follows from the general formula and relations (\ref{fundrelbis}).\ $\Box$\par\smallskip We recall the following generalized form of Kodaira vanishing (see \cite{EV}, page 56): \begin{thm}\label{kodaira} Let $Y$ be a smooth projective $n$-dimensional variety and let $L$ be an ample line bundle. Then: $${\rm H}^i(Y,\Omega^k_Y\!\otimes\! L^{-1})=0, \quad i+k<n.$$ Moreover, if $A+B$ is a reduced effective normal crossing divisor, then: $${\rm H}^i(Y,\Omega^k_Y(\log(A+B)\!\otimes\! L^{-1}(-B))=0, \quad i+k<n.$$ \end{thm} From prop.\ref{projform}, theorem \ref{kodaira} and assumption \ref{ampleness} it follows that the non-invariant part of the cohomology of $X$ is concentrated in dimension $n$. Thus we will be concerned only with the period map for the periods of $n$-forms. \section{Logarithmic forms and sheaf resolutions}\label{logarithmic} \setcounter{defn}{0} \setcounter{equation}{0} In this section we recall the definition and some properties of logarithmic forms and introduce a generalized notion of prolongation bundle. We would like to mention that Konno, when studying in \cite{konnovar} the global Torelli problem for complete intersections, has also introduced a generalization of the definition of prolongation bundle, which is however different from the one used here. Let $D$ be a normal crossing divisor with smooth components $D_1,\ldots D_r$ on the smooth $n$-dimensional variety $Y$. As usual, we denote by $\Omega_Y^k(\log D)$ the sheaf of $k$-forms with at most logarithmic poles along $D_1,\ldots D_r$ and by $T_Y(-\log D)$ the subsheaf of $T_Y$ consisting of the vector fields tangent to the components of $D$. Assume that $y\in Y$ lies precisely on the components $D_1,\ldots D_t$ of $D$, with $t\le n$. Let $x^1,\ldots x^t$ be local equations for $D_1, \ldots D_t$ and choose $x_{t+1},\ldots x_n$ such that $x_1,\ldots x_n$ are a set of parame\-tres at $y$. Then $\frac{dx_1}{x_1},\ldots \frac{dx_t}{x_t}$, $dx_{t+1},\ldots dx_n$ are a set of free gen\-er\-a\-tors for $\Omega^1_Y(\log D)$ and $x_1\frac{\partial}{\partial x_1},\ldots x_t\frac{ \partial}{\partial x_t}$, $\frac{\partial }{\partial x_{t+1}},\ldots\frac {\partial}{\partial x_n}$ are free generators for $T_Y(-\log D)$ in a neighbourhood of $y$. So the sheaves of logarithmic forms are locally free and one has the following canonical identifications: $\Omega_Y^k(\log D)=\wedge^k\Omega_Y^1(\log D)$ and $T_Y(-\log D)=\Omega_Y^1(\log D)^*$, duality being given by contraction of tensors. Moreover, we recall that, if ${\cal F}$ is a locally free sheaf of rank $m$ on $Y$, then the alternation map $\wedge^i{\cal F}\!\otimes\!\wedge ^{m-i}{\cal F}\to \det{\cal F}$ is a nondegenerate pairing, which induces a canonical isomorphism $\left(\wedge^i{\cal F}\right)^*\to \wedge^{m-i}{\cal F}\!\otimes\!(\det {\cal F})^{-1}$. So we have: \begin{eqnarray}\label{dualita'} T_Y(-\log D)\cong \Omega^{n-1}_Y(\log D)\!\otimes\! (\omega_Y(D))^{-1} \\ \Omega_Y^k(\log D)^*\cong \Omega_Y^{n-k}(\log D)\!\otimes\! (\omega_Y(D))^{-1} \nonumber \end{eqnarray} \paragraph{} The {\em (generalized) prolongation bundle} $P$ of $(D_1,\ldots D_r)$ is defined as the ex\-ten\-sion $0\to\Omega^1_Y\to P\to\oplus^r{\cal O}_Y\to 0$ as\-so\-cia\-ted to the class $(c_1(D_1),\ldots c_1(D_r))$ of ${\rm H}^1(Y,\oplus^r\Omega^1_Y)$. Let $\{U_{\alpha}\}$ be a finite affine covering of $Y$, let $x^i_{\alpha}$ be local equations for $D_i$ on $U_{\alpha}$, $i=1\ldots r$, and let $g^i_{\alpha\beta}=x^i_{\alpha}/x^i_{\beta}$. Denote by $e^1_{\alpha},\ldots e^r_{\alpha}$ the standard basis of $\oplus^r{\cal O}_Y|_{U_{\alpha}}$. The elements of $P|_{U_{\alpha}}$ are represented by pairs $(\sigma_{\alpha}, \sum_i z^i_{\alpha}e_i)$, where $\sigma_{\alpha}$ is a $1$-form and the $z^i_{\alpha}$'s are regular functions, satisfying the following transition relations on $U_{\alpha}\cap U_{\beta}$: $$\left(\sigma_{\alpha}, \sum_i z^i_{\alpha}e^i_{\alpha}\right)=\left(\sigma_{\beta}+\sum_i z^i_{\beta}\frac{dg^i_{\alpha\beta}}{g^i_{\alpha\beta}},\, \sum_iz^i_{\beta}e^i_{\beta}\right).$$ There is a natural short exact sequence: \begin{equation}\label{olog} 0\to\oplus_i{\cal O}_Y(-D_i)\to P\to\Omega^1_Y(\log D)\to 0 \end{equation} with dual sequence: \begin{equation}\label{tlog} 0\to T_Y(-\log D)\to P^*\to\oplus_i {\cal O}_Y(D_i)\to 0. \end{equation} In local coordinates the map $\oplus_i {\cal O}_Y(-D_i)\to P$ is defined by: $x^i_{\alpha}\mapsto (dx^i_{\alpha},x^i_{\alpha}e^i_{\alpha})$ and the map $P\to \Omega^1(\log D)$ is defined by: $(\sigma_{\alpha}, \sum_i z^i_{\alpha}e^i_{\alpha})\mapsto\sigma_{\alpha}-\sum z^i_{\alpha}dx^i_{\alpha}/x^i_{\alpha}$. We close this section by writing down a resolution of the sheaves of logarithmic forms that will be used in section \ref{ivhs}. Given an exact sequence $0\to A\to B\to C\to 0$ of locally free sheaves, for any $k\ge 1$ one has the following long exact sequence (see \cite{cime}, page 39): \begin{equation}\label{complex} 0\to S^kA\to B\!\otimes\! S^{k-1}A\to\ldots\to \wedge^{k-1}B\!\otimes\! A\to\wedge^k B\to \wedge^kC\to 0. \end{equation} Applying this to (\ref{olog}) and setting $V=\oplus_i{\cal O}_Y (D_i)$ yields: \begin{equation}\label{reslog} 0\to S^kV^*\to S^{k-1}V^*\!\otimes\! P\to\ldots \to V^*\!\otimes\!\wedge^{k-1}P\to\wedge^kP\to \Omega^k_Y(\log D)\to 0 \end{equation} \section{The algebraic part of the IVHS}\label{ivhs} \setcounter{defn}{0} \setcounter{equation}{0} The aim of this section is to give, in the case of abelian covers, a construction analogous to the Jacobian ring construction for hypersurfaces of \cite{suffampio}. Let ${\cal X}\to B$ be a flat family of smooth projective varieties of dimension $n$ and let $X$ be the fibre of ${\cal X}$ over the point $0\in B$; the differential of the period map for the periods of $k$-forms for ${\cal X}$ at $0$ is the composition of the Kodaira-Spencer map with the following universal map, induced by cup-product: \begin{equation} {\rm H}^1(X,T_X)\to \oplus_p{\rm Hom} ({\rm H}^p(X,\Omega_X^{k-p}),{\rm H}^{p+1}(X, \Omega_X^{k-p-1})), \label{inftor} \end{equation} This map is called the {\em algebraic part of the infinitesimal variation of Hodge structure} of $X$ (IVHS for short). Assume that $f:X\to Y$ is a smooth $G$-cover; then the $G$-action on the tangent sheaf and on the sheaves of differential forms is compatible with cup-product, so the map (\ref{inftor}) splits as the direct sum of the maps \begin{equation}\label{inftorab} \rho^k_{\chi,\phi} :{\rm H}^1(X,T_X)^{(\chi)}\to \oplus_p{\rm Hom}({\rm H}^p(X,\Omega^{k-p}_X)^{(\phi)},{\rm H}^{p+1}(X, \Omega_X^{k-p-1})^{(\chi\phi)}) \end{equation} As we have remarked at the end of section \ref{covers}, if $f:X\to Y$ satisfies the assumption \ref{ampleness}, then the non invariant part of the Hodge structure is concentrated in the middle dimension $n$, and so we will only describe the IVHS for $k=n$. We use the notation of section \ref{covers} and moreover, in order to keep formulas readable, we set: $$T^{inv}=\HH{1}(Y,T_Y(-\log D));\quad U^{k,inv}=\HH{k}(Y,\Omega_Y^{n-k})$$ $$U^{k,\chi}=\HH{k}(Y,\Omega^{n-k}_Y(\log D_{\chi,\chi^{-1}})\!\otimes\! L_{\chi}^{-1}),\qquad k=0,\ldots n,\quad \chi\in G^*\setminus\{1\}.$$ Given a character $\chi\in G^*$, let $D_{i_1},\ldots D_{i_s}$ be the components of $D_{\chi,\chi^{-1}}$; let $P^{\chi}$ be the generalized prolongation bundle of\, $(D_{i_1},\ldots D_{i_s})$ (see section \ref{logarithmic}) and let $V^{\chi}=\oplus_j{\cal O}_Y(D_{i_j})$. Consider the map $(P^{\chi})^*\to V^{\chi}$ defined in sequence (\ref{tlog}); tensoring this map with $S^{k-1}(V^{\chi})$ and composing with the symmetrization map $S^{k-1}(V^{\chi})\!\otimes\! V^{\chi}\to S^k(V^{\chi})$, one obtains a map: \begin{equation}\label{simm} S^{k-1}(V^{\chi})\!\otimes\! (P^{\chi})^*\to S^k(V^{\chi}) \end{equation} Given a line bundle $L$ on $Y$, we define $R^{k,\chi}_L$ to be the cokernel of the map: $$\HH{0}(Y,S^{k-1}(V^{\chi})\!\otimes\! (P^{\chi})^*\!\otimes\! L)\to\HH{0}(Y,S^k(V^{\chi})\!\otimes\! L),$$ obtained from (\ref{simm}) by tensoring with $L$ and passing to global sections. We set $R^{\chi}_L=\oplus_{k\ge 0}R^{k,\chi}_L$; for $L={\cal O}_Y$, $R^{\chi}=R^{\chi}_{{\cal O}_Y}$ is a graded ring and, in general $R^{\chi}_L$ is a module over $R^{\chi}$, that we call the {\em Jacobi module} of $L$. Moreover, if $L_1$ and $L_2$ are line bundles on $Y$, then there is an obvious multiplicative structure: $$R^{k,\chi}_{L_1}\!\otimes\! R^{h,\chi}_{L_2}\to R^{k+h,\chi}_{L_1\!\otimes\! L_2}.$$ In order to establish the relationship between the Jacobi modules and the IVHS of the cover $X$, we need some definitions. \begin{defn}\label{ipotesi} For a $G$-cover $f:X\to Y$ satisfying assumption \ref{ampleness}, let $\Gamma_f$ be the semigroup of ${\rm Pic}(Y)$ generated by the building data. We say that: \noindent $X$ has property $(A)$ iff $\HH{k}(Y,\Omega_Y^j\!\otimes\! L)=0$ and $\HH{k}(Y,\Omega_Y^j\!\otimes\!\omega_Y\!\otimes\! L)=0$ for $k>0$, $j\ge 0$, $L\in\Gamma_f\setminus\{0\}$; \noindent $X$ has property $(B)$ iff for $L_1$, $L_2$ in $\Gamma_f\setminus\{0\}$ the mul\-ti\-pli\-ca\-tion map $\HH{0}(Y,\omega_Y\!\otimes\! L_1)\otimes\HH{0}(Y,\omega_Y\!\otimes\! L_2)\to \HH{0}(Y,\omega_Y^2\!\otimes\! L_1\!\otimes\! L_2)$ is surjective. \end{defn} \begin{rem}\label{annullamento} If $M$ is an ample line bundle such that $\HH{k}(Y,\Omega_Y^j\!\otimes\! M)=0$ for $k>0$ and $j\ge 0$, then the cohomology groups $\HH{k}(Y,\wedge^jP\!\otimes\! M^{-1})$ vanish for $j\ge 0$ and $k<n$. \end{rem} {\bf Proof:\,} By Serre duality, it is equivalent to show that $\HH{r}(Y,\wedge^jP^*\!\otimes\! M\!\otimes\! \omega_Y)=0$ for $r>0$. In turn, this can be proven by induction on $j$, by looking at the hypercohomology of the complex obtained by applying (\ref{complex}) to the sequence $0\to \oplus_i{\cal O}_Y\to P^*\to T_Y\to 0$. \ $\Box$\par\smallskip We also introduce the following \begin{nota} Let $L$ and $M$ be line bundles on the smooth variety $Y$; if $L\otimes M^{-1}$ is ample, then we write $L>M$ and, if $L\!\otimes\! M^{-1}$ is nef, then we write $L\ge M$. We use the same notation for divisors. \end{nota} \begin{rem}\label{projective} Properties $(A)$ and $(B)$ are easily checked for coverings of certain varieties $Y$; for instance, if\/ $Y={\bf P}^n$, then by Bott vanishing theorem it is enough to require that $\omega_Y\!\otimes\! L_{\chi}>0$ and $\omega_Y(D_i)>0$ for every $\chi\ne 1$ and for $i=1,\ldots r$. \end{rem} The next proposition yields an effective criterion for $(A)$ and $(B)$ in case $Y$ is an arbitrary variety. \begin{prop}\label{effective} Let $f:X\to Y$ be a $G$-cover satisfying assumption \ref{ampleness} and let $E$ be a very ample divisor on $Y$. Define $c(n)=\left(\!\!\begin{array}{c} n-1\\ (n-1)/2\end{array}\!\!\right)$ if $n$ is odd and $c(n)= \left(\!\!\begin{array}{c} n-1 \\ n/2\end{array}\!\!\right)$ if $n$ is even, and set $E_n=\left(\omega_Y(2nE)\right)^{c(n)}$. i) if $D_i$, $L_{\chi}$, $\omega_Y(D_i)$, $\omega_Y\!\otimes\! L_{\chi}>E_n$ for $\chi\ne 1$ and for $i=1,\ldots r$, then $(A)$ is satisfied. ii) if $L_{\chi}$ and $D_i\ge (n+1)E$ for $\chi\ne 1$ and for $i=1,\ldots r$, then $(B)$ is satisfied. \end{prop} {\bf Proof:\,} The complete linear system $|E|$ embeds $Y$ in a projective space ${\bf P}$; since $\Omega_{{\bf P}}^{j}(j+1)$ is generated by global sections, the sheaf $W^j=\Omega_Y^j((j+1)E)$, being a quotient of the former bundle, is also generated by global sections. By Griffiths vanishing theorem (\cite{SS}, theorem 5.52), if $N$ is an ample line bundle, then the cohomology group $\HH{k}(Y,W^j\!\otimes\!\omega_Y\!\otimes\! \det(W^j)\!\otimes\! N)$ vanishes for $k>0$. We recall from adjunction theory that $\omega_Y((n+1)E)$ is base point free and therefore nef; using this fact, it is easy to check that $E_n\ge\det(W^j)$ for $j\ge 0$. Statement i) now follows immediately from Griffiths vanishing. In order to prove ii), set $V_1=\HH{0}(Y,\omega_Y\!\otimes\! L_1)$. The assumptions imply that $V_1$ is base-point free, so evaluation of sections gives the following short exact sequence of locally free sheaves: $0\to K_1\to{\cal O}_Y\!\otimes\! V_1\to \omega_Y\!\otimes\! L_1\to 0$. Twisting with $\omega_Y\!\otimes\! L_2$ and passing to cohomology, one sees that the statement follows if $\HH{1}(Y,K_1\!\otimes\!\omega_Y\!\otimes\! L_2)=0$. In turn, this is precisely case $k=q=1$ of theorem $2.1$ of \cite{el}. \ $\Box$\par\smallskip \begin{lem}\label{iso} Let $f:X\to Y$ be a $G$-cover satisfying property $(A)$ and let $L$ be a line bundle on $Y$; if $L$ or $L\!\otimes\!\omega_Y^{-1}$ belong to $\Gamma_f\setminus\{0\}$, then for every $\chi\in G^*\setminus\{1\}$ there is a natural isomorphism: $$\HH{k}(Y,\Omega_Y^{n-k}(\log D_{\chi,\chi^{-1}})\!\otimes\! L^{-1})\cong (R^{n-k,\chi}_{\omega_Y\!\otimes\! L})^*$$ In particular, there are natural isomorphisms: $$T^{inv}\cong(R^{n-1,1}_{\omega_Y^2(D)})^*;\qquad U^{k,\chi}\cong (R^{n-k,\chi}_{\omega_Y\!\otimes\! L_{\chi}})^*,\quad \chi\ne 1 $$ $$\HH{1}\left(Y,\Omega^{n-1}_Y(\log D_{\chi,\chi^{-1}})\!\otimes\!(L_{\chi}\!\otimes\! L_{\phi^{-1}}\!\otimes\!\omega_Y)^{-1}\right)\cong (R^{n-1,\chi}_{\omega_Y^2\!\otimes\! L_{\chi}\!\otimes\! L_{\phi^{-1}}})^*,\quad\chi\ne 1.$$ \end{lem} {\bf Proof:\,} For $k=n$, the statement is just Serre duality. For $k<n$, we compute $\HH{k}\left(Y,\Omega_Y^{n-k}(\log D_{\chi,\chi^{-1}})\!\otimes\! L^{-1}\right)$ by ten\-sor\-ing the res\-o\-lu\-tion (\ref{reslog}) of the sheaf $\Omega_Y^{n-k}(\log D_{\chi,\chi^{-1}})$ with $L^{-1}$ and breaking up the resolution thus obtained into short exact sequences. Remark \ref{annullamento} and theorem \ref{kodaira} imply that the cohomology groups $\HH{n-k+j}(Y,S^j(V_{\chi})^*\!\otimes\! L^{-1}\!\otimes\! \wedge^{n-k-j}P)$ vanish for $0\le j<n-k$; thus the group $\HH{k}(Y,\Omega_Y^{n-k}(\log D_{\chi,\chi^{-1}})\!\otimes\! L^{-1})$ and the kernel of the map $\HH{n}\left(Y,S^{n-k}(V^{\chi})^*\!\otimes\! L^{-1}\right)\to\HH{n}\left(Y,S^{n-k-1}(V^{\chi})^*\!\otimes\! P^{\chi}\!\otimes\! L^{-1}\right)$ are naturally isomorphic. By Serre duality, the latter group is dual to $R^{n-k,\chi}_{\omega_Y\!\otimes\! L}$. \ $\Box$\par\smallskip The next result is the analogue in our setting of Macaulay's duality theorem. \begin{prop}\label{macaulay} Assume that the cover $f:X\to Y$ satisfies property $(A)$. For $\chi\in G^*\setminus\{1\}$, set\, $\omega_{\chi}=\omega_Y^2(D_{\chi,\chi^{-1}})$\,; then: i) there is a natural isomorphism $R^{n,\chi}_{\omega_{\chi}}\cong {\bf C}$ ii) let $L$ be a line bundle on $Y$ such that $L$ and $L^{-1}(D_{\chi,\chi^{-1}})$ (or$L\!\otimes\! \omega_Y^{-1}$ and $(\omega_Y\!\otimes\! L)^{-1}(D_{\chi,\chi^{-1}})$) belong to $\Gamma_f\setminus\{0\}$; then the multiplication map $R^{k,\chi}_{\omega_Y\!\otimes\! L}\otimes R^{n-k,\chi}_{\omega_Y\!\otimes\! L^{-1}(D_{\chi,\chi^{-1}})}\to R^{n,\chi}_{\omega_{\chi}}$ is a perfect pairing, corresponding to Serre duality via the isomorphism of lemma \ref{iso}. In particular, one has natural isomorphisms: $$U^{k,\chi}\cong R^{k,\chi}_{\omega_Y\!\otimes\! L_{\chi^{-1}}}.$$ \end{prop} {\bf Proof:\,} In order to prove i), consider the complex (\ref{reslog}) for $k=n$: twisting it by $\omega_Y(D_{\chi,\chi^{-1}})^{-1}$ and arguing as in the proof of lemma \ref{iso}, one shows the existence of a natural isomorphism between $R^{n,\chi}_{\omega_{\chi}}$ and $\HH{0}(Y,{\cal O}_Y)={\bf C}$. In order to prove statement ii), one remarks that the group $\HH{k}(Y,\Omega_Y^{n-k}(\log D_{\chi,\chi^{-1}})\!\otimes\! L^{-1})$ is Serre dual to $\HH{n-k}(Y,\Omega^k_Y(\log D_{\chi,\chi^{-1}})\!\otimes\! L(-D_{\chi,\chi^{-1}}))$ by (\ref{dualita'}). By lemma \ref{iso} the latter group equals $( R^{k,\chi}_{\omega_Y\!\otimes\! L^{-1}(D_{\chi,\chi^{-1}})})^*$. Both these isomorphisms and the multiplication map are natural, and therefore compatible with Serre duality. The last claim follows in view of (\ref{fundrelbis}). \ $\Box$\par\smallskip \section{Infinitesimal Torelli}\label{infinitesimal} \setcounter{defn}{0} \setcounter{equation}{0} In this section we exploit the algebraic description of the IVHS of a $G$-cover to prove an infinitesimal Torelli theorem. We will use freely the notation introduced in section \ref{ivhs}. We recall that {\em infinitesimal Torelli} for the periods of $k$-forms holds for a variety $X$ if the map (\ref{inftor}) is injective. By the remarks at the beginning of section \ref{ivhs}, a $G$-cover $f:X\to Y$ satisfies infinitesimal Torelli property if for each character $\chi\in G^*$ the intersection, as $\phi$ varies in $G^*$, of the kernels of the maps $\rho^k_{\chi,\phi}$ of (\ref{inftorab}) is equal to zero. The next theorem shows that this is actually the case for $k=n$, under some ampleness assumptions on the building data of $f:X\to Y$. \begin{thm}\label{mainthm1} Let $X$, $Y$ be smooth complete algebraic varieties of dimension $n\geq 2$ and let $f:X\to Y$ be a $G$-cover with building data $L_{\chi}$, $D_i$, $\chi\in G^*\setminus\{1\}$, $i=1,\ldots r$. If properties $(A)$ and $(B)$ are satisfied, then the following map is injective: $${\rm H}^1(X,T_X)\to {\rm Hom} ({\rm H}^0(X,\omega_X),{\rm H}^1(X,\Omega_X^{n-1})),$$ and, as a consequence, infinitesimal Torelli for the periods of $n$-forms holds for $X$. \end{thm} Before giving the proof, we deduce two effective results from theorem \ref{mainthm1}. \begin{thm}\label{mainthm1p} Let $f:X\to{\bf P}^n$, $n\ge 2$, be a $G$-cover with building data $L_{\chi}$, $D_i$, $\chi\in G^*\setminus\{1\}$, $i=1,\ldots r$. Assume that $L_{\chi}\!\otimes\!\omega_{{\bf P}^n}>0$ and $D_i\!\otimes\!\omega_{{\bf P}^n}>0$ for $\chi\in G^*\setminus\{1\}$, $i=1,\ldots r$; then infinitesimal Torelli for the periods of $n$-forms holds for $X$. \end{thm} {\bf Proof:\,}: by remark \ref{projective}, properties $(A)$ and $(B)$ are satisfied in this case. \ $\Box$\par\smallskip \begin{thm}\label{mainthm1e} Let $X$, $Y$ be smooth complete algebraic varieties of dimension $n\geq 2$ and let $f:X\to Y$ be a $G$-cover with building data $L_{\chi}$, $D_i$, $\chi\in G^*\setminus\{1\}$, $i=1,\ldots r$. Let $E$ be a very ample divisor on $Y$ and let $E_n$ be defined as in prop. \ref{effective}; if $D_i, L_{\chi}, \omega_Y(D_i), \omega_Y\!\otimes\! L_{\chi}>E_n$ and $L_{\chi}, D_i\ge(n+1)E$ for $\chi\in G^*\setminus\{1\}$, $i=1,\ldots r$,\/then infinitesimal Torelli for the periods of $n$-forms holds for $X$. \end{thm} {\bf Proof:\,} Follows from theorem \ref{mainthm1} together with proposition \ref{effective}. \ $\Box$\par\smallskip \noindent{\bf Proof of theorem \ref{mainthm1}:} Since all cohomology groups appearing in this proof are computed on $Y$, we will omit $Y$ from the notation. \noindent By proposition \ref{projform} and by the discussion at the beginning of the section, we have to show that for every $\chi\in G^*$ the intersection of the kernels of the maps $$\rho_{\chi,\phi}: {\rm H}^1(T_Y(-\log \Delta_{\chi})\!\otimes\! L_{\chi}^{-1})\to{\rm Hom}(U^{0,\phi},U^{1,\chi\phi}),$$ as $\phi$ varies in $G^*$, is equal to zero. For $\chi,\phi\in G^*$, set $R_{\chi,\phi}=D_{\chi\phi,(\chi\phi)^{-1}}-D_{\chi\phi,\phi^{-1}}$. Notice that $R_{\chi,\phi}$ is effective. By (\ref{dualita'}) and (\ref{fundrel}) there is a natural identification: $$\Omega_Y^{n-1}(\log D_{\chi\phi,(\chi\phi)^{-1}})\!\otimes\!(\omega_Y\!\otimes\! L_{\chi\phi}\!\otimes\! L_{\phi^{-1}})^{-1} = T_Y(-\log D_{\chi\phi,(\chi\phi)^{-1}})\!\otimes\! L_{\chi}^{-1} (R_{\chi,\phi}).$$ So the map $\rho_{\chi,\phi}$ can be viewed as the composition of the map $$i_{\chi,\phi}:{\rm H}^1(T_Y(-\log \Delta_{\chi})\!\otimes\! L_{\chi}^{-1}) \to {\rm H}^1(T_Y(-\log D_{\chi\phi,(\chi\phi)^{-1}}) \!\otimes\! L_{\chi}^{-1} (R_{\chi,\phi})),$$ induced by inclusion of sheaves, and of the map $$r_{\chi\phi,\phi}:{\rm H}^1(\Omega_Y^{n-1}(\log D_{\chi\phi,(\chi\phi)^{-1}})\!\otimes\!(\omega_Y\!\otimes\! L_{\chi\phi}\!\otimes\! L_{\phi^{-1}})^{-1}) \to {\rm Hom}(U^{0,\phi},U^{1,\chi\phi})$$ induced by cup-product. Arguing as in the proof of thm. $3.1$ of \cite{torelli}, one can show that, for fixed $\chi\in G^*$, the intersection of the kernels of $i_{\chi,\phi}$, as $\phi$ varies in $G^*\setminus\{1,\chi^{-1}\}$, is zero. (Notice that lemma $3.1$ of \cite{torelli}, although stated for surfaces, actually holds for varieties of any dimension, and that the ampleness assumptions on the building data allow one to apply it, in view of thm. \ref{kodaira}.) So the statement will follow if we prove that the map $r_{\chi,\phi}:{\rm H}^1(\Omega^{n-1}_Y(\log D_{\chi,\chi^{-1}})\!\otimes\!(\omega_Y\!\otimes\! L_{\chi}\!\otimes\! L_{\phi^{-1}})^{-1}) \to$ ${\rm Hom}(U^{0,\phi}, U^{1,\chi})$ is injective for every pair $\chi$, $\phi$ of nontrivial characters. By lemma \ref{iso}, the map $r_{\chi,\phi}$ may be rewritten as: $r_{\chi,\phi}:(R^{n-1,\chi}_{\omega_Y^2\!\otimes\! L_{\chi}\!\otimes\! L_{\phi^{-1}}})^*\to (R^{0,\chi}_{\omega_Y\!\otimes\! L_{\phi^{-1}}})^*\otimes (R^{n-1,\chi}_{\omega_Y\!\otimes\! L_{\chi}})^*$. We prove that $r_{\chi,\phi}$ is injective by showing that the dual map $r_{\chi,\phi}^*:R^{0,\chi}_{\omega_Y\!\otimes\! L_{\phi^{-1}}}\otimes R^{n-1,\chi}_{\omega_Y\!\otimes\! L_{\chi}}\to R^{n-1,\chi}_{\omega_Y^2\!\otimes\! L_{\chi}\!\otimes\! L_{\phi^{-1}}}$, induced by multiplication, is surjective. In order to do this, it is sufficient to observe that the multiplication map $$\HH{0}(Y,\omega_Y\!\otimes\! L_{\phi^{-1}})\!\otimes\! \HH{0}(Y,S^{n-1}(V^{\chi})\!\otimes\!\omega_Y\!\otimes\! L_{\chi})\to\HH{0}(Y,S^{n-1}(V^{\chi})\!\otimes\!\omega_Y^2\!\otimes\! L_{\phi^{-1}}\!\otimes\! L_{\chi})$$ is surjective by property $(B)$. \ $\Box$\par\smallskip \begin{rem} In section 6 of \cite{moduli}, it is proven that for any abelian group $G$ there exist families of smooth $G$-covers with ample canonical class that are {\em generically complete}. In those cases, thm. \ref{mainthm1} means that the period map is \'etale on a whole component of the moduli space. \end{rem} \section{Sufficiently ample line bundles} \setcounter{defn}{0} \setcounter{equation}{0} We take up the following definition from \cite{suffampio} \begin{defn} A property is said to hold for a {\em sufficiently ample} line bundle $L$ on the smooth projective variety $Y$ if there exists an ample line bundle $L_0$ such that the property holds whenever the bundle $L\!\otimes\! L_0^{-1}$ is ample. We will denote this by writing that the property holds for $L>>0$. \end{defn} In this section, we collect some facts about sufficiently ample line bundles that will be used to prove our variational Torelli result. In particular, we prove a variant of proposition $5.1$ of \cite{ciclico} to the effect that, given sufficiently ample line bundles $L_1$ and $L_2$ on $Y$, it is possible to recover $Y$ from the kernel of the multiplication map ${\rm H}^0(Y,L_1)\!\otimes\! {\rm H}^0(Y,L_2)\to{\rm H}^0(Y,L_1\!\otimes\! L_2)$. The next lemma is ``folklore''. The proof given here has been communicated to the author by Mark Green. \begin{lem}\label{green} Let ${\cal F}$ be a coherent sheaf on $Y$ and let $L$ be a line bundle on $Y$. Then, if $L>>0$, $${\rm H}^i(Y,{\cal F}\!\otimes\! L)=0, \quad i>0.$$ \end{lem} {\bf Proof:\,} We proceed by descending induction on $i$. If $i>n$, then the statement is evident. Otherwise, fix an ample divisor $E$ and an integer $m$ such that ${\cal F}(mE)$ is generated by global sections. This gives rise to an exact sequence: $0\to{\cal F}_1\to \oplus {\cal O}_Y(-mE)\to {\cal F}\to 0$. Tensoring with $L$ and considering the corresponding long cohomology sequence, one sees that it is enough that ${\rm H}^i(Y,L(-mE))={\rm H}^{i+1}(Y,{\cal F}_1\!\otimes\! L)=0$, for $L>>0$. The vanishing of the former group follows from Kodaira vanishing and the vanishing of the latter follows from the inductive hypothesis. \ $\Box$\par\smallskip \begin{lem}\label{genproj} Let $|E|$ be a very ample lin\-ear sys\-tem on the smooth projective variety $Y$ of dimension $n$. Denote by ${\cal M}_y$ the ideal sheaf of a point $y\in Y$: if $L$ is a sufficiently ample line bundle on $Y$, then the multiplication map: $${\rm H}^0(Y,L)\!\otimes\!{\rm H}^0(Y,{\cal M}_y(E))\to{\rm H}^0(Y,{\cal M}_y(L+E))$$ is surjective for every $y\in Y$. \end{lem} {\bf Proof:\,} For $y\in Y$, set $V_y={\rm H}^0(Y,{\cal M}_y(E))$ and consider the natural exact sequence $0\to N_y\to V_y\!\otimes\!{\cal O}_Y\to{\cal M}_y(E)\to 0$; tensoring with $L$ and considering the corresponding cohomology sequence, one sees that if ${\rm H}^1(Y,N_y\!\otimes\! L)=0$ then the map ${\rm H}^0(Y,L)\!\otimes\! V_y\to{\rm H}^0(Y,{\cal M}_y(L+E))$ is surjective. By lemma \ref{green}, there exists an ample line bundle $L_y$ such that ${\rm H}^1(Y,N_y\!\otimes\! L)=0$ if $L> L_y$. In order to deduce from this the existence of a line bundle $L_0$ such that ${\rm H}^1(Y,N_y\!\otimes\! L)=0$ for every $y\in Y$ if $L> L_0$, we proceed as follows. Consider the product $Y\times Y$, with projections $p_i$, $i=1,2$, denote by ${\cal I}_{\Delta}$ the ideal sheaf of the diagonal in $Y\times Y$ and set ${\cal V}=p_{2*}(p_1^*{\cal O}_Y(E)\!\otimes\! {\cal I}_{\Delta})$. ${\cal V}$ is a fibre bundle on $Y$ such that the fibre of ${\cal V}$ at $y$ can be naturally identified with $V_y$. We define the sheaf ${\cal N}$ on $Y\times Y$ to be the kernel of the map $p_2^*{\cal V}\to p_1^*{\cal O}_Y(E)\!\otimes\! {\cal I}_{\Delta}$; the restriction of ${\cal N}$ to $p_2^{-1}(y)$ is precisely $N_y$. For any fixed line bundle $L$ on $Y$, $h^1(Y,N_y\!\otimes\! L)=h^1(p_2^{-1}(y),{\cal N}\!\otimes\! p_1^*L|_{p_2^{-1} (y)})$ is an upper-semicontinuous function of $y$. Thus, we may find a finite open covering $U_1,\ldots U_k$ of $Y$ and ample line bundles $L_1,\ldots L_k$ such that $h^1(Y,N_y\!\otimes\! L)=0$ for $y\in U_i$ if $L\!\otimes\! L_i^{-1}$ is ample. To finish the proof it is enough to set $L_0=L_1\!\otimes\!\ldots\!\otimes\! L_k$. \ $\Box$\par\smallskip \begin{lem}\label{veryample} Let $Y$ be a smooth projective variety of dimension $n\ge 2$, $Y\ne{\bf P}^n$. If $E$ is a very ample divisor on $Y$, then: i) if $L>\omega_Y((n-1)E)$, then $L$ is base point free. ii) if $L>\omega_Y(nE)$, then $L$ is very ample. \end{lem} {\bf Proof:\,} In order to prove the claim, it is enough to show that if $C$ is a smooth curve on $Y$ which is the intersection of $n-1$ divisors of $|E|$, then $L|_C$ is base point free (very ample) and $|L|$ restricts to the complete linear $L|_C$. In view of the assumption, the former statement follows from adjunction on $Y$ and Riemann-Roch on $C$, and the latter follows from the vanishing of ${\rm H}^1(Y,{\cal I}_C\!\otimes\! L)$, where ${\cal I}_C$ is the ideal sheaf of $C$. In turn, this vanishing can be shown by means of the Koszul complex resolution: $$0\to\L((1-n)E)\to L\!\otimes\!\wedge^{n-2}\left(\oplus_1^{n-1}{\cal O}_Y (-E)\right)\ldots\to \oplus_1^{n-1}L(-E)\to {\cal I}_C\!\otimes\! L\to 0.$$ \ $\Box$\par\smallskip The following proposition is a variant for sufficiently ample line bundles of prop. $5.1$ of \cite{ciclico}. Our statement is weaker, but we do not need to assume that one the line bundles involved is much more ample than the other one. \begin{prop}\label{11} Let $L_1$ and $L_2$ be very ample line bundles on a smooth projective variety $Y$. Let $\phi_1:Y\to{\bf P}_1$ and $\phi_2:Y\to{\bf P}_2$ be the corresponding embeddings into projective space, and let $f:Y\to{\bf P}_1\times{\bf P}_2$ be the composition of the diagonal embedding of $Y$ in $Y\times Y$ with the product map $\phi_1\times\phi_2$. If $L_1,L_2>>0$, then $f(Y)$ is the zero set of the elements of ${\rm H}^0({\bf P}_1\times{\bf P}_2,{\cal O}_{{\bf P}_1\times{\bf P}_2}(1,1))$ vanishing on it. \end{prop} {\bf Proof:\,} By \cite{ciclico}, prop. 5.1, we may find very ample line bundles $M_i$, $i=1,2$ such that, if $\psi_i:Y\to {\bf Q}_i$ are the corresponding embeddings in projective space and $g:Y\to{\bf Q}_1\times{\bf Q}_2$ is the composition of the diagonal embedding of $Y$ in $Y\times Y$ with $\psi_1\times\psi_2$, then $g(Y)$ is the scheme-theoretic intersection of the elements of ${\rm H}^0({\bf Q}_1\times{\bf Q}_2,{\cal O}_{{\bf Q}_1\times{\bf Q}_2}(1,1))$ vanishing on it. By prop. \ref{veryample}, we may assume that $L_i\!\otimes\! M_i^{-1}$ is very ample, $i=1,2$. To a divisor $D_i$ in $|L_i\!\otimes\! M_i^{-1}|$ there corresponds a projection $p_{D_i}:{\bf P}_i\cdots\to {\bf Q}_i$ such that $\psi_i=p_{D_i}\circ \phi_i$. The claim will follow if we show that for $(x_1,x_2)\notin f(Y)$ one can find $D_i\in |L_i\!\otimes\! M_i^{-1}|$ such that $p_{D_i}$ is defined at $x_i$, $i=1,2$, and $(p_{D_1}(x_1), p_{D_2}(x_2))\notin g(Y)$. In fact, this implies that there exists $s\in {\rm H}^0({\bf Q}_1\times {\bf Q}_2,{\cal O}_{{\bf Q}_1\times{\bf Q}_2}(1,1))$ that vanishes on $g(Y)$ and does not vanish at $(p_{D_1}(x_1), p_{D_2}(x_2))$, so that the pull-back of $s$ via $p_{D_1}\times p_{D_2}$ is a section of ${\rm H}^0({\bf P}_1\times {\bf P}_2,{\cal O}_{{\bf P}_1\times{\bf P}_2}(1,1))$ that vanishes on $f(Y)$ and does not vanish at $(x_1,x_2)$. By lemma \ref{genproj}, if $L_i>>0$, then the multiplication map ${\rm H}^0(Y,L_i\!\otimes\! M_i^{-1})\!\otimes\!{\rm H}^0(Y,M_i\!\otimes\!{\cal M}_y)\to{\rm H}^0(Y,L_i\!\otimes\!{\cal M}_y)$ is surjective $\forall y\in Y$ and for $i=1,2$. Notice that, in particular, this implies that the map ${\rm H}^0(Y,L_i\!\otimes\! M_i^{-1})\!\otimes\! {\rm H}^0(Y,M_i)\to\HH{0}(Y,M_i\!\otimes\! L_i)$ is surjective for $i=1,2$, so that, for a generic choice of $D_i$, the projection $p_{D_i}$ is defined at $x_i$. If either $(x_1,x_2)\in \phi_1(Y)\times\phi_2(Y)$ or there exists a divisor $D_i$, for $i=1$ or $i=2$, such that $p_{D_i}(x_i)\notin\psi_i(Y)$, then we are set. So assume that, say, $x_1\notin \phi_1(Y)$ and that $p_{D_i}(x_i)\in\psi_i(Y)$, for a generic choice of $D_i$, for $i=1,2$. Fix a divisor $D_2$ such that $p_{D_2}$ is defined at $x_2$ and write $p_{D_2}(x_2)=\psi_2(y_2)$, with $y_2\in Y$. Now it is enough to show that there exists $D_1$ such that $p_{D_1}$ is defined at $x_1$ and $p_{D_1}(x_1)\ne \psi_1(y_2)$. Since the multiplication map ${\rm H}^0(Y,L_1\!\otimes\! M_1^{-1})\!\otimes\!{\rm H}^0(Y,M_1\!\otimes\!{\cal M}_{y_2})\to{\rm H}^0(Y,L_1\!\otimes\!{\cal M}_{y_2})$ is surjective, there exist $\sigma\in {\rm H}^0(Y,L_1\!\otimes\! M_1^{-1})$ and $\tau \in {\rm H}^0(Y,M_1\!\otimes\!{\cal M}_{y_2})$ such that $\sigma\tau$ corresponds to a hyperplane of ${\bf P}_1$ passing through $\phi_1(y_2)$ but not through $x_1$. If $D_1$ is the divisor of $\sigma$, then the projection $p_{D_1}$ is defined at $x_1$ and $p_{D_1}(x_1)\ne\psi_1(y_2)$. \ $\Box$\par\smallskip \section{A variational Torelli theorem} \setcounter{defn}{0} \setcounter{equation}{0} In this section we prove a variational Torelli theorem for the family ${\cal X}\to W$ of abelian covers with fixed basis and fixed $L_{\chi}$'s. Let ${\cal Z}\to B$ be a flat family of smooth projective polarized varieties on which $G$ acts fibrewise, and let $Z$ be the fibre of ${\cal Z}$ over the point $0\in B$. It is possible to show that the monodromy action on the cohomology of $X$ preserves the group action; therefore one may define a {\em $G$-period map}, by dividing the period domain $D$ by the subgroup of linear transformations that, beside preserving the integral lattice and the polarization, are compatible with the $G$-action. In particular, let $f:X\to Y$ be a $G$-cover, with $X$ and $Y$ smooth projective varieties of dimension $n\ge 2$ and with building data $L_{\chi}$, $D_i$. In section \ref{covers}, we have introduced the family ${\cal X}\to W$ of $G$-covers of $Y$, obtained by letting the sections $s_i\in\HH{0}(Y,{\cal O}_Y(D_i))$ vary in equations (\ref{equazioni}). There is an obvious $G$-action on ${\cal X}$, and the choice of an ample divisor $E$ on $Y$ gives a $G$-invariant polarization of ${\cal X}$. Since two elements $(s_1,\ldots s_r)$ and $(s'_1,\ldots s'_r)$ of $W$ represent the same $G$-cover iff there exist $\lambda_i\in{\bf C}^*$ such that $s'_i=\lambda_is_i$, the $G$-period map can be regarded as being defined on the image $\tilde{W}$ of $W$ in $|D_1|\times\cdots\times|D_r|$. We denote by $s$ the image in $\tilde{W}$ of the point $(s_1,\ldots s_r)$, by $X_s$ the corresponding $G$-cover of $Y$, by $T_s$ the tangent space to $\tilde{W}$ at $s$ and, consistently with the notation of section \ref{ivhs}, by $U_s^{k,\chi}$ the subspace of $\HH{k}(X_s,\Omega^{n-k}_{X_s})$ on which $G$ acts via the character $\chi$. Remark that the space $T_s$ can be naturally identified with $\oplus_i\HH{0}(Y,{\cal O}_Y(D_i))/(s_i)$. Denote by $\Gamma$ the subgroup of $GL(T_s)\times GL(\HH{0}(X_s,\omega_{X_s}))\times GL(\HH{1}(X_s,\Omega^{n-1}_{X_s}))$ that preserves the $G$-action on the cohomology of $X_s$. \begin{thm}\label{mainthm2} Assume that the dimension $n$ of $Y$ is $\ge2$, that the canonical class $\omega_Y$ of $Y$ is very ample and that $L_{\chi}$, $D_i>>0$, $\chi\ne 1$, $i=1,\ldots r$. Assume that for $i=1,\ldots r$ the identity is the only automorphism of $Y$ that preserves the linear equivalence class of $D_i$; moreover assume that $\forall i=1,\ldots r$ there exists $\chi\in G^*$ (possibly depending on $i$) such that $\chi(g_i)\ne 1$ and $L_{\chi}>D_i$. Then a generic point $s\in\tilde{W}$ is determined by the $\Gamma$-class of the linear map: $$T_s\to\oplus_{\chi}{\rm Hom}(U^{0,\chi}_s,U^{1,\chi}_s)),$$ which represents the first piece of the algebraic part of the IVHS for $n$-forms. \end{thm} \begin{rem} The assumption, made in thm. \ref{mainthm2} and corollary \ref{mainthm2bis}, that for every $i=1,\ldots r$ there exists $\chi\in G^*$ such that $\chi(g_i)\ne 1$ and $L_{\chi}>D_i$ is not satisfied by simple cyclic covers, namely totally ramified covers branched on an irreducible divisor. Still there are many cases in which our results apply: for instance, construction $6.2$ of \cite{moduli} provides examples with $G={\bf Z}/_{m_1}\times\cdots\times{\bf Z}/_{m_{r-1}}$, $r-1\ge n$, branched on $r$ algebraically equivalent divisors $D_1,\dots D_r$. The ramification order over $D_i$ is equal to $m_i$ for $i<r$, and it is equal to the least common multiple of the $m_i$'s for $i=r$. (The assumption, made in \cite{moduli}, that $m_i|m_{i+1}$ is actually unnecessary in order to make the construction.) Apart from the case $r=m_1=m_2=2$, if the branch divisors are ample, then for every $i$ there exists $\chi$ such that $L_{\chi}(-D_i)$ is ample. \end{rem} Before giving the proof of thm. \ref{mainthm2}, we state: \begin{cor}\label{mainthm2bis} Under the same assumptions as in theorem \ref{mainthm2} the $G$-period map for $n$-forms has degree $1$ on $\tilde{W}$. \end{cor} {\bf Proof:\,} Let ${\cal X}\to B$ be a family of polarized varieties and let $X_b$ be the fibre of ${\cal X}$ over a point $b\in B$ and assume that, if there is an isomorphism of the IVHS's of $X_b$ and $X_{b'}$ preserving the polarization and the real structure, then the varieties $X_b$ and $X_{b'}$ are isomorphic ( this property is usually expressed by saying that ``variational Torelli holds''). In \cite{cdt} it is proven that in this case the period map is generically injective on $B$ up to isomorphism of varieties. Using exactly the same arguments, one can show that if $G$ acts on the family ${\cal X}\to B$ fibrewise and if the IVHS of a fibre $X_b$ determines $X_b$ up to isomorphisms preserving the $G$-action, then the $G$-period map is generically injective on $B$. \ $\Box$\par\smallskip \noindent{\bf Proof of theorem \ref{mainthm2}:} Whenever confusion is not likely to arise, we omit to write the space where cohomology groups are computed. By theorem $0.3$ of \cite{suffampio}, if $D_i>>0$ for $i=1,\ldots r$, then the period map $P$ for $n-1$ forms has degree $1$ on $|D_i|$. Denote by $U_i$ the open subset of $|D_i|$ consisting of the points $z$ such that $P^{-1}(P(z))=\{z\}$, and fix $s\in \tilde{W}\cap\left(U_1\times\ldots\times U_r\right)$. From now on we will drop the subscript $s$ and write $X$ for $X_s$, $T$ for $T_s$, and so on. For each index $i\in\{1,\ldots r\}$ set $S_i=\{\chi\in G^*|\chi(g_i)=1\}$; as a first step, we show that the subspace $\HH{0}({\cal O}_Y(D_i))/(s_i)$ of $T$ is the intersection of the kernels of the maps $\rho_{\chi}:T\to {\rm Hom}(U^{0,\chi},U^{1,\chi})$, as $\chi$ varies in $S_i\setminus\{1\}$. Since $\omega_Y$ is ample, $\HH{0}(T_Y)=0$ by thm. \ref{kodaira}, and so $T$ equals $R^{1,1}_{{\cal O}_Y}$. The map $\rho_{\chi}$ factors through the surjection $R^{1,1}_{{\cal O}_Y}\to R^{1,\chi}_{{\cal O}_Y}$, whose kernel is $\oplus_{\{i|\chi(g_i)= 1\}}\HH{0}(Y,{\cal O}_Y(D_i))/(s_i)$. In turn, by sequence \ref{tlog}, $R^{1,\chi}_{{\cal O}_Y}$ injects in $\HH{1}(Y,T_Y(-\log D_{\chi,\chi^{-1}}))$. We have shown in the proof of thm. \ref{mainthm1} that the map $\HH{1}(T_Y(-\log D_{\chi,\chi^{-1}}))\to {\rm Hom}(U^{0,\chi},U^{1,\chi})$ is injective. So $\ker\rho_{\chi}=\oplus_{\{i|\chi(g_i)= 1\}}\HH{0}(Y,{\cal O}_Y(D_i))/(s_i)$. As we have remarked in section \ref{covers}, if $i\ne j$, then the subgroups of $G$ generated by $g_i$ and $g_j$ intersect only in $\{0\}$, and so $i$ is the only index such that $\chi(g_i)=1$ for all $\chi\in S_i$. We conclude that $\cap_{\chi\in S_i\setminus\{1\}}\ker\rho_{\chi}=\HH{0}(Y,{\cal O}_Y(D_i))/(s_i)$. Now fix $i\in\{1,\ldots r\}$ and let $\chi\in G^*\setminus S_i$ be such that $L_{\chi}>D_i$: the restriction to $\HH{0}({\cal O}_Y(D_i))/(s_i)$ of the map $T\!\otimes\! U^{0,\chi^{-1}}\to U^{1,\chi^{-1}}$ is the multiplication map $\HH{0}({\cal O}_Y(D_i))/(s_i)\otimes\HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})\to\HH{0}(Y,\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}$, followed by the inclusion $\HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}\to U^{1,\chi^{-1}}$. {\em Claim:} the kernel of the latter map is equal to zero. If we assume that the claim holds, then we have recovered the multiplication map $\HH{0}({\cal O}_Y(D_i))/(s_i)\otimes \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})\to \HH{0}(Y,\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}$. The right kernel of this map is $(s_i)\HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})(-D_i))$. So we can reconstruct the map $\HH{0}({\cal O}_Y(D_i))|_{D_i}\otimes \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})|_{D_i}\to \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}$. Let $\phi_{1}:Y\to{\bf P}_1$ be the embedding defined by the linear system $|D_i|$, let $\phi_{2}:Y\to{\bf P}_2$ be the embedding defined by the linear system $|\omega_Y\!\otimes\! L_{\chi^{-1}}|$ and let $f:Y\to{\bf P}_1\times{\bf P}_2$ be the composition of the diagonal embedding $Y\to Y\times Y$ with the product map $\phi_1\times\phi_2$: by prop. \ref{11}, $f(Y)$ is the zero set of the elements of the kernel of $\HH{0}({\cal O}_Y(D_i))\otimes\HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})\to \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))$. This implies that $f(D_i)$ the zero set in ${\bf P}\left(\HH{0}({\cal O}_Y(D_i))|_{D_i}\right)\times {\bf P}\left(\HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})|_{D_i}\right)$ of the elements of the kernel of $\HH{0}({\cal O}_Y(D_i))|_{D_i}\otimes \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}})|_{D_i}\to \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}$. Thus it is possible to recover $D_i$ as an abstract variety, for every $i=1,\ldots r$. Since $s\in U_1\times\cdots\times U_r$, this is enough to determine the $D_i$'s as divisors on $Y$ and, in turn, the point $s$. In order to complete the proof, we have to prove the claim. Let $D_{i_1},\ldots D_{i_s}$ be the components of $D_{\chi,\chi^{-1}}$. (Recall that there exists $j_0$ such that $i=i_{j_0}$.) Let $V=\oplus_j{\cal O}_Y(D_{i_j})$, let $V_i=\oplus_{j\ne j_0}{\cal O}_Y(D_{i_j})$, let $P$ be the generalized prolongation bundle associated to $(D_{i_1}, \ldots D_{i_s})$ and let $P_i$ be the generalized prolongation bundle associated to $(D_{i_1},\ldots\hat{D_i},\ldots D_{i_s})$. There is a natural short exact sequence $0\to P_i\to P\to{\cal O}_Y\to 0$, with dual sequence $0\to{\cal O}_Y\to P^*\to P_i^*\to 0$. From this and sequence \ref{tlog}, tensoring with $\omega_Y\!\otimes\! L_{\chi^{-1}}$ and taking global sections, one deduces the following commutative diagram with exact rows: $$\begin{array}{ccccccccc} 0 &\!\!\!\!\!\! \to\!\!\!\!\!\! & \HH{0}(\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!) & \!\!\!\!\!\! \to\!\!\!\!\!\! & \HH{0}\!(P^*\!\otimes\!\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!) &\!\!\!\!\!\!\to\!\!\!\!\!\! & \HH{0}\!(P_i^*\!\otimes\!\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!) &\!\!\!\!\!\! \to\!\!\!\!\!\! & 0\\ \phantom{1} & \phantom{1} &\downarrow &\phantom{1} & \downarrow & \phantom{1}& \downarrow & \phantom{1}& \phantom{1}\\ 0 & \!\!\!\!\!\! \to\!\!\!\!\!\! & \HH{0}\!(\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!(D_i)) & \!\!\!\!\!\! \to\!\!\!\!\!\! & \HH{0}\!(V\!\otimes\!\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!) &\!\!\!\!\!\!\to\!\!\!\!\!\! & \HH{0}\!(V_i\!\otimes\!\omega_Y\!\otimes\! L_{\chi\!^{-1}}\!) & \!\!\!\!\!\! \to\!\!\!\!\!\! & 0 \end{array}$$ In view of sequence \ref{tlog}, by applying snake's lemma to this diagram one obtains the following exact sequence: $\HH{0}(T_Y(-\log(D_{\chi,\chi^{-1}}-D_i))\!\otimes\! \omega_Y\!\otimes\! L_{\chi^{-1}})\to \HH{0}(\omega_Y\!\otimes\! L_{\chi^{-1}}(D_i))|_{D_i}\to U^{1,\chi^{-1}}$. So it is enough to show that $\HH{0}(T_Y(-\log(D_{\chi,\chi^{-1}}-D_i)\!\otimes\! L_{\chi^{-1}})=0$. Using the isomorphism (\ref{dualita'}) and the relations (\ref{fundrelbis}), one has $T_Y(-\log(D_{\chi,\chi^{-1}}-D_i))\!\otimes\!\omega_Y\!\otimes\! L_{\chi^{-1}}\cong \Omega^{n-1}_Y(\log(D_{\chi,\chi^{-1}}-D_i)\!\otimes\! L_{\chi}^{-1}(D_i)$. In view of the assumptions, the required vanishing now follows from thm. \ref{kodaira}. \ $\Box$\par\smallskip

9507

alg-geom/9507010

en

https://arxiv.org/abs/alg-geom/9507010

alg-geom/9507010

Leonid Positselski

Leonid Positselski and Alexander Vishik

Koszul duality and Galois cohomology

AMS-LaTeX v.1.1, 10 pages, no figures. Replaced for tex code correction (%&amslplain added) by request of www-admin

Math. Research Letters 2 (1995), no.6, p.771-781

10.4310/MRL.1995.v2.n6.a8

It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.

alg-geom

\section{#1}\medskip} \newcommand{\operatorname{coker}}{\operatorname{coker}} \newcommand{\operatorname{id}}{\operatorname{id}} \newcommand{\operatorname{char}}{\operatorname{char}} \newcommand{\operatorname{Hom}}{\operatorname{Hom}} \newcommand{\operatorname{Tor}}{\operatorname{Tor}} \newcommand{\operatorname{Ext}}{\operatorname{Ext}} \newcommand{\subset}{\subset} \newcommand{\bigoplus\nolimits}{\bigoplus\nolimits} \newcommand{\bigcap\nolimits}{\bigcap\nolimits} \newcommand{\bigcup\nolimits}{\bigcup\nolimits} \newcommand{\{\,}{\{\,} \newcommand{\,\}}{\,\}} \newcommand{\langle}{\langle} \newcommand{\rangle}{\rangle} \newcommand{{\Bbb F}}{{\Bbb F}} \newcommand{{\Bbb Z}}{{\Bbb Z}} \newcommand{{\Bbb T}}{{\Bbb T}} \newcommand{{\Bbb N}}{{\Bbb N}} \newcommand{{\Bbb Q}}{{\Bbb Q}} \newcommand{{\Bbb R}}{{\Bbb R}} \newcommand{{\cal X}}{{\cal X}} \newcommand{{\cal W}}{{\cal W}} \renewcommand{\a}{\alpha} \renewcommand{\c}{\gamma} \renewcommand{\o}{\omega} \renewcommand{\O}{\Omega} \renewcommand{\d}{\partial} \newcommand{\varepsilon}{\varepsilon} \newcommand{\Delta}{\Delta} \newcommand{\operatorname{Gal}}{\operatorname{Gal}} \newcommand{\,\overline{\!F}}{\,\overline{\!F}} \newcommand{K^{\operatorname{M}}}{K^{\operatorname{M}}} \newcommand{{\operatorname{q}}}{{\operatorname{q}}} \newcommand{{\operatorname{gr}}}{{\operatorname{gr}}} \begin{document} \rightline{\scriptsize\hfill Preprint alg-geom/9507010} \vspace{0.75cm} \title{Koszul Duality and Galois Cohomology} \author{Leonid Positselski} \address{Independent University of Moscow} \email{posic@@ium.ips.ras.ru,\, posic@@math.harvard.edu} \author{Alexander Vishik} \address{Harvard University} \email{vishik@@math.harvard.edu} \maketitle \section*{Introduction} \smallskip Let $F$ be a field, $\,\overline{\!F}$ be its (separable) algebraic closure, and $G_F=\operatorname{Gal}(\,\overline{\!F}/F)$ be the absolute Galois group. Let $l\ne\operatorname{char} F$ be a prime number; assume that $F$ contains a $l$-root of unity $\zeta$. In this case, the Kummer pairing $$ \kappa\:G_F\times F^*\DOTSB\longrightarrow {\Bbb F}_l, \qquad \kappa(g,a)=s \text{ \ if \ } g(b)=\zeta^sb \text{ \ for \ } b=\sqrt[\uproot2\leftroot1 l] a\in\,\overline{\!F} $$ defines an isomorphism $F^*/(F^*)^l\;\widetilde\longrightarrow\; H^1(G_F,{\Bbb F}_l)$. The Milnor K-theory ring $K^{\operatorname{M}}(F)$ is a skew-commutative quadratic algebra over ${\Bbb Z}$ generated by $K^{\operatorname{M}}_1(F)=F^*$ with the Steinberg relations $\{a,1-a\}=0$. It is not difficult to show that the Kummer map can be extended to an algebra homomorphism $$ K^{\operatorname{M}}(F)\DOTSB\otimes{\Bbb F}_l\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, H^*(G_F,{\Bbb F}_l), $$ which is known as the {\it Galois symbol}, or the {\it norm residue homomorphism}. The well-known {\it Bloch--Kato conjecture\/} claimes that it is an isomorphism. It was proved by A.~Merkurjev and A.~Suslin~\cite{MS1,MS2} and M.~Rost~\cite{Ros} that this is true in degree~$2$ and for~$l=2$ in degree~$3$. The aim of this note is to show that the whole conjecture follows from its low-degree part provided the quadratic algebra $K^{\operatorname{M}}(F)\DOTSB\otimes{\Bbb F}_l$ is {\it Koszul\/} (see section~2 for the definition). We will assume that $F$ has no algebraic extensions of degree relatively prime to $l$. \begin{thm}{Theorem} Let $H=H^*(G,{\Bbb F}_l)$ be the cohomology algebra of a pro-$l$-group $G$. Assume that \begin{enumerate} \item $H^2$ is generated by $H^1$; \item in the subalgebra generated by $H^1$ in $H$, there are no nontrivial relations of degree~3; \item the quadratic algebra defined by $H^1$ and $H^2$ is Koszul. \end{enumerate} Then the whole algebra $H$ is quadratic. \end{thm} Actually, it is not essential that we consider the group cohomology here; the theorem is valid for any pro-nilpotent algebra. We will use the language of coalgebras in this paper in order to avoid dealing with projective limits and dualizations. We are grateful to V.~Voevodsky for stating the problem and numerous stimulating discussions and to J.~Bernstein who pointed out to us the necessity of using coalgebras in the Koszul duality. The first author is pleased to thank Harvard University for its hospitality which made it possible for this work to appear. \Section{Bar Construction} \subsection{The cohomology of augmented coalgebras} A {\it coalgebra\/} is a vector space $C$ over a field $\k$ equipped with a comultiplication map $\Delta\:C\DOTSB\longrightarrow C\DOTSB\otimes C$ and a counit map $\varepsilon\:C\DOTSB\longrightarrow\k$ satisfying the conventional associativity and counit axioms. An {\it augmented coalgebra\/} is a coalgebra $C$ equipped with a coalgebra homomorphism $\c\:\k\DOTSB\longrightarrow C$. The cohomology algebra of an augmented coalgebra $C$ is defined as the $\operatorname{Ext}$-algebra $H^*(C)=\operatorname{Ext}^*_C(\k,\k)$ in the category of left $C$-comodules, where $\k$ is endowed with the comodule structure by means of $\c$. We will calculate this cohomology using the following explicit comodule resolution $$ \k\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, C\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, C\DOTSB\otimes C^+\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, C\DOTSB\otimes C^+\DOTSB\otimes C^+\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\,\dotsb, $$ where $C^+=\operatorname{coker}(\c)$, the differential is $$ d(c_0\DOTSB\otimes\dots\DOTSB\otimes c_n)=\sum_{i=0}^n (-1)^{i-1} c_0\DOTSB\otimes\dots\DOTSB\otimes \Delta(c_i)\DOTSB\otimes\dots\DOTSB\otimes c_n, $$ and the coaction of $C$ is through the left components of these tensors. It is easy to check that the comodules $C\DOTSB\otimes W$ are injective, the differential is well-defined, and the operator $$ h\: c_0\DOTSB\otimes c_1\DOTSB\otimes\dots\DOTSB\otimes c_n \longmapsto \varepsilon(c_0)\sigma(c_1)\DOTSB\otimes c_2\DOTSB\otimes\dots\DOTSB\otimes c_n, $$ where $\sigma\:C^+\DOTSB\longrightarrow C$ is the splitting along $\varepsilon$, provides a $\k$-linear contracting homotopy, thus this is a resolution. Applying the functor $\operatorname{Hom}_C(\k,\cdot\,)$, we obtain $$ H^*(C)=H^*(\,\k\DOTSB\longrightarrow C^+\DOTSB\longrightarrow C^+\DOTSB\otimes C^+\DOTSB\longrightarrow\dotsb\,), $$ where the differential is given by the same formula and the multiplication on $H^*(C)$ is induced by the evident multiplication $$ (c_1\DOTSB\otimes\dots\DOTSB\otimes c_i)\cdot(c_{i+1}\DOTSB\otimes\dots\DOTSB\otimes c_{i+j})= c_1\DOTSB\otimes\dots\DOTSB\otimes c_i\DOTSB\otimes c_{i+1}\DOTSB\otimes\dots\DOTSB\otimes c_{i+j} $$ on this cobar-complex. \subsection{The homology of augmented algebras} An augmented algebra $A$ is an associative algebra over a field $\k$ endowed with an algebra homomophism $\a\:A\DOTSB\longrightarrow\k$. The homology coalgebra of an augmented algebra $A$ is by the definition $H_*(A)=\operatorname{Tor}_A(\k,\k)$, where the left and right module structures on $\k$ are defined by means of~$\a$. We will calculate it using the following explicit bar-resolution of the left $A$-module~$\k$ $$ \k\DOTSB\,\leftarrow\joinrel\relbar\joinrel\relbar\, A\DOTSB\,\leftarrow\joinrel\relbar\joinrel\relbar\, A\DOTSB\otimes A_+\DOTSB\,\leftarrow\joinrel\relbar\joinrel\relbar\, A\DOTSB\otimes A_+\DOTSB\otimes A_+\DOTSB\,\leftarrow\joinrel\relbar\joinrel\relbar\,\dotsb, $$ where $A_+=\ker(\a)$ and $$ \d(a_0\DOTSB\otimes\dots\DOTSB\otimes a_n)=\sum_{i=1}^n (-1)^i a_0\DOTSB\otimes\dots\DOTSB\otimes a_{i-1}a_i\DOTSB\otimes\dots\DOTSB\otimes a_n. $$ It is easy to check that the operator $$ h\: a_0\DOTSB\otimes a_1\DOTSB\otimes\dots\DOTSB\otimes a_{n-1} \longmapsto 1\DOTSB\otimes (a_0-\a(a_0))\DOTSB\otimes a_1\DOTSB\otimes\dots\DOTSB\otimes a_{n-1} $$ provides a $\k$-linear contracting homotopy. Applying the functor $\k\DOTSB\otimes_A\cdot\,$, we obtain $$ H_*(A)=H_*(\,\k\DOTSB\longleftarrow A_+\DOTSB\longleftarrow A_+\DOTSB\otimes A_+\DOTSB\longleftarrow\dotsb\,) $$ and the coalgebra structure on $H_*(A)$ is induced by the evident coalgebra structure $$ \Delta(a_1\DOTSB\otimes\dots\DOTSB\otimes a_n)=\sum_{i=0}^n (a_1\DOTSB\otimes\dots\DOTSB\otimes a_i)\DOTSB\otimes(a_{i+1}\DOTSB\otimes\dots\DOTSB\otimes a_n) $$ on this bar-complex. \Section{Koszul Duality} By a graded algebra (graded coalgebra) we mean a non-negatively graded vector space $A=\bigoplus\nolimits_{n=0}^\infty A_n$ ($C=\bigoplus\nolimits_{n=0}^\infty C_n$) over~a~field~$\k$ such that $A_0=\k$ ($C_0=\k$) equipped with an associative algebra (coalgebra) structure that respects the grading, i.~e., $A_i\cdot A_j\subset A_{i+j}$ and $1\in A_0\,$ ($\Delta(C_n)\subset\sum_{i+j=n}C_i\DOTSB\otimes C_j$ and $\varepsilon(C_{>0})=0$). A graded algebra (coalgebra) structure induces an augmented algebra (coalgebra) structure in an evident way. The homology coalgebra (cohomology algebra) of a graded algebra (coalgebra) is equipped with a natural second grading, as it can be seen from the explicit resolutions above: $$ H_*(A)=\bigoplus\nolimits_{i\le j}H_{ij}(A) \quad \text{and} \quad H^*(C)=\bigoplus\nolimits_{i\le j}H^{ij}(C). $$ In fact, all the results below in this section can be formulated in a more general setting of a graded algebra in a (semisimple abelian, not necessarily symmetric) tensor category, where the duality connects the algebras in the opposite categories; however, we prefer to deal with vector spaces here. \subsection{Quadratic algebras and coalgebras} \ \begin{rem}{Definition 1.} A graded coalgebra $C$ is called {\it one-cogenerated\/} if the iterated comultiplication maps $\Delta^{(n)}\:C_n\DOTSB\longrightarrow C_1^{\DOTSB\otimes n}$ are injective, or equivalently, all the maps $\Delta\:C_{i+j}\DOTSB\longrightarrow C_i\DOTSB\otimes C_j$ are injective. A graded coalgebra is called {\it quadratic\/} if it is isomorphic to the subcoalgebra of the form $$ \langle V,R\rangle=\bigoplus\nolimits_{n=0}^\infty\,\bigcap\nolimits_{i=1}^{n-1} V^{i-1}\DOTSB\otimes R\DOTSB\otimes V^{n-i-1} $$ of the tensor coalgebra ${\Bbb N}(V)=\bigoplus\nolimits_n V^{\DOTSB\otimes n}$ for some vector space $V$ and a subspace $R\subset V^{\ot2}$. With a graded coalgebra $C$, one can associate in a natural way a quadratic coalgebra ${\operatorname{q}} C$ and a morphism of graded coalgebras $r_C\:C\DOTSB\longrightarrow {\operatorname{q}} C$ that is an isomorphism on $C_1$ and an epimorphism on $C_2$. A graded algebra is called {\it quadratic\/} if it is isomorphic to the quotient algebra $\{V,R\}={\Bbb T}(V)/(R)$ of a tensor algebra ${\Bbb T}(V)=\bigoplus\nolimits_n V^{\DOTSB\otimes n}$ by the ideal generated by a subspace $R\subset V^{\ot2}$. With a graded algebra $A$, one can associate a quadratic algebra ${\operatorname{q}} A$ and a morphism of graded algebras $r_A\:{\operatorname{q}} A\DOTSB\longrightarrow A$ that is an isomorphism on $A_1$ and a monomorphism on $A_2$. \end{rem} \begin{remsec}{Definition 2.} The quadratic algebra $A=\{V,R\}$ and the quadratic coalgebra $C=\langle V,R\rangle$ are called {\it dual\/} to each other; we denote this as $C=A^!$ and $A=C^?$. Evidently, this defines an equivalence between the categories of quadratic algebras and quadratic coalgebras. \end{remsec} \begin{thmsec}{Proposition 1} A graded coalgebra $C$ is one-cogenerated iff one has $H^{1,j}(C)=0$ for $j>1$. A one-cogenerated coalgebra $C$ is quadratic iff $H^{2,j}(C)=0$ for $j>2$. Moreover, the morphism $r_C\:C\DOTSB\longrightarrow{\operatorname{q}} C$ is an isomorphism on $C_{\le n}$ iff $H^{2,j}(C)=0$ for $2<j\le n$. The analogous statements are true for graded algebras. \end{thmsec} \pr{Proof}: It is evident from the explicit form of the cobar-complex that $H^{1,>1}(C)=0$ for a one-cogenerated coalgebra $C$. Conversely, let $j>1$ be the minimal number for which $\Delta^{(j)}\:C_j\DOTSB\longrightarrow C_1^{\DOTSB\otimes j}$ is non-injective, then it is easy to see that the map $\Delta\:C_j\DOTSB\longrightarrow \bigoplus\nolimits^{s+t=j}_{s,t\ge1} C_s\DOTSB\otimes C_t$ is non-injective also, hence $H^{1,j}(C)\ne0$. Now let $C$ be one-cogenerated, then the map $r_C$ is an embedding. Let $z\in C_+\DOTSB\otimes C_+$ be a homogeneuos cocycle of degree $n$, thus $z=\sum^{s+t=n}_{s,t\ge1} z_{st}$, where $z_{st}\in C_s\DOTSB\otimes C_t$. The cocycle condition means that the images of $(\Delta\DOTSB\otimes\operatorname{id})(z_{u+v,w})$ and $(\operatorname{id}\DOTSB\otimes\Delta)(z_{u,v+w})$ in $C_u\DOTSB\otimes C_v\DOTSB\otimes C_w$ coincide for any $u,v,w\ge1$, $\,u+v+w=n$. Since the maps $\Delta^{(k)}$ are injective, it is equivalent to say that the elements $(\Delta^{(s)}\DOTSB\otimes\Delta^{(t)})(z_{st})\in C_1^{\DOTSB\otimes n}$ coincide for all $s$ and $t$. We have got an element in $C_1^{\DOTSB\otimes n}$; it is easy to see that it represents an element of ${\operatorname{q}} C$ which belongs to the image of $r_C$ iff $z$ is a coboundary. At last, if $r_C$ is an isomorphism in degree $<n$, then any element of ${\operatorname{q}} C$ corresponds to a cocycle $z$ in this way. \qed \begin{thm}{Proposition 2} For any graded coalgebra $C$, the diagonal subalgebra $\bigoplus\nolimits_i H^{i,i}(C)$ of the cohomology algebra $H^*(C)$ is a quadratic algebra isomorphic to $({\operatorname{q}} C)^?$. Analogously, for a graded algebra $A$, the diagonal quotient coalgebra $\bigoplus\nolimits_i H_{i,i}(A)$ of the homology coalgebra $H_*(A)$ is isomorphic to $({\operatorname{q}} A)^!$. \end{thm} \pr{Proof} is immediate. \qed \subsection{Koszul algebras and coalgebras} This definition is due to S.~Priddy~\cite{Pr}; see also~\cite{Lof,Bac,BF,BGS}. \begin{rem}{Definition 3.} A graded algebra $A$ is called {\it Koszul\/} if $H_{ij}(A)=0$ unless $i=j$. A graded coalgebra $C$ is called {\it Koszul\/} if $H^{ij}(C)=0$ unless $i=j$. It follows from Proposition~1 that any Koszul algebra (coalgebra) is quadratic. \end{rem} Now we are going to establish the criterion of Koszulity in the explicit linear algebra terms due to J.~Backelin~\cite{Bac}. In particular, we will see that the dual algebra and coalgebra are Koszul simultaneuosly. \begin{rem}{Definition 4.} A collection of subspaces $X_1$, \dots, $X_{n-1}$ in a vector space $W$ is called {\it distributive}, if there exists a (finite) direct decomposition $W=\bigoplus\nolimits_{\o\in\O}W_\o$ such that each subspace $X_k$ is the sum of a set of subspaces $W_\o$. Equivalently, the distributivity identity $(X+Y)\cap Z=X\cap Z+Y\cap Z$ should be satisfied for any triple of subspaces $X$, $Y$, $Z$ that can be obtained from the subspaces $X_k$ using the operations of sum and intersection. \end{rem} \begin{thmsec}{Lemma} Let $X_1$, \dots, $X_{n-1}\subset W$ be a collection of linear subspaces; assume that any its proper subcollection $X_1$, \dots, $\widehat{X}_k$, \dots, $X_{n-1}$ is distributive. Then the following three conditions are equivalent: \begin{enumerate} \item[(a)] the following complex $B^*(W,X)$ is exact everywhere outside its left term: $$ W\DOTSB\longrightarrow \bigoplus\nolimits_s W/X_s \DOTSB\longrightarrow \bigoplus\nolimits_{s<t}W/(X_s+X_t) \DOTSB\longrightarrow \dots \DOTSB\longrightarrow W/\textstyle\sum_k X_k \rarrow0; $$ \item[(b)] the following complex $B_*(W,X)$ is exact everywhere outside its left term: $$ W\DOTSB\longleftarrow \bigoplus\nolimits_s X_s \DOTSB\longleftarrow \bigoplus\nolimits_{s<t}X_s\cap X_t \DOTSB\longleftarrow \dots \DOTSB\longleftarrow \textstyle\bigcap\nolimits_k X_k \larrow0; $$ \item[(c)] the collection $X_1$, \dots, $X_{n-1}$ is distributive. \qed \end{enumerate} \end{thmsec} \begin{thmsec}{Proposition 3} Let $V$ be a vector space and $R\subset V\DOTSB\otimes V$ be a subspace, then the following three conditions are equivalent: \begin{enumerate} \item[(a)] the quadratic algebra $A=\{V,R\}$ is Koszul; \item[(b)] the quadratic coalgebra $C=\langle V,R\rangle$ is Koszul; \item[(c)] for any $n$, the collection of subspaces $V^{\DOTSB\otimes k-1}\DOTSB\otimes R\DOTSB\otimes V^{n-k-1}\subset V^{\DOTSB\otimes n}$, where $k=1$,~\dots,~$n-1$, is distributive. \end{enumerate} \end{thmsec} \pr{Proof}: Moreover, one has $H_{ij}(A)=0$ for $i<j\le n$ iff the collection of subspaces in $V^{\DOTSB\otimes n}$ is distributive, and the same for coalgebras. This follows immediately from Lemma by induction on $n$. \qed \Section{Cohomology of Nilpotent Coalgebras} \subsection{Nilpotent coalgebras} Let $C$ be an augmented coalgebra with the augmentation map $\c\:\k\DOTSB\longrightarrow C$. The {\it augmentation filtration\/} on an augmented coalgebra $C$ is an increasing filtration $N$ defined by the formula $$ N_nC=\{\, c\in C \mid \Delta^{(n+1)}(c)\in C^{\DOTSB\otimes n+1}_\c = \sum_{i=1}^{n+1} C^{\DOTSB\otimes i-1}\DOTSB\otimes\c(\k)\DOTSB\otimes C^{\DOTSB\otimes n-i+1}\subset C^{\DOTSB\otimes n+1} \,\}, $$ where $\Delta^{(m)}\:C\DOTSB\longrightarrow C^{\DOTSB\otimes m}$ denotes the iterated comultiplication map. In particular, we have $N_0C=\c(\k)$. \begin{thm}{Proposition 4} The filtration $N$ respects the coalgebra structure on $C$, that is $$ \Delta(N_nC)\subset \sum_{i+j=n} N_iC\DOTSB\otimes N_jC. $$ Furthermore, the associated graded coalgebra ${\operatorname{gr}}_NC=\bigoplus\nolimits_{n=0}^\infty N_nC/N_{n-1}C$ is one-cogenerated. \end{thm} \pr{Proof}: Let $\phi\:C\DOTSB\longrightarrow\k$ be a linear function annihilating $N_{k-1}C$, where $0\le k\le n$; then it can be factorized as $\phi=\psi\circ\Delta^{(k)}$, where $\psi\:C^{\DOTSB\otimes k}\DOTSB\longrightarrow\k$ is a function annihilating $C^{\DOTSB\otimes k}_\c$. We have to show that $(\phi\DOTSB\otimes\nobreak\operatorname{id})\Delta N_nC\subset N_{n-k}C$. Put for convenience $\Delta^{(0)}=\varepsilon$ and $\Delta^{(1)}=\operatorname{id}$; then one has $(\Delta^{(k)}\DOTSB\otimes\nobreak\Delta^{(n-k+1)})\circ\Delta=\Delta^{(n+1)}$, hence $(\phi\DOTSB\otimes\nobreak\Delta^{(n-k+1)})\Delta N_nC= (\psi\DOTSB\otimes\nobreak\operatorname{id}^{\DOTSB\otimes n-k+1})\Delta^{(n+1)}N_nC \subset C^{\DOTSB\otimes n-k+1}_\c$, so we are done. Since we have $\Delta^{(n)}(c)\notin C^{\DOTSB\otimes n}_\c$ for $c\notin N_{n-1}C$, the second assertion is immediate. \qed \begin{rem}{Definition 5.} An augmented coalgebra $C$ is called {\it nilpotent\/} if the augmentation filtration $N$ is full, that is $C=\bigcup_nN_nC$. \end{rem} \begin{remsec}{Example:} Let $G$ be a pro-$l$-group and $C={\Bbb F}_l(G)$ be the coalgebra of locally constant functions on $G$ with respect to the convolution; in other words, $C=\varinjlim{\Bbb F}_l(G/U)$, where the limit is taken over all open normal subgroups $U$ of $G$ and the coalgebra ${\Bbb F}_l(G/U)={\Bbb F}_l[G/U]^*$ is the dual vector space to the group algebra of~$G/U$. Let $\c:{\Bbb F}_l\DOTSB\longrightarrow{\Bbb F}_l(G)$ be the augmentation map that takes a constant from ${\Bbb F}_l$ to the corresponding constant function on $G$. Since the augmentation ideal of the group ring of a finite $l$-group over ${\Bbb F}_l$ is nilpotent, it follows by passing to the inductive limit that the augmented coalgebra ${\Bbb F}_l(G)$ is nilpotent also. It is easy to see that the category of ${\Bbb F}_l(G)$-comodules is equivalent to the category of discrete $G$-modules over ${\Bbb F}_l$ (and the same is true over ${\Bbb Z}$) for any pro-finite group $G$. \end{remsec} \subsection{Main theorem} Now we are ready to prove the theorem mentioned in Introduction. \begin{thm}{Theorem} Let $H=H^*(C)$ be the cohomology algebra of a nilpotent coalgebra $C$. Assume that \begin{enumerate} \item $H^2$ is generated by $H^1$; \item in the subalgebra generated by $H^1$ in $H$, there are no nontrivial relations of degree~3; \item the quadratic algebra ${\operatorname{q}} H$ defined by $H^1$ and $H^2$ is Koszul. \end{enumerate} Then the whole algebra $H$ is quadratic (and therefore, Koszul). In addition, there is an isomorphism $H^*(C)\simeq H^*({\operatorname{gr}}_NC)$. \end{thm} \pr{Proof}: The filtration $N$ on a coalgebra $C$ induces a filtration on the corresponding cobar-complex: $$ N_nC^{+\DOTSB\otimes i}=\bigoplus\nolimits_{j_1+\dots+j_i=n}N_{j_1}C^+\DOTSB\otimes\dots\DOTSB\otimes N_{j_i}C^+, $$ where $N_jC^+=N_jC/\c(\k)$, so that the filtration on $C^{+\DOTSB\otimes i}$ starts with $N_i$. Clearly, the associated graded complex coincides with the cobar-complex of ${\operatorname{gr}}_NC$, thus we obtain a multiplicative spectral sequence $$ E_1^{ij}=H^{ij}({\operatorname{gr}}_NC) \implies H^i(C), $$ which converges since the filtration is an increasing one. More exactly, the differentials have the form $d_r\:E_r^{i,j}\DOTSB\longrightarrow E_r^{i+1,j-r}$ and there is an induced increasing multiplicative filtration $N$ on $H^*(C)$ such that ${\operatorname{gr}}_N^jH^i(C)=E_\infty^{i,j}$. In particular, we see that the subalgebra $\bigoplus\nolimits N_iH^i(C)$ in $H^*(C)$ is isomorphic to the quotient algebra of the diagonal cohomology $\bigoplus\nolimits H^{i,i}({\operatorname{gr}}_NC)$ by the images of the differentials. By Propositions~4, the graded coalgebra ${\operatorname{gr}}_NC$ is one-cogenerated, hence (by Proposition~1) we have $E_1^{1,j}=H^{1,j}({\operatorname{gr}}_NC)=0$ for $j>1$, which implies $H^1(C)=N_1H^1(C)\simeq H^{1,1}({\operatorname{gr}}_NC)$ and $N_2H^2(C)\simeq H^{2,2}({\operatorname{gr}}_NC)$. Since (by Proposition~2) the diagonal cohomology algebra $\bigoplus\nolimits{}H^{i,i}({\operatorname{gr}}_NC)$ is quadratic, we conclude that it is isomorphic to~${\operatorname{q}} H^*(C)$. By Proposition~2 again, $\bigoplus\nolimits H^{i,i}({\operatorname{gr}}_NC)$ is the dual quadratic algebra to the coalgebra ${\operatorname{q}}{\operatorname{gr}}_NC$; since we suppose ${\operatorname{q}} H^*(C)$ is Koszul, the dual coalgebra ${\operatorname{q}}{\operatorname{gr}}_NC$ is Koszul also (Proposition~3). On the other hand, we have assumed that there are no cubic relations in the subalgebra generated by $H^1(C)$, hence all the differentials $d_r\:E_r^{2,3+r}\DOTSB\longrightarrow E_r^{3,3}$ targeting in $H^{3,3}({\operatorname{gr}}_NC)$ vanish. Now let us prove by induction that $H^{2,j}({\operatorname{gr}}_NC)=0$ for $j>2$. Assume that this is true for $2<j\le n-1$; by Proposition~1, it follows that the map $r_{{\operatorname{gr}}_NC}\:{\operatorname{gr}}_NC\DOTSB\longrightarrow{\operatorname{q}}{\operatorname{gr}}_NC$ is an isomorphism in degree $\le n-1$. Therefore, the induced map on the cobar-complex is an isomorphism in these degrees also, hence in particular $H^{3,j}({\operatorname{gr}}_NC)=H^{3,j}({\operatorname{q}}{\operatorname{gr}}_NC)$ for $j\le n-1$ (and even for $j\le n$). Since the coalgebra ${\operatorname{q}}{\operatorname{gr}}_NC$ is Koszul, it follows that $E_1^{3,j}=H^{3,j}({\operatorname{gr}}_NC)=0$ for $3<j\le n-1$ and the term $E_1^{2,n}=\allowbreak H^{2,n}({\operatorname{gr}}_NC)$ cannot die in the spectral sequence. But we have assumed that $H^2(C)$ is generated by $H^1(C)$, hence $H^2(C)=N_2H^2(C)$ and $E_\infty^{2,n}=0$, so we are done. We have seen that the coalgebra ${\operatorname{gr}}_NC$ is quadratic and ${\operatorname{q}}{\operatorname{gr}}_NC$ is Koszul, that is ${\operatorname{gr}}_NC$ is Koszul. It follows that $E_1^{i,j}=0$ for $i\ne j$, thus the spectral sequence degenerates and $H^*(C)=H^*({\operatorname{gr}}_NC)$. Therefore, $H^*(C)$ is Koszul also. \qed \begin{rem}{Remark:} This result is a formal analogue of some kind of Poincare--Birkhoff--Witt theorem for filtrations on quadratic algebras~\cite{PP}; in other words, it can be considered as reflecting the deformation properties of Koszul algebras. \end{rem} In the conclusion, recall the consequences we get for the Bloch--Kato conjecture. Since the conditions (1) and (2) of our Theorem are known to be satisfied for the coalgebra $C={\Bbb F}_2(G_F)$ of any absolute Galois pro-2-group $G_F$ and the quadratic part ${\operatorname{q}} H^*(C)$ of the corresponding cohomology algebra is exactly the Milnor K-theory algebra $K^{\operatorname{M}}(F)\DOTSB\otimes{\Bbb F}_2$, it suffices to establish the Koszul property of this quadratic algebra in order to prove the conjecture for $l=2$. The same would be true for the other $l$ if we know the norm residue homomorphism for that $l$ to be injective in degree~$3$. \clearpage \bigskip

9507

alg-geom/9507014

en

https://arxiv.org/abs/alg-geom/9507014

alg-geom/9507014

Leonid Positselski

Leonid Positselski

All strictly exceptional collections in $D^b_{coh}(P^m)$ consist of vector bundles

LaTeX 2e, 6 pages, no figures; replaced to correct formatting (amslatex to latex2e transition) and several misprints, no other changes

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

It is proved that any strictly exceptional collection generating the derived category of coherent sheaves on a smooth projective variety X with \rk K_0(X) = \dim X + 1 constists of locally free sheaves up to a common shift.

alg-geom

\section{Introduction} Let $\k$ be a field and ${\cal D}$ be a $\k$-linear triangulated category; we will denote, as usually, $\operatorname{Hom}^i(X,Y)=\operatorname{Hom}(X,Y[i])$ and $\operatorname{Hom^{\scriptscriptstyle\bullet}}(X,Y)=\bigoplus_i\operatorname{Hom}^i(X,Y)$. An object $E\in{\cal O}b\>{\cal D}$ is called {\it exceptional\/} if one has $\operatorname{Hom}^s(E,E)=0$ for $s\ne0$ and $\operatorname{Hom}^0(E,E)=\nobreak\k$. A finite sequence ${\cal E}$ of exceptional objects $E_1,\dots,E_n$ is called an {\it exceptional collection\/} if $\operatorname{Hom^{\scriptscriptstyle\bullet}}(E_i,E_j)=0$ for $i>j$. A collection ${\cal E}$ is called {\it full\/} if it generates ${\cal D}$ in the sence that any object of ${\cal D}$ can be obtained from $E_i$ by the operations of shift and cone. The Grothendieck group $K_0({\cal D})$ of a triangulated category ${\cal D}$ generated by an exceptional collection ${\cal E}$ is the free ${\Bbb Z}$-module generated by the classes of $E_1,\dots,E_n$, so any full exceptional collection consists of $n=\operatorname{rk} K_0({\cal D})$ objects. Moreover, it is explained in the paper~\cite{BK} that (under some technical restriction which is usually satisfied) a triangulated category ${\cal D}$ generated by an exceptional collection ${\cal E}$ is equivalent to the derived category of modules over the differential graded algebra corresponding to ${\cal E}$. Let $(E_1,\,E_2)$ be an exceptional pair; the {\it left\/} and {\it right mutated objects\/} $L_{E_1}E_2$ and $R_{E_2}E_1$ are defined as the third vertices of exceptional triangles \begin{gather*} E_2[-1]\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, L_{E_1}E_2\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\,\operatorname{Hom^{\scriptscriptstyle\bullet}}(E_1,E_2)\otimes E_1\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, E_2 \\ E_1\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\,\operatorname{Hom^{\scriptscriptstyle\bullet}}(E_1,E_2)^*\otimes E_2\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, R_{E_2}E_1\DOTSB\,\relbar\joinrel\relbar\joinrel\rightarrow\, E_1[1]. \end{gather*} This definition was given in the papers~\cite{Gor,Bon}; it was shown that the {\it mutated collections\/} \begin{gather*} E_1,\dots,E_{i-2},\,L_{E_{i-1}}E_i,\,E_{i-1},\, E_{i+1},\dots,E_n \\ E_1,\dots,E_{i-2},\,E_i,\,R_{E_i}E_{i-1},\, E_{i+1},\dots,E_n \end{gather*} remain exceptional (and full) and that the left and right mutations are inverse to each other. Mutations defined this way form an action of the Artin's braid group $B_n$ with $n$ strings on the set of all isomorphism classes of exceptional collections of $n$ objects. There is a central element $\phi\in B_n$ that corresponds to the rotation action of the circle on the space of $n$-point configurations in ${\Bbb C}$. Its action on exceptional collections can be described as follows. Let $E_{n+1}=R^{n-1}E_1$ be the object obtained by successive left mutations of $E_1$ through $E_2$, \dots, $E_n$. Then it follows that the collection $E_2,\dots,E_{n+1}$ is also exceptional. Proceeding in this way, we obtain the collection $E_3,\dots,E_{n+2}$, and so on, constructing an infinite sequence of exceptional objects $E_1$,~$E_2$,~$E_3$,~\dots with the property that any $n$ sequential objects $E_i,\dots,E_{i+n-1}$ form an exceptional collection. Using left mutations, we can continue it to the negative indices: $E_0=L^{n-1}E_n$, $\,E_{-1}=L^{n-1}E_{n-1}$, and so on. This sequence is called a {\it helix}. The action of $\phi$ on exceptional collections shifts it $n$ times to the left: $$ \phi(E_1,\dots,E_n)=(E_{-n+1},\dots,E_0). $$ The point is that this shift can be extended to an exact auto-equivalence of the category ${\cal D}$. Namely, the {\it Serre functor\/} for a triangulated category ${\cal D}$ is a covariant functor $F\:{\cal D}\DOTSB\longrightarrow{\cal D}$ for which there is a natural isomorphism $$ \operatorname{Hom^{\scriptscriptstyle\bullet}}(U,V)=\operatorname{Hom^{\scriptscriptstyle\bullet}}(V,FU)^*. $$ It is shown in~\cite{Bon} that one has $E_{i-n}=F(E_i)[-n+1]$ for a full helix ${\cal E}$ in ${\cal D}$. Now let us turn to exceptional collections in the derived category ${\cal D}^b_{\operatorname{coh}}(X)$ of coherent sheaves on a smooth projective algebraic variety $X$. In this case the Serre functor has the form $F(U)=U\otimes\omega_X[\dim X]$, where $\omega_X$ is the canonical line bundle. In the initial works of A.~Gorodentsev and A.~Rudakov~\cite{GR}, they considered exceptional collections consisting of pure sheaves, not complexes. Therefore, such mutations were not defined for any exceptional collections, but only under the conditions that some maps are injective or surjective. For example, we see that the helix generated by a full exceptional collection of sheaves will not consist of sheaves unless its period $n$ is equal to $\dim X+1$. Conversely, it was shown by A.~Bondal~\cite{Bon} that all mutations of a full exceptional collection of $\dim X+1$ sheaves in ${\cal D}^b_{\operatorname{coh}}(X)$ (that is, for a variety with $\operatorname{rk} K_0(X)=\dim X+1$) consist of pure sheaves again. Indeed, the statement that $R_{E_2}E_1$ is a sheave follows immediately from the isomorphism $R_{E_2}E_1=L_{E_3}\dotsm L_{E_{n}}E_{n+1}$, where all the objects $E_1,\dots,E_{n+1}$ are pure sheaves. It is also easy to see that in this case mutations preserve the property of a full exceptional collection to consist of locally free sheaves. Note that for any projective variety one has $\operatorname{rk} K_0(X)\ge\dim X+1$, since the cycles of self-intersection of $\O(1)$ are linearly independent over ${\Bbb Q}$; the equality holds for $\P^m$, odd-dimensional quadrics, and some others. The principal problem of the theory of mutations of exceptional bundles on $\P^m$ is to prove that their action on full exceptional collections of vector bundles is transitive. More generally, it was conjectured in~\cite{BP} that the action of the semidirect product group $B_n\righttimes\nobreak{\Bbb Z}^n$ generated by mutations and shifts on full exceptional collections in any triangulated category ${\cal D}$ is transitive. The second half of this latter conjecture for smooth projective varieties with $\operatorname{rk} K_0(X)=\dim X+1$ states that any full exceptional collection in ${\cal D}^b_{\operatorname{coh}}(X)$ consists of shifts of vector bundles. In this paper we prove this last statement under the following additional restriction. An exceptional collection is said to be {\it strictly exceptional\/} if one has $\operatorname{Hom}^s(E_i,E_j)=0$ for $s\ne0$. \begin{thm}{Theorem} Let $X$ be a smooth projective variety for which $n=\operatorname{rk} K_0(X)=\dim X+\nbk1$. Then for any strictly exceptional collection $E_1,\dots,E_n$ generating ${\cal D}^b_{\operatorname{coh}}(X)$ the objects $E_i$ are locally free sheaves shifted on the same number $a\in{\Bbb Z}$ in ${\cal D}$. \end{thm} Conversely, it was shown in~\cite{Bon} that any full exceptional collection of $\dim X+1$ sheaves on a smooth projective variety is strictly exceptional. In particular, if a full exceptional collection on a variety with $\operatorname{rk} K_0(X)=\dim X+1$ consists of pure sheaves, then these sheaves are locally free. On the other hand, it follows that the property of a full exceptional collection in a triangulated category of this kind to be strictly exceptional is preserved by mutations; moreover, all strictly exceptional collections in these categories are {\it geometric\/} in the sence of~\cite{BP}. At last, our methods provide an approach to the results on recovery of algebraic varieties from the derived categories of coherent sheaves, alternative to the one given by Bondal--Orlov~\cite{BO}. \begin{thm}{Corollary} Suppose the canonical sheave of a smooth projective variety $X$ is either ample or anti-ample. Then the standard $t$-structure on the derived category ${\cal D}^b_{{\operatorname{coh}}}(X)$ can be recovered (uniquely up to a shift) from the triangulated category structure. \end{thm} I am grateful to A.~Bondal who introduced me into the subject of triangulated categories and exceptional collections and to A.~Polishchuk and A.~N.~Rudakov for very helpful discussions. I am pleased to thank Harvard University for its hospitality during preparation of this paper. \smallskip \section{Reduction to a Local Problem} The next result is due to A.~Bondal and A.~Polishchuk~\cite{BP}. \begin{thm}{Proposition} Suppose a helix $\{E_i,\,i\in{\Bbb Z}\}$ in a triangulated category ${\cal D}$ is generated by a strictly exceptional collection $E_1,\dots,E_n$. Then one has $\operatorname{Hom}^s(E_i,E_j)=0$ for $s>0$ and $i\le j\in{\Bbb Z}$, as well as for $s<n-1$ and $i\ge j\in{\Bbb Z}$. \end{thm} \pr{Proof} First note that the Serre duality isomorphisms $$ \operatorname{Hom}^s(E_i,E_j)=\operatorname{Hom}^{n-1-s}(E_{j+n},E_i)^* $$ mean that two statements are equivalent to each other; let us prove the first one. The simplest way is to identify ${\cal D}$ with the derived category of modules over the homomorphism algebra $A=\bigoplus_{k,l=1}^nA_{kl}$, $\,A_{kl}=\operatorname{Hom}(E_k,E_l)$ of our strictly exceptional collection, so that the objects $E_l$ correspond to the projective $A$-modules $P_l=\bigoplus_kA_{kl}$ for $1\le l\le n$. Since the Serre functor provides $n$-periodicity isomorphisms $\operatorname{Hom}^s(E_i,E_j)=\operatorname{Hom}^s(E_{i+n},E_{j+n})$, we can assume that $1\le i\le n$. Let $j=k+Nn$ for some $1\le k\le n$; then we have $E_j=F^{-N}E_l\,@!@!@![N(n-1)]$. The Serre functor on ${\cal D}^b({\operatorname{mod}}{-}@! A)$ has the form $F(M)=\operatorname{Hom}_\k(\operatorname{RHom}_A(M,A),\k)$ and $F^{-1}(M)=\operatorname{RHom}_A(\operatorname{Hom}_\k(M,\k),A)$; since the homological dimension of $A$ is not greater than $n-1$, we obtain $E_j\in{\cal D}^{\le0}({\operatorname{mod}}{-}@! {A})$ for $j\ge1$. Since $E_i$ are projective for $1\le i\le n$, the assertion follows. A direct, but more complicated calculation from~\cite{BP} allows to avoid the additional condition on ${\cal D}$. \qed\smallskip \pr{Proof of Theorem} First let us show that $X$ is a Fano variety. We give a simple strengthening of the argument from~\cite{BP}. Since $\operatorname{rk}\operatorname{Pic}(X)=1$, there are only three types of invertible sheaves: ample ones, antiample ones, and sheaves of finite order. We have to prove that $\omega^{-1}$ is ample; it is enough to show that $H^0(\omega^N)=0$ for all $N>0$. Let us denote by ${\cal H}^s(U)$ the cohomology sheaves of a complex $U$. Since $E_1,\dots,E_n$ generate ${\cal D}$, it is clear that there exists $i$ and $s$ such that $\operatorname{supp} {\cal H}^s(E_i)=X$. Let we have a nonzero section $f\in H^0(\omega^N)$; it induces a morphism $E_i\DOTSB\longrightarrow E_i\otimes\omega^N$ which is nonzero since its restriction to ${\cal H}^s$ is. But we have $E_i\otimes\omega^N=E_{i-Nn}$ which provides a contradiction with Proposition. \begin{rem}{Remark 1:} More generally, one can see that the canonical sheave $\omega$ cannot be of finite order for a variety $X$ admitting a full exceptional colletion. Indeed, the action of invertible sheaves on $K_0(X)$ is unipotent with respect to the filtration by the dimensions of supports, thus in the case in question the action of $\omega$ on $K_0(X)$ must be trivial. But this action (skew-)symmetrizes the canonical bilinear form $\chi([U],[V])=\sum(-1)^s\dim\operatorname{Hom}^s(U,V)$ on $K_0(X)$. In the basis of $K_0$ corresponding to an exceptional collection, the matrix of this form is upper-triangular with units on the diagonal, so it cannot be skew-symmetric and if it is symmetric then it is positive. The latter is impossible since one has $\chi([\O_x],[\O_x])=0$ for the structure sheave $\O_x$ of a point $x\in X$. \end{rem} We will essentially use the tensor structure on ${\cal D}^b_{\operatorname{coh}}(X)$. Namely, let $$ \operatorname{\text{${\cal R}{\cal H}o@,m$}}\:{\cal D}^{\operatorname{opp}}\times{\cal D}\DOTSB\longrightarrow{\cal D} $$ be the derived functor of local homomorphisms of coherent sheaves; it can be calculated using finite locally free resolvents. We have $\operatorname{Hom}^s(U,V)=H^s(\operatorname{\text{${\cal R}{\cal H}o@,m$}}(U,V))$, where $H$ denotes the global sheave's cohomology. Let $i$, $j\in{\Bbb Z}$ be fixed and $N$ be large enough; one has $$ \operatorname{Hom}^s(E_i,E_{j+Nn})=H^s(\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E_i,E_j\otimes\omega^{-N})) =H^s(C_{ij}\otimes\omega^{-N}), $$ where we denote $C_{ij}=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E_i,E_j)$. Since $\omega^{-1}$ is ample, for large $N$ we have $H^{>0}{\cal H}^s(C_{ij}\otimes\omega^{-N})=0$, hence by the spectral sequence $$ H^s(C_{ij}\otimes\omega^{-N})=H^0{\cal H}^s(C_{ij}\otimes\nobreak\omega^{-N}). $$ Using the property of ample sheaves again, we see that $\operatorname{Hom}^s(E_i,E_{j+Nn})$ is nonzero iff ${\cal H}^s(C_{ij})$ is. Let ${\cal D}^{\le0}$ and ${\cal D}^{\ge0}$ denote the subcategories of ${\cal D}^b_{\operatorname{coh}}(X)$ defined in the standard way. Comparing the last result with Proposition, we finally obtain $C_{ij}\in{\cal D}^{\le0}$. \begin{rem}{Remark 2:} Now we can show easily that our exceptional collection is {\it geometric\/}~\cite{BP}. Indeed, using the duality $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(V,U)=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(\operatorname{\text{${\cal R}{\cal H}o@,m$}}(U,V),\,\O)$, one obtains $C_{ij}=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(C_{ji},\O)$ and since $\operatorname{\text{${\cal R}{\cal H}o@,m$}}({\cal D}^{\le0},\O)\subset{\cal D}^{\ge0}$, it follows that $C_{ij}$ are pure sheaves. Therefore $\operatorname{Hom}^{<0}(E_i,E_j)=H^{<0}(C_{ij})=0$ for any $i$ and $j$. \end{rem} The following local statement allows to finish the proof. \begin{thm}{Main Lemma} Let $E\in{\cal D}^b_{\operatorname{coh}}(X)$ be a coherent complex on a smooth algebraic variety $X$ such that $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E,E)\in{\cal D}^{\le0}$. Then $E$ is a (possibly shifted) locally free sheave. \end{thm} It only remains to show that all of $E_i$ are placed in the same degree in ${\cal D}$, which is true since they are locally free and $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E_i,E_j)$ is placed in degree~0. \qed\smallskip \pr{Proof of Corollary} The functor of twisting on $\omega$ on the derived category can be recovered in terms of the Serre functor. Let $\omega$ be anti-ample. According to Main Lemma, an object $E\in{\cal D}^b_{\operatorname{coh}}(X)$ is a shifted vector bundle iff $\operatorname{Hom}^s(E,E\otimes\omega^{-N})=0$ for $s\ne0$ and $N$ large enough. For a nonzero vector bundle $E$ and $U\in{\cal D}$ one has $U\in{\cal D}^{\ge0}$ iff $\operatorname{Hom}^{<0}(E_i,U\otimes\omega^{-N})=0$ for large $N$, and the same for ${\cal D}^{\le0}$. \qed \smallskip \section{The Proof of Main Lemma} \begin{thm}{Lemma 1} If $E\in\operatorname{Coh}(X)$ and $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E,\O)$ are pure sheaves placed in degree~0, then $E$ is locally free. \end{thm} \pr{Proof} Let $0\DOTSB\longrightarrow P_k\DOTSB\longrightarrow P_{k-1}\DOTSB\longrightarrow\dots\DOTSB\longrightarrow P_0\rarrow0$ be a locally free resolvent of $E$. Since $\hom^k(E,\O)=0$, we see that the morphism $\hom(P_{k-1},\O)\DOTSB\longrightarrow\hom(P_k,\O)$ is surjective. Thus, the inclusion $P_k\DOTSB\longrightarrow P_{k-1}$ is locally split and the quotient sheave $P_{k-1}/P_k$ is locally free, which allows to change our resolvent to a shorter one. \qed\smallskip Let $U\otimes^{\cal L} V$ denote the derived functor of tensor product over $\O_X$ on ${\cal D}^b_{\operatorname{coh}}(X)$; then one has $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(U,V)=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(U,\O)\otimes^{\cal L} V$. \begin{thm}{Lemma 2} Let $E$, $F\in{\cal D}^b_{\operatorname{coh}}(X)$; suppose $E\otimes^{\cal L} F\in{\cal D}^{\le0}$. Then for any $i+j\ge0$ one has $\operatorname{supp}{\cal H}^i(E)\cap\operatorname{supp}{\cal H}^j(F)=\empty$. \end{thm} \pr{Proof} Proceed by decreasing induction on $i+j$. Consider the K\"unneth spectral sequence $$ E_2^{pq}=\bigoplus_{i+j=q}\operatorname{Tor}_{-p}({\cal H}^iE,{\cal H}^jF) \implies {\cal H}^{p+q}(E\otimes^{\cal L} F). $$ If the intersection of supports is nonzero, then it is easy to see that ${\cal H}^iE\otimes{\cal H}^jF\ne0$, thus $E_2^{0,q}\ne0$. This term can be only killed by some $E^{-r,q+r-1}$, where $r\ge2$; but it follows from the induction hypothesis that $E_2^{p,\ge q+1}=0$. \qed\smallskip \pr{Proof of Main Lemma} Let $F=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E,\O)$; then one has $\operatorname{\text{${\cal R}{\cal H}o@,m$}}(E,E)=E\otimes^{\cal L} F$. Using a shift, we can assume that $E\in{\cal D}^{\le0}$ and ${\cal H}^0(E)\ne0$; then $F\in{\cal D}^{\ge0}$ and ${\cal H}^0F=\hom({\cal H}^0E,\O)$. By Lemma~2, we have $\operatorname{supp}{\cal H}^0E\cap\operatorname{supp}{\cal H}^{>0}F=\empty$. Clearly, one can assume that $X$ is irreducible. First let us show that $\operatorname{supp}{\cal H}^0(E)=X$. Indeed, in the other case it is clear that ${\cal H}^0F=0$ and the restriction of $F$ on $X\setminus\operatorname{supp}{\cal H}^{>0}F$ is acyclic while the restriction of $E$ is not, which contradicts the local nature of $\operatorname{\text{${\cal R}{\cal H}o@,m$}}$. Thus we have $\operatorname{supp}{\cal H}^0(E)=X$, which implies ${\cal H}^{>0}F=0$ and $F\in\operatorname{Coh}(X)$. It follows that $E=\operatorname{\text{${\cal R}{\cal H}o@,m$}}(F,\O)\in{\cal D}^{\ge0}$ and $E\in\operatorname{Coh}(X)$. By Lemma~1, $E$ is locally free. \qed\smallskip \smallskip

9507

alg-geom/9507002

en

https://arxiv.org/abs/alg-geom/9507002

alg-geom/9507002

Christoph Sorger

Yves Laszlo and Christoph Sorger

The line bundles on the stack of parabolic $G$-bundles over curves and their sections

LaTeX2e with package amsart, 31 pages, no figures. This is a revised version of our paper (mainly, the introduction and the section on pfaffians have been changed). The TeX file, as well as the .dvi and .ps files are also available at ftp://ftp.mathp7.jussieu.fr/pub/sorger

Let $X$ be a smooth, complete and connected curve and $G$ be a simple and simply connected algebraic group over $\comp$. We calculate the Picard group of the moduli stack of quasi-parabolic $G$-bundles and identify the spaces of sections of its members to the conformal blocs of Tsuchiya, Ueno and Yamada. We describe the canonical sheaf on these stacks and show that they admit a unique square root, which we will construct explicitly. Finally we show how the results on the stacks apply to the coarse moduli spaces and recover (and extend) the Drezet-Narasimhan theorem. We show moreover that the coarse moduli spaces of semi-stable $SO_r$-bundles are not locally factorial for $r\geq 7$.

alg-geom

\section{Introduction.} \subsection{}\label{th:Pic} Fix a simple and simply connected algebraic group $G$ over $k=\comp$ and a Borel subgroup $B\subset G$. Let $X$ be a smooth, complete and connected curve over $k$ and $p_{1},\dots,p_{n}$ be distinct points of $X$, labeled by standard (\ie containing $B$) parabolic subgroups $P_{1},\dots,P_{n}$ of $G$ (we allow $n=0$). Let $\Mpar$ be the moduli stack of quasi-parabolic $G$-bundles of type $\ul{P}=(P_{1},\dots,P_{n})$ at $\ul{p}=(p_{1},\dots,p_{n})$ and denote by $X(P_{i})$ the character group of $P_{i}$. \begin{th*}{} There is ${\scr{L}}\in\Pic(\M)$ such that we have an isomorphism $$\gamma:\Pic(\Mpar)\isom\reln{\scr{L}}\times\prod_{i=1}^{n}X(P_{i}).$$ If $G$ is of type $A$ or $C$, then ${\scr{L}}$ is the determinant of cohomology (\cf\ref{the-det-bundle}). If $G$ is of type $B$, $D$ or $G_{2}$, then ${\scr{L}}$ is the pfaffian of cohomology (\cf\ref{the-pf-bundle}). \end{th*} If $G$ is of type $E_{6},E_{7},E_{8},F_{4}$ we believe that we have ${\scr{L}}^{\otimes d(G)}={\scr{D}}_{\rho(G)}$ where respectively $d(G)=6,12,60,6$ and ${\scr{D}}_{\rho(G)}$ is the determinant of cohomology (\cf\ref{the-det-bundle}) associated to the fundamental representation $\rho(G)=\varpi_{6},\varpi_{7},\varpi_{8},\varpi_{4}$ (\cf the discussion in \ref{pb}). \subsection{} Suppose that the points $p_{1},\dots,p_{n}$ are instead labeled by finite dimensional simple representations $\lambda_{1},\dots,\lambda_{n}$ of $G$ and that an additional integer $\ell$, the {\em level}, is fixed. The choice of a representation $\lambda$ of $G$ is equivalent to the choice of a standard parabolic subgroup $P\subset G$ and a character $\chi\in X(P)$. Therefore, the labeling of the points $p_{1},\dots,p_{n}$ by the representations $\lambda_{1},\dots,\lambda_{n}$ defines the type $\ul{P}$ of a quasi-parabolic $G$-bundle, that is the stack $\Mpar$ {\em and}, by the above theorem, a line bundle ${\scr{L}}(\ell,\ul\chi)$ over $\Mpar$. The global sections of ${\scr{L}}(\ell,\ul\chi)$ give a vector space, the space of {\em generalized parabolic $G$-theta-functions of level $\ell$}, which is canonically associated to $(X,\ul{p},\ul\lambda)$. In mathematical physics, the rational conformal field theory of Tsuchiya, Ueno and Yamada \cite{TUY} associates also to $(X,\ul{p},\ul\lambda,\ell)$ a vector space: the space of {\em conformal blocks} $V_{X}(\ul{p},\ul\lambda,\ell)$ (\cf \cite{So3} for an overview). \begin{th*}{} Suppose that $G$ is classical or $G_2$. Then there is a canonical isomorphism \begin{equation}\label{Verlinde} H^{0}(\Mpar,{\scr{L}}(\ell,\ul\chi))\isom V_{X}(\ul{p},\ul\lambda,\ell). \end{equation} In particular, $\dim H^{0}(\Mpar,{\scr{L}}(\ell,\ul\chi))$ is given by the Verlinde dimension formula. \end{th*} For $n=0$, this has been proved independently by Beauville and the first author \cite{BL1} for $G=SL_{r}$ and by Faltings \cite{F} and Kumar, Narasimhan and Ramanathan \cite{KNR} for arbitrary simple and simply connected $G$. For arbitrary $n$ and $G=SL_r$ this has been proved by Pauly \cite{P} and can be proved for arbitrary simple and simply connected $G$ using (\ref{th:Uniformization}) (and therefore \cite{DS}) and (\ref{LGX-is-integral}) below following the lines of \cite{BL1} and \cite{P}. This is the subject of Section \ref{Identification}. \subsection{}\label{th:Uniformization} The above results are proved via the {\em uniformization} theorem: restrict for simplicity of the introduction to $n=0$. Suppose $p\in X$ and denote $X^{*}=X\moins p$. Define $D=\Spec(\hat{\cal{O}}_{p})$, where $\hat{\cal{O}}_{p}$ is the formal completion of the local ring ${\cal{O}}_{p}$ at $p$ and $D^{*}=\Spec(K_{p})$ where $K_{p}$ is the quotient field of $\hat{\cal{O}}_{p}$. Let $\LG$ (resp. $\LGp$, resp. $\LGX$) be the group of algebraic morphisms from $D^{*}$ (resp. $D$, resp. $X^{*}$) to $G$. \begin{th*}{} The algebraic stack $\M$ is canonically isomorphic to the double quotient stack $\LGX\backslash\LG/\LGp$. Moreover, the projection map $$\pi:\Q:=\LG/\LGp\ra \M$$ is locally trivial for the \'etale topology. \end{th*} This is proved in \cite{BL1} for $G=SL_{r}$. The extension to arbitrary $G$ has been made possible by Drinfeld and Simpson \cite{DS} in response to a question by the first author. They prove that if $S$ is a $k$-scheme and $E$ a $G$-bundle over $X\times S$ then, locally for the \'etale topology on $S$, the restriction of $E$ to $X^{*}\times S$ is trivial, which is essential for the proof. The above theorem is valid more generally for semi-simple $G$. In characteristic $p$, one has to replace ``\'etale'' by ``fppf'' if $p$ divides the order of $\pi_{1}(G(\comp))$. \subsection{}\label{pb} Consider the pullback morphism $$\pi^{*}:\Pic(\M)\efl{}{}\Pic(\Q).$$ The Picard group of $\Q$ is known (\cite{Ma},\cite{KNR}) to be canonically isomorphic to $\reln$, which reduces proving Theorem \ref{th:Pic} to proving that $\pi^{*}$ is an isomorphism. We will show that the injectivity of $\alpha$ will follow from the fact that $\LGX$ has no characters which in turn will follow from the fact that $\LGX$ is reduced and connected. Moreover, the surjectivity of $\alpha$ would follow from the simple connectedness of $\LGX$. Both topological properties, connectedness and simple connectedness of $\LGX$ are affirmed in \cite{KNR} and we believe them to be true. Whereas we will prove the connectedness of $\LGX$, following an idea of V. Drinfeld, we do not see how to prove the simple connectedness of $\LGX$. The injectivity is enough to prove the first part of Theorem \ref{th:Pic}, but to identify the generator ${\cal{L}}$ we should prove the surjectivity of $\alpha$. For classical $G$ and $G_{2}$ we do this by constructing a line bundle on $\M$ pulling back to the generator of $\Pic(\Q)$. \subsection{}\label{sr} The pfaffian construction (\ref{the-pf-bundle}) may be used to prove the following, valid over $k$ algebraically closed of characteristic $\not=2$. \begin{prop*}{} Suppose $G$ is semi-simple. Then, for every theta-characteristic $\kappa$ on $X$, there is a canonical square-root ${\cal{P}}_{\kappa}$ of the dualizing sheaf $\omega_{_{\M}}$ of $\M$. \end{prop*} \subsection{}\label{th:Pic(Mmod)} The last section will be devoted to Ramanathans moduli spaces $\Mod$ of semi-stable $G$-bundles. We will show how some of the results for the stack $\M$ will be true also for the moduli spaces $\Mod$. In particular we will recover (and extend) the Drezet-Narasimhan theorem. \begin{th*}{} There is a canonical isomorphism $\Pic(\Mod)\isom\reln L.$ If $G$ is of type $A$ or $C$ then $L$ is the determinant bundle and moreover, in this case $\Mod$ is locally factorial. If $G$ is of type $B_r$, $D_r$, $r\geq 4$ or $G_2$ then $L$ or $L^{\otimes 2}$ is the determinant bundle. \end{th*} This theorem has also been proved, independently and with a different method, by Kumar and Narasimhan \cite{KN}. The question whether $\Mod$ is locally factorial for $G$ of type other than the simply connected groups of type $A,C$ is the subject of a forthcoming paper. We show there for example that $\Pic(\ModSpin)$ is generated by the determinant of cohomology and in particular that $\ModSpin$ is not locally factorial for $r\geq 7$ by ``lifting'' to $\Spin_r$ the proof we give here (\ref{not-locally-factorial}) for the analogous statement for $\ModSO$. \bigskip {\em We would like to thank A. Beauville and C. Simpson for useful discussions and V. Drinfeld for his suggestion in (\ref{LGX-is-integral}) and for pointing out an inaccuracy in an earlier version of this paper.} \section{Some Lie theory.}\label{Lie-theory} Throughout this section $k$ will be an algebraically closed field of characteristic zero. \subsection{The general set up.} Let ${\goth{g}}$ be a simple finite dimensional Lie algebra over $k$. We fix a Cartan subalgebra ${\goth{h}}\subset{\goth{g}}$ and denote by $\Delta$ the associated root system. We have the root decomposition $\displaystyle{\goth{g}}={\goth{h}}\oplus\osum_{\alpha\in \Delta}{\goth{g}}_{\alpha}.$ The Lie subalgebra ${\goth{g}}_{-\alpha}\oplus[{\goth{g}}_{\alpha},{\goth{g}}_{-\alpha}]\oplus {\goth{g}}_{\alpha}$, isomorphic as a Lie algebra to ${\goth{sl}}_{2}$, will be denoted by ${\goth{sl}}_{2}(\alpha)$. Moreover we choose a basis $\Pi=\{\alpha_{1},\dots,\alpha_{r}\}$ of $\Delta$ and we denote by $\Delta_{+}$ the set of positive roots (with respect to $\Pi$). Put $\ds{\goth{b}}={\goth{h}}\oplus(\osum_{\alpha\in \Delta_{+}}{\goth{g}}_{\alpha})$. For each $\alpha\in \Delta_{+}$, we denote by $H_{\alpha}$ the coroot of $\alpha$, \ie the unique element of $[{\goth{g}}_{\alpha},{\goth{g}}_{-\alpha}]$ such that $\alpha(H_{\alpha})=2$, and we denote by $X_{\alpha}\in{\goth{g}}_{\alpha}$ and $X_{-\alpha}\in{\goth{g}}_{-\alpha}$ elements such that $$[H_{\alpha},X_{\alpha}]=2X_{\alpha},\hspace{1cm} [H_{\alpha},X_{-\alpha}]=-2X_{-\alpha},\hspace{1cm} [X_{\alpha},X_{-\alpha}]=H_{\alpha}.$$ When $\alpha$ is one of the simple roots $\alpha_{i}$, we write $H_{i},X_{i},Y_{i}$ instead of $H_{\alpha_{i}},X_{a_{i}},Y_{\alpha_{i}}$. Let $(\varpi_{i})$ be the basis of ${\goth{h}}^{*}$ dual to the basis $(H_{i})$. Denote by $P$ the weight lattice and by $P_{+}\subset P$ the set of dominant weights. Given a dominant weight $\lambda$, denote $L_{\lambda}$ the associated simple ${\goth{g}}$-module with highest weight $\lambda$ and $v_{\lambda}$ its highest weight vector. Finally $(\,,\,)$ will be the Cartan-Killing form normalized such that for the highest root $\theta$ we have $(\theta,\theta)=2$. \subsection{Loop algebras.} Let $\Lg={\goth{g}}\otimes_{k}k((z))$ be the {\em loop algebra} of $\g$ and $\Lgp={\goth{g}}\otimes_{k}k[[z]]$ its subalgebra of {\em positive} loops. There is a natural $2$-cocycle $$\begin{diagram} \psi_{\g}:&\Lg&\times&\Lg&\lra&k\\ &(X\otimes f&,&Y\otimes g)&\mapsto&(X,Y)\Res(gdf)\\ \end{diagram} $$ defining a central extension $\Lgh$ of $\g$: $$0\lra k\lra\Lgh\lra\Lg\lra 0.$$ Every other cocycle is a scalar multiple of $\psi_{\g}$ and the above central extension is universal. Let $\Lgph$ be the extension of $\Lgp$ obtained by restricting the above extension to $\Lgp$. As the cocycle is trivial over $\Lgp$ this extension splits. Let $\ell$ be a positive integer. A representation of $\Lgh$ is {\em of level $\ell$} if the center $c$ acts by multiplication by $\ell$. Such a representation is called integrable if $X\tensor f$ acts locally nilpotent for all $X\tensor f\in{\goth{g}}_{\alpha}\otimes_{k}k((z))$. The theory of affine Lie algebras \cite{Kac} affirms that the irreducible integrable representations of level $\ell$ of $\Lgh$ are classified (up to isomorphism) by the weights $P_{\ell}=\{\lambda\in P_{+}/(\lambda,\theta)\leq\ell\}.$ We denote by ${\cal{H}}_{\lambda,\ell}$ the irreducible integrable representation of level $\ell$ and highest weight $\lambda\in P_{\ell}$. If $\lambda=0$, the corresponding representation, which we denote simply by ${\cal{H}}_{\ell}$, is called the {\em basic representation of level $\ell$}. \subsection{The Dynkin index.}\label{section-Dynkin-index} Let $\rho:\g\ra{\goth{sl}}(V)$ be a representation of $\g$. Then $\rho$ induces a morphism of Lie algebras $\Lg\ra\LslV$. Pull back the universal central extension to $\Lg$: $$ \begin{diagram} 0&\lra&k&\lra&\widetilde{\Lg}&\lra&\Lg&\lra&0\\ &&\parallel&&\sfl{}{}&&\sfl{}{}\\ 0&\lra&k&\lra&\LslVh&\lra&\LslV&\lra&0\\ \end{diagram} $$ The cocycle of the central extension $\widetilde{\Lg}$ is of the type $d_{\rho}\psi_{\g}$. Define the {\em Dynkin index} of the representation $\rho$ of $\g$ by the number $d_{\rho}$. \begin{lem} Let $V=\sum_{\lambda}n_{\lambda}e^{\lambda}$ be the formal character of $V$. Then we have $$d_{\rho}=\frac{1}{2}\sum_{\lambda}n_{\lambda}\lambda(H_{\theta})^{2}$$ \end{lem} \begin{proof} By definition of the cocycle, we have $d_{\rho}=\Tr(\rho(X_{\theta})\rho(X_{-\theta}))$. Decompose the ${\goth{sl}}_{2}(\theta)$-module, $V$ as $\osum V^{(d_i)}$, where $V^{(d_i)}$ is the standard irreducible ${\goth{sl}}_{2}$-module with highest weight $d_i$. We may realize $V^{(d_i)}$ as the vector space of homogeneous polynomials in 2 variables $x$ and $y$ of degree $d_i$. Then $X_\theta$ acts as $x\partial/\partial y$, and $X_{-\theta}$ as $y\partial/\partial x$. Using the basis $x^ly^{d_i-l}, l=0,\ldots, d_i$ of $V(d_i)$, we see $$d_{\rho}=\sum_i\sum_{k=0}^{d_i}k(d_i+1-k).$$ The formal character of the ${\goth{sl}}_{2}(\theta)$-module $V^{(d)}$ is $\sum_{k=0}^{d}e^{d\rho_{\theta}-k\alpha_{\theta}}$ where $\alpha_{\theta}$ is the positive root of ${\goth{sl}}_{2}(\theta)$ and $\rho_{\theta}=\frac{1}{2}\alpha_{\theta}$. Therefore we are reduced to prove the equality $$\sum_{k=0}^{d}k(d+1-k)= \frac{1}{2}\sum_{k=0}^{d}[d\rho_{\theta}-k\alpha_{\theta})(H_{\theta})]^{2} =\frac{1}{2}\sum_{k=0}^{d}[d-2k]^{2} $$ which is easy. \end{proof} \begin{rem} The Dynkin index of a representation has been introduced to the theory of $G$-bundles over a curve by Faltings \cite{F} and Kumar, Narasimhan, Ramanathan \cite{KNR}. \end{rem} We are interested here in the {\em minimal Dynkin index} $d_{{\goth{g}}}$ defined to be as $\min d_{\rho}$ where $\rho$ runs over all representations $\rho:\g\ra{\goth{sl}}(V)$. \begin{prop}\label{minimal-Dynkin-index} The minimal Dynkin index $d_{{\goth{g}}}$ is as follows $$\begin{array}{c|c|c|c|c|c|c|c|c|c} \text{Type of }{\goth{g}}&A_{r}&B_{r},r\geq 3&C_{r}&D_{r},\geq 4&E_{6}&E_{7}&E_{8}&F_{4}&G_{2}\\\hline d_{{\goth{g}}}&1&2&1&2&6&12&60&6&2\\\hline \lambda\text{ s.t. } d_{{\goth{g}}}=d_{\rho(\lambda)}& \varpi_{1}&\varpi_{1}&\varpi_{1}&\varpi_{1}&\varpi_{6}&\varpi_{7}&\varpi_{8} &\varpi_{4}&\varpi_{1}\\ \end{array} $$ Moreover, for any representation $\rho:\g\ra{\goth{sl}}(V)$, we have $d_{\rho}=0\bmod d_{{\goth{g}}}$. \end{prop} {\it Proof.} It is enough to calculate the Dynkin index for the fundamental weights (note that $d_{V\otimes W}=r_{W}d_{V}+r_{V}d_{W}$ if $V$ and $W$ are two ${\goth{g}}$-modules of rank $r_{V}$ and $r_{W}$), which can be done explicitly \cite{D}. We give the values here for $E_8$, as not all of them in (\cite{D}, Table 5) are correct: $$\begin{array}{c|c|c|c|c|c|c|c|c|} &d_{\varpi_{1}}&d_{\varpi_{2}}& d_{\varpi_{3}}&d_{\varpi_{4}}&d_{\varpi_{5}}&d_{\varpi_{6}}& d_{\varpi_{7}}&d_{\varpi_{8}}\\\hline E_{8}&1500&85500&5292000&8345660400&141605100&1778400&14700&60\\\hline \end{array} $$ \section{The stack $\M$.}\label{the-stack-M} Throughout this section $k$ will be an algebraically closed field, $G$ will be semi-simple algebraic group over $k$. \subsection{} Let $X$ be a scheme over $k$. By a {\em principal $G$-bundle} over $X$ (or just $G$-bundle for short), we understand a scheme $E\ra X$ equipped with a right action of $G$ such that, locally in the flat topology, $E$ is trivial, \ie isomorphic to $G\times X$ as an $G$-homogeneous space. In particular, $E$ is affine, flat and smooth over $X$. Moreover, the above conditions imply that $E$ is even locally trivial for the \'etale topology. If $F$ is a quasi-projective scheme on which $G$ acts on the left and $E$ is a $G$-bundle, we can form $E(F)=E\times^{G}F$ the {\em associated bundle with fiber $F$}. It is the quotient of $E\times F$ under the action of $G$ defined by $g.(e,f)=(e.g,g^{-1}f)$. If $H$ is a subgroup of $G$, the associated $G/H$-bundle $E(G/H)$ will be denoted simply by $E/H$. Let $\rho:G\ra G^{\prime}$ be a morphism of algebraic groups. Then, as $G$ acts on $G^{\prime}$ via $\rho$, we can form the {\em extension of the structure group} of a $G$-bundle $E$, that is the $G^{\prime}$-bundle $E(G^{\prime})$. Conversely, if $F$ is a $G^{\prime}$-bundle, a {\em reduction of structure group} $F_{G}$ is a $G$-bundle $E$ together with an isomorphism $F_{G}(G^{\prime})\isom F$. If $\rho$ is a closed immersion, such reductions are in one to one correspondence with section of the associated bundle $F/G$. \subsection{} Let us collect some well known generalities on stacks for further reference. Let ${\tr A}\text{ff/$k$}$ be the flat affine site over $k$, that is the category of $k$-algebras equipped with the {\em fppf} topology. By {\em $k$-space} (resp. {\em $k$-group}) we understand a sheaf of sets (resp. groups) over ${\tr A}\text{ff/$k$}$. Any $k$-scheme can (and will) be considered as a $k$-space. We will view {\em $k$-stacks} from the pseudo-functorial point of view, \ie a $k$-stack ${\goth{X}}$ will associate to every $k$-algebra $R$ a groupoid ${\goth{X}}(R)$ and to every morphism of $k$-algebras $u:R\ra R^{\prime}$ a functor $u^{*}:{\goth{X}}(R^{\prime})\ra{\goth{X}}(R)$ together with isomorphisms of functors $(u\circ v)^{*}\simeq v^{*}\circ u^{*}$ satisfying the usual cocycle condition. The required topological properties are that for every $x,y\in\ob{\goth{X}}(R)$ the presheaf $\ul\Isom(x,y)$ is a sheaf and that all descent data are effective (\cite{LMB}, 2.1). Any $k$-space $X$ may be seen as a $k$-stack, by considering a set as a groupoid (with the identity as the only morphism). Conversely, any $k$-stack ${\goth{X}}$ such that ${\goth{X}}(R)$ is a discrete groupoid (\ie has only the identity as automorphisms) for all $k$-algebras $R$, is a $k$-space. A morphism $F:{\goth{X}}\ra{\goth{Y}}$ will associate, for every $k$-algebra $R$, a functor ${\goth{X}}(R)\ra{\goth{Y}}(R)$ satisfying the obvious compatibility conditions. Let $S=\Spec(R)$ and consider a morphism $\eta:S\ra{\goth{Y}}$, that is an object $\eta$ of ${\goth{Y}}(S)$. The fiber ${\goth{X}}_{\eta}$ is a stack over $S$. The morphism $F$ is {\em representable} if ${\goth{X}}_{\eta}$ is representable as a scheme for all $S=\Spec(R)$. A stack ${\goth{X}}$ is {\em algebraic} if the diagonal morphism ${\goth{X}}\ra{\goth{X}}\times{\goth{X}}$ is representable, separated and quasi-compact and if there is a scheme $X$ and a representable, smooth, surjective morphism of stacks $P:X\ra{\goth{X}}$. Suppose $X$ is a $k$-space and that the $k$-group $\Gamma$ acts on $X$. Then the quotient stack $[X/\Gamma]$ is defined as follows. Let $R$ be a $k$-algebra. The objects of $[X/\Gamma](R)$ are pairs $(E,\alpha)$ where $E$ is a $\Gamma$-bundle over $\Spec(R)$ and $\alpha:E\ra X$ is $G$-equivariant, the arrows are defined in the obvious way and so are the functors $[X/\Gamma](R^{\prime})\ra [X/\Gamma](R)$. \subsection{} Let $X$ be a smooth, complete and connected curve of genus $g$ over $k$. We denote by $\M$ the stack of principal $G$-bundles over $X$ which is defined as follows. For any $k$-algebra $R$ denote $X_R$ the scheme $X\times_{k}\Spec(R)$. Then objects of $\M(R)$ are $G$-bundles over $X_R$, morphisms of $\M(R)$ are isomorphisms of $G$-bundles. The following proposition is well known. \begin{prop} The stack $\M$ is algebraic and smooth. Moreover we have $\dim\M=(g-1)\dim G.$ \end{prop} \subsection{} Choose a closed point $p$ on $X$ and set $X^*=X\moins p$. Let ${\cal O}$ be the completion of the local ring of $X$ at $p$, and $K$ its field of fractions. Set $D=\Spec({\cal{O}})$ and $D^{*}=\Spec(K)$. We choose a local coordinate $z$ at $p$ and identify ${\cal O}$ with $k[[z]]$ and $K$ with $k((z))$. Let $R$ be a $k$-algebra. Define $X_R = X\times_k \Spec(R)$, $X_R^* = X^*\times_k \Spec(R)$, $D_R = \Spec\bigl(R[[z]]\bigr)$ and $D_R^* = \Spec\bigl(R((z))\bigr)$. Then we have the cartesian diagram $$\begin{diagram} D_R^*&\efl{}{}&D_{R}\\ \sfl{}{}&&\sfl{}{}\\ X_{R}^{*}&\efl{}{}&X_{R}\\ \end{diagram}$$ We denote by $A_{X_R}$ the $k$-algebra $\Gamma(X_{R}^*\, ,{\cal O}_{X^{*}_R})$. \subsection{Loop groups.}\label{Loop-groups} The category of $k$-spaces is closed under direct limits. A $k$-space ($k$-group) will be called an {\em ind-scheme} (resp. {\em ind-group}) if it is direct limit of a directed system of schemes. Remark that an ind-group is not necessarily an inductive limit of algebraic groups. We denote by $\LG$ the {\em loop group} of $G$ that is the $k$-group defined $R\mapsto G\bigl(R((z))\bigr)$, where $R$ is any $k$-algebra. The group of {\em positive loops}, that is the $k$-group $R\mapsto G\bigl(R[[z]]\bigr)$ will be denoted by by $\LGp$ and the group of {\em negative loops}, that is the $k$-group $R\mapsto G\bigl(R[z^{-1}]\bigr)$ will be denoted by $\LGm$. The group of {\em loops coming from $X^{*}$}, \ie the $k$-group defined by $R\mapsto G(A_{X_{R}})$, will be denoted by $\LGX$. Finally, we will use also the $k$-group $\LGmm$ defined by $R\mapsto G\bigl(z^{-1}R[z^{-1}]\bigr)$. Choose a faithful representation $G\subset SL_{r}$. For $N\ge 0$, we denote by $\LGN(R)$ the set of matrices $A(z)$ in $G\bigl(R((z))\bigr)\subset SL_{r}\bigl(R((z))\bigr)$ such that for both $A(z)$ and $A(z)^{-1}$, the coefficients have a pole of order $\le N$. This defines a subfunctor $\LGN$ of $\LG$ which is obviously representable by an (infinite dimensional) affine $k$-scheme. \begin{prop}\label{ind-groups} The $k$-group $\LGp$ is an affine group scheme. The $k$-group $\LG$ is an ind-group, direct limit of the sequence of the schemes $(\LGN)_{N\ge 0}$. Moreover, this ind-structure does not depend on the embedding $G\subset SL_{r}$. \end{prop} The $k$-group $\LGX$ has the structure of an ind-group induced by the one of $\LG$. The quotient $k$-space $\Q:=\LG/\LGp$ has equally the structure of an ind-scheme: define $\QN=\LGN/\LGp$ (note that $\LGN$ is stable under right multiplication by $\Qzero=\LGp$). \subsection{} Consider triples $(E,\rho,\sigma)$ where $E$ is a $G$-bundle on $X_R$, $\rho : G\times X_R^* \ra E_{|X_R^*}$ a trivialization of $E$ over $X_R^*$ and $\sigma : G\times D_R\ra E_{|D_R}$ a trivialization of $E$ over $D_R$. We let $T(R)$ be the set of isomorphism classes of triples $(E,\rho,\sigma)$. \begin{prop}\label{triples} The ind-group $\LG$ represents the functor $T$. \end{prop} \begin{proof} Let $(E,\rho,\sigma)$ be an element of $T(R)$. Pulling back the trivializations $\rho$ and $\sigma$ to $D^*_R$ provides two trivializations $\rho^*$ and $\sigma^*$ of the pull back of $E$ over ${D^*_R}$: these trivializations differ by an element $\gamma = \rho^{*-1}\circ\sigma^*$ of $ G\bigl(R((z))\bigr)$. Conversely, let us start from an element $\gamma$ of $G\bigl(R((z))\bigr)$. This element defines an isomorphism of the pullbacks over $D^*_R$ of the trivial $G$-bundle ${\cal{F}}$ over $X^*_R$ and the trivial $G$-bundle ${\cal{G}}$ over $D_R$. These two torsors glue together to a $G$-bundle $E$ in a functorial way by \cite{BL2} (in fact \cite{BL2} is written for $SL_r$ but the extension to arbitrary $G$ is straightforward). These constructions are inverse to each other by construction. \end{proof} \subsection{} Consider the functor $D_G$ which associates to a $k$-algebra $R$ the set $D_G(R)$ of isomorphism classes of pairs $(E,\rho)$, where $E$ is a $G$-bundle over $X_R$ and $\rho$ a trivialization of $E$ over $X^*_R$. \begin{prop}\label{pairs} The ind-scheme $\Q$ represents the functor $D_{G}$. \end{prop} \begin{proof} Let $R$ be a $k$-algebra and $q$ an element of $\Q(R)$. By definition there exists a faithfully flat homomorphism $R\rightarrow R'$ and an element $\gamma$ of $G\bigl(R'((z))\bigr)$ such that the image of $q$ in $\Q(R')$ is the class of $\gamma$. To $\gamma$ corresponds by Proposition \ref{triples} a triple $(E',\rho',\sigma')$ over $X_{R'}$. Let $R''=R'\otimes_R R'$, and let $(E''_1,\rho''_1)$, $(E''_2,\rho''_2)$ denote the pull-backs of $(E',\rho')$ by the two projections of $X_{R''}$ onto $X_{R'}$. Since the two images of $\gamma$ in $G\bigl(R''((z))\bigr)$ differ by an element of $G\bigl(R''[[z]]\bigr)$, these pairs are isomorphic. So the isomorphism $\rho^{\prime\prime}_2 \rho_1^{\prime\prime -1}$ over $X_{R''}^*$ extends to an isomorphism $u:E''_1\rightarrow E''_2$ over $X_{R''}$, satisfying the usual cocycle condition (it is enough to check this over $X^*$, where it is obvious). Therefore $(E',\rho')$ descends to a pair $(E,\rho)$ on $X_R$ as in the statement of the proposition. Conversely, given a pair $(E,\rho)$ as above over $X_R$, we can find a faithfully flat homomorphism $R\rightarrow R'$ and a trivialization $\sigma'$ of the pull back of $E$ over $D_{R'}$ (in fact, we can take $R'$ to be the product of henselization of each localized ring $R_x,\ x\in\Spec(R)$). By Proposition \ref{triples} we get an element $\gamma'$ of $G\bigl(R'((z))\bigr)$ such that the two images of $\gamma'$ in $G\bigl(R''((z))\bigr)$ (with $R''=R'\otimes_R R'$) differ by an element of $G\bigl(R''[[z]]\bigr)$; this gives an element of $\Q(R)$. The two constructions are clearly inverse one of each other. \end{proof} We will make use of the following theorem \begin{th}\label{Drinfeld-Simpson} (Drinfeld-Simpson \cite{DS}) Let $E$ be a $G$-bundle over $X_R$. Then the restriction of $E$ to $X^*_R$ is trivial {\it fppf} locally over $\Spec(R)$. If $char(k)$ does not divide the order of $\pi_{1}(G(\comp))$, then this is even true {\it \'etale} locally. \end{th} \subsection{Proof of Theorem \ref{th:Uniformization}.} The universal $G$-bundle $E$ over $X\times\Q$ (Proposition \ref{pairs}), gives rise to a map $\pi:\Q\rightarrow \M$. This map is $\LGX$-invariant, hence induces a morphism of stacks $\overline{\pi}:\LGX\bk\Q\rightarrow \M$. On the other hand we can define a map $\M\ra\LGX\bk\Q$ as follows. Let $R$ be a $k$-algebra, $E$ a $G$-bundle over $X_R$. For any $R$-algebra $R^{\prime}$, let $T(R^{\prime})$ be the set of trivializations $\rho$ of $E_{R^{\prime}}$ over $X^*_{R^{\prime}}$. This defines a $R$-space $T$ on which the group $\LGX$ acts. By Theorem \ref{Drinfeld-Simpson}, it is a torsor under $\LGX$. To any element of $T(R^{\prime})$ corresponds a pair $(E_{R^{\prime}},\rho)$, hence by Proposition \ref{pairs} an element of $\Q(R^{\prime})$. In this way we associate functorially to an object $E$ of $\M(R)$ a $\LGX$-equivariant map $\alpha:T\rightarrow \Q$. This defines a morphism of stacks $\M\rightarrow \LGX\bk\Q$ which is the inverse of $\overline{\pi}$. The second assertion means that for any scheme $S$ over $k$ (resp. over $k$ such that $char(k)$ does not divide the order of $\pi_{1}(G(\comp))$) and any morphism $f:T\rightarrow \M$, the pull back to $S$ of the fibration $\pi$ is {\it fppf} (resp. {\it\'etale}) locally trivial, i.e. admits local sections (for the {\it fppf} (resp. {\it \'etale}) topology). Now $f$ corresponds to a $G$-bundle $E$ over $X\times S$. Let $s\in S$. Again by Theorem \ref{Drinfeld-Simpson}, we can find an {\it fppf} (resp. {\it \'etale}) neighborhood $U$ of $s$ in $S$ and a trivialization $\rho$ of $E_{|X^*\times U}$. The pair $(E,\rho)$ defines a morphism $g:U\rightarrow \Q$ (Proposition \ref{pairs}) such that $\pi\rond g=f$, that is a section over $U$ of the pull back of the fibration $\pi$. \cqfd \section{The infinite grassmannian $\Q$} Let $G$ be semi-simple and $k$ be an algebraically closed field of characteristic $0$ in (\ref{ind-structures-are-the-same}). \subsection{}\label{mu-is-immersion} We will use the following two facts: $(a)$ We may write $\Q$ as direct limit of projective finite-dimensional $k$-schemes. $(b)$ The multiplication map $\mu:\LGmm\times \LGp \longrightarrow \LG$ is an open immersion. \noindent For the first statement, remark that it is enough to consider the $k$-space ${\cal{Q}}({\goth{g}})$ parameterizing isomorphism classes of sheaves of Lie algebra which are locally of the form $S\times{\rm Lie(G)}$ together with a trivialization over $\droitep^*$ (note that, looking at the adjoint group, $\Q$ may be seen as a connected component of ${\cal{Q}}({\goth{g}})$) then argue as in \cite{BL1}. For the second statement, the argument of (\cite{BL1} Proposition 1.11) generalizes to arbitrary $G$, once we know the following. {\em Claim.} Suppose $Y$ is a proper $S$-scheme and that the structural morphism has a section $\sigma: S\ra Y$. Suppose moreover that $G\bk H$ is a reductive subgroup of a reductive group $H$. Then, for any $G$-bundle $P$ trivial along $\sigma$ the following is true: if the associated $H$-bundle $P(H)$ is trivial, the $G$-bundle $P$ is so. Indeed, by assumption, there exists a section $\tau: Y\ra P(H)$. The quotient $G\bk H$ is affine. Therefore, the composite morphism from $Y$ to $G\bk H$ (which is the composition of $\tau$ and of the canonical projection $P(H)\ra G\bk H$ factors as $Y\ra S\hfl{p}{} G\setminus H.$ After an eventual translation of $\tau$ by an element of $H(S)$, we can assume that the restriction $\sigma^*(\tau)$ of $\tau$ along $\sigma$ is induced by the trivialization of $P$. Therefore, the morphism $p$ is the constant morphism with constant value $G\in G\bk H$. In other words, locally for the \'etale topology on $Y$, the section $\tau$ can be written $$\tau(y)=\bigl(p(y),h(y)\bigr) {\rm mod}\ G\ {\rm where\ }y\in Y\ {\rm and}\ p(y)\in P, h(y)\in G.$$ The expression $p(y)h(y)^{-1}$ is well defined and defines a section of $P$. \subsection{} \label{ind-structures-are-the-same} The quotient $\LG/\LGp$ has also been studied by Kumar and Mathieu. But the structure of ind-variety they put on the quotient is, a priori, not the same as the functorial one of section \ref{the-stack-M}. As we will use their results, we have to identify them. \begin{prop*}{} The ind-structure of $\Q$ defined in section \ref{the-stack-M} coincides with the ind-structure of Kumar and Mathieu. \end{prop*} \begin{proof} Recall that an ind-scheme is called {\em reduced} (resp. {\em irreducible, integral}) if it is a direct limit of an increasing sequence of reduced (resp. irreducible, integral) schemes. By Lemma 6.3 of \cite{BL1} an ind-scheme is integral if and only if it is irreducible and reduced. According to Faltings \cite{F} (see \cite{BL1} for the case $SL_{r}$), the ind-group $\LGm$ is integral. This may be seen by looking at $(\LGm)_{red}$ and using Shavarevich's theorem that a closed immersion of irreducible ind-affine groups which is an isomorphism on Lie algebras, is an isomorphism \cite{Sh}. Note that irreducibility is due to the fact that any element can be deformed to a constant in $G$; that $\Lie(\LGm)\rightarrow \Lie(\LGm)_{red}$ is an isomorphism can be seen by using the fact that $\Lie(G)$ is generated by nilpotent elements. It follows for semisimple $G$ that $\LGmm$, which is a semidirect product of $G$ and $\LGm$, is integral, and furthermore if $G$ is simply connected then $\Q$ is integral. Indeed by (\ref{mu-is-immersion} b) $\LGmm\ra\Q$ is an open immersion hence it is enough to show that $\Q$ is irreducible. Using that connected ind-groups are irreducible (\cite{Sh}, Proposition 3) and the quotient morphism $\LG\ra\Q$ we reduce to prove the connectedness of $\LG$ which is well known (and follows for example from uniformization for $\droitep$ and the corresponding statement for $\Mproj$ , \cf \cite{DS}). We are ready to deduce the identification of our ind-structure on $\Q$ with the one used by Kumar or Mathieu in their generalized Borel-Weil theory. Both Kumar and Mathieu define the structure of ind-variety on $\LG/\LGp$ using representation theory of Kac-Moody algebras; for instance Kumar, following Slodowy \cite{Sl}, considers the basic representation ${\cal{H}}_{\ell}$ for a fixed $\ell$, and a highest weight vector $v_{\ell}$. The subgroup $\LGp$ is the stabilizer of the line $kv_\ell$ in $\proj({\cal{H}}_{\ell})$, so the map $g\mapsto gv_\ell$ induces an injection $i_\ell:\LG/\LGp\mono\proj({\cal{H}}_{\ell})$. Let $U$ be the subgroup of $\LGp$ consisting of elements $A(z)$ such that $A(0)$ is in the unipotent part of a fixed Borel subgroup $B\subset G$; to each element $w$ of the Weyl group is associated a ``Schubert variety" $X_w$ which is a finite union of orbits of $U$. It turns out that the image under $i_\ell$ of $X_w$ is actually contained in some finite-dimensional projective subspace $\proj_w$ of $\proj({\cal{H}}_{\ell})$, and is Zariski closed in $\proj_w$. This defines on $X_w$ a structure of reduced projective variety, and a structure of ind-variety on $\LG/\LGp=\limind X_w$. By a result due to Faltings (\cf the Appendix of \cite{BL1} for $SL_{r}$), the irreducible integrable representation ${\cal{H}}_{\ell}$ of $\Lgh$ can be ``integrated" to an {\it algebraic} projective representation of $\LG$, that is a morphism of $k$-groups $\LG\rightarrow PGL({\cal{H}}_{\ell})$. It follows that the map $i_c$ is a morphism of ind-schemes of $\Q$ into $\proj({\cal{H}}_{\ell})$ (which is the direct limit of its finite-dimensional subspaces). But $i_\ell$ is even an {\it embedding}. It is injective by what we said above; let us check that it induces an injective map on the tangent spaces. Since it is equivariant under the action of $\LG$ it is enough to prove this at the origin $\omega$ of $\Q$. Then it follows from the fact that the annihilator of $v_\ell$ in the Lie algebra $\Lg$ is $\Lgp$. Therefore the restriction of $i_\ell$ to each of the subvarieties $\QN$ is proper, injective, and injective on the tangent spaces, hence is an embedding (in some finite-dimensional projective subspace of $\proj({\cal{H}}_{\ell})$). Each $X_w$ is contained in some $\QN$, and therefore is a closed subvariety of $\QNred$. Each orbit of $U$ is contained in some $X_w$; since the $X_w$'s define an ind-structure, each $\QN$ is contained in some $X_w$, so that $\QNred$ is a subvariety of $X_w$. Since $\Q$ is the direct limit of the $\QNred$, the two ind-structures coincide. \end{proof} \section{The ind-group $\LGX$} Throughout this section we suppose $k=\comp$ and $G$ semi-simple and simply connected. \begin{prop}\label{LGX-is-integral} The ind-group $\LGX$ is integral. \end{prop} \begin{cor}\label{LGX-has-no-characters} Every character $\chi:\LGX\ra G_m$ is trivial. \end{cor} \begin{proof} The differential of $\chi$, considered as a function on $\LGX$, is everywhere vanishing. Indeed, since $\chi$ is a group morphism, this means that the deduced Lie algebra morphism ${\goth{g}}\otimes A_{X}\ra k$ is zero. But as the derived algebra ${\cal D}({\goth{g}}\otimes A_X)$ is ${\cal D}({\goth{g}})\otimes A_X$ and therefore equal to ${\goth{g}}\otimes A_X$ because ${\goth{g}}$ is simple, any Lie algebra morphism ${\goth{g}}\otimes A_X\ra k$ is trivial. As $\LGX$ is integral we can write $\LGX$ as the direct limit of integral varieties $V_n$. The restriction of $\chi$ to $V_{n}$ has again zero derivative and is therefore constant. For large $n$, the varieties $V_{n}$ contain $1$. This implies $\chi_{\mid V_{n}}=1$ and we are done. \end{proof} \begin{proof} To see that the ind-group $\LGX$ is reduced, consider the \'etale trivial morphism $\bar\pi:{\cal{Q}}\ra\M$. Locally for the \'etale topology, $\bar\pi$ is a product $\Omega\times\LGX$ (where $\Omega$ is an \'etale neighborhood of $\M$). Then use that ${\cal{Q}}$ is reduced by (\ref{ind-structures-are-the-same}). As connected ind-groups are irreducible by Proposition 3 of \cite{Sh} it is enough to show that $\LGX$ is connected. The idea how to prove that $\LGX$ is connected is due to V. Drinfeld: consider distinct points $p_{1},\dots, p_{i}$ of $X$ which are all distinct from $p$. Define $X^{*}_i=X\moins\{p,p_{1},\dots,p_{i}\}$ and, for every $k$-algebra $R$, define $X_{i,R}^{*} = X^{*}_i\times_k\Spec(R)$. Denote by $A_{X_{i,R}}$ the $k$-algebra $\Gamma(X_{i,R}^{*},{\cal{O}}_{X^{*}_{i,R}})$ and by $\LGXi$ the $k$-group $R\mapsto G(A_{X_{i,R}})$. As $\LGX$, the $k$-group $\LGXi$ is an ind-group (\cf \ref{Loop-groups}). The natural inclusion $A_{X_{i,R}}\subset A_{X_{i+1,R}}$ defines a closed immersion $f:\LGXi\ra\LGXip$. \begin{lem}\label{pi_0} The closed immersion $\LGXi\ra\LGXip$ defines a bijection $$\pi_{0}(\LGXi)\isom\pi_{0}(\LGXip).$$ \end{lem} \begin{proof} Consider the morphism $\LGXip\ra\LG$ defined by the developpement in Laurent series at $p_{i+1}$. We get a morphism $\phi_{i+1}:\LGXip\ra\Qp$, where we denote $\Qp=\LG/\LGp$. (of course $\Q=\Qp$ but we emphasize here that we will consider the point $p_{i+1}$ and not $p$.) \medskip \noindent{\em Claim:} The morphism $\phi_{i+1}:\LGXip\ra\Qp$ induces an isomorphism on the level of stacks $\bar\phi_{i+1}:\LGXip/\LGXi\simeq\Qp$ and is locally trivial for the \'etale topology. \medskip The lemma reduces to the claim. Indeed, as $G$ is semi-simple and simply connected, we have $\pi_{i}([\Qp]^{an})=1$ for $i=0,1$ (by (\ref{ind-structures-are-the-same}) and Kumar and Mathieu) and the exact homotopy sequence associated of the (Serre)-fibration $\phi_{i+1}$ shows that $\pi_{0}([\LGXi]^{an})\isom\pi_{0}([\LGXip]^{an})$ ($[]^{an}$ means we consider the usual topology). From the bijection $\pi_{0}(\LGXiN)\isom\pi_{0}([\LGXiN]^{an})$ and Proposition 2 of \cite{Sh} it follows then that $\pi_{0}(\LGXi)\ra\pi_{0}([\LGXi]^{an})$ is bijective. \medskip \noindent{\em Proof of the claim:} Clearly $\phi_{i+1}:\LGXip\ra\Qp$ is $\LGXi$ invariant, hence defines a map $\bar\phi_{i+1}:\LGXip/\LGXi\ra\Qp$. Define a morphism $\Qp\ra\LGXip/\LGXi$ as follows. Let $R$ be a $k$-algebra. By Proposition \ref{pairs} to an element of $\Qp(R)$ corresponds a $G$-bundle $E\ra X_R$ together with a trivialization $\tau_{p_{i+1}}:G\times X_{p_{i+1},R}^*\ra E_{\mid X_{p_{i+1},R}^*}$. Here by $X_{p_{i+1},R}^*$ we denote $(X\moins\{p_{i+1}\})\times_k\Spec(R)$. For any $R$-algebra $R^\prime$, denote $T(R^\prime)$ the set of trivializations $\tau_i$ of $E_{R^\prime}$ over $X_{i,R}^{*}$. This defines a $R$-space $T$ on which $\LGXi$ acts. By Theorem \ref{Drinfeld-Simpson} it is a torsor under $\LGXi$. For any $\tau_i\in T(R^\prime)$ the composite $\tau_i^{-1}\circ\tau_{p_{i+1}}$ defines a morphism $X_{i+1,R}^{*}\ra G$ hence an element of $\LGXip(R)$. In this way we associate functorially to an object $(E,\tau_{p_{i+1}})$ of $\Qp(R)$ a $\LGXi$-invariant map $\alpha:T\ra\LGXip$, which defines the inverse of $\bar\phi$. The assertion concerning the local triviality is proved as in Theorem \ref{th:Uniformization}. \end{proof} Let us show that every element $g\in\LGX(k)$ is in the connected component of the unit of $\LGX(k)$. Using the well known fact that $G(K)$ is generated by the standard unipotent subgroups $U_{\alpha}(K)$, $\alpha\in\Delta$, we may suppose that $g$ is of the form $\prod_{j\in J}\exp(f_{j}n_{j})$ where the $n_{j}$ are nilpotent elements of $\g$ and $f_{j}\in K$. Let $\{p_{1},\dots,p_{i}\}$ be the poles of the functions $f_{j}, j\in J$. The morphism $$\begin{diagram} {\tr{A}}^{1}&\lra&\LGXi\\ t&\mapsto&\prod_{j\in J}\exp(tf_{j}n_{j})\\ \end{diagram} $$ is a path from $g$ to $1$ in $\LGXi$. By Lemma \ref{pi_0}, the morphism $\pi_{0}(\LGX)\ra\pi_{0}(\LGXi)$ is bijective which proves that $g$ and $1$ are indeed in the same connected component of $\LGX$. \end{proof} \section{Pfaffians} Let $k$ be an algebraically closed field of characteristic $\not=2$ and $S$ a $k$-scheme. \subsection{The Picard categories} Let $A$ be $\reln$ or $\reln/2\reln$. Denote by ${\goth{L}}_{A}$ the groupoid of $A$-graded invertible ${\cal{O}}_{S}$-modules. The objects of ${\goth{L}}_{A}$ are pairs $[L]=(L,a)$ of invertible ${\cal{O}}_{S}$-modules $L$ and locally constant functions $a:S\ra A$, morphisms $[f]:[L]\ra[M]$ are defined if $a=b$ and are isomorphisms $f:L\ra M$ of ${\cal{O}}_{S}$-modules. Denote $\idbb_{A}$ the object $({\cal{O}}_{S},0)$. The category ${\goth{L}}_{A}$ has tensor products, defined by $[L]\otimes[M]=(L\otimes M,a+b).$ Given $[L]$ and $[M]$ we have Koszul's symmetry isomorphism $\sigma_{_{[L],[M]}}: [L]\otimes[M]\ra[M]\otimes[L]$ defined on local sections $\ell$ and $m$ by $\sigma_{_{L,M}}(\ell\otimes m)=(-1)^{ab}m\otimes\ell$. Denote $\det_{A}$ the functor from the category of coherent locally free ${\cal{O}}_{S}$-modules with isomorphisms defined by $\det_{A}=(\Lambda^{max},\rang(V))$ and $\det_{A}(f)=\Lambda^{max}(f)$. In the following we drop the subscript $A$ for $A=\reln$ and replace it by $2$ for $A=\reln/2\reln$. \subsection{Pfaffians}\label{Pfaffians-generalities} Let $V$ be a coherent locally free ${\cal{O}}_{S}$-module of rank $2n$. Let $\Pf:\Lambda^{2}V^{*}\ra\Lambda^{2n}V^{*}$ be the unique map that commutes with base changes and such that if $(e^{*}_{1},\dots,e^{*}_{2n})$ is a local frame of $V^{*}$ and $\alpha=\Sigma_{i<j}a_{ij}e^{*}_{i}\wedge e^{*}_{j}$, then $$\Pf(\alpha)=\pf(a)e_{1}^{*}\wedge\dots\wedge e^{*}_{2n}$$ where $\pf(a)$ is the pfaffian [Bourbaki, Alg\`ebre 9.5.2] of the alternating matrix $a_{ij}=-a_{ji}$ for $i>j$ and $a_{ii}=0$ for $i=1, \dots, 2n$. Suppose $\alpha:V\ra V^{*}$ is skewsymmetric. View $\alpha$ as a section of $\Lambda^{2}V^{*}$ and define the {\em pfaffian} of $\alpha$ as the section $\Pf(\alpha):{\cal{O}}_{S}\ra\Lambda^{2n}V^{*}$. By [Bourbaki, Alg\`ebre 9.5.2] we know that \begin{equation}\label{square} \begin{diagram} {\cal{O}}_{S}\otimes{\cal{O}}_{S}&\lra&{\cal{O}}_{S}\\ \sfl{\Pf(\alpha)\otimes\Pf(\alpha)}{}&\swfl{}{\det(\alpha)}\\ \Lambda^{2n}V^{*}\otimes\Lambda^{2n}V^{*}\\ \end{diagram} \end{equation} commutes and that, if $u$ is an endomorphism of $V^{*}$, then \begin{equation}\label{sbc}\begin{diagram} {\cal{O}}_{S}&\efl{\Pf(\alpha)}{}&\Lambda^{2n}V^{*}\\ \sfl{\Pf(u\alpha u^{*})}{}&\swfl{}{\det(u)}\\ \Lambda^{2n}V^{*}\\ \end{diagram} \end{equation} commutes. \subsection{The pfaffian functor} We consider the following category ${\cal{A}}={\cal{A}}^{\bullet}(S)$: objects are complexes of locally free coherent ${\cal{O}}_{S}$-modules concentrated in degrees $0$ and $1$ of the form $$0\lra E\efl{\alpha}{} E^{*}\lra 0$$ with $\alpha$ skewsymmetric. Morphisms between two such complexes $E^{\bullet}$ and $F^{\bullet}$ are morphisms of complexes $f^{\bullet}:E^{\bullet}\lra F^\bullet$ such that $f^{\bullet*}[-1]$ is a homotopy inverse of $f^{\bullet}$, \ie $f^{\bullet*}[-1]\circ f^{\bullet}$ and $f^{\bullet}\circ f^{\bullet*}[-1]$ are homotopic to the identity. Let $\pi:{\goth{L}}\ra{\goth{L}}_{2}$ be the projection functor, $\Delta:{\goth{L}}_{2}\ra{\goth{L}}_{2}$ be the functor defined by $[L]\mapsto[L]\otimes[L]$ and $[f]\mapsto[f]\otimes[f]$ and $\Det:{\cal{A}}\ra{\goth{L}}$ be the determinant functor \cite{KM} . \begin{prop}\label{pfaffianfunctor} There is a natural functor, $\Pf:{\cal{A}}^{\bullet}\ra{\goth{L}}_{2}$, commuting with base changes, and a natural isomorphism of functors: $$\pi\circ\Det\isom\Delta\circ\Pf.$$ Moreover, if $f^{\bullet}:E^{\bullet}\lra E^\bullet$ is homotopic to the identity then $\Pf(f^{\bullet})=\id$. \end{prop} \begin{proof} Define $\Pf$ on the level of objects by $\Pf(E^{\bullet})=\det_{2}(E)$. On the level of morphisms we do the following. Let $f^{\bullet}=(f_{0},f_{1}):E^{\bullet}\lra F^{\bullet}$ be a morphism of ${\cal{A}}^{\bullet}$: $$\begin{diagram} E&\efl{\alpha_{E}}{}&E^{*}\cr \sfl{f_{0}}{}&&\sfl{f_{1}}{}\cr F&\efl{\alpha_{F}}{}&F^{*}\cr \end{diagram} $$ and consider the complex $C^{\bullet}_{f}$ (which is up to sign the cone of $f^{\bullet}$) \bigskip\bigskip $$C_{f}^{\bullet}= 0\lra E\efl{\begin{pmatrix}\alpha_{E}\cr -f_{0}\end{pmatrix}}{} E^{*}\oplus F\efl{(f_{1}\ \alpha_{F})}{}F^{*}\lra 0$$ As $f^{\bullet}$ is a quasi-isomorphism, $C^{\bullet}_{f}$ is acyclic. By the usual additivity property of determinants, we get a canonical isomorphism $$d(f):\Lambda^{max}E\otimes\Lambda^{max}F^{*}\ra\Lambda^{max}E^{*} \otimes\Lambda^{max}F.$$ Recall that this isomorphism is defined by taking a section $\begin{pmatrix}u\\ v\end{pmatrix}$ of $(f_{1}\ \alpha_{F})$ and calculating the determinant, which is independent of this choice, of the morphism $$M(f)=\begin{pmatrix}\alpha&u\cr -f_{0}&v\end{pmatrix}\in \Hom(E\oplus F^{*},E^{*}\oplus F)$$ \begin{lem}\label{compagnion} There is a skew-symmetric morphism $\gamma_{f}\in\Hom(F^{*},F)$ such that $\begin{pmatrix}f_{0}^{*}\\ \gamma\end{pmatrix}$ is a section of $(f_{1}\ \alpha_{F})$. \end{lem} \begin{proof} As $f\circ f^{*}[-1]$ is homotopic to $\Id$ there is a morphism $h$ such that $f_{0}f_{1}^{*}-1=h\alpha_{F}$ and $f_{1}f_{0}^{*}-1=\alpha_{F} h$. Now define $\gamma_{f}=\frac{h^{*}-h}{2}$. \end{proof} The pfaffian of the skew-symmetric morphism $$M(f,\gamma_{f})=\begin{pmatrix}\alpha&f_{0}^{*}\cr -f_{0}& \gamma_{f}\end{pmatrix}\in\Hom(E\oplus F^{*},E^{*}\oplus F)$$ defines a section $\pf(M(f,\gamma_{f})):{\cal{O}}_{S}\ra \Lambda^{max}E^{*}\otimes\Lambda^{max}F$. \begin{lem} The section $\pf(M(f,\gamma_{f}))$ is independent of the choice of $\gamma_{f}$. \end{lem} \begin{proof} Suppose $\gamma_{f}^{\prime}$ is another morphism satisfying (\ref{compagnion}). Then there is $g\in\Hom(F^{*},E)$ such that $\alpha_{E}g=0$ and $f_{0}g=-g^{*}f_{0}^{*}$ [use that $\gamma_{f}$ and $\gamma_{f}^{\prime}$ are skew]. These relations give $$M(f,\gamma_{f}^{\prime})= \begin{pmatrix}1&0\cr {\frac{g}{2}}^{*}&1\end{pmatrix} M(f,\gamma_{f})\begin{pmatrix}1&\frac{g}{2}\cr 0&1\end{pmatrix}$$ which in turn implies the required equality by (\ref{sbc}). \end{proof} As $\rank(E)=\rank(F)\bmod 2$, we get the isomorphism in ${\goth{L}}_{2}$: $$\pf(M(f)):\idbb_{2}\isom\det_{2}(E)^{*}\otimes\det_{2}(F).$$ Define the pfaffian of $f^{\bullet}$ by $$\Pf(f^{\bullet}):\det_{2}(E)\efl{1\otimes\pf(M(f,\gamma_{f}))}{} \det_{2}(E)\otimes\det_{2}(E)^{*}\otimes\det_{2}(F)\efl{\ev_{_{\det_{2}(E)}}}{} \det_{2}(F)$$ \begin{lem} $\Pf:{\cal{A}}\ra{\goth{L}}_{2}$ defines a functor. \end{lem} \begin{proof} As $\pf(M(\Id,0))=1$, we have $\Pf(\Id)=\Id$. Let $f^{\bullet}:E^{\bullet}\ra F^{\bullet}$ and $g:F^{\bullet}\ra G^{\bullet}$ be two morphisms of ${\cal{A}}$. Then the following diagram is commutative $$ \begin{diagram} \idbb_{2}&\efl{\pf(M(f,\gamma_{f}))\otimes\pf(M(g,\gamma_{g}))}{}& \det_{2}(E)^{*}\otimes\det_{2}(F)\otimes\det_{2}(F)^{*}\otimes\det_{2}(G)\\ \sfl{\Id}{}&&\sfl{}{1\otimes\ev_{\det_{2}(F)}\otimes 1}\\ \idbb_{2}&\efl{\pf(M(g\circ f,\gamma_{g\circ f}))}{}& \det_{2}(E)^{*}\otimes\det_{2}(G)\\ \end{diagram} $$ Indeed, remark that $\gamma_{gf}=g_{0}\gamma_{f}g_{0}^{*}+\gamma_{g}$ satisfies (\ref{compagnion}) for $g\circ f$ and make use of (\ref{sbc}) first with $$ \begin{pmatrix} \alpha_{E}&f_{0}^{*}&0&f_{0}^{*}g_{0}^{*}\\ -f_{0}&\gamma_{f}&1&\gamma_{f}g_{0}^{*}\\ 0&-1&0&0\\ -g_{0}f_{0}&g_{0}\gamma_{f}&0&\gamma_{gf}\\ \end{pmatrix} = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ -f_{1}&-\alpha_{F}&1&0\\ 0&g_{0}&0&1\\ \end{pmatrix} \begin{pmatrix} \alpha_{E}&f_{0}^{*}&0&0\\ -f_{0}&\gamma_{f}&0&0\\ 0&0&\alpha_{F}&g_{0}^{*}\\ 0&0&-g_{0}&\gamma_{g}\\ \end{pmatrix} \begin{pmatrix} 1&0&-f_{1}^{*}&0\\ 0&1&\alpha_{F}&g_{0}^{*}\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix}, $$ and then with $$ \begin{pmatrix} \alpha_{E}&0&0&f_{0}^{*}g_{0}^{*}\\ 0&\gamma_{f}&1&0\\ 0&-1&0&0\\ -g_{0}f_{0}&0&0&\gamma_{gf}\\ \end{pmatrix} = \begin{pmatrix} 1&0&f_{0}^{*}&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&g_{0}\gamma_{f}&1\\ \end{pmatrix} \begin{pmatrix} \alpha_{E}&f_{0}^{*}&0&f_{0}^{*}g_{0}^{*}\\ -f_{0}&\gamma_{f}&1&\gamma_{f}g_{0}^{*}\\ 0&-1&0&0\\ -g_{0}f_{0}&g_{0}\gamma_{f}&0&\gamma_{gf}\\ \end{pmatrix} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ f_{0}&0&1&-\gamma_{f}g_{0}^{*}\\ 0&0&0&1\\ \end{pmatrix} $$ The commutativity of the above diagram shows $\Pf(g\circ f)=\Pf(g)\circ\Pf(f)$. \end{proof} The statement on the natural transformation follows from the definitions and (\ref{square}). It remains to prove that if $f^{\bullet}:E^{\bullet}\ra E^{\bullet}$ is homotopic to the identity, then $\Pf(f)=\Id$. Indeed, let $h:E^{*}\ra E$ be such that $f_{0}-h\alpha_{E}=1$ and $f_{1}-\alpha h=1$. Then $\gamma_{f}:=-h+f_{0}h^{*}$ satisfies (\ref{compagnion}) and the statement follows from (\ref{sbc}) and $$ \begin{pmatrix}a_{E}&1\\-1&0\end{pmatrix}= \begin{pmatrix}1&0\\ h&1\end{pmatrix} \begin{pmatrix}a_{E}&f_{0}^{*}\\ -f_{0}^{*}&\gamma_{f}\end{pmatrix} \begin{pmatrix}1&h^{*}\\ 0&1\end{pmatrix}$$ This completes the proof of Proposition \ref{pfaffianfunctor}. \end{proof} \section{Line bundles on $\M$.} Let $k$ be an algebraically closed field of characteristic $\not=2$ in Sections (\ref{the-pf-bundle})-(\ref{dualizing-sheaf}). Denote $\Pic(\M)$ the group of isomorphism classes of line bundles on $\M$. (See (\cite{BL1}, 3.7) for a discussion of line bundles over $k$-spaces and stacks). We will construct special elements of $\Pic(\M)$. \subsection{The determinant line bundle.}\label{the-det-bundle} We start with the well known case of $G=GL_{r}$: let ${\cal{F}}$ be a family of vector bundle of rank $r$ parameterized by the locally noetherian $k$-scheme $S$. Recall that the complex $Rpr_{1*}({\cal{F}})$ may be represented by a perfect complex of length one $K^{\bullet}$ and define ${\scr{D}}_{{\cal{F}}}$ to be $\det(K^{\bullet})^{-1}$. This does not, up to canonical isomorphism, depend on the choice of $K^{\bullet}$. As the formation of the determinant commutes with base change, the fiber of ${\scr{D}}_{{\cal{F}}}$ over the point $s\in S$ is $\Lambda^{max}H^{0}(X,{\cal{F}}(s))^{*}\otimes \Lambda^{max}H^{1}(X,{\cal{F}}(s))$. The line bundle ${\scr{D}}_{{\cal{F}}}$ is called the {\em determinant of cohomology} line bundle associated to the family ${\cal{F}}$. Let ${\cal{U}}$ be the universal vector bundle on $\MGL\times X$ and define the determinant line bundle ${\scr{D}}=\det(Rpr_{1*}{\cal{U}})^{-1}$. It has the following universal property: for every family ${\cal{F}}$ of vector bundle parameterized by the locally noetherian $k$-scheme $S$, we have $f_{{\cal{F}}}^{*}({\scr{D}})={\scr{D}}_{{\cal{F}}}$ where $f_{{\cal{F}}}:S\ra\MGL$ is the deduced modular morphism. For the case of general $G$, consider a representation $\rho:G\ra GL_{r}$ and consider the morphism obtained by extension of structure group $f_{\rho}:\M\ra\MGL$. Then define the determinant of cohomology associated to $\rho$ by ${\scr{D}}_{\rho}=f_{\rho}^{*}({\scr{D}})$. \subsection{The Pfaffian bundle}\label{the-pf-bundle} Consider $G=\Spin_{r}$ with $r\geq 3$ (resp. $G=G_2$). Then the standard representation $\varpi_{1}$ factors through $SO_{r}$ (resp. $SO_{7}$). The stack $\MSO$ has two components: $\MSOzero$ and $\MSOone$. They are distinguished by the second Stiefel-Whitney class $$w_{2}: H^{1}_{\acute et}(X,SO_{r})\ra\H^{2}_{\acute et}(X,\reln/2\reln)=\reln/2\reln.$$ Let $\kappa$ be a theta-characteristic on $X$. Twisting by $\kappa$, we may and will see a $SO_{r}$-bundle as a vector bundle $F$ with trivial determinant together with a {\em symmetric} isomorphism $\sigma:F\ra F^{\vee}$, where $F^{\vee}=\ul\Hom_{{\cal{O}}_{X}}(F,\omega_{_{X}})$. The following Proposition shows the existence, for every $\kappa$, of a canonical square root ${\scr{P}}_{\kappa}$ of the determinant bundle ${\scr{D}}_{\varpi_{1}}$ over $\MSO$. \begin{prop} Let $(F,\sigma)$ be a family of vector bundles $F$ equipped with a quadratic form $\sigma$ with values in $\omega_{_{X}}$ parameterized by the locally noetherian $k$-scheme $S$. Then the determinant of cohomology ${\cal{D}}_{F}$ admits a canonical square root ${\cal{P}}_{(F,\sigma)}$. Moreover, if $f:S^{\prime}\ra S$ is a morphism of locally noetherian $k$-schemes then we have ${\cal{P}}_{(f^{*}F,f^{*}\sigma)}=f^{*}{\cal{P}}_{(F,\sigma)}$. \end{prop} \begin{proof} By (\cite{So2}, prop. 2.1 and proof of corollary 2.2, \cf also \cite{Ke}), Zariski locally on $S$, there are length $1$ complexes $M^{\bullet}$ of finite free ${\cal{O}}_{S}$-modules and quasi-isomorphisms $f:M^{\bullet}\ra Rpr_{1*}(F)$ such that the composition in the derived category $D(S)$ (use $\sigma$ and Grothendieck duality) $$M^{\bullet}\efl{f}{} Rpr_{1*}(F)\efl{\tau}{} \R\ul\Hom^{\bullet}( Rpr_{1*}(F),{\cal{O}}_{S})[-1]\efl{f^{*}[-1]}{} M^{\bullet*}[-1]$$ lifts to a symmetric isomorphism of complexes $\varphi:M^{\bullet}\ra M^{\bullet*}[-1]$: $$ \begin{diagram} 0&\lra&M^{0}&\efl{d_{M^{\bullet}}}{}&M^{1}&\lra&0\\ &&\sfl{\varphi_{0}}{\wr}&\sefl{}{\alpha}&\sfl{\wr}{\varphi_{0}^{*}}\\ 0&\lra&M^{1*}&\efl{-d_{M^{\bullet *}}}{}&M^{0*}&\lra&0\\ \end{diagram} $$ \comment{ [Indeed, for every $s\in S$ we may find, in a Zariski neighborhood $U$ of $s$, a length $1$ complex $(M^{\bullet},d_{M^{\bullet}})$ of finite free ${\cal{O}}_{S}$-modules such that $d_{M^{\bullet}}(s)=0$ and a quasi-isomorphism (over $U$) $f:M^{\bullet}\ra Rpr_{1*}(F)$. The isomorphism (in $D^{b}(U)$) $f^{*}[-1]\circ\tau\circ f:M^{\bullet}\ra M^{\bullet*}[-1]$ lifts to a morphism of complexes $\phi$, as the components of $M^{\bullet}$ are free. As $\tau$ is symmetric, the symmetrization $\varphi$ of $\phi$ still lifts $f^{*}[-1]\circ\tau\circ f$ and as $\varphi$ is an isomorphism at the point $s$, $\varphi$ remains an isomorphism, after eventually shrinking $U$, over $U$.] } Define $\widetilde{M}^{\bullet}$ by $0\ra M^{0}\efl{\alpha}{}M^{0*}\ra 0.$ Then $\alpha$ is skew and we have a natural isomorphism of complexes $\psi:M^{\bullet}\ra\widetilde{M}^{\bullet}$ such that $\psi^{*}[-1]\psi=f^{*}[-1]\tau f$ in $D^{b}(S)$. Cover $S$ by open subsets $U_{i}$ together with complexes $(M_{i}^{\bullet},d_{M^{\bullet}_{i}})$ and quasi-isomorphisms $f_{i}:M_{i}^{\bullet}\ra Rpr_{1*}(F)\restriction{U_{i}}$ as above. We define ${\cal{P}}_{(F,\sigma)}$ over $U_{i}$ by ${\cal{P}}_{i,(F,\sigma)}=\Pf(\widetilde{M}^{\bullet}_{i})$ and construct patching data $\rho_{ij}:{\cal{P}}_{i,(F,\sigma)}\isom{\cal{P}}_{j,(F,\sigma)}$ over $U_{ij}=U_{i}\cap U_{j}$ in the following way. \comment{ Consider the diagram of isomorphisms in $D^{b}(U_{ij})$, with $K_{ij}^{\bullet}=Rpr_{1*}(F)\restriction{U_{ij}}$ $$\begin{diagram}\widetilde{M_i}&\efl{{\psi_i^{-1}}}{}&M_{i}^{\bullet}&\\ &&\sfl{f_{i}}{}\cr M^{\bullet}_{j}&\efl{f_{j}}{}&K_{ij}^{\bullet}&\efl{\tau}{}& K_{ij}^{\bullet*}[-1]& \efl{f_{j}^{*}[-1]}{}& M^{\bullet*}_{j}[-1]\\ &&\sfl{\tau}{}&&&&\sfl{\wr}{\psi_j^*[-1]}\cr &&K_{ij}^{\bullet*}[-1] &&&&\widetilde{M_j^\bullet}{}^*[-1]\cr &&\sfl{f_{i}^{*}[-1]}{}\cr &&M^{\bullet*}[-1]_{i}\cr \end{diagram} $$ } Define first the morphism of complexes $\Sigma_{ij}:\ \widetilde{M}_{i}\ra\widetilde{M}_{j}$ as a lifting of the isomorphism in $D^{b}(U_{ij})$ $$\psi_{j}^{*-1}[-1]f_{j}^{*}[-1]\tau f_{i}\psi_{i}^{-1},$$ then $\rho_{ij}$ by $\Pf(\Sigma_{ij})$ [note that it follows from the symmetry of $\sigma$ (and that the components of the $\widetilde{M}_{i}$ are free) that $\Sigma_{ij}$ is a morphism of ${\cal{A}}(U_{ij})$]. By \ref{pfaffianfunctor}, $\rho_{ij}$ does not depend on the particular chosen lifting and the functoriality of $\Pf$ translates into $\rho_{ii}=\Id$, $\rho_{ij}=\rho_{ik}\rho_{kj}$ and also $\rho_{ij}=\rho_{ji}^{-1}$. Over $U_{i}$ we have ${\cal{P}}_{i,(F,\sigma)}\otimes{\cal{P}}_{i,(F,\sigma)} =\det(\widetilde{M}_{i}^{\bullet})$. As usual, the $\det(\widetilde{M}_{i}^{\bullet})$ path together (via $f_{i}\psi_{i}^{-1}$), to ${\scr{D}}_{F}$ and we get, again by Proposition \ref{pfaffianfunctor}, a canonical isomorphism ${\cal{P}}_{(F,\sigma)}\otimes {\cal{P}}_{(F,\sigma)}\isom {\cal{D}}_{F}.$ \end{proof} \subsection{} Considering the universal family over $\MSO\times X$, we get, by the above, for every theta-characteristic $\kappa$ a line bundle ${\scr{P}}_{\kappa}$ over $\MSO$. Consider $$e:\MSpin\lra\MSOzero$$ defined by extension of the structure group. This morphism defines a morphism on the level of Picard groups hence we can define a line bundle, denoted by ${\scr{P}}$, which is the pullback of the pfaffian line bundle ${\scr{P}}_{\kappa}$. We omit here the index $\kappa$ as we will see that ${\scr{P}}$ on $\MSpin$ does not depend on the choice of a particular theta-characteristic. In the same way, we define the line bundle ${\scr{P}}$ on $\MGtwo$. \subsection{The pfaffian divisor.} Let $r\geq 3$ and $({\cal E},q)$ be the universal quadratic bundle over $\MSOzero\times X$. For $\kappa$ a theta-characteristic, let us denote by $\Theta_\kappa$ the substack defined by $$\Theta_\kappa={\rm div}(Rpr_{1*}({\cal E}\otimes pr_2^*\kappa)).$$ {\em Claim: This substack is a divisor if and only if $r$ or $\kappa$ are even.} \begin{proof} Let $P=(E,q)$ be a $SO_r$-bundle, $r\geq 3$ and $\kappa$ be a theta-characteristic. Then \begin{equation}\label{w2} w_2(P)=\h^0(E\otimes\kappa)+r\h^0(\kappa)\bmod 2. \end{equation} Indeed, by Riemanns invariance mod $2$ theorem, the right hand side of (\ref{w2}) denoted $\bar w_2(P)$ in the following, is constant over the $2$ connected components of $\MSO$. Because (\ref{w2}) is true at the trivial $SO_{r}$-bundle ${\cal{T}}$, it is enough to prove that $\bar w_2$ is not constant. Let $L,M\in J_{2}$ (where $J_{2}=$ points of order 2 of the jacobian) such that for the Weyl pairing $<L,M>=1$. The choice of a trivialization of their square defines a non degenerated quadratic form on $E=(L\otimes M)\oplus L\oplus M\oplus (r-3){\cal{O}}_{X}$ hence a $SO_r$-bundle $P$. By \cite{Mu}, we know that we have $\bar w_2(P)=<L,M>\not=0=\bar w_2({\cal{T}})$, which proves (\ref{w2}). Choose an ineffective theta-characteristic $\kappa_0$ and set $L=\kappa_0\otimes\kappa^{-1}$. If $r$ is even, there exists a $SO_r$-bundle $P=(E,q)$ such that $H^0(E\otimes\kappa)=0$ and $w_2(P)=0$ (choose $E=rL$ with $L\in J_{2}$ and use (\ref{w2})). If $r$ is odd and $\kappa$ is even, there exists a $SO_r$-bundle $P=(E,q)$ such that $H^0(E\otimes\kappa)=0$ and $w_2(P)=0$ (by (\cite{Be}, lemme 1.5), there is a $SL_2$-bundle $F$ on $X$ such that $H^0(X,{\rm ad}(F)\otimes \kappa)=0$, then choose $E={\rm ad}(F)\oplus (r-3)L$ with the obvious quadratic form.) If $r$ and $\kappa$ are odd, then $H^0(E\otimes\kappa)$ is odd for all $P\in\MSOzero$. \end{proof} As the perfect complex $Rpr_{1*}({\cal E}\otimes pr_2^*\kappa)$ can be locally represented by a skew-symmetric perfect complex of length one $L\efl{\alpha}{}L^*$, the pfaffian of $\alpha$ defines a local equation of an effective divisor ${\Theta_\kappa\over 2}$ such that $2{\Theta_\kappa\over 2}=\Theta_\kappa$. This gives an easier way to define, by smoothness of $\M$, the pfaffian line bundle. The reason which motivated our construction above was to define this square root for arbitrary quadratic bundles (not only the even ones) and to make a construction for all theta-characteristics and not only the even ones (when $r$ is odd). \subsection{Proof of \ref{sr}.}\label{dualizing-sheaf} The dualizing line bundle $\omega_{_{\M}}$ on $\M$ is by definition the determinant line bundle of the cotangent complex of $\M$. Let $\Ad:G\ra\GL({\goth{g}})$ be the adjoint representation. Then $\omega_{_{\M}}={\scr{D}}_{\Ad}^{-1}$. Suppose that $G$ is semi-simple. Then the adjoint representation factors through the special orthogonal group because of the existence of the Cartan-Killing form. Choose a theta-characteristic $\kappa$ on $X$. Then, as in (\ref{the-pf-bundle}), we can define a square root $\omega^{\frac{1}{2}}_{_{\M}}(\kappa)$ of $\omega_{_{\M}}$. \section{The Picard group of $\M$.}\label{section-pic-of-M} Throughout the section suppose that $k=\comp$ and that $G$ is simple and simply connected. Let $\Pic_{\LGX}(\Q)$ the group of $\LGX$-linearized line bundles on $\Q$. Recall that a $\LGX$-linearization of the line bundle ${\scr{L}}$ on $\Q$ is an isomorphism $m^{*}{\scr{L}}\isom pr_{2}^{*}{\scr{L}}$, where $m:\LGX\times\Q\ra\Q$ is the action of $\LGX$ on $\Q$, satisfying the usual cocycle condition. Consider the projection $\pi:\Q\ra\M$ of Theorem $\ref{th:Uniformization}$. Let ${\cal{L}}$ be a line bundle on $\M$. As $\pi^{*}$ induces an isomorphism between the sections of ${\cal{L}}$ and $\LGX$-invariant sections of $\pi^{*}{\cal{L}}$ (\cite{BL1}, Lemma 7.2), we have \begin{prop}\label{pullback-is-injective} The projection $\pi:\Q\ra\M$ induces an injection $$\pi^{*}:\Pic(\M)\hookrightarrow\Pic_{\LGX}(\Q).$$ \end{prop} Any $\LGX$-linearization is necessarily unique: \begin{prop}\label{unique-linearization} The forgetful morphism $\Pic_{\LGX}(\Q)\ra\Pic(\Q)$ is injective. \end{prop} \begin{proof} The kernel of this morphism consists of the $\LGX$-linearizations of the trivial bundle. Any to such trivializations differ by an automorphisms of $pr_{2}^{*}{\cal{O}}_{\Q}$ that is by an invertible function on $\LGX\times\Q$. Since $\Q$ is integral (\ref{ind-structures-are-the-same}), it is the direct limit of the integral projective varieties $\Q=\limind\QNred$ and this function is the pull back of an invertible function $f$ on $\LGX$. The cocycle conditions on the linearizations imply that $f$ is a character, hence $f=1$ by Lemma \ref{LGX-has-no-characters}. \end{proof} \subsection{$\Pic(Q(G))$ and the canonical central extension of $LG$.} \label{Pic(Q)} Consider the embedding $\LG/\LGp\hookrightarrow\proj({\cal{H}}_1)$ of (\ref{ind-structures-are-the-same}) and define ${\cal{O}}_{\Q}(1)$ as the pullback of ${\cal{O}}_{\proj({\cal{H}}_1)}(1)$. By \cite{Ma} and \cite{KNR}, we know $\Pic(\Q)=\reln{\cal{O}}_{\Q}(1)$. The $k$-group $\LG$ acts on $\Q$ but the action does not lift to an action of $\LG$ on ${\cal{O}}_{\Q}(1)$. There is a canonical device to produce an extension $\LGh$ of $\LG$ such that the induced action of $\LGh$ lifts to an action of $\LGh$ on ${\cal{O}}_{\Q}(1)$: the Mumford group. This is the group of pairs $(g,f)$ where $g\in\LG$ and $f:g^{*}{\cal{O}}_{\Q}(1)\isom{\cal{O}}_{\Q}(1)$. We get a central extension (note that $\Q$ is direct limit of projective integral schemes) \begin{equation}\label{can-ext} 1\lra G_{m}\lra\LGh\lra\LG\lra 1 \end{equation} Note that the Mumford group of ${\cal{O}}_{\proj({\cal{H}}_1)}(1)$ is $GL({\cal{H}}_1)$. As the projective representation ${\cal{H}}_1$ of $\Lg$ can be integrated to a projective representation $\phi:\LG\ra PGL({\cal{H}}_1)$ (\cf \ref{ind-structures-are-the-same}), by functoriality of the Mumford group, (\ref{can-ext}) is also the pullback by $\phi$ of the central extension \begin{equation}\label{gl(H)-ext} 1\lra\G_{m}\lra GL({\cal{H}}_1)\lra PGL({\cal{H}}_1)\lra 1. \end{equation} Moreover, the restriction of ${\cal{H}}_1$ to $\Lgp$ can be integrated to a representation $\LGp\ra GL({\cal{H}}_{1})$. It follows that (\ref{gl(H)-ext}) splits {\em canonically} over $\LGp$, and ${\cal{O}}_{\Q}(1)$ is the line bundle on the homogeneous space $\Q=\widehat\LG/\widehat\LGp$ associated to the character $G_m\times\LGp\ra G_m$ defined by the first projection. \subsection{}\label{central-extensions-and-Dynkin-index} Consider the case of $G=SL_{r}$. In this case, we know from \cite{BL1}, that the pullback of the determinant line ${\cal{D}}$ bundle is ${\cal{O}}_{\QSL}(1)$. It follows from (\ref{pullback-is-injective}), (\ref{unique-linearization}) and (\ref{Pic(Q)}) that $\Pic(\M)=\reln{\scr{D}}$. If $\rho:G\ra SL_{r}$ is a representation of $G$ we get a commutative diagram $$\begin{diagram} \Q&\efl{}{}&\QSL\\ \sfl{}{}&\sefl{}{\varphi}&\sfl{}{}\\ \M&\efl{}{}&\MSL\\ \end{diagram} $$ \begin{lem}\label{Pullback-is-L-to-Dynkin} Let $d_\rho$ be the Dynkin index of $\rho$. Then the pullback of the determinant bundle under $\varphi$ is ${\cal{O}}_{\Q}(d_{\rho})$ \end{lem} \begin{proof} Consider the pullback diagram of (\ref{can-ext}) for $\LSL$: $$ \begin{diagram} 1&\lra& G_m&\lra&\widetilde\LG&\lra&\LG&\lra&1\\ &&\parallel&&\sfl{}{}&&\sfl{}{}\\ 1&\lra& G_m&\lra&\widehat{\LSL}&\lra& \LSL&\lra& 1 \end{diagram} $$ Looking at the differentials (note that $\Lie(\LSLh)=\Lslh$ by \cite{BL1}), on the level of Lie algebra, we restrict the universal central extension of $\Lsl$ to $\Lg$. The resulting extension $\widetilde\Lg$ is (\cf Section (\ref{section-Dynkin-index})) the Lie algebra of the Mumford group of ${\cal{O}}_{\Q}(d_{\rho})$ where $d_{\rho}$ is the Dynkin index of $d_{\rho}$, which proves the lemma. \end{proof} \begin{cor} As a pullback, the line bundle ${\cal{O}}_{\Q}(d_\rho)$ is $\LGX$-linearized. \end{cor} \subsection{Proof of Theorem \ref{th:Pic}.} By the above for series $A$ and $C$, as the Dynkin index for the standard representation is $1$, that all line bundles on $\Q$ are $\LGX$-linearized. For the series $B$ and $D$ (and also for $G_{2}$) the Dynkin index of the standard representation is $2$. But by the existence of the pfaffian line bundles we see also in this case, that all line bundles on $\Q$ are $\LGX$-linearized and Theorem \ref{th:Pic} for $n=0$ follows from (\ref{pullback-is-injective}), (\ref{unique-linearization}) and (\ref{Pic(Q)}) \begin{rem}\label{unique-splitting} The restriction to $\LGX$ of the canonical central extension $\LGh$ of $\LG$ splits (at least for classical $G$ and $G_2$). Moreover this splitting is unique. \end{rem} \begin{proof} As ${\cal{O}}_{\Q}(1)$ admits a $\LGX$-linearization, the action of $\LGX$ on $\Q$ induced by the embedding $\LGX\subset\LG$ lifts to an action to ${\cal{O}}_{\Q}(1)$. The central extension of $\LGX$ obtained by pullback of the canonical central extension $\LGh$ of $\LG$ is the Mumford group of $\LGX$ associated to ${\cal{O}}_{\Q}(1)$. But this extension splits as the action lifts and we are done. Two splittings differ by a character of $\LGX$. As there is only the trivial character (corollary \ref{LGX-has-no-characters}) the splitting must be unique. \end{proof} \section{Parabolic $G$-bundles.}\label{Gpar} Throughout this section $G$ is simple, simply connected and $k=\comp$. We will extend the previous sections to the moduli stacks of parabolic $G$-bundles. \subsection{}\label{Preparation-on-G/P} We use the notations of Section \ref{Lie-theory}. We will recall some standard facts for Lie groups, which we will use later. Let $G$ be the simple and simply connected algebraic group associated to $\g$. Denote by $T\subset G$ the Cartan subgroup associated to ${\goth{h}}\subset\g$ and by $B\subset G$ the Borel subgroup associated to ${\goth{b}}\subset\g$. Given a subset $\Sigma$ of the set of simple roots $\Pi$ (nodes of the Dynkin diagram), we can define a subalgebra ${\goth{p}}_{\Sigma}={\goth{b}}\oplus(\osum_{\alpha\in\Sigma} {\goth{g}}_{-\alpha})\subset\g,$ hence a subgroup $P_{\Sigma}\subset G$. Remark that $P_{\emptyset}=G/B$, $P_{\Pi}=G$ and that all $P_\Sigma$ contain $B$. The subgroup $P_\Sigma$ is parabolic and conversely any {\em standard } (\ie containing $B$) parabolic subgroup arises in this way. Fix $\Sigma\subset \Pi$ and let $\Gamma=\Pi\moins\Sigma$. Denote by $X(P_{\Sigma})$ the character group of $P_\Sigma$. Any weight $\lambda$ such that $\lambda(H_{\alpha})=0$ for all $\alpha\in\Sigma$ defines, via the exponential map, a character of $P_{\Sigma}$ and all characters arise in this way, \ie $X(P_{\Sigma})=\{\lambda\in P/\lambda(H_{\alpha})=0 \text{ for all }\alpha\in\Sigma\}.$ Given $\chi\in X(P_\Sigma)$ we can define the line bundle $L_\chi=G\times^{P_\Sigma}k_\chi$ on the homogeneous space $G/P_\Sigma$. In general, there is an exact sequence (\cite{FI}, prop. 3.1) $$1\lra X(G)\lra X(P_\Sigma)\lra\Pic(G/P_\Sigma)\lra\Pic(G)\lra\Pic(P)\lra 0.$$ As $G$ is simple, we have $X(G)=0$ and as $G$ is simply connected, we have $\Pic(G)=0$ (\cite{FI}, cor. 4.5). We get the isomorphism $X(P_{\Sigma})\isom\Pic(G/P_{\Sigma}).$ In particular, the Picard group of $G/P_{\Sigma}$ is isomorphic to the free abelian group generated over $\Gamma$. \subsection{} Consider closed points $p_{1},\dots,p_{n}$ of $X$, labeled with the standard parabolic subgroup $P_{1},\dots,P_{n}$. Let $\Sigma_{1},\dots,\Sigma_{n}$ be the associated subsets of simple roots and $\Gamma_{i}=\Pi\moins \Sigma_{i}$ for $i\in\{1,\dots,n\}$. In the following, underlining a character will mean that we consider the associated sequence, e.g. $\ul{P}$ will denote the sequence $(P_{1},\dots,P_{n})$, {\em etc.} Let $E$ be a $G$-bundle. As $G$ acts on $G/P_{i}$ we can define the associated $G/P_{i}$-bundle $E(G/P_{i})$. \begin{defi} (\cf \cite{MS}) A quasi-parabolic $G$-bundle of type $\ul{P}$ is a $G$-bundle $E$ on $X$ together with, for all $i\in\{1,\dots,n\}$, an element $F_{i}\in E(G/P_{i})(p_{i})$. A parabolic $G$-bundle of type $(\ul{P},\ul{m})$ is a quasi-parabolic $G$-bundle of type $\ul{P}$ together with, for $i\in\{1,\dots,n\}$, parabolic weights $(m_{i,j})_{j\in\Gamma_{i}}$ where the $m_{i,j}$ are strictly positive integers. \end{defi} \subsection{} Let $R$ be a $k$-algebra, $S=\Spec(R)$. A family of quasi-parabolic $G$-bundles of type $\ul{P}$ parameterized by $S$ is a $G$-bundle $E$ over $S\times X$ together with $n$ sections $\sigma_{i}:S\ra E(G/P_{i})_{\mid S\times\{p_{i}\}}$. A morphism from $(E,\ul\sigma)$ to $(E^{\prime},\ul\sigma^{\prime})$ is a morphism $f:E\ra E^{\prime}$ of $G$-bundles such that for all $i\in\{1,\dots,n\}$ we have $\sigma^{\prime}_{i}=f_{\mid S\times\{p_{i}\}}\circ\sigma_{i}$. We get a functor from the category of $k$-algebras to the category of groupoids by associating to $R$ the groupoid having as objects families of quasi-parabolic $G$-bundles of type $\ul{P}$ parameterized by $S=\Spec(R)$ and as arrows isomorphisms between such families. Moreover for any morphism $R\ra R'$ we have a natural functor between the associated groupoids. This defines the $k$-stack of quasi-parabolic $G$-bundles of type $\ul{P}$ which we will denote by $\Mpar$. The stack $\Mpar$ has, as $\M$, a natural interpretation as a double quotient stack. Define $$\Qpar=\Q\times\prod_{i=1}^{n} G/P_{i}.$$ The ind-group $\LGX$ acts on $\Q$ and by evaluation $ev(p_{i}):\LGX\ra G$ at $p_{i}$ also on each factor $G/P_{i}$. We get a natural action of $\LGX$ on $\Qpar$. The analogue of Theorem \ref{th:Uniformization}. for quasi-parabolic $G$-bundles is \begin{th}\label{th:ParUniformization} (Uniformization) There is a canonical isomorphism of stacks $$\overline{\pi}:\LGX\bk\Qpar\isom\Mpar.$$ Moreover the projection map is locally trivial for the \'etale topology. \end{th} \begin{proof} Let $R$ be a $k$-algebra, $S=\Spec(R)$. To an element $(E,\rho,\ul{f})$ of $\Qpar(R)$ (with $f_{i}\in\Mor(S,G/P_{i})$), we can associate a family of quasi-parabolic $G$-bundles of type $\ul{P}$ parameterized by $S$ in the following way. We only have to define the sections $\sigma_{i}$: $$\sigma_{i}:S\hfl{(\id,f_{i})}{} S\times G/P_{i}\hfl{\rho_i(G/P_{i})}{} E(G/P_{i})_{\mid S\times\{p_{i}\}}.$$ We get a $\LGX$ equivariant map $\pi:\Qpar\ra\Mpar$ which induces the map on the level of stacks $\overline{\pi}:\LGX\bk\Qpar\ra\Mpar$. Conversely, let $(E,\ul\sigma)$ be a family of quasi-parabolic $G$-bundles of type $\ul{P}$ parameterized by $S=\Spec(R)$. For any $R$-algebra $R^{\prime}$, let $T(R^{\prime})$ be the set of trivializations $\rho$ of $E_{R^{\prime}}$ over $X_{R^{\prime}}$. This defines a $R$-space $T$ which by Theorem \ref{Drinfeld-Simpson} is a $\LGX$-torsor. To any element in $T(R^{\prime})$, we can associate the family $\ul{f}$ by $$f_{i}:S\efl{\sigma_{i}}{} E(G/P_{i})_{\mid S\times\{p_{i}\}} \efl{\rho_i(G/P_{i})^{-1}}{} S\times G/P_{i}\efl{pr_{2}}{} G/P_{i}.$$ In this way we associate functorially to objects $(E,\ul\sigma)$ of $\Mpar(R)$ $\LGX$-equivariant maps $\alpha:T\rightarrow \Qpar$. This defines a morphism of stacks $$\Mpar\lra\LGX\bk\Qpar$$ which is the inverse of $\overline{\pi}$. The second statement is clear from the proof of Theorem \ref{th:Uniformization}. \end{proof} \subsection{} We study first line bundles over $\Qpar$. Using (\ref{Pic(Q)}), (\ref{Preparation-on-G/P}) and $H^{1}(G/P_{i},{\cal{O}})=0$, we obtain the following proposition, proving, as $\LGX$ has no characters, Theorem \ref{th:Pic}. \begin{prop} We have $$\Pic(\Qpar)=\reln{\cal{O}}_{\Q}(1)\times\prod_{i=1}^{n}\Pic(G/P_{i}) =\reln{\cal{O}}_{\Q}(1)\times\prod_{i=1}^{n}X(P_{i}).$$ \end{prop} \comment{Let $(E,\ul\sigma)$ be a family of quasi-parabolic $G$-bundles of type $\ul{P}$ parameterized by the $k$-scheme $S=\Spec(R)$. Fix $i\in\{1,\dots,n\}$ and $j\in\Gamma_{i}$. We may view $E\ra E(G/P_{i})$ as a $P_{i}$-bundle. Therefore the character of $P_{i}$ defined by $-\varpi_{j}$ defines a line bundle on $E(G/P_{i})$, hence by pullback, using the section $\sigma_{i}:S\ra E(G/P_{i})_{\mid S\times\{p_{i}\}}$, a line bundle ${\scr{L}}_{i,j}$ over $S$. This works for any $S$ and we get a line bundle over the stack $\Mpar$ which we denote again by ${\scr{L}}_{i,j}$.} \section{Conformal blocs and generalized theta functions.}\label{Identification} Throughout this section $G$ is simple and simply connected and $k=\comp$. \subsection{} Fix an integer $\ell\geq 0$ (the level) and let $p_{1},\dots,p_{n}$ be distinct closed points of $X$ (we allow $n=0$ \ie no points), each of it labeled with a dominant weight $\lambda_{i}$ lying in the fundamental alc\^ove $P_{\ell}$. Choose also another point $p\in X$, distinct from the points $p_{1},\dots,p_{n}$. Define $${\cal{H}}_{\ul\lambda}= {\cal{H}}_{\ell}\otimes(\omal_{i=1}^{n}L_{\lambda_{i}}).$$ and let $\LgX$ be $\g\otimes A_{X}$. We can map $\LgX$ via the Laurent developpement at the point $p$ to $\Lg$. The restriction to $\LgX$ of the universal central extension $\Lgh$ of $\Lg$ splits by the residue theorem, hence $\LgX$ may be considered as a {\em sub Lie-algebra} of $\Lgh$. In particular, ${\cal{H}}_{\ell}$ is a $\LgX$-module. Evaluating $X\otimes f\in\LgX$ at the point $p_{i}$, we may consider $L_{\lambda_{i}}$ as a $\LgX$-module. Therefore ${\cal{H}}_{\ul\lambda}$ is a (left) $\LgX$-module. Define the space of conformal blocks (or vacua) by $$V_{X}(\ul{p},\ul\lambda)=[{\cal{H}}_{\ul\lambda}^{*}]^{\LgX}:= \{\psi\in {\cal{H}}_{\ul\lambda}^{*}\ /\ \psi.(X\otimes f)=0\ \forall X\otimes f\in\LgX\}. $$ This definition is Beauville's description \cite{B} (see also \cite{So3}) of the space of conformal blocks of Tsuchiya, Ueno and Yamada \cite{TUY}. The labeling of the points $p_{i}$ induces $\Sigma_{i}=\{\alpha\in \Pi/\lambda_{i}(H_{\alpha})=0\}$, $\Gamma_{i}=\Pi\moins\Sigma_{i}$ and $m_{i,j}=\lambda_{i}(H_{\alpha_{j}})$ for $j\in\Gamma_{i}$, that is the type of a parabolic $G$-bundle. In particular we get, for $\ell\in\natn$, a natural line bundle on the moduli stack $\Mpar$ defined by $${\scr{L}}(\ell,\ul{m})={\scr{L}}^{\ell}\extern\bigl(\extern_{i=1}^{n} (\extern_{j\in\Gamma_{i}}\reln{\scr{L}}_{i,j}^{m_{i,j}})\bigr). $$ By construction, for the pull back of ${\scr{L}}(\ell,\ul{m})$ to $\Qpar$ we have $$\pi^{*}{\scr{L}}(\ell,\ul{m})={\cal{O}}_{\Q}(\ell)\extern\bigl( \extern_{i=1}^{n}{\scr{L}}_{-\lambda_{i}}\bigr)$$ where ${\scr{L}}_{-\lambda_{i}}$ is the line bundle on the homogeneous space $G/P_{i}$ defined by the character corresponding to the weight $-\lambda_{i}$. \subsection{Proof of (\ref{Verlinde}):} We extend the method of \cite{BL1} and \cite{P}. \medskip\noindent {\em Step 1:} As a pullback, $\pi^{*}{\scr{L}}(\ell,\ul{m})$ is canonically $\LGX$-linearized, that is equipped with $\varphi:m^{*}(\pi^{*}{\scr{L}}(\ell,\ul{m}))\isom pr_{2}^{*}(\pi^{*}{\scr{L}}(\ell,\ul{m}))$. Denote by $[H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))]^{\LGX}$ the space of $\LGX$-invariant sections, that is the sections $s$ such that $\varphi(m^{*}s)=pr_{2}^{*}s$. By Lemma 7.2 of \cite{BL1} we have the canonical isomorphism $$H^{0}(\Mpar,{\scr{L}}(\ell,\ul{m}))\isom [H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))]^{\LGX}$$ Denote by $[H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))]^{\LgX}$ the sections annihilated by $\Lie(\LGX)=\LgX$. By Proposition 7.4 of \cite{BL1}, using that $\LGX$ and $\Qpar$ are integral (\ref{LGX-is-integral} and \ref{ind-structures-are-the-same}), we have the canonical isomorphism $$ [H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))]^{\LGX}\isom [H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))]^{\LgX} $$ \medskip\noindent {\em Step 2:} By definition of $\LGh$, the space $H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))$ is naturally a $\LGh$-module. Moreover we know that $\LGh$ splits over $\LGX$ (at least for classical $G$ and $G_2$) and that this splitting is {\em unique}. The action of $\LgX\subset\Lgh$ deduced from this inclusion on $H^{0}(\Qpar,\pi^{*}{\scr{L}}(\ell,\ul{m}))$ is therefore the same as the preceding one. \medskip\noindent {\em Step 3:} We have the canonical isomorphism of $\Lgh$-modules $$H^0(\Qpar,\pi^{*}({\scr{L}}(\ell,\ul{m})))\isom H^{0}(\Q,{\cal{O}}_{\Q}(\ell))\otimes\bigl(\omal_{i=1}^{n} H^{0}(G/P_{i},{\scr{L}}_{-\lambda_i})\bigr)$$ To see this apply the Kunneth formula to the restriction of ${\scr{L}}(\ell,\ul{m})$ to the projective varieties $\Qpar^{(N)}=\QN\times\prod_{i=1}^{n}G/P_{i}$, then use that inverse limits commute with the tensor products by finite dimensional vector spaces. \medskip\noindent {\em Step 4:} We have the canonical isomorphism of $\LGh$-modules $$H^{0}(\Q,{\cal{O}}_{Q}(\ell))\otimes\bigl(\omal_{i=1}^{n} H^{0}(G/P_{i},{\scr{L}}_{-\lambda_i})\bigr)\isom {\cal{H}}_{\ell,0}^{*}\otimes \bigl(\omal_{i=1}^{n}L_{\lambda_i}^{*}\bigr)$$ This is Borel-Bott-Weil theory, in the version of Kumar-Mathieu (\cite{Ku}, \cite{Ma}) for the first factor, and the standard version \footnote{In \cite{Bott} only the case G/B (\ie $\Sigma=\emptyset$) is considered but the generalization to arbitrary $G/P_{\Sigma}$ is immediate (and well known)} for the others. The theorem follows from steps 1 to 4. As we know the dimensions (at least for classical $G$ and $G_{2}$) for the conformal blocks (\cite{F},\cite{B}, or \cite{So3} for an overview) we get the Verlinde dimension formula for the spaces of generalized parabolic theta-functions. \comment{ \begin{cor}\label{cor:Verlinde-formula}(Verlinde formula) The dimension of the space of generalized parabolic theta-functions $H^0(\Mpar,{\scr{L}}(\ell,\ul{m}))$ is $$(\# T_{\ell})^{g-1} \sum_{\mu\in P_{\ell}}\Tr_{L_{\ul\lambda}}(\exp \frac{2\pi i}{\ell+g^{*}}(\mu+\rho)) \prod_{\alpha\in\Delta_{+}} \left| 2\sin \frac{\pi}{\ell+g^{*}}(\alpha,\mu+\rho)\right|^{2-2g}$$ with $\#T_{\ell}=(\ell+g^{*})^{\rank{\goth{g}}}\#(P/Q)\#(Q/Q_{lg}).$ \end{cor} } \section{Moduli spaces.} \subsection{}\label{gen-on-CM} Suppose $char(k)=0$. We will show how the previous results apply to the {\em coarse moduli spaces} of principal $G$-bundles. We suppose that $G$ is reductive and that $g\geq 2$. Recall that a $G$-bundle $E$ over $X$ is {\em semi-stable} (resp. {\em stable}) if for every parabolic subgroup $P$ and for every reduction $E_{P}$ of $E$ to $G$, we have for every dominant character (with respect to some Borel $B\subset P$) $\chi$ of $P$, trivial over $Z_0(G)$, the following inequality $\deg(E_P(\chi))\leq 0 \text{ (resp. $<$)}.$ A stable $G$-bundle $E$ is called {\em regularly stable}, if moreover $\Aut(E)/Z(G)=\{1\}$. Topologicially, $G$-bundles over $X$ are classified by elements of $\pi_{1}(G)$. By Ramanathan's \cite{Ra} theorem, there are coarse moduli spaces $\Mt$ of semi-stable principal $G$-bundles of dimension $(g-1)\dim G+\dim Z_{0}(G)$, which are irreducible, once the topological type $\tau\in\pi_{1}(G)$ is fixed. Moreover $\Modt$ is normal and the open subset $\Modtreg\subset\Modt$ corresponding to regularly stable $G$-bundles is smooth. \subsection{}\label{locally-factorial} Denote $\Cl$ the group of Weil divisor classes. There is a commutative diagram $$\begin{diagram} \Pic(\Modt)&\efl{c}{}&\Cl(\Modt)\\ \sfl{r_{1}}{}&&\sfl{}{r_{2}}\\ \Pic(\Modtreg)&\efl{c_{reg}}{}&\Cl(\Modtreg)\\ \end{diagram} $$ By normality, the restriction $r_{1}$ is injective, by smoothness of $\Modtreg$, the canonical morphism $c_{reg}$ is an isomorphism and as (\cite{F1}, II.6) $\codim_{\Modt}\Modt\setminus\Modtreg\geq 2$ (except when $g=2$ and $G$ maps nontrivially to $PGL_2$) the restriction $r_{2}$ is an isomorphism. In particular, $\Modt$ is locally factorial \cite{DN} if and only if $r_{1}$ is surjective. \subsection{}\label{det-for-CM} Consider $G=GL_{r}$. Then we may present $\Modzero$ as the good quotient $\HG/GL(M)$ where $\HG$ is Grothendiecks Quot scheme $\HG=\Quot^{ss}(k^{M}\otimes{\cal{O}}_{X}(-N),P)$ parameterizing equivalence classes (with the obvious equivalence relation) of pairs $[E,\alpha]$ with $E$ a semi-stable vectorbundle of degree $0$ and $\alpha:k^{M}\isom E(N)$, where $N$ and $M=rN+\chi(E)$. Let ${\cal{E}}$ be the universal family over $\HG\times X$ and consider $D=\det(Rpr_{1*}({\cal{E}}\otimes pr_{2}^{*}(L)),$ with $L$ a line bundle. It is well known that $[E,\alpha]\in\HG$ has closed orbit exactly when $E$ is polystable, \ie direct sum of stable bundles: $E\simeq E_{1}^{\oplus n_{1}}\oplus\dots\oplus E_{\ell}^{\oplus n_{\ell}}$, and that the action of the stabilizer $GL(n_{1})\times\dots GL(n_{\ell})$ is given by the character $$(g_1,\dots,g_\ell) \mapsto\det(g_1)^{\chi(E_{1}\otimes L)}\cdot\dots\cdot \det(g_\ell)^{\chi(E_{1}\otimes L)}.$$ Choose a line bundle $L$ of degree $g-1$ on $X$. Then $\chi(E_{q}\otimes L)=0$ for $q\in\HG$ and the action is trivial. By Kempf's lemma \cite{DN}, $D$ descends to the determinant of cohomology line bundle on $\ModGLzero$. \subsection{Proof of \ref{th:Pic(Mmod)}.} Suppose $G$ is simple and simply connected. We have (except for $g=2$ and $G=SL_2$) $$\codim_{\M}(\M\setminus\Mreg)\geq 2.$$ To see this define the Harder-Narasimhan filtration in the case of $G$-bundles and calculate the codimension of the strata (\cite{LR}, Section 3) to show that for the open substack $\Mss\subset\M$ corresponding to semi-stable $G$-bundles we have $\codim_{\M}(\M\setminus\Mss)\geq 2$, then use (\cite{F1}, II.6). The smoothness of $\M$ implies $\Pic(\Mreg)=\Pic(\M)$ and it follows from Theorem \ref{th:Pic} that $\Pic(\Mod)$ is an infinite cyclic group (note that the canonical morphism $\Mreg\ra\Modreg$ induces an injection on the level of Picard groups). By (\ref{det-for-CM}) and (\ref{section-Dynkin-index}), we know that the generator is the determinant of cohomology for $G$ of type $A$ and $C$. Moreover, $\Mod$ is locally factorial by (\ref{locally-factorial}) in this case. \subsection{} Consider $G=SO_{r}$ with its standard (orthogonal) representation and suppose that $r\geq 7$. The moduli space $\Pic(\ModSO)$ is the good quotient of a parameter scheme $\Quad$ by $GL(H)$ with $H=k^{rN}$ (\cf \cite{So1}). The scheme $\Quad$ parameterizes equivalent (with the obvious equivalence relation) triples $([F,\sigma,\alpha])$, where $(F,\sigma)$ is a semistable $SO_{r}$-bundle and $\alpha:H^{0}(X,F(N))\isom H$. Choose a theta-characteristic $\kappa$ on $X$. Then on $\Quad$ there is the $GL(H)$-linearized pfaffian of cohomology line bundle ${\scr{P}}_{\kappa}$ deduced from the universal family over $\Quad\times X$. \begin{prop}\label{Descent-of-P-to-Mreg} The line bundle ${\scr{P}}_{\kappa}$ descends to $\ModSOreg$. \end{prop} \begin{proof} We use Kempf's lemma. If $r$ is even, the stabilizer at a point $q=[F,\sigma,\alpha]\in\Quad^{reg}$ is $\pm 1$; if $r$ is odd, the stabilizer is reduced to $1$. So in the latter case, there is nothing to prove. In the former case, by definition of the pfaffian of cohomology, using that its formation commutes with base change, the action $\pm 1$ is given by $g\mapsto g^{h^{1}(F\otimes\kappa)}$, so the action is trivial, as $h^{1}(F\otimes\kappa)$ is even. \end{proof} \begin{prop}\label{not-locally-factorial} If $r\geq 7$, the line bundle ${\scr{P}}_{\kappa}$ does not descend to $\ModSOzero$. In particular, $\ModSOzero$ is not locally factorial. \end{prop} \begin{proof} Let $(F_{1},\sigma_{1})$ be a regularly stable {\em odd} $SO_4$-bundle, and $(F_{2},\sigma_{2})$ be a regularly stable {\em odd} $SO_{r-4}$-bundle. If $r=8$, suppose that $(F_{1},\sigma_{1})$ and $(F_{2},\sigma_{2})$ are not isomorphic. Then the orthogonal sum $(F,\tau)=(F_{1}\oplus F_{2},\sigma_{1}\oplus\sigma_{2})$ is {\em even}. Let $[F,\tau,\alpha]\in\Quad$ be a point corresponding to $(F,\tau)$. Again, by definition of the pfaffian of cohomology, using that its formation commutes with base change, we see that the action of the stabilizer $\{\pm 1\}\times\{\pm 1\}$ is $$(g_{1},g_{2})\mapsto g_{1}^{h^{1}(F_{1}\otimes\kappa)} g_{2}^{h^{1}(F_{2}\otimes\kappa)}.$$ But then the element $(-1,1)$ acts nontrivially. \end{proof} \bigskip

9507

alg-geom/9507017

en

https://arxiv.org/abs/alg-geom/9507017

alg-geom/9507017

Ron Donagi

Ron Donagi and Eyal Markman

Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles

Latex, We restore the page numbers which were inadvertently omitted. The content stayed the same

This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid and the elliptic solitons. We then describe some systems in which the spectral curve is replaced by various higher dimensional analogues: a spectral cover of an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is that it is integrable analytically, but not algebraically: the Liouville tori (on which the system is linearized) are the intermediate Jacobians of a family of Calabi-Yau manifolds. Most of the results concerning these three types of non-curve-based systems are quite recent. Some of them, as well as the compatibility between spectral systems and the KP hierarchy, are new, while other parts of the story are scattered through several recent preprints. As best we could, we tried to maintain the survey style of this article, starting with some basic notions in the field and building gradually to the recent developments.

alg-geom

\section{Introduction} \label{ch1} The purpose of these notes is to present an algebro-geometric point of view on several interrelated topics, all involving integrable systems in symplectic-algebro-geometric settings. These systems range from some very old examples, such as the geodesic flow on an ellipsoid, through the classical hierarchies of $KP-$ and $KdV$-types, to some new systems which are often based on moduli problems in algebraic geometry. The interplay between algebraic geometry and integrable systems goes back quite a way. It has been known at least since Jacobi that many integrable systems can be solved explicitly in terms of {\it theta functions}. (There are numerous examples, starting with various {\it spinning tops} and the {\it geodesic flow on an ellipsoid}.) Geometrically, this often means that the system can be mapped to the total space of a family of Jacobians of some curves, in such a way that the flows of the system are mapped to linear flows along the Jacobians. In practice, these curves tend to arise as the spectrum (hence the name {\it `spectral'} curves) of some parameter-dependent operator; they can therefore be represented as branched covers of the parameter space, which in early examples tended to be the Riemann sphere ${\bf CP}^1$. In {\it Hitchin's system}, the base ${\bf CP}^1$ is replaced by an arbitrary (compact, non-singular) Riemann surface $\Sigma$. The cotangent bundle $T^*{\cal U}_\Sigma$ to the moduli space ${\cal U}_\Sigma$ of stable vector bundles on $\Sigma$ admits two very different interpretations:\ on the one hand, it parametrizes certain {\it Higgs bundles}, or vector bundles with a (canonically) twisted endomorphism; on the other, it parametrizes certain {\it spectral data}, consisting of torsion-free sheaves (generically, line bundles) on spectral curves which are branched covers of $\Sigma$. In our three central chapters (\ref{ch4},\ref{ch5},\ref{ch6}) we study this important system, its extensions and variants. All these systems are linearized on Jacobians of spectral curves. We also study some systems in which the spectral curve is replaced by a higher-dimensional geometric object: \ a {\it spectral variety} in Chapter \ref{ch9}, an algebraic {\it Lagrangian subvariety} in Chapter \ref{ch8}, and a {\it Calabi-Yau manifold} in Chapter \ref{ch7}. Our understanding of some of these wild systems is much less complete than in the case of the curve-based ones. We try to explain what we know and to point out some of what we do not. The Calabi-Yau systems seem particularly intriguing. Not only are the tori (on which these systems are linearized) not Jacobians of curves, they are in general not even abelian varieties. There are some suggestive relations between these systems and the conjectural mirror-symmetry for Calabi-Yaus. The first three chapters are introductory. In Chapter \ref{ch2} we collect the basic notions of {\it symplectic geometry} and {\it integrable systems} which will be needed, including some information about {\it symplectic reduction}. (An excellent further reference is \cite{AG}.) In Chapter \ref{ch3} we work out in some detail the classical theory of geodesic flow on an ellipsoid, which is integrable via hyperelliptic theta functions. We think of this both as a beautiful elementary and explicit example and as an important special case of the much more powerful results which follow. (Our presentation follows \cite{knorrer,reid,donagi-group-law}). Some of our main algebro-geometric objects of study are introduced in Chapter \ref{ch4}:\ vector bundles and their moduli spaces, spectral curves, and the {\it `spectral systems'} constructed from them. In particular, we consider the {\it polynomial matrix system} \cite{AHH,B} (which contains the geodesic flow on an ellipsoid as special case) and {\it Hitchin's system} \cite{hitchin,hitchin-integrable-system}. Each of the remaining five chapters presents in some detail a recent or current research topic. Chapter \ref{ch5} outlines constructions (from \cite{markman-higgs,botachin,tyurin-symplectic}) of the Poisson structure on the spectral system of curves. This is possible whenever the twisting line bundle $K$ is a non-negative twist $\omega_\Sigma(D)$ of the canonical bundle $\omega_\Sigma$, and produces an algebraically completely integrable Hamiltonian system. Following \cite{markman-higgs} we emphasize the deformation-theoretic construction, in which the Poisson structure on an open subset of the system is obtained via symplectic reduction from the cotangent bundle $T^*{\cal U}_{\Sigma ,D}$ of the moduli space ${\cal U}_{\Sigma ,D}$ of stable bundles with a {\it level-D structure}. In Chapter \ref{ch6} we explore the relation between these spectral systems and the $KP$-hierarchy and its variants (multi-component $KP$, Heisenberg flows, and their $KdV$-type subhierarchies). These hierarchies are, of course, a rich source of geometry:\ The Krichever construction (e.g. \cite{segal-wilson-loop-groups-and-kp}) shows that any Jacobian can be embedded in $KP$-space, and these are the only finite-dimensional orbits \cite{mulase-cohomological-structure, AdC, Sh}. Following \cite{adams-bergvelt,li-mulase-category} we describe some ``multi-Krichever'' constructions which take spectral data to the spaces of the $KP$, $mcKP$ and Heisenberg systems. Our main new result is that the flows on the spectral system which are obtained by pulling back the $mcKP$ or Heisenberg flows via the corresponding Krichever maps are {\it Hamiltonian} with respect to the Poisson structure constructed in Chapter \ref{ch5}. In fact, we write down explicitly the Hamiltonians for these $KP$ flows on the spectral system, as residues of traces of meromorphic matrices. (Some related results have also been obtained recently in \cite{li-mulase-compatibility}.) The starting point for Chapter \ref{ch7} is an attempt to understand the condition for a given family of complex tori to admit a symplectic structure and thus become an ACIHS. We find that the condition is a symmetry on the derivatives of the period map, which essentially says that the periods are obtained as partials of some field of symmetric cubic tensors on the base. In the rest of this Chapter we apply this idea to an analytically (not algebraically) integrable system constructed from any family of Calabi-Yau $3$-folds. Some properties of this system suggest that it may be relevant to a purely hodge-theoretic reformulation of the mirror-symmetry conjectures. (This chapter is based on \cite{cubics-calabi-yaus}.) Chapter \ref{ch8} is devoted to the construction of symplectic and Poisson structures in some inherently non-linear situations, vastly extending the results of Chapter \ref{ch5}. The basic space considered here is the moduli space parametrizing line-bundle-like sheaves supported on (variable) subvarieties of a given symplectic space $X$. It is shown that when the subvarieties are Lagrangian, the moduli space itself becomes symplectic. The spectral systems considered in Chapter \ref{ch5} can be recovered as the case where $X$ is the total space of $T^*\Sigma$ and the Lagrangian subvarieties are the spectral curves. (A fuller version of these results will appear in \cite{markman-lagrangian-sheaves}.) In the final chapter we consider extensions of the spectral system to allow a higher-dimensional base variety $S$, an arbitrary reductive group $G$, an arbitrary representation $\rho: G \to Aut V$, and values in an arbitrary vector bundle $K$. (Arbitrary reductive groups $G$ were considered, over a curve $S = \Sigma$ with $K = \omega_\Sigma$, by Hitchin \cite{hitchin-integrable-system}, while the case $K = \Omega_S$ over arbitrary base $S$ is Simpson's \cite{simpson-moduli}). We replace spectral curves by various kinds of spectral covers, and introduce the cameral cover, a version of the Galois-closure of a spectral cover which is independent of $K$ and $\rho$. It comes with an action of $W$, the Weyl group of $G$. We analyze the decomposition, under the action of $W$, of the cameral and spectral Picard varieties, and identify the distinguished Prym in there. This is shown to correspond, up to certain shifts and twists, to the fiber of the Hitchin map in this general setting, i.e. to moduli of Higgs bundles with a given $\widetilde{S}$. Combining this with some obvious remarks about existence of Poisson structures, we find that the moduli spaces of K-valued Higgs bundles support algebraically completely integrable systems. Our presentation closely follows that of \cite{MSRI} It is a pleasure to express our gratitude to the organizers, Mauro Francaviglia and Silvio Greco, for the opportunity to participate in the CIME meeting and to publish these notes here. During the preparation of this long work we benefited from many enjoyable conversations with M. Adams, M. Adler, A. Beauville, R. Bryant, C. L. Chai, I. Dolgachev, L. Ein, B. van Geemen, A. Givental, M. Green, P. Griffiths, N. Hitchin, Y. Hu, S. Katz, V. Kanev, L. Katzarkov, R. Lazarsfeld, P. van Moerbeke, D. Morrison, T. Pantev, E. Previato and E. Witten. \newpage \section{Basic Notions} \label{ch2} \label{sec-basic-notions} We gather here those basic concepts and elementary results from symplectic and Poisson geometry, completely integrable systems, and symplectic reduction which will be helpful throughout these notes. Included are a few useful examples and only occasional proofs or sketches. To the reader unfamiliar with this material we were hoping to impart just as much of a feeling for it as might be needed in the following chapters. For more details, we recommend the excellent survey \cite{AG}. \subsection{Symplectic Geometry} \label{subsec-symplectic-geometry} \noindent \underline{{\bf Symplectic structure}} A symplectic structure on a differentiable manifold $M$ of even dimension $2n$ is given by a non-degenerate closed 2-form $\sigma$. The non degeneracy means that either of the following equivalent conditions holds. \begin{itemize} \item $\sigma^n$ is a nowhere vanishing volume form. \item Contraction with $\sigma$ induces an isomorphism $\rfloor \sigma : TM \rightarrow T^*M$ \item For any non-zero tangent vector $v \in T_mM$ at $m \in M$, there is some $v' \in T_mM$ such that $\sigma(v,v') \ne 0$. \end{itemize} \begin{examples} \label{examples-symplectic-varieties} {\rm \SkipAfterTitle \begin{enumerate} \item \underline{Euclidean space} The standard example of a symplectic manifold is Euclidean space ${\bf R}^{2n}$ with $\sigma = \Sigma dp_i \wedge dq_i$, where $p_1,\cdots,p_n, \ q_1, \cdots,q_n$ are linear coordinates. Darboux's theorem says that any symplectic manifold is locally equivalent to this example (or to any other). \item \underline{Cotangent bundles} For any manifold $X$, the cotangent bundle $M := T^*X$ has a natural symplectic structure. First, $M$ has the tautological 1-form $\alpha$, whose value at $(x,\theta) \in T^*X$ is $\theta$ pulled back to $T^*M$. If $q_1 ,\cdots, q_n$ are local coordinates on $X$, then locally $\alpha = \Sigma p_i dq_i$ where the $p_i$ are the fiber coordinates given by $\partial / \partial q_i$. The differential $$ \sigma := d\alpha $$ is then a globally defined closed (even exact) 2-form on $M$. It is given in local coordinates by $\Sigma dp_i \wedge dq_i$, hence is non-degenerate. \item \underline{Coadjoint orbits} Any Lie group $G$ acts on its Lie algebra $\LieAlg{g}$ (adjoint representation) and hence on the dual vector space $\LieAlg{g}^*$ (coadjoint representation). Kostant and Kirillov noted that for any $\xi \in \LieAlg{g}^*$, the coadjoint orbit ${\cal O} = G \xi \subset \LieAlg{g}^*$ has a natural symplectic structure. The tangent space to $\cal O$ at $\xi$ is given by $\LieAlg{g}/\LieAlg{g}_\xi$, where $\LieAlg{g}_\xi$ is the stabilizer of $\xi$: $$ \LieAlg{g}_\xi := \{ x \in \LieAlg{g} \; |\; ad^*_x \xi = 0 \} = \{x \in \LieAlg{g} \; | \; (\xi,[x,y]) = 0 \quad \forall \ y \in \LieAlg{g} \}. $$ Now $\xi$ determines an alternating bilinear form on $\LieAlg{g}$ $$ x,y \longmapsto (\xi, [x,y]), $$ which clearly descends to $\LieAlg{g} / \LieAlg{g}_\xi$ and is non-degenerate there. Varying $\xi$ we get a non-degenerate 2-form $\sigma$ on $\cal O$. The Jacobi identity on $g$ translates immediately into closedness of $\sigma$. \end{enumerate} } \end{examples} \medskip \noindent \underline{{\bf Hamiltonians}} To a function $f$ on a symplectic manifold $(M,\sigma)$ we associate its {\it Hamiltonian vector field} $v_f$, uniquely determined by $$ v_f \; \rfloor \; \sigma = df. $$ A vector field $v$ on $M$ is Hamiltonian if and only if the 1-form $v \, \rfloor \, \sigma$ is exact. We say $v$ is {\it locally Hamiltonian} if $v \; \rfloor \; \sigma$ is closed. This is equivalent to saying that the flow generated by $v$ preserves $\sigma$. Thus on a symplectic surface $(n=1)$, the locally Hamiltonian vector fields are the area-preserving ones. \medskip \noindent {\bf Example:} (Geodesic flow) A Riemannian metric on a manifold $X$ determines an isomorphism of $M := TX$ with $T^*X$; hence we get on $M$ a natural symplectic structure together with a $C^\infty$ function $f =$ (squared length). The geodesic flow on $X$ is the differential equation, on $M$, given by the Hamiltonian vector field $v_f$. Its integral curves are the geodesics on $M$. \medskip \noindent \underline{{\bf Poisson structures}} The association $f \mapsto v_f$ gives a map of sheaves \begin{equation} \label{eq-functions-to-hamiltonian-vectorfields} v : C^\infty (M) \longrightarrow V(M) \end{equation} from $C^\infty$ functions on the symplectic manifold $M$ to vector fields. Now $V(M)$ always has the structure of a Lie algebra, under commutation of vector fields. The symplectic structure on $M$ determines a Lie algebra structure on $C^\infty(M)$ such that $v$ becomes a morphism of (sheaves of) Lie algebras. The operation on $C^\infty(M),$ called {\it Poisson bracket}, is $$ \{ f,g \} := (df, v_g) = -(dg, v_f) = {{n df \wedge dg \wedge \sigma^{n-1}} \over {\sigma^n}}. $$ More generally, a {\it Poisson structure} on a manifold $M$ is a Lie algebra bracket $\{\, ,\, \}$ on $C^\infty(M)$ which acts as a derivation in each variable: $$ \{f,gh\} = \{f,g\} h + \{f,h\}g, \ \ \ f,g,h \in C^\infty(M). $$ Since the value at a point $m$ of a given derivation acting on a function $g$ is a linear function of $d_mg$, we see that a Poisson structure on $M$ determines a global 2-vector $$ \psi \in H^0(M, \stackrel{2}{\wedge} TM). $$ or equivalently a skew-symmetric homomorphism $$ \Psi : T^*M \longrightarrow TM. $$ Conversely, any 2-vector $\psi$ on $M$ determines an alternating bilinear bracket on $C^\infty (M)$, by $$ \{f,g\} := (df \wedge dg, \psi), $$ and this acts as a derivation in each variable. An equivalent way of specifying a Poisson structure is thus to give a global 2-vector $\psi$ satisfying an integrability condition (saying that the above bracket satisfies the Jacobi identity, hence gives a Lie algebra). We saw that a symplectic structure $\sigma$ determines a Poisson bracket $\{\ ,\ \}$. The corresponding homomorphism $\Psi$ is just $(\rfloor \sigma)^{-1}$; the closedness of $\sigma$ is equivalent to integrability of $\psi$. Thus, a Poisson structure which is (i.e. whose 2-vector is) everywhere non-degenerate, comes from a symplectic structure. A general Poisson structure can be degenerate in two ways: first, there may exist non-constant functions $f \in C^\infty(M)$, called {\it Casimirs}, satisfying $$ 0 = df \rfloor \psi = \Psi(df), $$ i.e. $$ \{f,g\} = 0 \ \mbox{for all} \ g \in C^\infty(M). $$ This implies that the rank of $\Psi$ is less than maximal everywhere. In addition, or instead, rank $\Psi$ could drop along some strata in $M$. For even $r$, let $$ M_r := \{ m \in M | rank(\Psi) = r \}. $$ Then a basic result \cite{We} asserts that the $M_r$ are submanifolds, and they are canonically foliated into {\it symplectic leaves}, i.e. $r$-dimensional submanifolds $Z \subset M_r$ which inherit a symplectic structure. (This means that the restriction $\psi_{\mid_Z}$ is the image, under the inclusion $ Z \hookrightarrow M_r$, of a two-vector $\psi_Z$ on $Z$ which is everywhere nondegenerate, hence comes from a symplectic structure on $Z$.) These leaves can be described in several ways: \begin{itemize} \item The image $\Psi(T^*M_r)$ is an involutive subbundle of rank $r$ in $TM_r$; the $Z$ are its integral leaves. \item The leaf $Z$ through $m \in M_r$ is $Z = \{ z \in M_r | f(m) = f(z)\ \mbox{for all Casimirs} \ f \ \mbox{on} \ M_r \}$. \item Say that two points of $M$ are $\psi$-connected if there is an integral curve of some Hamiltonian vector field passing through both. The leaves are the equivalence classes for the equivalence relation generated by $\psi$-connectedness. \end{itemize} \noindent \begin{example}\label{example-coadjoint-orbits} {\rm The Kostant-Kirillov symplectic structures on coadjoint orbits of a Lie algebra $\LieAlg{g}$ extend to a Poisson structure on the dual vector space $\LieAlg{g}^*$. For a function $F \in C^\infty(\LieAlg{g}^*)$ we identify its differential $d_\xi F$ at $\xi \in \LieAlg{g}*$ with an element of $\LieAlg{g} = \LieAlg{g}^{**}$. We then set: $$ \{F,G\}(\xi) := (\xi, [d_\xi F,d_\xi G]). $$ This is a Poisson structure, whose symplectic leaves are precisely the coadjoint orbits. The rank of $\LieAlg{g}$ is, by definition, the smallest codimension $\ell$ of a coadjoint orbit. The Casimirs are the ad-invariant functions on $\LieAlg{g}^*$. Their restrictions to the largest stratum $\LieAlg{g}^*_{\dim \LieAlg{g} - \ell}$ foliate this stratum, the leaves being the {\it regular} (i.e. largest dimensional) coadjoint orbits.} \end{example} \subsection{Integrable Systems} We say that two functions $h_1,h_2$ on a Poisson manifold $(M,\psi)$ {\em Poisson commute} if their Poisson bracket $\{ h_1,h_2 \}$ is zero. In this case the integral flow of the Hamiltonian vector field of each function $h_i, \ i = 1,2$ is tangent to the level sets of the other. In other words, $h_2$ is a conservation law for the Hamiltonian $h_1$ and the Hamiltonian flow of $h_2$ is a symmetry of the Hamiltonian system associated with $(M, \psi, h_1)$ (the flow of the Hamiltonian vector field $v_{h_{1}}$ on $M$). A map $f : M \rightarrow B$ between two Poisson manifolds is a {\it Poisson map} if pullback of functions is a Lie algebra homomorphism with respect to the Poisson bracket $$ f^*\{F,G\}_B \, = \, \{f^*F,f^*G\}_M. $$ Equivalently, if $df(\psi_M)$ equals $f^*(\psi_B)$ as sections of $f^*(\stackrel{2}{\wedge} T_B)$. If $H:M \rightarrow B$ is a Poisson map with respect to the trivial (zero) Poisson structure on $B$ we will call $H$ a {\em Hamiltonian map}. Equivalently, $H$ is Hamiltonian if the Poisson structure $\psi$ vanishes on the pullback $H^*(T^*B)$ of the cotangent bundle of $B$ (regarding the latter as a subbundle of $(T^*M,\psi)$). In particular, the rank of the differential $dH$ is less than or equal to $\dim M - {1 \over 2} \mathop{\rm rank} (\psi)$ at every point. A Hamiltonian map pulls back the algebra of functions on $B$ to a commutative Poisson subalgebra of the algebra of functions on $M$. The study of a Hamiltonian system $(M,\psi,h)$ simplifies tremendously if one can extend the Hamiltonian function $h$ to a Hamiltonian map $H : M \rightarrow B$ of maximal rank $\dim M - {1 \over 2} \mathop{\rm rank}(\psi)$. Such a system is called a completely integrable Hamiltonian system. The Hamiltonian flow of a completely integrable system can often be realized as a linear flow on tori embedded in $M$. The fundamental theorem in this case is Liouville's theorem (stated below). \noindent \begin{definition} {\rm \begin{enumerate} \item \ Let $V$ be a vector space, $\sigma \in \stackrel{2}{\wedge} V^*$ a (possibly degenerate) two form. A subspace $Z \subset V$ is called {\em isotropic (coisotropic)} if it is contained in (contains) its symplectic complement. Equivalently, $Z$ is isotropic if $\sigma$ restricts to zero on $Z$. If $\sigma$ is nondegenerate, a subspace $Z \subset V$ is called {\em Lagrangian} if it is both isotropic and coisotropic. In this case $V$ is even (say $2n$) dimensional and the Lagrangian subspaces are the $n$ dimensional isotropic subspaces. \item Let $(M,\sigma)$ be a symplectic manifold. A submanifold $Z$ is {\em isotropic} (respectively {\em coisotropic, Lagrangian}) if the tangent subspaces $T_zZ$ are, for all $z \in Z$. \end{enumerate} } \end{definition} \begin{example} {\rm For every manifold $X$, the fibers of the cotangent bundle $T^*X$ over points of $X$ are Lagrangian submanifolds with respect to the standard symplectic structure. A section of $T^*X$ over $X$ is Lagrangian if and only if the corresponding $1$-form on $X$ is closed. } \end{example} We will extend the above definition to Poisson geometry: \noindent \begin{definition} {\rm \begin{enumerate} \item Let $U$ be a vector space, $\psi$ an element of $\stackrel{2}{\wedge}U$. Let $V \subset U$ be the image of the contraction $\rfloor \; \psi : U^* \rightarrow U$. Let $W \subset U^*$ be its kernel. $W$ is called the null space of $\psi$. $\psi$ is in fact a nondegenerate element of $\stackrel{2}{\wedge} V$ giving rise to a symplectic form $\sigma \in \stackrel{2}{\wedge} V^*$ (its inverse). A subspace $Z \subset U$ is {\em Lagrangian} with respect to $\psi$ if $Z$ is a Lagrangian subspace of $V \subset U$ with respect to $\sigma$. Equivalently, $Z$ is Lagrangian if $(U/Z)^*$ is both an isotropic and a coisotropic subspace of $U^*$ with respect to $\psi \in \stackrel{2}{\wedge} U \cong \stackrel{2}{\wedge} (U^*)^*$. \item Let $(M,\psi)$ be a Poisson manifold, assume that $\psi$ has constant rank (this condition will be relaxed in the complex analytic or algebraic case). A submanifold $Z \subset M$ is {\em Lagrangian} if the tangent subspaces $T_zZ$ are, for all $z \in Z$. Notice that the constant rank assumption implies that each connected component of $Z$ is contained in a single symplectic leaf. \end{enumerate} } \end{definition} \medskip \noindent \begin{theorem} \ (Liouville). \ Let $M$ be an m-dimensional Poisson manifold with Poisson structure $\psi$ of constant rank $2g$. Suppose that $H : M \rightarrow B$ is a proper submersive Hamiltonian map of maximal rank, i.e, dim $B = m - g$. Then \begin{description} \item[i)] \ The null foliation of $M$ is induced locally by a foliation of $B$ (globally if $H$ has connected fibers). \item[ii)] \ The connected components of fibers of $H$ are Lagrangian compact tori with a natural affine structure. \item[iii)] \ The Hamiltonian vector fields of the pullback of functions on $B$ by $H$ are tangent to the level tori and are translation invariant (linear). \end{description} \end{theorem} \medskip \noindent \begin {rem}: {\rm If $H$ is not proper, but the Hamiltonian flows are complete, then the fibers of $H$ are generalized tori (quotients of a vector space by a discrete subgroup, not necessarily of maximal rank). } \end{rem} \medskip \noindent \underline{Sketch of proof of Liouville's theorem:} \begin{description} \item[i)] Since $H$ is a proper submersion the connected components of the fibers of $H$ are smooth compact submanifolds. Since $H$ is a Hamiltonian map of maximal rank $m - g$, the pullback $H^*(T^*_B)$ is isotropic and coisotropic and hence $H$ is a Lagrangian fibration. In particular, each connected component of a fiber of $H$ is contained in a single symplectic leaf. \item[ii),iii)] Let $A_b$ be a connected component of the fiber $H^{-1}(b)$. Let $0 \rightarrow T_{A_b} \rightarrow T_{{M|}_{A_b}} \stackrel{dH}{\longrightarrow} (T_bB) \otimes {\cal O}_{A_b} \rightarrow 0$ be the exact sequence of the differential of $H$. Part i) implies that the null subbundle $W_{{|A}_b} := Ker [\Psi : T^*M \rightarrow TM]_{|A_b}$ is the pullback of a subspace $W_b$ of $T^*_bB$. Since $H$ is a Lagrangian fibration, the Poisson structure induces a surjective homomorphism $\phi_b : H^*(T^*_bB) \rightarrow T_{A_b}$ inducing a trivialization $\bar{\phi}_b : (T^*_bB / W_b) \otimes {\cal O}_{A_b} \stackrel{\sim}{\longrightarrow} \ T_{A_b}$. A basis of the vector space $T^*_bB/W_b$ corresponds to a frame of global independent vector fields on the fiber $A_b$ which commute since the map $H$ is Hamiltonian. Hence $A_b$ is a compact torus. \end{description} \EndProof \subsection{Algebraically Completely Integrable Hamiltonian Systems} All the definitions and most of the results stated in this chapter for $C^\infty$-manifolds translate verbatim and hold in the complex analytic and complex algebro-geometric categories replacing the real symplectic form by a holomorphic or algebraic $(2,0)$-form (similarly for Poisson structures). The (main) exception listed below is due to the differences between the Zariski topology and the complex or $C^\infty$ topologies. A Zariski open subset is the complement of the zero locus of a system of polynomial equations. It is hence always a dense open subset. The (local) foliation by symplectic leaves exists only local analytically. For example, a rank 2 translation invariant section $\psi \in H^0(A, \stackrel{2}{\wedge} TA)$ on a 3 dimensional abelian variety $A$ which is simple (does not contain any abelian subvariety) is an algebraic Poisson structure with a non algebraic null foliation. We will relax the definitions of a Lagrangian subvariety and integrable system in the algebro-geometric category: \begin{definition} {\rm Let $(M,\psi)$ be a Poisson smooth algebraic variety. An irreducible and reduced subvariety $Z \subset M$ is {\em Lagrangian} if the tangent subspace $T_zZ \subset T_zM$ is Lagrangian for a generic point $z \in Z$. } \end{definition} \begin{definition} {\rm An {\em algebraically completely integrable Hamiltonian system} consists of a proper flat morphism $H:M \rightarrow B$ where $(M,\psi)$ is a smooth Poisson variety and $B$ is a smooth variety such that, over the complement $B \smallsetminus \Delta$ of some proper closed subvariety $\Delta \subset B$, $H$ is a Lagrangian fibration whose fibers are isomorphic to abelian varieties. } \end{definition} Multiples of a theta line bundle embed an abelian variety in projective spaces with the coordinates being theta functions. Thus, a priori, the solutions of an algebraically completely integrable Hamiltonian system can be expressed in terms of theta functions. Finding explicit formulas is usually hard. In the next chapter we will study one example, the geodesic flow on ellipsoids, in some detail. Later we will encounter certain equations of $Kdv$ type, the Hitchin system, and a few other examples. Other classical integrable systems include various Euler-Arnold systems, spinning tops, the Neumann system of evolution of a point on the sphere subject to a quadratic potential. Most of these systems are the complexification of real algebraic systems. Given a real algebraic symplectic variety $(M,\sigma)$ and an algebraic Hamiltonian $h$ on $M$ we say that the system is {\em algebraically completely integrable} if its complexification $(M_{{\bf C}},\sigma_{{\bf C}}, \, h_{{\bf C}})$ is. A real completely integrable system $(M,\sigma,h)$ need not be algebraically completely integrable even if $(M,\sigma,h)$ are algebraic: \medskip \noindent {\bf A Counter Example:} \ \ Let $(M,\sigma)$ be $({\bf R}^2, \, dx \wedge dy)$ and $h:{\bf R}^2 \rightarrow {\bf R}$ a polynomial of degree $d$ whose level sets are nonsingular. The system is trivially completely integrable, but it is algebraically completely integrable if and only if $d=3$ because in all other cases the generic fiber of the complexification is a complex affine plane curve of genus $\frac{(d-1)(d-2)}{2} \neq 1$. \medskip \noindent \underline{{\bf Action Angle Coordinates}}: Let $(M,\sigma)$ be a $2n$-dimensional symplectic manifold, $H:M \rightarrow B$ a Lagrangian fibration by compact connected tori. \noindent \begin{theorem} (real action angle coordinates). \noindent In a neighborhood of a fiber of $H:M \rightarrow B$ one can introduce the structure of a direct product $({\bf R}^n / {\bf Z}^n) \times {\bf R}^n$ with action coordinates $(I_1 \, \cdots \, I_n)$ on the factor ${\bf R}^n$ and angular coordinates $(\phi_1, \, \cdots \, \phi_n)$ on the torus $({\bf R}^n/{\bf Z}^n)$ in which the symplectic structure has the form $\sum^n_{k=1} dI_k \wedge d\phi_k$. \end{theorem} The Local action coordinates on $B$ are canonical up to affine transformation on ${\bf R}^n$ with differential in $SL(n,{\bf Z})$. The angle coordinates depend canonically on the action coordinates and a choice of a Lagrangian section of $H:M \rightarrow B$. \noindent {\bf Remarks:} \begin{itemize} \begin{enumerate} \item In action angle coordinates the equations of the Hamiltonian flow of a function $h$ on $B$ becomes: $\dot{I}_k = 0$, $\dot{\varphi}_k = c_k(I_1,\cdots,I_n)$ where the slopes $c_k$ are $c_k = {{\partial h} \over {\partial I_k}}$. \item In the polarized complex analytic case, we still have local holomorphic action coordinates. They depend further on a choice of a Lagrangian subspace of the integral homology $H_1(A_b,{\bf Z})$ with respect to the polarization (a section of $\stackrel{2}{\wedge}H^1(A_b,{\bf Z}))$. \end{enumerate} \end{itemize} \subsection{Moment Maps and Symplectic Reduction} \label{subsec-moment-maps} \noindent \underline{{\bf Poisson Actions}} An action $\rho$ of a connected Lie group $G$ on a manifold $M$ determines an {\it infinitesimal action} $$ d \rho : \LieAlg{g} \longrightarrow V(M), $$ which is a homomorphism from the Lie algebra of $G$ to the Lie algebra of $C^\infty$ vector fields on $M$. When $(M,\sigma)$ is symplectic, we say that the action $\rho$ is {\it symplectic} if $$ (\rho(g))^* \sigma = \sigma, \quad \quad {\rm all} \quad g \in G, $$ or equivalently if the image of $d\rho$ consists of locally Hamiltonian vector fields. We say that the action $\rho$ is {\it Poisson} if it factors through the Lie algebra homomorphism (\ref{eq-functions-to-hamiltonian-vectorfields}) $v : C^\infty(M) \rightarrow V(M)$ and a Lie algebra homomorphism $$ H:\LieAlg{g} \longrightarrow C^\infty(M). $$ This imposes two requirements on $\rho$, each of a cohomological nature: the locally Hamiltonian fields $d\rho(X)$ should be globally Hamiltonian, $d\rho(X) = v(H(X))$; and it must be possible to choose the $H(X)$ consistently so that $$ H([X,Y]) \ = \ \{H(X), H(Y)\}. $$ (a priori the difference between the two terms is a constant function, since its $v$ is zero, so the condition is that it should be possible to make all these constants vanish simultaneously.) \medskip \noindent \underline{{\bf Moment Maps}} Instead of specifying the Hamiltonian lift $$ H : \LieAlg{g} \longrightarrow C^\infty(M) $$ for a Poisson action of $G$ on $(M,\sigma)$, it is convenient to consider the equivalent data of the {\it moment map} $$ \mu : M \longrightarrow \LieAlg{g}^* $$ defined by $$ (\mu(m),X) \, := \,H(X)(m). $$ It is a Poisson map with respect to the Kostant-Kirillov Poisson structure on $\LieAlg{g}$ (example \ref{example-coadjoint-orbits}), and is $G$-equivariant. \begin{examples} \label{examples-moment-maps} {\rm \begin{enumerate} \item Any action of $G$ on a manifold $X$ lifts to an action on $M := T^*X$. This action is Poisson. The corresponding moment map $T^*X \rightarrow \LieAlg{g}^*$ is the dual of the infinitesimal action $\LieAlg{g} \rightarrow \Gamma (TX)$. It can be identified with the pullback of differential forms from $X$ to $G$ via the action. \item The coadjoint action of $G$ on $\LieAlg{g}^*$ is Poisson, with the identity as moment map. \end{enumerate} } \end{examples} \medskip \noindent \underline{{\bf Symplectic Reduction}} Consider a Poisson action of $G$ on $(M,\sigma)$ for which a reasonable quotient $G/M$ exists. (We will remain vague about this for now, and discuss the properties of the quotient on a case-by-case basis. A general sufficient condition for the quotient to be a manifold is that the action is proper and free.) The Poisson bracket on $M$ then descends to give a Poisson structure on $M/G$. The moment map, $$\mu : M \longrightarrow \LieAlg{g}^*, $$ determines the symplectic leaves of this Poisson structure: \ let $\xi = \mu(m)$, let $\cal O$ be the coadjoint orbit through $\xi$ and let $G_\xi$ be the stabilizer of $\xi$. Assume for simplicity that $\mu^{-1}(\xi)$ is connected and $\mu$ is submersive at $\mu^{-1}(\xi)$. Then, the leaf through $m$ is $$ \mu^{-1}({\cal O}_\xi) / G \approx \mu^{-1}(\xi)/G_\xi. $$ These symplectic leaves are often called the Marsden-Weinstein reductions $M_{red}$ of $M$. As an example, consider a situation where $G$ acts on $X$ with nice quotient $X/G$. The lifted action of $G$ on $M = T^*X$ is Poisson, and has a quotient $M/G$ which is a vector bundle over $X/G$. The cotangent $T^*(X/G)$ sits inside $(T^*X)/G$ as the symplectic leaf over the trivial orbit ${\cal O}_0 = \{0\} \subset \LieAlg{g}^*.$ In contrast, the action of $G$ on $\LieAlg{g}^*$ does not in general admit a reasonable quotient. Its action on the dense open subset $\LieAlg{g}^*_{reg}$ of regular elements (cf. example \ref{example-coadjoint-orbits}) does have a quotient, which is a manifold. The Poisson structure on the quotient is trivial, so the symplectic leaves are points, in one-to-one correspondence with the regular orbits. We refer to this quotient simply as $\LieAlg{g}^*/G$. The map $\pi_{reg} \; : \; \LieAlg{g}^*_{reg} \rightarrow \LieAlg{g}^*/G$ extends to $\pi \, : \, \LieAlg{g}^* \rightarrow \LieAlg{g}^*/G$, and there is a sense in which $\LieAlg{g}^*/G$ really is the quotient of all $\LieAlg{g}^*$. Each coadjoint orbit $\cal O$ is contained in the closure of a unique regular orbit ${\cal O}'$ and $\pi({\cal O}) = \pi_{reg}({\cal O}')$. \medskip \noindent \underline{{\bf A Diagram of Quotients}} \nopagebreak[3] In the general situation of Poisson action (with a nice quotient $\pi$) of $G$ on a symplectic manifold $(M,\sigma)$, there is another, larger, Poisson manifold $\bar{M}$, which can also be considered as a reduction of $M$ by $G$. Everything fits together in the commutative diagram of Poisson maps: \begin{equation}\label{diagram-of-quotients} {\divide\dgARROWLENGTH by 2 \begin{diagram} \node[2]{M} \arrow{ssw,l}{\pi} \arrow{s} \arrow{sse,t}{\mu} \\ \node[2]{\bar{M}} \arrow{sw,r}{\bar{\pi}} \arrow{se,b}{\bar{\mu}} \\ \node{M/G} \arrow{sse} \arrow{se} \node[2]{\LieAlg{g}^{*}} \arrow{sw} \arrow{ssw} \\ \node[2]{\LieAlg{g}^{*}/G} \arrow{s} \\ \node[2]{(0)} \end{diagram} } \end{equation} $\bar{M}$ may be described in several ways: \begin{itemize} \item $\bar{M}$ is the quotient of $M$ by the equivalence relation $m \sim gm$ if $g \in G_{\mu(m)}$, i.e., if $g(\mu(m)) = \mu(m)$. \item $\bar{M}$ is the fiber product $\bar{M}$ = $(M/G) \times_{(\LieAlg{g}^*/G)} \LieAlg{g}^*$. \item $\bar{M}$ is the dual realization to the realization $M \rightarrow \LieAlg{g}^*/G.$ \end{itemize} A {\em realization} of a Poisson manifold $P$ is defined to be a Poisson map from a symplectic manifold $M$ to $P$ (see \cite{We}). The realization will be called {\it full} if it is submersive. A pair of realizations $P_2 \stackrel{f_2}{\longleftarrow} M \stackrel{f_1}{\longrightarrow} P_1$ is called a dual pair if functions on one induce vector fields along the fibers of the other (i.e., the two opposite foliations are symplectic complements of each other). We note that in the diagram of quotients, any two opposite spaces are a dual pair of realizations. Given a full dual pair with connected fibers, the symplectic leaf foliations on $P_1$ and $P_2$ induce the same foliation on $M$ ($P_1$ and $P_2$ have the ``same'' Casimir functions). The bijection between symplectic leaves on $P_1$ and $P_2$ is given by $$ P_1 \supset S_1 \mapsto f_2(f^{-1}_1(S_1)) = f_2(f^{-1}_1(x)) \ \ \ \forall \; x \in S_1. $$ Returning to moment maps, we have over a coadjoint orbit ${\cal O} \subset \LieAlg{g}^*$: \begin{itemize} \item $\mu^{-1}(\cal{O})$ is coisotropic in $M$ \item $\pi(\mu^{-1}(\cal{O}))$ is a symplectic leaf $M_{red}$ in $M/G$ \item $\bar{\mu}^{-1}(\cal{O})$ is also a symplectic leaf in $\bar{M}$. It is isomorphic to $\mu^{-1}(\cal{O})$/(null), or to $\mu^{-1}(\cal{O})/ \sim$, or to $M_{red} \times \cal O$. \end{itemize} \medskip \noindent \begin{example} {\rm Take $M$ to be the cotangent bundle $T^*G$ of a Lie group $G$. Denote by $\mu_L : T^*G \rightarrow \LieAlg{g}^*$ the moment map for the lifted left action of $G$. The quotient $\pi : M \rightarrow M/G$ is just the moment map $\mu_R : T^*G \rightarrow \LieAlg{g}^*$ for the lifted right action, and $\bar{M}$ is the fiber product $\LieAlg{g}^* \times_{(\LieAlg{g}^*/G)} \LieAlg{g}^*$.} \end{example} \noindent \begin{example} {\rm If $G$ is a connected commutative group $T$, the pair of nodes $\LieAlg{t}^*$ and $\LieAlg{t}^*/T$ coincide. Consequently, so do $M/T$ and $\bar{M}$. The diagram of quotients degenerates to} \begin{equation} {\divide\dgARROWLENGTH by 4 \begin{diagram}[M] \node{M} \arrow{s,r}{\pi} \\ \node{M/T} \arrow{s,r}{\bar{\mu}} \\ \node{\LieAlg{t}^{*}} \arrow{s} \\ \node{(0)} \end{diagram} } \end{equation} \end{example} \noindent \begin{example}\label{diagram-two-quotients} {\rm Consider two Poisson actions on $(M,\sigma)$ of two groups $G,T$ with moment maps $\mu_G, \mu_T$ with connected fibers. Assume that \begin{description} \item[i)] The actions of $G$ and $T$ commute. \end{description} \noindent It follows that $\mu_T : M \rightarrow \LieAlg{t}^*$ factors through $M/G$ and $\mu_G : M \rightarrow \LieAlg{g}^*$ factors through $M/T$. Assume moreover \begin{description} \item[ii)] $T$ is commutative, \item[iii)] $M \rightarrow \LieAlg{g}^*/G$ factors through $\LieAlg{t}^*$ \end{description} Then $\bar{\mu}_G : \bar{M}_G \rightarrow \LieAlg{g}^*$ factors through $M/T$ and the two quotient diagrams fit nicely together:} \begin{equation} {\divide\dgARROWLENGTH by 4 \begin{diagram} \node[3]{M} \arrow{s} \\ \node[3]{\bar{M}_{G}} \arrow[2]{sw} \arrow{se} \\ \node[4]{M/T} \arrow[2]{sw,b}{\bar{\mu}_{T}} \arrow{se} \\ \node{M/G} \arrow{se} \node[4]{\LieAlg{g}^*} \arrow[2]{sw} \\ \node[2]{\LieAlg{t}^*} \arrow{se} \\ \node[3]{\LieAlg{g}^*/G} \arrow{s} \\ \node[3]{(0)} \end{diagram} } \end{equation} \end{example} \begin{rem} \label{rem-acihs-implies-maximal-commutative-subalgebra} {\rm Note that condition iii in example \ref{diagram-two-quotients} holds whenever $M/G \rightarrow \LieAlg{t}^*$ is a completely integrable system (with connected fibers). In that case the map $M/G \rightarrow \LieAlg{t}^*$ pulls back $C^\infty(\LieAlg{t}^*)$ to a {\em maximal} commutative subalgebra ${\cal I}_T$ of $(C^\infty(M/G),\{,\})$. The map $M/G \rightarrow \LieAlg{g}^*/G$ pulls back $C^\infty(\LieAlg{g}^*/G)$ to a commutative Lie subalgebra ${\cal I}_G$ of $(C^\infty(M/G),\{,\})$. As the two group actions commute so do the subalgebras ${\cal I}_G$ and ${\cal I}_T$. By maximality, ${\cal I}_T$ contains ${\cal I}_G$ and consequently $M/G \rightarrow \LieAlg{g}^*/G$ factors through $\LieAlg{t}^*$. } \end{rem} The diagram of quotients for a Poisson action (diagram \ref{diagram-of-quotients}) generalizes to an analogous diagram for any full dual pair of realizations $P_2\stackrel{f_2}{\longleftarrow} M \stackrel{f_1}{\longrightarrow} P_1$. Denote by $\bar{M}$ the image of $M$ in the Poisson manifold $P_1 \times P_2$ under the diagonal Poisson map $f_1 \times f_2 : M \rightarrow P_1 \times P_2$. The realization dual to $f_1 \times f_2 : M \rightarrow \bar{M}$ is the pullback of the symplectic leaf foliations on $P_1$ or $P_2$ (they pull back to the same foliation of $M$). The following is the analogue of Example \ref{diagram-two-quotients} replacing the commutative $T$-action by a realization: \begin{example} \label{example-diagram-hexagon-plus-realization} {\rm \ Let $M/G \stackrel{\pi}{\longleftarrow} M \stackrel{\mu}{\longrightarrow} \LieAlg{g}^*$ be the full dual pair associated to a Poisson action of $G$ on $M$ and $N \stackrel{\ell}{\longleftarrow} M \stackrel{h}{\longrightarrow} B$ a full dual pair of realizations with connected fibers where: \begin{description} \item [(i)] $h$ is $G$-invariant \item [(ii)] $h : M \rightarrow B$ is a Hamiltonian map ($B$ is endowed with the trivial Poisson structure) and \item [(iii)] The composition $M \stackrel{\mu}{\longrightarrow} \LieAlg{g}^* \longrightarrow \LieAlg{g}^*/G$ factors through $h : M \rightarrow B$. \end{description} \noindent Then we get a diagram analogous to the one in example \ref{diagram-two-quotients}: \begin{equation} {\divide\dgARROWLENGTH by 4 \begin{diagram} \node[3]{M} \arrow{s} \\ \node[3]{\bar{M}_{G}} \arrow[2]{sw,t}{\bar{\pi}} \arrow{se} \\ \node[4]{N} \arrow[2]{sw,b}{\bar{h}} \arrow{se} \\ \node{M/G} \arrow{se} \node[4]{\LieAlg{g}^*} \arrow[2]{sw} \\ \node[2]{B} \arrow{se} \\ \node[3]{\LieAlg{g}^*/G} \arrow{s} \\ \node[3]{(0)} \end{diagram} } \end{equation} It follows that the Poisson map $M \stackrel{h \times \mu}{\longrightarrow} B \times_{(\LieAlg{g}^*/G)} \LieAlg{g}^*$ into the fiber product space factors through the realization $M \stackrel{\ell}{\rightarrow} N$ dual to $h : M \rightarrow B$. If, moreover, $M/G \rightarrow B$ is a Lagrangian fibration, then $M \stackrel{h \times \mu}{\longrightarrow} B \times_{(\LieAlg{g}^*/G)} \LieAlg{g}^*$ is itself a realization dual to $h : M \rightarrow B$. } \end{example} \subsection{Finite dimensional Poisson loop group actions} \label{sec-finite-dim-loop-group-actions} We present in this section two elementary constructions related to finite dimensional symplectic leaves in the Poisson quotient $Q_\infty$ of an infinite dimensional symplectic space $M$ by subgroups of loop groups. The material in this section will only be used in section \ref{sec-compatibility-of-heirarchies} so the reader may prefer to read it in conjunction with that section. We will not construct the quotient $Q_\infty$. The spaces involved are constructed independently. Rather, we will analyze the relationship between the Poisson action of the loop group on the infinite dimensional spaces and its descent to the finite dimensional symplectic leaves of $Q_\infty$. In fact, our main purpose in this section is to provide the terminology needed in order to study the Poisson loop group action in the finite dimensional setting (convention \ref{convention-abused-hamiltonian-language} and corollary \ref{cor-hamiltonians-on-the-base}). In section \ref{sec-finite-dim-approaximations} we note that the infinitesimal Hamiltonian actions of elements of the loop group descend to Hamiltonian vector fields on finite dimensional symplectic approximations $M_{(l,l)}$. The $M_{(l,l)}$'s dominate finite dimensional Poisson subvarieties $Q_l$ of $Q_{\infty}$ with positive dimensional fibers. In section \ref{sec-type-loci} the action of certain maximal tori in the loop group further descends to finite Galois covers of certain (type) loci in $Q$ and we examine the sense in which it is Hamiltonian. \subsubsection{Finite dimensional approximations} \label{sec-finite-dim-approaximations} The loop group $G_{\infty}$ is the group $GL(n,{\Bbb C}((z)))$. The level infinity group $G^{+}_{\infty}$ is its positive part $GL(n,{\Bbb C}[[z]])$. Let $(M,\sigma)$ be a symplectic variety with a Poisson loop group action whose moment map is \[ \mu : M \rightarrow \LieAlg{g}_{\infty}^*. \] In section \ref{sec-compatibility-of-heirarchies} $M$ will be the cotangent bundle of a projective (inverse) limit of finite dimensional smooth algebraic varieties (the cotangent bundles of the moduli spaces of vector bundles with level structure). It is thus the inductive (direct) limit of projective limits of finite dimensional varieties. All constructions (morphisms, group actions, symplectic structures etc ...) can be made precise as limits of the standard constructions on finite dimensional approximations. We will omit the technical details as our point is to transfer the discussion back to the finite dimensional symplectic leaves of the Poisson quotient $Q_\infty := M/G^{+}_{\infty}$. Let $\LevelInfinitySubgroup{l}$, $l \geq -1$, be the subgroup of $G^{+}_{\infty}$ of elements equal to $1$ up to order $l$. Denote by $\mu_{\LevelInfinitySubgroup{l}}$ its moment map. We assume that the subquotients \[ M_{(l,k)} := \mu_{\LevelInfinitySubgroup{l}}^{-1}(0)/\LevelInfinitySubgroup{k}, \ \ \ k \geq l, \] are smooth, finite dimensional and that they approximate $M$: \[M = \lim_{l \rightarrow \infty} \lim_{\infty \leftarrow k} M_{(l,k)}.\] Notice that $M_{(l,l)}$ is a symplectic reduction, hence symplectic. Let $a$ be an element of the loop algebra $\LieAlg{g}_{\infty}$ with poles of order at most $l_{0}$. The Hamiltonian vector field $\xi_a$ on $M$ is an infinite double sequence of Hamiltonian vector fields on $M_{(l,k)}$, $l \geq 0$, $k\geq \max\{l,l_0\}$ compatible with respect to projections and inclusions (by a Hamiltonian vector field on $M_{(l,k)} \subset M_{(k,k)}$ we mean, the restriction of a Hamiltonian vector field on $M_{(k,k)}$ which is tangent to $M_{(l,k)}$). The quotient $Q_\infty := M/G^{+}_{\infty}$ is the direct limit $\lim_{l \rightarrow \infty}Q_l$ of the finite dimensional Poisson varieties \[ Q_l := \mu_{\LevelInfinitySubgroup{l}}^{-1}(0)/G^{+}_{\infty} = M_{(l,k)}/ G_k = M_{(l,l)}/ G_l \] where $G_k:=G^{+}_{\infty}/\LevelInfinitySubgroup{k}$ is the finite dimensional level-$k$ group (we assume that the quotients $Q_l$ are smooth). \begin{example} {\rm The homogeneous $G^{+}_{\infty}$-space ${\cal U}_{\infty}:=G^{+}_{\infty}/ GL(n,{\Bbb C})$ is endowed with a canonical infinitesimal $G_{\infty}$-action via its embedding as the degree-$0$ component of the homogeneous $G_{\infty}$-space $G_{\infty} / GL(n,{\Bbb C}[[z^{-1}]])$ \[G^{+}_{\infty}/ GL(n,{\Bbb C}) \hookrightarrow G_{\infty} / GL(n,{\Bbb C}[[z^{-1}]])\] (the degree of $a\in G_{\infty}$ is the signed order of the pole/zero of $\det(a)$). Let $M$ be an open subset of the cotangent bundle $T^*{\cal U}_{\infty}$ for which the regularity assumptions on the approximating quotients $M_{(l,k)}$ hold. This will be made precise in section \ref{sec-polynomial-matrices} and the quotients $Q_l$ will be the spaces of conjugacy classes of polynomial matrices studied in that section. } \end{example} \bigskip Unfortunately, the action of $a \in \LieAlg{g}_{\infty}$ above is not defined on $Q_{l}$. It is well defined only when we retain at least the $l_0$-level structure, i.e., on $M_{(l,k)}$, $k \geq l_0$. In section \ref{sec-type-loci} we will see that the action of certain maximal tori in $G_{\infty}$ descends to {\em finite} Galois covers of certain loci in $Q$. \subsubsection{Type loci} \label{sec-type-loci} Let $(M,\sigma)$ be a smooth symplectic variety endowed with an infinitesimal Poisson action $\mu_{G}^*: \LieAlg{g} \rightarrow [\Gamma(M,\StructureSheaf{M}),\{,\}]$ of a group $G$. Consider a subgroup $G^+ \subset G$, a commutative subgroup $T \subset G$, and their intersection $T^+ := T \cap G^+$. Assume further that the following conditions hold: \smallskip \noindent i) The infinitesimal $G^+$-action integrates to a free action on $M$,\\ ii) $T^+$ is a maximal commutative subgroup whose Weyl group $W_{T^+}:= N_{G^+}(T^+)/T^+$ is finite. \begin{definition} \label{def-group-theoretic-definition-type-loci} {\rm The {\em type} $\tau$ of $T$ is the class of all commutative subgroups $T'$ of $G$ which are conjugate to $T$ via an element of $G^+$. } \end{definition} Let $W := [N_{G^+}(T^+)\cap N(T)]/T^+$ be the corresponding subgroup of both $W_{T^+}$ and $W_{T}$. Denote by \[ \LieAlg{g}^*_\tau \subset \LieAlg{g}^* \] (respectively, $\LieAlg{g}^*_T \subset \LieAlg{g}^*$) the subset of elements whose stabilizer (with respect to the coadjoint action) is a torus of type $\tau$ (respectively, precisely $T$). \begin{example} \label{example-loop-group-level-infinity-group} {\rm Let $G$ be the loop group, $G^+$ the level infinity group and $T \subset G$ a maximal torus of type $\underline{n}$ determined by a partition of the integer $n$ (see section \ref{sec-the-heirarchies}). In this case $G^+$ and $T$ generate $G$. It follows that $W=W_T=W_{T^+}$ and the type $\tau$ is invariant throughout a coadjoint orbit in $\LieAlg{g}^*$. } \end{example} Assume that a ``nice'' (Poisson) quotient $Q := M/G^+$ exists. Let \[M^\tau := \mu_G^{-1}(\LieAlg{g}^*_\tau), \ \ \ \mbox{and} \ \ \ Q^\tau := M^\tau/G^+ \subset Q\] be the loci of type $\tau$. Note that for each $T$ of type $\tau$ there is a canonical isomorphism \[ \mu^{-1}_G(\LieAlg{g}^*_T)/ [N_{G^+}(T^+) \cap N(T)] \stackrel{\cong}{\rightarrow} Q^{\tau} \subset Q. \] In particular, a choice of $T$ of type $\tau$ determines a canonical $W$-Galois cover of $Q^\tau$ \begin{equation} \label{eq-the-galois-cover} \tilde{Q}^T := \mu^{-1}_{G}(\LieAlg{g}^*_T)/T^+. \end{equation} All the $\tilde{Q}^T$ of type $\tau$ are isomorphic (not canonically) to a fixed abstract $W$-cover $\tilde{Q}^\tau$. Note that $\tilde{Q}^T$ is a subset of $M/T^+$. We get a canonical ``section'' (the inclusion) \begin{equation} \label{eq-the-section-from-the-galois-cover} s_T : \tilde{Q}^T \hookrightarrow M/T^+ \end{equation} into a $T$-invariant subset. Consequently, we get an induced $T$-action on the Galois cover $\tilde{Q}^T$. The moment map $\mu_T$ is $T$-invariant, hence, descends to $M/T^+$. Restriction to $s^T(\tilde{Q}^T)$ gives rise to a canonical map \begin{equation} \label{eq-loop-group-moment-map-on-galois-covers} \bar{\mu}_T : \tilde{Q}^T \rightarrow \LieAlg{t}^*. \end{equation} \bigskip The purpose of this section is to examine {\em the extent to which $\bar{\mu}_T$ is the moment map of the $T$-action with respect to the Poisson structure on $Q$}. In general, the $G^+$-equivariant projection \[ j: \LieAlg{g}^* \twoheadrightarrow (\LieAlg{g}^+)^* \] might {\em forget the type}. Coadjoint orbits $S \subset (\LieAlg{g}^+)^*$ may intersect nontrivially the images $j(\LieAlg{g}^*_\tau)$ of several types (e.g., take $S=0$ in example \ref{example-loop-group-level-infinity-group} and observe that the kernel of $j$ intersects coadjoint orbits of all types). Consequently, symplectic leaves $Q_S$ of $Q$ would intersect nontrivially several type loci $Q^\tau$. If $Q_S^{\tau^{open}}$ is an open subvariety of $Q_S$ of type $\tau$ (e.g., if $Q$ is the disjoint union of finitely many type loci and $\tau$ is a {\em generic type}) then the corresponding open subvariety $\tilde{Q}_S^{T^{open}}$ of $\tilde{Q}_S^T$ will be a symplectic variety. In this case the $T$-action on $\tilde{Q}_S^{T^{open}}$ is Poisson whose moment map $\bar{\mu}_T$ is given by (\ref{eq-loop-group-moment-map-on-galois-covers}). The Galois $W$-covers $\tilde{Q}_S^T$ of the nongeneric type loci in $Q_S$ are not symplectic. Nevertheless, motivated by the fact that $\bar{\mu}_T$ can be extended canonically to $M/T^+$ \begin{equation}\label{diag-extending-the-moment-map-from-the-galois-cover} {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{M} \arrow{s} \arrow[2]{e,t}{\mu_T} \node[2]{\LieAlg{t}^*} \\ \node{M/T^+} \arrow{s} \arrow{ene} \node{\tilde{Q}^T_S} \arrow{s} \arrow{w,t}{\supset} \arrow{ne,b}{\bar{\mu}_T} \\ \node{Q} \node{Q^\tau_S} \arrow{w,t}{\supset} \end{diagram} } \end{equation} we will adopt the: \begin{convention} \label{convention-abused-hamiltonian-language} {\rm i) Given an element $h$ of $\LieAlg{t}$ we will say that the corresponding vector field $\bar{\xi}_h$ on $\tilde{Q}_S^T$ is the {\em Hamiltonian vector field of $h$} (even if the type $\tau$ of $T$ is not generic in $Q_S$). ii) We will refer to the pair $(\bar{\mu}_T,\mu_T)$ as the moment map of the $T$-action on $\tilde{Q}_S^T$. } \end{convention} \begin{rems} {\rm Let $G$ be the loop group and $M,G^+,T$ as in section \ref{sec-finite-dim-approaximations}, \begin{enumerate} \item Diagram (\ref{diag-extending-the-moment-map-from-the-galois-cover}) has an obvious finite dimensional approximation in which $Q^\tau_S$, $\tilde{Q}^T_S$ and $T$ stay the same but with $M$ replaced by $M_{(l,l)}$ and $Q$ by $Q_{l}$. By $\mu_{T}$ we mean in this context a linear homomorphism $\mu_T^*:\LieAlg{t} \rightarrow \LieAlg{t}/\LoopAlgSubtorus{l} \rightarrow \Gamma(M_{(l,l)},\StructureSheaf{M_{(l,l)}}).$ \item (Relation with the diagram of quotients (\ref{diagram-of-quotients})) Let $S$ be a coadjoint orbit of level $l$, i.e., $S \subset \LieAlg{g}_l^* := (\LieAlg{g}^{+}_{\infty}/\LevelInfinitySubalg{l})^* \subset (\LieAlg{g}^{+}_{\infty})^*$. There is a rather subtle relationship between the Galois cover $\tilde{Q}_S^T \rightarrow Q^\tau_S$ and the space $\bar{M}_{(l,l)}$ dual to $\LieAlg{g}_l^*/G_{l}$ from the diagram of quotients (\ref{diagram-of-quotients}) of level $l$. The Galois cover $\tilde{Q}_S^T \rightarrow Q^\tau_S$ factors canonically through an intermediate subspace $\tilde{Q}_S^T/\sim$ of $\bar{M}_{(l,l)}$. Note that the loop group moment map $\mu_{G_\infty}$ descends to a map \[ \bar{\mu}_{G_\infty}:\tilde{Q}_S^T \rightarrow (\LieAlg{g}_{\infty}^*)_T \subset \LieAlg{g}_{\infty}^*. \] Two points $\tilde{x}_1, \tilde{x}_2 \in \tilde{Q}_S^T$ in a fiber over $x \in Q^\tau_S$ are identified in $\tilde{Q}_S^T/\sim$ if and only if $\bar{\mu}_{G_\infty}(\tilde{x}_1)$ and $\bar{\mu}_{G_\infty}(\tilde{x}_1)$ project to the same point in $S \subset (\LieAlg{g}^{+}_{\infty})^*$. The relation $\sim$ is a geometric realization of the partial type-forgetfullness of the projection $j:\LieAlg{g}_{\infty}^* \rightarrow (\LieAlg{g}^{+}_{\infty})^*$. The loci in $\bar{M}_{(l,l)}$ at which the type is not forgotten are precisely the loci to which the moment map of the infinitesimal {\em loop group} action descends (from $M^T:= \mu_{G_{\infty}}^{-1}((\LieAlg{g}_{\infty}^*)_T)$. Note that the moment map of the level infinity subgroup descends by definition of the quotient $\bar{M}_{(l,l)}$). In particular, the infinitesimal action of the maximal torus $\LieAlg{t}$ integrates to a Poisson action in these loci. (See section \ref{sec-compatibility-of-stratifications} for examples of such loci.) \end{enumerate} } \end{rems} \bigskip Assume further that we have a ``nice'' quotient $B := M/G$ and that $G$ is generated by $T$ and $G^+$. We get the type loci $B^\tau := M^\tau/G$. Fixing $T$ of type $\tau$ we get the $W$-cover $\tilde{B}^T := M^T/T$. The restriction of the moment map $\mu_T$ to $M^T$ descends further to $\phi_T:\tilde{B}^T \rightarrow \LieAlg{t}^*$ and we get the commutative diagram \begin{equation}\label{diag-factoring-the-moment-map-through-base} {\divide\dgARROWLENGTH by 2 \begin{diagram} \node[3]{M^T} \arrow{sw} \arrow{se,t}{\mu_G} \\ \node[2]{\tilde{Q}^T} \arrow{sw,t}{\tilde{h}} \node[2]{\LieAlg{g}^*_T} \arrow{se} \\ \node{\tilde{B}^T} \arrow[4]{e,t}{\phi_T} \node[4]{\LieAlg{t}^*} \end{diagram} } \end{equation} \begin{corollary} \label{cor-hamiltonians-on-the-base} i) The $T$-action on $M$ descends to a canonical action on the $W$-cover $\tilde{Q}^T$ of the type locus $Q^\tau \subset Q$. ii) Its moment map, in the sense of convention \ref{convention-abused-hamiltonian-language}, is $(\phi_T\circ \tilde{h},\mu_T)$. iii) If the type $\tau$ of $T$ is the generic type in a symplectic leaf $Q_S\subset Q$ and $Q_S^{\tau^{open}} \subset Q_S$ is an open subvariety, then the corresponding $W$-cover $\tilde{Q}_S^{T^{open}}$ is symplectic and $\phi_T\circ \tilde{h}$ is the moment map of the $T$-action in the usual sense. \end{corollary} \newpage \section{Geodesic flow on an ellipsoid} \label{ch3} Consider the geodesic flow on an ellipsoid $$ E = \{(x_1,\cdots,x_{n+1})|\sum^{n+1}_{i=1} \frac{1}{a_i} x^2_i = 1\} \subset {\bf R}^{n+1}, $$ where the metric is induced from the standard one on ${\bf R}^{n+1}$, and where the $a_i$ are distinct positive numbers, say $$ 0 < a_1 < \cdots < a_{n+1}. $$ For $n=1$, the problem is to compute arc length on an ellipse. It amounts to computing the integral $$ s = \int{ \sqrt{\frac{a^2_1 + (a_2-a_1)x^2}{a_1(a_1 - x^2)}}\ \ dx}. $$ (Hence the name {\it elliptic} for this and similar integrals.) For $n=2$, the problem was solved by Jacobi. Each geodesic $\gamma$ on $E$ determines a hyperboloid $E'$, intersecting $E$ in a pair of ovals. The geodesic $\gamma$ oscillates in the band between these ovals, meeting them tangentially. In fact, each tangent line of $\gamma$ is also tangent to the hyperboloid $E'$. The solutions can be parametrized explicitly in terms of hyperelliptic theta functions. The geodesic flow on an $n$-dimensional ellipsoid is integrable, in fact algebraically integrable. We will see this, first using some elementary geometric techniques to describe the geodesics concretely, and then again using the algebraic description of hyperelliptic jacobians which will be extended later to all spectral curves. \subsection{Integrability} The geodesic flow on the $2n$-dimensional symplectic manifold $TE \approx T^*E$ is given by the Hamiltonian function $h$=length square. We need $n-1$ farther, commuting, independent, Hamiltonians. Consider the family of quadrics confocal to $E$: $$ E_\lambda: \; \; \sum^{n+1}_{i=1} \frac{x^2_i}{a_i-\lambda} = 1, $$ depending on a parameter $\lambda$. (The name makes sense only when $n=1$: we get the family of ellipses $(\lambda < a_1)$, hyperbolas $(a_1 < \lambda < a_2)$, and empty (real) conics $(\lambda > a_2)$, with fixed foci.) Here is an intrinsic way to think of this family. Start with a linear pencil $$Q_\lambda = Q_0 + \lambda Q_\infty,\ \ \ \ \ (\lambda \in {\bf P}^1)$$ of quadrics in general position in projective space ${\bf P}^{n+1}$. By ``general position'' we mean that there are exactly $n+2$ values of $\lambda \in {\bf P}^1$ such that $Q_\lambda$ is singular, and for those $\lambda$, $Q_\lambda$ is a cone (i.e. its singular locus, or vertex, is a single point). \proclaim{Lemma}. A generic linear subspace $L \approx {\bf P}^{k-1}$ in ${\bf P}^{n+1}$ is tangent to $Q_\lambda$ for $k$ values of $\lambda$. The points of tangency $p_\lambda$ are pairwise harmonic with respect to each of the quadrics $Q_\mu$.\par \noindent {\bf Proof:} Four points of ${\bf P}^1$ are harmonic if their cross ratio is $-1$; e.g. $0,\infty,a,-a$. Two points $p_1,p_2 \in {\bf P}^1$ are harmonic with respect to a quadric $Q$ if the set $\{p_1,p_2\} \cup (Q \cap {\bf P}^1)$ is harmonic. For example, two points on the line at infinity in ${\bf P}^2$ (i.e. two directions in the affine plane) are harmonic with respect to some (hence every) circle, iff the directions are perpendicular. Since $Q$ is tangent to $L$ if and only if $Q \cap L$ is singular, the first part of the lemma follows by restriction of the pencil to $L$. The second part follows by restricting to the line $p_{\lambda_1},p_{\lambda_2}$, where in appropriate coordinates $Q_{\lambda_1} = x^2$ and $Q_{\lambda_2} = y^2$, so the points of tangency are $0,\infty$ and the quadric $Q_\lambda$ vanishes at $\pm a$, where $a^2 = \lambda$. \EndProof We choose the parameter $\lambda$ so that $Q_\infty$ is one of the singular quadrics. The dual $Q^*_\lambda$ of a non-singular $Q_\lambda$ is a non-singular quadric in $({\bf P}^{n+1})^*$. The dual of $Q_{\infty}$ is a hyperplane $H_{\infty} \subset ({\bf P}^{n+1})^*$ (corresponding to the vertex of $Q_\infty$), with a non-singular quadric $Q^*_\infty \subset H_{\infty}$. We get a family of confocal quadrics by restriction to the affine space ${\bf R}^{n+1}:= ({\bf P}^{n+1})^*\smallsetminus H_{\infty}$: $$ E_\lambda:= Q^*_\lambda|_{{\bf R}^{n+1}}. $$ If we choose coordinates so that $$ \begin{array}{lcl}Q_\infty & = & \sum^{n+1}_{i=1} x^2_i\{\Bbb Q}_0 & = & \sum^{n}_{i=0} a_i x^2_i\end{array} $$ (where $a_0 = -1$ and the other $a_i$ are as above), we retrieve the original $E_\lambda$. (Euclidean geometry in ${\bf R}^{n+1}$ is equivalent, in the sense of Klein's program, to the geometry of ${\bf P}^{n+1}$ with a distinguished ``light-cone'' $Q_\infty$. In this equivalence, $E = E_0$ corresponds to $Q_0$, which determines the pencil $\{Q_\lambda\}$, which corresponds to the confocal family $\{E_\lambda\}$.) Dualizing the lemma, for $k = n+1,n$, gives the following properties of the confocal family. (The reader is invited to amuse herself by drawing the case $n=1$ in the plane.) \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item Through a generic point $x$ of ${\bf R}^{n+1}$ pass $n+1$ of the $E_\lambda$. \item These $n+1$ quadrics intersect perpendicularly at $x$. \item A generic line $\ell$ in ${\bf R}^{n+1}$ is tangent to $n$ of the $E_\lambda$. \item The tangent hyperplanes to these $n$ quadrics (at their respective points of tangency to $\ell$) are perpendicular. \end{list} By property (3) we can associate to a generic line $\ell$ in ${\bf R}^{n+1}$ an unordered set of $n$ values $\lambda_i$, $1 \leq i \leq n$, such that $Q_{\lambda_i}$ is tangent to $\ell$. When $\ell$ comes from a point of $TE$, one of these, say $\lambda_n$, equals $0$. The remaining $n-1$ values $\lambda_i$ (or rather, their symmetric functions) give $n-1$ independent functions on $TE$; in fact, they can take an arbitrary $(n-1)$-tuple of values. These functions descend to the projectivized tangent bundle ${\bf P}(TE)$; so together with the original Hamiltonian $h$ (= length squared) they give $n$ independent functions on $TE$. The key to integrability is: \proclaim{Chasles' Theorem}. The $\lambda_i$ are flow invariants, i.e. they are constant along a geodesic $\gamma = \gamma(t)$.\par \noindent {\bf Proof}. For any curve $\gamma(t)$ in ${\bf R}^{n+1}$, the family of tangent lines $\ell(t)$ gives a curve $\Lambda$ in the Grassmannian $Gr(1,{\bf R}^{n+1})$ of affine lines in ${\bf R}^{n+1}$. This curve is developable, i.e. its tangent line $T_{\ell(t)}\Lambda$ is given by the pencil of lines through $\gamma(t)$ in the osculating plane of $\gamma$ at $t$. When $\gamma$ is a geodesic, this plane is the span of $\ell(t)$ and the normal vector $n(t)$ to $E$ at $\gamma(t)$. Write $\lambda_i(t)$ for the value of $\lambda_i$ at $\ell(t)$. Let $Z_i$ be the hypersurface in $Gr(1,{\bf R}^{n+1})$ parametrizing lines tangent to $E_{\lambda_i(t)}$, for some fixed $t$. The tangent space $T_{\ell(t)}Z_i$ contains all lines through $\gamma(t)$ in the tangent hyperplane $T_{p_i(t)}E_{\lambda_{i}(t)}$, and this hyperplane contains the normal $n(t)$, by property (4). Hence: $$ T_{\ell(t)}\Lambda \subset \{\mbox{lines\ through}\ \gamma(t),\ \mbox{in}\ T_{p_i(t)}E_{\lambda_{i(t)}}\} \subset T_{\ell(t)}Z_i\ \ \ \ \ i = 1,\cdots,n-1. $$ If the family $\Lambda$ of tangent lines to a geodesic meets $Z_i$, it must therefore stay in it. \EndProof \subsection{Algebraic integrability} Since a line $\ell$ determines two (opposite) tangent vectors of given non-zero length, we have identified the fiber of the geodesic flow as a double cover $\widetilde{K}$ of $$K := \{\ell \in Gr(1,{\bf R}^{n+1})|\ell \ \mbox{is\ tangent\ to}\ E_{\lambda_1},\cdots, E_{\lambda_{n}}\}. $$ Next we want to interpret this in terms of the real points of a complex abelian variety. We follow Kn\"{o}rrer's approach \cite{knorrer}, which in turn is based on \cite{moser},\cite{reid} and \cite{donagi-group-law}. Start with the pencil of quadrics in ${\bf P}^{2n+1}$ (over ${\bf C}$): $$ Y_\lambda := Y_0 + \lambda Y_\infty $$ with $$ \begin{array}{lcl} Y_0 & = & \sum^{2n+1}_{i=1} a_i x^2_i - x^2_0\\ Y_\infty & = & \sum^{2n+1}_{i=1} x^2_i.\end{array} $$ The base locus $X = Y_0 \cap Y_\infty$ is non-singular if the $a_i$ are distinct. We set $a_0 = \infty$. The family of linear subspaces ${\bf P}^n$ contained in a fixed quadric $Y_\lambda$ consists of two connected components, or rulings, for the non-singular $Y_\lambda$ $(\lambda \not\in \{a_i\})$, and of a single ruling for $\lambda = a_i$. We thus get a double cover $$\pi: C \to {\bf P}^1$$ of the $\lambda$-line, parametrizing the rulings. (More precisely, one considers the variety of pairs $$ {\cal P} = \{(A,\lambda)|A\ \ \mbox{is\ a}\ {\bf P}^n\ \ \mbox{contained\ in}\ Q_\lambda\}, $$ and takes the Stein factorization of the second projection.) Explicitly, $C$ is the hyperelliptic curve of genus $n$: $$C: \; s^2 = \Pi^{2n+1}_{i=1} (t-a_i).$$ Miles Reid \cite{reid} showed that the Jacobian $J(C)$ is isomorphic to the variety $$F:= \{A \in Gr(n-1,{\bf P}^{2n+1})|A \subset X\} $$ of linear subspaces in the base locus. An explicit group law on $F$ is given in \cite{donagi-group-law}, and corresponding results for rank $2$ vector bundles on $C$ are in \cite{DR}. Since we are interested in a family of varieties $F$ with varying parameters, we need some information about the isomorphism. Let $Pic^d(C)$ denote the variety parametrizing isomorphism classes of degree-$d$ line bundles on $C$. Then $Pic^0(C) = J(C)$ is a group, $Pic^1(C)$ is a torser (= principal homogeneous space) over it, but, these two have no natural identification; while $Pic^2(C) \approx J(C)$ canonically, using the hyperelliptic bundle on $C$. It turns out that $F$ is isomorphic to $Pic^0(C)$ and to $Pic^1(C)$, but neither isomorphism is canonical. Rather, we may think of $F$ as ``$Pic^{\frac{1}{2}}(C)$'': it is a torser over $J(C)$, and has a natural torser map $$ F \times F \to Pic^1(C). $$ All of this is based on the existence of a natural morphism $$ j: F \times C \to F . $$ The ruling $p$ (on the quadric $Y_{\pi(p)}$) contains a unique subspace ${\bf P}^n$ which contains a given ${\bf P}^{n-1}$-subspace $A \in F$. This ${\bf P}^n$ intersects $X$ in the union of $A$ and another element of $F$, which we call $j(A,p)$. We can also think of $j$ as a family of involutions of $F$, indexed by $p \in C$. This extends to a map $$ F \times J(C) \to F $$ which gives the torser structure on $F$. Once $F$ is thus identified with $J(C)$, the map $j$ becomes $$j(A,p) = p - A,$$ up to an additive constant. Since this is well defined globally, points $A \in F$ must behave as line bundles on $C$ of ``degree $\frac{1}{2}$''. In particular, we have for $0 \leq i \leq 2n+1$ the involution $$\begin{array}{cl}j_i: & F \to F\\& A \mapsto j(A,a_i),\\ \end{array}$$ where we set $a_i = \infty$ and identify the $a_i \in {\bf P}^1$ with the $2n+2$ Weierstrass points $\pi^{-1}(a_i) \in C$. Explicitly, each $j_i$ is induced by the linear involution $$\overline{j}_i: {\bf P}^{2n+1} \to {\bf P}^{2n+1}$$ flipping the sign of the $i$-th coordinate. Consider the linear projection $$ \begin{array}{rcl}\rho: {\bf P}^{2n+1} & \to & {\bf P}^{n+1}\\ (x_0,\cdots,x_{2n+1}) & \mapsto & (x_0,\cdots,x_{n+1}),\\ \end{array} $$ which commutes with the $\overline{j}_i,\ n + 2 \leq i$. Recall that in ${\bf P}^{n+1}$ we have the pencil of quadrics $Q_\lambda$, with dual quadrics $E_\lambda$ in $({\bf P}^{n+1})^*$. \proclaim{Proposition}. \begin{list}{{\rm(\roman{bacon})}}{\usecounter{bacon}} \item The projection $\rho$ maps $F$ to $$F' := \{B \in Gr(n- 1,{\bf P}^{n+1})|B\ \mbox{is\ tangent\ to}\ Q_{n+2},\cdots,Q_{2n+1}\}.$$ \item The induced $\rho: F \to F'$ is a finite morphism of degree $2^n$, and can be identified with the quotient of $F$ by the group $G \approx ({\bf Z}/2{\bf Z})^n$ generated by the involutions $j_i$, $n + 2 \leq i \in 2n+1$. \item Duality takes $F'$ isomorphically to the variety $K$ of lines in $({\Bbb P}^{n+1})^*$ tangent to $E_{\lambda_i}$, $\lambda_i = a_{n+1+i}$, $1 \leq i \leq n$. \end{list} \par We omit the straightforward proof. Let $\widetilde{G} \subset G$ be the index-$2$ subgroup generated by the products $j_{i_1} \circ j_{i_2}$, and set $$ \widetilde{K} := F/\widetilde{G}. $$ We obtain natural commuting maps, whose degrees are indicated next to the arrows: $$ \begin{array}{ccccc}F & \stackrel{2^{n-1}}{\longrightarrow} & \widetilde{K} & \stackrel{2^{n+1}}{\longrightarrow} & Pic^{1}(C)\\ & & \downarrow^2 & & ^2\downarrow\\ & & K & \stackrel{2^{n+1}}{\longrightarrow} & \mbox{Kummer}^{1}(C).\\ \end{array} $$ Here $\mbox{Kummer}^d(C)$ stands for the quotient of $Pic^dC$ by the involution $$ L \mapsto dH-L, $$ where $H$ is the hyperelliptic bundle $\in Pic^2(C)$. The composition of the maps in the top row is multiplication by $2$: $$ F \approx Pic^{\frac{1}{2}} (C) \stackrel{\cdot 2}{\longrightarrow} Pic^{1}(C). $$ In conclusion, the fiber of the geodesic flow on $E = E_0$ with invariants $h = 1$ (say) and $\lambda_i = a_{n+1+i}$, $1 \leq i \leq n$ can be identified with the real locus in $\widetilde{K} = \widetilde{K}(\lambda_1,\cdots,\lambda_n)$. The latter is a $2^{n+1}$-sheeted cover of $Pic^1(C)$, so up to translation by some points of order $2$, it is an abelian variety, isomorphic to a $2^{n+1}$ sheeted cover of the hyperelliptic Jacobian $J(C)$. \subsection{The flows} Two details of the above story are somewhat unsatisfactory: First, the asymmetry between the $n-1$ Hamiltonians $\lambda_i$ and the remaining Hamiltonian $H$ (length squared). And second, the fact that the complexified total space $TE$ of the system is not quite symplectic. Indeed, for an arbitrary algebraic hypersurface $M \subset {\bf C}^{n+1}$, given by $f = 0$, the complexified metric on ${\Bbb C}^{n+1}$ induces bundle maps $$ TM \hookrightarrow T{\bf C}^{n+1}|_M \stackrel{\sim}{\rightarrow} T^*{\bf C}^{n+1}|_M \rightarrow\!\!\!\rightarrow T^{*}M, $$ but the composition is not an isomorphism; rather, it is degenerate at points where $$0 = (\bigtriangledown f)^2 = \sum^{n+1}_{k=1} \left(\frac{\partial f}{\partial x_{k}}\right)^2. $$ For an ellipsoid $\sum x_k^2/a_k = 1$ (other than a sphere) there will be an empty real, but non-empty complex degeneracy locus, given by the equation $\sum(x_k/a_k)^2 = 0$. Both of these annoyances disappear if we replace the total space by the tangent bundle $TS$ of the sphere $$S = \{(x_1,\cdots,x_{n+1}) \in {\bf C}^{n+1}|\sum x_{k}^2 = 1\}, $$ i.e. $$TS = \{(x,y) \in {\bf C}^{2n+2}| \sum x_{k}^2 = 1, \sum x_k y_k = 0\}. $$ This is globally symplectic, and the $n$ (unordered) Commuting Hamiltonians can be taken to be the values $\lambda_i$, $1 \leq i \leq n$, such that the line $$\ell_{x,y} := (\mbox{line\ through}\ y\ \mbox{in\ direction}\ x\ )$$ is tangent to $E_{\lambda_i}$. The original system $TE$ can be recovered as a ${\bf C}^*$-bundle (where ${\bf C}^*$ acts by rescaling the tangent direction $x$) over the hypersurface $\lambda = 0$ in $TS$. Here is the explicit equation of the hypersurface: $$\begin{array}{rcl} \lambda = 0 & \Leftrightarrow & \ell_{x,y} \ \ \mbox{is\ tangent\ to}\ E = \{\sum \frac{x_{k}^2}{a_k} = 1\} \\ & \Leftrightarrow & -1+\sum \frac{(y_k + tx_k)^2}{a_k} = 0\ \mbox{has\ a\ unique\ solution}\ t \\ & \Leftrightarrow & (- 1+\sum \frac{y_{k}^2}{a_k} + 2t (\sum \frac{x_k y_k}{a_k}) + t^2(\sum \frac{x_{k}^2}{a_k}) = 0\ \mbox{has\ a\ unique\ solution} \\ & \Leftrightarrow & 0 = (\sum \frac{x_k y_k}{a_k})^2 - (\sum \frac{x_{k}^2}{a_k})(-1 + \sum \frac{y_{k}^2}{a_k}). \end{array} $$ More generally, this computation shows that $\ell_{x,y}$ is tangent to $E_{\lambda}$ if and only if \[ 0 = (\sum \frac{x_k y_k}{a_{k^{-\lambda}}})^2 - (\sum \frac{x_{k}^2}{a_{k^{-\lambda}}})(-1 + \sum \frac{y_{k}^2}{a_{k^- \lambda}}) = \sum_k \frac{x_k^2}{a_k-\lambda} + \sum_{k\neq l}\frac{x_k y_k x_l y_l- x_k^2 y_l^2}{(a_k-\lambda)(a_l-\lambda)}. \] As a function of $\lambda$, the last expression has first order poles at $\lambda=a_k$, $1\leq k \leq n+1$, so it can be rewritten as \[ \sum \frac{1}{a_k - \lambda} F_k(x,y) \] where the $F_k$ are found by taking residue at $\lambda=a_k$: \[ F_k (x,y) := x_{k}^2 + \sum_{\ell \neq k} \frac{(x_k y_\ell - x_\ell y_k)^2}{a_k - a_\ell}. \] We see that fixing the $n+1$ values $F_k(x,y)$, $1 \leq k \leq n+1$, subject to the condition $$ \sum^{n+1}_{k=1} F_k = 1, $$ is equivalent to fixing the $n$ (unordered) values $\lambda_i$, $1 \leq i \leq n$. This determines the hyperelliptic curve $$ C : \; s^2 = \prod^{n+1}_{k=1} (t - a_k) \cdot \prod^{n}_{i=1} (t - \lambda_i), $$ and the corresponding abelian variety $$ \widetilde{K} = \widetilde{K} (\lambda_1,\cdots,\lambda_n) = J(C)/\widetilde{G} \approx \{(x,y)| \ell_{x,y}\ \mbox{is\ tangent\ to} \ E_{\lambda_i}, \ \ \ 1 \leq i \leq n\}. $$ \vspace{0.1in} \proclaim{Theorem}. \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item Geodesic flow on the quadric $E_{\lambda_i}$ is the Hamiltonian vector field on $TS$ given by the (local) Hamiltonian $\lambda_i$. On $\widetilde{K}$ it is a constant vector field in the direction of the Weierstrass point $\lambda_i \in C$. \item The Hamiltonian vector field on $TS$ with Hamiltonian $F_k$ is constant on $\widetilde{K}$, in the direction of the Weierstrass point $a_k \in C$. \end{list} \par The direction at $\ell \in {\widetilde{K}}$ of geodesic flow on $E_{\lambda_i}$ was described in the proof of Chasles' theorem. The direction given by the Weierstrass point $\lambda_i$ is given at $A \in F$ as the tangent vector at $\lambda_i$ to the curve $$ p \mapsto j(A_{i},p)\ \ (\mbox{where}\ A_i = j(A,\lambda_i)). $$ The proof of (1) amounts to unwinding the definitions to see that these two directions agree. (For details, see \cite{knorrer} and \cite{donagi-group-law}.) Since the level sets of the $\lambda_i$ and the $F_k$ are the same, the Hamiltonian vector field of $F_k$ evolves on the same ${\widetilde{K}}$, and is constant there. A monodromy argument on the family of hyperelliptic curves then shows that its direction must agree with $a_k$. Mumford gives an explicit computation for this in \cite{Mum}, Theorem 4.7, following Moser \cite{moser}. We have identified the flows corresponding to $2n+1$ of the Weierstrass points. The remaining one, at $\lambda = \infty$, corresponds to the Hamiltonian $$ H = \frac{1}{2} \sum^{n+1}_{k=1} a_k F_k = \frac{1}{2} \sum a_k x_{k}^2 + \frac{1}{2} \sum y_{k}^2, $$ giving Neumann's system, which is the starting point for the analysis in \cite{moser} and \cite{Mum}. \subsection{Explicit parametrization} Fix the hyperelliptic curve of genus $n$ $$ C: \; s^2 = f(t) := \prod^{2n+1}_{i=1} (t - a_i), $$ with projection $$ \begin{array}{rcl}\pi:\ C & \rightarrow & {\bf P}^1\\(t,s) & \mapsto & t\\ \end{array} $$ and involution $$ \begin{array}{rcl}i:\ C & \rightarrow & C\\(t,s) & \mapsto & (t, - s).\\ \end{array} $$ We identify the various components $Pic^d(C)$ by means of the base point $\infty$, which is a Weierstrass point. The affine open subset $J(C) \smallsetminus \Theta$ can be described geometrically, by Riemann's theorem: $$ J(C) \smallsetminus \Theta \approx $$ $$\{L \in Pic^{n-1}(C)|h^0(L) = 0\} \approx $$ $$ \{(p_1,\cdots,p_n) \in Sym^nC| p_i \neq \infty, p_i \neq i(p_j)\}, $$ where the last identification sends $L$ to the unique effective divisor of $L(\infty)$. Mumford \cite{Mum} gives an explicit algebraic parametrization of the same open set, which he attributes to Jacobi: to the $n$-tuple $D = (p_1,\cdots,p_n)$ he associates three polynomials of a single variable $t$: \begin{list}{{\rm(\roman{bacon})}}{\usecounter{bacon}} \item $U(t) := \prod^{n}_{i=1} (t - t(p_i))$. \item $V(t)$ is the unique polynomial of degree $\leq n-1$ such that the meromorphic function $$ V \circ \pi - s \ \ : \ \ C \to {\bf P}^1 $$ vanishes on the divisor $D \subset C$. It is obtained by Lagrange interpolation of the expansions of $s$ at the $p_i$, e.g. when the $p_i$ are all distinct, $$ V(t) = \sum^{n}_{i=1} s(p_i) \prod_{j \neq i} \frac{t - t(p_j)}{t(p_i) - t(p_j)}. $$ \item $W(t) = \frac{f(t) - V(t)^2}{U(t)}$; the definition of $V$ and the equation $s^2 = f(t)$ guarantee that this is a monic polynomial of degree $n+1$. \end{list} Conversely, the polynomials $U,V,W$ determine the values $t(p_i), s(p_i)$, hence the divisor $D$. By reading off the coefficients, we obtain an embedding: $$(U,V,W): {\cal J}(C)\backslash \Theta \hookrightarrow {\bf C}^{3n+1}.$$ The image is $$\{(U,V,W)| V^2 + UW = f\}.$$ This description fits beautifully with the integrable system on $TS$ representing geodesic flow on the $E_\lambda$. We can rephrase our previous computation as: $$\ell_{x,y} \ \mbox{is\ tangent\ to}\ E_{\lambda_1},\cdots,E_{\lambda_n} \Leftrightarrow f_1(t)f_2(t) = UW + V^2,$$ where, for $(x,y) \in TS$, we set: $$ \begin{array}{lcl} f_1(t) = \prod^{n+1}_{k=1}(t-a_k) & \ \ \ \ & \mbox{(this\ is\ independent\ of}\ x,y) \\ f_2(t) = f_1(t) \cdot \sum_k \frac{F_k(x,y)}{t - a_k} & \ \ \ \ & (\mbox{this\ varies\ with}\ x,y;\ \mbox{the\ roots\ are\ the}\ \lambda_i) \\ U(t) = f_1(t) (\sum_k \frac{x_{k}^2}{t - a_k}) & & \\ W(t) = f_1(t) (1 + \sum_k \frac{y_{k}^2}{t - a_k}) & & \\ V(t) = \sqrt{-1} \cdot f_1(t) \cdot (\sum_k \frac{x_k y_k}{t - a_k}) & &\\ \end{array} $$ The entire system $TS$ is thus mapped to ${\bf C}^{3n+1}$. Each abelian variety ${\widetilde{K}} = {\widetilde{K}}(\lambda_1,\cdots,\lambda_n)$ is mapped to ${\cal J}(C) \backslash \Theta$ embedded in ${\bf C}^{3n+1}$ as before, where $C$ is defined by $s^2 = f(t)$, and $f = f_1 \cdot f_2$, with $f_1$ fixed (of degree $n+1$) and $f_2$ variable (of degree $n$). On each ${\widetilde{K}}$ the map is of degree $2^{n+1}$; the group $({\bf Z}/2{\bf Z})^{n+1}$ operates by sending $$ (x_k,y_k) \mapsto (\epsilon_k x_k, \epsilon_k y_k),\ \ \ \ \epsilon_k = \pm 1. $$ \newpage \section{Spectral curves and vector bundles} \label{ch4} We review in this chapter a general construction of an integrable system on the moduli space of Higgs pairs $(E,\varphi)$ consisting of a vector bundle $E$ on a curve and a meromorphic $1$-form valued endomorphism $\varphi$ (theorem \ref{thm-markman-botachin}). These moduli spaces admit a natural foliation by Jacobians of spectral curves. The spectral curves are branched covers of the base curve arising from the eigenvalues of the endomorphisms $\varphi$. We concentrate on two examples: \smallskip \noindent - The Hitchin system supported on the cotangent bundle of the moduli space of vector bundles on a curve (section \ref{sec-spectral-curves-and-the-hitchin-system}), and \noindent - An integrable system on the moduli space of conjugacy classes of polynomial matrices (section \ref{sec-polynomial-matrices}). \smallskip The latter is then used to retrieve the Jacobi-Moser-Mumford system which arose in chapter \ref{ch3} out of the geodesic flow on an ellipsoid. Both examples are endowed with a natural symplectic or Poisson structure. The general construction of the Poisson structure on the moduli spaces of Higgs pairs is postponed to chapter \ref{ch5}. We begin with a short survey of some basic facts about vector bundles on a curve. \subsection{Vector Bundles on a Curve} \label{sec-vector-bundles-on-a-curve} We fix a (compact, non-singular) curve $\Sigma$ of genus $g$. A basic object in these lectures will be the moduli space of stable (or semistable) vector bundles on $\Sigma$ of given rank $r$ and degree $d$. To motivate the introduction of this object, let us try to describe a ``general'' vector bundle on $\Sigma$. One simple operation which produces vector bundles from line bundles is the direct image: start with an $r$-sheeted branched covering $\pi : C \rightarrow \Sigma$, ramified at points of some divisor $R$ in the non-singular curve $C$. Then any line bundle $L \in Pic\; C$ determines a rank-$r$ vector bundle $E := \pi_*L$ on $\Sigma$. As a locally free sheaf of ${\cal O} _\Sigma$-modules of rank $r$, this is easy to describe $$ \Gamma({\cal U},\pi_*L) \ := \ \Gamma(\pi^{-1} {\cal U}, L), $$ for open subsets ${\cal U} \subset \Sigma$. As a vector bundle, the description is clear only at unbranched points of $\Sigma$: \ if $\pi^{-1}(p)$ consists of $r$ distinct points $p_1, \cdots , p_r$ then the fiber of $E$ at $p$ is naturally isomorphic to the direct sum of the fibers of $L$: $$ E_p \; \approx \; \bigoplus^r_{i=1} \; L_{p_i}. $$ At branch points of $\pi$, $E_p$ does not admit a natural decomposition, but only a filtration. This is reflected in a drop in the degree. Indeed, the Grothendieck-Riemann-Roch theorem says in our (rather trivial) case that $$ \chi(\pi_*L) = \chi(L), $$ where $\chi$ is the holomorphic Euler characteristic, $$ \chi(E) := \deg E - (g-1) \mathop{\rm rank} \; E = \deg E + \chi({\cal O}) \cdot \mathop{\rm rank} \; E. $$ Using Hurwitz' formula: $$ \chi({\cal O}_C) = r \, \chi({\cal O}_\Sigma) - {1 \over 2} \deg \; R, $$ this becomes: $$ \deg \; E = \deg \; L - {1 \over 2} \deg \; R. $$ \begin{example} \label{example-direct-image} {\rm Consider the double cover \begin{eqnarray*} \pi : {\Bbb P}^1 &\longrightarrow& {\Bbb P}^1 \\ w &\longrightarrow& z = w^2 \\ \end{eqnarray*} branched over $0,\infty$. The direct image of the structure sheaf is: $$ \pi_* {\cal O} \approx {\cal O} \oplus {\cal O}(-1). $$ We can think of this as sending a regular function $f = f(w)$ (on some invariant open set upstairs) to the pair $(f_+(z), f_-(z))$ downstairs, where $$ f(w) = f_+(w^2) \; + \; wf_-(w^2). $$ In the image we get all pairs with $f_+$ regular (i.e. a section of $\cal O$) and $f_-$ regular and vanishing at $\infty$ (i.e. a section of ${\cal O}(-1))$. (Similar considerations show that $$ \pi_* {\cal O}(-1) \; \approx \; {\cal O}(-1) \oplus {\cal O}(-1) $$ and more generally: $$ \pi_* {\cal O}(d) \approx {\cal O} ( [ {d \over 2} ]) \oplus {\cal O}([ {d-1 \over 2} ]). $$ Note that this has degree $d-1$, as expected). The structure of $\pi_*L$ near the branch point $z = 0$ can be described, in this case, by the action on the local basis $a,b$ (of even, odd sections) of multiplication by the section $w$ upstairs: $$ a \mapsto b , \quad \quad b \mapsto za, $$ i.e. $w$ is represented by the matrix $$ \left( \begin{array}{cc} 0 & z \\ 1 & 0 \\ \end{array} \right) $$ whose square is $z \cdot I$. At a branch point where $k$ sheets come together, the corresponding action (in terms of a basis indexed by the $k-th$ roots of unity) is given by the matrix: \begin{equation}\label{eq-ramification-matrix} P_k : = \left( \begin{array}{cccccc} 0 & & & & 0 & z \\ 1 & & & & \cdot & \cdot \\ \cdot & \cdot & & & \cdot & \cdot \\ \cdot & & \cdot & & \cdot & \cdot \\ \cdot & & & \cdot & 0 & 0 \\ 0 & & & & 1 & 0 \end{array} \right) \end{equation} whose $k$-th power is $z \cdot I$.} \end{example} \begin{example} \ \ {\rm Now consider a 2-sheeted branched cover $\pi : C \rightarrow {\Bbb P}^1$, where $g(C) > 0$. If we take $\chi(L) = 0$, i.e. $\deg L = g - 1$, we get $\deg(\pi_*L) = -2$. The equality $$ \ell := h^0(C,L) = h^0({\Bbb P}^1, \pi_*L) $$ implies $$ \pi_*L \approx {\cal O}(\ell - 1) \; \oplus \; {\cal O}(-\ell - 1). $$ In particular, we discover a very disturbing phenomenon: \ \ as the line bundle $L$ varies continuously, in $Pic^{g-1}C$, so should presumably $\pi_*L$; but if we consider a 1-parameter family of line bundles $L_t$ such that \begin{eqnarray*} L_0 &\in& \Theta \\ L_t &\notin& \Theta, \ \ \ t \not= 0 , \\ \end{eqnarray*} we see that the vector bundle $\pi_*L_t$ jumps from its generic value, ${\cal O}(-1) \oplus {\cal O}(-1)$ to ${\cal O} \oplus {\cal O}(-2)$ at $t=0$. Similar jumps can clearly be forced on a rank-$r$ bundle by considering $r$-sheeted branched covers.} \end{example} The moral of these examples is that if we want a moduli space parametrizing the ``general'' vector bundle on a curve and having a reasonable (say, separated) topology, we cannot consider {\it all} bundles. In the case of ${\Bbb P}^1$, we will end up with only the balanced bundles such as ${\cal O}(-1) \oplus {\cal O}(-1)$, thus avoiding the possibility of a discontinuous jump. The slope $\mu(E)$ of a vector bundle $E$ is defined by: $$ \mu(E) := {\deg E \over \mathop{\rm rank} E}. $$ A bundle $E$ is called stable (resp., semistable) if for every subbundle $F \subset E$ (other than $0,E$), $$ \mu(F) < \mu(E), \quad \quad (resp. \ \mu(F) \le \mu(E)). $$ The basic result due to Mumford and Seshadri \cite{seshadri-construction-moduli-vb}, is that reasonable (coarse) moduli spaces ${\cal U}^s_\Sigma(r,d) \subset {\cal U}_\Sigma(r,d)$ exist, with the following properties: \begin{itemize} \item ${\cal U}^s_\Sigma(r,d)$ is smooth; its points parametrize isomorphism classes of stable bundles of rank $r$ and degree $d$ on $\Sigma$; it is an open subset of ${\cal U}_\Sigma(r,d)$. \item ${\cal U}_\Sigma(r,d)$ is projective; its points parametrize equivalence classes of semistable bundles, where two bundles are equivalent, roughly, if they admit filtrations by semistable subbundles (of constant slope) with isomorphic graded pieces. \item Both are coarse moduli spaces; this means that any ``family'', i.e. vector bundle on a product $S \times \Sigma$, where $S$ is any scheme, whose restrictions $E_s$ to copies $s \times \Sigma$ of $\Sigma$ are (semi) stable of rank $r$ and degree $d$, determines a unique morphism of $S$ to ${\cal U}^s_\Sigma(r,d)$ (respectively, ${\cal U}_\Sigma(r,d))$ which sends each $s \in S$ to the isomorphism (resp. equivalence) class of $E_s$, and has the obvious functoriality properties. \end{itemize} \noindent \underline{Examples} \noindent \underline{$g = 0$}. \ The stable bundles are the line bundles ${\cal O}(d)$. The semi-stable bundles are the balanced vector bundles, ${\cal O}(d)^{\oplus r}$. Thus ${\cal U}_{P^1}(r,d)$ is a point if $r\mid d$, empty otherwise, while the stable subset is empty when $r \ne 1$. \smallskip \noindent \underline{$g = 1$}. \ Let $h := gcd(r,d)$. Atiyah \cite{atiyah-vb-on-elliptic-curves} shows that ${\cal U}_\Sigma(r,d)$ is isomorphic to the symmetric product $S^h\Sigma$, and that each semistable equivalence class contains a unique decomposable bundle $E = \oplus^h_{i=1} \; E_i$, where each $E_i$ is stable of rank $r/h$ and degree $d/h$. (Other bundles in this equivalence class are filtered, with the $E_i$ as subquotients.) Thus when $h=1, \ {\cal U}^s_\Sigma = {\cal U}_\Sigma$, and when $h > 1,\ {\cal U}^s_\Sigma$ is empty. The possibilities for semistable bundles are illustrated in the case $r=2$, $d=0$: given two line bundles $L_1,L_2 \in Pic^0 \ \Sigma$, the possible extensions are determined, up to non zero scalars, by elements of $$ Ext^1_{{\cal O}_\Sigma}(L_1,L_2) \approx H^1(L_2 \otimes L^{-1}_1). $$ The direct sum is thus the only extension when $L_1 \not\approx L_2$, while if $L_1 \approx L_2 \approx L$ there is, up to isomorphism, also a unique non-trivial extension, say $E_L$. There is, again, a jump phenomenon: \ \ by rescaling the extension class we get a family of vector bundles with generic member isomorphic to $E_L$ and special member $L \oplus L$. This explains why there cannot exist a moduli space parametrizing {\it isomorphism} classes of semistable bundles; neither $L \oplus L$ nor $E_L$ is excluded, and the point representing the former is in the closure of the latter, so they must be identified, i.e. $E_L$ and $L \oplus L$ must be declared to be equivalent. \smallskip \noindent \underline{Higher Genus}. \ \ The only other cases where an explicit description of ${\cal U}_\Sigma(r,d)$ is known are when $r=2$ and $g=3$ \cite{narasimhan-ramanan-rk2-genus3} or $r=2$ and $\Sigma$ is hyperelliptic of any genus \cite{DR}. In the latter case, the moduli space ${\cal U}_\Sigma(2,\xi)$ of rank $2$ vector bundles with a fixed determinant line bundle $\xi$ of odd degree is isomorphic to the family of linear spaces ${\Bbb P}^{g-2}$ in the intersection of the two quadrics in ${\Bbb P}^{2g+1}$ used in Chapter \ref{ch3}. In the even degree case, ${\cal U}_\Sigma(2,\xi)$ can also be described in terms of the same two quadrics; when $g = 2$, it turns out to be isomorphic to ${\Bbb P}^3$, in which the locus of semistable but non-stable points is the Kummer surface $K := {\cal J}(\Sigma) / \pm 1$, with its classical embedding in ${\Bbb P}^3$ as a quadric with 16 nodes. \bigskip Elementary deformation theory lets us make some general statements about ${\cal U} := {\cal U}_\Sigma(r,d)$ and ${\cal U}^s := {\cal U}^s_\Sigma(r,d)$: \begin{lem} For $g \ge 2$: \begin{itemize} \begin{enumerate} \item dim ${\cal U} = 1 + r^2(g-1)$, and ${\cal U}^s$ is a dense open subset. \item Stable bundles $E$ are simple, i.e. the only (global) endomorphisms of $E$ are scalars. \item Stable bundles are non-singular points of ${\cal U}$. \item At points of ${\cal U}^s$ there are canonical identifications \end{enumerate} \end{itemize} \end{lem} \begin{eqnarray*} T_E{\cal U}^s &\approx& H^1(End \; E) \\ T^*_E{\cal U}^s &\approx& H^0(w_\Sigma \otimes \; End \; E). \\ \end{eqnarray*} The proof of (2) is based on the observation that any nonzero $\alpha : E \rightarrow E$ must be invertible, otherwise either ker $\alpha$ or im $\alpha$ would violate stability. Therefore $H^0(End \; E)$ is a finite dimensional division algebra containing ${\bf C}$, hence equal to it. Since a vector bundle $E$ on $\Sigma$ is determined by a $1$-cocycle with values in $GL(r,{\cal O}_\Sigma)$ (= transition matrices), a first order deformation of $E$ is given by a 1-cocycle with values in the associated bundle of Lie algebras, i.e. (up to isomorphism) by a class in $H^1(End \; E)$. The functoriality property of $\cal U$ (``coarse moduli space'') implies that this is the Zariski tangent space, $T_E{\cal U}$. By Riemann-Roch $$ h^1 ({\rm End} \; E) = r^2(g-1) + h^0({\rm End} \; E). $$ so the minimal value is obtained at the simple points, and equals $1 + r^2(g-1)$ as claimed in (1). The identification of $T^*_E{\cal U}^s$ follows from that of $T_E{\cal U}^s$ by Serre duality. \EndProof \subsection{Spectral Curves and the Hitchin System} \label{sec-spectral-curves-and-the-hitchin-system} The relation between vector bundles and finite dimensional integrable systems arises from Hitchin's amazing result. \begin{theorem} \label{thm-hitchins-integrable-system} \cite{hitchin,hitchin-integrable-system} The cotangent bundle to the moduli space of semistable vector bundles supports a natural ACIHS. \end{theorem} At the heart of Hitchin's theorem is a construction of a spectral curve associated to a $1$-form valued endomorphism of a vector bundle. The spectral construction allows a uniform treatment of a wide variety of algebraically completely integrable Hamiltonian systems. We will concentrate in this section on the algebro-geometric aspects of these systems leaving their symplectic geometry to Chapter \ref{ch8}. We work with vector bundles over curves, other structure groups will be treated in Chapter \ref{ch9}. The reader is referred to \cite{B-N-R} and \cite{hitchin-integrable-system} for more details. The total space of the cotangent bundle $T^*{\cal U}^s_\Sigma(r,d)$ of the moduli space of stable vector bundles parametrizes pairs $(E,\varphi)$ consisting of a stable vector bundle $E$ and a covector $\varphi$ in $H^1(\Sigma, {\rm End} \; E)^* \simeq H^0(\Sigma, {\rm End} E \otimes \omega_\Sigma)$, i.e., a $1$-form valued endomorphism of $E$. Consider more generally a pair $(E,\varphi)$ of a rank $r$ vector bundle $E$ and a section $\varphi \in {\rm Hom}(E, E \otimes K)$ where $K$ is a line bundle on $\Sigma$. The $i$-th coefficient $b_i$ of the characteristic polynomial of $(E,\varphi)$ is a homogeneous polynomial of degree $i$ on $K^{-1}$, hence a section of $H^0(\Sigma, K^{\otimes i})$. In fact $b_i = (-1)^i\cdot {\rm trace}(\stackrel{i}{\wedge}\varphi)$. The Hamiltonian map of the Hitchin system is the characteristic polynomial map $$ H : T^* {\cal U}_\Sigma(r,d) \longrightarrow B_\omega := \bigoplus^r_{i=1} H^0(\Sigma, \omega^{\otimes i}). $$ The fibers of the Hitchin map $H$ turn out to be Jacobians of curves associated canonically to characteristic polynomials. Going back to the general $K$-valued pair $(E,\varphi)$, notice that a characteristic polynomial \ char $(\varphi) = y^r - {\rm tr}(\varphi) y^{r-1} + \cdots + (-1)^r\det \varphi$ in $B_K := \oplus^r_{i=1} H^0(\Sigma, K^{\otimes i})$ defines a morphism from the line bundle $K$ to $K^{\otimes r}$. The inverse image $C$ of the zero section in $K^{\otimes r}$ under a polynomial $P$ in $B_K$ is called a spectral curve. If $P$ is the characteristic polynomial of a pair $(E,\varphi)$ then indeed the fibers of $\pi : C \rightarrow \Sigma$ consist of eigenvalues of $\varphi$. If $K^{\otimes r}$ has a section without multiple zeroes (e.g., if it is very ample) then the generic spectral curve is smooth. Lagrange interpolation extends a function on the inverse image $\pi^{-1}(U) \subset C$ of an open set $U$ in $\Sigma$ to a unique function on the inverse image of $U$ in the surface $K$ which is a polynomial of degree $\le r-1$ on each fiber. It follows that the direct image $\pi_*{\cal O}_C$ is isomorphic to ${\cal O}_\Sigma \oplus K^{-1} \oplus \cdots \oplus K^{1-r}$. Assuming that $K^{\otimes i}$ has no sections for $i < 0$, the genus $h^1(C,{\cal O}_C) = h^1(\Sigma, \pi_*{\cal O}_C)$ of $C$ is equal to $\deg(K) \cdot r(r-1)/2 + r(g-1) + 1$. In particular, when $K = \omega_\Sigma$, the genus of $C$ is equal to half the dimension of the cotangent bundle. The data $(E,\varphi)$ determines moreover a sheaf $L$ on the spectral curve which is a line bundle if the curve is smooth. Away from the ramification divisor $R$ in $C$, $L$ is the tautological eigenline subbundle of the pullback $\pi^*E$. More precisely, the homomorphism $(\pi^*(\varphi) - y \cdot I) : \pi^*E \rightarrow \pi^*(E \otimes K)$, where $y \in H^0(C, \pi^*K)$ is the tautological eigenvalue section, has kernel $L(-R)$. Conversely, given a spectral curve $C$ and a line bundle $L$ on it we get a pair $(\pi_*L, \pi_*(\otimes y))$ of a rank $r$ vector bundle on $\Sigma$ and a $K$ valued endomorphism (see example \ref{example-direct-image}). The two constructions are the inverse of each other. \begin{proposition} \label{prop-ordinary-spectral-construction-higgs-pairs} \cite{hitchin-integrable-system,B-N-R} If $C$ is an irreducible and reduced spectral curve there is a bijection between isomorphism classes of \begin{description} \item [-] Pairs $(E,\varphi)$ with spectral curve $C$. \item [-] Rank 1 torsion free sheaves $L$ on $C$. \end{description} \end{proposition} Under this correspondence, line bundles on $C$ correspond to endomorphisms $\varphi$ which are regular in every fiber, i.e., whose centralizer in each fiber is an $r$-dimensional subspace of the corresponding fiber of End $E$. (This notion of regularity agrees with the one in Example \ref{example-coadjoint-orbits}.) We conclude that the fiber of the Hitchin map $H : T^*U^s_\Sigma(r,d) \rightarrow B_\omega$ over a characteristic polynomial $b \in B_\omega$ is precisely the open subset of the Jacobian $J_C^{d+r(1-g_\Sigma)+g_C-1}$ consisting of the line bundles $L$ whose direct image is a stable vector bundle. Moreover, the construction of the characteristic polynomial map and a similar description of its fibers applies to moduli spaces of pairs with $K$-valued endomorphism where $K$ need not be the canonical line bundle (Theorem \ref{thm-markman-botachin}). The missing line bundles in the fibers of the Hitchin map indicate that we need to relax the stability condition for the pair $(E,\varphi)$. \smallskip \noindent {\bf Definition:} \ \ A pair $(E,\varphi)$ is stable (semistable) if the slope of every $\varphi$-invariant subbundle of $E$ is less than (or equal) to the slope of $E$. As in the case of vector bundles we can define an equivalence relation for semistable pairs, where two bundles are equivalent, roughly, if they admit $\varphi$-invariant filtrations by semistable pairs (of constant slope) with isomorphic graded pieces. Two stable pairs are equivalent if and only if they are isomorphic. \begin{theorem} \cite{hitchin,simpson-moduli,Nit} There exists an algebraic coarse moduli scheme ${\rm Higgs}_K := {\rm Higgs}_\Sigma(r,d,K)$ parametrizing equivalence classes of semistable $K$-valued pairs. \end{theorem} The characteristic polynomial map $H : {\rm Higgs}_K \rightarrow B_K$ is a proper algebraic morphism. A deeper reason for working with the above definition of stability is provided by the following theorem from nonabelian Hodge theory: \begin{theorem} \label{thm-higgs-pairs-and-representations-of-pi1-for-curves} \cite{hitchin,simpson-higgs-bundles-and-local-systems} \ There is a canonical real analytic diffeomorphism between \begin{description} \item[-] The moduli space of conjugacy classes of semisimple representations of the fundamental group $\pi_1(\Sigma)$ in $GL(r,{\bf C})$ and \item[-] The moduli space of semistable $\omega$-valued (Higgs) pairs $(E,\varphi)$ of rank $r$ and degree $0$. \end{description} \end{theorem} In the case of Hitchin's system $(K = \omega_\Sigma)$, the symplectic structure of the cotangent bundle extends to the stable locus of the moduli space of Higgs pairs giving rise to an integrable system $H : {\rm Higgs}_\Sigma(r,d,\omega_\Sigma) \rightarrow B_\omega$ whose generic fiber is a complete Jacobian of a spectral curve. We will show in Chapter \ref{ch6} that the Hitchin system is, in fact, the lowest rank symplectic leaf of a natural infinite dimensional Poisson variety ${\rm Higgs}_\Sigma(r,d)$ obtained as an inductive limit of the moduli spaces $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ of $\omega(D)$-valued pairs as $D$ varies through all effective divisors on $\Sigma$. The basic fact, generalizing the results of \cite{hitchin-integrable-system,B-N-R,B} is: \begin{theorem} \label{thm-markman-botachin} \cite{botachin,markman-higgs} \ Let $D$ be an effective divisor (not necessarily reduced) on a smooth algebraic curve $\Sigma$ of genus $g$. Assume that $[\omega(D)]^{\otimes r}$ is very ample and if $g=0$ assume further that $\deg(D) > \max(2,\rho)$ where $0 \le \rho < r$ is the residue of $d$ mod $r$. Then \begin{itemize} \begin{enumerate} \item The moduli space ${\rm Higgs}^s_\Sigma(r,d,\omega(D))$ of stable rank $r$ and degree $d$ \ $\omega(D)$-valued Higgs pairs has a smooth component $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ of top dimension $r^2(2g - 2 + \deg(D)) + 1 + \epsilon_{D=0}$, where $\epsilon_{D=0}$ is $1$ if $D=0$ and zero if $D > 0$. $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ is the unique component which contains Higgs pairs supported on irreducible and reduced spectral curves. \item $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ has a canonical Poisson structure. \item The characteristic polynomial map $H : \HiggsModuli^{sm}_\Sigma(r,d,\omega(D)) \rightarrow B_{\omega(D)}$ is an algebraically completely integrable Hamiltonian system. The generic (Lagrangian) fiber is a complete Jacobian of a smooth spectral curve of genus $r^2(g-1) + 1 + (\deg D)( {r(r-1) \over 2})$. \item The foliation of $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ by closures of top dimensional symplectic leaves is induced by the cosets of $$ H^0 \left( \Sigma , \left[ \bigoplus^r_{i=1} \omega_\Sigma(D)^{\otimes i} \right] (-D) \right) \; {\rm in} \; B_{\omega(D)}. $$ \end{enumerate} \end{itemize} \end{theorem} \begin{definition} \label{def-good-component-of-higgs-pairs} As in the theorem, we will denote by $\HiggsModuli^{sm}_\Sigma(r,d,\omega(D))$ the unique component which contains Higgs pairs supported on irreducible and reduced spectral curves. \end{definition} In Chapter \ref{ch6} we will discuss the relationship of these integrable systems with flows of KdV type. In the next section we will discuss the example of geodesic flow on the ellipsoid as a Hamiltonian flow of a symplectic leaf of one of these spaces. See \cite{B,markman-higgs} for more examples. The Hitchin system has been useful in the study of the geometry of the moduli space of vector bundles. The main technique is to reduce questions about vector bundles to questions about spectral Jacobians. Hitchin used these ideas to compute the cohomology groups $H^i({\cal U}, S^kT)$, $i=0,1$, of the symmetric products of the tangent bundle of the moduli space $\cal U$ of rank $2$ and odd degree stable vector bundles. In \cite{B-N-R} these techniques provided the first mathematical proof that the dimensions of the space of sections of the generalized theta line bundle are \begin{eqnarray*} h^0({\cal U}_\Sigma(n, \, n(g-1)), \; \Theta) &=& 1, \\ h^0({\cal SU}_\Sigma(n), \; \Theta) &=& n^g, \\ \end{eqnarray*} where ${\cal SU}_\Sigma(n)$ denotes the moduli space of vector bundles with trivial determinant line bundle. (This of course is now subsumed in the Verlinde Formula for sections of powers of theta bundles.) These ideas were proven useful in the proof of the existence of a projectively flat connection on the bundles of level $k$ theta sections over the moduli space ${\cal M}_g$ of curve of genus $g$ \cite{hitchin-flat-connection}, an important fact in conformal field theory. Kouvidakis and Pantev applied these ideas to the study of automorphisms of the moduli space of vector bundles \cite{kouvidakis-pantev}. \subsection{ Polynomial Matrices} \label{sec-polynomial-matrices} Theorem \ref{thm-markman-botachin} has a concrete description when the base curve $\Sigma$ is ${\Bbb P}^1$. Let $K$ be the line bundle ${\cal O}_{{\Bbb P}^1}(d)$. Consider the moduli space ${\rm Higgs}_K := \HiggsModuli^{sm}_K(-r,r)$ of pairs $(E,\varphi)$ consisting of a vector bundle $E$ of rank $r$ and degree $-r$ with a $K$-valued endomorphism $\varphi : E \rightarrow E \otimes K$ (we also follow the notation of definition \ref{def-good-component-of-higgs-pairs} singling out a particular component). Choose a coordinate $x$ on ${\Bbb P}^1 - \{ \infty \}$. The space $B_K$ of characteristic polynomials becomes $$ \{ P(x,y) = y^r + b_1(x)y^{r-1} + \cdots + b_r(x) \; | \; b_i(x) \; {\rm is \; a \; polynomial \, in} \; x \; {\rm of \, degree} \; \le i \cdot d \}. $$ The total space of the line bundle ${\cal O}_{{\Bbb P}^1}(d \cdot \infty)$ restricted to the affine line ${\Bbb P}^1 - \{ \infty \}$ is isomorphic to the affine plane, and under this isomorphism $P(x,y)$ becomes the equation of the spectral curve as an affine plane curve. Denote by $B^0 \subset B_K$ the subset of smooth spectral curves. Let $Q := Q_r(d)$ be the subset of ${\rm Higgs}_K$ parametrizing pairs $(E,\varphi)$ with a smooth spectral curve and a vector bundle $E$ isomorphic to $E_0 := \oplus^r {\cal O}_{{\Bbb P}^1}(-1)$. $Q$ is a Zariski open (dense) subset of ${\rm Higgs}_K$ because: \begin{description} \item [i)] by definition \ref{def-good-component-of-higgs-pairs} $\HiggsModuli^{sm}_K(r,-r)$ is irreducible, \item [ii)] $E_0$ is the unique semistable rank $r$ vector bundle of degree $-r$ on ${\Bbb P}^1$ and semistability is an open condition. \end{description} The bundle ${\rm End} \; E_0$ is the trivial Lie algebra bundle $\goth{gl} _r({\bf C}) \otimes {\cal O}_{{\Bbb P}^1}$. Hence, every point in $Q$ is represented by an element $\varphi \in M_r(d) := H^0 ({\Bbb P}^1,\goth{gl} _r({\bf C}) \otimes {\cal O}_{{\Bbb P}^1}(d \cdot \infty))$, i.e., by an $r \times r$ matrix $\varphi$ with polynomial entries of degree $\le d$. Denote the inverse image of $B^0$ in $M_r(d)$ by $M^0_r(d)$. The subset $Q \subset {\rm Higgs}_K$ is simply the quotient of $M^0_r(d)$ by the conjugation action of $PGL_r({\bf C})$. \[ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{M^{0}_r(d)} \arrow{s} \\ \node{Q} \arrow{s} \arrow{e} \node{{\rm Higgs}_{K}} \arrow{s} \\ \node{B^0} \arrow{e} \node{B_K} \end{diagram} } \] In this setting, Theorem \ref{thm-markman-botachin} specializes to the following theorem of Beauville and Adams-Harnad-Hurtubise-Previato generalizing results of Mumford and Moser \cite{Mum} in rank $2$: \begin{theorem}\label{thm-beauville} \cite{B,AHH} \begin{itemize} \begin{enumerate} \item The quotient $Q$ of the action of $PGL_r({\bf C})$ by conjugation on $M^0_r(d)$ is a smooth variety. \item The fiber of the characteristic polynomial maps $H : Q \rightarrow B^0$ over the polynomial of a spectral curve $C$ is the complement $J^{g-1}_C - \Theta$ of the theta divisor in the Jacobian of line bundles on $C$ of degree $g-1$ \ $(g = {\rm genus \; o}f \, C)$. \item The choice of $d+2$ points $a_1, \cdots a_{d+2}$ on ${\Bbb P}^1$ determines a Poisson structure on $Q$. The characteristic polynomial map $H : Q \rightarrow B^0$ is an algebraically completely integrable Hamiltonian system with respect to each of these Poisson structures. \item The symplectic leaves of $Q$ are obtained by fixing the values (of the coefficients) of the characteristic polynomials at the points $\{ a_i \}^{d+2}_{i=1}$. \end{enumerate} \end{itemize} \end{theorem} We note that in \cite{B} the Poisson structure on $Q$ was obtained as the reduction of a Poisson structure on $M^0_r(d)$. The latter was the pullback of the Kostant-Kirillov Poisson structure via the embedding $$ M_r(d) \hookrightarrow \goth{gl}_r({\bf C})^{d+2} $$ by Lagrange interpolation at $a_1, \cdots , a_{d+2}$. This embedding will be used in section \ref{sec-geodesic-flow-via-polynomial-matrices} where geodesic flow on ellipsoids is revisited. A choice of a divisor $D = a_1 + \cdots + a_{d+2}$ of degree $d+2$ on ${\Bbb P}^1$ determines an isomorphism of ${\cal O}_{{\Bbb P}^1}(d \cdot \infty)$ with $\omega_{{\Bbb P}^1}(D)$. For example, if $a_1, \cdots , a_i$ are finite, $a_{i+1} = a_{a+2} = \cdots = a_{d+2} = \infty$ then we send a polynomial $f(x)$ of degree $\le d$ to the meromorphic $1$-form $$ {f(x) \over {\prod^i_{j=1}(x-a_j)}} dx. $$ When the $d+2$ points are distinct, Lagrange interpolation translates to the embedding $$ {\rm Res}: M_r(d) = H^0({\Bbb P}^1, \goth{gl}_r({\bf C}) \otimes \omega_{{\Bbb P}^1}(D)) \hookrightarrow \goth{gl}_r({\bf C})^{d+2} $$ via the residues of meromorphic $1$-form valued matrices at the points $a_i$ (if $a_i$ has multiplicity $2$ or higher, we replace the $i$-th copy of $\goth{gl}_r({\bf C})$ by its tangent bundle or higher order infinitesimal germs at $a_i$ of sections of the trivial bundle $\goth{gl}_r({\bf C}) \otimes {\cal O}_{{\Bbb P}^1})$. \subsubsection{Explicit Equations for Jacobians of Spectral Curves with a Cyclic Ramification Point} \label{sec-explicit-equations-for-jacobians} A further simplification occurs for matrices with a nilpotent leading coefficient (nilpotent at $\infty$). The projection $M^0_r(d) \rightarrow Q$ has a natural section over the image $N \subset Q$ of this locus. So $N$ can be described concretely as a space of polynomials (rather than as a quotient of such a space). As a consequence we obtain explicit equations in $M_r(d)$ for the complement $J_C - \Theta$ of the theta divisor in the Jacobian of every irreducible and reduced $r$-sheeted spectral curve over ${\Bbb P}^1$ which is totally ramified and smooth at $\infty$ (generalizing the equations for hyperelliptic curve (case $r=2$) obtained in \cite{Mum}). \begin{lem}\label{lemma-normal-form} Let $A = A_dx^d + \cdots + A_1x + A_0$ be an $r \times r$ traceless matrix with polynomial entries of degree $\le d$ \begin{description} \item[i)] whose spectral curve in ${\cal O}_{{\Bbb P}^1}(d \cdot \infty)$ is irreducible and reduced and smooth over $\infty$, and \item[ii)] whose leading coefficient $A_d$ is a nilpotent (necessarily regular) matrix. \end{description} Then there exists a unique element $g_0 \in PGL_r({\bf C})$ conjugating $A$ to a matrix $A' = x^d \cdot J + \sum ^{d-1}_{i=0} A'_i x^i$ of the form: \begin{equation}\label{eqn-normal-form} A' = x^d \left( \begin{array}{cccccc} 0 & & & & 0 & 0 \\ 1 & & & & 0 & 0 \\ 0 & \cdot & & & 0 & 0 \\ 0 & & \cdot & & 0 & 0 \\ 0 & & & \cdot & 0 & 0 \\ 0 & & & & 1 & 0 \end{array} \right) + x^{d-1} \left( \begin{array}{cccccc} \star & & \dots & & \star & \beta_{r} \\ \star & & \dots & & \star & 0 \\ \\ \vdots & & \vdots & & \vdots & \vdots \\ \\ \star & & \dots & & \star & 0 \end{array} \right) + \sum_{i=0}^{d-2}x^i A'_{i} \end{equation} \noindent where $(-1)^{r+1}\beta_r$ is the (leading) coefficient of $x^{dr-1}$ in the determinant $b_r(x)$ of $A(x)$. \end{lem} \noindent {\bf Remark:} \ Notice that the coefficients $b_i(x)$ in the characteristic polynomial $P(x,y) = y^r + b_1(x)y^{r-1} + \cdots + b_r(x)$ of $A(x)$ satisfy degree $b_i(x) \le d \cdot i-1$ since $A$ is nilpotent at $\infty$, and degree $b_r(x) = dr-1$ since the spectral curve is smooth over $\infty$. Thus $\beta_r \ne 0$. \smallskip \noindent {\bf Proof} (of lemma \ref{lemma-normal-form}): \ Let $J$ be the nilpotent regular constant matrix appearing as the leading coefficient of $A'(x)$ in the normalized form (\ref{eqn-normal-form}). Let ${\bf C}[J]$ be the algebra of polynomials in $J$ with constant coefficients. The proof relies on the elementary fact that ${\bf C}^r$, as a left ${\bf C}[J]$-module, is free. Any vector with non zero first entry is a generator. $A_d$ is conjugate to $J$. Thus we may assume that $A_d = J$ and it remains to show that there exists a unique element in the stabilizer of $J$ in $PGL_r({\bf C})$ conjugating $A(x)$ to the normal form (\ref{eqn-normal-form}). Since $A_d = J$, the first entry in the right column $R$ of $A_{d-1}$ is $\beta_r$. Thus $R$ is a generator of ${\bf C}^r$ as a ${\bf C}[J]$-module. Any element $g \in PGL_r({\bf C})$ in the commutator subgroup of $J$ is an invertible element in ${\bf C}[J]$ and can be written (up to scalar multiple) in the form $g = I + N$, $N$ nilpotent. The right column of $gA_{d-1}g^{-1}$ is $R + NR$ and there exists a unique nilpotent $N \in {\bf C}[J]$ such that $NR = \left( \begin{array}{c} \beta_r \\ 0 \\ \vdots\\ 0 \end{array} \right) - R.$ Thus $g$ is unique up to a scalar factor. \EndProof Denote the affine subvariety of $M^0_d(r)$ of matrices satisfying the $r^2+r - 1$ equations (\ref{eqn-normal-form}) by ${\tilde N}$. The subvariety ${\tilde N}$ is a section of the principal $PGL_r({\bf C})$ bundle $M^0_r(d) \rightarrow Q$ over the locus $N$ of conjugacy classes of polynomial matrices with a nilpotent leading coefficient. $N$ is a Poisson subvariety of $Q$ with respect to any Poisson structure on $Q$ determined by a divisor $D$ as in theorem \ref{thm-beauville}, provided that $D$ contains the point at infinity $\infty \in {\Bbb P}^1$. Choose a characteristic polynomial $P(x,y) = y^r + b_1(x)y^{r-1} + \cdots + b_r(x)$ in $B_{{\cal O}_{{\Bbb P}^1}(d)}$ of a smooth spectral curve $C$ with degree $b_i(x) \le id - 1$, $b_r(x)$ of degree $rd - 1$ with leading coefficient $(-1)^{r+1}\beta_r$. Theorem \ref{thm-beauville} implies that the equations \begin{description} \item[a)] $A_d = J,$ \item[b)] The $r$-th column of $A_{d-1}$ is $ \left( \begin{array}{c} {\cal \beta}_r \\ 0 \\ \vdots \\ 0 \\ \end{array} \right), $ \item[c)] char $(A(x)) = P(x,y)$ \end{description} define a subvariety of $M_r(d)$ isomorphic to the complement $J^{g-1}_C - \Theta_C$ of the theta divisor in the Jacobian of $C$. \subsubsection {Geodesic Flow on Ellipsoids via $2 \times 2$ Polynomial Matrices} \label{sec-geodesic-flow-via-polynomial-matrices} We use polynomial matrices to retrieve the Jacobi-Moser-Mumford system which arose in chapter \ref{ch3} out of the geodesic flow on an ellipsoid. Our presentation follows \cite{B}. Consider a spectral polynomial $P(x,y)$ in $B^0_{{\cal O}_{{\Bbb P}^1}(d \cdot \infty)}$ of the form \begin{description} \item[(i)] $P(x,y) = y^2 - f(x)$ where f(x) is monic of degree $2d - 1$. \end{description} \noindent The corresponding spectral curve $C$ is smooth, hyperelliptic of genes $g = d-1$ and ramified over $\infty$. Theorem \ref{thm-beauville} implies that the fiber $$ H^{-1}(P(x,y)) = \left\{ \left( \begin{array}{cc} V & U \\ W & -V \\ \end{array} \right) \; \mid \; V^2 + UW = f(x) \right\} / PGL_2({\bf C}) $$ of the characteristic polynomial map is isomorphic to the complement $J^{g-1}_C - \Theta$ of the theta divisor. Lemma \ \ref{lemma-normal-form} specializes in our case to the following statement (note that $\beta_r=1$ since $f$ is taken to be monic):\\ {\em The $PGL_2({\bf C})$ orbit of a matrix $\left( \begin{array}{cc} V & U \\ W & -V \\ \end{array} \right) $ over $H^{-1}(P(x,y)) \cong J_C^{g-1} - \Theta$ contains a unique matrix satisfying \begin{description} \item [(ii)] $W$ is monic of degree $d$, \\ $U$ is monic of degree $d-1$ and \\ deg $V \le d-2$. \end{description} } \noindent In other words, condition (ii) and \begin{description} \item [(iii)] $V^2 + UW = f(x)$ \end{description} are the equations of $J_C^{g-1} - \Theta$ as an affine subvariety of the subspace of traceless matrices in $M_2(d)$. In fact, condition (ii) defines a section $\varphi : N \rightarrow M_2(d)$ over the locus $N$ in $Q$ of conjugacy classes with characteristic polynomial satisfying condition (i). Recall that the Jacobi-Moser-Mumford integrable system linearizing the geodesic flow of the ellipsoid $\sum^d_{i=1} \; a^{-1}_i x^2_i = 1$ is supported on the tangent bundle $TS$ of the sphere $S \subset {\Bbb R}^d$. Our discussion ended by describing the quotient of $TS$ by the group $G \simeq ({\bf Z}/2{\bf Z})^d$ of involutions. We will describe in the next three steps an isomorphism of this quotient with a symplectic leaf $X$ of $Q$. \begin{description} \item[\underline{Step I:}] (Identification of the symplectic leaf $X$). Assume that the points $a_1, \cdots, a_d \in {\Bbb P}^1 - \{ \infty \}$ are distinct and let $a_{d+1} = a_{d+2} = \infty$. Let $X \subset Q$ be the symplectic leaf over the subspace of characteristic polynomials $P(x,y) = y^2 - f(x)$ satisfying $$f(a_i) = 0, \; 1 \le i \le d, \ \ \deg f = 2d-1 \ \ {\rm and}\; f \; {\rm is \; monic}.$$ The spectral curves of matrices in the leaf $X$ have genus $d-1$, and are branched over the fixed $g+2$ points $a_1, \cdots , a_d,\infty$ and $g$ varying points. \item[\underline{Step II:}] (Embedding of $X$ in the product ${\cal N}^d$ of the regular nilpotent orbit). The isomorphism ${\cal O}_{{\Bbb P}^1}(d \cdot \infty) \stackrel{\sim}{\rightarrow} \omega_{{\Bbb P}^1} (\sum^d_{i=1} a_i + 2 \cdot \infty)$ sending $F(x)$ to ${{F(x)dx} \over {\prod^d_{i=1}}(x-a_i)}$ translates the matrix $ \left( \begin{array}{cc} V(x) & U(x) \\ W(x) & -V(x) \\ \end{array} \right) $ to a matrix $\varphi$ of meromorphic $1$-forms. The residues of $\varphi$ satisfy:\\ \smallskip ${ R_\infty := Res_\infty(\varphi) = \left( \begin{array}{cc} 0 & -1 \\ s & 0 \end{array} \right) {\rm for \hspace{1ex} some} \ s \in {\bf C} \ \ \ ({\rm condition} \; (ii)), }$ ${ R_i := Res_{a_i}(\varphi) = \left( \begin{array}{cc} V(a_i) & U(a_i) \\ W(a_i) & -V(a_i) \\ \end{array} \right) {{1} \over {\prod^d_{\stackrel{j=1}{j \ne i}}} (a_i-a_j)}. }$ \smallskip \noindent The residues at the finite points $a_i$ can be calculated using Lagrange interpolation of polynomials of degree $d$ at the $d+1$ points $a_1, \cdots , a_d, \infty$ given by the formula \begin{equation} \label{eqn-lagrange-interpolation} F(x) = \sum^d_{i=1}F(a_i) \frac {\prod^d_{\stackrel{j=1}{j \ne i}}(x - a_j)} {\prod^d_{\stackrel{j=1}{j \ne i}}(a_i - a_j)} + F(\infty) \prod^d_{j=1} (x - a_j) \end{equation} where $F(\infty)$ is the leading coefficient of $F(x)$. \smallskip \noindent The residues $R_\infty$, $R_i$ are nilpotent regular $2 \times 2$ matrices and the residue theorem implies that $R_\infty = - \sum^d_{i=1} R_i$. The residue map $Res : X \rightarrow {\cal N}^d$ defines a symplectic embedding $\varphi \mapsto (R_1, \cdots, R_d)$ of the symplectic leaf $X$ of $Q$ in the Cartesian product of $d$ copies of the regular nilpotent orbit ${\cal N}$ in $\goth{gl}_2({\bf C})$. \item[\underline{Step III}] (The $2^d$ covering $TS \rightarrow X$). Endow ${\bf C}^2$ with the symplectic structure $2dx \wedge dy$. The map ${\bf C}^2 - \{(0,0)\} \rightarrow {\cal N}$ sending $(x,y)$ to $ \left( \begin{array}{cc} xy & -x^2 \\ y^2 & -xy \\ \end{array} \right) $ is a symplectic $SL_2({\bf C})$-equivariant double cover of the regular nilpotent orbit $\cal N$ (where $SL_2({\bf C}) \cong Sp_2({\bf C},2dx \wedge dy)$ acts on ${\bf C}^2$ via the standard representation). We obtain a $2^d$-covering $\tau : ({\bf C}^2 - \{(0,0) \})^d \rightarrow {\cal N}^d$. The residue theorem translates to the fact that the image $Res(X) \subset {\cal N}^d$ is covered by $$ \left\{ ({\bar x}, {\bar y}) = ((x_1,y_1), \cdots, (x_d,y_d)) \ | \ \sum \; x_iy_i = 0 \; {\rm and} \sum^d_{i=1}x^2_i = 1 \right\}. $$ This is exactly the tangent bundle $TS \subset ({\bf C}^2)^d$ of the sphere $S \subset {\bf C}^d$. \end{description} \newpage \section{Poisson structure via levels} \label{ch5} We construct a Poisson structure on the moduli space of meromorphic Higgs pairs in two steps (following \cite{markman-higgs}): \smallskip \noindent - First we realize a dense open subset of moduli as the orbit space of a Poisson action of a group on the cotangent bundle of the moduli space of vector bundles with level structures (sections \ref{sec-level-structures}, \ref{sec-the-cotangent-bundle} and \ref{the-poisson-structure}). \noindent - Next we exhibit a $2$-vector on the smooth locus of moduli, using a cohomological construction (section \ref{sec-linearization}). On the above dense open set this agrees with the Poisson structure, so it is a Poisson structure everywhere. \medskip We summarize the construction in section \ref{sec-hamiltonians-and-flows} in a diagram whose rotational symmetry relates dual pairs of realizations. \subsection{Level structures} \label{sec-level-structures} Fix a curve $\Sigma$, an effective divisor $D = \sum p_i$ in $\Sigma$, and a rank $r$ vector bundle $E$ on $\Sigma$. A level $D$ structure on $E$ is an ${\cal O}_D$-isomorphism $\eta: E \otimes {\cal O}_D \stackrel{\sim}{\rightarrow} {\cal O}^{\oplus r}_D$. Seshadri \cite{seshadri-construction-moduli-vb} constructs a smooth, quasi-projective moduli space ${\cal U}_\Sigma(r,d,D)$ parametrizing stable pairs $(E,\eta)$. Here stability means that for any subbundle $F \subset E$, $$\frac{deg F - deg D}{rank F} < \frac{deg E - deg D}{rank E}.$$ The level-$D$ group is the projectivized group of ${\cal O}_D$-algebra automorphisms, $$ G_D:= {\bf P} Aut_{{\cal O}_D}({\cal O}^{\oplus r}_{D}). $$ (i.e. the automorphism group modulo complex scalars ${\bf C}^*$.) It acts on ${\cal U}_\Sigma(r,d,D)$: an element $g \in G_D$ sends $$[(E,\eta)] \mapsto [(E,\overline{g} \circ \eta)],$$ where $\overline{g} \in Aut_{{\cal O}_D}({\cal O}^{\oplus r}_{D})$ lifts $g$, and $[\cdot]$ denotes the isomorphism class of a pair. The open set ${\cal U}^\circ_\Sigma (r,d,D)$, parametrizing pairs $(E,\eta)$ where $E$ itself is stable, is a principal $G_D$-bundle over ${\cal U}^s_\Sigma (r,d)$. The Lie algebra $\LieAlg{g}_D$ of $G_D$ is given by $\LieAlg{gl}_r({\cal O}_D)$/scalars. \subsection{The cotangent bundle} \label{sec-the-cotangent-bundle} We compute deformations of a pair $$(E,\eta) \in {\cal U}_D := {\cal U}_\Sigma(r,d,D)$$ as we did for the single vector bundle $$ E \in {\cal U} := {\cal U}_\Sigma (r,d). $$ Namely, $E$ is given (in terms of an open cover of $\Sigma$) by a $1$-cocycle with values in the sheaf of groups $GL_r({\cal O}_\Sigma)$. Differentiating this cocycle with respect to parameters gives a $1$-cocycle with values in the corresponding sheaf of Lie algebras, so we obtain the identification $$ T_E{\cal U} \approx H^1(End E). $$ Similarly, the pair $(E,\eta)$ is given by a $1$-cocycle with values in the subsheaf $$ GL_{r,D}({\cal O}_\Sigma) := \{f \in GL_r({\cal O}_\Sigma)|f-1 \in \LieAlg{gl}_r({\cal I}_D)\}. $$ Differentiating, we find the natural isomorphism $$ T_{(E,\eta)} {\cal U}_D \approx H^1({\cal I}_D \otimes End E), $$ so by Serre duality, $$ T^*_{(E,\eta)} {\cal U}_D \approx H^{0}(End E \otimes \omega(D)). $$ (we identify $End E$ with its dual via the trace.) We will denote a point of this cotangent bundle by a triple $(E,\varphi,\eta)$, where $(E,\eta) \in {\cal U}_D$ and $\varphi$ is a $D$-Higgs field, $$\varphi: E \to E \otimes \omega(D).$$ \subsection{The Poisson structure} \label{the-poisson-structure} The action of the level group $G_D$ on ${\cal U}_D$ lifts naturally to an action of $G_D$ on $T^*{\cal U}_D$. Explicitly, an element $g \in G_D$ with lift $\overline{g} \in GL_r({\cal O}_D)$ sends $$ (E,\varphi,\eta) \mapsto (E, \varphi, \overline{g} \circ \eta). $$ The lifted action has the following properties: \begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}} \item It is Poisson with respect to the standard symplectic structure on $T^*{\cal U}_D$ (holds for any lifted action, see example \ref{example-coadjoint-orbits}). \item The moment map $$ \mu: T^*{\cal U}_D \to \LieAlg{g}^*_D $$ is given by \begin{equation} \label{eq-moment-map-of-finite-dim-level-action} \mu(E,\varphi,\eta): A \mapsto \mbox{Res\ Trace}\ (A\cdot \varphi^\eta), \end{equation} where $$ \begin{array}{lcl} A & \in & \LieAlg{g}_D = (\LieAlg{gl}_r({\cal O}_D))/\mbox{(scalars)} \approx (\LieAlg{gl}_r({\cal O}_D))_{\mbox{traceless}},\\ \varphi^\eta & := & \eta \circ \varphi \circ \eta^{-1} \in H^0(\LieAlg{gl}_r(\omega(D) \otimes {\cal O}_D)),\\ \end{array} $$ and the residue map $$ Res : H^0(\omega(D) \otimes {\cal O}_D) \to H^1(\omega) \approx {\bf C} $$ is the coboundary for the restriction sequence $$ 0 \to \omega \to \omega(D) \to \omega(D) \otimes {\cal O}_D \to 0 $$ (cf. \cite{markman-higgs} Proposition 6.12). \item $G_D$ acts freely on the open subset $(T^*{\cal U}_D)^{\circ}$ parametrizing triples $(E,\varphi,\eta)$ where $(E,\eta)$ is stable and $(E,\varphi)$ is a stable Higgs bundle, since such bundles are simple. This makes $(T^*{\cal U}_D)^{\circ}$ into a principal $G_D$-bundle over an open subset ${\rm Higgs}^{\circ}_D$ of ${\rm Higgs}^s_D$. \end{list} We conclude that the symplectic structure on $(T^*{\cal U}_D)^{\circ}$ induces a Poisson structure on ${\rm Higgs}^{\circ}_D$. The symplectic leaves will then be the inverse images under $\mu$ of coadjoint orbits in $\LieAlg{g}^*_D$. \subsection{Linearization} \label{sec-linearization} The main remaining task is to find a two-vector on the non-singular locus ${\rm Higgs}^{ns}_D$ whose restriction to ${\rm Higgs}^{\circ}_D$ is the above Poisson structure. This two vector is then automatically Poisson. The algebraic complete integrability of the component $\HiggsModuli^{sm}_D$ (see definition \ref{def-good-component-of-higgs-pairs}) would then follow: The spectral curve $C_b$, for generic $b \in B_D$, is non-singular, so its Jacobian ${\cal J}(C)$ is contained in $\HiggsModuli^{sm}_D$. Thus any Hamiltonian vector field must be constant on the generic fiber ${\cal J}(C)$, hence on all fibers. A natural two-vector defined over all of ${\rm Higgs}^{ns}_D$ can be given in several ways. One \cite{markman-higgs} is to identify the tangent spaces to ${\rm Higgs}_D$ (and related spaces) at their smooth points as hypercohomologies, ${\Bbb H}^1$, of appropriate complexes: $$ \begin{array}{ccc} \mbox{{\underline{space}}} & \mbox{{\underline{at}}} & \mbox{{\underline{complex}}} \\ {\cal U} & E & End E \\ && \\ {\cal U}_D & (E,\eta) & End E(-D) \\ && \\ {\rm Higgs}_D & (E,\varphi) & \overline{{\cal K}}= [End E \stackrel{ad \varphi}{\longrightarrow} End E \otimes \omega(D)] \\ && \\ T^*{\cal U}_D & (E,\varphi,\eta) & {\cal K} := [End E(-D) \stackrel{ad \varphi \otimes i}{\longrightarrow} End E \otimes \omega(D)].\\ \end{array} $$ \vspace{0.1in} \noindent where $i: {\cal O}(-D) \hookrightarrow {\cal O}$ is the natural inclusion. These identifications are natural, and differentials of maps between these spaces are realized by maps of complexes. For example, the fibration $T^*{\cal U}_D \to {\cal U}_D$, with fiber $T^*_{(E,\eta)}{\cal U}_D$, gives the sequence \vspace{0.1in} $$ \begin{array}{ccccccccc} 0 & \to & T^*_{(E,\eta)} {\cal U}_D & \to & T_{(E,\varphi,\eta)}(T^*{\cal U}_D) & \to & T_{(E,\eta)}{\cal U}_D & \to & 0 \\ & & \parallel & & \parallel & & \parallel & \\ 0 & \to & H^0(End E \otimes \omega(D)) & \to & {\Bbb H}^1({\cal K}) & \to & H^1(End E(-D)) & \to & 0\\ \end{array} $$ \vspace{0.1in} \noindent derived from the short exact sequence of complexes, $$ 0 \rightarrow End E \otimes \omega(D)[-1] \rightarrow K \rightarrow End E (-D) \rightarrow 0 , $$ while the (rational) map $T^*{\cal U}_D \to {\rm Higgs}_D$ gives \vspace{0.1in} $$\begin{array}{ccccccccc}0 & \rightarrow & \LieAlg{g}_D & \rightarrow & T_{(E,\varphi,\eta)}(T^*{\cal U}_D) & \rightarrow & T_{(E,\varphi)}{\rm Higgs}_D & \rightarrow & 0 \\ & & \parallel & & \parallel & & \parallel & \\ 0 & \rightarrow & \frac{H^0(End E \otimes {\cal O}_D)}{H^0(End E)} & \rightarrow & {\Bbb H}^1({\cal K}) & \rightarrow & {\Bbb H}^1(\overline{{\cal K}}) & \rightarrow & 0\\ \end{array} $$ \vspace{0.1in} \noindent which derives from: $$0 \to {\cal K} \to \overline{{\cal K}} \to End E \otimes {\cal O}_D \to 0.$$ The dual of a complex ${\cal L}: A \to B$ of vector bundles is the complex $$ {\cal L}^\vee: B^* \otimes \omega \to A^* \otimes \omega. $$ Grothendieck duality in this case gives a natural isomorphism $$ {\Bbb H}^1({\cal L}) \approx {\Bbb H}^1({\cal L}^\vee)^*. $$ We note that ${\cal K}$ is self-dual, in the sense that there is a natural isomorphism of complexes, $J: {\cal K}^\vee \stackrel{\sim}{\rightarrow} {\cal K}$. For $\overline{{\cal K}}$ we obtain a natural isomorphism of complexes, $\overline{{\cal K}}^\vee \stackrel{\sim}{\rightarrow} \overline{{\cal K}} \otimes {\cal O}(-D)$, hence (composing with $i$) a morphism $$I: \overline{{\cal K}}^\vee \to \overline{{\cal K}}.$$ Combining with duality, we get maps $$ {\Bbb H}^1({\cal K})^* \approx {\Bbb H}^1({\cal K}^\vee) \stackrel{J}{\approx} {\Bbb H}^1({\cal K}) $$ and $$ {\Bbb H}^1(\overline{{\cal K}})^* \approx {\Bbb H}^1(\overline{{\cal K}}^\vee) \RightArrowOf{I} {\Bbb H}^1(\overline{{\cal K}}). $$ These give elements of $\otimes^2{\Bbb H}^1({\cal K})$ and $\otimes^2 {\Bbb H}^1(\overline{{\cal K}})$. Both are skew symmetric (since $ad_{\varphi}$, and hence $I,J$, are), so we obtain global two-vectors on $T^*{\cal U}^{ns}_D$ and ${\rm Higgs}^{ns}_D$. At stable points these agree with (the dual of) the symplectic form and its reduction modulo $G_D$, which is what we need. Another way to find the two-vector on ${\rm Higgs}_D$ is based on the interpretation of ${\rm Higgs}_D$ as a moduli space of sheaves on the total space $S$ of $\omega(D)$. At such a simple sheaf ${\cal E}$, with support on some spectral curve $C$, Mukai \cite{mukai} identifies the tangent space to moduli with $$Ext^1_{{\cal O}_S}({\cal E},{\cal E}),$$ and notes that any two-form $\sigma \in H^0(\omega_S)$ determines an alternating bilinear map: $$ Ext^1_{{\cal O}_S}({\cal E},{\cal E}) \times Ext^1_{{\cal O}_S}({\cal E},{\cal E}) \to Ext^2_{{\cal O}_S}({\cal E},{\cal E}) \stackrel{tr}{\rightarrow} H^2({\cal O}_S) \stackrel{\sigma}{\rightarrow} H^2(\omega_S) \approx {\bf C}, $$ hence a two-form on moduli. Mukai uses this argument to produce symplectic structures on the moduli spaces of sheaves on $K3$ and abelian surfaces. The same argument works, of course, for sheaves on $T^*\Sigma$; this reconstructs the symplectic form on Hitchin's system. Our surface $S$ (the total space of $\omega(D)$) is related to $T^*\Sigma$ by a birational morphism $\alpha: T^*\Sigma \to S$. The symplectic form $\sigma$ does not descend to $S$, but its inverse $\sigma^{-1}$ does give a two vector on $S$ which is non-degenerate away from $D$ and is closed there (since it is locally equivalent to the Poisson structure on $T^*\Sigma$). Tyurin notes \cite{tyurin-symplectic} that a variant of this argument produces a two-vector on moduli from a two-vector on $S$. Now the birational morphism $T^*\Sigma \to S$ takes the Poisson structure on $T^*\Sigma$ to one on $S$, so the Mukai-Tyurin argument gives the desired two-vector on ${\rm Higgs}_D$. In chapter \ref{ch8} this approach is generalized to higher dimensional varieties. A third argument for the linearization is given by Bottacin \cite{botachin}. He produces an explicit two-vector at stable points using a deformation argument as above, and then makes direct, local computations to check closedness of the Poisson structures and linearity of the flows. \subsection{Hamiltonians and flows in $T^*{\cal U}_D$} \label{sec-hamiltonians-and-flows} We saw that the level group $G_D$ acts on $T^*{\cal U}_D$, inducing the Poisson structure on the quotient ${\rm Higgs}_D$, and that the moment map is $$ \begin{array}{lcc} \mu: T^*{\cal U}_D & \rightarrow & \LieAlg{g}^*_D \\ \mu(E,\varphi, \eta)(A) & := & \mbox{Res\ Trace}\ (A \cdot \varphi^{\eta}).\\ \end{array} $$ The characteristic polynomial map $\widetilde{h}: T^*{\cal U}_D \to B_D$ is a composition of the Poisson map $T^*{\cal U}_D \to {\rm Higgs}_D$ with the Hamiltonian map $h: {\rm Higgs}_D \to B_D$. Hence $\widetilde{h}$ is also Hamiltonian. Clearly, $\widetilde{h}$ is $G_D$-invariant. The composition $T^*{\cal U}_D \stackrel{\mu}{\rightarrow} \LieAlg{g}^*_D \rightarrow \LieAlg{g}^*_D/G_D$ is a $G_D$-invariant Hamiltonian morphism and hence factors through ${\rm Higgs}_D$. It follows that it factors also through $B_D$ since $h: {\rm Higgs}_D \to B_D$ is a Lagrangian fibration whose generic fiber is connected (see remark \ref{rem-acihs-implies-maximal-commutative-subalgebra}). The conditions of example \ref{example-diagram-hexagon-plus-realization} in section \ref{subsec-moment-maps} are satisfied and we get a diagram with a $180^\circ$ rotational symmetry in which opposite spaces are dual pairs of realizations. The realization dual to $T^*{\cal U}_D \to \LieAlg{g}^*_D/G_D$ is the rational morphism $T^*{\cal U}_D \to G{\rm Higgs}_D := {\rm Higgs}_D \times_{(\LieAlg{g}^*_D/G_D)}\LieAlg{g}^*_D$ to the fiber product. The one dual to $\widetilde{h}: T^*{\cal U}_D \to B_D$ is the morphism $T^*{\cal U}_D \to G B_D:= B_D \times _{(\LieAlg{g}^*_D/G_D)}\LieAlg{g}^*_D$ to the fiber product. We write down the spaces and typical elements in them: \begin{equation} \label{diagram-hexagon-for-higgs-pairs} \begin{array}{ccc} {\divide\dgARROWLENGTH by 4 \divide\dgHORIZPAD by 2 \divide\dgCOLUMNWIDTH by 2 \begin{diagram}[TTT] \node[3]{T^*{\cal U}_D} \arrow{s} \\ \node[3]{G{\rm Higgs}_D} \arrow[2]{sw} \arrow{se} \\ \node[4]{GB_D} \arrow[2]{sw} \arrow{se} \\ \node{{\rm Higgs}_D} \arrow{se} \node[4]{\LieAlg{g}^*_D} \arrow[2]{sw} \\ \node[2]{B_D} \arrow{se} \\ \node[3]{\LieAlg{g}^*_D/G_D} \arrow{s} \\ \node[3]{(0)} \end{diagram} } & \hspace{3ex} & {\divide\dgARROWLENGTH by 4 \divide\dgHORIZPAD by 2 \divide\dgCOLUMNWIDTH by 2 \begin{diagram}[TTT] \node[3]{(E,\varphi,\eta)} \arrow{s} \\ \node[3]{(E,\varphi,\varphi^{\eta})} \arrow[2]{sw} \arrow{se} \\ \node[4]{({\rm char}\varphi,\varphi^{\eta})} \arrow[2]{sw} \arrow{se} \\ \node{(E,\varphi)} \arrow{se} \node[4]{\varphi^{\eta}} \arrow[2]{sw} \\ \node[2]{{\rm char}\varphi} \arrow{se} \\ \node[3]{{\rm char}\varphi^{\eta}} \arrow{s} \\ \node[3]{(0)} \end{diagram} } \end{array} \end{equation} \newpage \section {Spectral flows and $KP$} \label{ch6} \label{sec-spectral-flows-and-kp} Our aim in this section is to relate the general spectral system which we have been considering to the $KP$ and multi-component $KP$ hierarchies. We start by reviewing these hierarchies and their traditional relationship to curves and bundles via the Krichever map. We then reinterpret these flows as coming from Hamiltonians on the limit $T^*U_\infty$ of our previous symplectic spaces. We show that $\mbox{Higgs}_\infty$ can be partitioned into a finite number of loci, each of which maps naturally to one of the $mcKP$-spaces in a way which intertwines isospectral flows with $KP$ flows. As an example we consider the Elliptic solitons studied by Treibich and Verdier. \subsection{The hierarchies} \label{sec-the-heirarchies} \noindent \underline{KP} Following the modern custom (initiated by Sato, explained by Segal-Wilson \cite{segal-wilson-loop-groups-and-kp}, and presented elegantly in \cite{AdC,mulase-cohomological-structure,li-mulase-category} and elsewhere), we think of the $KP$ hierarchy as given by the action of an infinite-dimensional group on an infinite-dimensional Grassmannian: set $$\begin{array}{lrl}K & := & {\bf C}((z)) = \mbox{field\ of\ formal\ Laurent\ series\ in\ a\ variable}\ z\\ Gr & := & \{\mbox{subspaces}\ W \subset K| \mbox{projection}\ W \to K/{\bf C}[[z]] z\ \mbox{is\ Fredholm}\}\\ & = & \{\mbox{subspaces\ ``comparable\ to}\ {\bf C}[z^{-1}]"\}.\\ \end{array}$$ This can be given an algebraic structure which allows us to talk about vector fields on $Gr$, finite-dimensional algebraic subvarieties, etc. Every $a \in K$ determines a vector field $KP_a$ on $Gr$, whose value at $W \in Gr$ is the map $$W \hookrightarrow K \stackrel{a}{\rightarrow} K \rightarrow K/W,$$ considered as an element of $$Hom(W,K/W) \approx T_W Gr.$$ The (double) $KP$ hierarchy on $Gr$ is just this collection of commuting vector fields. For $a \in {\bf C}[[z]]$, this vector field comes from the action on $Gr$ of the one-parameter subgroup $exp(ta)$ in ${\bf C}[[z]]^*$, which we consider trivial. The $KP$ hierarchy itself thus consists of the vector fields $KP_a$, for $a \in {\bf C}[z^{-1}]z^{-1}$, on the quotient $Gr/({\bf C}[[z]]^*)$. This quotient is well-behaved: the action of ${\bf C} ^*$ is trivial, and the unipotent part $1 + z {\bf C} [[z]] $ acts freely and with transversal slices. One restricts attention to the open subset of this quotient ("the big cell") parametrizing $W$ of fixed index (the index of $W$ is the index of the Fredholm projection) and satisfying a general position condition with respect to the standard subspace $W_0 := {\bf C}[z^{-1}] $. Sato's construction identifies this subset with the space $\Psi$ of pseudo differential operators of the form $${\cal L} = D + \sum^{\infty}_{i=1} u_iD^{-i}$$ where $$u_i = u_i(t_1,t_2,\cdots)$$ and $D = \partial/\partial t_1$. The resulting flows on $\Psi$ have the familiar Lax form: $$\frac{\partial {\cal L}}{\partial t_i} = [({\cal L}^i) _{+}, {\cal L}],$$ where $({\cal L}^i) _{+}$ is the differential operator part of ${\cal L}^i$. \bigskip \noindent \underline{multi component KP} The $k^{th}$ multi-component $KP$ hierarchy ($mcKP$) is obtained by considering instead the Grassmannian $Gr_k$ of subspaces of $K^{\oplus k}$ comparable to $({\bf C}[z^{- 1}])^{\oplus k}$. The entire ``loop algebra'' $gl(k,K)$ acts here, but to obtain commuting flows we need to restrict to a commutative subalgebra. For the k-th multi-component KP we take the simplest choice, of diagonal matrices, i.e. we consider the action of $({\bf C}[z^{-1}]z^{-1})^{\oplus k}$ on the quotient $Gr_k/({\bf C}[[z]]^*)^k$. There is a big cell $\Psi_{k} \subset Gr_k/({\bf C}[[z]]^*)^k$, consisting as before of subspaces in general position with respect to a reference subspace $W_0$, on which the flow is given by a Lax equation (for vector-valued operators). \\ \noindent \underline{Heisenberg flows} More generally, for a partition $$\underline{n} = (n_1,\cdots,n_k)$$ of the positive integer $n$, we can consider, following \cite{adams-bergvelt}, the maximal torus $Heis_{\underline{n}}$ of type $\underline{n}$ in $GL(n,K)$, as well as $heis_{\underline{n}}$, the corresponding Lie subalgebra in $gl(n,K)$. These consist of matrices in block-diagonal form, where the $i^{th}$ block is a formal power series in the $n_i \times n_i$ matrix \begin{equation} \label{eq-the-generator-of-the-ith-heisenberg-block} P_{n_i}: = \left( \begin{array}{cccccc} 0 & & & & 0 & z \\ 1 & & & & 0 & 0 \\ 0 & \cdot & & & 0 & 0 \\ 0 & & \cdot & & 0 & 0 \\ 0 & & & \cdot & 0 & 0 \\ 0 & & & & 1 & 0 \end{array} \right) \end{equation} \noindent We recall that this matrix arises naturally when we consider a vector bundle which is the direct image of a line bundle, near a point where $n_i$ sheets come together: in terms of a natural local basis of the vector bundle, it expresses multiplication by a coordinate upstairs (see (\ref{eq-ramification-matrix})). The $\underline{n}^{th}$ $mcKP$ (or ``Heisenberg flows'' of type $\underline{n}$) lives on the quotient of $Gr_n$ by the non-negative powers of the $P_{n_i}$, and a basis for the surviving flows is indexed by $k$-tuples $(d_1,\cdots,d_k)$, $d_i > 0$. Again, this can all be realized by Lax equations on an appropriate space $\Psi_{\underline{n}}$ of pseudo differential operators. When $\underline{n} = (1,\cdots,1)$ we recover the $n^{th}$ $mcKP$. When $\underline{n} = (n)$, the flows are pulled back from the standard $KP$ flows on $Gr$, via the mixing map $$m_n: Gr_n \to Gr$$ sending $$\widetilde{W} \subset {\bf C}((\widetilde{z}))^{\oplus n}$$ to $$W := \{\sum^{n-1}_{i=0} a_i(z^n)z^i|(a_0(\widetilde{z}),\cdots,a_{n-1}(\widetilde{z})) \in \widetilde{W}\} \subset {\bf C}((z)).$$ An arbitrary $k$-part partition $\underline{n}$ of $n$ determines a map $$m_{\underline{n}}: Gr_n \to Gr_k,$$ and the $\underline{n}^{th}$ Heisenberg flows are pullbacks of the $k^{th}$ $mcKP$. The natural big cell in this situation is determined by the cartesian diagram: $$ \begin{array}{lcccc} Gr_n & \longrightarrow & Gr_n / Heis^+_{\underline{n}} & \hookleftarrow & \Psi _{\underline{n}}\\ \mbox{ } \downarrow m_{\underline{n}} & & \downarrow & & \downarrow \\ Gr_k & \longrightarrow & Gr_k / (C[[z]]^*)^k & \hookleftarrow & \Psi _k \end{array} $$ \subsection{ Krichever maps} \label{sec-krichever-maps} \noindent \underline{The data} A basic Krichever datum (for the $KP$ hierarchy) consists of a quintuple $$(C,p,z,L,\eta)$$ where: \begin{tabbing}..................\= \kill \> $C$ is a (compact, non-singular) algebraic curve \\ \> $p \in C$\\ \> $z$ is a local (analytic or formal) coordinate at $p$\\ \> $L \in Pic C$\\ \> $\eta: L \otimes \hat{{\cal O}}_p \stackrel{\approx}{\rightarrow} \hat{{\cal O}}_p \approx {\bf C}[[z]]$ is a (formal) trivialization of $L$ near $p$.\\ \end{tabbing} If we fix $C,p$ and $z$, we think of the Krichever datum as giving a point of $${\cal U}_C(1,\infty p) := \displaystyle{\lim_{\stackrel{\leftarrow}{\ell}}} {\cal U}_C(1,\ell p).$$ The Krichever map $$\{\mbox{Krichever\ data} \}\to Gr$$ sends the above datum to the subspace $$W := \eta (H^0(C,L(\infty p))) = \bigcup_k\eta (H^0(C,L(kp))) \subset {\bf C}((z)).$$ This subspace is comparable to ${\bf C}[z^{-1}]$, since it follows from Riemann-Roch that the dimension of $H^0(L(kp))$ differs from $k$ by a bounded (and eventually constant) quantity.\\ \noindent \underline{The flows} Let's work with a coordinate $z$ which is analytic, i.e. it actually converges on some disc. A line bundle $L$ on $C$ can be trivialized (analytically) on the Stein manifold $C\setminus p$. We can think of $(L,\eta)$ as being obtained from ${\cal O}_{C\setminus p}$ by glueing it to $\hat{{\cal O}}_p$ via a $1$-cocycle, or transition function, which should consist of an invertible function $g$ on a punctured neighborhood of $p$ in $C$. Conversely, we claim there is a map: $$ \exp{} : K \longrightarrow {\cal U}_C(1,\infty p) , $$ $$ f \longmapsto (L,\eta) . $$ For $f \in {\bf C}(z)$, this is defined by the above analytic gluing, using $g:=\exp{f}$, which is indeed analytic on a punctured neighborhood. For $f \in {\bf C}[[z]] \approx \hat{{\cal O}}_p$, on the other hand, we take $ (L, \eta) := ({{\cal O}} , \exp{f} ). $ These two versions agree on the intersection, $ f \in {\bf C}[z]_{(0)} $, so the map is uniquely defined as claimed. (The bundles we get this way all have degree 0, but we can also obtain maps $$ \exp{_{g_{0}}} : K \longrightarrow {\cal U}_C(1,d,\infty p)$$ to the moduli space of level-$\infty p$ line bundles of degree d, simply by fixing a meromorphic function $g_0$ on a neighborhood of $p$ which has order $d$ at $p$, and replacing the previous $g$ by $g_0 \exp{f}$. We will continue to suppress the degree $d$ in our notation.) Any $a \in K$ gives an additive flow on $K$, which at $f \in K$ is $$ t \longmapsto f+ta . $$ Under the composed map $$ K \stackrel{\exp{_{g_{0}}}} {\longrightarrow} {\cal U}_C(1,\infty p) \stackrel{{Krichever}} {\longrightarrow} Gr , $$ this is mapped to the double KP flow $KP_a$ on $Gr$. For $a \in {\bf C}[[z]]$ this flow does not affect the isomorphism class of $L$, and simply multiplies $\eta$ by $exp(ta)$. On the other hand, the $i^{th}$ $KP$ flow, given by $a = z^{-i}$, changes both $L$ and $\eta$ if $i > 0$. The projection to $Pic\ C$ is a linear flow, whose direction is the $i^{th}$ derivative at $p$, with respect to the coordinate $z$, of the Abel-Jacobi map $C \to Pic\ C$. Dividing out the trivial flows corresponds to suppressing $\eta$, so we obtain, for each $C,p,z$ and degree $d \in {\bf Z}$, a finite-dimensional orbit of the $KP$ flows in $Gr / {\bf C}[[z]]^*$, isomorphic to $Pic^dC$.\\ \noindent \underline{Multi-Krichever data} Several natural generalizations of the Krichever map to the multi-component KP can be found in \cite{adams-bergvelt,li-mulase-category} and elsewhere. Here are some of the possibilities. We can consider ``multi-Krichever'' data $$(C,D,z_i,L,\eta)$$ involving a curve $C$ with a divisor $D$ consisting of $k$ distinct points $p_i(1 \leq i \leq k)$, a coordinate $z_i$ at each $p_i$, a line bundle $L$, and a formal trivialization $\eta_i$ at each $p_i$. Fixing $C,p_i$ and $z_i$, we have a multi-Krichever map $$\{\mbox{multi-Krichever\ data}\} \approx {\cal U}_C(1,\infty D) \longrightarrow Gr_k$$ sending $$(L,\eta_i) \mapsto W := (\eta_1,\cdots,\eta_k)(H^0(C,L(\infty D))) \subset {\bf C}((z))^{\oplus k}.$$ The $k$-component KP flow on the right hand side given by $a=(a_1, \ldots, a_k) \in K^k$ restricts to the flow on the multi-Krichever data which multiplies the transition function at $p_i$ (for an analytic trivialization of $L$ on $C \setminus D$) by $\exp{a_i}$. We can also consider "vector-Krichever" data $(C,p,z,E,\eta)$ where the line bundle $L$ is replaced by a rank $n$ vector bundle $E$, and $$\eta: E \otimes \hat{{\cal O}}_p \stackrel{\approx}{\rightarrow} (\hat{{\cal O}}_p)^n \approx ({\bf C}[[z]])^n$$ is now a (formal) trivialization of $E$ near $p$. Not too surprisingly, the vector-Krichever map $$\{\mbox{vector-Krichever\ data} \}\to Gr_n$$ sends the above datum to the subspace $$W := \eta (H^0(C,E(\infty p))) \subset ({\bf C}((z)))^n.$$ In the next subsection we will see that the interesting interaction of these two types of higher Krichever maps occurs not by extending further (to objects such as $(C,D,z_i,E,{\eta}_i)$), but by restricting to those vector data on one curve which match some multi-data on another.\\ \noindent \underline{KdV-type subhierarchies} Among the Krichever data one can restrict attention to those quintuples where $z^{-n}$ (for some fixed $n$) happens to extend to a regular function on $C \setminus p$, i.e. gives a morphism $$f = z^{-n}: C \to {\bf P}^1$$ of degree $n$, such that the fiber $f^{-1}(\infty)$ is $n \cdot p_0$. The Krichever map sends such data to the $n^{th}$ $KdV$ hierarchy, the distinguished subvariety of $Gr$ (invariant under the (double) $KP$ flows) given by $$ \mbox{KdV}_n := \{ W \in Gr \ | \ z^{-n} W \subset W \} .$$ The corresponding subspace of $\Psi$ is $$\{{\cal L}|{\cal L}^n = {\cal L}^n_+ \ \mbox{is\ a\ \underline{differential}\ operator}\}.$$ Fixing a partition $\underline{n} = (n_1,\cdots,n_k)$ of $n$, we can similarly consider the covering data of type $\underline{n}$, consisting of the multi-Krichever data $(C,p_i,z_i,L,\eta_i)$ plus a map $f: C \to \Sigma$ of degree $n$ to a curve $\Sigma$ with local coordinate $z$ at a point $\infty \in \Sigma$, such that $$f^{-1}(p) = \Sigma n_kp_i,\ \ f^{-1}(z) = z_i^{n_i}\ \mbox{at}\ p_i.$$ Such a covering datum clearly gives a multi-Krichever datum on $C$, but it also determines a vector-Krichever datum $(E,\eta)$ on $\Sigma$: The standard $m$-sheeted branched cover $$\begin{array}{cccc} f_m: & {\bf C} & \rightarrow & {\bf C}\\ & \widetilde{z} & \mapsto & z = \widetilde{z}^m\\ \end{array}$$ of the $z$-line determines an isomorphism $$s_m: (f_m)_* {\cal O} \stackrel{\approx}{\rightarrow} {\cal O}^{\oplus m}$$ given by $$\sum^{m-1}_{i=0} a_i(\widetilde{z}^m)\widetilde{z}^i \mapsto (a_0(z),\cdots,a_{m-1}(z)).$$ To the covering datum above we can then associate the rank-$n$ vector bundle $E := f_*L$ on $\Sigma$, together with the trivialization at $p$ obtained by composing $$\oplus_i f_{*,p_i}(\eta_i): (f_*L)_p \stackrel{\sim}{\rightarrow} \oplus_i f_{*,p_i}({\cal O}_{p_i})$$ with the isomorphisms $$f_{*,p_i}({\cal O}_{p_i}) \stackrel{\sim}{\rightarrow} {\cal O}^{\oplus n_i}_{p}$$ which are conjugates of the standard isomorphisms $s_{n_i}$ by the chosen local coordinates $z,z_i$. Finally, we note that there are obvious geometric flows on these covering data: $L$ and $\eta _i$ flow as before, while everything else stays put. The compatibility of the two types of higher Krichever data is expressed by the commutativity of the diagram: \begin{equation} \label{eq-diagram-of-krichever-maps} \begin{array}{ccc} \{\underline{n}-\mbox{covering\ data}\} & \approx & \{ f:C \rightarrow \Sigma; \ p_i,z_i,L,\eta_i; \ z \ | \ldots \}\\ \downarrow & & \downarrow\\ \{vector \ Krichever \ data \ on \ \Sigma \} & & \{multi \ Krichever \ data \ on \ C \} \\ \parallel & & \parallel \\ \cup_{\Sigma,p,z}{\cal U}_\Sigma(n,\infty p) & & \cup_{C,D,z_i} {\cal U}_C(1,\infty D)\\ \downarrow & & \downarrow\\ Gr_n & \stackrel {m_{\underline{n}}} {\longrightarrow} & Gr_k\\ \downarrow & & \downarrow\\ Gr_n / Heis_{\underline{n}}^+ & \longrightarrow & Gr_k / ({\bf C}[[z]]^\times)^k.\\ \end{array} \end{equation} The mcKP flows on the bottom right pull back to the Heisenberg flows on the bottom left, and to the geometric flows on the $\underline{n}$-covering data.\\ \subsection{Compatibility of hierarchies} \label{sec-compatibility-of-heirarchies} \bigskip Fix a smooth algebraic curve $\Sigma$ of arbitrary genus and a point $P$ in it. The moduli space ${\rm Higgs}_D := \HiggsModuli^{sm}_{\Sigma}(n,d,\omega(D))$ (see definition \ref{def-good-component-of-higgs-pairs}) can be partitioned into type loci. We consider the Zariski dense subset consisting of the union of finitely many type loci ${\rm Higgs}_D^{\underline{n}}$ indexed by partitions $\underline{n}$ of $n$. A Higgs pair in ${\rm Higgs}_D^{\underline{n}}$ has a spectral curve $C \rightarrow \Sigma$ whose ramification type over $P \in \Sigma$ is $\underline{n}=(n_1,\dots,n_k)$. Fix a formal local parameter $z$ on the base curve $\Sigma$ at $P$. A Higgs pair in ${\rm Higgs}_D^{\underline{n}}$ (or rather its spectral pair $(C,L)$, see proposition \ref{prop-ordinary-spectral-construction-higgs-pairs}) can be completed to an ${\rm Heis}_{\underline{n}}^+$-orbit of an $\underline{n}$-covering data $(C,P_i,z_i,L,\eta_i) \rightarrow (\Sigma,P,z,E,\eta)$ in finitely many ways. These extra choices form a natural finite Galois cover $\widetilde{{\rm Higgs}}_D^{\underline{n}}$ of each type locus ${\rm Higgs}_D^{\underline{n}}$. We obtain Krichever maps (see diagram \ref{eq-diagram-of-krichever-maps}) from the Galois cover $\widetilde{{\rm Higgs}}_D^{\underline{n}}$ to the quotients $Gr_n/{\rm Heis}_{\underline{n}}^+$ and $Gr_k/({\Bbb C}[[z]]^\times)^k$. Both the mcKP and Heisenberg flows pull back to the same geometric flow on the Galois cover. It is natural to ask: \begin{question} \label{question-compatibility} {\bf (The compatibility question)} Is the Heisenberg flow Poisson with respect to the natural Poisson structure on ${\rm Higgs}_D$? \end{question} The Compatibility Theorem \ref{thm-compatibility} and its extension \ref{thm-compatibility-singular-case} provide an affirmative answer. We factor the moment map of the Heisenberg action through natural finite Galois covers of the ramification type loci in the space of characteristic polynomials (equations (\ref{eq-the-jth-hamiltonian-of-the-ith-component}) and (\ref{eq-the-moment-map-for-the-heisenberg-action})). The compatibility naturally follows from the construction of the Poisson structure via level structures. Recall the birational realization of the moduli space $\HiggsModuli^{sm}_{\Sigma}(n,d,\omega(lP+D))$ as a quotient of the cotangent bundle $T^*{\cal U}_{lP+D}$ of the moduli space ${\cal U}_{lP+D} := {\cal U}_{\Sigma}(n,d,lP+D)$ of vector bundles with level structures (Chapter \ref{ch5}). This realization is a finite dimensional approximation of the limiting realization of the moduli space \[ {\rm Higgs}_{\infty{P}+D} := \lim_{l\rightarrow \infty} {\rm Higgs}_{l{P}+D} \] as a quotient of (a subset of) the cotangent bundle $T^*{\cal U}_{\infty{P}+D}$. The ramification type loci ${\rm Higgs}_D^{\underline{n}}$, their Galois covers $\widetilde{{\rm Higgs}}_D^{\underline{n}}$ and the infinitesimal ${\rm Heis}_{\underline{n}}$-action on $\widetilde{{\rm Higgs}}_D^{\underline{n}}$ become special cases of those appearing in the construction of section \ref{sec-type-loci}. The compatibility theorem follows from corollary \ref{cor-hamiltonians-on-the-base} accompanied by the concrete identification of the moment maps in our particular example. \bigskip The rest of this section is organized as follows. In section \ref{sec-galois-covers-and-relative-krichever-maps} we emphasize the ubiquity of the setup of relative Krichever maps. They can be constructed for any family ${\cal J} \rightarrow B$ of Jacobians of branched covers of a fixed base curve $\Sigma$. The analogue of the compatibility question \ref{question-compatibility} makes sense whenever the family ${\cal J} \rightarrow B$ is an integrable system (see for example question \ref{question-compatibility-for-mukai-system-on-elliptic-k3}). Starting with section \ref{sec-compatibility-of-stratifications} we concentrate on the moduli spaces of Higgs pairs. Sections \ref{sec-compatibility-of-stratifications} and \ref{sec-the-compatibility-thm-smooth-case} consider the case of smooth spectral curves. Especially well behaved is the case where the point $P\in\Sigma$ of the $\underline{n}$-covering data is in the support of the polar divisor $D$. In this case the symplectic leaves foliation of the moduli space of Higgs pairs is a refinement of the type loci partition (lemma \ref{lemma-compatibility-of-stratifications}). Section \ref{sec-the-compatibility-thm-singular-case} is a generalization to singular cases. As an example, we consider in section \ref{sec-elliptic-solitons} the Elliptic Solitons studied by Treibich and Verdier. We conclude with an outline of the proof of the compatibility theorem in section \ref{sec-proof-of-compatibility-theorem}. \noindent {\em Note:} Type-$(1,1,\dots,1)$ relative Krichever maps were independently considered by Y. Li and M. Mulase in a recent preprint \cite{li-mulase-compatibility}. \subsubsection{Galois covers and relative Krichever maps} \label{sec-galois-covers-and-relative-krichever-maps} Let $B_D := \oplus_{i=1}^{n} H^{0}(\Sigma,(\omega_{\Sigma}(D))^{\otimes i})$ be the space of characteristic polynomials. For simplicity, we restrict ourselves to the Zariski open subset $Bsm_D$ of reduced and irreducible $n$-sheeted spectral curves $\pi:C\rightarrow \Sigma$ in $T^*_{\Sigma}(D)$ whose fiber over $P\in \Sigma$ consists of {\em smooth} points of $C$. Denote by ${\rm Higgsm}_{D}$ the corresponding open subset of ${\rm Higgs}_{D}$. The ramification type stratification of ${\rm Higgsm}_{D}$ is induced by that of $Bsm_D$ \[ Bsm_D = \cup_{\underline{n}} Bsm_D^{\underline{n}}. \] Given a Higgs pair $(E,\varphi)$ in ${\rm Higgsm}^{\underline{n}}_{D}$ corresponding to a torsion free sheaf $L$ on a spectral cover $C\rightarrow \Sigma$ we can complete it to an $\underline{n}$-covering data \[ (C,P_i,z_i,L,\eta_i) \rightarrow (\Sigma,P,z,E,\eta) \] by choosing i) a formal local parameter $z$ at $P$, ii) an $n_i$-th root $z_{i}$ of $\pi^*{z}$ at each point $P_i$ of $C$ over $P$ and iii) formal trivializations $\eta_{i}$ of the sheaf $L$ at $P_i$. The ${\rm Heis}_{\underline{n}}^+$-orbit of an $\underline{n}$-covering data consists precisely of all possible choices of the $\eta_i$'s. Thus, for fixed $P$ and $z$ only a finite choice is needed in order to obtain the points of the quotients of the Grassmannians $Gr_n/{\rm Heis}_{\underline{n}}^+$ and $Gr_k/({\Bbb C}[[z]]^\times)^k$ (see diagram \ref{eq-diagram-of-krichever-maps}). These choices are independent of the sheaf $L$. The choices are parametrized by the Galois cover $\widetilde{Bsm}^{\underline{n}}_D \rightarrow Bsm^{\underline{n}}_D$ consisting of pairs $(C,\lambda)$ of a spectral curve $C$ in $Bsm^{\underline{n}}$ and the discrete data $\lambda$ which amounts to: \smallskip \noindent i) (Parabolic data) An ordering $(P_1,P_2,\dots,P_k)$ of the points (eigenvalues) in the fiber over $P$ compatible with the fixed order of the ramification indices $(n_1,\dots,n_k)$ (say, $n_1\leq n_2 \leq \dots \leq n_k$). \noindent ii) A choice of an $n_i$-th root $z_{i}$ of $\pi^*{z}$ at $P_i$. \smallskip Denote by $\widetilde{{\rm Higgsm}}^{\underline{n}}_{D}$ the corresponding Galois cover of ${\rm Higgsm}^{\underline{n}}_{D}$. We get a canonical relative Krichever map \begin{equation} \label{eq-relative-krichever-map} \kappa_{\underline{n}}: \widetilde{{\rm Higgsm}}^{\underline{n}}_{D} \rightarrow Gr_n/{\rm Heis}_{\underline{n}}^+ \end{equation} from the Galois cover to the quotient Grassmannian. The Galois group of $\widetilde{Bsm}^{\underline{n}}_D \rightarrow Bsm^{\underline{n}}_D$ is the Weyl group $W_{\underline{n}} := N({\rm Heis}_{\underline{n}}^+)/{\rm Heis}_{\underline{n}}^+$ of the maximal torus of the level infinity group $G^{+}_{\infty}$. For example, $W_{(1,\dots,1)}$ is the symmetric group $S_n$, while $W_{(n)}$ is the cyclic group of order $n$. The discrete data $\lambda = [(P_1,P_2,\dots,P_k),(z_1,\dots,z_k)]$ of a point $(C,\lambda)$ in $\widetilde{Bsm}^{\underline{n}}_D$ is equivalent to a commutative ${\Bbb C}[[z]]$-algebras isomorphism \begin{equation} \label{eq-isomorphism-heis-to-structure-sheaf} \lambda:{\bf heis}_{\underline{n}}^+ \stackrel{\cong}{\rightarrow} \oplus_{i=1}^{k}\CompletedSheafOfAt{C}{(P_i)} \end{equation} of the torus algebra with the formal completion of the structure sheaf $\StructureSheaf{C}$ at the fiber over $P$. The inverse $\lambda^{-1}$ sends $z_i$ to the generator of the $i$-th block of the torus ${\bf heis}_{\underline{n}}^+$ given by (\ref{eq-the-generator-of-the-ith-heisenberg-block}). The finite Weyl group $W_{\underline{n}}$ acts on ${\bf heis}_{\underline{n}}^+$, hence on $\lambda$, introducing the $W_{\underline{n}}$-torsor structure on $\widetilde{Bsm}^{\underline{n}}_D$. (See also lemma \ref{lemma-two-type-loci-coincide} part \ref{lemma-item-two-galois-covers-coincide} for a group theoretic interpretation.) \bigskip We note that a Galois cover $\widetilde{B} \rightarrow B$ as above can be defined for any family ${\cal J} \rightarrow B$ of Jacobians of a family ${\cal C} \rightarrow B$ of branched covers with a fixed ramification type $\underline{n}$ of a fixed triple $(\Sigma,P,z)$. We obtain a relative Krichever map \[ \kappa_{\underline{n}}: \widetilde{{\cal J}} \rightarrow Gr_n/{\rm Heis}_{\underline{n}}^+ \] as above. The Heisenberg flow pulls back to an infinitesimal action, i.e., a Lie algebra homomorphism \[ d\rho : {\bf heis}_{\underline{n}} \rightarrow V(\widetilde{{\cal J}}) \] into a commutative algebra of vertical tangent vector fields. When ${\cal J}$ (and hence $\widetilde{{\cal J}}$) is an integrable system we are led to ask the compatibility question \ref{question-compatibility}: {\em is the action Poisson?} {\em i.e., can $d\rho$ be lifted to a Lie algebra homomorphism} \[ \mu^*_{{\bf heis}_{\underline{n}}}: {\bf heis}_{\underline{n}} \rightarrow \Gamma(\StructureSheaf{\widetilde{B}}) \hookrightarrow [\Gamma(\StructureSheaf{\widetilde{{\cal J}}}),\{,\}]? \] A priori, the vector fields $d\rho(a)$, $a\in{\bf heis}_{\underline{n}}$ may not even be {\em locally} Hamiltonian. Inherently nonlinear examples arise from the Mukai-Tyurin integrable system of a family of Jacobians of a linear system $B := {{\Bbb P}}H^{0}(S,{\cal L})$ of curves on a symplectic or Poisson surface $S$ (see chapter \ref{ch8}). Consider for example the: \begin{question} \label{question-compatibility-for-mukai-system-on-elliptic-k3} Let $\pi : S \rightarrow {\Bbb P}^1$ be an elliptic K3 surface and ${\cal L}$ a very ample line bundle on $S$. Fix $P\in {\Bbb P}^1$ and a local parameter $z$ and consider the Galois cover $\widetilde{B}^{(1,1,\dots,1)}$ of the generic ramification type locus. Is the Heisenberg action Poisson on $\widetilde{{\cal J}} \rightarrow \widetilde{B}^{(1,1,\dots,1)}$ (globally over $\widetilde{B}^{(1,1,\dots,1)}$)? \end{question} The compatibility question has an intrinsic algebro-geometric formulation: The $j$-th KP flow of the $P_i$-component is the vector field whose direction along the fiber over $(b,\lambda) \in \widetilde{B}^{(1,1,\dots,1)}$ is the $j$-th derivative of the Abel-Jacobi map at $P_i$. Using the methods of chapter \ref{ch8} it is easy to see that the Heisenberg action is symplectic. As we move the point $P(0):=P$ in ${\Bbb P}^1$ and its (Lagrangian) fiber in $S$ we obtain an analytic (or formal) family of {\em Lagrangian} sections ${\cal AJ}(P_i(z))$ of $\widetilde{{\cal J}} \rightarrow \widetilde{B}^{(1,1,\dots,1)}$ (see corollary \ref{cor-canonical-symplectic-str-on-albanese}). Translations by the sections ${\cal AJ}(P_i(z))-{\cal AJ}(P_i(0))$ is a family of symplectomorphisms of $\widetilde{{\cal J}}$. Thus, the vector field corresponding to its $j$-th derivative with respect to the local parameter $z$ is locally Hamiltonian. It seems unlikely however that the Heisenberg flow integrates to a global Poisson action for a general system as in question \ref{question-compatibility-for-mukai-system-on-elliptic-k3}. It is the {\em exactness} of the symplectic structure in a neighborhood of the fiber over $P$ in $T^*\Sigma$ which lifts the infinitesimal symplectic Heisenberg action to a Poisson action in the Hitchin's system case (see equation (\ref{eq-the-moment-map-for-the-heisenberg-action})). \subsubsection{Compatibility of stratifications} \label{sec-compatibility-of-stratifications} Prior to stating the compatibility theorem \ref{thm-compatibility} we need to examine the Poisson nature of the Galois covers $\widetilde{{\rm Higgsm}}^{\underline{n}}_{D}$. ${\rm Higgsm}^{(1,1,\dots,1)}_D$ is a Zariski {\em open Poisson} subvariety of ${\rm Higgsm}_D$. Hence, the unramified Galois cover $\widetilde{{\rm Higgsm}}^{(1,1,\dots,1)}_D$ is endowed with the canonical pullback Poisson structure. Though non generic, the other type strata are as important. The cyclic ramification type $(n)$, for example, corresponds to the single component KP-hierarchy (see \cite{segal-wilson-loop-groups-and-kp}). When the point $P$ of the $\underline{n}$-covering data is in the support of the divisor $D$, we obtain a {\em strict compatibility} between the $P$-ramification type stratification of ${\rm Higgsm}_D$ and its symplectic leaves foliation. All Galois covers $\widetilde{{\rm Higgsm}}^{\underline{n}}_D$, $P \in D$ are thus endowed with the canonical pullback Poisson structure: \begin{lem} \label{lemma-compatibility-of-stratifications} (conditional compatibility of stratifications) When the point $P \in \Sigma$ is in the support of $D$, the symplectic leaves foliation is a {\em refinement} of the ramification type stratification ${\rm Higgsm}_D = \cup_{\underline{n}} {\rm Higgsm}^{\underline{n}}_D$. \end{lem} \noindent {\bf Proof:} We need to show that the ramification type $\underline{n}$ of the spectral cover $\pi:C\rightarrow \Sigma$ over $P\in D$ is fixed throughout the symplectic leaf ${\rm Higgsm}_S \subset {\rm Higgsm}_D$ of a Higgs pair $(E_{0},\varphi_{0})$. The symplectic leaves of ${\rm Higgs}_D$ are determined by coadjoint orbits of the level $D$ algebra $\LieAlg{g}_D$. The coadjoint orbit $S$ is determined by the residue of the Higgs field, namely, the infinitesimal data $(\restricted{E}{D}, \restricted{\varphi}{D})$ encoded in the value of $\varphi$ at $D$ (see \cite{markman-higgs} Remark 8.9 and Proposition 7.17)). Thus, the Jordan type of the Higgs field $\varphi$ at $P$ is fixed throughout ${\rm Higgsm}_S$. In general, (allowing singularities over $P$), the Jordan type depends both on the ramification type of $C\rightarrow \Sigma$ and on the sheaf $L$ on $C$ corresponding to the Higgs pair. For Higgs pairs $(E,\varphi)$ in ${\rm Higgsm}_D$, the spectral curve $C$ is smooth over $P$, hence, its ramification type coincides with the Jordan type of $\varphi$ at $P$. \EndProof \subsubsection{The compatibility theorem, the smooth case} \label{sec-the-compatibility-thm-smooth-case} We proceed to introduce the moment map of the infinitesimal Poisson action \[d\rho : {\bf heis}_{\underline{n}} \rightarrow V(\widetilde{{\rm Higgsm}}^{\underline{n}}_D).\] Throughout the end of this subsection we will assume the \begin{condition}\label{condition-ramification-type} Ramification types $\underline{n}$ other than $(1,1,\dots,1)$ are considered only if $P$ is in the support of $D$. \end{condition} This condition will be relaxed later by conditions \ref{condition-only-generic-type} or \ref{condition-only-singularities-which-are-resolved-by-spectral-sheaf}. Let $b \in Bsm^{\underline{n}}_D$ be the polynomial of the spectral curve $\pi_b:C_b \rightarrow \Sigma$. Recall that spectral curves are endowed with a tautological meromorphic $1$-form $y_b \in H^0(C_b,\pi_b^*\omega_\Sigma(D))$ with poles over $D \subset \Sigma$ (see section \ref{sec-spectral-curves-and-the-hitchin-system}). Let $\phi^j_{P_i}$ be the function on $\widetilde{Bsm}^{\underline{n}}_D$ given at a pair $(b,\lambda) \in \widetilde{Bsm}^{\underline{n}}_D$ by \begin{equation} \label{eq-the-jth-hamiltonian-of-the-ith-component} \phi^j_{P_i}(b,\lambda) := Res_{P_i}((z_i)^{-j}\cdot y_b). \end{equation} The Lie algebra homomorphism $\mu_{{\bf heis}_{\underline{n}}}^*$ sends the inverse (in ${\bf heis}_{\underline{n}}$) of the generator of the $i$-th block of the torus ${\bf heis}_{\underline{n}}^+$ given in (\ref{eq-the-generator-of-the-ith-heisenberg-block}) to the function $\phi^1_{P_i}\circ \widetilde{{\rm char}}$ on $\widetilde{{\rm Higgsm}}^{\underline{n}}_D$. In other words, $\mu_{{\bf heis}_{\underline{n}}}^* :{\bf heis}_{\underline{n}} \rightarrow [\Gamma(\StructureSheaf{\widetilde{{\rm Higgsm}}^{\underline{n}}_D}),\{,\}]$ factors as a composition $\phi \circ \widetilde{{\rm char}}^*$ through $\Gamma(\StructureSheaf{\widetilde{Bsm}^{\underline{n}}_D})$. If we regard $\lambda$ also as an isomorphism from ${\bf heis}_{\underline{n}}$ to $\oplus_{i=1}^{k}\CompletedSheafOfAt{C}{(P_i)}$ (eq. (\ref{eq-isomorphism-heis-to-structure-sheaf})), then $\phi:{\bf heis}_{\underline{n}} \rightarrow \Gamma(\StructureSheaf{\widetilde{Bsm}^{\underline{n}}_D})$ is given by \begin{equation} \label{eq-the-moment-map-for-the-heisenberg-action} (\phi(a))(b,\lambda) = \sum_{\{P_i\}} Res_{P_i}(\lambda(a)\cdot y_{b}), \ \ \ a\in{\bf heis}_{\underline{n}}. \end{equation} \begin{theorem} \label{thm-compatibility} ({\bf The Compatibility Theorem, smooth case}) (Assuming condition \ref{condition-ramification-type}) The relative Krichever map \[ \kappa_{\underline{n}} : \widetilde{{\rm Higgsm}}^{\underline{n}}_D \rightarrow Gr_n/{\rm Heis}_{\underline{n}}^+ \] intertwines the Heisenberg flow on $Gr_n/{\rm Heis}_{\underline{n}}^+$ (and the mcKP flow on $Gr_k/({\Bbb C}[[z]]^\times)^k$) with an infinitesimal Poisson action of the maximal torus ${\bf heis}_{\underline{n}}$ on $\widetilde{{\rm Higgsm}}^{\underline{n}}_D$. The latter is induced by the Lie algebra homomorphism \[ \mu_{{\bf heis}_{\underline{n}}}^* = \phi \circ \widetilde{{\rm char}}^* :{\bf heis}_{\underline{n}} \RightArrowOf{\phi} \Gamma(\StructureSheaf{\widetilde{Bsm}^{\underline{n}}_D}) \HookRightArrowOf{\widetilde{{\rm char}}^*} [\Gamma(\StructureSheaf{\widetilde{{\rm Higgsm}}^{\underline{n}}_D}),\{,\}] \] which factors through the homomorphism $\phi$ given by (\ref{eq-the-moment-map-for-the-heisenberg-action}). \end{theorem} \begin{rems} {\rm \begin{enumerate} \item \label{rems-item-nonvanishing-casimirs} The subalgebra ${\bf heis}_{\underline{n}}^+$ acts trivially on $Gr_n/{\rm Heis}_{\underline{n}}^+$ hence also on $\widetilde{{\rm Higgsm}}^{\underline{n}}_D$. This corresponds to the fact that the functions $\phi^j_{P_i}$, $j\leq 0$ are Casimir. Indeed, if $P$ is not contained in $D$, the $1$-form $y_b$ is holomorphic at the fiber over $P$ and $\phi^j_{P_i}$ is identically zero for $j\leq 0$. If $P$ is in $D$ then the finite set of non-zero $\phi^j_{P_i}$, indexed by finitely many non-positive integers $j$, are among the Casimirs that induce the highest rank symplectic leaves foliation (see \cite{markman-higgs} Proposition 8.8). \item The multi-Krichever map $\kappa_{\underline{n}}$ depends on auxiliary parameters $P$ and $z$. In other words, it lives naturally on an infinite dimensional space $\cup_{P,z} \ \widetilde{{\rm Higgsm}}^{\underline{n}}_{D,P,z}$ in which $P$ and $z$ are allowed to vary. This is not as bad as it might seem, since the $j$-th flow on $\widetilde{{\rm Higgsm}}^{\underline{n}}_D$ really depends only on our finite dimensional choices of $P$ and the $j$-th order germ of $z$ there. Similarly, our Hamiltonians $\phi^j_{P_i}$ depend at most on the $(j+n\deg D)$-th germ of $z$, the shift arising, as in part \ref{rems-item-nonvanishing-casimirs} of this remark, from the possible poles above $P$ of the tautological $1$-form on the spectral curve. So we may think of $\cup_{P,z} \ \widetilde{{\rm Higgsm}}^{\underline{n}}_{D,P,z}$ as the inverse limit of a family of finite dimensional moduli spaces, indexed by the level. Each KP flow or Hamiltonian is defined for sufficiently high level. \end{enumerate} } \end{rems} \subsubsection{The compatibility theorem, singular cases} \label{sec-the-compatibility-thm-singular-case} The condition that the fiber over $P$ of the embedded spectral curve be smooth is too restrictive. The embedded spectral data $(\bar{C} \subset T^*_{\Sigma}(D),\bar{L})$ of a Higgs pair $(E,\varphi)$ may have singularities over $P$ which are canonically resolvable. The point is that the rank $1$ torsion free sheaf $\bar{L}$ on $\bar{C}$ determines a partial normalization $\nu: C \rightarrow \bar{C}$ and a unique rank $1$ torsion free sheaf $L$ on $C$ such that i) $\bar{L}$ is isomorphic to the direct image $\nu_{*}L$, and ii) $L$ is locally free at the fiber over $P$. We are interested in those Higgs pairs for which the fiber of $C$ over $P$ is smooth. Such data may also be completed in finitely many ways to an $\underline{n}$-covering data as in section \ref{sec-galois-covers-and-relative-krichever-maps}. \begin{definition}\label{def-non-essential-singularities} {\rm The singularities over $P$ of a spectral pair $(\bar{C} \subset T^*_{\Sigma}(D),\bar{L})$ are said to be {\em resolved by the spectral sheaf} $\bar{L}$ if i) $\bar{C}$ is irreducible and reduced. ii) The sheaf $\bar{L}$ is the direct image $\nu_{*}L$ of a rank $1$ torsion free sheaf $L$ on the normalization $\nu:C\rightarrow \bar{C}$ of the fiber of $\bar{C}$ over $P$. } \end{definition} Fixing a symplectic leaf ${\rm Higgs}_S$ we may consider the type sub-loci in the locus of Higgs pairs whose spectral curve has at worst singularities over $P$ which are resolved by the spectral sheaf. The topology of these type loci is quite complicated. As a result, the Galois covers of these type loci may not have a symplectic structure. Nevertheless, the construction of section \ref{sec-type-loci}, as used in section \ref{sec-proof-of-compatibility-theorem}, provides {\em canonical embeddings} of the Galois covers of these type loci in (finite dimensional) symplectic varieties. These embeddings realize the Heisenberg flow as a Hamiltonian flow. Control over the topology is regained below by restraining the singularities. In the smooth case (section \ref{sec-the-compatibility-thm-smooth-case}) it is the smoothness which assures that the generic ramification type locus in a symplectic leaf ${\rm Higgs}_S$ is open (rather than only Zariski dense). The point is that degenarations from a ramification type $\underline{n}$ through other types back to type $\underline{n}$ must end with a singular fiber over $P$ (and are thus excluded). If $P \in D$, there are symplectic leaves ${\rm Higgs}_S$, $S\subset \LieAlg{g}_D^*$ of ${\rm Higgs}_D$ for which the singularities over $P$ are encoded in the infinitesimal data associated to the coadjoint orbit $S$ and shared by the generic Higgs pair in ${\rm Higgs}_S$. Often, this is an indication that the Poisson surface $T^*_{\Sigma}(D)$ is not the best to work with. Moreover, a birational transformation $T^*_{\Sigma}(D) \rightarrow X_S$, centered at points of the fiber over $P$ and encoded in $S$, can simultaneously resolve the singularities (over $P$) of the spectral curve of the generic point in ${\rm Higgs}_S$ (see example \ref{example-the-coadjoint-orbit-of-elliptic-solitons}). In such a case, smoothness of the proper transform of the spectral curve in $X_S$ at points of the fiber over $P$ is an {\em open} condition and the corresponding locus in ${\rm Higgs}_S$ with generic ramification type is {\em symplectic}. When the multiplicity of $P$ in $D$ is greater than $1$ the correspondence between the coadloint orbits $S\subset \LieAlg{g}_D^*$ and their surfaces $X_S$ can be quite complicated. Instead of working the correspondence out, we will use the following notion of $S$-smoothness to assure (see condition \ref{condition-only-generic-type}) that the generic ramification type locus in a symplectic leaf ${\rm Higgs}_S$ is open (rather than only Zariski dense). \begin{definition}\label{def-S-smoothness-over-D} {\rm Let $S$ be a coadjoint orbit of $\LieAlg{g}_D$. An irreducible and reduced spectral curve $\pi:\bar{C}\rightarrow\Sigma$ is {\em $S$-smooth over $D$} if i) a line bundle $L$ on the resolution of the singularities $\nu: C \rightarrow \bar{C}$ of the fibers over $D$ results in a Higgs pair $(E,\varphi) := \pi\circ\nu_*(L,\otimes \nu^*(y))$ in ${\rm Higgs}_S$. and ii) the arithmetic genus of the normalization $C$ above is equal to half the dimension of the symplectic leaf ${\rm Higgs}_S$. } \end{definition} If $\bar{C}$ is an irreducible and reduced spectral curve then, by the construction of \cite{simpson-moduli}, the fiber of the characteristic polynomial map in $\HiggsModuli^{sm}_{\Sigma}(n,d,\omega(D))$ is its compactified Jacobian, the latter being the moduli space of all rank $1$ torsion free sheaves on $\bar{C}$ with a fixed Euler characteristic. The compactified Jacobian is known to be irreducible for irreducible and reduced curve on a surface (i.e., with planar singularities, \cite{a-i-k}). Moreover, the symplectic leaf ${\rm Higgs}_S$ intersects the compactified Jacobian of $\bar{C}$ in a union of strata determined by partial normalizations. If $(E,\varphi)$ in ${\rm Higgs}_S$ corresponds to $(\bar{C},\bar{L})$ and $\nu:(C,L)\rightarrow (\bar{C},\bar{L})$ is a partial normalization where $L$ is a locally free sheaf on $C$, then any twist $F$ of $L$ by a locally free sheaf in $Pic^0(C)$ (the component of $\StructureSheaf{C}$) will result in a Higgs pair $(E',\varphi') := (\pi\circ\nu_*(F),\pi_*(\otimes y))$ in ${\rm Higgs}_S$. The point is that the residue of $(E',\varphi')$ (with respect to any level-$D$ structure) will be in the same coadjoint orbit as that of $(E,\varphi)$. $S$-smoothness of $\bar{C}$ over $D$ is thus equivalent to the geometric condition: \bigskip \noindent {\em The fiber of the characteristic polynomial map over $\bar{C}$ intersects ${\rm Higgs}_S$ in a Lagrangian subvariety isomorphic to the compactified Jacobian of the resolution $C$ of the singularities of $\bar{C}$ over $D$. } The following example will be used in section \ref{sec-elliptic-solitons} to describe a symplectic leaf which parametrizes Elliptic solitons. \begin{example} \label{example-the-coadjoint-orbit-of-elliptic-solitons} {\rm Let $D = P$, $S \subset \LieAlg{g}^*_D \cong \LieAlg{gl(n)}^* \cong \LieAlg{gl(n)}$ be the coadjoint orbit containing the diagonal matrix \[ A = \left( \begin{array}{cccccc} -1 & 0 & & \dots & & 0 \\ 0 & -1 & & & & \\ & & \cdot & & & \vdots \\ \vdots & & & \cdot & & \\ & & & & -1 & 0 \\ 0 & & \dots & & 0 & n-1 \end{array} \right) \] $S$ is the coadjoint orbit of lowest dimension with characteristic polynomial $(x+1)^{n-1}(x-(n-1))$. Its dimension $2n-2$ is $(n-2)(n-1)$ less than the generic rank of the Poisson structure of $\LieAlg{gl(n)}^*$. If non-empty, each component of ${\rm Higgs}_S$ is a smooth symplectic variety of dimension $\dim {\rm Higgs}_{P}-(n-1)-(n-2)(n-1) = [n^2(2g-1)+1]-(n-1)^2$ (see theorem \ref{thm-markman-botachin} and \cite{markman-higgs} proposition 7.17). The spectral curves $\bar{C} \subset T^*_{\Sigma}(P)$ which are $S$-smooth over $P$ will have two points in the fiber over $P$, one smooth at the eigenvalue with residue $n-1$ and one (singular if $n\geq 3$) at the eigenvalue with residue $-1$. Assume $n \geq 3$. The resolution $C$ of the singularity of a typical (though not all) such $\bar{C}$ will be unramified over $P$ with $n-1$ points collapsed to one in $\bar{C}$. The sheaf $\bar{L}$ (of a Higgs pair in ${\rm Higgs}_S$ with spectral curve $\bar{C}$) will be a pushforward of a torsion free sheaf $L$ on $C$. In contrast, line bundles on that $\bar{C}$ will result in Higgs pairs in another symplectic leaf ${\rm Higgs}_{S_{reg}}$ corresponding to the {\em regular} coadjoint orbit $S_{reg}$ in $\LieAlg{gl(n)}^*\cong \LieAlg{gl(n)}$ with characteristic polynomial $(x+1)^{n-1}(x-(n-1))$. These Higgs pairs will {\em not} be $S_{reg}$-smooth. $S_{reg}$-smoothness coincides with usual smoothness of the embedded spectral curve which is necessarily ramified with ramification index $n-1$ at the point with residue $-1$ over $P$. The $S$-smooth spectral curves will be smooth on the blowup $\widehat{X_S}$ of $T^*_{\Sigma}(P)$ at residue $-1$ over $P$. If we blow down in $\widehat{X_S}$ the proper transform of the fiber of $T^*_{\Sigma}(P)$ we get a surface $X_S$ with a marked point $x_{n-1}$ over $P$. An $S$-smooth spectral curve $\bar{C}$ will correspond to a curve on $X_S$ through $x_{n-1}$. It will be smooth at $x_{n-1}$ if in addition it is of ramification type $(1,1,\dots,1)$ over $P$. Consider the compactification ${\Bbb P}(T^*_{\Sigma}(P)\oplus \StructureSheaf{\Sigma})$ of $T^*_{\Sigma}(P)$. Blowing it up and down as above we get a ruled surface $\bar{X}_S$ over $\Sigma$ which is isomorphic to the projectivization ${\Bbb P}W$ of the unique nontrivial extension \[ 0 \rightarrow \omega_{\Sigma} \rightarrow W \rightarrow \StructureSheaf{\Sigma} \rightarrow 0. \] In particular, the surface $\bar{X}_S$ is {\em independent of the point $P$}. The point is that blowing up and down the ruled surface ${\Bbb P}V := {\Bbb P}(T^*_{\Sigma}(P)\oplus \StructureSheaf{\Sigma})$ at residue $-1$ over $P$ results with the ruled surface of a rank $2$ vector bundle $W$ whose sheaf of sections is a subsheaf of $V := T^*_{\Sigma}(P)\oplus \StructureSheaf{\Sigma}$. This subsheaf consists of all sections which restrict at $P$ to the subspace of the fiber $\restricted{V}{P}$ spanned by $(-1,1)$ (i.e., $W$ is a Hecke transform of $V$, see \cite{tyurin-vb-survey}). Clearly, $\omega_{\Sigma}$ is a subsheaf of $W$ and the quotient $W / \omega_{\Sigma}$ is isomorphic to $\StructureSheaf{\Sigma}$. The resulting extension is non-trivial because $H^0(\Sigma,W)$ and $H^0(\Sigma,\omega_{\Sigma})$ are equal as subspaces of $H^0(\Sigma,V)$. } \end{example} \begin{rem} \label{rem-resolved-by-the-spectral-sheaf-vs-S-smoothness} {\rm $S$-smoothness over $D$ is stronger than having singularities over $D$ which are resolved by the spectral sheaf. They differ when the singularity appears at an infinitesimal germ of too high an order to be detected by $S$. E.g., take $n=3$ in example \ref{example-the-coadjoint-orbit-of-elliptic-solitons} and consider a pair $(\bar{C},\bar{L})$ with a tacnode at residue $-1$ over $P$ (two branches meet with a common tangent). The arithmetic genus of the normalization $\nu:C\rightarrow \bar{C}$ of the fiber over $P$ will drop by $2$ while the pushforward $\bar{L}:=\nu_*(L)$ of a line bundle $L$ on $C$ will belong to a symplectic leaf ${\rm Higgs}_{S}$ whose rank is $2$ less than the maximal rank (rather than $4$). Hence $(\bar{C},\bar{L})$ is $S$-singular. } \end{rem} We denote by ${\rm Higgsm}_{S/D}$ the subset of ${\rm Higgs}_S \subset {\rm Higgs}_D$ parametrizing Higgs pairs whose spectral curve is $S$-smooth over $D$. Unfortunately, the compatibility of stratifications (lemma \ref{lemma-compatibility-of-stratifications}) does not extend to the $S$-smooth case. To overcome this inconvenience we may either assume condition \ref{condition-only-generic-type} or condition \ref{condition-only-singularities-which-are-resolved-by-spectral-sheaf}. \begin{condition}\label{condition-only-generic-type} Consider the ramification locus ${\rm Higgsm}_{S/D}^{\underline{n}}$ in a component of ${\rm Higgsm}_{S/D}$ only if it is the generic ramification type in this component. \end{condition} Note that ${\rm Higgsm}_{S/D}$ parametrizes only Higgs pairs whose spectral curve is $S$-smooth over $D$. $S$-smoothness over $D$ assures that if the type $\underline{n}$ is a generic ramification type in a component of ${\rm Higgsm}_{S/D}$, then the corresponding component of ${\rm Higgsm}^{\underline{n}}_{S/D}$ is an {\em open} subset of ${\rm Higgsm}_{S/D}$ (i.e., it excludes degenerations of type $\underline{n}$ Higgs pairs through other types back to type $\underline{n}$). In particular, these components of ${\rm Higgsm}^{\underline{n}}_{S/D}$ are {\em symplectic}. Alternatively, we may relax condition \ref{condition-only-generic-type} even further at the expense of losing the symplectic nature of the loci and having to resort to convention \ref{convention-abused-hamiltonian-language}: \begin{condition} \label{condition-only-singularities-which-are-resolved-by-spectral-sheaf} Consider only Higgs pairs with a spectral curve whose singularities over $P$ are resolved by its spectral sheaf. Adopt convention \ref{convention-abused-hamiltonian-language}. \end{condition} \begin{theorem} \label{thm-compatibility-singular-case} The compatibility theorem \ref{thm-compatibility} holds for: i) The $W_{\underline{n}}$-Galois covers of the type loci in ${\rm Higgsm}_{S/D}$ satisfying the genericity condition \ref{condition-only-generic-type} (instead of condition \ref{condition-ramification-type}). ii) The $W_{\underline{n}}$-Galois covers of the locus in ${\rm Higgs}^{\underline{n}}_D$ consisting of Higgs pairs satisfying condition \ref{condition-only-singularities-which-are-resolved-by-spectral-sheaf}. In ii) however we adopt convention \ref{convention-abused-hamiltonian-language}. \end{theorem} \subsubsection{Elliptic Solitons} \label{sec-elliptic-solitons} In this subsection we illustrate the possibilities in the singular case with a specific example. Let $\Sigma$ be a smooth elliptic curve. A {\em $\Sigma$-periodic Elliptic KP soliton} is a finite dimensional solution to the KP hierarchy, in which the orbit of the first KP equation is isomorphic to $\Sigma$. Its Krichever data $(C,\tilde{P},\frac{\partial}{\partial z},L)$ consists of a reduced and irreducible curve $C$, a smooth point $\tilde{P}$, a nonvanishing tangent vector at $\tilde{P}$ and a rank $1$ torsion free sheaf $L$ on $C$ of Euler characteristic $0$ (we suppress the non essential formal trivialization $\eta$ and consider only the first order germ of $z$ which is equivalent to choosing a nonzero tangent vector at $P$). We will denote the global vector field extending $\frac{\partial}{\partial z}$ also by $\frac{\partial}{\partial z}$. The periodicity implies that the image of the Abel Jacobi map $AJ: C \hookrightarrow J_C$, $Q \mapsto Q-\tilde{P}$ is tangent at $0$ to a subtorus isomorphic to $\Sigma$. Composing the Abel Jacobi map with projection to $\Sigma$ we get a {\em tangential morphism} $\pi:(C,\tilde{P}) \rightarrow (\Sigma,P)$. Its degree $n$ is called the {\em order} of the Elliptic soliton. In general, a tangential morphism $\pi:(C,\tilde{P}) \rightarrow (\Sigma,P)$ is a morphism with the property that $AJ(C)$ is tangent at $AJ(\tilde{P})$ to $\pi^*J^0_\Sigma$. Notice that composing a tangential morphism $\pi:(C,\tilde{P}) \rightarrow (\Sigma,P)$ with a normalization $\nu:\tilde{C} \rightarrow C$ results in a tangential morphism. A tangential morphism is called {\em minimal} if it does not factor through another tangential morphism. The KP elliptic solitons enjoyed a careful and detailed study by A. Treibich and J.-L. Verdier in a series of beautiful papers (e.g., \cite {treibich-verdier-elliptic-solitons,treibich-verdier-krichever-variety}). Their results fit nicely with our picture: \begin{theorem} \label{thm-variety-of-kp-solitons} The variety of Krichever data of Elliptic KP solitons of order $n$ with a fixed pointed elliptic curve and a tangent vector $(\Sigma,P,\frac{\partial}{\partial z})$ is canonically birational to the divisor of traceless Higgs pairs in the symplectic leaf ${\rm Higgs}_S$ of $\HiggsModuli^{sm}_{\Sigma}(n,0,\omega_{\Sigma}(P))$ corresponding to the coadjoint orbit $S$ of example \ref{example-the-coadjoint-orbit-of-elliptic-solitons}. The KP flows are well defined on ${\rm Higgsm}_{S/P}\subset {\rm Higgs}_S$ as the Hamiltonian vector fields of the functions $\phi^j_{\tilde{P}}$ given in (\ref{eq-the-jth-hamiltonian-of-the-ith-component}). \end{theorem} \medskip Note that if non-empty (which is the case) ${\rm Higgs}_S$ is $2n$-dimensional (see example \ref{example-the-coadjoint-orbit-of-elliptic-solitons}). The correspondence between tangential covers and spectral covers is a corollary of the following characterization of tangential covers due to I. M. Krichever and A. Treibich. For simplicity we consider only the case in which the tangency point $\tilde{P}$ is not a ramification point of $\pi$. \begin{theorem} \label{thm-characterization-of-tangential-covers} \cite{treibich-verdier-krichever-variety} Assume that $\pi:C\rightarrow \Sigma$ is unramified at $\tilde{P}$. Then $\pi$ is tangential if and only if there exists a section $y \in H^0(C,\pi^*[\omega_{\Sigma}(P)])$ satisfying: \smallskip \noindent a) Near a point of $\pi^{-1}(P) - \tilde{P}$ (away from the tangency point $\tilde{P}$), $y- \pi^*(dz/z)$ is a holomorphic multiple of $\pi^*(dz)$, where $z$ is a local parameter at $P$. (If $\pi:C\rightarrow \Sigma$ is unramified over $P$, this is equivalent to saying that the residues $Res_{P_i}(y)$ are the same at all $P_i$ other than $\tilde{P}$ in the fiber over $P \in \Sigma$). \noindent b) The residue $Res_{\tilde{P}}(y)$ at $\tilde{P}$ does not vanish if $n \geq 2$. \end{theorem} It follows by the residue theorem that there is a unique such section which has residue $n-1$ at $\tilde{P}$ and which is moreover traceless $tr(y) = 0 \in H^0(\Sigma,\omega_{\Sigma}(P))$. Let $dz$ be a global holomorphic non zero $1$-form on $\Sigma$. The function $k := y/\pi^*(dz)$ is called a {\em tangential function} in \cite{treibich-verdier-krichever-variety}. It was also proven that a tangential morphism of order $n$ has arithmetic genus $\leq n$ and is minimal if and only if its arithmetic genus is $n$. (\cite{treibich-verdier-elliptic-solitons} Corollaire 3.10). \bigskip \noindent {\bf Sketch of proof of theorem \ref{thm-characterization-of-tangential-covers}:} (for $C$ smooth, $\pi:C\rightarrow \Sigma$ unramified over $P$.) \noindent \underline{Step 1}: (Cohomological identification of the differential of the Abel-Jacobi map) The differential $dAJ : T_{Q}C \cong H^{0}(Q,\StructureSheaf{Q}(Q)) \rightarrow H^{1}(C,\StructureSheaf{C})$ of the Abel-Jacobi map at $Q \in C$ is identified as the connecting homomorphism of the short exact sequence \[ 0 \rightarrow \StructureSheaf{C} \rightarrow \StructureSheaf{C}(Q) \rightarrow \StructureSheaf{Q}(Q) \rightarrow 0. \] Similarly, the differential $d(AJ \circ \pi^{-1})$ of the composition $ \Sigma \HookRightArrowOf{\pi^{-1}} \rm Sym^{n}C \RightArrowOf{AJ} J^{n}_C $ is given by \[ T_{P}\Sigma \cong H^{0}(P,\StructureSheaf{P}(P)) \HookRightArrowOf{\pi^*} H^{0}(\pi^{-1}(P),\StructureSheaf{\pi^{-1}(P)}(\pi^{-1}(P))) \stackrel{\cong}{\rightarrow} T_{[\pi^{-1}(P)]}\rm Sym^{n}C \RightArrowOf{dAJ} H^{1}(C,\StructureSheaf{C}), \] where the composition of the last two arrows is the connecting homomorphism of the short exact sequence \[ 0 \rightarrow \StructureSheaf{C} \rightarrow \StructureSheaf{C}(\pi^{-1}(P)) \rightarrow \StructureSheaf{\pi^{-1}(P)}(\pi^{-1}(P)) \rightarrow 0. \] \smallskip \noindent \underline{Step 2}: (residues as coefficients in a linear dependency of tangent lines) Clearly, the tangent line to $(AJ \circ \pi^{-1})(\Sigma)$ at the image of $P$ is in the span of the tangent lines to $AJ(C)$ at the points $P_i$. If, in addition, $\pi$ is tangential with tangency point $\tilde{P}\in C$, then the tangent lines to $AJ(C)$ at the points in the fiber over $P$ are linearly dependent. If $\pi:C\rightarrow \Sigma$ is unramified over $P$ we can write these observations in the form of two linear equations: \begin{equation} \label{eq-linear-condition-tangentiality} \sum dAJ_{P_i}(\frac{\partial}{\partial z_i}) = d(AJ\circ \pi^{-1})_{P}(\frac{\partial}{\partial z}), \end{equation} and \begin{equation} \label{eq-linear-dependency} \sum a_i dAJ_{P_i}(\frac{\partial}{\partial z_i}) = 0 \ \ \ \mbox{linear} \ \ \mbox{dependency}. \end{equation} Above, $\frac{\partial}{\partial z_i}$ is the lift of $\frac{\partial}{\partial z}$, i.e., $d\pi(\frac{\partial}{\partial z_i})=\frac{\partial}{\partial z}$. We claim that the coefficients $a_i$ in (\ref{eq-linear-dependency}) are residues of a meromorphic $1$-form $y$ at the points of the fiber. More precisely we have: \begin{lem} \label{lemma-residues-vs-differential-of-abel-jacobi} Assume that $\pi:C\rightarrow \Sigma$ is unramified over $P$. There exists a section $y \in H^0(C,\pi^*\omega_{\Sigma}(P))$ with residues $(a_1,a_2,\dots,a_n)$ at the fiber over $P$ if and only if $(a_1,a_2,\dots,a_n)$ satisfy equation (\ref{eq-linear-dependency}). \end{lem} \noindent {\bf Proof:} The global tangent vector field $\frac{\partial}{\partial z} \in H^{0}(\Sigma,T\Sigma)$ gives rise to the commutative diagram: \begin{equation}\label{diag-residues-and-differential-of-abel-jacobi} {\divide\dgARROWLENGTH by 4 \begin{diagram} \node{H^0(C,\pi^*\omega_{\Sigma}(P))} \arrow{s,lr}{\cong}{\rfloor\frac{\partial}{\partial z}} \arrow{e} \node{H^0(\pi^{-1}(P),\restricted{\pi^{*}\omega_{\Sigma}(P)}{\pi^{-1}(P)})} \arrow{s,lr}{\cong}{\rfloor\frac{\partial}{\partial z}} \arrow{e} \node{H^1(C,\pi^*\omega_{\Sigma})} \arrow{s,lr}{\cong}{\rfloor\frac{\partial}{\partial z}} \\ \node{H^0(C,\StructureSheaf{C}(\pi^{-1}(P)))} \arrow{e} \node{H^0(\pi^{-1}(P),\StructureSheaf{\pi^{-1}(P)}(\pi^{-1}(P))} \arrow{e,t}{dAJ} \node{H^1(C,\StructureSheaf{C}).} \end{diagram} } \end{equation} The middle contraction $\rfloor\frac{\partial}{\partial z}$ maps residues $(a_1,a_2,\dots,a_n) \in H^0(\pi^{-1}(P),\restricted{\pi^{*}\omega_{\Sigma}(P)}{\pi^{-1}(P)})$ to $(a_1\frac{\partial}{\partial z_1}, a_2\frac{\partial}{\partial z_2},\dots, a_n\frac{\partial}{\partial z_n})$. The lemma follows by the exactness of the horizontal sequences in the diagram. \EndProof \smallskip \noindent \underline{Step 3}: We conclude that $\pi:C\rightarrow \Sigma$ is tangential if and only if there is a $1$-form $y$ as in the theorem. If $\pi:C\rightarrow \Sigma$ is tangential then using lemma \ref{lemma-residues-vs-differential-of-abel-jacobi} we see that equation (\ref{eq-linear-condition-tangentiality}) gives rise to a $1$-form $y$ with residues $(-1,-1,\dots,n-1)$ as required. Conversely, given a $1$-form $y$ with residues $(-1,-1,\dots,n-1)$ lemma \ref{lemma-residues-vs-differential-of-abel-jacobi} and equations (\ref{eq-linear-dependency}) and (\ref{eq-linear-condition-tangentiality}) imply the tangentiality. This completes the proof of theorem \ref{thm-characterization-of-tangential-covers} in the generic case considered. \EndProof Theorem \ref{thm-variety-of-kp-solitons} would follow once the existence of either $n$-sheeted spectral covers $S$-smooth over $P$, or degree $n$ tangential covers smooth and unramified over $P$ is established for every choice of $(\Sigma,P)$. This was done in (\cite{treibich-verdier-elliptic-solitons} Theorem 3.11) by studying the linear system of transferred spectral curves on the surface $\bar{X}_{S}$ of example \ref{example-the-coadjoint-orbit-of-elliptic-solitons} and applying Bertini's theorem to show that the generic transferred spectral curve is smooth in $X_{S}$. It follows that the blow up of the point with residue $-1$ over $P$ resolves the generic $n$-sheeted spectral curve of Higgs pairs in ${\rm Higgs}_{S}$ to a smooth curve of genus $n$ unramified over $P$. Any $S$-smooth spectral curve $\bar{C}$ in $T^*\Sigma(P)$ of a Higgs pair $(E,\varphi)$ in ${\rm Higgs}_S$ admits a unique partial normalization $C$ of arithmetic genus $n$ by the spectral sheaf $\bar{L}$ corresponding to $(E,\varphi)$. The tautological $1$-form $\bar{y}$ pulls back to a $1$-form on $C$ of the type which characterize tangential covers by theorem \ref{thm-characterization-of-tangential-covers}. Conversely, a degree $n$ tangential cover $\pi:(C,\tilde{P})\rightarrow (\Sigma,P)$ of arithmetic genus $n$ which is smooth and unramified over $P$ is sent to the spectral curve $\bar{C}$ in $T^*\Sigma(P)$ of the Higgs pair \[ (E,\varphi):= (\pi_*(L),[\otimes y : \pi_*(L) \rightarrow \pi_*(L)\otimes\omega_{\Sigma}(P)]) \] for some, hence every, choice of a line bundle $L$ on $C$. Note that $\bar{C}$ is reduced since it is irreducible and the branch through residue $n-1$ over $P$ is reduced. The canonical morphism $\nu : C \rightarrow \bar{C}$ is the resolution by the spectral sheaf $\nu_*L$. Hence $(\bar{C},\nu_*L)$ is $S$-smooth. (The arithmetic genus of $\bar{C}$ is $\frac{1}{2}(n^2 -n+2)$, the common arithmetic genus to all $n$-sheeted spectral curves in $T^*\Sigma(P)$.) \medskip Finally we note that, as the tangency point $\tilde{P}$ over $P$ is marked by having residue $n-1$, it does not have monodromy and all the KP flows corresponding to it are well define on ${\rm Higgsm}_{S/P}$ as the Hamiltonian vector fields of the functions $\phi^j_{\tilde{P}}$ given in (\ref{eq-the-jth-hamiltonian-of-the-ith-component}). \subsubsection{Outline of the proof of the compatibility theorem} \label{sec-proof-of-compatibility-theorem} For simplicity we assume that $D = lP$, $l \geq 0$. The general case is similar replacing ${\cal U}_{\Sigma}(n,d,\infty P)$ by ${\cal U}_{\Sigma}(n,d,\infty P + D)$. Let $G_{\infty} := GL(n,K)$ be the loop group and $G_{\infty}^{+}$ the level infinity subgroup. (More canonically, $K \cong {\Bbb C}((z))$ should be thought of as the completion of the function field of $\Sigma$ at $P$, and we may postpone the choice of a coordinate $z$ until we need to choose generators for a maximal torus ${\rm Heis}_{\underline{n}}$ of $G_{\infty}$.) Denote by $M_{l,k}$, $k\geq l$ the pullback of $T^*{\cal U}_{\Sigma}(n,d,lP)$ to ${\cal U}_{\Sigma}(n,d,kP)$ via the rational forgetful morphism. $M:=T^*{\cal U}_{\Sigma}(n,d,\infty P)$ is defined as the limit of finite dimensional approximations (see \ref{sec-finite-dim-approaximations} for the terminology) \[ T^*{\cal U}_{\Sigma}(n,d,\infty P) := \lim_{l \rightarrow l}\lim_{\infty \leftarrow k} M_{l,k}. \] Denote by $M^s$ (resp. $M_{l,k}^s$) the subset of $T^*{\cal U}_{\Sigma}(n,d,\infty P)$ (resp. $M_{l,k}$) consisting of triples $(E,\varphi,\eta)$ with a {\em stable} Higgs pair $(E,\varphi)$. We arrive at the setup of section \ref{sec-finite-dim-approaximations} with the Poisson quotient $Q_{\infty} := M^s/G^{+}$ being the direct limit \[ Higgs_{\infty} := \lim_{l \rightarrow \infty} \HiggsModuli^{sm}_{\Sigma}(n,d,lp). \] We emphasize that the stability condition is used here for the morphism \[M_{l,k}^s \rightarrow \HiggsModuli^{sm}_{\Sigma}(n,d,lp)\] between the two {\em existing} coarse moduli spaces to be well defined, and {\em not} to define the quotient. The {\em infinitesimal} loop group action on ${\cal U}_{\infty P} := {\cal U}_{\Sigma}(n,d,\infty P)$ (the derivative of the action defined in section \ref{sec-krichever-maps} on the level of \v{C}ech $1$-cocycles) may be lifted to an infinitesimal action on its cotangent bundle. The point is that the infinitesimal action of $a \in \LieAlg{g}_{\infty}$, with poles of order $\leq l_0$, is well defined on the finite dimensional approximation ${\cal U}_{\Sigma}(n,d,l P)$ for $l \geq l_0$. Thus, it lifts to all cotangent bundles $M_{l,l}$, $l\geq l_0$. $M_{l,k}$ embeds naturally in $M_{k,k}$ as an invariant subvariety. This defines the action on the limit $T^*{\cal U}_{\Sigma}(n,d,\infty P)$. As a lifted action it is automatically Poisson. Its moment map \[\mu_{\infty}^* : \LieAlg{g}_{\infty} \rightarrow \Gamma[\StructureSheaf{T^*{\cal U}_{\infty P}},\{,\}] \] (the limit of the moment maps for the finite dimensional approximations) is given by the same formula that we have already encountered for the level groups (see \ref{eq-moment-map-of-finite-dim-level-action}): \begin{equation} \label{eq-moment-map-lifted-loop-group-action} (\mu_{\infty}^*(a))(E,\varphi,\eta) = Res_{P}trace(a\cdot(\varphi)^{\eta}), \ \ \ a \in \LieAlg{g}_{\infty}. \end{equation} Choosing a maximal torus ${\bf heis}_{\underline{n}} \subset \LieAlg{g}_{\infty}$ of type $\underline{n}$ we arrive at the setup of section \ref{sec-type-loci}. In particular, we obtain the type locus ${\rm Higgs}^{\underline{n}}_{\Sigma}(n,d,lp)$ in $\HiggsModuli^{sm}_{\Sigma}(n,d,lp)$. \begin{lem} \label{lemma-two-type-loci-coincide} \begin{enumerate} \item \label{lemma-item-two-type-loci-coincide} The algebro-geometric definition of the ramification type loci coincides with the group theoretic definition \ref{def-group-theoretic-definition-type-loci} when $char(E,\varphi)$ is an integral (irreducible and reduced) spectral curve. \item \label{lemma-item-two-galois-covers-coincide} A choice of generators for a maximal torus ${\bf heis}_{\underline{n}}$ as in (\ref{eq-the-generator-of-the-ith-heisenberg-block}) determines a canonical isomorphism between the group theoretic and the algebro-geometric $W_{\underline{n}}$-Galois covers. \end{enumerate} \end{lem} \noindent {\bf Proof:} \ref{lemma-item-two-type-loci-coincide}) The stabilizer $\LieAlg{t} \subset \LieAlg{g}_{\infty}$ of $(\varphi)^{\eta} \in \LieAlg{g}_{\infty}\otimes_{\StructureSheaf{(P)}} \omega_{\Sigma,(P)} \cong \LieAlg{g}_{\infty}^*$ with spectral curve $\pi : C = char(E,\varphi) \rightarrow \Sigma$ is precisely $U^{\eta}$ where $U$ is the stalk of $Ker[ad\varphi : \rm End E \rightarrow \rm End E\otimes \omega_{\Sigma}(lP)]$ at the formal punctured neighborhood of $P$. In addition, $U$ is canonically isomorphic to the stalk of the structure sheaf at the formal punctured neighborhood of the fiber of $C$ over $P$ via the completion of the canonical embedding: \[ \pi_*\StructureSheaf{C} \hookrightarrow \pi_*\rm End L \hookrightarrow \rm End E. \] Hence, the level infinity structure $\eta$ provides a {\em canonical} isomorphism \begin{equation} \label{eq-isomorphism-of-tori-induced-by-level-structure} \lambda : \LieAlg{t} \stackrel{\cong}{\rightarrow} U \end{equation} from the stabilizer algebra $\LieAlg{t}$ to the structure sheaf at the formal punctured neighborhood of the fiber of $C$ over $P$. \smallskip \noindent \ref{lemma-item-two-galois-covers-coincide}) As the types coincide, we may choose the level infinity structure $\eta$ so that the stabilizer $\LieAlg{t}$ coincides with the fixed torus ${\bf heis}_{\underline{n}}$. We may further require that the isomorphism $\lambda$ given by (\ref{eq-isomorphism-of-tori-induced-by-level-structure}) coincides with the one in (\ref{eq-isomorphism-heis-to-structure-sheaf}). This determines the ${\rm Heis}_{\underline{n}}^+$-orbit of $\eta$ uniquely, i.e., a point in the group theoretic Galois cover. \EndProof Theorems \ref{thm-compatibility} and \ref{thm-compatibility-singular-case} would now follow from corollary \ref{cor-hamiltonians-on-the-base} provided that we prove that the homomorphism $\phi$ of the theorems (given by \ref{eq-the-moment-map-for-the-heisenberg-action}) is indeed the factorization of the ${\bf heis}_{\underline{n}}$-moment map through the characteristic polynomial map. (Note that the existence of this factorization follows from diagram (\ref{diag-factoring-the-moment-map-through-base})). In other words, we need to prove the identity \begin{equation} \label{eq-equality-of-two-moment-maps} \sum_{\{P_i\}} Res_{P_i}(\lambda(a)\cdot y_b) = Res_{P}trace(a\cdot(\varphi)^{\eta}) \ \ \ a\in{\bf heis}_{\underline{n}}, \ b={\rm char}(E,\varphi) \end{equation} as functions on the set of all Higgs pairs $(E,\varphi,\lambda)$ in $\widetilde{{\rm Higgs}}_{lp}^{\underline{n}}$ for which the spectral sheaf resolves the singularities of their spectral curve over $P$ (see definition \ref{def-non-essential-singularities}). Above, $\eta$ is any level infinity structure in the ${\rm Heis}_{\underline{n}}^+$-orbit as in the proof of lemma \ref{lemma-two-type-loci-coincide} or, equivalently, $\lambda(a) = \eta^{-1} \circ a \circ \eta$ where we identify the structure sheaf of the formal punctured neighborhood with $U$ of that lemma. If the embedded spectral curve $\bar{C_b}$ is singular, the $P_i$ are the points over $P$ of its resolution $\nu: C_b \rightarrow \bar{C_b}$, and the tautological meromorphic $1$-form $y_b$ should be replaced by the pullback $\nu^*(y)$ of the tautological $1$-form $y$ on the surface $T^*_{\Sigma}(lP)$. Conjugating the right hand side of (\ref{eq-equality-of-two-moment-maps}) by $\eta$, we get \[ \sum_{\{P_i\}} Res_{P_i}(A \cdot \nu^*(y)) = Res_{P}trace((\pi\circ \nu)_*[A \cdot \nu^*(y)]) \] for $A$ a (formal) meromorphic function at the fiber over $P$. Working formally, we can consider only the ``parts'' with first order pole $r_i dlog{z_i}$ of $A \cdot \nu^*(y)$ at $P_i$. The equality follows from the identity $dlog{z_i} = (\pi\circ\nu)^*[\frac{1}{n_i}dlog{z}]$ which imply (projection formula) that $(\pi\circ\nu)_*(\otimes dlog{z_i})$ acts as $\frac{1}{n_i}e_{P_i}\otimes dlog{z}$ were $e_{P_i}$ is the projection onto the eigenspace of the point $P_i$. \EndProof \newpage \section{The Cubic Condition and Calabi-Yau threefolds} \label{ch7} We pose in section \ref{sec-families-of-tori} the general question: when does a family of polarized abelian varieties or complex tori support a completely integrable system? In section \ref{subsec-cubic-condition} we describe a general necessary infinitesimal symmetry condition on the periods of the family (the cubic condition of lemmas \ref{lemma-weak-cubic-cond-poisson} and \ref{lemma-weak-cubic-cond-symplectic}) and a sufficient local condition (lemmas \ref{lemma-strong-cubic-cond-local-coordinates} and \ref{lemma-strong-cubic-cond-coordinate-free}). In section \ref{sec-cy-threefolds} we use the Yukawa cubic to construct a symplectic structure (and an ACIHS) on the relative intermediate Jacobian over the moduli space of gauged Calabi-Yau threefolds (theorem \ref{thm-cy-acihs}). The symplectic structure extends to the bundle of Deligne cohomologies and we show that the image of the relative cycle map as well as bundles of sub-Hodge-structures are isotropic (corollary \ref{cor-contact-structure-extends-to-deligne-coho}). \subsection{Families of Tori} \label{sec-families-of-tori} Consider a Poisson manifold $(X,\psi)$ together with a Lagrangian fibration $$ \pi : {\cal X} \longrightarrow B $$ over a base $B$, whose fibers $$X_b := \pi^{-1}(b), \quad \quad b \in B $$ are tori. (We say $\pi$ is Lagrangian if each fiber $X_b$ is a Lagrangian submanifold of some symplectic leaf in $\cal X$.) All these objects may be $C^\infty$, or may be equipped with a complex analytic or algebraic structure. On $B$ we have the tangent bundle ${\bf T}_B$ as well as the vertical bundle $\cal V$, whose sections are vector fields along the fibers of $\pi$ which are constant on each torus. The pullback $\pi^*{\cal V}$ is the relative tangent bundle ${\bf T}_{{\cal X}/B}$; in the analytic or algebraic situations, we can define $\cal V$ as $\pi_*{\bf T}_{{\cal X}/B}$. The data $\pi$ and $\psi$ determine an injection $$ i : {\cal V}^* \hookrightarrow {\bf T}_B $$ or, equivalently, a surjection $$ i' : {\bf T}^*_B \twoheadrightarrow {\cal V} $$ sending a 1-form $\alpha$ on $B$ to the vertical vector field $$ i'(\alpha) := \pi^* \alpha \, \rfloor \, \psi. $$ The image $i({\cal V}^*) \subset {\bf T}_B$ is an integrable distribution on $B$. Its integral manifolds are the images in $B$ of symplectic leaves in $\cal X$. In this section we start with a family of tori $\pi : {\cal X} \rightarrow B$ and ask whether there is a {\it Lagrangian structure} for $\pi$, i.e. a Poisson structure on $\cal X$ making the map $\pi$ Lagrangian. More precisely, we fix $\pi : {\cal X} \rightarrow B$ and an injection $i : {\cal V}^* \hookrightarrow {\bf T}_B$ with integrable image, and ask for existence of a Lagrangian structure $\psi$ on $\cal X$ inducing the given $i$. In the $C^\infty$ category there are no local obstructions to existence of a Lagrangian structure: the fibration $\pi$ is locally trivial, so one can always find action-angle coordinates near each fiber, and use them to define $\psi$. In the analytic or algebraic categories, on the other hand, the fibers $X_b$ (complex tori, or abelian varieties) have invariants, given essentially by their {\it period matrix} $p(X_b)$, so the fibration may not be analytically locally trivial. We will see that there is an obstruction to existence of a Lagrangian structure for $\pi : {\cal X} \rightarrow B$, which we formulate as a symmetry condition on the derivatives of the period map $p$. These derivatives can be considered as a linear system of quadrics, and the condition is, roughly, that they be the polars of some cubic (= section of $\rm Sym^3{\cal V})$. Let $X$ be a $g$-dimensional complex torus, and $\gamma_1, \cdots , \gamma_{2g}$ a basis of the integral homology $H_1(X, {\bf Z})$. There is a unique basis $\alpha_1 , \cdots, \alpha_g$ for the holomorphic differentials $H^0(X,\Omega^1_X)$ satisfying $$ \int_{\gamma_{g+i}} \alpha_j = \delta_{ij}, \quad \quad 1 \le i,j \le g, $$ so we define the period matrix $P= p(X,\gamma)$ by $$ p_{ij} := \int_{\gamma_i} \alpha_j, \quad \quad 1 \le i,j \le g. $$ Riemann's first and second bilinear relations say that $X$ is a principally polarized abelian variety (PPAV) if and only if $P$ is in {\it Siegel's half space}: $$ {\Bbb H}_g := \{ {\rm symmetric} \; g \times g \; {\rm complex \, matrices \, whose \, imaginary \, part \, is \, positive \, definite} \}. $$ In terms of a dual basis $\gamma_1^*, \cdots , \gamma_{2g}^*$ of $H^1(X,{\Bbb Z})$, the integral class $\omega := \sum_{i=1}^{g}\gamma_i^*\wedge\gamma_{g+i}^* \in H^2(X,{\Bbb Z})$ is a K\"{a}hler class if and only if $P$ is in Siegel's half space. In this case we call $\omega$ a {\em principal polarization}. Given a family $\pi : {\cal X} \rightarrow B$ of PPAVs together with a continuously varying family of symplectic bases $\gamma_1 , \cdots , \gamma_{2g}$ for the fiber homologies, we then get a period map $$ p : B \longrightarrow {\Bbb H}_g. $$ If we change the basis $\gamma$ by a symplectic transformation $$ \left( \begin{array}{cc} A & B \\ C & D \\ \end{array} \right) \in Sp(2g, {\bf Z}), $$ the period matrix $P$ goes to $(AP + B)$ $(CP + D)^{-1}$. So given a family $\pi$ without the choice of $\gamma$, we get a multi-valued map of $B$ to ${\Bbb H}_g$, or a map $$ p : B \longrightarrow {\cal A}_g $$ to the moduli space of PPAV. The latter is a quasi projective variety, which can be described analytically as the quotient $$ {\cal A}_g = {\Bbb H}_g/\Gamma$$ of ${\Bbb H}_g$ by the action of the modular group $$ \Gamma := Sp(2g, {\bf Z}) / (\pm 1).$$ A PPAV $X$ determines a point $[X]$ (or carelessly, $X$) of ${\cal A}_g$. This point is non-singular if $X$ has no automorphisms other than $\pm 1$, and then the tangent space $T_{[X]}{\cal A}_g$ can be identified with $\rm Sym^2V_X$, where $V_X$ is the tangent space (at $0 \in X$) to $X$. This can be seen by identifying $T_{[X]}{\cal A}_g$ with $T_{[X]} {\Bbb H}_g$ and recalling that ${\Bbb H}_g$ is an open subset of $\rm Sym^2V_X$. More algebraically, this follows from elementary deformation theory: all first-order deformations of $X$ are given by $$ H^1(X,{\bf T}_X) \approx H^1(X,V_X \, \otimes_{{\bf C}} {\cal O}_X) \approx V_X \otimes H^1(X, {\cal O}_X) \approx \otimes^2 V_X, $$ and in there the deformations as abelian variety, i.e., the deformations preserving the polarization bilinear form on $H_1(X,{\bf Z})$, are given by the symmetric tensors $\rm Sym^2V_X$. \subsection{The Cubic Condition} \label{subsec-cubic-condition} Our condition for an analytic or algebraic family $\pi : {\cal X} \rightarrow B$ of PPAVs, given by a period map $p : B \rightarrow {\cal A}_g$, to have a Lagrangian structure $\psi$ inducing a given $i : {\cal V}^* \hookrightarrow {\bf T}_B$, can now be stated as follows. The differential of $p$ is a map of bundles: $$ dp : {\bf T}_B \longrightarrow \rm Sym^2{\cal V}, $$ so the composite $$ dp \circ i \; : \; {\cal V}^* \longrightarrow \rm Sym^2 {\cal V}$$ can be considered as a section of ${\cal V} \otimes \rm Sym^2{\cal V}$, and the condition is that it should come from the subbundle $\rm Sym^3{\cal V}$. In other words, there should exist a cubic $c \in H^0(B,\; Sym^3{\cal V})$ whose polar quadrics give the directional derivatives of the period map: if the tangent vector $\partial / \partial b \in T_bB$ equals $i(\beta)$ for some $\beta \in {\cal V}^*$, then: $$ {\partial p \over \partial b} = \beta \, \rfloor \, c.$$ We give two versions of this cubic condition. In the first, we check the existence of a two vector $\psi$, not necessarily satisfying the Jacobi identity, for which $\pi$ is Lagrangian, and which induces a given injection $i : {\cal V}^* \hookrightarrow {\bf T}_B$. (Note that neither the definition of the map $i$ induced by the two-vector $\psi$, nor the notion of $\pi$ being Lagrangian, require $\psi$ to be Poisson.) \begin{lem}\label{lemma-weak-cubic-cond-poisson} \ (Weak cubic condition, Poisson form). \ A family $\pi : {\cal X} \rightarrow B$ of polarized abelian varieties has a two vector $\psi$ satisfying \smallskip \noindent a) $\pi : {\cal X} \rightarrow B$ is Lagrangian\\ b) $\psi$ induces a given $i : {\cal V}^* \hookrightarrow {\bf T}_B$ \\ \smallskip \noindent if and only if $$ dp \circ i \in {\rm Hom}({\cal V}^*, Sym^2 {\cal V}) $$ comes from a cubic $$c \in H^0(B, \; Sym^3{\cal V}).$$ Moreover, in this case there is a unique such 2-vector $\psi$ which satisfies also \smallskip \noindent c) The zero section $z : B \rightarrow {\cal X}$ is Lagrangian, i.e., $(T^*_{{\cal X}/B})|_{z(B)}$ is $\psi$-isotropic (here we identify the conormal bundle of the zero section with $(T^*_{{\cal X}/B})|_{z(B)}$.) \end{lem} {\bf Proof:} (Note: we refer below to the vertical bundle ${\bf T}_{{\cal X}/B}$ by its, somewhat indirect, realization as the pullback $\pi^*{\cal V}$.) The short exact sequence of sheaves on $\cal X$: $$ 0 \rightarrow \pi^*{\cal V} \rightarrow {\bf T}_{\cal X} \rightarrow \pi^*{\bf T}_B \rightarrow 0 $$ determines a subsheaf $\cal F$ of $\Wedge{2}{\bf T}_{\cal X}$ which fits in the exact sequences: $$ \begin{array}{ccccccccc} 0 &\rightarrow& {\cal F} &\rightarrow& \Wedge{2}{\bf T}_{\cal X} &\rightarrow& \pi^*\Wedge{2}{\bf T}_B &\rightarrow& 0 \\ 0 &\rightarrow& \pi^*\Wedge{2}{\cal V} &\rightarrow& {\cal F} &\rightarrow& \pi^*({\cal V} \otimes {\bf T}_B) &\rightarrow& 0. \\ \end{array} $$ The map $\pi$ is Hamiltonian with respect to the two-vector $\psi \in H^0(B_{\cal X}, \Wedge{2}{\bf T}_{\cal X})$ if and only if $\psi$ goes to $0$ in $\Wedge{2}{\bf T}_B$, i.e., if and only if it comes from $H^0({\cal F})$. The question is therefore whether $i \in H^0(B, {\cal V} \otimes {\bf T}_B) \subset H^0({\cal X}, \pi^*({\cal V} \otimes {\bf T}_B))$ is in the image of $H^0({\cal X},{\cal F})$. Locally in $B$, this happens if and only if $i$ goes to $0$ under the coboundary map \[ \begin{array}{ccc} \pi_*\pi^*({\cal V} \otimes {\bf T}_B) &\longrightarrow& R^1 \pi_*\pi^* \Wedge{2}{\cal V} \\ \parallel & & \parallel \\ {\cal V} \otimes {\bf T}_B &\longrightarrow& \Wedge{2} {\cal V} \otimes {\cal V}. \\ \end{array} \] This latter map factors through the period map $$ 1 \otimes dp \; : \; {\cal V} \otimes {\bf T}_B \longrightarrow {\cal V} \otimes \rm Sym^2{\cal V} $$ and a Koszul map $$ {\cal V} \otimes \rm Sym^2{\cal V} \longrightarrow \Wedge{2}{\cal V} \otimes {\cal V}. $$ Now exactness of the Koszul sequence $$ 0 \rightarrow {\rm \rm Sym}^3 {\cal V} \rightarrow {\cal V} \otimes {\rm \rm Sym}^2 {\cal V} \rightarrow \Wedge{2} {\cal V} \otimes {\cal V} $$ shows that the desired $\psi$ exists if and only if $$ dp \circ i \; = \; (1 \otimes dp)(i) \in {\cal V} \otimes \rm Sym^2 {\cal V} $$ is in the subspace $\rm Sym^3{\cal V}$. (The Hamiltonian map $\pi$ will automatically be Lagrangian, since $i$ is injective.) We conclude that, locally on $B$, $i$ lifts to a $2$-vector $\psi$ satisfying conditions a), b), if and only if $dp \circ i$ is a cubic. If $\psi_1,\psi_2$ are two such lifts then $\psi_1 - \psi_2 \in H^0({\cal X},\Wedge{2} \pi^*{\cal V})$. Moreover, $\psi_1 - \psi_2$ is determined by its restriction to the zero section because $\Wedge{2}\pi^*{\cal V}$ restricts to a trivial bundle on each fiber. The zero section induces a splitting ${\bf T}_{{\cal X}_{|z(B)}} \simeq \pi^*{\bf T}_B \oplus (\pi^*{\cal V})_{|z(B)}$ and hence a well defined pullback $z^*(\psi) \in H^0(B, \stackrel{2}{\wedge} {\cal V})$ (locally on $B$). The normalizations $\psi - \pi^*(z^*(\psi))$ patch to a unique global section satisfying a), b) c). \EndProof \bigskip The symplectic version of this lemma is: \begin{lem}: \label{lemma-weak-cubic-cond-symplectic} (Weak cubic condition, quasi-symplectic form). A family $\pi : {\cal X} \rightarrow B$ of principally polarized abelian varieties has a $2$-form $\sigma$ satisfying \smallskip \noindent a) $\pi : {\cal X} \rightarrow B$ has isotropic fibers,\\ b) $\sigma$ induces a given (injective) homomorphism $j : {\bf T}_B \hookrightarrow {\cal V}^*,$ \smallskip \noindent if and only if $$ (1\otimes j^*)\circ dp \in \rm Hom({\bf T}_B,{\bf T}^*_B\otimes {\cal V}) \cong {\bf T}^*_B\otimes {\bf T}^*_B\otimes {\cal V} \ \ \mbox{is} \ \mbox{in} \ \ \rm Sym^2{\bf T}^*_B\otimes {\cal V}. $$ Moreover, in this case, there exists a unique 2-form $\sigma$ satisfying a), b), and the additional condition \smallskip \noindent c) the zero section is isotropic $(z^*\sigma = 0)$. \end{lem} \begin{rem} {\rm Riemann's first bilinear condition implies further that $(1\otimes j^*)\circ dp$ maps to $\rm Sym^3{\bf T}^*_B$, i.e., $(\rm Sym^{2}j^*)\circ dp \in {\rm Hom}({\bf T}_B,\rm Sym^2{\bf T}^*_B)$ comes from a cubic $c \in H^0(B, \rm Sym^3{\bf T}^*_B)$.} \end{rem} The cubic condition for an embedding $j : {\bf T}_B \hookrightarrow {\cal V}^*$ does not guarantee that the induced 2-form $\sigma$ on $\cal X$ is closed. In that sense, the cubic condition is a necessary condition for $j$ to induce a symplectic structure while the following condition is necessary and sufficient (but, in general, harder to verify). \bigskip \noindent \underline{Closedness Criterion for a Symplectic Form:} {\em Given a family $\pi : {\cal X} \rightarrow B$ of polarized abelian varieties and a surjective $j' : {\cal V} \rightarrow {\bf T}^*_B$, there exists a {\it closed} $2$-form $\sigma$ on $\cal X$ satisfying conditions a), b), c) of Lemma \ref{lemma-weak-cubic-cond-symplectic} if and only if $j'({\cal H}_1({\cal X}/B, {\bf Z})) \subset {\bf T}^*_B$ is a Lagrangian lattice in $T^*B$, i.e., if locally on $B$ it consists of closed $1$-forms. Moreover, the $2$-form $\sigma$ is uniquely determined by $j'$. } \smallskip \noindent {\bf Proof:} \ $j'({\cal H}_1({\cal X}/B, {\bf Z}))$ is Lagrangian $\Longleftrightarrow$ the canonical symplectic structure $\tilde{\sigma}$ on $T^*B$ is translation invariant under $j'({\cal H}_1({\cal X}/B, {\bf Z}))$ $\Longleftrightarrow$ $(j')^*(\tilde{\sigma})$ descends to the unique $2$-form $\sigma$ on ${\cal X} = {\cal V}/{\cal H}_1({\cal X}/B, {\bf Z})$ satisfying conditions a), b), c) of Lemma \ref{lemma-weak-cubic-cond-symplectic}. \EndProof \smallskip It is instructive to relate the cubic condition to the above criterion. This is done in lemma \ref{lemma-strong-cubic-cond-local-coordinates} in a down to earth manner and is reformulated in lemma \ref{lemma-strong-cubic-cond-coordinate-free} as a coordinate free criterion. \noindent \begin{lem} \label{lemma-strong-cubic-cond-local-coordinates} \ (``Strong Cubic Condition'') \noindent Let $V$ be a $g$-dimensional vector space, $\{e_1$ ,$\cdots$, $e_g \}$ a basis, $B \subset V^*$ an open subset, $p:B \rightarrow {\Bbb H}_g \hookrightarrow \rm Sym^2V$ a holomorphic map (${\Bbb H}_g$ is embedded in $\rm Sym^2V$ via the basis $\{e_j\}$), $\pi : {\cal X} \rightarrow B$ the corresponding family of principally polarized abelian varieties. Then the following are equivalent: \smallskip \noindent (i) There exists a symplectic structure $\sigma$ on ${\cal X}$ such that $\pi : ({\cal X},\sigma) \rightarrow B$ is a Lagrangian fibration and $\sigma$ induces the identity isomorphism $$ {\rm id} \in \rm Hom({\bf T}_{{\cal X}/B}, \pi^*{\bf T}^*_B) \simeq \rm Hom(\pi^*{\cal V},\pi^*{\cal V}). $$ (ii) $p : B \rightarrow \rm Sym^2V$ is, locally in $B$, the Hessian of a function on $B$,\\ (iii) $dp \in {\rm Hom}({\bf T}_B, \rm Sym^2{\cal V}) \simeq ({\cal V} \otimes \rm Sym^2{\cal V})$ is a section of $\rm Sym^3{\cal V}.$ \end{lem} \noindent {\bf Proof:} \ Let $\{e^*_j\}$ be the dual basis of $V$. \noindent \underline{(i) $\Leftrightarrow$ (iii):} By the closedness criterion above, there exists $\sigma$ as in (i) if and only if the subsheaf of lattices ${\cal H}^1({\cal X}/B, {\bf Z}) \subset T^*B$ is Lagrangian, i.e., if and only if its basis \[ \{e_1,\cdots,e_g, (p \; \rfloor \; e^*_1), \cdots, (p \; \rfloor \; e^*_g) \} \] consists of closed $1$-forms. The $e_i$'s are automatically closed. If we regard the differential $dp$ as a section of ${\bf T}^*_B \otimes \rm Sym^2{\cal V}$, then the two-form $d(p \; \rfloor \; e^*_j)$ is equal to the anti-symmetric part of the contraction $dp \rfloor \; e^*_j \in {\bf T}^*_B \otimes {\cal V} \cong {\cal V}\otimes {\cal V}$. Hence, closedness of $(p \; \rfloor \; e^*_j)$, $1 \le j \le g$, is equivalent to the symmetry of $dp \in {\cal V}\otimes \rm Sym^2{\cal V}$ also with respect to the first two factors, i.e., to $dp$ being a section of $\rm Sym^3{\cal V}$. \noindent \underline{(ii) $\Rightarrow$ (iii)}. \ \ Clear. \noindent \underline{(iii) $\Rightarrow$ (ii)}. \ \ Follows from the Poincare lemma. \EndProof \medskip The additional information contained in the ``Strong Cubic Condition'' and lacking in the ``Weak Cubic Condition'' is that a Lagrangian sublattice (with respect to the polarization) ${\cal L} \subset {\cal H}_1({\cal X}/B, {\bf Z})$ is mapped via $j' : {\cal V} \tilde{\rightarrow} {\bf T}^*_B$ to a sublattice $j'({\cal L}) \subset T^*B$ Lagrangian with respect to the holomorphic symplectic structure on $T^*B$. (In the above lemma, ${\cal L} = {\rm Sp} \{ e_1,\cdots,e_g \}$). The coordinate free reformulation of lemma \ref{lemma-strong-cubic-cond-local-coordinates} is: \begin{lem} \label{lemma-strong-cubic-cond-coordinate-free} (``Strong cubic condition'') \ Let $j' : {\cal V} \tilde{\rightarrow} {\bf T}^*_B$ be an isomorphism of the vertical bundle ${\cal V} = R^0_{\pi_*}(T_{{\cal X}/B})$ of the family $\pi : {\cal X} \rightarrow B$ of polarized abelian varieties with the cotangent bundle of the base. Assume only that $j'$ maps a sublattice ${\cal L} \subset {\cal H}_1({\cal X}/B, Z)$ Lagrangian with respect to the polarization to a sublattice $j'({\cal L}) \subset T^*B$ Lagrangian with respect to the holomorphic symplectic structure on $T^*B$. Then there exists a symplectic structure $\sigma$ on $\cal X$ s.t. $\pi : {\cal X} \rightarrow B$ is a Lagrangian fibration and inducing $j'$ if and only if $j'$ satisfies the weak cubic condition, i.e. $$ dp \circ i \in H^0(B,\rm Sym^3{\cal V}) \quad \quad \mbox{where} \ \ i = (j')^{*^{-1}}. $$ \end{lem} \begin{rem} {\rm In most cases however, $j'({\cal L})$ being Lagrangian implies $j'({\cal H}_1({\cal X}/B, {\bf Z}))$ being Lagrangian via the global monodromy action and without reference to the weak cubic condition.} \end{rem} Finally we remark that the above discussion applies verbatim to the case of polarized complex tori (not necessarily algebraic) since only the first Riemann bilinear condition was used. \subsection{An Integrable System for Calabi-Yau Threefolds} \label{sec-cy-threefolds} The {\it Hodge group} $H^{p,q}$ of an $n$-dimensional compact K\"{a}hler manifold $X$ is defined as the space of harmonic forms on $X$ of type $(p,q)$, i.e. involving $p$ holomorphic and $q$ antiholomorphic differentials. Equivalently, $H^{p,q}$ is isomorphic to the $q-th$ cohomology $H^q(X, \Omega^p)$ of the sheaf of holomorphic $p$-forms on $X$. The {\it Hodge theorem} gives a natural decomposition of the complex cohomology, $$ H^k(X,{\bf C}) \approx \oplus_{p+q=k} \; H^{p,q} \approx \oplus_{p+q=k} \; H^q(\Omega^p). $$ The {\it Hodge number} $h^{p,q}$ is the complex dimension of $H^{p,q}$. The {\it Hodge filtration} of $H^k(X,{\bf C})$ is defined by $$ F^iH^k(X,{\bf C}) := \oplus_{\stackrel{p+q=k}{p \geq i}} \; H^{p,q}. $$ The {\it k-th intermediate Jacobian} of $X$ \cite{c-g} is: $$J^k(X) := H^{2k-1}(X,{\bf C})/(F^kH^{2k-1}(X,{\bf C}) + H^{2k-1}(X,{\bf Z}))$$ $$\approx (F^{n-k+1} H^{2n-2k+1}(X,{\bf C}))^*/H_{2n-2k+1}(X,{\bf Z}).$$ Elementary properties of the Hodge filtration imply that this is a complex torus, but generally not an abelian variety unless $k=1$ or $k=n$: it satisfies Riemann's first bilinear condition (which expresses the skew symmetry of the cup product on $H^{2k-1}$), but not the second, since the sign of the product (on primitive pieces) will vary with the parity of $p$. The extreme cases correspond to the connected component of the Picard $(k=1)$ and Albanese $(k=n)$ varieties. The Hodge decomposition does not depend holomorphically on parameters, since both holomorphic and antiholomorphic differentials are involved. The advantage of the Hodge filtration is that it does vary holomorphically and even algebraically when $X$ is algebraic. The $F^p$ can be defined algebraically, as the hypercohomology of the complex \begin{equation} \label{eq-the-quotient-of-the-algebraic-derham-complex} 0 \rightarrow \Omega^p \rightarrow \Omega^{p+1} \rightarrow \cdots \rightarrow \Omega^n \rightarrow 0. \end{equation} In particular, the intermediate Jacobian $J^k(X)$ varies holomorphically with $X$. This means that a smooth analytic family ${\cal X} \rightarrow B$ of compact K\"{a}hler manifolds gives rise to analytic vector bundles $F^i{\cal H}^k({\cal X}/B)$ and to smooth analytic families ${\cal J}^k({\cal X}/B) \longrightarrow B$ of intermediate Jacobians of the fibers. The bundle ${\cal H}^k({\cal X}/B)$ is the complexification of a bundle ${\cal H}^k({\cal X}/B, {\bf Z})$ of discrete groups. In particular, it has a natural local trivialization. In other words, it admits a natural flat connection, called the {\it Gauss-Manin} connection. The holomorphic subbundles $F^i{\cal H}^k({\cal X}/B)$ are in general not invariant with respect to this connection, since the Hodge decomposition and filtration do change from point to point. {\it Griffiths' transversality} says that when a holomorphic section of $F^i{\cal H}^k$ is differentiated, it can move at most one step: $$ \nabla(F^i{\cal H}^k) \subset F^{i-1}{\cal H}^k \otimes \Omega^1_B. $$ An $n$-dimensional compact K\"{a}hler manifold $X$ is called {\it Calabi-Yau} if it has trivial canonical bundle, $$ \omega_X = \Omega_X^n \; \approx \; {\cal O}_X, $$ and satisfies $$ h^{p,0} = 0 \ {\rm for} \ 0 < p < n. $$ A {\it gauged Calabi-Yau} is a pair $(X,s)$ consisting of a Calabi-Yau manifold $X$ together with a non-zero volume form $$ s : {\cal O}_X \stackrel{\approx}{\longrightarrow} \omega_X. $$ A theorem of Bogomolov, Tian and Todorov \cite{bogomolov,Ti,To} says that $X$ has a smooth (local analytic) universal deformation space $M_X$. We say that a family $\chi: {\cal X} \rightarrow {\cal M}$ of Calabi-Yaus $X_t, t\in {\cal M}$, is {\em complete} if the local classifying map ${\cal M} \supset U_t \rightarrow M_{X_t}$ is an isomorphism for some neighborhood of every point $t \in {\cal M}$. It follows that ${\cal M}$ is smooth and that the tangent space at $t$ to ${\cal M}$ is naturally isomorphic to $H^1(X, {\bf T}_X)$. Typically, such families might consist of all Calabi-Yaus in some open subset of moduli, together with some ``level'' structure. The choice of gauge $s$ gives an isomorphism $$ \rfloor \;s : {\bf T}_X \longrightarrow \Omega^{n-1}_X, $$ hence an isomorphism $$ T_X{\cal M} \; \approx \; H^{n-1,1}(X). $$ Starting with a complete family $\chi : {\cal X} \rightarrow {\cal M}$, we can construct \begin{itemize} \item The bundle ${\cal J}^k \rightarrow {\cal M}$ of intermediate Jacobians of the Calabi-Yau fibers. \item The space $\tilde{\cal M}$ of gauged Calabi-Yaus, a ${\bf C}^*$-bundle over ${\cal M}$ obtained by removing the $0$-section from the line bundle $\chi_*(\omega_{{\cal X}/{\cal M}}).$ \item The fiber product $$\tilde{\cal J}^k := {\cal J}^k \; \times_{\cal M} \; \tilde{\cal M} , $$ which is an analytic family of complex tori $\pi : \tilde{\cal J}^k \rightarrow \tilde{\cal M}$. \end{itemize} \begin{theorem} \label{thm-cy-acihs} Let ${\cal X} \rightarrow {\cal M}$ be a complete family of Calabi-Yau manifolds of odd dimension $n=2k-1 \geq 3$. Then there exists a canonical closed holomorphic $2$-form $\sigma$ on the relative $k$-th intermediate Jacobian $\pi : \tilde{\cal J} \rightarrow \tilde{\cal M}$ with respect to which $\pi$ has maximal isotropic fibers. When $n=3$, the $2$-form $\sigma$ is a symplectic structure and $\pi : \tilde{\cal J} \rightarrow \tilde{\cal M}$ is an analytically completely integrable Hamiltonian system. \end{theorem} \noindent {\bf Proof.} \underline{Step I.} \ There is a canonical isomorphism $$T_{(X,s)} \tilde{\cal M} \; \approx \; F^{n-1} H^n(X,{\bf C}).$$ Indeed, the natural map $p : \tilde{\cal M} \rightarrow {\cal M}$ gives a short exact sequence $$ 0 \rightarrow T_{(X,s)}(\tilde{\cal M}/{\cal M}) \rightarrow T_{(X,s)} \tilde{\cal M} \rightarrow T_X{\cal M} \rightarrow 0, $$ in which the subspace can be naturally identified with $H^0(\omega_X) = H^0(\Omega^n_X)$, and the quotient with $H^1({\bf T}_X)$, which goes isomorphically to $H^1(\Omega^{n-1}_X)$ by $\rfloor \; s$. What we are claiming is that this sequence can be naturally identified with the one defining $F^{n-1}H^n$: $$ 0 \rightarrow H^0 (\omega_X) \rightarrow F^{n-1} \rightarrow H^1 (\Omega^{n-1}_X) \rightarrow 0,$$ i.e., that the extension data match, globally over $\tilde{\cal M}$. To see this we need a natural map $T_{(X,x)} \tilde{\cal M} \rightarrow F^{n-1} H^n$ inducing the identity on the sub and quotient spaces. Over $\tilde{\cal M}$ there is a tautological section $s$ of $F^n{\cal H}^n(\tilde{\cal X}/\tilde{\cal M},{\bf C})$. The Gauss-Manin connection defines an embedding $$ \nabla_{(\cdot)}s : T_{(X,s)} \tilde{\cal M} \longrightarrow H^n(X,{\bf C}). $$ Griffiths' transversality implies that the image is in $F^{n-1}H^n(X,{\bf C})$. Clearly $\nabla_{(\cdot)}s$ has the required properties. We will need also a description of the isomorphism in terms of Dolbeault cohomology. We think of a 1-parameter family $(X_t,s_t) \in \tilde{\cal M}$, depending on the parameter $t$, as living on a fixed topological model $X$ on which there are families $\bar{\partial}_t$ of complex structures (given by their $\bar{\partial}$-operator) and $s_t$ of $C^\infty$ $n$-forms, such that $s_t$ is of type $(n,0)$ with respect to $\bar{\partial}_t$, all $t$. Since the $s_t$ are now on a fixed underlying $X$, we can differentiate with respect to $t$: $$ s_t = s_0 + ta \quad \quad ({\rm mod}\;t^2). $$ Griffiths transversality now says that $a$ is in $F^{n-1}H^n(X_0)$. It clearly depends only on the tangent vector to $\tilde{\cal M}$ along $(X_t,s_t)$ at $t = 0$, so we get a map $T_{(X,s)} \tilde{\cal M} \longrightarrow F^{n-1}H^n$ with the desired properties. \medskip \underline{Step II}. Let ${\cal V}$ be the vertical bundle on $\tilde{\cal M}$ coming from $\pi:\tilde{\cal J} \rightarrow \tilde{\cal M}$. It is isomorphic to $$ F^k{\cal H}^n(\tilde{\cal X}/\tilde{\cal M})^* $$ (recall $n = 2k-1$). Combining with Step I, we get a natural injection $$j : {\bf T}_{\tilde{\cal M}} \hookrightarrow {\cal V}^*, $$ which above a given $(X,s)$ is the inclusion of $F^{n-1}H^n(X)$ into $F^kH^n(X)$. Its transpose $$ j' : {\cal V} \twoheadrightarrow T^*\tilde{\cal M} $$ determines a closed $2$-form $\sigma := (j')^* \tilde{\sigma}$ on $\cal V$, where $\tilde{\sigma}$ is the standard symplectic form on $T^* \tilde{\cal M}$ (see example \ref{examples-symplectic-varieties}). By construction, the fibers of $\cal V$ over $\tilde{\cal M}$ are maximal isotropic with respect to this form. \medskip \underline{Step III}. We need to verify that $\tilde{\sigma}$ descends to $${\cal J}^k(\tilde{\cal X} /\tilde{\cal M} ) = {\cal V}/{\cal H}_n (\tilde{\cal X}/\tilde{\cal M}, {\bf Z}).$$ Equivalently, a locally constant integral cycle $$ \gamma \in \Gamma(B,{\cal H}_n(\tilde{\cal X} / \tilde{\cal M},{\bf Z})), $$ defined over some open subset $B$ of $\tilde{\cal M}$, gives a section of $\cal V$ on $B$; hence through $j'$, a $1$-form $\xi$ on $B$, and we need this $1$-form to be closed. Explicitly, if $a$ is a section of ${\bf T}_{\tilde{\cal M}}$ over $B$, we have $$ a \, \rfloor s \in \Gamma (B, F^{n-1}{\cal H}^n(\tilde{\cal X} /\tilde{\cal M})) \subset \Gamma(B, {\cal H}^n(\tilde{\cal X} /\tilde{\cal M})) $$ and $\xi$ is defined by: $$ \langle \xi,a \rangle := \int_\gamma \; (a \rfloor s). $$ Consider the function $$ \begin{array}{c} g : B \longrightarrow {\bf C} \\ \\ g(X,s) := \int_\gamma \; s. \\ \end{array} $$ If we set $$ a = \left. {\partial \over {\partial t}} \right|_{t=0} (X_t,s_t) $$ as in Step I, we get: $$ \langle dg,a \rangle = \left. {\partial \over {\partial t}} \right|_{t=0} \; g(X_t,s_t) = \left. {\partial \over {\partial t}} \right|_{t=0} \; \int_\gamma \; s_t = \int_\gamma \; (a\, \rfloor s) = \langle \xi, a \rangle, $$ so $\xi = dg$ is closed. \EndProof \bigskip \begin{rem} \label{rem-to-thm-cy-acihs} {\rm \begin{enumerate} \item[(1)] The most interesting case is clearly $n=3$, when $\tilde{{\cal J}}$ has an honest symplectic structure. The cubic field on $\tilde{\cal M}$ corresponding to this structure by lemma \ref{lemma-weak-cubic-cond-poisson} made its first appearance in \cite{BG} and is essentially the {\it Yukawa coupling}, popular among physicists and mirror-symmetry enthusiasts. At $(X,x) \in \tilde{\cal M}$ there is a natural cubic form on $H^1({\bf T}_x)$: $$ c : \otimes^3H^{1}({\bf T}_X) \rightarrow H^3(\Wedge{3}{\bf T}_X) = H^3(\omega^{-1}_X) \stackrel{\cdot s^2}{\rightarrow} H^3(\omega_X) \RightArrowOf{\int} {\bf C}, $$ which pulls back to the required cubic on ${\bf T}_{(X,s)}\tilde{{\cal M}}$. Hodge theoretically, this cubic can be interpreted as the third iterate of the infinitesimal variation of the periods, or the Hodge structure, of $X$ c.f. \cite{IVHS-I} and \cite{BG}. By Griffiths transversality, each tangent direction on ${\cal M}$, $\theta \in H^1({\bf T}_X)$, determines a linear map $$ \theta_i : H^{i,3-i} \longrightarrow H^{i-1,4-i} \quad \quad i = 3,2,1, $$ and clearly the composition $$ \theta_1 \circ \theta_2 \circ \theta_3 : H^{3,0} \longrightarrow H^{0,3} $$ becomes $c(\theta)$ when we use $s$ to identify $H^{3,0}$ and its dual $H^{0,3}$ with ${\bf C}$. \item[(2)] For $n=2k-1 \geq 5$, we get a closed $2$-form on $\tilde{\cal J}$ which is in general not of maximal rank. The corresponding cubic is identically $0$. Hodge theoretically, the ``cubic'' multiplies the gauge $s \in H^0(\omega_X)$ by two elements of $H^1({\bf T}_X)$ (landing in $H^{n-2,2}$) and then with an element of $F^kH^n$. When $k > 2$ there are too many $dz's$, so the product vanishes. \item[(3)] The symplectic form $\sigma$ which we constructed on $\tilde{\cal J}$ is actually exact. Recall that the natural symplectic form $\tilde{\sigma}$ on $T^*\tilde{\cal M}$ is exact: $\tilde{\sigma} = d\tilde{\alpha}$, where $\tilde{\alpha}$ is the action $1$-form. We obtained $\sigma$ by pulling $\tilde{\sigma}$ back to $(j')^*\tilde{\sigma}$ on ${\cal V}$, and observing that the latter is invariant under translation by locally constant integral cycles $\gamma$, hence descends to $\tilde{{\cal J}}$. Now a first guess for the anti-differential of $\sigma$ would be the $1$-form $(j')^* \tilde{\alpha}$; but this is {\it not} invariant under translation: if the cycle $\gamma$ corresponds, as in Step III of the proof, to a $1$-form $\xi$ on $\tilde{\cal M}$, then the translation by $\gamma$ changes $(j')^* \tilde{\alpha}$ by $\pi^*\xi$, where $\pi : {\cal V} \rightarrow \tilde{\cal M}$ is the projection. To fix this discrepancy, we consider the tautological function $f \in \Gamma({\cal O}_{\cal V})$ whose value at a point $(X,s,v) \in {\cal V}$ (where $(X,s) \in \tilde{\cal M}$ and $v \in F^kH^n(X)^*)$ is given by \begin{equation} \label{eq-moment-map} f(X,s,v) = v(s). \end{equation} This $f$ is linear on the fibers of $\pi$, so $df$ is constant on these fibers, and therefore translation by $\gamma$ changes $df$ by $\pi^*$ of a $1$-form on the base $\tilde{\cal M}$. This $1$-form is clearly $\xi$, so we conclude that \begin{equation} \label{eq-contact-structure} (j')^* \tilde{\alpha} - df \end{equation} is a global 1-form on ${\cal V}$ which is invariant under translation by each $\gamma$, hence descends to a $1$-form $\alpha$ on $\tilde{{\cal J}}$. It satisfies $d\alpha = \sigma$, as claimed. \item[(4)] Another way to see the exactness of $\sigma$ on $\tilde{{\cal J}}$ is to note that it comes from a {\it quasi-contact structure} $\kappa$ on ${\cal J}$. By a quasi-contact structure we mean a line subbundle $\kappa$ of $T^*{\cal J}$. It determines a tautological $1$-form on the ${\bf C}^*$-bundle $\tilde{\cal J}$ obtain from $\kappa$ by omitting its zero section. Hence, it determines also an exact $2$-form $\sigma$ on $\tilde{\cal J}$. We refer to the pair $(\tilde{\cal J},\sigma)$ as the {\em quasi-symplectification} of $({\cal J},\kappa)$. Conversely, according to \cite{AG}, page 78, a $2$-form $\sigma$ on a manifold $\tilde{\cal J}$ with a ${\bf C}^*$-action $\rho$ is the quasi-symplectification of a line subbundle of the cotangent bundle of the quotient ${\cal J}$ if and only if $\sigma$ is homogeneous of degree $1$ with respect to $\rho$ (and the contraction of $\sigma$ with the vector field generating $\rho$ is nowhere vanishing). In our case, there are two independent ${\bf C}^*$-actions on the total space of $T^*\tilde{{\cal M}} \simeq [F^{n-1}{\cal H}^n(\tilde{\cal X} / \tilde{\cal M} , {\bf C})]^*$: the ${\bf C}^*$-action on $\tilde{\cal M}$ lifts to an action $\bar{\rho}'$ on $T^*\tilde{\cal M}$, and there is also the action $\bar{\rho}''$ which commutes with the projection to $\tilde{\cal M}$ and is linear on the fibers. The symplectic form $\tilde{\sigma}$ is homogeneous of weight $0$ with respect to $\bar{\rho}'$ and of weight $1$ with respect to $\bar{\rho}''$, hence of weight $1$ with respect to $\bar{\rho} := \bar{\rho}' \cdot \bar{\rho}''$. Hence, $\tilde{\sigma}$ is the symplectification of a contact structure on $T^*\tilde{\cal M}/\bar{\rho} \simeq [F^{n-1}{\cal H}^n({\cal X} / {\cal M}, {\bf C})]^*$ (suppressing the gauge). Denote a point in $[F^{n-1}{\cal H}^n(\tilde{\cal X} / \tilde{\cal M}, {\bf C})]^*$ by $(X,s,\xi)$. The actions, for $t \in {\bf C}^*$, are given by: \[ \begin{array}{lcl} \bar{\rho}' &:& (X,s,\xi) \longmapsto (X, ts, t^{-1} \xi) \\ \bar{\rho}'' &:& (X,s,\xi) \longmapsto (X, s, t\xi) \\ \bar{\rho} &:& (X,s,\xi) \longmapsto (X, ts, \xi). \\ \end{array} \] The function $f$ on ${\cal V}$, given by (\ref{eq-moment-map}), is the pullback $(j')^*(\bar{f})$ of the function $\bar{f}$ on $T^*\tilde{{\cal M}}$ given by $$ \bar{f}(X,s,\xi) = \xi(s). $$ The symplectic structure $\tilde{\sigma}$ on $T^*\tilde{\cal M}$ takes the vector fields generating the actions $\bar{\rho}', \bar{\rho}''$, and $\bar{\rho}$ to the $1$-forms $-d\bar{f}, \tilde{\alpha}$ and $\tilde{\alpha} - d \bar{f}$, respectively. The $1$-form $\tilde{\alpha} - d \bar{f}$, which is homogeneous of degree $1$ with respect to $\bar{\rho}$, is the $1$-form canonically associated to the contact structure on $T^*\tilde{\cal M}/\bar{\rho} \simeq [F^{n-1}{\cal H}^n({\cal X} / {\cal M}, {\bf C})]^*$ (namely, the contraction of $\tilde{\sigma}$ with the vector field of $\bar{\rho}$.) Similarly, we have three action $\rho', \rho''$ and $\rho = \rho' \cdot \rho'' $ on the total space of ${\cal V} \simeq [F^k {\cal H}^n( \tilde{\cal X}/ \tilde{\cal M}, {\bf C})]^*$. The surjective homomorphism $j' : {\cal V} \rightarrow T^* \tilde{\cal M}$ is $(\rho', \bar{\rho}' ), (\rho'', \bar{\rho}'')$, and $(\rho ,\bar{\rho})$-equivariant. The $1$-form $\tilde{\alpha} - d \bar{f}$ pulls back to the $1$-form $(j')^*(\tilde{\alpha}) - df$ given by (\ref{eq-contact-structure}). Clearly, the action $\rho$ commutes with translations by ${\cal H}_n(\tilde{\cal X} / \tilde{\cal M}, {\bf Z})$. Since the $2$-form $(j')^*\tilde{\sigma}$ is also ${\cal H}_n(\tilde{\cal X} / \tilde{\cal M}, {\bf Z})$-equivariant, $(j')^*(\alpha) - df$ descends to a $1$-form $\alpha$ on $\tilde{\cal J}$. Clearly, $d\alpha = \sigma$ and $\alpha$ is homogeneous of degree $1$ with respect to the ${\bf C}^*$-action on $\tilde{\cal J}$. Hence $\alpha$ comes from a quasi-contact structure $\kappa$ on ${\cal J}$. \end{enumerate} } \end{rem} The Abel-Jacobi map of a curve to its Jacobian has an analogue for intermediate Jacobians. Let $Z$ be a codimensional-$k$ cycle in $X$, i.e. a formal linear combination $Z = \sum m_i Z_i$, with integer coefficients, of codimension $k$ subvarieties $Z_i \subset X$. If $Z$ is homologous to $0$, we can associate to it a point $\mu(Z) \in {\cal J}^k(X)$, as follows. Choose a real $(2n-2k+1)$-chain $\Gamma$ in $X$ whose boundary is $Z$, and let $\mu(Z)$ be the image in $$ {\cal J}^k(X) \approx (F^{n-k+1} H^{2n-2k+1}(X,{\bf C}))^* / H_{2n-2k+1}(X,{\bf Z}) $$ of the linear functional $$ \int_\Gamma \in (F^{n-k+1} H^{2n-2k+1}(X,{\bf C}))^* $$ sending a cohomology class represented by a harmonic form $\alpha$ to $\int_\Gamma \alpha$. Changing the choice of $\Gamma$ changes $\int_\Gamma$ by an integral class, so $\mu(Z)$ depends only on $Z$. This construction depends continuously on its parameters: given a family $\pi:{\cal X} \rightarrow B$ and a family ${\cal Z} \rightarrow B$ of codimension-$k$ cycles in the fibers which are homologous to $0$ in the fibers, we get the {\it normal function}, or Abel-Jacobi map $$ \mu : B \longrightarrow {\cal J}^k({\cal X}/B) $$ to the family of intermediate Jacobians of the fibers. Abstractly, a normal function $\nu:B \rightarrow {\cal J}^k({\cal X}/B)$ is a section satisfying the infinitesimal condition: \smallskip \noindent {\em Any lift $$ \tilde{\nu} : B \longrightarrow {\cal H}^n({\cal X}/B, {\bf C}) $$ of $$ \nu:B \rightarrow J^k({\cal X}/B) \simeq {\cal H}^n({\cal X}/B, {\bf C}) /[F^k{\cal H}^n({\cal X}/B, {\bf C}) + {\cal H}^n({\cal X}/B, {\bf Z})] $$ satisfies \begin{equation} \label{eq-infinitesimal-cond-normal-fn} \nabla \tilde{\nu} \in F^{k-1}{\cal H}^n({\cal X}/B, {\bf C}) \otimes \Omega^1_B \end{equation} or equivalently $$ (\nabla \tilde{\nu}, s) = 0 \; {\rm for \; any \; section} \; s \; {\rm of} \; F^{k+1} {\cal H}^n({\cal X}/B, {\bf C}) $$ where $\nabla \tilde{\nu}$ is the Gauss-Manin derivative of $\tilde{\nu}$. } \smallskip \noindent This condition is independent of the choice of the lift $\tilde{\nu}$ by Griffiths' transversality. It is satisfied by the Abel-Jacobi image of a relative codimension $k$-cycle (see \cite{griffith-normal-functions}). More generally, we can consider maps $$ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{B} \arrow[2]{e,t}{\mu} \arrow{se,b}{q} \node[2]{{\cal J}^k} \arrow{sw} \\ \node[2]{{\cal M}} \end{diagram} } $$ The pullback ${\cal J}^k({\cal X}/B) \rightarrow B$ of the relative intermediate Jacobian to $B$ has a canonical section $\nu:B \rightarrow {\cal J}^k({\cal X}/B)$. We will refer to the subvariety $\mu(B)$ as a {\it multivalued normal function} if $\nu : B \rightarrow {\cal J}^k({\cal X}/B)$ is a normal function. \begin{theorem} \label{thm-integrality-of-normal-fn} Let $\tilde{{\cal X}} \rightarrow \tilde{{\cal M}}$ be a complete family of gauged Calabi-Yau manifolds of dimension $n = 2k - 1 \geq 3$, $\tilde{{\cal J}} \rightarrow \tilde{{\cal M}}$ the relative intermediate Jacobian, $B \RightArrowOf{q} \tilde{{\cal M}}$ a base of a family ${\cal Z} \rightarrow B$ of codimension-$k$ cycles homologous to $0$ in the fibers of $q^*\tilde{{\cal X}} \rightarrow B$. Then i) the Abel-Jacobi image in $\tilde{\cal J}$ of $B$ is isotropic with respect to the quasi-symplectic form $\sigma$ of theorem \ref{thm-cy-acihs}. ii) Moreover, the Abel-Jacobi image is also integral with respect to the $1$-form $\alpha$ given by (\ref{eq-contact-structure}). \end{theorem} \medskip \noindent {\bf Proof.} i) We follow Step III of the proof of theorem \ref{thm-cy-acihs}. We thus think, locally in $B$, of $X$ as being a fixed $C^\infty$ manifold with variable complex structure $\bar{\partial}_b$, $n$-form $s_b$, and cycle $Z_b$, subject to the obvious compatibility. We choose a family $\Gamma_b$, $b \in B$ of $n$-chains whose boundary is $Z_b$, and consider the $1$-form $\xi$ on $B$ given at $b \in B$ by $\int_{\Gamma_b}$; we need to show that $\xi$ is closed. (The new feature here is that instead of the cycles $\gamma_b \in H_n(X_b,{\bf Z})$ we have chains, or relative cycles $\Gamma_b \in H_n(X_b, |Z_b|, {\bf Z})$, where $|Z_b|$ is the support of $Z_b$, which varies with $b$.) As before, we consider the function \begin{eqnarray*} g:B &\longrightarrow& {\bf C} \\ g(X,s,Z,\Gamma) &:=& \int_\Gamma \, s, \\ \end{eqnarray*} and we claim $\xi = dg$. This time, in the integral $\int_{\Gamma_b} \; s_b$, both the integrand and the chain depend on $b$. So if we take a normal vector $v$ to the supports $|Z_b|$ along $\Gamma_b$, we obtain two terms: $$ {\partial \over {\partial b}} \; \int_{\Gamma_b} \; s_b = \int_{\Gamma_b} {{\partial s} \over {\partial b}} + \int_{\partial\Gamma_b} \; (v \, \rfloor s_b). $$ In the second term, however, $s_b$ is of type $(n,0)$ with respect to the complex structure $\bar{\partial}_b$, so the contraction $v \, \rfloor s_b$ is of type $(n-1,0)$ regardless of the type of $v$. Since $\partial \Gamma_b = Z_b$ is of the type $(k-1, k-1)$, the second term vanishes identically, so we have $dg = \xi$ as desired. \smallskip \noindent ii) Integration $\int_\Gamma(\cdot)$ defines a section of ${\cal V} \simeq [F^{k} {\cal H}^n]^*$. The function $g$ on $B$ is the pullback via $\int_\Gamma(\cdot)$ of the function $f$ on ${\cal V}$ given by the formula (\ref{eq-moment-map}). Similarly, integration $\int_\Gamma(\cdot)$ defines the section $\xi$ of $T^*\tilde{{\cal M}} \simeq [F^{n-1}{\cal H}^n]^*$. The pullback of the tautological $1$-form $\tilde{{\alpha}}$ by $\xi$ is $\xi$ itself. The equation $\xi - dg = 0$ translates to the statement that the $1$-form $(j')^*\tilde{{\alpha}}-df$ vanishes on the section $\int_\Gamma(\cdot)$ of ${\cal V}$ (see formula (\ref{eq-contact-structure})). In particular, its descent $\alpha$ vanishes on the Abel-Jacobi image of ${\cal Z} \rightarrow B$. \EndProof \medskip Again, the most interesting case is $n=3$. When $B$ dominates the moduli space $\tilde{\cal M}$, i.e. for a multivalued choice of cycles on the general gauged Calabi-Yau of a given type, the normal function produces a Lagrangian subvariety of the symplectic $\tilde{{\cal J}}$, generically transversal to the fibers of the completely integrable system. \begin{rem} \label{rem-integrality-of-normal-fn} \begin{enumerate} {\rm \item[(1)] The result of Theorem \ref{thm-integrality-of-normal-fn} holds for every multi-valued normal function $\mu : B \rightarrow \tilde{\cal J}^k(\tilde{\cal X}/\tilde{\cal M})$ , not only for those coming from cycles. Given a vector field ${\partial \over {\partial b}}$ on $B$, a lift $\tilde{\nu} : B \rightarrow {\cal H}^n$, and any section $s$ of $F^{k+1} {\cal H}^n(\tilde{\cal X}/\tilde{\cal M},{\bf C})$ , the infinitesimal condition for normal functions (\ref{eq-infinitesimal-cond-normal-fn}) becomes \begin{equation} \label{eq-derivative-of-normal-fn} 0 = \left( \nabla_{\frac{\partial}{\partial b}}\tilde{\nu}, s \right) = {\partial \over {\partial b}}(\tilde{\nu},s) - \left( \tilde{\nu}, \nabla_{\frac{\partial}{\partial b}}s \right). \end{equation} When $s$ is the tautological gauge, ${\partial \over {\partial b}} (\tilde{\nu}, s)$ is the pullback of $df$ by the projection of $\tilde{\nu}$ to ${\cal V} \cong {\cal H}^n / F^k{\cal H}^n$ (where $f$ is defined by the equation (\ref{eq-moment-map})). Similarly, $\left(\tilde{\nu}, \nabla_{\frac{\partial}{\partial b}} s \right)$ is the contraction $\xi \; \rfloor \; {\partial \over {\partial b}}$ of the pullback $\xi$ of the tautological $1$-form $\tilde{\alpha}$ on $T^*\tilde{\cal M}$ by the composition $$ \tilde{\mu} : B \rightarrow {\cal H}^n(\tilde{\cal X}/\tilde{\cal M}) \rightarrow {\cal H}^n / F^2 \simeq [F^{n-1}{\cal H}^n ]^* \simeq T^*\tilde{\cal M}. $$ Thus, the infinitesimal condition for a normal function (\ref{eq-derivative-of-normal-fn}) implies that the image $\mu(B) \subset \tilde{{\cal J}}^k(\tilde{\cal X}/\tilde{\cal M})$ is integral with respect to the 1-form $\alpha$ (defined in (\ref{eq-contact-structure})). In the case of $CY$ $3$-folds $(n=3, k = 2)$ we see that the Legendre subvarieties of ${\cal J}^2 \rightarrow {\cal M}$ (i.e., the $\kappa$-integral subvarieties of maximal dimension $h^{2,1}$ where $\kappa$ is the contact structure of Remark \ref{rem-to-thm-cy-acihs}(4)) are precisely the multivalued normal functions. \item[(2)] Both the infinitesimal condition for a normal function (\ref{eq-infinitesimal-cond-normal-fn}) and the (quasi) contact structure $\kappa$ on the relative Jacobian ${\cal J}^k \rightarrow {\cal M}$ (see Remark \ref{rem-to-thm-cy-acihs}(4)) are special cases of a more general filtration of Pfaffian exterior differential systems on the relative intermediate Jacobian ${\cal J}^k \rightarrow {\cal M}$ of any family ${\cal X} \rightarrow {\cal M}$ of $n = 2k-1$ dimensional projective algebraic varieties. The tangent bundle $T{\cal J}^k$ has a canonical decreasing filtration (defined by (\ref{eq-filtration}) below) $$ T{\cal J}^k = F^0T{\cal J}^k \supset F^1T{\cal J}^k \supset \cdots \supset F^{k-1}T{\cal J}^k \supset 0. $$ The quotient $T{\cal J}^k/F^iT{\cal J}^k$ is canonically isomorphic to the pullback of the Hodge bundle ${\cal H}^n/F^i{\cal H}^n$. The $F^{k-1}T{\cal J}^k$ integral subvarieties are precisely the multi-valued normal functions. When ${\cal J}^k$ is the relative intermediate Jacobian of a family of $CY$ $n$-folds, the subbundle $F^1T{\cal J}^k$ is a hyperplane distribution on ${\cal J}^k$ which defines the (quasi) contact structure $\kappa$ of Remark \ref{rem-to-thm-cy-acihs}(4). When $n=3$, $k=2$, the filtration is a two step filtration $$ T{\cal J}^2 \supset F^1T{\cal J}^2 \supset 0 $$ and the $F^1T{\cal J}^2$-integral subvarieties are precisely the normal functions. The filtration $F^iT{\cal J}^k$, $0 \leq i \leq k-1$ is defined at a point $(b,y) \in {\cal J}^k$ over $b \in {\cal M}$ as follows: Choose a section $\tilde{\nu} : {\cal M} \rightarrow {\cal J}^k$ through $(b,y)$ with the property that any lift $\tilde{\nu} : {\cal M} \rightarrow {\cal X}^n({\cal H}/{\cal M}, {\bf C})$ of $\nu$ satisfies the horizontality condition \begin{equation} \label{eq-generalized-normal-fn} \nabla \tilde{\nu} \in F^{i} {\cal H}^n({\cal X}/{\cal M}, {\bf C}) \otimes \Omega^1_{\cal M} \end{equation} The section $\nu$ defines a splitting $$ T_{(b,y)}{\cal J}^k = T_b{\cal M} \oplus \left[ H^n(X_b,{\bf C}) / F^k H^n(X_b,{\bf C}) \right] $$ and the $i$-th piece of the filtration is defined by \begin{equation} \label{eq-filtration} F^i T_{(b,y)}{\cal J}^k := T_b{\cal M} \oplus \left[ F^i H^n(X_b,{\bf C}) / F^k H^n(X_b,{\bf C}) \right]. \end{equation} The horizontality condition (\ref{eq-generalized-normal-fn}) implies that the subspace $F^{i}T{\cal J}^k_{(b,y)}$ is independent of the choice of the section $\nu$ through $(b,y)$. Moreover, the subbundle $F^iT{\cal J}^k$ is invariant under translations by its integral sections, namely, by sections $\nu : {\cal M} \rightarrow {\cal J}^k$ satisfying the i-th horizontality condition. } \end{enumerate} \end{rem} We noted above that when $k=1$ the intermediate Jacobian ${\cal J}^k(X)$ becomes the connected component $\rm Pic^{0}(X)$ of the Picard variety. The generalization of the Picard variety itself is the {\it Deligne cohomology} group $D^k(X)$, cf. \cite{EZ}. This fits in an exact sequence \begin{equation} \label{eq-exact-seq-deligne-coho} 0 \rightarrow {\cal J}^k(X) \rightarrow D^k(X) \stackrel{p}{\rightarrow} H^{k,k}(X,{\bf Z}) \rightarrow 0, \end{equation} where the quotient is the group of Hodge $(k,k)$-classes, $$ H^{k,k}(X,{\bf Z}) := H^{k,k}(X,{\bf C}) \cap H^{2k}(X,.{\bf Z}). $$ Any codimension-$k$ cycle $Z$ in $X$ has an Abel-Jacobi image, or cycle class $\mu(Z)$ in $D^k(X)$. Its image $p(\mu(Z))$ is the cycle class of $Z$ in ordinary cohomology. Formally, $D^k(X)$ is defined as the hypercohomology ${\Bbb H}^{2k}$ of the following complex of sheaves on $X$ starting in degree $0$. $$ 0 \rightarrow {\bf Z} \rightarrow {\cal O}_X \rightarrow \Omega^1_X \rightarrow \cdots \rightarrow \Omega^{k-1}_X \rightarrow 0. $$ The forgetful map to ${\bf Z}$ is a map of complexes, with kernel the complex $$ 0 \rightarrow {\cal O}_X \rightarrow \Omega^1_X \rightarrow \cdots \rightarrow \Omega^{k-1}_X \rightarrow 0. $$ The resulting long exact sequence of hyper cohomologies gives (\ref{eq-exact-seq-deligne-coho}). Let $H^{k,k}_{alg}$ be the subgroup of $H^{k,k}(X,{\bf Z})$ of classes of algebraic cycles. (The Hodge conjecture asserts that $H^{k,k}_{alg}$ is of finite index in $H^{k,k}(X,{\bf Z})$.) The inverse image $$ D^k_{alg}(X) := p^{-1} (H^{k,k}_{alg}) $$ has an elementary description: it is the quotient of $$ {\cal J}^k(X) \times \; \{ {\rm codimension-}k \; {\rm algebraic \; cycles} \} $$ by the subgroup of codimension-$k$ cycles homologous to $0$, embedded naturally in the second component and mapped to the first by Abel-Jacobi. As $X$ varies in a family, the rank of $H^{k,k}(X,{\bf Z})$ can jump up (at those $X$ for which the variable vector subspace $H^{k,k}(X,{\bf C})$ happens to be in special position with respect to the ``fixed'' lattice $H^{2k}(X,{\bf Z})$). To obtain a well-behaved family of Deligne cohomology groups, we require that $$ H^{k,k}(X,{\bf C}) = H^{2k}(X,{\bf C}) . $$ For example, this holds for $k=1$ or $k=n-1$ if $h^{2,0} = 0$. In this case we also have $H^{k,k}_{alg} = H^{k,k}(X,{\bf Z})$ and hence $D^k_{alg}(X) = D^k(X)$, by the Lefschetz theorem on $(1,1)$-classes \cite{griffiths-harris}. \begin{corollary} \label{cor-contact-structure-extends-to-deligne-coho} Let ${\cal X} \rightarrow {\cal M}$ be a complete family of $3$-dimensional Calabi-Yau manifolds, $\tilde{\cal X} \rightarrow \tilde{\cal M}$ the corresponding gauged family. Let ${\cal D} \rightarrow {\cal M}$, $\tilde{\cal D} \rightarrow \tilde{\cal M}$, be their families of (second) Deligne cohomology groups, ${\cal J}$, $\tilde{\cal J}$ their relative intermediate Jacobians. Then there is a natural contact structure $\kappa$ on ${\cal D}$ with symplectification $\sigma = d \alpha$ on $\tilde{\cal D}$ with the following properties: \begin{enumerate} \item[(a)] $\sigma$, $\alpha$, and $\kappa$ restrict to the previously constructed structures on $\tilde{\cal J}$ and ${\cal J}$. \item[(b)] The fibration $\tilde{\cal D} \rightarrow \tilde{\cal M}$ is Lagrangian. \item[(c)] The multivalued normal functions of ${\cal D}$ (resp. $\tilde{\cal D}$) are precisely the $\kappa$-integral (resp. $\alpha$-integral) subvarieties. In particular, all multi-valued normal functions in $\tilde{\cal D}$ are isotropic. \end{enumerate} \end{corollary} \medskip \noindent {\bf Proof:} The contact structure $\kappa$ on ${\cal J}$ defines one on ${\cal J} \times \{ cycles \}$, which descends to ${\cal D}$ since the equivalence relation is $\kappa$-integral by remark \ref{rem-integrality-of-normal-fn}. \EndProof \bigskip The mirror conjecture of conformal field theory predicts that to a family ${\cal X} \rightarrow {\cal M}$ of Calabi-Yau three folds, with some extra data, corresponds a ``mirror'' family ${\cal X}' \rightarrow {\cal M}'$, cf. \cite{morrison-guide} for the details. A first property of the conjectural symmetry is that for $X \in {\cal M}$, $X' \in {\cal M}'$, $$ h^{2,1}(X) = h^{1,1}(X'), h^{1,1} = h^{2,1}(X'). $$ The conjecture goes much deeper, predicting a relation between the Yukawa cubic of ${\cal M}$ and the numbers of rational curves of various homology classes in a typical $X' \in {\cal M}'$. This has been used spectacularly in \cite{candelas} and subsequent works, to predict those numbers on a non-singular quintic hypersurface in ${\Bbb P}^4$ and in a number of other families. We wonder whether the conjecture could be reformulated and understood as a type of Fourier transform between the integrable systems on the universal Deligne cohomologies $\tilde{\cal D}$ and $\tilde{\cal D}'$ of the mirror families $\tilde{\cal M}$ and $\tilde{\cal M}'$. Note that the dimensions $h^{2,1}$ and $h^{1,1}$ which are supposed to be interchanged by the mirror, can be read off the continuous and discrete parts of the fibers of $\pi : \tilde{\cal D} \rightarrow \tilde{\cal M}$, respectively. One may try to imagine the mirror as a transform, taking these Lagrangian fibers over $\tilde{\cal M}$ (which encode the Yukawa cubic, as in Section \ref{subsec-cubic-condition}) to Lagrangian sections over $\tilde{\cal M}'$, which should somehow encode the numbers of curves in $X'$ via their Abel-Jacobi images. \newpage \section{The Lagrangian Hilbert scheme and its relative Picard} \label{ch8} \label{sec-lagrangian-sheaves} \subsection{Introduction} The Lagrangian Hilbert scheme of a symplectic variety $X$ parametrizes Lagrangian subvarieties of $X$. Its relative Picard parametrizes pairs $(Z,L)$ consisting of a line bundle $L$ on a Lagrangian subvariety $Z$. We use the cubic condition of chapter \ref{ch7} to construct an integrable system structure on components of the relative Picard bundle over the Lagrangian Hilbert scheme. We interpret the generalized Hitchin integrable system, supported by the moduli space of Higgs pairs over an algebraic curve (see Ch V), as a special case of this construction. Other examples discussed include: \begin{description} \item [a)] Higgs pairs over higher dimensional base varieties (example \ref{moduli-higgs-pairs-as-lagrangian-sheaves}), and \item [b)] Fano varieties of lines on hyperplane sections of a cubic fourfold (example \ref{subsec-fanos-of-cubics}). \end{description} Understanding the {\em global} geometry of such an integrable system requires a compactification and a study of its boundary. Our compactifications of the relative Picard are moduli spaces of sheaves and we study the symplectic structure at (smooth, stable) points of the boundary. \bigskip Let $X$ be a smooth projective symplectic algebraic variety, $\sigma$ an everywhere non degenerate algebraic 2-form on $X$. A smooth projective Lagrangian subvariety $Z_0$ of $X$ determines a component $\bar{B}$ of the Hilbert scheme parametrizing deformations of $Z_0$ in $X$. The component $\bar{B}$ consists entirely of Lagrangian subschemes. Its dense open subset $B$, parametrizing smooth deformations of $Z_0$, is a smooth quasi-projective variety \cite{ziv-ran-lifting,voisin}. Choose a very ample line bundle ${\cal O}_X(1)$ on $X$ and a Hilbert polynomial $p$. The relative Picard $h:{\cal M}^{p} \rightarrow B$, parametrizing line bundles with Hilbert polynomial $p$ which are supported on Lagrangian subvarieties of $X$, is a quasi-projective variety (see \cite{simpson-moduli}). If the Chern class $c_1(L_0) \in H^2(Z_0,{\Bbb Z})$ of a line bundle on $Z_0$ deforms as a $(1,1)$-class over the whole of $B$, then $L_0$ belongs to a component ${\cal M}$ of ${\cal M}^p$ which {\it dominates} the Hilbert scheme $B$. (By Griffiths' and Deligne's Theorem of the Fixed Part, \cite{schmid-vhs-the-singularities} Corollary 7.23, this is the case for example, if $c_1(L_0)$ belongs to the image of $H^2(X,{\Bbb Q})$). Such components ${\cal M}$ are integrable systems, in other words: \begin{theorem} \label{thm-symplectic-structure-on-relative-picard} There exists a canonical symplectic structure $\sigma_{\cal M}$ on the relative Picard bundle ${\cal M} \stackrel{h}{\rightarrow} B$ over the open subset $B$ of the Hilbert scheme of smooth projective Lagrangian subvarieties of $X$. The support map $h:{\cal M} \rightarrow B$ is a Lagrangian fibration. \end{theorem} The relative Picard over the Hilbert scheme of curves on a $K3$ or abelian surface is an example \cite{mukai}. In example \ref{subsec-fanos-of-cubics}, $X$ is a symplectic fourfold. \begin{rem} \label{rem-conditions-for-thm-symp-case} {\rm Theorem \ref{thm-symplectic-structure-on-relative-picard} holds in a more general setting where $X$ is a smooth projective algebraic variety, $\sigma$ is a meromorphic, generically non degenerate closed 2-form on $X$. We let $D_0$ denote its degeneracy divisor, $D_\infty$ its polar divisor, and set $D = D_0 \cup D_\infty$. Let $Z_0$ be a smooth projective Lagrangian subvariety of $X$ which does not intersect $D$. Denote by $B$ the open subset of a component of the Hilbert scheme parametrizing smooth deformations of $Z_0$ which stay in $X-D$. Then $B$ is smooth and Theorem 1 holds. A special case is when $X$ has a generically non-degenerate Poisson structure $\psi$. In this case $D_{\infty}$, the polar divisor of the inverse symplectic structure, is just the degeneracy locus of $\psi$, while $D_{0}$ is empty. The case where the subvariety $Z_0$ does intersect the degeneracy locus $D_{\infty}$ of the Poisson structure is also of interest. It is discussed below under the category of Poisson integrable systems. } \end{rem} The moduli space of 1-form valued Higgs pairs is related to the case where $X = {\Bbb P}(\Omega^1_Y \oplus {\cal O}_Y)$ is the compactification of the cotangent bundle of a smooth projective algebraic variety $Y$, and $D = {\Bbb P} \Omega^1_Y$ is the divisor at infinity (see example \ref{moduli-higgs-pairs-as-lagrangian-sheaves}). \smallskip The relative Picard bundle ${\cal M}$ is in fact also a Zariski open subset of a component of the moduli space of stable coherent sheaves on $X$ (see \cite{simpson-moduli} for the construction of the moduli space). Viewed in this way, Theorem \ref{thm-symplectic-structure-on-relative-picard} extends a result of Mukai \cite{mukai} for sheaves on a $K3$ or abelian surface. \noindent {\bf Theorem} \cite{mukai}: {\it Any component of the moduli space of simple sheaves on $X$ is smooth and has a canonical symplectic structure.} Kobayashi \cite{kobayashi} generalized the above theorem to the case of simple vector bundles on a (higher dimensional) compact complex symplectic manifold $(X, \sigma)$:\\ {\it The smooth part of the moduli space has a canonical symplectic structure.} In view of Theorem \ref{thm-symplectic-structure-on-relative-picard} and Kobayashi's result one might be tempted to speculate that every component of the moduli space of (simple) sheaves on a symplectic algebraic variety has a symplectic structure. This is {\em false}. In fact, some components are odd dimensional (see example \ref{example-odd-dimensional-moduli-spaces}). \bigskip Returning to our symplectic relative Picard ${\cal M}$, it is natural to ask whether its {\em compactification} is symplectic. More precisely: \begin{description} {\it \item [(i)] Does the symplectic structure extend to the smooth locus of the closure of the relative Picard ${\cal M}$ in the moduli space of stable (Lagrangian) sheaves? \item[(ii)] Which of these components $\bar{{\cal M}}$ admits a smooth projective birational model which is symplectic? } \end{description} A partial answer to (i) is provided in Theorem \ref{thm-extension-of-symplectic-str}. We provide a cohomological identification of the symplectic structure which extends as a 2-form $\sigma_{\bar{{\cal M}}}$ over the smooth locus of $\bar{{\cal M}}$. We do not know at the moment if the 2-form $\sigma_{\bar{{\cal M}}}$ is {\em non-degenerate} at every smooth point of $\bar{{\cal M}}$. The cohomological identification of $\sigma_{\bar{{\cal M}}}$ involves a surprisingly rich {\em polarized Hodge-like structure} on the algebra $\rm Ext^*_X(L,L)$ of extensions of a Lagrangian line bundle $L$ by itself as an $\StructureSheaf{X}$-module. Much of the above generalizes to Poisson integrable systems. Tyurin showed in \cite{tyurin-symplectic} that Mukai's theorem generalizes to Poisson surfaces: \noindent {\it The smooth part of any component of the moduli space of simple sheaves on a Poisson surface has a canonical Poisson structure. } \smallskip \noindent When the sheaves are supported as line bundles on curves in the surface, we get an integrable system. More precisely: \begin{theorem} \label{thm-poisson-structure-on-relative-picard} Let $(X,\psi)$ be a Poisson surface, $D_\infty$ the degeneracy divisor of $\psi$. Let $B$ be the Zariski open subset of a component of the Hilbert scheme of $X$ parametrizing smooth irreducible curves on $X$ which are not contained in $D_\infty$. Then \begin{description} \item [i)] $B$ is smooth, \item [ii)] the relative Picard bundle $h:{\cal M} \rightarrow B$ has a canonical Poisson structure $\psi_M$, \item [iii)] The bundle map $h : {\cal M} \rightarrow B$ is a Lagrangian fibration and \item[iv)] The symplectic leaf foliation of ${\cal M}$ is induced by the canonical morphism $B \rightarrow \; {\rm Hilb}_{D_\infty}$ sending a curve $Z$ to the subscheme $Z \cap D_\infty$ of $D_\infty$. \end{description} \end{theorem} The generalization to higher dimensional Poisson varieties is treated here under rather restrictive conditions on the component of the Lagrangian Hilbert scheme (see condition \ref{setup-for-poisson-case}). These restrictions will be relaxed in \cite{markman-lagrangian-sheaves}. \smallskip The rest of this chapter is organized as follows: In section \ref{subsec-hilbert-schemes} we review the deformation theory of Lagrangian subvarieties. The construction of the symplectic structure is carried out in section \ref{subsec-construction} where we prove Theorems \ref{thm-symplectic-structure-on-relative-picard} and \ref{thm-poisson-structure-on-relative-picard}. In section \ref{subsec-extension-to-singular-lagrangian-sheaves} we outline the extension of the symplectic structure to the smooth locus of the moduli space of Lagrangian sheaves (Theorem \ref{thm-extension-of-symplectic-str}). We discuss the examples of Higgs pairs and of Fano varieties of lines on cubics in section \ref{subsec-examples-of-lagrangian-sheaves}. \subsection{Lagrangian Hilbert Schemes}\label{subsec-hilbert-schemes} Let $X$ be a smooth $n$-dimensional projective algebraic variety, $Z \subset X$ a codimension $q$ subvariety and ${\cal O}_X(1)$ a very ample line bundle. The Hilbert polynomial $p$ of $Z$ is defined to be $$ p(n) := \chi\Big( {\cal O}_Z(n) \Big) := \sum (-1)^i \dim H^i\Big( Z, {\cal O}_Z(n) \Big). $$ Grothendieck proved in \cite{grothendieck-existence} that there is a projective scheme ${\rm Hilb}^p_X$ parametrizing all algebraic subschemes of $X$ with Hilbert polynomial $p$ and having all the expected functoriality and naturality properties. The Zariski tangent space $T_{[Z]}{\rm Hilb}^p_X$ at the point $[Z]$ parametrizing a subvariety $Z$ is canonically identified with the space of sections $H^0(Z,N_{X/Z})$ of the normal bundle (normal sheaf if $Z$ is singular). The scheme ${\rm Hilb}^p_X$ may, in general, involve pathologies. In particular, it may be non-reduced. A general criterion for the smoothness of the Hilbert scheme at a point $[Z]$ parameterizing a locally complete intersection subscheme $Z$ is provided by: \begin{definition} The semi-regularity map $\pi : H^1(Z,N_{X/Z}) \longrightarrow H^{q+1}(X,\Omega^{q-1}_X)$ is the dual of the natural homomorphism $$ \pi^* : H^{n-q-1} \Big( X,\Omega^{n-q+1}_X \Big) \longrightarrow H^{n-q-1} \Big(Z, \omega_Z \otimes N^*_{Z/X} \Big). $$ Here $\omega_Z \simeq \; \stackrel{q}{\wedge} N_{Z/X} \otimes \omega_X$ is the dualizing sheaf of $Z$ and the homomorphism $\pi^*$ is induced by the sheaf homomorphism \begin{equation} \label{eq-sheaf-homomorphism-inducing-the-semiregularity-map} \Omega^{n-q+1}_X \simeq \omega_X \otimes \stackrel{q-1}{\wedge}T_X \longrightarrow \omega_X \otimes \Wedge{q-1}N_{Z/X} \cong \omega_{Z}\otimes\Normal{Z}{X}^{*}. \end{equation} \end{definition} \smallskip \noindent {\bf Theorem} {\em (Severi-Kodaira-Spencer-Bloch \cite{kawamata}) If the semi-regularity map $\pi$ is injective, then the Hilbert scheme is smooth at $[Z]$. } \smallskip Together with a result of Ran it implies: \begin{corollary} Let $(X,\psi)$ be a Poisson surface with a degeneracy divisor $D_\infty$ (possibly empty). Let $Z \subset X$ be a smooth irreducible curve which is {\it not} contained in $D_\infty$. Then the Hilbert scheme ${\rm Hilb}^p_X$ is smooth at $[Z]$. \end{corollary} {\bf Proof:} The Poisson structure induces an injective homomorphism $ \phi : N^*_{Z/X} \ \hookrightarrow \ T_Z. $ If $Z$ intersects $D_\infty$ non-trivially then $N_{Z/X} \simeq \omega_Z(Z \cap D_\infty)$ and hence $H^1(Z, N_{X/Z}) = (0)$ and the semi-regularity map is trivially injective. Note that in our case $n=2$, $q=1$ and the dual of the semi-regularity map $$ \pi^* : H^{0} \Big(X,\omega_{X} \Big) \longrightarrow H^{0} \Big(Z, \omega_Z \otimes N^*_{Z/X} \Big) $$ is induced by the sheaf homomorphism \[ \omega_{X} \rightarrow \omega_{\restricted{X}{Z}} \rightarrow \omega_Z \otimes N^*_{Z/X} \] given by (\ref{eq-sheaf-homomorphism-inducing-the-semiregularity-map}). If $D_\infty = \emptyset$ $(X$ is symplectic) then $\omega_{X}$, $\omega_{\restricted{X}{Z}}$ and $\omega_Z \otimes N^*_{Z/X}$ are all trivial line bundles and hence both $\pi^*$ and the semi-regularity map are isomorphisms. If $D_\infty \cap Z = \emptyset$ but $D_\infty \not= \emptyset$ then $\pi$ fails to be injective but the result nevertheless holds by a theorem of Ran which we recall below (Theorem \ref{thm-voisin-ziv-ran}). \EndProof \bigskip The condition that the curve $Z$ is not contained in $D_\infty$ is necessary as can be seen by the following counterexample due to Severi and Zappa: \begin{example} {\rm (\cite{mumford-curves-on-surface} Section 22) Let $C$ be an elliptic curve, $E$ a nontrivial extension $0 \rightarrow E_1 \rightarrow E \rightarrow E_2 \rightarrow 0$, \ $E_i \simeq {\cal O}_C$ and $\pi : X = {\Bbb P}(E) \rightarrow C$ the corresponding ruled surface over $C$. Denote by $Z$ the section $s:C \rightarrow X$ given by the line subbundle $E_1 \subset E$. Let ${\cal O}_X(-1)$ be the tautological subbundle of $\pi^*E$. Then ${\cal O}_X(1)$ is isomorphic to the line bundle ${\cal O}_X(Z)$ and the canonical bundle $\omega_X$ is isomorphic to $\pi^*\Big(\omega_C\Big) \otimes {\cal O}_X(-2) \simeq {\cal O}_X(-2)$. $H^0(X, \stackrel{2}{\wedge} T_X)$ is thus isomorphic to $H^0(C,\rm Sym^2 E^*)$ which is one dimensional. It follows that $X$ has a unique Poisson structure $\psi$ up to a scalar factor. The divisor $D_\infty = 2Z$ is the degeneracy divisor of $\psi$. Clearly, $N_{Z/X} \simeq \pi^* T_C \simeq T_Z$ and hence $H^0(Z, N_{Z/X})$ is one dimensional. On the other hand, $Z$ has no deformations in $X$ (its self intersection is $0$ and a deformation $Z'$ of $Z$ will contradict the nontriviality of the extension $0 \rightarrow E_1 \rightarrow E \rightarrow E_2 \rightarrow 0$). \EndProof } \end{example} A curve $Z$ on a symplectic surface $X$ is automatically Lagrangian. In the higher dimensional case we replace the curve $Z$ by a Lagrangian subvariety. Lagrangian subvarieties of symplectic varieties have two pleasant properties: \begin{description} {\it \item [i)] The condition of being Lagrangian is both open and closed, \item [ii)] Their deformations are unobstructed.} \end{description} More precisely, we have: \begin{theorem} (Voisin \cite{voisin}, Ran \cite{ziv-ran-lifting}) \label{thm-voisin-ziv-ran} Let $X$ be a smooth projective algebraic variety, $\sigma$ a generically non degenerate meromorphic closed $2$-form, $D_\infty$ its polar divisor, $D_0$ its degeneracy divisor. Assume that $Z_0 \subset X - D_\infty - D_0$ is a smooth {\it projective} Lagrangian subvariety. Then \begin{description} \item [(i)] The subset of the Hilbert scheme ${\rm Hilb}^p_X$ parametrizing deformations of $Z_0$ in $X - D_\infty$ consists entirely of Lagrangian subvarieties. \item [(ii)] The Hilbert scheme is smooth at $[Z_0]$. \end{description} \end{theorem} \noindent {\bf Sketch of Proof:} \ \ (i) \ \ The Lagrangian condition is closed. Thus, it suffices to prove that the open subset of smooth deformations of $Z_0$ is Lagrangian. If $Z \subset X-D_\infty$ then $\sigma_{|_Z}$ is a closed holomorphic $2$-form and the cohomology class $[\sigma_{|_Z}]$ in $H^{2,0}(Z)$ vanishes if and only if $\sigma_{|_Z}$ is identically zero. Since $\sigma$ induces a {\it flat} section of the Hodge bundle of relative cohomology with ${\bf C}$-coefficients, then $[\sigma_{|_Z}] = 0$ is an open and closed condition. (ii). The symplectic structure $\sigma$ induces a canonical isomorphism $N_{Z/X} \simeq \Omega^1_Z$ for any Lagrangian projective smooth subvariety $Z \subset X - D_\infty - D_0$. Ran proved a criterion for unobstructedness of deformations: the $T^1$-lifting property (see \cite{ziv-ran-lifting,kawamata}). Let $S_n = {\rm Spec}({\bf C}[t]/t^{n+1})$. Any flat $(n+1)$-st order infinitesimal embedded deformation $Z_{n+1} \rightarrow S_{n+1}$ of $Z_0 = Z$ restricts canonically to an n-th order deformation $Z_n \rightarrow S_n$. In our context, the $T^1$-lifting property amounts to the following criterion: \noindent { \it Given any $(n+1)$-st order flat embedded deformation $Z_{n+1} \rightarrow S_{n+1}$, every extension \\ (a) of $Z_n \rightarrow S_n$ to a flat embedded deformation $\tilde{Z}_n \rightarrow S_n \times_{{\Bbb C}} S_1$ \noindent \smallskip lifts to an extension \noindent \smallskip (b) of $Z_{n+1} \rightarrow S_{n+1}$ to $\tilde{Z}_{n+1} \rightarrow S_{n+1} \times_{{\Bbb C}} S_1$. } \smallskip \noindent Extensions in (a) and (b) are classified by $T^1(Z_i / S_i) \cong H^0 (Z_i, {\cal N}_{\varphi_i/S_i})$ where $\varphi_i : Z_i \rightarrow S_i \times X$ is the canonical morphism and ${\cal N}_{\varphi_i/S_i} $ is the relative normal sheaf. Recall that the De Rham cohomology and its Hodge filtration can be computed using the algebraic De Rham complex (\ref{eq-the-quotient-of-the-algebraic-derham-complex}). Consequently, the discussion of part (i) applies in the infinitesimal setting to show that $T_{{\cal Z}_i/S_i}$ is {\em Lagrangian} as a subbundle of the pullback $(\varphi^i)^*T_X$ with respect to the non-degenerate $2$-form $(\varphi^i)^*(\sigma)$ on $(\varphi^i)^*T_X$. The relative normal sheaf is the quotient \[ 0 \rightarrow T_{{\cal Z}_i/S_i} \rightarrow (\varphi^i)^*T_X \rightarrow {\cal N}_{\varphi_i/S_i} \rightarrow 0. \] Hence the symplectic structure induces an isomorphism ${\cal N}_{\varphi_i/S_i} \simeq \Omega^1_{Z_i/S_i}$. By a theorem of Deligne, $H^0\Big( \Omega^1_{Z_i/S_i}\Big)$, and hence also $H^0\Big({\cal N}_{\varphi_i/S_i}\Big)$, is a free ${\cal O}_{S_i}$-module \cite{deligne-leray-degenerates}. Thus, $H^0\Big({\cal N}_{\varphi_{n+1}/S_{n+1}}\Big) \longrightarrow H^0\Big( {\cal N} _{\varphi_n/S_n}\Big)$ is surjective and the $T^1$-lifting property holds. \EndProof \bigskip Note that the naive analogue of the above theorem fails for Poisson varieties. In general, deformations of Lagrangian subvarieties need not stay Lagrangian. Consider for example $({\Bbb P}^{2n},\psi)$ where the Poisson structure $\psi$ is the extension of the standard (non degenerate) symplectic structure on ${\Bbb A}^{2n}\subset {\Bbb P}^{2n}$. The Lagrangian Grassmannian has positive codimension in $Gr(n+1, \; 2n+1)$. \bigskip \subsection{The construction of the symplectic structure} \label{subsec-construction} The construction of the symplectic structure on the relative Picard bundle is carried out in three steps: In Step I we reduce it to the construction of the symplectic structure on the relative $Pic^0$-bundle. In Step II we verify the cubic condition and thus construct the 2-form (or the 2-tensor in the Poisson case). In Step III we prove the closedness of the 2-form. \bigskip \noindent {\bf Step I:}\ \ \underline{Reduction to the $Pic^0$-Bundle Case}: The construction of a 2-form on the relative Picard bundle ${{\cal M}} \stackrel{h}{\rightarrow} B$ reduces to constructing it on its zero component ${{\cal M}}^0 \stackrel{h}{\rightarrow} B$, namely the $Pic^0$-bundle, by the following: \begin{proposition} Any closed 2-form $\sigma_{{{\cal M}}^0}$ on ${{\cal M}}^0$, with respect to which the zero section of ${{\cal M}}^0$ is Lagrangian, extends to a closed 2-form $\sigma_{{\cal M}}$ on the whole Picard bundle $h :{{\cal M}} \rightarrow B$. The extension $\sigma_{{\cal M}}$ depends canonically on $\sigma_{{{\cal M}}^0}$ and the polarization ${\cal O}_X(1)$ of $X$. \end{proposition} {\bf Proof:} The point is that Picard bundles are rationally split. For any polarized projective variety $(Z, {\cal O}_Z(1))$, we have the Lefschetz map $$ Lef: Pic \ Z \longrightarrow Alb \ Z $$ $$ [D] \mapsto \Big[ D \cap [{\cal O}_Z(1)]^{n-1} \Big] $$ inducing an isogeny $$ Lef^0 : Pic^0 \; Z \ \longrightarrow Alb^0 \; Z. $$ We can set $$ L_Z := \left\{ s \in Pic \; Z\; | \; \exists \ \ell,m, \ \ \ell \not= 0, \ {\rm such\; that}\; \ell \cdot Lef(s) = m \cdot Lef({\cal O}(1)) \right\}. $$ This is an extension of $H^{1,1}_{\bf Z}(Z)$ by the torsion subgroup $L_Z^{tor}$ of $Pic(Z)$. In a family ${\cal Z} \rightarrow B$, these groups form a subsheaf ${\cal L}$ of ${{\cal M}} := Pic({\cal Z}/B)$, intersecting ${{\cal M}}^0 := Pic^0({\cal Z}/B)$ in its torsion subsheaf ${\cal L}^0$. In our situation, the 2-form $\sigma_{{{\cal M}}^0}$ is ${\cal L}^0$-invariant, so it extends uniquely to an ${\cal L}$-invariant closed 2-form $\sigma_{{\cal M}}$ on ${{\cal M}}$. \EndProof \bigskip \noindent {\bf Step II:} \ \ \underline{Verification of the Cubic Condition:} In this step we construct the $2$-form (or $2$-vector) on the relative Picard bundle. In the next step we will prove that it is closed (respectively, a Poisson structure). Let $(X,\psi)$ be a smooth projective variety, $\psi$ a generically non-degenerate holomorphic Poisson structure. Denote by $D_\infty$ the degeneracy divisor of $\psi$. We will assume throughout this step that $Z \subset X$ is a smooth subvariety, $Z \cap (X - D_\infty)$ is non empty and Lagrangian, and \begin{condition} \label{setup-for-poisson-case} \begin{description} {\it \item [i)] $[Z]$ is a smooth point of the Hilbert scheme, and \item [ii)] all deformations of $Z$ in $X$ are Lagrangian.} \end{description} \end{condition} \noindent As we saw in the previous section, conditions i) and ii) hold in case $\psi$ is everywhere non-degenerate $((X, \psi^{-1})$ is a symplectic projective algebraic variety), and also in case $X$ is a surface. Such $[Z]$ vary in a smooth Zariski open subset $B$ of the Hilbert scheme and we denote by $h:{{\cal M}} \rightarrow B$ the relative $Pic^0$-bundle. Condition \ref{setup-for-poisson-case} can be relaxed considerably (see \cite{markman-lagrangian-sheaves}). Let \begin{equation} \label{eq-sheaf-homomorphism-induced-by-poisson-str} \phi :N^*_{Z/X} \hookrightarrow T_Z \end{equation} be the injective homomorphism induced by the Poisson structure $\psi$. Its dual $\phi^* : T^*_Z \rightarrow N_{Z/X}$ induces an injective homomorphism. \begin{equation} \label{eq-global-sections-homomorphism-induced-by-poisson-str} i:H^0 \Big(Z,T^*_Z \Big) \hookrightarrow H^0 \Big( Z,{\cal N}_{Z/X} \Big). \end{equation} The vertical tangent bundle $V := h_* {\cal T}_{{{\cal M}}/B}$ is isomorphic to the Hodge bundle ${\cal H}^{0,1}({\cal Z}/B)$. The polarization induces an isomorphism $V^* \simeq {\cal H}^{1,0}$. We get a global injective homomorphism $i : V^* \hookrightarrow T_B$. \begin{proposition} \label{prop-verification-of-cubic-condition} The homomorphism $i$ is induced by a canonical $2$-vector $\psi_{{\cal M}} \in \\ H^0({{\cal M}}, \; \stackrel{2}{\wedge} \; T_{{\cal M}})$ with respect to which $h:{{\cal M}}\rightarrow B$ is a Lagrangian fibration. (We do not assert yet that $\psi_{{\cal M}}$ is a Poisson structure). \end{proposition} {\bf Proof:} It suffices to show that $i$ satisfies the (weak) cubic condition, namely, that $dp \circ i$ comes from a cubic. The derivative of the period map \begin{equation} \label{eq-differential-of-period-map} dp:H^0 \Big(Z,N_{Z/X}\Big) \longrightarrow \ {\rm Hom} \Big( H^{1,0}(Z), H^{0,1}(Z) \Big) \simeq \Big[ H^{1,0}(Z)^* \Big]^{\otimes 2} \end{equation} is identified by the composition $$ H^0 \Big(Z,N_{Z/X}\Big) \stackrel{{\rm K-S}}{\longrightarrow} H^1(Z,T_Z) \stackrel{VHS}{\longrightarrow} \rm Sym^2 H^{1,0}(Z)^* $$ where K-S is the Kodaira-Spencer map given by cup product with the extension class of $T_{\restricted{X}{Z}}$: \begin{equation} \label{eq-extension-class-related-to-kodaira-spencer-map} \tau \in \; {\rm Ext}^1 \Big(N_{Z/X}, T_Z \Big) \simeq H^1 \Big(Z, N^*_{Z/X} \otimes T_Z \Big), \end{equation} and the variation of Hodge structure map VHS is given by cup product and contraction $$ H^1(Z, T_Z) \otimes H^0(Z,T^*_Z) \longrightarrow H^1(Z,{\cal O}_Z). $$ The composition $({\rm K\!\!-\!\!S})\; \circ \; i:H^0(Z,T^*_Z) \rightarrow H^1(Z,T_Z)$ is then given by cup product with the class $(\phi \; \otimes \;{\rm id})(\tau) \in H^1(Z, \; T_Z \otimes T_Z)$. We will show that $(\phi \; \otimes \;{\rm id})(\tau)$ is symmetric, that is, an element of $H^1(Z, \rm Sym^2T_Z)$. This would imply that $dp$, regarded as a section of $H^0 \Big(Z,N_{Z/X}\Big)^* \otimes \rm Sym^2 H^{1,0}(Z)^* \stackrel{i}{\cong} H^{1,0}(Z)^* \otimes \rm Sym^2 H^{1,0}(Z)^*$, is symmetric also with respect to the first two factors. The cubic condition will follow. Lemma \ref{lemma-alternating-two-forms-and-symmetric-extensions} below implies that $(\phi \otimes \;{\rm id})(\tau)$ is in $H^1(Z, \rm Sym^2T_Z)$ if and only if $\phi$ is induced by a section $\psi$ in $H^0\Big(Z, \; \stackrel{2}{\wedge} T_{X_{|Z}}\Big)$ with respect to which $Z$ is Lagrangian (i.e., $N^*_{Z/X}$ is isotropic). This is indeed the way $\phi$ was defined. \EndProof \begin{lem} \label{lemma-alternating-two-forms-and-symmetric-extensions} Let $T$ be an extension \begin{equation} \label{eq-exact-seq-with-symmetric-extension-class} 0 \rightarrow Z \rightarrow T \rightarrow N \rightarrow 0 \end{equation} of a vector bundle $N$ by a vector bundle $Z$. Then the following are equivalent for any homomorphism $\phi : N^* \rightarrow Z$. \smallskip \noindent i) The homomorphism $\phi$ is induced by a section $\psi \in H^0(\Wedge{2}{T})$ with respect to which $N^*$ is isotropic. \smallskip \noindent ii) The homomorphism $\phi_* := H^1(\phi\otimes 1) : H^1(N^*\otimes Z) \rightarrow H^1(Z \otimes Z)$ maps the extension class $\tau \in H^1(N^*\otimes Z)$ of $T$ to a symmetric class $\phi_*(\tau)\in H^1(\rm Sym^2 Z) \subset H^1(Z \otimes Z)$. \end{lem} \noindent {\bf Proof:} We argue as in the proof of the cubic condition (lemma \ref{lemma-weak-cubic-cond-poisson}). The extension (\ref{eq-exact-seq-with-symmetric-extension-class}) induces an extension \[ 0 \rightarrow \Wedge{2}{Z} \rightarrow F \rightarrow Z\otimes N \rightarrow 0, \] where $F$ is the subsheaf of $\Wedge{2}{T}$ of sections with respect to which $N^*$ is isotropic. The homomorphism $\phi$, regarded as a section of $Z\otimes N$, lifts to a section $\psi$ of $F$ if and only if it is in the kernel of the connecting homomorphism \[ \delta: H^0(Z\otimes N) \rightarrow H^1(\Wedge{2}{Z}). \] The latter is given by a) pairing with the extension class $\tau$ \[ (\cdot)_{*}\tau : H^0(Z\otimes N) \rightarrow H^1(Z\otimes Z), \] followed by b) wedge product \[ H^1(Z\otimes Z) \RightArrowOf{\wedge} H^1(\Wedge{2}Z). \] Thus, $\delta(\phi)$ vanishes if and only if $\phi_{*}(\tau)$ is in the kernel of $\wedge$, i.e., in $H^1(\rm Sym^2 Z)$. \EndProof \bigskip The identification of the cubic is particularly simple in the case of a curve $Z$ on a surface $X$. In that case Serre's duality identifies VHS with the dual of the multiplication map $$ \rm Sym^2 H^0(Z,\omega_Z) \stackrel{VHS^*}{\longrightarrow} H^0(Z, \omega_Z^{\otimes 2}). $$ The cubic $c \in \rm Sym^3H^0(Z,\omega_Z)^*$ is given by composing the multiplication $$ \rm Sym^3 H^0(Z, \omega_Z) \longrightarrow H^0(Z, \omega^{\otimes 3}_Z) $$ with the linear functional $$ (\phi \otimes \;{\rm id})(\tau) \in H^1(Z,T^{\otimes 2}_Z) \simeq H^0(Z, \omega_Z^{\otimes 3})^* $$ corresponding to the extension class $\tau$. In higher dimension (say $n$), the cubic depends on the choice of a polarization $\alpha \in H^{1,1}(X)$: \[ \rm Sym^3 H^0(Z, \Omega^1_Z) \rightarrow H^0(Z, \rm Sym^3\Omega^1_Z) \LongRightArrowOf{\phi_*(\tau)} H^1(Z,\Omega^1_Z) \LongRightArrowOf{\restricted{\alpha}{Z}^{n-1}} H^{n,n}(Z) \cong {\Bbb C}. \] The choice of $\alpha$ is implicitly made in the proof of proposition \ref{prop-verification-of-cubic-condition} when we identify $H^{0,1}(Z)$ with $H^{1,0}(Z)^*$ via the Lefschetz isomorphism (see (\ref{eq-differential-of-period-map})) {}. \bigskip \noindent {\bf Step III:} \ \ \underline{Closedness}: In this step we prove that the canonical $2$-vector $\psi_{{\cal M}}$ constructed in the previous step is a Poisson structure. We first prove it in the symplectic case and later indicate the modifications needed for the Poisson case (assuming condition \ref{setup-for-poisson-case} of the previous step). This completes the proof of Theorems \ref{thm-symplectic-structure-on-relative-picard} and \ref{thm-poisson-structure-on-relative-picard} stated in the introduction to this chapter. \noindent \underline{Symplectic Case:} We assume, for simplicity of exposition, that $(X,\sigma)$ is a smooth projective symplectic algebraic variety. The arguments apply verbatim to the more general setup involving a smooth projective algebraic variety $X$, a closed generically non-degenerate meromorphic $2$-form $\sigma$ on $X$ with degeneracy divisor $D_0$ and polar divisor $D_\infty$, and Lagrangian smooth projective subvarieties which do not intersect $D_0 \cup D_\infty$. We then have a non-degenerate $2$-tensor $\psi_{{\cal M}}$ on $h: {\cal M} \rightarrow B$ and hence a $2$-form $\sigma_{{\cal M}}$. The closedness of $\sigma_{{\cal M}}$ follows from that of $\sigma_X$ as we now show. A polarization of X induces a relative polarization on the universal Lagrangian subvariety \[ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{{\cal Z}} \arrow{s,l}{\pi} \arrow{e} \node{B\times X} \arrow{sw,r}{p_{B}} \arrow{se,r}{p_{X}} \\ \node{B} \node[2]{X.} \end{diagram} } \] The relative polarization induces an isogeny \[ {\divide\dgARROWLENGTH by 2 \begin{diagram}[B] \node{{\cal M}} \arrow{se} \arrow[2]{e} \node[2]{{\cal A}} \arrow{sw} \\ \node[2]{B} \end{diagram} } \] between the relative $Pic^0$-bundle and the relative Albanese $h : {\cal A} \rightarrow B$. Hence, a $2$-form $\sigma_{\cal A}$ on ${\cal A}$. Clearly, closedness of $\sigma_{{\cal M}}$ is equivalent to that of $\sigma_{\cal A}$. Since the question is local, we may assume that we have a section $\xi : B \rightarrow {{\cal Z}}$. We then get for each positive integer $t$ a relative Albanese map \[ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{{\cal Z}^{t}} \arrow[2]{e,t}{a_{t}} \arrow{se,l}{\pi} \node[2]{{\cal A}} \arrow{sw,l}{h} \\ \node[2]{B} \end{diagram} } \] from the fiber product over $B$ of $t$ copies of the universal Lagrangian subvariety ${\cal Z} \rightarrow B$. For a fixed subvariety $Z_b$ and points $(z_1, \dots, z_t) \in Z^t_b, \ \ a_t$ is given by integration $$ \sum^t_{i=1} \ \int^{z_i}_{\xi(b)}(\cdot) \ \ ({\rm modulo} \ H_1(Z_b, {\bf Z})) \in H^{1,0}(Z)^* \Big/ H_1(Z_b,{\bf Z}). $$ We may assume, by choosing $t$ large enough, that $a_t$ is surjective. Thus, closedness of $\sigma_{\cal A}$ is equivalent to closedness of $a^*_t(\sigma_{\cal A})$. The closedness of $a^*_t(\sigma_{\cal A})$ now follows from that of $\sigma_{X}$ by lemma \ref{lemma-pullback-of-symplectic-structure-from-albanese}. \begin{lem} \label{lemma-pullback-of-symplectic-structure-from-albanese} Let $\ell : {\cal Z}^t \rightarrow X^t$ be the natural morphism; $\sigma_{X^t}$ the product symplectic structure on $X^t$. Then, \begin{equation}\label{eq-pulled-back-symplectic-str} a^*_t(\sigma_{\cal A}) = \ell^*(\sigma_{X^t}) - \pi^*(\xi^t)^* \; \ell^* (\sigma_{X^t}) . \end{equation} \end{lem} {\bf Proof:} The fibers of $\pi : {\cal Z}^t \rightarrow B$ are isotropic with respect to the $2$-forms on both sides of the equation (\ref{eq-pulled-back-symplectic-str}). Hence, these $2$-forms induce (by contraction) homomorphisms $$ f_{\cal A}, f_X : T_{Z^t_b} \longrightarrow N^*_{Z^t_b/{\cal Z}^t}. $$ The section $\xi^t (B) \subset {\cal Z}^t$ is also isotropic with respect to the $2$-forms on both sides of equation (\ref{eq-pulled-back-symplectic-str}). Thus, equality in (\ref{eq-pulled-back-symplectic-str}) will follow from equality of the induced homomorphisms $f_{\cal A}, f_X$. Proving the equality $f_{\cal A}= f_X$ is a straightforward, though lengthy, unwinding of cohomological identifications. The relative normal bundle is identified as the pullback of the tangent bundle of the Hilbert scheme \[ N_{Z^t_b/{\cal Z}^t} \simeq {\cal O}_{Z^t_b} \otimes (T_b B) \simeq {\cal O}_{Z^t_b} \otimes H^0 \Big(Z_b, N_{Z_b/X} \Big). \] We will show that the duals of both $f_{\cal A}$ and $f_X$ \[ f_{\cal A}^*,f_X^*: {\cal O}_{Z^t_b} \otimes H^0 \Big(Z_b, N_{Z_b/X} \Big) \rightarrow T_{Z^t_b}^* \] are identified as the composition of \noindent i) the diagonal homomorphism $$ {\cal O}_{Z^t_b} \otimes H^0(Z_b, N_{Z_b/X}) \stackrel{\Delta}{\hookrightarrow} {\cal O}_{Z^t_b} \otimes \Big[ H^0(Z_b, N_{Z_b/X})\Big]^t \ \ \ \ \mbox{followed} \ \mbox{by} $$ \noindent ii) the evaluation map $$ e_t : {\cal O}_{Z^t_b} \otimes \Big[ H^0(Z_b,N_{Z_b/X})\Big]^t \simeq {\cal O}_{Z^t_b} \otimes H^0 \left( Z^t_b, N_{Z^t_b/X^t} \right) \longrightarrow N_{Z^t_b/X^t} \ \ \ \ \mbox{followed} \ \mbox{by} $$ \noindent iii) contraction with the $2$-form $\sigma_{X^t}$ \[ (\phi^{-1^*})^t : N_{Z^t_b/X^t} \stackrel{\sim}{\rightarrow} T_{Z^t_b}^* \] ($\phi$ is given by contraction with the Poisson structure (\ref{eq-sheaf-homomorphism-induced-by-poisson-str})). \smallskip \noindent \underline{Identification of $f_{\cal A}$}: (for simplicity assume t=1). The 2-form $\sigma_{\cal A}$ is characterized as the unique 2-form with respect to which the three conditions of lemma \ref{lemma-weak-cubic-cond-poisson} hold, i.e., i) ${\cal A} \rightarrow B$ is a Lagrangian fibration, ii) the zero section is Lagrangian, and iii) $\sigma_{\cal A}$ induces the homomorphism $$ H^0(\phi^{-1^*}) = i^{-1} : H^0 (Z, N_{Z/X}) \stackrel{\sim}{\longrightarrow} H^0(Z,T^*_Z). $$ Thus, $a^*(\sigma_{\cal A})$ induces $$ f_{\cal A}^* = \left( {\cal O}_{Z_b} \otimes H^0 \Big(Z_b,N_{Z_b/X} \Big) \stackrel{(i^{-1})}{\longrightarrow} {\cal O}_{Z_b} \otimes H^0 \Big( Z_b,T^*_{Z_b} \Big) \stackrel{da^*}{\longrightarrow} T_{Z_b}^* \right) $$ and the codifferential $da^*$ of the Albanese map is the evaluation map. \noindent \underline{Identification of $f_X$}: $(t=1)$ Both 2-forms $\ell^*(\sigma_X)$ and $\ell^*(\sigma_X)-\pi^* \xi^* \ell^*(\sigma_X)$ induce the same homomorphism $f_X^*: {\cal O}_{Z_b} \otimes T_bB \rightarrow T^*_{Z_b}$. This homomorphism is the composition $\phi^{-1^*} \circ \overline{(d\ell)}$, where $\overline{d\ell}$ is the homomorphism $N_{Z_b/{\cal Z}} \rightarrow N_{Z_b/X}$ induced by the differential of $\ell : {\cal Z} \rightarrow X$: \[ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{0} \arrow{e} \node{T_{Z_b}} \arrow{e} \arrow{s,l}{=} \node{(T_{\cal Z})_{|Z_b}} \arrow{e} \arrow{s,l}{d\ell} \node{H^0( Z_b, N_{Z_b/X}) \otimes {\cal O}_{Z_b}} \arrow{s,l}{\overline{d\ell}} \arrow{e} \node{0} \\ \node{0} \arrow{e} \node{T_{Z_b}} \arrow{e} \node{(\ell^*TX)_{|Z_b}} \arrow{e} \node{N_{Z_b/X}} \arrow{e} \node{0.} \end{diagram} } \] Clearly $\overline{d\ell}$ is given by evaluation. This completes the proof of lemma \ref{lemma-pullback-of-symplectic-structure-from-albanese}. \EndProof \bigskip As a simple corollary of lemma \ref{lemma-pullback-of-symplectic-structure-from-albanese} we have: \begin{corollary} \label{cor-canonical-symplectic-str-on-albanese} There exists a canonical symplectic structure $\sigma_{{\cal A}^t}$ on the relative Albanese of degree $t\in {\Bbb Z}$, depending canonically on the symplectic structure $\sigma_X$ (independent of the polarization ${\cal O}_X(1)$!) and satisfying, for $t \geq 1$, $$ a_t^* (\sigma_{{\cal A}^t}) = \ell^*(\sigma_{X^t}). $$ (the pullback to the fiber product ${\cal Z}^t := \times^t_B {\cal Z}$ via the Albanese map coincides with the pullback of the symplectic structure $\sigma_{X^t}$ on $X^t$). \end{corollary} {\bf Proof:} The $t=0$ case is proven. We sketch the proof of the $t \geq 1$ case. The $t \leq -1$ case is similar. Let $\xi$ be a local section of ${\cal Z}^t \rightarrow B$. Translation by the section $-a_t(\xi)$ of ${\cal A}^{-t}$ defines a local isomorphism $$ \tau_\xi : {\cal A}^t \longrightarrow {\cal A}^0. $$ Let $$ \sigma_{{\cal A}^t} \ := \ \tau^*_\xi (\sigma_{{\cal A}^0}) + h^*\xi^*\ell^*(\sigma_{X^t}). $$ We claim that $\sigma_{{\cal A}^t}$ is independent of $\xi$. This amounts to the identity \[ \tau^*_{(\xi_1 - \xi_2)}(\sigma_{{\cal A}^0}) = \sigma_{{\cal A}^0} - h^*[a_0(\xi_1-\xi_2)]^* \sigma_{{\cal A}^0} \] for any two sections $\xi_1,\xi_2$ of ${\cal Z}^t \rightarrow B$. \EndProof \bigskip \noindent \underline{Poisson Case}: (assuming condition \ref{setup-for-poisson-case}) Showing that the $2$-vector $\psi_{{\cal M}}$ constructed in step II is a Poisson structure, amounts to showing that \begin{lem} \label{lemma-involutive-distribution} $\psi_{{\cal M}}(T^*_{{\cal M}}) \subset T_{{\cal M}}$ is an involutive distribution, \end{lem} and \begin{lem} \label{lemma-closedness-poisson-case} the induced $2$-form on each symplectic leaf is closed. \end{lem} {\bf Sketch of Proof of Lemma \ref{lemma-involutive-distribution}}: Since $h : {{\cal M}} \rightarrow B$ is a Lagrangian fibration with respect to $\psi_{{\cal M}}$ (by proposition \ref{prop-verification-of-cubic-condition}), the distribution is the pullback of the distribution on the base $B$. The latter is induced by the image of the injective homomorphism $i: V^* \hookrightarrow T_B$ identified by (\ref{eq-global-sections-homomorphism-induced-by-poisson-str}) $$ i = H^0(\phi^*) : H^0(Z,T^*_Z) \hookrightarrow H^0(Z,N_{Z/X}). $$ Recall (\ref{eq-sheaf-homomorphism-induced-by-poisson-str}) that $\phi$, in turn, is induced by the Poisson structure $\psi_X$ on $X$. The involutivity now follows from that of $\psi_X(T^*_X) \subset T_X$ by a deformation theoretic argument. The details are omitted. \EndProof \bigskip In case $X$ is a surface, the degeneracy divisor $D_\infty$ of $\psi_X$ is a curve and $iH^0(Z, T^*_Z) \subset H^0(Z, N_{Z/X})$ is the subspace of all infinitesimal deformations of $Z$ which {\it fix} the divisor $Z \cap D_\infty$. Thus, the distribution $i(V^*)$ on the Hilbert scheme $B$ (as in the proof of lemma \ref{lemma-involutive-distribution}) corresponds to the foliation by level sets of the algebraic morphism \begin{eqnarray*} R : B &\longrightarrow& \; {\rm Hilb}(D_\infty) \\ Z &\longmapsto& Z \cap D_\infty. \end{eqnarray*} The higher dimensional case is analogous. The degeneracy divisor $D_\infty$ has an algebraic rank stratification $$ D_\infty = \bigcup^{n-1}_{r=0} \ D_\infty[2r] \qquad \qquad (\dim X = 2n). $$ Each rank stratum is foliated, local analytically, by symplectic leaves. The subspace $$ i:H^0(Z,T^*_Z) \subset H^0(Z,N_{Z/X}) $$ is characterized as the subspace of all infinitesimal deformations of $Z$ which deform the subscheme $Z \cap D_\infty[2r]$ fixing the image $f(Z \cap D_\infty[2r])$ with respect to any Casimir function $f$ on $D_\infty[2r]$. As an illustration, consider the case where $X$ is the logarithmic cotangent bundle $T^*_M (log(D))$ and $Z$ is a $1$-form with logarithmic poles along a divisor $D$ with normal crossing. In this case the residues induce the symplectic leaves foliation. \noindent {\bf Sketch of Proof of Lemma \ref{lemma-closedness-poisson-case}:} The proof is essentially the same as in the symplectic case. We consider an open (analytic) subset $B_1$ of a leaf in $B$, the universal Lagrangian subvariety $ {\divide\dgARROWLENGTH by 2 \begin{diagram} \node{{\cal Z}_1} \arrow{e,t}{\ell} \arrow{s} \node{X} \\ \node{B_1} \end{diagram} } $, and the relative Albanese $\begin{array}{c} {\cal A}\\ \downarrow \\ B_1 \end{array}. $ One has to choose the section $\xi:B_1 \rightarrow {\cal Z}_1$ outside $\ell^{-1}(D_\infty)$ and notice that the identity (\ref{eq-pulled-back-symplectic-str}) implies that the pullback $\ell^*(\sigma^t_X)$ of the {\it meromorphic} closed $2$-form $\sigma^t_X$ (inverse of the generically non-degenerate Poisson structure on the product of $t$ copies of $X$) is a {\it holomorphic} $2$-form on ${\cal Z}^t_1$ (because $a^*_t(\sigma_{\cal A})$ is) and that $a^*_t(\sigma_{\cal A})$ is {\it closed} (because $\ell^*(\sigma_{X^t})$ is). \EndProof \subsection {Partial compactifications: a symplectic structure on the moduli space of Lagrangian sheaves} \label{subsec-extension-to-singular-lagrangian-sheaves} We describe briefly in this section the extension of the symplectic structure on the relative Picard ${{\cal M}}$ to an algebraic $2$-form on the smooth locus of a partial compactification. For details see \cite{markman-lagrangian-sheaves}. For simplicity, we assume that $(X,\sigma)$ is a smooth $2n$-dimensional projective symplectic variety. We note that with obvious modifications, the extension of the $2$-form will hold in the setup $(X,\sigma,D_0,D_\infty)$ as in remark \ref{rem-conditions-for-thm-symp-case} allowing $\sigma$ to degenerate and have poles away from the support of the sheaves. When $X$ is a symplectic surface, some of these extensions give rise to smooth projective symplectic compactifications \cite{mukai}. These projective symplectic compactifications appear also in the higher dimensional case: \begin{example} {\rm A somewhat trivial reincarnation of a relative Picard of a linear system on a K3 surface $S$ as a birational model of a relative Picard of a Lagrangian Hilbert scheme over a higher dimensional symplectic variety $X$ is realized as follows. Let $X$ be the Beauville variety $S^{[n]}$ which is the resolution of the $n$-th symmetric product of $S$ provided by the Hilbert scheme of zero cycles of length $n$ \cite{beauville-zero-first-chern-class}. The symmetric powers $C^{[n]}$ of smooth curves on $S$ are smooth Lagrangian subvarieties of $S^{[n]}$. Components of the relative Picard over the smooth locus in the linear system $|C|$ are isomorphic to Zariski open subsets of components of the relative Picard over the Lagrangian Hilbert scheme of $S^{[n]}$. } \end{example} This leads us to speculate that genuinely new examples of smooth symplectic {\em projective} varieties will arise as birational models of moduli spaces of Lagrangian line bundles. (see section \ref{subsec-fanos-of-cubics} for new {\em quasiprojective} examples). We worked so far with a component ${{\cal M}} \rightarrow B$ of the relative Picard of the universal smooth Lagrangian subvariety ${\cal Z} \rightarrow B$ which dominates the corresponding component $\bar{B}$ of the Lagrangian Hilbert scheme (i.e., if $L$ is supported on $Z$, $c_1(L) \in H^{1,1}_{\bf Z}(Z)$ remains of type $(1,1)$ over $B$). Let $p(n) := \chi\Big(L \otimes_{{\cal O}_X} {\cal O}_X(n)\Big)$ be the Hilbert polynomial of a Lagrangian line bundle $L$ parametrized by ${{\cal M}}$. A construction of C. Simpson enables us to compactify ${{\cal M}}$ as an open subset of a component ${{\cal M}}^{ss}$ of the moduli space of equivalence classes of coherent semistable sheaves on $X$ with Hilbert polynomial $p$ \cite{simpson-moduli}. Denote by ${{\cal M}}^s$ the open subset of ${{\cal M}}^{ss}$ parametrizing isomorphism classes of stable sheaves, ${{\cal M}}^{s,sm}$ the smooth locus of ${{\cal M}}^s$. Then ${\cal M} \subseteq {{\cal M}}^{s,sm} \subseteq {{\cal M}}^s \subseteq {{\cal M}}^{ss}$. In addition, the moduli space ${{\cal M}}^s$ embeds as a Zariski open subset of the moduli space of simple sheaves \cite{altman-kleiman-compactifying}. The Zariski tangent space $T_{[L]}{{\cal M}}^s$ at a stable sheaf $L$ is thus canonically isomorphic to the Zariski tangent space of the moduli space of simple sheaves. The latter is identified as the group ${\rm Ext}^1_{{\cal O}_X}(L,L)$ of extensions $0 \rightarrow L \rightarrow E \rightarrow L \rightarrow 0$ of $L$ by $L$ as an ${\cal O}_X$-module. When $X$ is a $K3$ or abelian surface, Mukai's symplectic structure is given by the pairing $$ {\rm Ext}^1_{{\cal O}_X}(L,L) \otimes {\rm Ext}^1_{{\cal O}_X}(L,L) \stackrel{{\rm Yoneda}}{\longrightarrow} {\rm Ext}^2_{{\cal O}_X}(L,L) \stackrel{{\rm S.D.}}{\longrightarrow} \ {\rm Hom}_X(L,L\otimes \omega_X)^* \stackrel{id\otimes \sigma}{\longrightarrow} {\bf C} $$ (Composition of the Yoneda pairing, Serre Duality, and evaluation at $$id \otimes \sigma \in {\rm Hom}_X(L,L\otimes \omega_X)).$$ The generalization of Mukai's pairing requires the construction of a homomorphism, depending linearly on the Poisson structure $\psi$, \begin{equation} \label{eq-homomorphism-lifting-the-polarization-to-a-two-extension-class} y : H^{1,1}(X) \rightarrow {\rm Ext}^2_{{\cal O}_X}(L,L). \end{equation} It sends the Kahler class $\alpha := c_1 ({\cal O}_X(1)) \in H^{1,1}(X)$ to a $2$-extension class $y(\alpha) \in {\rm Ext}^2_{{\cal O}_X}(L,L)$. Once this is achieved, the $2$-form $\sigma_{{\cal M}}$ will become: \begin{equation} \label{eq-generalized-mukai-pairing} \begin{array}{l} {\rm Ext}^1_{{\cal O}_X}(L,L) \otimes {\rm Ext}^1_{{\cal O}_X}(L,L) \LongRightArrowOf{{\rm Yoneda}} {\rm Ext}^2_{{\cal O}_X}(L,L) \LongRightArrowOf{y(\alpha)^{n-1}} {\rm Ext}^{2n}_{{\cal O}_X}(L,L) \stackrel{S.D.}{\rightarrow} \\ {\rm Hom}_X(L,L\otimes \omega_X)^* \LongRightArrowOf{id\otimes \sigma^n} {\bf C}. \end{array} \end{equation} \begin{rem} \label{rem-polarized-hodge-like-structure} {\rm When $L$ is a line bundle on a smooth Lagrangian subvariety $Z$ the construction involves a surprisingly rich polarized Hodge-like structure on the algebra $$ {\rm Ext}^*_{{\cal O}_X}(L,L) := \bigoplus^{2n}_{k=0} {\rm Ext}^k _{{\cal O}_X}(L,L). $$ Since ${\rm Ext}^k_{{\cal O}_X}(L,\cdot)$ is the right derived functor of the composition $\Gamma \circ \; {\cal H}{\rm om}_{{\cal O}_X}(L,\cdot)$ of the Sheaf Hom and the global sections functors, there is a spectral sequence converging to ${\rm Ext}^k_{{\cal O}_X}(L,L)$ with $$ E^{p,q}_2 = H^p\Big( Z, {\cal E}xt^q_{{\cal O}_X}(L,L) \Big) $$ (see \cite{hilton-stammbach}). The sheaf of $q$-extensions ${\cal E}xt^q_{{\cal O}_X}(L,L)$ is canonically isomorphic to $\stackrel{q}{\wedge} N_{Z/X}$ and thus, via the symplectic structure, to $\Omega^q_Z$. We obtain a canonical isomorphism $E^{p,q}_2 \simeq H^{q,p}(Z)$ with the Dolbeault groups of $Z$. Notice however, that the Dolbeault groups appear in {\it reversed order} compared to their order in the graded pieces of the Hodge filtration on the cohomology ring $H^*(Z,{\bf C})$. } \end{rem} The construction of the $2$-extension class $y(\alpha)$ and hence of the generalized Mukai pairing (\ref{eq-generalized-mukai-pairing}) can be carried out for all coherent sheaves parametrized by ${{\cal M}}^{s,sm}$. We obtain: \begin{theorem} \label{thm-extension-of-symplectic-str} \cite{markman-lagrangian-sheaves} The symplectic structure $\sigma_{{\cal M}}$ on the relative Picard ${{\cal M}}$ extends to an algebraic $2$-form over the smooth locus ${{\cal M}}^{s,sm}$ of the closure of ${{\cal M}}$ in the moduli space of stable sheaves on $X$. It is identified by the pairing (\ref{eq-generalized-mukai-pairing}). \end{theorem} The non-degeneracy of $\sigma_{{\cal M}}$ at a point $[L] \in {{\cal M}}$ parametrizing a line bundle on a smooth Lagrangian subvariety $Z$ follows from the Hard Lefschetz theorem. We expect $\sigma_{{\cal M}}$ to be non degenerate everywhere on ${{\cal M}}^{s,sm}$. Finally we remark that the pairing (\ref{eq-generalized-mukai-pairing}) can be used to define a $2$-form on other components of the moduli space of stable sheaves on $X$. This $2$-form will, in general, be degenerate. In fact, some components are odd dimensional: \begin{example} \label{example-odd-dimensional-moduli-spaces} {\rm Consider an odd dimensional complete linear system $|Z|$ whose generic element is a smooth ample divisor on an abelian variety $X$ of even dimension $\ge 4$, with a symplectic structure $\sigma$. The dimension of the component of the Hilbert scheme parameterizing deformations of $Z$ is $\dim ({\rm Pic} \; X) + \dim|Z| = \dim X + \dim|Z|$. Since $h^{1,0}(Z) = h^{1,0}(X)$, the component of the moduli space of sheaves parameterizing deformations of the structure sheaf ${\cal O}_Z$, as an ${\cal O}_X$-module, is of dimension $2 \cdot \dim X + \dim |Z|$ which is odd. \EndProof } \end{example} It is the Hodge theoretic interpretation of the graded pieces of the spectral sequence of ${\rm Ext}^k_{{\cal O}_X}(L,L)$ for {\em Lagrangian} line bundles which assures the non degeneracy of $\sigma_{{\cal M}}$. \subsection{Examples} \label{subsec-examples-of-lagrangian-sheaves} \subsubsection{Higgs Pairs} \label{moduli-higgs-pairs-as-lagrangian-sheaves} In chapter \ref{ch9} we define the notion of a 1-form valued Higgs pair $(E,\varphi)$ over a smooth n-dimensional projective algebraic variety $X$. It consists of a torsion free sheaf $E$ over $X$ and a homomorphism $\varphi : E \rightarrow E \otimes \Omega^1_X$ satisfying the symmetry condition $\varphi \wedge \varphi = 0$. The moduli space Higgs$_X$ of semistable Higgs pairs of rank $r$ with vanishing first and second Chern classes may be viewed as the Dolbeault non-abelian first $GL_r({\bf C})$-cohomology group of $X$ (cf. \cite{simpson-higgs-bundles-and-local-systems} and theorem \ref{thm-higgs-pairs-and-representations-of-pi1-for-curves} when $X$ is a curve): Non-abelian Hodge theory introduces a hyperkahler structure on the smooth locus of the space ${{\cal M}}_{{\rm Betti}}$ of isomorphism classes of semisimple $GL_r({\bf C})$-representations of the fundamental group $\pi_1(X)$ of $X$ \cite{deligne-twistors,hitchin,simpson-internetional-congress}. The hyperkahler structure consists of a Riemannian metric and an action of the quaternion algebra ${\Bbb H}$ on the real tangent bundle with respect to which \begin{description} \item [(i)] the (purely imaginary) unit vectors $\{a|a \bar{a} = 1\}$ in ${\Bbb H}$ correspond to a (holomorphic) ${\Bbb P}^1$-family of integrable complex structures, \item [(ii)] the metric is Kahler with respect to these complex structures. \end{description} All but two of the complex structures are isomorphic to that of ${{\cal M}}_{{\rm Betti}}$, the two special ones are that of ${\rm Higgs}_X$ and its conjugate (${{\cal M}}_{{\rm Betti}}$ and Higgs$_X$ are diffeomorphic). The hyperkahler structure introduces a holomorphic symplectic structure $\sigma$ on the smooth locus of Higgs$_X$. In case $X$ is a Riemann surface, that symplectic structure is the one giving rise to the Hitchin integrable system of spectral Jacobians. Our aim is to interpret the symplectic structure on Higgs$_X$ as an example of a Lagrangian structure over the relative Picard of a Lagrangian component of the Hilbert scheme of the cotangent bundle $T^*_X$ of $X$. This interpretation will apply to the Hitchin system (where $\dim X = 1$). For higher dimensional base varieties $X$ it will apply only to certain particularly nice cases. See also \cite{biswas-a-remark} for a deformation theoretic study of the holomorphic symplectic structure. The spectral construction (proposition \ref{prop-ordinary-spectral-construction-higgs-pairs}) can be carried out also for Higgs pairs over a higher dimensional smooth projective variety $X$ (cf. \cite{simpson-moduli}). We have a one to one correspondence between \begin{description} \item [(i)] (Stable) Higgs pairs $(E,\varphi)$ on $X$ (allowing $E$ to be a rank $r$ torsion free sheaf) and \item [(ii)] (Stable) sheaves $F$ on the cotangent bundle $T^*_X$ which are supported on (pure) $n$-dimensional projective subschemes of $T^*_X$ which are finite, degree $r$, branched coverings (in a scheme theoretic sense) of $X$. \end{description} Projective subvarieties of $T^*_X$ which are finite over $X$ are called {\it spectral coverings}. Spectral coverings $\tilde{X}$ are necessarily Lagrangian since the symplectic form $\sigma$ on $T^*_X$, which restricts to a global exact 2-form on $\tilde{X}$, must vanish on $\tilde{X}$. Let $B$ be the open subset of a component of the Hilbert scheme of ${\Bbb P}(T^*_X \oplus {\cal O}_X)$ parametrizing degree $r$ smooth spectral coverings (closed subvarieties of ${\Bbb P}(T^*_X \oplus {\cal O}_X)$ which are contained in $T^*_X$). The above correspondence embeds components of the relative Picard ${\cal M} \rightarrow B$ as open subsets of components of the moduli spaces of stable rank $r$ Higgs pairs over $X$. Theorem \ref{thm-symplectic-structure-on-relative-picard} of this chapter implies \begin{corollary} \begin{description} \item [(i)] The open subset ${{\cal M}}$ of the moduli Higgs$_X$ of Higgs pairs over $X$ which, under the spectral construction, parametrizes line bundles on smooth spectral covers, has a canonical symplectic structure $\sigma_{\cal M}$ (we do not require the Chern classes of the Higgs pairs to vanish). \item [(ii)] The support morphism $h:{\cal M} \rightarrow B$ is a Lagrangian fibration. \end{description} \end{corollary} \begin{rem} \label{rem-bad-components} {\rm In general, when $\dim X > 1$, there could be components of the moduli spaces of Higgs pairs for which the open set ${{\cal M}}$ above is empty, i.e., \begin{enumerate} \item the spectral coverings of all Higgs pairs in this component are singular, or \item the corresponding sheaves on the spectral coverings are torsion free but not locally free. \end{enumerate} } \end{rem} \subsubsection{Fano Varieties of Lines on Cubic Fourfolds} \label{subsec-fanos-of-cubics} We will use theorem \ref{thm-symplectic-structure-on-relative-picard} to prove: \begin{example} \label{fano-varieties-of-cubics} Let $Y$ be a smooth cubic hypersurface in ${\Bbb P}^5$. The relative intermediate Jacobian ${\cal J} \rightarrow B$ over the family $B \subset | {\cal O}_{{\Bbb P}^5} (1)|$ of smooth cubic hyperplane sections of $Y$ is an algebraically completely integrable Hamiltonian system. \end{example} The statement follows from a description of the family ${\cal J} \rightarrow B$ as an open subset of the moduli space of Lagrangian sheaves on the Fano variety $X$ of lines on $Y$. A. Beauville and R. Donagi proved \cite{beau-donagi} that $X$ is symplectic (fourfold). Clemens and Griffiths proved in \cite{c-g} that the intermediate jacobian $J_b$ of a smooth hyperplane section $Y \cap H_b$ is isomorphic to the Picard $Pic^0 Z_b$ of the $2$-dimensional Fano variety $Z_b$ of lines on the cubic $3$-fold $Y \cap H_b$. C. Voisin observed that $Z_b$ is a Lagrangian subvariety of $X$ \cite{voisin}. Since $h^{1,0}(Y \cap H_b) = 5$, $B$ is isomorphic to a dense open subset of a component of the Hilbert scheme. In fact, using results of Altman and Kleiman, one can show that the corresponding component is isomorphic to $| {\cal O}_{{\Bbb P}^5} (1)|$ (see \cite{altman-kleiman-fano} Theorem 3.3 (iv)). Theorem \ref{thm-symplectic-structure-on-relative-picard} implies that the relative Picard ${{\cal M}} \rightarrow B$ has a completely integrable Hamiltonian system structure. The symplectic structure $\sigma_{\cal M}$ is defined also at the fiber of the relative Picard corresponding to a Fano variety $Z_b$ of lines on a hyperplane section $Y\cap H_b$ with an ordinary double point $x_{b}\in Y\cap H_b$ (Theorem \ref{thm-extension-of-symplectic-str}). In that case, we have a genus $4$ curve $C_b$ in $Z_b$ parametrizing lines through $x_{b}$. $Z_b$ is isomorphic to the quotient $S^{2}C_b/(C_{1}\sim C_{2})$ of the second symmetric product of $C_b$ modulo the identification of two disjoint copies of $C_b$ \cite{c-g}. It is not difficult to check that $\sigma_{{\cal M}}$ is {\em non-degenerate} also on the fiber $Pic^0(Z_b)$ of the relative Picard which is a ${\Bbb C}^{\times}$-extension of the Jacobian of genus $4$. The non-degeneracy of the symplectic structure implies that we get an {\em induced boundary integrable system} on the relative Picard of the family of genus $4$ curves \begin{equation} \label{equation-integrable-system-of-genus-four-curves} {\cal P}ic({\cal C}) \rightarrow (Y^{*}-\Delta) \end{equation} over the complement of the singular locus $\Delta$ of the dual variety of the cubic fourfold. It is interesting to note that the boundary integrable system (\ref{equation-integrable-system-of-genus-four-curves}) can not be realized as the relative Picard of a family of curves on a symplectic surface. If this were the case, the generic rank of the pullback $a^{*}(\sigma_{{\cal P}ic^1({\cal C})})$ of the symplectic structure from ${\cal P}ic^1({\cal C}) \rightarrow (Y^{*}-\Delta)$ to ${\cal C} \rightarrow (Y^{*}-\Delta)$ via the Abel-Jacobi map would be $2$. On the other hand, $a^{*}(\sigma_{{\cal P}ic^1({\cal C})})$ is equal to the pullback of the symplectic structure $\sigma_X$ on $X$ via the natural dominant map ${\cal C} \rightarrow X$ (corollary \ref{cor-canonical-symplectic-str-on-albanese}). Thus, its generic rank is $4$. The importance of this rank as an invariant of integrable systems supported by families of Jacobians is illustrated in an interesting recent study of J. Hurtubise \cite{hurtubise-local-geometry}. More examples of nonrigid Lagrangian subvarieties can be found in \cite{voisin,ye}. \newpage \section{Spectral covers} \label{ch9} \subsection{Algebraic extensions} \label{intro}\ \indent We have seen that Hitchin's system, the geodesic flow on an ellipsoid, the polynomial matrices system of Chapter \ref{ch4} , the elliptic solitons, and so on, all fit as special cases of the spectral system on a curve. In this final chapter, we consider some algebraic properties of the general spectral system. We are still considering families of Higgs pairs $ (E \;, \; \varphi:E \longrightarrow E \otimes K) $, but we generalize in three separate directions: \begin{enumerate} \item The base curve $C$ is replaced by an arbitrary complex algebraic variety $S$. The spectral curve $\widetilde{C}$ then becomes a spectral cover $\widetilde{S}\longrightarrow S$. \item The line bundle $K$ in which the endomorphism $\varphi$ takes its values is replaced by a vector bundle, which we still denote by $K$. (this requires an integrability condition on $\varphi$.) Equivalently, $\widetilde{S}$ is now contained in the total space $\Bbb{K}$ of a vector bundle over $S$. \item Instead of the vector bundle $E$ we consider a principal $G$-bundle $\cal G$, for an arbitrary complex reductive group $G$. The $G$-vector bundle $E$ is then recovered as $E := {\cal G} \times^{G} V$, given the choice of a representation $\rho : G \longrightarrow Aut(V)$. The twisted endomorphism $\varphi$ is replaced by a section of $K \otimes \bdl{ad}(\cal G)$. Even in the original case of $G = GL(n)$ one encounters interesting phenomena in studying the dependence of $\widetilde{S} := \widetilde{S}_V$, for a given $ ({\cal G} , \varphi)$, on the representation $V$ of $G$. \end{enumerate} We will see that essentially all {\em algebraic} properties (but not the {\em symplectic} structure) of the Hitchin system, or of the (line-bundle valued, $G=GL(n)$) spectral system on a curve, survive in this new context. In fact, the added generality forces the discovery of some symmetries which were not apparent in the original: \begin{itemize} \item Spectral curves are replaced by spectral covers. These come in several flavors: $\widetilde{S}_V,\widetilde{S}_{\lambda},\widetilde{S}_P$, indexed by representations of $G$, weights, and parabolic subgroups. The most basic object is clearly the {\em cameral} cover $\widetilde{S}$; all the others can be considered as associated objects. In case $G=GL(n)$, the cameral cover specializes not to our previous spectral cover, which has degree $n$ over $S$, but roughly to its Galois closure, of degree $n!$ over $S$. \item The spectral Picards, $Pic(\widetilde{S}_V)$ etc., can all be written directly in terms of the decomposition of $Pic(\widetilde{S})$ into Prym-type components under the action of the Weyl group $W$. In particular, there is a distinguished Prym component common to all the nontrivial $Pic(\widetilde{S}_V)$. The identification of this component combines and unifies many interesting constructions in Prym theory. \item The Higgs bundle too can be relieved of its excess baggage. Stripping away the representation $V$ as well as the values bundle $K$, one arrives (in subsection \ref{abstract_objects}) at the notion of abstract, principal Higgs bundle. The abelianization procedure assigns to this a spectral datum, consisting of a cameral cover with an equivariant bundle on it. \item There is a Hitchin map (\ref{BigHitchin}) which is algebraically completely integrable in the sense that its fibers can be naturally identified, up to a "shift" and a "twist", with the distinguished Pryms (Theorem \ref{main} ). \item The "shift" is a property of the group $G$, and is often nonzero even when $\widetilde{S}$ is etale over $S$, cf. Proposition \ref{reg.ss.equivalence}. The "twist", on the other hand, arises from the ramification of $\widetilde{S}$ over $S$, cf. formula (\ref{G twist}). \item The resulting abelianization procedure is local in the base $S$, and does not require particularly nice behavior near the ramification, cf. example \ref{nilpo}. It does require that $\varphi$ be regular (this means that its centralizer has the smallest possible dimension; for $GL(n)$, this means that each eigenvalue may have arbitrary multiplicity, but the eigen-{\em space} must be 1-dimensional), at least over the generic point of $S$. At present we can only guess at the situation for irregular Higgs bundles. \end{itemize} The consideration of general spectral systems is motivated in part by work of Hitchin \cite{hitchin-integrable-system} and Simpson \cite{simpson-moduli}. In the remainder of this section we briefly recall those works. Our exposition in the following sections closely follows that of \cite{MSRI}, which in turn is based on \cite{D2}, for the group-theoretic approach to spectral decomposition used in section (\ref{deco}), and on \cite{D3} for the Abelianization procedure, or equivalence of Higgs and spectral data, in section(\ref{abelianization}). Some of these results, especially in the case of a base curve, can also be found in \cite{AvM, BK, F, K, Me, MS, Sc}. \\ \noindent \underline{\bf Reductive groups} \nopagebreak \noindent Principal $G$-bundles {\cal G} for arbitrary reductive $G$ were considered already in Hitchin's original paper \cite{hitchin-integrable-system}. Fix a curve $C$ and a line bundle $K$. There is a moduli space ${\cal M}_{G,K}$ parametrizing equivalence classes of semistable $K$-valued $G$-Higgs bundles, i.e. pairs $({\cal G}, \varphi)$ with $\varphi \in K \otimes \bdl{ad}(\cal G)$. The Hitchin map goes to $$B:=\oplus_{i} H^0(K^{\otimes d_i}),$$ where the $d_i$ are the degrees of the $f_i$, a basis for the $G$-invariant polynomials on the Lie algebra $\frak g$. It is: \[ h: ({\cal G}, \varphi) \longrightarrow (f_i (\varphi))_{i}. \] When $K=\omega_C$, Hitchin showed \cite{hitchin-integrable-system} that one still gets a completely integrable system, and that this system is algebraically completely integrable for the classical groups $GL(n), SL(n), SP(n), SO(n).$ The generic fibers are in each case (not quite canonically; one must choose various square roots! cf. sections \ref{reg.ss} and \ref{reg}) isomorphic to abelian varieties given in terms of the spectral curves $\widetilde{C}$: \begin{center} \begin{equation} \begin{array}{cl} \label{Pryms for groups} GL(n)& \widetilde{C} \mbox{ has degree n over C, the AV is Jac(} \widetilde{C}). \\ SL(n)& \widetilde{C} \mbox{ has degree n over C, the AV is Prym(} \widetilde{C} / C). \\ SP(n)& \widetilde{C} \mbox{ has degree 2n over C and an involution } x \mapsto -x. \\ & \mbox{ The map factors through the quotient } \overline{C}. \nonumber \\ & \mbox{ The AV is } Prym( \widetilde{C} / \overline{C}). \nonumber \\ SO(n)& \widetilde{C} \mbox{ has degree n and an involution , with: } \\ & \bullet \mbox{ a fixed component, when n is odd.} \\ & \bullet \mbox{ some fixed double points, when n is even.} \\ & \mbox{ One must desingularize } \widetilde{C} \mbox{ and the quotient } \overline{C}, \\ & \mbox{and ends up with the Prym of the} \\ & \mbox{desingularized double cover.} \ \end{array} \end{equation} \end{center} For the exceptional group $G_2$, the algebraic complete integrability was verified in \cite{KP1}. A sketch of the argument for any reductive $G$ is in \cite{BK}, and a complete proof was given in \cite{F}. We will outline a proof in section \ref{abelianization} below. \\ \noindent \underline{\bf Higher dimensions} \nopagebreak \noindent A sweeping extension of the notion of Higgs bundle is suggested by the work of Simpson \cite{simpson-moduli}, which was already discussed in Chapter \ref{ch8}. To him, a Higgs bundle on a projective variety S is a vector bundle (or principal $G$-bundle \ldots) $E$ with a {\em symmetric}, $\Omega^1_S$-valued endomorphism \[ \varphi : E \longrightarrow E \otimes \Omega^1_S. \] Here {\em symmetric} means the vanishing of: \[ \varphi\wedge\varphi : E \longrightarrow E \otimes \Omega^2_S, \] a condition which is obviously vacuous on curves. Simpson constructs a moduli space for such Higgs bundles (satisfying appropriate stability conditions), and establishes diffeomorphisms to corresponding moduli spaces of connections and of representations of $\pi_1(S)$ . In our approach, the $\Omega^1$-valued Higgs bundle will be considered as a particular realization of an abstract Higgs bundle, given by a subalgebra of $ad(\cal{G})$. The symmetry condition will be expressed in the definition \ref{princHiggs} of an abstract Higgs bundle by requiring the abelian subalgebras of $ad(\cal{G})$ to be abelian. \subsection {Decomposition of spectral Picards} \label{deco} \subsubsection{The question}\ \indent Throughout this section we fix a vector bundle $K$ on a complex variety $S$, and a pair $({\cal G},\varphi)$ where ${\cal G}$ is a principal $G$-bundle on $S$ and $\varphi$ is a regular section of $K \otimes \bdl{ad}(\cal G)$. (This data is equivalent to the regular case of what we call in section \ref{abstract_objects} a $K$-valued principal Higgs bundle.) Each representation \[ \rho : G \longrightarrow Aut(V) \] determines an associated $K$-valued Higgs (vector) bundle \[ ( {\cal V} := {\cal G} \times^{G} V, \qquad{\rho}(\varphi)\ ), \] which in turn determines a spectral cover $\widetilde{S}_V \longrightarrow S$. The question, raised first in \cite{AvM} when $S={\bf P}^1$, is to relate the Picard varieties of the $\widetilde{S}_V$ as $V$ varies, and in particular to find pieces common to all of them. For Adler and van Moerbeke, the motivation was that many evolution DEs (of Lax type) can be {\em linearized} on the Jacobians of spectral curves. This means that the "Liouville tori", which live naturally in the complexified domain of the DE (and hence are independent of the representation $V$) are mapped isogenously to their image in $\mbox{Pic}(\widetilde{S}_V)$ for each nontrivial $V$ ; so one should be able to locate these tori among the pieces which occur in an isogeny decomposition of each of the $\mbox{Pic}(\widetilde{S}_V)$. There are many specific examples where a pair of abelian varieties constructed from related covers of curves are known to be isomorphic or isogenous, and some of these lead to important identities among theta functions. \begin{eg} \begin{em} Take $G=SL(4)$ . The standard representation $V$ gives a branched cover $\widetilde{S}_V \longrightarrow S$ of degree 4. On the other hand, the 6-dimensional representation $\wedge ^2 V$ (=the standard representation of the isogenous group $SO(6)$) gives a cover $ \stackrel{\approx}{S} \longrightarrow S$ of degree 6, which factors through an involution: \[ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S. \] One has the isogeny decompositions: \[ Pic \, (\widetilde{S}) \sim Prym(\widetilde{S} / S) \oplus Pic \,(S) \] \[ Pic \,(\stackrel{\approx}{S}) \sim Prym(\stackrel{\approx}{S} / \overline{S}) \oplus Prym(\overline{S} / S) \oplus Pic \,(S). \] It turns out that \[ Prym(\widetilde{S} / S) \sim Prym(\stackrel{\approx}{S} / \overline{S}) . \] For $S={\bf P}^1$, this is Recillas' {\em trigonal construction} \cite{R}. It says that every Jacobian of a trigonal curve is the Prym of a double cover of a tetragonal curve, and vice versa. \end{em} \end{eg} \begin{eg} \begin{em} Take $G=SO(8)$ with its standard 8-dimensional representation $V$. The spectral cover has degree 8 and factors through an involution, $ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S.$ The two half-spin representations $V_1, V_2$ yield similar covers \[ \stackrel{\approx}{S} _i \longrightarrow \overline{S} _i \longrightarrow S, \qquad i=1,2. \] The {\em tetragonal construction} \cite{D1} says that the three Pryms of the double covers are isomorphic. (These examples, as well as Pantazis' {\em bigonal construction} and constructions based on some exceptional groups, are discussed in the context of spectral covers in \cite{K} and \cite{D2}.) \end{em} \end{eg} It turns out in general that there is indeed a distinguished, Prym-like isogeny component common to all the spectral Picards, on which the solutions to Lax-type DEs evolve linearly. This was noticed in some cases already in \cite{AvM}, and was greatly extended by Kanev's construction of Prym-Tyurin varieties. (He still needs $S$ to be ${\bf P}^1$ and the spectral cover to have generic ramification; some of his results apply only to {\em minuscule representations}.) Various parts of the general story have been worked out recently by a number of authors, based on either of two approaches: one, pursued in \cite{D2,Me,MS}, is to decompose everything according to the action of the Weyl group $W$ and to look for common pieces; the other, used in \cite{BK,D3,F,Sc}, relies on the correspondence of spectral data and Higgs bundles . The group-theoretic approach is described in the rest of this section. We take up the second method, known as {\em abelianization}, in section~\ref{abelianization}. \subsubsection{Decomposition of spectral covers} \label{decomp covers} \indent The decomposition of spectral Picards arises from three sources. First, the spectral cover for a sum of representations is the union of the individual covers $\widetilde{S}_V$. Next, the cover $\widetilde{S}_V$ for an irreducible representation is still the union of subcovers $\widetilde{S}_{\lambda}$ indexed by weight orbits. And finally, the Picard of $\widetilde{S}_{\lambda}$ decomposes into Pryms. We start with a few observations about the dependence of the covers themselves on the representation. The decomposition of the Picards is taken up in the next subsection. \\ \noindent \underline{\bf Spectral covers} \nopagebreak \noindent Whenever a representation space $V$ of $G$ decomposes, $$V = \oplus V_i,$$ there is a corresponding decomposition $$ \widetilde{S}_V =\cup \widetilde{S}_{V_i}, $$ so we may restrict attention to irreducible representations $V$. There is an \mbox{\em infinite} collection (of irreducible representations $V := V_{\mu}$, hence) of spectral covers $\widetilde{S}_V$, which can be parametrized by their highest weights $\mu$ in the dominant Weyl chamber $\overline{C}$ , or equivalently by the $W$-orbit of extremal weights, in $\Lambda / W$. Here $T$ is a maximal torus in $G$, $\Lambda := Hom(T, {\bf C}^*)$ is the {\em weight lattice } (also called {\em character lattice }) for $G$, and $W$ is the Weyl group. Now $V_{\mu}$ decomposes as the sum of its weight subspaces $V_{\mu}^{\lambda}$, indexed by certain weights $\lambda$ in the convex hull in $\Lambda$ of the $W$-orbit of $\mu$. We conclude that each $\widetilde{S}_{V_{\mu}}$ itself decomposes as the union of its subcovers $\widetilde{S}_{\lambda}$, each of which involves eigenvalues in a given $W$-orbit $W{\lambda}$ . ($\lambda$ runs over the weight-orbits in $V_{\mu}$.) \\ \noindent \underline{\bf Parabolic covers} \nopagebreak \noindent There is a {\em finite} collection of covers $\widetilde{S}_P$, parametrized by the conjugacy classes in $G$ of parabolic subgroups (or equivalently by arbitrary dimensional faces $F_P$ of the chamber $\overline{C}$) such that (for general $S$) each eigenvalue cover $\widetilde{S}_{\lambda}$ is birational to some parabolic cover $\widetilde{S}_{P}$, the one whose open face $F_P$ contains ${\lambda}$. \\ \noindent \underline{\bf The cameral cover} \nopagebreak \noindent There is a $W$-Galois cover $\widetilde{S} \longrightarrow S$ such that each $\widetilde{S}_{P}$ is isomorphic to $\widetilde{S} / W_P$, where $W_P$ is the Weyl subgroup of $W$ which stabilizes $F_P$. We call $\widetilde{S}$ the {\em cameral cover} , since, at least generically, it parametrizes the chambers determined by $\varphi$ (in the duals of the Cartans). Informally, we think of $\widetilde{S} \longrightarrow S$ as the cover which associates to a point $s \in S$ the set of Borel subalgebras of $ad({\cal G})_s$ containing $\phi(s)$. More carefully, this is constructed as follows: There is a morphism ${\frak g}\longrightarrow {\frak t}/W$ sending $g \in {\frak g}$ to the conjugacy class of its semisimple part $g_{ss}$. (More precisely, this is $Spec$ of the composed ring homomorphism ${\bf C} [ {\frak t} ] ^{W} { \stackrel{\simeq}{\leftarrow}} {\bf C}[{\frak g}]^{G} \label{t/W} \hookrightarrow {\bf C}[{\frak g}]$.) Taking fiber product with the quotient map ${\frak t}\longrightarrow {\frak t}/W$, we get the cameral cover ${\tilde{\frak g}}$ of ${\frak g}$. The cameral cover $\widetilde{S} \longrightarrow S$ of a $K$-valued principal Higgs bundle on $S$ is glued from covers of open subsets in $S$ (on which $K$ and $\cal G$ are trivialized) which in turn are pullbacks by $\varphi$ of ${\tilde{\frak g}} \longrightarrow {\frak g} $. \subsubsection{Decomposition of spectral Picards}\ \indent The decomposition of the Picard varieties of spectral covers can be described as follows:\\ \noindent \underline{\bf The cameral Picard} \nopagebreak \noindent From each isomorphism class of irreducible $W$-representations, choose an integral representative $\Lambda _i$. (This can always be done, for Weyl groups.) The group ring ${\bf Z} [W]$ which acts on $Pic(\widetilde{S}) $ has an isogeny decomposition: \begin{equation}\label{regular rep} {\bf Z} [W] \sim \oplus _i \Lambda _i \otimes_{\bf Z} \Lambda _i^{*}, \end{equation} \noindent which is just the decomposition of the regular representation. There is a corresponding isotypic decomposition: \begin{equation}\label{cameral Pic decomposition} Pic(\widetilde{S}) \sim \oplus _i \Lambda _i \otimes_{\bf Z} Prym_{\Lambda _i}(\widetilde{S}), \end{equation} \noindent where \begin{equation}\label{def of Prym_lambda} Prym_{\Lambda _i}(\widetilde{S} ):= Hom_W (\Lambda _i , Pic(\widetilde{S})). \end{equation}\\ \noindent \underline{\bf Parabolic Picards} \nopagebreak \noindent There are at least three reasonable ways of obtaining an isogeny decomposition of $Pic(\widetilde{S}_P) $, for a parabolic subgroup $P \subset G$: \begin{itemize} \item The `Hecke' ring $Corr_P$ of correspondences on $\widetilde{S}_P$ over $S$ acts on $Pic(\widetilde{S}_P) $, so every irreducible integral representation $M$ of $Corr_P$ determines a generalized Prym $$ Hom_{Corr_P} (M, Pic(\widetilde{S}_P)), $$ and we obtain an isotypic decomposition of $Pic(\widetilde{S}_P)$ as before. \item $Pic(\widetilde{S}_P)$ maps, with torsion kernel, to $Pic(\widetilde{S})$, so we obtain a decomposition of the former by intersecting its image with the isotypic components $\Lambda _i \otimes_{\bf Z} Prym_{\Lambda _i}(\widetilde{S})$ of the latter. \item Since $\widetilde{S}_P$ is the cover of $S$ {\em associated} to the $W$-cover $\widetilde{S}$ via the permutation representation ${\bf Z} [W_P \backslash W]$ of $W$, we get an isogeny decomposition of $Pic(\widetilde{S}_P)$ indexed by the irreducible representations in ${\bf Z} [W_P \backslash W]$. \end{itemize} It turns out (\cite{D2},section 6) that all three decompositions agree and can be given explicitly as \begin{equation} \label{multiplicity spaces} \oplus _i M _i \otimes Prym_{\Lambda _i}(\widetilde{S}) \subset \oplus _i \Lambda _i \otimes Prym_{\Lambda _i}(\widetilde{S}),\qquad M_i := (\Lambda_i)^{W_P}. \end{equation} \noindent \underline{\bf Spectral Picards} \nopagebreak \noindent To obtain the decomposition of the Picards of the original covers $\widetilde{S}_V$ or $\widetilde{S}_{\lambda}$, we need, in addition to the decomposition of $Pic(\widetilde{S}_P)$, some information on the singularities. These can arise from two separate sources: \begin{description} \item[Accidental singularities of the $\widetilde{S}_{\lambda}$. ] For a sufficiently general Higgs bundle, and for a weight $\lambda$ in the interior of the face $F_P$ of the Weyl chamber $\overline{C}$, the natural map: $$ i_{\lambda}: \widetilde{S}_P\longrightarrow \widetilde{S}_{\lambda} $$ is birational. For the {\em standard} representations of the classical groups of types $A_n, B_n$ or $C_n$, this {\em is} an isomorphism. But for general ${\lambda}$ it is {\em not}: In order for $i_{\lambda}$ to be an isomorphism, ${\lambda}$ must be a multiple of a fundamental weight, cf. \cite{D2}, lemma 4.2. In fact, the list of fundamental weights for which this happens is quite short; for the classical groups we have only: $\omega_1$ for $A_n, B_n$ and $C_n$, $\omega_n$ (the dual representation) for $A_n$, and $\omega_2$ for $B_2$. Note that for $D_n$ the list is {\em empty}. In particular, the covers produced by the standard representation of $SO(2n)$ are singular; this fact, noticed by Hitchin In \cite{hitchin-integrable-system}, explains the need for desingularization in his result~(\ref{Pryms for groups}). \item[Gluing the $\widetilde{S}_{V}$. ] In addition to the singularities of each $i_{\lambda}$, there are the singularities created by the gluing map $\amalg_{\lambda} \widetilde{S}_{\lambda} \longrightarrow \widetilde{S}_V$. This makes explicit formulas somewhat simpler in the case, studied by Kanev \cite{K}, of {\em minuscule} representations, i.e. representations whose weights form a single $W$-orbit. These singularities account, for instance, for the desingularization required in the $SO(2n+1)$ case in (\ref{Pryms for groups}). \end{description} \subsubsection{The distinguished Prym} \label{distinguished}\ \indent Combining much of the above, the Adler--van Moerbeke problem of finding a component common to the $Pic(\widetilde{S}_V)$ for all non-trivial $V$ translates into: \\ \begin{em} Find the non trivial irreducible representations $\Lambda_i $ of $W$ which occur in ${\bf Z} [W_P \backslash W] $ with positive multiplicity for all proper Weyl subgroups $W_P \subsetneqq W.$ \end{em} \\ It is easy to see that for arbitrary finite groups $W$, or even for Weyl groups $W$ if we allow arbitrary rather than Weyl subgroups $W_P$, there may be no common factors \cite{D2}. For example, when $W$ is the symmetric group $S_3$ (=the Weyl group of $GL(3)$) and $W_P$ is $S_2$ or $A_3$, the representations ${\bf Z} [W_P \backslash W] $ are 3 or 2 dimensional, respectively, and have only the trivial representation as common component. In any case, our problem is equivalent (by Frobenius reciprocity or (\ref{multiplicity spaces})) to \\ \begin{em} Find the irreducible representations $\Lambda_i $ of W such that for every proper Weyl subgroup $W_P \subsetneqq W, $ the space of invariants $M_i := (\Lambda_i)^{W_P} $ is non-zero. \end{em} \\ One solution is now obvious: the {\em{reflection representation}} of $W$ acting on the weight lattice $\Lambda$ has this property. In fact, $\Lambda^{W_P}$ in this case is just the face $F_P$ of $\overline{C}$. The corresponding component $Prym_{\Lambda }(\widetilde{S})$ , is called {\em{the distinguished Prym}.} We will see in section \ref{abelianization} that its points correspond, modulo some corrections, to Higgs bundles. For the classical groups, this turns out to be the only common component. For $G_2$ and $E_6$ it turns out (\cite{D2}, section 6) that a second common component exists. The geometric significance of points in these extra components is not known. As far as we know, the only component other than the distinguished Prym which has arisen `in nature' is the one associated to the 1-dimensional sign representation of $W$, cf. \cite{KP2}. \subsection {Abelianization}\label{abelianization} \subsubsection{Abstract vs. $K$-valued objects}\label{abstract_objects}\ \indent We want to describe the abelianization procedure in a somewhat abstract setting, as an equivalence between {\em{principal Higgs bundles}} and certain {\em spectral data}. Once we fix a {\em{values}} vector bundle $K$, we obtain an equivalence between {\em $K$-valued principal Higgs bundles} and {\em K-valued spectral data}. Similarly, the choice of a representation $V$ of $G$ will determine an equivalence of {\em $K$-valued Higgs bundles} (of a given representation type) with $K$-valued spectral data. As our model of a $W$-cover we take the natural quotient map $$G/T \longrightarrow G/N $$ and its partial compactification \begin{equation} \overline{G/T} \longrightarrow \overline{G/N}. \label{partial compactification} \end{equation} Here $T \subset G$ is a maximal torus, and $N$ is its normalizer in $G$. The quotient $G/N$ parametrizes maximal tori (=Cartan subalgebras) $\frak{t}$ in $\frak{g}$, while $G/T$ parametrizes pairs ${\frak t \subset \frak b}$ with ${\frak b \subset \frak g}$ a Borel subalgebra. An element $x \in {\frak g}$ is {\em regular} if the dimension of its centralizer ${\frak c \subset \frak g}$ equals $\dim{T}$ (=the rank of $\frak{g}$). The partial compactifications $ \overline{G/N}$ and $ \overline{G/T}$ parametrize regular centralizers ${\frak c }$ and pairs ${\frak c \subset \frak b}$, respectively. In constructing the cameral cover in section \ref{t/W}, we used the $W$-cover $\frak t \longrightarrow \frak t / W$ and its pullback cover ${ \widetilde{\frak g} \longrightarrow \frak g}$. Over the open subset $\frak g_{reg}$ of regular elements, the same cover is obtained by pulling back (\ref{partial compactification}) via the map $\alpha : \frak g_{reg} \longrightarrow \overline{G/N}$ sending an element to its centralizer: \begin{equation} \label{commutes} \begin{array}{lccccc} \frak t & \longleftarrow & \widetilde{\frak g}_{{reg}} & \longrightarrow & \overline{G/T} & \\ \downarrow & &\downarrow & & \downarrow & \\ \frak t /W & \longleftarrow & {\frak g}_{{reg}} & \stackrel{\alpha}{\longrightarrow} & \overline{G/N} &. \end{array} \end{equation} When working with $K$-valued objects, it is usually more convenient to work with the left hand side of (\ref{commutes}), i.e. with eigen{\em values}. When working with the abstract objects, this is unavailable, so we are forced to work with the eigen{\em vectors}, or the right hand side of (\ref{commutes}). Thus: \begin{defn} An abstract {\em cameral cover} of $S$ is a finite morphism $\widetilde{S} \longrightarrow S$ with $W$-action, which locally (etale) in $S$ is a pullback of (\ref{partial compactification}). \\ \end{defn} \begin{defn} A {\em $K$-valued cameral cover} ($K$ is a vector bundle on $S$) consists of a cameral cover $\pi : \widetilde{S} \longrightarrow S$ together with an $S$-morphism \begin{equation} \widetilde{S} \times \Lambda \longrightarrow \Bbb{K} \label{K-values} \end{equation} which is $W$-invariant ($W$ acts on $ \widetilde{S} , \Lambda,$ hence diagonally on $\widetilde{S} \times \Lambda $ ) and linear in $\Lambda$. \\ \end{defn} We note that a cameral cover $\widetilde{S}$ determines quotients $\widetilde{S}_P$ for parabolic subgroups $P \subset G$. A $K$-valued cameral cover determines additionally the $\widetilde{S}_{\lambda}$ for $\lambda \in \Lambda$, as images in $\Bbb{K}$ of $\widetilde{S} \times \{ \lambda \}$. The data of (\ref{K-values}) is equivalent to a $W$-equivariant map $\widetilde{S} \longrightarrow \frak{t}\otimes_{\bf C} K.$ \begin{defn} \label{princHiggs} A $G$-principal Higgs bundle on $S$ is a pair ($\cal{G}, \bdl{c})$ with $\cal{G}$ a principal $G$-bundle and $\bdl{c} \subset ad(\cal{G})$ a subbundle of regular centralizers. \\ \end{defn} \begin{defn} A $K$-valued $G$-principal Higgs bundle consists of $( \cal{G}, \bdl{c} )$ as above together with a section $\varphi$ of $\bdl{c} \otimes K$. \end{defn} A principal Higgs bundle $(\cal{G}, \bdl{c})$ determines a cameral cover $\widetilde{S}\longrightarrow S$ and a homomorphism $\Lambda \longrightarrow \mbox{Pic}(\widetilde{S}).$ Let $F$ be a parameter space for Higgs bundles with a given $\widetilde{S}$. Each non-zero $\lambda \in \Lambda$ gives a non-trivial map $F\longrightarrow \mbox{Pic}(\widetilde{S})$. For $\lambda$ in a face $F_P$ of $\overline{C}$, this factors through $\mbox{Pic}(\widetilde{S}_P)$. The discussion in section \ref{distinguished} now suggests that $F$ itself should be given roughly by the distinguished Prym, $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})). $$ It turns out that this guess needs two corrections. The first correction involves restricting to a coset of a subgroup; the need for this is visible even in the simplest case where $\widetilde{S}$ is etale over $S$, so $(\cal{G}, \bdl{c})$ is everywhere regular and semisimple (i.e. $ \bdl{c}$ is a bundle of Cartans.) The second correction involves a twist along the ramification of $\widetilde{S}$ over $S$. We explain these in the next two subsections. \subsubsection{The regular semisimple case: the shift} \label{reg.ss} \begin{eg} \label{unramified} \begin{em} Fix a smooth projective curve $C$ and a line bundle $K \in \mbox{Pic}(C)$ such that $K^{\otimes 2} \approx \cal{O}_C.$ This determines an etale double cover $\pi : \widetilde{C} \longrightarrow C$ with involution $i$, and homomorphisms \begin{center} $\begin{array}{cccccc} \pi^{*} &:& \mbox{Pic}(C) &\longrightarrow &\mbox{Pic}(\widetilde{C}) &, \\ \mbox{Nm} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow &\mbox{Pic}(C) &, \\ i^{*} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow &\mbox{Pic}(\widetilde{C}) &, \end{array}$ \end{center} satisfying $$ 1+i^{*} = \pi^* \circ \mbox{Nm}. $$ \begin{itemize} \item For $G = GL(2)$ we have $\Lambda = \bf{Z} \oplus \bf{Z}$, and $W = {\cal{S}}_{2}$ permutes the summands, so $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{C})) \approx \mbox{Pic}(\widetilde{C}). $$ And indeed, the Higgs bundles corresponding to $\widetilde{C}$ are parametrized by $\mbox{Pic}(\widetilde{C})$: send $L \in \mbox{Pic}(\widetilde{C})$ to $(\cal{G}, \bdl{c})$, where $\cal{G}$ has associated rank-2 vector bundle ${\cal V} := \pi_* L$, and $ \bdl{c} \subset \UnderlinedEnd{{\cal{V}}}$ is $\pi_* {\cal O}_{\widetilde{C}}.$ \item On the other hand, for $G=SL(2)$ we have $\Lambda=\bf{Z}$ and $W={\cal{S}}_2$ acts by $\pm 1$, so $$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})) \approx \{L \in \mbox{Pic}(\widetilde{C})\ | \ i^*L \approx L^{-1} \} = \mbox{ker}(1+i^*). $$ This group has 4 connected components. The subgroup $\mbox{ker(Nm)}$ consists of 2 of these. The connected component of 0 is the classical Prym variety, cf. \cite{MuPrym}. Now the Higgs bundles correspond, via the above bijection $L\mapsto \pi_*L$, to $$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} = {\mbox{Nm}}^{-1}(K). $$ Thus they form the {\em non-zero} coset of the subgroup $\mbox{ker(Nm)}$. (If we return to a higher dimensional $S$, there is no change in the $GL(2)$ story, but it is possible for $K$ not to be in the image of $\mbox{Nm}$, so there may be {\em no} $SL(2)$-Higgs bundles corresponding to such a cover.) \end{itemize} \end{em} \end{eg} This example generalizes to all $G$, as follows. The equivalence classes of extensions $$1 \longrightarrow T \longrightarrow N' \longrightarrow W \longrightarrow 1 $$ (in which the action of $W$ on $T$ is the standard one) are parametrized by the group cohomology $H^2(W,T)$. Here the 0 element corresponds to the semidirect product . The class $[N] \in H^2(W,T)$ of the normalizer $N$ of $T$ in $G$ may be 0, as it is for $G=GL(n) , {\bf P}GL(n) , SL(2n+1) $; or not, as for $G=SL(2n)$. Assume first, for simplicity, that $S,\widetilde{S}$ are connected and projective. There is then a natural group homomorphism \begin{equation} \label{c} c: Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow H^2(W,T). \end{equation} Algebraically, this is an edge homomorphism for the Grothendieck spectral sequence of equivariant cohomology, which gives the exact sequence \begin{equation} \label{c-edge}\qquad 0 \longrightarrow H^1(W,T) \longrightarrow H^1(S,{\cal{C}}) \longrightarrow Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})) \stackrel{c}{\longrightarrow} H^2(W,T). \end{equation} where ${\cal{C}} := \widetilde{S} \times _W T.$ Geometrically, this expresses a {\em Mumford group} construction: giving ${\cal{L}} \in \mbox{Hom}(\Lambda,\mbox{Pic}(\widetilde{S}))$ is equivalent to giving a principal $T$-bundle $\cal T$ over $\widetilde{S}$; for ${\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S}))$, $c({\cal{L}})$ is the class in $H^2(W,T)$ of the group $N'$ of automorphisms of $\cal T$ which commute with the action on $\widetilde{S}$ of some $w \in W$. To remove the restriction on $S, \widetilde{S}$, we need to replace each occurrence of $T$ in (\ref{c},\ref{c-edge}) by $\Gamma (\widetilde{S}, T)$, the global sections of the trivial bundle on $\widetilde{S}$ with fiber $T$. The natural map $H^2(W,T) \longrightarrow H^2(W,\Gamma (\widetilde{S}, T))$ allows us to think of $[N]$ as an element of $H^2(W,\Gamma (\widetilde{S}, T))$. \begin{prop} \cite{D3} \label{reg.ss.equivalence} Fix an etale $W$-cover $\pi: \widetilde{S}\longrightarrow S$. The following data are equivalent: \begin{enumerate} \item Principal $G$-Higgs bundles $(\cal{G}, \bdl{c})$ with cameral cover $\widetilde{S}$. \item Principal $N$-bundles $\cal N$ over $S$ whose quotient by $T$ is $\widetilde{S}.$ \item $W$-equivariant homomorphisms ${\cal{L}} : \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$ with $c({\cal L}) = [N] \in H^2(W,\Gamma (\widetilde{S}, T))$. \end{enumerate} \end{prop} We observe that while the shifted objects correspond to Higgs bundles, the unshifted objects $$ {\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S})), \qquad c({\cal L})=0 $$ \noindent come from the $\cal C$-torsers in $H^1(S, {\cal C} ).$ \subsubsection{The regular case: the twist along the ramification} \label{reg} \begin{eg} \label{ramified} \begin{em} Modify example \ref{unramified} by letting $K \in \mbox{Pic}(C) $ be arbitrary, and choose a section $b$ of $K ^{\otimes 2}$ which vanishes on a simple divisor $B \subset C$. We get a double cover $\pi : \widetilde{C} \longrightarrow C$ branched along $B$, ramified along a divisor $$ R \subset \widetilde{C}, \quad \pi(R)=B. $$ Via $L\mapsto \pi_*L$, the $SL(2)$-Higgs bundles still correspond to $$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} = {\mbox{Nm}}^{-1}(K). $$ But this is no longer in $ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))$; rather, the line bundles in question satisfy \begin{equation} \label{SL(2) twist} i^*L \approx L^{-1}(R). \end{equation} \end{em} \end{eg} For arbitrary $G$, let $\Phi$ denote the root system and $\Phi^+$ the set of positive roots. There is a decomposition $$ \overline{G/T} \ \smallsetminus \ G/T = \bigcup _{\alpha \in \Phi^+}R_{\alpha} $$ of the boundary into components, with $R_{\alpha}$ the fixed locus of the reflection $\sigma_{\alpha}$ in $\alpha$. (Via (\ref{commutes}), these correspond to the complexified walls in $\frak t$.) Thus each cameral cover $\widetilde{S} \longrightarrow S$ comes with a natural set of (Cartier) {\em ramification divisors}, which we still denote $R_{\alpha}, \quad \alpha \in \Phi^+.$ For $w \in W$, set $$ F_w := \left\{ \alpha \in \Phi^+ \ | \ w^{-1} \alpha \in \Phi^- \right\} = \Phi^+ \cap w \Phi^-, $$ and choose a $W$-invariant form $\langle , \rangle$ on $\Lambda$. We consider the variety $$ Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S})) $$ of $R$-twisted $W$-equivariant homomorphisms, i.e. homomorphisms $\cal L$ satisfying \begin{equation} \qquad \label{G twist} w^*{\cal L}(\lambda) \approx {\cal L}(w\lambda)\left( \sum_{\alpha \in F_w}{ {\langle-2\alpha,w\lambda \rangle \over \langle \alpha ,\alpha \rangle} R_{\alpha} } \right) , \qquad \lambda \in \Lambda, \quad w \in W. \end{equation} This turns out to be the correct analogue of (\ref{SL(2) twist}). (E.g. for a reflection $w=\sigma_{\alpha}$, \quad $F_w$ is $\left\{ \alpha \right\}$, so this gives $ w^*{\cal L}(\lambda) \approx {\cal L}(w\lambda)\left( {{\langle\alpha,2\lambda \rangle \over \langle \alpha,\alpha \rangle} R_{\alpha}} \right),$ which specializes to (\ref{SL(2) twist}).) As before, there is a class map \begin{equation} \label{c,R} c: Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow H^2(W,\ \Gamma (\widetilde{S}, T)) \end{equation} \noindent which can be described via a Mumford-group construction. To understand this twist, consider the formal object \begin{center} $\begin{array}{cccc} {1 \over 2} \mbox{Ram}: & \Lambda & \longrightarrow & {\bf Q}\otimes \mbox{Pic}\widetilde{S}, \\ & \lambda & \longmapsto & \sum_{ ( \alpha \in {\Phi^+} ) }{{\langle\alpha,\lambda \rangle \over \langle \alpha,\alpha \rangle} R_{\alpha}}. \end{array}$ \end{center} In an obvious sense, a principal $T$-bundle $\cal T$ on $\widetilde{S}$ (or a homomorphism ${\cal L}: \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$) is $R$-twisted $W$-equivariant if and only if ${\cal T} (-{1 \over 2} Ram)$ is $W$-equivariant, i.e. if ${\cal T}$ and ${1 \over 2} Ram$ transform the same way under $W$. The problem with this is that ${1 \over 2} Ram$ itself does not make sense as a $T$-bundle, because the coefficients ${\langle\alpha,\lambda\rangle \over \langle\alpha,\alpha\rangle} $ are not integers. (This argument shows that if $Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))$ is non-empty, it is a torser over the untwisted $Hom_{W} (\Lambda , \mbox{Pic}(\widetilde{S}))$.) \begin{thm} \cite{D3} \label{main} For a cameral cover $\widetilde{S} \longrightarrow S$, the following data are equivalent: \\ (1) $G$-principal Higgs bundles with cameral cover $\widetilde{S}$. \\ (2) $R$-twisted $W$-equivariant homomorphisms ${\cal L} \in c^{-1}([N]).$ \end{thm} The theorem has an essentially local nature, as there is no requirement that $S$ be, say, projective. We also do not need the condition of generic behavior near the ramification, which appears in \cite{F, Me, Sc}. Thus we may consider an extreme case, where $\widetilde{S}$ is `everywhere ramified': \begin{eg}\begin{em} \label{nilpo} In example \ref{ramified}, take the section $b=0$. The resulting cover $\widetilde{C}$ is a `ribbon', or length-2 non-reduced structure on $C$: it is the length-2 neighborhood of $C$ in $\Bbb{K}$. The SL(2)-Higgs bundles $({\cal G},\bdl{c})$ for this $\widetilde{C}$ have an everywhere nilpotent $\bdl{c}$, so the vector bundle ${\cal V} := {\cal G} \times^{SL(2)} V \approx \pi_* L$ (where $V$ is the standard 2-dimensional representation) fits in an exact sequence $$ 0 \longrightarrow {\cal S} \longrightarrow {\cal V} \longrightarrow {\cal Q} \longrightarrow 0 $$ with ${\cal S} \otimes K \approx {\cal Q}.$ Such data are specified by the line bundle ${\cal Q}$, satisfying ${\cal Q}^{\otimes 2} \approx K$, and an extension class in $\mbox{Ext}^1({\cal Q}, {\cal S}) \approx H^1(K^{-1})$. The kernel of the restriction map $ \mbox{Pic}(\widetilde{C}) \longrightarrow \mbox{Pic}(C) $ is also given by $H^1(K^{-1})$ (use the exact sequence $0 \longrightarrow K^{-1} \longrightarrow \pi_*{\cal O}_{\widetilde{C}}^{\times} \longrightarrow {\cal O}_C^{\times} \longrightarrow 0$), and the $R$-twist produces the required square roots of $K$. (For more details on the nilpotent locus, cf. \cite{L} and \cite{DEL}.) \end{em}\end{eg} \subsubsection{Adding values and representations}\ \indent Fix a vector bundle $K$, and consider the moduli space $ {\cal M}_{S,G,K} $ of $K$-valued $G$-principal Higgs bundles on $S$. (It can be constructed as in Simpson's \cite{simpson-moduli}, even though the objects we need to parametrize are slightly different than his. In this subsection we sketch a direct construction.) It comes with a Hitchin map: \begin{equation} \label{BigHitchin} h: {\cal M}_{S,G,K} \longrightarrow B_K \end{equation} \noindent where $B := B_K$ parametrizes all possible Hitchin data. Theorem \ref{main} gives a precise description of the fibers of this map, independent of the values bundle $K$. This leaves us with the relatively minor task of describing, for each $K$, the corresponding base, i.e. the closed subvariety $B_s$ of $B$ parametrizing {\em split} Hitchin data, or $K$-valued cameral covers. The point is that Higgs bundles satisfy a symmetry condition, which in Simpson's setup is $$ \varphi \wedge \varphi = 0, $$ and is built into our definition \ref{princHiggs} through the assumption that \bdl{c} is a bundle of regular centralizers, hence is abelian. Since commuting operators have common eigenvectors, this is translated into a splitness condition on the Hitchin data, which we describe below. (When $K$ is a line bundle, the condition is vacuous, $B_s = B$.) The upshot is: \begin{lem} \label{parametrization} The following data are equivalent: \\ (a) A $K$-valued cameral cover of $S$. \\ (b) A split, graded homomorphism $R{\bf \dot{\ }} \longrightarrow {Sym}{\bf \dot{\ }}K.$ \\ (c) A split Hitchin datum $b \in B_s$. \end{lem} Here $R{\bf \dot{\ }}$ is the graded ring of $W$-invariant polynomials on $\frak t$: \begin{equation} R{\bf \dot{\ }} := (\mbox{Sym}{\bf \dot{\ }} {\frak t}^*)^W \approx {\bf C}[\sigma_1,\ldots,\sigma_l], \qquad \deg (\sigma_i) = d_i \end{equation} \noindent where $l := \mbox{Rank}({\frak g})$ and the $\sigma_i$ form a basis for the $W$-invariant polynomials. The Hitchin base is the vector space $$ B := B_K := \oplus _{i=1}^l H^0(S, {Sym}^{d_i}K) \approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K). $$ \noindent For each $\lambda \in \Lambda$ (or $\lambda \in {\frak t}^*$, for that matter), the expression in an indeterminate $x$: \begin{equation} \label{rep poly} q_{\lambda}(x,t) := \prod_{w \in W}{(x-w\lambda(t))}, \qquad t \in {\frak t}, \end{equation} \noindent is $W$-invariant (as a function of $t$), so it defines an element $q_{\lambda}(x) \in R{\bf \dot{\ }}[x].$ A Hitchin datum $b \in B \approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K)$ sends this to $$ q_{\lambda,b}(x) \in \mbox{Sym}\dot{\ }(K)[x]. $$ We say that $b$ is {\em split} if, at each point of $S$ and for each $\lambda$, the polynomial $q_{\lambda,b}(x)$ factors completely, into terms linear in $x$. We note that, for $\lambda$ in the interior of $C$ (the positive Weyl chamber), $q_{\lambda,b}$ gives the equation in $\Bbb K$ of the spectral cover $\widetilde{S}_{\lambda}$ of section (\ref{decomp covers}): $q_{\lambda,b}$ gives a morphism $\Bbb K \longrightarrow \mbox{Sym}^N \Bbb K$, where $N:=\#W$, and $\widetilde{S}_{\lambda}$ is the inverse image of the zero-section. (When $\lambda$ is in a face $F_P$ of $\overline{C}$, we define analogous polynomials $q_{\lambda}^P(x,t)$ and $q_{\lambda,b}^P(x)$ by taking the product in (\ref{rep poly}) to be over $w \in W_P \backslash W.$ These give the reduced equations in this case, and $q_{\lambda}$ is an appropriate power.) Over $B_s$ there is a universal $K$-valued cameral cover $$ \widetilde{\cal S} \longrightarrow B_s $$ with ramification divisor $R \subset \widetilde{\cal S}$. From the relative Picard, $$ \mbox{Pic}( \widetilde{\cal S} / B_s) $$ we concoct the relative $N$-shifted, $R$-twisted Prym $$ \mbox{Prym}_{\Lambda ,R}( \widetilde{\cal S} / B_s). $$ By Theorem (\ref{main}), this can then be considered as a parameter space $ {\cal M}_{S,G,K} $ for all $K$-valued $G$-principal Higgs bundles on $S$. (Recall that our objects are assumed to be everywhere {\em regular}!) It comes with a `Hitchin map', namely the projection to $B_s$, and the fibers corresponding to smooth projective $\widetilde{S}$ are abelian varieties. When $S$ is a smooth, projective curve, we recover this way the algebraic complete integrability of Hitchin's system and its generalizations. More generally, for any $S$, one obtains an ACIHS (with symplectic, respectively Poisson structures) when the values bundle has the same (symplectic, respectively Poisson) structure, by a slight modification of the construction in Chapter \ref{ch8}. One considers only Lagrangian supports which retain a $W$-action, and only equivariant sheaves on them (with the numerical invariants of a line bundle). These two restrictions are symplecticly dual, so the moduli space of Lagrangian sheaves with these invariance properties is a symplectic (respectively, Poisson) subsp! ace of the total moduli space, and the fibers of the Hitchin map are Lagrangian as expected. \subsubsection{Irregulars?} \nopagebreak \noindent The Higgs bundles we consider in this survey are assumed to be everywhere regular. This is a reasonable assumption for line-bundle valued Higgs bundles on a curve or surface, but {\em not} in $\dim \geq 3$. This is because the complement of ${\frak g}_{{reg}}$ has codimension 3 in ${\frak g}$. The source of the difficulty is that the analogue of (\ref{commutes}) fails over ${\frak g}$. There are two candidates for the universal cameral cover: $\widetilde{\frak g}$, defined by the left hand side of (\ref{commutes}), is finite over ${\frak g}$ with $W$ action, but does not have a family of line bundles parametrized by $\Lambda$. These live instead on $\stackrel{\approx}{\frak g}$, the object defined by the right hand side, which parametrizes pairs $(x,{\frak b}), \qquad x \in {\frak b} \subset {\frak g}$ . This suggests that the right way to analyze irregular Higgs bundles may involve spectral data consisting of a tower $$ \stackrel{\approx}{S} \stackrel{\sigma}{\longrightarrow} \widetilde{S} \longrightarrow S $$ together with a homomorphism $ {\cal L} : \Lambda \longrightarrow \mbox{Pic}(\stackrel{\approx}{S})$ such that the collection of sheaves $$ \sigma_*({\cal L}(\lambda)), \qquad \lambda \in \Lambda $$ on $\widetilde{S}$ is $R$-twisted $W$-equivariant in an appropriate sense. As a first step, one may wish to understand the direct images $ R^i \sigma_*({\cal L}(\lambda)) $ and in particular the cohomologies $H^i(F, {\cal L}(\lambda))$ where $F$, usually called a {\em Springer fiber}, is a fiber of $\sigma$. For regular $x$, this fiber is a single point. For $x=0$, the fiber is all of $G/B$, so the fiber cohomology is given by the Borel-Weil-Bott theorem. The question may thus be considered as a desired extension of BWB to general Springer fibers. \newpage

9507

alg-geom/9507015

en

https://arxiv.org/abs/alg-geom/9507015

alg-geom/9507015

Brendan Hassett

Brendan Hassett

Correlation for Surfaces of General Type

AMSLaTeX. This version contains some minor corrections, and additions to the references

The main geometric result of this paper is that given any family of surfaces of general type f:X-->B, for sufficiently large n the fiber product X^n_B dominates a variety of general type. This result is especially interesting when it is combined with Lang's Conjecture. This states that for a variety V of general type over a number field K, the K rational points V(K) are not Zariski dense in V. Assuming Lang's Conjecture, we prove the existence of a uniform bound on the degree of the Zariski closure of the K-rational points of a surface of general type.

alg-geom

\section{Introduction} The purpose of this paper is to prove the following theorem: \begin{thm}[Correlation Theorem for Surfaces] Let $f:X \longrightarrow B$ be a proper morphism of integral varieties, whose general fiber is an integral surface of general type. Then for $n$ sufficiently large, $X^n_B$ admits a dominant rational map $h$ to a variety $W$ of general type such that the restriction of $h$ to a general fiber of $f^n$ is generically finite. \end{thm} This theorem has a number of geometric and number theoretic consequences that will be discussed in the final section of this paper. In particular, assuming Lang's conjecture on rational points of varieties of general type, we can prove a uniform bound on the number of rational points on a surface of general type not contained in rational or elliptic curves. \newline \indent This theorem is a special case of the following conjecture posed by Caporaso, Harris, and Mazur \cite{CHM}: \begin{conj}[Correlation Conjecture] Let $f:X \longrightarrow B$ be a proper morphism of integral varieties, whose general fiber is an integral variety of general type. \newline Then for $n$ sufficiently large, $X^n_B$ admits a dominant rational map $h$ to a variety $W$ of general type such that the restriction of $h$ to a general fiber of $f^n$ is generically finite. \end{conj} They prove this conjecture in the case where the general fiber is a curve of genus $g\geq 2$. This implies that if Lang's conjectures on the distribution of rational points on varieties of general type are true, then there is a uniform bound on the number of rational points on a curve of genus $g$ defined over a number field $K$. The paper \cite{CHM} contains most of the ingredients necessary for a proof of the general conjecture. However, at one point the argument relies heavily on the fact that the fibers of the map are curves: it invokes the existence of a `nice' class of singular curves, the stable curves. For the purposes of this discussion, `nice' means two things: \begin{enumerate} \item Given any proper morphism $f:X\longrightarrow B$ whose generic fiber is a smooth curve of genus $g \geq 2$, there exists a generically finite base change $B' \longrightarrow B$ so that the dominating component of $X \times_B B'$ is birational to a family of stable curves over $B'$. \item Let $f:X \rightarrow B$ be a family of stable curves, smooth over the generic point. Then the fiber products $X_B^n$ are canonical. \end{enumerate} For the purpose of generalizing to higher dimensions, we make the following definitions: \newline Let $\cal C$ be a class of singular varieties. \begin{quote} $\cal C$ is {\em inclusive} if for any proper morphism $f:X \rightarrow B$ whose generic fiber is a variety of general type, there is a generically finite base change $B' \rightarrow B$ such that $X \times_B B'$ is birational to a family $X' \rightarrow B'$ with fibers in $\cal C$. \end{quote} \begin{quote} $\cal C$ is {\em negligible} if for any family of varieties of general type ${f:X \rightarrow B}$ with singular fibers belonging to $\cal C$, the fiber products $X^n_B$ have canonical singularities. \end{quote} In a nutshell, the main obstruction to extending the results of \cite{CHM} is to find a class of higher dimensional singular varieties which is both negligible and inclusive. In this paper, we identify a class of surface singularities which is both inclusive and negligible, and prove that the class has these properties. This is the class of `stable surfaces', surfaces at the boundary of a compactification of the moduli space of surfaces of general type. \newline \indent In the second section of this paper, we describe these stable surfaces, and try to motivate their definition. In the third section, we prove that these stable surfaces actually form an inclusive class of singularities. In the fourth section, we prove that stable surfaces are negligible, i.e. that their fiber products have canonical singularities. In the fifth section, we sketch the proof of the Correlation Theorem for surfaces of general type outlined in \cite{CHM}. Finally, we state some consequences of the Correlation Theorem, assuming various forms of the Lang conjectures. \vskip.25in I would like to thank Joe Harris for suggesting this problem, and Dan Abramovich for his countless comments and corrections. J\'anos Koll\'ar also provided useful suggestions for the results in section four. \vskip.25in Thorughout this paper, we work over a field of characteristic zero. \section{Stable Surfaces} \indent In this section, we describe the class of stable surfaces and their singularities. Most of these results are taken from \cite{K-SB} and \cite{K}. \indent Stable surfaces are defined so that one has a stable reduction theorem for surfaces, analogous to stable reduction for curves. For motivation, we first review the curve case. Let $f:X \rightarrow \Delta$ be a flat family of curves of genus $g \geq 2$ over a disc. Assume that the fibers of the family are smooth, except for the fiber $f^{-1}(0)$ which may be singular. By Mumford semistable reduction (\cite{KKMS}), there is a finite base change $$\tilde{\Delta} \longrightarrow \Delta$$ ramified over $0$, and a resolution of singularities of the base-changed family $$d: Y \longrightarrow \tilde{X}$$ such that the fibers of the composed map $$F=d \circ \tilde{f}:Y \longrightarrow \tilde{\Delta}$$ are reduced normal crossings divisors. This semistable reduction is not unique, as we can always blow up $Y$ to get a `different' semistable reduction. These semistable reductions are all birational, and we can take a `canonical model' $\cal Y$ of the surface $Y$ by using the relative pluricanonical differentials to map $Y$ birationally into projective space. $\cal Y$ is the image of this map, and $\cal Y \rightarrow \tilde{\Delta}$ is called the stable reduction of our original family. Moreover, the birational map $Y \rightarrow {\cal Y}$ can be described quite explicity. It is the morphism that blows down all the $-1$ and $-2$ curves on $Y$. On the fibers, this corresponds to blowing down smooth rational components meeting the rest of the fiber in one or two points. {}From this, we see that the fibers of ${\cal Y} \rightarrow \tilde{\Delta}$ are just stable curves. \newline \indent For higher dimensional varieties, we can try to mimic the same procedure. We can still apply semistable reduction to obtain the family $Y \longrightarrow \tilde{\Delta}$, but this reduction is not unique. The problem is that it is not generally known how to obtain a canonical model $\cal Y$ for the birational equivalence class of $Y$. This canonical model $\cal Y$ would be our stable reduction, if it were well defined. In the case of families of surfaces where $Y$ is a threefold, we can use the minimal model program to construct the canonical model of a semistable family of surfaces (cf \cite{Ka}). The total space of our stable family will then have canonical singularities, and the singularities of the fibers of the family can then be described. \newline \indent Now we introduce the formal definitions. By definition, a variety $S$ is said to be $\Bbb Q$-Gorenstein if $\omega_S^{[k]}$ is locally free for some $k$. $\omega_S^{[k]}$ denotes the reflexive hull (i.e. the double dual) of the $k$th power of the dualizing sheaf. For a $\Bbb Q$-Gorenstein singularity, the smallest such $k$ is called the index of the singularity. A surface is {\em semi-smooth} if it has only the following singularities: \begin{enumerate} \item { $2$-fold normal crossings with equation $x^2=y^2$} \item { pinch points with equation $x^2=zy^2$ } \end{enumerate} A {\em good semi-resolution} resolution of $S$ is a proper map $g:T \longrightarrow S$ satisfying the following properties \begin{enumerate} \item { $T$ is semi-smooth} \item { $g$ is an isomorphism in the complement of a codimension two subscheme of $T$} \item { No component of the double curve of $T$ is exceptional for $g$.} \item {The components of the double curve of $T$ and the exceptional locus of $S$ are smooth, and meet transversally.} \end{enumerate} A surface $S$ is said to have semi-log-canonical singularities if \begin{enumerate} \item $S$ is Cohen-Macaulay and $\Bbb Q$-Gorenstein with index $k$ \item $S$ is semi-smooth in codimension one \item The discrepancies of a good semi-smooth resolution of $S$ are all greater than or equal to $-1$ (i.e. $\omega_T^k = g^*\omega^{[k]}_S(ka_1 E_1 +...+ ka_n E_N)$ where $a_i \geq-1$) \end{enumerate} In \cite{K-SB} a complete classification of semi-log-canonical singularities is given. \newline \indent The relevance of these definitions comes from a result proved in the same paper \begin{thm} Let $f: X \rightarrow \Delta$ be a family of surfaces over the disc. Then the following are equivalent: \begin{enumerate} \item The general fiber has rational double points, and the central fiber has semi-log-canonical singularities. \item For any base change $\tilde{\Delta} \rightarrow \Delta$, the base-changed family $$\tilde f: \tilde X \longrightarrow \tilde{\Delta}$$ has canonical singularities. \end{enumerate} In fact, if $X \rightarrow \Delta$ has a semistable resolution of singularities, then $X$ is canonical iff the general fiber has rational double points and the central fiber has semi-log-canonical singularities. \end{thm} In particular, this means that the `bad' fibers in a stable reduction of surfaces have only semi-log-canonical singularities. For the sake of this discussion, surfaces with only rational double point singularities are `good' fibers. This is reasonable, because we would like the canonical model of a smooth surface of general type to be `good'. This motivates the definition: \begin{quote} A surface $S$ is {\em stable} if $S$ has semi-log-canonical singularities, and for some sufficiently large $k$ $\omega_S^{[k]}$ is locally free and ample. \end{quote} Note that a smooth surface of general type is not stable if it contains $-1$ or $-2$ curves, but its canonical model will be stable. \begin{quote} A family of stable surfaces is defined to be a proper flat morphism $\cal S \rightarrow B$ whose fibers are stable surfaces, with the property that taking reflexive powers of the relative dualizing sheaf commutes with restricting to a fiber: $$\omega_{\cal S/B}^{[k]}|{\cal S}_b = \omega^{[k]}_{{\cal S}_b}$$ \end{quote} In particular, the reflexive powers of the relative dualizing sheaf are flat. This additional condition is necessary to guarantee that the moduli space in the next section is separated. Note that we can define $$K_S^2 = {1\over {k^2}}\#(\omega_S^{[k]},\omega_S^{[k]})$$ for any stable surface $S$, and that this number is constant in families. We also have the invariant $\chi_S=\chi(\cal O_S)$, which is also constant in families. Finally, stable surfaces are analogous to stable curves in one more important sense: \begin{thm} A stable surface has a finite automorphism group. \end{thm} The essence of the proof is easy to grasp. Let $S$ be stable, and let $\tilde S$ be its normalization. Let $\Delta$ be the double curve on $\tilde S$. The pair $(\tilde S,\Delta)$, is log-canonical (see \cite{K-SB}). Therefore, each component of $(\tilde S,\Delta)$ is of log-general type, and has a finite automorphism group by \cite{I}. \section{Stable Surface Singularities are Inclusive} \indent To show that the class of stable surfaces are inclusive, we need to invoke the existence of a proper coarse moduli space $\bar{\cal M}_{\chi,K^2}$ for the stable surfaces with invariants $\chi$ and $K^2$. We also need a finite covering $\phi:\Omega \rightarrow \bar{\cal M}_{\chi,K^2}$ of the moduli space that admits a tautological family $\cal S \rightarrow \Omega$: $$\begin{array}{ccc} \cal T & & \\ \downarrow & & \\ \Omega & \stackrel{\phi}{\rightarrow} & \bar{\cal M}_{\chi,K^2}\\ \end{array}$$ We have the following theorem: \begin{thm} For smoothable stable surfaces, there exists a coarse moduli space $\bar{\cal M}_{\chi,K^2}$ with these properties. \end{thm} By definition, a stable surface is {\em smoothable} if it is contained in a family of stable surfaces with $\Bbb Q$-Gorenstein total space, such that the general member has only rational double points. The proof of this theorem is scattered throughout the literature. The proof that the moduli space exists as a separated algebraic space is contained in \cite{K-SB} \S 5. This relies on the properties of semi-log-canonical singularities and the finite automorphism theorem. The proof that the moduli space has a functorial semipositive polarization is contained in \cite{K} \S 5. This paper also has a general argument for the existence of a finite covering of the moduli space possessing a tautological family (see also \cite{CHM} \S 5.1). The proof that the moduli space is of finite type for a given pair of invariants (and thus proper and projective by \cite{K}) is contained in \cite{A}. \newline \indent Using this moduli space, we can prove that the class of stable surface singularities is inclusive. \begin{prop} Let $f:X \rightarrow B$ be a proper morphism of integral varieties. Assume that the general fiber of this map is a smooth surface of general type. Then there exists a generically finite base change $B' \rightarrow B$ such that the dominating component of the fiber product $X \times_B B'$ is birational to a family of stable surfaces over $B'$. \end{prop} The proof of this follows the proof of the analogous result in \cite{CHM} \S 5.2 t quite closely. Since $\bar{\cal M}$ is a coarse moduli space, there is an induced rational map $B \rightarrow \bar{\cal M}$. Let $B_1$ be the closed graph of this map, $\Sigma_1$ its image in $\bar{\cal M}$, and $X_1\rightarrow B_1$ the dominating component of $X\times_B B_1$. Since there is no tautological family on $\bar{\cal M}$, we do not have a family of stable surfaces defined over $\Sigma_1$. However, we do have such a family over $\Sigma_2=\phi^{-1}(\Sigma_1)\subset \Omega$, which we denote $\cal T_2$. So we let $B_2=B_1 \times_{\Sigma_1} \Sigma_2$, $\mu:B_2 \rightarrow \Sigma_2$ the projection, and $X_2 \rightarrow B_2$ the main component of $X_1 \times_{B_1} B_2$. The family $\cal T_2$ of stable surfaces pulls back to a family $Y_2=\cal T_2 \times_{\Sigma_2}B_2 \rightarrow B_2$. For general $b\in B_2$, $(X_2)_b$ is birational to $(Y_2)_b$ by construction. \indent This is almost enough to prove that $X_2$ is birational to the family of stable surfaces $Y_2$. We just need one further finite base change $B_3 \rightarrow B_2$ to `straighten out' $X_2$. We have to get rid of isotrivial subfamilies that cannot be represented by pull backs of the tautological family on the moduli space. We describe the fiber of this base change over the generic point $b\in B_2$. Set theoretically, it will correspond to equivalence classes of birational morphisms $$ \{ \psi | \psi: (X_2)_b \rightarrow(\cal T_2)_{\mu(b)}\}$$ Two maps are equivalent if they induce the identity on the canonical model $(\cal T_2)_{\mu (b)}$. The algebraic structure is just the natural algebraic structure on the finite automorphism group of the stable surface. If our covering variety happens to be disconnected, choose a component $B''$ dominating $B_2$. Finally, we take $B_3 \rightarrow B$ to be the Galois normalization of $B'' \rightarrow B$, and let $G$ be Galois group $\operatorname{Gal}(k(B_3)/k(B))$. We let $X_3$ denote the principal component of $X_2 \times_{B_2}B_3$. We can represent a generic point of $X_3$ as a triple $$(p,b,\psi)$$ where $b \in B_2$, $p \in (X_2)_b$, and $\psi:(X_2)_b \rightarrow (\cal T_2)_{\mu(b)}$. The birational map from $X_3$ to $Y_3=\cal T_2 \times_{\Sigma_2}B_3$ is just the evaluation map $$(p,b,\psi) \longrightarrow (\psi(p),b,\psi)$$ Setting $B'=B_3$, we obtain the proposition that stable surface singularities are inclusive. $\square$ \newline \indent For the proof of the Correlation Theorem, we will need to elaborate a bit on this situation. Let $\Sigma_3 \rightarrow \Sigma_1$ denote the Galois normalization of $\Sigma_1$ in the function field $k(B_3)$. We have: $$\begin{array}{ccc} B_3 & \rightarrow & \Sigma_3 \\ \downarrow & \; & \downarrow\\ B & \dashrightarrow & \Sigma_1 \\ \end{array}$$ The bottom arrow is only birational. Note that the map $B_3 \rightarrow \Sigma_2$ factors naturally through $\Sigma_3$; this is just the Stein factorization. We claim that $G$ acts naturally on this diagram. $G$ consists of automorphisms of $k(B_3)$ fixing the subfield $k(B_1)$. Since $k(\Sigma_1)\subset k(B_1)$ these automorphisms fix $k(\Sigma_1)$ as well, and they restrict to automorphisms of the elements of $k(B_3)$ algebraic over $k(\Sigma_1)$, i.e. $k(\Sigma_3)$. \newline \indent Now let ${\cal T}_3$ denote ${\cal T}_2 \times_{\Sigma_2} \Sigma_3$. We shall show that $G$ also acts birationally and equivariantly on ${\cal T}_3 \rightarrow \Sigma_3$. Let $s \in \Sigma_3$ be a general point, and $({\cal T}_3)_s$ the corresponding fiber. For $g \in G$, we need to describe the map $$({\cal T}_3)_s \rightarrow ({\cal T}_3)_{g(s)}$$ By construction $({\cal T}_3)_s$ can be birationally identified with some corresponding fiber of $X_3 \rightarrow B_3$. The action of $g$ maps this surface to another fiber of $X_3 \rightarrow B_3$, which in turn can be birationally identifed with $({\cal T}_3)_{g(s)}$. This gives a commutative diagram of varieties with (birational) G-actions $$\begin{array}{ccc} X_3 & \rightarrow & {\cal T}_3 \\ \downarrow & \; & \downarrow \\ B_3 & \rightarrow & \Sigma_3 \\ \end{array}$$ {}From the arguments in the previous paragraph we see that $X_3$ is birational to $Y_3={\cal T}_3 \times_{\Sigma_3} B_3$, and that this birational map respects the Galois action of $G$. Taking quotients under this action gives a dominant rational map $$X\approx ({\cal T}_3 \times_{\Sigma_3} B_3) /G \longrightarrow {\cal T}_3 / G$$ This refined construction is crucial to the proof of the correlation theorem, so we summarize it below: \begin{cor} Let $f:X \rightarrow B$ be a proper morphism of integral varieties. Assume that the general fiber of $f$ is a surface of general type. Then there exists a generically finite Galois base extension $$B' \rightarrow B$$ with Galois group $G$, and a finite cover of the image of $B$ in the moduli space $$\Sigma' \rightarrow \Sigma$$ with the following properties: \begin{enumerate} \item {There is a tautological family of surfaces $${\cal T}' \rightarrow {\Sigma}'$$ over $\Sigma'$.} \item {G acts on $\Sigma'$, and this action lifts to a $G$ equivariant rational dominant map $$\begin{array}{ccc} X' & \rightarrow & {\cal T}' \\ \downarrow & \; & \downarrow \\ B' & \rightarrow & {\Sigma}' \\ \end{array}$$} \item { The pull back of ${\cal T}'$ to $B'$ is birational to $X'$, and the quotient of this variety under the $G$-action is birational to $X$.} \end{enumerate} \end{cor} \section{Stable Surface Singularities are Negligible} In this section, we will restrict our attention to families of stable surfaces $f:X \rightarrow B$ over a smooth base $B$, and their fiber products $f^n: X^n_B=X \times_B ... \times_B X \rightarrow B$. For such families, having canonical singularities (rational double points) is an open condition. That is, the locus $S \subset B$ corresponding to singularities worse than rational double points is Zariski closed. Here we will assume that it is a proper subvariety of $B$. We will prove the following: \begin{prop} Let $f: X \rightarrow B$ be a family of stable surfaces over a smooth proper base $B$. Assume the generic fiber has only canonical singularities. Then the fiber products of this family $$f^n: X_B^n \longrightarrow B$$ have canonical singularities. \end{prop} We prove this in three steps. First, we prove a general lemma on the singularities of fiber products. Then we establish the result in the case where $B=\Delta$ a complex disc. The third step is to reduce the general case to this special case. For this reduction, we will utilize results of Stevens \cite {St} on families of varieties with canonical singularities. \newline \indent Our first lemma gives some rough information on the singularities of fiber products: \begin{lm} Let $f: X\rightarrow B$ a family of stable surfaces, such that the general fiber is normal. Then the $n^{th}$ fiber product $$f^n: X_B^n \longrightarrow B$$ is a normal $\Bbb Q$-Gorenstein variety. \end{lm} $X_B^n$ is irreducible, because the family $X \rightarrow B$ is flat with general fiber irreducible. We show that $X_B^n$ is Cohen-Macaulay. $X_B$ itself is Cohen-Macaulay, as it is a flat family of Cohen-Macaulay varieties over a smooth base. In particular, the dualizing complex of the morphism $X \rightarrow B$ has only a single term, the relative dualizing sheaf $\omega_{X/B}$. This sheaf is flat over $B$, so the dualizing complex of $X_B^n \rightarrow B$ is just the tensor product of the dualizing complexes of each of the factors. In particular, this complex has only one term $$\omega_{X_B^n/B}=\pi_1^* \omega_{X/B} \otimes ... \otimes \pi_n^*\omega_{X/B}$$ and so $X^n_B$ is Cohen-Macaulay. Note that this implies that $X^n_B$ satisfies Serre's condition $S_r$ for every $r>0$. \newline \indent $X^n_B$ is reduced, because it is smooth at the generic point and satisfies the $S_1$ condition. We now prove that $X_B^n$ is normal. Because $X_B^n$ satisfies the $S_2$ condition, we just need to show that it is smooth in codimension one. Let $$\pi_j : X_B^n \longrightarrow X $$ be the $j^{th}$ projection map. The singularities of $X_B^n$ are contained in the set of points where $f^n$ fails to be a smooth morphism. But if $f^n$ fails to be smooth at $p$, then $f$ fails to be smooth at $\pi_j(p)$ for some $j$. Since $f$ is smooth on a set with codimension two complement, so is $f^n$. Thus the singularities of $X_B^n$ are in codimension two. \newline \indent Now we prove the $\Bbb Q$ Gorenstein assertion. First we check that $X$ itself is $\Bbb Q$ Gorenstein, i.e. $\omega^{[N]}_X$ is locally free for some $N$. Since $X$ is a family of stable surfaces, there exists an integer $N$ such that for each $b\in B$ $\omega^{[N]}_{X/B}|X_b$ is locally free. Since $\omega_{X/B}^{[N]}$ is free on every fiber of $X\rightarrow B$, $\omega_{X/B}^{[N]}$ is locally free. Since $B$ is smooth, $\omega_B$ is locally free, and $$\omega^{[N]}_X=\omega_{X/B}^{[N]} \otimes f^*\omega_B^N$$ This formula is not hard to prove. It is certainly true on the open set $U$ where $X \rightarrow B$ is smooth. The complement of $U$ has codimension two by hypothesis. Since $X$ is normal, and both sheaves are reflexive, the formula extends to all of $X$. For the basic properties of reflexive sheaves used here, see \cite{H}. \newline \indent Now we prove $X^n_B$ is $\Bbb Q$ Gorenstein. As in the previous paragraph, we have the formula $$\omega^{[N]}_{X_B^n}=\omega^{[N]}_{X_B^n/B} \otimes {f^n}^*\omega_B^{\otimes N}$$ so it suffices to prove that $\omega^{[N]}_{X_B^n/B}$ is locally free. Using the general formula: $$\omega_{X_B^n/B}= \pi_1^*\omega_{X/B}\otimes ...\otimes \pi_n^*\omega_{X/B}\quad (*)$$ we will prove $$\omega_{X_B^n/B}^{[N]}= \pi_1^*\omega_{X/B}^{[N]}\otimes ...\otimes \pi_n^*\omega_{X/B}^{[N]} \quad (**)$$ and so $\omega_{X_B^n/B}^{[N]}$ is locally free. The left hand side of (**) is reflexive by construction, and the right hand side is locally free because it is the tensor product of locally free sheaves. So we just need to prove the equivalence of the two sides of $(**)$ on an open set with codimension two complement. Again, we choose the open set where $f^n$ is smooth as a morphism. On this set, the formula follows immediately from (*), as the dualizing sheaves are already locally free. This completes the proof of the lemma. $\square$ \newline \indent Now we prove our proposition in the case where the base $B$ is one dimensional. In this special case it takes the following form: \begin{prop} Let $f: X \rightarrow \Delta $ be a family of stable surfaces over the disc. Assume that the general fiber has only rational double points. Then the fiber products of this family over $\Delta$ have canonical singularities. \end{prop} We apply semistable reduction to the family $X \rightarrow \Delta$. Let $$\tilde{\Delta} \longrightarrow \Delta$$ be the ramified base change, and $$d: Y \longrightarrow \tilde X$$ a resolution of singularities such that all the fibers of the induced map $$F=d \circ \tilde f : Y \longrightarrow \tilde \Delta$$ are reduced normal crossings divisors. We have the diagram: $$\begin{array}{ccccc} Y & \; & \; & \; & \; \\ \; & \stackrel{d}{\searrow} & \; & \; & \; \\ \; & \; & \tilde X & \rightarrow & X \\ \; & \; &{\scriptstyle {\tilde f}} \downarrow & \; & \downarrow \scriptstyle{f} \\ \; & \; & \tilde \Delta & \rightarrow & \Delta \end{array}$$ Using theorem 3, we find that $\tilde X$ still has canonical singularities. \newline \indent The next step is to take the nth fiber products of all the varieties in this diagram. We take the fiber products over the bases $\Delta $ and $\tilde \Delta$, and we use $f^n,{\tilde f}^n$, and $d^n$ to denote the maps on the fiber products induced by $f, {\tilde f}$, and $d$ respectively. We have the following diagram: $$\begin{array}{ccrcl} Y_{\tilde \Delta}^n & \; & \; & \; & \; \\ \; & \stackrel{d^n}{\searrow} & \; & \; & \; \\ \; & \; & \tilde X_{\tilde \Delta}^n & \longrightarrow & X_{\Delta}^n \\ \; & \; & {\scriptstyle{\tilde{f}^n}} \downarrow & \; & \downarrow \scriptstyle{f^n}\\ \; & \; & \tilde \Delta & \longrightarrow & \Delta \end{array}$$ \indent The general lemma implies that $X_{\Delta}^n$ and $\tilde X_{\tilde \Delta}^n$ are both $\Bbb Q$-Gorenstein and normal. As for $Y_{\tilde \Delta}^n$, recall that we constructed $Y$ so that its fibers over $\tilde \Delta$ have only reduced normal crossings. Using an argument of Viehweg \cite {V} \S 3.6, we see that the singularities of $Y_{\tilde \Delta}^n$ are canonical. (Using local analytic coordinates, we can see that the singularities are toroidal, and so are rational. The equations also show that the singularities are local complete intersections, hence Gorenstein. But rational Gorenstein singularities are canonical). \indent First, note that for any $M$ there is an inclusion map: $$d_*\omega_{Y}^{M} \hookrightarrow \omega_{\tilde X}^{[M]}$$ This is because the resolution $d:Y \rightarrow {\tilde X}$ is an isomorphism on an open set with codimension two in ${\tilde X}$, so pluricanonical forms on $Y$ yield sections of $\omega_{\tilde X}^{[M]}$. Moreover, since $\tilde X$ has canonical singularities we have that this is an isomorphism for some $M$, i.e. $$d_*\omega_{Y}^{M}=\omega_{\tilde X}^{[M]}$$ This expresses the fact that regular pluricanonical differentials on the smooth locus of $\tilde X$ lift to smooth differentials on the desingularization $Y$. We will show: $$\omega_{{\tilde X}_{\tilde \Delta}^n}^{[M]} =d^n_*\omega_{Y_{\tilde \Delta}^n}^{M} \quad (1)$$ This combined with the fact that ${\tilde X}_{\tilde \Delta}^n$ is $\Bbb Q$-Gorenstein and $Y_{\tilde \Delta}^n$ is canonical implies that ${\tilde X}_{\tilde \Delta}^n$ is canonical as well. \newline \indent We prove that $(1)$ holds. Again, we have projection maps, which fit into a commutative diagram $$\begin{array}{rcl} Y_{\tilde \Delta}^n & \stackrel{\phi_j}{\rightarrow} & Y\\ {\scriptstyle{d^n}} \downarrow & \; & \downarrow \scriptstyle{d} \\ {\tilde X}_{\tilde \Delta}^n & \stackrel{\pi_j}{\rightarrow} & {\tilde X}\\ \end{array}$$ An important element in the proof $(1)$ is the equation: $$d^n_*\phi_j^*\omega_{Y/{\tilde \Delta}}=\pi_j^*d_*\omega_{Y/{\tilde \Delta}} \quad (2)$$ For simplicity, we prove this for $j=1$. We begin factoring $\phi_1=q \circ p$ and $d^n=r\circ p$: $$\begin{array}{ccl} Y\times ...\times Y & \; & \;\\ {\scriptstyle{p}} \downarrow & \stackrel{\phi_1}{\searrow} & \;\\ Y\times \tilde X \times ....\times \tilde X & \stackrel{q}{\rightarrow} & Y\\ {\scriptstyle{r}} \downarrow & \; & \downarrow \scriptstyle{d}\\ {\tilde X}_{\tilde \Delta}^n & \stackrel{\pi_1}{\rightarrow} & \tilde X\\ \end{array}$$ We set $p=\text{Id} \times d^{n-1}$, $q$ the projection onto the first factor, and $r=d \times \text{Id}^{n-1}$. Note that $\pi_1$ is flat and the square part of the diagram is a flat base change of $d$, so $\pi_1^*d_*\omega_{Y/{\tilde \Delta}}= r_*q^*\omega_{Y/{\tilde \Delta}}$. The projection formula tells us that $p_*p^*(q^*\omega_{Y/{\tilde \Delta}}) =p_*{\cal O}_{Y_{\tilde \Delta}^n} \otimes q^*\omega_{Y/{\tilde \Delta}}$. Since $p$ is a birational map of normal varieties, we have $p_* {\cal O}_{Y_{\tilde \Delta}^n}= {\cal O}_{Y \times {\tilde X}\times ...\times {\tilde X}}$. Putting all this together gives \begin{eqnarray*} d^n_*\phi_1^*\omega_{Y/{\tilde \Delta}} &=&(r \circ p)_*(q \circ p)^* \omega_{Y/{\tilde \Delta}} \\ &=& r_*p_*p^*q^*\omega_{Y/{\tilde \Delta}} \\ &=& r_*(p_*{\cal O}_{Y_{\tilde \Delta}^n}\otimes q^*\omega_{Y/{\tilde \Delta}})\\ &=& r_*q^*\omega_{Y/{\tilde \Delta}}\\ &=& \pi_1^*d_* \omega_{Y/{\tilde \Delta}} \end{eqnarray*} This proves equation $(2)$. \newline \indent In the course of proving lemma 1, recall that we established the equation: $$\omega_{\tilde X_{\Delta}^n/{\tilde \Delta}}^{[M]}= \pi_1^*(\omega_{{\tilde X}/{\tilde \Delta}}^{[M]})\otimes ... \otimes \pi_n^*(\omega_{{\tilde X}/{\tilde \Delta}}^{[M]})\quad (*)$$ Using this along with $(2)$ gives \begin{eqnarray*} \omega_{{\tilde X}_{\tilde \Delta}^n/{\tilde \Delta}}^{[M]}&=& \pi_1^*(\omega_{{\tilde X}/{\tilde \Delta}}^{[M]})\otimes ... \otimes \pi_n^*(\omega_{{\tilde X}/{\tilde \Delta}}^{[M]})\\ &=&\pi_1^*(d_*\omega_{Y/{\tilde \Delta}}^{ M}) \otimes ... \otimes \pi_n^*(d_*\omega_{Y/{\tilde \Delta}}^{M})\\ &=&d^n_*\phi_1^*\omega_{Y/{\tilde \Delta}}^{M} \otimes ... \otimes d^n_*\phi_n^*\omega_{Y/{\tilde \Delta}}^{M}\\ &=&d^n_*\omega_{Y_{\tilde \Delta}^n/{\tilde \Delta}}^{M} \end{eqnarray*} The last step is just the formula for the dualizing sheaf of a fiber product. This completes the proof of equation $(1)$. We conclude that ${\tilde X}_{\tilde \Delta}^n$ has canonical singularities. \newline \indent Before completing the proof, we need to fix some additional notation. Set $$G :=\operatorname{Gal}({\tilde \Delta}/ \Delta)$$ and $j$ to be the map $$j: {\tilde X}_{\tilde \Delta}^n \longrightarrow X_{\Delta}^n$$ induced by the base change. Let $$ s: Z \longrightarrow {\tilde X}_{\tilde \Delta}^n$$ be an equivariant desingularization of ${\tilde X}_{\tilde \Delta}^n$ with respect to the Galois action of $G$ (see \cite {Hi}). Then we write the quotient map $$Q:Z \longrightarrow Z/G$$ Note that $Z/G$ may be singular. Finally, the map from $Z$ to $X_{\Delta}^n$ is $G$-equivariant, so it factors through $Z/G$ giving a map $$R: Z/G \longrightarrow X_{\Delta}^n$$ This is summarized in the following diagram: $$\begin{array}{rcl} \; & \; \atop Q & \; \\ Z & \longrightarrow & Z/G \\ {\scriptstyle s} \downarrow & \;\atop j & \downarrow {\scriptstyle R}\\ {\tilde X}_{\tilde \Delta}^n & \longrightarrow & X_{\Delta}^n \\ \downarrow & \; & \downarrow \\ {\tilde \Delta} & \rightarrow & \Delta \\ \end{array}$$ \newline \indent Now we show that $X_{\Delta}^n$ has canonical singularities. Let $$\alpha \in \Gamma(\omega_{X_{\Delta}^n}^{[m]})$$ be an $m$-pluricanonical form on $X_{\Delta}^n$. We want to show that $\alpha$ is a smooth form on $X_{\Delta}^n$, i.e. for any desingularization $V$ of $X_{\Delta}^n$, the pull back of $\alpha$ to $V$ is regular. It suffices to show that $R^*\alpha$ is a smooth form on $Z/G$, because a desingularization of $Z/G$ can also serve as a desingularization of $X_{\Delta}^n$. By an easy local computation, $j^*\alpha$ vanishes to order $m(|G|-1)$ along the central fiber of ${\tilde X}_{\tilde \Delta}^n$. Pulling back to the desingularization $Z$ (and increasing $m$ if necessary), we see that $s^*j^*\alpha$ is a smooth form vanishing to order $m(|G|-1)$ along the central fiber, because ${\tilde X}_{\tilde \Delta}^n$ has canonical singularities. The central fiber of $Z$ is precisely the fixed locus under the action of $G$. Therefore, we can apply the following lemma to $\theta=s^*j^*\alpha$ to show that it descends to a smooth form on $Z/G$: \begin{lm} Let $G$ be a finite group acting on the variety $Z$. Let $W$ denote a codimension $d$ subvariety of $Z$ fixed pointwise by a subgroup $G_W \subset G$, and $\theta$ an invariant $m$-pluricanonical form. If $\theta$ vanishes to order at least $m(|G_W|-d)$ at every such $W$, then $\theta$ descends to a smooth form on $Z/G$, \end{lm} For a proof of this, see \cite{CHM} \S 4.2. We use $\beta$ to denote this smooth form on $Z/G$. Using the commutative diagram above, one can see that $\beta = R^*\alpha$, i.e. the pluricanonical form $\alpha$ pulls back to a smooth form on $Z/G$. This proves that $X_{\Delta}^n$ has canonical singularities. $\square$ \newline \indent To summarize, this proves the proposition in the special case of a one dimensional base. The proposition is a local statement on the base (in the analytic topology), so if it is true for families over a disc then it is true for general one dimension families. We use this as the base case for induction on the dimension of the base. We now prove the inductive step. \newline \indent We will use the following result of Stevens(\cite {St}). \begin{thm} Let $g:V \rightarrow \Delta$ be a family of proper varieties. Assume: \begin{enumerate} \item $V$ is a $\Bbb Q$-Gorenstein integral variety, and the fibers of $g$ are integral varieties. \item The general fibers $g^{-1}(s)$ have only canonical singularities. \item The special fiber $g^{-1}(0)$ has log terminal singularities. \end{enumerate} Then $V$ has canonical singularities. \end{thm} We apply this theorem inductively to $V=X_B^n$ to reduce the dimension of the base. Note that $X_B^n$ has $\Bbb Q$-Gorenstein singularities by the lemma proven above. Choose a local analytic coordinate $y$ on $B$, and consider the level surfaces $$H_s=\{ b \in B : y(b)=s \}$$ for $s \in \Delta$. Assume that none of the $H_s$ are contained in the locus $S$ of surfaces with singularities worse than rational double points. Let $$h: X \rightarrow \Delta$$ be the map associating $X|_{H_s}=f^{-1}(H_s)$ to $s\in\ \Delta$. We also have the corresponding map $$g=h^n: X_B^n \rightarrow \Delta$$ For all $s$, $H_s \cap S$ is again a Zariski closed proper subset of $H_s$, so the family $$f^n_s: X|_{H_s}^n \rightarrow H_s$$ satisfies the hypotheses of Proposition 2. Applying the inductive hypothesis, we find that $$g^{-1}(s)=X_B^n|_{H_s}=X|_{H_s}^n$$ is canonical for all $s$. Since all the fibers of $g$ are canonical, and the total space $X_B^n$ is $\Bbb Q$-Gorenstein, we can apply Stevens' theorem to conclude that $X_B^n$ is also canonical. This concludes the proof that stable surface singularities are negligible. $\square$ \section{Proof of the Correlation Theorem} In this section, we prove the correlation theorem for surfaces of general type (Theorem 1). We first prove a special case where the family has maximal variation of moduli and the singularities are not too bad. By definition, a family has maximal variation of moduli if there are no isotrivial connected subfamilies through the generic point. \begin{thm}[Correlation for Families with Maximal Variation] Let $X\rightarrow B$ be a family of stable surfaces, with projective integral base and smooth general fiber. Assume that the associated map $\phi:B \rightarrow \bar{\cal M}$ is generically finite. Then there exists a positive integer $n$ such that $X^n_B$ is of general type. \end{thm} Being of general type is a birational property, so there is no loss of generality if we take the base $B$ to be smooth. To show that $X_B^n$ is of general type for some large $n$, we must verify two statements: \begin{enumerate} \item $X_B^n$ has canonical singularities \item $\omega_{X_B^n}$ is big \end{enumerate} Note that the first statement is equivalent to saying that stable surface singularities are negligible, which was proved in the last section. The second statement allows us to get lots of pluricanonical differentials on $X_B^n$, and which pull back to a desingularization of $X_B^n$. The key to the second statement is the following theorem: \begin{thm} Let $f:X \rightarrow B$ be a family of surfaces, such that the general fiber is a surface of general type. Assume this family has maximal variation. Then for $m$ sufficiently large, we have that $f_*\omega^m_{X/B}$ is big. \end{thm} This result is proven by Viehweg in \cite{V2} (and more generally for arbitrary dimensional fibers by Koll\'ar in \cite{K2}). We need the following consequence of this result: \begin{prop} Under the hypotheses of the theorem above, $\omega_{X^n_B}$ is big. \end{prop} We will show that Theorem 7 implies Proposition 4. Here $S^{[n]}$ will denote the reflexive hull of the $n$th symmetric power of a sheaf. To say that $f_*\omega_{X/B}^m$ is big means that for any ample line bundle $H$ on $B$ there exists an integer $n$ such that $$S^{[n]}(f_*\omega_{X/B}^m) \otimes H^{-1}$$ is generically globally generated, i.e. the global sections of this sheaf generate over an open set of $B$. It is equivalent to say that this sheaf is generically globally generated for sufficiently large $n$. Now let $T^{[n]}$ denote the reflexive hull of the $n$th tensor power of a sheaf. We claim that for sufficiently large $n$ $$T^{[n]}(f_*\omega_{X/B}^m) \otimes H^{-1}$$ is generically globally generated. To prove this, we need a result from representation theory: \begin{prop} Let $V$ be an $r$ dimensional vector space over a field of characteristic zero, and let $T^n(V)$ and $S^q(V)$ be the $n$th tensor power and $q$th symmetric power representations of $Gl(V)$ respectively, and write $t=r!$. Then each irreducible component of $T^n(V)$ is a quotient of a representation $$S^{q_1}(V) \otimes ... \otimes S^{q_t}(V)$$ where $q_i \geq {n\over{t+1}}$. \end{prop} This result is proved in \cite{H2} for arbitrary `positive' representations $T$ of $V$. This gives us a map $$\bigoplus S^{[q_1]}(f_*\omega_{X/B}^m)\otimes ... \otimes S^{[q_t]} (f_*\omega_{X/B}^m) \otimes H^{-1} \rightarrow T^{[n]} (f_*\omega_{X/B}^m) \otimes H^{-1}$$ which is surjective over an open set of $B$. Let $H$ be ample and globally generated, and choose $n$ large enough to guarantee that each of the $S^{[q_i]}(f_*\omega_{X/B}^m) \otimes H^{-1}$ is generically globally generated. This guarantees that the left hand side is generically globally generated, but then so is $T^{[n]}(f_*\omega_{X/B}^m) \otimes H^{-1}$. \newline \indent Now we will prove that for some large $n$ the dualizing sheaf $\omega_{X^n_B}$ is big, i.e. for large $m$ we have $$h^0(X^n_B,\omega^{[m]}_{X^n_B})\approx m^{(b+2n)}$$ where $b=\text{dim}(B)$. We restrict ourselves to values of $m$ for which \begin{enumerate} \item{$\omega^{[m]}_{X/B}$ is locally free.} \item{$f_*\omega^{m}_{X/B}$ is big.} \end{enumerate} First we compute the canonical bundle of $X^n_B$: $$\omega_{X_B^n}=\omega_{X_B^n/B} \otimes {f^n}^*\omega_B =\pi_1^*\omega_{X/B} \otimes ....\otimes \pi_n^*\omega_{X/B} \otimes {f^n}^* \omega_B$$ As in lemma 1, taking $m$th powers gives $$\omega^{[m]}_{X_B^n}=\pi_1^*\omega^{[m]}_{X/B} \otimes ...\otimes \pi_n^* \omega^{[m]}_{X/B} \otimes {f^n}^*\omega^m_B$$ Applying $f^n_*$ to this gives \begin{eqnarray*} f^n_*\omega^{[m]}_{X_B^n} &=&f^n_*(\pi_1^*\omega^{[m]}_{X/B} \otimes ... \otimes \pi_n^* \omega^{[m]}_{X/B}) \otimes \omega^m_B \\ &=&f^n_*\pi_1^*\omega^{[m]}_{X/B} \otimes ... \otimes f^n_*\pi_n^*\omega^{[m]}_{X/B} \otimes \omega^m_B \\ &=&T^{n}(f_*\omega^{[m]}_{X/B}) \otimes \omega^m_B\\ \end{eqnarray*} Note this is also a reflexive sheaf. The inclusion map $\omega^m_{X/B} \rightarrow \omega^{[m]}_{X/B}$ induces a map of reflexive sheaves $$T^{[n]}(f_*\omega^m_{X/B}) \rightarrow T^{n}(f_*\omega^{[m]}_{X/B})$$ which is an isomorphism at the generic point of $B$. \newline \indent Let $H$ be an invertible sheaf on $B$ so that $H \otimes \omega_B$ is very ample. By Viehweg's theorem and proposition 5, we can choose $n$ so that $T^{[n]}(f_*\omega^m_{X/B}) \otimes H^{-m}$ is generically globally generated for $m$ sufficiently large. The computations of the last paragraph show that $f^n_*\omega^{[m]}_{X^n_B}\otimes (H\otimes \omega_B)^{-m}$ is also generically globally generated for sufficiently large $m$. In particular, as this sheaf has rank on the order of $m^{2n}$, there at least this many global sections. By our assumption on $H$, we have that $(H\otimes \omega_B)^ m$ has on the order of $m^b$ sections varying horizontally along the base $B$. Tensoring, we get that $f^n_*\omega^{[m]}_{X^n_B}$ has on the order of $m^{2n+b}$ global sections. Thus we conclude that $$h^0(\omega^{[m]}_{X^n_B})\approx m^{2n+b}$$ This completes the proof of the proposition and the special case of the Correlation theorem. $\square$ \indent Now we extend this special case to prove correlation for arbitrary families $f:X \rightarrow B$ of surfaces of general type. Since stable surface singularities are inclusive, after a generically finite base extension $B' \rightarrow B$ every family of surfaces of general type dominates a family of stable surfaces $\psi:{\cal T}' \rightarrow {\Sigma}'$ with maximal variation: $$\begin{array}{cccc} X' & \rightarrow & {\cal T}' & \; \\ \downarrow & \; & \downarrow & (*)\\ B' & \rightarrow & {\Sigma}' & \; \\ \end{array}$$ Take $n^{th}$ fiber products, where $n$ is chosen to ensure that ${{\cal T}'}_{{\Sigma}'}^n$ is of general type. Use ${X'_{B'}}^n$ to denote the component of the fiber product dominating $B'$. We obtain a diagram: $$\begin{array}{cccc} {X'_{B'}}^n & \rightarrow & {{\cal T}'_{{\Sigma}'}}^n & \; \\ \downarrow & \; & \downarrow & (**) \\ B' & \rightarrow & {\Sigma}' & \; \\ \end{array}$$ with ${X'_{B'}}^n$ dominating ${{\cal T}'_{{\Sigma}'}}^n$, a variety of general type. This shows that the correlation result holds if we allow ourselves to make a finite base change before we take the fiber products. \newline \indent Now we show that the fiber products $X_B^N \rightarrow B$ dominate a variety of general type without taking a base change, provided $N$ is large enough. We will use the corollary at the end of \S 3 to construct our map. This corollary allows us to assume that the base extension $B' \rightarrow B$ is Galois with Galois group $G$, and that $G$ acts birationally on the entire diagram $(*)$. That is, $G$ acts birationally on each of the varieties in $(*)$, and this action commutes with the morphisms of the diagram. It follows that $G$ acts naturally and birationally on $(**)$, and taking quotients gives us $$\begin{array}{ccc} X_B^N & \rightarrow & {{\cal T}'_{{\Sigma}'}}^N /G \\ \downarrow & \; & \downarrow \\ B & \rightarrow & {\Sigma}' / G \\ \end{array}$$ Setting $W={{\cal T}'_{{\Sigma}'}}^N /G$, we obtain a rational dominant map $$h: X_B^n \longrightarrow W$$ {}From the construction, we see that this map is generically finite when restricted to a general fiber of $X_B^n \rightarrow B$. \newline \indent To conclude the proof, we need to show that $W={{\cal T}'_{{\Sigma}'}}^N /G$ is of general type, for some sufficiently large $N$. Specifically, we will show that enough of the $m$ pluricanonical differentials on $V={{\cal T}'_{{\Sigma}'}}^N$ descend to smooth differentials on $W=V/G$ to guarantee that $W$ is a variety of general type. First, note that $G$ acts faithfully on ${\cal T}'$, but does not necessarily act faithfully on the base ${\Sigma}'$. Let $G'$ be the maximal quotient of $G$ acting faithfully on ${\Sigma}'$, and set $g=|G|$. Let $\Phi_1 \subsetneq {\Sigma}'$ be the locus of points of ${\Sigma}'$ with nontrivial stabilizer under the $G'$ action, $\Phi_2 \subsetneq {\Sigma}'$ the locus on the base corresponding to fibers of ${\cal T}' \rightarrow {\Sigma}'$ fixed pointwise by a nontrivial subgroup of $G$. Let $D_0$ be an effective divisor on ${\Sigma}'$ containing $\Phi_1 \cup \Phi_2$, and $D$ the pullback of $g D_0$ to $V$. Note that for large $N$, the support of $D\subset V$ contains all the componenets of the fixed locus with codimension less than $g$. This is because the only components of the fixed point locus with small codimension correspond to fixed fibers of ${\cal T}'$. \newline \indent We repeat the proof we used in the maximum variation case, except that we choose positive $H$ so that $H \otimes \omega_{{\Sigma'}}(-gD_0)$ is very ample. Again, we can choose $N$ so that $$\psi^N_*\omega_V^{[m]}(-mD)= T^{[N]}(\psi^N_*\omega^m_{{\cal T}'/{\Sigma}'}) \otimes \omega^m_{{\Sigma}'} (-mgD_0)$$ has $m^{(\text{dim} V)}$ sections. This guarantees that $$h^0(\omega_V^{[m]}(-mD))\approx m^{(\text{dim} V)}$$ In other words, there are lots of $m$ pluricanonical differentials on $V$ vanishing to high order along subvarieties of $V$ nontrivial stabilizer and codimension less than $g$. We apply lemma 2 of \S 4 to conclude that these forms descend to smooth forms on $W$, i.e. forms on $W$ that pull back to regular forms on a desingularization of $W$. Therefore $W$ is of general type and the Correlation Theorem is proved. $\square$ \section{Consequences of the Correlation Theorem} We give some consequences of the Correlation Theorem. Many of these are stated in \S 6 of \cite{CHM}. The motivating conjectures can be found in \cite{L}. \newline \indent Recall the statement of the Geometric Lang Conjecture: \begin{conj} [Geometric Lang Conjecture] If $W$ is a variety of general type, the union of all irreducible, positive dimensional subvarieties of $W$ not of general type is a proper, closed subvariety $\Xi_W \subset W$. \end{conj} We will call $\Xi_W$ the {\em Langian exceptional locus} of $W$. The following theorem describes how the Langian exceptional locus varies in families, if the geometric Lang conjecture is true. \begin{thm} Assume the Geometric Lang Conjecture. \newline Let $f:X \rightarrow B$ be a flat family of surfaces in projective space, such that the general fiber is an integral surface of general type. Then there is a uniform bound on the degree of the Langian exceptional locus of fibers that are of general type i.e. $$deg(\Xi_{X_b})\leq D$$ \end{thm} By Noetherian induction, it suffices to prove the bound on an open subvariety of $B$. Using the Correlation Theorem, for sufficiently high fiber products $f^n: X_B^n \rightarrow B$ we obtain a dominant rational map to a variety of general type $$\psi:X_B^n \rightarrow W$$ We use $Y$ to denote the Langian exceptional locus of $W$, and $Z_1$ its preimage in $X_B^n$. Let $Z_2$ be the union of all positive dimensional fibers of the map $$X_B^n \rightarrow W \times B$$ We set $Z=Z_1 \cup Z_2$; $Z$ is a proper subvariety of $X_B^n$. \newline \indent Consider the projection map $$\pi_n:X_B^n \rightarrow X_B^{n-1}$$ with fiber $\pi_n^{-1}(p)$ isomorphic to the stable surface $X_{f^{n-1}(p)}$. Since $Z$ is a proper subvariety of $X_B^n$, over an open set $U_{n-1} \subset X_B^{n-1}$ the fibers of $\pi_n$ are not contained in $Z$. For all $p \in U_{n-1}$, the degree of $Z$ restricted to the fiber $\pi_n^{-1}(p)$ is bounded. At the same time, the Langian locus $\Xi_p$ of any of these fibers is contained in $Z$, because for any component $C\subset \Xi_p$ either $\psi(C)\subset Y$ or $\psi(C)$ is a point. This concludes the proof. $\square$ \newline \indent This yields many remarkable corollaries. One example is the following \begin{cor} Assume the Geometric Lang Conjecture. There exists a constant $D$ such that the sum of the degrees of all the rational and elliptic curves on a smooth quintic surface in $\Bbb P^3$ is less than $D$. In particular, there is a uniform bound on the number of rational and elliptic curves on a quintic surface. \end{cor} Recently, Abramovich \cite{AV} has found another proof of these results. \indent Now we shall discuss some number theoretic consequences of the Correlation Theorem. First, recall the Weak Lang Conjecture: \begin{conj}[Weak Lang Conjecture] If $W$ is a variety of general type defined over a number field $K$, then the $K$-rational points of $W$ are not Zariski dense in $W$. \end{conj} Assuming this conjecture, the Correlation Theorem implies the following: \begin{thm} Assume the Weak Lang Conjecture. \newline Let $X\rightarrow B$ be a flat family of surfaces in projective space defined over a number field $K$ such that the general fiber is an integral surface of general type. For any $b\in B(K)$ for which $X_b$ is of general type, let $N(b)$ be the sum of the degrees of the components of $\overline{X_b(K)}$. Then $N(b)$ is uniformly bounded; in particular, the number of $K$ rational points not contained in the Langian locus is uniformly bounded. \end{thm} The proof of this is similar to the proof of the previous theorem. Again we do induction on the dimension of the base $B$. First, we shall show that the rational points of the fibers must lie on a proper subscheme of bounded degree. Choose an integer $n$ so that there is a dominant rational map $$\psi:X^n_B \rightarrow W$$ to a variety of general type $W$. Let $Y$ denote be a proper subvariety of $W$ that contains its $K$ rational points, and let $Z$ be its preimage in $X^n_B$. All the $K$ rational points of $X^n_B$ are contained in $Z$. We use $$\pi_j:X_B^j \rightarrow X_B^{j-1}$$ to denote the projection morphisms. Finally, let $Z_j$ denote the maximal closed set in $X_B^j$ whose preimage in $X^n_B$ is $Z$, and let $U_j$ be the complement to $Z_j$. Note that $\pi_j^{-1}(Z_{j-1}) \subset Z_j$ by definition and that for $u\in U_{j-1}$ we have that $\pi_j^{-1}(u) \cap Z_j$ is a proper subvariety of $\pi_j^{-1}(u)$. We will use $d_j$ to denote the sum of the degrees of all the components of $Z_j \cap \pi_j^{-1}(u)$, regardless of their dimensions, and we set $$N=\operatorname{max}_j(d_j)$$ If all the $K$ rational points of $B$ are concentrated along a closed subset, we are done by induction. Otherwise, pick a general $K$ rational point $b\in B$. Let $j$ be the smallest integer for which $U_j \cap X_b^j(K)$ is empty, and let $u\in U_{j-1} \cap X_b^{j-1}(K)$. We have that $X_b=\pi_j^{-1}(u)$ and our set-up guarantees that $X_b(K) \subset Z_j \cap \pi_j^{-1}(u)$. In particular, since we have chosen everything generically, we find that $X_b(K)$ is contained in a subscheme of degree $N$. \newline \indent Now we complete the proof. We have shown that the rational points on each fiber are concentrated along a subscheme of degree $N$. The components of this subscheme consist of points, rational and elliptic curves, and curves of higher genus. The rational and elliptic curves are contained in the Langian locus, so we ignore them, and there are at most $N$ components of dimension zero. Therefore, we just need the following lemma: \begin{lm} Assume the Weak Lang Conjecture. Let $C$ be a (possibly singular) curve in projective space of degree $N$ defined over a number field $K$. Assume $C$ has no rational or elliptic components. Then there is a uniform bound on the number of $K$ rational points on $C$. \end{lm} First, because the degree is bounded there are only finitely many possibilities for the geometric genera of the components of $C$. By the hypothesis, these genera are all at least two, so we can apply the uniform boundedness results for curves in \cite{CHM}. This completes the proof of the theorem. $\square$ \newline \indent In the corollary that follows, quadratic points are points defined over some degree two extension of the base field. \begin{cor} Assume the Weak Lang Conjecture. Fix a number field $K$, and an integer $g>2$. Then there is a uniform bound on the number of quadratic points lying on a non-hyperelliptic, non-bielliptic curve $C$ of genus $g$ defined over $K$. \end{cor} Note that quadratic points on $C$ correspond to $K$ rational points on its symmetric square $\operatorname {Sym}^2(C)$. Moreover, a hyperelliptic (respectively bielliptic) system on $C$ corresponds to a rational (respectively elliptic) curve on $\operatorname{Sym}^2(C)$ (\cite{AH}). In particular, the curves described in the theorem are precisely those for which $\Xi_{\operatorname{Sym}^2(C)}=\emptyset$, and so by the theorem $\#\operatorname{Sym}^2(C)(K)$ is finite and uniformly bounded.

9507

alg-geom/9507011

en

https://arxiv.org/abs/alg-geom/9507011

alg-geom/9507011

Stephan Endrass

Stephan Endrass

A Projective Surface of Degree Eight with 168 Nodes

LaTeX 2.09 with amssymbols

The estimate for the maximal number of ordinary double points of a projective surface of degree eight is improved to $168\leq\mu(8)\leq 174$ by constructing a projective surface of degree eight with 168 nodes.

alg-geom

\section*{Introduction} Consider algebraic surfaces in complex projective threespace ${\Bbb P}_3$, denote by a {\em node} of such a surface an ordinary double point and by $\mu\left(d\right)$ the maximal number of nodes of an algebraic surface of degree $d$ in ${\Bbb P}_3$ with no further degeneracies. This note shows that $\mu\left(8\right)\geq 168$ by giving an example of an octic surface $X_8$ with $168$ nodes. $X_8$ is found within a seven parameter family of $112$--nodal octic surfaces admitting dihedral symmetry of order sixteen. This improves the estimate of $\mu\left(8\right)$ given by examples of Gallarati (\cite{gallarati}, $\mu\left(8\right)\geq 160$) and Kreiss (\cite{kreiss}, $\mu\left(8\right)\geq 160$). On the other hand, using Miyaokas upper bound \cite{miyaoka} for the number of nodes of a projective surface, we get $\mu\left(8\right)\leq 174$, thus \[ 168\leq\mu\left(8\right)\leq 174. \] I have made excessive use of the computer algebra system Maple V R3 computing $X_8$, in particular the calculations depend heavily on some new resp.\ improved features of release 3 (\cite{maple}, pp.\ 22--24). The construction of $X_8$ involves no free parameters, and in fact D.\ van Straten calculated that $X_8$ is rigid using MACAULAY. The octic $X_8$ is invariant under the group $D_8\times{\Bbb Z}_2$, therefore invariant under the reflection group ${\Bbb Z}_2^3$ of order eight. So $X_8$ is the eightfold cover of a projective quartic surface with thirteen nodes. This construction is referred to as the Segre--trick and it has been used extensively to construct sextic surfaces with a given number of nodes, see \cite{cataneseceresa} and \cite{barth}. {}From the existence of $X_8$ one however cannot deduce the existence of surfaces with $161$ up to $167$ nodes. It can be checked that the family of surfaces $X_8$ is constructed from contains no such surfaces. \medskip\\ {\em Acknowledgments: I would like to thank W.~Barth for suggesting to me that surfaces with many nodes can be found among $D_n$--symmetric surfaces.} \section*{The Dihedral Groups} Let the dihedral group $D_n$ of order $2n$ acting on ${\Bbb P}_3$ be generated by the rotation \[ \phi\colon\left(x:y:z:w\right)\longmapsto \left({\textstyle\cos\left({2\pi\over n}\right)} x-{\textstyle\sin\left({2\pi\over n}\right)} y: {\textstyle\sin\left({2\pi\over n}\right)} x+{\textstyle\cos\left({2\pi\over n}\right)} y:z:w\right) \] and the involution \[ \tau\colon\left(x:y:z:w\right)\longmapsto \left(x:-y:z:w\right). \] A surface $X=\left\{F=0\right\}\subset{\Bbb P}_3$ will be called $D_n$--symmetric if $F$ is $D_n$--invariant. The planes of reflection symmetry of $D_n$ are exactly the $n$ planes \[ E_j=\left\{ \sin\left({{j\pi\over n}}\right)x= \cos\left({{j\pi\over n}}\right)y\right\},\qquad j=0,\ldots,n-1. \] If now $X$ is $D_n$--symmetric and for some $j\in\left\{0,\ldots,n-1\right\}$ the plane curve $C=X\cap E_j$ has got a singularity in $p_0=\left(x_0:y_0:z_0:w_0\right)$, then (assuming that after a rotation $E_j=\left\{y=0\right\}$, thus $y_0=0$) \[ \frac{\partial F}{\partial\left\{x,z,w\right\}} \left(p_0\right)= \frac{\partial\left.F\right|_{E_j}}{\partial\left\{x,z,w\right\}} \left(p_0\right)=0. \] The reflection symmetry gives $F\left(x,y,z,w\right)=F\left(x,-y,z,w\right)$, so \[ \frac{\partial F}{\partial y}\left(p_0\right)= -\frac{\partial F}{\partial y}\left(p_0\right)=0, \] therefore $p_0$ induces an orbit of singularities on $X$ with length \[ \left|\hbox{orbit}\left(D_n,p_0\right)\right|=\left\{ \begin{array}{c@{\qquad}l} 1 & \hbox{if\ }x_0=0, \\ n/2 & \hbox{if $n$ is even and\ }z_0=w_0=0, \\ n & \hbox{otherwise}.\\ \end{array}\right. \] \section*{The Construction of $X_8$} Let us begin with the seven--parameter family of octic surfaces, therefore define $D_8$--invariant polynomials \begin{eqnarray*} P & = & \prod_{j=0}^7\left(\cos\left( {{j\pi\over 4}}\right)x+ \sin\left( {{j\pi\over 4}}\right)y-w\right)\\ & = & {\frac{1}{4}\left ({x}^{2}-{w}^{2}\right ) \left ({y}^{2}-{w}^{2}\right )\left (\left (x+y\right )^{2}-2 \,{w}^{2}\right )\left (\left (x-y\right )^{2}-2\, {w}^{2}\right )}\\ Q & = & \left (a\left (x^{2}+y^{2}\right )^{2} +\left (x^{2}+y^{2}\right )\left (b\,z^{2}+c\,zw+d\,w^{2} \right )\right.\\ & & \hspace{5ex}\left.+e\,z^{4}+f\,z^{3}w+g\,z^{2}w^{2} +h\,zw^{3}+i\,w^{4}\right )^{2}\\ \end{eqnarray*} with parameters $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $i\in{\Bbb C}$ and set $F=P-Q$, $X=\left\{F=0\right\}$. $P$ vanishes exactly on all eight planes $H_j=\left\{\cos\left({j\pi\over 4}\right)x +\sin\left({j\pi\over 4}\right)y=w\right\}$, $j=0,\ldots,7$. So $P$ vanishes to the second order on the $28$ lines $H_j\cap H_k$, $0\leq j<k\leq 7$ and $Q$ vanishes to the second order on a quartic surface. Therefore for general values of $a,\ldots,i$ the polynomial $F$ vanishes to the second order on $4\cdot 28=112$ points. Hence $X$ has got $112$ nodes, all of them lying on some symmetry plane $E_j$, $j\in\left\{0,\ldots,7\right\}$. Because of symmetry it is sufficient to consider only $E_0=\left\{y=0\right\}$ and $E_1=\left\{x=\left(1+\sqrt{2}\right)y\right\}$. Substituting the equations for $E_0$ and $E_1$ one gets homogeneous coordinates $\left(x:z:w\right)$ on $E_0$ and $\left(y:z:w\right)$ on $E_1$. Now consider the two curves $C_j=X\cap E_j$, $j=0,1$. Then, as divisors, we have: \begin{eqnarray*} \left\{\left.P\right|_{E_0}=0\right\} & = & L_1+L_2+2\left(L_3+L_4+L_5\right)\\ L_{1/2} & = & \left\{ x=\pm w\right\} \\ L_{3/4} & = & \left\{ x=\pm\sqrt{2}\,w\right\}\\ L_5 & = & \left\{ w=0\right\}\\ \end{eqnarray*} $C_0$ has got singularities in those twelve points where $\left\{\left.Q\right|_{E_0}=0\right\}$ meets one of the lines $L_3$, $L_4$ or $L_5$, so $C_0$ is a plane octic curve with twelve singularities admitting reflection symmetry ${\Bbb Z}_2$. Analogously, \begin{eqnarray*} \left\{\left.P\right|_{E_1}=0\right\} & = & 2\left(M_1+M_2+M_3+M_4\right)\\ M_{1/2} & = & \left\{y=\pm w\right\}\\ M_{3/4} & = & \left\{y=\pm\left(\sqrt{2}-1\right)w\right\}\\ \end{eqnarray*} so $C_1$ has got sixteen singularities and admits reflection symmetry ${\Bbb Z}_2$. The first step is to set $c=f=h=0$, then $X$ is $D_8\times{\Bbb Z}_2$--symmetric. Now both $C_0$ and $C_1$ admit reflection symmetry ${\Bbb Z}^2_2$ and therefore can be constructed from plane quartic curves $\tilde{C}_0$ and $\tilde{C}_1$ by applying the Segre--trick. The second step is to set $e=-1$ to mod out all projective transformations $z\mapsto\lambda\cdot z$, $\lambda\in{\Bbb C}^*$ from this family of octics. Now call a singularity of $\tilde{C}_0$ or $\tilde{C}_1$ outside of the coordinate axes a singularity in {\em general position}. Analogously call a point of contact of $\tilde{C}_0$ or $\tilde{C}_1$ to one of the coordinate axes outside the three points of intersection of those axes a point of contact in {\em general position}. Applying proposition 5 of \cite{cataneseceresa} gives: \begin{itemize} \item $\tilde{C}_0$ has got two nodes $s_1$ and $s_2$ and two contact points $t_1$ and $t_2$ to $\left\{w=0\right\}$ in general position. \item $\tilde{C}_1$ has got four nodes $u_1,\ldots,u_4$ in general position, no three of them collinear and thus splits into two conics. \item Every node in general position of $\tilde{C}_0$ or $\tilde{C}_1$ induces an orbit of $16$ singularities on $X$. \item Every point of contact in general position of either $\tilde{C}_0$ or $\tilde{C}_1$ to one of the coordinate axes $\left\{w=0\right\}$ or $\left\{z=0\right\}$ induces an orbit of $8$ singularities on $X$. \end{itemize} All points $s_1$, $s_2$, $t_1$, $t_2$, and $u_1,\ldots,u_4$ induce orbits of singularities on $X$, which will be denoted by the same letters. All those points are singularities of $X$, so if the determinant of the hesse matrix of $F$ in some point of such an orbit does not vanish, all points in the corresponding orbit are nodes. After substituting \begin{eqnarray*} i & = & -{1\over 4}\left(8\left(3-2\,\sqrt{2}\right) \left(4\,a+b^2\right) +4\left(2-\sqrt{2}\right) \left(bg+2\,d\right) +g^2 -i_1^2\right)\\ d & = & -{1\over 32}\left( 256\,a+64\,b^2+16\,bg-\sqrt{2}\,d_1^2+ \sqrt{2}\,i_1^2\right)\\ a & = & -{1\over 64}\left( 16\,b^2+a_1^2-\left(2-\sqrt{2}\right)d_1^2 -\left(2+\sqrt{2}\right)i_1^2\right)\\ \end{eqnarray*} the determinants of the hesse matrices of the above points can be computed (with Maple, of course): \begin{eqnarray*} \det\left(\mathop{\rm hesse}\nolimits\left(X,s_{1/2}\right)\right) & = & 8\,a_1^2\left(4\,b+2\,g\pm a_1\right )\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,t_{1/2}\right)\right) & = & const\cdot\left( a_1^2-\left(2-\sqrt{2}\right)d_1^2-\left(2+\sqrt{2}\right)i_1^2 \right )\\ & & \hspace{2ex}\left( 4\,b\pm\sqrt{ -a_1^2+\left(2-\sqrt{2}\right)d_1^2+\left(2+\sqrt{2}\right)i_1^2} \right)^{2}\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{1/2}\right)\right) & = & const\cdot\left( 4\,b+\left(2+\sqrt{2}\right)g \pm\left(2+\sqrt{2}\right)i_1\right)i_1^2\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{3/4}\right)\right) & = & const\cdot\left( 4\,b+\left(2-\sqrt{2}\right)g \pm\left(2-\sqrt{2}\right)d_1\right)d_1^2\\ \end{eqnarray*} The third step is to set the remaining parameters to: \[ \begin{array}{lll} a=-{1\over 4}\left(1+ \sqrt{2}\right) & b= {1\over 2}\left(2+ \sqrt{2}\right) & d= {1\over 8}\left(2+ 7\sqrt{2}\right)\\[2ex] g= {1\over 2}\left(1- 2\sqrt{2}\right) & i=-{1\over 16}\left(1+12\sqrt{2}\right) \\ \end{array} \] \smallskip\\ Now set $X_8=X$. It can be checked that $\tilde{C}_0$ admits an additional node $s_3=\left(8\left(\sqrt{2}-1\right):1:4\right)$ and an additional point of contact $t_3=\left(1:0:2\right)$ to $\left\{z=0\right\}$, whereas $\tilde{C}_1$ splits into one conic $K$ and two lines and therefore admits one additional node $u_5=\left( 2\left(3-2\,\sqrt{2}\right):3-2\,\sqrt{2}:4\right)$. Moreover $K$ has points of contact to $\left\{w=0\right\}$ in $v_1=\left( 1:3+2\,\sqrt{2}:0\right)$ and to $\left\{z=0\right\}$ in $v_2=\left( 1:0:4\right)$. Computing determinants of hesse matrices results in: \begin{eqnarray*} \det\left(\mathop{\rm hesse}\nolimits\left(X,s_{1}\right)\right) & = & 128 \\ \det\left(\mathop{\rm hesse}\nolimits\left(X,s_{2}\right)\right) & = & 1152 \\ \det\left(\mathop{\rm hesse}\nolimits\left(X,s_{3}\right)\right) & = & -128\left(239-169\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,t_{1}\right)\right) & = & {1\over 4}\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,t_{2}\right)\right) & = & {3\over 4}\left( 3+2\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,t_{3}\right)\right) & = & {\frac {9}{512}}\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{1}\right)\right) & = & 512\left(1451+1026\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{2}\right)\right) & = & 512\left(99-70\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{3}\right)\right) & = & 512\left(11243+7950\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{4}\right)\right) & = & 4608\left(331+234\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,u_{5}\right)\right) & = & 2\left(1451-1026\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,v_{1}\right)\right) & = & 512\left(3363+2378\,\sqrt{2}\right)\\ \det\left(\mathop{\rm hesse}\nolimits\left(X,v_{2}\right)\right) & = & {1\over 128}\left( 2979+2106\,\sqrt {2}\right)\\ \end{eqnarray*} This gives $16+16+8+8+8=56$ additional nodes, thus $168$ nodes altogether. \section*{$X_8$ is smooth away from the 168 nodes} Now one checks (again with Maple) that the 168 nodes are the only ones to appear on the eight planes $E_j$, $j=0,\ldots,7$: \begin{itemize} % \item points of contact to coordinate axes can be computed explicitly, % \item the fact that $K$ is non degenerate can be checked by computing its determinant, % \item the fact that $\tilde{C}_0$ is irreducible can be checked by projecting through a node onto some line not containing this node and computing the branch points with multiplicities. % \end{itemize} One also checks that the points of intersection of $X_8$ with the line $L=\left\{\left(x:y:0:0\right)\mid \left(x:y\right)\in{\Bbb P}_1\right\}$ are smooth. If $X_8$ would have a singularity $p_0$ outside of all eight planes $E_j$, $j=0,\ldots,7$ and outside the line $L$, then $p_0$ would induce an orbit of sixteen singularities of $X_8$. Now consider the following lemma: \begin{lemma} Let $Y=\left\{G=0\right\}$ be a $D_n$--symmetric surface of degree $n\geq 1$. Then $Y$ has got no orbit of $2n$ nodes. \end{lemma} \noindent{\bf Proof:}\ If $n\leq 3$ we have $\mu\left(n\right)<2n$, so let $n\geq 4$. Assume $O_{2n}$ is an orbit of $2n$ nodes of $Y$. Then $O_{2n}$ is contained in some plane curve $K$ of degree $\leq 2$. Let $E$ be the plane containing $K$. After a projective transformation we may assume $E=\left\{z=0\right\}$. Then $C=Y\cap E$ is a plane curve of degree $n$ with $C.K\geq 4n>2n$, so $C=K+C'$. But $C'.K\geq 2n>2\left(n-2\right)$, thus $C=2K+C''$. Therefore \[ \left.\frac{\partial G}{\partial\left\{x,y,w\right\}}\right|_K= \left.\frac{\partial \left.G\right|_E} {\partial\left\{x,y,w\right\}}\right|_K=0. \] Now $I=\left\{\partial G/\partial z|_E=0\right\}$ is a plane curve of degree $n-1$, meeting all $2n$ nodes, all of whose intersections with $K$ induce singularities on $Y$. So $I.K\geq 2n>2\left(n-1\right)$, thus $I=K+I'$ which means that \[ \left.\frac{\partial G}{\partial z}\right|_K= \left.\left.\frac{\partial G}{\partial z}\right|_E\right|_K=0. \] So $Y$ is singular along $K$, this contradicts that $K$ contains isolated singularities of $Y$. \medskip\\ \indent So $p_0$ would induce a singular curve on $X_8$ which itself would induce at least one singularity on every plane $E_j$, $j=0,\ldots,n-1$ which is not a node, contradiction. Therefore the surface \begin{eqnarray*} X_8 & = & \left\{64\left(x^2-w^2\right)\left(y^2-w^2\right) \left(\left(x+y\right)^2-2\,w^2\right) \left(\left(x-y\right)^2-2\,w^2\right)\right.\\ & & -\left[ -4\left(1+\sqrt{2}\right)\left(x^2+y^2\right)^2 +\left( 8\left(2+\sqrt{2}\right){z}^{2} +2\left(2+7\,\sqrt{2}\right)w^2 \right )\left(x^2+y^2\right) \right.\\ & & \left.\left. -16\,z^4 +8\left(1-2\,\sqrt{2}\right)z^2w^2 -\left(1+12\,\sqrt{2}\right)w^4\right]^2=0\right\} \end{eqnarray*} has exactly 168 nodes and no other singularities.

9507

alg-geom/9507012

en

https://arxiv.org/abs/alg-geom/9507012

alg-geom/9507012

Nakajima Hiraku

Hiraku Nakajima

Heisenberg algebra and Hilbert schemes of points on projective surfaces

AMS-LaTeX v. 1.1, 16 pages

I have just replaced the first line by %&amslplain in order to be compiled by AMS-LaTeX.

alg-geom

\section{Introduction} The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people (see e.g., \cite{Iar,ES,Got,Go-book}). The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras (see e.g., \cite{Kac}). We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain ``correspondences'' in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. Our construction has the same spirit with author's construction \cite{Na-quiver,Na-gauge} of representations of affine Lie algebras on homology groups of moduli spaces of ``instantons''\footnote{The reason why we put the quotation mark will be explained in Remark~\ref{sheaf}.} on ALE spaces which are minimal resolution of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist vector bundles along curves (irreducible components of the exceptional set), while ours twist around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank $1$ vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend \cite{Na-quiver,Na-gauge} to more general $4$-manifolds. The same program was also proposed by Ginzburg, Kapranov and Vasserot \cite{GKV}. Another motivation of our study is the conjecture about the generating function of the Euler number of the moduli spaces of instantons, which was recently proposed by Vafa and Witten~\cite{VW}. They conjectured that it is a modular form for $4$-manifolds under certain conditions. This conjecture was checked for various $4$-manifolds using various mathematicians' results. Among them, the most relevant to us is the case of K3 surfaces. G\"ottsche and Huybrechts \cite{GotHu} proved that the Betti numbers of moduli spaces of stable rank two sheaves are the same as those for Hilbert schemes. G\"ottsche \cite{Got} computed the Betti numbers of Hilbert schemes for general projective surfaces $X$. (The Hilbert schemes for $\operatorname{\C P}^2$ were studied earlier by Ellingsrud and Str\o mme's~\cite{ES}.) If $\Hilbn{X}$ is the Hilbert scheme parameterizing $n$-points in $X$, the generating function of the Poincar\'e polynomials is given by \begin{equation} \sum_{n=0}^\infty q^n P_t(\Hilbn{X}) = \prod_{m=1}^\infty \frac{(1 + t^{2m-1}q^m)^{b_1(X)}(1 + t^{2m+1}q^m)^{b_3(X)}} {(1 - t^{2m-2}q^m)^{b_0(X)}(1 - t^{2m}q^m)^{b_2(X)} (1 - t^{2m+2}q^m)^{b_4(X)}}\, , \label{Poincare}\end{equation} where $b_i(X)$ is the Betti number of $X$. Letting $t = -1$, we find the generating function of the Euler numbers is essentially the Dedekind eta function. In fact, the relation with the above formula and the Fock space was already pointed out in \cite{VW}. Our result should be considered as a geometric realization of their indication. The paper is organized as follows. In \secref{sec:pre} we give preliminaries. We recall the definition of the convolution product in \subsecref{subsec:conv} with some modifications and describe some properties of the Hilbert scheme $\HilbX{n}$ and the infinite Heisenberg algebra and its representations \S\S\ref{subsec:Hilb},\ref{subsec:Heisen}. The definition of correspondences and the statement of the main result are given in \secref{sec:main}. The proof will be given in \secref{sec:proof}. In the appendix, we study the particular case $X = {\Bbb C}^2$ in more detail. We give a description of $\Hilbn{({\Bbb C}^2)}$ as a hyper-K\"ahler quotient of finite dimensional vector space by a unitary group action. It is very similar to the definition of quiver varieties \cite{Na-quiver}. The only difference is that we have an edge joining a vertex with itself. Using this description as a hyper-K\"ahler quotient, we compute the homology group of $\Hilbn{({\Bbb C}^2)}$. We recover the formula \eqref{Poincare} for $X = {\Bbb C}^2$. The difference between our approach and Ellingsrud-Str\o mme's~\cite{ES} is only the description. Both use the torus action and study the fixed point set. But our presentation has a similarity in \cite{Na-homology}. The appendix is independent of the other parts of this paper, but those similarities with author's previous works explains motivation of this paper in part. While the author was preparing this manuscript, he learned that the similar result was announced by Grojnowski \cite{Gr}. He introduced exactly the same correspondence as ours. \subsection*{Acknowledgement} The author would like to thank C.~Vafa and E.~Witten, since it is clear that this work was not done unless he discussed with them. It is also a pleasure to acknowledge discussions with V.~Ginzburg and M.~Kapranov. His thanks go also to R.~Hotta, T.~Uzawa, K.~Hasegawa and G.~Kuroki who answered many questions on the representation theory. \section{Preliminaries} \label{sec:pre} \subsection{Convolution Algebras} \label{subsec:conv} We need a slight modification of the definition of the convolution product in the homology groups given by Ginzburg \cite{Gi} (see also \cite{Gi-book,Na-quiver}). For a locally compact topological space $X$, let $\Hlf_*(X)$ denote the homology group of possibly infinite singular chains with locally finite support (the Borel-Moore homology) with {\it rational\/} coefficients. The usual homology group of finite singular chains will be denoted by $H_*(X)$. If $\overline X = X\cup \{\infty\}$ is the one point compactification of $X$, we have $\Hlf_*(X)$ is isomorphic to the relative homology group $H_*(\overline X, \{\infty\})$. If $X$ is an $n$-dimensional oriented manifold, we have the Poincar\'e duality isomorphism \begin{equation} \Hlf_i(X) \cong H^{n-i}(X),\quad H_i(X) \cong H^{n-i}_c(X), \label{eq:PD}\end{equation} where $H^*$ and $H^*_c$ denote the ordinary cohomology group and the cohomology group with compact support respectively. Let $M^1$, $M^2$, $M^3$ be oriented manifolds of dimensions $d_1$, $d_2$, $d_3$ respectively, and $p_{ij}\colon M^1\times M^2\times M^3 \to M^i\times M^j$ be the natural projection. We define a convolution product \begin{equation*} \ast\colon \left(\Hlf_{i_1}(M^1)\otimes H_{i_2}(M^2)\right) \otimes \left(\Hlf_{d_2 - i_2}(M^2)\otimes H_{i_3}(M^3)\right) \to \Hlf_{i_1}(M^1)\otimes H_{i_3}(M^3) \end{equation*} by \begin{equation*} (c_1\otimes c_2) \ast (c'_2\otimes c_3) \overset{\operatorname{\scriptstyle def.}}{=} c_2\cap c'_2\; c_1\otimes c_3, \end{equation*} where $c_2\cap c'_2\in\Bbb Z$ is the natural pairing between $H_{i_2}(M^2)$ and $\Hlf_{d_2 - i_2}(M^2)\cong H^{i_2}(M^2)$. Suppose $Z$ is a submanifold in $M^1\times M^2$ such that \begin{equation} \text{the projection $Z \to M^1$ is proper.} \label{ass-proper} \end{equation} Then the fundamental class $[Z]$ defines an element in \begin{equation*} [Z] \in H_{\mathop{\text{\rm dim}}\nolimits_{\Bbb R} Z}(\overline M_1\times M_2, \{\infty\}\times M_2) = \bigoplus_{i + j = \mathop{\text{\rm dim}}\nolimits_{\Bbb R} Z} \Hlf_i(M^1)\otimes H_j(M^2), \end{equation*} where $\overline M_1 = M_1\cup\{\infty\}$ is the one point compactification of $M_1$ and we have used the K\"unneth formula. More generally, if $[Z]$ is a cycle whose support $Z$ satisfies \eqref{ass-proper}, the same construction works. Using \eqref{eq:PD}, we get an operator, which is denoted also by $[Z]$, \begin{equation*} [Z]\colon \Hlf_j(M^2)\to \Hlf_{j + \mathop{\text{\rm dim}}\nolimits_{\Bbb R} Z - d_2}(M^1). \end{equation*} \subsection{Hilbert Schemes of Points on Surfaces} \label{subsec:Hilb} Let $X$ be a nonsingular quasi-projective surface defined over the complex number ${\Bbb C}$. Let $\Hilbn{X}$ be the component of the Hilbert scheme of $X$ parameterizing the ideals of $\shfO_X$ of colength $n$. It is smooth and irreducible \cite{Fogarty}. Let $S^n X$ denotes the $n$-th symmetric product of $X$. It parameterizes formal linear combinations $\sum n_i [x_i]$ of points $x_i$ in $X$ with coefficients $n_i\in\Bbb Z_{> 0}$ with $\sum n_i = n$. There is a canonical morphism \begin{equation*} \pi\colon \Hilbn{X}\to S^n X; \quad \pi(\cal J) \overset{\operatorname{\scriptstyle def.}}{=} \sum_{x\in X} \operatorname{length}(\shfO_X/\cal J)_x [x]. \end{equation*} It is known that $\pi$ is a resolution of singularities. The symmetric power $S^n X$ has a natural stratification into locally closed subvarieties as follows. Let $\nu$ be a partition of $n$, i.e., a sequence $n_1, n_2, \dots, n_r$ such that \begin{equation*} n_1 \ge n_2 \ge \cdots \ge n_r, \qquad \sum n_i = n. \end{equation*} Then $S^n_\nu X$ is defined by \begin{equation*} S^n_\nu X \overset{\operatorname{\scriptstyle def.}}{=} \{ \sum_i n_i [x_i] \in S^n X\mid x_i \ne x_j \quad\text{for $i\ne j$}\, \}. \end{equation*} It is known that $\pi$ is semi-small with respect to the stratification $S^n X = \bigcup S^n_\nu X$ \cite{Iar}, that is \begin{enumerate} \item for each $\nu$, the restriction $\pi\colon \pi^{-1}(S^n_\nu X)\to S^n_\nu X$ is a locally trivial fibration, \item $\operatorname{codim} S^n_\nu X = 2\mathop{\text{\rm dim}}\nolimits \pi^{-1}(x)$ for $x\in S^n_\nu X$. \end{enumerate} Moreover, it is also known that $\pi^{-1}(x)$ is irreducible. \subsection{The Infinite Dimensional Heisenberg Algebra} \label{subsec:Heisen} We briefly recall the definition of the infinite dimensional Heisenberg algebra and its representations. See \cite[\S9.13]{Kac} for detail. The infinite dimensional Heisenberg algebra $\frak s$ is generated by $p_i$, $q_i$ ($i=1,2,\dots$) and $c$ with the following relations: \begin{gather} [p_i, p_j] = 0, \quad [q_i, q_j] = 0 \label{eq:rel1}\\ [p_i, q_j] = \delta_{ij} c. \label{eq:rel2} \end{gather} For every $a\in{\Bbb C}^*$, the Lie algebra $\frak s$ has an irreducible representation on the space $R = {\Bbb C}[x_1, x_2,\dots]$ of polynomials in infinitely many indeterminates $x_i$ defined by \begin{equation*} p_i \mapsto a \frac{\partial}{\partial x_i}, \quad q_i \mapsto x_i, \quad c \mapsto a \operatorname{Id}. \end{equation*} This representation has a highest weight vector $1$, and $R$ is spanned by elements \begin{equation*} x_1^{j_1} x_2^{j_2} \cdots x_n^{j_n} = q_1^{j_1} q_2^{j_2} \cdots q_n^{j_n} 1. \end{equation*} We extend $\frak s$ by a derivation $d_0$ defined by \begin{equation*} [d_0, q_j] = j q_j, \quad [d_0, p_j] = - j p_j. \end{equation*} The above representation $R$ extends by \begin{equation*} d_0 \mapsto \sum_j jx_j\frac{\partial}{\partial x_j}. \end{equation*} Then it is easy to see \begin{equation} \operatorname{tr}_R q^{d_0} = \prod_{j=1}^\infty \frac 1{(1 - q^j)}. \label{eq:char} \end{equation} The representation $R$ carries a unique bilinear form $B$ such that $B(1, 1) = 1$ and $p_i$ is the adjoint of $q_i$, provided $a\in {\Bbb R}$. In fact, distinct monomials are orthogonal and we have \begin{equation*} B(x_1^{j_1}\dots x_n^{j_n}, x_1^{j_1}\dots x_n^{j_n}) = a^{\sum j_k} \prod j_k ! \end{equation*} We also need the infinite dimensional Clifford algebra $\Cl$ (see e.g., \cite{Fr}). It is generated by $\psi_i$, $\psi^*_i$ ($i=1,2,\dots$) and $c$ with the relations \begin{gather} \psi_i\psi_j + \psi_j \psi_i = 0, \quad \psi_i^*\psi_j^* + \psi_j^* \psi_i^* = 0\label{eq:rel3}\\ \psi_i\psi_j^* + \psi_j^*\psi_i = \delta_{ij}c.\label{eq:rel4} \end{gather} This algebra has a representation on the exterior algebra $F = \bigwedge^* V$ of an infinite dimensional vector space $V = {\Bbb C} dx^1\oplus {\Bbb C} dx^2\oplus\cdots$ defined by \begin{equation*} \psi_i \mapsto dx^i \wedge\, , \quad \psi_i^* \mapsto \frac{\partial}{\partial x_i}\interior\, ,\quad c \mapsto \operatorname{Id}, \end{equation*} where $\interior$ denotes the interior product. This has the highest weight vector $1$ and spanned by \begin{equation*} dx_{i_1}\wedge\cdots\wedge dx_{i_n} = \psi_{i_1}\cdots \psi_{i_n} 1, \qquad (i_1 > i_2 > \cdots > i_n). \end{equation*} We extend $\Cl$ by $d$ defined by \begin{equation*} [d, \psi_i] = i\psi_i, \quad [d, \psi_i^*] = -i\psi_i. \end{equation*} It acts on $F$ by \begin{equation*} d(dx_{i_1}\wedge\cdots\wedge dx_{i_n}) = \left(\sum i_k\right) dx_{i_1}\wedge\cdots\wedge dx_{i_n}. \end{equation*} The character is given by \begin{equation*} \operatorname{tr}_F q^d = \prod_{j=1}^\infty (1 + q^j). \end{equation*} \section{Main Construction} \label{sec:main} \subsection{Definitions of Generators} Let $X$ as in \subsecref{subsec:Hilb}. Take a basis of $\Hlf_*(X)$ and assume that each element is represented by a (real) closed submanifold $C^a$. ($a$ runs over $1, 2, \dots, \mathop{\text{\rm dim}}\nolimits \Hlf_*(X)$.) Take a dual basis for $H_*(X)\cong H^{4 - *}_c(X)$, and assume that each element is represented by a submanifold $D^a$ which is compact. (Those assumptions are only for the brevity. The modification to the case of cycles is clear.) For each $a = 1,2,\dots,\mathop{\text{\rm dim}}\nolimits\Hlf_*(X)$, $n = 1,2,\dots$ and $i=1,2,\dots$, we introduce cycles of products of the Hilbert schemes by \begin{equation*} \begin{split} &E_i^a(n) \overset{\operatorname{\scriptstyle def.}}{=} \{\, (\cal J_1,\cal J_2)\in\HilbX{n-i}\times\HilbX{n} \mid \\ & \qquad\qquad\qquad\qquad\text{$\cal J_1\supset \cal J_2$ and $\Supp(\cal J_1/\cal J_2) = \{ p\}$ for some $p\in D^a$} \}, \\ &F_i^a(n) \overset{\operatorname{\scriptstyle def.}}{=} \{\, (\cal J_1,\cal J_2)\in\HilbX{n+i}\times\HilbX{n} \mid \\ & \qquad\qquad\qquad\qquad\text{$\cal J_1\subset \cal J_2$ and $\Supp (J_2/\cal J_1) = \{ p \}$ for some $p\in C^a$}\}. \end{split} \end{equation*} The dimensions are given by \begin{align*} & \mathop{\text{\rm dim}}\nolimits_{\Bbb R} E_i^a(n) = 4(n-i) + 2(i-1) + \mathop{\text{\rm dim}}\nolimits_{\Bbb R} D^a,\\ & \mathop{\text{\rm dim}}\nolimits_{\Bbb R} F_i^a(n) = 4n + 2(i-1) + \mathop{\text{\rm dim}}\nolimits_{\Bbb R} C^a. \end{align*} This follows from the fact $\Hilbn{X}\to S^n X$ is semi-small (see \subsecref{subsec:Hilb}). Since the projections $E_i^a(n)\to \HilbX{n-i}$ and $F_i^a(n)\to\HilbX{n+i}$ are proper, we have classes \begin{align*} & [E_i^a(n)]\in \bigoplus_{k,l} \Hlf_k(\HilbX{n-i})\otimes H_l(\HilbX{n}), \\ & [F_i^a(n)]\in \bigoplus_{k,l} \Hlf_k(\HilbX{n+i})\otimes H_l(\HilbX{n}). \end{align*} Our main result is the following: \begin{Theorem} The following relations hold in $\bigoplus_{k,l,m,n} \Hlf_k(\HilbX{m})\otimes H_l(\HilbX{n})$. \begin{gather} [E_i^a(n-j)]\ast [E_j^b(n)] = (-1)^{\mathop{\text{\rm dim}}\nolimits D^a\mathop{\text{\rm dim}}\nolimits D^b}[E_j^b(n-i)] \ast [E_i^a(n)]\label{eq:EE}\\ [F_i^a(n+j)]\ast [F_j^b(n)] = (-1)^{\mathop{\text{\rm dim}}\nolimits C^a\mathop{\text{\rm dim}}\nolimits C^b}[F_j^b(n-i)] \ast [F_i^a(n)]\label{eq:FF}\\ [E_i^a(n+j)]\ast [F_j^b(n)] = (-1)^{\mathop{\text{\rm dim}}\nolimits D^a\mathop{\text{\rm dim}}\nolimits C^b}[F_j^b(n-i)] \ast [E_i^a(n)] + \delta_{ab}\delta_{ij}c_i [\Delta(n)]\label{eq:EF}, \end{gather} where $\Delta(n)$ is the diagonal of $\Hilbn{X}$, and $c_i$ is a nonzero integer depending only on $i$ \rom(independent of $X$\rom). \label{th:main}\end{Theorem} In particular, for each fixed $a$, the map \begin{align*} & p_i \mapsto \sum_n [E_i^a(n)], \quad q_i \mapsto \sum_n [F_i^a(n)] \qquad\text{when $\mathop{\text{\rm dim}}\nolimits C^a$ is even} \\ & \psi_i^*\mapsto \sum_n [E_i^a(n)], \quad \psi_i\mapsto \sum_n [F_i^a(n)] \qquad\text{when $\mathop{\text{\rm dim}}\nolimits C^a$ is odd} \end{align*} defines a homomorphism from the Heisenberg algebra and the Clifford algebra respectively. Considering $[E_i^a(n)]$, $[F_i^a(n)]$ as operators on $\bigoplus_{k,n} \Hlf_k(\HilbX{n})$, we have a representation of the product of Heisenberg algebras and Clifford algebras. Comparing G\"ottsche's Betti number formula and the character formula, we get the following: \begin{Theorem} The direct sum $\bigoplus_{k,n} \Hlf_k(\HilbX{n})$ of homology groups of $\HilbX{n}$ is the highest weight module where the highest weight vector $v_0$ is the generator of $\Hlf_0(\HilbX{0})\cong\Bbb Q$. \end{Theorem} \begin{Remark} The author does not know the precise values of $c_i$'s. It is easy to get $c_1 = 1$, $c_2 = -2$, but general $c_i$ become difficult to calculate. \end{Remark} \section{Proof of {\protect\thmref{th:main}}} \label{sec:proof} \subsection{Proof of Relations (I)} Consider the product $\HilbX{n-i-j}\times\HilbX{n-j}\times\HilbX{n}$ and let $p_{12}$, etc. be as in \subsecref{subsec:conv}. The intersection $p_{12}^{-1}(E_i^a(n-j))\cap p_{23}^{-1}(E_j^b(n))$ consists of triples $(\idl_1, \idl_2, \idl_3)$ such that \begin{align} & \idl_1\supset\idl_2\supset\idl_3 \label{eq:incl}\\ & \text{$\Supp(\idl_1/\idl_2) = \{ p\}$, \; $\Supp(\idl_2/\idl_3) = \{ q\}$ for some $p\in D^a$, $q\in D^b$.} \label{eq:supp} \end{align} Replacing $D^b$ by $\widetilde D^b$ in the same homology class, we may assume $\mathop{\text{\rm dim}}\nolimits D^a\cap \widetilde D^b = \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits \widetilde D^b - 4$. (If the right hand side is negative, the set is empty.) Let $U$ be the open set in the intersection consisting points with $p\ne q$ in \eqref{eq:supp}. Outside the singular points of $p_{12}^{-1}(E_i^a(n-j))$, $p_{23}^{-1}(E_j^b(n))$, the intersection is transverse along $U$. The complement $p_{12}^{-1}(E_i^a(n-j))\cap p_{23}^{-1}(E_j^b(n))\setminus U$ consists of $(\idl_1, \idl_2, \idl_3)$ with \eqref{eq:incl} and \begin{equation*} \text{$\Supp(\idl_1/\idl_2) = \Supp(\idl_2/\idl_3) = \{p\}$ for some $p\in D^a\cap \widetilde D^b$.} \end{equation*} Its dimension is at most \begin{equation*} 4n - 2i - 2j - 4 + \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits D^b - 4, \end{equation*} which is strictly smaller than the dimension of the intersection \begin{equation*} 4n - 2i - 2j - 4 + \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits D^b. \end{equation*} Now consider the product $\HilbX{n-i-j}\times\HilbX{n-i}\times\HilbX{n}$. The intersection $p_{12}^{-1}(E_j^b(n-i))\cap p_{23}^{-1}(E_i^a(n))$ consists of triples $(\idl_1, \idl'_2, \idl_3)$ such that \begin{align} & \idl_1\supset\idl'_2\supset\idl_3 \label{eq:incl2}\\ & \text{$\Supp(\idl_1/\idl'_2) = \{q\}$ \; $\Supp(\idl'_2/\idl_3) = \{p\}$ for some $q\in D^b$, $p\in D^a$.} \label{eq:supp2} \end{align} Let $U'$ be the open set in the intersection consisting points with $p\ne q$ in \eqref{eq:supp2}. The intersection is again transverse along $U$ outside singular sets. The complement $p_{12}^{-1}(E_j^b(n-i))\cap p_{23}^{-1}(E_i^a(n))\setminus U'$ has dimension is at most \begin{equation*} 4n - 2i - 2j - 4 + \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits D^b - 4, \end{equation*} which is also strictly smaller than the dimension of the intersection. There exists a homeomorphism between $U$ and $U'$ given by \begin{equation*} U\ni (\idl_1, \idl_2, \idl_3) \mapsto (\idl_1, \idl'_2, \idl_3) \in U', \end{equation*} where $\idl'_2$ is a sheaf such that \begin{equation*} \idl_1 / \idl'_2 = \idl_2 / \idl_3, \quad \idl'_2 / \idl_3 = \idl_1 / \idl_2. \end{equation*} Such $\idl'_2$ exists since supports of $\idl_1 / \idl_2$ and $\idl_2 / \idl_3$ are different points $p$ and $q$. Taking account of orientations and the estimate of the dimension of the complements, we get the relation~\eqref{eq:EE}. The proof of \eqref{eq:FF} is exactly the same. \subsection{Proof of Relations (II)} The proof of \eqref{eq:EF} is almost similar to the above. Consider the product $\HilbX{n-i+j}\times\HilbX{n+j}\times\HilbX{n}$. The intersection $p_{12}^{-1}(E_i^a(n+j))\cap p_{23}^{-1}(F_j^b(n))$ consists of triples $(\idl_1, \idl_2, \idl_3)$ such that \begin{align} & \idl_1\supset\idl_2\subset\idl_3 \label{eq:incl3}\\ & \text{$\Supp(\idl_1/\idl_2) = \{p\}$, \; $\Supp(\idl_3/\idl_2) = \{q\}$ for some $p\in D^a$, $q\in C^b$.} \label{eq:supp3} \end{align} Replacing $C^b$ by $\widetilde C^b$ in the same homology class, we may assume $\mathop{\text{\rm dim}}\nolimits D^a\cap \widetilde C^b = \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits \widetilde C^b - 4$. (If the right hand side is negative, the set is empty.) Since $\{D^a\}$ and $\{C^b\}$ are dual bases each other, the equality holds if and only if $a = b$. Let $U$ be the open set in the intersection consisting points with $p\ne q$ in \eqref{eq:supp3}. Next consider the product $\HilbX{n-i+j}\times\HilbX{n-i}\times\HilbX{n}$. The intersection $p_{12}^{-1}(F_j^b(n-i))\cap p_{23}^{-1}(E_i^a(n))$ consists of triples $(\idl_1, \idl_2, \idl_3)$ such that \begin{align} & \idl_1\subset\idl'_2\supset\idl_3 \label{eq:incl4}\\ & \text{$\Supp(\idl'_2/\idl_1) = \{q\}$, \; $\Supp(\idl'_2/\idl_3) = \{p\}$ for some $q\in C^b$, $p\in D^a$.} \label{eq:supp4} \end{align} Let $U'$ be the open set in the intersection consisting points with $p\ne q$ in \eqref{eq:supp4}. There exists a homeomorphism between $U$ and $U'$ given by \begin{equation*} U\ni (\idl_1, \idl_2, \idl_3) \mapsto (\idl_1, \idl'_2, \idl_3) \in U', \end{equation*} where $\idl'_2$ is \begin{equation*} (\idl_1 \oplus \idl_3)/ \{ (f, f)\mid f\in \idl_2\} . \end{equation*} The inverse map is given by $\idl_2 = \idl_1 \cap \idl_3$. Let $U^c$, $U^{\prime c}$ be the complement of $U$ and $U'$ respectively. If $(\idl_1, \idl_3)$ is in the image $p_{13}(U^c)$ or $p_{13}(U^{\prime c})$, then $\idl_1$ and $\idl_3$ are isomorphic outside a point $p\in D^a\cap \widetilde C^b$. In particular, it is easy to check \begin{equation*} \mathop{\text{\rm dim}}\nolimits p_{13}(U^c), \, \mathop{\text{\rm dim}}\nolimits p_{13}(U^{\prime c}) \le 4n - 2i + 2j - 4 + \mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits C^b, \end{equation*} where the right hand side is the expected dimension of the intersection. The equality holds only if $i = j$ and $\mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits C^b = 4$. Moreover, $\{D^a\}$ and $\{C^b\}$ are dual bases, when $\mathop{\text{\rm dim}}\nolimits D^a + \mathop{\text{\rm dim}}\nolimits C^b = 4$, the intersection $D^a\cap \widetilde C^b$ is empty unless $a = b$. Thus we have checked \eqref{eq:EF} when $i\ne j$ or $a\ne b$. Now assume $i = j$ and $a = b$. Then $p_{13}(U^{\prime c})$ has smaller dimension and $p_{13}(U^c)$ is union of the diagonal $\Delta(n)$ and smaller dimensional sets. Hence the left hand side of \eqref{eq:EF} is a multiple of $[\Delta(n)]$. In order to calculate the multiple, we may restrict the intersection on the open set where \begin{enumerate} \item $\idl_1$ and $\idl_3$ are contained in the open stratum $\pi^{-1}(S^n_{1,1,\dots,1})$, \item $\Supp\shfO/\idl_1$, $\Supp\shfO/\idl_3$ do not intersect with $D^a\cap \widetilde C^a$. \end{enumerate} Then it is clear that the multiple is a constant independent of $n$ and $X$, which we denoted by $c_i$. The only thing left is to show $c_i \ne 0$. We may assume $X = {\Bbb C}^2$ and $n = i$. We consider the quotient of $\Hilb{({\Bbb C}^2)}{i}$ devided by the action of ${\Bbb C}^2$ which comes from the parallel translation. Thus $c_i$ is equal to the self-intersection number of $[\pi^{-1}(i[0])]$ in $\Hilb{({\Bbb C}^2)}{i}/{\Bbb C}^2$. Since $\pi\colon \Hilb{({\Bbb C}^2)}{i}\to S^i{\Bbb C}^2$ has irreducible fibers, $[\pi^{-1}(i[0])]$ is the generator of $H_{2i - 2}(\Hilb{({\Bbb C}^2)}{i}/{\Bbb C}^2)$. Now our assertion follows from a general result which holds for any semi-small morphism \cite[7.7.15]{Gi-book}. That is the non-degeneracy of the intersection form on the top degree of the fiber.

9507

alg-geom/9507004

en

https://arxiv.org/abs/alg-geom/9507004

alg-geom/9507004

Mikhail Zaidenberg

H. Flenner and M. Zaidenberg

On a class of rational cuspidal plane curves

LaTeX 30 pages, author-supplied DVI file available at http://www.math.duke.edu/preprints/95-00.dvi

Duke preprint DUKE-M-95-00

We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.

alg-geom

\section{On multiplicity sequences} \noindent {\bf 1.1. Definition.} Let $(C, \,P) \subset ({\bf C}^2,\,P)$ be an irreducible analytic plane curve germ, and let $${\bf C}^2 = V_0 \qquad {\stackrel{\sigma_1}{\longleftarrow}} \qquad V_1 \qquad {\stackrel{\sigma_2}{\longleftarrow}} \qquad \cdots \qquad {\stackrel{\sigma_n}{\longleftarrow}} \qquad V_n$$ be the sequence of blow ups over $P$ that yields the minimal embedded resolution of singularity of $C$ at $P$. Thus, the complete preimage of $C$ in $V_n$ is a simple normal crossing divisor $D = E + C_n$, where $E$ is the exceptional divisor of the whole resolution and $C_n$ is the proper preimage of $C$ in $V_n$. Denote by $E_n$ the only $-1$-component of $E$, so that $E_n \cdot (D_{\rm red} - E_n) \ge 3$. Let $E_i \subset V_i$ be the exceptional divisor of the blow up $\sigma_i$, $C_i \subset V_i$ be the proper transform of $C$ at $V_i$, and let $P_{i-1} = \sigma_i(E_i) \in E_{i-1} \cap C_{i-1}$ be the centrum of $\sigma_i$. Thus, $C = C_0 \subset V_0$ and $P = P_0 \in C_0$. Let $m_i$ denote the multiplicity of the point $P_i \in C_i$. The sequence ${\bar m}_P = (m_0,\,m_1,\dots,m_n)$, where $m_0\ge m_1 \ge \dots \ge m_n=1$, is called {\it the multiplicity sequence of $(C,\,P)$}. We have $$\mu = 2\delta = \sum\limits_{i=0}^n m_i(m_i - 1)\,,$$ where $\mu$ is the Milnor number of $(C,\,P)$ and $\delta$ is the virtual number of double points of $C$ at $P$ [Mil]. \\ The following proposition gives a characterization of the multiplicity sequences. \\ \noindent {\bf 1.2. Proposition.} {\it The multiplicity sequence ${\bar m}_P = (m_0,\,m_1,\dots,m_n)$ has the following two properties: \noindent i) for each $i = 1,\dots,n$ there exists $k \ge 0$ such that $$m_{i-1} = m_i + \dots +m_{i+k}\,,$$ where $$m_i = m_{i+1} =\dots =m_{i+k-1}\,,$$ and \noindent ii) if $$m_{n-r} > m_{n-r+1}=\dots=m_n=1\,,$$ then $m_{n-r} = r-1$. Conversely, if ${\bar m} = (m_0,\,m_1,\dots,m_n)$ is a non--increasing sequence of positive integers satisfying conditions i) and ii), then ${\bar m} = {\bar m}_P$ for some irreducible plane curve germ $(C,\,P)$. }\\ The proof is based on the following lemma.\\ \noindent {\bf 1.3. Lemma.} {\it Let ${\bar m}_P = (m_0,\,m_1,\dots,m_n)$ be the multiplicity sequence of an irreducible plane curve singularity $(C,\,P)$. Denote by $E_i^{(k)}$ the proper transform of the exceptional divisor $E_i$ of $\sigma_i$ at the surface $V_{i+k}$, so that, in particular, $E_i = E_i^{(0)}$. Then the following hold.\\ \noindent a) $E_iC_i = m_{i-1}$ and $$E_i^{(k)}C_{i+k} = {\rm max\,}\{0,\,m_{i-1} - m_i -\dots - m_{i+k-1}\}\,, \,\,k> 0\,.$$ In particular, $ E_i^{(1)} C_{i+1} = m_{i-1} - m_i$. \\ \noindent b) If $$m_{i-1} > m_i + \dots + m_{i+k-1}\,,$$ then $$m_i = m_{i+1} =\dots=m_{i+k-1}$$ and $$m_{i-1} \ge m_i + \dots + m_{i+k}\,.$$} \noindent {\bf Proof.} a) From the equalities $C_{i-1}^* := \sigma_i^*(C_{i-1}) = C_i + m_{i-1}E_i\,,\,\,E_i^2 = -1$ and $C_{i-1}^* E_i = 0$ it follows that $C_iE_i = m_{i-1}$. Assume by induction that a) holds for $k \le r-1$, where $r \ge 1$. If $C_{i+r}E_i^{(r)} > 0$, then $ C_{i+r-1}E_i^{(r-1)} > 0$ and $P_{i+r-1} \in C_{i+r-1} \cap E_i^{(r-1)}$. Therefore, by induction hypothesis we have $$C_{i+r-1} E_i^{(r-1)} = m_{i-1} - m_i - \dots - m_{i+r-2} > 0\,,$$ $C_{i+r} = C_{i+r-1}^* - m_{i+r-1} E_{i+r}$ and $E_i^{(r)} \cdot E_{i+r} = 1$. Hence, $E_i^{(r)} C_{i+r} = E_i^{(r)} C_{i+r-1}^* - m_{i+r-1} E_{i+r} E_i^{(r)} = E_i^{(r-1)} C_{i+r-1} - m_{i+r-1} = m_{i-1} - m_i -\dots -m_{i+r-1}$. This proves (a), and also proves that $$m_{i-1} \ge m_i +\dots+m_{i+r-1}$$ if $$m_{i-1} > m_i +\dots +m_{i+r-2}\,,$$ which is the second assertion of (b). To prove the first assertion of (b), note that $E_i^{(r-1)}$ is tangent to $C_{i+r-1}$ at the point $P_{i+r-1}$ iff $ E_i^{(r-1)} C_{i+r-1} > m_{i+r-1}$. As it was done in the proof of (a), one can easily show that the latter is equivalent to the inequality $$E_i^{(r)} C_{i+r} = m_{i-1} - m_i -\dots - m_{i+r-1} > 0\,,$$ and it implies in turn that $E_i^{(k)}$ is tangent to $C_{i+k}$ for each $k=0,\dots,r-1$. Since by (a) $C_{i+k} E_{i+k} = m_{i+k-1}$, the inequality $m_{i+k-1} > m_{i+k}$, where $1 \le k \le r-1$, would mean that the curve $E_{i+k}$ is tangent to $C_{i+k}$ at $P_{i+k}$, which is impossible, since it is transversal to $E_i^{(k)}$. Therefore, $m_{i+k-1} = m_{i+k}$ for all $k=1, \dots, r-1$. \hfill $\Box$\\ \noindent {\bf Proof of Proposition 1.2.} Let ${\bar m}_P = (m_0,\,m_1,\dots,m_n)$ be the multiplicity sequence of an irreducible plane curve singularity $(C,\,P)$. Write $m_{i-1} = k_im_i + r_i$ with $0 \le r_i < m_i$. It follows from Lemma 1.3(b) that $$m_i = m_{i+1} = \dots = m_{i+k_i -1}\,.$$ Thus, if $r_i=0$, then the condition i) is fulfilled. If $r_i>0$, then $m_{i-1} > k_im_i = m_i+\dots+ m_{i+k_i-1}$, so that by Lemma 1.3(b) we have $$m_{i-1} \ge k_im_i + m_{i+k_i}\,,$$ and whence $r_i \ge m_{i+k_i}$. But $r_i > m_{i+k_i}$ would imply that $$m_{i-1} > m_i+\dots+m_{i+k_i}\,,$$ which in turn implies by Lemma 1.3(b) that $$m_i =\dots=m_{i+k_i} < r_i\,,$$ which is a contradiction. Therefore, in this case $m_{i+k_i} = r_i$, and so $$m_{i-1} = m_i + \dots +m_{i+k_i-1}+m_{i+k_i}\,,$$ where $$m_i=\dots=m_{i+k_i-1}\,.$$ The proof of (ii) is easy, and so it is omited.\\ To prove the converse, we need the following lemma. For the moment we change the convention and define the multiplicity sequences to be infinite, setting $m_{\nu} = 1$ for all $\nu \ge n$. Thus, the sequence $(1,\,1,\,\dots)$ serves as multiplicity sequence of a smooth germ. \\ \noindent {\bf 1.4. Lemma.} {\it Let $(C,\,P)$ be an irreducible plane curve germ with multiplicity sequence ${\bar m}_P = (m_0,\,m_1,\dots,m_n, \dots)$. Then there exists a germ of a smooth curve $(\Gamma,\,P)$ through $P$ with $(\Gamma C)_P = k$ iff $k$ satisfies the condition \\ \noindent (*) $k = m_0 + m_1 + \dots + m_s\,\,\,$ for some $\,\,\,s > 0\,$ with $\,\,m_0 = m_1 = \dots = m_{s-1}\,.$}\\ \noindent {\it Proof.} We proceed by induction on the number of $m_{\nu}$ which are bigger than $1$. If it is equal to zero, i.e. if $(C,\,P)$ is a smooth germ, then our statement is evidently true. Let $(\Gamma,\,P) \subset (V_0,\,P)$ be a smooth curve germ through $P$, and let $\Gamma' \subset V_1$ be the proper transform of $\Gamma$. Then $C^* = C_1 + m_0 E_1$, and so $$k = (\Gamma C)_P = \Gamma' C_1 + m_0 \Gamma' E_1 = \Gamma' C_1 + m_0\,.$$ If $\Gamma' C_1 = 0$, then we are done. If not, then by induction hypothesis (applied to $C_1$) we have $$\Gamma' C_1 = m_1 + \dots + m_s$$ for some $s > 0$ and $m_1 = \dots = m_{s-1}$. If $s = 1$ then this proves the Lemma. If $s > 1$, i.e. $k = m_0 + m_1 + m_2 +\dots$, then we have to show that $m_0 = m_1$. Denote by $\Gamma''$ the proper transform of $\Gamma'$ on $V_2$. We have, as above, $$k - m_0 = \Gamma' C_1 = \Gamma'' C_2 + m_1\,,$$ which yields that $\Gamma'' C_2 = k - m_0 - m_1 > 0$, i.e. $\Gamma''$ meets $C_2$. Moreover, since $\Gamma' C_1 = m_1 + m_2 +\dots > m_1$, $\Gamma'$ is tangent to $C_1$ at $P_1 \in C_1$, and hence $P_2 \in \Gamma''$. Since $\Gamma'$ meets $E_1$ transversally, $\Gamma''$ does not meet the proper transform $E_1^{(1)}$ of $E_1$ in $V_2$. This means that $\Gamma''$ and $E_1^{(1)}$ meet $E_2$ in different points, and therefore $E_1^{(1)} C_2 = 0$. By Lemma 1.3(a) we have $E_1^{(1)} C_2 = m_0 - m_1$; thus, $m_0 = m_1$. This completes the proof in one direction. Conversely, assume that $k$ satisfies (*). Then $k - m_0$ satisfies (*) with respect to $(C_1,\,P_1)$. If $k = m_0$, then any generic smooth curve $\Gamma$ through $P=P_0$ satisfies the condition $(\Gamma C)_P = k = m_0$. If $k - m_0 > 0$, then by inductive hypothesis there is a smooth curve germ $\Gamma' \subset V_1$ through $P_1$ with $\Gamma' C_1 = k-m_0$. Let $\Gamma$ be the image of $\Gamma'$ in $V$. Then $\Gamma C = \Gamma' C_1 + m_0 \Gamma' E_1$. If $k - m_0 = m_1$, then $\Gamma'$ can be chosen generically, so transversally to $E_1$, and thus we have $\Gamma C = k$. If $k - m_0 > m_1$, then as above $\Gamma'' C_2 = k - m_0 - m_1 > 0$ and so $\Gamma'' E_1^{(1)} = 0$, which implies that $\Gamma' E_1 = 1$. Hence, $\Gamma C = k$ also in this case. The lemma is proven. \hfill $\Box$ \\ Returning to the proof of Proposition 1.2, fix a non-increasing sequence ${\bar m} = (m_0,\,m_1,\dots,m_n)$ that satisfies (i) and (ii). Note that the sequence ${\bar m}' := (m_1,\dots,m_n)$ satisfies the same assumptions. Let $\sigma_1\,:\,V_1 \to V_0 = {\bf C}^2$ be the blow up at the point $P \in {\bf C}^2$. Fix a point $P_1 \in E_1 = \sigma_1^{-1} (P) \subset V_1$. Consider first the case when $m_1 > 1$. We may assume by induction that there exists an irreducible plane curve germ $(C_1,\,P_1)$ with multiplicity sequence ${\bar m}_{P_1} = {\bar m}' = (m_1,\dots,m_n)$. Since $\bar m$ satisfies (i) and (ii), from Lemma 1.4 it easily follows that there is an embedding $(C_1,\,P_1) \hookrightarrow (V_1,\,P_1)$ such that $(E_1 C_1)_{P_1} = m_0$. Then obviously $C := \sigma_1 (C_1) \subset {\bf C}^2$ is a plane curve singularity with multiplicity sequence ${\bar m}_P = {\bar m} = (m_0,\,m_1,\dots,m_n)$. Finally, assume that $m_1 = 1$. Choose $C_1 \subset V_1$ to be a smooth curve with $(C_1 E_1)_{P_1} = m_0$. Then again $C := \sigma_1 (C_1) \subset {\bf C}^2$ has multiplicity sequence ${\bar m}_P = {\bar m} = (m_0,\,m_1,\dots,m_n)$, as desired. This proves Proposition 1.2. \hfill $\Box$ \\ \noindent {\bf 1.5. Remark.} It is well known that the multiplicity sequence carries the same information as the Puiseux characteristic sequence, i.e. each of them can be computed in terms of the other [MaSa]. Moreover, the multiplicity sequence determines the weighted dual graph of the embedded resolution of the cusp and vice versa. This easily follows from the proofs of (1.2) and (1.3), see also [EiNe] or [OZ1,2]. \\ \noindent {\bf 1.6. } Let $f\,:\,{\cal X} \to S$ be a flat family of irreducible plane curve singularities, i.e. there is a diagram \begin{center} \begin{picture}(1000,60) \thicklines \put(220,5){$S $} \put(174,45){${\cal X}$} \put(262,45){${\bf C}^2 \times S$} \put(220,45){$\hookrightarrow$} \put(185,38){\vector(1,-1){20}} \put(270,38){\vector(-1,-1){20}} \put(175,22){$f$} \put(270,22){pr} \end{picture} \end{center} \noindent and a subvariety $\Sigma \subset {\cal X}$ such that $f\,|\,\Sigma\,:\,\Sigma \to S$ is (set theoretically) bijective, $f\,|\,{\cal X} \setminus \Sigma\,:\, {\cal X} \setminus \Sigma \to S$ is smooth and the fibre $X_s := f^{-1}(s)$ has a cusp at the point $\{x_s\} = X_s \cap \Sigma$. We say that the family $f$ is {\it equisingular} if it possesses a simultaneous resolution, i.e. there is a diagram \begin{center} \begin{picture}(1000,80) \thicklines \put(168,75){$\tilde {\cal X}$} \put(268,75){$\cal Z$} \put(220,75){$\hookrightarrow$} \put(166,37){${\cal X}$} \put(250,37){${\bf C}^2 \times S$} \put(220,37){$\hookrightarrow$} \put(172,70){\vector(0,-1){20}} \put(272,70){\vector(0,-1){20}} \put(162,60){$\pi$} \put(277,60){$\pi$} \put(181,32){\vector(1,-1){19}} \put(262,32){\vector(-1,-1){19}} \put(220,3){$S$} \put(172,17){$f$} \put(269,17){pr} \end{picture} \end{center} \noindent where ${\cal Z}$ is smooth over $S$ and for each $s \in S$ the induced diagram of the fibres \begin{center} \begin{picture}(1000,65) \thicklines \put(168,55){$\tilde X_s$} \put(268,55){$ Z_s$} \put(220,55){$\hookrightarrow$} \put(167,17){$X_s$} \put(267,17){${\bf C}^2$} \put(220,17){$\hookrightarrow$} \put(173,50){\vector(0,-1){20}} \put(272,50){\vector(0,-1){20}} \put(162,40){$\pi$} \put(277,40){$\pi$} \end{picture} \end{center} \noindent yields an embedded resolution of $X_s$ in such a way that the weighted dual graphs of $\pi^{-1} (X_s)$ are all the same. Observe that if the family $f$ is equisingular, then all the cusps $(X_s,\, x_s)$ have the same multiplicity sequence, see (1.5). Vice versa, we have the following simple lemma, which will be useful in the next section.\\ \noindent {\bf 1.7. Lemma. } {\it Let $f\,:\,{\cal X} \to S$ be a flat family of irreducible plane curve singularities. Assume that $S$ is normal and all the cusps $(X_s,\,x_s)$, $s \in \Sigma$, have the same multiplicity sequence. Then the family $f$ is equisingular. } \\ \noindent {\bf Proof.} Note that $\Sigma$ is necessarily normal and $f\,|\,\Sigma \,:\,\Sigma \to S$ is an isomorphism. Blowing up $\Sigma$ gives a morphism $\pi_1\,:\,{\cal Z}_1 \to {\bf C}^2 \times S$ whose restriction to the fibre over $s$ yields the blowing up of ${\bf C}^2$ at $x_s$. Then the proper transform ${\cal X}_1$ of ${\cal X}$ in ${\cal Z}_1$ is the blowing up $\pi\,|\,{\cal X}_1 \,:\,{\cal X}_1 \to {\cal X}$ along $\Sigma$. The singular set of the induced map ${\cal X}_1 \to S$ is a subvariety $\Sigma_1$ mapped one--to--one onto $S$. Repeating the procedure and using the fact that all multiplicity sequences of the cusps $(X_s,\,x_s)$ are the same, leads to a simultaneous resolution of $f$ as above. \hfill $\Box$ \section{Computation of deformation invariants in terms of multiplicity sequences} \noindent {\bf 2.1. On the Rigidity Problem.} Consider a minimal smooth completion $V$ of an open surface $X = V \setminus D$ by a simple normal crossing (SNC for short) divisor $D$. Let $\Theta_V\langle \, D \, \rangle$ be the logarithmic tangent bundle. By [FZ] the groups $ H^i ( \Theta_V\langle \, D \, \rangle)$ control the deformations of the pair $(V,\,D)$; more precisely, $ H^0 ( \Theta_V\langle \, D \, \rangle)$ is the space of its infinitesimal automorphisms, $ H^1 ( \Theta_V\langle \, D \, \rangle)$ is the space of infinitesimal deformations and $ H^2 ( \Theta_V\langle \, D \, \rangle)$ gives the obstructions for extending infinitesimal deformations. In [FZ, Lemma 1.3] we proved that if $X$ is a $\bf Q$--acyclic surface, i.e. $H_i(X; {\bf Q}) = 0,\,i > 0$, then the Euler characteristic of $\Theta_V\langle \, D \, \rangle$ is equal to $K_V(K_V + D)$. If, in addition, $X$ is of log--general type, i.e. its log--Kodaira dimension $\bar {k} (X)= 2$, then $h^0 ( \Theta_V\langle \, D \, \rangle) = 0$ (indeed, by Iitaka's theorem [Ii,\,Theorem 6] the automorphism group of a surface $X$ of log--general type is finite). We conjectured in [FZ] that such surfaces are rigid and have unobstructed deformations, i.e. that for them $$ h^1 ( \Theta_V\langle \, D \, \rangle) = h^2 ( \Theta_V\langle \, D \, \rangle) = 0\,,$$ and thus also $$\chi ( \Theta_V\langle \, D \, \rangle) = 0\,.$$ This, indeed, is true in all examples that we know [FZ]. Let now $X = {\bf P}^2 \setminus C = V \setminus D$, where $C$ is an irreducible plane curve and $V \to {\bf P}^2$ is the minimal embedded resolution of singularities of $C$, so that the total transform $D$ of $C$ in $V$ is an SNC--divisor. In view of (1.6) and (1.7) the deformations of $(V,\,D)$ correspond to equisingular embedded deformations of the curve $C$ in ${\bf P}^2$. We say shortly that $C$ is {\it projectively rigid} (resp. {\it (projectively) unobstructed}) if the pair $(V,\,D)$ has no infinitesimal deformations, i.e. $ h^1 ( \Theta_V\langle \, D \, \rangle) =0$ (resp. $ h^2 ( \Theta_V\langle \, D \, \rangle) = 0$)\footnote{as an abstract curve, such $C$ may have non--trivial equisingular deformations, which might be obstructed.}. Observe that $C \subset {\bf P}^2$ is projectively rigid iff the only equisingular deformations of $C$ as a plane curve are those obtained via the action of the automorphism group ${\rm PGL}\,(3,\,{\bf C})$ on ${\bf P}^2$. Indeed, suppose that $C_t \subset {\bf P}^2,\,t \in T,$ is a family of deformations of $C_0 = C$ such that all the members $C_t$ have at the corresponding singular points the same multiplicity sequence. Then the singularities can be resolved simultaneously at a family of surfaces $(V_t,\,D_t),\,t \in T$, see (1.6), (1.7). In view of the rigidity, there is a local isomorphism with the trivial family $(V_0,\,D_0) \times T$, and so by blowing down this leads to a family of projective isomorphisms $C_t {\stackrel{\varphi_t}{\longrightarrow}}C_0$. The converse is evidently true. It is easily seen that if $C$ is a rational cuspidal curve, then the complement $X = {\bf P}^2 \setminus C$ is $\bf Q$--acyclic. If, in addition, $C$ has at least three cusps, then $X$ is also of log--general type [Wak]. Thus, the rigidity conjecture of [FZ] says that such a curve $C$ should be projectively rigid and unobstructed. Here we compute the deformation invariants of $X$ in terms of multiplicity sequences of the cusps of $C$. In the next section we apply these computations to check the above rigidity conjecture for the complements of rational cuspidal curves considered there (see Lemma 3.3; cf. also section 4). \\ \noindent {\bf 2.2. Definition} (cf. [MaSa, FZ]). Let the notation be as in Definition 1.1. The blowing up $\sigma_{i+1},\,i \ge 1$, of $V_i$ at the point $P_i \in C_i$ is called {\it inner} (or {\it subdivisional}) if $P_i \in E_i \cap E_{i-k}^{(k)}$ for some $k > 0$, and it is called {\it outer} (or {\it sprouting}) in the opposite case. Note that $\sigma_1$ is neither inner nor outer. Moreover, $\sigma_2$ is always outer, and so $\rho \ge 1$, where $\omega = \omega_P$ resp. $\rho = \rho_P$ denotes the number of inner resp. outer blowing ups. Denote also by $k = k_P$ the total number of blow ups, i.e. the length of the multiplicity sequence ${\bar m}_P = (m_0,\,m_1,\dots,m_{k_P})$ minus one. Clearly, $\omega + \rho = k-1$. By $\lceil a \rceil$ we denote the smallest integer $\ge a$. \\ \noindent {\bf 2.3. Lemma.} $$\omega_P = \sum\limits_{i=1}^{k_P} (\lceil{m_{i-1} \over m_i} \rceil - 1)$$ \noindent {\bf Proof.} It is clear that the total number of exceptional curves $E_i^{(j)} \subset V_{i+j}$, where $1 \le i+j < k$, passing through the centers $P_{i+j}$ of the blow ups $\sigma_{i+j+1}$ is $2\omega + \rho$. If $m_{i-1} = sm_i$, then by Lemma 1.3 $P_{i+j} \in E_i^{(j)}$ for $j=0,\,1,\dots,s-1$, i.e. exactly $s$ times, except in the case when $i = k_P$. If $m_{i-1} = sm_i + r$, where $0 < r < m_i$, then this happens for $j = 0,\,1,\dots,s$, so $(s+1)$ times. In any case, this happens $\lceil{m_{i-1} \over m_i} \rceil$ times, with the only exception when $i = k_P$. Therefore, $$2\omega + \rho = \sum\limits_{i=1}^k \lceil{m_{i-1} \over m_i} \rceil - 1 = \sum\limits_{i=1}^k (\lceil{m_{i-1} \over m_i} \rceil - 1) + (k - 1)\,.$$ Since $\omega + \rho = k-1$, we have the desired result. \hfill $\Box$\\ \noindent {\bf 2.4. Proposition.} {\it Let $V_0$ be a smooth compact complex surface, $C \subset V_0$ be an irreducible cuspidal curve, and $V \to V_0$ be the embedded resolution of singularities of $C$. Denote by $K_V$ resp. $K_{V_0}$ the canonical divisor of $V$ resp. $V_0$, by $D$ the reduced total preimage of $C$ at $V$, and by ${\bar m}_P = (m_{P,\,0},\,m_{P,\,1},\dots,m_{P,\,k_P})$ the multiplicity sequence at $P \in {\rm Sing}\,C$. Let, as before, $\omega_P$ be the number of inner blow ups over $P$. Set $$\eta_P = \sum\limits_{i=0}^{k_P} (m_{P,\,i}-1)\,.$$ Then $$K_V(K_V + D) = K_{V_0}(K_{V_0} + C) + \sum\limits_{P \in {\rm Sing}\,C} (\eta_P+\omega_P -1)\,.$$} \noindent {\bf Proof.} Let $\sigma_{i+1}\,:\,V_{i+1} \to V_i$ be a step in the resolution of singularities of $C$. Put $K_i = K_{V_i}$ and let $D_i$ be the reduced total preimage of $C$ at $V_i$. We have $$K_{i+1} = K_i^* + E_{i+1} \qquad {\rm and} \qquad D_i^* = \sigma_{i+1}^* (D_i) = D_{i+1} + (m_i -1) E_{i+1} + \delta_i E_{i+1}\,,$$ where \[\delta_i = \left\{ \begin{array}{ll} 0 & \mbox{if $\sigma_{i+1}$ is neither inner nor outer} \\ 1 & \mbox { if $\sigma_{i+1}$ is outer} \\ 2 & \mbox {if $\sigma_{i+1}$ is inner} \end{array} \right. \] It follows that $$K_i(K_i+D_i) = K_{i+1}(K_i^* + D_i^*) = K_{i+1} (K_{i+1} + D_{i+1} + (m_i + \delta_i -2)E_{i+1})$$ $$ = K_{i+1} (K_{i+1} + D_{i+1}) - (m_i + \delta_i -2)\,.$$ Thus, $$K_{i+1} (K_{i+1} + D_{i+1}) = K_i(K_i+D_i) +(m_i -1) + (\delta_i -1)\,.$$ Now the desired equality easily follows. \hfill $\Box$ \\ \noindent {\bf 2.5. Corollary.} {\it Let $C \subset {\bf P}^2$ be a plane cuspidal curve of degree $d \ge 3$, and let $\pi\,:\,V \to {\bf P}^2$ be the embedded resolution of singularities of $C$, $D$ be the reduced total preimage of $C$ in $V$ and $K = K_V$ be the canonical divisor. Then \begin{equation} \chi ( \Theta_V\langle \, D \, \rangle) = K(K+D) = -3(d-3) + \sum\limits_{P \in {\rm Sing}\,C} (\eta_P+\omega_P -1)\,.\end{equation} } \noindent {\bf 2.6. Remark.} In view of (2.5), in the case when $C \subset {\bf P}^2$ is a rational cuspidal curve with at least three cusps, the rigidity conjecture mentioned in (2.1) in particular yields the identity $$\sum\limits_{P \in {\rm Sing}\,C} (\eta_P+\omega_P -1) = 3(d-3)\,,$$ which, indeed, is true in all examples that we know (see e.g. Lemma 3.3 below). \section{Rational cuspidal plane curves of degree $d$ with a cusp of multiplicity $d-2$} \noindent {\bf 3.1. Lemma}. {\it Let $C \subset {\bf P}^2$ be a rational cuspidal curve of degree $d$ with a cusp $P \in C$ of multiplicity $m_P$ with multiplicity sequence ${\bar m}_P = (m_{P,\,0},\dots, m_{P,\,k_P})$. Then the projection $\pi_P\,:\,C \to {\bf P}^1$ from $P$ has at most $2(d-m-1)$ branching points. Furthermore, if $Q_1,\dots,Q_s$ are the other cusps of $C$ with multiplicities $m_1,\dots,m_s$, then $$\sum\limits_{j=1}^s (m_j - 1) + (m_{P,\,1} -1) \le 2(d-m-1)\,.$$} \noindent {\bf Proof.} By the Riemann--Hurwitz formula, applied to the composition ${\tilde \pi}_P \,:\,{\bf P}^1 = {\tilde C} \to {\bf P}^1$ of the normalization map ${\tilde C} \to C$ and the projection $\pi_P$, which has degree $d - m$, we obtain that $$2(d-m) = 2 + \sum\limits_{Q \in {\tilde C}} (v_Q - 1)\,,$$ where $v_Q$ is the ramification index of ${\tilde \pi}_P$ at $Q$. The singular point $Q_i$ of $C$ gives rise to a branching point with ramification index $\ge m_i$, and after blowing up at $P \in C$ the first infinitesimal point to $P$ gives rise to a branching point with ramification index $\ge m_{P,\,1}$. This proves the lemma. \hfill $\Box$\\ Denote by $(m_a)$, where $m > 1$, the following multiplicity sequence: $$(m_a) = ({\underbrace{m,\dots,m}_{a}},\,{\underbrace{1,\dots,1}_{m+1}})\,.$$ We write simply $(m)$ instead of $(m_1)$ for $a = 1$. Notice that $(2_k)$ is the multiplicity sequence of a simple plane curve singularity of type $A_{2k}\,\,(x^2 + y^{2k+1} = 0)$; thus, $(2)$ corresponds to an ordinary cusp $x^2 + y ^3 = 0$. \\ \noindent {\bf 3.2. Lemma}. {\it Let $C \subset {\bf P}^2$ be a rational cuspidal curve of degree $d$ with a cusp $P \in C$ of multiplicity $d-2$. Then $C$ has at most three cusps. Assume further that $C$ has three cusps. Then they are not on a line and have multiplicity sequences resp. $[(d-2),\,(2_a),\,(2_b)]$, where $a + b = d-2$. Each of these cusps has only one Puiseux characteristic pair; they are, respectively, $(d-1, \,d-2),\,(2a+1, \,2),\,(2b+1,\,2)$.} \\ \noindent {\bf Proof.} The projection $C \to {\bf P}^1$ from $P \in C$ being $2$--sheeted, by the preceding Lemma it has at most two ramification points. Thus, by Bezout's Theorem the multiplicities of other singular points are at most two and there are at most two of them. Moreover, it follows from Lemma 3.1 that in the case when there are two more singular points, the multiplicity sequence at $P$ should be $(d-2)$. Hence, the only multiplicity sequences in the case of three cusps are $[(d-2),\,(2_a),\,(2_b)]$. By the genus formula we have $${d-2 \choose 2} + a + b = {d-1 \choose 2}\,,$$ and thus $a+b = d-2$. That the three cusps do not lie on a line follows from Bezout's theorem. \hfill $\Box$\\ \noindent {\bf 3.3. Lemma}. {\it Let $C \subset {\bf P}^2$ satisfies the assumptions of Lemma 3.2. Then $C$ is projectively rigid and unobstructed \footnote{see (2.1) for the definitions.}.}\\ \noindent {\bf Proof.} Let $(V,\,D) \to ({\bf P}^2,\,C)$ be the minimal embedded resolution of singularities of $C$. Then, first of all, the Euler characteristic of the holomorphic tangent bundle $\chi = \chi(\Theta_V \langle \, D \, \rangle)$ vanishes. This follows from (1). Indeed, if $P$ has multiplicity sequence ${\bar m}_P=(m)$, then $$ \eta_P + \omega_P - 1 = 2m - 3\,,$$ whereas for the multiplicity sequence $(2_a)$ this quantity equals $a$. Thus, under the assumptions of Lemma 3.2 we have $$\chi = 9 - 3d + (a+b) + 2(d-2) - 3 = 0\,.$$ Furthermore, the projection from the point $P \in C$ of multiplicity $d-2$ yields a morphism $\pi_P\,:\,V \to {\bf P}^1$, which is a ${\bf P}^1$--ruling. Its restriction to $D$ is $3$-sheeted. Moreover, $X = V \setminus D = {\bf P}^2 \setminus C$ is a ${\bf Q}$--acyclic affine surface, i.e. $H_i (X; \,{\bf Q}) = 0,\,i>1$. By Proposition 6.2 from [FZ] it follows that $h^2 (\Theta_V \langle \, D \, \rangle) = 0$, and so $C$ is unobstructed. Since ${\bar k}\,(V \setminus D) = 2$ [Wak], due to Theorem 6 from [Ii] we also have $h^0 (\Theta_V \langle \, D \, \rangle) = 0$. Therefore, $h^1 (\Theta_V \langle \, D \, \rangle) = 0$, that means that $(V,\,D)$ is a rigid pair, and hence $C$ is projectively rigid (see (2.1). \hfill $\Box$\\ \noindent {\bf 3.4. Lemma}. {\it Let $(C,\,0) \subset ({\bf C}^2,\,0)$ be a plane curve germ given parametrically by $$t \longmapsto (f(t),\,g(t)) = (t^m ,\,\sum\limits_{\nu = 1}^{\infty} c_{\nu} t^{\nu} )\,.$$ Then the multiplicity sequence of $(C,\,0)$ has the form $$({\underbrace{m,\dots,m}_{r}},\dots)$$ iff (**) $c_i = 0$ for all $i$ with $i < mr$ such that $m\, \not\vert \,i$.\\ \noindent Furthermore, $(C,\,0)$ has multiplicity sequence $(2_r)$ iff $m=2$, the first $r$ odd coefficients vanish: $c_1 = c_3 =\dots=c_{2r-1} =0$ and, moreover, $c_{2r+1} \neq 0$. }\\ \noindent {\bf Proof.} After coordinate change of type $(f(t),\,g(t)) \longmapsto (f(t),\,g(t) - p (f(t)))$, where $p \in {\bf C} [z]$, we may assume that $c_m = c_{2m} =\dots =c_{rm} = 0$. Then $$g(t) = c_st^s + {\rm higher\,\,\,order\,\,\,terms}\,,$$ with $c_s \neq 0$ and either $s > rm$ or $m \not\vert \,s$. First of all, we show that if $(C,\,0)$ has multiplicity sequence $({\underbrace{m,\dots,m}_{r}},\dots)$, then $s > mr$, which is equivalent to (**). Let $s = \rho m + s_1$, where $0 \le s_1 < m$. If $\rho < r$, then after blowing up $\rho$ times we obtain the parametrized curve germ $$(f(t),\,g(t)/t^{\rho m})\,,$$ which still has multiplicity $m$. But since $g(t)/t^{\rho m}$ has multiplicity $s - \rho m = s_1$, this contradicts the assumption that $s_1 < m$. Thus, if $(C,\,0)$ has multiplicity sequence $({\underbrace{m,\dots,m}_{r}},\dots)$, then the condition (**) is satisfied. The converse is clear. Finally, assume that $m = 2,\,c_1=c_3=\dots =c_{2r-1}=0$ and $c_{2r+1} \neq 0$. Then after the above coordinate change we have $(f(t),\,g(t)) = (t^2,\,c_{2r+1} t^{2r+1} + \dots)$, and so due to the above criterion $(C,\,0)$ has multiplicity sequence $(2_r)$. Once again, the converse is clear. \hfill $\Box$\\ \noindent {\bf 3.5. Theorem}. {\it For any $d \ge 4, \,a\ge b \ge 1$ with $a+b = d-2$ there is a unique, up to projective equivalence, rational cuspidal curve $C = C_{d,\,a} \subset {\bf P}^2$ of degree $d$ with three cusps with multiplicity sequences $[(d-2),\,(2_a),\,(2_b)]$. In appropriate coordinates this curve can be parametrized as $$C_{d,\,a} = (P : Q : R) = (s^2(s-t)^{d-2}\,\,: \,\,t^2(s-t)^{d-2} \,\,:\,\, s^2t^2q_{d,\,a}(s,\,t))\,,$$ where $q_{d,\,a}(s,\,t) = \sum\limits_{i=0}^{d-4} c_i s^it^{d-4-i}$ and the polynomial ${\tilde q}_{d,\,a}(T) = \sum\limits_{i=0}^{d-4} c_i T^i$ is defined as $${\tilde q}_{d,\,a}(T) = {f_{d,\,a}(T^2) + T^{2a - 1} \over (1 + T)^{d-2}}\,.$$ Here $f_{d,\,a}(T)$ is a polynomial of degree $d-3$ uniquely defined by the divibisility condition $(1 + T)^{d-2} \,|\,(f_{d,\,a}(T^2) + T^{2a - 1})$.}\footnote{For the explicit equations, see Proposition 3.9 below.} \\ \noindent {\bf Proof.} Suppose that $C \subset {\bf P}^2$ is such a curve. Since by Lemma 3.2 its three cusps are not at a line, up to projective transformation we may assume that $C$ has cusps at the points $(0 : 0 : 1),\,(0 : 1 : 0),\,(1 : 0 : 0)$ with multiplicity sequences resp. $(d-2),\,(2_a),\,(2_b)$. Let $h = (P : Q : R)\,:\,{\bf P}^1 \to C \hookrightarrow {\bf P}^2$ be the normalization of $C$, where $(P : Q : R)$ is a triple of binary forms of degree $d$ without common zero such that $$h(1 : 1) = (0 : 0 : 1)$$ $$h(0 : 1) = (0 : 1 : 0)$$ $$h(1 : 0) = (1 : 0 : 0)\,.$$ Since $C$ is required to have cusps of multiplicity $d-2$ at $h(1 : 1)$ and of multiplicity $2$ at $h(0 : 1)$ and at $h(1 : 0)$, up to multiplication by constant factors we may write $$P(s,\,t) = (s-t)^{d-2}s^2$$ $$Q(s,\,t) = (s-t)^{d-2}t^2$$ $$R(s,\,t) = s^2t^2 q(s,t)\,,$$ where $$q(s,\,t) = \sum\limits_{i=0}^{d-4} c_is^i t^{d-4-i}\,\,\,\,{\rm and}\,\,\,\,c_0 \neq 0,\,\,c_{d-4} \neq 0,\,\,q(1,\,1) \neq 0\,.$$ We will show that under our assumptions $q$ is uniquely defined. To impose the conditions that there is a cusp of type $(2_a)$ at the point $h(0 : 1) = (0 : 1 : 0)$ resp. of type $(2_b)$ at the point $h(1 : 0) = (1 : 0 : 0)$, we rewrite the above parametrization in appropriate affine coordinates at the corresponding points. \\ \noindent At $(0 : 1)$ we set $\xi = s/t$ and we have $${\tilde f}(\xi) = {P \over Q} = {s^2 \over t^2} = \xi^2$$ $${\tilde g}(\xi) = {R \over Q} = {s^2 q(s,\,t) \over (s-t)^{d-2}} = {\xi^2 {\tilde q}(\xi) \over (\xi - 1)^{d-2}} \,,$$ where $${\tilde q} (\xi) = \sum\limits_{i=0}^{d-4} c_i \xi^i \,.$$ By Lemma 3.4 $C$ has a cusp of type $(2_a)$ at $h(0 : 1) = (0 : 1 : 0)$ iff the odd coefficients of $\xi^i$ of the function ${R \over \xi^2 Q} = {{\tilde q}(\xi) \over (\xi - 1)^{d-2}}$ vanish up to order $(2a - 3)$ (this imposes $(a-1)$ conditions) and the coefficient of $\xi^{2a-1}$ does not vanish. \\ \noindent At $(1 : 0)$ we set $\tau = t/s $ and we have $$ {\breve f}(\tau) = {Q \over P} = {t^2 \over s^2} = \tau^2$$ $${\breve g}(\tau) = {R \over P} = {\tau^2{\breve q} (\tau) \over (1-\tau)^{d-2}}\,,$$ where $${\breve q}(\tau) = \sum\limits_{i=0}^{d-4} c_i\tau^{d-4-i}\,.$$ By Lemma 3.4 $C$ has a cusp of type $(2_b)$ at $h(1 : 0) = (1 : 0 : 0)$ iff the odd coefficients of ${R \over \tau^2 P} = {{\breve q} (\tau) \over (1-\tau)^{d-2}}$ vanish up to order $(2b-3)$ (this imposes $(b-1)$ conditions) and the coefficient of $\tau^{2b+1}$ does not vanish. Note that the coefficients ${\tilde c}_i$ of $\xi_i$ in ${\tilde g}(\xi)/\xi^2$ and those ${\breve c}_i$ of $\tau_i$ in ${\breve g}(\tau)/\tau^2$ are linear functions in $c_0,\dots,c_{d-4}$. We must show that the system $${\tilde c}_1 = {\tilde c}_3 = \dots ={\tilde c}_{2a-3} = 0,\,\,\,{\tilde c}_{2a-1} = 1$$ $${\breve c}_1 = \dots {\breve c}_{2b-3} = 0$$ has the unique solution. Indeed, by symmetry then also the coefficient ${\breve c}_{2b-1}$ is uniquely defined and non--zero. This follows from the fact that the associate homogeneous system $${\tilde c}_1 = {\tilde c}_3 = \dots ={\tilde c}_{2a-3} = {\tilde c}_{2a-1} = 0$$ $${\breve c}_1 = \dots {\breve c}_{2b-3} = 0$$ has the unique solution, which corresponds to $q \equiv 0$. Observe that it has $$(a-1) + (b-1) + 1 = d - 3$$ equations and the same number of variables. To show the uniqueness we need the following lemma. Its proof is easy and can be omited. \\ \noindent {\bf 3.6. Lemma.} {\it Let $$h(T) = \sum_{\nu \ge 0} a_{\nu}T^{\nu} \in {\bf C} [T]$$ and $${\tilde h}(T) = h(T) (1 + T^2 u(T^2))$$ for some power series $u \in {\bf C} [[T]]$. Set ${\tilde h}(T) = \sum_{\nu \ge 0} {\tilde a}_{\nu}T^{\nu}$. Then $${\tilde a}_1 = {\tilde a}_3 =\dots ={\tilde a}_{2k+1} = 0$$ iff $$a_1 = a_3 =\dots = a_{2k+1} = 0\,.$$} Returning to the proof of the theorem, put $n = d-4$ and $$F(T) = {\tilde q}(T)(1+T)^{n+2} = {{\tilde q}(T) \over (1-T)^{n+2}}(1-T^2)^{n+2}$$ $$G(T) = {\breve q}(T) (1+T)^{n+2} = {{\breve q}(T) \over (1-T)^{n+2}} (1-T^2)^{n+2}\,.$$ By Lemma 3.6 the first $a$ (resp. $(b-1)$) odd coefficients of $F(T)$ (resp. of $G(T)$) vanish iff the same is true for ${\tilde q}(T) \over (1-T)^{n+2}$ (resp. for ${\breve q}(T) \over (1-T)^{n+2}$). Note that by definition ${\breve q}(T) = {\tilde q}({1 \over T})T^n$. Thus, we have that ${\rm deg}\,F = 2n+2$ is even and $$F({1 \over T})T^{2n+2} = {\tilde q}( {1 \over T})T^n (1 + {1 \over T})^{n+2}T^{n+2} = {\breve q}(T)(1+T)^{n+2} = G(T)\,.$$ Therefore, the conditions that the first $a$ odd coefficients of $F$ and the first $(b-1)$ odd coefficients of $G$ vanish are equivalent to $F$ being an even function: $F(T) = F(-T)$. Indeed, since $a+b-1 = d-3 = n+1$, the above conditions mean that all odd coefficients of $F$ vanish. Now we use the following elementary facts.\\ \noindent {\bf 3.7. Lemma.} {\it Assume that $p \in {\bf C}[T]$ and $(1+T)^kp(T)$ is even. Then $(1-T)^k \,|\,p(T)$. }\\ \noindent {\bf Proof.} By the condition we have $(1+T)^kp(T) = (1-T)^k p(-T)$, as the product is even. Thus $(1-T)^k \,|\,p(T)$. \hfill $\Box$\\ {}From this lemma immediatly follows\\ \noindent {\bf 3.8. Corollary.} {\it If ${\rm deg}\, p \le n$ and $(1+T)^{n+2}p(T)$ is even, then $p \equiv 0$.}\\ Being applied to $p = {\tilde q}$ and $F(T) = (1 + T)^{n+2}{\tilde q}(T)$, Corollary 3.8 implies that ${\tilde q} \equiv 0$ and so $q \equiv 0$, i.e. the above homogeneous system has a unique solution. This completes the proof of the first part of Theorem 3.5. As for the second one, we must prove the explicit presentation of ${\tilde q} = {\tilde q}_{d,\,a}$. As above, it follows from the assumptions that the first $(a-1)$ and the last $(b-1)$ odd coefficients of $F(T)$ vanish, while the coefficient of $T^{2a-1}$ is non--zero. Therefore, $F(T) = f(T^2) + T^{2a-1}$ with $f$ being a polynomial of degree $d-3$. Hence $${\tilde q}(T) = {f(T^2) + T^{2a-1} \over (1+T)^{d-2}}\,.$$ From the equality $F(T) = (1+T)^{d-2}{\tilde q}(T)$ we have that $$F(-1) = F'(-1) = \dots = F^{(d-3)}(-1) = 0\,.$$ These equations uniquely define the derivatives of the polynomial $f(\xi)$ at $\xi = 1$ up to order $(d-3)$, and therefore $f_{d,\,a}(\xi) = f(\xi) = \sum\limits_{k=0}^{d-3} {a_k \over k!} (\xi-1)^k$ is determined in a unique way. This completes the proof of Theorem 3.5. \hfill $\Box$ \\ \noindent {\bf 3.9. Proposition.} {\it a) The polynomial $f = f_{d,\,a}$ in Theorem 3.5 can be given as $$f(T) = \sum\limits_{k=0}^{d-3} {a_k \over k!} (T - 1)^k\,,$$ where $a_0 = 1, \,a_1 = a - {1 \over 2}$ and $$a_k = {1 \over 2^k}(2a-1)(2a-3)\dots(2a-(2k-1)) = a_1 (a_1 - 1)\dots (a_1 - (k-1)),\,\,\,k=1,\dots,d-3\,,$$ i.e. it coincides with the corresponding partial sum of the Taylor expansion at $T = 1$ of (the positive branch of) the function $T^{a_1}$. \\ \noindent b) In the affine chart $(X = x/z,\,Y = y/z)$ the curve $C_{d,\,a}$ as in Theorem 3.5 can be given by the equation $p(X,\,Y) = 0$, where $p = p_{d,\,a} \in {\bf Q}[X,\,Y]$ is defined as follows: $$p(X,\,Y) = {X^{2a+1}Y^{2b+1} - ((X - Y)^{d-2} - XY{\hat f}(X,\,Y))^2 \over (X - Y)^{d-2}}\,,$$ and where ${\hat f}(X,\,Y) = Y^{d-3}f({X \over Y})$ is the homogeneous polynomial which corresponds to $f(T)$. }\\ \noindent {\bf Proof.} We start with the proof of b). In the notation of Theorem 3.5 in the affine chart $\xi = s/t$ in ${\bf P}^1$ we have $${X \over Y} = {P \over Q} = \xi^2$$ and $$X = {(\xi - 1)^{d-2} \over {\tilde q}(\xi)}\,,$$ where $${\tilde q}(\xi) = {\tilde q}_{d,\,a}(\xi) = \sum\limits_{i=0}^{d-4} c_i \xi_i$$ is as above. Thus, $$(\xi^2 - 1)^{d-2} = X{\tilde q}(\xi)(\xi - 1)^{d-2} = X(f_{d,\,a}(\xi^2) + \xi^{2a-1})$$ by the definition of ${\tilde q}(\xi)$. Plugging here $\xi^2 = X/Y$ we obtain $$(X - Y)^{d-2} = XY(Y^{d-3}f({X\over Y}) + \xi X^{a-1}Y^b) = XY{\hat f}(X,\,Y) + \xi X^a Y^{b+1}\,.$$ Hence, $$\xi = {(X - Y)^{d-2} - XY{\hat f}(X,\,Y) \over X^aY^{b+1}}\,$$ and so $$\xi^2 = {X \over Y} = {((X - Y)^{d-2} - XY{\hat f}(X,\,Y))^2 \over X^{2a}Y^{2b+2}}\,.$$ Therefore, the curve $C_{d,\,a}$ in the affine chart $(X,\,Y)$ satisfies the equation ${\tilde p} = 0$, where $${\tilde p}(X,\,Y) = X^{2a+1}Y^{2b+1} - ((X - Y)^{d-2} - XY{\hat f}(X,\,Y))^2\,.$$ Since $C_{d,\,a}$ is an irreducible curve of degree $d$, b) follows from the next lemma.\\ \noindent {\bf 3.10. Lemma.} $$(X-Y)^{d-2}\,|\,{\tilde p}(X,\,Y)\,.$$ \noindent {\bf Proof.} We have $${\tilde p}(X,\,Y) \equiv \psi (X,\,Y)\,\,\,{\rm mod}\, (X-Y)^{d-2}\,,$$ where $$\psi (X,\,Y) := X^{2a+1}Y^{2b+1} - X^2Y^2{\hat f}^2 (X,\,Y)\,.$$ The polynomial $\psi$ is homogeneous of degree $2d-2$, and thus it is enough to show that \begin{equation} (X - 1)^{d-2} \,|\,\psi (X,\,1) \,,\end{equation} or equivalently, that $$ (X^2 - 1)^{d-2} \,|\,\psi (X^2,\,1) \,.$$ Since $\psi (X^2,\,1)$ is an even polynomial and $(X^2 - 1)^{d-2} = (X - 1)^{d-2} (X + 1)^{d-2}$, by (3.7) it is sufficient to check that $$(X + 1)^{d-2}\,|\,\psi (X^2,\,1)\,.$$ But $$\psi (X^2,\,1) = X^{4a+2} - X^4 {\hat f}^2 (X^2,\,1) \equiv 0 \,\,\,{\rm mod} \,(X+1)^{d-2}\,,$$ because by definition, $${\hat f} (X^2,\,1) \equiv -X^{2a-1} \,\,\,{\rm mod} \,(X+1)^{d-2}\,.$$ \hfill $\Box$ \\ \noindent {\bf Proof of Proposition 3.9, a).} From (2) it follows that $$ f^2(T) - T^{2a-1} = (f(T) - T^{a_1})(f(T) + T^{a_1}) \equiv 0 \,\,\,{\rm mod}\,(T-1)^{d-2} \,,$$ where by $T^{a_1}$ we mean those branch of the square root of $T^{2a-1}$ which is positive at $T=1$. Since $(T-1)^{d-2}$ does not divide the second factor, we have $$ f(T) - T^{a_1} \equiv 0 \,\,\,{\rm mod}\,(T-1)^{d-2} \,.$$ Thus, indeed, $f(T)$ is the (d-3)-th partial sum of the Taylor series of the function $T^{a_1} = T^{2a-1 \over 2}$ at the point $T=1$, and a) follows. This proves the Proposition. \hfill $\Box$ \\ \noindent {\bf 3.11. Remark.} By the way, it follows that any rational cuspidal plane curve $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$, can be defined over $\bf Q$. \\ \noindent {\bf 3.12. Examples.} Here we present the affine equations $p_{d,a} = 0$ of the curves $C_{d,\,a}$ for $4 \le d \le 7$\footnote{ they were found with "Maple".}.\\ \noindent $d=4$ and $a=1$ (Steiner's quartic) $$p_{4,3}(X,\,Y) = -{\frac {Y^{2}X^{2}}{4}}-\left (X-Y\right )^{2}+XY\left (Y+X\right ) $$ \noindent $d=5$ and $a=2$ $$p_{5,2}(X,\,Y) = {\frac {Y^{3}X^{2}}{64}}-{\frac {9\,Y^{2}X^{3}}{64}}-\left (X-Y\right )^{3}+XY\left ({\frac {3\,YX}{2}}-{\frac {Y^{2}}{4}}+{\frac {3\,X^{2}} {4}}\right ) $$ \noindent $d=6$ and $a=2$ $$p_{6,2}(X,\,Y) = {\frac {7\,Y^{3}X^{3}}{128}}-{\frac {Y^{2}X^{4}}{256}}-{\frac {Y^{4}X^ {2}}{256}}-\left (X-Y\right )^{4}$$ $$+XY\left ({\frac {9\,Y^{2}X}{8}}-{ \frac {Y^{3}}{8}}+{\frac {9\,YX^{2}}{8}}-{\frac {X^{3}}{8}}\right ) $$ \noindent $d=6$ and $a=3$ $$p_{6,3}(X,\,Y) = {\frac {3\,Y^{3}X^{3}}{128}}-{\frac {25\,Y^{2}X^{4}}{256}}-{\frac {Y^{ 4}X^{2}}{256}}-\left (X-Y\right )^{4}$$ $$+XY\left ({\frac {Y^{3}}{8}}-{ \frac {5\,Y^{2}X}{8}}+{\frac {15\,YX^{2}}{8}}+{\frac {5\,X^{3}}{8}} \right ) $$ \noindent $d=7$ and $a=3$ $$p_{7,3}(X,\,Y) = {\frac {475\,Y^{3}X^{4}}{16384}}-{\frac {25\,Y^{2}X^{5}}{16384}}-{ \frac {75\,Y^{4}X^{3}}{16384}}+{\frac {9\,Y^{5}X^{2}}{16384}}-\left (X -Y\right )^{5}$$ $$+XY\left ({\frac {3\,Y^{4}}{64}}-{\frac {5\,Y^{3}X}{16}} +{\frac {45\,Y^{2}X^{2}}{32}}+{\frac {15\,YX^{3}}{16}}-{\frac {5\,X^{4 }}{64}}\right ) $$ \noindent $d=7$ and $a=4$ $$p_{7,4}(X,\,Y) = {\frac {459\,Y^{3}X^{4}}{16384}}-{\frac {1225\,Y^{2}X^{5}}{16384}}-{ \frac {155\,Y^{4}X^{3}}{16384}}+{\frac {25\,Y^{5}X^{2}}{16384}}-\left (X-Y\right )^{5}$$ $$+XY\left ({\frac {7\,Y^{3}X}{16}}-{\frac {5\,Y^{4}}{64 }}-{\frac {35\,Y^{2}X^{2}}{32}}+{\frac {35\,YX^{3}}{16}}+{\frac {35\,X ^{4}}{64}}\right )\,. $$\\ \noindent {\bf 3.13. Remark.} The weighted dual graph of the resolution of a cusp with multiplicity sequence $(m)$ looks like $$ \begin{picture}(1000,90) \put(64,82){$-2$} \put(66,52){$E_2$} \put(70,70){\circle{10}} \put(77,70){\line(1,0){40}} \put(118,82){$-2$} \put(120,52){$E_3$} \put(125,70){\circle{10}} \put(132,70){\line(1,0){40}} \put(173,82){$-2$} \put(175,52){$E_4$} \put(180,70){\circle{10}} \put(187,70){\line(1,0){40}} \put(244,70){$\ldots$} \put(275,70){\line(1,0){40}} \put(315,82){$-1$} \put(300,52){$E_m$} \put(322,70){\circle{10}} \put(329,70){\vector(1,0){40}} \put(373,52){$C$} \put(376,70){\circle{10}} \put(322,62){\line(0,-1){40}} \put(322,15){\circle{10}} \put(330,14){$-m$} \put(317,-3){$E_1$} \end{picture} $$ while the dual resolution graph of a cusp $(2_a) = A_{2a}$ looks like $$ \begin{picture}(1000,90) \put(64,82){$-2$} \put(66,52){$E_1$} \put(70,70){\circle{10}} \put(77,70){\line(1,0){40}} \put(134,70){$\ldots$} \put(165,70){\line(1,0){40}} \put(206,82){$-2$} \put(206,52){$E_{a-1}$} \put(213,70){\circle{10}} \put(220,70){\line(1,0){40}} \put(260,82){$-3$} \put(260,52){$E_a$} \put(267,70){\circle{10}} \put(273,70){\line(1,0){40}} \put(315,82){$-1$} \put(297,52){$E_{a+2}$} \put(322,70){\circle{10}} \put(329,70){\vector(1,0){40}} \put(373,52){$C$} \put(376,70){\circle{10}} \put(322,62){\line(0,-1){40}} \put(322,15){\circle{10}} \put(330,14){$-2$} \put(317,-3){$E_{a+1}$} \end{picture} $$\\ Therefore, the dual graph of the total transform $D=D_{d,\,a}$ of $C_{d,\,a}$ in its minimal embedded resolution $V \to {\bf P}^2$ looks as follows: $$ \begin{picture}(1000,60) \put(175,32){$-(d-2)$} \put(192,1){${\tilde C}_{d,\,a}$} \put(200,20){\circle{10}} \put(150,20){\line(1,0){42}} \put(207,21){\line(2,1){40}} \put(207,19){\line(2,-1){40}} \put(105,10){\framebox(40,20){$(d-2)$}} \put(250,30){\framebox(40,20){$(2_a)$}} \put(250,-8){\framebox(40,20){$(2_b)$}} \end{picture} $$\\ where $b = d-a-2$ and boxes mean the corresponding local resolution trees, as above. \\ \noindent {\bf 3.14. Remark \footnote{This remark is due to a discussion with T. tom Dieck, who constructed examples of cuspidal plane curves starting from certain plane line arrangements, and with E. Artal Bartolo. We are grateful to both of them.}.} Here we show that each curve $C_{d,a}$ can be birationally transformed into a line. More precisely, let $P_0, P_a, P_b$ be the cusps of $C = C_{d,a}$ with multiplicity sequences resp. $(d-2),\, (2_a),\,(2_b)$. Let $l_0 = \{x = 0\}, \,l_{\infty} = \{y = 0\}$ be the lines through $P_0, P_a$, resp $P_0, P_b$, and $l_1 = \{x - y = 0\}$ be the cuspidal tangent line to $C$ at $P_0$. We will show that there exist three other rational cuspidal curves $C_1, \,C_2,\,C_3$, which meet $C$ only at the cusps of $C$, such that the curve $T = C \cup l_0 \cup l_1 \cup l_{\infty} \cup C_1, \cup C_2 \cup C_3$ can be transformed into a configuration $T'$ of $7$ lines in ${\bf P}^2$ by means of a birational transformation $\alpha\,:\, {\bf P}^2 \to {\bf P}^2$ which is biregular on the complements ${\bf P}^2 \setminus T$ and ${\bf P}^2 \setminus T'$. In fact, $\alpha$ consists of several birational transformations composed via the following procedure. \vspace{.1in} \noindent 1) Blowing up at $P_0$, we obtain the Hirzebruch surface $\pi \,:\, \Sigma (1) \to {\bf P}^1$ together with a two--sheeted section $C'$ (the proper preimage of $C$), the exceptional divisor $E$ (which is a section of $\pi$) and with three fibres $F_0 = {l'}_0, \, F_1 = {l'}_1, \, F_{\infty} = {l'}_{\infty}$ through three points of $C'$ which we still denote resp. as $P_a,\, P_0,\, P_b$. Observe that $C'$ is smooth at $P_0$ and by (1.3, a) $i(C',\, E;\, P_0) = d-2$. \vspace{.1in} \noindent 2) Perform $a$ resp. $b$ elementary transformations at $P_a \in C' \cap F_0$ resp. $P_b \in C' \cap F_{\infty}$, first blowing up at this point and then blowing down the proper preimage of the fibre $F_0$ resp. $F_{\infty}$. We arrive at another Hirzebruch surface $\Sigma (N)$ equiped with a smooth two--sheeted section $C''$, which is tangent to the fibres $F_0$ and $F_{\infty}$ and to the section $E'$, where now ${E'}^2 = d-3$. \vspace{.1in} \noindent 3) Performing further $d-2$ elementary transformations at $P_0 = E' \cap C'' \cap F_1$, we return back at $\Sigma (1)$ with $E^2 = -1$, this time the image $C'''$ of $C''$ being a smooth two-sheeted section which does not meet $E$. \vspace{.1in} \noindent 4) Contract $E$ back to a point $P_0 \in {\bf P}^2$. Then the image $\hat C$ of $C'''$ is a conic in ${\bf P}^2$, and the images of the fibres $F_0,\, F_1,\, F_{\infty}$ are resp. the lines $l_0, \,l_1,\,l_{\infty}$ through $P_0 \notin {\hat C}$, where $l_0,\,l_{\infty}$ are tangent to $\hat C$ resp. at the points $P_a,\, P_b \in {\hat C}$, and $l_1$ is a secant line passing, say, through a point $A \in {\hat C}$. \vspace{.1in} \noindent 5) Performing the Cremona transformation with centers at the points $A,\, P_a,\, P_b \in {\hat C}$, we obtain an arrangement $T'$ of $7$ lines in ${\bf P}^2$ with $6$ triple points. It can be described (in an affine chart) as a triangle together with its three medians and one more line through the middle points of two sides. It is easily seen that such a configuration $T'$ is projectively rigid. \vspace{.1in} \noindent The ${\bf Q}$--acyclic surface ${\bf P}^2 \setminus C$ can be reconstructed starting from the arrangement $T'$ by reversing the above procedure. In the tom Dieck-Petrie classification [tDP, Theorem D] this line configurations is denoted as $L(4)$. \\ \noindent {\bf 3.15. Remark.} E. Artal Bartolo has computed the fundamental groups $\pi_1 ({\bf P}^2 \setminus C_{d,\,a})$. Let, as always, $a + b = d - 2$, where $a \ge b \ge 1$. Set $2n + 1 = {\rm gcd}\,(2a+1, \,2b+1)$. Then $\pi_1 ({\bf P}^2 \setminus C_{d,\,a}) \approx G_{d,\,n}$, where $G_{d,\,n}$ is the group with presentation $$G_{d,\,n} = \,<u,\,v\,|\,u(vu)^n = (vu)^n v,\,(vu)^{d-1} = v^{d-2}>\,.$$ In particular, $G_{d,\,n}$ is abelian iff $n = 0$, i.e. ${\rm gcd}\,(2a+1, \,2b+1) =1$. Furthermore, among the non--abelian groups $G_{d,\,n}$ only $G_{4,\,1}$ and $G_{7,\,1}$ are finite. Note that, being non--isomorphic, the curves $C_{13,\,7}$ and $C_{13,\,10}$ have isomorphic fundamental groups of the complements, which are both infinite non--abelian groups isomorphic to $G_{13,\,1}$. Evidently, there are infinitely many such pairs. \section{Miscelleneous} Let $C \subset {\bf P}^2$ be an irreducible plane curve, $V \to {\bf P}^2$ the minimal embedded resolution of singularities of $C$, ${\tilde C} \subset V$ the proper transform of $C$ and $K = K_V$ the canonical divisor of $V$. Let also $D \subset V$ be the reduced total transform of $C$. Recall (see (2.1)) that $C$ being unobstructed simply means that $h^2(\Theta_V\langle \, D \, \rangle) = 0$. In the next lemma we give a sufficient condition for a plane curve to be unobstructed. \\ \noindent {\bf 4.1. Lemma}. {\it Let the notation be as above. \noindent a) If $K{\tilde C} <0$, then $H^2(\Theta_V\langle \, D \, \rangle) = 0$. \noindent b) Assume that $C$ is a cuspidal curve with cusps $P_1,\dots,P_s$ having multiplicity sequences $${\bar m}_{P_{\sigma}} = (m_{\sigma \,1},\dots,m_{\sigma \,r_{\sigma}}, {\underbrace{1,\dots,1}_{m_{\sigma \,r_{\sigma}} + 1}})\,,$$ where $m_{\sigma \,r_{\sigma}} \ge 2$. If $$K{\tilde C} < \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}}\,,$$ then $H^2(\Theta_V\langle \, D \, \rangle) = 0$.} \\ \noindent {\bf Proof. } a) Fix $\omega \in H^0(\Omega^1_V \langle \, D \, \rangle \bigotimes \omega_V)$. Then we have ${\rm Res}_{\tilde C}(\omega) = 0 \in H^0 ({\cal O}_{\tilde C} (K{\tilde C}))$, since by assumption the degree of ${\cal O}_{\tilde C} (K{\tilde C})$ is negative. Regarding $\omega$ as a meromorphic section in $H^0 ({\bf P}^2,\,\Omega^1_{{\bf P}^2} \otimes \omega_{{\bf P}^2})$ it follows that $\omega$ is holomorphic outside the cusps of $C$. Therefore, $\omega$ extends to a section in $\Omega^1_{{\bf P}^2} \otimes \omega_{{\bf P}^2}$, and hence $\omega = 0$. Thus, $H^0(\Omega^1_V \langle \, D \, \rangle \bigotimes \omega_V) = 0$. Now the result follows by Serre duality. For the proof of b) consider a factorization of the embedded resolution as $$V \to V' \to {\bf P}^2$$ such that $V' \to {\bf P}^2$ yields the minimal resolution of $C$ in the following sense: \noindent (i) The proper transform, say $C'$, of $C$ in $V'$ is smooth, and \noindent (ii) $C$ can not be resolved by fewer blowing ups. \noindent It is easily seen that $$K_{V'} C' = K_V {\tilde C} - \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}}\,$$ (cf. the proof of (4.3, b) below). By the above arguments, if $K_{V'} C' < 0$, then $H^0(\Omega^1_{V'} \langle \, D' \, \rangle \bigotimes \omega_{V'}) = 0$, where $D'$ is the reduced total transform of $C$ in $V'$. Hence also $ H^0(\Omega^1_V \langle \, D \, \rangle \bigotimes \omega_V) = 0$. \hfill $\Box$\\ \noindent {\bf 4.2. Corollary.} {\it With the notation as in (4.1, b), assume that $C$ is a rational cuspidal curve with ${\bar k} ({\bf P}^2 \setminus C) = 2$. If \begin{equation} \sum\limits_{\sigma = 1}^s \sum\limits_{j = 1}^{r_{\sigma}} m_{\sigma \,j} < 3d \,,\end{equation} then} $$\chi(\Theta_V\langle \, D \, \rangle) = K(K+D) = - h^1 (\Theta_V\langle \, D \, \rangle) \le 0\,.$$ {\bf Proof.} From Lemma 4.3,a) below it follows that $${\tilde C}^2 + \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}} = 3d - 2 - \sum\limits_{\sigma = 1}^s \sum\limits_{j = 1}^{r_{\sigma}} m_{\sigma \,j}\,. $$ Therefore, (3) is equivalent to the inequality $${\tilde C}^2 + \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}} \ge - 1\,.$$ Thus, we have $$K{\tilde C} = -{\tilde C}^2 - 2 < \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}} \,,$$ and hence by (4.1, b) $h^2(\Theta_V\langle \, D \, \rangle) = 0$. Since ${\bar k} ({\bf P}^2 \setminus C) = 2$, then also $h^0(\Theta_V\langle \, D \, \rangle) = 0$ (see [Ii, Theorem 6]), and the statement follows. \hfill $\Box$ \\ Note that in our examples, i.e. for $C = C_{d,\,a}$ being as in section 3, we have $K_V C = d-4$ (see (4.3, b)) and $ \sum_{\sigma} m_{\sigma \,r_{\sigma}} = d + 2$; furthermore, $\sum\limits_{\sigma = 1}^s \sum\limits_{j = 1}^{r_{\sigma}} m_{\sigma \,j} = 3(d-2) < 3d$. Thus, (4.1) or (4.2) gives another proof of unobstructedness of $C_{d,\,a}$ (cf. (3.3)). \\ \noindent {\bf 4.3. Lemma.} {\it Let $C \subset {\bf P}^2$ be a rational cuspidal curve, with cusps $P_1,\dots,P_s$ having multiplicity sequences ${\bar m}_{P_{\sigma}} = (m_{\sigma \,1},\dots,m_{\sigma \,k_{\sigma}})$. Then \noindent a) in the minimal embedded resolution $V \to {\bf P}^2$ of singularities of $C$ the proper transform $\tilde C$ of $C$ has selfintersection $${\tilde C}^2 = 3d + s - 2 - \sum\limits_{i,j} m_{ij} = 3d-2 - \sum\limits_{\sigma = 1}^s \sum\limits_{j = 1}^{r_{\sigma}} m_{\sigma \,j} - \sum\limits_{\sigma = 1}^s m_{\sigma \,r_{\sigma}}\,.$$ \noindent b) Furthermore, if $K = K_V$ is the canonical divisor, then $$K{\tilde C}= -3d - s + \sum\limits_{i,j} m_{ij}\,.$$ } \noindent {\bf Proof.} a) Clearly, $${\tilde C}^2 = C^2 - \sum\limits_{i,j} m_{ij}^2 + s = d^2 + s - \sum\limits_{i,j} m_{ij}^2\,.$$ The genus formula yields $$(d-1)(d-2) = \sum\limits_{i,j} m_{ij}(m_{ij} -1)\,.$$ Thus $$d^2 - \sum\limits_{i,j} m_{ij}^2 = 3d - 2 - \sum\limits_{i,j} m_{ij}\,,$$ and (a) follows. \\ b) follows from (a) and the equality $K{\tilde C} + {\tilde C}^2 = -2$. An alternative proof: we proceed by induction on the number of blow ups. First of all, for $K = K_{{\bf P}^2}$ and $C \subset {\bf P}^2$ we have $KC = -3d$. Furthermore, let $C \subset V$ be a curve on a surface $V$ and $K = K_V$ be the canonical divisor, $\sigma: V' \to V$ be the blow up at a cusp of $C$ of multiplicity $m$ and $K' = K_{V'}, \,C' \subset V'$ be the proper preimage of $C$. We have: $$KC = K'C^* = (C' + mE)K' = C'K' + mEK' = $$ $$= C'K' + m(E(K' + E) - E^2) = C'K' + m(-2+1) = K'C' - m\,,$$ hence $K'C' = KC + m$. This completes the proof. \hfill $\Box$\\ \noindent {\bf 4.4. Remark.} Let $E_P \subset V$ be the reduced exceptional divisor of the blow ups over $P \in {\rm Sing}\,C$. Then by Lemma 2 in [MaSa] $$E_P^2 = -\omega_P -1\,.$$ If $D = {\tilde C} + \sum\limits_{P \in {\rm Sing}\,C} E_P \subset V$ is the reduced total transform of $C$ in $V$, then we have (cf. [MaSa, Lemma 4]) $$D^2 = {\tilde C}^2 + 2 {\rm card}\,({\rm Sing}\,C) + \sum\limits_{P \in {\rm Sing}\,C} E_P^2 = {\tilde C}^2 - \sum\limits_{P \in {\rm Sing}\,C} (\omega_P -1) $$ $$ = 3d-2 - \sum\limits_{P \in {\rm Sing}\,C} (\sum\limits_{j=0}^{k_i} m_{P,\,j} + \omega_P - 1)\,.$$ \\ \noindent {\bf 4.5. Remark.} In [OZ2, Proposition 4] the following observation is done. \\ \noindent {\it A projectively rigid rational cuspidal curve $C \subset {\bf P}^2$ cannot have more than 9 cusps.} \\ \noindent The reason is quite simple. Denote by $\kappa$ the number of cusps of $C$. Assuming that $\kappa \ge 3$ we will have $\bar {k} ({\bf P}^2 \setminus C) = 2$ [Wak], and therefore due to Theorem 6 in [Ii], $h^0 = 0$, where $h^i := h^i(\Theta_V\langle \, D \, \rangle)\,,\, i = 0,\,1,\,2$. Let $K+D = H+N$ be the Zariski decomposition in the minimal embedded resolution $V \to {\bf P}^2$ of singularities of $C$. It can be shown that $N^2 = \sum_{P \in {\rm Sing}\,C} N_P^2$, where the local ingredient $N_P^2$ over a cusp $P \in {\rm Sing}\,C$ has estimate $-N_P^2 > 1/2$. Thus, \begin{equation} {\kappa} < 2 \sum\limits_{P \in {\rm Sing}\,C} (-N_P^2 ) = -2N^2 \,.\end{equation} We also have \begin{equation} (K+D)^2 = H^2 + N^2 \,\,\, \,\,\,{\rm and}\,\,\, \,\,\, (K+D)^2 = K(K+D) + D(K + D) = K(K+D) - 2 \,,\end{equation} where [FZ, (1.3)] \begin{equation} K(K + D) = \chi(\Theta_V\langle \, D \, \rangle) = h^2 - h^1 \,.\end{equation} {}From (4)--(6) and the logarithmic Bogomolov-Miyaoka-Yau inequality $H^2 \le 3$ [KoNaSa] we obtain $$ {\kappa} < -2N^2 = -2(K+D)^2 + 2H^2 \le 6 -2(K+D)^2 = 10 - 2K(K+D) = 10 - 2h^2 + 2h^1 \,.$$ Therefore, $${\kappa} < 10 $$ as soon as $h^1 = 0$, i.e. for a projectively rigid curve $C$. \\ Hence, once one constructs a rational cuspidal plane curve with 10 cusps or more, we know that it is not projectively rigid. The latter means that such a curve is a member of an equisingular \footnote{i.e. with cusps of the same type.} family of rational cuspidal plane curves, generically pairwise projectively non--isomorphic \footnote{i.e. non--equivalent under the action of the automorphism group ${\rm PGL}\,(3,\,{\bf C})$ on ${\bf P}^2$.} (see (2.1)). \\ \vspace{.2in} \newpage \centerline {\bf References} \vspace{.2in} {\footnotesize \noindent [EN] D. Eisenbud, W. D. Neumann. Three-dimensional link theory and invariants of plane curve singularities. {\it Ann. Math. Stud.} {\bf 110}, {\it Princeton Univ. Press}, Princeton 1985 \vspace{.1in} \noindent [FZ] H. Flenner, M. Zaidenberg. $\bf Q$--acyclic surfaces and their deformations. {\it Contemporary Mathem.} {\bf 162} (1964), 143--208 \vspace{.1in} \noindent [Ii] Sh. Iitaka. On logarithmic Kodaira dimension of algebraic varieties. In: {\it Complex Analysis and Algebraic Geometry, Cambridge Univ. Press}, Cambridge e.a., 1977, 175--190 \vspace{.1in} \noindent [KoNaSa] R. Kobayashi, S. Nakamura, F. Sakai. A numerical characterization of ball quotients for normal surfaces with branch loci. {\it Proc. Japan Acad.} {\bf 65(A)} (1989), 238--241 \vspace{.1in} \noindent [MaSa] T. Matsuoka, F. Sakai. The degree of rational cuspidal curves. {\it Math. Ann.} {\bf 285} (1989), 233--247 \vspace{.1in} \noindent [Mil] J. Milnor. Singular points of complex hypersurfaces. {\it Ann.Math.Stud.} {\bf 61}, {\it Princeton Univ. Press}, Princeton, 1968 \vspace{.1in} \noindent [Na] M. Namba. Geometry of projective algebraic curves. {\it Marcel Dekker}, N.Y. a.e., 1984 \vspace{.1in} \noindent [OZ1] S.Y. Orevkov, M.G. Zaidenberg. Some estimates for plane cuspidal curves. In: {\it Journ\'ees singuli\`eres et jacobiennes, Grenoble 26--28 mai 1993.} Grenoble, 1994, 93--116 \vspace{.1in} \noindent [OZ2] S.Y. Orevkov, M.G. Zaidenberg. On the number of singular points of plane curves. In: {\it Algebraic Geometry. Proc. Conf., Saintama Univ., March 15--17, 1995}, 22p. (to appear) \vspace{.1in} \noindent [Sa] F. Sakai. Singularities of plane curves. {\it Preprint} (1990), 1-10 \vspace{.1in} \noindent [tDP] T. tom Dieck, T. Petrie. Homology planes: An announcement and survey. In: {\it Topological methods in algebraic transformation groups, Progress in Mathem.} {\bf 80}, {\it Birkha\"user}, Boston e.a., 1989, 27--48 \vspace{.1in} \noindent [Ts] S. Tsunoda. The structure of open algebraic surfaces and its application to plane curves. {\it Proc. Japan Acad.} {\bf 57(A)} (1981), 230--232 \vspace{.1in} \noindent [Wak] I. Wakabayashi. On the logarithmic Kodaira dimension of the complement of a curve in ${\bf P}^2$. {\it Proc. Japan Acad.} {\bf 54(A)} (1978), 157--162 \vspace{.1in} \noindent [Wal] R. J. Walker. Algebraic curves. {\it Princeton Univ. Press}, Princeton, 1950 \vspace{.1in} \noindent [Y1] H. Yoshihara. On plane rational curves. {\it Proc. Japan Acad.} {\bf 55(A)} (1979), 152--155 \vspace{.1in} \noindent [Y2] H. Yoshihara. Rational curve with one cusp. I {\it Proc. Amer. Math. Soc.} {\bf 89} (1983), 24--26; II {\it ibid.} {\bf 100} (1987), 405--406 \vspace{.1in} \noindent [Y3] H. Yoshihara. Plane curves whose singular points are cusps. {\it Proc. Amer. Math. Soc.} {\bf 103} (1988), 737--740 \vspace{.1in} \noindent [Y4] H. Yoshihara. Plane curves whose singular points are cusps and triple coverings of ${\bf P}^2$. {\it Manuscr. Math.} {\bf 64} (1989), 169-187 \vspace{.2in} \noindent Hubert Flenner\\ \noindent Fakult\"at f\"ur Mathematik\\ Ruhr Universit\"at Bochum\\ Geb.\ NA 2/72\\ Universit\"atsstr.\ 150\\ 44180 BOCHUM, Germany\\ \noindent e-mail: [email protected] \vspace{.2in} \noindent Mikhail Zaidenberg\\ \noindent Universit\'{e} Grenoble I \\ Laboratoire de Math\'ematiques associ\'e au CNRS\\ BP 74\\ 38402 St. Martin d'H\`{e}res--c\'edex, France\\ \noindent e-mail: [email protected]} \end{document}

9507

alg-geom/9507006

en

https://arxiv.org/abs/alg-geom/9507006

alg-geom/9507006

Jeroen Spandaw

Jeroen G. Spandaw

A Noether-Lefschetz theorem for vector bundles

5 pages, no figures; LaTeX2e, should also work with LaTeX 2.09 with NFSS

In this note we use the monodromy argument to prove a Noether-Lefschetz theorem for vector bundles.

alg-geom

\section{Introduction}\pagenumbering{arabic} Let $X$ be a smooth complex projective manifold of dimension $n$ and let $E$ be a very ample vector bundle on $X$ of rank $r$. This means that the tautological quotient line bundle $L$ on the bundle $Y={\Bbb P}(E^\ast)$ of hyperplanes in $E$ is very ample. For almost all $s\in H^0(X,E)$ the zero-locus $Z$ is smooth, irreducible and of dimension $n-r$. In \cite[prop.~1.16]{S} Sommese proved that $H^{i}(X,Z;{\Bbb Z})$ vanishes for $i<n-r+1$ and is torsion free for $i=n-r+1$. Assume that $n-r$ is even, say $n-r=2p$. Let $\mathop{\mbox{\upshape Alg}}\nolimits\subset H^{n-r}(Z)$ be the space of algebraic classes and let $\mathop{\mbox{\upshape Im}}\nolimits=\mathop{\mbox{\upshape Im}}\nolimits(H^{n-r}(X)\hookrightarrow H^{n-r}(Z))$. (We always take coefficients in ${\Bbb C}$ unless other coefficients are mentioned explicitely (cf.~Remark~\ref{rmk}).) In this note we prove the following Noether-Lefschetz theorem for this situation. \begin{thm}\label{thm1} If $E$ is very ample and $s$ is general, then either $\mathop{\mbox{\upshape Alg}}\nolimits\subset \mathop{\mbox{\upshape Im}}\nolimits$ or $\mathop{\mbox{\upshape Alg}}\nolimits+\mathop{\mbox{\upshape Im}}\nolimits=H^{n-r}(Z)$. \end{thm} (With \lq\lq general\rq\rq we shall always mean general in the usual Noether-Lefschetz sense.) The following theorem, which generalizes the Noether-Lefschetz theorem for complete intersections in projective space (see \cite[pp.~328--329]{DK}) is an immediate corollary. \begin{thm}\label{thm2} If $h^{\alpha\beta}(X)<h^{\alpha\beta}(Z)$ for some pair $(\alpha,\beta)$ with $\alpha+\beta=n-r$ and $\alpha\neq \beta$, then every algebraic class on $Z$ is induced from $X$. \end{thm} \begin{rmk}\label{rmk}\normalshape Notice that the unique pre-images of algebraic classes are themselves Hodge classes, i.e.\ lie in $H^{p,p}(X)\cap H^{n-r}(X;{\Bbb Z})$. This follows from the fact that the cokernel of $H^{n-r}(X,{\Bbb Z})\to H^{n-r}(Z,{\Bbb Z})$ is torsion free. \end{rmk} It is not difficult to show that after replacing $E$ with $E\otimes L^k$, where $k\gg0$ and $L$ is an ample line bundle, the assumption of theorem~\ref{thm2} is satisfied. (E.g.\ the geometric genus of $X$ goes to infinity as $k$ goes to infinity.) In \cite{Sp} we used the notion of Castelnuovo-Mumford regularity (cf.~\cite[p.~99]{M}) to make the positivity assumption on $E$ more precise if $X={\Bbb P}^n$. Notations are as in theorem~\ref{thm1}. $\mathop{\mbox{\upshape Hdg}}\nolimits$ is defined to be the space of Hodge classes on $Z$ of codimension~$p$, i.e.\ $\mathop{\mbox{\upshape Hdg}}\nolimits=H^{p,p}(Z)\cap H^{n-r}(Z,{\Bbb Z})$. \begin{thm}\label{thm3} If $E$ is a $(-3)$-regular vector bundle of rank $r$ on $X={\Bbb P}^n$ and $Z$ is the zero-locus of a general global section of $E$, then $\mathop{\mbox{\upshape Hdg}}\nolimits\subset\mathop{\mbox{\upshape Im}}\nolimits$, unless $(X,E)=({\Bbb P}^3,{\cal O}(3))$. If $\dim Z=2$, then it suffices that $E$ be $(-2)$-regular unless $(X,E)=({\Bbb P}^3,{\cal O}(2))$, $({\Bbb P}^3,{\cal O}(3))$ or $({\Bbb P}^4,{\cal O}(2)\oplus{\cal O}(2))$. \end{thm} (Notice that $(-3)$-regularity $\Longrightarrow$ $(-1)$-regularity $\Longrightarrow$ very ampleness.) For the case $\dim Z=2$ theorem~\ref{thm3} is due to Ein \cite[thm.~3.3]{E}. The advantage of theorem~\ref{thm3} is that it applies to {\em Hodge\/} rather than {\em algebraic\/} classes on $Z$. For example, it implies that if all Hodge classes of codimension $n-r$ on ${\Bbb P}^n$ are algebraic, then the same holds for $Z$. The advantage of theorem~\ref{thm1} is that the positivity condition on $E$ is more geometric: the cohomological conditions from \cite{Sp} are replaced with the condition that $E$ be very ample plus a Hodge number inequality (cf.~theorem~\ref{thm2}). In other words, for very ample vector bundles, the Noether-Lefschetz property holds as soon as this is allowed by the Hodge numbers. However, this Hodge number inequality condition is of course a cohomological condition on $E$ in disguise. \smallskip \noindent {\em Acknowledgement\/} I am grateful to professor Sommese for the suggestion that I look at the bundle $\pi\colon {\Bbb P}(E^\ast)\to X$ of hyperplanes in $E$. \section{Proof of the main result} Let $V=H^0(X,E)$, let ${\Bbb P}(V)$ be the set of lines in $V$, let $N=\dim{\Bbb P}(V)=h^0(X,E)-1$ and set $X'={\Bbb P}(V)\times X$. Set $E'=p_1^\ast{\cal O}(1)\otimes p_2^\ast E$, where $p_i$ are the projections. $E'$ has a canonical section $s'$. Let ${\cal Z}$ be the zero locus of $s'$. The restriction $ p\colon{\cal Z}\to{\Bbb P}(V) $ of $p_1$ to ${\cal Z}$ is the universal family of zero loci of sections in $E$. We leave the proof of the following easy lemma to the reader. \begin{lem} If $E$ is very ample, then it is generated by its sections. If $E$ is generated by its sections, then ${\cal Z}$ is smooth, irreducible and of dimension $N+n-r$. \end{lem} Let $\Delta\subset{\Bbb P}(V)$ be the discriminant of $p$, i.r.\ \begin{eqnarray*} \Delta&=&p\{z\in{\cal Z}: \mathop{\mbox{\upshape rk}}\nolimits_z p\le N-1\}\\ &=&\{[s]\in{\Bbb P}(V): \hbox{$p^{-1}(s)$ is not smooth of dimension $n-r$}\}. \end{eqnarray*} Fix a point $[s_0]\in{\Bbb P}(V)\setminus\Delta$ and let $Z\subset X$ be the corresponding smooth fibre of $p$. Let $\Gamma$ the image of the monodromy representation $\pi_1({\Bbb P}(V) \setminus\Delta)\to \mathop{\mbox{\upshape Aut}}\nolimits(H^{n-r}(Z))$. Let $\mathop{\mbox{\upshape Im}}\nolimits^\perp$ be the orthogonal complement of $\mathop{\mbox{\upshape Im}}\nolimits$ with respect to the intersection form on $H^{n-r}(Z)$. Since for general $s\in H^0(X,E)$, $\mathop{\mbox{\upshape Alg}}\nolimits$ is a $\Gamma$-module (cf.\ \cite[p.~141]{H}), theorem~\ref{thm1} from the following proposition. \begin{pro} (Second Lefschetz Theorem) \begin{enumerate} \item $H^{n-r}(Z)=\mathop{\mbox{\upshape Im}}\nolimits\oplus\mathop{\mbox{\upshape Im}}\nolimits^\perp$ \item $\mathop{\mbox{\upshape Im}}\nolimits=H^{n-r}(Z)^\Gamma$ \item $\mathop{\mbox{\upshape Im}}\nolimits^\perp$ is an irreducible $\Gamma$-module \end{enumerate} \end{pro} \begin{pf} \begin{enumerate} \item Arguing as in the proof of \cite[thm.~6.1 (i)]{G} one shows that if $Z$ is submanifold of a compact K\"ahler manifold $X$ such that $H^{i}(X,Z)=0$ for $i\le m= \dim Z$, then the restriction of the intersection form to $\mathop{\mbox{\upshape Im}}\nolimits(H^m(X)\hookrightarrow H^m(Z))$ is non-degenerate. \item The inclusion $\mathop{\mbox{\upshape Im}}\nolimits\subset H^{n-r}(Z)^\Gamma$ is trivial. To prove that $H^{n-r}(Z)^\Gamma\subset\mathop{\mbox{\upshape Im}}\nolimits$, we argue as in \cite[thm.~6.1 (iii)]{G}. Consider the commutative diagram $$ \begin{CD} H^{n-r}({\Bbb P}(V)\times X) @>>> H^{n-r}({\cal Z})\\ @VVV @VVV\\ H^{n-r}(X) @>>> H^{n-r}(Z)^\Gamma. \end{CD} $$ By \cite[th\'eor\`eme 4.1.1 (ii)]{D} the map $H^{n-r}({\cal Z})\to H^{n-r}(Z)^\Gamma$ is surjective. By \cite[prop.~1.16]{S} the map $H^{n-r}({\Bbb P}(V)\times X) \to H^{n-r}({\cal Z})$ is surjective. \item Since the monodromy respects the intersection form, $I^\perp$ is a $\Gamma$-module. The standard argument using Lefschetz pencils and the theory of vanishing cycles reduces the problem of irreducibility to proposition~\ref{prop} below (cf. \cite[pp.~46--48]{L}). \end{enumerate} \end{pf} \begin{pro}\label{prop} \begin{enumerate} \item The discriminant $\Delta$ is an irreducible, closed, proper subvariety of ${\Bbb P}(V)$. \item Let $G\subset{\Bbb P}(V)$ be a general line. Then ${\cal Z}_G:=p^{-1}(G)$ is smooth, irreducible of dimension $n-r+1$ and the restricted family $p_G\colon{\cal Z}_G\to G$ is a holomorphic Morse function, i.e. all critical points are non-degenerate and no two lie in the same fibre (cf.~\cite[p.~34]{L}). $g\in G$ is a critical value of $p_G$ if and only if it is a critical value of $p$. \end{enumerate} \end{pro} \begin{pf} The statements about ${\cal Z}_G$ follow from Bertini. The remaining assertions are well-known if $\mathop{\mbox{\upshape rk}}\nolimits E=1$ (cf.~\cite[p.~19]{L}). In particular, they are true for $(Y,L)$, where $Y$ is the hyperplane bundle ${\Bbb P}(E)$ of $E$ and $L$ is the tautological quotient line bundle ${\cal O}_Y(1)$. The following proposition reduces the general case $(X,E)$ to this line bundle case $(Y,L)$, thus finishing the proof. \end{pf} Before we state the last proposition, notice that the natural map $s\mapsto \bar{s}\colon H^0(X,E)\to H^0(Y,L)$, where $\bar{s}(x,h):=\overline{s(x)}\in E(x)/h=L(x,h)$ for $(x,h)\in Y$, is an isomorphism. Indeed, the map is clearly injective and $h^0(Y,L)=h^0(X,\pi_\ast L)=h^0(X,E)$. For $s\in H^0(X,E)$ we denote by $Z_X(s)$ the zero-locus of $s$ in $X$ and by $Z_Y(\bar{s})$ the zero-locus of $\bar{s}$ in $Y$. \begin{pro} For $s\in H^0(X,E)\setminus\{0\}$, $Z=Z_X(s)$ is singular if and only if $W=Z_Y(\bar{s})$ is singular. More precisely, if $x\in\mathop{\mbox{\upshape Sing}}\nolimits Z$, then there exists a $y\in\mathop{\mbox{\upshape Sing}}\nolimits W$ with $\pi(y)=x$ and conversely, if $(x,h)\in\mathop{\mbox{\upshape Sing}}\nolimits Z$, then $x\in \mathop{\mbox{\upshape Sing}}\nolimits W$. Finally, if $(x,h)$ is a non-degenerate quadratic singularity, then so is $x$. \end{pro} \begin{pf} This is a calculation in local coordinates. Let $x_0\in Z$, i.e.\ $s(x_0)=0$. After choosing local coordinates $x_1,\ldots,x_n$ on $X$ and a local trivialization of $E$ near $x_0$ we may regard $s$ to be a function in $x_1,\ldots,x_n$. Then $x_0\in\mathop{\mbox{\upshape Sing}}\nolimits Z$ if and only if $\{\frac{\partial s}{\partial x_j}(x_0)\}_{j=1}^n$ does not span ${\Bbb C}^r$. Let $h_0\subset{\Bbb C}^r$ be a hyperplane containing $\mathop{\mbox{\upshape span}}\nolimits\{\frac{\partial s}{\partial x_j}(x_0)\}_{j=1}^n$. We claim that $y_0=(x_0,h_0)\in\mathop{\mbox{\upshape Sing}}\nolimits W$. We may assume that the local trivialization of $E$ has been chosen in such a way that $h_0$ is given by $z_r=0$, where $z_1,\ldots,z_r$ are coordinates on ${\Bbb C}^r$. Let $s=(f_1,\ldots,f_r)$. Local coordinates on $Y$ near $y_0$ are provided by the local coordinates $x_1,\ldots,x_n$ on $X$ near $x_0$ together with $(y_1,\ldots,y_{r-1})\in{\Bbb C}^{r-1}$: we let $(y_1,\ldots,y_{r-1})\in{\Bbb C}^{r-1}$ correspond to the hyperplane $\sum_{i=1}^r y_iz_i=0$, where $y_r:=1$. The point $y_0$ has coordinates $(x_0,0)$. In these local coordinates $\bar{s}(x,y)=\sum_{i=1}^ry_if_i(x)$. It now suffices to calculate $\frac{\partial \bar{s}}{\partial x_k}(x_0,0) =\frac{\partial f_r}{\partial x_k}(x_0)=0$ for $k=1,\ldots,n$ and $\frac{\partial \bar{s}}{\partial y_j}(x_0,0)=f_j(x_0)=0$ for $j=1,\ldots,r-1$. The converse is proven similarly. Let $y_0=(x_0,h_0)\in\mathop{\mbox{\upshape Sing}}\nolimits W$. We may again assume that $h_0$ is given by $z_r=0$. The Hessian of $\bar{s}$ in $y_0$ is of the form $\left(\begin{array}{cc} h & d^t\\ d & 0\end{array}\right)$, where the $n\times n$-matrix $h$ is the Hessian of $f_r$ and the $(r-1)\times n$-matrix $d$ is the Jacobian of $f':=(f_1,\ldots,f_{r-1})$ in $x_0$. Let $Z'=\{x\in X: f'(x)=0\}$. We have to check that the Hessian of $f_r|_{Z'}$ in $0$ is non-degenerate. Since we assume that the Hessian of $\bar{s}$ has maximal rank in $y_0$, so has $d$. Thus, after a change of coordinates, we may assume that $f_i(x)=x_i$ for $i<r$. Then $\bar{s}(x,y)=\sum_{i=1}^{r-1}x_iy_i+f_r(x)$, hence the Hessian of $\bar{s}$ in $y_0$ is $$ \left(\begin{array}{ccc} \ast & \ast & E_{r-1}\\ \ast & H & 0\\ E_{r-1} & 0 & 0, \end{array} \right), $$ where $H$ is the Hessian of $f_r|_{Z'}$ in $x_0$. It follows that $H$ is non-degenerate. \end{pf}

9507

alg-geom/9507001

en

https://arxiv.org/abs/alg-geom/9507001

alg-geom/9507001

Iwamoto Masayuki

Masayuki Iwamoto

General n-canonical divisors on two-dimensional smoothable semi-log-terminal singularities

AMSLaTeX v 1.1

In this paper we calculate genaral n-canonical divisors on smoothable semi-log-terminal singularities in dimension 2, in other words, the full sheaves associated to the double dual of the nth tensor power of the dualizing sheaves of these singularities. And as its application we give the inequality which bound the Gorenstein index by the local self intersection number of the n-canonical divisor of these singularities.

alg-geom

\section{Introduction} This paper is devoted to some fundumental calculation on 2-dimensional smoothable semi-log-terminal singularities. If we study minimal or canonical models of one parameter degeneration of algebraic surfaces, we must treat singularities that appear in the central fiber. Smoothable semi-log-terminal singularities are the singularities of the central fiber of the minimal model of degeneration, and the singularities of the central fiber of the canonical model of degeneration which may have large Gorenstein index. Koll\'{a}r and Shepherd-Barron caracterized these singularities in \cite{ksb}, but for numerical theory of degeneration, we need more detailed information. \par In this paper, we calculate general $n$-canonical divisors on these singularities, in other words, we calculate the full sheaves associated to the double dual of the $n$-th tensor power of the dualizing sheaves. And the application of this result, we bound the Gorenstein index by the local self intersection number of the $n$-canonical divisor. \par Notation: In this paper, \begin{align*} [q_1, q_2, q_3, \dots]&=q_1+ \cfrac{1}{q_2+ \cfrac{1}{q_3+{}_{\ddots} }}\\ [[q_1, q_2, q_3, \dots]]&=q_1- \cfrac{1}{q_2- \cfrac{1}{q_3-{}_{\ddots} }} \end{align*} If $p:\tilde{X}\to X$ is a birational morphism and $D$ is a divisor on $X$, we denote by $\tilde{D}$ the proper transform of $D$ on $\tilde{X}$. \section{Basic calculation} Let $Y$ be a cyclic quotient singularity of the form Spec$(\Bbb C[z_1,z_2])/\langle\alpha\rangle$, where $\langle\alpha\rangle$ is a cyclic group of order $r$ and $\alpha$ acts on Spec$(\Bbb C[z_1,z_2])/\langle\alpha\rangle$ as $(\alpha^* z_1,\alpha^* z_2)=(\eta^s z_1,\eta z_2)$ in which $\eta$ is a primitive $r$-th root of unity, $(r,s)=1$, and $0<s<r$. Let $\dfrac{r}{s}=[[q_1, q_2, \dots, q_k]]$ be an expansion into continued fraction, and $r_{i}$ be the $i$-th remainder of the Euclidean algorithm, i.e. $\{r_i\}_{i=0, 1, \dots, k+1}$ is a seqence determined by $r_0=r, r_1=s, r_{i-1}=q_i r_i-r_{i+1}$. Let $\dfrac{P_i}{Q_i}$ be the $i$-the convergent, i.e. $\{P_i\}_{i=-1, 0, \dots, k}$ is a sequence determined by $P_{-1}=0, P_0=1, P_i=q_i P_{i-1}-P_{i-2}$ and $\{Q_i\}_{i=-1, 0, \dots, k}$ is determined by $Q_{-1}=-1, Q_0=0, Q_i=q_iQ_{i-1}-Q_{i-2}$. Let $f:\Bbb C^{2}\rightarrow Y$ be a quotient map, and $p:\mbox{$\tilde{Y}$}\rightarrow Y$ the minimal desingularization. It is well known that the dual graph of the exceptional divisors of $p$ is a chain of rational curves $\cup_{1\le i\le k}E_i$ such that $E_i^2=-q_i$. We put $$\lambda_i=P_{i-1},\quad \mu_{i}=r_i,\quad \alpha^j_i= \begin{cases} \dfrac{1}{m}\mu_i\lambda_j\ &(\text{for}\ j\le i)\\ \dfrac{1}{m}\lambda_i\mu_j\ &(\text{for}\ i<j) \end{cases} $$ Note that $c_1z_1^{\lambda_i}+c_2z_2^{\mu_i}$ is a $\langle\alpha\rangle$-semi-invariant since $r_1P_i-r_0Q_i=r_{i+1}$. \begin{lem}\label{lemcurve} Let $C_i$ be a divisor on $Y$ such that $f^*C_i=(c_iz_1^{\lambda_i}+c_2z_2^{\mu_i}=0)$ in which $c_1,c_2\in \Bbb C^*$. Then $$ \text{\em{(i)}}\ \mbox{$\tilde{C}$}_i\cdot E_j=\delta_{i,j}\quad \text{\em{(ii)}}\ p^*C_i=\mbox{$\tilde{C}$}_i+\sum\alpha^i_jE_j$$ \end{lem} \begin{pf} If we write $Y=T_N\text{emb}(\sigma)$, where $N=\Bbb Z n_1+\Bbb Z n_2$ and $\sigma=\Bbb R_{\geq 0}n_1+\Bbb R_{\geq 0}[(r-s)n_1+n_2]$, then $E_i$ corresponds to $(P_{i-1}-Q_{i-1})n_1+P_{i-1}n_2$. The rest of proof is a direct calculation using the above description and the formula $P_iQ_{i-1}-Q_iP_{i-1}=-1$, and it can be easily done. \end{pf} Next lemma is a easy fact on continued fraction. We denote by $(2, q)$ the sequence $(2, 2, \dots, 2)$ of length $q$. \begin{lem}\label{lemcont1} Let $a$, $m$ be a natural number such that $(a, m)=1$, $\frac{m}{2}<a<m$. Put $b=m-a$. Let $\dfrac{a}{b}=[q_1, q_2, \dots, q_k]$. Then \begin{enumerate} \item If $k$ is even, \begin{align*} \dfrac{m}{a}=&[[(2, q_1), q_2+2, (2, q_3-1), q_4+2, \dots, (2, q_{k-3}-1), q_{k-2}+2,\\ & (2, q_{k-1}-1), q_k+1]]\\ \dfrac{m}{b}=&[[q_1+2, (2, q_2-1), q_3+2, (2, q_4-1), q_5+2, \dots, (2, q_{k-2}-1),\\ & q_{k-1}+2, (2, q_k+1)]] \end{align*} \item If $k$ is odd, \begin{align*} \dfrac{m}{a}=&[[(2, q_1), q_2+2, (2, q_3-1), q_4+2, \dots, q_{k-3}+2, (2, q_{k-2}-1),\\ &q_{k-1}+2, (2, q_k-1)]]\\ \dfrac{m}{b}=&[[q_1+2, (2, q_2-1), q_3+2, (2, q_4-1), q_5+2, \dots, (2, q_{k-3}-1), q_{k-2}+2,\\ & (2, q_{k-1}-1), q_k+1]] \end{align*} \end{enumerate} \end{lem} \begin{pf} Since this is elementary we left it for the reader. \end{pf} In the rest of this section we shall index the exceptional divisors of the minimal (semi-) resolution smoothable of the semi-log-terminal singularity. \par First we treat the normal case. Let $(a, d, m)$ be a triplet of positive integers such that $a<b$ and $a$ is prime to $m$. We denote by $X_{a, d, m}$ a 2-dimentional quotient singularity of the form Spec$(\Bbb C[z_1, z_2])/\langle\alpha\rangle$, where $\langle\alpha\rangle$ is a cyclic group of order $dm^2$ and $\alpha$ acts on Spec$(\Bbb C[z_1, z_2])$ as $\alpha^*(z_1, z_2)=(\varepsilon^{adm-1}z_1, \varepsilon z_2)$ in which $\varepsilon$ is a primitive $dm^2$-th root of unity. By [K-SB Proposition 3.10], a singularity of class T which is not RDP is analytically isomorphic to $X_{a, d, m}$ for some $(a, d, m)$. Let $f:\Bbb C^2\to X_{a, d, m}$ be a quotient map and $p:\mbox{$\tilde{X}$}_{a, d, m}\to X_{a, d, m}$ the minimal desingularization. We assume $2a>m$ since $X_{a, d, m}\simeq X_{m-a, d, m}$. Put $b=m-a$, and let $\dfrac{a}{b}=[q_1, q_2, \dots, q_k]$ be an expantion into continued fraction. Let $r_i$ be the $i$-th remainder of the Euclidean algorithm, i.e. $\{r_i\}_{i=0, 1, \dots, k+1}$ is a sequence determined by $r_0=a, r_1=m-a, r_{i-1}=q_i r_i+r_{i+1}$. Let $\dfrac{P_i}{Q_i}$ be an the $i$-th convergent, i.e. $\{P_i\}_{i=-1, 0, \dots, k}$ is a sequence determined by $P_{-1}=0, P_0=1, P_i=q_iP_{i-1}+P_{i-2}$ and $\{Q_i\}_{i=-1, 0, \dots, k}$ is determined by $Q_{-1}=1, Q_0=0, Q_i=q_iQ_{i-1}+Q_{i-2}$. \begin{lem}\label{lemcont2} Let $m$, $a$, $b$, $d$ be positive integers such that $m=a+b$, $a>b$, $(a, b)=1$. Let $\dfrac{a}{b}=[q_1, q_2, \dots, q_k]$ be an expansion into continued fraction. Then $$\dfrac{dma-1}{dmb+1}=[q'_1, q'_2, \dots, q'_{k'}]$$ where $q'_i$ is as follows. \begin{enumerate} \item If $d=1$ and $k$ is even, \begin{equation*} q'_i= \begin{cases} q_1\quad &(i=1)\\ q_i\quad &(2\le i\le k-1)\\ q_k+1\quad &(i=k)\\ q_k-1\quad &(i=k+1)\\ q_{2k-i+1}\quad &(k+2\le i\le 2k-1)\\ q_1+1\quad &(i=2k=k') \end{cases} \end{equation*} \item If $d=1$ and $k$ is odd, \begin{equation*} q'_i= \begin{cases} q_1\quad &(i=1)\\ q_i\quad &(2\le i\le k-1)\\ q_k-1\quad &(i=k)\\ q_k+1\quad &(i=k+1)\\ q_{2k-i+1}\quad &(k+2\le i\le 2k-1)\\ q_1+1\quad &(i=2k=k') \end{cases} \end{equation*} \item If $d\ge 2$ and $k$ is even, \begin{equation*} q'_i= \begin{cases} q_1\quad &(i=1)\\ q_i\quad &(2\le i\le k)\\ d-1\quad &(i=k+1)\\ 1\quad &(i=k+2)\\ q_k-1\quad &(i=k+3)\\ q_{2k+3-i}\quad &(k+4\le i\le 2k+1)\\ q_1+1\quad &(i=2k+2=k') \end{cases} \end{equation*} \item If $d\ge 2$ and $k$ is odd, \begin{equation*} q'_i= \begin{cases} q_1\quad &(i=1)\\ q_i\quad &(2\le i\le k-1)\\ q_k-1\quad &(i=k)\\ 1\quad &(i=k+1)\\ d-1\quad &(i=k+2)\\ q_{2k+3-i}\quad &(k+3\le i\le 2k+1)\\ q_1+1\quad &(i=2k+2=k') \end{cases} \end{equation*} \end{enumerate} \end{lem} By Lemma \ref{lemcont1} and \ref{lemcont2}, we can calculate the dual graph of the exceptional divisors of $p$ in terms of the continued fraction expansion of $\dfrac{a}{b}$. (See [K-SB]) {}From now, we assume $k$ is even since the calculation is the same for odd $k$. We shall index the exceptional divisors in the following manner. Set the index set $I_o, I_e, I$ as follows: \begin{align*} I_o&=\{(i, j)|1\leq i\leq k+1;i\ \text{odd}; 1\leq j\leq q_i(\text{for}\ i<k+1), 1\leq j\leq d(\text{for}\ i=k+1)\}\\ I_e&=\{(i, j)|1\leq i\leq k+1;i\ \text{even}; 1\leq j\leq q_i(\text{for}\ i<k), 1\leq j\leq q_k-1(\text{for}\ i=k)\}\\ I&=I_o\amalg I_e \end{align*} We define $\rho^i_j$, $\bar{\lambda}^i_j$, $\hat{\lambda}^i_j$, $\lambda^i_j$, $\bar{\mu}^i_j$, $\hat{\mu}^i_j$, $\mu^i_j$ for $(i, j)\in I$ as follows. \begin{equation*} \rho^i_j=r_{i-1}-(j-1)r_i \end{equation*} \begin{equation*} \bar{\lambda}^i_j= \begin{cases} P_{i-2}+(j-1)P_{i-1}\quad &((i, j)\in I_o)\\ -\{P_{i-2}+(j-1)P_{i-1}\}+da\rho^i_j\quad &((i, j)\in I_e) \end{cases} \end{equation*} \begin{equation*} \hat{\lambda}^i_j= \begin{cases} Q_{i-2}+(j-1)Q_{i-1}\quad &((i, j)\in I_o)\\ -\{Q_{i-2}+(j-1)Q_{i-1}\}+db\rho^i_j\quad &((i, j)\in I_e) \end{cases} \end{equation*} \begin{equation*} \bar{\mu}^i_j= \begin{cases} -\{P_{i-2}+(j-1)P_{i-1}\}+da\rho^i_j\quad &((i, j)\in I_o)\\ P_{i-2}+(j-1)P_{i-1}\quad &((i, j)\in I_e) \end{cases} \end{equation*} \begin{equation*} \hat{\mu}^i_j= \begin{cases} -\{Q_{i-2}+(j-1)Q_{i-1}\}+db\rho^i_j\quad &((i, j)\in I_o)\\ Q_{i-2}+(j-1)Q_{i-1}\quad &((i, j)\in I_e) \end{cases} \end{equation*} \begin{equation*} \lambda^i_j=\bar{\lambda}^i_j+\hat{\lambda}^i_j,\quad \mu^i_j=\bar{\mu}^i_j+\hat{\mu}^i_j \end{equation*} We can write $Y=T_N\text{emb}(\sigma)$, where $$N=\Bbb Z n_1+\Bbb Z n_2,\quad \sigma=\Bbb R_{\ge 0}n_1+\Bbb R_{\ge 0}[d(bm+1)n_1+dm^2n_2]$$ We denote by $E^i_j$ the exceptional divisor associated to $\hat{\lambda}^i_jn_1+\lambda^i_jn_2$. Note that by Lemma \ref{lemcurve}, for $\iota\in I$, the proper transform of $C_{\iota}=(c_1z_1^{\lambda_{\iota}}+c_2z_2^{\mu_{\iota}})/ \langle\alpha\rangle\in X_{a,d,m}$ intersects the exceptional locus transversely at $E_{\iota}$. We define the order `$\le$' in the index set $I$ by the lexicographic order. \par Next we treat the non-normal case. We denote by $NC^2=\text{Spec}\Bbb C [z_1, z_2, z_3]/(z_1 z_2)$ a 2-dimentional normal crossing point. Let $(a, m)$ be a pair of positive integers such that $0<a<m$ and $a$ is prime to $m$. Put $b=m-a$, and let $a'$ and $b'$ be integers such that $aa'\equiv bb'\equiv 1\pmod{m}$, $0<a'<m$, and $0<b'<m$. Let $\langle \alpha \rangle$ be a cyclic group of order $m$, and let $\langle \alpha \rangle$ act on $NC^2$ as $(\alpha^*z_1, \alpha^*z_2, \alpha^*z_3) =(\varepsilon^{a'}z_1, \varepsilon^{b'}z_2, \varepsilon z_3)$ where $\varepsilon$ is a primitive $m$-th root of unity. We denote by $X_{a, m}$ the quotient of $NC^2$ by this $\langle \alpha \rangle$-action. By [K-SB], 2-dimentional smoothable semi-log-terminal singularity which is neither normal nor $NC^2$ is analytically isomorphic to $X_{a, m}$ for some $(a, m)$. Put $X=X_{a, m}$. Let $f:NC^2\to X$ be the quotient map and $p:\tilde{X}\to X$ the minimal semi-resolution. Let $g:\Bbb C^2_o\amalg\Bbb C^2_e\to NC^2$, $g_X:X_o\amalg X_e\to X$, and $g_{\tilde{X}}:\tilde{X}_o\amalg\tilde{X}_e\to\tilde{X}$ be normalizations, where $\Bbb C^2_o=\text{Spec}\Bbb C [z_1, z_3]$, $\Bbb C^2_e=\text{Spec}\Bbb C [z_2, z_3]$, $X_o$ (resp. $X_e$) is the quotient of $\Bbb C^2_o$ (resp. $\Bbb C^2_e$), and $\tilde{X}_o$ (resp. $\tilde{X}_e$) is the minimal resolution of $X_o$ (resp. $X_e$). We get the following diagram. \begin{equation*} \begin{CD} \Bbb C^2_o\amalg \Bbb C^2_e @>{f_o\amalg f_e}>> X_o\amalg X_e @<{p_o\amalg p_e}<< \tilde{X}_o\amalg\tilde{X}_e \\ @V{g}VV @VV{g_X}V @VV{g_{\tilde{X}}}V \\ NC^2 @>>{f}> X @<<{p}< \tilde{X} \end{CD} \end{equation*} We denote by $\Delta$, $\Delta '$, and $\tilde{\Delta}$ the double curve of $X$, $NC^2$, and $\tilde{X}$ respectively. Let $\Delta_o$ (resp. $\Delta_e$) be the inverse image of $\Delta$ in $X_o$ (resp. $X_e$), and define $\Delta '_o$, $\Delta '_e$, $\tilde{\Delta}_o$, and $\tilde{\Delta}_e$ similarly. We assume $k$ is even. Set the index set $I_o$, $I_e$, $I$ as follows: \begin{align*} I_o&=\{(i, j)|1\le i\le k+1;i\ \text{odd}; 1\le j\le q_i(\text{for}\ i<k), j=1(\text{for}\ i=k+1)\}\\ I_e&=\{(i, j)|1\le i\le k+1;i\ \text{even}; 1\le j\le q_i\}\\ I&=I_o\amalg I_e \end{align*} We define $\lambda^i_j$, $\mu^i_j$ for $(i, j)\in I$ as follows: \begin{equation*} \lambda^i_j= \begin{cases} P_{i-2}+Q_{i-2}+(j-1)(P_{i-1}+Q_{i-1})\quad &((i, j)\in I_o)\\ r_{i-1}-(j-1)r_i\quad &((i, j)\in I_e) \end{cases} \end{equation*} \begin{equation*} \mu^i_j= \begin{cases} r_{i-1}-(j-1)r_i\quad &((i, j)\in I_o)\\ P_{i-2}+Q_{i-2}+(j-1)(P_{i-1}+Q_{i-1})\quad &((i, j)\in I_e) \end{cases} \end{equation*} We define $\bar{\lambda}^i_j$, $\hat{\lambda}^i_j$ for $(i, j)\in I_o$ as follows: $$\bar{\lambda}^i_j=P_{i-2}+(j-1)P_{i-1},\quad \hat{\lambda}^i_j=Q_{i-2}+(j-1)Q_{i-1}$$ We define $\bar{\mu}^i_j$, $\hat{\mu}^i_j$ for $(i, j)\in I_e$ as follows: $$\bar{\mu}^i_j=P_{i-2}+(j-1)P_{i-1},\quad \hat{\mu}^i_j=Q_{i-2}+(j-1)Q_{i-1}$$ We write $X_o=T_{N^o}\text{emb}(\sigma^o)$ and $X_e=T_{N^e}\text{emb}(\sigma^e)$ where $$N^o=\Bbb Z n^o_1+\Bbb Z n^o_2,\quad \sigma^o=\Bbb R_{\ge 0}n^o_1+\Bbb R_{\ge 0}(bn^o_1+mn^o_2)$$ $$N^e=\Bbb Z n^e_1+\Bbb Z n^e_2,\quad \sigma^e=\Bbb R_{\ge 0}n^e_1+\Bbb R_{\ge 0}(an^e_1+mn^e_2)$$ For $(i, j)\in I_o$ (resp. $\in I_e$), we denote by $E^i_j$ the exceptional divisor of $p_o$ (resp. $p_e$) which associated to $\hat{\lambda}^i_j n^o_1+\lambda^i_j n^o_2\in N^o$ (resp. $\hat{\mu}^i_j n^e_1+\mu^i_j n^e_2\in N^e$). Note that by Lemma \ref{lemcurve}, for $\iota\in I_o$ (resp.$I_e$), the proper transform of $C_{\iota}=(c_1z_3^{\lambda_{\iota}}+c_2z_1^{\mu_{\iota}})/ \langle\alpha\rangle\in X_o$ (resp. $C_{\iota}=(c_1z_2^{\lambda_{\iota}}+c_2z_3^{\mu_{\iota}})/ \langle\alpha\rangle\in X_e$) intersects the exceptional locus transversly at $E_{\iota}$. We define the order in $I$ as the same way as the normal case. \par In the rest of this paper, we treat $X_{a,d,m}$ and $X_{a,m}$ simultaneously, otherwise we specifically state the normal or non-normal case. \section{$\lambda$-expansion and $\mu$-expantion} In this section we introduce the notion of $\lambda$-expansion and $\mu$-expansion, which is the key in this paper. \begin{dfn}\label{dfnlambda} Let $L=(j_1, l_2, j_3, l_4, \dots, j_{k-1}, l_k)$ be a sequence of non-negative integers which is not $(0, 0, \dots, 0)$. We call $L$ a $\lambda$-sequence if it satisfies the following conditions. \begin{enumerate} \item $l_i\le q_i+1$ if $i\not= k$, and $l_k\le q_k$; $j_i\le q_i$ for all odd $i$, and $j_i\not= 1$ if $i\not= 1$ \item If $l_{i_0}=q_{i_0}+1$, then there exists odd $i_1$ and $i_2$ which satisfies the following conditions. \begin{enumerate} \item $i_1<i_0<i_2\le k-1$ \item $l_{i'}=q_{i'}$ for all even $i'$ such that $i_1<i'<i_2$ and $i'\not= i_0$; $j_{i'}=0$ for all odd $i'$ such that $i_1\le i'\le i_2$ \item $l_{i_1-1}<q_{i_1-1}$ if $i_1\ge 3$ and $l_{i_2+1}<q_{i_2}$ \end{enumerate} \item If $l_{i_0}=q_{i_0}$ and $j_{i_0+1}\ge 2$, then there exists odd $i_3$ which satisfies the following conditions. \begin{enumerate} \item $i_3<i_0$ \item $l_{i'}=q_{i'}$ for all even $i'$ such that $i_3<i'\le i_0$; $j_{i'}=0$ for all odd $i'$ such that $i_3\le i'<i_0$ \item $l_{i_3-1}<q_{i_3-1}$ if $i_3\ge 3$ \end{enumerate} \end{enumerate} \end{dfn} \begin{dfn} Let $M=(l_1, j_2, l_3, j_4, \dots, l_{k-1}, j_k)$ be a sequence of non-negative integers which is not $(0, 0, \dots, 0)$. We call $M$ a $\mu$-sequence if it satisfies the following conditions. \begin{enumerate} \item $l_i\le q_i+1$ for all odd $i$; $2\le j_i\le q_i$ for all even $i$ \item If $l_{i_0}=q_{i_0}+1$, then there exists even $i_1$ and $i_2$ which satisfies the following conditions. \begin{enumerate} \item $0\le i_1<i_0<i_2\le k$ \item $l_{i'}=q_{i'}$ for all odd $i'$ such that $i_1<i'<i_2$ and $i'\not= i_0$; $j_{i'}=0$ for all even $i'$ such that $i_1\le i'\le i_2$ \item $l_{i_1-1}<q_{i_1}$ if $i_1\ge 2$, $l_{i_2+1}<q_{i_2+1}$ if $i_2\le k-2$ \end{enumerate} \item If $l_{i_0}=q_{i_0}$ and $j_{i_0+1}\ge 2$, then there exists even $i_3$ which satisfies the following conditions. \begin{enumerate} \item $0\le i_3<i_0$ \item $l_{i'}=q_{i'}$ for all odd $i'$ such that $i_3<i'<i_0$; $j_{i'}=0$ for all even $i'$ such that $i_3\le i'<i_0$ \item $l_{i_3-1}<q_{i_3-1}$ if $i_3\ge 2$ \end{enumerate} \end{enumerate} \end{dfn} We denote by $\cal S_{\lambda}$ (resp. $\cal S_{\mu}$) the set of all $\lambda$- (resp. $\mu$-) sequences.\\ Let $L$ be a $\lambda$-sequence and $h$ an integer such that $1\le h\le k-1$. We say that {\it the condition $*(h)$ holds for $L$} if the following condions hold.\\ \begin{enumerate} \item If $h$ is odd, $$\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) <P_h+Q_h$$ \item If $h$ is even, $$\lambda^1_{j_i}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_h\lambda^{h+1}_1 <P_{h-1}+Q_{h-1}+P_h+Q_h$$ If $l_h<q_h$ or $j_{h+1}\ge 2$ also hold, $$\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_h\lambda^{h+1}_1 <P_h+Q_h$$ \end{enumerate} \begin{lem} $*(h)$ holds for all $L\in \cal S _{\lambda}$ and for all $h=1, 2, \dots, k-1$. \end{lem} \begin{pf} We use the induction on $h$. It is clear that $*(1)$ holds. Assume that $2\le h\le k-1$ and that $*(\tilde{h})$ holds for all $\tilde{h}$ such that $\tilde{h} <h$.\\ First we treat that the case where $h$ is odd.\\ If $j_h\ge 2$, then by $*(h)$, $$\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-2 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) l_{h-1}\lambda^h_1 <P_{h-1}+Q_{h-1}$$ Hence \begin{align*} \lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) &<P_{h-1}+Q_{h-1}+\lambda^h_{j_h}\\ &=P_{h-1}+Q_{h-1}+j_h(P_{h-1}+Q_{h-1})\\ &\le P_{h-2}+Q_{h-2}+q_h(P_{h-1}+Q_{h-1})\\ &=P_h+Q_h \end{align*} If $j_h=0$, by $*(h-1)$ \begin{align*} \lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) &=\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-2 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})+l_{h-1}\lambda^h_1\\ &<P_{h-2}+Q_{h-2}+P_{h-1}+Q_{h-1}\\ &\le P_h+Q_h \end{align*} Next we treat the case where $h$ is even. We divide the proof into four cases as follows\\ (1) $l_h<q_h$ (2) $l_h=q_h, j_h=0$ (3) $l_h=q_h, j_h\ge 2$ (4) $l_h=q_h+1$\\ (1) By $*(h-1)$, $$\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) <P_{h-1}+Q_{h-1}$$ Hence \begin{align*} \lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_k\lambda^{h+1}_1 &<P_{h-1}+Q_{h-1}+(q_h-1)(P_{h-1}+Q_{h-1})\\ &<P_h+Q_h \end{align*} (2) By $*(h-1)$, \begin{align*} \lambda^1_{j_1}+\sum \begin{Sb} 3\le i\ h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_h\lambda^{h+1}_1 &<P_{h-1}+Q_{h-1}+q_h(P_{h-1}+Q_{h-1})\\ &<P_{h-1}+Q_{h-1}+P_h+Q_h \end{align*} (3) By (iii) in Definition \ref{dfnlambda}, there exists odd $h'$ such that $l_{h'-1}<q_{h'-1}$ and $$(j_{h'}, l_{h'+1}, j_{h'+2}, l_{h'+3}, \dots, j_{h-1}, l_h) =(0, q_{h'+1}, 0, q_{h'+3}, \dots, 0, q_h)$$ Hence by $*(h'-1)$, \begin{align*} &\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) l_h\lambda^{h+1}_1\\ =&\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h'-2 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_{h'-1}\lambda^{h'}_1\\ &+\lambda^{h'}_{j_{h'}}+\sum \begin{Sb} h'+2\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_{h}\lambda^{h+1}_1\\ <&P_{h'-1}+Q_{h'-1}+\sum \begin{Sb} h'+2\le i\le h+1 \\ i\ \text{odd} \end{Sb} q_{i-1}(P_{i-2}+Q_{i-2})\\ =&P_h+Q_h \end{align*} (4) By (ii) in Definition \ref{dfnlambda}, there exists odd $h'$ such that $l_{h'-1}<q_{h'-1}$ and $$(j_{h'}, l_{h'+1}, j_{h'+2}, l_{h'+3}, j_{h'+4}, \dots, l_{h-2}, j_{h-1}, l_h)$$ $$=(0, q_{h'+1}, 0, q_{h'+3}, 0, \dots, q_{h-2}, 0, q_h)$$ Hence by $*(h'-1)$, \begin{align*} &\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_h\lambda^{h+1}_1\\ =&\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le h'-2 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_{h'-1}\lambda^{h'}_1\\ &+\lambda^{h'}_{j_{h'}}+\sum \begin{Sb} h'+2\le i\le h-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_h\lambda^{h+1}_1 \\ <&P_{h'-1}+Q_{h'-1}+\sum \begin{Sb} h'+2\le i\le h-1 \\ i\ \text{odd} \end{Sb} q_{i-2}(P_{i-2}+Q_{i-2}) (q_h+1)(P_{h-1}+Q_{h-1})\\ =&P_{h-1}+Q_{h-1}+P_h+Q_h \end{align*} \end{pf} We define the order in $\cal S_{\lambda}$ as $(j_1, l_2, \dots, l_k)<(j'_1, l'_2, \dots, l'_k)$ if and only if there exists $i$ such that $j_i<j'_i$ or $l_i<l'_i$ and that $j_h=j'_h$, $l_h=l'_h$ for all $h>i$. Let $v$ be a map from $\cal S_{\lambda}$ to $\Bbb Z$ defined by $$v(l_1, j_2, \dots, l_{k-1}, j_k) =\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le k-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_k\lambda^{k+1}_1$$ \begin{prop}\label{proplambda} The map $v$ is an order isomorphism from $\cal S_{\lambda}$ to $\{ n\in \Bbb Z |1\le n\le m-1\}$ \end{prop} \begin{pf} First we show that $L<L'$ implies $v(L)<v(L')$. Let $L=(j_1, l_2, \dots, j_{k-1}, l_k)$ and $L'=(j'_1, l'_2, \dots, j'_{k-1}, l'_k)$ be $\lambda$-sequences such that $L<L'$. Put $$i_0=\max\{ i| l_{\tilde{i}}=l'_{\tilde{i}}\ and\ j_{\tilde{i}}=j'_{\tilde{i}} \ \text{for all}\ \tilde{i} >i\}$$ If $i_0$ is odd, $$v(L')-v(L) \ge \lambda^{i_0}_{j'_{i_0}}-\lambda^{i_0}_{j_{i_0}} -\{\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le i_0-2 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i}) +l_{i_0-1}\lambda^{i_0}_1\}$$ Note that \begin{equation*} \lambda^{i_0}_{j'_{i_0}}-\lambda^{i_0}_{j_{i_0}}\ge \begin{cases} P_{i_0-1}+Q_{i_0-1} &(\text{if}\ j_{i_0}\ge 2)\\ P_{i_0-2}+Q_{i_0-2}+P_{i_0-1}+Q_{i_0-1} &(\text{if}\ j_{i_0}=0) \end{cases} \end{equation*} Hence by $*(i_0-1)$, $v(L')-v(L)>0$\\ If $i_0$ is even, \begin{align*} v(L')-v(L) &\ge (l'_{i_0}-l_{i_0})\lambda^{i_0+1}_1 -\{ \lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le i_0-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})\}\\ &\ge P_{i_0-1}+Q_{i_0-1}-\{\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le i_0-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})\} \end{align*} Hence by $*(i_0-1)$, $v(L')-v(L)>0$. Thus we have done.\\ Note that $\max\cal S_{\lambda} =(0, q_2, 0, q_4, 0, \dots, q_{k-2}, 0, q_k)$. Thus $v$ is an order-preserving injection into $\{ n\in\Bbb Z |1\le n\le m-1\}$. Hence the rest we must prove is that it is a injection into $\{ n\in\Bbb Z |1\le n\le m-1\}$. Note that $1=\lambda^1_1<\lambda^1_2<\dots <\lambda^1_{q_1} <\lambda^3_1<\lambda^3_2<\dots$. Thus it is sufficient to show that $\text{Im}(v)\supseteq \{n|1\le n<\lambda^h_1\}$ implies $\text{Im}(v)\supseteq \{n|1\le n<\lambda^{h+2}_1\}$. Let $n$ be an integer such that $\lambda^h_1\le n<\lambda^{h+2}_1$. We divide the proof into two cases.\\ (I) $\lambda^h_1\le n<\lambda^h_2$ or $q_h=1$ (II) $\lambda^h_2\le n<\lambda^{h+2}_1$\\ (I) Write $n=l_{h-1}\lambda^h_1+n'$ such that $0\le n'<\lambda^h_1$. It is clear that $l_{h-1}\le q_{h-1}+1$. By the induction hypothesis, there exists $L'=(j_1, l_2, \dots, l_{h-3}, j_{h-2}, 0, \dots, 0)$ such that $v(L')=n'$. Put $L=(j_1, l_2, \dots, l_{h-3}, j_{h-2}, l_{h-1}, 0, \dots, 0)$. Assume that $L$ is not a $\lambda$-sequence. Then by the definition of $\lambda$-sequence, we get $l_{h-1}=q_{h-1}+1$ and there exists $h'$ such that $$(l_{h'}, j_{h'+1}, l_{h'+2}, j_{h'+3}, l_{h'+4}, j_{h'+5}, \dots, l_{h-3}, j_{h-2}, l_{h-1})$$ $$=(q_{h'}+1, 0, q_{h'+2}, 0, q_{h'+4}, 0, \dots, q_{h-3}, 0, q_{h-1}+1)$$ or $$(j_{h'}, l_{h'+1}, j_{h'+2}, l_{h'+3}, j_{h'+4}, \dots, l_{h-3}, j_{h-2}, l_{h-1})$$ $$=(j_{h'}, q_{h'+1}, 0, q_{h'+3}, 0, \dots, q_{h-3}, 0, q_{h-1}+1)$$ and $j_{h'}>0$. In the both cases it is easily checked \begin{equation*} n\ge \begin{cases} \lambda^h_2 &(\text{if}\ q_h\ge 2)\\ \lambda^{h+2}_1 &(\text{if}\ q_h=1) \end{cases} \end{equation*} and this is a contradiction. Thus $L$ is a $\lambda$-sequence, so we have done.\\ (II) Put $j_h=\max\{ j|j\ge 2, \lambda^h_j\le n\}$. Clearly $j_h\le q_h$. Since $n-\lambda^h_{j_h}<q_h$, there exists $L'=(j_1, l_2, \dots, j_{h-2}, l_{h-1}, 0, \dots, 0)$ such that $v(L')=n-\lambda^h_{j_h}$ by the induction hypothesis and (I). Put $L=(j_1, l_2, \dots, j_{h-2}, l_{h-1}, j_h, 0, \dots, 0)$ We can check $L\in\cal S_{\lambda}$ by the definition of $\lambda$-sequence. \end{pf} When we write $n=\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le k-1 \\ i\ \text{odd} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})+l_k\lambda^{k+1}_1$ where $(j_1, l_2, \dots, l_{k-1}, j_k)$ is a $\lambda$-sequence, we call this expression a {\it $\lambda$-expansion of $n$}. By the above proposition, $n=1, 2, \dots, m-1$ has unique $\lambda$- expansion. Note that the proof of Proposition \ref{proplambda} shows how to calculate $\lambda$-expansion of actual number. \par When we write $n=\sum \begin{Sb} 2\le i\le k\\i\ \text{even} \end{Sb} (l_{i-1}\mu^i_1+\mu^i_{j_i})$ where $(l_1, j_2, \dots, l_{k-1}, j_k)$, we call this expression a {\it $\mu$-expansion of $n$}. Similarly to $\lambda$-expansion, we can prove that $n=1, 2, \dots, m-1$ has unique $\mu$-expansion. \section{General $n$-canonical divisors} \label{secdiv} Let $(Y,y)$ be a 2-dimensional rational singularity, and $p:\mbox{$\tilde{Y}$}\rightarrow Y$ be the minimal desingularization. Let $M$ be a reflexive module of rank 1 on $Y$, $F(M)$ the full sheaf associated to $M$. (For the definition of full sheaf, see \cite{esn}. ) In this situation, \begin{dfn} Let $D$ be a member of $|M|$. We call $D$ a general member of $|M|$ if \mbox{$\tilde{D}$}\ is a member of $|F(M)|$ and intersects the exeptional locus transversely. \end{dfn} Note that general members always exist since the full sheaf is generated by global sections. Let $E=\cup_i E_{i}$ be the exeptional locus, and write $p^{*}D=\mbox{$\tilde{D}$}+\sum_i\alpha(D)_{i}E_{i}$. \begin{lem}\label{lemfull} Let $D$ be a member of $|M|$ such that \mbox{$\tilde{D}$}\ and $E$ intersects transeversely. Then $D$ is a general member if and only if the inequality $\alpha(D)_{i}\leq\alpha(D^{\prime})_{i}$ holds for all $D^{\prime}\in|M|$ and all $E_{i}$. \end{lem} \begin{pf} Suppose that $D$ is a general member of $|M|$ and $D'$ is a member of $|M|$. The sequence $$H_{E}^{0}(\mbox{$\tilde{Y}$},\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))\rightarrow H^{0}(\mbox{$\tilde{Y}$},\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))\rightarrow H^{0}(\mbox{$\tilde{Y}$}\setminus E,\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))\rightarrow H_{E}^{1}(\mbox{$\tilde{Y}$},\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))$$ is exact. Since $\cal O_{\mbox{$\tilde{X}$}}(\mbox{$\tilde{D}$})$ is a full sheaf, $H_{E}^{0}(\mbox{$\tilde{Y}$},\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))=H_E^1(\mbox{$\tilde{Y}$},\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$}))=0$. Hence $H^0(\tilde{Y},\cal O_{\tilde{Y}}(\tilde{D}))\simeq H^0(Y, \cal O_Y(D))$. Thus $D$ is linearly equivalent to $D'+(\text{effective divisor})$. Hence the inequality holds since $\mbox{$\tilde{D}$}$ and $\mbox{$\tilde{D}$}^{\prime}+\sum\{\alpha(D^{\prime})_i-\alpha(D)_i\}E_i$ are linearly equivalent. Thus we have proved only if part. Next suppose that $D$ is a member of $|M|$ such that \mbox{$\tilde{D}$}\ and $E$ intersect transeversely and $\alpha(D)_i\leq\alpha(D^{\prime})_i$ for all $D^{\prime}\in|M|$. Let $D_0$ be a general member of $|M|$. By the assumption of $D$, $\alpha(D)_i\leq\alpha(D_0)_i$. By the fact that we have already showed, $\alpha(D)_i\geq\alpha(D_0)_i$. Hence $\alpha(D)_i=\alpha(D_0)_i$. Thus $D$ and $D_0$ are numerically equivalent, hence they are lenearly equivalent since $Y$ is a rational singularity. Hence $\cal O_{\mbox{$\tilde{Y}$}}(\mbox{$\tilde{D}$})$ is a full sheaf. \end{pf} \begin{cor}\label{corgenint} Let $M$ and $M^{\prime}$ be reflexive modules of rank 1 on $Y$. Let $D_0$(resp.$D_0^{\prime}$) be a general member of $|M|$(resp.$|M^{\prime}|$). Then $$D_0\cdot D_0^{\prime}=\min\{D\cdot D^{\prime}|D\in|M|,D^{\prime}\in|M^{\prime}|\}$$ \end{cor} \begin{pf} Let $D$(resp.$D^{\prime}$) be a member of $|M|$(resp.$|M^{\prime}|$). \begin{align*} D\cdot D^{\prime} & =(p^*D)\cdot\mbox{$\tilde{D}$}^{\prime} \\ & =(\mbox{$\tilde{D}$}+\sum\alpha(D)_iE_i)\cdot\mbox{$\tilde{D}$}^{\prime} \\ & \geq\sum\alpha(D)_i(E_i\cdot\mbox{$\tilde{D}$}^{\prime}) \\ & \geq\sum\alpha(D_0)_i(E_i\cdot\mbox{$\tilde{D}$}^{\prime}) \\ & =(\mbox{$\tilde{D}$}_0+\sum\alpha(D_0)_iE_i)\cdot\mbox{$\tilde{D}$}^{\prime} \\ & =p^*D_0\cdot\mbox{$\tilde{D}$}_0=D_0\cdot D^{\prime} \end{align*} Similarly we can show $D_0\cdot D'\ge D_0\cdot D_0'$. \end{pf} In the rest of this section we shall calculate the general element of the $n$-canonical system of semi-log-terminal singularities. We denote by $\cal L$ (resp. $\cal L_o$ resp. $\cal L_e$) the set of all functions from $I$ (resp. $I_o$ resp. $I_e$) to $\Bbb Z$. \par First we treat singularities of class T. We begin by purely arithmetical lemmas. Let $a$, $d$, $m$ and $n$ be positive integers such that $\dfrac{m}{2}<a<m$, $(a, m)=1$ and $n<m$. Let $\dfrac{a}{m-a} =[q_1, q_2, \dots, q_k]$ be the expantion into continued fraction, $r_i$ be the $i$-th remainder of the Euclidean algorithm, and $\dfrac{P_i}{Q_i}$ be the $i$-th convergent. \begin{lem}\label{lemrineq} Let $\{ t_i\}_{i=1, 2, \dots, k}$ be a sequence of non-negative integers such that $t_i\le q_i$ for $i\le k-1$ and $t_k\le q_k-1$. Assume that $t_{i_0}$ be positive. Then \begin{enumerate} \item If $i_0$ is even, $$-r_{i_0-1}<\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i<0$$ \item If $i_0$ is odd, $$0<\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i<r_{i_0-1}$$ \end{enumerate} \end{lem} \begin{pf} We use the induction on $k-i_0$. If $i_0=k$, it is clear that the inequalities hold. Assume that $i_0<k$ and that the inequalities hold for all $i'_0$ such that $i_0<i'_0\le k$. Let $i_0$ be even. First we show $\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i<0$. If $t_{i'}=0$ for all $i'$ such that $i'>i_0$ and $i'$ is odd, $$\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i\le -t_{i_0}r_{i_0}<0$$ Thus we assume that there exists $i'$ such that $i'>i_0$, $i'$ is odd, and $t_{i'}>0$. Let $i'_0$ be the minimum of such $i'$. By the induction hypothesis, $$\sum_{i'_0\le i\le k}(-1)^{i-1}t_ir_i<r_{i'_0}-1$$ Thus \begin{align*} \sum_{i'_0\le i\le k}(-1)^{i-1}t_ir_i &\le -t_{i_0}r_{i_0}+\sum_{i'_0\le i\le k}(-1)^{i-1}t_ir_i\\ &<-t_{i_0}r_{i_0}+r_{i'_0-1}\\ &=0 \end{align*} Next we show $\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i>-r_{i_0-1}$. If $t_{i'}=0$ for all $i'$ such that $i'>i_0$ and $i'$ is even, $$\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i\ge -t_{i_0}r_{i_0} \ge -q_{i_0}r_{i_0}>-r_{i_0-1}$$ Thus we assume that there exists $i'$ such that $i'>i_0$, $i'$ is even, and $t_{i'}>0$. By the induction hypothesis, $$\sum_{i'_0\le i\le k}(-1)^{i-1}t_ir_i>-r_{i'_0-1}$$ Thus \begin{align*} \sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i &\ge -t_{i_0}+\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i\\ &>-t_{i_0}-r_{i'_0-1}\\ &\ge -q_{i_0}r_{i_0}-r_{i_0+1}\\ &=-r_{i_0-1} \end{align*} The proof is similar for odd $i_0$. \end{pf} \begin{dfn} Let $(t_1, t_2, \dots, t_k)$ be a sequence of non-negative integers which is not $(0, 0, \dots, 0)$. We call this a $\tau$-sequence if it satisfies the following conditions. \begin{enumerate} \item $t_i\le q_i$ if $i\not= k$ and $t_k<q_k$. \item If $t_{i_0-1}>0$ and $t_{i_0}=q_{i_0}$ for some $i_0$ such that $1<i_0<k$, then $t_{i_0+1}=q_{i_0+1}$. If $t_{k-2}>0$ and $t_{k-1}=q_{k-1}$, then $t_k=q_k-1$. \end{enumerate} \end{dfn} \begin{lem}\label{lemtau} Let $(0, 0, \dots, 0, t_{i_0}, t_{i_0+1}, \dots, t_k)$ be a $\tau$-sequence such that $i_0$ is odd and $t_{i_0}>0$. Then \begin{enumerate} \item If $k$ is even, $$\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i=1\Leftrightarrow i_0=k-1, t_{k-1}=1, t_k=q_k-1$$ \item If $k$ is odd, $$\sum_{i_0\le i\le k}(-1)^{i-1}t_ir_i=1\Leftrightarrow i_0=k, t_k=1$$ \end{enumerate} \end{lem} \begin{pf} It can be easily checked and we left it for the reader. \end{pf} \begin{lem} For any integer $t$ such that $0<t<m-2$, there exists a $\tau$-sequence $(t_1, t_2, \dots, t_k)$ such that $t=\sum_{1\le i\le k}t_i(P_{i-1}+Q_{i-1})$.\\ (We call this expression of $t$ the $\tau$-expansion of $t$.) \end{lem} \begin{pf} (Step 1) We can write $t=\sum_{1\le i\le k}t_i(P_{i-1}+Q_{i-1})$ where $0\le t_i\le q_i$ for $i<k$ and $0\le t_k<q_k$.\\ (proof) We use the induction on $t$. By the induction hypothesis, we can write $t-1=\sum_{1\le i\le k}t_i(P_{i-1}+Q_{i-1})$, where $0\le t_i\le q_i$ for $i<k$ and $0\le t_k<q_k$. If $t_1<q_1$, we have done. Assume $t_1=q_1$. Note that $\sum_{1\le i\le k-1}q_i(P_{i-1}+Q_{i-1})+(q_k-1)(P_{k-1}+Q_{k-1})=m-2$. Hence there exists $i_0$ such that $i_0<k$, $t_i=q_i$ for all $i\le i_0$, and $t_{i_0+1}<q_{i_0+1}$ (if $i_0<k-1$); $t_k\le q_k-2$ (if $i_0=k-1$). Thus \begin{equation*} t= \begin{cases} \sum \begin{Sb} 1\le i\le i_0 \\ i\ \text{even} \end{Sb} q_i(P_{i-1}+Q_{i-1})+(t_{i_0+1}+1)(P_{i_0}+Q_{i_0}) +&\sum_{i_0+2\le i\le k}t_i(P_{i-1}+Q_{i-1}) \\ &(\text{if}\ i_0\ \text{is odd})\\ \sum \begin{Sb} 1\le i\le i_0 \\ i\ \text{odd} \end{Sb} q_i(P_{i-1}+Q_{i-1})+(t_{i_0+1}+1)(P_{i_0}+Q_{i_0}) +&\sum_{i_0+2\le i\le k}t_i(P_{i-1}+Q_{i-1}) \\ &(\text{if}\ i_0\ \text{is even}) \end{cases} \end{equation*} (Step 2) We write $t=\sum_{1\le i\le k}t^{(1)}_i(P_{i-1}+Q_{i-1})$ as in (Step 1). If there exists $i_0$ such that $t_{i_0-1}>0$, $t_{i_0}=q_{i_0}$, and $t_{i_0+1}<q_{i_0+1}$ (if $i_0<k-1$); $t_{i_0}\le q_{i_0+1}-2$ (if $i_0=k-1$), we transform $(t^{(1)}_1, \dots, t^{(1)}_k)$ to $(t^{(2)}_1, \dots, t^{(2)}_k) =(t_1, \dots, t_{i_0-2}, t_{i_0-1}-1, 0, t_{i_0+1}+1, t_{i_0+2}, \dots, t_k)$. We repeat this operation. Since $\sum_{1\le i\le k}t^{(j)}_i$ strictly decrease by this operation, we can get a $\tau$-sequence after finitely many operations. \end{pf} Remark. We can prove that the $\tau$-expansion is unique. Put $\cal T$, $v$ and $\cal T_{\min}$ as follows \begin{align*} \cal T &=\{ (s, t)\in \Bbb Z _{\ge 0} \times \Bbb Z _{\ge 0}| s+\{ dm(m-a)-1\} t\equiv dm(m-a)n\ (mod\ dm^2) \}\\ v&=\min\{s+t|(s, t)\in \cal T \}\\ \cal T _{\min}&=\{ (s, t)\in \cal T|s+t=v\} \end{align*} \begin{prop}\label{propsmin} \begin{enumerate} \item If $d=1$ and $k$ is even, then \begin{equation*} \begin{cases} \cal T _{\min}=\{ (n, n)\} &(n<m-(P_{k-1}+Q_{k-1}))\\ \cal T _{\min}=\{ (n+P_{k-1}+Q_{k-1}, n-m+P_{k-1}+Q_{k-1})\} \ &(n\ge m-(P_{k-1}+Q_{k-1})) \end{cases} \end{equation*} \item If $d=1$ and $k$ is odd, then \begin{equation*} \begin{cases} \cal T _{\min}=\{ (n, n)\} &(n<m-(P_{k-1}+Q_{k-1}))\\ \cal T _{\min}=\{ (n-m+P_{k-1}+Q_{k-1}, n+P_{k-1}+Q_{k-1})\} \ &(n\ge m-(P_{k-1}+Q_{k-1})) \end{cases} \end{equation*} \item If $d\ge 2$, then $\cal T_{\min} =\{ (n, n)\}$ \end{enumerate} \end{prop} \begin{pf} For a positive integer $t$, put $$s_t=\min\{ s|(s, t)\in \cal T \}$$ Since $(n, n)\in \cal T$, we get $v\le 2n$, thus $$v=\min\{ s_t+t|0\le t\le 2n\}$$ $$\cal T _{\min}=\{(s_t, t)|0\le t\le 2n, s_t+t=v\}$$ Hence we estimate $s_t$ for $t$ such that $0\le t\le 2n$. It is clear that $s_n=n$.\\ (Claim I) Assume $t<n$. Then \begin{enumerate} \item If $k$ is even, $d=1$, $n\ge m-(P_{k-1}+Q_{k-1})$ and $t=n-m+(P_{k-1}+Q_{k-1})$, then $s_t=n+P_{k-1}+Q_{k-1}$ \item Otherwise $s_t+t>2n$ \end{enumerate} (Proof of Claim I) It is clear that $s_0=dma>2n$, thus we assume $t>0$. Put $t'=n-t$. Let $t'=\sum_{1\le i\le k}t_i(P_{i-1}+Q_{i-1})$ be the $\tau$-expansion of $t'$. Put $i_0=\min\{i|t_i>0\}$ First let $i_0$ be even. We must prove $s_t+t>0$ in this case since $i_0=1$ when $k$ is even and $t=n-m+P_{k-1}+Q_{k-1}$. Let $(s, t)$ be an element in $\cal L$. We can write $s=n+\{dm(m-a)-1\}t'-s'$ in which $s'$ is an integer. Note that $(m-a)t'-m\sum_{1\le i\le k}t_iQ_{i-1} =\sum_{1\le i\le k}(-1)^{i-1}t_ir_i$. Assume $s'\ge \sum_{1\le i\le k}t_iQ_i$. Then by Lemma \ref{lemrineq}, \begin{align*} s&\le n-t'+dm\{ (m-a)t'-m\sum_{1\le i\le k}t_iQ_{i-1}\}\\ &=n-t'+dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i\\ &\le n-t'-dm\\ &<0 \end{align*} This is a contradiction. Thus $s'\le \sum_{1\le i\le k}t_iQ_{i-1}-1$.\par Next put $s'=\sum_{1\le i\le k}t_iQ_{i-1}-1$. Then by Lemma \ref{lemrineq}, \begin{align*} s&=n-t'+dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i+dm^2\\ &>n-t'+dma\\ &>2n \end{align*} Hence $s'_t=\sum_{1\le i\le k}t_iQ_{i-1}+1$ and $s_t>2n$. Thus we have done for even $i_0$.\\ Secondly let $i_0$ be odd. Assume that $s'\ge \sum_{1\le i\le k}t_iQ_{i-1}+1$. Then by Lemma \ref{lemrineq}, \begin{align*} s&\le n-t'+dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i-dm^2\\ &<n-t'+dma-dm^2\\ &<0 \end{align*} This is a contradiction, thus $s'\le \sum_{1\le i\le k}t_iQ_{i-1}+1$. Next assume $s'=\sum_{1\le i\le k}t_iQ_{i-1}$. Then by Lemma \ref{lemrineq}, $$s=n-t'+dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i\ge n-t'+dm>0$$ Hence $$s_t=n-t'+dm\sum_{1\le i\le k}t_ir_i$$ and $$s_t+t=2t+dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i$$ If $d\sum_{1\le i\le k}(-1)^{i-1}t_ir_i\ge 2$, then $s_t+t=2m>2n$. If $d\sum_{1\le i\le k}(-1)^{i-1}t_ir_i=1$, by Lemma \ref{lemtau},\\ (a) $d=1$, $k$ is even, and $t'=m-(P_{k-1}+Q_{k-1})$.\\ or (b) $d=1$, $k$ is odd, and $t'=P_{k-1}+Q_{k-1}$.\\ In the case (b), $$s_t+t=2n-2(P_{k-1}+Q_{k-1})+m>2n$$ In the case (a), $$t=n-m+P_{k-1}+Q_{k-1},\ s_t=n+P_{k-1}+Q_{k-1}$$ Thus we have done. (Claim II) Assume $t>n$. Then \begin{enumerate} \item If $k$ is odd, $d=1$, $n\ge m-(P_{k-1}+Q_{k-1})$, and $t=n+P_{k-1}+Q_{k-1}$, then $s_t=n-m+P_{k-1}+Q_{k-1}$ \item Otherwise $s_t+t>2n$ \end{enumerate} (Proof of Claim II) Put $t'=t-n$. If $t'=m-1$, then $n=m-1$ and $t=2m-2$. It is easy to see $$s_{2m-2}=dm(m-a)+m-1>2n$$ Thus we assume $t'\le m-2$. Let $t'=\sum_{1\le i\le k}t_i(P_{i-1}+Q_{i-1})$ be the $\tau$-expansion of $t'$, and put $i_0=\min\{i|t_i>0\}$. Write $s=n-\{ dm(m-a)-1\}t'+dm^2s'$ for $(s, t)\in \cal T$. For the case such that $i_0$ is even, we can get $$s_t=n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i>2n$$ by the similar way using Lemma \ref{lemrineq}. If $t=P_{k-1}+Q_{k-1}$, then $i_0=k$, thus we have done for this case. Assume that $i_0$ is odd, and let $(s, t)\in \cal T$. If $s'\le \sum_{1\le i\le k}t_iQ_{i-1}-1$, we can get $s<0$ by Lemma \ref{lemrineq}. Put $s'=\sum_{1\le i\le k}t_iQ_{i-1}$. Then we get $s=n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i$ If $d\sum_{1\le i\le k}(-1)^{i-1}t_ir_i\ge 2$, then $s_t=n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i+dm^2$, hence $s_t+t>2n$.\\ If $d\sum_{1\le i\le k}(-1)^{i-1}t_ir_i=1$, by Lemma \ref{lemrineq}\\ (a) $d=1$, $k$ is even, $t'=m-(P_{k-1}+Q_{k-1})$\\ or (b) $d=1$, $k$ is odd, $t'=P_{k-1}+Q_{k-1}$\\ In the case (a), $n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i =n-(P_{k-1}+Q_{k-1})$\\ If $n<P_{k-1}+Q_{k-1}$, then $s'_t\ge \sum_{1\le i\le k}t_iQ_{i-1}+1$, thus $s_t+t>2n$.\\ If $n\ge P_{k-1}+Q_{k-1}$, then $s_t=n-(P_{k-1}+Q_{k-1})$,\\ thus $s_t+t=2n+m-2(P_{k-1}+Q_{k-1})>2n$.\\ In the case (b), $$n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i =n+P_{k-1}+Q_{k-1}-m$$ If $n<m-(P_{k-1}+Q_{k-1})$, then $s_t=n+t'-dm\sum_{1\le i\le k}(-1)^{i-1}t_ir_i+dm^2$,\\ thus $s_t+t>2n$.\\ If $n\ge m-(P_{k-1}+Q_{k-1})$, then $s_t=n-m+P_{k-1}+Q_{k-1}$.\\ Thus we have done. Summarizing (I) and (II), we have proved the proposition. \end{pf} \par We define $\delta^{\iota}\in \cal L$ by $\delta^{\iota}_{\eta}=0$ for $\eta\not= \iota$ and $\delta^{\iota}_{\iota}=1$. And for $\nu\in\cal L$, we define $\alpha(\nu)\in\cal L\otimes\Bbb Q$ by $\alpha(\nu)_{\eta}=\sum_{\iota\in I}\alpha^{\iota}_{\eta}\nu_{\iota}$. \begin{dfn} Let $n$ be an integer such that $1\le n\le m-1$. Let \begin{equation*} n=\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le k-1 \\ i\ \text{\em{odd}} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})+l_k\lambda^{k+1}_1 =\sum \begin{Sb} 2\le i\le k\\i\ \text{\em{even}} \end{Sb} (l_{i-1}\mu^i_1+\mu^i_{j_i}) \end{equation*} be the $\lambda$- and $\mu$- expansion of $n$. We define $\nu(n)^o\in I_o$, $\nu(n)^e\in I_e$, $\nu(n)\in I$ as follows: \begin{align*} \nu(n)^o=&\delta^{1, j_1}+\sum \begin{Sb} 3\le i\le k-1\\i\ \text{\em{odd}} \end{Sb} (l_{i-1}\delta^{i, 1}+\delta^{i, j_i})+l_k\delta^{k+1, 1}\\ \nu(n)^e=&\sum \begin{Sb} 2\le i\le k\\i\ \text{\em{even}} \end{Sb} (l_{i-1}\delta^{i, 1}+\delta^{i, j_i})\\ \nu(n)=&\nu(n)^o+\nu(n)^e \end{align*} \end{dfn} \begin{thm}\label{thmdegfull1} Let $n$ be an integer such that $1\le n\le m-1$. Then $$\deg_{E_{\iota}}F(-nK_X)=\nu (n)_{\iota}\quad for\ all\ \iota\in I$$ \end{thm} \begin{pf} Let $\nu'(n)$ be an element of $\cal L$ such that $\nu'(n)_{\iota}=\deg_{E_{\iota}}F(-K_X)$. Put $d'=\lceil \dfrac{d}{2} \rceil$. Put $I'_o$ and $I'_e$ as follows \begin{align*} &I'_o=\{ (i, j)\in I|i \text{ is odd and } j\le d'\text{ if }i=k+1\}\\ &I'_e=I\smallsetminus I'_o \end{align*} For $\nu\in \cal L$, define $s(\nu)$ and $t(\nu)$ as follows $$s(\nu)=\sum_{\iota\in I'_o}\nu_{\iota}\lambda_{\iota},\quad t(\nu)=\sum_{\iota\in I'_e}\nu_{\iota}\mu_{\iota}$$ Let $\cal L (n)$ be the set of elements of $\cal L$ satisfying the following conditions\\ (i) $\nu_{\iota}\ge 0$ for all $\iota\in I$\\ (ii) $s(\nu )+\{ dm(m-a)-1\} t(\nu) \equiv dm(m-a)n \pmod{dm^2}$\\ By Lemma \ref{lemfull}, it is clear that $\nu'(n)$ is an element of $\cal L (n)$ which is characterized by the inequalities $\alpha (\nu'(n))_{\eta} =\alpha (\nu )_{\eta}$ for all $\nu\in\cal L$ and for all $\eta\in I$. First we show $$s(\nu'(n))=s(\nu (n)),\quad t(\nu'(n))=t(\nu (n))$$ For $\nu\in\cal L$, \begin{align*} dm^2\alpha (\nu)^{k+1}_{d'} =&\mu^{k+1}_{d'}\sum_{\iota\in I'_o}\nu_{\iota}\lambda_{\iota} +\lambda^{k+1}_{d'}\sum_{\iota\in I'_e}\nu_{\iota}\mu_{\iota}\\ =&\{ (d-d'+1)m-(P_{k-1}+Q_{k-1})\}(s(\nu )+t(\nu ))\\ &-\{(d-2d'+2)m-2(P_{k-1}+Q_{k-1})\}t(\nu )\\ =&:\alpha (s(\nu)+t(\nu))-\beta t(\nu) \end{align*} If $d\ge 2$, \begin{align*} dm^2\alpha (\nu )^{k+1}_{d'+1} =&\{ (d-d')m-(P_{k-1}+Q_{k-1})\}(s(\nu)+t(\nu))\\ &+\{ (2d'-d)m+2(P_{k-1}+Q_{k-1})\} t(\nu)\\ =&:\gamma (s(\nu )+t(\nu ))+\delta t(\nu ) \end{align*} Note that $\alpha$, $\beta$, $\gamma$, and $\delta$ are all positive. Thus by the Proposition \ref{propsmin} we have done for this case.\\ If $d=1$, $$dm^2\alpha (\nu)^k_{q_k -1}= \{m-2(P_{k-1}+Q_{k-1})\}(s(\nu)+t(\nu)) +4(P_{k-1}+Q_{k-1})t(\nu)$$ Thus we can use the same arguement as above.\\ Next we show $\nu (n)_{\eta}=\nu'(n)_{\eta}$ for $\eta\in I_o$ by induction. Let $\eta$ be an element of $I_o$ which is not (1, 1). Assume $\nu (n)_{\iota}=\nu'(n)_{\iota}$ for all $\iota\in I_o$ such that $\iota >\eta$. For $\nu\in\cal L$, \begin{align*} dm^2\alpha (\nu )_{\eta^l}=&-dm^2\nu_{\eta}+(s(\nu)- \sum_{\iota >\eta}\nu_{\iota}\lambda_{\iota})\mu_{\eta^l}\\ &+(t(\nu)+\sum_{\iota >\eta}\nu_{\iota}\mu_{\iota})\lambda_{\eta^l} \end{align*} Thus $$\nu'(n)_{\eta}\ge \nu(n)_{\eta}$$ Note that $$\sum_{\iota\in I'_o, \iota\le\eta}\nu'(n)_{\iota}\lambda_{\iota} =\sum_{\iota\in I'_o, \iota\le\eta}\nu (n)_{\iota}\lambda_{\iota}$$ Thus by the property of the $\lambda$-expansion, we get $\nu'(n)_{\eta}=\nu (n)_{\eta}$.\\ The same arguement shows $\nu (n)_{\eta}=\nu' (n)_{\eta}$ for all $\eta\in I_e$. \end{pf} \begin{cor} Let $n$ be an integer such that $1\le n\le m-1$. Let $$n=\lambda^1_{j_1}+\sum \begin{Sb} 3\le i\le k-1 \\ i\ \text{\em{odd}} \end{Sb} (l_{i-1}\lambda^i_1+\lambda^i_{j_i})+l_k\lambda^{k+1}_1 =\sum \begin{Sb} 2\le i\le k \\ i\ \text{\em{even}} \end{Sb} (l_{i-1}\mu^i_1+\mu^i_{j_i})$$ be the $\lambda$- and $\mu$- expansion.\\ Then $$C^1_{j_1}+\sum \begin{Sb} 3\le i\le k-1 \\ i\ \text{\em{odd}} \end{Sb} (\sum_{1\le h_{i-1}\le l_{i-1}}C^i_{1, h_{i-1}} +C^i_{j_i}) +\sum_{1\le h_k\le l_k}C^{k+1}_{1, h_k}$$ $$+\sum \begin{Sb} 2\le i\le k \\ i\ \text{\em{even}} \end{Sb} (\sum_{1\le h_{i-1}\le l_{i-1}}C^i_{1, h_{i-1}} +C^i_{j_i})$$ is a general member of $|-nK_X|$. \end{cor} Next we treat the non-normal case. It is easier than the normal case. \begin{thm} Let $n$ be an integer such that $1\le n\le m-1$. Then $$\deg_{E_{\iota}}F(-n(K_{X_o}+\Delta_o)) =\nu_o(n)_{\iota}\quad for\ all\ \iota\in I_o$$ $$\deg_{E_{\iota}}F(-n(K_{X_e}+\Delta_e)) =\nu_e(n)_{\iota}\quad for\ all\ \iota\in I_e$$ \end{thm} \begin{pf} We only show the first equality since the proof is similar for the second one. Let $\nu'_o(n)$ be an element of $\cal L_o$ such that $\nu'_o(n)_{\iota}=\deg_{E_{\iota}}F(-n(K_{X_o}+\triangle_o))$ for $\iota\in I_o$. Put $\sigma (\nu)=\sum_{\iota\in I_o}\nu_{\iota}\lambda_{\iota}$ for $\nu\in\cal L_o$, and put $$\cal L_o(n)=\{\nu\in\cal L_o|\nu\ \text{is nef}, \sigma (\nu)\equiv n\pmod{m}\}$$ By Lemma \ref{lemfull}, $\nu'_o(n)$ is an element of $\cal L_o(n)$ which is characterized by $\alpha(\nu'_o(n))_{\eta}\le\alpha(\nu)_{\eta}$ for all $\nu\in\cal L_o$ and for all $\eta\in I_o$. Since $m\alpha(\nu)^{k+1}_1=\sigma(\nu)$ and $\sigma(\nu_o(n))=n$, we get $\sigma(\nu'_o(n))=\sigma(\nu_o(n))$. Thus we can prove the theorem by the induction using the formula $$m\alpha(\nu)_{\eta^l}=-m\nu_{\eta}+ \mu_{\eta^l}(\sigma(\nu)- \sum_{\iota\in I_o, \iota >\eta}\nu_{\iota}\lambda_{\iota}) +\lambda_{\eta^l}\sum_{\iota\in I_o, \iota >\eta} \nu_{\iota}\lambda_{\iota}$$ similarly to the proof of Theorem \ref{thmdegfull1}. \end{pf} \section{Local intersection number} As the application of the result of Section \ref{secdiv}, we shall prove the following Thorem \ref{thmbound} in this section. \begin{dfn} For a sequence of positive integers $(L_1, L_2, \dots, L_J)$ and a positive integer $N$, we define the sequence $(N_{-1}, N_0, N_1, \dots, N_J)$ by $$N_{-1}=N_0=N,\ N_j=L_jN_{j-1}+N_{j-2}\ (1\le j\le J)$$ We define $B((L_1, L_2, \dots, L_J), N)$ by $B((L_1, L_2, \dots, L_J), N)=N_J$. \par For a pair of positive integers $(M, N)$, we define $B(M, N)$ by $$B(M, N)=\max\{B((L_1, L_2, \dots, L_J), N)| \sum_{1\le j\le J}L_j=M\}$$ \end{dfn} \begin{thm}\label{thmbound} Let $(X, x)$ be a 2-dimensional smoothable semi-log-terminal singularity, and $n$ a positive integer. Let $D$ and $D'$ be members in $|nK_X|$ which do not have common components. Then $$\text{\em{index}}(X, x)\le B(D\cdot D'+1, n)$$ \end{thm} We define $\Bbb Z$-valued symmetric bilinear forms $\cal O$ and $\cal E$ on $\cal L$ by \begin{equation*} \cal O(\delta^{\iota},\delta^{\eta})= \begin{cases} \lambda_{\iota}\bar{\lambda}_{\eta}\quad &(\iota,\eta\in I_o,\iota <\eta)\\ \bar{\lambda}_{\iota}\lambda_{\eta}\quad &(\iota,\eta\in I_o,\iota\ge\eta)\\ 0\quad &(\text{otherwise}) \end{cases} \end{equation*} \begin{equation*} \cal E(\delta^{\iota},\delta^{\eta})= \begin{cases} \mu_{\iota}\bar{\mu}_{\eta}\quad &(\iota,\eta\in \bar{I}_e,\iota<\eta)\\ \bar{\mu}_{\iota}\mu_{\eta}\quad &(\iota,\eta\in \bar{I}_e,\iota\ge\eta)\\ 0\quad &(\text{otherwise}) \end{cases} \end{equation*} where $\bar{I}_e=I_e\cup\{(k+1,d)\}$ in the normal case, and $\bar{I}_e=I_e$ in the non-normal case. \begin{lem}\label{lemint} Let $\nu=\nu^o+\nu^e$ and $\tilde{\nu}=\tilde{\nu}^o+\tilde{\nu}^e$ be members in $\cal L$. Then in the normal case, \begin{align*} dm^2(\nu\cdot\tilde{\nu}) =&(dma-1)\sigma (\nu^o)\sigma (\tilde{\nu}^o) +\sigma (\nu^o)\tau (\tilde{\nu}^e) +\tau (\nu^e)\sigma (\tilde{\nu}^o) -(dma+1)\tau (\nu^e) \tau(\tilde{\nu}^e)\\ &+dm^2 (\cal E (\nu^e , \tilde{\nu}^e) -\cal O (\nu^o, \tilde{\nu}^o )) \end{align*} and in the non-normal case, $$ m(\nu\cdot\tilde{\nu}) =a(\sigma(\nu^o)\sigma(\tilde{\nu}^o) -\tau(\nu^e)\tau(\tilde{\nu}^e)) +m(\cal E(\nu^e,\tilde{\nu}^e) -\cal O(\nu^o,\tilde{\nu}^o)) $$ \end{lem} \begin{pf} First, we treat the normal case. We shall calculate the each term of the left hand side of $$\nu \cdot \tilde{\nu} = \nu^o \cdot \tilde{\nu}^o +\nu^o \cdot \tilde{\nu}^e + \nu^e \cdot \tilde{\nu}^o + \nu^e \cdot \tilde{\nu}^e$$ We can easily get $dm^2 \nu^o \cdot \tilde{\nu}^e =\sigma (\nu^o)\tau (\tilde{\nu}^e)$ and $dm^2 \nu^e \cdot \tilde{\nu}^o = \tau (\nu^e)\sigma(\tilde{\nu}^o)$. By the formula $\mu_{\iota}=-\lambda_{\iota}+dm\rho_{\iota}$, \begin{align*} dm^2 \nu^o \cdot \tilde{\nu}^e &=\sum_{\iota \in I_o} \nu^o_{\iota} \{\lambda_{\iota} \sum_{\eta \in I_o , \eta \ge \iota} \tilde{\nu}^o_{\eta} \mu_{\eta} +\mu_{\iota}\sum_{\eta \in I_o, \eta <\iota} \tilde{\nu}^o_{\eta} \lambda_{\eta}\}\\ &=\sum_{\iota \in I_o} \nu^o_{\iota}[-\lambda_{\iota}\sigma (\tilde{\nu}^o) + dm\{\rho_{\iota}\sum_{\eta \in I_o, \eta <\iota}\tilde{\nu}^o_{\eta} \lambda_{\eta} +\lambda_{\iota}\sum_{\eta \in I_o, \eta \ge \iota} \tilde{\nu}^o_{\eta} \rho_{\eta}\}]\\ &=-\sigma (\nu^o)\sigma (\tilde{\nu}^o) +dm\sum_{\iota \in I_o} \nu^o_{\iota} \{ \rho_{\iota}\sum_{\eta \in I_o, \eta <\iota} \tilde{\nu}^o_{\eta} \lambda_{\eta} +\lambda_{\iota}\sum_{\eta \in I_o, \eta \ge \iota} \tilde{\nu}^o_{\eta} \rho_{\eta}\} \end{align*} By the formula $\rho_{\iota}=-m\bar{\lambda}_{\iota}+a\lambda_{\iota}$, we can get \begin{align*} &\rho_{\iota}\sum_{\eta \in I_o, \eta <\iota}\tilde{\nu}^o_{\eta}\lambda_{\eta} +\lambda_{\iota}\sum_{\eta\in I_o, \eta \ge \iota}\tilde{\nu}^o_{\eta}\rho_{\eta}\\ =&a\lambda_{\iota}\cdot\sigma(\tilde{\nu}^o) -m(\bar{\lambda}_{\iota}\sum_{\eta\in I_o, \eta <\iota} \tilde{\nu}^o_{\eta}\lambda_{\eta} +\lambda_{\iota}\sum_{\eta\in I_o, \eta\ge\iota}\tilde{\nu}^o_{\eta}\bar{\lambda}_{\eta} \end{align*} Hence we get $$dm^2\nu^o\cdot\tilde{\nu}^o = (dma-1)\sigma(\nu^o)\sigma (\tilde{\nu}^o) -dm^2 \cal O (\nu^o, \tilde{\nu}^o)$$ Simarlarly we get \begin{equation*} dm^2 \nu^e\cdot\tilde{\nu}^e =-(dma+1)\tau(\nu^e)\tau(\tilde{\nu}^e) +dm^2 \cal E (\nu^e, \tilde{\nu}^e) \end{equation*} Summarizing all the above formulas, we get the first equality. Next for the non-normal case, we can get $m\nu\cdot\tilde{\nu}= a\sigma(\nu)\sigma(\tilde{\nu})-m\cal O(\nu,\tilde{\nu})$ for $\nu,\tilde{\nu}\in\cal L_o$, $m\nu\cdot\tilde{\nu}= -a\tau(\nu)\tau(\tilde{\nu})+m\cal E(\nu,\tilde{\nu})$ for $\nu,\tilde{\nu}\in\cal L_e$. We leave the details for the reader. \end{pf} Note that $\nu\cdot\tilde{\nu}=\cal E (\nu^e, \tilde{\nu}^e)-\cal O (\nu^o, \tilde{\nu}^o)$ if $\sigma (\nu^o)=\tau (\nu^e)$ and $\sigma (\tilde{\nu}^o)=\tau (\tilde{\nu}^o)$ hold. \begin{cor}\label{corint} If $\nu^o\le\tilde{\nu}^o, \nu^e\le\tilde{\nu}^e, \sigma (\nu^o)=\tau (\nu^e), \sigma (\tilde{\nu}^o)=\tau (\tilde{\nu}^e)$ and $\bar{\sigma}(\tilde{\nu}^o)=\bar{\tau}(\tilde{\nu}^e)$ hold, then $\nu\cdot\tilde{\nu}=0$. \end{cor} \begin{pf} This can be easily checked by the above lemma. \end{pf} \begin{dfn} For $\iota =(i, j)\in I$ such that $i\not= k+1$, put $$\varphi (\iota)=-\delta^{i, 1}+\delta^{i, j}+(j-1)\delta^{i+1, j}$$ \end{dfn} \begin{lem}\label{lemvarphi} \begin{align*} &\sigma (\varphi (\iota )^o)=\tau(\varphi (\iota)^e)=(j-1)(P_{i-1}+Q_{i-1})\\ &\bar{\sigma}(\varphi(\iota)^o)=\bar{\tau}(\varphi (\iota)^e)=(j-1)P_{i-1}\\ &\varphi (\iota)^2=j-1 \end{align*} \end{lem} \begin{pf} We can get the first and the second formula by direct calculation. By the formula $P_{i-2}Q_{i-1}-Q_{i-2}P_{i-1}=(-1)^i$, we get $$\cal E (\varphi (\iota)^e, \varphi (\iota)^e)=(j-1)^2P_{i-1}(P_{i-1}+Q_{i-1})$$ and $$\cal O (\varphi (\iota)^o, \varphi (\iota)^o)=(j-1)^2 P_{i-1}(P_{i-1}+Q_{i-1})-j+1$$ Thus by the above corollary, we can get the third formula. \end{pf} \begin{dfn} Let $\iota =(i_1, j_1)$, $\eta=(i_2, j_2)$ be elements in $I$ such that $i_2\not= k+1$, the parity of $i_1$ coincides the one of $i_2$ and $\iota\le (i_2, 1)$. For such pair $(\iota, \eta)$, we define $\psi(\iota, \eta)$ as follows $$\psi (\iota, \eta)= \begin{cases} -\delta^{\iota^l}+\delta{\iota} +\sum \begin{Sb} \iota\le (i, 1)\le\eta \\ i\ \text{\em{odd}}, i\not= 1 \end{Sb} q_{i-1}\delta^{i,1} -\delta^{i_2, 1}+\delta^{\eta}+j_2 \delta^{i_2+1, 1}\quad & (\iota\in I_o)\\ -\delta^{\iota^r}+\delta{\iota} +\sum \begin{Sb} \iota\le (i, 1)\le\eta \\ i\ \text{\em{even}} \end{Sb} q_{i-1}\delta^{i,1} -\delta^{i_2, 1}+\delta^{\eta}+j_2 \delta^{i_2+1, 1}\quad & (\iota\in I_e) \end{cases} $$ \end{dfn} \begin{lem}\label{lempsi} Let $(\iota, \eta =(i'', 1))$ be the pair for which $\psi$ can be defined. Then $$\sigma (\psi (\iota, \eta)^o)=\tau (\psi (\iota, \eta )^e)=P_{i''+1}+Q_{i''+1}$$ $$\bar{\sigma}(\psi (\iota, \eta )^o)=P_{i''+1},\quad \bar{\tau}(\psi (\iota, \eta )^e)= \begin{cases} P_{i''-1}\qquad & (\iota \not= (2, 1))\\ P_{i''-1}+1\quad & (\iota = (2, 1)) \end{cases} $$ $$\psi (\iota, \eta )^2 = \begin{cases} 1+\sum \begin{Sb} i'+1\le i\le i''-1 \\ i\ \text{\em{even}} \end{Sb} q_i\qquad & (\iota\in I_o)\\ 1+\sum \begin{Sb} i'+1\le i\le i''-1 \\ i\ \text{\em{odd}} \end{Sb} q_i\qquad & (\iota\in I_e) \end{cases} $$ \end{lem} \begin{pf} Since the calculation is the same, we show the outline of it for the case $\iota \in I_o$. We can easily calculate $\sigma$, $\tau$, $\bar{\sigma}$, and $\bar{\tau}$. Hence $$\psi (\iota, \eta )^2 =\cal E (\psi (\iota, \eta)^e, \psi (\iota, \eta )^e)-\cal O (\psi (\iota, \eta)^o, ( \psi (\iota, \eta )^o)$$ By the definion, $$\cal E (\psi (\iota, \eta)^e, \psi (\iota, \eta)^e)=P_{i''-1}(P_{i''-1}+Q_{i''-1})$$ To calculate $\cal O (\psi (\iota, \eta)^o, \psi (\iota, \eta)^o)$, note that $$-\lambda_{\iota^l}+\lambda_{\iota} +\sum \begin{Sb} i'+2\le i\le h \\ i\ \text{odd} \end{Sb} q_{i-1}\lambda^i_1=P_{h-1}+Q_{h-1}$$ and $$\sum \begin{Sb} h+2\le i\le i''\\ i\ \text{odd} \end{Sb} q_{i-1}\bar{\lambda}^i_1=P_{i''-1}-P_{h-1}$$ for odd $h$ such that $i'+2\le h\le i''$. Then \begin{align*} \cal O (\psi (\iota, \eta)^o, \psi (\iota,\eta)^o)= &-\lambda_{\iota ^l}\bar{\sigma}(\psi (\iota, \eta)^o)+(-\lambda_{\iota^l}+\lambda_{\iota})\bar{\lambda}_{\iota}\\ &+\lambda (\sum \begin{Sb} i'+2\le i \le i''\\ i\ \text{odd} \end{Sb} q_{i-1}\bar{\lambda}^i_1)\\ &+\sum\begin{Sb} i'+2\le h \le i''\\ h\ \text{odd} \end{Sb} q_{h-1}\{ \bar{\lambda}^h_1 (P_{h-1}+Q_{h-1})+\lambda (P_{i''-1}-P_{h-1})\} \end{align*} Here \begin{align*} &-\lambda_{\iota^l}\bar{\sigma}(\psi (\iota, \eta)^o)+(-\lambda_{\iota^l}+\lambda_{\iota})\bar{\lambda}_{\iota} +\lambda_{\iota} (\sum \begin{Sb} i'+2\le i\le i''\\ i\ \text{odd} \end{Sb} q_{i-1}\bar{\lambda}^i_1 )\\ =& (-\lambda_{\iota}+P_{i'-1}+Q_{i'-1})P_{i''-1}+(P_{i'-1}+Q_{i'-1})\bar{\lambda}_{\iota} +\lambda_{\iota} (P_{i''-1}-P_{i'-1})\\ =&(P_{i'-1}+Q_{i'-1})P_{i''-1}+(P_{i'-1}+Q_{i'-1})\bar{\lambda}_{\iota}+\lambda_{\iota} P_{i-1}\\ =&(P_{i'-1}+Q_{i'-1})P_{i''-1}+Q_{i'-1}P_{i'-2}-P_{i'-1}Q_{i'-2}\\ =&(P_{i'-1}+Q_{i'-1})P_{i''-1}-1 \end{align*} and \begin{align*} &\bar{\lambda}^h_1 (P_{h-1}+Q_{h-1})+\lambda^h_1 (P_{i'-1}-P_{h-1})\\ =&(P_{h-2}+Q_{h-2})P_{i''-1}-1 \end{align*} Hence \begin{align*} &\cal O (\psi (\iota, \eta)^o, \psi (\iota, \eta)^o)\\ =&(P_{i'-1}+Q_{i'-1})P_{i''-1}-1+\sum \begin{Sb} i'+2\le h\le i''\\ h\ \text{odd} \end{Sb} q_{h-1}\{(P_{h-2}+Q_{h-2})P_{i''-1}-1\}\\ =&-1-\sum \begin{Sb} i'+2\le h \le i''\\ h\ \text{odd}\end{Sb} q_{h-1}+P_{i''}\{(P_{i'+1}+Q_{i'-1})+ \sum\begin{Sb} i'+2\le h\le i''\\ h\ \text{odd}\end{Sb} q_{h-1}(P_{h-2}+Q_{h-2})\}\\ =&-1-\sum \begin{Sb}i'+1\le i\le i''-1\\ i\ \text{even}\end{Sb} q_i+P_{i''-1}(P_{i''-1}+Q_{i''-1}) \end{align*} Thus we have done. \end{pf} \begin{cor}\label{corpsi} Let $(\iota,\eta)$ be a pair for which $\psi$ can be defined. Then $$\sigma (\psi (\iota, \eta)^o) =\tau (\psi (\iota, \eta)^e) =j''(P_{i''-1}+Q_{i''-1})$$ \begin{equation*} \bar{\sigma}(\psi (\iota, \eta)^o)= j''P_{i''-1 },\quad \bar{\tau}(\psi (\iota, \eta)^e)= \begin{cases} j''P_{i''-1}\qquad &(\iota\not= (2, 1))\\ j''P_{i''-1}+1\quad &(\iota =(2, 1)) \end{cases} \end{equation*} \begin{equation*} \psi (\iota, \eta)= \begin{cases} j''+\sum \begin{Sb}i'+1\le i\le i''-1\\ i\ \text{\em{even}}\end{Sb} q_i\quad &(if\ \iota \in I_o)\\ j''+\sum \begin{Sb}i'+1\le i\le i''-1\\ i\ \text{\em{odd}}\end{Sb} q_i\quad &(if\ \iota \in I_e) \end{cases} \end{equation*} \end{cor} \begin{pf} Note that $\psi (\iota, \eta)=\psi (\iota, (i'', 1))+\varphi (\eta)$. Hence $$\sigma (\psi (\iota, \eta)^o) =\sigma (\psi (\iota, \eta)^o)+\sigma (\varphi (\eta)^o) =j''(P_{i''-1}+Q_{i''-1})$$ $\tau$, $\bar{\sigma}$ and $\bar{\tau}$ are simarlarly calculated. By Corollary \ref{corint}, $\psi(\iota,(i'', 1))\cdot \varphi(\eta)=0$. Hence $$\psi(\iota, \eta)^2=\psi(\iota, (i'', 1))^2+\varphi(\eta)^2$$ Thus the formula follows from Lemma \ref{lemvarphi} and \ref{lempsi}. \end{pf} \begin{dfn} For $(i, j)\in I$ such that $1\le i\le k-1$ and $1\le j\le q_i$, we put $$\theta (i, j) =-\delta^{i, q_i-j+1}+\delta^{i+2, 1} +j\delta^{i+1, 1}$$ \end{dfn} \begin{lem}\label{lemtheta} $$\sigma (\theta (i, j)^o) =\tau (\theta (i, j)^e) =j(P_{i-1}+Q_{i-1})$$ $$\bar{\sigma}(\theta (i, j)^o) =\bar{\tau}(\theta (i, j)^e) =jP_{i-1}$$ $$\theta (i, j)=j$$ \end{lem} \begin{pf} We will only show the outline of the calculation for odd $i$ since it is similar for even $i$. We can easily get the formulas for $\sigma$, $\tau$, $\bar{\sigma}$ and $\bar{\tau}$. Thus by Lemma \ref{lemint}, $$\theta (i, j)^2 =\cal E (\theta (i,j)^e, \theta (i, j)^e) -\cal O (\theta (i, j)^o, \theta (i, j)^o)$$ By the definition, $$\cal E (\theta (i, j)^e, \theta (i, j)^e)= j^2P_{i-1}(P_{i-1}+Q_{i-1})$$ and \begin{align*} &\cal O (\theta (i, j)^o, \theta (i, j)^o)\\ =&\lambda^i_{q_i-j+1}(\bar{\lambda}^i_{q_i-j+1}-\bar{\lambda}^{i+2}_1) +(-\lambda^i_{q_i-j+1}+\lambda^{i+2}_1)\bar{\lambda}^{i+2}_1\\ =&-jP_{i-1}\lambda^i_{q_i-j+1} +j(P_{i-1}+Q_{i-1})\bar{\lambda}^{i+2}_1\\ =&j^2P_{i-1}(P_{i-1}+Q_{i-1})+j(P_iQ_{i-1}-Q_iP_{i-1})\\ =&j^2P_{i-1}(P_{i-1}+Q_{i-1})-j \end{align*} Hence we have done. \end{pf} For a positive integer $n$ which is smaller than $m-(P_{k-1}+Q_{k-1})$, we put $$i(n)=\max\{i|0\le i \le k-1, P_i+Q_i\le n\}$$ \begin{prop}\label{propnu1} If $0<n<m-(P_{k-1}+Q_{k-1})$, $$\nu (n)^2\le \dfrac{n}{P_{i(n)}+Q_{i(n)}}$$ \end{prop} \begin{pf} We use the induction on $i(n)$. If $i(n)=0$, it can be easily checked that $\nu (n)^2 =n$. Let $i$ be an integer such that $1\le i\le k-1$. Assume that the ineqality holds for all $n$ such that $i(n)<i$. We will show that the inequality holds for $n$ such that $i(n)=i$ under this assumption. We also assume $i$ is odd since the proof is similar for even $i$. Write $n=j(P_i+Q_i)+n'$ such that $0\le n'<P_i+Q_i$. If $n'=0$, we can check (by the definition of $\lambda$- and $\mu$- expansion) that $$\nu (n)=\psi ((2, 1), (i+1, j))$$ Thus by Corollary \ref{corpsi}, $$\nu (n)^2\le j$$ Hence we have done in this case. Thus we assume $n'>0$. We divide the proof into two cases as follows (i)$n'<P_{i-1}+Q_{i-1}$ (ii)$n'\ge P_{i-1}+Q_{i-1}$ (i) Put $$\eta =\min\{\iota\in\bar{I}_e|\iota '<\iota\ for\ all\ \iota '\in \bar{I}_e \ such\ that\ \nu (n')_{\iota '}\not= 0\}$$ We can check (by the definition of $\lambda$- and $\mu$- expantion) $$\nu (n)=\nu (n')+\psi (\eta, (i+1, j))$$ Since $\nu (n')^o\le \psi (\eta, (i+1, j))^o$ and $\nu (n')^e\le \psi (\eta, (i+1, j))^e$ hold, thus by Corollary \ref{corint}, $$\nu (n)^2=\nu (n')^2+\psi (\eta, (i+1, j))^2$$ By Corollary \ref{corpsi} and the induction hypothesis, \begin{align*} \nu (n)^2(P_{i}+Q_{i})&\ge \nu (n')^2(P_{i}+Q_{i}) +j(P_{i}+Q_{i})\\ &\ge \nu (n')^2(P_{i(n')}+Q_{i(n')})+j(P_{i}+Q_{i})\\ &\ge n'+j(P_i+Q_i)\\ &=n \end{align*} Thus we have done. \par (ii) We can check $$\nu (n)=\nu (n')+\varphi (i+1, j+1)$$ $$\nu (n')^o\le \varphi (i+1, j+1)^o,\quad \nu (n')^e\le \varphi (i+1, j+1)^e$$ Thus we can get the inequality by the similar way to (i) using Lemma \ref{lemvarphi} and the induction hypothesis, \end{pf} For $n$ such that $m-(P_{k-1}+Q_{k-1})\le n\le m-1$, we define $i(n)$ and $j(n)$ as follows $$i(n)=\min\{i|0\le i\le k-1, m-n\le P_i+Q_i\}$$ $$j(n)=\lceil \dfrac{m-n}{P_{i(n)-1}+Q_{i(n)-1}} \rceil -1$$ \begin{lem}\label{lemnutheta} Let $n$, $i$, $j$ be positive integers such that $m-(P_{k-1}+Q_{k-1})\le n\le m-1$, $i\le k-1$, $j\le q_i$ and $i(n)\le i-1$. Then $\nu (n)\cdot\theta (i, j)=j$. \end{lem} \begin{pf} We only show the proof for even $i$. By Lemma \ref{lemint}, $$\nu (n)\cdot\theta (i, j) =\cal E (\nu (n)^e, \theta (i, j)^e) -\cal O (\nu (n)^o, \theta (i, j)^o)$$ First we calculate $\cal E (\nu (n)^e, \theta (i, j)^e)$. Note that $m-(P_{i-1}+Q_{i-1})\le n\le m-1$. Thus the $\mu$-expantion of $n$ is as follows \begin{equation*} n=\sum\begin{Sb} 2\le h\le i-2\\ h\ \text{even}\end{Sb} (l_{h-2}\mu ^h_1 +\mu ^h_{j_h}) +l_{i-1}\mu ^i_1 +\sum\begin{Sb}i+2\le h\le k-2\\ h\ \text{even}\end{Sb} q_{h-1}\mu ^h_1 +q_{k-1}\mu ^k_1 +\mu ^k_{q_k} \end{equation*} Since $\sum\begin{Sb}\iota\in \bar{I}_e\\ \iota\ge (i+2, 1)\end{Sb} \nu (n)^e_{\iota}\mu_{\iota} =m-(P_{i-1}+Q_{i-1})$, we get $$\sum\begin{Sb} \iota\in \bar{I}_e\\ \iota\le (i, 1)\end{Sb} \nu (n)^e_{\iota} \mu_{\iota} =n-m+P_{i-1}+Q_{i-1}$$ Using this formula and the formula $\sum_{\iota\in \bar{I}_e, \iota\ge (i+2, 1)} \nu (n)^e_{\iota}\bar{\mu}_{\iota} =a-P_{i-1}$, we get \begin{align*} &\cal E (\nu (n)^e, \theta (i, j)^e)\\ =&\cal E (\sum_{\iota\in\bar{I}_e, \iota\le (i,1)} \nu (n)^e_{\iota}\delta^{\iota} +\sum_{\iota\in I_e, \iota\ge (i+2, 1)} \nu (n)^e_{\iota}\delta ^{\iota}, \theta (i, j)^e)\\ =&\tau (\sum _{\iota\in\bar{I}_e, \iota\le (i, 1)} \nu (n)^e_{\iota}\delta ^{\iota} \cdot\bar{\tau}(\theta (i, j)^e) +\bar{\tau}(\sum_{\iota\in\bar{I}_e, \iota\le (i+2, 1)} \nu (n)^e_{\iota}\delta^{\iota} \cdot\tau (\theta (i, j)^e)\\ =&(n-m+P_{i-1}+Q_{i-1})\cdot jP_{i-1} +(a-P_{i-1})\cdot j(P_{i-1}+Q_{i-1})\\ =&j(n-m)P_{i-1}+ja(P_{i-1}+Q_{i-1}) \end{align*} Next we calculate $\cal O (\nu (n)^o, \theta (i, j)^o)$. The $\lambda$-expansion of $n$ is as follows \begin{equation*} n=\lambda^1_{j_1}+\sum\begin{Sb} 3\le h\le i-1 \\ h\ \text{odd} \end{Sb} l_{h-1}\lambda^h_1+\lambda^h_{j_h}+l_{i+1}\lambda^{i+1}_1 +\sum\begin{Sb} i+3\le h\le k-1 \\ h\ \text{odd} \end{Sb} q_{h-1}\lambda^h_1+q_k\lambda^{k+1}_1 \end{equation*} Since $\sum\begin{Sb} \iota\in I_o \\ \iota>(i+1, 1)\end{Sb} \nu (n)^o_{\iota}\lambda_{\iota} =m-(P_{i-1}+Q_{i-1})$, we get $$\sum\begin{Sb} \iota\in I_o \\ \iota\le (i+1,1)\end{Sb} \nu (n)^o_{\iota}\lambda_{\iota}= n-m+P_i+Q_i$$ Using this formula and $\sum\begin{Sb}\iota\in I_o\\ \iota >(i+1, 1)\end{Sb} \nu (n)^o_{\iota}\bar{\lambda}_{\iota} =a-P_{i-1}$, we can get \begin{align*} &\cal O (\nu (n)^o, \theta (i, j)^o)\\ =&\cal O (\sum \begin{Sb} \iota\in I_o\\ \iota\le (i+1, 1)\end{Sb} \nu (n)^o_{\iota}\delta^{\iota}, \theta (i, j)^o ) +\cal O (\sum \begin{Sb}\iota\in I_o\\ \iota>(i+1, 1)\end{Sb} \nu (n)^o_{\iota}\delta^{\iota}, \theta (i, j)^o)\\ =&\sigma (\sum \begin{Sb} \iota\in I_o\\ \iota\le (i+1, 1)\end{Sb} \nu (n)^o_{\iota}\delta^{\iota}) \bar{\sigma}(\theta (i, j)^o) +\bar{\sigma}(\sum \begin{Sb} \iota\in I_o\\ \iota>(i+1, 1)\end{Sb} \nu (n)^o_{\iota}\delta^{\iota}) \sigma (\theta (i, j)^o)\\ =&(n-m+P_i+Q_i)\cdot jP_{i-1}+(a-P_i)\cdot j(P_{i-1}+Q_{i-1})\\ =&j(n-m)P_{i-1}+ja(P_{i-1}+Q_{i-1}) -j(P_iQ_{i-1}-Q_iP_{i-1})\\ =&j(n-m)P_{i-1}+ja(P_{i-1}+Q_{i-1})-j \end{align*} Thus we have done. \end{pf} \begin{lem} $$\nu (m-1)^2=\sum_{1\le h\le k}q_h$$ \end{lem} \begin{pf} Note that $\lambda$- and $\mu$- expansion of $m-1$ is as follows \begin{align*} m-1&=\lambda^1_0+\sum \begin{Sb}3\le h\le k-1\\ h\ \text{odd}\end{Sb} (q_{h-1}\lambda^h_1+\lambda^h_0)+q_k\lambda^{k+1}_1\\ &=\sum \begin{Sb}2\le h \le k-2\\ h\ \text{even}\end{Sb} (q_{h-1}\mu^h_1+\mu^h_0)+q_{k-1}\mu^k_1+\mu^k_{q_k} \end{align*} We leave the rest of calculation for the reader's exercise. \end{pf} \begin{prop}\label{propnu2} Let $n$ be an integer such that $m-(P_{k-1}+Q_{k-1})\le n\le m-1$. Then $$\nu (n)^2\ge \sum_{i(n)\le h\le k}q_h-j(n)$$ \end{prop} \begin{pf} We use the induction on $i(n)$. If $i(n)=0$, then $n=m-1$, Thus we have already done in the above lemma. Let $i$ be a positive integer and assume that the inequality holds for $n'$ such that $i(n')<i$. Let $n$ be an integer such that the inequality holds for this $n$. Put $n'=n+j(P_{i-1}+Q_{i-1})$. Then $i(n')<i$. We can check $$\nu (n)=\nu (n')-\theta (i, j)$$ Thus by Lemma \ref{lemtheta} and \ref{lemnutheta}, we can get \begin{align*} \nu (n)^2&=\nu (n')^2-2\nu (n')\cdot \theta (i, j) +\theta (i, j)^2\\ &=\nu (n')^2-j \end{align*} By the induction hypothesis, $$\nu (n')\ge \sum_{i(n')+1\le h\le k}q_h$$ Thus we have done. \end{pf} (Proof of the Theorem \ref{thmbound})\\ {}From Proposition \ref{propnu1} and Proposition \ref{propnu2}, we know the thorem holds if $D$ and $D'$ is general members in $|nK_X|$. Thus by Corollary \ref{corgenint}, we have proved the thorem.

9507

alg-geom/9507005

en

https://arxiv.org/abs/alg-geom/9507005

alg-geom/9507005

Mikhail Zaidenberg

S. Orevkov and M. Zaidenberg

On the number of singular points of plane curves

LaTeX, 24 pages with 3 figures, author-supplied DVI file available at http://www.math.duke.edu/preprints/95-00.dvi

Duke preprint DUKE-M-95-00

This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an irreducible cuspidal curve at 1992. Our proof is based on the localization, the local Zariski--Fujita decomposition and uses a graph discriminant calculus. The key point is a local analog of the BMY-inequality for a plane curve germ. As a corollary, a boundedness criterium for a family of plane curves has been obtained. Another application of our methods is the following fact: a rigid rational cuspidal plane curve cannot have more than 9 cusps.

alg-geom

\section{Asymptotics of the number of ordinary cusps} We start with a brief survey of known results in the simplest case of ordinary cusps. It is well known that for a nodal plane curve $D \subset {\bf P}^2$ of degree $d$ the number of nodes can be an arbitrary non--negative integer allowed by the genus formula, i.e. any integer from the interval $[0,\,{d-1 \choose 2}]$. If $D$ is a Pl\"ucker curve with only ordinary cusps as singularities, which has $\kappa$ cusps, then still $$\kappa \le {d-1 \choose 2}\,,$$ but this time the inequality is strict starting with $d = 5$. Indeed, by Pl\"ucker formulas $$0 < d^* = d(d-1) - 3\kappa$$ where $d^*$ is the class of $D$, and $$0 \le f = 3d(d-2) - 8\kappa$$ where $f$ is the number of inflexion points of $D$. Thus, we have \begin{equation} \kappa < {1 \over 3}d(d-1) \end{equation} and \begin{equation} \kappa \le {3 \over 8}d(d-2)\,, \end{equation} which is strictly less than ${d-1 \choose 2}$ for $d \ge 5$. Note that $d \le 4$ for a rational cuspidal Pl\"ucker curve $D$, due to (2) and the genus formula. Therefore, up to projective equivalence there exists only two such curves, namely the cuspidal cubic and the Steiner three-cuspidal quartic (we suppose here that at least one cusp really occurs, otherwise we have to add also the line and the smooth conic). {}From (2) it follows that \begin{equation} \limsup\limits_{d \to \infty} {\kappa \over d^2} \le {3 \over 8} \,\,.\end{equation} Using the spectrum of singularity (or, equivalently, the Mixed Hodge Structures) A. Varchenko [Va] found an estimate \begin{equation} \limsup\limits_{d \to \infty} {\kappa \over d^2} \le {23 \over 72} \,\,,\end{equation} which is better than (3) by ${1 \over 18}$. Another ${1 \over 144}$ was gained in the work of F. Hirzebruch and T. Ivinskis [H, Iv] by applying Miyaoka's logarithmic form of the Bogomolov-Miyaoka-Yau (BMY) inequality: \begin{equation} \limsup\limits_{d \to \infty} {\kappa \over d^2} \le {5 \over 16} \,\,.\end{equation} Furthermore, in this work an elegant example was given which shows that \begin{equation} \limsup\limits_{d \to \infty} {\kappa \over d^2} \ge {1 \over 4} \,,\end{equation} where $c$ means now the maximal number of cusps among all the cuspidal Pl\"ucker curves of degree $d$. \\ \noindent {\bf Example} [H, Iv]. Starting with a generic smooth cubic $C$, consider its dual curve $D = C^*$, which is an elliptic sextic with nine ordinary cusps as the only singularities. Let $F = 0$ be the defining equation of $D$. Set $D_k = \{F(x^k : y^k : z^k) = 0\}$. Then $D_k$ is again a cuspidal Pl\"ucker curve. It has degree $d_k = 6k$ and $\kappa_k = 9k^2$ cusps (indeed, $(x : y : z) \longmapsto (x^k : y^k : z^k)$ is a branched covering ${\bf P}^2 \to {\bf P}^2$ of degree $k^2$ ramified along the coordinate axes which meet $D$ normally). Thus, here $\kappa_k = d_k^2 / 4$. In fact, the lower bound $1/4$ can be improved, by a similar method, at least by $1/32$ (A. Hirano [Hi]). Together with (5) this yields \begin{equation} {10 \over 32} \ge \limsup\limits_{d \to \infty} {\kappa \over d^2} \ge {9 \over 32}\,, \end{equation} which is still far away from giving the exact asymptotic. See also [Sa] for a discussion on what is known for small values of $d$. \section{The main inequality. Bounded families of plane curves} Next we consider, more generally, plane curves with arbitrary singularities. By {\it a cusp} we mean below a locally irreducible singular point. We say that $D \subset {\bf P}^2$ is {\it a cuspidal curve } if all its singular points are cusps. The following theorem, which is the main result presented in the talk, was first proven by F. Sakai [Sa]. Independently and later it was also found by the authors in the special case of cuspidal curves [OZ] (actually, the proof in [OZ] goes through without changes for nodal--cuspidal curves, i.e. plane curves with nodes as the only reducible singular points). Both proofs are based on the logarithmic version of the BMY-inequality due to Miyaoka [Miy], but technically they are different. \\ \noindent {\bf Theorem 1.} {\it Let $D$ be a plane curve of degree $d$ with the singular points $P_1, \dots, P_s$. Let $\mu_i$ resp. $m_i$ be the Milnor number resp. the multiplicity of $P_i \in D$. If ${\bf P}^2 \setminus D$ has a non--negative logarithmic Kodaira dimension, then \begin{equation} \sum_{i=1}^s (1 + {1\over 2m_i})\,\mu_i \le d^2 - {3\over2}d\,\,. \end{equation} In particular, \begin{equation} \sum_{i=1}^s \mu_i \le {2m\over 2m+1}(d^2 - {3\over 2}d) \,\, , \end{equation} where $m = \max\limits_{1\le i \le s} \{m_i\}$.} \\ \noindent {\bf Remarks. } a) For $ m \le 3$ and $D$ irreducible Theorem 1 had been proved by Yoshihara [Yo1,2], whose work stimulated the later progress. \\ \noindent b) For an irreducible plane curve $D$ of degree $d \ge 4$ the logarithmic Kodaira dimension $\bar\kappa({\bf P}^2- D)$ is non--negative besides the case when $D$ is a rational cuspidal curve with one cusp; see [Wak]. \\ \noindent {\bf Corollary 1. } {\it If $D \subset {\bf P}^2$ is an irreducible cuspidal curve of geometric genus $g$, then under the assumptions of Theorem 1 one has \begin{equation} g\ge {d^2-3(m+1)d\over 2(2m+1)}+1 \,\, . \end{equation} In particular, a family of such curves is bounded iff $g$ and $m$ are bounded throughout the family. } \\ In general, the latter conclusion does not hold for non-cuspidal curves (indeed, the family of all the irreducible rational nodal plane curves is unbounded). However, it becomes true if one replaces the geometric genus $g = g(D)$ by the Euler characteristic $e(D)$ (thus, involving not only the topology of the normalization, but the topology of the plane curve itself). Moreover, in this form it works even for reducible curves. \\ \noindent {\bf Corollary 2. } {\it Under the assumptions of Theorem 1 one has \begin{equation} d(d-3(m+1)) \le (2m+1)(-e(D)) \,\,.\end{equation} Therefore, a family of (reduced) plane curves is bounded iff the absolute value of the Euler characteristic and the maximal multiplicity of the singular points are bounded throughout the family. } \\ The Corollary easily follows from Theorem 1 and the formula (see [BK]) $$\sum_{i=1}^s \mu_i = d(d-3) + e(D)\,\,.$$ In the case of irreducible curves, in the estimate (11) it is convinient to use the first Betti number $b_1(D) = 2 - e(D)$ instead of the Euler characteristic. In particular, for irreducible nodal curves it is the only parameter involved. An immediate consequence of (11) is that $d \le 3m+3$ if $e(D) \ge 0$. Furthermore, $d \le 3m+2$ if $e(D) > 0$; this is so, for instance, if $D$ is a rational cuspidal curve. In fact, in the latter case $d < 3m$ [MaSa], and also by the genus formula $$\sum\limits_{i=1}^s {\mu_i \over m_i} \le 3d-4$$ [OZ]. \section{BMY-inequalities} These inequalities provide the basic tool in the proof of Theorem 1. Let $\sigma \,:\,X \to {\bf P}^2$ be the minimal embedded resolution of singularities of $D$, and let ${\~ D} \subset X$ be the reduced total preimage of $D$. Thus, ${\~ D}$ is a reduced divisor of simple normal crossing type, and ${\~ D}=\sigma^{-1}(D)$. Let $K = K_X$ be the canonical divisor of $X$. If ${\bar k}({\bf P}^2 \setminus D) = {\bar k}(X \setminus {\~ D}) \ge 0$, i.e. if $|m(K+{\~ D})| \neq 0$ for $m$ sufficiently large, then (see [Fu]) there exists {\it the Zariski decomposition} $K + {\~ D} = H + N$, where $H,\,N$ are ${\bf Q}$--divisors in $X$ such that \\ \noindent i) the intersection form of $X$ is negatively definite on the subspace $V_N \subset {\rm Pic}X \otimes {\bf Q}$ generated by the irreducible componenets of $N$; \\ \noindent ii) $H$ is nef, i.e. $HC \ge 0$ for any complete irreducible curve $C \subset X$;\\ \noindent iii) $H$ is orthogonal to the subspace $V_N$.\\ \noindent By (iii) we have $$(K+{\~ D})^2 = H^2 + N^2\,,$$ where $N^2 \le 0$. Thus, $H^2 \ge (K+{\~ D})^2$. \\ \noindent {\bf Theorem} (Y. Miyaoka [Miy]; R. Kobayashi--S. Nakamura--F. Sakai [KoNaSa]). \\ {\it \noindent a) If ${\bar k}({\bf P}^2 \setminus D) \ge 0$, then \begin{equation} (K+{\~ D})^2 \le 3e({\bf P}^2 \setminus D)\,.\end{equation} b) If ${\bar k}({\bf P}^2 \setminus D) = 2$, then \begin{equation} H^2 \le 3e({\bf P}^2 \setminus D)\,.\end{equation}} \noindent {\bf Remark.} (13) holds, for instance, in the case when $D$ is an irreducible curve with at least three cusps [Wak]. \\ Next we describe an approach to the proof of Theorem 1, mainly following [OZ]. An advantage of this approach is that, in the particular case of irreducible cuspidal curves, we obtain formulas which express all the ingradients of the above BMY-inequalities in terms of the Puiseux characteristic sequences of the cusps. In fact, we prove a local version of Theorem 1 for the case of a cusp (see Theorem 2 in Section 11 below). Together with the BMY-inequality (12) this provides a proof of Theorem 1 in the cuspidal case. A similar local estimate participates in the proof in [Sa], which is, by the way, much shorter. Instead of the Puiseux data it deals with the multiplicity sequences of the singular points. Combining both approches, we give in Section 12 a proof of the local estimate for arbitrary singularity (Theorem 3), thus proving Theorem 1 in general case. Actually, this is the proof of F. Sakai, with more emphasize separately on the local and the global aspects. In the final Section 13 we apply the methods developed in the previous sections for pushing forward in the rigidity problem for rational cuspidal plane curves (see [FZ1,2]). \section{Localization} Let, as above, $D \subset {\bf P}^2$ be a plane curve of degree $d$ and let $\sigma\,:\,X \to {\bf P}^2$ be the minimal embedded resolution of the singular points $P_1,\dots,P_s$ of $D$. Let $D'$ be the proper preimage of $D$ in $X$, ${\~ D} = D' \cup E$ be the reduced total preimage of $D$ and $E = E_1 \cup \dots \cup E_k, \,\,E_i = \sigma^{-1}(P_i),$ be the exceptional divisor of $\sigma$. Let $E_i = \sum\limits_{j=1}^{k_i} E_{ij}$ be the decomposition of $E_i$ into irreducible components. Fix also a line $L \subset {\bf P}^2$ which meets $D$ normally; denote by ${L'}$ the proper preimage of $L$ in $X$. Then, clearly, $\{E_{ij}\}$ and $L'$ form a basis of the vector space ${\rm Pic}X \otimes {\bf Q}$. Let $V_i = V_{E_i}$ be the subspace generated by the irreducible components $E_{ij}$ of $E_i$ and $V_{L'}$ be the one--dimensional subspace generated by $L'$ in ${\rm Pic}X \otimes {\bf Q}$. Since the intersection form of $X$ is non-degenerate, we have the orthogonal decomposition $${\rm Pic}X \otimes {\bf Q} = V_{L'} \oplus (\bigoplus_{i=1}^s V_i)\,.$$ Therefore, for each $i=1,\dots,s$ there exists the unique orthogonal projection ${\rm Pic}X \otimes {\bf Q} \to V_i$, and also such a projection onto the line $V_{L'}$. For any ${\bf Q}$--divisor $Z$ denote by $Z_{L'}$ resp. $Z_i$ its projection into $V_{L'}$ resp. into $V_i$. Then we have $$Z^2 = Z_{L'}^2 + \sum\limits_{i=1}^s Z_i^2\,.$$ In particular, since $K_{L'} + {\~ D}_{L'} = (d-3){L'}$ we have $$(K + {\~ D})^2 = (d-3)^2 + \sum\limits_{i=1}^s (K_i + {\~ D}_i)^2\,,$$ where the summands in the last sum are all negative (indeed, $E_i$ being an exceptional divisor, the intersection form of $X$ is negatively definite on the subspace $V_i$). It is easily seen that (8) follows from (12) and the local estimates \begin{equation} -(K_i + {\~ D}_i)^2 \le (1 - {1 \over m_i}) \mu_i \,. \end{equation} In what follows we trace a way of proving (14). This is done in particular case of an irreducible singularity in Section 11 (Theorem 2), and in general in Section 12 (Theorem 3). Note that the assumption of local irredubicibility is important only in Sections 9, 10, 11. \section{Weighted dual graph} Let $E = E_1 \cup \dots \cup E_k$ be a curve with simple normal crossings in a smooth compact complex surface $X$. Assume, for simplicity, that all the irreducible components $E_i$ of $E$ are rational curves and that their classes in ${\rm Pic}X \otimes {\bf Q}$ are linearly independent. Let $A_E$ be the matrix of the intersection form of $X$ on the subspace $V_E = {\rm span}\,(E_1,\dots, E_k) \subset {\rm Pic}X \otimes {\bf Q}$ in the natural basis $E_1,\dots, E_k$ (we denote by the same letter a curve and its class in ${\rm Pic}X \otimes {\bf Q}$). Then $A_E$ is at the same time the incidence matrix of {\it the dual graph} $\Gamma_E$ of $E$, which is defined as follows. The vertices of $\Gamma_E$ correspond to the irreducible components $E_i$ of $E$; two vertices $E_i$ and $E_j$, where $i \neq j$, are joint by a link $[E_i, \,E_j]$ iff $E_i\cdot E_j > 0$. The weight of the vertex $E_i$ is defined to be the self--intersection index $E_i^2$. Let $C$ be another curve in $X$ which meets $E$ normally. Then we consider also {\it the dual graph $\Gamma_{E,\,C}$ of $E$ near $C$}; it is the graph obtained from $\Gamma_E$ by attaching $E_i\cdot C$ arrowheads to the vertex $E_i,\,\,i=1,\dots,k$. We denote by $\nu_i$ resp. ${\~ \nu}_i$ {\it the valency} of $E_i$ in $\Gamma_E$ resp. in $\Gamma_{E,\,C}$. By {\it a twig} of a graph $\Gamma$ one means an extremal linear branch of $\Gamma$; its end point is called {\it the tip} of the twig. \section{Local Zariski--Fujita decomposition} Since in the sequel we are working only locally over a fixed singular point $P=P_i$ of $D$, we change the notation. Omitting subindex $i$, from now on we denote by $E$ the corresponding exceptional divisor $E_i$ and by $V_E$ the corresponding subspace $V_i$. Thus, $K_E, \,{\~ D}_E, \,D'_E$ etc. mean the projections $K_i,\,{\~ D}_i,\,D'_i$... of the corresponding divisors into $V_E = V_i$. Set also $\mu = \mu_i$ and $m = m_i$. Note that in this case the dual graph $\Gamma_E$ is a tree. By {\it the local Zariski--Fujita decomposition} we mean the decomposition $$K_E + {\~ D}_E = H_E + N_E\,,$$ where $H_E,\,N_E \in V_E$ are effective ${\bf Q}$--divisors such that \\ \noindent i) the support of $N_E$ coincides with the union of all the twigs of $\Gamma_E$ which are not incident with the proper preimage $D'$ of $D$ in $X$, i.e. all the twigs of $\Gamma_{E,\,D'}$ without arrowheads, and \\ \noindent ii) $H_E$ is orthogonal to each irreducible component of ${\rm supp}\,N_E$. \\ Note that all the twigs in the ${\rm supp}\,N_E$ are {\it admissible}, i.e. all their weights are $\le -2$. Using non-degeneracy of the intersection form on an admissible twig, T. Fujita [Fu, (6.12)] proved that there exists the unique such decomposition. Moreover, he proved that up to certain exceptions the global Zariski decomposition $K + {\~ D} = H + N$ provides the local one via the projection (see [Fu, (6.20-6.24); OZ, Theorem 1.2]). Here we do not use this result, and so we do not give its precise formulation. What we actually use is the equality $$(K_E + {\~ D}_E)^2 = H_E^2 + N_E^2\,.$$ According to [Fu, (6.16); OZ, 1.1, 2.4], the latter summands can be computed in terms of the weighted graph $\Gamma_{E,\,D'}$. This is done in the next section. \section{Graph discriminants and inductances} By definition, {\it the discriminant} $d(\Gamma)$ of a weighted graph $\Gamma$ is $\det(-A)$, where $A$ is the incidence matrix of $\Gamma$ (or the intersection matrix of $E$, if $\Gamma=\Gamma_E$). It is easily seen that $d(\Gamma_E) = 1$, because in our case $E$ is a contractible divisor. {\it The inductance} of a twig $T$ of $\Gamma$ is defined as $$ {\rm ind\,}(T) = {d(T-{\rm tip\,}(T)) \over d(T)}\,. $$ Denote by $T_1,\dots,T_k$ the twigs of $\Gamma_E$ which are not incident with $D'$ , i.e. the twigs of $\Gamma_{E,\,D'}$. Then we have \\ \noindent {\bf Lemma 1} [Fu, (6.16)]. {\it \noindent a) \begin{equation} -N_E^2 = \sum\limits_{i=1}^k {\rm ind\,}(T_i)\,.\end{equation} b) Let $v_T$ be the first vertex of $T = T_i\,(1\le i \le k)$, i.e. the vertex of $T$ opposite to the tip of $T$. Then the coefficient of $v_T$ in the decomposition of the divisor $N_E$ is equal to $1/d(T)$.} \\ Since the graph $\Gamma_E$ is a tree, for given vertices $E_i$ and $E_j$ (not necessary distinct) there is the unique shortest path in $\Gamma_E$ which joins them. Denote by $\Gamma_{ij}$ the weighted graph obtained from $\Gamma_E$ by deleting of this path together with the vertices $E_i$ and $E_j$ themselves (and, of course, with all their incident links). So, in general the graph $\Gamma_{ij}$ is disconnected. Let $B_E = (b_{ij}) = A_E^{-1}$ be the inverse of the intersection matrix $A_E$. The following formula can be easily obtained by applying the Cramer rule. \\ \noindent {\bf Lemma 2} [OZ, (2.1)]. \begin{equation} b_{ij} = -d(\Gamma_{ij})\,\,.\end{equation} Recall that $\bar\nu_i$ resp. $\nu_i$ denotes the valency of the vertex $E_i$ of the graph $\Gamma_{E,\,D'}$ resp. $\Gamma_E$ and $\mu = \mu_i$ denotes the Milnor number of the singular point $P=P_i \in D$. We have\\ \noindent {\bf Lemma 3} [OZ, (2.4), (2.7), (4.1)]. {\it In the notation as above \begin{equation} H_E^2 = \sum\limits_{\bar\nu_i>2,\,\bar\nu_j>2} b_{ij} c_i c_j \,,\end{equation} where $$c_i = (\bar\nu_i-2)-\sum {1\over d(T_j)}$$ and the last sum is taken over all the twigs $T_j$ which are incident with the vertex $E_i$; \begin{equation} (K_E+E)^2 = -2-\sum_{i=1}^n b_{ii}(\nu_i-2)\,,\end{equation} and \footnote{Whereas (17) is valid for any rational SNC-tree $E$ with non-degenerate intersection form and admissible twigs on a smooth surface, (18) and (19) are true only when $E$ is the exceptional divisor of the resolution of singularity of a plane curve germ.} \begin{equation} \mu = 1-\sum_{i,j} b_{ij}(\bar\nu_i-2)(\bar\nu_j-\nu_j)\,. \end{equation}} The proof of (17) is based on the Adjunction Formula and Lemmas 1,2; (18) is proven by induction on the number of blow-ups; (19) follows from the adjunction formula and the formula $$\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mu = 1 - D'_E(K_E + {\~ D}_E) \,.\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,(19')$$ \section{Calculus of graph discriminants} To use the formulas from the preceding section we have to compute the entries $b_{ij}$ of the inverse $B_E = A_E^{-1}$, where $A_E$ is the intersection matrix of the exceptional divisor $E$; that means to compute corresponding graph discriminats (see Lemma 2). This section provides us with the necessary tools. They were developed in the work of Dr\"ucker-Goldschmidt [DG] (cf. also [Ra, Ne, Fu, (3.6)]) and afterwords interpreted by S. Orevkov [OZ] in the following way. Let $\Gamma$ be a weighted graph and let $A = A_{\Gamma}$ be its intersection form. Recall that $d(\Gamma) = {\rm det}\,(-A)$. For a vertex $v$ of $\Gamma$ let $\partial_v \Gamma$ denotes the graph obtained from $\Gamma$ by deleting $v$ together with all its incident links. If $X$ is a subgraph of $\Gamma$ such that $v \notin X$, then $\partial_vX$ denotes the subgraph of $\Gamma$ which is obtained from $X$ by deleting all the vertices in $X$ closest to $v$ together with their links. In what follows we suppose that $\Gamma$ is a tree and $X$ is a subtree; in this case there is always the unique vertex in $X$ closest to $v$. Let $X_1,\dots,X_N$ be all the non--empty subtrees of $\Gamma$, and put $P = {\bf Z} [ X_1,\dots,X_N]$, where we identify $1$ with the empty subtree and regard the disjoint union of subtrees as their product. Then the dicriminant $d$ extends to a ring homomorphism $$d\,:\,P \to {\bf Z}$$ and $\partial_v$ generates a ring derivation $$\partial_v\,:\,P \to P\,.$$ Denote also $d_v(\Gamma) = d(\partial_v \Gamma)$ and $d_{vv}(\Gamma) = d(\partial_v\partial_v \Gamma)$. Let $a_v$ be the weight of a vertex $v \in \Gamma$ . \\ \noindent {\bf Proposition 1.} {\it Let the notation be as above, and let $\Gamma$ be a weighted tree. Then \noindent a) For any vertex $v \in \Gamma$ we have \begin{equation} d(\Gamma) = -a(v)d_v(\Gamma) - d_{vv} (\Gamma) \,.\end{equation} b) If $\Gamma$ is a linear tree with the end vertices $v$ and $w$, then \begin{equation} d_v(\Gamma)d_w(\Gamma) - d(\Gamma)d_{vw}(\Gamma) = 1\,. \end{equation} c) Let $[v,\,w]$ be a link of $\Gamma$. Put $\Gamma \setminus \,]\,v,\,w\,[ \,= \Gamma_1 \cup \Gamma_2$, where $v \in \Gamma_1$ and $w \in \Gamma_2$. Then \begin{equation} d(\Gamma) = d(\Gamma_1)d(\Gamma_2) - d_v(\Gamma_1)d_w(\Gamma_2)\,. \end{equation} d) Let $T$ be a twig of $\Gamma$ incident with a branch vertex $v_0$ of $\Gamma$, and let $v$ be the tip of $T$. Put $d_T (\Gamma) = d(\Gamma - T - v_0)$. Then \begin{equation} d_T (\Gamma) = d_v(\Gamma)d(T) - d(\Gamma)d_v(T)\,. \end{equation}} {\bf Corollary.} {\it Let $T$ be a twig of $\Gamma$ such that $d(T) \neq 0$. Assume that $d(\Gamma) = 1$. Denote $a = d_T (\Gamma)\, / \, d(T)$. Let $\[a\]$ be the least integer bigger than or equal to $a$ and $\]a\[ = \[a\] - a$ be the upper fractional part of $a$. Then, in the notation of (d) above, we have} \begin{equation} d_v(\Gamma) = \[a\] \,\,\, \,\,\, {\rm and}\,\,\, \,\,\,{\rm ind\,}(T) = \]a\[ \,. \end{equation} \section{Puiseux data as graph discriminants $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,$ (after Eisenbud and Neumann)} Let $(C, \, 0)$ be a germ of an irreducible analytic curve, and let $$ x=t^m,\,\,\,\,\,\,\,\sp y=a_n t^n+a_{n+1}t^{n+1}+... ,\,\,\,\,\,\,\,\sp a_n\ne 0 \,, $$ be its analytic parametrization. We may assume that $m<n$ and $m$ does not divide $n$. Folowing [A] set $d_1 = m \,,\, m_1 = n;$ $$ d_i = \gcd( d_{i-1}, m_{i-1}) \,,\,\,\,\,\,\,\, m_i = \min \{\, j \mid a_j \ne 0 \,\,\,\,{\rm and}\,\,\,\, j \not\equiv 0 \,\,\,({\rm mod}\,d_i)\,\} \,,\,\, i>1\,. $$ Let $h$ be such that $d_h \ne 1$, $d_{h+1} =1$. Thus, $m_i$ resp. $d_i$ are defined for $i = 1, ..., h$ resp. for $i = 1, ..., h+1$, and $$ 0< n = m_1<m_2<...<m_h, \,\,\,\,\,\,\,\sp m=d_1>d_2>...>d_{h+1}=1\,\,. $$ Set $q_1 = m_1,\,\, q_i = m_i - m_{i-1}$ for $i = 2, ..., h$, and \begin{equation} r_i = (q_1 d_1 +...+ q_i d_i)/d_i, \,\,\, i=1,...,h\,\,. \end{equation} The sequence $(m; \,m_1,m_2,...,m_h)$ is called {\it the Puiseux characteristic sequence of the singularity $(C,\,0)$} [A, Mil]. The whole collection $(m_i),\,(d_i),\,(q_i),\,(r_i)$ we call {\it the Puiseux data}. We have the following \\ \noindent {\bf Proposition 2} [EN]. {\it a) Let $X \to {\bf C}^2$ be the embedded minimal resolution of the singularity $(C,\,0)$ with the exceptional divisor $E = \cup E_i$. The proper preimage of $C$ in $X$ we denote by the same letter. Then the dual graph $\Gamma_{E,\,C}$ of $E$ near $C$ looks like $$ \begin{picture}(1000,90) \put(66,82){$E_0$} \put(70,70){\circle{5}} \put(73,70){\line(1,0){50}} \put(120,82){$E_{h+1}$} \put(125,70){\circle{5}} \put(128,70){\line(1,0){50}} \put(175,82){$E_{h+2}$} \put(180,70){\circle{5}} \put(183,70){\line(1,0){50}} \put(244,70){$\ldots$} \put(271,70){\line(1,0){50}} \put(317,82){$E_{2h}$} \put(322,70){\circle{5}} \put(325,70){\vector(1,0){48}} \put(372,82){$C$} \put(125,68){\line(0,-1){50}} \put(125,15){\circle{5}} \put(120,-3){$E_1$} \put(180,68){\line(0,-1){50}} \put(180,15){\circle{5}} \put(175,-3){$E_2$} \put(322,68){\line(0,-1){50}} \put(322,15){\circle{5}} \put(317,-3){$E_h$} \end{picture} $$ where the edges mean linear chains of vertices of valency two, which are not shown. \\ b) Denote by $R_i$, $D_i$ and $S_i$ the connected components of $\Gamma_{E,\,C}-E_{h+i}$ which are to the left, to the bottom and to the right of the node $E_{h+i}$, respectively. Denote by $Q_i$ the linear chain between $E_{h+i-1}$ and $E_{h+i}$ (excluding $E_{h+i-1}$ and $E_{h+i}$). Then} $$ d(R_i)={r_i\over d_{i+1}}\,,\,\,\,\,\,\,\, d(D_i)={d_i\over d_{i+1}}\,,\,\,\,\,\,\,\, d(S_i)=1\,, \,\,\,\,\,\,\, d(Q_i)={q_i\over d_{i+1}}\,\, .\,\,\,\,\,\,\, $$ This graph is usually called {\it a comb} (see e.g. [FZ1]); M. Miyanishi suggested more pleasant name {\it a Christmas tree} (in this case it is drown in a slightly different manner). \section{Expressions of the local BMY-ingredients via the Puiseux data} \noindent {\bf Proposition 3} [OZ, (5.2), (5.4)]. {\it Let $(C,\,0)$ be the local branch of $D$ at a cusp $P = P_i$ of $D$. Then in the notation of Sections 6 and 9 we have \begin{equation} \mu = 1 - d_1 + \sum\limits_{i=1}^h r_i ({d_i\over d_{i+1}} -1) = 1 - d_1 + \sum_{i=1}^h q_i (d_i -1)\,; \end{equation} \begin{equation} 2\mu+H_E^2 = -{d_1\over r_1} +\sum_{i=1}^h {r_i\over d_{i+1}}({d_i\over d_{i+1}}-{d_{i+1}\over d_i}) = -{d_1\over q_1} +\sum_{i=1}^h q_i(d_i-{1\over d_i})\,;\end{equation} \begin{equation} - N_E^2 = \]{d_1\over r_1}\[ + \sum_{i=1}^h \]{r_i\over d_i}\[ = \[{d_1\over r_1}\] - {d_1\over r_1} +\sum_{i=1}^h (\[{r_i\over d_i}\] - {r_i\over d_i})\,; \end{equation} \begin{equation} 2\mu+(K_E+{\~ D}_E)^2 = -\[{d_1\over r_1}\] +\sum_{i=1}^h ({r_i d_i\over d_{i+1}^2}-\[{r_i\over d_i}\])\,\,. \end{equation} In particular, if $m$ and $n$ are coprime (i.e. there is the only one Puiseux characteristic pair), then $\rm (cf.\,\, [Mil,\, p. \,95])$ $\mu = (m-1)(n-1)$ and \begin{equation} -H_E^2 = (m-2)(n-2) + (m-n)^2/mn \,,\,\, \,\,\,\, - N_E^2 = \]{m\over n}\[ + \]{n\over m}\[\,\,\,.\end{equation}} The proof is based on the formulas in Lemma 3, where the corresponding entries $b_{ij}$ have been expressed in terms of the Puiseux data as it is done in Proposition 2 above, by using the graph discriminant calculus from Section 8. \section{Local inequality for irreducible $\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,$ singularities} Now we can prove the local inequality (14) for a cusp $P=P_i \in D$. As above, we regard the local branch of $D$ at $P=P_i$ as a germ $(C,\,0)$ of an analytic curve. \\ \noindent {\bf Theorem 2} [OZ, (6.2)]. {\it In the notation as above, for an irreducible plane curve germ $(C,\,0)$ one has \begin{equation} -(K_E + {\~ D}_E)^2 \le (1 - {1 \over m})\,\mu\,,\end{equation} where $m$ is the multiplicity and $\mu$ is the Milnor number of $(C,\,0)$. The equality in (31) holds iff $m=2$.} \\ The proof proceeds as follows. (31) is equivalent to the inequality $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mu + (K_E+ {\~ D}_E)^2 - {\mu\over m} \ge 0\,. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(31')$$ Using (25) and (26--29) we can express the quantity at the left hand side as \begin{equation} \mu + (K_E+ {\~ D}_E)^2 - {\mu\over m} = d_1(1-{1\over q_1})-{1\over d_1} + N_E^2 + \sum_{j=1}^h q_j(1-{d_j\over d_1})(1-{1\over d_j}) \,.\end{equation} It is easily varified that this quantity vanishes when $m = 2$. The last sum in (32) is always positive. Let us show that for $m > 2$ the rest at the right hand side of (32) is also positive. Indeed, by (28) we have $${d_1 \over q_1} - N_E ^2 = \[{d_1\over q_1}\] +\sum_{i=1}^h \]{r_i\over d_i}\[ < \[{m \over n}\] + h\ = 1+h\,.$$ Thus, it is enough to show that $d_1 - {1 \over d_1} - (1 +h) = m - {1 \over m} - (1+h) > 0$. It is true for $m \ge 4$ because $h\le\log_2m$; it is also true for $m = 3$ because then $h = 1$, and we are done. \\ \noindent {\bf Remark.} The estimate in Theorem 2 is asymptotically sharp in the following sense. For any positive integer $m$ and for any $\epsilon>0$ there exists an irreducible curve germ $(C,\,0)$ of multiplicity $m$ such that $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\mu + (K_E+ {\~ D}_E)^2 < \mu + H_E^2<(1+\epsilon)\mu/m\,\,. \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Indeed, consider the curve {$x^m = y^n$}, where gcd$(m,\,n) = 1$ and $n \gg m$. \section{Local inequality for arbitrary singularities} Here we prove (31) in general case when $(C,\,0)$ is not supposed to be irreducible, combining our approach with those of F. Sakai [Sa] (see the discussion at the end of Section 3). Let $r$ be the number of local branches of $C$, and let $$(m_1=m, \,m_2, \dots, m_n, {\underbrace{1,\dots,1}_{r}})$$ be {\it the multiplicity sequence} of $(C,\,0)$, i.e. the sequence of multiplicities of $(C,\,0)$ in all its infinitely near points (where $n$ is the total number of blow ups in the resolution process). Recall [Mil] that \begin{equation}\mu + r - 1 = \sum\limits_{j=1}^n m_j(m_j - 1)\,.\end{equation} Remind also that the blow-up at an infinitely near point which belongs to only one irreducible component of the exceptional divisor is called {\it sprouting} or {\it outer} blow-up, and the other blow-ups are called {\it subdivisional} or {\it inner} [MaSa, FZ1]. Following [Sa] denote by $\omega$ the number of subdivisional blow--ups, and set $$ \eta = \sum\limits_{j=1}^n (m_j -1)\,.$$ \noindent {\bf Lemma 4.} {\it In the notation as above, the following identities hold:} \begin{equation} -E^2 = \omega\,,\,\,\,\,\,\,EK_E = \omega - 2\,,\,\,\,\,\,\, E^2 + EK_E = -2 \,,\,\,\,\,\,\,\, -K_E^2=n \end{equation} \begin{equation} ED_E' = r \,,\,\,\,\,\,\, K_ED_E' = \sum\limits_{j=1}^n m_j \,,\,\,\,\,\, \, K_E(K_E + D_E') = \eta \end{equation} \begin{equation} -D_E'^2 = \sum\limits_{j=1}^n m_j^2 \,,\,\,\,\,\,\, -D_E'(K_E + D_E') = \sum\limits_{j=1}^n m_j(m_j -1) = 2 \delta\end{equation} \begin{equation} \mu + (K_E + {\~ D}_E)^2 = (\eta - 1) + (\omega - 1) + (r - 1) \end{equation} \noindent {\bf Proof.} The third equality in (34) resp. the second one in (36) immediately follows from the preceding ones. The first equality in (35) is evident. The other formulas in (34) - (36) are proven by an easy induction by the number of steps in the resolution process (cf. [MaSa, Lemma 2] for the first equality in (34)). To prove (37), transform its left hand side by using (19') and (34)--(36) as follows: $$ \mu + (K_E + {\~ D}_E)^2 = 1 + (K_E + {\~ D}_E)(K_E + E) = 1 + (K_E + D_E' + E)(K_E + E)$$ $$ = 1 + K_E (K_E + D_E') + 2 EK_E + E^2 + ED_E' = \eta + \omega + r - 3\,.$$ \hfill $\Box$ \\ The next theorem is a generalization of Theorem 2 to the case when $(C,\,0)$ is not necessarily irreducible. \\ \noindent {\bf Theorem 3.} {\it The inequality (31) is valid for any singular plane curve germ $(C,\,0)$, with the equality sign only for an irreducible singularity of multiplicity two.} \\ \noindent {\bf Proof.} Replace (31) by the equivalent inequality ($31'$). Applying (37) we obtain one more equivalent form of (31): \begin{equation} \eta + \omega + r - 3 \ge {\mu \over m}\,. \end{equation} This inequality was proven in [Sa]. For the sake of completeness we remind here the proof. From (33) it follows that \begin{equation} \eta \ge {\mu + r - 1 \over m}\,,\,\,\,\,\,\,{\rm or}\,\,\, \,\,\,\eta - {\mu \over m} \ge {r-1 \over m}\,. \end{equation} Therefore, it is enough to proof the inequality \begin{equation} \omega + r - 3 + {r-1 \over m} \ge 0\,, \end{equation} which is in turn a consequence of the following one \begin{equation} \omega + r \ge 3\,. \end{equation} Notice, following [Sa], that $\omega \ge 2$ as soon as at least one irreducible branch of $C$ at $0$ is singular, and $\omega = 1$ otherwise. But in the latter case $r \ge 2$, because $C$ is assumed being singular. This proves (41), and thus also (31). Due to (41) the inequality (40), and hence also (31), is strict if $r > 1$. In the case when $r = 1$ by Theorem 2 the equality sign in (31) corresponds to $m = 2$. This completes the proof. \hfill $\Box$ \section{On the rigidity problem for rational $\,\,\,\,\,\,\,\,\,\,\,\,$ cuspidal plane curves} Let $Y$ be a a smooth affine algebraic surface $/{\bf C}$. Assume that $Y$ is ${\bf Q}$--acyclic, i.e. $H_i (Y;\,{\bf Q}) = 0$ for all $i > 0$, and that $Y$ is of log--general type, i.e. ${\bar k}(Y) = 2$. In [FZ1] the problem was posed whether such a surface should be rigid. The latter means that $h^1 (\Theta_X\langle\,\~ D\,\rangle)=0$, where $X$ is a minimal smooth completion of $Y$ by a simple normal crossing divisor ${\~ D}$ and $\Theta_X\langle\,\~ D\,\rangle$ is the logarithmic tangent bundle of $X$ along $\~ D$. The rigidity holds in all known examples of ${\bf Q}$--acyclic surfaces of log--general type [FZ1]. Moreover, in all those examples $Y$ (or, more precisely, the logarithmic deformations of $Y$, see [FZ1]) is unobstructed, i.e. $h^2 (\Theta_X\langle\,\~ D\,\rangle)=0$, and therefore also the holomorphic Euler characteristic $\chi(\Theta_X\langle\,\~ D\,\rangle)$ vanishes (indeed, by Iitaka's Theorem [Ii, Theorem 6] $h^0 (\Theta_X\langle\,\~ D\,\rangle)=0$ as soon as $Y$ is of log--general type). We have the identity $\chi(\Theta_X\langle\,\~ D\,\rangle) = K(K + {\~ D})$ [FZ1, Lemma 1.3(5)], where $K = K_X$. Since ${\~ D}$ is a curve of arithmetic genus zero [FZ1, Lemma 1.2], the equality $\chi(\Theta_X\langle\,\~ D\,\rangle) = K(K + D) = 0$ is equivalent to the following one \begin{equation}(K + {\~ D})^2 = -2\,.\end{equation} Thus, if $Y$ is unobstructed, then it is rigid iff (42) holds. Consider now an irreducible plane curve $D$. It is easily seen (cf. [Ra]) that $Y = {\bf P}^2 \setminus D$ is a ${\bf Q}$--acyclic surface iff $D$ is a rational cuspidal curve. Furthermore, if $D$ has at least three cusps, then $Y$ is of log--general type [Wak]. The rigidity of $Y$ is equivalent to $D$ being projectively rigid in the following sense: any small deformation of $D$, which is a plane rational cuspidal curve with the same types of cusps (i.e. an equisingular embedded deformation), is projectively equivalent to $D$ [FZ2, (2.1)]. Once again, the rigidity holds in all known examples [FZ2, (3.3)], as well as the equality in (42) [FZ2, (2.1)]. Here we prove the following \\ \noindent {\bf Proposition 4.} {\it A projectively rigid rational cuspidal plane curve cannot have more than 9 cusps.} \\ Before giving the proof we remind the notation. Let $\sigma\,:\,X \to {\bf P}^2$ be the minimal embedded resolution of singularities of $D$, $K = K_X$ be the canonical divisor, ${\~ D} = \sigma^{-1}(D)$ and $K+{\~ D} = H+N$ be the Zariski decomposition. For a fixed cusp $P \in {\rm Sing}\,D$ let $K_E + {\~ D}_E = H_E + N_E$ be the local Zariski--Fujita decomposition, where $E = \sigma^{-1}(P)$ is the exceptional divisor. The proof of Proposition 4 is based on the following two observations.\\ \noindent {\bf Lemma 5.} {\it For the negative part $N_E$ of the local Zariski--Fujita decomposition over a cusp $P \in D$ the inequality $-N_E^2 > 1/2$ holds. } \\ \noindent {\bf Proof.} From (28) it follows that \footnote{ Recall that $\]a\[$ denotes $\[a\]-a$, where $\[a\] :=\min\{n\in{\bf Z}\,|\,n\ge a\}$.} \begin{equation} - N_E^2 \,= \,\]{d_1\over r_1}\[ + \sum_{i=1}^h \]{r_i\over d_i}\[\, \,\le \,\,\]{d_1\over r_1}\[ + \]{r_1\over d_1}\[ \,= \, \]{m\over n}\[ + \]{n \over m}\[ \,.\end{equation} By the definition of the Puiseux sequence (see section 9) we have $0 < {m \over n} < 1$ and ${m \over n} \neq {1 \over 2}$. Thus, the desired inequality follows from (43) and the next estimate, which is an easy exercise. \hfill $\Box$ \\ \noindent {\bf Claim.} {\it If $0< x <1$, then $\]x\[ + \]{1 \over x}\[ \ge {1 \over 2}$, where the equality holds only for $x = {1 \over 2}$. }\\ \noindent {\bf Lemma 6.} {\it Let $D$ be a rational cuspidal plane curve with at least three cusps. Then in the notation as above we have \begin{equation} H = (d-3)L' + \sum_{P \in {\rm Sing}\,D} H_E\,\,\,{\rm and }\,\,\,N = \sum_{P \in {\rm Sing}\,D} N_E,\,\end{equation} i.e. the global Zariski decomposition agrees with the local Zariski--Fujita ones.} \\ \noindent {\bf Proof.} By [Wak] we have ${\bar k}(Y) = 2$, where $Y = {\bf P}^2 \setminus D = X \setminus {\~ D}$. Thus, being a smooth ${\bf Q}$--acyclic surface of log--general type, $Y$ does not contain any simply connected curve (this was first proven in [Za] for acyclic surfaces and then generalized in [MT] to ${\bf Q}$--acyclic ones). In particular, $X$ does not contain any $(-1)$--curve $C$ with $C\cdot {\~ D} = 1$. Since $D$ has at least three cusps, the dual graph $\Gamma_{{\~ D}}$ of ${\~ D}$ has at least three branching points. Under these conditions the lemma follows from the results in [Fu, (6.20-6.24)] (see also [OZ, Theorem 1.2]). \hfill $\Box$ \\ \noindent {\bf Proof of Proposition 4.} Let $\kappa$ be the number of cusps of $D$. Evidently, we may suppose that $\kappa \ge 3$. It follows from Lemmas 5 and 6 that $$(K+{\~ D})^2 = H^2 + N^2 = H^2 + \sum_{P \in {\rm Sing}\,D} N_E^2 < H^2 - {1 \over 2}\,\kappa\,.$$ Due to BMY-inequality (13) we also have $H^2 \le 3$, and hence $$(K+{\~ D})^2 < 3 - {1 \over 2}\,\kappa\,.$$ Set $h^i = h^i (\Theta_X\langle\,\~ D\,\rangle)\,,\,\,i = 0,\,1,\,2$. The surface $Y = {\bf P}^2 \setminus D$ being of log--general type [Wak], by Iitaka's Theorem [Ii, Theorem 6] we have $h^0 = 0$. Since $D$ is assumed to be rigid, i.e. $h^1 = 0$, we also have $\chi(\Theta_X\langle\,\~ D\,\rangle) = h^2 = K(K+{\~ D}) \ge 0$, i.e. $(K+{\~ D})^2 \ge -2$. It follows that \begin{equation} \kappa < 6 - 2(K+{\~ D})^2 \le 10 \,,\end{equation} which completes the proof. \hfill $\Box$ \\ \noindent {\bf Remark.} Actually, for a rational cuspidal plane curve with at least three cusps we have proved the inequality \begin{equation} \kappa < 6 - 2(K+{\~ D})^2 = 10 - 2K(K+{\~ D}) \,.\end{equation} Therefore, $\kappa < 10$ as soon as $K(K+{\~ D}) \ge 0$, which is the case if $D$ is rigid. \vspace{.2in} \begin{center} {\bf REFERENCES} \end{center} \vspace{.2in} {\footnotesize \noindent [A] S. S. Abhyankar. Expansion technique in algebraic geometry. {\it Tata Inst. of Fund. Res.}, Bombay, 1977 \\ \noindent [BK] E. Briskorn, H. Kn\"orrer. Plane algebraic curves. {\it Birkh\"auser-Verlag,} Basel e.a., 1986 \\ \noindent [DG] D. Drucker, D.M. Goldschmidt. Graphical evaluation of sparce determinants. {\it Proc. Amer. Math. Soc.}, {\bf 77} (1979), 35 - 39\\ \noindent [EN] D. Eisenbud, W. D. Neumann. Three-dimensional link theory and invariants of plane curve singularities. {\it Ann.Math.Stud.} {\bf 110}, {\it Princeton Univ. Press}, Princeton 1985\\ \noindent [FZ1] H. Flenner, M. Zaidenberg. $\bf Q$--acyclic surfaces and their deformations. {\it Contemporary Mathem.} {\bf 162} (1964), 143--208 \\ \noindent [FZ2] H. Flenner, M. Zaidenberg. On a class of rational cuspidal plane curves. {\it Preprint} (1995), 1--28 \\ \noindent [Fu] T. Fujita. On the topology of non-complete algebraic surfaces, {\it J. Fac. Sci. Univ. Tokyo (Ser 1A)}, {\bf 29} (1982), 503--566 \\ \noindent [Hi] A. Hirano. Construction of plane curves with cusps, {\it Saitama Math. J.} {\bf 10} (1992), 21--24 \\ \noindent [H] F. Hirzebruch. Singularities of algebraic surfaces and characteristic numbers, {\it The Lefschetz Centennial Conf. Part.I (Mexico City 1984), Contemp. Math.} {\bf 58} (1985), 141-155 \\ \noindent [Ii] Sh. Iitaka. On logarithmic Kodaira dimension of algebraic varieties. In: {\it Complex Analysis and Algebraic Geometry, Cambridge Univ. Press}, Cambridge e.a., 1977, 175--190\\ \noindent [Iv] K. Ivinskis, Normale Fl\"achen und die Miyaoka--Kobayashi Ungleichung. {\it Diplomarbeit}, Bonn, 1985 \\ \noindent [KoNaSa] R. Kobayashi, S. Nakamura, F. Sakai. A numerical characterization of ball quotients for normal surfaces with branch loci. {\it Proc. Japan Acad.} {\bf 65(A)} (1989), 238--241 \\ \noindent [Ko] R. Kobayashi. An application of K\"ahler--Einstein metrics to singularities of plane curves. {\it Advanced Studies in Pure Mathem., Recent Topics in Differential and Anal. Geom.} {\bf 18-I} (1990), 321--326 \\ \noindent [MaSa] T. Matsuoka, F. Sakai. The degree of rational cuspidal curves. {\it Math. Ann.} {\bf 285} (1989), 233--247\\ \noindent [Mil] J. Milnor. Singular points of complex hypersurfaces. {\it Ann.Math.Stud.} {\bf 61}, {\it Princeton Univ. Press}, Princeton, 1968 \\ \noindent [MT] M. Miyanishi, S. Tsunoda. Abscence of the affine lines on the homology planes of general type. {\it J. Math. Kyoto Univ.} {\bf 32} (1992), 443--450 \\ \noindent [Miy] Y. Miyaoka. The minimal number of quotient singularities on surfaces with given numerical invariants, {\it Math. Ann.} {\bf 268} (1984), 159--171 \\ \noindent [Na] M. Namba. Geometry of projective algebraic curves. {\it Marcel Dekker}, N.Y. a.e., 1984 \\ \noindent [Ne] W.D. Neumann. On bilinear forms represented by trees. {\it Bull. Austral. Math. Soc.} {\bf 40} (1989), 303-321 \\ \noindent [OZ] S.Y. Orevkov, M.G. Zaidenberg. Some estimates for plane cuspidal curves. In: {\it Journ\'ees singuli\`eres et jacobiennes, Grenoble 26--28 mai 1993.} Grenoble, 1994, 93--116 (see also Preprint MPI/92-63, 1992, 1--13)\\ \noindent [Ra] C.P. Ramanujam. A topological characterization of the affine plane as an algebraic variety. {\it Ann. Math.} 94 (1971), 69-88 \\ \noindent [Sa] F. Sakai. Singularities of plane curves. {\it Preprint} (1990), 1-10\\ \noindent [Va] A.N. Varchenko. Asymptotics of integrals and Hodge structures. In: {\it Itogi Nauki i Techniki, Series "Contempor. Problems in Mathem."} {\bf 22} (1983), 130--166 (in Russian)\\ \noindent [Wak] I. Wakabayashi. On the logarithmic Kodaira dimension of the complement of a curve in ${\bf P}^2$. {\it Proc. Japan Acad.} {\bf 54(A)} (1978), 157--162 \\ \noindent [Wal] R. J. Walker. Algebraic curves. {\it Princeton Univ. Press}, Princeton, 1950 \\ \noindent [Yo1] H. Yoshihara. Plane curves whose singular points are cusps. {\it Proc. Amer. Math. Soc.} {\bf 103} (1988), 737--740 \\ \noindent [Yo2] H. Yoshihara. Plane curves whose singular points are cusps and triple coverings of ${\bf P}^2$. {\it Manuscr. Math.} {\bf 64} (1989), 169-187 \\ \noindent [Za] M. Zaidenberg. Isotrivial families of curves on affine surfaces and characterization of the affine plane. {\it Math. USSR Izvestiya} {\bf 30} (1988), 503-531; Addendum, {\it ibid.} {\bf 38} (1992), 435--437 \vspace{.2in} \noindent Stepan Orevkov\\ System Research Institute RAN\\ Moscow, Avtozavodskaja 23, Russia\\ e-mail: [email protected] \vspace{.2in} \noindent Mikhail Zaidenberg\\ Universit\'{e} Grenoble I \\ Laboratoire de Math\'ematiques associ\'e au CNRS\\ BP 74\\ 38402 St. Martin d'H\`{e}res--c\'edex, France\\ e-mail: [email protected]} \end{document}

9612

alg-geom/9612004

en

https://arxiv.org/abs/alg-geom/9612004

alg-geom/9612004

Ezra Getzler

Ezra Getzler (Northwestern University)

Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten invariants

25 pages. amslatex-1.2. This is the revised version which will appear in J. Amer. Math. Soc

MPI 96-161

The WDVV equation is satisfied by the genus 0 correlation functions of any topological field theory in two dimensions coupled to topological gravity, and may be used to determine the genus 0 (rational) Gromov-Witten invariants of many projective varieties (as was done for projective spaces by Kontsevich). In this paper, we present an equation of a similar universal nature for genus 1 (elliptic) Gromov-Witten invariants -- however, it is much more complicated than the WDVV equation, and its geometric significance is unclear to us. (Our prove is rather indirect.) Nevertheless, we show that this equation suffices to determine the elliptic Gromov-Witten invariants of projective spaces. In a sequel to this paper, we will prove that this equation is the only one other than the WDVV equation which relates elliptic and rational correlation functions for two-dimensional topological field theories coupled to topological gravity. It is unclear if there are any further equations of this type on the small phase space in higher genus, but we think it unlikely. (The genus 0 and 1 cases are special, since the correlation functions on the small phase space determine those on the large phase space.)

alg-geom

\section{Intersection theory on $\overline{\mathcal{M}}_{1,4}$} In this section, we calculate the relations among certain codimension two cycles in $\overline{\mathcal{M}}_{1,4}$; one such relation was known, and we find that there is one new one. First, we assign names to the codimension $1$ strata of $\overline{\mathcal{M}}_{1,4}$. Denote by $\Delta_0$ the boundary stratum of irreducible curves in $\overline{\mathcal{M}}_{1,4}$, associated to the stable graph $$ \begin{picture}(80,45)(40,745) \put( 20,765){$\Delta_0 =$} \put( 80,775){\circle{30}} \put( 80,760){\line(-3,-4){ 15}} \put( 80,760){\line(-1,-4){ 5}} \put( 80,760){\line( 1,-4){ 5}} \put( 80,760){\line( 3,-4){ 15}} \end{picture} $$ For each subset $S$ of $\{1,2,3,4\}$ of cardinality at least $2$, let $\Delta_S$ be the boundary stratum associated to the stable graph with two vertices, of genus $0$ and $1$, one edge connecting them, and with those tails labelled by elements of $S$ attached to the vertex of genus $0$; there are $11$ such graphs. In our pictures, we denote genus $1$ vertices by a hollow dot, leaving genus $0$ vertices unmarked. For example, $$ \begin{picture}(35,80)(60,722) \put( 20,760){$\Delta_{\{1,2\}} =$} \put( 80,780){\circle{5}} \put( 80,777){\line( 0,-1){ 38}} \put( 80,740){\line(-2,-3){ 10}} \put( 80,740){\line( 2,-3){ 10}} \put( 83,782){\line( 2, 3){ 10}} \put( 77,782){\line(-2, 3){ 10}} \put( 65,715){$1$} \put( 89,715){$2$} \put( 65,800){$3$} \put( 89,800){$4$} \end{picture} $$ We only need the three $\SS_4$-invariant combinations of these $11$ strata, which are as follows: \begin{align*} \Delta_2 &= \Delta_{\{1,2\}} + \Delta_{\{1,3\}} + \Delta_{\{1,4\}} + \Delta_{\{2,3\}} + \Delta_{\{2,4\}} + \Delta_{\{3,4\}} , \\ \Delta_3 &= \Delta_{\{1,2,3\}} + \Delta_{\{1,2,4\}} + \Delta_{\{1,3,4\}} + \Delta_{\{2,3,4\}} , \\ \Delta_4 &= \Delta_{\{1,2,3,4\}} . \end{align*} In summary, there are four invariant combinations of boundary strata: $\Delta_0$, $\Delta_2$, $\Delta_3$ and $\Delta_4$. We now turn to enumeration of the codimension two strata. These fall into two classes, distinguished by whether they are contained in the irreducible stratum $\Delta_0$ or not. We start by listing those which are not; each of them is the intersection of a pair of boundary strata $\Delta_S\*\Delta_T$. We give four examples: from these, the other strata may be obtained by the action of $\SS_4$: $$ \begin{picture}(100,95)(30,715) \put(-20,755){$\Delta_{\{1,2\}} \* \Delta_{\{3,4\}} =$} \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,762){\line( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 10}} \put( 80,780){\line( 1, 2){ 10}} \put( 80,780){\line(-1, 2){ 10}} \put( 67,803){$1$} \put( 87,803){$2$} \put( 67,710){$3$} \put( 87,710){$4$} \end{picture} \begin{picture}(100,85)(-20,700) \put(-30,740){$\Delta_{\{1,2\}} \* \Delta_{\{1,2,3\}} =$} \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,762){\line( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} \put( 56,690){$1$} \put( 77,690){$2$} \put( 87,710){$3$} \put( 77,783){$4$} \end{picture} $$ $$ \begin{picture}(100,75)(30,690) \put(-40,740){$\Delta_{\{1,2\}} \* \Delta_{\{1,2,3,4\}} =$} \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 56,690){$1$} \put( 77,690){$2$} \put( 77,710){$3$} \put( 87,710){$4$} \end{picture} \begin{picture}(100,75)(-40,690) \put(-50,740){$\Delta_{\{1,2,3\}} \* \Delta_{\{1,2,3,4\}} =$} \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} \put( 70,720){\line( 0,-1){ 20}} \put( 56,690){$1$} \put( 67,690){$2$} \put( 77,690){$3$} \put( 87,710){$4$} \end{picture} $$ The $\SS_4$-invariant combinations of these strata are as follows: \begin{align*} \Delta_{2,2} &= \Delta_{\{1,2\}} \* \Delta_{\{3,4\}} + \Delta_{\{1,3\}} \* \Delta_{\{2,4\}} + \Delta_{\{1,4\}} \* \Delta_{\{2,3\}} , \\ \Delta_{2,3} &= \Delta_{\{1,2\}} \* \Delta_{\{1,2,3\}} + \Delta_{\{1,2\}} \* \Delta_{\{1,2,4\}} + \Delta_{\{1,3\}} \* \Delta_{\{1,2,3\}} + \Delta_{\{1,3\}} \* \Delta_{\{1,3,4\}} \\ & + \Delta_{\{1,4\}} \* \Delta_{\{1,2,4\}} + \Delta_{\{1,4\}} \* \Delta_{\{1,3,4\}} + \Delta_{\{2,3\}} \* \Delta_{\{1,2,3\}} + \Delta_{\{2,3\}} \* \Delta_{\{2,3,4\}} \\ & + \Delta_{\{2,4\}} \* \Delta_{\{1,2,4\}} + \Delta_{\{2,4\}} \* \Delta_{\{2,3,4\}} + \Delta_{\{3,4\}} \* \Delta_{\{1,3,4\}} + \Delta_{\{3,4\}} \* \Delta_{\{2,3,4\}} , \\ \Delta_{2,4} &= \Delta_{\{1,2\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{1,3\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{1,4\}} \* \Delta_{\{1,2,3,4\}} \\ & + \Delta_{\{2,3\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{2,4\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{3,4\}} \* \Delta_{\{1,2,3,4\}} , \\ \Delta_{3,4} &= \Delta_{\{1,2,3\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{1,2,4\}} \* \Delta_{\{1,2,3,4\}} \\ & + \Delta_{\{1,3,4\}} \* \Delta_{\{1,2,3,4\}} + \Delta_{\{2,3,4\}} \* \Delta_{\{1,2,3,4\}} . \end{align*} Each of the intersections $\Delta_0\*\Delta_S$ is a codimension two stratum in $\Delta_0$; for example $$ \begin{picture}(80,80)(60,690) \put(0,740){$\Delta_0\*\Delta_{\{1,2\}} =$} \put( 80,755){\circle{30}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 56,690){$1$} \put( 77,690){$2$} \put( 77,710){$3$} \put( 87,710){$4$} \end{picture} \begin{picture}(80,80)(0,690) \put(-10,740){$\Delta_0\*\Delta_{\{1,2,3\}} =$} \put( 80,755){\circle{30}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} \put( 70,720){\line( 0,-1){ 20}} \put( 56,690){$1$} \put( 67,690){$2$} \put( 77,690){$3$} \put( 87,710){$4$} \end{picture} \begin{picture}(90,80)(-60,690) \put(-15,740){$\Delta_0\*\Delta_{\{1,2,3,4\}} =$} \put( 80,755){\circle{30}} \put( 80,740){\line( 0,-1){ 20}} \put( 80,720){\line(-3,-4){ 15}} \put( 80,720){\line(-1,-4){ 5}} \put( 80,720){\line( 1,-4){ 5}} \put( 80,720){\line( 3,-4){ 15}} \put( 61,690){$1$} \put( 72,690){$2$} \put( 83,690){$3$} \put( 93,690){$4$} \end{picture} $$ {}From these, we may form the $\SS_4$-invariant combinations \begin{align*} \Delta_{0,2} &= \Delta_0\*\Delta_{\{1,2\}} + \Delta_0\*\Delta_{\{1,3\}} + \Delta_0\*\Delta_{\{1,4\}} \\ & + \Delta_0\*\Delta_{\{2,3\}} + \Delta_0\*\Delta_{\{2,4\}} + \Delta_0\*\Delta_{\{3,4\}} , \\ \Delta_{0,3} &= \Delta_0 \* \Delta_{\{1,2,3\}} + \Delta_0 \* \Delta_{\{1,2,4\}} + \Delta_0 \* \Delta_{\{1,3,4\}} + \Delta_0 \* \Delta_{\{2,3,4\}} , \\ \Delta_{0,4} &= \Delta_0 \* \Delta_{\{1,2,3,4\}} . \end{align*} There remain seven strata which are not expressible as intersections, which we denote by $\Delta_{\alpha,i}$, $1\le i\le 4$, and $\Delta_{\beta,12|34}$, $\Delta_{\beta,13|24}$ and $\Delta_{\beta,14|24}$. We illustrate the stable graphs for two of these strata: $$ \begin{picture}(80,100)(40,720) \put( 20,775){$\Delta_{\alpha,1} =$} \put( 80,775){\circle{30}} \put( 80,760){\line(-1,-2){ 10}} \put( 80,760){\line( 0,-1){ 20}} \put( 80,760){\line( 1,-2){ 10}} \put( 80,790){\line( 0, 1){ 20}} \put( 77,815){$1$} \put( 66,727){$2$} \put( 77,727){$3$} \put( 87,727){$4$} \end{picture} \begin{picture}(100,100)(-20,720) \put( 05,775){$\Delta_{\beta,12|34}=$} \put( 80,775){\circle{30}} \put( 80,760){\line(-1,-2){ 10}} \put( 80,760){\line( 1,-2){ 10}} \put( 80,790){\line(-1, 2){ 10}} \put( 80,790){\line( 1, 2){ 10}} \put( 66,815){$1$} \put( 87,815){$2$} \put( 66,727){$3$} \put( 87,727){$4$} \end{picture} $$ Denote by $\Delta_\alpha$ and $\Delta_\beta$ the $\SS_4$-invariant combinations of strata: $$ \Delta_\alpha = \Delta_{\alpha,1} + \Delta_{\alpha,2} + \Delta_{\alpha,3} + \Delta_{\alpha,4} , \quad \Delta_\beta = \Delta_{\beta,12|34} + \Delta_{\beta,13|24} + \Delta_{\beta,14|24} . $$ For each of these strata, let $\delta_x=[\Delta_x]$ be the corresponding cycle in $H_\bullet(\overline{\mathcal{M}}_{1,4},\mathbb{Q})$, in the sense of orbifolds. (This is sometimes denoted $[\Delta_x]_Q$ instead, but we omit the letter $Q$ from the notation.) If the generic point of $\Delta_x$ has an automorphism group of order $e$, then $\delta_x$ is $e^{-1}$ times the scheme-theoretic fundamental class of $\Delta_x$; this occurs, with $e=2$, for the cycles $\delta_{2,3}$, $\delta_{2,4}$ and $\delta_{0,4}$. \begin{lemma} \label{trivial} The following relation among cycles holds in $H_4(\overline{\mathcal{M}}_{1,4},\mathbb{Q})$: $$ \delta_{0,2} + 3 \delta_{0,3} + 6 \delta_{0,4} = 3 \delta_\alpha + 4 \delta_\beta . $$ \end{lemma} \begin{proof} The two strata $$ {\setlength{\unitlength}{0.01in} \begin{picture}(90,85)(60,685) \put( 80,755){\circle{30}} \put( 80,740){\line( 0,-1){ 20}} \put( 80,720){\line(-1,-2){ 10}} \put( 80,720){\line( 1,-2){ 10}} \put( 63,683){$1$} \put( 87,683){$2$} \end{picture} \begin{picture}(65,85)(60,727) \put( 80,775){\circle{30}} \put( 80,760){\line( 0,-1){ 20}} \put( 80,790){\line( 0, 1){ 20}} \put( 77,814){$1$} \put( 77,725){$2$} \end{picture}} $$ define the same cycle, and are even rationally equivalent. (This is an instance of the WDVV equation.) We obtain the lemma by lifting this relation by the $6$ distinct projections $\overline{\mathcal{M}}_{1,4}\to\overline{\mathcal{M}}_{1,2}$ and summing the answers. \end{proof} We can now state the main result of this section. \begin{theorem} \label{main} The first seven rows of the intersection matrix of the nine $\SS_4$-in\-var\-i\-ant codimension two cycles in $\overline{\mathcal{M}}_{1,4}$ introduced above equals $$\begin{tabular}{C|CCCC|CCC|CC} & \delta_{2,2} & \delta_{2,3} & \delta_{2,4} & \delta_{3,4} & \delta_{0,2} & \delta_{0,3} & \delta_{0,4} & \delta_\alpha & \delta_\beta \\ \hline \delta_{2,2} & 1/8 & 0 & 0 & 0 & -3 & 0 & 3/2 & 0 & 3/2 \\ \delta_{2,3} & 0 & 0 & 0 & 0 & 0 & -6 & 6 & 6 & 0 \\ \delta_{2,4} & 0 & 0 & 0 & -1/2 & 0 & 6 & -3 & 0 & 0 \\ \delta_{3,4} & 0 & 0 & -1/2 & 1/6 & 6 & -2 & 0 & 0 & 0 \\ \hline \delta_{0,2} & -3 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \\ \delta_{0,3} & 0 & -6 & 6 & -2 & 0 & 0 & 0 & 0 & 0 \\ \delta_{0,4} & 3/2 & 6 & -3 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{tabular}$$ \end{theorem} \begin{proof} The following lemma shows that many of the intersection numbers vanish. (The use of this lemma simplifies our original proof of Theorem \ref{main}, and was suggested to us by C. Faber.) \begin{lemma} Let $\delta$ be a cycle in $\Delta_0$. Then $\delta_0\*\delta=0$. \end{lemma} \begin{proof} Consider the projection $\pi:\overline{\mathcal{M}}_{1,n}\to\overline{\mathcal{M}}_{1,1}$ which forgets all but the first marked point, and stabilizes the marked curve which results. The divisor $\Delta_0$ is the inverse image under $\pi$ of the compactification divisor of $\overline{\mathcal{M}}_{1,1}$; thus, we may replace it in calculating intersections by any cycle of the form $\pi^{-1}(x)$, where $x\in\mathcal{M}_{1,1}$. The resulting cycle has empty intersection with $\delta$, proving the lemma. \end{proof} This lemma shows that all intersections among the cycles $\delta_{0,2}$, $\delta_{0,3}$ and $\delta_{0,4}$, and between these and $\delta_\alpha$ and $\delta_\beta$ vanish. A number of other entries in the intersection matrix vanish because the associated strata do not meet: thus, \begin{align*} & \delta_{2,2}\*\delta_{2,3} = \delta_{2,2}\*\delta_{3,4} = \delta_{2,2}\*\delta_{0,3} = \delta_{2,2}\*\delta_\alpha = 0 , \\ & \delta_{2,3}\*\delta_{2,4} = \delta_{2,3}\*\delta_\beta = 0 , \\ & \delta_{2,4}\*\delta_\alpha = \delta_{2,4}\*\delta_\beta = \delta_{3,4}\*\delta_\alpha = \delta_{3,4}\*\delta_\beta = 0 . \end{align*} To calculate the remaining entries of the intersection matrix, we need the excess intersection formula (Fulton \cite{Fulton}, Section~6.3). \begin{proposition} \label{excess} Let $Y$ be a smooth variety, let $X\hookrightarrow Y$ be a regular embedding\xspace of codimension $d$, and let $V$ be a closed subvariety of $Y$ of dimension $n$. Suppose that the inclusion $W=X\cap V\hookrightarrow V$ is a regular embedding\xspace of codimension $d-e$. Then $$ [X]\*[V] = c_e(E) \cap [W] \in A_{n-d}(W) , $$ where $E=(N_XY)|_W/(N_WV)$ is the \emph{excess bundle} of the intersection. \end{proposition} Observe that in calculating the top four rows of our intersection matrix, at least one of the cycles which we intersect with has a regular embedding\xspace in $\overline{\mathcal{M}}_{1,4}$, since its dual graph is a tree. This makes the application of the excess intersection formula straightforward. It remains to give a formula for the normal bundles to the strata of $\overline{\mathcal{M}}_{1,4}$. \begin{definition} The \emph{tautological line bundles} are defined by $$ \omega_i = \sigma_i^*\omega_{\overline{\mathcal{M}}_{g,n+1}/\overline{\mathcal{M}}_{g,n}} , $$ where $\sigma_i:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_{g,n+1}$, $1\le i\le n$, are the $n$ canonical sections of the universal stable curve $\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$. Denote the Chern class $c_1(\omega_i)$ by $\psi_i$. \end{definition} To apply the excess intersection formula, we need to know the normal bundles of strata $\overline{\mathcal{M}}(G)\subset\overline{\mathcal{M}}_{g,n}$. The following result gives a partial answer to this question, and is all that we need for the calculations in this paper: a proof may be found in Section~4 of Hain-Looijenga \cite{HL}. \begin{proposition} \label{normal} Let $G$ be a stable graph of genus $g$ and valence $n$, and let $$ p: \prod_{v\in\VERT(G)} \overline{\mathcal{M}}_{g(v),n(v)} \to \overline{\mathcal{M}}_{g,n} $$ be the ramified cover (of degree $|\Aut(G)|$) of the closed stratum $\overline{\mathcal{M}}(G)$ of $\mathcal{M}_{g,n}$. Each edge $e$ of the graph determines two flags $s(e)$ and $t(e)$, and hence two tautological line bundles $\omega_{s(e)}$ and $\omega_{t(e)}$ on $\prod_{v\in\VERT(G)} \overline{\mathcal{M}}_{g(v),n(v)}$, and the normal bundle of $p$ is given by the formula $$ N_p = \bigoplus_{e\in\Edge(G)} \omega_{s(e)}^\vee\o\omega_{t(e)}^\vee . \qed$$ \end{proposition} In particular, if the graph $G$ has no automorphisms, so that $p$ is an embedding\xspace, the bundle $N_p$ may be identified with the normal bundle of the stratum $\overline{\mathcal{M}}(G)$. It is now straightforward to calculate the remaining entries of the intersection matrix. We will use the integrals \begin{equation} \label{tau} \int_{\overline{\mathcal{M}}_{0,4}} \psi_i = 1 , \quad \int_{\overline{\mathcal{M}}_{1,1}} \psi_1 = \int_{\overline{\mathcal{M}}_{1,2}} \psi_1 \cup \psi_2 = \frac{1}{24} , \end{equation} which are proved in Witten \cite{Witten}. In performing the calculations, it is helpful to introduce a graphical notation for the cycle obtained from a stratum by capping with a monomial in the Chern classes $-\psi_i$: we point a small arrow along each flag $i$ where we intersect by the class $-\psi_i$. (This notation generalizes that of Kaufmann \cite{Kaufmann}, who considers the case of trees where the genus of each vertex is $0$. The minus signs come from the inversion accompanying the tautological line bundles in the formula of Proposition \ref{normal}.) One then calculates the contribution of such a graph by multiplying together factors for each vertex equal to the integral over $\overline{\mathcal{M}}_{g(v),n(v)}$ of the appropriate monomial in the classes $-\psi_i$, and dividing by the order of the automorphism group $\Aut(G)$: in particular, this vanishes unless there are $3(g(v)-1)+n(v)$ arrows at each vertex $v$. We illustrate the sort of enumeration which arises with one of the most complicated of these calculations, that of $\delta_{2,4}\*\delta_{2,4}$. Two sorts of terms contribute: $6$~terms of the form $$ \bigl( \delta_{\{1,2\}}\*\delta_{\{1,2,3,4\}} \bigr)^2 = \frac{1}{24} , $$ and $6$ terms of the form $$ \delta_{\{1,2\}}\*\delta_{\{1,2,3,4\}}\*\delta_{\{3,4\}}\*\delta_{\{1,2,3,4\}} = - \frac{1}{24} . $$ Applying the excess intersection formula, we see that $$ \bigl(\delta_{\{1,2\}}\*\delta_{\{1,2,3,4\}}\bigr)^2 = c_2\bigl(N_{\Delta_{\{1,2\}}\cap\Delta_{\{1,2,3,4\}}}\overline{\mathcal{M}}_{1,4}\bigr) \cap \bigl( \delta_{\{1,2\}}\*\delta_{\{1,2,3,4\}} \bigr) . $$ Expanding the second Chern class of the normal bundle, we see that each term contributes the sum of four graphs: $$ \begin{picture}(80,90)(60,690) \put( 80,760){\circle{5}} \put( 80,740){\vector( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 70,720){\vector( 1, 2){ 10}} \put( 70,720){\line(-1,-2){ 10}} \put( 70,720){\line( 1,-2){ 10}} \end{picture} \begin{picture}(80,90)(60,690) \put( 80,760){\circle{5}} \put( 80,757){\vector( 0,-1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 70,720){\vector( 1, 2){ 10}} \put( 70,720){\line(-1,-2){ 10}} \put( 70,720){\line( 1,-2){ 10}} \end{picture} \begin{picture}(80,90)(60,690) \put( 80,760){\circle{5}} \put( 80,740){\vector( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 80,740){\vector(-1,-2){ 10}} \put( 70,720){\line(-1,-2){ 10}} \put( 70,720){\line( 1,-2){ 10}} \end{picture} \begin{picture}(50,90)(60,690) \put( 80,760){\circle{5}} \put( 80,757){\vector( 0,-1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line( 0,-1){ 20}} \put( 80,740){\vector(-1,-2){ 10}} \put( 70,720){\line(-1,-2){ 10}} \put( 70,720){\line( 1,-2){ 10}} \end{picture} $$ Only the first graph is nonzero, since in the other cases, the wrong number of arrows point towards the vertices. And the first graph contributes $$ \int_{\overline{\mathcal{M}}_{0,4}} (-\psi_1) \* \int_{\overline{\mathcal{M}}_{1,1}} (-\psi_1) = \frac{1}{24} . $$ In the case of terms of the form $\delta_{\{1,2\}}\*\delta_{\{1,2,3,4\}} \*\delta_{\{3,4\}}\*\delta_{\{1,2,3,4\}}$, the excess dimension $e$ equals $1$, and we must calculate the degree of the excess bundle on the stratum $\Delta_{\{1,2\}}\cap\Delta_{\{3,4\}}\cap\Delta_{\{1,2,3,4\}}$. Two graphs contribute: $$ \begin{picture}(120,90)(60,690) \put( 80,760){\circle{5}} \put( 80,740){\vector( 0, 1){ 18}} \put( 80,740){\line( 1,-1){ 20}} \put( 80,740){\line(-1,-1){ 20}} \put( 60,720){\line(-1,-2){ 10}} \put( 60,720){\line( 1,-2){ 10}} \put(100,720){\line(-1,-2){ 10}} \put(100,720){\line( 1,-2){ 10}} \end{picture} \begin{picture}(60,90)(60,690) \put( 80,760){\circle{5}} \put( 80,757){\vector( 0,-1){ 18}} \put( 80,740){\line( 1,-1){ 20}} \put( 80,740){\line(-1,-1){ 20}} \put( 60,720){\line(-1,-2){ 10}} \put( 60,720){\line( 1,-2){ 10}} \put(100,720){\line(-1,-2){ 10}} \put(100,720){\line( 1,-2){ 10}} \end{picture} $$ Only the first of these graphs gives a nonzero value, namely $$ \int_{\overline{\mathcal{M}}_{1,1}} (-\psi_1) = - \frac{1}{24} . $$ This completes our outline of the proof of Theorem \ref{main}. \end{proof} The intersection matrix of Theorem \ref{main} has rank $7$. We now apply the results of \cite{genus1}, where we calculated the character of the $\SS_n$-modules $H^i(\overline{\mathcal{M}}_{1,n},\mathbb{Q})$: these calculations show that $\dim H^4(\overline{\mathcal{M}}_{1,4},\mathbb{Q})^{\SS_4}=7$. This shows that our $9$ cycles span $H^4(\overline{\mathcal{M}}_{1,4},\mathbb{Q})^{\SS_4}$, and that the nullspace of the intersection matrix gives relations among them. We already know one such relation, by Lemma \ref{trivial}. Calculating the remaining null-vector of the intersection matrix, we obtain the main theorem of this paper. \begin{theorem} \label{relation} The following new relation among cycles holds: $$ 12\delta_{2,2} - 4\delta_{2,3} - 2\delta_{2,4} + 6\delta_{3,4} + \delta_{0,3} + \delta_{0,4} - 2\delta_\beta = 0 . \qed$$ \end{theorem} Using this theorem, it is easy to calculate the remaining intersections among our $9$ strata: $$ \delta_\alpha\*\delta_\alpha=16 , \quad \delta_\alpha\*\delta_\beta=-12 , \quad \delta_\beta\*\delta_\beta=9 . $$ C. Faber informs us that the direct calculation of these intersection numbers is not difficult. This would allow a different approach to the proof of Theorem \ref{relation}, using the theorem of \cite{elliptic3} that the strata of $\overline{\mathcal{M}}_{1,n}$ span the even-dimensional rational cohomology. \section{Gromov-Witten invariants} In the remainder of this paper, we apply the new relation to the calculation of elliptic Gromov-Witten invariants: we will do this explicitly for curves and for the projective plane $\mathbb{CP}^2$, and prove some general results in other cases. \subsection{The Novikov ring} Let $V$ be a smooth projective variety of dimension $d$. In studying the Gromov-Witten invariants, it is convenient to work with cohomology with coefficients in the Novikov ring $\Lambda$ of $V$, which we now define. Let $\NN_1(V)$ be the abelian group $$ \NN_1(V) = \ZZ_1(V) / \text{numerical equivalence} , $$ and let $\NE_1(V)$ be its sub-semigroup $$ \NE_1(V) = \ZE_1(V) / \text{numerical equivalence} , $$ where $\ZZ_1(V)$ is the abelian group of $1$-cycles on $V$, and $\ZE_1(V)$ is the semigroup of effective $1$-cycles. (Recall that two $1$-cycles $x$ and $y$ are numerically equivalent $x\equiv y$ when $x\*Z=y\*Z$ for any Cartier divisor $Z$ on $V$.) The Novikov ring is \begin{align*} \Lambda &= \mathbb{Q}[\NN_1(V)] \o_{\mathbb{Q}[\NE_1(V)]} \mathbb{Q}\[\NE_1(V)\] \\ &= \textstyle \bigl\{ a = \sum_{\beta\in\NN_1(V)} a_\beta q^\beta \mid \text{ $\supp(a) \subset \beta_0+\NE_1(V)$ for some $\beta_0\in\NN_1(V)$} \bigr\} , \end{align*} with product $q^{\beta_1}q^{\beta_2}=q^{\beta_1+\beta_2}$ and grading $|q^\beta|=-2c_1(V)\cap\beta$. That the product is well-defined is shown by the following proposition (Koll\'ar \cite{Kollar}, Proposition II.4.8). \begin{proposition} \label{Mori} If $V$ is a projective variety with K\"ahler form $\omega$, the set $$ \{\beta\in\NE_1(V)\mid \omega\cap\beta\le c\} $$ is finite for each $c>0$. \qed\end{proposition} For example, if $V=\mathbb{CP}^n$, then $\NN_1(\mathbb{CP}^n)=\mathbb{Z}\*[L]$, where $[L]$ is the cycle defined by a line $L\subset\mathbb{CP}^n$, and $\Lambda\cong\mathbb{Q}\(q\)$, with grading $|q|=-2(n+1)$, since $c_1(\mathbb{CP}^n)\cap[L]=n+1$. If $V=E$ is an elliptic curve, then $\NN_1(E)=\mathbb{Z}\*[E]$, and $\Lambda\cong\mathbb{Q}\(q\)$, concentrated in degree $0$. \subsection{Stable maps} The definition of Gromov-Witten invariants is based on the study of the moduli stacks $\overline{\mathcal{M}}_{g,n}(V,\beta)$ of stable maps of Kontsevich, which have been shown by Behrend and Manin \cite{BM} to be complete Deligne-Mumford stacks (though not in general smooth). For each $N\ge0$, let $\pi_{n,N}:\overline{\mathcal{M}}_{g,n+N}(V,\beta) \to \overline{\mathcal{M}}_{g,n}(V,\beta)$ be the projection which forgets the last $N$ marked points of the stable curve, and stabilizes the resulting map. In the special case $N=1$, we obtain a fibration $$ \pi: \overline{\mathcal{M}}_{g,n+1}(V,\beta) \to \overline{\mathcal{M}}_{g,n}(V,\beta) $$ which is shown by Behrend and Manin to be the universal curve; that is, its fibre over a stable map $(f:C\to V,x_i)$ is the curve $C$. Denote by $f:\overline{\mathcal{M}}_{g,n+1}(V,\beta)\to V$ the universal stable map, obtained by evaluation at $x_{n+1}$. \subsection{The virtual fundamental class} There are projections $\overline{\mathcal{M}}_{g,n}(V,\beta)\to\overline{\mathcal{M}}_{g,n}$, when $2(g-1)+n>0$, which send the stable map $(f:C\to V,x_i)$ to the stabilization of $(C,x_i)$. If the sheaf $R^1\pi_*f^*TV$ vanishes on $\overline{\mathcal{M}}_{g,n}(V,\beta)$, the Riemann-Roch theorem predicts that the fibres of the projection $\overline{\mathcal{M}}_{g,n}(V,\beta)\to\overline{\mathcal{M}}_{g,n}$ have dimension $$ d(1-g)+c_1(V)\cap\beta , $$ and hence that $\overline{\mathcal{M}}_{g,n}(V,\beta)$ has dimension $$ d(1-g)+c_1(V)\cap\beta + \dim\overline{\mathcal{M}}_{g,n} = (3-d)(1-g)+c_1(V)\cap\beta + n . $$ This hypothesis is only rarely true, and in any case only in genus $0$. However, Behrend-Fantecchi \cite{B,BF} and Li-Tian \cite{LT} show that there is a bivariant class $$ [\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n},R^\bullet\pi_*f^*TV] \in A^{d(1-g)+c_1(V)\cap\beta}(\overline{\mathcal{M}}_{g,n}(V,\beta)\to\overline{\mathcal{M}}_{g,n}) , $$ the virtual relative fundamental class, which stands in for $[\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n}]$ in the obstructed case. The following result is proved in \cite{B}, and sometimes permits the explicit calculation of Gromov-Witten invariants, as we will see later. \begin{proposition} \label{excess-virtual} If the coherent sheaf $R^1\pi_*f^*TV$ on $\overline{\mathcal{M}}_{g,n}(V,\beta)$ is locally trivial of dimension $e$ (the \emph{excess dimension}), then $\overline{\mathcal{M}}_{g,n}(V,\beta)$ is smooth of dimension $$ (3-d)(1-g)+c_1(V)\cap\beta+n+e , $$ and $[\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n},R^\bullet\pi_*f^*TV] = c_e(R^1\pi_*f^*TV) \cap [\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n}]$. \qed \end{proposition} \subsection{Gromov-Witten invariants} The Gromov-Witten invariant of genus $g\ge0$, valence $n\ge0$ and degree $\beta\in\NE_1(V)$ is a cohomology operation $$ I_{g,n,\beta}^V : H^{2d(1-g)+2c_1(V)\cap\beta+\bullet}(V^n,\mathbb{Q}) \to H^\bullet(\overline{\mathcal{M}}_{g,n},\mathbb{Q}) , $$ defined by the formula $$ I_{g,n,\beta}^V(\alpha_1,\dots,\alpha_n) = [\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n},R^\bullet\pi_*f^*TV] \cap \ev^*(\alpha_1\boxtimes\dots\boxtimes\alpha_n) , $$ where $\ev:\overline{\mathcal{M}}_{g,n}(V,\beta)\to V^n$ is evaluation at the marked points: $$ \ev : (f:C\to V,x_i) \mapsto (f(x_1),\dots,f(x_n)) \in V^n . $$ Capping $I_{g,n,\beta}^V$ with the fundamental class $[\overline{\mathcal{M}}_{g,n}]$, we obtain a numerical invariant $$ \<I_{g,n,\beta}^V\> : H^{2(d-3)(1-g)+2c_1(V)\cap\beta+2n}(V^n,\mathbb{Q}) \to \mathbb{Q} . $$ This is the $n$-point correlation function of two-dimensional topological gravity with the topological $\sigma$-model associated to $V$ as a background \cite{Witten}. Note that if $\beta\ne0$, $\<I_{g,n,\beta}^V\>$ may be defined even when $2(g-1)+n\le0$, even though $I_{g,n,\beta}^V$ does not exist. Introducing the Novikov ring, we may define the generating function $$ I_{g,n}^V = \sum_{\beta\in\NE_1(V)} q^\beta I_{g,n,\beta}^V : H^*(V,\Lambda)^{\o n} \to H^\bullet(\overline{\mathcal{M}}_{g,n},\Lambda) , $$ along with its integral over the fundamental class $[\overline{\mathcal{M}}_{g,n}]$ $$ \<I_{g,n}^V\> = \sum_{\beta\in\NE_1(V)} q^\beta \<I_{g,n,\beta}^V\> : H^*(V,\Lambda)^{\o n} \to \Lambda , $$ Note that $I_{g,n}^V$ and $\<I_{g,n}^V\>$ are invariant under the action of the symmetric group $\SS_n$ on $H^*(V,\Lambda)^{\o n}$. In the special case of zero degree, the moduli space $\overline{\mathcal{M}}_{g,n}(V,\beta)$ is isomorphic to $\overline{\mathcal{M}}_{g,n}\times V$. This allows us to calculate the Gromov-Witten invariants $\<I_{0,3,0}^V\>$ and $\<I_{1,1,0}^V\>$. The former is given by the explicit formula $$ \<I_{0,3,0}^V(\alpha_1,\alpha_2,\alpha_3)\> = \int_V \alpha_1\cup\alpha_2\cup\alpha_2 . $$ This formula is very simple to prove, since the moduli space $\overline{\mathcal{M}}_{0,3}(V,0)\cong V$ is smooth, with dimension equal to its virtual dimension $d$, and thus the virtual fundamental class $[\overline{\mathcal{M}}_{0,3}(V,0),R^\bullet\pi_*f^*TV]$ may be identified with the fundamental class of $V$. A similar proof shows that $\<I_{0,n,0}^V\>$ vanishes if $n>3$. The calculation of the Gromov-Witten invariant $\<I_{1,1,0}^V\>$ (see Bershadsky et al.\ \cite{BCOV}) is a good illustration of the application of Proposition \ref{excess-virtual}. \begin{proposition} \label{BCOV} $$ \<I_{1,1,0}^V(\alpha)\> = -\frac{1}{24} \int_V c_{d-1}(V)\cup\alpha , $$ while $\<I_{1,n,0}^V\>=0$ if $n>1$. \end{proposition} \begin{proof} The moduli stack $\overline{\mathcal{M}}_{1,n}(V,0)$ is isomorphic to $\overline{\mathcal{M}}_{1,n}\times V$, and the obstruction bundle $R^1\pi_*f^*TV$ is isomorphic to the vector bundle $\mathbb{E}^\vee\boxtimes TV$, of rank $d$, where $\mathbb{E}=\pi_*\omega_{\overline{\mathcal{M}}_{1,n+1}/\overline{\mathcal{M}}_{1,n}}$. Hence $R^1\pi_*f^*TV$ has top Chern class $$ c_d(\mathbb{E}^\vee\o f^*TV) = 1\boxtimes f^*c_d(V) - \lambda_1 \boxtimes f^*c_{d-1}(V) , $$ where $\lambda_1=c_1(\mathbb{E})$. By Proposition \ref{excess-virtual}, \begin{align*} \<I_{1,n,0}^V(\alpha_1,\dots,\alpha_n)\> &= \int_{\overline{\mathcal{M}}_{1,n}\times V} c_d(\mathbb{E}^\vee\o f^*TV) \boxtimes (\alpha_1\cup\dots\cup\alpha_n) \\ &= - \int_{\overline{\mathcal{M}}_{1,n}} \lambda_1 \* \int_V c_{d-1}(V) \cup \alpha_1\cup\dots\cup\alpha_n . \end{align*} On dimensional grounds, $\<I_{1,n,0}^V\>$ vanishes if $n>1$, while the formula follows when $n=1$ from $\lambda_1\cap[\overline{\mathcal{M}}_{1,1}]=\frac{1}{24}$. \end{proof} \subsection{The puncture axiom} One of the basic axioms satisfied by Gromov-Witten invariants is expressed in the relationship between virtual fundamental classes $$ [\overline{\mathcal{M}}_{g,n+1}(V,\beta)/\overline{\mathcal{M}}_{g,n+1},R^\bullet\pi_*f^*TV] = \pi^* [\overline{\mathcal{M}}_{g,n}(V,\beta)/\overline{\mathcal{M}}_{g,n},R^\bullet\pi_*f^*TV] . $$ Here, $\pi^*:A^k(\overline{\mathcal{M}}_{g,n}(V,\beta)\to\overline{\mathcal{M}}_{g,n}) \to A^k(\overline{\mathcal{M}}_{g,n+1}(V,\beta)\to\overline{\mathcal{M}}_{g,n+1})$ is the operation of flat pullback associated to the diagram $$\begin{CD} \overline{\mathcal{M}}_{g,n+1}(V,\beta) @>>> \overline{\mathcal{M}}_{g,n+1} \\ @V{\pi}VV @V{\pi}VV \\ \overline{\mathcal{M}}_{g,n}(V,\beta) @>>> \overline{\mathcal{M}}_{g,n} \end{CD}$$ This axiom implies that if $\alpha$ is a cohomology class on $V$ of degree at most $2$ and $2(g-1)+n>0$, \begin{equation} \label{low} I_{g,n+1,\beta}^V(\alpha,\alpha_1,\dots,\alpha_n) = \begin{cases} 0 , & |\alpha|=0,1 , \\ (\alpha\cap\beta) I_{g,n,\beta}^V(\alpha_1,\dots,\alpha_n) , & |\alpha|=2 . \end{cases} \end{equation} \subsection{Generating functions} Let $\Lambda\[H\]$ be the power series ring $\Lambda\[H_{\bullet+2}(V,\mathbb{Q})\]$. Let $\{\gamma^a\}_{a=0}^k$ be a homogeneous basis of the graded vector space $H^\bullet(V,\mathbb{Q})$, with $\gamma^0=1$, and let $\{t_a\}_{a=0}^k$ be the dual basis; the (homological) degree of $t_a$ equals the (cohomological) degree of $\gamma^a$ minus $2$. We may identify the ring $\Lambda\[H\]$ with $\Lambda\[t_0,\dots,t_k\]$. Let $F_g(V)$ be the generating function $$ F_g(V) = \sum_{n=0}^\infty \<I_{g,n}^V\> \in \Lambda\[H\] . $$ This is a power series of degree $2(d-3)(1-g)$. This suggests assigning to Planck's constant $\hbar$ the degree $2(d-3)(g-1)$, and forming the total generating function, homogeneous of degree $0$, $$ F(V) = \sum_{g=0}^\infty \hbar^{g-1} F_g(V) . $$ \subsection{The composition axiom} The composition axiom for Gromov-Witten invariants gives a formula for the integral of the Gromov-Witten invariant $I_{g,n}^V$ over the cycle $[\overline{\mathcal{M}}(G)]$ associated to a stable graph $G$ which bears a strong resemblance to the Feynman rules of quantum field theory: Let $\eta_{ab}$ be the Poincar\'e form of $V$ with respect to the basis $\{\gamma^a\}_{a=0}^k$ of $H^\bullet(V,\mathbb{Q})$. Then $$ \int_{\overline{\mathcal{M}}(G)} I_{g,n}^V(\alpha_1,\dots,\alpha_n) = \frac{1}{\Aut(G)} \sum_{\substack{a(e),b(e)=0 \\ e\in\Edge(G)}}^k \prod_{e\in\Edge(G)} \eta_{a(e),b(e)} \prod_{v\in\VERT(G)} \<I_{g(v),n(v)}^V(\dots)\> . $$ Here, the Gromov-Witten invariant $\<I_{g(v),n(v)}^V(\dots)\>$ is evaluated on the cohomology classes $\alpha_i$ corresponding to the tails of $G$ which meet the vertex $v$, on the $\gamma^{a(e)}$ corresponding to edges $e$ which start at the vertex $v$, and on the $\gamma^{b(e)}$ corresponding to edges $e$ which end at $v$. (The right-hand side is independent of the chosen orientation of the edges, by the symmetry of the Poincar\'e form.) \subsection{Relations among Gromov-Witten invariants} Let $G$ be a stable graph of genus $g$ and valence $n$. The subvariety $\pi_{n,N}^{-1}\bigl(\overline{\mathcal{M}}(G)\bigr) \subset \overline{\mathcal{M}}_{g,n+N}$ is the union of strata associated to the set of stable graphs obtained from $G$ by adjoining $N$ tails $\{n+1,\dots,n+N\}$ in all possible ways to the vertices of $G$. For example, consider the stratum $\Delta_{12|34}\subset\overline{\mathcal{M}}_{0,4}$, associated to the stable graph $$ \begin{picture}(35,85)(60,715) \put( 20,760){$\Delta_{12|34} =$} \put( 80,775){\line( 0,-1){ 30}} \put( 80,745){\line(-2,-3){ 10}} \put( 80,745){\line( 2,-3){ 10}} \put( 80,775){\line( 2, 3){ 10}} \put( 80,775){\line(-2, 3){ 10}} \put( 65,715){$1$} \put( 89,715){$2$} \put( 65,795){$3$} \put( 89,795){$4$} \end{picture} $$ The inverse image $\pi_{4,N}^{-1}(\Delta_{12|34})$ consists of the union of all strata in $\overline{\mathcal{M}}_{0,4+N}$ associated to stable graphs $$ \begin{picture}(35,85)(60,715) \put( 5,760){$\Delta_{12I|34J} =$} \put( 80,780){\line( 0,-1){ 40}} \put( 80,740){\line(-2,-3){ 10}} \put( 80,740){\line( 2,-3){ 10}} \put( 80,780){\line( 2, 3){ 10}} \put( 80,780){\line(-2, 3){ 10}} \put( 65,715){$1$} \put( 89,715){$2$} \put( 65,800){$3$} \put( 89,800){$4$} \put( 80,780){\line( 3,-1){ 30}} \put( 80,780){\line( 3, 1){ 30}} \put( 95,780){\dots} \put(115,777){$J$} \put( 80,740){\line( 3,-1){ 30}} \put( 80,740){\line( 3, 1){ 30}} \put( 95,740){\dots} \put(115,737){$I$} \end{picture} $$ where $I$ and $J$ form a partition of the set $\{5,\dots,N+4\}$. If $\delta$ is a cycle in $\overline{\mathcal{M}}_{g,n}$, define the generating function $$ F(\delta,V) = \sum_{N=0}^\infty \int_{\pi^{-1}(\delta)} I_{g,n+N}^V : H^{\bullet+2}(V,\Lambda)^{\o n} \to \Lambda\[H\] . $$ More explicitly, \begin{multline*} F(\delta,V)(\alpha_1,\dots,\alpha_n) \\ = \sum_{N=0}^\infty \frac{1}{N!} \sum_{a_1,\dots,a_N} t_{a_N}\dots t_{a_1} \int_\delta \bigl( \pi_{n,N} \bigr)_* I_{g,n+N}^V(\gamma^{a_1},\dots,\gamma^{a_N},\alpha_1,\dots,\alpha_n) . \end{multline*} In particular, if $\delta=[\overline{\mathcal{M}}(G)]$ where $G$ is a stable graph, we set $$ F(G,V)=F([\overline{\mathcal{M}}(G)],V) . $$ If $g>1$, $F_g(V)$ is a special case of this construction, with $\delta=[\overline{\mathcal{M}}_{g,0}]$. A little exercise involving Leibniz's rule shows that the composition axiom implies the following formula for these generatings functions: \begin{equation} \label{composition} F(G,V) = \frac{1}{\Aut(G)} \sum_{\substack{a(e),b(e)=0 \\ e\in\Edge(G)}}^k \prod_{e\in\Edge(G)} \eta_{a(e),b(e)} \prod_{v\in\VERT(G)} \partial^{n(v)}F_{g(v)}(V) (\dots) , \end{equation} where as before, the multilinear form $\partial^{n(v)}F_{g(v)}(V)$ is evaluated on the cohomology classes $\alpha_i$ corresponding to the tails of $G$ meeting the vertex $v$, on the $\gamma^{a(e)}$ corresponding to edges $e$ which start at the vertex $v$, and on the $\gamma^{b(e)}$ corresponding to edges $e$ which end at $v$. The composition axiom implies that any relation among the cycles $[\overline{\mathcal{M}}(G)]$ is reflected in a relation among Gromov-Witten invariants, which, by \eqref{composition} may be translated into a differential equation among generating functions $F_g(V)$. An example is the rational equivalence of the cycles associated to the three strata of $\overline{\mathcal{M}}_{0,4}$ of codimension $1$: $$ \begin{picture}(35,85)(60,715) \put( 80,775){\line( 0,-1){ 30}} \put( 80,745){\line(-1,-2){ 10}} \put( 80,745){\line( 1,-2){ 10}} \put( 80,775){\line( 2, 3){ 10}} \put( 80,775){\line(-2, 3){ 10}} \put( 65,710){$1$} \put( 89,710){$2$} \put( 65,795){$3$} \put( 89,795){$4$} \end{picture} \hskip0.5in \begin{picture}(35,85)(60,715) \put( 40,755){$\sim$} \put( 80,775){\line( 0,-1){ 30}} \put( 80,745){\line(-1,-2){ 10}} \put( 80,745){\line( 1,-2){ 10}} \put( 80,775){\line( 2, 3){ 10}} \put( 80,775){\line(-2, 3){ 10}} \put( 65,710){$1$} \put( 89,710){$3$} \put( 65,795){$2$} \put( 89,795){$4$} \end{picture} \hskip0.5in \begin{picture}(35,85)(60,715) \put( 40,755){$\sim$} \put( 80,775){\line( 0,-1){ 30}} \put( 80,745){\line(-1,-2){ 10}} \put( 80,745){\line( 1,-2){ 10}} \put( 80,775){\line( 2, 3){ 10}} \put( 80,775){\line(-2, 3){ 10}} \put( 65,710){$1$} \put( 89,710){$4$} \put( 65,795){$2$} \put( 89,795){$3$} \end{picture} $$ The equality of the Gromov-Witten invariant $F(\delta,V)$ when evaluated on these cycles is the WDVV equation. In order to express the relation among the Gromov-Witten invariants implied by Theorem \ref{relation}, it is useful to introduce certain operators which act on elements of $\Lambda[H]\o\Lambda\[H\]$ through differentiation in the first factor: the Laplacian $$ \Delta = \frac12 \sum_{a,b=0}^k \eta_{ab} \frac{\partial^2}{\partial t_a\partial t_b} , $$ and the sequence of bilinear differential operators $\Gamma_n$ by $\Gamma_0(f,g)=fg$ and $$ \Gamma_n(f,g) = \frac{1}{n} \bigl( \Delta\Gamma_{n-1}(f,g) - \Gamma_{n-1}(\Delta f,g) - \Gamma_{n-1}(f,\Delta g) \bigr) . $$ (We will abbreviate $\Gamma_1(f,g)$ to $\Gamma(f,g)$.) \begin{proposition} \label{Relation} Denote the derivative $\partial^{n(v)}F_{g(v)}(V)/n(v)!\in\Lambda[H]\o\Lambda\[H\]$ by $f_{g,n}$. (Note that $f_{g,n}=F([\overline{\mathcal{M}}_{g,n}],V)$.) Then \begin{align*} 6 \, \Gamma(\Gamma_1(f_{1,2},f_{0,3}),f_{0,3}) &- 5 \, \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) \\ & {} - 2 \, \Gamma(f_{0,3},\Gamma(f_{1,1},f_{0,4})) + 6 \, \Gamma(f_{0,4},\Gamma(f_{1,1},f_{0,3})) \\ & {} + \Gamma(f_{0,4},\Delta f_{0,4}) + \Gamma(f_{0,5},\Delta f_{0,3}) - \Gamma_2(f_{0,4},f_{0,4}) = 0 . \end{align*} \end{proposition} \begin{proof} This follows from the following table, which is obtained by application of \eqref{composition}. $$\begin{tabular}{|L|L||L|L|} \hline \delta & F(\delta,V) & & \\ \hline \delta_{2,2} & \frac12 \Gamma(\Gamma(f_{1,2},f_{0,3}),f_{0,3}) & \delta_{0,2} & \Gamma(f_{0,3},\Delta f_{0,5}) \\ & {} - \frac14 \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) & \delta_{0,3} & \Gamma(f_{0,4},\Delta f_{0,4}) \\ \delta_{2,3} & \frac12 \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) & \delta_{0,4} & \Gamma(f_{0,5},\Delta f_{0,3}) \\ \delta_{2,4} & \Gamma(f_{0,3},\Gamma(f_{1,1},f_{0,4})) & \delta_\alpha & \Gamma_2(f_{0,3},f_{0,5}) \\ \delta_{3,4} & \Gamma(f_{0,4},\Gamma(f_{1,1},f_{0,3})) & \delta_\beta & \frac12 \Gamma_2(f_{0,4},f_{0,4}) \\[1pt] \hline \end{tabular}$$ \end{proof} When we apply Proposition \ref{Relation} with $V=\mathbb{CP}^2$ and evaluate the resulting multilinear form to $\omega^{\boxtimes4}$, we obtain the recursion relation \eqref{recursion} for the elliptic Gromov-Witten invariants $N^{(1)}_n$ of $\mathbb{CP}^2$. \section{The symbol of the new relation} We may introduce a filtration on Gromov-Witten invariants with respect to which the leading order of our new relation takes a relatively simple form; by analogy with the case of differential operators, we call this leading order relation the symbol of the full relation. In some cases, this symbol may be used to prove that elliptic Gromov-Witten invariants are determined by rational ones. \begin{definition} The \emph{symbol} of a relation $\delta=0$ among cycles of strata in $\overline{\mathcal{M}}_{g,n}$ is the set of relations among Gromov-Witten invariants obtained by taking, for each $\beta\in\NE_1(V)$, the coefficient of $q^\beta$ in $I_{g,n}^V\cap[\delta]$, expanding in Feynman diagrams using the composition axiom, and setting all Gromov-Witten invariants $\<I_{g',n',\beta'}^V\>$ other than $\<I_{g,n,\beta}^V\>$ and $\<I_{0,3,0}^V\>$ to zero. \end{definition} We define a total order on the symbols $\<I_{g,n,\beta}^V\>$ by setting $\<I_{g',n',\beta'}^V\>\prec\<I_{g,n,\beta}^V\>$ if $g'<g$, or $g'=g$ and $n'<n$, or $g'=g$, $n'=n$ and $\beta=\beta'+\beta''$ where $\beta''\in\NE_1(V)$ is non-zero. Thus, knowledge of the symbol determines relations among Gromov-Witten invariants such that the error in the relation on $\<I_{g,n,\beta}^V\>$ involves invariants $\<I_{g',n',\beta'}^V\>$ with $\<I_{g',n',\beta'}^V\>\prec\<I_{g,n,\beta}^V\>$. (Here, we must of course exclude $\<I_{0,3,0}^V\>$.) We use the symbol $\sim$ to denote this equivalence relation. For example, the symbol of the WDVV equation is $$ (a,b,c\cup d) + (a\cup b,c,d) \sim (-1)^{|a|(|b|+|c|)} \bigl( (b,c,a\cup d) + (b\cup c,a,d) \bigr) , $$ where we have abbreviated $\<I_{0,n,\beta}^V (\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\dots,\alpha_n)\>$ to $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$. Next, consider the symbol of the relation $$ \pi_{4,n-4}^{-1}\bigl(12\delta_{2,2} - 4\delta_{2,3} - 2\delta_{2,4} + 6\delta_{3,4} + \delta_{0,3} + \delta_{0,4} - 2\beta\bigr) = 0 $$ in $H_{2n-4}(\overline{\mathcal{M}}_{1,n},\mathbb{Q})$. Only the cycles $\delta_{2,2}$ and $\delta_{2,3}$ contribute terms to the symbol. Abbreviate the Gromov-Witten class $\<I_{1,n,\beta}^V(\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n)\>$ to $\{\alpha_1,\alpha_2\}$. Up to a numerical factor to be determined, the cycle $\delta_{2,2}$ contributes the expression $$ \{a\cup b,c\cup d\} + (-1)^{|b|\,|c|} \{a\cup c,b\cup d\} + (-1)^{(|b|+|c|)|d|} \{a\cup d,b\cup c\} . $$ This numerical factor equals $$ \frac{1}{24} \* 3 \* 12 \* 8 = 12 . $$ The factor $1/24$ comes from symmetrization over the four inputs, the factor of $3$ from the three strata making up $\delta_{2,2}$, the factor of $12$ is the coefficient of the cycle in the relation, and the factor $8$ is illustrated by listing all of the graphs which contribute a term $\{a\cup b ,c\cup d\}$: $$ \def\nnn{\begin{picture}(48,95)(56,715) \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,762){\line( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 10}} \put( 80,780){\line( 1, 2){ 10}} \put( 80,780){\line(-1, 2){ 10}}} \nnn \put( 67,805){$a$} \put( 87,805){$b$} \put( 67,709){$c$} \put( 87,709){$d$} \end{picture} \nnn \put( 67,805){$a$} \put( 87,805){$b$} \put( 67,709){$d$} \put( 87,709){$c$} \end{picture} \nnn \put( 67,805){$b$} \put( 87,805){$a$} \put( 67,709){$c$} \put( 87,709){$d$} \end{picture} \nnn \put( 67,805){$b$} \put( 87,805){$a$} \put( 67,709){$d$} \put( 87,709){$c$} \end{picture} \nnn \put( 67,805){$c$} \put( 87,805){$d$} \put( 67,709){$a$} \put( 87,709){$b$} \end{picture} \nnn \put( 67,805){$d$} \put( 87,805){$c$} \put( 67,709){$a$} \put( 87,709){$b$} \end{picture} \nnn \put( 67,805){$c$} \put( 87,805){$d$} \put( 67,709){$b$} \put( 87,709){$a$} \end{picture} \nnn \put( 67,805){$d$} \put( 87,805){$c$} \put( 67,709){$b$} \put( 87,709){$a$} \end{picture} $$ Similarly, the cycle $\delta_{2,3}$ contributes the expression \begin{multline*} \{a,b\cup c\cup d\} + (-1)^{|a|\,|b|} \{b,a\cup c\cup d\} \\ + (-1)^{(|a|+|b|)|c|} \{c,a\cup b\cup d\} + (-1)^{(|a|+|b|+|c|)|d|} \{d,a\cup b\cup c\} , \end{multline*} with numerical factor $$ \frac{1}{24} \* 12 \* (-4) \* 6 = - 12 ; $$ the factor $12$ counts the strata making up $\delta_{2,3}$, $-4$ is the coefficient of the cycle in the relation, and we illustrate the factor $6$ by listing all of the graphs which contribute a term $\{a,b\cup c\cup d\}$: $$ \def\nnn{\begin{picture}(60,95)(50,695) \put( 80,760){\circle{5}} \put( 80,757){\line( 0,-1){ 18}} \put( 80,762){\line( 0, 1){ 18}} \put( 80,740){\line( 1,-2){ 10}} \put( 80,740){\line(-1,-2){ 20}} \put( 70,720){\line( 1,-2){ 10}} } \nnn \put( 56,688){$b$} \put( 77,688){$c$} \put( 87,708){$d$} \put( 77,783){$a$} \end{picture} \nnn \put( 56,688){$c$} \put( 77,688){$b$} \put( 87,708){$d$} \put( 77,783){$a$} \end{picture} \nnn \put( 56,688){$b$} \put( 77,688){$d$} \put( 87,708){$c$} \put( 77,783){$a$} \end{picture} \nnn \put( 56,688){$d$} \put( 77,688){$b$} \put( 87,708){$c$} \put( 77,783){$a$} \end{picture} \nnn \put( 56,688){$c$} \put( 77,688){$d$} \put( 87,708){$b$} \put( 77,783){$a$} \end{picture} \nnn \put( 56,688){$d$} \put( 77,688){$c$} \put( 87,708){$b$} \put( 77,783){$a$} \end{picture} $$ In conclusion, we obtain the following result. \begin{theorem} \label{symbol} Abbreviating $\<I_{1,n,\beta}^V(\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n)\>$ to $\{\alpha_1,\alpha_2\}$, we have \begin{align*} \Psi(a,b,c,d) & = \{a\cup b,c\cup d\} + (-1)^{|b|\,|c|} \{a\cup c,b\cup d\} + (-1)^{(|b|+|c|)|d|} \{a\cup d,b\cup c\} \\ & {} - \{a,b\cup c\cup d\} - (-1)^{|a|\,|b|} \{b,a\cup c\cup d\} \\ & {} - (-1)^{(|a|+|b|)|c|} \{c,a\cup b\cup d\} - (-1)^{(|a|+|b|+|c|)|d|} \{d,a\cup b\cup c\} \sim 0 . \end{align*} \end{theorem} Note that the linear form $\Psi(a,b,c,d)$ is (graded) symmetric in its four arguments, and vanishes if any of them equals $1$. \begin{corollary} \label{reduce} If $\omega\in H^2(V,\mathbb{Q})$ and $a,b\in H^\bullet(V,\mathbb{Q})$, then for $j>2$, $$\textstyle \{\omega^i\cup a,\omega^{j-i}\cup b\} = \binom{i+2}{2} \{a,\omega^j\cup b\} . $$ \end{corollary} \begin{proof} By Theorem \ref{symbol}, we have for $i\ge0$ and $j>2$, \begin{multline*} \Psi(\omega,\omega^{i+1}\cup a,\omega,\omega^{j-i-3}\cup b) - \Psi(\omega,\omega^i\cup a,\omega,\omega^{j-i-2}\cup b) \\ \begin{aligned} \sim & \quad \bigl( 2\{\omega^{i+2}\cup a,\omega^{j-i-2}\cup b\} + \{\omega^2,\omega^{j-2}\cup a\cup b\} \\ {} & {} - \{\omega^{i+1}\cup a,\omega^{j-i-1}\cup b\} - \{\omega^{i+3}\cup a,\omega^{j-i-3}\cup b\} \bigr) \\ {} & {} - \bigl( 2\{\omega^{i+1}\cup a,\omega^{j-i-1}\cup b\} + \{\omega^2,\omega^{j-2}\cup a\cup b\} \\ {} & {} - \{\omega^i\cup a,\omega^{j-i}\cup b\} - \{\omega^{i+2}\cup a,\omega^{j-i-2}\cup b\} \bigr) \\ \sim & \quad \{\omega^i\cup a,\omega^{j-i}\cup b\} - 3\{\omega^{i+1}\cup a,\omega^{j-i-1}\cup b\} \\ {} & \quad {} + 3\{\omega^{i+2}\cup a,\omega^{j-i-2}\cup b\} - \{\omega^{i+3}\cup a,\omega^{j-i-3}\cup b\} \sim 0 . \end{aligned} \end{multline*} This implies that the function $a(i,j)=\{\omega^i\cup a,\omega^{j-i}\cup b\}$ satisfies the difference equation $$ a(i,j) - 3 a(i+1,j) + 3a(i+2,j) - a(i+3,j) \sim 0 $$ with solution $a(i,j)\sim\binom{i+2}{2} a(0,j)$. \end{proof} We can now prove a weak analogue for elliptic Gromov-Witten invariants of the (first) Reconstruction Theorem of Kontsevich-Manin (Theorem 3.1 of \cite{KM}). For $0\le j\le d$, let $P_j(V)=\coker( H^{j-2}(V,\mathbb{Q}) \xrightarrow{\omega\cup\*} H^j(V,\mathbb{Q}) )$ be the $j^{\text{th}}$ primitive cohomology group of $V$. \begin{theorem} \label{reconstruction} If $P^i(V)=0$ for $i>2$, the elliptic Gromov-Witten invariants of $V$ are determined by its rational Gromov-Witten invariants together with the Gromov-Witten invariants $\<I_{1,1,\beta}(-)\>:H^{2i+2}(V,\mathbb{Q})\to\mathbb{Q}$ for $0\le c_1(V)\cap\beta=i<d$. (These are all of the non-vanishing Gromov-Witten invariants $\<I_{1,1,\beta}(\alpha)\>$.) \end{theorem} \begin{proof} We proceed by induction: by hypothesis, $\<I_{g,n,\beta}^V\>$ is known for $g=0$ or $g=1$ and $n=1$. Now consider the Gromov-Witten invariant $\<I_{1,n,\beta}^V(\alpha_1,\dots,\alpha_n)\>$, where $n>1$. By \eqref{low}, we may assume that $|\alpha_i|>2$, and under the hypotheses of the proposition, we may write it as $\omega^{p_i}\cup\gamma_i$ where $|\gamma_i|\le2$ is a primitive cohomology class. Step 1: If any two indices $p_i$ and $p_j$ satisfy $p_i+p_j>2$, we may apply Corollary \ref{reduce} to replace the pair $(\omega^{p_i}\cup\gamma_i,\omega^{p_j}\cup\gamma_j)$ by $(\gamma_i,\omega^{p_i+p_j}\cup\gamma_j)$. If $|\gamma_1|=1$, the result vanishes by \eqref{low}, while if $|\gamma_1|=2$, we may apply \eqref{low} to reduce $n$ by $1$. Step 2: We are reduced to considering $\<I_{1,n,\beta}^V(\omega\cup\gamma_1,\dots,\omega\cup\gamma_n)\>$, where the classes $\gamma_i$ have degree $1$ or degree $2$. Applying Theorem \ref{symbol}, we see that $$ \Psi(\omega,\gamma_1,\omega,\gamma_2) = 2\{\omega\cup\gamma_1,\omega\cup\gamma_2\} + \{\omega^2,\gamma_1\cup\gamma_2\} \sim 0 . $$ In particular, we may assume that $n=2$, since otherwise, we would be able to return to Step 1. There are two cases. Step 2a: If the classes $\gamma_i$ are both of degree $1$, we see that $\{\omega\cup\gamma_1,\omega\cup\gamma_2\}\sim0$, since in that case $\gamma_1\cup\gamma_2$ has degree $2$ and we may apply \eqref{low}. Step 2b: If the classes $\gamma_i$ are both of degree $2$, there is a class $\gamma\in H^2(V,\mathbb{Q})$ such that $\gamma_1\cup\gamma_2 = \omega\cup\gamma$, since $P^4(V)=0$. We must calculate $$ \<I_{1,2,\beta}^V(\omega^2,\gamma_1\cup\gamma_2)\> = \<I_{1,2,\beta}^V(\omega^2,\omega\cup\gamma)\> \sim 6 \<I_{1,2,\beta}^V(1,\omega^3\cup\gamma)\> = 0 , $$ where we have applied Corollary \ref{reduce} and \eqref{low}. \end{proof} Two special cases of this result are worth singling out: \begin{enumerate} \item If $V$ is a surface, the elliptic Gromov-Witten invariants are determined by the rational invariants together with $\<I_{1,1,\beta}(-)\>:H^2(V,\mathbb{Q})\to\mathbb{Q}$ for $c_1(V)\cap\beta=0$ and $\<I_{1,1,\beta}(-)\>:H^4(V,\mathbb{Q})\to\mathbb{Q}$ for $c_1(V)\cap\beta=1$. If $V$ is the blow-up of $\mathbb{CP}^2$ at a finite number of points, only $\beta=0$ satisfies $c_1(V)\cap\beta<2$, and by Proposition \ref{BCOV}, $\<I_{1,1,0}\>$ is determined by $c_1(V)$, while the rational Gromov-Witten invariants are determined by the WDVV equation (G\"ottsche-Pandharipande \cite{GP}). \item If $V=\mathbb{CP}^d$, the elliptic Gromov-Witten invariants are determined by the rational Gromov-Witten invariants. \end{enumerate} \section{Gromov-Witten invariants of curves} To illustrate our new relation, we start with the case where $V$ is a curve. We will only discuss curves of genus $0$ and $1$, since for curves of higher genus, $I_{g,n,\beta}^V=0$ if $\beta\ne0$, and the new relation is identically satisfied. \subsection{The projective line} When $V=\mathbb{CP}^1$, the potential $F_g$ is a power series of degree $4g-4$ in variables $t_0$ and $t_1$ (of degree $-2$ and $0$) and the generator $q$ of $\Lambda$, of degree $-4=-2c_1(\mathbb{CP}^1)\cap[\mathbb{CP}^1]$. By degree counting, together with \eqref{low}, we see that $$ F_g(\mathbb{CP}^1) = \begin{cases} \displaystyle t_0^2t_1/2 + q e^{t_1} , & g=0 , \\ \displaystyle -t_1/24 , & g=1 , \\ 0 , & g>1 ; \end{cases}$$ the only thing which is not immediate is the coefficient of $q$ in $F_0(\mathbb{CP}^1)$, which is the number of maps of degree $1$ from $\mathbb{CP}^1$ to itself, up to isomorphism, and clearly equals $1$. It is easy to calculate $F(\delta,\mathbb{CP}^1)$ for $\delta$ equal to one of our nine $2$-cycles: all of them vanish except $$ F(\delta_{3,4},\mathbb{CP}^1) = \frac{t_1^4}{24} \o(-qe^{t_1}/6) ; F(\delta_{0,4},\mathbb{CP}^1) = \frac{t_1^4}{24} \o qe^{t_1} ; F(\delta_\alpha,\mathbb{CP}^1) = \frac{t_1^4}{24} \o 2qe^{t_1} . $$ We see that the new relation holds among these potentials. \subsection{Elliptic curves} Let $E$ be an elliptic curve. Denote by $\xi,\eta$ variables of degree $-1$ corresponding to a basis of $H_1(E,\mathbb{Z})$ such that $\<\xi,\eta\>=1$. The ring $\Lambda$ has one generator $q$, of degree $0$ (since $c_1(V)=0$). Since there are no rational curves in $E$ of positive degree, we have $$ F_0(E) = t_0^2t_1/2 + t_0\xi\eta . $$ It is shown in \cite{BCOV} that \begin{equation} \label{Eisenstein} F_1(E) = - \frac{t_1}{24} + \sum_{\beta=1}^\infty \frac{\sigma(\beta)}{\beta} q^\beta \bigl(e^{\beta t_1} - 1\bigr) , \end{equation} since $\<I_{1,1,\beta}^E(\omega)\>=\sigma(\beta)$ counts the number of unramified covers of degree $\beta$ of the curve $E$ up to automorphisms, which are easily enumerated. An equivalent form of \eqref{Eisenstein} is $$ \frac{\partial F_1(E)}{\partial t_1} = G_2(qe^{t_1}) , $$ where $$ G_2(q) = - \frac{1}{24} + \sum_{\beta=1}^\infty \sigma(\beta) q^\beta $$ is the Eisenstein series of weight $2$. By degree counting, we also see that $F_g(E)=0$ for $g>1$. Note that the Gromov-Witten invariants of an elliptic curve are invariant under deformation; this is true for any smooth projective variety $V$ (Li-Tian \cite{LT}). In fact, the definition of Gromov-Witten invariants extends to any almost-K\"ahler manifold (a symplectic manifold with compatible almost-complex structure), and the resulting invariants are independent of the almost-complex structure (Li-Tian \cite{LT:symp}). It is simple to calculate the Gromov-Witten potentials $F(\delta,E)$ for our nine $2$-cycles in $\overline{\mathcal{M}}_{1,4}$. \begin{lemma} \label{elliptic} We have $$ F(\delta_{2,2},E) = \Bigl( \frac{5}{12} G_4(qe^{t_1}) - G_2(qe^{t_1})^2 \Bigr) (t_0t_1+\xi\eta)^2 = \frac{q}{2} (t_0t_1+\xi\eta)^2 + O(q^2) , $$ $F(\delta_{2,3},E)=3F(\delta_{2,2},E)$, while the remaining $7$ potentials vanish. \qed\end{lemma} Again, we see that the new relation holds. \section{The Gromov-Witten invariants of $\mathbb{CP}^2$} The Gromov-Witten potential $F_g(\mathbb{CP}^2)$ is a power series of degree $2g-2$ in variables $t_0$, $t_1$ and $t_2$, of degrees $-2$, $0$ and $2$, where $t_i$ is dual to $\omega^i$, and the generator $q$ of $\Lambda$, of degree $-6=-2c_1(\mathbb{CP}^2)\cap[L]$. By degree counting, together with \eqref{low}, we see that $$ F_g(\mathbb{CP}^2) = \begin{cases} \displaystyle \frac{1}{2} (t_0t_1^2 + t_0^2t_2) + \sum_{\beta=1}^\infty N^{(0)}_\beta q^\beta e^{\beta t_1} \frac{t_2^{3\beta-1}}{(3\beta-1)!} , & g=0 , \\ \displaystyle - \frac{t_1}{8} + \sum_{\beta=1}^\infty N^{(1)}_\beta q^\beta e^{\beta t_1} \frac{t_2^{3\beta}}{(3\beta)!} , & g=1 , \\ \displaystyle \sum_{\beta=1}^\infty N^{(g)}_\beta q^\beta e^{\beta t_1} \frac{t_2^{3\beta+g-1}}{(3\beta+g-1)!} , & g>1 , \end{cases} $$ where $N^{(g)}_\beta$ are certain rational coefficients. Using the Severi theory of plane curves, we will show that $N^{(g)}_\beta$ is the answer to an enumerative problem for plane curves; in particular, it is a non-negative integer. This phenomenon is special to del~Pezzo surfaces: we have already seen that the elliptic Gromov-Witten invariants of an elliptic curve are non-integral, while for $\mathbb{CP}^3$, they are not even positive. We apply the following result, which is Proposition 2.2 of Harris \cite{Harris}. \begin{proposition} \label{Harris} Let $S$ be a smooth rational surface. Let $\pi:\mathcal{C}\to\mathcal{M}$ be a family of curves of geometric genus $g$ with $\mathcal{M}$ irreducible, and let $f:\mathcal{C}\to\mathcal{M}$ be a map such that on each component of a general fibre $\mathcal{C}_z$ of $\pi$, the restriction $f_z$ of $f$ to $\mathcal{C}_z$ is not constant and $f_z^*\omega_S$ has negative degree. Let $W$ be the image of the map from $\mathcal{M}$ to the Chow variety of curves on $S$ defined by sending $z\in\mathcal{M}$ to the curve $\mathcal{C}_z$. Then $\dim(W)\le-\deg(f_z^*\omega_S)+g-1$, and if equality holds, then $f_z$ is birational for all $z\in\mathcal{M}$. \qed \end{proposition} \begin{corollary} \label{Severi} The coefficient $N^{(g)}_\beta$ equals the number of irreducible plane curves of arithmetic genus $g$ and degree $\beta$ passing through $3\beta+g-1$ general points in $\mathbb{CP}^2$. \end{corollary} \begin{proof} Let $\mathcal{M}$ be a component of the boundary $\overline{\mathcal{M}}_{g,n}(\mathbb{CP}^2,\beta)\setminus\mathcal{M}_{g,n}(\mathbb{CP}^2,\beta)$, and consider the family of curves $\mathcal{C}\to\mathcal{M}$ obtained by restricting the universal curve $\overline{\mathcal{M}}_{g+1,n}(\mathbb{CP}^2,\beta)\to\overline{\mathcal{M}}_{g,n}(\mathbb{CP}^2,\beta)$ to $\mathcal{M}$ and contracting to a point all components of the fibres on which $f$ has degree $0$. The geometric genus of the fibres of this family is bounded above by $g-1$. Applying Proposition \ref{Harris}, we see that the image of $\mathcal{M}$ in the Chow variety of plane curves has dimension at most $3\beta+g-2$. On the other hand, if $\mathcal{M}$ is a component of $\mathcal{M}_{g,n}(\mathbb{CP}^2,\beta)$, and $\mathcal{C}\to\mathcal{M}$ is the universal family of curves $\mathcal{C}\to\mathcal{M}$, we see that the image of $\mathcal{M}$ in the Chow variety of plane curves has dimension less than $3\beta+g-1$ unless the stable maps parametrized by $\mathcal{M}$ are birational to their image. The Gromov-Witten invariant $N^{(g)}_\beta$ equals the degree of the intersection of the image of $\overline{\mathcal{M}}_{g,3\beta+g-1}(\mathbb{CP}^2,\beta)$ in the Chow variety of curves in $\mathbb{CP}^2$ with the cycle of curves passing through $3\beta+g-1$ general points. By Bertini's theorem for homogenous spaces \cite{Kleiman}, we see that the points of intersection are reduced and lie in the components of $\mathcal{M}_{g,n}(\mathbb{CP}^2,\beta)$ on which the map $f$ is birational to its image, and hence an embedding\xspace. (This argument is borrowed from Section 6 of Fulton-Pandharipande \cite{FP}.) The result follows. \end{proof} \subsection{Comparison with the formulas of Caporaso and Harris} Caporaso and Harris \cite{CH} have calculated the numbers $N^{(g)}_\beta$ for all $g\ge0$, and we now turn the comparison of our results for $N^{(1)}_\beta$ . We have not been able to find a proof that our answers agree, but we have verified that this is so for $\beta\le6$. The recursion of Caporaso and Harris for the Gromov-Witten invariants of $\mathbb{CP}^2$ is more easily applied if it is recast in terms of generating functions. \begin{definition} If $\alpha$ is a partition, denote by $\ell(\alpha)$ the number of parts of $\alpha$ and by $|\alpha|$ the sum $\alpha_1+\dots+\alpha_{\ell(\alpha)}$ of the parts of $\alpha$. Let $\alpha!$ be the product $\alpha! = \alpha_1! \dots \alpha_{\ell(\alpha)}!$. \end{definition} Fix a line $L$ in $\mathbb{CP}^2$. If $\alpha$ and $\beta$ are partitions with $|\alpha|+|\beta|=d$, and $\Omega$ is a collection of $\ell(\alpha)$ general points of $L$, let $V^{d,\delta}(\alpha,\beta)(\Omega)=V^{d,\delta}(\alpha,\beta)$ be the generalized Severi variety: the closure of the locus of reduced plane curves of degree $d$ not containing $L$, smooth except for $\delta$ double points, having order of contact $\alpha_i$ with $L$ at $\Omega_i$, and to order $\beta_1,\dots,\beta_{\ell(\beta)}$ at $\ell(\beta)$ further unassigned points of $L$. For example, $V^{d,\delta}(0,1^d)$ is the classical Severi variety of plane curves of degree $d$ with $\delta$ double points, while $V^{d,\delta}(0,21^{d-1})$ is the closure of the locus of plane curves tangent to $L$ at a smooth point. Denote by $V_0^{d,\delta}(\alpha,\beta)$ the union of the components of $V^{d,\delta}(\alpha,\beta)$ whose general point is an irreducible curve. Let $N^{d,\delta}(\alpha,\beta)$ be the degree of $V^{d,\delta}(\alpha,\beta)$ and let $N_0^{d,\delta}(\alpha,\beta)$ be the degree of $V_0^{d,\delta}(\alpha,\beta)$. Form the generating functions \begin{align*} Z &= \sum \frac{z^{\binom{d+1}{2}-\delta+\ell(\beta)}} {\bigl(\binom{d+1}{2}-\delta+\ell(\beta)\bigr)!} \frac{p^\alpha}{\alpha!} q^\beta N^{d,\delta}(\alpha,\beta) , \\ F &= \sum \frac{z^{\binom{d+1}{2}-\delta+\ell(\beta)}} {\bigl(\binom{d+1}{2}-\delta+\ell(\beta)\bigr)!} \frac{p^\alpha}{\alpha!} q^\beta N_0^{d,\delta}(\alpha,\beta) . \end{align*} The integer $\binom{d+1}{2}-\delta+\ell(\beta)$ is the dimension of the variety $V^{d,\delta}(\alpha,\beta)$. The union of curves of degree $d_i$, $1\le i\le n$, with $\delta_i$ double points and partitions $\alpha_i$ and $\beta_i$ is a (reducible) curve has degree $d=d_1+\dots+d_n$ with $$ \delta = \delta_1 + \dots + \delta_n + \sum_{i<j} \delta_i\delta_j $$ double points and partitions $\alpha=(\alpha_1,\dots,\alpha_n)$ and $\beta=(\beta_1,\dots,\beta_n)$. This formula for $\delta$ amounts to the condition that the sum of the dimensions of the generalized Severi varieties $V_0^{d_i,\delta_i}(\alpha_i,\beta_i)$ equals the dimension of $V^{d,\delta}(\alpha,\beta)$. The proof of the relationship $Z = \exp(F)$ between these two generating functions is an exercise in the definition of degree (see Ran \cite{Ran}). Caporaso and Harris prove a recursion which in terms of the generating function $Z$ may be written $$ \frac{\partial Z}{\partial z} = \sum_{k=1}^\infty kq_k\frac{\partial Z}{\partial p_k} + \Res_{t=0} \biggl[ \exp\Bigl( \sum_{k=1}^\infty t^{-k} p_k + \sum_{k=1}^\infty k t^k \frac{\partial}{\partial q_k} \Bigr) \biggr] Z , $$ where $\Res_{t=0}$ is the residue with respect to the formal variable $t$, in other words, the coefficient of $t^{-1}$ when the exponential is expanded.% \footnote{The resemblance of the right-hand side to the Hamiltonian of the Liouville model is striking --- we have no idea why operators so closely resembling vertex operators make their appearance here.} Dividing by $Z$, we obtain $$ \frac{\partial F}{\partial z} = \sum_{k=1}^\infty kq_k\frac{\partial F}{\partial p_k} + \Res_{t=0} \biggl[ \exp\Bigl( \sum_{k=1}^\infty t^{-k} p_k + F|_{q_k\mapsto q_k+kt^k} - F \Bigr) \biggr] , $$ which clearly allows the recursive calculation of the coefficients $N_0^{d,\delta}(\alpha,\beta)$. As a special case of $Z=\exp(F)$, we have $$ 1 + \sum \frac{z^{\binom{d+2}{2}-\delta-1} q^d N^{d,\delta}} {\bigl(\binom{d+2}{2}-\delta-1\bigr)!} = \exp \biggl( \sum \frac{z^{\binom{d+2}{2}-\delta-1} q^d N_0^{d,\delta}} {\bigl(\binom{d+2}{2}-\delta-1\bigr)!} \biggr) , $$ since $\binom{d+1}{2}-\delta+d=\binom{d+2}{2}-\delta-1$. Expanding the exponential, we obtain $$ N^{d,\delta} = \sum_{n=1}^\infty \frac{1}{n!} \sum_{d=d_1+\dots+d_n} \\ \sum_{\substack{\delta=\sum_{i<j}\delta_i\delta_j\\+\delta_1+\dots+\delta_n}} \frac{\bigl(\binom{d+2}{2}-\delta-1\bigr)! N_0^{d_1,\delta_1} \dots N_0^{d_n,\delta_n}} {\bigl(\binom{d_1+2}{2}-\delta_1-1\bigr)! \dots \bigl(\binom{d_i+2}{2}-\delta_i-1\bigr)!} . $$ For example, with $d=5$, we obtain \begin{align*} N_0^{5,4} &= N^{5,4} - \frac{16!}{14!2!} N_0^{4,0}N_0^{1,0} = 36975 - 120 \* 1 = 36855 , \\ N_0^{5,5} &= N^{5,5} - \frac{15!}{13!2!} N_0^{4,1}N_0^{1,0} = 90027 - 105 \* 27 = 87192 , \end{align*} while with $d=6$ and $\delta=9$, we obtain \begin{align*} N_0^{6,9} &= N^{6,9} - 18! \biggl( \frac{N_0^{5,4}}{16!} \frac{N_0^{1,0}}{2!} - \frac{N_0^{4,1}}{13!} \frac{N_0^{2,0}}{3!} - \frac{1}{2} \Bigl( \frac{N_0^{3,0}}{9!} \Bigr)^2 - \frac{1}{2} \frac{N_0^{4,0}}{14!} \Bigl( \frac{N_0^{1,0}}{2!} \Bigr)^2 \biggr) \\ &= 63338881 - 153 \* 36855 \* 1 + 8568 \* 27 \* 1 + \tfrac12 \* 48620 \* 1^2 + \tfrac12 \* 18360 \* 1 \* 1^2 \\ &= 57435240 \end{align*} in agreement with the recursion \eqref{recursion}. By Proposition \ref{Severi}, the relation between the numbers $N_0^{d,\delta}$ and the Gromov-Witten invariants is very simple: $N^{(g)}_d=N_0^{d,\delta}$ where $g=\binom{d-1}{2}-\delta$. In terms of $F$, the Gromov-Witten potentials $F_g(\mathbb{CP}^2)$ are given by the formula $$ \sum_{g=0}^\infty \hbar^{g-1} F_g(\mathbb{CP}^2) = \frac{1}{2\hbar} (t_0^2t_2+t_0t_1^2) - \frac{t_1}{8} + F\big|_{\substack{(q_1,q_2,\dots)=(\hbar^{-3}qe^{t_1},0,\dots) \\ (p_1,p_2,\dots)=(0,0,\dots) , z=\hbar t_2}} . $$ \section{The elliptic Gromov-Witten invariants of $\mathbb{CP}^3$} For $g=0$ and $g=1$, the Gromov-Witten potentials of the projective space $\mathbb{CP}^3$ have the form $$ F_g(\mathbb{CP}^3) = \begin{cases} \bigl( \frac{1}{2} t_0^2t_3 + t_0t_1t_2 + \frac{1}{6} t_1^3 \bigr) + \displaystyle \sum_{4\beta=a+2b} N^{(0)}_{ab} q^\beta e^{\beta t_1} \frac{t_2^at_3^b}{a!b!} , & g=0 , \\ \displaystyle - \frac{t_1}{4} + \sum_{4\beta=a+2b} N^{(1)}_{ab} q^\beta e^{\beta t_1} \frac{t_2^at_3^b}{a!b!} , & g=1 . \end{cases}$$ Here, $t_i$ is the formal variable of degree $2i-2$ dual to $\omega^i\in H^{2i}(\mathbb{CP}^3,\mathbb{Q})$ and $q$ is the generator of the Novikov ring $\Lambda\cong\mathbb{Q}\(q\)$ of $\mathbb{CP}^3$. By Proposition \ref{BCOV}, the coefficient of $t_1$ in $F_1(\mathbb{CP}^3)$ equals $-c_2(\mathbb{CP}^3)/24$. Thus, the coefficient $N^{(g)}_{ab}$ is a rational number which ``counts'' the number of stable maps of degree $\beta$ from a curve of genus $g$ to $\mathbb{CP}^3$ meeting $a$ generic lines and $b$ generic points. It is shown by Fulton and Pandharipande \cite{FP} that $N^{(0)}_{ab}$ equals the number of rational space curves of degree $\beta$ which meet $a$ generic lines and $b$ generic points. In particular, they are non-negative integers. By contrast, the coefficients $N^{(1)}_{ab}$ are neither positive nor integral: for example, $N^{(1)}_{02}=-1/12$. In \cite{cp3}, we prove the following result. \begin{theorem} The number of elliptic space curves of degree $\beta$ passing through $a$ generic lines and $b$ generic points, where $4\beta=a+2b$, equals $N^{(1)}_{ab} + (2\beta-1)N^{(0)}_{ab}/12$. \end{theorem} By evaluating the equation of Proposition \ref{Relation} on $\omega\boxtimes\omega\boxtimes\omega\boxtimes\omega$, we obtain the following relation among the elliptic Gromov-Witten for $\mathbb{CP}^3$: if $a\ge2$, then \begin{multline*} 3 N^{(1)}_{ab} = 4 nN^{(1)}_{a-2,b+1} - \tfrac{1}{4} n^2 N^{(0)}_{ab} + \tfrac{1}{6} n^3 (n-3) N^{(0)}_{a-2,b+1} \\ \shoveleft{ {} - 2 \sum_{\substack{a-2=a_1+a_2\\b+1=b_1+b_2}} \textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} n_2^2 (n-3n_1) \binom{a-2}{a_1} \Bigr\{ n_1 \binom{b}{b_1} + n_2 \binom{b}{b_1-1} \Bigr\} } \\ \shoveleft{ {} + \sum_{\substack{a=a_1+a_2\\b=b_1+b_2}} N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} \textstyle \Bigl\{ n_1n_2 (n+3n_1) \binom{a-2}{a_1} + n_2^2 (3n_1-n) \binom{a-2}{a_1-1} - 6 n_2^3 \binom{a-2}{a_1-2} \Bigr\} \binom{b}{b_1} } \\ \shoveleft{{} + \tfrac{1}{12} {\displaystyle \sum_{\substack{a=a_1+a_2\\b=b_1+b_2}}} N^{(0)}_{a_1b_1} N^{(0)}_{a_2b_2} n_1 n_2^2 } \\ \textstyle \Bigl\{ n_1^2 (3-n_1) \binom{a-2}{a_1} + n_1n_2(n-3n_1-3) \binom{a-2}{a_1-1} + n_2^2 (-n_1+n_2-6) \binom{a-2}{a_1-2} \Bigr\} \binom{b}{b_1} \\ \shoveleft{{} + \tfrac{1}{2} \sum_{\substack{a=a_1+a_2+a_3\\b=b_1+b_2+b_3}} \textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} N^{(0)}_{a_3b_3} \Bigl\{ 2n_1n_2^3n_3(n+3n_1-3n_2) \binom{a-2}{a_2,a_3-2} - 6 n_2^3n_3^3 \binom{a-2}{a_2,a_3} } \\ \textstyle {} + n_2^2n_3^2 (3n_1-n) \Bigl( n_1 \binom{a-2}{a_2-1,a_3-1} + n_2 \binom{a-2}{a_2,a_3-1} + n_3 \binom{a-2}{a_2-1,a_3} \Bigr) \Bigr\} \binom{b}{b_2,b_3} . \end{multline*} This relation determines the elliptic coefficient $N^{(1)}_{ab}$ for $a>0$ in terms of $N^{(1)}_{0,\frac{1}{2}a+b}$, the elliptic coefficients of lower degree, and the rational coefficients. To determine the coefficients $N^{(1)}_{0,b}$, we need the relation obtained by evaluating Proposition \ref{Relation} on $\omega^2\boxtimes\omega^2\boxtimes\omega\boxtimes\omega$: if $b\ge2$, then \begin{multline*} 0 = N^{(1)}_{ab} + \tfrac{1}{24} n(2n-1) N^{(0)}_{a+2,b-1} + \tfrac{1}{48} N^{(0)}_{a+4,b-2} \\ \shoveleft{ {} + \sum_{\substack{a+2=a_1+a_2\\b-1=b_1+b_2}} \textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} \textstyle \Bigl\{ n_2 \Bigl( n \binom{a}{a_1} + n_2 \binom{a}{a_1-1} \Bigr) \binom{b-2}{b_1-1} } \\[-10pt] \shoveright{ \textstyle {} - \frac{1}{6} \Bigl( n_1(6n_1-n_2) \binom{a}{a_1} + n_2 (16n_1-n_2) \binom{a}{a_1-1} + 6n_2^2 \binom{a}{a_1-2} \Bigr) \binom{b-2}{b_1} \Bigr\} } \\ \shoveleft{{} - \tfrac{1}{12} \sum_{\substack{a+4=a_1+a_2\\b-2=b_1+b_2}} \textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} \Bigl( n_1 \binom{a}{a_1} + (2n_1-5n_2) \binom{a}{a_1-1} + 6n_2 \binom{a}{a_1-2} \Bigr) \binom{b-2}{b_1} } \\ \shoveleft{ {} - \tfrac{1}{48} \sum_{\substack{a+4=a_1+a_2\\b-2=b_1+b_2}} N^{(0)}_{a_1b_1} N^{(0)}_{a_2b_2} \textstyle \Bigl( n_1^3(n_1-1) \binom{a}{a_1} + n_1^2n_2(2n_1-2n_2+1) \binom{a}{a_1-1} } \\ \textstyle {} + n_1n_2^2(2n_1-2n_2+7) \binom{a}{a_1-2} + n_2^3(2n_1+5) \binom{a}{a_1-3} + n_2^4 \binom{a}{a_1-4} \Bigr) \binom{b-2}{b_1} \\ \shoveleft{{} - \tfrac{1}{12} \sum_{\substack{a+4=a_1+a_2+a_3\\b-2=b_1+b_2+b_3}} \textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} N^{(0)}_{a_3b_3} \textstyle \Bigl\{ 3n_2n_3 \Bigl( n_2^2 \binom{a}{a_2,a_3-2} + n_3^2 \binom{a}{a_2-2,a_3} \Bigr) } \\ \shoveleft{ {} + \textstyle n_1 \Bigl( n_2^3 \binom{a}{a_2,a_3-4} + n_2^2(6n_1-n_3) \binom{a}{a_2-1,a_3-3} - 7n_2n_3^2 \binom{a}{a_2-2,a_3-2} - 5n_3^3 \binom{a}{a_2-3,a_3-1} \Bigr) } \\ \shoveleft{\textstyle {} + \Bigl( n_2^3(n_1-5n_3) \binom{a}{a_2,a_3-3} + n_2^2n_3(5n_1-7n_3) \binom{a}{a_2-1,a_3-2} } \\ \textstyle {}+ n_2n_3^2 (5n_1-n_3) \binom{a}{a_2-2,a_3-1} + n_3^3(n_1+n_3) \binom{a}{a_2-3,a_3}\Bigr) \Bigr\} \binom{b-2}{b_2,b_3} . \end{multline*} This relation determine the coefficient $N^{(1)}_{0b}$ in terms of elliptic coefficients of lower order and the rational coefficients, and thus ultimately in terms of $N^{(0)}_{02}=1$, the number of lines between two points. Using these relation, we obtain the results of Table 2. Up to degree $3$, Theorem A is easily seen to hold, since there are no elliptic space curves of degrees $1$ and $2$, while all elliptic space curves of degree $3$ lie in a plane. It is well-known that there is one quartic elliptic space curve through $8$ general points, while the number of elliptic quartic space curves through $16$ general lines was calculated by Vainsencher and Avritzer (\cite{Vainsencher}; see also \cite{Avritzer}, which contains a correction to \cite{Vainsencher}, bringing it into agreement with our calculation!). \begin{table} \label{CP3} \caption{Rational and elliptic Gromov-Witten invariants of $\mathbb{CP}^3$} $$\begin{tabular}{|R|C|R|D{.}{}{2}|R|} \hline n & (a,b) & N^{(0)}_{ab} & N^{(1)}_{ab} & {\scriptstyle N^{(1)}_{ab}+(2n-1)N^{(0)}_{ab}/12} \\ \hline 1 & (0,2) & 1 & -.\frac{1}{12} & 0 \\ & (2,1) & 1 & -.\frac{1}{12} & 0 \\ & (4,0) & 2 & -.\frac{1}{6} & 0 \\[5pt] 2 & (0,4) & 0 & .0 & 0 \\ & (2,3) & 1 & -.\frac{1}{4} & 0 \\ & (4,2) & 4 & -1. & 0 \\ & (6,1) & 18 & -4.\frac{1}{2} & 0 \\ & (8,0) & 92 & -23. & 0 \\[5pt] 3 & (0,6) & 1 & -.\frac{5}{12} & 0 \\ & (2,5) & 5 & -2.\frac{1}{12} & 0 \\ & (4,4) & 30 & -12.\frac{1}{2} & 0 \\ & (6,3) & 190 & -78.\frac{1}{6} & 1 \\ & (8,2) & 1\,312 & -532.\frac{2}{3} & 14 \\ & (10,1) & 9\,864 & -3\,960. & 150 \\ & (12,0) & 80\,160 & -31\,900. & 1\,500 \\[5pt] 4 & (0,8) & 4 & -1.\frac{1}{3} & 1 \\ & (2,7) & 58 & -29.\frac{5}{6} & 4 \\ & (4,6) & 480 & -248. & 32 \\ & (6,5) & 4\,000 & -2\,023.\frac{1}{3} & 310 \\ & (8,4) & 35\,104 & -17\,257.\frac{1}{3} & 3\,220 \\ & (10,3) & 327\,888 & -156\,594. & 34\,674 \\ & (12,2) & 3259\,680 & -1\,515\,824. & 385\,656 \\ & (14,1) & 34\,382\,544 & -15\,620\,216. & 4\,436\,268 \\ & (16,0) & 383\,306\,880 & -170\,763\,640. & 52\,832\,040 \\[5pt] 5 & (0,10) & 105 & -36.\frac{3}{4} & 42 \\ & (2,9) & 1\,265 & -594.\frac{3}{4} & 354 \\ & (4,8) & 13\,354 & -6\,523.\frac{1}{2} & 3\,492 \\ & (6,7) & 139\,098 & -66\,274.\frac{1}{2} & 38\,049 \\ & (8,6) & 1\,492\,616 & -677\,808. & 441\,654 \\ & (10,5) & 16\,744\,080 & -7\,179\,606. & 5\,378\,454 \\ & (12,4) & 197\,240\,400 & -79\,637\,976. & 68\,292\,324 \\ & (14,3) & 2\,440\,235\,712 & -928\,521\,900. & 901\,654\,884 \\ & (16,2) & 31\,658\,432\,256 & -11\,385\,660\,384. & 12\,358\,163\,808 \\ & (18,1) & 429\,750\,191\,232 & -146\,713\,008\,096. & 175\,599\,635\,328 \\ & (20,0) & 6\,089\,786\,376\,960 & -1\,984\,020\,394\,752. & 2\,583\,319\,387\,968 \\ \hline \end{tabular}$$ \end{table}

9612

alg-geom/9612013

en

https://arxiv.org/abs/alg-geom/9612013

alg-geom/9612013

Misha S. Verbitsky

Misha Verbitsky

Desingularization of singular hyperkaehler varieties II

LaTeX 2e, 15 pages. This paper can be read independently from the first part. `Desingularization part I' appeared in alg-geom/9611015

This is a second part of alg-geom/9611015. We construct a natural hyperkaehler desingularization for all singular hyperkaehler varieties. The desingularization theorem was proven in alg-geom/9611015 under additional assumption of local homogeneity. Here we show that local homogeneity is redundant: every singular hyperkaehler variety has locally homogeneous singularities.

alg-geom

\section{Introduction} A hyperk\"ahler manifold is a Riemannian manifold with an action of a quaternion algebra $\Bbb H$ in its tangent bundle, such that for all $I\in \Bbb H$, $I^2=-1$, $I$ establishes a complex, K\"ahler structure on $M$ (see \ref{_hyperkahler_manifold_Definition_} for details). We extend this definition to singular varieties. The notion of a singular hyperk\"ahler variety has its origin in \cite{_Verbitsky:Hyperholo_bundles_} (see also \cite{_Verbitsky:Deforma_} and \cite{_Verbitsky:Desingu_}). Examples of singular hyperk\"ahler varieties are numerous, and come from several diverse sources (\ref{_singu_hype_Remark_}, \ref{_hyperho_defo_hyperka_Theorem_}; for additional examples see \cite{_Verbitsky:Deforma_}, Section 10). There is a weaker version of this definition: a notion of {\bf hypercomplex variety}. Singular hypercomplex varieties is what this paper primarily deals with. For a real analytic variety $M$, we say that $M$ is {\bf hypercomplex}, if $M$ is equipped with the complex structures $I$, $J$ and $K$, such that $I\circ J = - J\circ I =K$, and certain integrability conditions are satisfied (for a precise statement, see \ref{_hypercomplex_Definition_}). The paper \cite{_Verbitsky:Desingu_} dealt with the hypercomplex varieties with ``locally homogeneous singularities'' (LHS). A complex analytic variety $M$ is LHS if for each point $x\in M$, the completion $A$ of the local ring ${\cal O}_x M$ can be represented as a quotient of the power series ring by a homogeneous ideal (\ref{_SLHS_Definition_}, \ref{_locally_homo_coord_Claim_}). In \cite{_Verbitsky:Desingu_}, we described explicitly the singularities of hypercomplex LHS varieties. We have shown that every such variety, considered as a complex variety with a complex structure induced from the quaternions, is locally isomorphic to a union of planes in ${\Bbb C}^n$ (\ref{_singula_stru_Theorem_}). The normalization of such a variety is non-singular, which follows from this description of singularities. This gives a canonical, functorial way to desingularize hyperk\"ahler and hypercomplex varieties (\ref{_desingu_Theorem_}). The purpose of the present paper is to show that all hypercomplex varieties have locally homogeneous singularities (\ref{_hyperco_SLHS_Theorem_}). This is used to extend the desingularization results to all singular hyperk\"ahler or hypercomplex varieties (\ref{_hyperco_desingu_Corollary_}). In Sections \ref{_comple_with_au_Section_}--\ref{_homogeni_on_hype_Section_}, we prove that all hypercomplex varieties have locally homogeneous singularities. Section \ref{_comple_with_au_Section_} is purely a commutative algebra. We work with a complete local Noetherian ring $A$ over ${\Bbb C}$. By definition, an automorphism $e:\; A {\:\longrightarrow\:} A$ is called {\bf homogenizing} (\ref{_homogeni_automo_Definition_}) if its differential acts as a dilatation on the Zariski tangent space of $A$, with dilatation coefficient $|\lambda|<1$. As usual, by the Zariski tangent space we understand the space $\mathfrak m_A /\mathfrak m_A^2$, where $\mathfrak m_A$ is a maximal ideal of $A$. The main result of Section \ref{_comple_with_au_Section_} is the following. For a complete local Noetherian ring $A$ over ${\Bbb C}$ equipped with a homogenizing automorphism $e:\; A {\:\longrightarrow\:} A$, we show that $A$ has locally homogeneous singularities. In Section \ref{_homogeni_on_hype_Section_}, we construct a natural homogenizing automorphism of the ring of germs of complex analytic functions on a hypercomplex variety $M$ (\ref{_homogenizing_Proposition_}). Applying Section \ref{_comple_with_au_Section_}, we obtain that every hypercomplex variety has locally homogeneous singularities. \section{Preliminaries} \subsection{Definitions} This subsection contains a compression of the basic definitions from hyperk\"ahler geometry, found, for instance, in \cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}. \hfill \definition \label{_hyperkahler_manifold_Definition_} (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, a hyperk\"ahler manifold has the natural action of quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a complex manifold is denoted by $(M, L)$. \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. \subsection{Trianalytic subvarieties in compact hyperk\"ahler manifolds.} In this subsection, we give a definition and a few basic properties of trianalytic subvarieties of hyperk\"ahler manifolds. We follow \cite{_Verbitsky:Symplectic_II_}. \hfill Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$. \hfill \definition\label{_trianalytic_Definition_} Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is a complex analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$. Denote by $[N]\in H_{2n}(M)$ the homology class represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{_Verbitsky:Symplectic_II_}. \blacksquare \hfill \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. The non-singular part of a trianalytic subvariety is hyperk\"ahler. \subsection{Hypercomplex varieties} This subsection is based on the results and definitions from \cite{_Verbitsky:Desingu_}. Let $X$ be a complex variety, $X_{\Bbb R}$ the underlying real analytic variety. In \cite{_Verbitsky:Desingu_}, Section 2, we constructed a natural automorphism of the sheaf of K\"ahler differentials on $X_{\Bbb R}$ \[ I:\; \Omega^1X_{\Bbb R} {\:\longrightarrow\:} \Omega^1X_{\Bbb R}, \ \ I^2=-1. \] This endomorphism is a generalization of the usual notion of a complex structure operator, and its construction is straightforward. We called $I$ {\bf the complex structure operator on $X_{\Bbb R}$}. The operator $I$ is functorial: for a morphism $f:\; X {\:\longrightarrow\:} Y$ of complex varieties, the natural pullback map $df:\; f^*\Omega^1_{Y_{\Bbb R}} {\:\longrightarrow\:} \Omega^1X_{\Bbb R}$ commutes with the complex structure operators (see \cite{_Verbitsky:Desingu_} for details). The converse statement is also true: \hfill \theorem \label{_commu_w_comple_str_Theorem_} Let $X$, $Y$ be complex analytic varieties, and \[ f_{\Bbb R}:\; X_{\Bbb R}{\:\longrightarrow\:} Y_{\Bbb R}\] be a morphism of underlying real analytic varieties which commutes with the complex structure. Then there exist a unique morphism $f:\; X{\:\longrightarrow\:} Y$ of complex analytic varieties, such that $f_{\Bbb R}$ is its underlying morphism. {\bf Proof:} This is \cite{_Verbitsky:Desingu_}, Theorem 2.1. \blacksquare \hfill \definition Let $M$ be a real analytic variety, and \[ I:\; \Omega^1({\cal O}_M){\:\longrightarrow\:}\Omega^1({\cal O}_M) \] be an endomorphism satisfying $I^2=-1$. Then $I$ is called {\bf an almost complex structure on $M$}. If there exist a complex analytic structure $\mathfrak C$ on $M$ such that $I$ appears as the complex structure operator associated with $\mathfrak C$, we say that $I$ is {\bf integrable}. \ref{_commu_w_comple_str_Theorem_} implies that this complex structure is unique if it exists. \hfill \definition \label{_hypercomplex_Definition_} (Hypercomplex variety) Let $M$ be a real analytic variety equipped with almost complex structures $I$, $J$ and $K$, such that $I\circ J = -J \circ I = K$. Assume that for all $a, b, c\in {\Bbb R}$, such that $a^2 + b^2 + c^2=1$, the almost complex structure $a I + b J + c K$ is integrable. Then $M$ is called {\bf a hypercomplex variety}. \hfill \remark As follows from \cite{_Verbitsky:Desingu_}, Claim 2.7, every hyperk\"ahler manifold is hypercomplex, in a natural way. The proof is straightforward. \subsection{Singular hyperk\"ahler varieties} Throughout this paper, we never use the notion of hyperk\"ahler variety. For our present purposes, the hypercomplex varieties suffice. However, for the reader's benefit, we give a definition and a list of examples of hyperk\"ahler varieties. All hyperk\"ahler varieties are hypercomplex, and the converse is (most likely) false. However, it is difficult to construct examples of hypercomplex varieties which are not hyperk\"ahler, and all ``naturally'' occuring hypercomplex varieties come equipped with a singular hyperk\"ahler structure. This subsection is based on the results and definitions from \cite{_Verbitsky:Deforma_} and \cite{_Verbitsky:Desingu_}. For a more detailed exposition, the reader is referred to \cite{_Verbitsky:Deforma_}, Section 10. \hfill \definition\label{_singu_hype_Definition_} (\cite{_Verbitsky:Hyperholo_bundles_}, Definition 6.5) Let $M$ be a hypercomplex variety (\ref{_hypercomplex_Definition_}). The following data define a structure of a {\bf hyperk\"ahler variety} on $M$. \begin{description} \item[(i)] For every $x\in M$, we have an ${\Bbb R}$-linear symmetric positively defined bilinear form $s_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$ on the corresponding real Zariski tangent space. \item[(ii)] For each triple of induced complex structures $I$, $J$, $K$, such that $I\circ J = K$, we have a holomorphic differential 2-form $\Omega\in \Omega^2(M, I)$. \item[(iii)] Fix a triple of induced complex structure $I$, $J$, $K$, such that $I\circ J = K$. Consider the corresponding differential 2-form $\Omega$ of (ii). Let $J:\; T_x M {\:\longrightarrow\:} T_x M$ be an endomorphism of the real Zariski tangent spaces defined by $J$, and $Re\Omega\restrict x$ the real part of $\Omega$, considered as a bilinear form on $T_x M$. Let $r_x$ be a bilinear form $r_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$ defined by $r_x(a,b) = - Re\Omega\restrict x (a, J(b))$. Then $r_x$ is equal to the form $s_x$ of (i). In particular, $r_x$ is independent from the choice of $I$, $J$, $K$. \end{description} \noindent \remark \label{_singu_hype_Remark_} \nopagebreak \begin{description} \item[(a)] It is clear how to define a morphism of hyperk\"ahler varieties. \item[(b)] For $M$ non-singular, \ref{_singu_hype_Definition_} is equivalent to the usual one (\ref{_hyperkahler_manifold_Definition_}). If $M$ is non-singular, the form $s_x$ becomes the usual Riemann form, and $\Omega$ becomes the standard holomorphically symplectic form. \item[(c)] It is easy to check the following. Let $X$ be a hypercomplex subvariety of a hyperk\"ahler variety $M$. Then, restricting the forms $s_x$ and $\Omega$ to $X$, we obtain a hyperk\"ahler structure on $X$. In particular, trianalytic subvarieties of hyperk\"ahler manifolds are always hyperk\"ahler, in the sense of \ref{_singu_hype_Definition_}. \end{description} \hfill {\bf Caution:} Not everything which is seemingly hyperk\"ahler satisfies the conditions of \ref{_singu_hype_Definition_}. Take a quotient $M/G$ os a hyperk\"ahler manifold by an action of finite group $G$, acting in accordance with hyperk\"ahler structure. Then $M/G$ is, generally speaking, {\it not} hyperk\"ahler (see \cite{_Verbitsky:Deforma_}, Section 10 for details). \hfill The following theorem, proven in \cite{_Verbitsky:Hyperholo_bundles_} (Theorem 6.3), gives a convenient way to construct examples of hyperk\"ahler varieties. \hfill \theorem \label{_hyperho_defo_hyperka_Theorem_} Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $B$ a stable holomorphic bundle over $(M, I)$. Let $\operatorname{Def}(B)$ be the reduction\footnote{The deformation space might have nilpotents in the structure sheaf. We take its reduction to avoid this.} of the deformation space of stable holomorphic structures on $B$. Assume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect to the standard action of $SU(2)$ on $H^*(M)$. Then $\operatorname{Def}(B)$ has a natural structure of a hyperk\"ahler variety. \nopagebreak \blacksquare \section{Desingularization of hyperk\"ahler varieties} In this section, we recall the desingularization theorem for the hypercomplex varieties with locally homogeneous singularities, as it was proven in \cite{_Verbitsky:Desingu_}. In the last subsection, we state the main result of this paper, which is proven in Sections \ref{_comple_with_au_Section_}--\ref{_homogeni_on_hype_Section_}. \subsection{Spaces with locally homogeneous singularities.} \noindent \definition (local rings with LHS) Let $A$ be a local ring. Denote by $\mathfrak m$ its maximal ideal. Let $A_{gr}$ be the corresponding associated graded ring. Let $\hat A$, $\widehat{A_{gr}}$ be the $\mathfrak m$-adic completion of $A$, $A_{gr}$. Let $(\hat A)_{gr}$, $(\widehat{A_{gr}})_{gr}$ be the associated graded rings, which are naturally isomorphic to $A_{gr}$. We say that $A$ {\bf has locally homogeneous singularities} (LHS) if there exists an isomorphism $\rho:\; \hat A {\:\longrightarrow\:} \widehat{A_{gr}}$ which induces the standard isomorphism $i:\; (\hat A)_{gr}{\:\longrightarrow\:} (\widehat{A_{gr}})_{gr}$ on associated graded rings. \hfill \definition\label{_SLHS_Definition_} (SLHS) Let $X$ be a complex or real analytic space. Then $X$ is called {be a space with locally homogeneous singularities} (SLHS) if for each $x\in M$, the local ring ${\cal O}_x M$ has locally homogeneous singularities. \hfill The following claim might shed a light on the origin of the term ``locally homogeneous singularities''. \hfill \claim \label{_locally_homo_coord_Claim_} Let $A$ be a complete local Noetherian ring over ${\Bbb C}$. Then the following statements are equivalent \begin{description} \item[(i)] $A$ has locally homogeneous singularities \item[(ii)] There exist a surjective ring homomorphism $\rho:\; {\Bbb C}[[x_1, ... , x_n]] {\:\longrightarrow\:} A$, where ${\Bbb C}[[x_1, ... , x_n]]$ is the ring of power series, and the ideal $\ker \rho$ is homogeneous in ${\Bbb C}[[x_1, ... , x_n]]$. \end{description} {\bf Proof:} Clear. \blacksquare \subsection{Hyperk\"ahler varieties with locally homogeneous singularities} \noindent \noindent\proposition\label{_comple_LHS<=>real_LHS_Proposition_} Let $M$ be a complex variety, $M_{\Bbb R}$ the underlying real analytic variety. Then $M_{\Bbb R}$ is a space with locally homogeneous singularities (SLHS) if and only if $M$ is a space with locally homogeneous singularities. {\bf Proof:} This is \cite{_Verbitsky:Desingu_}, Proposition 4.6. \blacksquare \hfill \corollary \label{_hype_SLHS_for_diff_indu_c_str_Corollary_} \cite{_Verbitsky:Desingu_} Let $M$ be a hyperk\"ahler variety, $I_1$, $I_2$ induced complex structures. Then $(M, I_1)$ is a space with locally homogeneous singularities if and only is $(M, I_2)$ is SLHS. {\bf Proof:} The real analytic variety underlying $(M, I_1)$ coinsides with that underlying $(M, I_2)$. Applying \ref{_comple_LHS<=>real_LHS_Proposition_}, we immediately obtain \ref{_hype_SLHS_for_diff_indu_c_str_Corollary_}. \blacksquare \hfill \definition Let $M$ be a hyperk\"ahler or hypercomplex variety. Then $M$ is called a space with locally homogeneous singularities (SLHS) if the underlying real analytic variety is SLHS or, equivalently, for some induced complex structure $I$ the $(M, I)$ is SLHS. \hfill Some of the canonical examples of hyperk\"ahler varieties are spaces with locally homogeneous singularities {\it per se}. For instance, it is easy to prove the following theorem: \hfill \theorem (\cite{_Verbitsky:Desingu_}, Theorem 4.9) Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $B$ a stable holomorphic bundle over $(M, I)$. Assume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect to the standard action of $SU(2)$ on $H^*(M)$. Let $\operatorname{Def}(B)$ be a reduction of a deformation space of stable holomorphic structures on $B$, which is a hyperk\"ahler variety by \ref{_hyperho_defo_hyperka_Theorem_}. Then $\operatorname{Def}(B)$ is a space with locally homogeneous singularities (SLHS). \blacksquare \hfill However, for the other examples of hyperk\"ahler varieties, there is no easy {\it ad hoc} way to show that they are SLHS. The main aim of this paper, however, is to prove that every hypercomplex variety is SLHS (see Subsection \ref{_main_resu_Subsection_}). \subsection{Desingularization of hypercomplex varieties which are SLHS} \label{_desingu_for_SLHS_Subsection_} For hypercomplex varieties which are SLHS, we have a complete list of possible singularities (\cite{_Verbitsky:Desingu_}; see also \ref{_singula_stru_Theorem_}). This makes it possible to desingularize every hypercomplex (or hyperk\"ahler) variety in a natural way. The present paper shows that {\em every} hypercomplex variety is SLHS, thus extending the results of \cite{_Verbitsky:Desingu_} to all hypercomplex varieties. For the benefit of the reader, we relate in this Subsection the main results of \cite{_Verbitsky:Desingu_}. We don't use these results in the rest of the article, so the reader is free to skip this Subsection. \hfill Here is the theorem describing the shape of singularities. \hfill \theorem\label{_singula_stru_Theorem_} Let $M$ be a hypercomplex variety, and $I$ an induced complex structure. Assume that $M$ is SLHS. Then, for each point $x\in M$, there exists a neighbourhood $U$ of $x\in (M,I)$, which is isomorphic to $B \cap\left(\bigcup_i L_i\right)$, where $B$ is an open ball in ${\Bbb C}^n$ and $\bigcup_i L_i$ is a union of planes $L_i \in {\Bbb C}^n$ passing through $0\in {\Bbb C}^n$. In particular, the normalization of $(M,I)$ is smooth. {\bf Proof:} See Corollary 5.3 of \cite{_Verbitsky:Desingu_}. \blacksquare \hfill Here is the desingularization theorem. \hfill \theorem \label{_desingu_Theorem_} (\cite{_Verbitsky:Desingu_}, Theorem 6.1) Let $M$ be a hyperk\"ahler or a hypercomplex variety, $I$ an induced complex structure. Assume that $M$ is a space with locally homogeneous singularities. Let \[ \widetilde{(M, I)}\stackrel n{\:\longrightarrow\:} (M,I)\] be the normalization of $(M,I)$. Then $\widetilde{(M, I)}$ is smooth and has a natural hyperk\"ahler (respectively, hypercomplex) structure $\c H$, such that the associated map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} (M,I)$ agrees with $\c H$. Moreover, the hyperk\"ahler (hypercomplex) manifold $\tilde M:= \widetilde{(M, I)}$ is independent from the choice of induced complex structure $I$. \blacksquare \subsection{The main result: every hypercomplex variety is SLHS} \label{_main_resu_Subsection_} The proof of the following theorem is given in Sections \ref{_comple_with_au_Section_}--\ref{_homogeni_on_hype_Section_}. \theorem\label{_hyperco_SLHS_Theorem_} (the main result of this paper) Let $M$ be a hypercomplex variety. Then $M$ is a space with locally homogeneous singularities (SLHS). \hfill \ref{_hyperco_SLHS_Theorem_} has the following immediate corollary. \hfill \corollary \label{_hyperco_desingu_Corollary_} Let $M$ be a hypercomplex or a hyperk\"ahler variety. Then \ref{_singula_stru_Theorem_} (a theorem describing the shape of the singularities of $M$) and \ref{_desingu_Theorem_} (desingularization theorem) hold. \blacksquare \section{Complete rings with automorphisms} \label{_comple_with_au_Section_} \definition \label{_homogeni_automo_Definition_} Let $A$ be a local Noetherian ring over ${\Bbb C}$, equipped with an automorphism $e:\; A {\:\longrightarrow\:} A$. Let $\mathfrak m$ be a maximal ideal of $A$. Assume that $e$ acts on $\mathfrak m /\mathfrak m^2$ as a multiplication by $\lambda\in {\Bbb C}$, $|\lambda|< 1$. Then $e$ is called {\bf a homogenizing automorphism of $A$}. \hfill The aim of the present section is to prove the following statement. \proposition \label{_homogeni_LHS_Proposition_} Let $A$ be a complete Noetherian ring over ${\Bbb C}$, equipped with a homogenizing authomorphism $e:\; A {\:\longrightarrow\:} A$. Then there exist a surjective ring homomorphism $\rho:\; {\Bbb C}[[x_1, ... , x_n]] {\:\longrightarrow\:} A$, such that the ideal $\ker \rho$ is homogeneous in ${\Bbb C}[[x_1, ... , x_n]]$. In particular, $A$ has locally homogeneous singularities (\ref{_locally_homo_coord_Claim_}). \hfill This statement is well known. A reader who knows its proof should skip the rest of this section. \hfill \proposition \label{_homogeni_auto_then_basis_Proposition_} Let $A$ be a complete Noetherian ring over ${\Bbb C}$, equipped with a homogenizing authomorphism $e:\; A {\:\longrightarrow\:} A$. Then there exist a system of ring elements \[ f_1 , ..., f_m \in \mathfrak m, \ \ m = \dim_{\Bbb C}\mathfrak m /\mathfrak m^2, \] which generate $\mathfrak m /\mathfrak m^2$, and such that $e(f_i) = \lambda f_i$. \hfill {\bf Proof:} Let $\underline a\in\mathfrak m /\mathfrak m^2$. Let $a\in \mathfrak m$ be a representative of $\underline a$ in $\mathfrak m$. To prove \ref{_homogeni_auto_then_basis_Proposition_} it suffices to find $c \in \mathfrak m^2$, such that $e(a-c) = \lambda a -\lambda c$. Thus, we need to solve an equation \begin{equation}\label{_a_through_a_Equation_} \lambda c - e(c) = e(a) - \lambda(a). \end{equation} Let $r:= e(a)-\lambda a$. Clearly, $r\in \mathfrak m ^2$. A solution of \eqref{_a_through_a_Equation_} is provided by the following lemma. \hfill \lemma \label{_e-lambda_invertible_Lemma_} In assumptions of \ref{_homogeni_auto_then_basis_Proposition_}, let $r\in \mathfrak m^2$. Then, the equation \begin{equation}\label{_finding_eigen_Equation_} e(c) - \lambda c = r \end{equation} has a unique solution $c \in \mathfrak m^2$. \hfill {\bf Proof:} We need to show that the operator $P:= (e-\lambda)\restrict{\mathfrak m^2}$ is invertible. Consider the adic filtration $\mathfrak m^2 \subset \mathfrak m^3 \subset ...$ on $\mathfrak m^2$. Clearly, $P$ preserves this filtration. Since $\mathfrak m^2$ is complete with respect to the adic filtration, it suffices to show that $P$ is invertible on the successive quotients. The quotient $\mathfrak m^2/\mathfrak m^i$ is finite-dimensional, so to show that $P$ is invertible it suffices to calculate the eigenvalues. Since $e$ is an automorphism, restriction of $e$ to $\mathfrak m^i/\mathfrak m^{i-1}$ is a multiplication by $\lambda^i$. Thus, the eigenvalues of $e$ on $\mathfrak m^2/\mathfrak m^i$ range from $\lambda^2$ to $\lambda^{i-1}$. Since $|\lambda|>|\lambda|^2$, all eigenvalues of $P\restrict{\mathfrak m^2/\mathfrak m^i}$ are non-zero and the restriction of $P$ to $\mathfrak m^2/\mathfrak m^i$ is invertible. This proves \ref{_e-lambda_invertible_Lemma_}. $\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill {\bf The proof of \ref{_homogeni_LHS_Proposition_}.} Consider the map \[ \rho:\; {\Bbb C}[[x_1, ... x_m]] {\:\longrightarrow\:} A,\ \ \rho(x_i) = f_i,\] where $f_1, ... , f_m$ is the system of functions constructed in \ref{_homogeni_auto_then_basis_Proposition_}. By Nakayama, $\rho$ is surjective. Let $e_\lambda:\; {\Bbb C}[[x_1, ... x_m]] {\:\longrightarrow\:}{\Bbb C}[[x_1, ... x_m]] $ be the automorphism mapping $x_i $ to $\lambda x_i$. Then, the diagram \[\begin{CD} {\Bbb C}[[x_1, ... x_m]] @>{\rho}>> A \\ @V{e_\lambda}VV @VV{e}V\\ {\Bbb C}[[x_1, ... x_m]] @>{\rho}>> A \end{CD} \] is by construction commutative. Therefore, the ideal $I= \ker \rho$ is preserved by $e_\lambda$. Clearly, every $e_\lambda$-preserved ideal $I\subset {\Bbb C}[[x_1, ... x_m]]$ is homogeneous. \ref{_homogeni_LHS_Proposition_} is proven. \blacksquare \section[Authomorphisms of local rings of holomorphic functions on hyperk\"ahler varieties] {Authomorphisms of local rings of holomorphic functions on hyperk\"ahler varieties} \label{_homogeni_on_hype_Section_} Let $M$ be a hypercomplex variety, $x\in M$ a point, $I$ an induced complex structure. Let $A_I:= \hat {\cal O}_x(M,I)$ be the adic completion of the local ring ${\cal O}_x(M,I)$ of $x$-germs of holomorphic functions on the complex variety $(M,I)$. Clearly, the sheaf ring of the antiholomorphic functions on $(M,I)$ coinsides with ${\cal O}_x(M,-I)$. Thus, the corresponding completion ring is $A_{-I}$. As in \cite{_Verbitsky:Desingu_}, Claim 2.1, we have the natural isomorphism of completions: \begin{equation}\label{_co_ana_and_rea_isom_Equation_} \widehat{A_I \otimes_{\Bbb C} A_{-I}} =\widehat{{\cal O}_x(M_{\Bbb R})\otimes_{\Bbb R} {\Bbb C}}, \end{equation} where \[ \widehat{{\cal O}_x(M_{\Bbb R})\otimes_{\Bbb R} {\Bbb C}}\] is the $x$-completion of the ring of germs of real analytic complex-valued functions on $M$. Consider the natural quotient map \[ p:\;A_{-I}{\:\longrightarrow\:} {\Bbb C}.\] Denote the ring \[ \widehat{{\cal O}_x(M_{\Bbb R})\otimes_{\Bbb R} {\Bbb C}}\] by $A_{\Bbb R}$. Let $i_I:\; A_I \hookrightarrow A_{\Bbb R}$ be the natural embedding \[ a \mapsto a\times 1\in\widehat{A_I \otimes_{\Bbb C} A_{-I}},\] and $e_I:\; A_{\Bbb R} {\:\longrightarrow\:} A_I$ be the natural epimorphism associated with the surjective map \[ A_I \otimes_{\Bbb C} A_{-I} {\:\longrightarrow\:} A_I,\ \ a\otimes b \mapsto a\otimes p(b), \] where $a\in A_I$, $b\in A_{-I}$, and \[ a\otimes b\in{A_I \otimes_{\Bbb C} A_{-I}}\subset A_{\Bbb R}.\] For an induced complex structure $J$, we define $A_J$, $A_{-J}$, $i_J$, $e_J$ likewise. Let $\Psi_{I,J}:\; A_I {\:\longrightarrow\:} A_I$ be the composition \[ A_I \stackrel {i_I}{\:\longrightarrow\:} A_{\Bbb R}\stackrel {e_J}{\:\longrightarrow\:} A_J \stackrel {i_J}{\:\longrightarrow\:} A_{\Bbb R}\stackrel {e_I}{\:\longrightarrow\:} A_I. \] Clearly, for $I=J$, the ring morphism $\Psi_{I,J}$ is identity, and for $I=-J$, $\Psi_{I,J}$ is an augmentation map. \hfill \proposition \label{_homogenizing_Proposition_} Let $M$ be a singular hyperk\"ahler variety, $x\in M$ a point, $I$, $J$ be induced complex structures, such that $I\neq J$ and $I\neq -J$. Consider the map $\Psi_{I,J}:\; A_I {\:\longrightarrow\:} A_I$ defined as above. Then $\Psi_{I,J}$ is a homogenizing automorphism of $A_I$.\footnote{For the definition of a homogenizing automorphism, see \ref{_homogeni_automo_Definition_}.} \hfill {\bf Proof:} Let $d\Psi$ be differential of $\Psi_{I,J}$, that is, the restriction of $\Psi_{I,J}$ to $\mathfrak m/\mathfrak m^2$, where $\mathfrak m$ is the maximal ideal of $A_I$. By Nakayama, to prove that $\Psi_{I,J}$ is an automorphism it suffices to show that $d\Psi$ is invertible. To prove that $\Psi_{I,J}$ is homogenizing, we have to show that $d\Psi$ is a multiplication by a complex number $\lambda$, $|\lambda|<1$. As usually, we denote the real analytic variety underlying $M$ by $M_{\Bbb R}$. Let $T_I$, $T_J$, $\underline {T}_{\Bbb R}$ be the Zariski tangent spaces to $(M,I)$, $(M,J)$ and $M_{\Bbb R}$, respectively, in $x\in M$. Consider the complexification $T_{\Bbb R}:= \underline {T}_{\Bbb R}\otimes {\Bbb C}$, which is a Zariski tangent space to the local ring $A_{\Bbb R}$. To compute $d\Psi:\; T_I {\:\longrightarrow\:} T_I$, we need to compute the differentials of $e_I$, $e_J$, $i_I$, $i_J$, i. e., the restrictions of the homomorphisms $e_I$, $e_J$, $i_I$, $i_J$ to the Zariski tangent spaces $T_I$, $T_J$, $T_{\Bbb R}$. Denote these differentials by $de_I$, $de_J$, $di_I$, $di_J$. \hfill \lemma \label{_i_e_through_Hodge_Lemma_} Let $M$ be a hyperk\"ahler variety, $M_{\Bbb R}$ the associated real analytic variety, $x\in M$ a point. Consider the space $T_{\Bbb R} := T_x (M_{\Bbb R})\otimes {\Bbb C}$. For an induced complex structure $I$, consider the Hodge decomposition $T_{\Bbb R}= T^{1,0}_I \oplus T^{0,1}_I$. In our previous notation, $T_I^{1,0}$ is $T_I$. Then, $di_I$ is the natural embedding of $T_I = T_I^{1,0}$ to $T_{\Bbb R}$, and $de_I$ is the natural projection of $T_{\Bbb R}= T^{1,0}_I \oplus T^{0,1}_I$ to $T_I^{1,0}=T_I$. {\bf Proof:} Clear. \blacksquare \hfill We are able now to describe the map $d\Psi:\; T_I {\:\longrightarrow\:} T_I$ in terms of the quaternion action. Recall that the space $T_I$ is equipped with a natural ${\Bbb R}$-linear quaternionic action. For each quaternionic linear space $\underline V$ and each quaternion $I$, $I^2=-1$, $I$ defines a complex structure in $\underline V$. Such a complex structure is called {\bf induced by the quaternionic structure}. \hfill \lemma \label{_Psi_through_quate_Lemma_} Let $\underline V$ be a space with quaternion action, and $V:= \underline V \otimes {\Bbb C}$ its complexification. For each induced complex structure $I\in {\Bbb H}$, consider the Hodge decomposition $V:= V_I^{1,0} \oplus V_I^{0,1}$. For an induced complex structures $I, J\in \Bbb H$, let $\Phi_{I,J}(V)$ be a composition of the natural embeddings and projections \[ V_I^{1,0} {\:\longrightarrow\:} V {\:\longrightarrow\:} V_J^{1,0} {\:\longrightarrow\:} V {\:\longrightarrow\:} V_I^{1,0}. \] Using the natural identification $\underline V \cong V_I^{1,0}$, we consider $\Phi_{I,J}(V)$ as an ${\Bbb R}$-linear automorphism of the space $\underline V$. Then, applying the operator $\Phi_{I,J}(V)$ to the quaternionic space $T_I$, we obtain the operator $d\Psi$ defined above. {\bf Proof:} Follows from \ref{_i_e_through_Hodge_Lemma_} \blacksquare \hfill As we have seen, to prove \ref{_homogenizing_Proposition_} it suffices to show that $d\Psi$ is a multiplication by a non-zero complex number $\lambda$, $|\lambda| < 1$. Thus, the proof of \ref{_homogenizing_Proposition_} is finished with the following lemma. \hfill \lemma\label{_compu_of_Psi_for_qua_Lemma_} In assumptions of \ref{_Psi_through_quate_Lemma_}, consider the map \[ \Phi_{I,J}(V):\; V_I^{1,0} {\:\longrightarrow\:} V_I^{1,0}.\] Then $\Phi_{I,J}(V)$ is a multiplication by a complex number $\lambda$. Moreover, $\lambda$ is a non-zero number unless $I=-J$, and $|\lambda|< 1$ unless $I=J$. \hfill {\bf Proof:} Let $\underline V= \oplus \underline V_i$ be a decomposition of $V$ into a direct sum of $\Bbb H$-linear spaces. Then, the operator $\Phi_{I,J}(V)$ can also be decomposed: $\Phi_{I,J}(V) = \oplus \Phi_{I,J}(V_i)$. Thus, to prove \ref{_compu_of_Psi_for_qua_Lemma_} it suffices to assume that $\dim_{\Bbb H} \underline V=1$. Therefore, we may identify $\underline V$ with the space $\Bbb H$, equipped with the right action of quaternion algebra on itself. Consider the left action of $\Bbb H$ on $\underline V = \Bbb H$. This action commutes with the right action of $\Bbb H$ on $\underline V$. Consider the corresponding action \[ \rho:\; SU(2) {\:\longrightarrow\:} \operatorname{End}(\underline V) \] of the group of unitary quaternions ${\Bbb H}^{un}=SU(2)\subset \Bbb H$ on $\underline V$. Since $\rho$ commutes with the quaternion action, $\rho$ preserves $V^{1,0}_I \subset V$, for every induced complex structure $I$. By the same token, for each $g\in SU(2)$, the endomorphism $\rho(g)\in \operatorname{End}(V^{1,0}_I)$ commutes with $\Phi_{I,J}(V)$. Consider the 2-dimensional ${\Bbb C}$-vector space $V^{1,0}_I$ as a representation of $SU(2)$. Clearly, $V^{1,0}_I$ is an irreducible representation. Thus, by Schur's lemma, the automorphism $\Phi_{I,J}(V)\in \operatorname{End}(V^{1,0}_I))$ is a multiplication by a complex constant $\lambda$. The estimation $0< |\lambda| < 1$ follows from the following elementary argument. The composition $i_I \circ e_J$ applied to a vector $v\in V_I^{1,0}$ is a projection of $v$ to $V_J^{1,0}$ along $V_J^{0,1}$. Consider the natural Euclidean metric on $V = \Bbb H$. Clearly, the decomposition $V = V_J^{1,0}\oplus V_J^{0,1}$ is orthogonal. Thus, the composition $i_I \circ e_J$ is an orthogonal projection of $v\in V_I^{1,0}$ to $V_J^{1,0}$. Similarly, the composition $i_J \circ e_I$ is an orthogonal projection of $v\in V_J^{1,0}$ to $V_I^{1,0}$. Thus, the map $\Phi_{I,J}(V)$ is an orthogonal projection from $V_I^{1,0}$ to $V_J^{1,0}$ and back to $V_I^{1,0}$. Such a composition always decreases a length of vectors, unless $V_I^{1,0}$ coincides with $V_J^{1,0}$, in which case $I=J$. Also, unless $V_I^{1,0} = V_J^{0,1}$, $\Phi_{I,J}(V)$ is non-zero; in the later case, $I = -J$. \ref{_homogenizing_Proposition_} is proven. This finishes the proof of \ref{_hyperco_SLHS_Theorem_}. \blacksquare \hfill {\bf Acknowledgements:} A nice version of the proof of \ref{_e-lambda_invertible_Lemma_} was suggested by Roma Bez\-ru\-kav\-ni\-kov. I am grateful to A. Beilinson, R. Bez\-ru\-kav\-ni\-kov, P. Deligne, D. Kaledin, D. Kazhdan, M. Kontsevich, V. Lunts, T. Pantev and S.-T. Yau for enlightening discussions.

9612

alg-geom/9612007

en

https://arxiv.org/abs/alg-geom/9612007

alg-geom/9612007

Donu Arapura

Donu Arapura and Pramathanath Sastry

Intermediate Jacobians of moduli spaces

AMS-LaTeX, 16 pages

Let $SU_X(n,L)$ be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g curve X. Let $SU_X^s(n,L)$ denote the open subset parametrizing stable bundles. We show that if g>3 and n > 1, then the mixed Hodge structure on $H^3(SU_X^s(n, L))$ is pure of type ${(1,2),(2,1)}$ and it carries a natural polarization such that the associated polarized intermediate Jacobian is isomorphic J(X). This is new when deg L and n are not coprime. As a corollary, we obtain a Torelli theorem that says roughly that $SU_X^s(n,L)$ (or $SU_X(n,L)$) determines X. This complements or refines earlier results of Balaji, Kouvidakis-Pantev, Mumford-Newstead, Narasimhan-Ramanan, and Tyurin.

alg-geom

\section{Introduction}\label{s:intro} We work throughout over the complex numbers ${\Bbb C}$, i.e. all schemes are over ${\Bbb C}$ and all maps of schemes are maps of ${\Bbb C}$-schemes. A curve, unless otherwise stated, is a smooth complete curve. Points mean geometric points. We will, as is usual in such situations, toggle between the algebraic and analytic categories without warning. For a quasi-projective algebraic variety $Y$, the (mixed) Hodge structure associated with its $i$-th cohomology will be denoted $H^i(Y)$. For a curve $X$, ${\cal S}{\cal U}_X(n,\,L)$ will denote the moduli space of {\it semi-stable} vector bundles of rank $n$ and determinant $L$. The smooth open subvariety defining the {\it stable locus} will be denoted ${\cal S}{\cal U}_X^s(n,\,L)$. We assume familiarity with the basic facts about such a moduli space as laid out, for example in \cite{css-drez},\,pp.\,51--52,\,VI.A (see also Theorems 10, 17 and 18 of {\it loc.cit.}). Our principal result is the following theorem\,: \begin{thm}\label{thm:main} Let $X$ be a curve of genus $g\ge 3$, $n\ge 2$ an integer, and $L$ a line bundle of degree $d$ on $X$ with $d$ odd if $g=3$ and $n=2$. Let ${\cal S}^s={\cal S}{\cal U}_X^s(n,\,L)$. Then $H^3({\cal S}^s)$ is a pure Hodge structure of type $\{(1,2),\,(2,1)\}$, and it carries a natural polarization making the intermediate Jacobian $$J^2({\cal S}^s) = \frac{H^3({\cal S}^s,\,{\Bbb C})} {F^2+H^3({\cal S}^s,\,{\Bbb Z})} $$ into a principally polarized abelian variety. There is an isomorphism of principally polarized abelian varieties $J(X)\simeq J^2({\cal S}^s)$. \end{thm} The word ``natural'' above has the following meaning: an isomorphism between any two ${\cal S}^s$'s as above will induce an isomorphism on third cohomology which will respect the indicated polarizations. As an immediate corollary, we obtain the following Torelli theorem: \begin{cor}\label{cor:main} Let $X$ and $X'$ be curves of genus $g\ge 3$, $L$ and $L'$ line bundles of degree $d$ on $X$ and $X'$ respectively, and $n\ge 2$ an integer. If \begin{equation}\label{eqn:stable} {\cal S}{\cal U}_X^s(n,\,L) \simeq {\cal S}{\cal U}_{X'}^s(n,\,L') \end{equation} or if \begin{equation}\label{eqn:semistable} {\cal S}{\cal U}_X(n,\,L) \simeq {\cal S}{\cal U}_{X'}(n,\,L') \end{equation} then $$ X \simeq X' , $$ except when $g=3, n=2, (n,\,d)\ne 1$. \end{cor} \begin{pf} Since ${\cal S}{\cal U}_X^s(n,\,L)$ (resp. ${\cal S}{\cal U}_{X'}^s(n,\,L')$) is the smooth locus of ${\cal S}{\cal U}_X(n,\,L)$ (resp. ${\cal S}{\cal U}_{X'}(n,\,L')$), therefore it is enough to assume \eqref{eqn:stable} holds. By assumption $J^2({\cal S}{\cal U}^s_X(n,\,L))\simeq J^2({\cal S}{\cal U}^s_{X'}(n,\,L'))$ as polarized abelian varieties. Therefore $J(X)\simeq J(X')$, and the corollary follows from the usual Torelli theorem. \end{pf} The theorem is new for $(n,\,d) \ne 1$ (the so called ``non-coprime case"). When $(n,\,d)=1$ (the ``coprime case"), the theorem (and its corollary) has been proven by Narasimhan and Ramanan \cite{N-R}, Tyurin \cite{Ty} and (for $n=2$) by Mumford and Newstead \cite{M-N}. In the non-coprime case, Kouvidakis and Pantev \cite{KP} have proved the above corollary under the assumption \eqref{eqn:semistable}, and in fact the full result can be deduced from this case. \footnote{In fact, the the exceptional case in the corollary can be eliminated using the results in \cite{KP}} However the present line of reasoning is extremely natural, and is of a rather different character from that of Kouvidakis and Pantev. In particular, Theorem\, \ref{thm:main} will not follow from their techniques. In the special case where $n=2$ and $L= {\cal O}_X$, Balaji \cite{balaji} has shown a similar Torelli type theorem for Seshadri's canonical desingularization $N \to {\cal S}{\cal U}_X(2, {\cal O}_X)$ \cite{css-desing} in the range $g >3$. \footnote{Balaji states the result for $g \ge 3$, but his proof seems to work only for $g > 3$. (See Remark\,\ref{rmk:balaji}).} In the coprime case, the proofs in \cite{M-N} and \cite{N-R} rely on the fact that ${\cal S}{\cal U}_X^s(n,\,L) = {\cal S}{\cal U}_X(n,\,L)$, and hence ${\cal S}{\cal U}_X^s(n,\,L)$ is smooth projective, and most importantly the product $X \times {\cal S}{\cal U}_X(n,\,L)$ possesses a Poincar{\'e} bundle. In the non-coprime case ${\cal S}{\cal U}_X^s(n,\,L)$ is not complete and a result of Ramanan (see \cite{R}) says that there is no Poincar{\'e} bundle on $X \times U$ for any Zariski open subset $U$ of ${\cal S}{\cal U}_X(n,\,L)$. We concentrate primarily on the non-coprime case---the only remaining case of interest. Our strategy is to use a Hecke correspondence to relate the Hodge structure on $H^3({\cal S}{\cal U}^s_X(n,\,L))$ to that on $H^1(X)$. To this extent our proof resembles Balaji's in \cite{balaji}. We are able to deduce more than Balaji does by imposing a polarization (which varies well with ${\cal S}{\cal U}_X^s(n,\,L)$) on the Hodge structure of $H^3({\cal S}{\cal U}_X^s(n,\,L))$. This construction of the polarization needs a version of Lefschetz's Hyperplane Theorem (for quasi projective varieties. See Theorem\,\ref{thm:lefschetz}). There is however another approach to the problem of polarization, which uses M. Saito's theory of polarizations on Hodge modules (see Remark\,\ref{rmk:saito}). \section{The Main Ideas}\label{s:main-idea} For the rest of the paper, we fix a curve $X$ of genus $g$, $n \in {{\Bbb N}}$, $d \in {{\Bbb Z}}$ and a line bundle $L$ of degree $d$ on $X$. Assume, as in the main theorem, that if $n=2$, then $g \ge 4$, and that $g \ge 3$ otherwise. We shall assume, with one brief exception in step 3 below, that $(n,\,d)\ne 1$. We will also assume, for the rest of the paper, that $0 < d \le n$. This involves no loss of generality, for ${\cal S}{\cal U}_X(n,\,L)$ is canonically isomorphic to ${\cal S}{\cal U}_X(n,\,L\otimes\xi^n)$ for every line bundle $\xi$ on $X$. Let ${\cal S} = {\cal S}{\cal U}_X(n,\,L)$ and ${\cal S}^s = {\cal S}{\cal U}^s_X(n,\,L)$ and let $U\subseteq {\cal S}$ be a smooth open set containing ${\cal S}^s$. The broad strategy of our proof is as follows\,: Fix a set $\chi = \{x^1,\ldots,\,x^{d-1}\} \subset X$ of $d-1$ distinct points. \begin{step} First show that there are isomorphisms (modulo torsion), depending only on $(X,\,L,\,\chi)$, of Hodge structures $$ \psi_{X,L,\chi}\colon H^1(X)(-1) \stackrel{\sim}{\longrightarrow} H^3({\cal S}^s) $$ where $(-1)$ is the Tate twist. The isomorphism should vary well with the data $(X,\,L,\,\chi)$. More precisely, suppose $\widetilde{X}\overset{h}{\to} T$ is a family of curves of genus $g$, ${\cal L}$ a line bundle on $\widetilde{X}$, whose restrictions to the fibres of $h$ are of degree $d$, and $\widetilde{\chi}$ a set of $d-1$ mutually disjoint $T$-valued points on $\widetilde{X}$. Let the specialization of $(\widetilde{X},\,{\cal L},\,\widetilde{\chi})$ at $t\in T$ be $(X_t,\,L_t,\,\chi_t)$. Let $\widetilde{{\cal S}^s}\overset{g}{\to} T$ be the resulting family $\{{\cal S}{\cal U}_{X_t}^s(n,\,L_t)\}$. Then there is an isomorphism (modulo torsion) of variation of Hodge structures $$ \widetilde{\psi}\colon R^1h_*{\Bbb Z}(-1) \stackrel{\sim}{\longrightarrow} R^3g_*{\Bbb Z}, $$ which specializes at each $t\in T$ to $\psi_{X_t,L_t,\chi_t}$. Note that $\psi_{X,L,\chi}$ gives an isomorphism of complex tori $$ \varphi_{X,L,\chi}\colon J(X) \stackrel{\sim}{\longrightarrow} J^2({\cal S}^s) $$ also varies well with $(X,\,L,\,\chi)$. \end{step} \begin{step} Find a (possibly nonprincipal) polarization $\Theta({\cal S}^s)$ on $J^2({\cal S}^s)$ which depends only on ${\cal S}^s$, and varies well with ${\cal S}^s$. Let $\mu = \mu_{X,L,\chi}$ be the polarization on $J(X)$ induced by $\Theta({\cal S}^s)$ and $\varphi_{X,L,\chi}$. \end{step} \begin{step} In this step we relax the above assumptions, and no longer insist that $(n,d) \ne 1$. Suppose Steps 1 and 2 have been taken (see \cite{N-R} for the coprime case). Theorem\,\ref{thm:main} will follow by showing that there exists an integer $m$ such that $\frac{1}{m}\Theta$ is principal, and that $J^2({\cal S}^s)$ equipped with this polarization is isomorphic to $J(X)$ with its canonical polarization. The essence of the argument will be to show that any natural polarization on $J(X)$ must be a multiple of the standard one. The argument is lifted from \cite{balaji},\,\S5 where the idea is attributed to S. Ramanan. Pick a curve $X_{0}$ of genus $g$ such that the Neron-Severi group of its Jacobian, $NS(J(X_{0}))$ is ${\Bbb Z}$. By \cite{mori} such an $X_{0}$ exists. Pick a line bundle $L_{0}$ of degree $d$ on $X_{0}$, and a set of $d-1$ distinct points $\chi_{0} = \{x^1_{0},\ldots,\,x^{d-1}_{0}\}$ in $X_{0}$. One finds a family of curves $\widetilde{X} \to T$, a line bundle ${\cal L}$ on $\widetilde{X}$, and a set of $d-1$ mutually disjoint $T$-valued points $\widetilde{\chi} = \{\widetilde{x}^{\,1},\ldots,\,\widetilde{x}^{d-1} \}$, so that $(\widetilde{X},\,{\cal L},\,\widetilde{\chi})$ interpolates between $(X_{0},\,L_{0},\,\chi_{0})$ and $(X,\,L,\,\chi)$. To get such a triple, first observe that since the moduli space ${\cal{M}}_{g,d-1}$ of pointed curves is irreducible and quasi-projective, we can find $(\widetilde{X}\to T,\, \widetilde{\chi})$ interpolating between $(X_0,\,\chi_0)$ and $(X,\,\chi)$. Let $\widetilde{\operatorname{Pic}}^d \to T$ be the resulting family of degree $d$ components of the Picard groups. Since $L_{0}$ and $L$ are points on $\widetilde{\operatorname{Pic}}^d$, one can connect them by a (possibly singular, incomplete) curve $T'$. Base change everything to $T'$. Renaming $T'$ as $T$ and the resulting family of pointed curves as $(\widetilde{X},\,\widetilde{\chi})$ we get a $T$-valued point of the resulting bundle of degree $d$ components of the Picard groups. The line bundle ${\cal L}$ on $\widetilde{X}$ corresponding to this section completes the triple. We denote the specialization of $(\widetilde{X},\,{\cal L},\,\widetilde{\chi})$ at $t\in T$ by $(X_t,\,L_t,\,{\chi}_t)$. Let $t_0, t_1 \in T$ be points where $(X_0,\,L_0,\,{\chi}_0)$ and $(X, L, \chi)$ are realized. The ${\cal S}{\cal U}_{X_t}(n,\,L_t)$ string themselves into a family $\widetilde{{\cal S}} \to T$ (one uses Geometric Invariant Theory over the base $T$ to get $\widetilde{{\cal S}}$. The specializations behave well since we are working over ${\Bbb C}$). Similarly we have a family $\widetilde{{\cal S}^s} \to T$ specializing at $t \in T$ to ${\cal S}{\cal U}^s_{X_t}(n,\,L_t)$. The intermediate Jacobians $J^2({\cal S}{\cal U}^s_{X_t}(n,\,L_t))$ also string together into a family of abelian varieties ${\cal{A}} \to T$. Let ${\cal J}\to T$ be the family $\{J(X_t)\}$ of Jacobians. Step\,1 then gives an isomorphism of group schemes $$ \widetilde{\varphi}: {\cal J}\longrightarrow {\cal{A}} $$ which specializes at $t \in T$ to $\varphi_{X_t,L_t,\chi_t}$. By Step\,2 we get a family of polarizations $\{\mu_t=\mu_{X_t,L_t,\chi_t}\}_{t\in T}$ on ${\cal J}$. Since $NS(J(X_0)) = {\Bbb Z}$, therefore there exists an integer $m \ne 0$, such that $$ m\omega_{X_0} = \mu_{t_0} $$ where, for any curve $C$, $\omega_C$ denotes the principal polarization on $J(C)$. Since $\{\omega_{X_t}\}$ is a family of polarizations on ${\cal J}$ and since the Neron-Severi group is discrete, therefore $$ m\omega_{X_t} = \mu_t \qquad \qquad (t \in T). $$ Theorem\,\ref{thm:main} is now immediate. \end{step} \subsection{The isomorphism $\psi_{X,L,\chi}$.}\label{ss:psi} One produces $\psi_{X,L,\chi}$ as follows\,: Let $$ {\cal S}_1 = {\cal S}{\cal U}_X(n,\,L\otimes{\cal O}_X(-D)) $$ where $D$ is the divisor $\{x^1\}+\ldots +\{x^{d-1}\}$. Since the degree of $L\otimes{\cal O}_X(-D)$ is $1$, therefore ${\cal S}_1$ is smooth and there exists a Poincar{\'e} bundle ${\cal W}$ on $X\times{\cal S}_1$. Let ${\cal W}_1,\ldots,\,{\cal W}_{d-1}$ be the $d-1$ vector bundles on ${\cal S}_1$ obtained by restricting ${\cal W}$ to $\{x^1\}\times{{\cal S}_1}={\cal S}_1,\ldots,\,\{x^{d-1}\}\times{{\cal S}_1}={\cal S}_1$ respectively. Let ${\Bbb P}_k={\Bbb P}({\cal W}_k)$, $k=1,\ldots,\,d-1$, and ${\Bbb P}$ $(={\Bbb P}_{X,L,\chi})$ be the product ${\Bbb P}_1\times_{{\cal S}_1}\ldots\times_{{\cal S}_1}{\Bbb P}_{d-1}$. We will show (in \S\ref{s:hecke}) that there is a correspondence \begin{equation}\label{eqn:hecke} {\cal S}_1 \stackrel{\scriptstyle{\pi}}{\longleftarrow} {{\Bbb P}} \stackrel{\scriptstyle{f}}{\longrightarrow} {\cal S} \end{equation} where $\pi=\pi_{X,L,\chi}$ is the natural projection and $f=f_{X,L,\chi}$ is defined (via a generalized Hecke correspondence) in \ref{ss:map-f} (see \eqref{eqn:f}). We have isomorphisms of (integral, pure) Hodge structures \begin{equation}\label{eqn:tate} H^1(X,\,{\Bbb Z})(-1) \stackrel{\sim}{\longrightarrow} H^3({\cal S}_1,\,{\Bbb Z}) \stackrel{\sim}{\longrightarrow} H^3({\Bbb P},\,{\Bbb Z}). \end{equation} where the first isomorphism is that in \cite{N-R},\,p.\,392,\,Theorem\,3, and the second is given by Leray-Hirsch. Let ${\Bbb P}^s = f^{-1}({\cal S}^s)$. In \S\ref{s:hecke} (see Remark\,\ref{rmk:proj-bundle}, and \ref{ss:codim}) we will show \begin{prop}\label{prop:hecke} \begin{enumerate} \item[(a)] If $n\ge 3$ and $g \ge 3$, the codimension of ${\Bbb P}\setminus {\Bbb P}^s$ in ${\Bbb P}$ is at least $3$. \item[(b)] The map ${\Bbb P}^s \to {\cal S}^s$ is a ${\Bbb P}^{n-1}\times\ldots\times{\Bbb P}^{n-1}$ bundle, where the product is $(d-1)$-fold. \end{enumerate} \end{prop} Note that if $n=2$, the codimension of ${\Bbb P} \setminus \,{\Bbb P}^s$ in ${\Bbb P}$ is $g-1$ (see \cite{balaji-thesis},\,p.\,11,\,Prop.\,7), so that if $g\ge 4$ the codimension is at least $3$. This fact, along with and Proposition\,\ref{prop:hecke} implies that the codimension of ${\Bbb P}\setminus {\Bbb P}^s$ is greater than equal to $3$ for $n,\,g$ in the range of Theorem\,\ref{thm:main}. It then follows, from Lemma\,\ref{lem:codim} below, that the restriction maps \begin{align*} H^3({\Bbb P},\,{\Bbb Z}) & \longrightarrow H^3({\Bbb P}^s,\,{\Bbb Z}) \\ H^1({\Bbb P},\,{\Bbb Z}) & \longrightarrow H^1({\Bbb P}^s,\,{\Bbb Z}) \end{align*} are isomorphisms of Hodge structures. Note that this means: \begin{itemize} \item The Hodge structure of $H^3({\Bbb P}^s)$ is pure of weight $3$; \item The cohomology group $H^1({\Bbb P}^s,\,{\Bbb Z}) = 0$. Indeed, ${\Bbb P}$ is unirational (for ${\cal S}_1$ is --- see \cite{css-drez},\,pp.\,52--53,\,VI.B), whence $H^1({\Bbb P},\,{\Bbb Z}) = 0$. \end{itemize} We can now relate the Hodge structures on $H^1({\cal S}^s)$ and $H^3({\cal S}^s)$ with those on $H^1({\Bbb P}^s)$ and $H^3({\Bbb P}^s)$ using the map $f$ and part (b) of Proposition\,\ref{prop:hecke}. For the rest of this section let $f$ also denote the map ${\Bbb P}^s \to {\cal S}^s$. We claim that \begin{equation}\label{eqn:hodge} f^*: H^3({\cal S}^s) \to H^3({\Bbb P}^s) \end{equation} is an isomorphism of Hodge structures, modulo torsion. This implies that the Hodge structure on $H^3({\cal S}^s,\,{\Bbb Z})$ is pure of weight $3$, a fact that also follows from Corollary\,\ref{cor:lefschetz}. To prove that \eqref{eqn:hodge} is an isomorphism of Hodge structures, modulo torsion, we need: \begin{lem}\label{lem:pi1} ${\cal S}^s$ is simply connected. \end{lem} \begin{pf} ${\Bbb P}$ is unirational, therefore it is simply connected \cite{pi1}. Since $\operatorname{codim}{({\Bbb P}\setminus {\Bbb P}^s)} > 1$, it follows that ${\Bbb P}^s$ is also simply connected (purity of the branch locus). The lemma now follows from the homotopy exact sequence for $f$. \end{pf} \begin{cor} $H^1({\cal S}^s,\,{\Bbb Z}) = 0$. \end{cor} \begin{cor} $f_*{\Bbb Z} = {\Bbb Z}$, $R^1f_*{\Bbb Z} = R^3f_*{\Bbb Z} = 0$ and $R^2f_*{\Bbb Z} = {\Bbb Z}^{d-1}$. \end{cor} \begin{pf} As ${\cal S}^s$ is simply connected, $R^if_*{\Bbb Z}$ is just the constant sheaf associated to the $i$-th cohomology of ${\Bbb P}^{n-1}\times\ldots\times{\Bbb P}^{n-1}$. \end{pf} One can now verify \eqref{eqn:hodge} by using the Leray spectral sequence combined with the above isomorphisms. It follows that $H^3({\Bbb P}^s,\,{\Bbb Z})$ is isomorphic to the cokernel of the differential $$ H^0(R^2f_*{\Bbb Z}) \to H^3(f_*{\Bbb Z}) $$ but this vanishes mod torsion by \cite{deligne68}. The isomorphisms \eqref{eqn:tate} and the map \eqref{eqn:hodge}, give the desired mod-torsion isomorphism $$ \psi_{X,L,\chi}\colon H^1(X)(-1) \stackrel{\sim}{\longrightarrow} H^3({\cal S}^s). $$ \begin{rem}\label{rmk:var-hodge} This isomorphism varies well with $(X,\,L,\,\chi)$ as the construction of the correspondence \eqref{eqn:hecke} will show (see Remark\,\ref{rmk:variation}). \end{rem} Here then is the promised Lemma: \begin{lem}\label{lem:codim} If $Y$ is a smooth projective variety, $Z$ a codimension $k$ closed subscheme, and $U=Y\setminus Z$, then $$ H^j(Y,\,{\Bbb Z}) \stackrel{\sim}{\longrightarrow} H^j(U,\,{\Bbb Z}) $$ for $j < 2k-1$. \end{lem} \begin{pf} We have to show that $H^j_Z(X,\,{\Bbb Z})$ vanishes for $j < 2k$. By Alexander duality (see for e.g. \cite{iverson},\,p.\,381,\,Theorem\,4.7) we have $$ H^j_Z(Y,\,{\Bbb Z}) \stackrel{\sim}{\longrightarrow} H_{2m-j}(Z,\,{\Bbb Z}), $$ where $m=\dim{Y}$ and $H_*$ is Borel-Moore homology. Now use \cite{iverson},\,p.\,406,\,3.1 to conclude that the right side vanishes if $j < 2k$ (note that $``\dim"$ in {\it loc.cit} is dimension as an analytic space, and in {\it op.cit.} it is dimension as a topological (real) manifold). \end{pf} \begin{rem}\label{rmk:balaji} In view of the above Lemma, it seems that Balaji's proof of Torelli (for Seshadri's desingularization of ${\cal S}{\cal U}_X(2,\,{\cal O}_X)$) does not work for $g=3$, for in this case, the codimension of ${\Bbb P}\setminus {\Bbb P}^s =2$. (See \cite{balaji},\,top of p.\,624 and \cite{balaji-thesis},\,Remark\,9.) \end{rem} \subsection{The Polarization on $H^3({\cal S}^s)$.}\label{ss:polar} It remains to impose a polarization on the Hodge structure of $H^3({\cal S}^s)$ which varies well with ${\cal S}^s$. Note that the map $\psi_{X,L,\chi}$ tells us that the Hodge structure on $H^3({\cal S}^s)$ is pure. One knows from the results of Drezet and Narasimhan \cite{drez-nar}, that $\operatorname{Pic}({\cal S}^s)={\Bbb Z}$ (see p.\,89, 7.12 (especially the proof) of {\it loc.cit.}). Moreover, $\operatorname{Pic}({\cal S}) \to \operatorname{Pic}({\cal S}^s)$ is an isomorphism. Let $\xi'$ be the ample generator of $\operatorname{Pic}({\cal S}^s)$. It is easy to see that there exists a positive integer $r$, independent of $(X,\,L)$ (with genus $X=g$), such that $\xi = {\xi'}^r$ is very ample on ${\cal S}$ (we are not distinguishing between line bundles on ${\cal S}^s$ and their (unique) extensions to ${\cal S}$). Embed ${\cal S}$ in a suitable projective space via $\xi$. Let $e=\operatorname{codim}({\cal S}\setminus \,{\cal S}^s)$. Let $M$ be the intersection of $k = \dim{\cal S} -e +1$ hyperplanes (in general position) with ${\cal S}^s$. Then $M$ is smooth, projective and contained in ${\cal S}^s$. Let $p = \dim{{\cal S}}$ and $H^*_c$ --- cohomology with compact support. We then have a map $$ l\colon H^3({\cal S}^s) \longrightarrow H^{2p-3}_c({\cal S}^s) $$ defined by $$ x \mapsto x\cup c_1(\xi)^{p-k-3}\cup [M]. $$ If $M'$ is another $k$-fold intersection of general hyperplanes, then $[M'] = [M]$. Hence $l$ depends only on $\xi$. According to Proposition\,\ref{prop:polarization} (see also Remark\,\ref{rmk:polarization}), the pairing on $H^3({\cal S}^s,\,{\Bbb C})$ given by $$ <x,\,y> = \int_{{\cal S}^s}l(x)\cup\,y $$ gives a polarization on the Hodge structure of $H^3({\cal S}^s)$. Since $\xi$ ``spreads" (for $\xi'$ clearly does), therefore this polarization varies well with ${\cal S}^s$. Then by arguments already indicated in the beginning of this section, this polarization is a multiple of principal polarization (and the integer factor is necessarily unique). Thus one gets a natural principal polarization on $H^3({\cal S}^s)$. \begin{rem}\label{rmk:saito} There is another approach to this polarization, using Intersection Cohomology (middle perversity) and M. Saito's theory of Hodge modules \cite{saito}. The very ample bundle $\xi$ gives rise to Lefschetz operators $L^i\colon IH^q({\cal S}) \longrightarrow IH^{q+2i}({\cal S})$ (see \cite{del-beil-bern}). Our codimension estimates (see Remark\,\ref{rmk:codim}) are such that $IH^3({\cal S}) \stackrel{\sim}{\longrightarrow} H^3({\cal S}^s)$ and $IH^1({\cal S}) = H^1({\cal S}^s) = 0$. The group $IH^3({\cal S})$ has a pairing on it given by $$ <\alpha,\,\beta> = \int_SL^{p-3}\alpha\cup\beta $$ where $\int_S(\_)\cup\beta\colon IH^{2p-3}({\cal S})\to {\Bbb C}$ is the map given by the Poincar{\'e} duality pairing between $IH^{2p-3}({\cal S})$ and $IH^3({\cal S})$. According to M. Saito \cite{saito},\,5.3.2, this gives a polarization on the Hodge structure of $IH^3({\cal S})$ (since all classes in $IH^3({\cal S})$ are primitive). This polarization translates to one on $H^3({\cal S}^s)$. A little thought shows (say by desingularizing ${\cal S}$) that the pairing on $H^3({\cal S}^s)$ is $$ <x,\,y> = \int_{{\cal S}^s}c_1(\xi)^{p-3}\,\wedge\,x\,\wedge\,y. $$ Here, on the right side, we are using De Rham theory, and replacing the various elements in cohomology by forms which represent them. The integral above is the usual integral of forms. Note that we could not have defined the pairing by the above formula, for we have no {\it a priori} guarantee that the right side (which is an integral over an open manifold) is finite. \end{rem} \section{The correspondence variety ${\Bbb P}$}\label{s:hecke} In this section we define the map $f\colon {\Bbb P} \to {\cal S}$ and prove Proposition \ref{prop:hecke}. \subsection{The map $f\colon {\Bbb P} \to {\cal S}$.}\label{ss:map-f} We need some notations\,: \begin{itemize} \item For $1\le k\le d-1$, $\pi_k\colon {\Bbb P} \to {\Bbb P}_k$ is the natural projection; \item ${\imath}\colon Z \hookrightarrow X$ is the reduced subscheme defined by $\chi=\{x^1,\ldots,\,x^{d-1}\}$. \item ${\imath}_k\colon Z_k \hookrightarrow X$, the reduced scheme defined by $\{x_k\}$, $k= 1,\ldots,\,d-1$. \item For any scheme $S$, \begin{enumerate} \item[(i)] $p_S\colon X\times\,S \to S$ and $q_S\colon X\times\,S \to X$ are the natural projections; \item[(ii)] $Z^S = q_S^{-1}(Z)$; \item[(iii)] $Z_k^S = q_S^{-1}(Z_k)$, $k=1,\ldots,\,d-1$. Note that $Z^S_k$ can be identified canonically with $S$. \end{enumerate} \end{itemize} We will show --- in \ref{ss:u-exact} --- that there is an exact sequence \begin{equation}\label{eqn:u-exact} 0 \longrightarrow (1\times\pi)^*{\cal W} \longrightarrow {\cal V} \longrightarrow {\cal T}_0 \longrightarrow 0 \end{equation} on $X\times\,{\Bbb P}$, with ${\cal V}$ a vector bundle on $X\times{\Bbb P}$ and ${\cal T}_0$ a line bundle {\it on the subscheme $Z^{\Bbb P}$}, which is universal in the following sense\,: If $\psi\colon S \to {\cal S}_1$ is a ${\cal S}_1$-scheme and we have an exact sequence \begin{equation}\label{eqn:s-exact} 0 \longrightarrow (1\times\psi)^*{\cal W} \longrightarrow {\cal E} \longrightarrow {\cal T} \longrightarrow 0 \end{equation} on $X\times\,S$, with ${\cal E}$ a vector bundle on $X\times\,S$ and ${\cal T}$ a line bundle {\it on the subscheme} $Z^S$, then there is a unique map of ${\cal S}_1$-schemes $$ g\colon S \longrightarrow {\Bbb P} $$ such that, $$ (1\times\,g)^*\eqref{eqn:u-exact} \equiv \eqref{eqn:s-exact}. $$ The $\equiv$ sign above means that the two exact sequences are isomorphic, and the left most isomorphism $(1\times{g})^*{\scriptstyle{\circ}}(1\times{\pi})^* \stackrel{\sim}{\longrightarrow} (1\times\psi)^*$ is the canonical one. There is a way of interpreting this universal property in terms of quasi-parabolic bundles (see \cite{mehta-css},\,p.\,211--212,\,Definition\,1.5, for the definitions of quasi-parabolic and parabolic bundles). Taking $\chi$ as our collection of parabolic vertices, we can introduce a quasi-parabolic datum on $X$ by attaching the flag type $(1,\,n-1)$ to each point of $\chi$. From now onwards {\it quasi-parabolic structures will be with respect to this datum and on vector bundles of rank $n$ and determinant $L$}. One observes that for a vector bundle $V$ (of rank $n$ and determinant $L$), a surjective map $V\twoheadrightarrow {\cal O}_Z$ determines a unique quasi-parabolic structure, and two such surjections give the same quasi-parabolic strcuture if and only if they differ by a scalar multiple. The above mentioned universal property says that ${\Bbb P}$ is a (fine) moduli space for quasi-parabolic bundles. More precisely, the family of quasi-parabolic structures $$ {\cal V} \twoheadrightarrow {\cal T}_0 $$ parameterized by ${\Bbb P}$ is universal for families of quasi-parabolic bundles $$ {\cal E} \twoheadrightarrow {\cal T} $$ parameterized by $S$, whose kernel is a family of semi-stable bundles. The points of ${\Bbb P}$ parameterize quasi-parabolic structures $V \twoheadrightarrow {\cal O}_Z$ whose kernel is semi-stable. Let $\boldsymbol{\alpha} = (\alpha_1,\,\alpha_2)$, where $0 < \alpha_1 < \alpha_2 <1$, and let $\Delta = \Delta_{\boldsymbol{\alpha}}$ be the parabolic datum which attaches to each parabolic vertex (of our quasi-parabolic datum) weights $\alpha_1, \alpha_2$. We can choose $\alpha_1$ and $\alpha_2$ so small that \begin{itemize} \item a parabolic semi-stable bundle is parabolic stable\,; \item if $V$ is stable, then every parabolic structure on $V$ is parabolic stable\,; \item the underlying vector bundle of a parabolic stable bundle is semi-stable in the usual sense\,; \item if $V \twoheadrightarrow {\cal O}_Z$ is parabolic stable, then the kernel $W$ is semi-stable. \end{itemize} Showing the above involves some very elementary calculations. Denote the resulting moduli space of parabolic stable bundles ${\cal S}{\cal U}_X(n,\,L,\,\Delta)$. Let ${\Bbb P}^{ss}\subset {\Bbb P}$ be the locus on which ${\cal V}$ consists of parabolic semi-stable (=parabolic stable) bundles. One checks that ${\Bbb P}^{ss}$ is an open subscheme of ${\Bbb P}$ (this involves two things\,: (i) knowing that the scheme $\widetilde{R}$ of \cite{mehta-css},\,p.\,226 has a local universal property for parabolic bundles and (ii) knowing that the scheme ${\widetilde{R}}^{ss}$ of {\it loc.cit.} is open). Clearly ${\Bbb P}^{ss}$ is non-empty --- in fact if $V$ is stable of rank $n$ and determinant $L$, then any parabolic structure on $V$ is parabolic stable (see above). We claim that ${\Bbb P}^{ss} \simeq {\cal S}{\cal U}_X(n,\,L,\,\Delta)$. To that end, let $S$ be a scheme, and \begin{equation}\label{eqn:par-fly} {\cal E} \twoheadrightarrow {\cal T} \end{equation} a family of parabolic stable bundles parameterized by $S$. The kernel ${\cal W}'$ of \eqref{eqn:par-fly} is a family of stable bundles of rank $n$ and determinant $L\otimes{\cal O}_X(-D)$. Since ${\cal S}_1$ is a fine moduli space, we have a unique map $g\colon {\cal S}\to {\cal S}_1$ and a line bundle $\xi$ on $S$ such that $(1\times{g})^*{\cal W} = {\cal W}'\otimes\,p_S^*\xi$. By doctoring \eqref{eqn:par-fly} we may assume that $\xi={\cal O}_S$. The universal property of the exact sequence \eqref{eqn:u-exact} on ${\Bbb P}$ then gives us a unique map $$ g\colon\,S \longrightarrow {\Bbb P} $$ such that $(1\times{g})^*\eqref{eqn:u-exact}$ is equivalent to $$ 0 \longrightarrow {\cal W}' \longrightarrow {\cal E} \longrightarrow {\cal T} \longrightarrow 0. $$ Clearly $g$ factors through ${\Bbb P}^{ss}$. This proves that ${\Bbb P}^{ss}$ is ${\cal S}{\cal U}_X(n,\,L,\,\Delta)$. However, ${\cal S}{\cal U}_X(n,\,L,\,\Delta)$ is a projective variety (see \cite{mehta-css},\,pp.\,225--226,\,Theorem\,4.1), whence we have $$ {\Bbb P} = {\cal S}{\cal U}_X(n,\,L,\,\Delta). $$ It follows that ${\cal V}$ consists of parabolic stable bundles, and hence of (usual) semi-stable bundles (by our choice of $\boldsymbol{\alpha}$). Since ${\cal S}$ is a coarse moduli space, we get the map \begin{equation}\label{eqn:f} f\colon\,{\Bbb P} \longrightarrow {\cal S} . \end{equation} \begin{rem}\label{rmk:hecke} Note that the parabolic structure $\Delta$ is something of a red herring. In fact ${\cal S}{\cal U}_X(n,\,L,\,\Delta)$ parameterizes quasi-parabolic structures $V \twoheadrightarrow {\cal O}_Z$, whose kernel is semi-stable (cf. \cite{mehta-css},\,p.\,238,\,Remark\,(5.4), where this point is made for $n=2, d=2$). The space ${\Bbb P}$ should be thought of as the correspondence variety for a certain Hecke correspondence (cf. \cite{N-R-hecke}). \end{rem} \begin{rem}\label{rmk:proj-bundle} Let $V$ be a stable bundle of rank $n$, with $\det{V}=L$, so that (the isomorphism class of) $V$ lies in ${\cal S}^s$. Since any parabolic structure on $V$ is parabolic stable (by our choice of $\boldsymbol{\alpha}$), therefore we see that $f^{-1}(V)$ is canonically isomorphic to ${\Bbb P}(V_{x^1}^*)\times\ldots\times{\Bbb P}(V_{x^{d-1}}^*)$. \footnote{One can be more rigorous. Identifying $Z_k^{{\Bbb P}}$ with ${{\Bbb P}}$ for each $k = 1,\ldots,\,d-1$, we see that restricting the universal exact sequence to $Z_k^{{\Bbb P}}$ gives us $d-1$ quotients ${\cal O}_{{\Bbb P}}\otimes_{{\Bbb C}}V_{x^k} \twoheadrightarrow {\cal T}_0|Z^{{\Bbb P}}_k$. Let $S$ be a scheme which has $d-1$ quotients ${\cal O}_S\otimes_{{\Bbb C}}V_{x^k}\twoheadrightarrow {\cal L}_k$ $k = 1,\ldots,\,d-1$, on it, where the ${\cal L}_k$ are line bundles. These quotients extend to a family of parabolic structures $q_S^*V \twoheadrightarrow {\cal T}$ (on $V$) parameterized by $S$ in a unique way. The universal property of the exact sequence \eqref{eqn:u-exact} gives us a map $S \to {\Bbb P}$, and this map factors through $f^{-1}(V)$.} This gives us part (b) of Proposition\,\ref{prop:hecke}, for it is not hard to see that ${\Bbb P}^s \to {\cal S}^s$ is smooth (examine the effect on the tangent space of each point on ${\Bbb P}^s$). \end{rem} \subsection{Codimension estimates.}\label{ss:codim} We wish to estimate $\operatorname{codim}{({\Bbb P}\setminus {\Bbb P}^s})$. For any vector bundle $E$ on $X$, let $\mu(E)=\operatorname{rank}{E}/\deg{E}$. Let $\mu=d/n$. Let $V \twoheadrightarrow {\cal O}_Z$ be a parabolic bundle in ${\Bbb P}\setminus {\Bbb P}^s$. Then we have a filtration (see \cite{css-drez},\,p.\,18,\,Th{\'e}or{\`e}me\,10) $$ 0=V_{p+1} \subset V_p \subset \ldots \subset V_0=V $$ such that for $0\le i \le p$, $G_i=V_i/V_{i+1}$ is stable and $\mu(G_i)=\mu$. Moreover (the isomorphism class of) the vector bundle $\bigoplus{G_i}$ depends only upon $V$ and not on the given filtration. We wish to count the number of moduli at $[V\overset{\theta}{\twoheadrightarrow} {\cal O}_Z] \in {\Bbb P}\setminus {\Bbb P}^s$. There are three sources\,: \begin{enumerate} \item[a)] The choice of $\bigoplus_{i=0}^pG_i$\,; \item[b)] Extension data\,; \item[c)] The choice of parabolic structure $V\overset{\theta}{\twoheadrightarrow}{\cal O}_Z$, for fixed semi-stable $V$. \end{enumerate} The source c) is the easiest to calculate --- there is a codimension one subspace at each parabolic vertex, contributing $$ (n-1)(d-1) $$ moduli. Let $n_i=\operatorname{rank}{G_i}$. The number of moduli arising from a) is evidently $$ \sum_{i=0}^p(n_i^2-1)(g-1) + pg . $$ Indeed, the bundles $G_i$ have degree $n_i\mu$ and the product of their determinants must be $L$. They are otherwise unconstrained. It remains to estimate the number of moduli arising from extension data. Each extension $$ 0 \longrightarrow V_{i+1} \longrightarrow V_i \longrightarrow G_i \longrightarrow 0 \qquad i=0,\ldots,\,p $$ determines a class in $H^1(X,\,G_i^*\otimes{V_{i+1}})$. Note that \begin{equation*} \begin{split} h^0(G_i^*\otimes{V_{i+1}}) & = \dim{\operatorname{Hom}_{{\cal O}_X}(G_i,\,V_{i+1})} \\ & \le \sum_{j>i}\operatorname{Hom}_{{\cal O}_X}(G_i,\,G_j) \\ & \le p-i \end{split} \end{equation*} by the sub-additivity of $\dim{\operatorname{Hom}(G_i,\,\_)}$ and the stability of $G_i$. By the Riemann-Roch theorem \begin{equation*} \begin{split} h^1(G_i^*\otimes{V_{i+1}}) & = h^0(G_i^*\otimes{V_{i+1}}) - n_i(n_{i+1}+\ldots +n_p)(1-g) \\ & \le (p-i) - n_i(n_{i+1}+\ldots n_p)(1-g). \end{split} \end{equation*} The isomorphism class of $V_i$ depends only on a scalar multiple of the extension class. Therefore the number of moduli contributed by extensions is \begin{equation*} \begin{split} \sum_{i=0}^p\left [h^1(G_i^*\otimes{V_{i+1}} -1 \right] & \le \sum_{i=0}^p\left[p-i-n_i(n_{i+1}+\ldots n_p)(1-g) \right] - (p+1) \\ & = \dfrac{p(p+1)}{2} - \sum_{i=0}^{p-1}n_i(n_{i+1}+\ldots +n_p)(1-g) - (p+1) \\ & = \dfrac{(p+1)(p-2)}{2} - \sum_{i<j}n_in_j(1-g). \end{split} \end{equation*} Adding the contributions from a), b) and c) and subtracting from $$ \dim{\Bbb P}=(n^2-1)(g-1)+(n-1)(d-1)$$ we get \begin{equation*} \begin{split} \operatorname{codim}({\Bbb P}\setminus {\Bbb P}^s) & \ge (n^2-1)(g-1) - \sum_{i=o}^p(n_i^2-1)(g-1) - pg \\ & \quad - \sum_{i<j}n_in_j(g-1) - \dfrac{(p+1)(p-2)}{2} \\ & = \sum_{i<j}n_in_j(g-1) - \dfrac{(p-1)(p+2)}{2} \\ & = B \qquad (\text{say}). \end{split} \end{equation*} Now, $\sum_{i<j}n_in_j \ge {p(p+1)}/{2}$, therefore $$ B \ge \dfrac{p(p+1)}{2}(g-1) - \dfrac{(p+2)(p-1)}{2}. $$ It follows that $B\ge 3$ whenever $p\ge 2$ {\it and} $g\ge 3$. If $p=1$ and $n \ge 3$, then $$ B/(g-1) = \sum_{i<j}n_in_j \ge 2 $$ and one checks that $B\ge 3$ whenever $g\ge 3$. Proposition\,\ref{prop:hecke}(a) may now be considered as proved. \begin{rem}\label{rmk:codim} We could use similar techniques to estimate $\operatorname{codim}{({\cal S}\setminus {\cal S}^s)}$, but our task is made easier by the exact answers in \cite{css-drez},\,p.\,48,\,A. For just this remark, assume $d > n(2g-1)$, and let $a = (n,\,d)$. Then $a\ge 2$. Let $n_0 = n/a$. Then according to {\it loc.cit.}, \begin{equation*} \operatorname{codim}{({\cal S}\setminus {\cal S}^s)}= \begin{cases} (n^2-1)(g-1) - \dfrac{n^2}{2}(g-1) -2 +g & \text{if $a$ is even} \\ & {} \\ (n^2-1)(g-1) - \dfrac{n^2 + n_0^2}{2}(g-1) -2 + g & \text{if $a$ is odd}. \end{cases} \end{equation*} It now follows that $$ \operatorname{codim}{({\cal S}\setminus {\cal S}^s)} > 5 $$ if $n,\,g$ are in the range of Theorem\,\ref{thm:main}. \end{rem} \subsection{The universal exact sequence on $X\times{\Bbb P}$.}\label{ss:u-exact} We begin by reminding the reader of some elementary facts from commutative algebra. If $A$ is a ring (commutative, with $1$), $t\in A$ a non-zero divisor, and $M$ an $A$-module, then each element $m_0\in M$ gives rise to an equivalence class of extensions \begin{equation}\label{eqn:exact-mod} 0 \longrightarrow M \longrightarrow E_{m_0} \longrightarrow A/tA \longrightarrow 0 \end{equation} where $E_{m_0} = \left(A\bigoplus{M}\right)/A(t,\,m_0)$, and the arrows are the obvious ones. Moreover, if $m_0 - m_1 \in tM$, say $$ m_0 - m_1 = tm' $$ then the extension given by $m_0$ is equivalent to that given by $m_1$. In fact, one checks that \begin{equation}\label{eqn:patching} \begin{split} E_{m_0} & \stackrel{\sim}{\longrightarrow} E_{m_1} \\ (a,\,m) & \mapsto (a,\,m-am') \end{split} \end{equation} gives the desired equivalence of extensions. This is another way of expressing the well known fact that each element of $M/tM = \operatorname{Ext}^1(A/t,\,M)$ gives rise to an extension. One globalizes to get the following\,: Let $S$ be a scheme, $T\overset{{\imath}}{\hookrightarrow} S$ a closed immersion, ${\cal F}$ a quasi-coherent ${\cal O}_S$-module, $U$ an open neighbourhood of $T$ in $S$, and $t\in \Gamma(U,\,{\cal O}_S)$ an element which defines $T \hookrightarrow U$, and which is a non-zero divisor for $\Gamma(V,\,{\cal O}_S)$ for any open $V \subset U$. Then every global section $s$ of ${\imath}^*{\cal F} = {\cal F}\otimes{\cal O}_T$ gives rise to an equivalence class of extensions \begin{equation}\label{eqn:patched} 0 \longrightarrow {\cal F} \longrightarrow {\cal E} \longrightarrow {\cal O}_T \longrightarrow 0. \end{equation} Indeed, we are reduced immediately to the case $S = U$. We build up exact sequences \eqref{eqn:exact-mod} on each affine open subset $W \subset S$, by picking a lift $\tilde{s_W}\in\Gamma(W,\,{\cal F})$ of $s\,|\,W$. One patches together these exact sequences via \eqref{eqn:patching}. Now consider ${\Bbb P} = {\Bbb P}_1\times_{{\cal S}_1}\ldots\times_{{\cal S}_1}{\Bbb P}_{d-1}$. For each $k = 1,\ldots,\,d-1$, let $p_k\colon\,{\Bbb P}_k \to {\cal S}_1$ be the natural projection. We have a universal exact sequence $$ 0 \longrightarrow {\cal O}(-1) \longrightarrow p_k^*{\cal W}_k \longrightarrow B \longrightarrow 0 $$ whence a global section $s_k\in\Gamma({\Bbb P}_k,\,p_k^*{\cal W}_k(1))$. However, note that $$ p_k^*{\cal W}_k = (1\times p_k)^*{\cal W}\,|\,Z_k^{{\Bbb P}_k} $$ where we are identifying $Z_k^{{\Bbb P}_k}$ with ${\Bbb P}_k$. By \eqref{eqn:patched} we get exact sequences $$ 0 \longrightarrow (1\times\pi)^*{\cal W} \longrightarrow {\cal V}_k \longrightarrow {\cal O}_{Z_k^{\Bbb P}}\otimes{L_k} \longrightarrow 0 $$ where $L_k$ is the line bundle obtained by pulling up ${\cal O}_{{\Bbb P}_k}(-1)$. It is not hard to see that ${\cal V}_k$ is a family of vector bundles parameterized by ${\Bbb P}$. Glueing these sequences together --- the $k$-th and the $l$-th agree outside $Z^{{\Bbb P}}_k$ and $Z^{{\Bbb P}}_l$ --- we obtain \eqref{eqn:u-exact}. Now suppose we have a ${\cal S}_1$-scheme $\psi\colon\,S \to {\cal S}_1$ and the exact sequence \eqref{eqn:s-exact} $$ 0 \longrightarrow (1\times\psi)^*{\cal W} \longrightarrow {\cal E} \longrightarrow {\cal T} \longrightarrow 0 $$ on $X\times{S}$. Restricting \eqref{eqn:s-exact} to $Z^S_k$ ($1\le k \le d-1$) one checks that the kernel of $(1\times\psi)^*{\cal W}\,|\,Z^S_k \to {\cal E}\,|Z_k^S$ is a line bundle ${\cal L}_k$. Identifying $Z^S_k$ with $S$, we see that $(1\times\,\psi)^*{\cal W}\,|\,Z^S_k = \psi^*{\cal W}_k$. Thus ${\cal L}_k$ is a line sub-bundle of $\psi^*{\cal W}_k$. By the universal property of ${\Bbb P}_k$, we see that we have a unique map of ${\cal S}_1$-schemes $$ g_k\colon\,S \longrightarrow {\Bbb P}_k $$ such that ${\cal O}(-1)$ gets pulled back to ${\cal L}_k$. The various $g_k$ give us a map $$ g\colon S \longrightarrow {\Bbb P} $$ One checks that $g$ has the required universal property. The uniqueness of $g$ follows from the uniqueness of each $g_k$. \begin{rem}\label{rmk:variation} It is clear from the construction that the map $$ f = f_{X,L,\chi}\colon\,{\Bbb P}_{X,L,\chi} \longrightarrow {\cal S}{\cal U}_X(n,\,L) $$ varies well with $(X,\,L,\,\chi)$. This implies that the correspondence \eqref{eqn:hecke} also varies well with $(X,\,L,\,\chi)$ and hence so do $\psi_{X,L,\chi}$ and $\varphi_{X,L,\chi}$. \end{rem} \section{Polarizations}\label{s:polarization} Let $Y$ be an $m$-dimensional projective variety. Suppose that $U$ is a smooth Zariski open subset. One then has the following version of the Lefschetz theorem. \begin{thm}\label{thm:lefschetz} If $H$ is a hyperplane section of $Y$ such that $U\cap H$ is non-empty, then $$ H^i(U,\,{\Bbb Q}) \to H^i(U\cap{H},\,{\Bbb Q}) $$ is an isomorphism for $i < m-1$ and injective when $i = m-1$. \end{thm} \begin{pf} We need some results involving Verdier duality. The standard references are \cite{borel} and \cite{iverson}. Let $S$ be an analytic space and $p_S$ the map from $S$ to a point. For ${\cal F}\in D^b_{const}(S,\,{\Bbb Q})$ (the derived category of bounded complexes of ${\Bbb Q}_S$-sheaves whose cohomology sheaves are ${\Bbb Q}_S$-constructible), set $$ D_S({\cal F}) = {\Bbb R}{\cal H}\operatorname{om}_S({\cal F},\,p_S^!{\Bbb Q}). $$ We then have by Verdier duality \begin{equation}\label{eqn:verdier} {\Bbb H}^i(S,\,{\cal F}) \overset{\sim}{\longrightarrow} {\Bbb H}^{-i}(S,\,D_S({\cal F}))^*. \end{equation} Here ${\Bbb H}^*$ denotes ``hypercohomology". For an open immersion $h\colon S' \hookrightarrow S$, one has canonical isomorphisms \begin{align}\label{eqn:*!} {\Bbb R}{h}_*D_{S'}{\cal G} & \overset{\sim}{\longrightarrow} D_S(h_!{\cal G}) \\ \intertext{and} \label{eqn:!*} {{\Bbb R}}h_!D_{S'}{\cal G} & \overset{\sim}{\longrightarrow} D_S({\Bbb R}{h_*}{\cal G}) \end{align} Here ${\cal G}\in D^b_{const}(S',\,{\Bbb Q})$. The first isomorphism is easy (using Verdier duality for the map $h$) and the second follows from the first and from the fact that $D_{S'}$ is an involution. We have used (throughout) the fact that $h_!$ is an exact functor. If $S$ is smooth, we have \begin{equation}\label{eqn:dim} p_S^!{\Bbb Q} = {\Bbb Q}_S[2\dim{S}]. \end{equation} In order to prove the theorem, let $V=U\setminus H$ and $W=Y\setminus H$. We then have a cartesian square $$ \begin{array}{ccc} V & \stackrel{\scriptstyle{{{\imath}}'}}{\longrightarrow} & U \\ \vcenter{% \llap{$\scriptstyle{{{\jmath}}'}$}}\Big\downarrow & & \Big\downarrow\vcenter{% \rlap{$\scriptstyle{{\jmath}}$}} \\ W & \underset{\scriptstyle{{\imath}}}{\longrightarrow} & Y \end{array} $$ where each arrow is the obvious open immersion. We have, by \eqref{eqn:*!} and \eqref{eqn:!*}, the identity \begin{equation}\label{eqn:Dji} {{\jmath}}_!{\Bbb R}{{\imath}}'_*D_V{\Bbb Q}_V = D_Y({\Bbb R}{\jmath}_*{{\imath}}'_!{\Bbb Q}_V). \end{equation} Consider the exact sequence of sheaves $$ 0 \longrightarrow {{\imath}}'_!{\Bbb Q}_V \longrightarrow {\Bbb Q}_U \longrightarrow g_*{\Bbb Q}_{H\cap U} \longrightarrow 0 $$ where $g\colon H\cap{U}\to U$ is the natural closed immersion. It suffices to prove that $H^i(U,\,{{\imath}}'_!{\Bbb Q}_V)=0$ for $i\le m-1$. Now, $$ H^i(U,\,{{\imath}}'_!{\Bbb Q}_V)={\Bbb H}^i(Y,\,{\Bbb R}{\jmath}_*{{\imath}}'_!{\Bbb Q}_V). $$ Using \eqref{eqn:verdier}, \eqref{eqn:Dji} and \eqref{eqn:dim}, the above is dual to \begin{align*} {\Bbb H}^{-i}(Y,\,{\jmath}_!{\Bbb R}{{\imath}}'_*D_V{\Bbb Q}_V) & = {\Bbb H}^{2m-i}(Y,\,{\jmath}_!{\Bbb R}{{\imath}}'_*{\Bbb Q}_V) \\ \intertext{But ${\jmath}_!{\Bbb R}{{\imath}}'_* = {\Bbb R}{\imath}_*{{\jmath}}'_!$, and hence the above is} & = {\Bbb H}^{2m-i}(Y,\,{\Bbb R}{{\imath}}_*({{\jmath}}'_!{\Bbb Q}_V)) \\ & = {\Bbb H}^{2m-i}(W,\,{{\jmath}}'_!{\Bbb Q}_V) \\ & = H^{2m-i}(W,\,{{\jmath}}'_!{\Bbb Q}_V). \end{align*} Now, $W$ is an affine variety, and therefore, according to M. Artin, its constructible cohomological dimension is less than or equal to its dimension \cite{artin}. Consequently, the above chain of equalities vanish whenever $i<m$ (see also \cite{gor-mac}). \end{pf} We immediately have: \begin{cor}\label{cor:lefschetz} Let $e=\operatorname{codim}(Y\setminus U)$. For $i< e-1$, the Hodge structure $H^i(U)$ is pure of weight $i$. \end{cor} \begin{pf} This is true if $U$ is projective. In general proceed using Bertini's theorem, induction, Theorem\,\ref{thm:lefschetz} and the fact that submixed Hodge structures of pure Hodge structures are pure \cite{deligne-hodge}. \end{pf} Let $i\in{\Bbb N}$ and ${\cal L}$ a line bundle on $Y$ be such that \begin{enumerate} \item[(a)] $H^j(U,\,{\Bbb Q})=0$ for $j = i-2,\,i-4,\ldots $\,; \item[(b)] $i < e-1$\,; \item[(c)] ${\cal L}$ is very ample. \end{enumerate} \begin{rem}\label{rmk:polarization} Note that if $Y={\cal S}$, $U={\cal S}^s$, then $i=3$ and ${\cal L}=\xi$ ($\xi$= the very ample bundle of \ref{ss:polar}) satisfy the above conditions by the results of \ref{ss:psi} and Remark\,\ref{rmk:codim}. \end{rem} Let $M$ be the intersection of $k=m-e+1$ hyperplanes in general position. Then $M$ is a smooth variety contained in $U$. Let $$ l\colon H^i(U) \longrightarrow H^{2m-i}_c(U) $$ be the composite of \begin{equation*} \begin{split} H^i(U) & \longrightarrow H^i(M) \\ & \longrightarrow H^{2m-2k-i}(M) \\ & \longrightarrow H^{2m-i}_c(U) \end{split} \end{equation*} where the first map is restriction, the second is ``cupping with $c_1({\cal L})^{m-k-i}$" and the third is the Poincar{\'e} dual to restriction. The map $l$ is also described as $$ x \mapsto x\cup c_1({\cal L})^{m-k-i}\cup [M]. $$ One then has (easily) \begin{lem} If $M'$ is another $k$-fold intersection of general hyperplanes, then $[M'] = [M]$. Therefore $l$ depends only on ${\cal L}$. \end{lem} \begin{prop}\label{prop:polarization} The pairing $$ <x,\,y> = \int_Ul(x)\cup y $$ on $H^i(U,\,{\Bbb C})$ gives a polarization on the pure Hodge structure $H^i(U)$. \end{prop} \begin{pf} By Theorem\,\ref{thm:lefschetz}, we have an isomorphism $$ r\colon H^i(U) \longrightarrow H^i(M). $$ The latter Hodge structure carries a polarization given by $$ <\alpha,\,\beta> = \int_Mc_1({\cal L})^{m-k-i}\cup\alpha\cup\beta. $$ The conditions on $i$ and the Hodge-Riemann bilinear relations on the primitive part of $H^i(M,\,{\Bbb C})$, assure us that the above is indeed a polarization (see \cite{grif-period} or Chap.\,V,\,\S6 of \cite{wells}). In fact, our conditions on $i$ imply that the primitive part of $H^i(M)$ is everything. This translates to a polarization on $H^i(U)$ given by $$ <x,\,y> = \int_Ul(x)\cup y. $$ This gives the result. \end{pf} \begin{ack} We wish to thank Prof.\,M.\,S. Narasimhan and Prof.\,C.\,S. Seshadri for their encouragement and their help. Thanks to V. Balaji, L. Lempert, N. Raghavendra and P.\,A.Vishwanath for helpful discussions. Balaji made us aware of the problem, and generously discussed his proof (in \cite{balaji}) of the Torelli theorem for Seshadri's desingularization of ${\cal S}{\cal U}_X(2,\,{\cal O}_X)$. The second author gratefully acknowledges the four wonderful years he spent at the SPIC Science Foundation, Madras. \end{ack}

9612

alg-geom/9612008

en

https://arxiv.org/abs/alg-geom/9612008

alg-geom/9612008

Jim Bryan

Jim Bryan and Marc Sanders

Instantons on $S^{4}$ and $\cpbar $, rank stabilization, and Bott periodicity

20 pages, keywords: instantons, holomorphic bundles, Bott periodicity LaTeX2e

We study the large rank limit of the moduli spaces of framed bundles on the projective plane and the blown-up projective plane. These moduli spaces are identified with various instanton moduli spaces on the 4-sphere and $\cpbar $, the projective plane with the reverse orientation. We show that in the direct limit topology, these moduli spaces are homotopic to classifying spaces. For example, the moduli space of $Sp(\infty)$ or $SO(\infty)$ instantons on $\cpbar $ has the homotopy type of $BU(k)$ where $k$ is the charge of the instantons. We use our results along with Taubes' result concerning the $k\to \infty $ limit to obtain a novel proof of the homotopy equivalences in the eight-fold Bott periodicity spectrum. We give explicit constructions for these moduli spaces.

alg-geom

\section{Introduction} Let $\M{k}{G_{n}}(X)$ denote the space of (based) $G_{n}$-instantons on $X$ where $G_{n}$ is $SU(n)$, $SO(n)$, or $Sp(n/2)$. In 1989, Taubes \cite{Tau-stable} showed that there is a ``gluing'' map $\M{k}{G_{n}}(X)\hookrightarrow \M{k'}{G_{n}}(X)$ when $k'>k$. He proved that in the direct limit topology, the instantons capture all the topology of connections modulo gauge equivalence. In other words, there is a homotopy equivalence: $$ \operatorname{lim}_{k\to \infty }\M{k}{G_{n}}(X) \sim \operatorname{Map}_{0} (X,BG_{n}). $$ There is also an inclusion $\M{k}{G_{n}}(X)\hookrightarrow \M{k}{G_{n'}}(X)$ where $n'>n$ induced by the inclusion $G_{n}\hookrightarrow G_{n'}$. Not much is known about the homotopy type of $\M{k}{G}(X)=\operatorname{lim}_{n\to \infty }\M {k}{G_{n}}(X)$ for general $X$. In this paper we determine the homotopy type of $\M{k}{G}(X)$ when $X$ is $S^{4}$ or $\cpbar $ with their standard metrics. The results are \begin{eqnarray}\label{eqn:rank stable for S4} \M{k}{G}(S^{4})&\sim &\begin{cases} BU(k)&\text{if }G=SU,\\ BO(k)&\text{if }G=Sp,\\ BSp(k/2)&\text{if }G=SO; \end{cases}\\ \M{k}{G}(\cpbar )&\sim &\begin{cases} BU(k)\times BU(k)&\text{if }G=SU,\\ BU(k)&\text{if $G$ is $Sp$ or $SO$.} \end{cases}\nonumber \end{eqnarray} The results for the $S^{4}$ case were first proved in \cite{Sanders}, \cite{No-Sa}, and \cite{Kir} (c.f. \cite{Tian}) and we proved the $\M{k}{SU}(\cpbar )$ result in \cite{Br-Sa}. In this paper we are able to provide a unified approach to these moduli spaces and stabilization results. By employing Taubes' theorem and by utilizing the conformal map $f:\cpbar \to S^{4}$ to compare $\M{k}{G_{n}} (S^{4 })$ and $\M{k}{G_{n}} (\cpbar )$, and we are able to give a novel proof of the homotopy equivalences in the real and unitary Bott periodicity spectrums. Work in this direction has been done by Tian using instantons on $S^{4}$ (see \cite{Sanders},\cite{Tian}) where one can prove some of the 4-fold equivalences. By using the comparison with instantons on $\cpbar $ we are able to recover the finer 2-fold equivalences in the periodicity spectrum. The moduli spaces $\M{k}{G_{n}}(S^{4})$ and $\M{k}{G_{n}}(\cpbar )$ are known to be isomorphic to moduli spaces of certain holomorphic bundles and have been constructed in various guises (\cite{BU86}, \cite{King}, \cite{DonMonads} ). Using work of Donaldson and King we construct the spaces from a unified viewpoint (see Table 1). We describe the relevant moduli spaces of holomorphic bundles as follows: Let $H\subset \cnums \P ^{2}$ be a fixed hyperplane and let $\til{\mathbf{CP}}^{2}$ be the blow-up of $\cnums \P ^{2}$ at a point away from $H$. Donaldson showed \cite{DonMonads} that $\M{k}{SU(n)}(S^{4}) $ is isomorphic to the moduli space of pairs $(\mathcal{E} ,\tau )$ where $\mathcal{E} \to \cnums \P ^{2}$ is a rank $n$ holomorphic bundle with $c_{1}(\mathcal{E} )=0$, $c_{2}(\mathcal{E} )=k$ and $\tau :\mathcal{E} |_{H}\to \cnums ^{n}\otimes \O _{H}$ is a trivialization of $\mathcal{E} $ on $H$. In \cite{King}, King extended this result to $\cpbar $ by showing that $\M{k}{SU(n)}(\cpbar ) $ is isomorphic to the moduli space of pairs $(\mathcal{E} ,\tau )$ where $\mathcal{E} \to \til{\cnums \P }^{2}$ is a rank $n$ holomorphic bundle with $c_{1}(\mathcal{E} )=0$, $c_{2}(\mathcal{E} )=k$ and $\tau :\mathcal{E} |_{H}\to \cnums ^{n}\otimes \O _{H}$ is a trivialization of $\mathcal{E} $ on $H$. They also construct the moduli spaces in terms of ``linear algebra data.'' One can extend their results to $Sp(n/2) $ and $SO(n)$. Let $X$ denote $S^{4}$ or $\cpbar $. The moduli space of $Sp$-instantons (respectively $SO$-instantons) is isomorphic to the moduli space of triples $(\mathcal{E} ,\tau ,\phi )$ where $\phi $ is a symplectic (resp. real) structure: \begin{eqnarray*} \M{k}{Sp(n/2)}(X )&\cong &\{(\mathcal{E} ,\tau ,\phi ):(\mathcal{E},\tau )\in \M {k}{SU(n)} (X ), \phi :\mathcal{E}\stackrel{\cong }{\longrightarrow }\mathcal{E} ^{*}, \phi ^{*}= -\phi \},\\ \M{k}{SO(n)}(X )&\cong &\{(\mathcal{E} ,\tau ,\phi ):(\mathcal{E},\tau ) \in \M {k}{SU(n)} (X ),\phi :\mathcal{E}\stackrel{\cong }{\longrightarrow }\mathcal{E} ^{*}, \phi ^{*}= \phi \}. \end{eqnarray*} Our construction realizes these moduli spaces as quotients of affine varieties $A_{k}^{G_{n}}(X)$ (the ``linear algebra data'') by free actions. The key to proving our stability theorem is to show that in the large $n$ limit, the space of ``linear algebra data'' becomes contractible. The constructions also allow us to identify the universal bundles over $\M{k}{G}(X)$ inducing the homotopy equivalences of Equation \ref{eqn:rank stable for S4}. In the holomorphic setting they can be described as certain higher direct image bundles and in the connection setting they can be described as the index bundles of a certain family of coupled Dirac operators. In section \ref{sec: main result and BP} we fix notation, state the theorems, and prove Bott periodicity. In the subsequent sections we construct the moduli spaces and prove the theorems. We conclude with a short appendix discussing a more differentio-geometric construction of the universal bundles. The authors would like to thank John Jones and Ralph Cohen for suggesting that the homotopy equivalences of Equation \ref{eqn:rank stable for S4} should exist. \section{The main results and Bott periodicity}\label{sec: main result and BP} \subsection{Statement of the theorems}\label{subsec:statement of thms} Let $G_{n}\hookrightarrow P\to X$ be a principal bundle on a Riemannian $4$-manifold $X$ with structure group $G_{n}=SU(n)$, $SO(n)$, or $Sp(n/2)$. Using the defining representations for $SU(n)$ or $Sp(n/2)$ and the complexified standard representation for $SO(n)$, we associate to $P$ a rank $n$ complex vector bundle $E$ and we define the {\em charge} $k$ to be $c_{2}(E)[X]$. \footnote{Our definition of $k$ in the $SO(n)$ case differs from some of the literature by a factor of $2$.} A bundle isomorphism $\phi :E\to E^{*}$ is called a {\em real structure} if $\phi^{*}=\phi $ and it is called a {\em symplectic structure} if $\phi ^{*} =-\phi $. We can regard a $SO(n) $ or a $Sp(n/2)$ bundle as a $SU(n)$ bundle $E$ along with $\phi $, a real or symplectic structure respectively. Obviously, $n$ must be even for $E$ to have a symplectic structure and it is also not hard to see that if $E$ has a real structure, then our $k$ must be even. Let $\mathcal{A}(E)$ denote the space of connections on $E$ that are compatible with $\phi $ and let $F^{+}_{E} $ be the self-dual part of the curvature of a connection $A\in \mathcal{A}(E)$. Let $\mathcal{G}_{E}^{0}$ be the group of gauge transformations of $E$ commuting with $\phi $ and preserving a fixed isomorphism $E_{x_{0}}\cong \cnums ^{n}$ of the fiber over a base point $x_{0}\in X$. We define the (based) instanton moduli spaces to be (c.f. \cite{D-K}): $$ \M{k}{G_{n}}(X)=\{A\in \mathcal{A}(E):F_{A}^{+}=0 \}/\mathcal{G}_{E}^{0}. $$ >From here on let $X$ denote $S^{4}$ or $\cpbar $ with their standard metrics. We describe how the moduli spaces $\M{k}{G_{n}}(X)$ can be constructed from configurations of linear algebra data satisfying certain ``integrability'' conditions, modulo natural automorphisms. The configurations are laid out by Table 1 where we have adopted the following notations: Our vector spaces are always complex and our maps are always complex linear. We regard a map $f:U\to W$ as an element $f\in U^{*}\otimes W$. An isomorphism $\phi :W\to W^{*}$ is called a {\em symplectic structure} if $\phi \in \Lambda ^{2}W^{*}$ and a {\em real structure } if $\phi \in S^{2}W^{*}$. $Gl(W)$ denotes the group of isomorphism of $W$ and if $\phi $ is a symplectic (respectively real) structure on $W$, the let $Sp(W)$ (resp. $O(W)$) denote the group of isomorphisms of $W$ compatible with $\phi $ ({\em i.e. } $f^{*}\phi f=\phi $). When $n$ is even, let $J$ denote the standard symplectic structure on $\cnums ^{n}$. Unless otherwise noted, the vector spaces in Table 1 are $k$-dimensional. If $f:V_{1}\to V_{2}$, then $Gl(V_{1})\times Gl(V_{2})$ acts on $f$ by $f\mapsto g_{2}fg_{1}^{-1} $ and thus on $f^{*}$ by $(g_{1}^{-1})^{*}f^{*}g_{2}^{*}$. So the action of the automorphism group on $\cpbar $ configurations is given by \footnote{Note that if we fix bases for the vector spaces, $f^{*}$ is the transpose matrix and should not be confused with conjugate transpose.} \begin{eqnarray*} (g,h)\cdot (a_{1},a_{2},x,b,c)&=&(ga_{1}h^{-1},ga_{2}h^{-1},hxg^{-1},gb,ch^{-1})\\ g\cdot (\alpha _{1},\alpha _{2},\xi ,\gamma )&=&(g\alpha _{1}g^{*},g\alpha _{2}g^{*},(g^{*})^{-1}\xi g^{-1},\gamma g^{*}), \end{eqnarray*} and on $S^{4}$ configurations by \begin{eqnarray*} g\cdot (a_{1},a_{2},b,c)&=&(ga_{1}g^{-1},ga_{2}g^{-1},gb,cg^{-1})\\ g\cdot (\alpha _{1},\alpha _{2},\gamma )&=&(g\alpha _{1}g^{-1},g\alpha _{2}g^{-1},\gamma g^{-1}). \end{eqnarray*} The three main theorems of this paper are the following: \begin{table}\label{table} \begin{picture}(500,350) \put(-20,170){$ \begin{array}{||c|c|c|c|c|c||}\hline G_{n}& X& \text{Configurations}& \text{Integrability}& \text{Automorphism}&\dim _{\cnums }\M{}{}\\ &&& \text{Condition}&\text{Group}&\\ \hline \hline SU(n)& \cpbar &(a_{1},a_{2},x,b,c) &a_{1}xa_{2}-a_{2}xa_{1}+bc=0& Gl(W)\times Gl(U)&4nk\\ \cline{3-3}\cline{3-3} &&a_{i}\in U^{*}\otimes W&&&\\ &&x\in W^{*}\otimes U&&&\\ &&b\in \cnums^{n}\otimes W&&&\\ &&c\in U^{*}\otimes \cnums^{n}&&&\\ \hline Sp(n/2)&\cpbar &(\alpha _{1},\alpha _{2},\xi,\gamma )& \alpha _{1}\xi \alpha _{2}-\alpha _{2}\xi \alpha _{1}-\gamma ^{*}J\gamma =0 & Gl(W)&(n+2)k\\ \cline{3-3}\cline{3-3} && \alpha _{i}\in S^{2}W&&&\\ && \xi \in S^{2}W^{*}&&&\\ && \gamma\in W\otimes \cnums ^{n}&&&\\ \hline SO(n)&\cpbar &(\alpha _{1},\alpha _{2},\xi,\gamma )& \alpha _{1}\xi \alpha _{2}-\alpha _{2}\xi \alpha _{1}+\gamma ^{*}\gamma =0 & Gl(W)&(n-2)k\\ \cline{3-3}\cline{3-3} && \alpha _{i}\in \Lambda ^{2}W&&&\\ && \xi \in \Lambda ^{2}W^{*}&&&\\ && \gamma\in W\otimes \cnums ^{n}&&&\\ \hline SU(n)& S^{4}&(a_{1},a_{2},b,c)&[a_{1},a_{2}]+bc=0&Gl(W)&4nk\\ \cline{3-3} \cline{3-3} &&a_{i}\in W^{*}\otimes W&&&\\ &&b\in \cnums^{n}\otimes W&&&\\ &&c\in W^{*}\otimes \cnums^{n}&&&\\ \hline Sp(n/2)&S^{4}&(\alpha _{1},\alpha _{2},\gamma )&\Phi [\alpha _{1},\alpha _{2}]-\gamma ^{*}J\gamma =0&O(W)&(n+2)k\\ \cline{3-3}\cline{3-3} &&\text{Real str. }\Phi &&&\\ &&\Phi \alpha _{i}\in S^{2}W^{*}&&&\\ && \gamma\in W^{*}\otimes \cnums ^{n}&&&\\ \hline SO(n)&S^{4}&(\alpha _{1},\alpha _{2},\gamma )&\Phi [\alpha _{1},\alpha _{2}]+\gamma ^{*}\gamma =0&Sp(W)&(n-2)k\\ \cline{3-3}\cline{3-3} &&\text{Sympl. str. }\Phi &&&\\ &&\Phi \alpha _{i}\in \Lambda ^{2}W^{*}&&&\\ && \gamma\in W^{*}\otimes \cnums ^{n}&&&\\ \hline \end{array} $} \end{picture} \caption{Configurations that construct $\M{k}{G_{n}}(X)$.} \end{table} \begin{thm}[Moduli Construction]\label{thm:moduli construction} Let $\overline{A}^{G_{n}}_{k}(X)$ denote the space of integrable configurations as given by Table 1. There is an open dense set $A^{G_{n}}_{k}(X)\subset \overline{A}^{G_{n}}_{k}(X) $ (the ``non-degenerate'' configurations) such that the instanton moduli space $\M{k}{G_{n}}(X)$ is isomorphic to the quotient of $A^{G_{n}}_{k}(X)$ by the automorphism group. Furthermore, the action of the automorphism group on $A^{G_{n}}_{k}(X)$ is free and the vector spaces $W$ and $U$ of Table 1 are canonically isomorphic to $H^{1}(\mathcal{E} (-H))$ and $H^{1}(\mathcal{E} (-H+E))$ respectively. \end{thm} \begin{thm}[Lifts of instanton maps to configurations] \label{thm:lifts of maps to configurations} There are commuting inclusions of configurations (defined in section \ref{sec:lifts of maps}) \begin{eqnarray*} i&:&A^{G_{n}}_{k}(X)\hookrightarrow A^{G_{n'}}_{k}(X)\\ j&:&A^{G_{n}}_{k}(S^{4})\hookrightarrow A^{G_{n}}_{k}(\cpbar ) \end{eqnarray*} for $n<n'$ and $k<k'$. These maps intertwine the automorphisms and consequently descend to maps on the instanton moduli spaces. The map $i$ descends to the map induced by the inclusion $G_{n}\hookrightarrow G_{n'}$ and the map $j$ descends to the map induced by pulling back connections via $f:\cpbar \to S^{4}$. \end{thm} \begin{thm}[Rank Stabilization]\label{thm:rank stabilization} Let $A^{G}_{k}(X)$ be the direct limit space $$\operatorname{lim}_{n\to \infty }A^{G_{n}}_{k}(X)$$ defined by the inclusions $i$. Then $A^{G}_{k}(X)$ is contractible and consequently $$\M{k}{G}=\operatorname{lim}_{n\to \infty }\M{k}{G_{n}}$$ is homotopic to the classifying space for the associated automorphism group. This theorem implies the homotopy equivalences in Equation \ref{eqn:rank stable for S4}. \end{thm} \begin{rem}\label{rem:dimension count} A na\"{\i}ve dimension count for $\M{k}{G_{n}}(X)$ is obtained by subtracting the dimension of the automorphism group and the number of conditions imposed by integrability from the dimension of the configurations. This agrees with the dimension predicted by the Atiyah-Singer index formula and appears in the far right column of the table. \end{rem} We prove Theorems \ref{thm:moduli construction}, \ref{thm:lifts of maps to configurations}, and \ref{thm:rank stabilization} in Sections \ref{sec:construction}, \ref{sec:lifts of maps} , and \ref{sec:pf of stable thm} respectively. \subsection{Bott Periodicity}\label{subsec:Bott per.} In this subsection we show how Theorems \ref{thm:rank stabilization}, \ref{thm:lifts of maps to configurations}, and Taubes' stabilization leads to an alternative, relatively quick proof of the following homotopy equivalences in the periodicity spectrum: \begin{thm}[Bott]\label{thm:Bott periodicity} Let $SU$, $U$, $SO$, and $Sp$ denote the direct limit groups of $SU(n)$, $U(n)$, $SO(n)$, and $Sp(n)$ as $n\to \infty $. Let $\Omega ^{j}X$ denote the $j$-fold loop space of $X$. The following are homotopy equivalences: \begin{eqnarray*} \Omega ^{2}SU&\sim &U,\\ \Omega ^{2}Sp& \sim & U/O,\\ \Omega ^{2}SO& \sim & U/Sp,\\ \Omega ^{4}SO& \sim & Sp,\\ \Omega ^{4}Sp& \sim & O. \end{eqnarray*} \end{thm} \begin{rem} The first equivalence is Bott periodicity for the unitary group and the next four appear in the real periodicity spectrum. The only missing homotopy equivalences: $$ \Omega ^{2}(Sp/U)\sim BO\times \znums \text{ and } \Omega ^{2}(SO/U)\sim BSp\times \znums $$ are related to monopoles (see Cohen and Jones \cite{Co-Jo} and the thesis of Ernesto Lupercio \cite{Lupercio}). \end{rem} \proof Let $i'$, $j'$ and $t'$ denote the maps on the moduli spaces induced by rank inclusion, pull-back from $S^{4} $ to $\cpbar $, and Taubes' gluing respectively ($i'$ and $j' $ are the descent of the maps $i$ and $j$ in Theorem \ref{thm:lifts of maps to configurations}). We will argue that $i'$, $j'$, and $t'$ commute up to homotopy. The maps $i'$ and $j'$ commute (on the nose) from Theorem \ref{thm:lifts of maps to configurations}; and we can see that $t'$ commutes up to homotopy with $i' $ and $j'$ from some general properties of $t'$: The Taubes' map for any semi-simple compact Lie group $G$ is obtained from the Taubes map for $SU (2)$ via any homomorphism $SU (2)\to G$ generating $\pi _{3} (G)$. Since the inclusions $G_{n}\hookrightarrow G_{n'}$, $n'>n$ ($n>4$ if $G_{n}=SO (n)$) induce isomorphisms on $\pi _{3}$, $t'$ automatically commutes with $i'$. To see that $t'$ commutes up to homotopy with $j' $ we use almost instantons: connections with Yang-Mills energy smaller than a small constant $\epsilon $. The space of almost instantons $\M{k,\epsilon }{G_{n}} (X)$ has a strong deformation retract onto the space of instantons and there is a map $t'_{\epsilon }:\M{k,\epsilon }{G_{n}} (X)\to \M{k+1,\epsilon }{G_{n}} (X)$ homotopic to $t'$. It is local in the sense that $t_{\epsilon }' (A)$ agrees with $A$ up to gauge in the complement of a ball about the gluing point. On the other hand, the natural map $\cpbar \to S^{4}$ is a conformal isometry on the complement of the hyperplane that gets mapped to a point. Connections pulled back by this map have the same Yang-Mills energy and we get a map $j'_{\epsilon }$ on almost instantons. Thus as long as we choose our gluing point away from the hyperplane, $t'_{\epsilon }$ and $j'_{\epsilon }$ commute and so $t'$ and $j'$ commute up to homotopy. The maps $t'$, $j'$, and $j$ then induce commuting maps on the corresponding direct limit moduli and configurations spaces when $n\to \infty $ . We will assume that we have passed to that limit throughout the rest of this section. From Theorem \ref{thm:rank stabilization}, we have that $A^{G_{\infty }}_{k}(X)$ is contractible. We can thus identify the homotopy fibers of the $j'$ maps to get the following fibrations: \footnote{As we will see in Table 1, the structure groups of the various principle bundles $A^{G_{n}}_{k}(X)$ are the complex forms of the groups $U(k)$, $Sp(k/2)$, and $O(k)$ ({\em i.e. } $Gl(k,\cnums )$, $Sp(k/2,\cnums )$, and $O(k,\cnums )$). Since the complex forms of the groups are homotopic to their compact form, their classifying spaces are the same (up to homotopy). It is traditional to use the compact form when refering to classifying spaces so we will use the notation of the compact group for the rest of this section. } \begin{equation}\label{eqn:j-map fibrations} \begin{CD} U(k)\times U(k)/U(k)@>>>\M{k}{SU}(S^{4})@>{j'}>>\M{k}{SU}(\cpbar ),\\ U(k)/O(k)@>>>\M{k}{Sp}(S^{4})@>{j'}>>\M{k}{Sp}(\cpbar ),\\ U(k)/Sp(k/2)@>>>\M{k}{SO}(S^{4})@>{j'}>>\M{k}{SO}(\cpbar ) \end{CD} \end{equation} where $U(k)$ is included into $U(k)\times U(k)$ via the diagonal. Here we are using the fact that Theorem \ref{thm:lifts of maps to configurations} gives us $j$, the lift of $j'$ to the principle bundles $A^{G_{n}}_{k} (X)$ that intertwines the actions. Since $j'$ commutes with $t$, the above fibrations are valid for the direct limit spaces when $k\to \infty $. We now use Taubes' theorem to compare the above fibrations with the fibration on the space of connections induced by the cofibration $S^{2}\hookrightarrow \cpbar \to S^{4}$. Let $\mathcal{B}^{G_{n}}_{k}(X)$ denote the space of all $G_{n}$-connections of charge $k$ modulo based gauge equivalence. $\mathcal{B}^{G_{n}}_{k}(X)$ is homotopy equivalent to the mapping space $Map_{k}(X,BG_{n})$ and the cofibration $S^{2}\hookrightarrow \cpbar \to S^{4}$ gives rise to a fibration $$ \Omega _{k}^{4}BG_{n}\xrightarrow{j'} Map_{k}(\cpbar ,BG_{n})\to \Omega ^{2}BG_{n} . $$ Up to homotopy, the map $j'$ in the above sequence is induced by pulling back connections via $\cpbar \to S^{4}$ (thus justifying the notation). These maps also commute with the group inclusions $i$ and so give a fibration in the $n\to \infty $ limit. Also, $\mathcal{B}^{G_{n}}_{k}(X)$ and $\mathcal{B}^{G_{n}}_{k+1}(X)$ are naturally homotopy equivalent and so we implicitly identify them and drop the notational dependence; we have: \begin{equation}\label{eqn:mapping space fibration} \mathcal{B}^{G}(S^{4})\xrightarrow{j'}\mathcal{B}^{G}(\cpbar )\to \Omega G. \end{equation} Taubes' stabilization theorem states that the inclusions $\M{k}{G_{n}}(X)\hookrightarrow \mathcal{B}^{G_{n}}(X)$ induce a homotopy equivalence in the limit $k\to \infty $. Since the inclusion of the moduli spaces into $\mathcal{B}$ commutes with both $j'$ and $i'$ we can pass to the $k\to \infty $ and $n\to \infty $ limits and use Equation \ref{eqn:j-map fibrations} to identify the homotopy fiber of $j':\mathcal{B}^{G}(S^{4})\to\mathcal{B}^{G}(\cpbar ) $. This fiber is in turn homotopy equivalent to $\Omega ^{2}G$ by the sequence \ref{eqn:mapping space fibration}. Thus for $G=SU$, $Sp$, and $SO$ respectively, we get \begin{eqnarray*} \Omega ^{2}SU& \sim & U\times U/U\sim U,\\ \Omega ^{2}Sp& \sim & U/O,\text{ and}\\ \Omega ^{2}SO& \sim & U/Sp. \end{eqnarray*} The final two homotopy equivalences are arrived at by applying Theorem \ref{thm:rank stabilization} and Taubes' stabilization directly to $\M{k}{Sp}(S^{4})$ and $\M{k}{SO}(S^{4})$ . \qed \subsection{The algebro-geometric moduli spaces and $A _{k}^{G_{n}}$}\label{subsec: alg-geo and U's} >From now on, we will use the notation $X$ and $Y$ to denote $S^{4}$ and $\cnums \P ^{2} $ or $\cpbar$ and $\til{\cnums \P}^{2}$ (Recall from the introduction that $\til{\cnums\P}^{2}$ is the blown-up projective plane). Consider the moduli space $\M{alg}{n,k}(Y)$ consisting of pairs $(\mathcal{E} ,\tau )$ where $\mathcal{E} $ is a rank $n$ holomorphic bundle on $Y$ and $\tau :\mathcal{E}|_{H}\stackrel{\cong}{\longrightarrow} \cnums ^{n}\otimes \O _{H}$ is an isomorphism. Let $p:Y\to X $ be the smooth map that sends $H\mapsto x_{0}$ and is one-to-one elsewhere. The map $p$ is compatible with the natural orientations (we think of $p$ as a ``anti-holomorphic blowdown''). We can construct a natural map $$ \Xi :\M{k}{SU(n)}(X )\to \M{alg}{n,k}(Y) $$ by defining the holomorphic structure on $p^{*}(E)$ corresponding to $\Xi ([A])$ to be $(d_{p^{*}A})^{0,1}$ and $\tau $ is induced by the fixed isomorphism of $E_{x_{0}}$ (c.f. \cite{Bryan}). \begin{thm}[Donaldson \cite{DonMonads}, King \cite{King}]\label{thm:King's thm} The map $\Xi $ is an isomorphism of moduli spaces. \end{thm} Consider the moduli space $\M{alg,\pm }{n,k}(Y)$ of triples $(\mathcal{E} ,\tau ,\phi )$ where $(\mathcal{E} ,\tau )\in \M{alg}{n,k}(Y)$ and $\phi :\mathcal{E} \to \mathcal{E} ^{*}$ is a isomorphism such that $\phi ^{*}=\pm \phi $. We can construct maps $\Xi _{\pm }$ in the same fashion as $\Xi $. A consequence of Theorem \ref{thm:King's thm} is \begin{cor}\label{cor:Sp and SO bundles from SU } The maps \begin{eqnarray*} \Xi _{+}&:&\M{k}{SO(n)}(X )\to \M{alg,+}{n,k}(Y)\text{ and}\\ \Xi _{-}&:&\M{k}{Sp(n)}(X )\to \M{alg,-}{n,k}(Y) \end{eqnarray*} are moduli space isomorphisms ($(X,Y)$ is $(S^{4},\cnums \P ^{2})$ or $(\cpbar ,\til{\cnums \P }^{2})$). \end{cor} \proof Recall that we consider $SO(n)$ or $Sp(n/2)$ connections to be $SU(n)$ connections that are compatible with a real or symplectic structure $\phi $, {\em i.e. } $$ \nabla_{A^{*}}(\phi s)=\phi \nabla_{A}s $$ where $A\in \mathcal{A(E)}$ and $A^{*}$ is the induced connection in $\mathcal{A}(E^{*})$. Compatibility implies that $\phi $ will be a holomorphic map with respect to the holomorphic structures defined by $(d_{\pi^{*}A})^{0,1}$ and $(d_{\pi ^{*}(A^{*})})^{0,1}$. Conversely, let $(\mathcal{E} ,\tau ,\phi )$ be in $\M{k}{Sp(n/2)}(X)$ or $\M{k}{SO(n)}(X)$. Choose a hermitian structure on $\mathcal{E} $ compatible with $\phi $ and $\tau $. By Theorem \ref{thm:King's thm}, the unique hermitian connection is the pullback of an anti-self-dual $SU(n)$ connection on $E$ which is, by construction, compatible with $\phi $. \qed Henceforth we will drop the $\M{alg}{}$ notation and use $\M {k}{G_{n}}(X )$ to refer to either moduli space. The moduli space $\M{k}{G_{n}}(X) $ has a universal bundle (see Lemma 3.2 of \cite{Br-Sa}) $$ \begin{CD} \Bbb{E}\\ @VVV\\ \M{k}{G_{n}}(X)\times Y \end{CD} $$ so that $\Bbb{E}|_{\{\mathcal{E} \}\times Y}\cong \mathcal{E} $. Consider the cohomology groups $H^{i}(\mathcal{E} (-H))$. The fact that $\mathcal{E} $ is trivial on $H$ (and thus on nearby lines) implies that $H^{i}(\mathcal{E} (-H))=0$ for $i=0$ or $2$ (see \cite{Br-Sa}). The Riemann-Roch theorem then gives $\dim H^{1}(\mathcal{E} (-H))=k$. We will see from the construction of section \ref{sec:construction} that the vector space $W$ of Table 1 can be canonically identified with $H^{1}(\mathcal{E} (-H))$. In the case of $X=\cpbar $ and $G_{n}=SU(n)$, a similar argument shows $\dim H^{1}(\mathcal{E} (-2H+E))=k$ and $U$ is canonically identified with $H^{1}(\mathcal{E} (-2H+E))$. Let $\pi :\M{k}{G_{n}}(X)\times Y\to \M{k}{G_{n}}(X)$ be projection. One consequence of the above discussion is that the higher direct image sheaf $R^{1}\pi _{*}(\Bbb{E}(-H))$ is a rank $k$ bundle on $\M{k}{G_{n}}(X)$. Consequently, we have the following geometric interpretation of the configuration spaces $A^{G_{n}}_{k}(X)$ (c.f. Appendix \ref{subsec:diff-geo constr of univ bundles}): \begin{thm}\label{thm:config space is principal bundle assoc to R1pi*} The space of configurations $A^{G_{n}}_{k}(X) $ (see Table 1) is homeomorphic to the total space of the frame bundle of $R^{1}\pi _{*}(\Bbb{E}(-H))$ except for the case $A^{SU(n)}_{k}(\cpbar )$ which is homeomorphic to the frame bundle of $$R^{1}\pi _{*}(\Bbb{E}(-H))\oplus R^{1}\pi _{*}(\Bbb{E}(-2H+E)).$$ \end{thm} \proof The fiber of $A^{G_{n}}_{k}(X)\to \M{k}{G_{n}}(X)$ over a point $\mathcal{E} $ is the orbit of a representative configuration by the automorphism group. This can be identified with the space $\operatorname{Iso} (W,\cnums ^{k})$ (or $\operatorname{Iso} (W,\cnums ^{k})\times \operatorname{Iso} (U,\cnums ^{k})$ in the $X=\cpbar $, $G_{n}=SU(n)$ case) where we also understand $\operatorname{Iso} (W,\cnums ^{k})$ to be isomorphisms of symplectic or real vector spaces in the $X=S^{4}$, $G_{n}=SO(n)$ or $Sp(n/2)$ cases. \section{Construction of the moduli spaces} \label{sec:construction} \subsection{Preliminaries} To fill in Table 1, we rely heavily on the constructions of Donaldson and King; we will recall what we need from their constructions in subsection \ref{subsec:SU constructions}. Let us first begin by introducing some general notation. For an $n$-dimensional projective manifold $M$ and a coherent sheaf $\mathcal{E}$ on $M$ let $SD_{p,\mathcal{E}}$ denote the Serre duality isomorphism $$ SD_{p,\mathcal{E}}:H^{p}(\mathcal{E})\to H^{n-p}(\mathcal{E} ^{*}(K))^{*}. $$ Let $H^{i}(\phi ):H^{i}(\mathcal{E}\otimes \mathcal{G} )\to H^{i}(\mathcal{F}\otimes \mathcal{G})$ denote the map in cohomology induced by a sheaf map $\phi :\mathcal{E} \to \mathcal{F}$. If $s\in H^{0}(\O (D))$ is a section vanishing on $D$, we let $\delta _{s}:H^{0}(\mathcal{E} |_{D}) \to H^{1}(\mathcal{E} (-D))$ denote the coboundary map arising in the long exact sequence associated to $$ 0\to \mathcal{E} (-D)\xrightarrow{s}\mathcal{E} \xrightarrow{r}\mathcal{E} _{D}\to 0. $$ We will use the following elementary properties of Serre duality: \begin{enumerate} \item $SD_{p,\mathcal{E} }=(-1)^{p(n-p)}(SD_{n-p,\mathcal{E} ^{*}(K)})^{*}$, \item $SD_{p,\mathcal{E} }$ is natural in the sense that $$ \begin{CD} H^{p}(\mathcal{E} ) @>{SD_{p,\mathcal{E} }}>> H^{n-p}(\mathcal{E} ^{*}(K))^{*}\\ @VV{H^{p}(\phi )}V @VV{H^{n-p}(\phi ^{*})^{*}}V\\ H^{p}(\mathcal{F})@>{SD_{p,\mathcal{F}}}>> H^{n-p}(\mathcal{F^{*}}(K))^{*} \end{CD} $$ commutes. \end{enumerate} The sign in the first property arises from commuting the cup product. A {\em monad} is a three term complex of vector bundles over a complex manifold $$\mathcal{U} \xrightarrow{A}\mathcal{V}\xrightarrow{B}\mathcal{W}$$ such that $A$ is injective, $B$ is surjective, and $B\comp A$ is $0$. The monad determines its {\em cohomology bundle} $\mathcal{E}=\operatorname{Ker}(B)/\operatorname{Im}(A)$. The point is that one can build complicated holomorphic bundles from relatively simple bundles using monads. By fixing the bundles $(\mathcal{U} ,\mathcal{V},\mathcal{W})$ and allowing the maps $A$ and $B$ to vary, one parameterizes a family of bundles. We say that $(\mathcal{U},\mathcal{V},\mathcal{W})$ {\em effectively parameterizes bundles} if the morphisms of $(\mathcal{U},\mathcal{V},\mathcal{W})$-monads are in one-to-one correspondence with morphisms of the associated cohomology bundles. This will be the case under favorable cohomological conditions on $(\mathcal{U},\mathcal{V},\mathcal{W})$ (for details see \cite{King}, \cite{Horrocks}). The Chern character of the cohomology bundle can be computed by the formula $$ ch(\mathcal{E} )=ch(\mathcal{V})-ch(\mathcal{U})-ch(\mathcal{W}). $$ Note that the cohomology bundle associated to the dual monad $$ \mathcal{U}^{*}\xrightarrow{B^{*}}\mathcal{V}^{*} \xrightarrow{A^{*}}\mathcal{W}^* $$ is the dual bundle $E^{*}$. We call a monad {\em self-dual } (or {\em anti-self-dual} ) if it is of the form $$ \mathcal{U}\xrightarrow{A}\mathcal{V}\xrightarrow{A^{*}\beta ^{*}}\mathcal{U}^{*} $$ where $\beta :\mathcal{V}\to \mathcal{V}^{*}$ is a real (or symplectic) structure; {\em i.e. } $\beta ^{*}=\beta $ (or $\beta ^{*}=-\beta $). A self-dual monad is isomorphic to its dual by the isomorphism $({{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l ,\beta ,{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l)$ and an anti-self-dual monad is isomorphic to its dual by $({{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l ,\beta ,-{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l )$. Thus if $(\mathcal{U},\mathcal{V},\mathcal{U}^{*})$ effectively parameterizes bundles, then $({{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l,\beta ,\pm {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l )$ induces a real (or symplectic) structure $\phi :\mathcal{E} \to \mathcal{E} ^{*}$ on the cohomology bundle. \subsection{The $SU(n)$ constructions.} \label{subsec:SU constructions} We wish to show that the $SU(n)$ configurations of Table 1 give rise to bundles in $\M{k}{SU(n)}(X)$. For $(a_{1},a_{2},b,c)\in A^{SU(n)}_{k}(S^{4})$ consider the sequence of bundles on $Y=\cnums \P ^{2}$: \begin{equation}\label{eqn:S4 monad sequence} W\otimes \O (-H)\xrightarrow{A}(W\oplus W\oplus \cnums ^{n})\otimes \O \xrightarrow{B}W\otimes \O (H) \end{equation} where \begin{eqnarray*} A &=&\left(\begin{array}{c} x_{1}-a_{1}x_{3}\\ x_{2}-a_{2}x_{3}\\ cx_{3} \end{array} \right),\\ B&=&\left(\begin{array}{ccc} -x_{2}+a_{2}x_{3}&x_{1}-a_{1}x_{3}&bx_{3} \end{array} \right) \end{eqnarray*} and $\langle x_{1 },x_{2},x_{3} \rangle$ generates $H^{1}(\O (H))$ and $H$ is the zero set of $x_{3}$. The integrability condition is equivalent to $B\comp A=0$. We define $A^{SU(n)}_{k}(S^{4})$ to be the open dense set of the integrable configurations such that $A$ and $B$ are pointwise injective and surjective respectively. Thus for configurations in $A^{SU(n)}_{k}(S^{4})$, Sequence \ref{eqn:S4 monad sequence} is a monad. By computing Chern classes and restricting Sequence \ref{eqn:S4 monad sequence} to $H$, one can see that the cohomology bundle $\mathcal{E} $ lies in $\M{k}{SU(n)}(S^{4})$. In fact the converse is true: \begin{thm}[Donaldson] Every $\mathcal{E} \in \M{k}{SU(n)}(S^{4})$ is given by a monad of the form in Sequence \ref{eqn:S4 monad sequence} and the correspondence is unique up to the natural action of $Gl(W)$. Furthermore, $W$ is canonically identified with $H^{1}(\mathcal{E} (-H))$ (Okonek, et. al. \cite{oss} pg. 275). \end{thm} To finish the proof of Theorem \ref{thm:moduli construction} for $X=S^{4}$ and $G_{n}=SU(n)$ we only need to show that the automorphism group acts freely on $A^{SU(n)}_{k}(S^{4})$. This follows from the identification of $A^{SU(n)}_{k}(S^{4})$ with the frame bundle of $R^{1}\pi _{*}(\Bbb{E}(-H))$ (see Theorem \ref{thm:config space is principal bundle assoc to R1pi*}). For $(a_{1},a_{2},x,b,c)\in A^{SU(n)}_{k}(\cpbar )$ consider the sequence of bundles on $Y=\til{\cnums \P} ^{2}$: \begin{equation}\label{eqn:cpbar monad sequence} \begin{array}{c} {U\otimes \O (-H)}\\ \oplus\\ {W\otimes \O (-H+E)} \end{array} \stackrel{A}{\longrightarrow} V\otimes \mathcal{O}\stackrel{B}{\longrightarrow} \begin{array}{c} {W\otimes \O (H)}\\ \oplus\\ {U\otimes \O (H-E)} \end{array} \end{equation} where $V=W\oplus U\oplus W\oplus U\oplus \cnums ^{n}$ and \begin{eqnarray*} A&=&\left(\begin{array}{cc} a_1x_3 &-y_2\\ x_1-xa_1x_3 & 0 \\ a_2x_3 & y_1 \\ x_2-xa_2x_3 & 0 \\ cx_3 & 0 \end{array}\right)\label{eq:form of A}\\ B&=&\left(\begin{array}{ccccc} x_2 & a_2x_3 & -x_1 & -a_1x_3 & bx_3 \\ xy_1 & y_1 & xy_2 & y_2 & 0 \end{array}\right).\label{eq:form of B} \end{eqnarray*} We have chosen sections $\langle x_1,x_2,x_3 \rangle$ spanning $H^0(\O (H))$ and $\langle y_1,y_2\rangle$ spanning $H^0(\O (H-E))$ so that $x_{3} $ vanishes on $H$ and $x_{1}y_{1}+x_{2}y_{2}$ spans the kernel of $H^{0}(\O (H))\otimes H^{0}(\O (H-E))\to H^{0}(2H-E)$. The integrability condition is equivalent to $B\comp A=0$. We define $A^{SU(n)}_{k}(\cpbar )$ to be the open dense set of the integrable configurations that are such that $A$ and $B$ are pointwise injective and surjective respectively. Thus for configurations in $A^{SU(n)}_{k}(\cpbar )$, Sequence \ref{eqn:cpbar monad sequence} is a monad. By computing Chern classes and restricting Sequence \ref{eqn:cpbar monad sequence} to $H$, one can see that the cohomology bundle $\mathcal{E} $ lies in $\M{k}{SU(n)}(\cpbar )$. Once again the converse is true: \begin{thm}[King] Every $\mathcal{E} \in \M{k}{SU(n)}(\cpbar )$ is given by a monad of the form in Sequence \ref{eqn:cpbar monad sequence} and the correspondence is unique up to the natural action of $Gl(W)\times Gl(U)$. Furthermore, $W$ and $U$ are canonically identified with $H^{1}(\mathcal{E} (-H))$ and $H^{1}(\mathcal{E} (-H+E))$ respectively. \footnote{Historically, the correspondence between holomorphic bundles and instantons on $S^{4} $ or $\cpbar $ was proved by constructing the bundle moduli spaces as in this section and showing that the construction is equivalent to the ``twistor'' construction of instantons. Now there is a direct analytic proof of the correspondence due to Buchdahl \cite{Bu93} that also applies to $\cpbar \# \cdots \# \cpbar$.} \end{thm} Once again we see that automorphism group acts freely on $A^{SU(n)}_{k}(\cpbar )$ from the identification of $A^{SU(n)}_{k}(\cpbar )$ with the frame bundle of $$R^{1}\pi _{*}(\Bbb{E}(-H))\oplus R^{1}\pi_{*}(\Bbb{E}(-H+E)).$$ \begin{rem}\label{rem:dual configs} If $\mathcal{E} \in \M{k}{SU(n)}(X)$ then $\mathcal{E} ^{*}\in \M{k}{SU(n)}(X)$ and is given by the cohomology of the dual monad. To find the ``dual configuration'', we need to use a monad automorphism to put the dual monads into the form determined by a configuration. One can then see that the correspondence $\mathcal{E} \mapsto \mathcal{E} ^{*}$ is realized on the level of configurations by $$ (a_{1},a_{2}, b,c)\mapsto (a_{1}^{*},a_{2}^{*},-c^{*},b^{*}) $$ in the $S^{4}$ case and $$ (a_{1},a_{2},x, b,c)\mapsto (a_{1}^{*},a_{2}^{*},x^{*},-c^{*},b^{*}) $$ in the $\cpbar $ case. \end{rem} We also will need some finer information about these constructions. Namely, there is cohomological interpretations for the maps occurring in the configurations. For $(a_{1},a_{2},b,c)\in A^{SU(n)}_{k}(S^{4})$ the maps are given by the following compositions: \begin{eqnarray*} a_{i}&:&H^{1}(\mathcal{E} (-H))\xrightarrow{H^{1}(x_{3})^{-1}}H^{1}(\mathcal{E} (-2H))\xrightarrow{H^{1}(x_{i})} H^{1}(\mathcal{E} (-H)), \label{eqn:coh interp of ai in S4 case}\\ b&:&H^{0}(\mathcal{E} |_{H})\xrightarrow{\delta _{x_{3}}}H^{1}(\mathcal{E} (-H)),\\ c^{*}&:&H^{0}(\mathcal{E} ^{*}|_{H})\xrightarrow{\delta _{x_{3}}}H^{1}(\mathcal{E} ^{*}(-H))\xrightarrow{H^{1}(x_{3})^{-1}}H^{1}(\mathcal{E} ^{*}(-2H))\xrightarrow{SD}H^{1}(\mathcal{E} (-H))^{*}. \end{eqnarray*} With our definition of $\langle x_{1},x_{2},x_{3} \rangle$ and $\langle y_{1},y_{2} \rangle$ we get a well defined section $s=x_{2}/y_{1}=-x_{1}/y_{2}$ of $H^{0}(\O (E))$. For $(a_{1},a_{2},x,b,c)\in A^{SU(n)}_{k}(\cpbar )$ the maps are given by the following compositions: \begin{eqnarray*} a_{1}&:&H^{1}(\mathcal{E} (-H))\xrightarrow{H^{1}(x_{3})^{-1}}H^{1}(\mathcal{E} (-2H))\xrightarrow{H^{1}(-y_{2})} H^{1}(\mathcal{E} (-H)),\\ a_{2}&:&H^{1}(\mathcal{E} (-H))\xrightarrow{H^{1}(x_{3})^{-1}}H^{1}(\mathcal{E} (-2H))\xrightarrow{H^{1}(y_{1})} H^{1}(\mathcal{E} (-H)),\\ x&:&H^{1}(\mathcal{E} (-H+E))\xrightarrow{H^{1}(s)}H^{1}(\mathcal{E} (-H)),\\ b&:&H^{0}(\mathcal{E} |_{H})\xrightarrow{\delta _{x_{3}}}H^{1}(\mathcal{E} (-H)),\\ c^{*}&:&H^{0}(\mathcal{E} ^{*}|_{H})\xrightarrow{\delta _{x_{3}}}H^{1}(\mathcal{E} ^{*}(-H))\xrightarrow{H^{1}(x_{3})^{-1}}H^{1}(\mathcal{E} ^{*}(-2H))\xrightarrow{SD}H^{1}(\mathcal{E} (-H+E))^{*}. \end{eqnarray*} King gives a detailed discussion of this description. In the $S^{4} $ case, one can also ferret these maps out of the Beilinson spectral sequence derivation of the monads on $\cnums \P^{2}$ (\cite{oss} pg. 249-251,275) using the triviality of $\mathcal{E} $ on $H$. \begin{rem}\label{rem:jumping lines} In general, if $z$ is the defining section of a divisor $D$ and $D$ is geometrically a rational curve, then it is easy to see from the long exact sequence that $$ H^{1}(z):H^{1}(\mathcal{E} (-2D))\to H^{1}(\mathcal{E} (-D)) $$ is an isomorphism if and only if $\mathcal{E} |_{D}$ is trivial. Thus, in the above cohomological interpretations of configurations, $H^{1}(x_{3})$ is always an isomorphism. We see then, for example, that the map $x$ is singular if and only if $\mathcal{E} $ has the exceptional curve $E$ as a ``jumping line''. Likewise we can interpret $a_{1}$ and $a_{2}$: The complement of $H$ in $Y$ is either a complex plane or a complex plane blown-up at the origin. In either case lines through the origin are given by the zeros of $\mu _{1}x_{1}+\mu _{2}x_{2}$. Thus $\mathcal{E} $ will have jumping lines at those lines parameterized by $(\mu _{1},\mu _{2})$ for which $\mu _{1}a_{1}+\mu _{2}a_{2}$ is singular. This circle of ideas has been utilized heavily by Hurtubise, Milgram, {\it et. al.} who use jumping lines to give a filtration of the moduli spaces (\cite{Hurtubise}, \cite{Hur-Mil}, \cite{BHMM}). \end{rem} \subsection{Construction of $\M{k}{Sp(n/2)}(X)$ and $\M {k}{SO(n)}(X)$}\label{subs: construction of MSP and MSO} We now use the constructions of the previous subsection to construct the moduli spaces $\M{k}{Sp(n/2)}(X)$ and $\M{k}{SO(n)}(X)$. We first show that given $Sp(n/2)$ or $SO(n)$ configurations from Table 1, one produces an appropriate self-dual or anti-self-dual monad determining an element of the corresponding moduli space. We then show the converse, {\em i.e. } given an element $(\mathcal{E} ,\tau ,\phi )$ of $\M{k}{Sp(n/2)}(X)$ or $\M {k}{SO(n)}(X)$ we can get an equivalence class of the corresponding configurations from Table 1. For each of the four cases with $G_{n}=Sp(n/2)$ or $SO(n)$, we will use configurations to define a sequence $$ \mathcal{U}\xrightarrow{A}\mathcal{V}\xrightarrow{A^{*}\beta ^{*}}\mathcal{U^{*}} . $$ For integrable configurations (those in $\overline{A}^{G_{n}}_{k}(X)$), the sequence will satisfy $$ A^{*}\beta ^{*}A=0 $$ and for each of the cases we define $A^{G_{n}}_{k}(X)\subset \overline{A}^{G_{n}}_{k}(X)$ to be the open dense subset such that the corresponding map $A$ is pointwise injective. The sequence will then be a monad. For $(\alpha _{1},\alpha _{2},\gamma )\in {A}^{Sp(n/2)}_{k}(S^{4})$ we define an anti-self-dual monad by \begin{equation}\label{eqn:S4 Sp-monad sequence} W\otimes \O (-H)\xrightarrow{A}(W\oplus W\oplus \cnums ^{n})\otimes \O \xrightarrow{A^{*}\beta ^{*}}W^{*}\otimes \O (H) \end{equation} where \begin{eqnarray*} A &=&\left(\begin{array}{c} x_{1}-\alpha _{1}x_{3}\\ x_{2}-\alpha _{2}x_{3}\\ \gamma x_{3} \end{array} \right),\\ \beta &=&\left(\begin{array}{ccc} 0& \Phi & 0\\ -\Phi & 0& 0\\ 0& 0& J \end{array} \right). \end{eqnarray*} For $(\alpha _{1},\alpha _{2},\gamma )\in {A}^{SO(n)}_{k}(S^{4})$ we define an self-dual monad by \begin{equation}\label{eqn:S4 SO-monad sequence} W\otimes \O (-H)\xrightarrow{A}(W\oplus W\oplus \cnums ^{n})\otimes \O \xrightarrow{A^{*}\beta ^{*}}W^{*}\otimes \O (H) \end{equation} where \begin{eqnarray*} A &=&\left(\begin{array}{c} x_{1}-\alpha _{1}x_{3}\\ x_{2}-\alpha _{2}x_{3}\\ \gamma x_{3} \end{array} \right),\\ \beta &=&\left(\begin{array}{ccc} 0& \Phi & 0\\ -\Phi & 0& 0\\ 0& 0& {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l \end{array} \right). \end{eqnarray*} For $(\alpha _{1},\alpha _{2},\xi ,\gamma )\in {A}^{Sp(n/2)}_{k}(\cpbar )$ we define an anti-self-dual monad by \begin{equation}\label{eqn:cpbar Sp monad} \begin{array}{c} {W^{*}\otimes \O (-H)}\\ \oplus\\ {W\otimes \O (-H+E)} \end{array} \stackrel{A}{\longrightarrow} V'\otimes \mathcal{O}\xrightarrow{A^{*}\beta ^{*}} \begin{array}{c} {W\otimes \O (H)}\\ \oplus\\ {W^{*}\otimes \O (H-E)} \end{array} \end{equation} where \begin{eqnarray*} A&=&\left(\begin{array}{cc} \alpha _1x_3 &-y_2\\ x_1-\xi \alpha _1x_3 & 0 \\ \alpha _2x_3 & y_1 \\ x_2-\xi \alpha _2x_3 & 0 \\ \gamma x_3 & 0 \end{array}\right),\\ \beta &=&\left(\begin{array}{ccccc} 0& 0& \xi & 1& 0\\ 0& 0& 1& 0& 0\\ -\xi & -1& 0& 0& 0\\ -1& 0& 0& 0& 0\\ 0& 0& 0& 0& J \end{array}\right). \end{eqnarray*} and $V'=W\oplus W^{*}\oplus W\oplus W^{*}\oplus \cnums ^{n}$. For $(\alpha _{1},\alpha _{2},\xi ,\gamma )\in {A}^{SO(n)}_{k}(\cpbar )$ we define a self-dual monad by \begin{equation}\label{eqn:cpbar SO monad} \begin{array}{c} {W^{*}\otimes \O (-H)}\\ \oplus\\ {W\otimes \O (-H+E)} \end{array} \stackrel{A}{\longrightarrow} V'\otimes \mathcal{O}\xrightarrow{A^{*}\beta ^{*}} \begin{array}{c} {W\otimes \O (H)}\\ \oplus\\ {W^{*}\otimes \O (H-E)} \end{array} \end{equation} where \begin{eqnarray*} A&=&\left(\begin{array}{cc} \alpha _1x_3 &-y_2\\ x_1-\xi \alpha _1x_3 & 0 \\ \alpha _2x_3 & y_1 \\ x_2-\xi \alpha _2x_3 & 0 \\ \gamma x_3 & 0 \end{array}\right),\\ \beta &=&\left(\begin{array}{ccccc} 0& 0& \xi & 1& 0\\ 0& 0& -1& 0& 0\\ -\xi & -1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1 \end{array}\right). \end{eqnarray*} and $V'=W\oplus W^{*}\oplus W\oplus W^{*}\oplus \cnums ^{n}$. By computing the Chern characters and restricting the monads to $H$ one can see that the cohomology bundle of Monads \ref{eqn:S4 Sp-monad sequence} and \ref{eqn:S4 SO-monad sequence} lie in $\M{k}{SU(n)}(S^{4})$ and the cohomology bundle of Monads \ref{eqn:cpbar Sp monad} and \ref{eqn:cpbar SO monad} lie in $\M{k}{SU(n)}(\cpbar )$. Furthermore, since the Monads \ref{eqn:S4 Sp-monad sequence} and \ref{eqn:cpbar Sp monad} are anti-self-dual they induce a symplectic structure $\phi:\mathcal{E} \to \mathcal{E} ^{*} $ on the corresponding cohomology bundle which restricts to $J$ on $H$. The Monads \ref{eqn:S4 Sp-monad sequence} and \ref{eqn:cpbar Sp monad} thus define elements of $\M{k}{Sp(n/2)}(S^{4})$ and $\M{k}{Sp(n/2)}(\cpbar ) $ respectively. Similarly, the Monads \ref{eqn:S4 SO-monad sequence} and \ref{eqn:cpbar SO monad} define elements of $\M{k}{SO(n)}(S^{4})$ and $\M{k}{SO(n)}(\cpbar )$. Finally, the group of monad automorphisms that preserve the given form of the above monads is induced by the natural action of the configuration automorphism groups listed in Table 1. Now suppose that $(\mathcal{E} ,\tau ,\phi )$ is an element of $\M{k}{Sp(n/2)}(X)$ or $\M {k}{SO(n)}(X)$. We wish to produce an equivalence class of the corresponding configurations. Let $(a_{1},a_{2},b,c)$ or $(a_{1},a_{2},x,b,c)$ be a representative configuration for $(\mathcal{E} ,\tau )\in \M{k}{SU(n)}(X)$. We begin by defining the map $\Phi $ by the following composition of isomorphisms: \begin{equation}\label{eqn:defn of Phi} \Phi =H^{1}(\phi )^{*}\comp SD_{\mathcal{E} (-2H)}\comp H^{1}(x_{3})^{-1}. \end{equation} For $X=\cpbar $, $\Phi$ is a map from $W$ to $U^{*}$ and for $X=S^{4}$, $\Phi$ is a map from $W$ to $ W^{*}$. \begin{prop}\label{prop: commuting relations for Phi} When $X=S^{4}$ the map $\Phi $ satisfies the following relations: \begin{eqnarray*} \Phi ^{*}&=&\begin{cases} \Phi & \text{when $G_{n}=Sp(n/2)$,}\\ -\Phi &\text{when $G_{n}=SO(n)$,} \end{cases}\\ \Phi a_{i}&=&a_{i}^{*}\Phi,\\ c^{*}&=&\begin{cases} \Phi bJ& \text{if $G_{n}=Sp(n/2)$},\\ \Phi b&\text{if $G_{n}=SO(n)$ }. \end{cases} \end{eqnarray*} When $X=\cpbar $ the map $\Phi $ satisfies the following relations: \begin{eqnarray*} \Phi a_{i}&=&\begin{cases} -a_{i}^{*}\Phi ^{*}& \text{if $G_{n}=SO(n)$,}\\ a_{i}^{*}\Phi ^{*}& \text{if $G_{n}=Sp(n/2)$,} \end{cases}\\ x^{*}\Phi &=&\begin{cases} -\Phi ^{*}x& \text{if $G_{n}=SO(n)$,}\\ \Phi ^{*}x& \text{if $G_{n}=Sp(n/2)$,} \end{cases}\\ c^{*}&=&\begin{cases} \Phi bJ& \text{if $G_{n}=Sp(n/2)$},\\ \Phi b&\text{if $G_{n}=SO(n)$ }. \end{cases} \end{eqnarray*} \end{prop} \proof The proof is a straight forward application of the properties of Serre duality to the cohomological interpretation of the configuration maps and the definition of $\Phi $. For example, if $X=S^{4}$ we have a commutative diagram: $$ \begin{CD} H^{0}(\mathcal{E} ^{*}|_{H})@>{\delta _{x_{3}}}>>H^{1}(\mathcal{E} ^{*}(-H))@<{H^{1}(x_{3})}<<H^{1}(\mathcal{E} ^{*}(-2H))@>{SD}>>H^{1}(\mathcal{E} (-H))^{*}\\ @AA{H^{0}(\phi |_{H})}A@AA{H^{1}(\phi )}A@AA{H^{1}(\phi )}A@AA{H^{1}(\phi^{*} )^{*}}A\\ H^{0}(\mathcal{E}|_{H})@>{\delta _{x_{3}}}>>H^{1}(\mathcal{E} (-H))@<{H^{1}(x_{3})}<<H^{1}(\mathcal{E} (-2H))@>{SD}>>H^{1}(\mathcal{E}^{*} (-H))^{*} \end{CD} $$ Now follow the diagram from the lower left corner to the upper right using both directions along the perimeter. Since $\phi |_{H}$ is ${{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l $ in the $SO$ case, $J$ in the $Sp$ case, and $J^{-1}=-J$, we see that $c^{*}\comp \phi |_{H}=\pm \Phi \comp b$ where $\phi ^{*}=\pm \phi $. The result for $c^{*}$ follows and a similar diagram shows the $\cpbar $ case. If $X=S^{4}$ we wish to show that $\Phi ^{*}=\mp \Phi $ when $\phi ^{*}=\pm \phi $. Noting that $H^{1}(x_{3})^{*}=H^{1}(x_{3})$ we have the following commutative diagram: $$ \begin{CD} H^{1}(\mathcal{E} (-2H))@>{H^{1}(x_{3})}>>{H^{1}(\mathcal{E} (-H))}@>{H^{1}(\phi )}>> H^{1}(\mathcal{E} ^{*}(-H))\\ @VV{SD}V && @VV{SD}V\\ H^{1}(\mathcal{E} ^{*}(-H))^{*}@>{H^{1}(\phi ^{*})^{*}}>>H^{1}(\mathcal{E} (-H))^{*}@>{H^{1}(x_{3})}>>H^{1}(\mathcal{E} (-2H))^{*} \end{CD} $$ Following the diagram clockwise from the upper middle spot to the lower middle spot gives the map $-\Phi ^{*}$ since $SD=-SD^{*}$ in this case. Following the diagram counterclockwise yields $\pm \Phi $ when $\phi ^{*}=\pm \phi $ and so we have that $\Phi^{*} =\mp\Phi $. We can prove the relation $\Phi a_{i}=\mp a_{i}^{*}\Phi^{*}$ in a similar fashion. We write the relations algebraically and suppress the diagram: \begin{eqnarray*} \Phi a_{i}&=&H^{1}(\phi )^{*}\comp SD\comp H^{1}(x_{3})^{-1}\comp H^{1}(z_{i})\comp H^{1}(x_{3})^{-1}\\ &=&\pm H^{1}(\phi ^{*})^{*}\comp SD \comp H^{1}(x_{3})^{-1}\comp H^{1}(z_{i})\comp H^{1}(x_{3})^{-1}\\ &=&\pm SD\comp H^{1}(\phi )\comp H^{1}(x_{3})^{-1}\comp H^{1}(z_{i})\comp H^{1}(x_{3})^{-1}\\ &=&\mp H^{1}(x_{3})^{-1}\comp H^{1}(z_{i})\comp H^{1}(x_{3})^{-1}\comp SD^{*}\comp H^{1}(\phi )\\ &=&\mp a_{i}^{*} \Phi ^{*} \end{eqnarray*} where $z_{i}=x_{i}$ in the $S^{4}$ case and $(z_{1},z_{2})=(-y_{2},y_{1})$ in the $\cpbar $ case. Finally, we also have \begin{eqnarray*} x^{*}\Phi &=&H^{1}(s)^{*}\comp H^{1}(\phi )^{*}\comp SD\comp H^{1}(x_{3})^{-1} \\ &=&\pm (H^{1}(\phi ^{*})\comp H^{1}(s))^{*}\comp SD\comp H^{1}(x_{3})^{-1}\\ &=&\pm H^{1}(x_{3})^{-1}\comp SD \comp H^{1}(\phi )\comp H^{1}(s)\\ &=&\mp H^{1}(x_{3})^{-1}\comp SD^{*} \comp H^{1}(\phi )\comp H^{1}(s)\\ &=&\mp \Phi ^{*}x. \end{eqnarray*} \qed We are now in a position to define the inverse construction producing configurations from $(\mathcal{E} ,\tau ,\phi )$. Define a $Sp$ or $SO$ configuration $(\alpha _{1},\alpha _{2},\gamma )$ on $S^{4}$ by $\alpha _{i}=a_{i}$ and $\gamma =c$. Define a $Sp$ or $SO$ configuration $(\alpha _{1},\alpha _{2},\xi ,\gamma )$ on $\cpbar $ by $\alpha _{i}=a_{i}(\Phi ^{-1})^{*}$, $\xi =\Phi ^{*}x $ and $\gamma =c(\Phi ^{-1})^{*}$. The proposition then implies that these are integrable configurations. This correspondence intertwines the action of the automorphism group and is well defined on the quotient. It is also the inverse to the monad construction and so completes the proof of Theorem \ref{thm:moduli construction}. \section{Lifting of maps to configurations}\label{sec:lifts of maps} In this section we define the maps $i$ and $j$ and prove Theorem \ref{thm:lifts of maps to configurations}. They will be maps on configurations that descend to the maps on the moduli spaces. The map $i$ will descend to the map induced by the inclusion $G_{n}\hookrightarrow G_{n'}$ for $n'>n$ and $j$ will descend to the map induced by pulling back connections via the map $\cpbar \to S^{4}$. The maps will intertwine the action of the automorphism groups, {\em i.e. } $i$ will be equivariant (the automorphism groups are independent of the rank), and $j$ will intertwine the action with natural inclusions of the appropriate automorphism groups. \subsection{The rank inclusion map $i$.} We define $i$ on the various kinds of configurations by: \begin{eqnarray*} i&:&(a_{1},a_{2},b,c)\mapsto (a_{1},a_{2},b',c')\\ i&:&(a_{1},a_{2},x,b,c)\mapsto (a_{1},a_{2},x,b',c')\\ i&:&(\alpha _{1},\alpha _{2},\gamma )\mapsto (\alpha _{1},\alpha _{2},\gamma ')\\ i&:&(\alpha _{1},\alpha _{2},\xi ,\gamma )\mapsto (\alpha _{1},\alpha _{2},\xi ,\gamma ') \end{eqnarray*} where $c'=(\begin{array}{c}0\\c\end{array})$, $b'=\left(\begin{array}{cc}0& b\end{array} \right)$, $\gamma '=(\begin{array}{c}0\\ \gamma \end{array})$ and $0$ is the appropriate zero map to (or from) $\cnums ^{(n'-n)}$. The map is obviously equivariant with respect to the automorphism groups and from the monad constructions it is easy to see that $i$ descends to the map $\mathcal{E} \mapsto \mathcal{E} \oplus \O ^{(n'-n)}$. In terms of connections, this is the map $A\mapsto A\oplus \Theta $ where $\Theta $ is the trivial connection on the rank $n'-n$ bundle and this map is the natural one induced by the inclusion $G_{n}\hookrightarrow G_{n'}$. \subsection{The pullback map $j$.} We define the map $j$ as follows. For $G_{n}=Sp(n/2)$ or $SO(n)$ let $$ j:(\alpha _{1},\alpha _{2},\gamma )\mapsto ( \alpha _{1}(\Phi ^{-1})^{*},\alpha _{2}(\Phi ^{-1})^{*}, \Phi ^{*},\gamma (\Phi ^{-1})^{*}). $$ This map intertwines the actions of the automorphism groups and the natural inclusions $SO(W)\hookrightarrow Gl(W)$ or $Sp(W)\hookrightarrow Gl(W)$ so it descends to a map on the moduli spaces. For $G_{n}=SU(n)$ we have automorphism groups $Gl(W)$ and $Gl(W)\times Gl(U)$. Choose an isomorphism $\chi :W\to U$ so that we can define an inclusion $Gl(W)\hookrightarrow Gl(W)\times Gl(U)$ by $g\mapsto (g,\chi g\chi ^{-1})$. Define $j$ to be $$ j:(a_{1},a_{2},b,c)\mapsto (a_{1}\chi^{-1} ,a_{2}\chi^{-1} ,\chi ,b,c\chi ^{-1}). $$ We see that $j$ then intertwines the action of the automorphism groups with the inclusion $Gl(W)\hookrightarrow Gl(W)\times Gl(U)$ induced by $\chi $. It is clear that $j$ commutes with $i$. \begin{lem} The map $j$ induces the pull-back map on bundles. \end{lem} \proof We will proceed by (1) pulling back the $\cnums \P ^{2}$ monad defined by an $S^{4}$-configuration to a $\til{\cnums \P }^{2}$ monad via the blow-down map $\til{\cnums \P }^{2}\to \cnums \P ^{2}$ ({\it c.f.} subsection \ref{subsec: alg-geo and U's}); (2) we use $\chi $ and a direct sum of the monad with an exact sequence to get an equivalent monad of the form of sequence \ref{eqn:cpbar monad sequence}; then (3) we will use a monad automorphism to arrive at the monad defined by the $\cpbar $-configuration $(a_{1}\chi ^{-1},a_{2}\chi ^{-1},\chi ,b,c\chi ^{-1})$. The $Sp$ and $SO$ cases are similar and we leave them to the reader. (1) Since in our notation $\langle x_{1},x_{2},x_{3} \rangle$ and $\O (\pm H)$ on $\cnums \P ^{2}$ pull back to $\langle x_{1},x_{2},x_{3} \rangle$ and $\O (\pm H)$ on $\til{\cnums \P }^{2}$, the pull-back of sequence \ref{eqn:S4 monad sequence} does not change notationally. We apply the monad isomorphism $(\chi ,\chi \oplus \chi \oplus {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l ,{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l )$ to it to get \begin{equation} \O (-H)\xrightarrow{A_{1}}(W\oplus W\oplus \cnums ^{n})\otimes \O \xrightarrow{B_{1}}W\otimes \O (H) \end{equation} where \begin{eqnarray*} A_{1} &=&\left(\begin{array}{c} x_{1}-\chi a_{1}\chi ^{-1}x_{3}\\ x_{2}-\chi a_{2}\chi ^{-1}x_{3}\\ c\chi^{-1} x_{3} \end{array} \right),\\ B_{1}&=&\left(\begin{array}{ccc} -\chi ^{-1}x_{2}+a_{2}\chi ^{-1}x_{3}&\chi ^{-1}x_{1}-a_{1}\chi ^{-1}x_{3}&bx_{3} \end{array} \right). \end{eqnarray*} (2) Since $y_{1}$ and $y_{2}$ do not vanish simultaneously on $\til{\cnums \P }^{2}$, the sequence $$ W\otimes \O (-H+E)\xrightarrow{\left(\begin{array}{c}-y_{2}\\y_{1}\end{array} \right)} (W\oplus W)\otimes \O \xrightarrow{(\begin{array}{cc}\chi y_{1}&\chi y_{2}\end{array})} U\otimes \O (H-E) $$ is exact. We can thus direct sum this sequence to the previous monad to obtain a monad with the same cohomology bundle. We get \begin{equation} \begin{array}{c} {U\otimes \O (-H)}\\ \oplus\\ {W\otimes \O (-H+E)} \end{array} \stackrel{A_{2}}{\longrightarrow} V\otimes \mathcal{O}\stackrel{B_{2}}{\longrightarrow} \begin{array}{c} {W\otimes \O (H)}\\ \oplus\\ {U\otimes \O (H-E)} \end{array} \end{equation} where $V=W\oplus U\oplus W\oplus U\oplus \cnums ^{n}$ and \begin{eqnarray*} A_{2}&=&\left(\begin{array}{cc} 0 &-y_2\\ x_1-\chi a_1\chi ^{-1}x_3 & 0 \\ 0 & y_1 \\ x_2-\chi a_2\chi ^{-1}x_3 & 0 \\ c\chi ^{-1}x_3 & 0 \end{array}\right),\\ B_{2}&=&\left(\begin{array}{ccccc} 0 & -\chi ^{-1}x_{2}+a_2\chi ^{-1}x_3 &0&\chi ^{-1}x_{1} -a_1\chi ^{-1} x_3 & bx_3 \\ \chi y_1 &0 & \chi y_2 &0 & 0 \end{array}\right). \end{eqnarray*} (3) Finally we use an automorphism to put the monad into the form of the sequence \ref{eqn:cpbar monad sequence} . Recall that $s=-x_{1}/y_{2}=x_{2}/y_{1}$ is a well defined section in $H^{0}(\O (E))$. The automorphism we use is $(\eta _{1},\eta _{2},\eta _{3})$ where $$ \eta_{1}=\left(\begin{array}{cc}1&0\\-\chi ^{-1}s&1\end{array} \right) \text{, } \eta_{3}=\left(\begin{array}{cc}1&\chi ^{-1}s\\0&1\end{array} \right) , $$ $$ \eta _{2}=\left(\begin{array}{ccccc} 1& -\chi ^{-1}& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& -\chi ^{-1}& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1 \end{array} \right) $$ and matrix multiplication shows that $A=\eta _{2}A_{2}\eta _{1}^{-1}$ and $B=\eta _{3}B_{2}\eta _{2}^{-1}$ are exactly the monad maps defined by the $\cpbar $-configuration $(a_{1}\chi ^{-1},a_{2}\chi ^{-1},\chi ,b,c\chi ^{-1})$. \qed \section{Proof of the Stabilization theorem}\label{sec:pf of stable thm} In section we prove Theorem \ref{thm:rank stabilization}. We need to show that $\lim_{n\to \infty }A^{G_{n}}_{k}(X)$ is contractible. Since the $A^{G_{n}}_{k}$'s are all algebraic spaces and the inclusion maps are algebraic, they admit triangulations compatible with the maps. Thus $A^{G_{\infty }}_{k}$ inherits the structure of a CW-complex and so it suffices to show that the its homotopy groups are all zero. To this end we show that the inclusion $$ i:A^{G_{n}}_{k}(X)\to A^{G_{2k+n}}_{k}(X) $$ is null homotopic. The basic point is that in $A^{G_{2k}}_{k} (X)$ there are configurations whose only non-zero monad data consists of the maps to or from $\cnums ^{2k}$, in other words the data $a_{i},\alpha _{i},x,$ or $\xi $ are all zero. We will fix such a configuration in each case and show that the image of $A^{G_{n}}_{k} (X)$ in $A^{G_{n+2k}}_{k} (X)$ homotopes to the image of the fixed configuration. \begin{lem}\label{lem:existance of S1 invariant instantons} There are configurations of the form $(0,\dots ,0,b_{0},c_{0})\in A^{SU (2k)}_{k} (X)$ and $(0,\dots ,0,\gamma _{0})\in A^{Sp (2k)}_{k} (X)$ or $A^{SO (2k)}_{k} (X)$. \end{lem} \proof The integrability and non-degeneracy conditions for $SU $ configurations reduce to $b_{0}c_{0}=0$ with $c_{0}$ injective and $b_{0}$ surjective. This can be easily accomplished by having $c_{0}$ map isomorphically onto the first $k$ factors of $\cnums ^{2k} $ and $b_{0}$ an isomorphism on the remaining $k$ factors. For the $Sp$ and $SO$ cases we need a map $\gamma _{0}$ such that $\gamma_{0} ^{*}J\gamma_{0} =0$ or $\gamma_{0} ^{*}\gamma_{0} =0$ respectively, and so that $\gamma_{0} $ is injective. This is also easily done; for example, in the $SO$ case choose an isomorphism $Q:W\to \cnums ^{k}$ and let $\gamma _{0}= (Q,\sqrt{-1}Q)$ . We remark that configurations of this form correspond exactly to instantons on $S^{4}$ or $\cpbar $ that are invariant under the natural $S^{1}$ action. \qed Fix configurations as in the above lemma and define a homotopy $H_{t}:A^{G_{n}}_{k}(X)\to A^{G_{2k+n}}_{k}(X)$ by the following: For $X=S^{4}$ and $G_{n}=SU(n)$ $$ H_{t}(a_{1},a_{2},b,c)=((1-t)a_{1},(1-t)a_{2},(\begin{array}{cc}tb_{0}&(1-t)b \end{array}),\left(\begin{array}{c}tc_{0}\\(1-t)c \end{array} \right)) . $$ For $X=\cpbar $ and $G_{n}=SU(n)$ $$ H_{t}(a_{1},a_{2},x,b,c)=((1-t)^{2/3}a_{1},(1-t)^{2/3}a_{2},(1-t)^{2/3}x, (\begin{array}{cc}tb_{0}&(1-t)b \end{array}),\left(\begin{array}{c}tc_{0}\\(1-t)c \end{array} \right)) . $$ For $X=S^{4} $ and $G_{n}=Sp(n/2)$ or $SO(n)$ $$ H_{t}(\alpha _{1},\alpha _{2},\gamma )=((1-t)\alpha _{1},(1-t)\alpha _{2},\left(\begin{array}{c}t\gamma _{0}\\ (1-t)\gamma \end{array} \right) ) . $$ For $X=\cpbar $ and $G_{n}=Sp(n/2)$ or $SO(n)$ $$ H_{t}(\alpha _{1},\alpha _{2},\xi ,\gamma )=((1-t)^{2/3}\alpha _{1},(1-t)^{2/3}\alpha _{2},(1-t)^{2/3}\xi ,\left(\begin{array}{c}t\gamma _{0}\\ (1-t)\gamma \end{array} \right)) $$ Configurations in the image of $H_{t}$ are integrable and non-degenerate so $H_{t} $ is a well defined homotopy from the inclusion $i$ to a constant map. We can thus conclude that in the $n\mapsto \infty $ limit $A^{G_{n}}_{k} (X)$ is contractible and Theorem \ref{thm:rank stabilization} follows. \qed

9612

alg-geom/9612012

en

https://arxiv.org/abs/alg-geom/9612012

alg-geom/9612012

Rolf Schimmrigk

Rolf Schimmrigk

Scaling Behavior on the Space of Calabi-Yau Manifolds

11 pages, 4 eps figs Latex

BONN-TH-96-13

Recent work is reviewed which suggests that certain universal quantities, defined for all Calabi-Yau manifolds, exhibit a specific behavior which is not present for general K\"ahler manifolds. The variables in question, natural from a mathematical perspective, are of physical importance because they determine aspects of the low-energy string physics in four dimensions, such as Yukawa couplings and threshold corrections. It is shown that these quantities, evaluated on the complete class of Calabi-Yau hypersurfaces in weighted projective 4-space, exhibit scaling behavior with respect to a new scaling variable. (To appear in Mirror Symmetry II.)

alg-geom

\section*{Acknowledgment} It is a pleasure to thank Philip Candelas, Dimitrios Dais, Xenia de la Ossa, Ed Derrick, Michael Flohr, Ariane Frey, Jerry Hinnefeld, Vadim Kaplunovsky, Jack Morse, Werner Nahm, Steve Shore, and especially Andreas Honecker, Monika Lynker and Katrin Wendland for discussions. I'm grateful to the Theory Group at the University of Texas at Austin, the Department of Physics at Indiana University at South Bend, and Simulated Realities Inc., Austin, TX for hospitality. \vskip .1truein

9612

alg-geom/9612016

en

https://arxiv.org/abs/alg-geom/9612016

alg-geom/9612016

Dmitry Kaledin

D. Kaledin

Integrability of the twistor space for a hypercomplex manifold

9 pages, Latex2e

A hypercomplex manifold is by definition a smooth manifold equipped with two anticommuting integrable almost complex structures. For example, every hyperkaehler manifold is canonically hypercomplex (the converse is not true). For every hypercomplex manifold M, the two almost complex structures define a smooth action of the algebra of quaternions on the tangent bundle to M. This allows to associate to every hypercomplex manifold M of dimension 4n a certain almost complex manifold X of dimension 4n+2, called the twistor space of M. When M is hyperkaehler, X is well-known to be integrable. We show that for an arbitrary hypercomplex manifold its twistor space is also integrable.

alg-geom

\section*{Introduction} A {\em hyperk\"ahler manifold} is by definition a Riemannian manifold eqipped with a smooth parallel action of the algebra of quaternions on its tangent bundle. Hyperk\"ahler manifolds were introduced by Calabi in \cite{C} and have since been the subject of much research. They have been shown to possess many remarkable properties and a rich inner structure. (See \cite{Bes}, \cite{HKLR} for an overview.) In particular, there is a so-called {\em twistor space} associated to every hyperk\"ahler manifold. The twistor space is a complex manifold equipped with some additional structures, and many of the differential-geometric properties of a hyperk\"ahler manifold can be described in terms of holomorphic properties of its twistor space. Some of the properties of hyperk\"ahler manifolds do not actually depend on the Riemannian metric but only on the quaternion action. In particular, for every manifold equipped with a smooth action of quaternions (or, for brevity, a {\em quaternionic manifold}) one can construct an almost complex manifold which becomes the twistor space in the hyperk\"ahler case. Thus it would be very convenient to have a notion of ``a hyperk\"ahler manifold without a metric'', in the same sense as a complex manifold is a K\"ahler manifold without a metric. The analogy with the K\"ahler case suggests that this would require a certain integrability condition on the quaternionic action, automatic in the Riemannian case. One version of such a condition was suggested in \cite{Bes}, but the resulting notion of an integrable quaternionic manifold is too restrictive and excludes many interesting examples. A more convenient notion is that of a {\em hypercomplex manifold} (see \cite{Bo}). By definition a hypercomplex manifold is a smooth manifold equipped with two integrable anticommuting almost complex structures. (Note that two anticommuting almost complex structures induce an action of the whole quaternion algebra on the tangent bundle, and their integrability is in fact a condition on the resulting quaternionic manifold.) In this paper we show that this condition is in fact equivalent to the integrability of the almost complex twistor space associated to the quaternionic manifold in question. Here is a brief outline of the paper. In Section~\ref{1} we give the necessary definitions and formulate the result (Theorem~\ref{main}). In Section~\ref{2} we describe a linear-algebraic construction somewhat analogous to the Borel-Weyl localization of finite-dimensional representations of a reductive group. In Section~\ref{3} we use this version of localization to prove Theorem~\ref{main}. The paper is essentially self-contained and does not require any prior knowledge of the theory of hyperk\"ahler manifolds. \noindent {\bf Acknowledgements.} It is a pleasure to express my gratitude to Misha Verbitsky and Tony Pantev for many interesting and stimulating discussions and constant encouragement. I am especially grateful to Misha for his valuable suggestions on the present paper. \section{Preliminaries.}\label{1} \subsection{} Let ${\Bbb H}$ be the algebra of quaternions. \begin{defn} A {\em quaternionic manifold} is a smooth manifold $M$ equipped with a smooth action of the algebra ${\Bbb H}$ on the tangent bundle $\Theta(M)$ to $M$. \end{defn} Let $M$ be a quaternionic manifold. Every algebra embedding $I:{\Bbb C} \to {\Bbb H}$ defines by restriction an almost complex structure on $M$. Call it {\em the induced almost complex structure} and denote it by $M_I$. \subsection{}\label{maps} The set $\operatorname{Maps}({\Bbb C},{\Bbb H})$ of all algebra embeddings ${\Bbb C} \to {\Bbb H}$ can be given a natural structure of a complex manifold as follows. The algebra ${\Bbb H} \otimes_{\Bbb R} {\Bbb C}$ is naturally isomorphic to the $2 \times 2$-matrix algebra over ${\Bbb C}$. Every algebra embedding $I:{\Bbb C} \to {\Bbb H}$ defines a structure of a $2$-dimensional vector space ${\Bbb H}_I$ on ${\Bbb H}$ by means of left multiplication by $I({\Bbb C})$. It also defines a $1$-dimensional subspace $I({\Bbb C}) \subset {\Bbb H}_I$. The action of ${\Bbb H}$ on itself by right multiplication preserves the complex structure ${\Bbb H}_I$ and extends therefore to an action of the matrix algebra ${\Bbb H} \otimes_{\Bbb R} {\Bbb C}$. Let $\widehat{I} \subset {\Bbb H} \otimes_{\Bbb R} {\Bbb C}$ be the annihilator of $I({\Bbb C}) \subset {\Bbb H}_I$. The ideal $\widehat{I} \subset {\Bbb H}$ is a maximal right ideal, moreover, we have ${\Bbb H}_I = {\Bbb H} \otimes {\Bbb C} / \widehat{I}$. It is easy to check that every maximal right ideal of ${\Bbb H} \otimes_{\Bbb R} {\Bbb C}$ can be obtained in this way. This extablishes a bijection between $\operatorname{Maps}({\Bbb C},{\Bbb H})$ and the set of maximal right ideals in ${\Bbb H} \otimes_{\Bbb R} {\Bbb C}$. It is well-known that this set coincides with the complex projective line ${{\Bbb C}P^1}$. \subsection{} Let now $M$ be a quaternionic manifold, and let $X = M \times {{\Bbb C}P^1}$ be the product of $M$ with the smooth manifold underlying ${{\Bbb C}P^1}$. For every point $x = m \times I\in M \times {{\Bbb C}P^1}$ the tangent bundle $T_xX$ decomposes canonically as $T_xX = T_mM \oplus T_I{{\Bbb C}P^1}$. Define an endomorphism ${\cal I}:T_xX \to T_xX$ as follows: it acts as the usual complex structure map on $T_I{{\Bbb C}P^1}$, and on $T_mM$ it acts as the induced complex structure map $I:T_mM \to T_mM$ corresponding to $I \in {{\Bbb C}P^1} \cong \operatorname{Maps}({\Bbb C},{\Bbb H})$. The map ${\cal I}$ obviously depends smoothly on the point $x \in X$ and satisfies ${\cal I}^2 = -1$. Therefore, it defines an almost complex structure on the smooth manifold $X$. \begin{defn} The almost complex manifold $X = M \times {{\Bbb C}P^1}$ is called {\em the twistor space} of the quaternionic manifold $M$. \end{defn} \subsection{} The twistor space $X$ has the following obvious properties. \begin{enumerate} \item The canonical projection $\pi:X \to {{\Bbb C}P^1}$ is compatible with the almost complex structures. \item For every point $m \in M$ the embedding $\widetilde{m} = m \times \operatorname{id}: {{\Bbb C}P^1} \to M \times {{\Bbb C}P^1} = X$ is compatible with the almost complex structures. \end{enumerate} The embedding $\widetilde{m}:{{\Bbb C}P^1} \to X$ will be called {\em the twistor line} corresponding to the point $m \in M$. \subsection{} The goal of this paper is to prove the following theorem. \begin{theorem}\label{main} Let $M$ be a quaternionic manifold, and let $X$ be its twistor space. The following conditions are equivalent: \begin{enumerate} \item For two algebra maps $I,J:{\Bbb C} \to {\Bbb H}$ such that $I \neq J$ and $\overline{I} \neq J$ the induced almost complex structures $M_I$, $M_J$ on $M$ are integrable. \item For every algebra map $I:{\Bbb C} \to {\Bbb H}$ the induced almost complex structure $M_I$ on $M$ is integrable. \item The almost complex structure on $X$ is integrable. \end{enumerate} \end{theorem} The quaternionic manifold satisfying \thetag{i} is called {\em hypercomplex}. Note that $\thetag{iii} \Rightarrow \thetag{ii} \Rightarrow \thetag{i}$ is obvious, so it suffices to prove $\thetag{i} \Rightarrow \thetag{iii}$. \section{Localization of $\protect{\Bbb H}$-modules.}\label{2} \subsection{} We begin with some linear algebra. \begin{defn} A {\em quaternionic vector space} $V$ is a left module over the algebra ${\Bbb H}$. \end{defn} Let $V$ be a quaternionic vector space. Every algebra map $I:{\Bbb C} \to {\Bbb H}$ defines by restriction a complex vector space structure on $V$, which we will denote by $V_I$. Note that ${\Bbb H}$ is naturally a left module over itself. The associated $2$-dimensional complex vector space ${\Bbb H}_I$ is the same as in \ref{maps}, and we have $V_I = {\Bbb H}_I \otimes_{\Bbb H} V$. \subsection{}\label{loc} Let ${\operatorname{SB}}$ be the Severi-Brauer variety associated to the algebra ${\Bbb H}$, that is, the variety of maximal right ideals in ${\Bbb H}$. By definition ${\operatorname{SB}}$ is a real algebraic variety, an ${\Bbb R}$-twisted form of the complex projective line ${{\Bbb C}P^1}$. It is also equipped with a canonical maximal right ideal ${\cal I} \subset {\Bbb H} \otimes {\cal O}$ in the flat coherent algebra sheaf ${\Bbb H} \otimes {\cal O}$ on ${\operatorname{SB}}$. Let $V$ be a quaternionic vector space. Consider the flat coherent sheaf $V \otimes {\cal O}$ on ${\operatorname{SB}}$ of right ${\Bbb H} \otimes {\cal O}$-modules, and let $$ V_{loc} = V \otimes {\cal O} / {\cal I} \cdot V \otimes {\cal O} $$ be its quotient by the right ideal ${\cal I}$. Call the sheaf $V_{loc}$ {\em the localization} of the quaternionic vector space $V$. The localization is functorial in $V$ and gives a full embedding of the category of quaternionic vector spaces into the category of flat coherent sheaves on ${\operatorname{SB}}$. Say that a flat coherent sheaf ${\cal E}$ on ${\operatorname{SB}}$ is {\em of weight $k$} if the sheaf ${\cal E} \otimes {\Bbb C}$ on the complex projective line ${{\Bbb C}P^1} \cong {\operatorname{SB}} \otimes {\Bbb C}$ is isomorphic to a sum of several copies of the line bundle ${\cal O}(k)$. The essential image of the localization functor is the full subcategory of flat coherent sheaves of weight $1$. \subsection{} The set ${{\Bbb C}P^1} \cong {\operatorname{SB}}({\Bbb C})$ of ${\Bbb C}$-valued points of the variety ${\operatorname{SB}}$ was canonically identified in \ref{maps} with the set $\operatorname{Maps}({\Bbb C},{\Bbb H})$ of algebra maps ${\Bbb C} \to {\Bbb H}$. Let $I:{\Bbb C} \to {\Bbb H}$ be an algebra map, and let $\widehat{I} \in {{\Bbb C}P^1}$ be the corresponding ${\Bbb C}$-valued point of the variety ${\operatorname{SB}}$ or, equivalently, the maximal right ideal in the algebra ${\Bbb H} \otimes {\Bbb C}$. \begin{lemma}\label{hol} Consider a quaternionic vector space $V$, and let $V_{loc}$ be its localization. The fiber of the sheaf $V_{loc}$ at the point $\widehat{I} \in {{\Bbb C}P^1}$ is canonically isomorphic to the vector space $V_I$. \end{lemma} \noindent{\em Proof. } Indeed, $$ V_I \cong {\Bbb H}_I \otimes_{\Bbb H} V \cong {\Bbb H}_I \otimes_{{\Bbb H} \otimes {\Bbb C}} V \otimes {\Bbb C} \cong V \otimes {\Bbb C} / \widehat{I} \cdot V \otimes {\Bbb C} = V_{loc}|_{\widehat{I}}, $$ and all the isomorphisms are canonical. \ensuremath{\square} \subsection{} Consider now the set ${{\Bbb C}P^1} \cong \operatorname{Maps}({\Bbb C},{\Bbb H})$ as the smooth complex-analytic variety, and let ${\cal V}$ be the trivial bundle on ${{\Bbb C}P^1}$ with the fiber $V \otimes_{\Bbb R} {\Bbb C}$. Since ${\cal V}$ is trivial, we have a canonical holomorphic structure operator $\bar\partial:{\cal V} \to {\cal A}^{0,1}({\cal V})$ from ${\cal V}$ to the bundle ${\cal A}^{0,1}({\cal V})$ of ${\cal V}$-valued $(0,1)$-forms on ${{\Bbb C}P^1}$. The action of ${\Bbb H}$ on $V$ induces an operator ${\cal I}:{\cal V} \to {\cal V}$ which acts as $I(\sqrt{-1})$ on the fiber $V$ of ${\cal V}$ at a point $I \in \operatorname{Maps}({\Bbb C},{\Bbb H})$. The operator ${\cal I}$ obviously depends smoothly on the point $I$. It satisfies $I^2 = -1$ and induces therefore a smooth ``Hodge type'' decomposition ${\cal V} = {\cal V}^{1,0} \oplus {\cal V}^{0,1}$. \begin{lemma}\label{locc} The quotient ${\cal V}^{1,0}$ is compatible with the holomorphic structure $\bar\partial$ on ${\cal V}$. In other words, there exists a unique holomorphic structure operator $\bar\partial:{\cal V}^{1,0} \to {\cal A}^{0,1}({\cal V}^{1,0})$ making the diagram $$ \begin{CD} {\cal V} @>>> {\cal V}^{1,0} \\ @V{\bar\partial}VV @V{\bar\partial}VV \\ {\cal A}^{0,1}({\cal V}) @>>> {\cal A}^{0,1}({\cal V}^{1,0}) \end{CD} $$ commutative. \end{lemma} \noindent{\em Proof. } This follows directly from Lemma~\ref{hol} by the usual correspondence between flat coherent sheaves and holomorphic bundles on the underlying complex-analytic variety. \ensuremath{\square} \section{Proof of the theorem.}\label{3} \subsection{} Let $Z$ be a smooth almost complex manifold. Let ${\cal A}^1(Z,{\Bbb C})$ be the complexified cotangent bundle to $Z$, and let ${\cal A}^\cdot(Z,{\Bbb C})$ be its exterior algebra. The almost complex structure on $Z$ induces the Hodge type decomposition ${\cal A}^i(Z,{\Bbb C}) = \oplus_{p+q=i}{\cal A}^{p,q}(Z)$. Recall that the {\em Nijenhuis tensor} $N$ of the almost complex manifold $Z$ is the composition $$ N = P \circ d_Z \circ i:{\cal A}^{1,0}(Z) \to {\cal A}^1(Z,{\Bbb C}) \to {\cal A}^2(Z,{\Bbb C}) \to {\cal A}^{0,2}(Z), $$ where $d_Z$ is the de Rham differential, $i:{\cal A}^{\cdot,0}(Z) \to {\cal A}^\cdot(Z,{\Bbb C})$ is the canonical embedding, and $P:{\cal A}^\cdot(Z,{\Bbb C}) \to {\cal A}^{0,\cdot}(Z)$ is the canonical projection. Recall also that the almost complex manifold $Z$ is called {\em integrable} if its Nijenhuis tensor $N_Z:{\cal A}^{1,0}(Z) \to {\cal A}^{0,2}(Z)$ vanishes. \subsection{} We can now begin the proof of Theorem~\ref{main}. First we will prove a sequence of preliminary lemmas. Let $M$ be a smooth quaternionic manifold, and let $X$ be its twistor space. Since by definition $X = M \times {{\Bbb C}P^1}$ as a smooth manifold, the cotangent bundle ${\cal A}^1(X)$ decomposes canonically as \begin{equation}\label{eq.1} {\cal A}^1(X) = \sigma^*{\cal A}^1(M) \oplus \pi^*{\cal A}^1({{\Bbb C}P^1}), \end{equation} where $\sigma:X \to M$, $\pi:X \to {{\Bbb C}P^1}$ are the canonical projections. The almost complex structure ${\cal I}$ on $X$ preserves the decomposition \eqref{eq.1}. Therefore \eqref{eq.1} induces decompositions \begin{align*} {\cal A}^{1,0}(X) &= {\cal A}^{1,0}_M(X) \oplus {\cal A}^{1,0}_{{\Bbb C}P^1}(X), \\ {\cal A}^{0,2}(X) &= {\cal A}^{0,1}_M(X) \oplus {\cal A}^{0,1}_{{\Bbb C}P^1}(X), \end{align*} and, consequently, a decompositon \begin{equation}\label{dec} {\cal A}^{0,2}(X) = \left( {\cal A}^{0,1}_M(X) \otimes {\cal A}^{0,1}_{{\Bbb C}P^1}(X) \right) \oplus {\cal A}^{0,2}_M(X). \end{equation} (Note that ${\cal A}^{0,1}_{{\Bbb C}P^1}(X)$ is of rank $1$, therefore ${\cal A}^{0,2}_{{\Bbb C}P^1}(X)$ vanishes). More\-over, since the projection $\pi:X \to {{\Bbb C}P^1}$ is compatible with the almost complex structures, we have canonical isomorphisms $$ {\cal A}^{p,q}_{{\Bbb C}P^1}(X) \cong \pi^*{\cal A}^{p,q}({{\Bbb C}P^1}). $$ \subsection{} Let $N_X:{\cal A}^{1,0} \to {\cal A}^{0,2}(X)$ be the Nijenhuis tensor of the almost complex manifold $X$. We begin with the following. \begin{lemma} The restriction of the Nijenhuis tensor $N_X$ to the subbundle $$ \pi^*{\cal A}^{1,0}({{\Bbb C}P^1}) \cong {\cal A}^{1,0}_{{\Bbb C}P^1}(X) \subset {\cal A}^{1,0}(X) $$ vanishes. \end{lemma} \noindent{\em Proof. } Indeed, since the map $\pi:X \to {{\Bbb C}P^1}$ is compatible with the almost complex structures, the diagram $$ \begin{CD} \pi^*{\cal A}^{1,0}({{\Bbb C}P^1}) @>>> {\cal A}^{1,0}(X)\\ @VVV @VV{N_X}V \\ \pi^*{\cal A}^{0,2}({{\Bbb C}P^1}) @>>> {\cal A}^{0,2}(X) \end{CD} $$ is commutative, and ${\cal A}^{0,2}({{\Bbb C}P^1})$ vanishes. \ensuremath{\square} Therefore the Nijenhuis tensor $N_X$ factors through a map $$ N_X:{\cal A}^{1,0}_M(X) \to {\cal A}^{0,2}(X). $$ \subsection{} Let now $N_X = N_1 + N_2$ be the decomposition of the Nijenhuis tensor with respect to \eqref{dec}, so that $N_1$ is a map $$ N_1:{\cal A}^{1,0}_M(X) \to {\cal A}^{0,1}_M(X) \otimes \pi^*{\cal A}^{0,1}({{\Bbb C}P^1}), $$ and $N_2$ is a map $N_2:{\cal A}^{1,0}_M(X) \to {\cal A}^{0,2}_M(X)$. \begin{lemma} The component $N_1$ of the Nijenhuis tensor $N_X$ vanishes. \end{lemma} \noindent{\em Proof. } It suffices to prove that for every point $m \in M$ the restriction $\widetilde{m}^*N_1$ of $N_1$ onto the corresponding twistor line $\widetilde{m}:{{\Bbb C}P^1} \to X$ vanishes. Consider a point $m \in M$. Let $i:\widetilde{m}^*{\cal A}^{1,0}_M(X) \to \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C})$ be the canonical embedding, and let $$ P:{\cal A}^{0,1}({{\Bbb C}P^1}) \otimes \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \to {\cal A}^{0,1}({{\Bbb C}P^1}) \otimes \widetilde{m}^*{\cal A}_M^{0,1}(X) $$ be the canonical projection. Since the twistor line $\widetilde{m}:{{\Bbb C}P^1} \to X$ is compatible with the almost complex structures, we have $\widetilde{m}^*\pi^*{\cal A}^{0,1}({{\Bbb C}P^1}) \cong {\cal A}^{0,1}({{\Bbb C}P^1})$, and \begin{multline*} \widetilde{m}^*N_1 = P \circ \bar\partial \circ i:\widetilde{m}^*{\cal A}^{0,1}_M(X) \to \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \to \\ \to \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \otimes {\cal A}^{0,1}({{\Bbb C}P^1}) \to \widetilde{m}^*{\cal A}^{0,1}_M(X) \otimes {\cal A}^{0,1}({{\Bbb C}P^1}), \end{multline*} where $\bar\partial:\widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \to \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \otimes {\cal A}^{0,1}({{\Bbb C}P^1})$ is the trivial holomorphic structure operator on the constant bundle $\widetilde{m}^*{\cal A}^1_M(X,{\Bbb C})$. Let $V = T_mM$ be the tangent space to the manifold $M$ at the point $m$. Since $M$ is quaternionic, $V$ is canonically a quaternionic vector space. Let ${\cal V}$ and ${\cal V}^{1,0}$ be as in Lemma~\ref{locc}, and let ${\cal V}^*$ and $({\cal V}^{1,0})^*$ be the dual bundles on ${{\Bbb C}P^1}$. We have canonical bundle isomorphisms $$ {\cal V}^* \cong \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \qquad\qquad ({\cal V}^{1,0})^* \cong \widetilde{m}^*{\cal A}^{1,0}_M(X) $$ compatible with the natural embeddings. By the statement dual to Lemma~\ref{locc}, there exists a holomorphic structure operator $\bar\partial:\widetilde{m}^*{\cal A}^{1,0}_M(X) \to \widetilde{m}^*{\cal A}^{1,0}_M(X) \otimes {\cal A}^{0,1}({{\Bbb C}P^1})$ making the diagram $$ \begin{CD} \widetilde{m}^*{\cal A}^{1,0}_M(X) @>{i}>> \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \\ @V{\bar\partial}VV @V{\bar\partial}VV \\ \widetilde{m}^*{\cal A}^{1,0}_M(X) \otimes {\cal A}^{0,1}({{\Bbb C}P^1}) @>{i \otimes \operatorname{id}}>> \widetilde{m}^*{\cal A}^1_M(X,{\Bbb C}) \otimes {\cal A}^{0,1}({{\Bbb C}P^1}) \end{CD} $$ commutative. Therefore $N_1 = P \circ \bar\partial \circ i = P \circ (i \otimes \operatorname{id}) \circ \bar\partial$. But $P \circ (i \otimes \operatorname{id}) = 0$, hence $N_1$ vanishes. \ensuremath{\square} \subsection{} We can now prove Theorem~\ref{main}. As we have already proved, the Nijenhuis tensor $N_X$ of the twistor space $X$ reduces to a bundle map $$ N_X:{\cal A}^{1,0}_M(X) \to {\cal A}^{0,2}_M(X). $$ This map vanishes identically if and only if for every point $m \in M$ the restriction $\widetilde{m}^N_X$ of $N_X$ to the twistor line $\widetilde{m}:{{\Bbb C}P^1} \to X$ vanishes. Consider a point $m \in M$. By Lemma~\ref{locc} the restriction $\widetilde{m}^*{\cal A}^{1,0}_M(X)$ carries a natural holomorphic structure, and it is a holomorphic bundle of weight $-1$ with respect to this structure (in the sense of \ref{loc}). Consequently, the bundle $\widetilde{m}^*{\cal A}^{0,2}_M(X)$ is a holomorphic bundle of weight $2$. Moreover, the Nijenhuis tensor $$ \widetilde{m}^*N_X = P \circ d \circ i:\widetilde{m}^*{\cal A}^{1,0}_M(X) \to {\cal A}^{0,2}_M(X) $$ is a holomorphic bundle map. For every algebra map $I \in \operatorname{Maps}({\Bbb C},{\Bbb H}) \cong {{\Bbb C}P^1}$, the restriction of the Nijenhuis tensor $N_X$ to a fiber $M \times I \in M \times {{\Bbb C}P^1} = X$ of the projection $\pi:X \to {{\Bbb C}P^1}$ is the Nijenhuis tensor for the induced almost complex structure $M_I$ on the manifold $M$. Assume that Theorem~\ref{main}~\thetag{i} holds. Then at least four distinct induced almost complex structures on $M$ corresponding to $I,J,\overline{I},\overline{J} \in \operatorname{Maps}({\Bbb C},{\Bbb H})$ are integrable. Consequently, the Nijenhuis tensor $N_X$ vanishes identically on fibers of the projection $\pi:X \to {{\Bbb C}P^1}$ over at least four distinct points of ${{\Bbb C}P^1}$. Therefore the restriction $\widetilde{m}^*N_X$ has at least four distinct zeroes. But as we have proved, $\widetilde{m}^*N_X$ is a holomorphic map from a bundle of weight $-1$ to a bundle of weight $2$. Therefore it vanishes identically. Hence the almost complex manifold $X$ is integrable, which finishes the proof of Theorem~\ref{main}.

9612

alg-geom/9612011

en

https://arxiv.org/abs/alg-geom/9612011

alg-geom/9612011

Atsushi Moriwaki

Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves

Version 4.5 (33 pages). This paper will appear in Journal of AMS

Let f : X --> Y be a projective morphism of smooth algebraic varieties over an algebraically closed field of characteristic zero with dim f = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if X_y is smooth and E_y is semistable for some point y of Y, then f_* (2r c_2(E) - (r-1) c_1(E)^2) is weakly positive at y. We apply this result to obtain the following description of the cone of weakly positive $\QQ$-Cartier divisors on the moduli space of stable curves. Let M_g (resp. M_g^0) be the moduli space of stable (resp. smooth) curves of genus g >= 2. Let h be the Hodge class and d_i's (i = 0,...,[g/2]) the boundary classes. A Q-Cartier divisor x h + y_0 d_0 + ... + y_[g/2] d_[g/2] is weakly positive over M_g^0 if and only if x >= 0, g x + (8g + 4) y_0>= 0, and i(g-i) x + (2g+1) y_i>= 0 for all 1 <= i <= [g/2].

alg-geom

\section*{Introduction} \renewcommand{\theTheorem}{\Alph{Theorem}} Throughout this paper, we fix an algebraically closed field $k$. Let $f : X \to Y$ be a surjective and projective morphism of quasi-projective varieties over $k$ with $\dim f = 1$. Let $E$ be a vector bundle of rank $r$ on $X$. Then, we define the {\em discriminant divisor} of $E$ with respect to $f : X \to Y$ to be \[ \operatorname{dis}_{X/Y}(E) = f_*\left( \left(2rc_2(E) - (r-1)c_1(E)^2 \right) \cap [X] \right). \] Here $f_*$ is the push-forward of cycles, so that $\operatorname{dis}_{X/Y}(E)$ is a divisor modulo linear equivalence on $Y$. In this paper, we would like to show the following theorem (cf. Corollary~\ref{cor:nef:psudo:dis}) and give its applications. \begin{Theorem}[$\operatorname{char}(k) = 0$] \label{thm:intro:A} We assume that $Y$ is smooth over $k$. Let $y$ be a point of $Y$ and $\overline{\kappa(y)}$ the algebraic closure of the residue field $\kappa(y)$ at $y$. If $f$ is flat over $y$, the geometric fiber $X_{\bar{y}} = X \times_Y \operatorname{Spec}\left(\overline{\kappa(y)}\right)$ over $y$ is reduced and Gorenstein, and $E$ is semistable on each connected component of the normalization of $X_{\bar{y}}$, then $\operatorname{dis}_{X/Y}(E)$ is weakly positive at $y$, namely, for any ample divisors $A$ on $Y$ and any positive integers $n$, there is a positive integer $m$ such that \[ H^0(Y, {\mathcal{O}}_Y(m(n \operatorname{dis}_{X/Y}(E) + A))) \otimes {\mathcal{O}}_Y \to {\mathcal{O}}_Y(m(n \operatorname{dis}_{X/Y}(E) + A)) \] is surjective at $y$. Note that this theorem still holds in positive characteristic under the strong semistability of $E_{\bar{y}}$ \textup{(}cf. Corollary~\textup{\ref{cor:nef:psudo:dis:in:p}}\textup{)}. \end{Theorem} An interesting point of the above theorem is that even if the weak positivity of $\operatorname{dis}_{X/Y}(E)$ at $y$ is a global property on $Y$, it can be derived from the local assumption ``the goodness of $X_{\bar{y}}$ and the semistability of $E_{\bar{y}}$''. This gives a great advantage to our applications. In order to understand the intuition underlying the theorem, let us consider a toy case. Namely, we suppose that $f : X \to Y$ is a smooth surface fibred over a curve and the fiber is general. Bogomolov's instability theorem \cite{Bogo} says that if $E_{\bar{y}}$ is semistable, then the codimension two cycle $2r c_2(E) - (r-1)c_1(E)^2$ has non-negative degree. So if we push it down to a codimension one cycle on $Y$, then one can rephrase Bogomolov's theorem as saying that the semistability of $E_y$ implies the non-negativity of $\operatorname{dis}_{X/Y}(E)$. \medskip An immediate application of our inequality is a solution concerning the positivity of divisors on the moduli space of stable curves. Let $g \geq 2$ be an integer, and $\overline{\mathcal{M}}_g$ (resp. $\mathcal{M}_g$) the moduli space of stable (resp. smooth) curves of genus $g$ over $k$. The boundary $\overline{\mathcal{M}}_g \setminus \mathcal{M}_g$ is of codimension one and has $[g/2]+1$ irreducible components, say, $\Delta_0, \Delta_1, \ldots, \Delta_{[g/2]}$. The geometrical meaning of indexes is as follows. A general point of $\Delta_0$ represents an irreducible stable curve with one node, and a general point of $\Delta_i$ ($i > 0$) represents a stable curve consisting of a curve of genus $i$ and a curve of genus $g-i$ joined at one point. Let $\delta_i$ be the class of $\Delta_i$ in $\operatorname{Pic}(\overline{\mathcal{M}}_g) \otimes {\mathbb{Q}}$ (strictly speaking, $\delta_i = c_1({\mathcal{O}}(\Delta_i))$ for $i \not= 1$, and $\delta_1 = \frac{1}{2} c_1({\mathcal{O}}(\Delta_1))$), and $\lambda$ the Hodge class on $\overline{\mathcal{M}}_g$. A fundamental problem due to Mumford \cite{Mum} is to decide which ${\mathbb{Q}}$-divisor \[ a \lambda - b_0 \delta_0 - b_1 \delta_1 - \cdots - b_{[g/2]}\delta_{[g/2]} \] is positive, where $a, b_0, \ldots, b_{[g/2]}$ are rational numbers. Here, we can use a lot of types of positivity, namely, ampleness, numerical effectivity, effectivity, pseudo-effectivity, and so on. Besides them, we would like to introduce a new sort of positivity for our purposes. Let $V$ be a projective variety over $k$ and $U$ a non-empty Zariski open set of $V$. A ${\mathbb{Q}}$-Cartier divisor $D$ on $V$ is said to be {\em numerically effective over $U$} if $(D \cdot C) \geq 0$ for all irreducible curves $C$ on $V$ with $C \cap U \not= \emptyset$. A first general result in this direction was found by Cornalba-Harris \cite{CH}, Xiao \cite{Xi} and Bost \cite{Bo}. They proved that the ${\mathbb{Q}}$-divisor \[ (8g+4) \lambda - g (\delta_0 + \delta_1 + \cdots + \delta_{[g/2]}) \] is numerically effective over $\mathcal{M}_g$. As we observed in \cite{Mo4} and \cite{Mo5}, it is not sharp in coefficients of $\delta_i$ ($i > 0$). Actually, the existence of a certain refinement of the above result was predicted at the end of the paper \cite{CH}. Our solution for this problem is the following (cf. Theorem~\ref{thm:wpos:at:M:g} and Proposition~\ref{prop:samp:wp:U}). \begin{Theorem}[$\operatorname{char}(k) = 0$] \label{thm:intro:wp} The divisor \[ (8g+4) \lambda - g \delta_0 - \sum_{i=1}^{[g/2]} 4 i (g-i) \delta_i \] is weakly positive over $\mathcal{M}_g$, i.e., if we denote the above divisor by $D$, then for any ample ${\mathbb{Q}}$-Cartier divisors $A$ on $\overline{\mathcal{M}}_g$, there is a positive integer $n$ such that $n(D + A)$ is a Cartier divisor and \[ H^0(\overline{\mathcal{M}}_g, {\mathcal{O}}_{\overline{\mathcal{M}}_g}(n(D+A))) \otimes {\mathcal{O}}_{\overline{\mathcal{M}}_g} \to {\mathcal{O}}_{\overline{\mathcal{M}}_g}(n(D+A)) \] is surjective on $\mathcal{M}_g$. In particular, it is pseudo-effective, and numerically effective over $\mathcal{M}_g$. \end{Theorem} As an application of this theorem, we can decide the cone of weakly positive divisors over $\mathcal{M}_g$ (cf. Corollary~\ref{cor:wp:cone:equal}). \begin{Theorem}[$\operatorname{char}(k) = 0$] If we denote by $\operatorname{WP}(\overline{\mathcal{M}}_g; \mathcal{M}_g)$ the cone in $\operatorname{Pic}(\overline{\mathcal{M}}_g) \otimes {\mathbb{Q}}$ consisting of weakly positive ${\mathbb{Q}}$-Cartier divisors over $\mathcal{M}_g$, then \[ \operatorname{WP}(\overline{\mathcal{M}}_g; \mathcal{M}_g) = \left\{ x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i \ \left| \ \begin{array}{l} x \geq 0, \\ g x + (8g + 4) y_0 \geq 0, \\ i(g-i) x + (2g+1) y_i \geq 0 \quad (1 \leq i \leq [g/2]). \end{array} \right. \right\}. \] \end{Theorem} Moreover, using Theorem~\ref{thm:intro:wp}, we can deduce a certain kind of inequality on an algebraic surface. In order to give an exact statement, we will introduce types of nodes of semistable curves. Let $Z$ be a semistable curve over $k$, and $P$ a node of $Z$. We can assign a number $i$ to the node $P$ in the following way. Let $\iota_P : Z_P \to Z$ be the partial normalization of $Z$ at $P$. If $Z_P$ is connected, then $i=0$. Otherwise, $i$ is the minimum of arithmetic genera of two connected components of $Z_P$. We say the node $P$ of $Z$ is {\em of type $i$}. Let $X$ be a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a semistable curve of genus $g \geq 2$ over $Y$. By abuse of notation, we denote by $\delta_i(X/Y)$ the number of nodes of type $i$ in all singular fibers of $f$. Actually, $\delta_i(X/Y) = \deg(\pi^*(\delta_i))$, where $\pi : Y \to \overline{\mathcal{M}}_g$ is the morphism induced by $f : X \to Y$. Then, we have the following (cf. Corollary~\ref{cor:sharp:slope:inq}). \begin{Theorem}[$\operatorname{char}(k) = 0$] \label{thm:sharp:slope:inq:in:intro} With notation being as above, we have the inequality \[ (8g+4) \deg(f_*(\omega_{X/Y})) \geq g \delta_0(X/Y) + \sum_{i=1}^{[g/2]} 4 i (g-i) \delta_i(X/Y). \] \end{Theorem} As an arithmetic application of Theorem~\ref{thm:sharp:slope:inq:in:intro}, we can show the following answer for effective Bogomolov's conjecture over function fields (cf. Theorem~\ref{thm:bogomolov:function:field}). (Recently, Bogomolov's conjecture over number fields was solved by Ullmo \cite{Ul}, but effective Bogomolov's conjecture is still open.) \begin{Theorem}[$\operatorname{char}(k) = 0$] We assume that $f$ is not smooth and every singular fiber of $f$ is a tree of stable components, i.e., every node of type $0$ on the stable model of each singular fiber is a singularity of an irreducible component, then effective Bogomolov's conjecture holds for the generic fiber of $f$. Namely, let $K$ be the function field of $Y$, $C$ the generic fiber of $f$, $\operatorname{Jac}(C)$ the Jacobian of $C$, and let $j : C(\overline{K}) \to \operatorname{Jac}(C)(\overline{K})$ be the morphism given by $j(x) = (2g-2)x - \omega_C$. Then, the set $\{ x \in C(\overline{K}) \mid \Vert j(x) - P \Vert_{NT} \leq r \}$ is finite for any $P \in \operatorname{Jac}(C)(\overline{K})$ and any non-negative real numbers $r$ less than \[ \sqrt{\frac{(g-1)^2}{g(2g+1)}\left( \frac{g-1}{3}\delta_0(X/Y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i)\delta_i(X/Y) \right)}, \] where $\Vert \ \Vert_{NT}$ is the semi-norm arising from the Neron-Tate height paring on $\operatorname{Jac}(C)(\overline{K})$. \end{Theorem} Finally, we would like to express our hearty thanks to Institut des Hautes \'{E}tudes Scientifiques where all works of this paper had been done, and to Prof. Bost who pointed out a fatal error of the previous version of it. We are also grateful to referees for their wonderful suggestions. \section{Elementary properties of semi-ampleness and weak positivity} \label{sec:pef:pamp:div} \renewcommand{\theTheorem}{\arabic{section}.\arabic{Theorem}} In this section, we will introduce two kinds of positivity of divisors, namely semi-ampleness and weak positivity, and investigate their elementary properties. Let $X$ be a $d$-dimensional algebraic variety over $k$. Let $Z_{d - 1}(X)$ be a free abelian group generated by integral subvarieties of dimension $d - 1$, and $\operatorname{Div}(X)$ a group consisting of Cartier divisors on $X$. We denote $Z_{d - 1}(X)$ (resp. $\operatorname{Div}(X)$) modulo linear equivalence by $A_{d - 1}(X)$ (resp. $\operatorname{Pic}(X)$). An element of $Z_{d - 1}(X) \otimes {\mathbb{Q}}$ (resp. $\operatorname{Div}(X) \otimes {\mathbb{Q}}$) is called a {\em ${\mathbb{Q}}$-divisor} (resp. {\em ${\mathbb{Q}}$-Cartier divisor}) on $X$. We say a ${\mathbb{Q}}$-Cartier divisor $D$ is {\em the limit of a sequence $\{ D_m \}_{m=1}^{\infty}$ of ${\mathbb{Q}}$-Cartier divisors in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$}, denoted by ${\displaystyle D = \lim_{m \to \infty} D_m}$ in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$, if there are ${\mathbb{Q}}$-Cartier divisors $Z_1, \ldots, Z_{l}$ and infinite sequences $\{ a_{1, m} \}_{m=1}^{\infty}, \ldots, \{a_{l, m} \}_{m=1}^{\infty}$ of rational numbers such that (1) $l$ does not depend on $m$, (2) $D = D_m + \sum_{i=1}^{l} a_{i, m} Z_i$ in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$ for all $m \geq 1$, and (3) ${\displaystyle \lim_{m\to\infty} a_{i, m} = 0}$ for all $i=1, \ldots, l$. For example, a pseudo-effective ${\mathbb{Q}}$-Cartier divisor is the limit of effective ${\mathbb{Q}}$-Cartier divisors in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$. Let $x$ be a point of $X$. A ${\mathbb{Q}}$-Cartier divisor $D$ on $X$ is said to be {\em semi-ample at $x$} if there is a positive integer $n$ such that $nD \in \operatorname{Div}(X)$ and $H^0(X, {\mathcal{O}}_X(nD)) \otimes {\mathcal{O}}_X \to {\mathcal{O}}_X(nD)$ is surjective at $x$. Further, according to Viehweg, $D$ is said to be {\em weakly positive at $x$} if there is an infinite sequence $\{ D_m \}_{m=1}^{\infty}$ of ${\mathbb{Q}}$-Cartier divisors on $X$ such that $D_m$ is semi-ample at $x$ for all $m \geq 1$ and ${\displaystyle D = \lim_{m \to \infty} D_m}$ in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$. It is easy to see that if $D$ is weakly positive at $x$, then $(D \cdot C) \geq 0$ for any complete irreducible curves $C$ passing through $x$. As compared with the last property, weak positivity has an advantage that we can avoid bad subvarieties of codimension two (cf. Proposition~\ref{prop:wp:codim:2}). In order to consider properties of semi-ample or weakly positive divisors, let us begin with the following two lemma. \begin{Lemma}[$\operatorname{char}(k) \geq 0$] \label{lem:gen:by:global:sec:finite} Let $\pi : X \to Y$ be a proper morphism of quasi-projective varieties over $k$ and $y$ a point of $Y$ such that $\pi$ is finite over $y$. Let $F$ be a coherent ${\mathcal{O}}_X$-module and $H$ an ample line bundle on $Y$. Then there is a positive integer $n_0$ such that, for all $n \geq n_0$, \[ H^0(X, F \otimes \pi^{*}(H^{\otimes n})) \otimes {\mathcal{O}}_X \to F \otimes \pi^{*}(H^{\otimes n}) \] is surjective at each point of $\pi^{-1}(y)$. \end{Lemma} {\sl Proof.}\quad Let $n_0$ be a positive integer such that, for all $n \geq n_0$, $\pi_*(F) \otimes H^{\otimes n}$ is generated by global sections, i.e., \[ H^0(Y, \pi_*(F) \otimes H^{\otimes n}) \otimes {\mathcal{O}}_Y \to \pi_*(F) \otimes H^{\otimes n} \] is surjective. Thus, \[ H^0(X, F \otimes \pi^{*}(H^{\otimes n})) \otimes {\mathcal{O}}_X \to \pi^* \pi_*(F \otimes \pi^*(H^{\otimes n})) \] is surjective because $\pi_*(F \otimes \pi^{*}(H^{\otimes n})) = \pi_*(F) \otimes H^{\otimes n}$. On the other hand, since $\pi$ is finite over $y$, \[ \pi^* \pi_*(F \otimes \pi^*(H^{\otimes n})) \to F \otimes \pi^*(H^{\otimes n}) \] is surjective at each point of $\pi^{-1}(y)$. Thus, we get our assertion. \QED \begin{Lemma}[$\operatorname{char}(k) \geq 0$] \label{lem:pamp:plus:good:samp} Let $\pi : X \to Y$ be a proper morphism of quasi-projective varieties over $k$ and $x$ a point of $X$ such that $\pi$ is finite over $\pi(x)$. Let $D$ be a ${\mathbb{Q}}$-Cartier divisor on $X$ and $A$ a ${\mathbb{Q}}$-Cartier divisor on $Y$. If $D$ is weakly positive at $x$ and $A$ is ample, then $D + \pi^*(A)$ is semi-ample at $x$. \end{Lemma} {\sl Proof.}\quad By our assumption, there are ${\mathbb{Q}}$-Cartier divisors $Z_1, \ldots, Z_{l}$, an infinite sequence $\{ D_m \}_{m=1}^{\infty}$ of ${\mathbb{Q}}$-Cartier divisors, and infinite sequences $\{ a_{1, m} \}_{m=1}^{\infty}, \ldots, \{ a_{l, m} \}_{m=1}^{\infty}$ of rational numbers with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $D = D_m + \sum_{i=1}^{l} a_{i, m} Z_i$ in $\operatorname{Pic}(X) \otimes {\mathbb{Q}}$ for sufficiently large $m$. \item $\lim_{m\to\infty} a_{i, m} = 0$ for all $i=1, \ldots, l$. \item $D_m$ is semi-ample at $x$ for $m \gg 0$. \end{enumerate} By virtue of Lemma~\ref{lem:gen:by:global:sec:finite}, we can find a positive integer $n$ such that $n \pi^*(A) + Z_i$ and $n \pi^*(A) - Z_i$ are semi-ample at $x$. We choose a sufficiently large $m$ with $|nla_{i, m}| < 1$ ($i=1, \ldots, l$). Then, \[ \frac{1 + nla_{i, m}}{2ln} > 0 \quad\text{and}\quad \frac{1 - nla_{i, m}}{2ln} > 0. \] On the other hand, \[ D + \pi^*(A) \sim D_m + \sum_{i=1}^l \left( \frac{1 + nla_{i, m}}{2ln}(n\pi^*(A) + Z_i) + \frac{1 - nla_{i, m}}{2ln}(n\pi^*(A) - Z_i) \right). \] Thus, $D + \pi^*(A)$ is semi-ample at $x$. \QED As immediate consequences of Lemma~\ref{lem:pamp:plus:good:samp}, we have the following propositions. The first one is a characterization of weak positivity in terms of ample divisors. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:criterion:pamp} Let $X$ be a quasi-projective variety over $k$, $x$ a point of $X$, and $D$ a ${\mathbb{Q}}$-Cartier divisor on $X$. Then, the following are equivalent. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $D$ is weakly positive at $x$. \item For any ample ${\mathbb{Q}}$-Cartier divisors $A$ on $X$, $D + A$ is semi-ample at $x$. \item There is an ample ${\mathbb{Q}}$-Cartier divisor $A$ on $X$ such that $D + \epsilon A$ is semi-ample at $x$ for any positive rational numbers $\epsilon$. \end{enumerate} \end{Proposition} \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:wp:codim:2} Let $X$ be a normal quasi-projective variety over $k$, $X_0$ a Zariski open set of $X$, and $x$ a point of $X_0$. Let $D$ be a ${\mathbb{Q}}$-Cartier divisor on $X$ and $D_0 = \rest{D}{X_0}$. If $\operatorname{codim}(X \setminus X_0) \geq 2$, then we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $D$ is semi-ample at $x$ if and only if $D_0$ is semi-ample at $x$. \item $D$ is weakly positive at $x$ if and only if $D_0$ is weakly positive at $x$. \end{enumerate} \end{Proposition} Next, let us consider functorial properties of semi-ampleness and weak positivity under pull-back and push-forward. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:samp:pamp:pullback} Let $\pi : X \to Y$ be a morphism of quasi-projective varieties. Let $D$ be a ${\mathbb{Q}}$-Cartier divisor on $Y$ and $x$ a point of $X$. If $\pi^*(D)$ is defined, then we have the following. \textup{(}Note that even if $\pi^*(D)$ is not defined, there is a ${\mathbb{Q}}$-Cartier divisor $D'$ such that $D' \sim D$ and $\pi^*(D')$ is defined.\textup{)} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item If $D$ is semi-ample at $\pi(x)$, then $\pi^{*}(D)$ is semi-ample at $x$. \item If $D$ is weakly positive at $\pi(x)$, then $\pi^*(D)$ is weakly positive at $x$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) By our assumption, $H^0(Y, {\mathcal{O}}_Y(nD)) \otimes {\mathcal{O}}_Y \to {\mathcal{O}}_Y(nD)$ is surjective at $\pi(x)$ for a sufficiently large $n$. Thus, $H^0(Y, {\mathcal{O}}_Y(nD)) \otimes {\mathcal{O}}_X \to {\mathcal{O}}_X(n\pi^*(D))$ is surjective at $x$. Here let us consider the following commutative diagram: \[ \begin{CD} H^0(Y, {\mathcal{O}}_{Y}(nD)) \otimes {\mathcal{O}}_{X} @>{\alpha}>> {\mathcal{O}}_X(n\pi^*(D)) \\ @VVV @| \\ H^0(X, {\mathcal{O}}_{X}(n\pi^*(D))) \otimes {\mathcal{O}}_{X} @>{\alpha'}>> {\mathcal{O}}_X(n\pi^*(D)). \end{CD} \] Since $\alpha$ is surjective at $x$, so is $\alpha'$. Therefore, $\pi^*(D)$ is semi-ample at $x$. \medskip (2) Let $A$ be an ample divisor on $Y$ such that $\pi^*(A)$ is defined. Then, by Lemma~\ref{lem:pamp:plus:good:samp}, $D + (1/n)A$ is semi-ample at $\pi(x)$ for all $n > 0$. Thus, by (1), $\pi^*(D) + (1/n)\pi^*(A)$ is semi-ample at $x$ for all $n > 0$. Therefore, $\pi^*(D)$ is weakly positive at $x$. \QED \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:samp:pamp:push} Let $\pi : X \to Y$ be a surjective, proper and generically finite morphism of normal quasi-projective varieties over $k$. Let $D$ be a ${\mathbb{Q}}$-Cartier divisor on $X$ and $y$ a point of $Y$ such that $\pi_*(D)$ is a ${\mathbb{Q}}$-Cartier divisor on $Y$ and $\pi$ is finite over $y$. We set $\pi^{-1}(y) = \{x_1, \ldots, x_n \}$. Then, we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item If $D$ is semi-ample at $x_1, \ldots, x_n$, then $\pi_*(D)$ is semi-ample at $y$. \item If $D$ is weakly positive at $x_1, \ldots, x_n$, then $\pi_*(D)$ is weakly positive at $y$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) Clearly, we may assume that $D$ is a Cartier divisor. If we take a sufficiently large integer $m$, then $H^0(X, {\mathcal{O}}_X(mD)) \otimes {\mathcal{O}}_X \to {\mathcal{O}}_X(mD)$ is surjective at $x_1, \ldots, x_n$. Thus, there are sections $s_1, \ldots, s_n$ of $H^0(X, {\mathcal{O}}_X(m D))$ with $s_i(x_i) \not= 0$ for all $i=1, \ldots, n$. For $\alpha = (\alpha_1, \ldots\, \alpha_n) \in k^n$, we set $s_{\alpha} = \alpha_1 s_1 + \cdots + \alpha_n s_n$. Further, we set $V_i = \{ \alpha \in k^n \mid s_{\alpha}(x_i) = 0 \}$. Then, $\dim V_i= n-1$ for all $i$. Thus, since $\#(k) = \infty$, there is $\alpha \in k^n$ with $\alpha \not\in V_1 \cup \cdots \cup V_r$, i.e., $s_{\alpha}(x_i) \not= 0$ for all $i$. Let us consider a divisor $E = \operatorname{div}(s_{\alpha})$. Then, $E \sim m D$. Thus, $\pi_*(E) \sim m \pi_*(D)$. Here, $x_i \not\in E$ for all $i$. Hence, $y \not\in \pi_*(E)$. Therefore, we get our assertion. \medskip (2) Let $A$ be an ample divisor on $Y$. We set $D_m = D + (1/m)\pi^*(A)$. Then, by Lemma~\ref{lem:pamp:plus:good:samp}, $D_m$ is semi-ample at $x_1, \ldots, x_n$. Thus, by (1), $\pi_*(D_m) = \pi_*(D) + (1/m) \deg(\pi) A$ is semi-ample at $y$. Therefore, $\pi_*(D)$ is weakly positive at $y$. \QED \bigskip Finally, let us consider semi-ampleness and weak positivity over an open set. Let $X$ be a quasi-projective variety over $k$, $U$ a Zariski open set of $X$, and $D$ a ${\mathbb{Q}}$-Cartier divisor on $X$. We say $D$ is {\em semi-ample over $U$} (resp. {\em weakly positive over $U$}) if $D$ is semi-ample (resp. weakly positive) at all points of $U$. Then, we can easily see the following. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:samp:wp:U} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item If $D$ is semi-ample over $U$, then there is a positive integer $n$ such that $nD$ is a Cartier divisor and $H^0(X, {\mathcal{O}}_X(nD)) \otimes {\mathcal{O}}_X \to {\mathcal{O}}_X(nD)$ is surjective on $U$. \item If $D$ is weakly positive over $U$, then, for any ample ${\mathbb{Q}}$-Cartier divisors $A$ on $X$, there is a positive integer $n$ such that $n(D+A)$ is a Cartier divisor and \[ H^0(X, {\mathcal{O}}_X(n(D+A))) \otimes {\mathcal{O}}_X \to {\mathcal{O}}_X(n(D+A)) \] is surjective on $U$. \end{enumerate} \end{Proposition} \section{Proof of relative Bogomolov's inequality} Let $X$ be an algebraic variety over $k$, $x$ a point of $X$, and $E$ a coherent ${\mathcal{O}}_X$-module on $X$. We say $E$ is {\em generated by global sections at $x$} if $H^0(X, E) \otimes {\mathcal{O}}_X \to E$ is surjective at $x$. Let us begin with the following proposition. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:samp:det} Let $X$ be a smooth algebraic variety over $k$, $E$ a coherent ${\mathcal{O}}_X$-module, and $x$ a point of $X$. If $E$ is generated by global sections at $x$ and $E$ is free at $x$, then $\det(E)$ is generated by global sections at $x$, where $\det(E)$ is the determinant line bundle of $E$ in the sense of \cite{KM}. \end{Proposition} {\sl Proof.}\quad Let $T$ be the torsion part of $E$. Then, $\det(E) = \det(E/T) \otimes \det(T)$. If we set \[ D = \sum_{\substack{P \in X, \\ \operatorname{depth}(P) = 1}} \operatorname{lenght}(T_P) \overline{ \{ P \} }, \] then $\det(T) \simeq {\mathcal{O}}_X(D)$, where $\overline{ \{ P \} }$ is the Zariski closure of $\{ P \}$ in $X$. Here since $E$ is free at $x$, $x \not\in \operatorname{Supp}(D)$. Thus, $\det(T)$ is generated by global sections at $x$. Moreover, it is easy to see that $E/T$ is generated by global sections at $x$. Therefore, to prove our proposition, we may assume that $E$ is a torsion free sheaf. Let $r$ be the rank of $E$ and $\kappa(x)$ the residue field of $x$. Then, by our assumption, there are sections $s_1, \ldots, s_r$ of $E$ such that $\{ s_i(x) \}$ forms a basis of $E \otimes \kappa(x)$. Thus, $s = s_1 \wedge \cdots \wedge s_r$ gives rise to a section of $\det(E) = \left( \bigwedge^r E \right)^{**}$ with $s(y) \not= 0$. Hence, we get our proposition. \QED Next let us consider the following proposition. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:property:global:gen} Let $\pi : X \to Y$ be a proper and generically finite morphism of algebraic varieties over $k$. Let $y$ be a point of $Y$ such that $\pi$ is finite over $y$. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item Let $\phi : E \to Q$ be a homomorphism of coherent ${\mathcal{O}}_X$-modules. If $\phi$ is surjective at each point of $\pi^{-1}(y)$ and $\pi_*(E)$ is generated by global sections at $y$, then $\pi_*(Q)$ is generated by global sections at $y$. \item Let $E_1$ and $E_2$ be coherent ${\mathcal{O}}_X$-modules. If $\pi_*(E_1)$ and $\pi_*(E_2)$ are generated by global sections at $y$, then so is $\pi_*(E_1 \otimes E_2)$ at $y$. \item Let $E$ be a coherent ${\mathcal{O}}_X$-module. If $\pi_*(E)$ is generated by global sections at $y$, then so is $\pi_*(\operatorname{Sym}^n(E))$ at $y$ for every $n > 0$. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) We can take an affine open neighborhood $U$ of $y$ such that $\pi$ is finite over $U$ and $\phi : E \to Q$ is surjective over $\pi^{-1}(U)$. Thus, $\pi_*(E) \to \pi_*(Q)$ is surjective at $y$. Hence, considering the following diagram: \[ \begin{CD} H^0(Y, \pi_*(E)) \otimes {\mathcal{O}}_Y @>>> \pi_*(E) \\ @VVV @VVV \\ H^0(Y, \pi_*(Q)) \otimes {\mathcal{O}}_Y @>>> \pi_*(Q), \end{CD} \] we have our assertion. \medskip (2) Let $U$ be an affine open neighborhood of $y$ such that $\pi$ is finite over $U$. We set $U = \operatorname{Spec}(A)$ for some integral domain $A$. Since $\pi$ is finite over $U$, there is an integral domain $B$ with $\pi^{-1}(U) = \operatorname{Spec}(B)$. Here we take $B$-modules $M_1$ and $M_2$ such that $M_1$ and $M_2$ give rise to $\rest{E_1}{\pi^{-1}(U)}$ and $\rest{E_2}{\pi^{-1}(U)}$ respectively. Then, we have a natural surjective homomorphism $M_1 \otimes_A M_2 \to M_1 \otimes_B M_2$. This shows us that $\pi_*(E) \otimes \pi_*(E_2) \to \pi_*(E_1 \otimes E_2)$ is surjective at $y$. Here, let us consider the following diagram: \[ \begin{CD} H^0(Y, \pi_*(E_1)) \otimes H^0(Y, \pi_*(E_2)) \otimes {\mathcal{O}}_Y @>>> \pi_*(E_1) \otimes \pi_*(E_2) \\ @VVV @VVV \\ H^0(Y, \pi_*(E_1 \otimes E_2)) \otimes {\mathcal{O}}_Y @>>> \pi_*(E_1 \otimes E_2), \end{CD} \] where $H^0(Y, \pi_*(E_1)) \otimes H^0(Y, \pi_*(E_2)) \otimes {\mathcal{O}}_Y \to \pi_*(E_1) \otimes \pi_*(E_2)$ is surjective at $y$ by our assumption. Thus, we get (2). \medskip (3) This is a consequence of (1) and (2) because we have a natural surjective homomorphism $E^{\otimes n} \to \operatorname{Sym}^n(E)$. \QED Before starting the main theorem, we need to prepare the following formula derived from Grothendieck-Riemann-Roch theorem. \begin{Lemma}[$\operatorname{char}(k) \geq 0$] \label{lem:growth:c1:line:by:R:R} Let $X$ and $Y$ be algebraic varieties over $k$, and $f : X \to Y$ a surjective and projective morphism over $k$ of $\dim f = d$. Let $L$ and $A$ be line bundles on $X$. If $Y$ is smooth, then there are elements $Z_1, \ldots, Z_d$ of $A_{\dim Y - 1}(Y) \otimes {{\mathbb{Q}}}$ such that \[ c_1\left( Rf_*(L^{\otimes n} \otimes A) \right) \cap [Y] = \frac{ f_*(c_1(L)^{d+1} \cap [X])}{(d+1)!} n^{d+1} + \sum_{i=0}^{d} Z_i n^i \] for all $n > 0$. \end{Lemma} {\sl Proof.}\quad We use the same symbol as in \cite{Fu}. First of all, $Rf_*(L^{\otimes n} \otimes A) \in K^{\circ}(Y)$ because $Y$ is smooth. Thus, by \cite[Theorem~18.3, (1) and (2)]{Fu}, i.e., Riemann-Roch theorem for singular varieties, \addtocounter{Claim}{1} \begin{equation} \label{eqn:lem:growth:c1:line:by:R:R:1} \operatorname{ch}(Rf_*(L^{\otimes n} \otimes A)) \cap \tau_Y({\mathcal{O}}_Y) = f_*(\operatorname{ch}(L^{\otimes n} \otimes A) \cap \tau_X({\mathcal{O}}_X)). \end{equation} Since $\tau_X({\mathcal{O}}_X) = [X] + \text{terms of dimension $< \dim X$}$ by \cite[Theorem~18.3, (5)]{Fu}, it is easy to see that there are $T_0, \ldots, T_{d} \in A_{\dim Y - 1}(X) \otimes {\mathbb{Q}}$ such that \[ \left( \operatorname{ch}(L^{\otimes n} \otimes A) \cap \tau_X({\mathcal{O}}_X) \right)_{\dim Y - 1} = \frac{c_1(L)^{d+1} \cap [X]}{(d+1)!} n^{d+1} + \sum_{i=0}^d T_i n^i. \] Thus, \addtocounter{Claim}{1} \begin{equation} \label{eqn:lem:growth:c1:line:by:R:R:2} f_*(\operatorname{ch}(L^{\otimes n} \otimes A) \cap \tau_X({\mathcal{O}}_X))_{\dim Y - 1} = \frac{f_*(c_1(L)^{d+1} \cap [X])}{(d+1)!} n^{d+1} + \sum_{i=0}^d f_*(T_i) n^i. \end{equation} On the other hand, since $\tau_Y({\mathcal{O}}_Y) = [Y] + \text{terms of dimension $< \dim Y$}$, if we denote by $S$ the $(\dim Y - 1)$-dimensional part of $\tau_Y({\mathcal{O}}_Y)$, then \addtocounter{Claim}{1} \begin{equation} \label{eqn:lem:growth:c1:line:by:R:R:3} \left( \operatorname{ch}(Rf_*(L^{\otimes n} \otimes A)) \cap \tau_Y({\mathcal{O}}_Y) \right)_{\dim Y - 1} = c_1(Rf_*(L^{\otimes n} \otimes A)) \cap [Y] + \operatorname{rk} (Rf_*(L^{\otimes n} \otimes A)) S. \end{equation} Here, $\operatorname{rk} (Rf_*(L^{\otimes n} \otimes A)) = \chi(X_{\eta}, (L^{\otimes n} \otimes A)_{\eta})$ is a polynomial of $n$ with degree $d$ at most, where $\eta$ is the generic point of $Y$. Thus, combining \eqref{eqn:lem:growth:c1:line:by:R:R:1}, \eqref{eqn:lem:growth:c1:line:by:R:R:2} and \eqref{eqn:lem:growth:c1:line:by:R:R:3}, we have our lemma. \QED Let us start the main theorem of this paper. \begin{Theorem}[$\operatorname{char}(k) \geq 0$] \label{thm:nef:psudo:dis} Let $X$ be a quasi-projective variety over $k$, $Y$ a smooth quasi-projective variety over $k$, and $f : X \to Y$ a surjective and projective morphism over $k$ of $\dim f = 1$. Let $F$ be a locally free sheaf on $X$ with $f_*(c_1(F) \cap [X]) = 0$, and $y$ a point of $Y$. We assume that $f$ is flat over $y$, and that there are line bundles $L$ and $M$ on the geometric fiber $X_{\bar{y}}$ over $y$ such that \[ H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes L) = H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes M) = 0 \] for $m \gg 0$. Then, $f_*\left( (c_2(F) - c_1(F)^2) \cap [X]) \right)$ is weakly positive at $y$. \end{Theorem} {\sl Proof.}\quad Let $A$ be a very ample line bundle on $X$ such that $A_{\bar{y}} \otimes L$ and $A_{\bar{y}} \otimes M^{\otimes -1}$ are very ample on $X_{\bar{y}}$. First of all, we would like to see the following. \begin{Claim} \label{claim:thm:nef:psudo:dis:0} $H^0(X_y, \operatorname{Sym}^m(F_y) \otimes A_y^{\otimes -1}) = H^1(X_y, \operatorname{Sym}^m(F_y) \otimes A_y) = 0$ for $m \gg 0$. \end{Claim} In general, for a coherent sheaf $G$ on $X_y$, $H^i(X_{\bar{y}}, G \otimes_{\kappa(y)} \overline{\kappa(y)}) = H^i(X_y, G) \otimes_{\kappa(y)} \overline{\kappa(y)}$ for all $i \geq 0$. Thus, it is sufficient to show that \[ H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}^{\otimes -1}) = H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}) = 0 \] for $m \gg 0$. Since $A_{\bar{y}} \otimes L$ is very ample and $\#(\overline{\kappa(y)}) = \infty$, there is a section $s \in H^0(X_{\bar{y}}, A_{\bar{y}} \otimes L)$ such that $s \not= 0$ in $(A_{\bar{y}} \otimes L) \otimes \kappa(P)$ for any associated points $P$ of $X_{\bar{y}}$. Then, ${\mathcal{O}}_{X_{\bar{y}}} \overset{\times s}{\longrightarrow} A_{\bar{y}} \otimes L$ is injective. Thus, tensoring the above injection with $\operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}^{\otimes -1}$, we have an injection \[ \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}^{\otimes -1} \to \operatorname{Sym}^m(F_{\bar{y}}) \otimes L. \] Hence, $H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}^{\otimes -1}) = 0$ for $m \gg 0$. In the same way, there is a section $s' \in H^0(X_{\bar{y}}, A_{\bar{y}} \otimes M^{\otimes -1})$ such that $s' \not= 0$ in $(A_{\bar{y}} \otimes M^{\otimes -1}) \otimes \kappa(P)$ for any associated points $P$ of $X_{\bar{y}}$. Then, ${\mathcal{O}}_{X_{\bar{y}}} \overset{\times s'}{\longrightarrow} A_{\bar{y}} \otimes M^{\otimes -1}$ is injective and its cokernel $T$ has the $0$-dimensional support. Thus, tensoring an exact sequence \[ 0 \to {\mathcal{O}}_{X_{\bar{y}}} \to A_{\bar{y}} \otimes M^{\otimes -1} \to T \to 0 \] with $\operatorname{Sym}^m(F_{\bar{y}}) \otimes M$, we obtain an exact sequence \[ 0 \to \operatorname{Sym}^m(F_{\bar{y}}) \otimes M \to \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}} \to \operatorname{Sym}^m(F_{\bar{y}}) \otimes M \otimes T \to 0. \] Hence, we get a surjection \[ H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes M) \to H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}). \] Therefore, $H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A_{\bar{y}}) = 0$ for $m \gg 0$. \bigskip Since $X$ is an integral scheme over $k$ of dimension greater than or equal to $2$, and $X_y$ is a $1$-dimensional scheme over $\kappa(y)$, by virtue of \cite[Theorem~6.10]{JB}, there is $B \in |A^{\otimes 2}|$ such that $B$ is integral, and that $B \cap X_y$ is finite, i.e., $B$ is finite over $y$. Let $\pi : B \to Y$ be the morphism induced by $f$. Let $H$ be an ample line bundle on $Y$ such that $\pi_*(F_B) \otimes H$ and $\pi_*(A_B) \otimes H$ are generated by global sections at $y$, where $F_B = \rest{F}{B}$ and $A_B = \rest{A}{B}$. Let $\mu : P = \operatorname{Proj}\left( \bigoplus_{m=0}^{\infty} \operatorname{Sym}^m(F) \right) \to X$ be the projective bundle and ${\mathcal{O}}_{P}(1)$ the tautological line bundle on $P$. We set $h = f \cdot \mu : P \to Y$. Let us consider \[ c_1 \left( Rh_*(({\mathcal{O}}_{P}(1) \otimes h^*(H))^{\otimes m} \otimes \mu^*(A^{\otimes -1}) \otimes h^*(H)) \right) \cap [Y] \] for $m \gg 0$. By Lemma~\ref{lem:growth:c1:line:by:R:R}, there are elements $Z_0, \ldots, Z_{r}$ of $A_{\dim Y - 1}(Y) \otimes {\mathbb{Q}}$ such that \begin{multline*} c_1 \left( Rh_*(({\mathcal{O}}_{P}(1) \otimes h^*(H))^{\otimes m} \otimes \mu^*(A^{\otimes -1}) \otimes h^*(H) ) \right) \cap [Y] \\ = \frac{h_*( c_1({\mathcal{O}}_P(1) \otimes h^*(H))^{r+1} \cap [P])}{(r+1)!} m^{r + 1} + \sum_{i=0}^{r} Z_i m^i, \end{multline*} where $r$ is the rank of $F$. Here \[ \begin{cases} \mu_*(c_1({\mathcal{O}}_{P}(1))^{r+1} \cap [P]) = (c_1(F)^2 - c_2(F)) \cap [X], \\ \mu_*(c_1({\mathcal{O}}_{P}(1))^{r} \cap [P]) = c_1(F) \cap [X], \\ \mu_*(c_1({\mathcal{O}}_{P}(1))^{r-1} \cap [P]) = [X], \\ \mu_*(c_1({\mathcal{O}}_{P}(1))^{j} \cap [P]) = 0 \qquad (0 \leq j < r-1). \end{cases} \] Thus, by using projection formula, we have \begin{align*} h_*( c_1({\mathcal{O}}_P(1) \otimes h^*(H))^{r+1} \cap [P]) & = f_* \mu_*\left( \sum_{i=0}^{r+1} \mu^* f^* (c_1(H)^i) \cap (c_1({\mathcal{O}}_P(1))^{r+1-i} \cap [P]) \right) \\ & = f_* \left( (c_1(F)^2 - c_2(F)) \cap [X] \right) \\ & \qquad\qquad + r f_* \left( f^*(c_1(H)) \cap (c_1(F) \cap [X]) \right) \\ & \qquad\qquad\qquad\quad + \frac{r(r+1)}{2} f_* \left( f^*(c_1(H)^2) \cap [X] \right) \\ & = - f_* \left( (c_2(F) - c_1(F)^2) \cap [X] \right) \end{align*} because $f_* (c_1(F) \cap [X]) = 0$ and $f_*([X]) = 0$. Moreover, \[ R\mu_*(({\mathcal{O}}_{P}(1) \otimes h^*(H))^{\otimes m} \otimes \mu^*(A^{\otimes -1}) \otimes h^*(H)) = \operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H). \] Therefore, we get \begin{multline*} \sum_{i \geq 0} (-1)^i c_1\left( R^if_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right) \right) \cap [Y] \\ = -\frac{1}{(r+1)!} f_* \left( (c_2(F) - c_1(F)^2) \cap [X] \right) m^{r + 1} + \sum_{i=0}^{r} Z_i m^i. \end{multline*} Here we claim the following. \begin{Claim} \label{claim:thm:nef:psudo:dis:1} If $m \gg 0$, then we have the following. \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $c_1\left( R^if_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right) \right) \cap [Y] = 0$ for all $i \geq 2$. \item $f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right) = 0$. \item $R^1f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right)$ is free at $y$. \item $R^1 f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A \otimes f^*(H) \right) = 0$ around $y$. \end{enumerate} \end{Claim} (a) : Let $Y'$ be the maximal open set of $Y$ such that $f$ is flat over $Y'$. If $i \geq 2$, then the support of $R^if_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H) ) \otimes A^{\otimes -1} \otimes f^*(H) \right)$ is contained in $Y \setminus Y'$. Here $\operatorname{codim}(Y \setminus Y') \geq 2$. Thus, we get (a). (b) and (c) : By Claim~\ref{claim:thm:nef:psudo:dis:0}, $H^0(X_y, \operatorname{Sym}^m(F_{y}) \otimes A_{y}^{\otimes -1}) = 0$ for $m \gg 0$. Thus, using the upper-semicontinuity of dimension of cohomology groups, there is an open neighborhood $U_m$ of $y$ such that $f$ is flat over $U_m$ and $H^0(X_{y'}, \operatorname{Sym}^m(F_{y'}) \otimes A_{y'}^{\otimes -1}) = 0$ for all $y' \in U_m$, which implies (b) because $f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right)$ is torsion free. Here, since $f$ is flat over $U_m$, $\chi(X_{y'}, \operatorname{Sym}^m(F_{y'}) \otimes A_{y'}^{\otimes -1})$ is a constant with respect to $y' \in U_m$. Therefore, so is $h^1(X_{y'}, \operatorname{Sym}^m(F_{y'}) \otimes A_{y'}^{\otimes -1})$ with respect to $y' \in U_m$. Thus, we have (c). (d) : By virtue of Claim~\ref{claim:thm:nef:psudo:dis:0}, $H^1(X_{y}, \operatorname{Sym}^m(F_{y}) \otimes A_{y}) = 0$ for $m \gg 0$. Thus, there is an open neighborhood $U'_m$ of $y$ such that $f$ is flat over $U'_m$ and \[ H^1(X_{y'}, \operatorname{Sym}^m(F_{y'}) \otimes A_{y'}) = 0 \] for all $y' \in U'_m$. Hence, we can see (d). \bigskip By (a) and (b) of Claim~\ref{claim:thm:nef:psudo:dis:1}, \begin{multline*} \frac{1}{(r+1)!} f_* \left( (c_2(F) - c_1(F)^2) \cap [X] \right) \\ = \frac{c_1\left( R^1 f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right) \right) \cap [Y]}{m^{r+1}} + \sum_{i=0}^{r} \frac{Z_i}{m^{r+1-i}}. \end{multline*} Hence, it is sufficient to show that \[ c_1\left( R^1 f_*\left(\operatorname{Sym}^{m}(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \right) \right) \cap [Y] \] is semi-ample at $y$. \medskip Since $\pi_*(F_B \otimes \pi^*(H))$ and $\pi_*(A_B \otimes \pi^*(H))$ are generated by global sections at $y$, by (2) and (3) of Proposition~\ref{prop:property:global:gen}, $\pi_*(\operatorname{Sym}^m(F_B \otimes \pi^*(H)) \otimes A_B \otimes \pi^*(H))$ is generated by global sections at $y$. On the other hand, a short exact sequence \begin{multline*} 0 \to \operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H) \to \operatorname{Sym}^m(F \otimes f^*(H)) \otimes A \otimes f^*(H) \\ \to \operatorname{Sym}^m(F_B \otimes \pi^*(H)) \otimes A_B \otimes \pi^*(H) \to 0 \end{multline*} gives rise to an exact sequence \begin{multline*} 0 \to f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A \otimes f^*(H) ) \to \pi_*(\operatorname{Sym}^m(F_B \otimes \pi^*(H)) \otimes A_B\otimes \pi^*(H)) \\ \to R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H)) \to R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A \otimes f^*(H)). \end{multline*} Thus, by (d) of Claim~\ref{claim:thm:nef:psudo:dis:1}, \[ \phi_m : \pi_*(\operatorname{Sym}^m(F_B \otimes \pi^*(H)) \otimes A_B \otimes \pi^*(H)) \to R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H)) \] is surjective at $y$. Therefore, applying (1) of Proposition~\ref{prop:property:global:gen} to the case where $\operatorname{id}_Y: Y \to Y$ and $\phi = \phi_m$, $R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H))$ is generated by global sections at $y$. Moreover, by virtue of (c) of Claim~\ref{claim:thm:nef:psudo:dis:1}, $R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H))$ is free at $y$. Hence, by Proposition~\ref{prop:samp:det}, $c_1( R^1f_*(\operatorname{Sym}^m(F \otimes f^*(H)) \otimes A^{\otimes -1} \otimes f^*(H))) \cap [Y]$ is semi-ample at $y$. \QED As a corollary of Theorem~\ref{thm:nef:psudo:dis}, we have the following. \begin{Corollary}[$\operatorname{char}(k) = 0$] \label{cor:nef:psudo:dis} Let $X$ be a quasi-projective variety over $k$, $Y$ a smooth quasi-projective variety over $k$, and $f : X \to Y$ a surjective and projective morphism over $k$ with $\dim f = 1$. Let $E$ be a locally free sheaf on $X$ and $y$ a point of $Y$. If $f$ is flat over $y$, the geometric fiber $X_{\bar{y}}$ over $y$ is reduced and Gorenstein, and $E$ is semistable on each connected component of the normalization of $X_{\bar{y}}$, then $\operatorname{dis}_{X/Y}(E)$ is weakly positive at $y$. \end{Corollary} {\sl Proof.}\quad We set $F = \operatorname{\mathcal{E}\textsl{nd}}(E)$. First, we claim the following. \begin{Claim} \label{claim:cor:nef:psudo:dis} $H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes A^{\otimes -1}) = 0$ for any ample line bundles $A$ on $X_{\bar{y}}$ and any $m \geq 0$. \end{Claim} Let $\pi : Z \to X_{\bar{y}}$ be the normalization of $X_{\bar{y}}$. The semistability of tensor products of semistable vector bundles in characteristic zero was studied by a lot of authors \cite{Gi1}, \cite{Ha1}, \cite{Mi}, \cite{Ma1} and etc. (You can find a new elementary algebraic proof in \S\ref{sec:relative:bogomolov:inequality:positive:characteristic}, which works in any characteristic under strong semistability.) Thus, by virtue of our assumption, $\operatorname{Sym}^m(\pi^*(F_{\bar{y}}))$ is semistable and of degree $0$ on each connected component of $Z$. Hence, \[ H^0(Z, \pi^*( \operatorname{Sym}^m(F_{\bar{y}}) \otimes A^{\otimes -1} )) = 0. \] Here, since ${\mathcal{O}}_{X_{\bar{y}}} \to \pi_*({\mathcal{O}}_Z)$ is injective, the above implies our claim. \bigskip Let $L$ be an ample line bundle on $X_{\bar{y}}$ such that $L \otimes \omega_{X_{\bar{y}}}^{\otimes -1}$ is ample. Here, since $F_{\bar{y}}^* = F_{\bar{y}}$, by using Serre's duality theorem, $H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes L)$ is isomorphic to the dual space of $H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes (L \otimes \omega_{X_{\bar{y}}}^{\otimes -1})^{\otimes -1})$. Thus, by Claim~\ref{claim:cor:nef:psudo:dis}, \[ H^0(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes L^{\otimes -1}) = H^1(X_{\bar{y}}, \operatorname{Sym}^m(F_{\bar{y}}) \otimes L) = 0 \] for all $m \geq 0$. Hence, Theorem~\ref{thm:nef:psudo:dis} implies our corollary because $c_1(F) = 0$ and $c_2(F) = 2 \operatorname{rk}(E) c_2(E) - (\operatorname{rk}(E) -1) c_1(E)^2$. \QED \begin{Remark} Even if $\operatorname{rk}(F) = 1$, Theorem~\ref{thm:nef:psudo:dis} is a non-trivial fact. For, if $f : X \to Y$ is a smooth surface fibred over a projective curve, then the assertion of it is nothing more than the Hodge index theorem. \end{Remark} \section{A weakly positive divisor on the moduli space of stable curves} Throughout this section, we assume that $\operatorname{char}(k) = 0$. Fix an integer $g \geq 2$ and a polynomial $P_g(n) = (6n-1)(g-1)$. Let $H_g \subset \operatorname{Hilb}^{P_g}_{{\mathbb{P}}^{5g-6}}$ be a subscheme of all tri-canonically embedded stable curves over $k$, $Z_g \subset H_g \times {\mathbb{P}}^{5g-6}$ the universal tri-canonically embedded stable curves over $k$, and $\pi : Z_g \to H_g$ the natural projection. Let $\Delta$ be the minimal closed subset of $H_g$ such that $\pi$ is not smooth over a point of $\Delta$. Then, by \cite[Theorem~(1.6) and Corollary~(1.9)]{DM}, $Z_g$ and $H_g$ are quasi-projective and smooth over $k$, and $\Delta$ is a divisor with only normal crossings. Let $\Delta = \Delta_0 \cup \cdots \cup \Delta_{[g/2]}$ be the irreducible decomposition of $\Delta$ such that, if $x \in \Delta_i \setminus \operatorname{Sing}(\Delta)$, then $\pi^{-1}(x)$ is a stable curve with one node of type $i$. We set $U = H_g \setminus \Delta$, $H_g^0 = H_g \setminus \operatorname{Sing}(\Delta_1 + \cdots +\Delta_{[g/2]})$ and $Z_g^0 = \pi^{-1}(H_g^0)$. In \cite[\S3]{Mo5}, we constructed a reflexive sheaf $F$ on $Z_g$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $F$ is locally free on $Z_g^0$. \item For each $y \in H_g \setminus (\Delta_1 \cup \cdots \cup \Delta_{[g/2]})$, $\rest{F}{\pi^{-1}(y)} = \operatorname{Ker}\left(H^0(\omega_{\pi^{-1}(y)}) \otimes {\mathcal{O}}_{\pi^{-1}(y)} \to \omega_{\pi^{-1}(y)}\right)$. \item $\operatorname{dis}_{Z_g/H_g}(F) = (8g+4) \det(\pi_*(\omega_{Z_g/H_g})) - g \Delta_0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i$. \end{enumerate} Actually, $F$ can be constructed as follows. First of all, we set \[ E = \operatorname{Ker}\left( \pi^*(\pi_*(\omega_{Z_g/H_g})) \to \omega_{Z_g/H_g}\right). \] We would like to modify $E$ along singular fibers so that we can get our desired $F$. For this purpose, we consider $E^0 = \rest{E}{Z_g^0}$. It is easy to see that $E^0$ is a locally free sheaf on $Z_g^0$. For each $i \geq 0$, we denote $\Delta_i\cap H_g^0$ by $\Delta_i^0$. If $i \geq 1$, then there is the irreducible decomposition $\pi^{-1}(\Delta_i^0) = C_i^1 \cup C_i^2$ such that the generic fiber of $\rest{\pi}{C_i^1} : C_i^1\to \Delta_i^0$ (resp. $\rest{\pi}{C_i^2} : C_i^2 \to \Delta_i^0$) is of genus $i$ (resp. $g-i$). Moreover, if we set \[ Q_i^j = \operatorname{Ker}\left( \left(\rest{\pi}{C_i^j}\right)^*\left(\rest{\pi}{C_i^j}\right)_* (\omega_{C_i^j/\Delta_i^0}) \longrightarrow \omega_{C_i^j/\Delta_i^0}\right) \] for each $i \geq 1$ and $j = 1, 2$, then there is a natural surjective homomorphism \[ \alpha_i^j : \rest{E^0}{C_i^j} \to Q_i^j. \] Here let us consider \[ F^0= \operatorname{Ker}\left( \bigoplus_{i=1}^{\left[\frac{g}{2}\right]} \left( \alpha_i^1 \oplus \alpha_i^2 \right) \ : \ E^0 \longrightarrow \bigoplus_{i=1}^{\left[\frac{g}{2}\right]} \left( Q_i^1 \oplus Q_i^2 \right) \right). \] As we showed in \cite[\S3]{Mo5}, $F^0$ is a locally free sheaf on $Z_g^0$ with \[ \operatorname{dis}_{Z_g^0/H_g^0}(F^0) = (8g+4) \det(\pi_*(\omega_{Z_g^0/H_g^0})) - g \Delta_0^0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i^0. \] Let $\nu : Z_g^0 \to Z_g$ be the natural inclusion map. Then $F$ can be defined by $\nu_*(F^0)$. In order to see (2), note that $E = F$ over $H_g \setminus (\Delta_1 \cup \cdots \cup \Delta_{[g/2]})$ and $\pi^*(\pi_*(\omega_{Z_g/H_g})) \to \omega_{Z_g/H_g}$ is surjective on $H_g \setminus (\Delta_1 \cup \cdots \cup \Delta_{[g/2]})$ (cf. \cite[Proposition~2.1.3]{Mo5}). \bigskip Let $\overline{\mathcal{M}}_g$ (resp. $\mathcal{M}_g$) be the moduli space of stable (resp. smooth) curves of genus $g$ over $k$. Let $\phi : H_g \to \overline{\mathcal{M}}_g$ be the canonical morphism. Let $\lambda, \delta_0, \ldots, \delta_{[g/2]} \in \operatorname{Pic}(\overline{\mathcal{M}}_g) \otimes {\mathbb{Q}}$ such that $\phi^*(\lambda) = \det(\pi_*(\omega_{Z_g/H_g}))$ and $\phi^*(\delta_i) = \Delta_i$ for all $0 \leq i \leq [g/2]$. Let us begin with the following lemma. \begin{Lemma} \label{lem:criterion:pamp:on:M:g} Let $D$ be a ${\mathbb{Q}}$-Cartier divisor on $\overline{\mathcal{M}}_g$ and $x$ a point of $\overline{\mathcal{M}}_g$. If $\phi^*(D)$ is weakly positive at any points of $\phi^{-1}(x)$, then $D$ is weakly positive at $x$. \end{Lemma} {\sl Proof.}\quad It is well known that there are a surjective finite morphism $\pi : Y \to \overline{\mathcal{M}}_g$ of normal projective varieties and a stable curve $f : X \to Y$ of genus $g$ such that the induced morphism $Y \to \overline{\mathcal{M}}_g$ by $f : X \to Y$ is $\pi$. Since $\pi_*(\pi^*(D)) = \deg(\pi) D$, by Proposition~\ref{prop:samp:pamp:push}, it is sufficient to show that $\pi^*(D)$ is weakly positive at any points of $\pi^{-1}(x)$. Let $y$ be a point of $\pi^{-1}(x)$. Then, there is a Zariski open neighborhood $U$ of $y$ such that $\rest{f_*(\omega_{X/Y}^{\otimes 3})}{U}$ is free. Thus, \[ \operatorname{Proj}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n \left( \rest{f_*(\omega_{X/Y}^{\otimes 3})}{U} \right) \right) \simeq U \times {\mathbb{P}}^{5g-6}. \] Therefore, there is a morphism $\mu : U \to H_g$ with $\rest{\pi}{U} = \phi \cdot \mu$. By abuse of notation, the induced rational map $Y \dashrightarrow H_g$ is denoted by $\mu$. Let $\nu : Y' \to Y$ be a proper birational morphism of normal projective varieties such that $\mu' = \mu \cdot \nu : Y' \to H_g$ is a morphism and $\nu$ is an isomorphism over $\nu^{-1}(U)$. Then, we have the following diagram: \[ \begin{CD} Y' @>{\nu}>> Y \\ @V{\mu'}VV @VV{\pi}V \\ H_g @>>{\phi}> \overline{\mathcal{M}}_g. \end{CD} \] This diagram is commutative because $\phi \cdot \mu' = \pi \cdot \nu$ over $\nu^{-1}(U)$. Hence, $\nu^*(\pi^*(D)) = {\mu'}^*(\phi^*(D))$. Moreover, $\nu_*(\nu^*(\pi^*(D))) = \pi^*(D)$. Thus, by virtue of Proposition~\ref{prop:samp:pamp:push}, in order to see that $\pi^*(D)$ is weakly positive at $y$, it is sufficient to check that ${\mu'}^*(\phi^*(D))$ is weakly positive at $y \in \nu^{-1}(U)$. By our assumption, $\phi^*(D)$ is weakly positive at $\mu'(y)$ because $\phi(\mu'(y)) = x$. Hence, by Proposition~\ref{prop:samp:pamp:pullback}, ${\mu'}^*(\phi^*(D))$ is weakly positive at $y$. \QED \begin{Theorem} \label{thm:wpos:at:M:g} $(8g+4)\lambda - g \delta_0 - \sum_{i=1}^{[g/2]} 4i(g-i) \delta_i$ is weakly positive over $\mathcal{M}_g$. In particular, it is pseudo-effective, and numerically effective over $\mathcal{M}_g$. \end{Theorem} {\sl Proof.}\quad Let $y$ be a point of $U = H_g \setminus \Delta$. By virtue of \cite{PR}, $\rest{F}{\pi^{-1}(y)}$ is semistable. Thus, by Corollary~\ref{cor:nef:psudo:dis}, $\operatorname{dis}_{Z_g^0/H_g^0}(F^0)$ is weakly positive at $y$. Hence, by Proposition~\ref{prop:wp:codim:2}, so is $\operatorname{dis}_{Z_g/H_g}(F)$ at $y$ because $\operatorname{codim}(H_g \setminus H_g^0) = 2$. Thus, $\operatorname{dis}_{Z_g/H_g}(F)$ is weakly positive over $U = \phi^{-1}(\mathcal{M}_g)$. Therefore, by virtue of Lemma~\ref{lem:criterion:pamp:on:M:g}, we get our theorem. \QED As a corollary, we have the following. \begin{Corollary} \label{cor:sharp:slope:inq} Let $X$ be a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a semistable curve of genus $g \geq 2$ over $Y$. Then, we have the inequality \[ (8g+4) \deg(f_*(\omega_{X/Y})) \geq g \delta_0(X/Y) + \sum_{i=1}^{[g/2]} 4 i (g-i) \delta_i(X/Y), \] where $\delta_i(X/Y)$ is the number of nodes of type $i$ in all singular fibers of $f$. \end{Corollary} \begin{Remark} We don't know the proof of Corollary~\ref{cor:sharp:slope:inq} without using the moduli space $\overline{\mathcal{M}}_g$. Let $\mu : Y \to \overline{\mathcal{M}}_g$ be the morphism induced by $f : X \to Y$. Then, $\mu(Y)$ might pass through $\overline{\mathcal{M}}_g \setminus \phi(H_g^0)$. In this case, analyses of singular fibers only in $X$ seem to be very complicated. \end{Remark} \section{Cones of positive divisors on the moduli space of stable curves} \label{sec:cone:positive:divisor:moduli:spacc:stable:curve} Throughout this section, we assume that $\operatorname{char}(k) = 0$. Let $X$ be a projective variety over $k$ and $\mathcal{C}$ a certain family of complete irreducible curves on $X$. A ${\mathbb{Q}}$-Cartier divisor $D$ on $X$ is said to be {\em numerically effective for $\mathcal{C}$} if $(D \cdot C) \geq 0$ for all $C \in \mathcal{C}$. We set \[ \operatorname{Nef}(X, \mathcal{C}) = \left\{ D \in NS(X) \otimes {\mathbb{Q}} \mid \text{$D$ is numerically effective for $\mathcal{C}$} \right\}. \] Moreover, for subsets $A$ and $B$ in $X$, we denote by $\operatorname{Cur}^A_B$ the set of all irreducible complete curves $C$ on $X$ with $C \subseteq A$ and $C \cap B \not= \emptyset$. Let $g$ be an integer greater than or equal to $2$, $\mathcal{I}_g$ the locus of hyperelliptic curves in $\mathcal{M}_g$, $\overline{\mathcal{I}}_g$ the closure in $\overline{\mathcal{M}}_g$, and $\overline{\mathcal{M}}_g^{one}$ the set of all stable curves with at most one node, i.e., if we use the notation in the previous section, \[ \overline{\mathcal{M}}_g^{one} = \phi\left( H_g \setminus \operatorname{Sing}(\Delta_0 + \cdots + \Delta_{[g/2]}) \right). \] Let us begin with the following lemma. \begin{Lemma} \label{lem:existence:curve} There are complete irreducible curves $C, C_0, \ldots, C_{[g/2]}$ on $\overline{\mathcal{M}}_g$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $C, C_0, \ldots, C_{[g/2]} \in \operatorname{Cur}^{\overline{\mathcal{M}}_g^{one}}_{\mathcal{I}_g}$. \item $C \subset \mathcal{M}_g$. \item $C_i \subset \overline{\mathcal{I}}_g$ for all $0 \leq i \leq [g/2]$. \item For all $0 \leq i, j \leq [g/2]$, $(\delta_i \cdot C_j)$ is positive if $i=j$, and $(\delta_i \cdot C_j) = 0$ if $i \not= j$. \end{enumerate} \end{Lemma} {\sl Proof.}\quad Let $\overline{\mathcal{M}}^s_g$ be Satake's compactification of $\mathcal{M}_g$. Then, $\overline{\mathcal{M}}^s_g$ is projective and $\operatorname{codim}(\overline{\mathcal{M}}^s_g \setminus \mathcal{M}_g) \geq 2$. Pick up one point $P \in \mathcal{I}_g$. If we take general hyperplane sections $H_1, \ldots, H_{3g-4}$ passing through $P$, then $C = H_1 \cap \ldots \cap H_{3g-4}$ is a complete irreducible curve with $C \subseteq \mathcal{M}_g$ and $P \in C$. Applying Proposition~\ref{prop:hyperelliptic:fibration:2} to the case where $a=0$, and contracting all $(-2)$-curves in all singular fibers, we have a stable fibred surface $f_0 : X_0 \to Y_0$ such that $Y_0$ is projective, the generic fiber of $f_0$ is a smooth hyperelliptic curve of genus $g$, $f_0$ is not smooth, and that every singular fiber of $f_0$ is an irreducible nodal curve with one node. Let $\mu_0 : Y_0 \to \overline{\mathcal{M}}_g$ be the induced morphism by $f_0 : X_0 \to Y_0$. Then, $C_0 = \mu(Y_0)$ is our desired curve. Finally, we fix $i$ with $1 \leq i \leq [g/2]$. Using Proposition~\ref{prop:hyperelliptic:fibration}, there is a stable fibred surface $f_i : X_i\to Y_i$ such that $Y_i$ is projective, the generic fiber of $f_i$ is a smooth hyperelliptic curve of genus $g$, $f_i$ is not smooth, and that every singular fiber of $f_i$ is a reducible curve with one node of type $i$. Let $\mu_i : Y_i \to \overline{\mathcal{M}}_g$ be the induced morphism by $f_i : X_i \to Y_i$. If we set $C_i = \mu_i(Y_i)$, then $C_i$ satisfies our requirements. \QED By using curves in Lemma~\ref{lem:existence:curve}, we can show the following proposition. \begin{Proposition} \label{prop:cone:nef:include:special:cone} \[ \operatorname{Nef}\left(\overline{\mathcal{M}}_g, \operatorname{Cur}^{\overline{\mathcal{M}}_g^{one}}_{\mathcal{I}_g}\right) \subseteq \left\{ x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i \ \left| \ \begin{array}{l} x \geq 0, \\ g x + (8g + 4) y_0 \geq 0, \\ i(g-i) x + (2g+1) y_i \geq 0 \quad (1 \leq i \leq [g/2]). \end{array} \right. \right\}. \] \end{Proposition} {\sl Proof.}\quad Let $D = x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i$ be an arbitrary element of $\operatorname{Nef}(\overline{\mathcal{M}}_g, \operatorname{Cur}^{\overline{\mathcal{M}}_g^{one}}_{\mathcal{I}_g})$. Let $C, C_0, \ldots, C_{[g/2]}$ be irreducible complete curves as in Lemma~\ref{lem:existence:curve}. Then, $0 \leq (D \cdot C) = x (\lambda \cdot C)$. Hence $x \geq 0$. To get other inequalities, we need some facts of hyperelliptic fibrations. Details can be found in \cite[\S4, b]{CH}. For $i > 0$, $\Delta_i \cap \overline{\mathcal{I}}_g$ is irreducible. $\Delta_0 \cap \overline{\mathcal{I}}_g$ is however reducible and has $1 + [(g-1)/2]$ irreducible components, say, $\Sigma_0, \Sigma_1, \ldots, \Sigma_{[(g-1)/2]}$. Here a general point of $\Sigma_0$ represents an irreducible curve of one node, and a general point of $\Sigma_i$ ($i > 0$) represents a stable curve consisting of a curve of genus $i$ and a curve of genus $g - i - 1$ joined at two points. The class of $\Sigma_i$ in $\operatorname{Pic}(\overline{\mathcal{I}}_g) \otimes {\mathbb{Q}}$ is denoted by $\sigma_i$, and by abuse of notation, $\rest{\delta_i}{\overline{\mathcal{I}}_g}$ is denoted by $\delta_i$. Further, $\rest{\lambda}{\overline{\mathcal{I}}_g}$ is denoted by $\lambda$. Then, by virtue of \cite[Proposition~(4.7)]{CH}, \[ \delta_0 = \sigma_0 + 2 \left(\sigma_1 + \cdots + \sigma_{[(g-1)/2]} \right) \] and \[ (8g+4) \lambda = g \sigma_0 + \sum_{j=1}^{[(g-1)/2]} 2(j+1)(g-j) \sigma_j + \sum_{i=1}^{[g/2]} 4 i (g-i) \delta_i. \] Let us consider $D$ as a divisor on $\overline{\mathcal{I}}_g$. Using the above relations between $\lambda$, $\delta_i$'s and $\sigma_j$'s, we have \[ D = \left( \frac{g}{8g + 4} x + y_0 \right) \sigma_0 + 2 \sum_{j=1}^{[(g-1)/2]} \left( \frac{(j+1)(g-j)}{8g+4} x + y_0 \right) \sigma_j + \sum_{i=1}^{[g/2]} \left( \frac{i(g-i)}{2g+1} x + y_i \right)\delta_i. \] Note that $C_i \cap \Sigma_j = \emptyset$ for all $0 \leq i \leq [g/2]$ and $1 \leq j \leq [(g-1)/2]$ because $C_i \subset \overline{\mathcal{M}}_g^{one}$. Thus, considering $(D \cdot C_i)$, we have the remaining inequalities. \QED \begin{Corollary} \label{cor:nef:cone:equal} If $\mathcal{C}$ is a set of complete irreducible curves on $\overline{\mathcal{M}}_g$ with $\operatorname{Cur}^{\overline{\mathcal{M}}_g^{one}}_{\mathcal{I}_g} \subseteq \mathcal{C} \subseteq \operatorname{Cur}^{\overline{\mathcal{M}}_g}_{\mathcal{M}_g}$, then \[ \operatorname{Nef}\left(\overline{\mathcal{M}}_g, \mathcal{C} \right) = \left\{ x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i\ \left| \ \begin{array}{l} x \geq 0, \\ g x + (8g + 4) y_0 \geq 0, \\ i(g-i) x + (2g+1) y_i \geq 0 \quad (1 \leq i \leq [g/2]). \end{array} \right. \right\}. \] \end{Corollary} {\sl Proof.}\quad Since $\operatorname{Nef}\left(\overline{\mathcal{M}}_g, \mathcal{C} \right) \subseteq \operatorname{Nef}\left(\overline{\mathcal{M}}_g, \operatorname{Cur}^{\overline{\mathcal{M}}_g^{one}}_{\mathcal{I}_g}\right)$, the direction ``$\subseteq$'' is a consequence of Proposition~\ref{prop:cone:nef:include:special:cone}. Conversely, we assume that $D = x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i$ satisfies \[ \begin{cases} x \geq 0, \\ g x + (8g + 4) y_0 \geq 0, \\ i(g-i) x + (2g+1) y_i \geq 0 \quad (1 \leq i \leq [g/2]). \end{cases} \] Then, since \begin{multline*} D = \frac{x}{8g+4} \left( (8g+4)\lambda - g \delta_0 - \sum_{i=1}^{[g/2]} 4i(g-i) \delta_i \right) \\ + \left(y_0 + \frac{g}{8g+4} x\right)\delta_0 + \sum_{i=1}^{[g/2]} \left(y_i + \frac{i(g-i)}{2g+1} x \right) \delta_i \end{multline*} and $\mathcal{C} \subseteq \operatorname{Cur}^{\overline{\mathcal{M}}_g}_{\mathcal{M}_g}$, we can see that $D$ is numerically effective for $\mathcal{C}$ by using Theorem~\ref{thm:wpos:at:M:g}. \QED In the same way, we can see the following. \begin{Corollary} \label{cor:wp:cone:equal} If we set \[ \operatorname{WP}(\overline{\mathcal{M}}_g; \mathcal{M}_g) = \{ D \in \operatorname{Pic}(\overline{\mathcal{M}}_g) \otimes {\mathbb{Q}} \mid \text{$D$ is weakly positive over $\mathcal{M}_g$} \}, \] then \[ \operatorname{WP}(\overline{\mathcal{M}}_g; \mathcal{M}_g) = \left\{ x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i \ \left| \ \begin{array}{l} x \geq 0, \\ g x + (8g + 4) y_0 \geq 0, \\ i(g-i) x + (2g+1) y_i \geq 0 \quad (1 \leq i \leq [g/2]). \end{array} \right. \right\}. \] \end{Corollary} {\sl Proof.}\quad Note that \[ \operatorname{WP}(\overline{\mathcal{M}}_g; \mathcal{M}_g) \subseteq \operatorname{Nef}\left(\overline{\mathcal{M}}_g, \operatorname{Cur}^{\overline{\mathcal{M}}_g}_{\mathcal{M}_g}\right) \] and that $(8g+4)\lambda - g \delta_0 - \sum_{i=1}^{[g/2]} 4i(g-i)$ and $\delta_i$'s are weakly positive over $\mathcal{M}_g$. \QED \begin{Remark} In general, over an open set, weak positivity is stronger than numerical effectivity. Corollary~\ref{cor:nef:cone:equal} and Corollary~\ref{cor:wp:cone:equal} however say us that, on the moduli space of stable curves $\overline{\mathcal{M}}_g$, weak positivity over $\mathcal{M}_g$ coincides with numerical effectivity over $\mathcal{M}_g$. \end{Remark} \section{Effective Bogomolov's conjecture over function fields} \label{sec:bogo:conj} Let $X$ be a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$ over $Y$. Let $K$ be the function field of $Y$, $\overline{K}$ the algebraic closure of $K$, and $C$ the generic fiber of $f$. Let $j : C(\overline{K}) \to \operatorname{Jac}(C)(\overline{K})$ be the map given by $j(x) = (2g-2)x - \omega_C$ and $\Vert\ \Vert_{NT}$ the semi-norm arising from the Neron-Tate height pairing on $\operatorname{Jac}(C)(\overline{K})$. We set \[ B_C(P;r) = \left\{ x \in C(\overline{K}) \mid \Vert j(x) - P \Vert_{NT} \leq r \right\} \] for $P \in \operatorname{Jac}(C)(\overline{K})$ and $r \geq 0$, and \[ r_C(P) = \begin{cases} -\infty & \mbox{if $\#\left(B_C(P;0)\right) = \infty$}, \\ & \\ \sup \left\{ r \geq 0 \mid \#\left(B_C(P;r)\right) < \infty \right\} & \mbox{otherwise}. \end{cases} \] An effective version of Bogomolov's conjecture claims the following. \begin{Conjecture}[Effective Bogomolov's conjecture] \label{conj:effective:bogomolov} If $f$ is non-isotrivial, then there is an effectively calculated positive number $r_0$ with \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq r_0. \] \end{Conjecture} Recently, Ullmo \cite{Ul} proved that $r_C(P) > 0$ for all $P \in \operatorname{Jac}(C)(\overline{K})$ for the case where $K$ is a number field. As far as we know, the problem to find an effectively calculated $r_0$ is still open. The meaning of ``effectively calculated'' is that a concrete algorithm or formula to find $r_0$ is required. Here we need a rather technical condition coming from calculations of green functions along singular fibers. Let $\bar{f} : \overline{X} \to Y$ be the stable model of $f : X \to Y$. Let $X_y$ (resp. $\overline{X}_y$) be the singular fiber of $f$ (resp. $\bar{f}$) over $y \in Y$, and $S_y$ the set of nodes $P$ on $\overline{X}_y$ such that $P$ is not an intersection of two irreducible components of $\overline{X}_y$, i.e., a singularity of an irreducible component. Let $\pi : Z_y \to \overline{X}_y$ be the partial normalization of $\overline{X}_y$ at each node in $S_y$. We say $X_y$ is a {\em tree of stable components} if the dual graph of $Z_y$ is a tree graph. In other words, every node of type $0$ on $\overline{X}_y$ is a singularity of an irreducible component of $\overline{X}_y$. As an application of Corollary~\ref{cor:sharp:slope:inq}, we get the following solution of the above conjecture, which is a generalization of \cite[Theorem~5.2]{Mo5}. \begin{Theorem}[$\operatorname{char}(k) = 0$] \label{thm:bogomolov:function:field} If $f$ is not smooth and every singular fiber of $f$ is a tree of stable components, then \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq \sqrt{\frac{(g-1)^2}{g(2g+1)}\left( \frac{g-1}{3}\delta_0(X/Y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i)\delta_i(X/Y) \right)}. \] \end{Theorem} Before starting the proof of Theorem~\ref{thm:bogomolov:function:field}, let us recall several facts of green functions on a metrized graph. For details of metrized graphs, see \cite{Zh}. Let $G$ be a connected metrized graph and $D$ an ${\mathbb{R}}$-divisor on $G$. If $\deg(D) \not= -2$, then there are a unique measure $\mu_{(G,D)}$ on $G$ and a unique function $g_{(G,D)}$ on $G \times G$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item ${\displaystyle \int_{G} \mu_{(G,D)} = 1}$. \item $g_{(G,D)}(x, y)$ is symmetric and continuous on $G \times G$. \item For a fixed $x \in G$, $\Delta_y(g_{(G,D)}(x, y)) = \delta_x - \mu_{(G,D)}$. \item For a fixed $x \in G$, ${\displaystyle \int_G g_{(G,D)}(x, y) \mu_{(G,D)}(y) = 0}$. \item $g_{(G,D)}(D, y) + g_{(G,D)}(y, y)$ is a constant for all $y \in G$. \end{enumerate} The constant $g_{(G,D)}(D, y) + g_{(G,D)}(y, y)$ is denoted by $c(G, D)$. Further we set \[ \epsilon(G, D) = 2\deg(D)c(G, D) - g_{(G,D)}(D, D). \] We would like to calculate the invariant $\epsilon(G,D)$ for the metrized graph $G$ with the polarization $D$. First of all, let us see two examples, which will be elementary pieces for our calculations. \begin{Example}[{cf. \cite[Lemma~3.2]{Mo3}}] \label{exam:e:for:circle} Let $C$ be a circle of length $l$ and $O$ a point on $C$. Then, \[ g_{(C,0)}(O,O) = \frac{l}{12} \quad\text{and}\quad \epsilon(C, 0) = 0. \] \end{Example} \begin{Example}[{cf. \cite[Lemma~4.4]{Mo5}}] \label{exam:e:for:1:segment} Let $G$ be a segment of length $l$, and $P$ and $Q$ terminal points of $G$. Let $a$ and $b$ be real numbers with $a + b \not= 0$, and $D$ an ${\mathbb{R}}$-divisor on $G$ given by $D = (2a-1)P + (2b-1)Q$. Then, \[ \epsilon(G, D) = \left(\frac{4ab}{a+b} - 1\right)l,\quad g_{(G,D)}(P,P) = \frac{b^2}{(a+b)^2}l \quad\text{and}\quad g_{(G,D)}(Q,Q) = \frac{a^2}{(a+b)^2}l. \] \end{Example} Let $G_1$ and $G_2$ be metrized graphs. Fix points $x_1 \in G_1$ and $x_2 \in G_2$. The one point sum $G_1 \vee G_2$ with respect to $x_1$ and $x_2$, defined by $G_1 \times \{x_2\} \cup \{x_1\} \times G_2$ in $G_1 \times G_2$, is a metrized graph obtained by joining $x_1\in G_1$ and $x_2 \in G_2$. The joining point, which is $\{x_1\}\times\{x_2\}$ in $G_1 \times G_2$, is denoted by $j(G_1 \vee G_2)$. Any ${\mathbb{R}}$-divisor on $G_i$ ($i=1,2$) can be viewed as an ${\mathbb{R}}$-divisor on $G_1 \vee G_2$. Then, our basic tool for our calculations is the following. \begin{Proposition}[{cf. \cite[Proposition~4.2]{Mo5}}] \label{prop:e:for:join:graph} Let $G_1$ and $G_2$ be connected metrized graphs, and $D_1$ and $D_2$ ${\mathbb{R}}$-divisors on $G_1$ and $G_2$ respectively with $\deg(D_i) \not= -2$ \textup{(}$i=1,2$\textup{)}. Let $G = G_1 \vee G_2$, $O = j(G_1\vee G_2)$, and $D = D_1 + D_2$ on $G_1 \vee G_2$. If $\deg(D_1 + D_2) \not= -2$, then we have the following formulae, where $d_i = \deg(D_i)$ \textup{(}$i=1, 2$\textup{)} and $r_{G_2}(O,P)$ is the resistance between $O$ and $P$ on $G_2$. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item If $P \in G_2$, then {\allowdisplaybreaks \begin{align*} g_{(G, D)}(P,P) & = \frac{d_1}{d_1+d_2+2} r_{G_2}(O,P) +\frac{d_2 + 2}{d_1+d_2+2} g_{(G_2, D_2)}(P,P) \\ & \quad -\frac{d_1(d_2+2)} {(d_1+d_2+2)^2}g_{(G_2,D_2)}(O,O) +\frac{(d_1 +2)^2}{(d_1 + d_2 + 2)^2}g_{(G_1, D_1)}(O,O). \end{align*}} \item \yes \[ \epsilon(G, D) = \epsilon(G_1, D_1) + \epsilon(G_2, D_2) + \frac{2 d_2(d_1 + 2)g_{(G_1, D_1)}(O,O) + 2 d_1(d_2 + 2)g_{(G_2, D_2)}(O, O)}{d_1 + d_2 + 2}. \] \else \begin{multline*} \epsilon(G, D) = \epsilon(G_1, D_1) + \epsilon(G_2, D_2) \\ + \frac{2 d_2(d_1 + 2)g_{(G_1, D_1)}(O,O) + 2 d_1(d_2 + 2)g_{(G_2, D_2)}(O, O)}{d_1 + d_2 + 2}. \end{multline*} \fi \end{enumerate} \end{Proposition} Combining the above proposition and Example~\ref{exam:e:for:circle}, we have the following. \begin{Corollary} \label{cor:e:for:join:graph:circle} Let $G$ be a connected metrized graph and $D$ an ${\mathbb{R}}$-divisor on $G$ with $\deg(D) \not= -2$. Let $C$ be a circle of length $l$. Then, \[ \epsilon(G \vee C, D) = \epsilon(G, D) + \frac{\deg(D)}{3(\deg D + 2)} l. \] \end{Corollary} Let $G$ be a connected metrized graph. We assume that $G$ is a tree, i.e., there is no loop in $G$. Let $\operatorname{Vert}(G)$ (resp. $\operatorname{Ed}(G)$) be the set of vetexes (resp. edges) of $G$. For a function $\alpha : \operatorname{Vert}(G) \to {\mathbb{R}}$, we define the divisor $D(\alpha)$ on $G$ to be \[ D(\alpha) = \sum_{x \in \operatorname{Vert}(G)} (2\alpha(x) - 2 + v(x)) x, \] where $v(x)$ is the number of branches starting from $x$. It is easy to see that \[ \deg(D(\alpha)) + 2 = 2 \sum_{x \in \operatorname{Vert}(G)} \alpha(x). \] To give an exact formula for $\epsilon(G, D(\alpha))$, we need to introduce the following notation. Let $e$ be an edge of $G$, $P$ and $Q$ terminal points of $e$, and $e^{\circ} = e \setminus \{ P, Q \}$. Since $G$ is a connected tree, there are two connected sub-graphs $G'_e$ and $G''_e$ such that $G \setminus e^{\circ} = G'_e \coprod G''_e$. Then, we have the following. \begin{Proposition} \label{prop:cal:e:G:D:tree} With the same notation as above, if $\alpha(x) \geq 0$ for all $x \in \operatorname{Vert}(G)$ and $\sum_{x \in \operatorname{Vert}(G)} \alpha(x) \not= 0$, then \[ \epsilon(G, D(\alpha)) = \sum_{e \in \operatorname{Ed}(G)} \left( \frac{4 \left( \sum_{x \in \operatorname{Vert}(G'_e)} \alpha(x) \right) \left( \sum_{x \in \operatorname{Vert}(G''_e)} \alpha(x) \right)} {\sum_{x \in \operatorname{Vert}(G)} \alpha(x) } - 1 \right) l(e), \] where $l(e)$ is the length of $e$. \end{Proposition} {\sl Proof.}\quad For a positive number $t$, we set $\alpha_t(x) = \alpha(x) + t$. Then, it is easy to see that \[ \lim_{t \downarrow 0} \epsilon(G, D(\alpha_t)) = \epsilon(G, D(\alpha)). \] Thus, in order to prove our proposition, we may assume that $\alpha(x) > 0$ for all $x \in \operatorname{Vert}(G)$. We fix $P \in \operatorname{Vert}(G)$. For $e \in \operatorname{Ed}(G)$, we denote by $G_{P,e}$ the connected component of $G \setminus e^{\circ}$ not containing $P$, i.e., if $P \not\in G'_e$, then $G_{P,e} = G'_e$; otherwise, $G_{P,e} = G''_e$. With this notation, let us consider the following claim. \begin{Claim} \label{claim:cal:g:G:D:tree} \[ g_{(G, D(\alpha))}(P, P) = \sum_{e \in \operatorname{Ed}(G)} \frac{\left(\sum_{x \in \operatorname{Vert}(G_{P, e})} \alpha(x)\right)^2} {\left(\sum_{x \in \operatorname{Vert}(G)} \alpha(x)\right)^2} l(e). \] \end{Claim} We prove this claim by induction on $\#(\operatorname{Ed}(G))$. If $\#(\operatorname{Ed}(G)) = 0, 1$, then our assertion is obvious by Example~\ref{exam:e:for:1:segment}. Thus, we may assume that $\#(\operatorname{Ed}(G)) \geq 2$. First, we suppose that $P$ is not a terminal point. Let $G'$ be one branch starting from $P$, and $G''$ a connected sub-graph such that $G' \cup G'' = G$ and $G' \cap G'' = \{ P \}$. We define $\alpha' : \operatorname{Vert}(G') \to {\mathbb{R}}$ and $\alpha'' : \operatorname{Vert}(G'') \to {\mathbb{R}}$ by \[ \alpha'(x) = \begin{cases} 1 & \text{if $x = P$} \\ \alpha(x) & \text{otherwise} \end{cases} \] and $\alpha'' = \rest{\alpha}{\operatorname{Vert}(G'')}$. Then, we have $G = G' \vee G''$ and $D(\alpha) = D(\alpha') + D(\alpha'')$. Thus, using (1) of Proposition~\ref{prop:e:for:join:graph} and hypothesis of induction, we can easily see our claim. Next we suppose that $P$ is a terminal point. Pick up $e \in \operatorname{Ed}(G)$ such that $P$ is a terminal of $e$. Let $O$ be another terminal of $e$. We set $G' = e$ and $G'' = (G \setminus e) \cup \{ O \}$. Moreover, we define $\alpha' : \operatorname{Vert}(G') = \{P, O\} \to {\mathbb{R}}$ and $\alpha'' : \operatorname{Vert}(G'') \to {\mathbb{R}}$ by $\alpha'(P) = \alpha(P)$, $\alpha'(O) = 1$ and $\alpha'' = \rest{\alpha}{\operatorname{Vert}(G'')}$. Then, $G = G' \vee G''$ and $D(\alpha) = D(\alpha') + D(\alpha'')$. Thus, using (1) of Proposition~\ref{prop:e:for:join:graph}, Example~\ref{exam:e:for:1:segment} and hypothesis of induction, we can see our claim after easy calculations. \medskip Let us go back to the proof of Proposition~\ref{prop:cal:e:G:D:tree}. We prove it by induction on $\#(\operatorname{Ed}(G))$. If $\#(\operatorname{Ed}(G)) = 0, 1$, then our assertion comes from Example~\ref{exam:e:for:1:segment}. Thus, we may assume that $\#(\operatorname{Ed}(G)) \geq 2$. Let us pick up a terminal edge $e$ of $G$. Let $\{ O, P \}$ be terminals of $e$ such that $P$ gives a terminal of $G$. We set $G_1 = e$ and $G_2 = (G \setminus e) \cup \{ O \}$. Moreover, we define $\alpha_1 : \{ O, P \} = \operatorname{Vert}(G_1) \to {\mathbb{R}}$ and $\alpha_2 : \operatorname{Vert}(G_2) \to {\mathbb{R}}$ by $\alpha_1(O) = 1$, $\alpha_1(P) = \alpha(P)$, and $\alpha_2 = \rest{\alpha}{\operatorname{Vert}(G_2)}$. Then, $G = G_1 \vee G_2$ and $D(\alpha) = D(\alpha_1) + D(\alpha_2)$. Thus, if we set \[ \begin{cases} A = \sum_{x \in \operatorname{Vert}(G)} \alpha(x), \\ a = \alpha(P), \\ A_{e'} = \sum_{x \in \operatorname{Vert}(G_{O, e'})} \alpha(x) & \text{for $e' \in \operatorname{Vert}(G) \setminus \{ e \}$}, \end{cases} \] then, by (2) of Proposition~\ref{prop:e:for:join:graph}, Example~\ref{exam:e:for:1:segment}, Claim~\ref{claim:cal:g:G:D:tree} and hypothesis of induction, we have {\allowdisplaybreaks \begin{align*} \epsilon(G, D(\alpha)) & = \left(\frac{4a}{a+1} - 1 \right) l(e) + \sum_{e' \in \operatorname{Vert}(G) \setminus \{ e \}} \left(\frac{4 A_{e'}(A - a - A_{e'})}{A - a} - 1 \right) l(e') \\ & \qquad + \frac{4(A - a - 1)(a+1)}{A} \frac{a^2}{(a+1)^2} l(e) + \frac{4a(A-a)}{A} \sum_{e' \in \operatorname{Vert}(G) \setminus \{ e \}} \frac{A_{e'}^2}{(A - a)^2} l(e') \\ & = \left( \frac{4a(A-a)}{A} - 1 \right) l(e) + \sum_{e' \in \operatorname{Vert}(G) \setminus \{ e \}} \left( \frac{4 A_{e'} (A - A_{e'})}{A} - 1 \right) l(e'). \end{align*}} Therefore, we get our proposition. \QED \begin{Corollary}[$\operatorname{char}(k) \geq 0$] \label{cor:e:for:semistable:chain} Let $X$ be a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$ over $Y$. Let $X_y$ be the singular fiber of $f$ over $y \in Y$, and $X_y = C_1 + \cdots + C_n$ the irreducible decomposition of $X_y$. Let $G_y$ be the metrized graph given by the configuration of $X_y$, $v_i$ the vertex of $G_y$ corresponding to $C_i$, and $\omega_y$ the divisor on $G_y$ defined by $\omega_y = \sum_i (\omega_{X/Y} \cdot C_i) v_i$. If $X_y$ is a tree of stable components, then \[ \epsilon(G_y, \omega_y) = \frac{g-1}{3g} \delta_{0}(X_y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} \left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i}(X_y). \] \end{Corollary} {\sl Proof.}\quad Let $\bar{f} : \overline{X} \to Y$ be the stable model of $f : X \to Y$, and $S_y$ the set of nodes $P$ on $\overline{X}_y$ such that $P$ is a singularity of an irreducible component. Let $\pi : Z_y \to \overline{X}_y$ be the partial normalization of $\overline{X}_y$ at each node in $S_y$. Let $\overline{G}_y$ be the dual graph of $Z_y$. Let $l_1, \cdots, l_{r}$ be circles in $G_y$ corresponding to nodes in $S_y$. Then, $G_y = \overline{G}_y \vee l_1 \vee \cdots \vee l_{r}$. Moreover, if $g_i$ is the arithmetic genus of $C_i$ and $\alpha : \operatorname{Vert}(G_y) \to {\mathbb{R}}$ is given by $\alpha(v_i) = g_i$, then $\omega_y = D(\alpha)$. Here, by virtue of Proposition~\ref{prop:cal:e:G:D:tree}, \[ \epsilon(\overline{G}_y, \omega_y) = \sum_{i=1}^{\left[\frac{g}{2}\right]} \left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i}(X_y). \] Therefore, it follows from Corollary~\ref{cor:e:for:join:graph:circle} that \[ \epsilon(G_y, \omega_y) = \frac{g-1}{3g} \delta_{0}(X_y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} \left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i}(X_y). \] \QED Let us start the proof of Theorem~\ref{thm:bogomolov:function:field}. First of all, note the following fact (cf. \cite[Theorem 5.6]{Zh}, \cite[Corollary 2.3]{Mo3} or \cite[Theorem 2.1]{Mo4}). If $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a > 0$, then \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq \sqrt{(g-1)(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}, \] where $(\ \cdot \ )_a$ is the admissible pairing. By the definition of admissible pairing, we can see \[ \left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a \right)_a = \left(\omega_{X/Y} \cdot \omega_{X/Y} \right) - \sum_{y \in Y} \epsilon(G_y, \omega_y). \] On the other hand, by Corollary~\ref{cor:sharp:slope:inq}, we have \[ (8g + 4) \deg(f_*(\omega_{X/Y})) \geq g \delta_0(X/Y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i(X/Y). \] Thus, using Noether formula, the above inequality implies \[ (\omega_{X/Y} \cdot \omega_{X/Y}) \geq \frac{g-1}{2g+1} \delta_0(X/Y) + \sum_{i=1}^{\left[\frac{g}{2}\right]} \left(\frac{12i(g-i)}{2g+1} - 1 \right) \delta_i(X/Y). \] Therefore, we have our theorem by Corollary~\ref{cor:e:for:semistable:chain}. \QED Moreover, using Ullmo's result \cite{Ul} and Proposition~\ref{prop:cal:e:G:D:tree}, we have the following. \begin{Corollary} Let $K$ be a number field, $O_{K}$ the ring of integers, and $f : X \to \operatorname{Spec}(O_{K})$ a regular semistable arithmetic surface of genus $g \geq 2$ over $O_{K}$. Let $S$ be the subset of $\operatorname{Spec}(O_{K}) \setminus \{ 0 \}$ such that $P \in S$ if and only if the stable model of the geometric fiber $X_{\bar{P}}$ at $P$ is a tree of stable components. Then, we have \[ \left( \omega^{Ar}_{X/O_{K}} \cdot \omega^{Ar}_{X/O_{K}} \right) > \sum_{P \in S} \left\{ \frac{g-1}{3g}\delta_{0}(X_{\bar{P}}) + \sum_{i=1}^{[g/2]} \left(\frac{4i(g-i)}{g} - 1 \right)\delta_{i}(X_{\bar{P}}) \right\} \log\#(O_{K}/P). \] \end{Corollary} \section{Generalization to higher dimensional fibrations} In this section, we consider a generalization of Corollary~\ref{cor:nef:psudo:dis} to higher dimensional fibrations. First of all, let us recall the definition of semistability of vector bundles. Let $V$ be a smooth projective variety of dimension $d$ over $k$, and $H_1, \ldots, H_{d-1}$ ample line bundles on $V$. A vector bundle $E$ on $V$ is said to be semistable with respect to $H_1, \ldots, H_{d-1}$ if, for any non-zero subsheaves $G$ of $E$, \[ \frac{(c_1(G) \cdot c_1(H_1) \cdots c_1(H_{d-1}))}{\operatorname{rk} G} \leq \frac{(c_1(E) \cdot c_1(H_1) \cdots c_1(H_{d-1}))}{\operatorname{rk} E}. \] \bigskip Let $f : X \to Y$ be a surjective and projective morphism of quasi-projective varieties over $k$ with $\dim f = d \geq 1$. Let $H_1, \ldots, H_{d-1}$ be line bundles on $X$ and $E$ a vector bundle on $X$ of rank $r$. Then, $\left( (2r c_2(E) - (r-1)c_1(E)^2) \cdot c_1(H_1) \cdots c_1(H_{d-1}) \right) \cap [X]$ is a cycle of dimension $\dim Y - 1$ on $X$. Thus, $f_*\left( \left( (2r c_2(E) - (r-1)c_1(E)^2) \cdot c_1(H_1) \cdots c_1(H_{d-1}) \right) \cap [X]\right)$, denoted by $\operatorname{dis}_{X/Y}(E; H_1, \ldots, H_{d-1})$, is a divisor on $Y$. Then, we have the following theorem. \begin{Theorem}[$\operatorname{char}(k) = 0$] \label{thm:nef:psudo:dis:higher} We assume that $Y$ is smooth over $k$ and $H_1, \ldots, H_{d-1}$ are ample. If $y$ is a point of $Y$, $f$ is smooth over $y$, and $E_{\bar{y}}$ is semistable with respect to $(H_1)_{\bar{y}}, \ldots, (H_{d-1})_{\bar{y}}$ on each connected component of the geometric fiber $X_{\bar{y}}$ over $y$, then the discriminant divisor $\operatorname{dis}_{X/Y}(E; H_1, \ldots, H_{d-1})$ is weakly positive at $y$. \end{Theorem} {\sl Proof.}\quad We prove this theorem by induction on $d$. If $d = 1$, then our assertion is nothing more than Corollary~\ref{cor:nef:psudo:dis}. So we assume $d \geq 2$. We choose a sufficiently large integer $n$ so that $H_{d-1}^{\otimes n}$ is very ample, i.e., there is an embedding $\iota : X \hookrightarrow {\mathbb{P}}^N$ with $H_{d-1}^{\otimes n} \simeq \iota^*({\mathcal{O}}_{{\mathbb{P}}^N}(1))$. By Bertini's theorem, we can find a general member $\Gamma \in |{\mathcal{O}}_{{\mathbb{P}}^N}(1)|$ such that $X \cap \Gamma$ is integral and $f^{-1}(y) \cap \Gamma$ is smooth. We set $Z = X \cap \Gamma$ and $g = \rest{f}{Z} : Z \to Y$. Since $n$ is sufficiently large, $g^{-1}(y) \in \left|\rest{H_{d-1}^{\otimes n}}{f^{-1}(y)}\right|$ and $g^{-1}(y)$ is smooth, by virtue of \cite[Theorem 3.1]{Mo2}, $\rest{E}{Z_{\bar{y}}}$ is semistable with respect to $\rest{H_{1}}{Z_{\bar{y}}}, \ldots, \rest{H_{d-2}}{Z_{\bar{y}}}$ on each connected component of $Z_{\bar{y}}$. Therefore, by hypothesis of induction, $\operatorname{dis}_{Z/Y}(\rest{E}{Z}; \rest{H_1}{Z}, \ldots, \rest{H_{d-2}}{Z})$ is weakly positive at $y$. On the other hand, we have \begin{align*} \operatorname{dis}_{Z/Y}(\rest{E}{Z}; \rest{H_1}{Z}, \ldots, \rest{H_{d-2}}{Z}) & = \operatorname{dis}_{X/Y}(E; H_1, \ldots, H_{d-2}, H_{d-1}^{\otimes n}) \\ & = n \operatorname{dis}_{X/Y}(E; H_1, \ldots, H_{d-1}). \end{align*} Hence, $\operatorname{dis}_{X/Y}(E; H_1, \ldots, H_{d-1})$ is weakly positive at $y$. \QED \section{Relative Bogomolov's inequality in positive characteristic} \label{sec:relative:bogomolov:inequality:positive:characteristic} In this section, we will consider a similar result of Corollary~\ref{cor:nef:psudo:dis} in positive characteristic. The crucial point of the proof of Corollary~\ref{cor:nef:psudo:dis} is the semistability of tensor products of semistable vector bundles, which was studied by a lot of authors \cite{Gi1}, \cite{Ha1}, \cite{Mi}, \cite{Ma1} and etc. This however does not hold in positive characteristic \cite{Gi0}, so that we will introduce the strong semistability of vector bundles. \medskip Let $C$ be a smooth projective curve over $k$. For a vector bundle $F$ on $C$, we set $\mu(F) = \deg(F)/ \operatorname{rk} (F)$, which is called the {\em slope of $F$}. A vector bundle $E$ on $C$ is said to be {\em semistable} (resp. {\em stable}) if, for any proper subbundles $F$ of $E$, $\mu(F) \leq \mu(E)$ (resp. $\mu(F) < \mu(E)$). Moreover, $E$ is said to be {\em strongly semistable} if, for any finite morphisms $\pi : C' \to C$ of smooth projective curves over $k$, $\pi^*(E)$ is semistable. Then, we have the following elementary properties of semistable or strongly semistable vector bundles. \begin{Proposition}[$\operatorname{char}(k) \geq 0$] \label{prop:elem:prop:semistable} Let $E$ be a vector bundle of rank $r$ on $C$. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item \label{enum:prop:elem:prop:semistable:separable} Let $\pi : C' \to C$ be a finite separable morphism of smooth projective curves over $k$. If $E$ is semistable, then so is $\pi^*(E)$. \item \label{enum:prop:elem:prop:semistable:char:zero} Under the assumption of $\operatorname{char}(k) = 0$, $E$ is semistable if and only if $E$ is strongly semistable. \item \label{enum:prop:elem:prop:semistable:nef} Let $f : P = \operatorname{Proj}\left( \bigoplus_{m=0}^{\infty} \operatorname{Sym}^m(E) \right) \to C$ be the projective bundle of $E$ and ${\mathcal{O}}_P(1)$ the tautological line bundle on $P$. Then, $E$ is strongly semistable if and only if $\omega_{P/C}^{\otimes -1} = {\mathcal{O}}_P(r) \otimes f^*(\det E)^{\otimes -1}$ is numerically effective. \end{enumerate} \end{Proposition} {\sl Proof.}\quad (1) is nothing more than \cite[Lemma~1.1]{Gi1} and (2) is a consequence of (1). \medskip (3) First we assume that $E$ is strongly semistable. Let $Z$ be any irreducible curves on $P$. If $Z$ is contained in a fiber, then obviously $(\omega_{P/C}^{\otimes -1} \cdot Z) > 0$. So we may assume that $Z$ is not contained in any fibers. Let $C'$ be the normalization of $Z$ and $\pi : C' \to Z \to C$ the induced morphism. Let $E' = \pi^*(E)$, $f' : P' = \operatorname{Proj}\left( \bigoplus_{m=0}^{\infty} \operatorname{Sym}^m(E') \right) \to C'$ the projective bundle of $E'$, and ${\mathcal{O}}_{P'}(1)$ the tautological line bundle on $P'$. Then we have the following commutative diagram. \[ \begin{CD} P @<{\pi'}<< P' \\ @V{f}VV @VV{f'}V \\ C @<<{\pi}< C' \end{CD} \] By our construction, there is a section $Z'$ of $f'$ such that $\pi'(Z') = Z$. We set $Q' = \rest{{\mathcal{O}}_{P'}(1)}{Z'}$. Then, there is a surjective homomorphism $E' \to Q'$. Since $E'$ is semistable, we have $\mu(E') \leq \deg(Q')$, which means that $(\omega_{P'/C'}^{\otimes -1} \cdot Z') \geq 0$. Here, $\omega_{P'/C'} = {\pi'}^*(\omega_{P/C})$. Thus, we get $(\omega_{P/C}^{\otimes -1} \cdot Z) \geq 0$. Conversely, we assume that $\omega_{P/C}^{\otimes -1}$ is numerically effective on $P$. Let $\pi : C' \to C$ be a finite morphism of smooth projective curves over $k$. We set $f' : P' \to C'$ and $\pi' : P' \to P$ as before. Then, $\omega_{P'/C'}^{\otimes -1} = {\pi'}^*(\omega_{P/C}^{\otimes -1})$ is numerically effective on $P'$. Let $Q$ be a quotient vector bundle of $E' = \pi^*(E)$ with $s = \operatorname{rk} Q$. The projective bundle $\operatorname{Proj}\left( \bigoplus_{m=0}^{\infty} \operatorname{Sym}^m(Q) \right) \to C'$ gives a subvariety $V'$ of $P'$ with $\deg(Q) = ({\mathcal{O}}_{P'}(1)^s \cdot V')$ and $({\mathcal{O}}_{P'}(1)^{s-1} \cdot F' \cdot V') = 1$, where $F'$ is a fiber of $f'$. Since $\omega_{P'/C'}^{\otimes -1}$ is numerically effective, \[ 0 \leq \left( ({\mathcal{O}}_{P'}(r) \otimes {f'}^*(\det E')^{-1})^s \cdot V' \right) = r^{s-1}(r \deg(Q) - s \deg(E')). \] Thus, $\mu(E') \leq \mu(Q)$. \QED First, let us consider symmetric products of strongly semistable vector bundles. \begin{Theorem}[$\operatorname{char}(k) \geq 0$] \label{thm:semistable:sym} If $E$ is a strongly semistable vector bundle on $C$, then so is $\operatorname{Sym}^n(E)$ for all $n \geq 0$. \end{Theorem} {\sl Proof.}\quad Taking a finite covering of $C$, we may assume that $\deg(E)$ is divisible by $\operatorname{rk} E$. Let $\theta$ be a line bundle on $C$ with $\deg(\theta) = 1$. If we set $E_0 = E \otimes \theta^{\otimes -\frac{\deg(E)}{\operatorname{rk} E}}$, then $\deg(E_0) = 0$ and $\operatorname{Sym}^n(E_0) = \operatorname{Sym}^n(E) \otimes \theta^{\otimes -\frac{n\deg(E)}{\operatorname{rk} E}}$. Thus, to prove our theorem, we may assume $\deg(E) = 0$. We assume that $\operatorname{Sym}^n(E)$ is not strongly semistable for some $n \geq 2$. By replacing $C$ by a finite covering of $C$, we may assume that $\operatorname{Sym}^n(E)$ is not semistable. Let $f : P = \operatorname{Proj}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(E) \right) \to C$ be a projective bundle of $E$ and ${\mathcal{O}}_{P}(1)$ the tautological line bundle on $P$. Let $F$ be the maximal destabilizing sheaf of $\operatorname{Sym}^n(E)$. In particular, $F$ is semistable and $\mu(F) > 0$. We consider a composition of homomorphisms \[ \alpha : f^*(F) \to f^*(\operatorname{Sym}^n(E)) \to {\mathcal{O}}_{P}(n). \] Since $f_*(\alpha)$ induces the inclusion $F \to \operatorname{Sym}^n(E)$, $\alpha$ is a non-trivial homomorphism. Fix an ample line bundle $A$ on $C$. Let $l$ be a positive integer with $l\mu(F) > n(r-1)\deg(A)$ and $(l, p) = 1$, where $p = \operatorname{char}(k)$. Here we claim that ${\mathcal{O}}_P(l) \otimes f^*(A)$ is ample. Let $V$ be an $s$-dimensional subvariety of $P$. By virtue of Nakai's criterion, it is sufficient to show $(c_1({\mathcal{O}}_P(l) \otimes f^*(A))^s \cdot V) > 0$. If $V$ is contained in a fiber, our assertion is trivial. So we may assume that $V$ is not contained in any fibers. Then, \[ (c_1({\mathcal{O}}_P(l) \otimes f^*(A))^s \cdot V) = l^s(c_1({\mathcal{O}}_P(1))^s \cdot V) + s l^{s-1} (c_1({\mathcal{O}}_P(1))^{s-1} \cdot c_1(f^*(A)) \cdot V). \] Since ${\mathcal{O}}_P(1)$ is numerically effective on $P$ by (\ref{enum:prop:elem:prop:semistable:nef}) of Proposition~\ref{prop:elem:prop:semistable}, $(c_1({\mathcal{O}}_P(1))^s \cdot V) \geq 0$. Moreover, if $x$ is a general point of $C$, \[ (c_1({\mathcal{O}}_P(1))^{s-1} \cdot c_1(f^*(A)) \cdot V) = \deg(A)\deg(\rest{V}{f^{-1}(x)}) > 0. \] Therefore, we get our claim. Thus, there is a positive integer $m$ such that $({\mathcal{O}}_P(l) \otimes f^*(A))^{\otimes m}$ is very ample and $(m, p) = 1$. Take general elements $D_1 , \ldots , D_{r-1}$ of $\left| ({\mathcal{O}}_P(l) \otimes f^*(A))^{\otimes m} \right|$ such that $\Gamma = D_1 \cap \cdots \cap D_{r-1}$ is a non-singular curve and $\rest{f^*(F)}{\Gamma} \to \rest{{\mathcal{O}}_{P}(n)}{\Gamma}$ is generically surjective. If $x$ is a general point of $C$, \yes \[ \deg(\Gamma \to C) = (D_1 \cdots D_{r-1} \cdot f^{-1}(x)) = m^{r-1} (c_1({\mathcal{O}}_P(l) \otimes f^*(A))^{r-1} \cdot f^{-1}(x)) = (ml)^{r-1}. \] \else \begin{align*} \deg(\Gamma \to C) & = (D_1 \cdots D_{r-1} \cdot f^{-1}(x)) \\ & = m^{r-1} (c_1({\mathcal{O}}_P(l) \otimes f^*(A))^{r-1} \cdot f^{-1}(x)) = (ml)^{r-1}. \end{align*} \fi Thus, $k(\Gamma)$ is separable over $k(C)$ because $(p, (ml)^{r-1}) = 1$. Hence, $\rest{f^*(F)}{\Gamma}$ is semistable by (\ref{enum:prop:elem:prop:semistable:separable}) of Proposition~\ref{prop:elem:prop:semistable}. Therefore, \[ \frac{(c_1(f^*(F)) \cdot \Gamma)}{\operatorname{rk} F} \leq (c_{1}({\mathcal{O}}_{P}(n)) \cdot \Gamma), \] which implies \[ \frac{(c_1(f^*(F)) \cdot c_1({\mathcal{O}}_P(l) \otimes f^*(A))^{r-1})}{\operatorname{rk} F} \leq (c_{1}({\mathcal{O}}_{P}(n)) \cdot c_1({\mathcal{O}}_P(l) \otimes f^*(A))^{r-1}). \] This gives rise to \[ l^{r-1}\mu(F) \leq n(r-1)l^{r-2}\deg(A), \] which contradicts to the choice of $l$ with $l\mu(F) > n(r-1)\deg(A)$. \QED As a corollary of Theorem~\ref{thm:semistable:sym}, we have the following. \begin{Corollary}[$\operatorname{char}(k) \geq 0$] \label{cor:semistable:tensor} If $E$ and $F$ are strongly semistable vector bundles on $C$, then so is $E \otimes F$. \end{Corollary} {\sl Proof.}\quad Considering a finite covering of $C$ and tensoring line bundles, we may assume that $\deg(E) = \deg(F) = 0$ as in the same way of the beginning part of the proof of Theorem~\ref{thm:semistable:sym}. Then, $E \oplus F$ is strongly semistable. Thus, by Theorem~\ref{thm:semistable:sym}, $\operatorname{Sym}^2(E \oplus F)$ is strongly semistable. Here, \[ \operatorname{Sym}^2(E \oplus F) = (E \otimes F) \oplus \operatorname{Sym}^2(E) \oplus \operatorname{Sym}^2(F). \] Therefore, we can see that $E \otimes F$ is strongly semistable. \QED Thus, in the same way as the proof of Corollary~\ref{cor:nef:psudo:dis}, we have the following. \begin{Corollary}[$\operatorname{char}(k) \geq 0$] \label{cor:nef:psudo:dis:in:p} Let $X$ be a quasi-projective variety over $k$, $Y$ a smooth quasi-projective variety over $k$, and $f : X \to Y$ a surjective and projective morphism over $k$ with $\dim f = 1$. Let $E$ be a locally free sheaf on $X$ and $y$ a point of $Y$. If $f$ is flat over $y$, the geometric fiber $X_{\bar{y}}$ over $y$ is reduced and Gorenstein, and $E$ is strongly semistable on each connected component of the normalization of $X_{\bar{y}}$, then $\operatorname{dis}_{X/Y}(E)$ is weakly positive at $y$. \end{Corollary} \renewcommand{\thesection}{Appendix \Alph{section}} \renewcommand{\theTheorem}{\Alph{section}.\arabic{Theorem}} \renewcommand{\theClaim}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \renewcommand{\theequation}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}} \setcounter{section}{0} \section{A certain fibration of hyperelliptic curves} \label{sec:const:fib:hyperelliptic} In this section, we would like to construct a certain fibration of hyperelliptic curves, which is needed in \S\ref{sec:cone:positive:divisor:moduli:spacc:stable:curve}. Throughout this section, we assume that $\operatorname{char}(k) = 0$. Let us begin with the following lemma. \begin{Lemma} \label{lem:conic:fibration} For non-negative integers $a_1$ and $a_2$, there are a morphism $f_1 : X_1 \to Y_1$ of smooth projective varieties over $k$, an effective divisor $D_1$ on $X_1$, a line bundle $L_1$ on $X_1$, and a line bundle $M_1$ on $Y_1$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item $\dim X_1 = 2$ and $\dim Y_1 = 1$. \item Let $\Sigma_1$ be the set of all critical values of $f_1$, i.e., $P \in \Sigma_1$ if and only if $f_1^{-1}(P)$ is a singular variety. Then, for any $P \in Y_1 \setminus \Sigma_1$, $f_1^{-1}(P)$ is a smooth rational curve. \item $\Sigma_1 \not= \emptyset$, and for any $P \in \Sigma_1$, $f_1^{-1}(P)$ is a reduced curve consisting of two smooth rational curves $E_P^{1}$ and $E_P^{2}$ joined at one point transversally. \item $D_1$ is smooth over $k$ and $\rest{f_1}{D_1} : D_1 \to Y_1$ is \'{e}tale. \item $(E_P^{1} \cdot D_1) = a_1 + 1$ and $(E_P^{2} \cdot D_1) = a_2 + 1$ for any $P \in \Sigma_1$. Moreover, $D_1$ does not pass through $E_P^1 \cap E_P^2$. \item There is a map $\sigma : \Sigma_1 \to \{ 1, 2 \}$ such that \[ D_1 \in \left| L_1^{\otimes a_1 + a_2 + 2} \otimes f_1^*(M_1) \otimes {\mathcal{O}}_{X_1}\left(-\sum_{P \in \Sigma_1} (a_{\sigma(P)} + 1) E_P^{\sigma(P)}\right) \right|. \] \item $\deg(M_1)$ is divisible by $(a_1 + 1)(a_2 + 1)$. \end{enumerate} \yes \bigskip \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(110,70) \put(10,20){\framebox(80,50){}} \put(95,45){$X_1$} \put(10,10){\line(1,0){80}} \put(95,10){$Y_1$} \put(50,19){\vector(0,-1){8}} \put(53,15){$f_1$} \put(19,21){\line(0,1){48}} \put(81,21){\line(0,1){48}} \put(41,41){\line(1,2){14.3}} \put(41,49){\line(1,-2){14.3}} \put(26,41){\line(1,2){14.3}} \put(26,49){\line(1,-2){14.3}} \put(56,41){\line(1,2){14.3}} \put(61,48){$E_P^1$} \put(56,49){\line(1,-2){14.3}} \put(61,40){$E_P^2$} \put(10,22){\line(1,0){80}} \put(10,26){\line(1,0){80}} \put(10,30){\line(1,0){80}} \put(10,34){\line(1,0){80}} \put(10,38){\line(1,0){80}} \put(10,66){\line(1,0){80}} \put(10,59){\line(1,0){80}} \put(10,52){\line(1,0){80}} \put(0,43){$D_1 \begin{cases} \\ \\ \\ \\ \\ \\ \\ \\ \end{cases}$} \put(-10,63){$\begin{cases} a_1=2 \\ a_2=4 \end{cases}$} \put(67,10){\circle*{2}} \put(65,5){$P$} \end{picture} \end{center} \else\fi \end{Lemma} {\sl Proof.}\quad First of all, let us consider the function $\theta(x)$ defined by \[ \theta(x) = (a_1 + a_2 + 1) \int_{0}^{x} t^{a_1}(t-1)^{a_2} dt. \] Then, $\theta(x)$ is a monic polynomial of degree $a_1 + a_2 + 1$ over ${\mathbb{Q}}$. Moreover, it is easy to see that \[ \theta'(x) = (a_1 + a_2 + 1) x^{a_1}(x-1)^{a_2},\qquad \theta(0) = 0\quad\text{and}\quad \theta(1) = (-1)^{a_2}(a_1+a_2+1)\frac{(a_1)!(a_2)!}{(a_1 + a_2)!}. \] Thus, there are distinct non-zero algebraic numbers $\alpha_1, \ldots, \alpha_{a_2}$ and $\beta_1, \ldots, \beta_{a_1}$ such that \[ \theta(x) = x^{a_1+1}(x-\alpha_1) \cdots (x-\alpha_{a_2}) \] and \[ \theta(x) - \theta(1) = (x-1)^{a_2+1}(x-1-\beta_1) \cdots (x-1-\beta_{a_1}). \] Here we set \[ F(X,Y) = Y^{a_1 + a_2 + 1}\theta(X/Y) = X^{a_1 + 1}(X - \alpha_1 Y) \cdots (X - \alpha_{a_2} Y) \] and \[ G(X,Y,S,T) = T F(X,Y) - Y^{a_1 + a_2 + 1} S. \] Then, $F$ is a homogeneous polynomial of degree $a_1 + a_2 + 1$ over ${\mathbb{Q}}$, and $G$ is a bi-homogeneous polynomial of bi-degree $(a_1+ a_2 + 1, 1)$ in ${\mathbb{Q}}[X,Y] \otimes_{{\mathbb{Q}}} {\mathbb{Q}}[S,T]$. Let $D'$ (resp. $D''$) be the curve on ${\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)}$ given by the equation $\{ G = 0 \}$ (resp. $\{ Y = 0 \}$), where ${\mathbb{P}}^1_{(X,Y)} = \operatorname{Proj}(k[X, Y])$ and ${\mathbb{P}}^1_{(S,T)} = \operatorname{Proj}(k[S, T])$. Moreover, we set $D = D' + D''$. Let $p : {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)} \to {\mathbb{P}}^1_{(X,Y)}$ and $q : {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)} \to {\mathbb{P}}^1_{(S,T)}$ be the natural projections. Then, $D'$ (resp. $D''$) is an element of the linear system $\left| p^*({\mathcal{O}}_{{\mathbb{P}}^1}(a_1 + a_2 + 1)) \otimes q^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)) \right|$ (resp $\left| p^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)) \right|$). Thus, $D \in \left| p^*({\mathcal{O}}_{{\mathbb{P}}^1}(a_1 + a_2 + 2)) \otimes q^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)) \right|$, $(D' \cdot D'') = 1$ and $D' \cap D'' = \{ ((1 : 0), (1 : 0)) \}$. Here we claim the following. \begin{Claim} \label{claim:property:of:D:prime} \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $D'$ is a smooth rational curve. \item Let $\pi' : D' \to {\mathbb{P}}^1_{(S,T)}$ be the morphism induced by the projection $q : {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)} \to {\mathbb{P}}^1_{(S,T)}$. If we set $Q_1 = ((0 : 1), (0 : 1))$, $Q_2 = ((1 : 1), (\theta(1), 1))$ and $Q_3 = ((1 : 0), (1 : 0))$, then the set of ramification points of $\pi'$ is $\{ Q_1, Q_2, Q_3 \}$. Further, the ramification indexes at $Q_1$, $Q_2$ and $Q_3$ are $a_1 + 1$, $a_2 + 1$ and $a_1 + a_2 + 1$ respectively. \end{enumerate} \end{Claim} {\sl Proof.}\quad (a) Since $F(X, Y)$ has no factor of $Y$, the morphism $e : {\mathbb{P}}^1_{(X,Y)} \to D'$ given by \[ e(x:y) = \left( (x : y), (F(x,y) : y^{a_1+a_2+1}) \right) \] is well defined. Moreover, if we set $e' = \rest{p}{D'}$, then it is easy to see that $e \cdot e' = \operatorname{id}_{D'}$ and $e' \cdot e = \operatorname{id}_{{\mathbb{P}}^1}$. Thus, $D'$ is a smooth rational curve. \medskip (b) Pick up a point $(\lambda : \mu) \in {\mathbb{P}}^1_{(S, T)}$. Then, $G_{(\lambda, \mu)} = \mu F(X, Y) - Y^{a_1+a_2+1} \lambda$ is a homogeneous polynomial of degree $a_1 + a_2 +1$. First, we assume that $\mu \not= 0$, hence we may assume that $\mu=1$. Then, $Y$ is not a factor of $G_{(\lambda, 1)}(X, Y)$, which means that ${\pi'}^{-1}((\lambda : 1))$ sits in the affine open set $\operatorname{Spec}(k[X/Y,S/T])$. Thus, \[ {\pi'}^{-1}((\lambda : 1)) = \{ ((\gamma : 1), (\lambda : 1) ) \mid \theta(\gamma) - \lambda = 0 \}. \] Hence, in order to get ramification points of $\pi'$, we need to see multiple roots of $\phi(x) = \theta(x) - \lambda$. Here we will check that $\phi(x)$ has a multiple root if and only if $\lambda$ is either $0$ or $\theta(1)$. Moreover, if $\lambda$ is $0$ (resp. $\theta(1)$), then $0$ (resp. $1$) is the only multiple root of $\phi(x)$ with multiplicity $a_1 + 1$ (resp. $a_2+1$). Let $\gamma$ be a multiple root of $\phi(x) = 0$. Then, $\phi(\gamma) = \phi'(\gamma) = 0$. Here, \[ \phi'(x) = (a_1+a_2+1)x^{a_1}(x-1)^{a_2}. \] Thus, $\gamma$ is either $0$ or $1$. If $\gamma = 0$, then $\lambda = \theta(0) = 0$. If $\gamma = 1$, then $\lambda = \theta(1)$. In the same way, we can easily check the remaining part of our assertion. Therefore, we get two ramification points $Q_1$ and $Q_2$ whose ramification indexes are $a_1+1$ and $a_2 +1$ respectively. Next, we assume that $\mu = 0$, hence we may assume $\lambda=1$. Then, $G_{(\lambda, \mu)} = - Y^{a_1+a_2+1}$. Thus, $P_3$ is a ramification point whose ramification index is $a_1+a_2+1$. \QED \medskip Here we set $P_i = q(Q_i)$ ($i=1,2,3$), $b_1 = a_1+1$, $b_2 = a_2 + 1$, and $b_3 = a_1+a_2+1$. \begin{Claim} \label{claim:cyclic:cover} There is a cyclic covering $h_1 : Y_1 \to {\mathbb{P}}^1_{(S,T)}$ of smooth projective curves such that $\deg(h_1) = b_1 b_2 b_3$ and that, for any $i=1,2,3$ and any $P \in h_1^{-1}(P_i)$, the ramification index of $h_1$ at $P$ is $b_i$. \end{Claim} {\sl Proof.}\quad Since $b_1b_2 + b_2b_3 + b_3b_1 \leq 3 b_1b_2b_3$, there is an effective and reduced divisor $d$ on ${\mathbb{P}}^1_{(S,T)}$ such that $P_i \not\in \operatorname{Supp}(d)$ for each $i=1,2,3$ and \[ b_2b_3 P_1 + b_3b_1 P_2+ b_1b_2 P_3 + d \in \left| {\mathcal{O}}_{{\mathbb{P}}^1}(3b_1b_2b_3) \right|. \] Let $w$ be a section of $H^0({\mathcal{O}}_{{\mathbb{P}}^1}(3b_1b_2b_3))$ with $\operatorname{div}(w) = b_2b_3 P_1+ b_3b_1 P_2 + b_1b_2 P_3+ d$. Then, $w$ gives rise to the ring structure on $\bigoplus_{i=0}^{b_1b_2b_3 - 1} {\mathcal{O}}_{{\mathbb{P}}^1}(-3i)$. Let $Y_1$ be the normalization of \[ \operatorname{Spec}\left( \bigoplus_{i=0}^{b_1b_2b_3 - 1} {\mathcal{O}}_{{\mathbb{P}}^1}(-3i) \right) \] and $h_1: Y_1 \to {\mathbb{P}}^1$ the induced morphism. Then, by our choice of $w$, it is easy to see that $h_1 : Y_1 \to {\mathbb{P}}^1$ satisfies the desired properties. \QED Let $p_1 : {\mathbb{P}}^1_{(X,Y)} \times Y_1 \to {\mathbb{P}}^1_{(X,Y)}$ and $q_1 : {\mathbb{P}}^1_{(X,Y)} \times Y_1 \to Y_1$ be the natural projections, and $u_1 = \operatorname{id} \times h_1 : {\mathbb{P}}^1_{(X,Y)} \times Y_1 \to {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)}$. Then, we have a commutative diagram: \[ \begin{CD} {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)} @<{u_1}<< {\mathbb{P}}^1_{(X,Y)} \times Y_1 \\ @V{q}VV @VV{q_1}V \\ {\mathbb{P}}^1_{(S,T)} @<<{h_1}< Y_1 \end{CD} \] We set $h_1^{-1}(P_1)$, $h_1^{-1}(P_2)$ and $h_1^{-1}(P_3)$ as follows. \[ \begin{cases} h_1^{-1}(P_1) = \{ P_{1,1}, \ldots, P_{1,b_2b_3} \}, \\ h_1^{-1}(P_2) = \{ P_{2,1}, \ldots, P_{1,b_3b_1} \}, \\ h_1^{-1}(P_3) = \{ P_{3,1}, \ldots, P_{3,b_1b_2} \}. \end{cases} \] Then, there is a unique $Q_{i,j}$ on ${\mathbb{P}}^1_{(X,Y)} \times Y_1$ with $q_1(Q_{i,j}) = P_{i,j}$ and $u_1(Q_{i,j}) = Q_i$. \begin{Claim} \label{claim:sing:of:D} \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $u_1^*(D)$ is \'{e}tale over $Y_1$ outside $\{ Q_{i,j} \}_{i,j}$. In particular, $u_1^*(D)$ is smooth over $k$ outside $\{ Q_{i,j} \}_{i,j}$. \item If we set $c_1 = a_1+ 1$, $c_2 = a_2 + 1$ and $c_3 = a_1 + a_2 + 2$, then $u_1^*(D)$ has an ordinary $c_i$-fold point at $Q_{i,j}$ for every $i,j$. Moreover, each tangent of $u_1^*(D)$ at $Q_{i,j}$ is different from the fiber $q_1^{-1}(P_{i,j})$. \end{enumerate} \end{Claim} {\sl Proof.}\quad (a) is trivial because $\rest{q}{D} : D \to {\mathbb{P}}^1_{(S,T)}$ is \'{e}tale outside $\{ Q_1, Q_2, Q_3\}$. Since $u_1^*(D'') = p_1^{-1}((1:0))$, in order to see (b), it is sufficient to check the following. $u_1^*(D')$ has an ordinary $b_i$-fold point at $Q_{i,j}$ for every $i,j$. Moreover, for i=1,2, each tangent of $u_1^*(D')$ at $Q_{i,j}$ is different from the fiber $q_1^{-1}(P_{i,j})$, and each tangent of $u_1^*(D')$ at $Q_{3,j}$ is different from the fiber $q_1^{-1}(P_{3,j})$ and $p_1^{-1}((1:0))$. First we assume $i=1$. Let $z$ be a local parameter of $Y_1$ at $P_{1,j}$, $x = X/Y$, and $s = S/T$. Then, $(x,z)$ gives a local parameter of ${\mathbb{P}}^1_{(X,Y)} \times Y_1$ at $Q_{1,j}$. Since $s = v(z) z^{a_1+1}$ for some $v(z)$ with $v(0) \not= 0$, $u_1^*(D')$ is defined by \[ x^{a_1+1}(x - \alpha_1) \cdots (x - \alpha_{a_2}) - v(z) z^{a_1+1} = 0 \] around $Q_{1, j}$. Thus, since $\alpha_1 \cdots \alpha_{a_2} \not= 0$, $Q_{1,j}$ is an ordinary $(a_1+1)$-fold point and each tangent is different from $\{ z = 0 \}$. Next we assume $i=2$. Let $z$ be a local parameter of $Y_1$ at $P_{2,j}$, $x' = X/Y - 1$, and $s' = S/T - \theta(1)$. Then, $(x',z)$ gives a local parameter of ${\mathbb{P}}^1_{(X,Y)} \times Y_1$ at $Q_{2,j}$. Since $s' = v(z) z^{a_2+1}$ for some $v(z)$ with $v(0) \not= 0$, $u_1^*(D')$ is defined by \[ (x')^{a_2+1}(x' - \beta_1) \cdots (x' - \beta_{a_1}) - v(z) z^{a_2+1} = 0 \] around $Q_{2, j}$. Thus, we can see our assertion in this case because $\beta_1 \cdots \beta_{a_1} \not= 0$. Finally we assume that $i=3$. Let $z$ be a local parameter of $Y_1$ at $P_{3,j}$, $y = Y/X$, and $t = T/S$. Since $t = v(z) z^{a_1+a_2+1}$ for some $v(z)$ with $v(0) \not= 0$, $u_1^*(D')$ is defined by \[ v(z) z^{a_1+a_2+1} (1 - \alpha_1 y) \cdots (1 - \alpha_{a_2} y) - y^{a_1+a_2+1} = 0 \] around $Q_{3, j}$. Thus, $Q_{3,j}$ is an ordinary $(a_1+a_2+1)$-fold point and each tangent is different from $\{ z = 0 \}$ and $\{ y = 0 \}$. \QED Let $\mu_1 : Z_1 \to {\mathbb{P}}^1_{(X,Y)} \times Y_1$ be blowing-ups at all points $Q_{i,j}$, and $E_{i,j}$ $(-1)$-curve over $Q_{i,j}$. Let $\overline{D}_1$ be the strict transform of $u_1^*(D)$ by $\mu_1$, and $g_1 = q_1 \cdot \mu_1$. Then, by the previous claim, $\overline{D}_1$ is \'{e}tale over $Y_1$ and \[ \overline{D}_1 \in \left| \mu_1^*(p_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)))^{\otimes a_1+a_2+2} \otimes g_1^*(h_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1))) \otimes {\mathcal{O}}_{Z_1}\left(-\sum_{i,j} c_i E_{i,j}\right) \right|. \] Let $F_j$ be the strict transform of the fiber $q_1^{-1}(P_{3,j})$. Note that $F_j \cap \overline{D}_1 = \emptyset$ for all $j$. Since $F_j$'s are $(-1)$-curve, we can contract them. Let $\nu_1 : Z_1 \to X_1$ be the contraction of $F_j$'s, and $f_1 : X_1 \to Y_1$ the induced morphism. \[ \begin{CD} {\mathbb{P}}^1_{(X,Y)} \times {\mathbb{P}}^1_{(S,T)} @<{u_1}<< {\mathbb{P}}^1_{(X,Y)} \times Y_1 @<{\mu_1}<< Z_1 @>{\nu_1}>> X_1 \\ @V{q}VV @V{q_1}VV @V{g_1}VV @V{f_1}VV \\ {\mathbb{P}}^1_{(S,T)} @<<{h_1}< Y_1 @= Y_1 @= Y_1 \end{CD} \] Here we set $D_1 = (\nu_1)_*(\overline{D}_1)$, $L_1 = (\nu_1)_*(\mu_1^*(p_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1))))^{**}$, and \[ M_1 = h_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)) \otimes {\mathcal{O}}_{Y_1}\left(-\sum_j c_3 P_{3,j}\right) \simeq {\mathcal{O}}_{Y_1}\left(-\sum_j P_{3,j}\right). \] Then, since $(\nu_1)_*(g_1^*(h_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)))) = f_1^*(h_1^*({\mathcal{O}}_{{\mathbb{P}}^1}(1)))$ and $(\nu_1)_*(E_{3,j}) = f_1^*(P_{3,j})$ for all $j$, we can see \[ D_1 \in \left| L_1^{\otimes a_1+a_2+2} \otimes f_1^*(M_1) \otimes {\mathcal{O}}_{X_1}\left(-\sum_{\substack{i=1,2, \\ j \geq 1}} b_i \nu_1(E_{i,j})\right) \right|. \] Therefore, by our construction of $f_1 : X_1 \to Y_1$, $D_1$, $L_1$ and $M_1$, it is easy to see all properties (1) --- (6) in Lemma~\ref{lem:conic:fibration}. \QED \begin{Proposition} \label{prop:hyperelliptic:fibration} Let $g$ and $a$ be integers with $g \geq 1$ and $0 \leq a \leq [g/2]$. Then, there are a smooth projective surface $X$ over $k$, a smooth projective curve $C$ over $k$, and a surjective morphism $f : X \to Y$ over $k$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item The generic fiber of $f$ is a smooth hyperelliptic curve of genus $g$. \item $f$ is not smooth and every fiber is reduced. Moreover, every singular fiber of $f$ is a nodal curve consisting of a smooth curve of genus $a$ and a smooth curve of genus $g-a$ joined at one point. \end{enumerate} \end{Proposition} {\sl Proof.}\quad Applying Lemma~\ref{lem:conic:fibration} to the case where $a_1= 2a$ and $a_2 = 2g-2a$, we fix a conic fibration as in Lemma~\ref{lem:conic:fibration}. Adding one point to $\Sigma_1$, if necessarily, we can take an effective and reduced divisor $d$ on $Y_1$ such that $\Sigma_1 \subseteq \operatorname{Supp}(d)$ and $\deg(d)$ is even. Thus, there is a line bundle $\vartheta$ on $Y_1$ with ${\mathcal{O}}_{Y_1}(d) \simeq \vartheta^{\otimes 2}$, which produces a double covering $h_2 : Y \to Y_1$ of smooth projective curves such that $h_2$ is ramified over $\Sigma_1$. Let $\mu_2 : X_2 \to X_1 \times_{Y_1} Y$ be the minimal resolution of singularities of $X_1 \times_{Y_1} Y$. We set the induced morphisms as follows. \[ \begin{CD} X_1 @<{u_2}<< X_2 \\ @V{f_1}VV @VV{f_2}V \\ Y_1 @<{h_2}<< Y \end{CD} \] Let $\Sigma_2$ be the set of all critical values of $f_2$. Here, for all $Q \in \Sigma_2$, $f_2^{-1}(Q)$ is reduced, and there is the irreducible decomposition $f_2^{-1}(Q) = \overline{E}_Q^1 + \overline{E}_Q^2 + B_Q$ such that $u_2(\overline{E}_Q^i) = E_{h_2(Q)}^i$ for $i=1, 2$ and $B_Q$ is a $(-2)$-curves. We set $D_2 = u_2^*(D_1)$ and $B = \sum_{Q \in \Sigma_2} B_Q$. Then, $D_2$ is \'{e}tale over $Y$ and $D_2+ B$ is smooth over $k$ because $D_2 \cap B = \emptyset$. Moreover, \[ D_2 \in \left| u_2^*(L_1)^{\otimes 2g+2} \otimes f_2^*(h_2^*(M_1)) \otimes {\mathcal{O}}_{X_2}\left(-u_2^*\left(\sum_{P \in \Sigma_1} (a_{\sigma(P)} + 1) E_P^{\sigma(P)} \right) \right) \right|. \] Let $\sigma_2 : \Sigma_2 \to \{ 1, 2 \}$ be the map given by $\sigma_2 = \sigma \cdot h_2$. Then \[ u_2^*\left(\sum_{P \in \Sigma_1} (a_{\sigma(P)} + 1) E_P^{\sigma(P)} \right) = \sum_{Q \in \Sigma_2} (a_{\sigma_2(Q)} + 1) (2 \overline{E}_Q^{\sigma_2(Q)} + B_Q). \] Therefore, \[ D_2 + B \in \left| u_2^*(L_1)^{\otimes 2g+2} \otimes f_2^*(h_2^*(M_1)) \otimes {\mathcal{O}}_{X_2}\left(- \sum_{Q \in \Sigma_2} ( 2(a_{\sigma_2(Q)} + 1) \overline{E}_Q^{\sigma_2(Q)} + a_{\sigma_2(Q)} B_Q ) \right) \right|. \] Here, since $\deg(h_2^*(M_1)) = 2 \deg(M_1)$, $h_2^*(M_1)$ is divisible by $2$ in $\operatorname{Pic}(Y)$. Further, $a_i$ is even for each $i=1, 2$. Thus, \[ u_2^*(L_1)^{\otimes 2g+2} \otimes f_2^*(h_2^*(M_1)) \otimes {\mathcal{O}}_{X_2}\left(- \sum_{Q \in \Sigma_2} ( 2(a_{\sigma_2(Q)} + 1) \overline{E}_Q^{\sigma_2(Q)} + a_{\sigma_2(Q)} B_Q ) \right) \] is divisible by $2$ in $\operatorname{Pic}(X_2)$, i.e., there is a line bundle $H$ on $X_2$ with \[ H^{\otimes 2} \simeq u_2^*(L_1)^{\otimes 2g+2} \otimes f_2^*(h_2^*(M_1)) \otimes {\mathcal{O}}_{X_2}\left(- \sum_{Q \in \Sigma_2} ( 2(a_{\sigma_2(Q)}+1) \overline{E}_Q^{\sigma_2(Q)} + a_{\sigma_2(Q)} B_Q ) \right). \] Hence, we can construct a double covering $\mu_3 : X_3 \to X_2$ of smooth projective surfaces such that $\mu_3$ is ramified over $D_2 + B$. Let $f_3 : X_3 \to Y$ be the induced morphism. Then, there is the irreducible decomposition \[ f^{-1}(Q) = \overline{C}^1_Q + \overline{C}^2_Q + 2 \overline{B}_Q \] as cycles such that $\mu_3(\overline{C}^i_Q) = \overline{E}^i_Q$ ($i=1,2$) and $\mu_3(\overline{B}_Q) = B_Q$. Here it is easy to check that $\overline{B}_Q$ is a $(-1)$-curve. Thus, we have the contraction $\nu_3 : X_3 \to X$ of $\overline{B}_Q$'s, and the induced morphism $f : X \to Y$. \[ \begin{CD} X_1 @<{u_2}<< X_2@<{\mu_3}<< X_3 @>{\nu_3}>> X \\ @V{f_1}VV @V{f_2}VV @V{f_3}VV @V{f}VV \\ Y_1 @<{h_2}<< Y @= Y @= Y \end{CD} \] We denote $\nu_3(\overline{C}^i_Q)$ by $C_Q^i$. Then, $C^1_Q$ (resp. $C^2_Q$) is a smooth projective curve of genus $a$ (resp. $g-a$), $(C_Q^1 \cdot C_Q^2) = 1$, and $f^{-1}(Q) = C^1_Q + C^2_Q$. Thus, $f : X \to Y$ is our desired fibration. \QED In the same way, we can also show the following proposition. \begin{Proposition} \label{prop:hyperelliptic:fibration:2} Let $g$ and $a$ be integers with $g \geq 1$ and $0 \leq a \leq [(g-1)/2]$. Then, there are a smooth projective surface $X$ over $k$, a smooth projective curve $C$ over $k$, and a surjective morphism $f : X \to Y$ over $k$ with the following properties. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item The generic fiber of $f$ is a smooth hyperelliptic curve of genus $g$. \item $f$ is not smooth and every fiber is reduced. Moreover, every singular fiber of $f$ is a nodal curve consisting of a smooth curve of genus $a$ and a smooth curve of genus $g-a-1$ joined at two points. \end{enumerate} \end{Proposition} {\sl Proof.}\quad Applying Lemma~\ref{lem:conic:fibration} to the case where $a_1 = 2a+1$ and $a_2 = 2g-2a - 1$, we fix a conic fibration as in Lemma~\ref{lem:conic:fibration}. In this case, $\deg(M_1)$ is even. Thus, \[ L_1^{\otimes 2g+2} \otimes f_1^*(M_1) \otimes {\mathcal{O}}_{X_1}\left(-\sum_{P \in \Sigma_1} (a_{\sigma(P)} + 1) E_P^{\sigma(P)}\right) \] is divisible by $2$ in $\operatorname{Pic}(X_1)$. Therefore, there is a double covering $\mu : X \to X_1$ of smooth projective surfaces such that $\mu_3$ is ramified over $D_1$. Then, the induced morphism $f : X \to Y_1$ is a desired fibration. \QED \bigskip

9612

alg-geom/9612010

en

https://arxiv.org/abs/alg-geom/9612010

alg-geom/9612010

Wolfgang Ebeling

Wolfgang Ebeling

Strange duality, mirror symmetry, and the Leech lattice

LaTeX2e, 21 p. with 4 fig.; some corrections and additions

University of Hannover Preprint No. 279

We give a survey on old and new results concerning Arnold's strange duality. We show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.

alg-geom

\section*{Introduction} More than 20 years ago, V.~I.~Arnold \cite{Arnold75} discovered a strange duality among the 14 exceptional unimodal hypersurface singularities. A beautiful interpretation of this duality was given by H.~Pinkham \cite{Pinkham77} and independently by I.~V.~Dolgachev and V.~V.~Nikulin \cite{DN77, Dolgachev82}. I.~Nakamura related this duality to the Hirzebruch-Zagier duality of cusp singularities \cite{Nakamura80, Nakamura81}. In independent work in early 1982, C.~T.~C.~Wall and the author discovered an extension of this duality embracing on one hand series of bimodal singularities and on the other hand, complete intersection surface singularities in ${\mathbb{C}}^4$ \cite{EW85}. We showed that this duality also corresponds to Hirzebruch-Zagier duality of cusp singularities. Recent work has aroused new interest in Arnold's strange duality. It was observed by several authors (see \cite{Dolgachev95} and the references there) that Pinkham's interpretation of Arnold's original strange duality can be considered as part of a two-dimensional analogue of the mirror symmetry of families of Calabi-Yau threefolds. Two years ago, K.~Saito \cite{Saito94} discovered a new feature of Arnold's strange duality involving the characteristic polynomials of the monodromy operators of the singularities and he found a connection with the characteristic polynomials of automorphisms of the famous Leech lattice. Only shortly after, M.~Kobayashi \cite{Kobayashi95} found a duality of the weight systems associated to the 14 exceptional unimodal singularities which corresponds to Arnold's strange duality. He also related it to mirror symmetry. In this paper we first review these results. Then we consider our extension of this duality and examine which of the newly discovered features continue to hold. It turns out that with a suitable construction, Pinkham's interpretation can be extended to a larger class of singularities. In this way, one obtains many new examples of mirror symmetric families of K3 surfaces. We also associate characteristic polynomials to the singularities involved in our extension of the duality and show that Saito's duality continues to hold. Moreover, in this way we can realize further characteristic polynomials of automorphisms of the Leech lattice. The connection with the Leech lattice seems to be rather mysterious. We discuss some facts which might help to understand this connection. We conclude with some open questions. We thank the referee for his useful comments. \section{Arnold's strange duality} We first discuss Arnold's original strange duality among the 14 exceptional unimodal hypersurface singularities. We recall Dolgachev's construction \cite{Dolgachev74, Dolgachev75} (see also \cite{Looijenga83}) of these singularities. Let $b_1 \leq b_2 \leq b_3$ be positive integers such that $\frac{1}{b_1} + \frac{1}{b_2} + \frac {1}{b_3} < 1$. Consider the upper half plane ${\mathbb{H}} = \{ x+iy \in {\mathbb{C}} | y >0\}$ with the hyperbolic metric $\frac{1}{y^2}(dx^2+dy^2)$ and a solid triangle $\Delta \subset {\mathbb{H}}$ with angles $\frac{\pi}{b_1}$, $\frac{\pi}{b_2}$, $\frac{\pi}{b_3}$. Let $\Sigma$ be the subgroup of the group of isometries of ${\mathbb{H}}$ generated by the reflections in the edges of $\Delta$, and let $\Sigma_+$ be the subgroup of index 2 of orientation preserving isometries. Then $\Sigma_+ \subset {\rm PSL}_2({\mathbb{R}})$ and $\Sigma_+$ acts linearly on the total space $T{\mathbb{H}}$ of the tangent bundle on ${\mathbb{H}}$. The inclusion ${\mathbb{H}} \subset T{\mathbb{H}}$ as zero section determines an inclusion ${\mathbb{H}} / \Sigma_+ \subset T{\mathbb{H}} / \Sigma_+$ of orbit spaces. Collapsing ${\mathbb{H}} / \Sigma_+$ to a point yields a normal surface singularity $(X,x_0)$. This singularity is called a {\em triangle singularity}. The numbers $b_1$, $b_2$, $b_3$ are called the {\em Dolgachev numbers} ${\rm Dol}(X)$ of the singularity. The scalar multiplication in the fibres of the tangent bundle $T{\mathbb{H}}$ induces a good ${\mathbb{C}}^\ast$-action on $X$. A resolution of the singularity $(X,x_0)$ can be obtained by the methods of \cite{OW71}. A minimal good resolution consists of a rational curve of self-intersection number $-1$ and three rational curves of self-intersection numbers $-b_1$, $-b_2$, and $-b_3$ respectively intersecting the exceptional curve transversely. By \cite{Dolgachev74}, for exactly 14 triples $(b_1, b_2, b_3)$ the singularity $(X,x_0)$ is a hypersurface singularity. Thus it can be given by a function germ $f: ({\mathbb{C}}^3,0) \to ({\mathbb{C}},0)$ where $f$ is weighted homogeneous with weights $w_1$, $w_2$, $w_3$ and degree $N$. The corresponding weighted homogeneous functions, weights and degrees are indicated in Table~\ref{Table1}. It turns out that these singularities are unimodal and one gets in this way exactly the 14 exceptional unimodal hypersurface singularities in Arnold's classification \cite{Arnold75}. (The equations in Table~\ref{Table1} are obtained by setting the module equal to zero.) \begin{table}\centering \caption{The 14 exceptional unimodal singularities} \label{Table1} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Name & Equation & $N$ & Weights & Dol & Gab & $\mu$ & $d$ & Dual \\ \hline $E_{12}$ & $x^7+y^3+z^2$ & 42 & 6 14 21 & 2 3 7 & 2 3 7 & 12 & 1 & $E_{12}$ \\ \hline $E_{13}$ & $x^5y+y^3+z^2$ & 30 & 4 10 15 & 2 4 5 & 2 3 8 & 13 & $-2$ & $Z_{11}$ \\ \hline $E_{14}$ & $x^8+y^3+z^2$ & 24 & 3 8 12 & 3 3 4 & 2 3 9 & 14 & 3 & $Q_{10}$ \\ \hline $Z_{11}$ & $x^5+xy^3+z^2$ & 30 & 6 8 15 & 2 3 8 & 2 4 5 & 11 & $-2$ & $E_{13}$ \\ \hline $Z_{12}$ & $x^4y+xy^3+z^2$ & 22 & 4 6 11 & 2 4 6 & 2 4 6 & 12 & 4 & $Z_{12}$ \\ \hline $Z_{13}$ & $x^6+xy^3+z^2$ & 18 & 3 5 9 & 3 3 5 & 2 4 7 & 13 & $-6$ & $Q_{11}$ \\ \hline $Q_{10}$ & $x^4+y^3+xz^2$ & 24 & 6 8 9 & 2 3 9 & 3 3 4 & 10 & 3 & $E_{14}$ \\ \hline $Q_{11}$ & $x^3y+y^3+xz^2$ & 18 & 4 6 7 & 2 4 7 & 3 3 5 & 11 & $-6$ & $Z_{13}$ \\ \hline $Q_{12}$ & $x^5+y^3+xz^2$ & 15 & 3 5 6 & 3 3 6 & 3 3 6 & 12 & 9 & $Q_{12}$ \\ \hline $W_{12}$ & $x^5+y^4+z^2$ & 20 & 4 5 10 & 2 5 5 & 2 5 5 & 12 & 5 & $W_{12}$ \\ \hline $W_{13}$ & $x^4y+y^4+z^2$ & 16 & 3 4 8 & 3 4 4 & 2 5 6 & 13 & $-8$ & $S_{11}$ \\ \hline $S_{11}$ & $x^4+y^2z+xz^2$ & 16 & 4 5 6 & 2 5 6 & 3 4 4 & 11 & $-8$ & $W_{13}$ \\ \hline $S_{12}$ & $x^3y+y^2z+xz^2$ & 13 & 3 4 5 & 3 4 5 & 3 4 5 & 12 & 13 & $S_{12}$ \\ \hline $U_{12}$ & $x^4+y^3+z^3$ & 12 & 3 4 4 & 4 4 4 & 4 4 4 & 12 & 16 & $U_{12}$ \\ \hline \end{tabular} \end{table} Let $(X,x_0)$ be one of the 14 hypersurface triangle singularities, and denote by $X_t$ and $\mu$ its Milnor fibre and Milnor number respectively. We denote by $\langle \ , \ \rangle$ the intersection form on $H_2(X_t,{\mathbb{Z}})$ and by $H=(H_2(X_t,{\mathbb{Z}}),\langle \ , \ \rangle)$ the Milnor lattice. A.~M.~Gabrielov \cite{Gabrielov74} has shown that there exists a weakly distinguished basis of vanishing cycles of $H$ with a Coxeter-Dynkin diagram of the form of Fig.~\ref{Fig1}. The author \cite{Ebeling81} has shown that this diagram even corresponds to a distinguished basis of vanishing cycles (cf.\ also \cite{Ebeling96}). (For the notions of a distinguished and weakly distinguished basis of vanishing cycles see e.g.\ \cite{AGV88}). The numbers $p_1$, $p_2$, $p_3$ are called the {\em Gabrielov numbers} ${\rm Gab}(X)$ of the singularity. Here each vertex represents a sphere of self-intersection number $-2$, two vertices connected by a single solid edge have intersection number 1, and two vertices connected by a double broken line have intersection number $-2$. Using the results of K.~Saito (see \cite[Theorem~3.4.3]{Ebeling87}), one can see that the Gabrielov numbers are uniquely determined by the singularity. We denote by $d$ the discriminant of $H$, i.e.\ the determinant of an intersection matrix with respect to a basis of $H$. \begin{figure}\centering \unitlength1cm \begin{picture}(8.5,7.5) \put(0.5,0.5){\includegraphics{fig1.eps}} \put(0.2,0.4){$1$} \put(1.5,2.5){$p_1-1$} \put(2,3.5){$\mu-2$} \put(0,4.5){$p_1+p_2+p_3-3$} \put(0,6.9){$p_1+p_2-1$} \put(4.3,3.2){$p_1+p_2-2$} \put(7.7,3.2){$p_1$} \put(3.9,5){$\mu-1$} \put(3.9,6.5){$\mu$} \end{picture} \caption{Coxeter-Dynkin diagram of an exceptional unimodal singularity} \label{Fig1} \end{figure} Arnold has now observed: There exists an involution $X \mapsto X^\ast$ on the set of the 14 exceptional unimodal singularities, such that $${\rm Dol}(X) = {\rm Gab}(X^\ast), \quad {\rm Gab}(X)={\rm Dol}(X^\ast), \quad N=N^\ast, \quad \mu + \mu^\ast = 24. $$ This is called {\em Arnold's strange duality}. Note that also $d=d^\ast$. H.~Pinkham \cite{Pinkham77} has given the following interpretation of this duality. (This was independently also obtained by I.~V.~Dolgachev and V.~V.~Nikulin \cite{DN77, Dolgachev82}.) The Milnor fibre $X_t$ can be compactified in a weighted projective space to a surface with three cyclic quotient singularities on the curve at infinity; a minimal resolution of these singularities yields a K3 surface $S$. Denote by $G(p_1,p_2,p_3)$ the subgraph of the graph of Fig.~\ref{Fig1} which is obtained by omitting the vertices with indices $\mu -1$ and $\mu$. Let $M(p_1,p_2,p_3)$ be the lattice (the free abelian group with an integral quadratic form) determined by the graph $G(p_1,p_2,p_3)$. Then $H = M(p_1,p_2,p_3) \oplus U$, where $U$ is a unimodular hyperbolic plane (the lattice of rank 2 with a basis $\{e,e'\}$ such that $\langle e, e'\rangle = 1$, $\langle e,e \rangle = \langle e',e' \rangle =0$) and $\oplus$ denotes the orthogonal direct sum. The dual graph of the curve configuration of $S$ at infinity is given by $G(b_1,b_2,b_3)$. The inclusion $X_t \subset S$ induces a primitive embedding $H_2(X_t,{\mathbb{Z}}) \hookrightarrow H_2(S,{\mathbb{Z}})$ and the orthogonal complement is just the lattice $M(b_1,b_2,b_3)$. By \cite{Nikulin79} the primitive embedding of $M(p_1,p_2,p_3) \oplus U$ into the unimodular K3 lattice $L:=H_2(S,{\mathbb{Z}})$ is unique up to isomorphism. In this way, Arnold's strange duality corresponds to a duality of K3 surfaces. This is a two-dimensional analogue of the mirror symmetry between Calabi-Yau threefolds. This has recently been worked out by Dolgachev \cite{Dolgachev95}. We give an outline of his construction. Let $M$ be an even non-degenerate lattice of signature $(1,t)$. An {\em $M$-polarized} K3 surface is a pair $(S,j)$ where $S$ is a K3 surface and $j: M \hookrightarrow \mbox{Pic}(S)$ is a primitive lattice embedding. Here $\mbox{Pic}(S)$ denotes the Picard group of $S$. An $M$-polarized K3 surface $(S,j)$ is called {\em pseudo-ample} if $j(M)$ contains a pseudo-ample divisor class. We assume that $M$ has a unique embedding into the K3 lattice $L$ and the orthogonal complement $M^\perp$ admits an orthogonal splitting $M^\perp = U \oplus \check{M}$. (Dolgachev's construction is slightly more general.) Then we consider the complete family $\cal F$ of pseudo-ample $M$-polarized K3 surfaces and define its {\em mirror family} ${\cal F}^\ast$ to be any complete family of pseudo-ample $\check{M}$-polarized K3 surfaces. It is shown in \cite{Dolgachev95} that this is well defined and that there is the following relation between $\cal F$ and ${\cal F}^\ast$: The dimension of the family $\cal F$ is equal to the rank of the Picard group of a general member from the mirror family ${\cal F}^\ast$. In particular, this can be applied to $M=M(b_1,b_2,b_3)$ and $\check{M}= M(p_1,p_2,p_3)$ for one of the 14 Dolgachev triples $(b_1,b_2,b_3)$. See \cite{Dolgachev95} for further results and references. It was observed by I.~Nakamura \cite{Nakamura80, Nakamura81} that Arnold's strange duality corresponds to Hirzebruch-Zagier duality of hyperbolic (alias cusp) singularities. For details see \cite{Nakamura80, Nakamura81, EW85}. \section{Kobayashi's duality of weight systems} In his paper \cite{Kobayashi95}, M.~Kobayashi has observed a new feature of Arnold's strange duality which we now want to explain. A quadruple $W = (w_1, w_2, w_3; N)$ of positive integers with $N \in {\mathbb{N}} w_1+{\mathbb{N}} w_2+{\mathbb{N}} w_3$ is called a {\em weight system}. The integers $w_i$ are called the weights and $N$ is called the degree of $W$. A weight system $W = (w_1, w_2, w_3; N)$ is called {\em reduced} if $\gcd(w_1,w_2,w_3) = 1$. Let $W = (w_1, w_2, w_3; N)$ and $W' = (w'_1, w'_2, w'_3; N')$ be two reduced weight systems. An $3 \times 3$- matrix $Q$ whose elements are non-negative integers is called a {\em weighted magic square} for $(W,W')$, if $$(w_1,w_2,w_3)Q=(N,N,N) \quad \mbox{and} \quad Q \left( \begin{array}{c} w'_1 \\ w'_2 \\ w'_3 \end{array} \right) = \left( \begin{array}{c} N' \\ N' \\ N' \end{array} \right).$$ (In the case $w_1=w_2=w_3=w'_1=w'_2=w'_3=1$, $Q$ is an ordinary magic square.) $Q$ is called {\em primitive}, if $| \det Q | = N = N'$. We say that the weight systems $W$ and $W'$ are {\em dual} if there exists a primitive weighted magic square for $(W,W')$. Kobayashi now proves: \begin{theorem}[M.~Kobayashi] \label{thm:Kobayashi} Let $W = (w_1, w_2, w_3; N)$ be the weight system of one of the 14 exceptional unimodal singularities. Then there exists a unique (up to permutation) dual weight system $W^\ast$. The weight system $W^\ast$ belongs to the dual singularity in the sense of Arnold. \end{theorem} Moreover, Kobayashi shows that there is a relation between this duality of weight systems and the polar duality between certain polytopes associated to the weight systems. Such a polar duality was considered by V.~Batyrev \cite{Batyrev94} in connection with the mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. We refer to \cite{Ebeling98} for a more precise discussion of this relation. \section{Saito's duality of characteristic polynomials} Let $f: ({\mathbb{C}}^3,0) \to ({\mathbb{C}},0)$ be a germ of an analytic function defining an isolated hypersurface singularity $(X,x_0)$. A characteristic homeomorphism of the Milnor fibration of $f$ induces an automorphism $c:H_2(X_t,{\mathbb{Z}}) \to H_2(X_t,{\mathbb{Z}})$ called the {\em (classical) monodromy operator} of $(X,x_0)$. It is a well known theorem (see e.g. \cite{Brieskorn70}) that the eigenvalues of $c$ are roots of unity. This means that the characteristic polynomial $\phi(\lambda) = \det (\lambda I - c)$ of $c$ is a monic polynomial the roots of which are roots of unity. Such a polynomial can be written in the form $$\phi(\lambda)= \prod_{m \geq 0} (\lambda^m -1)^{\chi_m} \quad \mbox{for} \ \chi_m \in {\mathbb{Z}},$$ where all but finitely many of the integers $\chi_m$ are equal to zero. We note some useful formulae. \begin{proposition} \label{formulae} \begin{itemize} \item[{\rm (i)}] $\mu=\deg \phi = \sum_{m > 0} m\chi_m.$ \item[{\rm (ii)}] If $\sum_{m > 0} \chi_m = 0$ then $$\phi(1) = \prod_{m > 0} m^{\chi_m}.$$ \item[{\rm (iii)}] $\phi(1) = (-1)^\mu d$. \item[{\rm (iv)}] $ {\rm tr}\, c^k = \sum_{m | k} m \chi_m.$ \\ In particular $ {\rm tr}\, c = \chi_1$. \end{itemize} \end{proposition} \noindent {\em Proof.} (i) is obvious. For the proof of (ii) we use the identity $$(\lambda^m -1)=(\lambda -1)(\lambda^{m-1} + \ldots + \lambda +1).$$ To prove (iii), let $A$ be the intersection matrix with respect to a distinguished basis $\{\delta_1, \ldots , \delta_\mu \}$ of vanishing cycles. Write $A$ in the form $A=V+V^t$ where $V$ is an upper triangular matrix with $-1$ on the diagonal. Let $C$ be the matrix of $c$ with respect to $\{\delta_1, \ldots , \delta_\mu \}$. Then $C = - V^{-1}V^t$ (see e.g.\ \cite[Proposition~1.6.3]{Ebeling87}). Therefore $$\phi(1)=\det (I-C)=\det V^{-1}(V+V^t) = (-1)^\mu d.$$ Finally, (iv) is obtained as in \cite{A'Campo75} using the identity $$\det (I-tc) = \exp (\mbox{tr}\, (\log (I-tc))) = \exp (- \sum_{k \geq 1} \frac{t^k}{k} \mbox{tr}\, c^k).$$ This proves Proposition~\ref{formulae}. \addvspace{3mm} By A'Campo's theorem \cite{A'Campo73} $$ \mbox{tr}\, c = -1.$$ We assume that $c$ has finite order $h$. This is e.g.\ true if $f$ is a weighted homogeneous polynomial of degree $N$. In this case $h=N$. Then $\chi_m=0$ for all $m$ which do not divide $h$. K.~Saito \cite{Saito94} defines a {\em dual polynomial} $\phi^\ast(\lambda)$ to $\phi(\lambda)$: $$\phi^\ast(\lambda) = \prod_{k | h} (\lambda^k -1)^{-\chi_{h/k}}.$$ He obtains the following result. \begin{theorem}[K.~Saito] \label{thm:Saito} If $\phi(\lambda)$ is the characteristic polynomial of the monodromy of an exceptional unimodal singularity $X$, then $\phi^\ast(\lambda)$ is the corresponding polynomial of the dual singularity $X^\ast$. \end{theorem} For $\phi(\lambda)= \prod_{m|h} (\lambda^m -1)^{\chi_m}$ we use the symbolic notation $$\pi:= \prod_{m|h} m^{\chi_m}.$$ In the theory of finite groups, this symbol is known as a {\em Frame shape} \cite{Frame, CN79}. For, if one has a rational finite-dimensional representation of a finite group, then the zeros of the characteristic polynomials of each element of the group are also roots of unity. For a given rational representation, one can thus assign to each conjugacy class of the group its Frame shape. The number $$ \deg (\pi) = \sum m \chi_m$$ is called the {\em degree} of the Frame shape $\pi$. Let us denote the Frame shape of the dual polynomial $\phi^\ast(\lambda)$ by $\pi^\ast$. The Frame shapes of the monodromy operators of the 14 exceptional unimodal singularities are listed in Table~\ref{Table2}. \begin{table}\centering \caption{Frame shapes of the 14 exceptional unimodal singularities} \label{Table2} \begin{tabular}{|c|c|c|c|} \hline Name & $\pi$ & $\pi^\ast$ & Dual \\ \hline $E_{12}$ & $2 \cdot 3 \cdot 7 \cdot 42 / 1 \cdot 6 \cdot 14 \cdot 21$ & $2 \cdot 3 \cdot 7 \cdot 42 / 1 \cdot 6 \cdot 14 \cdot 21$ & $E_{12}$ \\ \hline $E_{13}$ & $2 \cdot 3 \cdot 30 / 1 \cdot 6 \cdot 15$ & $2 \cdot 5 \cdot 30 / 1 \cdot 10 \cdot 15$ & $Z_{11}$ \\ \hline $E_{14}$ & $2 \cdot 3 \cdot 24 / 1 \cdot 6 \cdot 8$ & $3 \cdot 4 \cdot 24 / 1 \cdot 8 \cdot 12$ & $Q_{10}$ \\ \hline $Z_{12}$ & $2 \cdot 22 / 1 \cdot 11$ & $2 \cdot 22 / 1 \cdot 11$ & $Z_{12}$ \\ \hline $Z_{13}$ & $2 \cdot 18 / 1 \cdot 6$ & $3 \cdot 18 / 1 \cdot 9$ & $Q_{11}$ \\ \hline $Q_{12}$ & $3 \cdot 15 / 1 \cdot 5$ & $3 \cdot 15 / 1 \cdot 5$ & $Q_{12}$ \\ \hline $W_{12}$ & $2 \cdot 5 \cdot 20 / 1 \cdot 4 \cdot 10$ & $2 \cdot 5 \cdot 20 / 1 \cdot 4 \cdot 10$ & $W_{12}$ \\ \hline $W_{13}$ & $2 \cdot 16 / 1 \cdot 4$ & $4 \cdot 16 / 1 \cdot 8$ & $S_{11}$ \\ \hline $S_{12}$ & $13 / 1$ & $13 / 1$ & $S_{12}$ \\ \hline $U_{12}$ & $4 \cdot 12 / 1 \cdot 3$ & $4 \cdot 12 / 1 \cdot 3$ & $U_{12}$ \\ \hline \end{tabular} \end{table} To two Frame shapes $\pi = \prod m^{\chi_m}$ and $\pi'=\prod m^{\chi'_m}$ of degree $n$ and $n'$ respectively one can associate a Frame shape $\pi\pi'$ of degree $n+n'$ by concatenation $$\pi\pi':=\prod m^{\chi_m}\prod m^{\chi'_m}=\prod m^{\chi_m + \chi'_m}.$$ If $\pi$ and $\pi'$ are the Frame shapes of the operators $c: H \to H$ and $c' : H' \to H'$ respectively, $\pi\pi'$ is the Frame shape of the operator $c \oplus c' : H \oplus H' \to H \oplus H'$. In the appendix of \cite{Saito94}, Saito notes the following observation: If $\pi$ is the Frame shape of the monodromy operator of an exceptional unimodal singularity, then the symbol $\pi\pi^\ast$ of degree 24 is a Frame shape of a conjugacy class of the automorphism group of the Leech lattice. The Leech lattice is a 24-dimensional even unimodular positive definite lattice which contains no roots (see e.g. \cite{Ebeling94}). It was discovered by J.~Leech in connection with the search for densest sphere packings. Its automorphism group $G$ is a group of order $2^{22}3^95^47^211\cdot 13 \cdot 23$. The quotient group $\mbox{Co}_1 := G/\{\pm 1\}$ is a famous sporadic simple group discovered by J.~Conway. The Frame shapes of the 164 conjugacy classes of $G$ have been listed by T.~Kondo \cite{Kondo85}. \section{An extension of Arnold's strange duality} C.~T.~C.~Wall and the author \cite{EW85} have found an extension of Arnold's strange duality. In order to consider this extension, we have to enlarge the class of singularities which we want to discuss. On the one hand, instead of restricting to the hypersurface case, we can also look at isolated complete intersection singularities (abbreviated ICIS in the sequel). Pinkham has shown \cite{Pinkham77b} that for exactly 8 triples $(b_1,b_2,b_3)$ the triangle singularities with these Dolgachev numbers are ICIS, but not hypersurface singularities. They are given by germs of analytic mappings $(g,f): ({\mathbb{C}}^4,0) \to ({\mathbb{C}}^2, 0)$. They are $\cal K$-unimodal singularities and appear in Wall's classification \cite{Wall83}. For certain values of the module, the equations are again weighted homogeneous. The corresponding 8 triples $(b_1,b_2,b_3)$, Wall's names and weighted homogeneous equations with weights $(w_1,w_2,w_3,w_4)$ and degrees $(N_1,N_2)$ are indicated in Table~\ref{Table3}. \begin{table}\centering \caption{The 8 triangle ICIS} \label{Table3} {\small \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Name & Equations & $N$ & Weights & Dol & Gab & $\mu$ & $d$ & Dual \\ \hline $J'_9$ & $\begin{array}{c} xw+y^2 \\ x^3+yw+z^2 \end{array}$ & $\begin{array}{c} 16 \\ 18 \end{array}$ & 6 8 9 10 & 2 3 10 & 2 2 2 3 & 9 & $-4$ & $J_{3,0}$ \\ \hline $J'_{10}$ & $\begin{array}{c} xw+y^2 \\ x^2y+yw+z^2 \end{array}$ & $\begin{array}{c} 12 \\ 14 \end{array}$ & 4 6 7 8 & 2 4 8 & 2 2 2 4 & 10 & 8 & $Z_{1,0}$ \\ \hline $J'_{11}$ & $\begin{array}{c} xw+y^2 \\ x^4+yw+z^2 \end{array}$ & $\begin{array}{c} 10 \\ 12 \end{array}$ & 3 5 6 7 & 3 3 7 & 2 2 2 5 & 11 & $-12$ & $Q_{2,0}$ \\ \hline $K'_{10}$ & $\begin{array}{c} xw+y^2 \\ x^3+z^2+w^2 \end{array}$ & $\begin{array}{c} 10 \\ 12 \end{array}$ & 4 5 6 6 & 2 6 6 & 2 3 2 3 & 10 & 12 & $W_{1,0}$ \\ \hline $K'_{11}$ & $\begin{array}{c} xw+y^2 \\ x^2y+z^2+w^2 \end{array}$ & $\begin{array}{c} 8 \\ 10 \end{array}$ & 3 4 5 5 & 3 5 5 & 2 3 2 4 & 11 & $-20$ & $S_{1,0}$ \\ \hline $L_{10}$ & $\begin{array}{c} xw+yz \\ x^3+yw+z^2 \end{array}$ & $\begin{array}{c} 11 \\ 12 \end{array}$ & 4 5 6 7 & 2 5 7 & 2 2 3 3 & 10 & 11 & $W_{1,0}$ \\ \hline $L_{11}$ & $\begin{array}{c} xw+yz \\ x^2y+yw+z^2 \end{array}$ & $\begin{array}{c} 9 \\ 10 \end{array}$ & 3 4 5 6 & 3 4 6 & 2 2 3 4 & 11 & $-18$ & $S_{1,0}$ \\ \hline $M_{11}$ & $\begin{array}{c} 2xw+y^2+z^2 \\ x^3+2yw \end{array}$ & $\begin{array}{c} 8 \\ 9 \end{array}$ & 3 4 4 5 & 4 4 5 & 2 3 3 3 & 11 & $-24$ & $U_{1,0}$ \\ \hline \end{tabular}} \end{table} By \cite{Hamm71}, the notion of Milnor fibre can also be extended to ICIS. We assume that $(g,f)$ are generically chosen such that $(X',0)=(g^{-1}(0),0)$ is an isolated hypersurface singularity of minimal Milnor number $\mu_1$ among such choices of $g$. Then the {\em monodromy operator} of $(X,0)$ is defined to be the monodromy operator of the function germ $f: (X',0) \to ({\mathbb{C}},0)$. By \cite{Ebeling87} there exists a distinguished set of generators consisting of $\nu:=\mu + \mu_1$ vanishing cycles, where $\mu$ is the rank of the second homology group of the Milnor fibre. Again the monodromy operator is the Coxeter element of this set, i.e.\ the product of the $\nu$ reflections corresponding to the vanishing cycles of the distinguished set of generators. For the 8 triangle ICIS, a Coxeter-Dynkin diagram corresponding to such a distinguished set is depicted in Fig.~\ref{Fig2} (cf.\ \cite{Ebeling87}). Let us call the characteristic numbers $p_1$, $p_2$, $p_3$, $p_4$ of these graphs the {\em Gabrielov numbers} ${\rm Gab}(X)$ of the singularity. They are also indicated in Table~\ref{Table3}. Again, using \cite[Theorem~3.4.3]{Ebeling87} one can see that these numbers are uniquely defined. \begin{figure}\centering \unitlength1cm \begin{picture}(8.5,8) \put(0.5,0.5){\includegraphics{fig2.eps}} \put(0.2,0.7){$1$} \put(8.4,0.7){$p_1$} \put(1.5,2.7){$p_1-1$} \put(6.4,2.7){$p_1+p_2-2$} \put(1.9,3.6){$p_1+p_2$} \put(5.8,3.6){$\nu-1$} \put(0.5,4.5){$p_1+p_2+p_3$} \put(2.1,5.4){$p_1+p_2-1$} \put(5.2,5.4){$\nu$} \put(6.4,4.5){$p_1+p_2+p_3+p_4-1$} \put(0,7){$p_1+p_2+2$} \put(3.6,6.4){$p_1+p_2+1$} \put(6.9,7){$p_1+p_2+p_3+1$} \end{picture} \caption{Coxeter-Dynkin diagram of a triangle ICIS} \label{Fig2} \end{figure} On the other hand, instead of starting with a hyperbolic triangle, we can start with a hyperbolic quadrilateral. Let $b_1$, $b_2$, $b_3$, $b_4$ be positive integers such that $$\frac{1}{b_1} + \frac{1}{b_2} + \frac{1}{b_3} + \frac{1}{b_4} < 2.$$ One can perform the same construction as above with a solid quadrilateral with angles $\frac{\pi}{b_1}$, $\frac{\pi}{b_2}$, $\frac{\pi}{b_3}$, $\frac{\pi}{b_4}$ instead of a triangle. The resulting normal surface singularities are called {\em quadrilateral singularities} (with {\em Dolgachev numbers} $b_1$, $b_2$, $b_3$, $b_4$) (cf.\ \cite{Looijenga84}). Again these singularities admit a natural good ${\mathbb{C}}^\ast$-action. A minimal good resolution consists of a rational curve of self-intersection number $-2$ together with four rational curves of self-intersection numbers $-b_1$, $-b_2$, $-b_3$, and $-b_4$ respectively intersecting the first curve transversely. For 6 quadruples $(b_1,b_2,b_3,b_4)$ these singularities are isolated hypersurface singularities. They are bimodal in the sense of Arnold. They can again be defined by weighted homogeneous equations. By \cite{EW85} they have a distinguished basis with a Coxeter-Dynkin diagram of the following form: It is obtained by adding a new vertex with number 0 to the graph of Fig.~\ref{Fig1} in one of the following two ways: \begin{itemize} \item[(1)] It is connected to the vertex $\mu$ and to the vertex $p_1+p_2$ by ordinary lines. We refer to this diagram by the symbol $(p_1, p_2, \underline{p_3})$. \item[(2)] It is connected to the vertex $\mu$ and to the vertices $p_1$ and $p_1+p_2-1$ by ordinary lines. We refer to this diagram by the symbol $(p_1, \underline{p_2}, \underline{p_3})$. \end{itemize} The corresponding data are indicated in Table~\ref{Table4}. The numbers $p_1$, $p_2$, $p_3$ defined by this procedure are unique except for the singularities $W_{1,0}$ and $S_{1,0}$, where we have two triples each. This follows from \cite[Supplement to Theorem~4.1]{Ebeling83}. One can also obtain an easier proof by considering the discriminants (and in one case the discriminant quadratic form) of the singularities and the possible determinants of the graphs (and in one case the discriminant quadratic form determined by the graph). The absolute values of the determinants in the respective cases are \begin{itemize} \item[(1)] $|d|=4(p_1 p_2 -p_1 - p_2)$, \item[(2)] $|d|=(p_1 -1)(p_2+p_3)$. \end{itemize} \begin{table}\centering \caption{The 6 quadrilateral hypersurface singularities} \label{Table4} {\small \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Name & Equation & $N$ & Weights & Dol & Gab & $\mu$ & $d$ & Dual \\ \hline $J_{3,0}$ & $x^9+y^3+z^2$ & 18 & 2 6 9 & 2 2 2 3 & 2 3 \underline{10} & 16 & 4 & $J'_9$ \\ \hline $Z_{1,0}$ & $x^7+xy^3+z^2$ & 14 & 2 4 7 & 2 2 2 4 & 2 4 \underline{8} & 15 & $-8$ & $J'_{10}$ \\ \hline $Q_{2,0}$ & $x^6+y^3+xz^2$ & 12 & 2 4 5 & 2 2 2 5 & 3 3 \underline{7} & 14 & 12 & $J'_{11}$ \\ \hline $W_{1,0}$ & $x^6+y^4+z^2$ & 12 & 2 3 6 & 2 2 3 3 & $\begin{array}{c} 2 \ 5 \ \underline{7} \\ 2 \ \underline{6} \ \underline{6} \end{array}$ & 15 & $-12$ & $\begin{array}{c} K'_{10} \\ L_{10} \end{array}$ \\ \hline $S_{1,0}$ & $x^5+y^2z+xz^2$ & 10 & 2 3 4 & 2 2 3 4 & $\begin{array}{c} 3 \ 4 \ \underline{6} \\ 3 \ \underline{5} \ \underline{5} \end{array}$ & 14 & 20 & $\begin{array}{c} K'_{11} \\ L_{11} \end{array}$ \\ \hline $U_{1,0}$ & $x^3y+y^3+z^3$ & 9 & 2 3 3 & 2 3 3 3 & 4 \underline{4} \underline{5} & 14 & 27 & $M_{11}$ \\ \hline \end{tabular}} \end{table} For another 5 quadruples $(b_1,b_2,b_3,b_4)$ the quadrilateral singularities with these Dolgachev numbers are ICIS. They are also $\cal K$-unimodal and appear in Wall's lists \cite{Wall83}. They can also be given by weighted homogeneous equations. These equations and Wall's names are listed in Table~\ref{Table5}. \begin{table}\centering \caption{The 5 quadrilateral ICIS} \label{Table5} \begin{tabular}{|c|c|c|c|c|} \hline Name & Equations & Restrictions & $N$ & Weights \\ \hline $J'_{2,0}$ & $\begin{array}{c} xw+y^2 \\ ax^5+xy^2+yw+z^2 \end{array}$ & $a \neq 0,-\frac{4}{27}$ & $\begin{array}{c} 8 \\ 10 \end{array}$ & 2 4 5 6 \\ \hline $L_{1,0}$ & $\begin{array}{c} xw+yz \\ ax^4+xy^2+yw+z^2 \end{array}$ & $a \neq 0,-1$ & $\begin{array}{c} 7 \\ 8 \end{array}$ & 2 3 4 5 \\ \hline $K'_{1,0}$ & $\begin{array}{c} xw+y^2 \\ ax^4+xy^2+z^2+w^2 \end{array}$ & $a \neq 0,\frac{1}{4}$ & $\begin{array}{c} 6 \\ 8 \end{array}$ & 2 3 4 4 \\ \hline $M_{1,0}$ & $\begin{array}{c} 2xw+y^2-z^2 \\ x^2y+ax^2z+2yw \end{array}$ & $a \neq \pm 1$ & $\begin{array}{c} 6 \\ 7 \end{array}$ & 2 3 3 4 \\ \hline $I_{1,0}$ & $\begin{array}{c} x^3+w(y-z) \\ ax^3+y(z-w) \end{array}$ & $a \neq 0,1$ & $\begin{array}{c} 6 \\ 6 \end{array}$ & 2 3 3 3 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|c|} \hline Name & Dol & Gab & $\mu$ & $d$ & Dual \\ \hline $J'_{2,0}$ & 2 2 2 6 & 2 2 2 \underline{6} & 13 & $-16$ & $J'_{2,0}$ \\ \hline $L_{1,0}$ & 2 2 3 5 & $\begin{array}{c} \mbox{2 2 3 \underline{5}} \\ \mbox{2 2 \underline{4} \underline{4}} \end{array}$ & 13 & $-28$ & $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ \\ \hline $K'_{1,0}$ & 2 2 4 4 & $\begin{array}{c} \mbox{2 3 2 \underline{5}} \\ \mbox{2 \underline{4} 2 \underline{4}} \end{array}$ & 13 & $-32$ & $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ \\ \hline $M_{1,0}$ & 2 3 3 4 & $\begin{array}{c} \mbox{2 3 \underline{3} \underline{4}} \\ \mbox{2 \underline{3} 3 \underline{4}} \end{array}$ & 13 & $-42$ & $M_{1,0}$ \\ \hline $I_{1,0}$ & 3 3 3 3 & $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ & 13 & $-54$ & $I_{1,0}$ \\ \hline \end{tabular} \end{table} The singularities $J'_{2,0}$, $L_{1,0}$, $K'_{1,0}$, and $M_{1,0}$ can be given by equations $(g,f)$ where $g$ has Milnor number $\mu_1 = 1$. Coxeter-Dynkin diagrams of these singularities are computed in \cite{Ebeling87}. Using transformations as in the proof of \cite[Proposition~3.6.1]{Ebeling87}, these graphs can be transformed to the following graphs. A Coxeter-Dynkin diagram corresponding to a distinguished set of generators is obtained by adding a new vertex to the graph of Fig.~\ref{Fig2}. It gets the number $p_1+p_2+2$ and the indices of the old vertices with numbers $p_1+p_2+2, p_1+p_2+3, \ldots , \nu$ are shifted by $1$. New edges are introduced in one of the following ways: \begin{itemize} \item[(1)] The new vertex is connected to the vertex $p_1+p_2+1$ and to the vertex with new index $p_1+p_2+p_3+3$ (old index $p_1+p_2+p_3+2$) by ordinary lines. We refer to this diagram by the symbol $(p_1, p_2, p_3, \underline{p_4})$. \item[(2)] The new vertex is connected to the vertex $p_1+p_2+1$ and to the vertices with new indices $p_1+p_2+3$ and $p_1+p_2+p_3+2$ (old indices $p_1+p_2+2$ and $p_1+p_2+p_3+1$ respectively) by ordinary lines. We refer to this diagram by the symbol $(p_1,p_2,\underline{p_3},\underline{p_4})$. \item[(3)] The new vertex is connected to the vertex $p_1+p_2+1$, to the vertex $p_1$, and to the vertex with new index $p_1+p_2+p_3+2$ (old index $p_1+p_2+p_3+1$) by ordinary lines. We refer to this diagram by the symbol $(p_1,\underline{p_2},p_3,\underline{p_4})$. \end{itemize} The absolute values of the determinants of the respective graphs are \begin{itemize} \item[(1)] $|d|=4(p_1p_2p_3 -p_1-p_3)$, \item[(2)] $|d|=p_1p_2(p_3+p_4+2)-p_1-p_2-p_3-p_4$, \item[(3)] $|d|=p_1p_3(p_2+p_4)$. \end{itemize} Comparing the values of these determinants with the discriminants of the above 4 quadrilateral ICIS, we find that the graphs listed in Table~\ref{Table5} are the only possible graphs of the types (1), (2), or (3) for these singularities. Again, in two cases the numbers $p_1$, $p_2$, $p_3$, $p_4$ are not uniquely defined. For the remaining singularity $I_{1,0}$, $\mu_1=2$. This singularity can be given by the following equations: \begin{eqnarray*} g(z) & = & z_1^2 + z_2^2 + z_3^2 + z_4^3, \\ f(z) & = & a_1z_1^2 + a_2z_2^2 + a_3z_3^2 + a_4z_4^3, \end{eqnarray*} where $a_i \in {\mathbb{R}}$, $a_1 < a_2 < a_3 < a_4$. For such a singularity H.~Hamm \cite{Hamm72} has given a basis of the complexified Milnor lattice $H_{\mathbb{C}} = H \otimes {\mathbb{C}}$. As in \cite[Sect.~2.3]{Ebeling87}, one can show that the cycles he constructs are in fact vanishing cycles and that there exists a distinguished set $\{\delta_1, \ldots , \delta_\nu\}$ of generators for this singularity with the following intersection numbers: $$\langle \delta_{i+1},\delta_{i+2} \rangle = -1, \quad i=0,2,4,6,8,10,$$ $$\langle \delta_{i+1},\delta_{i+3} \rangle =\langle \delta_{i+1},\delta_{i+4} \rangle = 0, \langle \delta_{i+2},\delta_{i+3} \rangle =\langle \delta_{i+2},\delta_{i+4} \rangle = 0,\quad i=0,4,8,$$ $$\langle \delta_{i+1},\delta_{j} \rangle =\langle \delta_{i+3},\delta_{j} \rangle = -1, \quad i=0,4,8, \ 1 \leq j\leq 12, j\neq i+1,i+2,i+3,i+4,$$ $$\langle \delta_{i+2},\delta_{2k} \rangle =\langle \delta_{i+4},\delta_{2k} \rangle = -1, \quad i=0,4,8, \ 1 \leq k \leq 6, 2k \neq i+2, i+4,$$ $$\langle \delta_{i+2},\delta_{2k-1} \rangle =\langle \delta_{i+4},\delta_{2k-1} \rangle = 0, \quad i=0,4,8, \ 1 \leq k \leq 6, 2k-1 \neq i+1, i+3,$$ $$\langle \delta_{2k-1} , \delta_{13} \rangle = -1, \langle \delta_{2k-1} , \delta_{15} \rangle = 0, \langle \delta_{2k} , \delta_{13} \rangle = 0, \langle \delta_{2k} , \delta_{15} \rangle = -1, \quad 1 \leq k \leq 6,$$ $$\langle \delta_{i} , \delta_{14} \rangle = -1, \quad 1 \leq i \leq 12,$$ $$\langle \delta_{13} , \delta_{14} \rangle = 0, \langle \delta_{13} , \delta_{15} \rangle = 0, \langle \delta_{14} , \delta_{15} \rangle = 0.$$ By the following sequence of braid group transformations (for the notation see e.g. \cite{Ebeling87}) the distinguished set $\{\delta_1, \ldots , \delta_\nu\}$ can be transformed to a distinguished set $\{\delta'_1, \ldots , \delta'_\nu\}$ where the Coxeter-Dynkin diagram corresponding to the subset $\{\delta'_1, \ldots , \delta'_{12}\}$ is the graph of Fig.~\ref{Fig3}: $$\beta_5,\beta_4,\beta_3,\beta_2;\alpha_6,\alpha_7,\alpha_8,\alpha_9,\alpha _{10}, \alpha_{11};\alpha_5,\alpha_6;\alpha_3,\alpha_4,\alpha_5;$$ $$\beta_8,\beta_7,\beta_6;\beta_{10},\beta_9,\beta_8,\beta_7;\kappa_4,\kappa_5, \kappa_6,\kappa_7,\kappa_8,\kappa_9.$$ We refer to the Coxeter-Dynkin diagram corresponding to $\{\delta'_1, \ldots , \delta'_\nu\}$ by the symbol $(\underline{3},\underline{3},\underline{3},\underline{3})$. This notation is motivated by Fig.~3. We admit, however, that it is somewhat arbitrary. \begin{figure}\centering \unitlength1cm \begin{picture}(5.5,5.5) \put(0.5,0.5){\includegraphics{fig3.eps}} \put(2.3,3.6){$1$} \put(1.6,3.9){$2$} \put(1.6,1.3){$3$} \put(3.5,3.9){$4$} \put(3.5,1.3){$5$} \put(2.3,1.6){$6$} \put(2.5,0.1){$7$} \put(0.2,4.6){$8$} \put(0.2,0.5){$9$} \put(4.9,4.6){$10$} \put(4.9,0.5){$11$} \put(2.5,5.1){$12$} \end{picture} \caption{Subgraph of a Coxeter-Dynkin diagram of the singularity $I_{1,0}$} \label{Fig3} \end{figure} The corresponding symbols are indicated in Table~\ref{Table5}. If one compares the Dolgachev and Gabrielov numbers of Tables \ref{Table3} and \ref{Table4} and of Table~\ref{Table5}, then one observes a correspondence between the 8 triangle ICIS and the 6 quadrilateral hypersurface singularities and between the 5 quadrilateral ICIS. The corresponding ''dual'' singularities are indicated in the last column of each table. Note that this correspondence is not always a duality in the strict sense. For the quadrilateral hypersurface singularity $W_{1,0}$ we have two corresponding ICIS $K'_{10}$ and $L_{10}$ and for $S_{1,0}$ we have the corresponding ICIS $K'_{11}$ and $L_{11}$. The quadrilateral ICIS $L_{1,0}$ and $K'_{1,0}$ are both self-dual and dual to each other. In the other cases the correspondence is one-to-one. Pinkham also defined "Gabrielov numbers" (in a weaker sense) for the triangle ICIS \cite{Pinkham77b} and he already made part of this observation (unpublished). This duality also corresponds to the Hirzebruch-Zagier duality of cusp singularities (see \cite{Nakamura81, EW85}). If one now compares the Milnor numbers of dual singularities, one finds \begin{itemize} \item for the triangle ICIS versus quadrilateral hypersurface singularities: $\mu + \mu^\ast = 25$. \item for the quadrilateral ICIS: $\mu + \mu^\ast = 26$. \end{itemize} (Note that also $d$ and $d^\ast$ do not coincide in each case.) So one still has to alter something. There are two alternatives: \begin{itemize} \item[(1)] subtract 1 for quadrilateral. \item[(2)] subtract 1 for ICIS. \end{itemize} In \cite{EW85} we considered the first alternative. The quadrilateral singularities are first elements ($l=0$) of series of singularities indexed by a non-negative integer $l$. We showed that to each such series one can associate a virtual element $l=-1$. We defined for these Milnor lattices, Coxeter-Dynkin diagrams, and monodromy operators, and showed that all features of Arnold's strange duality including Pinkham's interpretation continue to hold. For more details see below. A new discovery is that the second alternative works as well, and this also leads to an extension of Saito's duality. Recall that the triangle or quadrilateral ICIS with $\mu_1 = 1$ have a Coxeter-Dynkin diagram $D$ which is either the graph of Fig.~\ref{Fig2} or an extension of it. By similar transformations as in the proof of \cite[Proposition~3.6.2]{Ebeling87}, this graph can be transformed to a graph containing the subgraph $D^\flat$ depicted in Fig.~\ref{Fig4}. \begin{figure}\centering \unitlength1cm \begin{picture}(7.5,7.5) \put(0.5,0.5){\includegraphics{fig4.eps}} \put(0.2,0.5){$1$} \put(7,0.5){$p_1$} \put(1.4,2.6){$p_1-1$} \put(4.9,2.6){$p_1+p_2-2$} \put(4.5,3.5){$\mu^\flat-2$} \put(0,4.5){$p_1+p_2+p_3-3$} \put(4.9,4.5){$p_1+p_2+p_3+p_4-4$} \put(0,6.9){$p_1+p_2-1$} \put(5.4,6.9){$p_1+p_2+p_3-2$} \put(3.9,5.1){$\mu^\flat-1$} \put(3.9,6.4){$\mu^\flat$} \end{picture} \caption{Reduced Coxeter-Dynkin diagram $D^\flat$} \label{Fig4} \end{figure} (Unfortunately, this proof has to be modified slightly. The correct sequence of transformations is $$\beta_{\rho-1}, \beta_{\rho-2}, \beta_{\rho-3}, \beta_{\rho-3}, \beta_{\rho-2}, \kappa_{\rho-1}$$ and one has to consider the vertices $\lambda^\prime_{\rho-2}$ and $\lambda^\prime_{\rho}$ instead of $\lambda^\prime_{\rho-1}$ and $\lambda^\prime_{\rho}$.) By \cite[Remark~3.6.5]{Ebeling87}, the passage from $D$ to $D^\flat$ can be considered as a kind of "desuspension". For the singularity $I_{1,0}$, let $D^\flat$ be the graph of Fig.~\ref{Fig3}. In each case, the new Coxeter-Dynkin diagram $D^\flat$ has $\mu -1$ instead of $\mu + \mu_1$ vertices. Denote the corresponding Coxeter element (product of reflections corresponding to the vanishing cycles) by $c^\flat$. For the ICIS with $\mu_1=1$ one can compute (see \cite[Proposition~3.6.2]{Ebeling87}) that $$ \pi(c^\flat) = \pi(c)/1.$$ This means that $c^\flat$ has the same eigenvalues as $c$ but the multiplicities of the eigenvalue 1 differ by 1. For the singularity $I_{1,0}$ one has $\pi(c) = 6^3 / 1 \cdot 2^2$ (cf.\ \cite{Hamm72}), whereas $\pi(c^\flat) = 3^26^2 / 1^22^2$. Note that in any case $$\mbox{tr}\, c^\flat = -2.$$ The passage from $c$ to $c^\flat$ corresponds to the passage from the Milnor lattice $H$ of rank $\mu$ to a sublattice $H^\flat$ of rank $\mu^\flat = \mu -1$. The corresponding discriminants and discriminant quadratic forms of the lattices $H^\flat$ are listed in Table~\ref{Table6}. Here we use the notation of \cite{EW85}. \begin{table}\centering \caption{Discriminants and discriminant quadratic forms of the lattices $H^\flat$} \label{Table6} \begin{tabular}{|c|c|c|c|c|c|} \hline Name & $\mu^\flat$ & $d^\flat$ & $(H^\flat)^\ast/H^\flat$ & dual form & dual \\ \hline $J'_9$ & 8 & 4 & $q_{D_4}$ & $q_{D_4}$ & $J_{3,0}$ \\ \hline $J'_{10}$ & 9 &$-8$ & $q_{D_4} + q_{A_1}$ & $q_{D_4} + w^1_{2,1}$ & $Z_{1,0}$ \\ \hline $J'_{11}$ & 10 & 12 & $q_{D_4} + q_{A_2}$ & $q_{D_4} + w^{-1}_{3,1}$ & $Q_{2,0}$ \\ \hline $K'_{10}$ & 9 & $-12$ & $w^1_{2,2} + w^{-1}_{3,1}$ & $w^{-1}_{2,2} + w^1_{3,1}$ & $W_{1,0}$ \\ \hline $L_{10}$ & 9 & $-12$ & $w^1_{2,2} + w^{-1}_{3,1}$ & $w^{-1}_{2,2} + w^1_{3,1}$ & $W_{1,0}$ \\ \hline $K'_{11}$ & 10 & $20$ & $w^1_{2,1} + w^1_{2,1} + w^{-1}_{5,1}$ & $w^{-1}_{2,1} + w^{-1}_{2,1} + w^{-1}_{5,1}$ & $S_{1,0}$ \\ \hline $L_{11}$ & 10 & $20$ & $w^1_{2,1} + w^1_{2,1} + w^{-1}_{5,1}$ & $w^{-1}_{2,1} + w^{-1}_{2,1} + w^{-1}_{5,1}$ & $S_{1,0}$ \\ \hline $M_{11}$ & 10 & $27$ & $w^{-1}_{3,2} + w^{-1}_{3,1}$ & $w^1_{3,2} + w^1_{3,1}$ & $U_{1,0}$ \\ \hline $J'_{2,0}$ & 12 & 16 & $v_1 + v_1$ & $v_1 + v_1$ & $J'_{2,0}$\\ \hline $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ & 12 & 32 & $w^{1}_{2,2} + w^{-1}_{2,3}$ & $w^{1}_{2,2} + w^{-1}_{2,3}$ & $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ \\ \hline $M_{1,0}$ & 12 & 49 & $w^1_{7,1} + w^{-1}_{7,1}$ & $w^1_{7,1} + w^{-1}_{7,1}$ & $M_{1,0}$ \\ \hline $I_{1,0}$ & 12 & 81 & $\begin{array}{l} w^1_{3,1} + w^1_{3,1} \\ + w^{-1}_{3,1} + w^{-1}_{3,1} \end{array}$ & $\begin{array}{l} w^{-1}_{3,1} + w^{-1}_{3,1} \\ + w^1_{3,1} + w^1_{3,1} \end{array}$ & $I_{1,0}$ \\ \hline \end{tabular} \end{table} The Frame shapes of the corresponding operators $c^\flat$ are listed in Table~\ref{Table7}. \begin{table}\centering \caption{Frame shapes of the triangle ICIS and quadrilateral singularities} \label{Table7} \begin{tabular}{|c|c|c|c|} \hline Name & $\pi$ & $\pi^\ast$ & Dual \\ \hline $J_{3,0}$ & $2 \cdot 3 \cdot 18^2 / 1 \cdot 6 \cdot 9^2$ & $2^23 \cdot 18 / 1^26 \cdot 9$ & $J'_9$ \\ \hline $Z_{1,0}$ & $2 \cdot 14^2 / 1 \cdot 7^2$ & $2^2 14 / 1^27$ & $J'_{10}$ \\ \hline $Q_{2,0}$ & $3 \cdot 12^2 / 1 \cdot 6^2$ & $2^2 12 / 1^24$ & $J'_{11}$ \\ \hline $W_{1,0}$ & $2 \cdot 12^2 / 1 \cdot 4 \cdot 6$ & $2 \cdot 3 \cdot 12 / 1^26$ & $\begin{array}{c} K'_{10} \\ L_{10} \end{array}$ \\ \hline $S_{1,0}$ & $10^2 / 1 \cdot 5$ & $2 \cdot 10 / 1^2$ & $\begin{array}{c} K'_{11} \\ L_{11} \end{array}$ \\ \hline $U_{1,0}$ & $9^2 / 1 \cdot 3$ & $3 \cdot 9 / 1^2$ & $M_{11}$ \\ \hline \hline $J'_{2,0}$ & $2^210^2 / 1^25^2$ & $2^210^2 / 1^25^2$ & $J'_{2,0}$ \\ \hline $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ & $2 \cdot 8^2 / 1^24$ & $2 \cdot 8^2 / 1^24$ & $\begin{array}{c} L_{1,0} \\ K'_{1,0} \end{array}$ \\ \hline $M_{1,0}$ & $7^2 / 1^2$ & $7^2 / 1^2$ & $M_{1,0}$ \\ \hline $I_{1,0}$ & $3^26^2 / 1^22^2$ & $3^26^2 / 1^22^2$ & $I_{1,0}$ \\ \hline \end{tabular} \end{table} It turns out that the substitution $$\mu \mapsto \mu^\flat, \quad c \mapsto c^\flat, \quad H \mapsto H^\flat$$ for the ICIS yields $$ \mu + \mu^\ast = 24, \quad d = d^\ast, \quad \pi_{X^\ast} = \pi^\ast_X.$$ Moreover, the lattice $H$ admits an embedding into the even unimodular lattice $$K_{24} = (-E_8) \oplus (-E_8) \oplus U \oplus U \oplus U \oplus U$$ of rank $24$. This lattice can be considered as the full homology lattice of a K3 surface, $$K_{24} = H_0(S,{\mathbb{Z}}) \oplus H_2(S,{\mathbb{Z}}) \oplus H_4(S,{\mathbb{Z}}),$$ where the inner product on $H_0(S,{\mathbb{Z}}) \oplus H_4(S,{\mathbb{Z}})$ is defined in such a way that this lattice corresponds to a unimodular hyperbolic plane $U$. The orthogonal complement of $H$ is the lattice $\check{H}$ of the singularity $X^\ast$ (cf.\ Table~\ref{Table6}). Let us consider Pinkham's interpretation in the new cases. The Milnor fibre of a triangle or quadrilateral isolated hypersurface or complete intersection singularity can be compactified in such way that after resolving the singularities one gets a K3 surface $S$ \cite{Pinkham78}. We consider the dual graph of the curve configuration at infinity in each case. Let $G(p_1,p_2,p_3,p_4)$ and $\tilde{G}(p_1,p_2,p_3,p_4)$ be the subgraphs of the graphs of Fig.~\ref{Fig4} and Fig.~\ref{Fig2} respectively obtained by omitting the vertices $\mu^\flat-1$ and $\mu^\flat$, and $p_1+p_2-1$, $p_1+p_2+1$, and $\nu$ respectively. Denote by $M(p_1,p_2,p_3,p_4)$ and $\tilde{M}(p_1,p_2,p_3,p_4)$ the corresponding lattices. Recall that the homology lattice $H_2(S,{\mathbb{Z}})$ of the K3 surface is denoted by $L$. First start with a triangle ICIS $(X,x_0)$ with Dolgachev numbers $(b_1,b_2,b_3)$. Then the dual graph is the graph $G(b_1,b_2,b_3)$. This yields an embedding $M(b_1,b_2,b_3) \subset L$ and the orthogonal complement is the Milnor lattice $H=\tilde{M}(p_1,p_2,p_3,p_4) \oplus U$. By alternative (1) (cf.\ \cite{EW85}) the dual of $(X,x_0)$ is a bimodal series; the Milnor lattice of the "virtual" $l=-1$ element of the corresponding series is $M(b_1,b_2,b_3) \oplus U$. One can even associate a Coxeter element to the dual "virtual" singularity; it has order $\mbox{lcm}\,(N_1,N_2)$ where $N_1$, $N_2$ are the degrees of the equations of $(X,x_0)$ \cite{EW85}, whereas the monodromy operator of $(X,x_0)$ has order $N_2$. There is no Saito duality of characteristic polynomials. The dual singularities and orders of the monodromy operators are listed in Table~\ref{Table7b}. \begin{table}\centering \caption{The duality: 8 triangle ICIS versus 8 bimodal series} \label{Table7b} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Name & $\mu$ & Dol & Gab & $d$ & $h$ & $h^\ast$ & $\mu^\ast$ & Dual \\ \hline $J'_9$ & 9 & 2 3 10 & 2 2 2 3 & $-4$ & 18 & 144 & 15 & $J_{3,-1}$ \\ \hline $J'_{10}$ & 10 & 2 4 8 & 2 2 2 4 & 8 & 14 & 84 & 14 & $Z_{1,-1}$ \\ \hline $J'_{11}$ & 11 & 3 3 7 & 2 2 2 5 & $-12$ & 12 & 60 & 13 & $Q_{2,-1}$ \\ \hline $K'_{10}$ & 10 & 2 6 6 & 2 3 2 3 & 12 & 12 & 60 & 14 & $W_{1,-1}$ \\ \hline $K'_{11}$ & 11 & 3 5 5 & 2 3 2 4 & $-20$ & 10 & 40 & 13 & $S_{1,-1}$ \\ \hline $L_{10}$ & 10 & 2 5 7 & 2 2 3 3 & 11 & 12 & 132 & 14 & $W^\sharp_{1,-1}$ \\ \hline $L_{11}$ & 11 & 3 4 6 & 2 2 3 4 & $-18$ & 10 & 90 & 13 & $S^\sharp_{1,-1}$ \\ \hline $M_{11}$ & 11 & 4 4 5 & 2 3 3 3 & $-24$ & 9 & 72 & 13 & $U_{1,-1}$ \\ \hline \end{tabular} \end{table} On the other hand, we can start with a quadrilateral hypersurface singularity $(X,x_0)$ with Dolgachev numbers $(b_1,b_2,b_3,b_4)$. Then the dual graph is the graph $G(b_1,b_2,b_3,b_4)$. We obtain an embedding $M(b_1,b_2,b_3,b_4) \subset L$ and the orthogonal complement is the Milnor lattice of $(X,x_0)$ described in Table~\ref{Table4}. Here we use alternative (2) for the duality. The reduced Milnor lattice $H^\flat$ of the dual triangle ICIS according to Table~\ref{Table4} is the lattice $M(b_1,b_2,b_3,b_4) \oplus U$. Finally, let $(X,x_0)$ be one of the 5 quadrilateral ICIS. Then the dual graph is again the graph $G(b_1,b_2,b_3,b_4)$. One has an embedding $M(b_1,b_2,b_3,b_4) \subset L$ and the orthogonal complement is the Milnor lattice of $(X,x_0)$ described in Table~\ref{Table5}. Combining both alternatives (1) and (2), the lattice $M(b_1,b_2,b_3,b_4) \oplus U$ can be interpreted as follows: The 5 quadrilateral ICIS are the initial $l=0$ elements of 8 series of ICIS. To each such series one can again associate a virtual $l=-1$ element with a well-defined Milnor lattice. Then $M(b_1,b_2,b_3,b_4) \oplus U$ is the reduced Milnor lattice $\check{H}^\flat$ of the dual virtual singularity. This correspondence is indicated in Table~\ref{Table7c}. There is also a duality between the 8 virtual singularities as indicated in \cite{EW85} (see also \cite[Table~3.6.2]{Ebeling87}). There is no Saito duality of characteristic polynomials in both cases. But as we have seen above, using alternative (2) we get a third correspondence, for which Saito's duality of characteristic polynomials holds. \begin{table}\centering \caption{The duality between the quadrilateral ICIS} \label{Table7c} \begin{tabular}{|c|c|c|c|c|c|c|} \hline Name & $\mu$ & Dol & Gab & $d$ & $\mu^\ast$ & Dual \\ \hline $J'_{2,0}$ & 13 & 2 2 2 6 & 2 2 2 \underline{6} & $-16$ & 11 & $J'_{2,-1}$ \\ \hline $L_{1,0}$ & 13 & 2 2 3 5 & $\begin{array}{c} \mbox{2 2 3 \underline{5}} \\ \mbox{2 2 \underline{4} \underline{4}} \end{array}$ & $-28$ & 11 & $\begin{array}{c} L^\sharp_{1,-1} \\ K^\flat_{1,-1} \end{array}$ \\ \hline $K'_{1,0}$ & 13 & 2 2 4 4 & $\begin{array}{c} \mbox{2 3 2 \underline{5}} \\ \mbox{2 \underline{4} 2 \underline{4}} \end{array}$ & $-32$ & 11 & $\begin{array}{c} L_{1,-1} \\ K'_{1,-1} \end{array}$ \\ \hline $M_{1,0}$ & 13 & 2 3 3 4 & $\begin{array}{c} \mbox{2 3 \underline{3} \underline{4}} \\ \mbox{2 \underline{3} 3 \underline{4}} \end{array}$ & $-42$ & 11 & $\begin{array}{c} M^\sharp_{1,-1} \\ M_{1,-1} \end{array}$ \\ \hline $I_{1,0}$ & 13 & 3 3 3 3 & $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ & $-54$ & 11 & $I_{1,-1}$ \\ \hline \end{tabular} \end{table} By Dolgachev's construction \cite{Dolgachev95}, to each case of Pinkham's construction there corresponds a pair of mirror symmetric families of K3 surfaces. Moreover, also to each case where we only have a pair of lattices embedded as orthogonal complements to each other in the lattice $K_{24}$ (cf.\ Table~\ref{Table6}) there corresponds such a mirror pair. One can also investigate Kobayashi's duality of weight systems for our extension of Arnold's strange duality. As already observed by Kobayashi \cite{Kobayashi95}, only some of the weight systems of the quadrilateral hypersurface singularities have dual weight systems, the dual weight systems are in general not unique and they correspond again to isolated hypersurface singularities. Since an ICIS has two degrees $N_1$ and $N_2$, it is not quite clear how to generalize the notion of a weighted magic square. One possibility would be to work with the sum of the degrees $N := N_1 + N_2$ and to use also $3 \times 4$ and $4 \times 4$ matrices instead of $3 \times 3$ matrices. Then one finds again that in some cases there does not exist a dual weight system, the dual weight systems are in general not unique, most cases are self-dual, and only in the cases $J'_{10} \leftrightarrow Z_{1,0}$, $K'_{11} \leftrightarrow S_{1,0}$, $J'_{2,0} \leftrightarrow J'_{2,0}$, and $M_{1,0} \leftrightarrow M_{1,0}$ of our duality there exist weighted magic squares giving a duality of the corresponding weight systems. However, there is a relation between our extended duality and a polar duality between the Newton polytopes generalizing Kobayashi's observation for Arnold's strange duality. This can be used to explain Saito's duality of characteristic polynomials. For details see the forthcoming paper \cite{Ebeling98}. \section{Singular moonshine} Let us consider the symbols $\pi\pi^\ast$ of Table~\ref{Table7}. It turns out that they all occur in the list of Kondo, too. These pairs and the pairs from the original Arnold duality correspond to self-dual Frame shapes of the group $G$ with trace $-2$, $-3$, or $-4$. By examining Kondo's list one finds that there are 22 such Frame shapes and all but 3 occur. They are listed in Table~\ref{Table8}. Here we use the ATLAS notation \cite{ATLAS} for the conjugacy classes. For each value of the trace one symbol is missing. \begin{table}\centering \caption{Self-dual Frame shapes of $G$ with trace $-2$, $-3$ or $-4$} \label{Table8} \begin{tabular}{|c|c|c|c|c|} \hline ATLAS & Frame & & Niemeier & Duality \\ \hline 21A & $2^23^27^242^2 / 1^26^214^221^2$ & $*$ & & $E_{12} \leftrightarrow E_{12}$ \\ \hline 15E & $2^23 \cdot5 \cdot 30^2 / 1^26 \cdot 10 \cdot 15^2$ & $*$ & $D_{16} \oplus E_8$ & $E_{13} \leftrightarrow Z_{11}$ \\ \hline 24B & $2 \cdot 3^24 \cdot 24^2 / 1^26 \cdot 8^2 12$ & $*$ & & $E_{14} \leftrightarrow Q_{10}$ \\ \hline 11A & $2^222^2 / 1^211^2$ & $*$ & $D_{12}^2$ & $Z_{12} \leftrightarrow Z_{12}$ \\ \hline 18B & $2 \cdot 3 \cdot 18^2 / 1^26 \cdot 9$ & $*$ & $A_{17} \oplus E_7$ & $Z_{13} \leftrightarrow Q_{11}$ \\ \hline 15B & $3^215^2 / 1^25^2$ & $*$ & & $Q_{12} \leftrightarrow Q_{12}$ \\ \hline 20A & $2^25^220^2 / 1^24^210^2$ & $*$ & & $W_{12} \leftrightarrow W_{12}$ \\ \hline 16B & $2 \cdot 16^2 / 1^28$ & $*$ & $A_{15} \oplus D_9$ & $W_{13} \leftrightarrow S_{11}$ \\ \hline 13A & $13^2 / 1^2$ & $*$ & $A^2_{12}$ & $S_{12} \leftrightarrow S_{12}$ \\ \hline 12E & $4^212^2 / 1^23^2$ & $*$ & & $U_{12} \leftrightarrow U_{12}$ \\ \hline 9C & $2^33^218^3 /1^36^29^3$ & & $D_{10} \oplus E_7^2$ & $J'_9 \leftrightarrow J_{3,0}$ \\ \hline 7B & $2^314^3 / 1^37^3$ & $*$ & $D^3_8$ & $J'_{10} \leftrightarrow Z_{1,0}$ \\ \hline 12K & $2^23 \cdot 12^3 / 1^34 \cdot 6^2$ & $*$ & $A_{11} \oplus D_7 \oplus E_6$ & $\begin{array}{c} J'_{11} \leftrightarrow Q_{2,0} \\ K'_{10} \leftrightarrow W_{1,0} \\ L_{10} \leftrightarrow W_{1,0} \end{array}$ \\ \hline 10E & $2 \cdot 10^3 / 1^35$ & $*$ & $A_9^2 \oplus D_6$ & $\begin{array}{c} K'_{11} \leftrightarrow S_{1,0} \\ L_{11} \leftrightarrow S_{1,0} \end{array}$ \\ \hline 9A & $9^3 / 1^3$ & & $A_8^3$ & $M_{11} \leftrightarrow U_{1,0}$ \\ \hline 5B & $2^410^4 / 1^45^4$ & & $D_6^4$ & $J'_{2,0} \leftrightarrow J'_{2,0}$ \\ \hline 8C & $2^28^4 / 1^44^2$ & & $A_7^2 \oplus D_5^2$ & $\begin{array}{c} L_{1,0} \leftrightarrow L_{1,0} \\ L_{1,0} \leftrightarrow K'_{1,0} \\ K'_{1,0} \leftrightarrow K'_{1,0} \end{array}$ \\ \hline 7A & $7^4 / 1^4$ & & $A_6^4$ & $M_{1,0} \leftrightarrow M_{1,0}$ \\ \hline 6A & $3^46^4 / 1^42^4$ & & & $I_{1,0} \leftrightarrow I_{1,0}$ \\ \hline 10A & $5^210^2 / 1^22^2$ & $*$ & & \\ \hline 15A & $2^33^35^330^3 / 1^36^310^315^3$ & $*$ & $E_8^3$ & \\ \hline 12A & $2^43^412^4 /1^44^46^4$ & & $E_6^4$ & \\ \hline \end{tabular} \end{table} Special elements of $G$ correspond to the deep holes of the Leech lattice. A {\em deep hole} of the Leech lattice $\Lambda_{24}$ is a point in ${\mathbb{R}}^{24}$ which has maximal distance from the lattice points. It is a beautiful theorem of J.~H.~Conway, R.~A.~Parker, and N.~J.~A.~Sloane (\cite{CPS82}, see also \cite{Ebeling94}) that there are 23 types of deep holes in $\Lambda_{24}$ which are in one-to-one correspondence with the 23 isomorphism classes of even unimodular lattices in ${\mathbb{R}}^{24}$ containing roots, which were classified by H.-V.~Niemeier \cite{Niemeier73}. These lattices are characterized by the root systems which they contain. The Frame shape of the Coxeter element of such a root system is also the Frame shape of an automorphism of the Leech lattice. We have indicated in Table~\ref{Table8} the type of the root system of the Niemeier lattice, if a Frame shape corresponds to the Coxeter element of such a root system. The automorphism group $G$ of the Leech lattice contains the Mathieu group $M_{24}$ (see e.g.\ \cite{Ebeling94}). S.~Mukai \cite{Mukai88} has classified the finite automorphism groups of K3 surfaces (automorphisms which leave the symplectic form invariant) and shown that they admit a certain embedding into the Mathieu group $M_{24}$. He gives a list of 11 maximal groups such that every finite automorphism group imbeds into one of these groups. A table of the centralizers of the conjugacy classes of $G$ can be found in \cite{Wilson83}. For an element $g \in G$, denote its centralizer by $Z(g)$ and the finite cyclic group generated by $g$ by $\langle g \rangle$. We have marked by ($\ast$) in Table~\ref{Table8} the cases where there is an obvious inclusion of $Z(g) / \langle g \rangle$ in one of Mukai's groups. It follows that in these cases there is a K3 surface with an operation of $Z(g) / \langle g \rangle$ on it by symplectic automorphisms. To a Frame shape $$\pi = \prod_{m | N} m^{\chi_m}$$ one can associate a modular function \cite{Kondo85}. Let $$\eta(\tau) = q^{1/24} \prod^\infty_{n=1} (1-q^n), \quad q=e^{2\pi i \tau}, \tau \in {\mathbb{H}},$$ be the Dedekind $\eta$-function. Then define $$\eta_\pi(\tau) = \prod_{m / N} \eta(m\tau)^{\chi_m}.$$ Saito \cite{Saito94} has proved the identity $$\eta_\pi \left( - \frac{1}{N\tau} \right) \eta_{\pi^\ast}(\tau) \sqrt{d} = 1,$$ where $d = \prod m^{\chi_m}$ and $\pi^\ast$ is the dual Frame shape. From this it follows that $\eta_{\pi\pi^\ast}$ is a modular function for the group $$\Gamma_0(N) = \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in \mbox{SL}_2({\mathbb{Z}}) \left| \right. c \equiv 0 (N) \right\}.$$ \addvspace{3mm} \noindent {\bf Question~1} Let $\pi\pi^\ast$ be one of the self-dual Frame shapes of Table~\ref{Table8}. Is there any relation of $\eta_{\pi\pi^\ast}$ to the analogue of Dedekind's eta function for K3 surfaces considered in \cite{JT95} ? \addvspace{3mm} The Frame shape $\pi\pi^\ast$ is the Frame shape of the operator $c \oplus c^\ast$ which can be considered as an automorphism of a sublattice of finite index of the even unimodular 24-dimensional lattice $K_{24}$, which is the full homology lattice of a K3 surface. The lattice $K_{24}$ has the same rank as the Leech lattice, but contrary to the Leech lattice it is indefinite and has signature $(4,20)$. \addvspace{3mm} \noindent {\bf Question~2 } Is there an explanation for this strange correspondence between operators of different lattices? \addvspace{3mm} Is it only a purely combinatorial coincidence? One can try to classify finite sequences $(\chi_1, \chi_2, \ldots , \chi_N)$ with the following properties: \begin{itemize} \item[(1)] $\chi_m \in {\mathbb{Z}}$ for all $m=1, \ldots , N$, \item[(2)] $\sum m\chi_m = 24$, \item[(3)] $\chi_m = 0$ for $m \! \! \not| N$, \item[(4)] $\chi_m = - \chi_{N/m}$ for $m | N$, \item[(5)] $\prod m^{\chi_m} \in {\mathbb{N}}$, \item[(6)] $\chi_1 \in \{-2,-3,-4\}$, \item[(7)] $|\chi_m| \leq |\chi_N|$ for $m | N$. \end{itemize} By a computer search, one finds for $N \leq 119$ in addition to the 22 Frame shapes of Table~\ref{Table8} only the following Frame shapes: $$3^26 \cdot 12^2 /1^22 \cdot 4^2, \quad 2^43^44^424^4 / 1^46^48^412^4, \quad 2^24^25^240^2 / 1^28^210^220^2.$$ These Frame shapes also appear in Kondo's tables, namely in the tables of certain transforms of the Frame shapes of $G$ \cite[Table~III, 4C; Table~II, 12A; Table~II, 20A]{Kondo85}.

9612

alg-geom/9612005

en

https://arxiv.org/abs/alg-geom/9612005

alg-geom/9612005

Ezra Getzler

Ezra Getzler

The semi-classical approximation for modular operads

11 pages, amslatex-1.2

MPI 96-145

The semi-classical approximation is an explicit formula of mathematical physics for the sum of Feynman diagrams with a single circuit.In this paper, we study the same problem in the setting of modular operads (see dg-ga/9408003); instead of being a number, the interaction at a vertex of valence n is an S_n-module. As an application, we calculate the S_n-equivariant Hodge polynomials of the moduli spaces \Mbar_{1,n}.

alg-geom

\subsection*{Acknowledgments} I wish to thank the Department of Mathematics at the Universit\'e de Paris-VII the Max-Planck-Institut f\"ur Mathematik in Bonn for their hospitality during the inception and completion, respectively, of this paper. I am grateful to D. Zagier for showing me the asymptotic expansion of Corollary \ref{don}. This research was partially supported by a research grant of the NSF and a fellowship of the A.P. Sloan Foundation. \section{Wick's theorem and the semi-classical approximation} Let $\Gamma_{g,n}$ be the small category whose objects are isomorphism classes of stable graphs $G$ of genus $g(G)=g$ with $n$ totally ordered legs \cite{modular}, and whose morphisms are the automorphisms: if $G\in\Gamma_{g,n}$, its automorphism group $\Aut(G)$ is the subset of the permutations of the flags which preserve all the data defining the stable graph, including the total ordering of the legs. Because of the stability condition, $\Gamma_{g,n}$ is a finite category. Define polynomials $\{\mathsf{M} v_{g,n}\mid 2(g-1)+n>0\}$ of a set of variables $\{v_{g,n}\mid 2(g-1)+n>0\}$ by the following formula: \begin{equation} \label{M} \mathsf{M} v_{g,n} = \sum_{G\in\Ob\Gamma_{g,n}} \frac{1}{|\Aut(G)|} \prod_{v\in\VERT(G)} v_{g(v),n(v)} . \end{equation} Introduce the sequences of generating functions $$ a_g(x) = \sum_{2(g-1)+n>0} v_{g,n} \frac{x^n}{n!} , \quad\text{and}\quad b_g(x) = \sum_{2(g-1)+n>0} \mathsf{M} v_{g,n} \frac{x^n}{n!} . $$ Wick's theorem gives an integral formula for the generating functions $\{b_g\}$ in terms of $\{a_g\}$: $$ \sum_{g=0}^\infty b_g \hbar^{g-1} = \log \int_{-\infty}^\infty \exp \biggl( \sum_{g=0}^\infty a_g \hbar^{g-1} - \frac{(x-\xi)^2}{2\hbar} \biggr) \, \frac{dx}{\sqrt{2\pi\hbar}} . $$ As written, this is purely formal, since it involves the integration of a power series in $x$. It may be made rigourous by observing that the integral transform $$ f \DOTSB\mapstochar\to \int_{-\infty}^\infty f(\hbar,x) e^{-(x-\xi)^2/2\hbar} \, \frac{dx}{\sqrt{2\pi\hbar}} $$ induces a continuous linear map on the space of Laurent series $\mathbb{Q}\(\hbar\)\[x\]$ topologized by the powers of the ideal $(\hbar,x)$. The semi-classical expansion is a pair of formulas for $b_0$ and $b_1$ in terms of $a_0$ and $a_1$, which we now recall. \begin{definition} Let $R$ be a ring of characteristic zero. The Legendre transform $\mathcal{L}$ is the involution of the set $x^2/2+x^3R\[x\]$ characterized by the formula $$ (\mathcal{L} f)\circ f' + f = p_1 f' . $$ \end{definition} \begin{theorem} \label{legendre} The series $x^2/2+b_0$ is the Legendre transform of $x^2/2-a_0$. \end{theorem} The first few coefficients of $b_0$ may be calculated, either from the definition of $\mathsf{M} v_{0,n}$ or from Theorem \ref{legendre}: $$\begin{tabular}{|C|L|} \hline n & \mathsf{M} v_{0,n} \\[2pt] \hline 3 & v_{0,3} \\[5pt] 4 & v_{0,4} + 3v_{0,3}^2 \\[5pt] 5 & v_{0,5} + 10v_{0,4}v_{0,3} + 15v_{0,3}^3 \\[5pt] 6 & v_{0,6} + 15v_{0,5}v_{0,3} + 10v_{0,4}^2 + 105 v_{0,4}v_{0,3}^2 + 105 v_{0,3}^4 \\[3pt] \hline \end{tabular}$$ We now come to the formula for $b_1$, known as the semi-classical approximation. \begin{theorem} \label{semi} The series $b_1$ and $a_1$ are related by the formula $$ b_1 = \bigl( a_1 - \tfrac{1}{2} \log (1-a_0'') \bigr) \circ ( x + b_0' ) . $$ \end{theorem} By the definition of the Legendre transform, we see that $(\mathcal{L} f)'\circ f'=x$. It follows that Theorem \ref{semi} is equivalent to the formula $$ b_1 \circ ( x - a_0' ) = a_1 - \tfrac{1}{2} \log(1-a_0'') . $$ This formula expresses the fact that the stable graphs contributing to $b_1$ are obtained by attaching a forest whose vertices have genus $0$ to two types of graphs: \begin{enumerate} \item those with a single vertex of genus $1$ (corresponding to the term $a_1$); \item stable graphs with a single circuit, and all of whose vertices have genus $0$ --- we call such a graph a \emph{necklace}. \end{enumerate} The presence of a logarithm in the term which contributes the necklaces is related to the fact that there are $(n-1)!$ cyclic orders of $n$ objects. The first few coefficients of $b_1$ are also easily calculated: $$\begin{tabular}{|C|L|} \hline n & \mathsf{M} v_{1,n} \\[2pt] \hline 1 & v_{1,1} + \tfrac12 v_{0,3} \\[5pt] 2 & v_{1,2} + v_{1,1}v_{0,3} + \tfrac12\bigl( v_{0,4} + v_{0,3}^2 \bigr) \\[5pt] 3 & v_{1,3} + 3v_{1,2}v_{0,3} + v_{1,1}v_{0,4} + \tfrac12\bigl( v_{0,5} + 3 v_{0,4}v_{0,3} + 2v_{0,3}^3 \bigr) \\[5pt] 4 & v_{1,4} + 6v_{1,3}v_{0,3} + 3v_{1,2}v_{0,4} + 15v_{1,2}v_{0,3}^2 + v_{1,1}v_{0,5} \\ & \quad {}+ \tfrac12\bigl( v_{0,6} + 4v_{0,5}v_{0,3} + 3v_{0,4}^2 + 12v_{0,4}v_{0,3}^2 + 6v_{0,3}^4\bigr) \\[3pt] \hline \end{tabular}$$ \section{The semi-classical approximation for modular operads} In the theory of modular operads, one replaces the sequence of coefficients $\{v_{g,n}\}$ considered above by a stable $\SS$-module, that is, a sequence of $\SS_n$-modules $\mathcal{V}\(g,n\)$. The analogue of \eqref{M} is the functor on stable $\SS$-modules which sends $\mathcal{V}$ to \begin{equation} \label{MM} \mathbb{M}\mathcal{V}\(g,n\) = \colim_{G\in\Gamma_{g,n}} \bigotimes_{v\in\VERT(G)} \mathcal{V}\(g(v),n(v)\) . \end{equation} Thus, the coefficients in \eqref{M} are promoted to vector spaces, the product to a tensor product, the sum over stable graphs to a direct sum, and the weight $|\Aut(G)|^{-1}$ to $\colim_{\Aut(G)}$, that is, the coinvariants with respect to the finite group $\Aut(G)$. Note that this definition makes sense in any symmetric monoidal category ${\mathcal C}$ with finite colimits. We will need the Peter-Weyl theorem to hold for actions of the symmetric group $\SS_n$ on ${\mathcal C}$; thus, we will suppose that ${\mathcal C}$ is additive over a ring of characteristic zero. \begin{definition} The characteristic $\ch_n(\mathcal{V})$ of an $\SS_n$-module is defined by the formula $$ \ch_n(\mathcal{V}) = \frac{1}{n!} \sum_{\sigma\in\SS_n} \Tr_\sigma(\mathcal{V}) p_\sigma \in \Lambda_n\o K_0({\mathcal C}) , $$ where $p_\sigma$ is the product of power sums $p_{|\mathcal{O}|}$ over the orbits $\mathcal{O}$ of $\sigma$. \end{definition} Although this definition appears to require rational coefficients, this is an artifact of the use of the power sums $p_n$; it is shown in \cite{I} that the characteristic is a symmetric function of degree $n$ with values in the Grothendieck group of the additive category ${\mathcal C}$. If $\rk:\Lambda\to\mathbb{Q}[x]$ is the homomorphism defined by $h_n\DOTSB\mapstochar\to x^n/n!$, we have $$ \rk(\ch_n(\mathcal{V})) = [\mathcal{V}]/n!\in K_0({\mathcal C})\o\mathbb{Q} . $$ Note that $\rk(f)$ is obtained from $f$ by setting the powers sums $p_n$ to $0$ if $n>1$, and to $x$ if $n=1$. The place of the generating functions $a_g$ and $b_g$ is now taken by \begin{align*} \AA_g & = \sum_{2(g-1)+n>0} \ch_n(\mathcal{V}\(g,n\)) \in \Lambda \Hat{\otimes} K_0({\mathcal C}) , \\ \mathbf{b}_g & = \sum_{2(g-1)+n>0} \ch_n(\mathbb{M}\mathcal{V}\(g,n\)) \in \Lambda \Hat{\otimes} K_0({\mathcal C}) . \end{align*} Theorem (8.13) of \cite{modular}, whose statement we now recall, calculates $\mathbf{b}_g$ in terms of $\AA_h$, $h\le g$. Let $\Delta$ be the ``Laplacian'' on $\Lambda\(\hbar\)$ given by the formula $$ \Delta = \sum_{n=1}^\infty \hbar^n \left( \frac{n}{2} \frac{\partial^2}{\partial p_n^2} + \frac{\partial}{\partial p_{2n}} \right) . $$ \begin{theorem} \label{modular} If $\mathcal{V}$ is a stable $\SS$-module, then $$ \sum_{g=0}^\infty \mathbf{b}_g \hbar^{g-1} = \Log \biggl( \exp(\Delta) \Exp\Bigl( \sum_{g=0}^\infty \AA_g \hbar^{g-1} \Bigr) \biggr) . $$ \end{theorem} There is also a formula for $\mathbf{b}_0$ in terms of $\AA_0$. To state it, we must recall the definition of the Legendre transform for symmetric functions. Let $$ \Lambda_*\Hat{\otimes} K_0({\mathcal C}) = \{ f \in \Lambda\Hat{\otimes} K_0({\mathcal C}) \mid \rk(f) = x^2/2+O(x^3) \} . $$ If $f$ is a symmetric function, let $f'=\partial f/\partial p_1$; this operation may be expressed more invariantly as $p_1^\perp$ (Ex.\ I.5.3, Macdonald \cite{Macdonald}). \begin{definition} The Legendre transform $\mathcal{L}$ is the involution of $\Lambda_*\Hat{\otimes} K_0({\mathcal C})$ characterized by the formula $(\mathcal{L} f)\circ f' + f = p_1 f'$. \end{definition} The Legendre transform $\mathcal{L} f$ of a function $f$ is characterized by the formula $(\mathcal{L} f)'\circ f'=x$. For symmetric functions, although the analogue of this formula holds, in the form $$ (\mathcal{L} f)'\circ f' = h_1 , $$ the situation is not as simple, since there is no single notion of integral for symmetric functions (the ``constant'' term may be any function of the power sums $p_n$, $n>1$). Neverthless, there is a simple algorithm for calculating $\mathcal{L} f$ from $f$. Denote by $f_n$ and $g_n$ the coefficents of $f$ and $g=\mathcal{L} f$ lying in $\Lambda_n\o K_0({\mathcal C})$. \begin{enumerate} \item The formula $f'\circ(\mathcal{L} f)'=h_1$ may be rewritten as $$ \sum_{n=3}^N g_n' + \sum_{n=3}^N f_n' \circ \Bigl( h_1 + \sum_{k=3}^{N-1} g_k' \Bigr) \cong 0 \mod{\Lambda_N\o K_0({\mathcal C})} . $$ This gives a recursive procedure for calculating $g_n'$. \item Having determined $g'$, we obtain $g$ from the formula $f=\mathcal{L} g$, or $g = p_1 g' - f\circ g'$. \end{enumerate} We now recall Theorem (7.17) of \cite{modular}, which is the generalization to modular operads of Theorem \ref{legendre}. \begin{theorem} \label{Legendre} The symmetric function $h_2+\mathbf{b}_0$ is the Legendre transform of $e_2-\AA_0$. \end{theorem} The main result of this paper is a formula for $\mathbf{b}_1$ in terms of $\AA_1$ and $\AA_0$, generalizing Theorem \ref{semi}. If $f$ is a symmetric function, write $\dot{f}=\partial f/\partial p_2=\tfrac12 p_2^\perp f$. \begin{theorem} \label{Semi} $$ \mathbf{b}_1 = \biggl( \AA_1 - \frac{1}{2} \sum_{n=1}^\infty \frac{\phi(n)}{n} \log(1-\psi_n(\AA_0'')) + \frac{\dot{\AA}_0(\dot{\AA}_0+1)}{1-\psi_2(\AA_0'')} \biggr) \circ (h_1+\mathbf{b}_0') $$ Here, $\phi(n)$ is Euler's function, the number of prime residues modulo $n$. \end{theorem} \begin{remark} The first two terms inside the parentheses on the right-hand side of Theorem \ref{Semi} are analogues of the corresponding terms in the formula of Theorem \ref{semi}. In particular, the second of these terms is closely related to the sum over necklaces in the definition of $\mathbb{M}\mathcal{V}\(1,n\)$, as is seem from the formula $$ \sum_{n=1}^\infty \ch_n\bigl( \Ind_{\mathbb{Z}_n}^{\SS_n} {1\!\!1} \bigr) = - \sum_{n=1}^\infty \frac{\phi(n)}{n} \log(1-p_n) . $$ The remaining term may be understood as a correction term, which takes into account the fact that necklaces of $1$ or $2$ vertices have non-trivial involutions (while those with more vertices do not). A proof of the theorem could no doubt be given using this observation; however, we prefer to derive it directly from Theorem \ref{modular}. If we take the plethysm on the right of the formula of Theorem \ref{Semi} with the symmetric function $h_1-\AA_0'$, and apply the formula $(h_1+\mathbf{b}_0')\circ(h_1-\AA_0')=h_1$, we obtain the equivalent formulation of this theorem: $$ \mathbf{b}_1 \circ (h_1-\AA_0') = \AA_1 - \frac{1}{2} \sum_{n=1}^\infty \frac{\phi(n)}{n} \log(1-\psi_n(\AA_0'')) + \frac{\dot{\AA}_0(\dot{\AA}_0+1)}{1-\psi_2(\AA_0'')} . $$ \end{remark} \begin{proof}[Proof of Theorem \ref{Semi}] The symmetric function $\mathbf{b}_1$ is a sum over graphs obtained by attaching forests whose vertices have genus $0$ to either a vertex of genus $1$, or to a necklace. In other words, $$ \mathbf{b}_1 = \bigl( \AA_1 + \text{sum over necklaces} \bigr) \circ (h_1+\mathbf{b}_0') . $$ To prove the theorem, we must calculate the sum over necklaces. To do this, observe that a necklace is a graph with flags coloured red or blue, such that each vertex has exactly two red flags, each edge is red, and all tails are blue. Let $\mathcal{W}\(n\)$, $n\ge1$, be the sequence of representations of $\SS_2\times\SS_n$ $$ \mathcal{W}\(n\) = \Res^{\SS_{n+2}}_{\SS_n\times\SS_2} \mathcal{V}\(0,n+2\) ; $$ think of the first factor of the product $\SS_n\times\SS_2$ as acting on the blue flags at a vertex, and the second factor as acting on the red flags. Applying Theorem \ref{modular}, we see that $$ \Log \bigl( \exp(1\o\Delta) \Exp(\Ch(\mathcal{W})) \bigr) \in \Lambda\Hat{\otimes}\Lambda\Hat{\otimes} K_0({\mathcal C}) $$ is the sum over stable graphs all of whose edges are red. To impose the condition that all tails are blue, we set the variables $q_n$ to zero before taking the Logarithm. We now proceed to the explicit calculation. We set $\hbar=1$, since it plays no r\^ole when all graphs have genus $1$. In writing elements of $\Lambda\Hat{\otimes}\Lambda$, we will denote power sums in the first factor of $\Lambda$ by $p_n$, and in the second by $q_n$. \begin{lemma} The characteristic $\Ch(\mathcal{W})$ of $\mathcal{W}$ is the ``bisymmetric'' function $$ \Ch(\mathcal{W}) = \tfrac12 \AA_0''q_1^2 + \dot{\AA}_0 q_2 \in \Lambda\Hat{\otimes}\Lambda_2\Hat{\otimes} K_0({\mathcal C}) . $$ \end{lemma} \begin{proof} We have $\Ch(\mathcal{W})=h_2^\perp \AA_0\o h_2+e_2^\perp \AA_0\o e_2$. Expressing this in terms of power sums, we have \begin{align*} h_2^\perp \AA_0\o h_2+e_2^\perp \AA_0\o e_2 &= \bigl(\tfrac12(p_1^\perp)^2+p_2^\perp\bigr)\AA_0\o\tfrac12(q_1^2+q_2) + \bigl(\tfrac12(p_1^\perp)^2-p_2^\perp\bigr)\AA_0\o\tfrac12(q_1^2-q_2) \\ &= \tfrac12 (p_1^\perp)^2\AA_0\o q_1^2 + p_2^\perp \AA_0\o q_2 . \end{align*} \def{} \end{proof} From this lemma, it follows that $$ \Exp\bigl( \Ch(\mathcal{W}) \bigr) = \prod_{n=1}^\infty \exp\Bigl( \psi_n(\AA_0'') \frac{q_n^2}{2n} \Bigr) \prod_{n=1}^\infty \exp\Bigl( \psi_n(\dot{\AA}_0) \frac{q_{2n}}{n} \Bigr) \in \Lambda\Hat{\otimes}\Lambda\Hat{\otimes} K_0({\mathcal C}) , $$ We now apply the heat kernel and separate variables: \begin{multline*} \exp(1\o\Delta) \Exp\bigl( \Ch(\mathcal{W}) \bigr)\big|_{q_n=0} = \prod_{\text{$n$ odd}} \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} \right) \exp\Bigl( \psi_n(\AA_0'') \frac{q_n^2}{2n} \Bigr) \Big|_{q_n=0} \\ {} \times \prod_{\text{$n$ even}} \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} + \frac{\partial}{\partial q_n} \right) \exp\Bigl( \psi_n(\AA_0'') \frac{q_n^2}{2n} + \psi_{n/2}(\dot{\AA}_0) \frac{2q_n}{n} \Bigr) \Big|_{q_n=0} . \end{multline*} We now insert the explicit formulas for the heat kernel of the Laplacian, namely $$ \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} \right) f(q_n) \big|_{q_n=0} = \int_{-\infty}^\infty f(q_n) \exp\biggl( - \frac{q^2}{2n} \biggr) \frac{dq}{\sqrt{2\pi n}} . $$ For the odd variables, matters are quite straightforward: \begin{align*} \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} \right) \exp\Bigl( \frac{q_n^2}{2n} \psi_n(\AA_0'') \Bigr) \Big|_{q_n=0} &= \int_{-\infty}^\infty \exp\biggl( \psi_n(\AA_0'') \frac{q_n^2}{2n} - \frac{q_n^2}{2n} \biggr) \frac{dq_n}{\sqrt{2\pi n}} \\ &= \bigl( 1 - \psi_n(\AA_0'') \bigr)^{-1/2} . \end{align*} For the even variables, things become a little more involved: \begin{multline*} \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} + \frac{\partial}{\partial q_n} \right) \exp\Bigl( \psi_n(\AA_0'') \frac{q_n^2}{2n} + \psi_{n/2}(\dot{\AA}_0) \frac{2q_n}{n} \Bigr) \Big|_{q_n=0} \\ \begin{aligned} {} &= \exp\left( \frac{n}{2} \frac{\partial^2}{\partial q_n^2} \right) \exp\Bigl( \psi_n(\AA_0'') \frac{q_n^2}{2n} + \psi_{n/2}(\dot{\AA}_0) \frac{2q_n}{n} \Bigr) \Big|_{q_n=1} \\ {} &= \int_{-\infty}^\infty \exp\biggl( \psi_n(\AA_0'') \frac{q_n^2}{2n} + \psi_{n/2}(\dot{\AA}_0) \frac{2q_n}{n} - \frac{(q_n-1)^2}{2n} \biggr) \frac{dq_n}{\sqrt{2\pi n}} . \end{aligned} \end{multline*} To perform this gaussian integral, we complete the square in the exponent: \begin{multline*} \psi_n(\AA_0'') \frac{q_n^2}{2n} + \psi_{n/2}(\dot{\AA}_0) \frac{2q_n}{n} - \frac{(q_n-1)^2}{2n} \\ \begin{aligned} {} &= - \bigl( 1-\psi_n(\AA_0'') \bigr) \frac{q_n^2}{2n} + \bigl( 2\psi_{n/2}(\dot{\AA}_0) + 1 \bigr) \frac{q_n}{n} - \frac{1}{2n} \\ {} & = - \frac{1-\psi_n(\AA_0'')}{2n} \biggl( q_n - \frac{2\psi_{n/2}(\dot{\AA}_0)+1}{1-\psi_n(\AA_0'')} \biggr)^2 + \frac{2}{n} \Biggl( \frac{\psi_{n/2}(\dot{\AA}_0) \bigl(\psi_{n/2}(\dot{\AA}_0)+1\bigr)}{1-\psi_n(\AA_0'')} \Biggr) . \end{aligned} \end{multline*} Thus, the gaussian integral equals $$ \bigl( 1-\psi_n(\AA_0'') \bigr)^{-1/2} \exp\frac{2}{n} \Biggl( \frac{\psi_{n/2}(\dot{\AA}_0)\bigl(\psi_{n/2}(\dot{\AA}_0)+1\bigr)} {1-\psi_n(\AA_0'')} \Biggr) . $$ Putting these calculations together, we see that \begin{align*} \exp(1\o\Delta) \Exp\bigl( \Ch(\mathcal{W}) \bigr) |_{q_n=0} &= \prod_{n=1}^\infty \bigl( 1 - \psi_n(\AA_0'') \bigr)^{-1/2} \exp\frac{1}{n} \Biggl( \frac{\psi_n(\dot{\AA}_0)\bigl(\psi_n(\dot{\AA}_0)+1\bigr)} {1-\psi_{2n}(\AA_0'')} \Biggr) \\ &= \prod_{n=1}^\infty \bigl( 1 - \psi_n(\AA_0'') \bigr)^{-1/2} \Exp \biggl( \frac{\dot{\AA}_0 (\dot{\AA}_0+1)}{1-\psi_2(\AA_0'')} \biggr) , \end{align*} and, applying the operation $\Log$, that $$ \Log \bigl( \exp(1\o\Delta) \Exp\bigl( \Ch(\mathcal{W}) \bigr) |_{q_n=0} \bigr) = \Log \prod_{n=1}^\infty \bigl( 1 - \psi_n(\AA_0'') \bigr)^{-1/2} + \frac{\dot{\AA}_0(\dot{\AA}_0+1)}{1-\psi_2(\AA_0'')} . $$ The proof of the theorem is completed by the following lemma., applied to $f=1-\AA_0''$. \begin{lemma} Let $f\in\Lambda\Hat{\otimes} K_0({\mathcal C})$ have constant term equal to $1$; that is, $\rk(f)=1+O(x)$. Then $$ \Log \prod_{n=1}^\infty \psi_n(f)^{-1/2} = - \frac{1}{2} \sum_{n=1}^\infty \frac{\phi(n)}{n} \log(\psi_n(f)) . $$ \end{lemma} \begin{proof} By definition, $$ \Log \prod_{n=1}^\infty \psi_n(f)^{-1/2} = \sum_{k=1}^\infty \frac{\mu(k)}{k} \log \prod_{n=1}^\infty \psi_{nk}(f)^{-1/2} = - \frac{1}{2} \prod_{n=1}^\infty \Bigl( \sum_{d|n} \frac{\mu(d)}{d} \Bigr) \log(\psi_n(f)) . $$ The lemma follows from the formula $$ \sum_{d|n} \frac{\mu(d)}{d} = \frac{\phi(n)}{n} , $$ which follows by M\"obius inversion from $\sum_{d|n}\phi(d)=n$. \end{proof} \def{} \end{proof} \begin{corollary} \label{sEMI} Define $a_g=\rk(\AA_g)$, $b_g=\rk(\mathbf{b}_g)$, and $\dot{a}_0=\rk(\dot{\AA}_0)$. Then we have $$ a_1 \circ (x-a_0') = a_1 - \tfrac12 \log(1-a_0'') + \dot{a}_0 (\dot{a}_0+1) . $$ \end{corollary} \begin{example} Suppose $\mathcal{V}\(0,n\)={1\!\!1}$ is the trivial one-dimensional representation for all $n\ge3$, while $\mathcal{V}\(1,n\)=0$. Then $\mathbb{M}\mathcal{V}\(1,n\)$ is an $\SS_n$-module whose rank is the number of graphs in $\Gamma^0_{1,n}$, where $\Gamma^0_{1,n}\subset\Gamma_{1,n}$ is the subset of stable graphs all of whose vertices have genus $0$. We have $$ \AA_0 = \sum_{n=3}^\infty h_n = \exp\Bigl( \sum_{n=1}^\infty \frac{p_n}{n} \Bigr) - 1 - h_1 - h_2 . $$ Theorem \ref{Semi} leads to the following results; the calculations were performed using J.~Stembridge's symmetric function package \texttt{SF} for \texttt{maple} \cite{SF}. $$\begin{tabular}{|C|L|L|} \hline n & \ch_n\bigl(\mathbb{M}\mathcal{V}\(1,n\)\bigr) & |\Gamma_{1,n}^0| \\ \hline 1 & s_{1} & 1 \\[5pt] 2 & 3\,s_{2} & 3 \\[5pt] 3 & 7\,s_{3}+4\,s_{21} & 15 \\[5pt] 4 & 20\,s_{4}+17\,s_{31}+14\,s_{2^2}+4\,s_{21^2} & 111 \\[5pt] 5 & 52\,s_{5}+78\,s_{41}+71\,s_{32}+33\,s_{31^2}+34\,s_{2^21}+4\,s_{21^3}+s_{1^5} & 1104 \\ \hline \end{tabular}$$ An explicit formula for the generating function of the numbers $|\Gamma_{1,n}^0|$ may be obtained from Corollary \ref{sEMI}, using the formulas $a_0'=e^x-1-x$, $a_0''=e^x-1$ and $\dot{a}_0=\tfrac12(e^x-1)$. \begin{proposition} $$ \sum_{n=1}^\infty |\Gamma^0_{1,n}| \frac{x^n}{n!} = \Bigl( - \frac{1}{2} \log \bigl( 2 - e^x \bigr) + \frac{1}{4} (e^{2x}-1) \Bigr) \circ (1+2x-e^x)^{-1} . $$ \end{proposition} \end{example} \section{The $\SS_n$-equivariant Hodge polynomial of $\overline{\mathcal{M}}_{1,n}$} A more interesting application of Theorem \ref{Semi} is to the stable $\SS$-module in the category of $\mathbb{Z}$-graded mixed Hodge structures $$ \mathcal{V}\(g,n\) = H^\bullet_c(\mathcal{M}_{g,n},\mathbb{C}) . $$ Let ${\mathsf{KHM}}$ be the Grothendieck group of mixed Hodge structures. The $\SS_n$-equivariant Serre polynomial $\Serre^{\SS_n}(\mathcal{M}_{g,n})$ is by definition the characteristic $\ch_n(\mathcal{V}\(g,n\))\in\Lambda_n\o{\mathsf{KHM}}$. It follows from the usual properties of Serre polynomials (see \cite{I} or Proposition (6.11) of \cite{modular}) that $\ch_n(\mathbb{M}\mathcal{V}\(g,n\))$ is the $\SS_n$-equivariant Serre polynomial $\Serre^{\SS_n}(\overline{\mathcal{M}}_{g,n})$ of the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves. Since the moduli space $\overline{\mathcal{M}}_{g,n}$ is a complete smooth Deligne-Mumford stack, its $k$th cohomology group carries a pure Hodge structure of weight $k$; thus, the Hodge polynomial of $\overline{\mathcal{M}}_{g,n}$ may be extracted from $\Serre^{\SS_n}(\overline{\mathcal{M}}_{g,n})$. Using Theorem \ref{Semi}, we will calculate the Serre polynomials $\Serre^{\SS_n}(\overline{\mathcal{M}}_{1,n})$. It is shown in \cite{gravity} (see also \cite{I}) that $$ \AA_0 = \sum_{n=3}^\infty \Serre^{\SS_n}(\mathcal{M}_{0,n}) = \frac{\displaystyle \biggl\{ \prod_{n=1}^\infty (1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)(1+\mathsf{L}^d)} \biggr\} - 1}{\mathsf{L}^3-\mathsf{L}} - \frac{h_1}{\mathsf{L}^2-\mathsf{L}} - \frac{h_2}{\mathsf{L}+1} , $$ where $\mathsf{L}$ is the pure Hodge structure $\mathbb{C}(-1)$ of weight $2$. Theorem \ref{Legendre} implies that $$ h_2 + \mathbf{b}_0 = h_2 + \sum_{n=3}^\infty \Serre^{\SS_n}(\overline{\mathcal{M}}_{0,n}) $$ is the Legendre transform of $e_2-\AA_0$; this was used in \cite{gravity} to calculate $\Serre^{\SS_n}(\overline{\mathcal{M}}_{0,n})$. Let $\mathsf{S}_{2k+2}$ be the pure Hodge structure $\operatorname{gr}^W_{2k+1}H^1_c(\mathcal{M}_{1,1},\operatorname{Sym}^{2k}\mathsf{H})$, where $\mathsf{H}$ is the local system $R^1\pi_*\mathbb{Q}$ of rank $2$ over the moduli stack of elliptic curves. (Here, $\pi:\overline{\mathcal{M}}_{1,2}\to\overline{\mathcal{M}}_{1,1}$ is the universal elliptic curve.) This Hodge structure has the following properties: \begin{enumerate} \item $\mathsf{S}_{2k+2}=F^0\mathsf{S}_{2k+2}\oplus\overline{F^0\mathsf{S}_{2k+2}}$; \item there is a natural isomorphism between $F^0\mathsf{S}_{2k+2}$ and the space of cusp forms $S_{2k+2}$ for the full modular group $\SL(2,\mathbb{Z})$. (In particular, $\mathsf{S}_{2k+2}=0$ for $k\le4$.) \end{enumerate} It is shown in \cite{II} that \begin{multline*} \AA_1 = \sum_{n=1}^\infty \Serre^{\SS_n}(\mathcal{M}_{1,n}) = \res_0 \Biggl[ \left( \frac{\prod_{n=1}^\infty (1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)(1-\omega^d-\mathsf{L}^d/\omega^d+\mathsf{L}^d)} - 1} {1-\omega-\mathsf{L}/\omega+\mathsf{L}} \right) \\ \times \left( \sum_{k=1}^\infty \biggl( \frac{\mathsf{S}_{2k+2}+1}{\mathsf{L}^{2k+1}} \biggr) \omega^{2k} - 1 \right) \bigl( \omega-\mathsf{L}/\omega \bigr) d\omega \Biggr] , \end{multline*} where $\res_0[\alpha]$ is the residue of the one-form $\alpha$ at the origin. We may now apply Theorem \ref{Semi} to calculate the generating function of the $\SS_n$-equivariant Serre polynomials $\Serre^{\SS_n}(\overline{\mathcal{M}}_{1,n})$. We do not give the details, since they are quite straightforward, though the resulting formulas are tremendously complicated when written out in full. However, we do present some sample calculations, performed with the package \texttt{SF}. $$\begin{tabular}{|C|L|L|} \hline n & \Serre\bigl(\overline{\mathcal{M}}_{1,n}\bigr) & \chi(\overline{\mathcal{M}}_{1,n}) \\[2pt] \hline 1 & (\mathsf{L}+1)s_1 & 2 \\[5pt] 2 & (\mathsf{L}^2+2\mathsf{L}+1)s_2 & 4 \\[5pt] 3 & (\mathsf{L}^3+3\mathsf{L}^2+3\mathsf{L}+1)s_3+(\mathsf{L}^2+\mathsf{L})s_{21} & 12 \\[5pt] 4 & (\mathsf{L}^4+4\mathsf{L}^3+7\mathsf{L}^2+4\mathsf{L}+1)s_4+(2\mathsf{L}^3+4\mathsf{L}^2+2\mathsf{L})s_{31} +(\mathsf{L}^3+2\mathsf{L}^2+\mathsf{L})s_{2^2} & 49 \\[5pt] 5 & (\mathsf{L}^5+5\mathsf{L}^4+12\mathsf{L}^3+12\mathsf{L}^2+5\mathsf{L}+1)s_5 +(3\mathsf{L}^4+11\mathsf{L}^3+11\mathsf{L}^2+3\mathsf{L})s_{41} & 260 \\ & {}+(2\mathsf{L}^4+7\mathsf{L}^3+7\mathsf{L}^2+2\mathsf{L})s_{32}+(\mathsf{L}^3+\mathsf{L}^2)(s_{31^2}+s_{2^21}) & \\[2pt] \hline \end{tabular}$$ In a table at the end of the paper, we give a table of non-equivariant Serre polynomials of $\overline{\mathcal{M}}_{1,n}$ for $n\le15$; these give an idea of the way in which the Hodge structures $\mathsf{S}_{2k+2}$ typically enter into the cohomology. In particular, we see that the even-dimensional cohomology of the moduli spaces $\overline{\mathcal{M}}_{1,n}$ is spanned by Hodge structures of the form $\mathbb{Q}(\ell)$, while the odd dimensional cohomology is spanned by Hodge structures of the form $\mathsf{S}_{2k+2}(\ell)$. The rational cohomology groups of $\overline{\mathcal{M}}_{1,n}$ satisfy Poincar\'e duality: there is a non-degenerate $\SS_n$-equivariant pairing of Hodge structures $$ H^k(\overline{\mathcal{M}}_{1,n},\mathbb{Q}) \o H^{2n-k}(\overline{\mathcal{M}}_{1,n},\mathbb{Q}) \to \mathbb{Q}(-n) . $$ Unfortunately, our formula for $\Serre^{\SS_n}(\overline{\mathcal{M}}_{1,n})$ does not render this duality manifest. \section{The Euler characteristic of $\overline{\mathcal{M}}_{1,n}$} As an application of Corollary \ref{sEMI}, we give an explicit formula for the generating function of the Euler characteristics $\chi(\overline{\mathcal{M}}_{1,n})$. \begin{theorem} \label{funny} Let $g(x)\in x+x^2\mathbb{Q}\[x\]$ be the solution of the equation $$ 2g(x)-(1+g(x))\log(1+g(x))=x . $$ Then $$ \sum_{n=1}^\infty \chi(\overline{\mathcal{M}}_{1,n}) \frac{x^n}{n!} = - \frac{1}{12} \log\bigl(1+g(x)\bigr) - \frac{1}{2} \log\bigl(1-\log(1+g(x))\bigr) + \epsilon(g(x)) , $$ where $$ \epsilon(x) = \frac{1}{12} \bigl( 19\,x + 23\,x^2/2 + 10\,x^3/3 + x^4/2 \bigr) . $$ \end{theorem} \begin{proof} We apply Corollary \ref{sEMI} with the data \begin{align*} a_0' &= \sum_{n=2}^\infty \chi(\mathcal{M}_{0,n+1}) \frac{x^n}{n!} = \sum_{n=2}^\infty (-1)^n (n-2)! \frac{x^n}{n!} = (1+x)\log(1+x)-x , \\ a_0'' &= \log(1+x) , \quad \dot{a}_0 = \frac{1}{4} x(x+2) , \\ a_1 &= \chi(\mathcal{M}_{1,1}) x + \chi(\mathcal{M}_{1,2}) \frac{x^2}{2} + \chi(\mathcal{M}_{1,3}) \frac{x^3}{6} + \chi(\mathcal{M}_{1,4}) \frac{x^4}{24} + \frac{1}{12} \sum_{n=5}^\infty (-1)^n (n-1)! \frac{x^n}{n!} \\ &= x + \frac{x^2}{2} - \frac{1}{12} \log(1+x) + \frac{1}{12} \Bigl( x + \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \Bigr) , \\ \end{align*} where we have used that $\chi(\mathcal{M}_{1,1})=\chi(\mathcal{M}_{1,2})=1$ and $\chi(\mathcal{M}_{1,3})=\chi(\mathcal{M}_{1,4})=0$. The function $g(x)$ of the statement of the theorem is $x+\mathbf{b}_0'(x)$. \end{proof} The following corollary was shown us by D. Zagier. \begin{corollary} \label{don} $$ \chi(\overline{\mathcal{M}}_{1,n}) \sim \frac{(n-1)!}{4(e-2)^n} \Bigl( 1 + C n^{-1/2} + O\bigl(n^{-3/2}\bigr) \Bigr) , $$ where $$ C = \sqrt{\frac{e-2}{18\pi e}} ( 1 + 4e + 9e^2 + 4e^3 + 2e^4 ) \approx 18.31398807 . $$ \end{corollary} \begin{proof} To show this, we analytically continue $g(x)$ to the domain $\mathbb{C}\setminus[e-2,\infty)$. The resulting function has an asymptotic expansion of the form $$ g(x) \sim e - 1 - \sqrt{2e(e-2-x)} + \sum_{k=3}^\infty a_k (e-2-x)^{k/2} . $$ The asymptotics \eqref{don} follow by applying Cauchy's integral formula to the right-hand side of Theorem \ref{funny}, with contour the circle $|x|=e-2$. \end{proof} The peculiar polynomial $\epsilon(x)$ of Theorem \ref{funny} combines the error terms in the formula for $\chi(\mathcal{M}_{1,n})$ with the correction terms involving $\dot{a}_0$ in Corollary \ref{sEMI}. Omitting the term $\epsilon(g(x))$ in Theorem \ref{funny}, we obtain the generating function not of the Euler characteristics $\chi(\overline{\mathcal{M}}_{1,n})$, but rather of the virtual Euler characteristics $\chi_v(\overline{\mathcal{M}}_{1,n})$ of the underlying smooth moduli stack (orbifold). The asymptotic behaviour of the virtual Euler characteristics is the same as that of the Euler characteristics, with $C$ replaced by $\widetilde{C}=\bigl(\frac{e-2}{18\pi e}\bigr)^{1/2} \approx 0.06835794$. The ratio between these Euler characteristics has the asymptotic behaviour $$ \frac{\chi(\overline{\mathcal{M}}_{1,n})}{\chi_v(\overline{\mathcal{M}}_{1,n})} \sim (C-\widetilde{C}) n^{-1/2} + O(n^{-1}) , $$ giving a statistical measure of the ramification of $\overline{\mathcal{M}}_{1,n}$ for large $n$. \begin{sideways} $$\begin{tabular}{|C|L|} \hline n & \Serre(\overline{\mathcal{M}}_{1,n}) \\ \hline 1 & \mathsf{L}+1 \\ 2 & \mathsf{L}^2+2\,\mathsf{L}+1 \\ 3 & \mathsf{L}^3+5\,\mathsf{L}^2+5\,\mathsf{L}+1 \\ 4 & \mathsf{L}^4+12\,\mathsf{L}^3+23\,\mathsf{L}^2+12\,\mathsf{L}+1 \\ 5 & \mathsf{L}^5+27\,\mathsf{L}^4+102\,\mathsf{L}^3+102\,\mathsf{L}^2+27\,\mathsf{L}+1 \\ 6 & \mathsf{L}^6+58\,\mathsf{L}^5+421\,\mathsf{L}^4+756\,\mathsf{L}^3+421\,\mathsf{L}^2+58\,\mathsf{L}+1 \\ 7 & \mathsf{L}^7+121\,\mathsf{L}^6+1612\,\mathsf{L}^5+5077\,\mathsf{L}^4+5077\,\mathsf{L}^3+1612\,\mathsf{L}^2+12\,\mathsf{L}+1 \\ 8 & \mathsf{L}^8+248\,\mathsf{L}^7+5802\,\mathsf{L}^6+31072\,\mathsf{L}^5+52402\,\mathsf{L}^4+31072\,\mathsf{L}^3 +5802\,\mathsf{L}^2+248\,\mathsf{L}+1 \\ 9 & \mathsf{L}^9+503\,\mathsf{L}^8+19925\,\mathsf{L}^7+175036\,\mathsf{L}^6+480097\,\mathsf{L}^5+480097\,\mathsf{L}^4 +175036\,\mathsf{L}^3+19925\,\mathsf{L}^2+503\,\mathsf{L}+1 \\ 10 & \mathsf{L}^{10}+1014\,\mathsf{L}^9+66090\,\mathsf{L}^8+920263\,\mathsf{L}^7+3975949\,\mathsf{L}^6 +6349238\,\mathsf{L}^5+3975949\,\mathsf{L}^4+920263\,\mathsf{L}^3+66090\,\mathsf{L}^2+1014\,\mathsf{L}+1 \\ 11 & \mathsf{L}^{11}+2037\,\mathsf{L}^{10}+213677\,\mathsf{L}^9+4577630\,\mathsf{L}^8+30215924\,\mathsf{L}^7 +74269967\,\mathsf{L}^6+30215924\,\mathsf{L}^5+\ldots+1 - \mathsf{S}_{12} \\ 12 & \mathsf{L}^{12}+4084\,\mathsf{L}^{11}+677881\,\mathsf{L}^{10}+21793602\,\mathsf{L}^9 +213725387\,\mathsf{L}^8+784457251\,\mathsf{L}^7+1196288936\,\mathsf{L}^6+\ldots+4084\,\mathsf{L}+1 -11(\mathsf{L}+1)\mathsf{S}_{12} \\ 13 & \mathsf{L}^{13}+8179\,\mathsf{L}^{12}+2120432\,\mathsf{L}^{11}+100226258\,\mathsf{L}^{10} +1424858788\,\mathsf{L}^9+7603002045\,\mathsf{L}^8+17095248952\,\mathsf{L}^7+\ldots \\ & \quad {} - (66\,\mathsf{L}^2+429\,\mathsf{L}+66)\mathsf{S}_{12} \\ 14 & \mathsf{L}^{14}+16370\,\mathsf{L}^{13}+6563147\,\mathsf{L}^{12}+448463866\,\mathsf{L}^{11} + 9049174765\,\mathsf{L}^{10}+68547770726\,\mathsf{L}^9 +221071720149\,\mathsf{L}^8+324314241400\,\mathsf{L}^7+\ldots \\ & \quad {} - (286\,\mathsf{L}^3+6006\,\mathsf{L}^2+286\,\mathsf{L})\mathsf{S}_{12} \\ 15 & \mathsf{L}^{15}+32753\,\mathsf{L}^{14}+20153930\,\mathsf{L}^{13}+1963368663\,\mathsf{L}^{12} +55228789080\,\mathsf{L}^{11} + 581636563570\,\mathsf{L}^{10} +2627427327522\,\mathsf{L}^9+5488190927216\,\mathsf{L}^8+\ldots \\ & \quad {} - (1001\,\mathsf{L}^4+53053\,\mathsf{L}^3+186263\,\mathsf{L}^2+53053\mathsf{L}+1001)\mathsf{S}_{12}-\mathsf{S}_{16} \\ \hline \end{tabular}$$ \end{sideways} \newpage

9709

alg-geom/9709004

en

https://arxiv.org/abs/alg-geom/9709004

alg-geom/9709004

Ravi Vakil

Ravi Vakil

Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points

LaTeX2e

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). We then show that these numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we suggest a direction from which the remaining del Pezzo surfaces can be approached, and give a conjectural algorithm to compute the genus g Gromov-Witten invariants of the cubic surface.

alg-geom

\section{Introduction} In this note, we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). These numbers are the genus $g$ Gromov-Witten invariants of the surface (Subsection \ref{gwenumerative}). The genus $g$ Gromov-Witten invariants of $\mathbb P^2$ were already computed in [R] and [CH3], and those of $\mathbb P^1 \times \mathbb P^1$ and the blow-up of $\mathbb P^2$ at a point ($\mathbb F_1$) were computed in [V3]. Such classical enumerative questions have recently been the object of study by many people. Ideas from mathematical physics (cf. the inspiring [KM] and [DI]) have yielded formulas when $g=0$ on $\mathbb P^2$ (via associativity relations in quantum cohomology). Z. Ran solved the analogous (enumerative) problem for curves of arbitrary genus on $\mathbb P^2$ by degenerations methods (cf. [R]), and Caporaso and Harris gave a second solution by different degeneration methods (cf. [CH3]). These numbers for irreducible curves are also the genus $g$ Gromov-Witten invariants of $\mathbb P^2$ (Subsection \ref{gwenumerative}). P. Di Francesco and C. Itzykson calculated the genus 0 Gromov-Witten invariants of the plane blown up at up to six points in [DI], Subsection 3.3. Y. Ruan and G. Tian gave recursive formulas for the genus 0 Gromov-Witten invariants of Fano surfaces, and indicated their enumerative significance ([RT] Section 10). L. G\"{o}ttsche and R. Pandharipande later derived recursive formulas for the genus 0 Gromov-Witten invariants of the plane blown up at any number of points ([GP]). E. Kussell has recovered the Gromov-Witten invariants of $\mathbb P^2$ blown up at 2 points by Caporaso and Harris' ``rational fibration method'' ([Ku]). In another direction, extending work of I. Vainsencher ([Va]), Kleiman and Piene have examined systems with an arbitrary, but fixed, number $\delta$ of nodes ([K2]). The postulated number of $\delta$-nodal curves is given (conjecturally) by a polynomial, and they determine the leading coefficients, which are polynomials in $\delta$. L. G\"{o}ttsche has recently conjectured a surprisingly simple generating function ([G]) for these polynomials which reproduce the results of Vainsencher as well as Kleiman and Piene and experimentally reproduce the numbers of [CH3], [V3], S.T. Yau and E. Zaslow's count of rational curves on K3-surfaces ([YZ]), and others. The numbers of curves are expressed in terms of four universal power series, two of which G\"{o}ttsche gives explicitly as quasimodular forms. The philosophy here is that of Caporaso and Harris in [CH3]: we degenerate the point conditions to lie on $E$ one at a time. Our perspective, however, is different: we use the moduli space of stable maps rather than the Hilbert scheme. \subsection{Examples} Suppose $d$ and $g$ are integers ($d >0$, $g \geq 0$), and $s$ is a sequence of non-negative integers with no more than five terms. Let $N^g_{d,s}$ be the number of irreducible genus $g$ degree $d$ plane curves with fixed multiple points with multiplicities given by $s$, passing through the appropriate number of general points. As an example of the algorithm, Table \ref{table1} gives values of $N^g_{d,s}$ for $d \leq 5$. When $g=0$, these numbers agree with those found by Pandharipande and G\"{o}ttsche ([GP] Subsection 5.2; the numbers were called $N_{d,s}$ there). When $s=0$, these numbers agree with those found by Caporaso and Harris ([CH3]). A conjectural algorithm for computing the genus $g$ Gromov-Witten invariants of the cubic surface (i.e. the plane blown up at six points, no three collinear and not all on a conic) is given in Subsection \ref{cubic}. Based on that conjecture, we compute $N^0_{6,(2^6)}=3240$. \begin{table} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $N^0_1$ & $N^0_2$ & $N^1_3$ & $N^0_3$ & $N^0_{3,(2)}$ & $N^3_4$ & $N^2_4$ & $N^1_4$ & $N^0_4$ & $N^2_{4,(2)}$ & $N^1_{4,(2)}$ & $N^0_{4,(2)}$ \\ \hline 1 & 1 & 1 & 12 & 1 & 1 & 27 & 225 & 620 & 1 & 20 & 96 \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline $N^1_{4,(2^2)}$ & $N^0_{4,(2^2)}$& $N^0_{4,(2^3)}$ & $N^0_{4,(3)}$ & $N^6_5$ & $N^5_5$ & $N^4_5$ & $N^3_5$ & $N^2_5$ & $N^1_5$ & $N^0_5$ \\ \hline 1 & 12 & 1 & 1 & 1 & 48 & 882 & 7915 & 36855 & 87192 & 87304 \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline $N^5_{5,(2)}$ & $N^4_{5,(2)}$ & $N^3_{5,(2)}$ & $N^2_{5,(2)}$ & $N^1_{5,(2)}$ & $N^0_{5,(2)}$ \\ \hline 1 & 41 & 615 & 4235 & 13775 & 18132 \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $N^4_{5,(2^2)}$ & $N^3_{5,(2^2)}$ & $N^2_{5,(2^2)}$ & $N^1_{5,(2^2)}$ & $N^0_{5,(2^2)}$ & $N^3_{5,(2^3)}$ & $N^2_{5,(2^3)}$ & $N^1_{5,(2^3)}$ & $N^0_{5,(2^3)}$ \\ \hline 1 & 34 & 396 & 1887 & 3510 & 1 & 27 & 225 & 620 \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $N^2_{5,(2^4)}$ & $N^1_{5,(2^4)}$ & $N^0_{5,(2^4)}$ & $N^1_{5,(2^5)}$ & $N^0_{5,(2^5)}$ & $N^3_{5,(3)}$ & $N^2_{5,(3)}$ & $N^1_{5,(3)}$ & $N^0_{5,(3)}$ \\ \hline 1 & 20 & 96 & 1 & 12 & 1 & 28 & 240 & 640 \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline $N^2_{5,(3,2)}$ & $N^1_{5,(3,2)}$ & $N^0_{5,(3,2)}$ & $N^1_{5,(3,2^2)}$ & $N^0_{5,(3,2^2)}$ & $N^0_{5,(3,2^3)}$ & $N^0_{5,(4)}$ \\ \hline 1 & 20 & 96 & 1 & 12 & 1 & 1 \\ \hline \end{tabular} \end{center} \caption{Numbers of plane curves with fixed multiple points} \label{table1} \end{table} \subsection{Acknowledgements} The author is grateful to J. Harris, S. Kleiman, R. Pandharipande, and T. Graber for useful advice and conversations. This work was developed and largely written while the author was enjoying the hospitality of the Mittag-Leffler Institute in May 1997, and he is grateful to the organizers of the special year in Quantum Cohomology for this opportunity. This research was supported by a Sloan Dissertation Fellowship. \section{Statement of results} For any sequence $\alpha = (\alpha_1, \alpha_2, \dots)$ of nonnegative integers with all but finitely many $\alpha_i$ zero, set $$ | \alpha | = \alpha_1 + \alpha_2 + \alpha_3 + \dots $$ $$ I \alpha = \alpha_1 + 2\alpha_2 + 3\alpha_3 + \dots $$ $$ I^\alpha = 1^{\alpha_1} 2^{\alpha_2} 3^{\alpha_3} \dots $$ and $$ \alpha ! = \alpha_1 ! \alpha_2 ! \alpha_3! \dots . $$ We denote by $\operatorname{lcm}(\alpha)$ the least common multiple of the set $\# \{ i : \alpha_i \neq 0 \}$. The zero sequence will be denoted 0. We denote by $e_k$ the sequence $(0, \dots, 0, 1, 0 , \dots)$ that is zero except for a 1 in the $k^{\text{th}}$ term (so that any sequence $\alpha = (\alpha_1, \alpha_2, \dots)$ is expressible as $\alpha = \sum \alpha_k e_k$). By the inequality $\alpha \geq \alpha'$ we mean $\alpha_k \geq \alpha_k'$ for all $k$; for such a pair of sequences we set $$ \binom \alpha {\alpha'} = {\frac { \alpha!} {\alpha' ! (\alpha - \alpha')!}} = \binom {\alpha_1} {\alpha'_1} \binom {\alpha_2} {\alpha'_2} \binom {\alpha_3} {\alpha'_3} \dots . $$ This notation follows [CH3] and [V3]. Let $H$ be the divisor class of a line in $\mathbb P^2$. Fix a degree $d$, a genus $g$, sequences $\alpha$ and $\beta$, and a collection of points $\Gamma = \{ p_{i,j} \}_{1 \leq j \leq \alpha_i}$ (not necessarily distinct) of $E$. We define the {\em generalized Severi variety} $W^{d,g}(\alpha,\beta,\Gamma)$ to be the closure (in $|dH|$) of the locus of irreducible reduced curves in $\mathbb P^2$ in class $dH$ of geometric genus $g$, not containing the conic $E$, with (informally) $\alpha_k$ ``assigned'' points of contact of order $k$ and $\beta_k$ ``unassigned'' points of contact of order $k$ with $E$. Formally, we require that, if $\nu: C^\nu \rightarrow C$ is the normalization of $C$, then there exist $|\alpha|$ points $q_{i,j} \in C^\nu$, $j=1, \dots, \alpha_i$ and $|\beta|$ points $r_{i,j} \in C^\nu$, $j=1, \dots, \beta_i$ such that $$ \nu(q_{i,j}) = p_{i,j} \quad \text{and} \quad \nu^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j}. $$ If $I \alpha + I\beta \neq dH \cdot E = 2d$, $W^{d,g}(\alpha,\beta,\Gamma)$ is empty. For convenience, let \begin{eqnarray*} \Upsilon = \Upsilon^{d,g}(\beta) &:=& - (K_{\mathbb P^2} + E) \cdot (dH) + |\beta| + g-1 \\ &=& d + |\beta| + g-1. \end{eqnarray*} Then $W^{d,g}(\alpha,\beta,\Gamma)$ is a projective variety of pure dimension $\Upsilon$ (Proposition \ref{bigdim}). {\em Notational warning:} The notation $V^{d,g}(\alpha,\beta,\Gamma)$ and $W^{d,g}(\alpha,\beta,\Gamma)$ is used in [CH3] to refer to a slightly different notion: the sequences $\alpha$ and $\beta$ refer there to tangencies with a fixed line, rather than the fixed conic $E$. \begin{defn} \label{simple} If $\{ p_{i,j} \}$ are distinct points except $s_k$ points $\{ p_{1,j} \}$ are the same $(1 \leq k \leq l)$, we will say that $\Gamma$ is {\em simple}. \end{defn} In this case, if $\alpha'= \alpha + | s | e_1$, $W^{d,g}(\alpha',\beta,\Gamma)$ generically parametrizes curves that have multiple points of order $s_1$, \dots, $s_l$ on $E$ (with each branch transverse to $E$), $i$-fold tangent to $E$ at $\alpha_i$ fixed points of $E$, and $i$-fold tangent to $E$ at $\beta_i$ other points of $E$. When discussing properties of $W^{d,g}(\alpha',\beta,\Gamma)$ that depend only on $\alpha$, $\beta$, and $s$, we write $W^{d,g}(\alpha,\beta,s)$ for convenience. Let $N_{\text{irr}}^{d,g}\abG$ be the number of points of $W^{d,g}\abG$ whose corresponding curve passes through $\Upsilon$ fixed general points of $\mathbb P^2$. If $\Gamma$ is {\em simple} then $N^{d,g}_{\text{irr}}(\alpha',\beta,\Gamma)$ depends only on $(\alpha,\beta,s)$ (Subsection \ref{therecursiveformulas}), so we write $N^{d,g}_{\text{irr}}\abs$. Then $N_{\text{irr}}^{d,g}\abs$ is the degree of the generalized Severi variety (in the projective space $|dH|$). The main result of this note is the following. \begin{tm} \label{irecursion} If $\dim W^{d,g}\abs>0$, then \begin{eqnarray*} N_{\text{irr}}^{d,g}\abs &=& \sum_{\beta_k > 0} k N_{\text{irr}}^{d,g}(\alpha + e_k, \beta - e_k,s) \\ & & + \sum \frac 1 \sigma \binom {\alpha} {\alpha^1, \dots, \alpha^l, \alpha- \sum \alpha^i} \binom {\Upsilon^{d,g}(\beta)-1} {\Upsilon^{d^1,g^1}(\beta^1), \dots, \Upsilon^{d^l,g^l}(\beta^l)} \\ & & \cdot \prod_{i=1}^l \binom {\beta^i} {\gamma^i} I^{\beta^i - \gamma^i} N_{\text{irr}}^{d^i,g^i}(\alpha^i,\beta^i,s^i) \end{eqnarray*} where the second sum runs over choices of $d^i, g^i, \alpha^i, \beta^i, \gamma^i, s^i$ ($1 \le i \le l$), where $d^i$ is a positive integer, $g^i$ is a non-negative integer, $\alpha^i$, $\beta^i$, $\gamma^i$, $s^i$ are sequences of non-negative integers, $\sum_i d^i = d-2$, $\sum_i \gamma^i = \beta$, $\beta^i \gneq \gamma^i$, $\sum_i s^i_k = s_k - 1$ or $s_k$, and $\sigma$ is the number of symmetries of the set $\{ (d^i,g^i,\alpha^i,\beta^i,\gamma^i, s^i) \}_{1 \leq i \leq l}$. \end{tm} In the second sum, for the summand to be non-zero, one must also have $\sum \alpha^i \leq \alpha$, and $I \alpha^i + I \beta^i + |s^i|= 2 d^i$. If $\tilde{s}$ is the sequence $s$ with the zeros removed, then clearly \begin{equation} \label{fred} W^{d,g}\abs = W^{d,g}(\alpha,\beta,\tilde{s}) \end{equation} If $a$ is the number of ones in $s$, and $\tilde{s}$ is the sequence $s$ with the ones removed, then clearly \begin{equation} \label{barney} W^{d,g}\abs = W^{d,g}(\alpha + a e_1,\beta,\tilde{s}). \end{equation} (Requiring a curve to have a multiplicity-1 multiple point at a fixed point of $E$ is the same as requiring the curve to pass through a fixed point of $E$.) If $s_k>1$ for all $k$, then the variety $W^{d,g}\abs$ has dimension 0 if and only if $d=1$, $g=0$, $\beta=0$, $s=0$, and $\alpha = 2 e_1$ or $e_2$ (from Proposition \ref{bigdim} and simple case-checking). The first case is a line through two fixed points of the conic $E$, and the second is a line tangent to $E$ at a fixed point. In both cases, $N^{d,g}_{\text{irr}}\abs = 1$. Therefore, with this ``seed data'', Theorem \ref{irecursion} provides a means of recursively computing $N_{\text{irr}}^{d,g}\abs$ for all $d$, $g$, $\alpha$, $\beta$, $s$. \subsection{Relationship to Gromov-Witten invariants} Let $B$ be the blow-up of the plane at $l$ points ($0 \leq l \leq 5$), no three collinear. Let $E$ be a smooth conic through the $l$ points (unique if $l=5$). Let $H$ be the pullback of a line to $B$, and $E_1$, \dots, $E_l$ the exceptional divisors. The irreducible genus $g$ curves in a divisor class $D$ on $B$, through an appropriate number ($-K_B \cdot D + g-1$) of points, can now be counted. Call this number $GW^{D,g}_B$ for convenience. If $D$ is an exceptional divisor, or the proper transform of $E$ if $l=5$, then $$ GW^{D,g}_B = \begin{cases} 1 & \text{if $g=0$,} \\ 0 & \text{otherwise.} \end{cases} $$ If $D= dH - \sum_{k=1}^l m_k E_k$ is any other divisor class, then the (finite number of) genus $g$ curves in class $D$ on $B$ through $-K_B \cdot D + g-1$ general points correspond to the degree $d$ plane curves with $l$ fixed multiple points of multiplicity $m_1$, \dots, $m_5$ through the same number of points. Thus if $s_i = D \cdot E_i$, then $$ GW^{D,g}_B = N^{d,g}(0,(2d-\sum m_i) e_1, s). $$ In Subsection \ref{gwenumerative}, it will be shown that the numbers $GW^{D,g}_B$ give all the genus $g$ Gromov-Witten invariants of $B$. (It was previously known that the genus 0 invariants are enumerative, see [RT] Section 10 and [GP] Lemma 4.10.) \subsection{The strategy} In order to understand generalized Severi varieties, we will analyze certain moduli spaces of maps. Let $\overline{M}_g(\mathbb P^2,d)$ be the moduli space of maps $\pi: C \rightarrow \mathbb P^2$ where $C$ is irreducible, complete, reduced, and nodal, $(C, \pi)$ has finite automorphism group, and $\pi_* [C] = dH$. Let $d$, $g$, $\alpha$, $\beta$, $\Gamma$ be as in the definition of $W^{d,g}(\alpha,\beta,\Gamma)$ above. Define the {\em generalized Severi variety of maps} $W^{d,g}_m(\alpha,\beta,\Gamma)$ to be the closure in $\overline{M}_g(\mathbb P^2,d)$ of points representing maps $(C,\pi)$ where each component of $C$ maps birationally to its image in $\mathbb P^2$, no component maps to $E$, and $C$ has (informally) $\alpha_k$ ``assigned'' points of contact of order $k$ and $\beta_k$ ``unassigned'' points of contact of order $k$ with $E$. Formally, we require that there exist $|\alpha|$ smooth points $q_{i,j} \in C$, $j=1, \dots, \alpha_i$ and $|\beta|$ smooth points $r_{i,j} \in C$, $j=1, \dots, \beta_i$ such that $$ \pi(q_{i,j}) = p_{i,j} \quad \text{and} \quad \pi^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j}. $$ There is a natural rational map from each component of $W^{d,g}\abs$ to $W^{d,g}_m\abs$, and the dimension of the image will be $\Upsilon$. We will prove: \begin{pr} \label{idim} The components of $W^{d,g}_m\abs$ have dimension at most $\Upsilon$, and the union of those with dimension exactly $\Upsilon$ is the closure of the image of $W^{d,g}\abs$ in $W^{d,g}_m\abG$. \end{pr} (This will be an immediate consequence of Theorem \ref{bigdim}.) Fix $\Upsilon$ general points $s_1$, \dots, $s_\Upsilon$ on $\mathbb P^2$. The image of the maps in $W^{d,g}_m\abs$ whose images pass through these points are reduced. ({\em Proof:} Without loss of generality, restrict to the union $W$ of those components of $W^{d,g}_m\abs$ with dimension $\Upsilon$. By Proposition \ref{idim}, the subvariety of $W$ corresponding to maps whose images are {\em not} reduced contains no components of $W$ and hence has dimension less than $\Upsilon$. Thus no image of such a map passes through $s_1$, \dots, $s_{\Upsilon}$.) Therefore, if $H$ is the divisor class on $W^{d,g}_m\abs$ corresponding to requiring the image curve to pass through a fixed point of $\mathbb P^2$, then $$ N_{\text{irr}}^{d,g}\ab = H^\Upsilon. $$ Define the {\em intersection dimension} of a family $V$ of maps to $\mathbb P^2$ (denoted $\operatorname{idim} V$) as the maximum number $n$ of general points $s_1$, \dots $s_n$ on $\mathbb P^2$ such that there is a map $\pi: C \rightarrow \mathbb P^2$ in $V$ with $\{ s_1, \dots, s_n \} \subset \pi(C)$. Clearly $\operatorname{idim} V \leq \dim V$. Our strategy is as follows. Fix a general point $q$ of $E$. Let $H_q$ be the Weil divisor on $W^{d,g}_m\abs$ corresponding to maps with images containing $q$. We will find the components of $W^{d,g}_m\abs$ with intersection dimension $\Upsilon-1$ and relate them to $W^{d',g'}_m(\alpha',\beta',s')$ for appropriately chosen $d'$, $g'$, $\alpha'$, $\beta'$, $s'$. Then we compute the multiplicity with which each of these components appears. Finally, we derive a recursive formula for $N_{\text{irr}}^{d,g}\abs$ (Theorem \ref{irecursion}). \subsection{Counting reducible curves} Analogous definitions can be made of the space $V^{d,g}_m\abs$ parametrizing possibly reducible curves. (Then $W^{d,g}_m\abs$ is the union of connected components of $V^{d,g}_m\abs$ where the source curve $C$ is connected.) The arguments in this case are identical, resulting in a recursive formula for $N^{d,g}\abs$, the number of maps $(C,\pi)$ from genus $g$ curves with $\pi_*[C]=dH$, and intersection of $\pi_* C$ with $E$ determined by $\alpha$ and $\beta$, passing through $\Upsilon$ fixed general points of $\mathbb P^2$: \begin{tm} \label{rrecursion} If $\dim V^{d,g}_m\abs>0$, then \begin{eqnarray*} N^{d,g}\abs = \sum_{\beta_k > 0} k N^{d,g}(\alpha + e_k, \beta-e_k,s) \\ + \sum I^{\beta'-\beta} {\binom \alpha {\alpha'}} \binom {\beta'}{\beta} N^{d-2,g'}(\alpha',\beta',s') \end{eqnarray*} where the second sum is taken over all $\alpha'$, $\beta'$, $g'$ satisfying $\alpha' \leq \alpha$, $\beta' \geq \beta$, $s'_k= s_k$ or $s_k-1$, $g-g' = |\beta'-\beta| - 1$, $I \alpha' + I \beta' +|s'| = 2d-4$. \end{tm} Although the recurrence is simpler than in Theorem \ref{irecursion}, the seed data is more complicated. If $\dim V^{d,g}_m\abs=0$ and $N^{d,g}\abs \neq 0$, then $\beta = 0$, $\alpha_k = 0$ for $k>2$, $g=1-2d$, and $2d = |s| + I \alpha$. In this case, each point of $V^{d,g}_m\abs$ corresponds to a union of lines in the plane. Then if $s'$ is the sequence $s$ with $\alpha_1$ ones appended, $N^{d,g}\abs$ is the number of ways of expressing $s'$ as a sum of sequences that are 0 except for 1's in two places. (This involves chasing through the definitions: we are counting maps of $2d$ $\mathbb P^1$'s to the plane, each of degree 1, intersecting $E$ in a certain way.) Alternatively, $N^{d,g}\abs$ is the number of graphs with $|s'|$ labeled vertices with vertex $i$ having valence $s'_i$, where each edge joints two different vertices (although more than one edge can connect the same pair of vertices). \subsection{Simplifying formulas} Computationally, it is simpler to deal with the formulas of Theorems \ref{irecursion} and \ref{rrecursion} when $s=0$. (This was the form proved in [V3].) The following lemma allows us to quickly reduce to the case $s=0$ when $s_i=0$ for $i>3$. \begin{lm} \begin{enumerate} \item[(0)] $N^{d,g}_{\text{irr}}(\alpha,\beta,s \cup (0)) = N^{d,g}_{\text{irr}}\abs$ \item[(1)] $N^{d,g}_{\text{irr}}(\alpha,\beta,s\cup (1)) = N^{d,g}_{\text{irr}}(\alpha+e_1,\beta,s)$ \item[(2)] $N^{d,g}_{\text{irr}}(\alpha,\beta,s \cup (2)) = \frac 1 2 \left( N^{d,g}_{\text{irr}}(\alpha + 2 e_1,\beta,s) - N^{d,g}_{\text{irr}}(\alpha + e_2,\beta,s) \right)$ \item[(3)] $N^{d,g}_{\text{irr}}(\alpha,\beta,s\cup (3)) = \frac 1 6 N^{d,g}_{\text{irr}}(\alpha + 3 e_1,\beta,s) - \frac 1 2 N^{d,g}_{\text{irr}}(\alpha + e_1 + e_2,\beta,s) + \frac 1 3 N^{d,g}_{\text{irr}}(\alpha + e_3,\beta,s)$ \end{enumerate} \end{lm} Parts (0) and (1) are tautological, and were stated earlier (equations (\ref{fred}) and (\ref{barney})). There are analogous expressions for $N^{d,g}_{\text{irr}}(\alpha,\beta,s \cup (n))$ for all $n$. The result still holds when $N^{d,g}_{\text{irr}}$ is replaced by $N^{d,g}$. The lemma can be proved by induction (on $\alpha$, $\beta$, $s$) and Theorem \ref{irecursion}. It can also be proven by degeneration methods, and by the study of divisors on generalized Severi varieties. \subsection{Variations on a theme} The same arguments provide means of computing the number of curves in $V^{d,g}\abG$ through an appropriate number of points for $(X,E) = (\mathbb P^2,H)$, ($\mathbb P^2,E)$, or $(\mathbb F_n,E)$, even if $\Gamma$ is not a collection of reduced points. \section{Proof of results} \subsection{Dimension counts} The pair $(X,E) = (\mathbb P^2,E)$ satisfies properties P1--P4 of [V3], Subsection 1.1: \begin{enumerate} \item[{\bf P1.}] $X$ is a smooth surface and $E \cong \mathbb P^1$ is a divisor on $X$. \item[{\bf P2.}] The surface $X \setminus E$ is minimal, i.e. contains no (-1)-curves. \item[{\bf P3.}] The divisor class $K_X + E$ is negative on every curve on $X$. \item[{\bf P4.}] If $D$ is an effective divisor such that $-(K_X + E) \cdot D = 1$, then $D$ is smooth. \end{enumerate} Recall [V3] Theorem 2.1, whose proof depended only on properties P1--P4: \begin{tm} \label{bigdim} \begin{enumerate} \item[(a)] Each component $V$ of $V^{d,g}_m\abG$ is of dimension at most $$ \Upsilon = \Upsilon^{d,g}(\beta) = -(K_X + E) \cdot (dH) + | \beta | + g-1. $$ \item[(b)] The stable map $(C, \pi)$ corresponding to a general point of any component of dimension $\Upsilon$ satisfies the following properties. \begin{enumerate} \item[(i)] The curve $C$ is smooth, and the map $\pi$ is an immersion. \item[(ii)] The image is a reduced curve. If $\Gamma$ consists of distinct points, then the image is smooth along its intersection with $E$. \end{enumerate} \item[(c)] Conversely, any component whose general map satisfies property (i) has dimension $\Upsilon$. \end{enumerate} \end{tm} By ``the image is a reduced curve'', we mean $\pi_*[C]$ is a sum of distinct irreducible divisors on $X$. \subsection{Determining intersection components and their multiplicities} Fix $d$, $g$, $\alpha$, $\beta$, $s$, $\Gamma$, and a general point $q$ on $E$. Let $H_q$ be the divisor on $V^{d,g}_m\abs$ corresponding to maps whose image contains $q$. We will derive a list of subvarieties (which we will call {\em intersection components}) in which each component of $H_q$ of intersection dimension $\Upsilon-1$ appears, and then calculate the multiplicity of $H_q$ along each such component. In [V3], in a more general situation, a list of intersection components was derived and the multiplicities calculated. We recall these results, and apply them in this particular case. The potential components come in two classes that naturally arise from requiring the curve to pass through $q$. First, one of the ``moving tangencies'' $r_{i,j}$ could map to $q$. We will call such components {\it Type I intersection components}. Second, the curve could degenerate to contain $E$ as a component. We will call such components {\it Type II intersection components}. For any sequences $\alpha'' \leq \alpha$, $\gamma \geq 0$, and subsets $\{ p''_{i,1}, \dots, p''_{i,\alpha''_i} \}$ of $\{ p_{i,1}, \dots, p_{i,\alpha_i} \}$, let $g'' = g + |\gamma| + 1$ and $\Gamma'' = \{ p''_{i,j} \}_{1 \leq j \leq \alpha''_i}$. Define $K(\alpha'',\beta,\gamma,\Gamma'')$ as the closure in $\overline{M}_g(X,d)'$ of points representing maps $\pi: C' \cup C'' \rightarrow X$ where \begin{enumerate} \item[K1.] the curve $C'$ maps isomorphically to $E$, \item[K2.] the curve $C''$ is smooth, each component of $C''$ maps birationally to its image, no component of $C''$ maps to $E$, and there exist $|\alpha''|$ points $q_{i,j} \in C''$, $j = 1$, \dots, $\alpha_i''$, $|\beta|$ points $r_{i,j} \in C''$, $j = 1$, \dots, $\beta_i$, $|\gamma|$ points $t_{i,j} \in C''$, $j = 1$, \dots, $\gamma_i$ such that $$ \pi(q_{i,j}) = p''_{i,j} \quad \text{and} \quad (\pi|_{C''})^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j} + \sum i \cdot t_{i,j}, $$ and \item[K3.] the intersection of the curves $C'$ and $C''$ is $\{ t_{i,j} \}_{i,j}$. \end{enumerate} The variety $K(\alpha'',\beta,\gamma,\Gamma'')$ is empty unless $I(\alpha''+\beta+\gamma) = (dH-E) \cdot E = 2(d-2)$. The genus of $C''$ is $g''$, and there is a degree $\binom {\beta+\gamma} \beta$ rational map \begin{equation} \label{rratmap} K(\alpha'',\beta,\gamma,\Gamma'') \dashrightarrow V_m^{dH-E,g''}(\alpha'',\beta+\gamma,\Gamma'') \end{equation} corresponding to ``forgetting the curve $C'$''. Define the Type II intersection component $K_V(\alpha'',\beta,\gamma,\Gamma'')$ in the same way as $K(\alpha'',\beta,\gamma,\Gamma'')$, with an additional condition: \begin{enumerate} \item[K4.] The collection $\Gamma' = \Gamma \setminus \Gamma''$ consists of distinct points. \end{enumerate} If for the general $(C,\pi)$ in $V^{d,g}_m\abs$, $\pi(C)$ has a $k$-fold point at some fixed $p$ in $E$, then for a general $(C' \cup C'', \pi)$ in $K_V(\alpha'',\beta,\gamma,\Gamma'')$, $\pi(C'')$ has at least a $(k-1)$-fold point at $p$, by condition K4. \begin{tm} \label{rlist} Fix $d$, $g$, $\alpha$, $\beta$, $\Gamma$, and a point $q$ on $E$ not in $\Gamma$. Let $K$ be an irreducible component of $H_q$ with intersection dimension $\Upsilon - 1$. Then set-theoretically, either \begin{enumerate} \item[I.] $K$ is a component of $V^{d,g}_m(\alpha+e_k,\beta-e_k,\Gamma')$, where $\Gamma'$ is the same as $\Gamma$ except $p'_{k,\alpha_{k+1}} = q$, or \item[II.] $K$ is a component of $K_V(\alpha'',\beta,\gamma,\Gamma'')$ for some $\alpha''$, $\gamma$, $\Gamma''$. \end{enumerate} \end{tm} \noindent {\em Proof. } By [V3], Theorem 3.1, either \begin{enumerate} \item[I.] $K$ is a component of $V^{d,g}_m(\alpha+e_k,\beta-e_k,\Gamma')$, where $\Gamma'$ is the same as $\Gamma$ except $p'_{k,\alpha_{k+1}} = q$, or \item[II.] $K$ is a component of $K(\alpha'',\beta,\gamma,\Gamma'')$ for some $\alpha''$, $\gamma$, $\Gamma''$. \end{enumerate} Suppose the point $p$ appears in $\Gamma$ $n$ times. Then for a general map $(C,\pi)$ in $V^{d,g}_m\abG$, $\pi^*(p)$ is a length $n$ scheme by Theorem \ref{bigdim}(b) above. If $K$ is a component of $K(\alpha'',\beta,\gamma,\Gamma'')$ for some $\alpha''$, $\gamma$, and $\Gamma''$, and $(C,\pi) = (C' \cup C'', \pi)$ is a general map in $K$, then $(\pi |_{C'})^* p$ has length 1, so $(\pi |_{C''})^* p$ must have length at least $n-1$. As $\pi|_{C''}$ is an immersion (by Theorem \ref{bigdim}(b)(i)), $(\pi|_{C''})^* p$ is the number of times $p$ appears in $\Gamma''$. Thus if $p$ appears $n$ times in $\Gamma$ then it appears at least $n-1$ times in $\Gamma''$, so it appears at most once in $\Gamma' = \Gamma \setminus \Gamma''$. Hence $K$ is actually a component of $K_V(\alpha'',\beta,\gamma,\Gamma'')$. \qed \vspace{+10pt} There are other components of the divisor $H_q$ not counted in Theorem \ref{rlist}, but they will be enumeratively irrelevant. (See the end of [V3] Section 3 for example of such behavior.) There is an analogous result for $W^{d,g}_m\abs$. Let $K_W(\alpha'',\beta,\gamma,\Gamma'')$ be the union of components of $K_V(\alpha'',\beta,\gamma,\Gamma'')$ where the source curve $C$ is connected. Then $C''$ is a union of $l$ curves with image of degree $d^i$, or arithmetic genus $g^i$, with a subset $\Gamma^i$ of the points $\Gamma$ (and induced sequence $\alpha^i \leq \alpha$), and induced sequences $\beta^i$, $\gamma^i$, $s^i$. As the source curve is connected, $\gamma^i>0$ for all $i$. Let $\sigma$ be the symmetry group of the data $(d^i,g^i,\alpha^i,\beta^i,\gamma^i,s^i)$ (so for example $\sigma$ is the one-element group if no two $(d^i,g^i,\alpha^i,\beta^i,\gamma^i,s^i)$ are the same, and $\sigma = S_l$ is they are all the same). Then there is a generically $|\sigma|$-to-1 cover of this component $K_W$ of $K_W(\alpha'',\beta,\gamma,\Gamma'')$ which distinguishes $C^1$, \dots, $C^l$. Then there is a degree $\prod \binom { \beta^i + \gamma^i} {\gamma^i}$ map \begin{equation} \label{iratmap} K_V \dashrightarrow \prod V^{d^i,g^i} (\alpha^i, \beta^i + \gamma^i, \Gamma^i). \end{equation} \begin{tm} \label{ilist} Fix $d$, $g$, $\alpha$, $\beta$, $\Gamma$, and a point $q$ on $E$ not in $\Gamma$. Let $K$ be an irreducible component of $H_q$ on $W^{d,g}_m\abG$ with intersection dimension $\Upsilon - 1$. Then set-theoretically, either \begin{enumerate} \item[I.] $K$ is a component of $W^{d,g}_m(\alpha+e_k,\beta-e_k,\Gamma')$, where $\Gamma'$ is the same as $\Gamma$ except $p'_{k,\alpha_{k+1}} = q$, or \item[II.] $K$ is a component of $K_W(\alpha'',\beta,\gamma,\Gamma'')$ for some $\alpha''$, $\gamma$, $\Gamma''$. \end{enumerate} \end{tm} \noindent {\em Proof. } As $W^{d,g}_m\abG$ is the union of components of $V^{d,g}_m\abG$ where the source is connected, this follows immediately from Theorem \ref{rlist}. \qed \vspace{+10pt} \subsection{Multiplicity of $H_q$ along intersection components} The proofs of the multiplicity calculations are identical to those of [V3]. Let $K_k$ be the union of Type I intersection components of the form $V^{d,g}_m(\alpha+e_k, \beta-e_k, \Gamma')$ as described in Theorem \ref{rlist}. The following proposition is [V3] Proposition 4.1. \begin{pr} \label{multI} The multiplicity of $H_q$ along $K_k$ is $k$. \end{pr} Suppose $K = K_V(\alpha'', \beta, \gamma, \Gamma'')$ is a Type II component of $H_q$ (on $V^{d,g}_m\ab$). Let $m_1$, \dots, $m_{|\gamma|}$ be a set of positive integers with $j$ appearing $\gamma_j$ times ($j = 1$, 2, \dots), so $\sum m_i = I \gamma$. The following proposition is [V3] Proposition 5.2. \begin{pr} \label{multII} The multiplicity of $H_q$ along $K$ is $m_1 \dots m_{|\gamma|} = I^{\gamma}$. \end{pr} (The proof in [V3] assumed only that $\Gamma \setminus \Gamma''$ consisted of distinct points.) As $W^{d,g}_m\abG$ is a connected union of components of $V^{d,g}_m\abG$, analogous multiplicity results hold for $W^{d,g}_m\abG$. \subsection{The Recursive Formulas} \label{therecursiveformulas} Let $H_q$ be the divisor on $V^{d,g}_m(\alpha,\beta,\Gamma)$ corresponding to requiring the image to contain a general point $q$ of $E$. The components of $H_q$ of intersection dimension $\Upsilon - 1$ were determined in Theorem \ref{rlist}, and the multiplicities were determined in Propositions \ref{multI} and \ref{multII}: \begin{pr} In the Chow ring of $V^{d,g}_m\abG$, modulo Weil divisors of intersection dimension less than $\Upsilon - 1$, $$ H_q = \sum_{\beta_k>0} k \cdot V^{d,g}_m(\alpha+e_k,\beta-e_k,\Gamma \cup \{ q \}) + \sum I^{\gamma} \cdot K_V( \alpha'', \beta, \gamma, \Gamma'') $$ where the second sum is over all $\alpha'' \leq \alpha$, $\gamma \geq 0$, $\Gamma'' = \{ p''_{i,j} \}_{1 \leq j \leq \alpha''_i} \subset \Gamma$ such that $\Gamma \setminus \Gamma''$ consists of distinct points, $I(\alpha'' + \beta + \gamma ) = (dH) \cdot E$. \end{pr} Intersect both sides of the equation with $H_q^{\Upsilon - 1}$. As those dimension $\Upsilon - 1$ classes of intersection dimension less than $\Upsilon - 1$ are annihilated by $H_q^{\Upsilon - 1}$, we still have equality: \begin{eqnarray*} N^{d,g}\abG &=& H_q^{\Upsilon} \\ &=& \sum_{\beta_k>0} k V^{d,g}_m(\alpha+e_k,\beta-e_k,\Gamma \cup \{ q \}) \cdot H_q^{\Upsilon - 1}\\ & & + \sum I^{\gamma} \cdot K_V( \alpha'', \beta, \gamma, \Gamma'') \cdot H_q^{\Upsilon - 1}. \end{eqnarray*} From (\ref{rratmap}), each $K_V(\alpha'', \beta, \gamma, \Gamma'')$ admits a degree $\binom {\beta + \gamma} \beta$ rational map to $V^{dH-E,g''}_m(\alpha'', \beta+\gamma,\Gamma'')$ (where $g'' = g- |\gamma| + 1$) corresponding to ``forgetting the component mapping to $E$'', so $$ K_V(\alpha'', \beta, \gamma, \Gamma'') \cdot H_q^{\Upsilon - 1} = \binom {\beta + \gamma} \gamma N^{d-2,g''}(\alpha'', \beta+\gamma). $$ For each $\alpha''$, there are $\binom \alpha {\alpha''}$ choices of $\Gamma''$ (as this is the number of ways of choosing $\{ p''_{i,1} , \dots, p''_{i,\alpha''_i} \}$ from $\{ p_{i,1} , \dots, p_{i,\alpha_i} \}$). Thus \begin{eqnarray} \nonumber N^{d,g}\abG &=& \sum_{\beta_k > 0} k N^{d,g}(\alpha + e_k, \beta-e_k, \Gamma \cup \{ q \}) \\ & & + \sum I^{\gamma} {\binom \alpha {\alpha''}} \binom {\beta + \gamma}{\beta} N^{d-2,g''}(\alpha'',\beta + \gamma, \Gamma''). \label{donaldduck} \end{eqnarray} If $\Gamma$ is {\em simple} (Definition \ref{simple}), then so are $\Gamma \cup \{ q \}$ (as $q$ is general) and $\Gamma''$ (as $\Gamma'' \subset \Gamma$). By (\ref{donaldduck}) and induction, if $\Gamma$ is simple, $N^{d,g}(\alpha',\beta,\Gamma)$ depends only on $(\alpha, \beta, s)$. Rewriting (\ref{donaldduck}) in terms of $\alpha$, $\beta$, and $s$, this is Theorem \ref{rrecursion}. By the same argument for the irreducible case, using the rational map (\ref{iratmap}) rather than (\ref{rratmap}), yields Theorem \ref{irecursion}. \subsection{Genus $g$ Gromov-Witten invariants are enumerative on Fano surfaces} \label{gwenumerative} The definition of (genus $g$) Gromov-Witten invariants $I_{g,D}(\gamma_1 \cdots \gamma_n)$, where $g$ is the genus, $D \in A_1 X$, and $\gamma_i \in A^* X$ is given in [KM] and summarized in [V3]. As in [V3], to compute Gromov-Witten invariants of a surface, it suffices to deal with the case where $D$ is effective and non-zero and the $\gamma_i$ are points. Recall [V3] Lemma 7.1: \begin{lm} \label{gwlemma} Let $X$ be a Fano surface, and let $D$ be an effective divisor class on $X$. Suppose that $M$ is an irreducible component of $\overline{M}_g(X,d)$ with general map $(C,\pi)$. Then $$ \operatorname{idim} M \leq -K_X \cdot D + g-1. $$ If equality holds and $D \neq 0$, then $\pi$ is an immersion. \end{lm} Thus the genus $g$ Gromov-Witten invariants of ${\mathbb F}_n$ can be computed as follows. We need only compute $I_{g,D}(\gamma_1 \cdots \gamma_n)$ where $D$ is effective and nonzero, and the $\gamma_i$ are (general) points. By Lemma \ref{gwlemma}, this is the number of immersed genus $g$ curves in class $D'$ through the appropriate number of points of $X$. If $D$ is a (-1)-curve, the number is 1. Otherwise, the number is recursively calculated by Theorem \ref{irecursion}. \subsection{The cubic surface} \label{cubic} By the previous subsection, the algorithm of Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of the plane blown up at up to five points. As GW-invariants are deformation-invariant, one might hope to compute the invariants of the plane blown up at six points $P_1$, \dots, $P_6$ in general position (i.e. a general cubic surface in $\mathbb P^3$) by degenerating the six points $P_1$, \dots, $P_6$ to lie on a conic $E$. Call the resulting surface $B'$. (Then $B'$ has a (-2)-curve, the proper transform of $E$. The canonical map is an embedding away from $E$, and $E$ is collapsed to a simple double point.) If the enumerative significance of the genus $g$ Gromov-Witten invariants on this surface could be determined, Theorem \ref{irecursion} could be used to determine the invariants of $B'$, and hence (by deformation-invariance of Gromov-Witten invariants) the invariants of any cubic hypersurface. The rational ruled surface $\mathbb F_2$ ($\mathbb P( {\mathcal{O}}_{\mathbb P^1} \oplus {\mathcal{O}}_{\mathbb P^1}(2))$) is similar, in that $\mathbb F_2$ has a (-2)-curve, and the canonical map is an embedding away from the (-2)-curve, which is collapsed to a simple double point. In [K1], p. 22-23, Kleiman gives an enumerative interpretation for a particular genus 0 GW-invariant of $\mathbb F_2$, which was explained to him by Abramovich. This interpretation suggests the following conjecture. \begin{conj} \label{conj} Suppose $X$ is $\mathbb F_2$ or $B'$, and $E$ is the (-2)-curve on $X$. Let $D$ be an effective divisor class on $X$ (not 0 or $E$), and $\gamma$ the class of a point. Then the Gromov-Witten invariant $I_{g,D}(\gamma^{- K_X \cdot D + g-1})$ is the number of maps $\pi: C \rightarrow X$ with $\pi_*[C] = D$, where \begin{enumerate} \item[(i)] the curve $C$ has one component $C_0$ {\em not} mapping to $E$, and \item[(ii)] any other component $C'$ of $C$ maps isomorphically to $E$, and $C'$ intersections $\overline{C \setminus C'}$ at one point, which is contained in $C_0$. \end{enumerate} \end{conj} Simple tests on both $\mathbb F_2$ and $B'$ seem to corroborate this conjecture. As a more complicated test case, we compute the number of rational sextic curves in the plane with six nodes at fixed points $P_1$, \dots, $P_6$, and passing through five other fixed points $Q_1$, \dots, $Q_5$, where all the points are in general position. (This is the Gromov-Witten invariant $N^0_{6,2^6}$ of the cubic surface.) [DI] p. 119 gives this number as 2376, while [GP] p. 25 gives the number as 3240. G\"{o}ttsche and Pandharipande checked their number using different recursive strategies. According to the conjecture, this invariant is the sum of three contributions. \begin{enumerate} \item Those (irreducible) rational sextics with six fixed nodes $P_1$, \dots, $P_6$ lying on a conic, passing through $Q_1$, \dots, $Q_5$. By Theorem \ref{irecursion} (and some computation), this number is 2002. \item A stable map $\pi: C \rightarrow \mathbb P^2$ where $C$ has two irreducible rational components $C_0$ and $C_1$ joined at one point, $\pi$ maps $C_1$ isomorphically to $E$, and $\pi$ maps $C_0$ to an irreducible rational quartic through $P_1$, \dots, $P_6$ (which lie on a conic) and $Q_1$, \dots, $Q_5$. The image of the node $C_0 \cap C_1$ is one of the two points $\pi(C_0) \cap E \setminus \{ P_1, \dots, P_6 \}$. By Theorem \ref{irecursion}, there are 616 such quartics. There are two choices for the image of the node $C_0 \cap C_1$, so the contribution is 1232. \item A stable map $\pi: C \rightarrow \mathbb P^2$ where $C$ has three irreducible rational components $C_0$, $C_1$, $C_2$, where $C_1$ and $C_2$ intersect $C_0$, $\pi$ maps $C_1$ and $C_2$ isomorphically to $E$, and $\pi$ maps $C_0$ isomorphically to the conic through $Q_1$, \dots, $Q_5$. There are 12 choices of pairs of images of the nodes $C_0 \cap C_1$ and $C_0 \cap C_2$, and we must divide by 2 as exchanging $C_1$ and $C_2$ preserves the stable map. This this contribution is 6. \end{enumerate} Therefore, assuming Conjecture \ref{conj}, $$ N^0_{6,2^6} = 2002 + 1232 + 6 = 3240, $$ in agreement with [GP].

9709

alg-geom/9709020

en

https://arxiv.org/abs/alg-geom/9709020

alg-geom/9709020

Vladimir Masek

Vladimir Masek (Washington Univ. in St. Louis)

Very ampleness of adjoint linear systems on smooth surfaces with boundary

22 pages, AMS-LaTeX 1.2

Let M be a Q-divisor on a smooth surface over C. In this paper we give criteria for very ampleness of the adjoint of the round-up of M. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita's Conjecture in dimension 3.) In the last section we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.

alg-geom

\subsection*{Contents} \begin{enumerate} \item[0.] Introduction \item[1.] Base-point-freeness \item[2.] Separation of points \item[3.] Separation of tangent directions \item[4.] Example \end{enumerate} \subsection*{Notations} \begin{tabbing} 99\=9999999999\=9999999999999999999999999999\kill \>$\lceil \cdot \rceil$ \> round-up \\ \>$\lfloor \cdot \rfloor$ \> round-down \\ \>$\{ \cdot \}$ \> fractional part \\ \>$f^{-1}D$ \> strict transform (proper transform) \\ \>$f^*D$ \> pull-back (total inverse image) \\ \>$PLC$ \> partially log-canonical (Definition 1.7) \\ \>$\,\equiv$ \> numerical equivalence \\ \>$\,\sim$ \> linear equivalence \\ \>$\,\qle$ \> \ensuremath{\mathbb{Q}}-linear equivalence \\ \end{tabbing} \section{Introduction} Let $S$ be a nonsingular projective surface over $\ensuremath{\mathbb{C}}\,$, and let $H$ be a given line bundle on $S$. Consider the following natural questions regarding the complete linear system $|H|$: \emph{ \begin{enumerate} \item[(1)] Compute $\dim |H|$. \item[(2)] Is $|H|$ base-point-free? \item[(3)] Is $|H|$ very ample? \end{enumerate} } The answer to (1) is usually given in two parts: the Riemann-Roch theorem computes $\chi(S,H)$, and then we need estimates for $h^i(S,H),\; i>0$. In particular, we may ask the following question related to (1): \emph{ \begin{enumerate} \item[($1'$)] When are $h^1(S,H) \text{ and } h^2(S,H)$ equal to zero? \end{enumerate} } One classical answer to ($1'$) is provided by Kodaira's vanishing theorem: if $L$ is any ample line bundle on $S$, then $h^i(S,-L)=0$ for all $i<2$; therefore, by Serre duality, we have $h^i(S, K_S+L)=0$ for all $i > 0$. To answer ($1'$), write $H=K_S+L$ (thus defining $L$ as $H-K_S$); if $L$ is ample, then $h^i(S,H)=0$ for all $i>0$. \vspace{6pt} For questions (2) and (3), Reider \cite{rei} gave an answer which again considers $H$ in the form of an adjoint line bundle, $H=K_S+L$: {\bf Proposition} (cf. \cite[Theorem 1]{rei}){\bf .} \emph{ If $L$ is a line bundle on $S$, $L^2 \geq 5$ and $L \cdot C \geq 2$ for every curve $C \subset S$, then $|K_S+L|$ is base-point-free. If $L^2 \geq 10$ and $L \cdot C \geq 3$ for every curve $C$, then $|K_S+L|$ is very ample. } \vspace{6pt} We note here that Kodaira's theorem holds in all dimensions. Reider's criterion was tentatively extended in higher dimensions in the form of Fujita's conjecture (\cite{fuj}): if $X$ is a smooth projective variety of dimension $n$, and $L$ is an ample line bundle on $X$, then $|K_X+mL|$ is base-point-free for $m\geq n+1$ and very ample for $m\geq n+2$. Fujita's conjecture for base-point-freeness was proved in dimension 3 by Ein and Lazarsfeld (\cite{el}) and in dimension 4 by Kawamata (\cite{kaw}); more precise statements, which resemble Reider's criterion more closely, were also obtained. Very ampleness, however, is still open, even in dimension 3. Kodaira's vanishing theorem and Reider's criterion are already very useful as stated. However, the applicability of Kodaira's theorem was greatly extended, first on surfaces, by Mumford, Ramanujam, Miyaoka, and then in all dimensions by Kawamata and Viehweg, as follows. First, the ampleness condition for $L$ can be relaxed to $L\cdot C \geq 0$ for every curve $C$ and $L^2>0$ ($L$ \emph{nef} and \emph{big}). Second, and most important, assume that $L$ itself is not nef and big, but there is a nef and big \ensuremath{\mathbb{Q}}-divisor $M$ on $S$ ($M \in \Div(S)\otimes\ensuremath{\mathbb{Q}}$) such that $L = \rup{M}$ (i.e. $L-M$ is an effective \ensuremath{\mathbb{Q}}-divisor $B$ whose coefficients are all $<1$). Then we have $h^i(S,K_S+L)=0$ for all $i>0$, just as in Kodaira's theorem. (\ensuremath{\mathbb{Q}}-divisors were first considered in this context in connection with the Zariski decomposition of effective divisors.) In dimension $\geq 3$, the Kawamata--Viehweg vanishing theorem requires an extra hypothesis (the irreducible components of $\Supp(B)$ must cross normally); however, Sakai remarked that for surfaces this extra hypothesis is not necessary (see Proposition 1.2.1 in \S 1). \vspace{6pt} For base-point-freeness (question (2) above), Ein and Lazarsfeld (\cite{el}) proved a similar extension of Reider's criterion, expressing $H$ as $K_S+\rup{M}$ for a \ensuremath{\mathbb{Q}}-divisor $M$ on $S$; if $M^2>4$ and $M\cdot C \geq 2$ for every curve $C$, then $|H|$ is base-point-free. They used this result in their proof of Fujita's conjecture for base-point-freeness in dimension 3. (In fact they used a more precise local version, involving the local multiplicities of $B=L-M$; see \S 1 below). In this paper we give criteria for very ampleness of linear systems of the form $|K_S+B+M|$, $B=\rup{M}-M$, as above. In particular, we prove the following result: \begin{Theorem} Let $S$, $B$ and $M$ be as above, and assume that \begin{enumerate} \item[(0.1)] $M^2 > 2 (\beta_2)^2$, \item[(0.2)] $M \cdot C \geq 2\beta_1$ for every irreducible curve $C \subset S$, \newline where $\beta_2$, $\beta_1$ are positive numbers satisfying the following inequalities: \item[(0.3)] $\beta_2 \geq 2$, \item[(0.4)] $\beta_1 \geq \dfrac{\beta_2}{\beta_2-1}$. \end{enumerate} Then $|K_S+B+M|$ is very ample. \end{Theorem} An immediate consequence of Theorem 1 is the following: \begin{Corollary} Assume that (S,B) is as before, and $M$ is an ample \ensuremath{\mathbb{Q}}-divisor on $S$ such that $B=\rup{M}-M$, $M^2 > (2+\sqrt{2})^2$, and $M \cdot C > 2+\sqrt{2}$ for every curve $C \subset S$. Then $|K_S+B+M|$ is very ample. In particular, if $A$ is an ample divisor (with integer coefficients) on $S$, then $|K_S + \rup{aA}|$ is very ample for every $a \in \ensuremath{\mathbb{Q}}$, $a > 2+\sqrt{2}$. \end{Corollary} Note that Reider's criterion implies only that $|K_S+aA|$ is very ample for every \emph{integer} $a \geq 4$. \vspace{6pt} As in \cite[\S 2]{el} (where the analogue for base-point-freeness was proved), we prove a local version of Theorem 1, with the numerical conditions on $M$ relaxed in terms of local multiplicities of $B$. \vspace{6pt} As we mentioned earlier, the result for base-point-freeness on surfaces with boundary (i.e. for \ensuremath{\mathbb{Q}}-divisors $M$) was used in \cite{el} in the proof of Fujita's Conjecture in dimension 3. Similarly, we expect that the proof of the analogous result for very ampleness in dimension 3 will use very ampleness for \ensuremath{\mathbb{Q}}-divisors on surfaces. However, a natural and interesting question is whether or not the results for \ensuremath{\mathbb{Q}}-divisors on surfaces have any useful applications to the study of linear systems on surfaces. An example we discuss in \S 4 seems to indicate an affirmative answer. While the results proved in \S 4 can be obtained with other methods, our example shows how our \ensuremath{\mathbb{Q}}-Reider theorem extends the applicability of Reider's original result in the same way the Kawamata--Viehweg vanishing theorem extends the range of applicability of Kodaira's vanishing theorem. The usefulness of considering local multiplicities of $B$ is also evident in this example. \vspace{6pt} The paper is divided as follows: \S 1 is devoted to base-point-freeness. The results discussed in this section, with one exception, were proved in \cite{el}; I include a (slightly modified) proof to fix the ideas and notations for the later sections. As one might expect, separation of points is relatively easy (at least in principle); it is discussed in \S 2. Then we move on to separation of tangent directions in \S 3. This part is surprisingly delicate; in particular the ``multiplier ideal'' method of Ein--Lazarsfeld, or Kawamata's equivalent ``log-canonical threshold'' formalism, do not work in this context. We explain the geometric contents of our method in the beginning of \S 3. Theorem 1 follows from Proposition 4 in \S 2 and Proposition 5 in \S 3. Finally, \S 4 contains the example mentioned earlier. \vspace{6pt} The author is grateful to L. Ein, R. Lazarsfeld, S. Lee, and N. Mohan Kumar for their many useful suggestions. \section{Base-point-freeness} {\bf (1.1)} Let $S$ be a smooth projective surface over \ensuremath{\mathbb{C}}, and $B=\sum b_i C_i$ a fixed effective \ensuremath{\mathbb{Q}}-divisor on $S$ with $0 \leq b_i < 1$ for all $i$. (The pair $(S,B)$ is sometimes called a ``surface with boundary'', whence the title of this paper.) Let $M$ be a \ensuremath{\mathbb{Q}}-divisor on $S$ such that $B+M$ has integer coefficients. {\bf We assume throughout this paper that $M$ is nef and big,} i.e. that $M \cdot C \geq 0$ for every curve $C \subset S$ and $M^2 > 0$. \vspace{6pt} {\bf (1.2)} For convenience, we gather here two technical results which we use time and again in our proofs. {\bf (1.2.1)} We use the following variants of the Kawamata--Viehweg vanishing theorem, which hold on smooth surfaces: {\bf Theorem.} {\bf (a)} (cf. \cite[Lemma 1.1]{el}) Let $S$ be a smooth projective surface over \ensuremath{\mathbb{C}}, and let $M$ be a nef and big \ensuremath{\mathbb{Q}}-divisor on $S$. Then \begin{equation*} H^i(S, K_S+\rup{M})=0, \qquad \forall i > 0. \end{equation*} {\bf (b)} (cf. \cite[Lemma 2.4]{el}) Assume moreover that $C_1, \dots, C_k$ are distinct irreducible curves on $S$ which have integer coefficients in $M$. Assume that $M \cdot C_j > 0$ for all $j = 1, \dots, k$. Then \begin{equation*} H^i(S, K_S+\rup{M}+C_1+\cdots+C_k)=0, \qquad \forall i > 0. \end{equation*} \qed \vspace{6pt} {\bf (1.2.2)} We use the following criterion for base-point-freeness, respectively very ampleness, on a complete Gorenstein curve (cf. \cite{har}): {\bf Proposition.} Let $D$ be a Cartier divisor on the integral projective Gorenstein curve $C$. Then: {\bf (a)} $\deg(D) \geq 2 \implies$ the complete linear system $|K_C+D|$ is base-point-free; {\bf (b)} $\deg(D) \geq 3 \implies |K_C+D|$ is very ample. \begin{proof} See \cite[\S 1]{har} for the relevant definitions (generalized divisors on $C$, including $0$-dimensionals subschemes; degree; etc.) We prove (b); the proof of (a) is similar. By \cite[Proposition 1.5]{har}, it suffices to show that $h^0(C, K_C+D-Z)=h^0(C, K_C+D)-2$ for every $0$-dimensional subscheme $Z \subset C$ of length $2$. Consider the exact sequence: \begin{equation*} 0 \longrightarrow \mathcal{O}_C(K_C+D-Z) \longrightarrow \mathcal{O}_C(K_C+D) \longrightarrow \mathcal{O}_C(K_C+D) \otimes \mathcal{O}_Z \longrightarrow 0. \end{equation*} As $\mathcal{O}_C(K_C+D)\otimes\mathcal{O}_Z \cong \mathcal{O}_Z$ has length $2$, the conclusion will follow from the vanishing of $H^1(C, K_C+D-Z)$. By Serre duality (cf. \cite[Theorem 1.4]{har}), $H^1(C, K_C+D-Z) \cong H^0(C, Z-D)$, and $H^0(C, Z-D)=0$ due to $\deg(Z-D) = 2 - \deg(D) < 0$. \end{proof} \vspace{6pt} {\bf (1.3)} Fix a point $p \in S$. In this section we give sufficient conditions for $|K_S+B+M|$ to be free at $p$. \vspace{3pt} \noindent {\bf (1.3.1)} {\it Notation.} $\quad \mu = \ord_p(B) \overset{\text{def}}{=} \sum b_i \cdot \mult_p(C_i) \qquad (B = \sum b_i C_i).$ \vspace{4pt} \begin{Proposition} $|K_S+B+M|$ is free at $p$ in each of the following cases: \begin{enumerate} \item[{\bf 1.}] $\mu \geq 2$; \item[{\bf 2.}] $0 \leq \mu < 2$; $M^2 > (\beta_2)^2$, $M \cdot C \geq \beta_1$ for every irreducible curve $C \subset S$ such that $p \in C$, where $\beta_2$, $\beta_1$ are positive numbers which satisfy the inequalities: \end{enumerate} \begin{align*} \beta_2 &\geq 2-\mu , \tag{1.3.2} \\ \beta_1 &\geq \min \left\{ (2-\mu) ; \frac{\beta_2}{\beta_2-(1-\mu)} \right\} . \tag{1.3.3} \end{align*} \end{Proposition} \begin{Remark} Explicitly, the minimum in (1.3.3) is given by: \begin{equation*} \min \left\{ (2-\mu) ; \frac{\beta_2}{\beta_2-(1-\mu)} \right\} = \begin{cases} 2-\mu & \text{if $1 \leq \mu < 2$} \\ \dfrac{\beta_2}{\beta_2-(1-\mu)} & \text{if $0 \leq \mu < 1$.} \end{cases} \end{equation*} In other words, when $0 \leq \mu < 2$, the inequalities $\beta_2 \geq 2-\mu$ and $\beta_1\geq 2-\mu$ suffice. When $\mu < 1$ the inequality for $\beta_1$ can be relaxed to \begin{equation*} \beta_1 \geq \frac{\beta_2}{\beta_2-(1-\mu)} ; \tag{1.3.4} \end{equation*} this last part (which is useful in applications, cf. \S 4) is not contained in \cite{el}. \end{Remark} \vspace{6pt} \noindent {\bf Proof of Proposition 3.} \vspace{4pt} {\bf (1.4)} Let $f : S_1 \to S$ be the blowing-up of $S$ at $p$, and let $E \subset S_1$ be the exceptional divisor of $f$. We have $f^*B=f^{-1}B+\mu E$; $\rdn{f^{-1}B}=0$, and therefore \begin{equation*} \begin{split} K_{S_1}+\rup{f^*M} &= f^*K_S+E+\rup{f^*(B+M)-f^*B} \\ &= f^*(K_S+B+M)+E-\rdn{f^*B} \\ &= f^*(K_S+B+M)-(\rdn{\mu}-1)E. \end{split} \tag{1.4.1} \end{equation*} \vspace{4pt} {\bf (1.5)} If $\mu \geq 2$, then $p \notin \Bs |K_S+B+M|$. Indeed, in this case $t=\rdn{\mu}-1$ is a positive integer; since $f^*M$ is nef and big on $S_1$, the vanishing theorem (1.2.1)(a) yields \begin{equation*} H^1(S_1, K_{S_1}+\rup{f^*M})=0, \tag{1.5.1} \end{equation*} and therefore (using (1.4.1) and the projection formula) \begin{equation*} H^1(S, \mathcal{O}_S(K_S+B+M) \otimes {\frak{m}_p}^t)=0, \tag{1.5.2} \end{equation*} where $\frak{m}_p$ is the maximal ideal of $\mathcal{O}_S$ at $p$. The conclusion follows from the surjectivity of the restriction map \begin{equation*} H^0(S, K_S+B+M) \longrightarrow H^0(S, \mathcal{O}_S(K_S+B+M)\otimes \mathcal{O}_S/{\frak{m}_p}^t) \cong \mathcal{O}_S/{\frak{m}_p}^t . \end{equation*} \vspace{4pt} {\bf (1.5.3)} \emph{Remark:} In fact we proved that $|K_S+B+M|$ separates $s$-jets at $p$, if $\mu \overset{\text{def}}{=} \ord_p(B) \geq s+2$. \vspace{9pt} {\bf (1.6)} Now assume that $\mu < 2$, and $M^2 > (\beta_2)^2$ with $\beta_2 \geq 2-\mu$, etc. \vspace{6pt} {\bf (1.6.1) Claim:} We can find an effective \ensuremath{\mathbb{Q}}-divisor $D$ on $S$ such that $\ord_p(D) = 2-\mu$ and $D \qle tM$ for some $t \in \ensuremath{\mathbb{Q}}\,$, $0 < t < \dfrac{2-\mu}{\beta_2}$. ($\qle$ denotes \ensuremath{\mathbb{Q}}-linear equivalence, i.e. $mD$ and $mtM$ have integer coefficients and are linearly equivalent for some suitably large and divisible integer $m$.) \vspace{4pt} \emph{Proof of (1.6.1):} By Riemann--Roch, $\dim |nM|$ grows like $\frac{M^2}{2}n^2 > \frac{(\beta_2)^2}{2}n^2$ for $n$ sufficiently large and divisible (such that $nM$ has integer coefficients). Since $\dim (\mathcal{O}_{S,p}/{\frak{m}_p}^n)$ grows like $\frac{n^2}{2}$, for suitable $n$ we can find $G \in |nM|$ with $\ord_p(G) > \beta_2 n$. Take $D=rG$, $r=\dfrac{2-\mu}{\ord_p(G)}$; then $\ord_p(D) = 2-\mu$, and $D \qle tM$ for $t=rn < \dfrac{2-\mu}{\beta_2 n}n = \dfrac{2-\mu}{\beta_2}$. \qed \vspace{6pt} Note that $\dfrac{2-\mu}{\beta_2} \leq 1$, by (1.3.2), so that $t < 1$; therefore $M-D \qle (1-t)M$ is still nef and big. \vspace{8pt} {\bf (1.7)} Recall that $B = \sum b_i C_i$, for distinct irreducible curves $C_i \subset S$. Write $D = \sum d_i C_i$ (we allow some coefficients $b_i$ and $d_i$ to be zero); $d_i \in \ensuremath{\mathbb{Q}}$, $d_i \geq 0$, and $\ord_p(D) = \sum d_i \cdot \mult_p(C_i) = 2-\mu$. Let $D_i = f^{-1}C_i \subset S_1$ be the strict transform of $C_i$; then $f^*B = \sum b_i D_i + \mu E$, $f^*D = \sum d_i D_i + (2-\mu) E$, $K_{S_1}=f^*K_S+E$, and \begin{equation*} K_{S_1} - f^*(K_S+B+D) = -E - \sum (b_i + d_i) D_i. \end{equation*} \vspace{4pt} \emph{Definition.} $(S,B,D)$ is {\bf partially log-canonical at $p$} ($PLC$ at $p$) if $-(b_i + d_i) \geq -1$ (i.e. $b_i+d_i \leq 1$) for every $i$ such that $p \in C_i$. (The general definition requires the coefficient of $E$ to be $\geq -1$, too; in our case that coefficient is equal to $-1$.) Note that $PLC$ is not the same as \emph{log-canonical} (cf. \cite[Definition 0-2-10]{kmm}), because $f$ is not an embedded resolution of $(S, B+D)$. \vspace{6pt} {\bf (1.8)} If $(S,B,D)$ is $PLC$ at $p$, then the proof is almost as simple as in the case $\mu \geq 2$: \begin{equation*} \begin{split} K_{S_1}+\rup{f^*(M-D)} &= f^*K_S+E+f^*(B+M)-\rdn{f^*(B+D)} \\ &= f^*(K_S+B+M)+E-2E-\sum\rdn{b_i+d_i}D_i \\ &= f^*(K_S+B+M)-E-\textstyle\sum ' D_i -N_1, \end{split} \tag{1.8.1} \end{equation*} where $\sum ' D_i$ extends over those $i$ for which $p \in C_i$ and $b_i+d_i=1$ (if any), and $N_1$ is an effective divisor supported away from $E$. $f^*M \cdot D_i = M \cdot C_i > 0$ if $p \in C_i$; therefore (1.2.1)(b) yields: \begin{equation*} H^1(S_1, f^*(K_S+B+M)-E-N_1) = 0. \tag{1.8.2} \end{equation*} Arguing as in (1.5), we can show that $p \notin \Bs |K_S+B+M-N|$, where $N=f_*N_1$; i.e., $\exists \Lambda \in |K_S+B+M-N|$ with $p \notin \Supp(\Lambda)$. Then $\Lambda + N \in |K_S+B+M|$ and $p \notin \Supp(\Lambda + N)$, as required. Note that we haven't used (1.3.3) yet; all we needed so far was $\beta_1 > 0$. \vspace{8pt} {\bf (1.9)} Finally, assume that $(S,B,D)$ is not $PLC$ at $p$. Then $b_j+d_j > 1$ for some $j$ with $p \in C_j$. In fact, since $2 = \ord_p(B+D) = \sum (b_i+d_i) \cdot \mult_p(C_i)$, there can be at most one $C_j$ through $p$ with $b_j+d_j > 1$, and then that $C_j$ must be smooth at $p$ and also $b_i+d_i < 1$ for all $i \neq j$ with $p \in C_i$. Let that $j$ be $0$; thus $b_0+d_0 > 1$, $C_0$ is smooth at $p$, and $b_i+d_i < 1$ if $i \neq 0$ and $p \in C_i$. We say that $C_0$ is the {\bf critical curve} at $p$. Let $c$ be the {\bf $PLC$ threshold} of $(S,B,D)$ at $p$: \begin{equation*} c = \max \{ \lambda \in \ensuremath{\mathbb{Q}}_+ \mid (S,B,\lambda D) \text{ is $PLC$ at $p$} \}; \end{equation*} explicitly, $b_0+cd_0 = 1$, i.e. $c = \dfrac{1-b_0}{d_0}$. Note that $0<c<1$. $M-cD \qle (1-ct)M$ is still nef and big on $S$, and we have: \begin{equation*} K_S+\rup{M-cD} = K_S+B+M-\rdn{B+cD} = K_S+B+M-C_0-N , \end{equation*} with $p \notin \Supp(N)$. (If $p \in C_i$ and $i \neq 0$ then $b_i+d_i < 1$, and therefore $b_i+cd_i < 1$, too, because $c < 1$; hence $p \notin \Supp(N)$.) (1.2.1)(a) yields $H^1(S, K_S+B+M-C_0-N) = 0$, and therefore the restriction map $H^0(S, K_S+B+M-N) \to H^0(C_0, (K_S+B+M-N)|_{C_0})$ is surjective. Hence it suffices to show that $p \notin \Bs|(K_S+B+M-N)|_{C_0}|$. \vspace{3pt} We have \begin{equation*} K_S+B+M-N = K_S+\rup{M-cD}+C_0, \tag{1.9.1} \end{equation*} and therefore $(K_S+B+M-N)|_{C_0} = K_{C_0}+\rup{M-cD}|_{C_0}$; by (1.2.2)(a), it suffices to show that $\rup{M-cD} \cdot C_0 \geq 2$. In any event $\rup{M-cD}\cdot C_0$ is an integer; we will show that $\rup{M-cD} \cdot C_0 > 1$. \vspace{4pt} $\rup{M-cD} = (M-cD) + \Delta$, where $\Delta = \rup{M-cD}-(M-cD) = \rup{(M+B)-(B+cD)}-(M-cD) = (M+B)-\rdn{B+cD}-(M-cD) = (B+cD)-\rdn{B+cD} =\frp{B+cD}$. $\Delta$ is an effective divisor which intersects $C_0$ properly, because $C_0$ has integer coefficient (namely, 1) in $B+cD$. Moreover, in a neighborhood of $p$ we have $\frp{B+cD} = (B+cD) - C_0$, because $B+cD = C_0 + \sum_{i \neq 0} (b_i+cd_i) C_i$, and $0 \leq b_i+cd_i < 1$ for every $i \neq 0$ such that $p \in C_i$. In particular, we have \begin{equation*} \ord_p(\Delta) = \ord_p(B+cD)-1 = \mu+c(2-\mu)-1. \end{equation*} \vspace{4pt} \begin{align*} \rup{M-cD} \cdot C_0 &= (M-cD) \cdot C_0 + \Delta \cdot C_0 \geq (1-ct)M \cdot C_0 + \ord_p(\Delta) \tag{\bf 1.10} \\ &\geq (1-ct)\beta_1 + \mu + c(2-\mu) - 1. \end{align*} Therefore the inequality $\rup{M-cD} \cdot C_0 > 1$ follows from \begin{equation*} (1-ct)\beta_1 > (1-c)(2-\mu). \tag{1.10.1} \end{equation*} If $\beta_1 \geq 2-\mu$ then (1.10.1) is trivial, because $t < 1 \implies 1-ct > 1-c$. \vspace{6pt} {\bf (1.11)} When $\mu < 1$ the inequality we assume for $\beta_1$ (namely, (1.3.4)) is weaker than $\beta_1 \geq 2-\mu$. However, in this case the equation $B+cD = C_0 + \textit{other terms}$ yields a nontrivial lower bound for $c$: $\mu + c(2-\mu) = \ord_p(B+cD) \geq \ord_p(C_0) =1$, and therefore $c \geq \dfrac{1-\mu}{2-\mu} > 0$. The inequality (1.10.1) can also be written as \begin{equation*} c(2-\mu-t\beta_1) > 2-\mu-\beta_1. \tag{1.11.1} \end{equation*} We may assume that $\beta_1 < 2-\mu$ (or else (1.10.1) is already proved). We have $c \geq \dfrac{1-\mu}{2-\mu}$, $t < \dfrac{2-\mu}{\beta_2}$ (see (1.6.1)), and $\dfrac{1-\mu}{\beta_2} \leq 1 - \dfrac{1}{\beta_1}$ (by (1.3.4)); therefore \begin{equation*} \begin{split} c(2-\mu-t\beta_1) &>\frac{1-\mu}{2-\mu}(2-\mu-\frac{2-\mu}{\beta_2}\beta_1) = (1-\mu-\frac{1-\mu}{\beta_2}\beta_1) \\ &\geq (1-\mu)-(1-\frac{1}{\beta_1})\beta_1 = 2-\mu-\beta_1. \end{split} \end{equation*} \vspace{3pt} (1.11.1) is proved. This concludes the proof of Proposition 3. \section{Separation of points} {\bf (2.1)} Let $(S,B,M)$ be as in (1.1). Fix two distinct points $p,q\in S$. In this section we give criteria for $|K_S+B+M|$ to separate $(p,q)$. Note that in each case $|K_S+B+M|$ is free at $p$ and $q$, by Proposition 3, and therefore it suffices to find $s \in H^0(S, K_S+B+M)$ such that $s(p)=0, s(q) \neq 0$, \emph{or} vice-versa. \begin{Notation} $\mu_p = \ord_p(B), \quad \mu_q = \ord_q(B)$. \end{Notation} \vspace{6pt} \begin{Proposition} $|K_S+B+M|$ separates $(p,q)$ in each of the following cases: \begin{enumerate} \item[{\bf 1.}] $\mu_p \geq 2$ and $\mu_q \geq 2$; \item[{\bf 2.}] $\mu_q \geq 2$; $0 \leq \mu_p < 2$; $M^2 > (\beta_2)^2, M \cdot C \geq \beta_1$ for every irreducible curve $C \subset S$ passing through $p$, where $\beta_2, \beta_1$ are positive numbers which satisfy (1.3.2) and (1.3.3) for $\mu = \mu_p$; \item[{\bf 3.}] $0 \leq \mu_p < 2$ and $0 \leq \mu_q < 2$; $M^2 > (\beta_{2,p})^2+(\beta_{2,q})^2$, and \begin{enumerate} \item[(i)] $M \cdot C \geq \beta_{1,p}$ for every curve $C \subset S$ passing through $p$, \item[(ii)] $M \cdot C \geq \beta_{1,q}$ for every curve $C \subset S$ passing through $q$, \item[(iii)] $M \cdot C \geq \beta_{1,p}+\beta_{1,q}$ if $C$ passes through \emph{both} $p$ \emph{and} $q$, \end{enumerate} where $\beta_{2,p},\beta_{1,p}\,;\beta_{2,q},\beta_{1,q}$ are positive numbers which satisfy the inequalities \end{enumerate} \begin{align*} \beta_{2,p} &\geq 2-\mu_p ,\quad \beta_{2,q} \geq 2-\mu_q; \tag{2.1.1} \\ \beta_{1,p} &\geq \min \left\{ (2-\mu_p) ; \frac{\beta_{2,p}}{\beta_{2,p}-(1-\mu_p)} \right\} , \text{ and similarly for $\beta_{1,q}$}. \tag{2.1.2} \end{align*} \end{Proposition} \vspace{6pt} \noindent {\bf Proof of Proposition 4.} \vspace{4pt} {\bf (2.2)} Let $f:S_1 \to S$ be the blowing-up of $S$ at $p$ and $q$, with exceptional curves $E_p, E_q$. As in (1.4), we have: \begin{equation*} K_{S_1}+\rup{f^*M} = f^*(K_S+B+M) - (\rdn{\mu_p}-1)E_p - (\rdn{\mu_q}-1)E_q. \end{equation*} In particular, if $\mu_p \geq 2$ and $\mu_q\geq 2$ (case 1 of the proposition), we get \begin{equation*} H^1(S, \mathcal{O}_S(K_S+B+M) \otimes {\frak{m}_p}^{t_p} \otimes {\frak{m}_q}^{t_q})=0 \end{equation*} for positive integers $t_p, t_q$ (compare to (1.5.2)); the conclusion follows as in (1.5). \vspace{8pt} {\bf (2.3)} Next assume that $\mu_p < 2, \mu_q \geq 2,\, M^2 > (\beta_2)^2$ with $\beta_2 \geq 2-\mu_p,$ etc. (case 2 of the proposition). Write $\mu = \mu_p$. As in (1.6.1), we can find an effective \ensuremath{\mathbb{Q}}-divisor $D$ on $S$ such that $\ord_p(D) = 2-\mu$ and $D \qle tM$ for some $t \in \ensuremath{\mathbb{Q}}\, , 0 < t < \dfrac{2-\mu}{\beta_2}$. If $(S,B,D)$ is $PLC$ at $p$, the argument of (1.8) yields a vanishing \begin{equation*} H^1(S_1, f^*(K_S+B+M) - E_p - N_0) = 0 \tag{2.3.1} \end{equation*} where $N_0$ is an effective divisor supported away from $E_p$. Note that in this case $N_0 \geq E_q$, because $\mu_q \geq 2$. Indeed, (2.3.1) is obtained by applying (1.2.1)(b) to \begin{equation*} \begin{split} K_{S_1}+\rup{f^*(M-D)} &= f^*(K_S+B+M) - E_p - t_q E_q - \sum \rdn{b_i+d_i} D_i \\ &= f^*(K_S+B+M) - E_p - t_q E_q - \textstyle\sum ' D_i - N_1, \end{split} \tag{2.3.2} \end{equation*} where $\sum ' D_i$ and $N_1$ are as in (1.8.1) and $t_q = \rdn{\mu_q + \ord_q(D)}-1 $ is an integer, $t_q \geq 1$; then $N_0 = N_1 + t_q E_q \geq E_q$. \vspace{4pt} The vanishing (2.3.1) implies the surjectivity of the restriction map \begin{multline*} H^0(S_1, f^*(K_S+B+M)-N_0) \\ \longrightarrow H_0(E_p, (f^*(K_S+B+M)-N_0)|_{E_p}) \cong \ensuremath{\mathbb{C}} \end{multline*} (note that $f^*(K_S+B+M)|_{E_p}$ is trivial, and so is $N_0|_{E_p}$ because $N_0 \cap E_p = \emptyset$). Hence we can find $\Gamma \in |f^*(K_S+B+M)-N_0|$ such that $\Gamma \cap E_p = \emptyset$. As $\Gamma + N_0 \in |f^*(K_S+B+M)|$, we have $\Gamma + N_0 = f^*\Lambda$ for some $\Lambda \in |K_S+B+M|$. Moreover, $p \notin \Supp(\Lambda)$, because $f^*\Lambda \cap E_p = \emptyset$, but $q \in \Supp(\Lambda)$, because $f^*\Lambda = \Gamma + N_0 \geq E_q$. Thus $|K_S+B+M|$ separates $(p,q)$ in this case. \vspace{6pt} {\bf (2.4)} Now assume that $(S,B,D)$ is not $PLC$ at $p$. Let $c, C_0$ be the $PLC$ threshold and the critical curve at $p$, as in \S 1, (1.9)--(1.11). Let $\phi:S_2 \longrightarrow S$ be the blowing-up of $S$ at $q$ (only), with exceptional curve $F_q$. Let $C_0' \subset S_2$ be the proper transform of $C_0$ in $S_2$. Let $p' = \phi^{-1}(p)$. We have: \[ K_{S_2}+\rup{\phi^*(M-cD)} = \phi^*(K_S+B+M) - C_0' - N_0, \] where $p' \notin \Supp(N_0)$, as in (1.9), and $N_0 \geq F_q$, as in (2.3). The argument in (1.9)--(1.11) shows that there exists $\Gamma \in |\phi^*(K_S+B+M)-N_0|$ with $p' \notin \Supp(\Gamma)$. Now the proof can be completed as in the last part of (2.3). \vspace{10pt} {\bf (2.5)} Finally, consider the case $\mu_p < 2$ and $\mu_q < 2$, with $M^2 > (\beta_{2,p})^2 + (\beta_{2,q})^2$, etc. (case 3 of the proposition). \vspace{4pt} As in (1.6.1), we can find $G \in |nM|$ with $\ord_p(G) > \beta_{2,p} n$ and $\ord_q(G) > \beta_{2,q} n$. Let $r = \max \left\{ \dfrac{2-\mu_p}{\ord_p(G)}, \dfrac{2-\mu_q}{\ord_q(G)} \right\},$ and $D=rG$. Then $\ord_p(D) \geq 2-\mu_p$ and $\ord_q(G) \geq 2-\mu_q$, and at least one of the last two inequalities is an equality. Without loss of generality we may assume that $\ord_p(D) = 2-\mu_p$ and $m_q \overset{\text{def}}{=} \ord_q(D) \geq 2-\mu_q$. We have $D \qle tM$, with \begin{equation*} 0 < t = rn = \frac{2-\mu_p}{\ord_p(G)}\,n < \frac{2-\mu_p}{\beta_{2,p}} \leq 1; \tag{2.5.1} \end{equation*} also, $m_q = \ord_q(D) = r \cdot \ord_q(G) > r \cdot (\beta_{2,q} n) = t \beta_{2,q}$, and therefore \begin{equation*} t < \frac{m_q}{\beta_{2,q}} \tag{2.5.2} \end{equation*} (this is the analogue of (2.5.1) at $q$). \vspace{4pt} If $(S,B,D)$ is $PLC$ at $p$, then (1.2.1)(b) yields \begin{equation*} H^1(S_1, f^*(K_S+B+M) - E_p - N_0) =0, \tag{2.5.3} \end{equation*} with $N_0 \cap E_p = \emptyset, N_0 \geq E_q$ (the computation in (2.3.2) applies unchanged in this situation). In this case we conclude as in (2.3). \vspace{6pt} {\bf (2.6)} Now assume that $(S,B,D)$ is not $PLC$ at $p$. Let $c, C_0$ be the $PLC$ threshold and the critical curve at $p$. (1.2.1)(a) yields \begin{equation*} H^1(S_1, f^*(K_S+B+M) - D_0 - N_0) = 0, \qquad N_0 \cap E_p = \emptyset. \tag{2.6.1} \end{equation*} \vspace{4pt} If $N_0 \cap E_q \neq \emptyset$, we use (2.6.1) to find $\Gamma \in |f^*(K_S+B+M)-N_0|$ which does not pass through $\tilde{p} = D_0 \cap E_p$; the proof is the same as in (1.9)--(1.11). Then the conclusion follows as in (2.3). \vspace{4pt} Assume that $N_0 \cap E_q = \emptyset$. We discuss separately the subcases $q \in C_0$ and $q \notin C_0$. If $q \in C_0$, we separate $(p,q)$ on $C_0$. If $q \notin C_0$, we reverse the roles of $p$ and $q$. \vspace{6pt} {\bf (2.7)} First consider the subcase $q \in C_0$. The vanishing (2.6.1) implies \begin{equation*} H^1(S, K_S+B+M-C_0-N) = 0, \tag{2.7.1} \end{equation*} with $N=f_*N_0,\, \Supp(N) \cap \{ p,q \} = \emptyset$. Consequently, the restriction map \[ H^0(S, K_S+B+M-N) \longrightarrow H^0(C_0, (K_S+B+M-N)|_{C_0}) \] is surjective, and it suffices to show that $|(K_S+B+M-N)|_{C_0}|$ separates $(p,q)$ on $C_0$. As in (1.9.1), we have \[ (K_S+B+M-N)|_{C_0} = K_{C_0} + \rup{M-cD}|_{C_0}; \] by (1.2.2)(b) it is enough to show that $\rup{M-cD} \cdot C_0 > 2$ (and consequently $\geq 3$). We proceed as in \S 1: $\rup{M-cD}=(M-cD)+\Delta$, with $\Delta=\frp{B+cD}$; $\Delta$ and $C_0$ intersect properly, and $\ord_p(\Delta) = \mu_p+c(2-\mu_p)-1, \ord_q(\Delta) = \mu_q+c m_q-1$ (note that $N_0 \cap E_q = \emptyset \implies$ the only component with coefficient $\geq 1$ of $B+cD$ through $q$ is $C_0$, and moreover $C_0$ must be smooth at $q$). Therefore \begin{multline*} \rup{M-cD} \cdot C_0 = (M-cD) \cdot C_0 + \Delta \cdot C_0 \\ \geq (1-ct) M \cdot C_0 + \ord_p(\Delta) + \ord_q(\Delta) \\ \geq (1-ct)(\beta_{1,p}+\beta_{1,q}) + (\mu_p+c(2-\mu_p)-1) + (\mu_q + c m_q -1) \end{multline*} ($M \cdot C_0 \geq \beta_{1,p}+\beta_{1,q}$, because this time $C_0$ passes through both $p$ and $q$.) Hence $\rup{M-cD} \cdot C_0 > 2$ follows from \begin{equation*} (1-ct)(\beta_{1,p}+\beta_{1,q})+(\mu_p+c(2-\mu_p)-1)+(\mu_q+c m_q-1) > 2, \tag{2.7.2} \end{equation*} which in turn follows from the following two inequalities: \begin{align*} (1-ct)\beta_{1,p} + (\mu_p+c(2-\mu_p)-1) &> 1 \quad \text{and} \tag{2.7.3} \\ (1-ct)\beta_{1,q} + (\mu_q+c m_q-1) &> 1. \tag{2.7.4} \end{align*} (2.7.3) is proved like (1.10.1) in \S 1: if $\beta_{1,p} \geq 2-\mu_p$, then $t < 1 \implies (1-ct) \beta_{1,p} > (1-c)(2-\mu_p) \implies \text{(2.7.3)}$. If $\beta_{1,p} < 2-\mu_p$ (which can happen only if $\mu_p < 1$), then we have $c \geq \dfrac{1-\mu_p}{2-\mu_p}$ as in (1.11), $t < \dfrac{2-\mu_p}{\beta_{2,p}}$ by (2.5.1), and $\dfrac{1-\mu_p}{\beta_{2,p}} \leq 1 - \dfrac{1}{\beta_{1,p}}$ by (2.1.2), and therefore (2.7.3) follows as in (1.11). \vspace{4pt} (2.7.4) is proved similarly. First, since $m_q = \ord_q(D) \geq 2-\mu_q$, the inequality is true when $\beta_{1,q} \geq 2-\mu_q$, as in the proof of (2.7.3) above. When $\beta_{1,q} < 2-\mu_q$ we must have $\mu_q < 1$; then $B+cD\geq C_0 \implies \mu_q+ cm_q\geq 1 \implies c\geq\dfrac{1-\mu_q}{m_q}, \; t < \dfrac{m_q}{\beta_{2,q}}$ by (2.5.2), and $\dfrac{1-\mu_q}{\beta_{2,q}} \leq 1-\dfrac{1}{\beta_{2,q}}$ by (2.1.2); consequently \begin{equation*} \begin{split} c(m_q-t\beta_{1,q}) &> \frac{1-\mu_q}{m_q} \left( m_q-\frac{m_q}{\beta_{2,q}}\beta_{1,q} \right) = \\ &= (1-\mu_q) - \frac{1-\mu_q}{\beta_{2,q}}\beta_{1,q} \geq 2-\mu_q-\beta_{1,q}, \end{split} \end{equation*} which yields (2.7.4). Thus (2.7.2) is proved; this concludes the proof when $q \in C_0$. \vspace{6pt} {\bf (2.8)} To complete the proof of the proposition in case 3, consider the remaining subcase, $q \notin C_0$. In this subcase separation of $(p,q)$ is obtained by reversing the roles of $p$ and $q$. Namely, let $D'=\alpha D$, for the positive rational number $\alpha$ such that $\ord_q(D')=2-\mu_q$; that is, $\alpha = \dfrac{2-\mu_q}{\ord_q(D)} = \dfrac{2-\mu_q}{m_q}$. Note that $D' \qle t'M$, where $t' = \alpha t < \dfrac{2-\mu_q}{m_q} \cdot \dfrac{m_q}{\beta_{2,q}}$ (by (2.5.2)), i.e. \begin{equation*} 0 < t' < \frac{2-\mu_q}{\beta_{2,q}} \leq 1. \tag{2.8.1} \end{equation*} Let $c'$ be the $PLC$ threshold for $(S,B,D')$ at $q$; note that $c'\alpha > c$ ($c'\alpha$ is the $PLC$ threshold of $(S,B,D)$ at $q$, and therefore $c < c'\alpha$ follows from $N_0 \cap E_q = \emptyset$ in (2.6.1)). This, in turn, implies $B+c'D' = B+c'\alpha D \geq C_0$. If $(S,B,D')$ is $PLC$ at $q$ (i.e. if $c'=1$), then (1.2.1)(b) yields \begin{equation*} H^1(S_1, f^*(K_S+B+M)-E_q-N_0') = 0, \qquad N_0' \cap E_q = \emptyset \tag{2.5.$3'$} \end{equation*} (Compare to (2.5.3)). If $(S,B,D')$ is not $PLC$ at $q$ (i.e. if $c' < 1$), and $C_0'$ is the critical curve at $q$, then (1.2.1.)(a) yields \begin{equation*} H^1(S_1, f^*(K_S+B+M)-D_0'-N_0') = 0, \qquad N_0' \cap E_q = \emptyset \tag{2.6.$1'$} \end{equation*} (Compare to (2.6.1), noting that now $p$ and $q$ are interchanged.) In both cases, the arguments in (1.8) and, respectively, (1.9)--(1.11) show that there exists $\Lambda \in |K_S+B+M-N'|$ with $q \notin \Supp(\Lambda)$, where $N' = f_*N_0'$ is an effective divisor with $q \notin \Supp(N')$. Now, however, $N' \geq C_0$ (because $B+c'D' \geq C_0$, as noted earlier, and $q \notin C_0 \implies C_0$ is not discarded even when the vanishing theorem is used in the form (1.2.1)(b)); thus $\Gamma + N' \in |K_S+B+M|$ passes through $p$ but not through $q$. \vspace{6pt} This completes the proof of Proposition 4. \section{Separation of tangent directions} {\bf (3.1)} Let $(S,B,M)$ be as in \S 1. Fix a point $p \in S$. In this section we give criteria for $|K_S+B+M|$ to separate directions at $p$. The statements (and proofs) are somewhat similar to those in \S 2. The main difference is in the part of the proof corresponding to the discussion in (2.8). So far in our proofs we worked with $M-cD$, where $c$ was always the $PLC$ threshold at some point or another; this made the arguments relatively transparent. In (2.8), when we passed from $c$ = $PLC$ threshold at $p$ to $c'\alpha$ = $PLC$ threshold at $q$, the relevant fact was that $q \notin C_0$, where $C_0$ was the critical curve at $p$, and therefore $C_0$ did not affect the local computations around $q$. In separating tangent directions, the analogue is a curve $C_0$ through $p$, such that $\vec{v} \notin T_p(C_0)$ for some fixed $\vec{v} \in T_p(S), \vec{v} \neq \vec{0}$. Then we will have to increase $c$ to some larger value $c'$, but clearly in that case $(S, B, c'D)$ will no longer be $PLC$ at $p$. While this complicates the computations, the geometric idea is still the same: find a divisor $\Gamma \in |K_S+B+M-C_0-N|$, $p \notin \Supp(N)$, such that $\Gamma$ does not pass through $p$; then $\Gamma + C_0 +N$ has only one component through $p$, namely, $C_0$, and $\vec{v} \notin T_p(C_0)$ -- therefore $\Gamma + C_0 +N$ passes through $p$ and is not tangent to $\vec{v}$, as required. Another technical problem, which did not arise before, is that in some cases the ``minimizing'' curve $C_0$ may be singular at $p$. (This possibility is directly related to the need, in some cases, to increase $c$ beyond the $PLC$ threshold at $p$.) In those cases we separate the tangent directions on $C_0$, using (1.2.2)(b) (note that $C_0$ singular at $p$ \ $\implies T_p(S)=T_p(C_0)$); the vanishing (1.2.1) is then used to lift from $C_0$ to $S$. \vspace{6pt} {\bf (3.2)} Let $S$ be a smooth surface, as before; let $p$ denote a point on $S$, and fix $\vec{v} \in T_p(S), \vec{v} \neq \vec{0}$. Let $Z$ denote the zero-dimensional subscheme of length $2$ of $S$, corresponding to $(p,\vec{v})$; in local coordinates $(x,y)$ at $p$ such that $\vec{v}$ is tangent to $(y=0)$, $Z$ is defined by the ideal $\mathcal{I}_Z = (x^2,y) \cdot \mathcal{O}_S$. Let $f:S_1 \to S$ be the blowing-up of $S$ at $p$, with exceptional curve $E_p$, and let $V \in E_p$ correspond to (the direction of) $\vec{v}$. Let $g:S_2 \to S_1$ be the blowing-up of $S_1$ at $V$, with exceptional curve $F_{\vec{v}}$, and let $F_p = g^{-1}E_p$. Let $h = g \circ f$. Write \begin{equation*} h^*B = h^{-1}B + \mu_p F_p + \mu_{\vec{v}} F_{\vec{v}}; \tag{3.2.1} \end{equation*} $\mu_p = \ord_p(B)$, while (3.2.1) is the definition of $\mu_{\vec{v}}$. More generally, if $G$ is any effective \ensuremath{\mathbb{Q}}-divisor on $S$, denote the order of $h^*G$ along $F_{\vec{v}}$ by $o_{\vec{v}}(G)$; $o_{\vec{v}}(G) = \ord_p(G) + \ord_V(f^{-1}G)$. For convenience, let $o_V(G) \overset{\text{def}}{=} \ord_V(f^{-1}G)$, and let $\mu_V = o_V(B)$. Note that, in general, $o_{\vec{v}} = \ord_p + o_V$ and $o_V \leq \ord_p$; in particular: \begin{equation*} \mu_p \leq \mu_{\vec{v}} \leq 2\mu_p. \tag{3.2.2} \end{equation*} \vspace{6pt} {\bf (3.3)} Consider again $(S,B,M)$ as in \S 1, and fix $p, \vec{v}$ as in (3.2). Since the proofs will now be more complex, we will state the criteria for separating $\vec{v}$ at $p$ one by one, in increasing order of difficulty. The first (and easiest) case is: \begin{Proposition}[Case 1] If $\mu_p \geq 3$ or $\mu_{\vec{v}} \geq 4$, then $|K_S+B+M|$ separates $\vec{v}$ at $p$. ($M$ must still be nef and big.) \end{Proposition} \begin{proof} Recall that the conclusion means that the restriction map $$ H^0(S, K_S+B+M) \to H^0(Z, K_S+B+M|_Z) \cong \mathcal{O}_Z $$ is surjective. If $\mu_p \geq 3$, we use the vanishing theorem in the form (1.2.1)(a) for \begin{align*} K_{S_1} + \rup{f^*M} &= f^*(K_S+B+M) + E_p - \rdn{f^*B} \\ &= f^*(K_S+B+M) -t E_p, \end{align*} where $t = \rdn{\mu_p} - 1 \geq 2$, as in (1.4)--(1.5); then $H^0(S, K_S+B+M) \to \mathcal{O}_S/{\frak{m}_p}^t$ is surjective, and since $t \geq 2$, we have ${\frak{m}_p}^t \subset \mathcal{I}_Z$, i.e. $\mathcal{O}_S/{\frak{m}_p}^t \to \mathcal{O}_Z$ is also surjective. (See also Remark 1.5.3.) If $\mu_{\vec{v}} \geq 4$, the agrument is similar, starting on $S_2$: \begin{align*} K_{S_2}+\rup{h^*M} &= h^*(K_S+B+M)+F_p+2F_{\vec{v}}-\rdn{h^*B} \\ &= h^*(K_S+B+M)-t_pF_p - t_{\vec{v}}F_{\vec{v}} , \end{align*} where $t_{\vec{v}} = \rdn{\mu_{\vec{v}}}-2 \geq 2$, and $t_p = \rdn{\mu_p}-1 \geq 1$ (indeed, by (3.2.2), $\mu_p \geq \frac{1}{2}\mu_{\vec{v}} \geq 2$.) As in the previous case, we get a vanishing $H^1(S, \mathcal{O}_S(K_S+B+M) \otimes \mathcal{I}) = 0$ for $\mathcal{I} = h_*\mathcal{O}_{S_2}(-t_pF_p-t_{\vec{v}}F_{\vec{v}})$; $\Supp(\mathcal{O}_S/\mathcal{I}) = \{ p \}$ and $\mathcal{I} \subset \mathcal{I}_Z$, so the conclusion follows as before. \end{proof} \vspace{6pt} {\bf (3.4)} Now assume that $\mu_p < 3$ and $\mu_{\vec{v}} < 4$. First consider the case $2 \leq \mu_p < 3$. Then $2 \leq \mu_{\vec{v}} <4$, and therefore $0 < (4-\mu_{\vec{v}}) \leq 2$. \addtocounter{Theorem}{-1} \begin{Proposition}[Case 2] Let $2 \leq \mu_p < 3$ and $2 \leq \mu_{\vec{v}} < 4$. Assume that $M^2 > (4-\mu_{\vec{v}})^2$, $M \cdot C \geq \frac{1}{2}(4-\mu_{\vec{v}})$ for every curve $C \subset S$ through $p$, and $M \cdot C \geq (4-\mu_{\vec{v}})$ for every curve $C$ containing $Z$ -- i.e., such that $p \in C$ and $\vec{v} \in T_p(C)$. Then $|K_S+B+M|$ separates $\vec{v}$ at $p$. \end{Proposition} \vspace{4pt} \emph{Proof.} {\bf (3.5) Claim:} We can find an effective \ensuremath{\mathbb{Q}}-divisor $D$ on $S$ such that $o_{\vec{v}}(D) = 4-\mu_{\vec{v}}$ and $D \qle tM$ for some $t \in \ensuremath{\mathbb{Q}}\,, 0 < t < 1$. (See (3.2) for the definition of $o_{\vec{v}}(D)$.) \emph{Proof of (3.5):} Choose $a > (4-\mu_{\vec{v}})$ such that $M^2 > a^2$. Then $(h^*M-a F_{\vec{v}})^2 = M^2-a^2 >0$ and $(h^*M-a F_{\vec{v}}) \cdot h^*M=M^2>0$; therefore $h^*M-a F_{\vec{v}} \in N(S_2)^+$, the positive cone of $S_2$, and in particular it is big. (See, for example, \cite[(1.1)]{km}.) Therefore $\exists T$, effective \ensuremath{\mathbb{Q}}-divisor on $S_2$, such that $T \qle h^*M - a F_{\vec{v}}$. Put $D_1 = h_*(T+a F_{\vec{v}})$; then $D_1 \qle h_*(h^*M)=M$. Also, $h^*D_1= T + a F_{\vec{v}}$ (their difference has support contained in $F_p \cup F_{\vec{v}}$; on the other hand, $T + a F_{\vec{v}} \qle h^*M \implies \big( h^*D_1 - (T+a F_{\vec{v}}) \big) \cdot F_p = \big( h^*D_1 - (T+a F_{\vec{v}}) \big) \cdot F_{\vec{v}} = 0$, and $h^*D_1 = T+a F_{\vec{v}}$ follows from the negative definiteness of the intersection form on $h^{-1}(p) = F_p \cup F_{\vec{v}}$). We have $D_1 \qle M$ and $o_{\vec{v}}(D_1) \geq a > 4 - \mu_{\vec{v}}$. Take $D = tD_1$, $t = \dfrac{4-\mu_{\vec{v}}}{o_{\vec{v}}(D_1)} < 1$. \qed \vspace{4pt} \begin{Remark} The statement of (3.5) is similar to that of (1.6.1), and indeed, we could have proved it as in \S 1. However, the proof we gave here is easier to generalize, especially on \emph{singular} surfaces. \end{Remark} \vspace{4pt} {\bf (3.6)} We return to the proof of Proposition 5, Case 2. Choose $D$ as in (3.5). Write $B=\sum b_iC_i, D=\sum d_iC_i; D_i=f^{-1}C_i, T_i=g^{-1}D_i = h^{-1}C_i; h^*B=h^{-1}B + \mu_p F_p + \mu_{\vec{v}} F_{\vec{v}}, h^*D = h^{-1}D + m_p F_p + (4-\mu_{\vec{v}})F_{\vec{v}}$, where $m_p=\ord_p(D)$. We have $K_{S_2} = h^*K_S +F_p +2F_{\vec{v}}$. If $b_i+d_i \leq 1$ for every $C_i$ through $p$, then \begin{align*} K_{S_2}+\rup{h^*(M-cD)} &=h^*(K_S+B+M)+F_p+2F_{\vec{v}}-\rdn{h^*(B+D)} \\ &= h^*(K_S+B+M)-t_pF_p- 2 F_{\vec{v}}-\textstyle\sum'T_i-N_2, \tag{3.6.1} \end{align*} where $\sum ' T_i$ extends over all $i$ with $b_i+d_i=1$ and $p \in C_i$ (if any), $N_2$ is an effective divisor on $S_2$ such that $\Supp(N_2) \cap h^{-1}(p) = \emptyset$, and $t_p = \rdn{\mu_p + m_p}-1 \geq 1$ (because $\mu_p \geq 2$ by hypothesis). Then we conclude as in (3.3) (Case 1 of the Proposition), using the vanishing (1.2.1)(b) to dispose of $\sum ' T_i$ (if it is not zero). \vspace{4pt} {\bf (3.7)} Now assume that $b_i+d_i > 1$ for at least one $C_i$ through $p$. Let \begin{equation*} c \overset{\text{def}}{=} \min \left\{ \frac{3-\mu_p}{m_p} ; \frac{1-b_i}{d_i} : b_i+d_i>1 \text{ and } p \in C_i \right\}. \tag{3.7.1} \end{equation*} \vspace{4pt} If $c = \dfrac{3-\mu_p}{m_p}$, we finish again as in Case 1, using (1.2.1)(b) for \[ K_{S_1} + \rup{f^*(M-cD)} = f^*(K_S+B+M) -2E_p - \textstyle\sum ' D_i - N_1 \] on $S_1$, where $\sum ' D_i$ extends over all $i$ such that $b_i+cd_i=1$ and $p \in C_i$ (if any), and $\Supp(N_1) \cap E_p = \emptyset$. \vspace{4pt} {\bf (3.8)} If $c = \dfrac{1-b_0}{d_0} < \dfrac{3-\mu_p}{m_p}$ for some $C_0$ through $p$, then \[ \sum (b_i+cd_i) \cdot \mult_p(C_i) = \mu_p + cm_p <3 ; \] therefore $\mult_p(C_0) \leq 2$, and moreover, if $\mult_p(C_0) = 2$, then $b_i+cd_i < 1$ for all $C_i$ through $p$ with $i \neq 0$. Also, $\mu_{\vec{v}}+c(4-\mu_{\vec{v}}) < 4$ (since $c<1$), and therefore $B+cD \geq C_0 \implies o_{\vec{v}}(C_0) \leq 3$. \vspace{4pt} {\bf (3.9)} If $C_0$ is singular at $p$ and $\vec{v} \notin TC_p(C_0)$ (the \emph{tangent cone} to $C_0$ at $p$), then $o_{\vec{v}}(C_0) = 2$. We have $Z \subset C_0$, and \begin{align*} K_S + \rup{M-cD} &= (K_S+B+M) - \rdn{B+cD} \\ &= (K_S+B+M) - C_0 - N , \end{align*} with $p \notin \Supp(N)$. Using (1.2.1)(a), as in \S 1, it suffices to show that $\big( (K_S+B+M)-C_0-N \big) |_{C_0}$ separates $\vec{v}$ at $p$ on $C_0$; that, in turn, will follow from (1.2.2)(b), \emph{if} we can show that $\rup{M-cD} \cdot C_0 > 2$. As before, write $\rup{M-cD} = (M-cD)+\Delta$; $\Delta = \frp{B+cD}$ and $C_0$ intersect properly, and $\Delta = B+cD-C_0$ in an open neighborhood of $p$. We have: $\ord_p(\Delta) = \mu_p + cm_p - 2$, and therefore $\Delta \cdot C_0 \geq 2(\mu_p +cm_p -2)$. However, we get a better estimate if we consider orders along $F_{\vec{v}}$, as follows: $o_{\vec{v}}(\Delta) = \mu_{\vec{v}} + c(4-\mu_{\vec{v}}) - 2, \quad \text{because } o_{\vec{v}}(C_0) = 2 $; $\ord_p(\Delta) \geq \frac{1}{2} o_{\vec{v}}(\Delta)$, and therefore \[ \Delta \cdot C_0 \geq \frac{1}{2} o_{\vec{v}}(\Delta) \cdot 2 \geq \mu_{\vec{v}} + c(4-\mu_{\vec{v}}) - 2 . \] Finally, \begin{align*} \rup{M-cD} \cdot C_0 &= (M-cD) \cdot C_0 + \Delta \cdot C_0 = (1-ct) M \cdot C_0 + \Delta \cdot C_0 \\ &\geq (1-ct)(4-\mu_{\vec{v}}) + \mu_{\vec{v}} + c(4-\mu_{\vec{v}}) - 2 \tag{3.9.1} \\ &> 2 \qquad \text{(because $t<1$),} \end{align*} as required \vspace{4pt} {\bf (3.10)} If $C_0$ is singular at $p$ and $\vec{v} \in TC_p(C_0)$, then $o_{\vec{v}}(C_0) = 3$ ($ \geq 3$ is clear, and $\leq 3$ was shown in (3.8)). Working as in (3.9), we can show that \begin{equation*} \rup{M-cD} \cdot C_0 \geq (1-ct)(4-\mu_{\vec{v}})+\mu_{\vec{v}}+c(4-\mu_{\vec{v}})-3 > 1 \tag{3.10.1} \end{equation*} (now $o_{\vec{v}}(\Delta) = o_{\vec{v}}(B+cD-C_0) = \mu_{\vec{v}}+c(4-\mu_{\vec{v}})-3$); thus in this case we cannot use (1.2.2)(b) as in (3.9). We will modify the argument as follows: Start with $f^*(M-cD)$ on $S_1$; the vanishing theorem yields \begin{equation*} H^1(S_1, f^*(K_S+B+M)-E_p-D_0-N_1)=0, \qquad N_1 \cap E_p = \emptyset \tag{3.10.2} \end{equation*} (the coefficient of $E_p$ is $-1$ because $2 \leq \mu_p+cm_p < 3$; the first inequality follows from $\mu_p \geq 2$, and the second was shown in (3.8)). $\vec{v} \in TC_p(C_0) \implies V \in D_0$ (recall that $V \in E_p$ corresponds to $\vec{v} \in T_p(S)$). (3.10.2) implies the surjectivity of the restriction map \begin{multline*} H^0(S_1, f^*(K_S+B+M)-E_p-N_1) \\ \to H^0(D_0, f^*(K_S+B+M)-E_p-N_1 |_{D_0}). \tag{3.10.3} \end{multline*} We will show that $\exists \tilde{\Gamma} \in \left| f^*(K_S+B+M)-E_p-N_1 |_{D_0} \right|$ such that $V \notin \Supp(\tilde{\Gamma})$. Then we can lift $\tilde{\Gamma}$ to $\Gamma \in |f^*(K_S+B+M)-E_p-N_1|$, since (3.10.3) is surjective. $\Gamma+E_p+N_1 \in |f^*(K_S+B+M)|$ has the form $f^*\Lambda$ for some $\Lambda \in |K_S+B+M|$. Finally, $p \in \Supp(\Lambda)$, because $f^*\Lambda \geq E_p$, but $\vec{v} \notin T_p(\Lambda)$, because $V \notin \Supp(f^*\Lambda - E_p)$; this shows that $|K_S+B+M|$ separates $\vec{v}$ at $p$ on $S$. To prove the existence of $\tilde{\Gamma}$, note that $\big( f^*(K_S+B+M)-E_p-N_1 \big) |_{D_0} = K_{D_0} + \rup{f^*(M-cD)}|_{D_0}$; we will show that $\rup{f^*(M-cD)} \cdot D_0 > 1$ -- then (1.2.2)(a) implies the existence of $\tilde{\Gamma}$. As in (1.9), we can write $\rup{f^*(M-cD)} = f^*(M-cD) + \Delta_1$, where $\Delta_1 = \frp{f^*(B+cD)}$ and $D_0$ intersect properly, and $\Delta_1 = f^*(B+cD) - 2E_p - D_0 = f^*(B+cD-C_0)$ in a neighborhood of $E_p$ (the coefficient of $E_p$ in $f^*(B+cD)$ is $\mu_p+cm_p$, and $2 \leq \mu_p+cm_p <3$). We have: \begin{equation*} \begin{split} \rup{f^*(M-cD)} \cdot D_0 &= f^*(M-cD) \cdot D_0 + \Delta_1 \cdot D_0 \\ &\geq (M-cD) \cdot C_0 + \ord_p(B+cD-C_0) \cdot \mult_p(C_0) \\ &\geq (1-ct)(4-\mu_{\vec{v}})+\mu_{\vec{v}}+c(4-\mu_{\vec{v}})-3 \\ &> 1 \end{split} \tag{3.10.4} \end{equation*} as in (3.10.1) \vspace{4pt} {\bf (3.11)} Now consider the case: $C_0$ smooth at $p$ and tangent to $\vec{v}$, and $b_i+cd_i < 1$ for all $i \neq 0$ with $p \in C_i$. Write $\rup{M-cD} = (M-cD) + \Delta$, where $\Delta$ and $C_0$ intersect properly; then $\Delta \cdot C_0 = h^*\Delta \cdot T_0$ (projection formula: recall that $T_0=h^{-1}C_0$) $\geq o_{\vec{v}}(\Delta)$, because $F_{\vec{v}} \cdot T_0 = 1$; since $\Delta = \frp{B+cD} = B+cD-C_0$ in a neighborhood of $p$, we have $o_{\vec{v}}(\Delta) = \mu_{\vec{v}}+c(4-\mu_{\vec{v}})-2$, and therefore \begin{equation*} \rup{M-cD} \cdot C_0 \geq (1-ct)(4-\mu_{\vec{v}})+\mu_{\vec{v}}+c(4+\mu_{\vec{v}})-2 >2, \end{equation*} exactly as in (3.9.1). Thus $K_{C_0}+\rup{M-cD}|_{C_0}$ separates $\vec{v}$ on $C_0$; we conclude as in (3.9). \vspace{4pt} {\bf (3.12)} If $C_0$ is smooth at $p$ and tangent to $\vec{v}$, and moreover $b_i+cd_i = 1$ for at least one $i \neq 0$ with $p \in C_i$, then: such an $i$ is unique, say $i=1$, and $C_1$ must be smooth at $p$ and not tangent to $\vec{v}$; indeed, $B+cD \geq C_0+C_1$, while $\ord_p(B+cD) < 3$ and $o_{\vec{v}}(B+cD) < 4$. In this case reverse the roles of $C_0$ and $C_1$: thus $C_0$ will be smooth at $p$ and not tangent to $\vec{v}$. This situation is covered below, in (3.13). \vspace{4pt} {\bf (3.13)} Finally, assume that $C_0$ is smooth at $p$ and not tangent to $\vec{v}$. In this case we work with $M-c'D$ for some $c' \geq c$, namely: \[ c' \overset{\text{def}}{=} \min \left\{ 1; \frac{3-\mu_p}{m_p}; \frac{2-b_0}{d_0}; \frac{1-b_i}{d_i} \text{ with } i \neq 0, p \in C_i \text{ and } b_i+d_i \geq 1 \right\}. \] In all cases, $M-c'D \qle (1-c't)M$ is still nef and big; using the vanishing $H^1(S, K_S+\rup{M-c'D})=0$, or the corresponding vanishing on $S_1$ or $S_2$, we will show that $\exists \Lambda \in |K_S+B+M-C_0-N|, p \notin \Supp(N)$, such that $p \notin \Supp(\Lambda)$. Then $\Lambda+C_0+N \in |K_S+B+M|$ has the unique component $C_0$ through $p$ not tangent to $\vec{v}$, as required. \vspace{4pt} It remains to prove the existence of $\Lambda$. \vspace{4pt} {\bf (3.13.1)} If $c' = \dfrac{3-\mu_p}{m_p}$, then (1.2.1)(b) yields \[ H^1(S_1, f^*(K_S+B+M) - 2E_p - D_0 - N_1) = 0, \qquad N_1 \cap E_p = \emptyset; \] thus $H^1(S_1, f^*(K_S+B+M-C_0-N) - E_p) = 0$, where $N = f_*N_1$, and the existence of $\Lambda$ follows. \vspace{4pt} When $c' = 1$ the proof is similar, starting on $S_2$, as in the proof of Case~1 of the proposition. \vspace{4pt} {\bf (3.13.2)} If $c' = \dfrac{1-b_1}{d_1} < \dfrac{3-\mu_p}{m_p}$ for another curve $C_1$ through $p$ with $b_1+d_1 > 1$, then $C_1$ must be smooth at $p$ (because $B+c'D \geq C_0+C_1$, and $\ord_p(B+c'D) = \mu_p+c'm_p < 3$). We may have $c' = c$ (e.g., in the case discussed in (3.12)), or $c' > c$. In any event, $b_i+c'd_i < 1$ for all curves $C_i$ through $p$, $i \neq 0,1$. (1.2.1)(a) yields: \begin{equation*} H^1(S, K_S+B+M-C_0-C_1-N) = 0, \qquad p \notin \Supp(N). \tag{3.13.3} \end{equation*} We claim that $p \notin \Bs \left| K_S+B+M-C_0-N |_{C_1} \right|$, which in turn follows from (1.2.2)(a) once we show that $\rup{M-c'D} \cdot C_1 > 1$. Then we use (3.13.3) to lift from $C_1$ to $S$, proving the existence of $\Lambda$ as stated. \vspace{3pt} $\rup{M-c'D} = (M-c'D) + \Delta'$, with $\Delta' =\frp{B+c'D} = B+c'D-C_0-C_1$ in a neighborhood of $p$, and $\Delta', C_1$ intersect properly. \vspace{3pt} If $\vec{v} \notin T_p(C_1)$, then $M \cdot C_1 \geq \frac{1}{2}(4-\mu_{\vec{v}})$ by hypothesis, and $\ord_p(\Delta') \geq \frac{1}{2}o_{\vec{v}}(\Delta') \geq \frac{1}{2} \big( \mu_{\vec{v}}+c'(4-\mu_{\vec{v}})-2 \big)$; therefore \[ \rup{M-c'D} \cdot C_1 \geq \dfrac{1}{2}(1-c't)(4-\mu_{\vec{v}}) + \dfrac{1}{2} \big( \mu_{\vec{v}}+c'(4-\mu_{\vec{v}}) - 2 \big) > 1, \] as required (compare to (3.9.1)). The proof is the same when $c' = \dfrac{2-b_0}{d_0}$, i.e. $C_1 = C_0$; in that case $p \notin \Bs \left| K_S+B+M-C_0-N |_{C_0} \right|$. \vspace{3pt} If $\vec{v} \in T_p(C_1)$, then $M \cdot C_1 \geq 4-\mu_{\vec{v}}$ and $\Delta' \cdot C_1 \geq o_{\vec{v}}(\Delta') = \mu_{\vec{v}}+c'(4-\mu_{\vec{v}})-3$; all told, we have \[ \rup{M-c'D} \cdot C_1 \geq (1-c't)(4-\mu_{\vec{v}})+\mu_{\vec{v}}+c'(4-\mu_{\vec{v}})-3 >1, \] as claimed. \vspace{6pt} This concludes the proof of Proposition 5, Case 2. \vspace{10pt} {\bf (3.14)} Finally, consider the case $0 \leq \mu_p < 2$ (and therefore $0 \leq \mu_V < 2$ and $0 \leq \mu_{\vec{v}} = \mu_p + \mu_V < 4$). \vspace{4pt} \addtocounter{Theorem}{-1} \begin{Proposition}[Case 3] Assume that $0 \leq \mu_p < 2$. Assume, moreover, that $M^2 > (\beta_{2,p})^2 + (\beta_{2,V})^2$ and \begin{enumerate} \item[(i)] $M \cdot C \geq \beta_1$ for every curve $C \subset S$ passing through $p$, \item[(ii)] $M \cdot C \geq 2\beta_1$ for every curve $C$ containing $Z$ (i.e., passing through $p$ and with $\vec{v} \in T_p(C)$), \end{enumerate} where $\beta_{2,p}, \beta_{2,V}, \beta_1$ are positive numbers which satisfy: \begin{align*} \beta_{2,p} &\geq 2-\mu_p, \qquad \beta_{2,V} \geq 2-\mu_V; \tag{3.14.1} \\ \beta_1 &\geq \min \left\{ \frac{1}{2}(4-\mu_{\vec{v}}); \frac{\beta_{2,p}+\beta_{2,V}}{\beta_{2,p}+\beta_{2,V}-(2-\mu_{\vec{v}})} \right\} \tag{3.14.2} \end{align*} \end{Proposition} \begin{proof} The proof is very similar, in many respects, to that of Case 2. We indicate the main steps of the proof, and we provide explicit computations in a few cases, to show what kind of alterations are needed. \vspace{6pt} {\bf (3.15) Claim:} We can find $D$, an effective \ensuremath{\mathbb{Q}}-divisor on $S$, such that $o_{\vec{v}}(D) = 4-\mu_{\vec{v}}$ and $D \qle tM$ for some $t \in \ensuremath{\mathbb{Q}}\,, t > 0$, satisfying \begin{equation*} t < \frac{4-\mu_{\vec{v}}}{\beta_{2,p}+\beta_{2,V}} \tag{3.15.1} \end{equation*} -- and therefore, in particular, $t < 1$. \vspace{3pt} \emph{Proof of (3.15):} Choose $a > \beta_{2,p} , b > \beta_{2,V}$, such that $M^2 > a^2+b^2$. We have $\big( aF_p + (a+b)F_{\vec{v}} \big)^2 = -(a^2+b^2)$, and therefore $h^*M-\big( aF_p + (a+b)F_{\vec{v}} \big)$ is big, as in the proof of (3.5). Thus we can find $D_1 \qle M$ on $S$, $D_1 \geq 0$, such that $h^*D_1 \geq aF_p + (a+b)F_{\vec{v}}$. Then take $D = tD_1$, with \[ t = \frac{4-\mu_{\vec{v}}}{o_{\vec{v}}(D_1)} \leq \frac{4-\mu_{\vec{v}}}{a+b} < \frac{4-\mu_{\vec{v}}}{\beta_{2,p}+\beta_{2,V}}. \] \qed \vspace{6pt} {\bf (3.16)} If $D = \sum d_i C_i$, as before, and $b_i+d_i \leq 1$ for every $C_i$ through $p$, we conclude as in (3.6). If $b_i+d_i > 1$ for at least one $C_i$ through $p$, then define \begin{equation*} c = \min \left\{ \frac{3-\mu_p}{m_p}; \frac{1-b_i}{d_i} : b_i+d_i > 1 \text{ and } p \in C_i \right\}. \tag{3.16.1} \end{equation*} \vspace{4pt} If $c=\dfrac{3-\mu_p}{m_p}$, we finish as in (3.7). \vspace{3pt} If $c = \dfrac{1-b_0}{d_0} < \dfrac{3-\mu_p}{m_p}$ for some $C_0$ through $p$, then $\mult_p(C_0) \leq 2$ and $o_{\vec{v}}(C_0) \leq 3$; if $C_0$ is singular at $p$, then it is the only $C_i$ through $p$ with $b_i+cd_i \geq 1$, and we proceed as in (3.9) or (3.10), according to whether $\vec{v} \in TC_p(C_0)$ or not. Only the proof of $\rup{M-cD} \cdot C_0 > 2$ (if $\vec{v} \notin TC_p(C_0)$) or $ > 1$ (if $\vec{v} \in TC_p(C_0)$) needs adjustment. \vspace{3pt} Assume first that $\vec{v} \notin TC_p(C_0)$ (with $C_0$ singular at $p$). Then $\rup{M-cD} = (M-cD) + \Delta, \Delta = \frp{B+cD} = B+cD-C_0$ in a neighborhood of $p$, and $o_{\vec{v}}(\Delta) = \mu_{\vec{v}}+c(4-\mu_{\vec{v}})-2$; $\ord_p(\Delta) \geq \frac{1}{2} o_{\vec{v}}(\Delta)$ and $\mult_p(C_0)=2$, so that \begin{equation*} \rup{M-cD} \geq (1-ct)(2\beta_1) +\mu_{\vec{v}}+c(4-\mu_{\vec{v}})-2. \tag{3.16.2} \end{equation*} \vspace{3pt} If $\beta_1 \geq \frac{1}{2}(4-\mu_{\vec{v}})$, then $\rup{M-cD} \cdot C_0 > 2$ follows from (3.16.2) and $t < 1$. In particular, this is true if $\mu_{\vec{v}} \geq 2$. If $\mu_{\vec{v}} < 2$, the hypothesis is weaker than $\beta_1 \geq \frac{1}{2}(4-\mu_{\vec{v}})$, namely: \begin{equation*} \beta_1 \geq \frac{\beta_{2,p}+\beta_{2,V}}{\beta_{2,p}+\beta_{2,V} - (2-\mu_{\vec{v}})}. \tag{3.16.3} \end{equation*} Assume also that $\beta_1 < \frac{1}{2}(4-\mu_{\vec{v}})$ (otherwise we are done). Then $\rup{M-cD} \cdot C_0 > 2$ follows from (3.16.2), (3.16.3), (3.15.1), and $c \geq \dfrac{2-\mu_{\vec{v}}}{4-\mu_{\vec{v}}}$, exactly as in (1.11). \vspace{4pt} Now consider the case $\vec{v} \in TC_p(C_0)$ (with $C_0$ still singular at $p$). Using the strategy of (3.10), all we need to prove is $\rup{M-cD} \cdot C_0 > 1$, which follows from \begin{equation*} \rup{M-cD} \cdot C_0 \geq (1-ct)(2\beta_1) +\mu_{\vec{v}}+c(4-\mu_{\vec{v}})-3 \tag{3.16.4} \end{equation*} (same computation as in (3.10) -- see (3.10.4)). Using (3.16.4), the inequality $\rup{M-cD} \cdot C_0 > 1$ is proved exactly as in the previous paragraph. \vspace{6pt} {\bf (3.17)} If $C_0$ is smooth at $p$, $\vec{v} \in T_p(C_0)$, and $b_i+cd_i<1$ for every $C_i$ through $p$ with $i \neq 0$, then the proof goes as in (3.11); the inequality we need in this case, $\rup{M-cD} \cdot C_0 > 2$, is proved as above. As in the proof of Case 2 of the Proposition, if $C_0$ is smooth at $p$ and tangent to $\vec{v}$, and $b_1+cd_1 = 1$ for one more curve $C_1$ through $p$, then $C_1$ is unique with these properties, and is smooth at $p$ and $\vec{v} \notin T_p(C_1)$. Switching $C_0$ and $C_1$, we are in the situation discussed below. (Compare to (3.12).) \vspace{6pt} {\bf (3.18)} Finally, assume that $C_0$ is smooth at $p$ and $\vec{v} \notin T_p(C_0)$. Define \[ c' = \min \left\{ 1; \frac{3-\mu_p}{m_p} ; \frac{2-b_0}{d_0} ; \frac{1-b_i}{d_i}: i \neq 0, b_i+d_i > 1 \text{ and } p \in C_i \right\}. \] Consider, for example, the case $c' = \dfrac{2-b_0}{d_0} < 1$ and $< \dfrac{3-\mu_p}{m_p}$. In this case, we show that $p \notin \Bs|K_S+B+M-C_0-N|$, for some effective divisor $N$ supported away from $p$. Using the vanishing \[ H^1(S, K_S+\rup{M-c'D}) = H^1(S, K_S+B+M-2C_0-N) = 0, \] it suffices to show that $p \notin \Bs \left| K_S+B+M-C_0-N |_{C_0} \right|$; this, in turn, will follow from (1.2.2)(a) and $\rup{M-c'D} \cdot C_0 > 1$. Now $C_0$ passes through $p$ but is not tangent to $\vec{v}$, and therefore we have only $M \cdot C_0 \geq \beta_1$ (rather than $2\beta_1$). $\rup{M-c'D} = (M-c'D) + \Delta'$, with $\Delta' = B+c'D-2C_0$ in a neighborhood of $p$, and therefore $\ord_p(\Delta') \geq \frac{1}{2}(\mu_{\vec{v}}+c'(4-\mu_{\vec{v}})-2)$; it suffices to show that \[ (1-c't)\beta_1 + \frac{1}{2}(\mu_{\vec{v}}+c'(4-\mu_{\vec{v}})-2) > 1. \] An inequality equivalent to this one was already proved in (3.16). The proof in the remaining cases is a similar adaptation of the arguments in (3.13). \end{proof} \section{Example} Fix an integer $n$, $n \geq 1$. Let $S$ be the $n^{\text{th}}$ Hirzebruch surface, i.e. the geometrically ruled rational surface $\ensuremath{\mathbb{P}}(\mathcal{E})$, where $\mathcal{E}$ is the rank 2 vector bundle $\mathcal{O}_{\ensuremath{\mathbb{P}}^1}\oplus\mathcal{O}_{\ensuremath{\mathbb{P}}^1}(-n)$ on $\ensuremath{\mathbb{P}}^1$. Let $\pi:S\to\ensuremath{\mathbb{P}}^1$ be the ruling of $S$, and let $F$ denote a fiber of $\pi$. $S$ contains a unique irreducible curve $G$ with negative self-intersection, $G^2=-n$. $\Pic(S) \cong \ensuremath{\mathbb{Z}} \oplus \ensuremath{\mathbb{Z}}$, with generators $F$ and $G$; $F^2=0, F \cdot G = 1$. $K_S \sim -2G-(n+2)F$. If $C$ is any irreducible curve on $S$, then $C=G$, $C \sim F$, or $C \sim aG+bF$ with $a,b \in \ensuremath{\mathbb{Z}}, a \geq 1$ and $b \geq na$. All these properties are proved, for example, in \cite[Ch.V, \S 2]{harbook}. \vspace{4pt} Let $H_m=G+mF$. We will use the Reider-type results for \ensuremath{\mathbb{Q}}-divisors to prove the following facts: {\bf Claim.} (1) (See \cite[Ch.IV, Ex.1]{bv}.) \emph{ $|H_n|$ is base-point-free, and defines a morphism $\phi_n:S\to\ensuremath{\mathbb{P}}^{n+1}$. Moreover, $\phi_n$ is an isomorphism on $S \setminus G$, and $\bar{S} = \phi_n(S) \subset \ensuremath{\mathbb{P}}^{n+1}$ is a (projective) cone with vertex $x = \phi_n(G)$. (Thus $\bar{S}$ is the cone over a normal rational curve contained in a hyperplane $\ensuremath{\mathbb{P}}^n \subset \ensuremath{\mathbb{P}}^{n+1}$, because $G \cong \ensuremath{\mathbb{P}}^1$ and $G^2=-n$. $\phi_n$ is the blowing-up of $\bar{S}$ at $x$.) } (2) \emph{ $|H_m|$ is very ample for $m \geq n+1$, defining an embedding $\phi_m : S \to \ensuremath{\mathbb{P}}^{2m-n+1}$. } (See \cite[Ch.IV, Ex.2]{bv} for other properties of $|H_m|$.) \vspace{6pt} Certainly, these facts can be proved in many different ways. For instance, for (1): if $G \subset S$ is a smooth rational curve with negative self-intersection on \emph{any} smooth surface $S$, then there is a projective contraction $\phi : S \to \bar{S}$, which is an isomorphism on $S \setminus G$ and contracts $G$ to a normal point $x$. (This is a direct generalization of the ``easy'' part of Castelnuovo's criterion -- the ``hard'' part being the regularity of $\bar{S}$ at $x$ when $G^2=-1$.) The proof can be adapted to the situation of the Claim. Alternatively, most of the Claim is proved in \cite[Theorem 2.17]{harbook}. The methods used in these proofs are somewhat specialized (the ``normal contraction'' approach depends on $\Pic(G) \cong \ensuremath{\mathbb{Z}}$; the proof in \cite{harbook} is typical for ruled surfaces). From this point of view, Reider's theorem, which is based only on intersection numbers, is much more general. However, as we will see, Reider's theorem doesn't apply in the situation of the Claim. The proof we give below shows that there are instances where the scope of Reider's original results can be broadened by allowing \ensuremath{\mathbb{Q}}-divisors into the picture. \newpage \emph{Proof of the Claim.} (1) Write $H_n=K_S+L$, thus defining $L = H_n-K_S = 3G+(2n+2)F$. Then $L \cdot G = 3(-n)+(2n+2)=-n+2$; thus $L$ is not nef for $n \geq 3$, and therefore Reider's criterion does not apply. However, write $L$ as $B+M$, with $B=(1-\epsilon)G$ and $M=(2+\epsilon)G+(2n+2)F,\,\epsilon \in (0,1)$. Then \[ M \cdot F = (2+\epsilon), \quad M \cdot G = (2-\epsilon n), \quad M^2 = (2+\epsilon)(2n+4-\epsilon n). \] In particular, for $\epsilon \to 0$, we have $M \cdot F \to 2, M \cdot G \to 2, M^2 \to 2(2n+4) \geq 12$. Fix $\epsilon > 0, \epsilon \ll 1$, such that $M^2>9, M \cdot F \geq \frac{3}{2}, \text{ and }M \cdot G \geq \frac{3}{2} $. Since \emph{any} irreducible curve $C \subset S$ is either $C=G$, or $C \sim F$, or $C \sim aG+bF$ with $a \geq 1$ and $b \geq na$, we automatically have $M \cdot C \geq \frac{3}{2}$ for all such $C$. (We will use this observation again later: if $M\cdot G \geq 0$ and $C \neq G$ is an irreducible curve, then $M \cdot C \geq M \cdot F$.) Therefore $|H_n|$ is base-point-free by Proposition 3, part 2, with $\beta_2 = 3$ and $\beta_1 = \dfrac{3}{2} = \dfrac{\beta_2}{\beta_2 - 1}$. \vspace{4pt} Thus $|H_n|$ defines a morphism $\phi_n:S\to\ensuremath{\mathbb{P}}^{\nu}$, $\nu = \dim |H_n|$. We compute $\nu$. By Riemann--Roch, we have: \[ \chi(S,H_n) = \frac{H_n \cdot (H_n-K_S)}{2} + \chi(S,\mathcal{O}_S) = n+2. \] We get $\nu = h^0(S,H_n)-1=n+1$, as stated in the Claim, \emph{if} we can show that $h^i(S,H_n)=0$ for $i\geq 1$. If we write $H_n=K_S+L$, as before, Kodaira's vanishing theorem does not apply, because $L$ is not ample (it is not even nef). If we write $L=B+M$ as above, though, we get $h^i(S,H_n)=0$ for $i \geq 1$, by (1.2.1)(a). \vspace{4pt} Next we show that $\phi_n$ is an isomorphism on $S \setminus G$. Consider two distinct points $p,q \in F \setminus G$. Write $L=B'+M'$, with $B'=(1-\epsilon)G+(1-\alpha)F, \,M'=(2+\epsilon)G+(2n+1+\alpha)F, \; \epsilon, \alpha \in (0,1)$. (Note that we may use \emph{any} decomposition of $L$ of the form $B+M$, as long as $\rup{M}=L$.) We have: \begin{gather*} M' \cdot F = (2+\epsilon), \quad M' \cdot G = (1+\alpha-\epsilon n), \\ (M')^2 = (2+\epsilon)(2n+2+2\alpha - \epsilon n). \end{gather*} In particular, for $\epsilon, \alpha \to 0$, $M'$ is nef and big and $M' \to 2(2n+2) \geq 8$. Let $\mu \overset{\text{def}}{=} \mu_p = \mu_q = 1-\alpha$. Choose $\beta_2 = \beta_{2,p} = \beta_{2,q} = \frac{3}{2}$ (say); then $\beta_2 \geq 2-\mu = 1+\alpha$ and $(M')^2 > 2(\beta_2)^2$ for $\epsilon, \alpha \ll 1$. Fix $\epsilon \ll 1$, and then choose $\alpha \ll \epsilon$ such that $1+\dfrac{\epsilon}{2} \geq \dfrac{\beta_2}{\beta_2-(1-\mu)} = \dfrac{\beta_2}{\beta_2 - \alpha}$; this can be done, because $\dfrac{\beta_2}{\beta_2-\alpha} \to 1$ for $\alpha \to 0$. Then $M' \cdot F = 2+\epsilon = 2\beta_1$, with $\beta_1=1+\dfrac{\epsilon}{2}$ --- and therefore $M' \cdot C \geq 2\beta_1$ for every irreducible curve $C$ through $p$ or $q$. Hence Proposition 4, part 3, applies (with $\beta_{1,p} = \beta_{1,q} = \beta_1$): $|H_n|$ separates $(p,q)$. If $p,q \in S \setminus G$ are distinct points on another irreducible curve $\bar{F} \sim F$, the proof is similar --- take $B' = (1-\epsilon)G + (1-\alpha)\bar{F}$. (We say $\bar{F} \sim F$ instead of ``fiber of $\pi : S \to \ensuremath{\mathbb{P}}^1$'', to emphasize that the proof uses numerical arguments only.) Finally, if no such curve passes through both $p$ and $q$, the proof is even easier. Separation of tangent directions on $S \setminus G$ is proved exactly the same way; note that $\mu_p(B') = \mu_V(B') = 1-\alpha$ if $B' = (1-\epsilon)G+(1-\alpha)F,\, p \in F \setminus G, \text{ and } \vec{v} \in T_p(F) \setminus \{ \vec{0} \}$. $H_n\cdot G=0\text{ and } H_n\cdot F=1$; therefore $\phi_n$ contracts $G$ to a point $x \in \bar{S} = \phi_n(S) \subset \ensuremath{\mathbb{P}}^{n+1}$, and $\phi_n(\bar{F})$ is a straight line in $\ensuremath{\mathbb{P}}^{n+1}$ for every $\bar{F}\sim F$. \vspace{8pt} (2) As in part (1) of the Claim, we can show that $|H_m|$ is base-point-free for $m \geq n+1$, and defines a morphism $\phi_m : S \to \ensuremath{\mathbb{P}}^{2m-n+1}$ which is an isomorphism on $S \setminus G$. For $m \geq n+1$, we must show that $|H_m|$ separates $p,q$ even when $p$ (or $q$, or both) is on $G$, and also that $|H_m|$ separates tangent directions at every point $p \in G$. Let $\{p\}=F \cap G\text{ and }\vec{v}\in T_p(G) \setminus\{ \vec{0} \}$. We will show that $|H_{n+1}|$ separates $\vec{v}$ at $p$; the other properties have similar proofs. Write $H_{n+1} = K_S+L,\, L = 3G+(2n+3)F$. Write $L = B+M, \, B=(1-\epsilon)G, \, M = (2+\epsilon)G+(2n+3)F, \; \epsilon \in (0,1)$. We have: \[ M \cdot F = (2+\epsilon), \quad M \cdot G = (3-\epsilon n), \quad M^2 = (2+\epsilon)(2n+6- \epsilon n). \] For $\epsilon \to 0$ we have $M \cdot F \to 2, \, M \cdot G \to 3, \, \text{ and } M^2 \to 2(2n+6) \geq 16$; in particular $M$ is nef and big. (Note that $L$ itself is not nef, if $n \geq 4$; indeed, $L \cdot G = 3-n$.) We have $M \cdot C \geq 2+\epsilon$ for every irreducible curve $C \subset S$ (assuming that $\epsilon \ll 1$); also, if $\vec{v} \in T_p(C)$, then $M \cdot C \geq 3 - \epsilon n$, because in that case $C \sim aG+bF$ with $a \geq 1$ (proof: if $C \neq G$, then $C \cdot G \geq 2$, because $\vec{v} \in T_pC \cap T_pG$; therefore $C \not\sim F$.) We have $\mu_p = \mu_V = 1-\epsilon$, and $\mu_{\vec{v}} = 2(1-\epsilon)$. Choose $\beta_2 = \beta_{2,p} = \beta_{2,V} = 2$ (say), so that $M^2 > 2(\beta_2)^2, \, \beta_{2,p} \geq 2-\mu_p, \, \text{ and } \beta_{2,V} \geq 2-\mu_V$. Put $\beta_1 = \dfrac{2\beta_2}{2\beta_2-(2-\mu_{\vec{v}})} = \dfrac{\beta_2}{\beta_2-\epsilon}$. For $\epsilon \ll 1$, we have: \begin{align*} M\cdot C &= 2+\epsilon\geq\beta_1 \quad\text{for all curves $C\subset S$}, \\ M\cdot C &= 3-\epsilon n \geq 2\beta_1 \quad \text{ for all $C$ containing $(p,\vec{v})$}. \end{align*} (Note that $\beta_1 = \dfrac{\beta_2}{\beta_2-1} \to 1$ as $\epsilon \to 0$, so these inequalities are verified for all $\epsilon \ll 1$.) Now use Proposition 5, case 3. \qed \vspace{8pt} By inspecting the proof of the Claim, we can see that the only assumptions we used were that $\Pic(S) = \ensuremath{\mathbb{Z}} G \oplus \ensuremath{\mathbb{Z}} F, \, G^2=-n, \, F^2=0, \, G \cdot F = 1, \, \text{ and } K_S=-2G-(n+2)F$ (if the other hypotheses are satisfied, the last condition is equivalent to: $G$ and $F$ are smooth rational curves); this suggests the following \vspace{4pt} {\bf Exercise.} A surface $S$ with these properties is isomorphic to the $n^{\text{th}}$ Hirzebruch surface. \vspace{4pt} \emph{Hint.} There are several ways to see this. One, of course, is to use part (1) of the Claim: after all, we have shown that $S$ is the blowing-up of the cone over the normal rational curve of degree $n$. Another solution is to show that $|F|$ is base-point-free and $\dim |F| = 1$, as in the proof of part (1) of the Claim; thus $\phi = \phi_{|F|}$ realizes $S$ as a geometrically ruled surface over $\ensuremath{\mathbb{P}}^1$, as required. ($S$ is minimal, because $C^2 \geq 0$ for every irreducible curve $C \neq G$; this follows easily from the hypotheses.)

9709

alg-geom/9709003

en

https://arxiv.org/abs/alg-geom/9709003

alg-geom/9709003

Ravi Vakil

Ravi Vakil

Counting curves of any genus on rational ruled surfaces

LaTeX2e

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (in a future paper).

alg-geom

\section{Introduction} In this paper we study the geometry of the {\em Severi varieties} parametrizing curves on the rational ruled surface ${\mathbb F}_n = \mathbb P ({\mathcal{O}}_{\mathbb P^1} \oplus {\mathcal{O}}_{\mathbb P^1}(n))$ ($n \ge 0$) in a given divisor class. We compute the number of such curves through the appropriate number of fixed general points on ${\mathbb F}_n$ (Theorem \ref{recursion}), and the number of such curves which are irreducible (Theorem \ref{irecursion}). These numbers are known as {\em Severi degrees}; they are the degrees of unions of components of the Hilbert scheme. As (i) ${\mathbb F}_n$ can be deformed to $\mathbb F_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariants, and (iii) the Gromov-Witten invariants of $\mathbb F_0$ and $\mathbb F_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all ${\mathbb F}_n$ (Section \ref{gwenumerative}). (The genus 0 case is now well-known; it follows from the associativity of quantum cohomology. See [KM], [DI] and [K1] for discussion.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris (cf. [CH3]) and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (cf. [V3]). Such a classical enumerative question has recently been the object of study by many people. Ideas from mathematical physics (cf. the inspiring [KM] and [DI]) have yielded formulas when $g=0$ (via associativity relations in quantum cohomology). In March 1994, S. Kleiman and R. Piene found an elegant recursive formula for the genus 0 Gromov-Witten invariants for all ${\mathbb F}_n$, and found empirically that many of the invariants of ${\mathbb F}_n$ were enumerative and the same as those for $\mathbb F_{n+2}$, supporting the conjecture (since proved) that quantum cohomology exists and is a deformation invariant (see [K1] p. 22 for more information, and an introduction to applications of quantum cohomology to enumerative geometry). Z. Ran solved the analogous (enumerative) problem for curves of arbitrary genus on $\mathbb P^2$ by degenerations methods (cf. [R]), and Caporaso and Harris gave a second solution by different degeneration methods (cf. [CH3]). These numbers for irreducible curves are also the genus $g$ Gromov-Witten invariants of $\mathbb P^2$ (Section \ref{gwenumerative}). D. Abramovich and A. Bertram have used excess intersection and the moduli space of stable maps to calculate generalized Severi degrees for rational curves in all classes on $\mathbb F_2$, and for rational curves in certain classes on ${\mathbb F}_n$ ([AB]). (The author has used a similar idea for curves of arbitrary genus in certain classes on ${\mathbb F}_n$.) In [CH1] and [CH2], Caporaso and Harris found recursive formulas for these numbers when $g=0$ on $\mathbb F_0$, $\mathbb F_1$, $\mathbb F_2$, and $\mathbb F_3$, and on certain classes on general ${\mathbb F}_n$. D. Coventry has also recently derived a recursive formula for the number of rational curves in {\em any} class on ${\mathbb F}_n$ ([Co]), subsuming many earlier results. In another direction, extending work of I. Vainsencher ([Va]), Kleiman and Piene have examined systems with an arbitrary, but fixed, number $\delta$ of nodes ([K2]). The postulated number of $\delta$-nodal curves is given (conjecturally) by a polynomial, and they determine the leading coefficients, which are polynomials in $\delta$. L. G\"{o}ttsche has recently conjectured a surprisingly simple generating function ([G]) for these polynomials which reproduce the results of Vainsencher as well as Kleiman and Piene and experimentally reproduce the numbers of [CH3], [V3], S.T. Yau and E. Zaslow's count of rational curves on K3-surfaces ([YZ]), and others. The numbers of curves are expressed in terms of four universal power series, two of which G\"{o}ttsche gives explicitly as quasimodular forms. The philosophy here is that of Caporaso and Harris in [CH3]: we degenerate the point conditions to lie on $E$ one at a time. Our perspective, however, is different: we use the moduli space of stable maps rather than the Hilbert scheme. The author is grateful to J. Harris for originally suggesting this problem and for his contagious enthusiasm, and to D. Abramovich, E. Getzler, and T. Graber for fruitful discussions on several parts of the argument. The exposition was strengthened immeasurably thanks to advice from S. Kleiman. D. Watabe provided useful comments on earlier drafts. L. G\"{o}ttsche's maple program implementing the algorithm provided examples, and it is a pleasure to acknowledge him here. This research was supported (at different times) by a NSERC 1967 Fellowship and a Sloan Dissertation Fellowship. The bulk of this paper was written at the Mittag-Leffler Institute, and the author is grateful for the warmth and hospitality of the Institute staff. \subsection{Background} \label{background} We work over the complex numbers. Most of the arguments will be in some generality, so that they can be invoked in [V3] to count curves of arbitrary genus in any divisor class on the blow-up of the plane at up to five points. The Picard group of ${\mathbb F}_n$ is ${\mathbb{Z}}^2$, with generators corresponding to the fiber of the projective bundle $F$ and a section $E$ of self-intersection $-n$; $E$ is unique if $n>0$. Let $S$ be the class $E+nF$. (This class is usually denoted $C$, but we use nonstandard notation to prevent confusion with the source of a map $(C,\pi)$.) The canonical bundle $K_{\fn}$ is $-(S+E+2F)$. Throughout this paper, $X$ will be ${\mathbb F}_n$. Unless otherwise explicitly stated, we will use only the following properties of $(X,E)$. \begin{enumerate} \item[{\bf P1.}] $X$ is a smooth surface and $E \cong \mathbb P^1$ is a divisor on $X$. \item[{\bf P2.}] The surface $X \setminus E$ is minimal, i.e. contains no (-1)-curves. \item[{\bf P3.}] The divisor class $K_X + E$ is negative on every curve on $X$. \item[{\bf P4.}] If $D$ is an effective divisor such that $-(K_X + E) \cdot D = 1$, then $D$ is smooth. \end{enumerate} Property P2 could be removed by modifying the arguments very slightly, but there seems to be no benefit of doing so. Properties P3 and P4 would follow if $-(K_X + E)$ were very ample, which is true in all cases of interest here. Notice that if $L$ is a line on $\mathbb P^2$ then $(X,E) = (\mathbb P^2,L)$ also satisfies properties P1--P4. The resulting formulas for Severi degrees of $\mathbb P^2$ are then those of [CH3]. Theorem \ref{recursion} would become Theorem 1.1 of [CH3], and Theorem \ref{irecursion} would give a recursive formula for irreducible genus $g$ curves (which are the genus $g$ Gromov-Witten invariants). If $C$ is a smooth conic on $\mathbb P^2$, then $(X,E) = (\mathbb P^2,C)$ also satisfies properties P1--P4. This will be the basis of the computation of higher genus Gromov-Witten invariants of blow-ups of $\mathbb P^2$ at up to 5 points in [V3]. For any sequence $\alpha = (\alpha_1, \alpha_2, \dots)$ of nonnegative integers with all but finitely many $\alpha_i$ zero, set $$ | \alpha | = \alpha_1 + \alpha_2 + \alpha_3 + \dots $$ $$ I \alpha = \alpha_1 + 2\alpha_2 + 3\alpha_3 + \dots $$ $$ I^\alpha = 1^{\alpha_1} 2^{\alpha_2} 3^{\alpha_3} \dots $$ and $$ \alpha ! = \alpha_1 ! \alpha_2 ! \alpha_3! \dots . $$ We denote by $\operatorname{lcm}(\alpha)$ the least common multiple of the set $\# \{ i : \alpha_i \neq 0 \}$. The zero sequence will be denoted 0. We denote by $e_k$ the sequence $(0, \dots, 0, 1, 0 , \dots)$ that is zero except for a 1 in the $k^{\text{th}}$ term (so that any sequence $\alpha = (\alpha_1, \alpha_2, \dots)$ is expressible as $\alpha = \sum \alpha_k e_k$). By the inequality $\alpha \geq \alpha'$ we mean $\alpha_k \geq \alpha_k'$ for all $k$; for such a pair of sequences we set $$ \binom \alpha {\alpha'} = {\frac { \alpha!} {\alpha' ! (\alpha - \alpha')!}} = \binom {\alpha_1} {\alpha'_1} \binom {\alpha_2} {\alpha'_2} \binom {\alpha_3} {\alpha'_3} \dots . $$ This notation follows [CH3]. For any divisor class $D$ on $X$, genus $g$, sequences $\alpha$ and $\beta$, and collections of points $\Gamma = \{ p_{i,j} \}_{1 \leq j \leq \alpha_i}$ (not necessarily distinct) of $E$ we define the {\em generalized Severi variety} $V^{D,g}(\alpha,\beta,\Gamma)$ to be the closure (in $|D|$) of the locus of reduced curves $C$ in $X$ in divisor class $D$ of geometric genus $g$, not containing $E$, with (informally) $\alpha_k$ ``assigned'' points of contact of order $k$ and $\beta_k$ ``unassigned'' points of contact of order $k$ with $E$. Formally, we require that, if $\nu: C^\nu \rightarrow C$ is the normalization of $C$, then there exist $|\alpha|$ points $q_{i,j} \in C^\nu$, $j=1, \dots, \alpha_i$ and $|\beta|$ points $r_{i,j} \in C^\nu$, $j=1, \dots, \beta_i$ such that $$ \nu(q_{i,j}) = p_{i,j} \quad \text{and} \quad \nu^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j}. $$ If $I \alpha + I\beta \neq D \cdot E$, $V^{D,g}(\alpha,\beta,\Gamma)$ is empty. For convenience, let $$ \Upsilon = \Upsilon^{D,g}(\beta) := - (K_X + E) \cdot D + |\beta| + g-1. $$ We will see that $V^{D,g}\abG$ is a projective variety of pure dimension $\Upsilon$ (Prop. \ref{idim}). Let $N^{D,g}\abG$ be the number of points of $V^{D,g}\abG$ whose corresponding curve passes through $\Upsilon$ fixed general points of $X$. Then $N^{D,g}\abG$ is the degree of the generalized Severi variety (in the projective space $|D|$). When the points $\{ p_{i,j}\}$ are distinct, we will see that $N\abG$ is independent of $\Gamma$ (Section \ref{recursivesection}); for simplicity we will then write $N\ab$. The main result of this paper is the following. \begin{tm} \label{recursion} If $\dim V^{D,g}(\alpha,\beta)>0$, then \begin{eqnarray*} N^{D,g}\ab = \sum_{\beta_k > 0} k N^{D,g}(\alpha + e_k, \beta-e_k) \\ + \sum I^{\beta'-\beta} {\binom \alpha {\alpha'}} \binom {\beta'}{\beta} N^{D-E,g'}(\alpha',\beta') \end{eqnarray*} where the second sum is taken over all $\alpha'$, $\beta'$, $g'$ satisfying $\alpha' \leq \alpha$, $\beta' \geq \beta$, $g-g' = |\beta'-\beta| - 1$, $I \alpha' + I \beta' = (D-E) \cdot E$. \end{tm} When $X={\mathbb F}_n$, the condition $\Upsilon^{D,g}(\beta) = \dim V^{D,g}\ab>0$ is equivalent to $(D,g,\beta) \neq (k F, 1-k,0)$. With the ``seed data'' $$ N^{kF,1-k}(\alpha,0) = \begin{cases} 1 & \text{if $\alpha = k e_1$,} \\ 0 & \text{otherwise,} \end{cases} $$ this formula inductively counts curves of any genus in any divisor class of ${\mathbb F}_n$. In order to understand generalized Severi varieties, we will analyze certain moduli spaces of maps. Let $\overline{M}_g(X,D)'$ be the moduli space of maps $\pi: C \rightarrow X$ where $C$ is complete, reduced, and nodal, $(C, \pi)$ has finite automorphism group, and $\pi_* [C] = D$. (The curve $C$ is not required to be irreducible.) If $C'$ is any connected component of $C$, the map $(C', \pi)$ is stable. The moduli space $\overline{M}_g(X,D)$ is a union of connected components of $\overline{M}_g(X,D)'$. Let $D$, $g$, $\alpha$, $\beta$, $\Gamma$ be as in the definition of $V^{D,g}(\alpha,\beta,\Gamma)$ above. Define the {\em generalized Severi variety of maps} $V^{D,g}_m(\alpha,\beta,\Gamma)$ to be the closure in $\overline{M}_g(X,D)'$ of points representing maps $(C,\pi)$ where each component of $C$ maps birationally to its image in $X$, no component maps to $E$, and $C$ has (informally) $\alpha_k$ ``assigned'' points of contact of order $k$ and $\beta_k$ ``unassigned'' points of contact of order $k$ with $E$. Formally, we require that there exist $|\alpha|$ smooth points $q_{i,j} \in C$, $j=1, \dots, \alpha_i$ and $|\beta|$ smooth points $r_{i,j} \in C$, $j=1, \dots, \beta_i$ such that $$ \pi(q_{i,j}) = p_{i,j} \quad \text{and} \quad \pi^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j}. $$ As before, where the dependence on the points $p_{i,j}$ is not relevant --- for example, in the discussions of the dimensions or degrees of generalized Severi varieties --- we will suppress the $\Gamma$. There is a natural rational map from each component of $V^{D,g}\abG$ to $V^{D,g}_m\abG$, and the dimension of the image will be $\Upsilon$. We will prove: \begin{pr} \label{idim} The components of $V^{D,g}_m\abG$ have dimension at most $\Upsilon$, and the union of those with dimension exactly $\Upsilon$ is the closure of the image of $V^{D,g}\abG$ in $V^{D,g}_m\abG$. \end{pr} (This will be an immediate consequence of Theorem \ref{bigdim}.) Assume now that the $\{ p_{i,j} \}$ are distinct. Fix $\Upsilon$ general points $s_1$, \dots, $s_\Upsilon$ on $X$. The image of the maps in $V^{D,g}_m\ab$ whose images pass through these points are reduced. ({\em Proof:} Without loss of generality, restrict to the union $V$ of those components of $V^{D,g}_m\ab$ with dimension $\Upsilon$. By Proposition \ref{idim}, the subvariety of $V$ corresponding to maps whose images are {\em not} reduced contains no components of $V$ and hence has dimension less than $\Upsilon$. Thus no image of such a map passes through $s_1$, \dots, $s_{\Upsilon}$.) Therefore, if $H$ is the divisor class on $V^{D,g}_m\ab$ corresponding to requiring the image curve to pass through a fixed point of $X$, then $$ N^{D,g}\ab = H^\Upsilon. $$ Define the {\em intersection dimension} of a family $V$ of maps to $X$ (denoted $\operatorname{idim} V$) as the maximum number $n$ of general points $s_1$, \dots $s_n$ on $X$ such that there is a map $\pi: C \rightarrow X$ in $V$ with $\{ s_1, \dots, s_n \} \subset \pi(C)$. Clearly $\operatorname{idim} V \leq \dim V$. Our strategy is as follows. Fix a general point $q$ of $E$. Let $H_q$ be the Weil divisor on $V^{D,g}_m\abG$ corresponding to maps with images containing $q$. We will find the components of $V^{D,g}_m\abG$ with intersection dimension $\Upsilon-1$ and relate them to $V^{D',g'}_m(\alpha',\beta', \Gamma')$ for appropriately chosen $D'$, $g'$, $\alpha'$, $\beta'$, $\Gamma'$. Then we compute the multiplicity with which each of these components appears. Finally, we derive a recursive formula for $N^{D,g}\ab$ (Theorem \ref{recursion}). Analogous definitions can be made of spaces $W^{D,g}\abG$ and $W^{D,g}_m\abG$ parametrizing irreducible curves. The arguments in this case are identical, resulting in a recursive formula for $N_{\operatorname{irr}}^{D,g}\ab$, the number of irreducible genus $g$ curves in class $D$ intersecting $E$ as determined by $\alpha$ and $\beta$, passing through $\Upsilon$ fixed general points of $X$: \begin{tm} \label{irecursion} If $\dim W^{D,g}\ab>0$, then \begin{eqnarray*} N_{\operatorname{irr}}^{D,g}\ab &=& \sum_{\beta_k > 0} k N_{\operatorname{irr}}^{D,g}(\alpha + e_k, \beta - e_k) \\ & & + \sum \frac 1 \sigma \binom {\alpha} {\alpha^1, \dots, \alpha^l, \alpha- \sum \alpha^i} \binom {\Upsilon^{D,g}(\beta)-1} {\Upsilon^{D^1,g^1}(\beta^1), \dots, \Upsilon^{D^l,g^l}(\beta^l)} \\ & & \cdot \prod_{i=1}^l \binom {\beta^i} {\gamma^i} I^{\beta^i - \gamma^i} N_{\operatorname{irr}}^{D^i,g^i}(\alpha^i,\beta^i) \end{eqnarray*} where the second sum runs over choices of $D^i, g^i, \alpha^i, \beta^i, \gamma^i$ ($1 \le i \le l$), where $D^i$ is a divisor class, $g^i$ is a non-negative integer, $\alpha^i$, $\beta^i$, $\gamma^i$ are sequences of non-negative integers, $\sum D^i = D-E$, $\sum \gamma^i = \beta$, $\beta^i \gneq \gamma^i$, and $\sigma$ is the number of symmetries of the set $\{ (D^i,g^i,\alpha^i,\beta^i,\gamma^i) \}_{1 \leq i \leq l}$. \end{tm} In the second sum, for the summand to be non-zero, one must also have $\sum \alpha^i \leq \alpha$, and $I \alpha^i + I \beta^i = D^i \cdot E$. When $X={\mathbb F}_n$, the condition $\dim W^{D,g}\ab>0$ is equivalent to $(D,\beta) \neq (F, 0)$. Thus with the ``seed data'' $N_{\operatorname{irr}}^{F,0}(e_1,0) = 1$, this formula inductively counts irreducible curves of any genus in any divisor class of ${\mathbb F}_n$. \subsection{Examples} As an example of the algorithm in action, we calculate $N^{4S,1}(0,0) = 225$ on $\mathbb F_1$. (This is also the number of two-nodal elliptic plane quartics through 11 fixed general points.) There are a finite number of such elliptic curves through 11 fixed general points on $\mathbb F_1$. We calculate the number by specializing the fixed points to lie on $E$ one at a time, and following what happens to the finite number of curves. After the first specialization, the curve must contain $E$ (as $4S \cdot E = 0$, any representative of $4S$ containing a point of $E$ must contain all of $E$). The residual curve is in class $3S+F$. Theorem \ref{recursion} gives $$ N^{4S,1}(0,0) = N^{3S+F,1}(0,e_1). $$ After specializing a second point $q$ to lie on $E$, two things could happen to the elliptic curve. First, the limit curve could remain smooth, and pass through the fixed point $q$ of $E$. This will happen $N^{3S+F,1}(e_1,0)$ times. Second, the curve could contain $E$. Then the residual curve $C'$ has class $2S+2F$, and is a nodal curve intersecting $E$ at two distinct points. Of the two nodes of the original curve $C$, one goes to the node of $C'$, and the other tends to one of the intersection of $C'$ with $E$. The choice of the two possible limits of the node gives a multiplicity of 2. Theorem \ref{recursion} gives $$ N^{3S+F,1}(0,e_1) = N^{3S+F,1}(e_1,0) + 2 N^{2S+2F,1}(0,2e_1). $$ Now $N^{2S+2F,1}(0,2e_1)$, the number of nodal curves in the linear system $| 2S+2F|$, can be calculated to be 20 by further degenerations or by the well-known calculation of the degree of the hypersurface of singular sections in any linear system. This calculation is omitted. The number $N^{3S+F,1}(e_1,0)$ is calculated by specializing another point to be a general point of $E$. The limit curve will be of one of three forms; in each case the limit must contain $E$, and the residual curve $C''$ is in the class $2S+2F$. \begin{enumerate} \item The curve $C''$ could have geometric genus 0 and intersect $E$ at two points. There are two subcases: $C''$ could be irreducible, or it could consist of a fiber $F$ and a smooth elliptic curve in the class $2S+F$. These cases happen $N^{2S+2F,0}(0,2e_1)$ times. \item The curve $C''$ has geometric genus 1 and is tangent to $E$ at a general point. This happens $N^{2S+2F,1}(0,e_2)$ times. Each of these curves is the limit of {\em two} curves, so there is a multiplicity of 2. (This multiplicity is not obvious.) \item The curve $C''$ is smooth, and passes through the point $q \in E$. This happens $N^{2S+2F,1}(e_1,e_1)$ times. \end{enumerate} Theorem \ref{recursion} gives us $$ N^{3S+F,1}(e_1,0) = N^{2S+2F,0}(0,2e_1) + 2 N^{2S+2F,1}(0,e_2) + N^{2S+2F,1}(e_1,e_1). $$ One can continue and calculate $$N^{2S+2F,0}(0,2e_1) = 105, \: N^{2S+2F,1}(0,e_2) = 30, \: N^{2S+2F,1}(e_1,e_1)=20. $$ Then we can recursively calculate $N^{4S,1}(0,0)$: \begin{eqnarray*} N^{3S+F,1}(e_1,0) &=& N^{2S+2F,0}(0,2e_1) + 2 N^{2S+2F,1}(0,e_2) + N^{2S+2F,1}(e_1,e_1) \\ &=& 105 + 2 \cdot 30 + 20 \\ &=& 185 \end{eqnarray*} \begin{eqnarray*} \text{so } N^{4S,1}(0,0) &=& N^{3S+F,1}(0,e_1) \\ &=& N^{3S+F,1}(e_1,0) + 2 N^{2S+2F,1}(0,2e_1) \\ &=& 185 + 2 \cdot 20 \\ &=& 225. \end{eqnarray*} The calculation is informally summarized pictorially in Figure \ref{ruledfig}. The divisor $E$ is represented by the horizontal doted line, and fixed points on $E$ are represented by fat dots. Part of the figure, the calculation that $N^{2S+2F,0}(0,2e_1) = 105$, has been omitted. \begin{figure} \begin{center} \setlength{\unitlength}{.1\baseunit} {% \begingroup\makeatletter\ifx\SetFigFont\undefined \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \begin{picture}(11037,15348)(0,-10) \thicklines \dottedline{135}(1800,1821)(3900,1821) \dottedline{135}(4800,1821)(6900,1821) \path(5250,2121)(5250,771) \path(2100,546) (2112.093,591.278) (2124.028,635.601) (2135.810,678.974) (2147.441,721.407) (2158.926,762.906) (2170.267,803.479) (2181.467,843.132) (2192.532,881.873) (2214.264,956.649) (2235.491,1027.864) (2256.240,1095.577) (2276.539,1159.847) (2296.415,1220.732) (2315.895,1278.290) (2335.007,1332.579) (2353.779,1383.658) (2372.238,1431.586) (2390.411,1476.420) (2408.326,1518.219) (2426.010,1557.041) (2460.796,1625.990) (2494.990,1683.732) (2528.810,1730.734) (2562.476,1767.465) (2630.227,1811.976) (2700.000,1821.000) \path(2700,1821) (2772.748,1759.423) (2798.844,1694.793) (2809.760,1655.590) (2819.422,1612.395) (2827.954,1565.655) (2835.482,1515.822) (2842.131,1463.343) (2848.025,1408.669) (2853.289,1352.248) (2858.049,1294.531) (2862.430,1235.965) (2866.556,1177.001) (2870.553,1118.088) (2874.546,1059.675) (2878.659,1002.211) (2883.018,946.146) (2887.748,891.929) (2892.974,840.009) (2898.820,790.835) (2905.412,744.857) (2912.875,702.525) (2921.334,664.287) (2941.739,601.891) (3000.000,546.000) \path(3000,546) (3068.779,563.740) (3133.580,625.672) (3164.970,674.720) (3195.942,736.629) (3211.332,772.596) (3226.690,812.004) (3242.039,854.929) (3257.404,901.448) (3272.809,951.634) (3288.277,1005.565) (3303.834,1063.315) (3319.502,1124.960) (3335.306,1190.575) (3351.271,1260.236) (3367.419,1334.020) (3375.570,1372.480) (3383.775,1412.000) (3392.039,1452.587) (3400.364,1494.252) (3408.753,1537.005) (3417.208,1580.853) (3425.734,1625.808) (3434.333,1671.878) (3443.008,1719.073) (3451.762,1767.402) (3460.598,1816.875) (3469.519,1867.500) (3478.528,1919.289) (3487.629,1972.249) (3496.823,2026.391) (3506.114,2081.724) (3515.506,2138.257) (3525.000,2196.000) \path(4800,996) (4852.704,1004.433) (4903.528,1012.790) (4952.515,1021.083) (4999.711,1029.328) (5045.159,1037.537) (5088.906,1045.725) (5130.994,1053.904) (5171.470,1062.090) (5210.377,1070.295) (5247.760,1078.534) (5318.134,1095.165) (5382.948,1112.095) (5442.559,1129.432) (5497.324,1147.288) (5547.601,1165.771) (5593.747,1184.991) (5636.118,1205.059) (5710.966,1248.176) (5775.000,1296.000) \path(5775,1296) (5818.483,1342.054) (5857.892,1401.268) (5893.796,1471.381) (5910.611,1509.817) (5926.763,1550.131) (5942.322,1592.040) (5957.360,1635.260) (5971.948,1679.509) (5986.156,1724.505) (6000.056,1769.964) (6013.719,1815.605) (6027.215,1861.145) (6040.616,1906.301) (6053.993,1950.791) (6067.416,1994.331) (6080.957,2036.640) (6094.687,2077.435) (6108.676,2116.433) (6122.996,2153.352) (6152.911,2219.820) (6185.002,2274.580) (6219.835,2315.371) (6257.978,2339.931) (6300.000,2346.000) \path(6300,2346) (6370.180,2324.595) (6431.504,2271.241) (6459.188,2231.416) (6485.070,2182.203) (6509.288,2123.134) (6531.979,2053.744) (6542.794,2015.032) (6553.279,1973.564) (6563.451,1929.283) (6573.327,1882.129) (6582.924,1832.045) (6592.260,1778.972) (6601.351,1722.851) (6610.215,1663.624) (6618.868,1601.233) (6627.329,1535.620) (6635.613,1466.726) (6643.739,1394.493) (6647.748,1357.106) (6651.724,1318.862) (6655.668,1279.754) (6659.584,1239.775) (6663.473,1198.917) (6667.337,1157.174) (6671.179,1114.537) (6675.000,1071.000) \put(2865,9921){\blacken\ellipse{150}{150}} \put(2865,9921){\ellipse{150}{150}} \put(10081,7228){\blacken\ellipse{150}{150}} \put(10081,7228){\ellipse{150}{150}} \put(4283,4521){\blacken\ellipse{150}{150}} \put(4283,4521){\ellipse{150}{150}} \put(6480,4528){\blacken\ellipse{150}{150}} \put(6480,4528){\ellipse{150}{150}} \put(7530,4521){\blacken\ellipse{150}{150}} \put(7530,4521){\ellipse{150}{150}} \dottedline{135}(3900,15321)(6000,15321) \dottedline{135}(3900,12621)(6000,12621) \dottedline{135}(2250,9921)(4350,9921) \dottedline{135}(5400,9921)(7500,9921) \dottedline{135}(300,7221)(2400,7221) \dottedline{135}(3000,7221)(5100,7221) \path(3450,7821)(3450,6471) \dottedline{135}(5700,7221)(7800,7221) \dottedline{135}(8700,7221)(10800,7221) \dottedline{135}(3300,4521)(5400,4521) \dottedline{135}(6150,4521)(8250,4521) \dottedline{135}(8925,4521)(11025,4521) \path(4875,13746)(4875,12996) \path(4875,13746)(4875,12996) \path(4845.000,13116.000)(4875.000,12996.000)(4905.000,13116.000) \path(4725,11046)(3300,10071) \path(4725,11046)(3300,10071) \path(3382.096,10163.521)(3300.000,10071.000)(3415.977,10114.003) \path(4950,11046)(5850,10146) \path(4950,11046)(5850,10146) \path(5743.934,10209.640)(5850.000,10146.000)(5786.360,10252.066) \path(7425,8346)(9075,7596) \path(7425,8346)(9075,7596) \path(8953.342,7618.345)(9075.000,7596.000)(8978.170,7672.967) \path(2700,8346)(1200,7671) \path(2700,8346)(1200,7671) \path(1297.120,7747.601)(1200.000,7671.000)(1321.742,7692.886) \path(3300,8346)(3375,7896) \path(3300,8346)(3375,7896) \path(3325.680,8009.435)(3375.000,7896.000)(3384.864,8019.299) \path(3900,8346)(5925,7371) \path(3900,8346)(5925,7371) \path(5803.865,7396.028)(5925.000,7371.000)(5829.894,7450.088) \path(4500,8346)(8475,7446) \path(4500,8346)(8475,7446) \path(8351.338,7443.240)(8475.000,7446.000)(8364.587,7501.758) \path(6600,5646)(4650,4746) \path(6600,5646)(4650,4746) \path(4746.383,4823.526)(4650.000,4746.000)(4771.527,4769.048) \path(9150,5571)(7575,4896) \path(9150,5571)(7575,4896) \path(7673.480,4970.845)(7575.000,4896.000)(7697.115,4915.696) \path(9525,5571)(9525,5196) \path(9525,5571)(9525,5196) \path(9495.000,5316.000)(9525.000,5196.000)(9555.000,5316.000) \path(4125,2946)(3600,2346) \path(4125,2946)(3600,2346) \path(3656.443,2456.064)(3600.000,2346.000)(3701.598,2416.554) \path(4425,2946)(5175,2346) \path(4425,2946)(5175,2346) \path(5062.555,2397.537)(5175.000,2346.000)(5100.037,2444.389) \path(4125,15096) (4173.021,15088.102) (4219.346,15080.480) (4264.016,15073.127) (4307.074,15066.036) (4348.560,15059.200) (4388.514,15052.612) (4426.979,15046.266) (4463.996,15040.154) (4533.849,15028.607) (4598.404,15017.916) (4657.988,15008.026) (4712.932,14998.882) (4763.567,14990.430) (4810.220,14982.613) (4853.222,14975.378) (4892.903,14968.670) (4963.617,14956.612) (5025.000,14946.000) \path(5025,14946) (5063.401,14936.907) (5107.472,14922.657) (5156.372,14904.345) (5209.263,14883.071) (5265.307,14859.930) (5323.666,14836.020) (5383.499,14812.438) (5443.969,14790.281) (5504.236,14770.647) (5563.463,14754.633) (5620.811,14743.335) (5675.440,14737.851) (5726.512,14739.279) (5773.189,14748.714) (5814.631,14767.256) (5850.000,14796.000) \path(5850,14796) (5879.378,14862.562) (5881.163,14904.179) (5874.810,14948.033) (5860.669,14991.632) (5839.087,15032.486) (5775.000,15096.000) \path(5775,15096) (5716.131,15124.781) (5655.798,15142.307) (5594.390,15149.767) (5532.294,15148.348) (5469.900,15139.237) (5407.595,15123.621) (5345.770,15102.690) (5284.811,15077.629) (5225.109,15049.626) (5167.051,15019.870) (5111.026,14989.547) (5057.423,14959.845) (5006.630,14931.952) (4959.036,14907.055) (4915.030,14886.342) (4875.000,14871.000) \path(4875,14871) (4800.684,14836.037) (4758.083,14810.208) (4712.541,14780.471) (4664.577,14748.068) (4614.708,14714.241) (4563.450,14680.233) (4511.321,14647.286) (4458.839,14616.643) (4406.520,14589.546) (4354.881,14567.237) (4304.441,14550.959) (4255.715,14541.953) (4209.222,14541.464) (4165.478,14550.732) (4125.000,14571.000) \path(4125,14571) (4081.429,14630.966) (4070.234,14672.235) (4066.260,14717.014) (4069.629,14762.291) (4080.461,14805.055) (4125.000,14871.000) \path(4125,14871) (4173.647,14900.907) (4226.344,14919.008) (4282.438,14926.576) (4341.281,14924.886) (4402.222,14915.211) (4464.609,14898.824) (4527.794,14876.998) (4591.125,14851.009) (4653.952,14822.128) (4715.625,14791.630) (4775.493,14760.789) (4832.906,14730.877) (4887.214,14703.169) (4937.766,14678.937) (4983.911,14659.457) (5025.000,14646.000) \path(5025,14646) (5065.411,14636.250) (5114.876,14625.528) (5171.773,14613.919) (5234.480,14601.509) (5301.377,14588.382) (5370.842,14574.624) (5441.253,14560.319) (5510.989,14545.552) (5578.428,14530.409) (5641.950,14514.974) (5699.932,14499.333) (5750.753,14483.570) (5824.427,14452.018) (5850.000,14421.000) \path(5850,14421) (5817.470,14393.827) (5781.578,14383.685) (5734.870,14375.524) (5679.048,14369.108) (5615.812,14364.199) (5546.864,14360.561) (5473.905,14357.959) (5436.453,14356.971) (5398.637,14356.154) (5360.668,14355.477) (5322.760,14354.911) (5247.976,14353.993) (5175.987,14353.164) (5108.494,14352.186) (5047.197,14350.824) (4993.799,14348.841) (4950.000,14346.000) \path(4950,14346) (4893.660,14341.052) (4828.787,14335.278) (4792.396,14332.013) (4752.964,14328.460) (4710.191,14324.590) (4663.774,14320.376) (4613.411,14315.792) (4558.799,14310.808) (4499.637,14305.399) (4435.623,14299.536) (4366.455,14293.192) (4291.829,14286.340) (4252.376,14282.714) (4211.445,14278.952) (4168.999,14275.048) (4125.000,14271.000) \path(5475,9471) (5513.531,9520.056) (5551.095,9566.998) (5587.738,9611.859) (5623.504,9654.668) (5658.436,9695.458) (5692.581,9734.257) (5758.686,9806.012) (5822.175,9870.181) (5883.404,9927.009) (5942.731,9976.745) (6000.514,10019.636) (6057.108,10055.929) (6112.872,10085.871) (6168.162,10109.710) (6223.335,10127.692) (6278.748,10140.065) (6334.758,10147.076) (6391.723,10148.972) (6450.000,10146.000) \path(6450,10146) (6494.995,10139.015) (6538.468,10126.576) (6580.479,10109.077) (6621.084,10086.912) (6660.343,10060.474) (6698.314,10030.159) (6770.627,9959.467) (6805.085,9919.880) (6838.489,9877.990) (6870.898,9834.190) (6902.370,9788.876) (6932.962,9742.440) (6962.735,9695.277) (6991.745,9647.781) (7020.053,9600.345) (7047.715,9553.363) (7074.791,9507.230) (7101.338,9462.339) (7127.416,9419.084) (7153.083,9377.858) (7178.398,9339.057) (7228.202,9270.300) (7277.297,9215.965) (7326.150,9179.203) (7375.228,9163.164) (7425.000,9171.000) \path(7425,9171) (7454.407,9198.393) (7476.600,9248.546) (7490.984,9314.202) (7496.966,9388.103) (7493.953,9462.990) (7481.350,9531.608) (7458.563,9586.697) (7425.000,9621.000) \path(7425,9621) (7381.010,9636.301) (7334.352,9636.711) (7285.381,9624.048) (7234.454,9600.127) (7181.924,9566.763) (7128.149,9525.774) (7073.482,9478.975) (7018.279,9428.182) (6962.895,9375.212) (6907.687,9321.881) (6853.008,9270.004) (6799.215,9221.397) (6746.662,9177.878) (6695.705,9141.261) (6646.699,9113.363) (6600.000,9096.000) \path(6600,9096) (6562.475,9087.528) (6522.998,9080.683) (6481.184,9075.492) (6436.649,9071.982) (6389.008,9070.180) (6337.877,9070.115) (6282.871,9071.814) (6223.605,9075.304) (6159.696,9080.612) (6090.758,9087.767) (6016.407,9096.795) (5977.081,9102.020) (5936.258,9107.724) (5893.890,9113.910) (5849.928,9120.582) (5804.325,9127.742) (5757.031,9135.395) (5708.000,9143.544) (5657.183,9152.192) (5604.533,9161.343) (5550.000,9171.000) \path(2175,6921) (2135.635,6866.620) (2098.041,6817.007) (2062.026,6772.023) (2027.397,6731.531) (1961.529,6663.473) (1898.899,6611.734) (1837.969,6575.216) (1777.201,6552.820) (1715.058,6543.447) (1650.000,6546.000) \path(1650,6546) (1603.592,6554.652) (1560.036,6568.709) (1519.167,6587.786) (1480.815,6611.499) (1410.996,6671.294) (1349.243,6745.018) (1320.972,6786.141) (1294.215,6829.592) (1268.805,6874.986) (1244.575,6921.939) (1221.357,6970.066) (1198.983,7018.983) (1177.288,7068.304) (1156.102,7117.646) (1135.260,7166.624) (1114.593,7214.852) (1093.935,7261.946) (1073.118,7307.523) (1051.974,7351.196) (1030.336,7392.582) (984.911,7466.952) (935.504,7527.556) (880.775,7571.316) (819.387,7595.157) (750.000,7596.000) \path(750,7596) (698.415,7582.407) (648.115,7558.728) (599.629,7525.907) (553.487,7484.885) (510.216,7436.605) (470.346,7382.010) (434.405,7322.041) (402.923,7257.641) (376.427,7189.753) (355.446,7119.317) (340.510,7047.278) (332.147,6974.577) (330.886,6902.156) (337.255,6830.958) (351.783,6761.925) (375.000,6696.000) \path(375,6696) (421.514,6621.488) (485.143,6564.824) (521.972,6542.467) (561.398,6523.711) (602.858,6508.268) (645.791,6495.851) (689.638,6486.174) (733.836,6478.948) (777.825,6473.887) (821.044,6470.703) (862.932,6469.110) (902.928,6468.820) (975.000,6471.000) \path(975,6471) (1018.304,6477.499) (1062.261,6491.707) (1106.831,6512.455) (1151.978,6538.575) (1197.665,6568.901) (1243.852,6602.264) (1290.503,6637.496) (1337.580,6673.429) (1385.045,6708.895) (1432.860,6742.728) (1480.988,6773.758) (1529.392,6800.818) (1578.032,6822.740) (1626.872,6838.356) (1675.874,6846.499) (1725.000,6846.000) \path(1725,6846) (1766.541,6838.423) (1807.409,6824.055) (1848.702,6802.016) (1891.519,6771.427) (1936.958,6731.411) (1986.120,6681.086) (2040.100,6619.576) (2100.000,6546.000) \path(3075,7596) (3140.607,7585.208) (3204.839,7574.532) (3267.708,7563.968) (3329.224,7553.514) (3389.397,7543.165) (3448.239,7532.919) (3505.761,7522.771) (3561.972,7512.718) (3616.884,7502.758) (3670.507,7492.885) (3722.852,7483.098) (3773.931,7473.393) (3823.752,7463.765) (3872.329,7454.213) (3919.670,7444.732) (3965.787,7435.318) (4010.690,7425.969) (4054.391,7416.681) (4096.900,7407.451) (4138.227,7398.275) (4178.384,7389.149) (4217.381,7380.071) (4291.938,7362.042) (4361.985,7344.162) (4427.606,7326.401) (4488.889,7308.734) (4545.919,7291.132) (4598.781,7273.569) (4647.562,7256.016) (4692.347,7238.446) (4733.222,7220.831) (4803.586,7185.359) (4859.341,7149.381) (4929.770,7075.023) (4945.816,7036.204) (4950.000,6996.000) \path(4950,6996) (4942.997,6957.032) (4924.507,6920.278) (4850.319,6852.757) (4793.248,6821.659) (4721.944,6792.117) (4680.740,6777.877) (4635.720,6763.965) (4586.798,6750.360) (4533.889,6737.040) (4476.906,6723.986) (4415.764,6711.176) (4350.378,6698.590) (4280.660,6686.208) (4206.526,6674.009) (4167.775,6667.972) (4127.889,6661.972) (4086.855,6656.009) (4044.664,6650.077) (4001.304,6644.177) (3956.764,6638.303) (3911.035,6632.455) (3864.105,6626.630) (3815.963,6620.825) (3766.600,6615.036) (3716.003,6609.263) (3664.163,6603.502) (3611.068,6597.751) (3556.709,6592.007) (3501.073,6586.267) (3444.151,6580.529) (3385.932,6574.791) (3326.404,6569.050) (3265.558,6563.302) (3203.383,6557.547) (3139.867,6551.780) (3075.000,6546.000) \path(5700,6546) (5738.531,6595.056) (5776.095,6641.998) (5812.738,6686.859) (5848.504,6729.668) (5883.436,6770.458) (5917.581,6809.257) (5983.686,6881.012) (6047.175,6945.181) (6108.404,7002.009) (6167.731,7051.745) (6225.514,7094.636) (6282.108,7130.929) (6337.872,7160.871) (6393.162,7184.710) (6448.335,7202.692) (6503.748,7215.065) (6559.758,7222.076) (6616.723,7223.972) (6675.000,7221.000) \path(6675,7221) (6719.995,7214.015) (6763.468,7201.576) (6805.479,7184.077) (6846.084,7161.912) (6885.343,7135.474) (6923.314,7105.159) (6995.627,7034.468) (7030.085,6994.880) (7063.489,6952.990) (7095.898,6909.190) (7127.370,6863.876) (7157.962,6817.440) (7187.735,6770.277) (7216.745,6722.781) (7245.053,6675.345) (7272.715,6628.363) (7299.791,6582.230) (7326.338,6537.339) (7352.416,6494.084) (7378.083,6452.858) (7403.398,6414.057) (7453.202,6345.300) (7502.297,6290.965) (7551.150,6254.203) (7600.228,6238.164) (7650.000,6246.000) \path(7650,6246) (7679.407,6273.393) (7701.600,6323.546) (7715.984,6389.202) (7721.966,6463.103) (7718.953,6537.990) (7706.350,6606.608) (7683.563,6661.697) (7650.000,6696.000) \path(7650,6696) (7606.010,6711.301) (7559.352,6711.711) (7510.381,6699.048) (7459.454,6675.127) (7406.924,6641.763) (7353.149,6600.774) (7298.482,6553.975) (7243.279,6503.183) (7187.895,6450.212) (7132.687,6396.881) (7078.008,6345.004) (7024.215,6296.397) (6971.662,6252.878) (6920.705,6216.261) (6871.699,6188.363) (6825.000,6171.000) \path(6825,6171) (6787.475,6162.528) (6747.998,6155.683) (6706.184,6150.492) (6661.649,6146.982) (6614.008,6145.180) (6562.877,6145.115) (6507.871,6146.814) (6448.605,6150.304) (6384.696,6155.612) (6315.758,6162.767) (6241.407,6171.795) (6202.081,6177.020) (6161.258,6182.724) (6118.890,6188.910) (6074.928,6195.582) (6029.325,6202.742) (5982.031,6210.395) (5933.000,6218.544) (5882.183,6227.192) (5829.533,6236.343) (5775.000,6246.000) \path(8700,6846) (8738.531,6895.056) (8776.095,6941.998) (8812.738,6986.859) (8848.504,7029.668) (8883.436,7070.458) (8917.581,7109.257) (8983.686,7181.012) (9047.175,7245.181) (9108.404,7302.009) (9167.731,7351.745) (9225.514,7394.636) (9282.108,7430.929) (9337.872,7460.871) (9393.162,7484.710) (9448.335,7502.692) (9503.748,7515.065) (9559.758,7522.076) (9616.723,7523.972) (9675.000,7521.000) \path(9675,7521) (9719.995,7514.015) (9763.468,7501.576) (9805.479,7484.077) (9846.084,7461.912) (9885.343,7435.474) (9923.314,7405.159) (9995.627,7334.468) (10030.085,7294.880) (10063.489,7252.990) (10095.898,7209.190) (10127.370,7163.876) (10157.962,7117.440) (10187.735,7070.277) (10216.745,7022.781) (10245.053,6975.345) (10272.715,6928.363) (10299.791,6882.230) (10326.338,6837.339) (10352.416,6794.084) (10378.083,6752.858) (10403.398,6714.057) (10453.202,6645.300) (10502.297,6590.965) (10551.150,6554.203) (10600.228,6538.164) (10650.000,6546.000) \path(10650,6546) (10679.407,6573.393) (10701.600,6623.546) (10715.984,6689.202) (10721.966,6763.103) (10718.953,6837.990) (10706.350,6906.608) (10683.563,6961.697) (10650.000,6996.000) \path(10650,6996) (10606.010,7011.301) (10559.352,7011.711) (10510.381,6999.048) (10459.454,6975.127) (10406.924,6941.763) (10353.149,6900.774) (10298.482,6853.975) (10243.279,6803.183) (10187.895,6750.212) (10132.687,6696.881) (10078.008,6645.004) (10024.215,6596.397) (9971.662,6552.878) (9920.705,6516.261) (9871.699,6488.363) (9825.000,6471.000) \path(9825,6471) (9787.475,6462.528) (9747.998,6455.683) (9706.184,6450.492) (9661.649,6446.982) (9614.008,6445.180) (9562.877,6445.115) (9507.871,6446.814) (9448.605,6450.304) (9384.696,6455.612) (9315.758,6462.767) (9241.407,6471.795) (9202.081,6477.020) (9161.258,6482.724) (9118.890,6488.910) (9074.928,6495.582) (9029.325,6502.742) (8982.031,6510.395) (8933.000,6518.544) (8882.183,6527.192) (8829.533,6536.343) (8775.000,6546.000) \path(3375,3846) (3413.531,3895.056) (3451.095,3941.998) (3487.738,3986.859) (3523.504,4029.668) (3558.436,4070.458) (3592.581,4109.257) (3658.686,4181.012) (3722.175,4245.181) (3783.404,4302.009) (3842.731,4351.745) (3900.514,4394.636) (3957.108,4430.929) (4012.872,4460.871) (4068.162,4484.710) (4123.335,4502.692) (4178.748,4515.065) (4234.758,4522.076) (4291.723,4523.972) (4350.000,4521.000) \path(4350,4521) (4394.995,4514.015) (4438.468,4501.576) (4480.479,4484.077) (4521.084,4461.912) (4560.343,4435.474) (4598.314,4405.159) (4670.627,4334.468) (4705.085,4294.880) (4738.489,4252.990) (4770.898,4209.190) (4802.370,4163.876) (4832.962,4117.440) (4862.735,4070.277) (4891.745,4022.781) (4920.052,3975.345) (4947.715,3928.363) (4974.791,3882.230) (5001.338,3837.339) (5027.416,3794.084) (5053.083,3752.858) (5078.398,3714.057) (5128.202,3645.300) (5177.297,3590.965) (5226.150,3554.203) (5275.228,3538.164) (5325.000,3546.000) \path(5325,3546) (5354.407,3573.393) (5376.600,3623.546) (5390.984,3689.202) (5396.966,3763.102) (5393.953,3837.990) (5381.350,3906.608) (5358.563,3961.697) (5325.000,3996.000) \path(5325,3996) (5281.010,4011.301) (5234.352,4011.711) (5185.381,3999.048) (5134.454,3975.127) (5081.924,3941.763) (5028.149,3900.774) (4973.482,3853.975) (4918.279,3803.183) (4862.895,3750.212) (4807.687,3696.881) (4753.008,3645.004) (4699.215,3596.397) (4646.662,3552.878) (4595.705,3516.261) (4546.699,3488.363) (4500.000,3471.000) \path(4500,3471) (4462.475,3462.528) (4422.998,3455.683) (4381.184,3450.492) (4336.649,3446.982) (4289.008,3445.180) (4237.877,3445.115) (4182.871,3446.814) (4123.605,3450.304) (4059.696,3455.612) (3990.758,3462.767) (3916.407,3471.795) (3877.081,3477.020) (3836.258,3482.724) (3793.890,3488.910) (3749.928,3495.582) (3704.325,3502.742) (3657.031,3510.395) (3608.000,3518.544) (3557.183,3527.192) (3504.533,3536.343) (3450.000,3546.000) \path(6150,4146) (6188.531,4195.056) (6226.095,4241.998) (6262.738,4286.859) (6298.504,4329.668) (6333.436,4370.458) (6367.581,4409.257) (6433.686,4481.012) (6497.175,4545.181) (6558.404,4602.009) (6617.731,4651.745) (6675.514,4694.636) (6732.108,4730.929) (6787.872,4760.871) (6843.162,4784.710) (6898.335,4802.692) (6953.748,4815.065) (7009.758,4822.076) (7066.723,4823.972) (7125.000,4821.000) \path(7125,4821) (7169.995,4814.015) (7213.468,4801.576) (7255.479,4784.077) (7296.084,4761.912) (7335.343,4735.474) (7373.314,4705.159) (7445.627,4634.468) (7480.085,4594.880) (7513.489,4552.990) (7545.898,4509.190) (7577.370,4463.876) (7607.962,4417.440) (7637.735,4370.277) (7666.745,4322.781) (7695.053,4275.345) (7722.715,4228.363) (7749.791,4182.230) (7776.338,4137.339) (7802.416,4094.084) (7828.083,4052.858) (7853.398,4014.057) (7903.202,3945.300) (7952.297,3890.965) (8001.150,3854.203) (8050.228,3838.164) (8100.000,3846.000) \path(8100,3846) (8129.407,3873.393) (8151.600,3923.546) (8165.984,3989.202) (8171.966,4063.102) (8168.953,4137.990) (8156.350,4206.608) (8133.563,4261.697) (8100.000,4296.000) \path(8100,4296) (8056.010,4311.301) (8009.352,4311.711) (7960.381,4299.048) (7909.454,4275.127) (7856.924,4241.763) (7803.149,4200.774) (7748.482,4153.975) (7693.279,4103.183) (7637.895,4050.212) (7582.687,3996.881) (7528.008,3945.004) (7474.215,3896.397) (7421.662,3852.878) (7370.705,3816.261) (7321.699,3788.363) (7275.000,3771.000) \path(7275,3771) (7237.475,3762.528) (7197.998,3755.683) (7156.184,3750.492) (7111.649,3746.982) (7064.008,3745.180) (7012.877,3745.115) (6957.871,3746.814) (6898.605,3750.304) (6834.696,3755.612) (6765.758,3762.767) (6691.407,3771.795) (6652.081,3777.020) (6611.258,3782.724) (6568.890,3788.910) (6524.928,3795.582) (6479.325,3802.742) (6432.031,3810.395) (6383.000,3818.544) (6332.183,3827.192) (6279.533,3836.343) (6225.000,3846.000) \path(9075,3771) (9087.093,3816.278) (9099.028,3860.601) (9110.810,3903.974) (9122.441,3946.407) (9133.926,3987.906) (9145.267,4028.479) (9156.467,4068.132) (9167.532,4106.873) (9189.264,4181.649) (9210.491,4252.864) (9231.240,4320.577) (9251.539,4384.847) (9271.415,4445.732) (9290.895,4503.290) (9310.007,4557.579) (9328.779,4608.658) (9347.238,4656.586) (9365.411,4701.420) (9383.326,4743.219) (9401.010,4782.041) (9435.796,4850.990) (9469.990,4908.732) (9503.810,4955.734) (9537.476,4992.465) (9605.227,5036.976) (9675.000,5046.000) \path(9675,5046) (9748.058,4984.328) (9774.493,4919.594) (9785.610,4880.330) (9795.488,4837.068) (9804.247,4790.260) (9812.009,4740.354) (9818.894,4687.804) (9825.023,4633.057) (9830.517,4576.566) (9835.497,4518.781) (9840.084,4460.153) (9844.399,4401.131) (9848.562,4342.167) (9852.694,4283.711) (9856.916,4226.214) (9861.350,4170.126) (9866.115,4115.898) (9871.333,4063.981) (9877.124,4014.824) (9883.610,3968.879) (9890.911,3926.595) (9899.149,3888.425) (9918.915,3826.224) (9975.000,3771.000) \path(9975,3771) (10025.558,3785.301) (10072.866,3833.580) (10095.645,3871.630) (10118.024,3919.575) (10140.139,3977.882) (10162.129,4047.019) (10173.119,4085.794) (10184.130,4127.452) (10195.178,4172.050) (10206.280,4219.648) (10217.454,4270.303) (10228.717,4324.074) (10240.085,4381.020) (10251.577,4441.198) (10263.209,4504.667) (10274.998,4571.485) (10286.962,4641.711) (10299.117,4715.403) (10305.272,4753.567) (10311.482,4792.620) (10317.748,4832.568) (10324.072,4873.419) (10330.458,4915.181) (10336.906,4957.860) (10343.419,5001.464) (10350.000,5046.000) \path(4200,12921) (4240.636,12881.993) (4279.936,12844.450) (4354.679,12773.635) (4424.531,12708.309) (4489.794,12648.223) (4550.770,12593.130) (4607.762,12542.785) (4661.072,12496.938) (4711.001,12455.344) (4757.852,12417.754) (4801.927,12383.922) (4843.529,12353.601) (4882.958,12326.543) (4956.510,12281.228) (5025.000,12246.000) \path(5025,12246) (5063.159,12228.094) (5107.050,12207.302) (5155.826,12184.486) (5208.643,12160.509) (5264.656,12136.232) (5323.020,12112.518) (5382.888,12090.229) (5443.417,12070.226) (5503.762,12053.373) (5563.076,12040.531) (5620.515,12032.563) (5675.233,12030.330) (5726.386,12034.696) (5773.128,12046.521) (5814.615,12066.668) (5850.000,12096.000) \path(5850,12096) (5879.378,12162.562) (5881.163,12204.179) (5874.810,12248.033) (5860.669,12291.632) (5839.087,12332.486) (5775.000,12396.000) \path(5775,12396) (5716.131,12424.781) (5655.798,12442.307) (5594.390,12449.767) (5532.294,12448.348) (5469.900,12439.237) (5407.595,12423.621) (5345.770,12402.690) (5284.811,12377.629) (5225.109,12349.626) (5167.051,12319.870) (5111.026,12289.547) (5057.423,12259.845) (5006.630,12231.952) (4959.036,12207.055) (4915.030,12186.342) (4875.000,12171.000) \path(4875,12171) (4800.684,12136.037) (4758.083,12110.208) (4712.541,12080.471) (4664.577,12048.068) (4614.708,12014.241) (4563.450,11980.233) (4511.321,11947.286) (4458.839,11916.643) (4406.520,11889.546) (4354.881,11867.237) (4304.441,11850.959) (4255.715,11841.953) (4209.222,11841.464) (4165.478,11850.732) (4125.000,11871.000) \path(4125,11871) (4081.429,11930.966) (4070.234,11972.235) (4066.260,12017.014) (4069.629,12062.291) (4080.461,12105.055) (4125.000,12171.000) \path(4125,12171) (4173.647,12200.907) (4226.344,12219.008) (4282.438,12226.576) (4341.281,12224.886) (4402.222,12215.211) (4464.609,12198.824) (4527.794,12176.998) (4591.125,12151.009) (4653.952,12122.128) (4715.625,12091.630) (4775.493,12060.789) (4832.906,12030.877) (4887.214,12003.169) (4937.766,11978.937) (4983.911,11959.457) (5025.000,11946.000) \path(5025,11946) (5081.461,11931.132) (5146.421,11913.795) (5182.843,11903.995) (5222.296,11893.330) (5265.082,11881.718) (5311.504,11869.076) (5361.863,11855.323) (5416.461,11840.376) (5475.601,11824.152) (5539.585,11806.569) (5608.715,11787.545) (5683.292,11766.997) (5722.718,11756.126) (5763.620,11744.843) (5806.035,11733.138) (5850.000,11721.000) \path(2550,10221) (2590.636,10181.993) (2629.936,10144.450) (2704.679,10073.635) (2774.531,10008.309) (2839.794,9948.223) (2900.770,9893.130) (2957.762,9842.785) (3011.072,9796.938) (3061.001,9755.344) (3107.852,9717.754) (3151.927,9683.922) (3193.529,9653.601) (3232.958,9626.543) (3306.510,9581.228) (3375.000,9546.000) \path(3375,9546) (3413.159,9528.094) (3457.050,9507.302) (3505.826,9484.486) (3558.643,9460.509) (3614.656,9436.232) (3673.020,9412.518) (3732.888,9390.229) (3793.417,9370.226) (3853.762,9353.373) (3913.076,9340.531) (3970.515,9332.563) (4025.233,9330.330) (4076.386,9334.696) (4123.128,9346.521) (4164.615,9366.668) (4200.000,9396.000) \path(4200,9396) (4229.378,9462.562) (4231.163,9504.179) (4224.810,9548.033) (4210.669,9591.632) (4189.087,9632.486) (4125.000,9696.000) \path(4125,9696) (4066.131,9724.781) (4005.798,9742.307) (3944.390,9749.767) (3882.294,9748.348) (3819.900,9739.237) (3757.595,9723.621) (3695.770,9702.690) (3634.811,9677.629) (3575.109,9649.626) (3517.051,9619.870) (3461.026,9589.547) (3407.423,9559.845) (3356.630,9531.952) (3309.036,9507.055) (3265.030,9486.342) (3225.000,9471.000) \path(3225,9471) (3150.684,9436.037) (3108.083,9410.208) (3062.541,9380.471) (3014.577,9348.068) (2964.708,9314.241) (2913.450,9280.233) (2861.321,9247.286) (2808.839,9216.643) (2756.520,9189.546) (2704.881,9167.237) (2654.441,9150.959) (2605.715,9141.953) (2559.222,9141.464) (2515.478,9150.732) (2475.000,9171.000) \path(2475,9171) (2431.429,9230.966) (2420.234,9272.235) (2416.260,9317.014) (2419.629,9362.291) (2430.461,9405.055) (2475.000,9471.000) \path(2475,9471) (2523.647,9500.907) (2576.344,9519.008) (2632.438,9526.576) (2691.281,9524.886) (2752.222,9515.211) (2814.609,9498.824) (2877.794,9476.998) (2941.125,9451.009) (3003.952,9422.128) (3065.625,9391.630) (3125.493,9360.789) (3182.906,9330.877) (3237.214,9303.169) (3287.766,9278.937) (3333.911,9259.457) (3375.000,9246.000) \path(3375,9246) (3431.461,9231.132) (3496.421,9213.795) (3532.843,9203.995) (3572.296,9193.330) (3615.082,9181.718) (3661.504,9169.076) (3711.863,9155.323) (3766.461,9140.376) (3825.601,9124.152) (3889.585,9106.569) (3958.715,9087.545) (4033.292,9066.997) (4072.718,9056.126) (4113.620,9044.843) (4156.035,9033.138) (4200.000,9021.000) \put(3150,13896){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{4S,1}(0,0)=225$}}}}} \put(3150,11196){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{3S+F,1}(0,e_1)=225$}}}}} \put(5400,8421){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,1}(0,2e_1)=20$}}}}} \put(5250,5721){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,1}(0,e_2)=30$}}}}} \put(8850,5721){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,1}(e_1,e_1) = 20$}}}}} \put(9300,3021){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{S+3F,0}(0,3e_1)=1$}}}}} \put(5625,3021){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,1}(2e_1,0)=17$}}}}} \put(1725,3021){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,1}(e_2,0)=15$}}}}} \put(675,21){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{S+3F,0}(0,e_1+e_2)=4$}}}}} \put(4800,21){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{S+3F,-1}(0,3e_1)=7$}}}}} \put(0,5721){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{2S+2F,0}(0,2e_1) = 96+9=105$}}}}} \put(1500,8421){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$N^{3S+F,1}(e_1,0)=185$}}}}} \put(5550,10521){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$\times 2$}}}}} \put(4725,5046){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$\times 2$}}}}} \put(9675,5271){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$\times 3$}}}}} \put(3300,2571){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$\times 2$}}}}} \put(5925,7446){\makebox(0,0)[lb]{\smash{{{\SetFigFont{8}{9.6}{rm}$\times 2$}}}}} \end{picture} } \end{center} \caption{Calculating $N^{4S,1}(0,0) = 225$.} \label{ruledfig} \end{figure} Table \ref{class2c} gives the number of genus $g$ curves in certain classes on certain ${\mathbb F}_n$. Where the number of irreducible curves is different, it is given in brackets. Tables \ref{classf1} and \ref{classf2} give more examples; only the total number is given, although the number of irreducible curves could also be easily computed (using Theorem \ref{irecursion}). Many of these numbers were computed by a maple program written by L. G\"{o}ttsche to implement the algorithm of Theorem \ref{recursion}. \begin{scriptsize} \begin{table} \begin{center} \begin{tabular}{c|c|c|c|c|c|} & $\mathbb F_0$ & $\mathbb F_1$ & $\mathbb F_2$ & $\mathbb F_3$ & $\mathbb F_4$ \\ \hline $2S$ & & $g=0$: 1 & $g=1$: 1 & $g=2$: 1 & $g=3$: 1 \\ & & & $g=0$: 10 & $g=1$: 17 & $g=2$: 24 \\ & & & & $g=0$: 69 & $g=1$: 177 \\ & & & & & $g=0$: 406 \\ $2S+F$ & $g=0$: 1 & $g=1$: 1 & $g=2$: 1 & $g=3$: 1 \\ & & $g=0$: 12 & $g=1$: 20 & $g=2$: 28 \\ & & & $g=0$: 102 (93) & $g=1$: 246 (234) \\ & & & & $g=0$: 781 (594) \\ $2S+2F$ & $g=1$: 1 & $g=2$: 1 & $g=3$: 1 \\ & $g=0$: 12 & $g=1$: 20 & $g=2$: 28 \\ & & $g=0$: 105 (96) & $g=1$: 252 (240) \\ & & & $g=0$: 856 (636) \\ $2S+3F$ & $g=2$: 1 & $g=3$: 1 \\ & $g=1$: 20 & $g=2$: 28 \\ & $g=0$: 105 (96) & $g=1$: 252 (240) \\ & & $g=0$: 860 (640) \\ $2S+4F$ & $g=3$: 1 \\ & $g=2$: 28 \\ & $g=1$: 252 (240) \\ & $g=0$: 860 (640) \\ \end{tabular} \end{center} \caption{Number of genus $g$ curves in class $2S+kF$ on ${\mathbb F}_n$} \label{class2c} \end{table} \end{scriptsize} \begin{table} \begin{center} \begin{tabular}{|c|c|c|} \hline Class & Genus & Number \\ \hline \hline $3S$ & -2 & 15 \\ & -1 & 21 \\ & 0 & 12 \\ & 1 & 1 \\ \hline $3S+F$ & 0 & 675 \\ & 1 & 225 \\ & 2 & 27 \\ & 3 & 1 \\ \hline $3S+2F$ & 0 & 22647 \\ & 1 & 14204 \\ & 2 & 4249 \\ & 3 & 615 \\ & 4 & 41 \\ & 5 & 1 \\ \hline $3S+3F$ & 0 & 642434 \\ & 1 & 577430 \\ & 2 & 291612 \\ & 3 & 83057 \\ & 4 & 13405 \\ & 5 & 1200 \\ & 6 & 55 \\ & 7 & 1 \\ \hline \end{tabular} \end{center} \caption{Number of genus $g$ curves in various classes on $\mathbb F_1$} \label{classf1} \end{table} \begin{table} \begin{center} \begin{tabular}{|c|c|} \hline Genus & Number \\ \hline \hline -2 & 280 \\ -1 & 1200 \\ 0 & 2397 \\ 1 & 1440 \\ 2 & 340 \\ 3 & 32 \\ 4 & 1 \\ \hline \end{tabular} \end{center} \caption{Number of genus $g$ curves in class $3S$ on $\mathbb F_2$} \label{classf2} \end{table} We next review some earlier results. (This is only a partial summary of the voluminous research done on the subject.) In each case but the last two, the numbers have been checked to agree with those produced by the algorithm given here for ``small values''. Many of these verifications have been done by D. Watabe, and the author is grateful to him for this. For example, he has verified that the formula for the number of rational curves in the divisor class $2S$ on ${\mathbb F}_n$ passing through the appropriate number of general points for $n \leq 9$ agrees with the numbers obtained by Caporaso and Harris. In each case it seems difficult to directly prove that the numbers will always be the same, other than by noting that they count the same thing. \begin{itemize} \item The number of degree $d$ genus $g$ plane curves through $3d+g-1$ fixed general points was calculated by Ran (cf. [R]) and Caporaso and Harris (cf. [CH3]). This is also the number of genus $g$ curves in the divisor class $dS$ on $\mathbb F_1$ through $3d+g-1$ fixed general points, and the number of genus $g$ curves in the divisor class $(d-1)S+F$ on $\mathbb F_1$ through $3d+g-2$ fixed general points. \item The surfaces $\mathbb F_0$ and $\mathbb F_1$ are convex, so the ideas of [KM] allow one to count (irreducible) rational curves in all divisor classes on these surfaces (see [DI] for further discussion). These are known as the genus 0 Gromov-Witten invariants of $\mathbb F_0$ and $\mathbb F_1$. \item D. Abramovich and A. Bertram have proved several (unpublished) formulas counting (irreducible) rational curves in certain classes on $\mathbb F_n$ ([AB]). If $N^g_{\mathbb F_n}(aS+bF)$ is the number of irreducible genus $g$ curves in class $aS+bF$ through the appropriate number of points, then they have shown: \begin{enumerate} \item[(AB1)]$N^0_{\mathbb F_0}(aS+(b+a)F) = \sum_{i=0}^{a-1} \binom {b+2i} i N^0_{\mathbb F_2}(aS+bF-iE).$ \item[(AB2)]$N^0_{\mathbb F_n}(2S+bF) = N^0_{\mathbb F_{n-2}}(2S + (b+2)F) - \sum_{l=1}^{n-1} \binom {2 (n+b) + 3} {n-l-1} \left( l^2 (b+2) + \binom l 2 \right).$ \item[(AB3)] $N^0_{\mathbb F_n}(2S) = 2^{2n} (n+3) - (2n+3) \binom {2n+1} n.$ \item[(AB4)] $N^0_{\mathbb F_n}(2S+bF) = N^0_{\mathbb F_{n-1}}(2S+(b+1)F) - \sum_{l=1}^{n-1} \binom {2(n+b) + 2} {n-l-1} l^2 (b+2).$ \end{enumerate} Their method for (AB1) and (AB2) is to study the moduli space of stable maps of rational curves to $\mathbb F_n$, and deform the surface to $\mathbb F_{n-2}$. For (AB3) and (AB4), they relate curves on $\mathbb F_n$ to curves on $\mathbb F_{n-1}$. \item By undoubtedly similar methods, the author has obtained the formula \begin{eqnarray*} N^g_{\mathbb F_n}(2 S + k F) &=& N^g_{\mathbb F_{n+1}}(2S + (k-1) F) \\ & & + \sum_{f=0}^{n-g-1} \sum \binom {\alpha_1} {|\alpha| - g- 1} \binom { |\alpha|} { \alpha_1, \dots, \alpha_n} \binom { 2n+2k+2+g} f I^{2 \alpha} \end{eqnarray*} where the second sum is over all integers $f$ and sequences $\alpha$ such that $I \alpha = n+k-f$, $|\alpha| = k+1+g$, $k \leq \alpha_1$. \item Caporaso and Harris (in [CH1] and [CH2]) obtained recursive formulas for $N^0_{\mathbb F_n}(aS+bF)$ when $n \leq 3$, and the remarkable result that $N^0_{\mathbb F_n}(2S)$ is the co-efficient of $t^n$ in $(1+t)^{2n+3}/ (1-t)^3$. \item Coventry has also recently derived a recursive formula for the number of rational curves in {\em any} class in $\mathbb F_n$ ([Co]), using a generalization of the ``rational fibration'' method of [CH2]. \item Kleiman and Piene have examined systems with an arbitrary, but fixed, number $\delta$ of nodes ([K2]). The postulated number of $\delta$-nodal curves is given (conjecturally) by a polynomial, and they determine the leading coefficients, which are polynomials in $\delta$. Vainsencher determined the entire polynomial for $\delta \leq 6$ ([Va]). Kleiman and Piene extended his work so that a refinement of his computation of the polynomial for a given $\delta$ ought to yield the coefficients in the top $2\delta$ total degrees. They have done this explicitly for the plane for $\delta \leq 4$, in particular supporting and extending the conjecture on p. 86 of [DI], and expect to get it for $\delta \leq 6$. \end{itemize} \section{Dimension counts} In this section, we prove the main dimension count we need: \begin{tm} \label{bigdim} \begin{enumerate} \item[(a)] Each component of $V^{D,g}_m\abG$ is of dimension at most $$ \Upsilon = \Upsilon^{D,g}(\beta) = -(K_X + E) \cdot D + | \beta | + g-1. $$ \item[(b)] The stable map $(C, \pi)$ corresponding to a general point of any component of dimension $\Upsilon$ satisfies the following properties. \begin{enumerate} \item[(i)] The curve $C$ is smooth, and the map $\pi$ is an immersion. \item[(ii)] The image is a reduced curve. If the $\{ p_{i,j} \}$ are distinct, the image is smooth along its intersection with $E$. \end{enumerate} \item[(c)] Conversely, any component whose general map satisfies property (i) has dimension $\Upsilon$. \end{enumerate} \end{tm} By ``the image is a reduced curve'', we mean $\pi_*[C]$ is a sum of distinct irreducible divisors on $X$. Proposition \ref{idim} follows directly from Theorem \ref{bigdim}. \begin{lm}[Arbarello-Cornalba, Caporaso-Harris] \label{chlemma} Let $V$ be an irreducible subvariety of the moduli space $\overline{M}_g(Y,\beta)'$ where $Y$ is smooth, such that if $(C,\pi)$ corresponds to the general point of $V$ then $C$ is smooth $\pi$ is birational. Let $N = \operatorname{coker}(T_C \rightarrow \pi^* T_Y)$, and let $N_{\operatorname{tors}}$ be the torsion subsheaf of $N$. Then: \begin{enumerate} \item[(a)] $\dim V \leq h^0(C, N / N_{\operatorname{tors}})$. \item[(b)] Assume further that $Y$ is a surface. Fix a smooth curve $G$ in $Y$ and smooth points $\{ p_{i,j} \}$ of $G$, and assume that $$ \pi^* G = \sum_{i,j} i q_{i,j} + \sum_{i,j} i r_{i,j} $$ with $\pi(q_{i,j}) = p_{i,j}$. Then \begin{eqnarray*} \dim V &\leq& h^0 (C, N / N_{\operatorname{tors}} (- \sum_{i,j} i q_{i,j} - \sum_{i,j} (i-1) r_{i,j})) \\ &=& h^0(C, N / N_{\operatorname{tors}} (- \pi^* G + \sum_{i,j} r_{i,j})). \end{eqnarray*} \end{enumerate} \end{lm} This lemma appears (in a different guise) in Subsection 2.2 of [CH3]: (a) is contained in Corollary 2.4 and part (b) is Lemma 2.6. Part (a) was proven earlier by E. Arbarello and M. Cornalba in [AC], Section 6. Caporaso and Harris express (a) informally as: ``the first-order deformation of the map $\pi$ corresponding to a torsion section of $N$ can never be equisingular.'' Arbarello and Cornalba's version is slightly stronger: ``the first-order deformation of the map $\pi$ corresponding to a torsion section of $N$ can never preserve both the order and type of the singularities of the image.'' \begin{lm} \label{dimbd} Let $V$ be a component of $V_m^{D,g}\abG$ whose general point corresponds to a map $\pi: C \rightarrow X$ where $C$ is a smooth curve. Then $\dim V \leq \Upsilon$. If $\pi$ is not an immersion then the inequality is strict. \end{lm} \noindent {\em Proof. } Note that by the definition of $V^{D,g}_m\abG$, $\pi$ is a birational map from $C$ to its image in $X$, so we may invoke Lemma 2.2. The map $T_C \rightarrow \pi^* T_X$ is injective (as it is generically injective, and there are no nontrivial torsion subsheaves of invertible sheaves). If $N$ is the normal sheaf of $\pi$, then the sequence $$ 0 \rightarrow T_C \rightarrow \pi^* T_X \rightarrow N \rightarrow 0 $$ is exact. Let $N_{\operatorname{tors}}$ be the torsion subsheaf of $N$. The map $\pi$ is an immersion if and only if $N_{\operatorname{tors}} = 0$. Now \begin{eqnarray*} (\det N) (- \pi^* E + \sum r_{i,j}) &=& {\mathcal{O}}_C( - \pi^* K_X + K_C - \pi^* E + \sum r_{i,j} ) \\ &=& {\mathcal{O}}_C(- \pi^* K_X + K_C - \pi^* E + \sum r_{i,j} )\\ \end{eqnarray*} By property P3, the divisor $-\pi^*(K_X+E) + \sum r_{i,j}$ is positive on each component of $C$, so by Kodaira vanishing or Serre duality $$ H^1(C, (\det N)(-\pi^* E+\sum r_{i,j})) = 0. $$ As $N / N_{\operatorname{tors}}$ is a subsheaf of $\det N$, \begin{eqnarray} \label{fred} \lefteqn{h^0(C, N / N_{\operatorname{tors}} (- \pi^* E + \sum r_{i,j}))} \\ &\leq& h^0 (C, (\det N) (- \pi^* E + \sum r_{i,j})) \nonumber \\ &=& \chi (C, (\det N) (- \pi^* E + \sum r_{i,j})) \nonumber \\ &=& \deg N - E \cdot D + |\beta| - g + 1 \nonumber \\ &=& - (K_X+E) \cdot D + | \beta | + g - 1 \nonumber \\ &=& \Upsilon. \nonumber \end{eqnarray} If $C'$ is a component of $C$ with $- \pi^*(K_X + E) \cdot C' = 1$, then $\pi: C' \rightarrow X$ is an immersion by property P4. Thus if $N_{\operatorname{tors}} \neq 0$, then it is non-zero when restricted to some component $C''$ for which $-\pi^*(K_X + E) \cdot C'' \geq 2$. Let $p$ be a point on $C''$ in the support of $N_{\operatorname{tors}}$. Then $- \pi^*(K_X + E) + \sum r_{i,j} - p$ is positive on each component of $C$, so by the same argument as above, $N/N_{\operatorname{tors}}$ is a subsheaf of $(\det N)(-p)$, so \begin{eqnarray*} \lefteqn{h^0(C, N/N_{\operatorname{tors}} ( - \pi^* E + \sum r_{i,j} - p)) } \\ &\leq& h^0( C, (\det N) ( - \pi^* E + \sum r_{i,j} - p)) \\ &=& \Upsilon - 1. \end{eqnarray*} Therefore, equality holds at (\ref{fred}) only if $N_{\operatorname{tors}} = 0$, i.e. $\pi$ is an immersion. By Lemma \ref{chlemma}(a), the result follows. \qed \vspace{+10pt} {\noindent {\em Proof of Theorem \ref{bigdim}. }} Let $V$ be a component of $V^{D,g}_m \abG$ of dimension at least $\Upsilon$, and let $\pi: C \rightarrow X$ be the map corresponding to a general point of $V$. Let the normalizations of the components of $C$ be $C(1)$, $C(2)$, \dots, $C(s)$, so $p_a( \coprod_k C(k)) \leq p_a(C)$ with equality if and only if $C$ is smooth. Let $\beta = \sum_{k=0}^s \beta(k)$ be the partition of $\beta$ induced by $C = \cup_{k=1}^s C(k)$, let $g(k)$ be the genus of $C(k)$, and let $$ \Upsilon(k) = (K_X + E) \cdot \pi_*[C(k)] + | \beta(k) | + g(k) - 1. $$ By the definition of $V^{D,g}_m\abG$, $\pi|_{C(k)}$ is birational. By Lemma \ref{dimbd}, $C$ moves in a family of dimension at most \begin{eqnarray} \nonumber \sum_{k=1}^s \Upsilon(k) &=& \sum_{k=1}^s \left(- (K_X + E) \cdot \pi_*[C(k)] + | \beta(k) | + g(k) - 1 \right) \\ \nonumber &=& -(K_X+E) \cdot D + |\beta| + p_a \left( \coprod C(k) \right) - 1 \\ &\leq& -(K_X+E) \cdot D + |\beta| + p_a(C) - 1. \label{eqna} \end{eqnarray} This proves part (a). If $\dim V = \Upsilon$, then equality must hold in (\ref{eqna}), so $C$ is smooth, and by Lemma \ref{dimbd}, $\pi$ is an immersion. We next prove that the image is smooth along its intersection with $E$. Requiring $\pi(r_{i_0,j_0})$ to be a fixed point $p$ imposes an additional condition on $V^{D,g}_m\ab$, as the locus of such maps forms a variety of the form $V^{D,g}_m(\alpha+ e_{i_0}, \beta - e_{i_0}, \{ p_{i,j} \} \cup \{ p \})$ which has dimension $$ -(K_X+E) \cdot D + g-1 + ( |\beta| - 1) < \dim V^{D,g}_m(\alpha,\beta, \{ p_{i,j} \}) $$ by part (a). Thus if $\pi: C \rightarrow X$ is the map corresponding to a general point of $V^{D,g}_m \ab$ then \begin{enumerate} \item[(i)] $\pi(q_{i,j}) \neq \pi(q_{i',j'})$ for $(i,j) \neq (i',j')$ as $p_{i,j} \neq p_{i',j'}$, \item[(ii)] $\pi(r_{i,j}) \neq p_{i',j'}$ as requiring $\pi(r_{i,j})$ to be fixed imposes a nontrivial condition on $V^{D,g}_m\ab$, \item[(iii)] $\pi(r_{i,j}) \neq \pi(r_{i',j'})$ for $(i,j) \neq (i',j')$ as requiring both $r_{i,j}$ and $r_{i',j'}$ to be fixed imposes {\em two} independent conditions on $V^{D,g}_m\ab$. \end{enumerate} Thus the image is smooth along its intersection with $E$. If a component of the image curve is nonreduced (with underlying reduced divisor $D_1$), then this component cannot intersect $E$ (as the image is smooth along $E$). As each $C(k)$ is birational to its image, the image curve must be the image of two components $C(k_1)$ and $C(k_2)$ for which $\Upsilon(k_1) = \Upsilon(k_2) = 0$. Then \begin{eqnarray*} 0 &=& \Upsilon(k_1) \\ &=& \sum_{k=1}^s (- (K_X + E) \cdot \pi_*[C(k_1)] + | \beta(k_1) | + g(k_1) - 1) \\ &=& - K_X \cdot D_1 + g(k_1) - 1 \end{eqnarray*} As $-(K_X + E)$ is positive on all effective divisors, we must have $g(k_1)=0$ and $- (K_X+E) \cdot D_1 = 1$. Thus $D_1$ is rational, and by property P4, $D_1$ is smooth. Moreover, $E \cdot D_1 = 0$, so $- K_X \cdot D_1 = 0$ and therefore $D_1^2 = -1$. But then $D_1$ is an exceptional curve not intersecting $E$, contradicting property P2. Thus the image curve is reduced, so (b) is proved. For part (c), let $N$ be the normal bundle to the map $\pi$. As $\pi^* E$ contains no components of $C$, $$ N(- K_C) = {\mathcal{O}}_C(- \pi^*(K_X + E)) \otimes {\mathcal{O}}_C( \pi^* E). $$ is positive on every component of $C$ by property P3 , so $N$ is nonspecial. Therefore \begin{eqnarray*} h^0(N) &=& - K_X \cdot D + \deg K_C - g + 1 \\ &=& - K_X \cdot D + g-1 \end{eqnarray*} by Riemann-Roch. Requiring the curve to remain $i$-fold tangent to $E$ at the point $q_{i,j}$ of $C$ (where $\pi(q_{i,j})$ is required to be the fixed point $p_{i,j}$) imposes at most $i$ independent conditions. Requiring the curve to remain $i$-fold tangent to $E$ at the points $r_{i,j}$ of $C$ imposes at most $(i-1)$ independent conditions. Thus \begin{eqnarray*} \dim V &\geq& - K_X \cdot D + g-1 - I \alpha - I \beta + | \beta | \\ &=& - (K_X +E) \cdot D + | \beta | + g-1 \\ \end{eqnarray*} as $I \alpha + I \beta = D \cdot E$. \qed \vspace{+10pt} Let $V$ be an irreducible subvariety of $\overline{M}_g(X,D)'$, and let $\pi: C \rightarrow X$ be the map corresponding to a general point of a component of $V$. Assume that $\pi^* E = \sum m q_{i,j} + \sum m r_{i,j}$ where $\pi(q_{i,j})$ is required to be a fixed point $p_{i,j}$ of $E$ as $C$ varies. (In particular, no component of $C$ is mapped to $E$.) Define $\alpha$ by $\alpha_i = \# \{ q_{i,j} \}_j$, $\beta$ by $\beta_i = \# \{ r_{i,j} \}_j$, and $\Gamma= \{ p_{i,j} \}$. \begin{pr} \label{idimbound} The intersection dimension of $V$ is at most $$ - (K_X + E) \cdot D + | \beta| + g-1. $$ If equality holds then $V$ is a component of $V^{D,g}_m\abG$. \end{pr} The main obstacle to proving this result is that the map $\pi$ may not map components of $C$ birationally onto their image: the map $\pi$ may collapse components or map them multiply onto their image. \noindent {\em Proof. } If necessary, pass to a dominant generically finite cover of $V$ that will allow us to distinguish components of $C$. (Otherwise, monodromy on $V$ may induce a nontrivial permutation of the components of $C$.) For convenience, first assume that $C$ has no contracted rational or elliptic components. We may replace $C$ by its normalization; this will only make the bound worse. (The map from a component of the normalization of $C$ is also a stable map.) We may further assume that $C$ is irreducible, as $-(K_X+E) \cdot D + | \beta| + g-1$ is additive. Suppose $C$ maps with degree $m$ to the reduced irreducible curve $D_0 \subset X$. Then the map $\pi: C \rightarrow D_0$ factors through the normalization $\tilde{D}$ of $D_0$. Let $r$ be the total ramification index of the morphism $C \rightarrow \tilde{D}$. By Theorem \ref{bigdim}(a), \begin{eqnarray*} \operatorname{idim} V &\leq& \dim V \\ &\leq& - (K_X + E) \cdot D_0 + | \beta| + g(\tilde{D}) - 1 \\ &=& - \frac 1 m (K_X + E) \cdot \pi_* [C] + | \beta| + \frac 1 m ( g(C) - 1 - r/2) \\ &\leq& - (K_X + E) \cdot \pi_* [C] + | \beta| + g(C) - 1 \end{eqnarray*} where we use the Riemann-Hurwitz formula for the map $C \rightarrow \tilde{D}$ and the fact (property P3) that $-(K_X + E) \cdot D_0 > 0$. Equality holds only if $m=1$, so by Theorem \ref{bigdim}, equality holds only if $V$ is a component of $V^{D,g}_m\abG$ for some $g$, $\alpha$, $\beta$, $\Gamma$. If $C$ has contracted rational or elliptic components, replace $C$ with those components of its normalization that are not contracted elliptic or rational components (which reduces the genus of $C$) and follow the same argument. \qed \vspace{+10pt} \begin{pr} \label{nodal} If $X={\mathbb F}_n$, the $\{ p_{i,j} \}$ are distinct, and $(C,\pi)$ is a general curve in a component of $V^{D,g}_m\ab$, then $\pi(C)$ has at most nodes as singularities. \end{pr} Warning: To prove this, we will need more than properties P1--P4. However, this result will not be invoked later. \noindent {\em Proof. } By Theorem \ref{bigdim}, $\pi$ is an immersion and $\pi(C)$ is reduced. Thus we need only show that $\pi(C)$ has no triple points and that no two branches are tangent to each other. For the former, if $s$, $t$, and $u \in C$ are distinct points of $C$ with $\pi(s) = \pi(t) = \pi(u)$, it is enough to show that there is a section of the line bundle \begin{eqnarray*} L &:=& N ( - \sum i \cdot q_{i,j} - \sum (i-1) \cdot r_{i,j} ) \\ &=& {\mathcal{O}}_C( - \pi^*( K_X + E) + \sum r_{i,j} + K_C) \end{eqnarray*} vanishing at $s$ and $t$ but not at $u$, where $N$ is the normal bundle to the map $\pi$. As $\pi(C)$ is reduced, at most one of $s$, $t$, $u$ can lie on the component of $C$ mapping to the fiber $F$ through $\pi(s)$. If none of them lie on such a component, then $(\pi^* F- s - t - u)$ is effective, and $$ L(-s-t-u - K_C) = {\mathcal{O}}_C(\pi^*(S+F) + (\pi^* F-s-t-u) + \sum r_{i,j}) $$ is effective (consider the fiber through $\pi(s)$), so by Riemann-Roch and Kodaira vanishing, $$ h^0(C, L(-s-t-u)) = h^0(C,L)-3. $$ If $u$ lies on a component of $C$ mapping to $F$, and there is a point $r_{i,j}$ on the same component, then both $(\pi^*F-s-t)$ and $(r_{i,j} - u)$ are both effective, and the same argument holds. If there is no point $r_{i,j}$ on the same component as $u$, then all sections of $L$ vanish on $u$, and it suffices to find a section of $L$ vanishing at $t$ but not at $s$. But $(\pi^*F-s-t)$ is effective, so by the same argument $$ h^0(C, L(-s-t)) = h^0(C,L)-2. $$ To show that no two branches are tangent to each other, it is enough to show that if $s$, $t \in C$ are distinct points with $\pi(s)=\pi(t)$, there exists a section of $L$ vanishing at $s$ but not at $t$, which follows from a similar argument. \qed \vspace{+10pt} The following example shows that the analogue of Proposition \ref{nodal} does not hold for every $(X,E)$ satisfying properties P1--P4. Let $X=\mathbb P^2$ and $E$ be a smooth conic. Choose six distinct points $a$, \dots, $f$ on $E$ such that the lines $ab$, $cd$, and $ef$ meet at a point. Then $$ V^{D=3L, g=-2}(\alpha = 6 e_1, \beta=0, \Gamma = \{ a, \dots, f \}) $$ consists of a finite number of maps, one of which is the map sending three disjoint $\mathbb P^1$'s to the lines $ab$, $cd$, and $ef$. \section{Identifying Potential Components} \label{ipc} Fix $D$, $g$, $\alpha$, $\beta$, $\Gamma$, and a general point $q$ on $E$. Throughout this section, $\Gamma$ will be assume to consist of distinct points. (The more general case will be dealt with in [V3].) Let $H_q$ be the divisor on $V^{D,g}_m\abG$ corresponding to maps whose image contain $q$. In this section, we will derive a list of subvarieties (which we will call {\em potential components}) in which each component of $H_q$ of intersection dimension $\Upsilon-1$ appears. (In Sections~\ref{multI} and \ref{multII}, we will see that if the $\{ p_{i,j} \}$ are distinct, each potential component actually appears in $H_q$.) The potential components come in two classes that naturally arise from requiring the curve to pass through $q$. First, one of the ``moving tangencies'' $r_{i,j}$ could map to $q$. We will call such components {\it Type I potential components}. Second, the curve could degenerate to contain $E$ as a component. We will call such components {\it Type II potential components}. For any sequences $\alpha'' \leq \alpha$, $\gamma \geq 0$, and subsets $\{ p''_{i,1}, \dots, p''_{i,\alpha''_i} \}$ of $\{ p_{i,1}, \dots, p_{i,\alpha_i} \}$, let $g'' = g + |\gamma| + 1$ and $\Gamma'' = \{ p''_{i,j} \}_{1 \leq j \leq \alpha''_i}$. Define the Type II component $K(\alpha'',\beta,\gamma,\Gamma'')$ as the closure in $\overline{M}_g(X,D)'$ of points representing maps $\pi: C' \cup C'' \rightarrow X$ where \begin{enumerate} \item[K1.] the curve $C'$ maps isomorphically to $E$, \item[K2.] the curve $C''$ is smooth, $\pi$ maps each component of $C''$ birationally to its image, no component of $C''$ maps to $E$, and there exist $|\alpha''|$ points $q_{i,j} \in C''$, $j = 1$, \dots, $\alpha_i''$, $|\beta|$ points $r_{i,j} \in C''$, $j = 1$, \dots, $\beta_i$, $|\gamma|$ points $t_{i,j} \in C''$, $j = 1$, \dots, $\gamma_i$ such that $$ \pi(q_{i,j}) = p''_{i,j} \quad \text{and} \quad (\pi|_{C''})^*(E) = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j} + \sum i \cdot t_{i,j}, $$ and \item[K3.] the intersection of the curves $C'$ and $C''$ is $\{ t_{i,j} \}_{i,j}$. \end{enumerate} The variety $K(\alpha'',\beta,\gamma,\Gamma'')$ is empty unless $I(\alpha''+\beta+\gamma) = (D-E) \cdot E$. The genus of $C''$ is $g''$, and there is a degree $\binom {\beta+\gamma} \beta$ rational map $$ K(\alpha'',\beta,\gamma,\Gamma'') \dashrightarrow V_m^{D-E,g''}(\alpha'',\beta+\gamma,\Gamma'') $$ corresponding to ``forgetting the curve $C'$''. \begin{tm} \label{list} Fix $D$, $g$, $\alpha$, $\beta$, $\Gamma$, and a point $q$ on $E$ not in $\Gamma$. Let $K$ be an irreducible component of $H_q$ with intersection dimension $\Upsilon - 1$. Then set-theoretically, either \begin{enumerate} \item[I.] $K$ is a component of $V^{D,g}_m(\alpha+e_k,\beta-e_k,\Gamma')$, where $\Gamma'$ is the same as $\Gamma$ except $p'_{k,\alpha_{k+1}} = q$, or \item[II.] $K$ is a component of $K(\alpha'',\beta,\gamma,\Gamma'')$ for some $\alpha''$, $\gamma$, $\Gamma''$. \end{enumerate} \end{tm} \noindent {\em Proof. } Let $(C,\pi)$ be the map corresponding to a general point of $K$. Consider any one-parameter subvariety $({\mathcal{C}},\Pi)$ of $V^{D,g}_m\ab$ with central fiber $(C,\pi)$ and general fiber not in $H_q$. Then the total space of the curve ${\mathcal{C}}$ in the family is a surface, so the pullback of the divisor $E$ to this family has pure dimension 1. The components of $\Pi^* E$ not contained in a fiber ${\mathcal{C}}_t$ must intersect the general fiber and thus be the sections $q_{i,j}$ or multisections coming from the $r_{i,j}$. Therefore $\pi^{-1} E$ consists of components of $C$ and points that are limits of the $q_{i,j}$ or $r_{i,j}$. In particular: {\bf (*)} The number of zero-dimensional components of $\pi^* E$ not mapped to any $p_{i,j}$ is at most $\beta$, and {\bf (**)} If there are exactly $|\beta|$ such components, the multiplicities of $\pi^* E$ at these points must be given by the sequence $\beta$. {\em Case I.} If $C$ contains no components mapping to $E$, then $$ \pi^* E = \sum i \cdot a_{i,j} + \sum i \cdot b_{i,j} $$ where $\pi( \{ a_{i,j} \}_{i,j} ) = \{ p_{i,j} \}_{i,j} \cup \{ q \}$ and the second sum is over all $i$, $1 \leq j \leq \beta'_i$ for some sequence $\beta'$. By (*), $| \beta' | \leq |\beta| - 1$. Then by Proposition \ref{idimbound}, \begin{eqnarray*} \operatorname{idim} K &\leq& - (K_X + E) \cdot D + |\beta'| + g-1 \\ &\leq& - (K_X + E) \cdot D + |\beta|-1 + g-1 \\ &=& \Upsilon - 1. \end{eqnarray*} Equality must hold, so $|\beta'| = |\beta| - 1$ and $K$ is a generalized Severi variety of maps. The set $\pi^{-1} E$ consists of $|\alpha| + |\beta|$ points (which is also true of $\pi_0^{-1} E$ for a general map $(C_0,\pi_0)$ in $V^{D,g}_m\ab$) so the multiplicities at these points must be the same as for the general map (i.e. $\pi^* E |_{p_{i,j}}$ has multiplicity $i$, etc.) so $K$ must be as described in I. {\em Case II.} If otherwise a component of $C$ maps to $E$, say $C = C' \cup C''$ where $C' $ is the union of irreducible components of $C$ mapping to $E$ and $C''$ is the union of the remaining components. Define $m$ by $\pi_*[C'] = m E$, so $\pi_*[C''] = D-mE$. Let $s = \# (C' \cap C'' )$. Then $p_a(C') \geq 1-m$, so \begin{eqnarray*} p_a(C'') &=& g - p_a(C') + 1-s \\ &\leq& g+m-s. \end{eqnarray*} Assume $(\pi|_{C''})^* E = \sum i \cdot a_{i,j} + \sum i \cdot b_{i,j}$ where $\pi(a_{i,j})$ are fixed points of $E$ as $C''$ varies, and the second sum is over all $i$ and $1 \leq j \leq \beta''_i$ for some sequence $\beta''$. By (*), $|\beta''| \leq |\beta| + s$. By restricting to an open subset if necessary, the universal map may be written $({\mathcal{C}},\Pi)$ where ${\mathcal{C}} = {\mathcal{C}}' \cup {\mathcal{C}}''$, $\Pi_t({\mathcal{C}}'_t) \subset E$ for all $t$, and $\Pi_t({\mathcal{C}}''_t)$ has no component mapping to $E$. Let $K'$ be the family $({\mathcal{C}}'',\Pi|_{{\mathcal{C}}''})$. We apply Proposition \ref{idimbound} to the family $K'$: \begin{eqnarray*} \operatorname{idim} K &=& \operatorname{idim} K' \\ &\leq& - (K_X + E) \cdot (D-mE) + |\beta''| + p_a(C'') - 1 \\ &\leq& \left( -(K_X + E) \cdot D - 2 m \right) + ( |\beta| + s ) + (g+m-s) -1\\ &=& \left( -(K_X+E) \cdot D + | \beta| + g-1 \right) - 1 - (m-1) \\ &=& \Upsilon - 1 - (m-1) \\ &\leq& \Upsilon - 1. \end{eqnarray*} In the third line, we used property P1: $E$ is rational, so $(K_X+E) \cdot E = -2$. Equality must hold, so $m=1$ and $|\beta''| = |\beta| + s$. By (**), the multiplicity of $\pi^* E$ at the $|\beta|$ points of $C''$ not in $C' \cup \pi^* p_{i,j}$ is given by the sequence $\beta$. Let $\gamma$ be the sequence given by the multiplicities of $( \pi|_{C''})^* E$ at the $s$ points $C' \cap C''$. Let $\{ p''_{i,j} \}$ be the subset of $\{ p_{i,j} \}$ contained in $(\pi|_{C''})^* E$. The only possible limits of points $\{ q_{i,j} \}$, $\{ r_{i,j} \}$ that could be points of $(\pi|_{C''})^{-1} p_{i_0,j_0}$ is the section $q_{i_0,j_0}$, so $(\pi|_{C''})^* p_{i_0,j_0}$ consists of a single point $q_{i_0,j_0}$ with multiplicity $i_0$. In short, $K$ is a component of $K(\alpha'',\beta,\gamma,\Gamma'')$. \qed \vspace{+10pt} There are other components of the divisor $H_q$ not counted in Theorem \ref{list}. For example, if $X=\mathbb P^2$, and $E$ is a line $L$, $D=2L$, $g=0$, $\alpha=2e_1$, $\beta=0$, then $V^{D,g}_m\ab$ is a three-dimensional family (generically) parametrizing conics through 2 fixed points of $L$. One component of $H_q$ (generically) parametrizes a line union $L$; this is a Type II potential component. The other (generically) parametrizes degree 2 maps from $\mathbb P^1$ to $L$. This has intersection dimension 0, so it makes no enumerative contribution. \section{Multiplicity for Type I Intersection Components} \label{multI} We first solve a simpler analog of the problem. Fix points $\Gamma=(p_{i,j})_{i,j \in {\mathbb Z}, ij \leq d}$ on $E$, and let $\alpha$ and $\beta$ be sequences of non-negative integers. In $\mathbb P^d = \operatorname{Sym}^d E$ representing length $d$ subschemes of $E$, we have loci $v(\alpha,\beta,\Gamma)$ corresponding to the closure of the subvariety parametrizing $d$-tuples $$ \sum_i^d (\sum_{j=1}^{\alpha_i} i \cdot p_{i,j} + \sum_{k=1}^{\beta_i} i \cdot r_{i,j}) $$ where $(r_{i,j})$ are any points. Then $v(\alpha,\beta,\Gamma)$ is a smooth variety that is the image of $$ \prod_{i=1}^{\beta_d} \operatorname{Sym}^{\beta_d} E = \prod_{i=1}^{\beta_d} \mathbb P^{\beta_d} $$ embedded in $\mathbb P^d= \operatorname{Sym}^d E$ by the line bundle $h_1 + 2 h_2 + ... + d h_d$ where $h_i$ is the hyperplane class of $\mathbb P^{\beta_i}$. Let $q$ be a point on $E$ not in $\Gamma$, and let $H'_q$ be the hyperplane in $\mathbb P^d = \operatorname{Sym}^d E$ corresponding to $d$-tuples containing $q$. Then it is straightforward to check that, as divisors on $v(\alpha,\beta,\Gamma)$, \begin{equation} \label{babybabybaby} H'_q|_{v(\alpha,\beta,\Gamma)} = \sum_{\beta_k > 0} k \cdot v(\alpha+e_k,\beta-e_k,\Gamma_k) \end{equation} where $\Gamma_k$ is equal to $\Gamma$ in all positions except $p_{k,\alpha_k+1}=q$. (For example, start by observing that equality holds set-theoretically, and then find multiplicity by making the base change $E^{\beta_k} \rightarrow \operatorname{Sym}^{\beta_k} E$.) Let $K_k$ be the union of Type I potential components of the form $V^{D,g}_m(\alpha+e_k, \beta-e_k, \Gamma')$ as described in Theorem \ref{list}. \begin{pr} \label{multItm} The multiplicity of $H_q$ along $K_k$ is $k$. \end{pr} \noindent {\em Proof. } Let $U$ be the open subvariety of $V^{D,g}_m\abG$ where $\pi^* E$ contains no components of $C$. Then there is a rational map $$ r: V^{D,g}_m\abG \dashrightarrow \operatorname{Sym}^{D \cdot E} E $$ that is a morphism on $U$. Each component of $K_k$ intersects $U$, and $r^* H'_q = H_q$ as Cartier divisors and $r^* v(\alpha+e_k,\beta-e_k,\Gamma_k) = K_k \cap U$ as Weil divisors, so by (\ref{babybabybaby}) the result follows. \qed \vspace{+10pt} \section{Multiplicity for Type II Intersection Components} \label{multII} We translate the corresponding argument in [CH3] from the language of the Hilbert scheme to that of maps. \subsection{Versal deformation spaces of tacnodes} We first recall facts about versal deformation spaces of tacnodes. (This background is taken from [CH3], Section 4, and the reader is referred there for details.) Let $(C,p)$ be an $m^{\text{th}}$ order tacnode, that is, a curve singularity analytically equivalent to the origin in the plane curve given by the equation $y (y+x^m) = 0$. The Jacobian ideal ${\mathcal{J}}$ of $y^2+yx^m$ is $(2y+x^m, yx^{m-1})$, and the monomials $1, x, \dots, x^{m-1}$, $y, xy, \dots, x^{m-2} y$ form a basis for the vector space ${\mathcal{O}} / {\mathcal{J}}$ (see [A]). We can thus describe the versal deformation of $(C,p)$ space explicitly. The base is an \'{e}tale neighborhood of the origin in $\mathbb A^{2m-1}$ with co-ordinates $a_0, \dots, a_{m-2}$, and $b_0, \dots, b_{m-1}$, and the deformation space ${\mathcal{S}} \rightarrow \Delta$ is given by $$ y^2 + yx^m + a_0 y + a_1 xy + \dots + a_{m-2} x^{m-2} y + b_0 + b_1 x + \dots + b_{m-1} x^{m-1} = 0. $$ Call this polynomial $f(x,y,a_0, a_1,\dots,a_{m-2},b_0,b_1,\dots,b_{m-1})$. There are two loci in $\Delta$ of interest to us. Let $\Delta_m \subset \Delta$ be the closure of the locus representing a curve with $m$ nodes. This is equivalent to requiring that the discriminant of $f$, as a function of $y$, have $m$ double roots as a function of $x$: $$ (x^m+ a_{m-2} x^{m-2} + \dots + a_1 x + a_0)^2 - 4 (b_{m-1}x^{m-1} + \dots + b_1 x + b_0) = 0 $$ must have $m$ double roots. Thus $\Delta_m$ is given by the equations $b_0=\dots=b_{m-1}=0$; it is smooth of dimension $m-1$. (The locus $\Delta_m \subset \Delta$ corresponds to locally reducible curves.) Let $\Delta_{m-1} \subset \Delta$ be the closure of the locus representing a curve with $m-1$ nodes. This is equivalent to the discriminant being expressible as $$ (x^{m-1} + \lambda_{m-2} x^{m-2} + \dots + \lambda_1 x + \lambda_0)^2 (x^2 + \mu_1 x + \mu_0). $$ From this description, we can see that $\Delta_{m-1}$ is irreducible of dimension $m$, smooth away from $\Delta_m$, with $m$ sheets of $\Delta_{m-1}$ crossing transversely at a general point of $\Delta_m$. Let $m_1$, $m_2$, \dots be any sequence of positive integers, and $(C_j, p_j)$ be an $(m_j)^{\text{th}}$ order tacnode. Denote the versal deformation space of $(C_j,p_j)$ by $\Delta_j$, and let $(a_{j,m_j-2}, \dots, a_{j,0}, b_{j,m_j-1}, \dots, b_{j,0})$ be coordinates on $\Delta_j$ as above. For each $j$, let $\Delta_{j,m_j}$ and $\Delta_{j,m_j-1} \subset \Delta_j$ be as above the closures of loci in $\Delta_j$ over which the fibers of $\pi_j$ have $m_j$ and $m_j-1$ nodes respectively. Finally, set $$ \Delta = \Delta_1 \times \Delta_2 \times \dots, $$ $$ \Delta_m = \Delta_{1,m_1} \times \Delta_{2,m_2} \times \dots, $$ $$ \Delta_{m-1} = \Delta_{1,m_1-1} \times \Delta_{2,m_2-1} \times \dots . $$ Note that $\Delta$, $\Delta_m$ and $\Delta_{m-1}$ have dimensions $\sum (2 m_j - 1)$, $\sum (m_j - 1)$ and $\sum m_j$ respectively. Let $W \subset \Delta$ be a smooth subvariety of dimension $\sum (m_j-1) + 1$, containing the linear space $\Delta_m$. Suppose that the tangent plane to $W$ is not contained in the union of hyperplanes $\cup_j \{ b_{j,0}=0 \} \subset \Delta$. Let $\kappa := \prod m_j / \operatorname{lcm}(m_j)$. Then: \begin{lm} \label{tacnode} With the hypotheses above, in an \'{e}tale neighborhood of the origin in $\Delta$, $$ W \cap \Delta_{m-1} = \Delta_m \cup \Gamma_1 \cup \Gamma_2 \cup \dots \cup \Gamma_{\kappa} $$ where $\Gamma_1$, \dots, $\Gamma_{\kappa} \subset W$ are distinct reduced unibranch curves having intersection multiplicity exactly $\operatorname{lcm}(m_j)$ with $\Delta_m$ at the origin. \end{lm} This lemma arose in conversations with J. Harris, and appears (with proof) as part of [CH3] Lemma 4.3. The key ingredient is the special case of a single tacnode, which is [CH1] Lemma 2.14. Results of a similar flavor appear in [V2] Section 1. \subsection{Calculating the multiplicity} Suppose $K = K(\alpha'', \beta, \gamma, \Gamma'')$ is a Type II component of $H_q$ (on $V^{D,g}_m\abG$). Assume that the $\{ p_{i,j} \}$ are distinct. (However, the arguments carry through without change so long as $\{ p'_{i,j} \} = \Gamma \setminus \Gamma'$ are distinct; this will be useful in [V3].) Let $m_1$, \dots, $m_{|\gamma|}$ be a set of positive integers with $j$ appearing $\gamma_j$ times ($j = 1$, 2, \dots), so $\sum m_i = I \gamma$. \begin{pr} \label{multIItm} The multiplicity of $H_q$ along $K$ is $m_1 \dots m_{|\gamma|} = I^{\gamma}$. \end{pr} The proof of this proposition will take up the rest of this section. Fix general points $s_1$, \dots, $s_{\Upsilon-1}$ on $X$, and let $H_i$ be the divisor on $V^{D,g}_m\ab$ corresponding to requiring the image curve to pass through $s_i$. By Kleiman-Bertini, the intersection of $V^{D,g}_m\ab$ with $\prod H_i$ is a curve $V$ and the intersection of $K$ with $\prod H_i$ is a finite set of points (non-empty as $K$ has intersection dimension $\Upsilon -1$). Choose a point $(C, \pi)$ of $K \cap H_1 \cap \dots \cap H_{\Upsilon - 1}$. The multiplicity of $H_q$ along $K$ on $V^{D,g}_m\ab$ is the multiplicity of $H_q$ at the point $(C,\pi)$ on the curve $V$. For such $(C,\pi)$ in $K(\alpha'',\beta,\gamma,\Gamma'')$ there are unique choices of points $\{ q_{i,j} \}$, $\{ r_{i,j}\}$ on $C$ (up to permutations of $\{ r_{i,j} \}$ for fixed $i$): if $C = C' \cup C''$ (with $C'$ mapping isomorphically to $E$ and $\pi^{-1} E$ containing no components of $C''$), then the condition $(\pi |_{C''})^* E = \sum i \cdot q''_{i,j} + \sum i \cdot r_{i,j}$ with $\pi(q_{i,j}'') = p''_{i,j}$ specifies the points $\{ q''_{i,j} \}$, $\{ r_{i,j} \}$, and $q'_{i,j} = (\pi|_{C'})^{-1} p'_{i,j}$ specifies the points $\{ q'_{i,j} \} = \{ q_{i,j} \} \setminus \{ q''_{i,j} \}$. Define the map $(\tilde{C}, \tilde{\pi})$ as follows: $C \stackrel{\pi}{\rightarrow} X$ factors through $$ C \stackrel{\nu}{\rightarrow} \tilde{C} \stackrel{\tilde{\pi}}{\rightarrow} X. $$ where $\nu$ is a homeomorphism and $\tilde{\pi}$ is an immersion. Each node of $C$ is mapped to a tacnode (of some order) of $\tilde{C}$, and $\nu: C \rightarrow \tilde{C}$ is a partial normalization. Then $\tilde{C}$ has arithmetic genus $\tilde{g} := g + \sum (m_i - 1)$. Let $\operatorname{Def}(\tilde{C},\tilde{\pi})$ be the deformations of $(\tilde{C},\tilde{\pi})$ preserving the incidences through $s_1$, \dots, $s_{\Upsilon-1}$ and the tangencies ($\tilde{\pi}^* E = \sum i \cdot q_{i,j} + \sum i \cdot r_{i,j}$, $\tilde{\pi}(q_{i,j}) = p_{i,j}$). \begin{lm} The space $\operatorname{Def} (\tilde{C},\tilde{\pi})$ is smooth of dimension $\sum( m_j - 1) +1$. \end{lm} \noindent {\em Proof. } We will show the equivalent result: the vector space of first-order deformations of $(\tilde{C}, \tilde{\pi})$ preserving the tangency conditions (but not necessarily the incidence conditions $s_1$, \dots, $s_{\Upsilon- 1}$) has dimension $\Upsilon + \sum( m_i - 1)$, and they are unobstructed. As $(\tilde{C}, \tilde{\pi})$ is an immersion, there is a normal bundle to $\tilde{\pi}$ $$ N_{\tilde{C} / X} = {\mathcal{O}}_{\tilde{C}}( - \tilde{\pi}^* K_X + K_{\tilde{C}}). $$ By property P3, as $\tilde{\pi}^*( K_X + E - \sum r_{i,j})$ is negative on every component of $\tilde{C}$, \begin{equation} \label{unobstructed} h^1( \tilde{C}, N_{\tilde{C} / X} ( - \sum i \cdot q_{i,j} - \sum (i-1) \cdot r_{i,j})) = 0 \end{equation} so \begin{eqnarray*} & & h^0( \tilde{C}, N_{\tilde{C} / X} ( - \sum i \cdot q_{i,j} - \sum (i-1) \cdot r_{i,j})) \\ &=& \chi( \tilde{C}, N_{\tilde{C} / X} ( - \sum i \cdot q_{i,j} - \sum (i-1) \cdot r_{i,j})) \\ &=& \deg( \tilde{\pi}^*(-K_X - E + \sum r_{i,j})) + \deg K_{\tilde{C}} - \tilde{g} + 1 \\ &=& - (K_X + E) \cdot D + | \beta | + \tilde{g} - 1 \\ &=& - (K_X + E) \cdot D + | \beta | + g + \sum (m_i - 1) - 1 \\ &=& \Upsilon + \sum ( m_i - 1). \end{eqnarray*} Thus there are $\Upsilon + \sum (m_i - 1)$ first-order deformations, and by (\ref{unobstructed}) they are unobstructed. \qed \vspace{+10pt} For convenience, let $N := N_{\tilde{C}/X} (- \sum i \cdot q_{i,j} - \sum (i-1) \cdot r_{i,j})$. By the proof of the above lemma, $H^0( \tilde{C}, N)$ is naturally the tangent space to $\operatorname{Def}(\tilde{C}, \tilde{\pi})$. Now $-K_X$ restricted to $C'$ has degree $K_X \cdot E = 2 + E^2$; $K_{\tilde{C}}$ restricted to $C'$ has degree $I \gamma - 2$, which is $(\deg K_{C'})$ plus the length of the scheme-theoretic intersection of $C'$ and $C''$; and $- \sum i \cdot q'_{i,j}$ has degree $I \alpha'$. Therefore \begin{eqnarray*} \deg N|_{C'} &=& 2+ E^2 + I \gamma - 2 - I \alpha' \\ &=& D \cdot E - (D-E) \cdot E + I \gamma - I \alpha' \\ &=& (I \alpha + I \beta) - (I \alpha'' + I \beta + I \gamma) + I \gamma - I \alpha' \\ &=& 0 \end{eqnarray*} so the restriction of $N$ to $C'$ is the trivial line bundle. Also, if $q$ is a general point on $C'$ then $h^0(\tilde{C},N(-p)) = h^0(\tilde{C},N) - 1$. ({\em Proof:} From (\ref{unobstructed}), $h^1(\tilde{C},N) = 0$. By the same argument, as $\deg (K_X+E)|_E = -2$, $\tilde{\pi}^* ( K_X + E - \sum r_{i,j} + q)$ is negative on every component of $\tilde{C}$, so $h^1(\tilde{C}, N(-p)) = 0$. Thus $h^0(\tilde{C},N(-p)) - h^0(\tilde{C},N) = \chi(\tilde{C},N(-p)) - \chi(\tilde{C},N) = -1$.) Thus there is a section of $N$ that is nonzero on $C'$. Let ${\mathcal{J}}$ be the Jacobian ideal of $\tilde{C}$. In an \'{e}tale neighborhood of the $(C,\pi)$, there are natural maps $$ V \stackrel{\rho}{\rightarrow} \operatorname{Def} (\tilde{C},\tilde{\pi}) \stackrel{\sigma}{\rightarrow} \Delta $$ where the differential of $\sigma$ is given by the natural map \begin{equation} \label{differential} H^0(\tilde{C},N) \rightarrow H^0(\tilde{C},N \otimes ({\mathcal{O}}_{\tilde{C}} / {\mathcal{J}})). \end{equation} \begin{lm} In a neighborhood of the origin, the morphism $$ \sigma: \operatorname{Def}(\tilde{C}, \tilde{\pi}) \rightarrow \Delta $$ is an immersion, and the tangent space to $\sigma ( \operatorname{Def}(\tilde{C}, \tilde{\pi}))$ contains $\Delta_m$ and is not contained in the union of hyperplanes $\cup_j \{ b_{j,0} = 0 \}$. \end{lm} \noindent {\em Proof. } From (\ref{differential}), the Zariski tangent space to the divisor $\sigma^*( b_{j,0} = 0)$ is a subspace $Z$ of $H^0(\tilde{C},N)$ vanishing at a point of $C'$ (the $j^{\text{th}}$ tacnode). But $N |_{C'}$ is a trivial bundle, so this subspace of sections $Z$ must vanish on all of $C'$. As there is a section of $N$ that is non-zero on $C'$, $Z$ has dimension at most $h^0(\tilde{C},N) - 1 = \dim \operatorname{Def}(\tilde{C},\pi) - 1$. This proves that $\sigma$ is an immersion, and that the tangent space to $\sigma(\operatorname{Def}(\tilde{C},\tilde{\pi}))$ is not contained in $\{ b_{j,0} = 0 \}$. Finally, if $S$ is the divisor (on $\operatorname{Def}(\tilde{C},\tilde{\pi})$) corresponding to requiring the image curve to pass through a fixed general point of $E$, then $\sigma(S) \subset \Delta_m$, as the image curve must be reducible. As $\sigma$ is an immersion, \begin{eqnarray} \nonumber \sum ( m_i - 1) &=& \dim \operatorname{Def}(\tilde{C},\tilde{\pi}) - 1 \\ \nonumber &=& \dim S \\ \nonumber &=& \dim \sigma(S) \\ \label{bob} &\leq& \dim \Delta_m \\ \nonumber &=& \sum (m_i - 1) \end{eqnarray} so we must have equality at (\ref{bob}), and the linear space $\Delta_m= \sigma(S)$ is contained in $\sigma(\operatorname{Def}(\tilde{C},\tilde{\pi}))$, and thus in the tangent space to $\sigma(\operatorname{Def}(\tilde{C},\tilde{\pi}))$. \qed \vspace{+10pt} Thus the image $\sigma( \operatorname{Def}(\tilde{C},\tilde{\pi}))$ satisfies the hypotheses of Lemma \ref{tacnode}, so the closure of the inverse image $\sigma^{-1}( \Delta_{m-1} \setminus \Delta_m)$ will have $\prod m_i / \operatorname{lcm}(m_i)$ reduced branches, each having intersection multiplicity $\operatorname{lcm}(m_i)$ with $\sigma^{-1} ( \Delta_m)$ and hence with the hyperplane $H_q$. Since in a neighborhood of $(C, \pi)$ the variety $V$ is a curve birational with $\rho(V) = \overline{ \sigma^{-1} ( \Delta_{m-1} \setminus \Delta_m)}$, we conclude that the divisor $H_q$ contains $K(\alpha'', \beta, \gamma, \Gamma'')$ with multiplicity $m_1 \cdots m_{| \gamma| } = I^{\gamma}$. This completes the proof of Proposition \ref{multIItm}. As an added benefit, we see that $V^{D,g}_m\ab$ has $I^{\gamma} / \operatorname{lcm}(\gamma)$ branches at a general point of $K(\alpha'', \beta, \gamma, \Gamma'')$. \section{The Recursive Formulas} \label{recursivesection} We now collect what we know and derive a recursive formula for the degree of a generalized Severi variety. Fix $D$, $g$, $\alpha$, $\beta$, $\Gamma$ so that $\Upsilon>0$ (e.g. $(D,g,\beta) \neq (kF, 1-k,\vec 0)$ when $X={\mathbb F}_n$). Assume throughout this section that $\Gamma$ consists of distinct points. Let $H_q$ be the divisor on $V^{D,g}_m(\alpha,\beta,\Gamma)$ corresponding to requiring the image to contain a general point $q$ of $E$. The components of $H_q$ of intersection dimension $\Upsilon - 1$ were determined in Theorem \ref{list}, and the multiplicities were determined in Propositions \ref{multItm} and \ref{multIItm}: \begin{pr} In the Chow ring of $V^{D,g}_m\abG$, modulo Weil divisors of intersection dimension less than $\Upsilon - 1$, $$ H_q = \sum_{\beta_k>0} k \cdot V^{D,g}_m(\alpha+e_k,\beta-e_k,\Gamma \cup \{ q \}) + \sum I^{\gamma} \cdot K( \alpha'', \beta, \gamma, \Gamma'') $$ where the second sum is over all $\alpha'' \leq \alpha$, $\Gamma'' = \{ p''_{i,j} \}_{1 \leq j \leq \alpha''_i} \subset \Gamma$, $\gamma \geq 0$, $I(\alpha'' + \beta + \gamma ) = (D-E) \cdot E$. \end{pr} Intersect both sides of the equation with $H_q^{\Upsilon - 1}$. As those dimension $\Upsilon - 1$ classes of intersection dimension less than $\Upsilon - 1$ are annihilated by $H_q^{\Upsilon - 1}$, we still have equality: \begin{eqnarray*} N^{D,g}\abG &=& H_q^{\Upsilon} \\ &=& \sum_{\beta_k>0} k V^{D,g}_m(\alpha+e_k,\beta-e_k,\Gamma \cup \{ q \}) \cdot H_q^{\Upsilon - 1}\\ & & + \sum I^{\gamma} \cdot K( \alpha'', \beta, \gamma, \Gamma'') \cdot H_q^{\Upsilon - 1}. \end{eqnarray*} As remarked in Section \ref{ipc}, each $K(\alpha'', \beta, \gamma, \Gamma'')$ admits a degree $\binom {\beta + \gamma} \beta$ rational map to $V^{D-E,g''}_m(\alpha'', \beta+\gamma,\Gamma'')$ (where $g'' = g- |\gamma| + 1$) corresponding to ``forgetting the component mapping to $E$'', so $$ K(\alpha'', \beta, \gamma, \Gamma'') \cdot H_q^{\Upsilon - 1} = \binom {\beta + \gamma} \gamma N^{D-E,g''}(\alpha'', \beta+\gamma,\Gamma''). $$ Therefore \begin{eqnarray*} N^{D,g}\abG &=& \sum_{\beta_k>0} k V^{D,g}_m(\alpha+e_k,\beta-e_k,\Gamma \cup \{ q \}) \cdot H_q^{\Upsilon - 1}\\ & & + \sum I^{\gamma} \cdot \binom {\beta + \gamma} \gamma N^{D-E,g''}(\alpha'', \beta+\gamma,\Gamma''). \end{eqnarray*} Using this formula inductively, one sees that $N^{D,g}\abG$ is independent of $\Gamma$ (so long as the $\{ p_{i,j} \}$ are distinct). For each $\alpha''$, there are $\binom \alpha {\alpha''}$ choices of $\Gamma''$ (as this is the number of ways of choosing $\{ p''_{i,1} , \dots, p''_{i,\alpha''_i} \}$ from $\{ p_{i,1} , \dots, p_{i,\alpha_i} \}$). Thus \begin{eqnarray*} N^{D,g}\ab &=& \sum_{\beta_k > 0} k N^{D,g}(\alpha + e_k, \beta-e_k) \\ & & + \sum I^{\gamma} {\binom \alpha {\alpha''}} \binom {\beta + \gamma}{\beta} N^{D-E,g''}(\alpha'',\beta + \gamma). \end{eqnarray*} Renaming variables $\alpha' := \alpha''$, $\beta' := \beta + \gamma$, $g' := g''$, this is Theorem \ref{recursion}: \noindent {\bf Theorem \ref{recursion}.} {\em If $\dim V^{D,g}(\alpha,\beta)>0$, then \begin{eqnarray*} N^{D,g}\ab = \sum_{\beta_k > 0} k N^{D,g}(\alpha + e_k, \beta-e_k) \\ + \sum I^{\beta'-\beta} {\binom \alpha {\alpha'}} \binom {\beta'}{\beta} N^{D-E,g'}(\alpha',\beta') \end{eqnarray*} where the second sum is taken over all $\alpha'$, $\beta'$, $g'$ satisfying $\alpha' \leq \alpha$, $\beta' \geq \beta$, $g-g' = |\beta'-\beta| - 1$, $I \alpha' + I \beta' = (D-E) \cdot E$.} The numbers $N^{D,g}\ab$ can be easily inductively calculated. If $\Upsilon=0$, then a short calculation shows that $\beta=0$, $D=kF$, and $g=1-k$, so $N^{D,g}\ab$ is 1 is $\alpha=ke_1$ and 0 otherwise. If $\Upsilon>0$, then $N^{D,g}\ab$ can be calculated using Theorem \ref{recursion}. \subsection{Theorem \ref{recursion} as a differential equation} Define the generating function $$ G = \sum_{D,g,\alpha,\beta} N^{D,g}\ab v^D w^{g-1} \left( \frac {x^{\alpha}} {\alpha!} \right) y^{\beta} \left( \frac {z^{\Upsilon}} {\Upsilon!} \right) $$ (where $w$ and $z$ are variables, $x=(x_1, x_2, \dots)$, $y = (y_1, y_2, \dots)$, and $\{ v^D \}_{D \text{ effective}, D \neq E}$ is a semigroup algebra) Then Theorem \ref{recursion} is equivalent to the differential equation \begin{equation} \label{diffeq} \frac {\partial G} {\partial z} = \left( \sum k y_k \frac {\partial} {\partial x_k} + \frac {v^E} w e^{ \sum (x_k + k w \frac \partial {\partial y_k})} \right) G. \end{equation} The corresponding observation for the plane is due to E. Getzler (cf. [Ge] Subsection 6.3), and nothing essentially new is involved here. The notation is slightly different from Getzler's; the introduction of a variable $w$ corresponding to the arithmetic genus avoids the use of a residue. Define the generating function $$ G_{\operatorname{irr}} = \sum_{D,g,\alpha,\beta} {N^{D,g}_{\operatorname{irr}}\ab} v^D w^{g-1} \left( \frac {x^{\alpha}} {\alpha!} \right) y^{\beta} \left( \frac {z^{\Upsilon}} {\Upsilon!} \right). $$ Then by a simple combinatorial argument, $$ G = e^{G_{\operatorname{irr}}}. $$ Substituting this into (\ref{diffeq}) yields a differential equation satisfied by $G_{\operatorname{irr}}$: \begin{equation} \label{diffeq2} \frac {\partial G_{\operatorname{irr}}} {\partial z} = \sum k y_k \frac {\partial} {\partial x_k} G_{\operatorname{irr}} + \frac {v^E} w e^{ \sum (x_k + {G}_{\operatorname{irr}} \mid_{y_k \mapsto y_k + kw}) - G_{\operatorname{irr}} } \end{equation} where ${G}_{\operatorname{irr}} \mid_{y_k \mapsto y_k+kw}$ is the same as $G_{\operatorname{irr}}$ except $y_k$ has been replaced by $(y_k + k w)$. (Once again, this should be compared with Getzler's formula in [Ge].) However, $N_{\operatorname{irr}}^{D,g}\ab$ can also be calculated directly: \noindent {\bf Theorem \ref{irecursion}.} {\em If $\dim W^{D,g}\ab>0$, then \begin{eqnarray*} N_{\operatorname{irr}}^{D,g}\ab &=& \sum_{\beta_k > 0} k N_{\operatorname{irr}}^{D,g}(\alpha + e_k, \beta - e_k) \\ & & + \sum \frac 1 \sigma \binom {\alpha} {\alpha^1, \dots, \alpha^l, \alpha- \sum \alpha^i} \binom {\Upsilon^{D,g}(\beta)-1} {\Upsilon^{D^1,g^1}(\beta^1), \dots, \Upsilon^{D^l,g^l}(\beta^l)} \\ & & \cdot \prod_{i=1}^l \binom {\beta^i} {\gamma^i} I^{\beta^i - \gamma^i} N_{\operatorname{irr}}^{D^i,g^i}(\alpha^i,\beta^i) \end{eqnarray*} where the second sum runs over choices of $D^i, g^i, \alpha^i, \beta^i, \gamma^i$ ($1 \le i \le l$), where $D^i$ is a divisor class, $g^i$ is a non-negative integer, $\alpha^i$, $\beta^i$, $\gamma^i$ are sequences of non-negative integers, $\sum D^i = D-E$, $\sum \gamma^i = \beta$, $\beta^i \gneq \gamma^i$, and $\sigma$ is the number of symmetries of the set $\{ (D^i,g^i,\alpha^i,\beta^i,\gamma^i) \}_{1 \leq i \leq l}$. } (This recursion is necessarily that produced by the differential equation (\ref{diffeq2}).) The proof is identical, except that rather than considering all maps, we just consider maps from connected curves. The Type I components that can appear are analogous. The Type II components consist of maps from curves $C=C(0) \cup \dots \cup C(l)$ where $C(0)$ maps isomorphically to $E$, and $C(i)$ intersects $C(j)$ if and only if one of $\{i,j\}$ is 0. (In the previous ``possibly reducible'' case, we only required ``$C(i)$ intersects $C(j)$ only if one of $\{ i,j \}$ is 0.'') The numbers $N^{D,g}_{\operatorname{irr}}\ab$ can be easily inductively calculated. If $\Upsilon=0$, then a short calculation shows that $\beta=0$, $D=F$, and $g=0$, so $N^{D,g}_{\operatorname{irr}}\ab$ is 1 is $\alpha=e_1$ and 0 otherwise. If $\Upsilon>0$, then $N^{D,g}_{\operatorname{irr}}\ab$ can be calculated using Theorem \ref{irecursion}. \section{Theorem \ref{irecursion} for $\mathbb F_{n \text{ mod }2}$ computes genus $g$ Gromov-Witten invariants of $\mathbb F_n$} \label{gwenumerative} The results of this section are likely all well-known, but the author was unable to find them in the standard literature. By $\mathbb F_{n \text{ mod }2}$, we mean $\mathbb F_0$ if $n$ is even and $\mathbb F_1$ if $n$ is odd. (Genus $g$) Gromov-Witten invariants were defined by Kontsevich and Manin ([KM] Section 2). We recall their definition, closely following the discussion in [FP] Section 7 of the genus 0 case. The varieties $\overline{M}_{g,n}(X,D)$ come equipped with $n$ morphisms $\rho_1$, \dots, $\rho_n$ to $X$, where $\rho_i$ takes the point $[C, p_1, \dots, p_n, \mu] \in \overline{M}_{g,n}(X,D)$ to the point $\mu(p_i)$ in $X$. Given arbitrary classes $\gamma_1$, \dots, $\gamma_n$ in $A^* X$, we can construct the cohomology class $$ \rho_1^*(\gamma_1) \cup \cdots \cup \rho_n^*(\gamma_n) $$ on $\overline{M}_{g,n}(X,D)$, and we can evaluate its homogeneous component of the top codimension on the virtual fundamental class, to produce a number, call a {\em genus $g$ Gromov-Witten invariant}, that we denote by $I_{g,D}(\gamma_1 \dots \gamma_n)$: $$ I_{g,D}(\gamma_1 \cdots \gamma_n) = \int_{\overline{M}_{0,n}(X,D)} \rho_1^*(\gamma_1) \cup \cdots \cup \rho_n^*(\gamma_n) \cup F $$ where $F$ is the virtual fundamental class. If the classes $\gamma_i$ are homogeneous, this will be nonzero only if the sum of their codimensions is the ``expected dimension'' of $\overline{M}_{g,n}(X,D)$. By variations of the same arguments as in [FP] (p. 35): (I) If $D=0$, $I_{g,D}(\gamma_1 \cdots \gamma_n)$ is non-zero only if \begin{enumerate} \item[i)] $g=0$ and $n=3$, in which case it is $\int_X \gamma_1 \cup \gamma_2 \cup \gamma_3$, or \item[ii)] $g=1$, $n=1$, and $\gamma_1$ is a divisor class, in which case it is $(\gamma_1 \cdot K_X) / 24$. (The author is grateful to T. Graber for pointing out this fact, which is apparently well-known. This second case is the only part of the argument that is not essentially identical to the genus 0 presentation in [FP].) \end{enumerate} (II) If $\gamma_1 = 1 \in A^0 X$, $I_{g,D}(\gamma_1 \cdots \gamma_n)$ is nonzero unless $D = 0$, $g=0$, $n=3$, in which case it is $\int_X \gamma_2 \cup \gamma_3$. (III) If $\gamma_1 \in A^1 X$ and $D \neq 0$, then by the divisorial axiom ([KM] 2.2.4 or [FP] p. 35), $I_{g,D}(\gamma_1 \cdots \gamma_n) = \left( \int_D \gamma_1 \right) \cdot I_{g,D}(\gamma_2 \cdots \gamma_n)$. In light of these three observations, in order to compute the genus $g$ Gromov-Witten invariants for a surface, we need only compute $I_{g,D}(\gamma_1 \cdots \gamma_n)$ when each $\gamma_i$ is the class of a point. For the remainder of this section, we assume this to be the case. Now let $X$ be a Fano surface. The ``expected dimension'' of $\overline{M}_{g,n}(X,D)$ is $-K_X \cdot D + g - 1 + n$ ([KM] Section 2). Thus $I_{g,D}(\gamma_1 \cdots \gamma_n) = 0$ unless $n=-K_X \cdot D + g-1$, and the only components of $\overline{M}_g(X,D)$ contributing to the integral will be those with intersection dimension at least $-K_X \cdot D +g-1$. (On any other component, $\rho_1^*(\gamma_1) \cap \cdots \cap \rho_n^*(\gamma_n) = 0$: there are no curves in such a family passing through $n$ fixed general points.) \begin{lm} \label{gwlemma} Let $X$ be a Fano surface, and let $D$ be an effective divisor class on $X$. Suppose that $M$ is an irreducible component of $\overline{M}_g(X,D)$ with general map $(C,\pi)$. Then $$ \operatorname{idim} M \leq -K_X \cdot D + g-1. $$ If equality holds and $D \neq 0$, $\pi$ is an immersion. \end{lm} \noindent {\em Proof. } Note that if $D$ is an effective divisor that is not 0 or the class of a (-1)-curve, then $-K_X \cdot D > 1$. Assume first that $C$ is smooth and $\pi$ is birational from $C$ to its image. If $D$ is the class of a (-1)-curve, the result is immediate, so assume otherwise. Let $N= \operatorname{coker}(T_C \rightarrow \pi^* T_X)$ be the normal sheaf of $\pi$. Let $N_{\operatorname{tors}}$ be the torsion subsheaf of $N$, so $\pi$ is an immersion if and only if $N_{\operatorname{tors}} = 0$. Let $\det N$ be the determinant line bundle. By [AC] Section 6 or [CH3] Lemma 2.2, $\dim M \leq h^0(C, N/N_{\operatorname{tors}})$. (Caporaso and Harris express this informally as: ``the first-order deformations of the birational map $\pi$ corresponding to a torsion section of $N$ can never be equisingular.'' Arbarello and Cornalba's version, proved earlier, is slightly stronger: ``the first-order deformation of the birational map $\pi$ corresponding to a torsion section of $N$ can never preserve both the order and type of the singularities of the image.'') As $N/N_{\operatorname{tors}}$ is a subsheaf of $\det N$, \begin{eqnarray*} \operatorname{idim} M &\leq & \dim M \\ & \leq & h^0(C, N / N_{\operatorname{tors}}) \\ & \leq & h^0(C, \det N) \\ & = & h^0(C, \omega_C(- \pi^* K_X)) \\ & = & \chi(C, \omega_C(- \pi^* K_X)) \\ &=& \deg(K_C - \pi^* K_X) - g + 1 \\ &=& - K_X \cdot C + g-1 \end{eqnarray*} where equality in the fifth line comes from Kodaira vanishing or Serre duality, as $X$ is Fano. If $N_{\operatorname{tors}} \neq 0$ and $p$ is in the support of $N_{\operatorname{tors}}$, then $N/N_{\operatorname{tors}}$ is actually a subsheaf of $\det N(-p)$. But $-K_X \cdot D > 1$ (as $D \neq 0$, and $D$ is not the class of a (-1)-curve), so $- \pi^* K_X -p$ is positive on $C$, and the same argument gives \begin{eqnarray*} \operatorname{idim} M &\leq & \chi(C, \omega_C(- \pi^* K_X - p)) \\ &=& - K_X \cdot C + g-2. \end{eqnarray*} Thus the lemma is true if $C$ is smooth and $\pi$ is birational. Assume next that $C$ is smooth, but $\pi$ is not birational. If $D=0$, the result is immediate, so assume otherwise. The morphism $\pi$ factors through $$ C \stackrel {\pi_1} \rightarrow C' \stackrel {\pi_2} \rightarrow X $$ where $C'$ is a smooth curve birational to its image under $\pi_2$. Let $d>1$ be the degree of $\pi_1$. If $M'$ is any irreducible component of $\overline{M}_{g(C')}(X, \pi_{2*}[C'] = D/d)$ then, by the case already proved, \begin{eqnarray*} \operatorname{idim} M' &\leq& - K_X \cdot \pi_*[C'] + g(C') - 1 \\ &=& - \frac 1 d K_X \cdot D + g(C')-1. \end{eqnarray*} By Riemann-Hurwitz, $g-1 \geq d(g(C') - 1)$ with equality only if $\pi_1$ is unramified, so \begin{eqnarray*} \operatorname{idim} M & \leq & - \frac 1 d K_X \cdot D + \frac 1 d (g - 1) \\ & \leq & - K_X \cdot D + g - 1 \end{eqnarray*} with equality iff $-K_X \cdot D + g - 1 = 0$ and $\pi_1$ is unramified. As $X$ is Fano and $D$ is effective and non-zero, $-K_X \cdot D + g-1 = 0$ iff $g=0$ and $K_X \cdot D = -1$, i.e. $D$ is an exceptional curve and $C' \cong \mathbb P^1$. But there are no unramified maps to $C'$, so $$ \operatorname{idim} M < - K_X \cdot D + g-1. $$ Thus the lemma is true if $C$ is smooth. If $C$ has irreducible components with normalizations $C_1$, \dots, $C_{l'}$ (where $C_k$ has geometric genus $g_k$, and $\pi_*[C_k] = D_k$) and $C_1$, \dots, $C_l$ are those components that are not contracted rational or elliptic components, then by the smooth case above, \begin{eqnarray*} \operatorname{idim} M &\leq & \sum_{k=1}^l \left( - K_X \cdot D_k + g_k - 1 \right) \\ &=& - K_X \cdot D + \sum_{k=1}^l (g_k -1). \end{eqnarray*} But $\sum_{k=1}^l (g_k - 1) \leq g-1$ with equality if and only if $l=1$, $C_1$ is smooth, and there are no contracted rational or elliptic components. \qed \vspace{+10pt} Thus the genus $g$ Gromov-Witten invariants of ${\mathbb F}_n$ can be computed as follows. By our earlier comments, we need only compute $I_{g,D}(\gamma_1 \cdots \gamma_n)$ where $D$ is effective and nonzero, and the $\gamma_i$ are (general) points. Genus $g$ Gromov-Witten invariants are deformation-invariant ([LT] Theorem 6.1), and ${\mathbb F}_n$ degenerates to $\mathbb F_{n+2}$ with the classes $(E,F)$ on ${\mathbb F}_n$ transforming to $(E+F,F)$ on $\mathbb F_{n+2}$ (well-known; S. Katz has suggested the reference [N], p. 9-10). Hence if $D$ is the class $aE + bF$ on ${\mathbb F}_n$, $I_{g,D}(\gamma_1 \cdots \gamma_n)$ on ${\mathbb F}_n$ is $I_{g,D'}(\gamma_1 \cdots \gamma_n)$ on $\mathbb F_{n \text{ mod }2}$, where $$ D' = \left( a - [n/2] b \right) E + b F. $$ By Lemma \ref{gwlemma}, this is the number of immersed genus $g$ curves in class $D'$ through the appropriate number of points of $\mathbb F_{n \text{ mod }2}$. If $D' = E$, the number is 1. Otherwise, the number is recursively calculated by Theorem \ref{irecursion}.

9709

alg-geom/9709014

fr

https://arxiv.org/abs/alg-geom/9709014

alg-geom/9709014

Jean-Marc Drezet

Jean-Marc Dr\'ezet

Fibr\'es prioritaires g\'en\'eriques instables sur le plan projectif

LaTeX

The structure of the generic prioritary sheaf on the projective plane is given, when it cannot be semi-stable

alg-geom

\section{Introduction} Les faisceaux prioritaires sur \proj{2} ont \'et\'e introduits par A. Hirschowitz et Y. Laszlo dans \cite{hi_la}. Rappelons qu'un faisceau coh\'erent ${\cal E}$ sur \proj{2} est dit {\em prioritaire} s'il est sans torsion et si \ \m{\mathop{\rm Ext}\nolimits^2({\cal E},{\cal E}(-1))=0}. Par exemple les faisceaux semi-stables au sens de Gieseker-Maruyama sont prioritaires. On s'int\'eresse ici \`a la structure pr\'ecise du faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} lorsqu'il n'existe pas de faisceau semi-stable de m\^emes rang et classes de Chern. D'apr\`es \cite{hi_la}, le {\em champ} des faisceaux prioritaires est lisse et irr\'eductible. Les conditions d'existence des faisceaux prioritaires sont les suivantes : posons $$\mu \ = \ \q{c_1}{r}, \ \ \ \Delta \ = \ \q{1}{r}(c_2 - \q{r-1}{2r}c_1^2),$$ (si ${\cal E}$ est un faisceau coh\'erent ${\cal E}$ sur \proj{2} de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}, on appelle $\mu=\mu({\cal E})$ la {\em pente} de ${\cal E}$ et $\Delta=\Delta({\cal E})$ le {\em discriminant} de ${\cal E}$). Alors, si \ \m{-1\leq\mu\leq 0}, il existe un faisceau prioritaire de pente $\mu$ et de discriminant $\Delta$ si et seulement si on a $$\Delta \ \geq \ - \q{\mu(\mu+1)}{2}.$$ Les conditions d'existence des faisceaux semi-stables sur \proj{2} sont rappel\'ees ci-dessous. On peut voir qu'il existe beaucoup de triplets \m{(r,c_1,c_2)} tels qu'il existe un faisceau prioritaire de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} mais pas de faisceau semi-stable avec les m\^emes invariants. Les conditions d'existence des faisceaux semi-stables sur \proj{2} (cf \cite{dr_lp}) s'expriment en fonction des seules variables $\mu$ et $\Delta$. On montre qu'il existe une unique fonction $\delta(\mu)$ telle qu'on ait \ \m{\dim(M(r,c_1,c_2)) > 0} \ si et seulement si \ \m{\Delta\geq\delta(\mu)}. La fonction \m{\delta(\mu)} est d\'ecrite \`a l'aide des {\it fibr\'es exceptionnels}. On dit qu'un faisceau coh\'erent ${\cal E}$ sur \proj{2} est {\it exceptionnel} si ${\cal E}$ est {\it simple} (c'est-\`a-dire si les seuls endomorphismes de ${\cal E}$ sont les homoth\'eties), et si $$\mathop{\rm Ext}\nolimits^1({\cal E},{\cal E}) \ = \ \mathop{\rm Ext}\nolimits^2({\cal E},{\cal E}) \ = \ \lbrace 0\rbrace.$$ Un tel faisceau est alors localement libre et stable, et la vari\'et\'e de modules de faisceaux semi-stables correspondante contient l'unique point ${\cal E}$. Il existe une infinit\'e d\'enombrable de fibr\'es exceptionnels, et un proc\'ed\'e simple permet de les obtenir tous \`a partir des fibr\'es en droites (cf. \cite{dr1}). Notons qu'un fibr\'e exceptionnel est uniquement d\'etermin\'e par sa pente. Soit $F$ un fibr\'e exceptionnel. On note \m{x_F} la plus petite solution de l'\'equation $$X^2-3X+\q{1}{rg(F)^2} \ = \ 0.$$ Alors on montre que les intervalles \ \m{\rbrack\mu(F)-x_F,\mu(F)+x_F\lbrack} \ constituent une partition de l'ensemble des nombres rationnels. On va d\'ecrire la fonction \m{\delta(\mu)} sur cet intervalle. Posons $$P(X) = \q{X^2}{2}+\q{3}{2}X+1.$$ Sur l'intervalle \ \m{\rbrack\mu(F)-x_F,\mu(F)\rbrack}, on a $$\delta(\mu) \ = \ P(\mu-\mu(F))-\q{1}{2}(1-\q{1}{rg(F)^2}),$$ et sur \ \m{\lbrack\mu(F),\mu(F)+x_F\lbrack}, on a $$\delta(\mu) \ = \ P(\mu(F)-\mu)-\q{1}{2}(1-\q{1}{rg(F)^2}).$$ On obtient les courbes $D(F)$ et $G(F)$ repr\'esent\'ees sur la figure qui suit. Ce sont des segments de coniques. On consid\`ere maintenant la courbe \ \m{\Delta=\delta'(\mu)} \ d\'efinie de la fa\c con suivante : sur l'intervalle \ \m{\rbrack\mu(F)-x_F,\mu(F)+x_F\lbrack}, on a $$\delta'(\mu) = \delta(\mu) - \q{1}{rg(F)^2}(1-\q{1}{x_F}\mid\mu(F)-\mu\mid).$$ On obtient ainsi les segments de coniques $D'(F)$ et $G'(F)$. Le point \m{(\mu(F),\delta'(\mu(F)))} est la paire \m{(\mu,\Delta)} correspondant au fibr\'e exceptionnel $F$. Le point \m{(\mu(F),\delta(\mu(F)))} est le sym\'etrique de $F$ par rapport \`a la droite \ \m{\Delta=1/2}. Notons que si $\mu$ est un nombre rationnel diff\'erent de la pente d'un fibr\'e exceptionnel, le nombre $\delta'(\mu)$ est irrationnel. Ces courbes, sur l'intervalle \ \m{\rbrack\mu(F)-x_F,\mu(F)+x_F\lbrack} \ , sont repr\'esent\'ees ci-dessous : \vfill\eject \setlength{\unitlength}{0.012500in}% \begin{picture}(410,565)(200,235) \thicklines \multiput(400,800)(0.00000,-7.98561){70}{\line( 0,-1){ 3.993}} \multiput(610,520)(-7.96117,0.00000){52}{\line(-1, 0){ 3.981}} \put(400,760){\line(-2,-3){160}} \put(400,760){\line( 2,-3){160}} \put(560,520){\line(-2,-3){160}} \put(400,280){\line(-2, 3){160}} \multiput(240,520)(0.00000,-8.00000){33}{\line( 0,-1){ 4.000}} \multiput(560,520)(0.00000,-8.00000){33}{\line( 0,-1){ 4.000}} \put(285,630){\makebox(0,0)[lb]{\smash{$G(F)$}}} \put(495,630){\makebox(0,0)[lb]{\smash{$D(F)$}}} \put(280,395){\makebox(0,0)[lb]{\smash{$G'(F)$}}} \put(485,395){\makebox(0,0)[lb]{\smash{$ D'(F)$}}} \put(410,275){\makebox(0,0)[lb]{\smash{$F$}}} \put(410,760){\makebox(0,0)[lb]{\smash{$P$}}} \put(570,530){\makebox(0,0)[lb]{\smash{ $\Delta=1/2$}}} \put(400,235){\makebox(0,0)[lb]{\smash{$\mu=\mu(F)$}}} \put(215,245){\makebox(0,0)[lb]{\smash{$\mu=\mu(F)-x_F$}}} \put(535,245){\makebox(0,0)[lb]{\smash{$\mu=\mu(F)+x_F$}}} \end{picture} \bigskip \bigskip \bigskip Pour tout point $x$ de \proj{2}, soit \m{{\cal I}_x} le faisceau d'id\'eaux du point $x$. On a $$\mathop{\rm Ext}\nolimits^1({\cal I}_x,{\cal O})\simeq\cx{}.$$ Soit \m{{\cal V}_x} l'unique faisceau extension non triviale de \m{{\cal I}_x} par ${\cal O}$. On va d\'emontrer le \vfill\eject \noindent{\bf Th\'eor\`eme A : }{\em Soient $r$, \m{c_1}, \m{c_2} des entiers, avec \m{r\geq 1}, \m{-1<\mu\leq 0}, $$\Delta \ \geq \ \q{\mu(\mu+1)}{2},$$ et tels que la vari\'et\'e \m{M(r,c_1,c_2)} soit vide. \medskip \noindent 1 - Si \ \m{\Delta < \delta'(\mu)}, il existe des fibr\'es exceptionnels $E_0$, $E_1$, $E_2$, des espaces vectoriels de dimension finie $M_0$, $M_1$, $M_2$, dont un au plus peut \^etre nul, tels que le faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern $c_1$, $c_2$ soit isomorphe \`a $$(E_0\otimes M_0)\oplus(E_1\otimes M_1)\oplus(E_2\otimes M_2).$$ \medskip \noindent 2 - On suppose que \m{c_1\not = 0} ou \m{c_2>1}. Si \ \m{\Delta > \delta'(\mu)}, soit $F$ l'unique fibr\'e exceptionnel tel que \ \m{\mu\in \ \rbrack\mu(F)-x_F,\mu(F)+x_F\lbrack}. Alors si \ \m{\mu\leq\mu(F)}, l'entier $$p \ = \ r.rg(F)(P(\mu-\mu(F))-\Delta-\Delta(F))$$ est strictement positif, et le faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern $c_1$, $c_2$ est isomorphe \`a une somme directe $$(F\otimes \cx{p})\oplus{\cal E},$$ o\`u ${\cal E}$ est un fibr\'e semi-stable situ\'e sur la courbe $G(F)$. De m\^eme, si \ \m{\mu\geq\mu(F)}, l'entier $$p \ = \ r.rg(F)(P(\mu(F)-\mu)-\Delta-\Delta(F))$$ est strictement positif, et le faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern $c_1$, $c_2$ est isomorphe \`a une somme directe $$(F\otimes \cx{p})\oplus{\cal E},$$ o\`u ${\cal E}$ est un fibr\'e semi-stable situ\'e sur la courbe $D(F)$. \medskip \noindent 3 - Si \ \m{c_1=0}, \m{c_2=1}, le faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern $c_1$, $c_2$ est isomorphe \`a une somme directe du type $$({\cal O}\otimes\cx{r-2})\oplus{\cal V}_x.$$} \bigskip Le r\'esultat pr\'ec\'edent apporte des pr\'ecisions sur ce qui est d\'emontr\'e dans \cite{hi_la}, c'est-\`a-dire que s'il n'existe pas des faisceau semi-stable de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}, alors deux cas peuvent se produire : la filtration de Harder-Narasimhan du faisceau prioritaire g\'en\'erique de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} comporte deux termes, ou elle en comporte trois. Dans le premier cas, un des termes est semi-exceptionnel (c'est-\`a-dire de la forme \m{F\otimes\cx{k}}, avec $F$ exceptionnel), et dans le second cas les trois termes sont semi-exceptionnels. \bigskip Le th\'eor\`eme A permet aussi de conclure que s'il n'existe pas de faisceau semi-stable de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}, il n'existe pas non plus d'{\em espaces de modules fins} de faisceaux de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} contenant au moins un faisceau prioritaire. On appelle ici {\it espace de modules fin} de faisceaux de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} sur \proj{2} la donn\'ee d'une vari\'et\'e alg\'ebrique lisse $M$ non vide et d'un faisceau coh\'erent ${\cal F}$ sur \ \m{M\times\proj{2}} poss\'edant les propri\'et\'es suivantes : \medskip \noindent (i) Le faisceau ${\cal F}$ est plat sur $M$ et pour tout point ferm\'e $x$ de $M$, \ \m{{\cal F}_x={\cal F}_{\mid \lbrace x\rbrace\times\proja{{\ttf 2}}}} est un faisceau sans torsion sur \proj{2}, de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}. \noindent (ii) Pour tout point ferm\'e $x$ de $M$, le faisceau ${\cal F}_x$ est simple, on a \ \m{\mathop{\rm Ext}\nolimits^2({\cal F}_x,{\cal F}_x)=\lbrace 0\rbrace}, et le morphisme de d\'eformation infinit\'esimale de Koda\"ira-Spencer $$T_xM\longrightarrow\mathop{\rm Ext}\nolimits^1({\cal F}_x,{\cal F}_x)$$ est surjectif. \noindent (iii) Pour tous point ferm\'es distincts $x$ et $y$ de $M$, les faisceaux \m{{\cal F}_x} et \m{{\cal F}_y} ne sont pas isomorphes. \bigskip Par exemple, si $r$, \m{c_1} et $$\chi \ = \ r - c_2 + \q{c_1(c_1+3)}{2}$$ sont premiers entre eux, et s'il existe un faisceau stable de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}, la vari\'et\'e de modules de ces faisceaux stables, \'equip\'ee d'un {\it faisceau universel}, est un espace de modules fin. Ceci sugg\`ere la conjecture suivante : \bigskip \noindent{\bf Conjecture : }{\it Les seuls espaces de modules fins qui soient projectifs sont les vari\'et\'es de modules de faisceaux stables, lorsque $r$, \m{c_1} et $\chi$ sont premiers entre eux.} \bigskip Le th\'eor\`eme A entraine imm\'ediatement le \bigskip \noindent{\bf Th\'eor\`eme B : }{\em Soient $r$, \m{c_1}, \m{c_2} des entiers avec \ $r\geq 1$. On suppose que la vari\'et\'e modules \m{M(r,c_1,c_2)} des faisceaux semi-stables sur \proj{2} de rang $r$ et de classes de Chern \m{c_1}, \m{c_2} est vide. Alors il n'existe pas d'espace de modules fin de faisceaux de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}, et contenant un faisceau prioritaire.} \bigskip Il est possible de pr\'eciser le 1- du th\'eor\`eme A. On rappelle dans le \paragra~\hskip -2pt 2 la notion de {\em triade}, qui est un triplet particulier \m{(E,F,G)} de fibr\'es exceptionnels. On ne consid\`ere ici que des triades de fibr\'es exceptionnels dont les pentes sont comprises entre $-1$ et $0$. A la triade \m{(E,F,G)} correspond le {\em triangle} \m{{\cal T}_{(E,F,G)}} du plan (de coordonn\'ees \m{(\mu,\Delta)}), dont les c\^ot\'es sont des segments de paraboles et les sommets les points correspondant \`a $E$, $F$ et $G$. Ce triangle est d\'efini par les in\'equations $$\Delta\leq P(\mu-\mu(G))-\Delta(G), \ \ \Delta\geq P(\mu-\mu(H)+3)-\Delta(H), \ \ \Delta\leq P(\mu-\mu(E)+3)-\Delta(E),$$ $H$ \'etant le fibr\'e exceptionnel noyau du morphisme d'\'evaluation \ \m{E\otimes\mathop{\rm Hom}\nolimits(E,F)\longrightarrow F}. Soit {\bf T} l'ensemble des triades de fibr\'es exceptionnels dont les pentes sont comprises entre $-1$ et $0$. Soit ${\cal S}$ l'ensemble des points \m{(\mu,\Delta)} du plan tels que $$-1\leq\mu\leq 0, \ \ -\q{\mu(\mu+1)}{2}\leq\Delta\leq\delta'(\mu).$$ On d\'emontrera le \bigskip \bigskip \noindent{\bf Th\'eor\`eme C : }{\em 1 - Soient \m{(E,F,G)}, \m{(E',F',G')} des \'el\'ements distincts de {\bf T}. Alors les triangles \m{{\cal T}_{(E,F,G)}} et \m{{\cal T}_{(E',F',G')}} ont une intersection non vide si et seulement si cette intersection est un sommet commun ou un c\^ot\'e commun. Dans le premier cas, les fibr\'es exceptionnels correspondants sont identiques, et dans le second les paires de fibr\'es exceptionnels correspondantes le sont. \medskip \noindent 2 - On a \ \ \ \m{\displaystyle {\cal S}\ = \ \bigcup_{(E,F,G)\in{\bf T}}{\cal T}_{(E,F,G)}}. \medskip \noindent 3 - Soient \m{r,c_1,c_2} des entiers, avec \ \m{r\geq 1}, $$\mu=\q{r}{c_1}, \ \ \Delta=\q{1}{r}(c_2-\q{r-1}{2r}c_1^2).$$ On suppose que \ \m{(\mu,\Delta)\in{\cal T}_{(E,F,G)}}. Soit $H$ le noyau du morphisme d'\'evaluation \break \m{E\otimes\mathop{\rm Hom}\nolimits(E,F)\longrightarrow F}. Alors $$m \ = \ r.rg(E).(P(\mu-\mu(E)+3)-\Delta(E)),$$ $$n \ = \ r.rg(H).(P(\mu-\mu(H)+3)-\Delta(H)),$$ $$p \ = \ r.rg(G).(P(\mu-\mu(G))-\Delta(G))$$ sont des entiers positifs ou nuls, et le fibr\'e prioritaire g\'en\'erique de rang $r$ et de classes de Chern $c_1$, $c_2$ est de la forme $$(E\otimes\cx{m})\oplus(F\otimes\cx{n})\oplus(G\otimes\cx{p}).$$ } \bigskip \bigskip \bigskip \noindent{\bf Notations} \medskip Rappelons que le th\'eor\`eme de Riemann-Roch s'\'ecrit pour un faisceau coh\'erent $E$ de rang positif sur \proj{2} $$\chi(E) \ = \ rg(E).(P(\mu(E))-\Delta(E)),$$ \m{\chi(E)} d\'esignant la caract\'erisitique d'Euler-Poincar\'e de $E$. Si $E$, $F$ sont des faisceaux coh\'erents sur \proj{2}, on pose $$\chi(E,F) \ = \ \mathop{\hbox{$\displaystyle\sum$}}\limits_{0\leq i\leq 2}(-1)^i\dim(\mathop{\rm Ext}\nolimits^i(E,F)).$$ On a, si \ \m{rg(E)>0} \ et \ \m{rg(F)>0}, $$\chi(E,F) \ = \ rg(E).rg(E).(P(\mu(F)-\mu(E))-\Delta(E)-\Delta(F)).$$ On a en g\'en\'eral, pour tout entier $i$, un isomorphisme canonique $$\mathop{\rm Ext}\nolimits^i(E,F) \ \simeq \ \mathop{\rm Ext}\nolimits^{2-i}(F,E(-3))$$ (dualit\'e de Serre, cf. \cite{dr_lp}, prop. (1.2)). \section{Fibr\'es exceptionnels} \subsection{Construction des fibr\'es exceptionnels} Les r\'esultats qui suivent ont \'et\'e d\'emontr\'es dans \cite{dr_lp} ou \cite{dr1}. Un fibr\'e exceptionnel est enti\`erement d\'etermin\'e par sa pente. Soit ${\cal P}$ l'ensemble des pentes de fibr\'es exceptionnels. Si \m{\alpha\in{\cal P}}, on note \m{E_\alpha} le fibr\'e exceptionnel de pente $\alpha$, et \m{r_\alpha} son rang. On montre que \m{r_\alpha} et \m{c_1(E_\alpha)} sont premiers entre eux. Soit \ \m{\Delta_\alpha=\Delta(E_\alpha)}. Alors on a $$\Delta_\alpha\ = \ \q{1}{2}(1-\q{1}{r_\alpha^2}),$$ (ce qui d\'ecoule du fait que \ \m{\chi(E_\alpha,E_\alpha)=1}). Soit ${\cal D}$ l'ensemble des nombres rationnels diadiques, c'est-\`a-dire pouvant se mettre sous la forme \m{p/2^q}, $p$ et $q$ \'etant des entiers, \m{q\geq 0}. On a une bijection $$\epsilon : {\cal E}\longrightarrow{\cal P}.$$ Cette application est enti\`erement d\'etermin\'ee par les propri\'et\'es suivantes: \medskip \noindent - Pour tout entier $k$, on a \ \m{\epsilon(k)=k}. \noindent - Pour tout entier $k$ et tout \ \m{x\in{\cal D}}, on a \ \m{\epsilon(x+k)=\epsilon(x)+k}. \noindent - Pour tous entiers $p$, $q$, avec \ \m{q\geq 0}, on a $$\epsilon(\q{2p+1}{2^{q+1}}) \ = \ \epsilon(\q{p}{2^q})\times\epsilon(\q{p+1}{2^q}),$$ o\`u $\times$ est la loi de composition suivante : $$\alpha\times\beta\ = \ \q{\alpha+\beta}{2}+\q{\Delta_\alpha-\Delta_\beta} {3+\alpha-\beta}.$$ Cette relation signifie simplement que $$\chi(E_{\alpha\times\beta},E_\alpha) \ = \ \chi(E_\beta,E_{\alpha\times\beta}) \ = \ 0.$$ \bigskip La construction des pentes des fibr\'es exceptionnels comprises entre $-1$ et $0$ se fait donc en partant des pentes $-1$ et $0$, correspondant aux fibr\'es exceptionnels \m{{\cal O}(-1)} et ${\cal O}$. On appelle {\em triades} les triplets de fibr\'es exceptionnels de la forme \noindent\m{({\cal O}(k),{\cal O}(k+1),{\cal O}(k+2))}, \m{(E_\alpha,E_{\alpha\times\beta},E_\beta)}, \m{(E_{\alpha\times\beta},E_{\beta},E_{\alpha+3})} ou \m{(E_{\beta-3},E_\alpha,E_{\alpha\times\beta})}, \m{\alpha} et \m{\beta} \'etant des \'el\'ements de ${\cal P}$ de la forme $$\alpha \ = \ \epsilon(\q{p}{2^q}), \ \ \ \ \beta \ = \ \epsilon(\q{p+1}{2^q}).$$ o\`u $p$ et $q$ sont deux entiers avec \ \m{q\geq 0}. Les triades sont exactement les {\em bases d'h\'elice} de \cite{go_ru}. On donne maintenant la construction des triades de fibr\'es exceptionnels dont les pentes sont comprises entre \m{-1} et $0$. Ces triades sont du type \m{(E_\alpha,E_{\alpha\times\beta},E_\beta)}. La construction se fait de la fa\c con suivante, par r\'ecurrence : on part de la triade \m{({\cal O}(-1),Q^*,{\cal O})}, o\`u $Q$ est le fibr\'e exceptionnel quotient du morphisme canonique \ \m{{\cal O}(-1)\longrightarrow{\cal O}\otimes H^0({\cal O}(1))^*}. Supposons la triade \m{(E,F,G)} construite. Alors on construit les {\it triades adjacentes} \m{(E,H,F)} et \m{(F,K,G)}. Le fibr\'e $H$ est le noyau du morphisme canonique surjectif $$F\otimes\mathop{\rm Hom}\nolimits(F,G)\longrightarrow G$$ et $K$ est le conoyau du morphisme canonique injectif $$E\longrightarrow F\otimes\mathop{\rm Hom}\nolimits(E,F)^*.$$ De plus, le morphisme canonique $$E\otimes\mathop{\rm Hom}\nolimits(E,H)\longrightarrow H \ \ {\rm \ \ \ (resp. \ } K\longrightarrow G\otimes\mathop{\rm Hom}\nolimits(K,G)^* {\rm \ )}$$ est surjectif (resp. injectif) et son noyau (resp. conoyau) est isomorphe \`a \m{G(-3)} (resp. \m{E(3)}). \subsection{Suite spectrale de Beilinson g\'en\'eralis\'ee} A toute triade \m{(E,G,F)} et \`a tout faisceau coh\'erent ${\cal E}$ sur \proj{2} on associe une suite spectrale \m{E^{p,q}_r} de faisceaux coh\'erents sur \proj{2}, convergeant vers ${\cal E}$ en degr\'e 0 et vers 0 en tout autre degr\'e. Les termes \m{E^{p,q}_1} \'eventuellement non nuls sont $$E^{-2,q}_1\simeq H^q({\cal E}\otimes E^*(-3))\otimes E, \ \ E^{-1,q}_1\simeq H^q({\cal E}\otimes S^*)\otimes G, \ \ E^{0,q}_1\simeq H^q({\cal E}\otimes F^*)\otimes F,$$ $S$ d\'esignant le fibr\'e exceptionnel conoyau du morphisme canonique injectif \noindent\m{G\longrightarrow F\otimes\mathop{\rm Hom}\nolimits(G,F)}. \subsection{S\'erie exceptionnelle associ\'ee \`a un fibr\'e exceptionnel} Soit $F$ un fibr\'e exceptionnel. Les triades comportant $F$ comme terme de droite sont de la forme \m{(G_n,G_{n+1},F)}, o\`u la suite de fibr\'es exceptionnels \m{(G_n)} est enti\`erement d\'etermin\'ee par deux de ses termes cons\'ecutifs, par exemple \m{G_0} et \m{G_1}, par les suites exactes $$0\longrightarrow G_{n-1}\longrightarrow (G_n\otimes\mathop{\rm Hom}\nolimits(G_{n-1},G_n)^*)\simeq (G_n\otimes\mathop{\rm Hom}\nolimits(G_n,G_{n+1})) \longrightarrow G_{n+1}\longrightarrow 0.$$ On appelle \m{(G_n)} la {\it s\'erie exceptionnelle} \`a gauche associ\'ee \`a $F$. Les couples \m{(\mu(G_n),\Delta(G_n))} sont situ\'es sur la conique d'\'equation $$\Delta \ = \ P(\mu(F)-\mu)-\Delta(F),$$ (ce qui traduit le fait que \ \m{\chi(F,G_n)=0}). \bigskip \bigskip \setlength{\unitlength}{0.240900pt} \ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi \begin{picture}(1500,900)(0,0) \font\gnuplot=cmr10 at 10pt \gnuplot \sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% \put(176,877){\usebox{\plotpoint}} \multiput(176.58,872.50)(0.493,-1.250){23}{\rule{0.119pt}{1.085pt}} \multiput(175.17,874.75)(13.000,-29.749){2}{\rule{0.400pt}{0.542pt}} \multiput(189.58,840.16)(0.492,-1.358){21}{\rule{0.119pt}{1.167pt}} \multiput(188.17,842.58)(12.000,-29.579){2}{\rule{0.400pt}{0.583pt}} \multiput(201.58,808.63)(0.493,-1.210){23}{\rule{0.119pt}{1.054pt}} \multiput(200.17,810.81)(13.000,-28.813){2}{\rule{0.400pt}{0.527pt}} \multiput(214.58,777.75)(0.493,-1.171){23}{\rule{0.119pt}{1.023pt}} \multiput(213.17,779.88)(13.000,-27.877){2}{\rule{0.400pt}{0.512pt}} \multiput(227.58,747.75)(0.493,-1.171){23}{\rule{0.119pt}{1.023pt}} \multiput(226.17,749.88)(13.000,-27.877){2}{\rule{0.400pt}{0.512pt}} \multiput(240.58,717.57)(0.492,-1.229){21}{\rule{0.119pt}{1.067pt}} \multiput(239.17,719.79)(12.000,-26.786){2}{\rule{0.400pt}{0.533pt}} \multiput(252.58,688.88)(0.493,-1.131){23}{\rule{0.119pt}{0.992pt}} \multiput(251.17,690.94)(13.000,-26.940){2}{\rule{0.400pt}{0.496pt}} \multiput(265.58,660.14)(0.493,-1.052){23}{\rule{0.119pt}{0.931pt}} \put(268,673){$A$} \multiput(264.17,662.07)(13.000,-25.068){2}{\rule{0.400pt}{0.465pt}} \multiput(278.58,633.14)(0.493,-1.052){23}{\rule{0.119pt}{0.931pt}} \multiput(277.17,635.07)(13.000,-25.068){2}{\rule{0.400pt}{0.465pt}} \multiput(291.58,605.85)(0.492,-1.142){21}{\rule{0.119pt}{1.000pt}} \multiput(290.17,607.92)(12.000,-24.924){2}{\rule{0.400pt}{0.500pt}} \multiput(303.58,579.26)(0.493,-1.012){23}{\rule{0.119pt}{0.900pt}} \multiput(302.17,581.13)(13.000,-24.132){2}{\rule{0.400pt}{0.450pt}} \multiput(316.58,553.39)(0.493,-0.972){23}{\rule{0.119pt}{0.869pt}} \multiput(315.17,555.20)(13.000,-23.196){2}{\rule{0.400pt}{0.435pt}} \multiput(329.58,528.26)(0.492,-1.013){21}{\rule{0.119pt}{0.900pt}} \multiput(328.17,530.13)(12.000,-22.132){2}{\rule{0.400pt}{0.450pt}} \multiput(341.58,504.52)(0.493,-0.933){23}{\rule{0.119pt}{0.838pt}} \multiput(340.17,506.26)(13.000,-22.260){2}{\rule{0.400pt}{0.419pt}} \multiput(354.58,480.65)(0.493,-0.893){23}{\rule{0.119pt}{0.808pt}} \multiput(353.17,482.32)(13.000,-21.324){2}{\rule{0.400pt}{0.404pt}} \multiput(367.58,457.65)(0.493,-0.893){23}{\rule{0.119pt}{0.808pt}} \multiput(366.17,459.32)(13.000,-21.324){2}{\rule{0.400pt}{0.404pt}} \put(368,460){\circle*{20}} \put(383,460){$G_{n-1}$} \multiput(380.58,434.68)(0.492,-0.884){21}{\rule{0.119pt}{0.800pt}} \multiput(379.17,436.34)(12.000,-19.340){2}{\rule{0.400pt}{0.400pt}} \multiput(392.58,413.90)(0.493,-0.814){23}{\rule{0.119pt}{0.746pt}} \multiput(391.17,415.45)(13.000,-19.451){2}{\rule{0.400pt}{0.373pt}} \multiput(405.58,392.90)(0.493,-0.814){23}{\rule{0.119pt}{0.746pt}} \multiput(404.17,394.45)(13.000,-19.451){2}{\rule{0.400pt}{0.373pt}} \multiput(418.58,372.03)(0.493,-0.774){23}{\rule{0.119pt}{0.715pt}} \multiput(417.17,373.52)(13.000,-18.515){2}{\rule{0.400pt}{0.358pt}} \multiput(431.58,351.96)(0.492,-0.798){21}{\rule{0.119pt}{0.733pt}} \multiput(430.17,353.48)(12.000,-17.478){2}{\rule{0.400pt}{0.367pt}} \multiput(443.58,333.29)(0.493,-0.695){23}{\rule{0.119pt}{0.654pt}} \put(445,333){\circle*{20}} \put(445,333){$G_{n}$} \multiput(442.17,334.64)(13.000,-16.643){2}{\rule{0.400pt}{0.327pt}} \multiput(456.58,315.29)(0.493,-0.695){23}{\rule{0.119pt}{0.654pt}} \multiput(455.17,316.64)(13.000,-16.643){2}{\rule{0.400pt}{0.327pt}} \multiput(469.58,297.23)(0.492,-0.712){21}{\rule{0.119pt}{0.667pt}} \multiput(468.17,298.62)(12.000,-15.616){2}{\rule{0.400pt}{0.333pt}} \multiput(481.58,280.41)(0.493,-0.655){23}{\rule{0.119pt}{0.623pt}} \multiput(480.17,281.71)(13.000,-15.707){2}{\rule{0.400pt}{0.312pt}} \multiput(494.58,263.54)(0.493,-0.616){23}{\rule{0.119pt}{0.592pt}} \multiput(493.17,264.77)(13.000,-14.771){2}{\rule{0.400pt}{0.296pt}} \multiput(507.58,247.67)(0.493,-0.576){23}{\rule{0.119pt}{0.562pt}} \multiput(506.17,248.83)(13.000,-13.834){2}{\rule{0.400pt}{0.281pt}} \multiput(520.58,232.65)(0.492,-0.582){21}{\rule{0.119pt}{0.567pt}} \multiput(519.17,233.82)(12.000,-12.824){2}{\rule{0.400pt}{0.283pt}} \multiput(532.58,218.80)(0.493,-0.536){23}{\rule{0.119pt}{0.531pt}} \multiput(531.17,219.90)(13.000,-12.898){2}{\rule{0.400pt}{0.265pt}} \multiput(545.58,204.80)(0.493,-0.536){23}{\rule{0.119pt}{0.531pt}} \multiput(544.17,205.90)(13.000,-12.898){2}{\rule{0.400pt}{0.265pt}} \multiput(558.00,191.92)(0.539,-0.492){21}{\rule{0.533pt}{0.119pt}} \multiput(558.00,192.17)(11.893,-12.000){2}{\rule{0.267pt}{0.400pt}} \put(560,192){\circle*{20}} \put(575,192){$G_{n+1}$} \multiput(571.00,179.92)(0.496,-0.492){21}{\rule{0.500pt}{0.119pt}} \multiput(571.00,180.17)(10.962,-12.000){2}{\rule{0.250pt}{0.400pt}} \multiput(583.00,167.92)(0.590,-0.492){19}{\rule{0.573pt}{0.118pt}} \multiput(583.00,168.17)(11.811,-11.000){2}{\rule{0.286pt}{0.400pt}} \multiput(596.00,156.92)(0.590,-0.492){19}{\rule{0.573pt}{0.118pt}} \multiput(596.00,157.17)(11.811,-11.000){2}{\rule{0.286pt}{0.400pt}} \multiput(609.00,145.92)(0.600,-0.491){17}{\rule{0.580pt}{0.118pt}} \multiput(609.00,146.17)(10.796,-10.000){2}{\rule{0.290pt}{0.400pt}} \multiput(621.00,135.93)(0.728,-0.489){15}{\rule{0.678pt}{0.118pt}} \multiput(621.00,136.17)(11.593,-9.000){2}{\rule{0.339pt}{0.400pt}} \multiput(634.00,126.93)(0.824,-0.488){13}{\rule{0.750pt}{0.117pt}} \multiput(634.00,127.17)(11.443,-8.000){2}{\rule{0.375pt}{0.400pt}} \multiput(647.00,118.93)(0.824,-0.488){13}{\rule{0.750pt}{0.117pt}} \multiput(647.00,119.17)(11.443,-8.000){2}{\rule{0.375pt}{0.400pt}} \multiput(660.00,110.93)(0.758,-0.488){13}{\rule{0.700pt}{0.117pt}} \multiput(660.00,111.17)(10.547,-8.000){2}{\rule{0.350pt}{0.400pt}} \multiput(672.00,102.93)(1.123,-0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(672.00,103.17)(10.994,-6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(685.00,96.93)(1.123,-0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(685.00,97.17)(10.994,-6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(698.00,90.93)(1.123,-0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(698.00,91.17)(10.994,-6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(711.00,84.94)(1.651,-0.468){5}{\rule{1.300pt}{0.113pt}} \multiput(711.00,85.17)(9.302,-4.000){2}{\rule{0.650pt}{0.400pt}} \multiput(723.00,80.94)(1.797,-0.468){5}{\rule{1.400pt}{0.113pt}} \multiput(723.00,81.17)(10.094,-4.000){2}{\rule{0.700pt}{0.400pt}} \multiput(736.00,76.95)(2.695,-0.447){3}{\rule{1.833pt}{0.108pt}} \multiput(736.00,77.17)(9.195,-3.000){2}{\rule{0.917pt}{0.400pt}} \multiput(749.00,73.95)(2.472,-0.447){3}{\rule{1.700pt}{0.108pt}} \multiput(749.00,74.17)(8.472,-3.000){2}{\rule{0.850pt}{0.400pt}} \put(761,70.17){\rule{2.700pt}{0.400pt}} \multiput(761.00,71.17)(7.396,-2.000){2}{\rule{1.350pt}{0.400pt}} \put(774,68.67){\rule{3.132pt}{0.400pt}} \multiput(774.00,69.17)(6.500,-1.000){2}{\rule{1.566pt}{0.400pt}} \put(787,67.67){\rule{3.132pt}{0.400pt}} \multiput(787.00,68.17)(6.500,-1.000){2}{\rule{1.566pt}{0.400pt}} \put(812,67.67){\rule{3.132pt}{0.400pt}} \multiput(812.00,67.17)(6.500,1.000){2}{\rule{1.566pt}{0.400pt}} \put(825,68.67){\rule{3.132pt}{0.400pt}} \multiput(825.00,68.17)(6.500,1.000){2}{\rule{1.566pt}{0.400pt}} \put(838,70.17){\rule{2.700pt}{0.400pt}} \multiput(838.00,69.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}} \multiput(851.00,72.61)(2.472,0.447){3}{\rule{1.700pt}{0.108pt}} \multiput(851.00,71.17)(8.472,3.000){2}{\rule{0.850pt}{0.400pt}} \multiput(863.00,75.61)(2.695,0.447){3}{\rule{1.833pt}{0.108pt}} \multiput(863.00,74.17)(9.195,3.000){2}{\rule{0.917pt}{0.400pt}} \multiput(876.00,78.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}} \multiput(876.00,77.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}} \multiput(889.00,82.60)(1.651,0.468){5}{\rule{1.300pt}{0.113pt}} \multiput(889.00,81.17)(9.302,4.000){2}{\rule{0.650pt}{0.400pt}} \multiput(901.00,86.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(901.00,85.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(914.00,92.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(914.00,91.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(927.00,98.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}} \multiput(927.00,97.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}} \multiput(940.00,104.59)(0.758,0.488){13}{\rule{0.700pt}{0.117pt}} \multiput(940.00,103.17)(10.547,8.000){2}{\rule{0.350pt}{0.400pt}} \multiput(952.00,112.59)(0.824,0.488){13}{\rule{0.750pt}{0.117pt}} \multiput(952.00,111.17)(11.443,8.000){2}{\rule{0.375pt}{0.400pt}} \multiput(965.00,120.59)(0.824,0.488){13}{\rule{0.750pt}{0.117pt}} \multiput(965.00,119.17)(11.443,8.000){2}{\rule{0.375pt}{0.400pt}} \multiput(978.00,128.59)(0.728,0.489){15}{\rule{0.678pt}{0.118pt}} \multiput(978.00,127.17)(11.593,9.000){2}{\rule{0.339pt}{0.400pt}} \multiput(991.00,137.58)(0.600,0.491){17}{\rule{0.580pt}{0.118pt}} \multiput(991.00,136.17)(10.796,10.000){2}{\rule{0.290pt}{0.400pt}} \multiput(1003.00,147.58)(0.590,0.492){19}{\rule{0.573pt}{0.118pt}} \multiput(1003.00,146.17)(11.811,11.000){2}{\rule{0.286pt}{0.400pt}} \multiput(1016.00,158.58)(0.590,0.492){19}{\rule{0.573pt}{0.118pt}} \multiput(1016.00,157.17)(11.811,11.000){2}{\rule{0.286pt}{0.400pt}} \multiput(1029.00,169.58)(0.496,0.492){21}{\rule{0.500pt}{0.119pt}} \multiput(1029.00,168.17)(10.962,12.000){2}{\rule{0.250pt}{0.400pt}} \multiput(1041.00,181.58)(0.539,0.492){21}{\rule{0.533pt}{0.119pt}} \multiput(1041.00,180.17)(11.893,12.000){2}{\rule{0.267pt}{0.400pt}} \multiput(1054.58,193.00)(0.493,0.536){23}{\rule{0.119pt}{0.531pt}} \multiput(1053.17,193.00)(13.000,12.898){2}{\rule{0.400pt}{0.265pt}} \multiput(1067.58,207.00)(0.493,0.536){23}{\rule{0.119pt}{0.531pt}} \multiput(1066.17,207.00)(13.000,12.898){2}{\rule{0.400pt}{0.265pt}} \multiput(1080.58,221.00)(0.492,0.582){21}{\rule{0.119pt}{0.567pt}} \multiput(1079.17,221.00)(12.000,12.824){2}{\rule{0.400pt}{0.283pt}} \multiput(1092.58,235.00)(0.493,0.576){23}{\rule{0.119pt}{0.562pt}} \multiput(1091.17,235.00)(13.000,13.834){2}{\rule{0.400pt}{0.281pt}} \multiput(1105.58,250.00)(0.493,0.616){23}{\rule{0.119pt}{0.592pt}} \multiput(1104.17,250.00)(13.000,14.771){2}{\rule{0.400pt}{0.296pt}} \multiput(1118.58,266.00)(0.493,0.655){23}{\rule{0.119pt}{0.623pt}} \multiput(1117.17,266.00)(13.000,15.707){2}{\rule{0.400pt}{0.312pt}} \multiput(1131.58,283.00)(0.492,0.712){21}{\rule{0.119pt}{0.667pt}} \multiput(1130.17,283.00)(12.000,15.616){2}{\rule{0.400pt}{0.333pt}} \multiput(1143.58,300.00)(0.493,0.695){23}{\rule{0.119pt}{0.654pt}} \multiput(1142.17,300.00)(13.000,16.643){2}{\rule{0.400pt}{0.327pt}} \multiput(1156.58,318.00)(0.493,0.695){23}{\rule{0.119pt}{0.654pt}} \multiput(1155.17,318.00)(13.000,16.643){2}{\rule{0.400pt}{0.327pt}} \multiput(1169.58,336.00)(0.492,0.798){21}{\rule{0.119pt}{0.733pt}} \multiput(1168.17,336.00)(12.000,17.478){2}{\rule{0.400pt}{0.367pt}} \multiput(1181.58,355.00)(0.493,0.774){23}{\rule{0.119pt}{0.715pt}} \multiput(1180.17,355.00)(13.000,18.515){2}{\rule{0.400pt}{0.358pt}} \multiput(1194.58,375.00)(0.493,0.814){23}{\rule{0.119pt}{0.746pt}} \multiput(1193.17,375.00)(13.000,19.451){2}{\rule{0.400pt}{0.373pt}} \multiput(1207.58,396.00)(0.493,0.814){23}{\rule{0.119pt}{0.746pt}} \multiput(1206.17,396.00)(13.000,19.451){2}{\rule{0.400pt}{0.373pt}} \multiput(1220.58,417.00)(0.492,0.884){21}{\rule{0.119pt}{0.800pt}} \multiput(1219.17,417.00)(12.000,19.340){2}{\rule{0.400pt}{0.400pt}} \multiput(1232.58,438.00)(0.493,0.893){23}{\rule{0.119pt}{0.808pt}} \multiput(1231.17,438.00)(13.000,21.324){2}{\rule{0.400pt}{0.404pt}} \multiput(1245.58,461.00)(0.493,0.893){23}{\rule{0.119pt}{0.808pt}} \multiput(1244.17,461.00)(13.000,21.324){2}{\rule{0.400pt}{0.404pt}} \multiput(1258.58,484.00)(0.493,0.933){23}{\rule{0.119pt}{0.838pt}} \multiput(1257.17,484.00)(13.000,22.260){2}{\rule{0.400pt}{0.419pt}} \multiput(1271.58,508.00)(0.492,1.013){21}{\rule{0.119pt}{0.900pt}} \multiput(1270.17,508.00)(12.000,22.132){2}{\rule{0.400pt}{0.450pt}} \multiput(1283.58,532.00)(0.493,0.972){23}{\rule{0.119pt}{0.869pt}} \multiput(1282.17,532.00)(13.000,23.196){2}{\rule{0.400pt}{0.435pt}} \multiput(1296.58,557.00)(0.493,1.012){23}{\rule{0.119pt}{0.900pt}} \multiput(1295.17,557.00)(13.000,24.132){2}{\rule{0.400pt}{0.450pt}} \multiput(1309.58,583.00)(0.492,1.142){21}{\rule{0.119pt}{1.000pt}} \multiput(1308.17,583.00)(12.000,24.924){2}{\rule{0.400pt}{0.500pt}} \multiput(1321.58,610.00)(0.493,1.052){23}{\rule{0.119pt}{0.931pt}} \multiput(1320.17,610.00)(13.000,25.068){2}{\rule{0.400pt}{0.465pt}} \multiput(1334.58,637.00)(0.493,1.052){23}{\rule{0.119pt}{0.931pt}} \multiput(1333.17,637.00)(13.000,25.068){2}{\rule{0.400pt}{0.465pt}} \multiput(1347.58,664.00)(0.493,1.131){23}{\rule{0.119pt}{0.992pt}} \multiput(1346.17,664.00)(13.000,26.940){2}{\rule{0.400pt}{0.496pt}} \put(1310,673){$B$} \multiput(1360.58,693.00)(0.492,1.229){21}{\rule{0.119pt}{1.067pt}} \multiput(1359.17,693.00)(12.000,26.786){2}{\rule{0.400pt}{0.533pt}} \multiput(1372.58,722.00)(0.493,1.171){23}{\rule{0.119pt}{1.023pt}} \multiput(1371.17,722.00)(13.000,27.877){2}{\rule{0.400pt}{0.512pt}} \multiput(1385.58,752.00)(0.493,1.171){23}{\rule{0.119pt}{1.023pt}} \multiput(1384.17,752.00)(13.000,27.877){2}{\rule{0.400pt}{0.512pt}} \multiput(1398.58,782.00)(0.493,1.210){23}{\rule{0.119pt}{1.054pt}} \multiput(1397.17,782.00)(13.000,28.813){2}{\rule{0.400pt}{0.527pt}} \multiput(1411.58,813.00)(0.492,1.358){21}{\rule{0.119pt}{1.167pt}} \multiput(1410.17,813.00)(12.000,29.579){2}{\rule{0.400pt}{0.583pt}} \multiput(1423.58,845.00)(0.493,1.250){23}{\rule{0.119pt}{1.085pt}} \multiput(1422.17,845.00)(13.000,29.749){2}{\rule{0.400pt}{0.542pt}} \put(800.0,68.0){\rule[-0.200pt]{2.891pt}{0.400pt}} \put(176,661){\usebox{\plotpoint}} \put(176.00,661.00){\usebox{\plotpoint}} \put(196.76,661.00){\usebox{\plotpoint}} \multiput(201,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(217.51,661.00){\usebox{\plotpoint}} \put(238.27,661.00){\usebox{\plotpoint}} \multiput(240,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(259.02,661.00){\usebox{\plotpoint}} \multiput(265,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(279.78,661.00){\usebox{\plotpoint}} \put(300.53,661.00){\usebox{\plotpoint}} \multiput(303,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(321.29,661.00){\usebox{\plotpoint}} \multiput(329,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(342.04,661.00){\usebox{\plotpoint}} \put(362.80,661.00){\usebox{\plotpoint}} \multiput(367,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(383.55,661.00){\usebox{\plotpoint}} \put(404.31,661.00){\usebox{\plotpoint}} \multiput(405,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(425.07,661.00){\usebox{\plotpoint}} \multiput(431,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(445.82,661.00){\usebox{\plotpoint}} \put(466.58,661.00){\usebox{\plotpoint}} \multiput(469,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(487.33,661.00){\usebox{\plotpoint}} \multiput(494,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(508.09,661.00){\usebox{\plotpoint}} \put(528.84,661.00){\usebox{\plotpoint}} \multiput(532,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(549.60,661.00){\usebox{\plotpoint}} \put(570.35,661.00){\usebox{\plotpoint}} \multiput(571,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(591.11,661.00){\usebox{\plotpoint}} \multiput(596,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(611.87,661.00){\usebox{\plotpoint}} \put(632.62,661.00){\usebox{\plotpoint}} \multiput(634,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(653.38,661.00){\usebox{\plotpoint}} \multiput(660,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(674.13,661.00){\usebox{\plotpoint}} \put(694.89,661.00){\usebox{\plotpoint}} \multiput(698,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(715.64,661.00){\usebox{\plotpoint}} \multiput(723,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(736.40,661.00){\usebox{\plotpoint}} \put(757.15,661.00){\usebox{\plotpoint}} \multiput(761,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(777.91,661.00){\usebox{\plotpoint}} \put(798.66,661.00){\usebox{\plotpoint}} \multiput(800,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(819.42,661.00){\usebox{\plotpoint}} \multiput(825,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(840.18,661.00){\usebox{\plotpoint}} \put(860.93,661.00){\usebox{\plotpoint}} \multiput(863,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(881.69,661.00){\usebox{\plotpoint}} \multiput(889,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(902.44,661.00){\usebox{\plotpoint}} \put(923.20,661.00){\usebox{\plotpoint}} \multiput(927,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(943.95,661.00){\usebox{\plotpoint}} \put(964.71,661.00){\usebox{\plotpoint}} \multiput(965,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(985.46,661.00){\usebox{\plotpoint}} \multiput(991,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1006.22,661.00){\usebox{\plotpoint}} \put(1026.98,661.00){\usebox{\plotpoint}} \multiput(1029,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1047.73,661.00){\usebox{\plotpoint}} \multiput(1054,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1068.49,661.00){\usebox{\plotpoint}} \put(1089.24,661.00){\usebox{\plotpoint}} \multiput(1092,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1110.00,661.00){\usebox{\plotpoint}} \put(1130.75,661.00){\usebox{\plotpoint}} \multiput(1131,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1151.51,661.00){\usebox{\plotpoint}} \multiput(1156,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1172.26,661.00){\usebox{\plotpoint}} \put(1193.02,661.00){\usebox{\plotpoint}} \multiput(1194,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1213.77,661.00){\usebox{\plotpoint}} \multiput(1220,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1234.53,661.00){\usebox{\plotpoint}} \put(1255.29,661.00){\usebox{\plotpoint}} \multiput(1258,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1276.04,661.00){\usebox{\plotpoint}} \multiput(1283,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1296.80,661.00){\usebox{\plotpoint}} \put(1317.55,661.00){\usebox{\plotpoint}} \multiput(1321,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1338.31,661.00){\usebox{\plotpoint}} \put(1359.06,661.00){\usebox{\plotpoint}} \multiput(1360,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1379.82,661.00){\usebox{\plotpoint}} \multiput(1385,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1400.57,661.00){\usebox{\plotpoint}} \put(1421.33,661.00){\usebox{\plotpoint}} \multiput(1423,661)(20.756,0.000){0}{\usebox{\plotpoint}} \put(1436,661){\usebox{\plotpoint}} \put(1480,650){$\Delta=1/2$} \end{picture} \bigskip Dans la figure ci-dessus, les points $A$ et $B$ sont les intersections de cette conique avec la droite d'\'equation \ \m{\Delta=1/2}. On a $$\lim_{n\rightarrow -\infty}=A \ \ \ {\rm et} \ \ \ \lim_{n\rightarrow\infty}=B.$$ Remarquons que \ \m{\mu(B)-\mu(A)<3}. Si \m{F={\cal O}}, il existe une unique paire \m{(G_n,G_{n+1})} telle que \ \m{\mu(G_{n+1})-\mu(G_n)\geq 1}, c'est \m{({\cal O}(-2),{\cal O}(-1))}. Supposons que \ \m{-1<\mu(F)<0}. Il existe alors une unique triade de la forme \m{(E,F,G)}, avec \ \m{-1\leq\mu(E)<\mu(G)\leq 0}. On en d\'eduit que \m{(G(-3),E)} est une des paires \m{(G_n,G_{n+1})}. On peut supposer que \ \m{(G(-3),E)=(G_0,G_1)}. On a \ \m{\mu(G_1)-\mu(G_0)\geq 2}, et \m{(G_0,G_1)} est l'unique paire \m{(G_n,G_{n+1})} telle que \noindent\m{\mu(G_{n+1})-\mu(G_n)\geq 1}. On l'appelle la paire {\it initiale} de la s\'erie \m{(G_n)}. \bigskip \begin{xlemm} Le fibr\'e vectoriel \m{G_0^*\otimes G_1} est engendr\'e par ses sections globales. \end{xlemm} \noindent{\em D\'emonstration}. D'apr\`es la construction de \m{(G_0,G_1)}, il suffit de prouver le r\'esultat \hbox{suivant :} si \m{(A,B,C)} est une triade de fibr\'es exceptionnels telle que \ \m{\mu(C)-\mu(A)\leq 1}, les fibr\'es \m{B^*\otimes A(3)}, \m{C^*\otimes B(3)} et \m{C^*\otimes A(3)} sont engendr\'es par leurs sections globales. On d\'emontre cela par r\'ecurrence : il faut montrer que si c'est vrai pour une triade, c'est vrai pour les deux triades adjacentes. Supposons que ce soit vrai pour \m{(A,B,C)}. Soient $H$ le noyau du morphisme canonique surjectif $$B\otimes\mathop{\rm Hom}\nolimits(B,C)\longrightarrow C$$ et K le conoyau du morphisme canonique injectif $$A\longrightarrow B\otimes\mathop{\rm Hom}\nolimits(A,B)^*.$$ Il faut montrer que le r\'esultat est vrai pour les triades \m{(A,H,B)} et \m{(B,K,C)}. En consid\'erant la triade {\it duale} \m{(C^*(-1),B^*(-1),A^*(-1))}, on voit qu'il suffit de consid\'erer \m{(A,H,B)}. On a une suite exacte $$0\longrightarrow H\longrightarrow B\otimes\mathop{\rm Hom}\nolimits(B,C)\longrightarrow C\longrightarrow 0.$$ On en d\'eduit un morphisme surjectif $$B^*(3)\otimes A\otimes\mathop{\rm Hom}\nolimits(B,C)^*\longrightarrow H^*(3)\otimes A.$$ Puisque \m{B^*(3)\otimes A} est engendr\'e par ses sections globales (hypoth\`ese de r\'ecurrence), il en est de m\^eme de \m{H^*(3)\otimes A}. On a d'autre part une suite exacte $$0\longrightarrow C(-3)\longrightarrow A\otimes\mathop{\rm Hom}\nolimits(C(-3),A)^*\longrightarrow H\longrightarrow 0,$$ d'o\`u on d\'eduit un morphisme surjectif $$B^*(3)\otimes A\otimes\mathop{\rm Hom}\nolimits(C(-3),A)^*\longrightarrow B^*(3)\otimes H,$$ d'o\`u on d\'eduit que \m{B^*(3)\otimes H} est engendr\'e par ses sections globales. $\Box$ \bigskip \begin{xlemm} Pour tout entier $n$, on a \m{n\geq 1} si et seulement si pour tous entiers $a$, $b$, $c$ positifs ou nuls, le fibr\'e vectoriel $$(G_n\otimes\cx{a})\oplus(G_{n+1}\otimes\cx{b})\oplus(F\otimes\cx{c})$$ est prioritaire. \end{xlemm} \noindent{\em D\'emonstration}. Imm\'ediat. $\Box$ \bigskip On d\'efinit de m\^eme la {\em s\'erie exceptionnelle \`a droite} \m{(H_n)} associ\'ee \`a $F$. On a \break \m{H_n=G_n(3)} pour tout $n$. \subsection{\'Etude de {\bf T}} L'ensemble {\bf T} est construit comme une union croissante de sous-ensembles $$T_0=\lbrace({\cal O}(-1),Q^*,{\cal O})\rbrace\subset T_1\subset\ldots T_n\subset T_{n+1}\subset\ldots$$ $$T=\bigcup_{n\geq 0}T_n,$$ o\`u $T_n$ est l'ensemble des triades \m{(E_\alpha,E_{\alpha\times\beta}, E_\beta)}, $\alpha$, $\beta$ \'etant de la forme $$\alpha=\epsilon(\q{p}{2^n}), \ \ \beta=\epsilon(\q{p+1}{2^n}),$$ avec $p$ entier. Si $n>0$, les triades de \m{T_n\backslash T_{n-1}} forment une suite \m{t_0^{(n)}}, \ldots, \m{t_{2^n-1}^{(n)}}, $$t_i^{(n)}\ = \ (E_{\alpha(\q{i}{2^n})},E_{\alpha(\q{2i+1}{2^{n+1}})}, E_{\alpha(\q{i+1}{2^n})}).$$ On a $$\mu(E_{\alpha(\q{i}{2^n})}) \ < \ \mu(E_{\alpha(\q{2i+1}{2^{n+1}})}) \ < \ \mu(E_{\alpha(\q{i+1}{2^n})}),$$ et dans le plan de coordonn\'ees \m{(\mu,\Delta)}, \m{E_{\alpha(\q{2i+1}{2^{n+1}})}} est situ\'e au dessus de la droite\break \m{E_{\alpha(\q{i}{2^n})}E_{\alpha(\q{i+1}{2^n})}}. \setlength{\unitlength}{0.240900pt} \ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi \begin{picture}(1500,900)(0,0) \font\gnuplot=cmr10 at 10pt \gnuplot \sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% \put(233,68){\usebox{\plotpoint}} \multiput(233.00,68.58)(3.543,0.500){321}{\rule{2.930pt}{0.120pt}} \multiput(233.00,67.17)(1139.919,162.000){2}{\rule{1.465pt}{0.400pt}} \put(233,68){\usebox{\plotpoint}} \multiput(233.00,68.58)(0.591,0.500){967}{\rule{0.573pt}{0.120pt}} \multiput(233.00,67.17)(571.812,485.000){2}{\rule{0.286pt}{0.400pt}} \put(806,553){\usebox{\plotpoint}} \multiput(806.00,551.92)(0.887,-0.500){643}{\rule{0.810pt}{0.120pt}} \multiput(806.00,552.17)(571.320,-323.000){2}{\rule{0.405pt}{0.400pt}} \put(233,68){\usebox{\plotpoint}} \multiput(233,68)(9.029,18.689){20}{\usebox{\plotpoint}} \put(405,424){\usebox{\plotpoint}} \put(405,424){\usebox{\plotpoint}} \multiput(405,424)(19.758,6.356){21}{\usebox{\plotpoint}} \put(806,553){\usebox{\plotpoint}} \put(806,553){\usebox{\plotpoint}} \multiput(806,553)(20.608,-2.467){20}{\usebox{\plotpoint}} \put(1207,505){\usebox{\plotpoint}} \put(1207,505){\usebox{\plotpoint}} \multiput(1207,505)(11.006,-17.597){16}{\usebox{\plotpoint}} \put(1379,230){\usebox{\plotpoint}} \put(800,280){$t_i^{(n)}$} \put(430,330){$t_{2i}^{(n+1)}$} \put(1110,410){$t_{2i+1}^{(n+1)}$} \end{picture} Le segment de conique \m{E_{-1}E_0} de \m{{\cal T}_{(E_{-1},E_{\q{1}{2}},E_0)}} n'est autre que la courbe \ \m{\Delta=-\q{\mu(\mu+1)}{2}}. On en d\'eduit imm\'ediatement le \bigskip \begin{xlemm} Soit \ \m{Z = \bigcup_{(E,F,G)\in{\bf T}}{\cal T}_{(E,F,G)}}. Alors, si \ \m{(\mu,\Delta)\in Z}, on a \ \m{(\mu,\Delta')\in Z} \ si $$-\q{\mu(\mu+1)}{2} \ \leq \ \Delta' \ \leq \ \Delta.$$ \end{xlemm} \section{Fibr\'es prioritaires g\'en\'eriques} \subsection{Cohomologie naturelle} \begin{xlemm} Soient $F$ un fibr\'e exceptionnel, $r$, \m{c_1}, \m{c_2} des entiers tels que \m{r\geq 2},\break \m{\mu(F)-x_F<\mu\leq\mu(F)} \ et \ \m{\Delta=\delta(\mu)}. Alors il existe un fibr\'e vectoriel stable ${\cal E}$ de rang $r$ et de classes de Chern $c_1$, $c_2$, tel que \ \m{\mathop{\rm Ext}\nolimits^1({\cal E},F)=\lbrace 0\rbrace}. \end{xlemm} \noindent{\em D\'emonstration}. On consid\`ere la suite \m{(G_n)} de fibr\'es exceptionnels du \paragra~\hskip -2pt 2. Soient $n$ un entier et ${\cal E}$ un faisceau semi-stable de rang $r$ et de classes de Chern \m{c_1}, \m{c_2}. On pose $$k = \chi({\cal E},F), \ \ \ m_n \ = \ -\chi({\cal E}\otimes G_n^*(-3)),$$ qui sont ind\'ependants de ${\cal E}$. Ces entiers sont positifs : pour le premier, cela d\'ecoule du fait que le point correspondant \`a ${\cal E}$ est situ\'e sous la conique donnant l'\'equation de \m{\delta(\mu)} sur \m{\rbrack\mu(F),\mu(F)+x_F\lbrack}. Pour le second on utilise le fait que \m{H^0({\cal E}\otimes G_n^*(-3))} et \m{H^2({\cal E}\otimes G_n^*(-3))} sont nuls. On consid\`ere les triades \m{(F,G_{p-1}(3),G_p(3))}. Ceci sugg\`ere de trouver ${\cal E}$ comme noyau d'un morphisme surjectif ad\'equat $$\theta : (F\otimes\cx{k})\oplus(G_{p-1}(3)\otimes\cx{m_{p+1}})\longrightarrow G_p(3)\otimes\cx{m_p}.$$ Un tel fibr\'e a en effet les bons rang et classes de Chern, et de plus on a \ \m{\mathop{\rm Ext}\nolimits^1({\cal E},F)=\lbrace 0\rbrace}. Pour montrer que ${\cal E}$ se d\'eforme en fibr\'e stable, il suffit qu'il soit prioritaire, car le champ des faisceaux prioritaires est irr\'eductible (cf. \cite{hi_la}). On prend \ \m{p=1}, c'est-\`a-dire qu'on consid\`ere des morphismes $$(F\otimes\cx{k})\oplus(G_0(3)\otimes\cx{m_2})\longrightarrow G_1(3)\otimes\cx{m_1}.$$ Alors on a \ \m{\mu(G_1(3))-\mu(G_0(3))\geq 1}, donc \m{\mu(G_1(3))-\mu(F) > 1}, et la paire \m{(F,G_1(3))} est initiale dans la s\'erie qui la contient. Ceci entraine que le faisceau des morphismes pr\'ec\'edents est engendr\'e par ses sections globales. Comme \ \m{r\geq 2}, il existe un morphisme $$\theta : (F\otimes\cx{k})\oplus(G_0(3)\otimes\cx{m_2})\longrightarrow G_1(3)\otimes\cx{m_1}$$ qui est surjectif. Soit $${\cal E} \ = \ \ker(\theta).$$ Il reste \`a montrer que ${\cal E}$ est prioritaire, c'est-\`a-dire que \ \m{\mathop{\rm Hom}\nolimits({\cal E},{\cal E}(-2))=\lbrace 0\rbrace}. On a une suite exacte $$0\longrightarrow{\cal E}\longrightarrow (F\otimes\cx{k})\oplus(G_0(3)\otimes\cx{m_2})\longrightarrow G_1(3)\otimes\cx{m_1} \longrightarrow 0,$$ d'o\`u on d\'eduit que $$\mathop{\rm Hom}\nolimits({\cal E},{\cal E}(-2))\ \subset \ (\mathop{\rm Hom}\nolimits({\cal E},F(-2))\otimes\cx{k})\oplus (\mathop{\rm Hom}\nolimits({\cal E},G_0(1))\otimes\cx{m_2}).$$ Il faut montrer que $$\mathop{\rm Hom}\nolimits(({\cal E},F(-2))=\mathop{\rm Hom}\nolimits({\cal E},G_0(1))=\lbrace 0\rbrace.$$ Montrons d'abord que \ \m{\mathop{\rm Hom}\nolimits(({\cal E},F(-2))=\lbrace 0\rbrace}. D'apr\`es la suite exacte pr\'ec\'edente, on a une suite exacte $$(\mathop{\rm Hom}\nolimits(F,F(-2))\otimes\cx{k})\oplus(\mathop{\rm Hom}\nolimits(G_0(3),F(-2))\otimes\cx{m_2})\longrightarrow \mathop{\rm Hom}\nolimits(({\cal E},F(-2)) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \longrightarrow\mathop{\rm Ext}\nolimits^1(G_1(3),F(-2))\otimes\cx{m_1}.$$ On a \ \m{\mathop{\rm Hom}\nolimits(F,F(-2))=\mathop{\rm Hom}\nolimits(G_0(3),F(-2))=\lbrace 0\rbrace}, car \ \m{\mu(G_0(3))>\mu(F)>\mu(F(-2))}. D'autre part, $$\mathop{\rm Ext}\nolimits^1(G_1(3),F(-2))\ \simeq\ \mathop{\rm Ext}\nolimits^1(F(-2),G_1)^*$$ par dualit\'e de Serre. Pour montrer que \ \m{\mathop{\rm Ext}\nolimits^1(F(-2),G_1)= \lbrace 0\rbrace}, il suffit d'apr\`es \cite{dr1} de prouver que \ \m{\mu(F(-2))\leq\mu(G_1)}. Si \ \m{F={\cal O}} \ c'est \'evident car \ \m{G_1={\cal O}(-1)}. Sinon, on a \ \m{\mu(G_1)-\mu(G_0)\geq 2}, et si \ \m{\mu(F(-2))>\mu(G_1)}, on a \ \m{\mu(F)-\mu(G_0)>4}, ce qui est faux car \ \m{\mu(F)-\mu(G_0)<3}. Montrons maintenant que \ \m{\mathop{\rm Hom}\nolimits({\cal E},G_0(1))=\lbrace 0\rbrace}. On a une suite exacte $$(\mathop{\rm Hom}\nolimits(F,G_0(1))\otimes\cx{k})\oplus(\mathop{\rm Hom}\nolimits(G_0(3),G_0(1))\otimes\cx{m_2}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \longrightarrow \mathop{\rm Hom}\nolimits(({\cal E},G_0(1))\longrightarrow\mathop{\rm Ext}\nolimits^1(G_1(3),G_0(1))\otimes\cx{m_1}.$$ On a \ \m{\mathop{\rm Hom}\nolimits(F,G_0(1))=\lbrace 0\rbrace} \ car \ \m{\mu(F)>\mu(G_1)\geq\mu(G_0(1))}, et \m{\mathop{\rm Hom}\nolimits(G_0(3),G_0(1))=\lbrace 0\rbrace}. Il reste \`a prouver que \ \m{\mathop{\rm Ext}\nolimits^1(G_1(3),G_0(1))=\lbrace 0\rbrace}. On a $$\mathop{\rm Ext}\nolimits^1(G_1(3),G_0(1)) \ \simeq \ \mathop{\rm Ext}\nolimits^1(G_0(1),G_1)^* \ = \ \lbrace 0\rbrace$$ d'apr\`es \cite{dr1} et le fait que \ \m{\mu(G_0(1))\leq\mu(G_1)}. $\Box$ \subsection{d\'emonstration du th\'eor\`eme A} Soient $F$ un fibr\'e exceptionnel, $r$, \m{c_1}, \m{c_2} des entiers tels que \m{\mu(F)-x_F<\mu<\mu(F)+x_F}, \m{\Delta<\delta(\mu)} \ et \ \m{(\mu,\Delta)\not=(\mu(F),\Delta(F))}. On peut se limiter au cas o\`u \m{\mu(F)-x_F<\mu\leq\mu(F)}, l'autre cas s'en d\'eduisant par dualit\'e. On a alors $$p \ = \ r.rg(F)(P(\mu-\mu(F))-\Delta-\Delta(F)) \ \ > \ \ 0.$$ Supposons que \ \m{\mu > \delta'(\mu)}. Alors on a \ \m{p. rg(F) < r}. En effet, ceci \'equivaut \`a $$\delta(\mu)-\Delta \ < \ \q{1}{rg(F)^2}$$ (cf. la figure de l'Introduction). Il existe donc des entiers \m{r'}, \m{c'_1}, \m{c'_2}, tels que $r$, \m{c_1} et \m{c_2} soient le rang et le classes de Chern d'une somme directe d'un fibr\'e vectoriel ${\cal U}$ de rang \m{r'} et de classes de Chern \m{c'_1},\m{c'_2} et de \m{F\otimes\cx{p}}. Le point correspondant \`a ${\cal U}$ est situ\'e sur la conique d'\'equation $$\Delta = P(\mu-\mu(F))-\Delta(F)$$ et on a \ \m{\Delta \geq \delta'(\mu)} \ si et seulement si ce point est situ\'e sur le segment \m{G(F)} de la conique. Supposons que \ \m{\Delta \geq \delta'(\mu)} \ et \ \m{r'\geq 2}. Dans ce cas il existe d'apr\'es le lemme 3.1 un fibr\'e stable ${\cal U}$ de rang \m{r'} et de classes de Chern \m{c'_1},\m{c'_2} tel que \ \m{\mathop{\rm Ext}\nolimits^1({\cal U},F)=\lbrace 0\rbrace}. Le fibr\'e $${\cal E} \ = \ (F\otimes\cx{p})\oplus{\cal U}$$ est prioritaire, de rang \m{r} et de classes de Chern \m{c_1},\m{c_2}. Les fibr\'es prioritaires g\'en\'eriques sont de ce type, car les fibr\'es tels que ${\cal E}$ sont d\'efinis par la suite de conditions ouvertes suivante : \medskip \noindent (i) on a \ \m{\mathop{\rm Ext}\nolimits^2(F,{\cal E})=\lbrace 0\rbrace}. \noindent (ii) Le morphisme canonique d'\'evaluation $$ev : F\otimes\cx{p}=F\otimes\mathop{\rm Hom}\nolimits(F,{\cal E})\longrightarrow{\cal E}$$ est injectif. \noindent (iii) Si \ \m{{\cal U}=\mathop{\rm coker}\nolimits(ev)}, ${\cal U}$ est un fibr\'e stable tel que \ \m{\mathop{\rm Ext}\nolimits^1({\cal U},F)=\lbrace 0\rbrace}. \medskip Supposons maintenant que \ \m{r'=1}. Dans ce cas on doit avoir \ \m{F={\cal O}} \ et \m{c_2=1}. Les faisceaux de \m{M(r',c'_1,c'_2)} sont de la forme \m{{\cal I}_x} (id\'eal d'un point $x$ de \proj{2}). On a \ \m{\mathop{\rm Ext}\nolimits^1({\cal I}_x,{\cal O})=\cx{}}, d'o\`u le th\'eor\`eme A dans ce cas. Il reste \`a traiter le cas o\`u \ \m{\Delta < \delta'(\mu)}. C'est une cons\'equence du th\'eor\`eme C, dont la d\'emonstration suit. $\Box$ \subsection{{\cal D}\'emonstration du th\'eor\`eme C} Soit \m{(E,F,G)\in{\bf T}}. En consid\'erant la suite spectrale de Beilinson g\'en\'eralis\'ee associ\'ee \`a \m{(E,F,G)}, on voit imm\'ediatement que les points \m{(\mu,\Delta)} de \m{{\cal T}_{(E,F,G)}} (\`a coordonn\'ees rationnelles) sont les paires \m{(\mu({\cal E}),\Delta({\cal E}))}, o\`u ${\cal E}$ est de la forme $${\cal E}\ = (E\otimes\cx{a})\oplus(F\otimes\cx{b})\oplus(G\otimes\cx{c}),$$ avec \ $a,b,c\geq 0$ \ non tous nuls. Le fibr\'e pr\'ec\'edent est prioritaire et rigide, c'est donc un fibr\'e prioritaire g\'en\'erique. On pose comme dans le lemme 2.3, $$Z \ = \ \bigcup_{(E,F,G)\in{\bf T}}{\cal T}_{(E,F,G)}.$$ La partie 1- du th\'eor\`eme C est une cons\'equence imm\'ediate du \paragra~\hskip -2pt 2.4. Il reste donc \`a prouver que $$Z \ = \ {\cal S}.$$ Soit \ \m{(\mu,\Delta)\in Z}. Alors on a \ \m{\Delta\leq\delta'(\mu)}, car les fibr\'es prioritaires g\'en\'eriques ayant les invariants \m{\mu} et \m{\Delta} sont rigides, comme on vient de le voir. On a donc \ \m{Z\subset{\cal S}}. Soit $F$ un fibr\'e exceptionnel tel que \ \m{-1<\mu(F)\leq 0}, \m{(G_n)} la s\'erie exceptionnelle \`a gauche associ\'ee \`a $F$. On va montrer que lorsque $n$ tend vers l'infini, le segment de conique \m{G_nF} de \m{T_{(G_{n-1},G_n,F)}} tend vers le segment de conique $$\lbrace(\mu,\delta'(\mu)), \mu(F)-x_F<\mu\leq\mu(F)\rbrace.$$ On montrerait de m\^eme que si \ \m{-1\leq\mu(F)<0}, et si \m{(H_n)} est la s\'erie exceptionnelle \`a droite associ\'ee \`a $F$, alors lorsque $n$ tend vers moins l'infini, le segment de conique \m{FH_n} de \m{T_{(F,H_n,H_{n+1})}} tend vers le segment de conique $$\lbrace(\mu,\delta'(\mu)), \mu(F)\leq\mu<\mu(F)+x_F\rbrace.$$ D'apr\`es le lemme 2.3, ceci entraine que \ \m{{\cal S}\subset Z}. L'\'equation du segment de conique \m{G_nF} de \m{T_{(G_{n-1},G_n,F)}} est $$\Delta\ = \ P(\mu-\mu(G_{n-1})-3)-\Delta(G_{n-1}).$$ On a $$\lim_{n\rightarrow\infty}(\mu(G_{n-1})) \ = \ \mu(F)-x_F, \ \ \ \lim_{n\rightarrow\infty}(\Delta(G_{n-1})) \ = \ \q{1}{2}.$$ Donc le segment \m{G_nF} tend vers la courbe $$\lbrace(\mu,\phi(\mu)), \mu(F)-x_F<\mu\leq\mu(F)\rbrace.$$ avec $$\phi(\mu) \ = \ P(\mu-\mu(F)+x_F-3)-\q{1}{2}.$$ On v\'erifie imm\'ediatement que \ \m{\phi(\mu)=\delta'(\mu)}, ce qui ach\`eve la d\'emonstration du th\'eor\`eme C. $\Box$

9709

alg-geom/9709030

en

https://arxiv.org/abs/alg-geom/9709030

alg-geom/9709030

Brent Gordon

B. Brent Gordon

A Survey of the Hodge Conjecture for Abelian Varieties

68 pages, AMSTeX. To appear as Appendix B in the upcoming second edition of "A Survey of the Hodge Conjecture" by James D. Lewis

We review what is known about the Hodge conjecture for abelian varieties, with some emphasis on how Mumford-Tate groups have been applied to this problem.

alg-geom

\chapter{\let\savedef@\chapter \def\chapter##1{\let\chapter\savedef@ \leavevmode\hskip-\leftskip \rlap{\vbox to\z@{\vss\centerline{\tenpoint \frills@{CHAPTER\space\afterassignment\chapterno@ \global\chaptercount@=}% ##1\unskip}\baselineskip36pt\null}}\hskip\leftskip}% \nofrillscheck\chapter} \def\author{\let\savedef@\author \def\author##1\endauthor{\global\setbox\authorbox@ \vbox{\tenpoint\raggedcenter@ \frills@{\ignorespaces##1}\endgraf} \edef\next{\the\leftheadtoks}% \ifx\next\empty@\expandafter\uppercase{\leftheadtext{##1}}\fi} \nofrillscheck\author} \def\abstract{\let\savedef@\abstract \def\abstract{\let\abstract\savedef@ \setbox\abstractbox@\vbox\bgroup\noindent$$\vbox\bgroup\indenti=0pt \def\envir@end{\endabstract}\advance\hsize-2\indenti \def\usualspace{\enspace}\tenpoint \noindent \frills@{{\abstractfont@ Abstract.\enspace}}}% \nofrillscheck\abstract} \def\thanks#1\endthanks{% \gdef\thethanks@{\tenpoint\raggedcenter@#1\endgraf}} \def\keywords{\let\savedef@\keywords \def\keywords##1\endkeywords{\let\keywords\savedef@ \toks@{\def\usualspace{{\it\enspace}}\raggedcenter@\tenpoint}% \toks@@{##1\unskip.}% \edef\thekeywords@{\the\toks@\frills@{{\noexpand\it Key words and phrases.\noexpand\enspace}}\the\toks@@}}% \nofrillscheck\keywords} \def\subjclass{\let\savedef@\subjclass \def\subjclass##1\endsubjclass{\let\subjclass\savedef@ \toks@{\def\usualspace{{\rm\enspace}}\tenpoint\raggedcenter@}% \toks@@{##1\unskip.}% \edef\thesubjclass@{\the\toks@ \frills@{{\noexpand\rm1991 {\noexpand\it Mathematics Subject Classification}.\noexpand\enspace}}% \the\toks@@}}% \nofrillscheck\subjclass} \newskip\xcskip \newskip\afterxcskip \xcskip=10pt plus2pt minus0pt \afterxcskip=0pt \long\def\xcb#1{\par\ifnum\lastskip<\xcskip \removelastskip\penalty-100\vskip\xcskip\fi \noindent{\bf#1}% \nobreak\bgroup \xcbrosterdefs} \def\xcbrosterdefs{\normalparindent=0pt \setbox\setwdbox\hbox{0.}\rosteritemwd=\wd\setwdbox\relax \setbox\setwdbox\hbox{0.\hskip.5pc(c)}\rosteritemitemwd=\wd\setwdbox\relax \setbox\setwdbox\hbox{0.\hskip.5pc(c)\hskip.5pc(iii)}% \rosteritemitemitemwd=\wd\setwdbox\relax } \def\par\egroup{\par\egroup} \newif\ifplain@ \plain@false \def\output@{\shipout\vbox{% \ifplain@ \pagebody \else \iffirstpage@ \global\firstpage@false \pagebody \logo@ \baselineskip18\p@\line{\the\footline}% \else \ifrunheads@ \makeheadline \pagebody \else \pagebody \baselineskip18\p@\line{\the\footline} \fi \fi \fi}% \advancepageno \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} \outer\def\endtopmatter{\add@missing\endabstract \edef\next{\the\leftheadtoks}\ifx\next\empty@ \expandafter\leftheadtext\expandafter{\the\rightheadtoks}\fi \ifpart \global\plain@true \null\vskip9.5pc \thepart \vfill\eject \null\vfill\eject \fi \global\plain@false \global\firstpage@true \begingroup \topskip100pt \box\titlebox@ \endgroup \ifvoid\authorbox@\else \vskip2.5pcplus1pc\unvbox\authorbox@\fi \ifnum\addresscount@>\z@ \vfill Author addresses: \count@\z@ \loop\ifnum\count@<\addresscount@\advance\count@\@ne \csname address\number\count@\endcsname \csname email\number\count@\endcsname \repeat \vfill\eject\fi \ifvoid\affilbox@\else \vskip1pcplus\unvbox\affilbox@\fi \ifx\thesubjclass@\empty@\else \vfil\thesubjclass@\fi \ifx\thekeywords@\empty@\else \vfil\thekeywords@\fi \ifx\thethanks@\empty@\else \vfil\thethanks@\fi \ifvoid\abstractbox@\else \vfil\unvbox\abstractbox@\vfil\eject \fi \ifvoid\tocbox@\else\vskip1.5pcplus.5pc\unvbox\tocbox@\fi \prepaper \vskip22pt\relax } \def\raggedleft@{\leftskip\z@ plus.4\hsize \rightskip\z@ \parfillskip\z@ \parindent\z@ \spaceskip.3333em \xspaceskip.5em \pretolerance9999\tolerance9999 \exhyphenpenalty\@M \hyphenpenalty\@M \let\\\linebreak} \def\aufm #1\endaufm{\vskip12pt{\raggedleft@ #1\endgraf}} \widestnumber\key{M} \begingroup \let\head\relax \let\specialhead\relax \let\subhead\relax \let\subsubhead\relax \let\title\relax \let\chapter\relax \newbox\tocchapbox@ \gdef\newtocdefs{% \def\ptitle##1\endptitle {\penalty\z@ \vskip8\p@ \hangindent\wd\tocheadbox@\noindent{\bf ##1}\endgraf}% \def\title##1\endtitle {\penalty\z@ \vskip8\p@ \hangindent\wd\tocchapbox@\noindent{##1}\endgraf}% \def\chapter##1{\par Chapter ##1.\unskip\enspace}% \def\part##1{\par {\bf Part ##1.}\unskip\enspace}% \def\specialhead##1 ##2\endgraf\endgroup\nobreak\vskip\belowspecheadskip{\par \begingroup \hangindent.5em \noindent \if\notempty{##1}% \leftskip\wd\tocheadbox@ \llap{\hbox to\wd\tocheadbox@{\hfil##1}}\enspace \else \leftskip\parindent \fi ##2\endgraf \endgroup}% \def\head##1 ##2\endhead{\par \begingroup \hangindent.5em \noindent \if\notempty{##1}% \leftskip\wd\tocheadbox@ \llap{\hbox to\wd\tocheadbox@{\hfil##1}}\enspace \else \leftskip\parindent \fi ##2\endgraf \endgroup}% \def\subhead##1 ##2\endsubhead{\par \begingroup \leftskip4.5pc \noindent\llap{##1\enspace}##2\endgraf \endgroup}% \def\subsubhead##1 ##2\endsubsubhead{\par \begingroup \leftskip5pc \noindent\hbox{\ignorespaces##1 }##2\endgraf \endgroup}% }% \gdef\toc@#1{\relaxnext@ \DN@{\ifx\next\nofrills\DN@\nofrills{\nextii@}% \else\DN@{\nextii@{{#1}}}\fi \next@}% \DNii@##1{% \ifmonograph@\bgroup\else\setbox\tocbox@\vbox\bgroup \centerline{\headfont@\ignorespaces##1\unskip}\nobreak \vskip\belowheadskip \fi \def\page####1% {\unskip\penalty\z@\null\hfil \rlap{\hbox to\pagenumwd{\quad\hfil\rm ####1}}% \global\setbox\tocchapbox@\hbox{Chapter 1.\enspace}% \global\setbox\tocheadbox@\hbox{\hskip18pt \S0.0.}% \hfilneg\penalty\@M}% \leftskip\z@ \rightskip\leftskip \setboxz@h{\bf\quad000}\pagenumwd\wd\z@ \advance\rightskip\pagenumwd \newtocdefs }% \FN@\next@}% \endgroup \def\logo@{} \Monograph \catcode`\@=13 \def2.1{2.1} \refstyle{A} \define\dfn#1{{\it #1\/}} \define\dfnb#1{{\it #1}} \loadeusm \define\script{\eusm} \define\scr{\eusm} \catcode`\@=11 \def\hookrightarrowfill{$\m@th\mathord\lhook\mkern-3mu% \mathord-\mkern-6mu% \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill \mkern-6mu\mathord\rightarrow$} \def\hookleftarrowfill{$\m@th\mathord\leftarrow\mkern-6mu% \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill \mkern-6mu\mathord-\mkern-3mu\mathord\rhook$} \atdef@ C#1C#2C{\ampersand@\setbox\z@\hbox{$\ssize \;{#1}\;\;$}\setbox\@ne\hbox{$\ssize\;{#2}\;\;$}\setbox\tw@ \hbox{$#2$}\ifCD@ \global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi \ifdim\wd\z@>\bigaw@\global\bigaw@\wd\z@\fi \ifdim\wd\@ne>\bigaw@\global\bigaw@\wd\@ne\fi \ifCD@\hskip.5em\fi \ifdim\wd\tw@>\z@ \mathrel{\mathop{\hbox to\bigaw@{\hookrightarrowfill}}% \limits^{#1}_{#2}}\else \mathrel{\mathop{\hbox to\bigaw@{\hookrightarrowfill}}% \limits^{#1}}\fi \ifCD@\hskip.5em\fi\ampersand@} \atdef@ D#1D#2D{\ampersand@\setbox\z@\hbox{$\ssize \;\;{#1}\;$}\setbox\@ne\hbox{$\ssize\;\;{#2}\;$}\setbox\tw@ \hbox{$#2$}\ifCD@ \global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi \ifdim\wd\z@>\bigaw@\global\bigaw@\wd\z@\fi \ifdim\wd\@ne>\bigaw@\global\bigaw@\wd\@ne\fi \ifCD@\hskip.5em\fi \ifdim\wd\tw@>\z@ \mathrel{\mathop{\hbox to\bigaw@{\hookleftarrowfill}}% \limits^{#1}_{#2}}\else \mathrel{\mathop{\hbox to\bigaw@{\hookleftarrowfill}}% \limits^{#1}}\fi \ifCD@\hskip.5em\fi\ampersand@} \catcode`\@=13 \define\bs{\backslash} \define\compose{\circ} \define\dual{\spcheck} \define\hra{\hookrightarrow} \define\hla{\hookleftarrow} \define\intersect{\cap} \predefine\isom{\cong} \redefine\cong{\equiv} \predefine\imaginary{\Im} \redefine\Im{\operatorname{Im}} \define\lmt{\longmapsto} \define\lra{\longrightarrow} \define\ol{\overline} \define\onto{\twoheadrightarrow} \define\simto{\overset\sim\to\longrightarrow} \define\tensor{\otimes} \define\union{\cup} \define\ul{\underline} \define\varemptyset{\varnothing} \define\Qed{\hbox to 0.5em{ }\nobreak\hfill\hbox{$\square$}} \redefine\AA{{\Bbb A}} \define\Ad{\operatorname{Ad}} \define\ad{\operatorname{ad}} \define\angled#1{\langle #1 \rangle} \define\Aut{\operatorname{Aut}} \define\CC{{\Bbb C}} \define\Div{{\script D}} \define\End{\operatorname{End}} \define\Endo{\operatorname{End}^0} \define\EndoA{\Endo(A)} \define\GG{{\Bbb G}} \define\Gal{\operatorname{Gal}} \define\GHC{\operatorname{GHC}} \define\GL{\operatorname{GL}} \define\Gm{{\GG_{\text{m}}}{\vphantom{\GG}}} \define\GSp{\operatorname{GSp}} \define\Hdg{{\script H}} \define\Hg{\operatorname{Hg}} \define\hg{\frak{hg}} \define\Hom{\operatorname{Hom}} \define\Id{\operatorname{Id}} \define\id{\operatorname{id}} \define\ind{\operatorname{ind}} \define\Ker{\operatorname{Ker}} \define\Lie{\operatorname{Lie}} \define\Lf{\operatorname{Lf}} \define\MT{\operatorname{MT}} \define\MTA{\MT(A)} \define\mt{\frak{mt}} \define\mtA{\mt(A)} \define\NN{{\Bbb N}} \define\scO{{\script O}} \define\PP{{\Bbb P}} \redefine\phi{\varphi} \define\phibar{\ol{\phi}} \define\QQ{{\Bbb Q}} \define\Qbar{{\ol{\QQ}}} \define\RR{{\Bbb R}} \define\rank{\operatorname{rank}} \define\rdim{\operatorname{rdim}} \define\Res{\operatorname{Res}} \define\SS{{\Bbb S}} \define\Sbar{\ol{S}} \define\sgn{\operatorname{sgn}} \define\SL{\operatorname{SL}} \define\SMT{\operatorname{SMT}} \define\smt{\frak{smt}} \define\SMTA{\SMT(A)} \define\smtA{\smt(A)} \define\SO{\operatorname{SO}} \predefine\Sup{\Sp} \redefine\Sp{\operatorname{Sp}} \define\spec{\operatorname{spec}} \define\SU{\operatorname{SU}} \define\tr{\,\mathstrut^t} \define\U{\operatorname{U}} \define\V{{\script V}} \define\Vr{V_\RR} \define\twedge{{\tsize \bigwedge}} \define\Wr{W_\RR} \define\Weil{{\script W}} \define\X{\operatorname{X}} \define\ZZ{{\Bbb Z}} \pageno=-1 \topmatter \leftheadtext{appendix b. The Hodge conjecture for abelian varieties} \endtopmatter \pageno=306 \document {\bf Appendix B. A Survey of the Hodge Conjecture for Abelian Varieties} \smallskip \indent\hskip1.0in {\bf by B. Brent Gordon} \head Introduction \endhead The goal of this appendix is to review what is known about the Hodge conjecture for abelian varieties, with an emphasis on how Mumford-Tate groups have been applied to this problem. In addition to the book in which this appears, other survey or general articles that precede this one are Hodge's original paper \cite{B.53}, Grothendieck's modification of Hodge's general conjecture \cite{B.43}, Shioda's excellent survey article \cite{B.117}, Steenbrink's comments on the general Hodge conjecture \cite{B.120}, and van Geemen's pleasing introduction to the Hodge conjecture for abelian varieties \cite{B.35}. Naturally there is some overlap between this appendix and van Geemen's article, but since his emphasis is on abelian varieties of Weil type, we hope that this appendix will be a useful complement. \medpagebreak Since the language of linear algebraic groups and their Lie algebras, which cannot be avoided in any discussion of Mumford-Tate groups and their application to the Hodge conjecture for abelian varieties, may not be familiar to students of complex algebraic geometry and Hodge theory, we begin by recalling the definitions and facts we need and giving some examples. Towards the end of section one we also recall some basic facts about abelian varieties, including the Albert classification of their endomorphism algebras (Theorem~1.12.2), and give some example of abelian varieties to which we refer later. Most readers will find it more profitable to begin with section two, where we discuss the definitions and some general structural properties of the Hodge, Mumford-Tate and Lefschetz groups associated to an abelian variety, or section three, and refer to the first section as needed. Starting with section three we have tried to be comprehensive in summarizing the known results and indicating the main ideas involved in their proofs, while at the same time selecting some cross-section of proofs to discuss in more detail. In section three we follow Murty's exposition \cite{B.84} of the Hodge $(p,p)$ conjecture for arbitrary products of elliptic curves. In section four we summarize Shioda's results on abelian varieties of Fermat type \cite{B.116}, but only briefly consider the issues and results related to abelian varieties of Weil type, since \cite{B.35} treats this topic well. In section five we discuss the work of Moonen and Zarhin on four-dimensional abelian varieties \cite{B.75}, for this provides a nice illustration of the interplay between the endomorphism algebra and the Mumford-Tate group of an abelian variety. In section six we look at the work of Tankeev and Ribet on the Hodge conjecture for simple abelian varieties that satisfy some conditions on their dimension or endomorphism algebra \cite{B.124} \cite{B.125} \cite{B.126} \cite{B.93} \cite{B.94}; for example, the Hodge conjecture is true for simple abelian varieties of prime dimension. Here we look more closely at Ribet's approach, where he introduced and used the Lefschetz group of an abelian variety. Then the results of Murty and Hazama discussed in section seven build on and go beyond Ribet's methods to treat abelian varieties not assumed to be simple, but still assumed to satisfy some conditions on their dimensions or endomorphism algebras \cite{B.81} \cite{B.82} \cite{B.83} \cite{B.46} \cite{B.47} \cite{B.49}. In section eight we shift directions slightly, for here we have collected together examples of exceptional Hodge cycles, i.e., Hodge cycles not accounted for by linear combinations of intersections of divisor classes, and in this section, not known to be algebraic. Largely missing from section eight, but considered in section nine, are the particular problems posed by abelian varieties of complex multiplication type. Dodson \cite{B.28} \cite{B.29} \cite{B.30} and others have constructed numerous examples of such abelian varieties that support exceptional Hodge cycles. In section ten we examine what is known about the general Hodge conjecture for abelian varieties. The majority of the work on this problem is either a very geometric treatment of special abelian varieties in low dimension, for example \cite{B.12} or \cite{B.104}, or requires special assumptions about the endomorphism algebra, dimension or Hodge group, as in \cite{B.127}, \cite{B.128}, \cite{B.50} or \cite{B.4}. In the final section eleven we briefly mention three alternative approaches to proving the (usual) Hodge conjecture for arbitrary abelian varieties: First, a method involving the Weil intermediate Jacobian \cite{B.98}; then that the Tate conjecture for abelian varieties implies the Hodge conjecture for abelian varieties \cite{B.88} \cite{B.87} and \cite{B.27}; and thirdly, that the Hodge conjecture for abelian varieties would follow from knowing Grothendieck's invariant cycles conjecture (\cite{B.42}) for certain general families of abelian varieties, and moreover, that for these families, the invariant cycles conjecture would follow from the $L_2$-cohomology analogue of Grothendieck's standard conjecture~(A) that the Hodge $*$-operator is algebraic (\cite{B.44}) \cite{B.4}. The present state of our knowledge about the Hodge conjecture for abelian varieties is such that any or none of these approaches might ultimately work, or a counterexample might exist. Preceding the bibliography is a rough chronological table of the work that directly address some aspect of the Hodge conjecture for abelian varieties. I have tried to make sure that this table and this appendix as a whole mention all the relevant references through the end of 1996; if I have omitted something or otherwise not done it justice, that was quite unintentional. \head 1. Abelian varieties and linear algebraic groups \endhead The purpose of this section is to establish the language we use throughout the rest of this appendix to discuss abelian varieties and certain linear algebraic groups and Lie algebras associated with them. Although abelian varieties and linear algebraic groups are both algebraic groups, the issues surrounding them tend to be of a very different nature. It turns out to be most convenient to begin by recalling some of the definitions and basic properties of linear algebraic groups and their Lie algebras, and introducing some of the examples of these to which we will later refer, and then in the second half of the section review some of the definitions and basic properties of abelian varieties, and introduce some of the examples we will investigate later. \subhead 1.0. Notational conventions \endsubhead \nopagebreak \remark{1.0.1. Field of definition} Let $F$ be a field and $V$ and algebraic variety. Then we will write $V_F$ to signify or emphasize that $V$ is defined over $F$. When $V$ is an algebraic variety defined over $F$ and $K$ is a field containing $F$, then $V_K = V_F \times_{\spec F} \spec K$ is the base change to $K$, i.e., $V$ as a variety defined over $K$. We will generally try to distinguish the abstract variety $V_F$ defined over $F$ from its concrete set of $F$-points $V(F)$, and then $V(K) = V_F(K)$ is the set of $K$-points. \revert@envir\endremark\medskip \definition{1.0.2. Definition} Suppose $K$ is a separable algebraic extension of $F$ of finite degree~$d$, and $V$ is an algebraic variety defined over the larger field~$K$. Let $\{ \sigma_1, \ldots, \sigma_d\}$ be the set of distinct embeddings of $K$ into the algebraic closure $F^{\text{alg}}$ of $F$. Then the \dfn{restriction of scalars functor} $\Res_{K/F}$ from varieties over $K$ to varieties over $F$ is defined as follows: First let $V_{\sigma_i} = V_K \times_{\spec K,\,\sigma_i} F^{\text{alg}}$. Then for any variety $W$ defined over $F$ and a morphism $\phi:W\to V$ defined over $K$ there are morphisms $\phi_{\sigma_i} : W\to V_{\sigma_i}$. Then if $$ (\phi_{\sigma_1},\ldots , \phi_{\sigma_d}) : W \to V_{\sigma_1} \times \cdots \times V_{\sigma_d} $$ is an isomorphism, then \dfn{$W = \Res_{K/F} V$ is the variety obtained from $V$ by restriction of the field of definition from $K$ to $F$.} Its uniqueness is a consequence of the universal property that whenever $X$ is any variety defined over $F$ and $\psi: X \to V$ is a morphism defined over $K$, then there exists a unique $\Psi: X\to W$ defined over $F$ such that $\psi = \phi\circ\psi$. In practice it is often easiest to look at the $K$-points, then $$ \Res_{K/F}V(K) \simeq \prod_{\sigma\in \Hom_F(K,F^{\text{alg}})} V_{K,\sigma}(K) $$ together with the action of $\Gal(F^{\text{alg}}/F)$ permuting the factors according to its action on $\{\sigma_1 ,\ldots , \sigma_d\}$. For further details see \cite{B.136}~1.3. \enddefinition \definition{1.1. Definition} An \dfn{algebraic group} over $F$ is an algebraic variety $G$ defined over $F$ together with morphisms $$ \operatorname{mult} : G\times G \to G \qquad \text{ and } \qquad \operatorname{inv} : G\to G, $$ both defined over $F$, and an element $e \in G(F)$ such that $G$ is a group with identity~$e$, multiplication given by $\operatorname{mult}$, and inverses given by $\operatorname{inv}$. A \dfn{morphism of algebraic groups} is a morphism of algebraic varieties which is also a group homomorphism. \enddefinition As a variety an algebraic group is smooth, since it contains an open subvariety of smooth points and the group of translations $h\mapsto gh$ acts transitively. \definition{1.1.1. Definition} An \dfn{abelian variety} is a complete connected algebraic group. It follows from this definition that an abelian variety is a smooth projective variety and that its group law is commutative, see for example \cite{B.96}, \cite{B.121}, \cite{B.68}, \cite{B.95}, \cite{B.69}, \cite{B.74}, \cite{B.79} or \cite{B.134}. It also follows that every morphism of abelian varieties as varieties can be expressed as a composition of a homomorphism with a translation, though of course only homomorphisms are morphisms of abelian varieties as algebraic groups. In this appendix we will only be dealing with abelian varieties defined over $\CC$, that is, complex abelian varieties. When $A$ is a complex abelian variety then the manifold underlying $A(\CC)$ is a complex torus. \enddefinition \definition{1.1.2. Definition} An affine algebraic group is also called a \dfn{linear algebraic group.} This is justified by the fact that an affine algebraic group is isomorphic, over its field of definition, to a closed subgroup of $\GL(n)$ for some ~$n$, see \cite{B.14}, \cite{B.55}, \cite{B.52}, \cite{B.119}, \cite{B.133}. \enddefinition \definition{1.2. Definition} Any affine algebraic group that is isomorphic (as an algebraic group) to the diagonal subgroup of $\GL(n)$ for some $n$ is called an \dfn{algebraic torus.} For additional basic exposition on algebraic tori see \cite{B.14}~III.8 or \cite{B.55}~\S16. \enddefinition \example{1.2.1. Example} Our most basic and important example of an algebraic torus is $\Gm := \GL(1)$. {\it A priori\/} $\Gm = \Gm_{/\QQ}$ is defined over $\QQ$, and thus $\Gm(F) = F^\times$ for any field $F$ containing~$\QQ$. Similarly, with the conventions of ~1.0, $\Gm_{/F}(K) = K^\times$ when $K$ is a field containing $F$. \endexample \example{1.2.2. Example} We may also apply the restriction of scalars functor to an algebraic torus. For the purposes of this appendix, one of the most important examples that we will use later is $$ \SS := \Res_{\CC/\RR}\Gm_{/\CC} . $$ Then $\SS(\RR) = \CC^\times$ and $\SS(\CC) \simeq \CC^\times \times \CC^\times$, where these last two factors are interchanged by complex conjugation. In particular $\SS(\RR)$ embeds as the diagonal in $\SS(\CC)$. \endexample \definition{1.3. Definition} A connected linear algebraic group of positive dimension is said to be \dfn{semisimple} if it has no closed connected commutative normal subroups except the identity. A (Zariski-connected) linear algebraic group $G$ is said to be \dfn{reductive} if is the product of two (Zariski-connected) normal subgroups $G_{\text{ab}}$ and $G_{\text{ss}}$, where $G_{\text{ab}}$ is an algebraic torus and $G_{\text{ss}}$ is semisimple, and $G_{\text{ab}} \cap G_{\text{ss}}$ is finite. {\it A~fortiori\/} any semisimple group is reductive. \enddefinition \definition{1.4. Definition} Recall that a \dfn{representation} of a group $G$ is a homomorphism $\rho : G \to \GL(V)$ from $G$ to the automorphism group of a vector space~$V$. Such a representation may be referred to as $(\rho, V)$ or simply by $\rho$ or by~$V$. If $(\sigma, W)$ is another representation of $G$, a map $\psi: V\to W$ such that $\sigma(g)\circ\psi = \psi \circ \rho(g)$ for all $g\in G$ is said to be \dfn{$G$-linear} or \dfn{$G$-equivariant.} In this case, if $\psi$ is an isomorphism the representations $(\rho,V)$ and $(\sigma, W)$ are said to be \dfn{equivalent.} We frequently identify equivalent representations. A subrepresentation of a representation is defined in the natural way, and a representation is said to be \dfn{irreducible} if it contains no nontrivial subrepresentations. Further, given representations $(\rho,V)$ and $(\sigma,W)$ of $G$ we may form their direct sum or their tensor product. Thus the $r^{\text{th}}$ exterior power $(\twedge^r \rho, \twedge^r V)$ of a representation arises naturally as a subrepresentation of the $r$-fold tensor product of the representation $(\rho, V)$ with itself. Let $V\dual = \Hom(V,F)$ denote the dual space to $V$ (if $V$ is a vector space over $F$), and let $\angled{\ ,\ } :V\times V \to F$ be the natural pairing. Then $\rho$ induces a representation $\rho\dual: G\to \GL(V\dual)$, called the \dfn{dual,} or \dfn{contragredient representation,} of $G$. It is defined by requiring $$ \angled{\rho\dual(g)v\dual,\rho(g)v} = \angled{v\dual,v}; $$ concretely this means that $\rho\dual(g) = \tr\rho(g)^{-1}$. When $G$ is a subgroup of $\GL(V)$, for some vector space $V$, in particular when $G$ comes as a subroup of $\GL(n)$, the group of invertible $n\times n$ matrices, then by the \dfn{standard representation} of $G$ we mean the natural inclusion $G\hra \GL(V)$. \enddefinition \subhead 1.5. Examples of semisimple and reductive groups \endsubhead The examples that will be of interest to us are all classical groups, defined from the outset as subgroups of $\GL(n)$. \example{1.5.1. Example} The first basic example is $\SL(n)$, the subgroup of $\GL(n)$ of matrices of determinant~$1$. For $n\ge 2$, $\SL(n)$ is semisimple. It follows that $\GL(n)$ is reductive, as it is the product of its subgroup of diagonal matrices and $\SL(n)$. \endexample \example{1.5.2. Example} Let $F$ be a subfield of the real numbers, in particular $\RR$ itself, let $K$ be an imaginary quadratic extension of~$F$, and let $V$ be a vector space over $K$. Then a \dfn{Hermitian form} on $V$ is an $F$-bilinear form $H:V\times V \to K$ such that $H(v,u) = \sigma(H(u,v))$, where $\sigma$ is the nontrivial automorphism of $K$ over $F$, the restriction of complex conjugation. Then the \dfn{unitary group} $\U(V,H)$ is the subgroup of $g\in \GL(V)$ such that $H(gu,gv) = H(u,v)$, and the \dfn{special unitary group} $\SU(V,H)$ is the subgroup of $\U(V)$ of elements of determinant~$1$. Note that $\U(V,H)$ and $\SU(V,H)$ are algebraic groups defined over~$F$. When $F=\RR$ and $H$ can be represented by a diagonal matrix with $p$~$1$'s and $q$~$(-1)$'s then we may write $\U(p,q)$ or $\SU(p,q)$ for $\U(V,H)$ or $\SU(V,H)$, respectively; when $q=0$, that is when $H$ is equivalent to the standard form $(u,v) \mapsto \tr\bar u \cdot v$, we write $\U(n)$ or $\SU(n)$. As a particular special case, note that $U(1)$ is defined over $\RR$, and $\U(1,\RR)$ is the group of complex numbers of absolute value~$1$. Similarly as in the previous example, $\SU(n)$ is semisimple and $\U(n)$ is reductive, for $n\ge 2$. \endexample \example{1.5.3. Example} When $E$ is a skew-symmetric bilinear form on a vector space $V$, that is, $E(v,u) = -E(u,v)$, then the \dfn{symplectic group} $\Sp(V,E)$ is the subgroup of $g\in \GL(V)$ such that $E(gu,gv)= E(u,v)$. The \dfn{symplectic similitude group} $\GSp(V,E)$ is the group of $g\in\GL(V)$ such that there is a scalar $\nu(g)$ such that $E(gu,gv) = \nu(g)E(u,v)$. Thus $\GSp(V,E)$ contains $\Sp(E,V)$ as the subgroup of \dfn{similtude norm} $\nu(g) =1$. When $E$ can be represented by a matrix of the form $$ \pmatrix 0 & I_n \\ -I_n & 0 \endpmatrix , $$ then we may write $\Sp(2n)$ for $\Sp(V,E)$ and $\GSp(2n)$ for $\GSp(V,E)$. Moreover, $\Sp(2n)$ is semisimple. \endexample \example{1.5.4. Example} When $S$ is a symmetric bilinear form on a vector space $V$, then the \dfn{special orthogonal group} $\SO(V,S)$ is the subgroup of $g \in \SL(V)$ such that $S(gu,gv) = S(u,v)$. In particular, if $S$ can be represented by an identity matrix $I_n$ then we may write $\SO(n)$ instead of $\SO(V,S)$. Also $\SO(n)$ is semisimple. \endexample \subhead{1.6. Lie algebras} \endsubhead It will be useful later to have available some of the language of Lie algebras, so we briefly recall some of the definitions here. Our major references for this paragraph (and the next two) are \cite{B.34}, \cite{B.54}, \cite{B.14}, \cite{B.55} and \cite{B.103}. \definition{1.6.1. Definition} A \dfn{Lie algebra} is a vector space $\frak g$ together with a skew-symmetric bilinear map, the \dfn{bracket} operation, $$ [\ ,\ ]: \frak g \times \frak g \to \frak g $$ satisfying the \dfn{Jacobi identity} $$ [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] =0. $$ \enddefinition \example{Example} Let $V$ be a vector space over a field $F$. The fundamental example of a Lie algebra is $\frak{gl}(V)$, which is $\End_F(V)$ as a vector space with the bracket given by $[X,Y] := XY-YX$. \endexample \subsubhead{1.6.2. The Lie algebra of an algebraic group} \endsubsubhead Let $G$ be a linear algebraic group over a field $F$. In order to review how a Lie algebra $\Lie(G)$ is associated to $G$, first recall that when $A$ is any $F$-algebra then the Lie algebra of \dfn{$F$-derivations} from $A$ to $A$ can be described as $$ \operatorname{Der}_F(A,A) := \{ X\in \frak{gl}(A) : X(f\cdot g) = (Xf)\cdot g +f\cdot(Xg), \text{ for } f,g\in A\} $$ with the induced bracket $[X,Y] := XY-YX$. Now let $A = F[G]$, the coordinate ring of $G$ (as algebraic variety). Then $G$ acts on $F[G]$ by left translations: $(\lambda_g f)(x) := f(g^{-1}x)$, for $f\in F[G]$ and $g,x \in G$. Then the set of \dfn{left invariant} derivations $$ L(G) := \{X\in \operatorname{Der}_F(F[G],F[G]) : \lambda_g \circ X = X\circ \lambda_g \text{ for all }g\in G\} $$ is a Lie subalgebra of $\operatorname{Der}_F(A,A)$, and some authors take this as the definition of $\Lie(G)$. Next, recall that when $\scO_e$ is the local ring at $e$ and $\frak m_e$ its maximal ideal, the the \dfn{tangent space} to $G$ at the identity is $$ T(G)_e = \operatorname{Der}_F(\scO_e, \scO_e/\frak m_e) \isom \Hom_{F-\text{mod}}(\frak m_e/\frak m_e^2,F) . $$ Of course $\scO_e/\frak m_e$ is the residue field of the local ring at the identity~$e$. Then it turns out that evaluation at the identity $e$ of $G$ gives an isomorphism from $L(G)$ to $T(G)_e$ to $G$ at the identity (see references on linear algebraic groups cited above). Thus $\Lie(G)$ can also be defined as $T(G)_e$ with the bracket operation induced by the isomorphism with $L(G)$. \example{Example} As the notation suggests, $\frak{gl}(V) = \Lie(\GL(V))$. \endexample \remark{Remark} One motivation for working with Lie algebras is that for a connected linear group $G$ a homomorphism $\phi:G \to H$ to another group $H$ is determined by its differential at the identity. In this way Lie algebras linearize some of the problems of representation theory. More generally, when some property of $G$ is determined by an open neighborhood of the identity, it is often more effective work with $\Lie(G)$. \revert@envir\endremark\medskip \subsubhead{1.6.3. The adjoint representations} \endsubsubhead In general the \dfn{differential} of a morphism of (irreducible) algebraic varieties $\phi: X\to Y$ at $x\in X$ is the linear map on tangent spaces $d\phi_x:T(X)_x \to T(Y)_{\phi(x)}$ induced by $\phi^*: \scO_{\phi(x)} \to \scO_x$. In particular, $G$ acts on itself by inner automorphisms $$ \operatorname{Int}_g : h\mapsto g h g^{-1}, $$ and this action fixes the identity. Then the differential of this map $$ \Ad(g) := d(\operatorname{Int}_g)_e : T(G)_e \to T(G)_e $$ defines the \dfn{adjoint representation} of $G$ $$ \Ad: G\to \Aut(T(G)_e) : g\mapsto \Ad(g). $$ If we go one step further and take the differential of the adjoint representation, we get a Lie algebra morphism $$ \ad := d\Ad : T(G)_e \to \End(T(G)_e) , $$ with $\ad(X)(Y) = [X,Y]$. \definition{1.6.4. Definition} Now let $G$ be (the real points of) a connected algebraic group over {}~$\RR$, and let $K$ be a maximal compact subgroup. Then a \dfn{Cartan involution} of $G$ with respect to $K$ is an involutive automorphism of $G$ whose fixed point set is precisely $K$. The differential of a Cartan involution is a Cartan involution of $\Lie(G)$, and the decomposition $$ \Lie(G) = \frak k + \frak p $$ where $\frak k$ is the fixed point set and $\frak p$ is the $(-1)$-eigenspace, is called a \dfn{Cartan decomposition.} It follows that $$ [\frak k, \frak k] \subseteq \frak k, \qquad [\frak k, \frak p] \subseteq \frak p, \qquad [\frak p, \frak p] \subseteq \frak k. $$ \enddefinition \definition{1.6.5. Definition} A semisimple real Lie algebra $\frak g$ with a Cartan decomposition $\frak g = \frak k + \frak p$ is of \dfn{Hermitian type} if there exists an element $H_0$ in the center of $\frak k$ such that $(\ad(H_0))^2 = -1$ as endomorphisms of ~$\frak p$. Recall that a reductive group is an extension of a semisimple group by an algebraic torus. We will say that a real reductive algebraic group $G$ is of \dfn{Hermitian type} if the abelian part of $G$ is compact and the semisimple part of its Lie algebra, $\Lie(G)_{\text{ss}}$, is of Hermitian type in the previous sense.. \enddefinition \example{1.7. Examples} Let $V$ be a vector space over a field $F$. We have already noted that $\frak{gl}(V) = \Lie(\GL(V))$ is $\End(V)$ as a vector space with the bracket $[X,Y] = XY -YX$. The subgroup of $\frak{gl}(V)$ of endomorphisms with trace~$0$ is $\frak{sl}(V) = \Lie(\SL(V))$. Suppose $V$ is a symplectic space of dimension ~$2n$ whose skew-symmetric form is represented by $$ \pmatrix 0 & I_n \\ -I_n & 0 \endpmatrix $$ Then $\frak{sp}(2n) = \Lie(\Sp(2n))$ consists of matrices of the form $$ X = \pmatrix M & N \\ P & Q \endpmatrix $$ such that $N$ and $P$ are symmetric and $\tr M =-Q$. When $V$ has dimension ~$n$ and comes with a symmetric bilinear form represented by an identity matrix, then $\frak{so}(n) = \Lie(\SO(n))$ consists of $n\times n$ skew-symmetric matrices. \endexample \subhead{1.8. The spin representations of $\frak{so}(n)$} \endsubhead As a subgroup of $\frak{gl}(V) = \End(V)$ any of the Lie algebras above naturally acts on $V$, and we may think of this as the standard representation of the Lie algebra, cf.~1.4. Moreover, up to equivalence, all the representations of $\frak{sl}(n)$, respectively $\frak{sp}(2n)$, occur in some tensor power of the standard representation. However, that is not the case for $\frak{so}(n)$, so here we briefly recall the complex representation(s) that do not. Let $V$ be an $n$-dimensional vector space with a nondegenerate symmetric bilinear form ~$Q$. Then the quotient of the tensor algebra of $V$ by the ideal generated by all elements of the form $v\tensor v - Q(v,v)$ for $v\in V$ is called the \dfn{Clifford algebra} $C(V)$ of ~$V$. Since this ideal preserves the property that an element of the tensor algebra is the product of an even number of vectors, such products generate a subalgebra $C^+(V)$ of $C(V)$ called the \dfn{even Clifford algebra.} For more details about Clifford algebras some good sources are \cite{B.20} \cite{B.31} or \cite{B.19}. Now following \cite{B.34}~Chapter~20 we first observe that since $C(V)$ and $C^+(V)$ are associative algebras they determine Lie algebras with $[a,b] = a\cdot b - b \cdot a$. Moreover, $\frak{so}(V)$ embeds in $C^+(V)$ as a Lie subalgebra. Roughly speaking, on the one hand there is an embedding $\psi: \twedge^2V \to C^+(V)$, given by $\psi(a\wedge b) = a \cdot b - Q(a,b)$, while on the other hand there is an isomorphism $\phi: \twedge^2 V \simto \frak{so}(V) \subset \frak{gl}(V)$ given by $$ \phi(a\wedge b)(v) = 2(Q(b,v)a - Q(a,v)b) . $$ Now suppose $n=2m$ is even (and the underlying field $F =\CC$). Then $V$ can be written as the sum of two $m$-dimensional isotropic subspaces, $V= W\oplus W'$ (meaning that the restriction of $Q$ to $W$, respectively $W'$, is zero). Then the key lemma is that $C(V) \simeq \End(\twedge^*W)$, where $\twedge^*W$ signifies the exterior algebra of ~$W$. From this it follows that, if we write $\twedge^*W = \twedge^+W \oplus \twedge^-W$ corresponding to even and odd exterior powers, then $$ C^+(V) \simeq \End(\twedge^+W) \oplus \End(\twedge^-W). $$ Therefore the embedding of $\frak{so}(V)$ into $C^+(V)$ determines two (inequivalent) representations of $\frak{so}(V)$, namely its actions on $\twedge^+W$ and $\twedge^-W$ respectively. These are referred to as the \dfn{half-spin} representations of $\frak{so}(V)$, and their sum is the \dfn{spin} representation. When $n=2m+1$ is odd, then we may write $V = W \oplus W' \oplus U$, where $W$ and $W'$ are $m$-dimensional isotropic subspaces, as before, and $U$ is $1$-dimensional and orthogonal to both $W$ and $W'$. In this case $$ C(V) \simeq \End(\twedge^*W) \oplus \End(\twedge^*W') $$ and $C^+(V) \simeq \End(\twedge^*W)$. Thus the embedding of $\frak{so}(V)$ into $C^+(V)$ determines a single \dfn{spin} representation. \subhead 1.9. Quaternion algebras \endsubhead A \dfn{quaternion algebra} over a field $F$ (of characteristic not~$2$) is a simple $F$-algebra of rank~$4$ whose center is precisely~$F$. Over the complex numbers, or any algebraically closed field of characteristic not equal to ~$2$, there is up to isomorphism only one quaternion algebra, the $2\times 2$ matrix algebra. Over the real numbers, aside from the $2\times 2$ matrix algebra there is up to isomorphism only one other quaternion algebra, the \dfn{Hamiltonian} quaternion algebra ~$\Bbb H$, generated over $\RR$ by $1$ and elements $i$ and $j$ such that $$ i^2 = j^2 =-1, \qquad\quad ij =-ji . $$ Note that $\Bbb H$ is a division algebra. Over $\QQ$ there are infinitely many non-isomorphic quaternion algebras, all of which except the $2\times 2$ matrix algebra are (noncommutative) division algebras; see for example \cite{B.132}. A quaternion algebra over $\QQ$, or more generally a quaternion algebra over a subfield of $\RR$, is said to be \dfn{definite} or \dfn{indefinite} according as its tensor product with $\RR$ is isomorphic to $\Bbb H$ or to $M_2(\RR)$. In particular, an indefinite quaternion algebra can be embedded into $M_2(\RR)$. In general, a quaternion algebra over $F$ has a basis consisting of $1$ and elements $\alpha$, $\beta$ and $\alpha\beta$ such that $\alpha^2$ and $\beta^2$ are nonzero elements of $F$ and $\beta\alpha=-\alpha\beta$. There is also a \dfn{canonical involution} on a quaternion algebra given by $$ (a+b\alpha +c\beta +d\alpha\beta)' = a -b\alpha -c\beta +d\alpha\beta \quad \text{ or } \quad \pmatrix a& b\\c&d\endpmatrix' = \pmatrix d& -b \\ -c & a\endpmatrix . $$ Then the \dfn{reduced trace} and \dfn{reduced norm} of an element $\gamma$ are $\gamma + \gamma'$ and $\gamma\cdot \gamma'$ respectively. A subring of a quaternion algebra that is also a lattice, i.e., a free $\ZZ$-module of rank ~$4$, is called an \dfn{order} in the quaternion algebra. Very roughly, maximal orders play a similar role for quaternion algebras as rings of integers do for number fields, except that maximal orders in quaternion algebras need not be unique. \definition{1.10. Definition} A \dfn{complex structure} on a real vector space $\Wr$ of dimension~$2g$ is given by any of the following equivalent data: \roster \item"(i)" A scalar multiplication by $\CC$ with which $\Wr$ is a $g$-dimensional complex vector space. \item"(ii)" An endomorphism $J\in \End(\Wr)$ such that $J^2 =-\Id$. \item"(iii)" A homomorphism $h_1: \U(1) \to \GL(\Wr)$ of algebraic groups over ~$\RR$ such that for $u \in \U(1)$ the action of $h(u)$ on $W_\CC$ has only $u^{\pm1}$-eigenspaces, each occuring with equal multiplicity~$g$. \item"(iv)" A homomorphism $h: \SS \to \GL(\Wr)$ of algebraic groups over {}~$\RR$ such that for $(z,w)\in \SS(\CC)$ the action of $h(z,w)$ on $W_\CC$ has only $z^1w^0$- and $z^0w^1$-eigenspaces, each occuring with equal multiplicity~$g$. \endroster \enddefinition \demo\nofrills{} To see the equivalence of these conditions, if a complex vector space structure is given, let $J$ be the action of multiplication by $i=\sqrt{-1}$. Then $h_1(i) = J$ determines either $h_1$ or $J$ in terms of the other. Since $\SS = \Gm_{/\RR} \cdot \U(1)$, then $h$ is determined by $h_1$ and $\RR$-linearity, or by $h(a+bi) = a\Id +bJ$. And $h$ in turn defines a scalar multiplication by ~$\CC$. Also $h_1$ is the restriction of $h$ to $\U(1) \subset \SS$, on account of which we sometimes simply write $h$ instead of ~$h_1$. \enddemo \example{1.10.1. Example} In the notation of 1.6.5, $\ad(H_0)$ defines a complex structure on {}~$\frak p$, with which $\frak p$ becomes a complex vector space. \endexample \example{1.10.2. Example} To give a complex torus $V/L$, where $V$ is a $g$-dimensional complex vector space and $L \subset V$ is a lattice, is the same as giving a real $2g$-dimensional vector space $W$ together with a complex structure, say $J$, and a lattice $L \subset W$. We can go back and forth between these two points of view by thinking of $W$ as the real vector space underlying $V$ and $J$ as the induced complex structure, or by thinking of $V$ as the complex vector space defined by the pair $(W,J)$. In particular it will sometimes be convenient below to present a complex torus as a triple $(W, J, L)$ instead of in the form $V/L$. \endexample \subhead{1.11. Complex abelian varieties} \endsubhead After the definition given in 1.1.1, a complex abelian variety $A$ is a complete, connected algebraic group over~$\CC$ whose group law is necessarily commutative. A morphism of abelian varieties will always be taken to mean a morphism in the sense of algebraic groups (see~1.1). \definition{1.11.1. Definition} A \dfn{Riemann form} on a complex torus $V/L$ is a nondegenerate, skew-symmetric, real-valued, $\RR$-bilinear form $E:V\times V\to\RR$ such that \roster \item"(i)" $E(iv,iw) = E(v,w)$, \item"(ii)" $(v,w) \mapsto E(v,iw)$ is symmetric and positive definite, and \item"(iii)" $E(v,w)\in\ZZ$ whenever $v,w\in L$. \endroster \enddefinition For a proof of the following proposition, see the references cited in~1.1.1. \proclaim{1.11.2. Proposition} A complex torus is the underlying manifold of a complex abelian variety if and only if it admits a Riemann form. \Qed \endproclaim \definition{1.11.3. Definition} A morphism of complex abelian varieties is called an \dfn{isogeny} if it is surjective and has a finite kernel. Given an isogeny $\phi:A\to A'$ there exists a dual isogeny $\phi\dual: A'\to A$ such that $\phi\spcheck\circ\phi = m \Id_{A}$ and $\phi\circ\phi\spcheck = m \Id_{A'}$ for some positive integer~$m$ called the degree of~$\phi$. Thus two complex abelian varieties are said to be \dfn{isogenous} iff there exists an isogeny from one to the other, and being isogenous is an equivalence relation. An abelian variety is said to be \dfn{simple} iff it is not isogenous to a product of (positive dimensional) abelian varieties. \enddefinition The following proposition is proved in Lecture~12, 12.25, or see the references on abelian varieties cited above. \proclaim{1.11.4. Proposition {\rm (Poincar\'e Reducibility Theorem)}} If $A$ is an abelian variety and $A' \subset A$ is an abelian subvariety, then there exists an abelian subvariety $A'' \subset A$ such that $A' \cap A''$ is finite and $A$ is isogenous to $A'\times A''$. In particular, any abelian variety is isogenous to a product of simple abelian varieties. \endproclaim \definition{1.11.5. Definition} Two Riemann forms $E$ and $E'$ are said to be \dfn{equivalent} iff there exist positive integers $n$ and $n'$ such that $nE = n'E'$. A \dfn{polarization} of a complex torus $T$ is an equivalence class, say {}~$[E]$, of Riemann forms on~$T$. By a \dfn{polarized abelian variety} we mean an abelian variety together with a choice of polarization. In light of 1.10.2 and 1.11.2, a polarized abelian variety is determined by data $(W, J, L, E)$, where $W$ is an even-dimensional real vector space, $J$ is a complex structure on ~$W$, $L$ is a lattice in ~$W$, and $E$ is a Riemann form on the complex torus $(W,J,L)$. \enddefinition \subhead 1.12.. The endomorphism algebra of an abelian variety \endsubhead For a complex abelian variety $A$ let $\End(A)$ denote its endomorphism ring, and let $$ \EndoA := \End(A)\tensor_\ZZ \QQ . $$ \proclaim{1.12.1. Lemma} \roster \item When $A$ and $A'$ are isogenous abelian varieties, $\EndoA \simeq \Endo(A')$. \item $\EndoA$ is a semisimple $\QQ$-algebra with a positive involution. \endroster \endproclaim Recall that an involution $\iota$ of $\EndoA$ is said to be positive if for nonzero $\phi\in \EndoA$ the trace $\operatorname{Tr}(\phi\cdot \phi^\iota) > 0$. \demo{Proof} If $\phi:A\to A'$ is an isogeny and $\phi\dual:A'\to A$ is the dual isogeny, then $\phi^*:\Endo(A') \to \EndoA$ and $\frac 1 m (\phi\dual)^* : \EndoA \to \Endo(A')$ are mutually inverse ring homomorphisms. To prove part~2, first observe that in general the image of a homomorphism of complex tori is a subtorus, and the kernel is a closed subgroup whose connected component of the identity is a subtorus of finite index in the full kernel. Thus Schur's Lemma implies that $\EndoA$ is a division algebra when $A$ is a simple abelian variety, and then the semisimplicity of $\EndoA$ for general $A$ follows from the Poincar\'e Reducibility Theorem. To see that $\EndoA$ has a positive involution $\iota$, let $E$ be a Riemann form on $A$. Then $H(u,v) = E(u,iv) + iE(u,v)$ is a Hermitian form on $\Wr$, and if we take $\iota$ to be the antiautomorphism that takes $\phi\in\EndoA$ to its adjoint with respect to~$H$ then the conditions making $E$ a Riemann form imply that $\iota$ is a positive involution. \Qed \enddemo \definition{Definition} The involution $\iota$ of $\EndoA$ described in the proof above is called the \dfn{Rosati involution.} \enddefinition Thus when $A$ is simple, $\EndoA$ is a division algebra over~$\QQ$ which admits a positive involution. Such algebras were classified by Albert \cite{B.6}, \cite{B.7}, \cite{B.8}, see also \cite{B.109} and \cite{B.79}. The result is the following. \proclaim{1.12.2. Theorem {\rm (Albert classification)}} Let $A$ be a simple complex abelian variety. Let $K$ be the center of $\EndoA$ and let $K_0$ be the subfield of elements of $K$ fixed by the Rosati involution. Then $\EndoA$ is one of the following types: \roster \item"(I)" $\EndoA= K =K_0$ is a totally real algebraic number field, and the Rosati involution acts as the identity. \item"(II)" $K=K_0$ is a totally real number algebraic field, and $\EndoA$ is a division quaternion algebra over~$K$ such that every simple component of $\EndoA\tensor_\QQ \RR$ is isomorphic to $M_2(\RR)$; there is an element $\beta\in \EndoA$ such that $\vphantom{\beta}^{\operatorname{t}}\beta = -\beta$, and $\beta^2 \in K$ is totally negative; and the Rosati involution is given by $\alpha^\iota = \beta^{-1}\cdot\vphantom{\alpha}^{\operatorname{t}}\!\alpha\cdot\beta$. \item"(III)" $K=K_0$ is a totally real number algebraic field, and $\EndoA$ is a division quaternion algebra over~$K$ such that every simple component of $\EndoA\tensor_\QQ \RR$ is isomorphic to the Hamiltonian quaternion algebra $\Bbb H$ over~$\RR$; and $\alpha^\iota = \tr\alpha$. \item"(IV)" $K_0$ is a totally real number field, and $K$ is a totally imaginary quadratic extension of $K$, and $\EndoA$ is division algebra with center $K$, and the restriction of the Rosati involution to $K$ acts as the restriction of complex conjugation to ~$K$. \endroster \endproclaim Thus we will say that an abelian variety or an abelian manifold ~$A$ is \dfn{of type} (I), (II), (III) or (IV) if $A$ is simple and $\Endo(A)$ is of that type in the classification above, or when $A$ is not simple, if it is isogenous to a product of simple abelian varieties of that type. \subhead 1.13. Examples of abelian varieties \endsubhead We now introduce some basic constructions of complex abelian varieties with various endomorphism algebras. In the next sections we will more fully analyze their Hodge structures. \example{1.13.1. Elliptic curves} Let $E$ be a $1$-dimensional complex abelian variety, an \dfn{elliptic curve.} Then $E(\CC) \simeq \CC/(\tau\ZZ+\ZZ)$ for some $\tau\in\CC$ with $\Im\tau >0$. A Riemann form is given by the pairing $$ ((a\tau+b),(c\tau+d)) \mapsto \frac 1{\Im \tau} \Im((a\tau+b)(c\bar\tau+d)) = ad-bc $$ It is an elementary exercise to show that $\Endo(E)$ can only be isomorphic to $\QQ$ or to an imaginary quadratic field, say~$K$; in the latter case $E$ is said to have \dfn{complex multiplication} by $K$. Moreover $E$ has complex multiplication if and only if $\tau$ is quadratic over~$\QQ$, in which case $\QQ(\tau) = K$. The general elliptic curve, whose period $\tau$ has algebraically independent transcendental real and imaginary parts, has $\Endo(E)\simeq \QQ$. \endexample \example{1.13.2. Abelian varieties with multiplication by an imaginary quadratic field} We will say that an abelian variety has multiplication by an imaginary quadratic field $K$ if there is an embedding $K\hra \EndoA$. Following \cite{B.135}, to construct a simple complex abelian variety $A$ with multiplication $K$, let $W$ be an $n$-dimensional vector space over ~$K$. Then $\Wr = W\tensor_\QQ \RR$ may be identified with $\CC^n$, but we may also twist the complex structure as follows. Write $\Wr = \Wr' \oplus \Wr''$ as the direct sum of two subspaces over~$\CC$, and define a complex structure $J\in\End_\RR(\Wr)$ by $Jw' = i w'$ for $w'\in \Wr'$ and $J w'' = -i w''$ for $w''\in\Wr''$. Then with this complex structure, $\alpha \in K$ acts on $(\Wr,J)$ by $w'\mapsto \alpha w'$ for $w'\in \Wr'$ and $w'' \mapsto \bar\alpha w''$ for $w''\in \Wr''$. In particular, when $L\subset W$ is a lattice such that $L \tensor_\ZZ\QQ = W$, then the set of $\alpha \in K$ such that $\alpha\cdot L \subseteq L$ is a subring of $K$ commensurable with the ring of integers of~$K$. Thus $A(\CC)=(\Wr/L, J)$ is a complex torus with an embedding $K\hra \Endo(A)$. To exhibit a Riemann form for $A$, let $H$ be any $\QQ$-valued Hermitian form on $W\times W$ which as a $\CC$-valued form on $\Wr\times\Wr$ is positive definite on $\Wr'$ and negative definite on $\Wr''$, meaning in particular that these two subspaces are orthogonal with respect to ~$H$. Then $H = S + iE$ for $\RR$-valued forms $S$ and $E$, and the imaginary part~$E$ of {}~$H$ is a Riemann form for ~$A$. Let $n'= \dim_\CC \Wr'$ and $n'' = \dim_\CC \Wr''$. Then $n' +n'' =n$, and by \cite{B.109} Thm.5, when $n\ge 3$ then both $n'$ and $n''$ are positive, and $n=2$ does not occur. If $n' = n''$ the pair $(A,K)$ is said to be an abelian variety \dfn{of Weil type.} Equivalently, an abelian variety of Weil type is a pair $(A,K)$ consisting of an abelian variety $A$ and an imaginary quadratic field $K$ with an embedding $K\hra \Endo(A)$ such that for $\alpha\in K$ the corresponding endomorphism has the eigenvalues ~$\alpha$ and ~$\bar\alpha$ with equal multiplicity. The example of Mumford in \cite{B.88}, see Lecture~7, 7.23--7.28, is one example of an abelian variety of Weil type. Another case that will arise below is when $n'$ and $n''$ are relatively prime. We will refer to a pair $(A,K)$ satisfying this condition as an abelian variety \dfn{of Ribet type,} see \cite{B.94}~Thm.3. \endexample \example{1.13.3. Simple abelian varieties of odd prime dimension} Let $A$ be a simple complex abelian variety of odd prime dimension~$g$. Then reading off from the tables in \cite{B.85}, the cases that occur are: $\EndoA \simeq \QQ$, which is the general case; or $\EndoA$ is a totally real number field of degree~$g$ over ~$\QQ$; or $\EndoA$ is an imaginary quadratic field, in which case $A$ is of Ribet type; or $A$ is of CM-type, that is, a totally imaginary quadratic extension of a totally real field of degree~$g$ over~$\QQ$. \endexample \example{1.13.4. Simple abelian fourfolds} By reading the tables in \cite{B.85}, the endomorphism algebras that can occur for a simple abelian fourfold are, by Albert type: \roster \item"(I)" $\EndoA$ is $\QQ$, or a real quadratic field, or a totally real quartic field; \item"(II)" $\EndoA$ is an indefinite division quaternion algebra over $\QQ$ or a totally indefinite division quaternion algebra over a real quadratic field; \item"(III)" $\EndoA$ is a definite division quaternion over $\QQ$; \item"(IV)" $\EndoA$ is an imaginary quadratic field, in which case it can only be of Ribet type with $\{n',\,n''\} = \{1,\,3\}$ or of Weil type with $n'=n''=2$, or else $\EndoA$ is a CM-field of degree~$4$ or $8$ over~$\QQ$. \endroster \endexample \example{1.13.5. Abelian varieties with real multiplication} An abelian variety of type~(I) is sometimes said to have \dfn{real multiplication.} To construct an example of a simple abelian variety with real multiplication, let $K$ be a totally real number field with $[K:\QQ]=g$, and let $\scO$ be the ring of integers of ~$K$. Then there are $g$ distinct embeddings $\alpha\mapsto \alpha^{(j)}$ of $K$ into $\RR$. Let $\tau_j \in\CC$ with $\Im\tau_j>0$, for $1\le j\le g$. Then the image of $\scO \oplus \scO$ under the map $$ (\alpha,\beta)\mapsto (\alpha^{(1)}\tau_1 + \beta^{(1)}, \ldots , \alpha^{(g)}\tau_g + \beta^{(g)}) $$ is a lattice $L \subset \CC^g$, and $A = \CC^g/L$ is a complex abelian variety. A Riemann form is given by $$ E(z,w) = \sum_{j=1}^g (\Im\tau_j)^{-1} \Im(z_j\bar w_j), $$ where $z,w\in\CC^g$. Then $K\hra \Endo(A)$, and this is an isomorphism for general $(\tau_1,\dots,\tau_g)$. It can be shown that any simple abelian variety $A$ for which $\EndoA$ is a totally real number field is isogenous to one which can be constructed as we have here \cite{B.27}. \endexample \example{1.13.6. Abelian varieties of CM-type} Recall that an algebraic number field $K$ is said to be a \dfn{CM-field} iff it is a totally imaginary quadratic extension of a totally real number field~$K_0$. The embeddings of a CM-field $K$ into $\CC$ come in complex conjugate pairs. Then \dfn{CM-type} for $K$ is a subset $S\subset\Hom(K,\CC)$ containing exactly one from each pair of conjugate embeddings, so that $\Hom(K,\CC) = S \cup \ol S$. A simple abelian variety $A$ is said to be of \dfn{CM-type,} or to have \dfn{complex multiplication} by ~$K$, iff there exists a field $K\hra \Endo(A)$ such that $[K:\QQ] \ge 2\dim A$, in which case equality holds, $K\simeq \Endo(A)$ and $K$ is a CM-field, see \cite{B.115} or \cite{B.70}. More generally, an abelian variety may be said to be of \dfn{CM-type} if it is isogenous to a product of simple abelian varieties of CM-type, or equivalently if $\EndoA$ contains a commutative semisimple $\QQ$-algebra $R$ with $[R:\QQ] = 2\dim A$. To construct a simple abelian variety of CM-type, let $K$ be a CM-field with totally real subfield $K_0$ such that $[K:\QQ]=2g$, and let $S$ be a CM-type for ~$K$, and let $\scO$ be the ring of integers of ~$K$. Then $K\tensor_\QQ \RR \simeq \CC^g$. If we embed $\scO \hra \CC^g$ by $\alpha \mapsto (\sigma \alpha)_{\sigma\in S}$ and let $L$ be the image of this map, then $A=\CC^g/L$ is an abelian variety. To construct a Riemann form, choose an element $\beta\in \scO$ such that $K= K_0(\beta)$, and $-\beta^2$ is a totally positive element of $K_0$, and $\Im(\sigma\beta)>0$ for $\sigma \in S$. Then $$ E(z,w) = \sum_{j=1}^g \sigma_j(\beta) (z_j\ol w_j - \ol z_j w_j) $$ is a Riemann form, where $z,w\in\CC^g$. When $z= (\sigma \alpha_1)_{\sigma\in S}$ and $w= (\sigma \alpha_2)_{\sigma\in S}$ with $\alpha_1, \alpha_2 \in K$, then $E(z,w) = \operatorname{Tr}_{K/\QQ}(\beta \alpha_1 \ol \alpha_2)$. Moreover, $A$ has complex multiplication by $K$, with $\alpha \in \scO$ acting by $z_j \mapsto \sigma_j(\alpha) z_j$ for $1\le j\le g$; for more detail see \cite{B.70}~\S1.4. In addition, it can be shown that any simple abelian variety of CM-type is isogenous to one such as we constructed above \cite{B.27}. \endexample \example{1.13.7. Abelian surfaces with quaternionic multiplication} A simple abelian variety of type~(II) may be said to have \dfn{quaternionic multiplication,} or sometimes, to be of \dfn{QM-type.} The simplest example is an abelian surface $A$ whose endomorphism algebra is an indefinite division quaternion algebra $D$ over $\QQ$. To construct such an abelian surface, let $\scO$ be an order in ~$D$, fix an embedding $j: D \hra M_2(\RR)$, and let $\tau\in\CC$ with $\Im\tau >0$. Then the image of $\scO$ under the map $\psi: \alpha \mapsto j(\alpha)\left(\smallmatrix \tau \\ 1\endsmallmatrix \right)$ is a lattice $L \subset \CC^2$, and $A=\CC^2/L$ is a $2$-dimensional abelian variety. In the special case that $D\simeq M_2(\QQ)$, then $\scO$ is commensurable with $M_2(\ZZ)$ and $A$ is isogenous to the product of two isogenous elliptic curves. Otherwise $D$ is a division algebra and $A$ is a simple abelian surface. In this latter case there is an element $\beta\in \scO$ such that $\beta' = -\beta$ and $\beta^2<0$ in $\QQ$ (recall that $\beta\mapsto \beta'$ is the canonical involution on~$D$). Then a Riemann form on $A$ is given by $$ E((z_1,z_2),(w_1,w_2)) = \frac 1{\Im \tau} \Im \operatorname{Tr}\left( j(\beta)\cdot \pmatrix z_1\ol w_2 & \ol z_1 w_1 \\ z_2\ol w_2 & \ol z_2 w_1 \endpmatrix \right) = \operatorname{Tr} (\beta \cdot \alpha_1' \cdot \alpha_2 ) $$ when $\left(\smallmatrix z_1 \\ z_2 \endsmallmatrix \right) = \psi(\alpha_1)$ and $\left(\smallmatrix w_1 \\ w_2 \endsmallmatrix \right) = \psi(\alpha_2)$ for $\alpha_1,\alpha_2\in D\tensor_\QQ \RR$. Furthermore, for $\gamma\in \scO$, multiplication by $j(\gamma)$ on $\left(\smallmatrix z_1 \\ z_2\endsmallmatrix \right) \in\CC^2$ preserves $L$. Then by tensoring with $\QQ$ we get an inclusion $D\hra \Endo(A)$, which is an isomorphism for general~$\tau$. We leave it as an exercise for the reader to combine the construction of this example with that of 1.13.5 to obtain an arbitrary simple abelian variety of type~(II). \endexample \example{1.13.8. General abelian varieties} When $A$ is a $g$-dimensional complex abelian variety then $A(\CC) \simeq \CC^g/(T\ZZ^g + \ZZ^g)$ for some $T$ in the \dfn{Siegel upper half-space of genus~$g$,} consisting of symmetric complex $g\times g$ matrices with positive-definite imaginary part. Then generalizing the $1$-dimensional case, a Riemann form is given by $$ E(\bold z, \bold w) = \Im(\tr\bold z (\Im T)^{-1} \ol{\bold w}) = \tr \bold a \cdot \bold d - \tr \bold b \cdot \bold c $$ when $\bold z = T\bold a + \bold b$ and $\bold w = T \bold c + \bold d$, with $\bold a, \bold b, \bold c, \bold d \in \RR^g$. Then for general $T$, that is, when all the real and imaginary parts of the distinct entries of $T$ are algebraically independent real transcendental numbers, $\EndoA \simeq \QQ$. \endexample \head 2. The Hodge, Mumford-Tate and Lefschetz groups of an abelian variety \endhead \rightheadtext{The Hodge and Mumford-Tate groups} \nopagebreak \subhead 2.1. Rational Hodge structures \endsubhead A real Hodge structure is a natural generalization of a complex structure, and a rational Hodge structure is a real Hodge structure with an underlying $\QQ$-structure. For a rational vector space $V=V_\QQ$ we write $\Vr =V\tensor_\QQ \RR$ and $V_\CC = V\tensor_\QQ \CC$. \definition{2.1.1. Definition} A \dfn{rational Hodge structure of weight~$n$} consists of a finite-dimensional $\QQ$-vector space $V$ together with any of the following equivalent data: \roster \item"(i)" A decomposition $V_\CC = \bigoplus_{p+q=n} V^{p,q}$, called the {\rm Hodge decomposition,} such that $\ol{V^{p,q}} = V^{q,p}$. \item"(ii)" A decreasing filtration $F_H^rV_\CC$ of $V_\CC$, called the {\rm Hodge filtration,} such that $F_H^rV_\CC \oplus \ol{F_H^{n-r+1}V_\CC} =V_\CC$. \item"(iii)" A homomorphism $h_1:\U(1) \to \GL(\Vr)$ of real algebraic groups, and also specifying that the weight of the Hodge structure is {}~$n$. \item"(iv)" A homomorphism $h:\SS\to\GL(\Vr)$ of real algebraic groups such that via the composition $\Gm_{/\RR} \hra \SS \to \GL(\Vr)$ an element $t\in \Gm_{/\RR}$ acts as $t^{-n}\cdot\Id$. \endroster \demo\nofrills{} To see that the data (i)--(iv) are equivalent, the Hodge decomposition and the Hodge filtration are related by $F_H^rV_\CC = \bigoplus_{p\ge r}V^{p,n-p}$ and $V^{p,n-p} = F_H^pV_\CC \cap \ol{F_H^{n-p}V_\CC}$. The homomorphism $h$ and the Hodge decomposition are related by $h(z)\cdot v = z^{-p}\bar z^{-q} v$ for $v\in V^{p,q}$. The homomorphism $h_1$ can be obtained as the restriction of $h$; conversely, $V^{p,q}$ can be recovered as the subspace of $V_\CC$ on which $h_1(u)$ acts as $u^{q-p}$, provided $n=p+q$ is specified. \enddemo \enddefinition \example{2.1.2. Examples} 1. Let $V$ be a $\QQ$-vector space of even dimension, and suppose $h:\SS \to \GL(V_\RR)$ defines a complex structure on ~$V_\RR$. Then $V$ is a rational Hodge structure of weight~$-1$: criterion 1.10(iv) in the definition of complex structure immediately implies criterion 2.1.1(iv). Alternatively, the $z^1w^0$- and $z^0w^1$-eigenspaces of $h(z,w)$ acting on $V_\CC$ are $V^{-1,0}$ and $V^{0,-1}$ respectively, and these are complex conjugate as require by 2.1.1(i) because $h$ is defined over $\RR$. 2. When $X$ is (the analytic space underlying) a complex projective variety, or more generally, a compact K\"ahler manifold, then $H^n(X,\QQ)$ is a rational Hodge structure of weight~$n$. 3. In general, when $V$ is a rational Hodge structure of weight~$n$, its dual $V\dual$ is a rational Hodge structure of weight~$-n$, and $V^{\tensor r}\tensor_\QQ (V\dual)^{\tensor s}$ is a rational Hodge structure of weight $(r-s)n$. \endexample \remark{Remark} The convention that $(z,w)\in \SS(\CC)$ acts as $z^{-p}w^{-q}$ on $V^{p,q}$ follows \cite{B.25}~\S1 and \cite{B.27}. But as we will see below it really is the more natural choice in the context of Hodge structures associated to complex abelian varieties. \revert@envir\endremark\medskip \definition{2.1.3. Definition} The \dfn{type} of a rational Hodge structure $V$ is the set of pairs $(p,q)$ such that $V^{p,q}\ne 0$. \enddefinition \definition{2.1.4. Definition} When $V$ is a rational Hodge structure of even weight~$2p$, the subspace of \dfn{Hodge vectors} in $V$ is the subspace of $1$-dimensional rational sub-Hodge structures of~$V$, $$ \Hdg(V) : = V_\QQ \cap V^{p,p} . $$ \enddefinition \definition{2.1.5. Definition} A \dfn{morphism} of rational Hodge structures $\phi:V_1\to V_2$ is a $\QQ$-vector space map on the underlying vector spaces such that over $\CC$ \roster \item"(i)" $\phi(V_1^{p,q}) \subseteq V_2^{p,q}$, for all $p,q$; or \item"(ii)" $\phi(F_H^rV_{1\CC})\subseteq F_H^r V_{2\CC}$, for all ~$r$; or \item"(iii)" $\phi$ commutes with the action of $U(1)$, and preserves the common weight of $V_1$ and $V_2$; or \item"(iv)" $\phi$ commutes with the action of $\SS$, preserving the action of $\Gm_{/\QQ} \hra \SS$. \endroster \enddefinition \definition{2.1.6. Definition} A \dfn{polarization} of a rational Hodge structure $V$ is a morphism of rational Hodge structures $\psi : V\tensor V \to \QQ(-n)$ such that the real-valued form $(u,v) \mapsto \psi(u,h(i)v)$ on $\Vr$ is symmetric and positive-definite. Here $\QQ(m)$ is the is the vector space $\QQ$ as a one-dimensional rational Hodge structure of type~$(-m,-m)$, for $m\in\ZZ$. \enddefinition \subsubhead 2.1.7. The rational Hodge structure associated to an abelian variety \endsubsubhead When we speak of the rational Hodge structure associated to an abelian variety $A$ we always mean the rational Hodge structure $H^1(A,\QQ)$. Moreover, any morphism $\phi : A\to A'$ of abelian varieties induces a morphism of rational Hodge structures $\phi^*: H^1(A',\QQ) \to H^1(A,\QQ)$. In particular, it is easy to check that when $\phi$ is an isogeny it induces an isomorphism on the associated rational Hodge structures. Thus, up to isomorphism, the rational Hodge structure associated to an abelian variety depends only on its isogeny class. Similarly, an element of $\EndoA$ induces an endomorphism on the rational Hodge structure $H^1(A,\QQ)$. \remark{2.1.8. Notation} For a complex abelian variety $A$ we will regularly let $W= H_1(A,\QQ)$ and $V=H^1(A,\QQ) = W\dual$. Further, we denote the Hodge classes of $A$ by $$ \Hdg^p(A) := H^{2p}(A,\QQ) \cap H^{p,p}(A) \qquad\quad \Hdg(A) := \bigoplus_p \Hdg^p(A) , $$ and let $\Div^p(A)\subset H^{2p}(A,\QQ)$ be the $\QQ$-linear span of $p$-fold intersections of divisors on $A$, and $\Div(A) := \bigoplus_p \Div^p(A)$. \revert@envir\endremark\medskip \definition{2.2. Definition} Let $(V, h:\SS\to\GL(\Vr))$ be a rational Hodge structure. The \dfn{Hodge group} $\Hg(V)$ of~$V$, also called the \dfn{special Mumford-Tate group} of~$V$, is the smallest algebraic subgroup of $\GL(V)$ defined over $\QQ$ such that $h(\U(1)) \subset \Hg(\Vr)$. The \dfn{Mumford-Tate group} $\MT(V)$ of $V$ is the smallest algebraic subgroup of $\GL(V)$ defined over $\QQ$ such that $h(\SS)\subset \MT(\Vr)$. As a matter of notation we let $\hg(V) := \Lie(\Hg(V))$ and $\mt(V) := \Lie(\MT(V))$ denote their respective Lie algebras, as subalgebras of $\End_\QQ(V)$. When $A$ is a complex abelian variety then by the Hodge or the Mumford-Tate group of $A$ we mean $\Hg(A) := \Hg(H^1(A,\QQ))$ and $\MT(A) := \MT(H^1(A,\QQ))$ respectively. \enddefinition \remark{Remark} The Hodge group of an abelian variety was introduced in \cite{B.77}, see also \cite{B.78}. The general notion of the Mumford-Tate group of a rational Hodge structure seems to appear first in \cite{B.24}~\S7, and from a rather abstract point of view, in \cite{B.97}. A thorough analysis of the Mumford-Tate groups of Hodge structures that can be generated by the Hodge structures of abelian varieties can be found in \cite{B.25}~\S1, while the best place to find proofs of the basic properties of Mumford-Tate groups in general is \cite{B.27}~\S3. \revert@envir\endremark\medskip The first properties to follow directly from the definition are the following. \proclaim{2.3. Lemma} Let $V$ be a rational Hodge structure. Then \roster \item"(i)" $\Hg(V)$ and $\MT(V)$ are connected linear algebraic groups; \item"(ii)" $\Hg(V)\subseteq \SL(V)$; \item"(iii)" $\MT(V) = \Gm \cdot \Hg(V)$. \Qed \endroster \endproclaim The vital role of Mumford-Tate groups in analyzing Hodge structures comes from the following fact. \proclaim{2.4. Proposition} Let $V$ be a rational Hodge structure, and $r,s\in\NN$. Then $\MT(V)$ acts on the rational Hodge structure $V^{\tensor r}\tensor (V\dual)^{\tensor s}$, and the rational $\MT(V)$-subrepresentations in $V^{\tensor r}\tensor (V\dual)^{\tensor s}$ are precisely the rational sub-Hodge structures of $V^{\tensor r}\tensor (V\dual)^{\tensor s}$. \endproclaim The action of $\MT(V)$ on $V$ extends ``diagonally'' to an action on $V^{\tensor r}$, and the action of $\MT(V)$ on $V\dual$ is the contragredient of its action on $V$, as in~1.4. \demo{Proof} To simplify the notation, let $T:= V^{\tensor r}\tensor (V\dual)^{\tensor s}$. Then action of $\GL(V)$ on $V$ induces an action of $\GL(V)$, and thus of $\MT(V)$ on $T$. Now suppose first that $W\subset T$ is a $\QQ$-rational subspace preserved by the $\MT(V)$-action. Then over $\RR$ the composition $h:\SS \hra \MT(\Vr) \to \GL(\Wr)$ describes the sub-Hodge structure on $\Wr \subset T\tensor \RR$. Conversely, if $W\subset T$ is a rational sub-Hodge structure, then $W$ is a rational subspace of $T$ such that $\Wr \subset T_\RR$ is preserved by the action of $h(\SS)$. Therefore $W$ is preserved by the action of $\MT(V)$ on $T$. \Qed \enddemo \proclaim{Corollary} Let $V$ be a rational Hodge structure of weight~$n$. Then for any $r,s\in \NN$ such that $(r-s)n =2p$, $$ \Hdg^p(V^{\tensor r}\tensor (V\dual)^{\tensor s}) = (V^{\tensor r}\tensor (V\dual)^{\tensor s})^{\Hg(V)} $$ \endproclaim \remark{Remark} Hodge and Mumford-Tate groups have proved to be a powerful for studying the Hodge conjecture for abelian varieties. Typically the starting point is this corollary to Proposition~2.4, which implies that the Hodge cycles in $H^*(A^n,\QQ)$, say, for a complex abelian variety $A$, are precisely the invariants under the action of $\Hg(A)$. Then, the problem is to determine enough about $\Hg(A)$ to be able to describe its invariants, or at least determine their dimension. In many cases it is possible to show this way that the space of Hodge cycles is generated by those of degree~$2$, in other words, by divisors on $A$, and then in these cases the Hodge conjecture is verified. In other cases it is possible to show that the space of Hodge cycles is {\sl not\/} generated by divisors, but still something can be said about the dimension of the space of Hodge cycles. However, the actual computations are sometimes quite technical. There are many variously narrow results, and a few general results, as we will try to show in the following sections. \revert@envir\endremark\medskip Since a polarization of a complex abelian variety induces a polarization on its associated Hodge structure, the following proposition insures that all the Hodge and Mumford-Tate groups with which we will work in this appendix are reductive (definition~1.4); compare \cite{B.25}~Principe~1.1.9. \proclaim{2.5. Proposition} Let $V$ be a polarizable rational Hodge structure. Then $\MT(V)$ and $\Hg(V)$ are reductive. \endproclaim \demo{Proof {\rm (after \cite{B.27}~Prop.3.6)}} Suppose $h:\U(1) \to \GL(\Vr)$ defines the Hodge structure on $V$, and let $\psi : V\tensor V \to \QQ(-n)$ be a polarization. Then for $u,v \in V_\CC$ and $g \in \Hg(V,\CC)$ (the complexification, or complex points of $\Hg(V)$) $$ \psi(u,h(i) \bar v) = \psi(gu,gh(i)\bar v) = \psi(gu, h(i) (h(i)^{-1} g h(i)) \bar v) = \psi(gu, h(i)\ol{g^*v}), $$ where the first equality holds because $\psi$ is a Hodge structure morphism, and $g^* := h(i)^{-1} \bar g h(i)$. Therefore the positive-definite form on $\Vr$ given by $(u,v)\mapsto \psi(u,h(i)v)$ is invariant under the real form $\Hg^*(\Vr)$ of $\Hg(V,\CC)$ fixed by the involution $g\mapsto g^*$. It follows that $\Hg^*(\Vr)$ is a compact real form of $\Hg(V)$, and thus all its finite-dimensional representations are semisimple. This is equivalent to $\Hg^*(\Vr)$ being reductive, see for example \cite{B.103}~I.3. Then since the linear algebraic $\Hg(V)$ possesses a reductive real form, it is reductive (as an algebraic group), and then $\MT(V)$ is reductive as well. \Qed \enddemo \proclaim{2.5.1. Corollary} When $V$ is a polarizable rational Hodge structure, $\Hg(V)$ is the largest subgroup of $\GL(V)$ fixing all Hodge vectors in all $V^{\tensor r}\tensor (V\dual)^{\tensor s}$, for $r,s\in\NN$. \endproclaim \demo{Proof} A reductive subgroup of $\GL(V)$ is characterized by its invariants in the extended tensor algebra of ~$V$ (compare \cite{B.27}~Prop.3.1). \Qed \enddemo Once we know that $\Hg(V)$ and $\MT(V)$ are reductive, the next corollary follows from Lemma~2.3(iii). \proclaim{2.5.2. Corollary} Let $V$ be polarizable rational Hodge structure. Then $\Hg(V)$ is semisimple if and only if the center of $\MT(V)$ is $\Gm$, i.e., consists only of scalars. \endproclaim Now we turn specifically to the Hodge structures of complex abelian varieties. When combined with Proposition~2.4, the following lemma says that the endomorphism algebra of a complex abelian variety may be identified with the rational Hodge structure endomorphisms of its associated rational Hodge structure. \proclaim{2.6. Lemma} Let $A$ be a complex abelian variety. Then $$ \EndoA \isom \End_{\MT(A)}(H_1(A,\QQ)) = \End_{\Hg(A)}(H_1(A,\QQ)). $$ \endproclaim \demo{Proof} Let $\dim A =g$. Then $A(\CC) = \CC^g/L$ for some lattice $L$, and let $W=W_\QQ := L\tensor\QQ$. Then we can identify $W \simeq H_1(A,\QQ)$ and identify the universal covering space $\CC^g$ of $A(\CC)$ as the real $2g$-dimensional space $\Wr = W\tensor \RR$ together with the induced complex structure represented as a homomorphism $h:\SS\to\GL(\Wr)$. Now an element of $\EndoA$ is characterized by firstly being a $\QQ$-linear endomorphism of $W$ and secondly being a complex-linear endomorphism of $\Wr$, which precisely means that it commutes with $h(\SS(\RR))$. But a $\QQ$-linear endomorphism of $W$ that commutes with $h(\SS(\RR))$ must commute with all of $\MT(A)$ acting on {}~$W$. \Qed \enddemo \proclaim{2.7. Proposition {\rm (\cite{B.17})}} Let $A$ be a complex abelian variety. If $\EndoA$ is a simple $\QQ$-algebra with center $\QQ$, then $\Hg(A)$ is simple. \endproclaim \demo\nofrills Recall that $\EndoA$ is a simple $\QQ$-algebra precisely when $A$ is simple. Then following \cite{B.142}~p.66, the idea of the proof is that the center, say $\frak c$, of $\mt(A)$ contains $\QQ\cdot\Id$ and is contained in the center of $\EndoA$. So if the center of $\EndoA$ is $\QQ\cdot \Id$ then $\frak c =\QQ\cdot \Id$, and $\hg(A)$ is semisimple, and for simple $A$, it is simple. \enddemo As we have already begun to see, most of what can be said about the structure and classification of the Hodge and Mumford-Tate groups of abelian varieties is a consequence of the presence and properties of a polarization. The most fundamental fact is the following. \proclaim{2.8. Lemma} Let $A$ be a complex abelian variety, and let $[E]$ be a polarization of $A$ represented by the Riemann form~$E$. Further, let $W=W_\QQ =H_1(A,\QQ)$. Then $E$ is a skew-symmetric bilinear form on $W$, and there are natural representations $$ \Hg(A) \hra \Sp(W,E) \qquad \text{and}\qquad \MT(A) \hra \GSp(W,E) . $$ \endproclaim \demo{proof} This follows from the observation that the Riemann form $E$ is a polarization on the rational Hodge structure $W$. First, $E(h(i)u, h(i)v) = E(u,v)$, where $u,v\in\Wr$ and $h:\SS\to\GL(\Wr)$ represents the complex structure. Thus if we write $h(s) = a\Id + b h(i)$, then $E(h(s)u,h(s)v) = |a +bi|^2 E(u,v)$. Therefore $h(\SS) \hra \GSp(\Wr,E)$, and then by taking the Zariski-closure over~$\QQ$ we find $\MT(A) \hra \GSp(W,E)$. \Qed \enddemo The following criterion for the semisimplicity of the Hodge group is linked to whether there are any simple components of Weil type, see~1.13.2; cf.~also the discussions in sections four and five, below. \proclaim{2.9. Proposition \rm{(\cite{B.118})}} Suppose $A$ is an abelian variety defined over $\CC$. Then the Hodge group of $A$ is not semisimple if and only if for some simple component $B$ of $A$ the center of $\Endo(B)$ is a CM-field ~$K$ such that $(B,K)$, with $K$ embedded in $\Endo(B)$ by the identity map, is not of Weil type. \endproclaim The next proposition is that the real Lie groups $\Hg(A,\RR)$ and $\MT(A,\RR)$ are of Hermitian type (see~1.6.5). \proclaim{2.10. Proposition \rm{(\cite{B.77})}} Let $A$ be a complex abelian variety. Then $\Hg(A,\RR)$ is of Hermitian type. Further, letting $K = K_\RR$ denote the centralizer of $h(i)$, or equivalently of $h(\U(1))$, then the topologically connected component $K^+$ of $K$ is a maximal compact subgroup of the topologically connected component $\Hg(A,\RR)^+$ of $\Hg(A,\RR)$, and the quotient $\Hg(A,\RR)^+/K^+$ is a Hermitian symmetric space of noncompact type, i.e., a bounded symmetric domain. \endproclaim \demo{Proof {\rm (after \cite{B.25}, but see also \cite{B.103} and \cite{B.51}~Ch.VIII)}} First, the connected center of $\Hg(A,\RR)$ is compact, since the group itself is generated by compact subgroups, namely the $\Aut_\QQ(\CC)$-conjugates of $h(\U(1))$. From the proof that $\Hg(A)$ is reductive it follows that $\Ad(h(i))$ defines a Cartan involution on $\Hg(A,\RR)$, and thus $\ad(h(i))$ is a Cartan involution on $\frak g = \hg(A,\RR)_{\text{ss}}$, the semisimple part of the reductive Lie algebra $\hg(A,\RR)$. Let $\frak k + \frak p$ be the corresponding Cartan decomposition. Then the restriction $\tilde J$ of $h(i)$ to $\frak p$ is a derivation of $\frak g$. Since $\frak g$ is semisimple, there therefore exists $H_0 \in \frak g$ such that $\tilde J = \ad(H_0)$.. And since $\tilde J$ commutes with the Cartan decomposition $\ad(h(i))$, we have that $H_0$ is in $\frak k$, and in the center of $\frak k$, as required. \Qed \enddemo The following result points to how the complexified Lie algebras of the Hodge and Mumford-Tate groups of a complex abelian variety fit into the general classification of complex semisimple Lie algebras. This formulation of the result follows \cite{B.142}. \proclaim{2.11. Theorem {\rm (\cite{B.25}~\S1, but see also \cite{B.108}~\S3 and Appendix and \cite{B.141})}} Let $A$ be a complex abelian variety, let $\frak g$ be a simple factor of the complex semisimple Lie algebra $\mt(A,\CC)_{\text{ss}}$, and let $r$ denote the rank of~$\frak g$. Further, let $W = H_1(A,\QQ)$, and let $V \subset W_\CC$ be an irreducible subrepresentation for the action of $\frak g$ on $W_\CC$. Then $\frak g$ and $V$ must be one of the following: \roster \item"(A)" $\frak g \simeq \frak{sl}(r+1)$ and $V$ is equivalent to the $s$-th exterior power of the standard representation of dimension $r+1$, for some $1\le s \le r$. \item"(B)" $\frak g \simeq \frak{so}(2r+1)$, and $V$ is equivalent to the spin representation of dimension ~$2^r$. \item"(C)" $\frak g \simeq \frak{sp}(2r)$, and $V$ is equivalent to the standard representation of dimension~$2r$. \item"(D)" $\frak g \simeq \frak{so}(2r)$, and $V$ is equivalent to the standard representation of dimension~$2r$, or to one of the two half-spin representations of dimension~$2^{r-1}$. \endroster \endproclaim We do not give the proof here, but we just mention that the proof depends essentially on the symplectic representation in ~2.8. \proclaim{2.12. Proposition {\rm(\cite{B.78})}} A complex abelian variety is of CM-type if and only if $\Hg(A)$ is an algebraic torus. \endproclaim \demo{Proof} Suppose first that $A$ is of CM-type. Then $\EndoA$ contains a commutative semisimple $\QQ$-algebra of dimension $2\dim A$ over $\QQ$. {}From Lemma~2.6 it follows that $\Hg(A)$ commutes with a maximal commutative semisimple subalgebra $R'\subset \End(W)$, where $W=H_1(A,\QQ)$. Therefore $\Hg(A)$ is contained in the units of $R'$ and thus must be an algebraic torus. Conversely, if $\Hg(A)$ is an algebraic torus, then it is diagonalizable over $\CC$. Therefore its centralizer in $\End(W) \tensor \CC$, and thus its centralizer in $\End(W)$, contains a maximal commutative semisimple subalgebra $R'\subset \End(W)$. But then $[R':\QQ] = \dim W = 2\dim A$ and $R'\subset \EndoA$, so $A$ is of CM-type. \Qed \enddemo \remark{Remark} If $A$ is an abelian variety with complex multiplication by $K$, and $K_0$ is the maximal totally real subfield of $K$, then a more precise statement is that $$ \Hg(A) \subseteq \Ker\{\Res_{K/\QQ} \Gm_{/K} \lra \Res_{K_0/\QQ} \Gm_{/K_0} \} , $$ where the arrow is induced by the norm map from $K$ to $K_0$. The arguement above shows that $\Hg(A) \subseteq \Res_{K/\QQ} \Gm_{/K}$. Then observe that $h_1(\U(1))$ is contained in the real points of the indicated kernel, and recall that $\Hg(A)$ is the smallest algebraic subgroup defined over $\QQ$ which over $\RR$ contains $h_1(\U(1))$. \revert@envir\endremark\medskip \definition{2.13. Definition} Let $K$ be a CM field, $S$ a CM-type for $K$, and let $A$ be the corresponding abelian variety (up to isogeny), as described in~1.5.4. Then the CM-type $(K,S)$ or the abelian variety $A$ with that CM-type is said to be \dfn{nondegenerate} if $\dim\Hg(A) = \dim A = \frac12 [K:\QQ]$. \enddefinition \subhead The Lefschetz group of an abelian variety \endsubhead The Lefschetz group of an abelian variety was first studied by Ribet \cite{B.94} and further investigated by Murty \cite{B.82}~\S2. \definition{2.14. Definition} Let $A$ be a complex abelian variety, let $W= H_1(A,\QQ)$, and let $[E]$ be a polarization of $A$ represented by the Riemann form~$E$. The \dfn{Lefschetz group} of $A$ is the connected component of the identity in the centralizer of $\EndoA$ in $\Sp(W,E)$, $$ \Lf(A) := \{g\in\Sp(W,E) : g\circ \phi = \phi\circ g \enspace \text{for all } \phi\in\EndoA\}^\circ . $$ \enddefinition In this definition $\Lf(A)$ appears to depend on the choice of polarization, but if $[E']$ is another polarization, then there is an element $\psi \in \EndoA$ and positive $m\in \ZZ$ such that $mE' = E\psi$. To see this, we take the point of view that $E$ and $E'$ define isogenies, say $\phi$ and $\phi'$ respectively, from $A$ to its dual~$A\dual$. Then we can take $\psi = \phi\dual\circ\phi'$ and $m=\deg \phi$. Thus $\Lf(A)$ does not in fact depend on the choice of polarization. Furthermore, it is clear that $\Lf(A)$ is an algebraic group defined over $\QQ$, and $$ \Hg(A) \subseteq \Lf(A) . $$ The Lefschetz groups also has the following nice multiplicative property, that the Hodge and Mumford-Tate groups in general do not. \proclaim{2.15. Lemma {\rm (\cite{B.82}~Lem.2.1)}} If $A$ is isogenous to a product $B_1^{n_1} \times \dots \times B_r^{n_r}$, with the $B_i$ simple and non-isogenous, then $$ \Lf(A) \simeq \Lf(B_1) \times \dots \times \Lf(B_r) . $$ \endproclaim \demo{Proof} First let $A_i = B_i^{n_i}$, for $1\le i \le r$, and choose polarizations $[E_i]$ of $A_i$. Then $[E_1 \oplus \dots \oplus E_r]$ is a polarization of $A$, since $W = H_1(A,\QQ) \simeq \bigoplus_{i=1}^r H_1(A_i,\QQ)$. Further, since the $B_i$ are non-isogenous, $\Hom(A_i, A_j) =0$ for $i\ne j$, whence $\EndoA = \prod_{i=1}^r \Endo(A_i)$. Therefore any automorphism of $W$ that commutes with the action of $\EndoA$ must preserve each $ H_1(A_i,\QQ)$, and thus $\Lf(A) \simeq \Lf(A_1)\times \dots \times \Lf(A_r)$. Now fix $i$, and let $B=B_i$ and $A=B^n$. Then if $[E]$ is a polarization of $B$, then $[E\oplus \dots \oplus E]$ is a polarization of $A$. Further, $\EndoA \simeq M_n(\Endo(B))$, so the centralizer of $\EndoA$ in $\Sp(H_1(A,\QQ),E\oplus \dots \oplus E)$ can be identified with the centralizer of $\Endo(B)$ in $\Sp(H_1(B,\QQ),E)$. Therefore $\Lf(A)\simeq \Lf(B)$. \Qed \enddemo For later reference we state some variants of ``Goursat's Lemma'' that turn out to be useful, especially when extending results from simple abelian varieties to products of abelian varieties. The formulations given below come from \cite{B.90}, \cite{B.83} and \cite{B.75}. \proclaim{2.16. Proposition {\rm(Goursat's Lemma)}} \roster \item Let $G$ and $G'$ be groups and suppose $H$ is a subgroup of $G\times G'$ for which the projections $p:H\to G$ and $p': H\to G'$ are surjective. Let $N$ be the kernel of $p'$ and let $N'$ be the kernel of $p$. Then $N$ is a normal subgroup of $G$ and $N'$ is a normal subgroup of $G'$, and the image of $H$ in $G/N \times G'/N'$ is the graph of an isomorphism $G/N \simeq G'/N'$. \item Let $V_1$ and $V_2$ be two finite-dimen\-sion\-al complex vector spaces. Let $\frak s_1, \frak s_2$ be simple complex Lie subalgebras of $\frak{gl}(V_1)$, $\frak{gl}(V_2)$ respectively, of type A, B or C. Let $\frak s$ be a Lie subalgebra of $\frak s_1 \times \frak s_2$ whose projection to each factor is surjective. Then either $\frak s = \frak s_1 \times \frak s_2$ or $\frak s$ is the graph of an isomorphism $\frak s_1 \simeq \frak s_2$ induced by an $\frak s$-module isomorphism $V_2 \simeq V_1$ or $V_2 \simeq V_1\dual$. \item Let $\frak s_1, \dots , \frak s_d$ be simple finite-dimen\-sion\-al Lie algebras and let $\frak g$ be a subalgebra of the product $\frak s_1 \times \dots \times \frak s_d$. Assume that for $1\le i\le d$ the projection $\frak g \to \frak s_i$ is surjective, and that whenever $1\le i < j\le d$ the projection of $\frak g$ onto $\frak s_i \times \frak s_j$ is surjective. Then $\frak g = \frak s_1 \times \dots \times \frak s_d$. \item Let $I$ be a finite set and for each $\sigma\in I$, let $\frak s_\sigma$ be a finite-dimensional complex simple Lie algebra. Let $\frak g, \frak h$ be two algebras such that \itemitem"{(a)}" $\frak g \subseteq \frak h$. \itemitem"{(b)}" $\frak h$ is a subalgebra of $\prod_{\sigma\i I} \frak s_\sigma$ such that the projection to each $\frak s_\sigma$ is surjective. \itemitem"{(c)}" $\frak g, \frak h$ have equal images on $\frak s_\sigma \times \frak s_\tau$ for all pairs $(\sigma, \tau) \in I\times I$, $\sigma \ne \tau$. \item"" Then $\frak g = \frak h = \prod_{\sigma \in J} \frak s_j$ for some subset $J\subseteq I$. \item Let $V_1, \dots , V_n$ be finite-dimensional vector spaces over an algebraically closed field of characteristic zero, and let $\frak g$ be a semisimple Lie subalgebra of $\End(V_1) \times \dots \times \End(V_n)$. For $1\le i \le n$ let $\frak g_i\subseteq \End(V_i)$ be the projection of $\frak g$ onto the $i$-th factor. Assume that $\frak g_i$ is nonzero and simple for all ~$i$. Then for any simple Lie algebra $\frak h$ let $I(\frak h) \subset \{1,\dots, n\}$ be the set of indices for which $\frak g_i \simeq \frak h$. Assume that for any $\frak h$ with $\#I(\frak h) > 1$ the following conditions are satisfied: \itemitem"(a)" All automorphisms of $\frak h$ are inner. \itemitem"(b)" For $i\in I(\frak h)$ the representations $V_i$ are all isomorphic. \itemitem"(c)" $\End_{\frak g}(\bigoplus_{i\in I(\frak h)} V_i) = \prod_{i\in I(\frak h)} \End_{\frak g_i}(V_i)$. \item"" Then $\frak g \simto \frak g_1 \times \dots \times \frak g_n$. \endroster \endproclaim \head 3. Products of Elliptic curves \endhead Tate seems to be the first to have checked the (usual) Hodge conjecture for powers $E^n$ of an elliptic curve, see \cite{B.130}, \cite{B.43}~\S3, but he never published his proof. In \cite{B.80} Murasaki showed the $\Hdg^p(E^n) = \Div^p(E^n)$ for all $p$ by exhibiting explicit differential forms that give a basis for $\Hdg^1(E^n)$ and then carrying out explicit computations with them. In a different direction, Imai \cite{B.58} showed that when $E_1, \dots, E_n$ are pairwise non-isogenous elliptic curves, then $\Hg(E_1\times\dots \times E_n) \simeq \Hg(E_1)\times \dots \times \Hg(E_n)$. A unified approach to computing the Hodge and Mumford-Tate groups, and verifying the Hodge conjecture, for arbitrary products of elliptic curves can be found in \cite{B.84}. Since Murty's approach provides a nice example of how the Hodge and Mumford-Tate groups can be used to verify the usual Hodge conjecture, we summarize his exposition here. \proclaim{Theorem} Let $A= E_1^{n_1} \times \dots \times E_r^{n_r}$, where the $E_i$ are pairwise non-isogenous elliptic curves. Then \roster \item $\Hg(A) = \Hg(E_1) \times \dots \times \Hg(E_r)$. \item $\Hdg(A) = \Hdg(E_1^{n_1}) \tensor \dots \tensor \Hdg(E_r^{n_r}) = \Div(A)$. \endroster \endproclaim \demo{Proof {\rm (after \cite{B.84})}} Let $E$ be an elliptic curve. The cases where $E$ has or does not have complex multiplication have to be handled separately. If $E$ has complex multiplication, then $\Endo(E) =:K$ is an imaginary quadratic field, and (the proof of) Proposition~2.12 shows that $\MT(E,\QQ)\subseteq K^\times$, as algebraic groups over $\QQ$. Since the two-dimensional $\SS(\RR) \subset \MT(E,\RR)$, we see that $\MT(E) = \Res_{K/\QQ}(\Gm_{/K})$. When $E$ does not have complex multiplication, then $\hg(E)$ is a simple subalgebra of $\frak{sl}(V)$ which is already simple, so $\hg(E) = \frak{sl}(V)$ and thus $\Hg(E) =\SL(V)$ and $\MT(E) =\GL(V)$. Next consider $A= E^n$. Then $$ \align \Hdg(A) &= H^*(E^n,\QQ)^{\Hg(A)} \\ &= \twedge^*(H^1(E,\QQ) \oplus \dots \oplus H^1(E,\QQ))^{\Hg(A)} \\ &= \bigoplus (H^1(E,\QQ) \tensor \dots \tensor H^1(E,\QQ))^{\Hg(E)} . \endalign $$ Now if $E$ has complex multiplication then $\alpha \in K^\times = \MT(E)$ acts on $H^1(E,\QQ)\tensor_\QQ\CC \simeq \CC \oplus \CC$ by $\alpha(z,w) = (\alpha z, \bar\alpha w)$. Let $K^\times_1$ denote the elements of $K^\times$ of norm~$1$. Then $$ (H^1(E,\QQ) \tensor \dots \tensor H^1(E,\QQ))^{\Hg(E)} \tensor_\QQ \CC = \big( (H^1(E,\QQ)\tensor \CC) \tensor \dots \tensor (H^1(E,\QQ)\tensor \CC) \big)^{K^\times_1} , $$ in which any invariant class arises as a combination of products of elements of $$ \big( (H^1(E,\QQ)\tensor \CC) \tensor ((H^1(E,\QQ)\tensor \CC)) \big)^{K^\times_1} \subseteq (H^2(E\times E,\QQ)\tensor\CC)^{K^\times_1} . $$ Therefore the invariants are generated by those in $H^2(A,\QQ)$, which means that $\Hdg(A) =\Div(A)$ in this case. Next suppose that $E$ does not have complex multiplication. Then $\Hg(E) = \SL(2)$ and acts on $H^1(E,\QQ)$ by the standard representation. Now we invoke the well-known fact that the tensor invariants of $\SL(2)$ are generated by the determinant; see \cite{B.137} or \cite{B.10}~App.1 for this. Since the determinant is a representation of degree~$2$ lying in $H^1(E,\QQ) \tensor H^1(E,\QQ) \subset H^2(A,\QQ)$, again we find that all Hodge cycles of $A= E^n$ are generated by divisors. Finally let $A= E_1^{n_1} \times \dots \times E_r^{n_r}$, where the $E_i$ are pairwise non-isogenous elliptic curves. First suppose all the $E_i$ have complex multiplication by an imaginary quadratic field $K_i =\Endo(E_i)$. Since the field $\Endo(E_i)$ determines the isogeny class of $E_i$, all the $K_i$ are distinct. Then $$ \Hg(A) \subseteq K^\times_{1,1} \times \dots \times K^\times_{r,1}, $$ and moreover from the definition, $\Hg(A)$ surjects onto each factor. Therefore there is a surjection of character groups $$ \lambda: M\to \X(\Hg(A)) , $$ where $$ M:= \X(K^\times_{1,1}) \oplus \dots \oplus \X(K^\times_{r,1}) . $$ Now to see that $\lambda$ is an isomorphism and prove the theorem in the case where all the $E_i$ have complex multiplication, we observe that for each $i$ the composition $$ \X(K^\times_{i,1}) \hra M \to \X(\Hg(A)) $$ of $m\mapsto (0,\dots,0,m,0,\dots,0)$ with $\lambda$ is injective. In addition, all of these character groups are $\script G = \Gal(\QQ^{\text{ab}}/\QQ)$-modules and the maps $\script G$-module maps. Then since the fields are distinct there is some $\sigma\in\script G$ that acts as $+1$ on $\X(K^\times_{1,1})$ and $-1$ on the other components. Thus if $m = (m_1,\dots ,m_r)$ is in the kernel of $\lambda$ then $\sigma m + m = (2m_1, 0,\dots,0)$ must be as well. Then the injectivity of the composition above forces $m_1 =0$, and by induction the kernel of $\lambda$ is zero. Next suppose none of the $E_i$ has complex multiplication. Then we have $$ \hg(A) \subseteq \hg(E_1) \times \dots \times \hg(E_r) , $$ and mapping surjectively onto each factor. Then by Proposition~2.16.4, if it also maps surjectively onto each pair of factors, it is the entire product. But by Proposition~2.16.2, if $\hg(A)$ does not project onto $\hg(E_i) \times \hg(E_j)$ for all pairs $i\ne j$, then it projects to the graph of an isomorphism between them, which in turn could be used to produce an isogeny between $E_i$ and $E_j$, contrary to assumption. Finally it remains to see that if $A$ is an abelian variety isogenous to a product $B\times C$ with $\Hg(B)$ a torus and $\Hg(C)$ semisimple, then $\Hg(A) = \Hg(B) \times \Hg(C)$. However, this is a consequence of Proposition~2.16.1. This completes the proof of the theorem. \Qed \enddemo \head 4. Abelian varieties of Weil or Fermat type \endhead We have already defined abelian varieties of Weil type (1.13.2), and an abelian variety of Fermat type is one which is isogenous to a product of certain factors of the Jacobian variety of a Fermat curve $x^m + y^m + z^m =0$ \cite{B.116}. The important thing about these examples, insofar as the Hodge $(p,p)$ conjecture goes, is that they contain the only known examples where the conjecture has been verified for abelian varieties $A$ for which $\Hdg(A) \ne \Div(A)$. However, both types also provide explicit examples of Hodge cycles that are not known to be algebraic. Indeed, as is well-known, Weil has suggested that a place to look for a counterexample to the Hodge $(p,p)$ conjecture might be among what are now called abelian varieties of Weil type \cite{B.135}. We will begin by summarizing Shioda's results on abelian varieties of Fermat type. Then, since there is a nice presentation of the issues concerning abelian varieties in \cite{B.35}, we will just summarize the saliant points here. Finally we will recall the work of Schoen \cite{B.104} and van Geemen \cite{B.36} verifying the Hodge conjecture for special four-dimensional abelian varieties. \subhead Shioda's results on abelian varieties of Fermat type and Jacobians of hyperelliptic curves {\rm (\cite{B.116})} \endsubhead We will attempt to give a careful statement of the results, and refer the reader to the original for the proofs. \remark{4.1. Notation} Fix an integer $m>1$, and for $a\in\ZZ$ not congruent to zero modulo~$m$, let $1\le \bar a \le m-1$ be the unique integer such that $\bar a \cong a \pmod m$. Let $$\align \frak A_m^n &:= \{ \alpha =(a_0,\dots , a_{n+1}) : 1\le a_i \le m-1, \ \sum_{i=0}^{n+1} a_i \cong 0 \pmod m \} \\ \frak B_m^n &:= \{\alpha \in \frak A_m^n : |t\cdot \alpha| = (n/2)+1 \quad \text{for all } t\in (\ZZ/m\ZZ)^\times\} , \endalign $$ where in the latter case $n$ must be even, and where for $t\in(\ZZ/m\ZZ)^\times$ and $\alpha \in \frak A_m^n$, $$ t\cdot \alpha := (\ol{ta}_0, \dots , \ol{ta}_{n+1}), \qquad |\alpha| := \frac 1 m \sum_{i=0}^{n+1} a_i . $$ For $\alpha = (a_0,\dots , a_{r+1}) \in \frak A_m^r$ and $\beta = (b_0,\dots,b_{s+1}) \in \frak A_m^s$ let $$ \alpha * \beta := (a_0,\dots , a_{r+1}, b_0,\dots,b_{s+1}) \in \frak A_m^{r+s+2} . $$ Further, let $$\alignat2 M_m &:= \{ \xi=(x_1, \dots,x_{m-1};y) : \sum_{\nu=1}^{m-1} \ol{t\nu}\, x_\nu = my \enspace &&\text{for all }t\in(\ZZ/m\ZZ)^\times, \\ \vspace{-2\jot} &&& x_\nu,y\in\ZZ,\ x_\nu\ge 0,\ y>0\} \\ \vspace{2\jot} M_m(d) &:= \{ (x_1, \dots,x_{m-1};d) \in M_m\} . \endalignat $$ \revert@envir\endremark\medskip \definition{4.2. Definition} An element $\xi\in M_m$ is said to be \dfn{indecomposable} if $\xi \ne \xi' +\xi''$ for any $\xi',\xi''\in M_m$. An element $\xi\in M_m$ is called \dfn{quasi-decomposable} if there exists $\eta\in M_m(1)$ such that $\xi +\eta = \xi' +\xi''$ for some $\xi',\xi'' \in M_m$ with $\xi', \xi'' \ne \xi$. \enddefinition It is easy to see that the set $M_m$ is an additive semigroup with only a finite number of indecomposable elements. \definition{4.3. Definition} Let $X_m : x^m +y^m +z^m=0$ denote the Fermat curve of degree~$m$, let $J(X_m)$ be its Jacobian, and let $\frak S_m = (\ZZ/m\ZZ)^\times \bs \frak A_m^1$ be the orbit space. Then there is an isogeny $$ \pi :J(X_m) \to \prod_{S\in \frak S_m} A_S $$ where \roster \item"(i)" $A_S$ is an abelian variety of dimension $\phi(m')/2$ admitting complex multiplication by $\QQ(\zeta_{m'})$, where $\zeta_{m'} = e^{2\pi i/{m'}}$ and $m' = m/ \operatorname{gcd}(a,b,c)$ for $(a,b,c)\in \frak A_m^1$ belonging to the orbit~$S$. \item"(ii)" $H^1(A_S,\CC)$ has the eigenspace decomposition $$ H^1(A_S,\CC) = \bigoplus_{\alpha\in S} W(\alpha) $$ for the complex multiplication of~(i), where $\dim W(\alpha) =1$ and such that, if $\pi_S : J(X_m) \to A_S$ is the composition of $\pi$ and the projection to the $S$ factor, then $\pi_S^* W(\alpha) = U(\alpha)$ with $\alpha \in S$. Here $U(\alpha)$ is defined by $$ H^{1,0}(J(X_m)) = \bigoplus \Sb \alpha\in \frak A_m^1 \\ |\alpha|=1 \endSb U(\alpha) , \qquad H^{0,1}(J(X_m)) = \bigoplus \Sb \alpha\in \frak A_m^1 \\ |\alpha|=2 \endSb U(\alpha) $$ and $U(\alpha)$ is one-dimensional \cite{B.41} \cite{B.59}. \endroster Then an abelian variety will be said to be \dfn{of Fermat type of degree~$m$} if it is isogenous to a product of a finite number of factors $A_S$ satisfying~(i) and~(ii) as above. \enddefinition Thus an abelian variety of Fermat type of degree~$m$ is given by $$ A= \prod_{i=1}^k A_{S_i} \tag 4.3 $$ with $S_1,\dots,S_k \in \frak S_m$ not necessarily distinct. With this notation we can now state Shioda's results on the Hodge $(p,p)$ conjecture for these abelian varieties. \proclaim{4.4. Theorem {\rm (\cite{B.116}~Thm.4.3)}} Let $A$ be an abelian variety of Fermat type of degree~$m$. Assume that for any set $\{\alpha_1,\dots,\alpha_{2d}\}$ of distinct elements in the disjoint union of the $S_i$, $i=1,\dots,k$, such that $\alpha_1 * \dots * \alpha_{2d} = \gamma \in \frak B_m^{6d-2}$ there exists some $\beta_1,\dots , \beta_l \in \frak B_m^0 \cup \frak B_m^2 \cup (\frak B_m^4\cap \frak A_m^1 *\frak A_m^1)$ such that $\beta_1 * \dots * \beta_l$ coincides with $\gamma$ up to permutation. Then the Hodge $(p,p)$ conjecture is true for $A$ in codimension~$d$. \endproclaim \proclaim{4.5. Theorem {\rm (\cite{B.116}~Thm.4.4)}} If every decomposable element of $M_m(y)$ with $3\le y \le 3d$, if any, is quasi-decomposable, then the Hodge $(p,p)$ conjecture is true in codimension ~$d$ for all abelian varieties of Fermat type of degree~$m$. In particular, if $m$ is a prime or $m\le 20$, then the Hodge conjecture is true in any codimension for all abelian varieties of Fermat type of degree~$m$. \endproclaim \proclaim{4.6. Theorem {\rm (\cite{B.116}~Thm.5.6)}} For any given $d\ge 2$ there exists some abelian variety of Fermat type $A$ such that the Hodge ring $\Hdg^*(A)$ is not generated by $\sum_{r=1}^{d-1} \Hdg^r(A)$. \endproclaim \proclaim{4.7. Theorem {\rm (\cite{B.116}~Thm.5.3)}} Let $C_m : y^2 = x^m -1$ be the hyperelliptic curve of genus $g=[(m-1)/2]$, and let $J(C_m)$ be its Jacobian. If $m>2$ is a prime number, then the Hodge ring $\Hdg^*(J(C_m))$ is generated by $\Hdg^1(J(C_m))$, i.e., $\Hdg^*(J(C_m)) = \Div^*(J(C_m))$ and the Hodge conjecture is true for $J(C_m)$. The same result also holds for arbitrary powers of $J(C_m)$. \endproclaim \proclaim{4.8. Theorem {\rm (\cite{B.116}~Thm.5.4)}} For any odd $m\ge 3$ the Hodge conjecture is true for $J(C_m)$ in codimension~$2$. \endproclaim \subhead Abelian varieties of Weil type \endsubhead From 1.13.2, a Weil abelian variety of dimension $g=2n$ is an abelian variety $A$ together with an imaginary quadratic field $K$ embedded in $\EndoA$ such that the action of $\alpha \in K$ has the eigenvalues $\alpha$ and $\bar \alpha$ with equal multiplicity~$n$. Here we briefly review some of the important points about Weil abelian varieties, mainly following the exposition of \cite{B.35}, to which we refer the reader for more detail; the original source is \cite{B.135}. If $(A,K)$ is an abelian variety of Weil type, and $K=\QQ(\sqrt{-d})$, then it is possible to choose a polarization $[E]$ on $A$ normalized so that $(\sqrt{-d})^* E = d E$, where here we are viewing $\sqrt{-d} \in \scO_K \hra \End(A)$ as an endomorphism of $A$, so $(\sqrt{-d})^*$ and $(\sqrt{-d})_*$ denote the induced pullback and push-out maps, respectively. Hereafter when we speak of a polarized abelian variety of Weil type we will assume that its polarization is normalized this way. Let $W=H_1(A,\QQ)$. Then the $K$-valued Hermitian form $H: W\times W \to K$ associated to $E$ is given by $$ H(u,v) := E(u,(\sqrt{-d})_* v) + (\sqrt{-d})E(u,v) . $$ \proclaim{4.9. Theorem {\rm(\cite{B.135})}} The Hodge group of a general polarized abelian variety $(A,K,E)$ of Weil type is $\SU(W,H)$ (as an algebraic group over~$\QQ$). \endproclaim It can be shown that polarized abelian varieties of Weil type of dimension~$2n$ are parameterized by an $n^2$-dimensional space which can be described as the bounded symmetric domain assoicated to $\SU(W,H;\RR)\simeq \SU(n,n)$. Thus the word ``general'' in the statement of the theorem refers to a general point in this parameter space, analogously to the usage in~1.13.8. \definition{4.10. Definition} Let $(A,K)$ be an abelian variety of Weil type and dimension~$2n$. Then $V = H^1(A,\QQ)$ has the structure of a vector space over~$K$. The space of \dfn{Weil-Hodge cycles} on $A$ is the subset of $H^{2n}(A,\QQ)$ $$ \Weil(A) := \twedge_K^{2n} H^1(A,\QQ) , $$ where $\twedge_K^{2n}$ signifies the $2n^{\text{th}}$-exterior power of $H^1(A,\QQ)$ as a $K$@-vector space. \enddefinition \proclaim{Lemma} $\dim_\QQ \Weil(A) =2$, and $\Weil(A) \subset \Hdg^n(A)$. \endproclaim \proclaim{4.11. Theorem {\rm(\cite{B.135})}} Let $(A,K)$ be an abelian variety of Weil type of dimension $g=2n$. If the Hodge groups of $A$ is $\Hg(A) = \SU(W,H)$, then $$ \dim \Hdg^p(A) = \cases 1, & p\ne n, \\ 3, &p=n \endcases $$ and $\Hdg^n(A) = \Div^n(A) \oplus \Weil(A)$. \endproclaim Thus abelian varieties of Weil type provide examples of Hodge cycles which do not arise from products of those in codimension one. In the following the determinant of the Hermitian form $H$ is well-defined as an element of $\QQ^\times$ modulo the subgroup of norms from $K^\times$. \proclaim{4.12. Theorem {\rm(\cite{B.104})}} The Hodge $(2,2)$ conjecture is true for a general abelian variety of Weil type $(A,K)$ of dimension~$4$ with $K=\QQ(\sqrt{-3})$ or $K=\QQ(i)$, when $\det H =1$. \endproclaim A different proof for the case where $K=\QQ(i)$ and $\det H =1$ can be found in \cite{B.36}. In \cite{B.35}~7.3 it is pointed out that Schoen's methods together with a result of \cite{B.32} imply the Hodge $(3,3)$ conjecture for a general $6$-dimensional abelian variety of Weil type with $K=\QQ(\sqrt{-3})$ and $\det H =1$. \bigpagebreak Recently Moonen and Zarhin \cite{B.76} have considered the extent to which Weil's construction of exceptional Hodge classes can be generalized. For $K\hra \EndoA$ a subfield and $r= 2\dim A/[K:\QQ]$, let $$ \Weil_K(A) := \twedge_K^r H^1(A,\QQ) , $$ and call this the space of \dfn{Weil classes with respect to~$K$.} Let $V = H^1(A,\QQ)$. Then $K$ acts on $V$ and there is a decomposition $$ V\tensor\CC = \bigoplus_{\sigma\in\Hom(F,\CC)} V_{\CC,\sigma} = \bigoplus_{\sigma\in\Hom(F,\CC)} (V_{\CC,\sigma}^{1,0} \oplus V_{\CC,\sigma}^{0,1}). $$ \proclaim{4.13. Proposition {\rm(\cite{B.76})}} Let $A$ be a complex abelian variety, $K\hra \EndoA$ a subfield, and $r= 2\dim A/[K:\QQ]$. With the notation above, \roster \item If $\dim V_{\CC,\sigma}^{1,0} = V_{\CC,\ol\sigma}^{1,0}$ for all $\sigma \in \Hom(F,\CC)$, where $\ol\sigma$ denotes the complex conjugate of $\sigma$, then $\Weil_K(A)$ consists entirely of Hodge classes, i.e., $\Weil_K(A) \subset \Hdg^r(A)$; if $\dim V_{\CC,\sigma}^{1,0} \neq V_{\CC,\ol\sigma}^{1,0}$ for some $\sigma \in \Hom(F,\CC)$, then the zero class is the only Hodge class in~$\Weil_K(A)$. \item Suppose $A$ is isogenous to a power $B^m$ of a simple abelian variety~$B$, and suppose $K\hra \EndoA$ is a subfield such that $\Weil_K(A)$ consists of Hodge classes. Let $F$ be the center of $\Endo(B)$, and $F_0$ the maximal totally real subfield of~$F$. Then either $\Weil_K(A) \subset \Div(A)$, or all nonzero classes in $\Weil_K(A)$ are exceptional; this last possibility occurs precisely in the following cases: \itemitem"(a)" $B$ is of type~III, $m=1$ and $K\subsetneqq F$, \itemitem"(b)" $B$ is of type~III, $m\ge 2$ and $2m[F:\QQ]/[K:\QQ]$ is odd, \itemitem"(c)" $B$ is of type~IV, $(\dim_F(\Endo(B))^{1/2} =1$, $m=1$ and $K\subsetneqq F_0$, \itemitem"(d)" $B$ is of type~IV with $(\dim_F(\Endo(B))^{1/2} \ge 2$ or $m \ge 2$ and the map $$ \Lie(\U_F(1)) \hra \End_K(V) @> \operatorname{Tr} >> K $$ \item"" is nonzero. Here $\U_F(1) = \Res_{F/\QQ}\Gm_F \cap \U(1)$ (see~1.5.2). \endroster \endproclaim \head 5. Simple abelian fourfolds \endhead Since the Hodge $(p,p)$ conjecture is true for any smooth projective complex threefold, it might seem that fourfolds would be the next case to attack. However, as we've just seen, simple abelian fourfolds provide the first examples of abelian varieties of Weil type, for which the Hodge conjecture is mostly not known, and they also provide the first examples of abelian varieties of type~(III) in the Albert classification, as well as the first examples of abelian varieties that are not characterized by their endomorphism rings \cite{B.78}~\S4. Recently Moonen and Zarhin \cite{B.75} have analyzed the Hodge structures of simple abelian fourfolds, and their work makes an instructive example. The main result is the following, where a subalgebra of $\EndoA$ is said to be stable under all Rosati involutions if for every polarization it is stable under the associated Rosati involution. Recall that an exceptional Hodge class is one which is not accounted for by linear combinations of intersections of divisors. \proclaim{5.1. Theorem {\rm (\cite{B.75}~Thm.2.4)}} When $A$ is a simple abelian fourfold, then $A$ supports exceptional Hodge classes if and only if $\EndoA$ contains an imaginary quadratic field $K$ which is stable under all Rosati involutions and such that with the induced action of $K \subseteq \EndoA$ the complex Lie algebra $\Lie(A,\CC)$ of $A$ becomes a free $K\tensor\CC$-module. \endproclaim In the situation of the theorem, $\Lie(A)$ becomes a free $K\tensor \CC$-module if and only if $\alpha \in K$ acts as $\alpha$ and as $\bar\alpha$ with equal multiplicity~$2$. Thus a less precise but more simply stated corollary would be the following. \proclaim{Corollary} When $A$ is a simple abelian fourfold, if $\Hdg^2(A) \ne \Div^2(A)$ then $A$ must be an abelian variety of Weil type. \endproclaim For the abelian fourfolds $A$ for which $\Hdg^2(A) \ne \Div^2(A)$ Moonen and Zarhin prove the following theorem. \proclaim{5.2. Theorem {\rm (\cite{B.75}~Thm.2.12)}} Let $A$ be a simple abelian fourfold, and let $$ \V(A) := \sum_K \left( \twedge^4_K H^1(A,\QQ) \right) , $$ where the sum runs over all imaginary quadratic subfields $K\subset \EndoA$ that act on $A$ with multiplicities $(2,2)$. Then $\Hdg^2(A) = \Div^2(A) + \V(A)$. \endproclaim This theorem should be compared with Theorem~4.11: Theorem~5.1 applies to fourfolds, whereas in Theorem~4.11 it was assumed that $\Hg(A) = \SU(W,H)$, which is a sort of a generality assumption. \remark{5.3. Remark: Earlier work of Tankeev} In 1978 and 1979 Tankeev published two papers \cite{B.122} \cite{B.123} containing results about the Hodge structure and Hodge conjecture for abelian fourfolds. In particular, in \cite{B.123}~Thm.3.2 he proved that when $A$ is a simple abelian fourfold of type~(I) or type~(II), then $\Hdg(A) = \Div(A)$. First he showed that if the center of $\EndoA$ is a product of totally real fields, then $\Hg(A)$ is semisimple \cite{B.123}~Lemma~1.4, and then he proved the Hodge conjecture for simple abelian fourfolds of type~(I) or~(II) by considering the possible symplectic representations of the complexified Lie algebra $\hg_\CC(A)$ and showing that in each case that its invariants in $H^*(A,\CC)$ are generated by those of degree~$2$. The earlier paper \cite{B.122} considered the possible pairs $(\frak g, \rho)$, where $\frak g$ is the semisimple part of $\hg_\CC(A)$ and $\rho:\frak g \to \End_\CC(W_\CC)$ denotes its action on $W_\CC=H_1(A,\CC)$, under the assumption that that there does not exist an abelian variety $A_0$ defined over $\Qbar$ such that $A_0 \tensor_\Qbar \CC \simto A$; however, the proof there contains some gaps. Then he derived, under the same assumption that the abelian fourfold $A$ cannot be defined over $\Qbar$, that when $\EndoA$ is neither an imaginary quadratic field nor a definite quaternion algebra (type~(III)) then $\Hdg(A) = \Div(A)$. \revert@envir\endremark\medskip \subhead \nofrills \endsubhead We now proceed to sketch the outline of the proof of Theorem~5.1. One direction is covered by the following more general result. \proclaim{5.4. Theorem {\rm (\cite{B.75}~Thm.3.1)}} Let $A$ be a simple abelian variety, and assume that either \roster \item"(a)" $A$ is of type~\rom{(III)}, or \item"(b)" $\EndoA$ is a CM-field $K$ which contains a CM-subfield $F$ such that the multiplicity with which $\alpha \in F$ acts as $\sigma(\alpha)$ is the same for all $\sigma\in \Hom(F,\CC)$ \endroster Then $A$ supports exceptional Hodge classes \endproclaim \remark\nofrills The result for abelian varieties of type~(III) is due to Murty \cite{B.82}, see 8.6 below, although Moonen and Zarhin give a different proof. For case ~(b) what they show is that except for the zero element, $\twedge_F^m H^1(A,\QQ)$ consists entirely of exceptional Hodge classes, where $m= 2\dim A /[F:\QQ]$. \revert@envir\endremark\medskip The other direction of Theorem~5.2 is proved case by case, running through the different possible endomorphism algebras (see~1.13.4). It turns out that except when $\EndoA = \QQ$, knowing $\EndoA$ together with its action on $\Lie(A)$ suffices to determine $\hg(A)$, which in turn is enough to identify the absence or presence of exceptional cycles. We will run through the results, giving only some comments on the ingredients of the proofs. As usual, $W := H_1(A,\QQ)$. \proclaim{5.5. Type I(1)} Let $A$ be a simple abelian fourfold with $\EndoA =\QQ$. Then the Lie algebra $\hg(A)$ together with its representation on $W$ is isomorphic over $\Qbar$ to one of \roster \item"(i)" $\frak{sp}_4$ with the standard representation, or \item"(ii)" $\frak{sl}_2 \times \frak{sl}_2 \times \frak{sl}_2$ with the tensor product of the standard representation of each of the three factors. \endroster Both possibilities occur, and in both cases $\Hdg(A) = \Div(A)$. \endproclaim \remark\nofrills On the assumption that $\hg$ is simple, Theorem~2.11 can be used to show that case~(i) is the only possibility. When $\hg = \frak h_1 \times \dots \times \frak h_t$ is not simple, then $W= W_1\tensor \dots \tensor W_t$ and at least one $W_i$ must be $2$-dimensional. Then $\frak h_i = \frak{sl}_2$, and since the representation is symplectic, the complement must be $\frak{so}_4 \simeq \frak{sl}_2\times \frak{sl}_2$. In both cases $(\twedge^4 W)^{\mt}$ (the subspace of $\mt$-invariants in $\twedge^4 W$) is computed to be $1$-dimensional. \revert@envir\endremark\medskip \remark{5.6. Notation} Before treating the remaining type~(I) cases, suppose in general that $\EndoA$ contains a totally real field $F$, and suppose a polarization $[E]$ on $A$ is given. Then there is a unique $F$-bilinear alternating form $\psi: W\times W \to F$ whose trace $\operatorname{Tr}_{F/\QQ}(\psi(u,v)) = E(u,v)$. Then from the uniqueness of $\psi$ it follows that $\hg(A)$ is contained in $$ \frak{sp}_F(W,\psi) := \{ \phi\in \End_F(W) : \psi(\phi(u),v) + \psi(u,\phi(v)) =0 \enspace\text{for all } u,v \in W \} , $$ regarded as a Lie algebra over $\QQ$. \revert@envir\endremark\medskip \proclaim{5.7. Type I(2)} Let $A$ be a simple abelian fourfold with $\EndoA = F$ a real quadratic field. Then in the notation above, $\hg(A) \simeq \frak{sp}_F(W,\psi)$. In particular, $\Hdg(A^n) = \Div(A^n)$ for all ~$n$. \endproclaim \demo{Sketch of proof} The representation $\hg_\CC$ on $W_\CC$ splits as a direct sum $W_\CC = W_1 \oplus W_2$ with $\dim W_1 = \dim W_2 =4$. Further, the restriction of $\psi$ to $W_i$ is a nondegenerate skew-symmetric bilinear form $\psi_i: W_i\times W_i \to \CC$, and $\hg_\CC \subseteq \frak{sp}(W_1,\psi_1) \times \frak{sp}(W_2,\psi_2)$. Then the projection of $\hg_\CC$ onto $\frak{sp}(W_i,\psi_i)$ acting on $W_i$ must be on the list of Theorem~2.11, and since it is irreducible, symplectic, $4$-dimensional, it must be equal $\frak{sp}(W_i,\psi_i)$. Then since all automorphisms of $\frak{sp}_{4\,\CC}$ are inner, Proposition~2.16.5 implies that $\hg_\CC = \frak{sp}(W_1,\psi_1) \times \frak{sp}(W_2,\psi_2)$, and thus $\hg = \frak{sp}_F(W,\psi)$. \enddemo \proclaim{5.8. Type I(4)} Let $A$ be a simple abelian fourfold with $\EndoA = F$ a totally real field with $[F:\QQ] =4$. Then in the notation of~5.6, $\hg(A) \simeq \frak{sp}_F(W,\psi) \simeq \frak{sl}_{2\, F}$. In particular, $\Hdg(A^n) = \Div(A^n)$ for all ~$n$. \endproclaim \remark\nofrills The method of proof is similar to and easier than the previous case. Over $\Qbar$ or $\CC$ the representation of $\hg$ on $W$ splits into a sum of four mutually nonisomorphic, irreducible, symplectic, $2$-dimensional representations, and Proposition~2.16.3 applies. \revert@envir\endremark\medskip \proclaim{5.9. Type II} Let $A$ be a simple abelian fourfold of type~\rom{(II)}, i.e., $\EndoA$ is an indefinite quaternion algebra $D$ over a totally real field $F$ of degree $e\in\{1,2\}$ over~$\QQ$. Then $\hg(A)$ is the centralizer of $D$ in $\frak{sp}(W,E)$. In particular, $\Hdg(A^n) = \Div(A^n)$ for all ~$n$. \endproclaim For both $e=1$ and $e=2$ this is a special case of \cite{B.21}~Thm.4.10 and \cite{B.22}~Thm.7.4. Compare also with the method of Ribet \cite{B.94}, 6.3 below. \proclaim{5.10. Type III} Let $A$ be a simple abelian fourfold of type~\rom{(III)}, i.e., $\EndoA$ is a definite quaternion algebra $D$ over $\QQ$. Then $\hg(A)$ is the centralizer of $D$ in $\frak{sp}(W,E)$, which is a $\QQ$-form of $\frak{so}_4$. Moreover, $\dim \Hdg^2(A) =6$, and $\dim \Div^2(A) = 1$, and $\Hdg^2(A) = \Div^2(A) + \V(A)$, where $\V(A)$ is as in~5.2. \endproclaim \proclaim{5.11. Type IV(1,1)} Let $A$ be a simple abelian fourfold such that $\EndoA =K$ is an imaginary quadratic field. \roster \item"(i)" If $K$ acts with multiplicities $\{1,3\}$ then $\hg(A) = \frak u(W/K)$, and $\Hdg(A^n) = \Div(A^n)$ for all ~$n$. \item"(ii)" If $K$ acts with multiplicities $(2,2)$. In this case $\hg(A) = \frak{su}(W/K)$, and $\dim \Hdg^2(A) =3$, and $\dim \Div^2(A) = 1$, and $\Hdg^2(A) = \Div^2(A) \oplus \V(A)$, where $\V(A)$ is as in~5.2. \endroster \endproclaim \remark\nofrills In case~(i) $A$ is of Ribet type (1.13.2), see 6.3 below. In case~(ii) the equality $\hg(A) = \frak{su}(W/K)$ must be proved, and then Theorem~4.11 applies. \revert@envir\endremark\medskip \proclaim{5.12. Type IV(2,1)} Let $A$ be a simple abelian fourfold such that $\EndoA=K$ is a CM-field of degree~$4$ over $\QQ$. Then $\hg(A) \simeq \frak u_K(W,\psi)$. In particular, $\Hdg(A^n) = \Div(A^n)$ for all ~$n$. \endproclaim Similarly as in 5.6, here $\psi : W\times W \to K$ is the unique $K$-Hermitian form such that $\operatorname{Tr}_{K/\QQ}(\alpha\cdot \psi(u,v))$ is a Riemann form for $A$, for $\alpha \in K$ such that $\bar \alpha = -\alpha$ (The uniqueness is proved in \cite{B.27}). In this case $\hg(A)$ is contained in $$ \frak u_K(W,\psi) := \{ \phi \in \End_K(W) : \psi(\phi(u),v) + \psi(u,\phi(v)) =0 \enspace\text{for all } u,v \in W \} , $$ regarded as a Lie algebra over $\QQ$. \proclaim{5.13. Type IV(4,1)} Let $A$ be a simple abelian fourfold such that $\EndoA =K$ is a CM-field of degree~$8$ over $\QQ$. \roster \item"(i)" If $K$ does not contain an imaginary quadratic field $F$ acting on $A$ with multiplicities $(2,2)$, then $\hg(A) = \frak u_K$, which is a commutative Lie algebra of rank~$4$, and $\Hdg(A^n) = \Div(A^n)$ for all {}~$n$. \item"(ii)" If $K$ does contain an imaginary quadratic field $F$ acting on $A$ with multiplicities $(2,2)$, then $\hg(A) = \frak{su}_{K/F}$. In this case $\dim \Hdg^2(A) =8$, and $\dim \Div^2(A) = 6$, and $\Hdg^2(A) = \Div^2(A) + \V(A)$. \endroster \endproclaim We discuss abelian varieties with complex multiplication further below, see section nine. \head 6. Simple abelian varieties with conditions on dimension or endomorphism algebra \endhead \rightheadtext{simple abelian varieties with conditions} After Tankeev's early work on simple abelian fourfolds \cite{B.122} \cite{B.123}, the next progress on the Hodge $(p,p)$ conjecture was the work of Tankeev \cite{B.124} \cite{B.125} \cite{B.126} and Ribet \cite{B.93} \cite{B.94} on simple abelian varieties of types~(I), (II) or~(IV) whose dimension and endomorphism algebras satisfy various conditions. More precisely, Tankeev proved the following. \proclaim{6.1. Theorem {\rm (\cite{B.125} \cite{B.126})}} Let $A$ be a simple abelian variety of dimension~$d$. Then if \roster \item $A$ is of nondegenerate CM-type \rom{(2.13)}, or \item $\EndoA$ is a totally real field of degree~$e$ over $\QQ$, and $d/e$ is odd, or \item $\EndoA$ is a totally indefinite division quaternion algebra over a totally real field $K$ of degree~$e$ over $\QQ$, and $d/2e$ is odd, or \item $d$ is a prime, \endroster then $\Hdg(A) = \Div(A)$. \endproclaim However, soon afterward Ribet extended some of those results by first observing the following basic criterion, which he says was used implicitly in Tankeev's work, and then identifying some instances where it is satisfied. \proclaim{6.2. Theorem {\rm (\cite{B.94}~Theorem~0)}} Let $A$ be an abelian variety, and suppose \roster \item"(a)" $\EndoA$ is a commutative field, and \item"(b)" $\Hg(A) = \Lf(A)$ (the Lefschetz group, see~2.14). \endroster Then $\Hdg(A^n) =\Div(A^n)$ for $n\ge 1$. \endproclaim \proclaim{6.3. Theorem {\rm (\cite{B.94}~Theorems~1--3)}} Let $A$ be an abelian variety of dimension~$d$, and suppose \roster \item $\EndoA$ is a totally real field of degree~$e$ over $\QQ$, and $d/e$ is odd, or \item $d$ is prime and $A$ is of CM-type, or \item $\EndoA$ is an imaginary quadratic field $K$, and the multiplicities $n'$ and $n''$ with which $\alpha \in K$ acts as $\alpha$ and $\bar \alpha$ respectively are relatively prime. \endroster Then $\Hg(A) = \Lf(A)$ and thus $\Hdg(A^n) =\Div(A^n)$ for $n\ge 1$. \endproclaim \proclaim{Corollary} When $A$ is a simple abelian variety of prime dimension, then $\Hdg(A^n) =\Div(A^n)$ for $n\ge 1$. \endproclaim \demo\nofrills For if $A$ is simple and of prime dimension, then one of the conditions of Theorem~6.3 must be satisfied, see 1.13.3. \enddemo \remark{Remark} In \cite{B.139} Yanai showed that a prime-dimensional abelian variety of simple CM-type is nondegenerate (2.13). \revert@envir\endremark\medskip In a similar spirit as 6.3.2 above, Hazama proved the following. \proclaim{6.4. Theorem {\rm (\cite{B.45})}} Let $A$ be a simple abelian variety of CM-type. Then $\Hdg(A^n) =\Div(A^n)$ for all $n$ if and only if $\dim \Hg(A) =\dim A$. \endproclaim \demo{Sketch of proof of Theorem~6.2 {\rm (after \cite{B.94})}} In order to give some flavor of the techniques involved, consider the proof of Theorem~6.2. Since it is always the case that $\Hg(A) \subseteq \Lf(A)$, see 2.14, the condition that these two groups are equal should be thought of as saying that $\Hg(A)$ is as large as possible, whence has as few invariants as possible. Then the proof is separated into two cases, according as $\EndoA =K$ is a totally real or a CM-field. Consider first the case where $K$ is totally real. Then similarly as in 5.6 there is a unique $K$-bilinear alternating form $\psi: W\times W \to K$ whose trace $\operatorname{Tr}_{K/\QQ}(\psi(u,v))$ is a Riemann form $E(u,v)$ representing the chosen polarization on $A$, where $W=H_1(A,\QQ)$ \cite{B.27}~4.7. The uniqueness of $\psi$ implies that a $K$-automorphism of $W$ preserves $\psi$ if and only if it preserves $E$. Then from the definition 2.14 we get that $\Hg(A) = \Lf(A)$ is the symplectic group of $\psi$ acting on $W$ as a $K$-vector space, i.e., $$ \Hg(A) = \Res_{K/\QQ}\Sp(W_0, \psi), $$ where $W_0$ is $W$ as a $K$-vector space. Now what we need to prove is that $(\twedge_\QQ^*((W\dual)^n))^{\Hg(A)}$ is generated by its elements of degree~$2$, and for this question it suffices to extend scalars to $\CC$. Then $W\tensor \CC$ is a free $K\tensor_\QQ \CC$-module of rank $r= 2\dim A /[K:\QQ]$, and thus $$ W\tensor \CC \simeq \bigoplus_{\sigma\in\Hom(K,\CC)} U_\sigma, $$ where $U_\sigma$ is a complex vector space of dimension~$r$. Therefore $$ \Hg(A,\CC) \simeq \prod_{\sigma\in\Hom(K,\CC)} \Sp(U_\sigma, \psi_\sigma) , $$ from which we see that $$ (\twedge_\QQ^*((W\dual)^n))^{\Hg(A)} \tensor\CC \simeq \bigotimes_\sigma \ (\twedge_\CC^* (U_\sigma\dual)^n)^{\Sp(U_\sigma, \psi_\sigma)} . $$ So it suffices to know that this algebra of invariants is generated by its elements of degree~$2$, which is the case; see \cite{B.94}, or derive this fact using the methods of \cite{B.10}, \cite{B.137} or \cite{B.34}. The proof for the case where $\EndoA =K$ is a CM-field, either of degree $2\dim A$ or of degree $2$ over $\QQ$ follows a very similar pattern. As in 5.12 there is an element $\alpha \in K$ such that $\bar \alpha = -\alpha$ and a unique Hermitian form $\psi : W\times W \to K$ such that a Riemann form representing a polarization on $A$ is given by $E(u,v) = \operatorname{Tr}_{K/\QQ}(\alpha\psi(u,v))$. Then the centralizer $\Lf(A)$ of $K$ in $\Sp(W,E)$, which by hypothesis coincides with $\Hg(A)$, is $\Res_{K_0/\QQ}\U(W_0, \psi)$, where $K_0$ is the maximal totally real subfield of $K$ and $W_0$ is $W$ as a $K$-vector space. Now when we extend scalars to $\RR$, $$ W\tensor_\QQ \RR \simeq \bigoplus_{\sigma\in \Hom(K_0,\RR)} U_\sigma. $$ Moreover, $\psi$ induces a nondegenerate Hermitian form $\psi_\sigma$ on each $U_\sigma$, from which $$ \Hg(A,\RR) \simeq \prod_{\sigma\in \Hom(K_0,\RR)} \U(U_\sigma,\psi_\sigma) . $$ Thus in this case we need to know that for each $\sigma$ and for all~$n$ the algebra of invariants $(\twedge_\RR^*(U\dual_\sigma)^n)^{\U(U_\sigma,\psi_\sigma)}$ is generated by elements of degree~$2$, which is the case. The first step towards proving this is to extend scalars to $\CC$, so that the unitary group $\U(U_\sigma,\psi_\sigma)$ becomes a general linear group; we omit the invariant theory arguments here, see {\sl op.~cit.}\space for more details. \Qed \enddemo \demo{Sketch of proof of Theorem~6.3.3} We are assuming that $\EndoA =K$ is an imaginary quadratic field, and the multiplicities $n',~n''$, with which $\alpha\in K$ acts as $\alpha$ and acts as $\bar\alpha$ are relatively prime, and we want to show that $\Hg(A) =\Lf(A)$. In fact, it turns out to be more convenient to show that $\MT(A) = \Gm \cdot\Hg(A)$ coincides with $G = \Gm\cdot \Lf(A)$. This group $G$ may also be described as the largest connected subgroup of the symplectic similitude group $\GSp(W,E)$ that commutes with $K$, and in the present case $G=\Gm\cdot\Res_{K/\QQ}\U(W,\psi)$. It is clear that $\MT(A) \subseteq G$. Now, if $d= \dim A$, then $W$ is free of rank~$d$ as a vector space over~$K$, and thus $W\tensor_\QQ\CC$ is free of rank~$d$ over $K\tensor_\QQ \CC \simeq \CC\oplus \CC$, where the two copies are naturally indexed by the embeddings of $K$ into $\CC$. Therefore we may write $$ W\tensor_\QQ \CC = W' \oplus W'', $$ and this decomposition is compatible with the Hodge decomposition of and the action of $\MT(A)$ on $W\tensor_\QQ \CC$, since it is induced by endomorphisms of $A$. In particular, $$ W' = (W'\cap H^{-1,0}(A)) \oplus (W'\cap H^{0,-1}(A)) . $$ Now the action of $\MT(A,\CC)$ on $W\tensor_\QQ \CC$ induces an action of $\MT(A,\CC)$ on $W'$, and the next step of the proof is to see that the induced map $\MT(A,\CC) \to \GL(W')$ is surjective. However, this follows from \cite{B.106}~Prop.5; it is here that the relative primality of $n'$ and $n''$ is a required hypothesis. It follows that the commutator subgroup of $\MT(A,\CC)$ maps onto $\SL(W')$. What this means is that when we write $G$ as the product of its center~$C$ and its semisimple part $G_{\text{ss}}$, then $\MT(A) \supset G_{\text{ss}}$. Thus it remains to show that $C\subset \MT(A)$ as well. And since $\dim C =2$, it will suffice to show that the dimension of the center of $\MT(A)$ is at least~$2$, which in turn would follow from showing that $\MT(A)$ maps onto a $2$-dimensional torus. Since $\MT(A) \subseteq \Res_{K/\QQ}\GL(W_0)$, where $W_0$ is $W$ considered as a vector space over $K$, we may consider the determinant map $\Theta: \MT(A) \to \Res_{K/\QQ} \Gm_{/K} =: T$. Then from the fact that $\MT(A)$ contains $\Gm_{/\QQ}$ acting as homotheties on $W$, the image of $\Theta$ contains $\Gm_{/\QQ} \subset T$. If we extend scalars to $\CC$, then $T_\CC \simeq \Gm_{/\CC} \times \Gm_{/\CC}$, and the image of $\Gm_{/\QQ}$ in $T$ becomes the diagonal. On the other hand, $\MT(A)_\CC$ also contains $h(\SS_\CC) \simeq \Gm_{/\CC} \times \Gm_{/\CC}$. Then $\Theta(h(z,1)) = (z^{(n')}, z^{(n'')})$, and because $n' \ne n''$, this generates a torus distinct from the diagonal. Therefore $\Theta$ is surjective, which completes the proof. \Qed \enddemo \head 7. More abelian varieties with conditions on dimension or endomorphism algebra \endhead \rightheadtext{more abelian varieties with conditions} During the 1980's Hazama, Murty and others continued to generate results related to the Hodge conjecture by examining the interactions among the dimension, endomorphism algebra, and Hodge or Mumford-Tate group, in a spirit akin to the work of Tankeev and Ribet described in section~6. In particular, Hazama and Murty, working at about the same time but using different methods, produced a number of overlapping results about the Hodge conjecture for not-necessarily-simple abelian varieties extending the results of Tankeev and Ribet. \subhead Abelian varieties with generalized real multiplication \endsubhead A first set of results can be loosely grouped together as dealing with abelian varieties with generalized real multiplication. \definition{7.1. Definition} Several different definitions of what is meant by \dfn{real multiplication} appear in the literature. The narrowest would be that the abelian varietiey $A$ is of type~(I), i.e., every simple factor $A_s$ of $A$ has $\Endo(A_s)$ equal to a totally real field. A slightly broader definition would be to require that $\EndoA$ contains a product $R$ of totally real fields such that $[R:\QQ] = \dim A$ \cite{B.46}. Zarhin \cite{B.142} calls a $g$-dimensional abelian variety \dfn{of RM-type} if it contains a commutative semisimple $\QQ$-algebra of degree~$g$ over $\QQ$, and notes that this means that any abelian variety of CM-type is also automatically of RM-type. Murty variously considers the cases where a commutative semisimple $\QQ$-algebra $R\subseteq \EndoA$ is its own centralizer in $\EndoA$ \cite{B.81}, or $R$ is maximal among commutative semisimple subalgebras of $\EndoA$ and is a product of totally real fields \cite{B.83}. Altogether the most useful general definition might be to say that an abelian variety $A$ has generalized real multiplication if $\EndoA$ contains a commutative semisimple subalgebra $R$ with $[R:\QQ] =\dim A$, and $A$ is not of CM-type, i.e., $\EndoA$ does not contain a commutative semisimple subalgebra of degree~$2\dim A$ over $\QQ$. To avoid ambiguity we will try to give precise statements of results without using this terminology. \enddefinition The following theorem tries to summarize the main results concerning the Hodge $(p,p)$ conjecture for abelian varieties with some generalized real multiplication. \proclaim{7.2. Theorem {\rm (\cite{B.46} \cite{B.81} \cite{B.83})}} Let $A$ be an abelian variety. \roster \item Suppose $\EndoA$ contains a product $R$ of totally real fields with the property that $[R:\QQ] = \dim A$, and no simple component of $A$ of CM-type has dimension greater than~$1$. Then $\Hdg(A) =\Div(A)$. \item Suppose $\EndoA$ contains a commutative semisimple subalgebra $R$ that is its own centralizer in $\EndoA$, and $H^0(A,\Omega^1)$ is free of rank~$1$ over $R\tensor_\QQ \CC$. Then $\Hdg(A) =\Div(A)$. \item Suppose that a maximal commutative semisimple subalgebra $R$ of $\EndoA$ is a product of totally real fields, and that $W= H_1(A,\QQ)$ is free over $R$ of rank $2m$, where $m$ is odd. Then $\Hg(A) = \Lf(A)$ and thus $\Hdg(A^k) = \Div(A^k)$ for all $k\ge 1$. \endroster \endproclaim \remark{7.3.1. Remarks on Theorem 7.2.1 {\rm (\cite{B.46})}} The first observation about an abelian variety $A$ satisfying the conditions of 7.2.1 is that its simple isogeny factors must be of type~(I), or type~(II), or elliptic curves with complex mutliplication, as elliptic curves without complex multiplication are included in type~(I) \cite{B.37}. In particular, the condition is stable under taking products or abelian subvarieties. Without going into the proof at too great a length, some of the main ingredients include firstly a lemma, due to Tankeev \cite{B.123}~Lemma~1,4, that if the center of $\EndoA$ is a product of totally real fields, then $\Hg(A)$ is semisimple. Then Hazama proceeds to work out the Hodge Lie algebra $\hg(B)$, where $B$ is the isogeny factor of $A$ containing the simple factors of type~(I) or~(II). After complexifying and applying Goursat's Lemma he finds that $\hg(B,\CC) \simeq \frak{sl}_2 \times \dots \times \frak{sl}_2$. Finally, he observes, similarly as we did in section~3 on elliptic curves, that when $B$ is a abelian variety whose Hodge group is semisimple and $C$ is an abelian variety of CM-type, then $\Hg(B\times C) \simeq \Hg(B)\times \Hg(C)$. Then Theorem~7.2.1 follows from the invariant theory of $\frak{sl}_2$ and the known results for elliptic curves. \revert@envir\endremark\medskip \remark{7.3.2. Remarks on Theorem 7.2.2 {\rm (\cite{B.81})}} Murty calls a pair $(A,R)$ consisting of an abelian variety and a commutative semisimple subalgebra $R\subset \EndoA$ \dfn{of type~{\rm (H)}} when the hypotheses of Theorem~7.2.2 are satisfied. He observes that the product of two abelian varieties of type~(H) is again of type~(H), and further proves that in general when a commutative semisimple subalgebra of $\EndoA$ of degree $\dim A$ over $\QQ$ is a product $R$ of totally real fields, then $H^0(A,\Omega^1)$ is free of rank~$1$ over $R\tensor \CC$. Thus some examples of abelian varieties of type~(H) include $(E,\QQ)$, where $E$ is an elliptic curve without complex multiplication, and $(A,F)$ where $A$ is an abelian surface with quaternionic multiplication by a quaternion algebra $B$ over $\QQ$, as in 1.13.7, and $F$ is any real quadratic subfield of $B$ which splits~$B$. Although the analysis is somewhat different, some of the main ideas of the proof of Theorem~7.2.2 are similar to some of the main points of the proof of Theorem~7.2.1. In particular, under the hypothesis of type~(H) (and making use of Goursat's Lemma), Murty shows that $$ \hg(A) = \{ m \in \End_{\EndoA}(W) : \operatorname{tr}_R m =0\} , $$ and that not only is this semisimple, but over $\CC$ it is a product of $\frak{sl}_2$'s. Thus it is possible to deduce that all invariants are generated by those of degree~$2$. \revert@envir\endremark\medskip \example{7.3.3. Example: Jacobians of elliptic modular curves} Among the motivating examples for both Hazama and Murty were Jacobians of elliptic modular curves. To recall briefly, for $N\ge 3$ let $$ \Gamma_1(N) := \left\{ \pmatrix a&b\\ c&d\endpmatrix \in \SL(2,\ZZ) : c\cong 0 \text{ and } a\cong d \cong 1 \pmod N \right\} , $$ and let $\frak H = \{z\in \CC : \Im z >0 \}$ denote the upper half-plane. Then the quotient $$ X_1(N)(\CC) := \Gamma_1(N) \big\bs (\frak H \cup \QQ \cup \{ i\infty\}) $$ can be identified with the complex points of a nonsingular projective algebraic curve which can, in fact, be defined over~$\QQ$. Moreover, Shimura has shown that in the Jacobian $J_1(N) := \operatorname{Jac}(X_1(N))$ all the isogeny factors with complex multiplication are products of elliptic curves, and all the isogeny factors without complex multiplication are of real multiplication type in the sense that the endomorphism algebras of the simple factors contain a totally real number field whose degree over $\QQ$ is the dimension of that factor \cite{B.112} \cite{B.113} \cite{B.114} (see also \cite{B.89}). \endexample \remark{7.3.4. Remark on Theorem 7.2.3} Theorem 7.2.3 is an artifact of Murty's study \cite{B.83} of the semisimple parts of the Hodge groups of abelian varieties, and their relationship with the Lefschetz groups. We will return to this again briefly below. \revert@envir\endremark\medskip \subhead Stably nondegenerate abelian varieties \endsubhead A next group of results concerns conditions under which $\Hdg(A^k) =\Div(A^k)$ for all $k\ge 1$. Again we combine closely relatied results of Murty and Hazama; but first we need a definition. \definition{7.4. Definition {\rm (\cite{B.47})}} When $A$ is a simple abelian variety, the \dfn{reduced dimension} of $A$ is defined by $$ \rdim A := \cases \dim A, &\text{for $A$ of type~(I) or of type~(III),} \\ (\dim A)/2, &\text{for $A$ of type~(II),} \\ (\dim A)/d, &\text{for $A$ of type~(IV), and } [\EndoA: C(\EndoA)] = d^2 . \endcases $$ By $C(\EndoA)$ here we mean the center of $\EndoA$. When $A$ is isogenous to $\prod _i A_i^{m_i}$ with the $A_i$ simple and nonisogenous, then the \dfn{reduced dimension} of $A$ is $$ \rdim A := \sum_i \rdim A_i . $$ \enddefinition \proclaim{7.5. Theorem {\rm (\cite{B.82}, \cite{B.47})}} For an abelian variety $A$, the following are equivalent. \roster \item $\Hdg(A^k) =\Div(A^k)$ for all $k\ge 1$. \item $A$ has no factor of type~\rom{(III)}, and $\Hg(A) = \Lf(A)$. \item $\rank \Hg(A)_\CC = \rdim A$. \endroster \endproclaim \definition{7.6. Definition} An abelian variety satisfying the conditions of Theorem~7.5 may be called \dfn{stably nondegenerate.} \enddefinition \remark{7.6.1. Remarks} Hazama makes the following elementary observations about stable nondegeneracy \cite{B.47}: \roster \item If $A$ is stably nondegenerate, and $B$ is an abelian subvariety of $A$, then $B$ is stably nondegenerate. For up to isogeny $A\simeq B\times B'$, and thus if stable nondegeneracy (in the sense of 7.5.1) failed for $B$ it would fail for $A$. \item For any $k\ge 1$, $A$ is stably nondegenerate if and only if $A^k$ is stably nondegenerate. This follows from the definition 7.5.1 and the previous observation. \item For abelian varieties $A_i$ and integers $k_i$, the product $\prod_i A_i^{k_i}$ is stably nondegenerate if and only if $\prod_i A_i$ is stably nondegenerate. Observe that $$ \prod_i A_i^{k_i} \subset \big( \prod_i A_i \big)^{\max k_i} . $$ \endroster \revert@envir\endremark\medskip He also proves the following. \proclaim{7.6.2. Theorem {\rm (\cite{B.49)}}} If $A$ and $B$ are stably nondegenerate abelian varieties and contain no factors of type~(IV), then $A\times B$ is also stably nondegenerate. \endproclaim The difficulty with type~(IV) arises in taking products of, or with, abelian varieties of CM-type, see section nine below. Or from a different point of view, there is the theorem of Tankeev that if all the simple factors of an abelian variety $A$ are of types~(I), (II) or (III), then $\Hg(A)$ is semisimple \cite{B.123}, and this may fail for type~(IV). What can be said is that if $A$ is stably nondegenerate and has no factors of types~(IV), and $B$ is stably nondegenerate and of CM-type, then $A\times B$ is stably nondegenerate \cite{B.49}. On the other hand, since by 7.5.2 no abelian variety with a factor of type~(III) can be stably nondegenerate, Theorem~7.6.2 applies when all simple factors of $A$ and $B$ are of type~(I) or~(II). \remark{7.7. Some remarks on Theorem~7.5} In \cite{B.82} Murty proves (1) if and only if~(2). Much of the paper is devoted to a careful analysis of the structure of $\Lf(A)$. Given the multiplicativity of $\Lf(A)$, Lemma~2.15, we may assume $A$ is simple. Then fix a maximal commutative subfield $F \subset \EndoA$ which is totally real for type~(I) and a CM-field in the other three cases, and let $F_0$ be the maximal totally real subfield of $F$. Now extending scalars to $\RR$, there is a decomposition of $\Lf(A)$ into factors indexed by the embeddings $F_0 \hra \RR$. Then these factors are of the form: for type~(I), a symplectic group; for type~(II), the intersection of a unitary group and a symplectic group; for type~(III), the intersection of a unitary group and a special orthogonal group; for type~(IV), a unitary group \cite{B.82}~Lemma~2.3. Moreover, after complexifying, these act on the corresponding components of $W\tensor \RR$ as: for type~(I), as a standard symplectic repesentation; for type~(II), two copies of the standard representation of the complex symplectic group; for type~(III), two copies of the standard representation of the complex special orthogonal group; for type~(IV), the sum of a standard representation of the complex general linear group and its contragredient. Using this structural analysis, Murty is then able to prove the following. \proclaim{7.7.1. Proposition {\rm (\cite{B.82})}} If $A$ contains no simple factors of type~\rom{(III)}, then for all $k\ge 1$ $$ H^*(A^k, \QQ)^{\Lf(A)} = \Div(A^k) . $$ \endproclaim To complete the proof that 7.5.1 is equivalent to 7.5.2, Murty shows that a simple abelian variety of type~(III) supports an exceptional Hodge class, see~8.6 below for more discussion of this. \smallpagebreak Hazama's proof in \cite{B.47} that 7.5.1 is equivalent to 7.5.3 is based on a careful type by type analysis of the Lie algebra $\hg(A)_\CC$ and its action on $W_\CC$ using that all the possibilities are as listed in Theorem~2.11. For example, for a simple abelian variety of type~(I), the action of a simple component $\frak g_i$ of $\hg(A)_\CC$ on the corresponding component $W_i$ of $W_\CC$ is a symplectic representation, and indeed $\frak g_i \simeq \frak{sp}(W_i,\CC)$. A similar result holds for type~(II), whereas for type~(IV), the simple components of $\hg(A)_\CC$ are of the form $\frak{sl}_{d_i}$. In all these cases, careful invariant theory arguments using \cite{B.137} show that the invariants of $H^*(A^k,\CC)$ are generated by those of degree~$2$ if and only if the rank is as claimed. On the other hand, for type~(III) Hazama finds that $$ \rank \Hg(A)_\CC \le (\dim A)/2 < \rdim A , $$ i.e., equality never holds, and an abelian variety with a factor of type~(III) fails to be stably nondegenerate. It comes about in the proof that in general $$ \rank \Hg(A) \le \rdim A . $$ So both criteria 7.5.2 and 7.5.3 can be understood philosophically as saying that $A$ is stably nondegenerate when $\Hg(A)$ is as large as possible. \revert@envir\endremark\medskip \subhead Further work on Hodge and Mumford-Tate groups \endsubhead We conclude this section with certain additional results derived from close study of Hodge and Mumford-Tate groups. \subhead \nofrills \endsubhead In \cite{B.83} Murty examines the semisimple part of the Hodge group of an abelian variety, and finds the following. As usual, $W= H_1(A,\QQ)$. \proclaim{7.8. Theorem {\rm (\cite{B.83})}} If a maximal commutative subalgebra $R$ of $\EndoA$ is a product of CM-fields, and $W$ is free over $R$ of odd rank, and if $\Hdg(A) =\Div(A)$, then $\Hg(A)_{\text{ss}} = \Lf(A)_{\text{ss}}$. \endproclaim This together with 7.2.3 implies the following. \proclaim{Corollary {\rm (\cite{B.83})}} When $A$ is simple and of odd dimension, then $\Hdg(A) = \Div(A)$ implies that $\Hg(A)_{\text{ss}} = \Lf(A)_{\text{ss}}$. \endproclaim \subhead \nofrills \endsubhead In \cite{B.57} Ichikawa studies groups of Mumford-Tate type, and extending \cite{B.94} \cite{B.126} and his own earlier work \cite{B.56}, he obtains the following result. First we need some notation. \definition{7.9. Definition} Let $A$ be a simple abelian variety of dimension~$g$, let $K$ the center of $\EndoA$, let $e=[K:\QQ]$ and let $d^2=[\EndoA:K]$. Then the \dfn{relative dimension} of $A$ is defined by $$ \operatorname{rel\,dim}(A) := \cases g /e, &\text{if $A$ is of type~(I),}\\ g /2e, &\text{if $A$ is of type~(II) or type~(III),}\\ 2g/de, &\text{if $A$ is of type~(IV).} \endcases $$ \enddefinition \proclaim{7.10. Theorem {\rm (\cite{B.57})}} Let $A$ be an abelian variety all of whose simple factors have odd relative dimension. \roster \item When $A$ is isogenous to $A'\times A''$, where each simple factor of $A'$ is of type ~\rom{(I)}, \rom{(II)} or~\rom{(III)} and each simple factor of $A''$ is of type~\rom{(IV)}, then all Hodge cycles on $A$ are generated by the Hodge cycles on $A'$ and $A''$. \item When $A$ is isogenous to $\prod_j A_j^{m_j}$, where the $A_j$ are simple and mutually non-isogenous, then all Hodge cycles on $A$ are generated by the Hodge cycles on the $A_j$. \endroster \endproclaim \head 8. Exceptional Hodge cycles \endhead Thus far we have looked mainly at examples and conditions under which the Hodge $(p,p)$ conjecture is true. Now we consider the known examples of Hodge cycles that are not known to be algebraic, and thus might be considered potential counterexamples to the conjecture. \definition{8.1. Definition} By an \dfn{exceptional Hodge cycle} on $A$ we mean an element of $\Hdg^p(A) = H^{2p}(A,\QQ) \cap H^{p,p}(A)$, for some $p$, which is not in $\Div^p(A)$, that is to say, which cannot be written as a $\QQ$-linear combination of classes of $p$-fold intersections of divisors. \enddefinition \subhead 8.2. Mumford's CM fourfold \endsubhead Perhaps the first example of an abelian variety where $\Hdg(A) \ne \Div(A)$ was Mumford's example of the abelian fourfold with complex multiplication corresponding to a particular CM-type (see~1.13.6) for the splitting field of $(3X^4 - 6X^2 + X +1)(X^2 +1)$ \cite{B.88}. This example is described in Lecture~7, 7.23--7.28. \subhead 8.3. Abelian varieties of Weil type \endsubhead It was Weil's observation, however, that the crucial feature of Mumford's example was not that it was of CM-type, but rather that there was an imaginary quadratic field $F$ acting on $A$ in such a way that $\Lie(A)$ becomes a free $K\tensor \CC$-module, or equivalently, such that $\alpha \in K$ acts as $\alpha$ and as $\bar \alpha$ with equal multiplicity \cite{B.135}. Moreover, as we saw in Theorem~4.11, the general such abelian variety, what we now refer to as an abelian variety of Weil type, has a $2$-dimensional space of exceptional Weil-Hodge cycles in $\Hdg^n(A)$, where $\dim A =2n$. In Theorem~4.12 we recalled Schoen's examples of general abelian fourfolds with $K= \QQ(i)$ or $K=\QQ(\sqrt{-3})$ where he showed that the Weil-Hodge cycles are algebraic \cite{B.104}, and little else is known. \subhead 8.4. Abelian varieties of Fermat type \endsubhead Shioda's work on abelian varieties of Fermat type, see 4.1--4.8 above, provides examples of abelian varieties $A$ where, at least for some $p$, the space of Hodge cycles $\Hdg^p(A) \supsetneqq \Div^p(A)$ but is nonetheless generated by classes of algebraic cycles \cite{B.116}, see Theorems~4.4 and~4.5. In the same work he also shows the existence, for any $d\ge 2$, of an abelian variety $A$ of Fermat type whose Hodge ring $\Hdg^*(A)$ is not generated by $\sum_{r=1}^{d-1} \Hdg^r(A)$, let alone by $\Hdg^1(A)$. \subhead 8.5. Exceptional cycles in codimension~$2$ \endsubhead In \cite{B.124} Tankeev produced a family of abelian varieties of dimension~$4^m$ with exceptional cycles in codimension~$2$ when $m\ge 2$. \proclaim{Theorem {\rm (\cite{B.124}~Thm.5.6)}} For any $m\ge 1$ there exist abelian varieties $A$ such that \roster \item $\dim A = 4^m$, and \item $\EndoA = \QQ$, and \item $\hg_\CC(A) \simeq (\frak{sl}_{2\, \CC})^{2m+1}$, acting on $H^1(A,\CC) \simeq (\CC^2)^{\tensor(2m+1)}$ as the tensor product of a standard representation of each factor. \endroster Moreover, for any abelian variety satisfying these conditions, $\dim_\QQ \Hdg^2(A) \mathbreak = (4^m -1)/3$. In particular, if $m\ge 2$ then $\Hdg^2(A)$ is not generated by classes of intersections of divisors. \endproclaim The existence part of this theorem is obtained by generalizing Mumford's example in \cite{B.78} (not the example in \cite{B.88} mentioned in 8.2 above) of an abelian fourfold $A$ with $\EndoA = \QQ$ and thus not characterized by its endomorphism ring. The computation of $\dim_\QQ \Hdg^2(A)$ is proved by induction, and a computation with the roots of $\hg_\CC(A)$ and the character of its representation. \subhead 8.6. Abelian varieties of type~(III) \endsubhead In the same paper where he proved that $\Hdg(A^k) =\Div(A^k)$ for all $k\ge 1$ if and only if $A$ has no factor of type~(III) and $\Hg(A) = \Lf(A)$ \cite{B.82}, Murty also proved the existence of an exceptional Hodge cycle on abelian varieties of type~(III). \proclaim{Theorem {\rm (\cite{B.82})}} If an abelian variety $A$ has a factor of type~\rom{(III)}, then it supports an exceptional Hodge class $\omega$ with the property that $\pi_1^*(\omega) \tensor \pi_2^*(\omega) \in \Div(A^2)$, where $\pi_1$ and $\pi_2$ are the projections from $A^2 = A\times A$ to its first and second factors respectively. \endproclaim \remark{Remark} In \cite{B.135}, where he presented abelian varieties of Weil type as a place to look for counterexamples of the Hodge conjecture, Weil also asked whether a weaker statement might be true, that is, whether the presence of a Hodge cycle on $A$ might imply the presence of an algebraic cycle on some power of $A$. This result of Murty is the first example where the Hodge conjecture itself is not known to be true, but Weil's question is answered affirmatively. \revert@envir\endremark\medskip \demo\nofrills To get a flavor of the proof, suppose $A$ is simple and of type~(III), and let $F$ be the center of $\EndoA$, let $m= \dim_{\EndoA} W$, and let $d=(\dim A)/[F:\QQ]$. Then $d=2m$, and by \cite{B.109}~Prop.15, $m\ge 2$. Then as a consequence of his analysis of $\Lf(A)$, Murty shows that $$ \big( \twedge^* (W\dual)\big)^{\Lf(A)} \tensor_\QQ \RR = \bigotimes_{\sigma \in \Hom(F,\RR)} \big( \twedge^* X\dual_\sigma\big)^{\Lf(A)_\sigma} $$ with $\dim_\RR X_\sigma =4m$. Then $X_\sigma \tensor \CC$ becomes isomorphic to two copies of a standard representation of $\operatorname{SO}(V,\psi)$ for a suitable $V$ and $\psi$. Then by \cite{B.137}~p.53 the covariant tensors of $\operatorname{SO}(V,\psi)$ are generated by $\psi$ and the determinant, say~$\Delta$. Then $\Delta$ cannot be written as a polynomial in the degree~$2$ invariant $\psi$, but $\Delta^2$ can. Take $\omega$ to be the class corresponding to $\Delta$. \enddemo \subhead 8.7. Determinant cycles \endsubhead In \cite{B.57} Ichikawa uses the idea of Murty's construction to develop a certain extension of the work of Tankeev and Ribet on simple abelian varieties \cite{B.124} \cite{B.126} \cite{B.94}, see section~6. Firstly he observes that on any abelian variety of type~(I), (II) or~(III) there exist Hodge cycles that are $\CC$-linear combinations of the determinant forms on the spaces $V$ as in the last paragraph. He calls these \dfn{determinant Hodge cycles.} In this language, Murty's result above is that on an abelian variety of type~(III) no determinant cycle is generated by classes of divisors. Then Ichikawa proves the following result. Recall the definition of relative dimension from~7.9. \proclaim{Theorem {\rm (\cite{B.57})}} Let $A$ be an abelian variety whose simple factors are all of odd relative dimension, and suppose $A$ is isogenous to $A'\times A''$ where each simple factor of $A'$ is of type ~\rom{(I)}, \rom{(II)} or~\rom{(III)} and each simple factor of $A''$ is of type~\rom{(IV)}. Further, assume that the relative dimension of any simple factor of $A$ of type~\rom{(III)} is not equal to $\frac 1 2 \binom{2k}{k}$ for any power $k$ of~$2$, and that $A''$ is a power of a simple abelian variety of odd prime dimension. Then any Hodge cycle on $A$ is generated by classes of divisors and determinant Hodge cycles. \endproclaim \definition{8.8. Definition {\rm (\cite{B.47})}} Recall (definition~7.6) that a stably nondegenerate abelian variety is one which satisfies the conditions of Theorem~7.5, in particular, $\Hdg(A^k) =\Div(A^k)$ for all $k\ge 1$. Then a \dfn{stably degenerate} abelian variety $A$ is one which is not stably nondegenerate, that is, $\Hdg^p(A^n) \supsetneqq \Div^p(A^n)$ for some $p,n$. Then the least $n$ for which this occurs is called the \dfn{index of degeneracy,} which we will denote by $\ind(A)$. \enddefinition \subhead 8.9. Stably degenerate abelian varieties \endsubhead Hazama has given two examples of stably degenerate abelian varieties of type~(I) having index of degeneracy~$2$. \proclaim{8.9.1. Theorem {\rm (\cite{B.47})}} There exists a simple abelian variety $A$ of dimension~$4$ with the following properties: \roster \item"(a)" $A$ is of type~\rom{(I)}, \item"(b)" $\Hdg(A) = \Div(A)$, \item"(c)" $\Hdg(A^2) \supsetneqq \Div(A^2)$. \endroster \endproclaim \demo\nofrills To give a rough idea of the construction, let $K$ be a totally real number field of degree~$3$, let $B$ be a quaternion algebra over $K$ such that $B\tensor_\QQ \RR \simeq M_2(\RR) \times \Bbb H \times \Bbb H$, and let $G = \Res_{K/\QQ}\SL(1,B)$. Then $G(\RR) \simeq \SL(2,\RR) \times \SU(2) \times \SU(2)$, and there exists an $8$-dimensional $\QQ$-rational symplectic representation $\rho : G\to \Sp(8)$ satisfying the necessary analyticity conditions so that the induced map $\tau: X \to \frak H_4$ of Hermitian symmetric domains pulls back the universal family $\script A \to \frak H_4$ of polarized abelian fourfolds over the Siegel upper half-space to an analytic family of abelian fourfolds $A\to X$ \cite{B.64}. Moreover, if $A_0$ denotes a generic member of the family $A\to X$, then $\Hdg^p(A_0^k) = H^{2p}(A_0,\QQ)^G$ for all $k\ge 1$. Then computations with the complexified Lie algebra $\Lie(G,\CC) \simeq \frak{sl}_2 \times \frak{sl}_2 \times \frak{sl}_2$ show firstly that $\dim \Hdg^1(A_0) =1$, from which follows that $A_0$ is simple, of type~(I), and $\Hdg(A) =\Div(A)$, and secondly that $\Hdg^2(A^2)$ is not generated by the elements of $\Hdg^1(A^2)$. \enddemo \proclaim{8.9.2. Theorem {\rm (\cite{B.48})}} There exists a simple abelian variety $A$ of dimension~$10$ with the following properties: \roster \item"(a)" $\hg(A,\CC) \simeq \frak{sl}(6,\CC)$, \item"(b)" the representation $\hg(A,\CC) \to \End(H^1(A,\CC))$ is equivalent to the representation $\frak{sl}(6,\CC) \to \End(\twedge^3 \CC^6)$ induced by the natural action of $\frak{sl}(6,\CC)$ on~$\twedge^3 \CC^6$, \item"(c)" $\Hdg(A) = \Div(A)$, \item"(d)" $\Hdg(A^2) \supsetneqq \Div(A^2)$. \endroster \endproclaim \demo\nofrills The existence is worked out similarly as in the previous case, except that here $G$ is a $\QQ$-form of $\SU(5,1)$. Again $\dim \Hdg^1(A) = 1$, and $\EndoA =\QQ$. However, the actual computations are based on using Young diagrams and branching rules, see \cite{B.48} for the details. \enddemo \remark{Remark} In \cite{B.48} Hazama constructs a family $A_n$ of simple abelian varieties of dimension $\frac 1 2 \binom{4n+2}{2n+1}$, with $\Endo(A_n) =\QQ$, and $\hg(A_n,\CC) \simeq \frak{sl}(4n+2,\CC)$ and $H^1(A_n,\CC) \simeq \twedge^{2n+1}\CC^{(4n+2)}$ as a representation of $\hg(A_n,\CC)$. The abelian variety of Theorem~8.9.2 is the $A_1$ in this family. Then he also shows that the index of degeneracy $\ind(A_n) \le 2$ for $n\ge 2$, where the theorem shows that $\ind(A_1) =2$. \revert@envir\endremark\medskip \subhead 8.10. Invariants of partially indefinite quaternion algebras \endsubhead In the 1960's Kuga asked which semisimple algebraic groups $G$ defined over $\QQ$ together with which of their symplectic representations $\rho: G\to \Sp(W,\beta)$ satisfy the necessary and sufficient analyticity conditions to allow the construction of an algebraic family of polarized abelian varieties parameterized by $\Gamma\bs X$, where $\Gamma$ is a discrete subgroup of $G$ and $X$ is the Hermitian symmetric domain associated to~$G$ \cite{B.61} \cite{B.62} \cite{B.63}. Shortly thereafter Satake answered Kuga's question under the assumption that for each $\QQ$-simple factor $\rho$ comes from an absolutely irreducible representation of an absolutely simple factor of~$G$ \cite{B.99} \cite{B.100} \cite{B.102} \cite{B.103}. It turned out that the list was quite small, and nearly all cases had been considered by Shimura in his analysis of families of abelian varieties characterized by polarization, endomorphism ring and level structure \cite{B.109} \cite{B.110} \cite{B.112}; one more case was treated in \cite{B.101}. Some time later Addington considered Kuga's question without Satake's assumption, and for the groups corresponding to units of norm~$1$ in a partially indefinite quaternion algebras $B$ over a totally real field~$F$, i.e., $$ B\tensor_\QQ \RR \simeq M_2(\RR)^n \oplus \Bbb H^m \qquad \text{and} \qquad G = \Res_{F/\QQ}\SL(1,B) , $$ she developed a combinatorial scheme (called ``chemistry'') to describe which symplectic representations give rise to an algebraic family of abelian varieties \cite{B.5}. Then Tjiok \cite{B.131} and Abdulali \cite{B.1} \cite{B.2} \cite{B.3} showed that under certain reasonable hypotheses (``rigidity'' or ``condition~(H$_2$)'') the space of Hodge cycles in a generic fiber $A_0$ of the family is the space of $G$-invariants, $H^{2r}(A_0,\QQ)^G = (\twedge^{2r} W\dual)^G$. Thus, for the purposes of this appendix, where the issue is Hodge cycles on abelian varieties, statements about Hodge cycles on the generic fiber of such a family can be understood as statements about Hodge cycles on an abelian variety $A_0$ with specified semisimple Hodge group~$G$. Then the problem is to describe the invariants of $G$ in the exterior algebra $\twedge^*W\dual$. This is the problem taken up by Kuga in \cite{B.64}, \cite{B.65} and the series of papers \cite{B.66} \cite{B.67}, and Lee in \cite{B.71}. The results are rather involved to state precisely. In \cite{B.64} Kuga looks at some simple examples of the situation just described, and finds conditions (``totally disconnected triangular polymer'') where all the Hodge cycles in the abelian variety $A_0$ are generated by those of degree two or where only the Hodge cycles in codimension~$2$ or~$4$ are generated by those of degree two (``triangular polymer without double bond, short cycle or Hexatram''), which is to say that there are exceptional cycles in higher codimensions. At the end of \cite{B.65} is an example of a $16$-dimensional abelian variety $A$ for which the dimensions of the spaces $\Hdg^r(A)$ of Hodge cycles are determined, where $\dim \Hdg^2(A)= 82$ and $\dim \Div^2(A) = 10$. In \cite{B.66} and \cite{B.67} the focus is more on the complicated invariant theory in the exterior algebra for the groups and representations under consideration, in particular the latter papers examine the asymptotic behavior of $\dim \twedge^{2r}(\mu W\dual)^G$ as the multiplicity~$\mu$ grows. In \cite{B.71} Lee computes the dimensions of the spaces $\Hdg^r(A_0)$ for the $8$-dimensional abelian variety constructed from a particular form of $16$-dimensional representation of $G$. \head 9. The problem of complex multiplication \endhead In this section we look at what is known about the Hodge $(p,p)$ conjecture for abelian varieties with complex multiplication. We have already seen that the Hodge group of such an abelian variety is an algebraic torus, necessarily contained in $\Res_{K/\QQ}\Gm_{/K}$, where $K$ is the field of complex mutliplication (Proposition~2.12). For the general theory of complex multiplication, see \cite{B.115} \cite{B.70} and parts of \cite{B.112}. \definition{9.1. Definition} Recall (1.13.6) that a CM-type $(K,S)$ consists of a CM-field $K$ and a subset $S\subset \Hom(K,\CC)$ containing exactly one from each pair of conjugate embeddings. Moreover, given a CM-type $(K,S)$, there is a natural construction of an abelian variety $A$ with that CM-type, that is, with $\EndoA = K$ and $K$ acting on $H^{1,0}(A)$ as $\bigoplus_{\phi\in S} \phi$. Then we define the \dfn{rank} of the CM-type $(K,S)$ by $$ \rank (K,S) := \dim \MT(A) . $$ \enddefinition \remark{Remark} The rank of a CM-type seems to have originally been defined by Kubota \cite{B.60}, who defined it as $\dim_\QQ\{\sum_{\phi\in S} \phi(x) : x\in K\}$. The equality of this with the dimension of the Mumford-Tate group follows from the methods in \cite{B.91}, see also \cite{B.26} \cite{B.27}. \revert@envir\endremark\medskip \subhead 9.2. Pohlmann's criterion \endsubhead One of the first results about Hodge cycles on abelian varieties with complex multiplication is a theorem of Pohlmann \cite{B.88} that describes $\dim \Hdg(A)$ in terms of the Galois theory of~$K$. To state the theorem we need some notation. Let $A$ be an abelian variety with CM-type $(K,S)$, let $S = \{\phi_1, \dots , \phi_g\}$ and let $\Sbar = \{ \phibar_1, \dots ,\phibar_g\}$, so $\Hom_\QQ(K,\CC) = S\cup \Sbar$. Then $\alpha \mapsto (\phi_1(\alpha), \dots , \phi_g(\alpha))$, for $\alpha \in K$, induces an isomorphism of $K$ onto $H_1(A,\QQ)$ (in 1.13.6 we mapped $\script O_K$ onto $H_1(A,\ZZ)$), via which $H^1(A,\CC)$ can be identified with $\Hom_\QQ(K,\CC)$. Further, without loss of generality we may assume $K\subset \CC$, and let $L$ be the Galois closure of $K$ in $\CC$ and $G = \Gal(L/\QQ)$. Then we let $\sigma \in \Aut(\CC/\QQ)$ act on $f\in H^r(A,\CC)$, with $f:\twedge^r K \to \CC$, by $(\sigma f)(\lambda) = \sigma(f(\lambda))$ for $\lambda \in \twedge^r K$. Finally, for an ordered subset $\Delta \subset S$, let $|\Delta|$ denote the cardinality of $\Delta$ and let $\angled \Delta := \twedge_{\phi\in S} \phi$. \proclaim{Theorem {\rm (\cite{B.88}~Thm.1)}} When $A$ is an abelian variety with CM-type $(K,S)$, then $\Hdg^p(A) \tensor \CC$ has a basis consisting of those $\angled \Delta \in H^{2p}(A,\CC)$ such that $$ |\tau \Delta \cap S| = | \tau \Delta \cap \Sbar| \tag 9.2.1 $$ for every $\tau \in G$. Thus $\dim \Hdg^p(A)$ is the number of ordered subsets $\Delta \subset S$, with $|\Delta| = 2p$, that satisfy the condition~\rom{(9.2.1)}. \endproclaim \demo{Proof} If $f= \sum_i c_i\angled{\Delta_i} \in \Hdg^p(A)$ with $c_i\in\CC$, then $\sum_i \sigma(c_i)\angled{\sigma\Delta_i} = \sigma f = f$ is in $H^{p,p}(A)$ for all $\sigma\in \Aut(\CC/\QQ)$, hence $\Delta_i$ satisfies~(9.2.1). Then every element of $\Hdg^p(A)$ is a linear combination of $\Delta_i$ satisfying~(9.2.1). Conversely, let $\Delta$ be such taht $|\Delta | = 2p$ and (9.2.1) is satisfied. Let $\{u_1,\dots,u_s\}$ be a basis for $L$ over $\QQ$, and let $f_i = \sum_{\tau\in G} \tau(u_i) \angled{\tau\Delta}$ for $1\le i \le s$. Then $\sigma f_i = f_i$ for all $\sigma\in\Aut(\CC/\QQ)$, and by (9.2.1) $f_i \in H^{p,p}(A)$, so $f_i\in \Hdg^p(A)$. Further, since $\det(\tau(u_i))_{\tau, i} \ne 0$, we can solve the system of linear equations $f_i = \sum_{\tau \in G} \tau(u_i) \angled{\tau \Delta}$ and find that $\angled {\tau\Delta} \in \Hdg^p(A) \tensor \CC$ for $\tau \in G$. Thus $\angled \Delta \in \Hdg^p(A) \tensor \CC$, as required. \Qed \enddemo Pohlmann's theorem give a criterion for exceptional cycles, also see 9.3 below. \proclaim{9.2.2. Corollary {\rm (\cite{B.138})}} $\dim \Hdg^p(A) - \dim\Div^p(A)$ is the number of subsets $\Delta \subset \Hom(K,\CC)$ such that \roster \item"(a)" $\Delta - \ol\Delta \ne \varemptyset$, \item"(b)" $|\Delta \cap g S| = p$ for all $g\in G$. \endroster \endproclaim \subhead 9.3. Sporadic cycles \endsubhead In \cite{B.138} White observes that Pohlmann's criterion shows that when a CM abelian variety $A$ is nondegenerate, as defined in~2.13, then $\Hdg(A) =\Div(A)$. He then recounts that between 1977 and 1978 Ribet asked if these two conditions were equivalent, and that Lenstra was quickly able to show that they are, for a simple abelian variety of simple CM-type $(K,S)$, under the additional hypothesis that the CM-field $K$ is abelian over $\QQ$ (see \cite{B.138}~Thm.3). Then later Hazama showed that a simple abelian variety is nondegenerate if and only if $\Hdg(A^k) = \Div(A^k)$ for all $k\ge 1$, see Theorem~6.4 \cite{B.45} and Theorem~7.5 \cite{B.47}. Only recently, however, White showed the following. \proclaim{Theorem {\rm (\cite{B.138}~Thm.1)}} There exists an abelian variety of CM-type with $\Hdg(A) =\Div(A)$ and $\dim \Hg(A) \lneqq \dim A$. \endproclaim The argument involves a rather technical analysis of the $\QQ$-group ring of a non-abelian group. Eventually, however, the counterexample is a CM-type for a CM-field whose Galois group is $$ \ZZ/2\ZZ \times \ZZ/2\ZZ \times \ZZ/5\ZZ \times D_5, $$ where the last factor is the dihedral group and the first factor corresponds to complex conjugation. The splitting field of $$ X^5 - 10X^4 -70X^3 -25X^2 +190X +12 $$ is a totally real field with Galois group $D_5$, and it is easy to make disjoint totally real quadratic and quintic extensions, and then a totally imaginary quadratic extension. For the abelian variety $A$ with the requisite CM-type for this field, $\dim\Hg(A) = 84 < 100 =\dim A$. \subhead 9.4. Degenerate CM types \endsubhead Recall from definitions 2.13 and 9.1 that a CM-type $(K,S)$ is said to be nondegenerate if $\rank(K,S) = \dim A +1$, and is called degenerate otherwise, if $\rank (K,S) \le \dim A$, where $A$ an abelian variety CM-type $(K,S)$. A number of examples of degenerate CM-types and lower bounds for the rank as a function of $\dim A$ have been given by Ribet \cite{B.92}, Dodson \cite{B.28} \cite{B.29} \cite{B.30}, Mai \cite{B.72} and Yanai \cite{B.140}. \smallpagebreak The following proposition of Kubota can be a useful way of measuring the rank of a CM-type. Let $c$ denote complex conjugation. \proclaim{9.4.1. Proposition {\rm (\cite{B.60})}} $$ \rank (K,S) = 1 + \#\big\{\chi : \Gal(K/\QQ) \to \CC : \chi(c) =-1\enspace \&\enspace \sum_{s\in S} \chi(s) \ne 0 \big\}, $$ where only irreducible $\chi$ are included. \endproclaim \example{9.4.2. Examples {\rm (\cite{B.92})}} First, let $p \ge 5$ be a prime, let $K = \QQ(\zeta_p)$ be the field of $p^{\text{th}}$ roots of unity, and identify $G = \Gal(K/\QQ) \simeq (\ZZ/p\ZZ)^\times$. For $g\in G$ let $\angled g \cong g \pmod p$ with $1\le \angled g \le p-1$. Then for $1\le a \le p-2$ with $a^3\not \cong 1 \pmod p$ the set $$ S_a = \{g\in G : \angled g + \angled{ag} <p\} $$ is a simple CM-type. It is nondegenerate when $a=1$, but is degenerate for $p= 67$ and $a = 10$, $19$, $47$, $56$, $60$ \cite{B.40}. Lenstra and Stark also noticed that for $p\cong 7 \pmod {12}$ and sufficiently large there always exists a number~$a$ for which $S_a$ is degenerate, {\sl loc..~cit.} Next let $K=\QQ(\zeta_{32})$ and, identifying $\Gal(K/\QQ)$ with $(\ZZ/32\ZZ)^\times$, let $$ S = \{ 1,\ 7,\ 13,\ 21,\ 23,\ 27,\ 29\}, \qquad S'=\{1,\ 7,\ 9,\ 11,\ 13,\ 15,\ 27,\ 29\} . $$ Then $(K,S)$ and $(K,S')$ are both degenerate (simple) CM-types. This example is due to Lenstra. Let $K= \QQ(\zeta_{19})$ and, identifying $\Gal(K/\QQ)$ with $(\ZZ/19\ZZ)^\times$, let $$ S = \{ 1,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 10,\ 17\}. $$ Then again $(K,S)$ is a degenerate CM-type. This example is due to Serre. Finally, let $p$, $q$, $r$ be odd primes, let $G = \ZZ/2pqr\ZZ$ as cyclic group, and let $K$ be an extension of $\QQ$ with $\Gal(K/\QQ)\simeq G$. Then let $S$ be the subset of elements having order $1$, $pqr$, $2p$, $2q$, $2r$, $2pq$, $2pr$ or $2qr$. Then $(K,S)$ is a simple CM-type and $$ \rank(K,S) = 1 + pqr - (p-1)(q-1)(r-1) . $$ This example is due to Lenstra. All of these examples are verified in \cite{B.92} using Proposition~9.4.1 and exhibiting odd characters $\chi$ such that $\sum_{s\in S} \chi(s) = 0$. \endexample \example{9.4.3. Degenerate CM-types in composite dimension} In Theorem~6.3.2 we saw that an abelian variety of CM-type with prime dimension is always nondegenerate. Dodson has proved a number of theorems exhibiting the existence of degenerate abelian varieties in composite, i.e., non-prime, dimensions \cite{B.28} \cite{B.29} \cite{B.30}, and more recently Yanai has come up with a method for generating degenerate CM-types that encompasses some of the previous examples \cite{B.140}. \proclaim{Theorem{\rm (\cite{B.28}~Thm.3.2.1)}} When $n$ is composite, $n>4$, then there exist abelian varieties of CM-type with $\dim A =n$ and rank $n-l+2$ for a divisor $l\ge 2$ of $n$ such that $n/l >2$. \endproclaim \proclaim{Theorem{\rm (\cite{B.28}~Thm.3.2.2)}} Let $p$ be a prime and suppose the ideal $(2)$ decomposes in the cyclotomic field $\QQ(\zeta_p)$ into $g>1$ factors of degree~$f$. Note that $p\cong \pm1 \pmod 8$ is sufficient but not necessary to insure $g>1$. Then there exist abelian varieties of CM-type having dimension $2^{fs}p$ and rank~$p-1$, for $0\le s\le g-1$. In particular, for $s>0$ these abelian varieties are simple and degenerate. \endproclaim In the following $K_0^{\Gal}$ denotes the Galois closure of the field $K_0$, and $[(\ZZ/2\ZZ)^m]^+$ is the even subgroup of $(\ZZ/2\ZZ)^m$ \proclaim{Theorem{\rm (\cite{B.28}~Thm.3.3.1)}} \rom{(A)} Suppose there exists a totally real field $K_0^{\Gal}$ with Galois group isomorphic to the wreath product $(\ZZ/k\ZZ) \wr (\ZZ/l\ZZ)$, with $k>2$ and $l\ne 1$. Then there exist simple degenerate abelian varieties with complex multiplication by a non-Galois CM-field such that the varieties have dimension~$kl$ and rank~$n-l+2$. \par \rom{(B)} Under the same hypotheses except that $k=2$ is allowed, there exist simple abelian varieties of CM-type with dimension~$n' =k^2l$ having rank $\le kl+1$. \par \rom{(C)} Further, in the even case, the existence of totally real fields with Galois groups $[(\ZZ/2\ZZ)^m]^+\rtimes \ZZ/m\ZZ$ with $m\ge 3$, respectively $D_m$ with $m>5$ and odd, supplies simple abelian varieties of CM-type with dimension $n=4m$, respectively $n=2m$, and rank $\le \frac n 2 +1$ \endproclaim \proclaim{Theorem{\rm (\cite{B.29}~Thm.3.1)}} Let $d$ be a composite number. Then there exist simple abelian varieties of CM-type with dimension ~$d$ and rank $\le (d/2)+1$ whenever $d$ is \roster \item even; \item divisible by a square; \item of the form $d = \binom p m$ with $0\le m \le (p-1)/2$; or \item of the form $d= pt$ with $t\mid (p-1)$ and $t< p-1$. \endroster Further, the rank $p+1$ occurs in dimension~$d$ at least in the following cases: \roster \item when $d$ and $p$ are as in ~3 or ~4 above; \item when $d= 2^{(p-1)/2}pt$ with $t\mid (p-1)/2$; \item when $d=p q^2$ where $q$ is an odd prime and $p= q^2+ q +1$. \endroster \endproclaim The following is a nondegeneracy result \proclaim{Theorem{\rm (\cite{B.30}~Thm.2.1)}} Let $n$ be odd and suppose that $K$ is a CM-field of degree $2n$ such that the maximal totally real subfield $K_0$ has $\Gal(K_0^{\Gal}/\QQ)$ isomorphic to the symmetric groups or the alternating group on $n$~letters. Then every primitive CM-type $(K,S)$ is nondegenerate. \endproclaim Recently Yanai has developed a method for generating degenerate CM-types in higher dimension starting with degenerate CM-types in lower dimension. \proclaim{Theorem {\rm (\cite{B.140})}} Let $K$ be a CM-field with $[K:\QQ] = 2d$ and let $K_1$ be a proper subfield of $K$ with $[K_1:\QQ] =2d_1$. Further, let $\pi: X_K \to X_{K_1}$ be the canonical surjection from the character group of $\Res_{K/\QQ}\Gm_{/K}$ to the character group of $\Res_{K_1/\QQ}\Gm_{/K_1}$. Suppose that the CM-types $(K,S)$ and $K_1,S_1)$ satisfy the condition $$ \pi\big(\sum_{\sigma\in S}\sigma\big) = a \sum_{\sigma\in S_1}\sigma + b\sum_{\sigma\in S_1}\bar\sigma $$ with some nonnegative integers $a$ and $b$ such that $a+b =[K:K_1]$. Then $$ d+1 -\rank S \ge d_1 +1 -\rank S_1 . $$ Moreover, if $a=b$ then $$ d+1 - \rank S \ge d_1. $$ In particular, if the CM-type $(K_1,S_1)$ is degenerate or if $a=b$ then the CM-type $(K,S)$ is degenerate. \endproclaim \endexample \example{9.4.4. Lower bounds for CM-types} Ribet \cite{B.92} and Mai \cite{B.72} have given some lower bounds for the rank of a CM-type, and particularly the latter discusses how sharp these might be. \proclaim{Proposition {\rm (\cite{B.92})}} $\rank(K,S) \ge 2 + \log_2 (\dim A)$ \endproclaim Mai considers the case where the CM-filed $K$ is Galois over $\QQ$. For the next proposition, note that when $V =\ZZ[\Gal(K/\QQ)]\tensor\CC$ is considered as a $\Gal(K/\QQ)$-module, there is a decompositon $V= \bigoplus_\pi d_\pi V_\pi$, where $\pi$ ranges over the irreducible representations of $\Gal(K/\QQ)$ and $d_\pi = \dim V_\pi$. A representation $\pi$ is called \dfn{odd} if the value of its character at complex conjugation is~$-1$. \proclaim{Proposition {\rm(\cite{B.72}~Prop.1)}} When $K/\QQ$ is a Galois extension and $(K,S)$ is a simple CM-type, then $$ \rank (K,S) \ge 1 + \sum d_\pi , $$ where the sum ranges only over those odd irreducible representaions $\pi$ such that $\pi(\sum_{s\in S}s) \ne 0$. \endproclaim \proclaim{Proposition {\rm(\cite{B.72}~Prop.2)}} When $K/\QQ$ is a Galois extension and $(K,S)$ is a simple CM-type, then $$ \rank(K,S) \le \max \Big\{ \frac{(p-1)^2\alpha}{p} : p \text{ an odd prime, and } p^\alpha \,\big\|\, ([K:\QQ]/2) \Big\} . $$ \endproclaim The notation $p^\alpha \,\big\|\, N$ means $p^\alpha$ exactly divides~$N$, i.e., $p^\alpha \,\big|\,N$ and $p^{(\alpha+1)} \not{\big|}\,N$. \smallpagebreak In the following $S_a$ is the same CM-type that occured the first paragraph of 9.4.2. Such CM-types occur among factors of the Jacobians of Fermat curves. \proclaim{Proposition {\rm(\cite{B.72}~Prop.3)}} Let $K= \QQ(\zeta_p)$ and identify $\Gal(K/\QQ)$ with $(\ZZ/p\ZZ)^\times$. For $1\le a \le p-2$ let $S_a$ be the CM-type defined by $$ S_a = \{g\in \Gal(K/\QQ) : 1 \le \angled g + \angled{ag} < p \} , $$ where $1\le \angled g \le p-1$ and $\angled g \cong g \pmod p$. Then $$ \rank(K,S) \ge 1 + \frac{19}{21} d , \qquad d= \frac{p-1}2 . $$ \endproclaim \endexample \subhead 9.5. Andr\'e's description of CM-Hodge cycles as Weil cycles \endsubhead Finally we turn to a recent result of Andr\'e \cite{B.9} that every Hodge cycle on an abelian variety $A$ of CM-type is a linear combination of inverse images under morphisms $A\to B_J$ of Weil-Hodge cycles on various abelian varieties $B_J$ of CM-type. The following definition should be compared with 1.13.6, 4.10, and the discussion preceding Theorem~4.9. \definition{9.5.1. Definition} Let $A$ be an abelian variety, let $F$ be a CM-field contained in $\EndoA$, and let $V= H^1(A,\QQ)$. Then $A$ or $V$ is said to be \dfn{of Weil type relative to $F$} if there exists an $F$-Hermitian form $\psi$ on $V$ admitting a totally isotropic subspace whose dimension over $F$ is $\frac 1 2 \dim_F V$, and there exists a purely imaginary element $\alpha \in F$ such that $\operatorname{Tr}_{F/\QQ}(\alpha\cdot\psi(u,v))$ defines a polarization on~$A$. Then the elements of $\twedge_F^{2p}V$ are called \dfn{Weil-Hodge cycles relative to~$F$.} \enddefinition \proclaim{9.5.2. Theorem {\rm(\cite{B.9})}} Let $A$ be an abelian variety of CM-type, and $p$ a positive integer. Then there exists a CM-field $F$, a finite number abelian varieties $A_J$ with complex multiplication of Weil type relative to $F$, and morphisms $A\to A_J$, such that every Hodge cycle $\xi \in \Hdg^p(A)$ is a sum of inverse images of Weil-Hodge cycles $\xi_J \in \Hdg^p(A_J)$. \endproclaim \demo{Proof} We sketch Andr\'e's proof. Up to isogeny write $A = \prod_i A_i$ as a product of simple CM-abelian varieties, where $A_i$ is of CM-type $(K_i,S_i)$. Let $V = H^1(A,\QQ)$, let $V_{S_i} = H^1(A_i,\QQ)$, and let $F$ be the Galois closure of the compositum of all the $\Endo(A_i)$. Then $V = \bigoplus_{i\in I} V_{S_i}$ and $V_{S_i} \tensor F = \bigoplus_{\sigma \in \Hom(F,\CC)} V_{S_i,\sigma}$. Then $$ \spreadlines{1\jot} \align \big( \twedge_\QQ^{2p} \big) \tensor F & \simeq \sum_{\sum d_i = 2p} \big( \bigotimes_{i\in I} \twedge^{d_i} V_{S_i}\big) \tensor F \\ & \simeq \sum\Sb \sum d_{i,\sigma} =2p \\ d_{i,\sigma} \in \{0,1\} \endSb \bigotimes_{(i,\sigma)\in I\times \Hom(F,\CC)} V_{S_i,\sigma}^{\tensor d_{i,\sigma}} . \endalign $$ Let $T_F = \Res_{F/\QQ}\Gm_{/F}$. Then the action of $(T_F)^I$ on $\twedge^{2p}_\QQ V$ commutes with the action of $\Hg(V)$. Further, the action of $T_F$ can be extended by $F$-linearity to an action on $\bigotimes V_{S_i,\sigma}^{\tensor d_{i,\sigma}}$. It follows that every Hodge cycle $\xi \in \twedge^{2p}_\QQ$ can be written as $\xi = \sum \lambda_J \theta_J$, where $J$ indexes the set of sequences $(d_{i,\sigma})_{(i,\sigma)\in I\times \Hom(F,\CC)}$ with $d_{i,\sigma} \in \{0,1\}$ and $\sum d_{i,\sigma} =2p$, and where $\lambda_J \in F$, and $\theta_J \in \bigotimes_{(d_{i,\sigma})\in J} V_{S_i,\sigma}^{\tensor d_{i,\sigma}}$, and the restriction of $\Hg(V)$ acting on $F$ fixes~$\theta_J$. Now observe that each $\tau \in \Aut(F)$ induces an isomorphism of rational Hodge structures $V_{S_i,\sigma} \to V_{\tau S_i, \sigma\tau}$, although this isomorphism does not respect the action of $T_F$. Therefore we may write $V_J = \sum_{(d_{i,\sigma})\in J} V^{d_{i,\sigma}}_{S_i,\sigma}$, where $S_{i,\sigma} = \sigma ^{-1} S$, in such a way that via the isomorphisms induced by $\Aut(F)$ we get a morphism of rational Hodge structures of CM-type $V_J \to V$ and $\theta_J$ comes from an element $\zeta_J \in \bigotimes_{j\in J} V_{S_{i,\sigma},\id}^{\tensor d_{i,\sigma}} \subset \big( \twedge_F^{2p} V_J\big)\tensor_\QQ F$ which is invariant under the action of $\Hg(V_J)$ on~$F$. Note also that there is a natural basis $\chi_{(j,\sigma)}$ of characters of $T_{F^J}$, where $(j,\sigma)$ runs over $J\times \Hom(F,\CC)$. Let $\gamma_{(j,\sigma)}$ denote the dual basis of cocharacters. Then if $\zeta_J \ne 0$, it generates the character $\sum_{j\in J} j \chi_{(j,\sigma)}$ of~$T_{F^J}$. On the other hand, the Hodge structure of $V_J$ is determined by the cocharacter $h:\U(1) \to (T_{F^J})_{/\RR}$, whose complexification may be written out as $$ h_\CC = \sum_{(j,\sigma) \in J\times \Hom(F,\CC)} j(2S_j(\sigma) -1) \gamma_{(j,\sigma)}, $$ where $S_j(\sigma)$ is $1$ or $0$ according as $\sigma\in S_j$ or not. Then the fact that $\zeta_J$ is $\Hg(V_J)$-invariant implies that $\sum_{j\in J} j \angled{\tau h, \chi_{(j,\id)}} =0$ for all $\tau \in \Gal(F/\QQ)$. Then expanding this expression, we find $$ \sum_{j\in J} j(2S_j(\sigma) -1) =0 . $$ And since $\sum_{j\in J} j = 2p$, it follows that $$ \sum_{j\in J} j S_j({\ssize{\bullet}}) = p , $$ which implies that $V_J$ is a rational Hodge structure of Weil type. Moreover, $$ \xi_J := [F:\QQ]^{-1} \sum_{\tau \in \Gal(F/\QQ)} \lambda^\tau \zeta_J^\tau \in \twedge_F^{2p} V_J $$ is a Weil-Hodge cycle, and $\xi$ is a sum of images of $\xi_J$ under the maps $V_J \to V$, since $\xi = [F:\QQ]^{-1} \sum_{\tau \in \Gal(F/\QQ)} \lambda^\tau \theta^\tau_J$. \Qed \enddemo \head 10. The general Hodge conjecture \endhead In this section, when we speak of the general Hodge conjecture we always mean the Grothendieck amended version as in~(7.12) of the text, that the $r^{\text{th}}$ step of the arithmetic filtration $F_a^rH^i(A,\QQ)$ is the largest rational Hodge structure contained in $F^rH^i(Z,\CC) \cap H^i(A,\QQ)$, where $F^rH^i(A,\CC)$ is the Hodge filtration. In those cases where the stronger statement that $F_a^rH^i(A,\QQ) = F^rH^i(Z,\CC) \cap H^i(A,\QQ)$ we will speak of Hodge's original conjecture, or the strong form of the general Hodge conjecture. Based on the results assembled below, it would seem that when it is true, this stronger version is more amenable to being proved. \example{10.1. The general abelian variety} The earliest results about the general Hodge conjecture for abelian varieties are those of Comessatti \cite{B.23} and Mattuck \cite{B.73}, which show that Hodge's original conjecture is true for the general abelian variety described in~1.13.8. Mattuck's proof proceeds by induction and explicit computation with period matrices. Since the general $g$-dimensional abelian variety $A$ has $$ \Hg(A) = \Sp(H^1(A,\QQ),E) = \Lf(A), $$ (see~2.14) and also $\EndoA = \QQ$, it satisfies the hypotheses of Theorem~6.2. Thus $\Hdg(A^k) = \Div(A^k)$ for all $k\ge 1$, which is to say that the general abelian variety is stably nondegenerate, and Theorem~10.9 below applies. \endexample \subhead Abelian varieties of low dimension \endsubhead \nopagebreak \example{10.2. Various abelian threefolds} The first interesting case of the general Hodge conjecture is $\GHC(1,3,X)$ for a threefold $X$, and indeed, it was a special abelian threefold, the product of three copies of an elliptic curve whose period satisfies a cubic relation, that Grothendieck exhibited to show that Hodge's original conjecture needed to be modified, see (7.5) in the text or \cite{B.43}. In \cite{B.12} Bardelli took up the question of whether Grothendieck's counterexample to Hodge's original conjecture satisfies Grothendieck's amended version, and at the same time he considered a number of other abelian threefolds. \proclaim{10.2.1. Theorem {\rm (\cite{B.12}~Prop.3.8)}} The Grothendieck generalized Hodge conjecture holds for: \roster \item The generic abelian threefold; \item The generic member of the family of Jacobians of smooth genus three curves admitting a morphism onto some elliptic curve; \item The generic member of the family of Jacobians of smooth genus three curves admitting a morphism onto some genus two curve; \item The generic product of an abelian surface and an elliptic curve; \item The generic product of three elliptic curves; \item All products of three copies of the same elliptic curve, in particular when the period $\tau$ is quadratic over $\QQ$ (the CM case) or cubic over $\QQ$ (as in Grothendieck's counterexample). \endroster \endproclaim Bardelli's arguments are very geometric in nature. To give a hint of their flavor, let $J_H(A)$ denote the maximal subtorus of the intermediate Jacobian of an abelian threefold $A$ that is orthogonal, with respect to cup product, to $H^{3,0}(A)$. Then the key lemma has the following form. \proclaim{10.2.2. Lemma {\rm (\cite{B.12}~Lem.2.2)}} Let $T$ be an irreducible analytic subvariety of the Siegel upper half-space of genus three and let $p:\script A \to T$ be the restriction of the universal family of principally polarized abelian varieties over the Siegel half-space. Let $t_0 \in T$ be a generic point of $T$ at which $T$ is smooth. Then $$ \dim_\CC J_H(A_{t_0}) \le 9 - \dim T. $$ \endproclaim Recall (1.13..8) that the Siegel half-space of genus~$g$ consists of complex symmetric $g\times g$ matrices with positive-definite imaginary part, and thus represents the possible complex structures for an abelian variety of dimension~$g$. \medpagebreak Another example of abelian threefold for which the general Hodge conjecture is true comes up in \cite{B.86}. In the course of constructing a counterexample to a conjecture of Xiao, Pirola finds the following. \proclaim{10.2.3. Proposition} The general Hodge conjecture is true for the generic member of the family of abelian threefolds of the form $W=\operatorname{Jac}(C)/f^*E$, where $C$ is a smooth genus~$4$ curve, $E$ is an elliptic curve, and $f:C\to E$ is a $3$~to~$1$ cyclic Galois covering. \endproclaim \endexample \example{10.3. An abelian fourfold of Weil type} In \cite{B.105} Schoen proves the general Hodge conjecture for abelian fourfolds $A$ of Weil type with multiplication by $\EndoA = K =\QQ(i)$ (and determinant~$1$, associated to a Hermitian form of signature~$(3,1)$). Previously, in \cite{B.104}, see Theorem~4.12, he had proved the Hodge $(2,2)$ conjecture for these fourfolds. Thus in \cite{B.105} it only remained to verify $\GHC(1,4,A)$. Therefore the focus is on the rational Hodge substructure $U'\subset \twedge^4_K H^1(A,\QQ)$, where $U'$ is the unique $\Res_{K/\QQ}\Gm_{/K}$ subrepresentation of $H^4(A,\QQ)$ which after tensoring with $\CC$ becomes isomorphic to the sum of weight spaces $\alpha^4 \oplus \bar\alpha^4$. The Hodge type of $U'$ is $\{(3,1),\,(1,3)\}$. The highly geometric arguments are lengthy and intricate, so we will not go into them here, except to say that one of the main points is that the generic $A$ as above is a generalized Prym variety associated to a cyclic $4$-fold covering $\pi: C\to X$ of curves. See \cite{B.105} for the details. \endexample \example{10.4. Powers of elliptic curves or abelian surfaces with quaternionic multiplication} The simplest elliptic curves to deal with are those which entirely avoid the kind of problem in Grothendieck's counterexample to Hodge's original conjecture, namely where the period $\tau$ is either quadratic over $\QQ$, in which case the elliptic curve has complex mutliplication, or a general elliptic curve, whose period $\tau$ is transcendental over $\QQ$. \proclaim{10.4.1. Proposition {\rm (\cite{B.117})}} If $E$ is an elliptic curve with complex multiplication, then Hodge's original conjecture is true for $E^k$, for all $k\ge 1$. \endproclaim \proclaim{10.4.2. Proposition {\rm (\cite{B.38})}} If $E$ is a general elliptic curve, then Hodge's original conjecture is true for $E^k$, for all $k\ge 1$. \endproclaim \proclaim{10.4.3. Proposition {\rm (\cite{B.39})}} If $A$ is a general abelian surface with quaternionic multiplication, Hodge's original conjecture is true for $A^k$, for all $k\ge 1$. \endproclaim \proclaim{10.4.4. Proposition {\rm(\cite{B.4}~Thm.6.1)}} When $A$ is a product of elliptic curves, then the general Hodge conjecture is true for $A$. \endproclaim The first of these results is discussed in the text (7.18)--(7.20), and the next two are supersceded by Theorem~10.9 below. For the last, the multiplicativity of the Lefschetz group (Lemma 2.15) reduces the problem to powers of a single elliptic curve, see section three. Then the case where the curve is of CM-type is covered by 10.4.1 above, whereas the case where the curve is not of CM-type is a special case of Theorems~10.9 or~10.12 below. \endexample \subhead Abelian varieties with conditions on endomorphisms, dimension or Hodge group \endsubhead It is only quite recently that results about the general Hodge conjecture for abelian varieties of comparable generality to what has been proved for the usual Hodge conjecture have begun to appear. Here we collect together the main results before discussing some of what is involved in proving them. \proclaim{10.5. Theorem {\rm (\cite{B.127}~Thm.1)}} If $A$ be a simple abelian variety of type~\rom{(I)} such that $\dim A / [\EndoA :\QQ]$ is odd, then Hodge's original conjecture holds for~$A$. \endproclaim \proclaim{10.6. Theorem {\rm (\cite{B.127}~Thm.2)}} Let $A$ be a simple abelian variety of CM-type, with $\EndoA = K$ and $K_0$ the maximal totally real subfield of~$K$. If $[K^{\Gal}: K_0^{\Gal}] = 2^{\dim A}$, then Hodge's original conjecture holds for~$A$. \endproclaim \proclaim{10.7. Theorem {\rm (\cite{B.128}~Thm.1)}} Let $A$ be an abelian variety with $\EndoA =\QQ$. If \roster \item $\dim A \ne 4^l$, \item $\dim A \ne \frac 1 2 \binom{4l+2}{2l+1}^{2m-1}$, \item $\dim A \ne 2^{8lm+4l-4m-3}$, \item $\dim A \ne 4^l (m+1)^{2l+1}$, \item $\dim A \ne 2^{8ln +2n-4l-2} (8l+4)^{m-1}$, \endroster for any positive integers $l$, $m$, $n$, then the general Hodge conjecture holds for~$A$. Furthermore, $\Hdg(A) = \Div(A)$, and $\Hg(A) = \Sp(H^1(A,\QQ),E)$. \endproclaim \proclaim{10.8. Theorem {\rm (\cite{B.129}~Thm.1.1)}} Let $A$ be a simple complex abelian variety of dimension~$g$ with Hodge group $\Hg(A)$, let $\hg(A,\CC) = \Lie \Hg(A)\tensor \CC$, and let $\hg(A,\CC)_{\text{ss}}$ be the semisimple part of the reductive Lie algebra $\hg(A,\CC)$. Consider the following sets of natural numbers: $$ \operatorname{Ex}(1) := \Big\{ 4^l,\, \frac 12\binom{4l+2}{2l+1}^{2m-1},\, 2^{8lm+4l-4m-3},\, 4^l(m+1)^{2l+1} : l,m\in\ZZ_+ \Big\}; $$ $$\multline \operatorname{Ex}(3) := \Big\{ 46{l+1},\, 6^{l+1},\, \binom{4m+4}{2m+2}^l,\, \binom{4m+2}{2m+1}^{2l},\, 2^{(4m-1)}l , \\ 4^l(m+2)^{2l},\, 2^{l+1}(m+4)^{l+1} : l,m\in\ZZ_+ \Big\} ; \endmultline $$ $$\multline \operatorname{Ex}(4) := \Big\{ \binom{l+2}{m} \text{ for }1<m <(l+2)/2 , \\ \binom{l+2}{m}^{n+1} \text{ for }1\le m <(l+2)/2 : l,m,n\in\ZZ_+\Big\}. \endmultline $$ \roster \item If $\End(A)\tensor \RR =\RR$ and $g\notin \operatorname{Ex}(1)$, then $\hg(A,\CC) = \frak{sp}(2g)$ and the general Hodge conjecture is true for $A^k$, for $k\ge 1$. \item If $\End(A)\tensor \RR = M_2(\RR)$ and $g\notin 2\cdot\operatorname{Ex}(3)$, then $\hg(A,\CC) = \frak{sp}(g)$ and the general Hodge conjecture is true for $A^k$, for $k\ge 1$. \item If $\End(A)\tensor \RR = \Bbb H$, the Hamiltonian quaternion algebra, and $g\notin \operatorname{Ex}(3)$, then $\hg(A,\CC) = \frak{so}(g)$ and for $0\le r \le g$ we have $$ \dim_\QQ \Hdg^r = \cases 1&\text{if }r\neq g/2,\\ g+2&\text{if }r=g/2 \endcases $$ (in particular, if $r\neq g/2$, then $\Hdg^r = \Div^r$). \item If $\End(A)\tensor \RR =\CC$ and $g\notin \operatorname{Ex}(4)$, then $\hg(A,\CC)_{\text{ss}} = \frak{sl}(q)$ and for all integers $r\neq g/2$ we have $\Hdg^r =\Div^r$; in the case $rg/2$ we have the relations $$ \dim \Hdg^r = \cases 1&\text{if $\hg(A,\CC)$ is not semisimple,}\\ 3&\text{if $\hg(A,\CC)$ is semisimple.}\endcases $$ \endroster \endproclaim \proclaim{10.9. Theorem {\rm (\cite{B.50}~Thm.5.1)}} If $A$ is a stably nondegenerate abelian variety (see 7.5 and 7.6) all of whose simple components are of type~\rom{(I)} or~\rom{(II)}, the general Hodge conjecture holds for~$A$ and all powers $A^k$ of $A$, for $k\ge 1$. \endproclaim \remark{10.10. Remarks} 1.\enspace It follows from Theorem~6.3.1 that a simple abelian variety of type~\rom{(I)} such that $\dim A / [\EndoA :\QQ]$ is odd is stably nondegenerate, so Theorem~10.5 is a special case of Theorem~10.9. As remarked already, Propositions~10.1, 10.4.2 and 10.4.3 are included in 10.9, as well. Other examples of stably nondegenerate abelian varieties include arbitrary products of elliptic curves (section~3), simple abelian varieties of prime dimension (Theorem~6.3), or simple abelian varieties of odd dimension without complex multiplication (Theorems~6.3 and~7.5). Also an abelian variety $A$ is stably nondegenerate if and only if $A^k$ is stably nondegenerate, for any $k\ge 1$, by~7.6.1.2. 2.\enspace Proposition~10.4.1 is a special case of Theorem~10.6. 3.\enspace The methods used in the proof of Theorem~10.6 are similar to those in the proof of Theorem~9.2 \cite{B.88}. 4.\enspace The proof of Theorem~10.7 uses the classification result Theorem~2.11. Since $\EndoA=\QQ$, Theorem~2.7 applies, and over $\Qbar$ (or~$\CC$) the universal cover of $\Hg(A)$ is isomorphic to some number of copies of an almost simple $\Qbar$-group, say~$G_1$. Then if $G_1$ is any of the types in Theorem~2.11 other than $\frak{sp}(2d)$, where $d=\dim A$, then $d$ is one of the forbidden dimensions. Then the known representation theory of $\frak{sp}(2d)$ can be used to control the level of sub-Hodge structures, in a similar spirit though by a different argument as in the next paragraph. The proof of Theorem~10.8 applies similar ideas to the semisimple part of the Hodge group. \revert@envir\endremark\medskip The proof of Theorem~10.9 provides an illustrative example of how the representation theory of the symplectic group comes into proving the Hodge conjecture. \demo{10.11. Sketch of proof of Theorem~10.9 {\rm (following \cite{B.50})}} Recall that the \dfn{level} $l(W)$ of a rational Hodge structure~$W$, in particular a sub-Hodge structure of $H^m(A,\QQ)$, is the maximum of $|p-q|$ for which $W^{p,q}\ne 0$. Then it will suffice to prove that for any irreducible rational sub-Hodge structure $W$ of $H^m(A,\QQ)$ with $l(W) = m-2p$ there exists a Zariski-closed subset $Z$ of codimension~$p$ in $A$ such that $$ W \subset \Ker\{ H^m(A,\QQ) \to H^m(A-Z,\QQ)\} . $$ Now, it is a basic fact from the representation theory of $\frak{sp}(2n,\CC)$ that there is a one-to-one correspondence between its irreducible (finite-dimensional) representations and $n$-tuples $(\lambda_1, \dots , \lambda_n)$ of nonnegative integers with $\lambda_1\ge \lambda_2 \ge \dots \ge \lambda_n$, see \cite{B.33}, \cite{B.34}. Such an $n$-tuple s called a \dfn{Young diagram of length~$n$}. Then the crucial proposition, whose proof we omit here, is the following. \proclaim{10.11.1. Proposition {\rm (\cite{B.50})}} Let $A$ be an abelian variety with $\hg(A,\CC) \simeq \frak{sp}(2n,\CC)$, let $W$ be an irreducible rational sub-Hodge structure of $H^m(A^k,\QQ)$, and let $(\lambda_1,\dots,\lambda_n)$ be the associated Young diagram. Then $$ l(W) = \sum_{i=1}^n \lambda_i . $$ \endproclaim For a Young diagram $(\lambda_1,\dots,\lambda_n)$ the number $\sum_i \lambda_i$ is often referred to as \dfn{the number of boxes,} for the traditional representation of a Young diagram as $n$ rows with $\lambda_i$ boxes in the $i^{\text{th}}$ row. As a matter of notation, it is convenient to write $(1^a)$ for the Young diagram with $\lambda_1 = \dots = \lambda_a =1$ and $\lambda_{a+1} = \dots = \lambda_n =0$, and similarly $(2^c,1^d)$ for the diagram with $\lambda_1 = \dots = \lambda_c =2$ and $\lambda_{c+1} = \dots = \lambda_{c+d} = 1$ and $\lambda_{c+d+1} = \dots = \lambda_n =0$. By convention $(1^0)$ is the Young diagram for the trivial representation. Then we recall the following facts from representation theory. \proclaim{10.11.2. Lemma} \roster \item Let $V = \CC^{2n}$ as a standard representation of $\frak{sp}(2n,\CC)$. Then for $1\le i \le n$ $$ \twedge^i V \simeq (1^i) \oplus (1^{i-2}) \oplus \dots \oplus (1^{i-2[i/2]}) , $$ where the inclusion of $(1^a)$ into $\twedge^i V$ is defined by taking the exterior product $(i-a)/2$ times with $$ \Omega = \sum_{j=1}^n e_j\wedge e_{n+j} , $$ where $\{e_1,\ldots,e_{2n}\}$ is a standard symplectic basis. \item For nonnegative integers $a$, $b$ with $a\ge b$, $$ \align (1^a) \tensor (1^b) \simeq\ & \{(1^{a+b}) \oplus (2, 1^{a+b-2}) \oplus \dots \oplus (2^b, 1^{a-b})\} \\ & \quad \oplus \{ (1^{a+b-2}) \oplus (2, 1^{a+b-4}) \oplus \dots \oplus (2^{b-1}, 1^{a-b}) \} \\ & \quad \oplus \dots \oplus \{ (1^{a-b}) \} , \endalign $$ with the convention that Young diagrams on the right-hand side with more than $n$ rows are omitted. \endroster \endproclaim \demo\nofrills See \cite{B.18}~Ch.VIII,\S13 and \cite{B.33}. \enddemo Now the proof of the theorem is divided into three steps. First, consider the case where $A= B^k$, where $\hg(B,\CC) \simeq \frak{sp}(2n,\CC)$ acting on $V= H^1(B,\CC) \simeq \CC^{2n}$ as a standard representation. Then any irreducible rational sub-Hodge structure $W$ in $H^m(A,\QQ)$, say of level $l(W) = m-2p$, corresponds over $\CC$ to an irreducible $\hg(B,\CC)$ representation (see Proposition~2.4) occuring in one of the terms on the right-hand side of $$ H^m(A,\CC) \simeq \bigoplus_{m_1 +\dots + m_r =m} (\twedge^{m_1} V \tensor \dots \tensor \twedge^{m_r} V) . $$ By Proposition~10.11.1 the number of boxes in the Young diagram associated to $W_\CC$ is $m-2p$. Then Lemma~10.11.2 implies that the contraction, i.e., the reduction in the number of boxes, comes about only by taking the exterior product with $\Omega^p$. However, in the dictionary between the representation theory and the cohomology, $\Omega$ corresponds to a divisor, say $D$, and taking the exterior product with it corresponds to intersecting with~$D$. Thus $W$ is the cup product of a rational sub-Hodge structure in $H^{m-2p}(A,\QQ)$ with $D^p$, which verifies the general Hodge conjecture in this case. Secondly, consider the case where $A = B_1^{k_1} \times B_2^{k_2}$ with $B_1$ not isogenous to $B_2$, and as in the first case, $\frak g_i := \hg(B_i,\CC) \simeq \frak{sp}(2n_i,\CC)$ acting via a standard representation on $V_i = H^1(B_i,\CC)$. Then $\hg(A) \simeq \frak g_1 \times \frak g_2$, and any irreducible rational sub-Hodge structure $W_\QQ$ of $H^m(A,\QQ)$ must correspond to an irreducible $\hg(A,\CC)$-representation of the form $W_1\tensor W_2$ for some irreducible $\hg(B_i,\CC)$-representation $W_i \subset H^{m_i}(B_i^{k_i},\CC)$, with $m_1 + m_2 =m$. Moreover, as in Proposition~10.11.1, $l(W) = l(W_1) + l(W_2)$. However, by the previous case, if $l(W_i) = m_i -2p_i$, then $W_i$ is supported on a Zariski-closed subset $Z_i$ of codimension $p_i$ on $B_i^{k_i}$. Then $W_1\tensor W_2$ is supported on $Z_1\times Z_2$ of codimension $p_1 + p_2$ on $A$, which verifies the general Hodge conjecture in this case. Finally, if $A$ is an arbitrary abelian variety which satisfies the hypotheses of the theorem, then from \cite{B.47} and Theorem~2.11 it follows that $$ \hg(A,\CC) \simeq \frak{sp}(2n_1,\CC) \times \dots \times \frak{sp}(2n_r,\CC) $$ acting in the standard way on $$ V_1^{\oplus k_1} \oplus \dots \oplus V_r^{\oplus k_r}. $$ Now, for each $i$ the fundamental form $\Omega_i \in \twedge^2 V_i$ is $\frak{sp}(2n_i,\CC)$-invariant, thus by Lefschetz's theorem corresponds to a linear combination of divisor classes. Then arguing similarly as in the previous paragraph for $r=2$ shows that the general Hodge conjecture holds for~$A$, as was to be shown. \Qed \enddemo \example{10.12. Passing from the usual Hodge conjecture to the general Hodge conjecture} The last examples of abelian varieties for which the general Hodge conjecture has been proved come through a theorem of Abdulali \cite{B.4}. In the following statement recall that the derived group $G^{\text{der}} = (G,G)$ of a group $G$ is the (normal) subgroup generated by all elements of the form $ghg^{-1}h^{-1}$. \proclaim{Theorem {\rm (\cite{B.4})}} Let $A$ be an abelian variety whose Hodge group is semisimple and equal to the derived group of the Lefschetz group of $A$, i.e., $\Hg(A) = \Lf(A)^{\text{der}}$. Further suppose that for every simple factor $B$ of $A$ of type~\rom{(III)} the dimension of $H^1(B,\QQ)$ as a vector space over $\Endo(B)$ is odd. Then if the usual Hodge conjecture is true for $A^k$ for all $k\ge 1$, then the general Hodge conjecture is also true for $A$, and all $A^k$ for $k\ge 1$. \endproclaim \remark{10.12.1. Remarks} In the presence of the assumption that $\Hg(A)$ is semisimple, the hypothesis that $\Hg(A) = \Lf(A)^{\text{der}}$ can be alternately formulated as follows: There is a natural embedding $\Hg(A) \hra \Sp(H^1(A,\QQ))$ which induces a holomorphic embedding of the symmetric domain $D$ of $\Hg(A,\RR)$ into the symmetric domain $\frak H$ of $\Sp(H^1(A,\RR))$. Then the pull-back to $D$ of the universal family of polarized abelian varieties of dimension $\dim A$ natually lying over $\frak H$ determines a family of abelian varieties of \dfn{Hodge type} in the sense of \cite{B.78}. Then the hypothesis that $\Hg(A) = \Lf(A)^{\text{der}}$, or in the absence of the assumption that $\Hg(A)$ is semisimple, an assumption that $\Hg(A)^{\text{der}} = \Lf(A)^{\text{der}}$, is equivalent to requiring that the family of Hodge type be a family of abelian varieties of PEL-type, in the sense of \cite{B.109} \cite{B.110} \cite{B.111}. That is to say, a family of abelian varieties that is determined by polarization, endomorphism algebra and level structures. The essential use of this hypothesis in the proof of the theorem is in the multiplicativity of $\Lf(A)$. If $A$ is isogenous to $A_1^{k_1} \times \dots \times A_r^{k_r}$, then $\Lf(A) = \Lf(A_1) \times \dots \times \Lf(A_r)$, see Lemma~2.15, and thus under the assumptions at hand, $\Hg(A) = \Hg(A_1) \times \dots \times \Hg(A_r)$. This makes it possible to reduce the proof to the case where $A$ is isogenous to $A_0^k$ for a simple abelian variety $A_0$. The proof then proceeds by cases, according to whether a simple factor of $\Endo(A_0) \tensor \RR$ is $\RR$, or $\CC$ or $\Bbb H$. \revert@envir\endremark\medskip \example{10.12.2. Applications} What abelian varieties satisfy the hypotheses of Theorem~10.12? A stably nondegenerate abelian variety with a semisimple Hodge group cannot have factors of type~(III) or~(IV), so this is the same class as covered by Theorem~10.9. However, Abdulali observes that whenever the usual Hodge conjecture is true for an abelian four-fold $A$ of Weil type, then it is true for all powers $A^k$ of $A$ \cite{B.4}. Thus there is the following consequence of Theorem~4.12. \endexample \proclaim{10.12.3. Corollary} The general Hodge conjecture is true for all powers of a general abelian fourfold of Weil type $(A,K)$ with $K=\QQ(\sqrt{-3})$ or $K=\QQ(i)$, when the determinant of the associated Hermitian form is~$1$. \endproclaim \endexample \head 11. Other approaches to the Hodge conjecture \endhead In this section we look at three conditional results on the Hodge conjecture. \example{11.1. Higher Jacobians} In \cite{B.98}, Sampson outlines one possible approach to proving the Hodge conjecture for arbitrary abelian varieties. Given an abelian vareity $A$ over $\CC$, let $J^p(A)$ denote its $p^{\text{th}}$ Weil intermediate Jacobian, for odd $p$ with $1 < p \le \dim A$. Then Sampson gives an explicit but complicated construction of a surjective homomorphism $\pi: J^p(A) \to A$ which induces an isomorphism $f: \Hdg^p(A) \to \Div^1(J^p(A))$. By the Poincar\'e Reducibility Theorem 1.11.4, $J^p(A)$ splits, up to isogeny, as $\Ker(\pi) \times A'$. Thus a cycle class $[Z]$ of codimension~$r$ on $Ker(\pi)$ determines a cycle class $\operatorname{proj}_A((Z\times A')\cdot f(\phi))$ of codimension~$(r+1)$ on $A$, with $\phi \in \Hdg^p(A)$. Now if we fix $Z = H^{p-1}$ to be the $(p-1)$-fold self-intersection of a fixed hyperplane section, then $$ \phi \mapsto f^*(\phi) := \operatorname{proj}_A((H^{p-1}\times A')\cdot f(\phi)) $$ defines a homomorphism from $\Hdg^p(A)$ into the group of cohomology classes of algebraic cycles on $A$ of codimension~$p$. Then it is not hard to show that if $f^*$ were injective, then the Hodge $(p,p)$ conjecture would follow. However, the highly transcendental nature of the construction makes the connection between $f^*(\phi)$ and $\phi$ rather obscure, as well as apparently making it very difficult to determine whether $f^*$ is injective. \endexample \example{11.2. The Tate conjecture} The references for this are \cite{B.88} \cite{B.87} \cite{B.27}, and see also \cite{B.15} \cite{B.16} for related results. If $A$ is a complex abelian variety, then there is a subfield $F\subset \CC$ finitely generated over $\QQ$ and a model $A_0$ of $A$ over $F$, meaning that $A = A_0 \tensor_F \CC$. Then the $\ell$-adic \'etale cohomology of $A_0$ over the algebraic closure $F^{\text{alg}}$ of $F$, that is, $H_{\text{\'et}}^{2p}(A_0\tensor F^{\text{alg}}, \QQ_{\ell}(p))$, is naturally a $\Gal(F^{\text{alg}}/F)$-module. In \cite{B.130} Tate conjectured that the elements of $H_{\text{\'et}}^{2p}(A_0\tensor F^{\text{alg}}, \QQ_{\ell}(p))$ fixed by some open subgroup of $\Gal(F^{\text{alg}}/F)$, or equivalently by (the Zariski-closure of) the $\ell$-adic Lie subalgebra $\frak g_\ell \subset \End(H_{\text{\'et}}^{2p}(A_0\tensor F^{\text{alg}}, \QQ_{\ell}(p)))$ generated by the image of $\Gal(F^{\text{alg}}/F)$, is precisely the $\QQ_\ell$-span of the classes of algebraic cycles. In \cite{B.130} Tate himself observed that this conjecture has an air of compatibility with the Hodge conjecture, and already in \cite{B.77}, with the introduction of the Hodge group, Mumford reported that Serre conjectured that $$ \frak g_\ell = \mt(A) \tensor \QQ_\ell . $$ For an excellent early introduction to this conjecture and the relationships between the Tate and Hodge conjectures, see \cite{B.108}; the literature in the 20~years since then is extensive, it would take another appendix at least the size of this one to survey it. The main result of \cite{B.88} is that for abelian varieties of CM-type, the Hodge and Tate conjectures are equivalent. Then that the validity of the Tate conjecture for an abelian variety $A$ implies the validity of the Hodge conjecture for $A$ has been proved by Piatetskii-Shapiro \cite{B.87}, Deligne (unpublished) and \cite{B.27}. \cite{B.15} extends the result of \cite{B.87}, and \cite{B.16} contains a weaker version of the main theorem of \cite{B.27}, from which Tate implies Hodge for abelian varieties follows as a corollary. \endexample \example{11.3. Standard conjectures} In \cite{B.3} Abdulali shows that if one assumes Grothendieck's invariant cycles conjecture \cite{B.42} for families of abelian varieties of Hodge type in the sense of \cite{B.78}, then the Hodge conjecture for abelian varieties follows. He also formulates the $L_2$-cohomology analogue of Grothendieck's standard conjecture~(A) that the Hodge $*$-operator is algebraic \cite{B.44}, and shows that for the families of abelian varieties being considered that this conjecture implies the invariant cycles conjecture and thus the Hodge conjecture for abelian varieties. \proclaim{11.3.1. Conjecture {\rm (Invariant cycles conjecture \cite{B.42})}} Let $f: A\to V$ be a smooth and proper morphism of smooth quasiprojective varieties over~$\CC$. Let $P\in V$ and let $\Gamma:= \pi_1(V,P)$. Then the space of $s\in H^0(V, R^bf_*\QQ) \simeq H^b(A_P, \QQ)^\Gamma$ that represent algebraic cycles in $H^b(A_P, \QQ)^\Gamma$ is independent of~$P$. \endproclaim \definition{11.3.2. Families of abelian varieties of Hodge type {\rm (\cite{B.78})}} Let $A_0$ be a polarized abelian variety, let $W= H_1(A_0,\QQ)$, let $L = H_1(A_0,\ZZ)$, and let $E$ be a Riemann form on $W$ representing the polarization. Also, let $h:\U(1) \to \GL(\Wr)$ be the complex structure on $\Wr$, let $K^+$ be the connected component of the centralizer of $h(\U(1))$ in $\Hg(A_0,\RR)$, and let $D= \Hg(A_0,\RR)^+/K^+$ be the bounded symmetric domain associated to $\Hg(A_0)$, as in~2.10. Then to each point $x\in D$ we can associate the polarized abelian variety $A_x = (\Wr/L, ghg^{-1}, [E])$, where $x = g K^+$. Further, if $\Gamma \subset \Hg(A_0)$ is a torsion-free arithmetic subgroup that preserves $L$, then $\gamma \in\Gamma$ induces an isomorphism between $A_x$ and $A_{\gamma x}$. Thus we get a family $\{A_x : x\in V\}$ of polarized abelian varieties parameterized by $V=\Gamma \bs D$, which may be glued together into an analytic space $A \to V$ fibered over~$V$. Such a family of abelian varieties is said to be of \dfn{Hodge type} \cite{B.78}. Furthermore, $V$ has a canonical structure as a smooth quasiprojective algebraic variety \cite{B.11}, and the analytic map $A\to V$ is an algebraic morphism \cite{B.13}. \enddefinition \proclaim{11.3.3. Theorem {\rm (\cite{B.3}~Thm.6.1)}} If Conjecture~11.3.1 is true for all families of abelian varieties of Hodge type, then the Hodge conjecture is true for all abelian varieties. \endproclaim \demo\nofrills An outline of the proof may be sketched as follows. The first step is to deduce from Conjecture~11.3.1 that all Weil-Hodge cycles are algebraic. To do this, Abdulali shows that any abelian variety $A_1$ of Weil type is a member of a Hodge family whose general member $A_\eta$ has Hodge group equal to the full symplectic group. Then since $\Hdg(A_\eta) = \Div(A_\eta)$, the invariant cycles conjecture implies that all Weil cycles become algebraic in this family. The next point is to observe that Theorem~9.5.2 implies that if all Weil-Hodge cycles are algebraic, then the Hodge conjecture is true for all abelian varieties of CM-type. However, Mumford showed that every family of abelian varieties of Hodge type contains members of CM-type \cite{B.78}. Then the invariant cycles conjecture can be used again to deduce that a Hodge cycle on any member of the family is algebraic. See \cite{B.3} for more details. \enddemo \endexample \newpage \head Chronological listing of work on \\ the Hodge conjecture for abelian varieties \endhead \rightheadtext{Chronological listing} \medskip {\baselineskip=13.5pt \halign{\hfil#&\qquad#\hfil&\qquad#\hfil\cr \smc Year&\smc Author &\smc Topic\cr \noalign{\medskip} 1950&Hodge &Presented conjecture \cr 1958&Mattuck &GHC for general abelian variety \cr 1966&Mumford &Introduced Hodge group \cr 1968&Polhmann &Hodge if and only if Tate for CM-type \cr 1969&Mumford &Families of Hodge type \cr 1969&Grothendieck &Amended general Hodge conjecture \cr 1969&Murasaki &Elliptic curves \cr 1971&Piatetskii-Shapiro &Tate implies Hodge \cr 1974&Borovo\u\i &Tate implies Hodge \cr 1976&Imai &Elliptic curves \cr 1977&Serre &Connections between Hodge and Tate conj. \cr 1977&Borovo\u\i &Absolute Hodge cycles \cr 1977&Weil &Weil type \cr 1978&Tankeev &$4$-dimensional abelian varieties \cr 1979&Deligne &Classification of semisimple part of $\hg$ \cr 1979&Serre &Classification of semisimple part of $\hg$ \cr 1979&Tankeev &$4$- and $5$-dimensional \cr 1981&Borovo\u\i &Simplicity of Hodge group \cr 1981&Shioda &Fermat type \cr 1981&Tankeev &Simple abelian varieties \cr 1981&Tankeev &Simple ($5$-dimensional) abelian varieties \cr 1982&Tankeev &Simple abelian varieties, prime dimension \cr 1982&Deligne &Absolute Hodge cycles, Tate implies Hodge \cr 1982&Kuga &Exceptional cycles \cr 1982&Sampson &Alternate approach \cr 1982&Ribet &Simple abelian varieties \cr 1983&Shioda &Survey \cr 1983&Ribet &Simple abelian varieties, Lefschetz group \cr 1983&Hazama &Nondegenerage CM-type \cr 1983&Murty &Non-simple abelian varieties \cr 1983&Hazama &Non-simple abelian varieties \cr 1984&Kuga &Exceptional cycles \cr 1984&Hazama &Non-simple abelian varieties \cr 1984&Murty &Non-simple abelian var., exceptional cycles \cr 1984&Dodson &Degenerate CM-types \cr 1985&Zarhin &Classification of $\hg$, survey \cr 1985&Yanai &Nondegenerate CM-types \cr 1986&Dodson &Degenerate CM-types \cr 1987&Steenbrink &General Hodge conjecture, survey \cr 1987&Dodson &Degenerate CM-types \cr 1987&Bardelli &GHC, low dimension \cr 1987&Ichikawa &Non-simple abelian varieties MT groups \cr 1988&Hazama &Stably degenerate \cr 1988&Murty &Lefschetz group, semisimple part of $\Hg(A)$ \cr 1988&Schoen &Weil type \cr 1988&Gordon &GHC for powers of general QM-surfaces \cr 1989&Kuga &Exceptional cycles \cr 1989&Hazama &Non-simple abelian varieties \cr 1989&Schoen &GHC Weil type \cr 1989&Mai &Degenerate CM-types \cr 1990&Murty &Survey, Hodge group \cr 1990&Kuga, Perry, Sah &Exceptional cycles \cr 1991&Ichikawa &Non-simple abelian varieties, Hodge groups \cr 1992&Pirola &GHC special threefolds \cr 1992&Andr\'e &CM-type and Weil cycles \cr 1993&White &Degenerate CM-type \cr 1993&Gordon &GHC powers of general elliptic curve \cr 1993&Tankeev &GHC \cr 1994&Zarhin &Survey, connections with arithmetic \cr 1994&van Geemen &Survey, Weil type \cr 1994&Yanai &Degenerate CM-types \cr 1994&Hazama &GHC \cr 1994&Abdulali &Alternate approach \cr 1994&Tankeev &GHC \cr 1995&Moonen \& Zarhin &Abelian $4$-folds \cr 1996&Lee &Exceptional cycles \cr 1996&Abdulali &GHC \cr 1996&Tankeev &GHC\cr 1996&Silverberg and Zarhin&Hodge group, connection to arithmetic\cr 1996&Moonen \& Zarhin &Exceptional Weil cycles \cr }} \newpage \Refs\nofrills{} \rightheadtext{References} \tenpoint \parskip=\bigskipamount \widestnumber\key{\tenpoint [B.999]} {\bf REFERENCES} \ref \key B.1 \by Abdulali, S. \paper Zeta functions of Kuga fiber varieties \jour Duke Math. J. \vol 57 \yr 1988 \pages 333--345 \endref \ref \key B.2 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Conjugation of Kuga fiber varieties \jour Math. Ann. \vol 294 \yr 1992 \pages 225--234 \endref \ref \key B.3 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles in families of abelian varieties \jour Canad. J. Math. \vol 46 \yr 1994 \pages 1121--1134 \endref \ref \key B.4 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Abelian varieties and the general Hodge conjecture \toappear \jour Compositio Math. \endref \ref \key B.5 \by Addington, S. \paper Equivariant holomorphic maps of symmetric domains \linebreak \jour Duke Math. J. \vol 55 \yr 1987 \pages 65--88 \endref \ref \key B.6 \by Albert, A.A. \paper On the construction of Riemann matrices.~\rom{I, II} \jour Ann. of Math. \vol 35 \yr 1934 \pages 1--28 \moreref \vol 36 \yr 1935 \pages 376--394 \endref \ref \key B.7 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper A solution of the principal problem in the theory of Riemann matrices \jour Ann. of Math. \vol 35 \yr 1934 \pages 500--515 \endref \ref \key B.8 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Involutorial simple algebras and real Riemann matrices \jour Ann. of Math. \vol 36 \yr 1935 \pages 886--964 \endref \ref \key B.9 \by Andr\'e, Y. \paper Une remarque \`a propos des cycles de Hodge de type CM \inbook Seminaire de Th\'eorie des Nombres, Paris, 1989--1990 \bookinfo Progr. Math. 102 \publ Birkh\"auser \publaddr Boston \yr 1992 \pages 1--7 \endref \ref \key B.10 \by Atiyah, M., Bott, R., Patodi, V.K. \paper On the heat equation and the index theorem \jour Invent. Math. \vol 19 \yr 1973 \pages 279--330 \endref \ref \key B.11 \by Baily, W., Borel, A. \paper Compactifications of arithmetic quotients of bounded symmetric domains \jour Ann. of Math. \vol 84 \yr 1966 \pages 442--528 \endref \ref \key B.12 \by Bardelli, F. \paper A footnote to a paper by A. Grothendieck \rom{(}the Grothendieck generalized Hodge conjecture for some geometric families of abelian threefolds\rom{)} \jour Rend. Sem. Mat. Fis. Milano \finalinfo Proceedings of the Geometry Conference (Milan and Gargnano, 1987) \vol 57 \yr 1987 \pages 109--124 \endref \ref \key B.13 \by Borel, A. \paper Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem \jour J. Diff. Geom. \vol 6 \yr 1972 \pages 543--560 \endref \ref \key B.14 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Linear Algebraic Groups \bookinfo Second Enlarged Edition \publ Springer \publaddr New York etc. \yr 1991 \endref \ref \key B.15 \by Borovo\u\i, M.V. \paper On the Galois action on rational cohomology classes of type $(p,p)$ of abelian varieties \lang Russian \jour Mat. Sbornik \vol 94(136) \yr 1974 \pages 649--652, 656 \transl English transl. \jour Math. USSR Sbornik \vol 23 \yr 1974 \pages 613--616 \endref \ref \key B.16 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper The Shimura-Deligne schemes $M_\CC(G,h)$ and the rational cohomology classes of type $(p,p)$ of abelian varieties \lang Russian \inbook Problems in Group Theory and Homological Algebra \publ Yaroslav. Gos. Univ. \publaddr Yaroslavl \yr 1977 \pages 3--53 \endref \ref \key B.17 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper The Hodge group and the algebra of endomorphisms of an abelian variety \paperinfo Russian \inbook Problems in Group Theory and Homological Algebra \publ Yaroslav. Gos. Univ. \publaddr Yaroslavl \yr 1981 \pages 124--126 \endref \ref \key B.18 \by Bourbaki, N. \book Groupes et Alg\`ebras de Lie \publ Hermann \publaddr Paris \yr 1975 \endref \ref \key B.19 \by Cassels, J.W.S. \book Rational Quadratic Forms \publ Academic Press \publaddr London, New York, San Francisco \yr 1978 \endref \ref \key B.20 \by Chevalley, C.C. \book The Algebraic Theory of Spinors \publ Columbia Univ. Press \publaddr Morningside Heights, NY \yr 1954 \endref \ref \key B.21 \by Chi, W. \paper On the $\ell$-adic representations attached to some absolutely simple abelian varieties of type~\rom{II} \jour J. Fac. Sci. Univ. Tokyo Sect.~IA Math. \vol 37 \yr 1990 \pages 467--484 \endref \ref \key B.22 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper $\ell$-adic and $\lambda$-adic representations associated to abelian varieties defined over number fields \jour Amer. J. Math. \vol 114 \yr 1992 \pages 315--353 \endref \ref \key B.23 \by Comessatti, A. \paper Sugl'indici di singolarita a piu dimensioni della varieta abeliane \jour Recond. del Sem. Mat. della Univ. di Padova \vol 5 \yr 1934 \page 50 \endref \ref \key B.24 \by Deligne,~P. \paper La conjecture de Weil pour les surfaces $K3$ \jour Invent. Math. \vol 15 \yr 1972 \pages 206--226 \endref \ref \key B.25 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Vari\'et\'es de Shimura: interpretation modulaire et techniques de construction de mod\`eles canoniques \inbook Automorphic Forms, Representations, and $L$-functions \bookinfo Proc. Symp. Pure Math 33, part 2 \publ Amer. Math. Soc. \publaddr Providence \yr 1979 \pages 247--289 \endref \ref \key B.26 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Cycles de Hodge absolus et p\'eriodes des int\'egrales des vari\'et\'es ab\'eliennes \paperinfo (r\'edig\'e par J.-L. Brylinski) \jour Soc. Math. France 2e s\'er. M\'emoire \vol 2 \yr 1980 \pages 23--33 \endref \ref \key B.27 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Hodge cycles on abelian varieties \paperinfo (Notes by J.S.~Milne) \inbook Hodge Cycles, Motives, and Shimura Varieties \bookinfo Lecture Notes in Mathematics 900 \publ Springer-Verlag \publaddr New York \yr 1982 \pages 9--100 \endref \ref \key B.28 \by Dodson, B. \paper The structure of Galois groups of CM-fields \jour Trans. Amer. Math. Soc. \vol 283 \yr 1984 \pages 1--32 \endref \ref \key B.29 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Solvable and nonsolvable CM-fields \jour Amer. J. Math. \vol 108 \yr 1986 \pages 75--93 \endref \ref \key B.30 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper On the Mumford-Tate group of an abelian variety with complex multiplication \jour J. of Algebra \vol 111 \yr 1987 \pages 49--73 \endref \ref \key B.31 \by Eichler, M. \book Quadratische Formen und Orthogonale Gruppen \publ Springer \publaddr Berlin, Heidelberg, New York \yr 1974 \endref \ref \key B.32 \by Faber, C. \paper Prym varieties of triple cyclic covers \jour Math. Z. \vol 199 \yr 1988 \pages 61--79 \endref \ref \key B.33 \by Fischler, M. \paper Young-tableau methods for Kronecker products of representations of the classical groups \jour J. Math. Phys. \vol 22 \yr 1981 \pages 637--648 \endref \ref \key B.34 \by Fulton, W., Harris, J. \book Representation Theory: A First Course \bookinfo Graduate Texts in Math.129 \publ Springer \publaddr New York etc. \yr 1991 \endref \ref \key B.35 \by van Geemen, B. \paper An introduction to the Hodge conjecture for abelian varieties \inbook Algebraic Cycles and Hodge Theory, Torino, 1993 \eds A.~Albano and F.~Bardelli \bookinfo Lect. Notes in Math. 1594 \publ Springer \publaddr Berlin, etc. \yr 1994 \pages 233--252 \endref \ref \key B.36 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Theta functions and cycles on some abelian fourfolds \jour Math. Z. \vol 221 \yr 1996 \pages 617--631 \endref \ref \key B.37 \by Giraud, J. \paper Modules des vari\'et\'es ab\'eliennes et vari\'et\'es de Shimura \linebreak \inbook Vari\'et\'es de Shimura et fonctions~$L$ \bookinfo Publ. Math. de l'Univ. Paris,~VII, 6 \eds L. Breen, J.-P. Labesse \pages 21--42 \publ U.E.R. de Math\'ematiques \& L.A. 212 du C.N.R.S. \publaddr Paris \yr 1981 \endref \ref \key B.38 \by Gordon,~B.B. \paper Topological and algebraic cycles in Kuga-Shimura varieties \jour Math. Ann. \vol 279 \yr 1988 \pages 395--402 \endref \ref \key B.39 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles and the Hodge structure of a Kuga fiber variety \jour Trans. Amer. Math. Soc. \vol 336 \yr 1993 \pages 933--947 \endref \ref \key B.40 \by Greenberg, R. \paper On the Jacobian variety of some algebraic curves \jour Compositio Math. \vol 42 \yr 1980/81 \pages 345--359 \endref \ref \key B.41 \by Gross, B. \paper On the periods of integrals and a formula of Chowla and Selberg \jour Invent. math. \vol 45 \yr 1978 \pages 193--211 \endref \ref \key B.42 \by Grothendieck,~A. \paper On the deRham cohomology of algebraic varieties \jour Inst. Hautes \'Etudes Sci. Publ. Math. \vol 29 \yr 1966 \pages 96--103 \endref \ref \key B.43 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Hodge's general conjecture is false for trivial reasons \jour Topology \vol 8 \yr 1969 \pages 299--303 \endref \ref \key B.44 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Standard conjectures on algebraic cycles \inbook Algebraic Geometry \procinfo Papers Presented at the Bombay Colloquium, 1969 \bookinfo Tata Inst. Fund. Res. Studies Math. 4 \publ Oxford Univ. Press \publaddr Bombay, Oxford \yr 1969 \pages 193--199 \endref \ref \key B.45 \by Hazama, F. \paper Hodge cycles on abelian varieties of CM-type \jour Res. Act. Fac. Engrg. Tokyo Denki Univ. \vol 5 \yr 1983 \pages 31--33 \endref \ref \key B.46 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on abelian varieties with many real endomorphisms \jour T\^ohoku Math. J. (2) \vol 35 \yr 1983 \pages 303--308 \endref \ref \key B.47 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on certain abelian varieties and powers of special surfaces \jour J. Fac. Sci. Univ. Tokyo Sect.~IA Math. \vol 31 \yr 1984 \pages 487--520 \endref \ref \key B.48 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Branching rules and Hodge cycles on certain abelian varieties \jour Amer. J. Math. \vol 110 \yr 1988 \pages 235--252 \endref \ref \key B.49 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on nonsimple abelian varieties \jour Duke Math. J. \vol 58 \yr 1989 \pages 31--37 \endref \ref \key B.50 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper The generalized Hodge conjecture for stably nondegenerate abelian varieties \jour Compositio Math. \vol 93 \yr 1994 \pages 129--137 \endref \ref \key B.51 \by Helgason, S. \book Differential Geometry, Lie Groups, and Symmetric Spaces \publ Academic Press \publaddr San Diego etc. \yr 1978 \endref \ref \key B.52 \by Hochschild, G. \book Basic Theory of Algebraic Groups and Lie Algebras \linebreak \publ Springer \publaddr New York etc. \yr 1981 \endref \ref \key B.53 \by Hodge,~W.V.D. \paper The topological invariants of algebraic varieties \jour Proc. Int. Congr. Math. \yr 1950 \pages 182--182 \endref \ref \key B.54 \by Humphreys, J.E. \book Introduction to Lie Algebras and Representation Theory \publ Springer \publaddr New York etc. \yr 1972 \endref \ref \key B.55 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Linear Algebraic Groups \bookinfo Corrected Second Printing \publ Springer \publaddr New York etc. \yr 1981 \endref \ref \key B.56 \by Ichikawa, T. \paper On algebraic groups of Mumford-Tate type \inbook Galois Representations and Arithmetic Algebraic Geometry \bookinfo Advanced Studies in Pure Math. 12 \pages 249--258 \publ North-Holland \publaddr Amsterdam \yr 1987 \endref \ref \key B.57 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic groups associated with abelian varieties \jour Math. Ann. \vol 289 \yr 1991 \pages 133--142 \endref \ref \key B.58 \by Imai, H. \paper On the Hodge groups of some abelian varieties \jour Kodai Math. Sem. Rep. \vol 27 \yr 1976 \pages 367--372 \endref \ref \key B.59 \by Koblitz, N., Rohrlich, D. \paper Smple factors in the Jacobian of a Fermat curve \jour Canad. J. Math. \vol 30 \yr 1978 \pages 1183--1205 \endref \ref \key B.60 \by Kubota, T. \paper On the field extension by complex multiplication \jour Trans. Amer. Math. Soc. \vol 118 \yr 1965 \pages 113--122 \endref \ref \key B.61 \by Kuga, M. \book Fibre Varieties over a Symmetric Space whose Fibres are \linebreak Abelian Varieties \bookinfo vols.~I and~II \publ Univ. of Chicago \publaddr Chicago \yr 1964 \endref \ref \key B.62 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Fibre varieties over symmetric space whose fibres are abelian varieties \inbook Proc. of the U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965 \pages 72--81 \publ Nippon Hyoronsha \yr 1966 \endref \ref \key B.63 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Fiber varieties over symmetric space whose fibers are abelian varieties \inbook Algebraic Groups and Discontinuous Subgroups \bookinfo Proc. Symp. Pure Math. 9 \publ Amer. Math. Soc. \publaddr Providence \pages 338--346 \yr 1966 \endref \ref \key B.64 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles in gtfabv \jour J. Fac. Sci. Univ. Tokyo Sect. IA Math. \vol 29 \yr 1982 \pages 13--29 \endref \ref \key B.65 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Chemistry and GTFABV's \inbook Automorphic Forms of Several Variables \procinfo Taniguchi Symp., Katata, 1983 \bookinfo Progress in Math. 46 \publ Birkh\"auser \publaddr Boston \pages 269--281 \yr 1984 \endref \ref \key B.66 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Invariants and Hodge cycles \inbook Automorphic Forms and Geometry of Arithmetic Varieties \bookinfo Adv. Stud. Pure Math. 15 \publ North-Holland \publaddr Amsterdam \yr 1989 \pages 373--413 \moreref \paper \rom{II} \jour Papers of Coll. Arts and Sci. Univ. Tokyo \vol 40 \yr 1990 \pages 1--25 \endref \ref \key B.67 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ , Parry, W., Sah, C.-H. \paper Invariants and Hodge cycles, III \jour Proc. Japan Acad. \vol 66 \yr 1990 \pages 22--25 \moreref \paper \rom{IV} \jour ibid. \pages 49--52 \endref \ref \key B.68 \by Lang, S. \book Introduction to Algebraic and Abelian Functions \publ Springer \publaddr New York etc. \yr 1982 \endref \ref \key B.69 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Abelian Varieties \publ Springer-Verlag \publaddr New York Berlin Heidelberg Tokyo \yr 1983 \endref \ref \key B.70 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Complex Multiplication \bookinfo Grundlehren mathematischen Wissen- \linebreak schaften 255 \publ Springer \publaddr New York, etc. \yr 1983 \endref \ref \key B.71 \by Lee, M.-H. \paper Hodge cycles on Kuga fiber varieties \jour J. Austral. Math. Soc. Ser. A \vol 61 \yr 1996 \pages 1--13 \endref \ref \key B.72 \by Mai, L. \paper Lower bounds for the ranks of CM-types \jour J. Number Theory \vol 32 \yr 1989 \pages 192--202 \endref \ref \key B.73 \by Mattuck, A. \paper Cycles on abelian varieties \jour Proc. Amer. Math. Soc. \vol 9 \yr 1958 \pages 88--98 \endref \ref \key B.74 \by Milne, J.S. \paper Abelian varieties \inbook Arithmetic Geometry \eds G. Cornell, J.H. Silverman \publ Springer \publaddr New York etc. \yr 1986 \pages 103--150 \endref \ref \key B.75 \by Moonen, B.J.J., Zarhin, Yu.G. \paper Hodge classes and Tate classes on simple abelian fourfolds \jour Duke Math. J. \vol 77 \yr 1995 \pages 553--581 \endref \ref \key B.76 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Weil classes on abelian vaieties \yr 1996 \miscnote preprint \endref \ref \key B.77 \by Mumford,~D. \paper Families of abelian varieties \inbook Algebraic Groups and Discontinuous Subgroups \bookinfo Proc. Symp. Pure Math. 9 \publ Amer. Math. Soc \publaddr Providence \yr 1966 \pages 347--351 \endref \ref \key B.78 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper A note on Shimura's paper ``Discontinuous groups and abelian varieties'' \jour Math. Ann. \vol 181 \yr 1969 \pages 345--351 \endref \ref \key B.79 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Abelian Varieties \publ Oxford Univ. Press \publaddr Oxford \yr 1970 \endref \ref \key B.80 \by Murasaki, T. \paper On rational cohomology classes of type $(p,p)$ on an Abelian variety \jour Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A \vol 10 \yr 1969 \pages 66--74 \endref \ref \key B.81 \by Murty, V.K. \paper Algebraic cycles on abelian varieties \jour Duke Math. J. \vol 50 \yr 1983 \pages 487--504 \endref \ref \key B.82 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Exceptional Hodge classes on certain abelian varieties \jour Math. Ann. \vol 268 \yr 1984 \pages 197--206 \endref \ref \key B.83 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper The Hodge group of an abelian variety \jour Proc. Amer. Math. Soc. \vol 104 \yr 1988 \pages 61--68 \endref \ref \key B.84 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Computing the Hodge group of an abelian variety \inbook S\'eminaire de Th\'eorie des Nombres, Paris 1988--1989 \bookinfo Progr. Math. 91 \publ Birkh\"auser \publaddr Boston \yr 1990 \pages 141--158 \endref \ref \key B.85 \by Oort, F. \paper Endomorphism algebras of abelian varieties \inbook Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata~\rom{II} \eds H. Hijikata et al. \pages 469--502 \publ Kinokuniya \publaddr Tokyo \yr 1988 \endref \ref \key B.86 \by Pirola, G.P. \paper On a conjecture of Xiao \jour J. Reine angew. Math. \vol 431 \yr 1992 \pages 75--89 \endref \ref \key B.87 \by Piatetskii-Shapiro, I.I. \paper Interrelations between the Tate and Hodge conjectures for abelian varieties \lang Russian \jour Mat. Sbornik \vol 85(127) \yr 1971 \pages 610--620 \transl English transl. \jour Math. USSR Sbornik \vol 14 \yr 1971 \pages 615--625 \endref \ref \key B.88 \by Pohlmann, H. \paper Algebraic cycles on abelian varieties of complex multiplication type \jour Ann. of Math. (2) \vol 88 \yr 1968 \pages 161--180 \endref \ref \key B.89 \by Ribet, K.A. \paper Galois representations attached to eigenforms with Nebentypus \inbook Modular Forms of One Variable,~\rom{V} \eds J.-P. Serre, D.B. Zagier \bookinfo Lecture Notes in Math. 601 \pages 17--52 \publ Springer \publaddr Berlin etc. \yr 1976 \endref \ref \key B.90 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Galois action on division points of abelian varieties with real multiplication \jour Amer. J. Math. \vol 98 \yr 1976 \pages 751--804 \endref \ref \key B.91 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Twists of modular forms and endomorphisms of abelian varieties \jour Math. Ann. \vol 253 \yr 1980 \pages 43--62 \endref \ref \key B.92 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Division fields of abelian varieties with complex multiplication \jour Soc. Math. France 2e s\'er. M\'emoire \vol 2 \yr 1980 \pages 75--94 \endref \ref \key B.93 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Generalization of a theorem of Tankeev \inbook Seminar on Number Theory, 1981/1982 \publ Univ. Bordeaux I \publaddr Talence \yr 1982 \finalinfo Exp. No. 17, 4 pp \endref \ref \key B.94 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Hodge classes on certain types of Abelian varieties \jour Amer. J. Math. \vol 105 \yr 1983 \pages 523--538 \endref \ref \key B.95 \by Robert, A. \paper Introduction aux vari\'et\'es ab\'eliennes complexes \jour Enseign. \linebreak Math. \vol 28 \yr 1982 \pages 91--137 \endref \ref \key B.96 \by Rosen, M. \paper Abelian varieties over $\Bbb C$ \inbook Arithmetic Geometry \eds G.~Cornell and J.~Silverman \publ Springer \publaddr New York etc. \yr 1986 \endref \ref \key B.97 \by Saavedra Rivano,~N. \book Cat\'egories tannakiennes \bookinfo Lecture Notes in Math. 265 \publ Springer Verlag \publaddr Heidelberg etc. \yr 1972 \endref \ref \key B.98 \by Sampson, J.H. \paper Higher Jacobians and cycles on abelian varieties \jour Compositio Math. \vol 47 \yr 1982 \pages 133--147 \endref \ref \key B.99 \by Satake, I. \paper Holomorphic embeddings of symmetric domains into a Siegel space \jour Amer. J. Math. \vol 87 \yr 1965 \pages 425--461 \endref \ref \key B.100 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Symplectic representations of Algebraic groups \inbook Algebraic Groups and Discontinuous Subgroups \bookinfo Proc. Symp. Pure Math. 9 \publ Amer. Math. Soc \publaddr Providence \yr 1966 \pages 352--357 \endref \ref \key B.101 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Clifford algebras and families of abelian varieties \jour Nagoya Math. J. \vol 27 \yr 1966 \pages 435--446 \moreref \paper corrections \jour ibid. \vol 31 \yr 1968 \pages 295--296 \endref \ref \key B.102 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Symplectic representations of algebraic groups satisfying a certain analyticity condition \yr 1967 \vol 117 \jour Acta Math. \pages 215--299 \endref \ref \key B.103 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Algebraic Structures of Symmetric Domains \publ Princeton Univ. \linebreak Press \publaddr Princeton \yr 1980 \endref \ref \key B.104 \by Schoen, C. \paper Hodge classes on self-products of a variety with an automorphism \jour Compositio Math. \vol 65 \yr 1988 \pages 3--32 \endref \ref \key B.105 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Cyclic covers of $\Bbb P\sp v$ branched along $v+2$ hyperplanes and the generalized Hodge conjecture for certain abelian varieties \inbook Arithmetic of complex manifolds (Erlangen, 1988) \bookinfo Lecture Notes in Math. 1399 \publ Springer \publaddr Berlin etc. \yr 1989 \pages 137--154 \endref \ref \key B.106 \by Serre, J.-P. \paper Sur les groupes de Galois attach\'es aux groupes $p$-divisibles \inbook Proceedings of a Conference on Local Fields \ed T.A. Springer \pages 118--131 \publ Springer \publaddr Berlin etc. \yr 1967 \endref \ref \key B.107 \by Serre,~J.-P. \paper Repr\'esentations $l$@-adiques \inbook Algebraic Number Theory \linebreak \ed S. Iyanaga \bookinfo Papers contributed for the International Symposium, Kyoto, 1976 \publ Japan Society for the Promotion of Science \publaddr Tokyo \yr 1977 \pages 177--193 \endref \ref \key B.108 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Groupes alg\'ebriques associ\'e aux modules de Hodge-Tate \jour Ast\'erisque \vol 65 \yr 1979 \pages 155--188 \endref \ref \key B.109 \by Shimura, G. \paper On analytic families of polarized abelian varieties and automorphic functions \jour Ann. of Math. \vol 78 \yr 1963 \pages 149--192 \endref \ref \key B.110 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Moduli and fibre systems of abelian varieties \jour Ann. Math. \vol 83 \linebreak \yr 1966 \pages 294--338 \endref \ref \key B.111 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Moduli of abelian varieties and number theoru \inbook Algebraic Groups and Discontinuous Subgroups \bookinfo Proc. Symp. Pure Math. 9 \publ Amer. Math. Soc \publaddr Providence \yr 1966 \pages 312--332 \endref \ref \key B.112 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Introduction to the Arithmetic Theory of Automorphic Forms \bookinfo Publ. Math. Soc. Japan 11 \publ Princeton University Press \publaddr Princeton \yr 1971 \endref \ref \key B.113 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields \jour Nagoya Math. J. \vol 43 \yr 1971 \pages 199--208 \endref \ref \key B.114 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Class fields over real quadratic fields and Hecke operators \jour Ann. of Math. \vol 95 \yr 1972 \pages 130--190 \endref \ref \key B.115 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ , Taniyama, Y. \book Complex Multiplication of Abelian Varieties and its Applications to Number Theory \bookinfo Publ. Math. Soc. Japan 6 \yr 1961 \endref \ref \key B.116 \by Shioda, T. \paper Algebraic cycles on abelian varieties of Fermat type \jour Math. Ann. \vol 258 \yr 1981/1982 \pages 65--80 \endref \ref \key B.117 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper What is known about the Hodge conjecture? \inbook Algebraic Varieties and Analytic Varieties \bookinfo Adv. Stud. in Pure Math. 1 \publ Kinokuniya \publaddr Tokyo \yr 1983 \pages 55--68 \endref \ref \key B.118 \by Silverberg, A., Zarhin, Yu.G. \paper Hodge groups of abelian varieties with purely multiplicative reduction \lang Russian \jour Izv. Ross. Akad. Nauk Ser. Mat. \vol 60 \yr 1996 \pages 149--158 \transl English transl. \jour Izvestia: Math. \vol 60 \yr 1996 \pages 379--389 \endref \ref \key B.119 \by Springer, T.A. \book Linear Algebraic Groups \bookinfo Progress in Math. 9 \publ Birkh\"auser \publaddr Boston etc. \yr 1981 \endref \ref \key B.120 \by Steenbrink, J. \paper Some remarks about the Hodge conjecture \inbook Hodge Theory, Proceedings, Sant Cugat, Spain, 1985 \bookinfo Lect. Notes in Math. 1246 \eds Cattani, Guill\'en, Kaplan, Puerta \publ Springer \publaddr Berlin etc. \pages 165--175 \yr 1987 \endref \ref \key B.121 \by Swinnerton-Dyer, H.P.F. \book Analytic Theory of Abelian Varieties \publ Cambridge Univ. \publaddr Cambridge \bookinfo London Math. Soc. Lect. Note Series 14 \yr 1974 \endref \ref \key B.122 \by Tankeev, S.G. \paper Algebraic cycles on abelian varieties \lang Russian \jour Izv. Akad. Nauk SSSR Ser. Mat. \vol 42 \yr 1978 \pages 667--696 \transl English transl. \jour Math. USSR Izv. \vol 12 \yr 1978 \pages 617--643 \endref \ref \key B.123 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on abelian varieties.~\rom{II} \lang Russian \jour Izv. Akad. Nauk SSSR Ser. Mat. \vol 43 \yr 1979 \pages 418--429 \transl English transl. \jour Math. USSR Izv. \vol 14 \yr 1980 \pages 383--394 \endref \ref \key B.124 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on surfaces and abelian varieties \lang Russian \jour Izv. Akad. Nauk SSSR Ser. Mat. \vol 45 \yr 1981 \pages 398--434 \transl English transl. \jour Math. USSR Izv. \vol 18 \yr 1982 \pages 349--380 \endref \ref \key B.125 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles of simple $5$-dimensional abelian varieties \lang Russian \jour Izv. Akad. Nauk SSSR Ser. Mat. \vol 45 \yr 1981 \pages 793--823 \transl English transl. \jour Math. USSR Izv. \vol 19 \yr 1982 \pages 95--123 \endref \ref \key B.126 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Cycles on simple abelian varieties of prime dimension \lang Russian \jour Izv. Akad. Nauk SSSR Ser. Mat. \vol 46 \yr 1982 \pages 155--170 \transl English transl. \jour Math. USSR Izv. \vol 20 \yr 1983 \pages 157--171 \endref \ref \key B.127 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Abelian varieties and the general Hodge conjecture \lang Russian \jour Izv. Ross. Akad. Nauk Ser. Mat. \vol 57 \yr 1993 \pages 192--206 \transl English transl. \jour Russian Acad. Sci. Izv. Math. \vol 43 \yr 1994 \pages 179--191 \endref \ref \key B.128 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Algebraic cycles on an abelian variety without complex multiplication \lang Russian \jour Izv. Ross. Akad. Nauk Ser. Mat. \vol 58 \yr 1994 \pages 103--126 \transl English transl. \jour Russian Acad. Sci. Izv. Math. \vol 44 \yr 1995 \pages 531--553 \endref \ref \key B.129 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Cycles on abelian varieties and exceptional numbers \lang Russian \jour Izv. Ross. Akad. Nauk Ser. Mat. \vol 60 \yr 1996 \pages 159--194 \transl English transl. \jour Izvestia: Math. \vol 60 \yr 1996 \pages 391--424 \endref \ref \key B.130 \by Tate,~J. \paper Algebraic cycles and poles of zeta functions \inbook Arithmetical Algebraic Geometry \publ Harper and Row \publaddr New York \yr 1965 \pages 93--110 \endref \ref \key B.131 \by Tjiok, M.-C. \book Algebraic Cycles in a Certain Fiber Variety \bookinfo thesis \publ SUNY at Stoney Brook \yr 1980 \endref \ref \key B.132 \by Vigneras, M.-F. \book Arithmetique des Alg\'ebres de Quaternions \bookinfo Lecture Notes in Math. 800 \publ Springer-Verlag \publaddr Berlin etc. \yr 1980 \endref \ref \key B.133 \by Waterhouse,~W.C. \book Introduction to Affine Group Schemes \publ Springer Verlag \publaddr New York etc. \yr 1979 \endref \ref \key B.134 \by Weil, A. \book Introduction \`a l'\'Etude des Vari\'et\'es K\"ahl\'eriennes \publ Hermann \publaddr Paris \yr 1958 \endref \ref \key B.135 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Abelian varieties and the Hodge ring \inbook \OE vres Scientifiques Collected Papers \bookinfo vol.~3, [1977c] \publ Springer \publaddr New York etc. \yr 1979 \pages 421--429 \endref \ref \key B.136 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \book Adeles and Algebraic Groups \bookinfo Progress in Math. 23 \publ Birkh\"auser \publaddr Boston etc. \yr 1982 \endref \ref \key B.137 \by Weyl, H. \book The Classical Groups \publ Princeton University Press \publaddr Princeton \yr 1946 \endref \ref \key B.138 \by White, S.P. \paper Sporadic cycles on CM abelian varieties \jour Compositio Math. \vol 88 \yr 1993 \pages 123--142 \endref \ref \key B.139 \by Yanai, H. \paper On the rank of CM-type \jour Nagoya Math. J. \vol 97 \yr 1985 \pages 169--172 \endref \ref \key B.140 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper On degenerate CM-types \jour J. Number Theory \vol 49 \yr 1994 \pages 295--303 \endref \ref \key B.141 \by Zarhin, Yu.G. \paper Weights of simple Lie algebras in the cohomology of algebraic varieties \jour Math. USSR Izv. \vol 24 \yr 1985 \pages 245--281 \endref \ref \key B.142 \by\hbox to2pc{\hrulefill}\thinspace\kern\z@ \paper Abelian varieties and Lie algebras \jour Selecta Math. \vol 13 \yr 1994 \pages 55--95 \endref \endRefs \enddocument

9709

alg-geom/9709011

en

https://arxiv.org/abs/alg-geom/9709011

alg-geom/9709011

Jonathan Fine

Jonathan Fine

Local-global intersection homology

LaTeX 2e. 28 pages. This paper defines new intersection homology groups, that provide important new information

This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A local-global cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Central is a study of the cycles on a cone and a cylinder, in terms of those on the base. This leads to the topological definition of local-global intersection homology, and a formula for the expected Betti numbers of toric varieties. Various related questions are also discussed.

alg-geom

\section{Introduction} This paper defines new intersection homology groups. They record, in a global way, local information about the singularities. They give rise to new information, both globally and locally, and vanish on nonsingular varieties. Such groups are required, to obtain a satisfactory understanding of general convex polytopes. They also have other applications. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A local-global cycle is a family of such local cycles that is, at the same time, a global cycle. The chains, that produce the homology relations between the cycles, are to have a similar local-global nature. The theory of toric algebraic varieties, which associates an algebraic variety $\PDelta$ to each convex polytope $\Delta$ (provided $\Delta$ has rational vertices) establishes a dictionary between convex polytopes and algebraic varieties. Convex polytopes (or, if one prefers, the associated varieties) provide the simplest examples for these new concepts. The basic problem is to understand general polytopes in the same way as simple polytopes are already understood. Suppose $\Delta$ is a simple polytope. Loosely speaking, this means that the associated variety $\PDelta$ is nonsingular. The associated homology ring $H_\bullet\Delta$ has the following properties. It is generated by the facets of $\Delta$. It satisfies the Poincar\'e duality and strong Lefschetz theorems. The associated Betti numbers $h\Delta$ are a linear function of the face vector $f\Delta$, and vice versa. These facts are central to Stanley's proof \cite{bib.RS.NFSP} of the necessity of McMullen's numerical conditions \cite{bib.McM.NFSP} on the face vectors of simple polytopes. (An ingenious construction of Billera and Lee \cite{bib.LB-CL.SMC} proves sufficiency.) One would like to understand general polytopes in a similar way. The first results in this direction are due to Bayer and Billera \cite{bib.MB-LB.gDS}. They consider the flag vector, not the face vector. For simple polytopes Poincar\'e duality represents what are known as the Dehn-Somerville equations on the face vector. Bayer and Billera describe the generalised Dehn-Somerville equations on the flag vector. They also show that the flag vectors of $n$-dimensional polytopes span a space whose dimension is the $(n+1)$st Fibonacci number $F_{n+1}$. The problem of characterizing the flag vectors of general polytopes has guided the development of local-global intersection homology. The usual middle perversity intersection homology theory produces $\lfloor n/2 \rfloor + 1$ independent independent linear functions of the flag vector. (This is the Bernstein-Khovanskii-MacPherson formula for the mpih Betti numbers \cite{bib.JD-FL.IHNP,bib.KF.IHTV,bib.RS.GHV}.) Clearly, more Betti numbers are needed, to record the whole of the flag vector. In addition, some analogue or extension to the usual ring structure on the homology of a nonsingular variety is required. The general polytope problem makes it clear that some extension of intersection homology, and of the ring structure, is required. Topology by itself has failed to indicate clearly either the need for such an extension, or its form. (There are intersection homology groups for non-middle perversities, and `change of perversity' groups, but these have the same problems as ordinary homology.) Finally, there are polytopes whose combinatorial structure is such that it cannot be realised with a polytope that has rational vertices \cite[p94]{bib.BG.CP}. Thus, for general polytopes a theory that does not rely on algebraic geometry is required. The \emph{root problem} of this paper is as follows. \emph{Suppose $Z$ is a possibly singular projective algebraic variety. In terms of the cycles on $Z$, what are the cycles on $CZ$ and $IZ$?} Here, $CZ$ is the projective cone on $Z$, while $IZ$ is the product of $Z$ with $\bfP_1$. The answer depends on what one understands a cycle to be, or in other words on some perhaps implicit choice of a homology theory. Suppose this question has been answered. One will then have a wide range of examples. These will determine the corresponding definition of a cycle, in the same way that a number of points will determine a plane. These examples will also determine a linear function $h\Delta$ of the flag vector $f\Delta$ of convex polytopes. This is because of the following. There are operators $I$ and $C$ on polytopes, analogous to the $I$ and $C$ operators on varieties. An \emph{$IC$ polytope} is any polytope that can be obtained by repeatedly applying $I$ and $C$ to the point polytope. Any polytope flag vector can be written as a linear combination of the flag vectors of $IC$ polytopes. The examples determine $h\Delta$ on the $IC$ polytopes. This paper is organised as follows. First (\S2) notation and definitions are established, and some basic results stated. Next (\S3) the root problem is discussed and a solution presented. This serves to motivate the definition of the (extended) $h$-vector $h\Delta$ of convex polytopes (\S4), and the topological definition of local-global intersection homology (\S5). To finish (\S6), there is a summary, and a discussion of related questions. This paper considers the topological and combinatorial aspects of local-global homology. There are others, to be presented elsewhere. The \emph{linear algebra} \cite{bib.JF.CPLA} allows $h\Delta$ to be interpreted as the outcome of a `vector weighted inclusion-exclusion' construction. The \emph{intersection theory} \cite{bib.JF.IHRS} provides a structure that reduces, in the simple or nonsingular case, to the ring structure on ordinary homology. To date, the theory of local-global intersection homology consists of a series of definitions appropriate for each of the four aspects, together with examples and special cases, and various linking results. Much remains to be done, to fill in the `convex hull' of the four aspects. This paper has been written to be accessible to those who are unfamiliar with perhaps one or both of intersection homology and the combinatorics of convex polytopes. The reader who is in a hurry can find a summary in the final section. Formulae (\ref{eqn.Itilde})--(\ref{eqn.Abar-A}) define the extended $h$-vector. The topological definition is in \S5. Text in parentheses (except for short comments) can be omitted on a first reading. The reader who is having difficulties should first understand the mpih part of the theory (i.e.~ignore terms involving any of $A$, $\Abar$ or $\{k\}$). \section{Preliminaries} This section introduces notation and conventions. It also states results to be used later. This material is organised into six topics, namely language and conventions, cones and cylinders, local homology, the strong Lefschetz theorem, polytope flag vectors, and the index set for $h\Delta$. First, language and conventions. Ordinary homology fails to have suitable properties, and so the word `homology' when used without qualification will refer either to middle perversity intersection homology (mpih), or some local-global variant thereof. The abbreviation mpih will always refer to the usual intersection homology (as in \cite{bib.MG-RDM.IH}), with of course middle perversity. Each local-global homology group has an order, usually denoted by $r$. The usual mpih groups are order zero local-global groups. The higher order groups will be called \emph{strictly local-global}. Unless otherwise stated, homology will always be with rational (or real) coefficients. The analogy between convex polytopes and algebraic varieties is very important, particularly in \S3 and \S4. Throughout $\Delta$ will be a convex polytope of dimension~$n$, and $Z$ a (projective) algebraic variety, also of dimension~$n$. When $\Delta$ has rational vertices a projective toric variety $\PDelta$ (of dimension~$n$) can be constructed (as in, say, \cite{bib.VD.GTV}). If $\Delta$ is the $n$-simplex $\sigma_n$ then $\PDelta$ is projective $n$-space $\bfP_n$. To strengthen the analogy, for toric varieties homology will be indexed by the complex dimension (half the normally used real dimension). The mpih Betti numbers of $\PDelta$ are zero in the odd (real) dimensions, and the same is expected to hold for the local-global extension. Thus, this indexing convention amounts to ignoring the homology groups that are expected in any case to be zero. The concept of a cone is one of the most important in this paper. In fact, the same word will be used for three closely related constructions, that apply respectively to topological spaces, projective algebraic varieties, and convex polytopes. Suppose $B$ is a topological space. The \emph{cylinder} $IB$ on $B$ is the product $[0,1]\times B$ of $B$ with the interval $I=[0,1]$, equipped with the product topology. If $p=(\mu,l)$ is a point of $IB$ and $\lambda\in I$ is a scalar then $\lambda p = (\lambda\mu,l)$ is also a point on $IB$. The \emph{cone} $CB$ is the cylinder $IB$, with $\setzero\times B$ identified (collapsed) to a single point, the \emph{apex} of the cone. In \S5, local-global cycles will be described as global cycles that can be collapsed in some specified way. The locus $\{1\}\times B$ is called the \emph{base} of $CB$. There is a $\lambda$-action on $CB$ also. Now suppose $Z\subset \bfP_N$ is a projective algebraic variety. The (projective) \emph{cone} $CZ\subset \bfP_{N+1}$ with \emph{base} $Z$ is constructed as follows. Each point $p\in\bfP_N$ represents a line $l_p$ through the origin in $\bfA^{N+1}$. A point $v$ lies in the \emph{affine cone} $\Ztilde \subset \bfA^{N+1}$ just in case it lies on some $l_z$, with $z\in Z$. A `hyperplane at infinity' can be added to $\bfA^{N+1}$, to produce $\bfP_{N+1}$. This hyperplane is a copy of $\bfP_N$. The cone $CZ$ is the closure of $\Ztilde$. The base of $CZ$ is the copy $\setinfty\times Z$ of $Z$ that lies on the $\bfP_N$ at infinity. The origin of $\bfA^{N+1}$ is the \emph{apex} of the cone. The interaction between the cone structure and relations among cycles is central to \S3. The complex numbers act by multiplication on the `finite part' $\Ztilde$ of $CZ$. Thus, if $\eta$ is a cycle on $CZ$, lying entirely on $\Ztilde$, it can as be `coned away' by the \emph{$\lambda$-cone} $C_\lambda\eta$, where $\lambda$ ranges over $[0,1]$. This is a chain whose boundary is $\eta$ (unless $\eta$ has dimension zero). Now suppose $\eta$ on $CZ$ avoids the apex. In this case each point of $\eta$ lies on a unique line through the apex, and so there is a boundary that `moves' $\eta$ to an equivalent cycle $\eta_\infty$ that lies entirely on the base $Z$ of $CZ$. Finally, suppose that $\eta$ is a cycle lying on the base $Z$ of $CZ$. Each point of $\eta$ determines a line in $CZ$, and so $\eta$ determines a cycle $C\eta$ on $CZ$. However, it may not be possible to find an $\eta'$ lying entirely on $\Ztilde$, that is equivalent to $\eta$. (The reason for this is subtle. If it were always possible, then it would be possible to `cone away' the cycle due to the hyperplane at infinity in $\bfP_{n+1}$ (the cone on $\bfP_n$). But this cycle is not homologous to zero. Although $\eta$ can locally be moved away from the base of $CZ$, in a manner that is unique up to `phase', it may not be possible to get all the phases to match up.) The \emph{cylinder} $IZ$ is the product of $Z$ with $\bfP_1$, which via the Segre embedding is to be thought of as a subvariety of $\bfP_{2N+1}$. The variety $Z$ is the \emph{base} of the cylinder. If $Z$ is nonsingular, then so is $IZ$, whereas $CZ$ will in general have a singularity at its apex. Analogous operators $I$ and $C$ can be defined for convex polytopes. If $\Delta$ is a convex polytope then the \emph{cone} (or \emph{pyramid}) with \emph{base} $\Delta$ is the convex hull of $\Delta$ with a point (the \emph{apex}) that does not lie in the affine span of $\Delta$. Similarly, the \emph{cylinder} (or \emph{prism}) $I\Delta$ with \emph{base} $\Delta$ is the Cartesian product $[0,1]\times\Delta$ of $\Delta$ with an interval $I=[0,1]$. These operators respect the dictionary between convex polytopes and toric algebraic varieties. The symbol `$\sqdot$' will be used to denote both the projective variety $\bfP_0$ (a single point), and the single point convex polytope. Thus, $ICC\sqdot$ can denote either $\bfP_1\times\bfP_2$ or a triangular prism. An \emph{$IC$ polytope} is one obtained by successively applying $I$ and $C$ to the point polytope, and similarly for an \emph{$IC$ variety}. For every word in $I$ and $C$, the latter is the toric variety associated to the former. Sometimes the two concepts will be identified. In both cases one can also define the \emph{join} of two objects. Suppose that $Z_1$ and $Z_2$ are subvarieties of some $\bfP_N$, and that their affine linear spans are disjoint. In that case, their \emph{join} consists of all points that lie on some line $l(z_1,z_2)$ that joins a point $z_1\in Z_1$ to another $z_2\in Z_2$. Similarly, if $\Delta_1$ and $\Delta_2$ have disjoint affine linear spans, then their \emph{join} is the convex hull of $\Delta_1\cup\Delta_2$. For both polytopes and varieties, a cone is the join of the base to the apex. One could also join an object not to a point but to a projective line (respectively, an interval). This is the same as forming the cone on the cone. It will have an \emph{apex line} (resp.~\emph{apex edge}) rather than an apex. Intersection homology differs from ordinary homology in that for it nontrivial local cycles can exist. If $s$ is a point on a complex algebraic variety $Z$ any sufficiently small ball centered at $s$ is homeomorphic to the (topological) cone $CL_s$ on something. That something, which does not depend on the sufficiently small ball, is the \emph{link} $L_s$ at $s$. Now suppose $\eta$ is a cycle on $CL_s$. The $\lambda$-action on a cone can then be applied to $\eta$, to produce a chain $C_\lambda\eta$, whose boundary is $\eta$. (Strictly speaking, this is true only if $\dim \eta > 0$.) Local cycles of dimension zero are trivial, and will not be counted by $h\Delta$. Ordinary homology allows this \emph{coning away} of local cycles. The perversity conditions of intersection homology however can be used to prohibit the use of $C_\lambda\eta$ to generate a boundary. The local (intersection) homology groups can be defined as follows. A \emph{local cycle} $\eta$ at $s$ consists of a cycle $\eta_U$ for any sufficiently small open set $U$ containing $s$, such that if $U'\subset U$, one has on $U$ that $\eta_{U'}$ and $\eta_U$ are homologous. Similarly, a \emph{local boundary} $\xi$ at $s$ consists of a chain $\xi_U$ on each sufficiently small open set $U$, whose boundary $\eta$ (the system $\eta_U=d\xi_U$) is a local cycle. This definition avoids use of the cone structure. In \S4, another definition will be given. If $Z$ is a complex algebraic variety then it can be decomposed into a disjoint union of \emph{strata} $S_i$, where each $S_i$ is either empty or has complex dimension~$i$, and along $S_i$ the local topology of $Z$ is locally constant. From this it follows that the local homology groups are also locally constant along $S_i$, and so form what is known as a \emph{local system}. This concept is used only in \S5. However, it is closely related to an example of local-global homology. (This paragraph and the next can be omitted on a first reading.) Suppose that $i<j$ and that the stratum $S_i$ is in the closure of $S_j$. More particularly, suppose that $\gamma:[0,1]\to Z$ is a path, with $\gamma(0)\in S_i$ and $\gamma(\lambda)\in S_j$ otherwise. Now let $\eta_1$ be a local cycle at $\gamma(1)$. By local constancy, it can be moved along $\gamma$ until it is very close to $\gamma(0)$. At this point the translate $\eta_\lambda$ of $\eta_1$ can be thought of as a local cycle at $\gamma(0)$ on $S_i$. In other words, each path from $S_j$ to $S_i$ (with $j>i$) transfers local cycles from $S_j$ to $S_i$. Note that the reverse process will not in general be possible. For example, if $S_0$ is an isolated singularity, then a local cycle $\eta$ at $S_0$ cannot be moved away from $S_0$. Now consider $H_0(S_i,L_i)$, where $L_i$ is the local system formed from the local homology groups along $S_i$. A cycle $\eta\in H_0(S_i,L_i)$ is a formal sum of local homology cycles (about points on $S_i$) subject to the equivalence due to motion along paths. As already described, these groups can be `glued together' (certain elements identified) for different values of $i$. Provided one uses all $S_i$ whose dimension is at least some value value $j$, the result is independent of the stratification. (This is left to the reader. Then main point is that new strata have real codimension at least two, and so existing paths can be altered to avoid new strata.) These groups are examples of local-global homology. The \emph{strong Lefschetz theorem} is one of the central results in the homology of nonsingular algebraic varieties. It was stated by Lefschetz in 1924, but his proof was not satisfactory. The first proof is due to Hodge (1933--6, see \cite[p117]{bib.WVDH.TAHI}). It also follows from Deligne's proof of the Weil conjectures \cite{bib.NK.DPRH}. Strong Lefschetz also holds for middle perversity intersection homology. Here, Deligne's proof is the only method known. For more background see \cite{bib.SK.DIHT}. The infinitesimal form of Minkowski's facet area theorem for polytopes~\cite[p.332]{bib.HM.ALKP,bib.BG.CP} is a special case of both strong Lefschetz and the Riemann-Hodge inequalities. This seems not to have been noticed before. Suppose $Z\subset\bfP_N$ is a projective algebraic variety. For convenience, complex dimension will be used to index its homology groups $H_iZ$. If $i+j=n$ (the dimension of $Z$), then by Poincar\'e duality $H_iZ$ and $H_jZ$ have the same dimension, for they are dual vector spaces. The embedding $Z\subset \bfP_N$ determines a \emph{hyperplane class} $\omega=\omegaZ$ in $H_{n-1}Z$ with the following properties. First, the cap product $\omega\frown\eta$ is defined for any homology class $\eta$ on $Z$. This operation lowers degree by one. Now assume $i<j$ and also $i+j=n$. The \emph{strong Lefschetz theorem} asserts that the map \[ \omega^{j-i}: H_jZ \to H_iZ \] is an isomorphism. This result provides a decomposition of $H_0Z$. Suppose that the above isomorphism takes $\eta$ to $\eta'=\omega^{i-j}\eta$. Say that $\eta$ is \emph{primitive} and $\eta'$ is \emph{coprimitive}, if $\omega\eta'$ is zero. It is a standard result, that $H_\bullet Z$ is the direct sum of $\omega^iP_jZ$, where $P_jZ\subseteq H_jZ$ are the primitive classes, and $i+j\leq n$. The Lefschetz isomorphism allows an `inverse' $\omega^{-1}$ to $\omega$ to be defined. Define $\omega^{-1}$ to be the result of first applying the inverse of the Lefschetz isomorphism, then $\omega$, and then the Lefschetz isomorphism. It has degree $-1$. (The Riemann-Hodge inequality is that on $P_jZ$ the quadratic form $\eta\frown\omega^{j-i}\frown\eta$ is negative definite.) The primitive and coprimitive cycles have a special r\^ole in the study of the root problem, namely the cycles on $CZ$ in terms of those on $Z$. They can occur only in certain dimensions, for which it is useful to have special adjectives. Say that a cycle $\eta$ on $Z$ is \emph{upper} (respectively \emph{strictly upper}) if its dimension is at least (resp.~more than) half that of $Z$. Similarly, at most (resp.~less than) define \emph{lower} (resp.~\emph{stricly lower}). Primitives and coprimitives occur in the upper and lower dimensions respectively. A cycle dimension that is not lower is strictly upper, and vice versa. The \emph{middle dimension} is both upper and lower. The hyperplane class $\omegaZ$ on $Z$ can be represented as a Weil divisor (formal sum of codimension one subvarieties) on $Z$, namely the hyperplane section. For use in \S3, note that the hyperplane class $\omegaCZ$ on a cone can be represented either as the cone $C\omegaZ$ on the class of the base, or as the base $Z$ of the cone (by intersecting $CZ \subset \bfP_{N+1}$ with the $\bfP_N$ at infinity). If $\Delta$ is a \emph{simple} convex polytope (this means that at each vertex there are $n=\dim\Delta$ edges) then $\PDelta$ behaves like a nonsingular algebraic variety, so far as its homology (with rational coefficients) is concerned. Its Betti numbers $h_i\Delta=h_i\PDelta$ are then a linear function of the \emph{face vector} $f=f\Delta=(f_0,f_1,\ldots,f_n)$, where $f_i$ is the number of $i$-dimensional faces on $\Delta$. In fact, if one writes $f(x,y)=\sum f_ix^iy^{n-i}$, and $h(x,y)$ similarly, then the equation $h(x,x+y)=f(x,y)$ expresses the relation between $f$ and $h$. If $\Delta$ is a general convex polytope, then flags should be counted. A \emph{flag} is a sequence \[ \delta = ( \delta_1 \subset \delta_2 \subset \ldots \subset \delta_r \subset \Delta ) \] of faces, each stricly contained in the next. Its \emph{dimension vector} (or \emph{dimension} for short) is the sequence \[ d = ( d_1 < d_2 < \ldots < d_r < n ) \] of the dimensions $d_i$ of its terms $\delta_i$. Altogether, there are $2^n$ possible flag dimensions. The component $f_d\Delta$ of the \emph{flag vector} $f=f\Delta$ of $\Delta$ counts how many flags there on $\Delta$, whose dimension is $d$. (If $\Delta$ is simple, the flag vector is a linear function of the face vector, and so contains no new information.) For simple polytopes the \emph{Dehn-Somerville} equations state that $h(x,y)$ is equal to $h(y,x)$, or that the $h$-vector is \emph{palindromic}. (It is analogous to Poincar\'e duality.) For general polytopes the \emph{generalised Dehn-Somerville (gDS) equations} \cite{bib.MB-LB.gDS} imply that $f\Delta$ has the Fibonacci number $F_{n+1}$ linearly independent components. A similarly elegant interpretation of these equations is lacking. For flag vectors the \emph{$IC$ equation} \cite{bib.JF.MVIC} \[ (I-C)C \> I \> = \> I \> (I-C)C \] holds, in the following sense. Apply both sides to a convex polytope $\Delta$, to obtain convex polytopes $ICI\Delta$ etc. The corresponding equation then holds among the flag vectors of these polytopes. The flag vectors of the $IC$ polytopes span all polytope flag vectors, and those than contain neither `$II$' nor `$I\sqdot$' form a basis. It follows that if linear operators $\Itilde$ and $\Ctilde$ are given that satisfy the $IC$ equation, together with an initial value $\htilde(\sqdot)$ for which $\Itildehtilde(\sqdot)=\Ctilde\htilde(\sqdot)$, then there is a unique linear function $\htilde$ on polytope flag vectors, for which the equations \[ \htilde(I\Delta) = \Itilde h(\Delta) \>;\quad \htilde(C\Delta) = \Ctilde h(\Delta) \>; \] are satisfied. This is used in \S4, to define the extended $h$-vector. Finally, note that the polytope flag vectors span a proper subspace of the span of all flag vectors. To provide a linear function on this subspace is not the same as to provide such on the larger space. Conversely, different linear functions on the larger space can agree on the subspace. Related to this is the idea that equivalent homology theories (on projective varieties) can be given different definitions, and that different triangulations can be found for a given space. The last topic is the \emph{index set}. The extended $h$-vector $h\Delta$, to be defined in \S4, will be a formal sum of terms, of a particular type. Although the terms to be used will arise in a natural way, it is convenient to gather in one place a description of them. In some sense, their structure is a result of a geometric requirement (that the Betti numbers be organised into sequences, whose length grows with the dimension $n$ of the polytope) and a combinatorial requirement (the Fibonacci numbers). First, expressions such as $(a,b,c)$ will stand for the homogeneous polynomial $ax^2 +bxy + cy^2$ in commuting variables $x$ and $y$. To save space, commas will where possible be omitted. Thus, $(10)=x$, $(01)=y$, $(11)=x+y$ and $(1)=1$. Similarly, the expression $[abc]$ (short for $[a,b,c]$) will stand for $aX^2 +bXY +cY^2$, where $X$ and $Y$ are a different pair of commuting variables. Clearly, $[1]=(1)=1$. Each of $x$, $y$, $X$ and $Y$ will have degree one. Roughly speaking, to each coprimitive cycle on the base of a cone (or in the link along a face) the symbol $\{k\}$ will correspond, where $k$ is the dimension of the cycle. This symbol also corresponds to a local $k$-cycle. Because in the present context such can occur only in the strictly lower dimensions, $\{k\}$ will be given degree $2k+1$. The symbol $\{0\}$ will not be used. This corresponds to treating the class of a point as a trivial local cycle. The difference $b-a$, which `counts' dimension~$1$ coprimitives, will be denoted by $b'$. Similarly, $c'=c-b$, and so on up to halfway. In addition, a `padding' symbol $A$ (or $\Abar$) is required. It has degree one. The extended $h$-vector $h\Delta$ will be a sum of terms of the form $x^iy^jW$, where $W$ is a word in $A$ and $\{k\}$. Each such term will have degree equal to the dimension of $\Delta$. The word $W$ is allowed to be empty. This corresponds to the mpih part of $h\Delta$. The last symbol in $W$ is not to be an $A$. (This can be achieved by supposing that there is a terminating symbol `$\sqdot$' at the end of each word, and setting $A\sqdot$ equal to zero.) An auxiliary vector $\htilde\Delta$ is used in \S4. It is a sum of $X^iY^jW$ terms, where $W$ is a word in $\Abar$ and $\{k\}$. Its terms are otherwise the same as those of $h\Delta$. The numerology of $x^iy^jW$ is interesting. Recall that $h\Delta$ has $F_{n+1}$ independent components. There are $F_{n+2}$ terms satisfying the above conditions, whose degree is $n$. Of these $F_{n+1}$ satisfy $i\leq j$, (and they correspond to a maximal set of independent components in $h\Delta$). Thus, $F_n$ ($=F_{n+2}-F_{n+1}$) terms satisfy $i>j$. Similarly, $F_n$ terms satisfy $j>i$. Thus, $F_{n-1}$ terms satisfy $i=j$. The number of words $W$ of degree at most $n$ is $F_n$, for $n\geq 1$. (These results are not used, and so are stated without proof.) Also, the equation \[ 1 + F_1 + F_2 + \ldots + F_n = F_{n+2} \] can be interpeted as follows. Define $f_{(i)}\Delta$ to be the sum of the flag vectors of the $i$-faces (or $i$-links if one prefers) of $\Delta$. It follows from the flag vector concept that $f\Delta$ and $f_{(\bullet)}= (1, f_{(1)}, f_{(2)}, \ldots, f_{(n-1)})$ are linear functions of each other. (The `$1$' corresponds to the `empty' face, or to $\Delta$ itself.) Each $f_{(i)}$ has $F_{i+1}$ independent components, and so $f_{(\bullet)}\Delta$ (and hence $f\Delta$) has $F_{n+2}$ components whose dependence does not follow from the gDS equations on the faces. This helps justify $F_{n+2}$ as the number of components in $h\Delta$, for one wishes $h\Delta$ to permit an elegant expression of the generalised Dehn-Somerville equations. \section{Cycles on cones and cylinders} This section describes the mpih and local-global cycles on a cone and a cylinder in terms of those on the base. First, the mpih cycles are constructed and described. The local-global cycles are then a variant of the mpih cycles. They make use of information, that mpih ignores. Suppose that $\eta$ is a cycle on $Z$. Later, this statement will acquire a richer meaning, but for now suppose that $\eta$ is a formal sum of embedded simplices, whose boundary is zero. The cycle $\eta$ on $Z$ determines three cycles on $IZ$, which can be denoted by $\setzero\times\eta$, $\setinfty\times\eta$, and $I\eta$. The first two, which are equivalent, arise from the two `poles' $0$ and $\infty$ on $\bfP_1$, each of which determines an embedding of $Z$ in $IZ$. The third, $I\eta$, is the product of $\eta$ with $\bfP_1$. Similarly, on $CZ$ one will have $\setinfty\times\eta$ and $C\eta$. (There is also the apex, which will not be needed.) Relations, as well as cycles, must be considered. On $\bfP_1$ let $I_\lambda$ denote the chain that is a path from $0$ to $\infty$. Similarly, let $I_\lambda\eta$ denote the chain on $IZ$, whose boundary is $\setinfty\times\eta -\setzero\times\eta$. The relations on a cone are more complicated. As noted in \S2, in general it is not possible to cone a cycle $\eta$ on the base to form a relation $C_\lambda\eta$, whose boundary is $\eta$. However, if a cycle $\eta$ on $CZ$ is equivalent to an $\eta'$ that does not meet the base $Z$, then $\eta$ can (via $\eta'$) be coned away to produce a relation $C_\lambda\eta$. Note that if such an $\eta'$ can be found, then the cap product of $\eta$ with the base (if defined) will be equivalent to zero, for $\eta'$ does not meet the base. When no restrictions are places on the cycles and relations, ordinary homology is the result. Based on the preceeding discussion, one might expect the ordinary homology of $IZ$ to be the tensor product of that of $Z$ with that of $\bfP_1$ (the K\"unneth formula), while for $CZ$ one might expect the `cone' on the ordinary homology of the base. By this is meant the base homology raised by one in degree, with the class of a point appended in degree zero. There are similar expected formulae for the ordinary homology Betti numbers. However, ordinary homology does not in general satisfy Poincar\'e duality and strong Lefschetz. Also, its Betti numbers are not a linear function of the flag vector \cite{bib.McC.HTV}. This is discussed further in \S5. Now consider middle perversity intersection homology, or more precisely, a theory that satisfies the Poincar\'e and Lefschetz theorems. These properties, particularly strong Lefschetz, will leave one with little choice as to what the cycles on $CZ$ and $IZ$ are, and hence lead to the usual middle perversity conditions on the cycles. The task is to control the cycles and relations, so that the $I$ and $C$ operators preserve the Poincar\'e and Lefschetz properties. For $I$ the usual K\"unneth formula will do this, a result that is left to the reader. For $C$, more care is needed. Suppose $\setinfty\times\eta$ is a cycle on $CZ$. Now use the $C\omegaZ$ form of the hyperplane class. Clearly, one will have $C\omegaZ\frown(\setinfty\times\eta) = \setinfty\times(\omegaZ\frown\eta)$ as the hyperplane action. Similarly, if $C\eta$ is a cycle on $CZ$, use the `base' form $Z$ of the hyperplane class to obtain \[ \omegaCZ \frown C\eta \sim Z \frown C\eta = \setinfty\times\eta \] as the hyperplane action. Now suppose that $\eta$ is a primitive $j$-cycle on $C$, by virtue of $\omegaZ^i\eta \sim 0$. The relation \[ \omegaCZ^{i+1} C\eta \sim 0 \] follows from the above. Thus, \emph{$C\eta$ is a primitive on $CS$, whenever $\eta$ is a primitive on $Z$}. Moreover, $\setinfty\times\eta$ is equal to $\omegaCZ\frown C\eta$, and so cannot be primitive. All this assumes that $C\eta$ is allowed as a cycle, when $\eta$ on $Z$ primitive. These properties (K\"unneth and the coning of primitives) suffice to determine the homology of $IZ$ and $CZ$ respectively, in terms of that of $Z$. The task now is to express these groups in terms of topological cycles and relations. First consider $CZ$. By assumption, if $\eta$ is primitive on $Z$, then $C\eta$ is permitted on $CZ$ (and is there primitive). Primitive is not a topological concept; it depends on the projective embedding $Z\subset\bfP_N$. However, the primitive cycles have upper dimension, and that is a topological notion. Thus, permit $\xi=C\eta$ as a cycle on $CZ$ whenever $\eta$ on $Z$ is upper, or in other words when $\xi$ on $CZ$ is strictly upper. Now suppose that $\eta$ on $Z$ is coprimitive. It follows at once that \[ \setinfty\times\eta \frown \setinfty\times Z \sim 0 \>, \] and so there is nothing in the homology of $CZ$, that prevents $\setinfty\times\eta$ being moved away from the base $\setinfty\times Z$. Suppose that this can be done, to produce $\eta' \sim \setinfty\times\eta$. As noted in \S2, the $\lambda$-coning operation can be applied to $\eta$, via $\eta'$, to produce $C_\lambda\eta$. If $\eta$ on $Z$ is nonzero, then on $CZ$ it is also by assumption nonzero. Thus, the `coning away to a point' $C_\lambda\eta$ cannot be permitted (except perhaps if $\dim \eta =0$). As before, even though coprimitive is not a topological notion, the lower range of dimensions is. This leads to $\xi=C_\lambda\eta$ on $CZ$ being prohibited as a chain, whenever $\xi$ is lower. These two examples (primitive and coprimitive) establish the middle dimension as the cut-off point for cycles and chains being permitted or prohibited respectively. This applies to how they meet the $0$-strata. To obtain the remainder of the middle perversity conditions, study the cycles on $IZ$ due to $\eta$ on $Z$, where now $\eta$ is a cycle that satisfies the conditions that are already known. In this way, the rest can be built up, to produce the already known middle perversity intersection homology conditions on cycles and chains. This is left to the reader. The assumption, that if $\eta$ on $Z$ is coprimitive, then $\setinfty\times\eta$ is equivalent to an $\eta'$ that does not meet the base $Z$ of the cone, is quite strong. Previously, it was assumed that this might happen from time to time, and so the associated coning aways were prohibited. That there are such prohibited coning aways is a topological property, which is local to the apex of $CZ$. Local-global homology will count such `coning aways', but calls them local-global cycles. It will as a heuristic principle be assumed that any coprimitive can be moved to avoid the base, and so be coned away. Such assumptions support the calculation in \S4 of the expected values $h\Delta$ of the local-global Betti numbers. It is time to take stock. Recall that the purpose of this section is to describe the cycles on $IZ$ and $CZ$ in terms of those on $Z$. When no restrictions are imposed, ordinary homology is the result. Requiring Poincar\'e duality and strong Lefschetz produces middle perversity intersection homology. For mpih the cycles on $CZ$ are all of the form $C\eta$ or $\setinfty\times\eta$, for $\eta$ a cycle on $Z$. For the local-global extension, one also has the `coning away' or \emph{local-global cycle} $C_\lambda\eta$, for $\eta$ any coprimitive cycle on the base $Z$. (The apex of the cone $Z$ is also called the \emph{apex} of the local-global cycle $C_\lambda\eta$.) Now assume that on $Z$ itself there is such a local-global cycle. Further local-global cycles may arise on $CZ$ and $IZ$, as a result of this local-global cycle on $Z$. As in the mpih case, on $IZ$ it will be assumed that the K\"unneth formula continues to hold. In other words, any cycle on $IZ$ can be expressed using $\setzero\times\eta$ (or $\setinfty\times\eta$) and $I\eta$, where $\eta$ ranges over the cycles on $Z$. If $\eta$ is a local-global cycle, say $C_\lambda\xi$, then $I\eta$ is a new kind of local-global cycle. (Its apex locus is $I$ applied to that of $\eta$.) If $\eta$ is thought of as a local cycle, then $I\eta$ is a family of local cycles. In addition, note that by K\"unneth $\setzero\times\eta$ and $\setinfty\times\eta$ are equivalent cycles on $IZ$, and so local-global cycles must on occasion be allowed to move along the singular locus. (In fact, the rule will be that if they can move, then they are allowed to move.) At this point it is possible to give some examples. In dimensions $0$, $1$ and $2$ one has \[ h(\sqdot) = (1) \>\>; \quad h(C\sqdot) = h(I\sqdot) = (11) \>\>; \quad h(CC\sqdot) = (111) \>\>; \quad h(IC\sqdot) = (121) \>\>; \] of course. In dimension $3$ one has \[ h(CCC\sqdot) = (1111) \>\>;\quad h(ICC\sqdot) = (1221) \>\>;\quad h(IIC\sqdot) = (1331) \>\>; \] as the nonsingular (or simple) examples, while \[ h(CIC\sqdot) = (1221) + (1)\{1\} \>\>; \] is the only singular example. Here, $\{1\}$ counts the local-global cycles on $CZ$ due to the only nontrivial coprimitive on $Z$, where $Z$ is $\bfP_1\times\bfP_1$ (or a square). In dimension $4$ something new happens. The simple cases \[\begin{array}{ll} h(CCCC\sqdot) = (11111) \>\>;\quad & h(ICCC\sqdot) = (12221) \>\>;\\ h(IICC\sqdot) = (13431) \>\>;\quad & h(IIIC\sqdot) = (14641) \>\>;\\ \end{array}\] are just as before. The cones on the simple dimension $3$ examples come next. They are \[ h(CICC\sqdot) = (12221) + (1)\{1\}A \>\>;\quad h(CIIC\sqdot) = (13331) + (2)\{1\}A \>\>; \] where as before $\{1\}A$ counts the nontrivial coprimitives on the base. The remaining examples are $I$ and $C$ applied to $CIC\sqdot$, the only non-simple dimension $3$ example. The polytopes (or varieties) $ICIC\sqdot$ and $CCIC\sqdot$ are more similar than they might at first sight appear. The polytope $CIC\sqdot$ has an apex, and so $ICIC\sqdot$ has an \emph{apex edge}. Now consider $CCIC\sqdot$. From one point of view, this has two apexes, namely the apex of its base $CIC\sqdot$, and the apex of $CCIC\sqdot$ itself. However, $CCIC\sqdot$ is also the join of $IC\sqdot$ with an interval, and so there is no geometric way of distinguishing its two apexes. In other words, like $ICIC\sqdot$, it too has an apex edge. Combinatorially, the two polytopes are the same along their respective apex edges. (This fact is at the heart of the $IC$ equation.) The mpih parts of $h(ICIC\sqdot)$ and $h(CCIC\sqdot)$ are $(13431)$ and $(12221)$ respectively. The remaining contribution comes from the strictly local-global cycles along the apex edge. Clearly, if $\eta$ is the cycle that contributes $(1)\{1\}$ to $CIC\sqdot$, then $I\eta$ and $\setzero\times\eta$ will contribute $(01)\{1\}$ and $(10)\{1\}$ respectively to $ICIC\sqdot$. However, on $CIC\sqdot$ there is a single coprimitive cycle (it has dimension one), and so on $CCIC\sqdot$ there will be a local-global of type $\{1\}A$. Now note that if $h\Delta$ is to be a linear function function of the flag vector, the non-mpih parts of the $h$-vectors of the two polytopes should be the same. To achieve this, the values \[\begin{array}{l} h(ICIC\sqdot) = (13431) + (11)\{1\} + \{1\}A \>\>\;\\ h(CCIC\sqdot) = (12221) + (11)\{1\} + \{1\}A \>\>\; \end{array}\] will be postulated. This is to take any strictly local-global contribution that can occur for either $ICIC\sqdot$ or $CICC\sqdot$, and to insist that it can occur in the other. This forces $\setzero\times\eta$ on $ICIC\sqdot$ to contribute not only $(10)\{1\}$ as already noted, but also $(1)\{1\}A$. The cycle $\setinfty\times\eta$ on $CZ$ will make a similar contribution to $hCCIC\sqdot$. Also, some sort of coning $C\eta$ of the cycle $\eta$ must be allowed, to obtain on $CCIC\sqdot$ a contribution of $(10)\{1\}$. This discussion is an example of how topology and combinatorics work together to determine the structure of the theory of local-global homology. Now suppose that $\eta$ is a cycle (possibly of local-global type) on $Z$. Already, the associated cycles on $IZ$ have been described. The task now is to determine and describe the associated cycles on $CZ$. There are three basic possibilities. First, one can form $\setinfty\times\eta$, which is a cycle lying on the base $Z$ of $CZ$. Second, one can cone $\eta$ to form $C\eta$. Sometimes, as in the lower dimensions of mpih, this cycle is not needed. Finally, if $\setinfty\times\eta$ can be moved to an equivalent cycle $\eta'$, that does not meet the base $Z$ of $CZ$, one can form the `coning away' $C_\lambda\eta$. These possibilities will be considered, one at a time. First, the cycle $\setinfty\times\eta$ will always be admitted. There are no conditions imposed on $\eta$. One reason for this is that about their respective bases, the cylinder and the cone are combinatorially the same, and so that which is permitted for the one should be permitted for the other. But for the cylinder, the K\"unneth principle causes $\setinfty\times\eta$ to be admitted. As in the cylinder, this cycle may contribute to several distinct local-global homology groups. Next consider $C\eta$. The example of $CCIC\sqdot$ shows that this case requires more thought. As already noted, the cycle $\eta$ on $CIC\sqdot$ contributes $(11)\{1\}$ plus $(1)\{1\}A$ to $CCIC\sqdot$. Clearly, $\setinfty\times\eta$ contributes $(10)\{1\}$ and $(1)\{1\}A$. The remainder, $(01)\{1\}$ will have to come from $C\eta$. Thus, at least in this case, $C\eta$ must be allowed. But this seems to contradict the mpih case, where no use of $C\eta$ was made in the lower dimensions, and where any $C_\lambda\eta$ `coning away' was explicitly prohibited. However, it is possible to harmonise the two cases. This involves looking again at the mpih situation. Suppose that $\eta$ is an mpih cycle on $Z$. First consider $C\eta$ on $CZ$, as a purely formal object. Its main property is that $\omegaCZ\frown C\eta$ is equivalent to $\setinfty\times\eta$. In addition, if $\eta$ and $\xi$ on $Z$ have complementary dimensions (and so intersect to give a number), then $C\eta\frown\setinfty\times\xi = \eta\frown\xi$. These properties are not enough, in general, to determine $C\eta$ as a homology class on $CZ$. When $\xi$ is upper, $\omega$ is injective, and so $\omega\frown\xi$ determines $\xi$. This does not help in the other dimensions. Here, the Lefschetz isomorphism will be used. Write $\eta$ as $\omegaZ^r\eta'$, where $\eta$ and $\eta'$ have complementary dimensions. This representation is always possible and unique, provided $\eta$ is lower. Provided $r>0$, one can take $\setinfty\times (\omegaZ^{r-1}\eta')$ as the cycle on $CZ$ that represents the formal object $C\eta$. Between them, these two cover all the cases. Thus, there is no formal obstacle to thinking of $C\eta$ as a homology cycle on $CZ$. The following construction, at least in certain cases, leads to a geometric form for $C\eta$, in the lower dimensions. As motivation, think of $C\bfP_n$ (the cone on $\bfP_n$, not complex projective $n$-space). Here, each cycle $\eta$ on $\bfP_n$ can be coned to give a cycle $C\eta$ on $C\bfP_n$, \emph{provided the apex is not part of the stratification}. Adding the apex as a stratum will not however change the homology. It will thus be possible to move $\eta$ a little bit, so that it avoids the apex (at least in the lower dimensions). This change can be confined to a small ball centered at the apex. Think now of $C\eta$ on $CZ$ as follows. Cone $\eta$ to form $C\eta$, and form a small ball $B$ about the apex. Outside of $B$, there is no fault with $C\eta$. The task now is to change $C\eta$ within $B$, so as to avoid the apex. Consider now the intersection $R=S\cap C\eta$ of $C\eta$ with the boundary $S$ of $B$. For certain values of $R\subset S$, it will be possible to `fill-in' $R$ within $B$, to obtain part of an intersection homology cycle, and for others it will not. (The difference between any two solutions is, clearly, a local intersection homology cycle.) For certain $\eta$ it will be possible to solve the associated $R\subset S$ problem. For heuristic purposes, it will be assumed that this is always possible. Intersection properties can be used to resolve the indeterminacy due to local cycles at the apex. In this way it is possible (modulo some assumptions) to treat $C\eta$ as a cycle on $CZ$, when $\eta$ is any mpih cycle on $Z$. The key is to if necessary modify the geometric form of $C\eta$ within a small ball centered at the apex. Now consider $C\eta$ on $CZ$, where $\eta$ is a local-global cycle. The example of $CCIC\sqdot$ forces one to allow this cycle, in some form of the other. The previous paragraph shows how this might be done. The geometric form of $C\eta$ must be modified in a small ball centered about the apex of $CZ$, or perhaps more exactly, replaced by something else. In fact, this $C\eta$ problem need only be solved for mpih cycles. Each local-global cycle can be thought of as a $\lambda$-family of cycles. One can then solve this problem in the simpler case of $\lambda=1$, and then define $C\eta$ to be the result of applying scalar multiplication to the this solution. The third type of cycle on $CZ$ are those obtained by moving a cycle $\setinfty\times\eta$ away from the base $Z$, and then `coning it away'. This was the point of departure, for the local-global theory. (The mpih theory prohibits the use of such objects, to generate homology relations. The local-global theory treats such objects as a cycle, but of a new type.) This construction can be iterated. Here is an example. First, let $\eta$ be the local-global cycle on $CIC\sqdot$. Now let $Z$ be $IICIC\sqdot$, and on $II\sqdot$ let $\xi$ be a coprimitive cycle. On $Z$ there is a local-global cycle that can be written as $\xi\otimes\eta$. Now consider $CZ$. Provided $\setinfty\times(\xi\otimes\eta)$ can be moved within its class, so as to avoid the base $Z$, it can be `coned away'. This is an example of a second-order local-global intersection homology cycle. Earlier in this section, $h\Delta$ was presented for all the $IC$ polytopes of dimension at most $4$. To conclude, much the same will be done for dimension $5$. However, to save space this will be done only for certain polytopes, whose flag vectors provide a basis for all polytope flag vectors. They are the ones in which neither $II$ nor $I\sqdot$ occur. There are $F_6=8$ such polytopes. For these basis polytopes the $h$-vectors are as follows. \[ \begin{array}{rl} h(CCCCC\sqdot) &= (111111) \\ h(CCCIC\sqdot) &= (122221) +(111)\{1\} +(11)A\{1\} + (1)AA\{1\} \\ h(CCICC\sqdot) &= (122221) +\phantom{(111)\{1\}} +(11)A\{1\} + (1)AA\{1\} \\ h(CICCC\sqdot) &= (122221) +\phantom{(111)\{1\}} +\phantom{(11)A\{1\}} + (1)AA\{1\} \\ h(CICIC\sqdot) &= (134431) +(111)\{1\} +(11)A\{1\} + (2)AA\{1\} + (1)\{2\}\\ h(ICCCC\sqdot) &= (122221) \\ h(ICCIC\sqdot) &= (134431) +(121)\{1\} +(12)A\{1\} + (1)AA\{1\} \\ h(ICICC\sqdot) &= (134431) +\phantom{(111)\{1\}} +(11)A\{1\} + (1)AA\{1\} \end{array} \] Here is a summary of the discussion of the cycles on $IZ$ and $CZ$. On $IZ$ the cycles are as given by the K\"unneth principle. Each cycle on $IZ$ is a sum of products of a cycle on $I$ (or $\bfP_1$) with a cycle on $Z$. For $CZ$ the situation is more complicated. If $\eta$ is a cycle on $Z$, then one always has $\setinfty\times\eta$ on $CZ$. Provided the details are satisfied, as to what happens near to the apex, one will also have $C\eta$. (When $\eta$ is upper, these details are vacuous.) Finally, if $\setinfty\times\eta$ can be moved so as to avoid the base $Z$ of $CZ$, one also has its `coning away', the $\lambda$-cone $C_\lambda\eta$. (A necessary, and perhaps sufficient, condition for doing this is that $\omegaCZ\frown\setinfty\times\eta$, which is equal to $\setinfty\times(\omegaZ\frown\eta)$ be homologous to zero.) This description of cycles motivates both the definition of the extended $h$-vector (\S4), and the topological definition of local-global homology (\S5). In both cases, there are two aspects to the discussion. The first is the cycles themselves, the focus of this section. The second is how they are to be counted. Consider once again $ICIC\sqdot$ and $CCIC\sqdot$. There, it was seen that the same cycle may contribute to several different parts of $h\Delta$. This is something that is quite new. The basic idea is this. A local-global homology group is spanned by all cycles that satisfy certain conditions. If these conditions are relaxed, another local-global group is obtained. (The same happens in intersection homology, when the perversity is relaxed.) This is why the same local-global cycle may contribute to several components of $h\Delta$. The conditions are related to where the cycle may be found. For example ($\dim=4$), one can count all local $1$-cycles (subject to equivalence), or one can allow only those that have some degree of freedom, as to their location. The former are counted by $\{1\}A$, the later by $x\{1\}$. For example, $CICC\sqdot$ has the former but not the later, while $ICIC\sqdot$ has both. These conditions control both the cycles and the relations. Sometimes (the $4$-cross polytope for example), relaxing the conditions may allow new relations to appear amongst existing cycles. The varieties produced by $I$ and $C$ are special, in that this never happens. This makes the computation of their $h$-vectors much easier. This fact is exploited by the next section. \section{The extended $h$-vector} This section defines, for every convex polytope $\Delta$, an extended $h$-vector $h\Delta$. It does this by using rules $\Itilde$ and $\Ctilde$ that satisfy the $IC$ equation. These rules are motivated by the previous section. In the next section, local-global homology groups will be defined for algebraic varieties. Provided various assumptions are satisfied, for $\Delta$ an $IC$ polytope the extended $h$-vector $h\Delta$ will give the local-global Betti numbers of the associated toric variety $\PDelta$. (The same may not be true for other rational polytopes, and even it true will most likely be much harder to prove. Such would be both a formula for the local-global Betti numbers, and a system of linear inequalities on the flag vectors of rational polytopes.) There are two stages to the definition of $h\Delta$. The previous section described the local-global cycles on $IZ$ and $CZ$ in terms of those on $Z$. It also noted that the same cycle might contribute in several ways to the local-global homology. The first stage is to define operators $\Itilde$ and $\Ctilde$ that count the local-global cycles, but without regard to the multiple contributions. The second stage is to make a change of variable, to accomodate the multiple contributions. This corresponds to knowing the implications among the various conditions satisfied by local-global cycles. The second stage is vacuous for the mpih paart of the theory. (For the $IC$ polytopes, each local-global cycle is determined by the corresponding global cycle, together with a statement, as to the $\lambda$-coning conditions it satisfies. The first stage counts each local-global cycle only once, at the most stringent conditions it satisfies. This process is meaningful only for $IC$ and similar polytopes. In general, relaxation of conditions will admit new relations, as well as new cycles.) The first stage is to introduce an auxiliary vector $\htilde\Delta$, defined via rules $\Itilde$ and $\Ctilde$. The quantity $\htilde\Delta$ will be a sum of terms such as $[abcd]W$. As noted in \S2, $[abcd]$ stands for the homogeneous polynomial $aX^3+bX^2Y+cXY^2+dY^3$, while $W$ will be a word in $\{k\}$ and $\Abar$. The rules $\Itilde$ and $\Ctilde$ will be defined by their action on such terms. The rule for $\Itilde$ is to multiply by $[11]=X+Y$. It corresponds to the K\"unneth formula for cycles. The equation \begin{equation} \label{eqn.Itilde} \Itilde [abcd] W = [11] [abcd] W = [a,a+b,b+c,c+d,d] W \end{equation} is an example of this rule. Note that this rule preserves the property of being palindromic. If swapping $X$ and $Y$ leaves $\htilde\Delta$ unchanged, then the same is true of $\Itildehtilde\Delta$. The rule for $\Ctilde$ is more complicated. It has three parts. The first part $\Ctilde_1$ leaves $W$ unchanged. It corresponds to the idea, that if $\eta$ on $Z$ is primitive, then so is $C\eta$ on $CZ$. Here are some examples of the rule \[\begin{array}{ll} \Ctilde_1 [a]W = [aa]W \>;\quad & \Ctilde_1 [ab]W = [aab]W \>;\\ \Ctilde_1 [abc]W = [abbc]W \>;\quad & \Ctilde_1 [abcd]W = [abbcd]W \>;\\ \Ctilde_1 [abcde]W = [abccde]W \>;\quad & \Ctilde_1 [abcdef]W = [abccdef]W \>; \end{array}\] for this part. It is to repeat the exactly middle, or failing that the just before middle, term in the $[\ldots]$ sequence. The reader is asked to verify that the equation \[ (\Itilde - \Ctilde_1) \Ctilde_1 = [010] \] holds, in the sense the applying the left hand side to, say, $[abcde]$ will produce $[0abcde0]$. As $[11]$ and $[010]$ commute, $\Itilde$ and $\Ctilde_1$ satisfy the $IC$ equation. Using just this part of the rule for $\Ctilde$ (together with the rule for $\Itilde$, and $h(\sqdot)=[1]$ as an initial condition) will generate the mpih part of $h\Delta$. The second part $\Ctilde_2$ corresponds to the $\lambda$-coning of a cycle $\eta$ on the base. Such a cycle $\eta$ must be coprimitive. The numbers $b'=b-a$, $c'=c-b$ and so on count the coprimitives. New words will be obtained by prepending to $W$ a record, in the form $\Abar^j\{k\}$, of the coprimitive that has been $\lambda$-coned. Here are some examples of the rule \[\begin{array}{ll} \Ctilde_2 [a]W = 0 \>;\quad & \Ctilde_2 [ab]W = 0 \>;\\ \Ctilde_2 [abc]W = [b']\{1\}W \>;\quad & \Ctilde_2 [abcd]W = [b']\Abar\{1\}W \>;\\ \Ctilde_2 [abcde]W = [b']\Abar^2\{1\}W +[c']\{2\}W \>;\quad \hidewidth \\ \Ctilde_2 [abcdef]W = [b']\Abar^3\{1\}W +[c']\Abar\{2\}W \>;\quad \hidewidth \end{array}\] for this part. In $\Abar^j\{k\}$ the $k$ records the degree of the coprimitive, and the $j$ `takes up the slack', to ensure homogeneity. As noted the previous section, the trivial coprimitives (which correspond to $a'=a$) are not counted. The sum $\Ctilde_1 + \Ctilde_2$ of these two parts is not enough (or more exactly, is too much). One reason is that when used with $\Itilde$, the result does not satisfy the $IC$ equation. The third part is a correction, that balances the books. It is to subtract $[a]\Abar^j$, for the appropriate power of $j$. The geometric meaning of this correction will be presented later. Here now is the rule for $\Ctilde$. The examples \begin{equation} \begin{array}{rl} \Ctilde[a] &= [aa] - [a]\Abar \\ \Ctilde[ab] &= [aab] - [a]\Abar^2 \\ \Ctilde[abc] &= [abbc] - [a]\Abar^3 +[b']\{1\} \\ \Ctilde[abcd] &= [abbcd] - [a]\Abar^4 +[b']\Abar\{1\} \\ \Ctilde[abcde] &= [abccde]-[a]\Abar^5+[b']\Abar^2\{1\}+[c']\{2\}\\ \Ctilde[abcdef] &= [abccdef] -[a]\Abar^6 + [b']\Abar^3\{1\} +[c']\Abar\{2\} \\ \end{array} \end{equation} suffice to show the general rule. In the above, it is to be understood that both sides have been multiplied on the right by a word $W$ in the symbols $\Abar$ and $\{k\}$. This rule, and the rule for $\Itilde$, together satisfy the $IC$ equation. Here is an example. The calculation \[ \Itilde\Ctilde [abcde] = [11][abccde] - [aa]\Abar^5 +[b'b']\Abar^2\{1\} +[c'c']\{2\} \] follows immediately from the above. The calculation for $\Ctilde\Ctilde[abcde]$ is more involved. One has \[ \Ctilde[abccde] = [abcccde] - [a]\Abar^6 +[b']\Abar^3\{1\} +[c']\Abar\{2\} \] and also \begin{eqnarray*} -\Ctilde[a]\Abar^5 &=& -[aa]\Abar^5 + [a]\Abar \Abar^5 \\ \Ctilde[b']\Abar^2\{1\} &=& [b'b']\Abar^2\{1\} - [b']\Abar \Abar^2\{1\} \\ \Ctilde[c']\Abar^2\{1\} &=& [c'c']\{1\} - [c']\Abar\{1\} \end{eqnarray*} as the various contributions. Now compute the difference. All but two of the terms cancel. One has \[ (\Itilde\Ctilde-\Ctilde\Ctilde)[abcde] = [11][abccde] - [abcccde] \] which is, as for $\Ctilde_2$, is equal to $[010][abcde]$. As this example is completely typical, the result follows. To complete this definition of $\htilde\Delta$, one must supply an initial value $\htilde(\sqdot)$, such that $\Ctilde\htilde(\sqdot)$ and $\Itildehtilde(\sqdot)$ are equal. Here a problem arises. The value $\htilde(\sqdot)=[1]$ does not quite work. The quantities $\Ctilde[1]=[11]-[1]\Abar$ and $\Itilde[1]=[11]$ are not equal. Here is the solution. Recall that $h\Delta$ is to be a sum of terms of the form $h_W\Delta\cdot W$, for some family of symbols~$W$. Thus, one should really write $\htilde(\sqdot)=[1]W_0$, for some symbol $W_0$. Consistency is then equivalent to the equation $\Abar W_0 =0$. However, it is more convenient to use `$\sqdot$' as the initial value. The initial values \begin{equation} \htilde (\sqdot) = [1] \sqdot \> ; \qquad \Abar \sqdot = 0 \end{equation} conclude the definition. The multiple use of the symbol `$\sqdot$' in practice causes no confusion. (The geometric meaning of the correction $-[a]\Abar^i$ is as follows. Suppose that $\Delta$ is an $IC$ polytope, and that say $[abc]W$ appears in $\htilde\Delta$. This term $[abc]W$ is due to the $I$ and $C$ operations that constructed $\Delta$. In particular, going back in this case two steps, one obtains the polytope $\Delta_1$, from which $\Delta$ is derived by applying $I$ and $C$, and in $\htilde\Delta_1$ the term $[a]W$ appears. Going back another step, one has $\Delta_1=C\Delta_0$, and on $\Delta_0$ there are $a$ independent coprimitive cycles $\eta$, that contribute $[a]W$ to $\htilde\Delta_1$. Now consider $C\Delta$. These coprimitive cycles $\eta$ (multiplied by $\setinfty$ as appropriate) continue to exist on $\Delta$, and the $\Ctilde_2$ part of the rule for $\Ctilde$ will thus cause them to contribute $[a]\Abar^iW$. However, their contribution to $\htilde C\Delta$ has already been counted, as the `$a$' part of $\Ctilde_1[abc]W=[abbc]W$. This is because $\htilde\Delta$ counts cycles on $IC$ polytopes, at the most stringent condition they satisfy. Hence the correction $-[a]\Abar^i$.) Finally, note that if $\htilde\Delta$ is palindromic then so is $\Ctilde\htilde\Delta$. As the same is true for $\Itilde$, it follows that $\htilde\Delta$ is indeed palindromic, first for all $IC$ polytopes and then (by linearity) for all polytopes. This completes the definition of the auxiliary vector $\htilde\Delta$. The second stage in the definition of the extended $h$-vector is to apply a linear change of variable to the above quantity $\htilde\Delta$. The meaning of this transformation is as follows. Recall that for $\Delta$ an $IC$ polytope, $\htilde\Delta$ counts each local-global cycle once, according to the most stringent conditions it satisfies. The transformation is the process of relaxation of conditions, as it applies to cycles on $IC$ polytopes. Previously, little attention has been given to the conditions satisfied by the local-global cycles. These cycles have been generated from coprimitives via the $\lambda$-coning operation, but the topologicial properties of the cycles so obtained have not been explicitly formulated. Instead, the $X$, $Y$, $\{k\}$ and $\Abar$ symbols have been used in a somewhat formal way, to record the relevant facts relating to the construction of the associated cycles. Recall that in the discussion of mpih, one first obtained the cycles, and then the conditions that they satisfied. Something similar will be done with local-global homology. A local-global cycle on an $IC$ variety $\Delta$ consists of a cycle, as counted by $\htilde\Delta$, together with a condition that it satisfies. Such conditions, which control how the apex locus meets the strata, will be formulated in the next section. However, by assumption each word in $x$, $y$, $A$ and $\{k\}$ determines a condition, or in other words a type of local-global cycle. The present task is to describe how to pass from $\htilde\Delta$ to $h\Delta$. If done properly, it will implicitly determine the set of conditions, that will in the next section be explicitly stated. To each term such as $\Abar\{1\}$ or $X\{1\}$ in $\htilde\Delta$, there is to correspond a condition that applies to local-global cycles. The coefficient in $\htilde\Delta$ of such a term counts how many cycles there are on $\Delta$ that satisfy that condition, but not any condition that is more stringent. (This makes sense only for $IC$ polytopes. Other polytopes may produce negative coefficients.) The transformation from $\htilde$ to $h$ is thus determined once one knows the partial order on conditions, that is associated to stringency or implication. In other words, the partial order is that of implication among the associated conditions. Here is an example. In $\htilde(CCIC\sqdot)$ the term $X\{1\}$ occurs, with coefficient~$1$. (As noted in \S3, it also so occurs in $\htilde(ICIC\sqdot)$.) This term corresponds to a particular type of local-global cycle, namely one that can be found anywhere along the apex edge. Now consider $\htilde(CICC\sqdot)$. Here the term $\Abar\{1\}$ occurs, with coefficient~$1$. This too corresponds to a local-global cycle, but of a different type. It corresponds to a local-global cycle that can be found only about the apex of $CICC\sqdot$. (In the previous case, there was an apex edge, anywhere along which the local-global cycle could be found. In this case there is no apex edge.) Thus the condition $X\{1\}$ is more stringent than $\Abar\{1\}$. Each term such as $\Abar\{1\}$ or $X\{1\}$ in $\htilde\Delta$ will contribute $A\{1\}$ and $x\{1\}$ respectively to $h\Delta$. These `diagonal' terms arise because `the most stringent condition satisfied by a cycle' is also `a condition satisfied by a cycle'. In addition, as just noted, if a cycle satisfies $X\{1\}$ then it will also satisfy $\Abar\{1\}$ (but not as the most stringent condition), and so $X\{1\}$ will contribute $A\{1\}$, in addition to $x\{1\}$. The transformation of \[ \htilde (CCIC\sqdot) = [12221] + [11]\{1\} \] is therefore \[ h (CCIC\sqdot) = (12221) + (1)A\{1\} + (11)\{1\} \] which is as postulated in the previous section. Here is another example. In involves second order local-global cycles. The symbol $\{1\}\{1\}$ represents a local $1$-dimensional family of local $1$-cycles. In dimension~$7$ each of the terms $\Abar\{1\}$, $X\{1\}\{1\}$ and $\{1\}\Abar\{1\}$ is to represent a different condition on a local $1$-family of local $1$-cycles. To begin to understand these conditions, for each of these terms an $IC$ polytope will be produced, in whose $\htilde$-vector the term occurs. Here is a list \[\begin{tabular}{l} in $\htilde(CICCCIC\sqdot)$ the term $\Abar\{1\}\{1\}$ occurs \\ in $\htilde(CCICCIC\sqdot)$ the term $X\{1\}\{1\}$ occurs \\ in $\htilde(CICCICC\sqdot)$ the term $\{1\}\Abar\{1\}$ occurs \end{tabular}\] of such polytopes. This the reader is invited to verify. From this list the conditions, or more exactly the partial order, will be obtained. As $IC$ varieties, each of the above will have a minimal stratification. The conditions control how the apex loci of the cycle meets the strata. (The apex locus is due to the $\lambda$-coning or `locality' of the cycle.) Here is a list, which again the reader is invited to verify, \[\begin{tabular}{l} the strata for $\Abar\{1\}\{1\}$ have dimension $0$, $4$ and $7$ \\ the strata for $X\{1\}\{1\}$ have dimension $1$, $4$ and $7$ \\ the strata for $\{1\}\Abar\{1\}$ have dimension $0$, $3$ and $7$ \end{tabular}\] of the strata dimensions for the associated $IC$ varieties. The present discussion is of a local $1$-dimensional family of local $1$-cycles, and so there are two apex loci to consider. One of them lies on the strata of dimension $0$ or $1$, the other on strata of dimension $3$ or $4$. It is more stringent to require that an apex locus be found on a $1$-strata than on a $0$-strata, and similarly for $4$-strata and $3$-strata. Thus \[ X\{1\}\{1\} \implies \Abar\{1\}\{1\} \implies \{1\}\Abar\{1\} \] is the partial order on conditions. As in the example of $\Abar\{1\}$ and $X\{1\}$, this partial order allows $\htilde\Delta$ to be transformed into $h\Delta$. It is now possible to define the partial order on terms. As in the previous example, associate to each term an $IC$ polytope, whose $\htilde$-vector realises the term. Each such polytope, thought of as a variety, has a minimal stratification. The partial order on terms is then the partial order on the dimension vectors of the stratification associated to each term. Here are the details. The partial order does not compare, say, $\{1\}\{2\}$ and $\{2\}\{1\}$, or $X\{1\}$ and $Y\{1\}$. The associated cycles differ by more than a change in conditions. Say that two expressions in $X$, $Y$, $\Abar$ and $\{k\}$ are \emph{broadly similar} if, when $X$ and $\Abar$ have been deleted, they are identical. They should also have the same degree. The partial order applies only to broadly similar terms. Each term will appear in $\htilde\Delta$, for one or more $IC$ polytopes $\Delta$. Choose one of these polytopes (it does not matter which) for which the associated stratification has as few terms as possible. It will in fact have $r+1$ terms, where $r$ is the order of the term. This process associates a stratification dimension vector $d=(d_1<d_2\ldots<d_{r+1}=n)$ to each term. The partial order is that $d$ will imply $d'$ just in case $r=r'$, and $d_i\geq d_i'$ for $i=1,2,\ldots, r+1$. The above is a description, based on geometry, of the partial order on conditions. The stratification dimension vector $d$ associated to a term can also be computed directly. Suppose, for example, the term is $X^2Y^3\Abar^4\{5\}\Abar^2\{6\}$. The degree of this term is \[ 2 + 3 + 4 + (2\times5+1) + 2 + (2\times6+1) = 35 \] and so $35$ is the dimension of the top stratum. For the broadly similar term $X^i\{5\}\{6\}$, where $i$ makes the degree up to $35$, the dimension vector written backwards is \[ 35 > 35 - \deg \{6\} > 35 - \deg \{6\} - \deg \{5\} \] (or $11 < 22 < 35$ the normal way around). For the given term $X^2Y^3\Abar^4\{5\}\Abar^2\{6\}$ the dimension vector is \[ 35 > 35 - \deg \{6\} - 2 > 35 - \deg \{6\} - 2 - \deg \{5\} -4 \] and from this the partial order can be given a combinatorial description. (The proofs have been left to the reader.) The partial order on terms can be expressed in the following way. The terms implied by some given term, say $X^2Y^3\Abar^4\{5\}\Abar^2\{6\}$, can all be obtained in the following way. First of all, any number of occurences of $X$ can be replaced by $\Abar$. Such $\Abar$ should of course be placed after the $X^iY^j$ term. Next, any occurence of $\Abar$ can be `slid' rightwards over any $\{k\}$ symbol. Finally, note that the terminating symbol `$\sqdot$' is assumed to be at the end of each word, and that $\Abar\sqdot$ is zero. It is now possible to give an algebraic description of the rules that transform $\htilde\Delta$ to the extended $h$-vector $h\Delta$. First of all the change of variable \begin{equation} \label{eqn.XxYy} X = x + \Abar \>;\quad Y=y \>;\quad \end{equation} is made. To ensure that equations such as \[\begin{array}{l} X^2 = x^2 + x\Abar + \Abar^2 \> ; \\ X^3 = x^3 + x^2\Abar + x\Abar^2 + \Abar^3 \> ; \end{array}\] hold, the equation \begin{equation} \label{eqn.Abar-x} \Abar x = 0 \end{equation} is postulated. These rules allow $X$ and $Y$ to be replaced by $x$ and $y$. Of course, $X$ and $Y$ commute, as do $x$ and $y$. Thus, $y$ and $\Abar$ are also to commute. The equation \begin{equation} \label{eqn.Abar-i} \Abar \{k\} = A\{k\} + \{k\}\Abar \end{equation} allows the `rightward slide' of $\Abar$ over $\{k\}$. The $\Abar$ on the left hand side can remain where it is, to produce $A\{k\}$; or it can slide over the $\{k\}$. If so slid, it could be slid again. Thus, it is $\{k\}\Abar$ rather than $\{k\}A$ on the right hand side. Finally, in $\Abar A$ the second $A$ is `non-sliding'. This stops the $\Abar$ from sliding. Thus the equation \begin{equation} \label{eqn.Abar-A} \Abar A = AA \end{equation} is also postulated. The rules in the previous two paragraphs define the extended $h$-vector $h\Delta$ of any convex polytope $\Delta$, via the auxiliary vector $\htilde\Delta$. Implicit in this are topological conditions on local-global cycles. In the next section these conditions will be made explicit. \section{Topology and local-global homology} In this section $Z$ will be a complex algebraic variety, considered as a stratified topological space. Local-global intersection homology groups will be defined for $Z$. The starting point is a particular means of expressing the concept of a local cycle, and of course the basic concepts of intersection homology. By considering families of such cycles, the full concept of a local-global cycle is developed. This is done by extending the notion of a simplex. Recall that in the previous section there were two stages to the definition of the $h$-vector $h\Delta$. Loosely speaking the first stage, the definition of $\htilde\Delta$, corresponded to the definition of a local-global cycle. The second stage was concerned with the conditions (on how it meets the strata) satisfied by the cycle. The topological definition of local-global homology similarly has two stages, the definition of the cycles, and the definition of the conditions. Most of the justification for the cycle definition comes from \S3, while the conditions are those implicit in \S4. To begin with, consider the concept of a local cycle. In \S2 such was thought of as a cycle $\eta$ lying on any sufficiently small neighbourhood $U$ of the point $s$, about which the cycle is to be local. That the local topology of $Z$ about $s$ is the cone on the link can clarify this concept. In this section, all this will be incorporated into the definition of a local cycle. Some preliminaries are required. Homology can be defined using embedded simplices. Here is a review. An \emph{embedded $k$-simplex} is simply a continuous map $f:\sigma_k\to Z$ from the $k$-simplex $\sigma_k$. A \emph{$k$-chain} is a formal sum of (embedded) $k$-simplices. Each simplex $\sigma_k$ has a boundary $d\sigma_k$, which is a formal sum of $(k-1)$-simplices. In the same way, each $k$-chain $\eta$ has a \emph{boundary} $d\eta$, which is a $(k-1)$-chain. A \emph{$k$-cycle} is a $k$-chain $\eta$ whose boundary $d\eta$ is zero (as a formal sum of embedded simplices). Because $d\circ d = 0$, if $\xi$ is a chain then $\eta=d\xi$ is a cycle. Such a cycle $\eta$ will be called a \emph{boundary}. The \emph{$k$-th ordinary homology group} of $Z$ consists of the $k$-cycles modulo the $k$-boundaries. Intersection homology \cite{bib.MG-RDM.IH} places restrictions on how the embedded simplices can meet the strata. The conditions are expressed using a perversity, which is a sequence of numbers. In this paper only the middle perversity is used. It imposes the following condition on an embedded $k$-simplex $f:\sigma_k\to Z$. Let $S_i$ be the complex $i$-dimensional stratum of $Z$. Consider $f^{-1}(S_i)$, or more exactly its dimension. Let the empty set have dimension $-\infty$. If the inequality \[ \dim f^{-1}(S_i) \leq k - ( n-i ) \] holds for every stratum $S_i$, then $f:\sigma_k\to Z$ is \emph{allowed} (for the middle perversity). An \emph{allowed cycle} $\eta$ is a formal sum $\eta$ of embedded simplices, whose boundary is zero. An \emph{allowed boundary} $\eta=d\xi$ is a formal sum $\xi$ of allowed simplices, whose boundary $d\xi$ (which necessarily is a cycle) is also allowed. The $k$-th (middle perversity) \emph{intersection homology} group $z$ consists of the allowed cycles modulo the allowed boundaries. An important technical result \cite{bib.MG-RDM.IH2,bib.HK.TIIH} is that this group is independent of the stratification $S_i$ chosen for $Z$. Consider the concept of a local cycle. Each point $s$ on $Z$ has a neighbourhood that is homeomorphic to the cone $CL_s$ on something, namely the \emph{link} $L_s$ at $s$. One can use $CL_s$ as the neighbourhood $U$ in which the cycles $\eta$ local to $s$ can be found. Because $CL_s$ is a cone (and $\eta$ avoids the apex of the cone), the cycle $\eta$ is equivalent to a cycle $\eta'$, that is supported on the base $L_s$ of the cone. (Use the cone structure to move the cycle $\eta$ to the base of the cone.) In fact, for each $0<\lambda\leq 1$ one obtains a cycle $\eta_\lambda$, while the `limit' $\eta_0$ of the family is the apex $CL_s$, which is the point $s$ of $Z$. In other words, a local cycle is a cycle that can be `coned away' to a point, except that the perversity conditions may disallow this. The concept of a local cycle can be formulated without using either $U$ or $CL_s$. Define a \emph{coned $k$-simplex} to be the cone $C\sigma_k$ on a $k$-simplex. (Although isomorphic to $\sigma_{k+1}$, it will not be treated as such. Later, more complicated objects will be coned.) Local homology will be constructed using such simplices. An \emph{embedded coned $k$-simplex} is simply a continuous map $f:C\sigma_k\to Z$. The \emph{apex} of $f$ is the image under $f$ of the apex of $C\sigma_k$. A \emph{coned $k$-chain} is a formal sum of (embedded) coned $k$-simplices. Each coned simplex $C\sigma_k$ has a boundary, which is a formal sum of coned $(k-1)$-simplices. (The boundary is taken only the the $\sigma_k$ direction, and not in the $C$ direction.) In the same way, each coned $k$-chain $\xi$ has a \emph{boundary} $d\xi$, which is a coned $(k-1)$-chain. A \emph{coned $k$-cycle} is a coned $k$-chain whose boundary $d\eta$ is zero (as a formal sum of embedded coned $(k-1)$-simplices.) Because $d\circ d = 0$, if $\xi$ is a coned chain, then $d\xi$ is a coned cycle. Such a cycle $\eta=d\xi$ will be called a \emph{coned boundary}. Finally, only certain cycles and boundaries will be allowed. Use $\lambda$ and $p$ to denote the cone and simplex variable respectively. For each $0<\lambda\leq 1$, use $f_\lambda$ to denote the embedded simplex defined by the rule $f_\lambda(p)=f(\lambda,p)$, where $f$ is an embedded coned simplex. Say that a coned cycle $\eta$ is \emph{allowed} if $\eta_\lambda$ is similarly allowed. Say that a coned boundary $\eta=d\xi$ is \emph{allowed} if $\eta_\lambda=d\xi_\lambda$ is allowed, for $0<\lambda\leq1$. Now fix a point $z\in Z$. Say that an embedded coned simplex $f$ is \emph{local to $s$} if $s$ is the apex of $f$. In that case, $s$ will also be the apex of the boundary of $f$. The \emph{local $k$-homology at $s$} consists of the coned $k$-cycles modulo the coned $k$-boundaries, where the cycles and boundaries are allowed (by the perversity), \emph{and are constructed using only the embedded coned simplices local to $s$}. Local-global cycles differ from local cycles in that the base point or apex is allowed to move. To do this a new sort of simplex is required. Consider $\sigma_1\times C\sigma_k$. This can be thought of as a $1$-dimensional family of coned $k$-simplices. It has not a single point as its apex, but an \emph{apex edge}. Its boundary in the $\sigma_1$ direction is a pair of coned $k$-simplices, one at each end of the apex edge. It also has a boundary in the $\sigma_k$ direction, which is a formal sum of $\sigma_1\times C\sigma_{k-1}$ `simplices'. (As before, no boundary is taken in the cone direction.) The task now is to define the \emph{local-global $k$-homology} of $Z$. The cycles are as in local $k$-homology, except that it is not required that the embedded coned $k$-simplices be local to some fixed point $s$ of $Z$. Clearly, such a cycle can be written as a formal sum of local $k$-cycles, based at different points $s_1$, $\ldots$, $s_N$ of $Z$. The boundaries require more thought. Previously the simplices (possibly coned) were indexed by a single number $k$, and the boundary operator $d$ reduced the index by one. The present situation is that $C\sigma_k$ arises from both $C\sigma_{k+1}$ and $\sigma_1\times C\sigma_k$, and the latter also produces $\sigma_1\times C\sigma_{k-1}$. When $C\sigma_k$ is written as $\sigma_0\times C\sigma_k$, it becomes clear that the `simplices' are now indexed by a pair of numbers, and that the boundary operator $d$ can reduce either one or the other by one. Because of this, the concept of $\eta=d\xi$ being a \emph{local-global $k$-boundary} can be formulated in several, possibly inequivalent, ways. In each case $\xi$ will be a formal sum of embedded $\sigma_i\times C\sigma_j$ `simplices', with $i+j=k+1$. One also requires that $\eta$ is the boundary of $\xi$, and that for each $0<\lambda\leq 1$, one has the $\eta_\lambda=d\xi_\lambda$ is allowed (in the usual way). The different concepts arise, according to the values of $i$ and $j$ allowed. Say that $\xi$ is \emph{very pure} if all its `simplices' have the same type. Clearly, very pure boundaries should be allowed. In the present case, $\sigma_0\times C\sigma_{k+1}$ is required for local equivalence, while $\sigma_1\times C\sigma_k$ allows the apex to move. The sum of two very pure boundaries need not be very pure. Say that $\xi$ is \emph{pure} if all the `simplices' are capable, by virtue of their `dimension', of participating in a very pure boundary. In the present case, it means that $\xi$ is built out of a mixture of $\sigma_0\times C\sigma_{k+1}$ and $\sigma_1\times C\sigma_k$. Finally, say that $\xi$ is \emph{mixed} if it is built out of any mixture whatsoever of $\sigma_i\times C\sigma_j$ `simplices'. In principle, the three concepts are different. The author expects very pure and pure to give the same boundaries. (This means that if $\eta=d\xi$ is a pure boundary, then there are very pure $\eta_i=d\xi_i$, with $\eta=\sum\eta_i$. It is not required that $\xi=\sum\xi_i$.) The mixed concept permits $\sigma_{k+1}\times C\sigma_0$ to play a r\^ole. This seems to be wrong. In this paper, local-global homology will be defined using the pure concept. Experience will show if this is correct. (Already, the desired Betti numbers are known.) The general definition of local-global cycles and boundaries proceeds in the same way. (Recall that the conditions that control how the apex loci meets the strata have not yet been considered.) To begin with, let $k=(k_0,k_1,\ldots,k_r)$ be a sequence of nonnegative integers. Define the \emph{$k$-simplex} $\sigma_k$ to be the convex polytope \[ \sigma_k = \sigma_r \times C ( \sigma_{r-1} \times C ( \> \ldots\> (\sigma_1 \times C \sigma_0) \ldots )) \] where each $\sigma_i$ is an ordinary simplex of dimension $k_i$. In other words, $\sigma_k$ is $\sigma_r\times C\sigma_{k'}$, where $k'=(k_0,k_1,\ldots, k_{r-1})$. The number $r$ is the \emph{order} of $\sigma_k$ (and of $k$). It is also the number of coning operators. If $r=0$ then $k=(k_0)$, and $\sigma_k$ is an ordinary simplex, of dimension $k_0$. For each choice $\lambda=(\lambda_1, \ldots , \lambda_r)$ of a nonzero value for each of the coning variables in $\sigma_k$, one obtains a `simplex' $\sigma_{k,\lambda} \cong \sigma_r\times \ldots \times \sigma_0$. A formal sum $\eta$ of continuous maps $f:\sigma_k\to Z$ is a \emph{$k$-cycle} if the boundary $d\eta$ is zero, and for each such $\lambda$ the maps $f_\lambda:\sigma_{k,\lambda}\to Z$ associated to $\eta$ are allowed (by the perversity conditions). The $k$-cycle $\eta$ is a \emph{$k$-boundary} if there is a formal sum $\xi$ of $k'$-simplices such that $\eta=d\xi$, and again the $f_\lambda$ due to $\xi$ are also allowed (by the perversity conditions). Here, because pure boundaries are being used, the index $k'$ ranges over the $r+1$ indices obtained via choosing one of the $k_i$ in $k$, and raising it by~$1$. Note that the boundary components of $\xi$, whose index is not $k$, must all cancel to zero. This defines the local-global cycles and boundaries. Each local-global homology group is determined by the choice of an index $k$ (which gives the `dimension' of the cycles), and a choice of the conditions that control how the cycles and boundaries meet the strata. The rest of this section is devoted to the study of these conditions. Recall that $Z$ is a complex algebraic variety considered, as a stratified topological space. Each stratum $S_i$ has real dimension $2i$. The basic building block for local-global homology, the $k$-simplex $\sigma_k$, can also be thought of as a stratified object. However, instead of strata it has \emph{apex loci}. Altogether $\sigma_k$ will have $r$ (its order) apex loci. The $k$-simplex $\sigma_k$ is produced using $r$ coning operators. Each coning introduces an apex. Multiplication by the $\sigma_i$, and the subsequence coning operations, will convert this apex into an apex locus; except that any new apex does not belong to the already existing apex locus. The closure of the $i$-th apex locus is a $\sigma_{k'}$-simplex, where $k'=(k_i,k_{i+1}, \ldots, k_r)$. The conditions, that control how the apex loci meet the strata, are to give rise to groups that are independent of the stratification. From this, the nature of the conditions can be deduced. Here is an example. Suppose $A$ and $B$ are nonsingular projective varieties. Let $Z=A\times CB$ be the product of $A$ with the (projective) cone on $B$, with $Z$ and the apex locus $A$ as the closures of the strata. Each coprimitive $\eta$ on $B$ will (if it can be moved away from the base) determine a local-global cycle $C_\lambda\eta$ on $CB$ and then, by K\"unneth, the choice of a cycle $\xi$ on $A$ determines a local-global cycle $\xi\otimes C_\lambda\eta$ on $Z$. Suppose now that $\xi$ is the fundamental class $[A]$ on $A$, and that some condition permits $[A]\otimes C_\lambda\eta$. The apex locus of this cycle is the apex locus $A$ of $Z$. Now add strata to $Z$, that lie inside the apex locus $A$. There is no possibility of moving $[A]\otimes C_\lambda\eta$ within its equivalence class, in such a way that the apex locus of the cycle is changed. Thus, this addition of new strata to $A$ will not affect the admissability of $[A]\otimes C_\lambda\eta$. Return now to the general case. The previous example can be expressed in the following way. Use the stratification $S_i$ to $Z$ to define the filtration \[ U_i = Z - S_0 - S_1 \ldots -S_{i-1} = S_i \cup S_{i+1} \cup \ldots \cup S_n \] of $Z$ by open sets $Z=U_0 \supseteq U_1 \ldots \supseteq U_n$. For each apex locus $A$ of a local-global cycle, ask the following question: what is the largest $i$ such that $A\cap U_i$ is dense in $A$? Call this the \emph{$w$-codimension} $w(A)$ of $A$. (It tells one \emph{where} $A$ is generically to be found.) Clearly, adding strata as in the previous example will not change $w(A)$. Consider once again $Z=A\times CB$. Suppose one wants a local-global cycle on $Z$, whose $w$-codimension has some fixed value $i\leq\dim A$. Such can be achieved, provided $Z$ is suitably stratified. To begin with, let $A_i\subseteq A$ be a subvariety of dimension $i$. Now form the local-global cycle $[A_i]\otimes C_\lambda\eta$, where $[A_i]$ is the class of $A_i$ in $H_i(A)$. If $Z$ and $A$ are the only closures of strata, then $w(A_i)$ will be $\dim A$. However, one can always add $A_i$ to the stratification, and then $w(A_i)$ will be equal to $\dim A_i$, which by construction is equal to~$i$. The meaning of this example is as follows. Strata impose conditions on cycles. The more strata, the more conditions. Suppose a condition allows the $w$-codimension of the apex locus of a cycle to be some number $i$, smaller than $\dim A$. Cycles meeting this condition can be found, when a suitable stratum (the subvariety $A_i$) is added. Thus, the condition must allow the cycle $[A_i]\otimes C_\lambda\eta$, even when $Z$ is given its minimal stratification, if the result is to be stratification independent. Reducing the stratification can only reduce the $w$-codimension. Thus, if a condition allows $i$ as a $w$-codimension, it should also allow all values smaller than $i$. In other words, $i$ should be maximum allowed value for the $w$-codimension. (Another argument in favour of this conclusion is as follows. The apex locus of a local-global cycle need not be connected. Suppose the apex loci $A$ and $A'$ respectively of $\eta$ and $\eta'$ have $w$-codimensions $i$ and $i'$, with $i<i'$. Suppose also that $i$ is a permitted $w$-codimension. Thus, $\eta$ is permitted. Now consider $\eta+\eta'$. Although its apex locus need not in every case be $A\cup A'$, so far as $w$-codimension is concerned, it might as well be. Thus, $\eta +\eta'$ will also have $i$ as the $w$-codimension of its apex locus, and so is permitted. Because both $\eta$ and $\eta+\eta'$ are permitted, the difference $(\eta+\eta')-\eta=\eta'$ must also be allowed. In other words, $i$ is a minimum allowed value.) The \emph{local-global intersection homology groups} $H_{k,w}(Z)$ can now be defined. The subscript \[ k=(k_0,k_1,\ldots,k_r) \] is a sequence of positive integers. The cycles and boundaries of $H_{k,w}(Z)$ are constructed out of embedded $k$-simplices, as defined earlier in this section. The \emph{$w$-condition} $w$ is a sequence \[ w_1 \geq w_2 \geq \ldots \geq w_r \geq 0 \] of nonegative integers, which is used as follows. Each cycle (or boundary) will have $r$ apex loci, to be denoted by $A_1$, $A_2$, $\ldots$, $A_r$. Each apex locus $A_i$ will have a $w$-codimension $w(A_i)$. Only those cycles and boundaries for which \[ w_i(A) \leq w_i \qquad i = 1, \ldots, r \] holds are to be used in the construction of $H_{k,w}(Z)$. (In this construction, it is assumed that the middle perversity is used. If the $\eta_\lambda$ and $d\xi_\lambda=\eta_\lambda$ are to satisfy some other condition $p$, it can be added to the definition, and also to the notation $H^p_{k,w}(Z)$, like so. Only for certain values of $k$, $w$ and $p$ will nonzero groups be possible. The notation of the previous section takes this into account.) \section{Summary and Conclusions} To close this paper, its main points will be summarized, and then various questions arising are discussed. These are firstly, are the local-global intersection homology groups $H_{k,w}Z$ independent of the stratification of $Z$? Second, does $h\Delta$ compute the (local-global) Betti numbers of $\PDelta$. Third, can the (local-global) homology $H_\bullet\Delta$ of $\Delta$ be constructed without recourse to $\PDelta$? Fourth, what happens when integer coefficients are used? (This question is related to the resolution of singularities.) Fifth, does local-global homology have ring- or functor-like properties. Finally, there is a brief history of the genesis of this paper, and acknowledgements. The relative importance of the various parts of this paper depend on one's point of view. The construction (\S3) of local-global cycles on the $IC$ varieties was chosen as the starting point, from which both the formula for $h\Delta$ (\S4) and the topological definition (\S5) followed. For one interested in more general varieties the topological definition is perhaps most important. The examples in \S3 then become merely the application of a more general concept. Finally, not all general polytopes can be studied via topology, and so this gives $h\Delta$ (\S4) and the questions arising from it a special importance. In some sense, $\lambda$-coning and apex loci are the key new concepts, over and above perversity conditions on cycles and boundaries. At the risk of pleasing none, this paper has tried to please all. The concept of local-global homology has two aspects, namely the cycles and boundaries on the one side, and the strata conditions on the other. In the notation $H_{k,w}Z$, it is $k$ that controls the type of cycles and boundaries, while $w$ supplies the strata conditions. A local-global cycle $\eta$ is a global cycle $\eta_{(1)}$, which can be coned in any of $r$ (the order of $k$ and of $\eta$) $\lambda$-directions. The subscript in $\eta_{(1)}$ indicates that $(1,1,\ldots,1)=(1)$ are the values $\lambda_i$ of the coning variables. The cycle $\eta$ is to be a suitable family $\eta_\lambda$ of global cycles, which collapses in various ways as each coning variable $\lambda_i$ goes to zero. These $\lambda_i$ are not independent. If $\lambda_j=0$ then $\eta_\lambda$ does not depend on $\lambda_i$, for $i<j$. In other words, there is a sequence of \emph{collapsings} \[ \eta_0=\eta \gg \eta_1 \gg \eta_2 \gg \ldots \gg \eta_r \] of the family $\eta=\eta_\lambda$ of cycles. Each $\eta_i$ is the result of placing $\lambda_i=0$, and is called an \emph{apex locus}. The sequence $k=(k_0,k_1,\ldots,k_r)$ encodes the dimensions (more exactly the relative dimensions) of these families. These collapsings provide local information about the cycle $\eta$. Of interest is not only \emph{how} $\eta$ collapses to $\eta_i$, but also \emph{where}. By this is meant how $\eta_i$ meets the strata $S_j$ of $Z$. More exactly, $w(\eta_i)$ is defined to be the largest $j$ such that \[ \eta_i \cap ( S_j \cup S_{j+1} \cup \ldots \cup S_n) \] is dense in $\eta_i$. This is a measure of how special a locus is required, to generically effect the collapsing of $\eta$ to $\eta_i$. The index $w=(w_1,w_2,\ldots,w_r)$ in $H_{k,w}Z$ controls the cycles and boundaries used as follows. A cycle $\eta$ (or boundary $\xi$) is to be used only if $w(\eta_i)$ (or $w(\xi_i)$) is at least $w_i$. Thus, reducing $w$ will potentially allow more cycles (and more relations) to participate in $H_{k,w}Z$. Whether this increases or decreases the size of $H_{k,w}Z$ will just depend on the circumstances. Sections 3 and 4 apply these concepts to $IC$ and toric varieties. This ends the summary. The next topic is stratification independence. It is important that the $H_{k,w}Z$ not depend on the choice of the stratification. This is already known for mpih. First suppose $w$ is zero, so the `where' conditions are vacuous. Suppose also that $\eta$ is a $k$-cycle, namely a $\lambda$-coning of the global cycle $\eta_{(1)}$. Already known is that if the stratification is changed, an $\eta'$ equivalent to $\eta_{(1)}$ can be found. This is a local result, and so the $\eta'$ can be found close to $\eta_{(1)}$. Thus, in this special case stratification independence will follow if the $\lambda$-coning structure that converts $\eta_{(1)}$ to $\eta$ can be extended to a small neighbourhood of $\eta$. This is more a statement about the local conical structure of a stratified topological space, than about the particular cycle $\eta$. Now suppose $w$ is nonzero. As before, $\eta'$ can be found close to $\eta_{(1)}$. The new difficulty is the $w(\eta_i)$. By definition, this will cause no new problem, unless $w(\eta_i)$ is reduced. In this case, more must be done. Previously, $\eta_{(1)}$ was moved to $\eta'$, and it was assumed that $\eta'$ could be collapsed to the $\eta_i$ used for $\eta$. In this case, it is necessary to move the $\eta_i$. To do this, note that the collapsing $\eta \gg \eta_i$ will be a permissable coning away, \emph{for some perversity other that the middle}. As stratification independence is known for all perversities, this may provide a means of producing the $\eta'_i$. As before, one hopes that uniformity of the $\lambda$-coning will complete the proof. These arguments indicate that stratification independence for local-global homology will follow from uniformity of the $\lambda$-coning and some modification of the existing methods. There is however another approach. King \cite{bib.HK.TIIH} was able to prove stratification independence directly, without use of sheaves in the derived category. Habegger and Saper \cite{bib.NH-LS.IC} found that this leads to an intersection homology theory for generalised perversities and local systems. Usually, a local system on $Z$ is determined by its value on the generic stratum $S_n\subset Z$. When generalised, information about behaviour at the boundary (the smaller $S_i$) is recorded by the local system. It may be that local-global homology can be expressed as the homology of such a local system. From this, stratification independence would follow. (Certainly, the concepts are related. If $\eta$ is a local-global cycle, collapsing under $\lambda$ to $\eta_1$, and if $\eta_1\cap S_j$ is dense in $\eta_1$, then the following holds. There is a local system $L_j$ (in the usual sense) on $S_j$ such that $\eta$ both determines and is determined by a cycle $\eta'$ on $S_j$ with coefficients in $L_j$. (Because $S_j$ may not be closed, $\eta'$ might not be a compact cycle. This is a technical matter.) Consider this $\eta'$. The Goresky-MacPherson theory imposes conditions on the dimension in which the closure $\bar\eta'$ of $\eta'$ meets the strata. For local-global homology, $\eta'$ determines a local-global cycle $\eta_0$ in a neighbourhood of $S_j$. Whether or not such an $\eta_0$ can be closed up to produce an $\eta$ is a delicate matter, which depends not so much on how $\eta'$ meets the smaller strata, but on how the local topology of $Z$ about these strata interacts with $\eta_0$. The generalised concept of a local system may provide a place where this information can be stored, and used to control $\eta'$.) This discussion has assumed that existing techniques, perhaps adapted, will be applied to local-global homology. However, stratification independence is primarily a technical result on the local topology of stratified spaces. It may be that local-global homology will provide a suitable language for describing this local topology. One must show that it causes no obstruction to the motion of cycles and boundaries, that is required when the stratification is refined. If this holds, then the concept of local-global homology is already implicit in the proof of stratification independence, and the consequences of King's paper become less surprising. Each Betti number of $\PDelta$ is the dimension of a vector space, and so is nonnegative. Thus, a linear function that computes such a Betti number from $f\Delta$ is also a linear inequality on $f\Delta$, at least when $\Delta$ has rational vertices. Of special interest therefore are those parts of homology theory, for which the Betti numbers are indeed linear functions of $f\Delta$. Ordinary homology does not have this property~\cite{bib.McC.HTV}. Middle perversity intersection homology does \cite{bib.JD-FL.IHNP,bib.KF.IHTV}. This is a consequence of deep results in algebraic geometry, namely Deligne's proof~\cite{bib.NK.DPRH} of the Weil conjectures and the purity of mpih~\cite{bib.AB-JB-PD.FP}. It is possible that at least some part of the local-global homology theory will also have this purity property. An example of Bayer (personal communication) shows that certain components of the extended $h$-vector are sometimes negative, and so are not always Betti numbers. (Bayer's example is $\Delta=BICCC\sqdot$, where $B$ is the bipyramid operator, the combinatorial dual to the $I$ operator. In $h\Delta$ the term $xA\{1\}$ occurs with coefficient $-2$. The interpretation of this result requires some care. It does not show that the whole of $h\Delta$ is unsuitable, or that local-global homology is a flawed concept. The formula for $h\Delta$ is the extrapolation to all polytopes of the heuristically calculated formula for the various local-global homology Betti numbers of the $IC$ varieties. The component $xA\{1\}$ corresponds to the gluing together along paths of local cycles, a construction that is already known to yield a topological invariant. More exactly, $xA\{1\}$ counts local $1$-cycles under this equivalence, on a $4$-dimensional variety that has had its $0$-strata removed. The corresponding Betti number is as computed by $h\Delta$, for the $IC$ varieties, but clearly not in general.) (Ordinary homology can be approached in the same way. Use the simple case formula $h(x,x+y)=f(x,y)$ to define a `pseudo $h$-vector' $\hquest\Delta$ for general polytopes. The rules \[ \hquest (I\Delta) = (11) \hquest (\Delta) \> ;\quad \hquest (C\Delta) = (100\ldots0) + y\hquest (\Delta) \> ; \] give its transformation under $I$ and $C$. The author suspects, but does not known, that $\hquest\Delta$ gives the ordinary homology Betti numbers, when $\Delta$ is an $IC$ variety. However, when extrapolated to the octahedron, $(1,-1,5,1)$ is the result. Despite this negative value, ordinary homology is still a topological invariant. However, it is not suitable for the study of general polytope flag vectors.) For $\Delta$ rational the mpih part $(h_0,h_1,\ldots,h_n)$ of $h\Delta$ is not only nonnegative but also \emph{unimodal}. This means that $h_0 \leq h_1 \leq \ldots$ and so on up to halfway. It is a consequence of strong Lefschetz. The author suspects that the analogous result for general polytopes will be as follows. First compute $hC\Delta$. This is of course a linear function of $h\Delta$. Now look at the coefficients of $x^0y^0W$, where $W$ is a word in $A$ and $\{k\}$. These numbers are expected to be nonnegative. This is an extension of the mpih result. Of course, algebraic geometry can prove such results only when $\Delta$ is rational, for only then does $\PDelta$ exist. The only method that will produce such results for general polytopes, that the author can envision, is \emph{exact calculation of homology}. This means constructing a complex of vector spaces (of length at most $n$), whose Euler characteristic (alternating sum of dimensions) is the desired Betti number. Exact calculation then consists of proving that this complex is exact, except possibly at one location. The homology of this complex at that location is then a vector space which as a consequence of exactness has the desired dimension. In \cite{bib.JF.CPLA} the author introduces such complexes. The proof of exactness is expected to be difficult. McMullen's proof \cite{bib.McM.SP} of strong Lefschetz for simple polytopes, without the use of algebraic varieties, probably contains a prototype of the arguments that will be needed. There, the Riemann-Hodge inequalities were vital in supporting the induction on the dimension of the polytope. Thus, one would like local-global homology to support similar inequalities. The process of constructing the complexes for exact calculation requires that the linear function $h\Delta$ on polytope flag vectors be defined for all flag vectors, whether of a polytope or not. This allows one to talk about the contribution made by an individual flag to $h\Delta$. (One then interprets this contribution as the dimension of a vector space associated to the flag, and then assembles all these vector spaces into a complex. The boundary map is induced by the deletion of a single term from the flag. In \cite{bib.JF.CPLA}, the numerical contribution due to a flag is derived from a study of the associated vector spaces, which is the starting point. This approach better respects the inner logic of the exact calculation concept.) The extension of $h\Delta$ to all flag vectors (not just those of polytopes) can be done in the following way. Suppose $\delta$ is an $i$-face on $\Delta$. The local combinatorial structure of $\Delta$ along $\delta$ can be represented by an $(n-i-1)$ dimensional polytope, the \emph{link $L_\delta$ of $\Delta$ along $\delta$}. If $g_i$ is a linear function on flag vectors then the expression \begin{equation} \label{eqn.h-sumg} h\Delta = \sum \nolimits_{ \delta \subseteq \Delta } g_i B \end{equation} is a linear function on the flag vector of $\Delta$. (Here, $\delta$ runs over all faces of $\Delta$, $i$ is the dimension of $\delta$, and $B$ is the link along $\delta$.) In this paragraph, $h\Delta$ is a linear function that might or might not be the previously defined extended $h$-vector. Now use the rules \begin{eqnarray} \label{eqn.g_0B} g_0B &=& ChB -yhB \\ g_{i+1}B &=& yg_iB - g_i CB \end{eqnarray} and the initial value $g_0(\emptyset)=(1)$ to produce a recursive definition of $g$ and $h$. The $ChB$ in (\ref{eqn.g_0B}) stands for the rule $\Ctilde$ of \S4, translated into a $x$, $y$, $A$ and $\{k\}$ rule, and then applied to $hB$. (Here, $B$ has dimension less than that of $\Delta$, and so $hB$ is by induction already defined.) This defines a linear function $h\Delta$, which however can now be applied to non-polytope flag vectors. For polytope flag vectors, it agrees with the previously defined value. (Central to the proof of this is the expression of the links on $I\Delta$ and $C\Delta$ in terms of those on $\Delta$. This approach does not rely on the $IC$ equation.) The mpih part of (\ref{eqn.h-sumg}) is the usual formula \cite{bib.JD-FL.IHNP,bib.KF.IHTV,bib.RS.GHV} for these Betti numbers. The whole of (\ref{eqn.h-sumg}) can be `unwound' to express $h\Delta$ as a sum of numerical contributions due to individual flags. The mpih part has been presented in \cite{bib.MB.TASI}. Note that just as the topological space $\PDelta$ can be decomposed into cells in many ways, so its homology can be computed in various ways. To each suitable such method an extension of $h\Delta$ to all flag vectors will follow, and vice versa. An extension which on simple polytopes reduces to the $h(x,x+y)=f(x,y)$ formula could be very useful. Finally, note that when $h\Delta$ is a sum of flag contributions one can use `Morse theory' or shelling to compute $h\Delta$. Choose a linear `height' function, so that each vertex on $\Delta$ has a distinct height. Define the \emph{index} of a vertex $v$ to be the sum of the contributions due to the flag whose first term has $v$ as its highest point. It immediately follows that $h\Delta$ is the sum, over all vertices, of their index. Central to the problem of resolution of singularities (which is still open in finite characteristic) is the discovery of suitable invariants of singular varieties. These should be well behaved under monoidal transformation, and always permit the choice of a centre of transformation, which will reduce the invariant. When the invariant becomes zero, the variety should be nonsingular. It may be that some form of local-global homology will have such properties. One aspect of the problem is this. Suppose the singular locus consists of two lines meeting at a point. Along each line there is a locally constant singularity, whose resolution is essentially a lower dimensional problem, which can by induction be assumed solved. The difficulty is at the meeting point of the two lines. Along each line there is a resolution process. One needs to know whether and when a transformation should be centered at the common point. In the study of this process one should use not only the local cycles due to mpih, but also those due to the higher order local-global homology. (In addition, one should use integer rather than rational coefficients, but more on this later.) The local cycles generic along each of the two lines will influence if not control the resolution process along that line. The concepts of local-global homology allow the interaction to be studied at their common point, of these `controls' along the two lines of the resolution process. This argument indicates that local-global homology contains the right sort of information, for it to act as a suitable control on the resolution process. It does not show that it contains enough such information. Torsion is when a cycle is not a boundary, but some multiple of it is. Using rational or real coefficients sets all such torsion cycles equal to zero. This simplifies the theory, and for the study of Betti numbers and the flag vectors of convex polytopes, such complications are not needed. For resolution of singularities, and more subtle geometric problems, the situation is otherwise. Here is an example. Consider the affine surface $X=\{xy=z^k\}$, for $k\geq 2$. This has a singularity at the origin. However, there are no non-trivial local-global cycles on $X$, according to the definitions of \S5. The divisor class group ${\rm Cl}(X)$ however is nontrivial, and all its elements are torsion. For example, the line $L=\{x=z=0\}$ is not defined by a principal ideal, whereas $kL$ is defined by $\{x=0\}$. Such information can be recorded by local-global homology, provided integer coefficients and a different concept of a local cycle are used. It is known that duality has a local expression, which pairs compact cycles avoiding say the apex of a cone, and cycles of complementary dimension, that meet the apex. Duality, of course, ignores torsion cycles. However, non-trivial but torsion local cycles exist for $X=\{xy=z^k\}$, when the complementary concept of cycle is used. It is interesting that the construction of local-global cycles associated to the formula (\ref{eqn.h-sumg}) produces such cycles, rather than compact local-global cycles. It is natural to ask: does the vanishing of all such local-global cycles (except mpih of course) imply that the variety is either nonsingular, or a topological manifold? Because the Brieskorn hypersurface singularity $x_1^2 +x_2^2 +x_3^2 + y^3 + z^5=0$ is locally homeomorphic to the cone on an exotic $7$-sphere \cite{bib.JM.MH7S}, which is knotted inside a $9$-sphere, purely topological invariants are not enough to ensure the nonsingularity of a variety $Z$. The same may not be true for an embedded variety $Z\subset \bfP_n$. Ordinary homology and cohomology are functors. This distinguishes them from intersection homology. For mpih and, one hopes, a good part of the local-global homology, it is purity and formulae for Betti numbers that are the characteristic properties. Both concepts agree, of course, on nonsingular varieties. Thus, one of the geometric requirements on local-global homology is as follows. Suppose $f:Z_1 \to Z_2$ is a map between, say, two projective varieties. Suppose also that the strictly local-global homology of both $Z_1$ and $Z_2$ vanishes. It may be that this condition ensures that the $Z_i$ are, say, rational homology manifolds. If this is so, then $f$ induces a map $f_*:H_\bullet Z_1 \to H_\bullet Z_2$, which can be thought of as a point in $(H_\bullet Z_1)^*\otimes H_\bullet Z_2$. In the general case one would want $f$ to induce some similar map or object in a space, which reduces to the previous $f_*$ when the strictly local-global homology vanishes. Similarly, when $Z$ is nonsingular its homology carries a ring structure. (It is this structure that supplies the pseudopower inequalities \cite{bib.McM.NFSP} on the flag vector of a simple polytope.) Elsewhere \cite{bib.JF.IHRS}, the author provides for a general compact $Z$ a similar structure, that in the nonsingular case reduces to the homology ring. To close, some personal historical remarks, and acknowledgements. In 1985, upon reading the fundamental paper \cite{bib.MB-LB.gDS} of Bayer and Billera, it became clear to the author that a full understanding of general convex polytopes would require a far-reaching extension to the theory of intersection homology. The background to this insight came from Stanley's proof \cite{bib.RS.NFSP} of the necessity of McMullen's conjectured conditions \cite{bib.McM.NFSP} on the face vectors of simple polytopes, and the proof by Billera and Lee of their sufficiency. Danilov's exposition \cite{bib.VD.GTV} of toric varieties, and McConnell's result \cite {bib.McC.HTV}, also contributed. It was also clear that once the extended $h$-vector was known, the rest would soon follow. Already in the simple case there is an interplay between the topological definition of homology, $h$-vectors, and `combinatorial linear algebra' or exact calculation. The same holds for middle perversity intersection homology. In 1985 the Bernstein-Khovanskii-MacPherson formula was known, although not published until 1987~\cite{bib.RS.GHV}. Because of this circle of ideas, knowledge of say the $h$-vector is sufficient in practice to determine the other parts of the theory. This led the author to find the $IC$ equation \cite{bib.JF.MVIC} (again, known in 1985 but not published until much later). This moved the focus on to the rules for the transformation of $h\Delta$ under $I$ and $C$. In the early years of the search for these rule, two related and erroneous ideas were influential. The first is that the rule for $I$ (as via K\"unneth) should be multiplication by $(x+y)$. The second is that the generalised Dehn-Somerville equations should be expressed by $h\Delta$ being palindromic. (Also unhelpful was an undue concentration on the formula for $h\Delta$.) The crucial step that lead to these assumptions being dropped took place in 1993. Loosely speaking, it was the discovery of special cases of the `gluing local cycles together along paths' construction. At that time how to form families of local cycles, or in other words the `local-global' concept, was still mysterious. In late 1995 the present formula for $h\Delta$ was discovered, as a solution to the various geometric, topological and combinatorial constraints that were known. It was not at that time properly understood by the author. This definition was in 1996 pushed around the circle of ideas, to produce first a combinatorial linear algebra construction for $H_\bullet\Delta$, and then the topological definition. That all this can be done indicates that the definition of $h\Delta$ is correct. Once these advances had been understood, it was then possible to return to the derivation of the formula for $h\Delta$, and put it on a proper footing. It was only at this point that the fundamental concepts became clear. In 1997, the analogue to the ring structure was found. Its relation to the concepts presented here is, at the time of writing, still under investigation. Many of the results in this paper were first made available as preprints and the like, in 1996 and 1997. The difficulties encountered by the readers have lead to many revisions in the exposition, and clarification of the basic concepts, both on paper and in the author's understanding. Part of the difficulty is the extent of the circle of ideas, which passes through several areas of mathematics. Any one of topology, combinatorics, linear algebra and intersection theory can be chosen as the starting point. Another difficulty is that much of the intuition and guidance comes from perhaps uncomplicated examples and points of view that have, by and large, not yet been put into print. Finally, thanks are due to Marge Bayer, Lou Billera, Martin Hyland, Gil Kalai, Frances Kirwan, Carl Lee, Robert MacPherson, Mark McConnell, Peter McMullen, Rick Scott, Richard Stanley and David Yavin variously for their tolerant interest in the author's previous attempts to find $h\Delta$, to define $H_{k,w}Z$, and to express these results.

9709

alg-geom/9709016

en

https://arxiv.org/abs/alg-geom/9709016

alg-geom/9709016

James A. Carlson

Daniel Allcock, James A. Carlson, Domingo Toledo

A Complex Hyperbolic Structure for Moduli of Cubic Surfaces

Six pages, plain tex, available at http://www.math.utah.edu/~allcock

10.1016/S0764-4442(97)82711-5

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature.

alg-geom

\section{1. Main results} To a (marked) cubic surface corresponds a (marked) cubic threefold defined as the triple cover of ${\Bbb P}^3$ ramified along the surface. The period map $f$ for these threefolds is defined on the moduli space $M$ of marked cubic surfaces and takes its values in the quotient of the unit ball in ${\Bbb C}^4$ by the action of the projective monodromy group. This group $\Gamma_0$ is generated by complex reflections in a set of hyperplanes whose union we denote by ${\cal H}$. Then we have the following result: \proclaim{Theorem.} The period map defines a biholomorphism $$f: M \longrightarrow \left(B^4 - {\cal H}\right)/\Gamma_0.$$ \endproclaim \noindent From this identification we obtain results on the metric structure and the fundamental group: \proclaim{Corollaries.} (1) The moduli space of marked cubic surfaces carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature. (2) The fundamental group of the space of marked cubic surfaces contains a normal subgroup which is not finitely generated. (3) The fundamental group of the space of marked cubic surfaces is not a lattice in a semisimple Lie group. \endproclaim \medskip \noindent{\bf Remarks.} (1) Our methods also show that the metric completion of $(B^4 - {\cal H})/\Gamma_0$ is the complex hyperbolic orbifold $B^4/\Gamma_0$, which is isomorphic to the moduli space of marked stable cubic surfaces. (2) Recently E. Looijenga found a remarkable presentation of the orbifold fundamental group of the moduli space of smooth unmarked cubic surfaces. To make precise the notion of smooth marked cubic surface, fix the lattice $L$ to be the free ${\Bbb Z}$-module with basis $e_0 \ ,\ldots ,\ e_6$ endowed with the quadratic form for which the given basis is orthogonal and such that $\ip e_0/e_0/ = 1$, $\ip e_k/e_k/ = -1$ for $k > 0$. Let $\eta = 3e_0 - (e_1 + \cdots + e_6)$. Then a \emph marked cubic surface/ consists of a smooth cubic surface $S$ and an isometry $\psi: L \longrightarrow H^2(S,{\Bbb Z})$ which carries $\eta$ to the hyperplane class. The set $M$ of isomorphism classes of marked cubic surfaces has the structure of a variety and is a fine moduli space. A construction of it is described in \cite{9}, and a smooth compactification $C$ is given for which the points of $C - M$ constitute a normal crossing divisor. To define the group $\Gamma_0$, let ${\cal E}$ denote the ring of Eisenstein integers ${\Bbb Z}[\omega]$ where $\omega = (-1 + \sqrt{-3})/2$ is a cube root of unity, and consider the Cartesian product ${\cal E}^5$ endowed with the hermitian inner product $\hip v/w/ = -v_1\bar w_1 + v_2\bar w_2 + v_3\bar w_3 + v_4\bar w_4 + v_5\bar w_5$. Then $({\cal E}^5,h)$ is the unique self-dual lattice over the Eisenstein integers with signature $(4,1)$. Thus $Aut({\cal E}^5,h)$ is a discrete subgroup of the unitary group $U(h)$, which acts on $B^4 =\{\; \ell \in {\Bbb P}^4\; :\; h|\ell < 0 \;\}$. Observe that ${\cal E}/\sqrt{-3}{\cal E} \cong {\Bbb F}_3$ is a field of three elements and that there is a natural homomorphism $Aut({\cal E}^5,h) \longrightarrow Aut({\Bbb F}_3^5,q)$ where $q$ is the quadratic form obtained by reduction of $h$ modulo $\sqrt{-3}$. Let ``$P$'' denote projectivization, and define a group $\Gamma_0$ of automorphisms of $B^4$ by $$ 1 \longrightarrow \Gamma_0 \longrightarrow PAut({\cal E}^5,h) \longrightarrow PAut({\Bbb F}_3^5,q) \longrightarrow 1. $$ This is the discrete group of the main theorem. The hyperplanes of ${\cal H}$ are defined by the equations $h(x,v) = 0$ for vectors $v$ in ${\cal E}^5$ with $ h(v) = 1 $. Note that $PAut({\Bbb F}_3^5,q)$ is isomorphic to the Weyl group of the $E_6$ lattice. \section{2. Construction of a period mapping} To construct the period mapping, we examine in detail the Hodge structures for the cubic threefolds. The underlying lattice $H^3(T,{\Bbb Z})$ is ten-dimensional, carries a unimodular symplectic form $\Omega$, and admits a Hodge decomposition of the form $H^3(T,{\Bbb C}) = H^{2,1} \oplus H^{1,2}$. Choose a generator $\sigma$ for the group of automorphisms of $T$ over ${\Bbb P}^3$, and note that it operates without fixed points on $H^3(T,{\Bbb Z})$. This action gives $H^3(T,{\Bbb Z})$ the structure of a five-dimensional module over the Eisenstein integers. It carries a hermitian form $$ \hip x/y/ = {1\over 2}( \Omega((\sigma - \sigma^{-1})x,\, y) + (\omega - \omega^{-1}) \Omega( x,\, y) ) $$ which is unimodular and of signature $(4,1)$. Now consider the quotient module $H^3(T,{\Bbb Z})/(1-\omega)H^3(T,{\Bbb Z})$ and observe that it can be identified isometrically with $({\Bbb F}_3^5,q)$. We define a marking of $T$ to be choice of such an isometry, and we claim that a marking of a cubic surface determines a marking of the corresponding threefold. Indeed, if $\gamma$ is a primitive two-dimensional homology class on $S$ then it is the boundary of a three-chain $\Gamma$ on $T$. Since $\Gamma$ and $\sigma \Gamma$ have the same boundary, the three-chain $c(\gamma) = (1-\sigma)\Gamma$ is a cycle. However, it is well-defined only up to addition of elements $(1-\sigma)\Delta$ where $\Delta$ is a three-cycle on $T$. Thus a homomorphism $$ c: H_2^{prim}(S,{\Bbb Z}) \longrightarrow H_3(T,{\Bbb Z})/(1-\sigma) $$ is defined. Since a marking of $S$ can be viewed as a basis of $H_2^{prim}(S,{\Bbb Z})$, application of $c$ to the basis elements defines a basis of $H_3(T,{\Bbb Z})/(1-\sigma)$, and this gives the required marking of the threefold. The action of $\sigma$ decomposes $H^3(T,{\Bbb C})$ into eigenspaces $H^3_\lambda$ where $\lambda$ varies over the primitive cube roots of unity. Because $\sigma$ is holomorphic, the decomposition is compatible with the Hodge decomposition and one has $$ H^3_\omega = H^{2,1}_\omega \oplus H^{1,2}_\omega \qquad H^3_{\bar\omega} = H^{2,1}_{\bar\omega} \oplus H^{1,2}_{\bar\omega} . $$ The dimensions of the Hodge components can be found with the help of Griffiths' Poincar\'e residue calculus \cite{5}. Details for this case are found in \cite{3}, section 5. One finds that $$ \hbox{dim}\, H^{2,1}_\omega = 4, \quad \hbox{dim}\, H^{1,2}_\omega = 1, \qquad \hbox{dim}\, H^{2,1}_{\bar\omega} = 1, \quad \hbox{dim}\, H^{1,2}_{\bar\omega} = 4, $$ and from the Hodge-Riemann bilinear relations one finds that $h$ has signature $(4,1)$. Now let $\phi$ be a generator of the one-dimensional space $H^{2,1}_{\bar\omega}$ and let $\gamma_1 \ ,\ldots ,\ \gamma_5$ be a standard basis of $H^3(T,{\Bbb Z})$ considered as an ${\cal E}$-module. By this we mean that the $\gamma_k$ are orthogonal and that $h(\gamma_1,\gamma_1) = -1$ and $h(\gamma_k,\gamma_k) = 1$ for $k > 1$. Let $v(\phi,\gamma)$ be the vector in ${\Bbb C}^5$ with components $$ v_k = \int_{\gamma_k} \phi . $$ One verifies that $h(v,v) < 0$ where now $h$ is the hermitian form $-|v_1|^2 + |v_2|^2 + \cdots + |v_5|^2$. Thus the line generated by $v(\phi,\gamma)$ defines a point in $B^4 \subset {\Bbb P}^4$, and one checks that $v(\phi,\gamma) \not\in {\cal H}$. By well-known constructions (the work of Griffiths), the period vector defines a holomorphic map from the universal cover of $M$ to the ball which transforms according to the projectivized monodromy representation for marked cyclic cubic threefolds. The proof that $\Gamma_0$ is the projective monodromy group relies on the work of Libgober \cite{6} and the first author \cite{1}. Thus our construction yields a period map $f: M \longrightarrow (B^4 - {\cal H})/\Gamma_0$. \section{3. Properties of the period mapping} We must now show that $f$ is bijective. For injectivity, consider once again the period vector $v(\phi,\gamma)$. The vectors $\gamma_k$ can be decomposed into eigenvectors $\gamma_k'$ and $\gamma_k''$ for $\sigma$, with eigenvalues $\omega$ and $\bar\omega$, respectively. Let $\hat\gamma_k'$ and $\hat\gamma_k''$ denote elements of the corresponding dual basis. Because $\phi$ is an eigenvector with eigenvalue $\bar\omega$, its integral over $\gamma_k'$ vanishes, so that $$ \phi = \sum_k \hat\gamma_k''\int_{\gamma_k''}\phi = \sum_k \hat\gamma_k''\int_{\gamma_k}\phi . $$ Thus the components of $v(\phi,\gamma)$ determine $\phi$ as an element of $H^3_{\bar\omega}$. Consequently the line ${\Bbb C} v(\phi,\gamma)$ determines the complex Hodge structure $H^3_{\bar\omega}$. Viewing the Hodge components of $H^3_{\bar\omega}$ as subspaces of $H^3(T,{\Bbb C})$, we may take their conjugates to determine the complex Hodge structure $H^3_\omega$. These two complex Hodge structures determine the Hodge structure on $H^3(T,{\Bbb Z})$. Thus, by the Torelli theorem of Clemens-Griffiths \cite{4}, the period vector $v(\phi,\gamma)$ determines the cubic threefold $T$ up to isomorphism. It remains to show that $T$, which perforce is a cyclic cubic threefold, determines its ramification locus uniquely. This follows from the fact that the locus in question is a planar component of the Hessian surface. To prove surjectivity we first consider a smooth compactification $C$ of $M$ by a normal crossing divisor $D$, e.g., the one given by Naruki \cite{9}, as well as the Satake compactification $\overline{B^4/\Gamma_0}$, obtained by adding fourty points, the ``cusps,'' each corresponding to a null point of $P({\Bbb F}_3^5, q)$. By well-known results \cite{2} in complex variable theory, the period map has a holomorphic extension to a map $\bar f$ from $C$ to the Satake compactification. Since $C$ is compact, $\bar f$ is open, and $\overline{B^4/\Gamma_0}$ is connected, we conclude that $\bar f$ is surjective. \section{4. Boundary components} To pass from surjectivity of $\bar f$ to surjectivity of $f$, we must show that $\bar f$ maps the compactifying divisor $D$ to the complement of $(B^4 - {\cal H})/\Gamma_0$ in the Satake compactification. To this end write $D$ as a sum of irreducible components, $ D = \bigcup D_i' \cup \bigcup D''_j $, where $D_i'$ parametrizes nodal cubic surfaces via the map to the geometric invariant theory compactification of the moduli space of smooth cubics, and where in the same way the $D_j''$ parametrize cubics with an $A_2$ singularity. Now consider a one-parameter family of cubic surfaces with smooth total space acquiring a node. Its local equation near the node has the form $x^2 + y^2 + z^2 = t$ and the corresponding family of cyclic cubic threefolds has the form $x^2 + y^2 + z^2 + w^3 = t$. The local monodromy of the latter has order six, its eigenvalues are primitive sixth roots of unity, and the space of vanishing cycles is two-dimensional. (These facts are well-known and the relevant literature and arguments are summarized in \cite{3}, section 6). From \cite{7} we conclude that coefficients of the period vector on vanishing cycles are of the form $ A(t) t^{1/6} + B(t)t^{5/6} $ where $A$ and $B$ are holomorphic. Now the space of vanishing cycles is invariant under the action of $\sigma$ and so constitutes a rank one ${\cal E}$-submodule. One can choose a generator $\delta$ for it so that $h(\delta,\delta) = 1$, and then one has $$ \lim_{t \to 0} \int_\delta \phi = 0. $$ Thus the limiting value of the period vector lies in the orthogonal complement of $\delta$. In other words, $\bar f(D'_i)$ lies in ${\cal H}/\Gamma_0$, as required. Consider next a one-parameter family of cubic surfaces with smooth total space whose central fiber acquires an $A_2$ singularity. Its local equation is $x^2 + y^2 + z^3 = t$ and the corresponding family of cyclic cubic threefolds has local equation $x^2 + y^2 + z^3 + w^3 = t$. In this case the local monodromy is of infinite order. After replacing $t$ by $t^3$ one finds an expansion of the form $ \phi(t) = A(t)(\log t)\,\hat\gamma + \hbox{(terms bounded in $t$})$, where $A(0) \ne 0$ and where $\hat\gamma$ is an integer cohomology class which is isotropic for $h$. Consequently the line ${\Bbb C}\phi(t)$ converges to the isotropic line ${\Bbb C}\hat\gamma$ as $t$ converges to zero, hence converges to a cusp in the Satake compactification. \section{6. The corollaries} Finally, we comment on the collaries. Part (a) is immediate. For part (b) let $K$ denote the kernel of the map $\pi_1(M) \longrightarrow \Gamma_0$. Then $K$ is isomorphic to the fundamental group of $B^4 - {\cal H}$ and it is easy to see that its abelianization is not finitely generated. We remark that $K$ is not free: there are many sets of commuting elements corresponding to normal crossings of ${\cal H}$. For (c) we note first that for lattices in semisimple Lie group of real rank greater than one, the results of Margulis \cite{8} imply finite generation of all normal subgroups. The rank one case can be treated separately, as was shown to us by Michael Kapovich. \bibliography \bi{1} D. Allcock, New complex and quaternion-hyperbolic reflection groups, submitted, http://www.math.utah.edu/\~{}allcock \bi{2} A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays, J. Differential Geometry {\bf 6 } (1972), 543--560. \bi{3} J. Carlson and D. Toledo, Discriminant complements and kernels of monodromy representations, 22 pp, submitted, http://www.math.utah.edu/\~{}carlson \bi{4} C. H. Clemens and P. A. Griffiths, the intermediate Jacobian of the cubic threefold, Ann. of Math. {\bf 95 } (1972), 281--356. \bi{5} P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. {\bf 90} (1969), 460-541. \bi{6} A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. {\bf 162} (1978), 63--67. \bi{8} B. Malgrange, Int\'egrales asymptotiques et monodromie, Ann. Sci. Ecole Norm. Sup., ser. 4, tome { \bf 7} (1974), 405--430. \bi{8} G. A. Margulis, Quotient groups of discrete subgroups and measure theory, Funct. Anal. Appl. {\bf 12 } (1978), 295--305. \bi{9} I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) {\bf 45 } (1982), 1-30. \endbibliography \end

9709

alg-geom/9709013

en

https://arxiv.org/abs/alg-geom/9709013

alg-geom/9709013

Fernando Torres

Rainer Fuhrmann, Fernando Torres

On Weierstrass points and optimal curves

22 pages, Latex 2e

Rend. Circ. Mat. Palermo Suppl. 51, (1998) 25--46

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.

alg-geom

\section{Preliminaries}\label{1} In this section we summarize some background material concerning Weierstrass Point Theory, Frobenius orders and a rational divisor arising from the Zeta Function of a curve defined over a finite field. \subsection{Weierstrass Point Theory}\label{1.1} Here we repeat relevant material from St\"ohr-Voloch's \cite[\S1]{sv} (see also \cite[III.5]{fk}, \cite{lak} and \cite{sch}). Let $X$ be a curve of genus $g$ over an algebraically closed field $k$ and $k(X)$ the field of rational functions on $X$. Let $\mathcal D$ be a $g^r_d$ on $X$, say $$ \mathcal D=\{E+{\rm div}(f): f\in \mathcal D'\setminus\{0\}\}\subseteq |E|\, , $$ $E$ being an effective divisor on $X$ with ${\rm deg}(E)=d$ and $\mathcal D'$ an $(r+1)$-dimensional $k$-subspace of $L(E):=\{f\in k(X)^*: E+{\rm div}(f)\succeq 0\}$. To any $P\in X$ one then associates the sequence of {\it $(\mathcal D,P)$-orders} $$ \{j_0(P)<\ldots<j_r(P)\}:=\{v_P(D): D\in \mathcal D\}\, , $$ and on $X$ one has the so-called {\it $\mathcal D$-ramification divisor}, namely $$ R=R^{\mathcal D}={\rm div}({\rm det}((D^{\epsilon_i}_xf_j)))+\sum_{i=0}^{r}\epsilon_i{\rm div}(dx)+(r+1)E\, , $$ where $x$ is a separating variable of $k(X)|k$, $D^{i}_x$ is the $i$th Hasse derivative, $f_0,\ldots,f_r$ is a $k$-base of $\mathcal D'$, and $(\epsilon_0,\ldots,\epsilon_r)$ (called {\it the $\mathcal D$-orders}) is the minimum in the lexicographic order of the set $$ \{(j_0,\ldots,j_r)\in \mathbb N^{r+1}: j_0<\ldots<j_r;\ {\rm det}((D^{j_i}_xf_j))\neq 0\}\, . $$ One has \begin{equation}\label{eq1.1} \begin{split} & (a)\quad {\rm deg}(R)=\sum_{P\in X}v_P(R)=\sum_{i=0}^{r}\epsilon_i(2g-2)+(r+1)d\, ,\\ & (b)\quad j_i(P)\ge \epsilon_i\qquad \text{for each $i$ and for each $P$}\, ,\\ & (c)\quad v_P(R)\ge \sum_i(j_i(P)-\epsilon_i)\, ,\qquad \text{and} \\ & (d)\quad v_P(R)=\sum_i(j_i(P)-\epsilon_i) \Leftrightarrow {\rm det}(\binom{j_i(P)}{\epsilon_j})\not\equiv 0 \pmod{{\rm char}(k)}\, .\\ \end{split} \end{equation} Consequently the $(\mathcal D,P)$-orders are $\epsilon_0,\ldots,\epsilon_r$ if and only if $P\in X\setminus {\rm supp}(R)$. The points in ${\rm supp}(R)$ are the so-called {\it $\mathcal D$-Weierstrass points}. The $\mathcal K$-Weierstrass points, being $\mathcal K=\mathcal K_X$ the canonical linear system on $X$, are the {\it Weierstrass points} of $X$. In this case $H(P):= \mathbb N\setminus \{j_0(P)+1,\ldots,j_{g-1}(P)+1\}$ is the {\it Weierstrass semigroup} at $P$. We write $H(P)=\{m_0(P)=0<m_1(P)<\ldots\}$, the element $m_i(P)$ being the {\it $i$th non-gap} at $P$. The curve is called {\it classical} iff the $\mathcal K$-orders are $0,1,\ldots,g-1$ (i.e. $H(P)=\{0,g+1,g+2,\ldots\}$ for $P\not\in{\rm supp}(R^{\mathcal K})$). To any $P\in X$ one also associates the {\it $i$th osculating plane} $L_i(P)$: via the identification $\mathcal D\cong \mathbb P(\mathcal D')^*$ each hyperplane $H$ in $\mathbb P(\mathcal D')$ correspond to a divisor $D_H\in \mathcal D$; then $L_i(P)$ is the intersection of the hyperplanes $H$ in $\mathbb P(\mathcal D)^*$ such that $v_P(D_H)\ge j_{i+1}(P)$. Its (projective) dimension is $i$. In terms of projective coordinates $L_i(P)$ can be described as follows: let $f_0,\ldots,f_r$ be a $(\mathcal D,P)$-hermitian base of $\mathcal D$, i.e. a $k$-base of $\mathcal D'$ such that $v_P(t^{v_P(E)}f_i)=j_i(P)$ for $i=0,\ldots,r$; $t$ being a local parameter at $P$. Then for $i=0,\ldots,r-1$ \begin{equation}\label{eq1.2} L_i(P)=H_{i+1}\cap\ldots\cap H_{r}\quad \mbox{with}\ D_{H_j}=E+{\rm div}(f_j),\ j=i+1,\ldots,r\, . \end{equation} \subsection{Frobenius orders}\label{1.2} In the remaining part of this paper the ground field $k$ will be the algebraic closure of a finite field $\mathbb F_q$ with $q$ elements. In this subsection we summarize some results from St\"ohr-Voloch's \cite[\S2]{sv}. We keep the assumptions and the notations of the preceding subsection and we suppose that $X$ and $\mathcal D$ are defined over $\mathbb F_q$. We let ${\rm Fr}_{X}$ denote the Frobenius morphism (relative to $\mathbb F_q$) on $X$. Then $X$ is equipped with {\it the $\mathbb F_q$-Frobenius divisor associated to $\mathcal D$}, namely $$ S=S^{\mathcal D}={\rm div}( {\rm det} \begin{pmatrix} f_0\circ{\rm Fr}_{X} & \ldots & f_r\circ{\rm Fr}_{X} \\ D^{\nu_0}_xf_0 & \ldots & D^{\nu_0}_xf_r \\ \vdots & \vdots & \vdots \\ D^{\nu_{r-1}}_xf_0 & \ldots & D^{\nu_{r-1}f_r} \end{pmatrix})+\sum_{i=0}^{r-1}\nu_i{\rm div}(dx)+(q+r)E\, , $$ where $x$ is a separating variable of $\mathbb F_q(X)|\mathbb F_q$, $f_0,\ldots,f_r$ is a $\mathbb F_q$-base of $\mathcal D'$, and $(\nu_0,\nu_1,\ldots,\nu_{r-1})$, called {\it the $\mathbb F_q$-Frobenius orders of $\mathcal D$}, is the minimum in the lexicographic order of the set $$ \{(j_0,\ldots,j_{r-1})\in \mathbb N^r: j_0<\ldots<j_{r-1};\ {\rm det} \begin{pmatrix} f_0\circ{\rm Fr}_{X} &\ldots & f_r\circ{\rm Fr}_{X}\\ D^{j_0}_xf_0 &\ldots & D^{j_0}_xf_r\\ \vdots &\vdots & \vdots \\ D^{j_{r-1}}_xf_0 & \ldots & D^{j_{r-1}f_r} \end{pmatrix} \ \neq 0\}\, . $$ One has $X(\mathbb F_q)\subseteq {\rm supp}(S)$, \begin{equation}\label{eq1.3} {\rm deg}(S)=(\sum_{i=0}^{r-1}\nu_i)(2g-2)+(q+r)d\, , \end{equation} $\nu_i=\epsilon_i$ for $i<I$, $\nu_i=\epsilon_{i+1}$ for $i\ge I$, where $I=I^{(\mathcal D,q)}$ is the smallest integer such that $(f_0\circ{\rm Fr}_{X},\ldots,f_r\circ{\rm Fr}_{X})$ is a $\mathbb F_q(X)$-linear combination of the vectors $(D^{\epsilon_i}_xf_0,\ldots,D^{\epsilon_i}_xf_r)$ with $i=0,\ldots,I$. For $P\in X(\mathbb F_q)$ one also has \begin{equation}\label{eq1.4} \nu_i\le j_{i+1}(P)-j_1(P)\qquad i=0,\ldots,r-1\qquad \text{and} \end{equation} \begin{equation}\label{eq1.5} v_P(S)\ge \sum_{i=0}^{r-1}(j_{i+1}(P)-\nu_i)\, . \end{equation} \subsection{A $\mathbb F_q$-divisor from the Zeta Function}\label{1.3} In this subsection we generalize \cite[Lemma 1.1]{fgt} and its corollaries. Let $X\!\mid\!\mathbb F_q$ be a curve and $h(t)=h_{X,q}(t)$ its {\it $h$-polynomial}, i.e. the characteristic polynomial of the Frobenius endomorphism ${\rm Fr}_{\mathcal J}$ of the Jacobian $\mathcal J$ (over $\mathbb F_q$) of $X$. We let $\prod_{i=1}^{T}h^{r_i}_i(t)$ denote the factorization over $\mathbb Z [t]$ of $h(t)$. Because of the semisimplicity of ${\rm Fr}_{\mathcal J}$ and the faithfully of the representation of endomorphisms of $\mathcal J$ on the Tate module (cf. \cite[Thm. 2]{ta}, \cite[VI, \S3]{l}), we then have \begin{equation*} \prod_{i=1}^{T}h_i({\rm Fr}_{\mathcal J})=0\quad \mbox{on}\ \mathcal J\, .\tag{$*$} \end{equation*} Throughout this subsection we set $$ \sum_{i=1}^{U }\alpha_i t^{U-i}+t^U:= \prod_{i=1}^{T}h_i(t)\, , $$ we assume that $X$ has at least one $\mathbb F_q$-rational point, say $P_0$, and set $$ \mathcal D=\mathcal D^{(X,q,P_0)}:= ||m|P_0|\qquad \text{with}\qquad m:=\prod_{i=1}^{T}h_i(1)\, . $$ As $f\circ{\rm Fr}_{X}={\rm Fr}_{\mathcal J}{\rm Fr}_{X}$, $f=f^{P_0}$ being the natural map from $X$ to $\mathcal J$ given by $P\mapsto [P-P_0]$, Eq $(*)$ is equivalent to \begin{equation}\label{eq1.6} \sum_{i=1}^{U}\alpha_i{\rm Fr}_{X}^{U-i}(P)+{\rm Fr}_{X}^U(P)\sim mP_0\, . \end{equation} This suggests the \smallskip {\bf Problem.} Study the relation among the $\mathbb F_q$-rational points, the Weierstrass points, the $\mathcal D$-Weierstrass points, and the support of the $\mathbb F_q$-Frobenius divisor associated to $\mathcal D$. \begin{lemma}\label{l1.1} \begin{enumerate} \item If $P, Q\in X(\mathbb F_q)$, then $mP\sim mQ$. In particular, $|m|$ is a non-gap at each $P\in X(\mathbb F_q)$ provided that $\# X(\mathbb F_q)\ge 2$. \item If ${\rm char}(\mathbb F_q)\nmid m$ and $\#X(\mathbb F_q)\ge 2g+3$, then there exists $P_1\in X(\mathbb F_q)$ such that $|m|-1$ and $|m|$ are non-gap at $P_1$. \end{enumerate} \end{lemma} \begin{proof} (1) It follows inmediately from (\ref{eq1.6}). (2) (The proof is inspired by \cite[Prop.\! 1]{stix}.) By item (1) (and ${\rm char }(\mathbb F_q)\nmid m$) there exists a separable morphism $x:X \to \mathbb P^1(\bar \mathbb F_q)$ with ${\rm div}(x)=|m|P_0-|m|Q$, $P_0, Q\in X(\mathbb F_q)$. Let $n$ denote the number of unramified rational points for $x$. By the Riemann-Hurwitz formula we find that $n\ge \#X(\mathbb F_q)-2g-2$ so that $n>0$ by the hypothesis on $\#X(\mathbb F_q)$. Thus there exists $\alpha \in \mathbb F_q$, $P_1\in X(\mathbb F_q)\setminus \{P_0, Q\}$ such that ${\rm div}(x-\alpha)=P_1+D-mQ$ with $P_1, Q \not\in {\rm supp}(D)$. Let $y\in \mathbb F_q(X)$ be such that ${\rm div}(y)=|m|Q-|m|P_1$ (cf. item (1)). Then from the rational function $(x-\alpha)y$ we obtain item (2). \end{proof} It follows that the definition of $\mathcal D$ is independent of $P_0\in X(\mathbb F_q)$ and the \begin{corollary}\label{cor1.1} \begin{enumerate} \item If $\#X(\mathbb F_q)\ge 2$, then $\mathcal D$ is base-point-free. \item If ${\rm char}(\mathbb F_q)\nmid m$ and $\#X(\mathbb F_q)\ge 2g+3$, then $\mathcal D$ is simple. \end{enumerate} \end{corollary} Let us assume further that $\# X(\mathbb F_q)\ge 2$ and that \begin{equation*} m>0,\quad \alpha_1, \alpha_U\ge 1\quad\text{and}\quad \alpha_{i+1}\ge \alpha_i,\ \text{for}\ i=1,\ldots, U-1.\tag{Z} \end{equation*} \begin{remark*} There exist $h$-polynomials that do not satisfy $(Z)$; see, e.g. \cite{carb}. \end{remark*} We set $r:={\rm dim}(\mathcal D)$, i.e. $m_r(P)=m$ for each $P\in X(\mathbb F_q)$ (cf. Lemma \ref{l1.1}(1)). We keep the notations of the preceding subsections. \begin{lemma}\label{l1.2} \begin{enumerate} \item The $(\mathcal D,P)$-orders for $P\in X(\mathbb F_q)$ are $j_i(P)=m-m_{r-i}(P)$, $i=0,\ldots,r$. \item $j_1(P)=1$ for $P\not\in X(\mathbb F_q)$. In particular, $\epsilon_1=1$. \item The numbers $1, \alpha_1,\ldots,\alpha_U$ are $\mathcal D$-orders, so that $r\ge U+1$. \item If ${\rm Fr}_{X}^i(P)\neq P$ for $i=1,\ldots, U+1$, then $\alpha_U$ is a non-gap at $P$. In particular $\alpha_U$ is a generic non-gap of $X$. \item If ${\rm Fr}_{X}^i(P)\neq P$ for $i=1,\ldots, U$, but ${\rm Fr}_{X}^{U+1}(P)=P$, then $\alpha_U-1$ is a non-gap at $P$. \item If $g\ge \alpha_U$, then $X$ is non-classical. \end{enumerate} \end{lemma} \begin{proof} Items (1), (2) and (3) can be proved as in \cite[Thm. 1.4(iii), Prop. 1.5(iii)]{fgt}. To prove (4), (5) and (6) we apply ${({\rm Fr}_{X})}_*$, as in \cite[IV, Ex.\! 2.6]{har}, to the equivalence in (\ref{eq1.6}). Then $$ \sum_{i=1}^{U}\alpha_i{\rm Fr}_{X}^{U-i}(P)+{\rm Fr}_{X}^U(P)\sim \alpha_1{\rm Fr}_{X}^U(P)+\sum_{i=1}^{U-1}\alpha_{i+1}{\rm Fr}_{X}^{U-i}(P) +{\rm Fr}_{X}^{U+1}(P) $$ so that $$ \alpha_UP\sim {\rm Fr}_{X}^{U+1}(P)+(\alpha_1-1){\rm Fr}_{X}^U(P)+ \sum_{i=1}^{U-1}(\alpha_{i+1}-\alpha_i){\rm Fr}_{X}^{U-i}(P)\, . $$ Now the results follow. \end{proof} \begin{corollary}\label{cor1.2} If $\# X(\mathbb F_q)>q(m-\alpha_U)+1$, then $j_{r-1}(P)<\alpha_U$ for each $P\in X(\mathbb F_q)$. \end{corollary} \begin{proof} By Lemma \ref{l1.2}(1), $j_{r-1}(P)=m-m_1(P)$, and by \cite[Thm. 1(b)]{le}, $\# X(\mathbb F_q)\le 1+qm_1(P)$. Then $j_{N-1}(P)\ge \alpha_U$ implies $\# X(\mathbb F_q)\le q(m-\alpha_U)+1$. \end{proof} \begin{corollary}\label{cor1.3} \begin{enumerate} \item $\epsilon_r=\nu_{r-1}=\alpha_U$; \item $X(\mathbb F_q)\subseteq {\rm supp}(R^{\mathcal D})$. \end{enumerate} \end{corollary} \begin{proof} (1) By (\ref{eq1.1})(b), $\epsilon_{r-1}\le j_{r-1}$. Since $\alpha_U$ is a $\mathcal D$-order (Lemma \ref{l1.2}(3)) we conclude that $\epsilon_r=\alpha_U$. Now from (\ref{eq1.6}) and (\ref{eq1.2}) we have that ${\rm Fr}_{X}(P)$ belong to the $(r-1)$-th osculating hyperplane; thus $\nu_{r-1}=\epsilon_r$. (2) By Lemma \ref{l1.2}(1) for each rational point $j_r(P)=m$. Since $m>\alpha_U$, the result follows from (\ref{eq1.1})(c). \end{proof} \begin{corollary}\label{cor1.4} If $\# X(\mathbb F_q)\ge q\alpha_U+1$, then $m_1(P)=q$ for each $P\in X(\mathbb F_q)$. \end{corollary} \begin{proof} Let $P\in X(\mathbb F_q)$. By (\ref{eq1.1})(b), applied to the canonical linear system, we have $m_1(P)\le m_1(Q)$, $Q$ being a generic point of $X$. Then $m_1(P)\le \alpha_U$ by Lemma \ref{l1.2}(4). On the other hand, by \cite[Thm. 1(b)]{le} and the hypothesis on $\#X(\mathbb F_q)$ we get $m_1(P)\ge \alpha_U$ and we are done. \end{proof} \section{Maximal curves}\label{2} In this section we shall be dealing with maximal curves over $\mathbb F_q$ or equivalently with curves over $\mathbb F_q$ whose $h$-polynomial is $(t+\sqrt{q})^{2g}$, $g>0$ being the genus of the curve. We set $\ell:=\sqrt{q}$. Then, by \S\ref{1.3}, each maximal curve $X\!\mid\!\mathbb F_{\ell^2}$ is equipped with the linear system $$ \mathcal D:=|(\ell+1)P_0| \qquad P_0\in X(\mathbb F_{\ell^2})\, , $$ which will be fixed throughout the entire section. Notice that $X$ satisfies the hypotheses in Lemma \ref{l1.1} and $(Z)$ in \S\ref{1.3}. By Corollary \ref{cor1.1}, $\mathcal D$ is simple and base-point-free; by Lemma \ref{l1.2}(3), ${\rm dim}(\mathcal D)\ge 2$; for each $P\in X$ relation (\ref{eq1.6}) reads (\cite[Corollary 1.2]{fgt}) \begin{equation}\label{eq2.1} \ell P+ {\rm Fr}_{X}(P)\sim (\ell+1)P_0\, . \end{equation} Then, for each $P\in X$ \begin{equation}\label{eq2.2} m_0(P)=0<m_1(P)<\ldots <m_n(P)\le \ell<m_{n+1}(P)\, , \end{equation} where $n+1:={\rm dim}(\mathcal D)$. We keep the notations of the preceding section. \subsection{Known results}\label{2.1} The results of this subsection have been noticed in \cite[\S1]{fgt}. 2.1.1. By Corollary \ref{cor1.3}(1), $\epsilon_{n+1}=\nu_n=\ell$; this together with \cite{ft} and \cite[Proof of Thm.\! 1]{hv} imply: $\nu_1=\ell\Leftrightarrow n+1=2$, and $\nu_1=1\Leftrightarrow n+1\ge 3$. 2.1.2. Let $P\in X(\mathbb F_{\ell^2})$. By Lemma \ref{l1.2}(1), $j_i(P)=\ell+1-m_{n+1-i}(P)$, $i=0,1,\ldots,n+1$. Then $j_{n+1}(P)=m_{n+1}(P)=\ell+1$. The case $i=\ell$ in (\ref{eq1.5}) gives $j_1(P)=1$ so that $m_n(P)=\ell$. 2.1.3. Let $P\not\in X(\mathbb F_{\ell^2})$. We assume $n+1\ge 3$ (the case $n+1=2$ has been studied in \cite{ft}; see also Theorem \ref{t2.1} here). >From (\ref{eq2.1}), $j_1(P)=1$, and $m_n(P)=\ell$ whenever ${\rm Fr}_{X}^2(P)\neq P$. Furthermore, by (\ref{eq1.1}) and \cite[Thm.\! 1]{ho}, $m_{n-1}(P)=\ell-1$ if $P$ is not a Weierstrass point of $X$. Set $m_i:=m_i(P)$, $u_0:=1$; let $u=u(P), u_i=u_i(P)\in {\bar\mathbb F_{\ell^2}}(X)$ such that ${\rm div}(u_i)=D_i-m_iP$, $P\not\in{\rm supp}(D_i)$, and ${\rm div}(u)=\ell P+{\rm Fr}_{X}(P)-(\ell+1)P_0$. Then \begin{equation}\label{eq2.3} (\ell+1)P_0+{\rm div}(uu_i)=D_i+{\rm Fr}_{X}(P)+(\ell-m_i)P\, , \end{equation} and so $0,1,\ell-m_n,\ldots,\ell-m_0$ are $(\mathcal D,P)$-orders. 2.1.4. The $\mathcal D$-orders. Let $\tilde m_i$ denote the $i$th non-gap at a generic point. Then, by \S2.1.3, the $\mathcal D$-orders are \begin{align*} & \epsilon_0=\ell-\tilde m_n<\epsilon_1=1=\ell-\tilde m_{n-1}<\ldots< \epsilon_{n-J}=\ell-\tilde m_J<\epsilon_{n-J+1}< \\ & \epsilon_{n-J+2}=\ell-\tilde m_{J-1}<\ldots<\epsilon_{n+1}=\ell-\tilde m_0\, , \end{align*} whit $J\in \mathbb N$, $1\le J\le n-1$. \subsection{The $\mathbb F_{\ell^2}$-Frobenius orders of $\mathcal D$.}\label{2.2} \begin{proposition*} With the notations of \S2.1.4, the $\mathbb F_{\ell^2}$-Frobenius orders of $\mathcal D$ are $$ \{\epsilon_0,\ldots,\epsilon_{n+1}\}\setminus\{\epsilon_{n-J+1}\}. $$ \end{proposition*} \begin{proof} For $P$ a generic point of $X$ let $u, u_i \in {\bar\mathbb F_{\ell^2}}(X)$ be as in \S2.1.3, and $v\in {\bar\mathbb F_{\ell^2}}(X)$ such that $$ (\ell+1)P_0+{\rm div}(v)=\epsilon_{n-J+1}P+D_v\qquad P\not\in {\rm supp}(D_v)\, . $$ >From this equation and (\ref{eq2.3}) we have that $ \{uu_n,uu_{n-1},\ldots,uu_J,v,uu_{J-1},\ldots,uu_0\}$ is a $(\mathcal D,P)$-hermitian base of $\mathcal D$. Hence, by (\ref{eq1.2}), $ {\rm Fr}_{X}(P)\in L_{n-J+1}(P)$. Now the result follows from the \begin{claim*} \quad ${\rm Fr}_{X}(P)\not\in L_{n-J}(P)$. \end{claim*} Indeed, if ${\rm Fr}_{X}(P)\in L_{n-J}(P)$ then ${\rm Fr}_{X}(P)\in {\rm supp}(D_v)$; hence, by (\ref{eq2.1}), we would have $\ell-\epsilon_{n-J+1}\in H(P)$-a contradiction. \end{proof} \begin{remark*} A slight modification of the above proof shows that each point $P\in {\rm supp}(S^\mathcal D)\setminus X(\mathbb F_{\ell^2})$ is a Weierstrass point of $X$ (\cite{gt}). \end{remark*} \subsection{The morphism associated to $\mathcal D$}\label{2.3} Let $\pi:X\to \mathbb P^{n+1}(\bar\mathbb F_{\ell^2})$ be the morphism associated to $\mathcal D$. \begin{proposition*}\quad $\pi$ is a closed embedding, i.e. $X$ is $\mathbb F_{\ell^2}$-isomorphic to $\pi(X)$. \end{proposition*} \begin{proof} By \cite[Prop.\! 1.9]{fgt}, we have to show that $m_n(P)=\ell$ for each $P\in X$. By \S2.1.2 we can assume that $P\not\in X(\mathbb F_{\ell^2})$. For such a $P$, suppose that $m_n(P)<\ell$; then, as $j_1(P)=1$ and $j_{n+1}(P)=\ell$, the $(\mathcal D,P)$-orders would be $$ 0,1=\ell-m_n(P),\ell-m_{n-1}(P),\ldots,\ell-m_1(P),\ell\, . $$ Hence (\ref{eq1.2}) and (\ref{eq2.3}) would imply ${\rm Fr}_{X}(P)\in L_1(P)$. On the other hand, the hyperplane corresponding to the function $uu_n(P)$ in \S2.1.3 contains $P$ (with multiplicity 1) and ${\rm Fr}_{X}(P)$; thus it contains $L_1(P)$. This is a contradiction because the multiplicity of $L_1(P)$ at $P$ is at least $j_2(P)=\ell-m_{n-1}(P)\ge 2$. \end{proof} Now \cite[Prop.\! 1.10]{fgt} can be state without the hypothesis on $\pi$: \begin{corollary*} Let $X\!\mid\!\mathbb F_{\ell^2}$ be a maximal curve of genus $g$ . For some $P\in X(\mathbb F_{\ell^2})$ suppose that there exist $a, b\in H(P)$ such that all non-gaps less than or equal to $\ell+1$ are generated by $a$ and $b$. Then $H(P)=\langle a,b\rangle$, so that $g=(a-1)(b-1)/2$. \end{corollary*} \subsection{Weierstrass points and maximal curves}\label{2.4} In this section we show that each $\mathbb F_{\ell^2}$-rational point of $X$ is a Weierstrass point of the curve provided that $g$ is large enough. First we notice that (\ref{eq2.2}) implies $g\ge \ell-n$ and that $$ g=\ell-n\quad \Leftrightarrow\quad \{\ell+1,\ell+2,\ldots,\}\subseteq H(P)\quad \forall P\in X\, . $$ Since $\ell$ is a non-gap for a non-Weierstrass point, cf. \S2.1.3, (\ref{eq2.2}) also implies (\cite[Prop. 1.7(i)]{fgt}) $$ \text{$X$ classical}\quad \Rightarrow\quad g=\ell-n. $$ We remark that $g=\ell-n$ does not characterize classical maximal curves; see e.g. \cite[Prop. 1.8]{fgt}. The following results are contained in the proof of \cite[Satz II.2.5]{rai}. \begin{lemma}\label{l2.1} Let $X\!\mid\!\mathbb F_{\ell^2}$ be a maximal curve of genus $g$ and $P$ a non-Weierstrass point of $X$. If $\ell+1\in H(P)$, then $\ell+1,\ldots,2\ell\in H(P)$. In particular, since $\ell\in H(P)$, $\{\ell+1,\ell+2,\ldots\}\subseteq H(P)$ and $g=\ell-n$. \end{lemma} \begin{proof} Let $i\in \{1,\ldots,\ell\}$ such that $\ell+i\not\in H(P)$; then $\binom{\ell+i-1}{\ell} \not\equiv 0 \pmod{{\rm char}(\mathbb F_{\ell^2})}$. Hence, by the $p$-adic criterion \cite[Corollary 1.9]{sv}, $\ell+1\not\in H(P)$. \end{proof} \begin{corollary}\label{c2.1}\quad $ g=\ell-n\quad \Leftrightarrow\quad \text{$\ell+1$ is a non-gap at a non-Weierstrass point of $X$}.$ \end{corollary} \begin{corollary}\label{c2.2} If $g>\ell-n$, then $$ X(\mathbb F_{\ell^2})\subseteq \text{set of Weierstrass points of $X$}. $$ \end{corollary} \begin{proof} It follows from the above corollary and \S2.1.2. \end{proof} \begin{remark*} There exists maximal curves with $g=\ell-n$ where no $\mathbb F_{\ell^2}$-rational point is Weierstrass, see e.g. the remark after Proposition \ref{p2.1}. The hypothesis $g>\ell-n$ is satisfied if $g\ge \max{(2,\ell-1)}$; indeed, $g=\ell-1\le \ell-n$ implies $n=1$, i.e. $g=(\ell-1)\ell/2$ (\cite{ft}) and so $g\le 1$, a contradiction. \end{remark*} \begin{remark*} Let $X\!\mid\!\mathbb F_{\ell^2}$ be non-hyperelliptic and maximal of genus $g$. Denote by $\mathcal W$ the set of Weierstrass points of $X$ ($={\rm supp}(R^{K_X})$). Corollary \ref{c2.2} implies $$ \# \mathcal W > \begin{cases} 16 & \text{if $g=3$},\\ 25 & \text{if $g=6$},\\ \max{3(g+2), 4(g-1)} & \text{if $g\neq 3, 6$}. \end{cases} $$ Hence, we can use Pflaum's \cite[Corollary 2.6, Proof of Theorem 1.6]{pf} to describe the isomorphism-class (over $\bar \mathbb F_{\ell^2}$) and the automorphism group ${\rm Aut}(X)$ (also over $\bar\mathbb F_{\ell^2}$) of $X$ via Weierstrass points. In fact, we conclude that the isomorphism-class of maximal curves is determinated by their constellations of Weierstrass points and that $$ {\rm Aut}(X)\cong\{A\in PGL(g,\bar\mathbb F_{\ell^2}): A\rho(\mathcal W)=\rho(\mathcal W)\}, $$ where $\rho:X\to \mathbb P^{g-1}(\bar\mathbb F_{\ell^2})$ is the canonical embedding. Notice that, as the morphism $\pi:X\to \mathbb P^{n+1}$ associated to $\mathcal D$ is an embedding (\S\ref{2.3}), (\ref{eq2.1}) implies $$ {\rm Aut}_{\mathbb F_{\ell^2}}(X)\cong \{A\in PGL(n+1,\mathbb F_{\ell^2}): A\pi(X)=\pi(X)\}\, . $$ \end{remark*} \subsection{On the genus of maximal curves}\label{2.5} It has been noticed in \cite{ft} that the genus $g$ of a maximal curve $X\!\mid\!\mathbb F_{\ell^2}$ satisfies $$ g\le (\ell-1)^2/4\qquad\text{or}\qquad g=(\ell-1)\ell/2\, , $$ which was conjectured by Stichtenoth and Xing (cf. \cite{stix}). Moreover, we have the \begin{theorem}\label{t2.1} The following statements are equivalent \begin{enumerate} \item $X$ is $\mathbb F_{\ell^2}$-isomorphic to $y^\ell+y=x^{\ell+1}$ (the Hermitian curve over $\mathbb F_{\ell^2}$); \item $X\!\mid\!\mathbb F_{\ell^2}$ maximal with $g>(\ell-1)^2/4$; \item $X\!\mid\!\mathbb F_{\ell^2}$ maximal with ${\rm dim}(\mathcal D)=2$. \end{enumerate} \end{theorem} It is well known that $y^\ell+y=x^{\ell+1}$ is a maximal curve over $\mathbb F_{\ell^2}$ of genus $(\ell-1)\ell/2$; $(2)\Rightarrow (3)$ follows by Castelnuovo's genus bound for curves in projective spaces \cite[Claim 1]{ft}; $(3)\Rightarrow (1)$ is the main result of \cite{ft}. Next we write a new proof of this implication. \begin{proof} $(3)\!\Rightarrow\! (1):$ By \S\ref{2.1}, $(\epsilon_0,\epsilon_1,\epsilon_2)=(0,1,\ell)$, $(\nu_0,\nu_1)=(0,\ell)$, and $(j_0(P),j_1(P),j_2(P))=(0,1,\ell+1)$ for each $P\in X(\mathbb F_{\ell^2})$. Hence (\ref{eq1.1})(a)(c) imply $g=(\ell-1)\ell/2$. Let $x, y\in \mathbb F_{\ell^2}(X)$ with ${\rm div}_{\infty}(x)=\ell P_0$ and ${\rm div}_{\infty}(y)=(\ell+1)P_0$. Then $H(P_0)=\langle \ell,\ell+1\rangle$ and so ${\rm div}(dx)=(2g-2)P_0\ (*)$, because $H(P_0)$ is symmetric. By $\nu_1=\ell$ we have an equation of type (cf. \S\ref{1.2}) \begin{equation}\label{eq2.4} y^{\ell^2}-y=f(x^{\ell^2}-x)\ , \end{equation} with $f:=D^{1}y$ (derivation with respect to $x$); by $\epsilon_2=\ell$ we have the following two-rank matrices (cf. \S\ref{1.1}) $$ \left( \begin{array}{ccc} 1 & x & y\\ 0 & 1 & D^{(1)}y\\ 0 & 0 & D^{(j)}y \end{array} \right), \qquad 2\le j<\epsilon_2=\ell\, . $$ By $(*)$ and (\ref{eq2.4}), ${\rm div}_{\infty}(f)=\ell^2P_0$. Now $D^{(j)}y=0$ for $2\le j<\ell$ and (\ref{eq2.4}) imply $D^{(j)}f=0$ for $1\le j<\ell$. Thus, by \cite[Satz 10]{hasse}, there exists $f_1\in \mathbb F_{\ell^2}(X)$ such that $f=f_1^{\ell}$. Therefore $f_1=ax+b$ with $a, b\in \mathbb F_{\ell^2}$, $a\neq0$, and after some $\mathbb F_q$-linear transformations we obtain an equation of type $$ y_1^{\ell}+y_1-x_1^{\ell+1}=(y_1^\ell+y_1-x_1^{\ell+1})^{\ell}\, , $$ with $x_1, y_1\in \mathbb F_q(X)$. Now the proof follows. \end{proof} \begin{remark*} Let $X$ be the Hermitian curve over $\mathbb F_{\ell^2}$. From the above proof we have $\mathcal K_X=(\ell-2)\mathcal D$ and $(j_0(P),j_1(P),j_2(P))=(0,1,\ell)$ for each $P\not\in X(\mathbb F_{\ell^2})$. Then the $(\mathcal K_X,P)$-orders contains $ \{a+b\ell: a,b\ge 0, a+b\le \ell-2\}$ (resp. $\{a+b(\ell+1): a,b\ge 0, a+b\le \ell-2\}$) if $P\not\in X(\mathbb F_{\ell^2})$ (resp. $P\in X(\mathbb F_{\ell^2})$). Since these sets have cardinality equal to $g=(\ell-1)\ell/2$, these are the $(\mathcal K_X,P)$-orders; hence $$ X(\mathbb F_{\ell^2})={\rm supp}(R^\mathcal D)={\rm supp}(R^{\mathcal K_X} (= \text{set of Weierstrass points of $X$})\, , $$ and we have another proof of the fact that $X$ is non-classical (compare with Lemma \ref{l1.2}(6)). The above computations have been carried out in \cite{gv}. We mention that the first examples of non-classical curves were obtained from certain Hermitian curves (see \cite{sch}). \end{remark*} Now let us consider the following property for the maximal curve $X\!\mid\!\mathbb F_{\ell^2}$ of genus $g$ (recall that $n+1={\rm dim}(\mathcal D)$): \begin{equation}\label{eq2.5} \exists P_1\in X(\mathbb F_{\ell^2})\quad \exists m\in H(P_1)\quad \text{such that $mn\le \ell+1$}\, . \end{equation} Then $mn=\ell+1$ or $mn=\ell$. In both cases the hypothesis of the corollary in \S\ref{2.3} is satisfied; in particular, $g=(\ell-1)(m-1)/2$ or $g=\ell(m-1)/2$. {\bf Case $mn=\ell+1$.} This occurs iff $X$ is $\mathbb F_{\ell^2}$-isomorphic to $y^\ell+y=x^{(\ell+1)/n}$ \cite[Thm. 2.3]{fgt}) (so that $g= \frac{\ell-1}{2}(\frac{\ell+1}{n}-1)$). Hence there exists maximal curves of genus $(\ell-1)^2/4$ and indeed, $y^\ell+y=x^{(\ell+1)/2}$ is the unique maximal curve (up to $\mathbb F_{\ell^2}$-isomorphism) having such a genus (\cite[Thm. 3.1]{fgt}). Before we consider the case $mn=\ell$ we prove an analogue of $(1)\Leftrightarrow (2)$ of Theorem \ref{t2.1}. \begin{proposition}\label{p2.1} Let $X\!\mid\!\mathbb F_{\ell^2}$ be a maximal curve of genus $g$ and assume that $\ell$ is odd. Then the following statements are equivalent \begin{enumerate} \item $X$ is $\mathbb F_{\ell^2}$-isomorphic to $y^\ell+y=x^{(\ell+1)/2}$; \item $(\ell-1)(\ell-2)/4<g\le(\ell-1)^2/4$. \end{enumerate} Item (1) (or (2)) implies\quad $3.\, \, {\rm dim}(\mathcal D)=3$. \end{proposition} \begin{proof} We already noticed that $(1)\Rightarrow (2)$ and $(1)\Rightarrow (3)$. That $(2)\Rightarrow (3)$ follows by Castelnuovo's genus bound for curves in projective spaces \cite{c}, \cite[p.\! 116]{acgh}, \cite[Corollary 2.8]{ra}. $(2)\Rightarrow (1):$ The cases $\ell\le 5$ are trivial, so let $\ell>5$. According to \cite[Thm.\! 2.3]{fgt}, we look for a rational point $P$ such that there exists $m\in H(P)$ with $2m=\ell+1$. Let $m_1:=m_1(P)<\ell<\ell+1$ be the first three positive non-gaps at $P\in X(\mathbb F_{\ell^2})$. By \S2.1.2 the $(\mathcal D,P)$-orders are $0,1,j=\ell+1-m_1,\ell+1$. Notice that $\ell$ odd implies $2m_1\ge \ell+1$ and hence that $2j\le \ell+1$. Set $2\mathcal D:=|2(\ell+1)P_0|$; ${\rm dim}(\mathcal D)=3$ implies ${\rm dim}(2\mathcal D)\ge 8$; the lower bound on $g$ implies (once again via Castelnuovo's bound) ${\rm dim}(2\mathcal D)=8$. The $(2\mathcal D,P)$-orders ($P\in X(\mathbb F_{\ell^2})$) contains the set $\{0,1,2,j,j+1,2j,\ell+1,\ell+2,\ell+j+1,2\ell+2\}$; therefore ${\rm dim}(2\mathcal D)=8$ implies $j=2$ (i.e. $m_1(P)=\ell-1$) or $2j=\ell+1$ (i.e. $m_1(P)=(\ell+1)/2$) and we have to show that it is not possible to have $m_1(P)=\ell-1$ for each $P\in X(\mathbb F_{\ell^2})$. Suppose that $m_1(P)=\ell-1$ for each $P\in X(\mathbb F_{\ell^2})$. Then the $\mathcal D$-orders are $0,1,2,\ell$ and so $v_P(R_1)=1$ for each $P\in X(\mathbb F_{\ell^2})$ ($R_1$ being the $\mathcal D$-ramification divisor). Then, by (\ref{eq1.1}), $$ {\rm deg}(R_1)-\#X(\mathbb F_{\ell^2})=3(2g-2)-(\ell-3)(\ell+1)\, . $$ \begin{claim*} \quad For each $P\in {\rm supp}(R_1)\setminus X(\mathbb F_{\ell^2})$ the $(\mathcal D,P)$-orders are $0,1,(\ell+1)/2,\ell$. \end{claim*} In fact, for such a $P$ the $(\mathcal D,P)$-orders are $0,1,i,\ell$ with $i=i(P)>2$, and $$ \{0,1,2,i,i+1,2i,\ell,\ell+1,\ell+i,2\ell\} $$ is contained in the $(2\mathcal D,P)$-orders; thus ${\rm dim}(2\mathcal D)=8$ implies $i\in\{(\ell+1)/2,\ell-1\}$. Suppose that $i=\ell-1$; by (\ref{eq2.1}) there exists $Q_1, Q_2\in X\setminus\{P\}$ such that $P+{\rm Fr}_{X}(P)\sim Q_1+Q_2$, i.e. $X$ is hyperelliptic. This implies $g\le (\ell-1)/2$ (see e.g. \cite[Thm. 1(b)]{le}) and from the hypothesis on $g$ that $\ell< 4$, a contradiction. By the claim and (\ref{eq1.1})(d), for each $P\in {\rm supp}(R_1)\setminus X(\mathbb F_{\ell^2})$, $v_P(R_1)=(\ell-3)/2$ and $$ A:=\#({\rm supp}(R_1)\setminus X(\mathbb F_{\ell^2}))=\frac{6(2g-2)}{\ell-3}-2(\ell+1)\, . $$ With the above computations we analize $(2\mathcal D,P)$-orders for $P\in {\rm supp}(R_1)$. We have: $$ \text{$(2\mathcal D,P)$-orders}\ = \begin{cases} 0,1,2,3,4,\ell+1,\ell+2,\ell+3,2\ell+2 & \text{if $P\in X(\mathbb F_{\ell^2})$};\\ 0,1,2,(\ell+1)/2,(\ell+3)/2,\ell,\ell+1,(3\ell+1)/2,2\ell & \text{if $P\not\in X(\mathbb F_{\ell^2})$}. \end{cases} $$ Denote by $R_2$ the $2\mathcal D$-ramification divisor. Being $0,1,2,3,4,\ell,\ell+1,\ell+2,2\ell$ the $2\mathcal D$-orders we then have $$ v_P(R_2)\ge \begin{cases} 5 & \text{if $P\in X(\mathbb F_{\ell^2})$};\\ (3\ell-13)/2 & \text{if $P\in {\rm supp}(R_1)\setminus X(\mathbb F_{\ell^2})$}. \end{cases} $$ Then, again by (\ref{eq1.1}), $$ {\rm deg}(R_2)=(5\ell+13)(2g-2)+18(\ell+1)\ge 5\# X(\mathbb F_{\ell^2})+\frac{3\ell-13}{2} A\, , $$ which implies $2g-2\ge (\ell-3)(\ell+1)/2$, i.e. $2g-2=(\ell-3)(\ell+1)/2$ due to the upper bound on $g$. By \cite[Thm.\! 3.1]{fgt} we then conclude that $A=0$, i.e. $2g-2=(\ell-3)(\ell+1)/3$, a contradiction. This complete the proof. \end{proof} \begin{remark*} Let $X\!\mid\!\mathbb F_{\ell^2}$ be maximal of genus $g$ and suppose that $(\ell-1)(\ell-2)/6<g\ge (\ell-1)^2/4$. If $X \not\cong y^\ell+y=x^{(\ell+1)/2}$, then $(\ell-1)(\ell-2)/6<g\le (\ell-1)(\ell-2)/4$ by the last proposition. Cossidente and Korchmaros \cite{ck} constructed a maximal curve $X\!\mid\!\mathbb F_{\ell^2}$ with $g=(\ell+1)(\ell-2)/6$ and $\ell\equiv 2\pmod{3}$. By Castelnuovo's genus bound, the linears system $\mathcal D$ of this curve satisfies ${\rm dim}(\mathcal D)=3$; so this example shows that $(3)$ does not imply $(1)$. \end{remark*} \begin{remark*} For $\ell\equiv 3\pmod{4}$ there are at least two non $\bar\mathbb F_{\ell^2}$-isomorphism maximal curves over $\mathbb F_{\ell^2}$ with $g=\frac{\ell-1}{2}(\frac{\ell+1}{4}-1)$ and ${\rm dim}(\mathcal D)=5$, namely $$ (1)\ \ y^\ell+y=x^{(\ell+1)/4}\qquad\text{and}\qquad (2)\ \ x^{(\ell+1)/2}+y^{(\ell+1)/2}=1. $$ Indeed, the curve in (2) admits points $P$ with $H(P)=\langle (\ell-1)/2,(\ell+1)/2\rangle$ (e.g. $P$ over a root of $x^{\ell+1}=1$) and is well known that such semigroups cannot be realized by (1) (\cite{gv}). These examples show that one cannot expect the uniqueness of a maximal curve just by means of a given genus. It also shows that the hypothesis on non-gaps of \cite[Prop.\! 1.10, Thm.\! 2.3]{fgt} cannot be relaxed. The curve (2) have been considered by Hirschfeld and Korchmaros \cite{hk}. They noticed an interesting bound for the number of rational points of a curve; the curve in (2) attains such a bound. \end{remark*} \begin{remark*} In view of the above examples and \cite{ck}'s letter is reasonable to make the following conjectures. Let $X\!\mid\!\mathbb F_{\ell^2}$ be a maximal curve of genus $g$. (1) Let $\ell$ be odd. If $\ell\not\equiv 2\pmod{3}$, then $(\ell-1)(\ell-2)/6 <g\le (\ell-1)^2/4$ iff $g=(\ell-1)^2/4$ (i.e. $X$ is $\mathbb F_{\ell^2}$-isomorphic to $y^\ell+y=x^{(\ell+1)/2}$). If $\ell\equiv 2\pmod{3}$, then $(\ell-1)(\ell-2)/6<g\le (\ell-1)^2$ iff $g=(\ell+1)(\ell-2)/6$ or $g=(\ell-1)^2/4$. (2) With the exception of finitely many $\ell$'s and if $\ell\equiv 2 \pmod{3}$, then $(\ell-1)(\ell-3)/8<g\le (\ell-1)(\ell-2)/6$ iff $X$ is $\mathbb F_{\ell^2}$-isomorphic to $y^\ell+y=x^{(\ell+1)/3}$ (in particular $g=(\ell-1)(\ell-2)/6$). \end{remark*} {\bf Case $mn=\ell$.} Now we assume (\ref{eq2.5}) with $mn=\ell$. To begin we notice that the quotient of the Hermitian curve by a certain automorphism has a plane model of type $F(y)=x^{\ell+1}$, $F$ being an additive polynomial. These curves provide examples of maximal curves for this case. It has been conjectured in \cite{fgt} that $X$ is $\mathbb F_{\ell^2}$-isomorphic to the above plane model with ${\rm deg}(F)=m$; the fact that $g=\ell(m-1)/2$ may provide evidence for this conjecture. Next we state another proof of this fact, where is implicitely outlined a method to find a plane model for $X$: By Theorem \ref{t2.1} we can assume $n>1$. Let $x,y\in \mathbb F_{\ell^2}(X)$ such that ${\rm div}_{\infty}(x)=m$ and ${\rm div}_{\infty}(y)=\ell+1$. \begin{claim*} For each $\alpha\in \mathbb F_{\ell^2}$, $\# x^{-1}(\alpha)=m$ and $x^{-1}(\alpha)\subseteq X(\mathbb F_{\ell^2})$. \end{claim*} This implies $g=\ell(m-1)/2$ because ${\rm deg}({\rm div}(x^{\ell^2}-x))=0$ gives $ \ell^2+2\ell g=\ell^2m$.\newline The claim follows from two facts: \begin{fact}\label{f1} For each $P\neq P_1$, $\#x^{-1}(x(P))=m$. \end{fact} \begin{proof} {\it (Fact \ref{f1})} Let $P\neq P_1$ and for $x(P)=\alpha \in \bar\mathbb F_{\ell^2}$ set $e=v_P(x-\alpha)$. We have to show that $e=1$. Writing ${\rm div}(x-\alpha)=eP+D_P-mP_1$ with $P\not\in{\rm supp}(D_P)$, we then see that $e,\ldots,en$ are $(\mathcal D,P)$-orders. If $e>1$, then $j_{n+1}(P)=en$ because 1 is a $(\mathcal D,P)$-orders (cf. \S\ref{2.1}). This implies $P\not\in X(\mathbb F_{\ell^2})$ and so $e=m$ because $\ell=mn$. Consequently $mP\sim mP_1$ so that $\ell P\sim\ell P_1$. Then by (\ref{eq2.1}) we get ${\rm Fr}_{X}(P)\sim P_1$, a contradiction because $g>0$. This finish the proof of Fact 1. \end{proof} Let $P\neq P_1$. From the above proof, the $(\mathcal D,P)$-orders are $0,1,\ldots,n,\ell+1$ (resp. $0,1,\ldots,n,\ell$) if $P\in X(\mathbb F_{\ell^2})$ (resp. $P\not\in X(\mathbb F_{\ell^2})$). Hence the $\mathcal D$-orders are $0,1,\ldots,n,\ell$ so that ${\rm supp}(R^{\mathcal D})=X(\mathbb F_{\ell^2})$ with $v_P(R^{\mathcal D})=1$, $P\in X(\mathbb F_{\ell^2})\setminus\{P_1\}$ , cf. (\ref{eq1.1}). Now the morphism $\pi$ associated to $\mathcal D$ can be defined by $(1:x:\ldots:x^n:y)$; so $v_P(D^\ell y)=1$ for $P\in X(\mathbb F_{\ell^2})\setminus P_1$ and $v_P(D^\ell y)=0$ for $P\not\in X(\mathbb F_{\ell^2})$ (derivation with respect to $x$); cf. \S\ref{1.1}. \begin{fact}\label{f2} There exists $f\in \mathbb F_{\ell^2}(X)$ regular outside $P_1$ such that $D^\ell y= f(x^{\ell^2}-x)$. \end{fact} \begin{proof} {\it (Fact \ref{f2})} By (\ref{eq1.4}) the $\mathbb F_{\ell^2}$-Frobenius orders are $0,1,\ldots,n-1,\ell$. Hence (cf. \S\ref{1.2}) $$ {\rm det} \begin{pmatrix} 1 & x^{\ell^2} & \ldots & x^{\ell^2 n} & y^{\ell^2}\\ 1 & x & \ldots & x^n & y \\ 0 & D^1x & \ldots & D^1x^n & D^1y \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & D^nx & \ldots & D^nx^n & D^n \end{pmatrix} =0 $$ and so $$ y-y^{\ell^2}=\sum_{i=1}^{n}(x^i-x^{i\ell^2 })h_i $$ with $h_i\in \mathbb F_{\ell^2}(X)$ regular in $X\setminus\{P_1\}$ for each $i$. Fact \ref{f2} now follows by applying $D^\ell$ to the above equation and using the following properties: \begin{itemize} \item For $r$ a power of a prime we have $D^j f^r=(D^{j/r}f)^r$ if $r\mid j$ and $D^j f^r=0$ otherwise; \item the fact that $n$ is a power of a prime implies $D^i x^n=0$ for $i=1,\ldots,n-1$; \item $D^i\circ D^j=\binom{i+j}{i}D^{i+j}$\, . \end{itemize} \end{proof} \section{On the Deligne-Lusztig curve associated to the Suzuki group}\label{3} In this section we prove Theorem \ref{B} stated in the introduction. Throughout we let $q_0:=2^s$, $q:=2q_0$, $X\!\mid\!\mathbb F_q$ a curve of genus $g$ with $$ g=q_0(q-1)\qquad \text{and}\qquad \# X(\mathbb F_q)=q^2+1\, . $$ Then, by Serre-Weil's explicit formulae (cf. \cite{se}, \cite{h}), the $h$-polynomial of $X$ is $(t^2+2q_0t+q)^g$. Hence, by \S\ref{1.3}, $X$ is equipped with the base-point-free simple linear system $\mathcal D:=|(q+2q_0+1)P_0|$, with $P_0\in X(\mathbb F_q)$. Here for each $P\in X$, (\ref{eq1.6}) reads \begin{equation}\label{eq3.1} qP+2q_0{\rm Fr}_{X}(P)+{\rm Fr}_{X}^2(P)\sim (q+2q_0+1)P_0\, . \end{equation} We notice that $X$ satisfies the hypothesis in Lemma \ref{l1.1} and $(Z)$ in \S\ref{1.3}. As a first consequence we have the \begin{lemma}\label{l3.1} Let $X$ be as above. Then for each $P\in X(\mathbb F_q)$, $m_1(P)=q$. \end{lemma} Next we apply \cite{sv} to $\mathcal D$; we keep the notation in \S\ref{1} and set $r:={\rm dim}(\mathcal D)$. The key property of $\mathcal D$ will be the fact that ${\rm Fr}_{X}(P)$ belongs to the tangent line for $P$ generic (Lemma \ref{l3.4}(1)). For $P\in X(\mathbb F_q)$, Lemma \ref{l1.2}(1) gives $m_r(P)=q+2q_0+1$ and \begin{equation}\label{eq3.2} j_i(P)=m_r(P)-m_{r-i}(P)\quad \mbox{for}\ \ i=0,\ldots,r\, . \end{equation} This together with Lemma \ref{l3.1} imply \begin{equation}\label{eq3.3} j_r=(P)=q+2q_0+1,\quad j_{r-1}(P)=2q_0+1\qquad (P\in X(\mathbb F_q))\, . \end{equation} By Lemma \ref{l1.2}(3) $1, 2q_0, q$ are $\mathcal D$-orders, so $r\ge 3$ and $\epsilon_1=1$. By Corollary \ref{cor1.3}(1) and (\ref{eq1.1})(b), \begin{equation}\label{eq3.4} \epsilon_r=\nu_{r-1}=q\qquad\text{and}\qquad 2q_0\le\epsilon_{r-1}\le 1+2q_0\, . \end{equation} \begin{lemma}\label{l3.2}\quad $\epsilon_{r-1}=2q_0$. \end{lemma} \begin{proof} Suppose that $\epsilon_{r-1}> 2q_0$. Then, by (\ref{eq3.4}), $\epsilon_{r-2}=2q_0$ and $\epsilon_{r-1}=2q_0+1$. By (\ref{eq1.4}) and (\ref{eq3.3}), $\nu_{r-2}\le 2q_0=\epsilon_{r-2}$. Thus the $\mathbb F_q$-Frobenius orders of $\mathcal D$ would be $\epsilon_0,\epsilon_1,\ldots,\epsilon_{r-2}$, and $\epsilon_r$. By (\ref{eq1.5}) and (\ref{eq1.4}), for each $P\in X(\mathbb F_q)$ \begin{equation}\label{eq3.5} v_P(S)\ge \sum_{i=1}^{r}(j_i(P)-\nu_{i-1})\ge (r-1)j_1(P)+1+2q_0\ge r+2q_0\ , \end{equation} Thus ${\rm deg}(S)\ge (r+2q_0)\#X(\mathbb F_q)$, and from (\ref{eq1.3}), the identities $2g-2=(2q_0-2)(q+2q_0+1)$ and $\#X(\mathbb F_q)=(q-2q_0+1)(q+2q_0+1)$ we obtain $$ \sum_{i=1}^{r-2}\nu_i=\sum_{i=1}^{r-2}\epsilon_i\ge (r-1)q_0\, . $$ Now, as $\epsilon_i+\epsilon_j\le \epsilon_{i+j}$ for $i+j\le r$ \cite[Thm. 1]{e}, then we would have $$ (r-1)\epsilon_{r-2}\ge 2\sum_{i=0}^{r-2}\epsilon_i\, , $$ and hence $\epsilon_i+\epsilon_{r-2-i}=\epsilon_{r-2}$ for $i=0,\ldots,r-2$. In particular, $\epsilon_{r-3}=2q_0-1$ and by the $p$-adic criterion (cf. \cite[Corollary 1.9]{sv} we would have $\epsilon_i=i$ for $i=0,1,\ldots,r-3$. These facts imply $r=2q_0+2$. Finally, we are going to see that this is a contradiction via Castelnuovo's genus bound \cite{c}, \cite[p.\! 116]{acgh}, \cite[Corollary 2.8]{ra}. Castelnuovo's formula applied to $\mathcal D$ implies $$ 2g=2q_0(q-1)\le \frac{(q+2q_0-(r-1)/2)^2}{r-1}\, . $$ For $r=2q_0+2$ this gives $2q_0(q-1)< (q+q_0)^2/2q_0=q_0q+q/2+q_0/2$, a contradiction. \end{proof} \begin{lemma}\label{l3.3} There exists $P_1\in X(\mathbb F_q)$ such that $$ \left\{ \begin{array}{ll} j_1(P_1)=1 & {} \\ j_i(P_1)=\nu_{i-1}+1 & \mbox{if}\ i=2,\ldots, r-1. \end{array}\right. $$ \end{lemma} \begin{proof} By (\ref{eq3.5}), it is enough to show that there exists $P_1\in X(\mathbb F_q)$ such that $v_{P_1}(S)=r+2q_0$. Suppose that $v_P(S)\ge r+2q_0+1$ for each $P\in X(\mathbb F_q)$. Then by (\ref{eq1.3}) we would have that $$ \sum_{i=0}^{r-1}\nu_i \ge q+rq_0+1\ , $$ and, as $\epsilon_1=1$, $\nu_{r-1}=q$ and $\nu_i\le \epsilon_{i+1}$, that $$ \sum_{i=0}^{r-1}\epsilon_i \ge rq_0+2\, . $$ By \cite[Thm. 1]{e} (or \cite{ho}), we then would conclude that $r\epsilon_{r-1}\ge 2rq_0+4$, i.e. $\epsilon_{r-1}>2rq_0$, a contradiction with the previous lemma. \end{proof} \begin{lemma}\label{l3.4} \begin{enumerate} \item \ $\nu_1>\epsilon_1=1$. \item \ $\epsilon_2$ is a power of two. \end{enumerate} \end{lemma} \begin{proof} Statement (2) is consequence of the $p$-adic criterion \cite[Corollary 1.9]{sv}. Suppose that $\nu_1=1$. Then by Lemma \ref{l3.1}, Lemma \ref{l3.3}, (\ref{eq3.3}) and (\ref{eq3.2}) there would be a point $P_1\in X$ such that $H(P_1)$ would contain the semigroup $H:=\langle q, q+2q_0-1, q+2q_0, q+2q_0+1\rangle $. Then $g\le \#(\mathbb N\setminus H)$, a contradiction as follows from the remark below. \end{proof} \begin{remark*} Let $H$ be the semigroup defined above. We are going to show that $\tilde g:= \#(\mathbb N \setminus H)=g-q_0^2/4$. To begin, notice that $L:=\cup_{i=1}^{2q_0-1}L_i$ is a complete system of residues module $q$, where $$ \begin{array}{lll} L_i & = & \{iq+i(2q_0-1)+j: j=0,\ldots,2i\}\quad \mbox{if}\ \ 1\le i\le q_0-1,\\ L_{q_0} & = & \{q_0q+q-q_0+j:j=0,\ldots,q_0-1\},\\ L_{q_0+1} & = & \{(q_0+1)q+1+j:j=0,\ldots,q_0-1\},\\ L_{q_0+i} & = & \{(q_0+i)q+(2i-3)q_0+i-1+j: j=0,\ldots,q_0-2i+1\}\cup\\ & & \{(q_0+i)q+(2i-2)q_0+i+j: j=0,\ldots q_0-1\}\quad \mbox {if}\ \ 2\le i\le q_0/2,\\ L_{3q_0/2+i} & = & \{(3q_0/2+i)q+(q_0/2+i-1)(2q_0-1)+q_0+2i-1+j:\\ & & j=0,\ldots,q_0-2i-1\}\quad \mbox {if}\ \ 1\le i\le q_0/2-1. \end{array} $$ Moreover, for each $\ell \in L$, $\ell \in H$ and $\ell-q\not\in H$. Hence $\tilde g$ can be computed by summing up the coefficients of $q$ from the above list (see e.g. \cite[Thm. p.3]{sel}), i.e. $$ \begin{array}{lll} \tilde g & = & \sum_{i=1}^{q_0-1}i(2i+1)+q_0^2+(q_0+1)q_0+ \sum_{i=2}^{q_0/2}(q_0+i)(2q_0-2i+2)+\\ & & \sum_{i=1}^{q_0/2-1}(3q_0/2+i)(q_0-2i)=q_0(q-1)-q_0^2/4\, . \end{array} $$ \end{remark*} In the remaining part of this section let $P_1\in X(\mathbb F_q)$ be a point satisfying Lemma \ref{l3.3}; we set $m_i:= m_i(P_1)$ and denote by $v$ the valuation at $P_1$. The item (1) of the last lemma implies $\nu_i=\epsilon_{i+1}$ for $i=1,\ldots, r-1$. Therefore from (\ref{eq3.2}), (\ref{eq3.3}) and Lemma \ref{l3.3}, \begin{equation}\label{eq3.6} \left\{\begin{array}{ll} m_i=2q_0+q-\epsilon_{r-i} & \mbox{if}\ i=1,\ldots r-2\\ m_{r-1}=2q_0+q,\ \ m_r=1+2q_0+q. & {} \end{array}\right. \end{equation} Let $x, y_2,\ldots, y_r\in \mathbb F_q(X)$ be such that ${\rm div}_{\infty}(x)=m_1P_1$, and ${\rm div}_{\infty} (y_i)=m_i P_1$ for $i=2,\ldots, r$. The fact that $\nu_1>1$ means that the following matrix has rank two (cf. \S\ref{1.2}) $$ \left( \begin{array}{ccccc} 1 & x^q & y_2^q &\ldots &y_r^q\\ 1 & x & y_2 &\ldots &y_r\\ 0 & 1 & D^{(1)}y_2 &\ldots& D^{(1)}y_r \end{array} \right)\, . $$ In particular, \begin{equation}\label{eq3.7} y_i^q-y_i= D^{(1)}y_i(x^q-x) \quad \text{for}\ \ i=2,\ldots, r. \end{equation} \begin{lemma}\label{l3.5} \begin{enumerate} \item For $P\in X(\mathbb F_q)$, the divisor $(2g-2)P$ is canonical, i.e. the Weierstrass semigroup at $P$ is symmetric. \item Let $m\in H(P_1)$. If $m<q+2q_0$, then $m\le q+q_0$. \item For $i=2,\ldots,r$ there exists $g_i\in \mathbb F_q(X)$ such that $ D^{(1)}y_i=g_i^{\epsilon_2}$. Furthermore, ${\rm div}_{\infty}(g_i)=\frac{qm_i-q^2}{\epsilon_2}P_1$. \end{enumerate} \end{lemma} \begin{proof} (1) Since $2g-2=(2q_0-2)(q+2q_0+1)$, by (\ref{eq3.1}) we can assume $P=P_1$. Now the case $i=r$ of Eqs. (\ref{eq3.7}) implies $v(dx)=2g-2$ and we are done. (2) By (\ref{eq3.6}), $q, q+2q_0$ and $q+2q_0+1\in H(P_1)$. Then the numbers $$ (2q_0-2)q+q-4q_0+j\qquad j=0,\ldots,q_0-2 $$ are also non-gaps at $P_1$. Therefore, by the symmetry of $H(P_1)$, $$ q+q_0+1+j\qquad j=0,\ldots,q_0-2 $$ are gaps at $P_1$ and the proof follows. (3) Set $f_i:= D^{(1)}y_i$. By the product rule applied to (\ref{eq3.7}),\\ $D^{(j)}y_i=(x^q-x)D^{(j)}f_i+D^{(j-1)}f_i$ for $1\le j<q$. Then, $D^{(j)}f_i=0$ for $1\le j<\epsilon_2$, because the matrices $$ \left( \begin{array}{ccccc} 1 & x & y_2 &\ldots &y_r\\ 0 & 1 & D^{(1)}y_2 &\ldots& D^{(1)}y_r\\ 0 & 0 & D^{(j)}y_2 &\ldots& D^{(j)}y_r \end{array} \right), \quad 2\le j<\epsilon_2 $$ have rank two (cf. \S\ref{1.1}). Consequently, as $\epsilon_2$ is a power of 2 (Lemma \ref{l3.4}(2)), by \cite[Satz 10]{hasse}, $f_i=g_i^{\epsilon_2}$ for some $g_i\in \mathbb F_q(X)$. Finally, from the proof of item (1) we have that $x-x(P)$ is a local parameter at $P$ if $P\neq P_1$. Then, by the election of the $y_i$'s, $g_i$ has no pole but in $P_1$, and from (\ref{eq3.7}), $v(g_i)=-(qm_i-q^2)/\epsilon_2$. \end{proof} \begin{lemma}\label{l3.6}\quad $r=4$ and $\epsilon_2=q_0$. \end{lemma} \begin{proof} We know that $r\ge 3$; we claim that $r\ge 4$; in fact, if $r=3$ we would have $\epsilon_2=2q_0$, $m_1=q$, $m_2=q+2q_0$, $m_3=q+2q_0+1$, and hence $v(g_2)=-q$ ($g_2$ being as in Lemma \ref{l3.5}(3)). Therefore, after some $\mathbb F_q$-linear transformations, the case $i=2$ of (\ref{eq3.7}) reads $$ y_2^q-y_2=x^{2q_0}(x^q-x)\, . $$ Now the function $z:= y_2^{q_0}-x^{q_0+1}$ satisfies $z^q-z=x^{q_0}(x^q-x)$ and we find that $q_0+q$ is a non-gap at $P_1$ (cf. \cite[Lemma 1.8]{hsti}). This contradiction eliminates the case $r=3$. Let $r\ge 4$ and $2\le i\le r$. By Lemma \ref{l3.5}(3) $(qm_i-q^2)/\epsilon_2\in H(P_1)$, and since $(qm_i-q^2)/\epsilon_2\ge m_{i-1}\ge q$, by (\ref{eq3.6}) we have $$ 2q_0\ge \epsilon_2 +\epsilon_{r-i}\qquad \mbox{for}\ i=2,\ldots,r-2\, . $$ In particular, $\epsilon_2\le q_0$. On the other hand, by Lemma \ref{l3.5}(2) we must have $m_{r-2}\le q+q_0$ and so, by (\ref{eq3.6}), we find that $\epsilon_2\ge q_0$, i.e. $\epsilon_2=q_0$. Finally we show that $r=4$. $\epsilon_2=q_0$ implies $\epsilon_{r-2}\le q_0$. Since $m_2\le q+q_0$ (cf. Lemma \ref{l3.5}(2)), by (\ref{eq3.6}), we have $\epsilon_{r-2}\ge q_0$. Therefore $\epsilon_{r-2}=q_0=\epsilon_2$, i.e. $r=4$. \end{proof} {\bf Proof of Theorem \ref{B}.} Let $P_1\in X(\mathbb F_q)$ be as above. By (\ref{eq3.7}), Lemma \ref{l3.5}(3) and Lemma \ref{l3.6} we have the following equation $$ y_2^q-y_2=g_2^{q_0}(x^q-x)\ , $$ where $g_2$ has no pole except at $P_1$. Moreover, by (\ref{eq3.6}), $m_2=q_0+q$ and so $v(g_2)=-q$ (cf. Lemma \ref{l3.5}(3)). Thus $g_2=ax+b$ with $a,b\in \mathbb F_q$, $a\neq 0$, and after some $\mathbb F_q$-linear transformations we obtain Theorem \ref{B}. \begin{remarks*} (1) From the above computations we conclude that the Deligne-Lusztig curve associated to the Suzuki group $X$ is equipped with a complete simple base-point-free $g^4_{q+2q_)+1}$, namely $\mathcal D=|(q+2q_0+1)P_0|$, $P_0\in X(\mathbb F_q)$. Such a linear system is an $\mathbb F_q$-invariant. The orders of $\mathcal D$ (resp. the $\mathbb F_q$-Frobenius orders) are $0, 1, q_0, 2q_0$ and $q$ (resp. $0, q_0, 2q_0$ and $q$). (2) There exists $P_1\in X(\mathbb F_q)$ such that the $(\mathcal D,P_1)$-orders are $0,1,q_0+1, 2q_0+1$ and $q+2q_0+1$ (Lemma \ref{l3.3}). Now we show that the above sequence is, in fact, the $(\mathcal D,P)$-orders for each $P\in X(\mathbb F_q)$. To see this, notice that $$ {\rm deg}(S)=(3q_0+q)(2g-2)+(q+4)(q+2q_0+1)=(4+2q_0)\#X(\mathbb F_q). $$ Let $P\in X(\mathbb F_q)$. By (\ref{eq3.5}), we conclude that $v_P(S^\mathcal D)=\sum_{i=1}^{4}(j_i(P)-\nu_{i-1})=4+2q_0$ and so, by (\ref{eq1.4}), that $j_1(P)=1$, $j_2(P)=q_0+1$, $j_3(P)=2q_0+1$, and $j_4(P)=q+2q_0+1$. (3) Then, by (\ref{eq3.2}) $H(P)$, $P\in X(\mathbb F_q)$, contains the semigroup\newline $H:= \langle q,q+q_0,q+2q_0,q+2q_0+1\rangle$. Indeed, $H(P)=H$ since $\#(\mathbb N\setminus H)=g=q_0(q-1)$ (this can be proved as in the remark after Lemma \ref{l3.4}; see also \cite[Appendix]{hsti}). (4) We have $$ {\rm deg}(R^\mathcal D)=\sum_{i=0}^{4}\epsilon_i(2g-2)+5(q+2q_0+1)=(2q_0+3)\#X(\mathbb F_q)\, , $$ for $P\in X(\mathbb F_q)$, $v_P(R^\mathcal D)=2q_0+3$ as follows from items (1), (2) and (\ref{eq1.1}). Therefore the set of $\mathcal D$-Weierstrass points of $X$ is equal to $X(\mathbb F_q)$. In particular, the $(\mathcal D,P)$-orders for $P\not\in X(\mathbb F_q)$ are $0, 1, q_0, 2q_0$ and $q$. (5) We can use the above computations to obtain information on orders for the canonical morphism. By using the fact that $(2q_0-2)\mathcal D$ is canonical (cf. Lemma \ref{l3.5}(1)) and item (4), we see that the set $ \{a+q_0b+2q_0c+qd: a+b+c+d \le 2q_0-2\} $ is contained in the set of orders for $\mathcal K_X$ at non-rational points. (By considering first order differentials on $X$, similar computations were obtained in \cite[\S4]{gsti}.) (6) Finally, we remark that $X$ is non-classical for the canonical morphism: we have two different proofs for this fact: loc. cit. and \cite[Prop. 1.8]{fgt}). \end{remarks*} \begin{center} {\bf Appendix:} A remark on the Suzuki-Tits ovoid \end{center} \smallskip For $s\in \mathbb N$, let $q_0:=2^s$ and $q:=2q_0$. It is well known that the Suzuki-Tits ovoid $\mathcal O$ can be represented in $\mathbb P^4(\mathbb F_q)$ as $$ \mathcal O=\{(1:a:b:f(a,b):af(a,b)+b^2): a, b \in \mathbb F_q\}\cup\{(0:0:0:0:0:1)\}, $$ where $f(a,b):=a^{2q_0+1}+b^{2q_0}$ (see \cite{tits}, \cite[p.3]{pent}) Let $X$ be the Deligne-Lusztig curve associated to $Sz(q)$ and $\mathcal D=|(q+2q_0+1)P_0|$, $P_0\in X(\mathbb F_q)$ (see \S\ref{3}). By the Remark (item 3) in \S\ref{3}, we can associate to $\mathcal D$ a morphism $\pi=(1:x:y:z:w)$ whose coordinates satisfy ${\rm div}_{\infty}(x)=qP_0$, ${\rm div}_{\infty}(y)=(q+q_0)P_0$, ${\rm div}_{\infty}(z)=(q+2q_0)P_0$ and ${\rm div}_{\infty}(w)=q+2q_0+1$. \begin{claim*} (A. Cossidente)\quad $\mathcal O=\pi(X(\mathbb F_q))$. \end{claim*} \begin{proof} We have $\pi(P_0)=(0:0:0:0:1)$; we can choose $x$ and $y$ satisfying \newline $y^q-y=x^{q_0}(x^q-x)$, $z:= x^{2q_0+1}+y^{2q_0}$, and $w:= xy^{2q_0}+z^{2q_0}=xy^{2q_0}+x^{2q+2q_0}+y^{2q}$ (cf. \cite[\S1.7]{hsti}). For $P\in X(\mathbb F_q)\setminus\{P_0\}$ set $a:=x(P)$, $b:=y(P)$, and $f(a,b):= z(a,b)$. Then $w(a,b)=af(a,b)+b^2$ and we are done. \end{proof} \begin{remark*} The morphism $\pi$ is an embedding. Indeed, since $j_1(P)=1$ for each $P$ (cf. Remarks \S3(2)(4)), it is enough to see that $\pi$ is injective. By (\ref{eq3.1}), the points $P$ where $\pi$ could not be injective satisfy: $P\not\in X(\mathbb F_q)$$, {\rm Fr}_{X}^3(P)=P$ or ${\rm Fr}_{X}^2(P)=P$. Now from the Zeta function of $X$ one sees that $\#X(\mathbb F_{q^3})=\#X(\mathbb F_{q^2})=\#X(\mathbb F_q)$, and the remark follows. \end{remark*} \begin{remark*} From the claim, (\ref{eq3.1}) and \cite{he} we have $$ {\rm Aut}_{\bar\mathbb F_q}(X)={\rm Aut}_{\mathbb F_q}(X)\cong \{A\in PGL(5,q): A\mathcal O=\mathcal O\}\, . $$ \end{remark*}

9709

alg-geom/9709001

en

https://arxiv.org/abs/alg-geom/9709001

alg-geom/9709001

Mikhail Zaidenberg

H. Flenner and M. Zaidenberg

Rational cuspidal plane curves of type (d, d-3)

17 Pages. Latex

In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for the rational cuspidal plane curves C with a cusp of multiplicity deg C - 3.

alg-geom

\section*{Introduction} Let $C \subset {\bf P}^2$ be a rational cuspidal curve; that is, it has only irreducible singularities (called {\it cusps}). We say that $C$ is of type $(d,\,m)$ if $d =$deg$\,C$ is the degree and $m = \max_{P \in {\rm Sing}\,C} \{$mult$_P C\}$ is the maximal multiplicity of the singular points of $C.$ Topologically, $C$ is a 2-sphere $S^2$ (non-smoothly) embedded into ${\bf P}^2.$ Due to the Poincar\'e-Lefschetz dualities, the complement $X := {\bf P}^2 \setminus C$ to $C$ is a ${\bf Q}-$acyclic affine algebraic surface, i.e. ${\widetilde H}_*(X;\,{\bf Q})=0$ (see e.g. \cite{Ra,Fu,Za}). Furthermore, if $C$ has at least three cusps, then $X$ is of log-general type, i.e. ${\overline k}(X)=2,$ where ${\overline k}$ stands for the logarithmic Kodaira dimension \cite{Wa}. In \cite{FlZa 1} we conjectured that any ${\bf Q}-$acyclic affine algebraic surface $X$ of log-general type is rigid in the following sense. Let $V$ be a minimal smooth projective completion of $X$ by a simple normal crossing (SNC for short) divisor $D.$ We say that $X$ is {\it rigid} (resp. {\it unobstructed}) if the pair $(V,\,D)$ has no nontrivial deformations (resp. if the infinitesimal deformations of the pair $(V,\,D)$ are unobstructed). In the particular case when $X = {\bf P}^2 \setminus C$ with $C$ as above, the rigidity conjecture would imply that the curve $C$ itself is projectively rigid. This means that the only equisingular deformations of $C$ in ${\bf P}^2$ are those provided by automorphisms of ${\bf P}^2;$ in other words, all of them are projectively equivalent to $C$ (see \cite[sect.\ 2]{FlZa 2}). In turn, this would imply that there is only a finite number of non-equivalent rational cuspidal plane curves of a given degree with at least three cusps. Therefore, one may hope to give a classification of such curves. In \cite{FlZa 2} we obtained a complete list of rational cuspidal plane curves of type $(d,\,d-2)$ with at least three cusps, and showed that all of them are projectively rigid and unobstructed. In the theorem below we do the same for rational cuspidal plane curves of type $(d,\,d-3)$ with at least three cusps. The principal numerical invariant which characterizes a cusp up to equisingular deformation is its multiplicity sequence. Recall that, if $$ V_{n+1}\to V_n\to\dots\to V_1 \to V_0 = {\bf C}^2 $$ is a minimal resolution of an irreducible analytic plane curve germ $(C,\,{\overline 0})\subset ({\bf C}^2,\,{\overline 0}),$ and $(C_i,\,P_i)$ denotes the proper transform of $(C,\,{\overline 0})$ in $V_i,$ so that $(C_0,\,P_0) = (C,\,{\overline 0}),$ then $\underline{m}=(m^{(i)})_{i=0}^{n+1},$ where $m^{(i)} = $mult$_{P_i} C_i,$ is called {\it the multiplicity sequence} of the germ $(C,\,{\overline 0}).$ Thus, $m^{(i+1)} \le m^{(i)},\,\,\,m^{(n)} \ge 2$ and $m^{(n+1)} =1.$ A multiplicity sequence has the following characteristic property \cite[(1.2)]{FlZa 2}: \smallskip \noindent {\it for any $i=0,\dots,n-1$ either $m^{(i)} = m^{(i+1)},$ or there exists $k > 0$ such that $i+k \le n,$ and} $$m^{(i)} = m^{(i+1)} + \dots + m^{(i+k)} + m^{(i+k+1)}, \,\,\,\,\,\,\,\,\,\mbox{where}\,\,\,\,\,\,\,\,\, m^{(i+1)} =\dots = m^{(i+k)}\,.$$ We use the abbreviation $(m_k)$ for a (sub)sequence $m^{(i+1)} = m^{(i+2)} = \dots =m^{(i+k)}=m.$ Thus, we present a multiplicity sequence as $(m^{(1)}_{k_1},\,\dots,m^{(s)}_{k_s})$ with $m^{(i+1)} < m^{(i)};$ by abuse of notation, we assume here that $m^{(s)} \ge 2.$ For instance, $(2)$ means an ordinary cusp, and $(2_3)=(2,\,2,\,2,\,1)$ corresponds to a ramphoid cusp. With this notation we can formulate our main result as follows. \bigskip \noindent {\bf Theorem}. {\it (a) Let $C \subset {\bf P}^2$ be a rational cuspidal plane curve of type $(d,\,d-3),\,\,\,d\ge 6,$ with at least three cusps. Then $d = 2k + 3,$ where $k \ge 2,$ and $C$ has exactly three cusps, of types $(2k,\,2_k),\,\,(3_k),\,\,(2),$ respectively. \smallskip \noindent (b) For each $k \ge 1$ there exists a rational cuspidal plane curve $C_k$ of degree $d = 2k+3$ with three cusps of types $(2k,\,2_k),$ $(3_k)$ and $(2).$ \smallskip \noindent (c) Moreover, the curve $C_k$ as in (b) is unique up to projective equivalence. It can be defined over ${\bf Q}.$ } \bigskip \noindent {\bf Remarks.} (1) A classification of irreducible plane curves up to degree $5$ can be found e.g. in \ \cite{Nam}. In particular, there are, up to projective equivalence, only one rational cuspidal plane quartic with three cusps ({\it the Steiner quartic}) and only three rational cuspidal plane quintic curves with at least three cusps. Two of them have exactly three cusps, of types $(3),$ $(2_2),$ $(2)$ resp. $(2_2),$ $(2_2),$ $(2_2),$ and the third one has four cusps of types $(2_3),$ $(2),$ $(2),$ $(2)$ \ \cite[Thm. 2.3.10]{Nam}. \smallskip (2) In his construction of ${\bf Q}-$acyclic surfaces (see e.g. \cite{tD 1,tD 2}), T. tom Dieck found certain $(d,\,d-2)-$ and $(d,\,d-3)-$rational cuspidal curves, in particular, those listed in the theorem above, as well as some other series of rational cuspidal plane curves (a private communication\footnote{We are grateful to T. tom Dieck for communicating us the list of the multiplicity sequences of the constructed curves.}). Besides a finite number of sporadic examples, the curves with at least three cusps in the list of tom Dieck are organized in three series of $(d,\,d-2)-,$ $(d,\,d-3)-$ and $(d,\,d-4)-$type, respectively. It can be checked that all those curves are rigid and unobstructed. Following our methods, T.\ Fenske proved recently that the only possible numerical data of unobstructed rational cuspidal plane curves with at least three cusps and of type $(d,d-4)$ are those from the list of tom Dieck. He has also classified all rational cuspidal plane curves of degree 6 \cite{Fe}. It turns out that the only examples with at least 3 cusps are those described in \cite{FlZa 2}. \smallskip (3) For a rational cuspidal plane curve $C$ of type $(d,\,m)$ the inequality $m > d/3$ holds \cite{MaSa}. Recently, S. Orevkov obtained a stronger one\footnote{We are grateful to S. Orevkov for providing us with a preliminary version of his paper.}: If the complement ${\bf P}^2 \setminus C$ has logarithmic Kodaira dimension 2, then $d<\alpha m+\beta,$ where $\alpha:=(3+\sqrt{5})/2=2.6180\ldots$ and $\beta :=\alpha-1/\sqrt{5}=2.1708\ldots.$ \smallskip (4) It was shown in \cite{OrZa 1,OrZa 2} that a rational cuspidal plane curve with at least ten cusps cannot be projectively rigid. \bigskip Recall the Coolidge--Nagata Problem \cite{Co,Nag}: \smallskip \noindent {\it Which rational plane curves can be transformed into a line by means of Cremona transformations of} ${\bf P}^2?$ \smallskip \noindent It can be completed by the following question: \smallskip \noindent {\it Is this possible, in particular, for any rational cuspidal plane curve?} \smallskip \noindent Under certain restrictions, a positive answer was given in \cite{Nag,MKM,MaSa,Ii 2,Ii 3}. It can be verified that the last question has a positive answer for the rational cuspidal plane curves of degree at most five. In \cite{FlZa 2} we showed that any rational cuspidal plane curve of type $(d,\,d-2)$ with at least three cusps is rectifiable. Here we extend this result to $(d,\,d-3)-$curves. It will turn out to be an immediate consequence of our construction: \bigskip \noindent {\bf Corollary}. {\it Any rational cuspidal plane curve of type $(d,\,d-3)$ with at least three cusps is rectifiable, i.e.\ it can be transformed into a line by means of Cremona transformations.} \section{Proofs} Let $C \subset {\bf P}^2$ be a plane curve, and let $V \to {\bf P}^2$ be the minimal embedded resolution of singularities of $C,$ so that the reduced total transform $D$ of $C$ in $V$ is an SNC--divisor. By \cite{FlZa 1}, the cohomology groups $H^i(\Theta_V\langle \,D\,\rangle)$ of the sheaf of germs of holomorphic vector fields on $V$ tangent to $D$ control the deformations of the pair $(V,\,D)$; more precisely, $ H^0( \Theta_V\langle \, D \, \rangle)$ is the space of its infinitesimal automorphisms, $ H^1 ( \Theta_V\langle \, D \, \rangle)$ is the space of infinitesimal deformations and $ H^2 ( \Theta_V\langle \, D \, \rangle)$ gives the obstructions for extending infinitesimal deformations. The surface $X = V \setminus D = {\bf P}^2 \setminus C$ being of log-general type, the automorphism group Aut$X$ is finite \cite{Ii 1}, and hence $ h^0(\Theta_V\langle \,D\, \rangle) =0.$ Thus, the holomorphic Euler characteristic of the sheaf $\Theta_V\langle \, D \, \rangle$ is $$\chi ( \Theta_V\langle \, D \, \rangle) = h^2(\Theta_V\langle \,D\,\rangle) - h^1(\Theta_V\langle \,D\,\rangle).$$ \bigskip \noindent {\bf Lemma 1.1.} {\it If $C$ is a rational cuspidal plane curve of type $(d,\,d-3)$ with at least three cusps, then $h^2(\Theta_V\langle \,D\,\rangle) = 0,$ that is, $C$ is unobstructed\footnote{i.e.\ as a plane curve, it has unobstructed equisingular infinitesimal deformations.}, and so $\chi=\chi(\Theta_V\langle \,D\,\rangle) \le 0.$ } \bigskip \noindent {\it Proof.} Projecting from the cusp of multiplicity $d-3$ yields a fibration $V \to {\bf P}^1,$ which is three--sheeted when restricted to the proper transform of $C.$ Now \cite[(6.3)]{FlZa 1} shows that $h^2(\Theta_V\langle \, D \,\rangle) = 0.$ Since ${\overline k}(V \setminus D) = 2,$ we also have $h^0(\Theta_V\langle \, D \,\rangle) = 0.$ Hence $\chi=-h^1(\Theta_V\langle \, D \,\rangle) \le 0.$ \hfill $\Box$ \bigskip The next proposition proves part (a) of our main theorem. \bigskip \noindent {\bf Proposition 1.1.} {\it The only possible rational cuspidal plane curves $C$ of degree $d \ge 6$ with a singular point $Q$ of multiplicity $d-3$ and at least three cusps are those of degree $d = 2k+3,\,\,k=1,\dots,$ with three cusps of types $(2k,\,2_k),$ $(3_k)$ and $(2).$ Furthermore, these curves are projectively rigid.} \bigskip \noindent {\it Proof.} By \cite[(2.5)]{FlZa 2} and Lemma 1.1 above, we have: $$\chi=-3(d-3) + \sum_{P \in {\rm Sing}\,C} \chi_P \le 0\,,\eqno{(R_1)}$$ where $$\chi_P := \eta_P+\omega_P -1\,,$$ and where, for a singular point $P \in C$ with the multiplicity sequence $\underline{m}_P=(m^{(0)},\dots,m^{(k_P)}),$ $$ \eta_P = \sum\limits_{i=0}^{k_P} (m^{(i)}-1)\,\qquad{\rm and} \qquad \omega_P = \sum\limits_{i=1}^{k_P} (\lceil{m^{(i-1)} \over m^{(i)}} \rceil - 1)\, $$ (for $a \in {\bf R},\,\,\lceil {a} \rceil$ denotes the smallest integer $\ge a$). Observe that, by the Bezout theorem, $m_P^{(0)} + m_P^{(1)} \le d$ and $m_P^{(0)} + m_Q^{(0)} \le d.$ Thus $$ \mbox{for} \quad P\neq Q \quad\mbox{we have}\quad m_P^{(0)} \le 3;\quad \mbox{moreover we have} \quad m_Q^{(1)} \le 2, $$ since otherwise the tangent line $T_Q C$ would have the only point $Q$ in common with $C,$ and so, $C \setminus T_Q C$ would be an affine rational cuspidal plane curve with one point at infinity and with two cusps. But by the Lin-Zaidenberg Theorem \cite{LiZa}, up to biregular automorphisms of the affine plane ${\bf C}^2,$ the only irreducible simply connected affine plane curves are the curves $\Gamma_{k,\,l} = \{x^k - y^l = 0\},$ where $1 \le k \le l,$ and $(k,\,l) = 1.$ Hence, such a curve cannot have two cusps. Using the above restriction and the characteristic property of a multiplicity sequence cited above we obtain the following possibilities for the multiplicity sequence $\underline{m}_P$ at a singular point $P$: $$ \begin{array}{l}% \underline{m}_Q=(d-3)\mbox{ or }(d-3,2),\\[2pt] \underline{m}_P=(2_a) \mbox{ or } (3_a)\mbox{ or }(3_a,2)\quad \mbox{for } P\ne Q . \end{array}\eqno{(R_2)} $$ For different possible types of cusps of $C$ we have: \medskip \noindent (a) If $Q \in $\,Sing$\,C$ has the multiplicity sequence $(d-3),$ then $$ \eta_Q = d-4,\quad\omega_Q = d-4\quad\mbox{ and so}\quad\chi_Q = 2d-9. $$ \medskip \noindent (b) If $Q \in $\,Sing$\,C$ has the multiplicity sequence $(d-3,\,2_a)$ then, by the same characteristic property \cite[(1.2)]{FlZa 2}, $$ \quad \mbox{either}\quad d-3\le 2a \quad\mbox{is even or} \quad d-3 = 2a+1.\leqno (*) $$ In any case $$ \eta_Q = d-4 + a,\quad\omega_Q = \lceil{d-3\over 2} \rceil\quad\mbox{and so}\quad\chi_Q = d-5 + a + \lceil{d-3\over 2} \rceil. $$ \medskip \noindent (c) If $P \in $\,Sing$\,C$ has the multiplicity sequence $(2_a),$ then $$ \eta_P = a,\quad\omega_P = 1\quad \mbox{and so}\quad\chi_P = a. $$ \medskip \noindent (d) If $P \in $\,Sing$\,C$ has the multiplicity sequence $(3_a),$ then $$ \eta_P = 2a,\quad\omega_P = 2 \quad\mbox{and so}\quad\chi_P =2a+1. $$ \medskip \noindent (e) If $P \in $\,Sing$\,C$ has the multiplicity sequence $(3_a,\,2),$ then $$ \eta_P =2a+1,\quad\omega_P =2\quad\mbox{and so}\quad\chi_P =2a+2. $$ \medskip Furthermore, since $C$ rational, by the genus formula, we have $$ {d-1 \choose 2} = \sum_{P \in {\rm Sing}\,C}\delta_P \quad\mbox{where}\quad \delta_P:= \sum\limits_{i=1}^{k_P} {m_P^{(i)}\choose 2}\,. $$ Since $m_Q^{(0)} = d-3,$ we get $$ {d-1 \choose 2} = {d-3 \choose 2} + \sum_{(P,\,i) \neq (Q,\,0)} {m_P^{(i)} \choose 2}\,, $$ or, equivalently, $$ 2d-5 = \sum_{(P,\,i) \neq (Q,\,0)} {m_P^{(i)}(m_P^{(i)}-1)\over 2}\,.\eqno{(R_3)} $$ At last, consider the projection $\pi_Q\,:\,C \to {\bf P}^1$ from the point $Q.$ By the Riemann-Hurwitz Formula, it has at most four branching points. This gives the restriction (see \cite[(3.1)]{FlZa 2}) $$m^{(1)}_Q - 1 + \sum\limits_{P\neq Q} (m^{(0)}_P - 1) \le 4\,. \eqno{(R_4)}$$ Thus, if the curve $C$ has the numerical data $$ [(d-3,\,2_{a_1}),\,(2_{a_2}), \dots,(2_{a_k}),(3_{b_1}),\dots,(3_{b_l}), (3_{c_1},\,2),\dots,(3_{c_m},\,2)]\,, $$ then $k+2(l+m) \le 4.$ Hence, either $l+m=0$ and $3\le k \le 4,$ or $l+m=1$ and $k= 2,$ or $l+m=2$ and $k =0.$ Taking into account the above restrictions $(R_2) - (R_4)$ and $(*)$ from (b), the list of all possible data of rational cuspidal plane curves $C$ of degree $d\ge 6$ with a point of multiplicity $d-3$ and at least $3$ cusps is as follows, where $a,\,b,\,c,\,e > 0$: \begin{equation} [(d-3),\,\,\,(2_a),\,\,\,(2_b)] \quad \mbox{where} \quad a+b = 2d-5 \end{equation} \begin{equation} [(d-3),\,\,\,(2_a),\,\,\,(3_b)] \quad \mbox{where} \quad a+3b = 2d-5 \end{equation} \begin{equation} [(d-3),\,\,\,(3_a),\,\,\,(3_b)] \quad \mbox{where} \quad 3a+3b = 2d-5 \end{equation} \begin{equation} [(d-3),\,\,\,(3_a,\,2),\,\,\,(2_b)] \quad \mbox{where} \quad 3a+b = 2d-6 \end{equation} \begin{equation} [(d-3),\,\,\,(3_a,\,2),\,\,\,(3_b)] \quad \mbox{where} \quad 3a+3b = 2d-6 \end{equation} \begin{equation} [(d-3),\,\,\,(3_a,\,2),\,\,\,(3_b,\,2)] \quad \mbox{where} \quad 3a+3b = 2d-7 \end{equation} \begin{equation} [(d-3,\,2_a),\,\,\,(2_b),\,\,\,(2_c)] \quad \mbox{where} \quad a+b+c = 2d-5\mbox{ and } (*)\mbox{ holds}\end{equation} \begin{equation} [(d-3,\,2_a),\,\,\,(3_b),\,\,\,(2_c)] \quad \mbox{where} \quad a+3b +c= 2d-5\mbox{ and } (*)\mbox{ holds} \end{equation} \begin{equation} [(d-3,\,2_a),\,\,\,(3_b,\,2),\,\,\,(2_c)] \quad \mbox{where} \quad a+3b +c= 2d-6\mbox{ and } (*)\mbox{ holds}\end{equation} \begin{equation} [(d-3),\,\,\,(2_a),\,\,\,(2_b),\,\,\,(2_c)] \quad \mbox{where} \quad a+b +c= 2d-5\end{equation} \begin{equation} [(d-3),\,\,\,(3_a),\,\,\,(2_b),\,\,\,(2_c)] \quad \mbox{where} \quad 3a+b +c= 2d-5\end{equation} \begin{equation} [(d-3),\,\,\,(3_a,\,2),\,\,\,(2_b),\,\,\,(2_c)] \quad \mbox{where} \quad 3a+b +c= 2d-6\end{equation} \begin{equation} [(d-3,\,2_a),\,\,\,(2_b),\,\,\,(2_c),\,\,\,(2_e)] \quad \mbox{where}\quad a+b +c+e= 2d-5\mbox{ and } (*)\mbox{ holds}\end{equation} \begin{equation} [(d-3),\,\,\,(2_a),\,\,\,(2_b),\,\,\,(2_c),\,\,\,(2_e)] \quad \mbox{where} \quad a+b +c+e= 2d-5.\end{equation} We will examine case by case, computing $\chi=\chi ( \Theta_V\langle \, D \, \rangle).$ The genus formula and the restriction $(R_1)$ $\chi\le 0$ provided by Lemma 1.1 will allow to eliminate all the cases but one, namely, a subcase of (8). \medskip \noindent Case (1): $[(d-3),\,\,\,(2_a),\,\,\,(2_b)]$ where $a+b = 2d-5.$ By $(R_1),$ we have $\chi = (-3d+9)+(2d-9) + (a + b) = d-5 \le 0,$ a contradiction. \smallskip \noindent Case (2): $[(d-3),\,\,\,(3_b),\,\,\,(2_a)]$ where $a+3b = 2d-5.$ We have $\chi = (-3d+9)+(2d-9) + (a + 2b +1) = d-4-b \le 0,$ i.e.\ $b \ge d-4.$ On the other hand, $2d -5= a + 3b \ge 3b + 1,$ whence $b \le {2\over 3}d - 2.$ Therefore, $d - 4 \le {2\over 3}d - 2,$ i.e.\ $d \le 6.$ In the case $d = 6$ the only possibility would be $[(3),\,\,(2),\,\,(3_2)].$ Projecting from the cusp with the multiplicity sequence $(3_2),$ we get a contradiction to the Hurwitz formula (see ($R_4$)). \smallskip \noindent Case (3): $[(d-3),\,\,\,(3_a),\,\,\,(3_b)]$ where $ 3a+3b = 2d-5.$ We have $\chi = (-3d+9)+(2d-9) + (2a +1 + 2b + 1) = {d-4 \over 3} \le 0,$ i.e.\ $d \le 4,$ and we are done. \smallskip \noindent Case (4): $[(d-3),\,\,\,(3_a,\,2),\,\,\,(2_b)]$ where $3a+b = 2d-6.$ We have $\chi = (-3d+9)+(2d-9) + (2a +2 + b) = d - 4 - a \le 0,$ i.e.\ $a \ge d-4.$ But $2d-6 = 3a + b \ge 3a + 1,$ whence $a \le {2\over 3}d - {7\over 3},$ and thus $d - 4 \le {2\over 3}d - {7\over 3},$ or $d \le 5,$ a contradiction. \smallskip \noindent Case (5): $[(d-3),\,\,\,(3_a,\,2),\,\,\,(3_b)]$ where $3a+3b = 2d-6.$ We have $\chi = (-3d+9)+(2d-9) + (2a +2 + 2b+1) = {d\over 3} - 1 \le 0,$ i.e.\ $d\le 3,$ which is impossible. \smallskip \noindent Case (6): $[(d-3),\,\,\,(3_a,\,2),\,\,\,(3_b,\,2)]$ where $3a+3b = 2d-7.$ We have $\chi = (-3d+9)+(2d-9) + (2a + 2b+4)= {d\over 3} - {2\over 3} \le 0,$ which is impossible. \smallskip \noindent Case (7): $[(d-3,\,2_a),\,\,\,(2_b),\,\,\,(2_c)]$ where $a+b+c = 2d-5$ and $(*)$ holds. We have $\chi = (-3d+9)+ (d-5 +a +\lceil{d-3 \over 2} \rceil) + (b+c) = \lceil{d-3 \over 2} \rceil - 1 \le 0,$ or $d \le 5,$ and we are done. \smallskip \noindent Case (8): $[(d-3,\,2_a),\,\,\,(3_b),\,\,\,(2_c)]$ where $a+3b +c= 2d-5$ and $(*)$ holds. We have $\chi = (-3d+9)+ (d-5 +a +\lceil{d-3\over 2} \rceil) + (2b+1 +c) = \lceil{d-3 \over 2} \rceil - b \le 0,$ i.e.\ $b \ge \lceil{d-3 \over 2} \rceil.$ If $d-3$ is odd, then we get $2d-5 = a + 3b + c \ge 3b + 1 + {d-4 \over 2},$ as $a = {d-4 \over 2}$ by $(*).$ Hence, $b \le {d\over 2} - {4\over 3}.$ This leads to $ \lceil{d-3 \over 2} \rceil = {d-2 \over 2}\le {d\over 2} - {4\over3},$ which is a contradiction. If $d-3$ is even, then by $(*)$ we get $2d-5= a + 3b + c \ge 3b + 1 + {d-3 \over 2},$ hence $b \le {d\over 2} - {3\over 2}.$ Thus, $\lceil{d-3 \over 2} \rceil = {d-3 \over 2} \le b \le {d - 3 \over 2},$ which is only possible if $c = 1,\,\,a=b={d-3 \over 2}.$ With $k:= {d - 3 \over 2}$ we obtain that $d = 2k+3,\,\,a=b=k$ and $c = 1;$ that is, $C$ is as in the proposition. Observe that in this case $\chi = 0,$ and so $h^1(\Theta_V\langle \, D \,\rangle) = 0.$ Together with Lemma 1.1 this proves that the corresponding curve $C$ is projectively rigid and unobstructed (see [FZ 2, Sect. 2]). \smallskip \noindent Case (9): $[(d-3,\,2_a),\,\,\,(3_b,\,2),\,\,\,(2_c)]$ where $a+3b +c= 2d-6$ and $(*)$ holds. We have $\chi = (-3d+9)+ (d-5 +a +\lceil{d-3\over 2} \rceil) + (2b+2 +c) = \lceil{d-3 \over 2} \rceil - b \le 0,$ which gives $b \ge \lceil{d-3 \over 2} \rceil.$ If $d-3$ is odd, then we get $2d-6 = a + 3b + c \ge 3b + 1 + {d-4 \over 2},$ as $a = {d-4 \over 2}$ by $(*).$ Thus, $b \le {d\over 2} - {5\over 3},$ and so we have ${d-2\over 2}\le {d\over 2} - {5\over 3},$ which is a contradiction. If $d-3$ is even, then we get $2d-6 = a + 3b + c \ge 3b + 1 + {d-3\over 2}.$ Hence, $b \le {d\over 2} - {11\over 6}.$ This yields ${d-3\over 2}\le {d\over 2} - {11\over 6},$ which again gives a contradiction. \smallskip \noindent Case (10): $[(d-3),\,\,\,(2_a),\,\,\,(2_b),\,\,\,(2_c)]$ where $a+b +c= 2d-5.$ We have $\chi = (-3d+9)+(2d-9) +(a +b +c) = d-5 \le 0,$ and we are done. \smallskip \noindent Case (11) resp. (12), (13), (14) can be ruled out by the same computations as in case (2) resp. (4), (7), (10). This completes the proof of Proposition 1.1. \hfill $\Box$ \bigskip For the proof of part (b) and (c) the main theorem we need the following facts. \bigskip \noindent {\bf Lemma 1.2.} {\it Let $(C,\,{\overline 0}),\ (D,\,{\overline 0})\subseteq({\bf C}^2,\,{\overline 0})$ be two curve singularities which have no component in common. Then the following hold. \smallskip \noindent (a) $(CD)_{\overline 0}=\sum_P\mathop{\rm mult}\nolimits_PC \mathop{\rm mult}\nolimits_PD,$ where the sum is taken over ${\overline 0}$ and all its infinitesimally near points. \smallskip \noindent (b) Assume that $(D,\,{\overline 0})$ is a smooth germ and $(C,\,{\overline 0}= )$ is a cusp with the multiplicity sequence $\underline{m}=(m^{(0)},\ldots, m^{(n)}).$ Then $(CD)_{\overline 0}= m^{(0)}+\ldots+ m^{(s)}$ for some $s\ge 0,$ where $m^{(0)}=\ldots= m^{(s-1)}.$ \smallskip \noindent (c) Let $\pi:X\to {\bf C}^2$ be the blow up at ${\overline 0}.$ Denote by $E\subseteq X$ the exceptional curve, and by $C'$ the proper transform of $C.$ Then} $$ \mathop{\rm mult}\nolimits_{\overline 0}C=\sum_{P\in E}(EC')_P. $$ \smallskip \noindent {\it Proof.} The statements (a) and (c) are well known (see e.g.\ [Co]), whereas (b) is shown in [FlZa 2, (1.4)]. \hfill $\Box$ \bigskip The next result proves part (b) and (c) of the main theorem as well as the corollary from the introduction. \medskip \noindent {\bf Proposition 1.2.} \it (a) For each $k \ge 1$ there exists a rational cuspidal plane curve $C_k$ of degree $d = 2k+3$ with three cusps $Q_k,\ P_k,\ R_k$ of types $(2k,\,2_k),$ $(3_k)$ and $(2),$ respectively. (b) $C_k$ is unique up to a projective transformation of the plane. (c) $C_k$ is defined over ${\bf Q}$. (d) $C_k$ is rectifiable. \bigskip \rm \noindent {\it Proof.} We proceed by induction on $k.$ Namely, given a curve $C_k$ as in (a), we find a Cremona transformation $\psi_k\,:\,{\bf P}^2\to{\bf P}^2$ such that the proper transform $C_{k+1}=\psi_k(C_k)$ of $C_k$ under $\psi_k$ is a cuspidal curve of degree $2k+5$ with three cusps of type $(2k+2,\,2_{k+1}),\,\, (3_{k+1}),\,\, (2).$ Hence the existence follows. This construction will also show that (b)--(d) hold. We start with the rational cuspidal cubic $C_0\subseteq{\bf P}^2$ given by the equation $x^2z=y^3.$ Observe that $C_0$ is rectifiable. It has a simple cusp at $R_0:=(0:0:1)$ and the only inflectional tangent line $\ell_0$ at $P_0:=(1:0:0);$ that is, $\ell_0\cdot C_0=3P_0.$ Fix an arbitrary point\footnote{Observe that the projective transformation group $(x : y : z) \longmapsto (t^3x : t^2y : t^6z),\,\,\,t \in {\bf C}^*,$ acts transitively in $C_0 \setminus \{P_0,\,R_0\}.$ } $Q_0\in C_0 \setminus \{P_0,\,R_0\}.$ Let $t_0$ be the tangent line to $C_0$ at $Q_0;$ then we have $t_0\cdot C_0=2Q_0 + S_0,$ where, as it is easily seen, $S_0 \in C= _0$ is different from $P_0,\,Q_0$ and $R_0.$ Let $Q_0^*$ denote the intersection point $l_0 \cap t_0;$ clearly, $Q_0^* \notin C_0.$ Let, for a given $k > 0,$ $C_k$ denotes a curve with the cusps $Q_k,\ P_k,\ R_k$ as in the proposition, and let $C_0$ be the rational cubic with the distinguished points $Q_0,\ P_0,\ R_0, \ S_0$ as described above. For $k>0$ let $t_k$ be the tangent line of $C_k$ at $Q_k,$ and $\ell_k$ be the line $\overline{P_kQ_k},$ whereas for $k=0$ we choose $t_0$ and $\ell_0$ as above. In any case, using Bezout's Theorem and Lemma 1.2, we have $$ \ell_k\cdot C_k=(d-3)Q_k + 3P_k,\quad\mbox{and}\quad t_k\cdot C_k=(d-1)Q_k + S_k\,, $$ where $S_k \in C_k$ is different from $P_k,\,Q_k$ and $R_k.$ Indeed, the line $t_k$ intersects $C_k$ at the point $Q_k$ with multiplicity $d-1$ if $k>1$ (see Lemma 1.2 (b)) or $k=0.$ To show that this is also true for $k=1,$ assume that $t_1$ and $C_1$ only intersect in $Q_1$ with $(t_1C_1)_{Q_1}=d=5.$ The linear projection from $Q_1$ yields a 3-sheeted covering of the normalization of $C_1$ onto ${\bf P}^1.$ By the Riemann-Hurwitz formula, it must have four ramification points. But since $(t_1C_1)_{Q_1}=d=5,$ the point $Q_1$ would be a ramification point of index $\ge 2$ (see Lemma 1.2(a)), and so we would have three ramification points $Q_1,\, P_1,\, R_1$ of indices $2,\, 2,\, 1,$ respectively, which is a contradiction. Hence, for any $k\ge 0$ there is exactly one further intersection point $S_k\in C_k\cap t_k$ with $(t_kC_k)_{S_k}=1.$ Let $\sigma_k\,:\,X_k\to{\bf P}^2$ be the blow up at the point $t_k\cap\ell_k,$ which is $Q_k$ for $k>0$ and $Q_k^*$ for $k=0.$ Denote by $C'_k,$ $\ell_k',$ $t_k'$ the proper transforms in $X_k$ of the curves $C_k,$ $\ell_k,$ $t_k,$ respectively. Then $X_k\simeq \Sigma_1$ is a Hirzebruch surface with a ruling $\pi_k\,:\,X_k \to {\bf P}^1$ given by the pencil of lines through $Q_k$ resp.\ $Q_0^*,$ and with the exceptional section $E_k = \sigma_k^{-1}(Q_k),\,k>0,$ resp. $E_0 = \sigma_0^{-1}(Q_0^*),$ where $E_k^2 = -1.$ Thus, $\ell_k',$ $t_k= '$ are fibres of this ruling. By construction, the restriction $\pi_k\,|\,C_k'\,:\,C_k' \to {\bf P}^1$ is 3-sheeted, and we have $$ \ell_k'\cdot C_k'=3P_k',\quad t_k'\cdot C_k'=2Q_k' + S_k', \quad\mbox{and}\quad E_k'\cdot C_k'=(d-3)Q_k' = 2kQ_k'\,, $$ where $P_k',\,Q_k',\,R_k'$ and $S_k'$ are the points of $C_k'$ infinitesimally near to $P_k,\,Q_k,\,R_k$ and $S_k \in C_k,$ respectively (indeed, by Lemma 1.2(c), we have $(E_k'C_k')_{Q_k'} = $mult$_{Q_k^*}C_k = d-3,$ where for $k > 0$ we set $Q_k^* = Q_k$). Clearly, for $k > 0$ $P_k',\,Q_k'$ and $R_k'$ are cusps of $C_k'$ of types $(3_k),\,\,(2_k)$ and $(2),$ respectively, whereas $S_k'$ is a smooth point. Next we perform two elementary transformations\footnote{Recall that an elementary transformation of a ruled surface consists in blowing up at a point of a given irreducible fibre followed by the contraction of the proper transform of this fibre.} of $X_k,$ one at the point $S_k'$ and the other one at the intersection point $T_k':=\{E_k\cap \ell_k'\}.$ We arrive at a new Hirzebruch surface $X_{k+1}\simeq\Sigma_1,$ with the exceptional section $E_{k+1}$ being the proper transform of $E_k$ (indeed, since we perform elementary transformations at the points $S_k \notin E_k$ and $T_k' \in E_k,= $ we have $E_{k+1}^2=E_k^2=-1$). Denote by $C_{k+1}'$ the proper transform of $C_k',$ and by $t_{k+1}',$ $\ell_{k+1}'$ the fibres of the ruling $\pi_{k+1}\,:\,X_{k+1}\to {\bf P}^1$ which replace $t_k'$ resp.\ $\ell_k'.$ Using formal properties of the blowing up/down process we obtain, once again, the relations $$ \ell_{k+1}'\cdot C_{k+1}'=3P_{k+1}',\quad t_{k+1}'\cdot C_{k+1}'=2Q_{k+1= }' + S_{k+1}', \quad\mbox{and}\quad E_{k+1}'\cdot C_{k+1}'= 2(k+1)Q_{k+1}'\,, $$ where $P_{k+1}',\,Q_{k+1}',\,R_{k+1}'$ and $S_{k+1}'$ are the points of $C_{k+1}'$ infinitesimally near to $P_k',\,Q_k',\,R_k'$ and $S_k' \in C_k',$ respectively. It is easily seen that $P_{k+1}'$ resp. $Q_{k+1}',\,\,\,R_{k+1}'$ are cusps of $C_{k+1}'$ of types $(3_{k+1}),\,\,(2_{k+1})$ and $(2),$ respectively, whereas $S_{k+1}'$ is a smooth point. Blowing down the exceptional curve $E_{k+1}' \subset X_{k+1}$ we arrive again at ${\bf P}^2.$ Denote the images of $C_{k+1}',.$$ Q_{k+1}',$ $P_{k+1}',.$$ R_{k+1}'$ resp. by $C_{k+1},\ Q_{k+1},$ $P_{k+1},$ $R_{k+1}.$ We have constructed a rational cuspidal plane curve $C_{k+1}$ which has cusps at $Q_{k+1},\ P_{k+1},\ R_{k+1}$ with multiplicity sequences $(2(k+1),\, 2_{k+1}),\,\, (3_{k+1}),\,\, (2),$ respectively (see Lemma 1.2(c)). This completes the proof of existence. Note that the birational transformation $\psi_k:{\bf P}^2\to{\bf P}^2$, by which we obtained $C_{k+1}=\psi_k(C_k)$ from $C_k$, is just the Cremona transformation in the points $S_k$, $Q_k$ and the intersection point $E_k\cap \ell_k'$, which is infinitesimally near to $Q_k$. This transformation only depends upon $Q_k$, $S_k$ and the line $\ell_k;$ we denote it by $\psi(S_k,Q_k,\ell_k):=\psi_k.$ The inverse $\psi_k^{-1}$ is the transformation $\varphi_k=\psi(P_{k+1}, Q_{k+1},t_{k+1})$. Therefore, the curve $C_k$ is always transformable into the cuspidal cubic, and thus also into a line, by means of Cremona transformations, proving (d). In order to show (c) we note that, moreover, so constructed $C_k,$ as well as $P_k$, $Q_k$, $R_k$ and $S_k,$ are defined over ${\bf Q},$ as follows by an easy induction. \bigskip Finally, let us show that the curve $C_k$ is uniquely determined up to a projective transformation of the plane. We will again proceed by induction on $k$. Clearly, the cuspidal cubic is uniquely determined up to a projective transformation. Assume that uniqueness is shown for the curve $C_k$, and consider two curves $C_{k+1}$, $\tilde C_{k+1}$ as in (a). Let $P_{k+1}\in C_{k+1}$, $Q_{k+1}\in C_{k+1}$ and the tangent line $t_{k+1}$ of $C_{k+1}$ at $Q_{k+1}$ be as above; denote the corresponding data for $\tilde C_{k+1}$ by $\tilde P_{k+1},$ $\tilde Q_{k+1}$ and $\tilde t_{k+1}.$ Consider the Cremona transformations $\varphi_k:=\psi(P_{k+1}, Q_{k+1},t_{k+1})$ and $\tilde \varphi_k:=\psi(\tilde P_{k+1}, \tilde Q_{k+1},\tilde t_{k+1}),$ and also the proper transforms $C_k:=\varphi_k(C_{k+1})$ and $\tilde C_k:=\tilde \varphi_k(\tilde C_{k+1})$. Reversing the above arguments it i= s easily seen that the both curves $C_k,\,\tilde C_k$ are as in (a). By the induction hypothesis, they differ by a projective transformation $f:{\bf P}^2\to{\bf P}^2,$ i.e.\ $f(C_k)=\tilde C_k$. For $k>0$ the points $Q_k\in C_k$, $S_k\in C_k$ and the line $\ell_k$ are intrinsically defined by the curve $C_k,$ and so, $f$ maps these data onto the corresponding data $\tilde Q_k$, $\tilde S_k$ and $\tilde \ell_k$ for the curve $\tilde C_k$. Moreover, in the case $k=0$ it is easily seen that one can choose $f$ in such a way that $f(Q_0)=\tilde Q_0$. Then again $f(S_k)=f(\tilde S_k)$ and $f(\ell_k)=\tilde\ell_k$. Hence, the map $f$ is compatible with the Cremona transformations $\varphi_k^{-1}=\psi(S_k,Q_k,\ell_k)$ and $\tilde \varphi_k^{-1}= \psi(\tilde S_k,\tilde Q_k,\tilde \ell_k)$, i.e.\ there is a linear transformation $g$ of ${\bf P}^2$ such that $\varphi_k\circ g=f\circ \tilde \varphi_k$. Clearly, $g$ transforms $C_{k+1}$ into $\tilde C_{k+1}$. \hfill $\Box$ \bigskip \noindent {\bf Remarks.} (1) By the same approach as in the proof of Proposition 1.2, it is possible to show the existence and uniqueness of the rational cuspidal curves of type $(d,d-2)$ with at least three cusps, which was done by a different method in [FlZa 2]. By the result of loc.cit\ such a curve $C$ has exactly three cusps, say $Q,$ $P,$ $R,$ with the multiplicity sequences $(d-2),$ $(2_a),$ $(2_b),$ respectively, where $a+b=d-2.$ Set $\ell_P:=\overline{QP},$ $\ell_R:=\overline{QR}.$ and denote by $t_Q$ the tangent line at $Q.$ By Bezout's Theorem, $t_Q$ intersects $C$ in one further point $S$ different from $Q.$ Performing the Cremona transformation $\psi(S,Q,\ell_P)$ to the curve $C,$ we obtain a curve of degree $d+1$ with the multiplicity sequences $(d-1),$ $(2_{a+1}),$ $(2_b)$ at the cusps. Similarly, under the Cremona transformation $\psi(P,Q,\ell_R)$ the curve $C$ is transformed into a cuspidal curve of the same degree $d$ with the multiplicity sequences $(d-2),$ $(2_{a+1}),$ $(2_{b-1}).$ Thus, starting from the rational cuspidal quartic with three cusps, we can construct all such curves. It follows from this construction that these curves are rectifiable. \smallskip (2) Using the above arguments, it is also possible to classify the rational cuspidal curves of degree five with at least three cusps, which was done by M. Namba by a different method, see \cite[Thm.2.3.10]{Nam}. Indeed, if the largest multiplicity of a cusp is $3,$ then projecting $C$ from this point, say $Q,$ gives a two-sheeted covering $C \to {\bf P}^1$ with two ramification points. Hence, in this case $C$ has three cusps, with multiplicity sequences $(3)$ (at $Q$), $(2_2),\,\,(2),$ respectively. If all the cusps are of multiplicity $2,$ then $C$ has singular points $P,Q,R,\ldots$ with multiplicity sequences $(2_p),$ $(2_q),$ $(2_r),\ldots,$ where $p+q+r+\ldots=6.$ We may assume that $p\ge q\ge r\ldots.$ Projecting from $P$ gives a three-sheeted covering $C \to {\bf P}^1$ with four ramification points. Hence, $C$ has at most four cusps. The possibilities are as follows: (1) $C$ has 3 cusps of type $P=(2_2),$ $Q=(2_2),$ $R=(2_2).$ (2) $C$ has 3 cusps of type $P=(2_4),$ $Q=(2),$ $R=(2).$ (3) $C$ has 3 cusps of type $P=(2_3),$ $Q=(2_2),$ $R=(2).$ (4) $C$ has 4 cusps of type $P=(2_3),$ $Q=(2),$ $R=(2),$ $S=(2).$ (5) $C$ has 4 cusps of type $P=(2_2),$ $Q=(2_2),$ $R=(2),$ $S=(2).$ \noindent Curves as in (1) and (4) do exist and can be constructed by Cremona transformations. The other cases are not possible, as can be seen by the following arguments. (5) can be excluded since the dual curve would be a cubic with two cusps, which is impossible. To exclude (3), denote by $t_P$ the tangent line of $C$ at $P.$ By the Cremona transformation $\psi:= \psi(Q,P,t_P)$ a curve $C$ as in (3) is transformed into a quartic $C'$ with three simple cusps $P'$, $Q'$, $R'.$ It can be seen that there is a tangent line at a smooth point $S'$ of $C'$ passing through one of the cusps, say $Q'.$ Projecting from $Q'$ gives a two-sheeted covering $C' \to {\bf P}^1$ with three ramification points, namely $P',$ $R'$ and $S'.$ This contradicts the Hurwitz formula. In the case (2), consider the blow up at $P,$ and perform an elementary transformation at the point of the proper transform of $C$ over $P.$ Then the image of $P$ will be a point with the multiplicity sequence $(2_2).$ Performing at this point another elementary transformation and blowing down to ${\bf P}^2,$ we arrive at the same configuration as above. Hence, also (2) is impossible. (This last transformation may also be considered as a Cremona transformation, namely in the points $P$, $P'$ and $P'',$ where $P'$ is infinitesimally near to $P$ and $P''$ is infinitesimally near to $P'.$) Similarly, using Cremona transformations for the cases 1 and 4, one can construct these curves and show that they are rectifiable and projectively unique. It is also possible to treat in the same way the rational cuspidal quintics with one or two cusps. \medskip Finally, we give an alternative proof for the existence and uniqueness statements of Proposition 1.2. It provides a way of computing an explicit parameterization for these curves. \bigskip \noindent {\it Alternative proof of Proposition 1.2 (a)-(c). } For $k=1$ the result is known (see e.g. \cite{Nam}). Let $C_k\,\,(k > 1)$ be a rational cuspidal plane curve of degree $d = 2k+3$ with three cusps $P,\,Q,\,R$ of types $(3_k),$ $(2k,\,2_k)$ and $(2),$ respectively. Since, by Bezout's Theorem, they are not at the same line, we may chose them as $Q\,(0:0:1),\,P\,(0:1:0),\,R\,(1:0:0).$ We may also chose a parameterizatio= n ${\bf P}^1 \to C_k$ of $C_k$ such that $(0:1) \mapsto Q,\,(1:0) \mapsto P,\,(1:1) \mapsto R.$ Then, up to constant factors, this parameterization can be written as $$(x,\,y,\,z) = (s^{2k}t^3,\,\,\,\,s^{2k}(s-t)^2(as+bt),\,\,\,\,t^3(s-t)^2q_k(s,\,t))\,,$$ where $q_k \in {\bf C} [s,\,t]$ is a homogeneous polynomial of degree $2k-2.$ Let $\Gamma$ denotes a curve parameterized as above (with $q$ instead of $q_k$). It is enough to prove the following \smallskip \noindent {\bf Claim.} {\it There exists unique polynomials $as+bt$ and $q$ with rational coefficients, where $q(1,\,0) = 1,$ such that the multiplicity sequences of $\Gamma$ at the points $P,\,Q,\,R \in \Gamma= $ start, respectively, with $(3_k),\,(2k,\,2_k)$ and $(2)$}. \smallskip Indeed, if this is the case, then, by the genus formula, these multiplicity sequences actually coincide resp. with $(3_k),\,(2k,\,2_k)$ and $(2),$ and so, $C_k = \Gamma$ up to projective equivalence. This will prove the existence of the curves $C_k$ defined over ${\bf Q}$ for all $k > 1,$ as well as their uniqueness, up to projective equivalence. \smallskip \noindent {\it Proof of the claim.} It is easily seen that, after blowing up at $Q,$ the infinitesimally near point $Q'$ to $Q$ at the proper transform $\Gamma'$ of $\Gamma$ will be a singular point of multiplicity $2$ iff $as+bt = 2s+t.$ By \cite[(1.2)]{FlZa 2}, under this condition the multiplicity sequence of $\Gamma$ at $Q$ starts with $(2k,\,2_k).$ In the affine chart $(\widehat {x},\,\widehat {z}):=(x/y,\,z/y)$ centered at $P$ we have $$\widehat {x} = {t^3 \over (s-t)^2(2s+t)},\,\,\,\,\,\,\widehat {z}={t^3q(s,\,t) \over s^{2k}(2s+t)}\,.$$ In the sequel we denote by the same letter $t$ the affine coordinate $t/s$ in ${\bf P}^1 \setminus \{(0:1)\}.$ Thus, in this affine chart in ${\bf P}^1$ centered at $(1:0)$ we have $$(\widehat {x},\,\widehat {z})= \left({t^3 \over (t-1)^2(t +2)},\,\,{t^3 \over (t +2)}\,\widehat {q}(t)\right)\,,$$ where $\widehat {q}(t) = \sum_{i=0}^{2k-2} c_it^i$ and where, by the above assumption, $c_0 = 1.$ After blowing up at $P,$ in the affine chart with the coordinates $(u,\,v),$ where $(\widehat {x},\,\widehat {z}) = (u,\,uv),$ we will have $$(u,\,v) = (\widehat {x},\,\widehat {z}/\widehat {x}) = \left({t^3 \over (t-1)^2(t +2)},\,\,\,\,\widehat {q}(t)(t-1)^2\right)\,.$$ To move the origin to the infinitesimally near point $P' \in \Gamma'$ of $P,$ we set $$(\widehat {u},\, \widehat {v}) = (u,\,v-1) = \left({t^3 \over (t-1)^2(t +2)},\,\,\,\, \widehat {q}(t)(t-1)^2 - 1\right)\,.$$ The following conditions guarantee that the multiplicity of the curve $\Gamma'$ = at $P'$ is at least $3$: $$t^3 \,\vert\,\,[\widehat {q}(t)(t-1)^2 - 1] \Longleftrightarrow$$ $$[\widehat {q}(t)(t-1)^2 - 1]'_0 = [\widehat {q}(t)(t-1)^2 - 1]''_0 = 0\Longleftrightarrow$$ $$\widehat {q}'(0) = 2,\,\widehat {q}''(0) = 6 \Longleftrightarrow c_1 = 2, \,c_2=3\,.\eqno{(15)}$$ In the case when $k=2$ this uniquely determines the polynomial $q$: $$q(s,\,t) = s^2 + 2st + 3t^2\,.$$ In what follows we suppose that $k > 2.$ Assume that the conditions (15) are fulfilled. Then we have the following coordinate presentation of $\Gamma'$: $$(\widehat {u},\,\widehat {v}) = \left({t^3 \over (t-1)^2(t +2)},\,\,\,\,t^3h(t)\right)\,,$$ where $h(t):=[\widehat {q}(t)(t-1)^2 - 1] / t^3$ is a polynomial of degree $2k-3,$ which satisfies the conditions $$(t-1)^2 \,\vert\,\,[t^3h(t) + 1] \Longleftrightarrow h(1) = -1,\,h'(1) = 3\,.\eqno{(15')}$$ Once ($15'$) are fulfilled, one can find $\widehat {q}$ as $\widehat {q} = [t^3h(t) + 1]/(t-1)^2,$ and we have $\widehat {q} \in {\bf Q}[t]$ iff $h \in {\bf Q}[t= ].$ Let $\xi \in {\bf C}[[t]]$ be such that $\xi^3 = {t^3 \over (t-1)^2(t +2)}.$ By \cite[(3.4)]{FlZa 2}, the multiplicity sequence of $\Gamma'$ at $P'$ starts with $(3)_{k-1}$ iff $$t^3h(t) \equiv \widehat {f}(\xi^3) \,\,\,{\rm mod}\,\xi^{3(k-1)}\,,$$ where $\widehat {f} = \sum\limits_{i=0}^{k-1} \widehat {a}_ix^i \in {\bf C}[x]$ is a polynomial of degree $ \le k-1.$ Multiplying the both sides by the unit $[(t-1)^2(t +2)]^{k-1} \in {\bf C}[[t]]$, we will get $$[(t-1)^2(t +2)]^{k-1}t^3h(t) \equiv [(t-1)^2(t +2)]^{k-1} \sum\limits_{i=0}^{k-1} \widehat {a}_i\xi^{3i} \equiv \sum\limits_{i=0}^{k-1} \widehat {a}_it^{3i}[(t-1)^2(t +2)]^{k-1-i} \,\,\,\,{\rm mod}\,t^{3(k-1)}\,.$$ Since, by our assumption, $k > 1,$ we should have $\widehat {a}_0 = 0,$ and after dividing out the factor $t^3,$ we get $$[(t-1)^2(t +2)]^{k-1}h(t) \equiv \sum\limits_{i=0}^{k-2} \widehat {a}'_it^{3i}[(t-1)^2(t +2)]^{k-2-i} \,\,\,\,{\rm mod}\,t^{3(k-2)}\,,$$ where $\widehat {a}'_i = \widehat {a}_{i-1},\,\,i=1,\dots,k-2.$ In other words, we have $$[(t-1)^2(t +2)]^{k-1}h(t) = \widehat {f}(t^{3}, \,(t-1)^2(t +2)) + \widehat {g}(t)t^{3(k-2)}\,,$$ where $\widehat {f}(x,\,y) = \widehat {f}_k(x,\,y) := \sum_{i=0}^{k-2} \widehat {a}'_ix^iy^{k-2-i= }$ is a homogeneous polynomial of degree $k-2,$ and hence $\widehat {g}(t) = \widehat {g}_k(t) = \sum_{i=0}^{2k} \widehat {b}_it^i$ should be a polynomial of degree $2k.$ Denoting $\tau = t^3$ and $\lambda = (t-1)^2(t +2) = t^3 - 3t + 2$, we have $$\lambda^{k-1}h = \widehat {f}(\tau,\,\lambda) + \tau^{k-2}\widehat {g}\,.$$ Observe that $\widehat {f}(\tau,\,\lambda)$ (resp. $\tau^{k-2}\widehat {g}$) contains the monomial $\widehat {a}'_0\tau^{k+2}$ (resp. $\widehat {b}_0\tau^{k+2}$). To avoid indeterminacy, we may assume, for instance, that $\widehat {a}'_0=0.$ Then $\widehat {f} = \lambda f,$ where $f(x,\,y) := \sum_{i=0}^{k-2} a_ix^iy^{k-3-i},\,\,\, a_i := \widehat {a}'_{i-1},\,i=0,\dots, k-3,$ and so $$\lambda^{k-1}h = \lambda f(\tau,\,\lambda) + \tau^{k-2}\widehat {g}\,.$$ Since $(\tau,\,\lambda) = 1,$ we have $\lambda \,\vert \,\widehat {g},$ that is, $\widehat {g} = \lambda g,$ where $g(t) := \sum_{i=0}^{2k-3} b_it^i.$ Finally, we arrive at the relation $$\lambda^{k-2}h(t) = f(\tau,\,\lambda) + \tau^{k-2}g(t)\,,$$ where deg$\,f = k-3,$ deg$\,h= $deg$\,g = 2k-3,$ and $h$ should satisfy the conditions ($15'$). It follows that $$\lambda^{k-2}\,\vert\,\,[f(\tau,\,\lambda) + \tau^{k-2}g]\,,\eqno{(16)}$$ and $$\tau^{k-2}\,|\,\,[f(\tau,\,\lambda) -\lambda^{k-2}h]\,.\eqno{(16')}$$ Each of these conditions together with ($15'$) determines the triple of polynomials $f,\,g,\,h$ as above in a unique way. Indeed, once $f$ and $g$ satisfy ($15'$) and (16), we can find $h$ as $h = [f(\tau,\,\lambda) + \tau^{k-2}g]/ \lambda^{k-2}.$ Actually, (16) is equivalent to the vanishing of derivatives of the function $f(\tau,\,\lambda) + \tau^{k-2}g \in {\bf C}[t]$ at the point $t = 1$ up to order $2k-5$ and at the point $t = -2$ up to order $k-3.$ This yields a system of $3k-6$ linear equations in the $3k-4$ unknown coefficients of $f$ and $g$; ($15'$) provides another two linear equations. That is, we have the following system: $$ \left(f(\tau,\,\lambda) + \tau^{k-2}g\right)^{(m)}_{t=-2} = 0,\,\,\,m=0,\dots,k-3$$ $$\left(f(\tau,\,\lambda) + \tau^{k-2}g\right)^{(m)}_{t=1} = 0,\,\,\,m=0,\dots,2k-5$$ $$\left(f(\tau,\,\lambda) + \tau^{k-2}g\right)^{(2k-4)}_{t=1} = -3^{k-2}(2k-2)!\eqno{(S)}$$ $$\left(f(\tau,\,\lambda) + \tau^{k-2}g\right)^{(2k-3)}_{t=1} = -3^{k-3}(k-11)(2k-1)!$$ (Indeed, put $u = t-1$; in view of ($15'$) we have $$\lambda = (t-1)^2(t+2) = u^2(u+3),\,\,\,\,h(t) = -1 + 3u + \dots\,,$$ and hence $$f(\tau,\,\lambda) + \tau^{k-2}g(t) = \lambda^{k-2}h(t) = [u^2(u+3)]^{k-2}h(t) =$$ $$ u^{2k-4}(3^{k-2} + (k-2)3^{k-3}u + \dots)(-1 + 3u + \dots) = u^{2k-4}(-3^{k-2} - 3^{k-3}(k-11)u + \dots)\,.)$$ The system ($S$) has a unique solution iff it is so for the associated homogeneous system, say, ($S_0$). Passing from ($S$) to ($S_0$) actually corresponds to passing from $h$ to a polynomial $h_0$ of degree $\le 2k-3$ which satisfies, instead of ($15'$), the conditions $$h_0(1) = h'_0(1) = 0 \Longleftrightarrow (t-1)^2 \,\vert\,\,h_0(t) \Longleftrightarrow h_0(t) = (t-1)^2 {\widetilde h}(t),\,\,\,{\rm deg}\,{\widetilde h} \le 2k-5\,. \,\eqno{(15'')}\,.$$ Thus, we have to prove that the equality $$\lambda^{k-2}(t-1)^2 {\widetilde h}(t) = f(\tau,\,\lambda) + \tau^{k-2}g(t)\,,$$ where $f=0$ or deg$\,f = k-3,$ deg$\,g \le 2k-3,$ and ${\rm deg}\,{\widetilde h} \le 2k-5,$ is only possible for $f=g={\widetilde h}=0.$ Or, equivalently, we have= to show that the $5k-8$ polynomials in $t$ in the union $T$ of the three systems: $$T_1:= \left\{\tau^i\lambda^{k-3-i}\right\}_{i=0,\dots,k-3},\,\,\, T_2:=\left\{t^i(t-1)^2\lambda^{k-2}\right\}_{i=0,\dots,2k-5},\,\,\, T_3:=\left\{t^i\tau^{k-2}\right\}_{i=0,\dots,2k-3}$$ are linearly independent. After replacing the system $T_2$ by the equivalent one: $$T'_2:= \left\{(t-1)^{2k-2}(t+2)^{k-2+i}\right\}_{i=0,\dots,2k-5}\,,$$ we will present these three systems as follows: $$T_1=\left\{p_i:=\tau^{k-3-i}\lambda^i=t^{3(k-2-i)}(t-1)^{2i}(t+2)^i, \,\,\,\,i=0,\dots,k-3\right\}$$ $$T'_2=\left\{p_i:=(t-1)^{2k-2}(t+2)^i, \,\,\,\,i=k-2,\dots,3k-7\right\}$$ $$T_3=\left\{p_i:=t^i,\,\,\,\,i=3k-6,\dots,5k-9\right\}\,.$$ Denote $P= $ span$\,(T_1,\,T_2,\,T_3)= $ span$\,(T_1,\,T'_2,\,T_3).$ Note that deg$\,p \le 5k-9$ for all $p \in P,$ that is, dim$\,P \le 5k-8.$ Consider the following system of $5k-8$ linear functionals on $P$: $$\varphi_i\,:\,p \longmapsto p^{(i)}(-2),\,\,\,\,i=0,\dots,3k-7\,,$$ $$\varphi_i\,:\,p \longmapsto p^{(i)}(0),\,\,\,\,i=3k-6,\dots,5k-9\,.$$ It is easily seen that the matrix $M:=\left(\varphi_i(p_j)\right)_{i,\,j=0,\dots,5k-9}$ is triangular with non-zero diagonal entries. This proves that, indeed, rang$\,T =$dim$\,P=5k-8,$ as stated. The coefficients of the system ($S$) being integers, its unique solution is rational, i.e.\ the polynomials $f$ and $g$ are defined over ${\bf Q}.$ It follows as above that the polynomials $h$ and $q$ are also defined over ${\bf Q}.$ This completes the alternative proof of Proposition 1.2. \hfill $\Box$ \bigskip \noindent {\bf Remarks.} (1) In principle, the method used in the proof allows to compute explicitly parameterizations of the curves $C_k.$ For instance, we saw above that for $k=2$ a parameterization of $C_2$ is given by the choice $$ q_2(s,t):=s^2+2st+3t^2,\quad a:=2,\quad b:=1\,. $$ \noindent (2) We have to apologize for a pity mistake in Lemma 4.1(b) [FlZa 2, Miscellaneous] (this does not affect the other results of [FlZa 2], besides only the immediate Corollary 4.2).

9709

alg-geom/9709033

en

https://arxiv.org/abs/alg-geom/9709033

alg-geom/9709033

Daisuke Matsushita

Daisuke Matsushita

On fibre space structures of a projective irreducible symplectic manifold

In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that an 2n-dimensional projective irreducible symplectic manifold admits only an n-dimensional fibration over a Fano variety which has only Q-factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, a general fibre is an abelian surface for 4-fold.

alg-geom

\section{Introduction} We first define an {\it irreducible symplectic manifold}. \begin{defn} A complex manifold $X$ is called {\it irreducible symplectic} if $X$ satisfies the following three conditions: \begin{enumerate} \item $X$ is compact and K\"{a}hler. \item $X$ is simply connected. \item $H^{0}(X,\Omega^{2}_{X})$ is spanned by an everywhere non-degenerate two-from $\omega$. \end{enumerate} \end{defn} Such a manifold can be considered as an unit of compact K\"{a}hler manifold $X$ with $c_1 (X) = 0$ due to the following Bogomolov decomposition theorem. \begin{thm}[Bogomolov decomposition theorem \cite{bogomolov}] A compact K\"{a}hler manifold $X$ with $c_{1}(X) = 0$ admits a finite unramified covering of $\tilde{X}$ which is isomorphic to a product $T \times X_1 \times \cdots \times X_r \times A$ where $T$ is a complex torus, $X_i$ are irreducible symplectic manifolds and $A$ is a projective manifold with $h^{0}(A,\Omega^{p}) = 0$, $0 < p < \dim A$. \end{thm} In dimension 2, $K3$ surfaces are the only irreducible symplectic manifolds, and irreducible symplectic manifolds are considered as higher-dimensional analogies of $K3$ surfaces. In this note, we investigate {\it fibre space structures} of a projective irreducible symplectic manifolds. \begin{defn} For an algebraic variety $X$, a fibre space structure of $X$ is a proper surjective morphism $f : X \to S$ which satisfies the following two conditions: \begin{enumerate} \item $X$ and $S$ are normal varieties such that $0 < \dim S < \dim X$ \item A general fibre of $f$ is connected. \end{enumerate} \end{defn} Some of $K3$ surface $S$ has a fibre space structure $f : S \to {\mathbb P}^{1}$ whose general fibre is an elliptic curve. In higher dimensional analogy, we obtain the following results. \begin{thm} Let $f : X \to B$ be a fibre space structure of a projective irreducible symplectic $2n$-fold $X$ with projective base $B$. Then a general fibre $F$ of $f$ and $B$ satisfy the following three conditions: \begin{enumerate} \item $F$ is an abelian variety up to finite unramified cover and $K_F \sim {\cal O}_{F}$. \item $B$ is $n$-dimensional and has only ${\mathbb Q}$-factorial log-terminal singularities \item $-K_B$ is ample and Picard number $\rho (B)$ is one. \end{enumerate} Especially, if $X$ is $4$-dimensional, a general fibre of $f$ is an abelian surface. \end{thm} {\sc Example. \quad} Let $S$ be a $K3$ surface with an elliptic fibration $g: S \to {\mathbb P}^{1}$ and $S^{[n]}$ a $n$-pointed Hilbert scheme of $S$. It is known that $S^{[n]}$ is an irreducible symplectic $2n$-fold and there exists a birational morphism $\pi : S^{[n]} \to S^{(n)}$ where $S^{(n)}$ is the symmetric $n$-product of $S$ (cf. \cite{beauville}). We can consider $n$-dimensional abelian fibration $ g^{(n)} : S^{(n)} \to {\mathbb P}^{n}$ for the symmetric $n$-product of $S^{(n)}$. Then the composition morphism $g^{(n)} \circ \pi : S^{[n]} \to {\mathbb P}^{n}$ gives an example of a fibre space structure of an irreducible symplectic manifold. \vspace{5mm} \noindent {\sc Remark. \quad} Markushevich obtained some result of theorem 2 in \cite[Theorem 1, Proposition 1]{mark1} under the assumption $\dim X = 4$ and $f : X \to B$ is the moment map. In general, a fibre space structure of an irreducible symplectic manifold is not a moment map. Markushevich constructs in \cite[Remark 4.2]{mark2} counterexample. \vspace{5mm} \noindent {\sc Acknowledgment. \quad} The author express his thanks to Professors Y.~Miyaoka, S.~Mori and N.~Nakayama for their advice and encouragement. He also thanks to Prof. D.~Huybrechts \cite{hyubrechts} for his nice survey articles of irreducible symplectic manifolds. \section{Proof of Theorems} First we introduce the following theorem due to Fujiki \cite{fujiki} and Beauville \cite{beauville}. \begin{thm}[\cite{fujiki} Theorem 4.7, Lemma 4.11, Remark 4.12 \cite{beauville} Th\`{e}or\'{e}me 5] \label{quadtatic} Let $X$ be an irreducible symplectic $2n$-fold. Then there exists a nondegenerate quadratic form $q_{X}$ of signature $(3,b_{2}(X) - 3)$ on $H^{2}(X , {\mathbb Z})$ which satisfies \begin{eqnarray*} \alpha^{2n} &=& a_0 q_{X}(\alpha , \alpha)^{n} \\ c_{2i}(X)\alpha^{2n - 2i} &=& a_i q_{X}(\alpha , \alpha )^{n-i} \quad (i \ge 1), \end{eqnarray*} where $\alpha \in H^{2}(X, {\mathbb Z})$ and $a_i$'s are constants depending on $X$. \end{thm} We shall prove theorem 2 in five steps. \begin{enumerate} \item $\dim B = n$ and $B$ has only log-terminal singularities; \item A general fibre $F$ of $f$ is an abelian variety up to unramified finite cover and $K_F \sim {\cal O}_{F}$; \item $\rho (B) = 1$; \item $B$ is ${\mathbb Q}$-factorial; \item $-K_B$ is ample. \end{enumerate} \noindent {\sc Step 1. \quad} $\dim B = n$ and $B$ has only log-terminal singularities. \begin{lem}\label{pseado} Let $X$ be an irreducible symplectic projective $2n$-fold and $E$ be a divisor on $X$ such that $E^{2n} = 0$. Then, \begin{enumerate} \item If $E.A^{2n-1} = 0$ for some ample divisor $A$, $E \equiv 0$. \item If $E.A^{2n-1} > 0$ for an ample divisor $A$ on $X$, then $$ \left\{ \begin{array}{ccc} E^{m}A^{2n-m} & = 0 & (m > n) \\ & > 0 & (m \le n) \end{array} \right. $$ \end{enumerate} \end{lem} {\sc Proof of lemma. \quad} Let $V := \{ E \in H^{2}(X , {\mathbb Z}) | E.A^{2n-1} = 0 \}$. By \cite[Lemma 4.13]{fujiki}, $q_{X}$ is negative definite on $W$ where $V = H^{2,0} \oplus H^{0,2} \oplus W$. Thus, if $E.A^{2n-1} = 0$ and $E^{2n} = 0$, $E \equiv 0$. Next we prove (2). From Theorem \ref{quadtatic}, for every integer $t$, \begin{equation}\label{key} (tE + A)^{2n} = a_0 (q_{X}(tE+A , tE+A))^n . \end{equation} Because $E^{2n} = a_0 (q_{X}(E,E))^n = 0$, $$ q_{X}(tE+A, tE+A) = 2tq_{X}(E,A) + q_{X}(A,A). $$ Thus the right hand side of the equation (\ref{key}) has order at most $n$. Comparing the both hand side of the equation (\ref{key}), we can obtain $E^{m}.A^{2n-m} = 0$ for $m > n$. If $E.A^{2n-1} > 0$, comparing the first order term of $t$ of both hand of the equation (\ref{key}) we can obtain $q_{X}(E,A) > 0$. Because coefficients of other terms of left hand side of (\ref{key}) can be written $q_{X}(E,A)$ and $q_{X}(A,A)$, we can obtain $E^{m}.A^{2n-m} > 0$ for $0 < m \le n$. \hspace*{\fill} $\Box$ \noindent Let $H$ be a very ample divisor on $B$. Then $f^{*}H$ is a nef divisor such that $(f^{*}H)^{2n} = 0$, $(f^{*}H).A^{2n-1} > 0$ for an ample divisor $A$ on $X$. Thus $\dim B = n$. From \cite[Theorem 2]{nakayama}, $B$ has only log-terminal singularities. \vspace{5mm} \noindent {\sc Step 2. \quad} A general fibre $F$ of $f$ is an abelian variety up to unramified finite cover and $K_F \sim {\cal O}_{F}$. \vspace{5mm} \noindent By adjunction, $K_F \sim 0$. Moreover $$ c_2 (F) = c_2 (X)(f^{*}H)^{2n - 2} = a_1 (q_{X}(f^{*}H,f^{*}H))^{n-1} = 0, $$ by Theorem \ref{quadtatic}. Thus $F$ has an \'{e}tale cover $\tilde{F} \to F$ such that $\tilde{F}$ is an Abelian variety by \cite{yau}. \vspace{5mm} \noindent {\sc Step 3. \quad} $\rho (B) = 1$. \begin{lem} Let $E$ be a divisor of $X$ such that $E^{2n} = 0$ and $E^{n}.(f^{*}H)^{n} = 0$. Then $E \sim_{{\mathbb Q}} \lambda f^{*}H$ for some rational number $\lambda$. \end{lem} {\sc Proof of lemma. \quad} Considering the following equation \begin{eqnarray*} (E - \lambda f^{*}H)^{2n} &=& a_0 q_{X}(E - \lambda f^{*}H , E - \lambda f^{*}H )^n \\ &=& a_0 (2\lambda q_{X}(E,f^{*}H))^{n}, \end{eqnarray*} we can obtain $q_{X}(E,f^{*}H) = cE^{n}.(f^{*}H)^{n} = 0$ where $c$ is a constant. Thus $(E - \lambda f^{*}H)^{2n} = 0$. Because $f^{*}H . A^{2n-1} > 0$ for every ample divisor $A$ on $X$, we can take a rational number $\lambda$ such that $(E - \lambda f^{*}H ).A^{2n-1} = 0 $ Then $E -\lambda f^{*}H \equiv 0$ by lemma \ref{pseado}. \hspace*{\fill} $\Box$ \noindent Let $D$ be a Cartier divisor on $B$. Then $(f^{*}D)^{2n} = 0$ and $(f^{*}D)^{n}.(f^{*}H)^{n} = 0$, thus $E \sim_{{\mathbb Q}} \lambda H$ and $\rho (B) = 1$. \vspace{5mm} \noindent {\sc Step 4. \quad} $B$ is ${\mathbb Q}$-factorial. \vspace{5mm} \noindent Let $D$ be an irreducible and reduced Weil divisor on $B$ and $D_i$, $(1 \le i \le k)$ divisors on $X$ whose supports are contained in $f^{-1}(D)$. We construct a divisor $\tilde{D} := \sum \lambda_i D_i$, $(\tilde{D} \not\equiv 0)$ such that $\tilde{D}^{2n} = 0$. Let $A$ be a very ample divisor on $X$, $S := A^{n-1}.(f^{*}H)^{n-1}$ and $C := H^{n-1}$. Then there exists a surjective morphism $f' : S \to C$. If we choose $H$ and $A$ general, we may assume that $S$ and $C$ are smooth and $C \cap D$ are contained smooth locus of $B$. Because $D$ is a Cartier divisor in a neighborhood of $C \cap D$, we can define $f^{'*}D$ in a neighborhood $U$ of $S$. We can express $f^{'*}D = \sum \lambda_i D_i$ in $U$ and let $\tilde{D} := \sum \lambda_i D_i$. Note that if $\lambda_i > 0$, $f(D_i) = D$ because we choose $C$ generally. Compareing the $n$th order term of $t$ of the both hand side of the following equotion \begin{eqnarray*} (\tilde{D} + tf^{*}H )^{2n} &=& a_0 q_{X}(\tilde{D}+ t f^{*}H, \tilde{D} + t f^{*}H )\\ &=& a_0 (q_{X}(\tilde{D},\tilde{D}) + 2t q_{X}(\tilde{D},f^{*}H) )^{n}, \end{eqnarray*} we can see that $ \tilde{D}^{n}.(f^{*}H)^{n} = cq_{X}(\tilde{D},f^{*}H)$. Since $f(\tilde{D}) = D$, $\tilde{D}^{n}.(f^{*}H)^{n} = 0$ and $q_{X}(\tilde{D},f^{*}H) = 0$. Considering the following equation \begin{eqnarray*} (s\tilde{D} + tA + f^{*}H)^{2n} &=& a_0 q_{X}( s\tilde{D} + tA + f^{*}H , s\tilde{D} + tA + f^{*}H)^n \\ &=& a_0 ( s^2 q_{X}(\tilde{D},\tilde{D}) + t^2 q_{X}(A,A) + 2stq_{X}(\tilde{D},A) \\ & & + 2t q_{X}(A,f^{*}H))^{n}, \end{eqnarray*} we can obtain $q_{X}(\tilde{D},\tilde{D})q_{X}(A,f^{*}H) = c \tilde{D}^2 .A^{n-1}. (f^{*}H)^{n-1}$ where $c$ is a constant. Since $\tilde{D}.A^{n-1}.(f^{*}H)^{n-1}$ is a fibre of $f'$, $\tilde{D}^2 .A^{n-1}.(f^{*}H)^{n-1} = 0$. Thus $a_0 q_{X}(\tilde{D},\tilde{D}) = \tilde{D}^{2n} = 0$. Considering $\tilde{D}^{n}(f^{*}H)^{n} = 0$, we can obtain $\tilde{D} \sim_{\mathbb Q} \lambda f^{*}H$ by Lemma \ref{pseado}, and $D \sim_{{\mathbb Q}} \lambda H$ because $f(\tilde{D}) = D$. Therefore $B$ is ${\mathbb Q}$-factorial. \vspace{5mm} \noindent {\sc Step 5. \quad} $-K_B$ is ample. \vspace{5mm} From Step 3,4, we can write $-K_B \sim_{{\mathbb Q}} tH$. It is enough to prove $t > 0$. Because $K_X \sim {\cal O}_{X}$ and a general fibre of $f : X \to B$ is a minimal model, $\kappa (B) \le 0$ by \cite[Theorem 1.1]{kawamata} and $t \ge 0$. Assume that $t = 0$. If $K_B \not\sim {\cal O}_{B}$, we can consider the following diagram: $$ \begin{array}{ccccc} & X & \to & B & \\ \alpha & \uparrow & & \uparrow & \beta \\ & \tilde{X} & \to & \tilde{B} & , \end{array} $$ where $\beta$ is an unramified finite cover and $K_{\tilde{B}} \sim {\cal O}_{\tilde{B}}$. Because $\pi_{1}(X) = \{1 \}$, $\tilde{X}$ is the direct sum of $X$. Thus there exists a morphism from $X$ to $\tilde{B}$ and we may assume that $K_B \sim {\cal O}_{B}$. Then there exists a holomorphic $n$-form $\omega'$ on $X$ coming from $B$. However, if $n$ is odd, it is a contradiction because there exist no holomorphic $(2k-1)$-form on $X$. If $n$ is even, it is also a contradiction because $\omega'$ dose not generated by $\omega \in H^{0}(X , \Omega^{2})$. Thus $t > 0$ and we completed the proof of Theorem 2. \hspace*{\fill} Q.E.D.

9709

alg-geom/9709027

en

https://arxiv.org/abs/alg-geom/9709027

alg-geom/9709027

Masahiko Saito

Shinobu Hosono, Masa-Hiko Saito, and Jan Stienstra

On Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3 folds

LaTeX Version 2.09, 36 pages. Submitted to The Proceedings of Taniguchi Symposium 1997, "Integrable Systems and Algebraic Geometry, Kobe/Kyoto"

In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the Calabi-Yau 3-fold directly by means of a theta function and Dedekind's eta function. This gives infinitely many Gromov-Witten invariants, and equivalently infinitely many sets of rational curves in the Calabi-Yau 3-fold. Using the toric mirror construction, we also calculate the prepotential of the B-model Yukawa couplings of the mirror partner. Comparing the expansion of the B-model prepotential with that of the A-model prepotential, we check a part of the Mirror Symmetry Conjecture up to a high order.

alg-geom

\section{Introduction} \label{intro} Let $W$ be a generic complete intersection variety in $\P1 \times \P2 \times \P2$ which is defined by two equations of multi-degrees $(1, 3, 0)$ and $(1, 0, 3)$ respectively. A generic $W$ is a non-singular Calabi-Yau 3-fold, which we call {\em Schoen's Calabi-Yau 3-fold}~\cite{Sch}. The purpose of this paper is to verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-folds. In \cite{COGP}, Candelas et al. calculated the prepotential of the B-model Yukawa couplings of the mirror of generic quintic hypersurfaces $X$ in $\P4$ and under the mirror hypothesis they gave predictions of numbers of rational curves of degree $d$ in $X$. Their predictions have been verified mathematically only for $d \leq 3$, that is, for numbers of lines, conics and cubic curves (cf. e.g. \cite{E-S}). On the B-model side one can compute as many coefficients as one wants and thus conjecturally count curves of any degree. However it is very hard to calculate the Gromov--Witten invariants directly on the A-model side. In this paper we calculate, (directly on the A-model side), a part of the prepotential of Schoen's Calabi-Yau 3-fold $W$ which gives infinitely many Gromov--Witten invariants of $W$. The main strategy of our verification is as follows: \begin{itemize} \item We will calculate a part of the prepotential of the A-model Yukawa couplings (for genus zero) of Schoen's Calabi-Yau 3-fold by using a structure of fibration $h:W \lra \P1$ by abelian surfaces. The theory of Mordell-Weil lattices~\cite{Man-1,Shi-1,Saito} allows us to calculate that part of the prepotential coming from sections of $h$. Under very plausible assumptions, we can count the ``numbers of {\em pseudo-sections}'', which makes it possible for us to obtain a very explicit description of the 1-sectional part of the A-model prepotential (cf. Theorem~\ref{t:1-s-prep}) in 19 variables by using a lattice theta function for $E_8$ and Dedekind's eta function. \item According to Batyrev-Borisov~\cite{Batyrev-Borisov} we can construct a mirror partner $W^*$ of $W$. The prepotential of the B-model Yukawa couplings of $W^*$ can be defined by means of period integrals of $W^*$. Following the recipe in \cite{HKTY,Sti} we expand the B-model prepotential in 3 variables by using the toric data. These 3-variables correspond to 3 elements of the Picard group of $W$ coming from line bundles on the ambient space $\P1 \times \P2 \times \P2$. \item By identifying the 3 variables with the corresponding 3 variables on the A-model side we have two expansions which should be compared. By a simple computer calculation we can verify the conjecture up to a high order. \end{itemize} To state the results for the A-model side let $f_i:S_i \lra \P1$ $(i =1, 2)$ be two generic rational elliptic surfaces. Then Schoen's generic Calabi-Yau 3-fold can be obtained as the fiber product $h:W =S_1 \times_{\P1}S_2 \lra \P1$. A general fiber of $h$ is a product of two elliptic curves. Hence after fixing the zero section the set of sections of $h$ becomes an abelian group. In this case the group of sections $MW(W)$ is a finitely generated abelian group and admits a N\'{e}ron-Tate height pairing. Let $B$ be the symmetric bilinear form associated to the N\'{e}ron-Tate height pairing. According to Shioda, we call the pair $(MW(W), B)$ of the group and the symmetric bilinear form a {\em Mordell-Weil lattice}. Under the genericity condition for $W$ and a suitable choice of a N\'{e}ron-Tate height we can easily see that the Mordell-Weil lattice is isometric to $E_8 \times E_8$. (Cf. \cite{Saito}). There is a very suitable set of 19 generators $[F], [L_i], [M_j]$ $(0 \leq i, j \leq 8)$ for the Picard group of $W$. We introduce the parameters $p, q_i, s_j$ corresponding to these generators. The divisor class $[F]$, which is the class of the fiber, has a special meaning in our context. A homology class $\eta$ is called $k$-sectional if the intersection number $([F], [\eta]) = k$. Let $\Psi_A$ denote the prepotential of the A-model Yukawa couplings of $W$ and $\Psi_{A, k}$ its $k$-sectional part for $k \geq 0$. Then we have an expansion like $$ \Psi_{A} = \mbox{topological part} + \sum_{k=0}^{\infty} \Psi_{A, k}. $$ Our main theorem can be stated as follows. For detailed notation, see Section~\ref{s:prep}. \begin{Theorem} $($cf. Theorem~\ref{t:mw-prep}, \ref{t:1-s-prep}$).$ Assume that Conjecture~\ref{c:cont} in Section~\ref{s:pseudosection} holds. Then for a generic Schoen's Calabi-Yau 3-fold $W$ the 1-sectional prepotential is given by \begin{eqnarray*} \lefteqn{\Psi_{A,1}(p, q_0, \cdots q_8, s_0, \cdots, s_8)} \nonumber \\ = & \sum_{n =1}^{\infty} \frac{1}{n^3} \cdot(p \cdot \prod_{i=1}^8 (q_i \cdot s_i))^n \cdot A(n \tau_1, n z_1, \ldots, n z_8)\cdot A(n \tau_2, n y_1, \ldots, n y_8). \end{eqnarray*} where $$ A(\tau, x_1, \ldots, x_8) = \Theta_{E_8}^{root}(\tau, x_1, \ldots,x_8) \cdot [ \sum_{m = 0}^{\infty} p(m) \cdot \exp(2 \pi m i \tau)]^{12}. $$ Here $p(m)$ denotes the number of partitions of $m$. \end{Theorem} (For the definition of the various notations see Theorem~\ref{t:mw-prep}). Since we can prove $\Psi_{A, 0} \equiv 0$ (cf. \cite{Saito}) and $\Psi_{A,k}, k \geq 2$ consists of terms divisible by $p^2$, we obtain an asymptotic expansion of $\Psi_A$ with respect to $p$: $$ \Psi_{A} = \mbox{topological term} + p \cdot \prod_{i=1}^8(q_i s_i)\cdot A(\tau_1, {\bf z}) \cdot A(\tau_2, {\bf y}) + O(p^2) $$ where ${\bf z} = ( z_1, \cdots, z_8), {\bf y} = (y_1, \cdots, y_8)$. In Section~\ref{s:appendixII} we show that $\Theta_{E_8}^{root}$ has a very simple expression in terms of the standard Jacobi theta functions. On the other hand on the B-model side there are only 3 parameters involved in the calculation, because the Batyrev-Borisov construction can only deal with generators of the Picard group of $W$ coming from the ambient toric variety $\P1 \times \P2 \times \P2$. One can easily find the corresponding parameters $p = U_0, U_1 = \exp(2 \pi i t_1), U_2 = \exp(2 \pi i t_2)$ on the A-model side and one can obtain the following expansion: $$ \Psi_{A}^{res}(p, t_1, t_2) = \mbox{topological term} + p \cdot A( 3t_1, t_1 \gamma ) \cdot A(3 t_2, t_2 \gamma) + O(p^2), $$ where $\gamma=(1,1,1,1,1,1,1,-1)$. We also obtain an expansion of the $B$-model prepotential $$ \Psi_{B}(p, t_1, t_2) = \mbox{topological term} + p \cdot B(t_1) \cdot B(t_2) + O(p^2). $$ Therefore in this context the mirror symmetry conjecture can be stated as $$ \Psi_{A}^{res}(p, t_1, t_2) \equiv \Psi_{B}(p, t_1, t_2). $$ {From} the above asymptotic expansion, we come to a concrete mathematical conjecture: $$ A(3t, t \gamma) \equiv B(t). $$ At this moment we can calculate both sides up to high order of $U = \exp(2 \pi i t)$ by using computer programs. One can find the expansion of $A(t)$ up to the order of 50 at the end of Section~\ref{s:rest} and also the expansion of $B(t)$ at the end of Section~\ref{s:B-model}. The rough plan of this paper is as follows. In Section~\ref{s:schoen} we recall a basic property of Schoen's Calabi-Yau 3-fold and its toric description. In Section~\ref{s:mw} we recall the Mordell-Weil lattice which will be essential in the later sections. In Section~\ref{s:prep} we first recall the definition of Gromov-Witten invariants and the A-model prepotential. We calculate the Mordell-Weil part of the prepotential in terms of the lattice theta function $\Theta_{E_8}^{root}$ and state the main theorem (Theorem~\ref{t:1-s-prep}). Section~\ref{s:pseudosection} is devoted to counting the pseudo-sections in $W$. We also prove Theorem~\ref{t:1-s-prep} here. In Section~\ref{s:rest} we restrict the parameters of the A-model prepotential in order to compare the expansion with that of the B-model prepotential of the mirror. A table for the coefficients $\{a_n\}$ of $A(3t, t \gamma)$ is given up to order $50$(cf. Table 2 in Section~\ref{s:rest}). In Section~\ref{s:B-model} after recalling the formulation of the mirror symmetry conjecture we calculate the B-model prepotential of the mirror of Schoen's example following the recipe of \cite{HKTY,Sti}. We expand the function $B(t)$ whose coefficients $\{ b_n \}$ should coincide with $\{ a_n \}$ if the mirror symmetry conjecture is true. We check the coincidence up to order 50. In Appendix I (Section~\ref{s:B-model equations}) we derive the equation of the mirror according to the Batyrev-Borisov construction~\cite{Batyrev-Borisov}. In Appendix II (Section~\ref{s:appendixII}) we give a formula for the theta function of the lattice $E_8$. \vspace{5mm} Let us mention some papers which are related to our work. In the paper~\cite{D-G-W}, Donagi, Grassi and Witten calculate the non-perturbative superpotential in $F$-theory compactification to four dimensions on $\P1 \times S$, where $S$ is a rational elliptic surface. It is interesting enough to notice that the supperpotential in their case is also described by the lattice theta function for $E_8$. It is interesting that they also mention the contribution of Dedekind's eta function $\eta(\tau)$ to the superpotential, though we do not know any direct relation between $F$-theory and the Type II theory. In \cite{G-P}, G\"{o}ttsche and Pandharipande calculated the quantum cohomology of blowing-ups of $\P2$. Their calculation for the blowing-up of $9$-points in {\em general position} on $\P2$ is certainly related to our calculation for the rational elliptic surfaces. Moreover, in \cite{Y-Z} S.-T. Yau and Zaslow describe the counting of BPS states of Type II on K3 surfaces. In the paper, they treated rational curves with nodes, which may have some relation to our treatment of pseudo-sections. \vspace{5mm} \section{Schoen's Calabi-Yau 3-folds} \label{s:schoen} Let $f_i:S_i \lra {\bf P}^1$ ($i =1, 2$) be two smooth rational surfaces defined over ${\bf C}$. In this paper we always assume that an elliptic surface has a section. In \cite{Sch} C. Schoen showed that the fiber product of two rational elliptic surfaces $$ \begin{array}{ccccc} & W= & S_1 \times_{{\bf P}^1} S_2 & & \\ & & & & \\ & \swarrow p_1 & \quad & p_2 \searrow& \\ S_1 & & \downarrow h & & S_2 \\ & f_1 \searrow & & \swarrow f_2 & \\ & & {\bf P}^1 & & \end{array} $$ becomes a Calabi-Yau 3-fold after small resolutions of possible singularities of the fiber product. In what follows we consider such Calabi-Yau 3-folds which satisfy the following genericity assumption. \begin{Assumption} \label{as:generic} \begin{enumerate} \item The rational elliptic surfaces $f_i:S_i \lra {\bf P}^1$ ($i=1, 2$) are generic in the sense that the surfaces $S_i$ are smooth and every singular fiber of $f_i$ is of Kodaira type $I_1$, that is, a rational curve with one node. Then one can see that each fibration $f_i$ has exactly 12 singular fibers of type $I_1$. $($cf. $\cite{Kod})$. \item Let $\Sigma_i \subset {\bf P}^1$ be the set of critical values of $f_i$. Then we assume that $\Sigma_1 \cap \Sigma_2 = \emptyset$. \end{enumerate} \end{Assumption} Under Assumption~\ref{as:generic} the fiber product $W = S_1 \times_{{\bf P}^1} S_2 $ becomes a {\em nonsingular} Calabi-Yau 3-fold. The following facts are well-known. (See \cite{Kod} or \cite{Man-1}). \begin{Lemma} \label{l:el} Let $S_1, S_2, W$ be as above. \begin{enumerate} \item All fibers of $h:W \lra {\bf P}^1$ have vanishing topological Euler numbers. Hence we have $e(W) = 2(h^{1,1}(W) - h^{2,1}(W)) = 0$. \item A generic rational elliptic surface with section is obtained by blowing-up the 9 base points of a cubic pencil on ${\bf P}^2$. Let $\pi_1:S_1 \lra {\bf P}^2$ and $\pi_2:S_2 \lra {\bf P}^2$ be the blowing-ups and $E_i, i=1, \cdots, 9$ and $E'_j, j=1, \cdots, 9$ the divisor classes of the exceptional curves of $\pi_1$ and $\pi_2$ respectively. Set $H_i = \pi_i^*({\cal O}_{{\bf P}^2}(1))$. Then we have \begin{eqnarray} \Pic(S_1) & = & {\bf Z} H_1 \oplus {\bf Z} E_1 \oplus \cdots \oplus {\bf Z} E_9, \label{eq:pics1}\\ \Pic(S_2) & = & {\bf Z} H_2 \oplus {\bf Z} E'_1 \oplus \cdots \oplus {\bf Z} E'_9. \label{eq:pics2} \end{eqnarray} \item Let $F_1$ and $F_2$ be the divisor classes of the fibers of $f_1$ and $f_2$ respectively. Then we have \begin{eqnarray} F_1 = 3H_1 - \sum_{i=1}^{9} E_i, \quad F_2 = 3H_2 - \sum_{i=1}^{9} E'_i \label{eq:f-h} \end{eqnarray} \item We have the following isomorphism of groups. \begin{eqnarray} \Pic(W) \simeq p_1^*(\Pic(S_1)) \oplus p_2^*(\Pic(S_2))/ {\bf Z} [p_1^*(F_1) - p_2^*(F_2)] \label{eq:pic-w} \end{eqnarray} Hence the Picard number of $W$ is $h^{11}(W) = 19$. Also $h^{21}(W) = 19$ because $e(W) = 0$. \end{enumerate} \end{Lemma} {}\hfill$\Box$ We now show that Schoen's Calabi-Yau $3$-fold $W$ can also be realized as a complete intersection in the toric variety $\P1 \times \P2 \times \P2$. Let $z_0, z_1$, $x_0, x_1, x_2$, $y_0, y_1, y_2$ be the homogeneous coordinates of $\P1 \times \P2 \times \P2$ and let $$ a_0(x_0, x_1, x_2), a_1(x_0, x_1, x_2), b_0(y_0, y_1, y_2), b_1(y_0, y_1, y_2) $$ be generic homogeneous cubic polynomials. Then we can assume that the generic rational elliptic surfaces $S_1$ and $S_2$ in Lemma \ref{l:el} are obtained as hypersurfaces in $\P1 \times \P2$ as in the following way. \begin{eqnarray*} S_1 & =& \{P_1 = z_1\cdot a_0(x_0, x_1, x_2) - z_0 \cdot a_1(x_0, x_1, x_2) = 0 \} \subset \P1 \times \P2 \\ S_2 & =& \{P_2 = z_1 \cdot b_0(y_0, y_1, y_2) - z_0 \cdot b_1(y_0, y_1, y_2)= 0 \} \subset \P1 \times \P2 \end{eqnarray*} We have the natural morphisms $$ \begin{array}{cccclc} & & S_i & &\subset \P1 \times \P2 & \\ &f_i \swarrow & &\searrow\pi_i & & \\ & \P1 & &\P2 \ , & & \end{array} $$ where $f_1 =(p_1)_{|S_i}, \pi_i = (p_2)_{|S_i}$. Moreover, one can easily see that $W = S_1 \times_{{\bf P}^1} S_2 $ can be obtained as a complete intersection in the toric variety $ \P1 \times \P2 \times \P2$ of types $(1, 3, 0) $, $(1, 0, 3)$: $$ W = \left\{ \begin{array}{l|c} [z_0:z_1] \times [ x_0: x_1: x_2]\times[ y_0: y_1: y_2] & P_1 = 0 \\ \in \P1 \times \P2 \times \P2 & P_2 = 0 \end{array}\right\} $$ \section{Mordell-Weil lattices} \label{s:mw} The purpose of this section is a review of results on Mordell-Weil lattices which is needed to calculate a part of the prepotential of the A-model Yukawa couplings of Schoen's Calabi-Yau 3-folds. For more complete treatments the reader may refer to \cite{Man-1}, \cite{Shi-1}, \cite{Saito}. We keep the notation and assumptions of the previous section, that is, let $f_i:S_i \lra {\bf P}^1$ be rational elliptic surfaces and let $h: W = S_1 \times_{{\bf P}^1} S_2 \lra \P1$ be the fiber product. Let $MW(S_i), i =1, 2$ and $MW(W)$ denote the set of sections of $f_i$ and $h$ respectively. Since the exceptional curves of the blowing-ups $\pi_i : S_i \lra \P2$ are the images of sections of $f_i$, we denote by $e_1$ and $e'_1$ the sections of $f_1$ and $f_2$ respectively such that $e_1(\P1)=E_1 $ and $e'_1(\P2)= E_1'$. We take $e_1$ and $e'_1$ as zero sections of $f_1$ and $f_2$ respectively. Then $MW(S_1)$ and $MW(S_2)$ become finitely generated abelian groups with the identity elements $e_1$ and $e'_1$ respectively. The group $MW(S_i)$ is called the Mordell-Weil group of the rational elliptic surface $f_i:S_i \lra \P1$. Take the line bundles $$ L_0 = E_1 + F_1 \in \Pic(S_1),\;\; M_0 = E'_1 + F_2 \in \Pic(S_2). $$ Note that these line bundles are symmetric \footnote{A line bundle on a fibration of abelian varieties is called symmetric if it is invariant under the pull-back by the inverse automorphism ${\bf z} \rightarrow - {\bf z}$.} and numerically effective and $(L_0)^2 = (M_0)^2 = 1$. Hence $L_0$ and $M_0$ are nearly ample line bundles and $E_1$ (resp. $E_1'$) is the only irreducible effective curve on $S_1$ (resp. $S_2$) with $(L_0, E_1)_{S_1} = 0$ (resp. $(M_0, E'_1)_{S_2}= 0$). (Here $(C, D)_{S_i}$ denotes the intersection pairing of curves on the surface $S_i$. Later we sometimes identify this pairing with the natural pairing $H^2(S_i) \times H_2(S_i) \lra {\bf Z}$ via Poincar\'{e} duality. ) Thanks to Manin \cite{Man-1} we can define N\'{e}ron-Tate heights with respect to $2L_0$ and $2 M_0$, that is, quadratic forms on $MW(S_i)$ by \begin{eqnarray} Q_1(\sigma_1 ) = (2 L_0, \sigma_1(\P1) )_{S_1}, \quad Q_2(\sigma_2 ) = (2 M_0, \sigma_2(\P1) )_{S_2} \label{eq:height} \end{eqnarray} for $\sigma_1 \in MW(S_1)$ and $\sigma_2 \in MW(S_2)$. Let $B_i$ denote the positive definite symmetric bilinear form associated to the quadratic form $Q_i$, i.e. $B_i(\sigma,\sigma')= {1\over2}\{Q_i(\sigma+\sigma')-Q_i(\sigma)-Q_i(\sigma')\}.$ According to Shioda \cite{Shi-1} we call $(MW(S_i), B_i)$ {\em the Mordell-Weil lattice} of $f_i:S_i \lra \P1$. Noting that our N\'{e}ron-Tate height coincides with Shioda's \cite{Shi-1} we can show the following proposition. \begin{Proposition} Under Assumption~\ref{as:generic} in \S~\ref{s:schoen}, we have the following isometry of lattices. $$ (MW(S_i), B_i) \simeq E_8, \quad (i =1, 2) $$ where $E_8$ is the unique positive-definite even unimodular lattice of rank $8$. \end{Proposition} {}\hfill$\Box$ Next we consider the Mordell-Weil group $MW(W)$ of $h:W \lra \P1$, whose zero section corresponds to $(e_1, e'_1)$ (cf. (\ref{eq:isom})). {From} a property of the fiber product we have the following isomorphism: \begin{eqnarray} MW(W) & \stackrel{\sim}{\lra} & MW(S_1) \oplus MW(S_2) \label{eq:isom} \\ \sigma & \mapsto & (\sigma_1, \sigma_2) = ( p_1 \circ \sigma, \ p_2 \circ \sigma) \nonumber \end{eqnarray} Since the Picard group $\Pic(W)$ can be described as in (\ref{eq:pic-w}), we will use the following notation for the line bundles on $W$ pulled back by $p_1$ and $p_2$: $$ \begin{array}{lll} \ [F] = p_1^* (F_1)=p_2^* (F_2), & & \\ \ [H_1] = p_1^* (H_1), & [L_0] = p_1^* (L_0), & [E_i] = p_1^* (E_i), \quad (i =1, \cdots, 9), \\ \ [H_2]= p_2^* (H_2), & [M_0]=p_2^* (M_0), & [E_j'] = p_2^* (E_j'), \quad (j =1, \cdots, 9). \end{array} $$ We can easily see that $[J_0] :=[L_0] + [M_0] $ is a symmetric numerically effective line bundle on $W$. It defines a N\'{e}ron-Tate height on $MW(W)$ as follows. For each $\sigma \in MW(W)$ we set \begin{eqnarray} Q_W(\sigma) := ([2J_0], [\sigma(\P1)])_W. \end{eqnarray} Here $( \quad , \quad )_W$ denotes the natural pairing $H^2(W) \times H_2(W) \rightarrow {\bf Z}$. Note that the zero section of $MW(W)$ is $0_W =(e_1, e'_1) $ and $Q_W(0_W) = 0$. {From} this we obtain the Mordell-Weil lattice $(MW(W), B_W)$ where $B_W$ denotes the symmetric bilinear form associated to $Q_W$. Moreover we obtain the following relation for each section $\sigma \in MW(W)$: \begin{eqnarray} Q_W(\sigma)& =& ([2J_0], [\sigma(\P1)])_W \nonumber \\ &=& (2L_0, [\sigma_1(\P1)])_{S_1} + (2M_0, [\sigma_2(\P1)])_{S_2} = Q_1(\sigma_1) + Q_2(\sigma_2). \label{eq:qw} \end{eqnarray} Therefore we find the following \begin{Proposition} The N\'{e}ron-Tate height with respect to $[2J_0]$ on $MW(W)$ gives a lattice structure on $MW(W)$ which induces the isometry: $$ (MW(W), B_W) \simeq (MW(S_1), B_1) \oplus (MW(S_2), B_2) \simeq E_8 \oplus E_8. $$ \end{Proposition} {}\hfill$\Box$ There are natural maps $$ \begin{array}{cccl} j :&MW(S_i) &\lra & H_2(S_i) \\ & \sigma & \mapsto & j(\sigma) = [\sigma(\P1)]= \mbox{the homology class of the curve } \sigma(\P1) \\ & & & \\ j :&MW(W) &\lra & H_2(W) \\ & \sigma & \mapsto & j(\sigma) = [\sigma(\P1)]. \end{array} $$ For each section $\sigma_i \in MW(S_i)$, we can always find a birational morphism $\varphi_i:S_i \lra \overline{S_i}$ which contracts only the image of the section $\sigma_i(\P1)$. This implies the following lemma. \begin{Lemma}\label{l:j-inj} The maps $j$ are injective. \end{Lemma} {}\hfill$\Box$ Note that the maps $j$ are {\em not} homomorphisms of groups.\footnote{However, Shioda~\cite{Shi-1} obtained a way to modify the map $j$ to obtain a natural homomorphism. See \cite{Shi-1} or \cite{Saito}.} \vspace{5mm} Next we will choose other generators of $\Pic(S_i)$. These generators will be used for defining the parameters in which we will expand the prepotential of the A-model Yukawa coupling of Schoen's Calabi-Yau 3-folds. Let $(MW(S_i), B_i)$ be the Mordell-Weil lattices of $S_i$, which are isometric to the lattice $E_8$. We choose a set of simple roots $ \{ \alpha_1, \alpha_2, \cdots \alpha_8 \} $ of $E_8$ whose intersection pairing will be given by the following Dynkin diagram. \begin{picture}(315,120)(-50,0) \put(40,70){\circle{30}} \put(35,68){$\alpha_1$} \put(55,70){\line(1,0){14}} \put(85,70){\circle{30}} \put(81,68){$\alpha_2$} \put(100,70){\line(1,0){14}} \put(130,70){\circle{30}} \put(125,68){$\alpha_3$} \put(145,70){\line(1,0){14}} \put(130,25){\circle{30}} \put(130,54.5){\line(0, -1){14}} \put(125,23){$\alpha_8$} \put(175,70){\circle{30}} \put(170,68){$\alpha_4$} \put(190,70){\line(1,0){14}} \put(220,70){\circle{30}} \put(215,68){$\alpha_5$} \put(235,70){\line(1,0){14}} \put(265,70){\circle{30}} \put(260,68){$\alpha_6$} \put(280,70){\line(1,0){14}} \put(310,70){\circle{30}} \put(306,68){$\alpha_7$} \put(110,-15){Figure 1.} \end{picture} \vspace{5mm} \vspace{5mm} We also choose $a_1, \cdots, a_8 \in MW(S_1)$ (resp. $b_1, \cdots, b_8 \in MW(S_2)$) corresponding with the roots of $MW(S_1)$ (resp. $MW(S_2)$) so that the numbering of the roots is the same as in Figure 1. For each section $\sigma \in MW(S_i)$, one can define a translation automorphism $T_{\sigma}:S_i \lra S_i$: $$ \begin{array}{ccccc} S_i & & \stackrel{T_{\sigma}}{\lra} & & S_i \\ & f_i \searrow & & \swarrow f_i & \\ & & {\bf P}^1. & & \end{array} \ $$ Pulling back the line bundles $L_0$ and $M_0$ by the translation automorphisms $T_{a_i}$ and $T_{b_j}$ respectively, we define the line bundles \begin{eqnarray} L_i = T^*_{a_i} (L_0) \in \Pic(S_1), \quad M_j = T^*_{b_j} (M_0) \in \Pic(S_2), \end{eqnarray} for $1 \leq i, j \leq 8$. Then for each section $\sigma_i \in MW(S_i)$ we have \begin{eqnarray*} (L_i, j(\sigma_1))_{S_1} &= &(T_{a_i}^*(L_0), j(\sigma_1))_{S_1} =(L_0, j(\sigma_1 + a_i))_{S_1} = \frac{1}{2}Q_1(\sigma + a_i) \\ (M_i, j(\sigma_2))_{S_2} &= &(T_{b_i}^*(M_0), j(\sigma_2))_{S_2} =(M_0, j(\sigma_2 + b_i))_{S_2} = \frac{1}{2}Q_2(\sigma_2 + b_i) \end{eqnarray*} Now it is easy to see the following: \begin{Lemma}\label{l:gen} \begin{enumerate} \item The classes $F_1, L_0, L_1, \cdots, L_8 $ $($ resp. $F_2, M_0, M_1, \cdots, M_8 )$ are generators of $\Pic(S_1)$ (resp. $\Pic(S_2)$). \item $\Pic(W) $ is generated by $[F], [L_0], \cdots, [L_8], [M_0], [M_1], \cdots, [M_8]$. \item For $\sigma \in MW(W)$ set $\sigma_i = \sigma \circ p_i$. Then we have $$ \begin{array}{lcl} ([F], j(\sigma))_W & =& 1 \\[.5em] ([L_i], j(\sigma))_W & =& \frac{1}{2} Q_1(\sigma_1 + a_i), \\[.5em] ([M_i], j(\sigma))_W & =& \frac{1}{2} Q_2(\sigma_2 + b_i). \end{array} $$ \end{enumerate} \end{Lemma} {}\hfill$\Box$ Moreover, in order to see the relation between the A-model and the B-model later, we have to express $H_1$ and $H_2$ by $F_1, \{ L_i \}$ and $F_2, \{ M_j \}$. Obviously, we only have to see the case of $H_1$. Recall that the exceptional curves $\{ E_i \}$ in (\ref{eq:pics1}) are the images of sections of $f_1$. We denote by $e_i \in MW(S_1)$ the section corresponding to $E_i$; hence we have $e_i(\P1) = E_i$. In particular, $e_1$ is the zero section of $f_1:S_1 \lra \P1$. As for the system of roots, one can take the following elements: $$ a_1 = e_2, \ a_2 = e_3 - e_2, \ a_3 = e_4 - e_3, \cdots, a_7 = e_8 - e_7, $$ and $$ a_8 = e_2 + e_3 - \frac{1}{3} \sum_{i =2}^9 e_i. $$ Here all sums are taken in the Mordell-Weil group. We denote by $(\sigma) \in H^2(S_1, {\bf Z})$ the divisor class of the curve $\sigma(\P1) \subset S_1$. Since $L_0 = E_1 + F_1 = (e_1) + F_1$, we see that $$ L_i = T^*_{a_i}(L_0) = T^*_{a_i}((e_1) + F_1) = (-a_i) + F_1. $$ Moreover we can see the following relation. (For divisor classes $(-a_i)$ one may refer to \cite{Saito}). $$ \begin{array}{lclcl} L_0 & = & E_1 + F_1& & \\ L_1 & = & (-a_1) + F_1 & = & 2 E_1 -E_2 + 3 F_1 \\ L_2 & = & (-a_2) + F_1 & = & E_1 + E_2 - E_3 + 2 F_1 \\ L_3 & = & (-a_3) + F_1 & = & E_1 + E_3 -E_4 + 2 F_1 \\ L_4 & = & (-a_4) + F_1 & = & E_1 + E_4 - E_5 + 2F_1 \\ L_5 & = & (-a_5) + F_1 & = & E_1 + E_5 - E_6 + 2 F_1\\ L_6 & = & (-a_6) + F_1 & = & E_1 + E_6 - E_7 + 2 F_1\\ L_7 & = & (-a_7) + F_1 & = & E_1 + E_7 - E_8 + 2 F_1\\ L_8 & = & (-a_8) + F_1 & = & \frac{1}{3}\sum_{i=1}^9 E_i - (E_2 + E_3) +\frac{4}{3} F_1. \end{array} $$ Recall also the relation~(\ref{eq:f-h}): \begin{eqnarray*} F_1 = 3H_1 - \sum_{i=1}^{9} E_i. \end{eqnarray*} {From} these linear relations one easily derives the following: \begin{Lemma} One has the following relation in $\Pic(S_1)$: \begin{eqnarray} H_1 = 2 F_1 + 5 L_0 -2 L_1 - L_2 + L_8. \label{eq:h-f} \end{eqnarray} \end{Lemma} {}\hfill$\Box$ \section{The prepotential of the A-model Yukawa couplings and its 1-sectional part} \label{s:prep} In this section we summarize a result in (\cite{Saito}) on the Mordell-Weil part of the prepotential of the A-model Yukawa coupling of Schoen's Calabi-Yau 3-folds. The main theorems are Theorem~\ref{t:mw-prep} and Theorem~\ref{t:1-s-prep}. Following Section 3.3 in \cite{Mor-Math}, we define the (full) {\em A-model Yukawa coupling} for a Calabi-Yau 3-fold $X$ as a triple product on $H^2(X, {\bf Z})$: \begin{eqnarray} \Phi_A(M_1, M_2, M_3) = (M_1, M_2, M_3) + \sum_{0 \not\equiv \eta \in H_2(X, {\bf Z})} \Phi_{\eta}(M_1, M_2, M_3) \frac{q^{\eta}}{1 - q^{\eta}}. \label{eq:a-yukawa} \end{eqnarray} Here $M_1, M_2, M_3$ are elements of $ H^2(X, {\bf Z}) \cong \Pic(X)$ and $\Phi_{\eta}(M_1, M_2, M_3)$ is the Gromov-Witten Invariant for $ \eta \in H_2(X, {\bf Z})$ and $M_1, M_2, M_3$. Moreover, we have (cf. Section 3.2, \cite{Mor-Math}): \begin{eqnarray} \Phi_{\eta}(M_1, M_2, M_3) &= &n(\eta) (M_1, \eta)(M_2, \eta)(M_3, \eta). \label{eq:g-w.inv} \end{eqnarray} Here $(M_i, \eta)$ denote the natural pairing of $M_i \in H^2(X)$ and $\eta \in H_2(X) $ and $n(\eta)$ denotes the number of simple rational curves $\varphi:{\bf P}^1 \lra X $ with $\varphi_*([{\bf P}^1]) = \eta$. A more precise definition by $J$-holomorphic curves can be found in \cite{McD-S-1} and Lecture 3 of \cite{Mor-Math}. \vspace{5mm} The full Yukawa coupling $\Phi_A$ has the {\em prepotential} $\Psi_A$ defined by \begin{equation} \Psi_A = \mbox{(topological term)} + \sum_{0 \not\equiv \eta \in H_2(X, {\bf Z})} n(\eta) \Li_3( q^{\eta}), \end{equation} where \begin{equation} \Li_3( x ) = \sum_{n=1}^{\infty} \frac{x^n}{n^3} \end{equation} is the trilogarithm function. In general it is very difficult to calculate the prepotential of the A-model Yukawa coupling. Even for Schoen's Calabi-Yau 3-fold, we can not calculate the full prepotential, but by using the structure of its Mordell-Weil lattice, we can calculate a part of the prepotential $\Psi_A$ whose summation is taken just over the homology 2-cycles of sections of $h:W \lra \P1$. Later we will extend the summation to all homology classes of {\em pseudo-sections } (see Section~\ref{s:pseudosection}). (Cf. \cite{Saito}). \begin{Definition} \label{def:prep} {\rm For Schoen's generic Calabi-Yau 3-fold $W$ we define the {\em Mordell-Weil part of the prepotential of the A-model Yukawa coupling} by \begin{eqnarray} \Psi_{A, MW(W)} = \sum_{\sigma \in MW(X)} n(j(\sigma)) \Li_3 ( q^{j(\sigma)}). \label{eq:prepdef} \end{eqnarray} Here again $j(\sigma) $ denotes the homology class of the image $\sigma(\P1)$. } \end{Definition} \begin{Definition} \label{def:sec-prep} {\rm We define the $k$-sectional part of the prepotential of the A-model Yukawa coupling by \begin{eqnarray} \Psi_{A, k} = \sum_{0 \not\equiv \eta \in H_2(X, {\bf Z}), \ (F, \eta)= k} n(\eta) \Li_3( q^{\eta}). \label{eq:sec-prep} \end{eqnarray} Recall that we denote by $[F]$ the divisor class of the fiber of $h:W \lra \P1$.} \end{Definition} Obviously, we have the expansion \begin{eqnarray} \Psi_{A} = \mbox{topological term} + \sum_{k = 0}^{\infty} \Psi_{A, k}. \label{eq:expansion} \end{eqnarray} We are interested in calculating the functions $\Psi_{A, MW(W)}$ and $\Psi_{A, 1}$. \begin{Remark} {\rm We will find a difference in $\Psi_{A, MW(W)}$ and $ \Psi_{A, 1}$, which will be explained in the next section by introducing the notion {\it pseudo-section}. } \end{Remark} We first recall a result in \cite{Saito} on the calculation of $\Psi_{A, MW(W)}$ by using the theta function of the Mordell-Weil lattice. We need to introduce the special formal parameters in order to get explicit expansions of $\Psi_{A, MW(W)}$. Let $f_i:S_i \lra {\bf P}^1$ be two generic rational elliptic surfaces and let $h:W \lra \P1$ be the Calabi-Yau 3-fold as in Section~\ref{s:schoen}. Then from Lemma~\ref{l:gen} $\Pic(W) $ is generated by $[F], [L_0], \cdots, [L_8],$ $[M_0], [M_1], \cdots, [M_8]$. We introduce formal parameters $p, q_i, s_j$ for $0 \leq i, j \leq 8 $ corresponding to these generators: \begin{eqnarray} [F] \leftrightarrow p, \quad [L_i] \leftrightarrow q_i, \quad [M_j] \leftrightarrow s_j. \label{eq:parameter} \end{eqnarray} By using the formal parameters we can associate to $\sigma \in MW(W)$ the monomials \begin{eqnarray} q^{\sigma} = \prod_{i=0}^8 q_i^{([L_i], j(\sigma))_W}, \quad s^{\sigma} = \prod_{i=0}^8 s_i^{([M_i], j(\sigma))_W}. \label{eq:prod} \end{eqnarray} and \begin{equation} T^{\sigma} = p^{([F], j(\sigma))_W} \cdot q^{\sigma} \cdot s^{\sigma} = p \cdot q^{\sigma} \cdot s^{\sigma}. \label{eq:tsigma} \end{equation} Here $( \quad, \quad)_W:H^2(W) \times H_2(W) \lra {\bf Z} $ is the natural pairing. Note that all line bundles $[F], [L_i], [M_j]$ are numerically effective. Hence all exponents in $T^{\sigma}$ are non-negative. Now we can expand $\Psi_{A, MW(W)}$ in the parameters $p, q_i, s_j$. \begin{Theorem}\label{t:mw-prep} \begin{eqnarray} \lefteqn{\Psi_{A, MW(W)}(p, q_0, \cdots q_8, s_0, \cdots, s_8)} \nonumber \\ = & \sum_{n =1}^{\infty} \frac{1}{n^3} \cdot (p \cdot \prod_{i=1}^8 (q_i \cdot s_i))^n \cdot \Theta_{E_8}^{root}(n \tau_1, n \cdot {\bf z}) \cdot \Theta_{E_8}^{root}(n \tau_2, n \cdot {\bf y}) \label{eq:main} \end{eqnarray} Here, we set \begin{eqnarray*} {\bf z} = (z_1, \ldots, z_8), \quad {\bf y} = (y_1, \ldots, y_8), \end{eqnarray*} \begin{eqnarray} \exp( 2 \pi i \tau_1) = \prod_{i=0}^8 q_i, & \exp(2 \pi i z_i)= q_i \quad \mbox{for } 1 \leq i \leq 8 \\ \exp(2 \pi i \tau_2) = \prod_{i=0}^8 s_i, & \exp(2 \pi i y_j)= s_j \quad \mbox{ for } 1 \leq j \leq 8 \end{eqnarray} and \begin{eqnarray} \Theta_{E_8}^{root}(\tau, z_1, \cdots, z_8) = \sum_{\gamma \in E_8} \exp(2 \pi i ((\tau/2) Q(\gamma) + B(\gamma, \sum_{j =1}^{8} z_j \alpha_j)), \label{eq:roottheta} \end{eqnarray} where $\{\alpha_1, \cdots, \alpha_8 \}$ is the root system of $E_8$ as in Figure 1 and $B$ is the symmetric bilinear form on $E_8$. \end{Theorem} The following lemma is easy but essential to calculate the prepotential. \begin{Lemma} \label{l:mult=1} For each section $\sigma \in MW(W)$ the contribution of the homology $2$-cycle $j(\sigma) =[\sigma(\P1)]$ to the Gromov-Witten invariant~(\ref{eq:g-w.inv}) is one, that is, $ n(j(\sigma)) = 1 $ \end{Lemma} \noindent {\it Proof.} According to Lemma~\ref{l:j-inj} $MW(W)$ can be considered as a subset of $H_2(W, {\bf Z})$ via the map $j$. Moreover the rational curve $C = \sigma(\P1) \subset W$ has the normal bundle ${\cal O}_{\P1}(-1) \oplus {\cal O}_{\P1}(-1)$. Hence we have $n(j(\sigma)) =1$. {}\hfill$\Box$ \vspace{5mm} \noindent {\it Proof of Theorem~\ref{t:mw-prep}.} Recalling the isomorphism~(\ref{eq:isom}), one can write $\sigma \in MW(W)$ as $(\sigma_1, \sigma_2) \in MW(S_1) \oplus MW(S_2) \simeq E_8 \oplus E_8$. Since $Q_1(a_i) = Q_2(b_j) = 2$ for $1 \leq i, j \leq 8$ we obtain from Lemma~\ref{l:gen} \begin{eqnarray} ([L_i], [\sigma(\P1)])_W = \frac{1}{2} Q_1(\sigma_1 + a_i) = \frac{1}{2} Q_1(\sigma_1) + B_1(\sigma_1, a_i) + 1, \\ \quad ([M_j], [\sigma(\P1)])_W = \frac{1}{2} Q_2(\sigma_2 + b_j) = \frac{1}{2} Q_2(\sigma_2) + B_2(\sigma_2, b_j) + 1. \end{eqnarray} Therefore one has \begin{eqnarray*} \lefteqn{ q^{\sigma} = \prod_{i=0}^{8}(q_i)^{(1/2) Q_1(\sigma_1 + a_i) } } \\ & & = (\prod_{i=0}^{8}(q_i))^{(1/2) Q_1(\sigma_1)} \cdot (\prod_{i=1}^{8} (q_i)^{B_1(\sigma_1, a_i)}) \cdot (\prod_{i=1}^{8}q_i) \\ & & =(\prod_{i=1}^{8}q_i) \cdot \exp(2 \pi i ((1/2) Q_1(\sigma_1) \tau_1 + \sum_{i=1}^8 z_i B(\sigma_1, a_i) ), \end{eqnarray*} and a similar expression for $s^{\sigma}$. {From} these formulas one can obtain \begin{eqnarray} \lefteqn{ (T^{\sigma})^n = (p \cdot q^{\sigma} \cdot s^{\sigma})^n } \nonumber \\ \lefteqn{=( p \prod_{i=1}^8 (q_i \cdot s_i) )^n \times } \label{eq:tsigma1}\\ & \times \exp(2 \pi i n ((\tau_1/2) Q_1(\sigma_1) + B_1(\sigma_1, {\bf z} ) + (\tau_2/2) Q_2(\sigma_2) + B_2(\sigma_2, {\bf y} ) ) \nonumber \end{eqnarray} where we set ${\bf z} = \sum_{i=1}^8 z_i a_i$ and ${\bf y} = \sum_{i=1}^8 y_i b_i$. Therefore, if we take the summation of $(T^{\sigma})^n$ over $\sigma = (\sigma_1, \sigma_2) \in E_8 \oplus E_8$, we obtain the following formula: \begin{eqnarray} \lefteqn{\sum_{\sigma \in MW(W)} (T^{\sigma})^n} \nonumber \\ &= & ( p \prod_{i=1}^8 (q_i \cdot s_i) )^n \cdot \Theta_{E_8}^{root}(n \tau_1, n \cdot z ) \cdot \Theta_{E_8}^{root}(n \tau_2, n \cdot {\bf y}) \end{eqnarray} Now thanks to Lemma~\ref{l:mult=1}, we can calculate the prepotential as follows: $$ \begin{array}{lcl} \Psi_{A, MW(W)}& = & \sum_{\sigma \in MW(W)} \Li_3( T^{\sigma}) \\[.5em] & = & \sum_{\sigma \in MW(W)} ( \sum_{n = 1}^{\infty}\frac{(T^{\sigma})^n}{n^3} ) \\[.5em] &= & \sum_{n = 1}^{\infty} \frac{1}{n^3} \times [\sum_{\sigma \in MW(W)} (T^{\sigma})^n) ] \\[.6em] & = & \sum_{n = 1}^{\infty} \frac{1}{n^3} \cdot ( p \prod_{i=1}^8 (q_i \cdot s_i))^n \times \\[.5em] & & \hspace{1cm} \times \Theta_{E_8}^{root}(n \tau_1, n \cdot {\bf z}) \cdot \Theta_{E_8}^{root}(n \tau_2, n \cdot {\bf y}). \end{array} $$ This completes the proof of Theorem~\ref{t:mw-prep}. {}\hfill$\Box$ \vspace{5mm} For the 1-sectional part $\Psi_{A, 1}$ of the prepotential, we can show the following theorem, whose proof can be found in Section~\ref{s:pseudosection}. \begin{Theorem} \label{t:1-s-prep} Assume that Conjecture~\ref{c:cont} in Section~\ref{s:pseudosection} holds. Then, under the same notation as in Theorem~\ref{t:mw-prep}, for a generic Schoen's Calabi-Yau 3-fold $W$ the 1-sectional prepotential is given by \begin{eqnarray} \lefteqn{\Psi_{A,1}(p, q_0, \cdots q_8, s_0, \cdots, s_8)} \\ = & \sum_{n =1}^{\infty} \frac{1}{n^3}\cdot (p \cdot \prod_{i=1}^8 (q_i \cdot s_i))^n \cdot A(n \tau_1, n \cdot {\bf z})\cdot A(n \tau_2, n \cdot {\bf y}), \nonumber \label{eq:1-s-prep} \end{eqnarray} where \begin{eqnarray} A(\tau, {\bf x}) & = & \Theta_{E_8}^{root}(\tau, {\bf x} ) \cdot [ \sum_{m = 0}^{\infty} p(m) \cdot \exp(2 \pi m i \tau)]^{12}. \\ & = & \Theta_{E_8}^{root}(\tau, {\bf x} ) \cdot [\frac{1}{\prod_{n\geq1}( 1- \exp(2 \pi n i \tau))} ]^{12} \end{eqnarray} Here $p(m)$ denotes the number of partitions of $m$. \end{Theorem} \begin{Remark}\label{rem:root} {\rm In order to identify the theta function $\Theta_{E_8}^{root}(\tau, {\bf z})$ in (\ref{eq:roottheta}) with the theta function $\Theta_{E_8}(\tau, {\bf w})$ of (\ref{eq:e8}) in Appendix II we should apply the linear transformation ${\bf w} \lra {\bf z}$, for ${\bf w} = \sum_{i=1}^8 w_i \epsilon_i$ and ${\bf z} = \sum_{i=1}^8 z_i \alpha_i$. Fix an embedding $E_8 \subset {\bf R}^8$, that is, $\alpha_i $ should have coordinates in ${\bf R}^8$. For example, we can choose $$ \alpha_1 = \frac{1}{2}(\epsilon_1 + \epsilon_8) - \frac{1}{2}(\epsilon_2 + \epsilon_3 + \epsilon_4 + \epsilon_5 + \epsilon_6 + \epsilon_7) \hspace{1.2cm} $$ $$ \begin{array}{cllll} \hspace{1cm} &\alpha_2 = \epsilon_2 - \epsilon_1 & \alpha_3 = \epsilon_3- \epsilon_2 & \alpha_4 = \epsilon_4 - \epsilon_3 & \alpha_5 = \epsilon_5 - \epsilon_4\\ \hspace{1cm} & \alpha_6 = \epsilon_6 - \epsilon_5 & \alpha_7 = \epsilon_7 - \epsilon_6 & \alpha_8 = \epsilon_1 + \epsilon_2& \end{array} $$ (Note that the numbering of roots is the same as in Figure 1.)} \end{Remark} \begin{Remark} \label{r:first} {\rm In expansion~(\ref{eq:expansion}), we see that each term of the expansion of $\Psi_{A, k}$ for $k \geq 2$ is divisible by $p^2$. Moreover we can see that $\Psi_{A,0} \equiv 0$. (Cf. \cite{Saito}). Therefore, Theorem~\ref{t:1-s-prep} shows that if we expand the full A-model prepotential $\Psi_A$ in the variables in $p, q_i, s_j$, we have the following expansion: \begin{eqnarray} \lefteqn{\Psi_{A}(p, q_0, \cdots q_8, s_0, \cdots, s_8)} \nonumber \\ &= & \mbox{topological term} + (p \cdot \prod_{i=1}^8 (q_i \cdot s_i)) \cdot A( \tau_1, {\bf z} )\cdot A( \tau_2, {\bf y}) + O(p^2). \label{eq:1prep-exp} \end{eqnarray} } \end{Remark} \section{Counting Pseudo-Sections and Proof of Theorem 4.2} \label{s:pseudosection} In Section~\ref{s:prep}, we see differences between the two prepotentials $\Psi_{A, MW(W)}$ and $\Psi_{A, 1}$. Looking at the formulas~(\ref{eq:main}) and (\ref{eq:1-s-prep}) one can observe that $\Psi_{A, MW(W)}$ and $\Psi_{A, 1}$ are essentially produced by the functions \begin{eqnarray} \Psi_{A, MW(W)} & \leftrightarrow & \Theta_{E_8}^{root}(\tau, {\bf x}) \label{eq:theta}\\ \Psi_{A, 1} & \leftrightarrow & A(\tau, {\bf x}) = \Theta_{E_8}^{root}(\tau, {\bf x}) \cdot [ \sum_{m = 0}^{\infty} p(m) \cdot \exp(2 \pi m i \tau)]^{12} \label{eq:A} \end{eqnarray} As we see in Section~\ref{s:prep} the geometric meaning of the function $ \Theta_{E_8}^{root}$ is clear, that is, it is the generating function of the contributions of pure sections of $h:W \lra \P1$. However, the meaning of the factor $$ [ \sum_{m = 0}^{\infty} p(m) \cdot \exp(2 \pi m i \tau)]^{12} = \exp(\pi i \tau)\cdot \eta(\tau)^{-12} $$ was mysterious at least in the geometric sense.\footnote{ The similar factor are also discussed in the papers \cite{D-G-W} and \cite{Y-Z}.} In this section, we give a geometric explanation of this factor assuming one very plausible Conjecture~\ref{c:cont}, and we give a proof of Theorem~\ref{t:1-s-prep}. Our answer seems to be very simple and natural at least in a mathematical sense. For this purpose we give the following: \begin{Definition} {\rm We call a 1-dimensional connected subscheme $C$ of $W$ a {\em pseudo-section} if $C \subset W$ has no embedded point and \begin{eqnarray} ([F], C)_W = 1, \label{eq:p-section} \end{eqnarray} and the normalization $\tilde{C}_{red}$ of the reduced structure $C_{red}$ is a sum of $\P1$s.} \end{Definition} \begin{Example}{\rm The image $\sigma(\P1)$ of a section $\sigma \in MW(W)$ is a pseudo-section.} \end{Example} \begin{Example}{\rm Both rational elliptic surfaces $f_i:S_i \lra \P1 (i =1, 2)$ have 12 singular fibers of type $I_1$ (in Kodaira's notation~\cite{Kod}): \begin{eqnarray} D_1, D_2, \cdots, D_{12} \subset S_1, \\ D'_1, D'_2, \cdots, D'_{12} \subset S_2. \end{eqnarray} We set $d_i = f_1(D_i) \in \P1$ and $ d'_i = f_2(D'_i) \in \P1$, the supports of the singular fibers. By Assumption~\ref{as:generic} in Section~\ref{s:schoen}, the points $d_1, \cdots, d_{12}, d'_1, \cdots, d'_{12} $ are distinct on $\P1$. Take any $\sigma \in MW(W)$ and set $\sigma_1 =p_1 \circ \sigma \in MW(S_1), \sigma_2 = p_2 \circ \sigma \in MW(S_2)$. Hence $\sigma(\P1) \subset W_{|\sigma_2(\P1)}$. Now we take a singular fiber $D_1 \subset W_{|\sigma_2(\P1)} \simeq S_1$, then $$ \sigma(\P1) + D_1 \subset W_{|\sigma_2(\P1)} \subset W $$ is a pseudo-section. Since we have \begin{eqnarray} ([L_i], D_1)_W &=& (L_i, D_1)_{S_1} = (L_i, F_1)_{S_1} = 1 \label{eq:p-degree1} \\ ([M_j], D_1)_W &=& (M_j, \sigma_2(d_1))_{S_2} = 0, \quad \label{eq:p-degree2} \end{eqnarray} we obtain \begin{eqnarray} ([L_i], \sigma(\P1)+ D_1)_W = ([L_i], \sigma(\P1))_W + 1, \\ ([M_j],\sigma(\P1)+ D_1)_W = ([M_j], \sigma(\P1))_W. \end{eqnarray} } \end{Example} \begin{Example} {\rm More generally, to a pure section of $h:W \lra \P1$ we can add many rational curves coming from singular fibers of type $I_1$ and also with multiplicity. Fix a section $\sigma \in MW(W)$ and set $\sigma_1 =p_1 \circ \sigma$, $\sigma_2 = p_2 \circ \sigma$ as before. Consider the following (reduced) closed points: \begin{eqnarray} \sigma_1(d'_i) \in S_1, \quad \sigma_2(d_i) \in S_2. \end{eqnarray} Moreover, we set \begin{eqnarray} D'[\sigma_1, d'_i] = p_1^{-1}(\sigma_1(d'_i)) \subset W_{|\sigma_1(\P1)} ( \simeq S_2) \subset W \\ D[\sigma_2, d_i] = p_2^{-1}(\sigma_2(d_i)) \subset W_{|\sigma_2(\P1)} (\simeq S_1) \subset W \end{eqnarray} Note that $D'[\sigma_1, d'_i] $ and $D[\sigma_2, d_i]$ are reduced rational curves each of which has one node as its singularities. {From} (\ref{eq:p-degree1}), (\ref{eq:p-degree2}) it is easy to see that \begin{eqnarray} ([F], D[\sigma_2, d_i])_W = 0, \quad ([L_i], D[\sigma_2, d_i])_W = 1, ([M_j], D[\sigma_2, d_i])_W = 0, \label{eq:int-l}\\ ([F], D'[\sigma_1, d'_i])_W = 0, \quad ([L_i], D'[\sigma_1, d'_i])_W = 0, ([M_j], D'[\sigma_1, d'_i])_W = 1. \label{eq:int-m} \end{eqnarray} We denote by ${\cal I}(k_i, \sigma_2(d_i))$ an ideal sheaf on $S_2$ such that the quotient sheaf $$ {\cal O}_{S_2}/{\cal I}(k_i, \sigma_2(d_i)) $$ is supported on the point $\sigma_2(d_i)$ and $ \mathop{\rm length}\nolimits {\cal O}_{S_2}/ {\cal I}(k_i, \sigma_2(d_i))= k_i$. We call such an ideal ${\cal I}(k_i, \sigma_2(d_i))$ a punctual ideal of colength $k_i$ supported on $\sigma_2(d_i)$. And similarly for ${\cal I}( k'_j, \sigma_1(d'_j))$. For each of $1\leq i\leq 12$ (resp. $1 \leq j \leq 12$), let ${\cal I}(k_i, \sigma_2(d_i))$ (resp. ${\cal I}( k'_j, \sigma_1(d'_j))$) be a punctual ideal of colength $k_i$ (resp. $k'_j$) supported on $\sigma_2(d_i)$ (resp. $\sigma_1(d'_j)$). We denote by $$ D[{\cal I}(k_i, \sigma_2(d_i))] \quad (\mbox{resp.} D'[{\cal I}( k'_j,\sigma_1(d'_j))] ) $$ the one-dimensional subscheme of $W$ defined by the pullback of the ideal sheaf ${\cal I}(k_i, \sigma_2(d_i))$ (resp. ${\cal I}( k'_j, \sigma_1(d'_j))$) via $p_2$ (resp. $p_1$). Note that $$ D[{\cal I}(k_i, \sigma_2(d_i))]_{red} = D[\sigma_2, d_i], \quad D'[{\cal I}( k'_j,\sigma_1(d'_j))]_{red} = D'[\sigma_1, d'_j]. $$ Now we take the following subscheme of $W$: \begin{eqnarray} C = \sigma(\P1) + \sum_{i = 1}^{12} D[{\cal I}(k_i,\sigma_2(d_i))] + \sum_{j = 1}^{12} D'[{\cal I}( k'_j, \sigma_1(d'_j))]. \label{eq:p-general} \end{eqnarray} This one dimensional subscheme $C$ in (\ref{eq:p-general}) is actually a pseudo-section. } \end{Example} \begin{Definition}{\rm The pseudo-section $C$ in (\ref{eq:p-general}) is called {\em of type} $$ (\sigma, k_1, \cdots, k_{12}, k'_1, \cdots, k'_{12}) \in MW(W) \times ({\bf Z}_{+})^{24}. $$ } \end{Definition} \begin{Proposition} Every pseudo-section $C$ of $h:W \lra \P1$ can be written as in $(\ref{eq:p-general})$. \end{Proposition} {\it Proof.} Since $ ([F], C)_W = (F_1, (p_1)_* C))_{S_1} = (F_2, (p_2)_* C))_{S_2} = 1$, it is easy to see that there are sections $\sigma_i \in MW(S_i)$, $$ (p_1)_* (C) = \sigma_1(\P1) + \mbox{fibers} , \quad (p_2)_* (C) = \sigma_2(\P1) + \mbox{fibers}. $$ Then by definition of a pseudo-section, we can easily see that $C$ can be written in the form of (\ref{eq:p-general}) where $\sigma $ corresponds to $(\sigma_1, \sigma_2)$. {}\hfill$\Box$ \vspace{5mm} Fix a type $\mu = (\sigma, k_1, \cdots, k_{12}, k'_1, \cdots, k'_{12}) \in MW(W) \times ({\bf Z}_{+})^{24}$ of a pseudo-section of $h:W \lra \P1$. We would like to count the ``number''$n(\mu)$ of rational curves which gives the correct contribution to the Gromov-Witten invariant in the formula (\ref{eq:g-w.inv}). Since a pseudo-section of type $\mu$ is a non-reduced subscheme of $W$ if some $k_i$ or $k'_j$ is greater than 1, it is not easy to determine $n(\mu)$. Of course, the Gromov-Witten invariant should be defined as the number of $J$-holomorphic curves with a fixed homology class after perturbing the complex structure of $W$ to a generic almost complex structure $J$ (\cite{Mor-Math}, Theorem 3.3). However at this moment we do not know how to perturb the almost complex structure and how a pseudo-section $C$ of type $\mu$ arises as a limit of $J$-holomorphic curves. (Different $J$-holomorphic curves for generic $J$ may have the same limit in our complex structure of Schoen's Calabi-Yau 3-fold $W$.) Here we propose the following: \begin{Conjecture}\label{c:cont} The contribution $n(\mu) $ of all pseudo-sections of type $\mu$ is given by \begin{eqnarray} n(\mu) = e(\Hilb^{\mu}_W) = \mbox{ Topological Euler number of }( \Hilb^{\mu}_W), \label{eq:cont} \end{eqnarray} where $\Hilb^{\mu}_W $ is the Hilbert scheme parameterizing pseudo-sections $C \subset W$ of type $\mu$. \end{Conjecture} Let ${\bf C}^2$ be the complex affine space of dimension 2 and denote by $\Hilb^{k}({\bf C}^2, 0)$ the Hilbert scheme parameterizing the punctual ideal sheaves ${\cal I} \subset {\cal O}_{{\bf C}^2} $ of colength $k$ supported on the origin $0 \in {\bf C}^2$ \begin{Lemma} \label{l:hilb-1} Fix a type $\mu = (\sigma, k_1, \cdots, k_{12}, k'_1, \cdots, k'_{12})$ of pseudo-section. Then we have a natural isomorphism of schemes \begin{equation} \Hilb^{\mu}_W \simeq \prod_{i=1}^{12} (\Hilb^{k_i}({\bf C}^2, 0)) \times \prod_{j=1}^{12} (\Hilb^{k'_j}({\bf C}^2, 0)) \label{eq:hilb-isom} \end{equation} \end{Lemma} {\it Proof.} {From} the definition of pseudo-section $C$ of type $\mu$ in (\ref{eq:p-general}), we have the natural morphism $\varphi$ from $\Hilb^{\mu}_W$ to \begin{eqnarray*} \Hilb (\sigma(\P1) \subset W) \times \prod_{i=1}^{12} (\Hilb^{k_i}(S_2, \sigma_2(d_i))) \times \prod_{j=1}^{12} (\Hilb^{k'_j}(S_1, \sigma_1(d'_j))). \end{eqnarray*} of $W$ defined by \begin{eqnarray*} \varphi(C) & = & \varphi(\sigma(\P1) + \sum_{i = 1}^{12} D[{\cal I}(k_i, \sigma_2(d_i))] + \sum_{j = 1}^{12} D'[{\cal I}( k'_j, \beta[\sigma_1, d'_j])]) \\ & = & (\sigma(\P1), \ {\cal I}[k_i, \sigma_2(d_i)], \ {\cal I}[k'_j,\alpha[\sigma_1, d'_j]). \end{eqnarray*} (Here $\Hilb (\sigma(\P1) \subset W)$ denotes the connected component of the Hilbert scheme which contains the subscheme $\sigma(\P1) $ of $W$. ) Noting that $C$ is connected and $\sigma(\P1) \subset W$ has no deformation (in particular $ \Hilb(\sigma(\P1) \subset W) = 1 pt$) , we can easily see that $\varphi$ is an isomorphism and obtain (\ref{eq:hilb-isom}). {}\hfill$\Box$ \vspace{5mm} The following important lemma is a kind suggestion of Kota Yoshioka. \begin{Lemma} \label{l:hilb-2} The Hilbert scheme $\Hilb^{k}({\bf C}^2, 0)$ is irreducible scheme of dimension $k-1$ and $$ e(\Hilb^{k}({\bf C}^2, 0)) = p(k) $$ where $p(k)$ denotes the number of partitions of $k$. \end{Lemma} {\it Proof.} The irreduciblity of $\Hilb^{k}({\bf C}^2, 0) $ is due to Brian\c{c}on \cite{B}. Moreover $\Hilb^{k}({\bf C} \{ x, y \}) $ has a natural $S^1$-action induced by $(x, y) \rightarrow (t^a \cdot x, t^b \cdot y)$ for any weight $(a, b)$. Then for a general choice of a weight $(a, b)$ its fixed points set becomes just the set of monomial ideals of length $k$. Now a standard argument shows that the topological Euler number of $\Hilb^{k}({\bf C}^2, 0)$ is equal to the number of fixed points, and it is an easy exercise to see that the number of monomial ideals of ${\bf C}[x, y]$ with colength $k$ is equal to $p(k)$.{}\hfill$\Box$ \vspace{5mm} {From} Lemma~\ref{l:hilb-1} and Lemma~\ref{l:hilb-2}, we obtain the following result. \begin{Corollary} Let $\mu$ and $n(\mu)$ as in Conjecture~(\ref{c:cont}), then we have \begin{eqnarray} n(\mu) = e(\Hilb^{\mu}_W) = (\prod_{i=1}^{12} p(k_i)) \cdot (\prod_{j=1}^{12} p(k'_j)) \label{eq:cont-result} \end{eqnarray} {}\hfill$\Box$ \end{Corollary} \vspace{5mm} \begin{Lemma}\label{l:homology} Let $\mu = (\sigma, k_1, \cdots, k_{12}, k'_1, \cdots, k'_{12}), \mu' = (\sigma', l_1, \cdots, l_{12}, l'_1, \cdots, l'_{12}) $ be two types of pseudo-sections. Then a pseudo-section $C$ of type $\mu$ and $C'$ of type $\mu'$ have the same homology class in $H_2(W)$ if and only if \begin{eqnarray} \sigma = \sigma', \quad \sum_{i =1}^{12} k_i = \sum_{i =1}^{12} l_i, \quad \sum_{j =1}^{12} k'_j = \sum_{j =1}^{12} l'_j. \label{eq:iff} \end{eqnarray} \end{Lemma} \noindent {\it Proof.} The ``only if" part is obvious. Lemma~\ref{l:j-inj} shows that the first equality in (\ref{eq:iff}) is necessary. Moreover noting that $[D[{\cal I}(k_i, \sigma_2, d_i)]]$ is homologous to $k_i \cdot (p_2)^{-1}(\sigma_2(d_i)) $ and $[D'({\cal I}[k'_j, \sigma_1, d'_j)]] $ is homologous to $ k'_j \cdot [(p_1)^{-1}(\sigma_1(d'_j))] $ we have the other implication. {}\hfill$\Box$ \vspace{5mm} Let $\eta \in H_2(W)$ be such that $([F], \eta)_W = 1$, then in order to have non-vanishing contribution $n(\eta)$, $\eta$ must be the class of a pseudo-section, so write $\eta$ as $(\sigma, m \cdot p_2^{-1}( 1 pt), n p_1^{-1} ( 1 pt))$. We call $\eta$ {\em of type $(\sigma, m, n)$}. \begin{Proposition} For $\eta \in H_2(W)$ with $([F], \eta)_W = 1$ of type $(\sigma, m, n)$, we have \begin{eqnarray} n(\eta)= n(\sigma, m, n) := (\sum_{ \ k_1 + \cdots + k_{12} = n} \quad \prod_{i = 1}^{12}p(k_i )) (\sum_{ \ k'_1 + \cdots +k'_{12} = m} \quad \prod_{j = 1}^{12}p(k'_j)). \label{eq:coef} \end{eqnarray} Here $k_i$ and $k'_j$ run over the non-negative integers. \end{Proposition} \vspace{5mm} \noindent {\it Proof.} {From} Lemma~\ref{l:homology} and the remark above, we have \begin{eqnarray*} n(\eta) = \sum_{\mu} n(\mu) \end{eqnarray*} where the summation is taken over the types $\mu = (\sigma, k_1, \cdots, k_{12}, k'_1, \cdots, k'_{12})$ of pseudo-sections such that $$ n = \sum_{i=1}^{12} k_i, \quad m = \sum_{j =1}^{12} k'_j. $$ Combining this with (\ref{eq:cont-result}), we obtain the assertion~(\ref{eq:coef}). {}\hfill$\Box$ \vspace{5mm} \vspace{5mm} \noindent {\bf Proof of Theorem~\ref{t:1-s-prep}} \vspace{5mm} Let us fix a homology class $\eta \in H_2(W)$ with $([F], \eta) =1$ of type $(\sigma, n, m)$. From (\ref{eq:int-l}), (\ref{eq:int-m}) it is easy to see that \begin{eqnarray} ([F], \eta)_W & = &1, \\ ([L_i], \eta)_W & = & ([L_i], j(\sigma))_W + n \\ ([M_j], \eta)_W & = & ([M_j], j(\sigma) )_W + m. \end{eqnarray} We introduce parameters $z_0 = \log q_0, y_0 = \log s_0$. Note that we have set in Theorem~\ref{t:mw-prep} $$ \tau_1 = \sum_{j=0}^{8} z_j, \quad \tau_2 = \sum_{j=0}^8 y_j. $$ Moreover just for notation in the proof, we set $v_l = \exp(2 \pi i \tau_l)$ for $l =1, 2$. Recalling the definition of $T^{\sigma}$ (cf. (\ref{eq:tsigma})), we have \begin{eqnarray} T^{\eta} & = & \exp( 2 \pi i (t_0 [F] + \sum_{i=0}^8 z_i [L_i] + \sum_{j=1}^8 y_j [M_j], \eta)_W) \nonumber \\ & = & p \cdot \exp(2 \pi i (\sum_{i=0}^8 z_i [([L_i], j(\sigma))_W+ n] + \sum_{i=0}^8 y_j [([M_j], j(\sigma))_W + m])) \nonumber \\ & = & T^{\sigma} \cdot (v_1)^n \cdot (v_2)^m \end{eqnarray} Then we have \begin{eqnarray} \lefteqn{\Psi_{A, 1}(p, q_0, \cdots q_8, s_0, \cdots, s_8)} \nonumber \\ &= & \sum_{\eta \in H_2(W), ([F], \eta) = 1} n(\eta) \Li_3(T^{\eta}) \nonumber \\ &= & \sum_{\sigma \in MW(W), n \geq 0, m \geq 0} n(\sigma, n, m) \cdot \Li_3( T^{\sigma} \cdot (v_1)^n \cdot (v_2)^m) \nonumber \\ &= & \sum_{N = 1}^{\infty} [\sum_{\sigma \in MW(W), n \geq 0, m \geq 0} n(\sigma, n, m) \cdot \frac{(T^{\sigma})^N \cdot (v_1)^{Nn}\cdot (v_2)^{Nm}}{N^3}] \nonumber \\ & = & \sum_{N = 1}^{\infty} \frac{1}{N^3} \cdot [\sum_{\sigma \in MW(W)} (T^{\sigma})^N] \cdot [ \sum_{n \geq 0, m \geq 0} n(\sigma, n, m) (v_1)^{Nn} \cdot (v_2)^{Nm}]. \nonumber \\ & & \label{eq:final} \end{eqnarray} Note that the last equality follows from the fact that $n(\sigma, n, m)$ does not depend on $\sigma$. On the other hand, from equality~(\ref{eq:coef}) we have $$ [(\sum_{k =0}^{\infty} p(k) (v_1)^k )(\sum_{k' =0}^{\infty} p(k') (v_2)^{k'})]^{12} = \sum_{n \geq 0, m \geq 0} n(\sigma, n, m) (v_1)^n \cdot (v_2)^m. $$ Moreover as in the proof of Theorem~\ref{t:mw-prep} we can see that $$ [\sum_{\sigma \in MW(W)} (T^{\sigma})^N] = ( p \prod_{i=1}^8 (q_i \cdot s_i))^N \Theta_{E_8}^{root}(N \tau_1, N\cdot {\bf z}) \cdot \Theta_{E_8}^{root}(N \tau_2, N \cdot {\bf y}). $$ Combining these equalities with (\ref{eq:final}), we obtain the proof of Theorem~\ref{t:1-s-prep}. {}\hfill$\Box$ \vspace{5mm} \section{The restricted A-model Prepotential} \label{s:rest} In order to compare the prepotential of the A-model Yukawa coupling of $W$ with the B-model Yukawa coupling of the mirror partner $W^*$, which we obtain in Section~\ref{s:B-model}, we need to take a special restriction of the variables of the prepotential, that is, we have to specify the parameters which correspond to the line bundles which are induced from the ambient space $\P1 \times \P2 \times \P2$. Let $\iota:W \hookrightarrow \P1 \times \P2 \times \P2$ be the natural embedding. Then we set \begin{equation} [F] = \pi_1^*({\cal O}_{\P1}(1)), \quad [H_1]= \pi_2^*({\cal O}_{\P2}(1)), \quad [H_2] = \pi_3^*({\cal O}_{\P2}(1)), \label{eq:res-picard} \end{equation} and introduce corresponding parameters as follows: \begin{eqnarray} \quad [F] \leftrightarrow p = U_0 = \exp (2 \pi i t_0), \nonumber \\ \quad [H_1] \leftrightarrow U_1 = \exp (2 \pi i t_1), \label{eq:sp-coord} \\ \quad [H_2] \leftrightarrow U_2 = \exp (2 \pi i t_2). \nonumber \end{eqnarray} Now we consider the following restricted prepotential \begin{eqnarray} \Psi_{A}^{res} = \mbox{topological term} + \sum_{0 \neq \eta \in H_2(W)} n(\eta) \Li_3(U^{\eta}) \label{eq:res-fullprep} \end{eqnarray} where \begin{eqnarray} U^{\eta} & = & \exp( 2 \pi i (t_0 [F] + t_1 [H_1] + t_2 [H_2], \eta)_W) \\ & = & p^{([F], \eta )_W } \cdot (U_1)^{([H_1], \eta )_W } \cdot (U_2)^{ ([H_2], \eta )_W}. \end{eqnarray} Moreover, we can define the $k$-sectional part and the Mordell-Weil part of the restricted prepotential by \begin{eqnarray} \Psi_{A, k}^{res} = \sum_{0 \not\equiv \eta \in H_2(X, {\bf Z}), \ (F, \eta)= k} n(\eta) \Li_3( U^{\eta}), \label{eq:res-sec-prep} \end{eqnarray} \begin{eqnarray} \Psi_{A, MW(W)}^{res} = \sum_{\sigma \in MW(W)} \Li_3(U^{j(\sigma)}), \label{eq:res-prep} \end{eqnarray} respectively. \begin{Proposition} \begin{equation} \Psi_{A, MW(W)}^{res}(p, t_1, t_2) = \sum_{n=1}^{\infty} \frac{p^n}{n^3} \cdot \Theta_{E_8}(3n t_1, n t_1 \gamma) \cdot \Theta_{E_8}(3n t_2, n t_2 \gamma) \label{eq:resprep} \end{equation} \begin{equation} \Psi_{A, 1}^{res}(p, t_1, t_2) = \sum_{n=1}^{\infty} \frac{p^n}{n^3} \cdot A^{res}(n t_1) \cdot A^{res}(n t_2) \label{eq:A-res-prep} \end{equation} where $$ \gamma = (1, 1, 1, 1, 1, 1, 1, -1) $$ and \begin{eqnarray} A^{res}(t) &=& \Theta_{E_8}( 3 t, t \cdot \gamma) \cdot (\sum_{n=0}^{\infty} p(n)\exp (2\pi i n(3 t))^{12} \\ & = & \Theta_{E_8}( 3 t, t \cdot \gamma) \cdot \frac{\exp( 3 \pi i t)}{[\eta( 3t )]^{12}} \nonumber \\ & = & \Theta_{E_8}( 3 t, t \cdot \gamma) \cdot \frac{1}{[\prod_{m \geq 1 }(1 - \exp(2 \pi i m (3t)))]^{12}}. \nonumber \label{eq:ares} \end{eqnarray} \label{prop:res} \end{Proposition} \noindent {\it Proof.} {From} Relation~(\ref{eq:h-f}), we obtain for every $\sigma \in MW(W)$ \begin{eqnarray*} \lefteqn{([H_1], j(\sigma))_W = (H_1, j(\sigma_1))_{S_1} }\\ & & =(2 F_1 + 5 L_0 -2 L_1 - L_2 + L_8, j(\sigma_1))_{S_1} \\ & & = 2 + 1/2 (5 Q_1(\sigma_1) - 2 Q_1(\sigma_1 + a_1) - Q_1(\sigma_1 + a_2) + Q_1(\sigma_1 + a_8)) \\ & & = \frac{3}{2} Q_1(\sigma_1) + B_1(\sigma_1, -2 a_1 -a_2 + a_8), \end{eqnarray*} and a similar equation for $([H_2], j(\sigma))_W$. Then from Remark~\ref{rem:root}, we see that \begin{eqnarray*} \gamma = -2 a_1 - a_2 + a_8 & = & -[(\epsilon_1 + \epsilon_8) -(\epsilon_2 + \epsilon_3 + \epsilon_4 + \epsilon_5 + \epsilon_6 + \epsilon_7)] + \\ & & -(\epsilon_2 - \epsilon_1) + (\epsilon_1 + \epsilon_2) \\ &=& \epsilon_1 + \epsilon_2 + \epsilon_3 + \epsilon_4 + \epsilon_5 + \epsilon_6 + \epsilon_7 - \epsilon_8 \end{eqnarray*} Therefore we see that \begin{eqnarray*} \lefteqn{U^{j(\sigma)} = \exp (2 \pi i (t_0 + ([H_1],j(\sigma))_W t_1 + ([H_2],j(\sigma))_W t_2))} \\ &&= p \cdot \exp (2 \pi i(\frac{3t_1}{2} Q_1(\sigma_1)+B_1( \sigma_1, t_1 \gamma ) ) \exp ( 2 \pi i( \frac{3t_2}{2}Q_2(\sigma_2)+B_2( \sigma_2, t_2 \gamma ) ) ) \end{eqnarray*} Then as in the proof of Theorem~\ref{t:mw-prep}, we can obtain Assertion~(\ref{eq:resprep}). The proof of Assertion~(\ref{eq:A-res-prep}) is similar. {}\hfill$\Box$ \vspace{5mm} Now we consider the expansion of (\ref{eq:resprep}) and (\ref{eq:A-res-prep}) with respect to the variable $p$. \begin{equation} \Psi_{A, MW(W)}^{res} = p \cdot \Theta_{E_8}(3 t_1, t_1 \gamma) \cdot \Theta_{E_8}(3 t_2, t_2 \gamma) + O(p^2) \label{eq:2ptheta} \end{equation} \begin{equation} \Psi_A - \mbox{topological term} = \Psi_{A, 1}^{res} + O(p^2) = p \cdot A^{res} (t_1) \cdot A^{res}(t_2)+ O(p^2) \label{eq:2-A-prep} \end{equation} \vspace{1mm} Let us define sequences of positive integers $\{c_n \} $ and $\{ a_n \}$ by \begin{eqnarray} \Theta_{E_8}(3 t, t \gamma) = \sum_{n = 0}^{\infty} c_m \exp (2 \pi m i t) = \sum_{n = 0}^{\infty} c_m U^m \label{eq:expand} \\ A^{res}( t ) = \sum_{n = 0}^{\infty} a_m \exp (2 \pi m i t) = \sum_{n = 0}^{\infty} a_m U^m. \label{eq:expand-a} \end{eqnarray} Let us also expand functions as follows: \begin{eqnarray} p \cdot \Theta_{E_8}(3 t_1, t_1 \gamma) \cdot \Theta_{E_8}(3 t_2, t_2 \gamma) = \sum_{n_1 \geq 0, n_2 \geq 0} M_{1, n_1,n_2} \cdot p \cdot (U_1)^{n_1} (U_2)^{n_2}\label{eq:expandprod1} \\ p \cdot A^{res}(t_1) \cdot A^{res}(t_2) = \sum_{n_1 \geq 0, n_2 \geq 0} N_{1, n_1,n_2} \cdot p \cdot (U_1)^{n_1} (U_2)^{n_2}. \label{eq:expandprod2} \end{eqnarray} The proof of the following proposition follows from (\ref{eq:expandprod1}), (\ref{eq:expandprod2}) and Proposition~\ref{prop:res}. \begin{Proposition} \begin{enumerate} \item \begin{eqnarray} M_{1,n_1,n_2} = c_{n_1} \cdot c_{n_2}, \quad N_{1, n_1, n_2} = a_{n_1} \cdot a_{n_2} \label{eq:split} \end{eqnarray} \item Let $f:S \lra \P1$ be a generic rational elliptic surface as in section~\ref{s:schoen}. Then in the expansion~$(\ref{eq:expand})$ the coefficient $c_m$ is the number of sections of $f:S \lra \P1$ with degree $(H, [\sigma(\P1)])_S = m$, that is, $$ c_m = \sharp \{ \ \ \sigma \in MW(S) \quad | \quad (H, [\sigma(\P1)])_S = m \ \ \}, $$ where $H$ is the class of the total transform of a line on $\P2$. \item The integer $M_{1,n_1, n_2}$ $($resp.$N_{1,n_1, n_2}$$)$ is the number of sections $\sigma \in MW(W)$ (resp. pseudo-sections $\eta$) of $h:W \lra \P1$ with bidegree $(n_1, n_2) $ where $n_i = ([H_i], j(\sigma))_W$ $($resp. $n_i = ([H_i], \eta)_W$ $)$ is the degree with respect to $[H_i]$. \end{enumerate} \end{Proposition} {}\hfill$\Box$ \begin{Remark}{\rm The factorization property of $M_{1,n_1, n_2}$ and $N_{1, n_1, n_2}$ in $(\ref{eq:split})$ follows from the fact that sections and pseudo-sections of $h:W \lra \P1$ can be split as in $(\ref{eq:isom})$, (\ref{eq:p-general}). } \end{Remark} \begin{Remark}{\rm Note that the sequences $\{ c_m \}$ and $\{a_m \}$ are connected to each other by the formula: $$ \sum_{n = 0}^{\infty} a_m U^m = [\sum_{n = 0}^{\infty} c_m U^m][ \sum_{k =0}^{\infty} p(k) U^{3k}]^{12}. $$ The number $a_m$ can be considered as the number of pseudo-sections $C$ of a generic rational elliptic surface $f:S \lra \P1$ of degree $m$ with respect to the divisor class $[H]$. The term $$ [ \sum_{k =0}^{\infty} p(k) U^{3k}]^{12} $$ is nothing but the contribution of 12 singular fibers of type $I_1$, when we count the contribution of one singular fiber of type $I_1$ with multiplicity $k$ as $p(k)$. } \end{Remark} Here we will expand $\Theta_{E_8}(3 t, t \gamma)$ and $A^{res}(t)$ in the variable $U = \exp(2 \pi i t)$ and give the table of coefficients $c_n$ and $a_n$ up to order 50. (See also the last remark of Section~\ref{s:B-model}). We can use Proposition~\ref{pr:e8theta} to obtain the following expansion. \pagebreak \begin{center} Table. 1 \end{center} $$ \Theta_{E_8} (3 t , t \gamma ) = \sum_{m \geq 0} c_m U^m. $$ $$ \begin{array}{lclclcl} \begin{array}{r|l} n&c_n\\ &\\ 0& 9 \\ 1& 36\\ 2 & 126\\ 3 & 252 \\ 4 & 513 \\ 5 & 756 \\ 6 & 1332\\ 7 & 1764 \\ 8 & 2808 \\ 9& 3276 \\ 10 &4914 \\ 11 & 5616\\ 12 & 8190 \\ \end{array} &\hspace{5mm}& \begin{array}{r|l} n&c_n\\ &\\ 13 & 8892 \\ 14 & 12168 \\ 15 & 13104\\ 16 & 17766 \\ 17 & 18648 \\ 18 & 24390 \\ 19 & 25200\\ 20 & 33345 \\ 21 & 33516\\ 22 & 43344 \\ 23 & 43092 \\ 24 & 55692 \\ 25 & 54684\\ \end{array} &\hspace{5mm}& \begin{array}{r|l} n&c_n\\ &\\ 26 & 68922 \\ 27 & 68796 \\ 28 & 86580 \\ 29 & 84168 \\ 30 & 103824 \\ 31 & 101556 \\ 32 & 127647 \\ 33 & 121212 \\ 34 & 148878 \\ 35 & 143964 \\ 36 & 178776 \\ 37 & 170352 \\ 38 & 205380 \\ \end{array} &\hspace{5mm}& \begin{array}{r|l} n&c_n\\ &\\ 39 & 197136 \\ 40 & 241920\\ 41 & 227556 \\ 42 & 276948 \\ 43 & 262080 \\ 44 & 319410 \\ 45 & 298116 \\ 46 & 357912\\ 47 & 341460 \\ 48 & 410958 \\ 49 & 382356 \\ 50 & 458208 \\ & \\ \end{array} \end{array} $$ \begin{center} {\bf Table 2. (Table for $\{ a_n \} $).} \end{center} $$ A^{res}(t) = \sum_{m \geq 0} a_m U^m = (\sum_{n \geq 0} c_n U^n) \cdot (\sum_{k \geq 0} p(k) U^{3k} )^{12}. $$ $$ \begin{array}{lclcl} \begin{array}{r|l} n&a_n\\ &\\ 0&9\\ 1& 36\\ 2 & 126\\ 3 & 360\\ 4 & 945\\ 5 & 2268\\ 6 & 5166\\ 7 & 11160\\ 8 & 23220\\ 9 & 46620\\ 10 & 90972\\ 11 & 172872\\ 12 & 321237 \\ 13 & 584640\\ 14 & 1044810\\ 15 & 1835856\\ 16 & 3177153\\ \end{array} &\hspace{5mm}& \begin{array}{r|l} n&a_n\\ &\\ 17 & 5421132\\ 18 & 9131220\\ 19 & 15195600\\ 20 & 25006653\\ 21 & 40722840\\ 22 & 65670768\\ 23 & 104930280\\ 24 & 166214205\\ 25 & 261141300\\ 26 & 407118726\\ 27 & 630048384\\ 28 & 968272605\\ 29 & 1478208420\\ 30 & 2242463580\\ 31 & 3381344280\\ 32 & 5069259342\\ 33 & 7557818940\\ \end{array} &\hspace{5mm}& \begin{array}{r|l} n&a_n\\ &\\ 34 & 11208455370\\ 35 & 16538048640\\ 36 & 24282822798\\ 37 & 35487134928\\ 38 & 51626878470\\ 39 & 74779896240\\ 40 & 107861179482\\ 41 & 154945739844\\ 42 & 221711362038\\ 43 & 316042958880\\ 44 & 448856366490\\ 45 & 635216766732\\ 46 & 895854679650\\ 47 & 1259213600736\\ 48 & 1764210946995\\ 49 & 2463949037340\\ 50 & 3430694064888 \end{array} \end{array} $$ \vspace{5mm} \pagebreak \section{The prepotential of the B-model Yukawa coupling} \label{s:B-model} In this section we study the prepotential of the B-model Yukawa coupling for the mirror $W^*$ of Schoen's example in the sense of Batyrev-Borisov~\cite{Batyrev-Borisov} and compare it with the prepotential for the A-model Yukawa coupling of $W$. Formula (\ref{eq:Psi}) gives this B-model prepotential $\Psi_{B}$ explicitly. In order to determine the B-model prepotential we will basically follow the recipe of \cite{HKTY,Sti} which uses only the toric data of {the} A-model side. However in order to give an intuitive picture of the mirror $W^*$ we will put here the orbifold construction of the mirror $W^*$ of Schoen's example and the Picard-Fuchs equations of the periods of a holomorphic 3-form of $W^*$. Based on the Batyrev-Borisov mirror construction (cf. \cite{Batyrev-Borisov}, \cite{HKTY}) for complete intersection Calabi-Yau manifolds in toric varieties we can derive the following \begin{Proposition}\label{orbifold} The family of mirror Calabi-Yau 3-folds of $W$ is obtained by the orbifold construction with group ${\bf Z}_3\times{\bf Z}_3$ from the subfamily $W_{\alpha_0, \alpha_1, \beta_0, \beta_1}$ of $W$: \begin{eqnarray*} \lefteqn{W_{\alpha_0, \alpha_1, \beta_0, \beta_1} = } \\ && \left\{ \; [z_0:z_1] \times [ x_0: x_1: x_2]\times[ y_0: y_1: y_2] \in \P1 \times \P2 \times \P2 \;|\;P_1 = P_2 = 0 \; \right\} \end{eqnarray*} where $$ \begin{array}{rcl} P_1&=&(x_0^3+x_1^3+x_2^3+\alpha_0x_0x_1x_2)z_1+\alpha_1x_0x_1x_2 z_0 \;,\\ P_2&=&(y_0^3+y_1^3+y_2^3+\beta_0y_0y_1y_2)z_0+\beta_1y_0y_1y_2 z_1 \; \\ \end{array} $$ and the group ${\bf Z}_3\times{\bf Z}_3$ is generated by \begin{eqnarray} \label{eq:action} \lefteqn{g_1: ([z_0:z_1],[x_0:x_1:x_2],[y_0:y_1:y_2]) \hspace{2cm} } \nonumber \\ & & \hspace{1cm} \mapsto([z_0:z_1],[x_0:\omega x_1:\omega^2 x_2],[y_0:y_1:y_2]), \\ \lefteqn{ g_2: ([z_0:z_1],[x_0:x_1:x_2],[y_0:y_1:y_2]) \hspace{2cm} } \nonumber \\ && \hspace{1cm} \mapsto ([z_0:z_1],[x_0:x_1:x_2],[y_0:\omega y_1:\omega^2 y_2]), \nonumber \end{eqnarray} with $\omega=e^{2\pi i /3}$. That is, the mirror $W^*$ is $$ W^* = W_{\alpha_0, \alpha_1, \beta_0, \beta_1}/({\bf Z}_3 \times {\bf Z}_3). $$ \end{Proposition} \noindent {\it Proof.} See Section \ref{s:B-model equations}, Appendix I. {}\hfill$\Box$ In the equations $P_1$ and $P_2$ above we have kept four parameters $\alpha_0,\alpha_1, \beta_0,\beta_1$ for symmetry reasons. However only three of them are essential because of the scaling of the variables $z_0,z_1$. After the orbifoldization this three parameter deformation describes a three dimensional subspace in the complex structure (B-model) moduli space of $W^*$. The full complex structure moduli space has dimension 19. Under the mirror symmetry the three dimensional subspace will be mapped to the subspace in the complexified K\"ahler moduli space parameterized by $(t_0, t_1,t_2)$ in (\ref{eq:sp-coord}). The B-model calculations are local calculations based on the variation of the Hodge structure for the family $W^*$ about the {\it Large complex structure limit} (LCSL). A mathematical characterization of LCSL is given in \cite{Mor-II}. Here we simply follow a general recipe applicable to CICYs in toric varieties to find a LCSL and write the Picard-Fuchs differential equations \cite{HKTY}. We find that the origin of the local coordinate system $u=(u_0,u_1,u_2)$ with $u_0={\alpha_1\beta_1 \over \alpha_0 \beta_0}, \; u_1=-{1\over \alpha_0^3}$ and $u_2=-{1\over \beta_0^3}$ is a LCSL, and that the Picard-Fuchs (PF) differential operators about this point are \begin{equation} \begin{array}{rcl} D_1 &= & (3\theta_{u_1}-\theta_{u_0})\theta_{u_1} -9u_1(3\theta_{u_1}+\theta_{u_0}+2)(3\theta_{u_1}+\theta_{u_0}+1) \\ & & + u_0\theta_{u_1}(3\theta_{u_2}+\theta_{u_0}+1) \;\;, \\[.5em] D_2&=& (3\theta_{u_2}-\theta_{u_0})\theta_{u_2} -9u_2(3\theta_{u_2}+\theta_{u_0}+2)(3\theta_{u_2}+\theta_{u_0}+1) \\ & & + u_0\theta_{u_2}(3\theta_{u_1}+\theta_{u_0}+1) \;\;, \\[.5em] D_3 &=& \theta_{u_0}^2-u_0(3\theta_{u_1}+\theta_{u_0}+1) (3\theta_{u_2}+\theta_{u_0}+1) \;\;, \end{array} \label{eq:PF} \end{equation} with $\theta_{u_i}=u_i{\partial \; \over \partial u_i}$. We note that if we set $u_0=0$ in (\ref{eq:PF}), the operators $D_1$ and $D_2$ reduce to the PF equations for the Hesse pencil of elliptic curves. Local solutions about $u=0$ have several interesting properties. To state these, we denote the three elements $[F], [H_1]$ and $[H_2]$ in the Picard group Pic$(W)$ by $J_0, J_1$ and $J_2$, respectively. By the notation $K_{ijk} \; (i,j,k=0,1,2)$ we denote the classical intersection numbers among the corresponding divisors. Then the nonzero components are calculated, up to obvious permutations of the indices, by \begin{equation} K_{012}=9 \;\;,\;\; K_{112}=K_{122}=3 \;\;. \label{eq:Kcl} \end{equation} \begin{Proposition} \begin{enumerate} \item The Picard-Fuchs equation (\ref{eq:PF}) has only one regular solution, namely \begin{equation} \Omega^{(0)}(u):=\sum_{m_0,m_1,m_2\geq 0} {(m_0+3m_1)!\;(m_0+3m_2)! \over (m_0!)^2\, (m_1!)^3 \, (m_2!)^3 } u_0^{m_0}u_1^{m_1}u_2^{m_2} \label{eq:wo} \end{equation} \item All other solutions of (\ref{eq:PF}) have logarithmic regular singularities and have the following form in terms of the classical Frobenius method \begin{equation} \label{eq:wi} \begin{array}{rcl} &\displaystyle{ \Omega^{(1)}_i(u):= \drho{i} \Omega(u,\rho)\vert_{\rho=0}} \;,\hspace{4mm} \\ &\displaystyle{ \Omega^{(2)}_i(u):= {1\over2} \sum_{j,k=0,1,2}K_{ijk}\drho{j}\drho{k}\Omega(u,\rho)\vert_{\rho=0} \;,\; }\\ &\displaystyle{ \Omega^{(3)}(u) := -{1\over3!}\sum_{i,j,k=0,1,2}K_{ijk}\drho{i}\drho{j}\drho{k} \Omega(u,\rho)\vert_{\rho=0} \;,} \end{array} \end{equation} with \end{enumerate} $$ \Omega(u,\rho):= \hspace{10.5cm} $$ $$ \displaystyle{ \sum_{m_0,m_1,m_2\geq 0} { (1+\rho_0+3\rho_1)_{m_0+3m_1} (1+\rho_0+3\rho_2)_{m_0+3m_2} \over (1+\rho_0)_{m_0}^2 (1+\rho_1)_{m_1}^3 (1+\rho_2)_{m_2}^3 } u_0^{m_0+\rho_0}u_1^{m_1+\rho_1}u_2^{m_2+\rho_2}} $$ and $K_{ijk}$ being the coupling in $(\ref{eq:Kcl})$. The notation $(x)_m$ represents the Pochhammer symbol: $(x)_m:=x(x+1)\cdots(x+m-1)\,.$ {}\hfill$\Box$ \end{Proposition} Now we are ready to define the B-model prepotential and the mirror map: \begin{Definition} {\rm We define the {\em B-model prepotential} by \begin{equation}\label{eq:Psi} \Psi_B(u)={1\over2}\left({1\over\Omega^{(0)}(u)}\right)^2 \Bigl\{ \Omega^{(0)}(u)\Omega^{(3)}(u) + \sum_i \Omega_i^{(1)}(u)\Omega_i^{(2)}(u) \Bigr\} \;\;. \end{equation}} \end{Definition} \begin{Definition} \label{d:mirrormap}{\rm We define the {\em special coordinates on the B-model moduli space} by \begin{equation}\label{eq:coord} t_j={1\over 2\pi i}{\Omega^{(1)}_j(u) \over \Omega^{(0)}(u) } \:,\hspace{4mm}U_j:=e^{2\pi i t_j} \hspace{4mm}(j=0,1,2)\;. \end{equation} Then $U_0,U_1,U_2$ are functions of $u_0,u_1,u_2$ and $U_j=u_j+\mbox{higher order terms}$. The inverse map $(u_0(U),\,u_1(U),\,u_2(U))$ is called {\em the mirror map}.} \end{Definition} \begin{Conjecture} \label{c:mirror} {\bf (Mirror Conjecture)} The B-model prepotential $\Psi_B(u)$ combined with the mirror map has the expansion \begin{equation}\label{eq:mircon} \Psi_B(u(U)) ={(2\pi i)^3 \over 3!}\sum_{i,j,k=0,1,2} K_{ijk}t_it_jt_k +\sum_{n_0,n_1,n_2 \geq 0} N_{n_0,n_1,n_2} \Li_3(U_0^{n_0}U_1^{n_1}U_2^{n_2}) \end{equation} where $N_{n_0,n_1,n_2}$ is the number of rational curves $\varphi: {\bf P}^1 \mapsto W$ with $(J_i,\varphi_*([{\bf P}^1]))$ $= n_i, \;\;(i=0,1,2)$. In our context, we can state the conjecture in more precise form as follows: \begin{eqnarray} \Psi^{res}_A(U_0, U_1, U_2) = \Psi_B(u(U_0, U_1, U_2)) \label{eq:mirror} \end{eqnarray} where $\Psi^{res}_A(U_0, U_1, U_2)$ is the restricted $A$-model prepotential defined in $(\ref{eq:res-fullprep})$. \end{Conjecture} \vspace{5mm} Next we briefly sketch the approach of \cite{Sti} for calculating the B-model prepotential by using only toric data of the A-model side. This starts from the observation that Schoen's example $W$ can be embedded in $\P1\times \P2\times \P2$ as the intersection of a hypersurface of degree $(1,3,0)$ and a hypersurface of degree $(1,0,3)$. (cf. Section~\ref{s:schoen}). So $W$ is the zero locus of a (general) section of the rank $2$ vector bundle ${\cal O} (1,3,0)\oplus {\cal O} (1,0,3)$ on $\P1\times \P2\times \P2$. This vector bundle can be constructed as a quotient of an open part of ${\bf C}^{10}$ by a $3$-dimensional subtorus of $({\bf C}^\ast)^{10}$ acting by coordinatewise multiplication. The subtorus is the image of the homomorphism $({\bf C}^\ast)^{3}={\bf Z}^{3}\otimes{\bf C}^\ast\rightarrow {\bf Z}^{10}\otimes{\bf C}^\ast=({\bf C}^\ast)^{10}$ given by the $3\times 10$-matrix \begin{equation}\label{eq:Bmat} {\sf B}:=\left( \begin{array}{rrrrrrrrrr} -1&-1&1&1&0&0&0&0&0&0\\ -3&0&0&0&1&1&1&0&0&0\\ 0&-3&0&0&0&0&0&1&1&1 \end{array} \right) \end{equation} The open part of ${\bf C}^{10}$ is \begin{equation}\label{eq:union} \bigcup_{(i,j,k)\in\{3,4\}\times\{5,6,7\}\times\{8,9,10\}} \;{\bf C}^{10}_{(i,j,k)} \end{equation} with $$ {\bf C}^{10}_{(i,j,k)}:=\{(x_1,\ldots,x_{10})\in{\bf C}^{10}\;|\; x_i\neq 0\,,\:x_j\neq 0\,,\:x_k\neq 0\;\} $$ We view $\{3,4\}\times\{5,6,7\}\times\{8,9,10\}$ as a collection of 18 subsets of $\{1,\ldots,10\}$ and note that the complement of the union of these 18 subsets is $\{1,2\}$. As explained in \cite{Sti} this bit of combinatorial input suffices to explicitly write down the hypergeometric function from which one can subsequently compute the B-model prepotential. This hypergeometric function is a priori a function in 10 variables $v_1,\ldots,v_{10}$, which correspond to the a priori 10 coefficients in the equations $P_1$ and $P_2$ in proposition \ref{orbifold}: \begin{eqnarray*} \Phi&:=& (\bar J_0+3\bar J_1)(\bar J_0 + 3\bar J_2)\times v_1^{-1}v_2^{-1} \times\:u_1^{\bar J_0} u_2^{\bar J_1} u_3^{\bar J_2}\times\\ &\times& \sum_{m_0,m_1,m_2\geq 0} \frac{(1+\bar J_0+3\bar J_1)_{m_0+3m_1}\cdot (1+\bar J_0 + 3\bar J_2)_{m_0+3m_2}}{ {(1+\bar J_0)_{m_0}}\!^2\cdot {(1+\bar J_1)_{m_1}}\!^3\cdot {(1+\bar J_2)_{m_2}}\!^3}\:u_0^{m_0} u_1^{m_1} u_2^{m_2} \end{eqnarray*} with $$ u_0 \::=\,v_1^{-1}v_2^{-1}v_3v_4\:,\hspace{3mm} u_1 \::=-\,v_1^{-3}v_5v_6v_7\:,\hspace{3mm} u_2 \::=-\,v_2^{-3}v_8v_9v_{10} $$ and where $\bar J_0,\,\bar J_1,\,\bar J_2$ are elements in the ring $$ {\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}:= {\bf Z}[\bar J_0,\bar J_1, \bar J_2]/ (\bar J_0^2,\bar J_1^3,\bar J_2^3). $$ So, $v_1v_2\Phi$ is an element of $$ \left((\bar J_0+3\bar J_1)(\bar J_0 + 3\bar J_2) {\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}\right)\otimes {\bf Q}[[u_1,u_2,u_3]][\log u_1,\,\log u_2,\,\log u_3]\,. $$ The map multiplication by $(\bar J_0+3\bar J_1)(\bar J_0 + 3\bar J_2)$ on ${\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}$ induces an isomorphism of linear spaces from the ring $$ {\cal R}_{toric}:= {\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}/ Ann((\bar J_0+3\bar J_1)(\bar J_0 + 3\bar J_2)) $$ onto the ideal $(\bar J_0+3\bar J_1)(\bar J_0 + 3\bar J_2) {\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}$. ${\cal R}_{{\bf P}^1\times{\bf P}^2\times{\bf P}^2}$ is in fact the cohomology ring of the ambient toric variety $\P1\times\P2\times\P2$ and ${\cal R}_{toric}$ is a subring of the Chow ring of $W$. The classes of $\bar J_0,\bar J_1,\bar J_2$ in ${\cal R}_{toric}$ correspond to the elements $J_0,J_1,J_2$ of Pic$(W)$ defined earlier. One easily checks that ${\cal R}_{toric}$ is a free ${\bf Z}$-module of rank $8$ with basis $\{ 1\:,\;J_0\:,\;J_1\:,\;J_2\:,\; J_1^2\:,\;J_1J_2\:,\;J_2^2\:,\; J_1^2J_2 \}$ and that the following relations hold \begin{eqnarray*} && J_0^2=J_1^3=J_2^3=J_0J_1^2=J_0J_2^2=0\,, \\ && J_0J_1=3J_1^2\:,\hspace{3mm}J_0J_2=3J_2^2\:,\hspace{3mm} J_1J_2^2=J_1^2J_2\:,\hspace{3mm}J_0J_1J_2=3J_1^2J_2 \end{eqnarray*} Instead of $\bar\Phi$ we may as well work with \begin{eqnarray*} \Omega(u,J):= \hspace{9cm} \end{eqnarray*} $$ \sum_{m_0,m_1,m_2\geq 0} \frac{(1+ J_0+3 J_1)_{m_0+3m_1}\cdot (1+ J_0 + 3 J_2)_{m_0+3m_2}}{ {(1+ J_0)_{m_0}}\!^2\cdot {(1+ J_1)_{m_1}}\!^3\cdot {(1+ J_2)_{m_2}}\!^3}\:u_0^{m_0+J_0} u_1^{m_1+J_1} u_2^{m_2+J_2} $$ Using the notations (\ref{eq:Kcl}), (\ref{eq:wo}), (\ref{eq:wi}) and $J_0^\vee:={1\over9}J_1J_2-{1\over27}J_0J_1-{1\over27}J_0J_2, \;$ $J_1^\vee:={1\over9}J_0J_2,\;$ $ J_2^\vee:={1\over9}J_0J_1$ and $vol:={1\over9}J_0J_1J_2$ the relation between the two approaches may now be formulated as \begin{Proposition} $$ \Omega(u,J)=\Omega^{(0)}(u)+\sum_{i=0}^2 \Omega^{(1)}_i(u) J_i +\sum_{i=0}^2 \Omega^{(2)}_i(u) J_i^\vee - \Omega^{(3)}(u) vol $$ {}\hfill$\Box$ \end{Proposition} $\Omega(u,J)$ is an element of the ring ${\cal R}_{toric}\otimes {\bf Q}[[u_1,u_2,u_3]][\log u_1,\,\log u_2,\,\log u_3]\,.$ It is $1$ modulo $J_0,J_1,J_2,$ $u_0,u_1,u_2$ and hence its logarithm also exists in the ring ${\cal R}_{toric}\otimes {\bf Q}[[u_1,u_2,u_3]][\log u_1,\,\log u_2,\,\log u_3]\,.$ Expanding $\log\Omega(u,J)$ with respect to the basis $\{1,\,J_0,\,J_1,\,J_2, J_0^\vee,\,J_1^\vee,\,J_2^\vee,\,vol\}$ of ${\cal R}_{toric}$ one finds $$ \log\Omega(u,J)=\log\Omega^{(0)}(u)\,+\, \sum_{j=0}^2\log U_j\,J_j\,+\, \sum_{j=0}^2P_j\,J_j^\vee\,+\,P\,vol $$ with $U_j$ as in (\ref{eq:coord}) and hence $\log U_j= 2\pi i t_j$. A straightforward computation shows (see also (\ref{eq:Psi}) and (\ref{eq:mircon})) \begin{eqnarray} \nonumber P&=&- \left(\frac{\Omega^{(3)}(u)}{\Omega^{(0)}(u)}\,+\, \sum_{j=0}^2\frac{\Omega_j^{(1)}(u)}{\Omega^{(0)}(u)} \frac{\Omega_j^{(2)}(u)}{\Omega^{(0)}(u)}\,-\, \frac{(2\pi i)^3}{3}\sum_{m,j,k=0,1,2} K_{mjk}t_mt_jt_k \right)\\ \nonumber &=&-{2} \left(\Psi_B(u)\,-\, \frac{(2\pi i)^3}{3!}\sum_{m,j,k=0,1,2} K_{mjk}t_mt_jt_k \right)\\ \label{eq:P} &=&-{2} \sum_{n_0,n_1,n_2 \geq 0} N_{n_0,n_1,n_2} \Li_3(U_0^{n_0}U_1^{n_1}U_2^{n_2}) \end{eqnarray} \begin{Proposition} Let the numbers $ N_{n_0,n_1,n_2}$ be defined by (\ref{eq:P}). Then \begin{equation}\label{eq:N0} N_{0,n_1,n_2}=0\hspace{5mm}{\rm for\;all\;}\; n_1,n_2\geq 0 \end{equation} \end{Proposition} \noindent {\it Proof. } Note that modulo $u_0$ \begin{eqnarray*} \lefteqn{u_0^{-J_0} u_1^{-J_1} u_2^{-J_2}\Omega(u,J)\equiv} \\ && \equiv \left(\sum_{m_1\geq 0} \frac{(1+ J_0+3 J_1)_{3m_1}}{ {(1+ J_1)_{m_1}}\!^3} u_1^{m_1}\right) \left(\sum_{m_2\geq 0} \frac{(1+ J_0+3 J_2)_{3m_2}}{ {(1+ J_2)_{m_2}}\!^3} u_2^{m_2}\right) \end{eqnarray*} and take logarithms. The logarithms involve no mixed terms $J_1J_2$. This shows $P\equiv 0\bmod u_0$. {}\hfill$\Box$ \ \ As explained in \cite{F,Sti} a theorem of Bryant and Griffiths shows $$ P_j=-{1\over2}U_j\frac{\partial P}{\partial U_j} $$ for $j=0,1,2$. Hence \begin{equation}\label{eq:dP} P_j= \sum_{n_0,n_1,n_2 \geq 0} n_j N_{n_0,n_1,n_2} \Li_2(U_0^{n_0}U_1^{n_1}U_2^{n_2}) \end{equation} where $\Li_2(x):=\sum_{n\geq 1}\frac{x^n}{n^2}$ is the dilogarithm function. It follows from (\ref{eq:dP}) and (\ref{eq:N0}) that we can get all numbers $N_{n_0,n_1,n_2}$ from $P_0$. The computations are now greatly simplified by observing: \begin{Lemma} In ${\cal R}_{toric}$ the intersection of the ${\bf Z}$-module with basis \\ $\{1,J_1,J_2,J_1J_2\}$ and the ideal generated by $J_0,J_1^2,J_2^2$ is $0\,.$ {}\hfill$\Box$ \end{Lemma} So for studying $\log U_1\,,$ $\log U_2$ and $P_0$ we may reduce modulo the ideal $(J_0,J_1^2,J_2^2)\,;$ i.e. replace ${\cal R}_{toric}$ by ${\bf Z}[\bar J_1, \bar J_2]/(\bar J_1^2,\bar J_2^2)$. {From} now on we use $J_1$ resp. $J_2$ to denote the classes of $\bar J_1$ resp. $\bar J_2$ in the latter ring; so we have in particular from now on $$ J_1^2=J_2^2=0 $$ Let $$ \tilde\Omega(u,J_1,J_2):=\sum_{m_0,m_1,m_2\geq 0} \frac{(1+ 3 J_1)_{m_0+3m_1}\cdot (1+ 3 J_2)_{m_0+3m_2}}{ {m_0}!\,^2\cdot {(1+ J_1)_{m_1}}\!^3\cdot {(1+ J_2)_{m_2}}\!^3}\:u_0^{m_0} u_1^{m_1} u_2^{m_2} $$ Then clearly \begin{equation}\label{eq:logtildomexpand} \begin{array}{rcl} \log\tilde\Omega(u,J_1,J_2)&=& \log\Omega^{(0)}(u)\, + \, (\log U_1-\log u_1)\,J_1\,+\\[.4em] && +\,(\log U_2-\log u_2)\,J_2 \,+\,{1\over9}P_0\,J_1J_2 \end{array} \end{equation} We have the following expansion of $\tilde\Omega(u,J_1,J_2)$ w.r.t. $u_0$ $$ \tilde\Omega(u,J_1,J_2)=\phi_0(u_1,J_1)\phi_0(u_2,J_2) + \phi_1(u_1,J_1)\phi_1(u_2,J_2)u_0 + {\cal O}(u_0^2), $$ where we define \begin{eqnarray} \label{eq:phi0} \phi_0(w,\rho)&:=&\sum_{n\geq0} {(1+3\rho)_{3n} \over {(1+\rho)_n}^3 } w^n \;, \\ \phi_1(w,\rho)&:=&\sum_{n\geq0} {(1+3\rho)_{1+3n} \over {(1+\rho)_n}^3 } w^n . \nonumber \end{eqnarray} Note $$ \phi_1(w,\rho)=(1+3\rho)\phi_0(w,\rho)+ 3w{\partial\over\partial w}\phi_0(w,\rho)\,. $$ This shows that modulo $u_0^2$ \begin{equation}\label{eq:lophi} \displaystyle{ \log\tilde\Omega(u,J_1,J_2)} \equiv \displaystyle{\log\phi_0(u_1,J_1)\,+\,\log\phi_0(u_2,J_2)\,+ \hspace{2cm}} \end{equation} $$ + \displaystyle{ \left(1+3J_1+3u_1{\partial\over\partial u_1}\log\phi_0(u_1,J_1)\right) \left(1+3J_2+3u_2{\partial\over\partial u_2}\log\phi_0(u_2,J_2)\right) u_0} $$ Comparing (\ref{eq:logtildomexpand}) and (\ref{eq:lophi}) we see that we have proved: \begin{Proposition} Define for $j=1,2$ the function $\bar U_j$ by $$ \log \bar U_j:=\log u_j\,+\,{\partial\over\partial\rho}\log\phi_0(u_j,\rho) |_{\rho=0}\,. $$ Then \begin{eqnarray} \nonumber \log U_j&=&\log \bar U_j\,+\, {\cal O}(u_0) \\ \label{eq:P0} {1\over9}P_0&=&9\left(u_1{\partial\over\partial u_1}\log \bar U_1\right) \left(u_2{\partial\over\partial u_2}\log \bar U_2\right) u_0\,+\, {\cal O}(u_0^2) \end{eqnarray} {}\hfill$\Box$ \end{Proposition} \ Before we can draw conclusions for the numbers $N_{1,n_1,n_2}$ we must first analyse $U_0$ modulo $u_0^2$. Let $$ \tilde{\tilde\Omega}(u,J_0):=\sum_{m_0,m_1,m_2\geq 0} \frac{(1+ J_0)_{m_0+3m_1}\cdot (1+ J_0)_{m_0+3m_2}}{ {(1+ J_0)_{m_0}}\!^2\cdot m_1!\,^3\cdot m_2!\,^3}\:u_0^{m_0} u_1^{m_1} u_2^{m_2} $$ with as before $J_0^2=0$. Then $$ \log\tilde{\tilde\Omega}(u,J_0)\,=\,\log\Omega^{(0)}(u)+ (\log U_0-\log u_0)\,J_0 $$ Let \begin{equation}\label{eq:xi} \xi(w,\rho):=\sum_{n\geq0} {(1+\rho)_{3n} \over n!\,^3 } w^n \; \end{equation} Then $$ \tilde{\tilde\Omega}(u,J_0)= \xi (u_1,J_0)\cdot\xi (u_2,J_0) \,+\, {\cal O}(u_0) $$ and hence \begin{equation}\label{eq:U0} U_0=u_0\cdot\psi(u_1)\cdot\psi(u_2)\,+\, {\cal O}(u_0^2) \end{equation} where $$ \psi(w):=\exp({\partial\over\partial\rho}\log\xi(w,\rho)|_{\rho=0}) $$ By combining (\ref{eq:dP}), (\ref{eq:P0}) and (\ref{eq:U0}) we find \begin{Corollary} $$ \sum_{n_1,n_2 \geq 0} N_{1,n_1,n_2} \bar U_1^{n_1}\bar U_2^{n_2}\,=\,81 \left({1\over\psi(u_1)}u_1{\partial\over\partial u_1}\log \bar U_1\right) \left({1\over\psi(u_2)}u_2{\partial\over\partial u_2}\log \bar U_2\right) $$ The number $N_{1,n_1,n_2}$ factorizes as $$ N_{1,n_1,n_2}=b_{n_1} b_{n_2} \;\;, $$ where the numbers $b_n$ are defined by \begin{eqnarray} \sum_{n\geq 0} b_n \bar U_1^n \::=\:9 \left({1\over\psi(u_1)}u_1{\partial\over\partial u_1}\log \bar U_1\right). \label{eq:b-prep} \end{eqnarray} {}\hfill$\Box$ \end{Corollary} \begin{Corollary} Let $\{ b_n \}$ be the sequence of integers defined by the expansion $(\ref{eq:b-prep})$. We obtain the asymptotic expansion of the B-model prepotential as follows: \begin{eqnarray} \Psi_B(U_0, U_1, U_2) = \mbox{topological term} + U_0 B(t_1) B(t_2) + {\cal O}(U_0^2) \label{eq:b-asymp} \end{eqnarray} where $B(t)$ is defined by the series $$ B(t) = \sum_{n \geq 0} b_n \exp(2 \pi i n t) = \sum_{n \geq 0} b_n U^n. $$ \end{Corollary} {From} the asymptotic expansions of (\ref{eq:2-A-prep}) and (\ref{eq:b-asymp}) we obtain the following precise identity between two functions, which actually follows from the Mirror Conjecture~\ref{c:mirror}. \begin{Conjecture}\label{c:mirror-2} We will obtain the following identity $$ \fbox{$ A^{res}(t) \equiv B(t)$} $$ or equivalently $$ \fbox{$\sum_{n\geq 0} b_n U^n = \Theta_{E_8}(3t,t\gamma)\prod_{n\geq 1}(1- U^{3n})^{-12}$} $$ where $U = \exp(2 \pi i t) $ and $\gamma=(1,1,1,1,1,1,1,-1)$. \end{Conjecture} Unfortunately we are unable to prove Conjecture~\ref{c:mirror-2}. However since we can explicitly expand the right hand side of (\ref{eq:b-prep}), we can obtain the expansion of $B(t)$ by using a computer and compare the result with {the expansion} of $A^{res}(t)$. \begin{Proposition} Conjecture~\ref{c:mirror-2} is true up to order $U^{50}$. \end{Proposition} \vspace{5mm} To get started on the computer one may notice: \begin{eqnarray} \phi_0(u,0)=\xi(u,0)&=&\sum_{n\geq0} {(3n)!\over n!\,^3 } u^n \\ {\partial\over\partial u}\phi_0(u,\rho)|_{\rho=0}&=& \sum_{n\geq0} {(3n)!\over n!\,^3 } 3(g(3n)-g(n))u^n\\ {\partial\over\partial u}\xi(u,\rho)|_{\rho=0}&=& \sum_{n\geq0} {(3n)!\over n!\,^3 } g(3n) u^n \end{eqnarray} where $$ g(n)=\sum_{k=1}^n{1\over k}\;,\hspace{10mm} g(3n)=\sum_{k=1}^{3n}{1\over k} $$ A simple PARI program then yields: \begin{eqnarray} && \quad B(t) = 9{1\over\psi(u)}u{\partial\over\partial u}\log U = \nonumber \\ &&9+ 36U+ 126{U^2}+ 360{U^3}+ 945{U^4}+ 2268{U^5}+ 5166{U^6}+ 11160{U^7} \nonumber \\ && + 23220{U^8} +46620{U^9}+ 90972{U^{10}} + 172872{U^{11}} + 321237{U^{12}} \nonumber \\ &&+ 584640{U^{13}}+ 1044810{U^{14}}+1835856{U^{15}}+ 3177153{U^{16}}+5421132{U^{17}} \nonumber \\ && + 9131220{U^{18}} +15195600{U^{19}}+25006653{U^{20}} + 40722840{U^{21}} \nonumber \\ && + 65670768{U^{22}}+104930280{U^{23}}+166214205{U^{24}}+ 261141300{U^{25}} \nonumber \\ &&+ 407118726{U^{26}}+ 630048384{U^{27}} + 968272605{U^{28}} + 1478208420{U^{29}} \nonumber \\ && + 2242463580{U^{30}}+ 3381344280{U^{31}} + 5069259342{U^{32}} + 7557818940{U^{33}} \nonumber \\ && + 11208455370{U^{34}}+16538048640{U^{35}} + 24282822798{U^{36}} \nonumber \\ && +35487134928{U^{37}} + 51626878470{U^{38}}+ 74779896240{U^{39}} \nonumber \\ && + 107861179482{U^{40}} + 154945739844{U^{41}} + 221711362038{U^{42}} \nonumber \\ &&+316042958880{U^{43}} +448856366490{U^{44}}+ 635216766732{U^{45}} \nonumber \\ &&+895854679650{U^{46}} + 1259213600736{U^{47}} + 1764210946995{U^{48}} \nonumber \\ && +2463949037340{U^{49}}+3430694064888{U^{50}} + O(U^{51}) \nonumber \\ && \label{eq:b-expand} \end{eqnarray} Comparing this expansion~(\ref{eq:b-expand}) with Table 2 in Section~\ref{s:rest}, we see that $a_n= b_n$ for $n\leq 50$. {}\hfill$\Box$ \section{Appendix I: B-model equation} \label{s:B-model equations} In this appendix we derive the equations stated in proposition \ref{orbifold} for the mirror $W^*$ of Schoen's example $W$. We use the mirror construction of Batyrev-Borisov \cite{Batyrev-Borisov} by means of reflexive Gorenstein cones of index $2$. As explained in \cite{Sti} the story in \cite{Batyrev-Borisov} about split Gorenstein cones and NEF partitions can for examples like $W$ be reformulated in terms of triangulations of the polytope $\Delta$ on the mirror side; more specifically, $W$ can be embedded in $\P1\times \P2\times \P2$ as the intersection of a hypersurface of degree $(1,3,0)$ and a hypersurface of degree $(1,0,3)$. This leads to the matrix ${\sf B}$ in (\ref{eq:Bmat}) and to the set $\{3,4\}\times\{5,6,7\}\times\{8,9,10\}$ in (\ref{eq:union}). To get the reflexive Gorenstein cone $\Lambda$ from which the mirror of Schoen's example can be constructed one should take a $7\times 10$ -matrix ${\sf A}=(a_{ij})$ with rank $7$ and with integer entries such that ${\sf A}\cdot{\sf B}^t\:=\:0\,.$ We take $$ {\sf A}\::=\:\left( \begin{array}{rrrrrrrrrr} 1&0&1&0&1&1&1&0&0&0\\ 0&1&0&1&0&0&0&1&1&1\\ 0&0&1&-1&0&0&0&0&0&0\\ 0&0&0&0&1&-1&0&0&0&0\\ 0&0&0&0&1&0&-1&0&0&0\\ 0&0&0&0&0&0&0&1&-1&0\\ 0&0&0&0&0&0&0&1&0&-1 \end{array} \right) $$ Let ${\sf a}_1,\ldots,{\sf a}_{10}\in{\bf Z}^7$ be the columns of ${\sf A}$. Then $$ \Lambda:={\bf R}_{\geq 0}{\sf a}_1+\ldots+{\bf R}_{\geq 0}{\sf a}_{10}\;\subset{\bf R}^7 $$ The polytope $\Delta$ is the convex hull of the points ${\sf a}_1,\ldots,{\sf a}_{10}$ in ${\bf R}^7$. With $\{3,4\}\times\{5,6,7\}\times\{8,9,10\}$ viewed as a collection of subsets of $\{1,\ldots,10\}$ the complements of these 18 subsets are the index sets for the maximal simplices in a triangulation of $\Delta$. Let ${\bf S}_\Lambda$ denote the subalgebra of the algebra of Laurent polynomials $$ {\bf C}[u_1^{\pm 1},\ldots,u_7^{\pm 1}] $$ generated by the monomial $u_1^{m_1}\cdot\ldots\cdot u_7^{m_7}$ with $(m_1,\ldots,m_7)\in\Lambda\cap{\bf Z}^7$. Giving such a monomial degree $m_1+m_2$ makes ${\bf S}_\Lambda$ a graded ring. The scheme ${\bf P}_\Lambda\::=\:{\rm Proj }{\bf S}_\Lambda$ is a projective toric variety of dimension $6$. A global section of ${\cal O}_{{\bf P}_\Lambda}(1)$ is given by a Laurent polynomial (with coefficients $v_1,\ldots,v_{10}$) \begin{eqnarray*} {\sf s} & = & u_1(v_1+v_5u_4u_5+v_6u_4^{-1}+v_7u_5^{-1}+v_3u_3) + \\ & & u_2(v_2+v_8u_6u_7+v_9u_6^{-1}+v_{10}u_7^{-1}+v_4u_3^{-1}) \end{eqnarray*} For generic coefficients $v_1,\ldots,v_{10}$ the zero locus of ${\sf s}$ in ${\bf P}_\Lambda$ is a generalized Calabi-Yau manifold of dimension $5$ in the sense of \cite{Batyrev-Borisov}. This is one mirror of $W$ suggested by \cite{Batyrev-Borisov}. As in \cite{Batyrev-Borisov} Section 4, one can also realize a mirror as a complete intersection Calabi-Yau threefold in a $5$-dimensional toric variety, as follows. ${\bf P}_\Lambda$ is a compactification of the torus $({\bf C}^*)^7/{\bf C}^*$ where ${\bf C}^*:=\{(u,u,1,1,1,1,1)\in ({\bf C}^*)^7\}$. The morphism \begin{eqnarray*} ({\bf C}^*)^7&\rightarrow& \P1\times\P1\times\P3\times\P3 \end{eqnarray*} given by \begin{eqnarray*} \lefteqn{(u_1,\ldots,u_7)\mapsto ([u_1:u_1u_3],[u_2:u_2u_3^{-1}], } \\ && [u_1:u_1u_4u_5:u_1u_4^{-1}:u_1u_5^{-1}], [u_2:u_2u_6u_7:u_2u_6^{-1}:u_2u_7^{-1}]) \end{eqnarray*} extends to a morphism ${\bf P}_\Lambda\rightarrow \P1\times\P1\times\P3\times\P3$. The image is $$ V:=\left\{\left. \begin{array}{l} [p_0:p_1]\times [q_0:q_1]\times [s_0:s_1:s_2:s_3]\times [t_0:t_1:t_2:t_3] \\ \in \P1\times\P1\times\P3\times\P3 \\ p_0q_0= p_1q_1, \quad s_0^3= s_1s_2s_3, \quad t_0^3 = t_1t_2t_3 \end{array} \right.\right\} $$ As noted in \cite{Batyrev-Borisov} Cor.3.4 the complement of the generalized Calabi-Yau $5$-fold ${\sf s}=0$ in ${\bf P}_\Lambda$ is a ${\bf C}$-bundle over the complement in $V$ of the complete intersection Calabi-Yau $3$-fold with equations \begin{eqnarray*} (v_1s_0+v_5s_1+v_6s_2+v_7s_3)p_0+v_3s_0p_1 &=&0\\ (v_2t_0+v_8t_1+v_9t_2+v_{10}t_3)q_0+v_4t_0q_1&=&0 \end{eqnarray*} This complete intersection Calabi-Yau $3$-fold itself is another realization for a mirror of $W$. Now note that the morphism \begin{eqnarray*} \P1\times\P2\times\P2&\rightarrow&\P1\times\P1\times\P3\times\P3\;, \end{eqnarray*} $$ \begin{array}{l} ([z_0:z_1],[x_0:x_1:x_2],[y_0:y_1:y_2]) \mapsto \\ ([z_0:z_1],[z_1:z_0],[x_0x_1x_2:x_0^3:x_1^3:x_2^3] ,[y_0y_1y_2:y_0^3:y_1^3:y_2^3]) \end{array} $$ realizes $V$ also as the quotient of $ \P1 \times \P2 \times \P2$ by the group $ {\bf Z}_3\times{\bf Z}_3 $ acting as in Proposition~\ref{orbifold}. This completes the proof of Proposition~\ref{orbifold}. \section{Appendix II: The Theta function of the $E_8$ lattice } \label{s:appendixII} Let $\Lambda $ be a lattice of rank $d$ with positive definite quadratic form $Q:\Lambda \lra {\bf Z}$. We can fix an embedding $\Lambda \hookrightarrow {\bf R}^d$ such that the quadratic form $Q$ is induced by the usual Euclidean inner product $( \: , \: )$. Let ${\cal H} = \{ \tau \in {\bf C} | \mbox{\rm Im} (\tau) > 0 \}$ be the upper half plane. We denote by ${\bf w} = (w_1, \cdots, w_d)$ the standard complex coordinates of ${\bf C}^d = {\bf R}^d \otimes {\bf C}$. We define the theta function associated to the lattice $\Lambda$ by \begin{eqnarray} \Theta_{\Lambda}(\tau, {\bf w}) = \sum_{\sigma \in \Lambda} \exp(2 \pi i ((\tau/2) Q(\sigma) + (\sigma, {\bf w})). \end{eqnarray} For certain Calabi-Yau 3-folds with a fibration by abelian surfaces one can calculate a part of the prepotential of the Yukawa coupling arising from the sections of the fibration by using the theta function associated to the Mordell-Weil lattice \cite{Saito}. Since the Mordell-Weil lattice of a generic Schoen's example is isometric to $E_8 \times E_8$, we would like to calculate the theta function of $E_8$ and write it in an explicit form. For that purpose we fix a standard embedding of $D_8$ and $E_8$ into ${\bf R}^8$. (\cite{C-S} p. 117 $\sim$ p. 121). Let $e_1, e_2, \cdots, e_8$ be the standard orthonormal basis of ${\bf R}^8$. An element of ${\bf R}^8$ is written as $\sum_{i=1}^{8} x_i e_i$. We define lattices in ${\bf R}^8$ $$ {\bf Z}^{8} := \left\{ \sum_{i=1}^{8} x_i e_i, x_i \in {\bf Z} \right\} \supset D_8 := \left\{ \sum_{i=1}^{8} x_i e_i \in {\bf Z}^8, \\ \sum_{i=1}^{8} x_i \equiv 0\, (2) \right\}, $$ $$ E_8 = D_8 \cup (D_8 + s_0), \quad s_0 = \frac{1}{2} \sum_{i=1}^{8} e_i, $$ The inner product $( \:, \:)$ induces positive definite bilinear forms on these lattices and $E_8$ and $D_8$ have integral bases whose intersection matrices are the Cartan matrices of $E_8$ and $D_8$ respectively. The theta function for the one dimensional lattice $\Lambda = {\bf Z}$ with $Q(n) = n^2$ is the Jacobi theta function: \begin{eqnarray} \vartheta(\tau, w) :=\Theta_{{\bf Z}}(\tau, w) = \sum_{n \in {\bf Z}} \exp(\pi i n^2\tau + 2 \pi i n w). \end{eqnarray} We also have the following 4 theta functions (cf. \cite{Mum-Tata-1}): \begin{eqnarray} \vartheta_{0,0}(\tau, w) & = & \vartheta(\tau, w) \\ \vartheta_{0,1}(\tau, w) & = & \vartheta(\tau, w+\frac{1}{2}) \\ \vartheta_{1,0}(\tau, w) & = & \exp(\frac{\pi i \tau}{4} + \pi i w) \cdot \vartheta(\tau, w+\frac{\tau}{2}) \\ \vartheta_{1,1}(\tau, w) & = & \exp(\frac{\pi i \tau}{4} + \pi i(w+\frac{1}{2})) \cdot \vartheta(\tau, w+\frac{\tau+1}{2}) \end{eqnarray} \begin{Proposition} \label{pr:e8theta} Let ${\bf w} = (w_1, w_2, \cdots, w_8) \in {\bf C}^8$. \begin{eqnarray} \Theta_{{\bf Z}^8}(\tau, {\bf w}) & = & \prod_{i=1}^8 \vartheta_{0,0}(\tau, w_i) \label{eq:z8}\\ \Theta_{E_8}(\tau, {\bf w}) &= & \frac{1}{2} \sum_{(a, b) \in ({\bf Z}/2{\bf Z})^2} \prod_{i=1}^8 \vartheta_{a,b}(\tau, w_i) \label{eq:e8} \end{eqnarray} \end{Proposition} \noindent {\it Proof.} Straightforward exercise. See also \cite{D-G-W}. {}\hfill$\Box$ \ Recall $\gamma=(1,1,1,1,1,1,1,-1)$. The above formulas show (cf.\cite{Mum-Tata-1}): \begin{eqnarray*} \vartheta_{0,0}(\tau,-w)=\vartheta_{0,0}(\tau,w)\:,&\hspace{3mm}& \vartheta_{0,1}(\tau,-w)=\vartheta_{0,1}(\tau,w)\:,\\ \vartheta_{1,0}(\tau,-w)=\vartheta_{1,0}(\tau,w)\:,&\hspace{3mm}& \vartheta_{1,1}(\tau,-w)=-\vartheta_{1,1}(\tau,w) \end{eqnarray*} and hence \begin{equation} \Theta_{E_8}(3t, t\gamma)=\frac{1}{2}\{ \vartheta_{0,0}(3t,t)^8 +\vartheta_{0,1}(3t,t)^8+\vartheta_{1,0}(3t,t)^8 -\vartheta_{1,1}(3t,t)^8 \} \end{equation} Next note: \begin{eqnarray*} \vartheta_{0,0}(3t,t)&=& \exp(-\pi i t/3)\sum_{ m\equiv\pm 1\,(3)}\exp(\pi i t m^2/3) \\ \vartheta_{0,1}(3t,t)&=& -\exp(-\pi i t/3)\sum_{ m\equiv\pm 1\,(3)}(-1)^m\exp(\pi i t m^2/3) \\ \vartheta_{1,0}(3t,t)&=& \exp(-\pi i t/3) \sum_{ m\equiv\pm 1\,(6)}\exp(\pi i t m^2/12) \\ \vartheta_{1,1}(3t,t)&=& -i\exp(-\pi i t/3) \sum_{ m\equiv\pm 1\,(6)} \chi (m)\,\exp(\pi i t m^2/12) \end{eqnarray*} where the summations run over $m \in {\bf N}$ with the indicated restrictions and $\chi (m)=1$ (resp. $=-1$) if $m\equiv\pm 1\bmod 12$ (resp. $\equiv\pm 5\bmod 12$). Another useful observation is that the Jacobi product formula for $\vartheta_{1,1}(\tau,w)$ (see \cite{Mum-Tata-1}) implies $$ \vartheta_{1,1}(3t,t)=-i\exp(-\pi i t/4)\prod_{m\geq 1}(1-\exp(2\pi i m t)) $$ Now the computer can do its work and compute the expansion of $\Theta_{E_8}(3t, t\gamma)$. \section*{Acknowledgments} The first author would like to thank J.Bryan and N.C.Leung for notifying him of the paper\cite{G-P}. He would like to thank also to S.-T.Yau and the Mathematics Department of Harvard University for their hospitality when finishing this work. The second author would like to thank Taniguchi foundation for their generous support for the Symposium. He would like to thank also all participants in the symposium with whom he enjoyed fruitful discussion. In particular, He would like to thank Ron Donagi for the discussion about \cite{D-G-W}. He would like to thank the staff of Kobe University, where he is enjoying daily stimulating atmosphere and discussion about mathematics. Special thanks are due to Kota Yoshioka in Kobe University who kindly remarked Lemma~\ref{l:hilb-2}. The third author would like to thank the Japan Society for the Promotion of Science for a JSPS Invitation Fellowship in November-December 1996 and Kobe University for support for a visit in July 1997. He expresses special thanks to his host, Masa-Hiko Saito, for creating a very stimulating atmosphere during these two visits to Kobe. \vspace{5mm} \vspace{5mm}

9709

alg-geom/9709019

en

https://arxiv.org/abs/alg-geom/9709019

alg-geom/9709019

Vladimir Masek

Vladimir Masek (Washington Univ. in St. Louis)

Kawachi's invariant for normal surface singularities

16 pages, AMS-LaTeX 1.2

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are Q-Gorenstein). The main result is a criterion (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein-Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.

alg-geom

\subsection*{Contents} \begin{enumerate} \item[0.] Introduction \item[1.] Kawachi's invariant and log-canonical singularities \item[2.] A theorem of Reider type on normal surfaces with boundary \end{enumerate} \subsection*{Notations} \begin{tabbing} 99\=9999999999\=9999999999999999999999999999\kill \>$\lceil \cdot \rceil$ \> round-up \\ \>$\lfloor \cdot \rfloor$ \> round-down \\ \>$\{ \cdot \}$ \> fractional part \\ \>$f^{-1}D$ \> strict transform (proper transform) \\ \>$f^*D$ \> pull-back (total inverse image) \\ \>$\,\equiv$ \> numerical equivalence \\ \>$\,\sim$ \> linear equivalence \\ \end{tabbing} \section{Introduction} Let $Y$ be a normal algebraic surface over an algebraically closed field of arbitrary characteristic. Let $y \in Y$ be a fixed point on $Y$. Let $f : X \to (Y,y)$ be the minimal resolution of the germ $(Y,y)$ if $y$ is singular, resp. the blowing-up of $Y$ at $y$ if $y$ is smooth. Kawachi (\cite{kawachi1}) introduced the following numerical invariant of $(Y,y)$: \begin{definition} $\delta_y = -(Z-\Delta)^2$, where $Z$ is the fundamental cycle of $y$ and $\Delta = f^*K_Y - K_X$ is the canonical cycle (or antidiscrepancy) of $y$. \end{definition} In \S 1 we recall several definitions (including the fundamental and the canonical cycle, etc.); then we study Kawachi's invariant and we give a very short proof of the fact (well-known to the experts) that ``numerically'' log-canonical surface singularities are automatically \ensuremath{\mathbb{Q}\,}-Gorenstein. In \S 2 we prove several criteria for global generation of linear systems of the form $|K_Y+\rup{M}|$, $M$ a \ensuremath{\mathbb{Q}\,}-divisor on $Y$ such that $K_Y+\rup{M}$ is Cartier. This type of results was the original motivation for introducing the invariant $\delta_y$ (see \cite{kawachi1, kawachi2, kawachim}). The main interest in results of Reider type for \ensuremath{\mathbb{Q}\,}-divisors on normal surfaces comes from work related to Fujita's conjecture for (log-) terminal threefolds, cf. \cite{elm, matsushita}. Using the criterion proved in \S 2, together with other recent results, we can significantly improve the main theorem of \cite{elm}; we will do so in a forthcoming paper. The author has benefitted from numerous discussions with L.~B\u{a}descu, L.~Ein, T.~Kawachi, R.~Lazarsfeld, and N.~Mohan Kumar. \section{Kawachi's invariant and log-canonical singularities} In this section we recall several standard definitions and facts regarding normal surface singularities and we prove a number of elementary lemmas involving Kawachi's invariant. For convenience, we use M. Reid's recent notes \cite{reid} as our main reference. \vspace{6pt} {\bf (1.1)} Let $Y$ be a complete normal algebraic surface (in any characteristic), and let $f : X \to Y$ be a resolution of singularities of $Y$. We use Mumford's \ensuremath{\mathbb{Q}\,}-valued pullback and intersection theory on $Y$ (cf. \cite{mumford}): if $D$ is any Weil (or \ensuremath{\mathbb{Q}\,}-Weil) divisor on $Y$, then $f^*D = f^{-1}D+D_{\text{exc}}$, where $f^{-1}D$ is the strict transform of $D$ and $D_{\text{exc}}$ is the (unique) $f$-exceptional \ensuremath{\mathbb{Q}\,}-divisor on $X$ such that $f^*D \cdot F = 0$ for every $f$-exceptional curve $F \subset X$. (The existence and uniqueness of $D_{\text{exc}}$ follow from the negative definiteness of the intersection form on exceptional curves.) If $D_1, D_2$ are two \ensuremath{\mathbb{Q}\,}-Weil divisors on $Y$, then $D_1 \cdot D_2 \overset{\text{def}}{=} f^*D_1 \cdot f^*D_2$. See, e.g., \cite[\S 1]{elm} for a quick review of this theory. [Here are a few key points: $D \geq 0 \implies f^*D \geq 0$; $D_1 \cdot D_2$ is independent of resolution; if $C$ is a \ensuremath{\mathbb{Q}\,}-divisor on $X$, then $C \cdot f^*D = f_*C \cdot D$; if $D$ is \ensuremath{\mathbb{Q}\,}-Cartier, then the definition of $f^*D$ coincides with the usual one.] \begin{definition} (a) Let $M$ be a \ensuremath{\mathbb{Q}\,}-Weil divisor on $Y$. Then $M$ is \emph{nef} if $M \cdot C \geq 0$ for all irreducible curves $C \subset Y$. (Equivalently, $M$ is nef if and only if $f^*M$ is nef on the smooth surface $X$.) (b) Assume that $M$ is nef. Then $M$ is \emph{big} if in addition $M^2>0$ (i.e., if $f^*M$ is big on $X$). \end{definition} \vspace{6pt} {\bf (1.2)} Now let $y \in Y$ be a fixed point, and let $f : X \to (Y,y)$ be the \emph{minimal} resolution of the germ $(Y,y)$ if $y$ is singular, resp. the blowing-up of $Y$ at $y$ if $y$ is smooth. Let $f^{-1}(y) = \cup_{j=1}^{N} F_j$ (set-theoretically); $N = 1$ if $y$ is smooth. \begin{definition} The \emph{fundamental cycle} of $(Y,y)$ is the smallest nonzero effective divisor $Z = \sum z_j F_j$ on $X$ (with $z_j \in \ensuremath{\mathbb{Z}\,}$) such that $Z \cdot F_j\leq 0,\forall j$ (cf. \cite[4.5]{reid} or \cite[p.132]{artin}). \end{definition} Note that $z_j \geq 1, \forall j$, because $\cup F_j$ is connected. If $y$ is smooth, then $Z = F_1$ ($F_1$ is a $(-1)$-curve in this case). Let $p_a(Z) = \frac{1}{2} Z \cdot (Z+K_X) + 1$; then $p_a(Z) \geq 0$, cf. \cite[Theorem 3]{artin}. \vspace{6pt} {\bf (1.3)} Let $K_X$ be a canonical divisor on $X$; then $f_*K_X$ is a canonical divisor $K_Y$ on $Y$, and $\Delta \overset{\text{def}}{=} f^*K_Y - K_X$ is an $f$-exceptional \ensuremath{\mathbb{Q}\,}-divisor on $X$, $\Delta = \sum a_jF_j$. Note that $\Delta$ is $f$-numerically equivalent to $K_X$ ($\Delta \underset{f}{\equiv} -K_X$), i.e. $\Delta \cdot F_j = -K_X \cdot F_j, \forall j$; in particular, $p_a(Z) = \frac{1}{2} Z \cdot (Z-\Delta) + 1$, or $Z \cdot (Z-\Delta) = 2 p_a(Z) - 2$. If $y$ is smooth, then $\Delta = -F_1$. On the other hand, if $y$ is singular and $f$ is the minimal resolution of $(Y,y)$, then $\Delta \cdot F_j = -K_X \cdot F_j \leq 0, \forall j$, so that $\Delta$ is effective (see, e.g., \cite[Lemma 1.4]{elm}). In fact, $\Delta = 0$ if and only if $y$ is a canonical singularity (= $RDP$, = du Val singularity), cf. the last part of (1.4) below, and if $y$ is not canonical, then $a_j > 0$ for \emph{all} $j = 1,\dots ,N$ (again, because $\cup F_j$ is connected). \begin{definition} $\Delta$ is the \emph{canonical cycle} (or \emph{antidiscrepancy}) of $(Y,y)$. \end{definition} $\Delta$ is uniquely defined, even though $K_X$ (and, accordingly, $K_Y$) is defined only up to linear equivalence. This follows from $\Delta \cdot F_j = -K_X \cdot F_j, \forall j$, and the negative definiteness of $\Vert F_i \cdot F_j \Vert$. Assume that $y$ is Gorenstein and non-canonical. Then $\Delta$ has integer coefficients and $\Delta > 0$; by the definition of the fundamental cycle, $\Delta \geq Z$. On the other hand, if $(Y,y)$ is a log-canonical singularity, then $\Delta \leq Z$ (see below). \vspace{6pt} {\bf (1.4)} {\bf Definition.} $y$ is a \emph{rational} singularity if $R^1f_*\mathcal{O}_X = 0$, or equivalently (cf. \cite[Theorem 3]{artin}), if $p_a(Z)=0$. \vspace{4pt} If $y$ is any normal singularity and $D$ is a Cartier divisor on $Y$, then $f^*D$ has integer coefficients. If $y$ is a rational singularity, then the converse is also true: if $f^*D$ has integer coefficients, then $D$ is Cartier. (\emph{Proof:} $f^*D \cdot F_j = 0, \forall j$, and therefore $f^*D$ is trivial in an open neighborhood of $f^{-1}(y)$, cf. \cite[Lemma 5]{artin}; thus $D$ is trivial in a punctured open neighborhood of $y$ in $Y$, and therefore $D$ is Cartier, because $y$ is normal.) In particular, a rational surface singularity is always \ensuremath{\mathbb{Q}\,}-factorial. If $y$ is a rational singularity, then $\mult_y Y =-Z^2$ (cf. \cite[Corollary 6]{artin}, or \cite[4.17]{reid}); in particular, $\text{$y$ is a $RDP$} \iff Z^2 = -2 \iff Z \cdot \Delta = 0 \iff \Delta \cdot F_j = 0, \forall j \iff \Delta = 0$. (Therefore $RDP \implies$ Gorenstein, because $f^*K_Y = K_X$ has integer coefficients, and therefore $K_Y$ is Cartier. The converse is also true: rational Gorenstein $\implies RDP$. Indeed, if $y$ is Gorenstein, then either $\Delta = 0$ or $\Delta \geq Z$; but $\Delta \geq Z \implies Z \cdot (Z-\Delta) \geq 0 \implies p_a(Z) \geq 1$.) \vspace{6pt} {\bf (1.5)} {\bf Definition.} $y$ is an \emph{elliptic} singularity if $R^1f_* \mathcal{O}_X$ is 1-dimensional. It is \emph{elliptic Gorenstein} if in addition it is Gorenstein. \begin{Lemma} $y$ is elliptic Gorenstein if and only if $Z = \Delta$. \end{Lemma} \begin{proof} $\implies$ is proved in \cite[4.21]{reid}. Conversely, if $Z = \Delta$, then $K_X+Z = f^*K_Y$, so that $(K_X+Z) \cdot F_j = 0, \forall j$; the proof proceeds as in \cite[4.21]{reid} (go directly to Step 3 there). \end{proof} {\bf (1.6)} {\bf Definition.} $y$ is \emph{log-terminal} (resp. \emph{log-canonical}) if $a_j < 1$ (resp. $a_j \leq 1$), $\forall j$ -- where $\Delta = f^*K_Y-K_X = \sum a_jF_j$, as before. In dimension three or higher, one must assume that $y$ is \ensuremath{\mathbb{Q}\,}-Gorenstein before defining $\Delta$ (and log-terminal singularities, etc.) Using Mumford's definition of $f^*K_Y$, we don't need to make this assumption in the two-dimensional case. What's more, we will see in a moment that log-canonical (in our sense) automatically implies \ensuremath{\mathbb{Q}\,}-Gorenstein. (This is also clear from the complete list of all log-canonical singularities: the arguments in \cite{alexeev} do not use the \ensuremath{\mathbb{Q}\,}-Gorenstein condition.) \vspace{6pt} {\bf (1.7)} {\bf Definition.} $\delta_y = -(Z-\Delta)^2$. \vspace{4pt} Thus $\delta_y \in \ensuremath{\mathbb{Q}\,}, \delta_y \geq 0$, and $\delta_y = 0 \iff Z = \Delta \iff y$ is elliptic Gorenstein, by Lemma~1. \begin{Lemma} { \ \ \ } \begin{enumerate} \item[(a)] $\delta_y = 2-2p_a(Z)-\sum_{j=1}^N (z_j-a_j) K_X \cdot F_j$. \item[(b)] $\delta_y = 2-2p_a(Z)+\sum_{j=1}^N a_j (Z-\Delta) \cdot F_j$. \end{enumerate} \end{Lemma} \begin{proof} (a) $\delta_y = -(Z-\Delta)^2 = -Z \cdot (Z-\Delta) + \Delta \cdot (Z-\Delta) = 2 - 2p_a(Z) - K_X \cdot (Z-\Delta) = 2-2p_a(Z)-\sum_{j=1}^{N}(z_j-a_j) K_X\cdot F_j$ (because $\Delta\underset{f}{\equiv} K_X$). (b) is similar. \end{proof} \begin{Lemma} \textup{(cf. \cite[Theorem 1]{kawachim}) } \begin{enumerate} \item[(a)] $\delta_y = 4$ if $y$ is smooth; \item[(b)] $\delta_y = 2$ if $y$ is a $RDP$; \item[(c)] $0<\delta_y<2$ if $y$ is log-terminal but not smooth or a $RDP$; \item[(d)] $0 \leq \delta_y \leq 2$ if $y$ is log-canonical but not smooth. \end{enumerate} \end{Lemma} \begin{proof} (a) $y$ smooth $\implies Z-\Delta=2F_1$, and $F_1^2=-1$; thus $\delta_y=4$. (b) $y$ a $RDP \implies \Delta=0$, and $\delta_y = -Z^2 = \mult_y Y = 2$. (c), (d) $\delta_y$ is always $\geq 0$. If $\delta_y = 0$, then $\Delta = Z$, so that $a_j = z_j \geq 1, \forall j$; therefore $y$ log-terminal $\implies \delta_y > 0$. On the other hand, log-canonical $\implies a_j \leq 1 \leq z_j, \forall j$, so that $\delta_y \leq 2$ by Lemma~2 above ($K_X\cdot F_j\geq 0,\forall j$, because $f$ is the \emph{minimal} resolution). Finally, if $y$ is log-terminal, then $a_j < z_j, \forall j$; then $\delta_y = 2 \implies p_a(Z) = 0$ and $K_X \cdot F_j = 0, \forall j$ (recall that $p_a(Z) \geq 0$ for all normal singularities). Thus $y$ is a rational singularity, and the last paragraph of (1.4) shows that $y$ is a $RDP$. \end{proof} {\bf Corollary of the proof.} \emph{ If $y$ is log-terminal then it is rational. If $y$ is log-canonical then it is either elliptic Gorenstein or rational. In particular, log-canonical implies \ensuremath{\mathbb{Q}\,}-Gorenstein.} (We noted already in (1.4) that rational implies \ensuremath{\mathbb{Q}\,}-factorial.) Indeed, the proof above shows that $y$ log-canonical $\implies p_a(Z) \leq 1$ (because $\delta_y \geq 0, z_j-a_j \geq 0, \text{ and } K_X \cdot F_j \geq 0, \forall j$); moreover, $p_a(Z) = 1 \implies \delta_y = 0 \implies Z = \Delta \implies y$ is elliptic Gorenstein -- and this can happen only in the log-canonical, not in the log-terminal case. In all other cases, $p_a(Z) = 0$, and therefore $y$ is rational. \vspace{6pt} {\bf (1.8)} The invariant $-\Delta^2$ has been considered before; for example, Sakai \cite{sakai} proved results of Reider type on normal surfaces, using this invariant. In a sense, $-\Delta^2$ may be viewed as a local analogue of the Chern number $c_1^2$. \vspace{4pt} At least in the rational case, $-\Delta^2$ is closely related to $\mult_y Y$, via Kawachi's invariant: \begin{Lemma} $-\Delta^2 = -Z^2 + \delta_y + 4(p_a(Z)-1)$ \end{Lemma} \begin{proof} $\Delta = Z - (Z-\Delta)$; therefore \newline $-\Delta^2 = -Z^2 -(Z-\Delta)^2 +2 Z \cdot (Z-\Delta) = -Z^2 + \delta_y + 4(p_a(Z)-1)$ \end{proof} \begin{Corollary} If $y$ is rational, then $p_a(Z) = 0$ and $-Z^2 = \mult_y Y$, so that $$ -\Delta^2 = \mult_y Y - (4 - \delta_y). $$ \end{Corollary} In particular, if $y$ is log-terminal but not canonical, we get: $$ \mult_y Y - 4 < -\Delta^2 < \mult_y Y - 2. $$ \vspace{6pt} {\bf (1.9)} Since log-canonical surface singularities are classified (see, for example, \cite{alexeev, crepant}), one could compute $\delta_y$ explicitly in all cases. Indeed, this was Kawachi's original proof of part (c) in Lemma 3; see \cite{kawachi1} for the complete list in the log-terminal case. The computation of $Z, \Delta, \text{ and } \delta_y$ is an easy exercise in linear algebra. For illustration, we give here the values of $Z, \Delta, \text{ and } \delta_y$ for the ``truly'' log-canonical (i.e. non-log-terminal) singularities. The list of all such singularities can be found in \cite[p.58]{alexeev}. To simplify notation, we assume that the chains of smooth rational curves shown in \cite{alexeev} consist of just one curve each. In each case, we show the dual graph of $\cup F_j$, indicating the self-intersection numbers $F_j^2$. Notice that in the ``truly'' log-canonical case $\delta_y$ takes only the values 0, 1, and 2 (always an integer), and the value of $\delta_y$ distinguishes three types of log-canonical singularities (with one interesting exception, noted below): \begin{picture}(290,50)(0,0) \put(0,25){\makebox(0,0)[l]{Notation:}} \put(51,25){\circle{8}} \put(51,25){\circle{3}} \put(59,25){\makebox(0,0)[l]{= smooth elliptic curve; }} \put(174,25){\circle{8}} \put(182,25){\makebox(0,0)[l]{= smooth rational curve.}} \end{picture} \emph{Type 1:} elliptic Gorenstein (cases (4) and (5) in \cite{alexeev}) \begin{center} \begin{picture}(200,85)(0,10) \put(50,50){\circle{8}} \put(50,50){\circle{3}} \put(45,35){$F_1$} \put(70,50){\makebox(0,0)[l]{or}} \put(100,50){\circle{8}} \put(95,35){$F_1$} \put(124,26){\circle{8}} \put(119,11){$F_n$} \put(124,74){\circle{8}} \put(119,82){$F_2$} \put(102.82,52.82){\line(1,1){18.56}} \put(102.82,47.18){\line(1,-1){18.56}} \put(126.82,71.18){\line(1,-1){10}} \put(126.82,28.82){\line(1,1){10}} \qbezier[7](142,56)(148,50)(142,44) \end{picture} \end{center} $$ Z = \Delta = F_1, \text{ resp. } F_1+\cdots+F_n; \qquad \delta_y = 0. $$ \vspace{15pt} \emph{Type 2:} (case (6) in \cite{alexeev}) \begin{center} \begin{picture}(200,80)(0,18) \put(50,40){\circle{8}} \put(45,25){$F_2$} \put(45,46){\Small $-a$} \put(88,40){\circle{8}} \put(83,25){$F_1$} \put(89,46){\Small $-w$} \put(126,40){\circle{8}} \put(121,25){$F_4$} \put(121,46){\Small $-c$} \put(88,78){\circle{8}} \put(83,86){$F_3$} \put(89,67){\Small $-b$} \put(54,40){\line(1,0){30}} \put(92,40){\line(1,0){30}} \put(88,44){\line(0,1){30}} \end{picture} \end{center} \begin{gather*} w \geq 2;\quad (a,b,c)=(3,3,3),\,(2,2,4),\,\text{ or } (2,3,6) \\ Z = F_1+(F_2+F_3+F_4) \text{ if } w \geq 3, Z = 2F_1+(F_2+F_3+F_4) \text{ if } w = 2; \\ \Delta = F_1 + (1-\tfrac{1}{a})F_2 + (1-\tfrac{1}{b})F_3 + (1-\tfrac{1}{c})F_4; \qquad \delta_y = 1. \end{gather*} \vspace{15pt} \emph{Type 3:} (cases (7) -- (8) in \cite{alexeev}) \begin{center} \begin{picture}(200,120)(0,-20) \put(50,40){\circle{8}} \put(45,25){$F_2$} \put(45,46){\scriptsize $-2$} \put(88,40){\circle{8}} \put(89,25){$F_1$} \put(89,46){\scriptsize $-w$} \put(126,40){\circle{8}} \put(121,25){$F_4$} \put(121,46){\scriptsize $-2$} \put(88,78){\circle{8}} \put(83,86){$F_3$} \put(89,69){\scriptsize $-2$} \put(88,2){\circle{8}} \put(83,-13){$F_5$} \put(89,7){\scriptsize $-2$} \put(54,40){\line(1,0){30}} \put(92,40){\line(1,0){30}} \put(88,44){\line(0,1){30}} \put(88,36){\line(0,-1){30}} \end{picture} \end{center} $w \geq 3. \quad$ If $w \geq 4$, then: \[ Z = F_1 + (F_2 + \cdots + F_5); \;\; \Delta = F_1 + \tfrac{1}{2}(F_2 + \cdots + F_5); \;\; \delta_y = 2. \] However, if $w = 3$, then $Z = 2F_1 + (F_2 + \cdots + F_5)$ (while $\Delta$ is the same), and $\delta_y = 1$. This exceptional case illustrates an interesting property of the fundamental cycle; see (2.10) below. \vspace{6pt} {\bf (1.10)} {\bf Exercise.} Calculate $Z$, $\Delta$, and $\delta_y$ for the following dual graph (cf. \cite[p.350]{brieskorn}): \begin{center} \begin{picture}(180,180)(0,0) \put(91,95){\scriptsize $-4$} \put(90,90){\circle{8}} \put(85,75){$F_1$} \put(90,94){\line(0,1){30}} \put(85,135){\scriptsize $-3$} \put(90,128){\circle{8}} \put(91,114){$F_2$} \put(92.82,130.82){\line(1,1){21.21}} \put(111.87,160.87){\scriptsize $-2$} \put(116.87,154.87){\circle{8}} \put(111.87,139.87){$F_5$} \put(87.18,130.82){\line(-1,1){21.21}} \put(58.13,160.87){\scriptsize $-2$} \put(63.13,154.87){\circle{8}} \put(58.13,139.87){$F_6$} \put(87.18,87.18){\line(-1,-1){21.21}} \put(58.13,69.13){\scriptsize $-3$} \put(63.13,63.13){\circle{8}} \put(64.13,48.13){$F_3$} \put(59.13,63.13){\line(-1,0){30}} \put(20.13,69.13){\scriptsize $-2$} \put(25.13,63.13){\circle{8}} \put(20.13,48.13){$F_7$} \put(63.13,59.13){\line(0,-1){30}} \put(64.13,31.13){\scriptsize $-2$} \put(63.13,25.13){\circle{8}} \put(58.13,10.13){$F_8$} \put(92.82,87.18){\line(1,-1){21.21}} \put(111.87,69.13){\scriptsize $-3$} \put(116.87,63.13){\circle{8}} \put(117.87,48.13){$F_4$} \put(116.87,59.13){\line(0,-1){30}} \put(117.87,31.13){\scriptsize $-2$} \put(116.87,25.13){\circle{8}} \put(111.87,10.13){$F_9$} \put(120.87,63.13){\line(1,0){30}} \put(149.87,69.13){\scriptsize $-2$} \put(154.87,63.13){\circle{8}} \put(149.87,48.13){$F_{10}$} \end{picture} \end{center} \vspace{6pt} {\bf (1.11)} We conclude this section with another example. Assume that $f^{-1}(y)$ is a smooth curve $C$ of genus $g$, with $C^2=-w, w \geq 1$. This situation can be realized easily in practice: for example, $y$ could be the vertex of the cone $Y = \text{Proj} ( \bigoplus_{k \geq 0} H^0(C,kL) )$, with $C$ an arbitrary smooth curve of genus $g$ and $L$ an arbitrary divisor of degree $w$ on $C$. Then $Z=C, \;\; \Delta = ( \frac{2}{w}(g-1) + 1 ) C, \;\; \text{ and } \;\; \delta_y = \dfrac{4(g-1)^2}{w}$. If $g = 0$, then $\Delta = (1-\frac{2}{w})C$, $y$ is log-terminal, and $\delta_y = \frac{4}{w}$. If $w=1$ then $y$ is smooth. If $w=2$ then $y$ is an $A_1$ singularity (ordinary double point). If $w \geq 3$ then $y$ is a log-terminal singularity ``of type $A_1$''. If $g = 1$, then $y$ is log-canonical and $\delta_y = 0$. Such a singularity is known as \emph{simply elliptic}. If $g \geq 2$, then $y$ is not log-canonical. Note that $\delta_y$ can be arbitrarily large in this case (if $g$ is large relative to $w$). In particular, $\delta_y$ \emph{may} be greater than 4 (which is the value for smooth points). \section{A theorem of Reider type on normal surfaces with boundary} \vspace{6pt} In this section the ground field is \ensuremath{\mathbb{C}\,}. \vspace{6pt} {\bf (2.1)} Let $Y$ be a projective surface over \ensuremath{\mathbb{C}\,}, and let $y$ be a fixed point on $y$. Assume that $Y$ is smooth except perhaps at $y$, which may be either smooth or a $RDP$. Let $M$ be a nef and big \ensuremath{\mathbb{Q}\,}-divisor on $Y$, with the property that $\rup{M}$ is Cartier. Ein and Lazarsfeld proved the following criterion on global generation: \vspace{4pt} Let $B = \rup{M} - M$, and let $\mu = \mult_y B$ if $y$ is smooth, resp. $\mu = \max \{ t \geq 0 \mid f^*B \geq tZ \}$ when $y$ is a $RDP$, where $f:X\to(Y,y)$ is the minimal resolution and $Z$ is the fundamental cycle. {\bf Theorem. (\cite[Theorem 2.3]{el})} {\em Assume that $M^2 > (2-\mu)^2$ and $M \cdot \ensuremath{\mathbb{C}\,} \geq (2-\mu)$ for all curves $C$ through $y$ (when $y$ is smooth), resp. that $M^2 > 2 \cdot (1-\mu)^2$ and $M \cdot C \geq (1-\mu)$ for all $C$ through $y$ (when $y$ is a $RDP$). Then $y \notin \Bs |K_Y+ \ulc M \urc|$. } \vspace{4pt} This theorem was an important part of Ein and Lazarsfeld's proof of Fujita's conjecture on smooth threefolds. In extending that work to (log-) teminal threefolds (as required by the minimal model theory), it was necessary to extend the criterion mentioned above to arbitrary normal surfaces. Such extensions were obtained, e.g., in \cite[Theorem 1.6]{elm}, \cite[Theorem 7]{matsushita}. However, these generalizations, while effective, are not optimal. \vspace{4pt} In a somewhat different direction, Kawachi and the author proved the following criterion, of independent interest: \vspace{4pt} {\bf Theorem. (\cite[Theorem 2]{kawachim})} {\em Let $Y$ be a normal surface, and let $y$ be a fixed point on $Y$. Let $\delta = \delta_y$ if $y$ is log-terminal, $\delta = 0$ otherwise. Let $M$ be a nef divisor (with integer coefficients) on $Y$, such that $M^2>\delta$ and $K_Y+M$ is Cartier. If $y \in \Bs |K_Y+ \ulc M \urc|$, then there exists an effective divisor $C$ passing through $y$, such that $M \cdot C < \frac{1}{2}\delta$ and $C^2 \geq M \cdot C - \frac{1}{4} \delta$ } (in particular, $y$ must be log-terminal, because $M$ is nef). \vspace{4pt} \begin{Remark} When $y$ is smooth, this is equivalent to Reider's original criterion (\cite[Theorem 1]{reider}). When $y$ is a $RDP$, we recover the Ein--Lazarsfeld criterion, plus a lower bound on $C^2$. \end{Remark} While this result has several applications to linear systems on normal surfaces (cf. \cite{kawachim}), it cannot be used in the context of Fujita's conjecture on terminal threefolds, because $M$ is required to have integer coefficients. Kawachi formulated the following criterion for \ensuremath{\mathbb{Q}\,}-divisors on normal surfaces: \vspace{4pt} Let $Y,y$ be as before. Let $f:X \to (Y,y)$ be the minimal resolution if $y$ is singular, resp. the blowing-up at $y$ if $y$ is smooth. Let $Z$ and $\Delta$ be the fundamental and the canonical cycle, respectively. Let $\mu = \max \{ t \geq 0 \mid f^*B \geq t(Z-\Delta) \}$; note that $\mu = 2 \cdot \mult_y Y$ if $y$ is smooth. {\bf Open Problem. (cf. \cite{kawachi2}) } {\em Let $\delta = \delta_y$ if $y$ is log-terminal, $\delta = 0$ otherwise. Let $M$ be a nef \ensuremath{\mathbb{Q}\,}-Weil divisor on $Y$, such that $K_Y+B+M$ is Cartier, where $B = \rup{M} - M$. If $M^2 > (1-\mu)^2 \delta$ and $M \cdot C \geq (1-\mu) \frac{\delta}{2}$ for every curve $C$ through $y$, then $y \notin \Bs |K_Y+ \ulc M \urc|$. } \vspace{4pt} When $M$ has integer coefficients, this criterion is the same as the one mentioned above, minus the lower bound on $C^2$. Also, this criterion contains the Ein--Lazarsfeld results for smooth and rational double points. Kawachi formulated this as a theorem. Unfortunately his proof, based on a case-by-case analysis, is incomplete. In this section we prove a slightly weaker version, requiring that $M \cdot C \geq (1-\mu)$, rather than $\geq (1-\mu) \frac{\delta}{2}$, for all $C$ through $y$. For application to Fujita's conjecture on singular threefolds this makes little difference, though, because $\delta_y$ cannot be controlled in that situation anyway; the bound $\delta_y \leq 2$ (for $y$ singular) is used instead. \vspace{6pt} {\bf (2.2)} Let $Y$ be a normal surface (= compact, normal two-dimensional algebraic space over $\ensuremath{\mathbb{C}\,}$). Let $y \in Y$ be a given point, and let $B=\sum b_iC_i$ be an effective $\ensuremath{\mathbb{Q}\,}$-Weil divisor on $Y$ with all $b_i \in \ensuremath{\mathbb{Q}\,}$, $0 \leq b_i \leq 1$; the $C_i$ are distinct irreducible curves on $Y$. Since later we may need to consider more curves $C_i$ than there are in $\Supp(B)$, we allow some coefficients $b_i$ to be $0$. Let $f:X\to(Y,y)$ be the minimal resolution of singularities of the germ $(Y,y)$ -- resp. the blowing-up at $y$ if $y$ is a smooth point. Let $f^{-1}(y) = \cup F_j, Z=\sum z_jF_j, \Delta=\sum a_jF_j,$ as in \S 1. \vspace{6pt} {\bf (2.3)} Let $D_i = f^{-1}C_i$. Write $f^*B = f^{-1}B + B_{\text{exc}} = \sum b_i D_i + \sum b'_j F_j$. \begin{definition} $(Y,B,y)$ is log-terminal (respectively log-canonical) if $a_j+b_j' < 1$ (respectively $\leq 1$) for all $j$. Thus $(Y,y)$ is log-terminal (log-canonical) if $(Y,0,y)$ is. \end{definition} \begin{Remark} We do not require $K_Y+B$ to be $\ensuremath{\mathbb{Q}\,}$-Cartier at $y$ (unlike the similar definition in higher dimension); as in \S 1, this is a \emph{consequence} of the other conditions. Note that $B \geq 0 \implies f^*B \geq 0$, and therefore $(Y,B,y)$ log-terminal (log-canonical) $\implies (Y,y)$ log-terminal (log-canonical). Moreover, if $y \in \Supp(B)$, then $b'_j > 0,\; \forall j$, and therefore $(Y,B,y)$ log-canonical already implies $(Y,y)$ log-terminal. \end{Remark} \vspace{6pt} {\bf (2.4)} {\bf Definition.} Assume $(Y,B,y)$ is log-terminal. Define \[ \mu=\mu_{B,y}=\max\{ t\geq 0 \mid f^*B \geq t(Z-\Delta) \}. \] \begin{Remark} All the $z_j$ are $\geq 1$ and all the $a_j$ are $<1$; $\mu$ is given explicitly by \[ \mu = \min \left\{ \frac{b_j'}{z_j-a_j} \right\} . \] Of course, $\mu = 0$ if $B=0$ (or, more generally, if $y \notin \Supp(B)$). Note that $\mu = \tfrac{1}{2} \mult_y(B)$ if $y$ is a smooth point of $Y$. \end{Remark} \begin{Lemma} If $(Y,B,y)$ is log-terminal and $\mu$ is defined as above, then $0\leq \mu <1$. \end{Lemma} Indeed, $\mu \geq 0$ is clear. On the other hand, if $\mu \geq 1$ then we have $f^*B \geq (Z-\Delta)$, or $\Delta+f^*B \geq Z$, and therefore $a_j + b_j' \geq z_j \geq 1$ for every $j$, contradicting log-terminality. \qed \vspace{6pt} {\bf (2.5)} Let $Y$ be a normal surface and $y\in Y$ a given point, as in (2.1). Let $M$ be a {\bf nef and big} $Q$-Weil divisor on $Y$ such that $K_Y+\rup{M}$ is Cartier. Let $B = \rup{M} - M = \sum b_iC_i$; $B$ is an effective $Q$-Weil divisor on $Y$, with $0 \leq b_i <1,\; \forall i$. \vspace{4pt} We will prove the following criteria for freeness at $y$: \vspace{4pt} \begin{Theorem} \label{thm:notlt} If $(Y,B,y)$ is \textbf{not} log-terminal, then $y \notin \Bs |K_Y+ \ulc M \urc|$. \end{Theorem} \vspace{6pt} In the following two theorems, assume that $(Y,B,y)$ \emph{is} log-terminal. Then $(Y,y)$ is also log-terminal. Define $\mu$ as in (2.3) and $\delta_y$ as in \S 1. Also, assume that $y$ is \emph{singular}. \vspace{6pt} \begin{Theorem} \label{thm:kaw} Assume that $M^2>(1-\mu)^2\delta_y$ and $M \cdot C \geq (1-\mu)$ for every curve $C \subset Y$ passing through $y$. (Note that $M$ is still assumed to be nef, i.e. $M \cdot C \geq 0$ even for curves $C \subset Y$ not passing through $y$.) \newline Then $y \notin \Bs |K_Y+ \ulc M \urc|$. \end{Theorem} \vspace{6pt} In fact, if $y$ is a singularity of type $D_n$ or $E_n$ (see \cite[Remark 9.7]{crepant}), we don't even need the assumption on $M \cdot C$ for $C$ through $y$: \vspace{6pt} \begin{Theorem} \label{thm:dnen} Assume that $(Y,y)$ is a log-terminal singularity of type $D_n$ or $E_n$, and $M$ is nef and $M^2>(1-\mu)^2\delta_y$. If $(Y,y)$ is of type $D_n$, assume moreover that $M \cdot C > 0$ for every $C$ through $y$. \newline Then $y \notin \Bs |K_Y+ \ulc M \urc|$. \end{Theorem} \vspace{6pt} {\bf (2.6)} First we reduce the proof of Theorems \ref{thm:notlt}, \ref{thm:kaw} and \ref{thm:dnen} to the case when $y$ is the only singularity of $Y$: \begin{Lemma} \label{lemma:yonly} We may assume that $Y - \{y\}$ is smooth. \end{Lemma} \begin{proof} If $y$ is not the only singularity of $Y$, then let $g:S \to Y$ be a simultaneous resolution of all singularities of $Y$ \emph{except} $y$. Put $M' = g^*M$ and $y'=g^{-1}(y)$ (note that $g$ is an isomorphism of an open neighborhood of $y'$ onto an open neighborhood of $y$). $K_S+\rup{M'}$ is Cartier: outside $y'$ this is clear, because $S-\{y'\}$ is smooth and $K_S+\rup{M'}$ has integer coefficients, and in a certain open neighborhood of $y'$ this is also clear, because $g$ is an isomorphism there, and $K_Y+\rup{M}$ is Cartier by hypothesis. Also, all the numerical conditions on $M$ are satisfied by $M'$ (in each of the hypotheses of Theorems \ref{thm:notlt}, \ref{thm:kaw} and \ref{thm:dnen}). If the theorems are true for $S$, $y'$, $M'$, then we get $y' \notin \Bs|K_S+\rup{M'}|$. Write $\Delta' = g^*K_Y-K_S$; $\Delta'$ is an effective $\ensuremath{\mathbb{Q}\,}$-divisor on $S$, and we have: $K_S+\rup{M'} = \lceil g^*K_Y-\Delta'+g^*\rup{M}-g^*B \rceil = g^*(K_Y+ \rup{M}) - \rdn{\Delta' + g^*B}$ (note that $K_Y+\rup{M}$ is Cartier by hypothesis, and therefore $g^*(K_Y+\rup{M})$ has integer coefficients). Write $N = \rdn{\Delta'+g^*B}$; $N$ is a divisor with integer coefficients on $S$, and $y' \notin \Supp(N)$, because in a certain neighborhood of $y'$, $\Delta'$ is zero and $g^*B$ is identified to $B$ -- whose coefficients are all $<1$. In the first part of the proof we found a section $s \in H^0(S, g^*(K_Y+\rup{M})-N)$ which doesn't vanish at $y'$. Multiplying $s$ by a global section of $\mathcal{O}_S(N)$ whose zero locus is $N$, we find a new section $t \in H^0(S, g^*(K_Y+\rup{M}))$, which still doesn't vanish at $y'$. In turn, $t$ corresponds to a global section of $\mathcal{O}_Y(K_Y+\rup{M})$ which doesn't vanish at $y$. \end{proof} \vspace{6pt} {\bf (2.7)} Now assume that $y$ is the only singularity of $Y$. Let $f:X \to Y$ be the minimal \emph{global} desingularization of $Y$, $\Delta = f^*K_Y-K_X = \sum a_jF_j$, $B=\sum b_iC_i$, $f^*B=\sum b_iD_i + \sum b_j'F_j$, $Z=\sum z_j F_j$, etc. \vspace{6pt} \noindent {\bf Proof of Theorem \ref{thm:notlt}} \vspace{3pt} Assume that $(Y,B,y)$ is not log-terminal; then there is at least one $j$ such that $a_j+b_j' \geq 1$. $f^*M$ is nef and big on $X$, and therefore the Kawamata--Viehweg vanishing theorem (cf. \cite[Lemma 1.1]{el}) gives: $H^1(X, K_X+\rup{f^*M}) = 0$. $K_X+\rup{f^*M} = \lceil f^*K_Y-\Delta+f^*\rup{M}-f^*B \rceil = f^*(K_Y+\rup{M})- \rdn{\Delta + f^*B} = f^*(K_Y+\rup{M})- \sum \rdn{a_j+b_j'} F_j = f^*(K_Y+\rup{M})- G$, where $G$ is a nonzero effective divisor with integer coefficients on $X$ such that $f(G)=\{y\}$. ($G>0$ because $a_j+b_j'\geq 1$ for at least one $j$). We have $H^1(X, f^*(K_Y+\rup{M})-G)=0$, and therefore the restriction map \[ H^0(X, f^*(K_Y+\rup{M})) \to H^0(G, f^*(K_Y+\rup{M}) |_G) \] is surjective. As $f(G)=\{y\}$, $f^*(K_Y+\rup{M}) |_G$ is trivial, i.e. it has a global section which doesn't vanish anywhere on $G$. By surjectivity, this section lifts to a global section $s \in H^0(X,f^*(K_Y+\rup{M}))$ which doesn't vanish anywhere on $G$. In turn, $s$ corresponds to a global section of $\mathcal{O}_Y(K_Y+\rup{M})$ which doesn't vanish at $y$, or else $s$ would vanish everywhere on $f^{-1}(y)$, and in particular on $G$. \qed \vspace{6pt} {\bf (2.8)} Assume that $y$ is log-terminal but not smooth. Thus $\Delta=\sum a_j F_j$ with $0\leq a_j <1$ for every $j$. \vspace{6pt} \noindent {\bf Proof of Theorem \ref{thm:kaw}} \vspace{3pt} \begin{Lemma} If $M^2 >(1-\mu)^2\delta_y$, then we can find an effective $\ensuremath{\mathbb{Q}\,}$-Weil divisor $D$ on $Y$ such that $D\equiv M$ and $f^*D\geq(1-\mu)(Z-\Delta)$. \end{Lemma} \begin{proof} Since $M^2>(1-\mu)^2\delta_y$, $f^*M-(1-\mu)(Z-\Delta)$ is in the positive cone of $X$ (see \cite{kawachim}, (2.3) for a similar argument). In particular, $f^*M-(1-\mu)(Z-\Delta)$ is big. Let $G\in \big|\,k(f^*M-(1-\mu)(Z-\Delta)\,)\,\big|$ for some $k$ sufficiently large and divisible. Let $T=\frac{1}{k}\,G+(1-\mu)(Z-\Delta)$. Write $T=\sum d_i D_i+ \sum t_j F_j$. Define $D=f_*T=\sum d_iC_i$, and write $f^*D=\sum d_i D_i+ \sum d_j'F_j$. We have: \begin{enumerate} \item $T \equiv f^*M$, because $kT \sim kf^*M$; \item $T\geq 0$, and therefore $D \geq 0$; \item $T\cdot F_j=0$ for every exceptional curve $F_j$, because $T\equiv f^*M$; \item $f^*D=T$; indeed, the coefficients $d_j'$ are uniquely determined by the condition $f^*D \cdot F_j = 0$ for every $j$, and $T$ already satisfies this condition; \item $f^*D=T\geq(1-\mu)(Z-\Delta)$, because $G\geq 0$; \item finally, $D\equiv M$, because $f^*D=T\equiv f^*M$. \end{enumerate} \end{proof} \begin{Remark} $f^*D\geq(1-\mu)(Z-\Delta)$ means $d_j'\geq(1-\mu)(z_j-a_j)$ for every $j$. We may assume, however, that $d_j' > (1-\mu)(z_j-a_j)$ for every $j$. Indeed, since $M^2>(1-\mu)^2\delta_y$, we have $M^2>(1-\mu)^2\delta_y(1+\epsilon)^2$ for some small rational number $\epsilon>0$; then working as above we can find $D \equiv M$ such that $d_j'\geq(1-\mu)(z_j-a_j)(1+\epsilon)>(1-\mu)(z_j-a_j)$ for every $j$. \end{Remark} We'll assume that {\boldmath $d_j'>(1-\mu)(z_j-a_j)$} for all $j$. \vspace{8pt} For every rational number $c$, $0<c<1$, let $R_c=f^*(M-cD)$. $R_c\equiv(1-c)f^*M$, so that $R_c$ is nef and big, and we have \begin{equation} H^1(X,K_X+\rup{R_c})=0. \label{eq:van} \end{equation} \begin{align*} K_X+\rup{R_c} &= \rup{f^*K_Y-\Delta+f^*\rup{M}-f^*B-cf^*D} \\ &= f^*(K_Y+\rup{M})-\rdn{\Delta+f^*B+cf^*D} \\ &= f^*(K_Y+\rup{M})-\sum\rdn{b_i+cd_i}D_i-\sum\rdn{a_j+b_j'+cd_j'}F_j. \end{align*} Choose $c=\min\left\{\frac{1-a_j-b_j'}{d_j'}, \text{ all $j$}; \frac{1-b_i}{d_i}, \text{ all $i$ such that $d_i>0$ and $y\in C_i$}\right\}$. Note that $c > 0$, because $(Y,B,y)$ is log-terminal, and $c<1$, because for every $j$ we have $b_j'\geq\mu(z_j-a_j)$ and $z_j\geq 1$, and therefore $1-a_j-b_j' \leq z_j-a_j-\mu(z_j-a_j) = (1-\mu)(z_j-a_j) < d_j'$. Therefore (\ref{eq:van}) holds for this choice of $c$. Note also that $0<a_j+b_j'+cd_j'\leq 1$ for all $j$; so $F \overset{\text{def}}{=} \sum \rdn{a_j+b_j'+cd_j'} F_j$ is either zero or a sum of distinct irreducible components, $F=F_1+\cdots+F_s$ (after re-indexing the $F_j$ if necessary). Similarly, $\sum \rdn{b_i+cd_i}D_i = N+A$, where $\Supp(N) \cap f^{-1}(y)=\emptyset$, and $A$ is either zero or a sum of distinct irreducible components, $A=D_1+\cdots+D_t$, where $D_1, \ldots, D_t$ meet $f^{-1}(y)$. Each component $F_j$ of $F$ (if any), and each component $D_i$ of $A$ (if any), has coefficient $1$ in $\Delta + f^*B + c f^*D$. Also, $F$ and $A$ cannot both be equal to zero. \vspace{8pt} We will use the following form of the Kawamata--Viehweg vanishing theorem (see, for example, \cite{el}, Lemma 2.4): \begin{Lemma} \label{lemma:vt2} Let $X$ be a smooth projective surface over $\ensuremath{\mathbb{C}\,}$, and let $R$ be a nef and big $\ensuremath{\mathbb{Q}\,}$-divisor on $X$. Let $E_1,\ldots,E_m$ be distinct irreducible curves such that $\rup{R} \cdot E_i > 0$ for every $i$. Then $$ H^1(X,K_X+\rup{R}+E_1+\cdots+E_m)=0. $$ \end{Lemma} \vspace{8pt} We consider two cases, according to whether $F \neq 0$ or $F=0$. \vspace{6pt} \textbf{Case I:} $\mathbf{F \neq 0.}$ Using Lemma \ref{lemma:vt2} for $R_c$ in place of $R$ and $D_1,\ldots,D_t$ in place of $E_1,\ldots,E_m$, we get $H^1(X,K_X+\rup{R_c}+A)=0$, or $H^1(X,f^*(K_Y+\rup{M})-N-F)=0$. We conclude as in the proof of Theorem \ref{thm:notlt}; the `` $-N$ '' part is treated as in the proof of Lemma \ref{lemma:yonly}. \emph{Note:} $R_c \cdot D_i = (1-c) M \cdot C_i > 0$ for every $i$, and $D_i$ has integer coefficient in $R_c$ if $D_i$ is a component of $A$ -- and therefore $\rup{R_c} \cdot D_i > 0$ for such $D_i$. \vspace{6pt} \textbf{Case II:} $\mathbf{F = 0.}$ As noted earlier, in this case $A \neq 0$; using Lemma \ref{lemma:vt2} as in Case I above, we get $H^1(X,f^*(K_Y+\rup{M})-N-D_1)=0$, and therefore the restriction map \begin{equation} H^0(X,f^*(K_Y+\rup{M})-N) \to H^0(D_1,(f^*(K_Y+\rup{M})-N|_{D_1}) \label{eq:surj} \end{equation} is surjective. $D_1 \cap f^{-1}(y) \neq \emptyset$; let $x\in D_1\cap f^{-1}(y)$. Assume we can find a section $s'\in H^0(D_1, f^*(K_Y+\rup{M})-N|_{D_1})$ such that $s'(x)\neq 0$. Then by the surjectivity of (\ref{eq:surj}) we can find $s\in H^0(X,f^*(K_Y+\rup{M})-N)$ such that $s(x)\neq 0$; then we conclude as in the proof of Lemma \ref{lemma:yonly}. Hence the proof is complete if we show that $x \notin \Bs|\, f^*(K_Y+\rup{M})-N|_{D_1}\,|$. Note that $f^*(K_Y+\rup{M})-N|_{D_1}= K_X+\rup{R_c}+A|_{D_1}=(K_X+D_1)|_{D_1}+(\rup{R_c}+D_2+\cdots+D_t) |_{D_1}=K_{D_1}+(\rup{R_c}+D_2+\cdots+D_t)|_{D_1}$. By \cite{har}, Theorem 1.4 and Proposition 1.5, it suffices to show that $(\rup{R_c}+ D_2+\cdots+D_t)\cdot D_1 >1$ (then this intersection number is $\geq 2$, because it is an integer). $\rup{R_c} = R_c+\sum\{b_i+cd_i\}D_i+\sum\{a_j+b_j'+cd_j'\}F_j$. Note that $b_1 + c d_1 = 1$, and therefore $\{b_1+cd_1\}=0$. Also, since $F=0$, we have $0\leq a_j+b_j'+cd_j'<1$ for every $j$ -- and therefore $\{a_j+b_j'+cd_j'\}=a_j+b_j'+cd_j'$. Hence we have $$ (\rup{R_c}+D_2+\cdots+D_t)\cdot D_1 \geq R_c\cdot D_1 + \sum (a_j+b_j'+cd_j')F_j \cdot D_1. $$ $R_c \equiv (1-c)f^*M$, so that $R_c\cdot D_1 = (1-c)M\cdot C_1 \geq (1-c)(1-\mu)$, because $y \in C_1 = f_*D_1$. Also, $D_1$ meets at least one $F_j$, say $F_1$. Therefore we have \begin{align*} (\rup{R_c}+D_2+\cdots+D_t)\cdot D_1 &\geq(1-c)(1-\mu)+(a_1+b_1'+cd_1') \\ &> (1-c)(1-\mu) +a_1 +\mu(z_1-a_1) +c(1-\mu)(z_1-a_1) \\ &\geq (1-c)(1-\mu) +a_1 +\mu(1-a_1) +c(1-\mu)(1-a_1) \\ &= 1 +(1-c)(1-\mu)a_1 \\ &\geq 1. \end{align*} \qed \vspace{6pt} {\bf (2.9)} Finally, we prove Theorem \ref{thm:dnen}. We assume the reader is familiar with the classification of $RDP$'s. The classification of log-terminal singularities is similar: if $f:X\to (Y,y)$ is the minimal resolution of a log-terminal germ and $f^{-1}(y)=\cup F_j$, then the $F_j$ are smooth rational curves, and the dual graph is a graph of type $A_n$, $D_n$ or $E_n$ (see, e.g., \cite[\S 9]{crepant}, or \cite{alexeev}). The only difference is that in the log-terminal case the self-intersection numbers $-w_j = F_j^2$ are not necessarily all equal to $-2$. The classification of log-terminal singularities with non-zero reduced boundary is even simpler. We have: \begin{Lemma} \label{lemma:alexeev} Assume $(Y,y)$ is a normal surface germ, and $C_1$ is a reduced, irreducible curve on $Y$ such that $y \in C_1$. If $(Y,C_1,y)$ is log-terminal, then $(Y,y)$ is of type $A_n$. More explicitly, if $f:X\to (Y,y)$ is the minimal resolution, $f^{-1}(y) = F_1 \cup \ldots \cup F_n$, $-w_j=F_j^2$, and $f^{-1}C_1=D_1$, then the dual graph of the resolution is: \begin{center} \begin{picture}(190,60)(0,0) \put(14,30){\circle*{8}} \put(9,15){$D_1$} \put(18,30){\line(1,0){30}} \put(44,36){\scriptsize $-w_1$} \put(52,30){\circle{8}} \put(47,15){$F_1$} \put(56,30){\line(1,0){30}} \put(82,36){\scriptsize $-w_2$} \put(90,30){\circle{8}} \put(85,15){$F_2$} \put(94,30){\line(1,0){25}} \multiput(121,30)(4,0){6}{\line(1,0){2}} \put(145,30){\line(1,0){25}} \put(166,36){\scriptsize $-w_n$} \put(174,30){\circle{8}} \put(169,15){$F_n$} \end{picture} \end{center} (If $y$ is smooth, then $n=1$ and $w_1=1$; otherwise all $w_j\geq 2$.) \end{Lemma} Similarly, for log-canonical singularities with boundary we have: \begin{Lemma} \label{lemma:typeen} Let $(Y,y)$ and $C_1$ be as in the previous lemma. If $(Y,C_1,y)$ is log-canonical, then either $(Y,y)$ is of type $A_n$ and the dual graph is the one shown above, or $(Y,y)$ is of type $D_n$ and the dual graph of the resolution is: \begin{center} \begin{picture}(230,98)(0,-38) \put(14,30){\circle*{8}} \put(9,15){$D_1$} \put(18,30){\line(1,0){30}} \put(44,36){\scriptsize $-w_1$} \put(52,30){\circle{8}} \put(47,15){$F_1$} \put(56,30){\line(1,0){30}} \put(82,36){\scriptsize $-w_2$} \put(90,30){\circle{8}} \put(85,15){$F_2$} \put(94,30){\line(1,0){25}} \multiput(121,30)(4,0){6}{\line(1,0){2}} \put(145,30){\line(1,0){25}} \put(161,36){\scriptsize $-w_{n-2}$} \put(174,30){\circle{8}} \put(175,15){$F_{n-2}$} \put(178,30){\line(1,0){30}} \put(207,36){\scriptsize $-2$} \put(212,30){\circle{8}} \put(207,15){$F_n$} \put(174,26){\line(0,-1){30}} \put(175,-3){\scriptsize $-2$} \put(174,-8){\circle{8}} \put(164,-22){$F_{n-1}$} \end{picture} \end{center} \end{Lemma} All these facts can be found, for example, in \cite{alexeev}, and also in \cite{crepant}. \vspace{8pt} \noindent {\bf Proof of Theorem \ref{thm:dnen}} \vspace{3pt} Going back to the proof of theorem \ref{thm:kaw}, we will show that Case II of the proof is not possible if $y$ is of type $D_n$ or $E_n$. Therefore the condition $M \cdot C \geq (1-\mu)$ for $C$ through $y$ is no longer necessary. We still need $M \cdot C > 0$ for $C$ through $y$ if $A \neq 0$, to be able to use Lemma \ref{lemma:vt2} (see the \emph{Note} at the end of Case I). However, we will show that $A = 0$ if $y$ is of type $E_n$, so that in that case we don't even need the condition $M \cdot C > 0$ for $C$ through $y$. Assume that $A \neq 0$. Let $C_1$ be a component of $A$, and let $f^*C_1 = D_1 + \sum c'_j F_j$. Since $C_1$ is a component of $A$, we have $b_1+cd_1=1$, and therefore $B+cD=C_1+\textit{ other terms}\,$; consequently $f^*B+cf^*D \geq f^*C_1$, and in particular $b'_j+cd'_j \geq c'_j, \forall j$. If we end up in Case II in the proof of Theorem \ref{thm:kaw}, then $a_j+b'_j+cd'_j < 1$ for all $j$; consequently $a_j+c'_j < 1, \forall j$, so that $(Y,C_1,y)$ is log-terminal. By Lemma \ref{lemma:alexeev}, $(Y,y)$ must be of type $A_n$. Now assume that we end up in Case I, with $A \neq 0$. We still have $a_j+b'_j+cd'_j \leq 1, \forall j$, and therefore $(Y,C_1,y)$ is log-canonical. By Lemma \ref{lemma:typeen}, $(Y,y)$ is either of type $A_n$ or of type $D_n$. \qed \begin{Remark} For singularities of type $D_n$ and $E_n$, Theorem \ref{thm:dnen} is stronger than the Open Problem (see (2.1)). To complete the proof of the Open Problem, the only case to consider is that of a singularity of type $A_n$. In the proof of Theorem \ref{thm:kaw}, Case II (which is possible only when $y$ is of type $A_n$), we used the inequality $M \cdot C \geq (1-\mu)$ for $y \in C$. In fact, using Lemma \ref{lemma:alexeev} and modifying slightly the final computation in (2.8), we see that we need slightly less: $M \cdot C \geq (1-\mu)(1-a)$, where $a = \min \{ a_1,a_n \}$ (note that in Lemma \ref{lemma:alexeev}, $D_1$ could meet either $F_1$ or $F_n$). On the other hand, for $y$ of type $A_n$ we have $\delta_y = 2-(a_1+a_n)$ (for $n=1$ this follows from (1.11); for $n \geq 2$ use Lemma 2, (b), and the obvious formulae $(Z-\Delta) \cdot F_j = -1$ for $j=1$ and $j=n$, $(Z-\Delta) \cdot F_j = 0$ otherwise). Thus the Open Problem requires that $M \cdot C \geq (1-\mu)(1- \frac{a_1+a_n}{2})$ for $y \in C$. In particular, the Open Problem is proved if $a_1=a_n$ (e.g., if $y$ is of type $A_1$). \end{Remark} \vspace{6pt} {\bf (2.10)} Analyzing the proofs of Theorems \ref{thm:kaw} and \ref{thm:dnen}, we may ask: what was the relevance of $Z$ being the fundamental cycle of $y$? $\Delta$ arises naturally, as $f^*K_Y-K_X$; but $Z$ could have been any effective, $f$-exceptional cycle with integer coefficients such that $z_j \geq 1$ for all $j$ (i.e., such that $\Supp(Z) = f^{-1}(y)$). The answer is provided by the following proposition: \begin{Proposition} \label{prop:caract} Let $(Y,y)$ be a log-terminal singularity, with $f:X \to (Y,y)$ the minimal resolution (resp. the blowing-up at $y$ if $y$ is smooth). Let $Z$ and $\Delta$ be the fundamental, resp. the canonical cycle. Let $Z'=\sum z'_jF_j$ be any other effective, $f$-exceptional cycle (with integer coefficients), such that $z'_j \geq 1$ for all $j$. Then $\delta_y \leq \delta'$, where $\delta_y = -(Z-\Delta)^2$ and $\delta' = -(Z'-\Delta)^2$. Moreover, $Z$ is the (unique) largest cycle among all the $Z'$ for which $\delta' = \delta_y$. \end{Proposition} The proof depends on the detailed classification of log-terminal surface singularities, cf. \cite{crepant}. Explicitly, we need the following lemma, which can be proved by brute force (the computations are straightforward in all cases; for reference, they can be found in \cite{kawachi1}): \begin{Lemma} \label{lemma:tech} Let $(Y,y)$ be log-terminal. (a) There is at most one $F_j$ with $(Z-\Delta) \cdot F_j > 0$. If such an $F_j$ exists, then $(Z-\Delta) \cdot F_j = 1$ and the corresponding $w_j = -F_j^2 \geq 3$. (b) There is at most one $F_j$ with $z_j \geq 2$ and $(Z-\Delta)\cdot F_j<0$. If such an $F_j$ exists, then $z_j=2$ and $(Z-\Delta) \cdot F_j = -1$. \end{Lemma} \vspace{8pt} \noindent {\bf Proof of Proposition \ref{prop:caract}} \vspace{3pt} Let $Z' = Z+(P-N)$ with $P,N \geq 0$ without common components. Then \begin{align*} \delta' &= -(Z'-\Delta)^2 = -(Z+P-N-\Delta)^2 \\ &= -(Z-\Delta)^2 - (P-N)^2 - 2(Z-\Delta) \cdot (P-N) \\ &= \delta_y + (-P^2) - 2 (Z-\Delta) \cdot P + (-N^2) + 2 (Z-\Delta) \cdot N + 2 (P \cdot N). \end{align*} Since $P \cdot N \geq 0$, the Proposition is proved if we can prove that \begin{enumerate} \item[(a)] $(-P^2) - 2 (Z-\Delta) \cdot P > 0$ if $P>0$; \item[(b)] $(-N^2) + 2 (Z-\Delta) \cdot N \geq 0$. \end{enumerate} \vspace{5pt} \noindent \emph{Proof of (a).} \vspace{3pt} If $P = \sum t_jF_j$, then we have \[ (-P^2) - 2 (Z-\Delta) \cdot P = (-P^2) - 2\sum t_j (Z-\Delta) \cdot F_j. \] If $(Z-\Delta) \cdot F_j \leq 0$ for all $j$, then we are done (note that $(-P^2) > 0$ if $P>0$). Otherwise there is exactly one $j$, call it $j_0$, such that $(Z-\Delta) \cdot F_{j_0} > 0$. By Lemma \ref{lemma:tech}, we have $w_{j_0} = -F_{j_0}^2 \geq 3$ and $(Z-\Delta) \cdot F_{j_0} = 1$. Therefore \[ (-P^2) - 2\sum t_j (Z-\Delta) \cdot F_j \geq (-P^2) - 2t_{j_0}. \] We will show that $(-P^2) \geq t_{j_0}^2 + 2$; then (a) will follow. Write $F_j^2 = -w_j, F_i \cdot F_j = l_{ij}$ for $i<j$ ($l_{ij} = 1$ if $F_i$ meets $F_j$, $0$ otherwise). We have: \begin{align*} (-P^2) &= \sum_j w_j t_j^2 - \sum_{i<j} 2l_{ij} t_i t_j \\ &\geq t_{j_0}^2 + \sum_j 2t_j^2 - \sum_{i<j} 2 l_{ij} t_i t_j \end{align*} (note that $w_j \geq 2, \, \forall j$, and $w_{j_0} \geq 3$). Now consider a singularity $(Y',y')$, whose minimal resolution has a dual graph identical to that of $(Y,y)$, except that ${F'_j}^2 = -2,\,\forall j$ (thus $y'$ is a ``true'' $A_n, D_n,$ or $E_n$ rational double point). If $P' = \sum t_j F'_j$ (having the same coefficients as $P$), then $(-{P'}^2) > 0$; that is, \[ \sum_j 2t_j^2 - \sum_{i<j} 2l_{ij} t_i t_j > 0 \] --- and therefore $\geq 2$, because it is an \emph{even} integer. \qed \vspace{5pt} \noindent \emph{Proof of (b).} \vspace{3pt} If $N = \sum x_j F_j$, then we have \[ (-N^2) + 2 (Z-\Delta) \cdot N = (-N^2) + 2 \sum x_j (Z-\Delta) \cdot F_j. \] If $(Z-\Delta) \cdot F_j \geq 0$ for all $j$, or if $x_j = 0$ whenever $(Z-\Delta) \cdot F_j < 0$, then we are done. Note that $x_j = z_j - z'_j$ if $z_j > z'_j$, and $0$ otherwise. Thus $x_j \geq 1 \implies z_j \geq 2$. Therefore, by Lemma \ref{lemma:tech}, there can be at most one negative term in $2\sum x_j (Z-\Delta)\cdot F_j$; and if there is one, corresponding, say, to $j_1$, then $z_{j_1} = 2$ and $(Z-\Delta) \cdot F_{j_1} = -1$. Therefore $x_{j_1}=1$, and $2(Z-\Delta)\cdot N \geq -2$. Finally, $(-N^2) \geq 2$ (if $N \neq 0$), as in the proof of (a) above. \qed \vspace{5pt} \noindent \emph{Remarks.} \emph{1.} We showed that $\left[ -(Z'-\Delta)^2 = \delta_y \right] \implies [Z' \leq Z]$. The converse is not always true. For example, if $F_j$ is a component with $z_j=2$ and $(Z-\Delta)\cdot F_j = 0$ (such components exist in some cases of type $D_n$ and $E_n$, cf. \cite{kawachi1}), then taking $Z'=Z-F_j$ we get $-(Z'-\Delta)^2 > \delta_y$ (cf. the proof of (b) above). \emph{2.} If $(Y,y)$ is not log-terminal, then the statement of Proposition \ref{prop:caract} is no longer necessarily true, even if $y$ is rational. For example: \begin{center} \begin{picture}(100,110)(0,0) \put(51,56){\scriptsize $-5$} \put(50,50){\circle{8}} \put(45,35){$F_1$} \put(47.17,47.17){\line(-1,-1){21.21}} \put(18.13,29.13){\scriptsize $-2$} \put(23.13,23.13){\circle{8}} \put(18.13,8.13){$F_2$} \put(52.83,47.17){\line(1,-1){21.21}} \put(71.87,29.13){\scriptsize $-2$} \put(76.87,23.13){\circle{8}} \put(71.87,8.13){$F_3$} \put(54,50){\line(1,0){30}} \put(83,56){\scriptsize $-2$} \put(88,50){\circle{8}} \put(83,35){$F_4$} \put(50,54){\line(0,1){30}} \put(45,94){\scriptsize $-2$} \put(50,88){\circle{8}} \put(51,74){$F_5$} \put(46,50){\line(-1,0){30}} \put(7,56){\scriptsize $-2$} \put(12,50){\circle{8}} \put(7,35){$F_6$} \end{picture} \end{center} $Z = F_1 + (F_2 + \cdots + F_6), \; \Delta = \frac{6}{5} F_1 + \frac{3}{5} (F_2 + \cdots + F_6)$, and $\delta_y = \frac{13}{5}$; \newline but $\delta' = -(Z'-\Delta)^2 = \frac{8}{5} < \delta_y$ for $Z'=Z+F_1$. \vspace{5pt} \noindent \emph{Exercise.} Is the statement of Proposition \ref{prop:caract} true for log-canonical singularities? Hint: There is nothing to prove if $\delta_y \leq 1$. Notice that in the exceptional case of (1.9), Type 3 with $w=3$, we have $\delta' = 2$ for $Z' = F_1 + (F_2 + \cdots + F_5)$, while $\delta_y=1$ (in that case we have $Z = 2 F_1 + (F_2 + \cdots + F_5)$).

9709

alg-geom/9709031

en

https://arxiv.org/abs/alg-geom/9709031

alg-geom/9709031

Wolf Barth

W. Barth

K3 Surfaces with Nine Cusps

LaTeX

By a K3-surface with nine cusps I mean a surface with nine isolated double points A_2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown, that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin, that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.

alg-geom

\section{Introduction} If $E_1,...,E_{16}$ are 16 disjoint, smooth curves on a $K3$-surface $X$ then the divisor $\sum_1^{16} E_i$ is divisible by $2$ in $Pic(X)$. This was observed by V.V. Nikulin [N]. Equivalently: If $\bar{X}$ is the surface obtained from $X$ by blowing down the 16 rational curves to nodes $e_i \in \bar{X}$, there is a double cover $A \to \bar{X}$, with $A$ a complex torus, branched exactly over the 16 nodes $e_i$. The surface $\bar{X}$ is the Kummer surface of the complex torus $A$. The aim of this note is to prove an analog of Nikulin's theorem in the case of nine cusps (double points $A_2$) instead of 16 nodes (double points $A_1$): {\bf Theorem.} {\em Let $E_i,E_i', i=1,...,9$ be 18 smooth rational curves on a $K3$-surface $X$ with $$E_i.E_i'=1, \quad E_i.E_j=E_i.E_j'=E_i'.E_j'=0 \mbox{ for } i \not=j,$$ then there are integers $a_i,a_i'=1,2, a_i \not=a_i',$ such that the divisor $\sum_1^9 (a_iE_i+a_i'E_i')$ is divisible by $3$ in $Pic(X)$. Equivalently: If $\bar{X}$ is the surface obtained from $X$ by blowing down the nine pairs of rational curves to cusps $e_i \in \bar{X}$, then there is a cyclic cover $A \to \bar{X}$ of order three, with $A$ a complex torus, branched exactly over the nine cusps $e_i$.} The proof I give here essentially parallels Nikulin's proof in [N]. In the case of Nikulin's theorem of course each complex torus $A$ of dimension two appears (the covering involution is the map $a \mapsto -a$). But complex tori of dimension two admitting an automorphism of order three with nine fix-points are rarer. If the $K3$-surface $X$ is algebraic, then its Picard number is $\geq 19$. So in this case the surface $X$ and the covering surface $A$ can depend on at most one parameter. Examples of abelian surfaces with an automorphism of order three are given in [BH]: Each selfproduct $A=C \times C$, with $C$ an elliptic curve, admits the automorphism $$(x,y) \mapsto (-x,x-y).$$ It is shown in [BH] that the quotient $\bar{X}$ then is a double cover of the plane ${\rm I\!P}_2$, branched over the sextic $C^*$ dual to a plane cubic $C \subset {\rm I\!P}_2$, a copy of the elliptic curve $C$. The nine cusps of $\bar{X}$ of course ly over the nine cusps of $C^*$. By deformation theory of $K3$-surfaces, one may convince oneself, that there are also non-algebraic $K3$-surfaces with nine cusps. \vv Convention: Throughout this note the base field for algebraic varieties is $\C$. \vV \section{Cyclic triple covers of $K3$-surfaces} By a configuration of type $A_2$ on a smooth surface I mean a pair $E,E'$ of smooth rational curves with $E^2=(E')^2=-2, \quad E.E'=1$. Such a pair can be contracted to a double point $A_2$ (a cusp). {\bf Lemma 1.} {\em Let $X$ be a $K3$-surface carrying $p$ disjoint configurations of type $A_2$, and $\bar{X}$ the surface obtained from $X$ by contracting them to cusps. If there is a smooth complex surface $Y$ and a triple cover $Y \to \bar{X}$ branched (of order three) precisely over the $p$ cusps, then either $p=6$ with $Y$ a $K3$-surface or $p=9$ with $Y$ a torus.} Proof. First of all, $Y$ is k\"ahler: Indeed, $X$ is k\"ahler by [S]. Blow up $X$ in the $p$ points, where the $p$ pairs of curves in the $A_2$ configurations meet. The resulting surface $\tilde{X}$ is k\"ahler by [B, Theoreme II 6]. Pull back the covering $Y \to \bar{X}$ to a covering $\tilde{Y} \to \tilde{X}$. Here $\tilde{Y}$ is k\"ahler, since it is a smooth surface in some ${\rm I\!P}_1$-bundle over $\tilde{X}$, which is k\"ahler by [B, Theoreme principal II]. The surface $Y$ is obtained from $\tilde{Y}$ by blowing down $(-1)$-curves, so it is k\"ahler too by [F]. The canonical bundle of $Y$ admits a section with zeros at most in $p$ points. So there are no such zeros and $K_Y$ is trivial. By the classification of surfaces [BPV, p.188] the covering surface $Y$ therefore either is $K3$ with $e(Y)=24$ or a torus with $e(Y)=0$. The Euler number of $Y$ is computed in terms of $p$ as $$e(Y)= 3 \cdot e(\bar{X})- 2 \cdot p = 3 \cdot(24-2p)-2 \cdot p = 72-8 \cdot p.$$ The possibilities are $p=6$ with $Y$ a $K3$-surface and $p=9$ with $Y$ a complex torus. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par \vv Consider the $2p$ rational curves $E_i,E_i' \subset X$ forming the $p$ configurations of type $A_2$. The cyclic cover, lifted to $X$, is branched along all of these curves of order three. So there must be a divisor $$\sum_{i=1}^p a_i \cdot E_i+a_i' \cdot E_i', \qquad a_i,a_i'=1 \mbox{ or }2,$$ divisible by three in $Pic(X)$. This implies that all intersection numbers \begin{eqnarray*} (a_i E_i+a_i' E_i').E_i &=& -2 \cdot a_i+a_i' \\ (a_i E_i+a_i' E_i').E_i' &=& a_i -2 \cdot a_i' \end{eqnarray*} are divisible by three. Hence $$a_i=1 \Leftrightarrow a_i'=2 \quad \mbox{ and } \quad a_i=2 \Leftrightarrow a_i'=1.$$ \vV \section{The lattice generated by the nine cusps} Here let $X$ be a $K3$-surface and $L=H^2(X,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$})\simeq {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}^{22}$ its lattice provided with the (unimodular) intersection form. Assume that on $X$ there are nine disjoint $A_2$-configurations $E_i,E_i', \, i=1,...,9$. Following [N] we denote by $I \subset L$ the sublattice spanned (over ${\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$) by the 18 classes $[E_i], [E_i']$. Let me denote by $\bar{I} \subset L$ the {\em primitive} sublattice spanned by these classes over $\Q$ and let me put $$Q:=\bar{I}/I.$$ To study $Q$ we split $I$ in two sublattices by the base change $$ E_i,E_i' \quad \mbox{ replaced by } \quad E_i,F_i:=2 E_i+E_i' \mbox{ for }i=1,...,9.$$ The essential point is that the intersection numbers $$E_i.F_j=-3 \cdot \delta_{i,j}, \quad F_i.F_j= -6 \cdot \delta_{i,j}$$ are divisible by $3$. {\bf Lemma 2.} {\em If a class $$\sum_{i=1}^9 \epsilon_i [E_i] + \varphi_i[F_i], \quad \epsilon_i, \varphi_i \in \Q,$$ belongs to $\bar{I}$ then $$ \epsilon_i \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}, \quad 3 \cdot \varphi_i \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}.$$ In particular the order of the finite group $Q$ is $|Q| =3^n$ for some $n \geq 0.$} Proof. We just intersect the class with $E_k$ and $E_k'$ to find $$ \begin{array}{c}\D (\sum_{i=1}^9 \epsilon_i [E_i] + \varphi_i[F_i]).E_k = -2 \cdot \epsilon_k-3 \cdot \varphi_k \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} \\ \\ \D (\sum_{i=1}^9 \epsilon_i [E_i] + \varphi_i[F_i]).E_k' = \epsilon_k \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}. \\ \end{array}$$ This implies the assertion. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par Lemma 2 shows in particular $$\bar{I} = E + \bar{F}$$ with $E \subset I$ the lattice spanned by the classes $[E_i]$, with $F \subset I$ the lattice spanned by the classes $[F_i]$, and $\bar{F}$ the primitive sublattice of $L$ spanned over $\Q$ by $F$. \vv {\bf Lemma 3.} {\em The order $|Q|$ is $3^n$ with $n \geq 3$.} Proof. Choose a system of $n$ generators for $Q$. They are the residues of $n$ classes $q_1,...,q_n \in \bar{F}$. The set of these $n$ classes can be extended to a ${\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$-basis $$q_1,...,q_n, f_{n+1},...,f_9$$ of $\bar{F}$. So, if $n \leq 2$, the lattice $\bar{F}$ has a ${\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$-basis $q_1,q_2,f_3,...,f_9$ with $f_3,...,f_9$ integral linear combinations of the classes $[F_1],...,[F_9]$. We extend this basis to a ${\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$-basis of $\bar{I}$ with the classes $e_1=[E_1],...,e_9=[E_9]$, and to a basis of $L$ with some classes $t_{19},...,t_{22}$. In the basis $$f_3,...,f_9,e_1,...,e_9,q_1,q_2,t_{19},...,t_{22}$$ the intersection matrix is $$ \begin{array}{c|c|c} \multicolumn{1}{c}{7} & \multicolumn{1}{c}{9} & \multicolumn{1}{c}{6} \\ (f_i.f_j) & (f_i.e_j) & * \\ \hline (f_i.e_j) & (e_i.e_j) & * \\ \hline * & * & * \\ \end{array}$$ Each summand in the Leibniz expansion of the determinant contains at least ten factors $$f_i.f_j, \quad f_i.e_j \quad \mbox{ or } \quad e_i.e_j.$$ At most nine of them can be $e_i.e_j$. At least one of them must be a factor $ f_i.f_j$ or $f_i.e_j$ divisible by 3. This shows that the determinant of the $22 \times 22$ intersection matrix is divisible by $3$, a contradiction with unimodularity. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par \vV \section{The code of the nine cusps} Each class in $\bar{I}$ is of the form $$\sum_{i=1}^9 \epsilon_i \cdot E_i +\varphi_i \cdot F_i, \quad \epsilon_i \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}, \, \varphi_i \in \frac{1}{3} {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}.$$ By sending $$\varphi_i \mapsto \varphi_i \mbox{ mod } {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$$ we identify $Q$ with an ${\rm I\!F}_3$ sub-vector space of ${\rm I\!F}_3^9$. By lemma 3 the sub-vector space $Q \subset {\rm I\!F}_3^9$ has dimension $\geq 3$. In this section we want to identify this sub-vector space. In analogy with coding theory, we call each vector $q = (q_i)_{i=1,..,9} \in Q$ a {\em word}, and the number of its non-zero coefficients its {length} $|q|$. By lemma 1 all vectors $q \in Q$ have length $|q|=0,6$ or $9$. As $dim_{{\rm I\!F}_3}(Q) \geq 3$, the space $Q$ contains at least $3^3-3 =24$ words of length $6$. \vv We say that two words $q,q'$ overlap in $r$ places, if there are precisely $r$ ciphers $i$ such that both coefficients $q_i$ and $q_i'$ are nonzero. It is clear that any two nonzero words of length six overlap in at least three places. If they overlap in six places, they are linearly dependent: In fact, if $q+q' \not=0$, we have $q_i=q_i'$ for at least one $i$. Then $q+2q'$ has length $\leq 5$, hence $q+2q'=0$. {\em Claim 1. Any two linearly independent vectors $q,q'$ of length six overlap in three or in four places. } Proof. We have to exclude, that $q$ and $q'$ overlap in five places. Assume to the contrary that they do. By rescaling the basis vectors of ${\rm I\!F}^3$ we may assume $$q=(1,1,1,1,1,1,0,0,0)$$ and $$q'=(0,q_2',q_3',q_4',q_5',q_6',q_7',0,0), \quad q_i'=1 \mbox{ or }2.$$ Since $q+q'$ again is a word of length six, w.l.o.g. $$q'=(0,2,1,1,1,1,q_7',0,0).$$ Then $$q+2q' = (1,2,0,0,0,0,2q_7',0,0) \notin Q,$$ contradiction. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par \vv Now, let me call the nine ciphers $1,...,9$ 'points' and those triplets $\{i,j,k\}$ of ciphers 'lines', for which there is a word $q$ of length six with $q_i=q_j=q_k=0$. As there are at least 24 words of length six, there are at least twelve lines. As two linearly independent words of length six overlap in four or three places, two different lines intersect in one point, or not at all (parallel lines). This allows to count the number of lines: {\em Claim 2. There are precisely $12$ lines, and therefore the dimension of $Q$ is $n=3$.} Proof. Through each point, there are at most four distinct lines. So there are at most $9 \times 4 = 36$ incidences of lines with points. As on each line there are three points, we have indeed at most $36/3=12$ lines. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par This proof shows in particular, that through each point there are exactly four lines, or in other words: Each pair of points lies on a (uniquely determined) line. \vv {\em Claim 3. For each line there are precisely two parallel lines. These two parallel lines do not intersect.} Proof. Each line $L$ meets $3 \times 3=9$ other lines, hence there are two lines $L',L''$ parallel to it. If $L'$ and $L''$ would meet in a point, then through this point we would have five lines: the two lines $L'$ and $L''$ and the three lines joining this point with the three points on $L$. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par \vv {\em Claim 4. The code $Q$ contains a word of length nine.} Proof. Take three parallel lines $L,L',L''$ and two words $q,q'$ vanishing on the lines $L,L'$ respectively. These two words $q$ and $q'$ overlap in precisely three places (the points of $L''$). After replacing $q$ by $2 \cdot q$ if necessary, we may assume $q_i=q_i'$ for one $i \in L''$. Then $q-q'$ is a word of length six, i.e., $q_i=q_i'$ for all $i \in L''$. So $q+q'$ is a word of length nine. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par \vv Claim 4 proves the theorem from the introduction: The existence of a word of length nine shows that there is a linear combination $$D:=\sum_{i=1}^9 \varphi_i F_i \in Pic(X) \quad \mbox{with} \quad 0<\varphi_i<1, 3 \varphi_i \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}.$$ The divisor $$3 \cdot D - \sum_{\varphi_i=2} 3 \cdot E_i'$$ contains all curves $E_i$ and $E_i', \, i=1,...9,$ with multiplicity $1$ or $2$, and it is divisible by 3. \vV \section{The double cover branched over the dual cubic} A smooth cubic $C \subset {\rm I\!P}_2$ has nine flexes. On the dual cubic $C^* \subset {\rm I\!P}_2^*$ they yield nine cusps. So the double cover $\bar{X} \to {\rm I\!P}_2^*$ is an example of a $K3$-surface with nine cusps. Here I want to understand the $3$-torsion property on $\bar{X}$ in terms of plane projective geometry, independently of the theory in the preceding sections and of [BH]. The nine flexes of $C$ in a natural way have the structure of an affine plane over ${\rm I\!F}_3$. In fact, if $C$ is given in Hesse normal form $$x_0^3+x_1^3+x_2^3+3 \lambda x_0x_1x_2=0,$$ a transitive action of the vector space ${\rm I\!F}_3^2$ on the curve $C$ and thus on the set of its flexes is induced by the symmetries $$\sigma:x_i \mapsto x_{i+1}, \quad \tau:x_i \mapsto \omega^i \cdot x_i, \quad \omega \mbox{ a primitive third root of unity}.$$ Of course the 'lines' used in the preceding section must be the lines in this affine plane. This section will give a proof. Let me in this section denote by a line in the set of flexes, a line in the sense of the affine structure just mentioned. The flexes are cut out on $C$ by the coordinate triangle $x_0x_1x_2=0$. Two parallel lines are formed e.g. by the triplet of flexes $(0:1:-\omega^k)$ and the triplet $(1:0:-\omega^k)$. (All pairs of parallel lines are equivalent to this one, so let us restrict our attention to this pair.) The inflectional tangents there are $$ -\lambda \omega^k \cdot x_0+ x_1+\omega^{2k} \cdot x_2=0 \quad \mbox{ and } \quad x_0-\lambda \omega^k \cdot x_1+\omega^{2k} \cdot x_2=0.$$ The essential remark is, that they touch a nondegenerate conic, which in dual coordinates $(\xi_0:\xi_1:\xi_2)$ has the equation $$\xi_0 \cdot \xi_1 + \lambda \xi_2^2=0.$$ (Of course, here we have to exclude $\lambda=0$, the case of the Fermat cubic, where these triplets of inflectional tangents are concurrent.) This implies that the corresponding six cusps on the dual cubic $C^*$ in ${\rm I\!P}_2^*$ are cut out by the nondegenerate conic, whose equation was just given. This conic intersects $C^*$ in each cusp with multiplicity 2, so does not touch the tangent of the cusp. Clearly, the inverse image of this conic on $\bar{X}$ decomposes into two smooth rational curves $\bar{R},\bar{R}' \subset \bar{X}$ passing through our six distinguished cusps. Denote by $R,R'$ the proper transforms of these curves on the smooth surface $X$. A computation in local coordinates shows, that each curve $R$ or $R'$ meets just one of the two rational curves $E_i,E_i'$ from the $A_2$-configuration over each of the six distinguished cusps. Let me call $E_i$ those curves which meet $R$, and $E_i'$ the curves intersecting $R', \, i=1,...,6$. \vv {\bf Lemma 4.} {\em For general choice of $\lambda$, the $K3$ surface $X$ has Picard number 19.} By [PS, \S8] there is only a countable set of $K3$-surfaces with Picard number 20. So, all we have to show is that the structure of $X$ indeed varies with the elliptic curve $C$. In fact, a copy of $C$ (the proper transform of the branch locus) lies on $X$, where it passes through the intersection points in $E_i \cap E_i',\, i=1,...,6$. So, if the structure of $X$ would not vary with $C$, we would have on $X$ more than countably many elliptic pencils, a contradiction. \hspace*{\fill}\frame{\rule[0pt]{0pt}{8pt}\rule[0pt]{8pt}{0pt}}\par This implies that $NS(X)$ is generated (over $\Q$) by the classes of $E_i, E_i',\, i=1,...,9,$ and by the pullback $[H]$ of the class of a line on ${\rm I\!P}_2^*$. Now put $$R-R' \sim \sum_{i=1}^9 (n_i \cdot [E_i]+n_i' \cdot [E_i']) +n \cdot H, \; n_i,n_i',n \in \Q.$$ From $$(R-R').H = (R-R').E_i = (R-R').E_i' =0 \mbox{ for } i=7,8,9$$ we conclude $n=n_i=n_i'=0$ for $i=7,8,9$. The other intersection numbers are ($i=1,...,6$) $$\begin{array}{rcccr} 1 &=& (R-R').E_i &=& -2n_i+n_i' \\ -1 &=& (R-R').E_i' &=& n_i-2n_i' \\ \end{array}$$ This implies $$n_i' = -n_i = \frac{1}{3}.$$ We have shown that the class $$\frac{1}{3} \sum_{i=1}^6 (E_i'-E_i) = R-R'$$ is integral. This is equivalent to the fact that the classes $$\sum_{i=1}^6 (2 \cdot E_i+E_i') \quad \mbox{ and } \quad \sum_{i=1}^6 (E_i+2 \cdot E_i')$$ are divisible by 3. Finally, we remark: For a 3-divisible set of six cusps on $\bar{X}$ the pattern, in which the curves $E_i$ and $E_i'$ organize themselves into unprimed and primed ones, is given by their intersections with $R$ or $R'$. \vV \section{References} \noindent [BPV] Barth, W., Peters, C., Van de Van, A.: Compact complex surfaces, Ergebnisse der Math. (3), 4, Springer (1984) \noindent [BH] Birkenhake C., Lange, H.: A family of abelian surfaces and curves of genus four. manuscr. math. 85, 393-407 (1994) \noindent [B] Blanchard, A.: Sur les varietes analytiques complexes. Ann. Sci. ENS 73, 157-202 (1954) \noindent [F] Fujiki, A.: K\"{a}hlerian normal complex spaces. Tohoku Math. J., $2^{nd}$ series, 35, 101-118 (1983) \noindent [N] Nikulin, V.V.: On Kummer surfaces. Math. USSR Izv. 9, No 2, 261-275 (1975) \noindent [PS] Pjateckii-\v{S}apiro, I.I, \v{S}afarevi\v{c}, I.R.: A Torelli theorem for algebraic surfaces of type $K3$. Izv. Akad. Nauk SSSR, 35, 530-572 (1971) \noindent [S] Siu, Y.T.: Every $K3$-surface is k\"ahler. Invent. math. 73, 139-150 (1983) \end{document} \\

9709

alg-geom/9709017

en

https://arxiv.org/abs/alg-geom/9709017

alg-geom/9709017

Yavor Markov

Y. Markov, V. Tarasov, A. Varchenko

The Determinant of a Hypergeometric Period Matrix

21 pages, no figures, LaTeX2e

We consider a function $U=e^{-f_0}\prod_j^N f_j^{\alpha_j}$ on a real affine space, here $f_0,..,f_N$ are linear functions, $\alpha_1, ...,\alpha_N$ complex numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of hyperplanes in the affine space. We study the period matrix of the hypergeometric integrals associated with the arrangement and the function $U$ and compute its determinant as an alternating product of gamma functions and critical points of the functions $f_0,..., f_N$ with respect to the arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula takes the form $\int_0^\infty e^{-t} t^{\alpha -1} dt=\Gamma (\alpha).$ We also give a determinant formula for Selberg type exponential integrals. In this case the arangements of hyperplanes is special and admits a symmetry group, the period matrix is decomposed into blocks corresponding to different representations of the symmetry group on the space of the hypergeometric integrals associated with the arrangement. We compute the determinant of the block corresponding to the trivial representation.

alg-geom

\section{Introduction} The Euler beta function is an alternating product of Euler gamma functions, \begin{equation}\label{2} B (\alpha, \beta)\,=\, {\Gamma (\alpha)\,\Gamma (\beta) \over \Gamma (\alpha + \beta)} \end{equation} where the Euler gamma and beta functions are defined by \begin{equation}\label{1} \Gamma (\alpha)\,=\,\int_0^\infty\, t^{\alpha -1}\, e^{-t}\,dt, \qquad B (\alpha, \beta)\,=\,\int_0^1\, t^{\alpha -1}\,(1-t)^{\beta -1}\, \,dt. \end{equation} There is a generalization of formula (1) to the case of an arrangement of hyperplanes in an affine space, see [V1, V2, DT]. {\bf Example.} Consider an arrangement of three points $z_1, z_2, z_3$ in a line. The point $z_j$ is the zero of the function $f_j=t-z_j$. Set $$ \Delta_1= [z_1,z_2], \qquad \Delta_1= [z_2,z_3], $$ $$ U_\alpha\,=\,(t-z_1)^{\alpha_1}(t-z_2)^{\alpha_2}(t-z_3)^{\alpha_3}, $$ $$ \omega_1\,=\,\alpha_1 U_\alpha dt/(t-z_1), \qquad \omega_2\,=\,\alpha_2 U_\alpha dt/(t-z_2), $$ then $$ \det\Bigl(\;\!\int_{\Delta_i}\omega_j\Bigr)\,=\, {\Gamma (\alpha_1+1)\,\Gamma (\alpha_2+1)\,\Gamma (\alpha_3+1) \over \Gamma (\alpha_1+\alpha_2+\alpha_3+1)}\,\prod_{i\neq j} f_i^{\alpha_i}(z_j). $$ In this paper we describe a generalization of the first formula in (2) to the case of an arrangement of hyperplanes in an affine space. {\bf Example.} Consider an arrangement of two points $z_1, z_2$ in a line. Let $f_j=t-z_j$. Set $$ \Delta_1= [z_1,z_2], \qquad \Delta_2= [z_2,\infty], $$ $$ U_\alpha\,=\,(t-z_1)^{\alpha_1}(t-z_2)^{\alpha_2}, $$ $$ \omega_1\,=\,\alpha_1 U_\alpha dt/(t-z_1), \qquad \omega_2\,=\,\alpha_2 U_\alpha dt/(t-z_2). $$ Let $a$ be a positive number. Then $$ \det\Bigl(\;\!\int_{\Delta_i}e^{-at}\omega_j\Bigr)\,=\, \Gamma (\alpha_1+1)\Gamma (\alpha_2+1) \, e^{-a(z_1 + z_2)}\, a^{-(\alpha_1+\alpha_2)}\,\prod_{i\neq j} f_i^{\alpha_i}(z_j). $$ The determinant formulas are useful, in particular in applications to the Knizhnik-Zamolodchikov type of differential equations when a determinant formula allows one to conclude that a set of solutions to the equation given by suitable multidimensional hypergeometric integrals forms a basis of solutions, cf. [SV], [TV], [V3], see also [L], [LS], [V4], [V5]. \vskip\baselineskip The paper is organized as follows. Sections~\ref{arrangements} -- \ref{hg-section} contain definitions of the main objects: arrangements, critical values, and hypergeometric period matrices. The main result of the paper is Theorem~\ref{maintheorem}. The proofs of all statements are presented in Section~\ref{proofs}. In Sections~\ref{selberg} and \ref{pfselb} we discuss two determinant formulas for Selberg type integrals. In this case the configuration of hyperplanes is special and admits a symmetry group. The symmetry group acts on the domains of the configuration and on the hypergeometric differential forms associated with the configuration. Therefore the period matrix of the configuration $( \,\int_{\Delta_i} \omega_j\,)$ splits into blocks according to different representations of the symmetry group. We compute the determinant of the block corresponding to the trivial representation. \section{Arrangements} \label{arrangements} In this section we review results from [FT] and [V1]. \subsection{} Let $f_{1},\ldots ,f_{p}$ be linear polynomials on a real affine space $V$. Let $I$ denote $\{1,\ldots ,p\}$ and let $A$ be the arrangement $\{H_{i}\}_{i\in I}$, where $H_{i}= \ker f_{i}$ is the hyperplane defined by $f_{i}$. An {\it edge} of $A$ is a nonempty intersection of some of its hyperplanes. A {\it vertex} is a \mbox{$0$-}dimensional edge. Let $L(A)$ denote the set of all edges. An arrangement $A$ is said to be {\it essential} if it has vertices. Until the end of this paper we suppose that $A$ is essential. An arrangement $A$ is said to be in {\it general position} if, for all subarrangements $\{H_{i_{1}},\cdots ,H_{i_{k}}\}$ of $A$, we have $\mathop{\mathrm{codim}\;\!}\nolimits(H_{i_{1}}\cap\cdots\cap H_{i_{k}})=k$ if $1\leq k\leq\mathop{\mathrm{dim}\;\!}\nolimits V$ and $H_{i_{1}}\cap\cdots\cap H_{i_{k}}=\emptyset$ if $k> \mathop{\mathrm{dim}\;\!}\nolimits V$. Let \begin{equation} \label{defM} M(A)=V-\cup_{i\in I} H_{i}. \end{equation} The topological space $M (A)$ has finitely many connected components, which are called {\it domains}. Domains are open polyhedra, not necessary bounded. Their faces are precisely the domains of the arrangements induced by $A$ on the edges of $A$. More generally, in any subspace $U \subset V$ the arrangement $A$ cuts out a new arrangement $A_U$ consisting of the hyperplanes $\{ H_i\cap U \, | \, H_i \in A, \, U \not \subset H_i\}$. $A_U$ is called a {\it section} of the arrangement. For every edge $F$ of $A$ the domains of the section $A_F$ are called the {\it faces} of the arrangement $A$. Let $F$ be an edge of $A$ and $I(F)$ the set of all indices $i$ for which $F \in H_i$. The arrangement $A^F$ in $V$ consisting of the hyperplanes $\{H_i \, | \, H_i \in A, \, i\in I(F)\}$, is called the {\it localization of the arrangement at the edge $F$}. Every edge $F$ of codimension $l$ is associated to an arrangement in an \mbox{$(l-1)$--}dimensional projective space. Namely, let $L$ be a normal subspace to $F$ of the complementary dimension. Consider the localization at this edge and its section by the normal subspace. All of the hyperplanes of the resulting arrangement $(A^F)_L$ pass through the point $v=F\cap L$. We consider the arrangement which $(A^F)_L$ induces in the tangent space $T_vL$. It determines an arrangement in the projectivization of the tangent space, which is called the {\it projective normal arrangement} and denoted $PA^F$. The arrangements corresponding to different normal subspaces are naturally isomorphic. A face of an arrangement is said to be {\it bounded relative to a hyperplane} if the closure of the face does not intersect the hyperplane. It is known [V1, Theorem 1.5] that if $A=\{H_i\}_{i\in I}$ is an arrangement in a real projective space, then the number of domains bounded with respect to $H_i$ does not depend on $i$. This number is called the {\it discrete length of the arrangement}. The discrete length of the empty arrangement is set to be equal to 1. Let $F$ be an edge of an arrangement $A$ in a projective space. The {\it discrete length} of the edge is defined as the discrete length of the arrangement $A_F$; the {\it discrete width} of the edge is the discrete length of the arrangement $PA^F$; and the {\it discrete volume} of the edge is the product of its discrete length and discrete width. These numbers are denoted $l(F)$, $s(F)$, and $vol(F)$, respectively. If $F$ is a \mbox{$k$--}dimensional edge of an arrangement $A$ in an affine space, then its {\it discrete length} is the number of bounded \mbox{$k$--}dimensional faces of the arrangement $A_F$. Its {\it discrete width} is the discrete length of the arrangement $PA^F$, and its {\it discrete volume} is the product of its discrete length and discrete width. Another more invariant definition of the above quantities could be given as follows (see~[OT]). Consider the complexification and then the projectivization of the affine space $V$. Denote it $\mathbb{P}} %{ \hbox{\bf {\it I\hskip -2pt P}} V$. Let $\mathbb{H}} %{ \hbox{ I\hskip -2pt R} _i=\{f_i=0\}_{i\in I}$ be hyperplanes in $\mathbb{P}} %{ \hbox{\bf {\it I\hskip -2pt P}} V$,$\quad\mathbb{H}} %{ \hbox{ I\hskip -2pt R} _{\infty}$ the infinite hyperplane and $\mathbb{A}} %{ \hbox{\bf {\it A\hskip -10pt A}} $ the arrangement in $\mathbb{P}} %{ \hbox{\bf {\it I\hskip -2pt P}} V$ defined by all these hyperplanes. Let $\chi (\mathbb{A}} %{ \hbox{\bf {\it A\hskip -10pt A}} )$ denote the Euler characteristic of $\mathbb{P}} %{ \hbox{\bf {\it I\hskip -2pt P}} V - \cup_{i\in\overline{I}}\mathbb{H}} %{ \hbox{ I\hskip -2pt R} _i$, where $\overline{I} = 1,\ldots,p,\infty$. If $F$ is an edge of $\mathbb{A}} %{ \hbox{\bf {\it A\hskip -10pt A}} $, then $$l(F)=|\chi(\mathbb{A}} %{ \hbox{\bf {\it A\hskip -10pt A}} _F)|,\quad s(F)=|\chi(P\mathbb{A}} %{ \hbox{\bf {\it A\hskip -10pt A}} ^F)|,\quad vol(F)=l(F)s(F).$$ \subsection{The beta-function of an arrangement} An arrangement is called {\it weighted} if a complex number is assigned to every hyperplane of the arrangement. The complex numbers are called {\it weights}. The weight of a hyperplane $H_i$ is denoted $\alpha_i$. The {\it weight} of an edge $F$ of a weighted arrangement is the sum, $\alpha (F)$, of the weights of the hyperplanes which contain $F$. Let $A$ be a weighted arrangement in an affine space $V$. Make $V$ into a projective space by adding the hyperplane $H_{\infty}$ at infinity: $ \overline{V} = V \cup H_{\infty}$. For all $i\in I$ denote $\overline{H_i}$ the projective closure of $H_i$ in $\overline{V}$. Let $\overline{A}=\{\overline{H_i}\}_{i\in I}\cup\{H_{\infty}\}$ be the corresponding projective arrangement in $\overline{V}$. The arrangement $\overline{A}$ is called the {\it projectivization} of $A$. Set $\alpha_{\infty} = -(\alpha_1+\cdots+\alpha_p)$. Let $L_{-}$ denote the set of all edges at infinity of the arrangement $\overline{A}$ and $L_{+}$ the set consisting of all the other edges. \begin{definition} $(i)$ Let the weights $\alpha_1,\ldots ,\alpha_p$ of the hyperplanes be complex numbers with positive real part. The beta--function of an affine arrangement $A$ is defined by $$B(A;\alpha) = \prod_{F\in L_{+}} \Gamma (\alpha (F) +1)^{vol(F)} \big/ \prod_{F\in L_{-}} \Gamma (-\alpha (F) +1)^{vol(F)} ; $$ $(ii)$ In addition, let $H_0$ be a hyperplane in $V$ and $\overline{H_0}$ its closure in $\overline{V}$. The beta-function of an affine arrangement $A$ relative to the hyperplane $H_0$ is defined by $$B(A;\alpha ;H_0) = \prod_{F\in L_{+}} \Gamma (\alpha (F) +1)^{vol(F)} \big/ \prod_{F\in L_{-},\, F\subset \overline{H_0}} \Gamma (-\alpha (F) +1)^{vol(F)}.$$ \end{definition} {\sc Example.} Let $n=\mathop{\mathrm{dim}\;\!}\nolimits V$. For an arrangement $A$ of $p$ hyperplanes in general position the above formulas take the form $$B(A;\alpha) = \left(\frac{\Gamma(\alpha_1 +1)\ldots\Gamma (\alpha_p +1)} {\Gamma (\alpha_1+\cdots+\alpha_p +1)}\right)^{\binom{p-2}{n-1}}; \qquad B(A;\alpha;f_0) = \left(\Gamma(\alpha_1+1)\ldots\Gamma(\alpha_p+1)\right)^{\binom{p-2}{n-1}}.$$ \subsection{Trace of an arrangement at infinity relative to a hyperplane} \label{traceinf} Assume that an additional non-constant linear function $f_0$ on $V$ is given. Denote the hyperplane $\{f_0=0\}$ by $H_0$. Let A be an affine arrangement in the affine space $V$ . Consider the projectivized arrangement $\overline{A}$ in $\overline{V}$ and its section $\overline{A}_{H_{\infty}}$. The intersection of $\overline{A}$ with the affine space $W=H_{\infty} - \overline{H_0} \cap H_{\infty}$ is called the {\it trace of the arrangement $A$ at infinity} relative to the hyperplane $H_0$ and is denoted $tr(A)_{H_0}$. \begin{lemma} If the affine arrangement $A$ is given by the linear functions $\{f_i\}_{i\in I}$ on $V$, then the affine arrangement $tr(A)_{H_0}$ is given by the linear functions $\{h_i=f_i^0/f_0^0\, | \, i\in I;\,\, f_i^0/f_0^0 \not = const \}$ on $W$, where $f_i^0$ denotes the homogeneous part of $f_i$. \hfill~~\mbox{$\square$} \end{lemma} \section{Properties of an arrangement} \label{properties} In this section we examine the properties of the unbounded domains of an arrangement $A$ on which an additional linear function $f_0$ tends to $+\infty$. \vskip\baselineskip Let $A$ be an arrangement in an affine space $V$. Consider its projectivization $\overline{A}$ in the projective space $\overline{V}$. Let $\Delta$ be an unbounded face of the arrangement $A$. Take the closure $\overline{\Delta}$ of $\Delta$ in $\overline{V}$. Consider the intersection $\overline{\Delta}\cap H_{\infty}$. It is a union of faces of the arrangement $\overline{A}_{H_{\infty}}$. There is a unique one of highest dimension. We call it the {\it trace} of $\Delta$ at infinity and denote $tr(\Delta)$. An unbounded face $\Delta$ of $A$ is called a {\it growing face with respect to } $f_0$ if $f_0(x)$ tends to $+\infty$ whenever $x$ tends to infinity in $\Delta$. A face at infinity $\Sigma$ is called a {\it bounded face at infinity with respect to } $f_0$ if $\overline{\Sigma}\cap \overline{H_0}$ is empty. In other words, $\Sigma$ is a bounded face at infinity if and only if it is a bounded face of the affine arrangement $tr(A)_{H_0}$. If $\Sigma$ is a face of $A$, denote $F_{\Sigma}$ the unique edge of the smallest dimension which contains $\Sigma$. Define the discrete length, the discrete width and the volume of the face as the same quantities for the corresponding edge. \begin{theorem} \label{bijection} The trace map from the unbounded faces of $A$ to the faces of $\overline{A}$ in $H_{\infty}$ has the following properties: $(i)$ The trace of a growing face is a bounded face at infinity. $(ii)$ For any bounded face at infinity, $\Sigma$, there exist exactly $s(\Sigma )$ growing domains with trace $\Sigma$, where $s(\Sigma )$ denotes the discrete width of this face. $(iii)$ The number of growing domains of $A$ is equal to the sum of the volumes of all edges of $\overline{A}$ at infinity which do not lie in $\overline{H_0}$: $$ \mathrm{\#}\,\mathrm{ growing}\,\mathrm{ domains} = \sum_{F\subseteq H_{\infty},\, F \not\subseteq \overline{H_0}} vol(F).$$ \end{theorem} Theorem~\ref{bijection} is proved in Section~\ref{biject} \section{Critical values} \label{critical} The aim of this section is to define the critical values of the functions $f_1^{\alpha_1}, \ldots , f_p^{\alpha_p}$ on the bounded domains of an arrangement $A$ and the critical values of the same functions, with respect to an additional linear function $f_0$, on the bounded and growing domains of $A$. \medskip Let an arrangement $A$ be given by linear functions $\{f_i\}_{i\in I}$ and let $\alpha = \{\alpha_i\}_{i\in I}$ be a corresponding set of weights. For every $i\in I$, a face of the arrangement $A$ on which $f_i$ is constant is called a {\it critical face} with respect to $f_i$ and the value of $f_i$ on that face is called a {\it critical value}. In particular, each vertex is a critical face for every function $f_i$. Assume that a function $|f_i|$ is bounded on a face $\Sigma$ of $A$. The subset of $\overline{\Sigma}$ on which $|f_i|$ attains its maximum is a union of critical faces. Among them, there is a unique one of highest dimension. It is called the {\it external support} of the face $\Sigma$ with respect to $f_i$. Denote $\mathsf{Ch}(A)$ the set of all bounded domains of $A$. Let $\beta(A) = \left|\mathsf{Ch}(A)\right|$. Enumerate the bounded domains by numbers $1,\ldots,\beta(A)$. For every $i\in I$ and $j\in \{1,\ldots ,\beta (A)\}$, choose a branch of the multi--valued function $f_i^{\alpha_i}$ on the domain $\Delta_{j}$ and denote it $g_{i,j}$. Let $\Sigma_{i,j}$ be the external support of $\Delta_j$ with respect to $f_i$. Define the {\it extremal critical value of the chosen branch $g_{i,j}$ on $\Delta_j$} as the number $c(g_{i,j},\Delta_j)=g_{i,j}(\Sigma_{i,j})$. Denote $c(A;\alpha)$ the product of all extremal critical values of the chosen branches, $$ c(A;\alpha)=\prod_{j=1}^{\beta(A)}\prod_{i\in I}c(g_{i,j},\Delta_j).$$ Assume that an additional non-constant linear function $f_0$ on $V$ is given. Let $\Delta$ be bounded or growing domain of $A$. Then $f_0$ is bounded from below on $\Delta$. The subset of $\overline{\Delta}$ on which $e^{-f_0}$ attains its maximum coincides with the subset of $\overline{\Delta}$ where $f_0$ attains its minimum. This subset has a unique face of highest dimension; it is called the {\it support face} of $f_0$ on $\Delta$ and denoted $\Sigma_{\Delta}$. Define the {\it extremal critical value of $e^{-f_0}$ on $\Delta$} as the number $c(e^{-f_0},\Delta) = e^{-f_0(\Sigma_{\Delta})}$. Denote $\mathsf{Ch}(A;f_0)$ the set of all bounded or growing domains of $A$. Let $\gamma(A) = \left|\mathsf{Ch}(A;f_0)\right|$. Enumerate these domains by numbers $1,\ldots,\gamma(A)$. For every $i\in I$ and $j\in \{1,\ldots ,\gamma (A) \}$, choose a branch of the multi--valued function $f_i^{\alpha_i}$ on the domain $\Delta_{j}$ and denote it $g_{i,j}$. Assume that $|f_i|$ is bounded on $\Delta_j$. Let $\Sigma_{i,j}$ be the external support of $\Delta_j$ with respect to $f_i$. Define the {\it extremal critical value of the chosen branch $g_{i,j}$ on $\Delta_j$ with respect to $f_0$} as the number $c(g_{i,j},\Delta_j,f_0)=g_{i,j}(\Sigma_{i,j})$. Notice that, if $\Delta_j$ is a bounded domain of $A$, then $c(g_{i,j},\Delta_j)=c(g_{i,j},\Delta_j,f_0)$. Now assume that $|f_i|$ is unbounded on $\Delta_j$. Thus, $\Delta_j$ is a growing domain of $A$ and $tr(\Delta_j)$ is a bounded face of $tr(A)_{H_0}$. Denote $M=f_0(\Sigma_{\Delta_j})$. Consider the rational function $\widetilde{h_i}=f_i/(f_0-M)$ on $\Delta_j$. Notice that $\widetilde{h_i}|_{tr(\Delta_j)}$ coincides with the restriction of the linear function $h_i=f_i^0/f_0^0$ to the same set $tr(\Delta_j)$. Since the sign of $\widetilde{h_i}$ on $\Delta_j$ is the same as the sign of $f_i$ on $\Delta_j$ we can choose a branch of $\widetilde{h_i}^{\alpha_i}$ on $\Delta_j$ which has the same argument as $g_{i,j}$ and denote it $\widetilde{g_{i,j}}$. Let $\Sigma_j$ be the external support of $tr(\Delta_j)$ with respect to $h_i$ in the affine arrangement $tr(A)_{H_0}$. Define the {\it the extremal critical value of the chosen branch $g_{i,j}$ on $\Delta_j$ with respect to $f_0$} as the number $c(g_{i,j},\Delta,f_0)=\widetilde{g_{i,j}}(\Sigma_j)$. Denote $c(A;\alpha;f_0)$ the product of all extremal critical values with respect to $f_0$ of the chosen branches, $$ c(A;\alpha;f_0) = \prod_{j=1}^{\gamma(A)}\left(e^{-f_0(\Sigma_{\Delta_j})} \prod_{i\in I}c(g_{i,j},\Delta_j,f_0)\right).$$ \section{Hypergeometric period matrix}\label{hg-section} \subsection{$\beta${\bf nbc}-bases} \label{bnbcbases} Let $A$ be an essential arrangement in an \mbox{$n$--}dimensional real affine space $V$. Define a linear order $<$ in $A$ putting $H_{i}< H_{j}$ if $i<j$. A subset $\{H_{i}\}_{i\in J}$ of $A$ is called {\it dependent} if $\cap_{i\in J}H_{i}\neq\emptyset$ and $\mathop{\mathrm{codim}\;\!}\nolimits(\cap_{i\in J}H_{i})<\left|J\right|$. A subset of $A$ which has nonempty intersection and is not dependent is called {\it independent}. Maximal independent sets are called {\it bases}. An intersection of a basis defines a vertex. A $k$-tuple $S=(H_{1},\cdots ,H_{k})$ is called a {\it circuit} if $(H_{1},\cdots ,H_{k})$ is dependent and if for each $l$, $1\leq l\leq k$, the ($k-1$)-tuple $(H_{1},\cdots ,\widehat{H_{l}},\cdots ,H_{k})$ is independent. A $k$-tuple $S$ is called a {\it broken circuit} if there exists $H< min (S)$ such that $\{H\}\cup S$ is a circuit, where $min (S)$ denotes the minimal element of $S$ for $<$. \vskip\baselineskip The collection of subsets of $A$ having nonempty intersection and containing no broken circuits is denoted {\bf BC}. {\bf BC} consists of independent sets. Maximal (with respect to inclusion) elements of {\bf BC} are bases of $A$ called {\bf nbc}-bases. Recall that $n$ is the dimension of the affine space. An {\bf nbc}-basis $B=(H_{i_{1}},\cdots ,H_{i_{n}})$ is called {\it ordered} if $H_{i_{1}}< H_{i_{2}}< \cdots < H_{i_{n}}$. The set of all ordered {\bf nbc}-bases of $A$ is denoted {\bf nbc}$(A)$ A basis $B$ is called a $\beta{\textup{\textbf{nbc}}}$--{\it{basis}} if $B$ is an {\bf nbc}-basis and if \begin{equation} \label{bnbccond} \forall H\in B \,\exists H'<\, H \,\mathrm{such}\,\mathrm{ that}\, (B-\{H\})\cup \{H'\}\, \mathrm{is}\,\mathrm{ a}\,\mathrm{ base}. \end{equation} Denote $\beta{\textup{\textbf{nbc}}} (A)$ the set of all ordered $\beta{\textup{\textbf{nbc}}}$-bases. Put the lexicographic order on $\beta{\textup{\textbf{nbc}}} (A)$ The definition and basic properties of the $\beta{\textup{\textbf{nbc}}}$-bases are due to Ziegler [Z]. \vskip\baselineskip For a basis $B=(H_{i_{1}},\cdots,H_{i_{n}})$, let $F_{j}=\bigcap_{k=j+1}^{n}H_{i_{k}}$ for $0\leq j\leq n-1$ and $F_{n} = V$. Then $ \xi(B)= (F_{0}\subset F_{1} \subset \cdots\subset F_{n})$ is a flag of affine subspaces of $V$ with $\mathop{\mathrm{dim}\;\!}\nolimits F_{j} = j$ $(0\leq j\leq n)$. This flag is called the {\it flag associated with $B$}. For an edge $F$ of $A$, remind that $I(F) = \{i\in I \mid F \subseteq H_{i}\}$. Introduce a differential one-form $$\omega_{\alpha}(F,A)=\sum_{i\in I(F)} \alpha_{i}\frac{df_i}{f_i}.$$ For a basis $B=(H_{i_{1}},\cdots,H_{i_{n}})$, let $\xi(B)= (F_{0}\subset F_{1} \subset \cdots\subset F_{n})$ be the associated flag. Introduce a differential $n$--form $\Xi(B,A)= \omega_{\alpha}(F_{0},A) \wedge\cdots\wedge\omega_{\alpha}(F_{n-1},A)$. If $\beta{\textup{\textbf{nbc}}} (A)=\{B_{1},\cdots,B_{\beta (A)}\}$ and $\phi _{j} = \phi_{j}(A)=\Xi(B_{j},A)$ for $j\in\{1,\ldots,\beta(A)\}$, define \begin{equation} \Phi (A)= \{\phi_{1},\cdots,\phi_{\beta (A)}\}. \end{equation} {\sc Example.} For an arrangement $A$ of $p$ hyperplanes in general position, the set $\beta{\textup{\textbf{nbc}}}(A)$ coincides with the set $\{ (H_{i_1},\ldots,H_{i_n})\, | \, 2\leq i_1 <\cdots<i_n\leq p\}$. The latter corresponds to all vertices of $A$ away from the hyperplane $H_1$. The differential $n$--forms are $$ \Phi (A) = \{\alpha_{i_1}\ldots\alpha_{i_n}\frac{df_{i_1}}{f_{i_1}}\wedge \cdots\wedge\frac{f_{i_n}}{f_{i_n}}\, | \, 2\leq i_1 <\cdots<i_n\leq p\}.$$ \subsection{The definition of the hypergeometric period matrix} \label{defhg} Let $\xi = (F_{0} \subset F_{1} \subset \cdots \subset F_{n})$ be a flag of edges of $A$ with $\mathop{\mathrm{dim}\;\!}\nolimits F_{i} = i $ ($i = 0, \ldots, n-1;$ $F_n=V$). Let $\Delta$ be a domain of $A$ and $\overline{\Delta} $ its closure in $V$. We say that the flag is {\it adjacent to} the domain if $\mathop{\mathrm{dim}\;\!}\nolimits (F_{i} \cap \overline{\Delta}) = i$ for $i = 0,\ldots, n$. The following proposition from [DT, Proposition 3.1.2] allows us to enumerate the bounded domains of a configuration $A$ by means of $\beta{\textup{\textbf{nbc}}}(A)$. \begin{proposition} \label{bnbclabel} There exists a unique bijection $$ C : \beta{\textup{\textbf{nbc}}} (A) \longrightarrow \mathsf{Ch}(A) $$ such that for any $B\in \beta{\textup{\textbf{nbc}}}(A)$, the associated flag $\xi(B)$ is adjacent to the bounded domain $C(B)$. \end{proposition} Let $t>0$ be a number which is larger than the maximum of $f_0$ on the closure of any bounded domain of $A$. Then the hyperplane $H_t = \{f_0=t\}$ does not intersect the bounded domains of $A$. Consider the affine arrangement $A_t=A\cup \{H_t\}$. The set of its bounded domains consists of two disjoint subsets: the first is the subset of all bounded domains of $A$; the second is formed by the domains of $A_t$ which are intersections of unbounded domains of $A$ and the half--space $\{f_0<t\}$. Notice that the intersection of an unbounded domain $\Delta$ of $A$ and the half--space $\{f_0<t\}$ is nonempty and bounded if and only if $\Delta$ is a growing domain. Thus $\beta (A_t) =\gamma (A)$. Define an order $<$ on $A_t$ as $H_t<H_1<\ldots<H_p$. Consider the set $\beta{\textup{\textbf{nbc}}}(A_t)$ with respect to this order. If $B \in \beta{\textup{\textbf{nbc}}} (A_t)$, then $H_t \not \in B$ because of condition~(\ref{bnbccond}) and the minimality of $H_t$ with respect to the order $<$. This observation implies that $\beta{\textup{\textbf{nbc}}} (A_t)$ and $ \Phi (A_t)$ do not depend on $t$. Denote them $\beta{\textup{\textbf{nbc}}} (A;f_0)$ and $\Phi (A;f_0)$ respectively. Notice that $\beta{\textup{\textbf{nbc}}}(A)$ and $\Phi(A)$ are subsets of $\beta{\textup{\textbf{nbc}}}(A;f_0)$ and $\Phi(A;f_0)$ respectively because the order on $A$ is a restriction of the order on $A_t$ to its subset $A$ and because of condition~(\ref{bnbccond}). We also have an analog of Proposition~\ref{bnbclabel}. \begin{proposition} \label{bnbclabel1} There exists a unique bijection $$ \overline{C} : \beta{\textup{\textbf{nbc}}} (A;f_0) \longrightarrow \mathsf{Ch}(A;f_0) $$ such that for any $B\in \beta{\textup{\textbf{nbc}}} (A;f_0)$, the associated flag $\xi(B)$ is adjacent to the domain $\overline{C}(B)$. Moreover, $\overline{C}|_{\beta{\textup{\textbf{nbc}}}(A)} = C$. \hfill~~\mbox{$\square$} \end{proposition} Let the set $\beta{\textup{\textbf{nbc}}} (A;f_0) = \{B_{1}, \ldots, B_{\gamma}\}$ be lexicographically ordered as in section \ref{bnbcbases}. For $i = 1, \ldots, \gamma$, define a domain $\Delta_{i} \in \mathsf{Ch}(A;f_0)$ by $\Delta_{i} = C(B_{i})$. This gives us an order on the set of the growing and bounded domains of $A$. The order is called the $\beta{\textup{\textbf{nbc}}} $--{\it{order}}. We give an orientation to each domain $\Delta\in\mathsf{Ch}(A;f_0)$ as follows. Let $\Delta = C(B)$ with $B\in\beta{\textup{\textbf{nbc}}}(A;f_0). $ Let $\xi(B) = (F_{0}\subset F_{1} \subset \cdots\subset F_{n})$ be the associated flag. The flag $\xi(B)$ is adjacent to the domain $B$ and defines its {\it intrinsic orientation} [V2, 6.2]. The intrinsic orientation is defined by the unique orthonormal frame $\{e_{1},\ldots, e_{n}\}$ such that each $e_{i}$ is a unit vector originating from the point $F_{0}$ in the direction of $F_{i}\cap \overline{\Delta}$. Let $\beta = \beta(A)$. Assume that $\mathsf{Ch}(A)=\{\Delta_1, \ldots,\Delta_{\beta}\}$ is the $\beta{\textup{\textbf{nbc}}}$-ordered set of the bounded domains of $A$ and $\Phi (A) = \{\phi_1, \ldots, \phi_{\beta}\}$ is the $\beta{\textup{\textbf{nbc}}}$-ordered set of differential $n$--forms constructed in Section~\ref{bnbcbases}. Assume that the weights $\{\alpha_i\}_{i\in I}$ have positive real parts. For every $i\in I$ and $j\in \{1,\ldots,\beta\}$, choose a branch of $f_{i}^{\alpha_{i}}$ on the domain $\Delta_{j}$ and the intrinsic orientation of the domain $\Delta_{j}$. Let $U_{\alpha} :=f_{1}^{\alpha_{1}} \cdots f_{p}^{\alpha_{p}}$. The choice of branches of the functions $f_i^{\alpha_i}$ on all bounded domains defines a choice of branches of the function $U_\alpha$ on all bounded domains. Define the {\it hypergeometric period matrix} by \begin{equation} \label{period} \mathsf{PM}(A; \alpha) = \left[\int_{\Delta_j}U_{\alpha}\phi_k\right]_{k,j=1}^{\beta}. \end{equation} Since $\mathsf{Re}\,\alpha_i>0$, all elements of the period matrix are well defined. Let $\gamma = \gamma(A)$. Let $\mathsf{Ch}(A;f_0)=\{\Delta_1, \ldots,\Delta_{\gamma}\}$ be the $\beta{\textup{\textbf{nbc}}}$-ordered set of the bounded and growing domains of $A$ and let $\Phi (A;f_0) = \{\phi_1, \ldots, \phi_{\gamma}\}$ be the $\beta{\textup{\textbf{nbc}}}$-ordered set of differential $n$--forms constructed in Section~\ref{defhg}. Assume that the weights $\{\alpha_i\}_{i\in I}$ have positive real parts. For every $i\in I$ and $j\in \{1,\ldots,\gamma\}$, choose a branch of $f_{i}^{\alpha_{i}}$ on the domain $\Delta_{j}$ and the intrinsic orientation of each domain $\Delta_{j}$. The choice of branches of the functions $f_i^{\alpha_i}$ on all bounded and growing domains defines a choice of branches of the function $U_\alpha$ on all bounded and growing domains. Define the {\it hypergeometric period matrix with respect to $f_0$} by \begin{equation} \label{newperiod} \mathsf{PM}(A; \alpha; f_0) = \left[\int_{\Delta_j}e^{-f_0}U_{\alpha}\phi_k\right]_{k,j=1}^{\gamma}. \end{equation} Since $\mathsf{Re}\,\alpha_i>0$ and $f_0$ tends to $+\infty$ on the growing domains of $A$, all elements of the period matrix are well defined. \section{The main theorem}\label{mainth} In [DT], Douai and Terao proved the following theorem, cf. also [V1,V2]. \begin{theorem} \label{dt-main} Let $A$ be a weighted arrangement given by functions $\{f_i\}_{i\in I}$ and weights $\alpha=\{\alpha_i\}_{i\in I}$ such that $\mathsf{Re}\, \alpha_{i}>0$ for all $i\in I$. Fix branches of the multivalued functions $\{f_i^{\alpha_i}\}_{i\in I}$ on all bounded domains of $A$. Then \begin{equation}\label{dt-formula} \det \mathsf{PM} (A;\alpha)= c(A;\alpha) B(A;\alpha). \end{equation} \end{theorem} The main result of this paper is the following theorem. \begin{theorem} \label{maintheorem} Let $A$ be a weighted arrangement given by functions $\{f_i\}_{i\in I}$ and weights $\alpha=\{\alpha_i\}_{i\in I}$ such that $\mathsf{Re}\, \alpha_{i}>0$ for all $i\in I$. Let an additional non-constant linear function $f_0$ be given. Denote $H_0$ the hyperplane $\{f_0=0\}$. Fix branches of the multivalued functions $\{f_i^{\alpha_i}\}_{i\in I}$ on all bounded and growing domains of $A$. Then \begin{equation}\label{mainformula} \det \mathsf{PM} (A;\alpha;f_0)= c(A;\alpha;f_0)B(A;\alpha;H_0). \end{equation} \end{theorem} We will deduce this formula for the determinant of the period matrix with respect to $f_0$ from Theorem~\ref{dt-main} by passing to a limit. \section{Proofs} \label{proofs} \subsection{ Proof of Theorem~\ref{bijection}} \label{biject} \begin{lemma} Let $\Delta$ be a growing face. Then $\overline{tr(\Delta)}\cap \overline{H_0} = \emptyset$, i.e. $tr(\Delta)$ is a bounded face at infinity. \end{lemma} $\mathsf{Proof.}$ Let $x_0,\ldots,x_{n-1}$ be affine coordinates on $V$ such that $f_0(x)=x_0$. Let $(t_0:t_1:\cdots :t_n)$ be the corresponding projective coordinates in $\overline{V}$: $\{x_i=t_i/t_n\}_{i=0}^{n-1}$. Let $\Delta$ be a growing face. Assume that $\overline{tr(\Delta)}\cap \overline{H_0} \not = \emptyset$ and $P=(p_0:p_1:\cdots:p_n)$ is a point of this intersection. Thus, $p_0=p_n=0$. Let $Q=(q_0:q_1:\cdots:q_n)$ be any point inside $\Delta$. Thus $q_n\not = 0$. Since $\overline{\Delta}$ is a closed polyhedron in $\overline{V}$ it contains the segment $PQ$. This segment is parametrized by the points $P_{\lambda} = (\lambda p_0 + (1-\lambda)q_0:\lambda p_1 + (1-\lambda)q_1: \cdots:\lambda p_n + (1-\lambda)q_n)$, for $\lambda \in [0,1]$. The point $P_{\lambda}$ tends to $P\in H_{\infty}$ when $\lambda\mapsto 1$. We have $$f_0(P_{\lambda})=\frac{\lambda p_0 + (1-\lambda)q_0}{\lambda p_n + (1-\lambda)q_n}=\frac{q_0}{q_n} = \mathrm{constant}.$$ This contradicts to the assumption that $\Delta$ is a growing face. So $\overline{tr(\Delta)}\cap \overline{H_0} = \emptyset$. Part $(i)$ of Theorem~\ref{bijection} is proved. \hfill~~\mbox{$\square$} \begin{lemma}\label{supp-l} Let $\Delta$ be an unbounded domain of $A$. Let $tr(\Delta)$ be a bounded face at infinity with respect to $f_0$. Let $f_0$ be unbounded on $\Delta\cap\{f_0>0\}$. Then $\Delta$ is a growing domain of $A$. \end{lemma} $\mathsf{Proof.}$ Since $tr(\Delta)$ is bounded at infinity, we have $\overline{\Delta}\capH_{\infty}\cap\overline{H_0}=\emptyset$. For a real $t$, let $H_t=\{f_0=t\}$. Then \begin{equation}\label{support-c} \overline{\Delta}\capH_{\infty}\cap\overline {H_t}=\emptyset. \end{equation} Let $\{x_i\}_{i=1}^{\infty}$ be a sequence of points in $\Delta$ such that $x_i$ tends to $\infty$ when $i\mapsto\infty$. Choose a positive $T$. Assume that $T$ is larger than the supremum of $f_0$ on all bounded domains of $A$. Consider the arrangement $A_T=A\cup\{H_T\}$. Formula~(\ref{support-c}) implies that $H_T\cap\Delta$ is a bounded domain of the section $(A_T)_{H_T}$. Since $A_T$ is essential, Proposition~9.9 [BBR] is applicable. It implies that there is a bounded domain $\Delta_T$ of the arrangement $A_T$, such that $H_T\cap\Delta$ is a subset of the boundary of $\Delta_T$. This bounded domain must be $\Delta\cap\{f_0<T\}$, because of the choice of $T$. Since $\Delta_T$ is bounded, there exists a positive integer $N_T$ such that for every integer $n\geq N_T$ we have $x_n \in \Delta - \Delta_T$. Since $\Delta - \Delta_T = \Delta\cap\{f_0\geq T\}$, we have $f_0(x_n)\geq T$ for all $n\geq N_T$. This proves that $\Delta$ is a growing domain. \hfill~~\mbox{$\square$} \begin{lemma} For any bounded face at infinity, $\Sigma$, there exist exactly $s(\Sigma )$ growing domains with trace $\Sigma$, where $s(\Sigma )$ denotes the discrete width of this face. \end{lemma} $\mathsf{Proof.}$ Let $\Sigma$ be a bounded face at infinity of codimension $k$. Choose projective coordinates $(t_0:t_1:\cdots:t_n)$ on $\overline{V}$ such that $\overline{H_0} = \{t_0=0\}$, $H_{\infty} = \{t_n=0\}$, and $F_{\Sigma}$ is given by $t_1=\cdots =t_{k-1}= t_n= 0$. Let $v$ be a point in $\Sigma$ and $B$ an open ball around $v$. If the ball is sufficiently small, then the domains of $A$ which intersect $B$ are precisely those for which $v$ belongs to their closure in $\overline{V}$ and the hyperplanes of $\overline{A}$ which intersect $B$ are exactly those belonging to $\overline{A}^{F_{\Sigma}}$. Local affine coordinates on $B$ are given by $\{y_i=t_i/t_0\}_{i=1}^n$. Since $F_{\Sigma}$ is given by the equations $y_1=\cdots=y_{k-1}=y_n=0$, the subspace $L$ through $v$ spanned by the coordinate vectors $e_1,\ldots,e_{k-1},e_n$ is a normal subspace to $F_{\Sigma}$. Then the number of open domains in $B$ is equal to the number of open domains of the arrangement induced in the tangent space $T_vL$ by the arrangement $\overline{A}^{F_{\Sigma}}$. On $B$ the function $f_0$ has the form $f_0(y)=1/y_n$. We are interested in the domains in $B$ on which $f_0 \mapsto +\infty$ when $y_n \mapsto 0$. So, on this domains we must have $y_n >0$. If the codimension of $\Sigma$ in $H_{\infty}$ is $0$, then the number of such domains is equal to $1$, which is exactly the discrete width of the empty configuration. Assume that the above codimension is positive. Then the number of domains in $B$ on which $y_n>0$ is equal to the number of the domains of the projective normal arrangement $P\overline{A}^{F_{\Sigma}}$. Finally, we want to count only those domains for which $\overline{\Sigma}$ is the only part of their closure in $\overline{V}$, lying in $H_{\infty}$. Thus, they are the projective domains away from the hyperplane $y_n=0$. Their number is equal to the discrete length of $P\overline{A}^{F_{\Sigma}}$. By definition this number is equal to the discrete width of $\Sigma$. Lemma~\ref{supp-l} implies that the corresponding domains of $A$ are growing. \hfill~~\mbox{$\square$} \begin{lemma} The number of growing domains of $A$ is equal to the sum of the volumes of all edges of $\overline{A}$ at infinity which do not lie in $\overline{H_0}$. \end{lemma} $\mathsf{Proof.}$ Let $F$ be an edge at infinity with non-zero volume which do not lie in $\overline{H_0}$. Then, by definition, there are exactly $l(F)$ bounded faces at infinity which generate $F$. For each of them, $\Sigma$, there exist exactly $s(F)$ growing domains of $A$ with trace $\Sigma$. Thus there exist exactly $vol(F)=l(F)s(F)$ growing domains whose traces generate $F$. Finally, in order to count all growing domains of $A$, we have to sum over all edges at infinity which have non-zero volume and do not lie in $\overline{H_0}$. Theorem~\ref{bijection} is proved. \hfill~~\mbox{$\square$} \subsection{ Asymptotic behavior of critical values}\label{critasym} Let $A$ be an arrangement in the affine space $V$. Let $f_0$ be an additional non--constant linear function on $V$. Define $f_t=1-\frac{f_0}{t}$ and $H_t=\{f_t=0\}$. Consider a new weighted arrangement $A_t=A\cup \{H_t\}$ where we assume that the weight of $H_t$ is equal to $t$. For a sufficiently big $t$, the hyperplane $H_t$ intersects only some of the unbounded domains of the arrangement $A$. Moreover, the intersection creates a new bounded domain if and only if the intersected domain is a growing one. So if $\Delta$ is a growing domain, we will denote the corresponding bounded domain of $A_t$ by $\Delta_t$ and will call it a {\it growing bounded domain}. If $\Delta$ is a bounded domain of $A$, then it is also a bounded domain of $A_t$. This correspondence between the bounded domains of $A_t$ and the bounded or growing domains of $A$ is a bijection. \begin{lemma}\label{tcrit} Let $\Delta_t$ be a bounded domain of the arrangement $A_t$. Let $\Delta$ be the corresponding bounded or growing domain of $A$. If $t>0$ and $(1-f_0/t)$ is positive on $\Delta_t$ choose the positive branch $g_t$ of $(1-f_0/t)^t$ on $\Delta_t$. Then the external support of $\Delta_t$ with respect to $f_t$ is a face of the arrangement $A$. For every big enough $t$ this external support coincides with the support face, $\Sigma_{\Delta}$, of $f_0$ on $\Delta$. Moreover, $\lim_{t\mapsto +\infty} c(g_t,\Delta_t) = e^{-f_0(\Sigma_{\Delta})}$. \end{lemma} $\mathsf{Proof.}$ For a fixed $t$, the external support of $\Delta_t$ with respect to $f_t$ lies outside $H_t$. Thus, it is a face of the arrangement $A$. The set of all critical faces of $A$ with respect to $f_0$ is finite. Let $M$ be the maximum of $f_0$ on this set. Assume that $t > M$. Then all bounded domains of $A_t$ lie inside the positive half--space with respect to $f_t$. Let $f_{t}=1-f_0/t$ attains its maximum on a critical face $\Sigma_{\Delta}$ of $\overline{\Delta_{t}}$. This is equivalent to the condition that $f_0$ attains its minimum on the same face. So $\Sigma_{\Delta}$ is the support face of $\Delta$ with respect to $f_0$. On the other side, it is the external support of $\Delta_t$ with respect to $f_t$. Hence $$\lim_{t\mapsto +\infty} c(g_t,\Delta_t) = \lim_{t\mapsto +\infty} \left(1-\frac{f_0(\Sigma_{\Delta})}{t}\right)^t = e^{-f_0(\Sigma_{\Delta})}.$$ \hfill~~\mbox{$\square$} \begin{lemma} \label{unbcrit} Let the hyperplane $H=\{f=0\}$ belongs to the arrangement $A$. Let $\Delta$ be a growing domain of $A$ and $\Delta_t$ the corresponding growing bounded domain of $A_t$. Let $|f|$ be unbounded on the growing domain $\Delta$. Then there exists a unique face $\Sigma$ of highest dimension, belonging to the closure of $\Delta$, such that for every big enough $t$, the external support of the face $\Delta_t$ with respect to $f$ is $\Sigma_t=\Sigma \cap \Delta_t$. Moreover, $tr(\Sigma)$ is the external support of $tr(\Delta)$ with respect to $h$ in the affine space $W=H_{\infty} - \overline{H_0}\capH_{\infty}$, where $h=f^0/f^0_0$. The asymptotic behavior of $f(\Sigma_t)$ when $t$ tends to $+\infty$ is given by $f(\Sigma_t)=h(tr(\Sigma))t(1+o(1)).$ \end{lemma} $\mathsf{Proof.}$ The set of critical faces with respect to $f$ of the arrangement $A$ is finite. $|f|$ is bounded on this set. Since $|f|$ is unbounded on $\Delta$ the external support of $\Delta_t$ with respect to $f$ lies on $H_t$ for $t$ big enough. Let $\Sigma_{t_1}$ be a critical face of $\Delta_{t_1}$ which lies on $H_{t_1}$ for some $t_1$ fixed. Then $\Sigma_{t_1} = H_{t_1}\cap\Sigma$, where $\Sigma$ is a face of $\overline{\Delta}$. Consider the face $\Sigma_t=H_t\cap\Sigma$ of $\Delta_t$ for an arbitrary $t$. It is a critical face of $\Delta_t$ because $\Sigma_t$ is parallel to $\Sigma_{t_1}$ and the latter is parallel to $H$. Let us compute the asymptotic behavior of $f(\Sigma_t)$ when $t$ tends to $+\infty$. Choose affine coordinates $\{x_j\}_{j=0}^{n-1}$ in $V$ such that $f_0(x)=x_0$. Let $f=f^0 + b$ be the sum of the homogeneous part of $f$ and the constant term. Then $f=x_0(f^0/x_0 + b/x_0) = x_0(f^0/f_0^0 + b/x_0)$. Since $H_t=\{x_0=t\}$, $f(\Sigma_t) = t(h(\Sigma_t) + b/t) = h(tr(\Sigma))t(1+o(1))$. Let $\Sigma'$ be the external support of $tr(\Delta)$ relative to $tr(H)$. Let $\Sigma$ is the face of $\Delta$ for which $\Sigma'=tr(\Sigma)$. Then the previous computation shows that for every $t$ big enough $\Sigma_t$ is the external support of $\Delta_t$ with respect to $f$. \hfill~~\mbox{$\square$} \begin{corollary}\label{unb-cor} Let the conditions be as in Lemma~\ref{unbcrit}. In addition, assume that $\alpha$ is a complex number. Fix a branch of $f^{\alpha}$ on $\Delta$ and denote it $g$. Fix a branch of $(f/f_0)^{\alpha}$ on $\Delta$ as in Section~\ref{critical} and denote it $\tilde{g}$. Fix branches of $t^{\alpha}$ and $(1+o(1))^{\alpha}$ using the branch of the logarithm with zero argument. Then the asymptotic behavior of $c(g,\Delta_t)$ when $t$ tends to $+\infty$ is $c(g,\Delta_t)=c(g,\Delta,f_0)t^{\alpha}(1+o(1))$. \end{corollary} $\mathsf{Proof.}$ Use the notation of the previous proof. Since $c(g,\Delta_t)=g(\Sigma_t)$, $c(g,\Delta,f_0)=\tilde{g}(tr(\Sigma))$ and the arguments of $g$ and $\tilde{g}$ are the same on $\Delta$, the assymptotic formula for $f$ implies the statement of the corollary.\hfill~~\mbox{$\square$} \begin{lemma} \label{bndcrit} Let the hyperplane $H=\{f=0\}$ belongs to the arrangement $A$. Let $\alpha$ be a complex number. Let $\Delta$ be a growing domain of $A$ and $\Delta_t$ the corresponding growing bounded domain of $A_t$. Fix a branch of $f^{\alpha}$ on $\Delta$ and denote it $g$. Then $|f|$ is bounded on $\Delta$ if and only if $tr(\Delta)\subset tr(H)$. The latter condition is equivalent to the equation $h(tr(\Delta))=0$ where $h=f^0/f^0_0$ is a linear function of the arrangement $tr(A)_{H_0}$. Moreover, if $|f|$ is bounded on $\Delta$, then for every big enough $t$, $c(g,\Delta_t)$ equals the constant $c(g,\Delta,f_0)$. \end{lemma} $\mathsf{Proof.}$ $|f|$ is bounded on $\Delta$ if and only if $\Delta$ is placed between two hyperplanes $H'$ and $H''$ parallel to $H$. Denote the domain between these two hyperplanes by $D$. Since $\Delta\subset D$ we have $tr(\Delta)\subset tr(D)=tr(H)$. The reverse part is a consequence of Lemma~\ref{unbcrit}. Let $\Sigma$ be the external support of $\Delta$ with respect to $f$. Then for every big enough $t$, $c(g,\Delta_t)=g(\Sigma)$. The latter equals $c(g,\Delta,f_0)$. \hfill~~\mbox{$\square$} \subsection{ Proof of Theorem~\ref{maintheorem}} We prove Theorem~\ref{maintheorem} applying Theorem~\ref{dt-main} to the arrangement $A_t$ and then passing to the limit when $t\mapsto +\infty$. First study $B(A_t;\alpha,t)$. \begin{lemma} \label{b-asym} $(i)$ The only factor in the numerator of $B(A_t;\alpha,t)$ depending on $t$, when $t$ is big enough, is the factor corresponding to the edge $H_t$. It contributes $\Gamma(t+1)^{\mathrm{\#}},$ where \# is the number of growing domains of $A$. $(ii)$ The factors in the denominator depending on $t$ come from the edges at infinity with non-zero volume which do not lie in $\overline{H_0}$. Each of them, $F$, contributes $\Gamma(t+1+\alpha'(F))^{vol(F)}$, where $\alpha'(F) = \sum_{H\in A;\, F\not\subseteq \overline{H}} \alpha_H$. $(iii)$ The asymptotic behavior of $B(A_t;\alpha,t)$ when $t$ tends to $+\infty$ is given by $$ B(A_t;\alpha,t)=B(A;\alpha;H_0)\prod_{F\in L_{-};\, F\not\subset \overline{H_0}}t^{-\alpha'(F)vol(F)}(1+o(1)).$$ \end{lemma} $\mathsf{Proof.}$ Recall that $$B(A_t;\alpha,t) = \prod_{F\in L_{t+}} \Gamma (\alpha (F) +1)^{vol(F)} \big/ \prod_{F\in L_{t-}} \Gamma (-\alpha (F) +1)^{vol(F)} , $$ $$B(A;\alpha ;H_0) = \prod_{F\in L_{+}} \Gamma (\alpha (F) +1)^{vol(F)} \big/ \prod_{F\in L_{-}; F\subset H_0} \Gamma (-\alpha (F) +1)^{vol(F)} ,$$ where $L_{t-}$, $L_{-}$ denote the set of all edges at infinity of the arrangements $\overline{A_t}$ and $\overline{A}$, respectively, and $L_{t+}$, $L_{-}$ denote the set consisting of all the other edges of the same arrangements. $(i)$ Since the only weight depending on $t$ corresponds to $H_t$, the factors in the denominator that depend on $t$ correspond to the edges of $A_t$ lying in $H_t$. If such an edge $F$ is a proper subspace of $H_t$, then it is decomposable [STV, Section 2], that is the localization of the arrangement $A_t$ at the edge $F$ is a product of two nonempty subarrangements where one of the subarrangements is equal to $\{H_t\}$. According to [STV, Proposition 7], the discrete width of a decomposable edge is zero. Thus its discrete volume is zero. The volume of $H_t$ is the number of bounded domains of the section arrangement $(A_t)_{H_t}$ which is exactly the number of growing domains of the arrangement $A$. $(ii)$ If $F$ is an edge at infinity of $A_t$, then $\alpha(F) = -t-\sum_{i\in I}\alpha_p + \sum_{H\in A_t;\, F\subset \overline{H}}\alpha_H$. The last sum depends on $t$ if and only if $\sum_{H\in A_t;\, F\subset \overline{H}}\alpha_H$ does not depend on $t$, i.e. if and only if $F\not\subset \overline{H_t}$. Since $\overline{H_t}\cap H_{\infty} = \overline{H_0}\cap H_{\infty}$, the weight $\alpha(F)$ depends on $t$ if and only if $F\not\subset \overline{H_0}$. So $\alpha(F)= -t -\alpha'(F)$. Notice that such an edge is also an edge of the arrangement $A$. $(iii)$ According to Theorem~\ref{bijection} the number of growing domains of the arrangement $A$ is equal to the sum of the volumes of all edges at infinity of the arrangement $A$ which do not lie in $\overline{H_0}$. Thus the number of factors in the numerator and in the denominator containing $t$ is equal. Sterling's formula gives us $\Gamma(t+1)/\Gamma(t+1+a)=t^{-a}(1+o(1))$ when $t$ tends to $+\infty$. So we obtain the required formula. \hfill~~\mbox{$\square$} \vskip\baselineskip Now consider the limit of the product of the critical values, $c(A_t;\alpha,t)$. For every $i\in I$ and every bounded or growing domain $\Delta$ of the arrangement $A$, choose a branch of $f_i^{\alpha_i}$ on $\Delta$ and denote it $g_{i,\Delta}$. This also fixes branches of $f_i^{\alpha_i}$ on the bounded domains of $A_t$ independently on $t$. Notice that for every big enough $t$, $\,f_t$ is positive on all bounded domains of the arrangement $A_t$. Choose the positive branch of $(f_t)^t$ on this domains and denote it $g_t$. \begin{lemma} \label{c-asym} $c(A_t;\alpha,t)$ has the following asymptotic behavior when $t$ tends to $+\infty$: $$ c(A_t;\alpha,t)=c(A;\alpha;f_0)\prod_{F\in L_{-}; F\not\subset H_0}t^{\alpha'(F)vol(F)} (1+o(1)).$$ \end{lemma} $\mathsf{Proof.}$ \begin{eqnarray} C(A_t;\alpha;t) & = & \prod_{\Delta\in Ch(A_t)}\left( c(g_t,\Delta) \prod_{i\in I} c(g_{i,\Delta},\Delta)\right) \nonumber\\ & = & \left(\prod_{\Delta\in Ch(A_t)}c(g_t,\Delta)\right) \left(\prod_{\Delta\in Ch(A)}\prod_{i\in I}c(g_{i,\Delta},\Delta)\right) \left(\prod_{\Delta_t}\prod_{i\in I} c(g_{i,\Delta_t},\Delta_t)\right),\label{3prod} \end{eqnarray} where $\Delta_t$ in the last product ranges over the growing bounded domains of $A_t$. Describe the asymptotic behavior of each of the three products in formula~(\ref{3prod}). Assume that $\Delta$ is a bounded domain of $A_t$. Lemma~\ref{tcrit} asserts that $\lim_{t\rightarrow +\infty}c(g_t,\Delta)=e^{-f_0(\Sigma_{\Delta'})}$, where $\Delta'$ is the domain of $A$ (bounded or growing) which corresponds to the domain $\Delta$ of $A_t$ and $\Sigma_{\Delta'}$ is the support face of $f_0$ on $\Delta'$. If $\Delta$ is a bounded domain of $A$ and $i\in I$, then $c(g_{i,\Delta},\Delta)=c(g_{i,\Delta},\Delta,f_0)$ by definition. Let $\Delta_t$ be a growing bounded domain of $A_t$ and $\Delta$ the corresponding growing domain of $A$. Let $i\in I$. If $tr(\Delta)\not\subset \overline{H_i}$, then $c(g_{i,\Delta_t},\Delta_t)=c(g_{i,\Delta},\Delta,f_0)t^{\alpha_i}(1+o(1))$, by Corollary~\ref{unb-cor}. If $tr(\Delta)\subset \overline{H_i}$, then $c(g_{i,\Delta_t},\Delta_t)=c(g_{i,\Delta},\Delta,f_0)$ by Lemma~\ref{bndcrit}. Let $F$ be an edge at infinity. Assume that $F$ does not lie in $\overline{H_0}$ and has a non-zero volume. According to Theorem~\ref{bijection}, there exist exactly $vol(F)$ growing domains of the arrangement $A$, whose traces generate $F$. Every term in the last product of formula~(\ref{3prod}) depends on a growing bounded domain. Collect all the terms such that the trace of the corresponding growing domain generates $F$. Then for the product of the chosen factors we have \begin{eqnarray*} \prod_{\Delta,\, F_{tr(\Delta)}=F}\prod_{i\in I} c(g_{i,\Delta_t},\Delta_t) &=& \left( \prod_{\Delta,\, F_{tr(\Delta)}=F}\prod_{i\in I}c(g_{i,\Delta},\Delta,f_0)\right) \left( \prod_{\Delta,\, F_{tr(\Delta)}=F}\prod_{i,\, F\not\subset H_i} t^{\alpha_i}(1+o(1))\right)\\ &=& \left(\prod_{\Delta,\, F_{tr(\Delta)}=F}\prod_{i\in I} c(g_{i,\Delta},\Delta,f_0)\right) \left(\prod_{\Delta,\, F_{tr(\Delta)}=F} t^{\alpha'(F)}(1+o(1))\right)\\ &=& t^{\alpha'(F)vol(F)}(1+o(1)) \prod_{\Delta,\, F_{tr(\Delta)}=F}\prod_{i\in I} c(g_{i,\Delta},\Delta,f_0) \end{eqnarray*} Collecting the asymptotic behavior for the three products in (\ref{3prod}), we obtain the statement of the lemma.\hfill~~\mbox{$\square$} \vskip\baselineskip {\sf Proof of Theorem~\ref{maintheorem}:} Apply formula~(\ref{dt-formula}) to the weighted arrangement $A_t$. Lemmas~\ref{b-asym} and \ref{c-asym} show that the terms dependent on $t$ in the asymptotic formulas for $B(A_t;\alpha,t)$ and $c(A_t;\alpha,t)$ cancel out. Thus, $$\lim_{t\mapsto +\infty} c(A_t;\alpha,t) B(A_t;\alpha,t)= c(A;\alpha;f_0) B(A;\alpha;H_0),$$ which gives us the right hand side of formula~(\ref{mainformula}). Let us study the entries of the period matrix $\mathsf{PM} (A_t;\alpha,t)$: $\,\mathsf{PM}_{k,j}(t)=\int_{\Delta_j}U_{\alpha,t}\phi_k(A_t)$, where $\Delta_j$ is a bounded domain of $A_t$ and $\phi_k(A_t)$ is one of the $n$--forms constructed in Section~\ref{bnbcbases}. Remind that for a fixed $k$ and a big enough $t$, the form $\phi_k(A_t)$ is independent on $t$ and equals $\phi_k(A;f_0)$. Since $U_{\alpha,t}=(1-f_0/t)^tU_{\alpha}$, we have $\lim_{t\mapsto +\infty} U_{\alpha,t} = e^{-f_0}U_{\alpha}$. Since $\Delta_j$ is a bounded domain of $A_t$, there exists a unique bounded or growing domain, $\Delta$, of $A$ such that $\Delta_j=\Delta\cap \{f_0<t\}$. Extend $f_t$ as zero on $\Delta - \Delta_j$. Then $\mathsf{PM}_{k,j}(t)=\int_{\Delta}(f_t)^tU_{\alpha}\,\phi_k(A;f_0)$. Since $(f_t)^t<e^{-f_0}$ on $\Delta\cap\{f_0>0\}$, Lebesgue's convergence theorem is applicable and $\lim_{t\mapsto +\infty}\mathsf{PM}_{k,j}(t) = \int_{\Delta}e^{-f_0}U_{\alpha}\,\phi_k(A;f_0)= \mathsf{PM}_{k,j}(A;\alpha;f_0)$. These limits give us $\lim_{t\mapsto +\infty}\mathsf{PM} (A_t;\alpha,t) =\mathsf{PM} (A;\alpha;f_0)$. Theorem~\ref{maintheorem} is proved.\hfill~~\mbox{$\square$} \section{Determinant formulas for Selberg type integrals}\label{selberg} Let $z_1<\cdots<z_p$ be real numbers. Let $\alpha_1,\ldots,\alpha_p,\gamma$ be complex numbers with positive real parts. For $t\in \mathbb{R}^n$ define $$ \Phi(t,z) = \prod_{s=1}^p\prod_{i=1}^n(t_i-z_s)^{\alpha_s}\prod_{1\leq i<j\leq n} (t_j-t_i)^{2\gamma}.$$ The branches of $x^{\alpha_s}$ and $x^{2\gamma}$ are fixed by $-\pi/2< \arg\,x < 3\pi/2$ for all $s\in\{1,\ldots,p\}$. Let $\mathcal{Z}_n^p=\{ \mathbf{l}=(l_1,\ldots,l_p)\in \mathbb{Z}^p\,|\, l_i\geq 0,\,\,\,\, l_1+\cdots+l_p=n\}$. For every $s\in\{1,\ldots,p\}$ denote $\mathbf{l}^s=\sum_{i=1}^{s} l_i$, $\mathbf{l}^0=0$. Let $\mathbf{m} \in \mathcal{Z}_n^p$ and $s\in \{1,\ldots,p\}$. Denote $\Gamma_{\mathbf{m},s}$ the set of integers $\{\mathbf{m}^{s-1}+1,\ldots,\mathbf{m}^s\}$ and $d^nt=dt_1\wedge\ldots\wedge dt_n$. Define the following $n$--forms $$\omega_{\mathbf{m}}(t,z) =\left( \sum_{\sigma\in\mathbb{S}^n}\prod_{s=1}^{p}\frac{1}{m_s!} \prod_{j\in\Gamma_{\mathbf{m},s}}\frac{1}{(t_{\sigma_j}-z_s)}\right) d^nt.$$ If $\mathbf{m}\in \mathcal{Z}_n^{p-1}$, then we identify $\mathbf{m}$ with the $p$--tuple $(\mathbf{m},0)\in \mathcal{Z}_n^p$. For $\mathbf{l}\in\mathcal{Z}_n^{p-1}$, let $$\mathbb{U_{\mathbf{l}}}=\{t=(t_1,\ldots,t_n)\in\mathbb{R}^n \,|\,z_s\leq t_{\mathbf{l}^{s-1}+1}\leq\cdots\leq t_{\mathbf{l}^s}\leq z_{s+1} \mbox{ for all } s=1,\ldots,p-1\}.$$ Assume that all domains in the formulas below inherit the standard orientation from $\mathbb{R}^n$. \begin{theorem}\label{no-exp} $\mathrm{,}\,\,\mathrm{cf}\,\,\mathrm{[V6].}$ \begin{eqnarray} \lefteqn{\det\!\left[ \int_{\mathbb{U}_{\mathbf{l}}}\Phi(t,z)\omega_{\mathbf{m}}(t,z) \right]_{\mathbf{l},\mathbf{m}\in\mathcal{Z}_n^{p-1}} = } \label{no-exp-form}\\ & &\prod_{s=0}^{n-1}\left[ \frac{\Gamma((s+1)\gamma)^{p-1}}{\Gamma(\gamma)^{p-1}} \frac{\Gamma(1+\alpha_p+s\gamma)\prod_{j=1}^{p-1}\Gamma(\alpha_j+s\gamma)} {\Gamma(1+\sum_{j=1}^p\alpha_j +(2n-2-s)\gamma)} \right]^{\binom{p+n-s-3}{p-2}}\nonumber\\ & & \exp\Bigl(i\pi\binom{p+n-2}{p-1}\sum_{s=1}^{p} (s-1)\alpha_s\Bigr) \prod_{1\leq a<b\leq p} (z_b-z_a)^{(\alpha_a+\alpha_b) \binom{p+n-2}{p-1}+2\gamma\binom{p+n-2}{p}}. \nonumber \end{eqnarray} \end{theorem} Notice, that formula~(\ref{no-exp-form}) is not symmetric with respect to $\alpha_1,\ldots,\alpha_p$. To make it symmetric we introduce new differential $n$--forms, $\widetilde{\omega_{\mathbf{m}}}$, for $\mathbf{m}\in\mathcal{Z}_n^{p-1}$. Namely \begin{eqnarray*} \widetilde{\omega_{\mathbf{m}}}(t,z) & = & [\prod_{s=1}^{p-1}(m_s!)\alpha_s(\alpha_s+\gamma) \ldots(\alpha_s+(m_s-1)\gamma)]\omega_{\mathbf{m}}\\ & = & \sum_{\sigma\in\mathbb{S}^n}\prod_{s=1}^{p-1} \alpha_s(\alpha_s+\gamma)\ldots(\alpha_s+(m_s-1)\gamma) \prod_{j\in\Gamma_s} \frac{1}{(t_{\sigma_j}-z_s)}d^{n}t. \end{eqnarray*} Theorem~\ref{no-exp} implies \begin{eqnarray} \lefteqn{\det\!\left[ \int_{\mathbb{U}_{\mathbf{l}}}\Phi(t,z)\widetilde{\omega_{\mathbf{m}}}(t,z) \right]_{\mathbf{l},\mathbf{m}\in\mathcal{Z}_n^{p-1}} = }\label{no-exp-form1}\\ & &\prod_{s=0}^{n-1}\left[ \frac{\Gamma((s+1)\gamma+1)^{p-1}}{\Gamma(\gamma+1)^{p-1}} \frac{\prod_{j=1}^{p}\Gamma(\alpha_j+s\gamma+1)} {\Gamma(1+\sum_{j=1}^p\alpha_j +(2n-2-s)\gamma)} \right]^{\binom{p+n-s-3}{p-2}} \nonumber \\ & & \exp\Bigl(i\pi\binom{p+n-2}{p-1}\sum_{s=1}^{p} (s-1)\alpha_s\Bigr) \prod_{1\leq a<b\leq p} (z_b-z_a)^{(\alpha_a+\alpha_b)\binom{p+n-2}{p-1}+2\gamma\binom{p+n-2}{p}}. \nonumber \end{eqnarray} \begin{lemma}\label{eq-crit} For every $\mathbf{l}\in\mathcal{Z}_n^{p-1}$, $i,j \in \{1,\ldots,n\}$, $s\in\{1,\ldots.p\}$, fix branches $g_{j,s}$, $h_{j,i}$ of the multivalued functions $(t_j-z_s)^{\alpha_s}$ and $(t_j-t_i)^{2\gamma}$ ,respectively, on the domain $\mathbb{U}_{\mathbf{l}}$ as at the beginning of the current section. Then the product $$ \exp\Bigl(i\pi\binom{p+n-2}{p-1}\sum_{s=1}^{p} (s-1)\alpha_s\Bigr) \prod_{1\leq a<b\leq p} (z_b-z_a)^{(\alpha_a+\alpha_b)\binom{p+n-2}{p-1}+2\gamma\binom{p+n-2}{p}}$$ equals the product of critical values of the chosen branches \begin{equation}\label{prod-no-exp} \prod_{\mathbf{l}\in\mathcal{Z}_n^{p-1}}\left[ \prod_{j=1}^n\prod_{s=1}^p c(g_{j,s},\mathbb{U}_{\mathbf{l}})\prod_{1\leq i<j \leq n} c(h_{j,i},\mathbb{U}_{\mathbf{l}})\right]. \end{equation} The critical values were defined in Section~\ref{critical}. \end{lemma} Lemma~\ref{eq-crit} allow us to replace the last lines in formulas~(\ref{no-exp-form}) and (\ref{no-exp-form1}) by the product of critical values~(\ref{prod-no-exp}), cf. [V6]. For $\mathbf{l} \in \mathcal{Z}_n^p$, let $z_0=-\infty$ and $$\widetilde{\mathbb{U_{\mathbf{l}}}}=\{t=(t_1,\ldots,t_n)\in\mathbb{R}^n \,|\,z_{s-1}\leq t_{\mathbf{l}^{s-1}+1}\leq\cdots\leq t_{\mathbf{l}^s}\leq z_s \mbox{ for all } s=1,\ldots,p\}.$$ \begin{theorem}\label{expon} Let $a$ be a complex number with positive real part. Then \begin{eqnarray} \lefteqn{\det\!\left[ \int_{\widetilde{\mathbb{U}_{\mathbf{l}}}}\exp\Bigl(a\sum_{j=1}^n t_j\Bigr) \Phi(t,z)\omega_{\mathbf{m}}(t,z) \right]_{\mathbf{l},\mathbf{m}\in\mathcal{Z}_n^p} = (-1)^{n\binom{p+n-1}{p-1}} }\label{expon-form}\\ & & \prod_{s=0}^{n-1}\left[\frac{\Gamma((s+1)\gamma)^p}{\Gamma(\gamma)^p} \prod_{j=1}^p\Gamma(\alpha_j+s\gamma)\right]^{\binom{p+n-s-2}{p-1}} \prod_{1\leq a<b\leq p} (z_b-z_a)^{(\alpha_a+\alpha_b)\binom{p+n-1}{p}+2\gamma\binom{p+n-1}{p+1}} \nonumber \\ & & \exp\Bigl(i\pi\Bigl[\binom{p+n-1}{p}\sum_{s=1}^{p} s\alpha_s\Bigr]\Bigr) \exp\Bigl(a\pi\binom{p+n-1}{p}\sum_{s=1}^{p} z_s\Bigr)\, a^{-\binom{p+n-1}{p}\sum_{s=1}^{p}s\alpha_s-2p\binom{p+n-1}{p+1}\gamma} \nonumber \end{eqnarray} \end{theorem} The next lemma allow us to replace the last line in formula~(\ref{expon-form}) by the product of critical values~(\ref{prod-expon}). \begin{lemma}\label{eq-crit1} For every $\mathbf{l}\in\mathcal{Z}_n^p$, $i,j \in \{1,\ldots,n\}$, $s\in\{1,\ldots.p\}$, fix branches $g_{j,s}$, $h_{j,i}$ of the multivalued functions $(t_j-z_s)^{\alpha_s}$ and $(t_j-t_i)^{2\gamma}$ respectively, on the domain $\widetilde{\mathbb{U}_{\mathbf{l}}}$ as in the beginning of the current section. Then the product $$\exp\Bigl(i\pi[\binom{p+n-1}{p}\sum_{s=1}^{p} s\alpha_s]\Bigr) \exp\Bigl(a\pi\binom{p+n-1}{p}\sum_{s=1}^{p} z_s\Bigr)\, a^{-\binom{p+n-1}{p}\sum_{s=1}^{p}s\alpha_s-2p\binom{p+n-1}{p+1}\gamma}$$ equals the product of critical values of the chosen branches with respect to the linear function $-at_1$. \begin{equation}\label{prod-expon} \prod_{\mathbf{l}\in\mathcal{Z}_n^p}\left[ c(e^{a\sum_{j=1}^nt_j},\widetilde{\mathbb{U}_{\mathbf{l}}}) \prod_{j=1}^n\prod_{s=1}^p c(g_{j,s},\widetilde{\mathbb{U}_{\mathbf{l}}},-at_1)\prod_{1\leq i<j \leq n} c(h_{j,i},\widetilde{\mathbb{U}_{\mathbf{l}}},-at_1)\right]. \end{equation} The critical values were defined in Section~\ref{critical}. \end{lemma} \section{Proofs of Theorem~\ref{no-exp} and Theorem~\ref{expon}} \label{pfselb} Theorem~\ref{no-exp} is a direct corollary of Theorems~5.15 and 7.8 [TV]. The computations are long but straightforward.\hfill~~\mbox{$\square$} The correspondence in notation between the current paper and [TV] is as follows: \newpage \begin{table}[h] \begin{tabular}{l|c|c} Object & current notation & [TV] -- article\\ \hline Dimension of the vector space & n & l\\ Number of points & p & n\\ Coordinates & $t$ & $u$ \\ Weights & $\alpha$ & $2\Lambda/p$ \\ & $\gamma$ & $-1/p$ \\ Points & $z \in \mathbb{R}$ & $y/h=z \in i\mathbb{R}$\\ Parameter & a & $i\eta/p$ \end{tabular} \end{table} \vskip\baselineskip For $\mathbf{l}\in\mathcal{Z}_n^p$, let $z_0=-\infty$ and $$\mathbb{V_{\mathbf{l}}}=\{t=(t_1,\ldots,t_n)\in\mathbb{R}^n \,|\, z_{s-1}\leq t_{j_s}\leq z_{s} \mbox{ for all } s=1,\ldots,p \mbox{ and } j_s\in\Gamma_{\mathbf{l},s} \}.$$ Theorems~5.15 and 7.6 [TV] imply the following formula \begin{eqnarray} \lefteqn{\det\!\left[ \int_{\mathbb{V}_{\mathbf{l}}}\exp\Bigl(a\sum_{j=1}^n t_j\Bigr) \prod_{s=1}^p\prod_{i=1}^n(t_i-z_s)^{\alpha_s}\prod_{1\leq i<j\leq n} (t_i-t_j)^{2\gamma}\omega_{\mathbf{m}}(t,z) \right]_{\mathbf{l},\mathbf{m}\in\mathcal{Z}_n^p} = }\label{rect-exp} \\ & & (-1)^{n\binom{p+n-1}{p-1}} \prod_{\mathbf{l}\in\mathcal{Z}_n^p}\prod_{j=1}^p\prod_{s=1}^{l_j} \frac{\sin(-s\pi\gamma)}{\sin(-\pi\gamma)} \prod_{s=0}^{n-1} \left[\frac{\Gamma((s+1)\gamma)^p}{\Gamma(\gamma)^p} \prod_{j=1}^p\Gamma(\alpha_j+s\gamma)\right]^{\binom{p+n-s-2}{p-1}} \nonumber \\ & & \exp\Bigl(a\pi\binom{p+n-1}{p}\sum_{s=1}^{p} z_s\Bigr)\, a^{-\binom{p+n-1}{p}\sum_{s=1}^{p}s\alpha_s-2p\binom{p+n-1}{p+1}\gamma} \nonumber \\ & & \exp\Bigl(i\pi\Bigr[\binom{p+n-1}{p}\sum_{s=1}^{p} s\alpha_s + p^2\binom{p+n-1}{p+1}\gamma\Bigr]\Bigr) \prod_{1\leq a<b\leq p} (z_b-z_a)^{(\alpha_a+\alpha_b)\binom{p+n-1}{p}+2\gamma\binom{p+n-1}{p+1}}. \nonumber \end{eqnarray} In order to obtain Theorem~\ref{expon} we have to pass from "rectangular" domains $\mathbb{V}_{\mathbf{l}}$ to "triangular" domains $\widetilde{\mathbb{U}_{\mathbf{l}}}$. For any $\mathbf{l},\,\mathbf{m} \in\mathcal{Z}_n^p$ we have \begin{eqnarray*} \lefteqn{\int_{\mathbb{V}_{\mathbf{l}}}\exp\Bigl(a\sum_{j=1}^n t_j\Bigr)\prod_{s=1}^p \prod_{i=1}^n(t_i-z_s)^{\alpha_s}\prod_{1\leq i<j\leq n} (t_i-t_j)^{2\gamma}\omega_{\mathbf{m}}= } \\ & = & \sum_{\sigma\in\mathbb{S}^{l_1}\times\cdots\times\mathbb{S}^{l_p}} \int_{\sigma\widetilde{\mathbb{U}_{\mathbf{l}}}} \exp\Bigl(a\sum_{j=1}^n t_j\Bigr)\prod_{s=1}^p \prod_{i=1}^n(t_i-z_s)^{\alpha_s}\prod_{1\leq i<j\leq n} (t_i-t_j)^{2\gamma}\omega_{\mathbf{m}} \\ & = & e^{n(n-1)i\pi\gamma} [\prod_{j=1}^p1(1+e^{-2\pi i\gamma})\cdots(1+e^{-2\pi i\gamma} +\cdots + e^{-2\pi i \gamma(l_j-1)})]\\ & & \int_{\widetilde{\mathbb{U}_{\mathbf{l}}}} \exp\Bigl(a\sum_{j=1}^nt_j\Bigr)\Phi(t,z)\omega_{\mathbf{m}}(t,z)\\ &=& e^{n(n-1)i\pi\gamma} \prod_{j=1}^p\prod_{s=1}^{l_j}\frac{\sin(-s\pi\gamma)}{\sin(-\pi\gamma)} \prod_{j=1}^p e^{-i\pi\gamma l_j(l_j-1)/2} \int_{\widetilde{\mathbb{U}_{\mathbf{l}}}} \exp\Bigl(a\sum_{j=1}^nt_j\Bigr)\Phi(t,z)\omega_{\mathbf{m}}(t,z) \end{eqnarray*} This proves Theorem~\ref{expon}. \hfill~~\mbox{$\square$}

9709

alg-geom/9709012

en

https://arxiv.org/abs/alg-geom/9709012

alg-geom/9709012

Richard Earl

Richard Earl and Frances Kirwan

The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface

AMS-Latex, 15 pages, no figures

The cohomology of the moduli spaces of stable bundles M(n,d), of coprime rank n and degree d, over a Riemann surface (of genus g > 1) have been intensely studied over the past three decades. We prove in this paper that the Pontryagin ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is strict (i.e. there exists a non-zero element of degree 2n(n-1)(g-1) in Pont(M(n,d)).) This result is a generalisation of a 1967 Newstead conjecture that Pont(M(2,1)) vanished above 4(g-1) (or equivalently that \beta^g =0.) These results have been independently proved by Lisa Jeffrey and Jonathan Weitsman.

alg-geom

\section{Introduction} The cohomology of ${\cal M}(n,d)$, the moduli space of stable holomorphic bundles of coprime rank $n$ and degree $d$ and fixed determinant, over a Riemann surface $\Sigma$ of genus $g \geq 2$, has been widely studied and from a wide range of approaches. Narasimhan and Seshadri \cite{NS} originally showed that the topology of ${\cal M}(n,d)$ depends only on the genus $g$ rather than the complex structure of $\Sigma$. An inductive method to determine the Betti numbers of ${\cal M}(n,d)$ was first given by Harder and Narasimhan \cite{HN} and subsequently by Atiyah and Bott \cite{AB}. The integral cohomology of ${\cal M}(n,d)$ is known to have no torsion \cite{AB} and a set of generators was found by Newstead \cite{N} for $n=2$, and by Atiyah and Bott \cite{AB} for arbitrary $n$. Much work and progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of ${\cal M}(2,1)$ is now known to be complete \cite{K}; recently several authors have found a minimal complete set of relations for the `invariant' subring of the rational cohomology of ${\cal M}(2,1)$ \cite{Z,B,KN,ST}.\\ \indent Unless otherwise stated all cohomology in this paper will have rational coefficients.\\ \indent Let $V$ denote a normalised universal bundle over ${\cal M}(n,d) \times \Sigma$ \cite[p.582]{AB} and define classes \begin{equation} a_{r} \in H^{2r}({\cal M}(n,d)), \hspace{2mm} b_{r}^{k} \in H^{2r-1}({\cal M}(n,d)),\hspace{2mm} f_{r} \in H^{2r-2}({\cal M}(n,d)), \label{gen} \end{equation} for $2 \leq r \leq n$ and $1 \leq k \leq 2g$ by setting \[ c_{r}(V) = a_{r} \otimes 1 + \sum_{k=1}^{2g} b_{r}^{k} \otimes \alpha_{k} + f_{r} \otimes \Omega \indent (2 \leq r \leq n) \] where $\alpha_{1},...,\alpha_{2g}$ is a fixed basis for $H^{1}(\Sigma)$ and $\Omega$ is the standard generator for $H^{2}(\Sigma)$. Atiyah and Bott \cite[Prop. 2.20]{AB} showed that the rational cohomology ring $H^{*}({\cal M}(n,d))$ is generated as a graded algebra by the elements (\ref{gen}).\\ \indent The main results of this paper concern the vanishing of the Pontryagin ring of ${\cal M}(n,d)$ above a non-trivial degree. \begin{thm} \label{Pont} The Pontryagin ring of ${\cal M}(n,d)$ vanishes in degrees strictly greater than $2n(n-1)(g-1).$ \end{thm} \indent The real dimension of ${\cal M}(n,d)$ is $2(n^2-1)(g-1)$ and so Theorem 1 has consequence for $n \geq 2$ and $g \geq 2$. When $n =1$ or $g \leq 1$ the Pontryagin ring of ${\cal M}(n,d)$ is trivial. \begin{thm} \label{nonzero} There exists a non-zero element of degree $2n(n-1)(g-1)$ in the Pontryagin ring of ${\cal M}(n,d)$. \end{thm} \indent When $n=2$, Theorem 1 is the first Newstead-Ramanan conjecture \cite[p.344]{N}. In terms of the generators (\ref{gen}) above the Newstead-Ramanan conjecture is equivalent to showing that \[ (a_{2})^g = 0. \] This was first proved independently by Thaddeus \cite{T} and in \cite[$\S$4]{K}. Subsequently it has been proved in \cite{D, HS, JK, KN, W, WI2}. For the case $n=3$ Theorem 1 was also proved in \cite[$\S$5]{E}.\\ \indent Theorems 1 and 2 have recently been independently proved by Jeffrey and Weitsman \cite{JW} for the arbitrary rank case. When $n>2$ Theorems 1 and 2 are incompatible with a conjecture of Neeman \cite[p.458]{NE} which stated that the Pontryagin ring of ${\cal M}(n,d)$ should vanish in degree $2 g n^2 - 4 g(n-1) +2$ and above.\\[\baselineskip] \indent We shall prove these results by using formulas obtained in \cite{JK,JK2} for the intersection pairings in $H^{*}({\cal M}(n,d))$ between cohomology classes represented as polynomials in the generators $a_{r}, b_{r}^{k},f_{r}$. Knowing the intersection pairings of ${\cal M}(n,d)$ one can of course (in principal) determine the relations amongst the generators of $H^{*}({\cal M}(n,d))$, since by Poincar\'{e} duality an element $\zeta \in H^{*}({\cal M}(n,d))$ of degree $p$ is zero if and only if \[ \int_{{\cal M}(n,d)} \eta \zeta = 0 \] for every $\eta \in H^{*}({\cal M}(n,d))$ of complementary degree $2(n^{2}-1)(g-1) - p$. The results of \cite{JK,JK2} were inspired by Witten's paper \cite{WI2} and use the principle of nonabelian localization introduced in that paper and further developed in \cite{GK,JK3,M}.\\[\baselineskip] \indent The second Newstead-Ramanan conjecture states that the Chern classes of ${\cal M}(2,1)$ also vanish above degree $4(g-1)$. This was first proved geometrically by Gieseker \cite{G} and later by Zagier \cite{Z} using Thaddeus' intersection pairings. In $\S$5 we give explicit (though complicated) formulas for the pairings \[ \int_{{\cal M}(n,d)} \eta \cdot c({\cal M}(n,d))(t) \] of arbitrary $\eta \in H^{*}({\cal M}(n,d))$ with the Chern polynomial $c({\cal M}(n,d))(t)$ of ${\cal M}(n,d)$. When $n=2$ and $d$ is odd, a proof of the second Newstead-Ramanan conjecture may be easily rederived. Computer calculations for low values of $g$ and $n > 2$ suggest that in general the Chern classes of ${\cal M}(n,d)$ vanish above degree $2n(n-1)(g-1)$.\\[\baselineskip] \indent The last of the three Newstead-Ramanan conjectures states that \[ \chi({\cal M}(n,d), T{\cal M}(n,d)) = 3 - 3 g. \] This was proved (for general $n$) by Narasimhan and Ramanan in \cite{NR}. In fact they demonstrated the stronger result that \[ H^{i}({\cal M}(n,d),T{\cal M}(n,d)) = \left\{ \begin{array}{ll} 3g-3 & i=1\\ 0 & i \neq 1. \end{array} \right. \] \newpage \section{Residue formulas for the intersection pairings in $H^{*}({\cal M}(n,d))$} \indent In \cite{JK2} formulas are given for the intersection pairings in $H^{*}({\cal M}(n,d))$ between cohomology classes expressed as polynomials in the Atiyah-Bott generators (\ref{gen}). More precisely the evaluation $\int_{{\cal M}(n,d)} \eta$ of the formal cohomology class \begin{equation} \eta = \exp (f_{2} + \delta_{3} f_{3} + \cdots + \delta_{n} f_{n} ) \prod_{r=2}^{n} \left( (a_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (b_{r}^{k_{r}})^{p_{r,k_{r}}} \right) \label{eta} \end{equation} is considered where \begin{itemize} \item $\delta_{3},...,\delta_{n}$ are formal nilpotent parameters, \item $m_{2},...,m_{n}$ are non-negative integers, \item $p_{r,k_{r}} \in \{0,1\}$ for $2 \leq r \leq n$ and $1 \leq k_{r} \leq 2g$. \end{itemize} Note that each $b_{r}^{k}$ has odd degree and hence $(b_{r}^{k})^{2} = 0$. It is sufficient to consider $\int_{{\cal M}(n,d)} \eta$ for $\eta$ in the form (\ref{eta}); by varying the integers $m_{r}$ and $p_{r,k_{r}}$ and considering the coefficients of the monomials in $\delta_{3}, \ldots , \delta_{n}$ we may obtain the evaluation on the fundamental class $[{\cal M}(n,d)] \in H_{*}({\cal M}(n,d))$ of any polynomial in the generators (\ref{gen}), and hence the intersection pairing \[ \langle \zeta, \xi \rangle = \int_{{\cal M}(n,d)} \zeta \xi \] between any cohomology classes $\zeta,\xi \in H^{*}({\cal M}(n,d))$ expressed as polynomials in these generators.\\ \indent In \cite[Thm. 9.12]{JK2} the evaluation $\int_{{\cal M}(n,d)} \eta$ of $\eta$ on $[{\cal M}(n,d)]$ is equated to an iterated residue of a meromorphic function on the Lie algebra \[ \mbox{\bf \normalshape t} = \{ \mbox{diag}(X_{1},...,X_{n}): X_{1} + \cdots +X_{n} = 0 \} \] of the standard maximal torus $T$ of $SU(n)$. There is a co-ordinate system $(Y_{1},...,Y_{n-1})$ on $\mbox{\bf \normalshape t}$ given by the simple roots $e_{1}, \ldots, e_{n-1}$ of $SU(n)$, i.e. \[ Y_{j} = e_{j}(X) = X_{j} - X_{j+1} \mbox{ for } 1 \leq j \leq n-1, \] and the iterated residue is of the form \[ \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \quad g (Y_{1}, \ldots Y_{n-1}) \] where the variables $Y_{1},...,Y_{j-1}$ are held constant when calculating $\mbox{\normalshape Res}_{Y_{j}=0}$ which is the usual residue at $0$ of a meromorphic function in $Y_{j}$. If we use the Euclidean inner product \[ \langle X,X \rangle = (X_{1})^{2} + \cdots + (X_{n})^{2} \] to identify $\mbox{\bf \normalshape t}$ with its dual $\mbox{\bf \normalshape t}^{*}$ then the simple roots $e_{1},...,e_{n-1}$ correspond to generators \[ \hat{e}_{j} = (0,...,0,1,-1,0,...,0) \] for the integer lattice of $\mbox{\bf \normalshape t}$ (that is, the kernel of the exponential map from $\mbox{\bf \normalshape t}$ to $T$). Let \[ \{\zeta_{j}^{k} : 1 \leq j \leq n-1, \quad 1 \leq k \leq 2g \} \] be the induced basis for $H^{1}(T^{2g})$.\\ \indent For $2 \leq r \leq n$ let $\sigma_{r}$ be the polynomial function on $\mbox{\bf \normalshape t}$ given by the $r$th elementary symmetric function in $X_{1},...,X_{n}$ and for $X \in \mbox{\bf \normalshape t}$ let \[ q(X) = \sigma_{2}(X) + \delta_{3} \sigma_{3}(X) + \cdots + \delta_{n} \sigma_{n}(X) \] where $\delta_{3},...,\delta_{n}$ are the formal nilpotent parameters introduced above. We shall denote by $\mbox{\normalshape d} q_{X}:\mbox{\bf \normalshape t} \to \bold{R}$ the derivative of $q$ at $X \in \mbox{\bf \normalshape t}$, so that \[ \mbox{\normalshape d} q_{X} = \mbox{\normalshape d}(\sigma_{2})_{X} + \delta_{3} \mbox{\normalshape d}(\sigma_{3})_{X} + \cdots + \delta_{n} \mbox{\normalshape d}(\sigma_{n})_{X}. \] The Hessian $\partial^{2}q_{X}$ of $q$ at $X$ is the symmetric bilinear form on $\mbox{\bf \normalshape t}$ given in any co-ordinate system by the matrix of second partial derivatives of $q$ at $X$. Note that as $X_{1} + \cdots + X_{n} =0$ on $\mbox{\bf \normalshape t}$ we have \begin{eqnarray*} 0 = (X_{1} + \cdots +X_{n})^{2} & = & \sum_{j=1}^{n} (X_{j})^{2} + 2 \sum_{i<j} X_{i}X_{j}\\ & = & \langle X, X \rangle + 2 \sigma_{2}(X) \end{eqnarray*} so that \[ \sigma_{2}(X) = - \frac{1}{2} \langle X, X \rangle \] on $\mbox{\bf \normalshape t}$. Then \cite[Thm. 9.12]{JK2} gives us the following formula for $\int_{{\cal M}(n,d)} \eta$. \begin{thm} \label{RES} Let $\eta$ be the formal cohomology class given in (\ref{eta}). Then $\int_{{\cal M}(n,d)} \eta$ equals \[ \frac{(-1)^{n(n-1)(g-1)/2}}{n!} \sum_{w \in W_{n-1}} \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \left[ \frac{ \exp \left\{ \mbox{\normalshape d} q_{X}(w \tilde{c}) \right\} \prod_{r=2}^{n} \sigma_{r}(X)^{m_{r}} }{ {\cal D}(X)^{2g-2} \prod_{j=1}^{n-1} \left( 1 - \exp (\mbox{\normalshape d} q_{X}(\hat{e}_{j})) \right)} \times \right. \] \[ \times \left. \int_{T^{2g}} \exp \left\{ - \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} q_{X}( \hat{e}_{i}, \hat{e}_{j}) \right\} \prod_{r=2}^{n} \prod_{k_{r}=1}^{2g} \left( \sum_{j=1}^{n-1} \mbox{\normalshape d}(\sigma_{r})_{X}(\hat{e}_{j}) \zeta_{j}^{k_{r}} \right)^{p_{r,k_{r}}} \right] \] where \[ {\cal D}(X) = \prod_{i<j}(X_{i}-X_{j}) \] is the product of the positive roots of $SU(n)$, where $\tilde{c} \in \mbox{\bf \normalshape t}$ is the $n$-tuple with $j$th entry \[ \frac{d}{n} - \left[\frac{jd}{n} \right] + \left[\frac{(j-1)d}{n} \right], \] and where $W_{n-1} \cong S_{n-1}$ is the Weyl group of $SU(n-1)$ embedded in $SU(n)$ using the first $n-1$ co-ordinates. \end{thm} \begin{rem} \label{newrem} The formula of Theorem \ref{RES} is obtained by lifting the generators $a_{r},b_{r}^{k}$ and $f_{r}$ of $H^{*}({\cal M}(n,d))$ to $SU(n)$-equivariant cohomology classes $\tilde{a}_{r},\tilde{b}^{k}_{r},\tilde{f}_{r}$ on an `extended moduli space' (see \cite[$\S$4]{JK2}) with a Hamiltonian $SU(n)$-action whose symplectic quotient is ${\cal M}(n,d)$, and then using `nonabelian localisation' \cite{JK2,WI} to localise to components of the fixed point set of the maximal torus $T$. The restrictions of $\tilde{a}_{r},\tilde{b}^{k}_{r},\tilde{f}_{r}$ to these components, which are indexed by elements $\Lambda_{0}$ in the integer lattice of $\mbox{\bf \normalshape t}$, and which can be identified with copies of $T^{2g}$, are \[ \sigma_{r}(X), \indent \sum_{j=1}^{n-1} \mbox{\normalshape d} (\sigma_{r})_{X}(\hat{e}_{j}) \zeta_{j}^{k} \] and \[ \mbox{\normalshape d} (\sigma_{r})_{X}(\tilde{c} + \Lambda_{0}) + \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} (\sigma_{r})_{X}(\hat{e}_{i},\hat{e}_{j}) \] respectively. As the extended moduli space is not compact, nonabelian localisation cannot be applied directly, but instead one can exploit the periodicity of the exponential map and the extended moduli space's close relative \[ M_{\T}(c) = \{ (h_{1},...,h_{2g},\Lambda) \in SU(n)^{2g} \times \mbox{\bf \normalshape t} : \prod_{j=1} [h_{j},h_{j+g}] = c \exp(\Lambda) \} \] where \[ c = \exp(2 \pi i \tilde{c}) = e^{2 \pi i d / n} \mbox{diag}(1,1,\cdots ,1 ) \in SU(n). \] By \cite[Lemma 4.5]{JK2}, for any $\Lambda_{0}$ in the integer lattice in $\mbox{\bf \normalshape t}$ there is a homeomorphism $s_{\Lambda_{0}}: M_{\T}(c) \to M_{\T}(c)$ defined by \[ s_{\Lambda_{0}}(h,\Lambda) = (h, \Lambda + \Lambda_{0}). \] If as in \cite[$\S$ 3]{JK2} we represent $T$-equivariant cohomology classes on a manifold $M$, acted on by the torus $T$, by polynomial functions on the Lie algebra $\mbox{\bf \normalshape t}$ of $T$ with values in the De Rham complex of differential forms on $M$, then when $X$ is the co-ordinate on $\mbox{\bf \normalshape t}$, by (4.8) and (4.9) in \cite{JK2} we have \begin{eqnarray*} s^{*}_{\Lambda_{0}}(\tilde{a}_{r})(X) & = & \tilde{a}_{r}(X),\\ s^{*}_{\Lambda_{0}}(\tilde{b}_{r}^{k})(X) & = & \tilde{b}_{r}^{k}(X), \end{eqnarray*} and by \cite[Lemma 9.9]{JK2} \begin{equation} \label{three} s^{*}_{\Lambda_{0}} ( \tilde{f}_{2} + \sum_{r=3}^{n} \delta_{r} \tilde{f}_{r} )(X) = (\tilde{f}_{2} + \sum_{r=3}^{n} \delta_{r} \tilde{f}_{r} )(X) + \mbox{\normalshape d} \bar{q}_{X}(\Lambda_{0}) \label{cross} \end{equation} where \[ \bar{q}(X) = \sigma_{2}(X) + \sum_{r=3}^{n} (-1)^r \delta_{r} \sigma_{r}(X). \] (Note that $\bar{q}$ was denoted by $q_{0}$ in \cite{JK2}.)\\ \indent Thus the result of applying $s^{*}_{\Lambda_{0}}$ to the representative \[ \tilde{\eta} = \exp (\tilde{f}_{2} + \delta_{3} \tilde{f}_{3} + \cdots + \delta_{n} \tilde{f}_{n} ) \prod_{r=2}^{n} \left( (\tilde{a}_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (\tilde{b}_{r}^{k_{r}})^{p_{r,k_{r}}} \right) \] of $\eta$ is \begin{equation} \label{newchange} s^{*}_{\Lambda_{0}}(\tilde{\eta}) = \tilde{\eta} \exp ( \mbox{\normalshape d} \bar{q}_{X}(\Lambda_{0})). \end{equation} \indent Theorem \ref{RES} is now proved in \cite{JK2} using a version of nonabelian localisation due to Martin \cite{M} and Guillemin and Kalkman \cite{GK} which can be made to work in noncompact settings. First one reduces to working with the symplectic quotient of the extended moduli space by $T$ instead of by $SU(n)$ via the arguments of \cite{M}. Then one compares the integral over the $T$-quotient of the cohomology class induced by $\tilde{\eta}$ with the integral of the class induced by $s^{*}_{\Lambda_{0}}$, for $\Lambda_{0} = \hat{e}_{p}$ with $1 \leq p \leq n-1$, first using (\ref{three}) and secondly using nonabelian localisation as in \cite{GK} and \cite{M}. \end{rem} \begin{rem} \label{epsilon} The cohomology class $nf_{2}$ is represented by a symplectic form $\omega$ on ${\cal M}(n,d)$. If we replace $\omega$ by any non-zero scalar multiple $\epsilon\omega$ then the proof of \cite[Thm. 9.12]{JK2} shows that \[ \int_{{\cal M}(n,d)}\exp (\epsilon f_{2} + \delta_{3} f_{3} + \cdots + \delta_{n} f_{n} ) \prod_{r=2}^{n} \left( (a_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (b_{r}^{k_{r}})^{p_{r,k_{r}}} \right) \] is given by the same iterated residue as in Theorem \ref{RES} above except that $q(X)$ is replaced by \[ q^{(\epsilon)}(X) = \epsilon\sigma_{2}(X) + \delta_{3} \sigma_{3}(X) + \cdots + \delta_{n}\sigma_{n}(X) \] (see remarks 8.3 and 9.13 in \cite{JK2}). If we also multiply the formal parameters $\delta_{3},...,\delta_{n}$ by $\epsilon$ then we find that \[ \int_{{\cal M}(n,d)}\exp (\epsilon f_{2} + \epsilon\delta_{3} f_{3} + \cdots + \epsilon \delta_{n} f_{n} ) \prod_{r=2}^{n} \left( (a_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (b_{r}^{k_{r}})^{p_{r,k_{r}}} \right) \] is given by the same iterated residue as in Theorem \ref{RES} except that $q(X)$ is now replaced by $\epsilon q(X)$. \end{rem} \newpage \section{The Pontryagin Ring: Proof of Theorem \ref{Pont}} Any symmetric polynomial in $X_{1},...,X_{n}$ can be expressed as a polynomial in the elementary symmetric polynomials $\sigma_{1} ,...,\sigma_{n}$. Since $\sigma_{1}(X) = X_{1} + \cdots + X_{n}$ vanishes on $\mbox{\bf \normalshape t}$, any polynomial function on $\mbox{\bf \normalshape t}$ which is symmetric in $X_{1},...,X_{n}$ (or equivalently invariant under the action of the Weyl group of $SU(n)$) can be expressed as a polynomial $p(\sigma_{2}(X),\ldots,\sigma_{n}(X))$ and then represents the cohomology class $p(a_{2},...,a_{n})$ on ${\cal M}(n,d)$. \begin{prop} \label{Pontgen} The Pontryagin ring of ${\cal M}(n,d)$ is generated by the polynomials in $a_{2},...,a_{n}$ represented by the elementary symmetric polynomials in \[ \{ (X_{i} - X_{j})^{2} : 1 \leq i < j \leq n \}. \] \end{prop} {\bf Proof:} See \cite[Lemma 17]{E}.\\[\baselineskip] Hence Theorem 1 is an immediate corollary of \begin{thm} \label{alt} The subring of $H^{*}({\cal M}(n,d))$ generated by $a_{2},...,a_{n}$ vanishes in all degrees strictly greater than $2n(n-1)(g-1).$ \end{thm} Since ${\cal M}(n,d)$ is a compact manifold, by Poincar\'{e} duality Theorem \ref{alt} is itself an immediate corollary of \begin{prop} \label{main} Let $\eta$ be as given in (\ref{eta}). If $m_{2},...,m_{n}$ are non-negative integers such that \[ \mbox{\normalshape deg} \prod_{r=2}^{n} (a_{r})^{m_{r}} = \sum_{r=2}^{n} 2 r m_{r} > 2n(n-1)(g-1) \] then $\int_{{\cal M}(n,d)} \eta = 0$. \end{prop} {\bf Proof:} For $\epsilon \in {\bold R}$ let \[ G(\epsilon) = \int_{{\cal M}(n,d)} \exp (\epsilon f_{2} + \epsilon \delta_{3} f_{3} + \cdots + \epsilon \delta_{n} f_{n} ) \prod_{r=2}^{n} \left( (a_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (b_{r}^{k_{r}})^{p_{r,k_{r}}} \right). \] We shall prove that $G(\epsilon) = 0$ for all $\epsilon \in {\bold R}$; the result will then follow by taking $\epsilon =1$.\\ \indent First notice that \[ \mbox{deg } \left( \prod_{r=2}^{n} \left( (a_{r})^{m_{r}} \prod_{k_{r}=1}^{2g} (b_{r}^{k_{r}})^{p_{r,k_{r}}} \right) \right) = \sum_{r=2}^{n} 2 r m_{r} + \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} (2 r -1) p_{r,k_{r}}. \] Also $f_{r}$ has degree $2r-2$, which is at least 2 for $2 \leq r \leq n$, and the real dimension of ${\cal M}(n,d)$ is $2(n^{2}-1)(g-1)$. Thus $G(\epsilon)$ is a polynomial in $\epsilon$ of degree at most \[ \frac{1}{2} \left( 2(n^{2}-1)(g-1) - \sum_{r=2}^{n} 2 r m_{r} - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} (2 r -1) p_{r,k_{r}} \right). \] On the other hand Theorem \ref{RES} and Remark \ref{epsilon} show that $G(\epsilon)$ is a non-zero $\epsilon$-independent constant multiple of \[ \sum_{w \in W_{n-1}} \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \left[ \frac{ \exp \left\{ \epsilon \mbox{\normalshape d} q_{X}(w\tilde{c}) \right\} \prod_{r=2}^{n} \sigma_{r}(X)^{m_{r}} }{ {\cal D}(X)^{2g-2} \prod_{j=1}^{n-1} \left(1 - \exp (\epsilon \mbox{\normalshape d} q_{X}(\hat{e}_{j})) \right)} \times \right. \] \[ \times \left. \int_{T^{2g}} \exp \left\{ -\epsilon \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} q_{X}( \hat{e}_{i}, \hat{e}_{j}) \right\} \prod_{r=2}^{n} \prod_{k_{r}=1}^{2g} \left( \sum_{j=1}^{n-1} \mbox{\normalshape d}(\sigma_{r})_{X}(\hat{e}_{j}) \zeta_{j}^{k_{r}} \right)^{p_{r,k_{r}}} \right]. \] Now \[ q(X) = \sigma_{2}(X) + \delta_{3} \sigma_{3}(X) + \cdots + \delta_{n} \sigma_{n}(X) \] on $\mbox{\bf \normalshape t}$ where $\sigma_{2}(X) = - \langle X,X \rangle /2$ and $\delta_{3},...,\delta_{n}$ are formal nilpotent parameters. Since $\hat{e}_{j} \in \mbox{\bf \normalshape t}$ corresponds, under the identification of $\mbox{\bf \normalshape t}^{*}$ with $\mbox{\bf \normalshape t}$ defined by the inner product, to the simple root $e_{j} \in \mbox{\bf \normalshape t}^{*}$ given by $e_{j}(X) = X_{j} - X_{j+1} = Y_{j},$ we have $\mbox{\normalshape d}(\sigma_{2})_{X}(\hat{e}_{j}) = -Y_{j}$ and hence \[ \mbox{\normalshape d} q_{X}(\hat{e}_{j}) = - Y_{j} + N_{j} \] where \[ N_{j} = \delta_{3} \mbox{\normalshape d}(\sigma_{3})_{X}(\hat{e}_{j})+ \cdots + \delta_{n} \mbox{\normalshape d}(\sigma_{n})_{X}(\hat{e}_{j}) \] is nilpotent. Moreover \[ T(x) = \frac{x}{e^{x}-1} \] may be expressed as a formal power series in $x$ and then \begin{eqnarray*} \frac{1}{\exp(\epsilon \mbox{\normalshape d} q_{X}(\hat{e}_{j})) - 1} & = & \frac{1}{\exp(\epsilon N_{j} - \epsilon Y_{j}) -1}\\ & = & \frac{T(\epsilon N_{j} - \epsilon Y_{j})}{\epsilon N_{j} - \epsilon Y_{j}}\\ & = & - \frac{T(\epsilon N_{j} - \epsilon Y_{j})}{\epsilon Y_{j}} \sum_{m=0}^{\infty} \left( \frac{N_{j}}{Y_{j}} \right)^{m} \end{eqnarray*} The generators $\{ \zeta_{j}^{k} : 1 \leq j \leq n-1, 1 \leq k \leq 2g \}$ for $H^{1}(T^{2g})$ are of degree 1 and hence anticommute, and all have square 0. Their cup product is non-zero and hence spans the top cohomology group $H^{2(n-1)g}(T^{2g})$. Thus as a function of $\epsilon$ the integral \[ \int_{T^{2g}} \exp \left\{ -\epsilon \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} q_{X}( \hat{e}_{i}, \hat{e}_{j}) \right\} \prod_{r=2}^{n} \prod_{k_{r}=1}^{2g} \left( \sum_{j=1}^{n-1} \mbox{\normalshape d}(\sigma_{r})_{X}(\hat{e}_{j}) \zeta_{j}^{k_{r}} \right)^{p_{r,k_{r}}} \] is a polynomial in $\epsilon$ which is divisible by $\epsilon$ to the power \[ \frac{1}{2} \left( \dim T^{2g} - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} \right). \] Thus $G(\epsilon)$ is of the form \[ \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \left( \frac{F(X,\epsilon)}{\cal{D}(X)^{2g-2} \prod_{j=1}^{n-1} (\epsilon Y_{j})} \right) \] where, for $X \in \mbox{\bf \normalshape t}$ and $\epsilon$ any real number, $F(X, \epsilon)$ is a formal power series in $\epsilon$ and a formal Laurent series in the co-ordinates $Y_{1},...,Y_{n-1}$ on $\mbox{\bf \normalshape t}$, and $F(X,\epsilon)$ is divisible by $\epsilon$ raised to the power \[ \frac{1}{2} \left( 2(n-1)g - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} \right). \] Therefore $G(\epsilon)$ is a formal power series in $\epsilon$ which is divisible by $\epsilon$ raised to the power \[ \frac{1}{2} \left( 2(n-1)g - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} \right) - (n-1) = \frac{1}{2} \left( 2(n-1)(g-1) - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} \right) \] provided this is positive. On the other hand we saw earlier that $G(\epsilon)$ is a polynomial in $\epsilon$ of degree at most \[ \frac{1}{2} \left( 2(n^{2}-1)(g-1) - \sum_{r=2}^{n} 2 r m_{r} - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} (2 r -1) p_{r,k_{r}} \right). \] Hence $G(\epsilon)$ must be identically zero unless \[ 2(n-1)(g-1) - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} \leq 2(n^{2}-1)(g-1) - \sum_{r=2}^{n} 2 r m_{r} - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} (2 r -1) p_{r,k_{r}}. \] By hypothesis \[ \sum_{r=2}^{n} 2 r m_{r} > 2 n (n-1) (g-1). \] So as $2r-1 \geq 1$ when $r \geq 1$ we have \[ 2(n-1)(g-1) - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} p_{r,k_{r}} > 2(n^{2}-1)(g-1) - \sum_{r=2}^{n} 2 r m_{r} - \sum_{r=2}^{n} \sum_{k_{r}=1}^{2g} (2 r -1) p_{r,k_{r}} \] and hence $G(\epsilon)$ is identically zero. This completes the proof of proposition \ref{main} and hence of Theorems \ref{Pont} and \ref{alt}. \section{The Pontryagin Ring: Proof of Theorem \ref{nonzero}} By Proposition \ref{Pontgen} any symmetric polynomial in \[ \{(X_{i} - X_{j})^{2}: 1 \leq i < j \leq n\} \] represents an element of the Pontryagin ring of ${\cal M}(n,d)$. In particular the polynomial \[ {\cal D}(X)^{2g-2} = \prod_{i<j}(X_{i}-X_{j})^{2g-2} \] represents an element $\eta_{0}$ of degree $2n(n-1)(g-1)$ in the Pontryagin ring of ${\cal M}(n,d)$. Thus Theorem \ref{nonzero} follows from the following proposition. \begin{prop} \label{eta0} If $\eta_{0} \in H^{*}({\cal M}(n,d))$ is represented by ${\cal D}(X)^{2g-2}$ then \[ \int_{{\cal M}(n,d)} \eta_{0} \exp f_{2} \neq 0. \] \end{prop} {\bf Proof:} By Theorem \ref{RES} $\int_{{\cal M}(n,d)} \eta_{0} \exp f_{2}$ is a non-zero constant multiple of \[ \sum_{w \in W_{n-1}} \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \left[ \frac{ \exp \left\{ \mbox{\normalshape d} (\sigma_{2})_{X}(w\tilde{c}) \right\} }{ \prod_{j=1}^{n-1} \left( 1 - \exp (\mbox{\normalshape d} (\sigma_{2})_{X}(\hat{e}_{j})) \right)} \times \right. \] \[ \times \left. \int_{T^{2g}} \exp \left\{ - \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} (\sigma_{2})_{X}( \hat{e}_{i}, \hat{e}_{j}) \right\} \right] \] Now $\sigma_{2}$ is a quadratic form on $\mbox{\bf \normalshape t}$ so $\mbox{\normalshape d} (\sigma_{2})_{X}$ is linear in $X \in \mbox{\bf \normalshape t}$ and $\partial^{2}(\sigma_{2})_{X}$ is independent of $X$. Indeed we have already observed in the proof of Proposition \ref{main} that $\sigma_{2}(X) = - \langle X,X \rangle /2$ so that \[ \mbox{\normalshape d}(\sigma_{2})_{X}(\hat{e}_{j}) = - Y_{j} \] and \[ \mbox{\normalshape d} (\sigma_{2})_{X}(w\tilde{c}) = \beta_{1}(w)Y_{1} + \beta_{2}(w) Y_{2} + \cdots + \beta_{n-1}(w) Y_{n-1} \] for constants $\beta_{j}(w)$, while \[ \partial^{2}(\sigma_{2})_{X}(\hat{e}_{i},\hat{e}_{j}) = \left\{ \begin{array}{rl} -2 & \mbox{ if } i = j,\\ 1 & \mbox{ if } i-j = \pm 1,\\ 0 & \mbox{ if } |i-j|>1. \end{array} \right. \] Since $\{ \hat{e}_{1},...,\hat{e}_{n-1} \}$ is a basis for $\mbox{\bf \normalshape t}$ and since $H^{*}(T^{2g})$ is a free exterior algebra on \[ \{\zeta_{j}^{k}: 1 \leq j \leq n-1, 1 \leq k \leq 2g\} \] it follows that \[ \int_{T^{2g}} \exp \left\{ - \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \right\} \] is a non-zero constant independent of $X \in \mbox{\bf \normalshape t}$, which in fact equals one. Hence \[ \int_{T^{2g}} \exp \left\{ - \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} (\sigma_{2})_{X}( \hat{e}_{i}, \hat{e}_{j}) \right\} \] equals \[ (\det \{ \partial^{2} (\sigma_{2})_{X}( \hat{e}_{i}, \hat{e}_{j}) \})^{g} \int_{T^{2g}} \exp \left\{ - \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \right\} = (-1)^{(n-1)g}n^{g}. \] \indent Thus $\int_{{\cal M}(n,d)} \eta_{0} \exp f_{2}$ is \[ (-1)^{(n-1)g}n^{g} \sum_{w \in W_{n-1}} \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \prod_{j=1}^{n-1} \left( \frac{ \exp\{\beta_{j}(w) Y_{j}\}}{1 - \exp\{-Y_{j}\} } \right) = (-1)^{(n-1)g}n^{g} (n-1)! \] and the result follows. \indent $\Box$ \newpage \section{Chern Classes} The second Newstead-Ramanan conjecture states that the Chern classes $c_{r}({\cal M}(2,1))$ vanish for $r > 2(g-1)$. This was originally proved by Gieseker \cite{G} using a degeneration of the moduli space and subsequently by Zagier \cite{Z} using Thaddeus' pairings.\\ \indent The tangent bundle of ${\cal M}(n,d)$ equals \cite[p.582]{AB} \[ 1 - \pi_{!}(\mbox{End} V) \] where $\pi: {\cal M}(n,d) \times \Sigma \to {\cal M}(n,d)$ is the first projection and $V$ is a universal bundle over ${\cal M}(n,d) \times \Sigma$. In \cite[Prop. 10]{E} an expression for the Chern character of $V$ was found in terms of the generators (\ref{gen}), and hence using the Grothendieck-Riemann-Roch theorem an expression was determined for $\mbox{\normalshape ch}({\cal M}(n,d))$ in terms of these generators. If as in Remark \ref{newrem} we lift these generators, and thus $\mbox{\normalshape ch}({\cal M}(n,d))$, to $SU(n)$-equivariant cohomology classes $\tilde{a}_{r},\tilde{b}_{r}^{k},\tilde{f}_{r}$ on the extended moduli space and then restrict to the component indexed by $\Lambda_{0} \in \mbox{\bf \normalshape t}$ of the fixed point set of $T$, the result for $\mbox{\normalshape ch}({\cal M}(n,d))$ is \[ (1-g) + \sum_{i=1}^{n} \sum_{j=1}^{n} e^{X_{i}-X_{j}} (g-1 + \omega_{j} - \omega_{i}) - \sum_{1 \leq i < j \leq n} \sum_{k=1}^{g} Z_{i,j}^{k} Z_{i,j}^{k+g} (e^{X_{i}-X_{j}} + e^{X_{j}-X_{i}}) \] where $\tilde{c} + \Lambda_{0} = (\omega_{1},...,\omega_{n}),$ and \[ Z_{i,j}^{k} = - \zeta_{i-1}^{k} + \zeta_{i}^{k} + \zeta_{j-1}^{k} - \zeta_{j}^{k}, \] with the understanding that $\zeta_{0}^{k} = \zeta_{n}^{k} = 0$. Hence by \cite[Lemma 9]{E} the restriction of the lift of the Chern polynomial $c({\cal M}(n,d))(t) = \sum_{r \geq 0} c_{r}({\cal M}(n,d)) t^{r}$, which we shall denote by $\tilde{c}({\cal M}(n,d))(t)$, equals \[ \prod_{i<j} (1+(X_{i} - X_{j})t)^{g-1+\omega_{j}-\omega_{i}} (1+(X_{j} - X_{i})t)^{g-1+\omega_{i}-\omega_{j}} \exp \left\{ \frac{-2t \sum_{k=1}^{g} Z_{i,j}^{k} Z_{i,j}^{k+g}}{1 - (X_{i}-X_{j})^{2} t^{2}} \right\}. \] \indent The formulas in Remark \ref{newrem} show that when $\Lambda_{0} = \hat{e}_{p}$, \[ s_{\Lambda_{0}}^{*}(\tilde{c}({\cal M}(n,d))(t)) = V_{p}(X,t) \cdot \tilde{c}({\cal M}(n,d))(t) \] where \[ V_{p}(X,t) = \prod_{q=1}^{n} \frac{(1+(X_{q}-X_{p})t)(1+(X_{p+1}-X_{q})t)}{(1+(X_{p}-X_{q})t)(1+(X_{q}-X_{p+1})t)}. \] The proof of \cite[Thm. 9.12]{JK2} (see Remark \ref{newrem} above) with this formula replacing \cite[Lemma 9.9]{JK2} (that is equations (\ref{three}) and (\ref{newchange}) in Remark \ref{newrem}) shows that the pairing $\int_{{\cal M}(n,d)} \eta \cdot c({\cal M}(n,d))(t)$ equals $(-1)^{n(n-1)(g-1)/2}/n!$ times \begin{eqnarray*} \sum_{w \in W_{n-1}} \mbox{\normalshape Res}_{Y_{1}=0} \cdots \mbox{\normalshape Res}_{Y_{n-1}=0} \left\{ \frac{\exp \left\{ \mbox{\normalshape d} q_{X}(w\tilde{c}) \right\} \prod_{r=2}^{n} \sigma_{r}(X)^{m_{r}}}{ {\cal D}(X)^{2g-2} \prod_{p=1}^{n-1} \left( 1 - \exp ( \mbox{\normalshape d} q_{X}(\hat{e}_{p})) V_{p}(X,t) \right)} \times \right. \\ \nonumber \times \left( \prod_{p \neq q} (1+(X_{p}-X_{q})t)^{g-1+\tilde{c}_{q}-\tilde{c}_{p}} \right) \int_{T^{2g}} \left [ \exp \left( \sum_{i<j} \frac{ - 2 t \sum_{k=1}^{g} Z_{i,j}^{k} Z_{i,j}^{k+g} }{1 - (X_{i}-X_{j})^{2} t^{2}} \right) \right. \times \\ \nonumber \times \left. \left. \exp \left( \sum_{i,j=1}^{n-1} \sum_{k=1}^{g} \zeta_{i}^{k} \zeta_{j}^{k+g} \partial^{2} q_{X}(\hat{e}_{i}, \hat{e}_{j}) \right) \prod_{r=2}^{n} \prod_{k_{r}=1}^{2g} \left( \sum_{j=1}^{n-1} \mbox{\normalshape d} (\sigma_{r})_{X}(\hat{e}_{j}) \zeta_{j}^{k_{r}} \right)^{p_{r,k_{r}}} \right] \right\}. \end{eqnarray*} \newpage When $n=2$ it readily follows that the resulting expression is a polynomial in $t$ of degree at most $2g-2$ (see below). When $n \geq 3$, computer calculations for low values of $n$ and $g$ suggest the following (though we have thus far been unable to give proofs): \begin{itemize} \item $c_{r}({\cal M}(n,d))=0$ for $r > n(n-1)(g-1)$. \item $c_{n(n-1)(g-1)}({\cal M}(n,d))$ is a non-zero multiple of $\eta_{0}$, the cohomology class represented by ${\cal D}(X)^{2g-2}$ (see Proposition \ref{eta0}). \end{itemize} For $n=2$ the results of Gieseker and Zagier may be duplicated as follows. From \cite[p.144]{T} we need only consider those $\eta$ which are invariant under the induced action of the mapping class group, so for $\lambda \in {\bold C}$ let \[ \eta = (a_{2})^{r} \exp \left\{ f_{2} + \lambda \sum_{k=1}^{g} b_{2}^{k} b_{2}^{k+g} \right\}. \] For simplicity set $Y=Y_{1}$ and define \[ F(k,s) = \mbox{\normalshape Res}_{Y=0} \left( \frac{[(1 - \lambda Y^{2}/2)(1-Y^{2}t^{2}) + 4 t]^{k}}{Y^{2s} (e^{Y/2} (1+Y t)^{2} - e^{-Y/2} (1-Y t)^2)} \right). \] Simplifying, we see that $\int_{{\cal M}(2,1)} \eta c({\cal M}(2,1))(t)$ equals $(-1)^{g-1-r}2^{g-1-2r}F(g,g-1-r)$. We now claim that $F(k,s)$ is a rational function in $t$ of total degree at most $k+s-1$. We can show by a simple induction that if $F(k,s)$ has total degree at most $k+s-1$ for $k \leq K$ then $F(K+1,s)$ has total degree at most $K+s$. So consider $F(0,s)$. We may write \[ e^{Y/2} (1+Y t)^{2} - e^{-Y/2} (1-Y t)^2 = (1+ 4 t)Y \left( 1 + \sum_{i=1}^{\infty} r_{i}(t) Y^{2i} \right) \] where $r_{i}(t)$ is a rational function in $t$ of total degree 1. Hence $F(0,s)$ is the coefficient of $Y^{2s}$ in \[ \frac{1}{(1+ 4 t)} \sum_{j=1}^{\infty} \left( - \sum_{i=1}^{\infty} r_{i}(t) Y^{2i} \right)^{j} \] which is a rational function in $t$ of degree at most $s-1$, thus proving the claim.\\ \indent Therefore $c({\cal M}(2,1))(t)$ is a polynomial in $t$ of degree at most $2g-2$. It is in fact the case that the Chern polynomial is of precisely this degree and we may find an expression for $c_{2g-2}({\cal M}(2,1))$ as follows.\\ \indent Let $G(k,s)$ equal $F(k,s)$ modulo rational functions of degree strictly less than $k+s-1$. Then \[ G(0,s) = \frac{(-r_{1}(t))^{s}}{1+4t} = (-1)^{s} \frac{(t^2 +t/2 + 1/24)^{s}}{(1+4t)^{s+1}} = (-1)^{s} 2^{-2s-2} t^{s-1}, \] and from the recurrence relation \[ G(k+1,s) = 4 t G(k,s) - t^{2} G(k,s-1) \] we obtain \[ G(k,s) = (-1)^{s} 2^{3k - 2s -2} t^{k+s-1} \] for $k \leq s.$ Using the above recurrence relation again we find \[ G(s+1,s) = (-1)^{s} 2^{s} t^{2s} - t^{2} G(s,s-1) = (-1)^{s} t^{2s} \sum_{i=0}^{s} 2^{i} = (-1)^{s} (2^{s+1}-1) t^{2s}. \] Hence the coefficient of $t^{2g-2}$ in $\int_{{\cal M}(2,1)} \eta c({\cal M}(2,1))(t)$ equals $2^{g-1} (2^{g}-1)$.\\ \indent Let $\eta_{0}= (- 4 a_{2})^{g-1}$ be the class represented by ${\cal D}(X)^{2g-2} = Y^{2g-2}$. Then \[ \int_{{\cal M}(2,1)} \eta_{0}\eta =\frac{(-1)^{g-1}}{2} \mbox{\normalshape Res}_{Y=0} \left[ \frac{ e^{-Y/2}}{1-e^{-Y}} \int_{T^{2g}} \exp \left\{- 2 \sum_{k=1}^{g} \zeta_{1}^{k} \zeta_{1}^{k+g} \right\} \right] = (-2)^{g-1}. \] By Poincar\'{e} duality we find (cf. \cite[p.555]{Z}) \[ c_{2g-2}({\cal M}(2,1)) = (-1)^{g-1} (2^{g}-1) \eta_{0} = 2^{2g-2}(2^{g}-1) (a_{2})^{g-1}. \] \newpage

9709

alg-geom/9709034

en

https://arxiv.org/abs/alg-geom/9709034

alg-geom/9709034

Frank Sottile

Nantel Bergeron (York University, Toronto) and Frank Sottile (University of Toronto)

Skew Schubert functions and the Pieri formula for flag manifolds

24 pages, LaTeX 2e, with epsf.sty

Trans. Amer. Math. Soc., 354 No. 2, (2002), 651-673

10.1090/S0002-9947-01-02845-8

MSRI 1997-096

We show the equivalence of the Pieri formula for flag manifolds and certain identities among the structure constants, giving new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric function, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtaining a new combinatorial construction of Schubert polynomials.

alg-geom

\section*{Introduction} A fundamental open problem in the theory of Schubert polynomials is to find an analog of the Littlewood-Richardson rule. By this, we mean a bijective description of the structure constants for the ring of polynomials with respect to its basis of Schubert polynomials. Such a rule would express the intersection form in the cohomology of a flag manifold in terms of its basis of Schubert classes. Other than the classical Littlewood-Richardson rule (when the Schubert polynomials are Schur symmetric polynomials) little is known. Using geometry, Monk~\cite{Monk} established a formula for multiplication by linear Schubert polynomials (divisor Schubert classes). A Pieri-type formula for multiplication by an elementary or complete homogeneous symmetric polynomial (special Schubert class) was given in~\cite{LS82a}, but only recently proven~\cite{Sottile96} using geometry. There are now several proofs~\cite{C-F,Postnikov,Winkel_multiplication,Veigneau}, some of which~\cite{Postnikov,Winkel_multiplication,Veigneau} are purely combinatorial. In the more general setting of multiplication by a Schur symmetric polynomial, formulas for some structure constants follow from a family of identities which were proven using geometry~\cite{BS97a}. Also in ({\em ibid.}) are combinatorial results about intervals in the Bruhat order which are formally related to these identities. A combinatorial (but {\em not} a bijective) formula was given for these coefficients~\cite{BS_monoid} using the Pieri formula, which gave a direct connection between some of these order-theoretic results and identities. A first goal of this paper is to deduce another identity~\cite[Theorem~G({\em ii})]{BS97a} from the Pieri formula, and also to deduce the Pieri formula from these identities. This furnishes a new proof of the Pieri formula, shows its equivalence to these (seemingly) more general identities, and, together with the combinatorial proofs of the Pieri formula, gives a purely combinatorial proof of these identities. A key step is the definition of a symmetric function associated to any finite {\em symmetric labeled poset}, which is a poset whose Hasse diagram has edges labeled with integers with a symmetry condition satisfied by its maximal chains. This gives a unified construction of skew Schur functions (for intervals in Young's lattice of partitions), Stanley symmetric functions~\cite{Stanley84} (for intervals in the weak order on the symmetric group), and for intervals in a $k$-Bruhat order, {\em skew Schubert functions} (defined in another fashion in \S 1). In~\cite{LS82b}, Lascoux and Sch\"utzenberger show that if a Schubert polynomial is expressed as a univariate polynomial in the first variable, then the coefficients are (explicitly determined) multiplicity-free sums of Schubert polynomials in the remaining variables. This may be used to show that Schubert polynomials are sums of monomials with non-negative coefficients. We use a cohomological formula~\cite[Theorem 4.5.4]{BS97a} to generalize their result, obtaining a similar formula for expressing a Schubert polynomial as a polynomial in {\em any} variable. This also extends Theorem~C~({\em ii}) of~\cite{BS97a}, which identified the constant term of this expression. {}From this, we obtain a construction of Schubert polynomials purely in terms of chains in the Bruhat order, and a geometric proof that the monomials which appear in a Schubert polynomial have non-negative coefficients. The Pieri formula shows these coefficients are certain intersection numbers, recovering a result of Kirillov and Maeno~\cite{KM}. We found these precise formulas in terms of intersection numbers surprising; Other combinatorial constructions are either recursive~\cite[4.17]{Macdonald91} and do not give the coefficients, or are expressed in terms of combinatorial structures (the weak order on the symmetric group~\cite{BJS,FoSt,FK_YB} or diagrams of permutations~\cite{Kohnert,Be92,Winkel_kohnert_rule}) which are not geometric. Previously, we believed this non-negativity of monomials had no relation to geometry. Indeed, only monomials of the form $x^\lambda$ with $\lambda$ a partition are represented by positive cycles, other polynomial representatives of Schubert classes~\cite{BGG,De74} do not have this non-negativity, and polynomial representatives for the other classical groups cannot~\cite{FK_Bn} have such non-negativity. This paper is organized as follows. In Section 1, we give necessary background, define skew Schubert functions, and state our main results. In Section 2, we deduce the Pieri formula from the identities and results on the Bruhat order. In Section 3, we define a symmetric function $S_P$ associated to a symmetric labeled poset $P$ and complete the proof of the equivalence of the Pieri formula and these identities. We also show how this construction gives skew Schur and Schubert functions. In Section 4, we adapt an argument of Remmel and Shimozono~\cite{Remmel_Shimozono} to show that, for intervals in the weak order, this symmetric function is Stanley's symmetric function~\cite{Stanley84}. Finally, in Section 5, we use a geometric result of~\cite{BS97a} to generalize the result in~\cite{LS82b} and interpret the coefficient of a monomial in a Schubert polynomial in terms of chains in the Bruhat order. \section{Preliminaries} Let ${\mathcal S}_n$ be the symmetric group on $n$ letters and ${\mathcal S}_\infty:=\bigcup_n{\mathcal S}_n$, the group of permutations of ${\mathbb N}$ which fix all but finitely many integers. We let 1 be the identity permutation. For each $w\in{\mathcal S}_\infty$, Lascoux and Sch\"utzenberger~\cite{LS82a} defined a Schubert polynomial ${\mathfrak S}_w \in{\mathbb Z}[x_1,x_2,\ldots]$ with $\deg{\mathfrak S}_w = \ell(w)$. These satisfy the following: \begin{enumerate} \item $\{{\mathfrak S}_w\mid w\in{\mathcal S}_\infty\}$ is a ${\mathbb Z}$-basis for ${\mathbb Z}[x_1,x_2,\ldots]$. \item If $w$ has a unique descent at $k$ ($w(j)>w(j+1)\Rightarrow j=k$), then ${\mathfrak S}_w = S_\lambda(x_1,\ldots,x_k)$, where $\lambda_j = w(k+1-j)-k-1+j$. We write $v(\lambda,k)$ for this permutation and call $w$ a {\em Grassmannian permutation} with descent $k$. \end{enumerate} By the first property, there exist integral structure constants $c^w_{u\,v}$ for $w,u,v\in{\mathcal S}_\infty$ (non-negative from geometry) defined by the identity \begin{equation}\label{eq:cwuv} {\mathfrak S}_u \cdot {\mathfrak S}_v \ =\ \sum_w c^w_{u\,v}\,{\mathfrak S}_w. \end{equation} We are concerned with the coefficients $c^w_{u\;v(\lambda,k)}$ which arise when ${\mathfrak S}_v$ in~(\ref{eq:cwuv}) is replaced by the Schur polynomial $S_\lambda(x_1,\ldots,x_k)={\mathfrak S}_{v(\lambda,k)}$. It is well-known (see for example~\cite{Sottile96,BS97a}) that $c^w_{u\; v(\lambda,k)}\neq0$ only if $u\leq_k w$, where $\leq_k$ is the $k$-Bruhat order (introduced in~\cite{LS83}). In fact, $u\leq_k w$ if and only if there is some $\lambda$ with $c^w_{u\; v(\lambda,k)}\neq0$. This suborder of the Bruhat order has the following characterization: \begin{definition}[Theorem~A of~\cite{BS97a}]\label{def:1} {\sl Let $u,w\in{\mathcal S}_\infty$. Then $u\leq_k w$ if and only if \begin{enumerate} \item $a\leq k<b \Longrightarrow u(a)\leq w(a)$ and $u(b)\geq w(b)$, \item $a<b, u(a)<u(b)$, and $w(a)>w(b) \Longrightarrow a\leq k<b$. \end{enumerate} }\end{definition} For any infinite subset $P$ of ${\mathbb N}$, the order-preserving bijection ${\mathbb N}\leftrightarrow P$ and the inclusion $P\hookrightarrow {\mathbb N}$ induce a map $$ \varepsilon_P\ : \ {\mathcal S}_\infty\ \simeq\ {\mathcal S}_P \ \hookrightarrow\ {\mathcal S}_\infty. $$ {\em Shape-equivalence} is the equivalence relation generated by $\zeta \sim \varepsilon_P(\zeta)$ for $P\subset{\mathbb N}$. If $u\leq_k w$, let $[u,w]_k$ denote the interval between $u$ and $w$ in the $k$-Bruhat order. These intervals have the following property: \medskip \noindent{\bf Order 1} (Theorem~E({\em i}) of~\cite{BS97a}){\bf .} {\em Suppose $u,w,y,z\in{\mathcal S}_\infty$ with $u\leq_k w$, $y\leq_l z$, and $wu^{-1}$ shape-equivalent to $zy^{-1}$. Then $[u,w]_k\simeq[y,z]_l$. Moreover, if $zy^{-1}=\varepsilon_P(wu^{-1})$, then this isomorphism is induced by the map $v\mapsto \varepsilon_P(vu^{-1}) y$.}\medskip This has a companion identity among the structure constants $c^w_{u\;v(\lambda,k)}$: \medskip \noindent{\bf Identity 1} (Theorem~E({\em ii}) of~\cite{BS97a}){\bf .} {\em Suppose $u,w,y,z\in{\mathcal S}_\infty$ with $u\leq_k w$, $y\leq_k l$, and $wu^{-1}$ shape-equivalent to $zy^{-1}$. Then, for any partition $\lambda$, $$ c^w_{u\;v(\lambda,k)}\ =\ c^z_{y\;v(\lambda,l)}. $$ } This identity was originally proven using geometry~\cite{BS97a}. In~\cite{BS_monoid}, we showed how to deduce it from Order~1 and the Pieri formula for Schubert polynomials. Here, we use it to deduce the Pieri formula. By Identity 1, we may define a constant $c^\zeta_\lambda$ for any permutation $\zeta\in{\mathcal S}_\infty$ and any partition $\lambda$ by $c^\zeta_\lambda=c^w_{u\;v(\lambda,k)}$ for any $u\leq_k w$ with $w=\zeta u$. We also define the {\em skew Schubert function} $S_\zeta$ by \begin{equation}\label{eq:skew_Schubert} S_\zeta\ =\ \sum_\lambda c^\zeta_\lambda S_\lambda, \end{equation} where $S_\lambda$ is the Schur symmetric function~\cite{Macdonald95}. By Order~1, we may make the following definition: \begin{defn} Let $\eta,\zeta\in{\mathcal S}_\infty$. Then $\eta\preceq\zeta$ if and only if there is a $u\in{\mathcal S}_\infty$ and $k\in{\mathbb N}$ with $u\leq_k \eta u\leq_k \zeta u$. For $\zeta\in{\mathcal S}_\infty$, define $|\zeta|:= \ell(\zeta u)-\ell(u)$ for any $u,k$ with $u\leq_k \zeta u$. (There always is such a $u$ and $k$, see \S 2.) \end{defn} In \S2, $\preceq$ and $|\zeta|$ are given definitions that do not refer to $\leq_k$ or $\ell(w)$. Let $\zeta,\eta\in{\mathcal S}_\infty$. If we have $\eta\cdot\zeta=\zeta\cdot\eta$ with $|\zeta\cdot\eta|=|\zeta|+|\eta|$, and neither of $\zeta$ or $\eta$ is the identity, then we say that $\zeta\cdot\eta$ is the {\em disjoint product} of $\zeta$ and $\eta$. If a permutation cannot be written in this way, then it is {\em irreducible}. It is a consequence of~\cite[\S 3]{BS97a} that a permutation $\zeta$ factors uniquely into irreducibles as follows: Let $\Pi$ be the finest non-crossing partition~\cite{Kreweras} which is refined by the partition given by the cycles of $\zeta$. For each non-singleton part $p$ of $\Pi$, let $\zeta_p$ be the product of cycles which partition $p$. Each $\zeta_p$ is irreducible, and $\zeta$ is the disjoint product of the $\zeta_p$'s. See Remark~\ref{rem:cyclic} for a further discussion. \medskip \noindent{\bf Order 2} (Theorem~G({\em i}) of~\cite{BS97a}){\bf .} {\em Suppose $\zeta=\zeta_1\cdots\zeta_t$ is the factorization of $\zeta\in{\mathcal S}_\infty$ into irreducibles. Then the map $(\eta_1,\ldots,\eta_t)\mapsto \eta_1\cdots\eta_t$ induces an isomorphism $$ [1,\zeta_1]_\preceq\times\cdots\times[1,\zeta_t]_\preceq\ \stackrel{\sim}{\relbar\joinrel\longrightarrow}\ [1,\zeta]_\preceq. $$ } \noindent{\bf Identity 2} (Theorem~G({\em ii}) of~\cite{BS97a}){\bf .} {\em Suppose $\zeta=\zeta_1\cdots\zeta_t$ is the factorization of $\zeta\in{\mathcal S}_\infty$ into irreducibles. Then $$ S_\zeta\ =\ S_{\zeta_1}\cdots S_{\zeta_t}. $$ } Theorem G({\em ii}) in~\cite{BS97a} states that if $\zeta\cdot\eta$ is a disjoint product, then, for all partitions $\lambda$, $$ c^{\zeta\cdot\eta}_\lambda\ =\ \sum_{\mu,\,\nu} c^\lambda_{\mu\,\nu} c^\zeta_\mu c^\eta_\nu. $$ Thus we see that \begin{eqnarray*} S_\zeta \cdot S_\eta &=& \sum_{\mu,\,\nu} c^\zeta_\mu c^\eta_\nu S_\mu S_\nu\\ &=& \sum_{\lambda,\,\mu,\,\nu} c^\lambda_{\mu\,\nu}c^\zeta_\mu c^\eta_\nu S_\lambda\\ &=& \sum_\lambda c^{\zeta\cdot\eta}_\lambda S_\lambda\ \:=\ \: S_{\zeta\cdot\eta}. \end{eqnarray*} Iterating this shows the equivalence of Theorem G({\em ii}) of~\cite{BS97a} and Identity 2. \bigskip A {\em labeled poset} $P$ is a finite ranked poset together with an integer label for each cover. Its Hasse diagram is thus a directed labeled graph with integer labels. Write $u\stackrel{\mbox{\scriptsize $b$}}{\longrightarrow}w$ for a labeled edge in this Hasse diagram. In what follows, we consider four classes of labeled posets: \begin{enumerate} \item[] \hspace{-26pt}{\bf Intervals in a $k$-Bruhat order.} Labeling a cover $u\lessdot_k w$ in the $k$-Bruhat order with $b$, where $wu^{-1}=(a,\,b)$ and $a<b$ gives every interval in the $k$-Bruhat order the structure of a labeled poset. \item[] \hspace{-26pt}{\bf Intervals in the $\preceq$-order.} Likewise, a cover $\eta\prec\!\!\!\!\!\cdot\ \zeta$ in the $\preceq$-order gives a transposition $(a,\,b)=\zeta\eta^{-1}$ with $a<b$. Labeling such a cover with $b$ gives every interval in this order the structure of a labeled poset. Since $[\eta,\zeta]_\preceq \simeq [1,\zeta\eta^{-1}]_\preceq$, it suffices to consider intervals of the form $[1,\zeta]_\preceq$. \item[] \hspace{-26pt}{\bf Intervals in Young's lattice.} A cover $\mu\subset\!\!\!\!\!\cdot\ \lambda$ in Young's lattice of partitions gives a unique index $i$ with $\mu_i+1=\lambda_i$. Labeling such a cover with $\lambda_i-i$ gives every interval in Young's lattice the structure of a labeled poset. \item[] \hspace{-26pt}{\bf Intervals in the weak order.} Finally, labeling a cover $u\lessdot_{\mbox{\scriptsize\rm weak}}w$ in the weak order on ${\mathcal S}_\infty$ with the index $i$ of the transposition $wu^{-1}=(i,\,i{+}1)$ gives every interval in the weak order the structure of a labeled poset. Since, for $u\leq_{\mbox{\scriptsize\rm weak}}w$, $[u,w]_{\mbox{\scriptsize\rm weak}}\simeq [1,wu^{-1}]_{\mbox{\scriptsize\rm weak}}$, it suffices to consider intervals of the form $[1,w]_{\mbox{\scriptsize\rm weak}}$. \end{enumerate} The sequence of edge labels in a (maximal) chain of a labeled poset is the {\em word} of that chain. For a composition $\alpha=(\alpha_1,\ldots,\alpha_k)$ of $m=$ rank$P$, let $H_\alpha(P)$ be the set of maximal chains in $P$ whose word has descent set contained in $I(\alpha):=\{\alpha_1,\alpha_1+\alpha_2,\ldots,m-\alpha_k\}$. We say that $P$ is {\em symmetric} if the cardinality of $H_\alpha(P)$ depends only upon the parts of $\alpha$ and not their order. Each poset in the above classes is symmetric: For the $k$-Bruhat orders or $\preceq$ order, this is a consequence of the Pieri formula for Schubert polynomials. For Young's lattice, this is classical, and for intervals in the weak order, it is due to Stanley~\cite{Stanley84}. We wish to consider skew Young diagrams to be equivalent if they differ by a translation. This leads to the following notion of isomorphism for labeled posets. \begin{defn}\label{def:lposet} A map $f:P\rightarrow Q$ between labeled posets is an isomorphism if $f$ is an isomorphism of posets which preserves the relative order of the edge labels. \end{defn} That is, if $e,e'$ are edges of $P$ with respective labels $a\leq a'$, then the edge labels $b,b'$ of $f(e),f(e')$ in $Q$ satisfy $b\leq b'$. The isomorphisms of Order~1 and Order~2 are isomorphisms of labeled posets. We also see that the interval $[\mu,\lambda]_\subset$ in Young's poset is isomorphic to the interval $[v(\mu,k),v(\lambda,k)]_k$, since the difference between the label of a cover $v(\alpha,k)\lessdot_k v(\beta,k)$ in the $k$-Bruhat order and the corresponding cover $\alpha\subset\!\!\!\!\cdot\ \beta$ in Young's lattice is $k+1$. To every symmetric labeled poset $P$, we associate~(Definition~\ref{def:skew}) a symmetric function $S_P$ which has the following properties: \begin{thm}\label{thm:skew} \ \begin{enumerate} \item If $P\simeq Q$, then $S_P=S_Q$. \item If $u\leq_k w$, then $S_{[u,w]_k} = S_{wu^{-1}}$, the skew Schubert function. \item[2$'$\!.] For $\zeta\in{\mathcal S}_\infty$, $S_{[1,\zeta]_\preceq}= S_\zeta$, the skew Schubert function. \item Let $\mu\subset \lambda$ be partitions. Then $S_{[\mu,\lambda]_\subset} = S_{\lambda/\mu}$, the skew Schur function. \item For $w\in{\mathcal S}_\infty$, we have $S_{[1,w]_{\rm weak}} = F_w$, the Stanley symmetric function. \end{enumerate} \end{thm} Part 1 is Lemma~\ref{lem:coeff}(2), parts 2, 2$'$, and 3 are proven in \S3, and part 4 in \S4. A labeled poset $P$ is an {\em increasing chain} if it is totally ordered with increasing edge labels. A cycle $\zeta\in{\mathcal S}_\infty$ is {\em increasing} if $[1,\zeta]_\preceq$ is an increasing chain. Decreasing chains and cycles are defined similarly. For any positive integers $m,k$ let $r[m,k]$ denote the permutation $v((m,0,\ldots,0),\,k)$ which is the increasing cycle $(k{+}m,k{+}m{-}1,\ldots,k)$. It is an easy consequence (see Lemma~\ref{lem:perm_facts}) of the definitions of $\leq_k$ or $\preceq$ that any increasing cycle $\zeta$ of length $m{+}1$ is shape equivalent to $r[m,k]$ and hence $|\zeta|=m$. Likewise, the permutation $v(1^m,k)$ is the decreasing cycle $(k{+}1{-}m,\ldots,k,k{+}1)$ and any decreasing cycle of length $m{+}1$ is shape equivalent to $v(1^m,k)$ for any $k\geq m$. Here $1^m$ is the partition of $m$ into $m$ equal parts of size $1$. Note that $$ {\mathfrak S}_{r[m,k]}\ =\ h_m(x_1,\ldots,x_k) \qquad\mbox{and}\qquad {\mathfrak S}_{v(1^m,k)}\ =\ e_m(x_1,\ldots,x_k), $$ the complete homogeneous and elementary symmetric polynomials. \begin{prop}[Pieri formula for Schubert polynomials and flag manifolds]\label{Pieri_formula} Let $u\leq_k w$ with $m=\ell(w)-\ell(u)$. Then \begin{enumerate} \item ${\displaystyle c^w_{u\, r[m,k]}\ =\ \left\{ \begin{array}{ll} 1&\ \mbox{if $wu^{-1}$ is the disjoint product of increasing cycles}\\ 0&\ \mbox{otherwise.} \end{array}\right.}$ \item ${\displaystyle c^w_{u\, v(1^m,k)}\ =\ \left\{ \begin{array}{ll} 1&\ \mbox{if $wu^{-1}$ is the disjoint product of deceasing cycles}\\ 0&\ \mbox{otherwise.} \end{array}\right.}$ \end{enumerate} \end{prop} This is the form of the Pieri formula stated in~\cite{LS82a}, as such a disjoint products of increasing (decreasing) cycles are $k$-{\em soul\`evements droits} (respectively {\em gauches}) for $u$. By~\cite[Lemma 6]{Sottile96}, $wu^{-1}$ is a disjoint product of increasing cycles if and only if there is a maximal chain in $[u,w]_k$ with increasing labels, and such chains are unique. When this occurs, we write $u\stackrel{r[m,k]}{\relbar\joinrel\relbar\joinrel\longrightarrow} w$, where $m:= \ell(w)-\ell(u)$. Similarly, $wu^{-1}$ is a disjoint product of decreasing cycles if and only if there is a maximal chain in $[u,w]_k$ with decreasing labels, which is necessarily unique. Recall that \begin{eqnarray*} H^*(\mbox{\em Flags}({\mathbb C}\,^n)) &\simeq& {\mathbb Z}[x_1,x_2,\ldots]/ \langle {\mathfrak S}_w\mid w\not\in{\mathcal S}_n\rangle\\ &=& {\mathbb Z}[x_1,\ldots,x_n]/ \langle x^{\alpha} \mid \alpha_i\geq n-i,\mbox{ for some } i\rangle. \end{eqnarray*} The map defined by ${\mathfrak S}_w \mapsto {\mathfrak S}_{\overline{w}}$, where $\overline{w}=\omega_0 w\omega_0$, conjugation by the longest element $\omega_0$ in ${\mathcal S}_n$, is an algebra involution on $H^*(\mbox{\em Flags}({\mathbb C}\,^n))$. If $n\geq k+m$, then this involution shows the equivalence of the two versions of the Pieri formula. We state the main results of this paper: \begin{thm}\label{thm:main_equiv} Given the results Order 1 and 2 on the $k$-Bruhat orders/$\preceq$-order, the Pieri formula for Schubert polynomials is equivalent to the Identities 1 and 2. \end{thm} This is proven in \S2 and \S3. \begin{thm} If $w\in{\mathcal S}_n$ and $0\leq \alpha_i\leq n-i$ for $1\leq i\leq n-1$, then the coefficient of $x_1^{n-1-\alpha_1}x_2^{n-2-\alpha_2}\cdots x_{n-1}^{1-\alpha_{n-1}}$ in the Schubert polynomial ${\mathfrak S}_w(x)$ is the number of chains $$ w \stackrel{r[\alpha_1,1]}{\relbar\joinrel\relbar\joinrel\longrightarrow} w_1 \stackrel{r[\alpha_2,2]}{\relbar\joinrel\relbar\joinrel\longrightarrow} \cdots \stackrel{r[\alpha_{n-1},n-1]}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow} \omega_0 $$ between w and $\omega_0$, the longest element in ${\mathcal S}_n$. \end{thm} This is a restatement of Corollary~\ref{cor:chain_monomial}. \section{Proof of the Pieri formula for Schubert polynomials and flag manifolds} Here, we use Identities 1 and 2 to deduce the Pieri formula. We first establish some combinatorial facts about chains and increasing/decreasing cycles. Let $\zeta\in{\mathcal S}_\infty$. We give a $u\in{\mathcal S}_\infty$ and $k>0$ such that $u\leq_k\zeta u$ and $\zeta u$ is Grassmannian of descent $k$. Define up$(\zeta):=\{a\mid a<\zeta(a)\}$, down$(\zeta):=\{b\mid b>\zeta(b)\}$, fix$(\zeta):=\{c\mid c=\zeta(c)\}$, and set $k:=\#\mbox{up}(\zeta)$. If we have \begin{eqnarray*} \mbox{up}(\zeta)&=&\{a_1,\ldots,a_k\mid \zeta(a_1)<\zeta(a_2)<\cdots<\zeta(a_k)\},\\ \mbox{fix}(\zeta)\bigcup\mbox{down}(\zeta)&=&\{b_1,b_2,\ldots\mid \zeta(b_1)<\zeta(b_2)<\cdots\}, \end{eqnarray*} and define $u\in{\mathcal S}_\infty$ by $$ u\ :=\ \left\{\begin{array}{ll} a_i&\ \mbox{if } i\leq k\\ b_{i-k}&\ \mbox{if } i>k\end{array}\right., $$ then $u\leq_k\zeta u$. Set $w:=\zeta u$. This construction of $u\in{\mathcal S}_\infty$ is Theorem 3.1.5 ({\em ii}) of~\cite{BS97a}. There, we also show that $\eta\preceq\zeta$ if and only if \begin{enumerate} \item $a\in$up$(\zeta)\Longrightarrow\eta(a)\leq\zeta(a)$. \item $b\in$down$(\zeta)\Longrightarrow\eta(b)\geq\zeta(b)$. \item $a,b\in$up$(\zeta)$ (or $a,b\in$down$(\zeta)$) with $a<b$ and $\zeta(a)<\zeta(b)\Longrightarrow\eta(a)<\eta(b)$. \end{enumerate} \begin{lem}\label{lem:perm_facts} Let $\zeta\in{\mathcal S}_\infty$. The labeled poset $[1,\zeta]_\preceq$ is a chain if and only if $\zeta$ is either an increasing or a decreasing cycle. Moreover, if $\zeta$ is an increasing (decreasing) cycle of length $m+1$, then the chain $[1,\zeta]_\preceq$ is increasing (decreasing) and $\zeta$ is shape-equivalent to $r[m,1]$ ($v(1^m,m)$). \end{lem} \noindent{\bf Proof. } Let $\zeta\in{\mathcal S}_\infty$ and construct $u\leq_k\zeta u$ as above. Set $m:=\ell(\zeta u)-\ell(u)$, and consider any chain in $[u,w]_k$: $$ u=u_0\stackrel{b_1}{\longrightarrow} u_1 \stackrel{b_2}{\longrightarrow} u_2 \ \cdots\ u_{m-1}\stackrel{b_m}{\relbar\joinrel\longrightarrow} u_m=w. $$ Suppose that the poset $[1,\zeta]_\preceq\simeq[u,\zeta u]_k$ is a chain. By Order 2, $\zeta$ is irreducible. We show that $\zeta$ is either an increasing or a decreasing cycle by induction on $m$. Suppose $\eta=u_{m-1}u^{-1}$ is an increasing cycle. Then $\eta=(b_{m-1},b_{m-2},\ldots,b_1,a_1)$ where $u_1=(a_1,b_1)u$ and $u_i=(b_{i-1},b_i)u_{i-1}$ for $i>1$. Let $\zeta=(a_m,b_m)\eta$. Since $u_{m-1}^{-1}(b_{m-1})\leq k$ and $u_{m-1}^{-1}(b_m)> k$, we must have $b_{m-1}\neq b_m$. If $b_m>b_{m-1}$ so that $[1,\zeta]_\preceq$ is increasing, then, as $\zeta$ is irreducible, we must have $a_m=b_{m-1}$ and so $\zeta$ is the increasing cycle $$ (b_m,b_{m-1},\ldots,b_1,a_1). $$ Indeed, if either $a_m>b_{m-1}$ or $a_m<b_{m-2}$, then $[1,\zeta]_\preceq$ is not a chain, and $b_{m-1}>a_m\geq b_{m-2}$ contradicts $u_{m-2}\lessdot_k u_{m-1}\lessdot_k u_m$. Suppose now that $b_m<b_{m-1}$, then the irreducibility of $\zeta$ implies that $m=2$ and $b_m=a_1$, so that $[1,\zeta]_\preceq$ is decreasing and $\zeta$ is a decreasing cycle. Similar arguments suffice when $\eta=u_{m-1}u^{-1}$ is a decreasing cycle, and the other statements are straightforward. \QED \noindent{\bf Proof that Identities 1 and 2 imply the Pieri formula. } Let $\zeta\in{\mathcal S}_\infty$ and suppose $c^\zeta_{(m,0,\ldots,0)}\neq 0$. Then $m=|\zeta|$, by homogeneity. Replacing $\zeta$ by a shape-equivalent permutation if necessary, we may assume that $\zeta\in{\mathcal S}_n$ and $\zeta(i)\neq i$ for each $1\leq i\leq n$. Define $u$ and $w:=\zeta u$ as in the first paragraph of this section, so that $u,w\in{\mathcal S}_n$ and $c^\zeta_{(m,0,\ldots,0)}=c^w_{u\,r[m,k]}$. Since $c^w_{u\,r[m,k]}\neq 0$, we must have $m=n-k=\#$down$(\zeta)$: Consider any chain \begin{equation}\label{eq:beta-chain} u=u_0\stackrel{b_1}{\longrightarrow} u_1 \stackrel{b_2}{\longrightarrow} u_2 \ \cdots \ u_{m-1}\stackrel{b_m}{\relbar\joinrel\longrightarrow} u_m=w \end{equation} in $[u,w]_k$. Then down$(\zeta)\subset\{b_1,\ldots,b_m\}$ so that $m\geq n-k$. However, $c^w_{u\,r[m,k]}\neq 0$ and $w\in{\mathcal S}_n$ implies that $r[m,k]\in{\mathcal S}_n$, and hence $k{+}m\leq n$. It follows that down$(\zeta)=\{b_1,\ldots,b_m\}$. Thus if we have $u_i=u_{i-1}(c_i,\,d_i)$ with $c_i\leq k<d_i$, then by the construction of $u$, $\{d_1,\ldots,d_m\} = \{k{+}1,\ldots,k{+}m=n\}$. Consider the case when $\zeta$ is irreducible. Then we must have $c_1=c_2=\cdots=c_m$. This implies that $k=\#$up$(\zeta) = 1$, and $m=n-1$. By (1) of Definition~\ref{def:1} we must then have $b_1<b_2<\cdots<b_m$, and hence $\zeta=(n,\,n{-}1,\,\ldots,\,2,\,1)$, an increasing cycle. But this is $r[n{-}1,1]$, so $u=1$, the identity permutation. Since $c^w_{1\,v} = \delta_{w,\,v}$, the Kronecker delta, $c^\zeta_{\lambda}=\delta_{\lambda,\,(m,0,\ldots,0)}$ and so $S_\zeta = h_{n-1}$. If more generally we have $\eta\in{\mathcal S}_\infty$ with $\#$down$(\eta)=|\eta|=m$ and $\eta$ irreducible, then considering a shape-equivalent $\zeta\in{\mathcal S}_n$ with $n$ minimal, we see that $\eta$ is an increasing cycle and $S_{\eta}=h_m$. We return to the case of $\zeta\in{\mathcal S}_n$ with $c^\zeta_{(m,0,\ldots,0)}\neq 0$. Let $\zeta=\zeta_1\cdots\zeta_t$ be the disjoint factorization of $\zeta$ into irreducibles. Then each $\zeta_i$ is an increasing cycle. Suppose that $m_i=|\zeta_i|$. By Identity 2, we have that \begin{eqnarray*} S_\zeta&=& S_{\zeta_1}\cdots S_{\zeta_t}\\ &=& h_{m_1}\cdots h_{m_t}. \end{eqnarray*} This is equivalent to~\cite[Theorem 5]{Sottile96}. {}From this, we deduce that $c^\zeta_\lambda = c^\mu_{\nu\,\lambda}$, where $\mu/\nu$ is a horizontal strip with $m_i$ boxes in the $i$th row. By the classical Pieri formula for Schur polynomials, this implies that $c^\zeta_{(m,0,\ldots,0)}=1$. \QED \section{Skew Schur functions from labeled posets} In~\cite[Theorem 4.3]{BS_monoid}, we showed how the Pieri formula implies Identity 1. Here we complete the proof of Theorem~\ref{thm:main_equiv}, showing how the Pieri formula implies Identity 2. The first step is a reinterpretation of a construction in~\cite[\S4]{BS_monoid} from which we associate a symmetric function to any symmetric labeled poset. For intervals in Young's lattice, we obtain skew Schur functions, and for intervals in either a $k$-Bruhat order or the $\preceq$-order, skew Schubert functions. In Section 4, we show that for intervals in the weak order we obtain Stanley symmetric functions. Let $P$ be a labeled poset with total rank $m$. A (maximal) chain in $P$ gives a sequence of edge labels, called the {\em word} of that chain. A {\em composition} $\alpha:= (\alpha_1,\ldots,\alpha_k)$ of $m=\alpha_1+\cdots+\alpha_k$ ($\alpha_i\geq 0$), determines, and is determined by a (multi)subset $I(\alpha):=\{\alpha_1,\alpha_1+\alpha_2,\ldots,\alpha_1+\cdots+\alpha_k\}$ of $\{1,\ldots,m\}$. For a composition $\alpha$ of $m=$ rank$P$, let $H_{\alpha}(P)$ be the set of (maximal) chains in $P$ whose word $w$ has descent set $\{j\mid w_j>w_{j+1}\}$ contained in the set $I(\alpha)$. We adopt the convention that the last position of a word is a descent. If some $\alpha_i<0$, then we set $H_{\alpha}(P)=\emptyset$. We say that $P$ is {\em (label-) symmetric} if the cardinality of $H_{\alpha}(P)$ depends only upon the parts of $\alpha$ and not their order. Let $\Lambda$ be the ${\mathbb Z}$-algebra of symmetric functions. Recall that $\Lambda = {\mathbb Z}[h_1,h_2,\ldots]$, where $h_i$ is the complete homogeneous symmetric function of degree $i$, the sum of all monomials of degree $i$. For a composition $\alpha$, set $$ h_{\alpha}\ :=\ h_{\alpha_1}h_{\alpha_2}\cdots h_{\alpha_k}. $$ \begin{definition} {\sl Suppose $P$ is a symmetric labeled poset. Define the ${\mathbb Z}$-linear map $\chi_P : \Lambda \rightarrow {\mathbb Z}$ by $$ \chi_P\ :\ h_{\alpha} \longmapsto \# (H_{\alpha}(P)). $$ For any partition $\lambda$, define the skew coefficient $c^P_\lambda$ to be $\chi_P(S_\lambda)$, where $S_\lambda$ is the Schur symmetric function. } \end{definition} We point out some properties of these coefficients $c^P_\lambda$. For a partition $\lambda$ of $m$ ($\lambda\vdash m$) with $\lambda_{k+1}=0$ and a permutation $\pi\in {\mathcal S}_k$, let $\lambda_\pi$ be the following composition of $m$: $$ \pi(1)-1+\lambda_{k+1-\pi(1)},\,\pi(2)-2+\lambda_{k+1-\pi(2)},\, \ldots,\,\pi(k)-k+\lambda_{k+1-\pi(k)}. $$ \begin{lem}\label{lem:coeff} Let $P,Q$ be symmetric labeled posets. \begin{enumerate} \item For any partition $\lambda$, $$ c^P_\lambda\ :=\ \sum_{\pi\in {\mathcal S}_k} \varepsilon(\pi) \#( H_{\lambda_\pi}(P)) $$ where $\lambda_{k+1}=0$ and $\varepsilon :{\mathcal S}_k \to \{\pm 1\}$ is the sign character. \item If $P\simeq Q$ as labeled posets (Definition~\ref{def:lposet}) then for any partition $\lambda$, $c^P_\lambda = c^Q_\lambda$. \end{enumerate} \end{lem} The first statement follows from the Jacobi-Trudi formula, and the second by noting that the bijection $P\leftrightarrow Q$ induces bijections $H_{\alpha}(P)\leftrightarrow H_{\alpha}(Q)$. \begin{rem} By the Pieri formula for Schubert polynomials, the number $\#(H_{\alpha}([u,w]_k))$ is the coefficient of ${\mathfrak S}_w$ in the product ${\mathfrak S}_u\cdot h_{\alpha}(x_1,\ldots,x_k)$. It follows that intervals in a $k$-Bruhat order or in the $\preceq$-order are symmetric. For similar reasons, we see that intervals in Young's lattice are symmetric, as $\#(H_{\alpha}([\mu,\lambda]_\subset))$ is the skew Kostka coefficient $K_{\alpha,\,\lambda/\mu}$, which is the coefficient of $S_\lambda$ in $S_\mu\cdot h_\alpha$, equivalently, the number of semistandard Young tableaux of shape $\lambda/\mu$ and content $\alpha$. One may construct an explicit bijection with the second set as follows: A chain in $H_{\alpha}([\mu,\lambda]_\subset)$ is naturally decomposed into subchains with increasing labels of lengths $\alpha_1,\alpha_2,\ldots,\alpha_k$. Placing the integer $i$ in the boxes corresponding to covers in the $i$th such subchain furnishes the bijection. \end{rem} \begin{prop}[Theorem~4.3 of~\cite{BS_monoid}] Let $u\leq_k w$ and $\lambda\vdash \ell(w)-\ell(u)= m$. Then $c^w_{u\,v(\lambda,k)} = c^{[u,w]_k}_\lambda$. \end{prop} \noindent{\bf Proof. } By definition, $c^w_{u\,v(\lambda,k)}$ is the coefficient of ${\mathfrak S}_w$ in the expansion of the product ${\mathfrak S}_u\cdot S_\lambda(x_1,\ldots,x_k)$ into Schubert polynomials. By the Jacobi-Trudi formula, \begin{eqnarray*} {\mathfrak S}_u\cdot S_\lambda(x_1,\ldots,x_k) &=& {\mathfrak S}_u\cdot \sum_{\pi\in {\mathcal S}_k} \varepsilon(\pi) h_{\lambda_\pi}(x_1,\ldots,x_k) \\ &=& \sum_w \sum_{\pi\in {\mathcal S}_k} \varepsilon(\pi) \# (H_{\lambda_\pi}([u,w]_k))\: {\mathfrak S}_w\\ &=& \sum_w c^{[u,w]_k}_\lambda\; {\mathfrak S}_w. \qquad \QED \end{eqnarray*} \begin{prop}[Corollary 4.9 of\/~\cite{BS_monoid}]\label{prop:identity} If $u\leq_kw$ and $y\leq_l z$ with $wu^{-1}$ shape equivalent to $zy^{-1}$, then for all $\lambda$, $c^w_{u\,v(\lambda,k)} = c^z_{y\,v(\lambda,l)}$. \end{prop} \noindent{\bf Proof. } By Order 1, $[u,w]_k\simeq [y,z]_l$ is an isomorphism of labeled posets. \QED \begin{defn}\label{def:skew} Let $P$ be a ranked labeled poset with total rank $m$. Define the symmetric function $S_P$ by $$ S_P\ :=\ \sum_{\lambda\vdash m} c^P_\lambda S_\lambda, $$ where $S_\lambda$ is a Schur {\em function}. \end{defn} \noindent{\bf Proof of Theorem~\ref{thm:skew} (1), (2), and (3). } (1) is a consequence of Lemma~\ref{lem:coeff} (2). For (3), let $\mu\subset \nu$ in Young's lattice, suppose $\nu_{k+1}=0$, and consider the interval $[\mu,\nu]_\subset$ in Young's lattice. Then $[\mu,\nu]\simeq[v(\mu,k),v(\nu,k)]_k$, and so $c^{[\mu,\nu]}_\lambda = c^{v(\nu,k)}_{v(\mu,k)\,v(\lambda,k)}= c^\nu_{\mu\,\lambda}$. Hence $S_{[\mu,\nu]_\subset}=S_{\nu/\mu}$. Similarly, we see that for $u\leq_k w$ or $\zeta\in{\mathcal S}_\infty$, we have $S_{[u,w]_k}=S_{wu^{-1}}$ and $S_{[1,\zeta]_\preceq}=S_\zeta$, the skew Schubert functions of \S 1. \QED \begin{rem}\label{rem:cyclic} According to Proposition~\ref{prop:identity}, the skew Schubert function $S_\zeta$ depends only on the shape equivalence class of $\zeta$. In~\cite{BS97a} there is another identity:\medskip Theorem H of\/~\cite{BS97a}. {\em Suppose $\eta,\zeta\in{\mathcal S}_n$ with $\zeta=\eta^{(12\ldots n)}$. Then $S_\eta=S_\zeta$. }\medskip The example of $\eta=(1243)$ and $\zeta=(1243)$ in ${\mathcal S}_4$ (see Figure~\ref{fig:interval}) shows that in general $[1,\eta]_\preceq \not\simeq [1,\eta^{(12\ldots n)}]_\preceq$. However, these two intervals do have the same number of maximal chains~\cite[Corollary 1.4]{BS97a}. In fact, for $\eta\in{\mathcal S}_n$ and $\alpha$ a composition, $\# (H_{\alpha}([1,\eta]_\preceq))= \# (H_{\alpha}([1,\eta^{(12\ldots n)}]_\preceq))$. Thus if $\sim$ is the equivalence relation generated by shape equivalence and this `cyclic shift' ($\eta\sim\eta^{(12\ldots n)}$, if $\eta\in{\mathcal S}_n$), then $S_\zeta$ depends only upon the $\sim$-equivalence class of $\zeta$. (This is analogous to, but stronger than the fact that the skew Schur function $S_\kappa$ depends on $\kappa$ only up to a translation in the plane.) There is a combinatorial object $\Gamma_\zeta$ which determines the $\sim$-equivalence class of $\zeta$. First place the set $\{a\mid a\neq \zeta(a)\}$ at the vertices of a regular $\#\{a\mid a\neq \zeta(a)\}$-gon in clockwise order. Next, for each $a$ with $a\neq \zeta(a)$, draw a directed chord from $a$ to $\zeta(a)$. $\Gamma_\zeta$ is the resulting configuration of directed chords, up to rotation and dilation and without any vertices labeled ({\em cf.}~\cite[\S3.3]{BS97a}). The irreducible factors of $\zeta$ correspond to connected components of $\Gamma_\zeta$ (considered as a subset of the plane). The figure $\Gamma_{(1243)}=\Gamma_{(1423)}$ is also displayed in Figure~\ref{fig:interval}. \begin{figure}[htb] $$\epsfxsize=3.5in \epsfbox{interval.eps}$$ \caption{Intervals under cyclic shift and $\Gamma_\zeta$\label{fig:interval}} \end{figure} \end{rem} We conclude this section with the following Theorem: \begin{thm}\label{thm:product} Let $P$ and $Q$ be symmetric labeled posets with disjoint sets of edge labels. Then $$ S_{P\times Q}\ =\ S_P\cdot S_Q. $$ \end{thm} This will complete the proof of Theorem~\ref{thm:main_equiv}, namely that the Pieri formula and Order 2 imply Identity 2: If $\zeta\cdot\eta$ is a disjoint product, then $[1,\zeta]_\preceq$ and $[1,\eta]_\preceq$ have disjoint sets of edge labels. Together with Theorem~\ref{thm:skew}(4), this gives another proof of Theorem~3.4 in~\cite{Stanley84}, that $F_{w\times u}=F_w\cdot F_u$. \medskip To prove Theorem~\ref{thm:product}, we first study chains in $H_{\alpha}(P\times Q)$. Suppose that $P$ has rank $n$ and $Q$ has rank $m$. Note that a chain in $P\times Q$ determines and is determined by the following data: \begin{equation}\label{chain:data} \begin{array}{l} \bullet\ \mbox{A chain in each of $P$ and $Q$},\\ \bullet\ \mbox{A subset $B$ of $\{1,\ldots,n+m\}$ with $\#B=n$}. \end{array} \end{equation} Recall that covers $(p,q)\lessdot(p',q')$ in $P\times Q$ have one of two forms: either $p=p'$ and $q'$ covers $q$ in $Q$ or else $q=q'$ and $p'$ covers $p$ in $P$. Thus a chain in $P\times Q$ gives a chain in each of $P$ and $Q$, with the covers from $P$ interspersed among the covers from $Q$. If we set $B$ to be the positions of the covers from $P$, we obtain the description~(\ref{chain:data}). Define $$ \mbox{sort}\ :\ \mbox{\em chains}(P\times Q)\ \longrightarrow\ \mbox{\em chains}(P)\times\mbox{\em chains}(Q) $$ to be the map which forgets the positions $B$ of the covers from $P$. \begin{lem}\label{lem:bijection} Let $P$ and $Q$ be labeled posets with disjoint sets of edge labels and $\alpha$ be any composition. Then $$ \mbox{\rm sort}\ :\ H_{\alpha}(P\times Q)\ \longrightarrow\ \coprod_{\beta+\gamma=\alpha} H_{\beta}(P)\times H_{\gamma}(Q) $$ is a bijection. \end{lem} For integers $a<b$, let $[a,b]:=\{n\in{\mathbb Z}\mid a\leq n\leq b\}$. For a chain $\xi$, let $\xi|_{[a,b]}$ be the portion of $\xi$ starting at the $a$th step and continuing to the $b$th step. \noindent{\bf Proof. } Let $\xi\in H_{\alpha}(P\times Q)$ and set $I=I(\alpha)$ so that $I_i=\alpha_1+\cdots+\alpha_i$. Then sort$(\xi)\in H_{\beta}(P)\times H_{\gamma}(Q)$, where, for each $i$, $\beta_i$ counts the number of covers of $\xi|_{[I_{i-1},I_i]}$ from $P$ and $\gamma_i=\alpha_i-\beta_i$. To see this is a bijection, we construct its inverse. For chains $\xi^P\in H_{\beta}(P)$ and $\xi^Q\in H_{\gamma}(Q)$ with $\beta+\gamma=\alpha$, define the set $B$ by the conditions \begin{enumerate} \item $\beta_i=\# B\cap [I(\alpha)_{i-1},I(\alpha)_i]$. \item If $b_1\leq\cdots\leq b_{\beta_i}$ and $c_1\leq\cdots\leq c_{\gamma_i}$ are the covers in $\xi^P|_{[I(\beta)_{i-1},I(\beta)_i]}$ and $\xi^Q|_{[I(\gamma)_{i-1},I(\gamma)_i]}$ respectively, then, up to a shift of $I(\alpha)_{i-1}$, the set $B\cap [I(\alpha)_{i-1},I(\alpha)_i]$ records the positions of the the $b$'s in the linear ordering of $\{b_1,\ldots,b_{\beta_1},c_1,\ldots,c_{\gamma_i}\}$. \end{enumerate} This clearly gives the inverse to the map sort. \QED Recall that the comultiplication $\Delta:\Lambda \rightarrow \Lambda\otimes\Lambda$ is defined by $$ \Delta(h_a)\ =\ \sum_{b+c=a}h_b\otimes h_c. $$ Thus, for a composition $\alpha$, $$ \Delta(h_\alpha)\ =\ \sum_{\beta+\gamma=\alpha}h_\beta\otimes h_\gamma. $$ {}From Lemma~\ref{lem:bijection}, we immediately deduce: \begin{cor} Let $P,Q$ be symmetric labeled posets with disjoint sets of edge labels. Then $$ \begin{picture}(100,67) \put( 0,25){$\chi_{P\times Q}$} \put(9,43){$\Lambda$} \put(43,0){${\mathbb Z}$} \put(43,52){$\Delta$} \put(77,43){$\Lambda\otimes\Lambda$} \put(72,25){$\chi_P\otimes \chi_Q$} \put(18,40){\vector(1,-1){27}} \put(80,40){\vector(-1,-1){27}} \put(19,47){\vector(1,0){55}} \end{picture} $$ commutes. \end{cor} \begin{cor} Let $P,Q$ be symmetric labeled posets with disjoint sets of edge labels. Then, for any partition $\lambda$, $$ c^{P\times Q}_\lambda\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}\: c^P_\mu\; c^Q_\nu. $$ \end{cor} \noindent {\bf Proof. } Recall~\cite[I.5.9]{Macdonald95} that $\Delta(S_\lambda)\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}\:S_\mu\;S_\nu$. Hence $$ \chi_{P\times Q}(S_\lambda)\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}\: \chi_P(S_\mu)\:\chi_P(S_\nu).\qquad \QED $$ We complete the proof of Theorem~\ref{thm:product}: Let $P,Q$ be symmetric labeled posets with disjoint sets of edge labels. Then \begin{eqnarray*} S_P\cdot S_Q&=& \sum_{\mu,\nu}c^P_\mu\;S_\mu\: c^Q_\nu\; S_\nu\\ &=& \sum_{\lambda,\mu,\nu}c^\lambda_{\mu\,\nu}\;c^P_\mu\;c^Q_\nu\:S_\lambda\\ &=&\sum_\lambda c^{P\times Q}_\lambda S_{\lambda}\ \ =\ \ S_{P\times Q}.\qquad \QED \end{eqnarray*} \section{Stanley symmetric functions from labeled posets} We establish Theorem~\ref{thm:skew}(4) by adapting the proof of the Littlewood-Richardson rule in~\cite{Remmel_Shimozono} to obtain a bijective interpretation of the constants $c^{[1,w]_{\mbox{\scriptsize weak}}}_\lambda$, which shows $S_{[1,w]_{\mbox{\scriptsize weak}}}=F_w$ by the formulas in~\cite{LS82b,EG}. The main tool is a {\em jeu de taquin} for reduced decompositions. We use Cartesian conventions for Young diagrams and skew diagrams. Thus the first row is at the bottom. A filling of a diagram $D$ with positive integers which increase across rows and up columns is a {\em tableau} with {\em shape} $D$. The {\em word} of a tableau is the sequence of its entries, read across each row starting with the topmost row. A {\em reduced decomposition} $\rho$ for a permutation $w\in{\mathcal S}_\infty$ is the word of a maximal chain in $[1,w]_{\mbox{\scriptsize\rm weak}}$. Let $R(w)$ be the set of all reduced decompositions for $w$ and for a composition $\alpha$ of $\ell(w)$, write $H_\alpha(w)$ for $H_\alpha([1,w]_{\mbox{\scriptsize\rm weak}})$. Given any composition $\alpha$ and any reduced decomposition $\rho\in H_{\alpha}(w)$, there is a unique smallest diagram $\lambda/\mu$ with row lengths $\lambda_i-\mu_i=\alpha_{k+1-i}$ for which $\rho$ is the word of a tableau $T(\alpha,\rho)$ of shape $\lambda/\mu$. By this we mean that $\mu_j-\mu_{j+1}$ is minimal for all $j$. If $\mu_1=0$, then $T(\alpha,\rho)$ has {\em partition shape} $\lambda$ ($=\alpha$), otherwise $T(\alpha,\rho)$ has {\em skew shape}. Given a reduced decomposition $\rho\in R(w)$, define $T(\rho)$ to be the tableau $T(\alpha,\rho)$, where $I(\alpha)$ is the descent set of $\rho$. Stanley~\cite{Stanley84} defined a symmetric function $F_w$ for every $w\in{\mathcal S}_\infty$. (That $F_w$ is symmetric includes a proof that the intervals $[1,w]_{\mbox{\scriptsize weak}}$ are symmetric.) Thus there exists integers $a^w_\lambda$ such that $$ F_w\ =\ \sum_{\lambda\vdash l} a^w_\lambda S_\lambda. $$ A combinatorial interpretation for $a^w_\lambda$ was given (independently) in~\cite{LS82b} and~\cite{EG}: $$ a^w_\lambda\ =\ \#\{ \rho\in R(w)\mid T(\rho)\mbox{ has partition shape }\lambda\}. $$ (See~\cite[\S VII]{Macdonald91} for an account with proofs.) Theorem~\ref{thm:skew}(4) is a consequence of the following result: \begin{thm}\label{thm:weak_coefficients} For any $w\in{\mathcal S}_\infty$ and partition $\lambda\vdash\ell(w)$, $$ a^w_\lambda \ =\ c^{[1,w]_{\mbox{\scriptsize\rm weak}}}_\lambda. $$ \end{thm} Our proof is based on the proof of the Littlewood-Richardson rule given by Remmel and Shimozono~\cite{Remmel_Shimozono}. We define an involution $\theta$ on the set $$ \coprod_{\pi\in{\mathcal S}_k} \{\pi\}\times H_{\lambda_\pi}(w) $$ (here $\lambda\vdash \ell(w)$ and $\lambda_{k+1}=0$) such that \begin{enumerate} \item $\theta(\pi,\rho) = (\pi,\rho)$ if and only if $T(\rho)$ has shape $\lambda$, from which it follows that $\pi=1$. \item If $T(\rho)$ does not have shape $\lambda$, then $\theta(\pi,\rho) = (\pi',\rho')$ where $T(\rho')$ does not have shape $\lambda$ and $\rho'\in H_{\lambda_{\pi'}}(w)$ with $|\ell(\pi)-\ell(\pi')|=1$. \end{enumerate} Theorem~\ref{thm:weak_coefficients} is a corollary of the existence of such an involution $\theta$: By property 2, only the fixed points of $\theta$ contribute to the sum in Lemma~\ref{lem:coeff}(1). The involution $\theta$ will be defined using a {\em jeu de taquin} for tableaux whose words are reduced decompositions. Because we only play this {\it jeu de taquin} on diagrams with two rows, we do not describe it in full. \begin{defn} Let $T$ be a tableau of shape $(y+p,q)/(y,0)$ whose word is a reduced decomposition for a permutation $w$. If $y\neq 0$, we may perform an inward slide. This modification of an ordinary {\it jeu de taquin} slide ensures we obtain a tableau whose word is a reduced decomposition of $w$. Begin with an empty box at position $(y,1)$ and move it through the tableau $T$ according to the following local rules: \begin{enumerate} \item If the box is in the first row, it switches with whichever of its neighbors to the right or above is smaller. If both neighbors are equal, say they are $a$, then their other neighbor is necessarily $a+1$, as we have a reduced decomposition. Locally we will have the following configuration, where \raisebox{-2pt}{% \begin{picture}(10,10) \put( 0,10){\line(0,-1){10}}\put(10,10){\line(-1,0){10}} \put( 0, 0){\line(1, 0){10}}\put(10, 0){\line( 0,1){10}} \put( 0, 0){\line(1, 1){10}}\put(10, 0){\line(-1,1){10}} \end{picture}} denotes the empty box and $a+b+1<c$: $$ \begin{picture}(200,30) \thicklines \put( 0, 0){\line(1, 1){15}}\put(15, 0){\line(-1,1){15}} \put( 0, 0){\line(1,0){200}} \put( 0,15){\line(1,0){200}} \put( 0,30){\line(1,0){184}} \put( 0, 0){\line(0,1){30}} \put(15, 0){\line(0,1){30}} \put(42, 0){\line(0,1){30}} \put(72, 0){\line(0,1){30}} \put(114,15){\line(0,1){15}} \put(142, 0){\line(0,1){30}} \put(184, 0){\line(0,1){30}} \put(200, 0){\line(0,1){15}} \put(4,19){$a$} \put(26,4){$a$} \put(19,19){$a{+}1$} \put(46,4){$a{+}1$} \put(46,19){$a{+}2$} \put(86,4){$\cdots\cdots$} \put(86,19){$\cdots$} \put(118,19){$a{+}b$} \put(153,4){$a{+}b$} \put(146,19){$a{+}b{+}1$} \put(188,4){$c$} \end{picture} $$ The empty box moves through this configuration, transforming it into: $$ \begin{picture}(200,30) \thicklines \put( 0, 0){\line(1,0){198}} \put( 0,15){\line(1,0){198}} \put( 0,30){\line(1,0){184}} \put( 0, 0){\line(0,1){30}} \put(27, 0){\line(0,1){30}} \put(56, 0){\line(0,1){30}} \put(100,0){\line(0,1){30}} \put(142, 0){\line(0,1){30}} \put(184, 0){\line(0,1){30}} \put(198, 0){\line(0,1){15}} \put(142,15.5){\line(3, 1){42}}\put(142,29.5){\line(3,-1){42}} \put( 9, 4){$a$} \put( 4,19){$a{+}1$} \put(31,4){$a{+}1$} \put(31,19){$a{+}2$} \put(63,4){$\cdots\cdots$} \put(63,19){$\cdots\cdots$} \put(111, 4){$a{+}b$} \put(104,19){$a{+}b{+}1$} \put(146,4){$a{+}b{+}1$} \put(188,4){$c$} \end{picture} $$ This guarantees that we still have a reduced decomposition for $w$. \item If the box is in the second row, then it switches with its neighbour to the right. \end{enumerate} If $y+p>q$, then we may analogously perform an outward slide, beginning with an empty box at $(q+1,2)$ and sliding to the left or down according to local rules that are the reverse of those for the inward slide. \end{defn} We note some consequences of this definition. \begin{itemize} \item The box will change rows at the first pair of entries $b\leq c$ it encounters with $b$ at $(i,2)$ and $c$ immediately to its lower right at $(i+1,1)$. If there is no such pair, it will change rows at the end of the first row in an inward slide if $p+y=q$, and at the beginning of the second row in an outward slide if $y=0$. \item At least one of these will occur if $y$ is minimal given the word of the tableau and $p,q$. Suppose this is the case. Then the tableau $T'$ obtained from a slide will have another such pair $b'\leq c'$ with $b'$ at $(\imath',2)$ and $c'$ at $(\imath'+1,1)$. Hence, if we perform a second slide, the box will again change rows. \item The inward and outward slides are inverses. \end{itemize} Let $\overline{H}_\alpha(w)$ be the subset of $H_\alpha(w)$ consisting of chains $\rho$ such that $T(\alpha,\rho)$ has skew shape. The proof of the following lemma is straightforward. \begin{lem}\label{lem:bijections} Let $w\in {\mathcal S}_\infty$ and suppose $p<q$ with $p+q=\ell(w)$. Then $H_{(q,p)}(w)=\overline{H}_{(q,p)}(w)$ and \begin{enumerate} \item For every $\rho\in H_{(q,p)}(w)$, we may perform $q-p$ inward slides to $T((q,p),\rho)$. If $\rho'$ is the word of the resulting tableau, then the map $\rho\mapsto \rho'$ defines a bijection $$ H_{(q,p)}(w)\ \longleftrightarrow \ H_{(p,q)}(w). $$ The inverse map is given by the application of $q-p$ outward slides. \item If we now let $\rho'$ be the word of the tableau obtained after $q-p-1$ inward slides to $T((q,p),\rho)$ for $\rho\in H_{(q,p)}(w)$, then the map $\rho\mapsto \rho'$ defines a bijection $$ \overline{H}_{(q,p)}(w)\ \longleftrightarrow \ \overline{H}_{(p+1,q-1)}(w). $$ The inverse map is defined by the application of $q-1-p$ outward slides. \end{enumerate} \end{lem} The first part gives a proof that intervals in the weak order are symmetric: Let $\alpha=(\alpha_1,\ldots,\alpha_k)$ and $\alpha'=(\alpha_1,\ldots,\alpha_{r+1},\alpha_r,\ldots,\alpha_k)$ be compositions of $\ell(w)$. Then applying the bijection in Lemma~{lem:bijections}(1) to the segment $\rho_r$ of $\rho\in H_{\alpha}(w)$ between $I(\alpha)_{r-1}$ and $I(\alpha)_{r+1}$ defines a bijection $$ H_{\alpha}(w)\ \longleftrightarrow \ H_{\alpha'}(w). $$ \begin{rem} This bijection is different from the one used in~\cite{Stanley84} to prove symmetry of these intervals. Indeed, consider the example given there, which we write as a tableau: $$ \epsfxsize=3.5in \epsfbox{word1.eps} $$ In~\cite{Stanley84}, Stanley maps this to $$ \epsfxsize=4.03in \epsfbox{word2.eps} $$ But the bijection we define gives us this: $$ \epsfxsize=3.63in \epsfbox{word3.eps} $$ \end{rem} Now we may define $\theta$. By the definition of $\lambda_\pi$, if $\rho\in H_{\lambda_\pi}(w)$, then $T(\rho)$ has shape $\lambda$ if and only if $T(\lambda_\pi,\rho)$ has partition shape, which implies that $\pi=1$. \begin{defn}\label{def:theta} Suppose $w\in{\mathcal S}_\infty$ and $\lambda\vdash \ell(w)$ is a partition with $\lambda_{k+1}=0$. Let $\pi\in{\mathcal S}_k$. For $\rho\in H_{\lambda_\pi}(w)$, define $\theta(\pi,\rho)$ as follows: \begin{enumerate} \item If $T(\rho)$ has shape $\lambda$, set $\theta(\pi,\rho)=(\pi,\rho)$. In this case, $\pi=1$, so $\lambda_\pi=\lambda$ and $T(\rho)=T(\lambda_\pi,\rho)$. \item If $T(\rho)$ does not have shape $\lambda$, then $T(\lambda_\pi,\rho)$ has skew shape and we select $r=r(T(\lambda_\pi,\rho))$ with $1\leq r<k$ as follows:\smallskip Left justify the rows of $T(\lambda_\pi,\rho)$. Since $T(\lambda_\pi,\rho)$ has skew shape, there is an entry $a$ of this left-justified figure in postiton $(i,r+1)$ either with no entry in position $(i,r)$ just below it, or else with an entry $b\geq a$ just below it. Among all such $(i,r)$ choose the one with $i$ minimal, and for this $i$, $r$ maximal. \smallskip Let $\rho_r$ be the word given by the rows $r+1$ and $r$ of $T(\lambda_\pi,\rho)$ and $(q,p)$ the lengths of these two rows. Then $T((q,p),\rho_r)$ has skew shape, and we may apply the map of Lemma~\ref{lem:bijections}(2) to obtain the word $\rho_r'$. Define $\theta(\pi,\rho)=(\pi',\rho')$, where $\rho'$ is the word obtained from $\rho$ by replacing $\rho_r$ with $\rho_r'$ and $\pi'\pi^{-1}=(r,\,r{+}1)$. Note that $T(\lambda_{\pi'},\rho')$ also has skew shape and $T(\rho')$ does not have shape $\lambda$. \end{enumerate} \end{defn} \begin{ex} Let $w=4621357$ and $\lambda=(4,3,3,1)$. Then $\rho=5.345.236.1236\in H_{\lambda}(w)$ but $$ \begin{picture}(135,60) \thicklines \put( 0,22.5){$T(\lambda,\rho)\ =$} \put( 60,30){\line(0,1){30}} \put(60,60){\line(1,0){15}} \put( 75, 0){\line(0,1){60}} \put(60,45){\line(1,0){45}} \put( 90, 0){\line(0,1){45}} \put(60,30){\line(1,0){60}} \put(105, 0){\line(0,1){45}} \put(75,15){\line(1,0){60}} \put(120, 0){\line(0,1){30}} \put(75, 0){\line(1,0){60}} \put(135, 0){\line(0,1){15}} \put(64,49){5} \put(64,34){3}\put(79,34){4}\put(94,34){5} \put(79,19){2}\put(94,19){3}\put(109,19){6} \put(79, 4){1}\put(94, 4){2}\put(109, 4){3}\put(124, 4){5} \end{picture} $$ has skew shape. Left-justifying the rows of $T(\lambda,\rho)$, we obtain: $$ \begin{picture}(60,60) \thicklines \put( 0, 0){\line(0,1){60}} \put(0,60){\line(1,0){15}} \put(15, 0){\line(0,1){60}} \put(0,45){\line(1,0){45}} \put(30, 0){\line(0,1){45}} \put(0,30){\line(1,0){45}} \put(45, 0){\line(0,1){45}} \put(0,15){\line(1,0){60}} \put(60, 0){\line(0,1){15}} \put(0, 0){\line(1,0){60}} \put(4,49){5} \put(4,34){3}\put(19,34){4}\put(34,34){5} \put(4,19){2}\put(19,19){3}\put(34,19){6} \put(4, 4){1}\put(19, 4){2}\put(34, 4){3}\put(49, 4){5} \end{picture} $$ This is not a tableau, as the third column reads $365$, which is not increasing. Since this is the first such column and the last decrease is at position $2$, we have $r=2$. Since these two rows each have length 3, we perform one outward slide (by our choice of $r$, we can perform such a slide!) to obtain the tableau $T((4,2),\rho'_r)$ as follows: $$ \begin{picture}(60,30) \thicklines \put( 0,15){\line(0,1){15}} \put(15, 0){\line(1,0){45}} \put(15, 0){\line(0,1){30}} \put( 0,15){\line(1,0){60}} \put(30, 0){\line(0,1){30}} \put( 0,30){\line(1,0){60}} \put(45, 0){\line(0,1){30}} \put(60, 0){\line(0,1){30}} \put(45,15){\line(1,1){15}} \put(45,30){\line(1,-1){15}} \put(4,19){3}\put(19,19){4}\put(34,19){5} \put(19, 4){2}\put(34, 4){3}\put(49, 4){6} \end{picture} \qquad \raisebox{14pt}{$\relbar\joinrel\longrightarrow$} \qquad \begin{picture}(60,30) \thicklines \put( 0,15){\line(0,1){15}} \put(15, 0){\line(1,0){45}} \put(15, 0){\line(0,1){30}} \put( 0,15){\line(1,0){60}} \put(30, 0){\line(0,1){30}} \put( 0,30){\line(1,0){60}} \put(45, 0){\line(0,1){30}} \put(60, 0){\line(0,1){30}} \put(45, 0){\line(1,1){15}} \put(45,15){\line(1,-1){15}} \put(4,19){3}\put(19,19){4}\put(34,19){5}\put(49,19){6} \put(19, 4){2}\put(34, 4){3} \end{picture} \qquad \raisebox{14pt}{$\relbar\joinrel\longrightarrow$} \qquad \begin{picture}(45,30) \thicklines \put( 0,15){\line(0,1){15}} \put(15, 0){\line(1,0){45}} \put(15, 0){\line(0,1){30}} \put( 0,15){\line(1,0){60}} \put(30, 0){\line(0,1){30}} \put( 0,30){\line(1,0){60}} \put(45, 0){\line(0,1){30}} \put(60, 0){\line(0,1){30}} \put(15, 0){\line(1,1){15}} \put(15,15){\line(1,-1){15}} \put(4,19){3}\put(19,19){4}\put(34,19){5}\put(49,19){6} \put(34, 4){2}\put(49, 4){3} \end{picture} $$ Thus $\rho'=5.3456.23.1235\in H_{\lambda_{(2,\,3)}}(w)$. If we left justify $T(\lambda_{(2,\,3)},\rho')$, then we obtain: $$ \begin{picture}(60,60) \thicklines \put( 0, 0){\line(0,1){60}} \put(0,60){\line(1,0){15}} \put(15, 0){\line(0,1){60}} \put(0,45){\line(1,0){60}} \put(30, 0){\line(0,1){45}} \put(0,30){\line(1,0){60}} \put(45, 0){\line(0,1){15}} \put(0,15){\line(1,0){60}} \put(60, 0){\line(0,1){15}} \put(0, 0){\line(1,0){60}} \put(45,30){\line(0,1){15}} \put(60,30){\line(0,1){15}} \put(4,49){5} \put(4,34){3}\put(19,34){4}\put(34,34){5}\put(49,34){6} \put(4,19){2}\put(19,19){3} \put(4, 4){1}\put(19, 4){2}\put(34, 4){3}\put(49, 4){5} \end{picture} $$ The 5 in the third row has no lower neighbour, hence $2=r(\lambda,\rho)=r(\lambda_{(2,\,3)},\rho')$. \end{ex} We complete the proof of Theorem~\ref{thm:weak_coefficients} by showing that $\theta$ is an involution. This is a consequence of Lemma~\ref{lem:bijections}(2) and the following fact: \begin{lem} In (2) of Definition~\ref{def:theta}, if $\rho\in H_{\lambda_\pi}(w)$ and $T(\lambda_{\pi},\rho)$ has skew shape, then $r(T(\lambda_{\pi},\rho))=r(T(\lambda_{\pi'},\rho'))$. \end{lem} \noindent{\bf Proof. } Suppose we are in the situation of (2) in Definition~\ref{def:theta}. The lemma follows once we show that that $T((q,p),\rho_r)$ and $T((p+1,q-1),\rho'_r)$ agree in the first $i$ entries of their second rows, the first $i-1$ entries of their first rows, and the $i$th entry $c$ in the first row of $T((p+1,q-1),\rho'_r)$ satisfies $a\leq c$, or else there is no $i$th entry. In fact, we show this holds for each intermediate tableau obtained from $T((q,p),\rho)$ by some of the slides used to form $T((p+1,q-1),\rho')$. We argue in the case that $p<q$, that is, for inward slides. Suppose that $T$ is an intermediate tableau satisfying the claim, and that the tableau $T'$ obtained from $T$ by a single inward slide is also an intermediate tableau. It follows that $T'$ has skew shape, so that if $(y+s,t)/(y,0)$ is the shape of $T$, then $y>1$. Suppose that during the slide the box changes rows at the $j$th column. We claim that $j\geq i+y-1(>i)$. If this occurs, then the first $i$ entries in the second row and first $i-1$ entries in the first row of $T$ are unchanged in $T'$. Also, the $i$th entry in the first row of $T'$ is either the $i$th entry in the first row of $T$ (if $j\geq i+y$) or it is the $j$th entry in the second row of $T$, which is greater than the $i$th entry, $a$. Thus showing $j\geq i+y-1$ completes the proof. To see that $j\geq i+y-1$ note that if $j$ is the last column, then $j=t=s+y$. Since $s\geq i-1$, we see that $j\geq y+i-1$. If $j$ is not the last column, then the entries $b$ at $(j,2)$ $c$ at $(j+1,1)$ of $T$ satisfy $b\leq c$. Suppose that $j<i+y-1$. Then $c$ is the ($j-y+1$)th entry in the first row of $T$. Since $j-y+1<i$, our choice of $i$ ensures that $c$ is less than the entry at $(j-y+1,2)$ of $T$. Since $j-y+1<j$, this in turn is less than $b$, a contradiction. Similar arguments suffice for the case when $p\geq q$. \QED \begin{rem} While it may seem this proof has only a formal relation to the proof of Remmel and Shimozono~\cite{Remmel_Shimozono}, it is in fact nearly an exact translation---the only difference being in our choice of $r$. (Their choice of $r$ is not easily expressed in this setting.) We elaborate. The exact same proof, but with the ordinary {\em jeu de taquin}, shows that $c^{[\mu,\lambda]_\subset}_\nu$ counts the chains in $[\mu,\lambda]_\subset$ whose word is the word of a tableau of shape $\nu$. This is just the Littlewood-Richardson coefficient $c^\lambda_{\mu\,\nu}$. One way to see this is to consider the bijection between $H_\nu([\mu,\lambda]_\subset)$ and the set of semistandard Young tableaux of shape $\lambda/\mu$ and content $(\nu_k,\ldots,\nu_1)$. The chains whose word is the word of a tableau of shape $\nu$ correspond to {\em reverse} LR tableaux of shape $\lambda/\mu$, which are defined as follows: Let $f_{a,b}(T)$ be the number of $a$'s in the first $b$ positions of the word of $T$. A reverse LR tableau $T$ with largest entry $k$ is a tableau satisfying: $$ f_{1,b}(T)\ \leq\ f_{2,b}(T)\ \leq\ \cdots\ \leq\ f_{k,b}(T) $$ for all $b$. It is an exercise to verify that there are exactly $c^\lambda_{\mu\nu}$ reverse LR tableaux of shape $\lambda/\mu$ and content $\nu_k,\ldots,\nu_2,\nu_1$. The choice we make of $i$ and $r$ is easily expressed in these terms: $i$ is the minimum value of $f_{a,b}(T)$ among all violations $f_{a,b}(T)>f_{a+1,b}(T)$, and if $a$ is the minimal first index among all violations with $f_{a,b}(T)=i$, then $r=k-a$. The choice in~\cite{Remmel_Shimozono} for reverse LR tableaux would be $r=k-a$, where $f_{a,b}(T)$ is the violation with minimal $b$. The key step we used was the {\em jeu de taquin} whereas Remmel and Shimozono used an operation built from the $r$-pairing of Lascoux and Sch\"utzenberger~\cite{LS81}. In fact, this too is a direct translation. The reason for this is, roughly, that the passage from the word of a chain $\rho\in H_\alpha([\mu,\lambda]_\subset)$ to a semistandard Young tableau of shape $\lambda/\mu$ and content $(\alpha_k,\ldots,\alpha_1)$ (which interchanges shape with content) also interchanges Knuth equivalence and dual Knuth equivalence~\cite{Haiman_dual_equivalence}. The operators constructed from the $r$-pairing preserve the dual equivalence class of a 2-letter word but alter its content. In fact, this property characterizes such an operation. As shown in~\cite{Haiman_dual_equivalence}, there is at most one tableau in a given Knuth equivalence class and a given dual equivalence class. Also, for semistandard Young tableaux with at most 2 letters, there is at most one tableau with given partition shape and content. It follows that any operation on tableaux acting on the subtableau of entries $r,r+1$ which preserves the dual equivalence class of the subtableau, but reverses its content is uniquely defined by these properties. Thus the symmetrization operators in~\cite{LS81}, which generate an ${\mathcal S}_\infty$-action on tableaux extending the natural action on their contents, is unique. Expressed in this form, we see that this action coincides with one introduced earlier by Knuth~\cite{Knuth}. This action was the effect of permuting rows of a matrix on the $P$-symbol obtained from that matrix by Knuth's generalization of the Robinson-Schensted correspondence. The origin of these symmetrization operators in the work of Knuth has been overlooked by most authors, perhaps because Bender-Knuth~\cite{Bender_Knuth} later use a different operation to prove symmetry. \end{rem} For each poset $P$ in the classes of labeled posets we consider here, the symmetric function $S_P$ is Schur-positive. When $P$ is an interval in some $k$-Bruhat order, this follows from geometry, for intervals in Young's lattice, this is a consequence of the Littlewood-Richardson rule, and for intervals in the weak order, it is due to Lascoux-Sch\"utzenberger~\cite{LS82a} and Edelman-Greene~\cite{EG}. Is there a representation-theoretic explanation? In particular, we ask:\medskip \noindent{\bf Question:} {\em If $P$ is an interval in a $k$-Bruhat order, can one construct a representation $V_P$ of ${\mathcal S}_{\mbox{\scriptsize\rm rank}P}$ so that $S_P$ is its Frobenius character? More generally, for a labeled poset $P$, can one define a (virtual) representation $V_P$ so that $S_P$ is its Frobenius character? If so, is $V_{P\times Q}\simeq V_P\otimes V_Q$? }\medskip When P is an interval in Young's lattice this is a skew Specht module. For an interval $[1,w]_{\mbox{\scriptsize\rm weak}}$ in the weak order, Kr\'askiewicz~\cite{Kraskiewicz} constructs a ${\mathcal S}_{\ell(w)}$-representation of dimension $\# R(w)$. For general linear group represenations, such a construction is known. For intervals in the weak oder, this is due to Kr\'askiewicz and Pragacz~\cite{KP}. \section{The monomials in a Schubert polynomial} We give a new proof based upon geometry that a Schubert polynomial is a sum of monomials with non-negative coefficients. This analysis leads to a combinatorial construction of Schubert polynomials in terms of chains in the Bruhat order. It also shows these coefficients are certain intersection numbers, essentially the same interpretation found by Kirillov and Maeno~\cite{KM}. The first step is Theorem~\ref{thm:univariate}, which generalizes both Proposition 1.7 of~\cite{LS82b} and Theorem C ({\em ii}) of~\cite{BS97a}. Recall that $u\stackrel{r[m,k]}{\relbar\joinrel\relbar\joinrel\longrightarrow} w$ when one of the following equivalent conditions holds: \begin{itemize} \item $c^w_{u,\,r[m,k]}=1$. \item $u\leq_k w$ and $wu^{-1}$ is a disjoint product of increasing cycles. \item There is an chain in $[u,w]_k$: $$ u\ \stackrel{b_1}{\longrightarrow}\ u_1 \stackrel{b_2}{\longrightarrow}\ \cdots\ \stackrel{b_m}{\longrightarrow}\ u_m=w $$ with $b_1<b_2<\cdots<b_m$. \end{itemize} For $p\in{\mathbb N}$, define the map $\Phi_p:{\mathbb Z}[x_1,x_2,\ldots]\longrightarrow {\mathbb Z}[y]\otimes {\mathbb Z}[x_1,x_2,\ldots]$ by $$ \Phi_p(x_i)\ =\ \left\{\begin{array}{ll} x_i &\ \mbox{if}\ i<p\\ y &\ \mbox{if}\ i=p\\ x_{i-1}&\ \mbox{if}\ i>p \end{array}\right.. $$ For $w\in{\mathcal S}_\infty$ and $p,q\in{\mathbb N}$, define $\varphi_{p,q}(w)\in{\mathcal S}_\infty$ by $$ \varphi_{p,q} (w)(j) \quad =\quad \left\{\begin{array}{lcl} w(j) && j < p \mbox{ and } w(j) < q\\ w(j)+1 && j < p \mbox{ and } w(j)\geq q\\ q && j = p\\ w(j-1) && j > p \mbox{ and } w(j) < q\\ w(j-1)+1 && j > p \mbox{ and } w(j)\geq q \end{array}\right.. $$ Representing permutations as matrices, $\varphi_{p,q}$ adds a new $p$th row and $q$th column consisting mostly of zeroes, but with a 1 in the $(p,q)$th position. For example, $$ \varphi_{3,3}(23154)\ =\ 243165 \qquad\mbox{and}\qquad \varphi_{2,5}(2341)\ =\ 25342. $$ \begin{thm}\label{thm:univariate} For $u\in{\mathcal S}_n$, $$ \Phi_p {\mathfrak S}_u\ =\ \sum_{\stackrel{\mbox{\scriptsize $j, w$ \rm with}}% {u\stackrel{r[n{+}1{-}p{-}j,\,p]}% {\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow} \varphi_{p,n+1}(w)}} y^j\:{\mathfrak S}_w(x). $$ Moreover, if $n$ is not among $\{u(1),\ldots,u(p-1)\}$, then the sum may be taken over those $j,w$ with $u\stackrel{r[n{-}p{-}j,\,p]}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow} \varphi_{p,n}(w)$. \end{thm} Iterating this gives another proof that the monomials in a Schubert polynomial have non-negative coefficients. \begin{ex} Consider $\Phi_2{\mathfrak S}_{13542}$. We display all increasing chains in the 2-Bruhat order on ${\mathcal S}_5$ above $13542$ whose endpoint $w$ satisfies $w(2)=5$: $$\epsfxsize=2.in \epsfbox{univar.eps}$$ We see therefore that \begin{eqnarray*} 13542\ \stackrel{r[3,2]}{\relbar\joinrel\relbar\joinrel\longrightarrow} 25431&= & \varphi_{2,5}(2431),\\ 13542\ \stackrel{r[2,2]}{\relbar\joinrel\relbar\joinrel\longrightarrow} 25341&= & \varphi_{2,5}(2341),\\ 13542\ \stackrel{r[2,2]}{\relbar\joinrel\relbar\joinrel\longrightarrow} 15432&= & \varphi_{2,5}(1432),\\ 13542\ \stackrel{r[1,2]}{\relbar\joinrel\relbar\joinrel\longrightarrow} 15342&= & \varphi_{2,5}(1342). \end{eqnarray*} Then Theorem~\ref{thm:univariate} asserts that $$ \Phi_2{\mathfrak S}_{13542}\ =\ {\mathfrak S}_{2431}(x) + y{\mathfrak S}_{2341}(x) + y{\mathfrak S}_{1432}(x) + y^2{\mathfrak S}_{1342}(x), $$ which may also be verified by direct calculation. \end{ex} \noindent{\bf Proof of Theorem~\ref{thm:univariate}. } We make two definitions. For $p\leq n$, define another map $\psi_{p,[n]}:{\mathcal S}_n\times {\mathcal S}_m \hookrightarrow {\mathcal S}_{n+m}$ by \begin{equation}\label{eq:psi-map} \psi_{p,[n]}(w,z)(i)\ =\ \left\{\begin{array}{ll} w(i) &\ i<p\\ n+z(1) &\ i=p\\ w(i-1) &\ p<i\leq n+1\\ n+z(i-n)&\ n+1<i\leq n+m \end{array}\right.. \end{equation} Then $\psi_{p,[n]}(1,1)=r[n{+}1{-}p,p]$. Let $P\subset\{1,2,\ldots,n+m\}$ and suppose that \begin{eqnarray*} P&=&p_1<p_2<\cdots<p_n,\\ \{1,\ldots,n+m\}- P&=& q_1<q_2<\cdots<q_m. \end{eqnarray*} Define the map $\Psi_P:{\mathbb Z}[x_1,x_2,\ldots,x_{n+m}]\longrightarrow {\mathbb Z}[x_1,\ldots,x_n]\otimes{\mathbb Z}[y_1,\ldots,y_m]$ by $$ \Psi_P(x_i)\ =\ \left\{\begin{array}{rl} x_j&\ \mbox{if}\ i=p_j\\ y_j&\ \mbox{if}\ i=q_j \end{array}\right.. $$ Suppose now that $P=\{1,2,\ldots,p-1,p+1,\ldots,n+1\}$. Then for $u\in{\mathcal S}_{n+m}$, Theorem 4.5.4 of~\cite{BS97a} asserts that \begin{equation}\label{eq:coh} \Psi_P{\mathfrak S}_u\ \equiv\ \sum_{w\in{\mathcal S}_n,\ z\in{\mathcal S}_m} c^{\psi_{p,[n]}(w,z)}_{u\ r[n{+}1{-}p,p]} {\mathfrak S}_w(x)\otimes{\mathfrak S}_z(y), \end{equation} modulo the ideal $\langle {\mathfrak S}_w(x)\otimes 1, 1\otimes{\mathfrak S}_z(y) \mid w\not\in{\mathcal S}_n,z\not\in{\mathcal S}_m\rangle$ which is equal to the ideal $\langle x^{\alpha}\otimes 1, 1\otimes y^{\alpha} \mid \alpha_i\geq n-i\mbox{ for some }i\rangle$. (The calculation is in the cohomology of the product of flag manifolds {\em Flags}$({\mathbb C}\,^n)\times${\em Flags}$({\mathbb C}\,^m)$.) Suppose now that $u\in{\mathcal S}_n$ and $m\geq n$. Then~(\ref{eq:coh}) is an identity of polynomials, and not just of cohomology classes. We also see that $\Psi_P{\mathfrak S}_u=\Phi_p{\mathfrak S}_u$, since ${\mathfrak S}_u\in{\mathbb Z}[x_1,\ldots,x_n]$. By the Pieri formula, $$ c^{\psi_{p,[n]}(w,z)}_{u\ r[n{+}1{-}p,p]}\ =\ \left\{\rule{0pt}{20pt}\right. \begin{array}{ll} 1 & \ \mbox{if }u\stackrel{r[n{+}1{-}p,\,p]}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow} \psi_{p,[n]}(w,z),\\ 0&\ \mbox{otherwise}.\rule{0pt}{12pt} \end{array} $$ Since $u\leq_p \psi_{p,[n]}(w,z)$ and $u(n+i)=n+i$, Definition~\ref{def:1} (2) (for $u\leq_p \psi_{p,[n]}(w,z)$) implies that $$ \psi_{p,[n]}(w,z)(n+1)<\psi_{p,[n]}(w,z)(n+2)<\cdots. $$ Thus by the definition~(\ref{eq:psi-map}) of $\psi_{p,[n]}$, we have $z(2)<z(3)<\cdots$, and so $z$ is the Grassmannian permutation $r[z(1){-}1,1]$. Hence ${\mathfrak S}_z(y)=y^{z(1)-1}$. If we set $j=z(1)-1$, then $\psi_{P,[n]}(w,z)=\varphi_{p,n+1+j}(w)$. Thus, for $u\in{\mathcal S}_n$, we have $$ \Phi_p{\mathfrak S}_u\ =\ \sum_{\stackrel{\mbox{\scriptsize $j,w$ such that}}% {u\stackrel{r[n{+}1{-}p,\,p]}% {\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\longrightarrow} \varphi_{p,n+1+j}(w)}} y^j\,{\mathfrak S}_w(x). $$ Suppose that $u\stackrel{r[n{+}1{-}p,\,p]}% {\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel% \relbar\joinrel\longrightarrow} \varphi_{p,n+1+j}(w)$. Consider the unique increasing chain in the interval $[u,\ \varphi_{p,n+1+j}(w)]_p$: $$ u=u_0\stackrel{b_1}{\longrightarrow}\ \cdots\ \stackrel{b_{n-p-j}}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\longrightarrow} u_{n-p-j}\stackrel{b_{n+1-p-j}}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\longrightarrow} \ \cdots\ \stackrel{b_{n+1-p}}{\relbar\joinrel\relbar% \joinrel\relbar\joinrel\longrightarrow} \varphi_{p,n+1+j}(w). $$ Because $u\in{\mathcal S}_n$, we must have $b_{n+1-p-j}=n+1$ and so $u_{n+1-p-j}=\varphi_{p,n+1}(w)$. Moreover, if $n$ is not among $\{u(1),\ldots,u(p)\}$, then we have $b_{n-p-j}=n$ and so $u_{n-p-j}=\varphi_{p,n}(w)$. If $u(p)=n$, then we also have $u_{n-p-j}=\varphi_{p,n}(w)$. This completes the proof. \QED Define $\delta$ to be the sequence $(n-1,n-2,\ldots,1,0)$. \begin{cor}\label{cor:chain_monomial} For $w\in{\mathcal S}_n$ and $\alpha<\delta$, the coefficient of $x^{\delta-\alpha}$ in ${\mathfrak S}_w$ is the number of chains $$ w\lessdot w_1\lessdot w_2\lessdot\cdots\lessdot w_{\alpha_1+\cdots+\alpha_{n-1}} = \omega_0 $$ in the Bruhat order where, for each $1\leq k\leq n-1$, \begin{equation}\label{eq:ch-cond} w_{\alpha_1+\cdots+\alpha_{k-1}}\:\lessdot_k\: w_{1+\alpha_1+\cdots+\alpha_{k-1}}\: \lessdot_k\:\cdots\:\lessdot_k\: w_{\alpha_1+\cdots+\alpha_k} \end{equation} is an increasing chain in the $k$-Bruhat order. \end{cor} \begin{ex} Here are all such chains in ${\mathcal S}_4$ from $1432$ to $4321$, with the index $\alpha$ displayed above each chain: $$\epsfxsize=2.in \epsfbox{subs.eps}$$ {}From this, we see that \begin{eqnarray*} {\mathfrak S}_{1432}&=& x^{321-111}+x^{321-120}+x^{321-201}+x^{321-210}+x^{321-300}\\ &=&x_1^2x_2 + x_1^2x_3 + x_1x_2^2 + x_1x_2x_3 + x_2^2x_3. \end{eqnarray*} \end{ex} \noindent{\bf Proof. } Repeatedly applying $\Phi_1$ and iterating Theorem~\ref{thm:univariate}, we see that the coefficient of $x^{\delta-\alpha}$ in ${\mathfrak S}_w(x)$ is the number of chains $$ w\lessdot w_1\lessdot w_2\lessdot\cdots\lessdot w_{\alpha_1+\cdots+\alpha_{n-1}} = \omega_0 $$ which satisfy the conditions of the corollary, together with the (apparent) additional requirement that, for each $k<n$, \begin{equation} \label{eq:value} w_{\alpha_1+\cdots+\alpha_k}(j)\ =\ n+1-j\ \mbox{ for all }\ j \leq k. \end{equation} The corollary will follow, once we show this is no additional restriction. First note that if $u\stackrel{r[a,k]}{\relbar\joinrel\relbar\joinrel\longrightarrow}u'$ with $u'(j)=n+1-j$ for $1\leq j\leq k$, but $u(i)<n+1-i$ for some $1\leq i\leq k$, then $i=k$. To see this, note that since $u\leq_k u'$, the form of $u'$ and Definition~\ref{def:1} (2) implies that $u(1)>u(2)>\cdots>u(k)$. Set $\zeta=u'u^{-1}$. Since $u\stackrel{r[a,k]}{\relbar\joinrel\relbar\joinrel\longrightarrow}u'$, $\zeta$ is a disjoint product of increasing cycles, hence their supports are are non-crossing. Suppose $i<k$. Then $\{u(i),n+1-i=u'(i)\}$ and $\{u(i+1),n-i=u'(i+1)\}$ are in the support of distinct cycles. However, $u(i+1)<u(i)\leq n-i<n+1-i$ contradicts that these supports are non-crossing, so we must have $i=k$. Let $$ w\lessdot w_1\lessdot w_2\lessdot\cdots\lessdot w_{\alpha_1+\cdots+\alpha_{n-1}} = \omega_0 $$ be a chain which satisfies the conditions of the corollary. We prove that~(\ref{eq:value}) holds for all $k<n$ by downward induction. Since $\omega_0=w_{\alpha_1+\cdots+\alpha_{n-1}}$, we see that~(\ref{eq:value}) holds for $k=n-1$. Suppose that~(\ref{eq:value}) holds for some $k$. Set $u=w_{\alpha_1+\cdots+\alpha_{k-1}}$ and $u'=w_{\alpha_1+\cdots+\alpha_k}$. Then $u\stackrel{r[\alpha_k,k]}{\relbar\joinrel\relbar\joinrel\longrightarrow}u'$ with $u'(j)=n+1-j$ for $1\leq j\leq k$. By the previous paragraph, we must have $u(i)=n+1-i$ for all $i<k$, hence~(\ref{eq:value}) holds for $k-1$. \QED We could also have written the coefficient of $x^{\delta-\alpha}$ in ${\mathfrak S}_w(x)$ as the number of chains $$ w\ \stackrel{r[\alpha_1,1]}{\relbar\joinrel% \relbar\joinrel\relbar\joinrel\longrightarrow}\ w_1\ \stackrel{r[\alpha_2,2]}{\relbar\joinrel% \relbar\joinrel\relbar\joinrel\longrightarrow}\ w_2\ \stackrel{r[\alpha_3,3]}{\relbar\joinrel% \relbar\joinrel\relbar\joinrel\longrightarrow}\ \cdots\ \stackrel{r[\alpha_{n-1},n{-}1]}{\relbar\joinrel\relbar\joinrel% \relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow}\ \omega_0 $$ in ${\mathcal S}_n$. {}From this and the Pieri formula for Schubert polynomials, we obtain another description of these coefficients. First, for $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_{n-1})$ with $\alpha_i\geq 0$, let $h(\alpha)$ denote the product of complete homogeneous symmetric polynomials $$ h_{\alpha_1}(x_1) h_{\alpha_2}(x_1,x_2)\cdots h_{\alpha_{n-1}}(x_1,x_2,\ldots,x_{n-1}). $$ \begin{cor} For $w\in{\mathcal S}_n$, $$ {\mathfrak S}_w \ =\ \sum_{\alpha} d^w_{\alpha}x^{\delta-\alpha} $$ where $d^w_{\alpha}$ is the coefficient of ${\mathfrak S}_{\omega_0}$ in the product ${\mathfrak S}_w\cdot h(\alpha)$. \end{cor} This is essentially the same formula as found by Kirillov and Maeno~\cite{KM} who showed that the coefficient of $x^{\delta-\alpha}$ in ${\mathfrak S}_w$ to be the coefficient of ${\mathfrak S}_{\omega_0}$ in the product ${\mathfrak S}_{\omega_0 w\omega_0}\cdot e(\alpha)$, where $$ e(\alpha)=e_{\alpha_{n-1}}(x_1)e_{\alpha_{n-2}}(x_1,x_2)\cdots e_{\alpha_1}(x_1,\ldots,x_{n-1}). $$ To see these are equivalent, note that the algebra involution ${\mathfrak S}_w\mapsto{\mathfrak S}_{\overline{w}}$ on $H^*(\mbox{\em Flags}({\mathbb C}\,^n))$ interchanges $e(\alpha)$ and $h(\alpha)$. \section*{Acknowledgments} We thank Mark Shimozono and Richard Stanley for helpful comments. The second author is grateful to the hospitality of Universit\'e Gen\`eve and INRIA Sophia-Antipolis, where portions of this paper were developed and written.

9709

alg-geom/9709010

en

https://arxiv.org/abs/alg-geom/9709010

alg-geom/9709010

Yuri Tschinkel

Matthias Strauch and Yuri Tschinkel

Height zeta functions of toric bundles over flag varieties

64 pages, LaTeX

We investigate analytic properties of height zeta functions of toric bundles over flag varieties.

alg-geom

\section{Introduction} \label{1} \subsection* \noindent {\bf 1.1}\hskip 0,5cm Let $X$ be a nonsingular projective algebraic variety over a number field $F$. Let ${\cal L}=\left(L, (\|\cdot\|_v)_v\right)$ be a metrized line bundle on $X$, i.e., a line bundle $L$ together with a family of $v$-adic metrics, where $v$ runs over the set $\Val (F) $ of places of $F$. Associated to ${\cal L}$ there is a height function $$ H_{{\cal L}}\,:\, X(F) \rightarrow {\bf R}_{>0} $$ on the set $X(F)$ of $F$-rational points of $X$ (cf. \cite{ST,Peyre} for the definitions of $v$-adic metric, metrized line bundle and height function). For appropriate subvarieties $U\subset X$ and line bundles $L$ we have $$ N_U({\cal L},H):=\{x\in U(F)\,\,|\,\,H_{\cal L}(x)\le H\} <\infty $$ for all $H$ (e.g., this holds for any $U$ if $L$ is ample). We are interested in the asymptotic behavior of this counting function as $H \rightarrow \infty$. It is expected that the behavior of such asymptotics can be described in geometric terms (\cite{BaMa,FMT}). \noindent Let $$ \Lambda _{\rm eff}(X):=\sum_{H^0(X,L)\neq 0} {\bf R}_{\ge 0}[L]\subset {\rm Pic} (X)_{{\bf R}} $$ be the closed cone in ${\rm Pic}(X)_{{\bf R}}$ generated by the classes of effective divisors ($[L]$ denotes the class of the line bundle $L$ in ${\rm Pic}(X)$). Let $L$ be a line bundle on $X$ such that $[L]$ lies in the interior of $ \Lambda _{\rm eff}(X)$. Define $$ a(L):=\inf\{a\in {\bf R}\,\,|\,\, a[L]+[K_X]\in \Lambda _{\rm eff}(X)\}, $$ where $K_X$ denotes the canonical line bundle on $X$. Assume that $ \Lambda _{\rm eff}(X)$ is a finitely generated polyhedral cone. For $L$ as above we let $b(L)$ be the codimension of the minimal face of $ \Lambda _{\rm eff}(X)$ which contains $a(L)[L]+[K_X]$. By a Tauberian theorem (cf. \cite{De}, Th\'eor\`eme III), the asymptotic behavior of $ N_U({\cal L},H)$ can be determined if one has enough information about the height zeta function $$ Z_U({\cal L},s):= \sum_{x\in U(F)}H_{\cal L}(x)^{-s}. $$ More precisely, suppose that $Z_U({\cal L},s)$ converges for ${\rm Re}(s)\gg 0$, that it has an abscissa of convergence $a>0$ and that it can be continued meromorphically to a half-space beyond the abscissa of convergence. Suppose further that there is a pole of order $b$ at $s=a$ and that there are no other poles in this half-space. Then $$ N_U({\cal L},H)=cH^a(\log H)^{b-1}(1+o(1)) $$ for $H \rightarrow \infty$ and $$ c=\frac{1}{(b-1)!a}\cdot \lim_{s \rightarrow a} (s-a)^bZ_U({\cal L},s). $$ It is conjectured that for appropriate $U$ and ${\cal L}$ one has $a=a(L)$ and $b=b(L)$ (cf. \cite{BaMa,FMT}). Moreover, there is a conjectural framework how to interpret the constant $c$ (cf. \cite{Peyre,BaTschi5}). There are examples which demonstrate that this geometric ``prediction'' of the asymptotic cannot hold in complete generality, even for smooth Fano varieties (cf. \cite{BaTschi4}). Our goal is to show that the conjectures do hold for a class of varieties closely related to linear algebraic groups. Our results are a natural extension of corresponding results for flag varieties (cf. \cite{FMT}) and toric varieties (cf. \cite{BaTschi2,BaTschi3}). We proceed to describe the class of varieties under consideration. \subsection* \noindent {\bf 1.2}\hskip 0,5cm Let $G$ be a semi-simple simply connected split algebraic group over $F$ and $P\subset G$ an $F$-rational parabolic subgroup of $G$. Let $T$ be a split algebraic torus over $F$ and $X$ a projective nonsingular equivariant compactification of $T$. A homomorphism $\eta\,:\, P \rightarrow T$ gives rise to an action of $P$ on $X\times G$ and the quotient $Y:=(X\times G)/P$ is again a nonsingular projective variety over $F$. There is a canonical morphism $\pi\,:\, Y \rightarrow W:= P \backslash G$ such that $Y$ becomes a locally trivial fiber bundle over $W$ with fiber $X$. Corresponding to a character $ \lambda \in X^*(P)$ there is a line bundle $L_{ \lambda }$ on $W$ and the assignment $ \lambda \mapsto L_{ \lambda }$ gives an isomorphism $X^*(P) \rightarrow {\rm Pic}(W)$. The toric variety $X$ can be described combinatorially by a fan $\Sigma$ in the dual space of the space of characters $X^*(T)_{{\bf R}}$. Let $PL(\Sigma)$ be the group of $\Sigma$-piecewise linear integral functions on the dual space of $X^*(T)_{{\bf R}}$. Any $ \varphi \in PL(\Sigma)$ defines a line bundle $L_{ \varphi }$ on $X$ which is equipped with a canonical $T$-linearization and we get an isomorphism $PL(\Sigma)\simeq {\rm Pic}^T(X). $ There is a canonical exact sequence $$ 0 \rightarrow X^*(T) \rightarrow PL(\Sigma) \rightarrow {\rm Pic}(X) \rightarrow 0. $$ The $T$-linearization of $L_{ \varphi }$ allows us to define a line bundle $L^Y_{ \varphi }$ on $Y$ and this gives a homomorphism $PL(\Sigma) \rightarrow {\rm Pic}(Y)$. One can show that there is an exact sequence \begin{equation} \label{1.2.1} 0 \rightarrow X^*(T) \rightarrow PL(\Sigma)\oplus X^*(P) \rightarrow {\rm Pic}(Y) \rightarrow 0. \end{equation} Denote by $Y^{o}:=(T\times G)/P$ the open subvariety of $Y$ obtained as the twist of $T $ with $W$. \subsection* \noindent {\bf 1.3} \hskip 0,5cm By means of a maximal compact subgroup in the adelic group $G({\bf A})$ we can introduce metrics on the line bundles $L_{ \lambda }$. The corresponding height zeta functions are Eisenstein series: $$ \sum_{w\in W(F)}H_{{\cal L}_{ \lambda }}(w)^{-s}=E^G_P(s \lambda -\rho_P,1_G). $$ On the other hand, for any $ \varphi \in PL(\Sigma)$ there is a function $$ H_{\Sigma}(\,\cdot\, , \varphi )\,:\, T({\bf A}) \rightarrow {\bf R}_{>0} $$ such that $H_{\Sigma}(x, \varphi )^{-1}$ is the height of $x\in T(F)$ with respect to a metrization ${\cal L}_{ \varphi }$ of $L_{ \varphi }$. This metrization induces a metrization ${\cal L}_{ \varphi }^Y$ of the line bundle $L^Y_{ \varphi }$ on $Y$. Let $(x, \gamma)\in T(F)\times G(F)$ and let $y$ be the image of $(x, \gamma) $ in $Y(F)$. Then there is a $p_{ \gamma}\in P({\bf A})$ such that $$ H_{{\cal L}^Y_{ \varphi }}(y)=H_{\Sigma}(x\eta(p_{ \gamma}), \varphi )^{-1}. $$ Hence we may write formally \begin{equation} \label{1.3.3} Z_{Y^o}({\cal L}_{ \varphi }^Y\otimes \pi^*{\cal L}_{ \lambda },s) =\sum_{ \gamma\in P(F) \backslash G(F)}H_{{\cal L}_{ \lambda }}( \gamma)^{-s}\sum_{x\in T(F)} H_{\Sigma}(x\eta(p_{ \gamma}),s \varphi ). \end{equation} Now we apply Poisson's summation formula for the the torus and get \begin{equation} \label{1.3.4} \sum_{x\in T(F)} H_{\Sigma}(x\eta(p_{ \gamma}),s \varphi )=\int_{(T({\bf A})/T(F))^*}\hat{H}_{\Sigma}(\chi,s \varphi ) \chi(\eta(p_{ \gamma}))^{-1}d \chi , \end{equation} where $\hat{H}_{\Sigma}(\,\cdot\, ,s \varphi )$ denotes the Fourier transform of ${H}_{\Sigma}(\,\cdot\, ,s \varphi )$ and $(T({\bf A})/T(F))^*$ is the group of unitary characters of $T({\bf A})$ which are trivial on $T(F)$ equipped with the orthogonal measure $d \chi $. Actually, it is sufficient to consider only those characters which are trivial on the maximal compact subgroup ${\bf K}_T$ of $T({\bf A})$, because the function $H_{\Sigma}(\,\cdot\, , \varphi )$ is invariant under ${\bf K}_T$. The expression (\ref{1.3.4}) can now be put into (\ref{1.3.3}), and after interchanging summation and integration the result is \begin{equation} \label{1.3.5} Z_{Y^o}({\cal L}_{ \varphi }^Y\otimes \pi^* {\cal L}_{ \lambda },s)= \int_{(T({\bf A})/T(F){\bf K}_T)^*}\hat{H}_{\Sigma}( \chi ,s \varphi )E^G_P(s \lambda -\rho_P, (\chi\circ\eta)^{-1})d \chi \end{equation} where $E^G_P(s \lambda -\rho_P,\xi)= E^G_P(s \lambda -\rho_P,\xi,1_G)$ is the Eisenstein series twisted by a character $\xi$ of $ P({\bf A})$. This is the starting point for the investigation of the height zeta function. To get an expression which is more suited for our study we decompose the group of characters $(T({\bf A})/T(F){\bf K}_T)^*$ into a continuous and a discrete part, i.e., $$ (T({\bf A})/T(F){\bf K}_T)^*=X^*(T)_{{\bf R}}\oplus {\cal U}_T, $$ where $X^*(T)_{{\bf R}}$ is the continuous part and ${\cal U}_T$ is the discrete part. The right-hand side of (\ref{1.3.5}) is accordingly \begin{equation} \label{1.3.6} \int_{X^*(T)_{{\bf R}}}\left\{ \sum_{ \chi \in {\cal U}_T} \hat{H}_{\Sigma}( \chi ,s \varphi +ix)E^G_P(s \lambda -\rho_P -i\check{\eta}(x), (\chi\circ\eta)^{-1})\right\}dx. \end{equation} Recall that we would like to show that this function which is defined for ${\rm Re}(s)\gg 0$ (assuming that $( \varphi , \lambda )$ is contained in a convex open cone) can be continued meromorphically beyond the abscissa of convergence. To achieve this we need more information on the function under the integral sign in (\ref{1.3.6}). First we have to determine the singularities of $$ ( \varphi , \lambda )\mapsto \hat{H}_{\Sigma}( \chi , \varphi )E^G_P( \lambda -\rho_P,( \chi \circ\eta)^{-1}) $$ near the cone of absolute convergence. This is possible because $\hat{H}_{\Sigma}$ can be calculated rather explicitly and it is not so difficult to determine the singular hyperplanes of the Eisenstein series with characters. The next step consists in an iterated application of Cauchy's residue formula to the integral over the real vector space $X^*(T)_{{\bf R}}$. This can be done only if one knows that \begin{equation} \label{1.3.7} \sum_{ \chi \in {\cal U}_T} \hat{H}_{\Sigma}( \chi ,s \varphi +ix)E^G_P(s \lambda -\rho_P-i\check{\eta}(x), ( \chi \circ\eta)^{-1}) \end{equation} satisfies some growth conditions when $x\in X^*(T)_{{\bf R}}$ tends to infinity. This is true for the function $x\mapsto \hat{H}_{\Sigma}( \chi ,s \varphi +ix)$ thanks to the explicit expression mentioned above. The absolute value of the Eisenstein series $E^G_P(s \lambda -\rho_P -i\check{\eta}(x),( \chi \circ\eta)^{-1})$ will in general increase for $x \rightarrow \infty$ if ${\rm Re}(s) \lambda -\rho_P$ is not contained in the cone of absolute convergence. However, if ${\rm Re}(s) \lambda -\rho_P$ is sufficiently close to the boundary of that cone, this increasing behavior is absorbed by the decreasing behavior of $\hat{H}_{\Sigma}( \chi ,s \varphi +ix)$. Therefore, we may apply Cauchy's residue theorem and show that (\ref{1.3.5}) can be continued meromorphically to a larger half-space and that there are no poles (in $s$) with non-zero imaginary part. The Tauberian Theorem can now be used to prove asymptotic formulas for the counting function $N_{Y^o}({\cal L}^Y_{ \varphi }\otimes \pi^*{\cal L}_{ \lambda },H)$ provided that one knows the order of the pole of the height zeta function. This problem can be reduced to the question whether the ``leading term'' of the Laurent series of (\ref{1.3.7}) does not vanish. That this is indeed so will be shown in section 6. \subsection* \noindent {\bf 1.4}\hskip 0,5cm We have restricted ourself to the case of split tori and split groups because this simplifies some technical details. The general case can be treated similarly. We consider these results as an important step towards an understanding of the arithmetic of spherical varieties. For example, choosing $P=B$ a Borel subgroup, $T=B/U$ where $U$ is the unipotent radical of $B$ and $\eta\,:\, B \rightarrow T$ the natural projection, we obtain an equivariant compactification of $U \backslash G$, a horospherical variety. We close this introduction with a brief description of the remaining sections. Section 2 recalls the relevant facts we need concerning generalized flag varieties, i.e., description of line bundles on $W=P \backslash G$, the cone of effective divisors in ${\rm Pic}(W)_{{\bf R}}$, metrization of line bundles, height zeta functions. The exposition is based entirely on the paper \cite{FMT}. The next section contains the corresponding facts for toric varieties. It is a summary of a part of \cite{BaTschi1}. We give the explicit calculation of the Fourier transform $\hat{H}_{\Sigma}(\,\cdot \, , \varphi )$ and show that Poisson's summation formula can be used to give an expression of the height zeta function $Z_T({\cal L}_{ \varphi },s)$ . In section 4 we introduce twisted products, discuss line bundles on these, the Picard group (cf. (\ref{1.2.1})), metrizations of line bundles etc. It ends with the formula (\ref{1.3.5}) for the height zeta function $Z_{Y^o}({\cal L}_{ \varphi }^Y\otimes \pi^* {\cal L}_{ \lambda },s)$ in the domain of absolute convergence. The first part of section 5 explains the method for the proof that the height zeta function can be continued meromorphically to a half-space beyond the abscissa of absolute convergence. Moreover, we state a theorem which gives a description of the coefficient of the Laurent series at the pole in question. This coefficient will be the leading one, provided that it does not vanish. One can relate the coefficient to arithmetic and geometric invariants of the pair $(U,{\cal L})$ but we decided not to pursue this, since there are detailed expositions of all the necessary arguments in \cite{Peyre,BaTschi1,BaTschi5}. These two theorems (meromorphic continuation of certain integrals and the description of the coefficient) will be proved in a more general context in section 7. The second part of section 5 contains the proof that the hypothesis of these theorems are fulfilled in our case. It ends with the main theorem on the asymptotic behavior of the counting function $N_{Y^o}({\cal L},H)$, assuming that the coefficient of the Laurent series mentioned above does not vanish. Section 6 is devoted to the proof of this fact. In the last section we prove some statements on Eisenstein series (well-known to the experts) which are used in section 5. \bigskip \noindent {\bf Acknowledgements.} We are very grateful to V. Batyrev and J. Franke for helpful discussions and collaboration on related questions. The first author was supported by the DFG-Graduiertenkolleg of the Mathematics Institute of the University of Bonn. Part of this work was done while the second author was visiting the MPI in Bonn, ETH Z\"urich and ENS Paris. He would like to thank these institutions for their hospitality. \bigskip \noindent {\bf Some notations.} \hskip 0,5cm In this paper $F$ always denotes a fixed algebraic number field. The set of places of $F$ will be denoted by $\Val(F)$ and the subset of archimedean places by $\Val_{\infty}(F)$. We shall write $v\mid\infty$ if $v\in \Val_{\infty}(F)$ and $v\nmid \infty$ if $v\notin \Val_{\infty}(F)$ . For any place $v$ of $F$ we denote by $F_v$ the completion of $F$ at $v$ and by ${\cal O}_v$ the ring of $v$-adic integers (for $v\nmid \infty $). The local absolute value $|\cdot|_v$ on $F_v$ is the multiplier of the Haar measure, i.e., $d(ax_v)=|a|_vdx_v$ for some Haar measure $dx_v$ on $ F_v$. Let $q_v$ be the cardinality of the residue field of $F_v$ for non-archimedean valuations and put $q_v=e$ for archimedean valuations. We denote by ${\bf A}$ the adele ring of $F$. For any algebraic group $G$ over $ F$ we denote by $X^*(G)$ the group of (algebraic) characters which are defined over $F$. \bigskip \bigskip \bigskip \section{Generalized flag varieties} \label{2} \subsection* \noindent {\bf 2.1}\hskip 0,5cm Let $G$ be a semi-simple simply connected linear algebraic group which is defined and split over $F$. We fix a Borel subgroup $P_0$ over $F$ and a Levi decomposition $P_0=S_0U_0$ with a maximal $F$-rational torus $S_0$ of $G$. Denote by $P$ a standard (i.e., containing $P_0$) parabolic subgroup and by $W=P\backslash G$ the corresponding flag variety. The quotient morphism $G \rightarrow W$ will be denoted by $\pi_W$. Any character $ \lambda \in X^*(P)$ defines a line bundle $L_{ \lambda }$ on $W$ by $$ \Gamma(U,L_{ \lambda }):= \{f\in {\cal O}_G(\pi^{-1}_W(U))\,\,|\,\, f(pg)= \lambda (p)^{-1}f(g) \hskip 0,2cm \forall g\in \pi^{-1}_W(U), p\in P\, \}. $$ The assignment $ \lambda \mapsto L_{ \lambda }$ gives an isomorphism (because $G$ is assumed to be simply connected) $$ X^*(P) \rightarrow {\rm Pic}(W) $$ (cf. \cite{Sa}, Prop. 6.10). The anti-canonical line bundle ${\omega}^{\vee}_W$ corresponds to $2\rho_P$ (the sum of roots of $S_0$ occurring in the unipotent radical of $P$.) \subsection* \noindent {\bf 2.2}\hskip 0.5cm These line bundles will be metrized as follows. Choose a maximal compact subgroup ${\bf K}_G=\prod_{v}{\bf K}_{G,v}\subset G({\bf A})$ (${\bf K}_{G,v}\subset G(F_v)$), such that the Iwasawa decomposition $$ G({\bf A})=P_0({\bf A}){\bf K}_G $$ holds. Let $v\in {\rm Val}(F)$ and $w\in W(F_v)$. Choose $k\in {\bf K}_{G,v}$ which is mapped to $w$ by $\pi_W$. For any local section $s$ of $L_{ \lambda }$ at $w$ we define $$ \|w^*s\|_w:=|s(k)|_v. $$ This gives a $v$-adic norm $\|\cdot\|_w\,:\,w^*L_{ \lambda } \rightarrow {\bf R}$ and we see that the family $\|\cdot\|_v:=(\|\cdot\|_w)_{w\in W(F_v)}$ is a $v$-adic metric on $L_{ \lambda }$. The family $(\|\cdot\|_v)_{v\in {\rm Val}(F)}$ will then be an adelic metric on $L_{ \lambda }$ (cf. \cite{Peyre} for $ \lambda =2\rho_P$ and \cite{ST} for the definitions of ``$v$-adic metric'' and ``adelic metric''). The metrized line bundle $\left(L_{ \lambda }, (\|\cdot\|_v)_{v}\right)$ will be denoted by ${\cal L}_{ \lambda }$. \subsection* \noindent {\bf 2.3}\hskip 0,5cm Define a map $$ H_P=H_{P,{\bf K}_G}\,:\, G({\bf A}) \rightarrow \Hom_{{\bf C}}(X^*(P)_{{\bf C}},{\bf C}) $$ by $\langle \lambda ,H_P(g)\rangle=\log (\prod_v| \lambda (p_v)|_v)$ for $g=pk$ with $ p=(p_v)_v\in P({\bf A}), k\in {\bf K}_G$ and $ \lambda \in X^*(P)$. For $w=\pi_W( \gamma)\in W(F)$ and $ \gamma\in G(F)$ a simple computation (\cite{FMT}) shows that $$ H_{{\cal L}_{ \lambda }}(w)=e^{-\langle \lambda ,H_P( \gamma)\rangle}. $$ The height zeta function $$ Z_W({\cal L}_{ \lambda }, s)=\sum_{w\in W(F)}H_{{\cal L}_{ \lambda }}(w)^{-s} $$ is therefore an Eisenstein series $$ E^G_P(s \lambda -\rho_P,1_G)= \sum_{ \gamma\in P(F)\backslash G(F)}e^{\langle s \lambda ,H_P( \gamma)\rangle}. $$ To describe the domain of absolute convergence of this series we let $\Delta_0$ be the basis of positive roots of the root system $\Phi(S_0,G)$ which is determined by $P_0$. For any $ \alpha \in \Delta_0$ denote by $\check{ \alpha }$ the corresponding coroot. For $ \lambda \in X^*(P)=X^*(S_0)$ we define $\langle \lambda , \alpha \rangle$ by $( \lambda \circ\check{ \alpha })(t)=t^{\langle \lambda , \alpha \rangle}$ and extend this linearly in $ \lambda $ to $X^*(P_0)_{{\bf C}}$. Restriction of characters defines an inclusion $X^*(P) \rightarrow X^*(P_0)$. Let $$ \Delta_0^P=\{\, \alpha \in \Delta_0\,\,|\,\, \langle\,\cdot\,, \alpha \rangle\,\, {\rm vanishes}\,\,{\rm on}\,\, X^*(P)\,\}, \,\,\,\,\, \Delta _P=\Delta_0 - \Delta_0^P. $$ Put $$ X^*(P)^{+}=\{ \lambda \in X^*(P)_{{\bf R}}\,\,|\,\, \langle \lambda , \alpha \rangle > 0 \,\, {\rm for}\hskip 0,3cm {\rm all}\,\,\, \alpha \in \Delta_P\,\}. $$ By \cite{G}, Th\'eor\`eme 3, the Eisenstein series $$ E^G_P( \lambda ,g)=\sum_{ \gamma \in P(F)\backslash G(F)}e^{ \langle \lambda +\rho_P,H_P( \gamma g)\rangle} $$ converges absolutely for ${\rm Re} ( \lambda )-\rho_P$ in $X^*(P)^+ $ and it can be meromorphically continued to $X^*(P)_{{\bf C}}$ (cf. \cite{MW}, IV, 1.8). The closure of the image of $X^*(P)^{+}$ in ${\rm Pic}(W)_{{\bf R}}$ is the cone $ \Lambda _{\rm eff}(W)$ generated by the effective divisors on $W$ (\cite{J}, II, 2.6). \section{Toric varieties} \label{3} \subsection* \noindent {\bf 3.1}\hskip 0,5cm Let $T$ be a split algebraic torus of dimension $d$ over $F$. We put $M=X^*(T)$ and $N=\Hom(M,{\bf Z})$. Let $\Sigma$ be a complete regular fan in $N_{{\bf R}}$ such that the corresponding smooth toric variety $X=X_{\Sigma}$ is projective (cf. \cite{BaTschi1, Oda}). The variety $X$ is covered by affine open sets $$ U_{ \sigma }={\rm Spec}(F[M\cap \check{ \sigma }]), $$ where $ \sigma $ runs through $\Sigma$ and $\check{ \sigma }$ is the dual cone $$ \check{ \sigma }=\{\,m\in M_{{\bf R}}\,|\, n(m)\ge 0\,\, \forall\, n\in \sigma \,\}. $$ Denote by $PL(\Sigma)$ the group of $\Sigma$-piecewise linear integral functions on $N_{{\bf R}}$. By definition, a function $ \varphi \,:\, N_{{\bf R}} \rightarrow {\bf R}$ belongs to $PL(\Sigma)$ if and only if $ \varphi (N)\in {\bf Z}$ and the restriction of $ \varphi $ to every $ \sigma \in \Sigma$ is the restriction to $ \sigma $ of a linear function on $N_{{\bf R}}$. For $ \varphi \in PL(\Sigma)$ and every $d$-dimensional cone $ \sigma \in \Sigma$ there exists a unique $m_{ \varphi , \sigma }\in M$ such that for all $n\in \sigma $ we have $$ \varphi (n)=n(m_{ \varphi , \sigma }). $$ Fixing for any $ \sigma \in \Sigma$ a $d$-dimensional cone $ \sigma '$ containing $ \sigma $ we put $$ m_{ \varphi , \sigma }=m_{ \varphi , \sigma '}. $$ To any $ \varphi \in PL(\Sigma)$ we associate an invertible sheaf $L_{ \varphi }$ on $X$ as the subsheaf of rational functions on $X$ generated over $U_{ \sigma }$ by $\frac{1}{m_{ \varphi , \sigma }}$, considered as a rational function on $X$ ($L_{ \varphi }$ does not depend on the choice made above). The assignment $ \varphi \mapsto L_{ \varphi }$ gives an exact sequence $$ 0 \rightarrow M \rightarrow PL(\Sigma) \rightarrow {\rm Pic} (X) \rightarrow 0 $$ (cf. \cite{Oda}, Corollary 2.5). Denote by $\theta \,:\, X\times T \rightarrow X$ the action of $T$ on $X$ and by $p_1\,:\, X\times T \rightarrow X$ the projection onto the first factor. The induced $T$-action on the sheaf of rational functions restricts to any subsheaf $L_{ \varphi }$, i.e., there is a canonical $T$-linearization $$ \theta_{ \varphi }\,:\, \theta^*L_{ \varphi } \rightarrow p_1^*L_{ \varphi } $$ (cf. \cite{MFK}, Ch. 1, \S\, 3, for the notion of a $T$-linearization). In section four we will always consider $L_{ \varphi }$ not merely as a line bundle on $X$ but as a $T$-linearized line bundle with this $T$-linearization. In this sense $PL(\Sigma)$ is isomorphic to the group ${\rm Pic}^T(X)$ of isomorphism classes of $T$-linearized line bundles on $X$. Let $\Sigma_1\subset N$ be the set of primitive integral generators of the one-dimensional cones in $\Sigma$ and put $$ PL(\Sigma)^+:=\{ \varphi \in PL(\Sigma)_{{\bf R}}\,\, |\,\, \varphi (e)> 0\,\, {\rm for} \hskip 0,3cm {\rm all}\,\,\, e\in \Sigma_1\}. $$ It is well-known (cf. \cite{Reid}, \cite{BaTschi1} Prop. 1.2.11), that the cone of effective divisors $ \Lambda _{\rm eff}(X)\subset {\rm Pic}(X)_{{\bf R}}$ is the closure of the image of $PL(\Sigma)^+$ under the projection $PL(\Sigma)_{{\bf R}} \rightarrow {\rm Pic}(X)_{{\bf R}}$. Further, the anti-canonical line bundle on $X$ is isomorphic to $L_{ \varphi _{\Sigma}}$, where $ \varphi _{\Sigma}(e)=1$ for all $e\in \Sigma_1$ (cf. \cite{BaTschi1}, Prop. 1.2.12). \subsection* \noindent {\bf 3.2}\hskip 0,5cm We shall introduce an adelic metric on the line bundle $L_{ \varphi }$ as follows. For $ \sigma \in \Sigma$ and $v\in \Val (F)$ define $$ {\bf K}_{ \sigma ,v}:=\{\,x\in U_{ \sigma }(F_v)\,|\, |m(x)|_v\le 1\,\,\, \forall\,\, m\in \check{ \sigma }\cap M\,\}. $$These subsets cover $X(F_v)$ and we put for $x\in {\bf K}_{ \sigma ,v}$ and any local section $s$ of $L_{ \varphi }$ at $x$ $$ \|x^*s\|_x:=|s(x)m_{ \varphi , \sigma }(x)|_v. $$ The family $\|\cdot\|_v=(\|\cdot\|_x)_{x\in X(F_v)}$ is then a $v$-adic metric on $L_{ \varphi }$ and ${\cal L}_{ \varphi }=\left(L_{ \varphi },(\|\cdot\|_v)_v\right)$ is a metrization of $L_{ \varphi }$. Let ${\bf K}_{T,v}\subset T(F_v)$ be the maximal compact subgroup. Assigning to $x\in T(F_v)$ the map $$ M \rightarrow {\bf Z} \,\,({\rm resp.}\,\, {\bf R}\,\, {\rm if}\,\, v|\infty), $$ $$ m\mapsto -\frac{\log(|m(x)|_v)}{\log(q_v)}, $$ (where $q_v$ is the order of the residue field of $F_v$ for non-archimedean valuations and $\log(q_v)=1$ for archimedean valuations) we get an isomorphism $T(F_v)/{\bf K}_{T,v} \rightarrow N$ (resp. $N_{{\bf R}}$ if $v|\infty$). We will denote by $\overline{x}$ the image of $x\in T(F_v)$ in $N$ (resp. $N_{{\bf R}}$). For $ \varphi \in PL(\Sigma)_{{\bf C}}$ define $$ H_{\Sigma,v}(\,\cdot\,, \varphi )\,:\, T(F_v) \rightarrow {\bf C}, $$ $$ H_{\Sigma,v}(x, \varphi ):= e^{- \varphi (\overline{x})\log (q_v)}. $$ The corresponding global function $H_{\Sigma}(\,\cdot\,, \varphi )\, :\, T({\bf A}) \rightarrow {\bf C}$, $$ H_{\Sigma}(x, \varphi ):=\prod_v H_{\Sigma,v}(x_v, \varphi ), $$ is well defined since for almost all $v$ the local component $x_v$ belongs to ${\bf K}_{T,v}$. The functions $H_{\Sigma,v}(\,\cdot\,, \varphi )$, $ \varphi \in PL(\Sigma)$, are related to the $v$-adic metric on $L_{ \varphi }$ by the identity \begin{equation} \label{3.2.1} H_{\Sigma,v}(x, \varphi )=\|x^*s_{ \varphi }\|_x, \hskip 0,5cm (x\in T(F_v)), \end{equation} where $s_{ \varphi }\in H^0(T,L_{ \varphi })$ is the constant function $1$. In particular, for every $x\in T(F)$ we have $$ H_{{\cal L}_{ \varphi }}(x)=H_{\Sigma}(x, \varphi )^{-1}. $$ \subsection* \noindent {\bf 3.3}\hskip 0,5cm Let ${\bf K}_T=\prod_v{\bf K}_{T,v}\subset T({\bf A})$, and denote by $$ {\cal A}_T=(T({\bf A})/T(F){\bf K}_T)^* $$ the group of unitary characters of $T({\bf A})$ which are trivial on the closed subgroup $T(F){\bf K}_T$. For $m\in M$ we obtain characters $ \chi ^m$ defined by $$ \chi ^m(x):= e^{i\log(|m(x)|_{\bf A})}. $$ This gives an embedding $M_{{\bf R}} \rightarrow {\cal A}_T$. For any archimedean place $v$ and $\chi\in {\cal A}_T$ there is an $m_v=m_v(\chi)\in M_{{\bf R}}$ such that $\chi_v(x_v)=e^{-i\overline{x}_v(m_v)}$ for all $x_v\in T(F_v)$. We get a homomorphism \begin{equation} \label{3.3.1} {\cal A}_T \rightarrow M_{{\bf R},\infty}=\oplus_{v|\infty}M_{{\bf R}}, \end{equation} $$ \chi\mapsto m_{\infty}(\chi)=(m_v(\chi))_{v|\infty}. $$ Define $T({\bf A})^1$ to be the kernel of all maps $T({\bf A}) \rightarrow {\bf R}_{>0}, x\mapsto |m(x)|_{\bf A}$, for $m\in M$, and put $$ {\cal U}_T=(T({\bf A})^1/T(F){\bf K}_T)^*. $$ The choice of a projection ${\bf G}_m({\bf A}) \rightarrow {\bf G}_m({\bf A})^1$ induces by means of an isomorphism $T\stackrel{\sim}\longrightarrow {\bf G}_{m,F}^d$ a splitting of the exact sequence $$ 1 \rightarrow T({\bf A})^1 \rightarrow T({\bf A}) \rightarrow T({\bf A})/T({\bf A})^1 \rightarrow 1. $$ This gives decompositions \begin{equation} \label{3.3.2} {\cal A}_T=M_{{\bf R}}\oplus {\cal U}_T \end{equation} and $$ M_{{\bf R},\infty}=M_{{\bf R}}\oplus M^1_{{\bf R},\infty}, $$ where $M^1_{{\bf R},\infty}$ is the minimal ${\bf R}$-subspace of $M_{{\bf R},\infty}$ containing the image of ${\cal U}_T$ under the map (\ref{3.3.1}). >From now on we fix such a (non-canonical) splitting. By Dirichlet's unit theorem, the image of ${\cal U}_T \rightarrow M^1_{{\bf R},\infty}$ is a lattice of maximal rank. Its kernel is isomorphic to the character group of ${\rm Cl}_F^d$, where ${\rm Cl}_F$ is the ideal class group of $F$. For finite $v$ we let $dx_v$ be the Haar measure on $T(F_v)$ giving ${\bf K}_{T,v}$ the volume one. For archimedean $v$ we take on $T(F_v)/{\bf K}_{T,v}$ the pull-back of the Lebesgue measure on $N_{{\bf R}}$ (normalized by the lattice $N$) and on ${\bf K}_{T,v}$ the Haar measure with total mass one. The product measure gives an invariant measure $dx_v$ on $T(F_v)$. On $T({\bf A})$ we get a Haar measure $dx=\prod_vdx_v$. \subsection* \noindent {\bf 3.4}\hskip 0,5cm We will denote by $S^1$ the unit circle. For a character $\chi\,:\, T(F_v) \rightarrow S^1$ we define the Fourier transform of $H_{\Sigma,v}(\,\cdot\,, \varphi )$ by $$ \hat{H}_{\Sigma,v}(\chi, \varphi )=\int_{T(F_v)}H_{\Sigma,v}(x_v, \varphi )\chi(x_v)dx_v. $$ If $\chi$ is not trivial on ${\bf K}_{T,v}$ then $\hat{H}_{\Sigma,v}(\chi, \varphi )=0$ (assuming the convergence of the integral). We will show that these integrals do exists if ${\rm Re} ( \varphi ) $ is in $PL(\Sigma)^+$. Let $v$ be an archimedean place of $F$. Any $d$-dimensional cone $ \sigma \in \Sigma$ is simplicial (since $\Sigma$ is regular) and it is generated by the set $ \sigma \cap \Sigma_1$. Let $\chi$ be unramified, i.e., $\chi(x)=e^{-i\overline{x}(m)}$ with some $m\in M_{{\bf R}}$. Then we get \begin{equation} \label{3.4.1} \hat{H}_{\Sigma,v}(\chi, \varphi )=\sum_{\dim \sigma =d}\int_{ \sigma }e^{-( \varphi (n)+in(m))}dn= \sum_{\dim \sigma =d}\,\,\,\prod_{e\in \sigma \cap \Sigma_1}\frac{1}{ \varphi (e)+ie(m)}. \end{equation} To give the result for finite places we define rational functions $R_{ \sigma }$ in variables $u_e, e\in \Sigma_1$, for any $ \sigma \in \Sigma$ by $$ R_{ \sigma }((u_e)_e)=\prod_{e\in \sigma \cap \Sigma_1}\frac{u_e}{1-u_e}, $$ and put $$ R_{\Sigma}((u_e)_e)=\sum_{ \sigma \in \Sigma}R_{ \sigma }((u_e)_e), $$ $$ Q_{\Sigma}((u_e)_e)=R_{\Sigma}((u_e)_e)\prod_{e\in \Sigma_1}(1-u_e). $$ Although elementary, it is a very important observation that the polynomial $Q_{\Sigma}-1$ is a sum of monomials of degree not less than two (cf. \cite{BaTschi1}, Prop. 2.2.3). Let $\chi$ be an unramified unitary character of $T(F_v)$ and let ${\rm Re}( \varphi )$ be in $PL(\Sigma)^+$. Then we can calculate \begin{equation} \label{3.4.2} \hat{H}_{\Sigma,v}(\chi, \varphi )=\int_{T(F_v)}H_{\Sigma,v}(x_v, \varphi )\chi(x_v)dx_v= \sum_{n\in N}e^{- \varphi (n)\log(q_v)}\chi(n) \end{equation} $$ = \sum_{ \sigma \in \Sigma}\sum_{n\in { \sigma }^{o}\cap N}q_v^{- \varphi (n)}\chi(n) =\sum_{ \sigma \in \Sigma}R_{ \sigma }\left((\chi(e)q_v^{- \varphi (e)})_e\right) $$ $$ =Q_{\Sigma}\left((\chi(e)q_v^{- \varphi (e)})_e\right) \prod_{e\in \Sigma_1}\left(1-\chi(e)q_v^{- \varphi (e)}\right)^{-1}. $$ (Here we denoted by $ \sigma ^{o}$ the relative interior of the cone $ \sigma $.) Any $e\in \Sigma_1$ induces a homomorphism $F[M] \rightarrow F[{\bf Z}]$ and hence a morphism of tori ${\bf G}_m \rightarrow T$. For any character $\chi\in {\cal A}_T$ we denote by $\chi_e$ the Hecke character $$ {\bf G}_m({\bf A})\longrightarrow T({\bf A})\stackrel{\chi}\longrightarrow S^1 $$ thus obtained. The finite part of the Hecke $L$-function with character $\chi_e$ is by definition $$ L_f( \chi _e,s)=\prod_{v\nmid \infty}(1-\chi_e(\pi_v)q_v^s)^{-1} $$ and this product converges for ${\rm Re} (s)>1$ (here $\pi_v$ denotes a local uniformizing element). \noindent By (\ref{3.4.1}) and (\ref{3.4.2}) we know that the global Fourier transform $$ \hat{H}_{\Sigma}(\chi, \varphi )=\int_{T({\bf A})}H_{\Sigma}(x, \varphi )\chi(x)dx $$ exists (i.e., the integral on the right converges absolutely) if ${\rm Re}( \varphi )$ is contained in $ \varphi _{\Sigma}+PL(\Sigma)^+$, because $$ \prod_{v\nmid\infty } Q_{\Sigma}((\chi_v(e)q_v^{- \varphi (e)})_e) $$ is an absolutely convergent Euler product for ${\rm Re} ( \varphi (e))>1/2 $ (for all $e\in \Sigma_1$) and hence is bounded for ${\rm Re} ( \varphi )$ in any compact subset in $\frac{1}{2} \varphi _{\Sigma}+PL(\Sigma)^+$ (by some constant depending only on this subset). \bigskip \noindent {\bf Proposition 3.4}\,\,\, {\it The series $$ \sum_{x\in T(F)}H_{\Sigma}(xt, \varphi ) $$ converges absolutely and uniformly for $({\rm Re}( \varphi ),t)$ contained in any compact subset of $( \varphi _{\Sigma}+PL(\Sigma)^+)\times T({\bf A}).$ } \bigskip {\em Proof.} Let ${\bf K}$ be a compact subset of $ \varphi _{\Sigma}+PL(\Sigma)^+$ and let $C_v\subset T(F_v)$ (for every $v\in \Val (F)$) be a compact subset, equal to ${\bf K}_{T,v}$ for almost all $v$. Since any $ \varphi \in PL(\Sigma)_{{\bf C}}$ is a continuous piecewise linear function (with respect to a finite subdivision of $N_{{\bf R}}$ into simplicial cones) there exists a constant $c_v\ge 1$ (depending on ${\bf K}$ and $C_v$) such that for all $ \varphi $ with ${\rm Re}( \varphi )\in {\bf K}$, $x_v\in T(F_v)$ and $t_v\in C_v$ we have $$ \frac{1}{c_v}\le \left| \frac{H_{\Sigma,v}(x_vt_v, \varphi )}{H_{\Sigma,v}(x_v, \varphi )}\right|= \frac{H_{\Sigma,v}(x_vt_v,{\rm Re}( \varphi ))}{H_{\Sigma,v}(x_v,{\rm Re}( \varphi ))} \le c_v. $$ If $C_v={\bf K}_{T,v}$ we may assume $c_v=1$. Put $c=\prod_vc_v$. For all $ \varphi $ with ${\rm Re}( \varphi )\in {\bf K}$ and $t\in C:=\prod_v C_v$ we can estimate $$ \left| \sum_{x\in T(F)}H_{\Sigma}(xt, \varphi )\right|\le c\sum_{x\in T(F)} H_{\Sigma}(x,{\rm Re}( \varphi )). $$ Let $S$ be a finite set of places containing ${\rm Val}_{\infty}(F)$ and let $U_v\subset T(F_v)$ be a relatively compact open subset of $T(F_v)$ for each $v\in S$, such that for all $x_1\neq x_2\in T(F)$ $$ x_1U\cap x_2U=\emptyset, $$ where $U=\prod_{v\in S}U_v\prod_{v\notin S}{\bf K}_{T,v}. $ By the preceding argument, there exists a $c'>0$ such that for all $ \varphi \in {\bf K}$, $x\in T(F)$ and $u\in U$ $$ H_{\Sigma}(x, \varphi )\le c'H_{\Sigma}(xu, \varphi ). $$ Therefore, $$ \sum_{x\in T(F)}H_{\Sigma}(x, \varphi )\le \frac{c'}{{\rm vol}(U)}\sum_{x\in T(F)} \int_{U}H_{\Sigma}(xu, \varphi )du $$ $$ \le \frac{c'}{{\rm vol}(U)}\int_{T({\bf A})}H_{\Sigma}(x, \varphi )dx <\infty $$ by the discussion above. From the explicit expression for the integral (cf. (\ref{3.5.1})) we derive the uniform convergence in $ \varphi $ on ${\bf K}$. \hfill $\Box$ \subsection* \noindent {\bf 3.5}\hskip 0,5cm The aim is to apply Poisson's summation formula to the height zeta function. It remains to show that $\hat{H}(\,\cdot\,, \varphi )$ is absolutely integrable over ${\cal A}_T$. For $\chi\in {\cal A}_T$ and ${\rm Re} ( \varphi )$ contained in $\frac{1}{2} \varphi _{\Sigma}+PL(\Sigma)^+$ we put $$ \zeta_{\Sigma}(\chi, \varphi ):= \prod_{v|\infty}\hat{H}_{\Sigma,v}(\chi_v, \varphi ) \prod_{v\nmid\infty} Q_{\Sigma}((\chi_v(e)q_v^{- \varphi (e)})_e). $$ \bigskip \noindent {\bf Lemma 3.5}\hskip 0,5cm {\it Let ${\bf K}$ be a compact subset of $PL(\Sigma)_{{\bf C}}$ such that for all $ \varphi \in {\bf K}$ and $e\in \Sigma_1$ $$ {\rm Re} ( \varphi (e))>\frac{1}{2}. $$ Then there is a constant $c=c({\bf K})$ such that for all $ \varphi \in {\bf K}, \chi\in {\cal A}_T$ and $m\in M_{{\bf R}}$ we have $$ |\zeta_{\Sigma}(\chi, \varphi +im)|\le c\prod_{v|\infty}\left\{\sum_{\dim \sigma =d}\,\,\, \prod_{e\in \sigma \cap \Sigma_1}\frac{1}{(1+ |e(m+m_v(\chi))|)^{1+1/d}}\right\}. $$ } {\em Proof.} For ${\bf K}$ as above there exists a $c'>0$ such that for all $\chi\in {\cal A}_T$ and $m\in M_{{\bf R}}$ one has $$ \left|\prod_{v\nmid \infty} Q_{\Sigma}((\chi_v(e)q_v^{-( \varphi (e)+ie(m))})_e)\right|\le c' $$ for all $ \varphi \in {\bf K}$ (see the argument before Proposition 3.4. By \cite{BaTschi1}, Prop. 2.3.2, for all $v \mid {\infty}$ there is a constant $c_v$ such that for all $ \varphi \in {\bf K}$, $ \chi \in {\cal A}_T$ and $m\in M_{{\bf R}}$ $$ |\hat{H}_{\Sigma,v}(\chi_v, \varphi +im)|\le c_v\sum_{\dim \sigma =d} \,\,\,\prod_{e\in \sigma \cap \Sigma_1} \frac{1}{(1+|e(m+m_v(\chi))|)^{1+1/d}}. $$ Putting $c=c'\prod_{v|\infty}c_v$ we get the result. \hfill $\Box$ \bigskip \noindent For ${\rm Re} ( \varphi )$ contained in $ \varphi _{\Sigma}+PL(\Sigma)^+$ we can write \begin{equation} \label{3.5.1} \hat{H}_{\Sigma}(\chi, \varphi )= \zeta_{\Sigma}(\chi, \varphi )\prod_{e\in \Sigma_1}L_f(\chi_e, \varphi (e)). \end{equation} \noindent By the preceding lemma, we see that $\hat{H}_{\Sigma}(\,\cdot\,, \varphi )$ is absolutely integrable over ${\cal A}_T$. For $t\in T({\bf A})$ we have $$ \int_{T({\bf A}_F)}H_{\Sigma}(xt, \varphi ) \chi(x)dx=\chi^{-1}(t)\hat{H}_{\Sigma}(\chi, \varphi ). $$ Hence we can apply Poisson's summation formula (together with (\ref{3.3.2})) and obtain \begin{equation} \label{3.5.2} \sum_{x\in T(F)}H_{\Sigma}(xt, \varphi )= \mu_T\int_{M_{{\bf R}}} \left\{\sum_{ \chi \in {\cal U}_T} \hat{H}_{\Sigma}( \chi , \varphi +im)( \chi \c^m(t))^{-1}\right\}dm, \end{equation} where the Lebesgue measure $dm$ on $M_{{\bf R}}$ is normalized by $M$ and $$ \mu_T=\frac{1}{(2\pi \kappa)^d},\hskip 0,5cm \kappa=\frac{{\rm cl}_F\cdot {\rm R}_F}{{\rm w}_F} $$ with ${\rm cl}_F$ the class number, ${\rm R}_F$ the regulator and ${\rm w}_F$ the number of roots of unity in $F$. Note that $\hat{H}_{\Sigma}( \chi \cdot \chi ^m, \varphi )=\hat{H}_{\Sigma}( \chi , \varphi +im).$ \subsection* \noindent {\bf 3.6}\hskip 0,5cm In section 5 we need uniform estimates for $L$-functions in a neighborhood of the line ${\rm Re}(s)=1$. For any unramified character $\chi\,:\, {\bf G}_m({\bf A})/{\bf G}_m(F) \rightarrow S^1$ and any archimedean place $v$ there exists a $\tau_v\in {\bf R}$ such that $\chi_v(x_v)=|x_v|_v^{i\tau_v}$ for all $x_v\in {\bf G}_m(F_v)$. We put $$ \chi_{\infty}=(\tau_v)_{v|\infty}\in {\bf R}^{{\rm Val}_{\infty}(F)} \hskip 0,5cm {\rm and}\hskip 0,5cm \|\chi_{\infty}\|=\max_{v|\infty}|\tau_v|. $$ We will use the following theorem of Rademacher (\cite{Rademacher}, Theorems 4,5), which rests on the Phragm\'en-Lindel\"of principle. \bigskip \noindent {\bf Theorem 3.6}\hskip 0,5cm {\it For any $\epsilon>0$ there exists a $ \delta >0$ and a constant $c(\epsilon)>0$ such that for all $s$ with ${\rm Re}(s)> 1- \delta $ and all unramified Hecke characters which are non-trivial on ${\bf G}_m({\bf A})^1$ one has \begin{equation} \label{3.6.1} |L_f(\chi,s)|\le c(\epsilon)(1+|{\rm Im} (s)| +\|\chi_{\infty}\|)^{\epsilon}. \end{equation} For the trivial character $\chi=1$ one has \begin{equation} \label{3.6.2} |L_f(1,s)|\le c(\epsilon)\left|\frac{1+s}{1-s}\right|(1+|{\rm Im}(s)|)^{\epsilon}. \end{equation} } \bigskip \bigskip \section{Twisted products} \label{4} \subsection* \noindent {\bf 4.1}\hskip 0,5cm Let $G,P,W=P\backslash G$ etc. be as in section 2 and $T,\Sigma,X=X_{\Sigma}$ etc. be as in section 3. Let $\eta \,:\, P \rightarrow T$ be a homomorphism. Then $P$ acts from the right on $X\times G$ by $$ (x,g)\cdot p := (x\eta (p),p^{-1}g). $$ Since $\pi_W\,:\, G \rightarrow W$ is locally trivial, the quotient $$ Y=X\times^{P} G:= (X\times G)/P $$ exists as a variety over $F$. Moreover, the projection $X\times G \rightarrow G$ induces a morphism $\pi \,: \, Y \rightarrow W$ and $Y$ becomes a locally trivial fiber bundle over $W$ with fiber $X$ (compare \cite{J}, I.5.16). Hence, by the properties of $X$ (non-singular, projective), we see that $Y$ is a non-singular projective variety over $F$ (``projectivity'' requires a short argument, cf. \cite{Strauch}). The quotient morphism $X\times G \rightarrow Y$ will be denoted by $\pi_Y$. Let $ \varphi \in PL(\Sigma)$ and let $L_{ \varphi }$ be the invertible sheaf on $X$ defined in section 3.1. Denote by ${\bf L}_{ \varphi }$ the corresponding ${\bf G}_a$-bundle over $X$, i.e., ${\bf L}_{ \varphi }={\bf V}(L^{\vee}_{ \varphi })= {\bf V}(L_{- \varphi })$ (with the notation of \cite{Ha}, II, Exercise 5.18). The canonical $T$-linearization $$ \theta_{- \varphi }\,:\, \theta^*L_{- \varphi } \rightarrow p_1^*L_{- \varphi } $$ induces an action ${\bf L}_{ \varphi }\times T \rightarrow {\bf L}_{ \varphi }$ of $T$ on ${\bf L}_{ \varphi }$ which is compatible with the action of $T$ on $X$. The twisted product ${\bf L}_{ \varphi }\times^PG$ will then be a ${\bf G}_a$-bundle over $Y$ and we define $L^Y_{ \varphi }$ to be the sheaf of local sections of ${\bf L}_{ \varphi }\times^PG $ over $Y$. Note that $L^Y_{ \varphi }$ (and even its isomorphism class in ${\rm Pic}(Y) $) depends on the fixed $T$-linearization $\theta_{- \varphi }$. In fact, for $ \varphi \in PL(\Sigma)$ and $m\in M$ we have $$ L^Y_{ \varphi +m}\simeq L^Y_{ \varphi }\otimes \pi^* L_{m\circ \eta}. $$ Embedding $M$ in $PL(\Sigma)\oplus X^*(P)$ by $m\mapsto (m,-m\circ\eta)$ we see that $M$ is contained in the kernel of the homomorphism $$ \begin{array}{ccl} \psi \,:\, PL(\Sigma)\times X^*(P) & \rightarrow &{\rm Pic}(Y),\\ ( \varphi , \lambda ) &\mapsto & {\rm isomorphism}\,\, {\rm class}\,\, {\rm of}\,\, L^Y_{ \varphi }\otimes \pi^*L_{ \lambda }. \end{array} $$ \subsection* \noindent {\bf 4.2}\hskip 0,5cm In the following proposition we collect all relevant facts about the geometry of twisted products which we will need in the sequel. \bigskip \noindent {\bf Proposition 4.2}\hskip 0,5cm {\it a) The sequence $$ 0 \rightarrow M \rightarrow PL(\Sigma)\oplus X^*(P) \rightarrow {\rm Pic}(Y) \rightarrow 0 $$ is exact. b) The cone of effective divisors $ \Lambda _{\rm eff}(Y)\subset {\rm Pic}(Y)_{{\bf R}}$ is the image of the closure of $$ PL(\Sigma)^{+}\times X^*(P)^{+}\subset PL(\Sigma)_{{\bf R}}\oplus X^*(P)_{{\bf R}}. $$ c) The anti-canonical line bundle ${\omega}^{\vee}_Y$ is isomorphic to $L^Y_{ \varphi _{\Sigma}}\otimes \pi^*L_{2\rho_P}$. } \bigskip {\em Proof.} a) By \cite{Sa}, Proposition 6.10, there is an exact sequence $$ F[X\times G]^*/F^* \rightarrow X^*(P) \rightarrow {\rm Pic}(Y) \rightarrow {\rm Pic}(X\times G). $$ Denote by $\pi_X\,:\, X\times G \rightarrow X$ the canonical projection. Let $L$ be an invertible sheaf on $Y$. Then $$ \pi^*_YL\simeq \pi^*_XL_{ \varphi } $$ (for some $ \varphi \in PL(\Sigma)$) because ${\rm Pic}(X\times G)= {\rm Pic}(X)\oplus {\rm Pic}(G)={\rm Pic}(X)$ (cf. \cite{Sa}, Lemme 6.6 (i) and Lemme 6.9 (iv)). Note that $\pi^*_YL^Y_{ \varphi }\simeq \pi^*_XL_{ \varphi }$, so that $$ \pi^*_Y(L\otimes L^Y_{- \varphi }) $$ is trivial. Hence there exists a character $ \lambda $ of $P$ such that $L\otimes L^Y_{- \varphi }$ is isomorphic to $\pi^*L_{ \lambda } $ (the map $X^*(P) \rightarrow {\rm Pic}(Y)$ factorizes $X^*(P) \rightarrow {\rm Pic}(W) \rightarrow {\rm Pic}(Y)$). This shows surjectivity. Suppose now that for $ \varphi \in PL(\Sigma)$ and $ \lambda \in X^*(P)$ the sheaf $L_{ \varphi }^Y\otimes \pi^*L_{ \lambda }$ is trivial on $Y$. Then $\pi_Y^*(L^Y_{ \varphi }\otimes \pi^*L_{ \lambda })\simeq \pi^*_XL_{ \varphi }$ is trivial on $X\times G$, therefore $L_{ \varphi }\simeq {\cal O}_X, \varphi =m\in M$ and $L^Y_{ \varphi }\otimes \pi^*L_{ \lambda }=\pi^*L_{ \lambda +m\circ\eta}$. By Rosenlicht's theorem, $$ F[X\times G]^*/F^*=F[X]^*/F^*\oplus F[G]^*/F^*=X^*(G)=0, $$ therefore, the map $X^*(P) \rightarrow {\rm Pic}(Y)$ is injective, hence $ \lambda +m\circ \eta =0$ and $( \varphi , \lambda )$ is in the image of $M \rightarrow PL(\Sigma)\oplus X^*(P)$. b) For $ \varphi \in PL(\Sigma)$ denote by $\Box_{ \varphi }$ the set of all $m\in M$ such that for all $n\in N_{{\bf R}}$ $$ \varphi (n)+n(m)\ge 0. $$ By \cite{Oda}, Lemma 2.3, $\Box_{ \varphi }$ is a basis for $H^0(X,L_{ \varphi })$ (note the different sign conventions). It is easy to see that $$ \pi_*L_{ \varphi }^Y\simeq \oplus_{m\in \Box_{ \varphi }}L_{-m\circ\eta}. $$ Suppose $L^Y_{ \varphi }\otimes \pi^*L_{ \lambda }$ has a non-zero global section. Then $$ \pi_*(L_{ \varphi }^Y\otimes \pi^*L_{ \lambda })\simeq \oplus_{m\in \Box_{ \varphi }}L_{-m\circ \eta + \lambda } $$ has a non-zero global section, hence (cf. section 2.3) there is a $m'\in \Box_{ \varphi }$ such that $-m'\circ \eta + \lambda $ is contained in the closure of $ X^*(P)^{+}$. Putting $ \lambda '=-m'\circ \eta + \lambda , \varphi '= \varphi +m'$ we have $L^Y_{ \varphi '}\otimes \pi^*L_{ \lambda '}\simeq L^Y_{ \varphi }\otimes \pi^*L_{ \lambda }$ and $( \varphi ', \lambda ')$ is contained in the closure of $PL(\Sigma)^+\times X^*(P)^+$. On the other hand, if $( \varphi , \lambda )\in PL(\Sigma)\oplus X^*(P)$ is contained in the closure of $PL(\Sigma)^+\times X^*(P)^+$ then the trivial character corresponds to a global section of $L_{ \varphi }$. Hence $$ \pi_*(L_{ \varphi }^Y\otimes \pi^*L_{ \lambda }) = L_{ \lambda }\oplus \bigoplus_{m\in \Box_{ \varphi }-\{0\}}L_{-m\circ \eta + \lambda } $$ and $H^0(W,L_{ \lambda })\neq \{0\}$, i.e., $L_{ \varphi }^Y\otimes \pi^*L_{ \lambda }$ has a non-zero global section. c) Note first that the exact sequence $$ 0 \rightarrow \pi^*\Omega_W \rightarrow \Omega_Y \rightarrow \Omega_{Y/W} \rightarrow 0 $$ splits, and therefore $\omega_Y\simeq ( \Lambda ^d\Omega_{Y/W})\otimes \pi^*\omega_W$. Since $\omega_W\simeq L_{-2\rho}$ it remains to show that $( \Lambda ^d\Omega_{Y/W})^{\vee} \simeq L^Y_{ \varphi _{\Sigma}}$. Let ${\cal J}_{Y/W}$ be the ideal sheaf of the image of the diagonal morphism $$ \Delta_{Y/W}\,:\, Y \rightarrow Y\times_WY. $$ But $Y\times_WY$ is canonically isomorphic to $(X\times X)\times^PG$ and $\Delta_{Y/W}(Y)$ is just $\Delta_X(X)\times^PG$. Hence we see that $({\cal J}_{Y/W}/{\cal J}_{Y/W}^2)^{\vee}$ is the sheaf of local sections of $$ {\bf V}({\cal J}_X/{\cal J}_X^2)\times^PG, $$ where ${\cal J}_X$ is the ideal sheaf of $\Delta_X(X)\subset X\times X$. Pulling back to $Y$ and taking the $d$-th exterior power we get $$ {\bf V}( \Lambda ^d\Omega_{Y/W})\simeq {\bf V}( \Lambda ^d\Omega_X)\times^PG \simeq {\bf V}(\omega_X)\times^PG. $$ The canonical $T$-linearization of $\omega_X$ (induced by the action of $T$ on rational functions) corresponds to the $T$-linearization $\theta_{- \varphi _{\Sigma}}$ of $L_{- \varphi _{\Sigma}}\simeq \omega_X$, i.e., $$ {\bf L}_{ \varphi _{\Sigma}}\times^PG\simeq {\bf V}(\omega_X)\times^PG $$ and we get $L^Y_{ \varphi _{\Sigma}}\simeq ( \Lambda ^d\Omega_{Y/W})^{\vee}$. \hfill $\Box$ \subsection* \noindent {\bf 4.3}\hskip 0,5cm We are going to introduce an adelic metric on the sheaves $L^Y_{ \varphi }$. A section of $L^Y_{ \varphi }$ over an open subset $U\subset Y$ can be identified with a $P$-equivariant morphism $s\,:\, \pi^{-1}_Y(U) \rightarrow {\bf L}_{ \varphi }$ over $X$, i.e., $$ s(x\eta(p),p^{-1}g)=s(x,g)\cdot \eta(p). $$ Let $v$ be a place of $F$ and let $y\in Y(F_v)$ be the image of $(x,k)\in X(F_v)\times G(F_v)$ with $k\in {\bf K}_{G,v}$. Let $s\,:\, \pi^{-1}_Y(U) \rightarrow {\bf L}_{ \varphi }$ be a local section of $L^Y_{ \varphi }$ over $U\subset Y$ with $y\in U(F_v)$. Define $$ \|\cdot\|_y\,:\, y^*L^Y_{ \varphi } \rightarrow {\bf R} $$ by $$ \|y^*s\|_y=\|s\circ (x,k)\|_x. $$ Then $\|\cdot \|_v=(\|\cdot \|)_{y\in Y(F_v)}$ is a $v$-adic metric on $L^Y_{ \varphi }$ and ${\cal L}^Y_{ \varphi }=\left(L^Y_{ \varphi },(\|\cdot \|_v)_v\right)$ is a metrization of $L^Y_{ \varphi }$ (cf. \cite{Strauch}). Let $$ Y^o=T\times^P G\hookrightarrow X\times^PG=Y $$ be the twisted product of $T$ with $W$. Over $Y^o$ there is a canonical section of $L^Y_{ \varphi }$, namely $$ s^Y_{ \varphi }\,:\, \pi^{-1}(Y^o)=T\times G \rightarrow {\bf L}_{ \varphi }, $$ $ s^Y_{ \varphi }(x,g)=s_{ \varphi }(x)$, where $s_{ \varphi }\in H^0(T,L_{ \varphi })$ corresponds to the constant function $1$. Let $y=\pi_W(x,g)\in Y(F_v)$ where $ g=pk$ with $ p\in P(F_v)$ and $ k\in {\bf K}_{G,v}$. Then $$ \|y^*s_{ \varphi }^Y\|_y = \|s_{ \varphi }^Y\circ (x\eta(p),k)\|_{x\eta(p)} $$ $$ = \|(x\eta(p))^*s_{ \varphi }\|_{x\eta(p)}=e^{- \varphi (\overline{x\eta(p)})\log (q_v)} $$ (by (\ref{3.2.1})). Globally, for $y\in Y^o(F), y=\pi_Y(x, \gamma), x\in T(F), \gamma\in G(F), \gamma=p_{ \gamma}k_{ \gamma}$ and $ p_{ \gamma}, k_{ \gamma}$ as above, in $P({\bf A}), {\bf K}_G$, respectively, we get \begin{equation} \label{4.3.1} H_{{\cal L}^Y_{ \varphi }}(y)= \prod_v\|y^*_vs^Y_{ \varphi }\|_{y_v}^{-1}=H_{\Sigma}(x\eta(p_{ \gamma}),- \varphi ). \end{equation} \subsection* \noindent {\bf 4.4}\hskip 0,5cm Let $\xi \,:\, P({\bf A})/P(F) \rightarrow S^1$ be an unramified character, i.e., $\xi$ is trivial on $P({\bf A})\cap {\bf K}_G$. Using the Iwasawa decomposition we get a well defined function $$ \phi_{\xi}\,:\, G({\bf A}) \rightarrow S^1, $$ $$ \phi_{\xi}(g)=\xi(p), $$ if $g=pk$ as above. We denote by $$ E^G_P( \lambda ,\xi,g)= \sum_{ \gamma\in P(F)\backslash G(F)}\phi_{\xi}( \gamma g) e^{\langle \lambda +\rho_P,H_P( \gamma g)\rangle} $$ the corresponding Eisenstein series and we put $E^G_P( \lambda ,\xi)=E^G_P( \lambda ,\xi,1_G)$. This series converges absolutely for ${\rm Re}( \lambda )$ contained in the cone $\rho_P+X^*(P)^+$ (cf. (2.3)). A character $ \chi \in {\cal A}_T$ induces a character $ \chi _{ \eta }= \chi \circ \eta \,:\, P({\bf A})/P(F) \rightarrow S^1$. We denote by $\check{\eta}\,:\, X^*(T)_{{\bf R}} \rightarrow X^*(P)_{{\bf R}}$ the map on characters induced by $\eta$. \bigskip {\bf Proposition 4.4} {\it Let $L$ be a line bundle on $Y$ such that its class is contained in the interior of the cone $ \Lambda _{\rm eff}(Y)$. Let $( \varphi , \lambda )$ be in $PL(\Sigma)^+\times X^*(P)^+$ with $\psi ( \varphi , \lambda )=[L]$. There is a metrization ${\cal L}$ of $L$ such that for all $s$ with ${\rm Re}(s)( \varphi , \lambda )\in ( \varphi _{\Sigma},2\rho_P) + PL(\Sigma)^+\times X^*(P)^+$ the series $$ Z_{Y^o}({\cal L},s)=\sum_{y\in Y^{o}(F)}H_{{\cal L}}(y)^{-s} $$ converges absolutely. Moreover, for these $s$ $$ Z_{Y^o}({\cal L},s)=\mu_T\int_{M_{{\bf R}}} \left\{ \sum_{ \chi \in {\cal U}_T}\hat{H}_{\Sigma}( \chi ,s \varphi +im) E^G_P(s \lambda -\rho_P - i\check{\eta}(m),{\chi}_{\eta}^{-1}) \right\}dm, $$ where the sum and integral on the right converge absolutely too. } \bigskip {\em Proof.} Let $( \varphi ', \lambda ')\in PL(\Sigma)\oplus X^*(P)$ such that there is an isomorphism $L\simeq L^Y_{ \varphi '}\oplus \pi^* L_{ \lambda '}$. Denote by ${\cal L}$ the metrization of $L$ which is the pullback of ${\cal L}_{ \varphi '}^Y\otimes \pi^*{\cal L}_{ \lambda '}$ via this isomorphism. Let $m\in M_{{\bf R}}$ such that $$ ( \varphi , \lambda )=( \varphi '+m,-\check{\eta}(m) + \lambda ') $$ is contained in $PL(\Sigma)^+\times X^*(P)^+$. By (\ref{4.3.1}), we have for any $y\in Y^o(F), y=\pi_Y(x, \gamma )$ with $ x\in T(F),$ $ \gamma \in G(F)$ and $ \gamma =p_{ \gamma}k_{ \gamma}$ $$ H_{{\cal L}}(y)=H_{{\cal L}_{ \varphi '}^Y\otimes \pi^*{\cal L}_{ \lambda '}}(y)= e^{-\langle \lambda ',H_P( \gamma )\rangle}H_{\Sigma}(x\eta (p_{ \gamma}),- \varphi ') $$ $$ = e^{-\langle \lambda '-m\circ \eta,H_P( \gamma)\rangle} H_{\Sigma}(x\eta(p_{ \gamma}),-( \varphi '+m))= e^{-\langle \lambda ,H_P( \gamma)\rangle}H_{\Sigma}(x\eta(p_{ \gamma}),- \varphi ). $$ We consider $s=u+iv\in {\bf C}$ such that $u\cdot \varphi $ is contained in the shifted cone $ \varphi _{\Sigma}+PL(\Sigma)^+$ and $u\cdot \lambda $ is contained in the cone $2\rho_P+X^*(P)^+$. Then $$ \sum_{x\in T(F)}H_{\Sigma}(x\eta(p_{ \gamma}),u \varphi ) $$ converges by Proposition 3.4 and is equal to $$ \mu_T\int_{M_{{\bf R}}}\{\sum_{\chi\in {\cal U}_T} \hat{H}( \chi \c^m,u \varphi ) \chi \c^m(\eta(p_{ \gamma}))^{-1}\}dm $$ (cf. (\ref{3.5.2})). Moreover, $\hat{H}_{\Sigma}(\,\cdot\, , u \varphi )$ is absolutely convergent on ${\cal A}_T$ and therefore $$ \sum_{x\in T(F)}H_{\Sigma}(x\eta(p_{ \gamma}), u \varphi )\le \mu_T\int_{M_{{\bf R}}}\left\{\sum_{\chi\in {\cal U}_T} \left| \hat{H}_{\Sigma}(\chi \chi ^m,u \varphi )\right|\right\}dm $$ is bounded by some constant $c$ (which is independent of $\eta(p_{ \gamma})$). Thus we may calculate $$ \sum_{y\in Y^o(F)}\left| H_{\cal L}(y)^{-s}\right| =\sum_{ \gamma\in P(F) \backslash G(F)} e^{\langle u \lambda ,H_{P}( \gamma)\rangle}\sum_{x\in T(F)}H_{\Sigma}(x\eta(p_{ \gamma}), u \varphi ) $$ $$ \le c \sum_{ \gamma\in P(F) \backslash G(F)}e^{\langle u \lambda ,H_{P}( \gamma)\rangle}. $$ This shows the first assertion. Since $$ \mu_T\int_{M_{\bf R}}\left\{\sum_{\chi\in {\cal U}_T} \left|\hat{H}_{\Sigma}(\chi \chi ^m, u \varphi )\right|\right\}dm $$ converges, we can interchange the summation and integration and get $$ Z_{Y^o}({\cal L},s) =\sum_{ \gamma\in P(F)\backslash G(F)} e^{\langle s \lambda ,H_P( \gamma)\rangle}\mu_T\int_{M_{{\bf R}}} \left\{\sum_{ \chi \in {\cal U}_T}\hat{H}_{\Sigma}( \chi \c^m,s \varphi )( \chi \c^m)^{-1} (\eta(p_{ \gamma}))\right\}dm $$ $$ =\mu_T\int_{M_{{\bf R}}}\left\{\sum_{\chi\in {\cal U}_T} \hat{H}_{\Sigma}(\chi,s \varphi +im)E^G_P(s \lambda -\rho_P-i\check{\eta}(m), {\chi}_{\eta}^{-1})\right\}dm. $$ \hfill $\Box$ \bigskip \bigskip \section{Meromorphic continuation} \label{5} \subsection* \noindent {\bf 5.1}\hskip 0,5cm The proposition in section 4.4 gives an expression of the height zeta function (for the open subset $Y^o\subset Y$ ) which we will use to determine the asymptotic behavior of the counting function $N_{Y^o}({\cal L},H) $ (cf. sec. \ref{1}) by applying a Tauberian theorem. The first thing to do is to show that $Z_{Y^o}({\cal L},s)$ can be continued meromorphically to a half-space beyond the abscissa of convergence and that there is no pole on this line with non-zero imaginary part. Then it remains to prove that this abscissa is at ${\rm Re}(s)=a(L)$ and to determine the order of the pole in $s=a(L)$. We will see that this order is $b(L)$. The method which we will explain now consists in an iterated application of Cauchy's residue theorem. The proofs will be given in section 7. \subsection* \noindent {\bf 5.2}\hskip 0,5cm Let $E$ be a finite dimensional vector space over ${\bf R}$ and $E_{{\bf C}}$ its complexification. Let $V\subset E$ be a subspace and let $l_1,...,l_m\in E^{\vee} =\Hom_{{\bf R}}(E,{\bf R})$ be linearly independent linear forms. Put $H_j={\rm Ker}(l_j)$ for $j=1,...,m$. Let $B\subset E$ be an open and convex neighborhood of ${\bf 0}$ such that for all $x\in B$ and $j=1,...,m$ we have $l_j(x)>-1$. Let $T_B=B+iE\subset E_{{\bf C}}$ be the tube domain over $B$ and denote by ${\cal M}(T_B)$ the set of meromorphic functions on $T_B$. We consider meromorphic functions $f\in {\cal M}(T_B)$ with the following properties: The function $$ g(z)=f(z)\prod_{j=1}^m\frac{l_j(z)}{l_j(z)+1} $$ is holomorphic in $T_B$ and there is a sufficient function $c\,:\, V \rightarrow {\bf R}_{\ge 0}$ such that for all compacts ${\bf K}\in T_B$, all $z\in {\bf K}$ and all $v\in V$ we have the estimate $$ |g(z+iv)|\le \kappa ({\bf K})c(v). $$ (Cf. section 7.3 for a precise definition of a sufficient function. In particular, such a sufficient function is absolutely integrable over any subspace $U\subset V$.) In this case we call $f$ {\it distinguished} with respect to the data $(V;l_1,...,l_m)$. Let $C$ be a connected component of $B- \cup_{j=1}^mH_j$. By the conditions on $g$ the integral $$ \tilde{f}_C(z):=\frac{1}{(2\pi)^{\nu}}\int_V f(z+iv)dv $$ ($\nu =\dim V$ and $dv$ is a fixed Lebesgue measure on $V$) converges for every $z\in T_C$ and $\tilde{f}_C$ is a holomorphic function on $T_C$. \bigskip \noindent {\bf Theorem 5.2}\hskip 0,5cm {\it There is an open neighborhood $\tilde{B}$ containing $C$, and linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}$ which vanish on $V$ such that $$ z\mapsto \tilde{f}_{C}(z)\prod_{j=1}^{\tilde{m}}\tilde{l}_j(z) $$ has a holomorphic continuation to $T_{\tilde{B}}$. Moreover, for all $j\in \{1,...,\tilde{m}\}$ we have $\Ker (\tilde{l}_j)\cap C=\emptyset$. } \bigskip \noindent We shall give the proof of this theorem in sections 7.3 and 7.4. \subsection* \noindent {\bf 5.3}\hskip 0,5cm Put $E^{(0)}=\cap_{j=1}^m\Ker (l_j)$ and $E_0=E/E^{(0)}$. Let $\pi_0\,:\, E \rightarrow E_0$ be the canonical projection and suppose $V\cap E^{(0)} = \{{\bf 0}\}$. Let $$ E^+_0=\{x\in E_0\,|\, l_j(x)\ge 0\,\, {\rm for}\,\,{\rm all} \,\, j=1,...,m\} $$ and let $\psi_0\,:\, E_0 \rightarrow P:=E^+_0/\pi_0(V)$ be the canonical projection. We want to assume that $\pi_0(V)\cap E^+_0=\{ {\bf 0}\}$, so that $ \Lambda :=\psi_0(E^+_0)$ is a strictly convex polyhedral cone. Let $dy$ be the Lebesgue measure on $E_0^{\vee}$ normalized by the lattice $\oplus_{j=1}^m{\bf Z} l_j$. Let $A\subset V$ be a lattice and let $dv$ be the measure on $V$ normalized by $A$. On $V^{\vee}$ we have the Lebesgue measure $dy'$ normalized by $A^{\vee}$ and a section of the projection $E^{\vee}_0 \rightarrow V^{\vee}$ gives a measure $dy''$ on $P^{\vee}$ with $dy=dy'dy''$. Define the ${\cal X}$-function of the cone $ \Lambda $ by $$ {\cal X}_{ \Lambda }(x)=\int_{ \Lambda ^{\vee}}e^{-y''(x)}dy'' $$ for all $x\in P_{{\bf C}}$ with ${\rm Re}(x)$ contained in the interior of $ \Lambda $ (cf. section 7.1). Let $B\subset E$ be as above and let $f\in {\cal M}(T_B)$ be a distinguished function with respect to $(V;l_1,...,l_m)$. Put $$ g(z)=f(z)\prod_{j=1}^m\frac{l_j(z)}{l_j(z)+1}, $$ $$ B^+=B\cap \{x\in E\,|\, l_j(x)\ge 0, \,{\rm for}\,\, {\rm all}\,\, j = 1,...,m\}, $$ $$ \tilde{f}_{B^+}(z)=\frac{1}{(2\pi )^{\nu}}\int_Vf(z+iv)dv $$ (for $z\in T_B$). The function $\tilde{f}_{B^+}\,:\, T_{B^+} \rightarrow {\bf C}$ is holomorphic and has a meromorphic continuation to a neighborhood of ${\bf 0}\in E_{{\bf C}}$. In section 7.5 we will prove the following theorem. \bigskip \noindent {\bf Theorem 5.3}\hskip 0,5cm {\it For $x_0\in B^+$ we have $$ \lim_{s \rightarrow 0}s^{m-\nu}\tilde{f}_{B^+}(sx_0)= g({\bf 0}){\cal X}_{ \Lambda }(\psi_0(x_0)). $$ } \bigskip \subsection* \noindent {\bf 5.4}\hskip 0,5cm In this section we make some preparations in order to apply the general setting of 5.2. Let $L$ be a line bundle on $Y$ such that its class in ${\rm Pic}(Y)$ lies in the interior of $ \Lambda _{\rm eff}(Y)$. By the definition of $a(L)$ (cf. section 1 and Proposition 4.2), $$ a(L)[L]-\psi(\varphi_{\Sigma},2\rho_P)\in \Lambda (L) $$ where $ \Lambda (L)$ is the minimal face of $ \Lambda _{\rm eff}(Y)$ containing $a(L)[L]-\psi(\varphi_{\Sigma},2\rho_P)$. Define $ \varphi _e\in PL(\Sigma) $ (for $e\in \Sigma_1$) by $ \varphi _e(e')=\delta_{ee'}$, for all $e'\in \Sigma_1$ and put $$ \Sigma'_1:= \{e\in \Sigma_1\,|\, \psi( \varphi _e,{\bf 0})\in \Lambda (L)\}. $$ Let $P'\subset G$ be the standard parabolic subgroup with $$ \Delta_{P'}=\{ \alpha \in \Delta _P\,\,|\,\, \psi({\bf 0}, \varpi _{ \alpha })\in \Lambda (L)\}, $$ where $\langle\varpi_{ \alpha }, \beta \rangle =\delta_{ \alpha \beta }$ for all $ \alpha , \beta \in \Delta_0$. Let $$ ( \varphi _L, \lambda _L)\in (\sum_{e\in \Sigma'_1}{\bf R}_{>0} \varphi _e)\times (\sum_{ \alpha \in \Delta_{P'}}{\bf R}_{>0} \varpi _{ \alpha }) $$ such that $\psi( \varphi _L, \lambda _L)=a(L)[L]-\psi ( \varphi _{\Sigma},2\rho_P)$. Then $$ \hat{L}:=\frac{1}{a(L)}( \varphi _{\Sigma}+ \varphi _L,2\rho_P+ \lambda _L) $$ is mapped onto $[L]$ by $\psi$. Denote by \begin{equation} \label{hL} h_L( \varphi , \lambda ):=\prod_{e\in \Sigma_1-\Sigma_1'} \frac{ \varphi (e)}{ \varphi (e)+1}\prod_{ \alpha \in \Delta_P - \Delta_{P'} } \frac{\langle \lambda , \alpha \rangle}{ \langle \lambda , \alpha \rangle +1} \end{equation} \noindent and put $$ \tilde{\varphi}= \varphi + \varphi _{\Sigma}+ \varphi _L\hskip 0,5cm {\rm and} \hskip 0,5cm \tilde{\lambda}= \lambda +\rho_P+ \lambda _L. $$ \bigskip \noindent >From now on we will denote by ${\bf K}_G\subset G({\bf A})$ the maximal compact subgroup defined in section 8.2. \bigskip \noindent {\bf Lemma 5.4}\hskip 0,5cm {\it There exists a convex open neighborhood $B$ of ${\bf 0}$ in $PL(\Sigma)_{{\bf R}}\oplus X^*(P)_{{\bf R}}$ with the following property: For any compact subset ${\bf K}\subset T_B$ there is a constant $c=c({\bf K})>0$ such that for all $( \varphi , \lambda )\in {\bf K}, \chi \in {\cal U}_T$ and $m\in M_{{\bf R}}$ we have $$ \left|\hat{H}_{\Sigma}(\chi,\tilde{\varphi} +im)E^G_P(\tilde{\lambda}- i\check{\eta}(m), \chi _{\eta}^{-1})h_L( \varphi +im, \lambda -i\check{\eta}(m))\right| $$ $$ \le c\prod_{v|\infty}\left\{\sum_{\dim \sigma =d}\,\,\prod_{e\in \sigma \cap \Sigma_1} \frac{1}{(1+|e(m+m_v(\chi))|)^{1+1/2d}}\right\}. $$ } \bigskip {\em Proof.} Write as in (\ref{3.5.1}) $$ \hat{H}_{\Sigma}( \chi ,\tilde{\varphi}+im)=\zeta_{\Sigma}( \chi ,\tilde{\varphi}+im)\prod_{e\in \Sigma_1} L_f( \chi _e,1+( \varphi + \varphi _L+im)(e)). $$ For ${\rm Re}( \varphi )$ sufficiently small and $e\in \Sigma_1'$ we have $$ {\rm Re}( \varphi (e))+ \varphi _L(e)\ge \frac{1}{2} \varphi _L(e)>0. $$ Hence $$ \left| \prod_{e\in \Sigma_1'}L_f( \chi _e,1+( \varphi + \varphi _L+im)(e)) \right| $$ is bounded for ${\rm Re}( \varphi )$ sufficiently small. If $e\in \Sigma_1-\Sigma_1'$ then $ \varphi _L(e)=0$. By the estimates of Rademacher (cf. Theorem 3.6), we have for $ \chi _e\neq 1$ $$ \left| L_f( \chi _e,1+( \varphi +im)(e)) \right|\le c_e(1+|m(e)|+\|( \chi _e)_{\infty}\|)^{\epsilon} $$ for ${\rm Re}( \varphi (e))>-\delta$ and $ \varphi $ in a compact set ($ \delta$ depends on $\epsilon $, $c_e$ depends on this compact subset). \noindent If $ \chi _e=1$ (abusing notations we will denote from now on the trivial character by 1) then $$ \frac{( \varphi +im)(e)}{( \varphi +im)(e)+1} \left| L_f(1,1+( \varphi + im)(e)) \right|\le c_e(1+|m(e)|)^{\epsilon} $$ Now we use Proposition 8.7 concerning estimates for Eisenstein series. This proposition tells us that there is for given $\epsilon >0$ an open neighborhood of ${\bf 0}$ in $X^*(P)_{{\bf R}}$ such that for ${\rm Re}( \lambda )$ contained in this neighborhood $$ \left| \prod_{ \alpha \in \Delta _P} \frac{\langle \lambda + \lambda _L-i\check{\eta}(m), \alpha \rangle}{ \langle \lambda + \lambda _L-i\check{\eta}(m), \alpha \rangle +1} E^G_P(\tilde{\lambda} -i\check{\eta}(m),{\chi}_{\eta}^{-1}) \right| \le c_1(1+\|{\rm Im}( \lambda )+ \check{\eta}(m)\|+\|({{\chi}_{\eta}}^{-1})_{\infty}\|)^{\epsilon}. $$ (For the definition of $(\cdots )_{\infty}$ and the norms see section 8.5.) If we let $ \lambda $ vary in a compact subset in the tube domain over this neighborhood then there is a constant $c_2\ge c_1$ such that $$ c_1(1+\|{\rm Im}( \lambda )+\check{\eta}(m)\|+\|({{\chi}_{\eta}}^{-1})_{\infty}\|)^{\epsilon} \le c_2(1+\|\check{\eta}(m)\|+\|({{\chi}_{\eta}}^{-1})_{\infty}\|)^{\epsilon} $$ For ${\rm Re}( \lambda )$ sufficiently small and $ \alpha \in \Delta _{P'}$ we have $$ \langle {\rm Re}( \lambda )+ \lambda _L, \alpha \rangle\ge \frac{1}{2}\langle \lambda _L, \alpha \rangle >0. $$ Therefore, there are $c_3,c_4>0$ such that for all such $ \lambda $ and $m\in M_{{\bf R}}$ we have $$ c_3\le \left| \prod_{ \alpha \in \Delta _{P'}} \frac{\langle \lambda + \lambda _L-i\check{\eta}(m), \alpha \rangle}{ \langle \lambda + \lambda _L-i\check{\eta}(m), \alpha \rangle+1} \right| \le c_4 $$ Putting everything together, we can conclude that there is a neighborhood $B$ of ${\bf 0}$ in $ PL(\Sigma)_{{\bf R}}\oplus X^*(P)_{{\bf R}}$ such that for $( \varphi , \lambda ) $ in a compact subset ${\bf K}$ of the tube domain over $B$ we have $$ \left| \prod_{e\in \Sigma_1} L_f( \chi _e, 1+( \varphi + \varphi _L+im)(e)) \prod_{e\in \Sigma_1-\Sigma_1'} \frac{( \varphi +im)(e)}{( \varphi +im)(e)+1}\right| $$ $$ \times \left| E^G_P(\tilde{\lambda}-i\check{\eta}(m),{\chi}_{\eta}^{-1}) \prod_{ \alpha \in \Delta _P- \Delta _{P'}} \frac{ \langle \lambda -i\check{\eta}(m), \alpha \rangle}{\langle \lambda -i\check{\eta}(m), \alpha \rangle+1} \right| \le c'({\bf K})(1+\|m+m_{\infty}( \chi )\|)^{\epsilon}, $$ where $\|\cdot\|$ is a norm on $M_{{\bf R},\infty}$. On the other hand, by Lemma 3.5, we have $$ \left|\zeta_{\Sigma}( \chi ,\tilde{\varphi} +im)\right|\le c''({\bf K})\prod_{v|\infty} \left\{ \sum_{\dim \sigma =d}\,\,\prod_{e\in \sigma \cap \Sigma_1} \frac{1}{(1+|e(m+m_v( \chi ))|)^{1+1/d}} \right\} $$ Now we may choose $\epsilon $ and $c_5$ such that $$ (1+\|m+m_{\infty}( \chi )\|)^{\epsilon}\le c_5\prod_{v|\infty} \prod_{e\in \sigma _v\cap \Sigma_1}(1+|e(m+m_v( \chi ))|)^{1/2d} $$ for any system $( \sigma _v)_{v|\infty}$ of $d$-dimensional cones. This gives the claimed estimate. \hfill $\Box$ \subsection* \noindent {\bf 5.5}\hskip 0,5cm To begin with, we let $$ M'_{{\bf R}}:=\{ m\in M_{{\bf R}}\,\,|\,\, e(m)=0\, \forall e\in {\Sigma_1- \Sigma_1'}\,\, {\rm and}\,\, \langle \check{ \eta }(m), \alpha \rangle =0\,\, \forall\, \alpha \in \Delta _P - \Delta _{P'} \}, $$ $$ M'=M'_{{\bf R}}\cap M. $$ Then $M'_{{\bf R}}=M'\otimes {\bf R}$ and $M''=M/M'$ is torsion free. Put $d'={\rm rank}(M')$, $d''={\rm rank}(M'')$. The connection with sections 5.2 and 5.3 is as follows: $$ E=(PL(\Sigma)_{{\bf R}}\oplus X^*(P)_{{\bf R}})/M'_{{\bf R}}, $$ $$ V=M''\otimes {\bf R}, A=M'', \nu = d'', $$ the set of linear forms $l_1,...,l_m$ is given as follows \begin{equation} \label{5.5.1-0} ( \varphi , \lambda )+M'_{{\bf R}}\mapsto \varphi (e), \,\, e\in {\Sigma_1- \Sigma_1'}, \end{equation} \begin{equation} ( \varphi , \lambda )+M'_{{\bf R}}\mapsto \langle \lambda , \alpha \rangle, \alpha \in {\Delta_P -\Delta_{P'}} \label{5.5.1-1} \end{equation} The measure $dv=dm''$ on $V=M''_{{\bf R}}$ is normalized by $M''$, $dm=dm'dm''$, where $dm$ (resp. $dm'$) is the Lebesgue measure on $M_{{\bf R}}$ (resp. $M'_{{\bf R}}$) normalized by $M$ (resp. $M'$). Fix a convex open neighborhood of ${\bf 0}$ in $PL(\Sigma)_{{\bf R}}\oplus X^*(P)_{{\bf R}}$ for which Lemma 5.4 is valid. Denote by $B$ the image of this neighborhood in $E$. This is an open and convex neighborhood of ${\bf 0}$. Using Lemma 5.4 we see that $$ g( \varphi , \lambda )= \mu_{T'}\int_{M'_{{\bf R}}}\frac{1}{\kappa^{d''}} \left\{\sum_{\chi\in {\cal U}_T} \hat{H}_{\Sigma}(\chi,\tilde{\varphi}+im')E^G_P(\tilde{\lambda}-i\check{\eta}(m'),{\chi}_{\eta}^{-1}) h_L( \varphi , \lambda )\right\}dm' $$ is a holomorphic function on $T_B$ (here $ \mu_{T'}=1/(2\pi \kappa)^{d'}$). (We use the invariance of $g$ under $iM_{{\bf R}}'$ and Cauchy-Riemann differential equations to check that $g$ is actually a function on $T_B$.) Hence, $$ f( \varphi , \lambda ):=g( \varphi , \lambda )h_L( \varphi , \lambda )^{-1} $$ is a meromorphic function on $T_B$. Let $E^{(0)}$ be the common kernel of all maps (\ref{5.5.1-0}, \ref{5.5.1-1}). Note that there is an exact sequence $$ 0 \rightarrow M'_{{\bf R}} \rightarrow E^{(0)} \rightarrow \langle \Lambda (L)\rangle \rightarrow 0 $$ which implies \begin{equation} b(L)={\rm codim}\,\, \Lambda (L)=m-d'', \label{5.5.2} \end{equation} where $m=\#({\Sigma_1- \Sigma_1'} ) +\#({\Delta_P -\Delta_{P'}})$. By construction, $M_{{\bf R}}''\cap E^{(0)}=\{{\bf 0}\}$. Let $dy$ be the Lebesgue measure on $E_0^{\vee}$ normalized by the lattice generated by the linear forms (\ref{5.5.1-0}, \ref{5.5.1-1}). Denote by $E^+_0$ the closed simplicial cone in $E_0$ defined by these linear forms, and by $ \pi_0\,:\, E \rightarrow E_0$ the canonical projection. It is easily seen that $\pi_0(M_{{\bf R}}'')\cap E^+_0=\{{\bf 0}\}$ (using the exact sequence above). Let $$ \psi_0\,:\, E_0 \rightarrow P:=E_0/\pi_0(M_{{\bf R}}'') $$ be the canonical projection and put $$ \Lambda =\psi_0(E^+_0), $$ $$ B^+=B\cap \{ ( \varphi , \lambda )\in E\,|\, \varphi (e)>0\,\,\forall \,e\in {\Sigma_1- \Sigma_1'}, \langle \lambda , \alpha \rangle >0\,\,\forall\, \alpha \in {\Delta_P -\Delta_{P'}} \}. $$ By the following theorem the function $f\in {\cal M}(T_B)$ is distinguished with respect to $M_{{\bf R}}''$ and the set of linear forms $(\ref{5.5.1-0}, \ref{5.5.1-1})$. Therefore, we can define $\tilde{f}_{B^+}\,:\, T_{B^+} \rightarrow {\bf C}$ by $$ \tilde{f}_{B^+}(z)=\frac{1}{(2\pi)^{d''}}\int_{M_{{\bf R}}''} f\left(z+i(m'',-\check{\eta}(m''))\right)dm''. $$ \bigskip \noindent {\bf Theorem 5.5} \hskip 0,5cm {\it a) $f$ is a distinguished function with respect to $M_{{\bf R}}''$ and the set of linear forms $(\ref{5.5.1-0}, \ref{5.5.1-1})$. b) There exist an open neighborhood $\tilde{B}$ of ${\bf 0}$ containing $B^+$ and linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}$ which vanish on $M_{{\bf R}}''$ such that $$ \tilde{f}_{B^+}(z)\prod_{j=1}^{\tilde{m}}\tilde{l}_j(z) $$ has a holomorphic continuation to $T_{\tilde{B}}$ and $g({\bf 0})\neq 0$. } \bigskip {\em Proof.} a) Define $c_0\,:\, M_{{\bf R},\infty} \rightarrow {\bf R}_{\ge 0}$ by $$ c_0((m_v)_v)=\prod_{v|\infty} \{ \sum_{\dim \sigma =d }\,\,\prod_{e\in \sigma \cap \Sigma_1} \frac{1}{(1+|e(m_v)|)^{1+1/2d}}\}. $$ Let ${\cal F}\subset M_{{\bf R},\infty}^{1}$ be the cube spanned by a basis of the image of ${\cal U}_T$ in $M_{{\bf R},\infty}^{1}$. Let $c'>0$ such that for all $m_{\infty}(\chi)\in M_{{\bf R},\infty}^{1}$ ($\chi\in {\cal U}_T)$ and all $m^1\in {\cal F}$ $$ c_0(m_{\infty}(\chi))\le c'c_0(m_{\infty}(\chi)+m^1). $$ Let $dm^1$ be the Lebesgue measure on $M_{{\bf R},\infty}^{1}$ normalized by the image of ${\cal U}_T$. Then for $m\in M_{{\bf R}}$ $$ \int_{M'_{{\bf R}}}\left\{\sum_{ \chi \in {\cal U}_T}c_0(m_{\infty}( \chi )+m'+m) \right\}dm'\le c(m \hskip 0,2cm {\rm mod} \hskip 0,2cm M'_{{\bf R}}), $$ where $c\,:\, M^{''}_{{\bf R}} \rightarrow {\bf R}_{\ge 0}$ is defined by $$ c(m^{''}):={\rm cl}_F^dc'\int_{M'_{{\bf R}}}\int_{M_{{\bf R},\infty}^{1}} c_0(m'+m^1+m^{''})dm^1dm'. $$ \noindent By Lemma 5.4, for any compact subset ${\bf K}$ of $T_B$ there is a $c({\bf K})>0$ such that $$ |g(z+im'')|\le c(m'') $$ for all $z\in {\bf K}$ and $m''\in M_{{\bf R}}''$. Obviously, $c$ can be integrated over any subspace $U$ of $M_{{\bf R}}''$. It remains to show that for any $m''\in M_{{\bf R}}'' - U$ one has $$ \lim_{\tau \rightarrow \pm \infty}\int_Uc(\tau m''+u)du=0. $$ This exercise will be left to the reader. b) The first part concerning the meromorphic continuation and singularities of $\tilde{f}_{B^+}$ is the content of Theorem 5.2. The relation $$ \lim_{s \rightarrow 0}s^{b(L)}\tilde{f}_{B^+}(s\hat{L})= g({\bf 0}){\cal X}_{ \Lambda }(\psi_0(\hat{L})) $$ is satisfied by Theorem 5.3 and (\ref{5.5.2}). It will be shown in Section 6 that $g({\bf 0})\neq 0$. \hfill $\Box$ \subsection* \noindent {\bf 5.6}\hskip 0,5cm The main theorem of our paper is: \bigskip \noindent {\bf Theorem 5.6}\hskip 0,5cm {\it Let $L$ be a line bundle on $Y$ which lies in the interior of the cone of effective divisors. Then there exists a metrization ${\cal L}$ of $L$ with the following properties: a) The height zeta function $$ Z_{Y^o}({\cal L},s)=\sum_{y\in Y^o(F)}H_{\cal L}(y)^{-s} $$ is holomorphic for ${\rm Re} (s)>a(L)$ and it can be continued meromorphically to a half-space ${\rm Re}(s)>a(L)- \delta $ for some $ \delta >0$. In this half-space it has a pole of order $b(L)$ at $a(L)$ and no other poles. b) For the counting function one has the following asymptotic relation $$ N_{Y^{o}}({\cal L}, H)= c({\cal L})H^{a(L)}(\log H)^{b(L)-1}(1+o(1)) $$ for $H \rightarrow \infty$ with some constant $c({\cal L})>0$. } \bigskip {\em Proof.} a) By construction, $\hat{L}=\frac{1}{a(L)}( \varphi _{\Sigma}+ \varphi _L,2 \lambda _P+ \lambda _L)$ is mapped onto $[L]$ by $\psi$. Hence $Z_{Y^o}({\cal L},s)$ converges absolutely for ${\rm Re}(s)>a(L)$, where ${\cal L}$ is the metrization mentioned in Proposition 4.4. By the same proposition, $$ Z_{Y^o}({\cal L}, s+a(L))=\mu_T \int_{M_{{\bf R}}}\{\sum_{\chi\in {\cal U}_T}f_L(\chi,im)\}dm $$ where $$ f_L(\chi,im):= $$ $$ \hat{H}_{\Sigma}(\chi,\frac{s}{a(L)}( \varphi _{\Sigma}+ \varphi _L) + \varphi _{\Sigma}+ \varphi _L+im) E^G_P(\frac{s}{a(L)}(2\rho_P+ \lambda _L)+\rho_P+ \lambda _L-i\check{\eta}(m),{\chi}_{\eta}^{-1}) $$ for all $s$ with ${\rm Re}(s)>0$. However, this is just $$ \frac{1}{(2\pi)^{d''}} \int_{M_{{\bf R}}''}f\left(s\hat{L}+i(m'',-\check{\eta}(m''))\right)dm''= \tilde{f}_{B^+}(s\hat{L}) $$ with $f,B^+$ and $\tilde{f}_{B^+}$ introduced in the preceding section. By Theorem 5.5, $\tilde{f}_{B^+}$ extends to a meromorphic function on a tube domain over a neighborhood of ${\bf 0} $ and in this tube domain the only singularities are the hyperplanes defined over ${\bf R}$. Hence there is a $ \delta >0$ such that $Z_{Y^o}({\cal L},s+a(L))$ extends to a meromorphic function in the half-space ${\rm Re}(s)>- \delta $ and the only possible pole is in $s=0$ and its order is exactly $b(L)$ (Theorems 5.3 and 5.5). b) This result follows from a Tauberian theorem (cf. \cite{De}, Th\'eor\`eme III or \cite{Sh}, Problem 14.1 (in the constant stated there the factor $\frac{1}{k_0}$ is missing)). \hfill $\Box$ \section{Non-vanishing of asymptotic constants} \label{6} \subsection* \noindent {\bf 6.1} \hskip 0,5cm This section is devoted to the proof of the non-vanishing of $g({\bf 0})$ claimed in Theorem 5.5. All notations are as in sections 5.2-5.5. The function $g( \varphi , \lambda )$ which has been defined in 5.5 is given by $$ g( \varphi , \lambda )=\frac{\mu_{T'}}{\kappa^{d''}}\int_{M'} \left\{\sum_{\chi\in {\cal U}_T}\hat{H}_{\Sigma}(\chi,\tilde{\varphi}+im') E^G_P\left(\tilde{\lambda} -i\check{\eta}(m'),{\chi}_{\eta}^{-1}\right)h_L( \varphi , \lambda ) \right\}dm, $$ where $\tilde{\varphi},\tilde{\lambda}$ have been defined in 5.4. The function $h_L( \varphi , \lambda )$ was defined in 5.4: $$ h_L( \varphi , \lambda )=\prod_{e\in {\Sigma_1- \Sigma_1'}} \frac{ \varphi (e)}{ \varphi (e)+1}\prod_{ \alpha \in {\Delta_P -\Delta_{P'}}} \frac{\langle \lambda , \alpha \rangle}{ \langle \lambda , \alpha \rangle +1}. $$ The uniform convergency of the integral above in any compact subset of $T_B$ (cf. Lemma 5.4) allows us to compute the limit $$ \lim_{( \varphi , \lambda ) \rightarrow {\bf 0}}\prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e) \prod_{ \alpha \in {\Delta_P -\Delta_{P'}}}\langle \lambda , \alpha \rangle \hat{H}_{\Sigma}(\chi,\tilde{\varphi} +im') E^G_P\left( \tilde{\lambda} -i\check{\eta}(m'),{\chi}_{\eta}^{-1}\right) $$ first and then to integrate. We shall show that this limit vanishes if there are $e\in {\Sigma_1- \Sigma_1'}$ with $\chi_e\neq 1$ or $ \alpha \in {\Delta_P -\Delta_{P'}}$ with ${\chi}_{\eta}\circ\check{ \alpha }\neq 1$. Therefore, we may consider only $ \chi \in {\cal U}_T'$ where $$ {\cal U}_T':=\{ \chi \in {\cal U}_T\,|\, \chi _e=1\,\,\forall e\in {\Sigma_1- \Sigma_1'},\,\, {\chi}_{\eta}\circ\check{ \alpha }=1 \,\forall\,\, \alpha \in {\Delta_P -\Delta_{P'}}\}. $$ Let $\eta'\,:\, P' \rightarrow T$ be the uniquely defined homomorphism such that for all $ \alpha \in \Delta_{P'}$ we have $\eta'\circ\check{ \alpha }=\eta\circ\check{ \alpha }$. \bigskip \noindent {\bf Lemma 6.1}\hskip 0,5cm {\it \begin{equation} \label{6.1.1} g({\bf 0}) =\lim_{ \varphi \rightarrow {\bf 0},\, \varphi \in PL(\Sigma)^+ }\prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e)\cdot \frac{\mu_{T'}}{\kappa^{d''}}\cdot \frac{c_{P'}}{c_P} \int_{M'_{{\bf R}}}\{\sum_{ \chi \in {\cal U}_T'} \hat{H}_{\Sigma}( \chi ,\tilde{\varphi}+im') \end{equation} $$ \hskip 8,5cm \times E^G_{P'}\left( \lambda _L+\rho_{P'}-i\check{ \eta }'(m'),(\chi_{\eta'})^{-1}\right)\}dm' $$ (cf. 8.4 for the definition of $c_{P'}$ and $c_P$). } \bigskip {\em Proof.} Recall that (cf. (\ref{3.5.1})) $$ \hat{H}_{\Sigma}(\chi,\tilde{\varphi}+im')= \zeta _{\Sigma}(\chi, \tilde{\varphi} +im') \prod_{e\in \Sigma_1}L_f(\chi_e, 1+ \varphi (e)+ \varphi _L(e)+ie(m')) $$ and that $\zeta_{\Sigma}(\chi,\tilde{\varphi} +im')$ is regular for $ \varphi $ in a tube domain over a neighborhood of ${\bf 0}$ (Lemma 3.5). For $e\in \Sigma_1'$ we have $ \varphi _L(e)>0$, hence we see that the function $$ L_f(\chi_e, 1+ \varphi (e)+ \varphi _L(e)+ie(m')) $$ is holomorphic for ${\rm Re} ( \varphi )$ in a neighborhood of ${\bf 0}$. Let $e\in {\Sigma_1- \Sigma_1'}$. If $\chi_e\neq 1$ then the restriction of $\chi_e$ to ${{\bf G}_m({\bf A})^1}$ is non-trivial (by our construction of the embedding ${\cal U}_T \rightarrow {\cal A}_T$, cf. 3.3), hence $$ \varphi \mapsto L_f(\chi_e, 1+ \varphi (e)) $$ is an entire function and $ \varphi (e)L_f(\chi_e, 1+ \varphi (e))$ tends to $0$ as $ \varphi \rightarrow {\bf 0}$. For $ \alpha \in \Delta_{P'}$ we have $\langle \lambda _L, \alpha \rangle >0$, hence $ \lambda _L$ is contained in $ X^*(P')^+$. Let $ \alpha \in {\Delta_P -\Delta_{P'}}$. If ${\chi}_{\eta}\circ \check{ \alpha } \neq 1$ then ${\chi}_{\eta}\circ \check{ \alpha }$ restricted to ${{\bf G}_m({\bf A})^1}$ is non-trivial and therefore $$ \prod_{ \alpha \in {\Delta_P -\Delta_{P'}}}\langle \lambda , \alpha \rangle E^G_P\left( \lambda +\rho_P+ \lambda _L-i\check{\eta}(m'),{\chi}_{\eta}^{-1}\right) $$ vanishes as $ \lambda \rightarrow {\bf 0}$ (cf. Proposition 8.3). We have shown that it suffices to take the sum over all $\chi\in {\cal U}_T'$. To complete the proof, note that for $\chi\in {\cal U}_T'$ we have (cf. Proposition (8.4)) $$ \lim_{ \lambda \rightarrow {\bf 0}} \prod_{ \alpha \in {\Delta_P -\Delta_{P'}}}\langle \lambda , \alpha \rangle E^G_P\left( \tilde{\lambda}-i\check{\eta}(m'),{\chi}_{\eta}^{-1}\right ) = \frac{c_{P'}}{c_P}E^G_{P'}\left( \lambda _L+\rho_{P'}-i\check{\eta}'(m'), ({\chi}_{\eta}')^{-1}\right). $$ \hfill $\Box$ \subsection* \noindent {\bf 6.2}\hskip 0,5cm By the absolute and uniform convergence of $$ \int_{M'_{{\bf R}}}\left\{\sum_{\chi\in {\cal U}_T'} \hat{H}_{\Sigma}(\chi,\tilde{\varphi} +im') \prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e)\right\}dm' $$ (cf. Lemma 3.5 and Theorem 3.6) and the convergence of $$ \sum_{ \gamma \in P'(F) \backslash G(F)}e^{\langle \lambda _L+2\rho_{P'},H_{P'}( \gamma)\rangle} $$ we may change summation and integration in (\ref{6.1.1}) and get for all $ \varphi \in PL(\Sigma)^+$ $$ \frac{\mu_{T'}}{\kappa^{d''}}\cdot \frac{c_{P'}}{c_P} \int_{M'_{{\bf R}}}\sum_{ \chi \in {\cal U}_T'} \hat{H}_{\Sigma}( \chi ,\tilde{\varphi}+im') E^G_{P'}\left( \lambda _L+\rho_{P'}-i\check{\eta}'(m')(\chi_{\eta'})^{-1}\right)dm' $$ $$ =\frac{c_{P'}}{c_P\kappa^{d''}}\sum_{ \gamma\in P'(F) \backslash G(F)} e^{\langle \lambda _L+2\rho_{P'},H_{P'}({ \gamma}')\rangle} \mu_{T'} \int_{M'_{{\bf R}}} \sum_{ \chi \in {\cal U}_T^{'}}\hat{H}_{\Sigma}( \chi ,\tilde{\varphi}+im') ( \chi ^{m'} \chi )^{-1}( \eta '(p_{ \gamma}')) dm' $$ where $ \gamma =p_{ \gamma}'k_{ \gamma}$ as above. Let $I$ be the image of the homomorphism $$ \prod_{e\in {\Sigma_1- \Sigma_1'}}{\bf G}_m({\bf A})\times \prod_{ \alpha \in {\Delta_P -\Delta_{P'}}}{\bf G}_m({\bf A}) \rightarrow T({\bf A}) $$ induced by $M \rightarrow {\bf Z}^{{\Sigma_1- \Sigma_1'}}\oplus {\bf Z}^{{\Delta_P -\Delta_{P'}}}$, $$ m\mapsto \left((e(m))_{e\in {\Sigma_1- \Sigma_1'}},(\langle -m\circ\eta, \alpha \rangle)_{ \alpha \in {\Delta_P -\Delta_{P'}}})\right). $$ Then $M_{{\bf R}}'\oplus {\cal U}_T'$ is precisely the set of characters $T({\bf A}) \rightarrow S^1$ which are trivial on $T(F){\bf K}_TI$. Put $T'={\rm Spec} (F[M'])$ and $ T''={\rm Spec} (F[M'']).$ Then there is an exact sequence $$ 1 \rightarrow T'' \rightarrow T \rightarrow T' \rightarrow 1. $$ Note that $I\subset T''({\bf A})$. Denote by ${\bf K}_{T'}$ (resp. ${\bf K}_{T''}$) the maximal compact subgroup of $T'({\bf A})$ (resp. of $T''({\bf A})$). The linear forms $$ m\mapsto e(m),\,\,\, e\in {\Sigma_1- \Sigma_1'}, $$ $$ m\mapsto -\langle m\circ \eta, \alpha \rangle, \,\,\, \alpha \in {\Delta_P -\Delta_{P'}}, $$ when considered as functions on $M''$, generate a sublattice of finite index in $N''=\Hom (M'',{\bf Z})$. This shows that there is a $q>0$ such that the image of the $q$-th power homomorphism $T''({\bf A}) \rightarrow T''({\bf A})$, $t\mapsto t^q$, is contained in $I$. If $v$ is any archimedean place of $F$ the connected component of one in $T''(F_v)$ is therefore contained in $I$. Consequently, $$ T''(F)\cdot\prod_{v|\infty}T''(F_v) \prod_{v\nmid\infty }{\bf K}_{T'',v} \subset T''(F){\bf K}_{T''}\cdot I $$ and the left hand side is of finite index in $T''({\bf A})$. Put $$ {\cal A}_T'=\{ \chi \in {\cal A}_T\,\,|\,\, \chi =1 \,\, {\rm on}\,\, T(F){\bf K}_TT''({\bf A})\}. $$ We observe that $$ {\cal A}_T'\simeq {\cal A}_{T'}=(T'({\bf A})/T'(F){\bf K}_{T'})^*\subset M'_{{\bf R}}\oplus{\cal U}_T' $$ We denote by $$ \Ind(L)=(M'_{{\bf R}}\oplus{\cal U}_T')/{\cal A}_T' $$ and by $\ind(L)$ the order of $\Ind(L)$. Put ${\cal U}_{T'}={\cal U}_T\cap {\cal A}_T'$ (then $ {\cal A}_T'=M_{{\bf R}}'\oplus {\cal U}_{T'}$). Thus we may write $$ \int_{M'_{{\bf R}}}\left\{\sum_{ \chi \in {\cal U}'_T} \hat{H}_{\Sigma}( \chi ,\tilde{\varphi}+im')( \chi ^{m'} \chi )^{-1}( \eta (p_{ \gamma}')) \right\}dm' $$ $$ = \sum_{ \chi \in \Ind(L) } \int_{M'_{{\bf R}}}\left\{\sum_{ \chi '\in {\cal U}_{T'}} \hat{H}_{\Sigma}( \chi ' \chi ^{m'} \chi ,\tilde{\varphi}) ( \chi ' \chi ^{m'} \chi )^{-1}( \eta '(p'_{ \gamma}))\right\}dm' $$ For $ \chi \in M'_{{\bf R}}\oplus {\cal U}_T'$ and $x\in T'({\bf A})$ we consider the function $$ x\mapsto \int_{T''({\bf A})}H_{\Sigma}(xt \eta '(p_{ \gamma}'), \tilde{\varphi}) \chi (t)dt $$ (the Haar measure $dt$ on $T''({\bf A})$ is defined as $dx$ on $T({\bf A})$, cf. 3.3). The same argument as in the proof of Proposition 3.4 shows that this function is absolutely integrable over $T'(F)$ if $ \varphi \in PL(\Sigma)^+$. The Fourier transform for $ \chi '\in {\cal A}_T'$ $$ \int_{T'({\bf A})}\left(\int_{T''({\bf A})}H_{\Sigma}(xt\eta'(p_{ \gamma}'),\tilde{\varphi}) \chi (t)dt\right) \chi '(x)dx $$ is absolutely integrable over ${\cal A}'_T$. Using Poisson's summation formula twice we get $$ \sum_{ \chi \in \Ind(L)}\mu_{T'} \int_{M_{{\bf R}}'}\left\{\sum_{\chi'\in {\cal U}_{T'}} \hat{H}_{\Sigma}( \chi ^{m'} \chi ' \chi ,\tilde{\varphi}) ( \chi ' \chi ^{m'} \chi )^{-1}( \eta '(p'_{ \gamma}))\right\}dm' $$ $$ =\sum_{ \chi \in \Ind(L) } \sum_{x\in T'(F)}\int_{T''({\bf A})} H_{\Sigma}(xt \eta '(p_{ \gamma}',\tilde{\varphi}) \chi (t)dt= \sum_{x\in T'(F)} \ind(L) \int_{T''(F) {\bf K}_T I}H_{\Sigma}(xt \eta '(p_{ \gamma}'),\tilde{\varphi})dt. $$ \noindent Now we collect all the terms together. \bigskip \noindent {\bf Lemma 6.2}\hskip 0,5cm {\it The constant $g({\bf 0})$ is equal to $$ \frac{\ind(L) c_{P'}/c_{P}}{\kappa^{d''}} \sum_{ \gamma\in P'(F) \backslash G(F)}e^{\langle \lambda _L+2\rho_{P'},H_{P'}( \gamma)\rangle} \times $$ $$ \lim_{ \varphi \rightarrow {\bf 0},\, \varphi \in PL(\Sigma)^+} \prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e) \sum_{x\in T'(F)}\int_{T''(F){\bf K}_{T''}I} H_{\Sigma}(x \eta '(p_{ \gamma}')t,\tilde{\varphi})dt. $$ } \bigskip \noindent {\bf Lemma 6.3}\hskip 0,5cm {\it The limit $$ \lim_{ \varphi \rightarrow {\bf 0}, \varphi \in PL(\Sigma)^+} \prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e)\int_{T''({\bf A})} H_{\Sigma}(t,\tilde{\varphi})dt $$ exists and is positive. } \bigskip {\em Proof.} Consider the embedding $N''_{{\bf R}} \rightarrow N_{{\bf R}}$ and let $$ \Sigma'':=\{ \sigma \cap N_{{\bf R}}''\,|\, \sigma \in \Sigma\}. $$ This is a complete fan in $N_{{\bf R}}''$ which consists of rational polyhedral cones, but which is not necessary a regular fan. We can obtain a regular fan by subdivision of the cones into regular ones (cf. \cite{KKMS-D}, ch. I, \S 2, Theorem 11). This gives us a complete regular fan $\tilde{\Sigma}''$ such that any cone in $\tilde{\Sigma}''$ is contained in a cone of $\Sigma''$. Denote by $\tilde{\Sigma}_1''$ the set of primitive integral generators of the one-dimensional cones in $\tilde{\Sigma}''$. Computing the integral as in section 3.4 we get $$ \int_{T''({\bf A})}H_{\Sigma}(t,\tilde{\varphi} )dt= \zeta _{\tilde{\Sigma}''}(1,\tilde{\varphi} ) \prod_{\tilde{e}\in \tilde{\Sigma}_1''}L_f(1,\tilde{\varphi}(\tilde{e})) $$ (cf. (\ref{3.5.1})), where $\zeta_{\tilde{\Sigma}''}(1,\tilde{\varphi})$ is regular in a neighborhood of $ \varphi ={\bf 0}$ and positive for $ \varphi ={\bf 0}$. Let $\tilde{e}\in N$ and $ \sigma \in \Sigma'$ be a cone containing $\tilde{e}$. Write $\tilde{e}=\sum_{e\in \sigma \cap \Sigma_1}t_e\cdot e$ ($ t_e\in {\bf Z}_{\ge 0}$). Suppose $$ 1=( \varphi _{\Sigma}+ \varphi _L)(\tilde{e})=\sum_{e\in \sigma \cap \Sigma_1}t_e(1+ \varphi _L(e)). $$ Then $t_e=0$ for all $e\in \sigma \cap \Sigma_1'$ because $ \varphi _L\in \sum_{e\in \Sigma_1'} {\bf R}_{>0} \varphi _e$ (cf. 5.4). Hence, $$ \tilde{e}=\sum_{e\in \sigma \cap {\Sigma_1- \Sigma_1'}}t_e\cdot e $$ and $ \varphi _{\Sigma}(\tilde{e})=1$ implies $\tilde{e}\in {\Sigma_1- \Sigma_1'}$. Therefore, $$ \lim_{ \varphi \rightarrow {\bf 0},\, \varphi \in PL(\Sigma)^+}\prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e) \prod_{\tilde{e}\in \tilde{\Sigma}_1^{''}}L_f(1, \tilde{\varphi}(\tilde{e})) $$ $$ = \{\prod_{e\in {\Sigma_1- \Sigma_1'}}\lim_{ \varphi \rightarrow {\bf 0},\, \varphi \in PL(\Sigma)^+} \varphi (e) L_f(1,1+ \varphi (e))\} \prod_{\tilde{e}\in \tilde{\Sigma}_1''- ({\Sigma_1- \Sigma_1'}) } L_f\left(1,( \varphi _{\Sigma}+ \varphi _L)(\tilde{e})\right) $$ and this is a positive real number. \hfill $\Box$ \bigskip \noindent In theorem 5.5 we claimed the non-vanishing of $g({\bf 0})$. We are now in the position to prove \bigskip \noindent {\bf Corollary 6.4}\hskip 0,5cm {\it $$ g({\bf 0})>0. $$ } \bigskip {\em Proof.} By Lemma 6.2 it is enough to show that $$ \lim_{ \varphi \rightarrow {\bf 0}, \varphi \in PL(\Sigma)^+}\prod_{e\in {\Sigma_1- \Sigma_1'}} \varphi (e) \int_{T''(F){\bf K}_{T''}\cdot I}H_{\Sigma}(t,\tilde{\varphi})dt $$ is positive. Let $t_1,...,t_{\nu}\in T({\bf A})$ such that $$ T''({\bf A})=\bigcup_{j=1}^{\nu}t_jT''(F){\bf K}_{T''}\cdot I. $$ Then there exists a constant $c>0$ such that for all $t\in T''({\bf A})$ and $j=1,...,\nu$ we have $$ H_{\Sigma}(tt_j,\tilde{\varphi})\le \frac{c}{\nu}H_{\Sigma}(t,\tilde{\varphi}). $$ Hence we can estimate $$ \int_{T''({\bf A})}H_{\Sigma}(t,\tilde{\varphi})dt=\sum_{j=1}^{\nu} \int_{T''(F){\bf K}_{T''} I}H_{\Sigma}(tt_j,\tilde{\varphi})dt $$ $$ \le c\int_{T''(F){\bf K}_{T''} I}H_{\Sigma}(t,\tilde{\varphi})dt. $$ Lemma 6.3 allows us to conclude that the limit above is indeed positive. \hfill $\Box$ \section{Technical theorems} \label{7} \subsection* \noindent {\bf 7.1}\hskip 0,5cm Let $(A, V, \Lambda ) $ be a triple consisting of a free abelian group $A$ of rank $d$, a $d$-dimensional real vector space $V:= A \otimes {\bf R}$ containing $A$ as a sublattice of maximal rank, and a closed strongly convex polyhedral $d$-dimensional cone $\Lambda \subset A_{{\bf R}}$ such that $\Lambda \cap - \Lambda =\{{\bf 0}\}$. Denote by $ \Lambda ^{\circ}$ the interior of $ \Lambda $. Let $( A^{\vee} , V^{\vee }, \Lambda ^{\vee }) $ be the triple consisting of the dual abelian group $A^{\vee } = {\rm Hom}(A, {\bf Z})$, the dual real vector space $V^{\vee } = {\rm Hom}(V, {\bf R})$ and the dual cone $ \Lambda ^{\vee } \subset V^{\vee }$. We normalize the Haar measure $ dy$ on $V^{\vee }$ by the condition: ${\rm Vol}(V^{\vee }/A^{\vee })=1$. We denote by $\chi_{ \Lambda }(v)$ the set-theoretic characteristic function of the cone $ \Lambda $ and by ${\cal X}_{ \Lambda }(v)$ the Laplace transform of the set-theoretic characteristic function of the dual cone $$ {\cal X}_{ \Lambda }(v) =\int_{V^{\vee }}\chi_{ \Lambda ^{\vee}}(y)e^{-\langle v,y \rangle } dy= \int_{{ \Lambda }^{\vee }} e^{- \langle v, y \rangle} dy, $$ where ${{\rm Re} }(v) \in { \Lambda }^{\circ}$ (for these $v$ the integral converges absolutely). Consider a complete regular fan $\Sigma$ on $V$, that is, a subdivision of the real space $V$ into a finite set of convex rational simplicial cones, satisfying certain conditions (see \cite{BaTschi1}, 1.2). Denote by $\Sigma_1$ the set of primitive generators of one dimensional cones in $\Sigma$. Denote by $PL(\Sigma)_{{\bf R}}$ the vector space of real valued piecewise linear functions on $V$ and by $PL(\Sigma)_{{\bf C}}$ its complexification. \begin{prop}(\cite{BaTschi1}, Prop. 2.3.2, p. 614) For any compact ${\bf K}\subset PL(\Sigma)_{{\bf C}}$ with the property that ${\rm Re}(\varphi(v))>0$ for all $\varphi \in {\bf K}$ and $v\neq {\bf 0}$ there exists a constant $\kappa({\bf K})$ such that for all $\varphi\in {\bf K}$ and all $y\in V^{\vee}$ the following inequality holds: $$ \left|\int_{V}e^{-\varphi(v) -i \langle v,y \rangle}dv\right| \le \kappa({\bf K})\sum_{\dim \sigma =d}\frac{1}{\prod_{e\in \sigma }(1+ |\langle e,y \rangle |)^{1+1/d}}. $$ \label{estimate-fan} \end{prop} \bigskip \noindent {\bf 7.2}\hskip 0,5cm Let $H\subset V$ be a hyperplane with $H\cap \Lambda =\{{\bf 0}\}$. Let $y_0\in V^{{\vee }}$ with $H=\Ker (y_0)$, such that for all $x\in \Lambda \,:\, y_0(x)\ge 0$. Then $y_0$ is in the interior of $ \Lambda ^{\vee}\subset V^{\vee}$. Let $x_0\in \Lambda ^{\circ}$ and let $$ H'=\{y'\in V^{\vee }\,|\, y'(x_0)=0\}. $$ We have $V^{\vee}= H'\oplus {\bf R} y_0$. Define $ \varphi \,:\, H' \rightarrow {\bf R}$ by $$ \varphi (y')=\min \{ t\,|\, y'+ty_0\in \Lambda ^{{\vee }}\}. $$ The function $ \varphi $ is piecewise linear with respect to a complete fan of $H'$. Taking a subdivision, if necessary, we may assume it to be regular. \begin{prop} The function ${\cal X}_{ \Lambda }(u)$ is absolutely integrable over any linear subspace $U\subset H$. \label{estimate-chi} \end{prop} {\em Proof.} For $h\in H$ we have $$ {\cal X}_{ \Lambda }(x_0+ih)=\int_{ \Lambda ^{\vee}}e^{-y(x_0+ih)}dy= \int_{H'}\int_{ \varphi (y')}^{\infty}e^{-(y'+ty_0)(x_0+ih)}dtdy' $$ $$ = \int_{H'}e^{- \varphi (y')}e^{-iy'(h)}dy' $$ Therefore, $h\mapsto {\cal X}_{ \Lambda }(x_0+ih)$ is the Fourier transform of the function $y'\mapsto e^{- \varphi (y')}$ on $H'\simeq H^{\vee}$. The statement follows now from 7.1. \hfill $\Box$ \bigskip \noindent {\bf 7.3}\hskip 0,5cm The rest of this section is devoted to the proof of the meromorphic continuation of certain functions which are holomorphic in tube domains over convex finitely generated polyhedral cones. In section \ref{5} we have already introduced the terminology and explained how this technical theorem is applied to height zeta functions. Let $E$ be a finite dimensional vector space over ${\bf R}$ and $E_{{\bf C}}$ its complexification. Let $V\subset E$ be a subspace. We will call a function $c\,:\, V \rightarrow {\bf R}_{\ge 0}$ sufficient if it satisfies the following conditions: (i) For any subspace $U\subset V$ and any $v\in V$ the function $U \rightarrow {\bf R}$ defined by $u \rightarrow c(v+u)$ is measurable on $U$ and the integral $$ c_U(v):= \int_U c(v+u)du $$ is always finite ($du$ is a Lebesgue measure on $U$). (ii) For any subspace $U\subset V$ and every $v\in V - U$ we have $$ \lim_{\tau \rightarrow \pm \infty} c_U(\tau \cdot v)=0. $$ Let $l_1,...,l_m\in E^{\vee}=\Hom_{{\bf R}}(E,{\bf R})$ be linearly independent linear forms. Put $H_j={\Ker } (l_j)$ for $j=1,..., m$. Let $B\subset E$ be an open and convex neighborhood of ${\bf 0}$, such that for all $x\in B$ and all $j=1,...,m$ we have $l_j(x)>-1$. We denote by $T_B:=B+iE \subset E_{{\bf C}}$ the complex tube domain over $B$. We denote by ${\cal M}(T_B)$ the set of meromorphic functions on $T_B$. A meromorphic function $f\in {\cal M}(T_B)$ will be called {\it distinguished } with respect to the data $(V; l_1,...,l_m)$ if it satisfies the following conditions: (i) The function $$ g(z):=f(z)\prod_{j=1}^m\frac{l_j(z)}{l_j(z)+1} $$ is holomorphic in $T_B$. (ii) There exists a sufficient function $c\,:\, V \rightarrow {\bf R}_{\ge 0}$ such that for any compact ${\bf K}\subset T_B$ there is a constant $\kappa({\bf K})\ge 0$ such that for all $z\in {\bf K}$ and all $v \in V$ we have $$ |g(z+iv)|\le \kappa({\bf K})c(v). $$ Let $C$ be a connected component of $B - \bigcup_{j=1}^m H_j$ and $T_C$ a tube domain over $C$. We will consider the following integral: $$ \tilde{f}_C(z):=\frac{1}{(2\pi )^d}\int_Vf(z+iv)dv. $$ Here we denoted by $d=\dim V$ and by $dv$ a fixed Lebesgue measure on $V$. \begin{prop} Assume that $f$ is an distinguished function with respect to $(V;l_1,...,l_m)$. Then the following holds: a) $\tilde{f}_C\,:\, T_{C} \rightarrow {\bf C}$ is a holomorphic function. b) There exist an open and convex neighborhood $\tilde{B}$ of ${\bf 0}$, containing $C$, and linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}$, which vanish on $V$, such that $$ z \rightarrow \tilde{f}_C(z)\prod_{j=1}^{\tilde{m}}\tilde{l}_j(z) $$ has a holomorphic continuation to $T_{\tilde{B}}$. \label{cont} \end{prop} {\em Proof.} a) Let ${\bf K}\subset T_C$ be a compact subset and let $\kappa({\bf K})\ge 0$ be a real number such that for all $z\in {\bf K} $ and all $v\in V$ we have $|g(z+iv)|\le \kappa({\bf K})c(v)$. Since ${\bf K}$ is a compact and $C$ doesn't intersect any of the hyperplanes $H_j$ there exist real numbers $c_j\ge 0$ for any $j=1,...,m$, such that for all $z\in {\bf K}$ and $v\in V$ the following inequalities hold $$ \left|\frac{l_j(z+iv)+1}{l_j(z+iv)}\right|\le c_j. $$ Therefore, for $z\in {\bf K}$ and $v\in V$ we have $$ |f(z+iv)|\le c_1\cdots c_m\kappa({\bf K})c(v). $$ It follows that on every compact ${\bf K}\subset T_C$ the integral converges absolutely and uniformly to a holomorphic function $\tilde{f}_C$. \medskip b) The proof proceeds by induction on $d=\dim V$. For $d=0$ there is nothing to prove. Assume that $d\ge 1$ and let $v_0\in V - \{{\bf 0}\}$ be a vector such that both $v_0,-v_0\in B$. We define $B_1\subset B$ as the set of all vectors $x\in B$ which satisfy the following two conditions: the vector $x\pm v_0\in B $ and $|\frac{l_j(x)}{l_j(v_0)}|\le\frac{1}{2}$ for all $j\in \{1,...,m\}$ with $l_j(v_0)\neq 0$. The set $B_1$ is a convex open neighborhood of ${\bf 0}\in E$. Fix a vector $x_0\in C$. Without loss of generality we can assume that $$ \{1,...,m_0\}:=\{j\in \{1,...,m\}\,|\, l_j(v_0)l_j(x_0)<0\} $$ with $0\le m_0\le m$. For $j\in \{1,...,m_0\}, k\in \{1,...,{\widehat{j} },...,m\}$ we define $$ l_{j,k}(x):= l_k(x)-l_j(x)\frac{l_k(v_0)}{l_j(v_0)},\hskip 1cm H_{j,k}:={\Ker} (l_{j,k})\subset E. $$ For all $j\in \{1,...,m_0\}$ we have that $(l_{j,k})_{1\le k\le m, k\neq j}$ is a set of linearly independent linear forms on $E$. Moreover, for all $x\in B_1$ and $j\in \{1,...,m_0\}$ we have $$ x-\frac{l_j(x)}{l_j(v_0)}v_0 = \left(1-\frac{l_j(x)}{l_j(v_0)}\right)x+\frac{l_j(x)}{l_j(v_0)}(x-v_0)\in B, $$ in the case that $\frac{l_j(x)}{l_j(v_0)}\ge 0$ and, similarly, in the case that $\frac{l_j(x)}{l_j(v_0)}< 0$ $$ x-\frac{l_j(x)}{l_j(v_0)}v_0 =\left(1+\frac{l_j(x)}{l_j(v_0)}\right)x + \left(-\frac{l_j(x)}{l_j(v_0)}\right)(x+v_0)\in B. $$ Therefore, for all $x\in B_1$ and $j\in \{1,...,m_0\}, k\in \{1,...,{\hat{ j}},...,m\}$ we have $$ l_{j,k}(x)=l_k\left(x-\frac{l_j(x)}{l_j(v_0)}v_0\right)>-1. $$ Let $C_1$ be a connected component of $$ B_1 -\left( \bigcup_{j=1}^{m_0}\bigcup_{1\le k\le m, k\neq j } H_{j,k} \cup \bigcup_{j=1}^mH_j\right), $$ which is contained in $C$. For $z\in T_{C}$ we define $$ h_C(z):=\frac{1}{2\pi}\int_{-\infty}^{+\infty}f(z+i\tau v_0)d\tau = \frac{1}{2\pi i}\int_{{\rm Re}( \lambda )=0}f(z+ \lambda v_0)d \lambda . $$ As in (i) one shows that $h_{C}$ is a holomorphic function on $T_{C}$. For $x\in B_1$ and $ \lambda \in [0,1]$ we have $$ x+ \lambda v_0=(1- \lambda )x+ \lambda (x+v_0)\in B. $$ If for some $z = x+iy \in T_{C_1}$ ($x\in C_1$) and $ \lambda \in [0,1]+i{\bf R}, j\in \{1,...,m\}$ we have $$ l_j(z+ \lambda v_0)=0, $$ then it follows that $l_j(x)+{\rm Re}( \lambda )l_j(v_0)=0$, and therefore, $l_j(x)l_j(v_0)<0$ (since $l_j(x)$ has the same sign as $l_j(x_0)$). Consequently, $j\in \{1,...,m_0\}$. For $z\in T_{C_1}$ and $j\in \{1,...,m_0\}$ we put $$ \lambda _j(z):=-\frac{l_j(z)}{l_j(v_0)}. $$ By our assumptions, we have $ 0<{\rm Re}( \lambda _j(z))<\frac{1}{2}$. >From $ \lambda _j(z)= \lambda _{j'}(z),$ with $j,j'\in \{1,...,m_0\}$ and $j\neq j'$ it follows now that $$ l_{j'}\left(z-\frac{l_j(z)}{l_j(v_0)}v_0\right)=0. $$ In particular, we have $l_{j,j'}({\rm Re}(z))=0$. This is not possible, because $z\in T_{C_1}$. Assume now that $x\in B_1$. We have, assuming that $l_k(v_0)\neq 0$, that $$ |l_k(x+v_0)|=|l_k(v_0)|\cdot \left|\frac{l_k(x)}{l_k(v_0)}+1\right|\ge |l_k(v_0)|\cdot |-\frac{1}{2}+1|=\frac{1}{2}|l_k(v_0)|. $$ If $l_k(v_0)=0$ then we have $l_k(x+v_0)=l_k(x)$. Fix a $z\in T_{C_1}$. Then there exist some numbers $c_1(z)\ge 0, c_2(z)\ge 0$ such that for all $ \lambda \in [0,1], \tau \in {\bf R}$ with $|\tau |\ge c_1(z)$ we have $$ |f(z+ \lambda v_0+i\tau v_0)| $$ $$ =|g(z+ \lambda v_0+i\tau v_0)|\cdot |\prod_{j=1}^{m}(1+\frac{1}{ l_j(z+ \lambda v_0 +i\tau v_0)})|\le c_2(z)\kappa (z+[0,1]v_0)c(\tau v_0). $$ For any $z\in T_{C_1}$ we have therefore $$ h_C(z)=-\sum_{j=1}^{m_0}{\rm Res}_{ \lambda = \lambda _j(z)}( \lambda \rightarrow f(z+ \lambda v_0)) + \frac{1}{2\pi i}\int_{{\rm Re} ( \lambda )=1}f(z+ \lambda v_0)d \lambda . $$ Since $ \lambda _j(z)\neq \lambda _{j'}(z)$ for $j,j'\in \{1,...,m_0\}$, $j\neq j'$, $z\in T_{C_1}$, we have $$ {\rm Res}_{ \lambda = \lambda _j(z)}( \lambda \rightarrow f(z+ \lambda v_0)) $$ $$ = \lim_{ \lambda \rightarrow \lambda _j(z)} ( \lambda - \lambda _j(z)) \prod_{k=1}^m \frac{ l_k(z+ \lambda v_0)+1}{l_k(z+ \lambda v_0)} g(z+ \lambda v_0) $$ $$ = \frac{1}{l_j(v_0)}g\left(z-\frac{ l_j(z)}{l_j(v_0)}v_0\right) \prod_{1\le k\le m, k\neq j} \frac{l_{j,k}(z)+1}{l_{j,k}(z)}. $$ Put now for $j\in \{1,...,m_0\}, z\in T_{B_1}$ $$ f_j(z):= -\frac{1}{l_j(v_0)}g\left(z-\frac{l_j(z)}{l_j(v_0)}v_0\right)\cdot \prod_{1\le k\le m,\, k\neq j}\frac{l_{j,k}(z)+1}{l_{j,k}(z)} $$ and $$ g_j(z):= -\frac{1}{l_j(v_0)}\cdot g\left(z-\frac{l_j(z)}{l_j(v_0)}v_0\right). $$ Let $V_1\subset V$ be a hyperplane which does not contain $v_0$. We want to show that the function $f_j(z)$ is distinguished with respect to $(V_1;(l_{j,k})_{1\le k\le m,\, k\neq j})$. The function $f_j$ is meromorphic on $T_{B_1}$. Also, for all $x\in B_1$ and all $k\in \{1,...,{\hat j},...,m\}$ we have $l_{j,k}(x)>-1$. Further, we have that $$ g_j(z)=f_j(z)\cdot \prod_{1\le k\le m,\, k\neq j}\frac{l_{j,k}(z)}{l_{j,k}(z)+1} $$ is a holomorphic function on $T_{B_1}$. Let ${\bf K}_1\subset T_{B_1}$ ($\subset T_B$) be a compact, and let $$ {\bf K}(j):=\{z-\frac{l_j(z)}{l_j(v_0)}v_0\,|\, z\in {\bf K}_1 \}. $$ This is a compact subset in $T_B$. Put $$ \kappa_j({\bf K}_1)=\frac{1}{|l_j(v_0)|}\kappa({\bf K}(j)), $$ where $\kappa({\bf K}(j)) $ is a constant such that $|g(z+iv)|\le \kappa ({\bf K}(j))c(v)$ for all $z\in {\bf K}(j)$ and all $v\in V$. For $v_1\in V_1$ we put $$ c_j(v_1)=c\left(v_1-\frac{l_j(v_1)}{l_j(v_0)}v_0\right). $$ Then for all $z\in {\bf K}_1$ and $v\in V_1$ we have $$ |g_j(z+iv_1)|\le \kappa_j({\bf K}_1)c_j(v_1). $$ Moreover, for any subspace $U_1\subset V_1$ and all $v_1\in V_1$ the function $U_1 \rightarrow {\bf R}, \,\, u_1 \rightarrow c_j(u_1+v_1)$ is measurable and we have $$ c_{j,U_1}(v_1):=\int_{U_1}c_j(v_1+u_1)du_1\, < \infty . $$ For all $v_1\in V_1 - U_1$ we have $$ \lim_{\tau \rightarrow \pm \infty} c_{j,U_1}(\tau v_1)=0. $$ This shows that $f_j$ is distinguished with respect to $(V_1; (l_{j,k})_{1\le k\le m,\, k\neq j} )$. For $z\in T_{B_1}$ we put $$ f_0(z):=\frac{1}{2\pi i} \int_{{\rm Re} ( \lambda )=1} f(z+ \lambda v_0)d \lambda . $$ If $l_k(v_0)\neq 0$ we have (as above) for all $x\in B_1$ the following inequality $$ |l_k(x+v_0)|\ge \frac{1}{2} |l_k(v_0)|. $$ Therefore, we conclude that the function $$ g_0(z):= f_0(z)\prod_{1\le k\le m,\, l_k(v_0)= 0}\frac{l_k(z)}{l_k(z)+1} $$ $$ = \frac{1}{2\pi }\int_{-\infty}^{+\infty} \{\prod_{1\le k\le m, l_k(v_0)\neq 0} \frac{l_k(z+v_0+i\tau v_0)+1}{l_k(z+v_0+i\tau v_0)} \} g(z+v_0+i\tau v_0)d\tau $$ is holomorphic in $T_{B_1}$. Further, we have for $z\in {\bf K}_1 $ (with ${\bf K}_1\subset T_{B_1}$ a compact) and $v_1\in V_1$ the inequality $$ |g_0(z+iv_1)|\le \kappa_0({\bf K}_1)c_0(v_1), $$ where $\kappa_0({\bf K}_1)$ is some suitable constant and $$ c_0(v_1):=\int_{-\infty}^{+\infty}c(v_1+\tau v_0)d\tau. $$ Again, for any subspace $U_1\subset V_1$ and any $u_1\in V_1$ we have that the map $U_1 \rightarrow {\bf R}$ given by $u_1 \rightarrow c_0(v_1+u_1)$ is measurable, and that $$ c_{0,U_1}(v_1)=\int_{U_1}c_0(v_1+u_1)du_1 \,<\infty. $$ For all $v_1\in V_1 - U_1$ we have $$ \lim_{\tau \rightarrow \pm \infty} c_{0,U_1}(\tau v_1) =0. $$ Therefore, $f_0$ is distinguished with respect to $(V_1; (l_k)_{1\le k\le m, \, l_k(v_0)\neq 0})$. The Cauchy-Riemann equations imply that $g_0$ is invariant under ${\bf C} v_0$, that is, for all $z_1,z_2\in T_{B_1}$ with $z_1-z_2\in {\bf C} v_0$ we have $g_0(z_1)=g_0(z_2)$. We see that $f_0$ is also invariant under ${\bf C} v_0$ (in this sense), as well as $f_1,...,f_{m_0}$ (this can be seen from the explicit representation of these functions). For $z\in T_{C_1}$ we have $$ h_C(z)=f_0(z)+\sum_{j=1}^{m_0} f_j(z)=\sum_{j=0}^{m_0}f_j(z). $$ Moreover, for such $z$ we have $$ \tilde{f}_C(z)=\frac{1}{(2\pi)^{d-1}}\int_{V_1}h_C(z+iv_1)dv_1 =\sum_{j=0}^{m_0}\frac{1}{(2\pi)^{d-1}}\int_{V_1}f_j(z+iv_1)dv_1, $$ where $dv_1d\tau =dv $ (and ${\rm Vol}_{d\tau}( \{ \lambda v_0\,| \lambda \in [0,1]\} )=1)$. By our induction hypothesis, there exists an open and convex neighborhood $B'$ of ${\bf 0}$ in $E$ and linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde m}$, which vanish on $V$, such that $$ \tilde{f}_C(z)\prod_{j=1}^{\tilde m} \tilde{l}_j(z) $$ has a holomorphic continuation to $T_{B'}$. (Strictly speaking, from the induction hypothesis it follows only that the linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}\in E^{\vee }$ vanish on $V_1$. But since the functions $f_0,...,f_{m_0}$ ``live'' already on a tube domain in $(E/{\bf R} v_0)_{{\bf C}}$, it follows that the linear forms are also ${\bf R} v_0$-invariant.) Now we notice that $\tilde{f}_C(z)\prod_{j=1}^{\tilde m} \tilde{l}_j(z)$ is holomorphic on $T_C$. Let $\tilde{B}$ be the convex hull of $B'\cup C$. Then we have that $$ \tilde{f}_C(z)\prod_{j=1}^{\tilde m} \tilde{l}_j(z) $$ is holomorphic on $T_{\tilde B}$ (cf. \cite{Hor}, Theorem 2.5.10). \hfill $\Box$ \bigskip \noindent {\bf 7.4}\hskip 0,5cm Let $E,V,l_1,...,l_m$ and $ B\subset \{x\in E\,|\, l_j(x)>-1\hskip 0,2cm \forall \hskip 0,2cm j=1,...,m\}$ be as above. Let $C$ be a connected component of $B - \bigcup_{j=1}^m H_j$. Let $f\in {\cal M}(T_B)$ be an distinguished function with respect to $(V; l_1,...,l_m)$. \begin{prop} There exist an open convex neighborhood $\tilde B$ of ${\bf 0}$ in $E$ containing $C$ and linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}\in E^{\vee }$ vanishing on $V$ such that a) for all $j\in \{1,...,\tilde{m} \} $ we have $ {\rm Ker}(\tilde{l}_j)\cap C=\emptyset$ b) $\tilde{f}_C(z)\prod_{j=1}^{\tilde{m}} \tilde{l}_j(z)$ has a holomorphic continuation to $T_{\tilde{B}}$. \end{prop} {\em Proof.} By the proposition above, there exist linear forms $\tilde{l}_1,...,\tilde{l}_{\tilde{m}}$ such that $ V\subset \cap_{j=1}^{\tilde{m}} {\rm Ker}(\tilde{l}_j)$ and $\tilde{f}_C(z)\prod_{j=1}^{\tilde{m}}\tilde{l}_j(z)$ has a holomorphic continuation to a tube domain $T_{\tilde{B}}$ over a convex open neighborhood $\tilde{B}$ of ${\bf 0}\in E$ containing $C$. Suppose that there exist an $x_0\in C$ and a $j_0\in \{1,...,\tilde{m}\}$ such that $\tilde{l}_{j_0}(x_0)=0$. Then the function $$ \tilde{f}_C(z)\prod_{j\neq j_0}\tilde{l}_j(z) $$ is still holomorphic in $T_{\tilde{B}'}$ with $\tilde{B}'= (B- {\rm Ker}(\tilde{l}_{j_0}))\cup C$. It is easy to see that $\tilde{B}'$ is connected. The convex hull of $\tilde{B}'$ is equal to $\tilde{B}$. Therefore, already the function $$ \tilde{f}_C(z)\prod_{j\neq j_0}\tilde{l}_j(z) $$ is holomorphic on $T_{\tilde{B}}$ (cf. \cite{Hor}, loc. cit.). \subsection* \noindent {\bf 7.5}\hskip 0,5cm As above, let $E$ be a finite dimensional vector space over ${\bf R}$ and let $l_1,...,l_m$ be linearly independent linear forms on $E$. Put $H_j:={\rm Ker}(l_j)$, for $j=1,...,m$, $E^{(0)}=\bigcap_{j=1}^m H_j$ and $E_{0}=E/E^{(0)}$. Let $\pi_0\,:\, E \rightarrow E_0$ be the canonical projection. Let $V\subset E$ be a subspace with $V\cap E^{(0)}=\{ {\bf 0}\}$, such that $\pi_0 |_V\,:\, V \rightarrow E_0$ is an injective map. Let $$ E^+_0:=\{ x\in E_0 \,|\, l_j(x)\ge 0, \hskip 0,1cm j=1,...,m\} $$ and let $\psi\,:\, E_0 \rightarrow P := E_0/\pi_0 (V)$ be the canonical projection. We want to assume that $\pi_0(V)\cap E^+_0=\{{\bf 0}\}$, so that $ \Lambda :=\psi (E^+_0)$ is a strictly convex polyhedral cone. Let $dy$ be the Haar measure on $E_0^{\vee }$ normalized by ${\rm Vol}_{dy}(E^{\vee }_0/\oplus_{j=1}^m{\bf Z} l_j)=1$. Let $A\subset V$ be a lattice, and let $dv$ be a measure on $V$ normalized by ${\rm Vol}_{dv}(V/A)=1$. On $V^{\vee }$ we have a measure $dy'$ normalized by $A^{\vee }$ and a section of the projection $E_0^{\vee } \rightarrow V^{\vee }$ gives a measure $dy''$ on $P^{\vee }$ with $dy=dy'dy''$. Let $B\subset E$ be an open and convex neighborhood of ${\bf 0}$, such that for all $x\in B$ and $j\in \{1,...,m\}$ we have $l_j(x)>-1$. Let $f\in {\cal M}(T_B)$ be a meromorphic function in the tube domain over $B$ which is distinguished with respect to $(V;l_1,...,l_m)$. Put $$ B^+=B\cap \{x\in E\,|\, l_j(x)>0, \hskip 0,1cm j=1,...,m \}, $$ $$ \tilde{f}_{B^+} (z)=\frac{1}{(2\pi)^d}\int_Vf(z+iv)dv $$ where $d=\dim V$. By 7.3, the function $\tilde{f}_{B^+}\,:\, T_{B^+} \rightarrow {\bf C}$ is holomorphic and it has a meromorphic continuation to a neighborhood of ${\bf 0}\in E_{{\bf C}}$. Put $$ g(z)=f(z)\prod_{j=1}^m\frac{l_j(z)}{l_j(z)+1}. $$ \begin{prop} For $ x_0\in B^+ $ we have $$ \lim_{s \rightarrow 0}s^{m-d}\tilde{f}_{B^+}(sx_0)=g({\bf 0}){\cal X}(\psi (x_0)). $$ \end{prop} {\em Proof.} For $j\in J:=\{1,...,m \}$ we define $$ H_{j,+}:=\{v\in V\,\,|\,\, l_j(v)=1\}, $$ $$ H_{j,-}:=\{v\in V\,\,|\,\, l_j(v)=-1\}. $$ Let ${\cal C}$ be the set of connected components of $V- \bigcup_{j=1}^m(H_{j,+}\cup H_{j,-})$. For a $C\in {\cal C}$ we put $$ J_C:=\{j\in J \,|\, |l_j(v)|< 1\hskip 0,1cm {\rm for}\hskip 0,2cm {\rm all} \hskip 0,2cm v\in C\} $$ and $$ V^C:=V\cap \bigcap_{j\in J_C}H_j. $$ Denote by $V_C$ the complement to $V^C$ in $V$ and let $\pi_C\, :\, V=V_C\oplus V^C \rightarrow V_C$ be the projection. Since the map $V_C \rightarrow {\bf R}^{J_C}, v \rightarrow (l_j(v))_{j\in J_C}$, is injective we see that $\pi_C(C)$ is a bounded open subset of $V_C$. For $v_1\in \pi_C(C)$ we put $$ C(v_1):=\{v'\in V^C\,|\, v_1+v'\in C\}. $$ The set $C(v_1)$ is a convex open subset of $V^C$. Let $dv_1,dv'$ be measures on $V_C$ (resp. on $V^C$), with $dv_1 dv'=dv$. For all $s\in (0,1]$ we have $$ s^{m-d}\int_{C}f(sx_0+iv)dv=s^{m-d}\int_{\pi_C(C)}\int_{C(v_1)} f(sx_0+iv_1+iv')dv'dv_1 $$ $$ = s^{m-d^C}\int_{\frac{1}{s}\pi_C(C)}\int_{C(sv_1)}f(sx_0+isv_1+iv')dv'dv_1 $$ $$ =\int_{V}\chi_{C_s}(v)s^{m-d^C}f_{C,s}(v)dv. $$ Here we denoted by $d^C:=\dim V^C$ and by $$ C_s:=\{v=v_1+v'\in V_C\oplus V^C\,|\, sv_1+v'\in C\} $$ and by $\chi_{C_s}$ the set-theoretic characteristic function of $C_s$. We have put for any $s\in (0,1]$ and any $v=v_1+v'\in V$ $$ f_{C,s}(v):=f(sx_0+isv_1+iv'). $$ \noindent The set $$ {\bf K}_C:=\{ sx_0+iv_1\,|\, s\in [0,1], v_1\in \overline{\pi_C(C)}\} $$ is contained in $T_B$ and is compact. Further, there exist $c',c''\ge 0$ such that for all $s\in [0,1]$ and all $v=v_1+v'\in C$ we have $$ \left| \prod_{j\in J_C}(l_j(sx_0+iv)+1)\prod_{j\notin J_C}\frac{l_j(sx_0+iv)+1}{ l_j(sx_0 +iv)}\right|\le c', $$ $$ 1\le c''\cdot \left|\prod_{\j\notin J_C}\frac{1}{l_j(x_0+iv_1)}\right|. $$ Therefore, for $s\in (0,1]$ and $v\in V$ we have $$ |\chi_{C_s}(v)s^{m-d^C}f_{C,s}(v)|= $$ $$ \left|\chi_{C_s}(v)g(sx_0+isv_1+iv')s^{m-d^C} \prod_{j\in J}(1+\frac{1}{l_j(sx_0+isv_1+iv')})\right| $$ $$ \le \chi_{C_s}(v)s^{m-d^C}\left|\prod_{j\in J_C}\frac{1}{l_j(sx_0+isv_1)} \prod_{j\notin J_C}\frac{1}{l_j(x_0+iv_1)}\right| c' c''\kappa ({\bf K}_C)c(v') $$ $$ \le c'c''\kappa({\bf K}_C)\left|\prod_{j\in J}\frac{1}{l_j(x_0+iv_1)}\right| c(v')s^{m-d^C-\#J_C} $$ $$ \le c'c''\kappa({\bf K}_C) \left|\prod_{j\in J}\frac{1}{l_j(x_0+iv_1)}\right|c(v'), $$ since $m-d^C-\#J_C\ge 0$. (The constant $\kappa({\bf K}_C)$ and the function $c\,:\,V \rightarrow {\bf R}_{\ge 0}$ were introduced above.) The ${\cal X}$-function corresponding to the cone $E^+_0\subset E_0$ (and the measure $dy$) is given by $$ {\cal X}_{E^+_0}(x_0+iv_1)=\prod_{j=1}^m\frac{1}{l_j(x_0+iv_1)}. $$ Since the map from $V_C$ to $E_0$ is injective and since $\pi_0(V_C)\cap E^+_0=\{{\bf 0}\}$ we know that the function $v_1 \rightarrow {\cal X}_{E^+_0}(x_0+iv_1)$ is absolutely integrable over $V_C$ (by 7.2). Therefore, $$ v=v_1+v'\mapsto c'c''\kappa({\bf K}_C)|\prod_{j\in J} \frac{1}{l_j(x_0+iv_1)}|c(v') $$ is integrable over $V$. For a fixed $v\in V$ we consider the limit $$ \lim_{s \rightarrow 0}\chi_{C_s}(v)s^{m-d^C}f_{C,s}(v). $$ The estimate above shows that this limit is $0$ if $m-d^C-\# J_C>0$. Therefore, we assume that $m=d^C+\# J_C$. Then the map $V^C \rightarrow {\bf R}^{J - J_C}$ is an isomorphism. Since $\pi_0 (V)\cap E^+_0=\{{\bf 0}\}$, it follows that $J_C=J$. There exists exactly one $C\in {\cal C}$ with $J_C=J$ and we denote it by $C^{\circ}$. This $C^{\circ}$ contains ${\bf 0}$ and for all sufficiently small $s>0$ we have $s\cdot v\in C^{\circ}$, and therefore, $v\in C^{\circ}_s$. Moreover, we have $$ \lim_{s \rightarrow 0}s^mf_{C,s}(v)=\lim_{s \rightarrow 0}s^mg(sx_0+isv)\prod_{j=1}^m \frac{l_j(sx_0+isv)+1}{l_j(sx_0+isv)} $$ $$ =g({\bf 0})\prod_{j=1}^m\frac{1}{l_j(x_0+iv)}. $$ Using the theorem of dominated convergence (Lebesgue's theorem), we obtain $$ \begin{array}{rcl} \lim_{s \rightarrow 0}s^{m-d}\tilde{f}_{B^+}(sx_0) & = & \lim_{s \rightarrow 0} \sum_{C\in {\cal C}}\frac{1}{(2 \pi )^d}\int_V \chi_{C_s}(v) s^{m-d}f_{C,s}(v)dv\\ & & \\ & = & \frac{1}{(2\pi )^d}g({\bf 0}) \int_V \prod_{j=1}^m \frac{1}{l_j(x_0+iv)}dv\\ & & \\ & = & g({\bf 0}){\cal X}_{ \Lambda }(\psi (x_0)). \end{array} $$ \hfill $\Box$ \section{Some statements on Eisenstein series} \label{8} \subsection* \noindent {\bf 8.1}\hskip 0,5cm Let $G$ be a semi-simple simply connected algebraic group which is defined and split over $F$. Fix a Borel subgroup $P_0$ (defined over $F$) and a Levi decomposition $P_0=S_0U_0$, where $S_0$ is a maximal $F$-rational torus of $G$. Denote by ${\bf g}$ (resp. ${\bf a}_0$) the Lie algebra of $G$ (resp. $S_0$). We are going to define a certain maximal compact subgroup ${\bf K}_G\subset G({{\bf A}})$. This maximal compact subgroup will have the advantage that the constant term of Eisenstein series, more precisely, certain intertwining operators, can be calculated explicitly, uniformly with respect to all places of $F$. In general, i.e., for an arbitrary maximal compact subgroup, there will be some places where such an explicit expression is not available. In principle, this should cause no problems. Any statement in this section should be valid for an arbitrary maximal compact subgroup. \subsection* \noindent {\bf 8.2} \hskip 0,5cm Let $\Phi=\Phi(G,S_0)$ be the root system of $G$ with respect to $S_0$. We denote by $\Delta_0$ the basis of simple roots determined by $P_0$. For $ \alpha \in \Phi$ let $$ {\frak g}_{ \alpha }:= \{ X\in {\frak g}\,\,|\,\, [H,X]= \alpha (H)X\} $$ be the corresponding root space. Let $((H_{ \alpha })_{ \alpha \in \Delta_0}, (X_{ \alpha })_{ \alpha \in \Phi})$ be the Chevalley basis of ${\frak g}$. In particular, this means that $$ {\frak g}_{ \alpha }=FX_{ \alpha }\,( \alpha \in \Phi),\,\,\,\,\,\,[X_{ \alpha },X_{- \alpha }]=H_{ \alpha } \,\, ( \alpha \in \Delta_0 ), $$ $$ {\frak a}_0= \oplus_{ \alpha \in \Delta_0}FH_{ \alpha }. $$ Put $$ {\frak g}_{{\bf Q}}=\sum_{ \alpha \in \Delta_0}{\bf Q} H_{ \alpha } + \sum_{ \alpha \in \Phi}{\bf Q} X_{ \alpha }\subset {\frak g}, $$ This is a ${\bf Q}$-structure for ${\frak g}$ and for any $v\in \Val (F)$ the Lie algebra of $G(F_v)$ is ${\frak g}\otimes_{{\bf Q}}F_v$. We put $$ {\frak k}:=\oplus_{ \alpha \in \Phi^+}{\bf R}(X_{ \alpha }-X_{- \alpha }), $$ $$ {\frak p}:=\bigoplus_{ \alpha \in \Delta_0}{\bf R} H_{ \alpha } \oplus\bigoplus_{ \alpha \in \Phi^+}{\bf R} (X_{ \alpha }+X_{- \alpha }), $$ where $\Phi^+$ is the set of positive roots of $\Phi$ determined by $\Delta_0$. Then ${\frak k}\oplus {\frak p}$ is a Cartan decomposition of ${\frak g}_{{\bf Q}}\otimes_{{\bf Q}}{\bf R}$, ${\frak g}_{c}:={\frak k} \oplus i{\frak p}\subset {\frak g}_{{\bf Q}}\otimes_{{\bf Q}}{\bf C}$ is a compact form of ${\frak g}_{{\bf Q}}\otimes_{{\bf Q}}{\bf C}$ and ${\frak g}_{{\bf Q}}\otimes_{{\bf Q}}{\bf C}={\frak g}_{c}\oplus i{\frak g}_{c} $ is a Cartan decomposition of ${\frak g}_{{\bf Q}}\otimes_{{\bf Q}}{\bf C}$. For any complex place $v$ of $F$ we define ${\bf K}_v$ to be $\langle \exp({\frak g}_{c})\rangle\subset G(F_v)$ (identifying $F_v$ with ${\bf C}$ via a corresponding embedding $F\hookrightarrow {\bf C}$). If $v$ is a real place of $F$ we define ${\bf K}_v=G(F_v)\cap\langle \exp({\frak g}_c)\rangle$ (identifying $F_v(\sqrt{-1})\simeq {\bf C}$ via a corresponding embedding $F\hookrightarrow {\bf R}$). In this case, ${\bf K}_v$ contains $\langle \exp({\frak k})\rangle$. In both cases ${\bf K}_v$ is a maximal compact subgroup of $G(F_v)$. Now let $v$ be a finite place of $F$ and let ${\bf K}_v$ be the stabilizer of the lattice $$ \sum_{ \alpha \in \Delta_0}{\cal O}_v\cdot H_{ \alpha }+ \sum_{ \alpha \in \Phi}{\cal O}_v\cdot X_{ \alpha }\subset {\frak g}\otimes_FF_v. $$ By \cite{Bruhat}, sec. 3, Example 2, ${\bf K}_v $ is a maximal compact subgroup of $G(F_v)$. In any case, the Iwasawa decomposition $G(F_v)=P_0(F_v){\bf K}_v$ holds (for non-archimedean $v$, cf. \cite{Bruhat}, loc. cit.). Then ${\bf K}_G=\prod_v{\bf K}_v$ is a maximal compact subgroup of $G({\bf A})$ and $G({\bf A})=P_0({\bf A}){\bf K}_G$. \subsection* \noindent {\bf 8.3} \hskip 0,5cm As in section 2.3 we defined for any standard parabolic subgroup $P\subset G$ $$ H_P=H_{P,{\bf K}_G}\,:\, G({\bf A}) \rightarrow \Hom_{{\bf C}}(X^*(P)_{{\bf C}},{\bf C}) $$ by $\langle \lambda , H_P(g)\rangle=\log (\prod_v| \lambda (p_v)|_v)$ for $ \lambda \in X^*(P)$ and $g=pk$, $p=(p_v)_v\in P({\bf A}), k\in {\bf K}_G$. The restriction of $H_{P_0}$ to $S_0({\bf A})$ is a homomorphism, its kernel will be denoted by $ S_0({\bf A})^1$. The choice of a projection ${\bf G}_m({\bf A}) \rightarrow {\bf G}_m({\bf A})^1$ induces by means of an isomorphism $S_0 \rightarrow {\bf G}_{m,F}^{\# \Delta_0}$ a projection $S_0({\bf A}) \rightarrow S_0({\bf A})^1$ and this in turn gives an embedding $$ {\cal U}_0:=(S_0({\bf A})^1/S_0(F)(S_0({\bf A})\cap {\bf K}_G))^*\hookrightarrow (S_0({\bf A})/S_0(F)(S_0({\bf A})\cap {\bf K}_G))^*. $$ Let $( \varpi _{ \alpha })_{ \alpha \in \Delta_0}$ be the basis of $X^*(S_0)$ which is determined by $\langle \varpi _{ \alpha }, \beta \rangle = \delta _{ \alpha \beta }$ for all $ \alpha , \beta \in \Delta _0$. Let $P\subset G$ be a standard parabolic subgroup. Then $ \varpi _{ \alpha }$ for all $ \alpha \in \Delta _P$ lifts to a character of $P$ and $( \varpi _{ \alpha })_{ \alpha \in \Delta _P}$ is a basis of $X^*(P)$. Put $$ {\cal U}_P:=\{ \chi \,\,|\,\, \chi = \chi _0\circ\prod_{ \alpha \in \Delta _P}\check{ \alpha }\circ \varpi _{ \alpha } \,\,{\rm with}\,\, \chi _0\in {\cal U}_0\}. $$ Any $ \chi \in {\cal U}_P$ is a character of $P({\bf A})/P(F)(P({\bf A})\cap {\bf K}_G).$ Define $$ \phi_{ \chi }\,:\, G({\bf A}) \rightarrow S^1 $$ by $\phi_{ \chi }(pk)= \chi (p)$ for $p\in P({\bf A}), k\in {\bf K}_G$. The Eisenstein series $$ E^G_P( \lambda , \chi ,g)=\sum_{g\in P(F) \backslash G(F)}\phi_{ \chi }( \gamma g)e^{\langle \lambda +\rho_P,H_P( \gamma g)\rangle} $$ converges absolutely and uniformly for ${\rm Re}( \lambda )$ contained in any compact subset of the open cone $\rho_P+X^*(P)^+$ (cf. \cite{G}, Th\'eor\`eme III) and can be continued meromorphically to the whole of $X^*(P)_{{\bf C}}$. For the Eisenstein series corresponding to $P_0$ a proof is given in \cite{MW}, chapitre IV. In section 8.4 we will give an explicit expression for the Eisenstein series $E^G_P, $ with $P\neq P_0$ as an iterated residue of $E^G_{P_0}$ which shows the claimed meromorphy on $X^*(P)_{{\bf C}}$. Let $\chi\in {\cal U}_0$. The constant term of $E_{P_0}^G ( \lambda ,\chi)$ along $P=LU$ is by definition $$ E^G_{P_0}( \lambda ,\chi)_P(g)=\int_{U(F)\backslash U({\bf A})}E_{P_0}^G( \lambda ,\chi,ug)du, $$ where the Haar measure on $U({\bf A})$ is normalized such that $U(F)\backslash U({\bf A})$ gets volume one. It is an elementary calculation to show that for any parabolic subgroup $P\not\supsetneq P_0$ the constant term $E^G_{P_0}( \lambda ,\chi)_P$ is orthogonal to all cusp forms in $A_0(L(F)U({\bf A}) \backslash G({\bf A}))$ (cf. \cite{MW}, I.2.18, for the definition of this space). More precisely, for any parabolic subgroup $P\not\supseteq P_0$ the cuspidal component of $E^G_{P_0}( \lambda ,\chi)$ along $P$ vanishes (cf. \cite{MW}, I.3.5, for the definition of ``cuspidal component''). By Lemme I.4.10 in \cite{MW}, the singularities of the Eisenstein series $E^G_{P_0}( \lambda ,\chi)$ and the singularities of $E^G_{P_0}( \lambda ,\chi)_{P_0}$ coincide. Let $$ {\cal W}={\rm Norm}_{G(F)}\left(S_0(F)\right)/S_0(F) $$ be the Weyl group of $G$ with respect to $S_0$. For any $w\in {\cal W}$ we normalize the Haar measures such that $$ \int_{(U_0(F)\cap wU_0(F)w^{-1}) \backslash (U_0({\bf A})\cap wU_0({\bf A})w^{-1})}du=1 $$ and on $(U_0({\bf A})\cap wU_0({\bf A})w^{-1}) \backslash U_0({\bf A})$ we take the quotient measure. Using Bruhat's decomposition $G(F)=\bigcup_{w\in {\cal W}} P_0(F)w^{-1}P_0(F)$ we can calculate \begin{equation} \label{8.3.1} \int_{U_0(F) \backslash U_0({\bf A})}E^G_{P_0}( \lambda , \chi ,ug)du=\sum_{w\in {\cal W}} c(w, \lambda , \chi )\phi_{w \chi }(g)e^{\langle w \lambda +\rho_0,H_{P_0}(g)\rangle}, \end{equation} \noindent where $\rho_0=\rho_{P_0}$, $(w\chi)(t)=\chi(w^{-1}tw)$ for all $t\in S_0({\bf A})$, and the functions $c(w, \lambda ,\chi)$ are given by $$ c(w, \lambda ,\chi):=\int_{(U_0({\bf A})\cap wU_0({\bf A})w^{-1}) \backslash U_0({\bf A})} \phi_{ \chi }(w^{-1})e^{\langle \lambda +\rho_P,H_{P_0}(w^{-1}u)\rangle}du $$ (cf. \cite{MW}, Prop. II.1.7). They satisfy functional equations: \begin{equation} \label{8.3.2} E^G_{P_0}( \lambda , \chi ,g)=c(w, \lambda , \chi )E^G_{P_0}(w \lambda ,w \chi ,g) \end{equation} \begin{equation} c(w'w, \lambda , \chi )=c(w',w \lambda ,w \chi )c(w, \lambda , \chi ) \end{equation} \noindent (cf. \cite{MW}, Th\'eor\`eme IV.1.10). Therefore, it suffices to calculate $c(w_{ \alpha }, \lambda , \chi ) $ for $ \alpha \in \Delta _0$ ($w_{ \alpha }$ corresponds to the reflection along $ \alpha $). Put $S_{ \alpha }=\Ker ( \alpha )^{0}\subset S_{0}$ and $G_{ \alpha }=Z_{G}(S_{ \alpha })$. The Lie algebra of $G_{ \alpha }$ is ${\frak a}_0\oplus {\frak g}_{- \alpha }\oplus {\frak g}_{ \alpha }$. There is a homomorphism $\varphi_{ \alpha }\,:\, SL_{2,F} \rightarrow {\cal D}G_{ \alpha }$ ($=$ derived group of $G_{ \alpha }$) such that $d \varphi _{ \alpha }$ maps the matrices $$ \left(\begin{array}{cc}0&1\\ 0 & 0\end{array}\right), \left(\begin{array}{cc}1&0\\ 0 & -1\end{array}\right), \left(\begin{array}{cc}0&0\\ 1 & 0\end{array}\right) $$ to $X_{ \alpha }, H_{ \alpha },X_{- \alpha }$, respectively. On ${\bf A}$ we take the measure $dx$ that is described in Tate's thesis (then ${\rm Vol} (F \backslash {\bf A})=1$). We have $$ c(w_{ \alpha }, \lambda ,\chi)=\int_{{\bf A}}\phi_{\chi}( \varphi _{a} ( \begin{array}{cc} \scriptstyle{0}&\scriptstyle{-1}\\ \scriptstyle{1}&\scriptstyle{x}\end{array}) ) \exp\left(\langle \lambda +\rho_0,H_{P_0}( \varphi _{ \alpha }(\begin{array}{cc} \scriptstyle{0}&\scriptstyle{-1}\\ \scriptstyle{1}&\scriptstyle{x}\end{array}))\rangle \right)dx. $$ It is an exercise to compute this integral. The result is \begin{equation} \label{8.3.3} c(w_{ \alpha }, \lambda ,\chi)=\frac{L( \chi \circ\check{ \alpha },\langle \lambda , \alpha \rangle)}{ L( \chi \circ\check{ \alpha },1+\langle \lambda , \alpha \rangle)}. \end{equation} \noindent The Hecke $L$-functions are defined as follows. Let $ \chi \,:\, {\bf G}_m({\bf A})/{\bf G}_m(F) \rightarrow S^1$ be an unramified character. For any finite place $v$ we put $$ L_v( \chi _v,s)=(1- \chi _v(\pi_v)|\pi_v|^s_v)^{-1} $$ and $$ L_f( \chi ,s)=\prod_{v\nmid \infty }L_v( \chi _v,s). $$ For any archimedean place $v$ there is a $\tau_v\in {\bf R}$ such that $ \chi _v(x_v)=|x_v|^{i\tau_v}_v$ for all $x_v\in F_v^*$. Then $$ L_v( \chi _v,s):=\left\{ \begin{array}{rccl} \pi^{-\frac{(s+i\tau_v)}{2}}\Gamma(\frac{s+i\tau_v}{2}), & {\rm if} & v & {\rm is}\,\,{\rm real}\\ (2\pi)^{-(s+i\tau_v)}\Gamma(s+i\tau_v), & {\rm if} & v & {\rm is}\,\,{\rm complex} \end{array} \right. $$ We define the complete Hecke $L$-function by $$ L( \chi ,s)=D^{s/2}L_{\infty}( \chi ,s)L_f(c,s), $$ where $L_{\infty}( \chi ,s)=\prod_{v|\infty}L_v( \chi _v,s)$ and $D=D(F/{\bf Q})$ is the absolute value of the discriminant of $F/{\bf Q}$. If the restriction of $ \chi $ to ${\bf G}_m({\bf A})^1$ is not trivial then $L( \chi ,s)$ is an entire function. If $ \chi =1$ then $ L( \chi ,s)$ has exactly two poles of order one at $s=1 $ and $s=0$. To state the functional equation we let $(\pi_v^{d_v})$ (with $d_v \ge 0$ and $d_v=0$ for almost all $v$) be the local discriminant of $F_v$ over the completion of ${\bf Q}\in F_v$ (for non-archimedean places $v$). Put $\delta=(\delta_v)_v\in {\bf G}_m({\bf A})$ with $\delta_v=1$ for all archimedean places and $\delta_v=\pi_v^{d_v}$ for all non-archimedean places. Then \begin{equation} \label{8.3.4} L( \chi ,s)= \chi (\delta)L( \chi ^{-1},1-s). \end{equation} Using the functional equations (\ref{8.3.2}) and (\ref{8.3.3}) we get \begin{equation} \label{8.3.4-1} c(w, \lambda ,\chi)=\prod_{ \alpha >0,\,\, w \lambda <0}\frac{L(\chi\circ\check{ \alpha },\langle \lambda , \alpha \rangle)}{L(\chi\circ\check{ \alpha },1+\langle \lambda , \alpha \rangle)}. \end{equation} \bigskip \noindent {\bf Proposition 8.3}\hskip 0,5cm {\it Let $ \Delta _0(\chi)$ be the set of $ \alpha \in \Delta _0$ such that $\chi\circ\check{ \alpha }=1$. Then $$ \prod_{ \alpha \in \Delta _0(\chi)}\langle \lambda , \alpha \rangle E^G_{P_0}( \lambda +\rho_0,\chi) $$ has a holomorphic continuation to the tube domain over $-\frac{1}{4}\rho_0 + X^*(P_0)^+$. } \bigskip {\em Proof.} For $ \alpha \in \Phi^+- \Delta _0$ we have $\langle \rho_0, \alpha \rangle \ge 2$ and therefore $$ c(w_{ \alpha }, \lambda +\rho_0, \chi )=\frac{L(\chi\circ\check{ \alpha },\langle \lambda +\rho_0, \alpha \rangle)}{L(\chi\circ\check{ \alpha },1+\langle \lambda +\rho_0, \alpha \rangle)} $$ is holomorphic in this domain. If $ \alpha \in \Delta _0- \Delta _0(\chi)$ then $ \chi \circ\check{ \alpha }$ restricted to ${\bf G}_m({\bf A})^1$ is nontrivial, hence $c(w_{ \alpha }, \lambda +\rho_0, \chi )$ is holomorphic in this domain too. For $ \alpha \in \Delta _0(\chi)$ the function $$ \langle \lambda , \alpha \rangle L(\chi\circ\check{ \alpha },\langle \lambda +\rho_0, \alpha \rangle)=\langle \lambda , \alpha \rangle L(1,1+\langle \lambda , \alpha \rangle ) $$ is also holomorphic in this domain. This shows the holomorphy of $$ \prod_{ \alpha \in \Delta _0(\chi)}\langle \lambda , \alpha \rangle E^G_{P_0}( \lambda +\rho_0,\chi)_{P_0}(g) $$ $$ = \sum_{w\in {\cal W}}\prod_{ \alpha \in \Delta _0(\chi)} \langle \lambda , \alpha \rangle c(w, \lambda +\rho_0,\chi)\phi_{w\chi}(g)e^{\langle w( \lambda +\rho_0)+\rho_0,H_{P_0}(g)\rangle } $$ for ${\rm Re} ( \lambda )$ contained in $-\frac{1}{4}\rho_0+ X^*(P_0)^+$. By \cite{MW}, Lemme I.4.10, we conclude that the same holds for $\prod_{ \alpha \in \Delta _0(\chi)}E^G_{P_0}( \lambda +\rho_0,\chi)$. \hfill $\Box$ \subsection* \noindent {\bf 8.4} Let $P=LU$ be a standard parabolic subgroup and $\chi\in {\cal U}_P$. For $ \lambda \in X^*(P)_{{\bf C}}$ with ${\rm Re} ( \lambda )$ contained in the interior of $ X^*(P)^+ $ and $\vartheta$ contained in $X^*(P_0)^+ $ we have $$ E^G_{P_0}(\vartheta+ \lambda +\rho_0,\chi,g) =\sum_{ \gamma\in P(F) \backslash G(F)} \phi_{\chi}( \gamma g)e^{\langle \lambda ,H_P( \gamma g)\rangle}\sum_{\delta\in (L\cap P_0)(F) \backslash L(F)}e^{\langle \vartheta+2\rho_0,H_{P_0}( \delta \gamma g)\rangle}. $$ Let $w_L$ be the longest element of the Weyl group of $L$ (with respect to $S_0$) and define $$ c_P:=\lim_{\vartheta \rightarrow {\bf 0}, \vartheta\in X^*(P_0)^+ } \left(\prod_{ \alpha \in \Delta _0^P} \langle \vartheta , \alpha \rangle\right )c(w_L,\vartheta+\rho_0,1). $$ By (\ref{8.3.4-1}) this limit exists and is a positive real number. \bigskip \noindent {\bf Proposition 8.4}\hskip 0,5cm {\it a) $$ \lim_{\vartheta \rightarrow {\bf 0}, \,\vartheta\in X^*(P_0)^+ } \prod_{ \alpha \in \Delta _0^P} \langle \vartheta , \alpha \rangle E^G_{P_0}(\vartheta+ \lambda +\rho_0,\chi,g)= c_PE^G_P( \lambda +\rho_P,\chi,g). $$ b) Let $P'=L'U'$ be a standard parabolic subgroup containing $P$ and suppose that $\chi\circ\check{ \alpha }=1$ for all $ \alpha \in \Delta ^{P'}_0$. Then $\chi\in {\cal U}_{P'}$ and for all $ \lambda \in X^*(P')_{{\bf C}}$ we have $$ \lim_{\vartheta \rightarrow {\bf 0},\, \vartheta\in X^*(P)^+}\prod_{ \alpha \in \Delta _P- \Delta _{P'}} \langle \vartheta , \alpha \rangle E^G_{P}(\vartheta+ \lambda +\rho_P,\chi,g)= \frac{c_{P'}}{c_P}E^G_{P'}( \lambda +\rho_{P'},\chi,g). $$ } \bigskip {\em Proof.} a) The proof rests on the fact that a measurable function of moderate growth on $L(F) \backslash L({\bf A})$ for which all cuspidal components vanish vanishes almost everywhere. (For a proof cf. \cite{MW}, Prop. I.3.4.) We claim that $$ \lim_{\vartheta \rightarrow {\bf 0},\, \vartheta\in X^*(P_0)^+}\,\,\prod_{ \alpha \in \Delta _0^P} \langle \vartheta , \alpha \rangle \left\{ \sum_{\delta\in (L\cap P_0)(F) \backslash L(F)} e^{\langle \vartheta+2\rho_0,H_{P_0}( \delta \gamma g)\rangle} \right\} = c_Pe^{\langle 2\rho_P,H_{P}( \gamma g)\rangle }. $$ In fact, the cuspidal components of both sides along all non-minimal standard parabolic subgroups of $L$ vanish. To compare the constant terms along $P_0\cap L$ we can use (\ref{8.3.1}) (for $L$ instead of $G$ and $P_0\cap L$ instead of $P_0$) and the explicit expression of the functions $c(w,\vartheta+\rho_0,1)$ to get the identity stated above (note that $w_L\rho_0+\rho_0=2\rho_P$). b) Write $\chi=\chi_0\cdot\prod_{ \alpha \in \Delta _P}\check{ \alpha }\circ \varpi _{ \alpha }$ with $\chi_0 \in {\cal U}_{P'}$. For $ \alpha \in \Delta _P- \Delta _{P'}$ we have $1=\chi\circ\check{ \alpha }=\chi_0\circ\check{ \alpha }$. Thus $ \chi=\chi_0\circ\prod_{ \alpha \in \Delta _{P'}}\check{ \alpha }\circ \varpi _{ \alpha }\in {\cal U}_{P'}$. Using a) we get $$ \lim_{\vartheta \rightarrow {\bf 0},\, \vartheta\in X^*(P)^+}\prod_{ \alpha \in \Delta _P- \Delta _{P'}} \langle \vartheta , \alpha \rangle E^G_P(\vartheta+ \lambda +\rho_P,\chi,g) $$ $$ = \frac{1}{c_P} \lim_{\vartheta \rightarrow {\bf 0}, \,\vartheta\in X^*(P)^+}\prod_{ \alpha \in \Delta _0^{P'}} \langle \vartheta , \alpha \rangle E^G_P(\vartheta+ \lambda +\rho_0,\chi,g) = \frac{c_{P'}}{c_P}E^G_{P'}(\chi+\rho_{P'},\chi,g). $$ \hfill $\Box$ \subsection* \noindent {\bf 8.5}\hskip 0,5cm For $ \chi \in (S_0({\bf A})/S_0(F)(S_0({\bf A})\cap {\bf K}_G))^*$ and $v\in \Val_{\infty}(F) $ there is a character $ \lambda _v= \lambda _v( \chi )\in X^*(S_0)_{{\bf R}}$ such that for all $x\in S_0(F_v)$ $$ \chi (x)=e^{i\log(| \lambda _v(x)|_v)}. $$ This gives a homomorphism $$ (S_0({\bf A})/S_0(F)(S_0({\bf A})\cap {\bf K}_G))^* \rightarrow X^*(S_0)_{{\bf R}}^{\Val_{\infty}(F)} $$ $$ \lambda \mapsto \chi _{\infty}=( \lambda _v( \chi ))_{v|\infty} $$ which has a finite kernel. The image of ${\cal U}_0$ under this map is a lattice of rank $$ (\#\Val_{\infty}(F) -1)\dim S_0. $$ Fix a norm $\|\cdot\|$ on $X^*(S_0)_{{\bf R}}$ and denote by the same symbol the induced maximum norm on $X^*(S_0)_{{\bf R}}^{\Val_{\infty}(F)}$ (i.e., $\|( \lambda _v)_{v|\infty}\|=\max_{v|\infty}\|l_v\|$). Let $a,b>0$ and put $$ B_{a,b}:=\{ \lambda \in X^*(P_0)_{{\bf R}}\,\,|\,\, - \lambda +\frac{a}{2}\rho_0\in X^*(P_0)^+, \,\,\, \lambda +\frac{b}{2}\rho_0\in X^*(P_0)^+\}. $$ This is a bounded convex open neighborhood of ${\bf 0}$ in $X^*(P_0)_{{\bf R}}$. Note that if $ \lambda \in B_{a,b}$ then $w_0 \lambda +\frac{a}{2}\rho_0 \in X^*(P_0)^+$, where $w_0$ is the longest element of ${\cal W}$. Fix an $A>0$ such that $$ {\rm Re} \left(\langle \lambda +s\rho_0, \alpha \rangle (\langle \lambda +s\rho_0, \alpha \rangle-1)\right)+A\ge 1 $$ for all $ \lambda \in X^*(P_0)_{{\bf C}}$ with ${\rm Re}( \lambda )\in B_{a,b}$, $ -1-a\le {\rm Re}(s)\le 1+b$ and $ \alpha \in\Phi^+$. Fix $ \lambda \in X^*(P_0)_{{\bf C}}$ with ${\rm Re}( \lambda )\in B_{a,b}$ and $ \chi \in {\cal U}_0$. Denote by $\Phi^+(\chi)$ the set of all positive roots $ \alpha $ such that $ \chi \circ\check{ \alpha }=1$. Then $$ f_{ \lambda , \chi }(s,g):= $$ $$ \prod_{ \alpha \in \Phi^+( \chi )} \frac{\langle \lambda +s\rho_0, \alpha \rangle (\langle \lambda +s\rho_0, \alpha \rangle-1)}{ \langle \lambda +s\rho_0, \alpha \rangle(\langle \lambda +s\rho_0, \alpha \rangle-1)+A} \prod_{ \alpha >0}L_f(\chi\circ\check{ \alpha },1+\langle \lambda +s\rho_0, \alpha \rangle) E^G_{P_0}( \lambda +s\rho_0, \chi ,g) $$ is for any fixed $s$ in this strip an automorphic form on $G(F) \backslash G({\bf A})$. Indeed, we observe that all cuspidal components of $f_{ \lambda ,c}(s,\,\cdot\,)$ along non-minimal standard parabolic subgroups vanish. Now we can use (\ref{8.3.1}) and the explicit formulas for the functions $c(w, \lambda , \chi )$ in (\ref{8.3.4}) to see that the constant term of $f_{ \lambda ,c}(s,\,\cdot\,)$ along $P_0$ is holomorphic for $s$ in this domain. By Lemme I.4.10 in \cite{MW} we can conclude that the same is true for $f_{ \lambda , \chi }(s,\,\cdot\,)$. It is our aim to apply a version of the Phragm\'en-Lindel\"of principle due to Rademacher (cf. \cite{Rademacher}, Theorem 2) to the function $$ s\mapsto f_{ \lambda , \chi }(s,g) $$ in the strip $-1-a\le {\rm Re}(s)\le 1+b$. Using the functional equations of Eisenstein series (\ref{8.3.2}) and $L$-functions (\ref{8.3.4}) we get $$ f_{ \lambda ,c}(-1-a-it,g) =\prod_{ \alpha \in \Phi^+( \chi )}\frac{\langle \lambda -(1+a+it)\rho_0, \alpha \rangle (\langle \lambda -(1+a+it)\rho_0, \alpha \rangle -1)}{\langle \lambda -(1+a+it)\rho_0, \alpha \rangle (\langle \lambda -(1+a+it)\rho_0, \alpha \rangle -1)+A} $$ $$ \times \prod_{ \alpha >0}( \chi \circ\check{ \alpha })(\delta)D^{-\langle \lambda -(1+a+it)\rho_0, \alpha \rangle}\prod_{ \alpha >0}L_f((w_0\chi)\circ\check{ \alpha }, 1+ \langle w_0 \lambda +(1+a+it)\rho_0, \alpha \rangle) $$ $$ \times \prod_{ \alpha >0} \frac{L_{\infty}\left((w_0\chi)\circ\check{ \alpha }, 1+ \langle w_0 \lambda +(1+a+it)\rho_0, \alpha \rangle\right)}{ L_{\infty}((w_0\chi)^{-1}\circ\check{ \alpha }, 1-\langle w_0 \lambda +(1+a+it)\rho_0, \alpha \rangle)}E^G_{P_0}( w_0 \lambda +(1+a+it)\rho_0,w_0 \chi ,g). $$ Note that $L_{\infty}(\cdots) $ is a product of $\Gamma$-functions. Using the functional equation of the $\Gamma$-function we can derive the following estimate: There is $c>0$ depending only on $a$ and $b$ such that for ${\rm Re}( \lambda )\in B_{a,b}$ and $ \chi \in {\cal U}_0$ we have \begin{equation} \label{8.5.1} |f_{ \lambda , \chi }(-1-a-it,g)| \end{equation} $$ \le cE^G_{P_0}({\rm Re}(w_0 \lambda )+(1+a)\rho_0,g)\times (1+\|{\rm Im}( \lambda )\|+\| \chi _{\infty}\|)^{\delta_{\mu}}|1+it|^{\delta_{\mu}}. $$ where $\delta_{\mu}:=\mu(2+a+b)$ and $\mu>0$ depends only on $F$ and $G$. Moreover, assuming $c$ to be big enough, we have also \begin{equation} \label{8.5.2} |f_{ \lambda , \chi }(1+b+it,g)|\le cE^G_{P_0}({\rm Re}( \lambda )+(1+b)\rho_0,g). \end{equation} The proof of the following lemma was suggested to us by J. Franke. \bigskip \noindent {\bf Lemma 8.5}\hskip 0,5cm {\it For ${\rm Re}( \lambda )\in B_{a,b}, \chi \in {\cal U}_0$ and $-1-a\le \sigma \le 1+b$ the following estimate holds: $$ |f_{ \lambda , \chi }( \sigma +it,g)| \le cE^G_{P_0}({\rm Re}( \lambda )+(1+b)\rho_0,g)^{\frac{ \sigma +1+a}{2+a+b}} $$ $$ \times \left\{ E^G_{P_0}({\rm Re}(w_0 \lambda )+(1+a)\rho_0,g)(1+\|{\rm Im} ( \lambda )\|+\|\chi_{\infty}\|)^{ \delta_{\mu}}|2+a+ \sigma +it|^{\delta_{\mu}}\right\}^{\frac{1+b- \sigma }{2+a+b}}. $$ } \bigskip {\em Proof.} This follows immediately from Theorem 2 in \cite{Rademacher} once we have shown that for $-1-a\le \sigma \le 1+b$ we have \begin{equation} \label{*} |f_{ \lambda , \chi }( \sigma +it,g)|\le c_1e^{|t|^{c_2}} \end{equation} for some $c_1,c_2>0$. By (\ref{8.5.1}) and (\ref{8.5.2}), the function $$ s\mapsto e^{s^2}f_{ \lambda , \chi }(s,g) $$ can be integrated over the lines ${\rm Re}(s)=-1-a-\epsilon$ and ${\rm Re}(s)=1+b+\epsilon$ for some $\epsilon >0$. We claim that for all $s$ with $-1-a\le {\rm Re}(s)\le 1+b$ \begin{equation} \label{**} e^{s^2}f_{ \lambda , \chi }(s,g) \end{equation} $$ =-\frac{1}{2\pi i}\int_{{\rm Re}(z)=-1-a-\epsilon} \frac{e^{z^2}f_{ \lambda , \chi }(z,g)}{z-s}dz + \frac{1}{2\pi i}\int_{{\rm Re}(z)=1+b+c} \frac{e^{z^2}f_{ \lambda , \chi }(z,g)}{z-s}dz. $$ Denote the right-hand side by $h(s,g)$. This is a measurable function of moderate growth on $G(F) \backslash G({\bf A})$ (cf. (\ref{8.5.1}) and (\ref{8.5.2})). All cuspidal components of $h(s,\,\cdot\,)$ along non-minimal standard parabolic subgroups vanish. The same is true for the left-hand side. It remains to compare the constant terms of both sides along $P_0$. By the absolute and uniform convergence of the integrals over the vertical lines we see that the constant term of $h(s,\,\cdot\,)$ along $P_0$ is \begin{equation} \label{***} -\frac{1}{2\pi i}\int_{{\rm Re}(z)=-1-a-\epsilon} \frac{g_{ \lambda , \chi }(z,g)}{z-s}dz+ \frac{1}{2\pi i}\int_{{\rm Re}(z)=1+b+\epsilon} \frac{g_{ \lambda , \chi }(z,g) }{z-s}dz. \end{equation} where $$ g_{ \lambda , \chi }(z,g):= \int_{U_0(F) \backslash U({\bf A})} e^{z^2} f_{ \lambda , \chi }(z,ug)du. $$ The explicit expression of the constant term of $E^G_{P_0}$ along $P_0$ in (\ref{8.3.1}) and uniform estimates for $L$-functions as in \cite{Rademacher}, Theorem 5 (but for a larger strip), allow us to conclude that (\ref{***}) is just the constant term of $e^{s^2}f_{ \lambda , \chi }(s,\,\cdot\,)$ along $P_0$. Thus, by Proposition I.3.4 in \cite{MW}, we have established (\ref{**}). >From (\ref{**}) it follows that $ |e^{s^2}f_{ \lambda , \chi }(s,\,\cdot\,)|$ is bounded by some constant in the strip $-1-a\le {\rm Re}(s)\le 1+b$ and this in turn implies (\ref{*}). \hfill $\Box$. \bigskip \noindent {\bf Proposition 8.6}\hskip 0,5cm {\it Let $a>0$. For any $\epsilon >0$ there exist constants $b,c>0$ such that for all $ \lambda \in X^*(P_0)_{{\bf C}}$ with ${\rm Re}( \lambda )\in B_{a,b}$ and $ \chi \in {\cal U}_0$ we have $$ \left| \prod_{ \alpha \in \Delta _0( \chi )}\frac{\langle \lambda , \alpha \rangle}{\langle \lambda , \alpha \rangle +1} E^G_{P_0}( \lambda +\rho_0, \chi ) \right| \le c(1+\|{\rm Im}( \lambda )\|+\|\chi_{\infty}\|)^{\epsilon}. $$ } \bigskip {\em Proof.} Note that if $ \alpha \in \Phi^+ - \Delta _0$ then $\langle \rho_0, \alpha \rangle\ge 2$ and hence $\langle \lambda +\rho_0, \alpha \rangle (\langle \lambda +\rho_0, \alpha \rangle -1)$ does not vanish for ${\rm Re}( \lambda )\in B_{a,b}$ and $b>0$ sufficiently small. For such $b$ there is a constant $c_1$ such that $$ \left|\prod_{ \alpha \in \Delta _0( \chi )}\frac{\langle \lambda , \alpha \rangle}{\langle \lambda , \alpha \rangle +1}E^G_{P_0}( \lambda +\rho_0, \chi )\right|\le c\left|f_{ \lambda , \chi }(1,1_G)\right|. $$ Now we use the estimate for $|f_{ \lambda , \chi }(1,1_G)|$ in Lemma 8.5 and require that $\mu b\le \epsilon$. This gives the desired result. \hfill $\Box$ \bigskip \noindent {\bf Proposition 8.7}\hskip 0,5cm {\it Let $P$ be a standard parabolic subgroup of $G$. Let $a,\epsilon >0$. Then there exist $b,c >0$ such that for all $ \chi \in {\cal U}_P$ and $ \lambda \in X^*(P)_{{\bf C}}$ with $-{\rm Re}( \lambda )+\frac{a}{2}\rho_0\in X^*(P)^+$ and ${\rm Re}( \lambda )+\frac{b}{2}\rho_0\in X^*(P)^+$ we have $$ \left| \prod_{ \alpha \in \Delta _P( \chi )}\frac{\langle \lambda , \alpha \rangle}{\langle \lambda , \alpha \rangle +1} E^G_{P}( \lambda +\rho_P, \chi ) \right| \le c(1+\|{\rm Im}( \lambda )\|+\|\chi_{\infty}\|)^{\epsilon}, $$ where $ \Delta _P( \chi )= \Delta _0( \chi )\cap \Delta _P$. } \bigskip {\em Proof.} By the preceding proposition, there exist $b,c'>0$ such that for all $ \lambda '\in X^*(P_0)_{{\bf C}}$ with ${\rm Re}( \lambda ')\in B_{a,b}$ we have $$ \left|\prod_{ \alpha \in \Delta _0( \chi )}\frac{\langle \lambda ', \alpha \rangle}{\langle \lambda ', \alpha \rangle +1}E^G_{P_0}( \lambda '+\rho_0, \chi )\right| \le c'(1+\|{\rm Im}( \lambda ')\|+\|\chi_{\infty}\|)^{\epsilon}. $$ Note that $\chi\circ \check{ \alpha }=1 $ for all $ \alpha \in \Delta _0^P$ and hence $ \Delta _0( \chi )= \Delta _P( \chi )\cup \Delta _0^P$. Now let $ \lambda \in X^*(P)_{{\bf C}}$ be as in the proposition, i.e., $-{\rm Re}( \lambda )+\frac{a}{2}\rho_P\in X^*(P)^+$ and ${\rm Re}( \lambda )+\frac{b}{2}\rho_0\in X^*(P)^+$. Then for all sufficiently small $\vartheta\in X^*(P_0)^+$ we have $$ -(\vartheta +{\rm Re}( \lambda ))+\frac{a}{2}\rho_0\in X^*(P_0)^+ $$ $$ \vartheta +{\rm Re}( \lambda )+\frac{b}{2}\rho_0\in X^*(P_0)^+, $$ i.e., $\vartheta +{\rm Re}( \lambda )\in B_{a,b}$. Hence for those $\vartheta$ we have $$ \left| \prod_{ \alpha \in \Delta _0( \chi )}\frac{\langle \vartheta + \lambda , \alpha \rangle}{\langle \vartheta+ \lambda , \alpha \rangle +1} E^G_{P_0}(\vartheta+ \lambda +\rho_0, \chi ) \right| \le c'(1+\|{\rm Im}( \lambda )\|+\|\chi_{\infty}\|)^{\epsilon}. $$ Letting $\vartheta $ tend to ${\bf 0}$ and using Proposition 8.4 we can conclude that $$ \left| \prod_{ \alpha \in \Delta _P( \chi )}\frac{\langle \lambda , \alpha \rangle}{\langle \lambda , \alpha \rangle +1}E^G_{P}( \lambda +\rho_P, \chi ) \right| \le \frac{c'}{c_P}(1+\|{\rm Im}( \lambda )\|+\|\chi_{\infty}\|)^{\epsilon}. $$ \hfill $\Box$

9709

alg-geom/9709018

en

https://arxiv.org/abs/alg-geom/9709018

alg-geom/9709018

Lakshmibai

V. Lakshmibai and Peter Magyar

Degeneracy Schemes and Schubert Varieties

16 pp, Northeastern University, Latex

A result of Zelevinsky states that an orbit closure in the space of representations of the equioriented quiver of type $A_h$ is in bijection with the opposite cell in a Schubert variety of a partial flag variety $SL(n)/Q$. We prove that Zelevinsky's bijection is a scheme-theoretic isomorphism, which shows that the universal degeneracy schemes of Fulton are reduced and Cohen-Macaulay in arbitrary characteristic.

alg-geom

\section{Zelevinsky's bijection} \subsection{Quiver varieties} Fix an $h$-tuple of non-negative integers ${\bf n} = (n_1,\ldots,n_h)$ and a list of vector spaces $V_1,\ldots, V_h$ over an arbitrary field ${\bf k}$ with respective dimensions $n_1,\ldots,n_h$. Define the {\it variety of quiver representations} (of dimension ${\bf n}$, of the equioriented quiver of type $A_h$) to be the affine space $Z$ of all $(h\!-\!1)$-tuples of linear maps $(f_1,\ldots,f_{h\!-\!1}):$ $$ V_1 \stackrel{f_1}{\to} V_2 \stackrel{f_2}{\to} \cdots \stackrel{f_{h\!-\!2}}{\to} V_{h\!-\!1} \stackrel{f_{h\!-\!1}}{\to} V_h \ . $$ If we endow each $V_i$ with a basis, we get $V_i \cong {\bf k}^{n_i}$ and $$ Z \cong M(n_2 \times n_1) \times \cdots \times M(n_{h} \times n_{h\!-\!1}) , $$ where $M(k\times l)$ denotes the affine space of matrices over ${\bf k}$ with $k$ rows and $l$ columns. The group $$ G_{{\bf n}} = GL(n_1) \times \cdots \times GL(n_h) $$ acts on $Z$ by $$ (g_1,g_2,\cdots,g_h) \cdot (f_1,f_2,\cdots,f_{h\!-\!1}) = (g_2 f_1 g_1^{-1}, g_3 f_2 g_2^{-1},\cdots, g_{h}f_{h\!-\!1} g_{h\!-\!1}^{-1}), $$ corresponding to change of basis in the $V_i$. Now, let ${\bf r} = (r_{ij})_{1 \leq i \leq j \leq h}$ be an array of non-negative integers with $r_{ii} = n_i$, and define $r_{ij} = 0$ for any indices other than $1\leq i\leq j \leq h$. Define $$ Z^{\circ}({\bf r}) = \{(f_1,\cdots,f_{h\!-\!1}) \in Z \ \mid\ \forall\, i\!<\!j,\ \mbox{\rm rank} (f_{j\!-\!1} \cdots f_i : V_i \to V_j) = r_{ij} \}. $$ (This set might be empty for a bad choice of ${\bf r}$.) \\[1em] {\bf Proposition.} {\it The $G_{{\bf n}}$-orbits of $Z$ are exactly the sets $Z^{\circ}({\bf r})$ for ${\bf r}=(r_{ij})$ with $$ r_{i,j\!-\!1} - r_{i,j} - r_{i\!-\!1,j\!-\!1} + r_{i\!-\!1,j} \geq 0,\quad \forall\ 1\! \leq\! i\! <\! j\! \leq\! h. $$ } \noindent {\bf Proof.} This is a standard result of algebraic quiver theory \cite{BGP}, \cite{G}, \cite{W}. Since this theory is not well known among geometers, we recall it here. Consider the abelian category ${\cal R}$ of quiver representations whose objects are sequences of linear maps $(V_1 \stackrel{f_1}{\to} \cdots \stackrel{f_{h\!-\!1}}{\to} V_h)$, where the $V_i$ are {\it any} vector spaces of {\it arbitrary} dimension. A morphism of ${\cal R}$ from the object $(V_1 \stackrel{f_1}{\to} \cdots \stackrel{f_{h\!-\!1}}{\to} V_h)$ to the object $(V'_1 \stackrel{f'_1}{\to} \cdots \stackrel{f'_{h\!-\!1}}{\to} V'_h)$ is defined to be an $h$-tuple of linear maps $(\phi_i:V_i \to V'_i)$ such that each square $$ \begin{array}{ccc} V_i & \stackrel{f_i}{\to} & V_{i+1} \\ \mbox{\tiny $\phi_i$} \downarrow && \downarrow \mbox{\tiny $\phi_{i+1}$} \\ V'_i & \stackrel{f'_i}{\to} & V'_{i+1} \end{array} $$ commutes. Direct sum of objects is defined componentwise, and it is known (Krull-Remak-Schmidt Theorem) that any object $R \in {\cal R}$ can be written uniquely as a direct sum of the indecomposable objects $$ \begin{array}{ccccc} R_{ij} = (0 \to \cdots \to 0\to & \!\!\!\!{\bf k}\!\!\!\! & \stackrel{\rm \sim}{\to} \cdots \stackrel{\rm \sim}{\to} {\bf k} \to & \!\!\!\!0\!\!\!\! & \to \cdots \to 0) \\ &\!\!\!\! V_i \!\!\!\!&& \!\! \!\!V_j\!\! \!\! & \end{array} $$ for $1 \leq i<j \leq h+1$ (corresponding to the positive roots of the root system $A_h$). That is, $$ R \cong \bigoplus_{1 \leq i<j \leq h+1} m_{ij} R_{ij} $$ for unique multiplicities $m_{ij} \in {\Bbb Z}^+$. Our variety $Z$ consists of representations with fixed $(V_i)$ and all possible $(f_i)$. Two points of $Z$ are in the same $G_{{\bf n}}$-orbit exactly if they are isomorphic as objects in ${\cal R}$. So the orbits correspond to arrays $(m_{ij})_{1 \leq i<j \leq h+1}$ with $m_{ij} \in {\Bbb Z}^+$ and $n_i = \sum_{k\leq i< l} m_{kl}$. We can compute the rank numbers $(r_{ij})$ from the multiplicities $(m_{ij})$: $$ r_{ij} = \sum_{k\leq i<j<l} m_{kl}, $$ and conversely $$ m_{ij}= r_{i,j\!-\!1} - r_{i,j} - r_{i\!-\!1,j\!-\!1} + r_{i\!-\!1,j} . $$ Hence the arrays $(r_{ij})$ with the stated conditions classify the $G_{{\bf n}}$-orbits on $Z$. $\bullet$ \\[1em] We define the {\it quiver variety} $$ Z({\bf r}) =\{(f_1,\cdots,f_{h\!-\!1}) \in Z \mid \forall i,j,\ \mbox{\rm rank} (f_{j\!-\!1} \cdots f_i : V_i \to V_j) \leq r_{ij}\}. $$ Finally, we have the dimension formula due to Abeasis and Del Fra \cite{AF}. \\[1em] {\bf Propsition.} $$ \dim Z({\bf r}) = \dim G_{{\bf n}} -\!\!\! \sum_{1\leq i\leq j \leq h} (r_{ij}-r_{i,j+1}) (r_{ij}-r_{i-1,j}). $$ \subsection{Schubert varieties} Given ${\bf n}=(n_1,\cdots,n_h)$, for $1 \leq i \leq h$ let $$ a_i = n_1 + n_2 + \cdots +n_i \qquad \mbox{and} \qquad n = n_1 + \cdots + n_h \ . $$ For positive integers $i \leq j$, we shall frequently use the notations $$ [i,j] = \{ i, i+1, \ldots, j\}, \qquad\qquad [i] = [1,i]\ . $$ Let ${\bf k}^n$ be a vector space (over our arbitrary field ${\bf k}$) with standard basis $e_1,\ldots,e_n$. Consider its general linear group $GL(n)$, the subgroup $B$ of upper-triangular matrices, and the parabolic subgroup $Q$ of block upper-triangular matrices $$ Q = \{ (a_{ij} \in GL(n) \mid a_{ij}=0 \ \mbox{whenever}\ j\leq a_k <i \ \mbox{for some}\ k \}\ . $$ A {\it partial flag of type $(a_1<a_2<\cdots <a_h=n)$ } (or simply a {\it flag}) is a sequence of supspaces $U. = (U_1 \subset U_2 \subset \cdots \subset U_h = {\bf k}^n)$ with $\dim U_i = a_i$. Let $E_i = \langle e_1,\ldots,e_{a_i}\rangle$ the span of the first $a_i$ coordinate vectors, and $E'_i = \langle e_{a_i+1},\ldots,e_n\rangle$ the natural complementary subspace to $E_i$, so that $E_i \oplus E'_i = {\bf k}^n$. Call $E. = (E_1 \subset E_2 \subset \cdots)$ the {\it standard flag}. Let $\mbox{\rm Fl}$ denote the set of all flags $U.$ as above. $\mbox{\rm Fl}$ has a transitive $GL(n)$-action induced from ${\bf k}^n$, and $Q = \mbox{\rm Stab}_{GL(n)}( E.)$, so we may identify $\mbox{\rm Fl} \cong GL(n)/Q$, \ $g\!\! \cdot\!\! E. \leftrightarrow gQ$\ . The {\it Schubert varieties} are the closures of $B$-orbits on $\mbox{\rm Fl}$. Such orbits are usually indexed by certain permutations of $[n]$, but we prefer to use {\it flags of subsets} of $[n]$, of the form $$ \tau = (\tau_1 \subset \tau_2 \subset\cdots \subset \tau_h = [n]), \qquad \#\tau_i=a_i\ . $$ (A permutation $w: [n]\to[n]$ corresponds corresponds to the subset-flag with $\tau_i = w[a_i] = \{w(1),w(2),\ldots,w(a_i)\}$. This gives a one-to-one correspondence between cosets of the symmetric group $S_n$ modulo the Young subgroup $S_{n_1} \times \cdots \times S_{n_h}$, and subset-flags.) Given such $\tau$, let $E_i(\tau) = \langle e_j \mid j \in \tau_i \rangle$ be a coordinate subspace of ${\bf k}^n$, and $E.(\tau) = (E_1(\tau) \subset E_2(\tau) \subset \cdots) \in \mbox{\rm Fl}$. Then we may define the {\it Schubert cell} $$ \begin{array}{rcl} X^{\circ}(\tau) &= &B\cdot E(\tau)\\ &=& \left\{(U_1\subset U_2\subset\cdots)\in \mbox{\rm Fl}\ \ \left|\ \begin{array}{c} \dim U_i \cap {\bf k}^j = \#\, \tau_i \cap [j]\\[.2em] 1\leq i \leq h,\ 1\leq j \leq n \end{array} \right.\right\} \end{array} $$ and the {\it Schubert variety} $$ \begin{array}{rcl} X(\tau) &= &\overline{X^{\circ}(\tau)}\\ &=& \left\{(U_1\subset U_2\subset\cdots)\in \mbox{\rm Fl}\ \ \left|\ \begin{array}{c} \dim U_i \cap {\bf k}^j \geq \#\, \tau_i \cap [j]\\[.2em] 1\leq i \leq h,\ 1\leq j \leq n \end{array} \right.\right\} \end{array} $$ where ${\bf k}^j = \langle e_1,\ldots,e_j\rangle \subset {\bf k}^n$. We define the {\it opposite cell} ${\bf O} \subset \mbox{\rm Fl}$ to be the set of flags in general position with respect to the spaces $E'_1 \supset \cdots \supset E'_{h-1}$: $$ {\bf O} = \{(U_1\subset U_2\subset\cdots)\in \mbox{\rm Fl}\ \mid\ U_i \cap E'_{i}=0\}. $$ We also define $Y(\tau) = X(\tau) \cap {\bf O}$, an open subset of >$X(\tau)$. By abuse of language, we call $Y(\tau)$ the {\it opposite cell} of $X(\tau)$, even though it is not a cell. \subsection{The bijection $\zeta$} We define a special subset-flag $ \tau^{ \mbox{\tiny max} } = ( \tau^{ \mbox{\tiny max} } _1 \subset \cdots \subset \tau^{ \mbox{\tiny max} } _h = [n])$ corresponding to ${\bf n} = (n_1,\ldots,n_h)$. We want $ \tau^{ \mbox{\tiny max} } _i$ to contain numbers as large as possible given the constraint $[a_{i\!-\!1}]\subset \tau^{ \mbox{\tiny max} } _i$. Namely, we define $ \tau^{ \mbox{\tiny max} } _i$ recursively by $$ \tau^{ \mbox{\tiny max} } _h = [n];\quad \tau^{ \mbox{\tiny max} } _{i} = [a_{i\!-\!1}] \cup \{ \mbox{largest $n_i$ elements of $ \tau^{ \mbox{\tiny max} } _{i+1}$}\}. $$ Furthermore, given ${\bf r} = (r_{ij})_{1\leq i\leq j\leq h}$ indexing a quiver variety, define a subset-flag $\tau^{\rr}$ to contain numbers as large as possible given the constraints $$ \#\, [a_j]\cap \tau^{\rr}_i = \left\{ \begin{array}{cl} a_i -r_{i,j+1} & \mbox{for}\ i\leq j \\ a_j& \mbox{for}\ i> j \\ \end{array} \right. $$ Namely, $$ \tau^{\rr}_i = \{\, \underbrace{1\ldots a_{i\!-\!1}}_ {\mbox{\small $a_{i\!-\!1}$}} \ \underbrace{. \ldots\ldots a_{i}}_ {\mbox{\small $r_{ii}\!-\!r_{i,i+1}$}} \ \underbrace{.\ldots\ldots a_{i+1}}_ {\mbox{\small $r_{i,i+1}\!-\!r_{i,i+2}$}} \ \underbrace{.\ldots\ldots a_{i+2}}_ {\mbox{\small $r_{i,i+2}\!-\!r_{i,i+3}$}}\ \ldots\ \ \underbrace{.\ldots\ldots n_{\mbox{}}}_ {\mbox{\small $r_{i,h}$}} \} $$ where we use the visual notation $$ \underbrace{\cdots\cdots a}_{\mbox{\small $b$}} = [a-b+1,a]. $$ Note that $r_{ij} -r_{i,j+1} \leq n_j$, so that each $\tau^{\rr}_i$ is a list of increasing integers, and that $r_{ij}-r_{i,j+1}\leq r_{i+1,j}-r_{i+1,j+1}$, so that $\tau^{\rr}_i \subset \tau^{\rr}_{i+1}$. Thus $ \tau^{ \mbox{\tiny max} } $ and $\tau^{\rr}$ are indeed subset-flags. Now define the Zelevinsky map $$ \begin{array}{rccc} \zeta: & Z & \to &\mbox{\rm Fl} \\ & (f_1,\ldots,f_{h\!-\!1}) & \mapsto &(U_1\subset U_2 \subset \cdots) \end{array} $$ where $$ U_i = \{ (u_1,\ldots,u_h)\in {\bf k}^{n_1}\!\oplus\! \cdots \!\oplus\! {\bf k}^{n_h} = {\bf k}^n \mid \forall\, j>i,\ u_{j+1} = f_j(u_j)\}. $$ In terms of coordinates, if we identify the linear maps $(f_1,\ldots,f_{h\!-\!1})$ with the matrices $(A_1,\ldots,A_{h\!-\!1})$, and identify $\mbox{\rm Fl} \cong GL(n)/Q$, we have $$ \zeta(A_1,\ldots,A_{h-1}) = \left( \begin{array}{ccccc} I_1 & 0 & 0 & 0& \cdots \\ A_1 & I_2 &0 & 0& \cdots \\ A_2 A_1 & A_2 & I_3 & 0& \cdots \\ A_3 A_2 A_1 & A_3 A_2 &A_3 & I_4 & \cdots \\[-.4em] \vdots & \vdots & \vdots & \vdots& \end{array} \right) \ \ \mbox{\rm mod} \ \ Q $$ where $I_i$ is an identity matrix of size $n_i$. \\[1em] {\bf Theorem.} {\it (Zelevinsky \cite{Z})\\ (i) $\zeta$ is a bijection of $Z$ onto its image $Y( \tau^{ \mbox{\tiny max} } )$: \quad $\zeta : Z \stackrel{\rm \sim}{\to} Y( \tau^{ \mbox{\tiny max} } )$.\\ Also, \\[-1em] $$ \mbox{}\hspace{-2em} (*)\qquad Y( \tau^{ \mbox{\tiny max} } )= \{(U_1\subset U_2 \subset \cdots) \ \mid \ \forall\ i,\ \ E_{i-1} \subset U_i, \ \ U_i \cap E'_{i} = 0 \}. $$ (ii) $\zeta$ restricts to a bijection from $Z({\bf r})$ onto $Y(\tau^{\rr})$:\quad $\zeta : Z({\bf r}) \stackrel{\rm \sim}{\to} Y(\tau^{\rr})$.\\ Also, \\[-1em] $$ \mbox{} \hspace{-2em} (**) \quad Y(\tau^{\rr})=\left\{(U_1\subset U_2 \subset \cdots) \ \left| \ \begin{array}{c} \forall\ i\leq j,\quad \dim \, E_j \cap U_i \geq a_i-r_{i,j\!+\!1} ,\\[.3em] E_{i-1} \subset U_i, \quad U_i \cap E'_{i} = 0\end{array} \right.\right\}\ . $$ } \noindent{\bf Proof.} Obviously $\zeta$ is injective. To prove (i), we first show that $\zeta(Z)$ is equal to the right hand side of equation $(*)$. One inclusion is clear. To show the other inclusion, consider any $U.$ with $E_{i\!-\!1} \subset U_i$ and $U_i\cap E'_{i}=0$ for all $i$. Let $\pi_i:{\bf k}^n = E_i \oplus E'_i \to E_i$ be the projection. Then $\pi_{h\!-\!1}$ restricts to an isomorphism $U_{h\!-\!1} \stackrel{\rm \sim}{\to} E_{h\!-\!1}$, so there exists an inverse linear map $$ \mbox{id} \oplus f_{h\!-\!1} : E_{h\!-\!1} \to E_{h\!-\!1}\oplus{\bf k}^{n_h} $$ such that $$ U_{h\!-\!1} = \mbox{Graph}(f_{h\!-\!1}) \subset E_{h\!-\!1} \oplus {\bf k}^{n_h} ={\bf k}^n\ . $$ Since $E_{h\!-\!2}\subset U_{h\!-\!1}$, we have $f_{h\!-\!1}(E_{h\!-\!2})=0$. Next, $\pi_{h\!-\!2}$ restricts to an isomorphism $U_{h-2} \stackrel{\rm \sim}{\to} E_{h-2}$, and there exists a linear map $\tilde{f}_{h\!-\!2}: E_{h\!-\!2} \to E'_{h\!-\!2}$ with $\tilde{f}_{h\!-\!2}(E_{h\!-\!3})=0$ such that $$ U_{h\!-\!2} = \mbox{Graph}(\tilde{f}_{h\!-\!2}) \subset E_{h\!-\!2} \oplus E'_{h\!-\!2} ={\bf k}^{n}. $$ Since $U_{h\!-\!2}\subset U_{h-1}$, we have $$ \tilde{f}_{h\!-\!2} = (f_{h\!-\!2},\, f_{h\!-\!1}f_{h\!-\!2}) $$ for some $f_{h\!-\!2}:E_{h\!-\!2} \to {\bf k}^{n_{h\!-\!1}}$. Continuing in this way, we find that $U.\in \zeta(Z)$. Thus it suffices to show that $(*)$ is valid. Again, the inclusion $\subset$ is clear. Now consider a flag $U.$ satisfying $E_{i\!-\!1} \subset U_i$ for all $i$. Then we will show that $U.$ must satisfy $\dim({\bf k}^i \cap U_j) \geq \#\, [i] \cap \tau^{ \mbox{\tiny max} } _j$ for all $1\leq i\leq n$,\, $1 \leq j\leq h$. Acting by $B$ does not change $\dim U_i \cap {\bf k}^j$, so we may assume our $U.$ is a flag of coordinate subspaces $U. = E(\mu)$ for some $\mu=(\mu_1\subset\cdots\subset \mu_{h}=[n])$ with $[a_{i\!-\!1}]\subset \mu_i$ for all $i$, so that $\dim U_i \cap {\bf k}^j = \# \mu_i \cap [j]$. Then by the definition of $ \tau^{ \mbox{\tiny max} } $, we must have $$\forall\, j, \qquad \# \mu_{h\!-\!1} \cap [j] \geq \# \tau^{ \mbox{\tiny max} } _{h\!-\!1}\cap [j],\qquad \# \mu_{h\!-\!2} \cap [j] \geq \# \tau^{ \mbox{\tiny max} } _{h\!-\!2}\cap [j],\ \ldots\ldots $$ This proves $(*)$, and hence part (i). The proof of (ii) is similar. Clearly $Y(\tau^{\rr}) \subset Y( \tau^{ \mbox{\tiny max} } ) = \zeta(Z)$. For any flag $U.=\zeta(f_1,\ldots,f_{h\!-\!1})$, we have $$ \begin{array}{rcl} \dim\, E_j \cap U_i &=& \dim E_{i\!-\!1} + \dim \mbox{Ker}(f_j f_{j\!-\!1} \cdots f_i)\\ &=& \dim E_{i\!-\!1} + \dim V_i - \mbox{\rm rank}( f_j f_{\!j-\!1}\cdots f_i)\\ &=& a_i -\mbox{\rm rank}( f_j f_{j\!-\!1}\cdots f_i). \end{array} $$ Hence $\dim\, E_{j} \cap U_i \geq a_i - r_{i,j\!+\!1}$ if and only if $U. \in \zeta(Z({\bf r}))$, so that $\zeta(Z({\bf r}))$ is equal to the right hand side of $(**)$. But the conditions on the right side of $(**)$ are enough to force the flag $U.$ to lie in the Schubert variety $X(\tau^{\rr})$ on the left hand side, as in part (i). $\bullet$ \subsection{The actions of $B$, $Q$ and $G_{{\bf n}}$} Let $W = S_n$ and $W_{{\bf n}} = S_{n_1} \times \cdots \times S_{n_h}$ a Young subgroup. Let $W_{{\bf n}}$ act on the the coset space $W/W_{{\bf n}}$ by left multiplication. Then we may consider $ \tau^{ \mbox{\tiny max} } $ as a coset in $W/W_{{\bf n}}$ which is Bruhat-maximal within its $W_{{\bf n}}$ orbit. Since $W_{{\bf n}}$ is the Weyl group of $Q$, this means that the $B$-action on the Schubert variety $X( \tau^{ \mbox{\tiny max} } )$ extends to a $Q$-action. We may embed $G_{{\bf n}}$ into $Q$ as the block diagonal matrices, so that $G_{{\bf n}}$ acts on $X( \tau^{ \mbox{\tiny max} } )$ and in fact on the open subvariety $Y( \tau^{ \mbox{\tiny max} } )$. Then $\zeta: Z \to Y( \tau^{ \mbox{\tiny max} } )$ is equivariant with respect to the $G_{{\bf n}}$-action. \\[1em] Now we relate our combinatorial formalism to that in Zelevinsky's original paper \cite{Z}. We have just seen that our $\tau^{\rr}$ correspond to certain double cosets in $W_{{\bf n}}\backslash W / W_{{\bf n}}$. Following Zelevinsky, we may index such double cosets by {\it block permutation matrices}, which are defined to be the $h\times h$ arrays $T=(t_{ij})$ of non-negative integers with row and column sums equal to the $n_i$ , so that for all $1\leq i,j\leq h$, $$ \sum_{i=1}^h t_{ij} = n_j \qquad \sum_{j=1}^h t_{ij} = n_i\ . $$ (If all $n_i=1$, this defines an ordinary permutation matrix.) A permutation $w \in W$ correponds to the block permutation matrix $\mbox{\rm Block}(w)$ defined by partitioning the ordinary permutation matrix of $w$ into blocks, and summing all entries in each block: $$ \mbox{\rm Block}(w) = (t_{ij}) \qquad t_{ij} = \#\ [a_{i\!-\!1}+1, a_i] \cap w[a_{j\!-\!1}+1,a_j]\ . $$ The block map induces a one-to-one correspondence between double cosets $W_{{\bf n}}\backslash W / W_{{\bf n}}$ and block permutation matrices. Zelevinsky's map takes $Z({\bf r})$ to $Y(\tau^{\rr})$ for each ${\bf r}=(r_{ij})$. Recall from the proof of Proposition 1.1 that the rank numbers $r_{ij}$, $1\leq i\leq j \leq h$, can be computed from certain multiplicities $m_{ij}$, $1\leq i<j \leq h+1$. Then the block permutation matrix corresponding to $\tau^{\rr}$ is given by $$ \left( \begin{array}{ccccc} m_{12} & m^*_{1} & 0 & 0 &\cdots \\ m_{13} & m_{23} & m^*_2 & 0 & \cdots \\ m_{14} & m_{24} & m_{34} & m^*_3 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \end{array} \right) $$ where $$ m^*_{i} = \sum_{k < i+1 < l} m_{kl}\ . $$ \section{Plucker coordinates and determinantal ideals} For a variety $X$ embedded in an affine space $V$ over an infinite field ${\bf k}$, the {\it vanishing ideal} ${\cal I}$ of $X$ is the set of polynomial functions on $V$ which restrict to zero on $X$. However, if ${\bf k}$ is a finite field, we modify this definition in the usual way: the vanishing ideal is the the set of polynomials on $V$ which are zero on the points of $X$ over the algebraic closure of ${\bf k}$: $$ {\cal I}=\{f \in {\bf k}[V] \mid f(x)=0\ \, \forall\, x\in X(\overline{{\bf k}})\}. $$ The ideal ${\cal I}$ is necessarily reduced (radical). \subsection{Coordinates on the opposite big cell} Consider the opposite cell ${\bf O} \subset GL(n)/Q$. It is easily seen that ${\bf O}$ consists of those cosets which have a unique representative $A$ of the form $$ A = (a_{kl}) = \left( \begin{array}{ccccc} I_1& 0 & 0& \cdots & 0\\ A_{21} & I_2 & 0 & \cdots &0\\ A_{31}& A_{32} & I_3 &\cdots & 0\\ \vdots&\vdots&\vdots &&\vdots\\ A_{h1}& A_{h2}& A_{h3}&\cdots& I_{h} \end{array} \right)\ \mbox{\rm mod} \ Q, $$ where $I_i$ is the identity matrix of size $n_i$, and $A_{ij}$ is an arbitrary matrix of size $n_i \times n_j$. That is, ${\bf O}$ is an affine space with coordinates $a_{kl}$ for those positions $(k,l)$ with $1 \leq l \leq a_i <k \leq n$ for some $i$. Its coordinate ring is the polynomial ring $$ {\bf k}[{\bf O}] = {\bf k}[a_{kl}]. $$ For a matrix $M \in M(k\times l)$ and subsets $\lambda\subset [k]$, $\mu \subset [l]$, let $\det M_{\lambda\times \mu}$ be the minor with row indices $\lambda$ and column indices $\mu$. Now let $\sigma \subset [n]$ be a subset of size $\#\sigma = a_i$ for some $i$. Define the {\it Plucker coordinate} $p_{\sigma} \in {\bf k}[{\bf O}]$ to be the $a_i$-minor of our matrix $A$ with row indices $\sigma$ and column indices the interval $[a_i]$: $$ p_{\sigma}=p_{\sigma}(A)=\det A_{\sigma \times [a_i]}. $$ Define a partial order on Plucker coordinates by: $$ \sigma \leq \sigma' \quad \Longleftrightarrow \quad \begin{array}{c} \sigma = \{\sigma(1)<\sigma(2)<\cdots<\sigma(a_i)\},\\ \sigma' = \{\sigma'(1)<\sigma'(2)<\cdots<\sigma'(a_i)\},\\ \sigma(1)\leq \sigma'(1),\ \sigma(2) \leq \sigma'(2), \cdots, \sigma(a_i) \leq \sigma'(a_i). \end{array} $$ This is a version of the Bruhat order. \\[1em] {\bf Proposition.} {\it Let $\tau = (\tau_1 \subset \cdots \subset \tau_{h} = [n])$ be a subset-flag and $Y(\tau)$ the intersection of the Schubert variety $X(\tau)$ with the opposite cell ${\bf O}$. Then the vanishing ideal ${\cal I}(\tau) \subset {\bf k}[{\bf O}]$ of $Y (\tau) \subset {\bf O}$ is generated by those Plucker coordinates $p_{\sigma}$ which are incomparable with one of the $p_{\tau_i}$: $$ {\cal I}(\tau) = \langle p_{\sigma} \mid \exists\, i,\ \#\sigma = a_i, \ \sigma \not\leq \tau_i \rangle. $$ } \noindent {\bf Proof.} This follows from well-known results of Lakshmibai-Musili-Seshadri in Standard Monomial Theory (see e.g.~\cite{MS},\cite{LS}). \subsection{The main theorem} Denote a generic element of the quiver space $ Z = M(n_2\times n_1) \times \cdots \times M(n_{h}\times n_{h\!-\!1})$ by $(A_1,\ldots,A_{h-1})$, so that the coordinate ring of $Z$ is the polynomial ring in the entries of all the matrices $A_i$. Let ${\bf r} = (r_{ij})$ index the quiver variety $Z({\bf r}) = \{(A_1,\ldots,A_{h-1}) \mid \mbox{\rm rank}\, A_{j-1}\cdots A_i \leq r_{ij}\}$. Let ${\cal J}({\bf r}) \subset {\bf k}[Z]$ be the ideal generated by the determinantal conditions implied by the definition of $Z({\bf r})$: $$ {\cal J}({\bf r}) = \left\langle \det(A_{j-1} A_{j-2} \cdots A_i)_ {\lambda\times\mu} \ \left| \ \begin{array}{c} i\leq j,\ \lambda \subset [n_j],\ \mu \subset [n_i] \\[.2em] \#\lambda = \#\mu = r_{ij}+1 \end{array} \right. \right\rangle\ . $$ Clearly ${\cal J}({\bf r})$ defines $Z({\bf r})$ set-theoretically. \\[1em] {\bf Theorem.} {\it ${\cal J}({\bf r})$ is a prime ideal and is the vanishing ideal of $Z({\bf r})\subset Z$. There are isomorphisms of reduced schemes $$ Z({\bf r}) = \mbox{Spec}({\bf k}[Z]\,/\,{\cal J}({\bf r})) \cong \mbox{Spec}({\bf k}[{\bf O}]\,/\,{\cal I}(\tau^{\rr})) = Y(\tau^{\rr}). $$ That is, the quiver scheme $Z({\bf r})$ defined by ${\cal J}({\bf r})$ is equal to the reduced, irreducible variety $Y(\tau^{\rr})$, the opposite cell of a Schubert variety. } \vspace{.5em} \noindent {\bf Proof.} Consider the map of \S1.3, $\zeta: Z \stackrel{\rm \sim}{\to} Y( \tau^{ \mbox{\tiny max} } ) \subset {\bf O}$. It is clear that $\zeta$ is an algebraic isomorphism onto its image, since it is injective on points and on tangent vectors. (In fact, in appropriate coordinates ${\bf O} \cong Z\times V$ for some affine space $V$, and for a certain polynomial function $\phi:Z\to V$, \, $\zeta$ is equivalent to the map $Z \to Z\times V$, \ $z \mapsto (z,\phi(z))$.\,) Thus by Proposition 2.1 we have the exact sequence $$ 0\to {\cal I}( \tau^{ \mbox{\tiny max} } ) \to {\bf k}[{\bf O}] \stackrel{\zeta^*}{\to} {\bf k}[Z] \to 0\ . $$ Let $\widetilde{\JJ}({\bf r}) \subset {\bf k}[Z]$ be the (reduced) vanishing ideal of $Z({\bf r}) \subset Z$. Clearly ${\cal J} \subset \widetilde{\JJ}$. Since $\zeta$ maps $Z({\bf r})$ isomorphically onto $Y(\tau^{\rr})$ by Theorem 1.3, we have $(\zeta^*)^{-1}\widetilde{\JJ}({\bf r}) = {\cal I}(\tau^{\rr})$ by Proposition 2.1. Hence $$ \begin{array}{rcl} Z({\bf r}) = \mbox{Spec}({\bf k}[Z]\,/\,\widetilde{\JJ}({\bf r})) &\cong& \mbox{Spec}({\bf k}[{\bf O}]\,/\,(\zeta^*)^{-1}\widetilde{\JJ}(\tau^{\rr}))\\ &=&\mbox{Spec}({\bf k}[{\bf O}]\,/\,{\cal I}(\tau^{\rr})) \ = \ Y(\tau^{\rr}). \end{array} $$ Furthermore, ${\cal J}({\bf r}) = \widetilde{\JJ}({\bf r})$ if and only if $(\zeta^*)^{-1}{\cal J}({\bf r}) = (\zeta^*)^{-1}\widetilde{\JJ}({\bf r})$; and ${\cal J}({\bf r})$ is prime if and only if $(\zeta^*)^{-1}{\cal J}({\bf r})$ is prime. Thus to show the Theorem, it suffices to prove $$ (\zeta^*)^{-1}{\cal J}({\bf r}) = {\cal I}(\tau^{\rr}). $$ But clearly $(\zeta^*)^{-1} {\cal J}({\bf r}) \subset (\zeta^*)^{-1}\widetilde{\JJ}({\bf r}) = {\cal I}(\tau^{\rr})$, so we are left with the opposite inclusion $$ (\zeta^*)^{-1} {\cal J}({\bf r}) \supset {\cal I}(\tau^{\rr}), $$ which we will prove in the next section. \subsection{Proof of the main theorem: determinant identities} We define ideals ${\cal I}_0, {\cal I}_1, {\cal I}_2 \subset {\bf k}[{\bf O}]$ generated by certain minors of the generic matrix $A \in {\bf O}$ at the end of \S1.2: $$ {\cal I}_0 =(\zeta^*)^{-1}{\cal J}({\bf r}) \hspace{3.5in} \mbox{} $$ \vspace{-1.8em} $$ = {\cal I}( \tau^{ \mbox{\tiny max} } ) + \left\langle \det (A_{j,j\!-\!1} A_{j\!-\!1,j\!-\!2}\!\cdots\! A_{i\!+\!1,i} )_{\lambda\times\mu} \left| \begin{array}{c} i\!<\!j,\ \ \lambda\! \subset\! [n_j],\ \ \mu\!\subset\! [n_i]\\[.2em] \# \lambda = \#\mu = r_{ij}+1 \end{array} \right. \right\rangle $$ \vspace{.4em} $$ {\cal I}_1\ =\ {\cal I}( \tau^{ \mbox{\tiny max} } ) + \left\langle \det A_{\lambda\times\mu}\ \left|\ \begin{array}{c} i<j,\ \ \lambda \subset [a_j\!+\!1,n],\ \ \mu\subset [a_i]\\[.2em] \# \lambda = \#\mu = r_{ij}\!+\!1 \end{array} \right. \right\rangle $$ \vspace{.4em} $$ {\cal I}_2\ =\ {\cal I}(\tau^{\rr})\ =\ \left\langle \det A_{\sigma\times [a_i]}\ \left|\ \begin{array}{c} 1\leq i \leq h\!-\!1,\ \ \sigma \subset [n]\\[.2em] \# \sigma = a_i,\ \ \sigma \not \leq \tau^{\rr}_i \end{array} \right. \right\rangle $$ To finish the proof of Theorem 2.2, we will show $$ {\cal I}_0 \supset {\cal I}_1 \supset {\cal I}_2\ . $$ \noindent {\bf Lemma 1.} {\it Let $X = (x_{ij})$ and $Y= (y_{kl})$ be matrices of variables $x_{ij}$, $y_{kl}$ generating a polynomial ring. Let ${\cal J}_{X}$ (resp.~${\cal J}_{Y}$) be the ideal generated by all $r\!+\!1$-minors of $X$ (resp.~$Y$). Then ${\cal J}_X$ and ${\cal J}_Y$ both contain all $r\!+\!1$-minors of the product $XY$. } \vspace{.5em} \noindent {\bf Proof.} $$ \det (XY)_{\lambda\times\mu} = \sum_{\nu} \det X_{\lambda\times \nu}\, \det Y_{\nu\times\mu}. \quad \bullet $$ \noindent {\bf Lemma 2.} {\it Let $(A_1,\ldots,A_{h-1})$ be a generic element of $Z$, and for $i \leq j$ let ${\cal J}_{ij}$ be the ideal generated by all $r\!+\!1$-minors of the $n_j \times n_i$ product matrix $A_j\cdots A_i$. Then ${\cal J}_{ij}$ contains all $r\!+\!1$-minors of the $(n\!-\!a_{j\!-\!1}) \times a_i$ matrix $$ \widetilde{A} = \left( \begin{array}{cccc} A_j\!\! \cdots\! A_{1} & A_j\!\!\cdots\! A_2& \cdots& A_j\!\! \cdots\! A_i \\ A_{j\!+\!1}\!\! \cdots\! A_{1} & A_{j\!+\!1}\!\!\cdots \!A_2& \cdots& A_{j\!+\!1}\!\! \cdots\! A_i \\ \vdots&\vdots&&\vdots\\ A_h\!\! \cdots\! A_{1} & A_h\!\!\cdots\! A_2& \cdots& A_h\!\! \cdots A_i\! \\ \end{array} \right) $$ } \vspace{.5em} \noindent {\bf Proof.} Note that we can factor the matrix $$ \widetilde{A} = \left(\!\!\!\begin{array}{c} I_{j} \\ A_{j\!+\!1} \\ \vdots \\ A_h\! \cdots\! A_{j\!+\!1} \end{array} \!\!\!\right) \cdot\, A_j\!\!\cdots\!\! A_i\, \cdot\ ( A_{i\!-\!1}\!\!\cdots\!\! A_1,\ A_{i\!-\!1}\!\!\cdots\!\! A_2,\ \cdots\ ,\ A_{i\!-\!1},\ I_i) $$ Now apply Lemma 1 twice. \\[1em] {\bf Lemma 3.}\qquad $ {\cal I}_0 \supset {\cal I}_1\ .$ \\[.5em] {\bf Proof.} Let $\lambda \subset [a_j\!+\!1]$,\ $\mu \subset [a_i]$, $\#\lambda = \#\mu = r_{ij}+1$. Then clearly $$ \det A_{\lambda\times \mu} \in (\zeta^*)^{-1}( \det \widetilde{A}_{\lambda\times\mu} )\ . $$ Hence by Lemma 2, the generators of ${\cal I}_1$ lie in ${\cal I}_0$. $\bullet$ \\[1em] {\bf Lemma 4.} {\it (Gonciulea-Lakshmibai)\ Let $A$ be a generic element of ${\bf O}$. Let $1 \leq t\leq a_i$, \ $1\leq s \leq n$, and $\sigma = \{ \sigma(1)<\sigma(2)<\cdots<\sigma(a_i)\} \subset [n]$ with $\sigma(a_i-t+1) \geq s$. Then $p_{\sigma}(A)$ belongs to the ideal of ${\bf k}[{\bf O}]$ generated by $t$-minors of $A$ with row indices $\geq s$ and column indices $\leq a_i$. } \vspace{.5em} \noindent{\bf Proof.} Choose $\sigma'\subset [s,n] \cap \sigma$ with $\#\sigma' =t$, and let $\sigma'' = \sigma \setminus \sigma'$. Then the Laplace expansion of $p_{\sigma}(A)$ with respect to the rows $\sigma'$, $\sigma''$, gives $$ p_{\sigma}(A) = \det A_{\sigma\times[a_i]} = \sum_{\lambda' \cup \lambda''= [a_i]} \!\!\!\pm \det A_{\sigma'\times \lambda'} \det A_{\sigma''\times\lambda''}, $$ where the sum is over all partitions of the interval $[a_i]$. The first factor of each term in the sum is of the form required. $\bullet$ \\[1em] {\bf Lemma 5.}\qquad ${\cal I}_1 \supset {\cal I}_2\ .$ \\[.5em] {\bf Proof.} Let $\sigma \subset [n]$ with $\# \sigma = a_i$,\ $\sigma \not \leq \tau^{\rr}_i$ for some $i$,\ $1 \leq i\leq h\!-\!1$. Now, $\tau^{\rr}_i$ has the largest possible entries such that $$ \tau^{\rr}_i(a_i-r_{i,j+1})\leq a_j,\qquad \forall \,j\geq i , $$ so $\sigma \not \leq \tau^{\rr}_i$ must violate this condition for some $j$: $$ \sigma(a_i-r_{i,j+1}) \geq a_j+1,\qquad \exists\, j\geq i. $$ Hence by Lemma 4, $p_{\sigma}(A)$ is in ${\cal I}_1$. $\bullet$ \\[1em] The Main Theorem 2.2 is therefore proved. \subsection{Degeneracy schemes} Fulton \cite{F} defines the universal degeneracy scheme $\Omega_w$ associated to a permutation $w \in S_{m+1}$ as follows. Fix $2m$ vector spaces $F_1,F_2,\ldots,F_m,E_m,\ldots,E_2,E_1$ with $\dim F_i = \dim E_i = 1$, and let $$ Z = M_{2\times 1}\times M_{3\times 2} \times \cdots \times M_{m\times m\!-\!1} \times M_{m\times m} \times M_{m\!-\!1\times m} \times\cdots\times M_{1\times 2} $$ be the quiver space of all maps of the form $$ F_1 \to F_2 \to \cdots \to F_m \to E_m \to \cdots \to E_2 \to E_1. $$ (For convenience we will refer to these maps and their compositions by symbols such as $F_i\to F_j$ and $F_i\to E_j$.) Define rank numbers $$ r(F_i,E_j) = \# \,[i] \cap w[j], \quad 1\leq i,j\leq m $$ $$ r(F_i,F_j) = i\quad r(E_j,E_i)=i\quad 1\leq i<j\leq m $$ and let ${\bf r}_w$ be the array of these numbers. Then let $$ \Omega_w = Z({\bf r}_w) \subset Z, $$ the variety of all quiver representations satisfying $$ \mbox{\rm rank}(F_i\to E_j) \leq \#\, [i]\cap w[j], \quad 1\leq i,j\leq m $$ $$ \mbox{\rm rank}(F_i\to F_j) \leq i,\quad \mbox{\rm rank}(E_j\to E_i)\leq i,\quad 1\leq i<j\leq m. $$ (The latter conditions are clearly superfluous.) More precisely, define $\Omega_w$ as a scheme by the same determinantal equations defining $Z({\bf r}_w)$ in \S2.2. \\[1em] {\bf Proposition.} {\it The scheme $\Omega_w$ over an arbitrary field ${\bf k}$ is reduced and is isomorphic to the opposite cell of a Schubert variety in $\mbox{\rm Fl} = GL(n)/Q$, a partial flag variety of >${\bf k}^n$, where $n = 2(1+\cdots+m) = m(m+1)$. In particular, $\Omega_w$ is irreducible, Cohen-Macaulay, and normal, and has rational singularities.} \\[1em] {\bf Proof.} This follows since Schubert varieties are known to have these properties (see e.g.~\cite{R}). $\bullet$ \\[1em] {\bf Proposition.} $$ \mbox{\rm codim}_Z\, \Omega_{w} = \ell(w), $$ where $$\ell(w)=\#\left\{ (i,j) \ \left|\ \begin{array}{c} 1\leq i,j\leq m\!+\!1 \\[.4em] i<j,\quad w(i) > w(j) \end{array}\right.\right\} $$ is the Bruhat length. \\[1em] {\bf Proof.} By the dimension formula of Abeasis and Del Fra \cite{AF} (c.f.~\S1.1 above), we have $$ \begin{array}{rcl} \dim \Omega_w & = & \dim G_{{\bf n}} - \sum_{1\leq i,j\leq m} (r(F_i,E_j)-r(F_i,E_{j\!-\!1})) (r(F_i,E_j)-r(F_{i\!-\!1},E_{j})) \\[.5em] &&\quad -\sum_{1 \leq i\leq j\leq m} (r(F_i,F_j)-r(F_i,F_{j\!+\!1})) (r(F_i,F_j)-r(F_{i\!-\!1},F_{j})) \\[.5em] &&\quad -\sum_{1 \leq i\leq j\leq m} (r(E_j,E_i)-r(E_j,E_{i\!-\!1})) (r(E_j,E_i)-r(E_{j\!+\!1},E_{i})) \\[.5em] &&\quad -(r(F_m,F_m)-r(F_m,E_m))(r(F_m,F_m)-r(F_{m-1},F_m)) \\[.9em] &=& 2(1^2 + 2^2 +\cdots+m^2)\\[.5em] &&- \sum_{1\leq i,j\leq m} \#\,([i]\cap w[j])\setminus ([i]\cap w[j\!\!-\!\!1]) \cdot \#\,([i]\cap w[j])\setminus ([i\!\!-\!\!1]\cap w[j]) \\[.9em] &=& 2(1^2 + 2^2 +\cdots+m^2) \ -\ \#\left\{\ (i,j) \ \left|\ \begin{array}{c} 1\leq i,j\leq m +1 \\[.4em] w^{-1}(i) \leq j,\quad i \geq w(j)\ \end{array}\right.\right\} \\[1em] &=& 2(1^2 + 2^2 +\cdots+m^2) - m - \ell(w)\ . \end{array} $$ On the other hand, $$ \dim Z = 2(1\cdot 2 + 2 \cdot 3 + \cdots + (m\!-\!1) \cdot m) + m^2. $$ Hence $$ \begin{array}{rcl} \mbox{\rm codim}_Z\, \Omega_w &=& 2(1\cdot 2 + \cdots + (m\!-\!1) \cdot m) + m^2 - 2(1^2 + \cdots m^2) + m + \ell(w)\\[.3em] &= & \ell(w). \end{array} $$ \vspace{-2em} \mbox{}$\hfill \bullet$ \\[1em] {\bf Concluding remarks} \\[.5em] Denote $Z=Z(m)$ and $\Omega_w= \Omega_w(m)$ to emphasize the dependence on $m$. Consider $S_{m+1} \subset S_{m+2}$ in the usual way. Then there is a natural map $\pi: Z(m\!+\!1) \to Z(m)$ given by forgetting the middle two spaces, and we may easily see the stability property: $$ \Omega_w(m\!+\!1) = \pi^{-1} \Omega_w(m). $$ The map $\pi$ is a fiber bundle over some open set of $\Omega_w(m)$, and >from the previous proposition, the generic fibers of $\pi: \Omega_w(m\!+\!1)\to \Omega_w(m)$ have the same dimension as the generic fibers of $\pi: Z(m+1) \to Z(m)$. Finally, we note that $\Omega_w(m\!+\!1)$ is closely related to a Schubert variety of $\mbox{\rm Fl}' = GL(m\!+\!1)/B$, the complete flag variety of ${\bf k}^{m\!+\!1}$, a much smaller flag variety than that of Zelevinsky's bijection (c.f.~Fulton \cite{F} \S 3). Namely, consider the open set $Z^{\circ}(m\!+\!1)$ of elements $F_1 \to \cdots\to E_1$ with $F_{i}\to F_{i\!+\!1}$ injective, $E_{i\!+\!1}\to E_i$ surjective, and $F_{m\!+\!1}\to E_{m\!+\!1}$ bijective. Then we have a principal fiber bundle $$ \begin{array}{cccc} \psi: & Z^{\circ} & \to & \mbox{\rm Fl}' \times \mbox{\rm Fl}' \\ & (F_1 \to \cdots \to E_1) & \mapsto & (V.\, ,U.) \end{array} $$ where $$ V_i = \mbox{Im}(F_i \to E_{m+1}) \qquad U_i = \mbox{Ker}(E_{m+1} \to E_{m+1-i})\ . $$ Now, letting $\Omega_w^{\circ}(m\!+\!1) = \Omega_w(m\!+\!1) \cap Z^{\circ}(m\!+\!1)$, an open subet of $\Omega_w(m\!+\!1)$, we have $$ \Omega_w^{\circ}(m\!+\!1) = \psi^{-1}\ \{\, (V.,U.) \in \mbox{\rm Fl}' \times \mbox{\rm Fl}'\ \mid \ V_i \cap U_j \leq \#\, w w_0 [i] \cap [j]\, \}. $$ where $w_0$ is the longest element of $S_{m\!+\!1}$. It is well-known that the subset of $\mbox{\rm Fl}' \times \mbox{\rm Fl}'$ on the right is a fiber bundle over $\mbox{\rm Fl}'$ with fiber equal to the Schubert variety $X(w w_0) \subset \mbox{\rm Fl}'$.

9709

alg-geom/9709032

fr

https://arxiv.org/abs/alg-geom/9709032

alg-geom/9709032

Laurent Evain

L. Evain

Dimension of linear systems: a combinatorial and differential approach

17 pages, in french, also available at http://193.49.162.129/~evain/home.html

UA 45

We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension of linear systems of hypersurfaces in a projective space $\PP^n$ with generically prescribed singularities, and the calculus of collisions of fat points in $\PP^2$. These applications will be treated independently but a simple example in the introduction explains how the theorem will be used.

alg-geom

\section{Introduction par un exemple} Consid\'erons le syst\`eme lin\'eaire ${\cal L}_t$ des courbes projectives planes de degr\'e $d$ passant par trois points fixes $p_1,p_2,p_3$ et par un point $p_4(t)$ avec multiplicit\'es respectives $m_1,m_2,m_3$ et $m_4$. Supposons que $p_1,p_2,p_3$ soient align\'es sur une droite, et que $p_4(t)$ soit g\'en\'erique dans le plan. Le syst\`eme est de dimension projective au moins $\frac{d(d+3)}{2} -\sum \frac{m_i(m_i+1)}{2}$. Et la dimension est exactement $\frac{d(d+3)}{2} -\sum \frac{m_i(m_i+1)}{2}$ si les conditions impos\'ees par les points multiples sont ind\'ependantes. \ \\[2mm] Choisissons dans notre exemple les conditions num\'eriques $m_1=m_2=m_3=1$, $m_4=3$ et $d=5$. On veut montrer que le syst\`eme ${\cal L}_t$ est de dimension onze, et il suffit de voir qu'il est de dimension au plus onze. On sp\'ecialise le point g\'en\'erique $p_4(t)$ en un point $p_4(0)$ de la droite $D$ joignant $p_1,p_2$ et $p_3$, ce qui d\'efinit un syst\`eme lin\'eaire ${\cal L}_0$. Par semi-continuit\'e, $dim {\cal L}_t \leq dim {\cal L}_0$. Les diviseurs de ${\cal L}_0$ sont des courbes de degr\'e cinq qui coupent la droite $D$ le long d'un sch\'ema ponctuel de degr\'e six, donc ils contiennent $D$. En soustrayant $D$ \`a chaque diviseur de ${\cal L}_0$, on voit que la dimension de ${\cal L}_0$ est la m\^eme que celle du syst\`eme lin\'eaire des courbes de degr\'e quatre passant par $p_4(0)$ avec multiplicit\'e deux, c'est \`a dire onze. On avait donc bien $dim {\cal L}_t=11$. \ \\[2mm] Il existe de nombreuses situations dans lesquelles on essaie d'appliquer la strat\'egie pr\'ec\'edente: on sp\'ecialise des points g\'en\'eriques sur des diviseurs de sorte que le probl\`eme se simplifie en position sp\'eciale et on conclut par un argument de semi-continuit\'e. Bien s\^ur les conditions num\'eriques de l'exemple ont \'et\'e choisies pour que la strat\'egie s'applique sans difficult\'e. En revanche, il existe en g\'en\'eral des difficult\'es num\'eriques, comme l'illustre le cas suivant. \ \\[2mm] Choisissons dans notre exemple introductif $m_1=m_2=m_3=2$, $m_4=3$, et $d=6$. On veut montrer que le syst\`eme ${\cal L}_t$ est de dimension douze. Sp\'ecialisons le point $p_4(t)$ en un point $p_4(0)$ de la droite $D$ joignant $p_1,p_2$ et $p_3$. Comme pr\'ec\'edemment, $D$ est une composante du syst\`eme lin\'eaire ${\cal L}_0$ dans cette position sp\'eciale, donc ${\cal L}_0$ a la m\^eme dimension que le syst\`eme des courbes de degr\'e cinq passant par $p_1,p_2,p_3,p_4(0)$ avec multiplicit\'e un, un, un et deux, c'est \`a dire au moins quatorze. On ne peut pas conclure. En fait, la dimension du syst\`eme lin\'eaire en position sp\'eciale a saut\'e car on a mis ``trop de conditions sur la droite'': il suffit qu'un diviseur $\Delta$ de degr\'e six coupe la droite $D$ le long d'un sch\'ema de degr\'e sept pour que $D$ soit inclus dans $\Delta$, or un diviseur de ${\cal L}_0$ coupe la droite le long d'un sch\'ema de degr\'e neuf. \ \\[2mm] La m\'ethode d'Horace [AH1,AH2,H] propose un ensemble de techniques pour g\'erer les probl\`emes num\'eriques qui apparaissent lorsqu'on traite des exemples pr\'ecis. On se propose dans ce papier d'enrichir la m\'ethode d'Horace d'un nouveau th\'eor\`eme (Th\'eor\`eme \ref{theo}). \ \\[2mm] Alors que l'\'enonc\'e g\'en\'eral n\'ecessite quelques notations, on peut illustrer facilement le th\'eor\`eme sur l'exemple pr\'ec\'edent. \ \\[2mm] Quand $p_4(t)$ n'est pas sur $D$, un diviseur $\Delta$ du syst\`eme lin\'eaire ${\cal L}_t$ coupe $D$ deux fois en $p_1$, deux fois en $p_2$ et deux fois en $p_3$. Il manque encore une condition sur la droite pour que $D$ soit composante fixe de ${\cal L}_t$. Raisonnons malgr\'e tout comme si $D$ \'etait composante fixe. Alors, un diviseur de ${\cal L}_t(-D)$ serait une courbe de degr\'e cinq qui couperait $D$ en $p_1,p_2,p_3$. Il manquerait cette fois-ci trois conditions pour que $D$ soit composante du syst\`eme ${\cal L}_t(-D)$, c'est \`a dire pour que $2D$ soit dans le lieu fixe de ${\cal L}$. \ \\[2mm] On va ``prendre les conditions dont on a besoin sur le point $p_4$'', qui est un point multiple d'ordre trois, lorsque celui-ci approche de $D$. On pr\'el\`eve les conditions par l'op\'eration combinatoire suivante. Passer par un point de multiplicit\'e trois \'equivaut \`a contenir un gros point de taille trois de $\PP^2$. Un gros point de taille trois est un sch\'ema mon\^omial i.e. d\'efini par des \'equations mon\^omiales dans un bon syst\`eme de coordonn\'ees. On associe des objets combinatoires aux sch\'emas mon\^omiaux: des escaliers. Dans le cas du gros point de taille trois, l'escalier associ\'e est dessin\'e dans la figure ci-apr\`es. \begin{center} \setlength{\unitlength}{0.01250000in}% \begingroup\makeatletter\ifx\SetFigFont\undefined \def\ensuremath{\times}#1#2#3#4#5#6#7\relax{\def\ensuremath{\times}{#1#2#3#4#5#6}}% \expandafter\ensuremath{\times}\fmtname xxxxxx\relax \def\y{splain}% \ifx\ensuremath{\times}\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\ensuremath{\times}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\ensuremath{\times} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \begin{picture}(316,127)(13,685) \thinlines \put( 15,810){\line( 0,-1){ 90}} \put( 15,720){\line( 1, 0){ 75}} \put(120,810){\line( 0,-1){ 90}} \put(120,720){\line( 1, 0){ 90}} \put(240,810){\line( 0,-1){ 90}} \put(240,720){\line( 1, 0){ 87}} \put( 15,765){\line( 1, 0){ 15}} \put( 30,765){\line( 0,-1){ 15}} \put( 30,750){\line( 1, 0){ 15}} \put( 45,750){\line( 0,-1){ 15}} \put( 45,735){\line( 1, 0){ 15}} \put( 60,735){\line( 0,-1){ 15}} \put( 15,750){\line( 1, 0){ 15}} \put( 30,750){\line( 0,-1){ 30}} \put( 15,735){\line( 1, 0){ 30}} \put( 45,735){\line( 0,-1){ 15}} \put(120,750){\line( 1, 0){ 30}} \put(150,750){\line( 0,-1){ 15}} \put(150,735){\line(-1, 0){ 30}} \put(240,735){\line( 1, 0){ 30}} \put(270,735){\line( 0,-1){ 15}} \put(255,735){\line( 0,-1){ 12}} \put(135,750){\line( 0,-1){ 15}} \put(255,726){\line( 0,-1){ 6}} \put( 15,705){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}Escalier du }}} \put( 15,687){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}gros point.}}} \put(132,705){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}Suppression }}} \put(132,687){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}des lignes}}} \put(261,705){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}Escalier }}} \put(261,687){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}de $Z$}}} \end{picture} \end{center} On a vu que l'on avait besoin successivement d'une puis de trois conditions sur la droite $D$. On effectue alors la proc\'edure suivante. On supprime dans l'escalier les lignes de longueur un et trois, puis on ``pousse'' les cubes restant vers le bas. On obtient ainsi un nouvel escalier, associ\'e \`a un sous-sch\'ema mon\^omial $Z$ de $\PP^2$. Notons ${\cal C}$ le syst\`eme lin\'eaire form\'e des diviseurs $\Delta$ de degr\'e six contenant deux fois $D$ et pour lesquels $\Delta-2D$ est une courbe de degr\'e quatre contenant $Z$ (contenir $Z$ s'interpr\`ete g\'eom\'etriquement par le fait que $\Delta-2D$ a un ordre de contact d'ordre deux avec $D$ en $p_4(0)$). Le th\'eor\`eme \ref{theo} \'etablit l'in\'egalit\'e $dim {\cal L}_t \leq dim \cal C$. Puisque $\cal C$ est de dimension douze, on a bien $dim {\cal L}_t=12$. \ \\[2mm] Plus g\'en\'eralement, notre th\'eor\`eme s'int\'eresse \`a la dimension de certains syst\`emes lin\'eaires ${\cal L} _t$. On associe \`a ${\cal L}_t$ un syst\`eme $\ensuremath{\mathcal C}$ au moyen d'op\'erations combinatoires et on \'etablit l'in\'egalit\'e $dim {\cal L}_t \leq dim \ensuremath{\mathcal C}$. \ \\[2mm] En fait, notre th\'eor\`eme ne s'appliquera pas uniquement \`a $\PP^2$ et \`a la sp\'ecia\-li\-sa\-tion de gros points sur des droites, comme cela a \'et\'e le cas dans l'exemple. Il s'appliquera \`a toute vari\'et\'e projective irr\'eductible $X$ sur un corps al\-g\'e\-bri\-que\-ment clos de caract\'eristique quelconque, et \`a la sp\'ecialisation de sch\'emas mon\^omiaux en un point $p$ d'un diviseur de Weil irr\'eductible $D$ de $X$ en lequel $D$ et $X$ sont lisses. \ \\[2mm] La d\'emonstration consiste essentiellement \`a contr\^oler le syst\`eme lin\'eaire limite (th\'eor\`eme \ref{propfond}) lorsqu'on sp\'ecialise le sch\'ema mon\^omial , ce qui est obtenu par une \'etude diff\'erentielle. \ \\[2mm] Notre \'enonc\'e est tr\`es similaire \`a la m\'ethode d'Horace diff\'erentielle introduite dans [AH1]. D'un c\^ot\'e, le th\'eor\`eme pr\'esent\'e ici est plus g\'en\'eral puisque la m\'ethode d'Alexander et Hirschowitz ne permet d'utiliser qu'une seule tranche d'un sch\'ema mon\^omial (i.e. avec les notations du th\'eor\`eme \ref{theo}, ils se limitent au cas $r=1$). Mais d'un autre c\^ot\'e, Alexander et Hirschowitz s'autorisent \`a bouger simultan\'ement plusieurs sch\'emas mon\^omiaux, alors que la m\'ethode pr\'esent\'ee ici ne permet de sp\'ecialiser qu'un unique sch\'ema mon\^omial. Avec quelques adaptations dans les d\'emonstrations, il aurait \'et\'e possible de donner un \'enonc\'e qui englobe l'\'enonc\'e d'Alexander-Hirschowitz et le notre. Mais un tel \'enonc\'e serait beaucoup plus technique et ne donnerait pas lieu \`a de nouvelles appplications. Sous la forme pr\'esent\'ee dans ce travail, le th\'eor\`eme \'etait pressenti non seulement par Alexander et Hirscho\-witz, mais aussi par Simpson qui avait fait une conjecture en ce sens d\`es 1995. \ \\[2mm] Signalons aussi que Joe Harris a donn\'e une conf\'erence \`a Alghero en Juin dernier (1997) dans laquelle il a annonc\'e avoir obtenu avec Lucia Caporaso des r\'esultats similaires \`a ceux de cet article quand $X$ est de dimension deux, et quand la caract\'eristique du corps de base est nulle ou assez grande. \ \\[2mm] On trouvera des applications du th\'eor\`eme dans [E] o\`u on montre qu'il n'existe pas de courbe plane de degr\'e cent soixante quatorze contenant dix points singuliers d'ordre cinquante cinq (ce qui, en un sens \`a pr\'eciser, constitue le premier cas ``critique'' pour lequel la postulation de points singuliers ordinaires n'est pas connue). \ \\[2mm] Le plan de l'article est le suivant. Dans la section \ref{monom}, on explique le lien entre les sch\'emas mon\^omiaux et les escaliers. La section \ref{dechargeable} est une section technique d'alg\`ebre commutative utile pour la d\'emonstration du th\'eor\`eme. Le th\'eor\`eme est \'enonc\'e et d\'emontr\'e dans la section \ref{thm}. \section{Sch\'emas mon\^omiaux} \label{monom} \subsection{D\'efinition des sch\'emas mon\^omiaux} On appelle escalier une partie $E$ de $\NN^n$ dont le compl\'ementaire $C$ v\'erifie $C + \NN^n \ensuremath{\subset} C$. Dans la suite, nous ne manipulerons que des escaliers finis. On dira par abus de langage qu'un mon\^ome $m=x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$ de $k[[x_1,\dots,x_n]]$ est dans $E$ si $(a_1,a_2,\dots,a_n)$ est dans $E$. L'escalier $E$ d\'efinit un id\'eal $I^E$ de $k[[x_1,\dots,x_n]]$ qui est l'id\'eal engendr\'e par les mon\^omes hors de $E$. \ \\[2mm] Soit $p$ un point lisse d'une vari\'et\'e $X$ de dimension $n$. Le compl\'et\'e $\hat{O_p}$ de l'anneau local de $X$ en $p$ est isomorphe \`a l'anneau de s\'eries formelles $k[[x_1,\dots,x_n]]$. Le choix d'un isomorphisme induit un syst\`eme de coordonn\'ees locales en $p$, not\'e $\phi: Spec\; k[[x_1, \dots, x_n]] \ensuremath{\rightarrow} X$. Moyennant ce choix, tout sous-sch\'ema ponctuel de $X$ support\'e par $p$ peut \^etre vu comme un sous-sch\'ema de $Spec\; k[[x_1,\dots,x_n]]$. \begin{defi} Un sous-sch\'ema ponctuel $Z$ de $X$ support\'e par $p$ est dit mon\^o\-mial d'escalier $E$ si on peut choisir un isomorphisme entre $\hat {O_p} $ et $ k[[x_1,\dots,x_n]]$ tel que l'id\'eal d\'efinissant $Z$ dans $Spec\; k[[x_1,\dots,x_n]]$ soit $I^E$. On notera $X_{\phi}(E)$ le sch\'ema mon\^omial d\'efini par $\phi$ et $E$. \end{defi} \begin{ex} Les gros points de taille $m$ de $X$ sont les sch\'emas mon\^omiaux d'escalier $E_m$, avec $E_m=\{(a_1,a_2,\dots,a_n),\ a_1+a_2 +\dots +a_n<m\}$. \end{ex}\noindent \subsection{ D\'ecoupage d'un escalier en tranches. Suppression de tranches} \begin{defi} \normalfont Un escalier $E$ de $\NN^n$ d\'efinit une famille d'escaliers $T(E,k)$ de $\NN^{n}$ index\'ee par $\NN-\{0\}$: $$T(E,k):= \{(0,a_2,a_3,\dots,a_n) \mbox{ pour lesquels } (k-1,a_2,a_3,\dots,a_n) \in E\}$$ L'escalier $T(E,k)$ est appel\'e $k^{\mbox{\`eme}}$ tranche de $E$. \ \\[2mm] Un escalier fini peut \^etre caract\'eris\'e par une application hauteur $h_E$ de $\NN^{n-1}$ dans $\NN$ qui v\'erifie $h_E(a+b)\leq h_E(a)$ pour tout couple $(a,b)$ de $(\NN^{n-1})^2$: l'escalier d\'efini par $h_E$ est l'ensemble des $n$-uplets $(a_1,\dots,a_n)$ v\'erifiant $a_1 < h_E(a_2,\dots,a_n)$. \ \\[2mm] Pour un escalier $E$ d\'efini par une fonction $h_E$ et un entier $n_i>0$, on appelle escalier r\'esiduel apr\`es suppression de la tranche $n_i$ l'escalier $S(E,n_i)$ d\'efini par la fonction hauteur $h_{S(E,n_i)}$: \begin{eqnarray*} h_{S(E,n_i)}(a_2,\dots,a_n)&=&h_E(a_2,\dots,a_n)\mbox{ si } n_i > h_E(a_2,\dots,a_n) \\ &=&h_E(a_2,\dots,a_n)-1 \mbox{ si } n_i \leq h_E(a_2,\dots,a_n) \end{eqnarray*} \hspace{4cm} \setlength{\unitlength}{0.00083300in}% \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(2274,1856)(6889,-1605) \thicklines \put(6901,-699){\line( 1,-1){187.500}} \put(7089,-886){\line( 0,-1){188}} \put(7464,-1074){\line( 0, 1){188}} \put(7464,-886){\line( 1, 1){187}} \put(7464,-886){\line( 0, 1){187}} \put(7464,-699){\line( 1, 1){187.500}} \put(7089,-886){\line( 1, 1){187}} \put(7276,-699){\line( 0,-1){187}} \put(7089,-511){\line( 1,-1){375}} \put(7276,-699){\line( 0, 1){188}} \put(7276,-511){\line( 1, 1){187.500}} \put(7089,-324){\line( 1,-1){375}} \put(7089,-136){\line( 1,-1){187.500}} \put(7276,-324){\line( 1, 1){188}} \put(7089, 51){\line( 1,-1){187}} \put(7276,-136){\line( 1, 1){187.500}} \put(7276,-136){\line( 0,-1){375}} \put(7089,-324){\line( 0,-1){187}} \put(7089,-511){\line( 1,-1){375}} \put(7464,-886){\line( 0, 1){187}} \put(7464,-699){\line(-1, 1){375}} \put(7464,-699){\line( 1, 1){187.500}} \put(7651,-511){\line( 0,-1){188}} \put(7651,-699){\line(-1,-1){187}} \put(7276,-511){\line( 1, 1){187.500}} \put(7464,-324){\line( 1,-1){187}} \put(7651,-511){\line(-1,-1){187.500}} \put(7464,-699){\line(-1, 1){188}} \put(6901,-699){\line( 1, 1){188}} \put(7089,-511){\line( 0, 1){562}} \put(7089, 51){\line( 1, 1){187.500}} \put(7276,239){\line( 1,-1){188}} \put(7464, 51){\line( 0,-1){375}} \put(7464,-324){\line( 1,-1){187}} \put(7651,-511){\line( 0,-1){375}} \put(7651,-886){\line(-1,-1){187.500}} \put(7464,-1074){\line(-1, 1){188}} \put(7276,-886){\line(-1,-1){187.500}} \put(7089,-1074){\line(-1, 1){188}} \put(6901,-886){\line( 0, 1){187}} \put(7276,-511){\line( 0,-1){188}} \put(8401,-1589){\makebox(0,0)[lb]{{de la deuxi\`eme tranche}}} \put(7464,-699){\line( 0,-1){187}} \put(8401,-699){\line( 1, 1){188}} \put(8589,-511){\line( 0, 1){375}} \put(8589,-136){\line( 1, 1){187}} \put(8776, 51){\line( 1,-1){187.500}} \put(8964,-136){\line( 0,-1){375}} \put(8964,-511){\line( 0, 1){ 0}} \put(8964,-511){\line( 1,-1){187.500}} \put(9151,-699){\line( 0,-1){187}} \put(9151,-886){\line(-1,-1){187.500}} \put(8964,-1074){\line(-1, 1){188}} \put(8776,-886){\line(-1,-1){187.500}} \put(8589,-1074){\line(-1, 1){188}} \put(8401,-886){\line( 0, 1){187}} \put(8401,-699){\line( 1,-1){187.500}} \put(8589,-886){\line( 0,-1){188}} \put(8589,-886){\line( 1, 1){187}} \put(8776,-699){\line( 0,-1){187}} \put(8776,-699){\line( 1,-1){187.500}} \put(8964,-886){\line( 1, 1){187}} \put(8964,-1074){\line( 0, 1){188}} \put(8589,-511){\line( 1,-1){187.500}} \put(8776,-699){\line( 1, 1){188}} \put(8589,-136){\line( 1,-1){187.500}} \put(8776,-324){\line( 1, 1){188}} \put(8776,-324){\line( 0,-1){375}} \put(8589,-324){\line( 1,-1){187}} \put(8776,-511){\line( 1, 1){187.500}} \put(8401,-1449){\makebox(0,0)[lb]{Escalier apr\`es suppression}} \put(6901,-1449){\makebox(0,0)[lb]{Escalier }} \end{picture} \ \\[2mm] Si $(n_1,n_2,\dots,n_r)$ est un $r$-uplet d'entiers v\'erifiant $n_1>n_2 \dots >n_r > 0$, on d\'efinit l'escalier $S(E,n_1,\dots, n_r)$ obtenu \`a partir de $E$ par suppression des tranches $n_i$ r\'ecursivement: $S(E,n_1,\dots,n_r):=S(S(E,n_1\dots,n_{r-1}),n_r)$. \end{defi} \noindent \section{Id\'eaux et transporteurs} \label{dechargeable} Dans la section pr\'ec\'edente, nous avons d\'efini un id\'eal $I^E$ dans $k[[x_1,\dots,x_n]]$, qui correspond g\'eom\'etriquement \`a un sch\'ema ponctuel. Consid\'erons le morphisme de translation $T$: \begin{eqnarray*} T: k[[x_1,\dots,x_n]] &\ensuremath{\rightarrow} & k[[x_1,\dots,x_n]]\otimes k[[t]]\\ x_1&\mapsto& x_1\otimes 1 -1 \otimes t\\ x_i&\mapsto& x_i\otimes 1 \mbox{ si $i>1$} \end{eqnarray*} L'id\'eal $$J(E):=T(I^E)k[[x_1,\dots,x_n]] \otimes k[[t]]$$ d\'efinit une famille plate de sous-sch\'emas de $Spec\; k[[x_1,\dots,x_n]]$ param\'etr\'ee par $Spec\; k[[t]]$ qui correspond g\'eom\'etriquement \`a une translation du sch\'ema ponctuel dans la direction $x_1$. \ \\[2mm] Au cours de la d\'emonstration du th\'eor\`eme, nous serons grosso modo amen\'es \`a effectuer les calculs suivants: partant de $J_1:=J(E)$, d\'eterminer $J_2:=(J_1:x_1)$, $J_3:=(J_2:x_1)$ \dots On aimerait en outre que tous les $J_i$ soient de la forme $J(F_i)$ pour un escalier $F_i$ de sorte que les id\'eaux soient faciles \`a d\'ecrire et \`a manipuler via leur escalier. Ce n'est malheureusement pas le cas. Il est n\'eanmoins possible de donner une notion d'id\'eal associ\'e \`a un escalier de sorte que tous les id\'eaux soient contr\^ol\'es par le fait que ce sont des id\'eaux associ\'es \`a un escalier. C'est l'objet de la d\'efinition suivante. \ \\[2mm] Pour des raisons techniques, nous ne travaillerons pas dans $k[[x_1,\dots,x_n]] \otimes k[[t]]$, mais dans $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[[t]]/t^q$ pour diff\'erents $s$ et diff\'erents $q$ (o\`u $\goth m$ d\'esigne l'id\'eal maximal de $k[[x_1,\dots,x_n]]$ et $s$ et $q$ sont des entiers). Le probl\`eme reste n\'eanmoins le m\^eme, \`a savoir contr\^oler des calculs de transporteurs \`a l'aide d'escaliers. \ \\[2mm] On notera \begin{itemize} \item $r_{qp}^{su}: k[t]/t^{q}\otimes k[[x_1,\dots,x_n]]/\goth m^s \ensuremath{\rightarrow} k[t]/t^{p}\otimes k[[x_1,\dots,x_n]]/\goth m^u$ la projection naturelle, o\`u $p,q,s,u$ sont quatre entiers v\'erifiant$0< p\leq q$ et $0< u \leq s$ \item $J(E,q,s)$ la projection de $J(E)$ dans $k[t]/t^{q}\otimes k[[x_1,\dots,x_n]]/\goth m^s$ \item $I^E$ l'id\'eal de $k[t]/t^{q}\otimes k[[x_1,\dots,x_n]]/\goth m^s$ engendr\'e par les mon\^omes hors de $E$. \end{itemize} \begin{defi} \label{def ideal d'esc E} Soient $q\geq 1$ et $s\geq 1$ deux entiers, et $E$ un escalier de $\NN^n$. Un id\'eal $J$ de $k[t]/t^{q}\otimes k[[x_1,\dots,x_n]]/\goth m^s$ est dit id\'eal d'escalier $E$ s'il v\'erifie: \begin{itemize} \item $J=I^E$ si $q=1$ \item si $q>1$ \begin{itemize} \item $J\ensuremath{\subset} I^{T(E,q)}$ \item pour tout couple $(p,u)$ avec $0<p<q$, et $0<u <s$, $r_{qp}^{su}(J:x_1)$ est un id\'eal d'escalier $S(E,q)$ dans $k[t]/t^p\otimes k[[x_1,\dots,x_n]]/\goth m^{u}$. \end{itemize} \end{itemize} \end{defi} \noindent Dans cette d\'efinition, les deux premi\`eres propri\'et\'es sont les propri\'et\'es vou\-lues pour un id\'eal d'escalier $E$ tandis que la troisi\`eme nous assure que la notion est stable par calcul de transporteurs. \ \\[2mm] L'id\'eal d'escalier $E$ que nous int\'eresse est le suivant: \begin{prop} \label{Jd'escE} \ L'id\'eal $J(E,q,s)$ de $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^q$ est un id\'eal d'es\-ca\-lier $E$. \end{prop} \noindent Le reste de la section est consacr\'e \`a la d\'emonstration de cette proposition. Commen\c cons par le faire dans le cas $n=1$. Notons $E_h$ l'escalier de $\NN$ contenant les \'el\'ements inf\'erieurs strictement \`a $h$. Puisque tout escalier de $\NN$ est de la forme $E_h$ pour un certain $h$, la proposition pour $n=1$ dit que l'id\'eal $((x_1-t)^h)$ de $k[[x_1]]/x_1^s\otimes k[t]/t^q$ est un id\'eal d'escalier $E_h$. Pour v\'erifier ce fait, la d\'efinition \ref{def ideal d'esc E} nous invite \`a effectuer des calculs de transporteurs et des restrictions. Lors des calculs, les id\'eaux successifs apparaissant se ressemblent au sens o\`u ils admettent tous des syst\`emes de g\'en\'erateurs similaires. Nous introduisons dans la prochaine d\'efinition la notion d'id\'eaux d\'echargeables de hauteur $H$, qui sont des id\'eaux admettant un ``bon'' syst\`eme de g\'en\'erateurs (et bien \'evidemment, tous les id\'eaux apparaissant dans les calculs sont des id\'eaux d\'echargeables). Et la propri\'et\'e fondamentale est que tout id\'eal d\'echargeable de hauteur $H$ est un id\'eal d'escalier $E_H$. \ \\[2mm] La raison pour laquelle nous avons introduit la notion d'id\'eal d'escalier $E$ alors que finalement nous travaillons dans une classe d'id\'eaux plus petite, \`a savoir la classe des id\'eaux d\'echargeables est la suivante: lors de la d\'emonstration du th\'eor\`eme, les propri\'et\'es qui nous int\'eresseront vraiment pour un id\'eal sont celles qui en font un id\'eal d'escalier $E$. On a donc mis en \'evidence ces propri\'et\'es dans une d\'efinition. N\'eanmoins, pour montrer que $J(E,q,s)$ est un id\'eal d'escalier $E$, les calculs sont plus commodes dans une classe d'id\'eaux plus petite (les d\'echargeables) dans laquelle les id\'eaux sont contr\^ol\'es par un syst\`eme de g\'en\'erateurs. \ \\[2mm] En tant que $k$-espace vectoriel, $k[[x_1]]/x_1^s\otimes k[t]/t^q$ s'identifie au sous-espace vectoriel de $k[x_1,t]$ form\'e par les polyn\^omes de degr\'e en $x_1$ plus petit que $s$ et de degr\'e en $t$ plus petit que $q$. On dit qu'un \'el\'ement $x_1^{\beta}$ divise un \'el\'ement $Q$ de $k[[x_1]]/x_1^s\otimes k[t]/t^q$, et on \'ecrira $e=\frac{Q}{x_1^{\beta}}$ si, moyennant l'identification pr\'ec\'edente, $Q$ est une combinaison lin\'eaire $\sum \lambda_i x_1^{a_i}t^{b_i}$ de mon\^omes o\`u chaque $a_i$ est plus grand que $\beta$, et $e=\sum \lambda_i x_1^{a_i-\beta}t^{b_i}$ \begin{defi} Un id\'eal $I$ de $k[[x_1]]/x_1^s \otimes k[t]/t^q$ est dit d\'echargeable de hauteur $H$ s'il est engendr\'e par des \'el\'ements $(e_1,\dots,e_r)$ avec \begin{itemize} \item $e_1=\frac{(x_1-t)^h}{x_1^{\beta_1}}$ pour des entiers $h$ et $\beta_1$ v\'erifiant $H=h-\beta_1$, et ($\beta_1=0$ si $q>H$) \item pour $i\geq 2$, $e_i=\frac{t^{\alpha_i}(x_1-t)^h}{x_1^{\beta_i}}$ avec: $\alpha_i \geq 1$ et, $\forall p\leq q, \ x_1^{q-p+1}$ divise $r_{qp}^{ss}(e_i)$. \end{itemize} \end{defi} \noindent \begin{prop} Soit $I=(e_1,\dots,e_r)$ un id\'eal d\'echargeable de hauteur $H$ de $k[[x_1]]/x_1^s \otimes k[t]/t^q$. Si $q\leq H$, alors $(I:x_1)=(x_1^{s-1},\frac{e_1}{x_1}, \frac{e_2}{x_1}, \dots,\frac{e_r}{x_1})$. Si $q>H$, alors $(I:x_1)=(x_1^{s-1},e_1,\frac{t^{q-h}e_1}{x_1},\frac{e_2}{x_1}, \frac{e_3}{x_1}, \dots,\frac{e_r}{x_1})$. \end{prop} \noindent \textit{D\'emonstration: \\ Le cas $q\leq H$}: $e_1$, vu comme polyn\^ome en $x_1$, admet comme terme constant un multiple de $t^H$. Donc ce terme est nul et $e_1$ est bien divisible par $x_1$. Les \'el\'ements $e_2,\dots,e_r$ sont divisibles par $x_1$ par d\'efinition des id\'eaux d\'echargeables. L'id\'eal $(x_1^{s-1},\frac{e_1}{x_1}, \frac{e_2}{x_1}, \dots,\frac{e_r}{x_1})$ est donc bien d\'efini. L'inclusion $(I:x_1) \supset (x_1^{s-1},\frac{e_1}{x_1}, \frac{e_2}{x_1}, \dots,\frac{e_r}{x_1})$ \'etant \'evi\-den\-te, il nous reste \`a voir qu'un \'el\'ement $m$ de \!$(I:x_1)$ est dans $(x_1^{s-1}\!\!, \frac{e_1}{x_1}, \frac{e_2}{x_1}, \dots,\frac{e_r}{x_1})$. L'\'el\'ement $x_1m$, qui est dans $I$, s'\'ecrit $\sum \lambda _i e_i$ o\`u les $\lambda_i$ sont des \'el\'ements de $k[[x_1]]/x_1^s \otimes k[t]/t^q$. D'o\`u la relation $$x_1(m-\sum \lambda_i \frac{e_i}{x_1})=0$$ Le noyau de la multiplication par $x_1$ \'etant l'id\'eal $(x_1^{s-1})$, $m$ est bien dans l'id\'eal $(x_1^{s-1},\frac{e_1}{x_1},\frac{e_2}{x_1}, \frac{e_3}{x_1}, \dots,\frac{e_r}{x_1})$ \ \\[2mm] \textit{ Le cas $q>H$}: commme pr\'ec\'edemment, la seule chose non imm\'ediate est qu'un \'el\'ement $m$ de $(I:x_1)$ est dans l'id\'eal $(x_1^{s-1},e_1,\frac{t^{q-h}e_1}{x_1},\frac{e_2}{x_1}, \frac{e_3}{x_1}, \dots,\frac{e_r}{x_1})$. Toujours comme pr\'ec\'edemment, on a l'\'egalit\'e \begin{eqnarray} \label{eq7} x_1.m=\sum \lambda_i {e_i}. \end{eqnarray} En utilisant l'identification expliqu\'ee plus haut, $\lambda_1$ peut \^etre vu comme un \'el\'ement de $k[x_1,t]$ et on peut \'ecrire la division $$\lambda_1=x_1.Q+R$$ o\`u $R$ est un \'el\'ement de $k[t]$. Cette expression et l'expression (\ref{eq7}) fournissent l'\'egalit\'e: $$x_1(m-\sum_{i\geq 2}\lambda_i \frac{e_i}{x_1}-Qe_1)=R e_1.$$ Donc $x_1$ divise $R e_1$, ce qui n'est possible que si $R$ est un multiple de $t^{q-h}$: $R=\mu t^{q-h}$. Finalement, l'\'egalit\'e $$ x_1(m-\sum_{i\geq 2} \lambda_i \frac{e_i}{x_1}- Qe_1 - {\mu} \frac{t^{q-h}e_1}{x_1})=0 $$ et le fait que le noyau de la multiplication par $x_1$ est l'id\'eal $(x_1^{s-1})$, nous assurent que $m$ est dans l'id\'eal $(x_1^{s-1},e_1,\frac{t^{q-h}e_1}{x_1},\frac{e_2}{x_1}, \frac{e_3}{x_1}, \dots,\frac{e_r}{x_1})$. \begin{flushright}\rule{2mm}{2mm \begin{coro} \label{calcul transp} Si $I$ est un id\'eal d\'echargeable de hauteur $H$ de $k[[x_1]]/x_1^s \otimes k[t]/t^q$ et si $q \leq H$, alors pour tout couple $(p,u)$ v\'erifiant $p<q$ et $u<s$, $r_{qp}^{s u}(I:x_1)$ est un id\'eal d\'echargeable de hauteur $H-1$ de $k[[x_1]]/x_1^u \otimes k[t]/t^p$. \\ Si $I$ est d\'echargeable de hauteur $H$ et si $q > H$, $r_{qp}^{su}(I:x_1)$ est un id\'eal d\'echargeable de hauteur $H$ de $k[[x_1]]/x_1^u \otimes k[t]/t^p$. \end{coro} \noindent \textit{D\'emonstration:} si $q\leq H$, $r_{qp}^{su}(I:x_1)$ admet $(e'_1,\dots,e'_r)$ comme g\'en\'erateurs avec $e'_i=r_{qp}^{su}(\frac{e_i}{x_1})$. L'\'el\'ement $e'_1$ v\'erifie trivialement la premi\`ere condition demand\'ee aux g\'en\'erateurs d'un id\'eal d\'echargeable de hauteur $H-1$. Pour la deuxi\`eme condition, il faut voir que pour tout $p'\leq p$ et $i\geq 2$, $x_1^{p-p'+1}$ divise $r_{pp'}^{uu}\circ r_{qp}^{su}(\frac{e_i}{x_1})=r_{qp'}^{su}(\frac{e_i}{x_1})$. Il suffit pour cela de voir que $x_1^{p-p'+2}$ divise $r_{qp'}^{ss}(e_i)$. Or, par hypoth\`ese, $I=(e_1,\dots,e_r)$ est un id\'eal d\'echargeable donc $x_1^{q-p'+1}$ divise $r_{qp'}^{ss}(e_1)$, et $q-p'+1\geq p-p'+2$. \\ Dans le cas $q>H$, $r_{qp}^{su}(I:x_1)$ est de la forme $(e'_1,\dots,e'_{r+1})$ avec $e'_1=r_{qp}^{su}(e_1)$, $e'_i=r_{qp}^{su}(e_i/x_1)$ pour $2\leq i \leq r$ et $e'_{r+1}=r_{qp}^{su}(\frac{t^{q-h}(x_1-t)^h}{x_1})$. Toutes les v\'erifications, sauf une, sont les m\^emes qu'au cas pr\'ec\'edent: il nous faut en outre montrer que pour tout $p'<p$, $x_1^{p-p'+1}$ divise $r_{pp'}^{uu}(e'_{r+1})$. Ceci est vrai car le coefficient en $x_1^k$ de $r_{pp'}^{uu}(e'_{r+1})$ est un multiple de $t^{q-1-k}$: si $k$ est inf\'erieur ou \'egal \`a $p-p'$, il est strictement plus petit que $q-p'$, l'exposant $q-1-k$ de $t$ est strictement plus grand que $p'-1$ donc $t^{q-1-k}$ est nul dans $k[[x_1]]/x_1^u \otimes k[t]/t^{p'}$. \begin{flushright}\rule{2mm}{2mm \begin{coro} Si $I=(e_1,\dots,e_r)$ est un id\'eal de $k[[x_1]]/x_1^s\otimes k[t]/t^q$ d\'echar\-gea\-ble de hauteur $H$, alors $I$ est un id\'eal d'escalier $E_H$. \end{coro} \noindent \textit{D\'emonstration}: par r\'ecurrence sur $q$. Pour $q=1$, tous les termes $e_i$ avec $i\geq 2$ d'un id\'eal d\'echargeable $I=(e_1,\dots,e_r)$ sont nuls car ils sont de la forme $\frac{t^{\alpha_i}(x_1-t)^h}{x_1^{\beta_i}}$ avec $\alpha_i\geq 1$. Donc $I=(e_1)$ et $e_1=\frac{(x_1-t)^h}{x_1^{\beta_1}}=\frac{x_1^h}{x_1^{\beta_1}} =x_1^{H}$. On a bien $I=I^{E_H}$. \ \\[2mm] Pour $q>1$, il faut voir que $I$ est inclus dans $I^{T(E_H,q)}$ et que, pour $p<q$ et $u<s$, $r_{qp}^{su}(I:x_1)$ est un id\'eal d'escalier $S(E_H,q)$. \\ Si $q$ est plus grand que $H$, $I^{T(E_H,q)}$ est l'id\'eal unit\'e donc la premi\`ere condition est trivialement v\'erifi\'ee. Dans ce cas, $S(E_H,q)=E_H$. D'apr\`es la proposition \ref{calcul transp}, $r_{qp}^{su}(I:x_1)$ est un id\'eal d\'echargeable de hauteur $H$, donc c'est un id\'eal d'escalier $E_H$ par hypoth\`ese de r\'ecurrence. \\ Si $q$ est inf\'erieur ou \'egal \`a $H$, la premi\`ere condition dit que $I$ est inclus dans l'id\'eal $(x_1)$. V\'erifions que c'est le cas pour chacun des g\'en\'erateurs de $I$. C'est vrai pour les \'el\'ements $e_2,\dots,e_r$ par d\'efinition des g\'en\'erateurs d'un id\'eal d\'echargeable. C'est \'egalement vrai pour $e_1= \frac {(x_1-t)^h}{x_1^{\beta_1}}$ car son terme constant est un multiple de $t^H$, donc est nul. \\ Pour la deuxi\`eme condition, il faut voir que $r_{qp}^{su}(I:x_1)$ est un id\'eal d'escalier $S(E_H,q)=E_{H-1}$. Or, d'apr\`es la proposition \ref{calcul transp}, $r_{qp}^{su}(I:x_1)$ est un id\'eal d\'echargeable de hauteur $H-1$. C'est donc aussi un id\'eal d'escalier $E_{H-1}$ par l'hypoth\`ese de r\'ecurrence. \begin{flushright}\rule{2mm}{2mm \begin{coro} \label{J d'esc E cas n=1} Soit $E_h \ensuremath{\subset} \NN$ un escalier. L'id\'eal $J(E_h,s,q)$ de $k[[x_1]]/x_1^s\otimes k[t]/t^q$ est un id\'eal d'escalier $E_h$. \end{coro} \noindent \textit{D\'emonstration}: $J(E_h,s,q)=((x_1-t)^h)$ est trivialement un id\'eal d\'echar\-gea\-ble de hauteur $h$. C'est donc un id\'eal d'escalier $E_h$ d'apr\`es le corollaire pr\'ec\'edent. \begin{flushright}\rule{2mm}{2mm Soit $E$ un escalier de $\NN^n$. On va maintenant montrer pour $n$ quelconque que $J(E,s,q)$ est un id\'eal d'escalier $E$ de $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^q$ en se ramenant au cas $n=1$. Identifions pour cela ensemblistement l'anneau $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^q$ au produit $$ \prod_{(\alpha_2,\dots,\alpha_n)\; t.q.\ s-\alpha_2-\dots-\alpha_n\geq 1}k[[x_1]]/x_1^{s-\alpha_2-\dots-\alpha_n}\otimes k[t]/t^q $$ o\`u l'identification envoie un terme $m$ de la composante d'indice $(\alpha_2,\dots,\alpha_n)$ sur le produit $mx_2^{\alpha_2}x_3^{\alpha_3}\dots x_n^{\alpha_n}$. \begin{lm} \label{J est gradue} Soient $E$ un escalier fini de $\NN^n$ d\'efini par une fonction hauteur $h_E$ et $I_{\alpha_2,\dots,\alpha_n}$ l'id\'eal de $k[[x_1]]/x_1^{s-\alpha_2-\dots-\alpha_n}\otimes k[t]/t^q$ engendr\'e par $(x_1-t)^{h_E(\alpha_2,\dots,\alpha_n)}$. L'id\'eal $J(E,s,q)$ co\"\i ncide avec le produit $\prod_{\alpha_2,\dots,\alpha_n}I_{\alpha_2,\dots,\alpha_n}$ \end{lm} \noindent \textit{D\'emonstration}: puisque chaque $I_{\alpha_2,\dots,\alpha_n}$ est inclus dans $J(E,s,q)$, on a l'inclusion $$ \prod_{\alpha_2,\dots,\alpha_n}I_{\alpha_2,\dots,\alpha_n} \ensuremath{\subset} J(E,s,q)$$ Les \'el\'ements $(x_1-t)^{h_E(\alpha_2,\dots,\alpha_n)} x_2^{\alpha_2}.x_3^{\alpha_3}.\dots.x_n^{\alpha_n} $ engendrent $J(E,s,q)$ et sont dans $ \prod_{\alpha_2,\dots,\alpha_n}I_{\alpha_2,\dots,\alpha_n}$ . Il suffit donc pour montrer l'inclusion inverse de v\'erifier que le produit $\prod_{\alpha_2,\dots,\alpha_n}I_{\alpha_2,\dots,\alpha_n}$ est un id\'eal de $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^q$. Ce produit est clairement un $k[t]/t^q$-module. Utilisant alors la lin\'earit\'e, il suffit de v\'erifier que le produit d'un \'el\'ement $e_0$ de $I_{\alpha_2^0,\dots,\alpha_n^0}$ et d'un mon\^ome $m=x_1^{\beta_1}.x_2^{\beta_2}.x_3^{\beta_3}.\dots.x_n^{\beta_n}$ est dans $\prod_{\alpha_2,\dots,\alpha_n}I_{\alpha_2,\dots,\alpha_n}$. Par d\'efinition de $I_{\alpha_2^0,\dots,\alpha_n^0}$, $$e_0=(x_1-t)^{h_E(\alpha_2^0,\dots,\alpha_n^0)}. x_2^{\alpha_2^0}.x_3^{\alpha_3^0}.\dots.x_n^{\alpha_n^0}.\mu$$ o\`u $\mu$ est un \'el\'ement de $k[[x_1]]/x_1^s\otimes k[t]/t^q$. On a donc $$m.e_0= (x_1-t)^{h_E(\alpha_2^0,\dots,\alpha_n^0)+\beta_1}. x_2^{\alpha_2^0+\beta_2}.x_3^{\alpha_3^0+\beta_3}.\dots. x_n^{\alpha_n^0+\beta_n}.\mu$$ Le terme $me_0$ est donc aussi un multiple de $$(x_1-t)^{h_E(\alpha_2^0+\beta_2,\dots,\alpha_n^0+\beta_n)}. x_2^{\alpha_2^0+\beta_2}.x_3^{\alpha_3^0+\beta_3}.\dots. x_n^{\alpha_n^0+\beta_n}$$ en vertu de l'in\'egalit\'e $$h_E(\alpha_2^0+\beta_2, \alpha_3^0+\beta_3,\dots, \alpha_n^0+\beta_n)\leq h_E(\alpha_2^0, \alpha_3^0,\dots, \alpha_n^0).$$ Par suite $m.e_0$ est dans $I_{\alpha_2^0+\beta_2, \dots, \alpha_n^0+\beta_n}$. \begin{flushright}\rule{2mm}{2mm \begin{lm} \label{etre d'esc E est une prop graduee} Soit $E$ un escalier de $\NN^n$ donn\'e par une fonction hauteur $h_E$. Soit $J$ un id\'eal de $k[[x_1,\dots,x_n]] /\goth m^s \otimes k[t]/t^q$ tel que $J= \prod_{\alpha_2,\dots,\alpha_n}J_{\alpha_2,\dots,\alpha_n}$, o\`u chaque $J_{\alpha_2,\dots,\alpha_n}$ est un id\'eal de $k[[x_1]] /x_1^{s-\alpha _2-\dots-\alpha_n} \otimes k[t]/t^q$ d'escalier $E_{h_E(\alpha_2,\dots,\alpha_n)}$ . Alors $J$ est un id\'eal d'escalier $E$. \end{lm} \noindent \textit{D\'emonstration}: appelons id\'eal gradu\'e de $k[[x_1,\dots,x_n]] /\goth m^s \otimes k[t]/t^q$ un id\'eal $K$ qui s'\'ecrit comme produit d'id\'eaux $K=\prod K_{\alpha_2, \dots,\alpha_n}$. On dira que les $K_{\alpha_2,\dots,\alpha_n}$ sont les parties gradu\'ees de $K$. Deux id\'eaux gradu\'es $L$ et $K$ v\'erifient $L\ensuremath{\subset} K$ si et seulement si pour tout $(\alpha_2,\dots,\alpha_n)$, $L_{\alpha_2,\dots,\alpha_n} \ensuremath{\subset} K_{\alpha_2,\dots,\alpha_n}$. De plus, si $K$ est gradu\'e, les id\'eaux $(K:x_1)$ et $r_{qp}^{su}(K)$ sont gradu\'es et, plus pr\'ecis\'ement, $(K:x_1)=\prod (K_{\alpha_2,\dots,\alpha_n}:x_1)$ et $r_{qp}^{su}(K)=\prod r_{qp}^{su}(K_{\alpha_2,\dots,\alpha_n})$. En d\'efinitive, dans la d\'efinition \ref{def ideal d'esc E}, toutes les v\'erifications \`a faire concernent des id\'eaux gradu\'es, et les calculs de transporteur et les restrictions respectent la graduation. Donc le fait d'\^etre un id\'eal d'escalier $E$ se v\'erifie sur chaque partie gradu\'ee. \begin{flushright}\rule{2mm}{2mm \noindent \textit{D\'emonstration de la proposition \ref{Jd'escE}}: d'apr\`es le lemme \ref{J est gradue}, l'id\'eal $J(E,s,q)$ est un produit d'id\'eaux $I_{\alpha_2,\dots,\alpha_n}$. Chacun de ces id\'eaux $I_{\alpha_2,\dots,\alpha_n}$ est un id\'eal d'escalier $E_ {h(\alpha_2,\dots,\alpha_n)}$ d'apr\`es le corollaire \ref{J d'esc E cas n=1}. On conclut enfin avec le lemme \ref{etre d'esc E est une prop graduee} que $J(E,s,q)$ est un id\'eal d'escalier $E$. \begin{flushright}\rule{2mm}{2mm \section{Le th\'eor\`eme} \label{thm} Le th\'eor\`eme traite de syst\`emes lin\'eaires. Comme dans l'exemple introductif, les syst\`emes consid\'er\'es seront des sous-syst\`emes ${\cal L}_t$ d'un syst\`eme lin\'eaire ${\cal L}$; les diviseurs de ${\cal L}_t$ seront des diviseurs de ${\cal L}$ qui contiennent un sch\'ema mon\^omial $X(t)$ variant avec le temps $t$. Au temps $t=0$, le sch\'ema $X(0)$ se sp\'ecialise sur un diviseur de Weil irr\'eductible $D$. Le trajet du sch\'ema mon\^omial sera une translation relativement \`a un syst\`eme de coordonn\'ees locales ``compatible'' avec le diviseur $D$. Expliquons ce que cela signifie. \ \\[2mm] Soient $X$ une vari\'et\'e projective irr\'eductible de dimension $n$, $D$ une sous-vari\'et\'e irr\'eductible de $X$ de dimension $n-1$. Soit $p$ un point de $D$ en lequel $X$ et $D$ sont lisses. Choisissons une fois pour toutes un syst\`eme de coordonn\'ees locales $\phi:Spec\; k[[x_1,\dots,x_n]] \ensuremath{\rightarrow} X $ en $p$ de sorte que $D$ soit localement d\'efini par $x_1=0$. L'id\'eal $J(E)$ introduit au d\'ebut de la section pr\'ec\'edente d\'efinit une famille plate de sous-sch\'emas de $Spec\; k[[x_1,\dots,x_n]]$ param\'etr\'ee par $Spec\; k[[t]]$. On peut \'egalement voir cette famille comme une famille plate de sous-sch\'emas de $X$ moyennnant le morphisme de coordonn\'ees locales $\phi$. On note $X_{\phi}(E,t)$ la fibre g\'en\'erique de cette famille plate. La fibre sp\'eciale de cette famille est $X_{\phi}(E,0)=X_{\phi}(E)$. Cette famille plate est associ\'ee \`a un morphisme $Spec\; k[[t]]\ensuremath{\rightarrow} Hilb(X)$ qui correspond au trajet du sch\'ema mon\^omial d\'efini par la translation. \ \\[2mm] Soient ${\cal L}$ un syst\`eme lin\'eaire de diviseurs de Cartier sur $X$ et $Y$ un sous-sch\'ema de $X$. On note ${\cal L}(-Y)$ le sous-syst\`eme lin\'eaire de ${\cal L}$ form\'e par les diviseurs qui contiennent $Y$. Si $Y$ et $Z$ sont deux sous-sch\'emas de $X$, le produit des id\'eaux $I(Y)$ et $I(Z)$ de $Y$ et $Z$ d\'efinit un sous-sch\'ema de $X$ not\'e $Y+Z$. En particulier, ${\cal L}(-Y-Z)$ est bien d\'efini, m\^eme si $Y$ est un diviseur de $X$ et $Z$ un sous-sch\'ema de dimension z\'ero. \ \\[2mm] Notons $Z_k$ le sous-sch\'ema de $X$ d\'efini par le syst\`eme de coordonn\'ees locales $\phi$ et la tranche $T(E,k)$: $Z_k:=X_{\phi}(T(E,k))$. Les sch\'emas $Z_k$ sont inclus dans le diviseur $D$. \\ Les sch\'emas mon\^omiaux d'escalier $E$ s'organisent en une vari\'et\'e irr\'eductible [H] et on peut donc parler du sch\'ema g\'en\'erique d'escalier $E$, qu'on note $X(E)$. \begin{thm} \label{theo} Soient ${\cal L}$ un syst\`eme lin\'eaire sur $X$ et $n_1,n_2,\dots,n_r$ des entiers v\'erifiant $n_1>n_2>\dots>n_r>0$. Supposons que pour tout $i$ compris entre un et $r$, ${\cal L}(-(i-1) D-Z_{n_i})={\cal L}(-iD)$. Alors $$dim \; {\cal L}(-X(E)) \leq dim\; {\cal L}(-rD-X_{\phi} (\;S(E,n_1,\dots, n_r)\;))$$ \end{thm} \noindent Le th\'eor\`eme est une cons\'equence imm\'ediate du th\'eor\`eme suivant: \begin{thm} \label{propfond} Soient ${\cal L}$ un syst\`eme lin\'eaire sur $X$ et $n_1,n_2,\dots,n_r$ des entiers v\'erifiant $n_1>n_2>\dots>n_r>0$. Supposons que pour tout $i$ compris entre un et $r$, ${\cal L}(-(i-1) D-Z_{n_i})={\cal L}(-iD)$. Alors on a l'inclusion $$\lim_{t \ensuremath{\rightarrow} 0} {\cal L}(-X_{\phi}(E,t)) \ensuremath{\subset} {\cal L}(-rD-X_{\phi} (\;S(E,n_1,\dots, n_r)\;))$$ \end{thm} \noindent \textit{D\'emonstration du th\'eor\`eme \ref{theo}}: puisque $X_{\phi}(E,t)$ est une sp\'ecialisation de $X(E)$, on a par semi-continuit\'e $$dim\;{\cal L}(-X(E))\leq dim\; {\cal L}(-X_{\phi}(E,t)) $$ La limite \'etant par d\'efinition une limite dans une Grassmannienne, on a: $$ dim\;{\cal L}(-X_{\phi}(E,t)) = dim\;lim_{t \ensuremath{\rightarrow} 0}{\cal L}(-X_{\phi}(E,t)) $$ Enfin, la proposition \ref{propfond} implique: $$ dim\; lim_{t \ensuremath{\rightarrow} 0}{\cal L}(-X_{\phi}(E,t)) \leq dim\; {\cal L}(-rD-X_{\phi} (\;S(E,n_1,\dots, n_r)\;)) $$ Ces in\'egalit\'es mises bout \`a bout donnent l'in\'egalit\'e du th\'eor\`eme. \begin{flushright}\rule{2mm}{2mm \textit{D\'emonstration du th\'eor\`eme \ref{propfond}}: \ \\[2mm] Le syst\`eme lin\'eaire ${\cal L}$ est de la forme $I\!\!P(V)$ pour un fibr\'e en droites $F$ sur $X$ et un espace vectoriel $V$ de sections de $F$. Notons $n-1$ la dimension projective du syst\`eme lin\'eaire ${\cal L}(-X_{\phi}(E,t))$. Il existe un unique morphisme $$f: Spec\; k[[t]] \ensuremath{\rightarrow} \mathbb{G}(n,V)$$ qui envoie le point g\'en\'erique sur le point (non ferm\'e) de la grassmannienne param\'etrant le syst\`eme lin\'eaire ${\cal L}(-X_{\phi}(E,t))$. L'image du point sp\'ecial d\'efinit un sous-espace vectoriel $W$ de $V$ et, par d\'efinition, $I\!\!P(W)=\lim_{t \ensuremath{\rightarrow} 0} {\cal L}(-X_{\phi}(E,t))$ \ \\[2mm] Restreignons la base du fibr\'e $F$ \`a $Spec\; \hat O_{X,p}$, o\`u $\hat O_{X,p}$ est le compl\'et\'e de l'anneau local de $X$ en $p$. Au dessus de cette base, le faisceau localement libre $F$ est trivial et on peut en choisir un g\'en\'erateur local $g$. Une fois $g$ choisi, on peut r\'ealiser toute section de $F$ comme une fonction de $\hat O_{X,p}$. Le syst\`eme $\phi$ de coordonn\'ees locales en $p$ \'etant donn\'e, toute fonction de $\hat O_{X,p}$ s'identifie \`a un \'el\'ement de $k[[x_1,\dots,x_n]]$. On dispose donc d'un morphisme, injectif car $X$ est irr\'eductible: $$i:V \ensuremath{\rightarrow} k[[x_1,\dots,x_n]]$$ Notons $p_s$ la projection de $k[[x_1,\dots,x_n]]$ dans $k[[x_1,\dots,x_n]]/ \goth m^s$. Puisque $V$ est de dimension finie, le morphisme $$p_s \circ i: V \ensuremath{\rightarrow} k[[x_1,\dots,x_n]]/\goth m ^s$$ est \'egalement injectif pour $s$ assez grand. Un \'el\'ement $f$ de $V$ s'annule $n$ fois sur $D$ si et seulement si $i(f)$ est divisible par $x_1^n$. Toujours pour $s$ assez grand, $f$ s'annule $n$ fois sur $D$ si et seulement si $p_s\circ i(f)$ est un multiple de $x_1^n$. \ \\[2mm] Pour $q \geq 0$, notons $f_q$ la restriction du morphisme $f$ \`a $Spec\; k[t]/t^{q}$: $$f_q: Spec\; k[t]/t^{q} \ensuremath{\rightarrow} \mathbb{G}(n,V)$$ L'image inverse par $$f_q\ensuremath{\times} Id:Spec\; k[t]/t^{q} \ensuremath{\times} V \ensuremath{\rightarrow} \mathbb{G}(n,V)\ensuremath{\times} V$$ du fibr\'e universel au dessus de $\mathbb{G}(n,V)$ est un sous-fibr\'e $F_q$ de rang $n$ de $Spec\; k[t]/t^{q} \ensuremath{\times} V$. Expliquons comment associer un id\'eal $I(s_q,s)$ de $k[[x_1,\dots,x_n]]/\goth m^s\otimes k[t]/t^{q}$ \`a une section $s_q$ de $F_q$. \\ Toute section $s_q$ de $F_q$ est aussi une section de $Spec\; k[t]/t^{q} \ensuremath{\times} V$, et est d\'efinie par un morphisme de $Spec\; k[t]/t^{q}$ dans $V$. Par composition \`a droite avec le morphisme $p_s \circ i$, la section $s_q$ d\'efinit un morphisme $f(s_q,s)$: $$f(s_q,s):Spec\; k[t]/t^{q} \ensuremath{\rightarrow} k[[x_1,\dots,x_n]]/ \goth m^s.$$ Il existe un ferm\'e $U$ de $k[[x_1,\dots,x_n]]/ \goth m^s \ensuremath{\times} Spec\; k[[x_1,\dots,x_n]]/ \goth m^s$ dont la fibre au dessus d'un point $f$ est le sous-sch\'ema de $Spec\; k[[x_1,\dots,x_n]]/ \goth m^s$ d\'efini par l'id\'eal $(f)$. L'image inverse de $U$ par $f(s_q,s)\ensuremath{\times} Id$ est un ferm\'e $U(s_q,s)$ de $Spec\; k[t]/t^{q} \ensuremath{\times} Spec\; k[[x_1,\dots,x_n]]/ \goth m^s$. On note $I(s_q,s)$ l'id\'eal de $k[t]/t^{q} \otimes k[[x_1,\dots,x_n]]/ \goth m^s$ d\'efinissant $U(s_q,s)$. \ \\[2mm] La signification g\'eom\'etrique de $I(s_q,s)$ est la suivante. La section $s_q$ d\'efinit une famille de diviseurs de $X$ param\'etr\'ee par $Spec\; k[t]/t^{q}$, donc un sous-sch\'ema $Z$ de $Spec\; k[t]/t^{q} \ensuremath{\times} X$. La trace de $Z$ sur $$Spec\; k[t]/t^{q} \ensuremath{\times} Spec\; k[[x_1,\dots,x_n]]/\goth m^s$$ est un sous-sch\'ema d\'efini par l'id\'eal $I(s_q,s)$. \ \\[2mm] Le comportement par restriction des id\'eaux $I(s_q,s)$ est agr\'eable: si $p,q,s,u$ sont quatre entiers avec $q\geq p$, $s\geq u$, et si $s_p$ est la restriction de $s_q$ au dessus de $Spec\; k[t]/t^{p}$, alors $I(s_p,u)=r_{qp} ^{su}(I(s_q,s))$. \ \\[2mm] Pour montrer le th\'eor\`eme, il nous faut voir $(**)$ que pour toute section $s_1$ de $F_1$ au dessus du point ferm\'e et pour tout entier $s$ assez grand, $I(s_1,s)\ensuremath{\subset} x_1^r. I^{S(E,n_1,\dots,n_r)}$. \ \\[2mm] Toute section $s_1$ de $F_1$ au dessus du point ferm\'e est la restriction d'une section $s_{n_1}$ de $F_{n_1}$ au dessus de $Spec\; k[t]/t^{n_1}$. Notons $s_{n_i}$ la restriction de $s_{n_1}$ \`a $Spec\; k[t]/t^{n_i}$. \ \\[2mm] Montrons la proposition $(*)$ suivante, qui impliquera facilement $(**)$ et donc le th\'eor\`eme \ref{propfond}: pour $s$ assez grand, l'id\'eal $I(s_{n_i},s)$ est inclus dans un id\'eal $x_1^i. M(n_i,s)$, o\`u pour tout $p<n_i$ et $u<s$, $r_{n_ip}^{su}M(n_i,s)$ est un id\'eal d'escalier $S(E,n_1,\dots,n_i)$. \\ On proc\`ede par r\'ecurrence sur $i$. \ \\[2mm] Pour $i=1$, on peut dire informellement que $s_{n_1}$ est une famille de sections de $F$ param\'etr\'ee par un temps $t$ dans $Spec\; k[t]/t^{n_1}$, et que cette famille de sections s'annule ``\`a tout instant $t$ sur le translat\'e par $t$ dans la direction $x_1$ du sch\'ema $X_{\phi}(E)$''. Plus rigoureusement, on a l'inclusion \begin{eqnarray} \label{eq1} I(s_{n_1},s) \ensuremath{\subset} J(E,n_1,s) \end{eqnarray} De plus, $J(E,n_1,s)$ est un id\'eal d'escalier $E$ de $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^q$ d'apr\`es la proposition \ref{Jd'escE}, donc \begin{eqnarray} \label{eq2} J(E,n_1,s)\ensuremath{\subset} I^{Tr(E,n_1)} \end{eqnarray} Pour $s$ assez grand, les inclusions (\ref{eq1}) et (\ref{eq2}) montrent que $s_{n_1}$ d\'efinit une famille de sections de $F$ s'annulant \`a tout instant sur $Z_{n_1}$. Donc, par hypoth\`ese, c'est \'egalement une famille de sections s'annulant sur $D$. Remarquons qu'\`a priori, l'hypoth\`ese dit qu'une section de $F$ qui s'annule sur $Z_{n_1}$ s'annule sur $D$ mais ne dit rien pour les familles de sections. Cependant, si on note $W_{n_1}$ le lieu sch\'ematique dans $V$ form\'e par les sections de $F$ qui s'annulent sur $Z_{n_1}$ et $W_D$ le lieu sch\'ematique form\'e par les sections qui s'annulent sur $D$, $W_{n_1}$ et $W_D$ sont r\'eduits car ce sont des espaces vectoriels. En particulier, l'inclusion ensembliste de $W_{n_1}$ dans $W_D$, v\'erifi\'ee par hypoth\`ese, implique l'inclusion sch\'ematique. Les familles de sections de $F$ param\'etr\'ees par une base $B$ et s'annulant sur $Z_{n_1}$ correspondent aux morphismes de $B$ dans $W_{n_1}$, qui sont aussi des morphismes de $B$ dans $W_D$. Les familles de sections s'annulant sur $Z_{n_1}$ s'annulent donc sur $D$. \ \\[2mm] Puisque $s_{n_1}$ est une famille de sections de $F$ s'annulant sur $D$, tout \'el\'ement $e$ de $I(s_{n_1},s)$ est divisible par $x_1$: $e=x_1.f$, et d'apr\`es la relation \ref{eq1}, $f \in (J(E,n_1,s):x_1)$, ce qui s'\'ecrit aussi $$ I(s_{n_1},s) \ensuremath{\subset} x_1.(J(E,n_1,s):x_1) $$ Posons $M(n_1,s):=(J(E,n_1,s):x_1)$. Puisque $J(E,n_1,s)$ est un id\'eal d'escalier $E$ et par d\'efinition des id\'eaux d'escaliers $E$, pour tout $p<n_1$ et $u<s$, $r_{n_1p}^{su}(M(n_1,s))=r_{n_1p}^{su}(J(E,n_1,s):x_1)$ est bien un id\'eal d'escalier $S(E,n_1)$. La proposition $(*)$ est vraie pour $i=1$. \ \\[2mm] Supposons $(*)$ vraie au rang $q-1$. L'inclusion \begin{eqnarray*} I(s_{n_{q-1}},s+q)\ensuremath{\subset} x_1^{q-1}.M(n_{q-1},s+q) \end{eqnarray*} est v\'erifi\'ee pour $s$ assez grand et implique par la restriction $r_{n_{q-1}n_{q}}^{(s+q)(s+q-1)}$ \begin{eqnarray} \label {eq22} I(s_{n_q},s+q-1) \ensuremath{\subset} x_1^{q-1}. r_{n_{q-1}n_{q}}^{(s+q)(s+q-1)}M(n_{q-1},s+q) \end{eqnarray} Puisque $r_{n_{q-1}n_{q}}^{(s+q)(s+q-1)}M(n_{q-1},s+q)$ est un id\'eal d'escalier $S(E,n_1,\dots,n_{q-1})$ de $k[[x_1,\dots,x_n]]/\goth m^s \otimes k[t]/t^{n_q}$, il est inclus dans $I^{T(S(E,n_1,\dots,n_{q-1}),n_q)}=I^{T(E,n_q)}$. Pour $s$ assez grand, l'in\-clu\-si\-on (\ref{eq22}) montre alors que $s_{n_q}$ est une famille de sections de $F$ s'an\-nu\-lant sur \!$(q-1)D+Z_{n_q}$, ce qui par hypoth\`ese est aussi une famille de sections de $F$ s'annulant sur $qD$. Tout \'el\'ement $e$ de $I(s_{n_q},s+q-1)$ est donc un multiple de $x_1^q$: \begin{eqnarray} \label{eq222} e=x_1^q.f. \end{eqnarray} Par l'inclusion (\ref{eq22}), $e$ s'\'ecrit aussi $$ e=x_1^{q-1}.g $$ o\`u $g$ est dans $r_{n_{q-1}n_{q}}^{(s+q)(s+q-1)}M(n_{q-1},s+q)$. On en d\'eduit l'\'egalit\'e $$x_1^{q-1}(g-x_1f)=0 $$ Puisque le noyau de la multiplication par $x_1^{q-1}$ dans $k[[x_1,\dots,x_n]]/ \goth m^{q+s-1}\otimes k[t]/t^{n_q}$ est inclus dans $\goth m^{s}$, on a donc $$ r_{n_q,n_q}^{q+s-1,s}(f).x_1=r_{n_q,n_q}^{q+s-1,s}(g). $$ Le terme $r_{n_q,n_q}^{q+s-1,s}(g)$ est dans $$r_{n_q,n_q}^{q+s-1,s} \circ r_{n_{q-1}n_{q}}^{(s+q)(s+q-1)}M(n_{q-1},s+q)= r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q) $$ d'o\`u $$ r_{n_q,n_q}^{q+s-1,s}(f)\in (r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q):x_1) $$ L'image de l'\'egalit\'e (\ref{eq222}) par $r_{n_q,n_q}^{q+s-1,s}$ montre alors que $$ I(s_{n_q},s)\ensuremath{\subset} x_1^q.(r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q)\;:\;x_1). $$ Posons $M(n_q,s)=(\ r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q):x_1\ ) $. On a bien $I(s_{n_q},s)$ qui est inclus dans l'id\'eal $x_1^q. M(n_q,s)$. Il reste \`a voir que pour tout $p<n_q$ et $u<s$, $r_{n_qp}^{su}M(n_q,s)$ est un id\'eal d'escalier $S(E,n_1,\dots,n_q)$. Ce qui est vrai car $r_{n_qp}^{su}M(n_q,s) = r_{n_qp}^{su}(\ r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q):x_1\ )$ et $ r_{n_{q-1}n_q}^{s+q,s}M(n_{q-1},s+q)$ est un id\'eal d'escalier $S(E,n_1,\dots,n_{q-1})$ de $k[[x_1,\dots,x_n]] \otimes k[t]/t^{n_q}$. \ \\[2mm] La d\'emonstration de la r\'ecurrence est termin\'ee. D\'eduisons maintenant $(**)$ de la proposition $(*)$, ce qui ach\`evera la d\'emonstration du th\'eor\`eme \ref{propfond}. \ \\[2mm] Si $n_r\neq 1$, la proposition $(*)$ appliqu\'ee \`a $i=r$ dit que $I(s_{n_r},s+1)$ est inclus dans un id\'eal produit $x_1^r.M(n_r,s+1)$. L'image de cette inclusion par l'application de restriction $r_{n_r1}^{(s+1)s}$ donne $I(s_1,s)\ensuremath{\subset} x_1^r. I^{S(E,n_1, \dots,n_r)}$ car la restriction de $M(n_r,s+1)$ est $I^{S(E,n_1,\dots,n_r)}$ par d\'efinition de $M(n_r,s+1)$ et des id\'eaux d'escalier $ S(E,n_1,\dots,n_r)$. La proposition $(**)$ est donc d\'emontr\'ee pour $n_r\neq 1$. \ \\[2mm] Si $n_r=1$, la proposition $(*)$ appliqu\'ee \`a $i={r-1}$ dit que $I(s_{n_{r-1}},s+r+1)$ est inclus dans un id\'eal $x_1^{r-1}.M(n_{r-1},s+r+1)$. L'image de cette inclusion par l'application de restriction $r_{n_{r-1}1}^{(s+r+1)(s+r)}$ est \begin{eqnarray} \label{eq33} I(s_1,s+r)\ensuremath{\subset} x_1^{r-1}. I^{S(E,n_1, \dots,n_{r-1})}. \end{eqnarray} Puisque $I^{S(E,n_1, \dots,n_{r-1})}\ensuremath{\subset} I^{T(E,1)}$, on a \'egalement l'inclusion $I(s_1,s+r)\ensuremath{\subset} x_1^{r-1}.I^{T(E,1)}$, ce qui signifie pour $s$ assez grand que $s_1$ est section de $F$ qui s'annule sur $(r-1)D+Z_1$. Par hypoth\`ese, $s_1$ est alors une section de $F$ qui s'annule sur $rD$. Tout \'el\'ement $e$ de $I(s_{1},s+r)$ est donc divisible par $x_1^r$: \begin{eqnarray} \label{eq333} e=x_1^r.f. \end{eqnarray} Cette \'egalit\'e, la relation (\ref{eq33}), et le fait que le noyau de la multiplication par $x_1^{r-1}$ dans $k[[x_1,\dots,x_n]]/\goth m^{s+r}$ soit inclus dans $\goth m^{s+1}$ montrent que $$ r_{11}^{s+r,s+1}(f) \in (I^{S(E,n_1, \dots,n_{r-1})}:x_1)=I^{S(E,n_1, \dots,n_r)}+(x_1^s). $$ Cette appartenance et la relation (\ref{eq333}) donnent finalement $$I(s_1,s) \ensuremath{\subset} x_1^r.I^{S(E,n_1, \dots,n_r)}.$$ La d\'emonstration de $(**)$ est termin\'ee. \begin{flushright}\rule{2mm}{2mm Bibliographie:\ \\[2mm] [AH1]: Alexander J. et Hirschowitz A., An asymptotic vanishing theorem for generic unions of multiple points, duke e-print 9703037 \ \\[2mm] [AH2]: Alexander J. et Hirschowitz A., La m\'ethode d'Horace \'eclat\'ee: application \`a l'interpolation en degr\'e quatre, Invent. Math. 107, (1992), 586-602 \ \\[2mm] [CM]: Ciliberto C. et Miranda R., On the dimension of linear systems of plane curves with general multiple base points, duke e-print 9702015 \ \\[2mm] [E]: Evain L., Une g\'en\'eralisation de la conjecture de Harbourne-Hirschowitz aux points infiniment voisins, pr\'eprint en pr\'eparation. \ \\[2mm] [H]: Hirschowitz A., La m\'ethode d'Horace pour l'interpolation \`a plusieurs variables, Manuscripta math, vol. 50, (1995), 337-388 \end{document}

9709

alg-geom/9709028

en

https://arxiv.org/abs/alg-geom/9709028

alg-geom/9709028

Karin Smith

Edward Bierstone and Pierre D. Milman (University of Toronto)

Resolution of Singularities

45 pages, 7 Postscript figures, LATEX. To appear in Current Developments in Several Complex Variables, MSRI Proceedings, ed. M. Schneider and Y.-T. Siu, Cambridge University Press

This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by local computations that provide equations for the centres of blowing up. We describe the origin of our approach and present the proof (in the hypersurface case) in parallel with a worked example.

alg-geom

\section{Introduction} Resolution of singularities has a long history that goes back to Newton in the case of plane curves. For higher-dimensional singular spaces, the problem was formulated toward the end of the last century, and it was solved in general, for algebraic varieties defined over fields of characteristic zero, by Hironaka in his famous paper [H1] of 1964. ([H1] includes the case of real-analytic spaces; Hironaka's theorem for complex-analytic spaces is proved in [H2], [AHV1], [AHV2].) But Hironaka's result is highly non-constructive. His proof is one of the longest and hardest in mathematics, and it seems fair to say that only a handful of mathematicians have fully understood it. We are not among them! Resolution of singularities is used in many areas of mathematics, but even certain aspects of the theorem (for example, {\em canonicity}; see 1.11 below) have remained unclear. This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero. Our proof was sketched in the hypersurface case in [BM4] and is presented in detail in [BM5]. When we started thinking about the subject almost twenty years ago, our aim was simply to understand resolution of singularities. But we soon became convinced that it should be possible to give simple direct proofs of at least those aspects of the theorem that are important in analysis. In 1988, for example, we published a very simple proof that any real-analytic variety is the image by a proper analytic mapping of a manifold of the same dimension [BM1]. The latter statement is a real version of a local form of resolution of singularities, called {\em local uniformization}. It is the idea of [BM1, Section 4] that we have developed (via [BM2]) to define a new local invariant for desingularization that is the main subject of this exposition. Our invariant ${\rm inv}_X(a)$ is a finite sequence (of nonnegative rational numbers and perhaps $\infty$, in the case of a hypersurface), defined at each point $a$ of our space $X$. Such sequences can be compared lexicographically. ${\rm inv}_X(\cdot)$ takes only finitely many maximum values (at least locally), and we get an algorithm for canonical resolution of singularities by successively blowing up its maximum loci. Moreover, ${\rm inv}_X(\cdot)$ can be described by local computations that provide equations for the centres of blowing up. We begin with an example to illustrate the meaning of resolution of singularities: \medskip\noindent {\em Example\ } {\em 1.1.}\quad Let $X$ denote the quadratic cone $x^2-y^2-z^2=0$ in affine 3-space --- the simplest example of a singular surface. \bigskip \begin{center} {\hskip .3in{\epsfxsize=2.5in \epsfbox{ed1.eps}{\vskip -2.18in\hskip .02in{\smit z}}{\vskip .35in \hskip 2.4in{\smit y}}} {\vskip .95in\hskip 2.8in{\smit x}}{\vskip -1.6in\hskip 1.31in{${\mbox{Sing}}\, X$}}} \end{center} \vspace{1.4truein} \begin{center} $X:\ x^2-y^2-z^2=0$ \end{center} $X$ can be desingularized by making a simple quadratic transformation of the ambient space: \begin{displaymath} \sigma:\quad x=u,\ y=uv,\ z=uw . \end{displaymath} The inverse image of $X$ by this mapping $\sigma$ is given by substituting the formulas for $x$, $y$ and $z$ into the equation of $X$: \begin{displaymath} \sigma^{-1}(X):\quad u^2(1-v^2-w^2) = 0. \end{displaymath} Thus $\sigma^{-1}(X)$ has two components: The plane $u=0$ is the set of critical points of the mapping $\sigma$; it is called the {\em exceptional hypersurface}. (Here $E':=\{u=0\}$ is the inverse image of the singular point of $X$.) The quotient after completely factoring out the ``exceptional divisor'' $u$ defines what is called the {\em strict transform\/} $X'$ of $X$ by $\sigma$. Here $X'$ is the cylinder $v^2+w^2=1$. \medskip \begin{center} {\hskip .1in{\epsfxsize=2.5in \epsfbox{ed2.eps}{\vskip -2.18in\hskip -.2in{\smit{w}}} {\vskip .17in\hskip 1in {$\smit{E':\ u=0}$}} {\vskip .04in\hskip 2.1in{\smit{v}}} {\vskip .4in\hskip 3.18in{$\smit{X':\ v^2+w^2=1}$}} {\vskip .29in\hskip 2.43in{\smit{u}}}}} \end{center} \vspace{.4truein} In this example, $\sigma|X'$ is a {\em resolution of singularities\/} of $X$: $X'$ is smooth and $\sigma|X'$ is a proper mapping onto $X$ that is an isomorphism outside the singularity. But the example illustrates a stronger statement, called {\em embedded resolution of singularities}: $X$ is desingularized by making a simple transformation of the ambient space, after which, in addition, the strict transform $X'$ and the exceptional hypersurface $E'$ have only {\em normal crossings}; this means that each point admits a coordinate neighbourhood with respect to which both $X'$ and $E'$ are coordinate subspaces. \medskip The quadratic transformation $\sigma$ in Example 1.1 is also called a {\em blowing-up\/} with {\em centre\/} the origin. (The centre is the set of critical values of $\sigma$.) More accurately, the blowing-up of affine 3-space with centre a point is covered in a natural way by three affine coordinate charts, and $\sigma$ above is the formula for the blowing-up restricted to one chart. Sequences of quadratic transformations, or point blowings-up, were first used to resolve the singularities of curves by Max Noether in the 1870's [BN]. The more general statement of ``embedded resolution of singularities'' seems to have been formulated precisely first by Hironaka. But it is implicit already in the earliest rigorous proofs of local desingularization of surfaces, as a natural generalization prerequisite to the inductive step of a proof by induction on dimension (cf. Sections 2,3 below). For example, in one of the earliest proofs of local desingularization or uniformization of surfaces, Jung used embedded desingularization of curves by sequences of quadratic transformations (applied to the branch locus of a suitable projection) to prove uniformization for surfaces [Ju]. Similar ideas were used in the first proofs of global resolution of singularities of algebraic surfaces, by Walker [Wa] and Zariski [Z1] in the late 1930's. (The latter was the first algebraic proof, by sequences of normalizations and point blowings-up.) From the point of view of subsequent work, however, Zariski's breakthrough came in the early 1940's when he localized the idea of the centre of blowing-up, thus making possible an extension of the notion of quadratic transformation to blowings-up with centres that are not necessarily $0$-dimensional [Z2]. This led Zariski to a version of embedded resolution of singularities of surfaces, and to a weaker (non-embedded) theorem for 3-dimensional algebraic varieties [Z3]. It was the path that led to Hironaka's great theorem and to most subsequent work in the area, including our own. (See References below.) From a general viewpoint, some important features of our work in comparison with previous treatments are: (1) It is canonical. (See 1.11.) (2) We isolate simple local properties of an invariant (Section 4, Theorem B) from which global desingularization is automatic. (3) Our proof in the case of a hypersurface (a space defined locally by a single equation) does not involve passing to higher codimension (as in the inductive procedure of [H1]). Very significant results on resolution of singularities over fields of nonzero characteristic have recently been obtained by de Jong [dJ] and have been announced by Spivakovsky. \medskip \noindent {\em 1.2. Blowing up.} We first describe the blowing-up of an open subset $W$ of $r$-dimensional affine space with centre a point $a$. (Say $a=0\in W$.) The {\em blowing-up\/} $\sigma$ with {\em centre\/} $0$ is the projection onto $W$ of a space $W'$ that is obtained by replacing the origin by the $(r-1)$-dimensional projective space ${\Bbb P}^{r-1}$ of all lines through $0$: \[ W' = \{\, (x,\lambda)\in W\times {\Bbb P}^{r-1}:\ x\in\lambda\,\} \] and $\sigma$: $W'\rightarrow W$ is defined by $\sigma(x,\lambda)=x$. (Outside the origin, a point $x$ belongs to a unique line $\lambda$, but $\sigma^{-1}(0) = {\Bbb P}^{r-1}$. Clearly, $\sigma$ is a proper mapping.) $W'$ has a natural algebraic structure: If we write $x$ in terms of the affine coordinates $x=(x_1,\ldots,x_r)$, and $\lambda$ in the corresponding homogeneous coordinates $\lambda=[\lambda_1,\ldots,\lambda_r]$, then the relation $x\in\lambda$ translates into the system of equations $x_i\lambda_j = x_j\lambda_i$, for all $i,j$. These equations can be used to see that $W'$ has the structure of an algebraic manifold: For each $i=1,\ldots,r$, let $W_i'$ denote the open subset of $W'$ where $\lambda_i\ne 0$. In $W_i'$, $x_j=x_i\lambda_j/\lambda_i$, for each $j\ne i$, so we see that $W_i'$ is smooth: it is the graph of a mapping in terms of coordinates $(y_1,\ldots,y_r)$ for $W_i'$ defined by $y_i=x_i$ and $y_j=\lambda_j/\lambda_i$ if $j\neq i$. In these coordinates, $\sigma$ is a quadratic transformation given by the formulas \[ x_i=y_i,\quad x_j=y_iy_j\ \ \mbox{for all $j\ne i$,} \] as in Example 1.1. Once blowing up with centre a point has been described as above, it is a simple matter to extend the idea to blowing up a manifold, or smooth space, $M$ with centre an arbitrary smooth closed subspace $C$ of $M$: Each point of $C$ has a product coordinate neighbourhood $V\times W$ in which $C = V\times\{0\}$; over this neighbourhood, the blowing-up with centre $C$ identifies with ${\mbox{id}}_V\times\sigma$: $V\times W'\rightarrow V\times W$, where ${\mbox{id}}_V$ is the identity mapping of $V$ and $\sigma$: $W'\rightarrow W$ is the blowing-up of $W$ with centre $\{0\}$. The blowing-up $M'\rightarrow M$ with centre $C$ is an isomorphism over $M{\backslash} C$. The preceding conditions determine $M'\rightarrow M$ uniquely, up to an isomorphism of $M'$ commuting with the projections to $M$. \medskip\noindent {\em Example\ } {\em 1.3.}\quad \begin{center} {\hskip .1in{\epsfxsize=2.5in \epsfbox{ed3.eps}{\vskip -1.59in\hskip 2.19in{${\mbox{Sing}}\, X$}}} {\vskip .45in\hskip 2.6in{\smit x}} {\vskip 0.6in \hskip .11in{\smit z}} {\vskip -.91in \hskip -2.4in{\smit y}}} \end{center} \vspace{.5truein} \begin{center} {$X:\ z^3-x^2yz-x^4=0$} \end{center} \vspace{.5truein} This surface is particularly interesting in the real case because, as a {\em subset\/} of ${\Bbb R}^3$, it is singular only along the nonnegative $y$-axis. But resolution of singularities is an algebraic process: it applies to spaces that include a functional structure (given here by the equation for $X$). As a {\em subspace\/} of ${\Bbb R}^3$, $X$ is singular along the entire $y$-axis. \medskip In general for a hypersurface $X$ --- say that $X$ is defined locally by an equation $f(x)=0$ --- to say that a point $a$ is {\em singular\/} means there are no linear terms in the Taylor expansion of $f$ at $a$; in other words, the order $\mu_a(f) > 1$. (The {\em order\/} or {\em multiplicity\/} $\mu_a(f)$ of $f$ at $a$ is the degree of the lowest-order homogeneous part of the Taylor expansion of $f$ at $a$. We will also call $\mu_a(f)$ the {\em order\/} $\nu_{X,a}$ of the hypersurface $X$ at $a$.) \medskip The general philosophy of our approach to desingularization (going back to Zariski [Z3]) is the blow up with smooth centre as large as possible inside the locus of the most singular points. In our example here, $X$ has order $3$ at each point of the $y$-axis. In general, order is not a delicate enough invariant to determine a centre of blowing-up for resolution of singularities, even in the hypersurface case. (We will refine order in our definition of ${\rm inv}_X$.) But here let us take the blowing-up $\sigma$ with centre the $y$-axis: \[ \sigma:\quad x=u,\ y=v,\ z=uw. \] (Again, this is the formula for blowing up in one of two coordinate charts required to cover our space. But the strict transform of $X$ in fact lies completely within this chart.) The inverse image of $X$ is \[ \sigma^{-1}(X):\quad u^3(w^3-vw-u)=0; \] $\{u=0\}$ is the exceptional hypersurface $E'$ (the inverse image of the centre of blowing up) and the strict transform $X'$ is smooth. (It is the graph of a function $u=w^3-vw$.) \begin{center} {\hskip .1in{\epsfxsize=2.5in \epsfbox{ed4.eps}}{\vskip -1.08in\hskip -2.08in{\smit v}} {\vskip .55in\hskip .18in{\smit w}} {\vskip -1.04in\hskip 2.39in{\smit u}} {\vskip -.6in\hskip 3.05in{$X':\ u=w^3-vw$}} {\vskip -.9in\hskip 1.7in{$E':\ u=0$}}} \end{center} \vspace{2truein} $X'$ is a desingularization of $X$, but we have not yet achieved an embedded resolution of singularities because $X'$ and $E'$ do not have normal crossings at the origin. Further blowings-up are needed for embedded resolution of singularities. \medskip \noindent {\em 1.4. Embedded resolution of singularities.} Let $X$ denote a (singular) space. We assume, for simplicity, that $X$ is a closed subspace of a smooth ambient space $M$. (This is always true locally.) The goal of embedded desingularization, in its simplest version, is to find a proper morphism $\sigma$ from a smooth space $M'$ onto $M$, in our category, with the following properties: (1) $\sigma$ is an isomorphism outside the singular locus ${\mbox{Sing}}\, X$ of $X$. (2) The strict transform $X'$ of $X$ by $\sigma$ is smooth. (See 1.6 below.) $X'$ can be described geometrically (at least if our field ${\Bbb K}$ is algebraically closed; cf. [BM5, Rmk. 3.15]) as the smallest closed subspace of $M'$ that includes $\sigma^{-1}(X{\backslash}{\mbox{Sing}}\, X)$. (3) $X'$ and $E'=\sigma^{-1}({\mbox{Sing}}\, X)$ simultaneously have only normal crossings. This means that, locally, we can choose coordinates with respect to which $X'$ is a coordinate subspace and $E'$ is a collection of coordinate hyperplanes. We can achieve this goal with $\sigma$ the composite of a sequence of blowings-up; a finite sequence when our spaces have a compact topology (for example, in an algebraic category), or a locally-finite sequence for non-compact analytic spaces. (A sequence of blowings-up over $M$ is {\em locally finite\/} if all but finitely many of the blowings-up are trivial over any compact subset of $M$. The composite of a locally-finite sequence of blowings-up is a well-defined morphism $\sigma$.) \medskip \noindent {\em 1.5. The category of spaces.} Our desingularization theorem applies to the usual spaces of algebraic and analytic geometry over fields ${\Bbb K}$ of characteristic zero --- algebraic varieties, schemes of finite type, analytic spaces (over ${\Bbb R}$, ${\Bbb C}$ or any locally compact ${\Bbb K}$) --- but in addition to certain categories of spaces intermediate between analytic and $C^\infty$ (See [BM5].) In any case, we are dealing with a category of local-ringed spaces $X=(|X|,{\cal O}_X)$ over ${\Bbb K}$, where ${\cal O}_X$ is a coherent sheaf of rings. We are intentionally not specific about the category in this exposition because we want to emphasize the principles involved, and the main requirement for our desingularization algorithm is simply that a smooth space $M=(|M|,{\cal O}_M)$ in our category admit a covering by {\em (regular) coordinate charts\/} in which we have analogues of the usual operations of calculus of analytic functions; namely: The coordinates $(x_1,\ldots,x_n)$ of a chart $U$ are {\em regular functions\/} on $U$ (i.e., each $x_i\in {\cal O}_M(U)$) and all partial derivatives $\partial^{|\alpha|}/ \partial x^\alpha = \partial^{\alpha_1+\cdots+\alpha_n}/ \partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}$ make sense as transformations ${\cal O}_M(U)\rightarrow {\cal O}_M(U)$. Moreover, for each $a\in U$, there is an {\em injective\/} ``Taylor series homomorphism'' $T_a$: ${\cal O}_{M,a}\rightarrow {\Bbb F}_a[[X]]= {\Bbb F}_a[[X_1,\ldots,X_n]]$, where ${\Bbb F}_a$ denotes the residue field ${\cal O}_{M,a}/{\underline m}_{M,a}$, such that $T_a$ induces an isomorphism ${\widehat\cO}_{M,a}\stackrel{\cong}{{\rightarrow}} {\Bbb F}_a [[X]]$ and $T_a$ commutes with differentiation: $T_a\circ (\partial^{|\alpha|}/ \partial x^\alpha)=(\partial^{|\alpha|}/\partial X^\alpha)\circ T_a$, for all $\alpha\in{\Bbb N}^n$. (${\underline m}_{M,a}$ denotes the maximal ideal and ${\widehat\cO}_{M,a}$ the completion of ${\cal O}_{M,a}$. ${\Bbb N}$ denotes the nonnegative integers.) \medskip In the case of real- or complex-analytic spaces, of course, ${\Bbb K}={\Bbb R}$ or ${\Bbb C}$, ${\Bbb F}_a={\Bbb K}$ at each point, and ``coordinate chart'' means the classical notion. Regular coordinate charts for schemes of finite type are introduced in [BM5, Section 3]. Suppose that $M=(|M|,{\cal O}_M)$ is a manifold (smooth space) and that $X=(|X|,{\cal O}_X)$ is a closed subspace of $M$. This means there is a coherent sheaf of ideals ${\cal I}_X$ in ${\cal O}_M$ such that $|X|={\mbox{supp}}\,{\cal O}_M/{\cal I}_X$ and ${\cal O}_X$ is the restriction to $|X|$ of ${\cal O}_M/{\cal I}_X$. We say that $X$ is a {\em hypersurface\/} in $M$ if ${\cal I}_{X,a}$ is a principal ideal, for each $a\in |X|$. Equivalently, for every $a\in |X|$, there is an open neighbourhood $U$ of $a$ in $|M|$ and a regular function $f\in{\cal O}_M(U)$ such that $|X|U|=\{\, x\in U:\ f(x)=0\,\}$ and ${\cal I}_X|U$ is the principal ideal $(f)$ generated by $f$; we write $X|U = V(f)$. \medskip \noindent {\em 1.6. Strict transform.} Let $X$ denote a closed subspace of a manifold $M$, and let $\sigma$: $M'\rightarrow M$ be a blowing-up with smooth centre $C$. If $X$ is a hypersurface, then the strict transform $X'$ of $X$ by $\sigma$ is a closed subspace of $M'$ that can be defined as follows: Say that $X=V(f)$ in a neighbourhood of $a\in|X|$. Then, in some neighbourhood of $a'\in \sigma^{-1} (a)$, $X'=V(f')$, where $f'=y_{\rm exc}^{-d} f\circ \sigma$, $y_{\rm exc}$ denotes a local generator of ${\cal I}_{\sigma^{-1}(C)}\subset{\cal O}_{M'}$, and $d=\mu_{C,a}(f)$ denotes the {\em order\/} of $f$ {\em along\/} $C$ at $a$: $d=\max\{\,k:\ (f)\subset {\cal I}_{C,a}^k\,\}$; $d$ is the largest power to which $y_{\rm exc}$ factors from $f\circ\sigma$ at $a'$. The strict transform $X'$ of a general closed subspace $X$ of $M$ can be defined locally, at each $a'\in \sigma^{-1}(a)$, as the intersection of all hypersurfaces $V(f')$, for all $f\in{\cal I}_{X,a}$. We likewise define the strict transform by a sequence of blowings-up with smooth centres. Each of the categories listed in 1.5 above is closed under blowing up and strict transform [BM5, Prop. 3.13 ff.]; the latter condition is needed to apply the desingularization algorithm in a given category. \medskip \noindent {\em 1.7. The invariant.} Let $X$ denote a closed subspace of a manifold $M$. To describe ${\rm inv}_X$, we consider a sequence of transformations $$ \begin{array}{cccccccccc} {\rightarrow} & M_{j+1} & \stackrel{\sigma_{j+1}}{{\rightarrow}} & M_j & {\rightarrow} & \cdots & {\rightarrow} & M_1 & \stackrel{\sigma_1}{{\rightarrow}} & M_0=M \\ & X_{j+1} & & X_j & & & & X_1 & & X_0 = X \\ & E_{j+1} & & E_j & & & & E_1 & & E_0 = \emptyset \end{array} \leqno(1.8) $$ where, for each $j$, $\sigma_{j+1}$: $M_{j+1}\rightarrow M_j$ denotes a blowing-up with smooth centre $C_j\subset M_j$, $X_{j+1}$ is the strict transform of $X_j$ by $\sigma_{j+1}$, and $E_{j+1}$ is the set of exceptional hypersurfaces in $M_{j+1}$; i.e., $E_{j+1}=E_j'\cup \{\sigma_{j+1}^{-1} (C_j)\}$, where $E_j'$ denotes the set of strict transforms by $\sigma_{j+1}$ of all hypersurfaces in $E_j$. Our invariant ${\rm inv}_X(a)$, $a\in M_j$, $j=0,1,2,\ldots$, will be defined inductively over the sequence of blowings-up; for each $j$, ${\rm inv}_X(a)$, $a\in M_j$, can be defined provided that the centres $C_i$, $i<j$, are {\em admissible\/} (or ${\rm inv}_X$-{\em admissible}) in the sense that: (1) $C_i$ and $E_i$ simultaneously have only normal crossings. (2) ${\rm inv}_X(\cdot)$ is locally constant on $C_i$. The condition (1) guarantees that $E_{i+1}$ is a collection of smooth hypersurfaces having only normal crossings. We can think of the desingularization algorithm in the following way: $X\subset M$ determines ${\rm inv}_X(a)$, $a\in M$, and thus the first admissible centre of blowing up $C=C_0$; then ${\rm inv}_X(a)$ can be defined on $M_1$ and determines $C_1$, etc. The notation ${\rm inv}_X(a)$, where $a\in M_j$, indicates a dependence not only on $X_j$, but also on the original space $X$. In fact ${\rm inv}_X(a)$, $a\in M_j$, is invariant under local isomorphisms of $X_j$ that preserve $E(a)=\{\, H\in E_j:\ H\ni a\,\}$ and certain subcollections $E^r(a)$ (which will be taken to encode the history of the resolution process). To understand why some dependence on the history should be needed, let us consider how, in principle, it might be possible to determine a {\em global\/} centre of blowing up using a {\em local\/} invariant: \medskip\noindent {\em Example\ } {\em 1.9.}\quad It is easy to find an example of a surface $X$ whose singular locus, in a neighbourhood of a point $a$, consists of two smooth curves \begin{minipage}[t]{2in} with a normal crossing at $a$, and where $X$ has the property that, if we blow up with centre $\{ a\}$, then there are points $a'$ in the fibre $\sigma^{-1}(a)$ where the strict transform $X'$ has the {\em same\/} local equation (in suitable\break \vspace{-.15in} \end{minipage}\hfill \raisebox{.4cm}{\begin{minipage}[t]{3in} \hskip 4in{\epsfxsize=2.5in \epsfbox{ed5.eps}{\vskip -1in\hskip .87in{$a$}}{\vskip .5in\hskip .3in{${\mbox{Sing}}\, X$}}} \end{minipage}} coordinates) as that of $X$ at $a$, or an even more complicated equation (as in Example 3.1 below). This suggests that to simplify the singularities in a neighbourhood of $a$ by blowing up with smooth centre in ${\mbox{Sing}}\, X$, we should choose as centre one of the two smooth curves. But our surface may have the property that neither curve extends to a global smooth centre, as illustrated. So there is no choice but to blow up with centre $\{ a\}$, although it seems to accomplish nothing: The figure shows the \break {\begin{minipage}[t]{2in} singular locus of $X'$; there are two points $a'\in \sigma^{-1} (a)$ where the singularity is the same as or worse than before. But what has changed at each of these points is the status of one of the curves, which is now {\em exceptional}. The moral is that,\break \vspace{-.13in} \end{minipage}}\hfill \raisebox{.1cm}{\begin{minipage}[t]{3in} \hskip 4in{\epsfxsize=2.5in\epsfbox{ed6.eps}{\vskip -1.7in\hskip .62in{$X'\cap E'$}}{\vskip .86in\hskip .61in{$a'$}}} \end{minipage}} \noindent although the singularity of $X$ at $a$ has not been simplified in the strict transform, an invariant which takes into account the history of the resolution process as recorded by the accumulating exceptional hypersurfaces might nevertheless measure some improvement. \medskip Consider a sequence of blowings-up as before. For simplicity, {\em we will assume that\/} $X\subset M$ {\em is a hypersurface}. Then ${\rm inv}_X(a)$, $a\in M_j$, is a finite sequence beginning with the order $\nu_1(a)=\nu_{X_j,a}$ of $X_j$ at $a$: \[ {\rm inv}_X(a)\ =\ \big( \nu_1(a), s_1(a);\, \nu_2(a),s_2(a);\, \ldots,\, s_t(a);\, \nu_{t+1}(a)\big) . \] (In the general case, $\nu_1(a)$ is replaced by a more delicate invariant of $X_j$ at $a$ --- the Hilbert-Samuel function $H_{X_j,a}$ (see [BM5]) --- but the remaining entries of ${\rm inv}_X(a)$ are still rational numbers (or $\infty$) as we will describe, and the theorems below are unchanged.) The $s_r(a)$ are nonnegative integers counting exceptional hypersurfaces that accumulate in certain blocks $E^r(a)$ depending on the history of the resolution process. And the $\nu_r(a)$, $r\ge 2$, represent certain ``higher-order multiplicities'' of the equation of $X_j$ at $a$; $\nu_2(a),\ldots,\nu_t(a)$ are quotients of positive integers whose denominators are bounded in terms of the previous entries of ${\rm inv}_X(a)$. (More precisely, $e_{r-1}! \nu_r (a)\in {\Bbb N}$, $r=1,\ldots,t$, where $e_0=1$ and $e_r = \max\{ e_{r-1}!, e_{r-1}! \nu_r (a)\}$.) The pairs $\big( \nu_r(a), s_r(a)\big)$ can be defined successively using data that depends on $n-r+1$ variables (where $n$ is the ambient dimension), so that $t\le n$ by exhaustion of variables; the final entry $\nu_{t+1}(a)$ is either $0$ (the order of a nonvanishing function) or $\infty$ (the order of the function identically zero). \medskip\noindent {\em Example\ } {\em 1.10.}\quad Let $X\subset{\Bbb K}^n$ be the hypersurface $x_1^{d_1} + x_2^{d_2}+\cdots + x_t^{d_t} =0$, where $1<d_1\le \cdots \le d_t$, $t\le n$. Then \[ {\rm inv}_X(0) = \left( d_1, 0;\, \frac {d_2}{d_1}, 0;\, \ldots;\, \frac{d_t}{d_{t-1}},0;\, \infty\right) . \] This is ${\rm inv}_X(0)$ in ``year zero'' (before the first blowing up), so there are no exceptional hypersurfaces. \medskip\noindent{\bf Theorem\ } {\bf A.}\quad {\rm (Embedded desingularization.)} {\em There is a finite sequence of blowings-up (1.8) with smooth ${\rm inv}_X$-admissible centres $C_j$ (or a locally finite sequence, in the case of noncompact analytic spaces) such that: {\rm (1)} For each $j$, either $C_j\subset{\mbox{Sing}}\, X_j$ or $X_j$ is smooth and $C_j\subset X_j\cap E_j$. {\rm (2)} Let $X'$ and $E'$ denote the final strict transform of $X$ and exceptional set, respectively. Then $X'$ is smooth and $X'$, $E'$ simultaneously have only normal crossings.} \medskip If $\sigma$ denotes the composite of the sequence of blowings-up $\sigma_j$, then $E'$ is the critical locus of $\sigma$ and $E'=\sigma^{-1} ({\mbox{Sing}}\, X)$. In each of our categories of spaces, ${\mbox{Sing}}\, X$ is closed in the {\em Zariski topology\/} of $|X|$ (the topology whose closed sets are of the form $|Y|$, for any closed subspace $Y$ of $X$; see [BM5, Prop. 10.1]). Theorem A resolves the singularities of $X$ in a meaningful geometric sense provided that $|X|{\backslash} {\mbox{Sing}}\, X$ is (Zariski-)dense in $|X|$. (For example, if $X$ is a {\em reduced\/} complex-analytic space or a scheme of finite type.) More precise desingularization theorems (for example, for spaces that are not necessarily reduced) are given in [BM5, Ch. IV]. This paper contains an essentially complete proof of Theorem A in the hypersurface case, presented though in a more informal way than in [BM5]. We give a constructive definition of ${\rm inv}_X$ in Section 3, in parallel with a detailed example. In Section 4, we show that ${\rm inv}_X$ is indeed an invariant, and we summarize its key properties in Theorem B. (The terms $s_r(a)$ of ${\rm inv}_X(a)$ can, in fact, be introduced immediately in an invariant way; see 1.12 below.) It follows from Theorem B(3) that the maximum locus of ${\rm inv}_X$ has only normal crossings and, moreover, each of its local components extends to a global smooth subspace. (See Remark 3.6.) The point is that each component is the intersection of the maximum locus of ${\rm inv}_X$ with those exceptional hypersurfaces containing the component; the exceptional divisors serve as global coordinates.) We can obtain Theorem A by successively blowing up with centre given by any component of the maximum locus. \medskip \noindent {\em 1.11. Universal and canonical desingularization.} The exceptional hypersurfaces (the elements of $E_j$) can be ordered in a natural way (by their ``years of birth'' in the history of the resolution process). We can use this ordering to extend ${\rm inv}_X(a)$ by an additional term $J(a)$ that will have the effect of picking out one component of the maximum locus of ${\rm inv}_X(\cdot)$ in a canonical way; see Remark 3.6. We write ${\rm inv}_X^{\rm e} (\cdot)$ for the extended invariant $\big( {\rm inv}_X(\cdot); J(\cdot)\big)$. Then our embedded desingularization theorem A can be obtained by the following: \medskip\noindent{\bf Algorithm.\quad } {\em Choose as each successive centre of blowing up $C_j$ the maximum locus of ${\rm inv}_X^{\rm e}$ on $X_j$.} \medskip The algorithm stops when our space is ``resolved'' as in the conclusion of Theorem A. In the general (not necessarily hypersurface) case, we choose more precisely as each successive centre $C_j$ the maximum locus of ${\rm inv}_X^{\rm e}$ on the non-resolved locus $Z_j$ of $X_j$; in general, $\{ x:\ {\rm inv}_X(x)={\rm inv}_X(a)\} \subset Z_j$ (as germs at $a$), so that again each $C_j$ is smooth, by Theorem B(3), and the algorithm stops when $Z_j=\emptyset$. The algorithm applies to a category of spaces satisfying a compactness assumption (for example, schemes of finite type, restrictions of analytic spaces to relatively compact open subsets), so that ${\rm inv}_X(\cdot)$ has global maxima. Since the centres of blowing up are completely determined by an invariant, our desingularization theorem is automatically {\em universal\/} in the following sense: To every $X$, we associate a morphism of resolution of singularities $\sigma_X$: $X'\rightarrow X$ such that any local isomorphism $X|U \rightarrow Y|V$ (over open subsets $U$ of $|X|$ and $V$ of $|Y|$) lifts to an isomorphism $X'|\sigma_X^{-1} (U)\rightarrow Y'|\sigma_Y^{-1} (V)$ (in fact, lifts to isomorphisms throughout the entire towers of blowings-up). For analytic spaces that are not necessarily compact, we can use an exhaustion by relatively compact open sets to deduce {\em canonical\/} resolution of singularities: Given $X$, there is a morphism of desingularization $\sigma_X$: $X'\rightarrow X$ such that any local isomorphism $X|U \rightarrow X|V$ (over open subsets of $|X|$) lifts to an isomorphism $X'|\sigma_X^{-1} (U) \rightarrow X'| \sigma_X^{-1} (V)$. (See [BM5, Section 13].) \medskip \noindent {\em 1.12. The terms $s_r(a)$.} The entries $s_1(a),\, \nu_2(a),\, s_2(a),\,\ldots$ of ${\rm inv}_X(a) = \big( \nu_1(a), s_1(a);\, \ldots, s_t(a);\, \nu_{t+1}(a)\big)$ will themselves be defined recursively. Let us write ${\rm inv}_r$ for ${\rm inv}_X$ truncated after $s_r$ (with the convention that ${\rm inv}_r(a) = {\rm inv}_X(a)$ if $r>t$). We also write ${\rm inv}_{r+\frac{1}{2}}=({\rm inv}_r;\nu_{r+1})$ (with the same convention), so that ${\rm inv}_{\frac{1}{2}}(a)$ means $\nu_1(a)=\nu_{X_j,a}$ (in the hypersurface case, or $H_{X_j,a}$ in general). For each $r$, the entries $s_r$, $\nu_{r+1}$ of ${\rm inv}_X$ can be defined over a sequence of blowings-up (1.8) whose centres $C_i$ are $(r-\frac{1}{2})$-{\em admissible\/} (or ${\rm inv}_{r-\frac{1}{2}}$-{\em admissible}) in the sense that: (1) $C_i$ and $E_i$ simultaneously have only normal crossings. (2) ${\rm inv}_{r-\frac{1}{2}}(\cdot)$ is locally constant on $C_i$. The terms $s_r(a)$ can be introduced immediately, as follows: Write $\pi_{ij}=\sigma_{i+1}\circ\cdots\cdot \sigma_j$, $i=0,\ldots, j-1$, and $\pi_{jj}=$ identity. If $a\in M_j$, set $a_i=\pi_{ij}(a)$, $i=0,\ldots,j$. First consider a sequence of blowings-up (1.8) with $\frac{1}{2}$-admissible centres. (${\rm inv}_{\frac{1}{2}}=\nu_1$ can only decrease over such a sequence; see, for example, Section 2 following.) Suppose $a\in M_j$. Let $i$ denote the ``earliest year'' $k$ such that $\nu_1(a)=\nu_1(a_k)$, and set $E^1(a)=\{\, H\in E(a):$ $H$ is the strict transform of some hypersurface in $E(a_i)\,\}$. We define $s_1(a)=\# E^1(a)$. The block of exceptional hypersurfaces $E^1(a)$ intervenes in our desingularization algorithm in a way that can be thought of intuitively as follows. (The idea will be made precise in Sections 2 and 3.) The exceptional hypersurfaces passing through $a$ but not in $E^1(a)$ have accumulated during the recent part of our history, when the order $\nu_1$ has not changed; we have good control over these hypersurfaces. But those in $E^1(a)$ accumulated long ago; we have forgotten a lot about them in the form of our equations (for example, if we restrict the equations of $X$ to these hypersurfaces, their orders might increase) and we recall them using $s_1(a)$. In general, consider a sequence of blowings-up (1.8) with $(r+\frac{1}{2})$-admissible centres. (${\rm inv}_{r+\frac{1}{2}}$ can only decrease over such a sequence; see Section 3 and Theorem B.) Suppose that $i$ is the smallest index $k$ such that ${\rm inv}_{r+\frac{1}{2}}(a)= {\rm inv}_{r+\frac{1}{2}}(a_k)$. Let $E^{r+1}(a)=\{ \, H\in E(a){\backslash} \bigcup_{q\le r} E^q(a):$ $H$ is transformed from $E(a_i)\,\}$. We define $s_{r+1}(a) = \# E^{r+1}(a)$. It is less straightforward to define the multiplicities $\nu_2(a), \nu_3(a),\ldots$ and to show they are invariants. Our definition depends on a construction in local coordinates that we present in Section 3. But we first try to convey the idea by describing the origin of our algorithm. \section{The origin of our approach} Consider a hypersurface $X$, defined locally by an equation $f(x)=0$. Let $a\in X$ and let $d=d(a)$ denote the order of $X$ (or of $f$) at $a$; i.e., $d=\nu_1(a)=\mu_a(f)$. We can choose local coordinates $(x_1,\ldots,x_n)$ in which $a=0$ and $(\partial^d f/\partial x_n^d)(a)\neq 0$; then we can write \[ f(x) = c_0({\tilde x}) + c_1({\tilde x}) x_n + \cdots + c_{d-1} ({\tilde x}) x_n^{d-1} +c_d (x) x_n^d \] in a neighbourhood of $a$, where $c_d(x)$ does not vanish. (${\tilde x}$ means $(x_1,\ldots,x_{n-1})$.) Assume for simplicity that $c_d(x)\equiv 1$ (for example, by the Weierstrass preparation theorem, but see Remark 2.3 below). We can also assume that $c_{d-1}({\tilde x})\equiv 0$, by ``completing the $d$'th power'' (i.e., by the coordinate change $x_n'=x_n+c_{d-1} ({\tilde x})/d$); thus $$ f(x) = c_0({\tilde x}) + \cdots + c_{d-2}({\tilde x}) x_n^{d-2} + x_n^d . \leqno(2.1) $$ Our aim is to simplify $f$ by blowing up with smooth centre in the {\em equimultiple locus\/} of $a=0$; i.e., in the locus of points of order $d$, \[ S_{(f,d)} = \{\, x:\ \mu_x(f) = d\,\} . \] The representation (2.1) makes it clear that the equimultiple locus lies in a smooth subspace of codimension $1$; in fact, by elementary calculus, $$ S_{(f,d)} = \{\, x:\ x_n=0\ \mbox{and}\ \mu_{\tilde x} (c_q)\ge d-q, \ q = 0,\ldots,d-2\,\} . \leqno(2.2) $$ The idea now is that the given data $\big( f(x),d\big)$ involving $n$ variables should be equivalent, in some sense, to the data ${\cal H}_1(a)=\big\{ \big( c_q({\tilde x}),d-q\big)\big\}$ in $n-1$ variables, thus making possible an induction on the number of variables. (Here in ``year zero'', before we begin to blow up, $\nu_2(a) = \min_q \mu_a (c_q) / (d-q)$.) \medskip\noindent {\em Remark\ } {\em 2.3.}\quad For the global desingularization algorithm, the Weierstrass preparation theorem must be avoided for two important reasons: (1) It may take us outside the given category (for example, in the algebraic case). (2) Even in the complex-analytic case, we need to prove that ${\rm inv}_X$ is semicontinuous in the sense that any point admits a coordinate neighbourhood $V$ such that, given $a\in V$, $\{\, x\in V:\ {\rm inv}_X(x)\le {\rm inv}_X(a)\,\}$ is Zariski-open in $V$ (i.e., is the complement of a closed analytic subset). We therefore need a representation like (2.2) that is valid in a Zariski-open neighbourhood of $a$ in $V$. This can be achieved in the following simple way that involves neither making $c_d(x)\equiv 1$ nor explicitly completing the $d$'th power: By a linear coordinate change, we can assume that $(\partial^d f/\partial x_n^d)(a)\ne 0$. Then in the Zariski-open neighbourhood of $a$ where $(\partial^d f/\partial x_n^d)(x)\ne 0$, we let $N_1=N_1(a)$ denote the submanifold of codimension one (in our category) defined by $z=0$, where $z=\partial^{d-1} f/\partial x_n^{d-1}$, and we take ${\cal H}_1(a)=\big\{ \,\big( (\partial^q f/\partial x_n^q) |N_1,\ d-q\big)\,\big\}$. As before, we have $S_{(f,d)}=\{\, x:\ x\in N_1$ and $\mu_x(h)\ge \mu_h$, for all $(h,\mu_h)=\big( (\partial^q f/\partial x_n^q)|N_1,\ d-q\big) \in {\cal H}_1(a)\,\}$. \medskip We now consider the effect of a blowing-up $\sigma$ with smooth centre $C\subset S_{(f,d)}$. By a transformation of the variables $(x_1,\ldots,x_{n-1})$, we can assume that in our local coordinate neighbourhood $U$ of $a$, $C$ has the form $$ Z_I = \{\, x:\ x_n=0\ \hbox{ and }\ x_i=0,\ i\in I\,\} , \leqno(2.4) $$ where $I\subset \{ 1,\ldots,n-1\}$. According to 1.2 above, $U'=\sigma^{-1} (U)$ is covered by coordinate charts $U_i'$, $i\in I\cup \{ n\}$, where each $U_i'$ has coordinates $y=(y_1,\ldots,y_n)$ in which $\sigma$ is given by $$ \begin{array}{ll} x_i = y_i &\\ x_j = y_i y_j ,&\qquad j\in (I\cup\{ n\}){\backslash} \{i\}, \\ x_j = y_j\ \, ,&\qquad j\not\in I\cup \{ n\}. \end{array} $$ In each $U_i'$, we can write $f\big(\sigma(y)\big) = y_i^d f'(y)$; the strict transform $X'$ of $X$ by $\sigma$ is defined in $U_i'$ by the equation $f'(y)=0$. (To be as simple as possible, we continue to assume $c_d(x)\equiv 1$, though we could just as well work with the set-up of Remark 2.3; see [BM5, Prop. 4.12].) By (2.1), if $i\in I$, then $$ f'(y) = c_0'({\tilde y}) + \cdots + c_{d-2}' ({\tilde y}) y_n^{d-2} + y_n^d , \leqno(2.5) $$ where $$ c_q'({\tilde y}) = y_i^{-(d-q)} c_q\big({\tilde\sigma}({\tilde y})\big),\qquad q=0,\ldots,d-2. \leqno(2.6) $$ The analogous formula for the strict transform in the chart $U_n'$ shows that $f'$ is invertible at every point of $U_n'{\backslash} \bigcup_{i\in I} U_i' = \{ \, y\in U_n':\ y_i=0,\ i\in I\, \}$; in other words, $X'\cap U'\subset \bigcup_{i\in I} U_i'$. The formula for $f'(y)$ above shows that the representation (2.2) of the equimultiple locus (or that of Remark 2.3) is stable under $\nu_1$-admissible blowing up when the order does not decrease; i.e., at a point $a'\in U_i'$ where $d(a')=d$, $S_{(f',d)} = \{ \, y:\ y_n=0$ and $\mu_{\tilde y}(c_q')\ge d-q$, $q=0,\ldots,d-2\,\}$, where $N_1(a')=\{y_n=0\}$ is the strict transform of $N_1(a)=\{ x_n=0\}$ and the $c_q'$ are given by the transformation law (2.6). The latter is not strict transform, but something intermediate between strict and total transform $c_q\circ\sigma$. It is essentially for this reason that some form of embedded desingularization will be needed for the coefficients $c_q$ (i.e., in the inductive step) even to prove a weaker form of resolution of singularities for $f$. $N_1(a)$ is called a smooth hypersurface of {\em maximal contact} with $X$; this means a smooth hypersurface that contains the equimultiple locus of $a$, stably (i.e., even after admissible blowings-up as above). The existence of $N_1(a)$ depends on characteristic zero. A maximal contact hypersurface is crucial to our construction by increasing codimension. (In 1.12 above, $E^1(a)$ is the block of exceptional hypersurfaces that do not necessarily have normal crossings with respect to a maximal contact hypersurface; the term $s_1(a)$ in ${\rm inv}_X(a)$ is needed to deal with these exceptional divisors.) We will now make a simplifying assumption on the coefficients $c_q$: Let us assume that one of these functions is a monomial (times an invertible factor) that divides all the others, but in a way that respects the different ``multiplicities'' $d-q$ associated with the transformation law (2.6); in other words, let us make the monomial assumption on the $c_q^{1/(d-q)}$ (to equalize the ``assigned multiplicities'' $d-q$) or on the $c_q^{d!/(d-q)}$ (to avoid fractional powers). We assume, then, that $$ c_q({\tilde x})^{d!/(d-q)} = ({\tilde x}^\Omega)^{d!} c_q^* ({\tilde x}),\qquad q=0,\ldots,d-2, \leqno(2.7) $$ where $\Omega=(\Omega_1,\ldots,\Omega_{n-1})$ with $d!\Omega_i\in{\Bbb N}$ for each $i$, ${\tilde x}^\Omega = x_1^{\Omega_1}\cdots x_{n-1}^{\Omega_{n-1}}$, and the $c_q^*$ are regular functions on $\{ x_n=0\}$ such that $c_q^*(a)\neq 0$ for some $q$. We also write $\Omega=\Omega(a)$. We can regard (2.7) provisionally as an assumption made to see what happens in a simple test case, but in fact we can reduce to this case by a suitable induction on dimension (as we will see below). (Assuming (2.7) in year zero, $\nu_2(a)=|\Omega|$, where $|\Omega|=\Omega_1+\cdots+\Omega_{n-1}$. But from the viewpoint of our algorithm for canonical desingularization as presented in Section 3, the argument following is analogous to a situation where the variables $x_i$ occurring in ${\tilde x}^\Omega$ are exceptional divisors in $E(a)\backslash E^1(a)$; in this context, $|\Omega|$ is an invariant we call $\mu_2(a)$ (Definition 3.2) and $\nu_2 (a)=0$.) Now, by (2.2) and (2.7), \[ S_{(f,d)} = \{\, x:\ x_n=0\ \mbox{and}\ \mu_{\tilde x}({\tilde x}^\Omega)\ge 1\,\} . \] (The order of a monomial with rational exponents has the obvious meaning.) Therefore (using the notation (2.4)), $S_{(f,d)} = \bigcup Z_I$, where $I$ runs over the {\em minimal\/} subsets of $\{1,\ldots,n-1\}$ such that $\sum_{j\in I} \Omega_j \ge 1$; i.e., where $I$ runs over the subsets of $\{ 1,\ldots,n-1\}$ such that $$ 0 \le \sum_{j\in I} \Omega_j -1 < \Omega_i,\qquad \mbox{ for all $i\in I$}. \leqno(2.8) $$ Consider the blowing-up $\sigma$ with centre $C=Z_I$, for one such $I$. By (2.7), in the chart $U_i'$ we have $$ c_q'({\tilde y})^{d!/(d-q)} = \big( y_1^{\Omega_1}\cdots y_i^{\sum_I\Omega_j-1}\cdots y_{n-1}^{\Omega_{n-1}}\big)^{d!} c_q^* \big({\tilde\sigma}({\tilde y})\big) , \leqno(2.9) $$ $q=0,\ldots,d-2$. Suppose $a'\in \sigma^{-1} (a)\cap U_i'$. By (2.5), $d(a')\le d(a)$. Moreover, if $d(a')=d(a)$, then by (2.8) and (2.9), $1\le |\Omega(a')| < |\Omega(a)|$. In particular, the order $d$ must decrease after at most $d!|\Omega|$ such blowings-up. The question then is whether we can reduce to the hypothesis (2.7) by induction on dimension, replacing $(f,d)$ in some sense by the collection ${\cal H}_1(a)=\{(c_q,d-q)\}$ on the submanifold $N_1=\{x_n=0\}$. To set up the induction, we would have to treat from the start a collection ${\cal F}_1=\{ (f,\mu_f)\}$ rather than a single pair $(f,d)$. (A general $X$ is, in any case, defined locally by several equations.) Moreover, since the transformation law (2.6) is not strict transform, we would have to reformulate the original problem to not only desingularize $X$: $f(x)=0$, but also make its total transform normal crossings. To this end, suppose that $f(x)=0$ actually represents the strict transform of our original hypersurface in that year in the history of the blowings-up involved where the order at $a$ first becomes $d$. (We are following the transforms of the hypersurface at a sequence of points ``$a$'' over some original point.) Suppose there are $s=s(a)$ accumulated exceptional hypersurfaces $H_p$ passing through $a$; as above, we can also assume that $H_p$ is defined near $a$ by an equation \[ x_n + b_p({\tilde x}) = 0, \] $1\le p\le s$. (Each $\mu_a(b_p)\ge 1$.) The transformation law for the $b_p$ analogous to (2.6) is \[ b'_p({\tilde y}) = y_i^{-1} b_p \big({\tilde\sigma} ({\tilde y})\big),\qquad p=1,\ldots,s. \] Suppose now that in (2.7) we also have \[ b_p({\tilde x})^{d!} = ({\tilde x}^\Omega)^{d!} b_p^* ({\tilde x}),\qquad p=1,\ldots,s \] (and assume that either some $c_q^*(a)\neq 0$ or some $b_p^*(a)\neq 0$). Then the argument above shows that $\big( d(a'),s(a')\big)\le \big( d(a),s(a)\big)$ (with respect to the lexicographic ordering of pairs), and that if $\big( d(a'),s(a')\big)= \big( d(a),s(a)\big)$ then $1\le |\Omega(a')| < |\Omega(a)|$. ($s(a')$ counts the exceptional hypersurfaces $H_p'$ passing through $a'$. As long as $d$ does not drop, the new exceptional hypersurfaces accumulate simply as $y_i=0$ for certain $i=1,\ldots,n-1$, in suitable coordinates $(y_1,\ldots,y_{n-1})$ for the strict transform $N'=\{y_n=0\}$ of $N=\{ x_n=0\}$.) The induction on dimension can be realized in various ways. The simplest --- the method of [BM1, Section 4] --- is to apply the inductive hypothesis within a coordinate chart to the function of $n-1$ variables given by the product of all nonzero $c_q^{d!/(d-q)}$, all nonzero $b_p^{d!}$, and all their nonzero differences. The result is (2.7) and (2.10) (with $c_q^*(a)\neq 0$ or $b_p^*(a)\neq 0$ for some $q$ or $p$; see [BM1, Lemma 4.7]). Pullback of the coefficients $c_q$ by a blowing-up in $(x_1,\ldots,x_{n-1})$ with smooth centre $C$, corresponds to strict transform of $f$ by the blowing-up with centre $C\times \{ x_n-\mbox{axis}\}$. Thus we sacrifice the condition that each centre lie in the equimultiple locus (or even in $X$!). But we do get a very simple proof of local uniformization. In fact, we get the conclusion (2) of our desingularization theorem A, using a mapping $\sigma$: $M'\rightarrow M$ which is a composite of mappings that are either blowings-up with smooth centres or surjections of the form $\coprod_j U_j \rightarrow\bigcup_j U_j$, where the latter is a locally-finite open covering of a manifold and $\coprod$ means disjoint union. To prove our canonical desingularization theorem, we repeat the construction above in increasing codimension to obtain ${\rm inv}_X(a)=\big(\nu_1(a)$, $s_1(a);\, \nu_2(a),\ldots\,\big)$ --- $\big(\nu_1(a),s_1(a)\big)$ is $\big( d(a),s(a)\big)$ above --- together with a corresponding local ``presentation''. The latter means a local description of the locus of constant values of the invariant in terms of regular functions with assigned multiplicities, that survives certain blowings-up. ($N_1(a), {\cal H}_1(a)$ above is a presentation of $\nu_1$ at $a$.) \section{The desingularization algorithm} In this section we give a constructive definition of ${\rm inv}_X$ together with a corresponding presentation (in the hypersurface case). We illustrate the construction by applying the desingularization algorithm to an example --- a surface whose desingularization involves all the features of the general hypersurface case. We will use horizontal lines to separate from the example the general considerations that are needed at each step. \medskip\noindent {\em Example\ } {\em 3.1.}\quad Let $X\subset {\Bbb K}^3$ denote the hypersurface $g(x)=0$, where $g(x)=x_3^2 -x_1^2 x_2^3$. \begin{center} {\hskip .2in{\epsfxsize=2.5in\epsfbox{ed7.eps}{\vskip -2.25in\hskip -.11in{$x_3$}} {\vskip .35in\hskip 2in{$X$}} {\vskip .07in\hskip -.88in{$a=0$}} {\vskip .25in\hskip -2.04in{$x_1$}} {\vskip -.13in\hskip 2.47in{$x_2$}}}} \end{center} \vspace{.75in} Let $a=0$. Then $\nu_1(a)=\mu_a(g)=2$. Of course, $E(a)=\emptyset$, so that $s_1(a)=0$. (This is ``year zero''; there are no exceptional hypersurfaces.) Thus ${\rm inv}_1(a)=\big(\nu_1(a),s_1(a)\big)=(2,0)$. Let ${\cal G}_1(a)=\{ (x_3^2-x_1^2 x_2^3,2)\}$. We say that ${\cal G}_1(a)$ is a {\em codimension\/} 0 {\em presentation of\/} ${\rm inv}_{\frac{1}{2}}=\nu_1$ {\em at\/} $a$. (Here where $s_1(a)=0$, we can also say that ${\cal G}_1(a)$ is a codimension $0$ presentation of ${\rm inv}_1=(\nu_1,s_1)$ at $a$.) \linee In general, consider a hypersurface $X\subset M$. Let $a\in M$ and let $S_{{\rm inv}_{\frac{1}{2}}}(a)$ denote the germ at $a$ of $\{\, x:\ {\rm inv}_{\frac{1}{2}}(x)\ge{\rm inv}_{\frac{1}{2}}(a)\,\}$ $=$ the germ at $a$ of $\{\, x:\ {\rm inv}_{\frac{1}{2}}(x)= {\rm inv}_{\frac{1}{2}}(a)\,\}$. If $g\in{\cal O}_{M,a}$ generates the local ideal ${\cal I}_{X,a}$ of $X$ and $d=\nu_1(a)=\mu_a(g)$, then ${\cal G}_1(a)=\{(g,d)\}$ is a codimension $0$ presentation of ${\rm inv}_{\frac{1}{2}}=\nu_1$ at $a$. This means $S_{{\rm inv}_{\frac{1}{2}}}(a)$ coincides with the germ of the ``equimultiple locus'' $S_{{\cal G}_1(a)} = \{\, x:\ \mu_x(g)=d\,\}$, and that the latter condition survives certain transformations. More generally, suppose that ${\cal G}_1(a)$ is a finite collection of pairs $\{ (g,\mu_g)\}$, where each $g$ is a germ at $a$ of a regular function (i.e., $g\in{\cal O}_{M,a}$) with an ``assigned multiplicity'' $\mu_g\in{\Bbb Q}$, and where we assume that $\mu_a(g)\ge\mu_g$ for every $g$. Set \[ S_{{\cal G}_1(a)}\ =\ \{\, x:\ \mu_x(g)\ge\mu_g,\ \mbox{for all}\ (g,\mu_g)\in{\cal G}_1(a)\,\} ; \] $S_{{\cal G}_1(a)}$ is well-defined as a germ at $a$. To say that ${\cal G}_1(a)$ is a {\em codimension\/} $0$ {\em presentation of\/} ${\rm inv}_{\frac{1}{2}}$ {\em at\/} $a$ means that \[ S_{{\rm inv}_{\frac{1}{2}}}(a) = S_{{\cal G}_1(a)} \] and that this condition survives certain transformations: To be precise, we will consider triples of the form $\big( N=N(a)$, ${\cal H}(a)$, ${\cal E}(a)\big)$, where: $N$ is a germ of a submanifold of codimension $p$ at $a$ (for some $p\ge 0$). ${\cal H}(a)=\{(h,\mu_h)\}$ is a finite collection of pairs $(h,\mu_h)$, where $h\in{\cal O}_{N,a}$, $\mu_h\in{\Bbb Q}$ and $\mu_a(h)\ge\mu_h$. ${\cal E}(a)$ is a finite set of smooth (exceptional) hyperplanes containing $a$, such that $N$ and ${\cal E}(a)$ simultaneously have normal crossings and $N\not\subset H$, for all $H\in {\cal E}(a)$. A {\em local blowing-up\/} $\sigma$: $M'\rightarrow M$ over a neighbourhood $W$ of $a$, with smooth centre $C$, means the composite of a blowing-up $M'\rightarrow W$ with centre $C$, and the inclusion $W\hookrightarrow M$. \medskip\noindent {\em Definition\ } {\em 3.2.}\quad We say that $\big( N(a),{\cal H}(a),{\cal E}(a)\big)$ is a {\em codimension\/} $p$ {\em presentation of\/} ${\rm inv}_{\frac{1}{2}}$ {\em at\/} $a$ if: \smallskip (1) $S_{{\rm inv}_{\frac{1}{2}}}(a) = S_{{\cal H}(a)}$, where $S_{{\cal H}(a)}=\{\, x\in N:\ \mu_x(h)\ge\mu_h$, for all $(h,\mu_h)\in{\cal H}(a)\,\}$ (as a germ at $a$). (2) Suppose that $\sigma$ is a $\frac{1}{2}$-admissible local blowing-up at $a$ (with smooth centre $C$). Let $a'\in\sigma^{-1}(a)$. Then ${\rm inv}_{\frac{1}{2}}(a')={\rm inv}_{\frac{1}{2}}(a)$ if and only if $a'\in N'$ (where $N'=N(a')$ denotes the strict transform of $N$) and $\mu_{a'}(h')\ge \mu_{h'}$ for all $(h,\mu_h)\in{\cal H}(a)$, where $h'=y_{\rm exc}^{-\mu_h} h\circ\sigma$ and $\mu_{h'}=\mu_h$. ($y_{\rm exc}$ denotes a local generator of ${\cal I}_{\sigma^{-1}(C)}$.) In this case, we will write ${\cal H}(a')=\{\, (h',\mu_{h'}):\ (h,\mu_h)\in {\cal H}(a)\,\}$ and ${\cal E}(a')=\{\, H':\ H\in{\cal E}(a)\,\}\cup \{\sigma^{-1} (C)\}$. (3) Conditions (1) and (2) continue to hold for the transforms $X'$ and $\big( N(a'),{\cal H}(a'),{\cal E}(a')\big)$ of our data by sequences of morphisms of the following three types, at points $a'$ in the fibre of $a$ (to be also specified). \medskip The three types of morphisms allowed are the following. (Types (ii) and (iii) are not used in the actual desingularization algorithm. They are needed to prove invariance of the terms $\nu_2(a), \nu_3(a), \ldots$ of ${\rm inv}_X(a)$ by making certain sequences of ``test blowings-up'', as we will explain in Section 4; they are not explicitly needed in this section.) \medskip (i) $\frac{1}{2}$-{\em admissible local blowing-up\/} $\sigma$, and $a'\in\sigma^{-1}(a)$ such that ${\rm inv}_{\frac{1}{2}}(a')={\rm inv}_{\frac{1}{2}}(a)$. (ii) {\em Product with a line.\/} $\sigma$ is a projection $M'=W\times{\Bbb K}\rightarrow W\hookrightarrow M$, where $W$ is a neighbourhood of $a$, and $a'=(a,0)$. (iii) {\em Exceptional blowing-up.\/} $\sigma$ is a local blowing-up $M'\rightarrow W\hookrightarrow M$ over a neighbourhood $W$ of $a$, with centre $H_0\cap H_1$, where $H_0,H_1\in{\cal E}(a)$, and $a'$ is the unique point of $\sigma^{-1}(a)\cap H_1'$. \medskip The data is transformed to $a'$ in each case above, as follows: \medskip (i) $X'=$ strict transform of $X$; $\big( N(a'),{\cal H}(a'),{\cal E}(a')\big)$ as defined in 3.2(2) above. (ii) and (iii) $X'=\sigma^{-1}(X)$, $N(a')=\sigma^{-1}(N)$, ${\cal H}(a')=\{(h\circ\sigma,\mu_h)\}$. ${\cal E}(a') = \{\,\sigma^{-1}(H):\ H\in{\cal E}(a)\,\}\cup \{W\times 0\}$ in case (ii); ${\cal E}(a')=\{\, H':\ H\in{\cal E}(a),\ a'\in H'\,\}\cup \{\sigma^{-1}(C)\}$ in case (iii). \medskip If $\big( N(a),{\cal H}(a),{\cal E}(a)\big)$ is a presentation of ${\rm inv}_{\frac{1}{2}}$ at $a$, then $N(a)$ is called a subspace of {\em maximal contact} (cf. Section 2). Suppose now that ${\cal G}_1(a)$ is a codimension $0$ presentation of ${\rm inv}_{\frac{1}{2}}$ at $a$. (Implicitly, $N(a)=M$ and ${\cal E}(a)=\emptyset$.) Assume, moreover, that there exists $(g,\mu_g)=(g_*,\mu_{g_*})\in {\cal G}_1(a)$ such that $\mu_a(g_*)=\mu_{g_*}$ (as in Example 3.1). We can always assume that each $\mu_g\in{\Bbb N}$, and even that all $\mu_g$ coincide: Simply replace each $(g,\mu_g)$ by $(g^{e/\mu_g},e)$, for suitable $e\in{\Bbb N}$. Then, after a linear coordinate change if necessary, we can assume that $(\partial^d g_*/\partial x_n^d)(a)\neq 0$, where $d=\mu_{g_*}$. Set \begin{eqnarray*} z & = & \frac{\partial^{d-1} g_*}{ \partial x_n^{d-1}}\in {\cal O}_{M,a}\\ N_1 = N_1(a) & = &\{ z=0\} \\ {\cal H}_1(a) & = &\left\{ \left( \frac{\partial^q g}{\partial x_n^q} \bigg| _{N_1}, \mu_g-q\right):\ 0\le q < \mu_g,\ (g,\mu_g)\in{\cal G}_1(a)\right\} . \end{eqnarray*} Then $\big( N_1(a),{\cal H}_1(a),{\cal E}_1(a)=\emptyset\big)$ is a codimension $1$ presentation of ${\rm inv}_{\frac{1}{2}}$ at $a$: This is an assertion about the way our data transforms under sequences of morphisms of types (i), (ii) and (iii) above. The effect of a transformation of type (i) is essentially described by the calculation in Section 2. The effect of a transformation of type (ii) is trivial, and that for type (iii) can be understood in a similar way to (i): see [BM5, Props. 4.12 and 4.19] for details. \medskip\noindent {\em Definition\ } {\em 3.3.}\quad We define \[ \mu_2(a) = \min_{{\cal H}_1(a)}\, \frac{\mu_a(h)}{\mu_h}. \] Then $1\le \mu_2(a)\le\infty$. If ${\cal E}(a)=\emptyset$ (as in year zero), we set \[ \nu_2(a)=\mu_2(a) \] and ${\rm inv}_{1\frac{1}{2}}(a)=\big({\rm inv}_1(a);\nu_2(a)\big)$. Then $\nu_2(a)\le\infty$. Moreover, $\nu_2(a)=\infty$ if and only if ${\cal G}_1(a)\sim \{(z,1)\}$. (This means that the latter is also a presentation of ${\rm inv}_{\frac{1}{2}}$ at $a$.) If $\nu_2(a)=\infty$, then we set ${\rm inv}_X(a)={\rm inv}_{1\frac{1}{2}}(a)$. ${\rm inv}_X(a)=(d,0,\infty)$ if and only if $X$ is defined (near $a$) by the equation $z^d=0$; in this case, the desingularization algorithm can do no more, unless we blow-up with centre $|X|$! \linee In Example 3.1, $\mu_a(g)=2=\mu_g$, and by the construction above we get the following codimension $1$ presentation of ${\rm inv}_{\frac{1}{2}}$ (or ${\rm inv}_1$) at $a$: \[ N_1(a)=\{x_3=0\},\qquad {\cal H}_1(a) = \{ (x_1^2 x_2^3,2)\}. \] Thus $\nu_2(a)=\mu_2(a)=5/2$. As a codimension $1$ presentation of ${\rm inv}_{1\frac{1}{2}}$ (or ${\rm inv}_2$) at $a$, we can take \[ N_1(a),\qquad {\cal G}_2(a) = \{ (x_1^2 x_2^3,5)\} . \] \linee In general, ``presentation of ${\rm inv}_r$'' (or ``of ${\rm inv}_{r+\frac{1}{2}}$'') means the analogue of ``presentation of ${\rm inv}_{\frac{1}{2}}$'' above. Suppose that $\big( N_1(a),{\cal H}_1(a)\big)$ is a codimension $1$ presentation of ${\rm inv}_1$ at $a$ $\big({\cal E}_1(a)=\emptyset\big)$. Assume that $1\le\nu_2(a)<\infty$. (In year zero, we always have $\nu_2(a)=\mu_2(a)\ge 1$.) Let \[ {\cal G}_2(a) = \big\{\, \big(h,\nu_2(a)\mu_h\big):\ (h,\mu_h)\in{\cal H}_1(a)\big\} . \] Then $\big( N_1(a),{\cal G}_2(a)\big)$ is a codimension $1$ presentation of ${\rm inv}_{\frac{1}{2}}$ at $a$ (or of ${\rm inv}_2$ at $a$, when $s_2(a)=0$ as here). Clearly, there exists $(g_*,\mu_{g_*})\in {\cal G}_2(a)$ such that $\mu_a(g_*)=\mu_{g_*}$. This completes a cycle in the recursive definition of ${\rm inv}_X$, and we can now repeat the above constructions: Let $d=\mu_{g_*}$. After a linear transformation of the coordinates $(x_1,\ldots,x_{n-1})$ of $N_1(a)$, we can assume that $(\partial^d g_* /\partial x_{n-1}^d) (a)\neq 0$. We get a codimension $2$ presentation of ${\rm inv}_2$ at $a$ by taking \begin{eqnarray*} N_2(a) & = & \left\{\, x\in N_1(a):\ \frac{\partial^{d-1} g_*} {\partial x_{n-1}^{d-1}} (x) = 0\,\right\} ,\\ {\cal H}_2(a) & = & \left\{\,\left( \frac{\partial^q g}{\partial x_{n-1}^q} \bigg|_{N_2(a)} , \mu_g -q\right):\ 0\le q<\mu_g,\ (g,\mu_g)\in {\cal G}_2(a)\,\right\} . \end{eqnarray*} In our example, the calculation of a codimension $2$ presentation can be simplified by the following useful observation: Suppose there is $(g,\mu_g)\in{\cal G}_2(a)$ with $\mu_a(g)=\mu_g$ and $g=\prod g_i^{m_i}$. If we replace $(g,\mu_g)$ in ${\cal G}_2(a)$ by the collection of $(g_i,\mu_{g_i})$, where each $\mu_{g_i}=\mu_a(g_i)$, then we obtain an (equivalent) presentation of ${\rm inv}_2$. \linee In our example, therefore, \[ N_1(a) = \{ x_3=0\},\qquad {\cal G}_2(a) = \{ (x_1,1), (x_2,1)\} \] is a codimension $1$ presentation of ${\rm inv}_2$ at $a$. It follows immediately that \[ N_2(a)=\{x_2=x_3=0\},\qquad {\cal H}_2(a) = \{ (x_1,1)\} \] is a codimension $2$ presentation of ${\rm inv}_2$ at $a$. Then $\nu_3(a)=\mu_3(a)=1$ and, as a codimension $3$ presentation of ${\rm inv}_{2\frac{1}{2}}$ (or of ${\rm inv}_3$) at $a$, we can take \[ N_3(a) = \{ x_1=x_2=x_3=0\},\qquad {\cal H}_3(a) = \emptyset . \] We put $\nu_4(a)=\mu_4(a)=\infty$. Thus we have \[ {\rm inv}_X (a) = (2,0;\, 5/2,0;\, 1,0;\,\infty) \] and $S_{{\rm inv}_X}(a)=S_{{\rm inv}_3}(a)=N_3(a)=\{a\}$. The latter is the centre $C_0$ of our first blowing-up $\sigma_1$: $M_1\rightarrow M_0={\Bbb K}^3$; $M_1$ can be covered by three coordinate charts $U_i$, $i=1,2,3$, where each $U_i$ is the complement in $M_1$ of the strict transform of the hyperplane $\{ x_i=0\}$. The strict transform $X_1=X'$ of $X$ lies in $U_1\cup U_2$. To illustrate the algorithm, we will follow our construction at a sequence of points over $a$, choosing after each blowing-up a point in the fibre where ${\rm inv}_X$ has a maximum value in a given coordinate chart. \medskip {\em Year one.}\quad $U_1$ has a coordinate system $(y_1,y_2,y_3)$ in which $\sigma_1$ is given by the transformation \[ x_1=y_1,\quad x_2=y_1y_2,\quad x_3=y_1y_3 . \] Then $X_1\cap U_1 =V(g_1)$, where \[ g_1 = y_1^{-2} g\circ\sigma_1 = y_3^2 - y_1^3 y_2^3 . \] Consider $b=0$. Then $E(b)=\{ H_1\}$, where $H_1$ is the exceptional hypersurface $H_1=\sigma_1^{-1}(a) = \{ y_1=0\}$. Now, $\nu_1(b)=2=\nu_1(a)$. Therefore $E^1(b)=\emptyset$ and $s_1(b)=0$. We write ${\cal E}_1(b)=E(b){\backslash} E^1(b)$, so that ${\cal E}_1(b)=E(b)$ here. Let ${\cal F}_1(b)={\cal G}_1(b)=\{ (g_1,2)\}$. Then $\big( N_0(b)=M_1, {\cal F}_1(b),{\cal E}_1(b)\big)$ is a codimension $0$ presentation of ${\rm inv}_1$ at $b$. Set \[ N_1(b) = \{ y_3=0\} = N_1(a)',\qquad {\cal H}_1(b) = \{ (y_1^3 y_2^3,2)\} ; \] $\big( N_1(b), {\cal H}_1(b),{\cal E}_1(b)\big)$ is a codimension $1$ presentation of ${\rm inv}_1$ at $b$. As before, \[ \mu_2(b) = \min_{{\cal H}_1(b)} \frac{\mu_b(h)}{\mu_h} = \frac{6}{2} = 3 . \] But, here, in the presence of nontrivial ${\cal E}_1(b)$, $\nu_2(b)$ will involve first factoring from the $h\in {\cal H}_1(b)$ the exceptional divisors in ${\cal E}_1(b)$ (taking, in a sense, ``internal strict transforms'' at $b$ of the elements of ${\cal H}_1(a)$). \linee In general, we define \[ {\cal F}_1(b) = {\cal G}_1(b)\cup \big(E^1(b),1\big) , \] where $\big( E^1(b),1\big)$ denotes $\{ \,(y_H,1):\ H\in E^1(b)\,\}$, and $y_H$ means a local generator of the ideal of $H$. Then $\big( N_0(b),{\cal F}_1(b),{\cal E}_1(b)\big)$ is a codimension $0$ presentation of ${\rm inv}_1=(\nu_1,s_1)$ at $b$, and there is a codimension $1$ presentation $\big( N_1(b),{\cal H}_1(b),{\cal E}_1(b)\big)$ as before. The construction of Section 2 above shows that we can choose the coordinates $(y_1,\ldots,y_{n-1})$ of $N_1(b)$ so that each $H\in{\cal E}_1(b)=E(b){\backslash} E^1(b)$ is $\{y_i=0\}$, for some $i= 1,\ldots,n-1$; we again write $y_H=y_i$. (In other words, ${\cal E}_1(b)$ and $N_1(b)$ simultaneously have normal crossings, and $N_1(b)\not\subset H$, for all $H\in{\cal E}_1(b)$.) \medskip\noindent {\em Definition\ } {\em 3.4.}\quad For each $H\in {\cal E}_1(b)$, we set \[ \mu_{2H}(b) = \min_{(h,\mu_h)\in{\cal H}_1(b)} \frac {\mu_{H,b}(h)}{\mu_h} , \] where $\mu_{H,b}(h)$ denotes the {\em order\/} of $h$ {\em along\/} $H$ at $b$; i.e., the order to which $y_H$ factors from $h\in{\cal O}_{N,b}$, $N=N_1(b)$, or $\max\{\, k: \ h\in{\cal I}_{H,b}^k\,\}$, where ${\cal I}_{H,b}$ is the ideal of $H\cap N$ in ${\cal O}_{N,b}$. We define \[ \nu_2(b) = \mu_2(b) - \sum_{H\in{\cal E}_1(b)} \mu_{2H} (b) . \] \linee In our example, \[ \nu_2(b) = \mu_2(b) - \mu_{2H_1}(b) = 3 - \frac{3}{2} = \frac{3}{2} . \] \linee Write \[ D_2(b) = \prod_{H\in {\cal E}_1(b)} y_H^{\mu_{2H}(b)} . \] Suppose, as before, that all $\mu_h$ are equal: say all $\mu_h = d\in{\Bbb N}$. Then $D^d=D_2(b)^d$ is the greatest common divisor of the $h$ that is a monomial in the exceptional coordinates $y_H$, $H\in{\cal E}_1(b)$. For each $h\in {\cal H}_1(b)$, write $h=D^d g$ and set $\mu_g = d\nu_2 (b)$; then $\mu_b(g)\ge\mu_g$. Clearly, $\nu_2(b) = \min_g \mu_b (g)/d$. Moreover, $0\le \nu_2(b)\le\infty$, and $\nu_2(b)=\infty$ if and only if $\mu_2(b)=\infty$. If $\nu_2(b)=0$ or $\infty$, we put ${\rm inv}_X(b)={\rm inv}_{1\frac{1}{2}}(b)$. If $\nu_2(b)=\infty$, then $S_{{\rm inv}_X}(b)=N_1(b)$. If $\nu_2(b)=0$ and ${\cal G}_2(b) = \big\{\big( D_2(b),1\big)\big\}$, then $\big( N_1(b),{\cal G}_2(b),{\cal E}_1(b)\big)$ is a codimension $1$ presentation of ${\rm inv}_X$ at $b$; in particular, \[ S_{{\rm inv}_X}(b) = \big\{\, y\in N_1(b):\ \mu_y\big( D_2(b)\big)\ge 1\big\} \] (cf. Section 2). Consider the case that $0<\nu_2(b)<\infty$. Let ${\cal G}_2(b)$ denote the collection of pairs $(g,\mu_g)= \big( g, d\nu_2(b)\big)$ for all $(h,\mu_h)=(h,d)$, as above, together with the pair $\big( D_2(b)^d,\big(1-\nu_2(b)\big)d\big)$ {\em provided that\/} $\nu_2(b)<1$. Then $\big( N_1(b),{\cal G}_2(b),{\cal E}_1(b)\big)$ is a codimension $1$ presentation of ${\rm inv}_{1\frac{1}{2}}$ at $b$. In the latter case, we introduce $E^2(b)\subset {\cal E}_1(b)$ as in 1.12, and we set $s_2(b) = \# E^2(b)$, ${\cal E}_2(b)= {\cal E}_1(b){\backslash} E^2(b)$. Set \[ {\cal F}_2(b) = {\cal G}_2(b) \cup \big( E^2(b),1\big) . \] Then $\big( N_1(b),{\cal F}_2(b),{\cal E}_1(b)\big)$ is a codimension $1$ presentation of ${\rm inv}_2$ at $b$, and we can pass to a codimension $2$ presentation $\big( N_2(b),{\cal H}_2(b),{\cal E}_2(b)\big)$. Here it is important to replace ${\cal E}_1(b)$ by the subset ${\cal E}_2(b)$, to have the property that ${\cal E}_2(b)$, $N_2(b)$ simultaneously have normal crossings and $N_2(b)\not\subset H$, for all $H\in{\cal E}_2(b)$. (Again, the main r\^ole of ${\cal E}$ in a presentation is to prove invariance of the $\mu_{2H} (\cdot)$ and in general of the $\mu_{3H}(\cdot),\ldots,$ as in Section 4.) \linee Returning to our example (in year one), we have ${\cal H}_1(b)=\{ (y_1^3 y_2^3,2)\}$, so that $D_2(b)=y_1^{3/2}$. We can take ${\cal G}_2(b)=\{(y_2^3,3)\}$ or, equivalently, ${\cal G}_2(b)=\{ (y_2,1)\}$ to get a codimension $1$ presentation $\big( N_1(b), {\cal G}_2(b),{\cal E}_1(b)\big)$ of ${\rm inv}_{1\frac{1}{2}}$ at $b$. Now, $E^2(b)=\{H_1\}$, so that $s_2(b)=1$. We set \[ {\cal F}_2(b) = {\cal G}_2(b)\cup \big( E^2(b),1\big) = \{(y_1,1),(y_2,1)\} \] and ${\cal E}_2(b)={\cal E}_1(b){\backslash} E^2(b)=\emptyset$. Then $\big( N_1(b),{\cal F}_2(b),{\cal E}_1(b)\big)$ is a codimension $1$ presentation of ${\rm inv}_2$ at $b$, and we can get a codimension $2$ presentation $\big( N_2(b),{\cal H}_2(b), {\cal E}_2(b)\big)$ of ${\rm inv}_2$ at $b$ by taking $N_2(b)= \{ y_2=y_3=0\}$ and ${\cal H}_2(b)=\{ (y_1,1)\}$. It follows that $\nu_3(b)=1$. Since $E^3(b)=\emptyset$, $s_3(b)=0$. We get a codimension $3$ presentation of ${\rm inv}_3$ at $b$ by taking \[ N_3(b) = \{ y_1=y_2=y_3=0\} = \{b\},\qquad H_3(b) = \emptyset . \] Therefore, \[ {\rm inv}_X(b) = \left( 2,0;\, \frac{3}{2},1;\, 1,0;\, \infty\right) \] and $S_{{\rm inv}_X}(b)=S_{{\rm inv}_2}(b)=\{b\}$. The latter is the centre of the next blowing-up $\sigma_2$. $\sigma_2^{-1}(U_1)$ is covered by 3 coordinate charts $U_{1i}=\sigma_2^{-1}(U_1){\backslash} \{ y_i=0\}'$, $i=1,2,3$. For example, $U_{12}$ has coordinates $(z_1,z_2,z_3)$ with respect to which $\sigma_2$ is given by \[ y_1 = z_1 z_2,\qquad y_2 = z_2,\qquad y_3 = z_2 z_3 . \] \linee \medskip\noindent {\em Remark\ } {\em 3.5.}\quad {\em Zariski-semicontinuity of the invariant.} Each point of $M_j$, $j=0,1,\ldots$, admits a coordinate neighbourhood $U$ such that, for all $x_0\in U$, $\{ \, x\in U:\ {\rm inv}_{\!\!\lower2pt\hbox{$\displaystyle\cdot$}} (x) \le {\rm inv}_{\!\!\lower2pt\hbox{$\displaystyle\cdot$}} (x_0)\,\}$ is Zariski-open in $U$ (i.e., the complement of a Zariski-closed subset of $U$): For ${\rm inv}_{\frac{1}{2}}$, this is just Zariski-semicontinuity of the order of a regular function $g$ (a local generator of the ideal of $X$). For ${\rm inv}_1$, the result is a consequence of the following semicontinuity assertion for $E^1(x)$: There is a Zariski-open neighbourhood of $x_0$ in $U$, in which $E^1(x)=E(x)\cap E^1(x_0)$, for all $x\in S_{{\rm inv}_ {\frac{1}{2}}(x_0)} = \{\, x\in U:\ {\rm inv}_{\frac{1}{2}}(x)\ge {\rm inv}_{\frac{1}{2}}(x_0)\,\}$. (See [BM5, Prop. 6.6] for a simple proof.) For ${\rm inv}_{1\frac{1}{2}}$: Suppose that $\mu_k=d\in{\Bbb N}$, for all $(h,\mu_h)\in{\cal H}_1(x_0)$, as above. Then, in a Zariski-open neighbourhood of $x_0$ where $S_{{\rm inv}_{1}(x)}=\{\, x:\ {\rm inv}_1(x)={\rm inv}_1(x_0)\,\}$, we have \[ d\nu_2(x) = \min_{{\cal H}_1(x_0)} \mu_x \left( \frac{h}{D_2(x_0)^d}\right) ,\qquad x\in S_{{\rm inv}_{1}(x_0)} . \] Semicontinuity of $\nu_2(x)$ is thus a consequence of semicontinuity of the order of an element $g=h/D_2(x_0)^d$ such that $\mu_{x_0}(g)=d\nu_2(x_0)$. Likewise for ${\rm inv}_2$, ${\rm inv}_{2\frac{1}{2}}$, $\ldots$. \linee {\em Year two.}\quad Let $X_2$ denote the strict transform $X_1'$ of $X_1$ by $\sigma_2$. Then $X_2\cap U_{12}=V(g_2)$, where \[ g_2 = z_2^{-2} g_1\circ \sigma_2 = z_3^2 -z_1^3 z_2^4 . \] Let $c$ be the origin of $U_{12}$. Then $E(c)=\{ H_1,H_2\}$ where \begin{eqnarray*} &&H_1 = \{ y_1=0\}' =\{z_1=0\} ,\\ &&H_2 = \sigma_2^{-1} (b) = \{ z_2 =0\} . \end{eqnarray*} We have $\nu_2(c)=2=\nu_2(a)$. Therefore, $E^1(c)=\emptyset$, $s_1(c)=0$, ${\cal E}_1(c)=E(c)$. ${\cal F}_1(c)={\cal G}_1(c)=\{ (g_2,2)\}$ provides a codimension $0$ presentation of ${\rm inv}_1$ at $c$, and we get a codimension $1$ presentation by taking \[ N_1(c) = \{ z_3=0\},\qquad {\cal H}_1(c) = \{ (z_1^3 z_2^4,2)\} . \] Therefore $\mu_2(c)=7/2$, $\mu_{2H_1}(c) = 3/2$ and $\mu_{2H_2}(c) = 4/2 =2$, so that $\nu_2(c)=0$ and \[ {\rm inv}_X(c) = (2,0;\,0) . \] Moreover, $D_2(c)=z_1^{\frac{3}{2}} z_2^2$, and we get a codimension $1$ presentation of ${\rm inv}_X={\rm inv}_{1\frac{1}{2}}$ at $c$ using \[ N_1(c) = \{ z_3=0\},\qquad {\cal G}_2(c) = \{ (z_1^{\frac{3}{2}} z_2^2,1)\} . \] Therefore, \[ S_{{\rm inv}_X}(c) = S_{{\rm inv}_{1\frac{1}{2}}} (c) = \{ z_1=z_3=0\} \cup \{ z_2=z_3=0\} ; \] of course, $\{ z_1=z_3=0\} = S_{{\rm inv}_X} (c)\cap H_1$ and $\{ z_2=z_3=0\} = S_{{\rm inv}_X}(c)\cap H_2$. \linee \medskip\noindent {\em Remark\ } {\em 3.6.}\quad In general, suppose that ${\rm inv}_X(c)={\rm inv}_{t+\frac{1}{2}}(c)$ and $v_{t+1}(c)=0$. (We assume $c\in M_j$, for some $j=1,2,\ldots$.) Then ${\rm inv}_X$ has a codimension $t$ presentation at $c$: $N_t(c) = \{ z_{n-t+1} = \cdots = z_n = 0\}$, ${\cal G}_{t+1}(c) = \big\{ \big( D_{t+1} (c),1\big)\big\}$, where $D_{t+1}(c)$ is a monomial with rational exponents in the exceptional divisors $z_H$, $H\in {\cal E}_t(c)$; $N_t(c)$ has coordinates $(z_1,\ldots,z_{n-t})$ in which each such $z_H=z_i$, for some $i=1,\ldots,n-t$. It follows that each component $Z$ of $S_{{\rm inv}_X}(c)$ has the form \[ Z = S_{{\rm inv}_X} (c) \cap \bigcap \{\, H\in E(c):\ Z\subset H\,\} ; \] we will write $Z=Z_I$, where $I=\{ \,H\in E(c):\ Z\subset H\,\}$. It follows that, if $U$ is any open neighbourhood of $c$ on which ${\rm inv}_X(c)$ is a maximum value of ${\rm inv}_X$, then every component $Z_I$ of $S_{{\rm inv}_X}(c)$ extends to a global smooth closed subset of $U$: First consider any total order on $\{\, I:\ I\subset E_j\,\}$. For any $c\in M_j$, set \begin{eqnarray*} J(c) & = &\max\{\, I:\ Z_I\ \mbox{is a component of}\ S_{{\rm inv}_X}(c)\,\} ,\\ {\rm inv}_X^{\rm e} (c) & = & \big( {\rm inv}_X (c);\, J(c)\big) . \end{eqnarray*} Then ${\rm inv}_X^{\rm e}$ is Zariski-semicontinuous (again comparing values of ${\rm inv}_X^{\rm e}$ lexicographically), and its locus of maximum values on any given open subset of $M_j$ is smooth. Of course, given $c\in M_j$ and a component $Z_I$ of $S_{{\rm inv}_X}(c)$, we can choose the ordering of $\{\, J:\ J\subset E_j\,\}$ so that $I=J(c)=\max\{\, J:\ J\subset E_j\,\}$. It follows that, if $U$ is any open neighbourhood of $c$ on which ${\rm inv}_X(c)$ is a maximum value of ${\rm inv}_X$, then $Z_I$ extends to a smooth closed subset of $U$. To obtain an algorithm for canonical desingularization, we can choose as each successive centre of blowing up the maximum locus of ${\rm inv}_X^{\rm e}(\cdot) = \big({\rm inv}_X(\cdot), J(\cdot)\big)$, where $J$ is defined as above using the following total ordering of the subsets of $E_j$: Write $E_j=\{ H_1^j,\ldots,H_j^j\}$, where each $H_i^j$ is the strict transform in $M_j$ of the exceptional hypersurface $H_i^i=\sigma_i^{-1} (C_{i-1})\subset M_i$, $i=1,\ldots,j$. We can order $\{\, I:\ I\subset E_j\,\}$ by associating to each subset $I$ the lexicographic order of the sequence $(\delta_1,\ldots,\delta_j)$, where $\delta_i=0$ if $H_i^j\not\in I$ and $\delta_i=1$ if $H_i^j\in I$. \linee In our example, year two, we have \[ S_{{\rm inv}_X}(c) = \big( S_{{\rm inv}_X}(c)\cap H_1\big) \cup \big( S_{{\rm inv}_X}(c)\cap H_2\big) . \] (Each $H_i$ is $H_i^2$ in the notation preceding.) The order of $H_1$ (respectively, $H_2$) is $(1,0)$ (respectively, $(0,1)$), so that $J(c)=\{ H_1\}$ and the centre of the third blowing-up $\sigma_3$ is $C_2=S_{{\rm inv}_X}(c)\cap H_1 = \{ z_1=z_3=0\}$. Thus $\sigma_3^{-1}(U_{12})=U_{121}\cup U_{123}$, where $U_{12i}=\sigma_3^{-1}(U_{12}){\backslash} \{ z_i=0\}'$, $i=1,3$. The strict transform of $X_2\cap U_{12}$ lies in $U_{121}$; the latter has coordinates $(w_1,w_2,w_3)$ in which $\sigma_3$ can be written \[ z_1=w_1,\qquad z_2=w_2,\qquad z_3=w_1w_3 . \] \medskip {\em Year three.}\quad Let $X_3$ denote the strict transform of $X_2$ by $\sigma_3$. Then $X_3\cap U_{121} = V(g_3)$, where $g_3(w)=w_3^2 - w_1 w_2^4$. Let $d=0$ in $U_{121}$. There are three exceptional hypersurfaces $H_1=\{ z_1=0\}'$, $H_2=\{ z_2=0\}' = \{ w_2=0\}$ and $H_3=\sigma_3^{-1} (C_2) = \{ w_1=0\}$; since $H_1\not\ni d$, $E(d)=\{ H_2,H_3\}$. We have $\nu_1(d)=2=\nu_1(a)$. Therefore, $E^1(d)=\emptyset$, $s_1(d)=0$ and ${\cal E}_1(d)=E(d)$. ${\cal F}_1(d)={\cal G}_1(d)=\{ (g_3,2)\}$ provides a codimension $0$ presentation of ${\rm inv}_1$ at $d$, and we get a codimension $1$ presentation by taking \[ N_1(d) = \{ w_3=0\},\qquad {\cal H}_1(d) = \{ (w_1 w_2^4 ,2)\} . \] Therefore, $\mu_2(c)=\frac{5}{2}$ and $D_2(d)=w_1^{\frac{1}{2}} w_2^2$, so that $\nu_2(d)=0$ and \[ {\rm inv}_X (d) = (2,0,0) = {\rm inv}_X(c) ! \] However, \[ \mu_2(d) = \frac{5}{2} < \frac{7}{2} = \mu_2(c) ; \] i.e., $1\le\mu_X(d) < \mu_X(c)$, where $\mu_X=\mu_2$ (cf. (2.8) ff.). We get a codimension $1$ presentation of ${\rm inv}_X={\rm inv}_{1\frac{1}{2}}$ at $d$ by taking \[ N_1(d) = \{ w_3=0\},\qquad {\cal G}_2(d) = \big\{ \big( D_2(d),1\big)\big\} . \] Therefore, \[ S_{{\rm inv}_X}(d) = S_{{\rm inv}_1}(d) = \{ w_2 = w_3 = 0\} , \] so we let $\sigma_4$ be the blowing-up with centre $C_3=\{ w_2 = w_3 = 0\}$. Then $\sigma_4^{-1}(U_{121}) = U_{1212} \cup U_{1213}$, where $U_{121i}=\sigma_4^{-1} (U_{121}){\backslash} \{ w_i = 0\}'$, $i=2,3$; $U_{1212}$ has coordinates $(v_1,v_2,v_3)$ in which $\sigma_4$ is given by \[ w_1=v_1,\qquad w_2=v_2,\qquad w_3=v_2 v_3 . \] \medskip {\em Year four.}\quad Let $X_4$ be the strict transform of $X_3$. Then $X_4\cap U_{1212}=V(g_4)$, where $g_4(v)=v_3^2-v_1 v_2^2$. Let $e=0$ in $U_{1212}$. Then $E(e)=\{ H_3,H_4\}$, where $H_3=\{ w_1=0\}'=\{v_1=0\}$ and $H_4=\sigma_4^{-1} (C_3)=\{v_2=0\}$. Again $\nu_1(e)=2=\nu_1(a)$, so that $E^1(e)=\emptyset$, $s_1(e)=0$ and ${\cal E}_1(e)=E(e)$. Calculating as above, we obtain $\mu_2(e)=\frac{3}{2}$ and $D_2(e)=v_1^{\frac{1}{2}} v_2$, so that $\nu_2(e)=0$ and ${\rm inv}_X(e)=(2,0;\, 0)$ again. But now $\mu_X(e)=\mu_2(e)=3/2$. Our invariant ${\rm inv}_X$ is presented at $e$ by \[ N_1(e) = \{ v_3=0\},\qquad {\cal G}_2(e) = \{ (v_1^{\frac{1}{2}} v_2,1)\} . \] Therefore, $S_{{\rm inv}_X}(e)=\{ v_2=v_3=0\}$. Taking as $\sigma_5$ the blowing-up with centre $C_4=S_{{\rm inv}_X}(e)$, the strict transform $X_5$ becomes smooth (over $U_{1212}$). ($\mu_2(e)-1<1$, so $\nu_1(\cdot)$ must decrease over $C_4$.) Further blowings-up are still needed to obtain the stronger assertion of embedded resolution of singularities. \medskip\noindent {\em Remark\ } {\em 3.7.}\quad The hypersurface $V(g_4)$ in year four above is called ``Whitney's umbrella''. Consider the same hypersurface $X=\{x_3^2-x_1x_2^2=0\}$ but without a history of blowings-up; i.e., $E(\cdot)=\emptyset$. Let $a=0$. In this case, ${\rm inv}_{1\frac{1}{2}}(a)=(2,0;\, \frac{3}{2})$, and we get a codimension $1$ presentation of ${\rm inv}_{1\frac{1}{2}}$ at $a$ using \[ N_1(a) =\{ x_3=0\},\qquad {\cal G}_2(a) = \{ (x_1 x_2^2 ,3)\} \] or, equivalently, ${\cal G}_2(a)=\{ (x_1,1),(x_2,1)\}$, as in year zero of Example 3.1. Therefore, \[ {\rm inv}_X(a) = (2,0;\, \frac{3}{2},0;\, 1,0;\, \infty) . \] As a centre of blowing up we would choose $C=S_{{\rm inv}_X}(a)= \{ a\}$ --- not the $x_1$-axis as in year four above, although the singularity is the same! \section{Key properties of the invariant} Our main goal in this section is to explain why ${\rm inv}_X(a)$ is indeed an invariant. Once we establish invariance, the Embedded Desingularization Theorem A follows directly from local properties of ${\rm inv}_X$. The crucial properties have already been explained in Section 3 above; we summarize them in the following theorem. \medskip\noindent{\bf Theorem\ } {\bf B.}\quad {\rm ([BM5, Th. 1.14].)} {\em Consider any sequence of ${\rm inv}_X$-admissible (local) blowings-up (1.8). Then the following properties hold: {\rm (1) Semicontinuity.} (i) For each $j$, every point of $M_j$ admits a neighbourhood $U$ such that ${\rm inv}_X$ takes only finitely many values in $U$ and, for all $a\in U$, $\{ x\in U:\ {\rm inv}_X(x)\le{\rm inv}_X(a)\}$ is Zariski-open in $U$. (ii) ${\rm inv}_X$ is {\rm infinitesimally upper-semicontinuous} in the sense that ${\rm inv}_X(a)\le{\rm inv}_X\big(\sigma_j(a)\big)$ for all $a\in M_j$, $j\ge 1$. {\rm (2) Stabilization.} Given $a_j\in M_j$ such that $a_j=\sigma_{j+1}(a_{j+1})$, $j=0,1,2,\ldots$, there exists $j_0$ such that ${\rm inv}_X(a_j)={\rm inv}_X(a_{j+1})$ when $j\ge j_0$. (In fact, any nonincreasing sequence in the value set of ${\rm inv}_X$ stabilizes.) {\rm (3) Normal crossings.} Let $a\in M_j$. Then $S_{{\rm inv}_X}(a)$ and $E(a)$ simultaneously have only normal crossings. Suppose ${\rm inv}_X(a)=\big(\ldots; \nu_{t+1} (a)\big)$. If $\nu_{t+1}(a)=\infty$, then $S_{{\rm inv}_X}(a)$ is smooth. If $\nu_{t+1}(a)=0$ and $Z$ denotes any irreducible component of $S_{{\rm inv}_X}(a)$, then \[ Z=S_{{\rm inv}_X} (a) \cap \bigcap \{ H\in E(a):\ Z\subset H\} . \] {\rm (4) Decrease.} Let $a\in M_j$ and suppose ${\rm inv}_X(a)=\big(\ldots; \nu_{t+1}(a)\big)$. If $\nu_{t+1}(a)=\infty$ and $\sigma$ is the local blowing-up of $M_j$ with centre $S_{{\rm inv}_X}(a)$, then ${\rm inv}_X(a')<{\rm inv}_X(a)$ for all $a'\in\sigma^{-1}(a)$. If $\nu_{t+1}(a)=0$, then there is an additional invariant $\mu_X(a)=\mu_{t+1}(a)\ge 1$ such that, if $Z$ is an irreducible component of $S_{{\rm inv}_X}(a)$ and $\sigma$ is the local blowing-up with centre $Z$, then $\big( {\rm inv}_X(a'),\mu_X(a')\big)< \big( {\rm inv}_X(a), \mu_X(a)\big)$ for all $a'\in\sigma^{-1}(a)$. ($e_t! \mu_X(a)\in{\Bbb N}$, where $e_t$ is defined as in Section 1 or in the proof following.)} \medskip\noindent{\em Proof.\quad } The semicontinuity property (1)(i) has been explained in Remark 3.5. Infinitesimal upper-semicontinuity (1)(ii) is immediate from the definition of the $s_r(a)$ and from infinitesimal upper-semicontinuity of the order of a function on blowing up locally with smooth centre in its equimultiple locus. (The latter property is an elementary Taylor series computation, and is also clear from the calculation in Section 2 above.) The stabilization property (2) for ${\rm inv}_{\frac{1}{2}}$ is obvious in the hypersurface case because then ${\rm inv}_{\frac{1}{2}} (a)= \nu_1(a)\in{\Bbb N}$. (In the general case, we need to begin with stabiization of the Hilbert-Samuel function; see [BM2, Th. 5.2.1] for a very simple proof of this result due originally to Bennett [Be].) The stabilization assertion for ${\rm inv}_X$ follows from that for ${\rm inv}_{\frac{1}{2}}$ and from infinitesimal semicontinuity because, although $\nu_{r+1}(a)$, for each $r>0$, is perhaps only rational, our construction in Section 3 shows that $e_r! \nu_{r+1} (a)\in{\Bbb N}$, where $e_1=\nu_1(a)$ and $e_{r+1}=\max \{ e_r !, e_r! \nu_{r+1} (a)\}$, $r>0$. (In the general case, the Hilbert-Samuel function $H_{X_j,a}(\ell)$ coincides with a polynomial if $\ell\ge k$, for $k$ large enough, and we can take as $e_1$ the least such $k$.) The normal crossings condition (3) has also been explained in Section 3; see Remark 3.6, in particular, for the case that $\nu_{t+1}(a)=0$. The calculation in Section 2 then gives the property of decrease (4), as is evident also in the example of Section 3. \hfill$\Box$ \medskip When our spaces satisfy a compactness assumption (so that ${\rm inv}_X$ takes maximum values), it follows from Theorem B that we can obtain the Embedded Desingularization Theorem A by simply applying the algorithm of 1.11 above, stopping when ${\rm inv}_X$ becomes (locally) constant. To be more precise, let ${\rm inv}_X^{\rm e}$ denote the extended invariant for canonical desingularization introduced in Remark 3.6. Consider a sequence of blowings-up (1.8) with ${\rm inv}_X$-admissible centres. Note that if $X_j$ is not smooth and $a\in {\mbox{Sing}}\, X_j$, then $S_{{\rm inv}_X}(a)\subset {\mbox{Sing}}\, X_j$ because $\nu_1$ (or, in general, $H_{X_j,a}$) already distinguishes between smooth and singular points. Since ${\mbox{Sing}}\, X_j$ is Zariski-closed, it follows that if $C_j$ denotes the locus of maximum values of ${\rm inv}_X^{\rm e}$ on ${\mbox{Sing}}\, X_j$, then $C_j$ is smooth. By Theorem B, there is a finite sequence of blowings-up with such centres, after which $X_j$ is smooth. On the other hand, if $X_j$ is smooth and $a\in S_j$, where $S_j=\{ x\in X_j:\ s_1(x)>0\}$, then $S_{{\rm inv}_X}(a) \subset S_j$. Since $S_j$ is Zariski-closed, it follows that if $C_j$ denotes the locus of maximum values of ${\rm inv}_X^{\rm e}$ on $S_j$, then $C_j$ is smooth. Therefore, after finitely many further blowings-up $\sigma_{j+1},\ldots,\sigma_k$ with such centres, $S_k=\emptyset$. It is clear from the definition of $s_1$ that, if $X_k$ is smooth and $S_k=\emptyset$, then each $H\in E_k$ which intersects $X_k$ is the strict transform in $M_k$ of $\sigma_{i+1}^{-1} (C_i)$, for some $i$ such that $X_i$ is smooth along $C_i$; therefore, $X_k$ and $E_k$ simultaneously have only normal crossings, and we have Theorem A. \medskip We will prove invariance of ${\rm inv}_X$ using the idea of a ``presentation'' introduced in Section 3 above. It will be convenient to consider ``presentation'' in an abstract sense, rather than associated to a particular invariant: Let $M$ denote a manifold and let $a\in M$. \medskip \noindent {\em Definitions 4.1.}\quad An abstract {\em (infinitesimal) presentation} of {\em codimension} $p$ at $a$ means simply a triple ($N=N_p(a)$, ${\cal H}(a)$, ${\cal E}(a)$) as in Section 3; namely: $N$ is a germ of a submanifold of codimension $p$ at $a$, ${\cal H}(a)$ is a finite collection of pairs $(h,\mu_h)$, where $h\in{\cal O}_{N,a}$, $\mu_h\in{\Bbb Q}$ and $\mu_a(h)\ge \mu_h$, and ${\cal E}(a)$ is a finite set of smooth hypersurfaces containing $a$, such that $N$ and ${\cal E}(a)$ simultaneously have normal crossings and $N\not\subset H$, for all $H\in{\cal E}(a)$. A local blowing-up $\sigma$ with centre $C\ni a$ will be called {\em admissible} (for an infinitesimal presentation as above) if $C\subset S_{{\cal H}(a)}=\{ x\in N:\ \mu_x (h)\ge \mu_h$, for all $(h,\mu_h)\in{\cal H}(a)\}$. \medskip\noindent {\em Definition\ } {\em 4.2.}\quad We will say that two infinitesimal presentations ($N=N_p(a)$, ${\cal H}(a)$, ${\cal E}(a)$) and ($P=P_q(a)$, ${\cal F}(a)$, ${\cal E}(a)$) with given ${\cal E}(a)$, but not necessarily of the same codimension, are {\em equivalent} if (in analogy with Definition 3.2): (1) $S_{{\cal H}(a)}=S_{{\cal F}(a)}$, as germs at $a$ in $M$. (2) If $\sigma$ is an admissible local blowing-up and $a'\in\sigma^{-1}(a)$, then $a'\in N'$ and $\mu_{a'} (y_{\rm exc}^{-\mu_h} h\circ\sigma)\ge\mu_h$ for all $(h,\mu_h)\in {\cal H}(a)$ if and only if $a'\in P'$ and $\mu_{a'}(y_{\rm exc}^{-\mu_f} f\circ\sigma)\ge\mu_f$ for all $(f,\mu_f)\in{\cal F}(a)$. (3) Conditions (1) and (2) continue to hold for the transforms ($N_p(a')$, ${\cal H}(a')$, ${\cal E}(a')$) and ($P_q(a')$, ${\cal F}(a')$, ${\cal E}(a')$) of our data by sequences of morphisms of types (i), (ii) and (iii) as in Definition 3.2. \medskip We will, in fact, impose a further condition on the way that exceptional blowings-up (iii) are allowed to occur in a sequence of transformations in condition (3) above; see Definition 4.5 below. Our proof of invariance of ${\rm inv}_X$ follows the constructive definition outlined in Section 3. Let $X$ denote a hypersurface in $M$, and consider any sequence of blowings-up (or local blowings-up) (1.8), where we assume (at first) that the centres of blowing up are $\frac{1}{2}$-admissible. Let $a\in M_j$, for some $j=0,1,2,\ldots$. Suppose that $g\in {\cal O}_{M_j,a}$ generates the local ideal ${\cal I}_{X_j,a}$ of $X_j$ at $a$, and let $\mu_g=\mu_a(g)$. Then, as in Section 3, ${\cal G}_1(a)=\{ (g,\mu_g)\}$ determines a codimension zero presentation ($N_0(a)$, ${\cal G}_1(a)$, ${\cal E}_0(a)$) of ${\rm inv}_{\frac{1}{2}}=\nu_1$ at $a$, where $N_0(a)$ is the germ of $M_j$ at $a$, and ${\cal E}_0(a)=\emptyset$. In particular, the equivalence class of ($N_0(a)$, ${\cal G}_1(a)$, ${\cal E}_0(a)$) in the sense of Definition 4.2 depends only on the local isomorphism class of $(M_j,X_j)$ at $a$. We introduce $E^1(a)$ as in 1.12 above, and let $s_1(a)=\# E^1(a)$, ${\cal E}_1(a)=E(a)\backslash E^1(a)$. Let \[ {\cal F}_1(a) = {\cal G}_1(a) \cup \big( E^1 (a),1\big) , \] where $\big( E^1(a),1\big)$ denotes $\{ (x_H,1):\ H\in E^1(a)\}$ and $x_H$ means a local generator of the ideal of $H$. Then ($N_0(a)$, ${\cal F}_1(a)$, ${\cal E}_1(a)$) is a codimension zero presentation of ${\rm inv}_1=(\nu_1,s_1)$ at $a$. Clearly, the equivalence class of ($N_0(a)$, ${\cal F}_1(a)$, ${\cal E}_1(a)$) depends only on the local isomorphism class of ($M_j$, $X_j$, $E_j$, $E^1(a)$). Moreover, ($N_0(a)$, ${\cal F}_1(a)$, ${\cal E}_1(a)$) has an equivalent codimension one presentation ($N_1(a)$, ${\cal H}_1(a)$, ${\cal E}_1(a)$) as described in Section 3. For example, let $a_k=\pi_{kj} (a)$, $k=0,\ldots,j$, as in 1.12, and let $i$ denote the ``earliest year'' $k$ such that ${\rm inv}_{\frac{1}{2}}(a)={\rm inv}_{\frac{1}{2}}(a_k)$. Then ${\cal E}_1(a_i)=\emptyset$. As in Section 3, we can take $N_1(a_i) =$ any hypersurface of maximal contact for $X_i$ at $a_i$. If $(x_1,\ldots,x_n)$ are local coordinates for $M_i$ with respect to which $N_1(a_i)=\{ x_n=0\}$, then we can take \[ {\cal H}_1(a_i) = \left\{ \left( \frac{\partial^q f}{\partial x_n^q} \bigg|_{N_1(a_i)},\, \mu_f-q\right):\ 0\le q<\mu_f,\ (f,\mu_f) \in {\cal F}_1(a_i)\right\} . \] A codimension one presentation ($N_1(a)$, ${\cal H}_1(a)$, ${\cal E}_1(a)$) of ${\rm inv}_1$ at $a$ can be obtained by transforming ($N_1(a_i)$, ${\cal H}_1(a_i)$, ${\cal E}_1(a_i)$) to $a$. The condition that $N_1(a)$ and ${\cal E}_1(a)$ simultaneously have normal crossings and $N_1(a)\not\subset H$ for all $H\in {\cal E}_1(a)$ is a consequence of the effect of blowing with smooth centre of codimension at least $1$ in $N(a_k)$, $i\le k<j$ (as in the calculation in Section 2). Say that ${\cal H}_1(a)=\{ (h,\mu_h)\}$; each $h\in {\cal O}_{N_1(a),a}$ and $\mu_h\le \mu_a(h)$. Recall that we define \begin{eqnarray*} \mu_2(a) & = & \min_{{\cal H}_1(a)} \frac{\mu_a(h)}{\mu_h}\\ \mu_{2H} (a) & = & \min_{{\cal H}_1(a)} \frac{\mu_{H,a}(h)}{\mu_h} , \qquad H \in {\cal E}_1 (a) ,\\ \hbox{and}\qquad \nu_2(a) & = &\mu_2(a) - \sum_{H\in{\cal E}_1(a)} \mu_{2H}(a) . \end{eqnarray*} (Definitions 3.2, 3.4). Propositions 4.4 and 4.6 below show that each of $\mu_2(a)$ and $\mu_{2H}(a)$, $H\in{\cal E}_1(a)$, depends only on the equivalence class of ($N_1(a)$, ${\cal H}_1(a)$, ${\cal E}_1(a)$), and thus only on the local isomorphism class of ($M_j$, $X_j$, $E_j$, $E^1(a)$). If $\nu_2(a)=0$ or $\infty$, then we set ${\rm inv}_X(a)={\rm inv}_ {1\frac12}(a)$. If $0<\nu_2(a)<\infty$, then we construct a codimension one presentation ($N_1(a)$, ${\cal G}_2(a)$, ${\cal E}_1(a)$) of ${\rm inv}_{1\frac12}$ at $a$, as in Section 3. From the construction, it is not hard to see that the equivalence class of ($N_1(a)$, ${\cal G}_2(a)$, ${\cal E}_1(a)$) depends only on that of ($N_1(a)$, ${\cal H}_1(a)$, ${\cal E}_1(a)$). (See [BM5, 4.23 and 4.24] as well as Proposition 4.6 ff. below.) This completes a cycle in the inductive definition of ${\rm inv}_X$. Assume now that the centres of the blowings-up in (1.8) are $1\frac12$-admissible. We introduce $E^2(a)$ as in 1.12, and let $s_2(a)=\#E^2(a)$, ${\cal E}_2(a)={\cal E}_1(a)\backslash E^2(a)$. If ${\cal F}_2(a)={\cal G}_2(a)\cup \big( E^2(a),1\big)$, where $\big( E^2(a),1\big)$ denotes $\{ (x_H|_{N_1(a)},1):\ H\in E_2(a)\}$, then ($N_1(a), {\cal F}_2(a), {\cal E}_2(a)$) is a codimension one presentation of ${\rm inv}_2 = ({\rm inv}_{1\frac12},s_2)$ at $a$, whose equivalence class depends only on the local isomorphism class of ($M_j$, $X_j$, $E_j$, $E^1(a)$, $E^2(a)$). It is clear from the construction of ${\cal G}_2(a)$ that $\mu_{{\cal G}_2(a)}=1$, where \[ \mu_{{\cal G}_2(a)} = \min_{(g,\mu_g)\in{\cal G}_2(a)} \frac{\mu_a(g)}{\mu_g} . \] Therefore ($N_1(a)$, ${\cal F}_2(a)$, ${\cal E}_2(a)$) admits an equivalent codimension two presentation ($N_2(a)$, ${\cal H}_2(a)$, ${\cal E}_2(a)$), and we define $\nu_3(a)=\mu_3(a)-\sum_{H\in{\cal E}_2(a)} \mu_{3H}(a)$, as above. By Propositions 4.4 and 4.6, $\mu_3(a)$ and each $\mu_{3H}(a)$ depend only on the equivalence class of ($N_2(a)$, ${\cal H}_2(a)$, ${\cal E}_2(a)$), $\ldots$. We continue until $\nu_{t+1}(a)=0$ or $\infty$ for some $t$, and then take ${\rm inv}_X(a)={\rm inv}_{t+\frac12}(a)$. Invariance of ${\rm inv}_X$ thus follows from Propositions 4.4 and 4.6 below, which are formulated purely in terms of an abstract infinitesimal presentation. Let $M$ be a manifold, and let ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) be an infinitesimal presentation of codimension $r\ge 0$ at a point $a\in M$. We write ${\cal H}(a)=\{ (h,\mu_h)\}$, where $\mu_a(h)\ge\mu_h$ for all $(h,\mu_h)$. \medskip \noindent {\em Definitions 4.3.}\quad We define $\mu(a)=\mu_{{\cal H}(a)}$ as \[ \mu_{{\cal H}(a)} = \min_{{\cal H}(a)} \frac{\mu_a(h)}{\mu_h} . \] Thus $1\le \mu(a)\le\infty$. If $\mu(a)<\infty$, then we define $\mu_H(a)=\mu_{{\cal H}(a),H}$, for each $H\in{\cal E}(a)$, as \[ \mu_{{\cal H}(a),H} = \min_{{\cal H}(a)} \frac{\mu_{H,a}(h)}{\mu_h} . \] \medskip We will show that each of $\mu(a)$ and the $\mu_H(a)$ depends only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) (where we consider only {\em presentations of the same codimension} $r$). The main point is that $\mu(a)$ and the $\mu_H(a)$ can be detected by ``test blowings-up'' (test transformations of the form (i), (ii), (iii) as allowed by the definition 4.2 of equivalence). For $\mu(a)$, we show in fact that if ($N^i(a)$, ${\cal H}^i(a)$, ${\cal E}(a)$), $i=1,2$, are two infinitesimal presentations of the same codimension $r$, then $\mu_{{\cal H}^1(a)}=\mu_{{\cal H}^2(a)}$ if the presentations are equivalent with respect to transformations of types (i) and (ii) alone (i.e., where we allow only transformations of types (i) and (ii) in Definition 4.2). This ia a stronger condition than invariance under equivalence in the sense of Definition 4.2 (using all three types of transformations) because the equivalence class with respect to transformations of types (i) and (ii) alone is, of course, larger than the equivalence class with respect to transformations of all three types (i), (ii) and (iii). \medskip\noindent{\bf Proposition\ } {\bf 4.4.} [BM5, Prop. 4.8]. {\em $\mu(a)$ depends only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) (among presentations of the same codimension $r$) with respect to transformations of types (i) and (ii).} \medskip\noindent{\em Proof.\quad } Clearly, $\mu(a)=\infty$ if and only if $S_{{\cal H}(a)}=N(a)$; i.e., if and only if $S_{{\cal H}(a)}$ is (a germ of) a submanifold of codimension $r$ in $M$. Suppose that $\mu(a)<\infty$. We can assume that ${\cal H}(a)=\{ (h,\mu_h)\}$ where all $\mu_h=e$, for some $e\in {\Bbb N}$. Let $\sigma_0$: $P_0=M\times{\Bbb K}\to M$ be the projection from the product with a line (i.e., a morphism of type (ii)) and let ($N(c_0)$, ${\cal H}(c_0)$, ${\cal E}(c_0)$) denote the transform of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) at $c_0= (a,0)\in P_0$; i.e., $N(c_0)=N(a)\times{\Bbb K}$, ${\cal E}(c_0)= \{ H\times{\Bbb K}$, for all $H\in{\cal E}(a)$, and $M\times\{0\}\}$ and ${\cal H}(c_0)=\{ (h\circ\sigma_0,\mu_h)$: $(h,\mu_h)\in{\cal H}(a)\}$. We follow $\sigma_0$ by a sequence of admissible blowings-up (morphisms of type (i)), \[ {\rightarrow}\ P_{\beta+1}\ \stackrel{\sigma_{\beta+1}}{{\rightarrow}}\ P_\beta\ {\rightarrow}\ \cdots\ {\rightarrow}\ P_1\ \stackrel{\sigma_1}{{\rightarrow}}\ P_0 , \] where each $\sigma_{\beta+1}$ is a blowing-up with centre a point $c_\beta\in P_\beta$ determined as follows: Let $\gamma_0$ denote the arc in $P_0$ given by $\gamma_0(t)=(a,t)$. For $\beta\ge 1$, define $\gamma_{\beta+1}$ inductively as the lifting of $\gamma_\beta$ to $P_{\beta+1}$, and set $c_{\beta+1}=\gamma_{\beta+1}(0)$. We can choose local coordinates $(x_1,\ldots,x_n)$ for $M$ at $a$, in which $a=0$ and $N(a)=\{ x_{n-r+1}=\cdots= x_n=0\}$. Write $(x,t)=(x_1,\ldots,x_{n-r},t)$ for the corresponding coordinate system of $N(c_0)$. In $P_1$, the strict transform $N(c_1)$ of $N(c_0)$ has a local coordinate system $(x,t)=(x_1,\ldots,x_{n-r},t)$ at $c_1$ with respect to which $\sigma_1(x,t)=(tx,t)$, and $\gamma_1(t)=(0,t)$ in this coordinate chart; moreover, ${\cal H}(c_1)=\{ (t^{-e}h(tx),e)$, for all $(h,\mu_h)= (h,e)\in{\cal H}(a)\}$. After $\beta$ blowings-up as above, $N(c_\beta)$ has a local coordinate system $(x,t)=(x_1,\ldots,x_{n-r},t)$ with respect to which $\sigma_1\circ\cdots\circ\sigma_\beta$ is given by $(x,t)\mapsto (t^\beta x,t)$, $\gamma_\beta (t)=(0,t)$ and ${\cal H}(c_\beta)=\{ (h',\mu_{h'}=e)\}$, where \[ h' = t^{-\beta e} h(t^\beta x) , \] for all $(h,\mu_h)=(h,e)\in{\cal H}(a)$. By the definition of $\mu(a)$, each \[ h(t^\beta x) = t^{\beta \mu(a)e} \tilde h'(x,t) , \] where the $\tilde h'(x,t)$ do not admit $t$ as a common divisor; for each $(h,\mu_h)\in{\cal H}(a)$, we have $$ h' = t^{\beta (\mu(a)-1) e} \tilde h' . $$ We now introduce a subset $S$ of ${\Bbb N}\times{\Bbb N}$ depending only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) (with respect to transformations of types (i) and (ii)) as follows: First, we say that $(\beta,0)\in S$, $\beta\ge 1$, if after $\beta$ blowings-up as above, there exists (a germ of) a submanifold $W_0$ of codimension $r$ in the exceptional hypersurface $H_\beta = \sigma_\beta^{-1} (c_{\beta-1})$ such that $W_0\subset S_{{\cal H}(c_\beta)}$. If so, then necessarily $W_0=H_\beta \cap N(c_\beta)=\{t=0\}$, and the condition that $W_0\subset S_{{\cal H}(c_\beta)}$ means precisely that $\mu_{W_0,c_\beta}(h')\ge e$, for all $h'$; i.e., that $\beta\big(\mu(a)-1\big)e\ge e$, or $\beta\big(\mu(a)-1\big)\ge 1$. (In particular, since $\mu(a)\ge 1$, $(\beta,0)\not\in S$ for all $\beta\ge 1$ if and only if $\mu(a)=1$.) Suppose that $(\beta,0)\in S$, for some $\beta\ge 1$, as above. Then we can blow up $P_\beta$ locally with centre $W_0$. Set $Q_0=P_\beta$, $d_0=c_\beta$ and $\delta_0=\gamma_\beta$. Let $\tau_1$: $Q_1\to Q_0$ denote the local blowing-up with centre $W_0$, and let $d_1=\delta_1(0)$, where $\delta_1$ denotes the lifting of $\delta_0$ to $Q_1$. (Then $\tau_1|N(d_1)$: $N(d_1)\to N(d_0)$ is the identity.) We say that $(\beta,1)\in S$ if there exists a submanifold $W_1$ of codimension $r$ in the hypersurface $H_1=\tau_1^{-1} (W_0)$ such that $W_1\subset S_{{\cal H}(d_1)}$. If so, then again necessarily $W_0=H_1\cap N(d_1)=\{ t=0\}$. Since ${\cal H}(d_1)=\{ (h',e)\}$, where each $h'=t^{\beta (\mu(a)-1)e-e}\tilde h'$ and the $\tilde h'$ do not admit $t$ as a common factor, it follows that $(\beta,1)\in S$ if and only if $\beta\big(\mu(a)-1\big) e-e\ge e$. We continue inductively: If $\alpha\ge 1$ and $(\beta,\alpha-1)\in S$, let $\tau_\alpha$: $Q_\alpha\to Q_{\alpha-1}$ denote the local blowing-up with centre $W_{\alpha-1}$, and let $d_\alpha=\delta_\alpha (0)$, where $\delta_\alpha$ is the lifting of $\delta_{\alpha-1}$ to $Q_\alpha$. We say that $(\beta,\alpha)\in S$ if there exists (a germ of) a submanifold $W_\alpha$ of codimension $r$ in the exceptional hypersurface $H_\alpha=\tau_\alpha^{-1} (W_{\alpha-1})$ such that $W_\alpha\subset S_{{\cal H}(d_\alpha)}$. Since ${\cal H}(d_\alpha)=\{ (h',e)\}$, where each $h'=t^{\beta(\mu(a)-1)e-\alpha e}\tilde h'$ and the $\tilde h'$ do not admit $t$ as a common factor, it follows as before that $(\beta,\alpha)\in S$ if and only if $\beta\big(\mu(a)-1\big)-\alpha \ge 1$. Now $S$, by its definition, depends only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) (with respect to transformations of types (i) and (ii)). On the other hand, we have proved that $S=\emptyset$ if and only if $\mu(a)=1$, and, if $S\ne\emptyset$, then \[ S = \big\{ (\beta,\alpha)\in{\Bbb N}\times{\Bbb N}:\ \beta\big( \mu(a)-1\big) -\alpha\ge 1\big\} . \] Our proposition follows since $\mu(a)$ is uniquely determined by $S$; in the case that $S\ne \emptyset$, $$ \mu(a) = 1 + \sup_{(\beta,\alpha)\in S} \frac{\alpha+1}{\beta} . \eqno{\Box} $$ \medskip Suppose that $\mu(a)<\infty$. Then we can also use test blowings-up to prove invariance of $\mu_H(a)=\mu_{{\cal H}(a),H}$, $H\in{\cal E}(a)$: Fix $H\in{\cal E}(a)$. As before we begin with the projection $\sigma_0$: $P_0=M\times{\Bbb K}\to M$ from the product with a line. Let ($N(a_0)$, ${\cal H}(a_0)$, ${\cal E}(a_0)$) denote the transform of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) at $a_0= (a,0)\in P_0$ by the morphism $\sigma_0$ (of type (ii)), and let $H_0^0 = M\times\{0\}$, $H_1^0 =\sigma_0^{-1} (H) = H\times{\Bbb K}$. Thus $H_0^0,H_1^0\in {\cal E}(a_0)$. We follow $\sigma_0$ by a sequence of exceptional blowings-up (morphisms of type (iii)), \[ {\rightarrow}\ P_{j+1}\ \stackrel{\sigma_{j+1}}{{\rightarrow}}\ P_j\ {\rightarrow}\ \cdots\ {\rightarrow}\ P_1\ \stackrel{\sigma_1}{{\rightarrow}}\ P_0 , \] where each $\sigma_{j+1}$, $j\ge 0$, has centre $C_j=H_0^j\cap H_1^j$ and $H_0^{j+1}=\sigma_{j+1}^{-1} (C_j)$, $H_1^{j+1}=$ the strict transform of $H_1^j$ by $\sigma_{j+1}$. Let $a_{j+1}$ denote the unique intersection point of $C_{j+1}$ and $\sigma_{j+1}^{-1} (a_j)$, $j\ge 0$. ($a_{j+1}=\gamma_{j+1}(0)$, where $\gamma_0$ denotes the arc $\gamma_0(t)=(a,t)$ in $P_0$ and $\gamma_{j+1}$ denotes the lifting of $\gamma_j$ by $\sigma_{j+1}$, $j\ge 0$.) We can choose local coordinates $(x_1,\ldots,x_n)$ for $M$ at $a$, in which $a=0$, $N(a)=\{ x_{n-r+1} = \cdots = x_n=0\}$, and each $K\in{\cal E}(a)$ is given by $x_i=0$, for some $i=1,\ldots,n-r$. (Set $x_i=x_K$.) Write $(x,t)=(x_1,\ldots,x_m,t)$, where $m=n-r$, for the corresponding coordinate system of $N(a_0)=N(a)\times{\Bbb K}$. We can assume that $x_H=x_1$. In $P_1$, the strict transform $N(a_1)$ of $N(a_0)$ has a chart with coordinates $(x,t)=(x_1,\ldots,x_m,t)$ in which $\sigma_1$ is given by $\sigma_1(x,t)=(tx_1,x_2,\ldots,x_m,t)$ and in which $a_1=(0,0)$, $\gamma_1(t)=(0,t)$ and $x_1=x_H$. ($x_H$ now means $x_{H_1^1}$.) Proceeding inductively, for each $j$, $N(a_j)$ has a coordinate system $(x,t)=(x_1,\ldots,x_m,t)$ in which $a_j=(0,0)$ and $\sigma_1\circ\cdots\circ\sigma_j$: $N(a_j)\to N(a_0)$ is given by \[ (x,t)\mapsto (t^j x_1, x_2,\ldots,x_m,t) . \] We can assume that $\mu_h=e\in{\Bbb N}$, for all $(h,\mu_h)\in{\cal H}(a)$. Set $$ D = \prod_{K\in{\cal E}(a)} x_K ^{\mu_K(a)} . $$ Thus $D^e$ is a monomial in the coordinates $(x_1,\ldots,x_m)$ of $N(a)$ with exponents in ${\Bbb N}$, and $D^e$ is the greatest common divisor of the $h$ in ${\cal H}(a)$ which is a monomial in $x_K$, $K\in{\cal E}(a)$ (by Definitions 4.3). In particular, for some $h=D^e g$ in ${\cal H}(a)$, $g=g_H$ is not divisible by $x_1=x_H$. Therefore, there exists $i\ge 1$ such that \[ \mu_{a_j} (g_H\circ\pi_j) = \mu_{a_i} (g_H\circ\pi_i) , \] for all $j\ge i$, where $\pi_j=\sigma_0\circ\sigma_1\circ \cdots\circ\sigma_j$. (We can simply take $i$ to be the least order of a monomial not involving $x_H$ in the Taylor expansion of $g_H$.) On the other hand, for each $h=D^e g$ in ${\cal H}(a)$, $\mu_{a_j}(g\circ\pi_j)$ increases as $j\to\infty$ unless $g$ is not divisible by $x_H$. Therefore, we can choose $h=D^e g_H$, as above, and $i$ large enough so that we also have $\mu(a_j)=\mu_{a_j} (h\circ\pi_j)/e$, for all $j\ge i$. Clearly, if $j\ge i$, then \[ \mu_H (a) = \mu(a_{j+1}) - \mu(a_j) . \] Since $\mu(a)$ depends only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) among presentations of the same codimension $r$, as defined by 4.2, the preceding argument shows that each $\mu_H (a)$, $H\in{\cal E}(a)$, is also an invariant of this equivalence class. But the argument shows more precisely that the $\mu_H(a)$ depend only on a larger equivalence class obtained by allowing in Definition 4.2 only certain sequences of morphisms of types (i), (ii) and (iii): \medskip\noindent {\em Definition\ } {\em 4.5.}\quad We weaken the notion of equivalence in Definition 4.2 by allowing only the transforms induced by certain sequences of morphisms of types (i), (ii) and (iii); namely, \[ \begin{array}{cccccccccccc} \rightarrow & M_j &\stackrel{\sigma_j}{\rightarrow} &M_{j-1} &\rightarrow &\cdots &\stackrel{\sigma_{i+1}}{\rightarrow} & M_i &\rightarrow &\cdots &\rightarrow &M_0=M \\ &{\cal E}(a_j)&&{\cal E}(a_{j-1})&&&&{\cal E}(a_i)&&&&{\cal E}(a_0)={\cal E}(a) \end{array} \] where, if $\sigma_{i+1},\ldots,\sigma_j$ are exceptional blowings-up (iii), then $i\ge 1$ and $\sigma_i$ is of either type (iii) or (ii). In the latter case, $\sigma_i$: $M_i=M_{i-1}\times{\Bbb K}\to M_{i-1}$ is the projection, each $\sigma_{k+1}$, $k=i,\ldots,j-1$, is the blowing-up with centre $C_k=H_0^k \cap H_1^k$ where $H_0^k$, $H_1^k\in{\cal E}(a_k)$, $a_{k+1}=\sigma_{k+1}^{-1}(a_k)\cap H_1^{k+1}$, and we require that the $H_0^k$, $H_1^k$ be determined by some fixed $H\in {\cal E}(a_{i-1})$ inductively in the following way: $H_0^i=M_{i-1}\times\{0\}$, $H_1^i=\sigma_i^{-1} (H)$, and, for $k=i+1,\ldots,j-1$, $H_0^k=\sigma_k^{-1}(C_{k-1})$, $H_1^k=$ the strict transform of $H_1^{k-1}$ by $\sigma_k$. \medskip In other words, with this notion of equivalence, we have proved: \medskip\noindent{\bf Proposition\ } {\bf 4.6.}\quad [BM5, Prop. 4.11]. {\em Each $\mu_H(a)$, $H\in{\cal E}(a)$, and therefore also $\nu(a)=\mu(a)-\Sigma \mu_H(a)$ depends only on the equivalence class of ($N(a)$, ${\cal H}(a)$, ${\cal E}(a)$) (among presentations of the same codimension).} \medskip Recall that in the $r$'th cycle of our recursive definition of ${\rm inv}_X$, we use a codimension $r$ presentation ($N_r(a)$, ${\cal H}_r(a)$, ${\cal E}_r(a)$) of ${\rm inv}_r$ at $a$ to construct a codimension $r$ presentation ($N_r(a)$, ${\cal G}_{r+1}(a)$, ${\cal E}_r(a)$) of ${\rm inv}_{r+\frac12}$ at $a$. The construction involved survives transformations as allowed by Definition 4.5, but perhaps not an arbitrary sequence of transformations of types (i), (ii) and (iii) (cf. [BM5, 4.23 and 4.24]; in other words, we show only that the equivalence class of ($N_r(a)$, ${\cal G}_{r+1}(a)$, ${\cal E}_r(a)$) as given by Definition 4.5 depends only on that of ($N_r(a)$, ${\cal H}_r(a)$, ${\cal E}_r(a)$). It is for this reason that we need Proposition 4.6 as stated. \vskip .25in \noindent {\it Acknowledgement.}\quad We are happy to thank Paul Centore for the line drawings in this paper. \vskip .25in

9709

alg-geom/9709007

en

https://arxiv.org/abs/alg-geom/9709007

alg-geom/9709007

Ravi Vakil

Ravi Vakil

The enumerative geometry of rational and elliptic curves in projective space

LaTeX2e, 95 pages with 18 figures

We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)

alg-geom

\section{Introduction} \label{intro} In this article, we study the geometry of varieties (over $\mathbb{C}$) parametrizing degree $d$ rational and elliptic curves in $\mathbb P^n$ intersecting fixed general linear spaces and tangent to a fixed hyperplane $H$ with fixed multiplicities along fixed general linear subspaces of $H$. As an application, we derive recursive formulas (Theorem \ref{rrecursiveX2} and Theorem \ref{erecursiveW}) for the number of such curves when the number is finite. As with M. Kontsevich's marvelous formula of [KM], these recursive formulas require as ``seed data'' only one input: there is one line in $\mathbb P^1$ through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree $d$ rational or elliptic curves in $\mathbb P^n$, or on ${\overline{M}}_0(\mathbb P^n,d)$ or certain components of ${\overline{M}}_1(\mathbb P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. The recursive formulas are convenient to program or use by hand, and provide quick diagramatic enumerative calculations. For example, Figure \ref{rcubics} shows a diagramatic enumeration of the 80,160 twisted cubics in $\mathbb P^3$ intersecting 12 fixed general lines. The methods used are surprisingly elementary. Little is assumed about Kontsevich's space of stable maps. The low genus of the curves under consideration makes dimension counting straightforward. \begin{figure} \begin{center} \getfig{rcubics}{.1} \end{center} \caption{Counting $80,160$ cubics in $\mathbb P^3$ through 12 general lines} \label{rcubics} \end{figure} Interest in such classical enumerative problems has been revived by recent ideas from quantum field theory leading to the definition of quantum cohomology and Gromov-Witten invariants, and by the subsequent discovery by Kontsevich in 1993 of an elegant recursion solving the problem in genus 0 when no tangencies are involved (cf. [KM]; another proof, using different techniques, was given independently by Y. Ruan and G. Tian in [RT]). The enumerative results here are a generalization of [KM], and the methods seem more likely to generalize further than those of [KM]. In particular, such ideas could apply to certain nonconvex rational spaces, in the same way that the ideas of Caporaso and Harris in $\mathbb P^2$ extend to rational ruled surfaces (cf. [CH3] and [V1]). In Section \ref{elliptic}, the same ideas are applied to the genus 1 case. One of the motivations for this study was to gain more information about higher genus Gromov-Witten invariants. The enumerative results for elliptic curves in $\mathbb P^3$ have been independently derived by E. Getzler (cf. [G3]), without tangency conditions, by determining the genus 1 Gromov-Witten invariants of $\mathbb P^3$ and relating them to the enumerative problem. (T. Graber and R. Pandharipande ([GrP]) have also proposed a programme to determine these numbers for all $\mathbb P^n$ as well.) There is some hope of unifying the two methods, and getting some information about Gromov-Witten invariants through degeneration methods. There is also some hope that this method will apply to curves of higher genus. The enumerative geometry of curves (of any genus) in the plane as described in [CH3] can be seen as a variant of this perspective. In [V1], the corresponding problem (for curves of any genus in any divisor class) on any rational ruled surface is solved by the same method. In Subsection \ref{ehighgenus}, we will briefly discuss the possibilities and potential obstructions to generalization to curves of higher genus. The results should also carry over to other highly symmetric rational varieties, especially Flag manifolds and towers of $\mathbb P^1$-bundles, in the same way as the enumerative geometry of curves in $\mathbb P^2$ as described in [CH3] was generalized to rational ruled surfaces in [V1]. This article contains the majority of the author's 1997 Harvard Ph.D. thesis, and was partially supported by an NSERC 1967 Fellowship and a Sloan Fellowship. Tha author is extremely grateful to his advisor, Joe Harris, and also to Dan Abramovich, Michael Thaddeus, Tony Pantev, Angelo Vistoli, Rahul Pandharipande, Michael Roth, Lucia Caporaso, and Ezra Getzler for useful discussions. \subsection{Notation and basic results.} The base field will always be assumed to be $\mathbb{C}$. If $S$ is a set, define $\operatorname{Aut}(S)$ to be the automorphisms (or symmetries) of $S$. The main results of this article will be about varieties, but it will be convenient to occasionally use the language of algebraic stacks (in the sense of Deligne and Mumford, cf. [DM]). Stacks have several advantages: calculating Zariski-tangent spaces to moduli stacks is simpler than calculating tangent spaces to moduli spaces, and moduli stacks (and morphisms between them) are smooth ``more often'' than the corresponding moduli spaces (and morphisms between them). In general, for all definitions of varieties made, there is a corresponding definition in the language of stacks, and the corresponding stack will be written in a calligraphic font. For example, $M_g$ is the moduli space of smooth genus $g$ curves, and ${\mathcal{M}}_g$ is the corresponding moduli stack. Stacks are invoked as rarely as possible, and the reader unfamiliar with stacks should have no problem following the arguments. An introduction to the theory of algebraic stacks is given in the appendix to [Vi1]. \subsubsection{The moduli space of stable maps} \label{itmsosm} For the convenience of the reader we recall certain facts about moduli spaces (and stacks) of stable maps to $\mathbb P^n$, without proofs. A {\em stable marked curve}, denoted $(C, \{ p_1, \dots, p_m \} )$, of genus $g$ with $m$ marked points $p_1$, \dots, $p_m$ is a connected nodal complete marked curve with finite automorphism group; the marked points are required to be smooth points of $C$. The set of {\em special points} on (the normalization of) a component of $C$ is the set of marked points union the set of branches of nodes. A marked curve is stable if each rational component has at least three special points and each elliptic component has at least one special point. A degree $d$ {\em stable map} $(C, \{ p_1, \dots, p_m \}, \pi)$ to $\mathbb P^n$ consists of a connected nodal complete marked curve $(C, \{ p_i \} )$ and a morphism $\pi: C \rightarrow \mathbb P^n$ such that $\pi_*[C] = d[L]$ (where $[L]$ is the class of a line in the first Chow group $A_1(\mathbb P^n)$ of $\mathbb P^n$, or equivalently in $H_2(\mathbb P^n, \mathbb{Z})$), such that the map $\pi$ has finite automorphism group. This last condition is equivalent to requiring each collapsed rational component to have at least three special points and each contracted elliptic component to have at least one special point. There is a coarse projective moduli space ${\overline{M}}_{g,m}(\mathbb P^n,d)$ for such stable maps with $p_a(C) = g$. There is an algebraic stack ${\overline{\cm}}_{g,m}(\mathbb P^1,d)$ that is a fine moduli space for stable maps. The stack ${\overline{\cm}}_{g,m+1}(\mathbb P^1,d)$ can be considered as the universal curve over ${\overline{\cm}}_{g,m}(\mathbb P^1,d)$. There is an open subvariety $M_{g,m}(\mathbb P^n,d)$ of ${\overline{M}}_{g,m}(\mathbb P^n,d)$ that is a coarse moduli space of stable maps from smooth curves, and an open substack ${\mathcal{M}}_{g,m}(\mathbb P^n,d)$ of ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$ that is a fine moduli space of stable maps from smooth curves. The stack ${\overline{\cm}}_{0,m}(\mathbb P^n,d)$ is smooth of dimension $(n+1)(d+1)+m -4$. The versal deformation space to the stable map $(C, \{ p_i \}, \pi)$ in ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$ is obtained from the complex $$ \underline{\Omega}_\pi = \left( \pi^* \Omega_{\mathbb P^n} \rightarrow \Omega_C(p_1 + \dots + p_m)\right). $$ (The versal deformation space depends only on the image of $\underline{\Omega}_{\pi}$ in the derived category of coherent sheaves on $C$.) The vector space $\operatorname{{\mathbb H}om} (\underline{\Omega}_\pi, {\mathcal{O}}_C)$ parametrizes infinitesimal automorphisms of the map $\pi$; as $(C, \{ p_i \}, \pi)$ is a stable map, $\operatorname{{\mathbb H}om} (\underline{\Omega}_\pi, {\mathcal{O}}_C) = 0$. The space of infinitesimal deformations to the map $(C, \{ p_i \}, \pi)$ (i.e. the Zariski tangent space to ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$ at the point representing this stable map), denoted $\operatorname{Def}_{(C, \{ p_i \}, \pi )}$, is given by $\operatorname{{\mathbb E}xt}^1(\underline{\Omega}_\pi, {\mathcal{O}}_C)$ and the obstruction space, denoted $\operatorname{Ob}_{(C, \{ p_i \},\pi)}$, is given by $\operatorname{{\mathbb E}xt}^2(\underline{\Omega}_\pi, {\mathcal{O}}_C)$. By applying the functor $\operatorname{Hom}( \cdot, {\mathcal{O}}_C)$ to the exact sequence of complexes $$ 0 \rightarrow \Omega_C(p_1 + \dots + p_m)[-1] \rightarrow \underline{\Omega}_\pi \rightarrow \pi^* \Omega_{\mathbb P^n} \rightarrow 0 $$ we obtain the long exact sequence \begin{eqnarray} \nonumber 0 \rightarrow \operatorname{{\mathbb H}om}( \underline{\Omega}_\pi, {\mathcal{O}}_C) &\rightarrow& \operatorname{Hom}( \Omega_C(p_1 + \dots + p_m), {\mathcal{O}}_C) \rightarrow H^0(C,\pi^* T_{\mathbb P^n}) \\ \nonumber \rightarrow \operatorname{{\mathbb E}xt}^1( \underline{\Omega}_\pi, {\mathcal{O}}_C) &\rightarrow& \operatorname{Ext}^1( \Omega_C(p_1 + \dots + p_m), {\mathcal{O}}_C) \rightarrow H^0(C,\pi^* T_{\mathbb P^n}) \\ \nonumber \rightarrow \operatorname{{\mathbb E}xt}^2( \underline{\Omega}_\pi, {\mathcal{O}}_C) &\rightarrow& 0. \end{eqnarray} By the identifications given in the previous paragraph, and using $\operatorname{Hom}(\Omega_C(p_1 + \dots + p_m), {\mathcal{O}}_C) = \operatorname{Aut}(C, \{ p_i \})$ and $\operatorname{Ext}^1(\Omega_C(p_1 + \dots + p_m), {\mathcal{O}}_C) = \operatorname{Def}(C, \{ p_i \})$, this long exact sequence can be rewritten as \begin{eqnarray} \nonumber 0 &\longrightarrow& \operatorname{Aut} (C, \{ p_i \}) \longrightarrow H^0(C,\pi^* T_{\mathbb P^n}) \\ \nonumber \longrightarrow \operatorname{Def} (C, \{ p_i \}, \pi ) & \longrightarrow & \operatorname{Def} (C, \{ p_i \}) \longrightarrow H^1(C,\pi^* T_{\mathbb P^n} ) \\ \longrightarrow \operatorname{Ob} (C, \{ p_i \}, \pi ) &\longrightarrow & 0. \nonumber \end{eqnarray} The construction of the versal deformation space from $\underline{\Omega}_\pi$ is discussed in [R3], [Vi2], and [LT2]. All other facts described here appear in the comprehensive introduction [FP]. \subsection{Divisors on subvarieties of $\mbar_{g,m}(\proj^1,d)$} \label{ikey} The results proved here will be invoked repeatedly in Sections \ref{rational} and \ref{elliptic}. \subsubsection{A property of stable maps from curves to curves} We will make repeated use of a special property of maps from curves to curves. Let $(C, \{ p_i\}, \pi)$ be a stable map of a complete marked curve to $\mathbb P^1$. The scheme $\pi^{-1}(p)$ consists of reduced unmarked points for almost all $p$. Let $A = A_1 \coprod \dots \coprod A_l$ be the union of the connected components of fibers of $\pi$ that are {\em not} reduced unmarked points. Call the $A_j$ the {\em special loci} of the map $\pi$. Then each special locus $A_j$ is either a ramification of $\pi$, a labeled point (that may also be a ramification), or a union of contracted components (possibly containing labeled points, and possibly attached to other components at their ramification points). Let $\operatorname{Def}_{(C,\{ p_i \}, \pi)}$ be the versal deformation space to the stable map $(C, \{ p_i \} , \pi)$. Let $\operatorname{Def}_{A_j}$ be the versal deformation space of the map $$ (C \setminus \cup_{i \neq j} A_i, \{ p_i \} \cap A_j, \pi ); $$ the space $\operatorname{Def}_{A_j}$ parametrizes formal deformations of the map $\pi$ that are trivial away from $A_j$. \begin{pr} The versal deformation space $\operatorname{Def}_{(C,\{ p_i \}, \pi)}$ is naturally $\prod_j \operatorname{Def}_{A_j}$. \label{ilocal} \end{pr} An informal argument in the analytic category is instructive. Let $U_j \subset C$ (for $1 \leq j \leq l$) be an open (analytic) neighborhood of the special locus $A_j$ whose closure does not intersect the other special loci $\cup_{i \neq j} A_i$. Let $U$ be an open subset of $C$ whose closure does not intersect the special loci $\cup A_i$, and such that $$ U \cup \left( \bigcup_{j=1}^l U_j \right) = C. $$ Then ``small'' deformations of $(C, \{ p_i \}, \pi)$ are trivial on $U$, and thus the deformations of $\pi$ on $U_1$, \dots, $U_l$ are mutually independent: $$ \operatorname{Def}_{C,\{ p_i \}, \pi} = \prod_j \operatorname{Def}_{(U_j, \{ p_i \} \cap U_j, \pi)}. $$ This argument can be carried out algebraically on the level of formal schemes, which will give a rigorous proof. {\noindent {\em Proof of the proposition. }} The versal deformation to the stable map $(C, \{ p_i \}, \pi)$ is constructed using the complex $$ \underline{\Omega}_{(C, \{ p_i \}, \pi)} = \left( \pi^* \Omega_{\mathbb P^1} \rightarrow \Omega_C \left( \sum p_i \right) \right) $$ (see Subsubsection \ref{itmsosm}). The complex $\underline{\Omega}_{(C, \{ p_i \}, \pi)}$ is exact on $C \setminus \cup A_j$, so it splits in the derived category as a direct sum of objects $$ \underline{\Omega}_{ (C, \{ p_i \}, \pi)} = \oplus \underline{\Omega}_{A_j} $$ where the cohomology sheaves of $\underline{\Omega}_{A_j}$ are supported on $A_j$. Then the entire construction of the versal deformation space naturally factors through this direct sum, and the result follows. \qed \vspace{+10pt} Thus to understand the versal deformation space of a map to $\mathbb P^1$ we need only understand the versal deformations of the special loci $\operatorname{Def}_{A_j}$. Let $(C, \{ p_i \}, \pi)$ (resp. $(C', \{ p'_i \}, \pi')$) be a stable map to $\mathbb P^1$ and let $A_1$ (resp. $A'_1$) be one of its special loci consisting of a connected union of components contracted by $\pi$ (resp. $\pi'$). Let $\tilde{C}$ be the closure of $C \setminus A_1$ in $C'$, and $\tilde{C'}$ the closure of $C' \setminus A'_1$ in $C'$. Consider $A_1$ (resp. $A_1'$) as a marked curve, where the markings are the intersection of $\tilde{C}$ with $A_1$ (resp. $\tilde{C'}$ with $A'_1$) and the labeled points of $C$ (resp. $C'$). Assume that $A_1$ and $A_1'$ are isomorphic as marked curves, and that the ramification orders of the points of $\tilde{C} \cap A_1$ on $\tilde{C}$ are the same as those of the corresponding points of $\tilde{C'} \cap A_1'$ on $\tilde{C'}$. \begin{pr} \label{ilocal2} There is an isomorphism of versal deformation spaces $\operatorname{Def}_{A_1} \cong \operatorname{Def}_{A_1'}$. \end{pr} As before, an analytic perspective is instructive. The hypotheses of the theorem imply that there is an analytic neighborhood $U_{\text{an}}$ of $A_1$ that is isomorphic to an analytic neighborhood $U'_{\text{an}}$ of $A_1'$, and $\pi|_{U_{\text{an}}} = \pi'|_{U'_{\text{an}}}$. \noindent {\em Proof. } The hypotheses of the proposition imply that a formal neighborhood $U$ of $A_1$ is isomorphic to a formal neighborhood $U'$ of $A_1'$, and $\pi|_U$ agrees with $\pi'|_{U'}$ (via this isomorphism). As the cohomology sheaves of $\underline{\Omega}_{A_1}$, $\underline{\Omega}_{A'_1}$ are supported in $U$ and $U'$ respectively, the entire construction of $\operatorname{Def}_{A_1}$ and $\operatorname{Def}_{A'_1}$ depends only on $(U, \pi|_U)$ and $(U', \pi'|_{U'})$. \qed \vspace{+10pt} \subsubsection{Subvarieties of ${\overline{M}}_{g,m}(\mathbb P^1,d)$} Fix a positive integer $d$, and let $\vec{h} = (h_1,h_2,\dots)$ represent a partition of $d$ with $h_1$ 1's, $h_2$ 2's, etc., so $\sum_m m h_m = d$. Fix a point $z$ on $\mathbb P^1$. Let $X = X^{d,g}(\vec{h})$ be the closure in ${\overline{M}}_{g,\sum h_m + 1}(\mathbb P^1,d)$ of points representing stable maps $(C, \{ p^j_m \}, q, \pi)$ where $C$ is an irreducible curve with $\sum h_m + 1$ (distinct) marked points $\{ p^j_m \}_{1 \leq j \leq h_m}$ and $q$, and $\pi^*(z) = \sum_{j,m} m p^j_m$. (In short, we have marked all the pre-images of $z$ and one other point. The map $\pi$ has $h_m$ ramifications of order $m$ above the point $z$.) Then by Riemann-Hurwitz, \begin{equation} \label{idimX} \dim X = d+2g-1+\sum h_m, \end{equation} which is $d+2g-1$ plus the number of pre-images of $z$. (We are implicitly invoking the Riemann existence theorem here.) Notice that for the map corresponding to a general point in $X$, each special locus $A_j$ is either a marked ramification above the point $z$, a simple unmarked ramification, or the point $q$ (at which $\pi$ is smooth). In these three cases, the formal deformation space $\operatorname{Def} A_j$ is 0, $\operatorname{Spec} \mathbb{C} [[t]]$, and $\operatorname{Spec} \mathbb{C} [[t]]$ respectively. There is a corresponding stack ${\mathcal{X}} = {\mathcal{X}}^{d,g}(\vec{h}) \subset {\overline{\cm}}_{g,\sum h_m + 1}(\mathbb P^1,d)$ as well. Let $D$ be the divisor $\{ \pi(q) = z \}$. There are three natural questions to ask. \begin{enumerate} \item What are the components of the divisor $D$? \item With what multiplicity do they appear? \item What is the local structure of $X$ near these components? \end{enumerate} We will partially answer these three questions. Fix a component $Y$ of the divisor $D$ and a map $(C, \{ p^j_m \}, q, \pi)$ corresponding to the general element of $Y$. Notice that $\pi$ collapses a component of $C$ to $z$, as otherwise $\pi^{-1}(z)$ is a union of points, and $$ \deg \pi^* (z) \geq \sum_{j,m} \deg \pi^* (z) |_{p^j_m} + \deg \pi^* (z) |_q \geq \sum_{j,m} m + 1 > d. $$ Let $C(0)$ be the connected component of $\pi^{-1} (z)$ containing $q$, and let $\tilde{C}$ be the closure of $C \setminus C(0)$ in $C$ (see Figure \ref{iegC}; $C(0)$ are those curves contained in the dotted rectangle, and $\tilde{C}$ is the rest of $C$). \begin{figure} \begin{center} \getfig{iegC}{.1} \end{center} \caption{The map $(C, \{p^j_m \}, q, \pi \} ) \in Y$} \label{iegC} \end{figure} Let $h_m(0)$ be the number of $\{ p^j_m \}_j$ in $C(0)$, and $\tilde{h}_m = h_m - h_m(0)$ the number in $\tilde{C}$. Let $\{ p^j_m(0) \}_{m,1 \leq j \leq h_m(0)}$ and $\{ \tilde{p}^j_m \}_{m,1 \leq j \leq \tilde{h}_m}$ be the partition of $\{ p^j_m \}_{1 \leq j \leq h_m}$ into those marked points lying on $C(0)$ and those lying on $\tilde{C}$. Let $s$ be the number of intersections of $C(0)$ and $\tilde{C}$, and label these points $r^1$, \dots, $r^s$. Thus $g = p_a(C(0)) + p_a(\tilde{C}) + s - 1$. Let $m^k$ be the multiplicity of $(\pi|_{\tilde{C}})^*(z)$ at $r^k$. The data $(m^1, \dots, m^s)$ must be constant for any choice of $(C, \{p^j_m \}, q, \pi)$ in an open subset of $Y$. \begin{pr} The stable map $(\tilde{C}, \{ p^j_m(0) \}, \{ r^k \}, \pi)$ has no collapsed components, and only simple ramification away from $\pi^{-1}(z)$. The curve $\tilde{C}$ is smooth. \end{pr} The map $(\tilde{C}, \{ p^j_m(0) \}, \{ r^k \}, \pi)$ will turn out to correspond to a general element in $X^{d,g'}(\vec{h'})$ for some $\vec{h'}$, $g'$. Note that $\tilde{C}$ may be reducible. \noindent {\em Proof. } Let $A_1$, \dots, $A_l$ be the special loci of $\pi$, and say $q \in A_1$. The map $(C, \{ p^j_m \}, q, \pi)$ lies in $X$ and hence can be deformed to a curve where each special locus is either a marked ramification above $z$, a simple unmarked ramification, or an unramified marked point. If $A_k$ ($k>1$) is not one of these three forms then by Proposition \ref{ilocal} there is a deformation of the map $(C, \{ p^j_m \}, q, \pi)$ preserving $\pi$ at $A_i$ ($i \neq k$) but changing $A_k$ into a combination of special loci of these three forms. Such a deformation (in which $A_1$ is preserved and thus still smoothable) is actually a deformation in the divisor $D = \{ \pi(q) = z \}$, contradicting the generality of $(C, \{ p^j_m \}, q, \pi)$ in $Y$. \qed \vspace{+10pt} The map $(\tilde{C}, \{ p^j_m(0) \}, \{ r^k \}, \pi)$ must lie in $X^{d, p_a(\tilde{C})}( \vec{h'})$ where $\vec{h'}$ is the partition corresponding to $(\pi |_{\tilde{C}})^*(z)$. By Riemann-Hurwitz, $\tilde{C}$ moves in a family of dimension at most \begin{equation*} d + 2 p_a(\tilde{C}) - 2 + \left( \sum \tilde{h}_m + s \right) , \end{equation*} and the curve $C(0)$ (as a nodal curve with marked points $\{ r^k \}_{1 \leq k \leq s}$, $\{ p^j_m(0) \}_{m, 1 \leq j \leq h_m(0)}$, and $q$) moves in a family of dimension at most $$ 3 p_a (C(0)) - 3 + \sum h_m(0) + s +1 $$ so $Y$ is contained in a family of dimension \begin{eqnarray} \nonumber d + 2 p_a(\tilde{C}) - 2 + \sum \tilde{h}_m + s &+& 3 p_a(C(0)) - 3 + \sum h_m(0) + s +1\\ \nonumber &=& d + 2g-1 + \sum h_m - 1 + p_a(C(0))\\ &=& \dim X - 1 + p_a(C(0)) \label{inaive} \end{eqnarray} by (\ref{idimX}). We can now determine all components $Y$ of $D$ satisfying $p_a(C(0)) = 0$. For each choice of a partition of $\{ p^j_m \} $ into $\{ p^j_m(0) \} \cup \{ \tilde{p}^j_m \}$ (inducing a partition of $h_m$ into $h_m(0) + \tilde{h}_m$ for all $m$), a positive integer $s$, and $(m^1, \dots, m^s)$ satisfying $\sum m^s + \sum m \tilde{h}^j_m = d$, there is a variety (possibly reducible) of dimension $\dim X - 1$ that is the closure in ${\overline{M}}_{g, \sum h_m + 1}(\mathbb P^1,d)$ of points corresponding to maps $$ (C(0) \cup \tilde{C}, \{ p^j_m \}, q, \pi) $$ where \begin{enumerate} \item[A1.] The curve $C(0)$ is isomorphic to $\mathbb P^1$ and has labeled points $$ \{ p^j_m(0) \} \cup \{ r^k \} \cup \{ q \}, $$ and $\pi(C(0)) = z$. \item[A2.] The curve $\tilde{C}$ is smooth of arithmetic genus $g-s+1$ with marked points $\{ \tilde{p}^j_m \} \cup \{ r^k \}$. The map $\pi$ is degree $d$ on $\tilde{C}$, and $$ ( \pi|_{\tilde{C}})^*(z) = \sum m \tilde{p}^j_m + \sum m^k r^k. $$ \item[A3.] The curve $C(0) \cup \tilde{C}$ is nodal, and the curves $C(0)$ and $\tilde{C}$ intersect at the points $\{ r^k \}$. \end{enumerate} Let $U$ be the union of these varieties. An irreducible component $Y$ of the divisor $D$ satisfying $p_a(C(0)) = 0$ has dimension $\dim X - 1$ and is a subvariety of $U$, which also has dimension $\dim X - 1$. Hence $Y$ must be a component of $U$ and the stable map corresponding to a general point of $Y$ satisfies properties A1--A3 above. (We don't yet know that all such $Y$ are subsets of $X$, but this will follow from Proposition \ref{imultg0} below.) For example, if $d=2$, $g=0$, and $h_1 = 2$, then there are four components (see Figure \ref{ideg0}; $\pi^{-1}(z)$ is indicated by a dotted line). The components (from left to right) are a subset of the following. \begin{enumerate} \item The curve $\tilde{C}$ is irreducible and maps with degree 2 to $\mathbb P^1$, ramifying over general points of $\mathbb P^1$. The marked points $q$ and $p^1_1$ lie on $C(0)$, and $p^2_1$ lies on $\tilde{C}$. The curve $C(0)$ is attached to $\tilde{C}$ at the point $$ ( \pi|_{\tilde{C}} )^{-1}(z) \setminus \{ p^2_1 \}. $$ \item This case is the same as the previous one with $p^1_1$ and $p^2_1$ switched. \item The curve $\tilde{C}$ is the disjoint union of two $\mathbb P^1$'s, each mapping to $\mathbb P^1$ with degree 1. Both intersect $C(0)$, which contains all the marked points. \item The curve $\tilde{C}$ is irreducible and maps with degree 2 to $\mathbb P^1$, and one of its branch points is $z$. All of the marked points lie on $C(0)$. \end{enumerate} \begin{figure} \begin{center} \getfig{ideg0}{.1} \end{center} \caption{The possible components of $D$ on $X^{2,0}(h_1=2)$} \label{ideg0} \end{figure} Given a component $Y$ of $U$, we can determine the multiplicity of the divisor $D = \{ \pi(q) = z \}$ along $Y$. As this multiplicity will turn out to be positive, we will have the corollary that, as sets, $U \subset D$. For technical reasons, we use the language of stacks. \begin{pr} \label{imultg0} Fix such a component ${\mathcal{Y}}$ with $p_a(C(0)) = 0$. The multiplicity of $D$ along ${\mathcal{Y}}$ is $\prod_{k=1}^s m^k$. \end{pr} In particular, $Y$ is a subset of $X$. \noindent {\em Proof. } We make a series of reductions to simplify the proof. {\em Step 1: The deformations of $A_1$.} Let $\operatorname{Def}_{(C, \{ p^j_m \}, q, \pi), {\mathcal{X}}}$ be the versal deformation space of $(C, \{ p^j_m \}, q, \pi)$ in ${\mathcal{X}}$; it is a subspace of $\operatorname{Def}_{(C, \{ p^j_m \}, q, \pi)}$. We can compute the multiplicity on the versal deformation space of a stable map $\operatorname{Def}_{(C, \{ p^j_m \}, q, \pi), {\mathcal{X}}}$ corresponding to a general point in ${\mathcal{Y}}$. If $A_1$ is the special locus of $\pi$ containing $q$, then the divisor corresponding to $D$ on $\operatorname{Def}_{(C, \{ p^j_m \}, q, \pi), {\mathcal{X}}}$ is the pullback of a divisor $D_{\operatorname{Def}_{A_1,{\mathcal{X}}}}$ on $\operatorname{Def}_{A_1,{\mathcal{X}}}$. We will study $\operatorname{Def}_{A_1}$ by analyzing a simpler map. {\em Step 2: A simpler map.} Consider the stable map $$ (C'(0) \cup \tilde{C}', \{ p^j_m(0) \}, q, \{r^k \}, \pi) \in {\overline{\cm}}_{0, \sum h_m(0)+1+s}(\mathbb P^1,\sum m h_m(0)) $$ where: \begin{enumerate} \item[B1.] The marked curve $(C'(0), \{ p^j_m(0) \}, q, \{r^k \})$ is isomorphic to the marked curve $(C(0), \{ p^j_m(0) \}, q, \{r^k \})$, and is collapsed to $z$ by $\pi$. \item[B2.] The stable map $(\tilde{C}', \pi)$ consists of $s$ rational curves $C'(1)$, \dots, $C'(s)$ of degrees $m^1$, \dots, $m^s$ respectively, each ramifying completely over $z$. \item[B3.] The point of ramification of $C'(k)$ over $z$ is glued to $r^k$ on $C'(0)$. \end{enumerate} Let $A'_1$ be the special locus $C'(0)$ of this stable map. By Proposition \ref{ilocal2}, $\operatorname{Def}_{A_1} \cong \operatorname{Def}_{A'_1}$. Thus without loss of generality we may assume that $g=0$, $\vec{h} = \vec{h}(0)$, and the point in ${\mathcal{Y}}$ is of the form $$ (C = C(0) \cup \tilde{C}, \{ p^j_m \}, q, \{r^k \}, \pi) \in {\overline{\cm}}_{0, \sum h_m+1+s}(\mathbb P^1,d) $$ with properties B1--B3. Note that in this case $d = \sum m h_m = \sum m^k r^k$. {\em Step 3: Fixing the other special loci.} Since we are interested in $\operatorname{Def}_{A_1}$, it will be to our advantage to hold the other special loci constant. Fix a point $y \neq z$ on the target $\mathbb P^1$. Let ${\mathcal{X}}'$ be the closed substack that is the stack-theoretic closure (in ${\overline{\cm}}_{0,\sum h_m + 1 + s}(\mathbb P^1,d)$) of the points representing stable maps $$ (C, \{ p^j_m \}, q, \{ y^k \}, \pi ) $$ where $\pi^*(z) = \sum m p^j_m(0)$ and $\pi^*(y) = \sum m^k y^k$. Let ${\mathcal{Y}}'$ be the closure of points representing maps $$ (C = C(0) \cup C(1) \cup \dots \cup C(s), \{ p^j_m \}, q, \{ y^k \},\pi ) $$ where: \begin{enumerate} \item[(i)] The curve $C(0)$ is rational, contains $\{ p^j_m \}$, intersects all of the other $C(k)$, and is collapsed to $z$ by $\pi$. \item[(ii)] The curve $C(k)$ ($k>0$) is rational. The map $\pi$ is degree $m^k$ on $C(k)$, and $C(k)$ ramifies totally above $z$ (where it intersects $C(0)$) and $y$ (where it is labeled $y^k$). \end{enumerate} If $(C, \{ p^j_m \}, q, \{y^k \}, \pi)$ is a map corresponding to a general point in ${\mathcal{Y}}'$, $$ \operatorname{Def}_{(C, \{ p^j_m \}, q, \{y^k \}, \pi), {\mathcal{X}}'} = \operatorname{Def}_{A_1, {\mathcal{X}}}: $$ the only deformations of such a map preserving the ramifications above $y$ are deformations of $A_1$. {\em Step 4: Fixing the marked curve.} We next reduce to the case where $(C, \{ p^j_m \}, q, \{y^k \})$ is a fixed stable marked curve. There is a morphism of stacks $\alpha: {\mathcal{X}} \rightarrow {\overline{\cm}}_{0, \sum h_m + 1 + s}$ that sends each map to stable model of the underlying pointed nodal curve. Given any smooth marked curve $(C(0), \{ p^j_m \}, q, \{ y^k \} )$ in ${\overline{\cm}}_{0, \sum h_m + 1 + s}$, the stable map $(C, \{ p^j_m \}, q, \{ y^k \},\pi )$ defined in Step 3 (where $C$ is a union of irreducible curves $C(0) \cup \dots \cup C(s)$) corresponds to a point in $\alpha^{-1}(C(0), \{ p^j_m \}, q, \{ y^k \} )$. (The stable curve $\alpha(C, \{ p^j_m \}, q, \{ y^k \},\pi )$ is constructed by forgetting $\pi$ and contracting the rational tails $C(1)$, \dots, $C(s)$.) Hence $\alpha|_{{\mathcal{Y}}'}$ is surjective. Let ${\mathcal{F}}_\alpha$ be a general fiber of $\alpha$. By Sard's theorem, $\alpha |_{{\mathcal{Y}}'}$ is regular in a Zariski-open subset of ${\mathcal{Y}}'$, so $[{\mathcal{Y}}'] \cap [{\mathcal{F}}_\alpha] = [{\mathcal{Y}}' \cap {\mathcal{F}}_\alpha]$ in the Chow group of $[{\mathcal{X}}']$. In order to determine the multiplicity of $D|_{{\mathcal{X}}'}$ along ${\mathcal{Y}}'$, it suffices to determine the multiplicity of the Cartier divisor $D|_{{\mathcal{F}}_\alpha}$ along ${\mathcal{Y}}' \cap {\mathcal{F}}_\alpha$ (in the Chow group of ${\mathcal{F}}_\alpha$). ({\em Proof:} As $D$ is a Cartier divisor, $[D|_{{\mathcal{F}}_\alpha}] = [D|_{{\mathcal{X}}'}] \cdot [{\mathcal{F}}_\alpha]$. Thus if $[D|_{{\mathcal{X}}'}] = m [ {\mathcal{Y}}']$ in $A^1 {\mathcal{X}}'$ then, intersecting with $[{\mathcal{F}}_\alpha]$, $[D|_{{\mathcal{F}}_\alpha}] = [D|_{{\mathcal{X}}'}] \cdot [{\mathcal{F}}_\alpha] = m [{\mathcal{Y}}'] \cdot [{\mathcal{F}}_\alpha] = m [{\mathcal{Y}}' \cap {\mathcal{F}}_\alpha ]$ in $A^1 {\mathcal{F}}_\alpha$.) With this in mind, fix a general $(C, \{ p^j_m \}, q, \{ y^k \} )$ in ${\overline{\cm}}_{0, \sum h_m + 1 + s}$ and let ${\mathcal{X}}''_o$ be the points of ${\overline{\cm}}_{0, \sum h_m + 1 + s}(\mathbb P^1,d)$ representing stable maps $(C, \{ p^j_m \}, q, \{ y^k \}, \pi )$ where $\pi^*(z)= \sum m p^j_m$ and $\pi^*(y) = \sum m^k y^k$. Let ${\mathcal{X}}'' = {\mathcal{X}}' \cap {\mathcal{F}}_\alpha$ be the closure of ${\mathcal{X}}''_o$, let $X''_o$, $X''$ be the corresponding varieties, and define ${\mathcal{Y}}'' = {\mathcal{Y}} \cap {\mathcal{F}}_\alpha$ and $Y''$ similarly. {\em Step 5: The variety $X''$ is actually $\mathbb P^1$!} Let $f$ and $g$ be sections of ${\mathcal{O}}_C(d)$ with $$ (f=0) = \sum m p^j_m, \quad (g=0) = \sum m^k y^k; $$ the maps in ${\mathcal{X}}''_o$ are those of the form $[\beta f, \gamma g ]$ where $$ [\beta, \gamma] \in \mathbb P^1 \setminus \{ [0,1], [1,0] \} $$ where $z = [0,1]$ and $y = [1,0]$. The variety $X''$ is proper, so the normalization of the variety $X''$ is $\mathbb P^1$. The curve $X''$ has a rational map to $\mathbb P^1$ given by $$ (C, \{ p^j_m \}, q, \{ y^k \}, \pi ) \rightarrow \pi(q) $$ and this map is an isomorphism from $X''_o$ to $\mathbb P^1 \setminus \{ [0,1], [1,0] \}$, so it must be an isomorphism from $X''$ to $\mathbb P^1$. {\em Step 6: Calculating the multiplicity.} Let $z'$ be a general point of the target $\mathbb P^1$. Then the divisor $\{ \pi(q) = z' \}$ is linearly equivalent to $D|_{X''} = \{ \pi(q) = z \}|_{X''}$, and is ${\mathcal{O}}_{X''}(1)$. Thus, {\em as varieties}, $D|_{X''} = [1,0] = Y''$. But the limit map has automorphism group $$ \mathbb{Z} / {m^1 \mathbb{Z}} \oplus \dots \oplus \mathbb{Z} / m^s \mathbb{Z} $$ (as $\operatorname{Aut}(C(k), \pi|_{C(k)}) = m^k$) so {\em as stacks} $[D |_{{\mathcal{X}}''} ] = \left( \prod m^k \right) [{\mathcal{Y}}'']$. Therefore $[D] = \prod m^k [{\mathcal{Y}}]$. \qed \vspace{+10pt} The above argument can be refined to determine the local structure of ${\mathcal{X}}$ near ${\mathcal{Y}}$ (and thus $X$ near $Y$ if the map corresponding to a general point of $Y$ has no automorphisms): \begin{co} \label{ilocalst} Let ${\mathcal{Y}}$ be the same component as in Proposition \ref{imultg0}. an \'{e}tale neighborhood of a general point of ${\mathcal{Y}}$, the stack ${\mathcal{X}}$ is isomorphic to $$ \operatorname{Spec} \mathbb{C} [[a, b_1,\dots,b_s,c_1,\dots,c_{\dim X -1 }]] / (a = b_1^{m^1} = \dots = b_s^{m^s}) $$ with $D$ given by $\{ a = 0 \}$, and ${\mathcal{Y}}$ given set-theoretically by the same equation. \end{co} In particular, if $\gcd(m^i,m^j)>1$ for some $i$ and $j$, ${\mathcal{X}}$ fails to be unibranch at a general point of ${\mathcal{Y}}$. \noindent {\em Proof. } In the proof of Proposition \ref{imultg0} above, at the end of Step 4 we had reduced to considering a fixed marked curve $(C, \{ p^j_m \}, q, \{ y^k \} )$ in ${\overline{\cm}}_{0, \sum h_m + 1 + s}$ and maps $\pi$ from this marked curve to $\mathbb P^1$ where $\pi^*(z) = \sum m p^j_m$ and $\pi^*(y) = \sum m^k y^k$. These maps are parametrized by the stack ${\mathcal{X}}''$. {\em Step 5${}'$: Rigidifying the moduli problem.} It will be more convenient to work with varieties than stacks, so we rigidify the moduli problem to eliminate nontrivial automorphisms. Fix a point $x \in \mathbb P^1$ distinct from $y$ and $z$. We will mark the $d$ pre-images of $x$ with the labels $\{ x^1, \dots, x^d \}$. Let ${\mathcal{X}}''_x$ be the moduli stack parametrizing maps $(C, \{p^j_m \}, q, \{ y^k \}, \{ x^l \}, \pi)$ where \begin{enumerate} \item[C1.] The pointed curve $(C, \{p^j_m \}, q, \{ y^k \})$ is fixed. \item[C2.] $(C, \{p^j_m \}, q, \{ y^k \}, \pi)$ is a stable map, \item[C3.] $\pi(x^i) = x$ for all $i$, and $x^i \neq x^j$ for $i \neq j$. \end{enumerate} The moduli stack ${\mathcal{X}}''_x$ is actually a variety (as none of the maps para\-metrized by this stack have automorphisms), and over an open neighborhood of ${\mathcal{Y}}''$ in ${\mathcal{X}}''$ the natural morphism $$ \eta: {\mathcal{X}}''_x \rightarrow {\mathcal{X}}'' $$ is an \'{e}tale (degree $d!$) morphism of proper stacks at a point of ${\mathcal{Y}}''$. (The variety ${\mathcal{X}}''_x$ is an atlas for the stack ${\mathcal{X}}''$.) Define the Weil divisor ${\mathcal{Y}}''_x$ on ${\mathcal{X}}''_x$ similarly; it is a union of points. We can now reprove Proposition \ref{imultg0} using the variety ${\mathcal{X}}''_x$: if ${\mathcal{Y}}''$ is the point in ${\mathcal{X}}''$ corresponding to the point in ${\mathcal{Y}}$ (i.e. the curve $C(0)$ with $l$ rational tails $C(1)$, \dots, $C(l)$ ramifying completely over $y$ and $z$) then $\eta^{-1}({\mathcal{Y}}'')$ is set-theoretically $$ {\binom d {m^1, \dots, m^s} } \prod_{k=1}^s (m^k - 1)! $$ points: there are $\binom d {m^1, \dots, m^s}$ ways to divide the $d$ points $\{ x^1, \dots, x^d \}$ above $x$ among $C(1)$, \dots, $C(s)$ and $(m^k - 1)!$ possible choices of the markings above $x$ on $C(k)$ up to automorphisms of $\pi |_{C(k)}$. This is the number of partitions of $\{ x^1, \dots, x^d \}$ into cyclically-ordered subsets of sizes $m^1$, \dots, $m^s$. Hence the multiplicity at each one of these points must be $$ {\frac {d!} { {\binom {d} {m^1, \dots, m^s} } \prod_{k=1}^s (m^k - 1)! } } = \prod_{k=1}^s m^k. $$ {\em Step 6${}'$: The calculation.} We now determine the local structure at one of these points. Fix sections $f$ and $g$ of ${\mathcal{O}}_C(d)$ with $$ (f=0) = \sum m p^j_m, \quad (g=0) = \sum m^k y^k $$ and let $\pi$ be the morphism to $\mathbb P^1$ given by $\pi = [f,g]$. Rather than considering elements of ${\mathcal{X}}''_x$ as maps $[\beta f, \gamma g]$ (with $x$ fixed, and $\pi(q)$ varying), we now consider them as maps $[f,g]$ with $x$ moving (and $\pi(q)$ fixed). (We are degenerating the point $(\mathbb P^1, x, y, z, \pi(q)) \in {\overline{M}}_{0,4}$ in two ways. Originally we fixed $x$, $y$, $z$ and let $\pi(q)$ degenerate to $z$. Now we fix $y$, $z$, $\pi(q)$ and let $x$ tend to $y$. They are equivalent as they represent the same point in the curve ${\overline{M}}_{0,4} = \mathbb P^1$.) The Weil divisor is now defined (set-theoretically) by $\{ x = y \}$, not $\{ \pi(q) = z \}$. Then $$ {\mathcal{X}}''_x = \overline{\underbrace{C \times_{\pi} \dots \times_{\pi} C}_{d} \setminus \Delta} $$ where $\Delta$ is the big diagonal (where any two of the factors are the same), and the closure is in $$ \underbrace{C \times_{\pi} \dots \times_{\pi} C}_{d}. $$ Fix one of the points of ${\mathcal{Y}}''_x$, which corresponds to a partition of $\{ x^1, \dots, x^d \}$ into subsets of sizes $m^1$, \dots, $m^s$ and cyclic orderings of these subsets. Consider a neighborhood of this point in ${\mathcal{X}}''_x$. By relabeling if necessary, we may assume that $x^k$ is in the $k^{\text{th}}$ subset for $1 \leq k \leq s$ (so, informally, $x^k$ is close to $t_k$; see Figure \ref{inap}). \begin{figure} \begin{center} \getfig{inap}{.1} \end{center} \caption{Near a point of ${\mathcal{Y}}''_x$} \label{inap} \end{figure} In an \'{e}tale neighborhood of point in ${\mathcal{Y}}''_x$, $$ {\mathcal{X}}''_x = \overline{\underbrace{C \times_{\pi} \dots \times_{\pi} C}_{d} \setminus \Delta} = \overline{\underbrace{C \times_{\pi} \dots \times_{\pi} C}_s \setminus \Delta} $$ where the second product consists of the first $s$ factors of the first. Let $a$ be a local co-ordinate for $x$ near $y$. As the ramification of $\pi$ at $y^k$ is $m^k - 1$, there is an \'{e}tale-local co-ordinate $b_k$ for $x^k$ near $y^k$ where $\pi$ is given by $a = b_k^{m^k}$. Therefore in an \'{e}tale neighborhood of our point of ${\mathcal{Y}}''_x$, \begin{eqnarray*} {\mathcal{X}}''_x &=& \overline{\underbrace{C \times_{\pi} \dots \times_{\pi} C}_s \setminus \Delta} \\ &=& \operatorname{Spec} \mathbb{C} [[ a,b_1, \dots, b_s ]] / (a = b_1^{m^1}= \dots = b_s^{m^s}). \end{eqnarray*} Hence the deformation space $\operatorname{Def}_{A_1}$ is isomorphic to $$ \operatorname{Spec} \mathbb{C} [[ a,b_1, \dots, b_s ]] / (a = b_1^{m^1}= \dots = b_s^{m^s}). $$ The divisor $D=(y=x)$ is given by $(a=0)$. {\em Step 7${}'$: Returning to the original problem.} For $j>1$, $A_j$ is either a marked point $p^j_m$ with a ramification of order $m$ over $z$ or a simple ramification. In these cases we have $\operatorname{Def} A_j = 0$ or $\operatorname{Def} A_j = \operatorname{Spec} \mathbb{C}[[c]]$ respectively. Hence the deformation space $\operatorname{Def}_{(C, \{ p^j_m \}, q, \pi)}$ is isomorphic to $$ \operatorname{Spec} \mathbb{C} [[a, b_1,\dots,b_s,c_1,\dots,c_{\dim X-1}]] / (a = b_1^{m^1} = \dots = b_s^{m^s}). $$ \qed \vspace{+10pt} Proposition \ref{imultg0} and Corollary \ref{ilocalst} above are statements about varieties, so long as $d \neq 2$. In order to extend these results to components for which $p_a(C(0)) = 1$, we will need the following result. \begin{pr} Let $Y$ be a component of $D$, with $(C,\{p^j_m \}, q, \pi)$ the map corresponding to a general point of $Y$, $\{ r^1, ..., r^s \} = C(0) \cap \tilde{C}$, and $m^k$ the multiplicity of $\pi^*(z)$ along $\tilde{C}$ at $r^k$. Then $$ {\mathcal{O}}_{C(0)}\left(\sum_{m,j} m p^j_m(0)\right) \cong {\mathcal{O}}_{C(0)}\left(\sum_{k=1}^s m^k r^k \right) $$ where $\{ p^j_m(0) \}^{h_m(0)}_{j=1} \subset \{ p^j_m \}_{j=1}^{h_m}$ are the marked points whose limits lie in $C(0)$. \end{pr} \noindent {\em Proof. } For a map $(C, \{ p^j_m \}, q ,\pi)$ corresponding to a general point in $X$, we have the following relation in the Picard group of $C$: $$ \pi^*({\mathcal{O}}_{\mathbb P^1}(1)) \cong {\mathcal{O}}_C(\sum_{m,j} m p^j_m). $$ Thus for the curve corresponding to a general point of our component of $D$ the invertible sheaf ${\mathcal{O}}_C (\sum_{m,j} m p^j_m)$ must be a possible limit of $\pi^*({\mathcal{O}}_{\mathbb P^1}(1))$. The statement of the lemma depends only on an analytic neighborhood of $C(0)$, so we may assume (as in Step 2 of the proof of Proposition \ref{imultg0}) that $\vec{h} = \vec{h}(0)$, $p_a(C) = 1$, and $\tilde{C}$ consists of $k$ rational tails $C(1)$, \dots, $C(k)$ each totally ramified where they intersect $C(0)$. As the dual graph of $C$ is a tree, $C$ is of compact type (which means that $\operatorname{Pic} C$ is compact). One possible limit of $\pi^*({\mathcal{O}}_{\mathbb P^1}(1))$ is the line bundle that is trivial on $C(0)$ and degree $m^k$ on $C(k)$. If a curve $C'$ is the central fiber of a one-dimensional family of curves, and $C' = C_1 \cup C_2$, and a line bundle ${\mathcal{L}}$ is the limit of a family of line bundles, then the line bundle ${\mathcal{L}}'$ whose restriction to $C_i$ is ${\mathcal{L}}|_{C_i}( (-1)^i C_1 \cap C_2)$ is another possible limit. Thus the line bundle that is trivial on $\tilde{C}$ and ${\mathcal{O}}_C(\sum m^k r^k)$ on $C(0)$ is a possible limit of $\pi^*({\mathcal{O}}_{\mathbb P^1}(1))$. If two line bundles on a nodal curve $C$ of compact type are possible limits of the same family of line bundles, and they agree on all components but one of $C$, then they must agree on the remaining component. But ${\mathcal{O}}_C(\sum m p^j_m)$ is another limit of $\pi^*({\mathcal{O}}_{\mathbb P^1}(1))$ that is trivial on $\tilde{C}$, so the result follows. \qed \vspace{+10pt} We are now in a position to determine all components $Y$ of $D$ satisfying $p_a(C(0)) = 1$. For each choice of a partition of $\{ p^j_m \} $ into $\{ p^j_m(0) \} \cup \{ \tilde{p}^j_m \}$, a positive integer $s$, and $(m^1, \dots, m^s)$ satisfying $\sum m^s + \sum m \tilde{h}^j_m = d$, there is a variety (possibly reducible) that is the closure in ${\overline{M}}_{g, \sum h_m + 1}(\mathbb P^1,d)$ of points corresponding to maps $$ (C(0) \cup \tilde{C}, \{ p^j_m \}, q, \pi) $$ where \begin{enumerate} \item[D1.] The curve $C(0)$ is a smooth elliptic curve with labeled points $$ \{ p^j_m(0) \} \cup \{ r^k \} \cup \{ q \}, $$ where ${\mathcal{O}}_{C(0)}(\sum m p^j_m(0)) \cong {\mathcal{O}}_{C(0)}( \sum m^k r^k )$, and $\pi(C(0)) = z$. \item[D2.] The curve $\tilde{C}$ is smooth of arithmetic genus $g-s$ with marked points $\{ \tilde{p}^j_m \} \cup \{ r^k \}$. The map $\pi$ is degree $d$ on $\tilde{C}$, and $$ ( \pi|_{\tilde{C}})^*(z) = \sum m \tilde{p}^j_m + \sum m^k r^k. $$ \item[D3.] $C(0) \cup \tilde{C}$ is nodal, and $C(0)$ and $\tilde{C}$ are glued at the points $\{ r^k \}$. \end{enumerate} Let $U$ be the union of these varieties. The divisorial condition ${\mathcal{O}}_{C(0)}(\sum m p^j_m(0)) \cong {\mathcal{O}}_{C(0)}( \sum m^k r^k )$ defines a substack ${\mathcal{M}}'$ of pure codimension 1 in ${\mathcal{M}}_{1,\sum h_m + 1 + s}$: for any $$ (C, \{ p^j_m \}, q, \{ r^k \}_{k>1})\in {\mathcal{M}}_{1, \sum h_m + 1 + (s-1)} $$ the subscheme of points $r^1 \in C$ satisfying $$ {\mathcal{O}}(m^1 r^1) \cong {\mathcal{O}} \left( \sum m p^j_m(0) - \sum_{k>1} m^k r^k \right) $$ is reduced of degree $(m^1)^2$. The stack ${\mathcal{M}}'$ is a degree $(m^1)^2$ \'{e}tale cover of ${\mathcal{M}}_{1, \sum h_m + 1 + (s-1)}$. By this observation and (\ref{inaive}), $U$ has pure dimension $\dim X - 1$. An irreducible component $Y$ of the divisor $D$ satisfying $p_a(C(0)) = 1$ has dimension $\dim X - 1$ and is a subvariety of $U$, which also has dimension $\dim X - 1$. Hence $Y$ must be a component of $U$ and the stable map corresponding to a general point of $Y$ satisfies properties D1--D3 above. The determination of multiplicity and local structure is identical to the genus 0 case. \begin{pr} \label{imultg1} Fix such a component ${\mathcal{Y}}$ with $p_a(C(0)) = 1$. If $m^1$, \dots, $m^s$ are the multiplicities of $\pi^* (z)$ along $\tilde{C}$ at the $s$ points $C(0) \cap \tilde{C}$, then this divisor appears with multiplicity $\prod_k m^k$. \end{pr} \noindent {\em Proof. } The proof is identical to that of Proposition \ref{imultg0}. We summarize the steps here. {\em Step 1.} If $A^1$ is the special locus of $\pi$ containing $q$, then it suffices to analyze $\operatorname{Def} A_1$. {\em Step 2.} We may assume that the map corresponding to a general point in ${\mathcal{Y}}$ consists of $C(0)$ and $s$ rational tails ramifying completely over $z$. {\em Step 3.} We require the $s$ rational tails to ramify completely over another point $y$, and we label these ramifications $y^1$, \dots, $y^s$. {\em Step 4.} Let ${\overline{\cm}}'_{1, \sum h_m + 1 + s}$ be the substack of ${\overline{\cm}}_{1, \sum h_m + 1 + s}$ that is the closure of the set of points representing smooth stable curves where ${\mathcal{O}}(\sum m p^j_m) \cong {\mathcal{O}}(\sum m^k y^k)$. If $\alpha$ is defined by $$ \alpha: {\mathcal{X}}' \rightarrow {\overline{\cm}}'_{1, \sum h_m + 1 + s} $$ then $\alpha |_{{\mathcal{Y}}}$ is dominant, so we may consider a fixed stable curve $$ (C, \{ p^j_m \}, q, \{ y^k \}) \in {\overline{\cm}}'_{1, \sum h_m + 1 + s}. $$ {\em Step 5.} The variety $X''$ is $\mathbb P^1$. {\em Step 6.} The multiplicity calculation is identical. \qed \vspace{+10pt} Once again, we get the \'{e}tale or formal local structure of ${\mathcal{X}}$ as a corollary. The statement and proof are identical to those of Corollary \ref{ilocalst}. \begin{co} Let ${\mathcal{Y}}$ be the same component as in Proposition \ref{imultg1}. In an \'{e}tale neighborhood of a general point of ${\mathcal{Y}}$, the stack ${\mathcal{X}}$ is isomorphic to $$ \operatorname{Spec} \mathbb{C} [[a, b_1,\dots,b_s,c_1,\dots,c_{\dim X - 1}]] / (a = b_1^{m^1} = \dots = b_s^{m^s}) $$ with $D$ given by $\{ a = 0 \}$, and ${\mathcal{Y}}$ given set-theoretically by the same equation. \end{co} If $g=0$ or $1$, then $p_a(C(0)) = 0$ or $1$, so we have found all components of $D=\{ \pi(q) = z \}$ and the multiplicity of $D$ along each component. We summarize this in two theorems which will be invoked in Sections \ref{rational} and \ref{elliptic}. \begin{tm} \label{igenus0} If $g=0$, the components of $D = \{ \pi(q) = z \}$ on $X^{d,g}(\vec{h})$ are of the following form. Fix a positive integer $l$, $\{ d(k) \}_{k=1}^l$ with $\sum_{k=0}^l d(k) = d$, and a partition of the points $\{ p^j_m \}$ into $l$ subsets $\cup_{k=1}^l \{ p^j_m(k) \}$. This induces a partition of $\vec{h}$ into $\sum_{k=0}^l \vec{h}(k)$. Let $m^k = d(k) - \sum m h_m(k)$. Then the general member of the component is a general map from $C(0) \cup \dots \cup C(l)$ to $\mathbb P^1$, where: \begin{itemize} \item The irreducible components $C(0)$, \dots, $C(l)$ are rational. \item The curve $C(0)$ contains the marked points $\{ p^j_m(0) \}$, $q$, $\{ r^k \}$, and $\pi(C(0)) = z$. \item For $k>0$, $C(k)$ maps to $\mathbb P^1$ with degree $d(k)$ and $$ \pi^*(z) |_{C(k)} = \sum_{m,j} m p^j_m(k) + m^k r^k. $$ The curves $C(0)$ and $C(k)$ intersect at $r^k$. \end{itemize} This component appears with multiplicity $\prod_k m^k$. \end{tm} \begin{tm} \label{igenus1} If $g=1$, the components of $D= \{ \pi(q) = z \}$ on $X^{d,g}(\vec{h})$ are of the following form. Fix a positive integer $l$, $\{ d(k) \}_{k=1}^l$ with $\sum_{k=0}^l d(k) = d$, and a partition of the points $\{ p^j_m \}$ into $l$ subsets $\cup_{k=1}^l \{ p^j_m(k) \}$. This induces a partition of $\vec{h}$ into $\sum_{k=0}^l \vec{h}(k)$. Let $m^k = d(k) - \sum m h_m(k)$. Then the general member of the component is a general map from $C(0) \cup \dots \cup C(l)$ to $\mathbb P^1$, where: \begin{itemize} \item The curve $C(0)$ contains the marked points $\{ p^j_m(0) \}$ and $q$, and $\pi(C(0)) = z$. \item For $k>0$, $C(k)$ maps to $\mathbb P^1$ with degree $d(k)$. \end{itemize} Furthermore, one of the following cases holds: \begin{enumerate} \item[a)]The curve $C(1)$ is elliptic and the other components are rational. When $k>0$, the curves $C(0)$ and $C(k)$ intersect at the point $r^k$, and $\pi^* (z) |_{C(k)} = \sum_{m,j} p^j_m(k) + m^k r^k$. \item[b)] All components are rational. When $k>1$, the curves $C(0)$ and $C(k)$ intersect at the point $r^k$, and $\pi^* (z) |_{C(k)} = \sum_{m,j} m p^j_m(k) + m^k r^k$. The curves $C(1)$ and $C(0)$ intersect at two points $r^1_1$ and $r^1_2$, and the ramifications $m^1_1$ and $m^1_2$ at these two points sum to $m^1$. $\pi^*(z) |_{C(1)} = \sum_{m,j} m p^j_m(k) + m^1_1 r^1_1 + m^1_2 r^1_2$. \item[c)]The curve $C(0)$ is elliptic and the other components are rational. When $k>0$, the curves $C(0)$ and $C(k)$ intersect at $r^k$, and $\pi^* (z) |_{C(k)} = \sum_{m,j} m p^j_m(k) + m^k r^k$. Also, ${\mathcal{O}}_{C(0)}( \sum_{m,j} m p^j_m(0)) \cong {\mathcal{O}}_{C(0)}( \sum m^k r^k)$. \end{enumerate} The components of type a) and c) appear with multiplicity $\prod_{k=1}^l m^k$ and those of type b) appear with multiplicity $m^1_1 m^1_2 \prod_{k=2}^l m^k$. \end{tm} In all three cases, the multiplicity is the product of the ``new ramifications'' of the components not mapped to $z$. For general $g$, the above argument identifies some of the components, but further work is required to determine what happens when $p_a(C(0)) > 1$. \subsection{Pathological behavior of $\mbar_g(\proj^1,d)$} When $d>2$, $g>0$, $\mbar_g(\proj^1,d)$ has more than one component. The most interesting one consists (generically) of irreducible genus $g$ curves. Call this one $\mbar_g(\proj^1,d)^o$. A second consists (generically) of two intersecting components, one of genus $g$ and mapping to a point, and the other rational and mapping to $\mathbb P^1$ with degree $d$. The first has dimension $2d+2g-2$, and the second has dimension $2d+3g-3$, so the second is not in the closure of the first. The local structure of $\mbar_g(\proj^1,d)^o$ may be complicated where it intersects the other components. One might hope that $\mbar_g(\proj^1,d)^o$ is smooth (at least as a stack, or equivalently on the level of deformation spaces). This is not the case; $\mbar_g(\proj^1,d)^o$ can be singular and even fail to be unibranch. Let $g=3$ (so a general degree 3 divisor has one section, but all degree 4 divisors have two) and $d=4$. Then ${\overline{M}}_3(\mathbb P^1,4)^o$ has dimension 12. Consider the family $Y$ of stable maps whose general element parametrizes a smooth genus 3 curve $C(0)$ with two rational tails $C(1)$ and $C(2)$. The curve $C(0)$ maps with degree 0 to $\mathbb P^1$, and the rational tails each map with degree 2 to $\mathbb P^1$, both ramifying at their intersection with $C$. The subvariety $Y$ has dimension 11: 8 for the two-pointed genus 3 curve $C(0)$, 1 for the image of $C(0)$ in $\mathbb P^1$, and 2 for the other ramification points of $C(1)$ and $C(2)$. Thus if $Y$ is contained in ${\overline{M}}_3(\mathbb P^1,4)^o$, it is a Weil divisor. \begin{pr} ${\overline{M}}_3(\mathbb P^1,4)^o$ has two smooth branches along $Y$, intersecting transversely. \end{pr} This gives an example of a map that could be smoothed in two different ways. \noindent {\em Proof. } For convenience, we use the language of stacks. Let ${\mathcal{X}}$ be the stack corresponding to $X$, and ${\mathcal{Y}}$ the stack corresponding to $Y$. Let $(C, \pi)$ be the map corresponding to a general point of ${\mathcal{Y}}$ (where $C = C(0) \cup C(1) \cup C(2)$ where $C(0)$, $C(1)$, $C(2)$ are as described above), and let $A_1$ be the special locus consisting of the collapsed genus 3 curve. By Proposition \ref{ilocal}, it suffices to consider $\operatorname{Def}_{A_1}$. As in Step 3 of Proposition \ref{imultg0}, we fix the other special loci. Fix a point $y \in \mathbb P^1$. We may restrict attention to maps ramifying at two points above $y$, labeled $y^1$ and $y^2$. Denote by ${\mathcal{X}}'$ and ${\mathcal{Y}}'$ the substacks of ${\overline{\cm}}_{3,2}(\mathbb P^1,4)$ (with ramification above $y$ at $y^1$ and $y^2$) corresponding to ${\mathcal{X}}$ and ${\mathcal{Y}}$. It suffices to prove the corresponding result for ${\mathcal{X}}'$ and ${\mathcal{Y}}'$. Next, fix another point $z \in \mathbb P^1$ ($z \neq y$) and mark the four pre-images of $z$ with the labels $p^1$, \dots, $p^4$. Denote by ${\mathcal{X}}''$ and ${\mathcal{Y}}''$ the substacks of ${\overline{\cm}}_{3,6}(\mathbb P^1,4)$ corresponding to ${\mathcal{X}}'$ and ${\mathcal{Y}}'$. It suffices to prove the corresponding result for ${\mathcal{X}}''$ and ${\mathcal{Y}}''$. Let ${\mathcal{M}}$ be the substack of ${\overline{\cm}}_{3,6}$ that is the closure of the points representing stable marked curves $(C, \{ p^i \}, y^1, y^2)$ where $C$ is smooth and ${\mathcal{O}}_C(p^1 + \dots + p^4) \cong {\mathcal{O}}_C(2 y^1 + 2 y^2)$. If $\alpha$ is the natural map $\alpha: {\overline{\cm}}_{3,6}(\mathbb P^1,4) \rightarrow {\overline{\cm}}_{3,6}$ then $\alpha({\mathcal{X}}'') \subset {\mathcal{M}}$ as for a general map $(C, \{ p^i \}, y^1, y^2, \pi) \in {\mathcal{X}}''$, $$ {\mathcal{O}}_C(p^1 + \dots + p^4) \cong \pi^* ( {\mathcal{O}}_{\mathbb P^1}(1)) \cong {\mathcal{O}}_C(2 y^1 + 2 y^2). $$ Moreover $\alpha |_{{\mathcal{Y}}''}$ surjects onto ${\mathcal{M}}$: for any curve $(C, \{ p^i \}, r^1, r^2 ) \in {\mathcal{M}}$ consider the map $(C(0) \cup C(1) \cup C(2) , \{ p^i \}, y^1, y^2, \pi)$ where $C(0)$ and $C(k)$ intersect at $r^k$ ($k=1,2$), $$ (C(0), \{ p^i \}, r^1, r^2) \cong (C, \{ p^i \}, r^1, r^2 ) $$ as marked curves, $\pi(C(0)) = z$, and $C(k)$ ($k = 1,2$) maps to $\mathbb P^1$ with degree 2 ramifying over $y$ (at $y^k$) and $z$ (at $r^k$). The stable model of the underlying curve of such a map is indeed isomorphic to $(C, \{ p^i \}, r^1, r^2 )$. Thus we may restrict attention to a fixed (general) marked curve $(C, \{ p^i \}, y^1, y^2)$ in ${\mathcal{M}}$. We may now directly follow steps 5${}'$ and $6'$ in the proof of Corollary \ref{ilocalst}. The steps are summarized here. {\em Step 5${}'$.} Rigidify the moduli problem by eliminating automorphisms. Fix a point $x \in \mathbb P^1$ distinct from $y$ and $z$, and mark the points of $\pi^{-1}(x)$ with the labels $\{ x^1, \dots, x^d \}$. Call the resulting stack ${\mathcal{X}}''_x$. {\em Step 6${}'$.} Observe that $$ {\mathcal{X}}''_x = \overline{\underbrace{C \times_{\pi} \dots \times_{\pi} C}_{d} \setminus \Delta} $$ where $\Delta$ is the big diagonal, and the closure is in $$ \underbrace{C \times_{\pi} \dots \times_{\pi} C}_{d}. $$ Then show that $$ \operatorname{Def}_{A_1} = \operatorname{Spec} \mathbb{C} [[ a, b_1, b_2]] / (a = b_1^2 = b_2^2). $$ \qed \vspace{+10pt} By a similar argument, we can find a codimension 1 unibranch singularity of ${\overline{M}}_4(\mathbb P^1,5)^o$ and singularities of ${\overline{M}}_6(\mathbb P^1,7)^o$) with several codimension 1 singular branches. \subsection{Possible applications of these methods} Ideas involving degenerations and the space of stable maps can be used in other cases besides those dealt with in this article, and we will briefly mention them here. In [V1] the same methods are used for genus $g$ curves in a divisor class $D$ on the rational ruled surface ${\mathbb F}_n$: the curves through $-K_{{\mathbb F}_n} \cdot D + g - 1$ general points are enumerated; these also repreoduce the calculations in [CH3] of genus $g$ Gromov-Witten invariants of the plane (i.e. the enumerative geometry of plane curves) in the language of maps. The genus $g$ Gromov-Witten invariants of the plane blown up at up to five points are calculated in [V2]. In [V3], the analogous question for plane curves with certain allowed singularities is incompletely addressed. It also seems possible that the classical question of characteristic numbers of rational and elliptic curves in $\mathbb P^n$ would be susceptible to such an approach. (The rational case has already been settled by L. Ernstr\"{o}m and G. Kennedy by different methods in [EK1].) \section{Rational Curves in Projective Space} \label{rational} In this section, we use the ideas and results of Section \ref{intro} to study the geometry of varieties parametrizing degree $d$ rational curves in $\mathbb P^n$ intersecting fixed general linear spaces and tangent to a fixed hyperplane $H$ with fixed multiplicities along fixed general linear subspaces of $H$. We employ two general ideas. First, we specialize a linear space (that the curve is required to intersect) to lie in the hyperplane $H$, and analyze the limit curves. It turns out that the limit curves are of the same form, and are in some sense simpler. Enumerative results have been proved using such specialization ideas since the nineteenth century (see [PZ], for example, especially pp. 268--275 and pp. 313--319). The second general idea we use is Kontsevich's moduli spaces of stable maps, particularly the spaces ${\overline{M}}_{0,m}(\mathbb P^n,d)$ and ${\overline{M}}_{0,m}(\mathbb P^1,d)$ (and the stacks ${\overline{\cm}}_{0,m}(\mathbb P^n,d)$ and ${\overline{\cm}}_{0,m}(\mathbb P^1,d)$). The calculations in Section \ref{intro} on the space ${\overline{M}}_{0,m}(\mathbb P^1,d)$ will give the desired results in $\mathbb P^n$. L. Caporaso and J. Harris' results on plane curves of any degree and genus (cf. [CH3]) can also be reinterpreted in this light. The reader can verify that the argument here for $n=2$ is in essence the same as that in [CH3] for genus 0 curves. \subsubsection{Example: 2 lines through 4 general lines in $\mathbb P^3$} We can follow through these ideas in a classical special case. Fix four general lines $L_1$, $L_2$, $L_3$, $L_4$ in $\mathbb P^3$, and a hyperplane $H$. There are a finite number of lines in $\mathbb P^3$ intersecting $L_1$, $L_2$, $L_3$, $L_4$. Call one of them $l$. We will specialize the lines $L_1$, $L_2$, $L_3$, and $L_4$ to lie in $H$ one at a time and see what happens to $l$. First, specialize the line $L_1$ to (a general line in) $H$, and then do the same with $L_2$ (see Figure \ref{r2lines}). If $l$ doesn't pass through the intersection of $L_1$ and $L_2$, it must still intersect both $L_1$ and $L_2$, and thus lie in $H$. Then $l$ is uniquely determined: it is the line through $L_3 \cap H$ and $L_4 \cap H$. Otherwise, if $l$ passes through the point $L_1 \cap L_2$, it is once again uniquely determined (as only one line in $\mathbb P^3$ can pass through two general lines and one point --- this can also be seen through further degeneration). \begin{figure} \begin{center} \getfig{r2lines}{.15} \end{center} \caption{Possible positions of $l$ after $L_1$ and $L_2$ have degenerated to $H$} \label{r2lines} \end{figure} The above argument can be tightened to rigorously show that there are two lines in $\mathbb P^3$ intersecting four general lines. The only information one needs to know in advance is that there is one line through two distinct points. This is the same as the seed data for Kontsevich's recursive formula in [KM], and it is all we will need in this section. \subsubsection{Example: 92 conics through 8 general lines in $\mathbb P^3$} \label{r92subsection} The example of conics in $\mathbb P^3$ is a simple extension of that of lines in $\mathbb P^3$, and gives a hint as to why stable maps are the correct way to think about these degenerations. Consider the question: How many conics pass through 8 general lines $L_1$, \dots, $L_8$? (For another discussion of this classical problem, see [H1] p. 26.) We introduce a pictorial shorthand that will allow us to easily follow the degenerations (see Figure \ref{r92conics}); the plane $H$ is represented by a parallelogram. \begin{figure} \begin{center} \getfig{r92conics}{.1} \end{center} \caption{Counting 92 conics in $\mathbb P^3$ through 8 general lines} \label{r92conics} \end{figure} We start with the set of conics through 8 general lines (the top row of the diagram --- the label 92 indicates the number of such conics, which we will calculate last) and specialize one of the lines $L_1$ to $H$ to get row 7. (The line $L_1$ in $H$ is indicated by the dotted line in the figure.) When we specialize another line $L_2$, one of two things can happen: the conic can intersect $H$ at the point $L_1 \cap L_2$ and one other (general) point, or it can intersect $H$ once on $L_1$ and once on $L_2$ (at general points). (The requirement that the conics must pass through a fixed point in the first case is indicated by the thick dot in the figure.) In this second case (the picture on the right in row 6), if we specialize another line $L_3$, one of three things can happen. \begin{enumerate} \item The conic can stay smooth, and not lie in $H$, in which case it must intersect $H$ at $\{ L_1 \cap L_3, L_2 \}$ or $\{L_1, L_2 \cap L_3 \}$ (hence the ``$\times 2$'' in the figure). \item The conic could lie in $H$. In this case, there are eight conics through five fixed points $L_4 \cap H$, \dots, $L_8 \cap H$ with marked points on the lines $L_1$, $L_2$, and $L_3$. \item The conic can degenerate into the union of two intersecting lines, one ($l_0$) in $H$ and one ($l_1$) not. These lines must intersect $L_4$, \dots, $L_8$. (The line $l_0$ already intersects $L_1$, $L_2$, $L_3$, so we don't have to worry about these conditions.) Either three or four of $\{ L_4, \dots, L_8 \}$ intersect $l_1$. In the first case, there are $\binom 5 3$ choices of the three lines, and two configurations $(l_0,l_1)$ once the three lines are chosen. In the second case there are a total of $\binom 5 4 \times 2$ configurations by similar reasoning. Thus the total number of such configurations is 30. \end{enumerate} We fill out the rest of the diagram in the same way. Then, using the enumerative geometry of lines in $\mathbb P^3$ and conics in $\mathbb P^2$ we can work our way up the table, attaching numbers to each picture, finally deducing that there are 92 conics through 8 general lines in $\mathbb P^3$. To make this argument rigorous, precise dimension counts and multiplicity calculations are needed. The algorithm described in this section is slightly different: we will parametrize rational curves with various conditions {\em and marked intersections with $H$}. In the case of conics through 8 lines, for example, we would count 184 conics through 8 lines with 2 marked points on $H$, and then divide by 2. The argument will then be cleaner. The resulting pictorial table is almost identical to Figure \ref{r92conics}; the only difference is in the first two lines (see Figure \ref{r92conics2}). \begin{figure} \begin{center} \getfig{r92conics2}{.1} \end{center} \caption{Counting 184 conics with two marked points on $H$ through 8 general lines} \label{r92conics2} \end{figure} \subsubsection{Example: Cubics in $\mathbb P^3$} The situation in general is not much more complicated than our calculations for conics in $\mathbb P^3$. Two additional twists come up, which are illustrated in the case of the $80,160$ twisted cubics through 12 general lines in $\mathbb P^3$. This calculation is indicated pictorially in Figure \ref{rcubics} at the beginning of the introduction. The third figure in row 8 represents a nodal (rational) cubic in $H$. There are 12 nodal cubics through 8 general points in $\mathbb P^2$ (well-known, e.g. [DI] p. 85). The algorithm of this section will actually calculate $80,160 \times 3!$ cubics with marked points on $H$ through 12 general lines. On the left side of row 8 we see a new degeneration (from twisted cubics through nine general lines intersecting $H$ along three fixed general lines in $H$): a conic tangent to $H$, intersecting a line in $H$. (The tangency of the conic is indicated pictorially by drawing its lower horizontal tangent inside the parallelogram representing $H$.) We also have an unexpected multiplicity of 2 here. The appearance of these new degenerations indicate why, in order to enumerate rational curves through general linear spaces by these degeneration methods, we must expand the set of curves under consideration to include those required to intersect $H$ with given multiplicity, along linear subspaces. \subsubsection{The algorithm (informally)} The algorithm in general is (informally) as follows. (Theorem \ref{rrecursiveX2} describes the algorithm rigorously.) Fix positive integers $n$ and $d$, and fix a hyperplane $H$ in $\mathbb P^n$. Let $\vec{h} = ( h_{m,e} )_{m \geq 1, e \geq 0}$ and $\vec{i} = ( i_e )_{e \geq 0}$ be sets of non-negative integers. Fix a set of general linear spaces $\Gamma = \{ \Gamma^j_{m,e} \}_{m,e,1 \leq j \leq h_{m,e}}$ in $H$ where $\dim \Gamma^j_{m,e} = e$. Fix a set of general linear spaces $\Delta = \{ \Delta^j_e \}_{e,1 \leq j \leq i_e}$ in $\mathbb P^n$ where $\dim \Delta^j_{e} = e$. Let $X_n(d, \Gamma, \Delta)$ be the closure in ${\overline{M}}_{0, \sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ of points representing maps $$ (\mathbb P^1, \{ p^j_{m,e} \}_{m,e,1 \leq j \leq h_{m,e}}, \{ q^j_e \}_{e, 1 \leq j \leq i_e }, \pi) $$ where \begin{itemize} \item $\pi(p^j_{m,e}) \in \Gamma^j_{m,e}$, $\pi(q^j_e) \in \Delta^j_e$. \item $\pi^{-1}H$ is a set of points, and as divisors $\pi^* H = \sum_{m,e,j} m p^j_{m,e}$. \end{itemize} Assume $X(d,\Gamma,\Delta)$ is a finite set. We will count the points of $X(d,\Gamma, \Delta)$. Specialize one of the linear spaces $\Delta^{j_1}_{e_1}$ to lie in $H$, and consider the limits of the $\# X(d,\Gamma, \Delta)$ stable maps. One of the two following types of limits will appear. \begin{enumerate} \item The limit map is of the form $$ (C(0) \cup C(1),\{ p^j_{m,e} \}_{m,e,j}, \{ q^j_e \}_{e, j}, \pi) $$ where the curves $C(0)$ and $C(1)$ are both smooth and rational, $\pi(C(0))$ is a point, for some $(m_0, e_0, j_0)$ the curve $C(0)$ contains the marked points $q^{j_1}_{e_1}$ and $p^{j_0}_{m_0, e_0}$ (and $C(1)$ contains the other marked points), and $$ (\pi \mid_{C(1)})^* H = \sum_{ \substack{ {m,e,j} \\ {(m,e,j) \neq (m_0, e_0, j_0)}}} m p^j_{m,e} + m_0 (C(1) \cap C(0)). $$ Also, $\pi ( p^j_{m,e}) \in \Gamma^j_{m,e}$, $\pi(q^j_e) \in \Delta^j_e$, and (as a consequence) $\pi(C(0))$ is contained in $\Delta^{j_1}_{e_1} \cap \Gamma^{j_0}_{m_0,e_0}$. We can ignore the rational tail, replacing it with another marked point, and continue the process. There are $m_0$ curves of $X(d,\Gamma,\Delta)$ tending to this limit. \item The limit map is of the form $$ (C = C(0) \cup C(1) \cup \dots \cup C(l),\{ p^j_{m,e} \}_{m,e,j}, \{ q^j_e \}_{e, j }, \pi) $$ where $C(k)$ ($0 \leq k \leq l$) is smooth and rational. The points $ \{ p^j_{m,e} \}$, $\{ q^j_e \}$ are partitioned into sets $\{ p^j_{m,e}(k) \}$, $\{ q^j_e(k) \}$, where the $k^{\text{th}}$ subset lies in $C(k)$; this induces partitions $\vec{h} = \sum_{k=0}^l \vec{h}(k)$ and $\vec{i} = \sum_{k=0}^l \vec{i}(k)$. The marked point $q^{j_1}_{e_1}$ lies on $C(0)$; that is, $q^{j_1}_{e_1} \in \{ q^j_e(0) \}$. The component $C(0)$ intersects all other components $C(1)$, \dots, $C(l)$. The map $\pi$ sends $C(0)$ to $H$ with positive degree, and sends no other component of $C$ to $H$. If $m^k = \deg (\pi \mid_{C(k)}) - \sum_{m,e} m h_{m,e}(k)$, then $$ ( \pi \mid_{C(k)})^* H = \sum_{m,e,j} m p^j_{m,e}(k) + m^k (C(0) \cap C(k)) $$ as divisors for $1 \leq k \leq l$. Finally, $\pi(p^j_{m,e}) \in \Gamma^j_{m,e}$ for all $m$, $e$, $j$, and $\pi(q^j_e) \in \Delta^j_e$ for all $e$, $j$. There are $\prod_{k=1}^l m^k$ curves in $X(d,\Gamma, \Delta)$ tending to this limit. \end{enumerate} Examples of both types of limits can be seen in Figure \ref{rcubics}. Given the results of Subsection \ref{ikey}, the algorithm and multiplicities are not completely unexpected. \subsection{Notation and summary} For convenience, let $\vec{\epsilon}_e$, $\vec{\epsilon}_{m,e}$ be the natural basis vectors: $(\vec{\epsilon}_e)_{e'} = 1$ if $e = e'$ and 0 otherwise; and $(\vec{\epsilon}_{m,e})_{m',e'} = 1$ if $(m,e)=(m',e')$, and 0 otherwise. Fix a hyperplane $H$ in $\mathbb P^n$, and a hyperplane $A$ of $H$. \subsubsection{The schemes $X({\mathcal{E}})$} \label{r21} We now define the primary objects of interest to us. Let $n$ and $d$ be positive integers, and let $H$ be a hyperplane in $\mathbb P^n$. Let $\vec{h} = ( h_{m,e} )_{m \geq 1, e \geq 0}$ and $\vec{i} = ( i_e )_{e \geq 0}$ be sets of non-negative integers. Let $\Gamma = \{ \Gamma^j_{m,e} \}_{m,e,1 \leq j \leq h_{m,e}}$ be a set of linear spaces in $H$ where $\dim \Gamma^j_{m,e} = e$. Let $\Delta = \{ \Delta^j_e \}_{e,1 \leq j \leq i_e}$ be a set of linear spaces in $\mathbb P^n$ where $\dim \Delta^j_{e} = e$. \begin{defn} \label{defnX} The scheme $X_n(d,\Gamma,\Delta)$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying $\pi(p_{m,e}^j) \in \Gamma_{m,e}^j$, $\pi(q_e^j) \in \Delta_e^j$, and $\pi^* H = \sum_{m,e,j} m p^j_{m,e}$. \end{defn} In particular, $\sum_{m,e} m h_{m,e} = d$, and no component of $C$ is contained in $\pi^{-1}H$. The incidence conditions define closed subschemes of ${\overline{M}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$, so the union of these conditions indeed defines a closed subscheme of ${\overline{M}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$. Define ${\mathcal{X}}_n(d,\Gamma,\Delta)$ in the same way as a substack of $$ {\overline{\cm}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d). $$ When we speak of properties that are constant for general $\Gamma$ and $\Delta$ (such as the dimension of $X(d,\Gamma,\Delta)$), we will write $X_n(d,\vec{h}, \vec{i})$. For convenience, write ${\mathcal{E}}$ (for ${\mathcal{E}}$verything) for the data $d, \vec{h}, \vec{i}$, so $X_n({\mathcal{E}}) = X_n(d,\vec{h},\vec{i})$. Also, the $n$ will often be suppressed for convenience. The variety $X(d, \Gamma, \Delta)$ can be loosely thought of as parametrizing degree $d$ rational curves in projective space intersecting certain linear subspaces of $\mathbb P^n$, and intersecting $H$ with different multiplicities along certain linear subspaces of $H$. For example, if $n=3$, $d=3$, $h_{2,0}=1$, $h_{1,2}=1$, $X$ parametrizes twisted cubics in $\mathbb P^3$ tangent to $H$ at a fixed point. In the special case where $h_{m,e}=0$ when $e<n-1$ and $\vec{i}=\vec{\epsilon}_n$, define $\hat{\ce}$ by $\hat{d} = d$, $\hat{i}_1 = 1$, $\hat{h}_{m,0} = h_{m,n-1}$. We will relate the geometry of ${\mathcal{X}}_n({\mathcal{E}})$ to that of ${\mathcal{X}}_1(\hat{\ce})$, which was studied in Subsection \ref{ikey}. (The general point of ${\mathcal{X}}_n({\mathcal{E}})$ corresponds to a general degree $d$ map from $\mathbb P^1$ to $\mathbb P^n$ with $\pi^*H$ consisting of points with multiplicity given by the partition $(h_{1,n-1}, h_{2,n-1}, \dots)$. The general point of ${\mathcal{X}}_1(\hat{\ce})$ corresponds to a general degree $d$ map from $\mathbb P^1$ to $\mathbb P^1$ with $\pi^*z$ consisting of points with multiplicity given by the partition $(h_{1,n-1}, h_{2,n-1}, \dots)$.) The geometry of ${\mathcal{X}}_n({\mathcal{E}})$ for general ${\mathcal{E}}$ can be understood from this special case. For example, consider ${\mathcal{X}} = {\mathcal{X}}_3(d = 2, h_{2,0} = 1, i_1 = 2)$, the stack parametrizing conics in $\mathbb P^3$ through two general lines, tangent to $H$ at a fixed point of $H$. To analyze ${\mathcal{X}}$, we study the stack ${\mathcal{X}}_1 \subset {\overline{\cm}}_{0,2}(\mathbb P^3,2)$ parametrizing conics tangent to $H$ (where the tangency is labeled $p^1_{2,0}$) with a marked point $q^1_1$ (with no other incidence conditions). We take the universal curve over this stack ${\mathcal{X}}_2$ (which can be seen as a substack of ${\overline{\cm}}_{0,3}(\mathbb P^3,2)$), and label the point of the universal curve $q^2_1$. Then we require $\pi(p^1_{2,0})$ to lie on two general hyperplanes $H_1$ and $H_2$ (thus requiring $\pi(p^1_{2,0})$ to be a fixed general point $H \cap H_1 \cap H_2$ of $H$), $\pi(q^1_1)$ to lie on two general hyperplanes $H_3$ and $H_4$ (thus requiring $\pi(q^1_1)$ to lie on a fixed general line $H_3 \cap H_4$ of $\mathbb P^3$), and $\pi(q^2_1)$ to lie on two general hyperplanes $H_5$ and $H_6$ (thus requiring $\pi(q^2_1)$ to lie on a fixed general line $H_5 \cap H_6$ of $\mathbb P^3$). We shall prove (in the next section) that the resulting stack is indeed ${\mathcal{X}}$. By these means we show that if the linear spaces $\Gamma$, $\Delta$ are general, these varieties have the dimension one would naively expect. The family of degree $d$ rational curves in $\mathbb P^n$ has dimension $(n+1)d+(n-3)$. Requiring the curve to pass through a fixed $e$-plane should be a codimension $(n-1-e)$ condition. Requiring the curve to be $m$-fold tangent to $H$ along a fixed $e$-plane of $H$ should be a codimension $(m-1)+(n-1-e)$ condition. Thus we will show (Theorem \ref{rdimX}) that when the linear spaces in $\Gamma$, $\Delta$ are general, each component of $X({\mathcal{E}}, \Gamma, \Delta)$ has dimension $$ (n+1) d + (n-3) - \sum_{m,e} (n+m-e-2)h_{m,e} - \sum_e (n-1-e)i_e. $$ Moreover, $X({\mathcal{E}})$ is reduced. When the dimension is 0, $X({\mathcal{E}})$ consists of a finite number of reduced points. We call their number $\# X({\mathcal{E}})$ --- these are the numbers we want to calculate. Define $\# X({\mathcal{E}})$ to be zero if $\dim X({\mathcal{E}}) > 0$. For example, when $n=3$, $d=3$, $h_{1,2} = 3$, $i_1 = 12$, $\# X({\mathcal{E}})$ is 3! times the number of twisted cubics through 12 general lines. (The ``3!'' arises from the markings of the three points of intersection of the cubic with $H$.) \subsubsection{The schemes $Y ( {\mathcal{E}}(0); \dots; {\mathcal{E}}(l) )$} We will be naturally led to consider subvarieties of $X({\mathcal{E}},\Gamma,\Delta)$ which are similar in form. Fix $n$, $d$, $\vec{h}$, $\vec{i}$, $\Gamma$, $\Delta$, and a non-negative integer $l$. Let $\sum_{k=0}^l d(k)$ be a partition of $d$. Let the points $\{ p^j_{m,e} \}_{m,e,j}$ be partitioned into $l+1$ subsets $\{ p^j_{m,e}(k) \}_{m,e,j}$ for $k= 0$, \dots, $l$. This induces a partition of $\vec{h}$ into $\sum_{k=0}^l \vec{h}(k)$ and a partition of the set $\Gamma$ into $\coprod_{k=0}^l \Gamma(k)$. Let the points $\{ q^j_e \}_{e,j}$ be partitioned into $l+1$ subsets $\{ q^j_{e}(k) \}_{e,j}$ for $k= 0$, \dots, $l$. This induces a partition of $\vec{i}$ into $\sum_{k=0}^l \vec{i}(k)$ and a partition of the set $\Delta$ into $\coprod_{k=0}^l \Delta(k)$. Define $m^k$ by $m^k = d(k) - \sum_m m h_m(k)$, and assume $m^k>0$ for all $k = 1$, \dots, $l$. \begin{defn} \label{rdefY} The scheme $$ Y_n(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying the following conditions \begin{enumerate} \item[Y1.] The curve $C$ consists of $l+1$ irreducible components $C(0)$, \dots, $C(l)$ with all components intersecting $C(0)$. The map $\pi$ has degree $d(k)$ on curve $C(k)$ ($0 \leq k \leq l$). \item[Y2.] The points $\{ p^j_{m,e}(k) \}_{m,e,j}$ and $\{ q^j_e(k)\}_{e,j}$ lie on $C(k)$, and $\pi(p^j_{m,e}(k)) \in \Gamma^j_{m,e}(k)$, $\pi(q^j_e(k)) \in \Delta^j_e(k)$. \item[Y3.] As sets, $\pi^{-1}H = C(0) \cup \{ p^j_{m,e} \}_{m,e,j}$, and for $k>0$, $$ ( \pi \mid_{C(k)} )^* H = \sum_{m,e,j} m p^j_{m,e}(k) + m^k (C(0) \cap C(k)). $$ \end{enumerate} \end{defn} A pictorial representation of such a map is given in Figure \ref{rtype2eg}. Note that $d(k)>0$ for all positive $k$ by the last condition. \begin{figure} \begin{center} \getfig{rtype2eg}{.1} \end{center} \caption{An example of a map corresponding to a general point of some $Y({\mathcal{E}}(0); {\mathcal{E}}(1); {\mathcal{E}}(2))$} \label{rtype2eg} \end{figure} When discussing properties that hold for general $\{ \Gamma^j_{m,e} \}_{m,e,j}$, $\{ \Delta^j_e \}_{e,j}$, we will write $$ Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) = Y(d(0),\vec{h}(0),\vec{i}(0); \dots; d(l), \vec{h}(l),\vec{i}(l)). $$ If $\vec{h}(k) + \vec{i}(k) \neq \vec{0}$ for all $k>0$, $Y(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l))$ is isomorphic to a closed subscheme of $$ {\overline{M}}_{0,\sum h_{m,e}(0) + \sum i_e(0)+l}(H,d(0)) \times \prod_{k=1}^l X(d(k),Ga'(k),\Delta(k)). $$ where $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k,n-1}$ and $\Gamma'(k)$ is the same as $\Gamma(k)$ except $\Gamma^{h_{m^k,n-1}}_{m^k,n-1}=H$ Define ${\mathcal{Y}}(d(0), \Delta(0), \Gamma(0); \dots; d(l), \Delta(l), \Gamma(l))$ as the analogous stack. \subsubsection{Enumeratively meaningful subvarieties of ${\overline{M}}_{g,m}(\mathbb P^n,d)$} \label{rgeomean} For any irreducible substack ${\mathcal{V}}$ of ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$, there is an open subset ${\mathcal{U}}$ such that for the stable maps $(C, \{ p_i \}, \pi)$ corresponding to points of ${\mathcal{U}}$, the reduced image curve $\pi(C)^{\text{red}}$ has constant Hilbert polynomial $f(t)$. This gives a morphism $\xi$ from ${\mathcal{U}}$ to the Hilbert scheme $H_{f(t)}$. \begin{defn} The substack ${\mathcal{V}}$ is {\em enumeratively meaningful} if the dimension of $\xi({\mathcal{U}})$ is the same as that of ${\mathcal{U}}$ (i.e. $\xi$ is generically finite onto its image). \end{defn} Define {\em enumeratively meaningful subvarieties} of ${\overline{M}}_{g,m}(\mathbb P^n,d)$ in the same way. When making enumerative calculations, we are counting reduced points of ${\overline{M}}_{g,m}(\mathbb P^n,d)$, which are obviously enumeratively meaningful. This definition will be important in Section 3. \subsection{Preliminary results} The following proposition is an analog of Bertini's theorem. \begin{pr} \label{rgeneral} Let ${\mathcal{A}}$ be a reduced irreducible substack of ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$, and let $p$ be one of the labeled points. Then there is a Zariski-open subset $U$ of the dual projective space $(\mathbb P^n)^*$ such that for all $[H'] \in U$ the intersection ${\mathcal{A}} \cap \{ \pi(p) \in H' \}$, if nonempty, is reduced of dimension $\dim {\mathcal{A}} - 1$. \end{pr} Loosely, this result states that the requirement that a marked point lie on a general hyperplane imposes one transverse condition on an irreducible substack of ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$. To prove this proposition, we will invoke Theorem 2 of [Kl]. \begin{tm}[Kleiman] Let $G$ be an integral algebraic group scheme, $X$ an integral algebraic scheme with a transitive $G$-action. Let $f: Y \rightarrow X$ and $g: Z \rightarrow X$ be two maps of integral algebraic schemes. For each rational point $s$ of $G$, let $sY$ denote $Y$ considered as an $X$-scheme via the map $y \mapsto sf(y)$. \begin{enumerate} \item[(i)] Then, there exists a dense open subset $U$ of $G$ such that, for each rational point $s$ in $U$, either the fibered product, $(sY)\times_X Z$ is empty or it is equidimensional and its dimension is given by the formula, $$ \dim(( sY) \times_X Z) = \dim(Y) + \dim(Z) - \dim(X). $$ \item[(ii)] Assume the characteristic is zero, and $Y$ and $Z$ are regular. Then, there exists a dense open subset $U$ of $G$ such that, for each rational point $s$ in $U$, the fibered product $(sY) \times_X Z$, is regular. \end{enumerate} \end{tm} The proof of Kleiman's theorem carries through without change if $Z$ is an algebraic stack. \noindent {\em Proof of Proposition \ref{rgeneral}.} Let $G = PGL(n)$, $X= \mathbb P^3$. Let $Y$ be a hyperplane of $X$ with $f: Y \rightarrow X$ the immersion. Let $Z$ be the smooth points of ${\mathcal{A}}$, with $g: Z \rightarrow X$ given by evaluation at $p$. Then the result follows immediately from Kleiman's theorem. \qed \vspace{+10pt} The next proposition is a variation of Proposition \ref{rgeneral}. \begin{pr} \label{rgeneralproper} With the hypotheses of Proposition \ref{rgeneral}, let ${\mathcal{B}}$ be a proper closed substack of ${\mathcal{A}}$. Then there is a Zariski-open subset $U$ of the dual projective space $(\mathbb P^n)^*$ such that for all $[H'] \in U$, each component of ${\mathcal{B}} \cap \{ \pi(p) \in H' \}$ is a proper closed substack of a component of ${\mathcal{A}} \cap \{ \pi(p) \in H' \}$. \end{pr} \noindent {\em Proof. } The components of ${\mathcal{A}} \cap \{ \pi(p) \in H' \}$ are each of dimension $\dim {\mathcal{A}} - 1$ (by Proposition \ref{rgeneral}), and the components of ${\mathcal{B}} \cap \{ \pi(p) \in H' \}$ are each of dimension less than $\dim {\mathcal{A}} - 1$ (by Proposition \ref{rgeneral} applied to the irreducible components of ${\mathcal{B}}$). \qed \vspace{+10pt} We now summarize the results of the remainder of this section. There are many relations among the various spaces $X({\mathcal{E}})$ as ${\mathcal{E}}$ varies. Some are universal curves over others (Proposition \ref{runiversal}). Some are the intersections of others with a divisor (Proposition \ref{rgeneral2}). The variety $X({\mathcal{E}})$ can be identified with $Y({\mathcal{E}}(0);{\mathcal{E}}(1))$ for appropriately chosen ${\mathcal{E}}(0)$ and ${\mathcal{E}}(1)$ (Proposition \ref{rXY}). A smoothness result (Proposition \ref{rbig}) allows us to use results about stable maps to $\mathbb P^1$ proved in the previous section. A first application of Propositions \ref{runiversal}, \ref{rgeneral2}, and \ref{rbig} is a calculation of the dimension of $X({\mathcal{E}})$ and $Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ (Proposition \ref{rdimX}); more will follow in subsequent sections. Finally, Proposition \ref{rXnice} ensures that the image of the stable map corresponding to a general point of $X({\mathcal{E}})$ is smooth. Let $A$ be a general $(n-2)$-plane in $H$. The projection $p_A$ from $A$ induces a rational map $\rho_A: {\overline{\cm}}_{0,m}(\mathbb P^n,d) \dashrightarrow {\overline{\cm}}_{0,m}(\mathbb P^1,d)$, that is a morphism (of stacks) at points representing maps $(C, \{ p_i \}, \pi)$ whose image $\pi(C)$ does not intersect $A$. Via ${\overline{\cm}}_{0,m}(\operatorname{Bl}_A \mathbb P^n,d)$, the morphism can be extended over the set of maps $(C, \{ p_i \}, \pi)$ where $\pi^{-1} A$ is a union of reduced points distinct from the $m$ marked points $\{ p_i \}$. The image of such curves in ${\overline{\cm}}_{0,m}(\mathbb P^1,d)$ is a stable map $$ (C \cup C_1 \cup \dots \cup C_{\# \pi^{-1} A }, \{ p_i \}, \pi') $$ where $C_1$, \dots, $C_{\# \pi^{-1} A }$ are rational tails attached to $C$ at the points of $\pi^{-1} A$, $$ \pi' \mid_{ \{ C \setminus \pi^{-1} A \} } = ( p_A \circ \pi ) \mid_{ \{ C \setminus \pi^{-1} A \} } $$ (which extends to a morphism from all of $C$) and $\pi' \mid_{C_k}$ is a degree 1 map to $\mathbb P^1$ ($1 \leq k \leq \# \pi^{-1} A $). \begin{pr} If $(C, \{ p_i \}, \pi) \in {\overline{\cm}}_{0,m}(\mathbb P^n,d)$ and $\pi^{-1} A$ is a union of reduced points disjoint from the marked points, then at the point $(C,\{ p_i \}, \pi)$, $\rho_A$ is a smooth morphism of stacks of relative dimension $(n-1)(d+1)$. \label{rbig} \end{pr} \noindent {\em Proof. } To show that a morphism of stacks ${\mathcal{A}} \rightarrow {\mathcal{B}}$ is smooth at a point $a \in {\mathcal{A}}$, where ${\mathcal{B}}$ is smooth and ${\mathcal{A}}$ is equidimensional, it suffices to show that the fiber is smooth at $a$, or equivalently that the Zariski tangent space to the fiber at $a$ is of dimension $\dim {\mathcal{A}} - \dim {\mathcal{B}}$. Recall that ${\overline{\cm}}_{0,m}(\mathbb P^n,d)$ is a smooth stack of dimension $(n+1)d+m-1$ (see Subsubsection \ref{itmsosm}). The first order deformations of $(C, \{ p_i \}, \pi)$ in the fiber of $\rho_A$ can be identified with sections of the vector bundle $\pi^*({\mathcal{O}}(1)^{n-1})$: these are deformations of a map $$ (C, \{ p_i \},\pi) {\stackrel {(s_0, s_1, \dots, s_n)} \longrightarrow} \mathbb P^n $$ keeping the marked curve $(C, \{ p_i \}, \pi)$ and the sections $(s_0,s_1)$ constant. But \begin{eqnarray*} h^0(C, \pi^* {\mathcal{O}}_{\mathbb P^n}(1)^{n-1}) &=& (n-1) h^0(C, \pi^* {\mathcal{O}}_{\mathbb P^n}(1)) \\ &=& (n-1) (d+1) \\ &=& \dim {\overline{\cm}}_{0,m}(\mathbb P^n,d) - \dim {\overline{\cm}}_{0,m} (\mathbb P^1,d) \end{eqnarray*} as desired. \qed \vspace{+10pt} The following two propositions give relationships among the spaces $X({\mathcal{E}})$ as ${\mathcal{E}}$ varies. \begin{pr} \label{runiversal} Given $d$, $\vec{h}$, $\vec{i}$, $\Gamma$, $\Delta$, let $\vec{i'} = \vec{i} + \vec{\epsilon}_n$, and define $\Delta'$ to be the same as $\Delta$ except $\Delta^{i'_n}_n = \mathbb P^n$. Then ${\mathcal{X}}(d, \Gamma, \Delta')$ is the universal curve over ${\mathcal{X}}(d, \Gamma, \Delta)$. \end{pr} \noindent {\em Proof. } The moduli stack ${\overline{\cm}}_{0, \sum h_{m,e} + \sum i_e + 1} (\mathbb P^n,d)$ is the universal curve over ${\overline{\cm}}_{0, \sum h_{m,e} + \sum i_e } (\mathbb P^n,d)$ (see Subsubsection \ref{itmsosm}). The proposition is a consequence of the commutativity of the following diagram: $$\begin{CD} {\mathcal{X}}(d, \Gamma, \Delta') @>>> {\overline{\cm}}_{0, \sum h_{m,e} + \sum i_e + 1}(\mathbb P^n,d) \\ @V{p}VV @VV{p}V \\ {\mathcal{X}}(d, \Gamma, \Delta) @>>> {\overline{\cm}}_{0,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d) \end{CD}$$ \qed \vspace{+10pt} \begin{pr} \label{rgeneral2} Let $H'$ be a general hyperplane of $\mathbb P^n$. \begin{enumerate} \item[a)] The divisor $$ \{ \pi(p^{j_0}_{m_0,e_0}) \in H' \} \subset {\mathcal{X}}(d, \Gamma, \Delta) $$ is ${\mathcal{X}}(d, \Gamma', \Delta)$ where \begin{itemize} \item $\vec{h'}=\vec{h} - \vec{\epsilon}_{m_0,e_0} +\vec{\epsilon}_{m_0,e_0-1}$ \item For $(m,e) \neq (m_0,e_0), (m_0,e_0-1)$, ${\Gamma'}^j_{m,e} = \Gamma^j_{m,e}$. \item $\{ {\Gamma'}^j_{m_0,e_0} \}_j = \{ \Gamma^j_{m_0,e_0} \}_j \setminus \{ \Gamma^{j_0}_{m_0,e_0} \}$, $\{ {\Gamma'}^j_{m_0,e_0-1} \} = \{ \Gamma^j_{m_0,e_0-1} \}_j \cup \{ \Gamma^{j_0}_{m_0,e_0} \cap H' \} $ \end{itemize} \item[b)] The divisor $$ \{ \pi(q^{j_0}_{e_0}) \in H' \} \subset {\mathcal{X}}(d, \Gamma, \Delta) $$ is ${\mathcal{X}}(d, \Gamma, \Delta')$ where \begin{itemize} \item $\vec{i'} = \vec{i} - \vec{\epsilon}_{e_0} + \vec{\epsilon}_{e_0-1}$ \item For $e \neq e_0, e_0-1$, ${\Delta'}^j_{e} = \Delta^j_{e}$. \item $\{ {\Delta'}^j_{e_0} \}_j = \{ \Delta^j_{e_0} \}_j \setminus \{ \Delta^{j_0}_{e_0} \}$, $\{ {\Delta'}^j_{e_0-1} \}_j = \{ \Delta^j_{e_0-1} \}_j \cup \{ \Delta^{j_0}_{e_0} \cap H' \} $ \end{itemize} \end{enumerate} \end{pr} \noindent {\em Proof. } We prove a) first. Every point of $\{ \pi(p^{j_0}_{m_0,e_0}) \in H' \}$ represents a map where $\pi(p^j_{m,e}) \subset \Gamma^j_{m,e}$, $\pi(q^j_e) \subset \Delta^j_e$, $\pi(p^{j_0}_{m_0,e_0}) \in H'$. Clearly $$ {\mathcal{X}}(d, \Gamma', \Delta) \subset \{ \pi(p^{j_0}_{m_0,e_0}) \in H' \}; $$ each component of ${\mathcal{X}}(d, \Gamma', \Delta)$ appears with multiplicity one by Proposition \ref{rgeneral}. The only other possible components of $\{ \pi(p^{j_0}_{m_0,e_0}) \in H' \}$ are those whose general point represents a map where $\pi^{-1} H$ is not a union of points (i.e. contains a component of $C$). But such maps form a union of proper subvarieties of components of ${\mathcal{X}}(d, \Gamma, \Delta)$, and by Proposition \ref{rgeneralproper} such maps cannot form a component of $$ \{ \pi(p^{j_0}_{m_0,e_0}) \in H' \} \cap {\mathcal{X}}(d, \Gamma, \Delta). $$ Replacing $p^{j_0}_{m_0,e_0}$ with $q^{j_0}_{e_0}$ in the previous paragraph gives a proof of b). \qed \vspace{+10pt} The next observation is analogous to a well-known fact about the moduli space of stable marked curves. Consider the stack ${\overline{\cm}}_{g,m}$ where the $m$ marked points are labeled $a_1$, $a_2$, $b_1$, \dots, $b_{m-2}$. Then the closed substack ${\mathcal{V}}$ of ${\overline{\cm}}_{g,m}$ parametrizing marked curves $A \cup B$ with $a_i \in A$, $b_i \in B$, $p_a(A) = 0$, $p_a(B) = g$ is isomorphic to ${\overline{\cm}}_{g,m-1}$ where the $m-1$ marked points are labeled $c$, $b_1$, \dots, $b_{m-2}$. The isomorphism ${\overline{\cm}}_{g,m-1} \rightarrow {\mathcal{V}}$ involves gluing a rational tail (with marked points $a_1$, $a_2$) at $c$. Fix ${\mathcal{E}}$, integers $m_0$, $e_0$, $e_1$, and general $\Gamma$, $\Delta$. Let $j_0 = h_{m_0,e_0}$, $j_1 = i_{e_1}$, $e' = e_0 + e_1 - n$, and $j' = h_{m_0,e'} + 1$. There is a subvariety $Y$ of $X(d,\Gamma,\Delta)$ where $\pi( p^{j_0}_{m_0,e_0}) = \pi(q^{j_1}_{e_1})$. The general point of $Y$ represents a map $\pi: C(0) \cup C(1) \rightarrow \mathbb P^n$ where $C(0)$ and $C(1)$ are both isomorphic to $\mathbb P^1$, $\pi$ collapses $C(0)$ to a point, $p^{j_0}_{m_0,e_0}$ and $q^{j_1}_{e_1}$ are on $C(0)$, and the rest of the marked points are on $C(1)$. Necessarily $\pi(C(0)) \subset \Gamma^{j_0}_{m_0,e_0} \cap \Delta^{j_1}_{e_1}$. Such maps form a dense open subset of $Y(d(0),\Gamma(0),\Delta(0); d(1),\Gamma(1),\Delta(1))$ where \begin{itemize} \item $(d(0),\vec{h}(0),\vec{i}(0)) = (0,\vec{\epsilon}_{m_0,e_0},\vec{\epsilon}_{e_1})$, $\Gamma(0) = \{ \Gamma^{j_0}_{m_0,e_0} \}$, $\Delta(0) = \{ \Delta^{j_1}_{e_1} \}$. \item ${\mathcal{E}}(1) = {\mathcal{E}} - {\mathcal{E}}(0)$, $\Gamma(1) = \Gamma \setminus \Gamma(0)$, $\Delta(1) = \Delta \setminus \Delta(0)$. \end{itemize} Now let $d' = d$, $\vec{h'} = \vec{h}(1) + \vec{\epsilon}_{m_1,e'}$, $\vec{i'} = \vec{i}(1)$, $\Gamma' = \Gamma(1) \cup \{ \Gamma^{j_0}_{m_0,e_0} \cap \Delta^{j_1}_{e_1} \}$, and $\Delta' = \Delta(1)$. The stable map corresponding to a general point of $Y$ can also be identified with a stable map $(C(1), \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ in $X(d',\Gamma',\Delta')$ where $C(1)$ is smooth and not contained in $\pi^{-1}H$, by attaching a rational tail $C(0)$ (with two marked points $p^{j_0}_{m_0,e_0}$ and $q^{j_1}_{e_1}$) at the point $p^{j'}_{m_0,e'}$ of $C(1)$. In this way we get an isomorphism of $X(d',\Gamma',\Delta')$ with $$ Y(d(0),\Gamma(0),\Delta(0);d(1),\Gamma(1),\Delta(1)): $$ \begin{pr} There is a natural isomorphism $$ \phi: X(d', \Gamma', \Delta') \rightarrow Y(d(0), \Gamma(0), \Delta(0); d(1),\Gamma(1),\Delta(1)). $$ \label{rXY} \end{pr} \noindent {\em Proof. } The points in a dense open set of $X(d', \Gamma', \Delta')$ represent degree $d$ stable maps $\pi$ from a smooth curve $C$ to $\mathbb P^n$ with incidences $\pi({p'}^j_{m,e}) \in \Gamma^j_e$, $\pi( {q'}^j_e) \in \Delta^j_e$ and an equality of divisors $\pi^* H = \sum m {p'}^j_{m,e}$ on $C$. To each such map, consider the map to $\mathbb P^n$ where the marked point ${p'}^j_{m,e}$ is replaced by $p^j_{m,e}$ for $(m,e,j) \neq (m_0, e', h'_{m_0,e'})$, ${q'}^j_e$ is replaced by $q^j_e$, and $p^{h'_{m_0,e'}}_{m_0,e'}$ is replaced by a rational tail with additional marked points $p^{j_0}_{m_0,e_0}$ and $q^{j_1}_{e_1}$. The resulting stable maps corresponds to points in a dense open set of $Y(d(0), \Gamma(0), \Delta(0); d(1),\Gamma(1),\Delta(1))$. \qed \vspace{+10pt} \begin{pr} \label{rdimX} Every component of $X({\mathcal{E}})$ is reduced of dimension $$ (n+1) d + (n-3) - \sum_{m,e} (n+m-e-2)h_{m,e} - \sum_e (n-1-e)i_e. $$ The general element of each component is (a map from) an irreducible curve. If $\sum_{k=0}^l {\mathcal{E}}(k) = {\mathcal{E}}$, then every component of $Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) $ is reduced of dimension $\dim X({\mathcal{E}}) - 1$. \end{pr} \noindent {\em Proof. } We will prove the result about $\dim X({\mathcal{E}})$ in the special case $\vec{i} = \vec{0}$ and $h_{m,e}=0$ when $e<n-1$. Then the result holds when $\vec{i} = i_n \vec{\epsilon}_n$ by Proposition \ref{runiversal} (applied $i_n$ times), and we can invoke Proposition \ref{rgeneral2} repeatedly to obtain the result in full generality. (This type of reduction will be used often.) In this special case, we must prove that each component of $X({\mathcal{E}})$ is reduced of dimension $$ (n+1) d + (n-3) - \sum_m (m-1) h_{m,n-1}. $$ The natural map ${\mathcal{X}}({\mathcal{E}}) \dashrightarrow {\mathcal{X}}_1(\hat{\ce})$ induced by $$ \rho_A: {\overline{\cm}}_{0,\sum h_{m,n-1}}(\mathbb P^n,d) \dashrightarrow {\overline{\cm}}_{0,\sum h_{m,n-1}}(\mathbb P^1,d) $$ is smooth of relative dimension $(n-1)(d+1)$ at a general point of any component of ${\mathcal{X}}({\mathcal{E}})$ by Proposition \ref{rbig}. The stack ${\mathcal{X}}_1(\hat{\ce})$ is reduced of dimension $2d-1 - \sum(m-1) h_{m,n-1}$ by Subsection \ref{ikey}, so ${\mathcal{X}}({\mathcal{E}})$ is reduced of dimension $$ (n-1)(d+1) + \dim {\mathcal{X}}_1(\hat{\ce}) = (n+1)d + (n-3) - \sum_m (m-1) h_{m,n-1} $$ as desired. As the general element of ${\mathcal{X}}_1(\hat{\ce})$ is (a map from) an irreducible curve, the same is true of ${\mathcal{X}}({\mathcal{E}})$, and thus $X({\mathcal{E}})$. The same argument works for $Y$, as in Subsection \ref{ikey} it was shown that $\dim Y_1(\hat{\ce}) = \dim X_1 (\hat{\ce}) - 1$. \qed \vspace{+10pt} The following proposition is completely irrelevant to the rest of the argument. It is included to ensure that we are actually counting what we want. \begin{pr} \label{rXnice} If $n \geq 3$, and $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ is the stable map corresponding to a general point of a component of $X({\mathcal{E}})$, then $C \cong \mathbb P^1$ (with distinct marked points), and $\pi$ is a closed immersion. \end{pr} \noindent {\em Proof. } By Propositions \ref{runiversal} and \ref{rgeneral2} again, we can assume $\vec{i} = \vec{0}$ and $h_{m,e}=0$ when $e<n-1$. By the previous proposition, the curve $C$ is irreducible. We need only check that $\pi$ is a closed immersion. The line bundle ${\mathcal{O}}_C(d)$ is very ample, so a given non-zero section $s_0$ and three general sections $t_1$, $t_2$, $t_3$ will separate points and tangent vectors. If $\pi = (s_0, s_1, s_2,s_3, \dots)$ then the infinitesimal deformation $(s_0, s_1 + \varepsilon t_1, s_2 + \varepsilon t_2, s_3 + \varepsilon t_3, s_4, \dots)$ will separate points and tangent vectors and still lie in $X({\mathcal{E}})$. As $(C,\{ p^j_{m,e} \}, \{ q^j_e \},\pi)$ corresponds to a general point in $X({\mathcal{E}})$, the map $\pi$ must be an immersion at this point. \qed \vspace{+10pt} \subsection{Degenerations set-theoretically} \label{rdst} Fix ${\mathcal{E}}=(d,\vec{h},\vec{i})$ and a non-negative integer $E$, and let $\Gamma$ and $\Delta$ be sets of general linear spaces of $\mathbb P^n$ (as in the definition of $X(d,\Gamma,\Delta)$). Let $q$ be the marked point corresponding to one of the (general) $E$-planes $Q$ in $\Delta$. Let $D_H = \{ \pi(q) \in H \}$ be the divisor on $X({\mathcal{E}})$ that corresponds to requiring $q$ to lie on $H$. In this section, we will determine the components of $D_H$. That is, we will give a list of subvarieties, and show that the components of $D_H$ are a subset of this list. In the next section, we will determine the multiplicity with which each component appears. In particular, we will see that the multiplicity of each component is at least one, so each element of the list is indeed a component of $D_H$. \begin{figure} \begin{center} \getfig{rspecialize}{.1} \end{center} \caption{Specializing $H' \cap Q$ to lie in $H$} \label{rspecialize} \end{figure} But first, let us relate this result to the enumerative problem we wish to solve. If $X({\mathcal{E}}^-)$ is a finite set of reduced points, we can determine $\# X({\mathcal{E}}^-)$ by specializing one of the linear spaces of $\Delta$, of dimension $E-1$, to lie in the hyperplane $H$ (see Figure \ref{rspecialize}). Define ${\mathcal{E}}$ by $$ (d, \vec{h}, \vec{i}) = (d^-, \vec{h^-}, \vec{i^-} + \vec{\epsilon}_E - \vec{\epsilon}_{E-1}). $$ Then $X({\mathcal{E}})$ is a dimension 1 variety by Proposition \ref{rdimX}. Let $q$ be the marked point on one of the $E$-planes $Q$ in $\Delta^-$. The conditions of ${\mathcal{E}}$ are weaker than those of ${\mathcal{E}}^-$: in ${\mathcal{E}}$ we allow $q$ to lie on a linear space of dimension one more than in ${\mathcal{E}}^-$. Let $D_{H'}$ be the divisor on $X({\mathcal{E}})$ that corresponds to requiring $q$ to lie on a fixed general hyperplane $H'$ in $\mathbb P^n$. Then the divisor $D_{H'}$ on $X({\mathcal{E}})$ is $X({\mathcal{E}}^-)$ by Proposition \ref{rgeneral2}. If we specialize $H'$ to $H$, $H' \cap Q$ will specialize to a general $(E-1)$-plane $H \cap Q$ in $H$. As $(D_H) \sim (D_{H'})$ as divisor classes on the complete curve $X({\mathcal{E}})$, $\deg D_H = \deg D_{H'}$. So to calculate $\# X({\mathcal{E}}^-)$, we can simply enumerate the points $D_H$ on $X({\mathcal{E}})$, with the appropriate multiplicity. Only enumeratively meaningful divisors on $X({\mathcal{E}})$ are relevant to such enumerative calculations: we are counting points on $X({\mathcal{E}})$, which are obviously enumeratively meaningful. The components of $D_H$ on $X({\mathcal{E}})$ are given by the following result. \begin{tm} \label{rlist1} If $\Gamma$ and $\Delta$ are general, each component of $D_H$ (as a divisor on $X(d,\Gamma,\Delta)$) is a component of $$ Y(d(0), \Gamma(0), \Delta(0); \dots; d(l), \Gamma(l), \Delta(l)) $$ for some $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$, with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \cup_{k=0}^l \Gamma(k)$, $\Delta = \cup_{k=0}^l \Delta(k)$, $Q \in \Gamma(0)$. \end{tm} \noindent {\em Proof. } We may assume that $h_{m,e} = 0$ unless $e=n-1$, and that $\vec{i} = \vec{\epsilon}_n$ (and that $E=n$ and $q=q^1_n$). The general case follows by adding more marked points (Proposition \ref{runiversal}) and requiring each marked point to lie on a certain number of general hyperplanes (Proposition \ref{rgeneral2}). With these assumptions, the result becomes much simpler. The stack ${\mathcal{X}}(d,\vec{h},\vec{i})$ is the universal curve over ${\mathcal{X}}(d,\vec{h},\vec{0})$, and we are asking which points of the universal curve lie in $\pi^{-1}H$. Let $(C, \{ p^j_{m,n-1} \}, q, \pi)$ be the stable map corresponding to a general point of a component of $D_H$. Choose a general $(n-2)$-plane $A$ in $H$. The set $\pi^{-1} A$ is a union of reduced points on $C$, so by Proposition \ref{rbig} $\rho_A$ is smooth (as a morphism of stacks) at the point representing $(C, \{ p^j_{m,n-1} \}, q, \pi)$ (by Proposition \ref{rbig}). As a set, $D_H$ contains the entire fiber of $\rho_A$ above $\rho_A(C, \{ p^j_{m,n-1} \}, q, \pi)$, so $\rho_A(D_H)$ is a Weil divisor on $X_1(\hat{\ce})$ that is a component of $\{\pi(q) = z\}$ where $z = p_A(H)$. By Theorem \ref{igenus0}, the curve $C$ is a union of irreducible components $C(0) \cup \dots \cup C(l')$ with $\rho_A \circ \pi(C(0)) = z$ (i.e. $\pi(C(0)) \subset H$), $C(0) \cap C(k) \neq \phi$, and the marked points split up among the components: $\vec{h} = \sum_{k=0}^{l'} \vec{h}(k)$. If $d(0) = \deg \pi |_{C(0)}$, then $d(0)$ of the curves $C(1)$, \dots, $C(l')$ are rational tails that are collapsed to the $d(0)$ points of $C(0) \cap A$; they contain no marked points. Let $l = l' - d(0)$. Also, $\vec{i}(k) = \vec{0}$ for $k>0$, as the only incidence condition in $\vec{i}$ was $q \in Q$, and $q \in C(0)$. Therefore this component of $D_H$ is contained in $$ Y = Y(d(0), \Gamma(0), \Delta(0); \dots; d(l), \Gamma(l), \Delta(l) ). $$ But $\dim Y = \dim X({\mathcal{E}}) - 1$ (by Proposition \ref{rdimX}), so the result follows. \qed \vspace{+10pt} For enumerative calculations, we need only consider enumeratively meaningful components. With this in mind, we restate Theorem \ref{rlist1} in language reminiscent of [CH3]. Let $\phi$ be the isomorphism of Proposition \ref{rXY}. The following theorem will be more convenient for computation. \begin{tm} \label{rlist} If $\Gamma$ and $\Delta$ are general, each enumeratively meaningful component of $D_H$ (as a divisor on $X(d,\Gamma,\Delta)$) is one of the following. \begin{enumerate} \item[(I)] A component of $\phi(X(d',\Gamma', \Delta'))$, where, for some $m_0, e_0$, $1 \leq j_0 \leq h_{m_0, e_0}$, $e' := e_0 + E - n \geq 0$: \begin{itemize} \item $d' = d$, $\vec{h'} = \vec{h} - \vec{\epsilon}_{m_0,e_0} + \vec{\epsilon}_{m_0, e'}$, $\vec{i'} = \vec{i} - \vec{\epsilon}_E$ \item ${\Gamma'}^j_{m,e} = \Gamma^j_{m,e} \quad \text{if $(m,e) \neq (m_0, e_0)$}$ \item $ \{ {\Gamma'}^j_{m_0, e_0} \}_j = \{ \Gamma_{m_0,e_0}^j \}_j \setminus \{ \Gamma^{j_0}_{m_0,e_0} \}$ \item ${\Gamma'}_{m_0, e'}^{h'_{m_0, e'}} = \Gamma^{j_0}_{m_0,e_0} \cap Q$ \item ${\Delta'}^j_e = \Delta^j_e$ if $e \neq E$, and $\{ {\Delta'}_E^j \}_j = \{ \Delta^j_E \}_j \setminus \{ Q \}$. \end{itemize} \item[(II)] A component of $Y(d(0),\Gamma(0),\Delta(0);\dots;d(l),\Gamma(l),\Delta(l))$ for some $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$, with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \cup_{k=0}^l \Gamma(k)$, $\Delta = \cup_{k=0}^l \Delta(k)$, $Q \in \Gamma(0)$, and $d(0)>0$. \end{enumerate} \end{tm} Call these components {\em Type I components} and {\em Type II components} respectively. \noindent {\em Proof. } Consider a component $Y$ of $D_H$ that is not a Type II component (so $d(0) = 0$). Let $\{ C(0) \cup \dots \cup C(l), \{ p^j_{m,e} \}, q, \pi \}$ be the stable map corresponding to a general point of $Y$. The curve $C(0)$ has at least 3 special points: $q$, one of $\{ p^j_{m,e}\}$ (call it $p^{j_0}_{m_0,e_0}$), and $C(0) \cap C(1)$. If $C(0)$ had more than 3 special points, then the component would not be enumeratively meaningful, due to the moduli of the special points of $C(0)$. Thus $l=1$, and $Y$ is a Type I component. \qed \vspace{+10pt} \subsection{Multiplicity calculations} \label{rmultgen} In Subsection \ref{rdst}, we saw that, in a neighborhood of a general point of a component of the divisor $D_H$, there was a smooth morphism $\rho_A$ to $X_1(\hat{\ce})$ whose behavior at the corresponding divisor we understood well. For this reason, the multiplicity will be easy to calculate. We will see that the multiplicity with which the component $Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ appears is $\prod_{k=1}^l m^k$, where $m^k = d(k) - \sum_{m,e} m h_{m,e}(k)$ as defined earlier. As usual, Propositions \ref{runiversal} and \ref{rgeneral2} allow us to assume that $h_{m,e} = 0$ unless $e=n-1$, and $\vec{i} = \vec{\epsilon}_n$. Recall that $X_1(\hat{\ce})$ is the closure of the (locally closed) subvariety of ${\overline{M}}_{0,\sum h_{m,n-1}+1}(\mathbb P^1, d)$ parametrizing stable maps from pointed rational curves with points $\{ p^j_{m,n-1} \}_{1 \leq j \leq h_m}$, $q$ such that $\pi^* z = \sum_m m p_{m,n-1}^j$. By (\ref{idimX}) (or a quick count of ramification points away from $z$), $\dim X_1(\hat{\ce}) = d-1 + \sum_m h_{m,n-1}$. Consider $Y_1 = Y_1(\hat{\ce}(0),\dots,\hat{\ce}(l),\hat{{\mathcal{E}}'}(1),\dots,\hat{{\mathcal{E}}'}(d(0)))$ where $\hat{{\mathcal{E}}'}(i) = (1,\vec{0},\vec{0})$. The general point of $Y$ is a map from a tree of rational curves $A^0$, $A^1$, \dots, $A^l$, $B^1$, \dots, $B^{d(0)}$ with $A^0$ intersecting the other components and mapping to $z$, $B^1, \dots, B^{d(0)}$ mapping to $\mathbb P^1$ with degree 1, and $A^k$ mapping to $\mathbb P^1$ with degree $d(k)$ (for $k>0$), with $$ (\pi\mid_{A^k}) ^* z = \sum_m \left( \sum_j m p^j_{m,n-1}(k) \right) + m^k (A^k \cap A_1). $$ The rest of the marked points (including $q$) are on $A^0$. By Theorem \ref{igenus0}, $Y_1$ is a Weil divisor on $X_1(\hat{\ce})$. Let $D$ be the Cartier divisor on $X_1(\hat{\ce})$ defined by $\{ \pi(q) = z \}$. Choose a general point $(C, \{ p^j_{m,n-1} \}, q, \pi)$ of our component. Note that that the image $\rho_A(C, \{ p^j_{m,n-1} \}, q, \pi)$ is a general point of $Y_1$, with $A^k = C(k)$ (for $0 \leq k \leq l$). The additional components $B^1$, \dots, $B^{d(0)}$ come from the $d(0)$ intersections of $C(0)$ with the general $(n-2)$-plane $A$ of $H$. \begin{lm} In a neighborhood of $(C, \{ p^j_{m,n-1} \}, q,\pi)$: \begin{enumerate} \item[(a)] ${\overline{\cm}}_{0,\sum h_{m,n-1}+1}(\mathbb P^n,d) \rightarrow {\overline{\cm}}_{0,\sum h_{m,n-1}+1}(\mathbb P^1,d)$ is a smooth morphism of smooth stacks or algebraic spaces. Equivalently, the morphism is smooth on the level of deformation spaces. \item[(b)] The diagram $$\begin{CD} {\mathcal{X}}({\mathcal{E}}) @>>> {\overline{\cm}}_{0,\sum h_{m,n-1}+1}(\mathbb P^n,d) \\ @V{p}VV @VV{p}V \\ {\mathcal{X}}_1(\hat{\ce}) @>>> {\overline{\cm}}_{0,\sum h_{m,n-1}+1}(\mathbb P^1,d) \end{CD}$$ is a fiber square. \item[(c)] The component ${\mathcal{Y}}({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ is $p^* {\mathcal{Y}}_1$ on ${\mathcal{X}}({\mathcal{E}})$. \item[(d)] As Cartier divisors, $p^* D = D_H$. \end{enumerate} \end{lm} \noindent {\em Proof. } Part (a) is Proposition \ref{rbig}. Both (b) and (c) are clearly true set-theoretically, and the fact that they are true stack-theoretically follows from (a). Part (d) is clear: the divisor $D_H$ is $\{ \pi(q) \in H \}$, and the divisor $D$ is $\{ p_A(\pi(q)) = z \}$, where $p_A$ is the projection from $A$ in $\mathbb P^n$. \qed \vspace{+10pt} Combining these four statements with Theorem \ref{igenus0} and Corollary \ref{ilocalst}, we have: \begin{tm} \label{rmult2} The multiplicity of $D_H$ along the component $Y' = Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ is the multiplicity of $D$ along $Y$, which is $\prod_{k=1}^l m^k$. In an \'{e}tale or formal neighborhood of a general point of $Y'$, $X({\mathcal{E}})$ is isomorphic to $$ \operatorname{Spec} \mathbb{C}[[a, b_1,\dots,b_l,c_1,\dots,c_{\dim X({\mathcal{E}}) -1 } ]] / ( a = b_1^{m^1} = \dots = b_l^{m^l}) $$ where $D_H$ is given by $a=0$. \end{tm} Thus if $\lambda = \operatorname{lcm}(m^1,\dots,m^l)$, then $X({\mathcal{E}})$ has $\prod m^k / \lambda$ distinct reduced branches in an \'{e}tale neighborhood of a general point of $Y$, all smooth if and only if $\lambda = m^k$ for some $k$. \subsubsection{Multiplicity of $D_H$ along Type I components} \label{rmultI} Recall that a Type I component parametrizes those stable maps in $X({\mathcal{E}})$ where one of the marked points $p^{j_0}_{m_0,e_0}$ is mapped to the linear space $Q$; call this component $Z = Z(m_0,e_0,j_0)$. By the above argument, $Z$ appears with multiplicity $m_0$. But the following argument is more direct. By Propositions \ref{runiversal} and \ref{rgeneral2}, we may assume $h_{m,e} = 0$ unless $e=n-1$, and $\vec{i} = \vec{\epsilon}_n$. The stack ${\mathcal{X}}({\mathcal{E}})$ is the universal curve over ${\mathcal{X}}(d,\vec{h},\vec{0})$, and the Type I component $Z(m_0,e_0,j_0)$ corresponds to the section $p^{j_0}_{m_0,e_0}$ of the universal curve. On the general fiber $C$ of the family ${\mathcal{X}}({\mathcal{E}}) \rightarrow {\mathcal{X}}(d,\vec{h},\vec{0})$, $D_H = \sum m p^j_{m,e}$. Hence $D_H$ contains $Z(m_0,e_0,j_0)$ with multiplicity $m_0$. \subsection{Recursive formulas} \label{rrecursive} \subsubsection{The enumerative geometry of $Y$ from that of $X$} Now that we inductively understand the enumerative geometry of varieties of the form $X({\mathcal{E}})$, we can compute $\#Y({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$. The method can be seen through a simple example. Fix a hyperplane $H \subset \mathbb P^4$. In $\mathbb P^4$ the number of ordered pairs of lines $(L_0, L_1)$ consisting of lines $L_0 \subset H$ and $L_1 \subset \mathbb P^4$, with $L_0$ intersecting 3 fixed general lines $a_1$, $a_2$, $a_3$ in $H$, $L_1$ intersecting 5 fixed general 2-planes $b_1$, \dots, $b_5$ in $\mathbb P^4$, and $L_0$ intersecting $L_1$ (see Figure \ref{rYeg}), can be determined as follows. \begin{figure} \begin{center} \getfig{rYeg}{.1} \end{center} \caption{How many $(L_0,L_1)$ satisfy the desired conditions?} \label{rYeg} \end{figure} There is a one-parameter family of lines $L_0$ in $H$ intersecting the general lines $a_1$, $a_2$, $a_3$, and this family sweeps out a surface $S \subset H$ of some degree $d_0$. The degree $d_0$ is the number of lines $l_0$ intersecting the lines $a_1$, $a_2$, and $a_3$ {\em and another general line in $H$}, so this is $\# X_3({\mathcal{E}}'(0))$ for $d'(0) = 1$, $\vec{h'}(0) = \vec{\epsilon}_{1,2}$, $\vec{i'}(0) = 4 \vec{\epsilon}_1$. There is also a one-parameter family of lines $L_1$ intersecting the general 2-planes $b_1$, \dots, $b_5$, and the intersection point of such $L_1$ with $H$ sweeps out a curve $C \subset H$ of some degree $d_1$. The degree $d_1$ is the number of lines intersecting the 2-planes $b_1$, \dots, $b_5$ in $\mathbb P^4$ {\em and another general 2-plane in $H$}. Thus $d_1 = \# X_4({\mathcal{E}}'(1))$ for $d'(1) = 1$, $\vec{h'}(1) = \vec{\epsilon}_{1,2}$, $\vec{i'}(1) = 5 \vec{\epsilon}_1$. The answer we seek is $\# (C \cap S) = d_0 d_1$. The same argument in general yields: \begin{pr} \label{rrecursiveY} $$ \# Y_n ( {\mathcal{E}}(0); \dots; {\mathcal{E}}(l) ) = \frac { \# X_{n-1} ( {\mathcal{E}}'(0)) }{ d(0)!} \prod_{k=1}^l \# X_n({\mathcal{E}}'(k)) $$ where \begin{itemize} \item $h'(0) = d(0) \vec{\epsilon}_{1,n-2}$ \item $i'_e(0) = i_{e+1}(0) + \# \{ \dim X_n({\mathcal{E}}(k)) = e \}_{1 \leq k \leq l} + \sum_m h_{m,e}(0)$ \item for $1 \leq k \leq l$, $\vec{i'}(k) = \vec{i}(k)$ and $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k, n-1-\dim X_n({\mathcal{E}}(k))}$. \end{itemize} \end{pr} The $d(0)!$ is included to account for the $d(0)!$ possible labelings of the intersection points of a degree $d(0)$ curve in $H$ with a fixed general hyperplane $H'$ of $H$. The following result is trivial but useful. \begin{pr} \label{rtrivial} If the data ${\mathcal{E}}'$ is the same as ${\mathcal{E}}$ except $\vec{i'} = \vec{i}+ \vec{\epsilon}_{n-1}$, then $\# X({\mathcal{E}}') = d \cdot \# X({\mathcal{E}})$. \end{pr} \noindent {\em Proof. } The stable maps in $X({\mathcal{E}}')$ are just the stable maps in $X({\mathcal{E}})$ along with a marked point mapped to a fixed general hyperplane. There are $d$ choices of this marked point. (This is analogous to the divisorial axiom for Gromov Witten invariants, cf. [FP] p. 35 (III).) \qed \vspace{+10pt} We now summarize the results of Subsections \ref{rdst} and \ref{rmultgen}. Along with Propositions \ref{rrecursiveY} and \ref{rtrivial}, this will give an algorithm to compute $\# X({\mathcal{E}})$ for any ${\mathcal{E}}$. (Proposition \ref{rtrivial} isn't strictly necessary, but will make the algorithm faster.) The only initial data needed is the ``enumerative geometry of $\mathbb P^1$'': the number of stable maps to $\mathbb P^1$ of degree 1 is 1. Given ${\mathcal{E}}$, fix an $E$ such that $i_E>0$. Partitions of ${\mathcal{E}}$ are simultaneous partitions of $d$, $\vec{h}$, and $\vec{i}$. Define multinomial coefficients with vector arguments as the product of the multinomial coefficients of the components of the vectors: $$ \binom {\vec{h} }{ { \vec{h}(0), \dots, \vec{h}(l)} } = \prod_{m,e} \binom { h_{m,e} }{ {h_{m,e}(0), \dots, h_{m,e}(l)}}, $$ $$ \binom { \vec{i} }{ { \vec{i}(0), \dots, \vec{i}(l)} } = \prod_e \binom { i_e }{{i_e(0), \dots, i_e(l)}}. $$ Define ${\mathcal{E}}^-$ by $(d^-, \vec{h^-}, \vec{i^-}) = ( d, \vec{h}, \vec{i} - \vec{\epsilon}_E + \vec{\epsilon}_{E-1})$, and let $\Gamma^- = \Gamma$ and $\Delta^- = \Delta \cup \{ \Delta^{i_E}_E \cap H' \} \setminus \{ \Delta^{i_E}_E \}$ where $H'$ is a general hyperplane. (This notation was used earlier, in Subsection \ref{rdst}.) \begin{tm} \label{rrecursiveX1} In $A^1(X_n(d, \Gamma, \Delta))$, the cycle $X_n(d^-, \Gamma^-, \Delta^-)$ is rationally equivalent to $$ \sum \left( \prod_{k=1}^l m^k \right) Y_n(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ where the sum is over all $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$, with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \coprod_{k=0}^l \Gamma(k)$, $\Delta = \coprod_{k=0}^l \Delta(k)$, $\Gamma_{E}^{i_E} \in \Gamma(0)$. \end{tm} \noindent {\em Proof. } The left side is rationally equivalent (in ${\mathcal{X}}({\mathcal{E}})$) to $D_H = \{ \pi(q^{i_E}_E) \in H \}$. The right side is set-theoretically $D_H$ by Theorem \ref{rlist1}, and the multiplicity $\prod m^k$ was determined in Subsection \ref{rmultgen}. \qed \vspace{+10pt} If $\# X_n({\mathcal{E}}^-)$ is finite, the following statement is more suitable for computation. \begin{tm} \label{rrecursiveX2} $$ \# X_n({\mathcal{E}}^-) = \sum_{m,e} m h_{m,e} \cdot \# X_n({\mathcal{E}}'(m,e)) $$ $$ + \sum \left( \prod_{k=1}^l m^k \right) \binom { {\vec{h}} }{ { \vec{h}(0), \dots, \vec{h}(l)} } \binom { {\vec{i} - \vec{\epsilon}_E} }{ { \vec{i}(0)-\vec{\epsilon}_E,\vec{i}(1), \dots, \vec{i}(l)} } $$ $$ \quad \quad \quad \cdot \frac { \# Y_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) }{ \operatorname{Aut}( {\mathcal{E}}(1),\dots,{\mathcal{E}}(l))} $$ where, in the first sum, ${\mathcal{E}}'(m,e) = (d,\vec{h} -\vec{\epsilon}_{m,e}+\vec{\epsilon}_{m,e+E-n}, \vec{i} - \vec{\epsilon}_E)$; the second sum is over all $l$ and all partitions ${\mathcal{E}}(0)$,\dots, ${\mathcal{E}}(l)$ of ${\mathcal{E}}$ with $d(0)>0$. \end{tm} This follows from Theorem \ref{rlist} and the multiplicity calculations of Subsection \ref{rmultgen}. The only new points requiring explanation are the combinatorial aspects: the $h_{m,e}$ in the first sum, and the ``$\operatorname{Aut}$'' and various multinomial coefficients in the second. In Theorem \ref{rlist}, the Type I components were indexed by $(m_0, e_0, j_0)$. But for fixed $(m_0,e_0)$, $\# X({\mathcal{E}}'(m_0,e_0))$ is independent of $j_0$, so the above formula eliminates this redundancy. Similarly, in Theorem \ref{rlist}, the Type II components were indexed by partitions of the points $\{ p^j_{m,e} \}_{m,e,j}$ and $\{ q^j_e \}_{e,j} \setminus \{ q \}$, but the value of $\# Y_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ depends only on $\{ \vec{h}(k), \vec{i}(k) \}_{k=0}^l$ and not on the actual partitions. The multinomial coefficients in the second line eliminate this redundancy. Finally, we divide the last term by $\operatorname{Aut}({\mathcal{E}}(1); \dots; {\mathcal{E}}(l))$ to ensure that we are counting each Type II component once. \subsubsection{Transposing these results to subvarieties of the Hilbert scheme} Our original result was an equality of divisors on $X({\mathcal{E}})$. We will briefly sketch the analogous equality in the Chow ring of the Hilbert scheme. Assume for convenience that $n \geq 3$, and that $h_{m,e} = i_e = 0$ when $e > n-2$. By Proposition \ref{rXnice}, there is a dense open subset $U$ of $X({\mathcal{E}})$ such that the image of the map corresponding to a point on $U$ is smooth. We can take a smaller $U$ such that the images of the corresponding maps intersect each $\Gamma^j_{m,e}$ and $\Delta^j_e$ in one point. Define the closed subscheme $X^{\operatorname{Hilb}}({\mathcal{E}})$ of the Hilbert scheme to be the closure of the points $U^{\operatorname{Hilb}}$ representing the images of the maps corresponding to points of $U$. The subvarieties $Y^{\operatorname{Hilb}}({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ can be defined analogously. There is a rational map $$ \psi: X^{\operatorname{Hilb}}({\mathcal{E}}) \dashrightarrow X({\mathcal{E}}) $$ that restricts to an isomorphism from $U^{\operatorname{Hilb}}$ to $U$. (The map $\psi$ is the inverse of the rational map $\xi$ defined in Subsubsection \ref{rgeomean}.) Let $\Phi_1$, $\Phi_2$ be the projection of the graph of $\psi$ to $X^{\operatorname{Hilb}}({\mathcal{E}})$ and $X({\mathcal{E}})$ respectively. The exceptional divisors of $\psi$ are defined to be the image under $\Phi_1$ of the divisors on the graph collapsed by $\psi_2$. (It is not clear to the author if such divisors exist.) Then Theorem \ref{rrecursiveX1} can be reinterpreted as follows. \begin{tm} \label{rrecHilb} In $A^1( X^{\operatorname{Hilb}}(d,\Gamma,\Delta))$, modulo the exceptional divisors of $\psi$, $$ X^{\operatorname{Hilb}}(d^-,\Gamma^-,\Delta^-) = \sum \left( \prod_{k=1}^l m^k \right) Y^{\operatorname{Hilb}}(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ where the sum is over all $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$, with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \coprod_{k=0}^l \Gamma(k)$, $\Delta = \coprod_{k=0}^l \Delta(k)$, $\Gamma_{E}^{i_E} \in \Gamma(0)$. \end{tm} This result follows from Theorem \ref{rrecursiveX1} and the multiplicity calculations of Subsection \ref{rmultgen}. \section{Elliptic Curves in Projective Space} \label{elliptic} In this section, we extend our methods to study the geometry of varieties $W({\mathcal{E}})$ parametrizing degree $d$ elliptic curves in $\mathbb P^n$ intersecting fixed general linear spaces and tangent to a fixed hyperplane $H$ with fixed multiplicities along fixed general linear subspaces of $H$. We use the same general ideas as in the preceding section: we work with the variety ${\overline{M}}_{1,m}(\mathbb P^n,d)$ (and the stack ${\overline{\cm}}_{1,m}(\mathbb P^n,d)$) and specialize linear spaces (which the curve is required to intersect) to lie in $H$ one at a time. Many arguments will carry over wholesale. The main additions deal with new types of degenerations. \subsubsection{Example: Cubic elliptic space curves} \label{ecubics} The example of smooth elliptic cubics in $\mathbb P^3$ illustrates some of the degenerations we will see, and shows a new complication. There are 1500 smooth elliptic cubics in $\mathbb P^3$ through 12 general lines, and we can use the same degeneration ideas to calculate this number. Figure \ref{e1500cubics} is a pictorial table of the degenerations; smooth elliptic curves are indicated by an open circle. \begin{figure} \begin{center} \getfig{e1500cubics}{.1} \end{center} \caption{Counting 1500 elliptic cubics through 12 general lines in $\mathbb P^3$} \label{e1500cubics} \end{figure} The degenerations marked with an asterisk have a new twist. For example, consider the cubics through 9 general lines $L_1$, \dots, $L_9$ and 3 lines $L_{10}$, $L_{11}$, $L_{12}$ in $H$ (in row 9) and specialize $L_9$ to lie in $H$. The limit cubic could be a smooth plane curve in $H$ (the left-most picture of row 8 in the figure). In this case, it must pass through the eight points $L_1 \cap H$, \dots, $L_8 \cap H$. But there is an additional restriction. The cubics (before specialization) intersected $L_{10}$, $L_{11}$, $L_{12}$ in three points $p_{10}$, $p_{11}$, $p_{12}$ ($p_i \in L_i$), and as elliptic cubics are planar, these three points must have been collinear. Thus the possible limits are those curves in $H$ through $L_1 \cap H$, \dots, $L_8 \cap H$ and passing through collinear points $p_{10}$, $p_{11}$, $p_{12}$ (with $p_i \in L_i$). (There is also a choice of a marked point of the curve on $L_9$, which will give a multiplicity of 3.) This collinearity condition can be written as $\pi^*({\mathcal{O}}(1)) = p_{10} + p_{11} + p_{12}$ in the Picard group of the curve. The existence of such degenerations is analogous to the divisorial condition of Theorem \ref{igenus1}. We will have to count elliptic curves with such a divisorial condition involving the marked points; this locus forms a divisor on a variety of the form $W({\mathcal{E}})$. Fortunately, we can express this divisor in terms of divisors we understand well (Subsubsection \ref{eevalZ}). Thus as a side benefit, we get enumerative data about elliptic curves in $\mathbb P^n$ with certain incidence and tangency conditions, and a divisorial condition as well. \subsection{Notation and summary} For convenience, let $\vec{\epsilon}_e$, $\vec{\epsilon}_{m,e}$ be the natural basis vectors: $(\vec{\epsilon}_e)_{e'} = 1$ if $e = e'$ and 0 otherwise; and $(\vec{\epsilon}_{m,e})_{m',e'} = 1$ if $(m,e)=(m',e')$, and 0 otherwise. Fix a hyperplane $H$ in $\mathbb P^n$, and a hyperplane $A$ of $H$. From the previous section, recall the definitions of ``enumeratively meaningful'', $X({\mathcal{E}})$, ${\mathcal{X}}({\mathcal{E}})$, $Y({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))$, and ${\mathcal{Y}}({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))$. Motivated by the analysis in Subsection \ref{ikey} of divisors on subvarieties of ${\overline{M}}_{1,m}(\mathbb P^1,d)$, we define five new classes of varieties, labeled $W$, $Y^a$, $Y^b$, $Y^c$, and $Z$ and corresponding stacks, labeled ${\mathcal{W}}$, ${\mathcal{Y}}^a$, ${\mathcal{Y}}^b$, ${\mathcal{Y}}^c$, and ${\mathcal{Z}}$. \subsubsection{The schemes $W({\mathcal{E}})$} The objects of primary interest to us are smooth degree $d$ elliptic curves in $\mathbb P^n$ ($n \geq 2$) intersecting a fixed hyperplane $H$ with various multiplicities along various linear subspaces of $H$, and intersecting various general linear spaces in $\mathbb P^n$. We will examine these objects as stable maps from marked curves to $\mathbb P^n$ (where the markings will be the various intersections with $H$ and incidences). Let $n$ and $d$ be positive integers, and let $H$ be a hyperplane in $\mathbb P^n$. Let $\vec{h} = ( h_{m,e} )_{m \geq 1, e \geq 0}$ and $\vec{i} = ( i_e )_{e \geq 0}$ be sets of non-negative integers. Let $\Gamma = \{ \Gamma^j_{m,e} \}_{m,e,1 \leq j \leq h_{m,e}}$ be a set of linear spaces in $H$ where $\dim \Gamma^j_{m,e} = e$. Let $\Delta = \{ \Delta^j_e \}_{e,1 \leq j \leq i_e}$ be a set of linear spaces in $\mathbb P^n$ where $\dim \Delta^j_{e} = e$. \begin{defn} The scheme $W_n(d,\Gamma,\Delta)$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the marked points are labeled $\{p_{m,e}^j\}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying $\pi(p_{m,e}^j) \in \Gamma_{m,e}^j$, $\pi(q_e^j) \in \Delta_e^j$, $\pi^* H = \sum_{m,e,j} m p^j_{m,e}$, and where no components of $C$ are collapsed by $\pi$. \end{defn} In particular, $\sum_{m,e} m h_{m,e} = d$, and no component of $C$ is contained in $\pi^{-1}H$. The incidence conditions define closed subschemes of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$, so the union of these conditions indeed defines a closed subscheme of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$. Define ${\mathcal{W}}_n(d,\Gamma,\Delta)$ in the same way as a substack of ${\overline{\cm}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$. When we speak of propertis that are constant for general $\Gamma$ and $\Delta$ (such as the dimension), we will write $W_n(d,\vec{h},\vec{i})$. For convenience, write ${\mathcal{E}}$ (for ${\mathcal{E}}$verything) for the data $d, \vec{h}, \vec{i}$, so $W_n({\mathcal{E}}) = W_n(d,\vec{h},\vec{i})$. Also, the $n$ will often be suppressed for convenience. The variety $W(d, \Gamma, \Delta)$ (analogous to $X(d,\Gamma,\Delta)$ defined in Section \ref{rational}) can be loosely thought of as parametrizing degree $d$ elliptic curves in projective space intersecting certain linear subspaces of $\mathbb P^n$, and intersecting $H$ with different multiplicities along certain linear subspaces of $H$. For example, if $n=3$, $d=3$, $h_{2,0}=1$, $h_{1,2}=1$, $W$ parametrizes elliptic cubics in $\mathbb P^3$ tangent to $H$ at a fixed point. In the special case where $h_{m,e}=0$ when $e<n-1$ and $\vec{i}=\vec{\epsilon}_n$, define $\hat{\ce}$ by $\hat{d} = d$, $\hat{i}_1 = 1$, $\hat{h}_{m,0} = h_{m,n-1}$. We will relate the geometry of ${\mathcal{W}}_n({\mathcal{E}})$ to that of ${\mathcal{W}}_1(\hat{\ce})$, which was studied in Subsection \ref{ikey}. The geometry of ${\mathcal{W}}_n({\mathcal{E}})$ for general ${\mathcal{E}}$ can be understood from this special case. If the linear spaces $\Gamma$, $\Delta$ are general, these varieties have the dimension one would naively expect. The family of degree $d$ elliptic curves in $\mathbb P^n$ has dimension $(n+1)d$. Requiring the curve to pass through a fixed $e$-plane should be a codimension $(n-1-e)$ condition. Requiring the curve to be $m$-fold tangent to $H$ along a fixed $e$-plane of $H$ should be a codimension $(m-1)+(n-1-e)$ condition. Thus we will show (Theorem \ref{edimW}) that when the linear spaces in $\Gamma$, $\Delta$ are general, each component of $W(d, \Gamma, \Delta)$ has dimension $$ (n+1) d - \sum_{m,e} (n+m-e-2)h_{m,e} - \sum_e (n-1-e)i_e. $$ Moreover, $W({\mathcal{E}})$ is reduced. When the dimension is 0, $W({\mathcal{E}})$ consists of reduced points. We call this number $\# W({\mathcal{E}})$ --- these are the numbers we want to calculate. Define $\# W({\mathcal{E}})$ to be zero if $\dim W({\mathcal{E}}) > 0$. For example, when $n=3$, $d=3$, $h_{1,2} = 3$, $i_1 = 12$, $W({\mathcal{E}})$ consists of a certain number of reduced points: 3! times the number of elliptic cubics through 12 general lines. (The 3! arises from the markings of the three intersections of the cubic with $H$.) \subsubsection{The schemes $Y^a$, $Y^b$, and $Y^c$} We will be naturally led to consider subvarieties of $W(d,\Gamma,\Delta)$ which are similar in form to the varieties $$ Y(d(0), \Gamma(0), \Delta(0); \dots; d(l), \Gamma(l), \Delta(l) ) $$ of the previous section. Fix $n$, ${\mathcal{E}}$, $\Gamma$, $\Delta$, and a non-negative integer $l$. Let $\sum_{k=0}^l d(k)$ be a partition of $d$. Let the points $\{ p^j_{m,e} \}_{m,e,j}$ be partitioned into $l+1$ subsets $\{ p^j_{m,e}(k) \}_{m,e,j}$ for $k= 0$, \dots, $l$. This induces a partition of $\vec{h}$ into $\sum_{k=0}^l \vec{h}(k)$ and a partition of the set $\Gamma$ into $\coprod_{k=0}^l \Gamma(k)$. Let the points $\{ q^j_e \}_{e,j}$ be partitioned into $l+1$ subsets $\{ q^j_{e}(k) \}_{e,j}$ for $k= 0$, \dots, $l$. This induces a partition of $\vec{i}$ into $\sum_{k=0}^l \vec{i}(k)$ and a partition of the set $\Delta$ into $\coprod_{k=0}^l \Delta(k)$. Define $m^k$ by $m^k = d(k) - \sum_m m h_m(k)$, and assume $m^k>0$ for all $k = 1$, \dots, $l$. \begin{defn} The scheme $$ Y^a_n(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying the following conditions \begin{enumerate} \item[Y1.] The curve $C$ consists of $l+1$ irreducible components $C(0)$, \dots, $C(l)$ with all components intersecting $C(0)$. The map $\pi$ has degree $d(k)$ on curve $C(k)$ ($0 \leq k \leq l$). \item[Y2.] The points $\{ p^j_{m,e}(k) \}_{m,e,j}$ and $\{ q^j_e(k)\}_{e,j}$ lie on $C(k)$, and $\pi(p^j_{m,e}(k)) \in \Gamma^j_{m,e}(k)$, $\pi(q^j_e(k)) \in \Delta^j_e(k)$. \item[Y3.] As sets, $\pi^{-1}H = C(0) \cup \{ p^j_{m,e} \}_{m,e,j}$, and for $k>0$, $$ ( \pi \mid_{C(k)} )^* H = \sum_{m,e,j} m p^j_{m,e}(k) + m^k (C(0) \cap C(k)). $$ \item[Y${}^{\text{a}}$4.] The curve $C(1)$ is elliptic and the other components are rational. \end{enumerate} \end{defn} Conditions Y1--Y3 appeared in the definition of $Y$ (Definition \ref{rdefY}). Note that $d(k)>0$ for all positive $k$ by condition Y3. When discussing properties that hold for general $\{ \Gamma^j_{m,e} \}_{m,e,j}$, $\{ \Delta^j_e \}_{e,j}$, we will write $$ Y^a_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) = Y^a_n(d(0),\vec{h}(0),\vec{i}(0); \dots; d(l),\vec{h}(l),\vec{i}(l)) . $$ The $n$ will often be suppressed for convenience. If $\vec{h}(k) + \vec{i}(k) \neq \vec{0}$ for all $k>0$, $$ Y^a({\mathcal{E}}(0),\Gamma(0),\Delta(0); \dots; {\mathcal{E}}(l),\Gamma(0),\Delta(0)) $$ is isomorphic to a closed subscheme of $$ {\overline{M}}_{0,\sum h(0) + \sum i(0)+l}(H,d(0)) \times W(d(1),\Gamma(1),\Delta(1)) $$ $$ \times \prod_{k=2}^l X(d(k),\Gamma'(k),\Delta(k)), $$ where for $k= 1, \dots, l$, $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k,n-1}$ and $\Gamma'(k)$ is the same as $\Gamma(k)$ except $\Gamma^{h_{m^k,n-1}+1}_{m^k,h_{m^k,n-1}+1} = H$. Define ${\mathcal{Y}}^a({\mathcal{E}}(0), \Delta(0), \Gamma(0); \dots; {\mathcal{E}}(l), \Delta(l), \Gamma(l))$ as the analogous stack. \begin{defn} The scheme $$ Y^b_n(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying the conditions Y1--Y2 above, and \begin{enumerate} \item[Y${}^{\text{b}}$3.] As sets, $\pi^{-1}H = C(0) \cup \{ p^j_{m,e} \}_{m,e,j}$, and for $k>1$, $$ ( \pi \mid_{C(k)} )^* H = \sum_{m,e,j} m p^j_{m,e}(k) + m^k (C(0) \cap C(k)). $$ \item[Y${}^{\text{b}}$4.] All components of $C$ are rational. The curves $C(0)$ and $C(1)$ intersect at two distinct points $\{ a_1, a_2 \}$. (These points are not marked; monodromy may exchange them.) Also, $$ (\pi \mid_{C(1)}) ^* H = \sum_{m,e} \sum_{j=1}^{h^k_{m,e}} m p^j_{m,e} + m^1_1 a_1 + m^1_2 a_2 $$ where $m^1_1 + m^1_2 = m^1$. \end{enumerate} \end{defn} Thus $Y^b({\mathcal{E}}(0); \dots; {\mathcal{E}}(l) )$ is naturally the union of $[m^1/2]$ (possibly reducible) schemes (where $[\cdot]$ is the greatest-integer function), indexed by $m^1_1$. For convenience, label these varieties $$ \{ Y^b({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m_1^1}\}_{1 \leq m^1_1 < m^1}, $$ so $\{ Y^b({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m_1^1}\}_{m^1_1} = \{ Y^b({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m_1^1}\}_{m^1 - m^1_1}$. For enumerative reasons, we define a slightly different variety. \begin{defn} The scheme $$ \tilde{Y}^b_n(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l))_{m^1_1} $$ is the (scheme-theoretic) closure of the locally closed subset of the universal curve over ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{1 \leq j \leq i_e}$, and the point on the universal curve is labeled $a_1$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ (with additional point $a_1$) satisfying the conditions Y1, Y2, Y${}^{\text{b}}$3, and Y${}^{\text{b}}$4 above (for some other point $a_2$). \end{defn} There is a morphism $\tilde{Y}^b_n(d(0), \dots, \Delta(l))_{m^1_1} \rightarrow Y^b_n(d(0), \dots, \Delta(l))_{m^1_1}$ corresponding to forgetting the point $a_1$. This morphism is an isomorphism if $m^1_1 \neq m^1_2$ and it is generically two-to-one when $m^1_1 = m^1_2$. When discussing properties that hold for general $\{ \Gamma^j_{m,e} \}_{m,e,j}$, $\{ \Delta^j_e \}_{e,j}$, we will write $$ Y^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) =Y^b_n(d(0),\vec{h}(0),\vec{i}(0); \dots; d(l),\vec{h}(l),\vec{i}(l)) $$ and $$ \tilde{Y}^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) =\tilde{Y}^b_n(d(0),\vec{h}(0),\vec{i}(0); \dots; d(l),\vec{h}(l),\vec{i}(l)) $$ The $n$ will often be suppressed for convenience. If $\vec{h}(k) + \vec{i}(k) \neq \vec{0}$ for all $k>0$, $\tilde{Y}^b(d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l), \Delta(l))$ is isomorphic to a closed subscheme of $$ {\overline{M}}_{0,\sum h(0) + \sum i(0)+l+1}(H,d(0)) \times \prod_{k=1}^l X(d(k), \Gamma'(k),\Delta(k)) $$ for appropriately chosen $\Gamma'(k)$, $k = 1$, \dots, $l$. Define ${\mathcal{Y}}^b(d(0), \Delta(0), \Gamma(0); \dots; d(l), \Delta(l), \Gamma(l))$ as the analogous stack. \begin{defn} \label{eYcdef} The scheme $$ Y^c_n( d(0),\Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{1,\sum h_{m,e} + \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{ p_{m,e}^j \}_{1 \leq j \leq h_{m,e}}$ and $\{ q_e^j \}_{ 1 \leq j \leq i_e}$) representing stable maps $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ satisfying conditions Y1--Y3, and \begin{enumerate} \item[Y${}^{\text{c}}$4.] The curve $C(0)$ is elliptic and the other components are rational. The morphism $\pi$ has positive degree on every component. \item[Y${}^{\text{c}}$5.] In $\operatorname{Pic}(C(0))$, \begin{eqnarray*} \pi^*({\mathcal{O}}_{\mathbb P^n}(1)) & \otimes & {\mathcal{O}}_{C(0)} \left( \sum_{k=1}^l m^k (C(0) \cap C(k)) \right) \\ & \cong & {\mathcal{O}}_{C(0)} \left( \sum_{m,e} \sum_{j=1}^{h_{m,e}(0)} m p_{m,e}^j(0) \right). \end{eqnarray*} \end{enumerate} \end{defn} The divisorial condition Y${}^{\text{c}}$5 is motivated by the ideas of Subsection \ref{ikey}. If ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, and $\Gamma$ and $\Delta$ are general with $\Gamma = \coprod_{k=0}^l \Gamma(k)$, $\Delta = \coprod_{k=0}^l \Delta(k)$, then the variety $$ Y^c_n(d(0), \Gamma(0),\Delta(0); \dots; d(l),\Gamma(l),\Delta(l)) $$ will turn out to be a Weil divisor on $W(d, \Gamma, \Delta)$. The stable map $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ corresponding to a general point of $W(d,\Gamma,\Delta)$ satisfies $\pi^* ({\mathcal{O}}_{\mathbb P^n}(1)) \cong {\mathcal{O}}_C ( \sum_{m,e,j} m p^j_{m,e})$, and this condition must in some sense be inherited by the map corresponding to a general point on the Weil divisor. This condition was actually present in $Y^a$ and $Y^b$ (and the Type II component $Y$ of the previous section), but as $C(0)$ was rational in each of these cases, the requirement reduced to $$ d(0) + \sum_{k=1}^l m^k = \sum_{m,e} m h_{m,e}(0) $$ which was always true. When discussing properties that hold for general $\{ \Gamma^j_{m,e} \}_{m,e,j}$, $\{ \Delta^j_e \}_{e,j}$, we will write $Y^c_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$. The $n$ will often be suppressed for convenience. If $\vec{h}(k) + \vec{i}(k) \neq \vec{0}$ for all $k>0$, $$ Y^c(d(0), \Gamma(0), \Delta(0); \dots; d(l), \Gamma(l), \Delta(l)) $$ is isomorphic to a closed subscheme of $$ {\overline{M}}_{1,\sum h_{m,e}(0) + \sum i_e(0)+l}(H,d(0)) \times \prod_{k=1}^l X(d(k), \Gamma'(k), \Delta(k)) $$ for appropriately chosen $\Gamma'(k)$. Define ${\mathcal{Y}}^c(d(0), \Delta(0), \Gamma(0); \dots; d(l), \Delta(l), \Gamma(l))$ as the analogous stack. The five classes of varieties $W$, $X$, $Y^a$, $Y^b$, $Y^c$ are illustrated in Figure \ref{ewxy}. In the figure, the dual graph of the curve corresponding to a general point of the variety is given. Vertices corresponding to components mapped to $H$ are labeled with an $H$, and vertices corresponding to elliptic components are open circles. \begin{figure} \begin{center} \getfig{ewxy}{.1} \end{center} \caption{Five classes of varieties} \label{ewxy} \end{figure} \subsubsection{The scheme $Z(d,\vec{i})_{{\mathcal{D}}}$} \label{esubZ} Because of the divisorial condition Y${}^{\text{c}}$5 in the definition of $Y^c({\mathcal{E}})$, we will also be interested in the variety parametrizing smooth degree $d$ elliptic curves in $\mathbb P^n$ ($n \geq 2$) with a condition in the Picard group of the curve involving the marked points and $\pi^*({\mathcal{O}}_{\mathbb P^n}(1))$. Let $d$ and $n$ be positive integers and $\vec{i} = (i_e)_{e \geq 0}$ a set of non-negative integers. Let $\Delta = \{ \Delta^j_e \}_{e, 1 \leq j \leq i_e}$ be a set of linear spaces in $\mathbb P^n$ where $\dim \Delta^j_e = e$. Let ${\mathcal{D}}$ be a linear equation in formal variables $\{ q^j_e \}_{e,j}$ with integral coefficients summing to $d$. \begin{defn} The scheme $Z_n(d,\Delta)_{{\mathcal{D}}}$ is the (scheme-theoretic) closure of the locally closed subset of ${\overline{M}}_{1, \sum i_e}(\mathbb P^n,d)$ (where the points are labeled $\{q^j_e \}_{1 \leq j \leq i_e}$) representing stable maps $(C, \{ q^j_e \}, \pi)$ satisfying the following conditions: \begin{enumerate} \item[(i)] The curve $C$ is smooth, \item[(ii)] $\pi(q^j_e) \in \Delta^j_e$, and \item[(iii)] in $\operatorname{Pic}(C)$, $\pi^* ({\mathcal{O}}_{\mathbb P^n}(1)) \cong {\mathcal{O}}_C({\mathcal{D}})$. \end{enumerate} \end{defn} When discussing properties that hold for general $\{ \Delta^j_e \}_{e,j}$, we will write $Z_n(d,\vec{i})_{{\mathcal{D}}}$. The $n$ will often be suppressed for convenience. Define ${\mathcal{Z}}(d,\vec{i})_{{\mathcal{D}}}$ as the analogous stack. For example, $$ Z_2(d=4,i_0 = 11)_{q^1_0 + q^2_0 + q^3_0 + q^4_0}. $$ parametrizes the finite number of smooth two-nodal quartic plane curves through 11 fixed general points $\{ q^j_0 \}_{1 \leq j \leq 11}$ satisfying $$ \pi^*({\mathcal{O}}_{\mathbb P^2}(1)) \cong {\mathcal{O}}(q^1_0 + q^2_0 + q^3_0 + q^4_0) $$ in the Picard group of the normalization of the curve. When $\Gamma$ and $\Delta$ are general, all of the varieties $W$, $Y^a$, $Y^b$, $Y^c$, $Z$ defined above will be seen to be reduced (Propositions \ref{edimW} and \ref{edimZ}). When the dimension is 0 (and, as before, $\Gamma$ and $\Delta$ are general), they consist of reduced points, and the number of points is independent of $\Gamma$ and $\Delta$. We call this number $\# W({\mathcal{E}})$, $\# Y^a({\mathcal{E}})$, etc. We will calculate all of these values for all $n$ and ${\mathcal{E}}$. \subsection{Preliminary results} In this section, we prove preliminary results we will need. Recall Propositions \ref{rgeneral} and \ref{rgeneralproper}, which are collected in the following proposition: \begin{pr} \label{egeneral} Let ${\mathcal{A}}$ be a reduced irreducible substack of ${\overline{\cm}}_{g,m}(\mathbb P^n,d)$, and let $p$ be one of the labeled points. Then there is a Zariski-open subset $U$ of the dual projective space $(\mathbb P^n)^*$ such that for all $[H'] \in U$ the intersection ${\mathcal{A}} \cap \{ \pi(p) \in H' \}$, if nonempty, is reduced of dimension $\dim {\mathcal{A}} - 1$. Let ${\mathcal{B}}$ be a proper closed substack of ${\mathcal{A}}$. Then there is a Zariski-open subset $U'$ of the dual projective space $(\mathbb P^n)^*$ such that for all $[H'] \in U'$, each component of ${\mathcal{B}} \cap \{ \pi(p) \in H' \}$ is a proper closed substack of a component of ${\mathcal{A}} \cap \{ \pi(p) \in H' \}$. \end{pr} The following two propositions are variants of Propositions \ref{runiversal} and \ref{rgeneral2} of the previous section. The proofs are identical once ${\mathcal{X}}$ is replaced with ${\mathcal{W}}$. \begin{pr} \label{euniversal} Given $d$, $\vec{h}$, $\vec{i}$, $\Gamma$, $\Delta$, define $\vec{i'} = \vec{i} + \vec{\epsilon}_n$ and $\Delta'$ the same as $\Delta$ except $\Delta^{i'_n}_n = \mathbb P^n$. Then ${\mathcal{W}}(d, \Gamma, \Delta')$ is the universal curve over ${\mathcal{W}}(d,\Gamma, \Delta)$. \end{pr} \begin{pr} \label{egeneral2} Let $H'$ be a general hyperplane of $\mathbb P^n$. \begin{enumerate} \item[a)] The divisor $$ \{ \pi(p^{j_0}_{m_0,e_0}) \in H' \} \subset {\mathcal{W}}(d, \Gamma, \Delta) $$ is ${\mathcal{W}}(d, \Gamma', \Delta)$ where \begin{itemize} \item $\vec{h'}= \vec{h} - \vec{\epsilon}_{m_0,e_0} + \vec{\epsilon}_{m_0,e_0-1}$ \item For $(m,e) \neq (m_0,e_0), (m_0,e_0-1)$, ${\Gamma'}^j_{m,e} = \Gamma^j_{m,e}$. \item $\{ {\Gamma'}^j_{m_0,e_0} \}_j = \{ \Gamma^j_{m_0,e_0} \}_j \setminus \{ \Gamma^{j_0}_{m_0,e_0} \}$ \item $\{ {\Gamma'}^j_{m_0,e_0-1} \} = \{ \Gamma^j_{m_0,e_0-1} \}_j \cup \{ \Gamma^{j_0}_{m_0,e_0} \cap H' \} $ \end{itemize} \item[b)] The divisor $$ \{ \pi(q^{j_0}_{e_0}) \in H' \} \subset {\mathcal{W}}(d, \Gamma, \Delta) $$ is ${\mathcal{W}}(d, \Gamma, \Delta')$ where \begin{itemize} \item $\vec{i'} = \vec{i} - \vec{\epsilon}_{e_0} + \vec{\epsilon}_{e_0-1}$ \item For $e \neq e_0, e_0-1$, ${\Delta'}^j_{e} = \Delta^j_{e}$. \item $\{ {\Delta'}^j_{e_0} \}_j = \{ \Delta^j_{e_0} \}_j \setminus \{ \Delta^{j_0}_{e_0} \}$, $\{ {\Delta'}^j_{e_0-1} \}_j = \{ \Delta^j_{e_0-1} \}_j \cup \{ \Delta^{j_0}_{e_0} \cap H' \} $ \end{itemize} \end{enumerate} \end{pr} An analogous proposition holds for ${\mathcal{Z}}_n(d,\Delta)$. The proof is essentially the same, and is omitted. \begin{pr} \label{ezgeneral2} Let $H'$ be a general hyperplane of $\mathbb P^n$. The divisor $\{ \pi(q^{j_0}_{e_0}) \in H' \}$ on ${\mathcal{Z}}(d,\Delta)_{{\mathcal{D}}}$ is ${\mathcal{Z}}(d, \Delta')_{{\mathcal{D}}}$ where \begin{itemize} \item $\vec{i'} = \vec{i} - \vec{\epsilon}_{e_0} + \vec{\epsilon}_{e_0-1}$ \item For $e \neq e_0, e_0-1$, ${\Delta'}^j_{e} = \Delta^j_{e}$. \item $\{ {\Delta'}^j_{e_0} \}_j = \{ \Delta^j_{e_0} \}_j \setminus \{ \Delta^{j_0}_{e_0} \}$, $\{ {\Delta'}^j_{e_0-1} \}_j = \{ \Delta^j_{e_0-1} \}_j \cup \{ \Delta^{j_0}_{e_0} \cap H' \} $ \end{itemize} \end{pr} For a given ${\mathcal{E}}'$, $\Gamma'$, $\Delta'$, we also have an isomorphism $\phi_1$ between $W(d', \Gamma', \Delta')$ and $Y^a(d(0),\Gamma(0),\Delta(0);d(1),\Gamma(1),\Delta(1))$ (for appropriately chosen ${\mathcal{E}}(0)$, \dots, $\Delta(1)$) that is similar to the isomorphism $\phi$ of Proposition \ref{rXY}. The notation used in this proposition is the same, and the proof is also identical: the morphism involves attaching a rational tail with two marked points. \begin{pr} Fix ${\mathcal{E}}$, integers $m_0$, $e_0$, $e_1$, and general $\Gamma$, $\Delta$. Let $j_0 = h_{m_0,e_0}$, $j_1 = i_{e_1}$, $e' = e_0+e_1-n$, and $j' = h_{m_0,e'}+1$. Let $(d(0), \vec{h}(0), \vec{i}(0)) = ( 0, \vec{\epsilon}_{m_0,e_0}, \vec{\epsilon}_{e_1})$, $\Gamma(0) = \{ \Gamma^{j_0}_{m_0,e_0} \}$, $\Delta(0) = \{ \Delta^{j_1}_{e_1} \}$, ${\mathcal{E}}(1) = {\mathcal{E}} - {\mathcal{E}}(0)$, $\Gamma(1) = \Gamma \setminus \Gamma(0)$, $\Delta(1) = \Delta \setminus \Delta(0)$. Let $(d', \vec{h'}, \vec{i'}) = ( d, \vec{h}(1)+ \vec{\epsilon}_{m_1,e'}, \vec{i}(1))$, $\Gamma' = \Gamma(1) \cup \{ \Gamma^{j_0}_{m_0,e_0} \cap \Delta^{j_1}_{e_1} \}$, and $\Delta' = \Delta(1)$. Then there is a natural isomorphism $$ \phi_1: W(d', \Gamma', \Delta') \rightarrow Y^a(d(0), \Gamma(0), \Delta(0); d(1),\Gamma(1),\Delta(1)). $$ \label{eWY} \end{pr} Proposition \ref{ebig} is a variation of the smoothness result (Proposition \ref{rbig}) that was so useful in Section \ref{rational}. To prove it, we will need some preliminary results about stable maps from elliptic curves to $\mathbb P^1$. \begin{lm} \label{ecomb} Let $C$ be a complete reduced nodal curve of arithmetic genus 1. Let $\pi$ be a morphism $\pi: C \rightarrow \mathbb P^1$ contracting no component of $C$ of arithmetic genus 1. Then $$ H^1(C,\pi^*({\mathcal{O}}_{\mathbb P^1}(1))) = H^1(C,\pi^*({\mathcal{O}}_{\mathbb P^1}(2))) = 0. $$ \end{lm} By ``contracting no component of $C$ of arithmetic genus 1'' we mean that all connected unions of contracted irreducible components of $C$ have arithmetic genus 0. \noindent {\em Proof. } By Serre duality, it suffices to show that $$ H^0(C,K_C \otimes \pi^*({\mathcal{O}}(-1))) = 0. $$ Assume otherwise that such $(C,\pi)$ exists, and choose one with the fewest components, and choose a nonzero global section $s$ of $K_C \otimes \pi^*({\mathcal{O}}(-1))$. If $C = C' \cup R$ where $R$ is a rational tail (intersecting $C'$ at one point), then $s=0$ on $R$ as $$ \deg_R ( K \otimes \pi^* ( {\mathcal{O}}(-1))) = -1 - \deg_{\pi} R < 0. $$ Then $s|_{C'}$ is a section of $(K_C \otimes \pi^*( {\mathcal{O}}(-1)))|_{C'}$ that vanishes on $C' \cap R$. But $K_{C'} = K_C(-C' \cap R) |_{C'}$, so this induces a non-zero section of $K_{C'} \otimes (\pi|_{C'})^*({\mathcal{O}}(-1))$, contradicting the minimality of the number of components. Thus $C$ has no rational tails, and $C$ is either an irreducible elliptic curve or a cycle of rational curves. If $C$ is an irreducible elliptic curve, then $C$ isn't contracted by hypothesis, so $K_C \otimes \pi^*({\mathcal{O}}(-1))$ is negative on $C$ as desired. If $C$ is a cycle $C_1 \cup \dots \cup C_s$ of $\mathbb P^1$'s, then $$ \deg_{C_i} ( K_C \otimes \pi^* ({\mathcal{O}}(-1) )) = - \deg C_i \leq 0. $$ As one of the curves has positive degree, there are no global sections of $K_C \otimes \pi^* ({\mathcal{O}}(-1))$. \qed \vspace{+10pt} \begin{lm} \label{esm1} Let $(C, \{ p_i \}_{i=1}^m, \pi)$ be a stable map in ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ having no contracted component of arithmetic genus 1. Then ${\overline{\cm}}_1(\mathbb P^1,d)$ is smooth of dimension $2d+m$ at $(C,\{ p_i \},\pi)$. \end{lm} \noindent {\em Proof. } As $H^1(C,\pi^* T_{\mathbb P^1}) = 0$ by the previous lemma, ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ is smooth of dimension $\deg \pi^*T_{\mathbb P^1} + m = 2d+m$. The argument is well-known, but for completeness we give it here. From the exact sequence for infinitesimal deformations of stable maps (see Subsubsection \ref{itmsosm}), we have \begin{eqnarray} \label{edefsm} 0 &\longrightarrow& \operatorname{Aut} (C, \{ p_i \}) \longrightarrow H^0(C,\pi^* T_{\mathbb P^1}) \\ \nonumber \longrightarrow \operatorname{Def} (C, \{ p_i \}, \pi ) & \longrightarrow & \operatorname{Def} (C, \{ p_i \}) \longrightarrow H^1(C,\pi^* T_{\mathbb P^1} ) \\ \longrightarrow \operatorname{Ob} (C, \{ p_i \}, \pi ) &\longrightarrow & 0 \nonumber \end{eqnarray} where $\operatorname{Aut} (C,\{ p_i \}) = \operatorname{Hom} (\Omega_C(p_1 + \dots + p_m),{\mathcal{O}}_C)$ (resp. $\operatorname{Def} (C, \{ p_i \}) = \operatorname{Ext}^1(\Omega_C(p_1 + \dots + p_m),{\mathcal{O}}_C )$) are the infinitesimal automorphisms (resp. infinitesimal deformations) of the marked curve, and $\operatorname{Def} (C, \{ p_i \}, \pi)$ (resp. $\operatorname{Ob} (C, \{ p_i \}, \pi)$) are the infinitesimal deformations (resp. obstructions) of the stable map. As $H^1(C,\pi^* T_{\mathbb P^1}) = 0$, $\operatorname{Ob}(C, \{ p_i \}, \pi) = 0$ from (\ref{edefsm}). Thus the deformations of $(C,\{ p_i \} , \pi)$ are unobstructed, and the dimension follows from: \begin{eqnarray*} \dim \operatorname{Def} (C, \{ p_i \}, \pi) &-& \dim \operatorname{Ob} (C, \{ p_i \}, \pi) \\ &=& ( \dim \operatorname{Def} (C, \{ p_i \}) - \dim \operatorname{Aut} (C, \{ p_i \})) \\ & & + (h^0(C,\pi^* T_{\mathbb P^1}) - h^1(C,\pi^* T_{\mathbb P^1}) ) \\ &=& m + 2d. \end{eqnarray*} \qed \vspace{+10pt} The next lemma will be useful for studying the behavior of the space ${\overline{M}}_{1,m}(\mathbb P^n,d)$ at points representing maps with contracted elliptic components. \begin{lm} \label{etancondition} Let $C$ be a complete reduced nodal curve of arithmetic genus 1, and let $\pi: C \rightarrow \mathbb P^n$. Assume $(C,\pi)$ can be smoothed. If $B$ is a connected union of contracted components of $C$ of arithmetic genus 1, intersecting $\overline{C \setminus B}$ in $k$ points, and $T_1$, \dots, $T_k$ are the tangent vectors to $\overline{C \setminus B}$ at those points, then $\{ \pi( T_i ) \}_{i=1}^k$ are linearly dependent in $T_{\pi(B)} \mathbb P^n$. \end{lm} More generally, this result will hold whenever $\pi$ is a map to an $n$-dimensional variety $X$, and $B$ is contracted to a smooth point of $X$. It is a variation of [V2] Theorem 1 in higher dimensions. \noindent {\em Proof. } Let $\Delta$ be a smooth curve parametrizing maps $({\mathcal{C}}_t, \pi)$ (with total family $({\mathcal{C}},\pi)$) to $\mathbb P^n$, with $({\mathcal{C}}_0,\pi) = (C,\pi)$ and general member a map from a smooth curve. Blowing up points of the central fiber changes $C$, but does not change the hypotheses of the proposition, so we may assume without loss of generality that the total family ${\mathcal{C}}$ is a smooth surface. The following diagram is commutative. $$\begin{array}{rcccl} {\mathcal{C}} & \; & {\stackrel \pi \longrightarrow} & \; & \mathbb P^n \times \Delta \\ \; & \searrow & \; & \swarrow & \; \\ \; & \; & \Delta & \; & \; \end{array}$$ There is an open neighborhood $U$ of $B \subset {\mathcal{C}}$ such that $\pi \mid_{U \setminus B}$ is an immersion. Thus $\pi$ factors through a family ${\mathcal{C}}'$ that is the same as ${\mathcal{C}}$ except $B$ is contracted. Let $\pi'$ be the contraction $\pi': {\mathcal{C}} \rightarrow {\mathcal{C}}'$. The family ${\mathcal{C}}'$ is also flat, and its general fiber has genus 1. The central fiber is a union of rational curves, at most nodal away from the image of $B$. If the images of $T_1$, \dots, $T_k$ in ${\mathcal{C}}'_0$ are independent, the reduced fiber above 0 would have arithmetic genus 0, so the central fiber (reduced away from the image of $B$) would have arithmetic genus at most zero, contradicting the constancy of arithmetic genus in flat families. Thus the images of $T_1$, \dots, $T_k$ in $T_{\pi'(B)} {\mathcal{C}}'_0$ must be dependent, and hence their images in $T_{\pi(B)}\mathbb P^n$ must be dependent as well. \qed \vspace{+10pt} In Lemma \ref{esm2}, we will prove that the moduli stack ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ is smooth even at some points with contracted components of arithmetic genus 1. Let $(C,\{ p_i \}_{i=1}^m, \pi )$ be a stable map in ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ with $\pi^{-1}(z)$ containing (as a connected component) a curve $E$ of arithmetic genus 1, where $E$ intersects the rest of the components $R$ at two points $p$ and $q$ (and possibly others) with the $\pi|_R$ \'{e}tale at $p$. (This result should be true even without the \'{e}tale condition.) \begin{lm} \label{esm2} The moduli stack ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ is smooth at $(C, \{ p_i \}, \pi)$ of dimension $2d+m$. \end{lm} \noindent {\em Proof. } For convenience (and without loss of generality) assume $m=0$. The calculations of Lemma \ref{ecomb} show that $h^1(C,\pi^* T_{\mathbb P^1}) = 1$, so our proof of Lemma \ref{esm1} will not carry through. However, $\operatorname{Def} (C,\pi)$ does not surject onto $\operatorname{Def}(C)$ in long exact sequence (\ref{edefsm}), as it is not possible to smooth the nodes independently: one cannot smooth the node at $p$ while preserving the other nodes even to first order. (This is well-known; one argument, due to M. Thaddeus, is to consider a stable map $(C, \pi)$ in ${\overline{\cm}}_1(\mathbb P^1,1)$ and express the obstruction space $\operatorname{{\mathbb E}xt}^2 ( \underline{\Omega}_{\pi}, {\mathcal{O}}_C)$ as the dual of $H^0(C, {\mathcal{F}})$ for a certain sheaf ${\mathcal{F}}$.) Thus the map $\operatorname{Def}(C) \rightarrow H^1(C,\pi^* T_{\mathbb P^1})$ is not the zero map, so $\operatorname{Def}(C)$ surjects onto $H^1(C,\pi^* T_{\mathbb P^1})$. Therefore $\operatorname{Ob}(C, \pi) = 0$, so the deformations are unobstructed. The rest of the proof is identical to that of Lemma \ref{esm1}. \qed \vspace{+10pt} With these lemmas in hand we are now ready to prove an important smoothness result. Let $A$ be a general $(n-2)$-plane in $H$. Projection from $A$ induces a rational map $\rho_A: {\overline{\cm}}_{1,m}(\mathbb P^n,d) \dashrightarrow {\overline{\cm}}_{1,m}(\mathbb P^1,d)$, that is a morphism (of stacks) at points representing maps $(C, \{ p_i \}, \pi)$ whose image $\pi(C)$ doesn't intersect $A$. Via ${\overline{\cm}}_{1,m}(\operatorname{Bl}_A \mathbb P^n,d)$, the morphism can be extended over the set of maps $(C,\{ p_i \}, \pi)$ where $\pi^{-1} A$ is a union of reduced points distinct from the $m$ marked points $\{ p_i \}$. The image of such curves in ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ is a stable map $$ (C \cup C_1 \cup \dots \cup C_{\# \pi^{-1} A }, \{ p_i \}, \pi') $$ where $C_1$, \dots, $C_{\# \pi^{-1} A }$ are rational tails attached to $C$ at the points of $\pi^{-1} A$, $$ \pi' \mid_{ \{ C \setminus \pi^{-1} A \} } = ( p_A \circ \pi ) \mid_{ \{ C \setminus \pi^{-1} A \} } $$ (which extends to a morphism from all of $C$) and $\pi' \mid_{C_k}$ is a degree 1 map to $\mathbb P^1$ ($1 \leq k \leq \# \pi^{-1} A$). \begin{pr} If $(C,\{ p_i \}, \pi) \subset {\overline{\cm}}_{1,m}(\mathbb P^n,d)$, the scheme $\pi^{-1} A$ is a union of reduced points disjoint from the marked points, and $\pi$ collapses no components of arithmetic genus 1, then at $(C,\{ p_i \}, \pi)$, $\rho_A$ is a smooth morphism of stacks of relative dimension $(n-1)d$. \label{ebig} \end{pr} \noindent {\em Proof. } If no components of $C$ of arithmetic genus 1 are mapped to $H$, then $\rho_A(C,\pi)$ is a smooth point of ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ by Lemma \ref{esm1}. If a component of $C$ of arithmetic genus 1 is mapped to $H$, it must intersect $A$ in at least two points. In this case $\rho_A(C,\pi)$ consists of a curve with a contracted elliptic component, and this elliptic component has at least two rational tails that map to $\mathbb P^1$ with degree 1. Thus by Lemma \ref{esm2}, $\rho_A(C,\pi)$ is a smooth point of ${\overline{\cm}}_{1,m}(\mathbb P^1,d)$ as well. By Lemma \ref{ecomb}, $H^1(C,\pi^*({\mathcal{O}}(1))) = 0$, so $h^0(C,\pi^*({\mathcal{O}}(1))) = d$ by Riemann-Roch. The proof is then identical to that of Proposition \ref{rbig}. \qed \vspace{+10pt} We now calculate the dimension of the varieties $W$, $Y^a$, $Y^b$, $Y^c$, and $Z$. \begin{pr} \label{edimW} Every component of $W({\mathcal{E}})$ is reduced of dimension $$ (n+1) d - \sum_{m,e} (n+m-e-2)h_{m,e} - \sum_e (n-1-e)i_e. $$ The general element of each component is (a map from) a smooth curve. If $\sum_{k=1}^l {\mathcal{E}}(k) = {\mathcal{E}}$, then every component of $Y^a = Y^a({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))$ (respectively $Y^b$, $Y^c$) is reduced of dimension $\dim W({\mathcal{E}})-1$. \end{pr} \noindent {\em Proof. } We will prove the result about $\dim W({\mathcal{E}})$ in the special case $\vec{i} = \vec{0}$ and $h_{m,e}=0$ when $e<n-1$. Then the result holds when $\vec{i} = i_n \vec{\epsilon}_n$ by Proposition \ref{euniversal} (applied $i_n$ times), and we can invoke Proposition \ref{egeneral2} repeatedly to obtain the result in full generality. (As in the previous section, this type of reduction will be used often.) In this special case, we must prove that each component of $W({\mathcal{E}})$ is reduced of dimension $$ (n+1) d - \sum_m (m-1) h_{m,n-1}. $$ Consider any point $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ on $W({\mathcal{E}})$ where no component maps to $H$ and $\pi$ collapses no component of arithmetic genus 1. The natural map ${\mathcal{W}}({\mathcal{E}}) \dashrightarrow {\mathcal{W}}_1(\hat{\ce})$ induced by $\rho_A: {\overline{\cm}}_{1,\sum h_{m,n-1}}(\mathbb P^n,d) \dashrightarrow {\overline{\cm}}_{1,\sum h_{m,n-1}}(\mathbb P^1,d)$ is smooth of relative dimension $(n-1)d$ at the point $(C, \{ p^j_{m,e} \}, \{ q^j_e \}, \pi)$ by Proposition \ref{ebig}. The stack ${\mathcal{W}}_1(\hat{\ce})$ is reduced of dimension $2d+1 - \sum(m-1) h_{m,n-1}$ by Subsection \ref{ikey}, so ${\mathcal{W}}({\mathcal{E}})$ is reduced of dimension $$ (n-1)d + \dim ( {\mathcal{W}}_1(\hat{\ce})) = (n+1)d - \sum_m (m-1) h_{m,n-1} $$ as desired. As the general element of ${\mathcal{W}}_1(\hat{\ce})$ is (a map from) an irreducible curve, the same is true of ${\mathcal{W}}({\mathcal{E}})$, and thus $W({\mathcal{E}})$. The same argument works for $Y^a$, $Y^b$, and $Y^c$, as in Subsection \ref{ikey}, it was shown that $Y^a_1(\hat{\ce})$, $Y^b_1(\hat{\ce})$, and $\dim Y^c_1(\hat{\ce})$ are reduced divisors of $W_1(\hat{\ce})$. \qed \vspace{+10pt} \begin{pr} \label{edimZ} Every component of $Z_n(d,\vec{i})_{\mathcal{D}}$ is reduced of dimension $$ (n+1)d - \sum_e (n-1-e) i_e - 1. $$ \end{pr} \noindent {\em Proof. } It suffices to prove the result for the generically degree $d!$ cover $Z'_n(d,\vec{i})_{\mathcal{D}}$ obtained by marking the points of intersection with a fixed general hyperplane $H$. This is a subvariety of $W(d,d\vec{\epsilon}_{1,n-1},\vec{i})$, and as $$ \dim W_n(d,d\vec{\epsilon}_{1,n-1},\vec{i}) = (n+1)d - \sum_e (n-1-e) i_e, $$ we wish to show that $Z'_n(d,\vec{i})_{\mathcal{D}}$ is a reduced Weil divisor of the variety $W_n(d,d\vec{\epsilon}_{1,n-1},\vec{i})$. By Proposition \ref{ezgeneral2}, we may assume that $i_e = 0$ unless $e=n$. By relabeling if necessary, assume $q^{i_n}_n$ appears in ${\mathcal{D}}$ with non-zero coefficient $\alpha$ (so ${\mathcal{D}} - \alpha q^{i_n}_n$ is a sum of integer multiples of $q^1_n$, \dots, $q^{i_n-1}_n$). Let ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i} - \vec{\epsilon}_n)^o$ be the open subset of ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i} - \vec{\epsilon}_n)$ representing maps from smooth elliptic curves. On the universal curve over ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i} - \vec{\epsilon}_n)^o$ there is a reduced divisor ${\mathcal{Z}}$ corresponding to points $q$ such that $$ \alpha q = ({\mathcal{D}} - \alpha q^{i_n}_n) - \pi^*({\mathcal{O}}(1)) $$ in the Picard group of the fiber. The universal curve over the stack ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i} - \vec{\epsilon}_n)$ is ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i})$ by Proposition \ref{euniversal}, so by definition the closure of ${\mathcal{Z}}$ in ${\mathcal{W}}(d_1, d \vec{\epsilon}_{1,n-1}, \vec{i})$ is ${\mathcal{Z}}(d,\vec{i})_{{\mathcal{D}}}$. \qed \vspace{+10pt} We will need to avoid the locus on $W({\mathcal{E}})$ where an elliptic component is contracted. Lemma \ref{etancondition} identifies which such stable maps could lie in $W({\mathcal{E}})$. It is likely that every stable map of the form described in the lemma can be smoothed, which would suggest (via a dimension estimate) that when $k \leq n+1$ those maps with a collapsed elliptic component intersecting $k$ noncontracted components (with linearly dependent images of tangent vectors) form a Weil divisor of $W({\mathcal{E}})$. Because of the moduli of ${\overline{M}}_{1,k}$, none of these divisors would be enumeratively meaningful. Thus the following result is not surprising. \begin{pr} \label{ecodim2} If $W'$ is an irreducible subvariety of $W({\mathcal{E}})$ whose general map has a contracted elliptic component (or more generally a contracted connected union of components of arithmetic genus 1) and $W'$ is of codimension 1, then $W'$ is not enumeratively meaningful. \end{pr} \noindent {\em Proof. } By Proposition \ref{egeneral2}, we may assume $i_e = 0$ unless $e=n$, and $h_{m,e} = 0$ unless $e=n-1$. We could proceed naively by using the previous lemma and simply counting dimensions, but the following argument is slightly cleaner. Let $(C, \{ p^j_{m,n-1} \}, \{ q^j_n \}, \pi)$ be a general point of $W'$, and let $E$ be the contracted component of $C$. Say $E$ has $s$ special points (markings or intersections with noncontracted components) including $k$ intersections with noncontracted components. Replace $E$ by a rational $R=\mathbb P^1$, with the $s$ special points distinct, to obtain a new stable map $(C', \{ p^j_{m,n-1} \}, \{ q^j_n \}, \pi') \in X({\mathcal{E}})$. The family of such $(C',\pi')$ forms a subvariety $X'$ of $X({\mathcal{E}})$, and $X'$ is contained in $X''$ where in the latter we don't impose the dependence of tangent vectors required by the previous lemma. Let $\xi$ be the natural rational map to the Hilbert scheme of Subsubsection \ref{rgeomean}. If $s \geq 2$, $X''$ is codimension at least 1 in $X({\mathcal{E}})$. Due to the moduli of $s$ points on $R$, $\xi(X'')$ is codimension at least $1 + (s-3) = s-2$ in $\xi(X({\mathcal{E}}))$. The previous lemma imposes an additional $\max(n+1-k,0)$ conditions, which are independent as the rational curves intersecting $R$ can move freely under automorphisms of $\mathbb P^n$ preserving $H$. Thus the codimension of $\xi(X')$ in $\xi(X({\mathcal{E}}))$ is at least $n-1+(s-k) \geq n-1$. But $\dim X({\mathcal{E}}) - \dim W({\mathcal{E}}) = n-3$, so $\dim \xi(W') < \dim W({\mathcal{E}})-2 = \dim W'$, as desired Otherwise, $k=s=1$. By Proposition \ref{euniversal}, we may assume that $\vec{i} = \vec{0}$ as there are no marked points on the contracted component $E$. Then $X''$ can be identified with the subvariety of $X(d, \vec{h}, \vec{\epsilon}_n)$ where the corresponding map $\pi: (C, \{ p^j_{m,n-1} \}, q^1_n) \rightarrow \mathbb P^n$ is singular at $q^1_n$. As the singularity condition imposes $n$ conditions, \begin{eqnarray*} \dim \xi(W') & \leq & \dim X(d, \vec{h}, \vec{\epsilon}_n) - n \\ &=& \dim X({\mathcal{E}})+1-n \\ &=& (\dim W({\mathcal{E}}) + n - 3 ) + 1-n \\ &=& \dim W({\mathcal{E}}) - 2 \\ &=& \dim W' - 1 \end{eqnarray*} as desired. \qed \vspace{+10pt} \subsection{Degenerations set-theoretically} The theorem listing the possible degenerations follows the same pattern as the corresponding results (Theorems \ref{rlist1} and \ref{rlist}) of the previous section. Fix ${\mathcal{E}}$ and a non-negative integer $E$, and let $\Gamma$ and $\Delta$ be sets of general linear spaces of $\mathbb P^n$ (as in the definition of $W(d,\Gamma,\Delta)$). Let $q$ be the marked point corresponding to one of the (general) $E$-planes $Q$ in $\Delta$. Let $D_H = \{ \pi(q) \in H \}$ be the divisor on $W(d, \Gamma,\Delta)$ that corresponds to requiring $q$ to lie on $H$. In this section, we will determine the enumeratively meaningful components of $D_H$. That is, we will give a list of subvarieties, and show that the enumeratively meaningful components of $D_H$ are a subset of this list. In the subsequent section, we will determine the multiplicity with which each enumeratively meaningful component appears. In particular, we will see that the multiplicity of each component on the list is at least one, so each element of the list is indeed a component of $D_H$. As before, we can relate this result to the enumerative problem we wish to solve. If $W({\mathcal{E}}^-)$ is a union of points, we can determine $\# W({\mathcal{E}}^-)$ by specializing one of the linear spaces of $\Delta$, of dimension $E-1$, to the hyperplane $H$. Define ${\mathcal{E}}$ by $(d,\vec{h},\vec{i}) = (d^-,\vec{h^-},\vec{i^-} + \vec{\epsilon}_E - \vec{\epsilon}_{E-1})$. To calculate $\# W({\mathcal{E}}^-)$, we simply enumerate the points $D_H$ on $W({\mathcal{E}})$, with the appropriate multiplicity. Let $\phi_1$ be the isomorphism of Proposition \ref{eWY}. \begin{tm} \label{elist} If $\Gamma$ and $\Delta$ are general, each enumeratively meaningful component of $D_H$ (as a divisor on $W(d,\Gamma,\Delta)$) is one of the following. \begin{enumerate} \item[(I)] A component of $\phi_1(W(d',\Gamma', \Delta'))$, where, for some $m_0, e_0$, $1 \leq j_0 \leq h_{m_0, e_0}$, $e' := e_0 + E - n \geq 0$: \begin{itemize} \item $(d',\vec{h'}, \vec{i'}) = (d,\vec{h} - \vec{\epsilon}_{m_0, e_0} + \vec{\epsilon}_{m_0, e'}, \vec{i} - \vec{\epsilon}_E )$ \item ${\Gamma'}^j_{m,e} = \Gamma^j_{m,e} \quad \text{if $(m,e) \neq (m_0, e_0)$}$ \item $ \{ {\Gamma'}^j_{m_0, e_0} \}_j = \{ \Gamma_{m_0,e_0}^j \}_j \setminus \{ \Gamma^{j_0}_{m_0,e_0} \}$ \item ${\Gamma'}_{m_0, e'}^{h'_{m_0, e'}} = \Gamma^{j_0}_{m_0,e_0} \cap Q$ \item ${\Delta'}^j_e = \Delta^j_e \quad \text{if $e \neq E$}$, $\{ {\Delta'}_E^j \}_j = \{ \Delta^j_E \}_j \setminus \{ Q \}$. \end{itemize} \item[(II)] A component of $Y^a(d(0),\dots, \Delta(l))$, $Y^b(d(0),\dots,\Delta(l))$, or $Y^c(d(0),\dots,\Delta(l))_{m^1_1}$ for some $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$, with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \cup_{k=0}^l \Gamma(k)$, $\Delta = \cup_{k=0}^l \Delta(k)$, $Q \in \Gamma(0)$, and $d(0)>0$. \end{enumerate} \end{tm} Call the components of (I) Type I components, and call the three types of components of (II) Type IIa, IIb, and IIc components respectively. \noindent {\em Proof. } We follow the proofs of Theorems \ref{rlist1} and \ref{rlist}. By Propositions \ref{euniversal} and \ref{egeneral2}, we may assume that $\vec{i} = \vec{\epsilon}_n$, $E=n$, and $h_{m,e} = 0$ unless $e=n-1$. With these assumptions, the result becomes much simpler. The stack ${\mathcal{W}}(d,\vec{h},\vec{i})$ is the universal curve over ${\mathcal{W}}(d,\vec{h},\vec{0})$, and we are asking which points of the universal curve lie in $\pi^{-1}H$. Let $(C, \{ p^j_{m,n-1} \}, q, \pi)$ be the map corresponding to a general point of a enumeratively meaningful component $Y$ of $D_H$. By Proposition \ref{ecodim2}, the morphism $\pi$ doesn't contract any component of $C$ of arithmetic genus 1. Choose a general $(n-2)$-plane $A$ in $H$. The set $\pi^{-1} A$ is a union of reduced points on $C$, so $\rho_A$ is smooth (as a morphism of stacks) at $(C, \{ p^j_{m,n-1} \}, q, \pi)$ by Proposition \ref{ebig}. As a set, $D_H$ contains the entire fiber of $\rho_A$ above $\rho_A(C,\pi)$, so $\rho_A(D_H)$ is a Weil divisor on $W_1(\hat{\ce})$ that is a component of $\{\pi(q) = z\}$. By Theorem \ref{igenus1}, the curve $C$ is a union of irreducible components $C(0) \cup \dots \cup C(l')$ with $\rho_A \circ \pi(C(0)) = z$ (i.e. $C(0) \subset \pi^{-1}H$), $C(0)\cap C(k) \neq \phi$, and the marked points split up among the components: $\vec{h} = \sum_{k=0}^{l'} \vec{h}(k)$. If $d(0) = \deg \pi |_{C(0)}$, then $d(0)$ of the curves $C(1)$, \dots, $C(l')$ are rational tails that are collapsed to the $d(0)$ points of $C(0) \cap A$; they contain no marked points. Let $l = l' - d(0)$. Also, $\vec{i}(0) = \vec{\epsilon}_n$, and $\vec{i}(k) = \vec{0}$ for $k>0$, as the only incidence condition was $q \in Q$, and $q(0) \in C(0)$. {\em Case $d(0)>0$.} By Theorem \ref{igenus1}, the component $Y$ is contained in $$ Y^a(d, \dots, \Delta(l)), \quad Y^b (d(0), \dots, \Delta(l)), \quad \text{or} \quad Y^c (d(0), \dots, \Delta(l)). $$ As the dimensions of each of these three is $\dim W(d,\Gamma,\Delta) - 1 = \dim Y$, $Y$ must be a Type II component as described in the statement of the theorem. {\em Case $d(0)=0$.} As the morphism $\pi$ contracts no elliptic components, the curve $C(0)$ is rational. Also, $C(0)$ has at least 3 special points: $q$, one of $\{ p^j_{m,e} \}$ (call it $p^{j_0}_{m_0,e_0}$), and $C(0) \cap C(1)$. If $C(0)$ had more than 3 special points, then the component would not be enumeratively meaningful, due to the moduli of the special points of $C(0)$. Thus $l=1$, and $Y$ is a Type I component. \qed \vspace{+10pt} In fact, this argument determines {\em all} the components of $D_H$ except those representing maps with collapsed elliptic tails. (It is not clear whether such components exist.) When $n=2$, the only enumeratively meaningful Type II divisors are Type IIa and Type IIb with $d(0)=1$. This agrees with the genus 1 case of [CH3]. \subsection{Multiplicity calculations} \label{emultgen} The proof of multiplicities of $D_H$ along the enumeratively meaningful components are the same as in the genus 0 case (Subsection \ref{rmultgen}). The Type I component $\phi_1(W(d',\Gamma',\Delta'))$ appears with multiplicity $m_0$, where $m_0$ was defined in Theorem \ref{elist}. (The argument of Subsubsection \ref{rmultI} also proves this.) The Type IIa component $Y^a({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))$ appears with multiplicity $\prod_{k=1}^l m^k$, where $m^k = d(k) - \sum_{m,e,j} m h^j_{m,e}(k)$ as defined earlier. The Type IIc component $Y^c({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))$ appears with multiplicity $\prod_{k=1}^l m^k$ as well. The Type IIb component $Y^b({\mathcal{E}}(0);\dots;{\mathcal{E}}(l))_{m^1_1}$ appears with multiplicity $$ m^1_1 m^1_2 \prod_{k=2}^l m^k = m^1_1 (m^1- m^1_1) \prod_{k=2}^l m^k . $$ By Corollary \ref{ilocalst} of Section \ref{intro}, we also get the same results about the structure of $W({\mathcal{E}})$ in a formal, \'{e}tale, or analytic neighborhood of these components. \subsection{Recursive formulas} We now enumerate the points of our varieties when the number is finite. The only initial data needed is the ``enumerative geometry of $\mathbb P^1$'': the number of stable maps to $\mathbb P^1$ of degree 1 is 1. \subsubsection{A recursive formula for $\# W$} Given ${\mathcal{E}}$, fix an $E$ such that $i_E>0$. Partitions of ${\mathcal{E}}$ are simultaneous partitions of $d$, $\vec{h}$, and $\vec{i}$. Define multinomial coefficients with vector arguments as the product of the multinomial coefficients of the components of the vectors. Define ${\mathcal{E}}^-$ by $(d^-, \vec{h^-}, \vec{i^-}) = ( d, \vec{h}, \vec{i} - \vec{\epsilon}_E + \vec{\epsilon}_{E-1})$, and let $\Gamma^- = \Gamma$ and $\Delta^- = \Delta \cup \{ \Delta^{i_E}_E \cap H' \} \setminus \{ \Delta^{i_E}_E \}$ where $H'$ is a general hyperplane. The following theorem is an analog of Theorem \ref{rrecursiveX1}. \begin{tm} \label{erecursiveW1} In $A^1 (W(d,\Gamma, \Delta))$, modulo enumeratively meaningful divisors, $$ W_n(d^-, \Gamma^-, \Delta^-) = \sum m_0 \cdot W_n(d'(m_0,e_0,j_0), \Gamma', \Delta') $$ $$ + \sum \left( \prod_{k=1}^l m^k \right) Y^a_n(d(0), \dots, \Delta(l) ) $$ $$ + \sum \left( \prod_{k=2}^l m^k \right) \left( \sum_{m^1_1 = 1}^{ [ m^1/2]} m^1_1 (m^1 - m^1_1) Y^b_n(d(0), \dots, \Delta(l))_{m^1_1} \right) $$ $$ + \sum \left( \prod_{k=1}^l m^k \right) Y^c_n(d(0), \dots, \Delta(l) ) $$ where the first sum is over all $(m_0,e_0,j_0)$, and $(d', \Gamma', \Delta')$ is as defined in Theorem \ref{elist}; and the last three sums are over all $l$, ${\mathcal{E}}(0)$, \dots, $\Delta(l)$ with ${\mathcal{E}} = \sum_{k=0}^l {\mathcal{E}}(k)$, $\Gamma = \coprod_{k=0}^l \Gamma(k)$, $\Delta = \coprod_{k=0}^l \Delta(k)$, $\Gamma_E^{i_E} \in \Gamma(0)$, and $d(0)>0$. \end{tm} \noindent {\em Proof. } The left side is linearly equivalent (in ${\mathcal{W}}(d,\Gamma,\Delta)$) to $D_H = \{ \pi(q^{i_E}_E) \in H \}$. The right side is set-theoretically $D_H$ by Theorem \ref{elist}, and the multiplicities were determined in Subsection \ref{emultgen}. \qed \vspace{+10pt} If $\# W_n({\mathcal{E}}^-)$ is finite, the following statement is more suitable for computation. It is an analog of Theorem \ref{rrecursiveX2}. \begin{tm} \label{erecursiveW} { \scriptsize $$ \# W_n({\mathcal{E}}^-) = \sum_{m,e} h_{m,e} \cdot m \cdot \# W_n({\mathcal{E}}'(m,e)) $$ $$ + \sum \left( \prod_{k=1}^l m^k \right) { \binom {\vec{h}} { \vec{h}(0), \dots, \vec{h}(l)} } { \binom {\vec{i} - \vec{\epsilon}_E} { \vec{i}(0)-\vec{\epsilon}_E,\vec{i}(1), \dots, \vec{i}(l)} } \frac { \# Y^a_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) }{ \operatorname{Aut}( {\mathcal{E}}(2),\dots,{\mathcal{E}}(l))} $$ $$ + \frac 1 2 \sum \left( \prod_{k=2}^l m^k \right){ \binom {\vec{h}} { \vec{h}(0), \dots, \vec{h}(l)} } { \binom {\vec{i} - \vec{\epsilon}_E} {\vec{i}(0)-\vec{\epsilon}_E,\vec{i}(1), \dots, \vec{i}(l)} } $$ $$ \quad \quad \quad \cdot \left( \sum_{m^1_1 = 1}^{ m^1-1} m^1_1 (m^1 - m^1_1) \frac { \# \tilde{Y}^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1} }{ {\operatorname{Aut}( {\mathcal{E}}(2),\dots,{\mathcal{E}}(l))}} \right) $$ $$ + \sum \left( \prod_{k=1}^l m^k \right){ \binom {\vec{h}} { \vec{h}(0), \dots, \vec{h}(l)} } { \binom {\vec{i} - \vec{\epsilon}_E} { \vec{i}(0)-\vec{\epsilon}_E,\vec{i}(1), \dots, \vec{i}(l)} } \frac { \# Y^c_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) }{ \operatorname{Aut}( {\mathcal{E}}(1),\dots,{\mathcal{E}}(l))} $$ } where, in the first sum, ${\mathcal{E}}'(m,e) = (d, \vec{h} -\vec{\epsilon}_{m,e}+\vec{\epsilon}_{m,e+E-n}, \vec{i} - \vec{\epsilon}_E)$; the last three sums are over all $l$ and all partitions ${\mathcal{E}}(0)$, \dots, ${\mathcal{E}}(l)$ of ${\mathcal{E}}$ with $d(0)>0$. \end{tm} \noindent {\em Proof. } When $\#W_n({\mathcal{E}}^-)$ is finite, all components of $D_H$ are enumeratively meaningful. Take degrees of both sides of the equation in Theorem \ref{erecursiveW1}. As $$ \# \tilde{Y}^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1} = \begin{cases} 2 (\# Y^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1}) & \text{if $2 m^1_1 = m^1$,} \\ \# Y^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1} & \text{otherwise,} \end{cases} $$ it follows that \begin{eqnarray*} & & \sum_{m_1^1 = 1}^{[m^1/2]} m_1^1 ( m^1 - m^1_1) \# Y^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1} \\ & & \quad \quad \quad \quad \quad \quad = \frac 1 2 \sum_{m_1^1 = 1}^{m^1-1} m_1^1 ( m^1 - m^1_1) \# \tilde{Y}^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1}. \end{eqnarray*} The only additional points requiring explanation are the combinatorial aspects: the $h_{m,e}$ in the first sum, and the ``$\operatorname{Aut}$'' and various multinomial coefficients in the last three. In Theorem \ref{elist}, the Type I components were indexed by $(m_0, e_0, j_0)$. But for fixed $(m_0,e_0)$, $\# W({\mathcal{E}}'(m_0,e_0))$ is independent of $j_0$, so the above formula eliminates this redundancy. Similarly, in Theorem \ref{rlist}, the Type IIa, IIb, and IIc components were indexed by partitions of the points $\{ p^j_{m,e} \}_{m,e,j}$ and $\{ q^j_e \}_{e,j} \setminus \{ q \}$, but the values of $\# Y^a_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$, $\# Y^b_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$, and $\# Y^c_n({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))$ depend only on $\{ \vec{h}(k), \vec{i}(k) \}_{k=0}^l$ and not on the actual partitions of marked points. The multinomial coefficients eliminate this redundancy. We divide by $\operatorname{Aut}({\mathcal{E}}(1); \dots; {\mathcal{E}}(l))$ (resp. $\operatorname{Aut}({\mathcal{E}}(2); \dots; {\mathcal{E}}(l))$, $\operatorname{Aut}({\mathcal{E}}(1); \dots; {\mathcal{E}}(l))$) to ensure that we are counting each Type IIa (resp. Type IIb, Type IIc) component once. \qed \vspace{+10pt} Theorem \ref{erecursiveW1} can be strengthened to be true modulo those divisors whose general map has a collapsed elliptic component. There is a similar statement in the Chow ring of the Hilbert scheme modulo exceptional divisors of $\phi_1$ (cf. Theorem \ref{rrecHilb}). \subsubsection{A recursive formula for $Y^a$} \label{eYarec} In the previous section, we found a formula for $\# Y$ in terms of $\# X$ (Proposition \ref{rrecursiveY}). By the same argument, we have $$ \# Y^a_n( {\mathcal{E}}(0);\dots;{\mathcal{E}}(l)) = \frac { \# X_{n-1}({\mathcal{E}}'(0)) }{ d(0)!} \cdot \# W_n({\mathcal{E}}'(1)) \cdot \prod_{k=2}^l \# X_n({\mathcal{E}}'(k)) $$ where \begin{itemize} \item $d'(0) = d(0)$, $\vec{h'}(0) = d(0) \vec{\epsilon}_{1,n-2}$ \item $\vec{i'}(0) = \sum_e i_{e+1}(0) \vec{\epsilon}_e + \vec{\epsilon}_{\dim W_n({\mathcal{E}}(1))} + \sum_{k=2}^l \vec{\epsilon}_{\dim X_n({\mathcal{E}}(k))} + \sum_{m,e} h_{m,e}(0) \vec{\epsilon}_e$ \item $\vec{h'}(1) = \vec{h}(1) + \vec{\epsilon}_{m^1, n-1-\dim W_n({\mathcal{E}}(1))}$ and $\vec{i'}(1) = \vec{i}(1)$. \item For $2 \leq k \leq l$, $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k, n-1-\dim X_n({\mathcal{E}}(k))}$ and $\vec{i'}(k) = \vec{i}(k)$. \end{itemize} \subsubsection{Computing $\# Y^b$ and $\# \tilde{Y}^b$} The difficulty in computing $\# Y^b$ and $\# \tilde{Y}^b$ comes from requiring the curves $C(0)$ and $C(1)$ to intersect twice (at marked points of each curve, on $H$), so a natural object of study is the blow-up of $H \times H$ along the diagonal $\Delta$, $\operatorname{Bl}_\Delta H \times H$. But when $n=2$, the situation is simpler. The curve $C(0)$ is $H$, and $C(1)$ will always intersect it. In this case, for $C(0) \cup \dots \cup C(l)$ to be determined by the incidence conditions (up to a finite number of possibilities), each of $C(1)$, \dots, $C(l)$ must also be determined (up to a finite number). The analogous formula to that for $\# Y^a$ (and $\# Y$ in the previous section) is $$ \# \tilde{Y}^b_2({\mathcal{E}}(0); \dots; {\mathcal{E}}(l))_{m^1_1} = \prod_{k=2}^l \# X_2(d(k), \vec{h'}(k), \vec{i}(k)) $$ where $\vec{h'}(1) = \vec{h}(1) + \vec{\epsilon}_{m^1_1,1} + \vec{\epsilon}_{m^1_2,1}$, and for $2 \leq k \leq l$, $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k,1}$. This is in agreement with Theorem 1.3 of [CH3]. We now calculate $\# Y^b$ and $\# \tilde{Y}^b$ when $n=3$; the same method will clearly work for $n>3$. As an illustration of the method, consider the following enumerative problem. {\em Fix seven general lines $L_1$, \dots, $L_7$ in $\mathbb P^3$ and a point $p$ on a hyperplane $H$. How many pairs of curves $(C(0),C(1))$ are there with $C(0)$ a line in $H$ through $p$ and $C(1)$ a conic intersecting $L_1$, \dots, $L_7$ and intersecting $C(0)$ at two distinct points, where the intersections are labeled $a_1$ and $a_2$?} The answer to this enumerative problem is $$ \# Y^b = \# Y^b_3(1, \vec{\epsilon}_{1,0},\vec{0}; 2, \vec{0}, 7 \vec{\epsilon}_1); $$ we will calculate instead $\# \tilde{Y}^b = 2 ( \# Y^b )$. The space of lines in $H$ passing through $p$ is one-dimensional, and this defines a three-dimensional locus in $\operatorname{Bl}_\Delta H \times H$ (which is the Fulton-Macpherson configuration space of 2 points in $\mathbb P^3$; alternately, it is a degree 2 \'{e}tale cover of $\operatorname{Sym}^2 H$). The space of conics in $\mathbb P^3$ passing through 7 general lines is one-dimensional, and thus defines a one-dimensional locus in $\operatorname{Bl}_\Delta H \times H$ parametrizing the points of intersection of the conic with $H$. Then $\# \tilde{Y}^b$ is the intersection of these two classes, and $\# Y^b$ (and the answer to the enumerative problem) is half this. Let $h_i$ be the class (in the Chow group) of the hyperplane on the $i^{th}$ factor of $\operatorname{Bl}_{\Delta} H \times H$ ($i = 1,2$), and let $e$ be the class of the exceptional divisor. Then the Chow ring of $\operatorname{Bl}_\Delta H \times H$ is generated (as a ${\mathbb Z}$-module) by the classes listed below with the relations $$ h_1^3 = 0, \quad h_2^3 = 0, \quad e^2 = 3 h_1 e - h_1^2 - h_1 h_2 - h_2^2. $$ \begin{center} \begin{tabular}{c|c} Codimension & Classes \\ \hline 0 & 1 \\ 1 & $h_1$, $h_2$, $e$ \\ 2 & $h_1^2$, $h_1 h_2$, $h_2^2$, $h_1e = h_2e$ \\ 3 & $h_1^2 h_2$, $h_1 h_2^2$, $h_1^2 e = h_1 h_2 e = h_2^2 e$ \\ 4 & $h_1^2 h_2^2$\\ \end{tabular} \end{center} Let the image of possible pairs of points on $C(0)$ be the class ${\mathcal{C}}(0) = \alpha (h_1 + h_2) + \beta e$. Then ${\mathcal{C}}(0) \cdot h_1^2 h_2 = \alpha$ and ${\mathcal{C}}(0) \cdot e h_1^2 = - \beta$. But ${\mathcal{C}}(0) \cdot h_1^2 h_2$ is the number of lines in $H$ passing through $p$ (class ${\mathcal{C}}(0)$) and another fixed point (class $h_1^2$) with a marked point on a fixed general line (class $h_2$), so $\alpha =1$. Also, ${\mathcal{C}}(0) \cdot e h_1^2$ is the number of lines in the plane through $p$ (class ${\mathcal{C}}(0)$) and another fixed point $q$ (class $h_1^2$) with a marked point mapping to $q$, so $\beta = -1$. Thus ${\mathcal{C}}(0) = h_1 + h_2 - e$. Let the image of possible pairs of points on $C(1) \cap H$ be the class ${\mathcal{C}}(1) = \alpha ( h_1^2 h_2 + h_1 h_2^2) + \beta h_1 h_2 e$. Thus ${\mathcal{C}}(1) \cdot h_1 = \alpha$ and ${\mathcal{C}}(1) \cdot e = - \beta$. Then ${\mathcal{C}}(1) \cdot h_1$ is the number of conics in $\mathbb P^3$ through 7 general lines in $\mathbb P^3$ and a general line in $H$, which is 92 from Subsection \ref{r92subsection} of the previous section. Also, ${\mathcal{C}}(1) \cdot e$ counts the number of conics in $\mathbb P^3$ through 7 general lines in $\mathbb P^3$ and tangent to $H$, which is 116 using the methods of the previous section. Thus ${\mathcal{C}}(1) = 92 (h_1^2 h_2 + h_1 h_2^2) - 116 e h_1 h_2$. Finally, the number of pairs of curves is $$ (h_1 + h_2 - e) \left( 92 (h_1^2 h_2 + h_1 h_2^2 \right) - 116 e h_1 h_2) = 92 + 92 - 116 = 68. $$ Thus $\# \tilde{Y}^b$ is 68, and $\# Y^b$ (and the answer to the enumerative problem) is 34. When $n=3$ in general, there are three cases to consider. Let $C=C(0) \cup C(1) \cup \dots \cup C(l)$ as usual. {\em Case i).} If the incidence conditions ${\mathcal{E}}(1)$ specify $C(1)$ up to a finite number of possibilities, then: $$ \# \tilde{Y}^b ({\mathcal{E}}(0); {\mathcal{E}}(1); \dots; {\mathcal{E}}(l)) \quad \quad \quad $$ $$ \quad \quad \quad = \# X (d(1),\vec{h'}(1),\vec{i}(1)) \cdot \# Y (d(0),\vec{h'}(0),\vec{i}(0); {\mathcal{E}}(2);\dots;{\mathcal{E}}(l)) $$ where \begin{itemize} \item $\vec{h'}(1) = \vec{h}(1) + \vec{\epsilon}_{m^1_1,2} + \vec{\epsilon}_{m^1_2,2}$ (the curve $C(1)$ intersects $H$ at two points $a_1$ and $a_2$ with multiplicity $m^1_1$ and $m^1_2$ respectively; these will be the intersections with $C(0)$) \item $\vec{h'}(0) = \vec{h}(0) + 2 \vec{\epsilon}_{1,0}$ (the curve $C(0)$ must pass through two points of intersection $a_1$ and $a_2$ of $C(1)$ with $H$). \end{itemize} {\em Case ii).} If the incidence conditions ${\mathcal{E}}(1)$ specify $C(1)$ up to a one-parameter family, we are in the same situation as in the enumerative problem above. Then $$ \# \tilde{Y}^b ({\mathcal{E}}(0); {\mathcal{E}}(1); \dots; {\mathcal{E}}(l)) = \quad \quad \quad $$ $$ \Bigl( d(0) \left( \# X (d(1),\vec{h'},\vec{i}(1)) + d(0) \# X (d(1),\vec{h''},\vec{i}(1)) \right) $$ $$ \quad \quad \quad \quad - \# X (d(1),\vec{h'''},\vec{i}(1)) \Bigr) \cdot \# Y (d(0),\vec{h'}(0),\vec{i}(0); {\mathcal{E}}(2);\dots;{\mathcal{E}}(l)) $$ where \begin{itemize} \item $\vec{h'} = \vec{h}(1) + \vec{\epsilon}_{m^1_1,1} + \vec{\epsilon}_{m^1_2,2}$ (the curve $C(1)$ intersects $H$ with multiplicity $m^1_1$ at $a_1$ along a fixed general line and with multiplicity $m^1_2$ at $a_2$ at another point of $H$), \item $\vec{h''} = \vec{h}(1) + \vec{\epsilon}_{m^1_2,1} + \vec{\epsilon}{m^1_1,2}$ (the curve $C(1)$ intersects $H$ with multiplicity $m^1_2$ at $a_2$ along a fixed general line and with multiplicity $m^1_1$ at $a_1$ at another point of $H$), \item $\vec{h'''} = \vec{h}(1) + \vec{\epsilon}_{m^1,2}$ (the points $a_1$ and $a_2$ on the curve $C(1)$ coincide, and $C(1)$ is required to intersect $H$ at this point with multiplicity $m^1 = m^1_1 + m^1_2$), \item $\vec{h'}(0) = \vec{h}(0) + \vec{\epsilon}_{1,0}$ (the curve $C(0)$ is additionally required to pass through a fixed point in $H$). \end{itemize} {\em Case iii).} If the incidence conditions on $C(0) \cup C(2) \cup \dots \cup C(l)$ specify the union of these curves up to a finite number of possibilities (and the incidence conditions on $C(1)$ specify $C(1)$ up to a two-parameter family), a similar argument gives $$ \# \tilde{Y}^b ({\mathcal{E}}(0); {\mathcal{E}}(1); \dots; {\mathcal{E}}(l)) = \quad \quad \quad $$ $$ \left( d(0) \# X (d(1), \vec{h'}(1),\vec{i}(1)) - \# X (d(1), \vec{h''}(1), \vec{i}(1)) \right) $$ $$ \quad \quad \quad \cdot \# Y ({\mathcal{E}}(0); {\mathcal{E}}(2);\dots;{\mathcal{E}}(l)) $$ where \begin{itemize} \item $\vec{h'}(1) = \vec{h}(1) + \vec{\epsilon}_{m^1_1,1} + \vec{\epsilon}_{m^1_2,1}$ (the curve $C(1)$ must intersect $H$ along two fixed general lines at the points $a_1$ and $a_2$ with multiplicity $m^1_1$ and $m^1_2$ respectively) \item $\vec{h''}(1) = \vec{h}(1) + \vec{\epsilon}_{m^1,1}$ (the points $a_1$ and $a_2$ coincide on the curve $C(1)$, and $C(1)$ is required to intersect $H$ at that point with multiplicity $m^1 = m^1_1 + m^1_2$ along a fixed general line of $H$). \end{itemize} These three cases are illustrated pictorially in Figure \ref{eYb} for the special case of conics in $\mathbb P^3$ intersecting a line in $H$ at two points, with the entire configuration required to intersect 8 general lines in $\mathbb P^3$. One of the intersection points of the conic with $H$ is marked with an ``$\times$'' to remind the reader of the marking $a_1$. The distribution of the line conditions (e.g. the number of line conditions on the conic) is indicated by a small number. The bigger number beside each picture is the actual solution to the enumerative problem corresponding to the picture. For example, there are 116 conics in $\mathbb P^3$ tangent to a general hyperplane $H$ intersecting 7 general lines. \begin{figure} \begin{center} \getfig{eYb}{.07} \end{center} \caption{Calculating $\# \tilde{Y}^b$: A pictorial example} \label{eYb} \end{figure} \subsubsection{A recursive formula for $Y^c$} By the same method as in Subsubsection \ref{eYarec} for $Y^a$ (and $Y$ in the previous section), we have $$ \# Y^c({\mathcal{E}}(0); \dots; {\mathcal{E}}(l)) = \quad \quad \quad \quad $$ $$ \# Z_{n-1} (d(0), \vec{i'}(0) )_{\sum_k m^k {q''}^k - \sum_{m,e,j} m {q'}^j_{m,e}} \cdot \prod_{k=1}^l \# X_n(d(k), \vec{h'}(k), \vec{i}(k)) $$ where \begin{itemize} \item For $1 \leq k \leq l$, $\vec{h'}(k) = \vec{h}(k) + \vec{\epsilon}_{m^k, n-1-\dim X({\mathcal{E}}(k))}$. \item $i'_e(0) = i_{e+1}(0) + \# \{ \dim X({\mathcal{E}}(k)) = e \}_{1 \leq k \leq l} + \sum_m h_{m,e}(0)$ \item the marked points on the curves of $Z_{n-1}(d(0),\vec{i'}(0))$ have been relabeled $$ \{ {q'}^1_{m,e}, \dots, {q'}^{h_{m,e}(0)}_{m,e} \}_{m,e} \cup \{ {q''}^1, \dots, {q''}^l \} \cup \{ {q'''}^1_e, \dots, {q'''}^{i_{e+1}(0)}_e \}_e $$ where ${q'}^j_{m,e} = p^j_{m,e}(0)$, ${q''}^k = C(0) \cap C(k)$, ${q'''}^j_e = q^j_{e+1}(0)$. \end{itemize} This formula is merely a restatement of the divisorial condition Y${}^{\text{c}}$5 in Definition \ref{eYcdef}. \subsubsection{Evaluating $\#Z$} \label{eevalZ} We will use intersection theory on elliptic fibrations over a curve --- the Chow ring modulo algebraic equivalence or numerical equivalence will suffice. Let ${\mathcal{F}}$ be an elliptic fibration over a smooth curve whose fibers are smooth elliptic curves, except for a finite number of fibers that are irreducible nodal elliptic curves. Let $F$ be the class of a fiber. Then the self-intersection of a section is independent of the choice of section. ({\em Proof:} $K_{\mathcal{F}}$ restricted to the generic fiber is trivial, so $K_{\mathcal{F}}$ is a sum of fibers. Let $S_1$, $S_2$ be two sections. Using adjunction, $S_1^2 + K_{\mathcal{F}} \cdot S_1 = (K_{\mathcal{F}} + S_1) \cdot S_1 = 0$, so $S_1^2 = - K_{\mathcal{F}} \cdot S_1 = - K_{\mathcal{F}} \cdot S_2 = S_2^2$.) For convenience, call the self-intersection of a section $S^2$. The parenthetical proof above shows that $K_{\mathcal{F}} = - S^2 F$. \begin{pr} Let $S$ be a section, and $C$ a class on ${\mathcal{F}}$ such that $S=C$ on the general fiber. Then $S = C + (\frac { S^2 - C^2 }{ 2})F$. \end{pr} \noindent {\em Proof. } As all fibers are irreducible, $S = C + kF$ for some $k$. By adjunction, \begin{eqnarray*} 0 &=& S \cdot (K_{\mathcal{F}} + S) \\ &=& (C + kF) (C + (k - S^2)F) \\ &=& C^2 + 2k-S^2. \end{eqnarray*} Hence $k = (S^2 - C^2)/2$. \qed \vspace{+10pt} If the dimension of $Z_n(d,\vec{i})_{\sum m^j_e q^j_e}$ is 0, then consider the universal family over the curve parametrizing maps to $\mathbb P^n$ with the incidence conditions of $\vec{i}$. This is $W_n(d, d \vec{\epsilon}_{1,n-1},\vec{i})$ modulo the symmetric group $S_d$. The general point of $W_n(d, d \vec{\epsilon}_{1,n-1},\vec{i})$ represents a smooth elliptic curve. The remaining points of $W_n(d, d \vec{\epsilon}_{1,n-1},\vec{i})$ represent curves that are either irreducible and rational or elliptic with rational tails. (This can be proved by simple dimension counts on $W_1(d, d \vec{\epsilon}_{1,0}, \vec{0})$.) Normalize the base (which will normalize the family), and blow down (-1)-curves in fibers. The curves blown down come from maps from nodal curves $C(0) \cup C(1)$, where $C(0)$ is rational and $C(1)$ is elliptic. Call the resulting family ${\mathcal{F}}$. Let $H$ be the pullback of a hyperplane to ${\mathcal{F}}$, and let $Q^j_e$ be the section given by $q^j_e$. \begin{tm} Let $D = H - \sum m^j_e Q^j_e$. Then $$ \# Z_n(d,\vec{i})_{\sum m^j_e q^j_e} = S^2 - D^2 / 2. $$ \end{tm} \noindent {\em Proof. } Let $Q$ be any section. Let $S$ be the section given by $Q + \pi^*({\mathcal{O}}(1)) - \sum m^j_e q^j_e$ in the Picard group of the generic fiber. Then $$ \# Z_n (d, \vec{i})_{\sum m^j_e q^j_e} = S \cdot Q. $$ (The sections $S$ and $Q$ intersect transversely from Subsubsection \ref{esubZ}.) By the previous proposition, as $S^2 = Q^2$, \begin{eqnarray*} S &=& Q + D + \left( \frac {S^2 - (Q+D)^2 }{ 2} \right) F \\ \text{so } S \cdot Q &=& \left(Q + D + \left( \frac {S^2 - (Q+D)^2 }{ 2} \right)F \right) \cdot Q \\ &=& Q^2 + D \cdot Q + \frac {S^2 - Q^2 - D^2 }{ 2} - D \cdot Q \\ &=& S^2 - D^2 / 2. \end{eqnarray*} \qed \vspace{+10pt} To calculate $$ \#Z_n(d,\vec{i})_{\sum m^j_e q^j_e} = S^2 - (H - \sum m^j_e Q^j_e)^2 / 2, $$ we need to calculate $H^2$, $H \cdot Q^j_e$, and $Q^j_e \cdot Q^{j'}_{e'}$, and these correspond to simpler enumerative problems. If $(e,j) \neq (e',j')$, $Q^j_e$ could intersect $Q^{j'}_{e'}$ in two ways. If $e' + e \geq n$, the elliptic curve could pass through $\Delta^j_e \cap \Delta^{j'}_{e'}$, which will happen $\# W ({\mathcal{E}}') / d!$ times (where $d' = d$, $\vec{h'} = d \vec{\epsilon}_{1,n-1}$, $\vec{i'} = \vec{i} - \vec{\epsilon}_e - \vec{\epsilon}_{e'} + \vec{\epsilon}_{e+e'-n}$). Or the curve could break into two intersecting components, one rational (call it $R$) containing $Q^j_e$ and $Q^{j'}_{e'}$ (which will be blown down in the construction of ${\mathcal{F}}$), and the other (call it $E$) smooth elliptic. This will happen $$ \sum_{\substack{{d(0) + d(1) = d }\\{ \vec{i}(0) + \vec{i}(1) = \vec{i}}}} (d(0) d(1))^{\delta_{n,2}} \binom { \vec{i} - \vec{\epsilon}_e - \vec{\epsilon}_{e'}} {\vec{i}(1)} \left( \frac {\# X({\mathcal{E}}(0)) }{ d(0)!} \right) \left( \frac {\# W({\mathcal{E}}(1)) }{ d(1)!} \right) $$ times where $h_{n-1}(0) = d(0)$, $h_{n-1}(1) = d(1)$. The factor of $(d(0) d(1))^{\delta_{n,2}}$ corresponds to the fact when $n=2$, $\pi(C) = \pi(R \cup E)$ is a plane curve, and the point $R \cap E$ could map to any node of the plane curve $\pi(R \cup E)$. Transversality in both cases is simple to check, and both possibilities are of the right dimension. Thus $$ Q^j_e \cdot Q^{j'}_{e'} = \frac {\# W({\mathcal{E}}') }{ d!} \quad \quad \quad \quad \quad \quad $$ $$ + \sum_{ \substack{{ d(0) + d(1) = d }\\ { \vec{i}(0) + \vec{i}(1) = \vec{i}}}} (d(0) d(1))^{\delta_{n,2}} \binom { \vec{i} - \vec{\epsilon}_e - \vec{\epsilon}_{e'} }{ \vec{i}(1)} \left( \frac {\# X({\mathcal{E}}(0))}{ d(0)!} \right) \left(\frac {\# W({\mathcal{E}}(1)) }{d(1)!} \right). $$ To determine $H \cdot Q^j_e$, fix a general hyperplane $h$ in $\mathbb P^n$, and let $H$ be its pullback to the fibration ${\mathcal{F}}$. Then $H$ is a multisection of the elliptic fibration. The cycle $H$ could intersect $Q^j_e$ in two ways. Either $\pi(q^j_e) \in h \cap \Delta^j_e$ --- which will happen $\# W({\mathcal{E}}') / d!$ times with $(d', \vec{h'}, \vec{i'}) = (d, d \vec{\epsilon}_{1,n-1}, \vec{i} - \vec{\epsilon}_e + \vec{\epsilon}_{e-1})$ --- or the curve breaks into two pieces, one rational containing a point of $h$ and $p^j_e$, which will happen $$ \sum_{ \substack{{ d(0) + d(1) = d }\\{ \vec{i}(0) + \vec{i}(1) = \vec{i}}}} (d(0) d(1))^{\delta_{n,2}} d(0) \binom { \vec{i} - \vec{\epsilon}_e }{ \vec{i}(1)} \left( \frac{ \# X({\mathcal{E}}(0))}{d(0)!} \right) \left(\frac {\# W({\mathcal{E}}(1)) }{ d(1)!} \right) $$ times where $h_{1,n-1}(0) = d(0)$, $h_{1,n-1}(1) = d(1)$. (The second $d(0)$ in the formula comes from the choice of point of $h$ on the degree $d(0)$ rational component.) Thus $$ H \cdot Q^j_e = \frac { \# W({\mathcal{E}}') }{ d! } \quad \quad \quad \quad \quad \quad $$ $$ + \sum_{ \substack {{d(0) + d(1) = d }\\{ \vec{i}(0) + \vec{i}(1) = \vec{i}}}} (d(0)d(1))^{\delta_{n,2}} d(0) \binom { \vec{i} - \vec{\epsilon}_e }{\vec{i}(1)} \left( \frac {\# X({\mathcal{E}}(0))}{ d(0)!} \right) \left(\frac {\# W({\mathcal{E}}(1)) }{ d(1)!} \right). $$ To determine $H^2$, fix a second general hyperplane $h'$ in $\mathbb P^n$, and let $H'$ be its pullback to ${\mathcal{F}}$. Once again, $H$ could intersect $H'$ in two ways depending on if the curve passes through $h \cap h'$, or if the curve breaks into two pieces. Thus $$ H^2 = \frac {\# W ({\mathcal{E}}') }{ d! } \quad \quad \quad \quad \quad \quad $$ $$ + \sum_{ \substack{{ d(0) + d(1) = d }\\{ \vec{i}(0) + \vec{i}(1) = \vec{i}}}} (d(0)d(1))^{\delta_{n,2}} d(0)^2 \binom { \vec{i} }{ \vec{i}(1)} \left( \frac {\# X({\mathcal{E}}(0))}{ d(0)!} \right) \left( \frac {\# W({\mathcal{E}}(1)) }{ d(1)!} \right) $$ where $\vec{h'} = d \vec{\epsilon}_{1,n-1}$, $\vec{i'} = \vec{i} + \vec{\epsilon}_{n-2}$, $\vec{h}(0) = d(0) \vec{\epsilon}_{1,n-1}$, $\vec{h}(1) = d(1) \vec{\epsilon}_{1,n-1}$. The self-intersection of a section $S^2$ ($= (Q^j_e)^2$) can be calculated as follows. Fix $e$ such that $i_e>0$. We can calculate $H \cdot Q^1_e$, so if we can evaluate $(H - Q^1_e) \cdot Q^1_e$ then we can find $S^2 = (Q^1_e)^2$. Fix a general hyperplane $h$ containing $\Delta^1_e$, and let $(H-Q^1_e)$ be the multisection that is the pullback of $h$ to ${\mathcal{F}}$, minus the section $Q^1_e$. The cycle $(H-Q^1_e)$ intersects $Q^1_e$ if the curve is tangent to $h$ along $\Delta^1_e$ or if the curve breaks into two pieces, with $Q^1_e$ on the rational piece. Thus $$ (H-Q^1_e) \cdot Q^1_e = \frac { \# W({\mathcal{E}}') }{ (d-2)! } \quad \quad \quad \quad \quad \quad $$ $$ + \sum_{ \substack{{ d(0) + d(1) = d }\\{ \vec{i}(0) + \vec{i}(1) = \vec{i} - \vec{\epsilon}_e}}} (d(0)d(1))^{\delta_{n,2}} \binom { \vec{i} - \vec{\epsilon}_e }{ \vec{i}(1)} \left( \frac { \# X({\mathcal{E}}(0)) }{ ( d(0)-2)!} \right) \left(\frac{\# W({\mathcal{E}}(1))}{ d(1)!} \right) $$ where $\vec{h'} = (d-2) \vec{\epsilon}_{1,n-1} + \vec{\epsilon}_{2,e}$, $\vec{h}(0) = ( d(0) - 1) \vec{\epsilon}_{1,n-1} + \vec{\epsilon}_{1,e}$, $\vec{h}(1) = d(1) \vec{\epsilon}_{1,n-1}$. The denominator $(d(0)-2)!$ arises because we have a degree $d(0)$ (rational) curve passing through an $e$-plane on $h$, and various incidence conditions $\vec{i}(0)$. The number of such curves with a choice of one of the other intersections of $C(0)$ with $h$ is $(d(0)-1) \# X({\mathcal{E}}(0)) / (d(0) - 1)!$. As an example, consider the elliptic quartics in $\mathbb P^2$ passing through 11 fixed points, including $q^1_0$, $q^2_0$, $q^3_0$, $q^4_0$. How many such two-nodal quartics have ${\mathcal{O}}(1) = q^1_0 + \dots + q^4_0$ in the Picard group of the normalization of the curve? We construct the fibration ${\mathcal{F}}$ over the (normalized) variety of two-nodal plane quartics through 11 fixed points. We have sections $Q^1_0$, \dots, $Q^{11}_0$ and a multisection $H$. If $i \neq j$, $Q^i_0 \cdot Q^j_0 = 3$, $H \cdot Q^j_0 = 30$, $H ^2 = 225 + 3 {\binom {11} 2} = 390$, $(H-Q^1_0) \cdot Q^1_0 = 185$, so $$ S^2 = H \cdot Q^1_0 - (H - Q^1_0) \cdot Q^1_0 = -155. $$ Let $D = H - Q^1_0 - \dots - Q^4_0$. Then \begin{eqnarray*} D^2 &=& H^2 + 4 S^2 - 8 H \cdot Q^1_0 + 12 Q^1_0 \cdot Q^2_0 \\ &=& 390 + 4 (-155) - 8 (30) + 12(3) \\ &=& -434 \end{eqnarray*} so the answer is $S^2 - D^2/2 = 62$. To determine the enumerative geometry of quartic elliptic space curves (see Subsubsection \ref{eqesc}), various $\# Z_2(d,\vec{i})_{\mathcal{D}}$ were needed with $d=3$ and $d=4$. When $d=3$, $i_0$ is necessarily 8, and the results are given in the Table \ref{ez3}. For convenience, we write $p^j = q^j_0$ for the base points and $l^j = q^j_1$ for the marked points on lines. These values were independently confirmed by M. Roth ([Ro]). When $d=4$, $i_0$ must be 11, and the results are given in the Table \ref{ez4}. For convenience again, we write $p^j = q^j_0$ and $l^j = q^j_1$. \begin{table} \begin{center} \begin{tabular}{c|c|c} $i_1$ & ${\mathcal{D}}$ & $\# Z_2(d,\vec{i})_{\mathcal{D}}$ \\ \hline 0 & $p^1 + p^2 + p^3$ & 0 \\ 1 & $p^1 + p^2 + l^1$ & 1 \\ 2 & $p^1 + l^1 + l^2$ & 5 \\ 3 & $l^1 + l^2 + l^3$ & 18 \\ 0 & $p^1 + 2 p^2 $ & 1 \\ 1 & $2p^1 + l^1 $ & 4 \\ 1 & $p^1 + 2l^1 $ & 5 \\ 2 & $l^1 + 2l^2 $ & 16 \\ 0 & $3p^1 $ & 3 \\ 1 & $3l^1 $ & 14 \\ 0 & $p^1 + p^2 + p^3 + p^4 - p^5 $& 1 \\ 1 & $p^1 + p^2 + p^3 + p^4 - l^1 $& 2 \\ 1 & $p^1 + p^2 + p^3 + l^1 - p^4 $& 4 \\ 2 & $p^1 + p^2 + p^3 + l^1 - l^2 $& 10 \\ 2 & $p^1 + p^2 + l^1 + l^2 - p^3 $& 14 \\ 3 & $p^1 + p^2 + l^1 + l^2 - l^3 $& 39 \\ 3 & $p^1 + l^1 + l^2 + l^3 - p^2 $& 45 \\ 4 & $p^1 + l^1 + l^2 + l^3 - l^4 $& 135 \\ 4 & $l^1 + l^2 + l^3 + l^4 - p^1 $& 135 \\ 5 & $l^1 + l^2 + l^3 + l^4 - l^5 $& 432 \end{tabular} \end{center} \caption{Counting cubic elliptic plane curves with a divisorial condition} \label{ez3} \end{table} \begin{table} \begin{center} \begin{tabular}{c|c|c} $i_1$ & ${\mathcal{D}}$ & $\# Z_2(d,h,\vec{i})_{\mathcal{D}}$ \\ \hline 0 & $p^1 + p^2 + p^3 + p^4 $ & 62 \\ 1 & $p^1 + p^2 + p^3 + l^1 $ & 464 \\ 2 & $p^1 + p^2 + l^1 + l^2 $ & 2,522 \\ 3 & $p^1 + l^1 + l^2 + l^3 $ & 11,960 \\ 4 & $l^1 + l^2 + l^3 + l^4 $ & 52,160 \end{tabular} \end{center} \caption{Counting quartic elliptic plane curves with a divisorial condition} \label{ez4} \end{table} \subsection{Examples} \subsubsection{Plane curves} Type IIc components in this case are never enumeratively meaningful, as the elliptic curve $C(0)$ must map to the line $H$ with degree at least two. The recursive formulas we get are identical to the genus 1 recursive formulas of Caporaso and Harris in [CH3]. \subsubsection{Cubic elliptic space curves} The number of smooth cubic elliptic space curves through $j$ general points and $12-2j$ general lines is 1500, 150, 14, and 1 for $j= 0$, 1, 2, and 3 respectively. (The number is 0 for $j>3$ as cubic elliptic space curves must lie in a plane.) The degenerations involved in calculating the first case appeared in Subsubsection \ref{ecubics}. As the Chow ring of the space of smooth elliptic cubics is not hard to calculate (see [H1], p. 36), these results may be easily verified. The number of cubic elliptics tangent to $H$, through $j$ general points and $11-2j$ general lines is 4740, 498, 50, and 4 for $j=0$, 1, 2, and 3 respectively. The number of cubic elliptics triply tangent to $H$ through $j$ general points and $10-2j$ general lines is 2790, 306, 33, and 3 for $j=0$, 1, 2, and 3 respectively. These numbers are needed for the next examples. \begin{table} \begin{center} \begin{tabular}{c|r} $j$ & \# quartics \\ \hline 0 & 52,832,040 \\ 1 & 4,436,208 \\ 2 & 385,656 \\ 3 & 34,674 \\ 4 & 3,220 \\ 5 & 310 \\ 6 & 32 \\ 7 & 4 \\ 8 & 1 \end{tabular} \end{center} \caption{Number of quartic elliptic space curves through $j$ general points and $16-2j$ general lines} \label{eqescnums} \end{table} \begin{table} \begin{center} \begin{tabular}{r|c|c|c|r} &$(i_0,i_1,h_0,h_1)$ & point & line & \# curves\\ & & degen. & degen. & \\ \hline 1 & (16,0,0,0) & & 10 & 52,832,040 \\ 2 & (14,1,0,0) & 40 & 11 & 4,436,268 \\ 3 & (12,2,0,0) & 41 & 12 & 385,656 \\ 4 & (10,3,0,0) & 42 & 13 & 34,674 \\ 5 & (8,4,0,0) & 43 & 14 & 3,220 \\ 6 & (6,5,0,0) & 44 & 15 & 310 \\ 7 & (4,6,0,0) & 45 & 16 & 32 \\ 8 & (1,7,0,0) & 46 & 17 & 4 \\ 9 & (0,8,0,0) & 47 & & 1 \\ 10 & (15,0,1,0) & & 18, 40 & 52,832,040 \\ 11 & (13,1,1,0) & 48 & 19, 41 & 4,436,268 \\ 12 & (11,2,1,0) & 49 & 20, 42 & 385,656 \\ 13 & (9,3,1,0) & 50 & 21, 43 & 34,674 \\ 14 & (7,4,1,0) & 51 & 22, 44 & 3,220 \\ 15 & (5,5,1,0) & 52 & 23, 45 & 310 \\ 16 & (3,6,1,0) & 53 & 24, 46 & 32 \\ 17 & (1,7,1,0) & 54 & 25, 47 & 4 \\ 18 & (14,0,2,0) & & 26, 48+ & 48,395,772 \\ 19 & (12,1,2,0) & 55+ & 27, 49+ & 4,050,612 \\ 20 & (10,2,2,0) & 56+ & 28, 50+ & 350,982 \\ 21 & (8,3,2,0) & 57+ & 29, 51+ &31,454 \\ 22 & (6,4,2,0) & 58+ & 30, 52 & 2,910 \\ 23 & (4,5,2,0) & 59 & 31, 53 & 278 \\ 24 & (2,6,2,0) & 60 & 32, 54 & 28 \\ 25 & (0,7,2,0) & 61 & & 3 \\ 26 & (13,0,3,0) & & 33, 55+ & 39,347,736 \\ 27 & (11,1,3,0) & 62+ & 34, 56+ & 3,266,100 \\ 28 & (9,2,3,0) & 63+ & 35, 57+ & 280,752 \\ 29 & (7,3,3,0) & 64+ & 36, 58+ & 24,972 \\ 30 & (5,4,3,0) & 65+ & 37, 59+ & 2,290 \\ 31 & (3,5,3,0) & 66+ & 38, 60+ & 214 \\ 32 & (1,6,3,0) & 67+ & 39, 61+ & 20 \\ 33 & (12,0,4,0) & & 62+ & 23,962,326 \\ 34 & (10,1,4,0) & + & 63+ & 1,939,857 \end{tabular} \end{center} \caption{Quartic elliptic space curves with incidence conditions} \label{eqesc1} \end{table} \begin{table} \begin{center} \begin{tabular}{r|c|c|c|r} &$(i_0,i_1,h_0,h_1)$ & point & line & \# curves\\ & & degen. & degen. & \\ \hline 35 & (8,2,4,0) & + & 64+ & 161,735 \\ 36 & (6,3,4,0) & + & 65+ & 13,908 \\ 37 & (4,4,4,0) & + & 66+ & 1,222 \\ 38 & (2,5,4,0) & + & 67+ & 104 \\ 39 & (0,6,4,0) & + & & 8 \\ 40 & (14,0,0,1) & & 48+ & 4,436,268 \\ 41 & (12,1,0,1) & 68 & 49 & 385,656 \\ 42 & (10,2,0,1) & 69 & 50 & 34,674 \\ 43 & (8,3,0,1) & 70 & 51 & 3,220 \\ 44 & (6,4,0,1) & 71 & 52 & 310 \\ 45 & (3,5,0,1) & 72 & 53 & 32 \\ 46 & (2,6,0,1) & 73 & 54 & 4 \\ 47 & (0,7,0,1) & 74 & & 1 \\ 48 & (13,0,1,1) & & 55, 68+ & 4,436,268 \\ 49 & (11,1,1,1) & 75+ & 56, 69+ & 385,656 \\ 50 & (9,2,1,1) & 76+ & 57, 70+ & 34,674 \\ 51 & (7,3,1,1) & 77+ & 58, 71+ & 3,220 \\ 52 & (5,4,1,1) & 78+ & 59, 72 & 310 \\ 53 & (3,5,1,1) & 79 & 60, 73 & 32 \\ 54 & (1,6,1,1) & 80 & 61, 74 & 4 \\ 55 & (12,0,2,1) & & 62, 75 & 4,028,112 \\ 56 & (10,1,2,1) & 81+ & 63, 76+ & 349,032 \\ 57 & (8,2,2,1) & 82+ & 64, 77+ & 28,340 \\ 58 & (6,3,2,1) & 83+ & 65, 78+ & 2,901 \\ 59 & (4,4,2,1) & 84+ & 66, 79+ & 278 \\ 60 & (2,5,2,1) & 85+ & 67, 80+ & 28 \\ 61 & (0,6,2,1) & 86+ & & 3 \\ 62 & (11,0,3,1) & & 81+ & 2,849,436 \\ 63 & (9,1,3,1) & + & 82+ & 243,507 \\ 64 & (7,2,3,1) & + & 83+ & 21,310 \\ 65 & (5,3,3,1) & + & 84+ & 1,909 \\ 66 & (3,4,3,1) & + & 85+ & 172 \\ 67 & (1,5,3,1) & + & 86+ & 14 \\ 68 & (12,0,0,2) & & 75+ & 385,656 \end{tabular} \end{center} \caption{Quartic elliptic space curves with incidence conditions, cont'd} \end{table} \begin{table} \begin{center} \begin{tabular}{r|c|c|c|r} &$(i_0,i_1,h_0,h_1)$ & point & line & \# curves\\ & & degen. & degen. & \\ \hline 69 & (10,1,0,2) & 87 & 76+ & 34,674 \\ 70 & (8,2,0,2) & 88 & 77+ & 3,220 \\ 71 & (6,3,0,2) & 89 & 78+ & 310 \\ 72 & (4,4,0,2) & 90 & 79 & 32 \\ 73 & (2,5,0,2) & 91 & 80 & 4 \\ 74 & (0,6,0,2) & 92 & & 1 \\ 75 & (11,0,1,2) & & 81, 87+ & 384,156 \\ 76 & (9,1,1,2) & 93+ & 82, 88+ & 34,524 \\ 77 & (7,2,1,2) & 94+ & 83, 89+ & 3,206 \\ 78 & (5,3,1,2) & 95+ & 84, 90+ & 309 \\ 79 & (3,4,1,2) & 96 & 85, 91+ & 32 \\ 80 & (1,5,1,2) & 97 & 86, 92+ & 4 \\ 81 & (10,0,2,2) & & 93+ & 312,348 \\ 82 & (8,1,2,2) & + & 94+ & 28,340 \\ 83 & (6,2,2,2) & + & 95+ & 2,612 \\ 84 & (4,3,2,2) & + & 96+ & 246 \\ 85 & (2,4,2,2) & + & 97+ & 24 \\ 86 & (0,5,2,2) & + & & 2 \\ 87 & (10,0,0,3) & & 93+ & 34,674 \\ 88 & (8,1,0,3) & 98+ & 94+ & 3,220 \\ 89 & (6,2,0,3) & 99+ & 95+ & 310 \\ 90 & (4,3,0,3) & 100+ & 96 & 32 \\ 91 & (2,4,0,3) & 101 & 97 & 4 \\ 92 & (0,5,0,3) & 102 & & 1 \\ 93 & (9,0,1,3) & & 98+ & 31,056 \\ 94 & (7,1,1,3) & + & 99+ & 3,052 \\ 95 & (5,2,1,3) & + & 100+ & 304 \\ 96 & (3,3,1,3) & + & 101+ & 32 \\ 97 & (1,4,1,3) & + & 102+ & 4 \\ 98 & (8,0,0,4) & & + &2,519 \\ 99 & (6,1,0,4) & + & + & 277 \\ 100 & (4,2,0,4) & + & + & 31 \\ 101 & (2,3,0,4) & + & + & 4 \\ 102 & (0,4,0,4) & + & & 1 \end{tabular} \end{center} \caption{Quartic elliptic space curves with incidence conditions, cont'd} \label{eqesc3} \end{table} \subsubsection{Quartic elliptic space curves} \label{eqesc} The number of smooth quartic elliptic space curves through $j$ general points and $16-2j$ general lines is given in the Table \ref{eqescnums}. These numbers agree with those recently found by Getzler by means of genus 1 Gromov-Witten invariants (cf. [G3]). \begin{figure} \begin{center} \getfig{qesc1}{.1} \caption{Counting quartic elliptic space curves through 16 general lines} \label{qesc1} \end{center} \end{figure} The space of smooth quartic elliptic space curves is birational to pencils in the space of space quadrics (as a quartic elliptic space curve is the base locus of a unique pencil, and a general pencil defines a smooth quartic elliptic). By this means the last four numbers in Table \ref{eqescnums} may be easily calculated. D. Avritzer and I. Vainsencher used this method (cf. [AV]) to calculate the top number, although they likely misprinted their answer ([G5]). \begin{figure} \begin{center} \getfig{qesc2}{.1} \caption{Additional degenerations of quartic elliptic space curves} \label{qesc2} \end{center} \end{figure} Other enumerative data can also be found. For example, Tables \ref{eqesc1} to \ref{eqesc3} give the number of smooth quartic elliptic space curves through $i_0$ general points and $i_1$ general lines, and $h_0$ general points and $h_1$ general lines in $H$, with $$ 2 i_0 + i_1 + 2 h_0 + h_1= 16. $$ At each stage, the number may be computed by degenerating a point or a line (assuming there is a point or line to degenerate). Each row is labeled, and the labels of the different degenerations that are also smooth quartics are given in each case, and a ``+'' is added if there are other degenerations. (This will help the reader to follow through the degenerations.) Keep in mind that these numbers are not quite what the algorithm of this section produces; in the algorithm, the intersections with $H$ are labeled, so the number computed for $(i_0,i_1,h_0,h_1)$ will be $(4-h_0-h_1)!$ times the number in the table. These computations are not as difficult as one might think. For example, if $i_0$ and $i_1$ are both positive, it is possible to degenerate a point and then a line, or a line and then a point. Both methods must yield the same number, providing a means of double-checking. \begin{figure} \begin{center} \getfig{qesc3}{.1} \caption{Additional degenerations of quartic elliptic space curves, cont'd} \label{qesc3} \end{center} \end{figure} As an example, the degenerations used to compute the 52,832,040 quartic space curves through 16 general lines are displayed in Figures \ref{qesc1} to \ref{qesc4}, using the pictorial shorthand described earlier. In Figure \ref{qesc1}, degenerations involving nondegenerate quartic elliptic space curves are given (as well as a few more). The remaining degenerations are given in Figures \ref{qesc2} to \ref{qesc4}. The boldfaced numbers indicated the corresponding rows in Tables \ref{eqesc1} to \ref{eqesc3}. \subsection{Curves of higher genus} \label{ehighgenus} The genus 2 case seems potentially tractable. An analog of Proposition \ref{ecodim2} is needed, showing that in the space of stable maps of the desired sort (with general assigned incidences and intersections with $H$), no divisor representing maps with collapsed components of positive genus is enumeratively meaningful. New types of components arise, including one in which $C(0)$ and $C(1)$ are both rational, and intersect each other in 3 points (which will require the intersection theory of a blow-up of $H^3$), and one in which $C(0)$, $C(1)$, and $C(2)$ are rational and $C(0)$ intersects $C(i)$ ($i = 1,2$) in 2 points (which will require the intersection theory of $(\operatorname{Bl}_\Delta H^2)^2$). The main difficulty will arise from components where $p_a(C(0)) = 2$, as the divisorial condition analogous to that of Type IIc components is a codimension 2 condition, and the calculation of $\# Z$ (when $p_a(C(0))=1$) using elliptic fibrations now involves fibrations of abelian surfaces. \begin{figure} \begin{center} \getfig{qesc4}{.1} \caption{Additional degenerations of quartic elliptic space curves, cont'd} \label{qesc4} \end{center} \end{figure} For genus greater than 2, the situation is more grave. The map $$ \rho_A: {\overline{M}}_{g,m}(\mathbb P^n,d) \dashrightarrow {\overline{M}}_{g,m}(\mathbb P^1,d) $$ induced by projection from a general $(n-2)$-plane $A$ in $H$ is not dominant, so the multiplicity calculations are no longer immediate from the situation on ${\overline{\cm}}_{g,m}(\mathbb P^1,d)$. Second, the divisorial condition is even more complicated than for genus 2, and the other computations (involving the intersection theory of products of repeated blow-ups of powers of $H$ along various diagonals) will be horrendous. It is also awkward that the dimensions of these spaces may not be what one would naively expect. For example, the space of genus 3 quartic space curves is of dimension 17 (as all genus 3 quartic space curves must lie in a plane, so the dimension is $\dim \mathbb P^{3*}$ plus the dimension of the space of plane quartics), not 16. (This is because the normal bundle of the general such map has non-zero $H^1$.) But all is not necessarily lost. Even in the case of genus 3 quartic space curves we can successfully follow through the degenerations (see Figure \ref{egenus3}; genus 3 curves are indicated by 3 open circles). The unexpectedly high dimension of the space is compensated by unexpectedly high dimensions of degenerations. For example, one would naively expect that requiring two curves to intersect in three points would impose three conditions, but when one of the curves is cubic elliptic (and hence planar) and the other is a line, the cost is only two conditions (as a line intersecting the cubic at two points necessarily intersects it at a third). This should work in general when $d$ is small enough (for fixed $g$) that the curve must be planar, and the degenerations will look very much like those of [CH3] and Figure \ref{egenus3}. Perhaps when $d$ is small enough that $C(0)$ must be genus 0 or 1 this analysis can still be carried through. \begin{figure} \begin{center} \getfig{egenus3}{.1} \end{center} \caption{Genus 3 quartic space curves.} \label{egenus3} \end{figure}

9709

alg-geom/9709002

en

https://arxiv.org/abs/alg-geom/9709002

alg-geom/9709002

Vicente Munoz Velazquez

Vicente Mu\~noz

Wall-crossing formulae for algebraic surfaces with $q>0$

Latex2e, 20 pages

We extend the ideas of Friedman and Qin (Flips of moduli spaces and transition formulae for Donaldson polynomial invariants of rational surfaces) to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with geometrical genus zero, positive irregularity and anticanonical divisor effective, for any wall $\zeta$ with $l_{\zeta}=(\zeta\sp{2}-p_1)/4$ being zero or one.

alg-geom

\section{Introduction} \label{sec:intro} The Donaldson invariants of a smooth oriented $4$-manifold $X$ depend by definition on a Riemannian metric $g$. In the case $b^+>1$ they however turn out to be independent of $g$. When $b^+=1$, they depend on $g$ through a structure of walls and chambers, that we recall briefly here (we refer to~\cite{Kotschick1}~\cite{KM2} for more details). Fix $w \in H^2(X;{\Bbb Z})$. Then for any $p_1 \leq 0$ with $p_1 \equiv w^2 \pmod 4$, we set $d=-p_1-{3 \over 2}(1-b_1 +b^+)$, for half of the dimension of the moduli space ${\cal M}_{X,g}^{w,d}$ of $g$-antiselfdual connections on the $SO(3)$-principle bundle over $X$ with second Stiefel-Whitney class the reduction mod $2$ of $w$, and first Pontrjagin number $p_1$. The corresponding Donaldson invariant will be denoted $D_{X,g}^{w,d}$. This is a linear functional on the elements of degree $2d$ of $\AA(X)=\text{Sym}^*(H_0(X) \oplus H_2(X)) \otimes \bigwedge^*(H_1(X)\oplus H_3(X))$, where the degree of elements in $H_i(X)$ is $4-i$ ($H_i(X)$ will always denote homology with rational coefficients, and similarly for $H^i(X)$). This invariant is only defined in principle for generic metrics. From now on let $X$ be a compact smooth oriented $4$-manifold with $b^+=1$. Let ${\Bbb H}$ be the image of the positive cone $\{ x \in H^2(X; {\Bbb R})/ x^2>0 \}$ in ${\Bbb P}(H^2(X; {\Bbb R}))$, which is a model of the hyperbolic disc of dimension $b^-$. The period point of $g$ is the line $\o_g \in {\Bbb H} \subset {\Bbb P}(H^2(X; {\Bbb R}))$ given by the selfdual harmonic forms for $g$. A {\bf wall} of type $(w,p_1)$ is a non-empty hyperplane $W_{\zeta}=\{ x \in {\Bbb H} / x\cdot \zeta =0 \}$ in ${\Bbb H}$, with $\zeta \in H^2(X; {\Bbb Z})$, such that $\zeta \equiv w\pmod 2$ and $p_1 \leq \zeta^2 < 0$. The connected components of the complement of the walls of type $(w,p_1)$ in ${\Bbb H}$ are the {\bf chambers} of type $(w,p_1)$. Let ${\frak M}$ denote the space of metrics of $X$. Then we have a map ${\frak M} \to {\Bbb H}$ which sends every metric $g$ to its period point $\o_g$. The connected components of the preimage of the chambers of ${\Bbb H}$ are, by definition, the chambers of ${\frak M}$. A wall $W'_{\zeta}$ for ${\frak M}$ is a non-empty preimage of a wall $W_{\zeta}$ for ${\Bbb H}$. When $g$ moves in a chamber ${\cal C}'$ of ${\frak M}$ the Donaldson invariants do not change. But when it crosses a wall they change. So for any chamber ${\cal C}'$ of ${\frak M}$, we have defined $D_{X,{\cal C}'}^{w,d}$, by choosing any generic metric $g \in {\cal C}'$, so that the moduli space is smooth, and computing the corresponding Donaldson invariants (to avoid flat connections we might have to use the trick in~\cite{MM}). For a path of metrics $\{ g_t \}_{t \in [-1,1]}$, with $g_{\pm 1} \in {\cal C}'_{\pm}$, we have the difference term $\d_{X}^{w,d}({\cal C}'_-,{\cal C}'_+)=D_{X,{\cal C}'_+}^{w,d}-D_{X,{\cal C}'_-}^{w,d}$. When $b_1=0$, Kotschick and Morgan~\cite{KM2} prove that the invariants only depend on the chamber ${\cal C}$ of ${\Bbb H}$ in which the period point of the metric lies. For this, they find that the change in the Donaldson invariant when the metric crosses a wall $W'_{\zeta}$ depends only on the class $\zeta$ and not on the particular metric having the reducible antiselfdual connection (Leness~\cite{Leness} points out that their argument is not complete and checks that it is true at least for the case $l_{\zeta}={1 \over 4} (\zeta^2-p_1) \leq 2$). In this case, the difference term is defined as $$ \d_{X}^{w,d}({\cal C}_-,{\cal C}_+)=D_{X,{\cal C}_+}^{w,d}-D_{X,{\cal C}_-}^{w,d}, $$ for chambers ${\cal C}_{\pm}$ of ${\Bbb H}$. Then $\d_{X}^{w,d}({\cal C}_-,{\cal C}_+)= \sum \d_{S,\zeta}^{w,d}$, where the sum is taken over all $\zeta$ defining walls separating ${\cal C}_-$ and ${\cal C}_+$. Moreover $\d_{S,\zeta}^{w,d}=\varepsilon(\zeta,w) \d_{S,\zeta}^d$, with $\d^d_{S,\zeta}$ not dependent on $w$, $\varepsilon(\zeta,w)=(-1)^{({\zeta-w \over 2})^2}$. Now suppose $S$ is a smooth algebraic surface (not necessarily with $b_1=0$), endowed with a Hodge metric $h$ corresponding to a polarisation $H$. Let ${\frak M}_H(c_1,c_2)$ be the Gieseker compactification of the moduli space of $H$-stable rank two bundles $V$ on $X$ with $c_1(V)={\cal O}(L)$ (a fixed line bundle with topological first Chern class equal to $w$) and $c_2={1 \over 4}(c_1^2 -p_1)$. The Donaldson invariants (for the metric $h$) can be computed using ${\frak M}_H(c_1,c_2)$ (see~\cite{FM}) whenever the moduli spaces ${\frak M}_H(c_1,c_2)$ are generic (i.e. $H^0(\text{End}_0E)=H^2(\text{End}_0E)=0$, for every stable bundle $E \in {\frak M}_H(c_1,c_2)$). The period point of $h$ is the line spanned by $H \in H^2(X;{\Bbb Z}) \subset H^2(X;{\Bbb R})$. Now let $C_S \subset {\Bbb H}$ be the image of the {\bf ample cone} of $S$, i.e. the subcone of the positive cone generated by the ample classes (polarisations). We have walls and chambers in $C_S$ in the same vein as before (actually they are the intersections of the walls and chambers of ${\Bbb H}$ with $C_S$, whenever this intersection is non-empty). Now ${\frak M}_H(c_1,c_2)$ is constant on the chambers of $C_S$ (and so the invariant stays the same), and when $H$ crosses a wall $W_{\zeta}$, ${\frak M}_H(c_1,c_2)$ changes (see~\cite{Qin}). From the point of view of the Donaldson invariants, this corresponds to restricting our attention from the positive cone of $S$ to its ample cone. When the irregularity $q$ of $S$ is zero, the wall-crossing terms have been found out in~\cite{flips}~\cite{Gottsche-notengo}~\cite{Gottsche-bott}. In~\cite{flips} Friedman and Qin obtain some wall-crossing formulae for algebraic surfaces $S$ with $-K$ being effective ($K=K_S$ the canonical divisor) and the irregularity $q=0$ (equivalently, $b_1=0$). We want to adapt their results to the case $q >0$ modifying their arguments where necessary. If $-K$ is effective then the change of ${\frak M}_H(c_1,c_2)$ when $H$ crosses a wall $W$ can be described by a number of flips. We shall write the change of the Donaldson invariant as a sum of contributions $\d_{S,\zeta}^{w,d}$, for the different $\zeta$ defining $W$. \begin{rem} \label{rem:kld} The condition of $-K$ being effective can be relaxed for the case $q=0$ to the following two conditions: $K$ is not effective, $\pm \zeta +K$ are not effective for any $\zeta$ defining the given wall (we call such a wall a {\bf good wall}, see~\cite{Gottsche-bott}~\cite{Gottsche-notengo}). Probably the same is true for the case $q>0$, since these two conditions ensure that the change in ${\frak M}_H(c_1,c_2)$ when crossing a wall is described by flips. Nonetheless we will suppose $-K$ effective, which allows us to define the Donaldson invariants for any polarisation. Note that when $-K$ is effective, all walls are good. \end{rem} The paper is organised as follows. In section~\ref{sec:2} we extend the arguments of~\cite{flips} to the case $q>0$. In sections~\ref{sec:lz=0} and~\ref{sec:lz=1} we compute the wall crossing formulae for any wall with $l_{\zeta}={1 \over 4}(\zeta^2-p_1)$ being $0$ and $1$ respectively. Then in section~\ref{sec:5}, we give the two leading terms of the wall crossing difference for any wall $\zeta$. As a consequence of our results, we propose a conjecture on the shape of the wall crossing terms. In the appendix we give, for the convenience of the reader, a list of all the algebraic surfaces with $p_g=0$ and $-K$ effective, i.e. the surfaces to which the results from this paper apply. \noindent {\em Acknowledgements:\/} I am very grateful to my D. Phil.\ supervisor Simon Donaldson, for many good ideas. Conversations with Lothar G\"ottsche have been very useful for a checking of the formulae here obtained. Also I am indebted to the Mathematics Department in Universidad de M\'alaga for their hospitatility and financial support. \noindent {\em Note:\/} After the completion of this work, L.\ G\"ottsche provided me with a copy of~\cite{Gottsche-notengo}. The arguments for computing the wall-crossing terms in~\cite{Gottsche-notengo} can also be extended to the case $q>0$, in a similar fashion to the work carried out in this paper. \section{Wall-crossing formulae} \label{sec:2} From now on, $S$ is a smooth algebraic manifold with irregularity $q \geq 0$ and $p_g=0$ (equivalently $b^+=1$) and with anticanonical divisor $-K$ effective. Let $w \in H^2(S;{\Bbb Z})$, $p_1 \equiv w^2 \pmod 4$. Put $$ d=-p_1- {3\over 2}(1-b_1+b^+)=-p_1-3(1 -q) $$ and let $\zeta$ define a wall of type $(w,p_1)$. In every chamber ${\cal C}$ of the ample cone, we have well-defined the Donaldson invariant $D^{w,d}_{S,{\cal C}}$ associated to polarisations in that chamber. For two different chambers ${\cal C}_+$ and ${\cal C}_-$, there is a {\bf wall-crossing difference term\/} $$ \d_S^{w,d}({\cal C}_-,{\cal C}_+)=D^{w,d}_{S,{\cal C}_+}- D^{w,d}_{S,{\cal C}_-}, $$ which can be written as a sum $$ \d_S^{w,d}({\cal C}_-,{\cal C}_+)=\sum_{\zeta} \d_{S,\zeta}^{w,d}, $$ where $\zeta$ runs over all walls of type $(w, p_1)$ with ${\cal C}_- \cdot \zeta <0 < {\cal C}_+ \cdot \zeta$. Suppose from now on that ${\cal C}_-$ and ${\cal C}_+$ are two adjacent chambers separated by a single wall $W_{\zeta}$ of type $(w,p_1)$. For simplicity, we will assume that the wall $W_{\zeta}$ is only represented by the pair $\pm \zeta$ since in the general case we only need to add up the contributions for every pair representing the wall. Then the wall-crossing term is $\d_{S,\zeta}^{w,d}$. Set $$ l_{\zeta}=(\zeta^2 -p_1)/4 \in {\Bbb Z}. $$ Let $\zeta$ define the wall separating ${\cal C}_-$ from ${\cal C}_+$ and put, as in~\cite[section 2]{flips}, $E_{\zeta}^{n_1,n_2}$ to be the set of all isomorphism classes of non-split extensions of the form $$ \exseq{{\cal O}(F) \otimes I_{Z_1}}{V}{{\cal O}(L-F) \otimes I_{Z_2}}, $$ where $F$ is a divisor such that $2F-L$ is homologically equivalent to $\zeta$, and $Z_1$ and $Z_2$ are two zero-dimensional subschemes of $S$ with $l(Z_i)=n_i$ and such that $n_1+n_2=l_{\zeta}$. Let us construct $E_{\zeta}^{n_1,n_2}$ explicitly. Consider $H_i =\text{Hilb}_{n_i}(S)$ the Hilbert scheme of $n_i$ points on $S$, $J=\text{Jac}^F(S)$ the Jacobian parametrising divisors homologically equivalent to $F$, ${\cal Z}_i \subset S \times H_i$ the universal codimension $2$ scheme, and ${\cal F} \subset S \times J$ the universal divisor. Then we define ${\cal E}_{\zeta}^{n_1,n_2} \to J\times H_1 \times H_2$ to be $$ {\cal E}={\cal E}_{\zeta}^{n_1,n_2}={\cal E}{\text{xt}}_{\pi_2}^1({\cal O}_{S \times (J\times H_1 \times H_2)} (\pi_1^* L -{\cal F}) \otimes I_{{\cal Z}_2}, {\cal O}_{S \times (J\times H_1 \times H_2)} ({\cal F}) \otimes I_{{\cal Z}_1}), $$ for $\pi_1:S \times (J\times H_1 \times H_2) \rightarrow S$ and $\pi_2:S \times (J\times H_1 \times H_2) \rightarrow J\times H_1 \times H_2$, the projections (we do not denote all pull-backs of sheaves explicitly). This is a vector bundle over $J\times H_1 \times H_2$ of rank $$ \text{rk}({\cal E})=l_{\zeta}+h^1({\cal O}_S(2F-L))=l_{\zeta}+h(\zeta)+q, $$ where $$ h(\zeta)= {\zeta \cdot K_S \over 2} -{\zeta^2 \over 2} -1, $$ by Riemann-Roch~\cite[lemma 2.6]{flips}. Note that $l_{\zeta} \geq 0$ and $h(\zeta)+q \geq 0$. Put $N_{\zeta}=\text{rk}({\cal E})-1$. Then $E_{\zeta}^{n_1,n_2} ={\Bbb P}(({\cal E}_{\zeta}^{n_1,n_2})^{\vee})$ (we follow the convention ${\Bbb P}({\cal E})= \text{Proj} (\oplus_i S^i({\cal E}))$), which is of dimension $q+2l_{\zeta}+(l_{\zeta}+h(\zeta)+q)$. Also $N_{\zeta}+N_{-\zeta}+q+2l_{\zeta}=d-1$. We will have to treat the case $\text{rk}({\cal E})=0$ (i.e. $l_{\zeta}=0$ and $h(\zeta)+q=0$) separately. We can modify the arguments in sections 3 and 4 of~\cite{flips} to get intermediate moduli spaces ${\frak M}_0^{(k)}$ together with embeddings $E_{\zeta}^{l_{\zeta}-k,k} \hookrightarrow {\frak M}_0^{(k)}$ and $E_{-\zeta}^{k,l_{\zeta}-k} \hookrightarrow {\frak M}_0^{(k-1)}$, fitting in the following diagram $$ \begin{array}{ccccccccccccc} && \widetilde{{\frak M}}_0^{(l_{\zeta})} & & & & \cdots & & & & \widetilde{{\frak M}}_0^{(0)} & &\\ & \swarrow & & \searrow && \swarrow & &\searrow & & \swarrow & & \searrow & \\ {\frak M}_0^{(l_{\zeta})} & & & & {\frak M}_0^{(l_{\zeta}-1)} & & & & {\frak M}_0^{(0)} & & & & {\frak M}_0^{(-1)} \\ \parallel &&&&&&&&&&&& \parallel \\ {\frak M}_- &&&&&&&&&&&& {\frak M}_+ \end{array} $$ where $\widetilde{{\frak M}}_0^{(k)} \rightarrow {\frak M}_0^{(k)}$ is the blow-up of ${\frak M}_0^{(k)}$ at $E_{\zeta}^{l_{\zeta}-k,k}$ and $\widetilde{{\frak M}}_0^{(k)} \rightarrow {\frak M}_0^{(k-1)}$ is the blow-up of ${\frak M}_0^{(k-1)}$ at $E_{-\zeta}^{k,l_{\zeta}-k}$. This is what is called a flip. Basically, the space $E_{\zeta}= \sqcup E_{\zeta}^{l_{\zeta}-k,k}$ parametrises $H_-$-stable sheaves which are $H_+$-unstable. Analogously, $E_{-\zeta}= \sqcup E_{-\zeta}^{k,l_{\zeta}-k}$ parametrises $H_+$-stable sheaves which are $H_-$-unstable. Hence one could say that ${\frak M}_+$ is obtained from ${\frak M}_-$ by removing $E_{\zeta}$ and then attaching $E_{-\zeta}$. The picture above is a nice description of this fact and allows us the find the universal sheaf for ${\frak M}_+$ out of the universal sheaf for ${\frak M}_-$ by a sequence of elementary transforms. The point is that whenever $-K_S$ is effective, we have an embedding $E_{\zeta}^{0,l_{\zeta}} \rightarrow {\frak M}_-$ (the part of $E_{\zeta}$ consisting of bundles) and rational maps $E_{\zeta}^{k,l_{\zeta}-k} \dashrightarrow {\frak M}_-$, $k>0$, but if we blow-up ${\frak M}_-$ at $E_{\zeta}^{0,l_{\zeta}}$, we have already an embedding from $E_{\zeta}^{1,l_{\zeta}-1}$ to this latter space. Now we can proceed inductively for $k=0,\ldots, l_{\zeta}$. Analogously, we can have started from ${\frak M}_+$ blowing-up $E_{-\zeta}^{k,l_{\zeta}-k}$ one by one. The diagram above says that we can perform these blow-ups and blow-downs alternatively, instead of first blowing-up $l_{\zeta}+1$ times and then blowing-down $l_{\zeta}+1$ times. We see that the exceptional divisor in $\widetilde{{\frak M}}_0^{(k)}$ is a ${\Bbb P}^{N_{\zeta}} \times {\Bbb P}^{N_{-\zeta}}$-bundle over $J \times H_{l_{\zeta}-k} \times H_{k}$. When adapting the arguments of~\cite[sections 3 and 4]{flips}, the only place requiring serious changes is proposition 3.7 in order to prove proposition 3.6. \begin{prop}{\em {\bf (\cite[proposition 3.6]{flips})}} \label{prop:lod} The map $E^{l_{\zeta}-k,k}_{\zeta} \rightarrow {\frak M}_0^{(\zeta, {\bf k})}$ is an immersion. The normal bundle ${\cal N}_{\zeta}^{l_{\zeta}-k,k}$ to $E^{l_{\zeta}-k,k}_{\zeta}$ in ${\frak M}_0^{(\zeta, {\bf k})}$ is exactly $\rho^*{\cal E}^{k,l_{\zeta}-k}_{-\zeta} \otimes {\cal O}_{E^{l_{\zeta}-k,k}_{\zeta}}(-1)$, where $\rho: E^{l_{\zeta}-k,k}_{\zeta} \rightarrow J \times H_{l_{\zeta}-k} \times H_k$ is the projection. Here we have defined ${\cal E}_{-\zeta}^{k,l_{\zeta}-k}={\cal E}{\text{xt}}_{\pi_2}^1({\cal O}_{S \times (J\times H_1 \times H_2)} ({\cal F}) \otimes I_{{\cal Z}_1}, {\cal O}_{S \times (J\times H_1 \times H_2)} (\pi_1^* L -{\cal F}) \otimes I_{{\cal Z}_2})$. \end{prop} Proposition~\ref{prop:lod} is proved as~\cite[proposition 3.6]{flips} making use of the following analogue of~\cite[proposition 3.7]{flips} \begin{prop} For all nonzero $\xi \in \text{Ext}^1=\text{Ext}^1({\cal O}(L-F) \otimes I_{Z_2},{\cal O}(F) \otimes I_{Z_1})$, the natural map from a neighbourhood of $\xi$ in $E^{l_{\zeta}-k,k}_{\zeta}$ to ${\frak M}_0^{(\zeta, {\bf k})}$ is an immersion at $\xi$. The image of $T_{\xi} E^{l_{\zeta}-k,k}_{\zeta}$ in $\text{Ext}^1_0(V,V)$ (the tangent space to ${\frak M}_0^{(\zeta, {\bf k})}$ at $\xi$, where $V$ is the sheaf corresponding to $\xi$) is exactly the kernel of the natural map $\text{Ext}_0(V,V) \rightarrow \text{Ext}^1({\cal O}(F) \otimes I_{Z_1},{\cal O}(L-F) \otimes I_{Z_2})$, and the normal space to $E^{l_{\zeta}-k,k}_{\zeta}$ at $\xi$ in ${\frak M}_0^{(\zeta, {\bf k})}$ may be canonically identified with $\text{Ext}^1({\cal O}(F) \otimes I_{Z_1},{\cal O}(L-F) \otimes I_{Z_2})$. \end{prop} \begin{pf} We have that $\text{Ext}^1 (I_Z, I_Z)$ parametrises infinitesimal deformations of $I_Z$ as a sheaf. The deformations of $I_Z$ are of the form $I_{Z'} \otimes {\cal O}(D)$ for $D \equiv 0$. The universal space parametrising these sheaves is $\text{Hilb}_r(S) \times \text{Jac}^0(S)$, where $r$ is the length of $Z$. There is an exact sequence $$ \exseq{H^0 ({\cal E} \text{xt}^1(I_Z,I_Z))}{\text{Ext}^1 (I_Z, I_Z)}{H^1({\cal H} \text{om}(I_Z,I_Z))}, $$ where $H^0 ({\cal E} \text{xt}^1(I_Z,I_Z))=H^0 ({\cal H}\text{om}(I_Z,{\cal O}_Z))= \text{Hom} (I_Z,{\cal O}_Z)$ is the tangent space to $\text{Hilb}_r(S)$ and $H^1({\cal H} \text{om}(I_Z,I_Z))=H^1({\cal O})$ is the tangent space to the Jacobian. Analogously, $\text{Ext}^1(V,V)$ is the space of infinitesimal deformations of $V$ (but the determinant is not preserved). The infinitesimal deformations preserving the determinant are given by the kernel $\text{Ext}_0^1(V,V)$ of a map $\text{Ext}^1(V,V) \rightarrow H^1({\cal H}\text{om}(V,V)) \rightarrow H^1({\cal O})$. Now $E=E_{\zeta}^{l_{\zeta}-k,k}$ sits inside the bigger space $\tilde{E}= \tilde{E}_{\zeta}^{l_{\zeta}-k,k}$ given as $$ {\Bbb P}({\cal E}{\text{xt}}_{\pi_2}^1({\cal O}_{S \times (J_1\times H_1 \times J_2 \times H_2)} (\pi_1^* L -{\cal F}_2) \otimes I_{{\cal Z}_2}, {\cal O}_{S \times (J_1\times H_1 \times J_2\times H_2)} ({\cal F}_1) \otimes I_{{\cal Z}_1})^{\vee}), $$ for $J_1=J_2=J$, ${\cal F}_i \subset S \times J_i$ the universal divisor, and $H_i$ the Hilbert scheme parametrising $Z_i$. The arguments in~\cite[proposition 3.7]{flips} go through to prove that for every non-zero $\xi \in \text{Ext}^1=\text{Ext}^1({\cal O}(L -F) \otimes I_{Z_2}, {\cal O}(F) \otimes I_{Z_1})$ we have the following commutative diagram with exact rows and columns $$ \begin{CD} T_{\xi} E @>>> \text{Ext}^1_0(V,V) @>>> \text{Ext}^1({\cal O}(F) \otimes I_{Z_1},{\cal O}(L -F) \otimes I_{Z_2}) \\ @VVV @VVV @| \\ T_{\xi} \tilde{E} @>>> \text{Ext}^1(V,V) @>>>\text{Ext}^1({\cal O}(F) \otimes I_{Z_1},{\cal O}(L -F) \otimes I_{Z_2}) \\ @VVV @VVV @. \\ H^1({\cal O}) @= H^1({\cal O}) @. \end{CD} $$ So the natural map from a neighbourhood of $\xi$ in $E$ to ${\frak M}_0^{(\zeta, {\bf k})}$ is an immersion at $\xi$ and the normal space may be canonically identified with $\text{Ext}^1({\cal O}(F) \otimes I_{Z_1},{\cal O}(L -F) \otimes I_{Z_2})$. \end{pf} Therefore proposition~\ref{prop:lod} is true for $q>0$. The set up is now in all ways analogous to that of~\cite{flips}. We fix some notations~\cite[section 5]{flips}: \begin{notation} \label{not:wall} Let $\zeta$ define a wall of type $(w,p_1)$. \begin{itemize} \setlength{\itemsep}{0pt} \item $\l_k$ is the tautological line bundle over $E_{\zeta}^{l_{\zeta}-k,k} ={\Bbb P}(({\cal E}_{\zeta}^{l_{\zeta}-k,k})^{\vee})$. $\l_k$ will also be used to denote its first Chern class. \item $\rho_k:S \times E_{\zeta}^{l_{\zeta}-k,k} \rightarrow S \times (J \times H_{l_{\zeta}-k} \times H_k)$ is the natural projection. \item $p_k: \widetilde{{\frak M}}_0^{(k)} \rightarrow {\frak M}_0^{(k)}$ is the blow-up of ${\frak M}_0^{(k)}$ at $E_{\zeta}^{l_{\zeta}-k,k}$. \item $q_{k-1}: \widetilde{{\frak M}}_0^{(k)} \rightarrow {\frak M}_0^{(k-1)}$ is the contraction of $\widetilde{{\frak M}}_0^{(k)}$ to ${\frak M}_0^{(k-1)}$. \item The normal bundle of $E_{\zeta}^{l_{\zeta}-k,k}$ in ${\frak M}_0^{(k)}$ is ${\cal N}_k = \rho_k^* {\cal E}_{-\zeta}^{k,l_{\zeta}-k} \otimes \l_k^{-1}$, where ${\cal E}_{-\zeta}^{k,l_{\zeta}-k}={\cal E}{\text{xt}}_{\pi_2}^1({\cal O}_{S \times (J\times H_1 \times H_2)} ({\cal F}) \otimes I_{{\cal Z}_1}, {\cal O}_{S \times (J\times H_1 \times H_2)} (\pi_1^* L -{\cal F}) \otimes I_{{\cal Z}_2})$. \item $D_k={\Bbb P}({\cal N}_k^{\vee})$ is the exceptional divisor in $\widetilde{{\frak M}}_0^{(k)}$. \item $\xi_k={\cal O}_{\widetilde{{\frak M}}_0^{(k)}}(-D_k)|_{D_k}$ is the tautological line bundle on $D_k$. \item $\mu^{(k)}(\a)=-{1 \over 4}p_1({\frak g}_{{\cal U}^{(k)}}) \backslash \a$, for $\a \in H_2(S;{\Bbb Z})$ and ${\cal U}^{(k)}$ a universal sheaf over $S \times {\frak M}_0^{(k)}$. Let $\mu^{(l_{\zeta})}(\a)=\mu_-(\a)$ and $\mu^{(-1)}(\a)=\mu_+(\a)$. \item Let $z=x^r\a^s \gamma_1 \cdots \gamma_a A_1 \cdots A_b$ be any element in $\AA(S)$, where $x\in H_0(S;{\Bbb Z})$ is the generator of the $0$-homology, $\gamma_i \in H_1(S;{\Bbb Z})$, $\a \in H_2(S;{\Bbb Z})$, $A_i \in H_3(S;{\Bbb Z})$. Then we define $\mu^{(k)}(z)$ as $\mu^{(k)}(x)^r \mu^{(k)}(\a)^s \mu^{(k)}(\gamma_1) \cdots \mu^{(k)}(\gamma_a) \mu^{(k)}(A_1)\cdots \mu^{(k)}(A_b)$. \end{itemize} \end{notation} Although ${\cal U}^{(k)}$ might not exist, there is always a well-defined element $p_1({\frak g}_{{\cal U}^{(k)}})$. As in~\cite{flips}, we are using the natural complex orientations for the moduli spaces. These differ from the natural ones used in the definition of the Donaldson invariants by a factor $\epsilon_S(w)=(-1)^{K \cdot w+w^2 \over 2}$. The analogues of lemma 5.2 and lemma 5.3 of~\cite{flips} are \begin{lem} \label{lem:ezlk} Let $\gamma \in H_1(S;{\Bbb Z})$, $\a \in H_2(S;{\Bbb Z})$, $A \in H_3(S;{\Bbb Z})$. Put $a=(\zeta\cdot \a)/2$. Then $$ \left\{ \begin{array}{l} p_k^*\mu^{(k)}(\a)|_{D_k}=(p_k|_{D_k})^* \left( [{\cal Z}_{l_{\zeta}-k}] \backslash \a + [{\cal Z}_k] \backslash \a -a \l_k - c_1({\cal F})^2 \backslash \a \right) \vspace{1mm}\\ p_k^*\mu^{(k)}(\gamma)|_{D_k}=(p_k|_{D_k})^* \left( [{\cal Z}_{l_{\zeta}-k}] \backslash \gamma + [{\cal Z}_k] \backslash \gamma -\l_k (c_1({\cal F}) \backslash \gamma) \right) \vspace{1mm}\\ p_k^*\mu^{(k)}(A)|_{D_k}=(p_k|_{D_k})^* \left( [{\cal Z}_{l_{\zeta}-k}] \backslash A + [{\cal Z}_k] \backslash A - (\zeta c_1({\cal F})) \backslash A\right) \vspace{1mm}\\ p_k^*\mu^{(k)}(x)|_{D_k}=(p_k|_{D_k})^* \left( [{\cal Z}_{l_{\zeta}-k}] \backslash x + [{\cal Z}_k] \backslash x -{1 \over 4} \l_k^2 \right) \end{array} \right. $$ \end{lem} \begin{lem} \label{lem:pkmu} Let $\gamma \in H_1(S;{\Bbb Z})$, $\a \in H_2(S;{\Bbb Z})$, $A \in H_3(S;{\Bbb Z})$. Put $a=(\zeta\cdot \a)/2$. Then $$ \left\{ \begin{array}{l} q_{k-1}^*\mu^{(k-1)}(\a) = p_k^*\mu^{(k)}(\a) -aD_k \vspace{1mm}\\ q_{k-1}^*\mu^{(k-1)}(\gamma) = p_k^*\mu^{(k)}(\gamma) - (c_1({\cal F}) \backslash \gamma) D_k \vspace{1mm}\\ q_{k-1}^*\mu^{(k-1)}(A) = p_k^*\mu^{(k)}(A) \vspace{1mm}\\ q_{k-1}^*\mu^{(k-1)}(x) = p_k^*\mu^{(k)}(x) -{1 \over 4} (D_k^2 +2\l_k D_k) \end{array} \right. $$ \end{lem} We immediately see that it is important to understand the cohomology classes $e_{\a}=c_1({\cal F})^2 \backslash \a$, $e_{\gamma}=c_1({\cal F}) \backslash \gamma$, and $e_{S}=c_1({\cal F})^4 \backslash [S]$. We write $c_1({\cal F})= c_1(F) + \sum \b_i \otimes \b_i^{\#}$, the K\"unneth decomposition of $c_1({\cal F}) \in H^2(S\times J)$, where $\{\b_i\}$ is a basis for $H^1(S)$ and $\{\b_i^{\#}\}$ is the dual basis for $H^1(J) \cong H^1(S)^*$. Now we have more explicit expressions \begin{equation} \left\{ \begin{array}{l} e_{\a} = -2\sum\limits_{i<j} <\b_i \wedge \b_j, \a> \b_i^{\#}\wedge \b_j^{\#} \in H^2(J) \\ e_{\gamma} = \sum <\gamma, \b_i> \b_i^{\#} \in H^1(J) \vspace{1mm}\\ e_{\zeta A}= (\zeta c_1({\cal F})) \backslash A = \sum <\text{P.D.} [A] \wedge \b_i, \zeta> \b_i^{\#} \in H^1(J) \vspace{1mm}\\ e_S= \sum\limits_{i,j,k,l}<\b_i \wedge \b_j \wedge \b_k \wedge \b_l, [S]> \b_i^{\#}\wedge \b_j^{\#}\wedge \b_k^{\#}\wedge\b_l^{\#} \in H^4(J) \end{array} \right. \label{eqn:the-e's}\end{equation} \vspace{1mm} \begin{thm} \label{thm:wall-formula} Let $\zeta$ define a wall of type $(w,p_1)$ and $d=-p_1-3(1-q)$. Suppose $l_{\zeta}+ h(\zeta)+q >0$. For $\a \in H_2(S;{\Bbb Z})$, put $a=(\zeta\cdot \a)/2$. Let $z=x^r\a^s \gamma_1 \cdots \gamma_a A_1 \cdots A_b \in \AA(S)$ be of degree $2d$. Then $\d_{S,\zeta}^{w,d}(\a)$ is $\epsilon_S(w)$ times $$ \sum_{0 \leq k \leq l_{\zeta}} ([{\cal Z}_{l_{\zeta}-k}]\backslash x+[{\cal Z}_k] \backslash x - {1 \over 4} X^2)^r ([{\cal Z}_{l_{\zeta}-k}]\backslash \a+[{\cal Z}_k] \backslash \a -e_{\a} +a X)^s ([{\cal Z}_{l_{\zeta}-k}]\backslash \gamma_1+ $$ $$ +[{\cal Z}_k] \backslash \gamma_1 +e_{\gamma_1}X) \cdots ([{\cal Z}_{l_{\zeta}-k}]\backslash A_b+[{\cal Z}_k] \backslash A_b -e_{\zeta A_b}) $$ \vspace{1mm} where $X^N=(-1)^{N-N_{-\zeta}} \, s_{N-1-N_{\zeta}-N_{-\zeta}} ({\cal E}_{\zeta}^{l_{\zeta}-k,k}\oplus ({\cal E}_{-\zeta}^{k,l_{\zeta}-k})^{\vee})$, $s_i(\cdot)$ standing for the Segre class. \end{thm} \begin{pf} By lemma~\ref{lem:pkmu}, $\mu^{(k-1)}(z)$ is equal to (we omit the pull-backs) $$ (\mu^{(k)}(x) -{1 \over 4} (D_k^2 +2\l_k D_k))^r (\mu^{(k)}(\a) -aD_k)^s (\mu^{(k)}(\gamma_1) - e_{\gamma_1} D_k) \cdots \mu^{(k)}(A_b) $$ which is $\mu^{(k)}(z)$ plus things containing at least one $D_k$. So $\mu^{(k-1)}(z) = \mu^{(k)}(z) + D_k \cdot s$, where $s$ is formally (recall $\xi_k=-D_k|_{D_k}$) $$ {1 \over -\xi_k} \Big( (\mu^{(k)}(x)|_{D_k} -{1 \over 4} (\xi_k^2 -2\l_k \xi_k))^r (\mu^{(k)}(\a)|_{D_k} +a\xi_k)^s (\mu^{(k)}(\gamma_1)|_{D_k} + e_{\gamma_1} \xi_k) \cdots $$ $$ \cdots (\mu^{(k)}(\gamma_a)|_{D_k} + e_{\gamma_1} \xi_k) \mu^{(k)}(A_1)|_{D_k} \cdots \mu^{(k)}(A_b)|_{D_k} \Big)_0 $$ where the subindex $0$ means ``forgetting anything not containing at least one $\xi_k$''. So $s$ is (we drop the subindices) $$ -{1 \over \xi} \Big( ([{\cal Z}]\backslash x+[{\cal Z}] \backslash x - {1 \over 4}(\xi-\l)^2)^r ([{\cal Z}]\backslash \a+[{\cal Z}] \backslash \a -e_{\a} +a(\xi-\l))^s ([{\cal Z}]\backslash \gamma_1 + $$ $$ + [{\cal Z}] \backslash \gamma_1 +e_{\gamma_1}(\xi-\l)) \cdots ([{\cal Z}]\backslash A_b+[{\cal Z}] \backslash A_b -e_{\zeta A_b}) \Big)_0 $$ We need the easy formula (which can be proved by induction) $$ {1 \over \xi} \left( (\xi-\l)^N \right)_0={ (\xi-\l)^N -(-\l)^N \over \xi}= \sum_{i=0}^{N-1} (-\l)^i (\xi-\l)^{N-i-1} $$ As $\xi-\l$ is the tautological bundle corresponding to ${\cal E}_{-\zeta}^{k,l_{\zeta}-k}$ (see items 5 to 7 in notation~\ref{not:wall}), we have $$ \left\{ \begin{array}{l} \l^u=s_{u-N_{\zeta}}({\cal E}_{\zeta}^{l_{\zeta}-k,k}) \cdot \l^{N_{\zeta}} + O(\l^{N_{\zeta}-1}) \\ (\xi-\l)^u=s_{u-N_{-\zeta}}({\cal E}_{-\zeta}^{k,l_{\zeta}-k}) \cdot (\xi-\l)^{N_{-\zeta}} + O((\xi-\l)^{N_{-\zeta}-1}) \end{array} \right. $$ Evaluating (and doing the sum from $k=0$ to $k=l_{\zeta}$) we get the statement of the theorem where \begin{eqnarray*} X^N &=&-\sum (-1)^i \, s_{i-N_{\zeta}}({\cal E}_{\zeta}^{l_{\zeta}-k,k}) \cdot s_{N-i-1-N_{-\zeta}}({\cal E}_{-\zeta}^{k,l_{\zeta}-k}) =\\ &=& \sum (-1)^{N-N_{-\zeta}} \, s_{i-N_{\zeta}}({\cal E}_{\zeta}^{l_{\zeta}-k,k}) \cdot s_{N-i-1-N_{-\zeta}}(({\cal E}_{-\zeta}^{k,l_{\zeta}-k})^{\vee})= \\ &=& (-1)^{N-N_{-\zeta}}\, s_{N-1-N_{\zeta}-N_{-\zeta}}({\cal E}_{\zeta}^{l_{\zeta}-k,k}\oplus ({\cal E}_{-\zeta}^{k,l_{\zeta}-k})^{\vee}). \end{eqnarray*} \end{pf} An immediate corollary which generalises~\cite[theorem 5.4]{flips} is \begin{cor} \label{cor:wall-formula} Let $\zeta$ define a wall of type $(w,p_1)$ and $d=-p_1-3(1-q)$. Suppose $l_{\zeta}+ h(\zeta)+q >0$. For $\a \in H_2(S;{\Bbb Z})$, put $a=(\zeta\cdot \a)/2$. Then $\mu_+(\a^d)-\mu_-(\a^d)$ is equal to $$ \sum (-1)^{h(\zeta)+l_{\zeta}+j} \, {d! \over j! b! (d-j-b)!} a^{d-j-b} ([{\cal Z}_{l_{\zeta}-k}]\backslash \a+[{\cal Z}_k]\backslash \a)^j \cdot e_{\a}^b \cdot s_{2l_{\zeta}-j +q-b}({\cal E}_{\zeta}^{l_{\zeta}-k,k}\oplus ({\cal E}_{-\zeta}^{k,l_{\zeta}-k})^{\vee}), $$ where the sum runs through $0 \leq j \leq 2l_{\zeta}$, $0 \leq b \leq q$, $0 \leq k \leq l_{\zeta}$. As $\mu_+(\a^d)-\mu_-(\a^d)$ is computed using the complex orientation, we have that $\d_{S,\zeta}^{w,d}(\a^d)=\epsilon_S(w)(\mu_+(\a^d)-\mu_-(\a^d))$. \end{cor} \begin{rem} \label{rem:ko} In Kotschick notation~\cite{Kotschick1}, $\varepsilon(\zeta,w)=(-1)^{({\zeta-w \over 2})^2}$. So $\epsilon_S(w) (-1)^{h(\zeta)}= (-1)^{d+q}\varepsilon(\zeta,w)$. \end{rem} \begin{thm} \label{thm:lz=0} Let $\zeta$ define a wall of type $(w,p_1)$ and $d=-p_1-3(1-q)$. Suppose $l_{\zeta}+h(\zeta)+q=0$ i.e. $l_{\zeta}=0$ and $h(\zeta)+q=0$. For $\a \in H_2(S;{\Bbb Z})$, put $a=(\zeta\cdot \a)/2$. Let $z=x^r\a^s \gamma_1 \cdots \gamma_a A_1 \cdots A_b \in \AA(S)$ be of degree $2d$. Then $\d_{S,\zeta}^{w,d}(\a)$ is $\epsilon_S(w)$ times $$ (- {1 \over 4} X^2)^r (-e_{\a} +a X)^s (e_{\gamma_1}X) \cdots (-e_{\zeta A_b}) $$ where $X^N=s_{N-N_{-\zeta}}({\cal E}_{-\zeta}^{0,0}) = (-1)^{N-N_{-\zeta}}\, s_{N-1-N_{\zeta}-N_{-\zeta}} ({\cal E}_{\zeta}^{0,0}\oplus ({\cal E}_{-\zeta}^{0,0})^{\vee})$. \end{thm} \begin{pf} Now ${\frak M}_+$ is ${\frak M}_-$ with an additional connected component $D=E^{0,0}_{-\zeta}$ which is a ${\Bbb P}^{d-q}$-bundle over $J$, since $E^{0,0}_{\zeta} = \emptyset$. The universal bundle over $E^{0,0}_{-\zeta}$ is given by an extension $$ \exseq{\pi^*{\cal O}_{S \times J}(\pi_1^* L-{\cal F}) \otimes p^* \l} {{\cal U}}{\pi^*{\cal O}_{S \times J}({\cal F})}, $$ where $\pi: S \times E^{0,0}_{-\zeta} \rightarrow S \times J$ and $p:S \times E^{0,0}_{-\zeta} \rightarrow E^{0,0}_{-\zeta}$ are projections and $\l$ is the tautological line bundle. From this $$ \left\{ \begin{array}{l} \mu (\a)|_D= a\l - e_{\a} \vspace{1mm}\\ \mu (\gamma)|_D= \l e_{\gamma} \vspace{1mm}\\ \mu (A)|_D= -e_{\zeta A}\vspace{1mm}\\ \mu (x)|_D= -{1 \over 4} \l^2 \end{array} \right. $$ with notations as in theorem~\ref{thm:wall-formula}. As in the proof of theorem~\ref{thm:wall-formula}, $\l^u=s_{u-N_{-\zeta}}({\cal E}_{-\zeta}^{0,0}) \cdot \l^{-N_{\zeta}} + O(\l^{-N_{\zeta}-1})$, so the expression of the statement of the theorem follows with $X^N=s_{N-N_{-\zeta}}({\cal E}_{-\zeta}^{0,0})$. \end{pf} The next step is to find more handy expressions for the set of classes given by~\eqref{eqn:the-e's}. \begin{lem} \label{lem:wedge} Let $S$ be a manifold with $b^+=1$. Then there is a (rational) cohomology class $\S \in H^2(S)$ such that the image of $\wedge: H^1(S) \otimes H^1(S) \rightarrow H^2(S)$ is ${\Bbb Q}[\S]$. Also $e_S=0$. \end{lem} \begin{pf} Let $\b_1,\b_2 ,\b_3 ,\b_4 \in H^1(S)$. If $\b_1 \wedge \b_2 \wedge \b_3 \wedge \b_4 \ne 0$ then the image of $\wedge:H^1(S) \otimes H^1(S) \rightarrow H^2(S)$ contains the subspace $V$ generated by $\b_i \wedge \b_j$, which has dimension $6$, with $b^+=3$ and $b^-=3$. This is absurd, so $\b_1 \wedge \b_2 \wedge \b_3 \wedge \b_4 = 0$. Then $e_S=0$. Now let $\S_1 =\b_1 \wedge \b_2$, $\S_2 =\b_3 \wedge \b_4 \in H^2(S)$. Then $\S_1^2=\S_2^2=0$ together with the fact that $b^+=1$ imply that $\S_1 \cdot \S_2 \neq 0$ unless $\S_1$ and $\S_2$ are proportional. Since $\S_1 \cdot \S_2 = 0$ by the above, it must be the case that $\S_1$ and $\S_2$ are proportional. \end{pf} \begin{rem} If $S \to C_g$ is a ruled surface with $q>0$ and fiber class $f$, then $\S=f$. Note also that the class $\S$ does not change under blow-ups. \end{rem} Now write $\seq{\b}{1}{2q}$ for a basis of $H^1(S)$ and fix a generator $\S$ of the image of $\wedge: H^1(S) \otimes H^1(S) \rightarrow H^2(S)$. Let $\seq{\d}{1}{2q}$ be the dual basis for $H_1(S)$. Put $\b_i \wedge \b_j=a_{ij} \S$. The Jacobian of $S$ is $J= H^1(S;{\Bbb R})/H^1(S;{\Bbb Z})$, so naturally $H^1(J) \stackrel{\sim}{\ar} H^1(S)^*$. Let ${\cal L} \rightarrow S \times J$ be the universal bundle parametrising divisors homologically equivalent to zero. Then $E=c_1({\cal L})=\sum \b_i \otimes \b_i^{\#}$, with $\b_i^{\#}$ corresponding to $\d_i$ under the isomorphism $H^1(J) \cong H_1(S)$. So \begin{equation} \left\{ \begin{array}{l} e_{\a}=-2 \sum\limits_{i<j} a_{ij} (\S \cdot \a) \b_i^{\#}\wedge \b_j^{\#}=- 2 ( \S \cdot \a) \o \\ e_{\d_i} =\b_i^{\#} \vspace{1mm}\\ e_{\zeta \b_i} = \sum (\S\cdot \zeta) a_{ij} \b_j^{\#} = (\S\cdot\zeta) i_{\b_i} \o \end{array} \right. \label{eqn:the-e's2}\end{equation} where we write $\o = \sum\limits_{i<j} a_{ij}(\b_i^{\#}\wedge \b_j^{\#}) \in H^2(J)$, which is an element independent of the chosen basis. We also have implicitly assumed $H_3(S) \cong H^1(S)$ through Poincar\'e duality, in the third line. We define $$ F: \AA(S) \rightarrow \L^*H_1(S) \otimes \L^*H_3(S) \rightarrow {\Bbb Q} $$ given by projection followed by the map sending $\gamma_1 \wedge \cdots \wedge \gamma_a \otimes A_1 \wedge \cdots \wedge A_b$ to zero when $a+b$ is odd and to $$ \int_J (\gamma_1 \wedge \cdots \wedge \gamma_a \wedge i_{A_1} \o \wedge \cdots \wedge i_{A_b} \o \wedge \o^{q-(a+b)/2}) $$ when $a+b$ is even. We note that we always can find a basis $\seq{\b}{1}{2q}$ with $$ \o=a_1 \b_1^{\#}\wedge \b_2^{\#} + a_2 \b_3^{\#}\wedge \b_4^{\#} + \cdots a_r \b_{2r-1}^{\#}\wedge \b_{2r}^{\#}, $$ where $a_i \neq 0$ are integers and $r \leq q$. So if $\o$ is degenerate, $F(1) =\int_J \o^q=0$. In general, for a basis element $z=x^r\a^s\d_{i_1}\cdots \d_{i_a}\b_{j_1} \cdots \b_{j_b}$, $F(z)$ is zero unless $z$ contains $\d_{2r+1}\cdots \d_{2q}$, and for every pair $(2i-1,2i)$, $1 \leq i \leq r$, either $\d_{2i-1}\d_{2i}$, $\b_{2i-1}\b_{2i}$, $\d_{2i-1}\b_{2i-1}$, $\d_{2i}\b_{2i}$ or nothing. In any case, for subsequent use, we set $$ \text{vol} = { 1\over q!} \int_J \o^q. $$ The number $\text{vol}$ depends on the choice of $\S$, as when $\S$ is changed to $r\S$, $\text{vol}$ is changed to $r^{-q}\text{vol}$. The final expressions we get for the wall-crossing terms are (as expected) independent of this choice. Also we are going to need the following \begin{prop} \label{prop:segre} For any sheaf ${\cal F}$ on any complex variety, the Segre classes of ${\cal F}$ are given by $s_t({\cal F})=c_t({\cal F})^{-1}$. For the relationship between the Chern classes of ${\cal F}$ and its Chern character, write $a_i$ for $i!$ times the $i$-th component of $\text{ch}\: {\cal F}$. Then $$ c_n({\cal F})={1 \over n!}\left|{\begin{array}{ccccc} a_1 & n-1 & 0 & \cdots & 0 \\ a_2 & a_1 & n-2 & \cdots & 0 \\ \vdots & & \ddots & & \vdots \\ \vdots & & & \ddots & 1 \\ a_n & a_{n-1} &a_{n-2} & \cdots & a_1 \end{array}}\right| $$ and $$ s_n({\cal F})={1 \over n!}\left|{\begin{array}{ccccc} -a_1 & -(n-1) & 0 & \cdots & 0 \\ a_2 & -a_1 & -(n-2) & \cdots & 0 \\ \vdots & & \ddots & & \vdots \\ \vdots & & & \ddots & 1 \\ (-1)^na_n & (-1)^{n-1}a_{n-1} &(-1)^{n-2}a_{n-2} & \cdots & -a_1 \end{array}}\right| $$ \end{prop} \section{The case $l_{\zeta}=0$} \label{sec:lz=0} In this section we are going to compute $\d_{S,\zeta}^{w,d}$ in the case $l_{\zeta}=0$, i.e. when $\zeta^2 =p_1$. We have the following theorem which extends~\cite[theorems 6.1 and 6.2]{flips}~\cite{Kotschick1}. \begin{thm} \label{thm:mainl0} Let $\zeta$ be a wall with $l_{\zeta}=0$. Then $\d_{S,\zeta}^{w,d}(x^r\a^{d-2r})$ is equal to $$ \varepsilon(\zeta,w) \sum_{0 \leq b \leq q} (-1)^{r+d} 2^{3q-b-d}{q! \over (q-b)!} {d-2r \choose b} ( \zeta \cdot \a)^{d-2r-b} ( \S \cdot \a)^b ( \S \cdot \zeta)^{q-b} \text{vol}, $$ where terms with negative exponent are meant to be zero. \end{thm} \begin{pf} For simplicity of notation, let us do the case $r=0$ (the other case is very similar). Recall that $F$ is a divisor such that $2F-L$ is homologically equivalent to $\zeta$ and $J=\text{Jac}^F(S)$ is the Jacobian parametrising divisors homologically equivalent to $F$. Then ${\cal F} \subset S \times J$ denotes the universal divisor. Now ${\cal E}_{\zeta}={\cal E}_{\zeta}^{0,0}=R^1 \pi_*({\cal O}_{S\times J}(2{\cal F} - \pi_1^*L))$ (with $\pi: S \times J \rightarrow J$ the projection) is a vector bundle over $J$. We note that $H^0({\cal O}_S (2 F - L)) =0$ and $H^0({\cal O}_S (L-2 F) \otimes K) =0$, as $\zeta$ is a good wall, so $R^0 \pi_*$ and $R^2 \pi_*$ vanish. Then $$ \text{ch}\: {\cal E}_{\zeta} = -\text{ch}\: \pi_! ({\cal O}_{S\times J}(2{\cal F} - \pi^*L)) = -\pi_*(\text{ch}\: {\cal O}_{S\times J}(2{\cal F} - \pi^*L) \cdot \text{Todd }T_S) = $$ \begin{equation} = - ({\zeta^2 \over 2} -{\zeta\cdot K \over 2} +1 -q) + e_{K-2\zeta} -{2 \over 3} e_S = \text{rk}({\cal E}_{\zeta}) + e_{K-2\zeta}, \label{eqn:rar}\end{equation} since $e_S=0$ (lemma~\ref{lem:wedge}). A fortiori $\text{ch}\: {\cal E}^{\vee}_{-\zeta} =- ({\zeta^2 \over 2} +{\zeta\cdot K \over 2} +1 -q) - e_{K+2\zeta}$ and $$ \text{ch}\:( {\cal E}_{\zeta} \oplus {\cal E}_{-\zeta}^{\vee}) = (-\zeta^2 +2q-2)- 4e_{\zeta}. $$ From proposition~\ref{prop:segre}, $s_i({\cal E}_{\zeta} \oplus {\cal E}_{-\zeta}^{\vee})= {4^i \over i!} e_{\zeta}^i$. This together with theorem~\ref{thm:lz=0} or theorem~\ref{thm:wall-formula} (depending on whether $h(\zeta)+q$ is zero or not) gives \begin{eqnarray*} \d_{S,\zeta}^{w,d}(\a^d) &=& \epsilon_S(w)\sum_{0 \leq b \leq q} (-1)^{h(\zeta)} {d \choose b} a^{d-b} e_{\a}^b \cdot s_{q-b}({\cal E}_{\zeta}\oplus {\cal E}_{-\zeta}^{\vee})= \\ &=& \epsilon_S(w)\sum_{0 \leq b \leq q} (-1)^{h(\zeta)} {d \choose b} a^{d-b} e_{\a}^b \cdot {4^{q-b} \over (q-b)!} e_{\zeta}^{q-b}= \\ &=& \epsilon_S(w)\sum_{0 \leq b \leq q} (-1)^{h(\zeta)+q} {2^{3q-b-d} \over (q-b)!} {d \choose b} ( \zeta \cdot \a)^{d-b} ( \S \cdot \a)^b ( \S \cdot \zeta)^{q-b} \o^q, \end{eqnarray*} using~\eqref{eqn:the-e's2}. Now we substitute $\epsilon_S(w) (-1)^{h(\zeta)}= (-1)^{d+q}\varepsilon(\zeta,w)$ (remark~\ref{rem:ko}), to get the desired result. \end{pf} We can also generalise introducing classes of odd degree. If $z=x^r\a^s\gamma_1 \cdots \gamma_a A_1 \cdots A_b$ with $d=s+2r+{3 \over 2}a +{1 \over 2}b$, then theorem~\ref{thm:lz=0} or theorem~\ref{thm:wall-formula} gives \begin{eqnarray*} \d_{S,\zeta}^{w,d}(z) &=& \epsilon_S(w)(- {1 \over 4} X^2)^r (-e_{\a} +a X)^s (e_{\gamma_1}X) \cdots (-e_{\zeta A_b}) \\ &=& \epsilon_S(w) \sum_{j} (- {1 \over 4})^r (-1)^b {s \choose j} a^{s-j} 2^j ( \S \cdot \a)^j (\S\cdot\zeta)^b \o^j \, \gamma_1 \cdots \gamma_a \, i_{A_1}\o \cdots i_{A_b} \o \cdot \\ & & \cdot (-1)^{2r+a+s-j-N_{-\zeta}} \, s_{2r+a+s-j-1-N_{\zeta}-N_{-\zeta}} ({\cal E}_{\zeta} \oplus {\cal E}_{-\zeta}^{\vee}). \end{eqnarray*} Now $2r+a+s-1-N_{\zeta}-N_{-\zeta}=q-(a+b)/2$, so \begin{eqnarray*} \d_{S,\zeta}^{w,d}(z) &=& \varepsilon(\zeta,w) \sum_{j} (-1)^{r+d+b} \, 2^{3q-d-b-j} \, {s \choose j} {F(z) \over (q-(a+b)/2-j)!} \cdot \\ & & \cdot(\zeta\cdot\a)^{s-j} ( \S \cdot \a)^j (\S\cdot\zeta)^{q+(b-a)/2-j}\text{vol}. \end{eqnarray*} \section{The case $l_{\zeta}=1$} \label{sec:lz=1} Now we want to compute $\d_{S,\zeta}^{w,d}$ in the case $l_{\zeta}=1$, i.e. when $\zeta^2 =p_1 + 4$. In this case, $J \times H_{l_{\zeta}-k}\times H_k \cong J \times S$, both for $k=0$ and $k=1$. The universal divisor ${\cal Z}_1 \subset S \times H_1 =S \times S$ is the diagonal $\Delta$. Let again ${\cal L} \to S \times J$ be the universal bundle parametrising divisors homologically equivalent to zero, so ${\cal F}= \pi_1^* F + {\cal L}$. With this understood, we have the following easy extension of~\cite[lemma 5.11]{flips} \begin{lem} \label{lem:5.11} Let $\text{Hom}=\text{Hom}(I_{{\cal Z}_k},I_{{\cal Z}_{l_{\zeta}-k}})$ and $\text{Ext}^1=\text{Ext}^1(I_{{\cal Z}_k},I_{{\cal Z}_{l_{\zeta}-k}})$, $\pi_1$, $p$ and $\pi_2$ be the projections from $S \times (J \times H_{l_{\zeta}-k}\times H_k)$ to $S$, $S \times J$ and $J \times H_{l_{\zeta}-k}\times H_k$, respectively. Let $E=c_1({\cal L})$. Then we have the following exact sequences $$ \exseq{R^1\pi_{2*}(p^*(\zeta+2E)\otimes \text{Hom})}{{\cal E}_{\zeta}^{l_{\zeta}-k,k}}{\pi_{2*}(p^*(\zeta+2E)\otimes \text{Ext}^1)} $$ $$ \exseq{\pi_{2*}(p^*(\zeta+2E)\otimes {\cal O}_{{\cal Z}_{l_{\zeta}-k}})}{R^1\pi_{2*}(p^*(\zeta+2E)\otimes \text{Hom})}{R^1\pi_{2*}(p^*(\zeta+2E))} $$ where the last sheaf is $M_{\zeta}=R^1 \pi_{2*}({\cal O}_{S\times J}(2{\cal F} - \pi_1^*L))$, which is a line bundle over $J$ with $ch M_{\zeta} = rk M_{\zeta} + e_{K-2\zeta}$ (computed in equation~\eqref{eqn:rar}). \end{lem} We apply this lemma to our case $l_{\zeta}=1$. Then for $k=0$, $\text{Hom}=\text{Hom}({\cal O},I_{\Delta})=I_{\Delta}$, $\text{Ext}^1=\text{Ext}^1 ({\cal O},I_{\Delta})=0$, and for $k=1$, $\text{Hom}=\text{Hom}(I_{\Delta},{\cal O})={\cal O}_{S\times S}$, $\text{Ext}^1=\text{Ext}^1(I_{\Delta},{\cal O})={\cal O}_{\Delta}(\Delta)$. Using lemma~\ref{lem:5.11} and the fact $\pi_{2*}({\cal O}_{\Delta}(\Delta))={\cal O}_S(-K)$, we get $$\text{ch}\: {\cal E}^{1,0}_{\zeta}=\text{ch}\: M_{\zeta} + \text{ch}\: \zeta \, \text{ch}\: 2E$$ $$\text{ch}\: {\cal E}^{0,1}_{\zeta}=\text{ch}\: M_{\zeta} + \text{ch}\: (\zeta-K)\, \text{ch}\: 2E$$ We recall from notation~\ref{not:wall}, \begin{eqnarray*} {\cal E}_{\zeta}^{l_{\zeta}-k,k}&=& {\cal E}{\text{xt}}_{\pi_2}^1( {\cal O}_{S \times (J\times H_1 \times H_2)} (\pi_1^* L -{\cal F}) \otimes I_{{\cal Z}_2}, {\cal O}_{S \times (J\times H_1 \times H_2)} ({\cal F}) \otimes I_{{\cal Z}_1}) \\ &=& {\cal E}{\text{xt}}_{\pi_2}^1( I_{{\cal Z}_2}, {\cal O}_{S \times (J\times H_1 \times H_2)} (\zeta + 2E) \otimes I_{{\cal Z}_1}), \\ {\cal E}_{-\zeta}^{k,l_{\zeta}-k} &=& {\cal E}{\text{xt}}_{\pi_2}^1(I_{{\cal Z}_1}, {\cal O}_{S \times (J\times H_1 \times H_2)} (-\zeta - 2E) \otimes I_{{\cal Z}_2}). \end{eqnarray*} Then we have $\text{ch}\: {\cal E}^{1,0}_{-\zeta}=\text{ch}\: M_{-\zeta} + \text{ch}\: (-\zeta) \, \text{ch}\: (-2E)$ and $\text{ch}\: {\cal E}^{0,1}_{-\zeta}=\text{ch}\: M_{-\zeta} + \text{ch}\: (-\zeta-K)\, \text{ch}\: (-2E)$. So $$ \text{ch}\: ({\cal E}^{1,0}_{\zeta} \oplus ({\cal E}^{0,1}_{-\zeta})^{\vee}) = (-\zeta^2+2q-2) -4e_{\zeta} +2\text{ch}\: \zeta \text{ch}\: 2E + {K^2\over 2} +K\zeta +K(1+2E+2E^2) $$ $$ \text{ch}\: ({\cal E}^{0,1}_{\zeta} \oplus ({\cal E}^{1,0}_{-\zeta})^{\vee}) = (-\zeta^2+2q-2) -4e_{\zeta} +2\text{ch}\: \zeta \text{ch}\: 2E + {K^2\over 2} -K\zeta - K(1+2E+2E^2) $$ \begin{equation} \label{eqn:what} \end{equation} We shall compute $s_i=s_i ({\cal E}^{1,0}_{\zeta} \oplus ({\cal E}^{0,1}_{-\zeta})^{\vee}) + s_i ({\cal E}^{0,1}_{\zeta} \oplus ({\cal E}^{1,0}_{-\zeta})^{\vee})$, as a class on $J \times S$. From proposition~\ref{prop:segre}, $s_i$ is an polynomial expression on $a_i^{(k)}=i! \, \text{ch}\:_i ({\cal E}^{1-k,k}_{\zeta} \oplus ( {\cal E}^{k,1-k}_{-\zeta})^{\vee})$, $k=0,1$. Furthermore, $s_i$ is invariant under $K \mapsto -K$, and hence an even function of $K\zeta$, $K$, $KE$ and $KE^2$. Now the only non-zero even combinations of $K\zeta$, $K$, $KE$ and $KE^2$ are $1$ and $K \cdot K$. The first consequence is that we can ignore $K\zeta$, $KE$ and $KE^2$ in $a_i^{(k)}$ for the purposes of computing $s_i$. So we can suppose $$ \left\{ \begin{array}{l} a_1^{(k)} = -4e_{\zeta}+2\zeta +4E+ (-1)^k K \vspace{1mm} \\ a_2= 2\zeta^2 +8E^2 +K^2 +8E\zeta \vspace{1mm} \\ a_3= 24 E^2 \zeta =24 e_{\zeta}[S] \end{array} \right. $$ where $a_i=a_i^{(0)}=a_i^{(1)}$ for $i \geq 2$, and $a_i=0$ for $i \geq 4$ (here we have used that $E^3=0$ and $E^4=0$ as a consequence of lemma~\ref{lem:wedge}). Put $a_1 = -4e_{\zeta}+2\zeta+4E$ and define $$ I_n = \left|{\begin{array}{ccccc} -a_1 & -(n-1) & \cdots & 0 \\ a_2 & -a_1 & \cdots & 0 \\ -a_3 & a_2 & \cdots & 0 \\ 0 & -a_3 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0& \cdots & -a_1 \end{array}}\right| $$ and $I_n^{(k)}$ defined similarly with $a_1^{(k)}$ in the place of $a_1$. Then by proposition~\ref{prop:segre}, $n !\,s_n= I_n^{(0)}+I_n^{(1)}$. Easily we have $n! \, s_n=2I_n +2 {n \choose 2}K^2 (4e_{\zeta})^{n-2}$. Now we can look for an inductive formula for $I_n$. For $n \geq 2$, \begin{eqnarray*} I_n &=& -a_1 I_{n-1} + (n-1) (2\zeta^2 +K^2 +8E\zeta) (4e_{\zeta}-4E)^{n-2} +(n-1)8 E^2 I_{n-2} \\ & & -6(n-1)(n-2)[S](4e_{\zeta})^{n-2} = \\ &=& -a_1 I_{n-1} + (n-1) (2\zeta^2 +K^2 +8E\zeta) (4e_{\zeta})^{n-2} -(n-1)8E\zeta (n-2)(4E)(4e_{\zeta})^{n-3} \\ & & +(n-1)8 E^2 I_{n-2} -6(n-1)(n-2)[S](4e_{\zeta})^{n-2} = \\ &=& -a_1 I_{n-1} + (n-1) (2\zeta^2 +K^2 +8E\zeta) (4e_{\zeta})^{n-2} -8(n-1)(n-2)[S](4e_{\zeta})^{n-2} \\ & & +(n-1)8 E^2 \big( (4e_{\zeta})^{n-2} -(n-2)(4e_{\zeta})^{n-3}2\zeta \big) -6(n-1)(n-2)[S](4e_{\zeta})^{n-2} = \\ & =& -a_1 I_{n-1} + (n-1) (4e_{\zeta})^{n-2} (2\zeta^2 +K^2 +8E\zeta +8E^2-18(n-2)[S]). \end{eqnarray*} In the first equality we have used that $I_n P =(4e_{\zeta}-4E)^n P$, for any $P \in H^i(S) \otimes H^j(J)$ with $i=3,4$. In the third equality we use that $I_n E^2 =(4e_{\zeta}-2\zeta)^n E^2$. With this inductive formula for $I_n$, we get, for $n \geq 2$, $$ I_n = (4e_{\zeta}-2\zeta-4E)^n +\sum_{i=2}^n (4e_{\zeta}-2\zeta-4E)^{n-i} (i-1)(4e_{\zeta})^{i-2} \Big( 2\zeta^2 +K^2 $$ \begin{equation} +8E\zeta +8E^2-18(i-2)[S] \Big). \label{eqn:lamia} \end{equation} Now for any $k \geq 0$ (we always understand ${k \choose i}=0$ if either $i<0$ or $i>k$), \begin{eqnarray*} (4e_{\zeta}-2\zeta -4E)^k &=& (4e_{\zeta})^k +k (-2\zeta -4E)(4e_{\zeta})^{k-1} + {k \choose 2} (-2\zeta -4E)^2(4e_{\zeta})^{k-2} \\ & & + {k \choose 3}3(-2\zeta)(-4E)^2 (4e_{\zeta})^{k-3} = \\ & = & (4e_{\zeta})^k +k (-2\zeta -4E)(4e_{\zeta})^{k-1} \\ & & + {k \choose 2} (4\zeta^2 +16E\zeta +16E^2) (4e_{\zeta})^{k-2} -24 {k \choose 3} [S](4e_{\zeta})^{k-2}. \end{eqnarray*} Substituting this into~\eqref{eqn:lamia}, we have \begin{eqnarray*} I_n &=& (4e_{\zeta})^n +n (-2\zeta -4E)(4e_{\zeta})^{n-1} + {n \choose 2} (4\zeta^2 +16E\zeta +16E^2) (4e_{\zeta})^{n-2} \\ & & -24 {n \choose 3} [S](4e_{\zeta})^{n-2} +\sum_{i=2}^n \Big( (4e_{\zeta})^{n-i} (i-1)(4e_{\zeta})^{i-2} (2\zeta^2 +K^2 -18(i-2)[S]) \\ & & + \big( (4e_{\zeta})^{n-i} + (n-i)(-4E)(4e_{\zeta})^{n-i-1} \big) (i-1)(4e_{\zeta})^{i-2} 8E\zeta \\ & & + \big( (4e_{\zeta})^{n-i} + (n-i)(-2\zeta)(4e_{\zeta})^{n-i-1} \big) (i-1)(4e_{\zeta})^{i-2} 8E^2 \Big) =\\ &=& (4e_{\zeta})^n -n (2\zeta +4E)(4e_{\zeta})^{n-1} + (4e_{\zeta})^{n-2} \Big[ {n \choose 2} (4\zeta^2 +16E\zeta +16E^2) -24 {n \choose 3} [S] \\ & & +\sum_{i=2}^n (i-1) \big( 2\zeta^2 +K^2 -18(i-2)[S] + 8E\zeta +8E^2 - 8(n-i)[S] -4(n-i)[S] \big) \Big]. \end{eqnarray*} Putting this into the expression for $s_n$, we get \begin{eqnarray*} s_n &=& {2 \over n!} \Big( (4e_{\zeta})^n -n(2\zeta+4E)(4e_{\zeta})^{n-1} +(4e_{\zeta})^{n-2} \big[ {n \choose 2} (6\zeta^2 + 24E\zeta +24 E^2 +K^2) \\ & & - 24 {n \choose 3}[S] +\sum_{i=2}^n (i-1) (36-12n-6i)[S] \big] \Big) +{2 \over n!} {n \choose 2}K^2 (4e_{\zeta})^{n-2}. \end{eqnarray*} The expression in the summatory adds up to $-48{n \choose 3}$, so finally $$ s_n =2{(4e_{\zeta})^n \over n!} -(4\zeta+8E) {(4e_{\zeta})^{n-1} \over (n-1)!} + (6\zeta^2 + 2 K^2 + 24E\zeta+24E^2 ) {(4e_{\zeta})^{n-2} \over (n-2)!} $$ \begin{equation} - 24[S]{(4e_{\zeta})^{n-2} \over (n-3)!}, \label{eqn:sn} \end{equation} for $n \geq 2$ (where the last summand is understood to be zero when $n=2$). This expression is actually valid for $n \geq 0$ under the proviso that the terms with negative exponent are zero. \begin{thm} \label{thm:mainl1} Let $\zeta$ be a wall with $l_{\zeta}=1$. Then $\d_{S,\zeta}^{w,d}(x^r\a^{d-2r})$ is equal to $\varepsilon(\zeta,w)$ times $$ \sum_{b=0}^q (-1)^{r+d+1} 2^{3q-b-d} \Big[ ( \zeta \cdot \a)^{d-2r-b} \Big( {d-2r \choose b}(6\zeta^2+2K^2-24q-8r) + 8{d-2r \choose b+1}{b+1 \choose 1} \Big) + $$ $$ + 8 ( \zeta \cdot \a)^{d-2r-b-2}\a^2 {d-2r \choose b+2}{b+2 \choose 2} \Big] ( \S \cdot \a)^b ( \S \cdot \zeta)^{q-b} {q!\over (q-b)!} \text{vol}, $$ where terms with negative exponent are meant to be zero. \end{thm} \begin{pf} By theorem~\ref{thm:wall-formula}, $\d_{S,\zeta}^{w,d}(x^r\a^{d-2r})= \epsilon_S(w) ([S]-{1 \over 4}X^2)^r(\a -e_{\a}+aX)^{d-2r}$ evaluated on $J\times S$, where \begin{eqnarray*} X^N &=&(-1)^{N-N_{-\zeta}} \, \left( s_{N-1-N_{\zeta}-N_{-\zeta}} ({\cal E}^{1,0}_{\zeta} \oplus ({\cal E}^{0,1}_{-\zeta})^{\vee}) + s_{N-1-N_{\zeta}-N_{-\zeta}} ({\cal E}^{0,1}_{\zeta} \oplus ({\cal E}^{1,0}_{-\zeta})^{\vee}) \right) = \\ &=& (-1)^{N-N_{-\zeta}} \, s_{N-1-N_{\zeta}-N_{-\zeta}}. \end{eqnarray*} Hence $$ \d_{S,\zeta}^{w,d}(x^r\a^{d-2r})=\epsilon_S(w) \sum_b (-1)^{h(\zeta)+1} {d-2r \choose b}a^{d-2r-b}(-{1 \over 4})^r e_{\a}^{b-2} \cdot $$ $$ \cdot \Big[ -4r[S]e_{\a}^2 \cdot s_{q-b} +{b \choose 2} \a^2\cdot s_{q-b+2}+{b \choose 1} \a (-e_{\a})\cdot s_{q-b+2}+e_{\a}^2 \cdot s_{q-b+2} \Big]. $$ Substituting the values of $s_n$ from~\eqref{eqn:sn} and using remark~\ref{rem:ko}, we get \begin{eqnarray*} \d_{S,\zeta}^{w,d}(x^r\a^{d-2r}) &=& \varepsilon(\zeta,w)\sum_b (-1)^{r+d+1} 2^{3q-b-d} \, {d-2r \choose b} ( \zeta \cdot \a)^{d-2r-b} \Big[(6\zeta^2+2K^2 \\ & & -24q-8r) ( \S \cdot \a)^b {( \S \cdot \zeta)^{q-b}\over (q-b)!} + 16 (\zeta \cdot \a) {b\choose 1} ( \S \cdot \a)^{b-1} {( \S \cdot \zeta)^{q-b+1}\over (q-b+1)!} + \\ & & + 32 \a^2 {b \choose 2} ( \S \cdot \a)^{b-2} {( \S \cdot \zeta)^{q-b+2}\over (q-b+2)!} \Big]\, \int_J \o^q. \end{eqnarray*} Reagrouping the terms we get the desired result. \end{pf} This result agrees with theorems 6.4 and 6.5 in~\cite{flips} particularising for $q=0$ and $r=0,1$. We see from theorem~\ref{thm:mainl1} that the difference terms $\d_{S,\zeta}^{w,d}$ do not satisfy in general the simple type condition~\cite{KM}. \begin{rem} L.\ G\"ottsche and the author have obtained the same formula of theorem~\ref{thm:mainl1} in some examples, like ${\Bbb C \Bbb P}^1 \times C_1$ ($C_1$ being an elliptic curve) using the simple type condition in limiting chambers. These arguments will appear elsewhere. \end{rem} \section{General case} \label{sec:5} We do not want to enter into more detailed computations of the wall-crossing formulae, but just to remark that the pattern laid in~\cite{flips} together with theorem~\ref{thm:wall-formula} can be used here to obtain partial information of $\d_{S,\zeta}^{w,d}$. For instance, we write $$ S_{j,b}= \sum_k([{\cal Z}_{l_{\zeta}-k}]\backslash \a+[{\cal Z}_k] \backslash \a)^j \cdot e_{\a}^b \cdot s_{2l_{\zeta}-j +q-b}({\cal E}_{\zeta}^{l_{\zeta}-k,k}\oplus ({\cal E}_{-\zeta}^{k,l_{\zeta}-k})^{\vee}), $$ so that corollary~\ref{cor:wall-formula} says $\d_{S,\zeta}^{w,d}(\a^d)=\varepsilon(\zeta,w) \sum (-1)^{d+q+l_{\zeta}+j}{d! \over j!b!(d-j-b)!}a^{d-j-b} S_{j,b}$. Then we can obtain (compare~\cite[proposition 5.12]{flips}) \begin{eqnarray*} S_{2l_{\zeta},q} &=& {(2l_{\zeta})! \over l_{\zeta}!} (\a^2)^{l_{\zeta}} \, e_{\a}^q \\ S_{2l_{\zeta}-1,q} &=& (-4){(2l_{\zeta})! \over l_{\zeta}!} (\a^2)^{l_{\zeta}-1} \, a \, e_{\a}^q \\ S_{2l_{\zeta},q-1} &=& 4 {(2l_{\zeta})! \over l_{\zeta}!} (\a^2)^{l_{\zeta}} \, e_{\a}^{q-1} \, e_{\zeta} \end{eqnarray*} As an easy consequence of this we get (compare~\cite[theorems 5.13 and 5.14]{flips}) \begin{cor} Let $\zeta$ be a wall of type $(w,p_1)$. Let $\a \in H_2(S;{\Bbb Z})$ and $a=(\zeta\cdot\a) /2$. Then $\d_{S,\zeta}^{w,d}(\a^d)$ is congruent (modulo $a^{d-2l_{\zeta}-q+2}$) with $$ \varepsilon(\zeta,w) (-1)^{d+l_{\zeta}} \, 2^q \Big[ a^{d-2l_{\zeta}-q} { d! \over l_{\zeta}! (d-2l_{\zeta}-q)!} (\a^2)^{l_{\zeta}} (\S\cdot\a)^q + $$ $$ + 4a^{d-2l_{\zeta}-q+1} { d! \, q \over l_{\zeta}! (d-2l_{\zeta}-q+1)!} (\a^2)^{l_{\zeta}} (\S\cdot\a)^{q-1}(\S\cdot\zeta) \Big] \text{vol}. $$ \end{cor} \begin{cor} In the conditions of the previous corollary, suppose furthermore $d-2r \geq 2l_{\zeta}+q$. Then $\d_{S,\zeta}^{w,d}(x^r\a^{d-2r})$ is congruent (modulo $a^{d-2r-2l_{\zeta}-q+2}$) with $$ \epsilon(\zeta,w)(-1)^{d+l_{\zeta}+r} \, 2^{q-2r} (-{1 \over 4})^r \Big[ a^{d-2r-2l_{\zeta}-q} { (d-2r)! \over l_{\zeta}! (d-2r-2l_{\zeta}-q)!} (\a^2)^{l_{\zeta}} (\S\cdot\a)^q + $$ $$ + 4 a^{d-2r-2l_{\zeta}-q +1} { (d-2r)! \, q \over l_{\zeta}! (d-2r-2l_{\zeta}-q+1)!} (\a^2)^{l_{\zeta}} (\S\cdot\a)^{q-1}(\S\cdot\zeta) \Big] \text{vol}. $$ \end{cor} \section{Conjecture} \label{sec:conj} It is natural to propose the following \noindent {\bf Conjecture.} Let $X$ be an oriented compact four-manifold with $b^+=1$ and $b_1=2q$ even. Let $w \in H^2(X;{\Bbb Z})$. Choose $\S \in H^2(X)$ generating the image of $\wedge: H^1(X) \otimes H^1(X) \rightarrow H^2(X)$. Define $\o \in H^2(J)$ such that $e_{\a}=-2(\S\cdot \a)\o$ and put $\text{vol}=\int_J \frac{\o^n}{n!}$. If $\zeta$ defines a wall, then the wall-crossing difference term $\d_{X,\zeta}^{w,d} (x^r\a^{d-2r})$ only depends on $w$, $d$, $r$, $b_1=2q$, $b_2$, $\zeta^2$, $\a^2$, $(\zeta \cdot \a)$ and $(\S\cdot \a)^i(\S\cdot \zeta)^{q-i} \text{vol}$, $0 \leq i \leq q$. The coefficients are universal on $X$. This is quite a strong conjecture and one can obviously write down weaker versions. It would allow one to carry out similar arguments to those in~\cite{Gottsche} and therefore to find out the general shape of the wall-crossing formulae for arbitrary $X$, involving modular forms. One should be able to determine then all wall-crossing formulae from particular cases. This and applications to computing the invariants of ${\Bbb C \Bbb P}^1 \times C_g$ ($C_g$ the genus $g$ Riemann surface) will be carried out in following joint work with L.\ G\"ottsche. \section*{Appendix. Algebraic surfaces with $p_g=0$ and $-K$ effective} From~\cite{BPV}, the algebraic surfaces with $p_g=0$ and $-K$ effective are ${\Bbb C \Bbb P}^2$, ruled surfaces and blow-ups of these. For the case $q=0$, we have thus ${\Bbb C \Bbb P}^2$, the Hirzebruch surfaces and their blow-ups. Not all blow-ups have $-K$ effective, but they are always deformation equivalent to one with $-K$ effective. For the case $q>0$, the minimal models are ruled surfaces over a surface $C_g$ of genus $g \geq 1$. They have $c_1^2=8(1-g)$. Let $S \to C_g$ be a ruled surface. It has $b_2=2$ and $b_1=2g$, so $g=q$. Let $f$ be the class of the fibre and $\sigma=\sigma_{-N}$ the class of the section with negative self-intersection $\sigma_{-N}^2=-N \leq 0$. Then there is a section $\sigma_N$ homologically equivalent to $\sigma_{-N}+Nf$ with square $N$. Write $X={\Bbb P}(V^{\vee})$, for $V \to C_g$ a rank two bundle. Then $K={\frak a} f-2\sigma$, with ${\frak a}=\sigma^2 +K_{C_g}$ a divisor on $C_g$ (see~\cite[section 5.2]{Hartshorne}). Therefore $-K$ is effective if and only if $-{\frak a}$ is effective. The section $\sigma$ corresponds to a sub-line bundle $L \hookrightarrow V$ with ${\cal O}_{C_g}(\sigma^2)=L^{-2}\otimes \det(V)$. Then $-{\frak a}$ is effective when $L^2 \otimes \det(V)^{-1} \otimes K_{C_g}^{-1}$ has sections. We can find examples for any $N$ as long as $N \geq 2(g-1)$. Again, the non-minimal examples are blow-ups of these, and can be found to have $-K$ effective. For fixed $q=g >0$, there are only two deformation classes of minimal ruled surfaces, corresponding to two diffeomorphism types, the two different ${\Bbb S}^2$-bundles over $C_g$, one with even $w_2$, the other with odd $w_2$. \begin{itemize} \item {\bf $N$ even:} $S$ is diffeomorphic to $S_0={\Bbb C \Bbb P}^1 \times C_g$ (and the canonical classes correspond). Let $C$ be the homology class of\, $\pt \times C_g$ coming from the diffeomorphism. Then $\sigma$ is homologous to $C -{N \over 2}f$. The ample cone $C_S$ of $S$ is generated by $f$ and $\sigma_N= C + {N \over 2}f$ (i.e. it is given by ${\Bbb R}^+ f + {\Bbb R}^+ \sigma_N$). Note that the bigger $N$, the smaller the ample cone. The wall-crossing terms $\d_{S,\zeta}^{w,d}$ do not depend on the complex structure of $S$, so our results for the case $-K$ effective give the wall-crossing terms for $S_0$ for any wall inside $C_S$. Letting $N=2(g-1)$, we actually compute $\d_{S,\zeta}^{w,d}$ for any $\zeta= a\, {\Bbb C \Bbb P}^1 -b \,C$ with $a,b >0$, $a> b (g-1)$ (note that all these walls are good). \item {\bf $N$ odd:} $S$ is diffeomorphic to the non-trivial ${\Bbb S}^2$-bundle over $C_g$. Arguing as above, we compute the wall-crossing terms $\d_{S,\zeta}^{w,d}$ for any $\zeta= a \, {\Bbb C \Bbb P}^1 -b \, \sigma_{-(2g-1)}$ with $a,b >0$, $a> b {2g-1 \over 2}$. \end{itemize}

9709

alg-geom/9709029

en

https://arxiv.org/abs/alg-geom/9709029

alg-geom/9709029

Robert Friedman

Robert Friedman, John W. Morgan, and Edward Witten

Vector Bundles over Elliptic Fibrations

101 pages, AMS-TeX, amsppt style

This paper gives various methods for constructing vector bundles over elliptic curves and more generally over families of elliptic curves. We construct universal families over generalized elliptic curves via spectral cover methods and also by extensions, and then give a relative version of the construction in families. We give various examples and make Chern class computations.

alg-geom

\section{Introduction} Let $\pi \: Z \to B$ be an elliptic fibration with a section. The goal of this paper is to study holomorphic vector bundles over $Z$. We are mainly concerned with vector bundles $V$ with trivial determinant, or more generally such that $\det V$ has trivial restriction to each fiber, so that $\det V$ is the pullback of a line bundle on $B$. (The case where $\det V$ has nonzero degree on every fiber is in a certain sense simpler, since it usually reduces to the case considered here for a bundle of smaller rank.) We give two constructions of vector bundles, one based on the idea of a spectral cover of $B$ and the other based on the idea of extensions of certain fixed bundles over the elliptic manifold $Z$. Each of these constructions has advantages and the combination of the two seems to give the most comprehensive information. Vector bundles over a single elliptic curve were first classified by Atiyah \cite{1}; however, he did not attempt to construct universal bundles or work in families. The case of rank two bundles over an elliptic surface was studied in \cite{3}, \cite{6}, \cite{4} with a view toward making computations in Donaldson theory. The motivation for this paper and the more general study of the moduli of principal $G$-bundles over families of elliptic curves (which will be treated in another paper) grew out of questions arising in the recent study of $F$-theory by physicists. The explanation of these connections was given in \cite{7}. For these applications $Z$ is assumed to be a Calabi-Yau manifold, usually of dimension two or three. However, most of the results on vector bundles and more generally $G$-bundles are true with no assumptions on $Z$. The case of a general simple and simply connected complex Lie group $G$ involves a fair amount of algebraic group theory and will be treated elsewhere, but the case $G=SL_n(\Cee)$ can be done in a quite explicit and concrete way, and that is the subject of this paper. For both mathematical and physical reasons, we shall be primarily interested in constructing stable vector bundles on $Z$. Of course, stability must be defined with respect to a suitable ample divisor. Following well-established principles, the natural ample divisors to work with are those of the form $H_0 + N\pi^*H$ for $N\gg 0$, where $H_0$ is some fixed ample divisor on $Z$ and $H$ is an ample divisor on $B$. If $V$ is stable with respect to such a divisor, then $V|f$ is semistable with respect to almost all fibers $f$. (However the converse is not necessarily true.) One special feature of vector bundles $V$ with trivial determinant on an elliptic curve is that, if the rank of $V$ is at least two, then $V$ is never properly stable. Moreover, if $V$ has rank $n>1$, then $V$ is never simple; in fact, the endomorphism algebra of $V$ has dimension at least $n$. But there is still a relative coarse moduli space $\Cal M_{Z/B}$, which turns out to be a $\Pee^{n-1}$-bundle over $B$. A stable vector bundle on $Z$ defines a rational section of $\Cal M_{Z/B}$. Conversely, a regular section of $\Cal M_{Z/B}$ defines a vector bundle over $Z$, and in fact it defines many such bundles. Our goal will be to describe all such bundles, to see how the properties of the section are reflected in the properties of these bundles, and to find sufficient conditions for the bundles in question to be stable. In the first three sections we consider a single (generalized) elliptic curve $E$. In Section 1 we construct a coarse moduli space for $S$-equivalence classes of semistable $SL_n(\Cee)$-bundles over $E$. It is a projective space $\Pee^{n-1}$, in fact it is the projective space of the complete linear system $|np_0|$ where $p_0\in E$ is the origin of the group law. It turns out that each $S$-equivalence class of semistable bundles has a ``best" representative, the so-called regular representative. The defining property of these bundles, at least when $E$ is smooth, is that their automorphism groups are of the smallest possible dimension, namely $n$. We view them as analogues of regular elements in the group $SL_n(\Cee)$. The moduli space we construct is also the coarse moduli space for isomorphism classes of regular semistable $SL_n(\Cee)$-bundles over $E$. As we shall see, the regular bundles are the bundles which arise if we try to fit together the $S$-equivalence classes in order to find universal holomorphic bundles over $\Pee^{n-1}\times E$. In Section 2, assuming that $E$ is smooth, we construct a tautological bundle $U$ over $\Pee^{n-1}\times E$ which is regular semistable and with trivial determinant on each slice $\{x\}\times E$ and such that $U|\{x\}\times E$ corresponds to the regular bundle over $E$ whose $S$-equivalence class is $x$. There is not a unique such bundle over $\Pee^{n-1}\times E$, and we proceed to construct all such. The idea is that there is an $n$-sheeted covering $T\to \Pee^{n-1}$ called the {\sl spectral cover\/}, such that $U$ is obtained by pushing down a Poincar\'e line bundle ${\Cal P}\to T\times E$ under the covering map. It turns out that every bundle over $\Pee^{n-1}\times E$ which is of the correct isomorphism class on each slice $\{x\}\times E$ is obtained by pushing down ${\Cal P}\otimes p_1^*M$ for some line bundle $M$ on $T$. There is a generalization of this result to cover the case of families of regular semistable bundles on $E$ parameterized by arbitrary spaces $S$. In Section 3 we turn to a different construction of ``universal" bundles over $\Pee^{n-1}\times E$. Here we consider the space of extensions of two fixed bundles with determinants $\scrO_E(\pm p_0)$. For a fixed rank $d$, there is a unique stable bundle $W_d$ of rank $d$ such that $\det W_d \cong \scrO_E(p_0)$. For $1\leq d\leq n-1$, we consider the space of all nonsplit extensions $V$ of the form $$0 \to W_d\spcheck \to V \to W_{n-d} \to 0.$$ The moduli space of all such extensions is simply $\Pee H^1(W_{n-d}\spcheck \otimes W_d\spcheck) \cong \Pee^{n-1}$. Over $\Pee^{n-1} \times E$ there is a universal extension whose restriction to each fiber is regular semistable. There is thus an induced map from the $\Pee^{n-1}$ of extensions to the coarse moduli space defined in Section 1, which is $|np_0|\cong \Pee^{n-1}$. By a direct analysis we show that this map is an isomorphism. Actually, there are $n-1$ different versions of this construction, depending on the choice of the integer $d$, but the projective spaces that they produce are all canonically identified. On the other hand, the universal extensions associated with different versions of the construction are non-isomorphic universal bundles. Finally, we relate these families of bundles to the ones arising from the spectral cover construction, which we can then extend to the case where $E$ is singular. We remark here that we can interpret the construction of Section 3 as parametrizing those bundles whose structure group can be reduced to a maximal parabolic subgroup $P$ of $SL_n$, such that the induced bundle on the Levi factor is required to be $W_d\spcheck \oplus W_{n-d}$ in the obvious sense. This interpretation can then be generalized to other complex simple groups \cite{8}. In Section 4 we generalize the results of the first three sections to a family $\pi \: Z\to B$ of elliptic curves with a section $\sigma$. By taking cohomology along the fibers of $\pi$, we produce a vector bundle over the base, namely $\pi _*\scrO_Z(n\sigma)=\Cal V_n$, which globalizes $H^0(E,\scrO_E(np_0))$. The associated projective bundle $\Pee\pi _*\scrO_Z(n\sigma)=\Pee\Cal V_n$ then becomes the appropriate relative coarse moduli space. We show that $\pi _*\scrO_Z(n\sigma)$ has a natural splitting as a direct sum of line bundles. This decomposition is closely related to the fact that the coefficients of the characteristic polynomial of an element in $\frak{sl}_n$ are a polynomial basis for the algebra of polynomial functions on $\frak{sl}_n$ invariant under the adjoint action. Having constructed the relative coarse moduli space, we give a relative version of the constructions of Sections 2 and 3 to produce bundles over $\Pee\Cal V_n\times _BZ$. The extension construction generalizes easily. The bundles we used over a single elliptic curve have natural extensions to any elliptic fibration. We form the relative extension bundle and the universal relative extension in direct analogy with the case of a single elliptic curve. Relative versions of results from Section 3 show that the relative extension space is identified with $\Pee\Cal V_n$. Following the pattern of Section 3, we use the extension picture to define a universal spectral cover of $\Pee\Cal V_n$, and in turn use this spectral cover to construct new universal vector bundles. Finally, we calculate the Chern classes of the universal bundles we have constructed. In Section 5, using the theory developed in the first four sections, we study vector bundles $V$ over an elliptic fibration $\pi\: Z\to B$ such that the restriction of $V$ to every fiber is regular and semistable. To such a bundle $V$, we associate a section $A(V)$ of $\Pee\Cal V_n$ and a cover $C_A\to B$ of degree $n$, the {\sl spectral cover\/} of $B$ determined by $V$. Conversely, $V$ is determined by $A$ and by the choice of a line bundle on $C_A$. After computing some determinants and Chern classes, we discuss the possible line bundles which can exist on the spectral cover. Then we turn to specific types of bundles. After describing symmetric bundles, which are interesting from the point of view of $F$-theory, we turn to bundles corresponding to a degenerate section. First we consider the most degenerate case, and then we consider reducible sections where the restriction of $V$ to every fiber has a section. Finally, we relate reducible sections to the existence of certain subbundles of $V$. In Section 6, we consider bundles $V$ whose restriction to a generic fiber is regular and semistable, but such that there exist fibers $E_b$ where $V|E_b$ is either unstable or it is semistable but not regular. If $V$ fails to be regular or semistable in codimension one, it can be improved by elementary modifications to a reflexive sheaf whose restriction to every fiber outside a codimension two set is regular and semistable. We describe this process and, as an illustration, analyze the tangent bundle to an elliptic surface. On the other hand, if the locus of bad fibers has codimension at least two, no procedure exists for improving $V$, and we must analyze it directly. The case of instability in codimension two or higher corresponds to the case where the rational section $A$ determined by $V$ does not actually define a regular section (this case can also lead to reflexive but non-locally free sheaves). The case where $V$ has irregular restriction to certain fibers in codimension at least two corresponds to singular spectral covers. We give some examples of such behavior, without trying to be definitive. Our construction can be viewed as a generalization of the method of Section 3 to certain non-maximal parabolic subgroups of $SL_n$. Finally, we turn in Section 7 to the problem of deciding when the bundles $V$ constructed by our methods are stable. This is the most interesting case for both mathematical and physical reasons. While we do not try to give necessary and sufficient conditions, we show that, in case the spectral cover $C_A$ of $B$ determined by $V$ is irreducible, then $V$ is stable with respect to all ample divisors of the form $H_0 + N\pi ^*H$, where $H_0$ is an ample divisor on $Z$ and $H$ is an ample divisor on $B$, and $N\gg 0$. We are only able to give an effective bound on $N$ in case $\dim B =1$, i\.e\. $Z$ is an elliptic surface, but it seems likely that such an effective bound exists in general. We will have to deal systematically with singular fibers of $Z\to B$, and the price that must be paid for analyzing this case is a heavy dose of commutative algebra. In an attempt to make the paper more readable, we have tried to isolate these arguments where possible. We collect here some preliminary definitions and technical results. While these results are well-known, we could not find an adequate reference for many of them. \ssection{Notation and conventions.} All schemes are assumed to be separated and of finite type over $\Cee$. A sheaf is always a coherent sheaf. We will identify a vector bundle with its locally free sheaf of sections, covariantly. If $V$ is a vector bundle, then $\Pee V$ is the projective space bundle whose associated sheaf of graded algebras is $\bigoplus _{k\geq 0}\Sym ^kV\spcheck$; thus these conventions are opposite to those of EGA or \cite{10}. Given sheaves $\Cal S, \Cal S'$, we denote by $Hom(\Cal S, \Cal S')$ the sheaf of homomorphisms from $\Cal S$ to $\Cal S'$ and by $\Hom(\Cal S, \Cal S') = H^0(Hom(\Cal S, \Cal S'))$ the group of all such homomorphisms. Likewise $Ext ^k(\Cal S, \Cal S')$ is the Ext sheaf and $\Ext ^k(\Cal S, \Cal S')$ is the global Ext group (related to the local Ext groups by the local to global spectral sequence). \ssection{0.1. Elliptic curves and elliptic fibrations.} Recall that a {\sl Weierstrass equation\/} is a homogeneous cubic equation of the form $$Y^2Z=4X^3-g_2XZ^2-g_3Z^3,\tag{0.1}$$ with $g_2,g_3$ constants. We will refer to the curve $E$ in $\Pee^2$ defined by such an equation, together with the marked point $p_0=[0,1,0]$ at infinity, as a {\sl Weierstrass cubic}. Setting $$\Delta(g_2,g_3)=g_2^3- 27g_3^2,$$ if $\Delta(g_2,g_3)\neq 0$, then (0.1) defines a smooth cubic curve in $\Pee^2$ with the marked point $[0,1,0]$, i\.e\., defines the structure of an elliptic curve. If $\Delta(g_2,g_3) =0$, then the corresponding plane cubic $E$ is a singular curve with arithmetic genus $p_a(E) =1$. If $(g_2, g_3)$ is a smooth point of the locus $\Delta(g_2,g_3) =0$, then the corresponding plane cubic curve is a rational curve with a single node. The smooth points of such a curve form a group isomorphic to $\Cee^*$ with identity element $p_0$. The point $g_2= g_3=0$ is the unique singular point of $\Delta(g_2,g_3) =0$ and the corresponding plane curve is a rational curve with a single cusp. Once again its smooth points form a group, isomorphic to $\Cee$, with identity element $p_0$. These are all possible reduced and irreducible curves of arithmetic genus one. Next we consider the relative version of a Weierstrass equation. Let $\pi \: Z \to B$ be a flat morphism of relative dimension one, such that the general fiber is a smooth elliptic curve and all fibers are isomorphic to reduced irreducible plane cubics. Here we will assume that $B$ is a smooth variety (although the case of a complex manifold is similar). We shall always suppose that $\pi$ has a section $\sigma$, i\.e\. there exists a divisor $\sigma$ contained in the smooth points of $Z$ such that $\pi|\sigma$ is an isomorphism. Let $L = R^1\pi_*\scrO_Z \cong \scrO_Z(-\sigma)|\sigma$, viewed as a line bundle on $B$. Then there are sections $G_2 \in H^0(B; L^{\otimes 4})$ and $G_3 \in H^0(B; L^{\otimes 6})$ such that $\Delta (G_2, G_3)\neq 0$ as a section of $L^{\otimes 12}$, and $Z$ is isomorphic to the subvariety of $\Pee(\scrO_B\oplus L^2\oplus L^3)$ defined by the Weierstrass equation $Y^2Z=4X^3-G_2XZ^2-G_3Z^3$. Conversely, given the line bundle $L$ on $B$ and sections $G_2 \in H^0(B; L^{\otimes 4})$, $G_3 \in H^0(B; L^{\otimes 6})$ such that $\Delta (G_2, G_3)\neq 0$, the equation $Y^2Z=4X^3-G_2XZ^2-G_3Z^3$ defines a hypersurface $Z$ in $\Pee(\scrO_B\oplus L^2\oplus L^3)$, such that the projection to $B$ is a flat morphism whose fibers are reduced irreducible plane curves, generically smooth. We will not need to assume that $Z$ is smooth; it is always Gorenstein and the relative dualizing sheaf $\omega_{Z/B}$ is isomorphic to $ L$. Thus, the dualizing sheaf $\omega _Z$ is isomorphic to $\pi^*K_B\otimes L$. Let us describe explicitly the case where the divisors associated to $G_2$ and $G_3$ are smooth and meet transversally. This means in particular that if $G_2$ and $G_3$ are chosen generically, then $G_2^3 - 27G_3^2$ defines a section of $L^{12}$. We shall denote by $\overline{\Gamma}$ the zero set of this section. Then $\overline{\Gamma}$ is smooth except where $G_2 = G_3 = 0$, where it has singularities which are locally trivial families of cusps. The fiber of $\pi$ over a smooth point of $\overline{\Gamma}$ is a nodal plane cubic, and over a point where $G_2 = G_3 = 0$ the fibers of $\pi$ are cusps. Let $\Gamma$ be the locus of points where $\pi$ is singular. Thus $\Gamma$ maps bijectively onto $\overline{\Gamma}$. There are local analytic coordinates on $B$ so that, near a cuspidal fiber $Z$ has the local equation $y^2 = x^3 + sx+t$. Here $x,y$ are a set of fiber coordinates for $\Pee(\scrO_B\oplus L^2\oplus L^3)$ away from the line at infinity and $x,y,s,t$ form part of a set of local coordinates for $\Pee(\scrO_B\oplus L^2\oplus L^3)$. Thus $x,y,s$ are coordinates for $Z$. The local equation for $\overline{\Gamma}$ is $4s^3 + 27 t^2 = 0$. The equations for the singular point of the fiber over $\overline{\Gamma}$ are as follows: $y=0, s = -3x^2, t= 2x^3$. In particular, $\Gamma$ is smooth, and is the normalization of $\overline{\Gamma}$. The morphism from $Z$ to $B$ is given locally by $(s,t)$, where $t= y^2-x^3-sx$. \ssection{0.2. Rank one torsion free sheaves.} Let $E$ be a singular Weierstrass cubic and let $E_{\text{reg}}$ be the set of smooth points of $E$. The arithmetic genus $p_a(E)$ is one. We let $n\: \tilde E \to E$ be the normalization map. The generalized Jacobian $J(E)$ is the group of line bundles of degree zero on $E$, and (as in the smooth case) is isomorphic to $E_{\text{reg}}$ via the map $e\in E_{\text{reg}}\mapsto \scrO_E(e-p_0)$. Just as we can compactify $E_{\text{reg}}$ to $E$ by adding the singular point, we can compactify $J(E)$ to the {\sl compactified generalized Jacobian\/} $\bar J(E)$, by adding the unique rank one torsion free sheaf which is not locally free. Here a sheaf $\Cal S$ over $E$ is {\sl torsion-free\/} if it has no nonzero sections which are supported on a proper closed subset (i\.e\. a finite set). In particular, the restriction of $\Cal S$ to the smooth points of $E$ is a vector bundle, and so has a well-defined rank, which we also call the rank of $\Cal S$. If $\Cal S$ is a torsion-free sheaf on $E$ we let $\deg \Cal S = \chi (\Cal S) + (p_a(E)-1)(\operatorname{rank} \Cal S) = \chi (\Cal S)$. (This agrees with the usual Riemann-Roch formula in case $E$ is smooth.) Thus the degree of such sheaves is additive in exact sequences, and if $\Cal S'\subseteq \Cal S$ such that the quotient is supported at a finite set of points, then $\deg \Cal S' \leq \deg \Cal S$ with equality if and only if $\Cal S' = \Cal S$. If $\Cal S$ is torsion free and $V$ is locally free, then $\deg (V\otimes \Cal S)= (\deg V)(\operatorname{rank}\Cal S) + (\deg \Cal S)(\operatorname{rank} V)$. To see this, first use the fact that there is a filtration of $V$ by subbundles whose successive quotients are line bundles, so by the additivity of degree we can reduce to the case where $V$ is a line bundle. In this case, we may write $V= \scrO_E(d_1- d_2)$, where $d_1$ and $d_2$ are effective divisors supported on the smooth points of $E$, and then use the exact sequences $$0 \to \Cal S \otimes \scrO_E(-d_2) \to \Cal S \to \Cal S\otimes \scrO_{d_2} \to 0$$ and $$0 \to \Cal S \otimes \scrO_E(-d_2) \to \Cal S \otimes \scrO_E(d_1-d_2)\to \Cal S\otimes \scrO_{d_1} \to 0,$$ together with the usual properties, to conclude that $\deg (V\otimes \Cal S)= (\deg V)(\operatorname{rank}\Cal S) + (\deg \Cal S)$ in case $V$ is a line bundle. Thus we have established the formula in general. Next let us show that there is a unique torsion free rank one sheaf $\Cal F$ which compactifies the generalized Jacobian. \lemma{0.2} There is a unique rank one torsion free sheaf $\Cal F$ on $E$ of degree zero which is not locally free. It satisfies: \roster \item"{(i)}" $Hom(\Cal F, \Cal F) =n_*\scrO_{\tilde E}$. \item"{(ii)}" $\Cal F\spcheck \cong \Cal F$. \item"{(iii)}" For all line bundles $\lambda$ of degree zero, $Hom(\lambda, \Cal F) = Hom (\Cal F, \lambda) = \Cal F$ and $\Hom(\lambda, \Cal F) = \Hom (\Cal F, \lambda) = 0$. Likewise $\Ext^1(\Cal F, \lambda) = \Ext ^1(\lambda, \Cal F) = 0$. \endroster \endstatement \proof The first statement is essentially a local result. Let $R$ be the local ring of $E$ at the singular point and let $\tilde R$ be the normaliztion of $R$. If locally $\Cal F$ corresponds to the $R$-module $M$, let $\tilde M$ be the $\tilde R$-module $M\otimes _R\tilde R$ modulo torsion. Then by construction $\tilde M$ is a torsion free rank one $\tilde R$-module, so that we may choose a $\tilde R$-module isomorphism from $\tilde M$ to $\tilde R$. Since $M$ is torsion free, the natural map from $M$ to $\tilde M \cong \tilde R$ is injective, identifying $M$ as an $R$-submodule of $\tilde R$ which generates $\tilde R$ as a $\tilde R$-module. Thus $M$ contains a unit of $\tilde R$, which after a change of basis we may assume to be $1$, and furthermore $M$ contains $R\cdot 1=R\subseteq \tilde R$. But since the singularity of $E$ is a node or a cusp, $\ell(\tilde R/R) =1$, and so either $M=R$ or $M=\tilde R$. Note that there are two isomorphic non-locally free $R$-modules of rank one: $\tilde R$ and $\frak m$, where $\frak m$ is the maximal ideal of $R$. The ideal $\frak m$ is the conductor of the extension $\tilde R$ of $R$, and $\Hom_R(\tilde R, R) \cong \frak m$, where the isomorphism is canonical. By the above, every rank one torsion free sheaf on $E$ is either a line bundle or of the form $n_*L$, where $L$ is a line bundle on $\tilde E$. Now $\tilde E\cong \Pee^1$, and $\deg n_*\scrO_{\Pee^1}(a) = a+1$. Thus $n_*\scrO_{\Pee^1}(-1)$ is the unique rank one torsion free sheaf on $E$ of degree zero which is not locally free. Note that, if $\frak m_x$ is the ideal sheaf of the singular point $x\in E$, then $\deg \frak m_x = -1$, by using the exact sequence $$0 \to \frak m_x \to \scrO_E \to \Cee _x \to 0.$$ Thus $\frak m_x=n_*\scrO_{\tilde E}(-2)$. To see (i), note that $$Hom (n_*\scrO_{\Pee^1}(-1), n_*\scrO_{\Pee^1}(-1)) = n_*Hom(n^*n_*\scrO_{\Pee^1}(-1), \scrO_{\Pee^1}(-1)).$$ Since $n$ is finite, the natural map $n^*n_*\scrO_{\Pee^1}(-1) \to \scrO_{\Pee^1}(-1)$ is surjective, and its kernel is torsion. Thus $$Hom(n^*n_*\scrO_{\Pee^1}(-1), \scrO_{\Pee^1}(-1)) \cong Hom(\scrO_{\Pee^1}(-1), \scrO_{\Pee^1}(-1)) = \scrO_{\Pee^1}= \scrO_{\tilde E},$$ proving (i). To see (ii), we have invariantly that $$Hom(n_*\scrO_{\tilde E}, \scrO_E) = \frak m_x=n_*\scrO_{\tilde E}(-2).$$ Thus tensoring with $\scrO_E(-p_0)$ and using $n_*\scrO_{\tilde E}\otimes \scrO_E(-p_0) = n_*n^*\scrO_E(-p_0) = n_*\scrO_{\tilde E}(-1)$ gives $$\gather Hom(n_*\scrO_{\tilde E}(-1), \scrO_E)= Hom(n_*\scrO_{\tilde E}\otimes \scrO_E(-p_0),\scrO_E)= \\ =n_*\scrO_{\tilde E}(-2)\otimes \scrO_E(p_0) = n_*\scrO_{\tilde E}(-1), \endgather$$ which is the statement that $\Cal F\spcheck \cong \Cal F$. To see (iii), if $\lambda$ is a line bundle of degree zero, then $Hom(\lambda, \Cal F) =\lambda^{-1}\otimes \Cal F$ is a nonlocally free sheaf of degree zero, and hence it is isomorphic to $\Cal F$ by uniqueness. Likewise $$Hom (\Cal F, \lambda) \cong \lambda \otimes \Cal F\spcheck \cong \lambda \otimes \Cal F \cong \Cal F.$$ Moreover, $\Hom(\lambda, \Cal F) = H^0(\Cal F) = 0$, since by degree considerations a nonzero map $\lambda ^{-1}\to \Cal F$ would have to be an isomorphism, contradicting the fact that $\Cal F$ is not locally free. The proof that $\Hom (\Cal F, \lambda)=0$ is similar. Now $\Ext^1(\Cal F, \lambda)$ is Serre dual to $\Hom(\lambda,\Cal F)=0$, since $\lambda$ is locally free. Also, $\Ext ^1(\lambda, \Cal F) = H^1(\lambda^{-1}\otimes \Cal F) = H^1(\Cal F) =0$, since $h^0(\Cal F) = \deg \Cal F =0$. \endproof \remark{Remark} In case $E$ is nodal, $\Ext ^1(\Cal F, \Cal F)$ is not Serre dual to $\Hom (\Cal F, \Cal F)$, and in fact $\Ext ^1(\Cal F, \Cal F)\cong H^0(Ext ^1(\Cal F, \Cal F))$ has dimension two. In this case $\Pee\Ext ^1(\Cal F, \Cal F)\cong \Pee^1$ can be identified with the normalization of $E$. The preimages $\{x_1, x_2\}$ of the singular point give two different non-locally free extensions, and the remaining locally free extensions $V$ of $\Cal F$ by $\Cal F$ are parametrized by $\Pee^1-\{x_1, x_2\}\cong \Cee^*$. The set of such $V$ is in $1-1$ correspondence with $J(E) \cong E$ via the determinant. \endremark \medskip Next we define the compactified generalized Jacobian of $E$. Let $\Delta _0$ be the diagonal in $E\times E$ and let $I_{\Delta _0}$ be its ideal sheaf. We let $\scrO_{E\times E}(\Delta _0) = I_{\Delta _0}\spcheck$ and $$\Cal P_0 = \scrO_{E\times E}(\Delta _0-(E\times \{p_0\})) = I_{\Delta _0}\spcheck\otimes \pi _2^*\scrO_E(-p_0).$$ \lemma{0.3} In the above notation, \roster \item"{(i)}" $\Cal P_0$ is flat over both factors of $E\times E$, and $\Cal P_0\spcheck$ is locally isomorphic to $I_{\Delta _0}$. \item"{(ii)}" If $e$ is a smooth point of $E$, the restriction of $\Cal P_0$ to the slice $\{e\}\times E$ is $\scrO_E(e-p_0)$. If $x$ is the singular point of $E$, the restriction of $\Cal P_0$ to the slice $\{x\}\times E$ is $\Cal F$. \item"{(iii)}" Suppose that $S$ is a scheme and that $\Cal L$ is a coherent sheaf on $S\times E$, flat over $S$, such that for every slice $\{s\}\times E$, the restriction of $\Cal L$ to $\{s\}\times E$ is a rank one torsion free sheaf on $E$ of degree zero. Then there exists a unique morphism $f\: S \to E$ and a line bundle $M$ on $S$ such that $\Cal L = (f\times \Id)^*\Cal P_0\otimes \pi _1^*M$. \endroster \endstatement \proof We shall just outline the proof of this essentially standard result. The proofs of (i) and (ii) in case $\Cal P_0$ is replaced by $I_{\Delta _0}$, with the necessary changes in (ii), are easy: From the exact sequence $$0 \to I_{\Delta _0} \to \scrO_{E\times E} \to \scrO_{\Delta _0} \to 0,$$ and the fact that both $E\times E$ and $\Delta _0$ are flat over each factor, we see that $I_{\Delta _0}$ is flat over both factors, and the restriction of $I_{\Delta _0}$ to the slice $\{e\}\times E$ is $\scrO_E(-e)$, if $e\neq x$, and is $\frak m_x$ in case $e=x$. To handle the case of $\Cal P_0$, the main point is to check that $\scrO_{E\times E}(\Delta _0) = I_{\Delta _0}\spcheck$ is locally isomorphic to $I_{\Delta _0}$, and that the inclusion $I_{\Delta _0} \to \scrO_{E\times E}$ dualizes to give an exact sequence $$0 \to \scrO_{E\times E} \to \scrO_{E\times E}(\Delta _0) \to \scrO_{\Delta _0} \to 0.$$ This may be checked by hand, by working out a local resolution of $I_{\Delta _0}$. We omit the details. Another, less concrete, proof which generalizes to a flat family $\pi\: Z \to B$ is as follows. Dualizing the inclusion of $I_{\Delta _0} \to \scrO_{E\times E}$ gives an exact sequence $$0 \to \scrO_{E\times E} \to I_{\Delta _0}\spcheck \to Ext^1(\scrO_{\Delta_0},\scrO_{E\times E}) \to 0.$$ To check flatness of $I_{\Delta _0}\spcheck$ and the remaining statements of (ii), it suffices to show that, locally, $Ext^1(\scrO_{\Delta_0},\scrO_{E\times E}) \cong \scrO_{\Delta_0}$. Clearly $Ext^1(\scrO_{\Delta_0},\scrO_{E\times E})$ is a sheaf of $\scrO_{\Delta_0}$-modules and thus it is identified with a sheaf on $E$ via the first projection. If $\pi_1, \pi_2\: E\times E\to E$ are the projections, we have the relative Ext sheaves $Ext^i_{\pi_1}(\scrO_{\Delta_0},\scrO_{E\times E})$. (See for example \cite{2} for properties of these sheaves.) The curve $E$ is Gorenstein and thus $\Ext^1(\Cee_x, \scrO_E) \cong \Cee$ for all $x\in E$. By base change, $Ext^1_{\pi_1}(\scrO_{\Delta_0},\scrO_{E\times E})$ is a line bundle on $E$. On the other hand, by the local to global spectral sequence, $$Ext^1_{\pi_1}(\scrO_{\Delta_0},\scrO_{E\times E})= \pi_1{}_*Ext^1(\scrO_{\Delta_0},\scrO_{E\times E}).$$ Thus $Ext^1(\scrO_{\Delta_0},\scrO_{E\times E})$ can be identified with a line bundle on $\Delta _0$, and so it is locally isomorphic to $\scrO_{\Delta_0}$. Dualizing this argument gives an exact sequence (locally) $$0 \to \Cal P_0\spcheck \to \scrO_{E\times E} \to Ext^1(\scrO_{\Delta_0},\scrO_{E\times E}) \to 0,$$ and so (locally again) $\Cal P_0\spcheck = (I_{\Delta _0})\ddual \cong I_{\Delta _0}$. In particular $\Cal P_0\spcheck$ is also flat over $E$. To see (iii), suppose that $S$ and $\Cal L$ are as in (iii). By base change, $\pi_1{}_*(\Cal L\otimes\pi_2^*\scrO_E(p_0)) = M^{-1}$ is a line bundle on $S$, and the morphism $$\pi _1^* \pi_1{}_*(\Cal L\otimes\pi_2^*\scrO_E(p_0)) =\pi _1^*M ^{-1}\to \Cal L\otimes\pi_2^*\scrO_E(p_0)$$ vanishes along a subscheme $\Cal Z$ of $S\times E$, flat over $S$ and of degree one on every slice. Thus $\Cal Z$ corresponds to a morphism $f\: S\to E= \operatorname{Hilb}^1E$, such that $\Cal Z$ is the pullback of $\Delta_0\subset E\times E$ by $(f\times \Id)^*$. This proves (iii). \endproof A very similar argument proves the corresponding result for the dual of the ideal of the diagonal in $Z\times _BZ$, where $\pi\: Z \to B$ is a flat family of Weierstrass cubics. In this case, we let $\Delta _0$ be the ideal sheaf of the diagonal in $Z\times _BZ$, and set $\Cal P_0 = I_{\Delta _0}\spcheck \otimes \pi _2^*\scrO_Z(-\sigma)$, where $\sigma$ is the section. Then $\Cal P_0$ is flat over both factors $Z$, and has the properties (i)--(iii) of (0.3). We leave the details of the formulation and the proof to the reader. Finally we discuss a local result which will be needed to handle semistable sheaves on a singular $E$. (In the application, $R$ is the local ring of $E$ at a singular point.) \lemma{0.4} Let $R$ be a local Cohen-Macaulay domain of dimension one and let $Q$ be a finitely generated torsion free $R$-module. Then $\Ext^1_R(Q, R) = 0$. \endstatement \proof By a standard argument, if $Q$ has rank $n$ there exists an inclusion $Q\subseteq R^n$. Thus necessarily the quotient $R^n/Q$ is a torsion $R$-module $T$. Now $\Ext^1_R(Q, R) \cong \Ext^2_R(T, R)$. Since $R$ is Cohen-Macaulay, if $\frak m$ is the maximal ideal of $R$, then $\Ext^2_R(R/\frak m, R) = 0$. An induction on the length of $T$ then shows that $\Ext^2_R(T, R)=0$ for all $R$-modules $T$ of finite length. Hence $\Ext^1_R(Q, R) = 0$. \endproof \ssection{0.3. Semistable bundles and sheaves on singular curves.} Let $E$ be a Weierstrass cubic and let $\Cal S$ be a torsion free sheaf on $E$. The {\sl normalized degree\/} or {\sl slope\/} $\mu (\Cal S)$ of $\Cal S$ is defined to be $\deg \Cal S/\operatorname{rank} \Cal S$. A torsion free sheaf $\Cal S$ is {\sl semistable\/} if, for every subsheaf $\Cal S'$ of $\Cal S$ with $0< \operatorname{rank} \Cal S' <\operatorname{rank} \Cal S$, then we have $\mu (\Cal S') \leq \mu (\Cal S)$, and it is {\sl unstable\/} if it is not semistable. Equivalently, $\Cal S$ is semistable if, for all surjections $\Cal S \to \Cal S''$, where $\Cal S''$ is torsion free and nonzero, we have $\mu (\Cal S'') \geq \mu (\Cal S)$. A torsion free rank one sheaf is semistable. Given an exact sequence $$0 \to \Cal S' \to \Cal S \to \Cal S'' \to 0,$$ with $\mu(\Cal S') =\mu (\Cal S) = \mu (\Cal S'')$, $\Cal S$ is semistable if and only if both $\Cal S'$ and $\Cal S''$ are semistable. If $\Cal S$ is a torsion free semistable sheaf of negative degree, then (for $E$ of arithmetic genus one) $h^0(\Cal S) = 0$ and hence $h^1(\Cal S) = -\deg \Cal S$, and if $\Cal S$ is a torsion free semistable sheaf of strictly positive degree, then since $h^1(\Cal S)$ is dual to $\Hom (\Cal S, \scrO_E)$, it follows that $h^1(\Cal S) = 0$ and that $h^0(\Cal S) = \deg \Cal S$. Every torsion free sheaf $\Cal S$ has a canonical Harder-Narasimhan filtration, in other words a filtration by subsheaves $F^0 \subset F^1\subset \cdots$ such that $F^{i+1}/F^i$ is torsion free and semistable and $\mu (F^i/F^{i-1}) > \mu (F^{i+1}/F^i)$ for all $i\geq 1$. \definition{Definition 0.5} Let $V$ and $V'$ be two semistable torsion free sheaves on $E$. We say that $V$ and $V'$ are {\sl $S$-equivalent\/} if there exists a connected scheme $S$ and a coherent sheaf $\Cal V$ on $S\times E$, flat over $S$, and a point $s'\in S$ such that $V\cong \Cal V|\{s\}\times E$ if $s\neq s'$ and $V'\cong \Cal V|\{s'\}\times E$. We define {\sl $S$-equivalence\/} to be the equivalence relation on semistable torsion free sheaves generated by the above relation. Suppose that $V$ and $V'$ are two semistable bundles on $E$. We say that $V$ and $V'$ are {\sl restricted $S$-equivalent\/} if there exists a connected scheme $S$, a vector bundle $\Cal V$ on $S\times E$, and a point $s'\in S$ such that $V\cong \Cal V|\{s\}\times E$ if $s\neq s'$ and $V'\cong \Cal V|\{s'\}\times E$. We define {\sl restricted $S$-equivalence\/} to be the equivalence relation on semistable bundles generated by the above relation. \enddefinition \section{1. A coarse moduli space for semistable bundles over a Weierstrass cubic.} Fix a Weierstrass cubic $E$ with an origin $p_0$ and consider semistable vector bundles of rank $n$ and trivial determiniant over $E$. Our goal in this section will be to construct a coarse moduli space of such bundles, which we will identify with the linear system $|np_0|$. Given a vector bundle $V$, we associate to $V$ a point $\zeta(V)$ in the projective space $|np_0|$ associated to the linear system $\scrO_E(np_0)$ on $E$. In case $E$ is smooth, $\zeta(V)$ records the unordered set of degree zero line bundles that occur as Jordan-H\"older quotients of any maximal filtration of $V$. More generally, if ${\Cal V}\to S\times E$ is an algebraic (or holomorphic) family of bundles of the above type on $E$, then the function $\Phi\colon S\to |np_0|$ defined by $\Phi(s)=\zeta\left({\Cal V}|\{s\}\times E\right)$ is a morphism. If $E$ is smooth, two semistable bundles $V$ and $V'$ are are $S$-equivalent if and only $\zeta (V) =\zeta (V')$. This identifies $|np_0|$ as a (coarse) moduli space of $S$-equivalence classes of semistable rank $n$ bundles with trivial determinant on $E$. A similar result holds if $E$ is cuspidal. In case $E$ is nodal, however, there exist $S$-equivalent bundles $V$ and $V'$ such that $\zeta (V) \neq\zeta (V')$. It seems likely that, in case $E$ is nodal, $\zeta (V) =\zeta (V')$ if and only if $V$ and $V'$ are restricted $S$-equivalent (0.5). The moduli space $|np_0|$ is not a fine moduli space, for two reasons. One problem is the issue of $S$-equivalence versus isomorphism. To deal with this problem, we will attempt to choose a ``best" representative for each $S$-equivalence class, the regular representative. In case $E$ is smooth, a regular bundle $V$ is one whose automorphism group has dimension equal to its rank, the minimum possible dimension. Even after choosing the regular representative, however, $|np_0|$ fails to be a fine moduli space because the bundles $V$ are never simple. This allows us to twist universal bundles by line bundles on an $n$-sheeted cover of $|np_0|$, the {\sl spectral cover}. This construction will be described in Section 2. \ssection{1.1. The Jordan-H\"older constituents of a semistable bundle.} The two main results of this section are the following: \theorem{1.1} Let $V$ be a semistable torsion free sheaf of rank $n$ and degree zero over $E$. Then $V$ has a Jordan-H\"older filtration $$0\subset F^0\subset F^1\subset \cdots\subset F^n=V$$ so that each quotient $F^i/F^{i-1}$ is a rank one torsion free sheaf of degree zero. For $\lambda$ a rank one torsion free sheaf of degree zero, define $V(\lambda)$ to be the sum of all the subsheaves of $V$ which have a filtration such that all of the successive quotients are isomorphic to $\lambda$. Then $V =\bigoplus _\lambda V(\lambda)$. In particular, if $V$ is locally free, then $V(\lambda)$ is locally free for every $\lambda$. \endstatement \theorem{1.2} Let $V$ be a semistable torsion free sheaf of rank $n$ and degree zero on $E$. Then \roster \item"{(i)}" $h^0(V\otimes \scrO_E(p_0))=n$ and the natural evaluation map $$ev\colon H^0(V\otimes \scrO_E(p_0))\otimes_\Cee \scrO_E\to V\otimes \scrO_E(p_0)$$ is an isomorphism over the generic point of $E$. \item"{(ii)}" Suppose that $V$ is locally free with $\det V =\scrO_E(e-p_0)$. The induced map on determinants defines a map $$\wedge ^nev\: \det H^0(V\otimes \scrO_E(p_0))\otimes_\Cee \scrO_E \cong \scrO_E\to \det \left(V\otimes \scrO_E(p_0)\right) \cong \scrO_E((n-1)p_0+e).$$ Thus $\wedge ^nev$ defines a non-zero section of $\scrO_E((n-1)p_0+e)$ up to a nonzero scalar multiple, i\.e\. a point of $|(n-1)p_0+e|$. We denote this element by $\zeta(V)$. In particular, if $e=p_0$, then $\zeta (V)\in |np_0|$. \endroster \endstatement \demo{Proof of Theorem \rom{1.2}} Let $V$ be a semistable sheaf of degree zero and rank $n$ on $E$. The degree of $V\otimes \scrO_E(p_0)$ is $n$. By definition, $h^0(V\otimes \scrO_E(p_0)) - h^1(V\otimes \scrO_E(p_0)) = n$. By Serre duality, $h^1(V\otimes \scrO_E(p_0)) =\dim \Hom(V, \scrO_E(-p_0))$. Since $V$ is semistable, $\Hom(V, \scrO_E(-p_0)) =0$. Thus $h^0(V\otimes \scrO_E(p_0)) = n$. Next we claim that the induced map $ev\colon H^0(V\otimes \scrO_E(p_0))\otimes_\Cee \scrO_E \to V\otimes \scrO_E(p_0)$ is an isomorphism over the generic point of $E$; equivalently, its image $I\subset V\otimes \scrO_E(p_0)$ has rank $n$. To prove this, we use the following lemma. \lemma{1.3} Let $E$ be a Weierstrass cubic, let $I$ be a torsion free sheaf on $E$ and let $\mu _0(I)$ be the maximal value of $\mu (J)$ as $J$ runs over all torsion free subsheaves of $I$. Then $$h^0(I) \leq \max(\mu _0(I), 1)\operatorname{rank}I.$$ \endstatement \proof If $0 \subset F^0 \subset \cdots \subset F^k = I$ is the Harder-Narasimhan filtration of $I$, then $\mu _0(I) = \mu (F^0)$, $F^{i+1}/F^i$ is semistable, and $\mu (F^{i+1}/F^i) < \mu _0(I)$ for all $i \geq 1$. Furthermore $$h^0(I) \leq \sum _ih^0(F^{i+1}/F^i).$$ Now if $\mu (F^{i+1}/F^i) > 0$, then since $h^1(F^{i+1}/F^i)= \dim \Hom (F^{i+1}/F^i, \scrO_E) = 0$, it follows that $h^0(F^{i+1}/F^i) = \deg (F^{i+1}/F^i) = \mu (F^{i+1}/F^i)\cdot \operatorname{rank}(F^{i+1}/F^i) \leq \mu_0(I) \operatorname{rank}(F^{i+1}/F^i)$. If $\mu (F^{i+1}/F^i) < 0$, then $$h^0(F^{i+1}/F^i) = 0 \leq \operatorname{rank}(F^{i+1}/F^i).$$ There remains the case that $\mu (F^{i+1}/F^i) = 0$. In this case, we claim that $$h^0(F^{i+1}/F^i) \leq \operatorname{rank}(F^{i+1}/F^i).$$ In fact since $F^{i+1}/F^i$ is semistable, this follows from the next claim. \lemma{1.4} If $V$ is a semistable torsion free sheaf on $E$ with $\mu (V) = 0$, then $h^0(V) \leq \operatorname{rank} V.$ \endstatement \proof Argue by induction on $\operatorname{rank}V$. If $\operatorname{rank} V = 1$ and $h^0(V) \geq 1$, then there exists a nonzero map $\scrO_E\to V$, and since $\mu (\scrO_E) = \mu (V)$, this map must be an isomorphism. Thus $h^0(V) = 1$. In general, if $\operatorname{rank}V = n+1$ and $h^0(V) \neq 0$, choose a nonzero map $\scrO_E \to V$. Since $V$ is semistable, the cokernel $Q$ of this map is torsion free and thus is also semistable, with $\mu (Q) =0$. Since the rank of $Q$ is $n$, by induction we have $h^0(V) \leq 1 + h^0(Q) \leq n+1$. \endproof Returning to the proof of (1.3), we see that in all cases $$h^0(F^{i+1}/F^i) \leq \max(\mu _0(I), 1)\operatorname{rank}(F^{i+1}/F^i).$$ Summing over $i$ gives the statement of (1.3). \endproof We continue with the proof of Theorem 1.2. There is the map $$ev\colon H^0(V\otimes \scrO_E(p_0))\otimes_\Cee\scrO_E \to V\otimes \scrO_E(p_0).$$ Let $I$ be its image. By construction $I$ is a subsheaf of a locally free sheaf and hence is torsion free. Also, by construction the map $H^0(I) \to H^0(V\otimes \scrO_E(p_0))$ is an isomorphism, and thus $h^0(I) = n$. Since $V\otimes \scrO_E(p_0)$ is semistable and $\mu (V\otimes \scrO_E(p_0)) = 1$, we have $\mu _0(I) \leq 1$. Thus, by (1.3), $n = h^0(I) \leq \operatorname{rank} I \leq n$, and so $\operatorname{rank} I = n$. Equivalently, the image of $ev$ is equal to $V\otimes \scrO_E(p_0)$ at the generic point. From this, the remaining statements in Theorem 1.2 are clear. \endproof \demo{Proof of Theorem \rom{1.1}} Let us first show that $V$ has a Jordan-H\"older filtration as described. The proof is by induction on the rank $n$ of $V$. If $n=1$, there is nothing to prove. For arbitrary $n$, we shall show that there exists a nonzero map $\lambda \to V$, where $\lambda$ is a rank one torsion free sheaf of degree at least zero. By semistability, the degree of $\lambda$ is exactly zero and $V/\lambda$ is torsion free. We can then apply induction to $V/\lambda$. The proof of Theorem 1.2 above shows that, if $V$ is a semistable torsion free sheaf of rank $n$ and degree zero, then there is an injective map $\scrO_E^{\oplus n} \to V\otimes \scrO_E(p_0)$ whose image has rank $n$. Thus there is a map $\scrO_E(-p_0)^{\oplus n} \to V$ whose image has rank $n$. The cokernel of this map must be a torsion sheaf $\tau$. Note that, in case $V$ is locally free, $\tau$ is supported exactly at the points in the support of $\zeta (V)$. Since $\deg V=0$, $\tau \neq 0$. Choose a point $x$ in the support of $\tau$. If $R$ is the local ring of $E$ at $x$ and $\frak m$ is the maximal ideal of $x$, then $\tau_x$ is annihilated by some power of $\frak m$. Let $k$ be such that $\frak m^k\tau \neq 0$ but $\frak m^{k+1}\tau =0$. Choosing a section of $\frak m^k\tau$ produces a subsheaf $\tau_0$ of $\tau$ which is isomorphic to $\Cee_x$, in other words is isomorphic to $R/\frak m$ as an $R$-module. Let $V_0\subseteq V$ be the inverse image of $\tau _0$. Then $V_0$ corresponds to an extension of $\Cee_x$ by $\scrO_E(-p_0)^{\oplus n}$, and hence to an extension class in $$\Ext^1(\Cee_x, \scrO_E(-p_0)^{\oplus n}) \cong H^0(Ext^1(\Cee_x, \scrO_E(-p_0)^{\oplus n}) \cong \Ext^1_R(R/\frak m, R^n).$$ The ring $R$ is a Gorenstein local ring of dimension one, and so $\Ext^1_R(R/\frak m, R) \cong \Cee$. (Of course, this could be verified directly for the local rings $R$ under consideration.) In fact, if $x$ is a smooth point of $E$ and $t$ is a local parameter at $x$, then the unique nontrivial extension of $R/\frak m$ by $R$ corresponds to the exact sequence $$0 \to R @>{\times t}>> R \to R/\frak m \to 0,$$ whereas if $x$ is a singular point then the nontrivial extension is given by $$0 \to R \to \tilde R \to R/\frak m \to 0.$$ Let $\xi$ be the extension class corresponding to $V_0$ in $$\Ext^1(\Cee_x, \scrO_E(-p_0)^{\oplus n}) \cong \Ext^1_R(R/\frak m, R^n) \cong \Cee^n.$$ In the local setting, let $M$ be the $R$-module corresponding to $V_0$, and suppose that we are given an extension $$0\to R \to N \to R/\frak m \to 0,$$ with a corresponding extension class $\eta \in \Ext^1_R(R/\frak m, R)$ and a homomorphism $f\: R\to R^n$ such that $f_*(\eta) =\xi$. By a standard result, there is a homomorphism $N\to M$ lifting $f$, viewed as a homomorphism $R\to M$. In particular, this says that the image of $R$ in $M$ is contained in a strictly larger rank one torsion free $R$-module. Returning to the global situation, let $\lambda$ be the unique nontrivial extension of $\Cee_x$ by $\scrO_E(-p_0)$, and let $\eta$ be the corresponding extension class, well-defined up to a nonzero scalar. Thus $\lambda$ is a rank one torsion free sheaf of degree zero. Since $Hom(\scrO_E(-p_0), \scrO_E(-p_0)^{\oplus n})$ is generated by its global sections, there exists a homomorphism $f\: \scrO_E(-p_0)\to \scrO_E(-p_0)^{\oplus n}$ such that the image of $\eta$ under $f_*$ in $\Ext^1(\Cee_x, \scrO_E(-p_0)^{\oplus n})$ is $\xi$. Then the inclusion $\scrO_E(-p_0)\to \scrO_E(-p_0)^{\oplus n} \to V_0 \to V$ factors through a nonzero map $\lambda \to V$, necessarily an inclusion with torsion free cokernel. Thus we have proved the existence of the Jordan-H\"older filtration by induction. By using the fact that $\Ext^1(\lambda, \lambda') =0$ if $\lambda\neq \lambda'$, an easy argument left to the reader shows that $V(\lambda)\neq 0$ if and only if $\Hom (V,\lambda) \neq 0$ if and only if $\Hom (\lambda, V)\neq 0$. Thus we can always arrange that, if $\lambda$ is a sheaf appearing as one of the quotients in Theorem 1.1, then there exists a filtration for which $\lambda =F^0$ is the first such sheaf which appears, and also one for which $\lambda = F^n/F^{n-1}$ is the last such sheaf which appears. Fix a rank one torsion free sheaf $\lambda$ of degree zero, and let $V'(\lambda)$ be the sum of all subsheaves of $V$ which have a filtration by rank one torsion free sheaves of degree zero which are not isomorphic to $\lambda$. Let $V(\lambda) = V/V'(\lambda)$. Clearly $V(\lambda)$ is a torsion free semistable sheaf, such that all of the quotients in a Jordan-H\"older filtration of $V$ are isomorphic to $\lambda$. Again using $\Ext^1(\lambda, \lambda') =0$ if $\lambda\neq \lambda'$, one checks that $\Ext^1(V(\lambda), V'(\lambda)) =0$. Thus, by induction on the rank, $V$ is isomorphic to the direct sum of the $V(\lambda)$. This concludes the proof of Theorem 1.1. \endproof The construction of Theorem 1.2 works well in families. \theorem{1.5} Let $E$ be a Weierstrass cubic, and let $S$ be a scheme or analytic space. Let $\Cal V$ be a rank $n$ vector bundle over $S\times E$ such that on each slice $\{s\}\times E$, $\Cal V$ restricts to a semistable vector bundle $V_s$ of trivial determinant. Then there exists a morphism $\Phi\: S \to |np_0| = \Pee ^{n-1}$ such that, for all $s\in S$, we have $\Phi(s)=\zeta(V_s)$. In particular, if $V$ and $V'$ are restricted $S$-equivalent, then $\zeta (V) =\zeta (V')$. \endstatement \proof Let $p_1, p_2$ be the projections from $S\times E$ to $S$ and $E$. To construct a morphism from $S$ to $|np_0|$ we shall construct a homomorphism $\Psi\colon p_1^*L_0 \to p_1^*L_1\otimes p_2^*\scrO_E(np_0)$, where $L_0,L_1$ are line bundles on $S$, with the property that the restriction of $\Psi$ to each slice $\{s\}\times E$ determines a nonzero section of $\scrO_E(np_0)$ (which is thus well-defined mod scalars), agreeing with $\wedge^nev$. The map $\Psi$ is defined in the next lemma. \lemma{1.6} The sheaf $p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0))$ is a locally free sheaf of rank $n$ on $S$. Let $L_0$ be its determinant line bundle. If $\hat\Psi\colon p_1^*p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0))\to\Cal V\otimes p_2^*\scrO_E(p_0)$ is the natural evaluation map, then its restriction to each slice $\{s\}\times E$ is generically an isomorphism, agreeing with $ev$. Thus $$\Psi=\det\hat\Psi\colon p_1^*L_0\to \det{\Cal V}\otimes p_2^*\scrO_E(np_0)$$ has the property that its restriction to each slice $\{s\}\times E$ is is nonzero and agrees with $\wedge^nev$. \endstatement \proof It follows from Theorem 1.2 that, if $\Cal V_s$ is the restriction of $\Cal V$ to the slice $\{s\}\times E$, then $h^0({\Cal V}_s\otimes \scrO_E(p_0))=n$ is independent of $s$. Standard base change arguments \cite{10, Theorem 12.11, pp\. 290--291} show that, even if $S$ is nonreduced, $p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0))$ is a locally free sheaf of rank $n$ on $S$, and the natural map $p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0))_s\to H^0(V_s\otimes \scrO_E(p_0))$ is an isomorphism for every $s\in S$. Thus the induced morphism $\hat\Psi\: p_1^*p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0)) \to \Cal V \otimes p_2^*\scrO_E(p_0)$ is a morphism between two vector bundles of rank $n$. Let $V_s$ be the restriction of $\Cal V$ to the slice $\{s\}\times E$. Again by base change, the natural map $p_1{}_*(\Cal V \otimes p_2^*\scrO_E(p_0))_s \to H^0(V_s\otimes \scrO_E(p_0))\otimes_\Cee\scrO_E$ is surjective. The result is now immediate from Theorem 1.2. \endproof Next notice that since for every $s\in S$, $\det{\Cal V}|(\{s\}\times E)$ is trivial, it follows that $\det {\Cal V}$ is isomorphic to $p_1^*L_1$ for some line bundle $L_1$ on $S$. To complete the proof of Theorem 1.5 we need to check that the section $\Psi(s)=\zeta(V_s)$ for all $s\in S$. This is immediate from the corresponding statement in Theorem 1.2. \endproof In fancier terms, Theorem 1.5 says that there is a morphism of functors from the deformation functor of semistable vector bundles of rank $n$ and trivial determinant on $E$ to the functor represented by the scheme $|np_0|$. In general, this morphism is far from smooth; for example, at the trivial bundle $\scrO_E^n$, the derivative of the morphism is identically zero. However, if we restrict to regular semistable bundles (to be defined in \S 1.2 below), then it will follow from (v) in Theorem 3.2 that the derivative is always an isomorphism. The sheaves $\lambda$ which appear as successive quotients of $V$ in Theorem 1.1 are the {\sl Jordan-H\"older quotients\/} or {\sl Jordan-H\"older constituents\/} of $V$. They appear with multiplicities and the multiplicity of $\lambda$ in $V$ is independent of the choice of the filtration. The summands $V(\lambda)$ of $V$ are canonically defined. It is easy to see from the construction that $\zeta (V) = \sum _\lambda \zeta (V(\lambda))$. More generally, $\zeta$ is additive over exact sequences of semistable vector bundles of degree zero. Also, if $\det V = \scrO_E(e-p_0)$ and $e'$ is a smooth point of $E$, then $e'$ lies in the support of $\zeta (V)$ as a divisor in $|(n-1)p_0 +e|$ if and only if $\lambda =\scrO_E(e'-p_0)$ is a Jordan-H\"older constituent of $V$. Thus, if the rank of $V(\lambda)$ is $d_\lambda$, then $$\zeta (V) = \sum _{\lambda \neq \Cal F}d_\lambda e_\lambda + e_{\Cal F},$$ where $\lambda \cong \scrO_E(e_\lambda -p_0)$ and $e_{\Cal F}$ is a divisor of degree $d_{\Cal F}$ supported at the singular point of $E$. In this way we can associate a point of the $n^{\text{th}}$ symmetric product of $E$ with such a $V$: namely $$\sum_{\lambda\neq \Cal F} \operatorname{rank}(V(\lambda))e_\lambda+d_{\Cal F}\cdot s$$ where $\lambda\cong\scrO_E(e_\lambda-p_0)$ and $s\in E$ is the singular point. Note that there is a morphism $|np_0| \to \Sym ^nE$, which is a closed embedding if $E$ is smooth, or more generally away from the elements of $|np_0|$ whose support meets the singular point of $E$. Suppose that $E$ is smooth. Since a degree zero line bundle on $E$ is identified with a point of $E$ via the correspondence $\lambda \mapsto q$ if $\lambda\cong \scrO_E(q-p_0)$, the map which assigns to a semistable bundle $V$ the unordered $n$-tuple of its Jordan-H\"older quotients, including multiplicities, is the same as the map assigning to $V$ an unordered $n$-tuple $\zeta(V)$ of points of $E$, i.e., a point $$\zeta(V)\in\underbrace{(E\times \cdots \times E)}_{\text{$n$ times}}/{\frak S}_n,$$ where ${\frak S}_n$ is the symmetric group on $n$ letters. If $\zeta(V)=(e_1,\ldots,e_n)$, then the condition that the determinant of $V$ is trivial means that $\sum_{i=1}^ne_i=0$ in the group law of $E$, or equivalently that the divsior $\sum_{i=1}^ne_i$ is linearly equivalent to $np_0$. Thus, the unordered $n$-tuple $(e_1,\ldots,e_n)$ associated to $V$ can be identified with a point in the complete linear system $|np_0|$, and this point is exactly $\zeta (V)$. An important difference in case $E$ is singular is that, while a point of $|np_0|$ determines a point on the symmetric $n$-fold product of $E$, in general it contains more information at the singular point than just its multiplicity. Thus, the function $\Phi$ should be viewed not as a point in the $n$-fold symmetric product but as a point in the linear system $|np_0|$. For example, if $E$ is nodal and $n>2$, then an element of $|np_0|$ supported entirely at the singular point corresponds to a hyperplane in $\Pee^{n-1}$ meeting the image of $E$ embedded by the complete linear system $|np_0|$ just at the singular point. As such, it is specified by two positive integers $a$ and $b$ with $a+b = n$, the orders of contact of the hyperplane with the two branches of $E$ at the node. \ssection{1.2. Regular bundles over a Weierstrass cubic.} Let $E$ be a Weierstrass cubic. Every semistable bundle is of the form $$V\cong \bigoplus_\lambda V(\lambda)$$ where $\lambda$ ranges over the isomorphism classes of rank one torsion free sheaves on $E$ of degree zero. Let us first analyze $V(\lambda)$ in case $\lambda$ is a line bundle. If $V(\lambda)$ is a semistable bundle with the property that all Jordan-H\"older quotients of $V$ are isomorphic to $\lambda$, or in other words $H^0(\lambda'\otimes V(\lambda))=0$ for all $\lambda'\not=\lambda^{-1}$, then the associated graded to every Jordan-H\"older filtration of $V(\lambda)$ is a direct sum of line bundles isomorphic to $\lambda$. Of course, one possibility for $V(\lambda)$ is the split one: $$V(\lambda)\cong\lambda ^{\oplus r}.$$ At the other extreme we have the maximally non-split case: \lemma{1.7} Let $E$ be a Weierstrass cubic, possibly singular. For each natural number $r>0$ and each line bundle $\lambda$ of degree zero there is a unique bundle $I_r(\lambda)$ up to isomorphism with the following properties: \roster \item"{(i)}" the rank of $I_r(\lambda)$ is $r$. \item"{(ii)}" all the Jordan-H\"older quotients of $I_r(\lambda)$ are isomorphic to $\lambda$. \item"{(iii)}" $I_r(\lambda)$ is indecomposable under direct sum. \endroster Furthermore, for all $r>0$ and all line bundles $\lambda$, $I_r(\lambda)$ is semistable, $I_r(\lambda)\spcheck = I_r(\lambda^{-1})$, $\det I_r(\lambda)=\lambda ^r$, and $\dim \Hom (I_r(\lambda), \lambda)=\dim \Hom (\lambda, I_r(\lambda))=1$. \endstatement \proof We first construct the bundle $I_r=I_r(\scrO_E)$ by induction on $r$. For $r=1$ we set $I_r=\scrO_E$. Suppose inductively that we have constructed $I_{r-1}$ with the properties given in the lemma. Suppose in addition that $H^0(I_{r-1})\cong\Cee$. Since the degree of $I_{r-1}$ is zero, it follows that $H^1(I_{r-1})\cong\Cee$ and hence there is exactly one non-trivial extension, up to scalar multiples, of the form $$0\to I_{r-1}\to I_r\to \scrO_E\to 0.$$ One checks easily all the inductive hypotheses for the total space of this extension. This proves the existence of $I_r$ for all $r>0$. Uniqueness is easy and is left to the reader. We define $I_r(\lambda)=I_r\otimes\lambda$. The statements of (1.7) are then clear. \endproof The bundle $I_r(\lambda)$ has an increasing filtration by subbundles isomorphic to $I_k(\lambda)$, $k\leq r$. We denote this filtration by $$\{0\} \subset F_1I_r(\lambda) \subset \cdots \subset F_rI_r(\lambda) = I_r(\lambda),$$ and refer to $F_iI_r(\lambda)$ as the {\sl $i^{\text{th}}$ filtrant\/} of $I_r(\lambda)$. When the bundle is clear from the context, we denote the subbundles in this filtration by $F_i$. Notice that $F_t\cong I_t(\lambda)$ and that $I_r(\lambda)/F_{r-t}\cong I_t(\lambda)$. Let us note some of the basic properties of the bundles $I_r(\lambda)$. \lemma{1.8} Let $J$ be a proper degree zero subsheaf of $I_r(\lambda)$. Then $J$ is contained in $F_{r-1}$. In fact, $J = F_t$ for some $t<r$. \endstatement \proof By the semistability of $I_r(\lambda)$, $I_r(\lambda)/J$ is a nonzero semistable torsion free sheaf of degree zero. Clearly all of its Jordan-H\"older quotients are isomorphic to $\lambda$. In particular there is a nonzero map $I_r(\lambda)/J \to \lambda$. The composition $I_r(\lambda)\to I_r(\lambda)/J \to \lambda$ defines a nonzero map from $I_r(\lambda)$ to $\lambda$ containing $J$ in its kernel. By (1.7), there is a unique such nonzero map mod scalars, and its kernel is $F_{r-1}$. Thus $J\subseteq F_{r-1}$. Applying induction to the inclusion $J\subseteq F_{r-1}\cong I_{r-1}(\lambda)$, we see that $J = F_t$ for some $t<r$. \endproof Our next result is that the filtration is canonical, i.e., invariant under any automorphism of $I_r(\lambda)$. \corollary{1.9} If $\varphi\colon I_r(\lambda)\to I_r(\lambda)$ is a homomorphism, then, for all $i\le r$, $$\varphi(F_i)\subseteq F_i.$$ It follows that if $\varphi$ is an automorphism, then for all $i$ we have $\varphi(F_i)=F_i$. More generally, if $\varphi\colon I_r(\lambda)\to I_t(\lambda)$ is a homomorphism, then $$\varphi(F_s(I_r(\lambda)))\subseteq F_s(I_t(\lambda)).$$ \endstatement \proof It suffices to prove the last statement. By the semistability of $I_r(\lambda)$ and $I_t(\lambda)$, $\varphi(F_s(I_r(\lambda)))$ is a degree zero subsheaf of $I_t(\lambda)$ of rank at most $s$. Thus it is contained in $F_s(I_t(\lambda))$. \endproof \lemma{1.10} Fix $0\le t\le r$ and let $q_{r,t}\colon I_r(\lambda)\to I_t(\lambda)$ be the natural quotient map. Then $q_{r,t}$ induces a surjective homomorphism from the endomorphism algebra of $I_r(\lambda)$ to that of $I_t(\lambda)$. A similar statement holds for the automorphism groups. Finally, as a $\Cee$-algebra, $\Hom(I_r(\lambda), I_r(\lambda)) \cong \Cee[t]/(t^r)$. \endstatement \proof That $q_{r,t}$ induces a map on endomorphism algebras is immediate from (1.9). Let us show that it is surjective. We might as well assume that $t=r-1>0$ since the other cases will then follow by induction. Let $A\colon I_{r-1}(\lambda)\to I_{r-1}(\lambda)$ be an endomorphism. Since the map $q_{r,r-1}\colon I_r(\lambda)\to I_{r-1}(\lambda)$ induces the zero map on $\Hom(\lambda,\cdot)$, it follows by duality that the map $q_{r,r-1}^* \colon \Ext^1(I_{r-1}(\lambda), \lambda)\to \Ext^1(I_r(\lambda), \lambda)$ is zero. Hence the composition $A\circ q_{r,r-1}\colon I_r(\lambda)\to I_{r-1}(\lambda)$ lifts to a map $\hat A\colon I_r(\lambda)\to I_r(\lambda)$. Thus the restriction map on endomorphism algebras is surjective. To see the statement on automorphism groups, suppose that $A$ is an isomorphism. We wish to show that $\hat A$ is an isomorphism. To see this, perform the construction for $A^{-1}$ as well, obtaining a map $\widehat{A^{-1}}\colon I_r(\lambda)\to I_r(\lambda)$. The composition $B=\widehat{A^{-1}}\circ \hat A\colon I_r(\lambda)\to I_r(\lambda)$ projects to the identity on $I_{r-1}(\lambda)$. This means that $B-\operatorname{Id}\colon I_r(\lambda)\to F_{1}(I_\lambda))$. Since $r>1$, this map is nilpotent, and hence $B=\operatorname{Id}+(B-\operatorname{Id})$ is an isomorphism. Finally we prove the last statement. Let $A_r\: I_r(\lambda) \to I_r(\lambda)$ be any endomorphism defined by a composition $$I_r(\lambda) \twoheadrightarrow I_{r-1}(\lambda) \hookrightarrow I_r(\lambda).$$ Note that $A_r^r =0$ and that the restriction of $A_r$ to $I_r(\lambda)/F_1\cong I_{r-1}(\lambda)$ is of the form $A_{r-1}$. Suppose by induction that $\Hom (I_{r-1}(\lambda), I_{r-1}(\lambda)) =\Cee[A_{r-1}]$. Then every endomorphism $T$ of $I_r(\lambda)$ is of the form $T = p(A_r) + T'$, where $p$ is a polynomial of degree at most $r-2$ in $A_r$ and $T'$ induces the zero map on $I_r(\lambda)/F_1$. In this case $T'$ is given by a map from $I_r(\lambda)$ to $F_1$, necessarily zero on $F_{r-1}$, and it is easy to check that $T'$ must in fact be a multiple of $A_r^{r-1}$. Thus $\Hom (I_r(\lambda), I_r(\lambda)) =\Cee[A_r] \cong \Cee[t]/(t^r)$. \endproof We need to define an analogue of $I_r(\lambda)$ in case the Jordan-H\"older quotients are all isomorphic to the non-locally free sheaf $\Cal F$. We say that a semistable degree zero bundle $I(\Cal F)$ concentrated at the singular point of $E$ is {\sl strongly indecomposable} if $\Hom(I(\Cal F),{\Cal F})\cong\Cee$. Notice that since $\Hom(V(\Cal F),\Cal F)\not= 0$ for any non-trivial semistable bundle $V(\Cal F)$ concentrated at the singular point, it follows that if $I(\Cal F)$ is strongly indecomposable, then it is indecomposable as a vector bundle in the usual sense. However, the converse is not true: there exist indecomposable vector bundles which are not strongly indecomposable. It is natural to ask if every vector bundle supported at $\Cal F$ is an extension of strongly indecomposable bundles. Unlike the smooth case, it is also not true that $I(\Cal F)$ is determined up to isomorphism by its rank and the fact that it is strongly indecomposable. Nor is it true that $I(\Cal F)$ always has a unique filtration with successive quotients isomorphic to $\Cal F$. As we shall show in Section 3, $I(\Cal F)$ is determined up to isomorphism by its rank and the point $\zeta(I(\Cal F))$. There is the following analogue for $I(\Cal F)$ of (1.8): \lemma{1.11} Suppose that $I(\Cal F)$ is strongly indecomposable. Let $\rho\: I(\Cal F)\to \Cal F$ be a nonzero homomorphism, unique up to scalar multiples, and let $X=\Ker\rho$. If $J$ is a proper degree zero subsheaf of $I(\Cal F)$, then $J$ is contained in $X$. \endstatement \proof Let $J\subset I(\Cal F)$ be a subsheaf of degree zero. The quotient $Q=I(\Cal F)/J$ must be torsion-free, for otherwise $J$ would be contained in a larger subsheaf $\hat J$ of the same rank and bigger degree, contradicting the semistability of $I(\Cal F)$. This means that $Q$ is semistable of degree zero. Clearly, it is concentrated at the singular point. Thus, there is a nontrivial map $Q\to {\Cal F}$. By the strong indecomposability of $I(\Cal F)$, the composition $I(\Cal F)\to Q\to {\Cal F}$ is some nonzero multiple of $\rho$. In particular, the kernel of this composition is $X$. This proves that $J\subset X$. \endproof \definition{Definition 1.12} Let $V$ be a semistable bundle with trivial determinant over a Weierstrass cubic. We say that $V$ is {\sl regular} or {\sl maximally nonsplit\/} if, $$V\cong \bigoplus_iI_{r_i}(\lambda_i)\oplus I(\Cal F)$$ where the $\lambda_i$ are pairwise distinct line bundles and $I(\Cal F)$ is a strongly indecomposable bundle concentrated at the singular point. \enddefinition For $E$ smooth, Atiyah proved \cite{1} that every vector bundle $V$, all of whose Jordan-H\"older quotients are isomorphic to $\lambda$, can be written as a direct sum $\bigoplus _iI_{r_i}(\lambda)$. The argument carries over to the case where $E$ is singular, provided that $\lambda$ is a line bundle $\scrO_E(e-p_0)$. Thus, in this case there is a unique regular bundle $V$ of rank $r$ such that the support of $\zeta(V)$ is $e$. More generally, given a divisor $e_1+ \cdots + e_n\in |np_0|$ supported on the smooth points, there is a unique regular semistable rank $n$ vector bundle $V$ of trivial determinant over $E$ such that $\zeta(V)=(e_1,\ldots,e_n)$. An analogue of Atiyah's theorem for the singular points has been established by T. Teodorescu \cite{12}. In this paper, we shall show in Section 3 that, given a Cartier divisor $D$ in $|np_0|$ whose support is the singular point, then there is a unique regular semistable rank $n$ vector bundle $V$ of trivial determinant such that $\zeta (V) = D$. Regular bundles have an extremely nice property: Their automorphism groups have minimal possible dimension. We shall show this for smooth $E$ in the next lemma. To put this property in context, let us consider first the centralizers of elements in $GL_n(\Cee)$. The centralizer of any element has dimension at least $n$. Elements in $GL_n(\Cee)$ whose centralizers have dimension exactly $n$ are said to be {\sl regular} elements. Every element in $GL_n(\Cee)$ is $S$-equivalent to a unique regular element up to conjugation. Here two elements $A,B\in GL_n(\Cee)$ are said to be $S$-equivalent if every algebraic function on $GL_n(\Cee)$ which is invariant under conjugation takes the same value on $A$ and $B$. Said another way, $A$ and $B$ are $S$-equivalent if there is an element $C\in GL_n(\Cee)$ which is in the closure of the orbits of both $A$ and $B$ under the conjugation action of $GL_n(\Cee)$ on itself. From our point of view regular bundles are the analogue of regular elements. In fact, for a smooth elliptic curve $E$, one way to construct a holomorphic vector bundle over $E$ is to fix an element $u$ in the Lie algebra of $SL_n(\Cee)$. Define a holomorphic connection on the trivial bundle $$\overline\partial_u=\overline\partial+ud\overline z$$ where $\overline\partial$ is the usual operator on the trivial bundle. If $u$ is close to the origin in the Lie algebra, then the automorphism group of this new holomorphic bundle will be the centralizer of $u$ in $GL_n(\Cee)$. In particular, this bundle will be regular and have trivial determinant if and only if $U=\operatorname{exp}(u)$ is a regular element in $SL_n(\Cee)$. For example, if $U$ is a regular semisimple element of $SL_n(\Cee)$ then the corresponding vector bundle over $E$ will be a sum of distinct line bundles of degree zero. More generally, the decomposition of $U$ into its generalized eigenspaces will correspond to the decomposition of $V$ into its components $V(\lambda)$. Clearly, $S$-equivalent elements of $GL_n(\Cee)$ yield $S$-equivalent bundles. Here is the analogue of the dimension statements for vector bundles over a smooth elliptic curve. \lemma{1.13} Let $V$ be a semistable rank $n$ vector bundle over a smooth elliptic curve $E$. \roster \item"{(i)}" $\dim \Hom (V,V) \geq n$. \item"{(ii)}" $V$ is regular if and only if $\dim \Hom (V,V) = n$. In this case, if $V = \bigoplus _iI_{d_i}(\lambda _i)$, then the $\Cee$-algebra $\Hom (V,V)$ is isomorphic to $\bigoplus _i\Cee[t]/(t^{d_i})$. In particular, $\Hom (V,V)$ is an abelian $\Cee$-algebra. \item"{(iii)}" $V$ is regular if and only if, for all line bundles $\lambda$ of degree zero on $E$, $h^0(V\otimes \lambda ^{-1})\leq 1$. \endroster \endstatement \proof It is easy to check that $\Hom(V(\lambda), V(\lambda ')) \neq 0$ if and only if $\lambda = \lambda '$, and (using Corollary 1.9 and Lemma 1.10) that $\Hom(I_d, I_d) \cong \Cee[t]/(t^d)$. The statements (i) and (ii) follow easily from this and from Atiyah's theorem. To see (iii), note that, for a line bundle $\mu$ of degree zero, $V(\mu)$ is regular if and only if $h^0(V(\mu) \otimes \mu ^{-1}) = 1$, which implies that $h^0(V\otimes \lambda ^{-1})\leq 1$ for all $\lambda$ since $h^0(V(\mu) \otimes \lambda ^{-1}) = 0$ if $\lambda \neq \mu$. \endproof We will prove a partial analogue of (ii) in Lemma 1.13 for singular curves in Section 3. Very similar arguments show: \lemma{1.14} Let $E$ be a Weierstrass cubic and let $V$ be a semistable rank $n$ vector bundle over $E$. Then $V$ is regular if and only if, for every rank one torsion free sheaf $\lambda$ of degree zero on $E$, $\dim \Hom (V, \lambda)\leq 1$. Moreover, suppose that $V$ is regular and that $\Cal S$ is a semistable torsion free sheaf of degree zero on $E$. Then $\dim\Hom (V, \Cal S) \leq \operatorname{rank}\Cal S$. \qed \endstatement \section{2. The spectral cover construction.} In this section we shall construct families of regular semistable bundles over a smooth elliptic curve $E$. The main result is Theorem 2.1, which gives the basic construction of a universal bundle over $|np_0|\times E$, where $|np_0|\cong \Pee^{n-1}$ is the coarse moduli space of the last section. We prove that the restriction of the universal bundle to every slice is in fact regular, and that every regular bundle occurs in this way. By twisting by a line bundle on the spectral cover, we construct all possible families of universal bundles (Theorem 2.4) and show how they are all related by elementary modifications. In Theorem 2.8, we generalize this result to families of regular semistable bundles parametrized by an arbitrary base scheme. In case $E$ is singular, we establish slightly weaker versions of these results. Most of this material will be redone from a different perspective in the next section. Finally, we return to the smooth case and give the formulas for the Chern classes of the universal bundles. {\bf Throughout this section, unless otherwise noted, $E$ denotes a smooth elliptic curve with origin $p_0$.} \ssection{2.1. The spectral cover of $|np_0|$.} Let $E^{n-1}$ be embedded in $E^n$ as the set of $n$-tuples $(e_1, \dots, e_n)$ such that $\sum _ie_i = 0$ in the group law on $E$, or equivalently, such that the divisor $\sum _ie_i$ on $E$ is linearly equivalent to $np_0$. The natural action of the symmetric group $\frak S_n$ on $E^n$ thus induces an action of $\frak S_n$ on $E^{n-1}$. As we have seen, the quotient $E^{n-1}/\frak S_n$ is naturally the projective space $|np_0| \cong \Pee ^{n-1}$. View $\frak S_{n-1}$ as the subgroup of $\frak S_n$ fixing $n$, and let $T= E^{n-1}/\frak S_{n-1}$. Corresponding to the inclusion $\frak S_{n-1}\subset \frak S_n$ there is a morphism $\nu \: T \to \Pee ^{n-1}$ which realizes $T$ as an $n$-sheeted cover of $\Pee ^{n-1}$. Here $\nu$ is unbranched over $e_1+ \dots + e_n\in |np_0|$ if and only if the $e_i$ are distinct. The branch locus of $\nu$ in $\Pee ^{n-1}$ is naturally the dual hypersurface to the elliptic normal curve defined by the embedding of $E$ in the dual projective space (except in case $n=2$, where it corresponds to the four branch points of the map from $E$ to $\Pee ^1$). The map $\nu\: T\to |np_0|$ is called {\sl the spectral cover of $|np_0|$}. We will discuss the reason for this name later. The sum map $(e_1, \dots, e_n) \mapsto -\sum _{i=1}^{n-1}e_i$ is a surjective homomorphism from $E^n$ to $E$, and its restriction to $E^{n-1}$ is again surjective, with fibers invariant under $\frak S_{n-1}$. Thus there is an induced morphism $r\: T \to E$. In fact, $r(e_1, \dots, e_n) = e_n$ and $$F_e=r^{-1}(e) = \{\, (e_1, \dots, e_{n-1}, e): \sum _{i=1}^{n-1}e_i + e = np_0\,\},$$ modulo the obvious $\frak S_{n-1}$-action. Thus the fiber of $r$ over $e$ is the projective space $|np_0-e|$, of dimension $n-2$. Globally, $T$ is the projectivization of the rank $n-1$ bundle $\Cal E$ over $E$ defined by the exact sequence $$0 \to \Cal E \to H^0(E; \scrO_E(np_0)) \otimes _\Cee \scrO_E \to \scrO_E(np_0) \to 0,$$ where the last map is evaluation and is surjective since $\scrO_E(np_0)$ is generated by its global sections. The fiber of $r$ over a point $e\in E$ consists of those sections of $\scrO_E(np_0)$ vanishing at $e$, and the corresponding projective space is just $|np_0 - e|$. We see that there is an induced morphism on projective bundles $$g\: \Pee \Cal E \to \Pee \left(H^0(E;\scrO_E( np_0)) \otimes _\Cee \scrO_E\right) = |np_0| \times E \cong \Pee ^{n-1} \times E,$$ such that $g$ is a closed embedding of $T$ onto the incidence divisor in $|np_0| \times E$, and that $r$ is just the composition of this morphism with the projection $|np_0|\times E \to E$. Clearly $\nu$ is the composition of the morphism $g\: \Pee \Cal E \to |np_0| \times E$ with projection to the first factor, or equivalently $g=(\nu, r)$. Given $e\in E$, let $F_e = r^*e$ be the fiber over $e$ and let $\zeta$ be the divisor class corresponding to $c_1(\scrO_T(1))$, viewing $T$ as $\Pee \Cal E $. Since $\Cal E$ sits inside the trivial bundle, it follows that $\zeta = g^*\pi _1^*h$, where $h = c_1(\scrO_{\Pee ^{n-1}}(1))$, and thus $\zeta = \nu ^*h$. Note also that each fiber $F_e$ of $T= \Pee \Cal E \to E$ is mapped linearly into the corresponding hyperplane $H_e = |np_0 - e|$ of $\Pee ^{n-1} = |np_0|$ consisting of divisors containing $e$ in their support. Thus as divisor classes $\nu _*[F_e] = h$. There is a special point $\bold o = \bold o_E= np_0 \in |np_0|$. (In terms of regular semistable bundles, $\bold o$ corresponds to $I_n$.) It is one of the $n^2$ points of ramification of order $n$ for the map $T\to |np_0|$, corresponding to the $n$-torsion points of $E$. \ssection{2.2. A universal family of regular semistable bundles.} Next we turn to the construction of a universal family of regular semistable bundles $E$. It will be given by a bundle $U_0$ over $|np_0|\times E$. Over $E^{n-1}\times E$, we have the diagonal divisor $$\{\, (e_1, \dots, e_n, e): e = e_n, \sum _{i=1}^ne_i = 0\,\},$$ which is invariant under the $\frak S_{n-1}$-action and so descends to a divisor $\Delta$ on $T\times E$, which is the graph of the map $r\colon T\to E$. . Note that $\Delta \cong T$ and that $$\Delta = (r\times \Id)^*\Delta _0,$$ where $\Delta _0$ is the diagonal $\{\, (e, e): e\in E\,\}$. Let $G = T\times \{p_0\}$. Then the divisor $\Delta - G$ has the property that its restriction to a slice $\{(e_1, \dots, e_{n-1}, e_n)\}\times E$ can be identified with the line bundle $\scrO_E(e_n - p_0)$. We define ${\Cal L}_0\to T\times E$ to be the line bundle $\scrO_{T\times E}(\Delta - G)$, and we set $$U_0=\left(\nu\times\operatorname{Id}\right)_*{\Cal L}_0.$$ \theorem{2.1} Let $E$ be a smooth elliptic curve. The sheaf $U_0$ over $|np_0|\times E$ constructed above is a vector bundle of rank $n$. For each $x\in |np_0|$ the restriction of $U_0$ to $\{x\}\times E$ is a regular semistable bundle $V_x$ with trivial determinant and with the property that $\zeta(V_x)=x$. \endstatement \proof Since $\nu\times\operatorname{Id}$ is an $n$-sheeted covering of smooth varieties, it is a finite flat morphism and hence $U_0$ is a vector bundle of rank $n$. If $\{(e_1, \dots, e_{n-1}, e_n)\}$ is not a branch point of $\nu$, or in other words if the $e_i$ are pairwise distinct, then $U_0 = (\nu\times \Id)_*\scrO_{T\times E}(\Delta - G)$ restricts over the slice $\{(e_1, \dots, e_{n-1}, e_n)\}\times E$ to a bundle isomorphic to the direct sum $$\bigoplus_{e_i}{\Cal L}_0|\{e_i\}\times E $$ which is clearly isomorphic to $$\scrO_E(e_1 - p_0) \oplus \cdots \oplus \scrO_E(e_n - p_0).$$ This shows that for a generic point $s\in |np_0|$ the restriction $U_0|\{s\}\times E$ is as claimed: it is the unique regular semistable bundle with the given Jordan-H\"older quotients. In general, consider a point $x\in |np_0|$ of the form $\sum _{i=1}^\ell r_ie_i$, where the $e_i\in E$ and the $r_i$ are positive integers with $\sum _ir_i = n$. We claim that the Jordan-H\"older quotients of the corresponding bundle are $\scrO_E(e_i-p_0)$, with multiplicity $r_i$. The preimage of $x$ in $T$ consists of $\ell$ points $y_1, \dots, y_\ell$, each of multiplicity $r_i$. Viewing $T$ as the incidence correspondence in $\Pee^{n-1}\times E$, the point $y_i$ corresponds to $\left(\sum _{i=1}^\ell r_ie_i, e_i\right)$. If $R = \scrO_T/\frak m_x\scrO_T$ is the coordinate ring of the fiber over $x$, then $R$ is the product of $\ell$ local rings $R_i$ of lengths $r_1, \dots, r_\ell$. It clearly suffices to prove the following claim. \claim{2.2}In the above notation, $\Cal L_0\otimes R_i$ has a filtration all of whose successive quotients are isomorphic to $\lambda_i$ where $\lambda_i\cong\scrO_E(e_i-p_0)$. In particular, the restriction of $U_0$ to this slice is semistable and has determinant $\lambda ^{r_i}$. \endstatement \proof The ring $R_i$ has dimension $r_i$ and is filtered by ideals whose successive quotients are isomorphic to $\Cee_{y_i}$. Thus $\Cal L_0\otimes R_i$ is filtered by subbundles whose quotients are all isomorphic to the line bundle $\Cal L_0|\{y_i\}\times E$. But by construction this restriction is $\scrO_E(e_i-p_0)$. \endproof At this point, we have seen that $U_0$ is a family of semistable bundles on $E$ whose restriction to every fiber has trivial determinant and with the ``correct" Jordan-H\"older quotients. It remains to show that $U_0$ is a family of regular bundles over $E$. \claim{2.3} The restriction of $U_0$ to every slice $\{e\}\times E$ is regular. \endstatement \demo{Proof of the claim} To see that the restriction to each slice is regular, note that a semistable $V$ of degree $0$ on $E$ is regular if and only if, for all line bundles $\lambda$ on $E$ of degree zero, $h^0(V\otimes \lambda ^{-1})\leq 1$. By Riemann-Roch on $E$, $h^0(V\otimes \lambda ^{-1}) = h^1(V\otimes \lambda ^{-1})$. Thus we must show that $h^1(V\otimes \lambda ^{-1})\leq 1$. First we calculate $R^1\pi _1{}_*(U_0\otimes \pi _2^*\lambda ^{-1})$, where $\pi _1\: \Pee ^{n-1} \times E\to \Pee^{n-1}$ is the projection to the first factor. Let $q_1\: T\times E \to T$ be the first projection. Consider the diagram $$\CD T\times E @>{\nu \times \Id}>> \Pee ^{N-1} \times E\\ @V{q_1}VV @VV{\pi _1}V\\ T @>{\nu}>> \Pee ^{n-1}. \endCD$$ Since $\nu$ and $\nu \times \Id$ are affine, we obtain $$R^1\pi _1{}_*\left[(\nu \times \Id)_*\scrO_{T\times E }(\Delta - G) \otimes \pi _2^*\lambda ^{-1}\right] = \nu _*R^1q_1{}_*\left[\scrO_{T\times E}(\Delta - G ) \otimes q_2^* \lambda ^{-1}\right].$$ Now apply flat base change to the Cartesian diagram $$\CD T\times E @>{r \times \Id}>> E \times E\\ @V{q_1}VV @VV{p _1}V\\ T @>{r}>> E. \endCD$$ We have $\scrO_{T\times E }(\Delta - G) = (r\times \Id)^*\scrO_{E\times E}(\Delta _0 - (E \times \{p_0\}) )$, and thus the sheaf $R^1q_1{}_*\left[\scrO_{T\times E }(\Delta - G) \otimes q_2^* \lambda ^{-1}\right]$ is isomorphic to $$r^*R^1p_1{}_*\left[\scrO_{E\times E }(\Delta _0 -( E \times \{p_0\}) )\otimes p_2^* \lambda ^{-1}\right].$$ Rrestricting to the slice $\{e\}\times E$, we see that $$R^1p_1{}_*\left(\scrO_{E\times E }(\Delta _0 - (E \times \{p_0\}))\otimes p_2^* \lambda ^{-1}\right)$$ is supported at the point $e$ of $E$ corresponding to the line bundle $\lambda$ (i\.e\. $\lambda = \scrO_E(e-p_0)$), and the calculation of \cite{6}, Lemma 1.19 of Chapter 7, shows that the length at this point is one. Thus taking $r^*$ gives the sheaf $\scrO_{F_e}$, and $\nu_*\scrO_{F_e}= \scrO_{H_e}$, where $H_e$ is a reduced hyperplane in $\Pee ^{n-1}$. Thus we have seen that $R^1\pi _1{}_*(U\otimes \pi _2^*\lambda ^{-1})$ is (up to twisting by a line bundle) $\scrO_{H_e}$, where $H_e$ is the hyperplane in $\Pee ^{n-1}$ corresponding to $|np_0 - e|$. Since $\pi_1$ has relative dimension one, $R^2\pi _1 {}_*(U\otimes \pi _2^*\lambda ^{-1})=0$. It follows by the theorem on cohomology and base change \cite{10} Theorem 12.11(b) that the map $R^1\pi _1 {}_*(U\otimes \pi _2^*\lambda ^{-1})\to H^1(V\otimes \lambda ^{-1})$ is surjective, and thus $h^1(V\otimes \lambda ^{-1})\leq 1$ as desired. \endproof \enddemo \ssection{2.3. All universal families of regular semistable bundles.} We have constructed a bundle $U_0$ over $|np_0|\times E$ with given restriction to each slice. Our next goal is to understand all such bundles. \theorem{2.4} Let $E$ be a smooth elliptic curve. Let $\pi_1\colon |np_0|\times E\to |np_0|$ be the projection onto the first factor, and let $U_0$ be the bundle constructed in Theorem \rom{2.1}. Then: \roster \item"{(i)}" The sheaf $\pi _1{}_*Hom(U_0,U_0)$ is a locally free sheaf of algebras of rank $n$ over $|np_0|$ which is isomorphic to $\nu _*\scrO_T$. \item"{(ii)}" Let $U'$ be a rank $n$ vector bundle over $|np_0|\times E$ with the following property. For each $x\in |np_0|$ the restriction of $U'$ to $\{x\}\times E$ is isomorphic to the restriction of $U_0$ to $\{x\}\times E$. Then $U'=(\nu \times \Id)_*\left[\scrO_{T\times E }(\Delta-G)\otimes q _1^*L\right]$ for a unique line bundle $L$ on $T$. \endroster \endstatement \proof In view of Claim 2.3 and the definition of a regular bundle, $\pi _1{}_*Hom(U_0,U_0)$ is a locally free sheaf of algebras of rank $n$ over $|np_0|$. To see that it is isomorphic to $\nu _*\scrO_T$, note that multiplication by functions defines a homomorphism $\nu _*\scrO_T \to \pi _1{}_*Hom(U_0,U_0)$ which is clearly an inclusion of algebras. Since both sheaves of algebras are rank $n$ vector bundles over $\Pee^{n-1}$, they agree at the generic point of $|np_0|$. Thus, over every affine open susbet of $|np_0|$ the rings corresponding to $\pi _1{}_*Hom(U_0,U_0)$ and $\nu _*\scrO_T$ are two integral domains with the same quotient fields. Since $T$ is normal and $\pi _1{}_*Hom(U_0,U_0)$ is finite over $\nu _*\scrO_T$ (since it is finite over $\scrO_{\Pee^{n-1}}$), the two sheaves of algebras must coincide. This proves (i). Now suppose that $U'$ satisfies the hypotheses (ii) of (2.4). By base change $\pi _1{}_*Hom(U',U_0)$ is a locally free rank $n$ sheaf over $|np_0|$. Composition of homomorphisms induces the structure of a $\pi _1{}_*Hom(U_0,U_0)$-module on $\pi _1{}_*Hom(U',U_0)$. Thus $\pi _1{}_*Hom(U',U_0)$ corresponds to a $\nu_*\scrO_T$-module. We claim that, as an $\scrO_T$-module, $\pi _1{}_*Hom(U',U_0)$ is locally free rank of rank one. To see this, fix a point $x$ in $|np_0|$ and let $V', V$ be the vector bundles corresponding to the restrictions of $U', U_0$ to the slice $\{x\}\times E$. Of course, by hypothesis $V'$ and $V$ are isomorphic. Choose an isomorphism $s\: V'\to V$ and extend it to a local section of $\pi _1{}_*Hom(U',U_0)$ in a neighborhood of $x$, also denoted $s$. The map $\pi _1{}_*Hom(U_0,U_0) \to \pi _1{}_*Hom(U',U_0)$ defined by multiplying against the section $s$ is then surjective at $s$, and hence in a neighborhood. Viewing both sides as locally free rank $n$ sheaves over $|np_0|$, the map is then a local isomorphism. But this exactly says that $\pi _1{}_*Hom(U',U_0)$ is a locally free $\pi _1{}_*Hom(U_0,U_0)$-module of rank one. Thus $\pi _1{}_*Hom(U',U_0)$ corresponds to a line bundle on $T$, which we denote by $L^{-1}$. Of course, for any line bundle $M$ on $T$, setting $$U_0[M] = (\nu \times \Id)_*\left[\scrO_{T\times E }(\Delta-G)\otimes q _1^*M\right].$$ we have $$\align \pi _1{}_*Hom (U', U_0[M]) &= \pi _1{}_*(\nu \times \Id)_*Hom((\nu \times \Id)^*U', \scrO_{T\times E}(\Delta-G)\otimes q _1^*M))\\ &=\nu _*q_1{}_*\left[q_1^*M\otimes Hom((\nu \times \Id)^*U', \scrO_{T\times E}(\Delta-G))\right]\\ &= \nu _*\left[M\otimes q_1{}_* Hom((\nu \times \Id)^*U', \scrO_{T\times E}(\Delta-G))\right], \endalign$$ The case $M=\scrO_T$ tells us that, as $\nu_*\scrO_T$-modules, $$\nu_*L^{-1}=\pi_1{}_*Hom (U', U_0)= \nu _*\left[q_1{}_* Hom((\nu \times \Id)^*U', \scrO_{T\times E}(\Delta-G))\right].$$ Hence, we have $$\pi _1{}_*Hom (U', U_0[M]) =\nu_*(M\otimes L^{-1}).$$ Taking $M=L$ we have $$\pi _1{}_*Hom (U', U_0[L]) \cong\nu_*\scrO_T.$$ Via this identification, the section $1\in H^0(\scrO_T)$ then defines an isomorphism from $U'$ to $U_0(L)$, as claimed. \endproof In view of the previous result, we need to describe all line bundles on $T$. Since $T$ is a $\Pee^{n-2}$ bundle over $E$, we have: \lemma{2.5} The projection mapping $r\colon T\to E$ induces an injection $$r^*\Pic E\to \Pic T.$$ If $n=2$, $r$ is an isomorphism and thus $\Pic T\cong \Pic E$. For $n> 2$, since $T$ is included in $\Pee \Cal E \subset \Pee^{n-1}\times E$, we can define by restriction the line bundle $\scrO_{\Pee^{n-1}}(1)|T=\scrO_T(1)$ on $T$. Then $$\Pic T =r^*\Pic E\oplus \Zee[\scrO_T(1)]. \qquad \qed$$ \endstatement In view of Lemma 2.5, we make the following definition. For $p\in E$, let $F_p\subset T$ be the divisor which is the preimage of $p$. \definition{Definition 2.6} For every integer $a$, let $U_a =(\nu\times \Id)_*\scrO_{T\times E }(\Delta - G - a(F_{p_0}\times E))$. More generally, given $e\in E$, define $$U_a[e] = (\nu\times \Id)_*\scrO_{T\times E }(\Delta - G - (a+1)(F_{p_0}\times E)+(F_e\times E)).$$ Thus $U_a[p_0] = U_a$. By Lemma 2.5, every vector bundle obtained from $U_0$ by twisting by a line bundle on the spectral cover is of the form $U_a[e]\otimes \pi _1^*\scrO_{\Pee^{n-1}}(b)$ for some $b\in \Zee$ and $e\in E$. (For $n=2$, we have the relation $\nu ^*\scrO_{\Pee^1}(1) = \scrO_E(2p_0)$, and thus $U_a\otimes \pi _1^*\scrO_{\Pee ^{1}}(b)\cong U_{a-2b}$.) \enddefinition The next lemma says that the $U_a$ are all elementary modifications of each other: \lemma{2.7} Let $H =\nu(F_{p_0})$ be the hyperplane in $\Pee^{n-1} =|np_0|$ of divisors whose support contains $p_0$, and let $i\: H \to \Pee^{n-1}$ be the inclusion. Then there is an exact sequence $$0 \to U_a \to U_{a-1} \to (i\times \Id)_*\scrO_{H\times E} \to 0.$$ Moreover $\dim \Hom (U_{a-1}|H\times E, \scrO_{H\times E}) = 1$, so that the above exact sequence is the unique elementary modification of this type. Likewise, $U_a[e]$ is given as an elementary modification: $$0 \to U_{a+1} \to U_a[e] \to (i\times \Id)_*\scrO_{H_e\times E}\otimes \pi_2^*\scrO_E(e-p_0) \to 0.$$ \endstatement \proof Consider the exact sequence $$\gather 0 \to \scrO_{T\times E }(\Delta - G - a(F_{p_0}\times E)) \to \\ \to \scrO_{T\times E }(\Delta - G - (a-1)(F_{p_0}\times E))\to \scrO_{F_{p_0}\times E }(\Delta - G - (a-1)(F_{p_0}\times E))\to 0. \endgather$$ Clearly the restriction of the line bundle $\scrO_{T\times E }(F_{p_0}\times E)$ to $F_{p_0}\times E$ is trivial, and $G$ and $\Delta$ both restrict to the divisor $F_{p_0}\times \{p_0\}\subset F_{p_0}\times E$. Hence the last term in the above sequence is $\scrO_{F_{p_0}\times E }$. Applying $(\nu\times \Id)_*$ to the sequence gives the exact sequence of (2.7). For $V$ a bundle corresponding to a point of $H$, $\dim \Hom(V, \scrO_E) = 1$. Thus $\pi _1{}_*Hom(U_a|H\times E, \scrO_{H\times E})$ is a line bundle on $H$. The given map $U_a|H\times E \to \scrO_{H\times E}$ constructed above is an everywhere generating section of this line bundle, so that $\pi _1{}_*Hom(U_a|H\times E, \scrO_{H\times E})$ is trivial and $\dim \Hom (U_a|H\times E, \scrO_{H\times E}) = 1$. The proof of the exact sequence relating $U_{a+1}$ and $U_a[e]$ is similar. \endproof In fact, suppose that we have an elementary modification $$0 \to U' \to U_a \to \scrO_{D\times E}\otimes \pi _2^*\lambda \to 0,$$ where $D$ is a hypersurface in $|np_0|$ and $\lambda$ is a line bundle of degree zero on $E$. Then it is easy to check that necessarily $D=H_e$ for some $e$ and $\lambda =\scrO_E(e-p_0)$. Of course, it is also possible to make elementary modifications along certain hyperplanes corresponding to taking higher rank quotients of $U_a$. \ssection{2.4. Families of bundles over more general parameter spaces.} Now let us examine in what sense the bundles $U\to |np_0|\times E$ that we have constructed are universal. \theorem{2.8} Let $E$ be a smooth elliptic curve and let $S$ be a scheme or analytic space. Suppose that ${\Cal U}\to S\times E$ is a rank $n$ holomorphic vector bundle whose restriction to each slice $\{s\}\times E$ is a regular semistable bundle with trivial determinant. Let $\Phi\colon S\to |np_0|$ be the morphism constructed in Theorem \rom{1.5}. Let $\nu_S\colon\tilde S\to S$ be the pullback via $\Phi$ of the spectral covering $T\to |np_0|$\rom: $$\tilde S=S\times_{|np_0|}T,$$ and let $\tilde\Phi\colon \tilde S\to T$ be the map covering $\Phi$. Let $q_1\colon \tilde S\times E\to \tilde S$ be the projection onto the first factor. Then there is a line bundle ${\Cal M}\to \tilde S$ and an isomorphism of $\Cal U$ with $$(\nu_S\times\operatorname{Id})_* \left((\tilde\Phi\times \Id)^*(\scrO_{T\times E}(\Delta-G))\otimes q_1^*{\Cal M}\right).$$ \endstatement \proof By construction the bundle $$(\nu_S\times\operatorname{Id})_* (\tilde\Phi\times \Id)^*(\scrO_{T\times E}(\Delta-G))$$ is a family of regular semistable bundles with trivial determinant $E$, which fiber by fiber have the same Jordan-H\"older quotients as the family ${\Cal U}$. But regular semistable bundles are determined up to isomorphism by their Jordan-H\"older quotients. This means that the two families are isomorphic on each slice $\{s\}\times E$. Now the argument in the proof of Theorem 2.4 applies to establish the existence of the line bundle $\Cal M$ on the spectral covering $\tilde S$ as required. \endproof We can also construct the spectral cover $\tilde S$ of $S$ directly. This construction will also the explain the origin of the name {\sl spectral cover\/}. If $p_1, p_2$ are the projections of $S\times E$ to the first and second factors, then by standard base change results $p_1{}_*Hom (\Cal U, \Cal U)$ is a locally free sheaf of coherent $S$-algebras. Moreover, by the classification of regular semistable bundles, it is commutative. Thus there is a well-defined space $\tilde S=\bold{Spec}\,p_1{}_*Hom (\Cal U, \Cal U)$ and a morphism $\nu\: \tilde S \to S$ such that $\scrO_{\tilde S} = p_1{}_*Hom (\Cal U, \Cal U)$. It is easy to check directly that $\tilde S = S\times _{|np_0|}T$. By construction, there is an action of $\scrO_{\tilde S}$ on $\Cal U$ that commutes with the action of $\scrO_E$, and thus $\Cal U$ corresponds to a coherent sheaf $\Cal L$ on $\tilde S \times E$. Again by the classification of regular semistable bundles, it is straightforward to check directly that $\Cal L$ is locally free of rank one. Clearly, $(\nu\times \Id)_*\Cal L = \Cal U$. We can view Theorem 2.8 as allowing us to replace a family of possibly non-regular, semistable bundles with trivial determinant on $E$ with a family of regular semistable bundles without changing the Jordan-H\"older quotients on any slice. Suppose that ${\Cal V}\to S\times E$ is any family of semistable bundles with trivial determinant over $E$. We have the map $\Phi\colon S\to |np_0|$ of Theorem 1.5, and $(\Phi\times \operatorname{Id})^*U_0\to S\times E$ is a family of regular semistable bundles with the same Jordan-H\"older quotients as ${\Cal V}$ along each slice $\{s\}\times E$. Of course, the new bundle will not be isomorphic to ${\Cal V}$ (even after twisting with a line bundle on the spectral cover) unless the original family is a family of regular bundles. \ssection{2.5. The case of singular curves.} There is an analogue of these constructions for singular curves. Let $E$ be a Weierstrass cubic. The constuction given at the beginning of this section is valid in this context and produces a $\Pee^{n-2}$-bundle $T=\Pee\Cal E$ over $E$ and an $n$-fold covering map $\nu\colon T\to |np_0|$. By the description of $T$ as $\Pee\Cal E$, the projection $T\to \Pee^{n-1}$ is a finite flat morphism. Let $\Omega\subset |np_0|$ be the Zariski open subset of all divisors whose support does not contain the singular point of $E$, and let $T_\Omega\subset T$ be $\nu^{-1}(\Omega)$. We denote by $\nu_\Omega$ the restriction of $\nu$ to $T_\Omega$. It is a finite surjective morphism of degree $n$ between smooth varieties. As before, we have the divisor $\Delta\subset T\times E$. We denote by $\Delta_\Omega\subset T_\Omega\times E$ the restriction of $\Delta$ to this open subset. We form the line bundle $${\Cal L}_0^\Omega=\scrO_{T_\Omega\times E}(\Delta_\Omega-G),$$ where $G$ is the divisor $T_\Omega\times \{p_0\}$. Let $U_0^\Omega$ be the sheaf $(\nu_\Omega\times \operatorname{Id})_*({\Cal L}_0^\Omega)$ over $\Omega\times E$. It is a vector bundle of rank $n$. The arguments in the proof of Claim 2.3 apply in this context and show that $U_0^\Omega$ is a family of regular bundles on $E$ parametrized by $\Omega$. The arguments in the proof of Theorem 2.4 apply to yield the following result. \proposition{2.9} Let $E$ be a Weierstrass cubic. Let $\Omega\subset |np_0|$ be the Zariski open subset defined in the previous paragraph. Let $\pi^\Omega_1\colon \Omega\times E\to \Omega$ be the projection onto the first factor, and let $U_0^\Omega=(\nu_\Omega\times \operatorname{Id})_*{\Cal L}_0^\Omega$ be the bundle over $\Omega\times E$ constructed in the previous paragraph. Then: \roster \item"{(i)}" The sheaf $(\pi_1^\Omega)_* Hom(U_0^\Omega,U_0^\Omega)$ is a locally free sheaf of algebras of rank $n$ over $\Omega\subset |np_0|$ which is isomorphic to $\nu _*\scrO_{T_\Omega}$. \item"{(ii)}" Let $U'$ be a rank $n$ vector bundle over $\Omega\times E$ with the following property. For each $x\in \Omega$ the restriction of $U'$ to $\{x\}\times E$ is isomorphic to the restriction of $U_0$ to $\{x\}\times E$. Then $U'=(\nu_\Omega \times \Id)_*\left[\scrO_{T_\Omega\times E }(\Delta_\Omega-G)\otimes q _1^*L\right]$ for a unique line bundle $L$ on $T_\Omega$. \endroster \endstatement In Section 3, we shall show how to extend this construction over the singular points of $E$. \ssection{2.6. Chern classes.} Finally, we return to the case where $E$ is smooth and give the Chern classes of the various bundles over $|np_0|\times E$ in case $E$ is smooth. The proof of (2.10) will be given in the next section, and we will prove the remaining results assuming (2.10). \proposition{2.10} Identify $h \in H^2(\Pee ^{n-1})$ with its pullback to $\Pee ^{n-1} \times E$. Then the total Chern class $c(U_0)$ and the Chern character $\ch(U_0)$ of $U_0$ are given by the formulas\rom: $$\align c(U_0) &= (1-h+ \pi _2^*[p_0]\cdot h)(1-h)^{n-2}\\ \ch U_0 &= ne^{-h} + (1- \pi _2^*[p_0])(1- e^{-h}). \endalign$$ \endstatement Once we have (2.10), we can calculate the Chern classes of all the universal bundles. \proposition{2.11} Let $U_a$ and $U_a[e]$ be defined as in \rom{(2.6)}. Let $h$ be the class of a hyperplane in $\Pic\Pee^{n-1}$, which we also view by pullback as an element of $\Pic (\Pee^{n-1}\times E)$. \roster \item"{(i)}" $c(U_a) = (1-h+ \pi _2^*[p_0]\cdot h)(1-h)^{a+n-2}$. \item"{(ii)}" $\ch (U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)) = ne^{(b-1)h} + (1-a- \pi _2^*[p_0])(e^{bh}- e^{(b-1)h})$. \item"{(iii)}" $\det U_a[e] = -(a+n-1)h$. \item"{(iv)}" Let $\tilde c_2$ denote the refined Chern class of a vector bundle in the Chow group $A^2(\Pee^{n-1}\times E)$. Let $A^2_0(\Pee^{n-1}\times E)$ be the subgroup of $A^2(\Pee^{n-1}\times E)$ of all cycles homologous to zero, so that $A^2_0(\Pee^{n-1}\times E) \cong E$. Then $$\tilde c_2(U_a[e]\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)) - \tilde c_2(U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)) = e$$ as an element of $A^2_0(\Pee^{n-1}\times E) \cong E$. \endroster \endstatement \proof By (2.7), $$c(U_a) = c(U_{a-1})c((i\times \Id)_*\scrO_{H\times E})^{-1}$$ and likewise $$\ch U_a = \ch U_{a-1} - \ch ((i\times \Id)_*\scrO_{H\times E}).$$ Using the exact sequence $$0 \to \scrO_{\Pee^{n-1}\times E}(-H\times E) \to \scrO_{\Pee^{n-1}\times E} \to (i\times \Id)_*\scrO_{H\times E} \to 0,$$ we have $$\align c((i\times \Id)_*\scrO_{H\times E}) &= (1-h)^{-1};\\ \ch ((i\times \Id)_*\scrO_{H\times E}) &= 1-e^{-h}. \endalign$$ A little manipulation, starting with (2.10), gives (i) and (ii). To see (iii), note that by construction $\det U_a[e]$ is the pullback of a class in $\Pic\Pee^{n-1}$. Moreover, it is independent of the choice of $e\in E$. Thus we may as well take $e=p_0$, in which case $U_a[p_0]=U_a$. In this case, the result is immediate from (i). (iv) follows by using the elementary modification relating $U_a[e]$ and $U_{a+1}$. \endproof Note that $$c_1\left(U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b))\right) = 0$$ if and only if $a-1 = n(b-1)$. A natural solution to this equation is $a=b=1$.The bundle $U = U_1\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b) = (\nu\times \Id)_*\scrO_{T\times E}(\Delta - G -(F_{p_0}\times E))$ is singled out in this way as $(\nu \times \Id)_*\Cal P$, where $\Cal P$ is the pullback to $T\times E$ of the symmetric line bundle $\scrO_{E\times E }(\Delta _0 - \{p_0\}\times E -f\times \{p_0\})$, which is a Poincar\'e line bundle for $E\times E$. In this case $\ch U = n + \pi _2^*[p_0](1- e^h)$. Moreover, one can check that $c_1(U) = 0$ and $c_k(U) =(-1)^kh^{k-1}\pi _2^*[p_0]$ for $k\geq 2$. It is easy to check that, for $n>2$, $U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b) = U_{a'}\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b')$ if and only if $a=a'$ and $b=b'$. It is also possible to vary $aF$ within its algebraic equivalence class, which is a family isomorphic to $E$, and this difference is detectable by looking at $c_2\left(U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)\right)$ in the Chow group $A^2(\Pee ^{n-1}\times E)$. More precisely, we have the following: \proposition{2.12} Given two vector bundles $U' =(\nu\times \Id)_*\left(\scrO_{T\times E }(\Delta - G )\otimes M'\right)$ and $U'' =(\nu\times \Id)_*\left(\scrO_{T\times E }(\Delta - G )\otimes M''\right)$, where $M'$ and $M''$ are line bundles on $E$, then $U'$ and $U''$ are isomorphic if and only if they have the same Chern classes as elements of $A^*(\Pee ^{n-1}\times E)$. \endstatement \proof For simplicity, we shall just consider the case $n>2$. Using the notation of (2.6) and the description of $\Pic T$, it suffices to show that, if the Chern classes of $U_a[e]\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)$ and of $U_{a'}[e']\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b')$ are equal in the Chow ring, then $a=a', b=b'$, and $e=e'$. Following the above remarks, the Chern classes of $U_a[e]\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)$ in rational cohomology, which are of course the same as those of $U_a\otimes \pi _1^*\scrO_{\Pee ^{n-1}}(b)$, determine $a$ and $b$ (use $c_3$ to find $b$ and $c_1$ to find $a$). By (iii) of (2.11), the class $\tilde c_2$ then determines $e$. \endproof \section{3. Moduli spaces via extensions.} In this section we shall describe a completely different approach to constructing universal bundles over $\Pee ^{n-1}\times E$. The idea here is to consider the space of extensions of fixed (and carefully chosen) bundles over $E$. From this point of view the projective space is the projective space of the relevant extension group, which is {\it a priori\/} a very different animal from $|np_0|$. We shall show however (Proposition 3.13) that this projective space is naturally identified with $|np_0|$. There are several reasons for considering this alternative approach. First of all it works as well for singular curves as for smooth ones, so that the restrictions of the last section to smooth curves or to bundles concentrated away from the singularities of a singular curve can be removed. Also, this method works well for a family of elliptic curves, not just a single elliptic curve. Lastly, this approach has a natural generalization to all holomorphic principal bundles with structure group an arbitrary complex simple group $G$, something which so far is not clear for the spectral cover approach. The generalization to $G$-bundles is discussed in \cite{8}. The disadvantange of the approach of this section is that it constructs some but not all of the families that the spectral cover approach gives. The reason is that from this point of view one cannot see directly the analogue of twisting by a general line bundle on the spectral cover to produce the general family of regular bundles. The main results of this section are as follows. In Theorem 3.2, we consider the set of relevant extensions and show that every such extension is a regular semistable bundle with trivial determinant. Conversely, every regular semistable bundle with trivial determinant arises as such an extension. In the construction of bundles of rank $n$ over $E$ we must choose an integer $d$ with $1\le d<n$. We show that constructions for different $d$ are related to one another (Proposition 3.11 and Theorem 3.12). Next, we compare the extension moduli space, which is a $\Pee^{n-1}$, to the coarse moduli space which is $|np_0|$. We find a natural cohomological identification of these two projective spaces (Theorem 3.13) and check that it corresponds to the morphism $\Phi$ of Section 1 (Proposition 3.16). Next we show how the universal bundles defined via the extension approach lead to the spectral covers of Section 2 (Theorem 3.21). In this way, we can both identify the universal bundles constructed here with those constructed via spectral covers (Theorem 3.23 and Corollary 3.24), and extend the spectral cover construction to the case of a singular $E$. {\bf Throughout this section, $E$ denotes a Weierstrass cubic with origin $p_0$.} \ssection{3.1. The basic extensions.} We begin by recalling a result, essentially due to Atiyah, which produces the basic bundles for our extensions: \lemma{3.1} For each $d\geq 1$, there is a stable bundle $W_d$ of rank $d$ on $E$ whose determinant is isomorphic to $\scrO_E(p_0)$. It is unique up to isomorphism. For every rank one torsion free sheaf $\lambda$ of degree zero, $h^0(W_d\otimes \lambda )=1$ and $h^1(W_d\otimes \lambda )=0$. \endstatement \proof We briefly outline the proof. An inductive construction of $W_d$ is as follows: set $W_1 =\scrO_E(p_0)$. Assume inductively that $W_{d-1}$ has been constructed and that $h^0(W_{d-1}) = 1$. It then follows by Riemann-Roch that $h^1(W_{d-1}) = 0$, and thus that $h^0(W_{d-1}\spcheck) = 0$, $h^1(W_{d-1}\spcheck) = 1$. We then define $W_d$ by taking the unique nonsplit extension $$0 \to \scrO_E \to W_{d} \to W_{d-1} \to 0.$$ By construction $W_d$ has a filtration whose successive quotients, in increasing order, are $\scrO_E, \dots, \scrO_E$, $\scrO_E(p_0)$, and such that all of the intermediate extensions are not split. It is the unique bundle with this property. An easy induction shows that $W_d$ is stable. To see this, note that $W_d$ is stable if and only if every proper subsheaf $J$ of $W_d$ has degree at most zero. But if $J$ is a proper subsheaf of $W_d$ of positive degree, then the image of $J$ in $W_{d-1}$ also has positive degree, and hence $J\to W_{d-1}$ is surjective. But since the rank of $J$ is at most $d-1$, the projection of $J$ to $W_{d-1}$ is an isomorphism. This says that $W_d$ is a split extension of $W_{d-1}$ by $\scrO_E$, a contradiction. Thus $W_d$ is stable. The uniqueness statement is clear in the case of rank one. Now assume inductively that we have showed that, for $d<n$, every stable bundle of rank $d$ whose determinant is isomorphic to $\scrO_E(p_0)$ is isomorphic to $W_d$. Let $W$ be a stable bundle of rank $n$ such that $\det W = \scrO_E(p_0)$. By stability, $h^1(W) = \dim \Hom (W, \scrO_E) = 0$, and so $h^0(W) = 1$. If $\scrO_E \to W$ is the map corresponding to a nonzero section, then by stability the cokernel $Q$ is torsion free. An argument as in the proof that $W_d$ is stable shows that $Q$ is stable. If $E$ is smooth, then $Q$ is automatically locally free. When $E$ is singular, Lemma 0.4 implies that $W$ is locally isomorphic to $Q\oplus \scrO_E$. Thus, if $W$ is locally free, then $Q$ is locally free as well. Once we know that $Q$ is locally free, we are done by induction. To see the final statement, first note that, since $\deg (W_d\otimes \lambda )=1$, we have by definition that $$h^0(W_d\otimes \lambda ) - h^1(W_d\otimes \lambda ) = 1.$$ It will thus suffice to show that $h^1(W_d\otimes \lambda ) = 0$. By Serre duality, $$h^1(W_d\otimes \lambda )=\dim \Hom (W_d\otimes \lambda , \scrO_E) =\dim \Hom(W_d, \lambda\spcheck).$$ Since $\lambda\spcheck$ is also a rank one torsion free sheaf of degree zero, $\Hom(W_d, \lambda\spcheck) =0$ by stability. \endproof \remark{Exercise} We have defined $\Cal E$ in the previous section as the rank $n-1$ vector bundle which is the kernel of the evaluation map $H^0(\scrO_E(np_0))\otimes \scrO_E \to \scrO_E(np_0)$. Show that $$\Cal E \cong W_{n-1}\spcheck \otimes \scrO_E(-p_0).$$ \endremark Now we are ready to see how extensions of the $W_d$ can be used to make regular semistable bundles. \theorem{3.2} Let $V$ be an extension of the form $$0 \to W_d\spcheck \to V \to W_{n-d} \to 0.$$ Then: \roster \item"{(i)}" $V$ has trivial determinant \item"{(ii)}" $V$ is semistable if and only if the above extension is not split. In this case $V$ is regular. \item"{(iii)}" Suppose that $V$ is semistable, i\.e\. that the above extension is not split. Then $\dim \Hom (V,V) = n$ and $\Hom(V,V)$ is an abelian $\Cee$-algebra. Moreover, every homomorphism $W_d\spcheck \to V$ is of the form $\phi \circ \iota$, where $\phi \in \Hom (V,V)$ and $\iota$ is the given inclusion $W_d\spcheck \to V$. If $V$ and $V'$ are given as extensions as above, then $V$ and $V'$ are isomorphic if and only if their extension classes in $\Ext^1(W_{n-d},W_d\spcheck)$ are multiples of each other. \item"{(iv)}" If $V$ is a regular semistable vector bundle of rank $n>1$ with trivial determinant, then $V$ can be written as an extension as above. \item"{(v)}" If $V$ is a nontrivial extension of $W_{n-d}$ by $W_d\spcheck$, and $ad(V)$ is the sheaf of trace free endomorphisms of $V$, then $H^0(ad(V)) \cong \Ker\{\, \Hom (W_d\spcheck, W_{n-d})\to H^1(Hom (W_d\spcheck, W_d\spcheck))\cong \Cee\,\}$ and $H^1(ad(V)) \cong \Ext^1(W_{n-d},W_d\spcheck)/\Cee\xi$, where $\xi$ is the extension class corresponding to $V$. \endroster \endstatement \proof (i) This is clear since $\det W_d \cong\det W_{n-d}$. \smallskip \noindent (ii) If $V$ is unstable, let $W$ be the maximal destabilizing subsheaf. Then $W$ is stable of positive degree and rank $r$ for some $r< n$. Since $\Hom (W, W_d\spcheck) = 0$, the induced map $W \to W_{n-d}$ is nonzero. Now it is easy to see by the stability of $W_s$ that if there is a nonzero map $W \to W_s$, where $W$ has positive degree and rank $r$, then $r\geq s$, and every nonzero such map is surjective. (From this it follows in particular that, for $r\geq s$, $\Hom(W_r, W_s) \cong \Hom (W_s, W_s) = \Cee$.) If $r> n-d$, the kernel of the map $W \to W_{n-d}$ is a subsheaf of degree at least zero of $W_d\spcheck$, and since $W_d\spcheck$ is a stable bundle of degree $-1$, the kernel is zero. Hence $W\cong W_{n-d}$, which means that the extension is split. Conversely, if the extension is split then $V$ is unstable. Next we show that $V$ is regular. Since $W_{n-d}$ is a stable bundle of degree $1$, $\Hom (W_{n-d}, \lambda) =0$ for every rank one torsion free sheaf $\lambda$ of degree zero. Moreover, with $\lambda$ as above, $h^0((W_d\otimes \lambda) = \dim \Hom (W_d\spcheck, \lambda) = 1$ by the last sentence in Lemma 3.1. Thus $\dim \Hom (V,\lambda)\leq 1$ for every $\lambda$ of degree zero, so that, by (1.14), $V$ is regular. \smallskip \noindent (iii) Consider the exact sequence $$\Hom (W_{n-d}, W_d\spcheck) \to \Hom (W_{n-d}, V) \to \Hom (W_{n-d}, W_{n-d}) \to \Ext ^1(W_{n-d}, W_d\spcheck).$$ Note that $\Hom (W_{n-d}, W_d\spcheck) = 0$ by stability. Since $W_{n-d}$ is stable, it is simple, and so $\Hom (W_{n-d}, W_{n-d}) = \Cee \cdot \Id$. But the image of $\Id$ in $\Ext ^1(W_{n-d}, W_d\spcheck)$ is the extension class. Since this class is nonzero, $\Hom (W_{n-d}, V) = 0$ as well. Next consider the exact sequence $$0= \Hom (W_{n-d}, V) \to \Hom (V, V) \to \Hom (W_d\spcheck, V) \to H^1(W_{n-d}\spcheck \otimes V).$$ Since $\Hom (W_{n-d}, V) = 0$, the map $\Hom (V, V) \to \Hom (W_d\spcheck, V)$ is an injection. Moreover, we have a commutative diagram $$\minCDarrowwidth{.175 in} \CD \Cee\cdot \Id @>>> \Hom (V,V) @>>> H^0(ad(V)) @>>> 0\\ @V{\cong}VV @VVV @VVV @.\\ \Hom(W_d\spcheck, W_d\spcheck) @>>> \Hom (W_d\spcheck, V) @>>> \Hom (W_d\spcheck, W_{n-d}) @>>> H^1(Hom(W_d\spcheck, W_d\spcheck)). \endCD$$ Thus we see that $H^0(ad(V)) \cong \Ker\{\, \Hom (W_d\spcheck, W_{n-d})\to H^1(Hom (W_d\spcheck, W_d\spcheck))\cong \Cee\,\}$ and by duality $H^1(ad(V)) \cong \Ext^1(W_{n-d},W_d\spcheck)/\Cee\xi$, where $\xi$ is the extension class corresponding to $V$. This proves (v). Let us assume that $\dim \Hom (V,V) \geq n$, which we have already checked in case $E$ is smooth. (We will establish this for singular curves after proving Part (iv), as well as checking the fact that $\Hom (V,V)$ is abelian. These results are not used in the proof of Part (iv) of the theorem.) If we can show that $\dim \Hom (W_d\spcheck, V) = n$, then $\Hom (V, V) \to \Hom (W_d\spcheck, V)$ is an isomorphism, and in particular $\dim \Hom (V,V) = n$ as well. We compute the dimension of $\Hom(W_d\spcheck,V)$. Consider the exact sequence $$0 \to \Hom (W_d\spcheck, W_d\spcheck) \to \Hom (W_d\spcheck, V) \to \Hom (W_d\spcheck, W_{n-d}) \to H^1(Hom (W_d\spcheck, W_d\spcheck)) $$ Since $W_d\spcheck$ is stable, $\dim \Hom (W_d\spcheck, W_d\spcheck) = 1$. Next we claim that $$\dim \Hom (W_d\spcheck, W_{n-d}) = h^0(W_d\otimes W_{n-d}) = n.$$ \claim{3.3} If $E$ is a Weierstrass cubic, then $h^0(W_d\otimes W_{n-d}) = n$ and $h^1(W_d\otimes W_{n-d}) = 0$. Dually, $h^0(W_d\spcheck\otimes W_{n-d}\spcheck) = 0$ and $h^1(W_d\spcheck\otimes W_{n-d}\spcheck) = n$. \endstatement \proof If $d=1$, this follows from the exact sequence $$0 \to W_{n-1} \to W_{n-1}\otimes \scrO_E(p_0) \to (\Cee _{p_0}) ^{n-1} \to 0,$$ together with the fact that $h^1(W_{n-1}) = 0$ by stability. The general case follows by induction on $n$, by tensoring the exact sequence $$0 \to \scrO_E \to W_d \to W_{d-1}\to 0$$ by $W_{n-d}$. \endproof By Riemann-Roch, $h^1(Hom (W_d\spcheck, W_d\spcheck))=1$. Thus by counting dimensions, to show that $\dim \Hom (W_d\spcheck, V) = n$ it will suffice to show that $\Hom (W_d\spcheck, W_{n-d}) \to H^1(Hom (W_d\spcheck, W_d\spcheck))$ is surjective. Equivalently we must show that the map from $H^1(Hom (W_d\spcheck, W_d\spcheck)) \to H^1(Hom (W_d\spcheck, V))$ is zero. But this map is dual to the map $\Hom (V, W_d\spcheck) \to \Hom (W_d\spcheck, W_d\spcheck) = \Cee \cdot \Id$. A lifting of $\Id$ to a homomorphism $V\to W_d\spcheck$ would split the exact sequence, contrary to assumption. This completes the proof of all of Part (iii) except for the last sentence. We turn to the last statement in Part (iii). If $V$ is a split extension, then it is unstable and so $V'$ is unstable and therefore a split extension as well. Thus we may suppose that $V$ and $V'$ are nontrivial extensions of the given type and that $\psi\colon V'\to V$ is an isomorphism. Using $\psi$ to identify $V$ and $V'$, suppose that we are given two inclusions $\iota _1, \iota _2\: W_d\spcheck \to V$ such that both quotients are isomorphic to $W_{n-d}$. By the first part of (iii), there is an endomorphism $A$ of $V$ such that $A\circ \iota _1 = \iota_2$. Since $W_{n-d}$ is simple, the induced map on the quotient $W_{n-d}$ factors must be a multiple $\alpha\in \Cee$ of the identity. This multiple $\alpha$ cannot be zero, since otherwise $A$ would define a splitting of the extension corresponding to $\iota _2$. In particular, $A$ is an automorphism of $V$. Furthermore, we see that the extension class for $V'$ is $\alpha$ times the extension class for $V$. This completes the proof of (iii). \smallskip To prove Part (iv) of the theorem, given a semistable $V$, we seek subbundles of $V$ isomorphic to $W_d\spcheck$ such that the quotient is isomorphic to $W_{n-d}$. \lemma{3.4} Fix $d>0$. For any $r>0$ and any line bundle $\lambda$ of degree zero there is a map $$W_d\spcheck \to I_r(\lambda)$$ whose image is not contained in a proper degree zero subsheaf of $I_r(\lambda)$. Likewise, for any strongly indecomposable, degree zero, semistable bundle $I(\Cal F)$ concentrated at the singular point of a singular curve, there is a map $W_d\spcheck\to I(\Cal F)$ whose image is not contained in a proper degree zero subsheaf. \endstatement \proof We consider case of $I_r(\lambda)$ first. It suffices by (1.8) to show that there is a map $W_d\spcheck\to I_r(\lambda)$ whose image is not contained in $F_{r-1}\cong I_{r-1}(\lambda)$. Tensoring the exact sequence $$0\to I_{r-1}(\lambda)\to I_r(\lambda)\to \lambda\to 0$$ with $W_d$, we see that there is an exact sequence $$0 \to \Hom (W_d\spcheck,I_{r-1}(\lambda))\to \Hom (W_d\spcheck,I_r(\lambda))\to \Hom (W_d\spcheck,\lambda)\to 0$$ By the last statement in (3.1), there is a nonzero element of $\Hom (W_d\spcheck,\lambda)$, and by induction on $r$, $H^1(W_d\otimes I_{r-1}(\lambda)) =0$. Thus there is a map $W_d\spcheck\to I_r(\lambda)$ not in the image of a homomorphism into $I_{r-1}(\lambda)$. Now let us consider the case of a strongly indecomposable bundle $I(\Cal F)$. Since every semistable bundle concentrated at the singular point is filtered with associated gradeds isomorphic to ${\Cal F}$, we have a short exact sequence $$0\to X\to I(\Cal F)\to {\Cal F}\to 0.$$ Direct cohomology computations as above show that there is a map $W_d\spcheck\to I(\Cal F)$ which has nontrivial image in the quotient ${\Cal F}$. Clearly, the image of this map is not contained in $X$. But, by (1.11), every proper degree zero subsheaf of $I(\Cal F)$ is contained in $X$, proving the result in this case as well. \endproof We can generalize (3.4) to every regular semistable bundle $V$. \corollary{3.5} Let $V$ be a regular semistable bundle and let $d$ be a positive integer. Then there is a map $W_d\spcheck\to V$ whose image is not contained in any proper degree zero subsheaf of $V$. \endstatement \proof This is immediate from the previous result and the fact that $V$ decomposes uniquely as a direct sum $\bigoplus_iI_{r_i}(\lambda_i)\oplus I(\Cal F)$, where the $\lambda_i$ are pairwise distinct line bundles of degree zero and $I(\Cal F)$ is a strongly indecomposable bundle concentrated at the node. Since the $\lambda_i$ are pairwise distinct, any degree zero subsheaf of $V$ is a direct sum of subsheaves of the factors. Thus, for each summand $I_{r_i}(\lambda_i)$ or $I(\Cal F)$, choose a map $W_d\spcheck$ to the corresponding summand whose image is not contained in any proper degree zero subbundle of the summand. The induced map of $W_d\spcheck$ into the direct sum is as desired. \endproof Note that, if instead $\lambda _i = \lambda _j$ for some $i\neq j$, then there would exist degree zero subsheaves of the direct sum which were not a direct sum of subsheaves of the summands, and in fact (3.5) always fails to hold in this case. Now let us show that the quotient of a map satisfying the conclusions of (3.5) is $W_{n-d}$. \proposition{3.6} Let $V$ be a semistable regular bundle of rank $n$ with trivial determinant and let $\iota\: W_d\spcheck\to V$ be a map whose image is not contained in any proper degree zero subsheaf of $V$. If the rank of $V$ is strictly greater than $d$, then $\iota$ is an inclusion and the quotient $V/W_d\spcheck$ is isomorphic to $W_{n-d}$. Conversely, if $\iota$ is the inclusion of $W_d\spcheck$ in $V$ so that the quotient $V/W_d\spcheck$ is isomorphic to $W_{n-d}$, then the image of $\iota$ is not contained in a proper subsheaf of degree zero. \endstatement \proof Let $V$ have rank $n\ge d+1$, and suppose that we have a map $\iota\:W_d\spcheck\to V$ whose image is not contained in a proper degree zero subsheaf of $V$. In particular, $\iota$ is nontrivial. If $\iota$ is not injective, then by the stability of $W_d\spcheck$, the image of $\iota$ is a subsheaf of $V$ of rank $\leq d-1$ and degree $\geq 0$, and hence is a proper subsheaf of $V$ of degree zero, contrary to assumption. Likewise, if the cokernel of $\iota$ is not torsion free, then the image of $\iota$ is contained in a proper subsheaf of $V$ whose degree is strictly larger than $-1$, and thus the degree is at least zero. This again contradicts our assumption about the map and the fact that the rank of $V$ is at least $d+1$. Thus $\iota$ is injective and its cokernel $W$ is torsion free. Using (0.4), $W$ is locally a direct summand of $V$, and thus $W$ is locally free. It follows that $W$ is a rank $n-d$ vector bundle whose determinant is $\scrO_E(p_0)$. To conclude that the quotient $W$ is isomorphic to $W_{n-d}$, it suffices to show that $W$ is stable. If $W$ is not stable, then there is a proper subsheaf $U$ of $W$ with degree at least one. Let $U''\subset V$ be the preimage of $U$. The degree of $U''$ is at least zero, and hence, by the semistablility of $V$ is of degree zero. Clearly, $U''$ contains the image of $\iota$. Hence by our hypothesis on $\iota$, $U''=V$, and consequently, $U=W$. This is a contradiction, so that $W$ is stable. Finally we must show that, if $V$ is written as an extension of $W_{n-d}$ by $W_d\spcheck$, then the subbundle $W_d\spcheck$ cannot be contained in a proper subsheaf $U$ of $V$ of degree zero. If $U$ is a such a subsheaf, then $U/W_d\spcheck$ would be a proper subsheaf of $W_{n-d}$ of degree at least one, contradicting the stability of $W_{n-d}$. \endproof Corollary 3.5 and Proposition 3.6 show that any regular semistable bundle over $E$ can be written as an extension of $W_{n-d}$ by $W_d\spcheck$. This completes the proof of Part (iv). Now let us return to the point in the proof of (iii) where it is claimed that $\dim \Hom (V,V) \geq n$ for all $V$ which are given as a nonsplit extension of $W_d$ by $W_{n-d}\spcheck$. In order to establish this result, we first describe the space of all such extensions, which is an immediate consequence of (3.3): \lemma{3.7} The space $\Ext^1(W_{n-d}, W_d\spcheck) = H^1(W_{n-d}\spcheck \otimes W_d\spcheck)$ has dimension $n$, and thus the associated projective space is a $\Pee ^{n-1}$. \qed \endstatement By general properties, there is a universal extension $\bold U(d;n)$ over $\Pee_d^{n-1}\times E$ of the form $$0 \to \pi _2^*W_d\spcheck \otimes \pi _1^*\scrO_{\Pee_d ^{n-1}}(1) \to \bold U(d;n) \to \pi _2^*W_{n-d} \to 0,$$ with the restriction of $\bold U(d;n)$ restricted to any slice $\{x\}\times E$ being isomorphic the bundle $V$ given by the line $\Cee\cdot x\subset \Ext^1(W_{n-d},W_d\spcheck)$. When $n$ is clear from the context, we shall abbreviate $\bold U(d;n)$ by $\bold U(d)$. Next we claim that there is a nonempty open subset of $\Pee ^{n-1}_d$ such that for $V$ a vector bundle corresponding to a point of this subset, $\dim \Hom (V,V) = n$. In fact, suppose that $V = \bigoplus _{i=1}^n\lambda _i$, where the $\lambda _i$ are distinct line bundles of degree zero whose product is trivial. By Part (iv) of the theorem, $V$ can be written as a nonsplit extension of $W_d$ by $W_{n-d}\spcheck$, and we have seen that $\dim \Hom (V,V) = n$. A straightforward argument by counting dimensions shows that the set of such $V$ is an open subset of $\Pee^{n-1}_d$; indeed, we will identify this set more precisely in (3.17) below as corresponding to the set of all sections in $|np_0|$ consisting of $n$ smooth points on $E$. Thus there is a nonempty open subset of bundles $V$ such that $\dim \Hom (V,V) = n$. By upper semicontinuity applied to the bundle $Hom(\bold U(d;n), \bold U(d;n))$ over $\Pee_d^{n-1}\times E$, it follows that $\dim \Hom (V,V) \geq n$ for all bundles $V$ corresponding to a point of $\Pee_d^{n-1}$. Finally, we must show that $\Hom (V,V)$ is abelian. Using the universal extension $\bold U(d)$ as above, we can fit together the $\Hom (V,V)$ to a rank $n$ vector bundle $\pi_1{}_*Hom (\bold U(d), \bold U(d))$, which is a coherent sheaf of algebras over $\Pee^{n-1}$. Consider the map $$\pi_1{}_*Hom (\bold U(d), \bold U(d))\otimes \pi_1{}_*Hom (\bold U(d), \bold U(d)) \to \pi_1{}_*Hom (\bold U(d), \bold U(d))$$ defined by $(A,B)\mapsto AB-BA$. Since $\Hom(V,V)$ is abelian for $V$ in a Zariski open subset of $\Pee^{n-1}_d$, namely those $V$ which are a direct sum of $n$ distinct line bundles of degree zero, this map is identically zero. By base change, the fiber of $\pi_1{}_*Hom (\bold U(d), \bold U(d))$ at a point corresponding to $V$ is $\Hom(V,V)$. Thus $\Hom (V,V)$ is abelian. \endproof The following was checked directly in Lemma 1.13 if $E$ is smooth, but is by no means obvious in the singular case: \corollary{3.8} Let $V$ be a regular semistable bundle of rank $n$ over a Weierstrass cubic. Then: \roster \item"{(i)}" $\Hom(V,V)$ is an abelian $\Cee$-algebra of rank $n$. \item"{(ii)}" The dual bundle $V\spcheck$ is a regular semistable bundle. \item"{(iii)}" For all rank one torsion free sheaves $\lambda$ of rank zero on $E$, $\dim \Hom (\lambda, V) \leq 1$. \endroster \endstatement \proof The first part is immediate from Parts (iv) and (iii) of Theorem 3.2. (ii) is clear since if $V$ is a nonsplit extension of $W_{n-d}$ by $W_d\spcheck$, then $V\spcheck$ is a nonsplit extension of $W_d$ by $W_{n-d}\spcheck$. (iii) follows from (ii), since $\Cal F\spcheck \cong \Cal F$ and $\Hom (\lambda, V)\cong \Hom ( V\spcheck,\lambda\spcheck)$ for all rank one torsion free sheaves $\lambda$. \endproof \remark{Question} For if $V$ is a semistable bundle of degree zero whose support is concentrated at a smooth point of $E$, then $V$ is regular if and only if $\dim \Hom (V,V)= \operatorname{rank}V$. Does this continue to hold at the singular point of a singular curve? For $V$ strongly indecomposable, what is the structure of the algebra $\Hom(V,V)$? \endremark \ssection{3.2. Relationship between the constructions for various $d$.} For each $d$ with $1\le d<n$ we have a family of regular semistable bundles parametrized by the projective space $\Pee_d^{n-1}=\Pee(\Ext^1(W_{n-d},W_d\spcheck) )$, and given as a universal extension $$0 \to \pi _2^*W_d\spcheck \otimes \pi _1^*\scrO_{\Pee_d ^{n-1}}(1) \to \bold U(d) \to \pi _2^*W_{n-d} \to 0.$$ In this section we shall identify the $\Pee_d^{n-1}$ for the various $d$, although under this identification the bundles $\bold U(d)$ are different for different $d$. Using the universal bundle $\bold U(d)$ and Theorem 1.5, there is a morphism $\Pee_d^{n-1} \to |np_0|$, which is easily checked to be of degree one and thus an isomorphism. Thus all of the $\Pee_d^{n-1}$ are identified with $|np_0|$ and hence with each other, but we want to find a direct identification here so as to be able to compare universal bundles. \lemma{3.9} Let $d,n-d\ge 1$. The natural injection $$W_d\spcheck\otimes W_{n-d}\spcheck\to W_{d+1}\spcheck\otimes W_{n-d}\spcheck $$ induces an injective map on $H^1$. The image of this map on $H^1$ is the kernel of the map induced by the tensor products of the projections $$H^1(W_{d+1}\spcheck\otimes W_{n-d}\spcheck)\to H^1(\scrO_E\otimes\scrO_E)=H^1(\scrO_E).$$ The extensions $X$ of $W_{n-d}$ by $W_{d+1}\spcheck$ which are in the image of the above map are exactly the extensions $X$ such that $\Hom (X, \scrO_E)\neq 0$. \endstatement \proof We have a short exact sequence $$0\to W_d\spcheck\otimes W_{n-d}\spcheck \to W_{d+1}\spcheck\otimes W_{n-d}\spcheck \to \scrO_E\otimes W_{n-d}\spcheck\to 0.$$ By (3.3), all the $H^0$ terms vanish. Thus, the injectivity of the map on $H^1$ is immediate. Furthermore, the image is identified with the kernel of the map $$H^1(W_{d+1}\spcheck\otimes W_{n-d}\spcheck)\to H^1(\scrO_E\otimes W_{n-d}\spcheck).$$ The last term is one-dimensional and the projection $\scrO_E\otimes W_{n-d}\spcheck\to \scrO_E\otimes\scrO_E$ induces an isomorphism on $H^1$. Finally, a bundle $X$ corresponds to an extension in the image of the map on $H^1$'s if and only if $X$ is the pushout of an extension of $W_{n-d}$ by $W_d\spcheck$ under the inclusion $W_d\spcheck \to W_{d+1}\spcheck$. Thus, if $X$ is the image of an extension $V$, the quotient of the inclusion $V\to X$ is $\scrO_E$. Conversely, if there is a nontrivial map $X\to \scrO_E$, then the induced map $W_{d+1}\spcheck \to \scrO_E$ is nonzero and thus surjective, and the kernel of $X\to \scrO_E$ is then an extension $V$ of $W_{n-d}$ by $W_d\spcheck$ such that $X$ is the pushout of $V$. \endproof The symmetry of the situation with respect to the two factors leads immediately to the following. \corollary{3.10} If $n-d\ge 2$, then the natural inclusions of bundles induce the maps $$H^1(W_d\spcheck\otimes W_{n-d}\spcheck)\to H^1(W_{d+1}\spcheck\otimes W_{n-d}\spcheck)$$ and $$H^1(W_{d+1}\spcheck\otimes W_{n-d-1}\spcheck)\to H^1(W_{d+1}\spcheck\otimes W_{n-d}\spcheck)$$ which are injections with the same images. In particular, this produces a natural identification of $\Ext^1(W_{n-d},W_d\spcheck)$ with $\Ext^1(W_{n-d-1},W_{d+1}\spcheck)$, and hence of the projective spaces $\Pee_d^{n-1}\cong \Pee_{d+1}^{n-1}$. Finally, an extension $X$ of $W_{n-d}$ by $W_{d+1}\spcheck$ is obtained from an extension $V$ of $W_{n-d-1}$ by $W_{d+1}\spcheck$ via pullback if and only if $\Hom (X, \scrO_E)\neq 0$. \qed \endstatement Now let us see how the bundles described by extensions which are identified under this isomorphism are related. \proposition{3.11} Let $\epsilon_d\in \Ext^1(W_{n-d},W_d\spcheck)$ and $\epsilon_{d+1}\in \Ext^1(W_{n-d-1},W_{d+1}\spcheck)$ be nonzero classes that correspond under the identification given in Corollary \rom{3.10}. Let $V$ and $V'$, respectively, be the total spaces of these extensions. Then $V$ and $V'$ are isomorphic bundles. \endstatement \proof Let $X$ be the bundle of rank $n+1$ obtained by pushing out the extension $V$ by the map $W_d\spcheck\to W_{d+1}\spcheck$. Clearly, we have a short exact sequence $$0\to V\to X\to \scrO_E\to 0.$$ Similarly, let $X'$ be the rank $n+1$-bundle obtained by pulling back the extension $V'$ along the map $W_{n-d}\to W_{n-d-1}$. Dually, we have an exact sequence $$0\to \scrO_E\to X'\to V'\to 0.$$ It follows easily from Theorem 3.2 that writing $V=Y\oplus I_r(\scrO_E)$ with $H^0(Y)=0$, we have $X\cong Y\oplus I_{r+1}(\scrO_E)$. Similarly, writing $V'=Y'\oplus I_s(\scrO_E)$ we have $X'\cong Y'\oplus I_{s+1}(\scrO_E)$. The fact that $\epsilon_d$ and $\epsilon_{d+1}$ are identified means that the extensions for $X$ and $X'$ are isomorphic. In particular, $X$ and $X'$ are isomorphic bundles. This implies that $r=s$ and that $Y$ and $Y'$ are isomorphic. But then $V$ and $V'$ are isomorphic as well. \endproof Notice that the isomorphism produced by the previous result is canonical on $Y\subseteq V$ but is not canonical on the $I_r(\scrO_E)$ factor. We shall see later that the families of bundles $\bold U(d)$ and $\bold U(d+1)$ are not isomorphic, which means that there cannot be a canonical isomorphism in general between corresponding bundles. In practice, this means the following: suppose that $V$ is given as an extension of $W_{n-d}$ by $W_d\spcheck$, with $n-d >1$. Then $W_{n-d}$ has the distinguished subbundle isomorphic to $\scrO_E$. Let $W'$ be the preimage in $V$ of this bundle, so that $W'$ is an extension of $\scrO_E$ by $W_d\spcheck$. Then $W'\cong W_{d+1}\spcheck$ if and only if $h^0(V) = 0$, if and only if the support of $V$ does not contain $\scrO_E$, but otherwise $W' \cong W_d\spcheck\oplus \scrO_E$. The direct comparison of the extension classes given above leads to a comparison of universal bundles. \theorem{3.12} Let $H$ be the divisor in $\Pee^{n-1}_d$ such that, if $x\in H$ and $V$ is the corresponding extension, then $h^0(V) = 1$. Let $i\:H\to \Pee^{n-1}_d$ be the inclusion. Then there is an exact sequence $$0 \to \bold U(d-1) \to \bold U(d) \to (i\times \Id)_*\scrO_{H\times E}(1) \to 0,$$ which expresses $\bold U(d-1;n)$ as an elementary modification of $\bold U(d;n)$. Moreover, this elementary modification is unique in an appropriate sense. \endstatement \proof Let $H'$ be the hyperplane in $\Pee^n_d$ corresponding to the set of extensions $X$ of $W_{n-d+1}$ by $W_d\spcheck$ such that $\Hom (X, \scrO_E) \neq 0$. By the last statements of (3.9) and (3.10), $H'$ is the image of $\Pee^{n-1}_d$ in $\Pee^n_d$ as well as the image of $\Pee^{n-1}_{d+1}$. By base change, $\pi_1{}_*Hom (\bold U(d;n+1)|H'\times E, \scrO_{H'\times E})$ is a line bundle on $H'$. By looking at the exact sequence $$\gather 0 \to Hom (\pi _2^*W_{n-d+1}, \scrO_{\Pee^n_d\times E}) \to Hom (\bold U(d;n+1)|H'\times E, \scrO_{\Pee^n_d\times E}) \to \\ \to Hom (\pi _2^*W_d\spcheck\otimes \pi _1^*\scrO_{\Pee^n_d}(1), \scrO_{\Pee^n_d\times E})\to 0, \endgather$$ and restricting to $H'\times E$, we see that this line bundle is $\scrO_{H'\times E}(-1)$. Thus there is a surjection $$\bold U(d;n+1)|H'\times E \to \scrO_{H'\times E}(1).$$ We claim that the kernel of this surjection is identified with $\bold U(d-1;n)$. In fact, if $\bold U'$ denotes the kernel, we have a commutative diagram with exact rows and columns $$\minCDarrowwidth{.3 in} \CD @. 0 @. 0 @. @.\\ @. @VVV @VVV @. @.\\ 0@>>> \pi_2^*W_{d-1}\spcheck\otimes \scrO_{H'\times E}(1) @>>>\bold U' @>>>\pi_2^*W_{n-d+1} @>>> 0\\ @. @VVV @VVV @| @.\\ 0@>>> \pi_2^*W_d\spcheck\otimes \scrO_{H'\times E}(1) @>>>\bold U(d;n+1)|H'\times E @>>>\pi_2^*W_{n-d+1} @>>> 0\\ @. @VVV @VVV @. @.\\ @. \scrO_{H'\times E}(1) @= \scrO_{H'\times E}(1) @. @.\\ @. @VVV @VVV @. @.\\ @. 0 @. 0 @. @.\\ \endCD$$ and tracing through the diagram identifies $\bold U'$ with $\bold U(d-1;n)$, compatibly with the identification of $H'$ with $\Pee^{n-1}_d$. Now we can also consider the line bundle $\pi _1{}_*(\bold U(d;n+1)|H'\times E)$. A very similar argument shows that this line bundle is isomorphic to $\scrO_{H'}$, and that the quotient $\bold U''$ of $\bold U(d;n+1)|H'\times E$ via the natural map $$\scrO_{H'\times E} = \pi^*\pi _1{}_*(\bold U(d;n+1)|H'\times E) \to \bold U(d;n+1)|H'\times E$$ is isomorphic to $\bold U(d;n)$. Putting these two constructions together, we see that we have found $$\bold U(d-1;n) \to \bold U(d;n+1)|H'\times E \to \bold U(d;n).$$ Away from $H$, which is the image of $\Pee^{n-2}_d$ in $\Pee^{n-1}_d$, the inclusion $\bold U(d-1;n) \to \bold U(d;n)$ is an equality. To summarize, then, there is a commutative diagram $$\CD @. @. 0 @. @.\\ @. @. @VVV @. @.\\ @. @. \bold U(d-1;n) @= \bold U(d-1;n) @.\\ @. @. @VVV @VVV @.\\ 0@>>> \scrO_{H'\times E} @>>>\bold U(d;n+1)|H'\times E @>>>\bold U(d;n) @>>> 0\\ @. @| @VVV @. @.\\ @. \scrO_{H'\times E} @>>> \scrO_{H'\times E}(1) @. @.\\ @. @. @VVV @. @.\\ @. @. 0 @. @.\\ \endCD$$ The map $\scrO_{H'\times E} \to \scrO_{H'\times E}(1)$ can only vanish along $H$. Thus it vanishes exactly along $H$, and the quotient of $\bold U(d;n)$ by $\bold U(d-1;n)$ is a line bundle supported on $H\times E$. By what we showed above for $\bold U(d;n+1)$, this line bundle is necessarily $\scrO_{H\times E}(1)$. Hence we have found an exact sequence $$0 \to \bold U(d-1;n) \to \bold U(d;n) \to (i\times \Id)_* \scrO_{H\times E}(1) \to 0,$$ realizing $\bold U(d-1;n)$ as an elementary modification of $\bold U(d;n)$. The uniqueness is straightforward and left to the reader. This completes the proof of Theorem 3.12. \endproof \ssection{3.3. Comparison of coarse moduli spaces.} We have succeeded in identifying the $\Pee_d^{n-1}$ for the various $d$, $1\le d<n$, in a purely cohomological way and in showing that extension classes in different groups which are identified produce isomorphic vector bundles. Next we wish to identify these projective spaces with the projective space $|np_0|$ which is the parameter space of regular semistable rank $n$ bundles with trivial determinant in the spectral cover construction of these bundles. Of course, the existence of the bundle $\bold U(d)$ and Theorem 1.5 give us one such identification. However, although we shall not need this in what follows, we want to find a direct cohomological comparison between $\Ext^1(W_{n-d}, W_d\spcheck)$ and $H^0(\scrO_E(np_0))$. We have two identifications: one purely cohomological and the other using the bundles to identify the spaces. We shall show that these identifications agree. Let us begin with the purely cohomological identification. (We will be pedantic here about identifying various one-dimensional vector spaces with $\Cee$ in order to carry out the discussion in families in the next section.) \proposition{3.13} Let $H^0_{n-1}=H^0(\scrO_E(p_0)\otimes W_{n-1})$. It is an $n$-dimensional vector space. Let $D=H^1(\scrO_E)$ be the dualizing line. Let $$I\colon H^1(\scrO_E(-p_0)\otimes W_{n-1}\spcheck)\to H^0(\scrO_E(np_0))\otimes \det(H^0_{n-1})^{-1}\otimes D$$ be the composition $$\gather H^1(\scrO_E(-p_0)\otimes W_{n-1}\spcheck) @>{S}>> (H^0_{n-1})^*\otimes D @>{A}>> \bigwedge^{n-1}H^0_{n-1}\otimes \operatorname{det}(H^0_{n-1})^{-1}\otimes D \to \\ @>{ev\otimes\Id\otimes\Id}>> H^0(\det(\scrO_E(p_0)\otimes W_{n-1}))\otimes \det(H^0_{n-1})^{-1}\otimes D \\ = H^0(\scrO_E(np_0))\otimes \det(H^0_{n-1})^{-1}\otimes D, \endgather$$ where $S$ is Serre duality, $A$ is the map induced by taking adjoints from the natural pairing $$H^0_{n-1}\otimes \bigwedge^{n-1}H^0_{n-1}\to \det(H^0_{n-1}),$$ $ev$ is the map $$ev\colon \bigwedge^{n-1}H^0(\scrO_E(p_0)\otimes W_{n-1})\to H^0(\bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1})).$$ Then $I$ is an isomorphism. \endstatement \proof On general principles $S$ is an isomorphism. Since $H^0_{n-1}$ is $n$-dimensional, the adjoint map $A$ is clearly an isomorphism. What is less obvious that the map $ev$ is an isomorphism, which follows from the next claim. \claim{3.14} The evaluation map $$ev\colon \bigwedge^{n-1}H^0(\scrO_E(p_0)\otimes W_{n-1})\to H^0(\bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1}))$$ is an isomorphism. \endstatement \proof We prove this by induction. The case $n=2$ is clear since $ev$ is the identity. Assume the result for $n-1\ge 2$. There is a short exact sequence $$0\to \scrO_E(p_0)\to \scrO_E(p_0)\otimes W_{n-1}\to \scrO_E(p_0)\otimes W_{n-2}\to 0\tag{$*$}$$ leading to a short exact sequence $$0\to H^0(\scrO_E(p_0))\to H^0(\scrO_E(p_0)\otimes W_{n-1})\to H^0(\scrO_E(p_0)\otimes W_{n-2})\to 0,$$ since by (3.3) the $H^1$ terms vanish. Since the first term has dimension one, we have a short exact sequence $$0\to H^0(\scrO_E(p_0))\otimes \bigwedge^{n-2}H^0_{n-2}\to \bigwedge^{n-1}H^0_{n-1}\to \bigwedge^{n-1}H^0_{n-2}\to 0.$$ Taking determinants in ($*$) yields an isomorphism $$\bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1})\cong \scrO_E(p_0)\otimes \bigwedge^{n-2}(\scrO_E(p_0)\otimes W_{n-2}).$$ Tensoring the inclusion $\scrO_E\to \scrO_E(p_0)$ with $\bigwedge^{n-2}(\scrO_E(p_0)\otimes W_{n-2})$ and using the above isomorphism leads to a short exact sequence $$0\to \bigwedge^{n-2}(\scrO_E(p_0)\otimes W_{n-2})\to \bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1})\to \bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1})|_{p_0}\to 0.$$ Unraveling the definitions one sees that the map $ev$ induces a commutative diagram, with exact columns, $$\CD 0 @. 0\\ @VVV @VVV\\ H^0(\scrO_E(p_0))\otimes \bigwedge^{n-2}H^0_{n-2} @>>> H^0(\scrO_E(p_0))\otimes H^0(\bigwedge^{n-2}(\scrO_E(p_0)\otimes W_{n-2}))\\ @VVV @VVV \\ \bigwedge^{n-1}H^0_{n-1} @>{ev}>> H^0(\bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1})) \\ @VVV @VVV \\ \bigwedge^{n-1}H^0_{n-2} @>{e}>> (\bigwedge^{n-1}(\scrO_E(p_0)\otimes W_{n-1}))|_{p_0}\\ @VVV @VVV\\ 0 @. 0 , \endCD$$ and that the restriction of $ev$ to the first term is simply the tensor product of the identity on $H^0(\scrO_E(p_0))$ and the evaluation map, with $n-2$ replacing $n-1$. By induction, the top horizontal map is an isomorphism. To finish the proof of (3.14), and thus of (3.13), it suffices by the $5$-lemma to show that $e$ is an isomorphism. Now $e$ is the $(n-1)$-fold wedge product of a map $$\overline{e}\colon H^0(\scrO_E(p_0)\otimes W_{n-2})\to (\scrO_E(p_0)\otimes W_{n-1})|_{p_0} $$ defined as follows. For any section $\psi$ of $\scrO_E(p_0)\otimes W_{n-2}$, lift to a section $\tilde \psi$ of $\scrO_E(p_0)\otimes W_{n-1}$, and then restrict $\tilde \psi$ to $p_0$. Thus, it suffices to prove: \claim{3.15} The map $\overline{e}$ described above is an isomorphism. \endstatement \proof First notice that if $\tilde \psi$ and $\tilde \psi'$ are lifts of $\psi$ to sections of $\scrO_E(p_0)\otimes W_{n-1}$, then they differ by a section of $\scrO_E(p_0)\subset \scrO_E(p_0)\otimes W_{n-1}$. But any section of $\scrO_E(p_0)$ vanishes at $p_0$ so that $\tilde \psi$ and $\tilde \psi'$ have the same restriction to $p_0$. This shows that $\overline{e}$ is well-defined. From the diagram $$\CD @. @. 0 @. @. \\ @. @. @VVV @. @. \\ @. @. W_{n-1} @. @. \\ @. @. @VVV @. @. \\ 0 @>>> \scrO_E(p_0) @>>> \scrO_E(p_0)\otimes W_{n-1} @>>> \scrO_E(p_0)\otimes W_{n-2} @>>> 0, \\ @. @. @VVV @. @. \\ @. @. \scrO_E(p_0)\otimes W_{n-1}|_{p_0} @. @. \\ @. @. @VVV @. @. \\ @. @. 0 @. @. \\ \endCD$$ the fact that all the $H^1$ terms vanish, and the fact that both $H^0(\scrO_E(p_0))$ and $H^0(W_{n-1})$ are one dimensional, the claim comes down to the statement that the images in $H^0(\scrO_E(p_0)\otimes W_{n-1})$ of $H^0(\scrO_E(p_0))$ and of $H^0(W_{n-1})$ are equal. But we also have a commutative square $$\CD \scrO_E @>>> \scrO_E\otimes W_{n-1} \\ @VVV @VVV \\ \scrO_E(p_0) @>>> \scrO_E(p_0)\otimes W_{n-1} \endCD$$ with the top arrow and the left arrow inducing isomorphisms on $H^0$. Claim 3.14 and hence (3.13) now follow. \endproof \enddemo\enddemo Proposition 3.13 and Corollary 3.10 have produced cohomological isomorphisms between the extension groups $\Ext^1(W_{n-d},W_d\spcheck)$ and $H^0(\scrO_E(np_0))$. On the other hand, as remarked previously, from the existence of the bundle $\bold U(d)\to \Pee_d^{n-1}\times E$, Theorem 1.5 produces isomorphisms $$\Phi_d\colon \Pee_d^{n-1}\to |np_0|$$ sending $x\in \Pee_d^{n-1}$ to the point $\zeta(V_x)$, where $V$ is the the extension determined by the point $x$. By Proposition 3.11 the identification of $\Pee_d^{n-1}$ with $\Pee_{d+1}^{n-1}$ given in Corollary 3.10 identifies $\Phi_d$ with $\Phi_{d+1}$. Still, it remains to compare the map $\Phi_1$ with the projectivization of the map produced by Proposition 3.13. \proposition{3.16} The map $\Phi_1\colon \Pee_1^{n-1}\to |np_0|$ is the projectivization of the identification $$I\colon H^1(\scrO_E(-p_0)\otimes W_{n-1}\spcheck)\to H^0(\scrO_E(np_0))\otimes M,$$ where $M$ is the line $\det(H^0(\scrO_E(p_0)\otimes W_{n-1}))^{-1}\otimes D$. In other words, if $V$ is a nontrivial extension corresponding to $\alpha \in \Ext^1(W_{n-1},\scrO_E(-p_0))$, then the point $\zeta(V)\in |np_0|$ corresponds to the line $$\Cee\cdot I(\alpha)\subset H^0(\scrO_E(np_0))\otimes M.$$ In particular, $\Phi_1$ is an isomorphism, and hence so is $\Phi_d$ for every $1\le d<n$. \endstatement \proof Let $\bar I\: \Pee^{n-1}_1\to |np_0|$ be the projectivization of $I$. We begin by determining when a line bundle $\lambda$ of degree zero is in the support of $V$. \claim{3.17} Let $V$ be given by an extension class $\alpha\in H^1(\scrO_E(-p_0)\otimes W_{n-1}\spcheck)$. Then $\Hom(V,\lambda)\not=0$ if and only if the image of $\alpha$ in $H^1(\lambda \otimes W_{n-1}\spcheck)$ under the map induced by the inclusion $\scrO_E(-p_0)\to \lambda$ is zero. \endstatement \proof There is a nonzero map $\scrO_E(-p_0) \to \lambda$, and it is unique up to a scalar. Let $V'$ be the pushout of the extension $V$ by the map $\scrO_E(-p_0)\to \lambda$. Then the pushout extension is trivial, i\.e\. the image of $\alpha$ in $H^1(\lambda \otimes W_{n-1}\spcheck)$ is zero, if and only if there is a map $V'\to \lambda$ splitting the inclusion of $\lambda$ into $V'$. Such a map is equivalent to a map $V\to \lambda$ so that the composition $\scrO_E(-p_0)\to V\to \lambda$ is the inclusion. By (3.6), if $V$ has a nontrivial map to $\lambda$, then this map restricts to $\scrO_E(-p_0)$ to be the inclusion (since otherwise the image of $\scrO_E(-p_0)$ would be contained in a proper subbundle of degree zero). Thus, there is such a map if and only if the $\lambda$ component of $V$ is nonzero, which is equivalent to the existence of a nonzero map $V\to \lambda$. \endproof \claim{3.18} Suppose that $\lambda\cong\scrO_E(q-p_0)$ for some $q\in E$. Let $V$ be given by an extension class $\alpha\in H^1(\scrO_E(-p_0)\otimes W_{n-1}\spcheck)$. Then $\lambda$ is in the support of $V$ if and only if $q$ is in the support of $\bar I(\alpha) \in |np_0|$. \endstatement \proof Applying Serre duality $S$ and the adjoint map $A$ to the previous claim, and tracing through the identifications, we see that $\lambda=\scrO_E(q-p_0)$ is in the support of $V$ if and only if the corresponding map $$\bigwedge^{n-1}H^0_{n-1}\otimes L\to \bigwedge^{n-1}(\scrO_E(p)\otimes W_{n-1}\otimes L)|_q$$ vanishes, if and only if the section giving $I(\alpha)$ in $\det(\scrO_E(p_0)\otimes W_{n-1})\otimes L$ vanishes at $q$, if and only if $q$ is in the support of $\bar I(\alpha)$. \endproof Now we can prove Proposition 3.16. We have two maps $\bar I$, the projectivization of the linear map $I$, and $\Phi_1$, mapping $\Pee_1^{n-1}\to |np_0|$. We wish to show $\bar I$ and $\Phi_1$ are equal. We know that $\bar I$ is an isomorphism. Thus, it suffices to show that, for an open dense subset $U$ of $|np_0|$, $\Phi_1(x)=\bar I(x)$ for all $x\in \Phi_1^{-1}(U)$. Choose $U$ to be the open subset of divisors in $|np_0|$ whose support is $n$ distinct smooth points of $E$. Let $x=\sum _ie_i\in U$. The extension determined by $\alpha =\Phi_1^{-1}(x)$ is a semistable bundle $V$ which is written as $\bigoplus_i\lambda_i$ for $n$ distinct line bundles $\lambda_i$, where $\lambda_i=\scrO_E(e_i-p_0)$. According to Claim 3.18, $\bar I(\alpha)$ contains $e_i,\ 1\le i\le n$ in its support, and hence $\bar I\circ \Phi_1^{-1}(x)=x$. This completes the proof of Proposition 3.16. \endproof \ssection{3.4. From universal bundles to spectral covers.} We now take another look at the spectral cover construction,and generalize it to singular curves. Fix an integer $d$, $1\leq d\leq n-1$, and consider the sheaf $\bold A = \pi_1{}_*Hom (\bold U(d), \bold U(d))$. It is a locally free rank $n$ sheaf of commutative algebras over $\Pee^{n-1}$, and in case $E$ is smooth we have identified this sheaf with $\nu_*\scrO_T$ in (2.4). (There is nothing special about taking $\bold U(d)$; we could replace $\bold U(d)$ by any ``universal" bundle, once we know how to construct one.) We propose to reverse this procedure: starting with $\bold A$, define $T$ to be the corresponding space $\bold{Spec}\,\bold A$. In particular, this gives a definition of $T$ in case $E$ is singular. \lemma{3.19} Let $E$ be a Weierstrass cubic. With $T$ as defined above, there is a finite flat morphism $\nu\:T \to \Pee^{n-1}$. Moreover, $T$ is reduced. \endstatement \proof By construction, there is a finite morphism $\nu\: T\to \Pee^{n-1}$. In fact, since $\bold A$ is locally free, $T$ is flat over $\Pee^{n-1}$ of degree $n$. The fact that $T$ is reduced follows from the fact that $\Pee^{n-1}$ is reduced and that $T$ is generically reduced, and as such is a general fact concerning finite flat morphisms. Cover $\Pee^{n-1}$ by affine open sets $\Spec R$ such that $\nu^{-1}(\Spec R) = \Spec R'$, where $R'$ is a free rank $n$ $R$-module. It will suffice to show that $R'$ is reduced for every such $R$. If $f\in R'$, then $f$ does not vanish on a Zariski open set, since $R'$ is locally free and $R$ is reduced. Thus the restriction of $f$ to a general fiber of $\nu$, consisting of $n$ distinct (reduced) points, is nonzero. It follows that $f^k\neq 0$ for every $k>0$. Thus $R'$ is reduced, and so $T$ is reduced. \endproof In case $E$ is smooth, Theorem 2.4 shows that the $T$ defined above is the same as the spectral cover $T$ defined in Section 2, although even in this case it will be useful to define $T$ as we have above in order to compare $\bold U(d)$ with the bundles $U_a$. The points of $T$ are by definition in one-to-one correspondence with pairs $(V, \frak m)$, where $V$ is a regular semistable bundle with trivial determinant and $\frak m$ is a maximal ideal in $\Hom(V,V)$. Let us describe such maximal ideals: \lemma{3.20} If $V$ is a regular semistable bundle of rank $n$, then the maximal ideals $\frak m$ of $\Hom(V,V)$ are in one-to-one correspondence with nonzero homomorphisms $\rho\: V\to \lambda$ mod scalars, where $\lambda$ is a torsion free rank one sheaf of degree zero. The correspondence is as follows: given $\rho$, we set $$\frak m= \{\, A\in \Hom (V,V):\rho\circ A = 0\,\},$$ and given a maximal ideal $\frak m$, we define $\lambda = V/\frak m \cdot V$ and take $\rho$ to be the obvious projection. \endstatement \proof If $V = \left(\bigoplus _iI_{r_i}(\lambda _i)\right) \oplus I(\Cal F)$, then $\Hom (V,V)$ is a direct sum $$\left(\bigoplus _i\Hom (I_{r_i}(\lambda _i),I_{r_i}(\lambda _i))\right) \oplus \Hom (I(\Cal F),I(\Cal F)),$$ and it will clearly suffice to consider the case where $V$ is either $I_{r_i}(\lambda _i)$ or $I(\Cal F)$. For simplicity, we assume that $V=I(\Cal F)$. Thus there is a unique $\rho$ mod scalars by definition. If we set $\frak m= \{\, A\in \Hom (V,V):\rho\circ A = 0\,\}$, then $\frak m$ is an ideal of $\Hom (V,V)$. In fact, there is an induced homomorphism $\Hom (V,V) \to \Hom (\Cal F, \Cal F) =H^0(\scrO_{\tilde E}) = \Cee$, and $\frak m$ is the kernel of this homomorphism. Thus $\frak m$ is a maximal ideal. Next we claim that $\frak m$ is the unique maximal ideal in $\Hom(V,V)$. It suffices to show that $\frak m$ contains every non-invertible element of $\Hom(V,V)$. If $A\in \Hom(V,V)$ is not invertible, then $\operatorname{Im}A$ is a proper torsion free subsheaf of $V$ of rank smaller than $n$ and degree at least zero. It follows that $\deg \operatorname{Im}A = 0$. But then, by Lemma 1.11, $\operatorname{Im}A\subseteq \Ker \rho$. It follows that $\rho\circ A =0$, so that by definition $A\in \frak m$. Thus $\frak m$ is the unique maximal ideal of $\Hom(V,V)$. Finally we claim that $V/\frak m\cdot V \cong \Cal F$. By definition, the surjection $\rho\: V \to \Cal F$ factors through the quotient $V/\frak m\cdot V$, so that $\frak m\cdot V\subseteq \Ker \rho $. Choosing a basis $A_1, \cdots, A_{n-1}$ for $\frak m$, we see that $\frak m\cdot V$ is of the form $A_1(V) +\cdots + A_{n-1}(V)$, and thus it is a subsheaf of $V$ of degree at least zero. Hence it has degree exactly zero, and thus it is filtered by subsheaves whose quotients are isomorphic to $\Cal F$. If $\frak m\cdot V$ has rank $r$, it follows by Lemma 1.14 that $\dim \Hom (V, \frak m\cdot V) \leq r$. But clearly $\Hom (V,V) = \Hom (V,\frak m\cdot V)\oplus \Cee\Id$, and since $\dim \Hom (V,V)= n$, we must have $r=n-1$. Since both $\frak m\cdot V$ and $\Ker \rho$ have degree zero and rank $n-1$, and $\frak m\cdot V\subseteq \Ker \rho $, $\frak m\cdot V = \Ker \rho$. Thus $V/\frak m\cdot V \cong \Cal F$. \endproof Next, given the spectral cover $T$, by construction $\bold U(d)$ is a module over $\scrO_T= \pi _1{}_*Hom(\bold U(d), \bold U(d))$, and thus $\bold U(d)$ corresponds to a coherent sheaf $\Cal L_d$ over $T\times E$. By construction, $(\nu\times \Id)_*\Cal L_d =\bold U(d)$. In case $E$ is smooth, or more generally in case $(V, \frak m)$ is a point of $T$ such that the support of $V$ does not contain the singular point of $E$, then it is easy to check directly that $\Cal L_d$ is a line bundle near $(V, \frak m)\times E$. We can now summarize our description of $T$ as follows: \theorem{3.21} There is an isomorphism $(\nu, r)$ of $T$ onto the incidence correspondence in $|np_0|\times E$ with the following property: Let $\Delta_0$ be the diagonal in $E\times E$, with ideal sheaf $I_{\Delta _0}$, and let $\Cal P_0$ be the sheaf on $E\times E$ defined by $Hom(I_{\Delta _0}, \scrO_{E\times E}(-E\times \{p_0\})$. Then there exists a line bundle $M$ on $T$ such that $\Cal L_d = ( r\times \Id)^*\Cal P_0\otimes \pi _1^*M$. \endstatement \proof We have shown in (0.3) that $\Cal P_0$ is flat over the first factor in the product $E\times E$ and identifies the first factor with $\bar J(E)$, the compactified generalized Jacobian of $E$. Let $T'$ be the incidence correspondence in $|np_0|\times E$. Note that $T'$ is irreducible; in fact, projection onto the second factor makes $T'$ a $\Pee^{n-2}$-bundle over $E$, namely $T'=\Pee\Cal E$ as in Section 2. We will first find a morphism from $T'$ to $T$ which is a bijection as a set-valued function. Let $\nu'\: T'\to \Pee^{n-1}$ and $r'\: T' \to E$ be the projections to the first and second factors. By construction, for a point $(x, e)\in T'$, if $V$ is the vector bundle over $E$ corresponding to $x$ and $\lambda$ is the rank one torsion free sheaf of degree zero corresponding to $e$, $\dim \Hom(V, \lambda) = 1$. Thus, by base change, with $\pi _1\: T'\times E\to T'$ the projection, $$\pi _1{}_*\left[(\nu'\times \Id)^*\bold U(d)\spcheck \otimes (r'\times \Id)^*\Cal P_0\right] =M$$ is a line bundle on $T'$. After replacing $(r'\times \Id)^*\Cal P_0$ by $(r'\times \Id)^*\Cal P_0\otimes M^{-1}=\Cal P'$, we can assume that there is a surjection $$\rho\: (\nu'\times \Id)^*\bold U(d) \to \Cal P'.$$ On every fiber, the homomorphism $V \to \lambda$ is preserved up to scalars by every endomorphism of $V$. Thus, there is an induced homomorphism $$\pi_1{}_*Hom ((\nu'\times \Id)^*\bold U(d), (\nu'\times \Id)^*\bold U(d)) \to \pi_1{}_*Hom (\Cal P', \Cal P').$$ Now by the flatness of $\Cal P_0$, it is easy to check that $Hom (\Cal P', \Cal P')$ is flat over $T'$ and that the natural multiplication map $\scrO_{T'} \to \pi_1{}_*Hom (\Cal P', \Cal P')$ is an isomorphism. By base change, the first term $\pi_1{}_*Hom ((\nu'\times \Id)^*\bold U(d), (\nu'\times \Id)^*\bold U(d)) $ in the above homomorphism is the pullback to $T'$ of the sheaf of algebras $\bold A = \pi _1{}_*Hom (\bold U(d), \bold U(d))$ over $\Pee^{n-1}$, and hence it is just the structure sheaf $\scrO_{T'\times _{\Pee^{n-1}}T}$ of the fiber product $T'\times _{\Pee^{n-1}}T$. The homomorphism $\scrO_{T'\times _{\Pee^{n-1}}T}\to \scrO_{T'}$ corresponds to a morphism $T'\to T'\times _{\Pee^{n-1}}T$, which is a section of the natural projection $T'\times _{\Pee^{n-1}}T \to T'$. Such a section is the same thing as a morphism $T' \to T$ (covering the given maps to $\Pee^{n-1}$). On the level of points, this morphism is as follows: take an element $D$ in $|np_0|$ and a point $e$ in the support of $D$. Pass to the corresponding vector bundle $V$ and the morphism $V\to \lambda$, where $\lambda$ is the rank one degree zero torsion free sheaf corresponding to $e$, and then set $\frak m$ to be the maximal ideal corresponding to $V\to \lambda$. It is then clear that $T'\to T$, as constructed above, is a bijection of sets. In particular, $T$ is irreducible. Now we want to construct a morphism which is the inverse of the morphism $T'\to T$. It suffices to find the morphism $r\: T \to E$. Viewing $E$ as isomorphic to the compactified generalized Jacobian of $E$, we can find such a morphism once we know that the sheaf $\Cal L_d$ is flat over $E$: \lemma{3.22} The sheaf $\Cal L_d$ is flat over $T$. If $t\in T$ corresponds to the the pair $(V, \frak m)$, and $\lambda$ is the rank one torsion free sheaf of degree zero given by $V/\frak m\cdot V$, then the restriction of $\Cal L_d$ to the slice $\{t\}\times E$ is $\lambda$. Thus $\Cal L_d$ is a flat family of rank one torsion free sheaves on $T\times E$. \endstatement \proof First let us show that, in the above notation, the restriction of $\Cal L_d$ to the slice $\{t\}\times E$ is $\lambda$. In fact, suppose that $t$ corresponds to the pair $(V, \frak m)$ and view $V$ as a rank one module over $\Hom(V,V)$. Then the restriction of $V$ to $\{t\}\times E$ is given by $V/\frak m \cdot V =\lambda$. Now the Hilbert polynomial $P_\lambda (n) = \chi(E; \lambda\otimes \scrO_E(np_0))$ is independent of the choice of $\lambda$. As we have proved above, $T$ is irreducible since it is the image of $T'$, and thus, since it is reduced, it is integral. The proof of Theorem 9.9 on p\. 261 of \cite{10} then shows that $\Cal L_d$ is flat over $T$. The last statement is then clear. \endproof By (0.3), as $\Cal L_d$ is flat over $T$, it defines a morphism $ r\: T\to E$ (viewing $E$ as $\bar J(E)$). Thus we also have the product morphism $(\nu, r)\: T \to \Pee^{n-1}\times E$, whose image is $T'$. Clearly, on the level of sets, the morphism $T\to T'$ is the inverse of the morphism $T'\to T$ constructed above. Since both $T$ and $T'$ are reduced, in fact the two maps are inverses as morphisms. By the functorial properties of the compactified Jacobian (0.3), $\Cal L_d = ( r\times \Id)^*\Cal P\otimes \pi _1^*M$. This then concludes the proof of (3.21). \endproof We have now lined up the spectral covers, and proceed to identify the bundles $\bold U(d)$ in terms of $T$. It suffices to identify the bundle $\pi _1^*M$ in (3.21). We do this first for $d=1$. In order to do so, we first make the following preliminary remarks. Let $\scrO_{T\times E}(\Delta)$ denote the rank one torsion free sheaf $( r\times \Id)^*Hom(I_{\Delta _0}, \scrO_{E\times E})$. Suppose that $\Cal L$ is any flat family of rank one torsion free sheaves on $T\times E$ such that there exists an injection $\scrO_{T\times E} \to \Cal L$, with the cokernel exactly supported along $\Delta$ and with multiplicity one at a nonempty Zariski open subset of the smooth points. We claim that in this case $\Cal L = \scrO_{T\times E}(\Delta)$. First, the universal property of the compactified Jacobian implies that $\Cal L = (\alpha \times \Id)^*Hom(I_{\Delta _0}, \scrO_{E\times E}) \otimes \pi_1^*M$ for some morphism $\alpha \: T \to E$ and line bundle $N$ on $T$. By hypothesis, $\alpha = r$ on a Zariski open dense subset of $T$, and thus everywhere. Next, since $\scrO_T \to \pi_1{}_*\scrO_{T\times E}(\Delta)$ is an isomorphism, $H^0(N) =H^0(\pi_1{}_*\scrO_{T\times E}(\Delta) \otimes N ) = H^0(\Cal L)$, and every section of $\Cal L$ is given by multiplying the natural section of $\scrO_{T\times E}(\Delta)$ by a section $s$ of $N$. In this case, the cokernel is supported at $\Delta \cup \pi_1^{-1}(D)$, where $D$ is the divisor of zeroes of $s$. Thus, if the support of the cokernel is $\Delta$, then $N$ must have a nowhere vanishing section, and so is trivial. We may thus conclude that $\Cal L = \scrO_{T\times E}(\Delta)$. \theorem{3.23} $\bold U(1) = (\nu\times \Id)_*\scrO_{T\times E}(\Delta - G)\otimes \pi _1^*\scrO_{\Pee^{n-1}}(1)$. \endstatement \proof An equivalent formulation is: $$(\nu\times \Id)_*\scrO_{T\times E}(\Delta) \cong \bold U(1) \otimes \pi_2^*\scrO_E(p_0) \otimes \pi _1^*\scrO_{\Pee^{n-1}}(-1).$$ We will find a section of the torsion free rank one sheaf $\Cal L$ on $T\times E$ corresponding to $\bold U(1) \otimes \pi_2^*\scrO_E(p_0) \otimes \pi _1^*\scrO_{\Pee^{n-1}}(-1)$ which vanishes to order one along $\Delta$. By the remarks before the proof, this will imply that $\Cal L = \scrO_{T\times E}(\Delta)$. Now there is an inclusion $$\scrO_{\Pee^{n-1}\times E} \to \bold U(1) \otimes \pi_2^*\scrO_E(p_0) \otimes \pi _1^*\scrO_{\Pee^{n-1}}(-1)$$ whose cokernel is $\pi_2^*(\scrO_E(p_0)\otimes W_{n-1})\otimes \pi _1^*\scrO_{\Pee^{n-1}}(-1)$. Thus $h^0(\Cal L) = h^0(\bold U(1) \otimes \pi_2^*\scrO_E(p_0) \otimes \pi _1^*\scrO_{\Pee^{n-1}}(-1)) = 1$. To see where the unique section of $\Cal L$ vanishes, fix a point $x\in \Pee^{n-1}$ corresponding to a regular semistable $V$, and consider where the corresponding section of $V\otimes \scrO_E(p_0)$ vanishes. This section arises from a homomorphism $\scrO_E(-p_0) \to V$ constructed in (3.4) and (3.5). For example, if $V=\bigoplus _i\scrO_E(e_i-p_0)$, then, up to the action of $(\Cee^*)^n$, the map is the direct sum of the natural inclusions $\scrO_E(-p_0)\to \scrO_E(e_i-p_0)$. At each fiber $\{(V, e_i)\}\times E$ of $T\times E$ lying over $\{V\}$, the section therefore vanishes simply at $((V, e_i), e_i)$. For a general point $t=(V, \frak m)$ of $T$, the restriction of the section of $\Cal L$ to the fiber $\{t\}\times E$ vanishes at the point of $E$ where the corresponding section of the composite map $$\scrO_E(-p_0) \to V \to V/\frak m\cdot V \cong \lambda$$ vanishes. By the construction of (3.4), the composite map $\scrO_E(-p_0) \to V \to \lambda$ is not identically zero, and hence vanishes exactly at the point $e$ of $E$ corresponding to $\lambda$. Thus the section of $\Cal L$ vanishes exactly along $\Delta$, with multiplicity one on a Zariski open and dense subset, proving (3.23). \endproof \corollary{3.24} Suppose that $E$ is smooth. For all $d\in \Zee$ with $1\leq d\leq n-1$, $\bold U(d) = U_{1-d}\otimes \pi _1^*\scrO_{\Pee^{n-1}}(1)$. \endstatement \proof This follows by writing both sides as successive elementary modifications of $\bold U(1)$, resp. $U_0\otimes \pi _1^*\scrO_{\Pee^{n-1}}(1)$, and applying (3.23). \endproof \ssection{3.5. The general spectral cover construction.} For every Weierstrass cubic $E$, we have now constructed a finite cover $T\to \Pee^{n-1}$ and a torsion free rank one sheaf $\Cal L_0=\scrO_{T\times E}(\Delta)\otimes \pi _2^*\scrO_E(-p_0)$. The proof of Theorem 2.4 goes over word-for-word to show: \theorem{3.25} Let $E$ be a Weierstrass cubic and let $U'$ be a rank $n$ vector bundle over $|np_0|\times E$ with the following property. For each $x\in |np_0|$ the restriction of $U'$ to $\{x\}\times E$ is isomorphic to the restriction of $U_0$ to $\{x\}\times E$. Then $U'=(\nu \times \Id)_*\left[\scrO_{T\times E }(\Delta-G)\otimes q _1^*L\right]$ for a unique line bundle $L$ on $T$. \qed \endstatement We may define $U_a$ and, for a smooth point $e\in E$, $U_a[e]$ exactly as in Definition 2.6, and the proof of Lemma 2.7 shows that $U_a$ is an elementary modification of $U_{a-1}$, and similarly for $U_a[e]$. Since $\Pic T\cong r^*\Pic E\oplus \Zee$, every bundle $U'$ as described in Theorem 3.25 is of the form $U_a[e]\otimes \scrO_{\Pee^{n-1}}(b)$ for integers $a,b$ and a smooth point $e\in E$. \remark{Question} In case $E$ is singular, $T$ is singular as well. Is there an analogue of twisting by Weil divisors on $T$ which are not Cartier, which produces bundles which are not regular, or perhaps sheaves which are not locally free, over points of $\Pee^{n-1}$ corresponding to the singular points of $T$? See Section 6 for a related construction in the smooth case. \endremark \medskip The following is proved as in the proof of Theorem 2.8. \theorem{3.26} Let $E$ be a Weierstrass cubic and let $S$ be a scheme or analytic space. Suppose that ${\Cal U}\to S\times E$ is a rank $n$ holomorphic vector bundle whose restriction to each slice $\{s\}\times E$ is a regular semistable bundle with trivial determinant. Let $\Phi\colon S\to |np_0|$ be the morphism constructed in Theorem \rom{1.5}. Let $\nu_S\colon\tilde S\to S$ be the pullback via $\Phi$ of the spectral covering $T\to |np_0|$: $$\tilde S=S\times_{|np_0|}T,$$ and let $\tilde\Phi\colon \tilde S\to T$ be the map covering $\Phi$. Let $q_1\colon \tilde S\times E\to \tilde S$ be the projection onto the first factor. Then there is a line bundle ${\Cal M}\to \tilde S$ and an isomorphism of ${\Cal U}$ with $$(\nu_S\times\operatorname{Id})_* \left((\tilde\Phi\times \Id)^*(\scrO_{T\times E}(\Delta-G))\otimes q_1^*{\Cal M}\right).\qquad\qed$$ \endstatement \ssection{3.6. Chern classes.} \theorem{3.27} For all $d$ with $1\leq d \leq n-1$, the total Chern class and the Chern character of $\bold U(d)$ are given by\rom: $$\align c(\bold U(d)) &= (1+h+\pi_2^*[p_0])(1+h)^{d-1};\\ \ch \bold U(d) &= (d -\pi _2^*[p_0])e^h + (n-d) + [p_0]. \endalign$$ Thus $c(U_0) = (1-h+ \pi _2^*[p_0]\cdot h)(1-h)^{n-2}$ and $\ch U_0 = ne^{-h} + (1- \pi _2^*[p_0])(1- e^{-h})$. \qed \endstatement \proof In $K$-theory, $W_d$ is the same as the direct sum of $d-1$ trivial bundles and the line bundle $\scrO_E(p_0)$. Thus $$\align c(\bold U(d)) &= (1-\pi_2^*[p_0] + h)(1+h)^{d-1}(1+\pi_2^*[p_0])\\ &=(1+h+\pi_2^*[p_0])(1+h)^{d-1}, \endalign$$ and $$\align \ch \bold U(d) &= \pi _2^*\ch (W_d\spcheck)\cdot \pi _1^*\ch (\scrO_{\Pee ^{n-1}}(1)) + \pi _2^*\ch W_{n-d}\\ & = (d-1+ \pi _2^*e^{-[p_0]})e^h + (n-d-1 + \pi _2^*e^{[p_0]}) \\ &=(d -\pi _2^*[p_0])e^h + (n-d) + [p_0], \endalign$$ since $e^{-[p_0]}= 1-[p_0]$ and similarly for $e^{[p_0]}$. The formulas for $c(U_0)$ and $\ch U_0$ then follow from (3.23). \endproof \section{4. A relative moduli space for elliptic fibrations.} Our goal in this section is to do the constructions of the last three sections in the relative setting of a family $\pi\: Z \to B$ of elliptic curves (possibly with singular fibers), in order to produce families of bundles whose restriction to every fiber of $\pi$ is regular semistable and with trivial determinant. First we identify the relative coarse moduli space as a projective bundle over $Z$. The extension picture generalizes in a straightforward way to give $n-1$ ``universal" bundles $\bold U(d)$, $1\leq d\leq n-1$, and they are related via elementary modifications. Using these bundles, we can generalize the spectral covers picture as well. Finally, we compute the Chern classes of the universal bundles. Let $\pi \: Z \to B$ be an elliptic fibration with a section $\sigma$. Following the notational conventions of the introduction, we shall always let $L^{-1} = R^1\pi _*\scrO_Z$, which we can also identify with the normal bundle $\scrO_Z(\sigma)|\sigma$. \ssection{4.1. A relative coarse moduli space.} Our first task is to find a relative version of $|np_0|$ for a single elliptic curve. The relative version of the vector space $H^0(E; \scrO_E(np_0))$ is just the rank $n$ vector bundle $\pi _*\scrO_Z(n\sigma)=\Cal V_n$, and the relative moduli space will then be the associated projective bundle. From the exact sequence $$0 \to \scrO_Z((n-1)\sigma) \to \scrO_Z(n\sigma) \to \pi ^*L^{-n}|\sigma \to 0,$$ we obtain for $n\geq 2$ an exact sequence $$0 \to \Cal V_{n-1} \to \Cal V_n \to L^{-n} \to 0.$$ (For $n=1$ the corresponding sequence identifies $\pi_*\scrO_Z(\sigma)$ with $\scrO_B$ and shows that there is an isomorphism $\scrO_Z(\sigma)|\sigma\to R^1\pi _*\scrO_Z=L^{-1}$.) Thus $\Cal V_n$ is naturally filtered by subbundles such that the successive quotients are decreasing powers of $L$. The following well-known lemma shows that this filtration is split: \lemma{4.1} $\pi _*\scrO_Z(\sigma) = \scrO_B$, and, for $n\geq 2$, $$\pi _*\scrO_Z(n\sigma) = \Cal V_n = \scrO_B\oplus L^{-2} \oplus \cdots \oplus L^{-n}.$$ \endstatement \proof Since $h^0(E;\scrO_E(np_0))=n$ for all the fibers $f$ of $\pi$, it follows from base change that $\pi_*\scrO_Z(n\sigma)$ is a vector bundle of rank $n$. Furthermore, the local sections of this bundle over an open subset $U\subset B$ are simply the meromorphic functions on $\pi^{-1}(U)$ which have poles of order at most $n$ along $\sigma\cap U$. For $U$ sufficiently small, there are functions $X$ with a pole of order $2$ along $\sigma$ and $Y$ with a pole of order $3$. Moreover, if we require that $X$ and $Y$ satisfy a Weierstrass equation, then $X$ and $Y$ are unique up to nowhere vanishing functions in $U$ and transform as sections of $L^{-2}, L^{-3}$ respectively. We can also use the defining equation of $Z$ to write $Y^2$ as a cubic polynomial in $X$. Now every section of $\pi_*\scrO_Z(n\sigma)$ can be written uniquely as $$(\alpha_0+\alpha_1X+\cdots+\alpha_kX^k)+ Y(\beta_0+\beta_1X+\cdots+\beta_\ell X^\ell)$$ where the $\alpha_i$ are holomorphic sections of $L^{-2i}$ and the $\beta_j$ are holomorphic sections of $L^{-2j-3}$ and $2k\le n$ and $2\ell+3\le n$. The $\alpha_i,\beta_j$ determine the isomorphism claimed in the statement of the lemma. \endproof Notice that the inclusion $\Cal V_{n-1}\subset \Cal V_n$ corresponds to the natural inclusion $$ \scrO_B\oplus L^{-2} \oplus \cdots \oplus L^{-(n-1)}\subset \scrO_B\oplus L^{-2} \oplus \cdots \oplus L^{-n}.$$ In particular, the distinguished points $\bold o_E=np_0\in |np_0|$ corresponding to the bundles with all Jordan-H\"older quotients trivial fit together to make a section $\bold o_Z$ of $\Pee\Cal V_n$. This section is the projectivization $\Pee \scrO_B$ of the first factor $\scrO_B$ in the above decomposition. We call the above splitting the {\sl $X$-$Y$ splitting} of $\pi_*\scrO_Z(n\sigma)$. While this decomposition of $\pi_*\scrO_Z(n\sigma)$ is natural it is not the only possible decomposition, even having the property described in the previous paragraph. For example, another splitting was suggested to us by P. Deligne. There is a global holomorphic differential $\omega$ on $E$ which is given on a Zariski open subset of $E$ by $dX/Y$. There is a local complex coordinate $\zeta$ for $E$ centered at $p_0$ with the property that on the open set on which this local coordinate is defined we have $\omega=d\zeta$. Of course, there is a homomorphism $\Cee\to E$ which pulls $\zeta$ back to the usual coordinate on $\Cee$. Every meromorphic function on $E$ with a pole of order at most $n$ at $p_0$ can be expanded as a Laurent series in $\zeta$: $$f=\sum_{i=-n}^\infty b_i\zeta^i.$$ The coefficient $b_i$ in this expansion is a section of $L^i$. We can then use the coefficients $b_{-n},\ldots,b_{-2},b_0$ to define a splitting of $\pi_*\scrO_Z(n\sigma)$. (If $f$ is a meromorphic function on $E$ whose only pole is at $p_0$ then $b_{-1}$ is determined by the $b_{-i}$ for $-n\le -i\le -2$.) This splitting is different from the $X$-$Y$ splitting, but both splittings induce the same filtration on $\pi_*\scrO_Z(n\sigma)$. In Theorem 1.2 we showed how a semistable bundle of rank $n$ and trivial determinant on a smooth elliptic curve $E$ determines a point of the linear series $\scrO_E(np_0)$. This works well for bundles over families of elliptic curves. \lemma{4.2} Let $p\:\Pee \Cal V_n \to B$ be the projection. Thus, the fiber of $p$ over $b\in B$ is the complete linear system $|np_0|$, where $E_b=\pi ^{-1}(b)$ and $p_0$ is the smooth point $\sigma \cap E_b$. If $V\to Z$ is a rank $n$ vector bundle whose restriction to each fiber of $Z\to B$ is a semistable bundle with trivial determinant, then $V$ determines a section $$A(V)\colon B\to \Pee \Cal V_n ,$$ with the property that, for each $b\in B$, $$A(V)(b)=\zeta(V| E_b ).$$ \endstatement \proof Arguing as in (1.6), there is an induced morphism $$\Psi\: \pi^*\pi_*(V \otimes \scrO_Z(\sigma)) \to V \otimes \scrO_Z(\sigma).$$ The determinant of this morphism is a section of $\pi^* M \otimes \scrO_Z(n\sigma)$, for some line bundle $M$ on $B$, and it gives a well-defined section $A(V)$ of $\Pee \Cal V_n $ over $B$. \endproof We note that the proof of (4.2) does not require that $B$ be smooth, or even reduced. There is also an analogue for families of elliptic curves of Theorem 1.5. \lemma{4.3} Let $S$ be a scheme or analytic space over $B$ and let $\Cal V$ be a rank $n$ vector bundle over $S\times _BZ$, such that the restriction of $\Cal V$ to every fiber $p_1^{-1}(s) \cong \pi ^{-1}(b)$ is semistable with trivial determinant, where $p_1, p_2$ are the projections of $S\times _BZ$ to the first and second factors and $b$ is the point of $B$ lying under $s$. Then there is an induced morphism $\Phi \: S\to \Pee \Cal V_n $ of spaces over $B$, which agrees over each $b\in B$ with the morphism defined in \rom{(1.5)}. \endstatement \proof Let $\hat Z=S\times _BZ$, with $\hat\pi \: \hat Z \to S$ the first projection. Then $\hat Z$ is an elliptic scheme over $S$ which maps naturally to $Z$ covering the map of $S\to B$. Let $\hat\sigma$ be the induced section. Set $\widehat {\Cal V}_n = \hat\pi _*\scrO_{\hat Z}(n\hat \sigma)$. Clearly $\Pee \widehat {\Cal V}_n$ is identified with the pullback of $\Pee \Cal V_n $. Now apply the above result to this elliptic scheme to produce a section $S\to \Pee \widehat {\Cal V}_n$ which when composed with the natural map $\Pee \widehat {\Cal V}_n \to \Pee\Cal V_n$ is the morphism $\Phi$ of the proposition. \endproof \ssection{4.2. Construction of bundles via extensions.} Our goal for the remainder of this section is to construct various ``universal" bundles over $\Pee \Cal V_n \times _BZ$. The first and easiest construction of the universal moduli space is via the extension approach, generalizing what we did in Section 3 for a single elliptic curve. In order to make the extension construction in families, we first need to extend the basic bundle $W_k$ over $E$ to bundles over the elliptic scheme $Z$. \proposition{4.4} There is a vector bundle $\Cal W_d$ on $Z$ such that $\Cal W_d$ is filtered, with successive quotients $\pi ^*L^{d-1}, \pi ^*L^{d-2}, \dots, \scrO_Z(\sigma)$, and such that on every fiber $\Cal W_d$ restricts to $W_d$. Moreover, $\Cal W_d$ is uniquely specified by the above properties. In fact, if $\Cal W$ is a vector bundle on $Z$ such that $\Cal W$ restricts to $W_d$ on every fiber, then there exists a line bundle $M$ on $B$ such that $\Cal W = \Cal W_d\otimes \pi ^*M$. Finally, $R^0\pi _*\Cal W_d = L^{(d-1)}$ and $R^1\pi _*(\Cal W_d\spcheck) = L^{-d}$. \endstatement \proof In case $d=1$, take $\Cal W_1= \scrO_Z(\sigma)$. Now suppose inductively that $\Cal W_{d-1}$ has been defined, and that $R^1\pi _*(\Cal W_{d-1}\spcheck) = L^{-(d-1)}$. We seek an extension of $\Cal W_{d-1}$ by a line bundle trivial on ever fiber of $\pi$, and thus of the form $\pi^*M$ for some line bundle $M$ on $B$, and such that $H^0(R^1\pi _*(\Cal W_{d-1}\spcheck\otimes \pi^*M))$ has an everywhere generating section. Now $$R^1\pi _*(\Cal W_{d-1}\spcheck\otimes \pi^*M) = R^1\pi _*(\Cal W_{d-1}\spcheck) \otimes M = L^{-(d-1)}\otimes M.$$ Thus we must have $M=L^{d-1}$. With this choice, noting that $R^0\pi _*(\Cal W_{d-1}\spcheck\otimes \pi ^*L^{d-1}) = 0$ since $W_d\spcheck$ has no sections, the Leray spectral sequence gives an isomorphism $$H^1(\Cal W_{d-1}\spcheck\otimes \pi ^*L^{d-1}) \cong H^0(R^1\pi _*(\Cal W_{d-1}\spcheck\otimes \pi ^*L^{d-1}))=H^0(\scrO_B)$$ and thus a global extension of $\Cal W_{d-1}$ by $\pi ^*L^{d-1}$ restricting to $W_d$ on every fiber. Since the unique section of $W_d$ is given by the inclusion of the canonical subbundle $\scrO_f \to W_d$, we must have $R^0\pi _*\Cal W_d = L^{(d-1)}$, and a similar argument (or relative duality) evaluates $R^1\pi _*(\Cal W_d\spcheck)$. Finally suppose that $\Cal W$ is another bundle on $Z$ restricting to $W_d$ on every fiber. Then since $W_d$ is simple, $\pi _*\Hom (\Cal W_d, \Cal W)$ is a line bundle $M$ on $B$, and thus $\pi _*\Hom (\Cal W_d\otimes \pi ^*M, \Cal W)\cong \scrO_B$. The element $1\in H^0(\scrO_B)$ then defines an isomorphism from $\Cal W_d\otimes \pi ^*M$ to $\Cal W$. \endproof Note that the formation of $\Cal W_d$ is compatible with base change, in the following sense. Given a morphism $g\: B'\to B$, let $Z'=Z\times _BB'$, with $f\: Z'\to Z$ the induced morphism, and let $\sigma '$ be the induced section of $\pi'\: Z'\to B'$. Then the bundle $\Cal W_d'$ constructed for $\pi'\:Z'\to B'$ and the section $\sigma'$ is $f^*\Cal W_d$. Next we construct a universal bundle via extensions. First we identify the relevant bundles to use as the parameter space of the extension: \lemma{4.5} For $1\leq d \leq n-1$, the sheaves $R^1\pi _*(\Cal W_{n-d}\spcheck \otimes \Cal W_d\spcheck)=\Cal V_{n,d}$ are locally free of rank $n$ over $B$, and are all canonically identified. \endstatement \proof The local freeness and the rank statement follow from Claim 3.3 and base change. The canonical identifications follow from Corollary 3.10. \endproof Let $\Cal V_{n,d} = R^1\pi _*(\Cal W_{n-d}\spcheck \otimes \Cal W_d\spcheck)$ as above, and let ${\Cal P}_{n-1,d}$ be the associated projective space bundle $\Pee(\Cal V_{n,d}) \to B$. By the general properties of extensions, there is a universal extension over ${\Cal P}_{n-1,d}\times_BZ$ of the form $$0\to \pi_2^*{\Cal W}_d\spcheck\otimes \pi_2^*\scrO_{{\Cal P}_{n-1,d}}(1)\to {\bold U}(d)\to \pi_2^*{\Cal W}_{n-d}\to 0.$$ Applying Lemma 4.3 to these bundles produces bundle maps over $B$ $$\Phi_d\colon {\Cal P}_{n-1,d}\to \Pee \Cal V_n .$$ The projective space bundles ${\Cal P}_{n-1,d}$ over $B$ are all canonically isomorphic. Under these isomorphisms, the universal bundles ${\bold U}(d)$ are all distinct. Nevertheless, the result in Proposition 3.10 shows that there is an isomorphism $I$ which identifies $R^1\pi_*(\scrO_Z(-\sigma)\otimes {\Cal W}\spcheck_{n-1})$ with $$R^0\pi_*(\det(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1}))\otimes \det(R^0\pi_*(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1}))^{-1}\otimes R^1\pi_*\scrO_Z.$$ Let us identify the various factors on the right-hand-side of this expression. First of all, it is straightforward given the inductive definition of the ${\Cal W}_{n-1}$ to show that. $$\det(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1})\cong \scrO_Z(n\sigma)\otimes \pi^*L^{(n-1)(n-2)/2}.$$ It follows that $R^0\pi_*(\det(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1}))\cong R^0\pi_*\scrO_Z(n\sigma)\otimes L^{(n-1)(n-2)/2}$. Next, we have exact sequences $$0\to R^0\pi_*(\scrO_Z(\sigma)\otimes \pi^*L^{n-2})\to R^0\pi_*(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1})\to R^0\pi_*(\scrO_Z(\sigma)\otimes {\Cal W}_{n-2})\to 0.$$ Since by Proposition 4.4 we have $R^0\pi_*(\pi^*L^a\otimes \scrO_Z(\sigma))\cong L^a$, and since $$R^0\pi_*(\scrO_Z(2\sigma))\cong L^{-2}\oplus \scrO_B,$$ an easy inductive argument shows that $$\det(R^0\pi_*(\scrO_Z(\sigma)\otimes {\Cal W}_{n-1}))\cong L^{((n-1)(n-2)/2)-2}.$$ Lastly, $R^1\pi_*\scrO_Z\cong L^{-1}$. Putting all this together, we get: \theorem{4.6} There is an isomorphism of vector bundles over $B$ $$I\colon R^1\pi_*(\scrO_Z(-\sigma)\otimes {\Cal W}_{n-1})\cong R^0\pi_*\scrO_Z(n\sigma)\otimes L,$$ which fiber by fiber agrees with the map $I$ of Proposition \rom{3.13}. In other words, $$\Cal V_{n,1} \cong \Cal V_n\otimes L.$$ Furthermore, the map induced by projectivizing $I$ agrees with the map $\Phi_1$ produced by applying Lemma \rom{4.3} to the family $\bold U(1)$ over ${\Cal P}_{n-1,1}\times_BZ$. Let $\Phi_d$ be the map ${\Cal P}_{n-1,d}\to \Pee \Cal V_n $ obtained by applying Lemma \rom{4.3} to the family $\bold U(d)$. Then, the maps $\Phi_d$ for $1\le d<n$ are compatible with the identifications coming from Corollary \rom{3.10}, and hence each of these maps is an isomorphism of projective bundles over $B$. \endstatement Note that, while the $\Pee^{n-1}$-bundles ${\Cal P}_{n-1,d}$ and $\Pee \Cal V_n $ are isomorphic, the tautological bundles $\scrO_{{\Cal P}_{n-1,d}}(1)$ and $\scrO_{\Pee \Cal V_n}(1)$ differ by a twist by $p^*L$. We shall use $\Cal P_{n-1}$ to denote the bundle $\Pee \Cal V_n $ together with its tautological line bundle. If $\zeta =c_1(\scrO_{\Pee \Cal V_n}(1))$ and $\zeta' = c_1(\scrO_{{\Cal P}_{n-1,d}}(1))$, then $\zeta = \zeta ' + L$. \corollary{4.7} Via the isomorphism of \rom{(4.6)} and \rom{(4.6)}, $$\Cal V_{n,d} = R^1\pi _*(\Cal W_{n-d}\spcheck \otimes \Cal W_d\spcheck) \cong L\oplus L^{-1} \oplus \cdots \oplus L^{-(n-1)}.$$ This splitting is compatible with the inclusion of $\Cal V_{n-1, d}$ in $\Cal V_{n,d}$ as well as that of $\Cal V_{n-1, d-1}$ in $\Cal V_{n,d}$. \qed \endstatement \corollary{4.8} Under the isomorphism $\pi |\sigma\: \sigma \cong B$, there is a natural splitting $$\Cal W_n|\sigma \cong L^{n-1} \oplus L^{n-2}\oplus \cdots \oplus L \oplus L^{-1}.$$ In fact, the extension $$0 \to \pi ^*L^{n-1} \to \Cal W_n \to \Cal W_{n-1} \to 0$$ restricts to the split extension over $\sigma$. \endstatement \proof Let us first show that the restriction of $\Cal W_n\spcheck$ to $\sigma$ is split. Begin with the exact sequence $$0 \to \scrO_Z(-\sigma) \otimes \Cal W_n\spcheck \to \Cal W_n\spcheck \to \Cal W_n\spcheck|\sigma \to 0,$$ and apply $R^i\pi _*$. We get an exact sequence $$\CD 0 @>>> \pi _*(\Cal W_n\spcheck|\sigma) @>>> R^1\pi _*(\scrO_Z(-\sigma) \otimes \Cal W_n\spcheck) @>>> R^1\pi _*\Cal W_n\spcheck @>>> 0\\ @. @| @| @| @.\\ 0 @>>> \pi _*\Cal W_n\spcheck|\sigma @>>> L\oplus L^{-1}\oplus \cdots \oplus L^{-n} @>>> L^{-n} @>>> 0. \endCD$$ Tracing through the identifications shows that the map $R^1\pi _*(\scrO_Z(-\sigma) \otimes \Cal W_n\spcheck) \to R^1\pi _*\Cal W_n\spcheck$ is the same as the map $$R^1\pi _*(\scrO_Z(-\sigma) \otimes \Cal W_n\spcheck) \to R^1\pi _*\scrO_Z(-\sigma) \otimes L^{-n+1}= L^{-1}\otimes L^{-n+1} = L^{-n}$$ coming from the long exact sequence associated to $$0 \to \scrO_Z(-\sigma) \otimes \Cal W_{n-1}\spcheck \to \scrO_Z(-\sigma) \otimes \Cal W_n\spcheck \to \scrO_Z(-\sigma)\otimes \pi^*L^{-n+1} \to 0.$$ This identifies the map $L\oplus L^{-1}\oplus \cdots \oplus L^{-n} \to L^{-n}$ with projection onto the last factor. Hence $\Cal W_n\spcheck|\sigma$ is identified with $L\oplus L^{-1}\oplus \cdots \oplus L^{-n+1}$. Dualizing gives the splitting of $\Cal W_n|\sigma$. The splitting of the extension $$0 \to \pi ^*L^{n-1} \to \Cal W_n \to \Cal W_{n-1} \to 0$$ is similar. \endproof Now let us relate the bundles $\bold U(d)$ via elementary modifications. \proposition{4.9} Let $\Cal H$ be the smooth divisor which is the image of $\Pee \Cal V_{n-1} =\Cal P_{n-2}$ in $\Cal P_{n-1}$ under the natural inclusion $\pi _*\scrO_Z((n-1)\sigma) \subset\pi _*\scrO_Z(n\sigma)$, and let $i\: \Cal H \to \Cal P_{n-1}$ be the inclusion. Then there is an exact sequence $$0 \to \bold U(d) \to \bold U(d+1) \to (i\times \Id)_*\scrO_{\Cal H\times _BZ}(1)\otimes \pi^*L^{-d} \to 0,$$ where $\scrO_{\Cal H\times _BZ}(1)$ denotes the restriction of $\scrO_{{\Cal P}_{n-1,d}}(1)= \scrO_{\Pee\Cal V_{n,d}}(1)$ to $\Cal H\times _BZ$. Thus $\bold U(d)$ is an elementary modification of $\bold U(d+1)$, and it is the only possible such modification along $\Cal H\times_BZ$. \endstatement \proof The construction of the proof of Theorem 3.12 gives an inclusion $\bold U(d) \to \bold U(d+1)$ whose cokernel is the direct image of a line bundle supported along $\Cal H\times _BZ$. As in the first paragraph of the proof of (3.12), this line bundle is the inverse of $\pi _1{}_*Hom(\bold U(d+1)|\Cal H\times _BZ, \scrO_{\Cal H\times _BZ})$ (where for the rest of the proof we let $\pi_1$ be the first projection $\Cal H\times _BZ \to \Cal H$). From the defining exact sequence for $\bold U(d+1)$, $$\gather \pi _1{}_*Hom(\bold U(d+1)|\Cal H\times _BZ, \scrO_{\Cal H\times _BZ})\cong \pi _1{}_*Hom(\pi _2^*\Cal W_{d+1}\spcheck \otimes \scrO_{\Cal H\times _BZ}(1), \scrO_{\Cal H\times _BZ})\\ = \pi _1{}_*\pi _2^*\Cal W_{d+1}\otimes \scrO_{\Cal H\times _BZ}(-1). \endgather$$ Here by base change $\pi _1{}_*\pi _2^*\Cal W_{d+1}$ is a line bundle on $\Cal H$ whose restriction to every fiber is the nonzero section of $W_{d+1}$ on that fiber. Now $\Cal W_{d+1}$ is filtered by subbundles with successive quotients $\pi^*L^d, \pi ^*L^{d-1}, \dots, \scrO_Z(\sigma)$, and the inclusion of $\pi ^*L^d$ in $\Cal W_{d+1}$ defines a map $L^d \to \pi _1{}_*\pi _2^*\Cal W_{d+1}$ which restricts to the nonzero section on every fiber. Thus $\pi _1{}_*\pi _2^*\Cal W_{d+1} \cong L^d$. Hence $$\pi _1{}_*Hom(\bold U(d+1)|\Cal H\times _BZ, \scrO_{\Cal H\times _BZ}) \cong L^d\otimes \scrO_{\Cal H\times _BZ}(-1),$$ and thus the cokernel of the map $\bold U(d)\to \bold U(d+1)$ is as claimed. The uniqueness is clear. \endproof This completes the construction of ``universal" bundles over $\Pee\Cal V_n\times_BZ$. However, we have constructed only $n-1$ bundles ${\bold U}(d)$ for $1\le d<n$. Note that the formation of the universal bundles $\bold U(d)$ over $\Cal P_{n-1,d}\times _BZ$ is also compatible with base change $B'\to B$ in the obvious sense. \ssection{4.3. The spectral cover construction.} Now we turn to the generalization of the spectral covering construction. First let us define the analogues of $E^{n-1}, T$, and $\nu$. By repeating the construction on each smooth fiber, we could take the $(n-1)$-fold fiber product $Z\times _BZ\times _B \cdots \times _BZ$ and its quotient under $\frak S_n$ and $\frak S_{n-1}$. However this construction runs into trouble at the singular fibers, reflecting the difference between the $n$-fold symmetric product of $E$ and the linear system $|np_0|$ for a singular fiber. Instead, we construct the spectral cover in families as follows: Let $\Cal E$ be defined by the exact sequence of vector bundles over $Z$ $$0 \to \Cal E \to \pi ^*\pi _*\scrO_Z(n\sigma) \to \scrO_Z(n\sigma) \to 0,$$ where the last map is the natural evaluation map and is surjective. We set $\Cal T = \Pee \Cal E$, with $r\: \Cal T \to Z$ the projection. By construction $\Cal T$ is a $\Pee^{n-2}$-bundle over $Z$. There is an inclusion $$\Cal T \to \Pee (\pi ^*\pi _*\scrO_Z(n\sigma)) = \Pee \Cal V_n \times _BZ,$$ and we let $\nu$ be the composition of this morphism with the projection $q_1\: \Pee \Cal V_n \times _BZ\to \Pee \Cal V_n $. It is easy to see that $\nu\: {\Cal T}\to \Pee \Cal V_n$ is an $n$-sheeted covering, which restricts to the spectral cover described in Section 2 on each smooth fiber of $Z\to B$. By analogy with the case of a single elliptic curve, we would like to consider the sheaf $${\Cal U}_0=(\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta - \Cal G),$$ where $\Delta = (r\times \Id)^*(\Delta_0)$, for $\Delta_0$ the diagonal in $Z\times _BZ$, and $\Cal G=(r\times \Id)^*p_2^*\sigma$ for $p_1, p_2$ the projections of $Z\times _BZ$ to the first and second factors. Here we can define $\scrO_{Z\times _BZ}(\Delta_0)$ to be the dual of the ideal sheaf of $\Delta_0$ in $Z\times _BZ$. It is an invertible sheaf away from the singularities of $Z\times _BZ$. The proof of (0.4) shows that $\scrO_{Z\times _BZ}(\Delta_0)$ is flat over both factors of $Z\times_BZ$, and identifies the first factor, say, with the relative compactified generalized Jacobian. As we shall see, $\Cal U_0$ is indeed a vector bundle of rank $n$, and that its restriction over each smooth fiber is the bundle $U_0$ described in Proposition 2.9. In particular, ${\Cal U}_0$ is a family of regular, semistable bundles with trivial determinant over the family $Z\to B$ of elliptic curves. Although we shall not need this in what follows, for concreteness sake let us describe the singularities of $Z\times_BZ$ and $\Cal T$ explicitly the case where the divisors associated to $G_2$ and $G_3$ are smooth and meet transversally as in the introduction. In this case $Z$ has the local equation $y^2 = x^3 + sx+t$. The morphism to $B$ is given locally by $(s,t)$, where $t= y^2-x^3-sx$. Using $x,y,s$ as part of a set of local coordinates for $Z$, the fiber product has local coordinates $x,y,s,x',y', \dots$ and a local equation $$y^2-x^3-sx = (y')^2-(x')^3-s(x').$$ Rewrite this equation as $$y^2- (y')^2 = (x-x')(s+ x^2 + xx' + (x')^2) = h_1h_2, $$ say, where a local calculation shows that $h_1(s,x,x')$ and $h_2(s,x,x')$ define two smooth hypersurfaces meeting transversally along $\Gamma \times _B\Gamma$. It follows that the total singularity in $Z\times _BZ$ is a locally trivial fibration of ordinary threefold double points, and $\Delta$ is a smooth divisor which fails to be Cartier at the singularities. We return now to the case of a general $Z$. Fix a $d$ with $1\leq d\leq n-1$, and consider the sheaf of algebras $\pi _1{}_*Hom (\bold U(d), \bold U(d)) =\bold A$ over $\Cal P_{n-1}$. Arguing as in Lemma 3.19, the space $\bold {Spec}\, \bold A$ is reduced and there is a finite flat morphism $\nu\: \bold {Spec}\, \bold A \to \Cal P_{n-1}$ restricts over each fiber to give $\nu\: T\to \Pee^{n-1}$. Moreover, $\bold U(d) = (\nu\times \Id)_*\Cal L_d$ for some sheaf $\Cal L_d$ on $\bold {Spec}\, \bold A\times _BZ$. The method of proof of Theorem 3.21 then shows: \theorem{4.10} There is an isomorphism from $\bold {Spec}\, \bold A$ to $\Cal T$. Under this isomorphism, there is a line bundle $\Cal M$ over $\Cal T$ such that $\bold U(d) =(\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G)\otimes \pi _1^*\Cal M$. \qed \endstatement Since $\bold U(1)\otimes \scrO_{\Pee\Cal V_{n,d}}(-1) \otimes \pi_2^*\scrO_Z(\sigma)$ has a section vanishing exactly along $\Delta$, the proof of 3.23 identifies this line bundle in case $d=1$: \theorem{4.11} In the above notation, $$\align \bold U(1) &\cong (\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G) \otimes \scrO_{\Pee\Cal V_{n,d}}(1)\\ &=(\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G) \otimes \scrO_{\Pee\Cal V_n}(1)\otimes L^{-1}.\qed \endalign$$ \endstatement For every $a\in \Zee$, we can then define $\Cal U_a = (\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G -a(r^*\sigma\times _BZ))$. It follows that $\Cal U_a$ is a vector bundle for every $a\in \Zee$. \theorem{4.12} With $\Cal U_a$ defined as above, there is an exact sequence $$0 \to \Cal U_a \to \Cal U_{a-1} \to (i\times \Id)_*\scrO_{\Cal H\times _BZ}\otimes \pi ^*L^{a-1} \to 0,$$ which realizes $\Cal U_a$ as an elementary modification of $\Cal U_{a-1}$. Thus, for $1\leq d\leq n-1$, $$\bold U(d) \cong\Cal U_{1-d}\otimes \pi_1^*\scrO_{\Pee\Cal V_n}(1)\otimes L^{-1}.$$ \endstatement \proof From the definition of $\Cal U_a$, there is an exact sequence $$\align 0 &\to (\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G -a(r^*\sigma\times _BZ)) \\ &\to (\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G -(a-1)(r^*\sigma\times _BZ)) \\ &\to (\nu\times \Id)_*\scrO_{r^*\sigma\times _BZ}(\Delta -\Cal G -(a-1)(r^*\sigma\times _BZ))\to 0. \endalign$$ The divisor $r^*\sigma$ is a $\Pee^{n-2}$-bundle over $B$ which intersects each fiber of $\Cal T\to B$ in the $\Pee^{n-2}$ fiber $r^{-1}(p_0)$. This fiber is mapped linearly via $\nu$ to the hyperplane $H_{p_0}$ in $|np_0|$. Thus, $\nu_*r^*\sigma = \Cal H$. Now both $\Delta$ and $\Cal G$ have the same restriction to $r^*\sigma\times _BZ$, namely $r^*\sigma\times _B\sigma$. Also, $\scrO_{r^*\sigma\times _BZ}(r^*\sigma\times _BZ)$ is the pullback of the line bundle $\scrO_Z(\sigma)|\sigma = L^{-1}$. It follows that the quotient of $\Cal U_{a-1}$ by the image of $\Cal U_a$ is exactly the direct image of $\scrO_{\Cal H\times _BZ}\otimes \pi ^*L^{a-1}$, as claimed. The final statement in (4.12) then follows by comparing elementary modifications. \endproof Finally we shall need the analogue of Proposition 2.4 for a single elliptic curve. It is proved exactly as in (2.4). \theorem{4.13} Let $\Cal U'$ be a rank $n$ vector bundle over $\Pee \Cal V_n\times _BZ$ such that, for all $x\in \Pee \Cal V_n$, $\Cal U'|q_1^{-1}(x) \cong \Cal U_0|q_1^{-1}(x)$. Then there is a unique line bundle $\Cal M$ on $\Cal T$ such that, if $\pi _1\: \Cal T\times_BZ \to \Cal T$ is projection onto the first factor, then $\Cal U' \cong (\nu\times \Id)_*\left(\scrO_{\Cal T\times _BZ}(\Delta - \Cal G)\otimes \pi _1^*\Cal M\right)$. \qed \endstatement \ssection{4.4. Chern class calculations.} Recall that we let $\zeta = c_1(\scrO_{\Cal P_{n-1}}(1))$, viewed as a class in $H^2(\Cal P_{n-1})$. By pullback, we can also view $\zeta$ as an element of $H^2(\Cal P_{n-1}\times _BZ)$. We also have the line bundle $\scrO_{\Cal P_{n-1,d}}(1))$, and its first Chern class $\zeta'$ is given by $\zeta ' =\zeta - L$ (where we identify $L$ with its first Chern class in $H^2(B)$ and then by pullback in any of the relevant spaces). \theorem{4.14} The Chern characters of the bundles $\bold U(d)$ and $\Cal U_a$ are given by: $$\gather \ch \bold U(d) = (e^{-\sigma} + e^{-L} + \dots + e^{-(d-1)L})e^{\zeta - L} + (e^{\sigma} + e^L + \dots + e^{(n-d-1)L}); \\ \ch \Cal U_a = e^{-\zeta}\fracwithdelims(){1 - e^{(a+n)L}}{1-e^L}-\frac{1 - e^{aL}}{1-e^L} +e^{-\sigma}(1-e^{-\zeta}). \endgather$$ \endstatement \proof The first statement is clear from the filtration on the $\Cal W_k$ and the definition of $\zeta$. To see the second, we use (4.12) for $1\leq d\leq n-1$ and calculate $$\gather \ch(\bold U (d)\otimes \scrO_{\Pee\Cal V_n}(-1)\otimes L) = \ch\bold U (d)\cdot e^{-\zeta + L}=\\ (e^{-\sigma} + e^{-L} + \dots + e^{-(d-1)L}) + (e^{\sigma +L} + e^{2L} + \dots + e^{(n-d)L})e^{-\zeta}\\ =(e^{-\sigma}-1) + (1+e^{-L} + \dots + e^{-(d-1)L}) + \\ +(e^{\sigma +L-\zeta}-e^{L-\zeta}) + (e^L + e^{2L} + \dots + e^{(n-d)L})e^{-\zeta}. \endgather$$ Let $a = 1-d$. A little manipulation shows that we can write: $$\align 1+e^{-L} + \dots + e^{-(d-1)L} &= - \frac{e^L - e^{aL}}{1-e^L};\\ e^L + e^{2L} + \dots + e^{(n-d)L} &= \frac{e^L - e^{(a+n)L}}{1-e^L};\\ (e^{-\sigma}-1) + (e^{\sigma +L-\zeta}-e^{L-\zeta}) &= -(1-e^{-\sigma})(1-e^{\sigma + L-\zeta}). \endalign$$ In the last term, note that $1-e^{-\sigma}$ is a power series without constant term in $\sigma$ and thus annihilates every power series without constant term in $\sigma +L$, since $\sigma ^2 = -L\cdot \sigma$. Thus we can replace the last term by $-(1-e^{-\sigma})(1-e^{-\zeta})$. It follows that $$\align \ch\bold U(d)\cdot e^{-\zeta+L}& =e^{-\zeta}\fracwithdelims(){e^L - e^{(a+n)L}}{1-e^L}-\frac{e^L - e^{aL}}{1-e^L} - (1-e^{-\sigma})(1-e^{-\zeta})\\ & =e^{-\zeta}\fracwithdelims(){1 - e^{(a+n)L}}{1-e^L}-\frac{1 - e^{aL}}{1-e^L} +e^{-\sigma}(1-e^{-\zeta}). \endalign$$ In particular, we have established the formula in (4.14) for $\ch \Cal U_a$ provided $a = 1-d$ with $1\leq d\leq n-1$. On the other hand, the formula for $\Cal U_a$ as an elementary modification shows that $$\ch \Cal U_a = \ch \Cal U_{a-1} - \ch (\scrO_{\Cal H\times _BZ}\otimes L^{a-1}).$$ Now from the exact sequence $$0 \to \scrO_{\Cal P_{n-1}\times _BZ}(-\Cal H\times _BZ) \to \scrO_{\Cal P_{n-1}\times _BZ} \to \scrO_{\Cal H\times _BZ}\to 0,$$ we see that $$\align \ch (\scrO_{\Cal H\times _BZ}\otimes L^{a-1}) &= \ch (\scrO_{\Cal H\times _BZ})\cdot e^{(a-1)L}\\ &= e^{(a-1)L}(1- e^{-\Cal H}). \endalign$$ Next we claim: \lemma{4.15} $[\Cal H] = \zeta - nL$. \endstatement \proof We have identified $\Cal H$ with the image of $\Pee\Cal V_{n-1}$ in $\Pee\Cal V_n$. The lemma now follows from the more general statement below, whose proof is left to the reader: \enddemo \lemma{4.16} Let $\Cal V$ be a vector bundle over a scheme $B$, and suppose that there is an exact sequence $$0 \to \Cal V' \to \Cal V \to M \to 0,$$ where $M$ is a line bundle on $B$. Let $\Cal H$ be the Cartier divisor $\Pee(\Cal V') \subset \Pee (\Cal V)$. Then, if $p\: \Pee(\Cal V) \to B$ is the projection, $$\scrO_{\Pee (\Cal V)}(\Cal H) = \scrO_{\Pee (\Cal V)}(1) \otimes p^*M. \qed$$ \endstatement Plugging in the expression for $[\Cal H]$, we see that $$\ch \Cal U_a - \ch \Cal U_{a-1} = - e^{(a-1)L}(1- e^{-(\zeta - nL)}).$$ Comparing this difference with the formula of (4.14) shows that (4.14) holds for one value of $a$ if and only if it holds for all values of $a$. Since we have already checked it for $a=0$, we are done. \endproof Similar computations give the Chern class of $\bold U(d)$ and $\Cal U_a$. We leave the calculations to the reader. \theorem{4.17} The total Chern class of $\bold U(d)$ is given by the formula: $$c(\bold U(d)) = (1+\zeta -L +\zeta\cdot \sigma)\prod_{r=1}^{d-1}(1-(r+1)L+\zeta)\prod_{s=1}^{n-d-1}(1+sL).$$ If $a\geq 0$, then $$c(\Cal U_a) = (1-\zeta +L +\zeta\cdot \sigma)\prod_{s=1}^{n+a-2}(1+(s+1)L-\zeta)\prod_{r=1}^{a-1}(1+ rL)^{-1}.$$ If $-(n-1)\leq a < 0$, then $$c(\Cal U_a) = (1-\zeta +L +\zeta\cdot \sigma)\prod_{s=1}^{n+a-2}(1+(s+1)L-\zeta)\prod_{r=1}^{-a}(1- rL).$$ If $ a < -(n-1)$, then $$c(\Cal U_a) = (1-\zeta +L +\zeta\cdot \sigma)\prod_{s=0}^{1-n-a}(1-(s-1)L-\zeta)^{-1}\prod_{r=1}^{-a}(1- rL).\qed$$ \endstatement Let us work out explicitly the first two Chern classes of ${\Cal U}_a$. First, $$c_1(\Cal U_a) = \left[an + \fracwithdelims(){n^2-n}{2}\right]L - (n+a-1)\zeta.$$ To give $c_2(\Cal U_a)$, write $$\frac{1-e^{cx}}{1-e^x} = c + \fracwithdelims(){c^2-c}2x + P(c)x^2 + \cdots,$$ where $$P(c) = \frac{c(2c-1)(c-1)}{12} = \frac{2c^3 -3c^2 +c}{12}$$ (if $c$ is a positive integer then $P(c) = \frac12\sum _{i=1}^{c-1}i^2$). A little manipulation shows that $c_2(\Cal U_a)$ is equal to $$\gather \frac{(a+n-1)(a+n-2)}2\zeta ^2 -(n^2+2an-2n -a)\left(\frac{a+n-1}{2}\right) \zeta\cdot L+\\ \left[\frac{1}{2}\left(an+\frac{n^2-n}{2}\right)^2-P(a+n)+P(a)\right] L^2+(\sigma\cdot \zeta). \endgather$$ Finally, we remark that it is possible to work out the first two terms in $\ch \Cal U_a$ by applying the Grothendieck-Riemann-Roch theorem directly to the description of $\Cal U_a$ as $(\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta -\Cal G -a(r^*\sigma\times _BZ))$. This calculation is somewhat long and painful, and does not give the full calculation of $\ch \Cal U_a$ because $\Delta$ is not a Cartier divisor. \section{5. Bundles which are regular and semistable on every fiber.} So far in this paper we have been working universally with the moduli space of all regular semistable bundles with trivial determinant over an elliptic curve or an elliptic fibration. In this section we wish to study bundles $V$ over an elliptic fibration $\pi\: Z\to B$ with the property that the restriction of $V$ to every fiber is a regular semistable bundle with trivial determinant. \ssection{5.1. Sections and spectral covers.} Suppose that $V\to Z$ is a vector bundle of rank $n$ whose restriction to each fiber is a regular semistable bundle with trivial determinant. Then for each $b\in B$ the bundle $V|_{E_b}$ determines a point in the fiber of ${\Cal P}_{n-1}$ over $b$. This means that $V$ determines a section $A(V) = A\colon B\to {\Cal P}_{n-1}$, as follows from (4.2). We shall usually identify $A$ with the image $A(B)$ of $A$ in ${\Cal P}_{n-1}$. Conversely, given a section $A$ of ${\Cal P}_{n-1}$ we can construct a bundle $V$ over $Z$ which is regular semistable with trivial determinant on each fiber and such that the section determined by $V$ is $A$. There are many bundles with this property and we shall analyze all such. We first begin by describing all sections of $\Cal P_{n-1}$. \lemma{5.1} A section $A\colon B\to {\Cal P}_{n-1}$ is equivalent to a line bundle $M\to B$ and an inclusion of $M^{-1}$ into $\Cal V_n$, or equivalently to sections of $M\otimes L^{-i}$ for $i=0,2,3,\ldots,n$ which do not all vanish at any point of $B$, modulo the diagonal action of $\Cee^*$. Under this correspondence, the normal bundle of $A$ in $\Cal P_{n-1}$ is isomorphic to $(\Cal V_n\otimes M)/\scrO_B$, where the inclusion of $\scrO_B$ in $\Cal V_n\otimes M$ corresponds to the inclusion of $M^{-1}$ into $\Cal V_n$. Finally, if either $h^1(\scrO_B) =0$ or $h^1(\Cal V_n\otimes M) = 0$, then the deformations of $A$ in $\Cal P_{n-1}$ are unobstructed. \endstatement \proof Let $A$ be a section, which we identify with its image in $\Cal P_{n-1}$. Of course, $A\cong B$ via the projection $p\: \Cal P_{n-1} \to B$. We have the inclusion of $\scrO_{\Pee\Cal V_n}(-1)$ in $p^*\Cal V_n$. Pulling back via $A$, we set $M = \scrO_{\Pee\Cal V_n}(1)|A$, which is a line bundle such that $M^{-1}$ is a subbundle of $p^*\Cal V_n|A = \Cal V_n$. An inclusion $$M^{-1}\to \Cal V_n = \scrO_B \oplus L^{-2} \oplus \cdots \oplus L^{-n}$$ is given by a nowhere vanishing section of $(M \otimes \scrO_B) \oplus (M \otimes L^{-2}) \oplus \cdots \oplus (M \otimes L^{-n})$, or equivalently by sections of the bundles $(M\otimes \scrO_B)$, $(M\otimes L^{-2})$, \dots, $(M\otimes L^{-n})$ which do not all vanish simultaneously, and these sections are well-defined modulo the diagonal $\Cee^*$ action. Conversely, a nowhere vanishing section of $\Cal V_n\otimes M$ defines an inclusion $M^{-1} \to \Cal V_n$ and thus a section of $\Cal P_{n-1}$, and the two constructions are inverse to each other. The normal bundle $N_{A/\Cal P_{n-1}}$ to $A$ in $\Cal P_{n-1}$ is just the restriction to $A$ of the relative tangent bundle $T_{\Cal P_{n-1}/B}$, and thus it is isomorphic to $(\Cal V_n\otimes M)/\scrO_B$. The deformations of the subvariety $A$ are unobstructed if every element of $H^0(N_{A/\Cal P_{n-1}})$ corresponds to an actual deformation of $A$. From the exact sequence $$0\to H^0(\scrO_B) \to H^0(\Cal V_n\otimes M) \to H^0(N_{A/\Cal P_{n-1}}) \to H^1(\scrO_B) \to H^1(\Cal V_n\otimes M),$$ we see that, if $H^1(\scrO_B)=0$, then every section of the normal bundle lifts to a section of $\Cal V_n\otimes M$, unique mod the image of $H^0(\scrO_B) = \Cee$, and thus gives an actual deformation of $A$. If $H^1(\Cal V_n\otimes M) = 0$, then viewing the deformations of $M$ as parametrized by $\Pic B$, if $M'$ is sufficiently close to $M$ in $\Pic B$, then $H^1(\Cal V_n\otimes M') = 0$ as well and by standard base change results the groups $H^0(\Cal V_n\otimes M')$ fit together to give a vector bundle over a neighborhood of $M$ in $\Pic B$. The associated projective space bundle then gives a smooth family of deformations of $A$ such that the associated Kodaira-Spencer map is an isomorphism onto $H^0(N_{A/\Cal P_{n-1}})$. Thus $A$ is unobstructed in this case as well. \endproof \definition{Definition 5.2} Let $A\: B\to {\Cal P}_{n-1}$ be a section, and let $(A,\Id)$ be the corresponding section of ${\Cal P}_{n-1} \times _BZ \to Z$. For all $a\in \Zee$, let $$V_{A,a} = (A,\Id)^*\Cal U_a.$$ For every pair $(A,a)$, the bundle $V_{A,a}$ is of rank $n$ and the restriction of $V_{A,a}$ to every fiber of $\pi$ is regular and semistable with trivial determinant. Furthermore, for all $a\in \Zee$, the section determined by $V_{A,a}$ is $A$. \enddefinition More generally, we could take any bundle ${\Cal U}$ over ${\Cal P}_{n-1}\times_BZ$ obtained by twisting ${\Cal U}_a$ by a line bundle on the universal spectral cover ${\Cal T}$ over ${\Cal P}_{n-1}$, and form $V_{A,{\Cal U}}=(A,\text{Id})^*{\Cal U}$ to produce a bundle with these properties. However, these will not exhaust all the possibilities in general. To describe all possible bundles $V$ corresponding to $A$, we shall need to define the spectral cover associated to $A$. \definition{Definition 5.3} Let $A\subseteq {\Cal P}_{n-1}$ be a section. The scheme-theoretic inverse image $\nu ^*A$ of $A$ in $\Cal T$ is a subscheme $C_A$ of $\Cal T$, not necessarily reduced or irreducible. The morphism $g_A = \nu|A\: C_A\to A\cong B$ is finite and flat of degree $n$. We call $C_A$ the {\sl spectral cover\/} associated to the section $A$. \enddefinition In the notation of (5.1), we shall show below that $C_A$ is smooth for $M$ sufficiently ample and for a general section corresponding to $M$. In general, however, no matter how bad the singularities of $C_A$, we have the following: \lemma{5.4} The restriction of $r$ to $C_A$ embeds $C_A$ as a subscheme of $Z$ which is a Cartier divisor. In fact, if $V$ is a vector bundle with semistable restriction to every fiber and $A$ is the associated section, then $C_A$ is the scheme of zeroes of $\det \Psi$, where $$\Psi\: \pi^*\pi_*(V\otimes \scrO_Z(\sigma)) \to V\otimes \scrO_Z(\sigma)$$ is the natural map. The line bundle $\scrO_Z(C_A)$ corresponding to $C_A$ is isomorphic to $\scrO_Z(n\sigma) \otimes \pi ^*M$, where $M$ is the line bundle corresponding to the section $A$. Moreover, the image of $C_A$ in $Z$ determines $A$. Finally, if $C\subset Z$ is the zero locus of a section of $\scrO_Z(n\sigma) \otimes \pi ^*M$ and the induced morphism from $C$ to $B$ is finite, then $C = C_A$ for a unique section $A$ of $\Cal P_{n-1}$. \endstatement \proof Let $i\colon C_A\to \Cal T$ be the natural embedding. We claim that $r\circ i\colon C_A\to Z$ is a scheme-theoretic embedding. To see this, recall that we have $\Cal T \subset {\Cal P}_{n-1}\times _BZ$ via $(\nu, r)$. In fact, from the defining exact sequence $$0 \to \Cal E \to \pi^*\pi_*\scrO_Z(n\sigma) \to \scrO_Z(n\sigma) \to 0,$$ we see that $\Cal T=\Pee\Cal E$ is a Cartier divisor in $\Pee(\pi^*\pi_*\scrO_Z(n\sigma)) ={\Cal P}_{n-1}\times _BZ$ defined by the vanishing of a section of $\pi_2^*\scrO_Z(n\sigma)\otimes \pi_1^*\scrO_{\Cal P_{n-1}}(1)$. Clearly, the image of $i(C_A)$ under the map $C_A\to \Cal T \to {\Cal P}_{n-1}\times _BZ$ is an embedding of $C_A$ in $A\times _BZ\cong Z$. Thus $r\circ i$ is an embedding of $C_A$ into $Z$. Moreover, $C_A$ is the restriction of $\Cal T\subset {\Cal P}_{n-1}\times _BZ$ to $A\times _BZ$, and thus $C_A$ is a Cartier divsior in $Z$. Essentially by definition, $C_A$ is defined by the vanishing of $\det \Psi$ (since this holds on every fiber $E_b$). Moreover, $\scrO_Z(C_A)$ is the restriction to $A\times _BZ$ of $\pi_2^*\scrO_Z(n\sigma)\otimes \pi_1^*\scrO_{\Cal P_{n-1}}(1)$, namely $\scrO_Z(n\sigma) \otimes \pi ^*M$. Since the hypersurface $\Cal T\subset {\Cal P}_{n-1}\times _BZ$ is the incidence correspondence, the line bundle $\scrO_{{\Cal P}_{n-1}\times _BZ}(\Cal T)$ restricts on every fiber $E$ of $\pi$ to $\scrO_E(np_0)$, and the effective divisor $\Cal T$ restricts to the tautological divisor in $|np_0|\times E$ corresponding to the inclusion $\Cal T\subset {\Cal P}_{n-1}\times _BZ$. Thus, by restriction, if $\scrO_Z(C_A)$ is the line bundle in $Z$ corresponding to the Cartier divisor $C_A$, then for every fiber $E=E_b$ of $\pi$, $\scrO_Z(C_A)|E\cong \scrO_E(np_0)$, and the section of $\scrO_E(np_0)$ defined by $C_A$ is $A(b)$. Thus the image of $C_A$ in $Z$ determines $A$. Finally, let $C$ be the zero locus of a section of $\scrO_Z(n\sigma) \otimes \pi ^*M$. Note that $$H^0(Z; \scrO_Z(n\sigma) \otimes \pi ^*M) = H^0(B; \pi_*(\scrO_Z(n\sigma) \otimes \pi ^*M)) = H^0(B; \Cal V_n \otimes M),$$ so that sections $s$ of $\scrO_Z(n\sigma) \otimes \pi ^*M$ mod $\Cee^*$ correspond to sections $s'$ of $\Cal V_n\otimes M$. Under this correspondence, $s'$ vanishes at a point of $B$ if and only if $s$ vanishes along the complete fiber $\pi^{-1}(b)$. Thus we see that the subschemes $C$ mapping finitely onto $B$ are in $1-1$ correspondence with sections $A$ of $\Cal P_{n-1}$ whose associated line bundle is $M$. \endproof We define $T_A= C_A\times _BZ\subseteq \Cal T\times _BZ$, and let $\rho _A\: T_A \to C_A$ be the natural map. There is an induced map $\nu _A\: T_A \to Z$ such that the following diagram is Cartesian: $$\CD T_A @>{\nu _A}>> Z\\ @V{\rho _A}VV @VV{\pi}V\\ C_A@>{g _A}>> B. \endCD$$ Thus, $T_A$ is an elliptic scheme over $C_A$ pulled back from the elliptic scheme $Z\to B$ via the natural projection mapping $C_A\to B$. Even if $C_A$ is smooth, however, $T_A$ is singular along the intersection of $C_A\times _BZ$ with $\Gamma\times _B\Gamma\subset Z\times _BZ$, at points corresponding to $\Gamma \cap C_A\subset Z$. If $\dim B =1$, the generic section $A$ will be such that $C_A\cap \Gamma =\emptyset$. However, if $\dim B \geq 2$ and $A$ is sufficiently ample, $C_A\cap \Gamma$ is nonempty. In the generic situation described in the last section, where $G_2$ and $G_3$ are smooth and meet transversally, the singularities of $T_A$ are locally trivial families of threefold double points. In general, if no component of $\Gamma$ is contained in $C_A$, the codimension of $C_A\cap \Gamma$ in $C_A$ is two and the codimension of the corresponding subset of $T_A$ is three. If a component of $\Gamma$ is contained in $C_A$, then the codimension of $C_A\cap \Gamma$ in $C_A$ is one and the codimension of the corresponding subset of $T_A$ is two. Note that $\Delta$ is a Cartier divisor in the complement of the subset of $T_A$ consisting of singular points of singular fibers lying over $C_A\cap \Gamma$. Let us examine the pullback to $T_A= C_A\times _BZ$ of the divisors in ${\Cal T}$. The section $\sigma \subset Z$ pulls back via $\nu_A^*$ to a section $\Sigma _A$ of the elliptic fibration $\nu _A\: T_A\to C_A$. Clearly $\Sigma _A =\nu_A^*\sigma = \Cal G|T_A$, where as in the last section $\Cal G$ is the pullback to $\Cal T\times _BZ$ of $\sigma \subset Z$ by the second projection. The diagonal $\Delta_0$ in $Z\times _BZ$ pulls back to a hypersurface in $T_A$, which is the restriction of $\Delta\subset \Cal T\times _BZ$ to $C_A\times _BZ = T_A$. We shall continue to denote this subvariety by $\Delta$. However $\Delta$ is not a Cartier divisor along the singular set of $T_A$. On the other hand, the restriction of $\rho_A$ to $\Delta$ is an isomorphism from $\Delta$ to $C_A$, so that in a formal sense $\Delta$ is a section. There is also the class $\zeta$, which is obtained as follows: take the class $\zeta$ on $\Cal P_{n-1}$, pull it back to $\Cal T$, and then restrict to $C_A$. In the notation of (5.1), this class is just $\alpha = c_1(M)$, pulled back from $B$. The remaining ``extra" class $r^*\sigma\times _BZ|T_A$ corresponds to $\sigma \cdot C_A=F$ in $Z$, and in particular it is pulled back from a class on $C_A$. Note that $F$ maps isomorphically to its image in $B$. Using $\nu_*r^*\sigma = \Cal H$, we see that the image of $F$ in $B$ corresponds to $A\cap \Cal H$. If $D$ is the divisor in $B$ corresponding to $A\cap \Cal H$ and $V$ is a bundle with semistable restriction to every fiber whose associated section $A(V)$ is $A$, then $V|E_b$ has $\scrO_E$ as a Jordan-H\"older quotient if and only if $b\in D$. The above classes, together with the pullbacks of classes from $B$, are the only divisor classes that exist ``universally" on $C_A\times _BZ = T_A$ for all sections $A$. Using these classes, let us realize the bundles $V_{A,a}$ as pushforwards from $T_A$. Note that, from the definition, it is not {\it a priori\/} clear that $(\nu_A)_*\scrO_{T_A}(\Delta-\Sigma _A)$ is locally free, since $\Delta$ need not be Cartier. \lemma{5.5} For every section $A$ of ${\Cal P}_{n-1}$ and for every $a\in \Zee$, we have $$V_{A,a}=(\nu_A)_*\scrO_{T_A}(\Delta-\Sigma _A-aF).$$ \endstatement \proof There is a commutative diagram, which is in fact a Cartesian square: $$\CD T_A @>>> {\Cal T}\times_BZ \\ @V{\nu_A}VV @VV{\nu\times \Id}V \\ Z @>{(A,\Id)}>> {\Cal P}_{n-1}\times_BZ. \endCD$$ Moreover, by definition $V_{A,a} = (A,\Id)^*(\nu\times \Id)_*\scrO_{\Cal T\times _BZ}(\Delta - \Cal G-aF)$. The morphism $\nu\times \Id$ is finite. Pulling back by the top horizontal arrow, the sheaf $\scrO_{\Cal T\times _BZ}(\Delta - \Cal G-aF)$ restricts to $\scrO_{T_A}(\Delta-\Sigma _A-aF)$. Thus, (5.5) is a consequence of the following general result: \enddemo \lemma{5.6} Let $$\CD X'@>{f}>> X\\ @V{\pi'}VV @VV{\pi}V\\ Y'@>{g}>> Y \endCD$$ be a Cartesian diagram of schemes, with $\pi$ a finite morphism. Let $\Cal S$ be a sheaf on $X$. Then the natural map $g^*\pi_*\Cal S \to (\pi')_*f^*\Cal S$ is an isomorphism. \endstatement \proof The question is local in $Y$ and $Y'$, so that we may assume that $Y =\Spec R$ and $Y' =\Spec R'$ are affine. Since $\pi$ and $\pi'$ are finite, and thus affine, we may thus assume that $X=\Spec S$ and $X' =\Spec S'$, with $S'= S\otimes _RR'$. Suppose that $\Cal S$ corresponds to the $S$-module $M$. Let $M_R$ be the $S$-module $M$, viewed as an $R$-module. The assertion of the lemma is the statement that $$(M_R)\otimes _RR' \cong (M\otimes _SS')_{R'}.$$ But $M\otimes _SS' = M\otimes _S(S\otimes _RR')$, and a standard argument now identifies $(M\otimes _S(S\otimes _RR'))_{R'}$ with $(M_R)\otimes _RR'$. This proves the lemma. \endproof Once we know that the sheaf $\scrO_{T_A}(\Delta-{\Sigma _A}-aF)$ pushes down to a vector bundle on $Z$, the same will be true for the twist of this sheaf by any line bundle on $C_A$. Conversely, we have the following: \proposition{5.7} Let $V$ be a vector bundle of rank $n$ on $Z$ such that $V|E_b$ is a regular semistable bundle with trivial determinant for every fiber $E_b$. Let $A=A(V)$ be the section determined by $V$ and let $C_A\to A$ be the induced spectral cover. Then there is a unique bundle $N$ on $C_A$, such that $V\cong (\nu_A)_*\left[\scrO_{T_A}(\Delta - \Sigma _A)\otimes \rho_A^*N\right]$. \qed \endstatement The proof of this result is similar to the proof of Part (ii) of Theorem 2.4 and will be omitted. Next we look at the deformation theory of $V$. \proposition{5.8} applying the Leray spectral sequence for $\pi\: Z \to B$ to compute $H^1(Z; Hom(V,V))$, there is an exact sequence $$0\to H^1(B; \pi_*Hom (V,V)) \to H^1(Z; Hom(V,V)) \to H^0(B; R^1\pi_*Hom (V,V)).$$ \roster \item"{(i)}" The first term is $H^1(\scrO_{C_A})$ and corresponds to first order deformations of a line bundle on the spectral cover $C_A$; \item"{(ii)}" If $L$ is not trivial, then $H^0(B; R^1\pi_*Hom (V,V))$ is the tangent space to $A$ in the space of all sections of $\Cal P_{n-1}$, and the restriction map $$H^1(Z; Hom(V,V)) \to H^0(B; R^1\pi_*Hom (V,V))$$ is the natural one which associates to a first order deformation of $V$ a first order deformation of the section $A(V)$. \item"{(iii)}" Suppose that $L$ is nontrivial and that $C_A$ is smooth, or more generally that $h^1(\scrO_{C_A})$ is constant in a neighborhood of $A$. Suppose also either that $h^1(\scrO_B) = 0$ or that $h^1(\Cal V_n\otimes M)=0$, which will hold as soon as $M$ is sufficiently ample. Then the local moduli space of deformations of $V$ is smooth of dimension equal to $h^1(Z; Hom(V,V))$. In other words, all first order deformations of $V$ are unobstructed. \endroster \endstatement \proof By construction $\pi_*Hom (V,V) = (g_A)_*\scrO_{C_A}$, and we leave to the reader the check that the inclusion $H^1(B; \pi_*Hom (V,V)) \to H^1(Z; Hom(V,V))$ corresponds to deforming the line bundle on $C_A$. Next, let us fix for a moment a regular semistable bundle $V$ over a single Weierstrass cubic $E$. Applying (1.5) with $S=\Cee[\epsilon]$, the dual numbers, for every deformation of $V$ over $S$, there is an induced morphism $S\to |np_0|$ which restricts over $S_{\text{red}}$ to $\zeta (V)$. Thus there is an intrinsic homomorphism from $H^1(ad(V))$ to the tangent space $H^0(\scrO_E(np_0))/\Cee\cdot \zeta (V)$ of $|np_0|$ at $\zeta(V)$. By (v) of Theorem 3.2, if $V$ is a regular semistable bundle, then there is an exact sequence $$0 \to \Cee \to H^1(W_{n-d}\spcheck\otimes W_d\spcheck) \to H^1(ad(V)) \to 0.$$ which identifies $H^1(ad(V))$ with the tangent space to $|np_0|$ at $\zeta(V)$. Using the parametrized version of this construction (Lemma 4.3, with $S$ equal to $\Cee[\epsilon]\times B$), there is an induced morphism from $H^0(R^1\pi_*ad (V))$ to $\Hom(\Cee[\epsilon]\times B, \Cal P_{n-1};A)$, the space of morphisms from $\Cee[\epsilon]\times B$ to $\Cal P_{n-1}$ extending the section $A$. This gives an isomorphism from $R^1\pi_*ad (V)$ to the relative tangent bundle $T_{\Cal P_{n-1}/B}$ restricted to $A$. As we have seen in Lemma 5.1, this restriction is just the normal bundle $N_{A/\Cal P_{n-1}}$ to $A$ in $\Cal P_{n-1}$. Clearly the map $H^0(R^1\pi _*ad(V)) \to H^0(N_{A/\Cal P_{n-1}})$ is the natural map from the tangent space of deformations of $V$ to the tangent space to deformations of the section $A$ in $\Cal P_{n-1}$. Now $Hom (V,V) = ad(V) \oplus \scrO_Z$, and so $R^1\pi_*Hom (V,V) = R^1\pi_*ad (V)\oplus L^{-1}$. Either $L^4$ or $L^6$ has a nonzero section, so that $L^{-1}$ has a nonzero section if and only if $L$ is trivial. Thus, if $L$ is not trivial, then $H^0(L^{-1}) =0$, and so $$H^0(B; R^1\pi_*Hom (V,V)) = H^0(R^1\pi_*ad (V))$$ as claimed in (ii). To prove (iii), begin by using Lemma 5.1 to find a smooth space $Y$ parametrizing small deformations of the section $A$, of dimension $h^0(N_{A/\Cal P_{n-1}})$. If $\Cal A \to Y$ is the total space of this family, there is an induced family of spectral covers $\Cal C \to Y$. By assumption, the relative Picard scheme $\Pic(\Cal C/Y)$ is smooth in a neighborhood of the fiber over $A$. Use this smooth space of dimension $h^1(\scrO_{C_A})+ h^0(N_{A/\Cal P_{n-1}})$ to find a family of bundles parametrized by a smooth scheme $S$, which is an open subset of $\Pic(\Cal C/Y)$ and thus is fibered over the open subset $Y$ of sections of $\Cal P_{n-1}$. This implies that the Kodaira-Spencer map of this family, followed by the map from $H^1(Z; Hom(V,V))$ to $H^0(B; R^1\pi_*Hom(V,V))$ is onto, and then that the Kodaira-Spencer map is an isomorphism onto $H^1(Z; Hom(V,V))$. Thus, the first order deformations of $V$ are unobstructed. \endproof \ssection{5.2. Relationship to the extension point of view.} Next we relate the description of bundles constructed out of sections $A$ of ${\Cal P}_{n-1}$ with the point of view of extensions. As usual, this will enable us to construct some of the bundles previously constructed via spectral covers, but not all. We have already constructed the bundles $\Cal W_k$ over $Z$ as well as the universal extension $\bold U(d)$, $1\le d<n$, which sits in an exact sequence $$0 \to \pi _2^*\Cal W_d\spcheck \otimes \pi _1^*\scrO_{\Cal P_{n-1,d}}(1) \to \bold U(d) \to \pi _2^*\Cal W_{n-d} \to 0.$$ Here the projective space $\Cal P_{n-1,d}$ of the vector space of extensions is identified with ${\Cal P}_{n-1}$, but, by Theorem 4.6, under this identification $$\scrO_{\Cal P_{n-1,d}}(1) \otimes \pi^*L =\scrO_{{\Cal P}_{n-1}}(1).$$ Finally, we have $$\bold U(d) = \Cal U_{1-d}\otimes \pi _1^*\scrO_{{\Cal P}_{n-1}}(1)\otimes L^{-1}.$$ Thus there is an exact sequence $$0 \to \pi _2^*\Cal W_d\spcheck \to \Cal U_{1-d} \to \pi _2^*\Cal W_{n-d}\otimes \pi _1^*\scrO_{{\Cal P}_{n-1}}(-1)\otimes L \to 0.$$ Given a section $A$ of $\Cal P_{n-1,d}={\Cal P}_{n-1}$ such that $\scrO_{\Cal P_{n-1,d}}(1)|A = M'$, we can pull back the defining extension for $\bold U(d)$ to obtain an extension $$0 \to \Cal W_d\spcheck \otimes \pi^*M' \to U_A \to \Cal W_{n-d} \to 0.$$ (Of course, $M'$ is $M\otimes L^{-1}$.) Conversely, suppose that we are given an extension of $\Cal W_{n-d}$ by $\Cal W_d\spcheck \otimes \pi^*M'$, where $M'$ is a line bundle on $B$ which we can write as $M\otimes L^{-1}$. In this case, by the Leray spectral sequence $$\gather H^1(\Cal W_{n-d}\spcheck \otimes \Cal W_d\spcheck \otimes \pi^*M') \cong H^0(R^1\pi _*(\Cal W_{n-d}\spcheck \otimes \Cal W_d\spcheck) \otimes M')\\= H^0(\Cal V_{n,d}\otimes M\otimes L^{-1})=H^0(\Cal V_n\otimes M). \endgather$$ Thus nontrivial extensions of $\Cal W_{n-d}$ by $\Cal W_d\spcheck \otimes \pi^*M'$ which restrict to nontrivial extensions on every fiber can be identified with sections of $\Cal P_{n-1,d}$ corresponding to the line bundle $M$. Finally, we see that, for $1\leq d\leq n-1$, we can write $V_{A,1-d}$ as an extension $$0 \to \Cal W_d\spcheck \to V_{A, 1-d} \to \Cal W_{n-d}\otimes \pi^*(M^{-1}\otimes L)\to 0.$$ We can also relate the deformation theory of $U_A$ above to the bundles $\Cal W_d$ and $\Cal W_{n-d}$. Thus, the tangent space to $\Ker\{\,(g_a)_*\: \Pic C_A \to \Pic B\,\}$ is $H^1(B; \pi _*(\Cal W_d\otimes \Cal W_{n-d})\otimes M^{-1}\otimes L)$, and the tangent space to deformations of the section $A$ is $H^0(B; R^1\pi _*(\Cal W_d\spcheck\otimes \Cal W_{n-d}\spcheck)\otimes M\otimes L^{-1})$, provided that $L$ is not trivial. \ssection{5.3. Chern classes and determinants.} Let $A$ be a section of $\Cal P_{n-1}$. Corresponding to $A$, there is the line bundle $M$ on $B$ which is the restriction to $A$ of $\scrO_{\Cal P_{n-1}}(1)$. We denote by $\alpha$ the class $c_1(M) \in H^2(B;\Zee)$. Our goal is to express the Chern classes of $V_{A,a}$ in terms of $\alpha$ and the standard classes on $Z$. We will also consider more general bundles arising from twisting by a line bundle on the spectral cover. First we shall determine the Chern classes of $V_{A,a}$. We begin with the following lemma: \lemma{5.9} Let $A$ be a section of $\Cal P_{n-1}$ corresponding to the inclusion of a line bundle $M^{-1}$ in $\Cal V_n$. Then, for $k\geq 0$, we have $p_*([A]\cdot \zeta ^k) =\alpha ^k\in H^{2k}(B;\Zee)$. \endstatement \proof Note that by definition $\zeta|_A=c_1(M)=\alpha$ when we identify $A$ and $B$ in the obvious way. It follows that $\zeta^k|_A=\alpha^k$. This means that $p_*([A]\cdot \zeta^k)=\alpha^k$. \endproof Using (5.9), we can compute the Chern classes $c_i(V_{A,a})$ by taking the formula for $c_i(\Cal U_a)$ and replacing $\zeta ^i$ by $\alpha^i$. Thus \theorem{5.10} Suppose that $A$ is a section of ${\Cal P}_{n-1}$ such that the corresponding line bundle $M$ has $c_1(M)=\alpha \in H^2(B)$ \rom(or $\Pic B$\rom). Then $$\ch(V_{A,a}) = e^{-\alpha}\fracwithdelims(){1 - e^{(a+n)L}}{1-e^L}-\frac{1 - e^{aL}}{1-e^L} +e^{-\sigma}(1-e^{-\alpha}).$$ Moreover, in $\pi^*\Pic B \subset \Pic Z$, $$\det (V_{A,a}) = -(n+a-1)\alpha + \left[an + \fracwithdelims(){n^2-n}{2}\right]L.\qed$$ \endstatement There is also a formula for $c(V_{A,a})$ which follows similarly from the formula for $c(\Cal U_a)$. Now let us consider the effect of twisting by a line bundle on the spectral cover. If $N$ is a line bundle on the spectral cover $C_A$ associated to $A$, let $$V_{A, 0}[N] = (\nu_A)_*\left[\scrO_{T_A}(\Delta - \Sigma _A )\otimes \rho_A^* N\right].$$ For example, suppose that $N$ is of the form $\scrO_{C_A}(-aF)\otimes g_A^*N_0$, where $N_0$ is a line bundle on $B$. Then $$V_{A, 0}[N] = V_{A, a}\otimes \pi^*N_0.$$ In particular, we see that if $N=\scrO_{C_A}(-aF)\otimes g_A^*N_0$, for some line bundle $N_0$ on $B$ and some integer $a$, then $$\ch(V_{A, 0}[N])= \left[e^{-\alpha}\fracwithdelims(){1 - e^{(a+n)L}}{1-e^L}-\frac{1 - e^{aL}}{1-e^L} +e^{-\sigma}(1-e^{-\alpha})\right]\cdot e^{c_1(N_0)}.$$ For more general line bundles $N$ on $C_A$, we can calculate the determinant of $V_{A, 0}[N]$. In what follows, we identify $\Pic B$ with a subgroup of $\Pic Z$ via $\pi^*$ and write the group law additively. \lemma{5.11} With $V_{A, 0}[N]$ as defined above, the following formula holds in $\Pic B$: $$c_1(V_{A, 0}[N]) = -(n-1)\alpha + \fracwithdelims(){n^2-n}{2}L + (g_A)_*c_1(N).$$ Thus, for a fixed section $A$ of $\Cal P_{n-1}$ and a fixed line bundle $\Cal N$ on $B$, the set of bundles $V$ on $Z$ which are regular semistable on every fiber, with $A(V) = A$ and $\det V =\pi^*\Cal N$ is a principal homogeneous space over $\Ker \{g_A{}_*\: \Pic C_A \to \Pic B\}$, which is a generalized abelian variety times a finitely generated abelian group. \endstatement \proof Since it is enough to compute the determinant in the complement of a set of codimension two, we may restrict attention to the open subset of $T_A$ where $\Delta$ is a Cartier divisor. Now it is a general formula that, for a Cartier divisor $D$ on $T_A$, $$c_1 \left[(\nu_A)_*\scrO_{T_A}(D)\right] = c_1 \left[(\nu_A)_*\scrO_{T_A}\right]+ (\nu_A)_*D.$$ Thus, applying this formula to $\scrO_{T_A}(\Delta -\Sigma _A)$ and $\scrO_{T_A}(\Delta -\Sigma _A)\otimes \rho_A^* N$, we see that $$c_1(V_{A, 0}[N]) = c_1(V_{A, 0}) + (\nu_A)_*\rho_A^* c_1(N).$$ But we have calculated $c_1(V_{A, 0}) = \dsize -(n-1)\alpha + \fracwithdelims(){n^2-n}{2}L$, and $(\nu_A)_*\rho_A^* c_1(N) = \pi^*(g_A)_*c_1(N)$ since $T_A = C_A\times _BZ$. Putting these together gives the formula in (5.11). \endproof If $\dim B \geq 2$ and $M$ is sufficiently ample, we we will see in the next subsection that the generalized abelian variety $\Ker \{g_A{}_*\: \Pic C_A \to \Pic B\}$ is in fact a finitely generated abelian group, with no component of positive dimension. Using (5.11), let us consider the following problem: Given the section $A$, when can we find a line bundle $N$ such that $V_{A, 0}[N]$ actually has trivial determinant? We are now in position to answer this question in this case if we consider twisting only by line bundles which exist universally for all spectral covers. \proposition{5.12} Given a section $A$, suppose that $N=\scrO_{C_A}(-aF)\otimes g_A^*N_0$ for a line bundle $N_0$ on $B$ and an integer $a$. Then $V_{A, 0}[N]$ has trivial determinant for some choice of an $N$ as above if at least one of the following conditions holds: \roster \item"{(i)}" $n$ is odd, \item"{(ii)}" $L$ is divisible by $2$ in $\Pic B$, or \item"{(iii)}" $\alpha\equiv L\bmod 2$ in $\Pic B$. \endroster \endstatement \proof It suffices to show that there exists an $a\in \Zee$ such that $\det (V_{A,a})$ is divisible by $n$. For then, for an appropriate line bundle $N_0$ on $B$, we can arrange that $V=V_{A,a}\otimes N$ has trivial determinant. By (5.10), we must have $$(a-1)\alpha \equiv \frac{n(n-1)}2L \bmod n.$$ In the first two cases we simply take $a\equiv 1 \bmod n$. Lastly, let us suppose that $n$ is even and that $L$ is not divisible by $2$. Then the condition $\dsize (a-1)\alpha \equiv \frac{n(n-1)}2L \bmod n$ is a nontrivial condition on $\alpha$. It is satisfied for the appropriate $a$ if $\alpha\equiv L\bmod 2$ in $\Pic B$. \endproof We leave it to the reader to write out necessary and sufficient conditions for the equation $\dsize (a-1)\alpha \equiv \frac{n(n-1)}2L \bmod n$ to have a solution in general. For a general line bundle $N$ on $C_A$, we can use the Grothendieck-Riemann-Roch theorem to calculate the higher Chern classes of $\ch(V_{A, 0}[N])$, but only in the range where $\Delta$ is a Cartier divisor. Thus, we are essentially only able to compute $c_2$ by this method for a general line bundle $N$: \proposition{5.13} Suppose that no component of $\Gamma$ is contained in $C_A$. Let $\ch_2$ be the degee two component of the Chern character. Then $$\gather \ch_2(V_{A, 0}[N])- \ch_2(V_{A,0})= \\ (\nu_A)_*\left(\left( \Delta -\Sigma _A + \frac12(\nu_A^*K_Z- K_{T_A})\right)\cdot (\rho_A)^*( N) \right)+(\pi_A)^*(g_A)_*\frac{( N)^2}{2}. \endgather$$ \endstatement \proof Working where $\Delta$ is Cartier, we can apply the Grothendieck-Riemann-Roch theorem to the local complete intersection morphism $\nu_A\: T_A\to Z$ to determine the Chern character of $V_{A,0} = (\nu_A)_*\scrO_{T_A}(\Delta - \Sigma _A)$: $$\ch (V_{A,a}) = (\nu_A)_*\left(e^{\Delta -\Sigma}\Todd(T_A/Z)\right),$$ valid under our assumptions through terms of degree two. Applying the same method to calculate the Chern character of $V_{A, 0}[N]$, we find that, at least through degree two, $$\ch (V_{A, 0}[N])- \ch (V_{A,0})= (\nu_A)_*\left((e^N-1)(e^{\Delta -\Sigma}\Todd(T_A/Z)\right).$$ Expanding this out gives (5.13). \endproof \ssection{5.4. Line bundles on the spectral cover.} In this section, we look at the problem of finding extra line bundles on the spectral cover $C_A$, under the assumption that $C_A$ is smooth and that $M$ is sufficiently ample. As we shall see, the discussion falls naturally into three cases: $\dim B =1$, $\dim B = 2$, $\dim B \geq 3$. First let us consider the case that $B$ is a curve, with $M$ arbitrary but $C_A$ assumed to be smooth, or more generally reduced. Let $A$ correspond to the line bundle $M$ on $B$. Given $V=V_{A, 0}[N]$, we seek $\det V$ and $c_2(V)$. First, by (5.11), working in $\Pic B$ written additively, $$\det V_{A, 0}[N]= -(n -1)M + \fracwithdelims(){n^2-n}{2}L+ (g_A)_*N.$$ Since $g_A{}_*\: \Pic C _A\to \Pic B$ is surjective in case $C_A$ is reduced, we can arrange that the determinant is in fact trivial, and then the line bundle $\scrO_{C_A}(D)$ is determined up to the subgroup $\Ker \{g_A{}_*\: \Pic C_A \to \Pic B\}$. If $C_A$ is smooth, then this subgroup is the product of an abelian variety and a finite group We may summarize this discussion as follows: \theorem{5.14} Suppose that $\dim B =1$. Given a section $A$ of $\Cal P_{n-1}$ such that $C_A$ is reduced, the set of bundles $V$ with trivial determinant such that $A(V) = A$ is a nonempty principal homogeneous space over $\Ker \{g_A{}_*\: \Pic C_A \to \Pic B\}$. The same statement holds if we replace the condition that $V$ has trivial determinant by the condition that the determinant of $V$ is $\pi ^*\lambda$ for some fixed line bundle $\lambda$ on $B$. \qed \endstatement The remaining Chern class is $c_2(V)$. In this case, in $H^4(Z; \Zee)$, with no assumptions on $\Gamma$, we have (as computed in \cite{3} in case $n=2$): \proposition{5.15} For every line bundle $N$ on $C_A$, $$c_2(V_{A,0}[N]) = c_2(V) = \sigma \cdot \alpha = \deg M.$$ \endstatement \proof First assume that $C_A$ is reduced. Write $N \cong \scrO_{C_A}(\sum _ip_i)$, where the $p_i$ are points in the smooth locus of $C_A$ which lie under smooth fibers. Thus $\rho_A^{-1}(p_i) = f_i$ is a smooth fiber of $T_A$. In this case, we can obtain $V_{A,0}[N]$ as a sequence of elementary modifications of the form $$0 \to V_{A,0}[N_j] \to V_{A,0}[N_{j+1}] \to (i_j)_*\lambda _j \to 0,$$ where $E_j$ is the fiber on $Z$ corresponding to $f_j\subset T_A$, $i_j\: E_j \to Z$ is the inclusion, and $\lambda _j = \scrO_{T_A}(\Delta - \Sigma _A)|f_j$ is a line bundle of degree zero. By standard calculations, $$c_2(V_{A,0}[N_j]) = c_2(V_{A,0}[N_{j+1}])$$ and so $c_2(V_{A,0}[N]) = c_2(V_{A,0}) = \sigma \cdot \alpha$. In case $C_A$ is not reduced, a similar argument applies, where we replace $p_i$ by a Cartier divisor whose support is contained in the smooth locus of $(C_A)_{\text{red}}$ and $E_j$ by a thickened fiber. \endproof \remark{Remark} On the level of Chow groups, the refined Chern class $\tilde c_2(V_{A,0}[N])$ essentially records the extra information coming from the natural map $\Pic C_A\to A^2(Z)$. \endremark \medskip Next we consider the case where $\dim B > 1$. First we have the following result, with no assumption on $C_A$, concerning the connected component of $\Pic C_A$. \lemma{5.16} Suppose that $\dim B \geq 2$ and that $M$ is sufficiently ample. More precisely, suppose that $$H^i(B; L^{-1}\otimes M^{-1}) = H^i(B; L\otimes M^{-1}) = \cdots = H^i(B; L^{n-1}\otimes M^{-1}) = 0$$ for $i=0,1$. Then the natural map from $H^1(Z; \scrO_Z)$ to $H^1(C_A; \scrO_{C_A})$ is an isomorphism. Finally, if in addition $L$ is not trivial, then the norm map from $\Pic^0C_A$ to $\Pic ^0B$ is surjective with finite kernel. Thus $\Ker \{g_A{}_*\: \Pic C_A \to \Pic B\}$ is a finitely generated abelian group. \endstatement \proof From the exact sequence $$0 \to \scrO_Z(-n\sigma) \otimes \pi^*M^{-1} \to \scrO_Z \to \scrO_{C_A} \to 0,$$ we see that there is a long exact sequence $$H^1(\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}) \to H^1(\scrO_Z) \to H^1(\scrO_{C_A}) \to H^2(\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}).$$ Applying the Leray spectral sequence to $\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}$, we have that $$H^i(\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}) = H^{i-1}(R^1\pi _*\left[\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}\right]).$$ Now, by duality, $$\gather R^1\pi_*\left[\scrO_Z(-n\sigma) \otimes \pi^*M^{-1}\right] = R^1\pi _*\scrO_Z(-n\sigma) \otimes M^{-1}\\ = \left(L^{-1}\oplus L\oplus \cdots \oplus L^{n-1}\right) \otimes M^{-1}. \endgather$$ Thus by our assumptions the map $H^1\scrO_Z) \to H^1(\scrO_{C_A})$ is an isomorphism. By applying the Leray spectral sequence to $\scrO_Z$, we see that there is an exact sequence $0\to H^1(\scrO_B) \to H^1\scrO_Z) \to H^0(L^{-1})$. As we saw in the proof of (ii) of (5.8), if $L$ is not trivial, then $H^0(L^{-1}) = 0$ and the pullback map $H^1(\scrO_B) \to H^1\scrO_Z)$ is an isomorphism. The last statement of the lemma is then clear. \endproof \lemma{5.17} If $M$ is sufficiently ample on $B$, then $C_A$ is an ample divisor in $Z$. \endstatement \proof Equivalently, we must show that for $M$ sufficiently ample on $B$, $\pi ^*M\otimes \scrO_Z(n\sigma)$ is ample. But $\scrO_Z(n\sigma)$ is relatively ample, and thus by a standard result $\pi ^*M\otimes \scrO_Z(n\sigma)$ is ample for $M$ sufficiently ample (compare \cite{10, p\. 161. (7.10)(b)} for the case where $\scrO_Z(n\sigma)$ is relatively very ample). \endproof \corollary{5.18} If $\dim B \geq 3$, $M$ is sufficiently ample, and $Z$ and $C_A$ are smooth, then $\Pic Z \cong \Pic C_A$. If $\dim B =2$, $M$ is sufficiently ample, and $Z$ and $C_A$ are smooth, then the restriction mapping $\Pic Z \to \Pic C_A$ is injective. \endstatement \proof This is immediate from the Lefschetz theorem and (5.17). \endproof \remark{Remark} If $\dim B = 2$ and $M$ is sufficiently ample, it is natural to expect an analogue of the Noether-Lefschetz theorem to hold: for generic sections $C_A$ of $\pi ^*M\otimes \scrO_Z(n\sigma)$, $\Pic Z \cong \Pic C_A$. However, in the next section, we will see how to construct sections $A$ such that the spectral cover $C_A$ is smooth but has larger Picard number than expected. \endremark \ssection{5.5. Symmetric bundles.} Next we turn to bundles with a special invariance property. \definition{Definition 5.19} Let $\iota\: Z \to Z$ be the involution which is $-1$ in every fiber. A bundle $V$ is {\sl symmetric\/} if $\iota^*V \cong V\spcheck$. \enddefinition We shall now analyze when a bundle $V$ is symmetric. We fix a section $A$, corresponding to the class $\alpha$ and denote $C_A, \nu _A$, $T_A$, $g_A$ simply by $C, \nu, T,g$. \proposition{5.20} For a suitable choice of $N\in \Pic C$ the bundle $V_{A,0}[N]$ is symmetric if and only if $g^*(L+\alpha) + nF$ is divisible by $2$ in $\Pic C$. In this case, for a fixed section $A$, the set of all symmetric bundles whose section is $A$ is a principal homogeneous space over the $2$-torsion in $\Pic C$. \endstatement \proof Suppose that $V=V_{A,0}[N] = \nu _*\left[\scrO_T(\Delta - \Sigma_A )\otimes \rho^*\scrO_{C}(N)\right]$, where $N$ is a divisor on $C$. For our purposes, since both $\iota^*V$ and $V\spcheck$ are bundles, they are isomorphic if and only if they are isomorphic outside the complement of a set of codimension two in $Z$. Thus, we shall work as if $\Delta$ is a Cartier divisor. There is an induced involution on $T$, also denoted by $\iota$, for which $\nu$ is equivariant. Thus $$\align \iota ^*V&= \iota^*\nu _*\left[\scrO_T(\Delta - \Sigma _A )\otimes \rho^*\scrO_{C}(N)\right]\\ &=\nu_*\iota^*\left[\scrO_T(\Delta - \Sigma _A)\otimes \rho^*\scrO_{C}(N)\right]. \endalign$$ Now $\iota^*\Sigma _A = \Sigma _A$ and $\iota ^*\rho^*\scrO_{C}(N) = \rho^*\scrO_{C}(N)$. One the other hand, $\iota ^*\Delta$ is linearly equivalent to $2\Sigma _A -\Delta$ on a generic fiber. This says that $$\iota ^*\Delta = 2\Sigma _A -\Delta + \rho^*D$$ for some divisor $D$ on $C$. To determine $D$, restrict both sides above to $\Sigma _A$ where $\iota$ acts trivially. We find that $D = 2\Delta \cdot \Sigma _A -2\Sigma _A^2$, viewed in the obvious way as a divisor class on $C$. Thus $$\iota ^*\Delta = 2\Sigma _A -\Delta + 2\rho^*D_0$$ where $D_0$ is the fixed divisor class $\Delta \cdot \Sigma _A -\Sigma _A^2$, viewed as a divisor on $C$. Here the main point will be the factor of $2$. However we note that $\Sigma _A^2 = -[L']$, where $L'= g^*L$ is the line bundle for the elliptic scheme $T$, and $$\Delta \cdot \Sigma _A = \Delta \cdot \nu^*\sigma = \nu^*(\nu_*\Delta )\cdot \sigma =\nu ^*(C\cdot \sigma),$$ which after pullback corresponds to the divisor class $F$ on $T$. (Here $\Delta = C\times _BC\subset C\times _BZ$, and so $\nu_*\Delta =C$ since $\nu$ is just the natural projection of $T=C\times _BZ$ to $Z$.) Next we calculate $V\spcheck$. Relative duality for the finite flat morphism $\nu$ says that, for every Cartier divisor $D$ on $T$, $\left[\nu _*\scrO_T(D)\right]\spcheck = \nu _*\left[\scrO_T(-D)\otimes K_{T/Z}\right]$, where $K_{T/Z} = K_T\otimes \nu^*K_Z^{-1}$ is the relative dualizing sheaf of the morphism $\nu$. Thus we must have $$\Sigma _A -\Delta -\rho ^*N + K_T-\nu ^*K_Z = \Sigma _A -\Delta + 2\rho^*D_0 +\rho ^*N.$$ Equivalently, we must have $$K_T-\nu ^*K_Z = 2\rho ^*N + 2\rho^*D_0= 2\rho ^*(N +[L'] + F).$$ Conversely, given that the above equality holds, the corresponding vector bundles will be symmetric. To see if this equality holds for the appropriate choice of $N$, we must calculate $K_T-\nu ^*K_Z$. Since $Z$ is an elliptic fibration, $K_Z = \pi ^*(K_B+L)$, and likewise $K_T = \rho ^*(K_C+L')$, where $L' = g^*L$. Thus $K_T-\nu ^*K_Z = \rho^*(K_C - g^*K_B)$. To calculate $K_C$, we use (5.4), which says that $K_C = K_Z+C|C = K_Z + \pi^*L + n\sigma |C$. On the other hand, $K_Z - \pi ^*K_B = \pi ^*L$. Restricting to $C$ gives: $$K_C - g^*K_B = g^*(L+\alpha) + nF.$$ Putting this together, we see that, if $V$ is symmetric, then we must have $\rho^*(g^*(L+\alpha) + nF)$ divisible by $2$ in $\rho^*\Pic C$, and conversely. Next we claim that $\rho^*\: \Pic C \to \Pic T$ is injective. It suffices to show that $\rho_*\scrO_T =\scrO_C$, for then $\rho_*\rho^*N = N$ for every line bundle $N$ on $C$. But by flat base change $g^*\pi_*\scrO_Z = \rho_*\nu ^*\scrO_Z = \rho_*\scrO_T$. Since $\pi_*\scrO_Z =\scrO_B$, we have that $g^*\pi_*\scrO_Z = g^*\scrO_B = \scrO_C = \rho_*\scrO_T$. Hence $\rho_*\scrO_T =\scrO_C$, and so $\rho^*$ is injective. Thus, $V$ is symmetric if and only if $g^*(L+\alpha) + nF$ divisible by $2$ in $\Pic C$. Moreover the set of possible line bundles $N$ for which $V_{A,0}[N]$ is symmetric is a principal homogeneous space over the $2$-torsion in $\Pic C$, as claimed. This concludes the proof of (5.20). \endproof If $\dim B \geq 3$, $Z$ and $C$ are smooth, and $M$ is sufficiently ample, then $g^*(L+\alpha) + nF$ is divisible by $2$ in $\Pic C$ if and only if $\pi^*(L+\alpha) + n\sigma$ is divisible by $2$ in $\Pic Z$. This can only happen if $n$ is even and $\alpha \equiv L \bmod 2$. A similar statement is likely to hold if $\dim B =2$ and $A$ is also assumed to be general. We can see the conditions $n$ is even and $\alpha \equiv L \bmod 2$ clearly in terms of extensions. In this case $n=2d$, and we can write $V_{A,1-d}$ as an extension $$0 \to \Cal W_d\spcheck \to V_{A, 1-d} \to \Cal W_d\otimes M^{-1}\otimes L\to 0.$$ Under the assumption that $M^{-1}\otimes L =M_0^{\otimes 2}$ for some line bundle $M_0$, we can write $V_{A, 1-d}\otimes M_0^{-1}$ as an extension of $\Cal W_d\otimes M_0$ by the dual bundle $\Cal W_d\spcheck \otimes M_0^{-1}$, and then check directly that the corresponding bundles are symmetric. \ssection{5.6. The case of the trivial section.} We turn to bundles which have reducible or non-reduced spectral covers. We begin with the extreme case of the trivial section $\bold o = \bold o_Z =\Pee \scrO_B \subset {\Cal P}_{n-1}$. To construct this section we take $M=\scrO_B$ and take a nowhere vanishing section of $\scrO_B$ and the zero section of $L^{-a}$ for all $a>0$. Since $M=\scrO_B$, the class $\alpha$ is zero. The spectral cover $C=C_{\bold o}\subset Z$ is simply the nonreduced scheme $n\sigma$, and the associated reduced subscheme $C_{\text{red}}$ is identified with $B$. The bundles associated to this section have the property that their restrictions to each fiber of $Z$ are isomorphic to $I_n(\scrO)$. Conversely, if we have such a bundle $V$ over $Z$, then the section it determines is $\bold o$. By our general existence theorem we immediately conclude: \corollary{5.21} For every $n\ge 1$ there is a vector bundle $V\to Z$ whose restriction to each fiber $E_b\subset Z$ is isomorphic to $I_n(\scrO_{E_b})$. \qed \endstatement The structure sheaf $\scrO_C$ is filtered by subsheaves with successive quotients $$L^{n-1}, L^{n-2}, \dots, \scrO_B.$$ The restriction of $\scrO_{C\times _BZ}(\Delta -\Sigma _A -aF)$ to $C_{\text{red}}\times _BZ\cong Z$, is isomorphic to $$\scrO_{C\times _BZ}(\Delta -\Sigma _A -aF)|(C_{\text{red}}\times _BZ) \cong \scrO_Z(\sigma -\sigma)\otimes L^a) =L^a.$$ From this it follows that $V_{\bold o, a}$ has a filtration by subbundles with successive quotients $L^{a+n-1}, L^{a+n-2}, \dots, L^a$. Consequently, $$\ch (V_{\bold o, a}) = \frac{e^{aL} - e^{(a+n)L}}{1-e^L},$$ which agrees with the formula in Theorem 5.10 since $\alpha = 0$. We have the inclusion $B=C_{\text{red}}\subset C$ and the projection $C\to B$ so that $\scrO_C$ splits as a module over $\scrO_B$ into $\Cal S\oplus \scrO_B$ with $\Cal S$ a locally free sheaf of rank $n-1$ over $\scrO_B$. From the filtration of $\scrO_C$ as an $\scrO_B$-module, we see that $\Cal S$ has a filtration with successive quotients $L^{n-1}, L^{n-2}, \dots,L$. Thus, $\Pic C \cong\Pic B \oplus H^1(\Cal S)$, and $H^1(\Cal S)$ is a vector group. In particular, as far as Chern classes are concerned, we may as well just twist by line bundles $N$ on $C$ which are pulled back from $B$. Even if the line bundle $N$ on $C$ is not pulled back from $B$, if $N_0$ is the restriction of $N$ to $C_{\text{red}} \cong B$, it is still clear that $V_{(\bold o, 0)}[N]$ has a filtration with successive quotients $L^{n-1}\otimes N_0, L^{n-2}\otimes N_0, \dots, L\otimes N_0$. We have $$\ch(V_{\bold o, 0}[N])=\frac{1 - e^{nL}}{1-e^L}\cdot e^{N_0}.$$ \remark{Remark} (1) Note that, unless $L$ is a torsion line bundle, the bundles $V_{\bold o, 0}[N]$ are unstable with respect to every ample divisor. \noindent (2) By contrast with (5.14), even if $\dim B =1$, we cannot always arrange trivial determinant for $V_{\bold o, 0}[N]$. \endremark \medskip If instead we try to construct $V_{\bold o, a}$ directly as a sequence of global extensions on $Z$, we run into the following type of question. Suppose for simplicity that $n=2$ and that $a=0$. In this case we try to find a bundle on $Z$ which restricts over every fiber $f$ of $Z$ to be the nontrivial extension of $\scrO_f$ by $\scrO_f$, in other words to $I_2$. We may as well try to write it as an extension of $\scrO_Z$ by the pullback of a line bundle $N$ on $B$. To do this we need a class $H^1(\pi^*N)$ whose restriction to every fiber is non-trivial. That is to say, we need an element in $H^1(\pi^*N)$ whose image under the natural map $\psi$ in the Leray spectral sequence (which is an exact sequence in this case) $$H^1(\pi _*\pi ^*N) \to H^1(\pi ^*N) \buildrel\psi\over\longrightarrow H^0(R^1\pi _*\pi ^*N) \to H^2(\pi _*\pi ^*N)$$ is a nowhere zero section of $R^1\pi _*\pi ^*N$. Of course, $\pi _*\pi ^*N \cong N$ and $R^1\pi _*\pi ^*N \cong N\otimes R^1\pi _*\scrO_Z= N\otimes L^{-1}$. Thus if there is to exist a nowhere vanishing section of $H^0(R^1\pi _*\pi ^*N)$, it must be the case that $N=L$. But we also need the condition that the map $H^1(\pi ^*L) \to H^0(L\otimes L^{-1})= H^0(\scrO_B)$ is surjective. This is not immediately obvious from the spectral sequence since there is no reason for $H^2(B;L)$ to vanish. Nevertheless, it follows from our construction of $V_{\bold o, 0}$ that the map $\psi$ is onto in the case $N=L$. Finally, the set of possible extensions is a principal homogeneous space over $H^1(B;L)$, which is identified with the kernel of the natural map $\Pic (2\sigma) \to \Pic B$. \ssection{5.7. Deformation to a reducible spectral cover.} For every choice of a rank $n > \dim B$ and for all sections $A$ of $\Cal P_{n-1}$ which correspond to a sufficiently ample line bundle, we have constructed vector bundles $V_{A,a} = V_{A,a}(n)$. In this subsection, we try to relate the $V_{A,a}(n)$ for various choices of $n$. To this end, let $\Cal H = \Cal P^{n-2} = \Pee (\scrO_B \oplus L^{-2} \oplus \cdots \oplus L^{-n+1}) \subset {\Cal P}_{n-1}$. We begin by considering what happens when the section $A$ lies in the subbundle $\Cal H$, but is otherwise generic. To insure that there are actually sections of $\Cal H$ as opposed to just rational sections, it is reasonable to assume that $n \geq \dim B +2$. A section $A$ of $\Cal H$ is given by a line bundle $M$ and by $n$ sections $\sigma_0, \dots, \sigma_{n-1}$ of $M, M\otimes L^{-2}, \dots, M\otimes L^{-(n-1)}$ which have no common zeroes. If $M$ is sufficiently ample, the section $A=A_0$ will then move in a family $A_t$ of sections of ${\Cal P}_{n-1}$, by choosing a nonzero section $\sigma_n$ of $M\otimes L^{-n}$ and considering the family defined by the sections $A_t=(\sigma_0, \dots, \sigma_{n-1}, t\sigma_n)$. Roughly speaking, $V_{A_0,a}(n)$ is obtained from the bundle $V'$ of rank $n-1$ corresponding to $A_0$, viewed as a section of $\Cal P_{n-2}$. Along each fiber $f$ we add a trivial $\scrO_f$ factor to the restriction of $V'$. This statement is correct as long as the restriction of $V'$ to the fiber does not itself contain an $\scrO_f$ factor, or more generally a summand of the form $I_d(\scrO_f)$ for some $d\leq n$. The simplest possibility would be that $V_{A_0,a}(n)$ is a deformation of $V_{A,a}(n-1)\oplus \scrO_Z$, but a calculation with Chern classes rules this out. Likewise, $V_{A_0,a}(n)$ is not a deformation of $V_{A,a}(n-1)\oplus \pi^*N$ for any line bundle $N$ on $B$. Instead, we shall see that $V_{A_0,a}(n)$ is a deformation of a suitable elementary modification of $V_{A,a}(n-1)\oplus \pi^*L^a$. Finally, we shall use the construction to check the Chern class calculations. To make this construction, it is best to begin by working universally again. We have the $n$-to-$1$ map $\nu\: \Cal T \to {\Cal P}_{n-1}$. Inside ${\Cal P}_{n-1}$, there is the smooth divisor $\Cal H = \Cal P_{n-2}$. Now in $\Cal T= \Cal T_{n-1}$ there is a smooth divisor $\Cal T'\cong \Cal T_{n-2}$ defined by the diagram $$\CD 0 @>>> \Cal E @>>> \pi ^*\pi _*\scrO_Z(n\sigma) @>>> \scrO_Z(n\sigma) @>>> 0\\ @. @AAA @AAA @AAA @.\\ 0 @>>> \Cal E' @>>> \pi ^*\pi _*\scrO_Z((n-1)\sigma) @>>> \scrO_Z((n-1)\sigma) @>>> 0. \endCD$$ We take $\Cal T' = \Pee (\Cal E') \subset \Pee (\Cal E) =\Cal T$. The restriction of $\nu$ to $\Cal T'$ defines the corresponding map $\Cal T_{n-2} \to \Cal P_{n-2}$, and in particular $\nu |\Cal T'$ has degree $n-1$. Clearly, we have an equality of smooth divisors in $\Cal T$: $$\nu ^*\Cal H = \Cal T' + r^*\sigma.$$ The intersection $\Cal T' \cap r^*\sigma$ is clearly the smooth divisor $\Cal P_{n-3} \subset r^*\sigma \cong \Cal P_{n-2}$; it lies over $\Cal P_{n-3}$. A local calculation shows that $\Cal T'$ and $ r^*\sigma$ meet transversally at the generic point of $\Cal P_{n-3}$ and thus everywhere. Note that $\Cal T_1\cong Z$, $r\: \Cal T_1 \to Z$ is the identity, and the intersection of $\Cal T_1$ and $r^*\sigma$ in $\Cal T_2$ is $\sigma \subset \Cal T_1$. This is compatible with the convention $\bold o \cong \Cal P_0 \cong B$. Let $\Cal D = \Cal T'\times _BZ$ and, as usual, let $F = r^*\sigma \times _BZ$. Then $F$ is a smooth divisor and $\Cal D$ is smooth away from the singularities of $\Cal T\times _BZ$. The divisors $\Cal D$ and $F$ meet in a reduced divisor $\Cal P_{n-3}\times _BZ$. We thus have an exact sequence: $$0 \to \scrO_{\Cal D + F} \to \scrO_{\Cal D}\oplus \scrO_F \to \scrO_{\Cal D\cap F} \to 0.$$ Tensoring the above exact sequence by the sheaf $\scrO_{\Cal T\times _BZ}(\Delta - \Sigma _A -aF)$, using the fact that $\Delta \cap F = \Sigma _A\cap F$, gives a new exact sequence $$0 \to \scrO_{\Cal D + F}(\Delta - \Sigma _A -aF) \to \scrO_{\Cal D}(\Delta - \Sigma _A -aF)\oplus \scrO_F(-aF)\to \scrO_{\Cal D\cap F}( -aF) \to 0.$$ (In a neighborhood of $F$, $\Delta$ is Cartier, and so the above sequence is still exact.) Of course, $F|F = -L|F$. Now apply $(\nu \times \Id)_*$ to the above exact sequence. To keep track of the ranks, we shall write $\Cal U_a(n)$ when we want to denote the appropriate vector bundle of rank $n$, and similarly for $V_{A,a}(n)$. (However, in the notation, $V_{A,a}(n-1)$ will be a general rank $(n-1)$-bundle but $V_{A,a}(n)$ will be the special rank $n$ bundle corresponding to a reducible section. Of course, this will not affect Chern class calculations.) We have: $$0 \to \Cal U_a(n)|\Cal P_{n-2}\times _BZ \to \Cal U_a(n-1) \oplus (L^a|\Cal P_{n-2}\times _BZ) \to L^a|\Cal P_{n-3}\times _BZ \to 0.$$ Let $A$ be a section of ${\Cal P}_{n-1}$ lying in $\Cal P_{n-2}$ and otherwise general. Pulling back the above exact sequence via $A$, we get an exact sequence relating the special rank $n$ bundle $V_{A,a}(n)$ with a general rank $(n-1)$-bundle $V_{A,a}(n-1)$ obtained by viewing $A$ as a section of $\Cal P_{n-2}$: $$0\to V_{A,a}(n) \to V_{A,a}(n-1)\oplus \pi ^*L^a \to (\pi ^*L^a)|D \to 0,$$ where $D$ is the divisor in $Z$ corresponding to $\Cal P_{n-3} \cap A$. In particular $D$ is pulled back from $B\cong A$. Thus we have realized the special bundle $V_{A,a}(n)$ as an elementary modification of $V_{A,a}(n-1)\oplus \pi ^*L^a$ along the divisor $D$. To calculate the cohomology class of $D$, note that the class of $\Cal P_{n-3}$ in $\Cal P_{n-2}$ is given by $\zeta - (n-1)L$ (by applying (4.15) with $n$ replaced by $n-1$), and so the class of $D$ is given by $p_*([A]\cdot (\zeta - (n-1)L))$. By (5.9), $$[D] = \alpha - (n-1)L. \tag5.22$$ For $M$ sufficiently ample and $A$ general, $D$ is a smooth divisor, and we get $V_{A,a}(n)$ by an elementary modification of the direct sum $V_{A,a}(n-1)\oplus \pi ^*L^a$ along $D$. Here, of course, the surjection from $V_{A,a}(n-1)$ to $\pi ^*L^a|D$ arises because on every fiber $f$ over a point of $D$, $V_{A,a}(n-1)$ has a trivial quotient $\scrO_f$. Note that, assuming we are the range where the calculations are correct, we obtain an inductive formula for $\ch V_{A,a}(n)$: $$\ch V_{A,a}(n) = \ch V_{A,a}(n-1) + \ch (L^a) -\ch (L^a|D).$$ Now from the exact sequence $$0 \to L^a\otimes \scrO_Z(-D) \to L^a \to L^a|D \to 0,$$ we see that $\ch (L^a|D) = \ch (L^a) - \ch (L^a\otimes \scrO_Z(-D))$, and thus using (5.22) $$\ch V_{A,a}(n)=\ch V_{A,a}(n-1)+e^{(a+n-1)L+\alpha}.$$ Note that this is consistent with the formula given in (5.10) for $\ch V_{A,a}$. This inductive picture must be modified for small values of $n$. For example, in case $\dim B =3$, a general section in $\Cal P_2$ degenerates to a rational section of $\Cal P_1$ plus some exceptional fibers, and there is a further problem in the passage from $\Cal P_1$ to $\Cal P_0 = \bold o$. However, we will not discuss these matters further. \ssection{5.8. Subsheaves of $V$ and reducible spectral covers.} \proposition{5.23} Let $V$ be a rank $n$ bundle on $Z$ whose restriction to every fiber is regular and semistable with trivial determinant. Then the spectral cover $C=C_A$ associated to $V$ is reduced and irreducible if and only if there is no subsheaf $V'\subset V$ whose restriction to the generic fiber is a semistable bundle of degree zero and rank $r$ with $0<r<n$, if and only if there is no quotient sheaf $V''$ of $V$ which is torsion free and whose restriction to the generic fiber is a semistable bundle of degree zero and rank $r$ with $0<r<n$. \endstatement \proof Clearly, $V$ has a subsheaf $V'$ as in the statement of the proposition if and only if it has a quotient sheaf $V''$ as described above. If $C$ is not reduced and irreducible, then there is a proper closed subvariety $C'\subset C$ which maps surjectively onto $B$ and is finite of degree $r,\ 0<r<n$ over $B$. We may assume that $C'$ is reduced. Let $T'=T\times _BC'$ be the corresponding subscheme of $T=T_A$. The surjection $\scrO_T \to \scrO_{T'}$ and the fact that $\nu =\nu_A$ is finite leads to a surjection $$V=(\nu\times \Id)_*\left[\scrO_T(\Delta - \Sigma _A)\otimes \rho^*N\right] \twoheadrightarrow (\nu\times \Id)_*\left[\scrO_{T'}(\Delta - \Sigma _A)\otimes \rho^*N\right]=V''.$$ By construction, $V''$ is a torsion free sheaf on $Z$ of rank $r$ with $0< r < n$. Restrict to a generic smooth fiber $\pi ^{-1}(b)=E_b$ of $\pi$ such that the fiber of the projection $C'\to B$ has $r$ distinct points $e_1, \dots, e_r \in E_b$ over $b$. By Lemma 5.6, the restriction of $V''$ to $E_b$ is a direct sum of the $r$ line bundles $\scrO_{E_b}(e_i-p_0)$, and in particular it is semistable (and in fact regular). Conversely, suppose that there is an exact sequence $$0 \to V' \to V \to V'' \to 0,$$ where both $V'$ and $V''$ are nonzero torsion free sheaves whose restrictions to a generic fiber are semistable. Let $r'$ be the rank of $V'$ and $r''$ be the rank of $V''$. After restricting to a nonempty Zariski open subset of $Z$, we may assume that $V'$ and $V''$ are locally free. Consider now the commutative diagram $$\minCDarrowwidth{.2 in} \CD 0 @>>> \pi^*\pi_*(V'\otimes \scrO_Z(\sigma)) @>>> \pi^*\pi_*(V\otimes \scrO_Z(\sigma)) @>>> \pi^*\pi_*(V''\otimes \scrO_Z(\sigma)) @>>> 0\\ @. @V{\Psi'}VV @V{\Psi}VV @V{\Psi''}VV @.\\ 0 @>>> V'\otimes \scrO_Z(\sigma) @>>> V\otimes \scrO_Z(\sigma) @>>> V''\otimes \scrO_Z(\sigma) @>>> 0. \endCD$$ By definition, $C$ is the Cartier divisor which is the scheme of zeroes of $\det\Psi$. On the other hand, we clearly have $\det \Psi = \det \Psi'\cdot \det \Psi''$. If $C'$ is the scheme of zeroes of $\det \Psi '$, and $C''$ is the scheme of zeroes of $\det \Psi ''$, then $C = C'+C''$ on a nonempty Zariski open subset of $Z$. Furthermore, $C'$ maps to $B$ with degree $r'$ and $C''$ maps to $B$ with degree $r''$, so that neither of $C', C''$ is trivial. It follows that the restriction of $C$ to a nonempty Zariski open subset of $Z$ is either nonreduced or reducible, and so the same is true for $C$ as well. \endproof Finally, let us remark that if $V$ is merely assumed to be regular and semistable on a generic fiber, so that $A(V)$ is just a rational section, the above proof still goes through. \section{6. Bundles which are not regular and semistable on every fiber.} Let $\pi\: Z \to B$ be an elliptic fibration with $\dim B =d$, and let $E_b =\pi^{-1}(b)$. In this section, we consider some examples of bundles $V$, such that $\det V$ has trivial restriction to each fiber, which fail to be regular or semistable on every fiber $E_b$. From the general principles mentioned in the introduction, it is reasonable to consider only those bundles whose restriction to the generic fiber is semistable. We shall further assume here that the restriction to the generic fiber is regular (this will exclude, for example, the tangent bundle of an elliptic fibration whose base $B$ has dimension at least two). Thus, we shall consider bundles $V$ such that, for a nonempty proper closed subset $Y$ of $B$ and for all $y\in Y$, either $V|E_y$ is unstable or it is semistable but not regular. There is an important difference between the case $\dim Y =d-1$ and $\dim Y<d-1$. In the first case, $V$ is not determined by its restriction to $\pi ^{-1}(B-Y)$ and can be obtained via elementary modifications from a ``better" bundle (or reflexive sheaf). In this case, there is a lot of freedom in creating such $V$ where $V|E_y$ is unstable along a hypersurface. By contrast, it is more difficult to arrange that $V|E_y$ is semistable but not regular along a hypersurface. If $\dim Y < d-1$, then, since $V$ is a vector bundle, it is determined by its restriction to $\pi ^{-1}(B-Y)$ and the behavior of $V$ is much more tightly controlled by the rational section $A(V)$ of $\Cal P_{n-1}$. Here the case where $V|E_y$ is unstable for $y\in Y$ (as well as the case where $V$ is reflexive but not locally free) corresponds to the case where $A(V)$ is just a quasisection, i\.e\. where the projection $A(V) \to B$ has degree one but is not an isomorphism. The case where $V|E_y$ is semistable but not regular for $y\in Y$ corresponds to the case where there are singularities in the spectral cover $C_A$, and $V$ is obtained by twisting by a line bundle on $C_A|B-Y$ which does not extend to a line bundle on $B$. As will be clear from the examples, a wide variety of behavior is possible, and we shall not try to give an exhaustive discussion of all that can occur. \ssection{6.1. Codimension one phenomena and elementary modifications.} First we shall discuss the phenomena which occur in codimension one, and which amount to generalized elementary modifications. As will be clear, when we make the most general elementary modifications, we lose control in codimension two on $B$. Thus for example many of the constructions lead to reflexive sheaves which are not locally free. For this reason, we shall concentrate to a certain extent on the case $\dim B = 1$, which will suffice for the generic behavior in codimension one when $\dim B$ is arbitrary. The first very general lemma says that, locally, every possible bundle with a given restriction to the generic fiber arises as an elementary modification. \lemma{6.1} Let $V$ be a vector bundle on $Z$ whose restriction to every fiber $E_b$ is semistable and whose restriction to the generic fiber is regular. Suppose that $A(V)=A$ is the section of $\Cal P_{n-1}$ corresponding to $V$. Let $$Y = \{\, b\in B: \text{ $V|E_b$ is not regular}\,\}.$$ Then $Y$ is a Zariski closed subset of $B$. For every $y\in Y$, there exists a Zariski neighborhood $\Omega$ of $y$ in $B$ and a morphism $\varphi\: V_{A,0}|\pi ^{-1}(\Omega)\to V|\pi ^{-1}(\Omega)$ which is an isomorphism over a nonempty Zariski open subset of $\Omega$. Moreover, we can choose a $\varphi$ which extends to a homomorphism $V_{A,0} \otimes \pi ^*M^{-1} \to V$, where $M$ is a sufficiently ample line bundle on $B$. More generally, suppose that $V$ is merely assumed to have regular semistable restriction to the generic fiber, so that $V|E_b$ may be unstable for some fibers. Then there exists a closed subset $X$ of $B$ of codimension at least two such that the section $A(V)$ extends over $B-X$ and, with $Y$ as above, for every $y\in Y-X$, there exists a Zariski neighborhood $\Omega$ of $y$ in $B$ and a morphism $\varphi\: V_{A,0}|\pi ^{-1}(\Omega)\to V|\pi ^{-1}(\Omega)$ which is an isomorphism over a nonempty Zariski open subset of $\Omega$. Finally, we can choose a $\varphi$ which extends to a homomorphism $V_{A,0} \otimes \pi ^*M^{-1} \to V|B-X$, where $M$ is a sufficiently ample line bundle on $B-X$. \endstatement \proof Let us first consider the case where the restriction of $V$ to every fiber is semistable. In this case the section $A=A(V)$ is defined over all of $B$. Consider the sheaf $\pi_*Hom (V_{A,0}, V)$. On $B-Y$, this sheaf is locally free of rank $n$. On a sufficiently small open set $\Omega$, we can thus find a section $\varphi$ of $\pi_*Hom (V_{A,0}, V)|\Omega$ which restricts to an isomorphism on a general fiber. Since this is an open condition, the set of points $b\in \Omega$ such that $\varphi$ fails to be an isomorphism on $E_b$ is a proper Zariski closed subset of $\Omega$, as claimed. Finally, if $M$ is sufficiently ample, then $\pi_*Hom (V_{A,0}, V)\otimes M$ is generated by its global sections. Choosing such a section which restricts to an isomorphism from $V_{A,0} \otimes \pi ^*M^{-1}|E_b$ to $V|E_b$ for a fiber $E_b$ defines a map $\varphi$ which extends to a homomorphism $V_{A,0} \otimes \pi ^*M^{-1}\to V$, as claimed. In case $V$ has unstable restriction to some fibers, the above proof goes through as long as we are able to define the section $A(V)$. Now the rational section of $\Cal P_{n-1}$ defined by $V$ extends to a closed irreducible subvariety of $\Cal P_{n-1}$, which we shall also denote by $A(V) =A$. The morphism $p|A\: A\to B$ is birational, and thus over the complement of a codimension two set $X$ in $B$ it is an isomorphism. Thus $A$ is a well-defined section over $B-X$, and so defines a bundle $V_{A,0}$ over $\pi ^{-1}(B-X)$. We may then apply the first part of the proof. \endproof Let $V$ be a vector bundle on $Z$ whose restriction to the generic fiber $E_b$ is semistable. Let $$Y = \{\, b\in B: \text{ $V|E_b$ is not semistable}\,\}.$$ Then $Y$ is a Zariski closed subset of $B$. Suppose that $W=\pi^{-1}(Y)\subset Z$. We can restrict $V$ to the elliptic fibration $W\to Y$. For simplicity, we shall assume that $W$ is irreducible (otherwise we would need to work one irreducible component at a time). By general theory, there exists a torsion free sheaf $\Cal S$ over $W$ and a surjection $V|W \to \Cal S$, such that at a generic point $w$ of $W$, the map $V|E_w \to \Cal S|E_w$ is the maximal destabilizing quotient of $V|E_w$. Let $i\: W\to Z$ be the inclusion and let $V'$ be the kernel of the surjection $V \to i_*\Cal S$. If $W$ is a hypersurface in $Z$, i\.e\. if $Y$ is a hypersurface in $B$, then $V'$ is a reflexive sheaf. However, if $W$ has codimension greater than one, $V'$ fails to be reflexive, and in fact $(V')\ddual = V$. For example, if $\dim B=1$, $W$ is a finite set of points. Choosing one such point $w$, we have that $V|E_w$ is unstable. Let $Q$ be the maximal destabilizing quotient sheaf for $V|E_w$, and suppose that $\deg Q = e< 0$. Then $V'$ fits into an exact sequence $$0 \to V' \to V \to i_*Q \to 0,$$ where $i$ is the inclusion of the fiber $E_w$ in $Z$. Such elementary modifications of $V$ are {\sl allowable\/} in the terminology of \cite{4}, \cite{5}. As opposed to the general construction of (6.1), allowable elementary modifications are canonical, subject to a choice of an irreducible component of $W$. For the above allowable elementary modification over an elliptic surface, we have $$c_2(V') = c_2(V) + e< c_2(V).$$ Thus an allowable elementary modification always decreases $c_2$. \lemma{6.2} A sequence of allowable elementary modifications terminates. The end result is a torsion free reflexive sheaf $V'$ such that the set $$\{\, b\in B: \text{ $V'|E_b$ is not semistable}\,\}$$ has codimension at least two. \endstatement \proof We shall just write out the proof in the case $\dim B = 1$. In this case, by (6.1), we can fix a bundle $V_0 = V_{A,0}\otimes \pi^*M^{-1}$ for some section $A$, together with a morphism $\varphi\: V_0 \to V$ which is an isomorphism over a general fiber. Thus $\det \varphi$ defines an effective Cartier divisor, not necessarily reduced, supported on a union of fibers of $\pi$. Denote this divisor by $D$. Clearly $D$ is the pullback of a divisor $\bold d$ on $B$, and thus has a well-defined length $\ell$, namely the degree of $\bold d$. We claim that every sequence of allowable elementary modifications has length at most $\ell$. This is clearly true if $\ell =0$, since then $V_0\to V$ is an isomorphism and every fiber of $V$ is already semistable. Since a sequence of allowable elementary modifications will stop only when the restriction of $V$ to every fiber is semistable, we will get the desired conclusion. Let $V'$ be an allowable elementary modification of $V$ at the fiber $E_w$. We claim that $\varphi$ factors through the map $V'\to V$. In this case, it follows that $E_w$ is in the support of $D$. Thus, if $\varphi'\: V_0 \to V'$ is the induced map, then $(\det \varphi ') = D-E_w$, which has length $\ell -1$, and we will be done by induction on the length $\ell$. It suffices to prove that the induced map $V_0 \to i_*Q$ is zero in the above notation. Equivalently, we must show that the induced map $V_0|E_w \to Q$ is zero. But $V_0|E_w$ is semistable and $\deg Q <0$, and so we are done. \endproof As a corollary, we have the following Bogomolov type inequality: \corollary{6.3} Let $V$ be a vector bundle on $Z$ such that the restriction of $V$ to a generic fiber $E_b$ is regular and semistable. Suppose that $\dim B = d$. Then, for every ample divisor $H$ on $B$, $c_2(V)\cdot \pi^*H^{d-1} \geq 0$. Moreover, equality holds if and only if $V$ is semistable in codimension one and the line bundle $M$ corresponding to the rational section $A(V)$ is a torsion line bundle. Finally, $M$ is a torsion line bundle if and only if either the rational section $A(V) =\bold o$ or $L$ is a torsion line bundle and $M$ is a power of $L$. \endstatement \proof We may assume that $H$ is very ample. By choosing a general curve which is a complete intersection of $d-1$ divisors linearly equivalent to $H$, we can further assume that $\dim B = 1$, and must show that $c_2(V) \geq 0$. Since an allowable elementary modification strictly decreases $c_2$, we can further assume that the restriction of $V$ to every fiber is semistable. Choose a nonzero map $V_0 \to V$, where $V_0$ is regular semistable on every fiber. Defining $Q$ by the exact sequence $$0 \to V_0 \to V \to Q \to 0,$$ $Q$ is a torsion sheaf supported on some (possibly nonreduced) fibers whose restriction to a $b\in B$ has a filtration by degree zero sheaves on $E_b$. It then follows that $c_2(V) = c_2(V_0)$. Now if $M$ is the line bundle corresponding to the section $A(V)$ of $\Cal P_{n-1}$, then by (5.15) $c_2(V_0) = \deg M$. On the other hand, at least one of $M, M\otimes L^{-2}, \dots, M\otimes L^{-n}$ has a nonzero section. Thus, for some $i=0, 2, \dots, n$, $\deg M \geq i\deg L$. Now $\deg L \geq 0$, and $\deg L = 0$ only if $L$ is a torsion line bundle. Thus, $\deg M \geq 0$, and $\deg M = 0$ only if $i=0$, in which case $M$ is trivial, or $L$ is torsion and there is a nowhere vanishing section of $M\otimes L^{-i}$. In all cases $M$ is a torsion line bundle and we have proved the statements of the lemma. \endproof \remark{Remark} (1) If $c_2(V)\cdot \pi^*H^{d-1} =0$ above, in other words we have equality, it follows that the rational section $A(V)$ is actually a section. \smallskip \noindent (2) If $A$ is a rational section and $A \neq \bold o$, we get better inequalities along the lines of $$c_2(V)\cdot\pi ^*H^{d-1} \geq 2L\cdot H^{d-1},$$ since we must have nonzero sections of $M\otimes L^{-i}$, $i=0,2, \dots, n$ for at least two values of $i$. If $A$ is a section, then except for a small number of exceptional cases we will actually have $c_2(V)\cdot\pi ^*H^{d-1} \geq (d+1)L\cdot H^{d-1}$. \endremark \medskip The process of taking allowable elementary modifications is in a certain sense reversible: we can begin with a bundle $V_0$ such that the restriction of $V_0$ to every fiber is semistable and introduce instability by making elementary modifications. Let us first consider the case where $\dim B =1$. At the first stage, fixing a fiber $E_b$ and a stable sheaf $Q$ on $E_b$ of positive degree, we seek a surjection $V_0|E_b \to Q$. To analyze when such surjections exist is beyond the scope of this paper. However, in case $V_0|E_b$ is regular and $Q=W_k$, then we have seen in Section 3 that such a surjection always exists; indeed, the set of all surjections is an open subset in $\Hom (V, W_k)$ which has dimension $n$. Note however that while allowable elementary modifications are canonical, their inverses are not. To be able to continue to make elementary modifications along the same fiber, we would also have to analyze when there exist surjections from $V|E_b$ to $Q$, where $V$ is a rank $n$ bundle on $E_b$ of degree zero, $Q$ is a torsion free sheaf of rank $r<n$ on $E_b$, and $\mu(Q)$ is larger than the maximum of $\mu(\Cal S)$ as $\Cal S$ ranges over all proper torsion free subsheaves of $V|E_b$. In case $\dim B > 1$, further complications can ensue in codimension two. For example, suppose that $V_0$ has regular semistable restriction to every fiber of $\pi$. Let $D$ be a divisor in $B$ and let $W=\pi^{-1}(D)$, with $\pi'=\pi|W$. Even though we can find a surjection $V|E_b \to W_k$ for every $b\in D$, we can only find a global surjection $V|W \to \Cal W_k\otimes (\pi')^*N$, for some line bundle $N$ on $D$, under special circumstances. We can find a nonzero such map in general, but it will vanish in general in codimension two, leading to a reflexive but not locally free sheaf. We turn next to the issue of bundles which are semistable on every fiber, but which are not regular in codimension one. It turns out that we do not have the freedom that we did before in introducing instability on a fiber; there is a condition on the spectral cover in order to be able to make a bundle not be regular. (See \cite{3}, \cite{6} for the rank two case.) We shall just state the result in the case where $\dim B =1$. The result is that, if the spectral cover is smooth, it is not possible to create a non-regular but semistable bundle over any fiber. \proposition{6.4} Let $\dim B =1$ and let $V$ be a vector bundle over $Z$ whose restriction to every fiber is semistable and whose restriction to the generic fiber is regular. Let $A=A(V)$ be the corresponding section and $C=C_A$ be the spectral cover. If $b\in B$ and $C$ is smooth at all points lying over $b\in B$, then $V|E_b$ is regular. \endstatement \proof Using Lemma 6.1, write $V$ as a generalized elementary modification $$0 \to V_0 \to V \to Q \to 0,$$ where $V_0$ is regular and semistable on every fiber, and $Q$ is a torsion sheaf supported on fibers. Looking just at the part of $Q$ which is supported on $E_b$, this sheaf (as a sheaf on $Z$) has a filtration whose successive quotients are direct images of torsion free rank one sheaves of degree zero on $E_b$. By induction on the length of $Q$, as in the proof of Lemma 6.2, it will suffice to show the following: if $V_0$ has regular semistable restriction to $E_b$, if $i\: E_b \to Z$ is the inclusion, and if $C$ is smooth over all points lying over $b$, then for every exact sequence $$0 \to V_0 \to V \to i_*\lambda \to 0,$$ where $\lambda$ is a rank one torsion free sheaf on $E_b$ of degree zero, $V|E_b$ is again regular. After shrinking $B$, we can assume that $V_0$ is regular everywhere and that $V_0 \to V$ is an isomorphism away from $b$. It will suffice to show that $V\spcheck|E_b$ is regular. There is the dual exact sequence $$0 \to V\spcheck \to V_0\spcheck \to i_*\lambda^{-1} \to 0.$$ By assumption, $\dim \Hom (V_0\spcheck, i_*\lambda^{-1}) = \dim \Hom (V_0\spcheck|E_b, \lambda^{-1}) = 1$. Thus there is a unique possible elementary modification. On the other hand, there is a unique point $b'\in C$ lying above $b$ and corresponding to the surjection $V_0\spcheck|E_b \to \lambda^{-1}$. Since by assumption $b'$ is a smooth point of $C$, it is a Cartier divisor, and the ideal sheaf of $b'$ is the line bundle $\scrO_C(-b')$. Now we know that $V_0\spcheck$ is of the form $(\nu\times \Id)_*\left[\scrO_C(\Delta - \Sigma)\otimes \rho^*N\right]=V_{A,0}[N]$ for a line bundle $N$ on $C$. Let $i'$ be the inclusion of the fiber over $b'$ (which is just $E_b$) into $T$. Applying $(\nu\times \Id)_*$ to the exact sequence $$0 \to \scrO_C(\Delta - \Sigma)\otimes \rho^*(N\otimes \scrO_C(-b')) \to \scrO_C(\Delta - \Sigma)\otimes \rho^*N \to (i')_*\lambda^{-1} \to 0,$$ we get an exact sequence $$0 \to V_{A,0}[N\otimes \scrO_C(-b')] \to V_0\spcheck \to i_*\lambda^{-1} \to 0.$$ By the uniqueness of the map $V_0\spcheck \to i_*\lambda^{-1}$, it then follows that $$V \spcheck = V_{A,0}[N\otimes \scrO_C(-b')]$$ and in particular it is regular. Thus the same is true for $V$. \endproof \remark{Remark} (1) Of course, Proposition 6.4 gives conditions in case $\dim B > 1$ as well. \noindent (2) The condition that $C$ is singular at the point corresponding to $b$ and $V_0 \to \lambda$ is not a sufficient condition for there to exist an elementary modification such that the result is not regular over $E_b$. \endremark \ssection{6.2. The tangent bundle of an elliptic surface.} As an example of the preceding discussion, we analyze the tangent bundle of an elliptic surface. Let $\pi\: Z \to B$ be an elliptic surface over the smooth curve $B$, with $g(B) = g$. We suppose that $Z$ is generic in the following sense: $Z$ is smooth, the line bundle $L$ has positive degree $d$, so that the Euler characteristic of $Z$ is $12d$, all the singular fibers of $\pi$ are nodal curves (and thus there are $12d$ such curves), and the $j$-function $B\to \Pee^1$ has generic branching behavior in the sense of \cite{6, p\. 63}. The assumption of generic branching behavior implies that the Kodaira-Spencer map associated to the deformation $Z$ of the fibers of $\pi$ is an isomorphism at the curves with $j=0, 1728, \infty$ and that the Kodaira-Spencer map vanishes simply where it fails to be an isomorphism. By the Riemann-Hurwitz formula, if $b$ is equal to the number of points where the Kodaira-Spencer map is not an isomorphism, then $b= 10d+2g-2$ \cite{6, p\. 68}. Quite generally, we have the following lemma: \lemma{6.5} Let $\pi\: Z\to B$ be a smooth elliptic surface, and suppose that $V$ is a a vector bundle on $Z$ whose restriction to a general fiber is $I_2$. Then there is an exact sequence $$0 \to \pi^*M_1\to V \to \pi^*M_2\otimes I_X \to 0,$$ where $M_1$ and $M_2$ are line bundles on $B$ and $X$ is a zero-dimensional local complete intersection subscheme of $Z$. Here $\det V = \pi ^*(M_1\otimes M_2)$ and $c_2(V) =\ell (X)$. \endstatement \proof By assumption, $\pi_*V =M_1$ is a rank one torsion free sheaf on $B$, and thus it is a line bundle. We have the natural map $\psi\: \pi^*\pi_*V =\pi^*M_1\to V$. If this map were to vanish along a divisor, the divisor would have to be a union of fibers. But this is impossible since the induced map $$\pi _*\pi^*M_1 = M_1 \to \pi_*V = M_1$$ is the identity. Thus $\psi$ only vanishes in codimension two. The remaining statements are clear. \endproof Of course, in the case of the tangent bundle, we can identify this sequence precisely as follows: \lemma{6.6} With $\pi\: Z\to B$ a smooth elliptic surface as before, there is an exact sequence $$0 \to T_{Z/B} \to T_Z \to \pi ^*T_B \otimes I_X \to 0.$$ Here $T_{Z/B}=L^{-1}$ is the sheaf of relative tangent vectors and $I_X$ is the ideal sheaf of the $12d$ singular points of the singular fibers. \endstatement \proof Begin with the natural map $T_Z \to \pi ^*T_B$. This map is surjective except at a singular point of a singular fiber, where it has the local form $$h_1\frac{\partial}{\partial z_1} + h_2\frac{\partial}{\partial z_2} \mapsto (z_2h_1+ z_1h_2)\frac{\partial}{\partial t}.$$ Thus the image of $T_Z$ in $\pi ^*T_B$ is exactly $\pi ^*T_B \otimes I_X$. The kernel of the map $T_Z \to \pi ^*T_B$ is by definition $T_{Z/B}$, which can be checked directly to be a line bundle in local coordinates. Moreover, $T_{Z/B}$ is dual to $K_{Z/B}=L$, and thus $T_{Z/B} =L^{-1}$. \endproof \corollary{6.7} If $E$ is a singular fiber of $\pi$, there is an exact sequence $$0 \to n_*\scrO_{\tilde E} \to T_Z|E \to \frak m_x \to 0,$$ where $n\: \tilde E \to E$ is the normalization and $x$ is the singular point of $E$. In particular $T_Z|E$ is unstable. If $E$ is a smooth fiber where the Kodaira-Spencer map is zero, then $T_Z|E \cong \scrO_E\oplus \scrO_E$. For all other fibers $E$, $T_Z|E\cong I_2$. \endstatement \proof If $E$ is a singular fiber, then by restriction we have a surjection $T_Z|E \to \frak m_x$. The kernel must be a non-locally free rank one torsion free sheaf of degree one, and thus it is isomorphic to $ n_*\scrO_{\tilde E}$. For a smooth fiber $E$, restricting the tangent bundle sequence to $E$ gives an exact sequence $$0\to \scrO_E \to T_Z|E \to \scrO_E \to 0,$$ such that the coboundary map $$\theta\: H^0(\scrO_E) = H^0(N_{E/Z})\to H^1(\scrO_E) = H^1(T_E)$$ is the Kodaira-Spencer map. This map is nonzero, then, if and only if $T_Z|E\cong I_2$, and it is zero if and only if $T_Z|E\cong \scrO_E\oplus \scrO_E$. \endproof To go from $T_Z$ to one of our standard bundles, begin by making the allowable elementary modifications along the singular fibers, by taking $V'$ to be the kernel of the induced map $T_Z \to \bigoplus _x(i_x)_*\frak m_x$. Here the sum is over the the singular points, i\.e\. the $x\in X$, and $i_x$ is the inclusion of the singular fiber containing $x$ in $Z$. Note that $c_2(V') =0$, so that no further allowable elementary modifications are possible, and the restriction of $V'$ to every fiber is semistable. Let $F$ be the union of the singular fibers. Thus as a divisor on $Z$, $F =\pi^*\bold f$, where $\bold f$ is a divisor on $B$ of degree $12d$ which is a section of $L^{12}$. If $I_F$ is the ideal of $F$, then there is an inclusion $I_F\subset I_X$ and thus an inclusion $\pi ^*T_B \otimes I_F \subset \pi ^*T_B \otimes I_X$. Clearly $V'$ is the result of pulling back the extension $T_Z$ of $\pi ^*T_B \otimes I_X$ by $L^{-1}$ via the inclusion $\pi ^*T_B \otimes I_F \subset \pi ^*T_B \otimes I_X$. Thus there is an exact sequence $$0 \to \pi^*L^{-1} \to V' \to \pi^*(T_B \otimes \scrO_B(-\bold f)) \to 0.$$ Taking the map $$\gather \Ext^1(\pi ^*T_B \otimes I_F, L^{-1}) =H^1(\pi ^*T_B^{-1} \otimes\scrO_Z(F)\otimes L^{-1}) \\ \to H^0(R^1\pi_*(\pi ^*T_B^{-1} \otimes\scrO_Z(F)\otimes L^{-1})) = H^0(B; K_B\otimes L^{-2}\otimes \scrO_B(\bold f)), \endgather$$ and using the fact that $\scrO_B(\bold f)\cong L^{12}$, we see that the extension restricts to the trivial extension over a section of $K_B\otimes L^{10}$, and thus at $10d+2g-2$ points, confirming the numerology above. Note that the passage from $T_Z$ to $V'$ was canonical. Next we want to go from $V'$ to a bundle $V_0$ which is regular semistable on every fiber, and thus is isomorphic to $I_2$ on every fiber. We claim that a further elementary modification of $V'$ will give us back a bundle which restricts to $I_2$ on every fiber. Quite generally, suppose that $V'$ is given as an extension $$0 \to \pi^*L_1\to V' \to \pi^*L_2 \to 0,$$ where the image of the extension class in $H^0(B; L_2^{-1}\otimes L_1\otimes L^{-1})$ vanishes simply at $k$ points $x_1, \dots, x_k$. After twisting $V'$ by the line bundle $\pi^*L_2^{-1}$, we may assume that $L_2$ is trivial. Thus in the case where we began with the tangent bundle, and after relabeling $V'$, we wind up with a bundle $V'$ which fits into an exact sequence $$0 \to \pi^*L_1\to V' \to \scrO_Z \to 0,$$ where $L_1 = K_B\otimes L^{11}$. The extension class for $V'$ defines an element of $H^1(Z; \pi^*L_1)$. Via the Leray spectral sequence, there is a homomorphism from $H^1(Z; \pi^*L_1)$ to $H^0(B; R^1\pi_*\scrO_Z\otimes L_1) = H^0(B; L_1\otimes L^{-1})$. Thus there is a section of $L_1\otimes L^{-1}$, well-defined up to a nonzero scalar, and it defines a homomorphism $\pi^*L \to \pi^*L_1$ and thus a homomorphism $H^1(\pi^*L) \to H^1(\pi^*L_1)$. Consider the commutative diagram $$\CD H^1(B; L) @>>> H^1(\pi^*L) @>>> H^0(R^1\pi_*\pi^*L) = H^0(\scrO_B)\\ @VVV @VVV @VVV\\ H^1(B; L_1) @>>> H^1(\pi^*L_1) @>>> H^0(R^1\pi_*\pi^*L_1) = H^0(L_1\otimes L^{-1}). \endCD$$ The induced map $H^0(\scrO_B) \to H^0(L_1\otimes L^{-1})$ is just the given section of $L_1\otimes L^{-1}$. We have seen in \S 5.6 that there is a class $\xi_0 \in H^1(\pi^*L)$ mapping to $1\in H^0(\scrO_B)$. Since the map $H^1(B; L) \to H^1(B; L_1)$ is surjective, we can modify $\xi_0$ by an element in $H^1(B; L)$ so that its image in $H^1(\pi^*L_1)$ is the same as the extension class for $V'$, and the resulting element $\xi$ of $H^1(\pi^*L)$ is unique up to adding an element of the kernel of the map $H^1(B; L) \to H^1(B; L_1)$. Let $V_0$ be the extension of $\scrO_Z$ by $\pi^*L$ corresponding to $\xi$. Thus $V_0$ is some bundle of the form $V_{\bold o, 0}[N]$. There is an induced map of extensions $$\CD 0 @>>> \pi^*L_1 @>>> V' @>>> \scrO_Z @>>> 0\\ @. @| @VVV @VVV @.\\ 0 @>>> \pi^*L_1 @>>> V_0\otimes \pi^*(L_1\otimes L^{-1}) @>>> \pi^*(L_1\otimes L^{-1}) @>>> 0. \endCD$$ Thus there is an exact sequence $$0 \to V' \to V_0\otimes \pi^*(L_1\otimes L^{-1}) \to \bigoplus _i\scrO_{E_{x_i}} \to 0,$$ and we have realized the tangent bundle as obtained from $V_0$ by elementary modification and twisting. Of course, we can construct many other bundles this way, starting from $V_0$, not just the tangent bundle. Begin with $V_0$ which has restriction $I_2$ to every fiber. Normalize so that there is an exact sequence $$0 \to \pi^*L \to V_0 \to \scrO_Z \to 0$$ as in \S 5.6. Here $L =\pi_*V_0$. The bundle $\pi^*L$ is destabilizing. Choose $r$ fibers $E_{x_i}$ lying over $x_i\in B$, where we make elementary modifications by taking the unique quotient $\scrO_{E_{x_i}}$ of $V_0|E_{x_i}$. The result is a new bundle $V'$. The subbundle $\pi^*L$ still maps into $V'$, in fact we continue to have $L=\pi_*V'$, and the quotient is $\pi^*\scrO_B(-\bold r)$, where $\bold r$ is the divisor $\sum _ix_i$ of degree $r$ on $B$. The bundle $V'$ is the pullback of the extension $V_0$ by the morphism $\pi^*\scrO_B(-\bold r) \to \pi^*\scrO_B$. In particular, by reversing the arguments above, we see that the restriction of the extension to $E_{x_i}$ becomes split. Thus $V'|E_{x_i} \cong \scrO_{E_{x_i}} \oplus \scrO_{E_{x_i}}$ and the restriction of $V'$ to all other fibers is $I_2$. Note that $\pi^*L$ continues to destabilize $V'$. Choose $s$ fibers lying over points $y_j\in B$ distinct from the $x_i$, and let $\bold s$ be the divisor $\sum _jy_j$. Choose rank one torsion free sheaves $\mu_j$ on $E_{y_j}$ of degree $d_j > 0$ and surjections from $I_2$ to $\mu _j$. (Such surjections always exist.) Take the bundle $V$ defined to be the kernel of the given surjection $V' \to \bigoplus _j\mu _j$. Now $\det V = (d- r-s)f$ and $c_2(V) =\sum _j\deg \mu _j$. The bundle $\pi^*L$ no longer maps into $V$, since the composed morphism $\pi^*L|E_{y_j} \to \mu_j$ is nontrivial for every $j$. In fact, $\pi^*(L\otimes \scrO_B(-\bold s))$ maps to $V$, and $\pi_*V = L\otimes \scrO_B(-\bold s)$. Note that this subbundle fails to be destabilizing exactly when $2(d-s) < d-r-s$, or equivalently $d+r< s$. In this case, for a suitable ample divisor $H$ as defined in \cite{6}, $V$ is $H$-stable. \ssection{6.3. Quasisections and unstable fibers.} For the rest of this section, we shall assume that $V$ is regular and semistable in codimension one and consider the phenomena that arise in higher codimension. Over a Zariski open subset of $B$, we have defined $A(V)$, and it extends to a subvariety of $\Cal P_{n-1}$ mapping birationally to $B$, in other words to a {\sl quasisection\/} of $\Cal P_{n-1}$. Of course, if the restriction of $V$ to every fiber is semistable, then $A(V)$ is a section. \remark{Question} Suppose that $V$ is a vector bundle over $Z$ and that there exists a closed subset $Y$ of $B$ of codimension at least two such that, for all $b\notin Y$, $V|E_b$ is semistable. Suppose further that $A(V)$ is actually a section. Does it then follow that $V|E_b$ is semistable for all $b\in B$? \endremark \medskip For the remainder of this subsection, we shall assume that $A(V)$ is an honest quasisection, in other words that the morphism $A(V) \to B$ is not an isomorphism, and see what kind of behavior is forced on $V$. For example, if $n \leq \dim B$, then with a few trivial exceptions there are no honest sections of $\Cal P_{n-1}$ and we are forced to consider quasisections. We will analyze the case where $\dim B =2$ and see that two kinds of behavior are possible: either $V$ has unstable restriction to some fibers or $V$ fails to be locally free at finitely many points $Z$. For example, suppose that $1\leq d\leq n-1$ and consider $V_{A,1-d}$, defined over the complement of a set of codimension $2$ in $B$. Then as we have seen in \S 5.2, $V_{A, 1-d}$ is given as an extension of $\Cal W_{n-d}\otimes \pi^*(M^{-1} \otimes L)$ by $\Cal W_d\spcheck$. This extension extends over $B$, but it induces the split extension of $W_{n-d}$ by $ W_d\spcheck$ wherever the section of $\Cal V_n\otimes M$ vanishes. Assume that $\dim B =2$ and let $s$ be a section of $\Cal V_n \otimes M$ which vanishes simply at finitely many points, but which is otherwise generic. The corresponding quasisection $A=A(V)$ will contain a line inside the full fiber of $\Cal P_{n-1}$ at these points, which is a $\Pee^{n-1}$, and will simply be the blowup of $B$ over the corresponding points. Pulling back the $\Pee^{n-1}$-bundle $\Cal P_{n-1}$ by the morphism $A\to B$, we get an honest section over $A$. Let $\tilde Z = Z\times _BA$. Clearly $\tilde Z$ is the blowup of $Z$ along the fibers over the exceptional points of $B$, and the exceptional divisors of $\tilde Z \to Z$ are of the form $\Pee^1\times E_b$, where the $\Pee^1$ is linearly embedded in the $\Pee^{n-1}$ fiber. The section $A$ of $\Cal P_{n-1}\times _BA$ defines a vector bundle $\tilde \Cal U_a \to \tilde Z$ for every $a\in \Zee$. To decide what happens over the exceptional points of $B$, we need to understand the restriction of $\tilde \Cal U_a$ to the exceptional fibers $\Pee^1\times E_b$. Of course, this is just the restriction of the universal bundle $U_a$ defined over $\Pee^{n-1}\times E_b$ to the subvariety $\Pee^1\times E_b$. Thus we need to know the restriction of $U_a$ to $\Pee^1\times \{e\}$. We shall be able to find this restriction in case $-(n-2) \leq a \leq 1$, but for arbitrary $a$ we shall further need to assume that the $\Pee^1$ is a generic line in $\Pee^{n-1}$. \proposition{6.8} Let $E$ be a smooth elliptic curve and let $e\in E$. Suppose that $-(n-2) \leq a \leq 1$. Then $$U_a|\Pee ^{n-1}\times \{e\} \cong \cases \scrO_{\Pee ^{n-1}}^{1-a}\oplus \scrO_{\Pee ^{n-1}}(-1)^{n-1+a}, &\text{if $a \neq 1$ or $e\neq p_0$;}\\ \scrO_{\Pee ^{n-1}} \oplus \Omega^1_{\Pee^{n-1}}, &\text{if $a = 1$ and $e= p_0$,} \endcases$$ where $\Omega ^1_{\Pee^{n-1}}$ is the cotangent bundle of $\Pee^{n-1}$. \endstatement \proof Let $i_e$ be the inclusion of $\Pee ^{n-1}$ in $\Pee ^{n-1}\times E$ via the slice $\Pee ^{n-1}\times \{e\}$. Then $$i_e^*(\nu \times \Id)_*(\Delta - G-aF) = \nu _*\scrO_T(F_e - aF_{p_0}) = \nu _*r^*\scrO_E(e-ap_0).$$ Set $d=1-a$, then $U_a =\bold U(d) \otimes \pi_1^*\scrO_{\Pee^{n-1}}(-1)$. Thus, for $e\in E$, the restriction of $\pi_2^*W_d\spcheck$ to $\Pee^{n-1}\times \{e\}$ is trivial, and similarly for $\pi_2^*W_{n-d}$, and the defining exact sequence $$0 @>>> \pi _2^*W_d\spcheck @>>> U_a @>>> \pi_2^*W_{n-d} \otimes \pi_1^*\scrO_{\Pee^{n-1}}(-1) @>>> 0$$ restricts to the exact sequence $$ 0 @>>> \scrO_{\Pee^{n-1}}^d @>>> U_a |\Pee^{n-1}\times \{e\} @>>> \scrO_{\Pee^{n-1}}(-1)^{n-d} @>>> 0.$$ Since $\Ext ^1(\scrO_{\Pee^{n-1}}(-1)^{n-d},\scrO_{\Pee^{n-1}}^d) = H^1(\scrO_{\Pee^{n-1}}(1) )^{d(n-d)} =0$, this extension splits and we see that $$\align U_a |\Pee^{n-1}\times \{e\} &\cong \scrO_{\Pee^{n-1}}^d \oplus \scrO_{\Pee^{n-1}}(-1)^{n-d}\\ &\cong \scrO_{\Pee^{n-1}}^{1-a} \oplus \scrO_{\Pee^{n-1}}(-1)^{n+a-1}. \endalign$$ Now suppose that $a=1$. In this case $U_1 |\Pee^{n-1}\times \{e\} = \nu _*r^*\scrO_E(e-p_0)$, and thus $h^0(U_1 |\Pee^{n-1}\times \{e\})$ is zero if $e\neq p_0$ and one if $e=p_0$. We have the elementary modification $$0 \to U_1|\Pee^{n-1}\times \{e\} \to U_0 |\Pee^{n-1}\times \{e\} \to \scrO_H \to 0,$$ where $H$ is a hyperplane in $\Pee^{n-1}$. Thus we may write $$0 \to U_1 |\Pee^{n-1}\times \{e\}\to \scrO_{\Pee^{n-1}} \oplus \scrO_{\Pee^{n-1}}(-1)^{n-1}\to \scrO_H \to 0.$$ Clearly $h^0(U_1 |\Pee^{n-1}\times \{e\})=0$ if and only if the induced map $\scrO_{\Pee^{n-1}} \to \scrO_H$ is nonzero, or equivalently onto. In this case, we can choose a summand $\scrO_{\Pee^{n-1}}$ of $\scrO_{\Pee^{n-1}} \oplus \scrO_{\Pee^{n-1}}(-1)^{n-1}$ such that the map $\scrO_{\Pee^{n-1}} \oplus \scrO_{\Pee^{n-1}}(-1)^{n-1} \to \scrO_H$ is zero on the factor $\scrO_{\Pee^{n-1}}(-1)^{n-1}$ and is the obvious map on the first factor. Thus the kernel is $\scrO_{\Pee^{n-1}}(-1)^n$. In the remaining case, corresponding to $e=p_0$ and $a=1$, the map $\scrO_{\Pee^{n-1}} \oplus \scrO_{\Pee^{n-1}}(-1)^{n-1} \to \scrO_H$ is zero on the first factor. Now $H\cong \Pee^{n-2}$, and modulo automorphisms of $\scrO_{\Pee^{n-1}}(-1)^{n-1}$ there is a unique surjection $\scrO_{\Pee^{n-1}}(-1)^{n-1} \to \scrO_H$. We must therefore identify the kernel of this surjection with $\Omega ^1_{\Pee^{n-1}}$. Begin with the Euler sequence $$0 \to \Omega ^1_{\Pee^{n-1}} \to \bigoplus _{i=1}^n\scrO_{\Pee^{n-1}}(-1) \to \scrO_{\Pee^{n-1}} \to 0.$$ After a change of basis in the direct sum, we can assume that the right hand map restricted to the $n^{\text{th}}$ factor vanishes along $H$. Thus there is an induced surjection $$\bigoplus _{i=1}^{n-1}\scrO_{\Pee^{n-1}}(-1) \to \scrO_{\Pee^{n-1}}/\scrO_{\Pee^{n-1}}(-1) =\scrO_H$$ whose kernel is $\Omega ^1_{\Pee^{n-1}}$, as claimed. \endproof We remark that, in case $\dim B$ is arbitrary, $a=1$ and $A$ is a quasisection corresponding to a simple blowup of $B$, then one can show directly from (6.8) that $V_{A,1}$ does not extend to a vector bundle over $Z$. When we are not in the range $-(n-2) \leq a \leq 1$, we do not identify explicitly the bundle $U_a|\Pee^{n-1}\times \{e\}$, except in case $n=2$. However, the next result identifies its restriction to a generic line. \proposition{6.9} Let $\ell \cong \Pee ^1$ be a line in $\Pee ^{n-1}$, and suppose that $\ell$ is not contained in any of the one-dimensional family of hyperplanes $H_e$. Write $a = a' + nk$, where $-(n-2) \leq a' \leq 1$. Then $$U_a|\ell\times E \cong (U_{a'} \otimes \pi_1^*\scrO_{\Pee^1}(-k))|\Pee^1\times E.$$ In particular $$U_a|\ell\times \{e\} \cong \cases \scrO_{\Pee ^1}(-k)^{1-a'}\oplus \scrO_{\Pee ^1}(-k-1)^{n-1+a'}, &\text{if $a' \neq 1$ or $e\neq p_0$;}\\ \scrO_{\Pee ^1}^{n-2}(-k-1) \oplus \scrO_{\Pee ^1}(-k) \oplus \scrO_{\Pee ^1}(-k-2), &\text{if $a' = 1$ and $e= p_0$.} \endcases$$ \endstatement \proof Let $C$ be the preimage of $\ell$ in $T$. If $\ell$ is not contained in any of the hyperplanes $H_e$, then it will meet each $H_e$ in exactly one point. Thus the map $r|C\: C \to E$ has degree one, and $F_{p_0}\cdot C = p_0$. We claim that, under the morphism $\nu \: E \to \Pee ^1$, $\scrO_{\Pee ^1}(1)$ pulls back to $\scrO_E(np_0)=nF_{p_0}|C$. To see this, let $\nu^*\scrO_{\Pee^{n-1}}(1)=\zeta\in \Pic T$. Then the class of $\nu^*\ell$ lies in $\zeta ^{n-1}$. Now $T= \Pee\Cal E$ with $c_1(\Cal E) = -np_0$. Thus, in $A^{n-1}(T)$, $$\zeta ^{n-1} =r^*(np_0)\cdot \zeta^{n-2}.$$ Hence $\zeta |C = r^*(np_0)|C = nF_{p_0}|C$. Write $a = a'+nk$ with $-(n-2)\leq a' \leq 1$. Then $$\align &(\nu \times \Id)_*\scrO_{C\times E}(\Delta -G -aF_{p_0}) =\\ &= (\nu \times \Id)_*\left(\scrO_{C\times E}(\Delta -G -a'F_{p_0}) \otimes \pi _1^*\scrO_E(-nkp_0)\right)\\ &=(\nu \times \Id)_* \left(\scrO_{C\times E}(\Delta -G -a'F_{p_0})\otimes (\nu \times \Id) ^*\scrO_{\Pee ^1}(-k)\right)\\ &=(\nu \times \Id)_*\scrO_{C\times E}(\Delta -G -a'F_{p_0}) \otimes \pi_1^*\scrO_{\Pee ^1}(-k), \endalign$$ proving the first claim. The second statement follows from the special case $-(n-2)\leq a \leq 1$ proved in (6.8), and the well-known fact (which follows from the conormal sequence) that $\Omega ^1_{\Pee^{n-1}} |\Pee^1 \cong \scrO_{\Pee^1}(-1)^{n-2} \oplus \scrO_{\Pee^1}(-2)$. \endproof Now we can analyze what happens to $V_{A,a}$ when $\dim B =2$ and $A$ is a quasisection, under a slight genericity condition on $A$, generalizing the case (for $\dim B$ arbitrary) where $-(n-2) \leq a \leq 0$: \theorem{6.10} Suppose that $\dim B =2$. Let $A$ be a quasisection of $\Cal P_{n-1}$, and suppose that $a\not \equiv 1 \bmod n$. Suppose that $A$ is smooth and is the blowup of $B$ at a finite number of points $b_1, \dots, b_r$, and that the image of the exceptional $\Pee^1$ is a generic line in the fiber $\Pee^{n-1}$ as in \rom{(6.9)}, in other words it is not contained in one of the hyperplanes $H_e$. Then the rank $n$ bundle $V_{A,a}$, which is defined on $Z-\bigcup _iE_{b_i}$, extends to a vector bundle over $Z$, which we continue to denote by $V_{A,a}$. The restriction of $V_{A,a}$ to a fiber $E_{b_i}$ is the unstable bundle $W_d\spcheck \oplus W_{n-d}$, where $a = a'+nk$ with $-(n-2)\leq a' \leq 1$, and $d=1-a'$. \endstatement \proof By assumption, $A$ is the blowup of $B$ at a finite number of points $b_1, \dots, b_r$, where the quasisection $A$ contains a $\Pee^1$ lying in the $\Pee^{n-1}$-fiber of $p\: \Cal P_{n-1} \to B$. As we have defined earlier, let $\tilde Z = Z\times _BA$, so that $\tilde Z$ is a blowup of $Z$ at the fibers $E_{b_i}$. Let $D_i \cong \Pee^1\times E_{b_i}$ be the exceptional divisor of the blowup $q\: \tilde Z \to Z$ over $E_{b_i}$. There is a section of $\tilde B\to A$ corresponding to the inclusion of $A$ in $\Cal P_{n-1}$, and hence by pulling back $\Cal U_a$ there is a bundle corresponding to $A$, which we shall denote by $\tilde V$. Using (6.8) and (6.9), the restiction of $\tilde V$ to an exceptional divisor $D_i =\Pee^1\times E_{b_i}$, which is the same as the restriction of $\Cal U_a$, namely $U_a$, fits into an exact sequence $$0 \to \pi_2^*W_d\spcheck \otimes \pi _1^*\scrO_{\Pee ^1}(-k+1) \to \tilde V|D_i \to \pi_2^*W_{n-d} \otimes \pi _1^*\scrO_{\Pee ^1}(-k)\to 0.$$ Make the elementary modification along the divisor $D_i$ corresponding to the surjection $\tilde V|D_i \to \pi_2^*W_{n-d} \otimes \pi _1^*\scrO_{\Pee ^1}(-k)$. The result is a new bundle $V'$ over $\tilde Z$, such that over $D_i$ we have an exact sequence $$0 \to \pi_2^*W_{n-d} \otimes \pi _1^*\scrO_{\Pee ^1}(-k+1)\to V' |D_i \to \pi_2^*W_d\spcheck \otimes \pi _1^*\scrO_{\Pee ^1}(-k+1) \to 0.$$ Now, since $H^1(E_{b_i}; W_d \otimes W_{n-d}) = H^1(\scrO_{\Pee^1}) =0$, it follows from the K\"unneth formula that $$\Ext^1(\pi_2^*W_d\spcheck \otimes \pi _1^*\scrO_{\Pee ^1}(-k+1), \pi_2^*W_{n-d} \otimes \pi _1^*\scrO_{\Pee ^1}(-k+1)) = 0.$$ Thus $V'\otimes \scrO_{\tilde Z}(-(k+1)D_i)|D_i = \pi_2^*(W_d\spcheck \oplus W_{n-d})$. It follows by standard blowup results that $q_*V'\otimes \scrO_{\tilde Z}(-(k+1)\sum _iD_i)$ is locally free on $Z$ and its restriction to each fiber $E_{b_i}$ is $W_d\spcheck \oplus W_{n-d}$. This completes the proof. \endproof Finally we must deal with the case $a\equiv 1 \bmod n$. \theorem{6.11} Suppose that $\dim B =2$. Let $A$ be a quasisection of $\Cal P_{n-1}$, and suppose that $a\equiv 1 \bmod n$. Suppose that $A$ is smooth and is the blowup of $B$ at a finite number of points $b_1, \dots, b_r$, and that the image of the exceptional $\Pee^1$ is a generic line in the fiber $\Pee^{n-1}$ as in \rom{(6.9)}, in other words it is not contained in one of the hyperplanes $H_e$. Then the rank $n$ bundle $V_{A,a}$, which is defined on $Z-\bigcup _iE_{b_i}$, extends to a reflexive non-locally free sheaf on $Z$, which we continue to denote by $V_{A,a}$. The sheaf $V_{A,a}$ is locally free except at the points $\sigma \cap E_{b_i}$. Near such points, $V_{A,a}$ has the local form $$R^{n-2}\oplus M,$$ where $R=\Cee\{z_1, z_2, z_3\}$, and $M$ is the standard rank two reflexive non-locally free sheaf given by the exact sequence $$0 \to R\to R^3 \to M \to 0,$$ where the map $R\to R^3$ is given by $1\mapsto (z_1, z_2, z_3)$. \endstatement \proof We shall just work near a single fiber $E_b = E_{b_i}$ for some $i$. Thus let $\tilde Z$ be the blowup of $Z$ along $E_b$, with exceptional divisor $D \cong \Pee^1\times E_b$. The basic birational picture to keep in mind is the following: if we blow up the subvariety $\Pee^1\times \{p_0\}\subset D$, we get a new exceptional divisor $D_1$ in $Z_1 = \operatorname{Bl}_{\Pee^1\times \{p_0\}}\tilde Z$. Here $D_1\cong \Pee(\scrO_{\Pee^1} \oplus \scrO_{\Pee^1}(-1))$, and so $D_1$ is isomorphic to the blowup $\Bbb F_1$ of $\Pee^2$ at one point. The proper transform $D'$ of $D$ in $Z_1$ meets $D_1$ along the exceptional divisor in $D_1$, and can be contracted in $Z_1$. The result is a new manifold $Z_2$, isomorphic to the blowup of $Z$ at the point $\sigma \cap E_b$, where $D_1$ blows down to the exceptional divisor $P$ in $Z_2$. The quasisection $A$ defines a section of the pullback of $\Cal P_{n-1}$ to $B$, and thus a bundle $\tilde V$ over $\tilde Z$, which we can then pull back to $Z_1$. The next step is to show that, after appropriate elementary modifications, $\tilde V$ corresponds to a bundle over $Z_2$ whose restriction to $P$ is just $(T_P\otimes \scrO_P(-1))\oplus \scrO_P^{n-2}$, where $T_P$ is the tangent bundle to $P$. Finally, a local lemma shows that every such bundle has a direct image on $Z$ which has the local form $M\oplus R^{n-2}$. Since each of these steps is somewhat involved, we divide the proof into three parts. First we describe the basic geometry of the blowups involved. Let $\tilde Z$ be the blowup of $Z$ along $E_b$, with exceptional divisor $D \cong \Pee^1\times E_b$. Let $Z_1$ be the blowup of $\tilde Z$ along $\Pee^1\times \{p_0\}\subset D$, with exceptional divisor $D_1$. Let $D'$ be the proper transform of $D$ in $Z_1$. The divisor $D_1 =\Pee(\scrO_{\Pee^1} \oplus \scrO_{\Pee^1}(-1))$ is isomorphic to $\Bbb F_1$. Let $j\: D_1\to Z_1$ be the inclusion and $q\: D_1 \to \Pee^1$ be the morphism induced by projection from a point. Let $\ell = \Pee^1\times \{p_0\} = D'\cap D_1$, so that $\ell$ is the exceptional divisor in $D_1$ viewed as the blowup of $\Pee^2$. Finally we let $s\: D_1\to \Pee^2$ be the blowup map. On a fiber $\Pee^1\times \{e\}$ with $e\neq p_0$, $\tilde V \otimes \scrO_{\tilde Z}(-D')$ restricts to $\scrO_{\Pee^1}^n$, whereas it restricts on $\Pee^1\times \{p_0\}$ to $\scrO_{\Pee^1}(1) \oplus \scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}$. Thus, if $V_0$ is the pullback to $Z_1$ of $\tilde V \otimes \scrO_{\tilde Z}(-D')$, then $V_0$ restricts on $D_1$ to $q^*\left[\scrO_{\Pee^1}(1) \oplus \scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}\right]$. \claim{1} Let $V_0$ be the pullback to $Z_1$ of $\tilde V \otimes \scrO_{\tilde Z}(-D')$. Make the elementary modification $$0 \to V'\to V_0 \to j_*q^*\scrO_{\Pee^1}(-1) \to 0.$$ Then $V'$ restricted to $\ell$ is the trivial bundle $\scrO_{\Pee^1}^n$. It follows that $V'|D'$ is pulled back from the factor $E_b$. \endstatement \proof We have an exact sequence $$0 \to V'|D'\to V_0|D' \to j_*\scrO_{\Pee^1}(-1) \to 0,$$ where we write $j$ also for the inclusion of the fiber $\ell =\Pee^1\times \{p_0\}$ in the ruled surface $D'\cong \Pee^1\times E_b$. By standard formulas for elementary modifications, it is straightforward to compute that $c_2(V'|D') = c_2(V_0|D') -1$. But $c_2(V_0|D') =h\pi_2^*[p_0]=1$ by the formulas of \S 2.6. Thus $c_2(V'|D') =0$. Now by a sequence of allowable elementary modifications $V_0|D', V'|D'=V_1, \dots, V_r$, we can reach a vector bundle $V_r$ over $D'$ whose restriction to every fiber $\Pee^1\times \{e\}$ is semistable and thus trivial; this happens if and only if $V_r$ is pulled back from the base, and so has $c_2=0$. But each allowable elementary modification along the fiber $\Pee^1\times \{p_0\}$ drops $c_2$ by a positive integer. Since $V'|D'$ already has $c_2=0$, no further elementary modifications are possible. Hence $V'|\ell$ is already semistable and therefore trivial, and thus $V'|D'$ is pulled back from $E_b$ as claimed. \endproof By construction, $V'|\ell$ is given as an extension $$0 \to \scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2} \to V'|\ell \to \scrO_{\Pee^1}(1) \to 0.$$ Now $\Ext^1( \scrO_{\Pee^1}(1), \scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}) \cong H^1(\scrO_{\Pee^1}(-2)) \cong \Cee$, so there is a unique nonsplit extension of this type, which is clearly the trivial bundle $\scrO_{\Pee^1}^n$. \claim{2} With $V'$ as in Claim \rom1, the restriction of $V'$ to $D_1$ is the pullback $s^*(T_{\Pee^2}(-1)\oplus \scrO_{\Pee^2}^{n-2})$. \endstatement \proof By definition, there is an exact sequence $$0 \to q^*[\scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}] \otimes \scrO_{D_1}(-D_1) \to V'|D_1 \to q^* \scrO_{\Pee^1}(1) \to 0.$$ Next, a straightforward calculation shows that $\scrO_{D_1}(-D_1)=\scrO_{D_1}(\ell)\otimes q^*\scrO_{\Pee^1}(1)$. Thus the extensions of $q^*\scrO_{\Pee^1}(1)$ by $q^*[\scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}]\otimes \scrO_{D_1}(-D_1))$ are classified by $$H^1(D_1; q^*[\scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}] \otimes\scrO_{D_1}(\ell)).$$ It is easy to check that $H^1(D_1; \scrO_{D_1}(\ell)) = 0$ and that $h^1( \scrO_{D_1}(\ell)\otimes q^*\scrO_{\Pee^1}(-1) ) = 1$. Thus the dimension of the Ext group in question is one, so that there just one nontrivial extension up to isomorphism. Note that $V'|D_1$ is itself such an extension: it cannot be the split extension since the restriction of $V'|D_1$ to $\ell$ is trivial. Thus, to complete the proof of Claim 2, it will suffice to show that $s^*(T_{\Pee^2}(-1)\oplus \scrO_{\Pee^2}^{n-2})$ is also given as an extension of $q^*\scrO_{\Pee^1}(1)$ by $q^*[\scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1}^{n-2}]\otimes \scrO_{D_1}(-D_1)$. It clearly suffices to do the case $n=2$, i\.e\. show that $s^*T_{\Pee^2}(-1)$ is an extension of $q^*\scrO_{\Pee^1}(1)$ by $q^*\scrO_{\Pee^1}(-1)$, necessarily nonsplit since the restriction to $\ell$ is trivial. To see this, note that $T_{\Pee^2}(-1)$ has restriction $\scrO_{\Pee^1}\oplus \scrO_{\Pee^1}(1)$ to every line. Thus by the standard construction (cf\. \cite{11}, p\. 60) there is an exact sequence $$0 \to \scrO_{\Bbb F_1}(\ell) \otimes q^*\scrO_{\Pee ^1}(t) \to s^*T_{\Pee ^2}(-1) \to q^*\scrO_{\Pee ^1}(1-t)\to 0$$ for some integer $t$. By looking at $c_2$, we must have $t=0$ and thus $s^*T_{\Pee ^2}(-1)$ is an extension of $q^*\scrO_{\Pee ^1}(1)$ by $\scrO_{\Bbb F_1}(\ell)$, which is nonsplit because its restriction to $\ell$ is trivial. Thus we have identified $V'|D_1$ with $s^*(T_{\Pee^2}(-1)\oplus \scrO_{\Pee^2}^{n-2})$. \endproof Let $Z_2$ be the result of contracting $D'$ in $Z_1$. This has the effect of contracting $\ell\subset D_1$ to a point, so that the image of $D_1$ in $Z_2$ is an exceptional $\Pee^2$, which we denote by $P$. Moreover, by the above claims $V'$ induces a vector bundle on $Z_2$ whose restriction to $P$ is identified with $T_{\Pee^2}(-1)\oplus \scrO_{\Pee^2}^{n-2}$. Thus, the proof of (6.11) will be complete once we prove the following: \claim{3} Let $X$ be a manifold of dimension $3$ and let $\tilde X$ be the blowup of $X$ at a point $x$, with exceptional divisor $P\cong \Pee^2$. Suppose that $W$ is a vector bundle on $\tilde X$ such that $W|P \cong T_{\Pee^2}(-1)\oplus \scrO_{\Pee^2}^{n-2}$. Let $\rho\: \tilde X \to X$ be the blowup map. Then $\rho_*W$ is locally isomorphic to $M\oplus R^{n-2}$ in the notation above. In particular, $\rho_*W$ is reflexive but not locally free. \endstatement \noindent {\it Proof.} We shall just do the case $n=2$, the other cases being similar. By the formal functions theorem, the completion of the stalk of the direct image $\rho_*W$ at $x$ is $M'= \varprojlim H^0(W \otimes \scrO_{nP})$. Now from the exact sequences $$0 \to \scrO_{\Pee^2}(-1) \to \scrO_P^3 \to W|P \to 0$$ and the sequence $$0 \to W\otimes \scrO_{\tilde X}(-(n+1)P) \to W\otimes \scrO_{(n+1)P}\to W\otimes \scrO_{nP}\to 0,$$ it is easy to check that the three sections of $ W|P$ lift to give three generators of $M'$ as an $R$-module. Hence there is a surjection $\scrO_{\tilde X}^3 \to\rho^*\rho_*W\to W$, and by checking determinants the kernel is $\scrO_{\tilde X}(P)$. Now up to an change of coordinates in $\Cee^3$ the only injective homomorphism from $\scrO_{\tilde X}(P)$ to $\scrO_{\tilde X}^3$ is given by the three generators of the maximal ideal of $\Cee ^3$ at the origin. Taking direct images of the exact sequence $$0 \to \scrO_{\tilde X}(P)\to \scrO_{\tilde X}^3 \to W \to 0$$ and using the vanishing for the first direct image of $\scrO_{\tilde X}(P)$ gives $M'\cong M$ as previously defined. So we have established Claim 3, and hence (6.11). \endproof We give a brief and inconclusive discussion of how the above constructions begave in families, assuming $\dim B =2$ for simplicity. Let $D$ be the unit disk in $\Cee$. Suppose that we are given a general family of nowhere vanishing sections $s_t$ of $\Cal V_n$ which at a special point $t=0$ acquires a simple zero at $b\in B$. We can view the family $s=\{s_t\}$ as a section of the pullback of $\Cal V_n$ to $B\times D$, where it has a simple zero at $(b,0)$. Thus, for an integer $a$, there is a bundle $\Cal V_{s,a}$ over $Z\times D-\{(b,0)\}$, which completes uniquely to a reflexive sheaf over $Z\times D$, which we continue to denote by $\Cal V_{s,a}$. For example, if $-(n-2)\leq a \leq 0$, then it is easy to see that $\Cal V_{s,a}$ is a bundle over $Z\times D$, whose restriction to $Z\times \{0\}$ is everywhere regular semistable except over $E_b$ where it restricts to $W_d\spcheck \oplus W_{n-d}$ for the appropriate $d$. One can ask if this holds for all $a \not\equiv 1 \bmod n$. Note that, if we consider the relative deformation theory of the unstable bundle $W_d\spcheck \oplus W_{n-d}$ over the base $B$, for $n=2$ the codimension of the locus of unstable bundles forces every deformation of $V$ to have unstable restriction to some fibers, whereas for $n> 2$ we expect that in the general deformation $V_t$ we can arrange that the restriction of $V_t$ to every fiber is semistable. If $a\equiv 1 \bmod n$, then $\Cal V_{s,a}$ is a flat family of coherent sheaves. However, there is no reason {\it a priori\/} why $\Cal V_{s,a}|Z\times \{0\}$ is reflexive. In fact, preliminary calculations suggest that, for $a=1$, the restriction $\Cal V_{s,a}|Z\times \{0\}$ has the local form $M\oplus \frak m^{n-2}$, where $\frak m$ is the maximal ideal of the point $\sigma \cap E_b$. Note that the $R$-module $M$ is not smoothable, even locally, but that $R^k\oplus M$ is smoothable to a free $R$-module for all $k\geq 1$. One can also show that the more complicated $R$-module $\frak m^k\oplus M$ is smoothable to a free $R$-module for all $k\geq 1$. This agrees with the picture for sections of the bundle $\Cal V_n$: for $n=2$, if a section has a simple isolated zero, that zero must remain under deformation, but for $n>2$ we expect in general that we can deform to an everywhere nonzero section in general. \ssection{6.4. Bundles which are not regular in high codimension.} In this subsection we consider bundles $V$ such that $V|E_b$ is semistable for all $b$, and $Y= \{\, b\in B: \text{ $V|E_b$ is not regular}\,\}$ has codimension at least $2$ in $B$. The first lemma shows that, if the spectral cover $C_A$ is smooth, then $V$ is in fact everywhere regular. \lemma{6.12} Let $V$ be a vector bundle over $Z$ such that $V|E_b$ is semistable for all $b$, and $Y= \{\, b\in B: \text{ $V|E_b$ is not regular}\,\}$ has codimension at least $2$ in $B$. Suppose that the associated spectral cover $C_A$ is smooth. Then $V|E_b$ is regular for all $b\in B$. More generally, suppose that $V$ is a vector bundle over $Z$ such that $Y= \{\, b\in B: \text{ $V|E_b$ is either not semistable or not regular}\,\}$ has codimension at least $2$ in $B$, that the section $A$ defined by $V$ over $B-Y$ extends to a section over all of $B$, and that the associated spectral cover $C_A$ of $B$ is smooth. Then $V|E_b$ is semistable and regular for all $b\in B$. \endstatement \proof We have seen in (5.7) that there is a line bundle $N$ on $C_A-g_A^{-1}(Y)$ such that $V|Z-\pi^{-1}(Y) \cong V_{A,0}[N]$. Since $C_A$ is smooth, and $g_A^{-1}(Y)$ has codimension at least two in $C_A$, the line bundle $N$ on $C_A-g_A^{-1}(Y)$ extends to a line bundle over $C_A$, which we continue to denote by $N$. We now have two vector bundles on $Z$, namely $V$ and $V_{A,0}[N]$, which are isomorphic over $Z-\pi^{-1}(Y)$. Since the codimension of $\pi^{-1}(Y)$ in $Z$ is at least two, $V$ and $V_{A,0}[N]$ are isomorphic. But $V_{A,0}[N]$ restricts to a regular bundle on every fiber, and so the same must be true for $V$. \endproof We turn to methods for constructing bundles which are semistable on every fiber but which are not regular in codimension two. Of course, by the above lemma, the corresponding spectral covers will not be smooth. The idea is to find such bundles by using a three step filtration, as opposed to the two-step extensions which have used from Section 3 onwards in our constructions. Such constructions correspond to nonmaximal parabolic subgroups in $SL_n$. Consider first the case of a single Weierstrass cubic $E$. We seek bundles of rank $n+1$ which have a filtration $0\subset F^0\subset F^1 \subset V$, where $F^0\cong W_k\spcheck, F^1/F^0\cong \scrO_E$, and $V/F^1\cong W_{n-k}$. Such extensions can be described by a nonabelian cohomology group as in \cite{8}. However, it is also easy to describe them directly. Note that a fixed $F^1$ is described by an extension class $\alpha _0$ in $\Ext^1(\scrO_E, W_k\spcheck) \cong H^1(W_k\spcheck) \cong \Cee$. If $\alpha _0 = 0$, then $F^1 = W_k\spcheck \oplus \scrO_E$, and if $\alpha _0 \neq 0$ then $F^1\cong W_{k+1}\spcheck$. Having determined $F^1$, the extension $F^2$ corresponds to a class in $\Ext^1(W_{n-k}, F^1)$. Since $\Hom (W_{n-k}, \scrO_E) = \Ext^2(W_{n-k}, W_k\spcheck) =0$, there is a short exact sequence $$0 \to \Ext ^1(W_{n-k}, W_k\spcheck) \to \Ext^1(W_{n-k}, F^1) \to \Ext^1(W_{n-k}, \scrO_E) \to 0,$$ and so $\dim \Ext^1(W_{n-k}, F^1) = n+1$. Thus roughly speaking the moduli space of filtrations as above is an affine space $\Cee^{n+2}$. In fact, by general construction techniques there is a universal bundle $\Cal F^1$ over $\Ext^1(\scrO_E, W_k\spcheck)\times E =\Cee\times E$. We can then form the relative Ext sheaf $$Ext^1_{\pi_1}(\pi _2^*W_{n-k}, \Cal F^1)= R^1\pi_1{}_*(\pi _2^*W_{n-k}\spcheck\otimes \Cal F^1).$$ It is a vector bundle of rank $n+1$ over $\Cee$, which is necessarily trivial, and thus the total space of this vector bundle is $\Cee^{n+2}$. There is a universal extension of $\pi_2^*W_{n-k}$ by $\Cal F^1$ defined over $\Cee^{n+2}\times E$. It follows that the set of filtrations is indeed parametrized by a moduli space isomorphic to $\Cee^{n+2}$, although there is not a canonical linear structure. What is canonical is the exact sequence $$0 \to \Ext^1(W_{n-k}, W_k\spcheck) \to \Cee^{n+2} \to \Ext^1(\scrO_E, W_k\spcheck) \oplus \Ext^1(W_{n-k}, \scrO_E) \to 0.$$ We understand this sequence to mean that the first term, which is a vector space, acts on the middle term, which is just an affine space, via affine translations, and the quotient is the last term, which is again a vector space. Here the projection to $\Ext^1(\scrO_E, W_k\spcheck) \oplus \Ext^1(W_{n-k}, \scrO_E)$ measures the extensions $F^1$ of $\scrO_E$ by $W_k\spcheck$ and $V/F^0$ of $W_{n-k}$ by $\scrO_E$. We denote the image of $\xi\in \Cee^{n+2}$ in $\Ext^1(\scrO_E, W_k\spcheck) \oplus \Ext^1(W_{n-k}, \scrO_E) \cong \Cee\oplus \Cee$ by $(\alpha _0, \alpha_1)$. Here $\alpha _0 \neq 0$ if and only if $F^1\cong W_{k+1}\spcheck$ and $\alpha_1 \neq 0$ if and only if $V/F^0\cong W_{n-k+1}$. In case $\alpha _0 = 0$, say, $F^1\cong W_k\spcheck \oplus \scrO_E$, and $\Ext^1(W_{n-k}, F^1)$ naturally splits as $\Ext ^1(W_{n-k}, W_k\spcheck)\oplus \Ext^1(W_{n-k}, \scrO_E)$. In this case, both the class $\alpha_1$ and the class $e\in \Ext ^1(W_{n-k}, W_k\spcheck)$ are well-defined. A similar statement holds if $\alpha _1 =0$. Note that the affine space $\Cee^{n+2}$ parametrizes filtrations $F^i$ together with {\sl fixed\/} isomorphisms $F^0\to W_k\spcheck$, $F^1/F^0\to \scrO_E$, $V/F^1 \to W_{n-k}$. The subspace $\Ext^1(W_{n-k}, W_k\spcheck)$, namely where both $\alpha_0$ and $\alpha _1$ vanish, corresponds to those $V$ of the form $V'\oplus \scrO_E$, where $V'$ is an extension of $W_{n-k}$ by $W_k\spcheck$. There is a hyperplane $H$ in $\Ext^1(W_{n-k}, W_k\spcheck)$ where such $V'$ contain a Jordan-H\"older quotient isomorphic to $\scrO_E$, and thus over the locus $\alpha _0 = \alpha _1 = 0, e\in H$, $V=V'\oplus \scrO_E$ has a subbundle of the form $\scrO_E\oplus \scrO_E$. Hence, over a affine subspace of $\Cee^{n+2}$ of codimension three, the $V$ we have constructed are not regular. \lemma{6.13} Suppose that $V$ corresponds to a class $\xi \in \Cee^{n+2}$, and that $\alpha_0, \alpha _1$ are as above. \roster \item"{(i)}" $V$ is unstable if and only $\alpha_0 \alpha _1=0$ and $e=0$ \rom(this statement is well-defined by the above remarks\rom). \item"{(ii)}" If $\alpha_0 \alpha _1 \neq 0$, then $h^0(V) = 0$ and conversely. \endroster \endstatement \proof (i) Let us assume for example that $\alpha _0 = e=0$. Then $F^1 = W_k\spcheck \oplus \scrO_E$ and $V$ is isomorphic either to $W_k\spcheck \oplus W_{n-k+1}$ or to $W_k\spcheck \oplus \scrO_E \oplus W_{n-k}$, and in either case it is unstable. Conversely, if $V$ is unstable, then it has a maximal destabilizing subsheaf $W$ of positive degree, which is stable and which must map nontrivially onto $W_{n-k}$. Thus $\deg W =1$. Now if $W\cap W_k\spcheck\neq 0$, then $W\cap W_k\spcheck$ has degree $\leq -1$ and is contained in the kernel of the map $W\to W_{n-k}$. This would force the image of $W$ to have degree at most zero, which is impossible. So $W\cap W_k\spcheck=0$ and thus the map $W\to V/F^0$ is injective. Now either $V/F^0 \cong W_{n-k+1}$ or $V/F^0 \cong W_{n-k}\oplus \scrO_E$. In the first case, $W\cong W_{n-k+1}$ by the stability of $ W_{n-k+1}$ and $V\cong W_k\spcheck \oplus W_{n-k+1}$. In this case $\alpha _0 = e=0$. In the remaining case, $V/F^0 \cong W_{n-k}\oplus \scrO_E$ and $W\cong W_{n-k}$. In this case $\alpha _1=e=0$. In both cases we must have $\alpha_0 \alpha _1=0$ and $e=0$. \smallskip \noindent (ii) First suppose that $\alpha_0 \alpha _1 \neq 0$. Since $\alpha _0\neq0$, $F^1\cong W_{k+1}\spcheck$. From the exact sequence $$0\to F^1\to V \to W_{n-k} \to 0,$$ and the fact that $H^0(F^1) = 0$, there is an exact sequence $H^0(V) \to H^0(W_{n-k}) \to H^1(F^1)$. If we compose the map $H^0(W_{n-k}) \to H^1(F^1)$ with the natural map $ H^1(F^1) \to H^1(\scrO_E)$, the result is $\alpha _1$ up to a nonzero scalar. Thus, if $\alpha _1\neq 0$, the map $H^0(W_{n-k}) \to H^1(F^1)$ is injective and so $H^0(V) =0$. Conversely, suppose that either $\alpha _0$ or $\alpha _1$ is zero. If for example $\alpha _1=0$, then $V/F^0 \cong W_{n-k}\oplus \scrO_E$, so that $h^0(V/F^0)=2$. Since $H^0(V)$ is the kernel of the map $H^0(V/F^0) \to H^1(F^0) = H^1(W_k\spcheck)\cong \Cee$, $H^0(V) \neq 0$. The case $\alpha _0 \neq 0$ is similar and simpler. Thus, if $h^0(V) =0$, then $\alpha_0 \alpha _1 \neq 0$. \endproof The group $\Cee^*\times \Cee^*$ (or more precisely $\Cee^3/\Cee$) acts on the affine space $\Cee^{n+2}$ by acting on the identifications of the quotients $F^{i+1}/F^i$ with the standard bundles. The quotient by this action (which is not in fact separated) is the set of bundles $V$ of rank $n+1$, together with filtrations on $V$ with the appropriate graded object. The action of $\Cee^*\times \Cee^*$ is compatible with the projection to $\Cee^2$. If we normalize the action so that $(\lambda, \mu)\cdot (\alpha_0, \alpha _1) = (\lambda\alpha_0, \mu\alpha _1)$, then $(\lambda, \mu)$ acts on the distinguished subspace $\Ext^1(W_{n-k}, W_k\spcheck) \cong \Cee^n$ by $e\mapsto \lambda\mu e$. Clearly, the action is free over the set $\alpha_0\alpha _1\neq 0$. The quotient of the points where $\alpha _0\neq 0$, $\alpha _1 = 0$, $e\neq 0$ is a $\Pee^{n-1}$, and this $\Pee^{n-1}$ is identified with the corresponding $\Pee^{n-1}$ where $\alpha _1\neq 0$, $\alpha _0 = 0$, $e\neq 0$; in fact, the $\Cee^*\times \Cee^*$-orbits intersect along the subspace where $\alpha _0=\alpha _1 = 0$, $e\neq 0$. The points $\alpha_0\alpha _1=e=0$ are unstable points and do not appear in a GIT quotient for the action. We also have the map (1.5) $\Phi\: \Cee^{n+2}-(\Cee \cup\Cee)$ to the coarse moduli space $\Pee^n$ of semistable bundles of rank $n+1$ on $E$. By Lemma 6.13, the image of the two subsets $\{\,\alpha _0 = 0, e\neq 0\,\}$ and $\{\,\alpha _1 = 0, e\neq 0\,\}$ is exactly the hyperplane in $\Pee^n$ corresponding to bundles $V$ such that $h^0(V) \neq 0$, or in other words such that $V$ has $\scrO_E$ as a Jordan-H\"older quotient. \lemma{6.14} The map $\Phi\: \Cee^{n+2}-(\Cee \cup\Cee) \to \Pee^n$ is the geometric invariant theory quotient of $\Cee^{n+2}-(\Cee \cup\Cee)$ by the action of $\Cee^*\times \Cee^*$. \endstatement \proof First suppose that the point $x\in \Cee^{n+2}-(\Cee \cup\Cee)$ lies in the open dense subset $\alpha_0\alpha _1\neq 0$ where $\Cee^*\times \Cee^*$ acts freely. Thus if $V$ is the vector bundle corresponding to $x$, then $h^0(V) =0$; equivalently, $V$ has no Jordan-H\"older quotient equal to $\scrO_E$, and $V$ is a regular semistable bundle. If $\Phi(x) = \Phi(x')$, then $x'$ also lies in the set $\alpha_0\alpha _1\neq 0$, and the bundle $V'$ corresponding to $x'$ is also regular and semistable. Thus $V\cong V'$, and we must determine if the filtration $F^i$ on $V$ exists is unique up to isomorphism. First, if $V$ is a regular semistable bundle of rank $n+1$, then it is an extension of $W_{n-k}$ by $W_{k+1}\spcheck$, where the subbundle $W_{k+1}\spcheck$ of $V$ is unique modulo automorphisms of $V$, and taking the further filtration of $W_{k+1}\spcheck$ by the subbundle $W_k\spcheck $, with quotient $\scrO_E$. Thus $V=\Phi(x)$ for some $x$. Conversely, if $V$ has on it a filtration $F^i$ with $\alpha_0\alpha_1 \neq 0$, then $F^1\cong W_{k+1}\spcheck$. Moreover, if $H^0(V) = 0$, then every subbundle of $V$ isomorphic to $W_k\spcheck$ is contained in a subbundle isomorphic to $W_{k+1}\spcheck$ (whereas if $H^0(V) \neq 0$, this is no longer the case; cf\. \S 3.2). Thus the filtration $F^i$ is unique up to automorphisms of $V$. The above argument shows that $\Phi$ induces an isomorphism $$\left(\Cee^{n+2} - \{\, \alpha_0\alpha _1= 0\,\}\right)/\Cee^*\times \Cee^* \to \Pee^n - H.$$ In case $x$ lies in the set $\alpha _0 = 0, , \alpha _1\neq 0, e\neq 0$, a straightforward argument identifies the quotient by $\Cee^*\times \Cee^*$ with $H\subset \Pee^n$, and likewise for $\alpha _0 \neq 0, \alpha _1= 0, e\neq 0$, $\alpha _0 =\alpha _1= 0, e\neq 0$. \endproof The coarse moduli space $\Pee^n$ has its associated spectral cover $T$, which is an $(n+1)$-sheeted cover of $\Pee^n$. Let $\tilde T \to \Cee^{n+2}-(\Cee \cup\Cee)$ be the pulled back cover of $\Cee^{n+2}-(\Cee \cup\Cee)$ via the morphism $\Phi$. Using Lemma 6.14, we can see directly that $\tilde T$ is singular, with the generic singularities a locally trivial family of threefold double points. In fact, the inverse image of $H$ in $T$ is of the form $H\cup T'$, where $T'$ is the spectral cover of $H\cong \Pee^{n-1}$. The intersection of $H$ and $T'$ is transverse (see \S 5.7), and $H\cap T'$, viewed as a subset of $H\subset \Pee^n$, corresponds to those bundles which have $\scrO_E$ as a Jordan-H\"older quotient with multiplicity at least two. If $t$ is a local equation for $H$ in $\Pee^n$ near a generic point of $H\cap T'$, there are local coordinates on $T$ for which $t=uv$, since $H$ splits into $H\cup T'$. Thus the local equation for $\tilde T$ is $\alpha_0\alpha _1= uv$, which is the equation for a family of threefold double points. We can also do the above constructions in families $\pi\: Z \to B$. We could take the point of view of \cite{8} and realize the relative nonabelian cohomology groups as a bundle of affine spaces over $B$. However, it is also possible to proceed directly as in \S 5.2. We seek vector bundles $V$ which have a filtration $0\subset F^0\subset F^1 \subset V$, where $F^0\cong \Cal W_k\spcheck \otimes \pi^*M_0$, $F^1/F^0\cong \pi^*M_1$, and $V/F^1\cong \Cal W_{n-k} \otimes \pi^*M_2$ for line bundles $M_0, M_1, M_2$ on $B$. Of course, we can normalize by twisting $V$ so that one of the $M_i$ is trivial. The analysis of such extensions parallels the analysis for a single $E$. We begin by constructing $F^0$. It is described by an extension class in $$\gather H^1(Z; \pi^*M_1^{-1}\otimes \Cal W_k\spcheck \otimes \pi^*M_0) \cong H^0(B; R^1\pi_*(\Cal W_k\spcheck) \otimes M_1^{-1}\otimes M_0) \\ =H^0(B;L^{-k}\otimes M_1^{-1}\otimes M_0). \endgather$$ If the difference line bundle $M_1^{-1}\otimes M_0$ is sufficiently ample, then there will be nonzero sections $\alpha _0$ of $L^{-k}\otimes M_1^{-1}\otimes M_0$ vanishing along a divisor $D_0$ in $B$. Next, we seek extensions of $F^1$ by $\Cal W_{n-k} \otimes \pi^*M_2$. Now $H^0(\Cal W_{n-k}\spcheck \otimes \pi^*M_2^{-1}\otimes \pi^*M_1) = 0$, and by the Leray spectral sequence $$H^2(\Cal W_{n-k}\spcheck \otimes \pi^*M_2^{-1}\otimes \Cal W_k\spcheck \otimes \pi^*M_0) \cong H^1(B; R^1\pi_*(\Cal W_{n-k}\spcheck \otimes \Cal W_k\spcheck)\otimes M_2^{-1}\otimes M_0).$$ We assume that $M_2^{-1}\otimes M_0$ is so ample that the above group is zero. In this case there is an exact sequence $$\gather 0 \to H^1(\Cal W_{n-k}\spcheck \otimes \pi^*M_2^{-1}\otimes \Cal W_k\spcheck \otimes \pi^*M_0) \to \Ext^1(\Cal W_{n-k} \otimes \pi^*M_2, F^1) \to \\ H^1(\Cal W_{n-k}\spcheck \otimes \pi^*M_2^{-1}\otimes \pi^*M_1) \to 0. \endgather$$ The left-hand group is $H^0(R^1\pi_*(\Cal W_{n-k}\spcheck \otimes \Cal W_k\spcheck)\otimes M_2^{-1}\otimes M_0)$, and the right-hand group is $H^0(L^{n-k} \otimes M_2^{-1}\otimes M_1)$. Thus, for $ M_2^{-1}\otimes M_1$ sufficiently ample, there will exist sections $\alpha _1$ of $L^{n-k} \otimes M_2^{-1}\otimes M_1$, vanishing along a divisor $D_1$ in $B$, and we will be able to lift these sections to extension classes in $\Ext^1(\Cal W_{n-k} \otimes \pi^*M_2, F^1)$. Moreover, if we restrict, say, to the divisor $D_0 = 0$, then there is also a well-defined class $e$ in $H^0(R^1\pi_*(\Cal W_{n-k}\spcheck \otimes \Cal W_k\spcheck)\otimes M_2^{-1}\otimes M_0)$. There is a divisor $D$ on $B$ corresponding to such extensions which have a factor $\scrO_{E_b}$ for $b\in D$. In fact, if $\alpha = c_1(L\otimes M_2^{-1}\otimes M_0)$, then it follows from (4.15) and (5.9) that $[D] = \alpha - nL$. (Compare also (5.21).) As long as $M_2^{-1}\otimes M_0$ is also sufficiently ample, we can assume that the divisors $D_0, D_1$ and $D$ are smooth and intersect transversally in a subvariety of $B$ of codimension three. Along this subvariety, $V$ fails to be regular. Note that the $V$ constructed above are a deformation of $V'\oplus \scrO_Z$, where $V'$ is a twist of a bundle of the form $V_{A, 1-k}$; it suffices for example to take $\alpha _0 = 0$ and $e\neq 0$. For generic choices, the spectral cover $C_A$ will acquire singularities in codimension three, which will generically be families of threefold double points. In particular, there are Weil divisors on $C_A$ which do not extend to Cartier divisors, as predicted by Lemma 6.12. It is also amusing to look at the case $\dim B =2$, where for generic choices the spectral cover will be smooth. The construction then deforms $V'\oplus \scrO_Z$ to a bundle $V$ which has regular semistable restriction to every fiber. Starting with a generic $V' = V_{A,a}(n)$ of rank $n$, we cannot in general deform $V'\oplus \scrO_Z$ to a standard bundle $V_{A,b}(n+1)\otimes \pi^*N_0$. Instead, the spectral cover $C_A$ has Picard number larger than expected. In fact, we have the divisor $F = (r^*\sigma \times _BZ)\cap C_A$, which is mapped isomorphically onto its image in $B$, and this image is the same as $A\cap \Cal H\subset \Cal P_{n-1}$. Now $A\cap \Cal H$ corresponds to the bundles $V$ such that $h^0(V) \neq 0$, and thus by Lemma 6.13 this locus is just $D_0\cup D_1$. Thus in $C_A$ the divisor $F$ splits into a sum of two divisors, which we continue to denote by $D_0$ and $D_1$. Using these extra divisors, we can construct more vector bundles over $Z$, of the form $V_{A,0}[N]$ for some extra line bundle $N$, which enable us to deform $V'\oplus \scrO_Z$ to a bundle which is everywhere regular and semistable. Let us just give the details in a symmetric case. Let $M$ be a sufficiently ample line bundle on $B$. There exist bundles $V$ on $Z$ which have regular semistable restriction to every fiber and also have a filtration $F^0\subset F^1 \subset V$, with $$V/F^1 \cong \Cal W_k\otimes \pi^*M^{-1}; \qquad F^1/F^0\cong \scrO_Z; \qquad F^0 \cong \Cal W_k\spcheck \otimes \pi^*M.$$ The bundle $V$ is a deformation of a bundle of the form $V'\oplus \scrO_Z$. Thus, there must exist a line bundle $N$ on the spectral cover $C_A$ such that $V_{A,0}[N]$ has the same Chern classes as $V$. Direct calculation with the Grothendieck-Riemann-Roch theorem shows that this happens for $$N = M \otimes \scrO_{C_A}(-F + (k+1)D_0 + kD_1)$$ as well as for $$N = M \otimes \scrO_{C_A}(-F + kD_0 + (k+1)D_1),$$ and that these are the only two ``universal" choices for $N$. \remark{Question} Suppose that $\dim B = 3$ and consider spectral covers which have an ordinary threefold double point singularity. The local Picard group of the singularity is $\Zee$. Given $a\in \Zee$, we can twist by a line bundle over the complement of the singularity which correspond to the element $a\in \Zee$. The result is a vector bundle on $Z$, defined in the complement of finitely many fibers, and thus the direct image is a coherent reflexive sheaf on $Z$. What is the relationship of local behavior of this sheaf at the finitely many fibers to the integer $a$? \endremark \section{7. Stability.} Our goal in this final section will be to find sufficient conditions for $V_{A,a}$, or more general bundles constructed in the previous two sections, to be stable with respect to a suitable ample divisor. Here suitable means in general a divisor of the form $H_0+ N\pi^*H$, where $H_0$ is an ample divisor on $Z$ and $H$ is an ample divisor on $B$, and $N\gg 0$. As we have already see in \S 5.6, for $A =\bold o$, the bundle $V_{\bold o, a}$ is essentially always unstable with respect to every ample divisor. Likewise, suppose that $A$ is a section lying in $\Cal H$ as in \S 5.7, so that $V_{A,a}$ has a surjection to $\pi^*L^a$. If the line bundle corresponding to $A$ is sufficiently ample, it is easy to see that for appropriate choices of $a$ we can always arrange $\mu_H(\pi^*L^a) < \mu_H(V_{A,a})$, so that $V_{A,a}$ is unstable. Thus, we shall have to make some assumptions about $A$. More generally, let $V$ be a bundle whose restriction to the generic fiber is regular and semistable, and let $A$ be the associated quasisection. It turns out that, if the spectral cover $C_A$ is irreducible, then $V$ is stable with respect to all divisors of the form $H_0+ N\pi^*H$, provided that $N\gg 0$. A similar result holds in families. However, we are only able to give an effective estimate for $N$ in case $\dim B = 1$. In particular, whether there is an effective bound for $N$ which depends only on $Z$, $H_0$, $H$, $c_1(V)$, and $c_2(V)$ is open in case $\dim B > 1$. We believe that such a bound should exist, and can give such an explicit bound for a general $B$ in the rank two case for an irreducible quasisection $A$. (Of course, when $\dim B > 2$, an irreducible quasisection $A$ will almost never be an actual section.) However, we shall not give the details in this paper. \ssection{7.1. The case of a general $Z$.} Let $\pi\: Z\to B$ be a flat family of Weierstrass cubics with a section. We suppose in fact that $Z$ is smooth of dimension $d+1$. Fix an ample divisor $H_0$ on $Z$ and an ample divisor $H$ on $B$, which we will often identify with $\pi^*H$ on $Z$. \theorem{7.1} Let $V$ be a vector bundle of rank $n$ over $Z$ whose restriction to the generic fiber is regular and semistable, and such that the spectral cover of the quasisection corresponding to $V$ is irreducible. Then there exists an $\epsilon_0 > 0$, depending on $V, H_0, H$, such that $V$ is is stable with respect to $\epsilon H_0 + H$ for all $0< \epsilon < \epsilon_0$. \endstatement \proof Let $W$ be a subsheaf of $V$ with $0 < \operatorname{rank} W < r$. The semistability assumption on $V|f$, for a generic $f$, and the fact that $W|f \to V|f$ is injective for a generic $f$ imply that $c_1(W) \cdot f \leq 0$. If however $c_1(W) \cdot f = 0$, then $W$ and $V/W$ are also semistable on the generic fiber. By Proposition 5.22, the spectral cover corresponding to $V$ would then be reducible (the proof in (5.22) needed only that $V$ has regular semistable restriction to the generic fiber), contrary to hypothesis. Thus in fact $c_1(W) \cdot f < 0$. Equivalently, $c_1(W) \cdot H^d < 0$. For a torsion free sheaf $W$, define $\mu _H(W) = \dsize \frac{c_1(W)\cdot H^d}{\operatorname{rank} W}$, by analogy with an ample $H_0$. If $W$ is a subsheaf of $V$ such that $0 < \operatorname{rank} W < n$, then $\mu _H(W)$ is a strictly negative rational number with denominator bounded by $n-1$. \lemma{7.2} There is a constant $A$, depending only on $V, H_0, H$, such that $$\frac{c_1(W)\cdot H^i\cdot H_0^{d-i}}{\operatorname{rank} W} \leq A$$ for all $i$ with $0\leq i\leq d$ and all nonzero subsheaves $W$ of $V$. \endstatement \proof There exists a filtration $$0\subset F^0\subset F^1\subset \cdots \subset F^{n-1}=V$$ such that $F^j/F^{j-1}$ is a torsion free rank one sheaf, and thus is of the form $L_j\otimes I_{X_j}$ for $L_j$ a line bundle on $Z$ and $X_j$ a subscheme of codimension at least two (possibly empty). Suppose that $W$ has rank one. Then there is a nonzero map from $W$ to $L_j\otimes I_{X_j}$ for some $j$, and thus $W$ is of the form $L_j\otimes \scrO_Z(-D)\otimes I_X$ for some effective diviisor $D$ on $Z$ and subscheme $X$ of codimension at least two (possibly empty). Thus $$c_1(W)\cdot H^i\cdot H_0^{d-i} \leq c_1(L_j)\cdot H^i\cdot H_0^{d-i}.$$ Thus these numbers are bounded independently of $W$. In case $W$ has arbitrary rank $r$, $1\leq r\leq n-1$, find a similar filtration of the bundle $\bigwedge ^rV$ by subsheaves whose successive quotients are rank one torsion free sheaves, and use the existence of a nonzero map $\bigwedge ^rW\to \bigwedge ^rV$ to argue as before. \endproof Returning to the proof of Theorem 7.1, if $W$ is a subsheaf of $V$ such that $0 < \operatorname{rank} W < n$, it follows that $$\mu_{\epsilon H_0 + H}(W) = \frac{c_1(W)\cdot (\epsilon H_0 + H)^d}{\operatorname{rank} W}\leq -\frac{1}{n-1}+O(\epsilon).$$ On the other hand, since $\det V$ is pulled back from $B$, $c_1(V) \cdot H^d = 0$ and so $$\mu_{\epsilon H_0 + H}(V) = \frac{c_1(V)\cdot (\epsilon H_0 + H)^d}{n}= O(\epsilon).$$ Thus, for $\epsilon$ sufficiently small, for every subsheaf $W$ of $V$ with $0 < \operatorname{rank} W < n$, $$\mu_{\epsilon H_0 + H}(W) < \mu_{\epsilon H_0 + H}(V).$$ In other words, $V$ is stable with respect to $\epsilon H_0 + H$. \endproof \corollary{7.3} Let $\Cal V$ be a family of vector bundles over $S\times Z$, such that, for each $s\in S$, the restriction $V_s=\Cal V|\{s\}\times Z$ has regular semistable restriction to the generic fiber of $\pi$ and the corresponding spectral cover is irreducible. Then there exists an $\epsilon_0 > 0$, depending on $\Cal V, H_0, H$, such that, for every $s\in S$, $V_s$ is is stable with respect to $\epsilon H_0 + H$ for all $0< \epsilon < \epsilon_0$. \endstatement \proof We may assume that $S$ is irreducible. The proof of Theorem 7.1 goes through as before as long as we can uniformly bound the integers $c_1(W)\cdot H^i\cdot H_0^{d-i}$ as $W$ ranges over subsheaves of $V_s$ over all $s\in S$. But there exists a filtration $$0\subset F^0\subset F^1\subset \cdots \subset F^{n-1}=\Cal V$$ such that $F^j/F^{j-1}$ is a torsion free rank one sheaf on $S\times Z$, and thus is of the form $\Cal L_j\otimes I_{\Cal X_j}$ for $\Cal L_j$ a line bundle on $S\times Z$ and $\Cal X_j$ a subscheme of $S\times Z$ of codimension at least two, such that, at a generic point $s$ of $S$, $(\{s\}\times Z)\cap\Cal X_j $ has codimension at least two in $Z$. On a nonempty Zariski open subset $\Omega$ of $S$, the filtration restricts to a filtration of $V_s$ of the form used in the proof of Lemma 7.2, and $c_1(\Cal L_j| \{s\}\times Z)$ is independent of $s$. Similar filtrations exist for the exterior powers $\bigwedge ^r\Cal V$. This bounds $c_1(W)\cdot H^i\cdot H_0^{d-i}$ as $W$ ranges over subsheaves of $V_s$ over all $s$ in a nonempty Zariski open subset of $S$. By applying the same construction to the components of $S-\Omega$ and induction on $\dim S$, we can find the desired bound for all $s\in S$. \endproof \ssection{7.2. The case of an elliptic surface.} In case $\dim B =1$, there is a more precise result. \theorem{7.4} Let $\pi\: Z \to B$ be an elliptic surface and let $H_0$ be an ample divisor on $Z$. Let $f$ be the numerical equivalence class of a fiber. Let $V$ be a vector bundle of rank $n$ on $Z$ which is regular and semistable on the generic fiber, with $\det V$ the pullback of a line bundle on $B$, and with $c_2(V) = c$, and such that the spectral cover of $V$ is irreducible. Then for all $\dsize t\geq t_0= \frac{n^3}{4}c_2(V)$, $V$ is stable with respect to $H_0+tf=H_t$. \endstatement \proof If $V$ is $H_{t_0}$-stable, then as it is $f$-stable (here stability is defined with respect to the nef divisor $f$) it is stable with respect to every convex combination of $H_{t_0}$ and $f$ and thus for every divisor $H_t$ with $t\geq t_0$. Thus we may assume that $V$ is not $H_{t_0}$-stable for some $t_0\geq 0$. \lemma{7.5} Suppose that $V$ is not $H_{t_0}$-stable for some $t_0\geq 0$. Then there exists a $t_1 \geq t_0$ and a divisor $D$ such that $D\cdot H_{t_1} = 0$ and $$-\frac{n^3}{2}c_2(V) \leq D^2 < 0.$$ \endstatement \proof By Theorem 7.1, for all $t\gg 0$, $V$ is $H_t$-stable. Let $t_1$ be the greatest lower bound of the $t$ such that, for all $t'\geq t$, $V$ is $H_{t'}$-stable. Thus $t_1\geq 0$. The condition that $V$ is $H_t$-unstable is clearly an open condition on $t$. It follows that $V$ is strictly $H_{t_1}$-semistable, so that there is an exact sequence $$0 \to V' \to V \to V'' \to 0,$$ with both $V'$, $V''$ torsion free and of strictly smaller rank than $V$, and with $\mu_{H_{t_1}}(V') = \mu_{H_{t_1}}(V'') = \mu_{H_{t_1}}(V)$. Thus, both $V'$ and $V''$ are $H_{t_1}$-semistable. Let $D = r'c_1(V'') - r''c_1(V')$. Then the equality $\mu_{H_{t_1}}(V') = \mu_{H_{t_1}}(V'')$ is equivalent to $D\cdot H_{t_1} = 0$. Note that $D$ is not numerically trivial, for otherwise $V$ would be strictly $H_t$-semistable for all $t$, contradicting the fact that it is $H_t$-stable for $t\gg 0$. Thus, by the Hodge index theorem, $D^2<0$. Now, for a torsion free sheaf of rank $r$, define the Bogomolov number (or discriminant) of $W$ by $$B(W) = 2rc_2(W) - (r-1)c_1(W)^2.$$ If $W$ is semistable with respect to some ample divisor, then $B(W) \geq 0$. Finally, we have the identity (\cite{5}, Chapter 9, ex\. 4): $$B(V) = 2nc_2(V) = \frac{n}{r'}B(V') + \frac{n}{r''}B(V'') - \frac{D^2}{r'r''},$$ and thus, as $B(V')\geq 0$ and $B(V'')\geq 0$ by Bogomolov's inequality, $$D^2\geq -(r'r'')2nc_2(V).$$ Since $r'+r'' = n$, $r'r''\leq n^2/4$, and plugging this in to the above inequality proves the lemma. \endproof Returning to the proof of Theorem 7.4, the proof of Lemma 1.2 in Chapter 7 of \cite{6} (see also \cite{5}, Chapter 6, Lemma 3) shows that, if $\dsize t\geq t_0 = \frac{n^3}{4}c_2(V)$, then for every divisor $D$ such that $\dsize D^2 \geq -\frac{n^3}{2}c_2(V)$ and $D\cdot f > 0$, we have $D\cdot H_t> 0$. Now, if $V$ is not $H_{t_0}$-stable, we would be able to find a $t_1 \geq t_0$ and an exact sequence $0 \to V' \to V \to V'' \to 0$ as above, with $\mu_{H_{t_1}}(V') = \mu_{H_{t_1}}(V'')$. Now setting $D = r'c_1(V'') - r''c_1(V')$ as before, we have $$\gather 0< \mu_f(V) -\mu _f(V') = \frac{c_1(V')\cdot f + c_1(V'')\cdot f}{n} - \frac{c_1(V')\cdot f}{r'} \\ = \frac{\left(r'c_1(V'') - r''c_1(V')\right)\cdot f}{r'n} = \frac{D\cdot f}{r'n}, \endgather$$ so that $D\cdot f > 0$, and likewise $D\cdot H_{t_1}= 0$. Thus $D\cdot H_{t_0} <0$, contradicting the choice of $t_0$. It follows that, for all $\dsize t\geq t_0 = \frac{n^3}{4}c_2(V)$, $V$ is $H_t$-stable. This completes the proof of (7.5). \endproof As a final comment, the difficulty in finding an effective bound in case $\dim B > 1$ is the following: For a torsion free sheaf $W$, we can define $B(W)$ as before, but it is an element of $H^4(Z)$, not an integer. In the notation of the proof of Lemma 7.5, Bogomolov's inequality can be used to give a bound for $B(V')\cdot H_t^{n-2}$ and $B(V'')\cdot H_t^{n-2}$ for some (unknown) value of $t$, and thus there is a lower bound for $D^2\cdot H_t^{n-2}$, again for one unknown value of $t$. However this does not seem to give enough information to complete the proof of the theorem, except in the rank two case where the lower bound can be made explicit for all $t$. \Refs \ref \no 1\by M. Atiyah \paper Vector bundles over an elliptic curve \jour Proc. London Math. Soc. \vol 7\yr 1957 \pages 414--452\endref \ref \no 2 \by C. B\u anic\u a, M. Putinar, and G. Schumacher \paper Variation der globalen Ext in Deformationen kompakter komplexer R\"aume \jour Math. Annalen \vol 250 \yr 1980 \pages 135--155 \endref \ref \no 3\by R. Friedman \paper Rank two vector bundles over regular elliptic surfaces \jour Inventiones Math. \vol 96 \yr 1989 \pages 283--332 \endref \ref \no 4 \bysame \paper Vector bundles and $SO(3)$-invariants for elliptic surfaces \jour J. Amer. Math. Soc. \vol 8 \yr 1995 \pages 29--139 \endref \ref \no 5 \bysame \book Algebraic Surfaces and Holomorphic Vector Bundles \publ Springer-Verlag \publaddr Berlin Heidelberg New York \yr 1998 \endref \ref \no 6\by R. Friedman and J.W. Morgan \book Smooth Four-Manifolds and Complex Surfaces \bookinfo Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge \vol 27 \publ Springer-Verlag \publaddr Berlin Heidelberg New York \yr 1994 \endref \ref \no 7 \by R. Friedman, J.W. Morgan and E. Witten \paper Vector bundles and $F$ theory \paperinfo hep-th/9701166 \endref \ref \no 8 \bysame\paper Principal $G$-bundles over an elliptic curve \paperinfo alg-geom/9707004 \endref \ref \no 9\by W. Fulton \book Intersection Theory \bookinfo Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge \vol 2\publ Springer-Verlag \publaddr Berlin Heidelberg \yr 1984 \endref \ref \no 10\by R. Hartshorne \book Algebraic Geometry \publ Springer Verlag \publaddr New York Heidelberg Berlin \yr 1977 \endref \ref \no 11\by C. Okonek, M. Schneider and H. Spindler \book Vector Bundles on Complex Projective Spaces \bookinfo Progress in Mathematics {\bf 3}\publ Birkh\"auser \publaddr Boston, Basel, Stuttgart \yr 1980 \endref \ref \no 12\by T. Teodorescu \paperinfo Columbia University thesis \yr 1998 \endref \endRefs \enddocument

9709

alg-geom/9709025

en

https://arxiv.org/abs/alg-geom/9709025

alg-geom/9709025

Christoph Sorger

Christoph Sorger

On Moduli of G-bundles over Curves for exceptional G

Plain TeX, 6 p. Reason for resubmission: proof of main result has been simplified

Let $G$ be a simple and simply connected complex Lie group, ${\goth{g}}$ its Lie algebra. I remove the restriction ``$G$ is of classical type or $G_2$'' made on $G$ in the papers of Beauville, Laszlo and myself [L-S] and [B-L-S] on the moduli of principal G-bundles over a curve. As I will just "patch" the missing technical points, this note should be seen as an appendix to the above cited papers.

alg-geom

\section{Introduction} \par\hskip 1truecm\relax Let $G$ be a simple and simply connected complex Lie group, $\g$ its Lie algebra. In the following, I remove the restriction ``$G$ is of classical type or $G_2$'' made on $G$ in the papers of Beauville, Laszlo and myself \cite{L-S:verlinde},\cite{B-L-S:picard} on the moduli of principal $G$-bundles on a curve. As I will just ``patch" the missing technical points, this note should be seen as an appendix to the above cited papers. \par\hskip 1truecm\relax Let $\M$ be the stack of $G$-bundles on the smooth, connected and projective algebraic curve $X$ of genus $g$. If $\rho:G\ra\SL_{r}$ is a representation of $G$, denote by ${\cal{D}}_{\rho}$ the pullback of the determinant bundle \cite{D-N:picard} under the morphism $\M\ra\MSL$ defined by extension of the structure group. Associate to $G$ the number $d(G)$ and the representation $\rho(G)$ as follows: $${\eightpoint\vbox{\offinterlineskip\def\noalign{\hrule}{\noalign{\hrule}} \halign{\cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&% \cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&% \cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&\cc{$#$}&\tvi\vrule#&% \cc{$#$}\cr \text{Type of }G&&A_{r}&&B_{r}\,(r\geq 3)&&C_{r}&&D_{r}\, (r\geq 4)&&E_{6}&&E_{7}&&E_{8}&&F_{4}&&G_{2}\cr\noalign{\hrule} d(G)&&1&&2&&1&&2&&6&&12&&60&&6&&2\cr\noalign{\hrule} \rho(G)&& \varpi_{1}&&\varpi_{1}&&\varpi_{1}&&\varpi_{1}&&\varpi_{6} &&\varpi_{7}&&\varpi_{8} &&\varpi_{4}&&\varpi_{1}\cr}}}$$ \begin{th}{Theorem}\label{th:pic} There is a line bundle ${\cal{L}}$ on $\M$ such that $\mathop{\rm Pic}\nolimits(\M)\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\reln{\cal{L}}$. Moreover we may choose ${\cal{L}}$ in such a way that ${\cal{L}}^{\otimes d(G)}={\cal{D}}_{\rho(G)}$. \end{th} The above theorem is proved, for classical $G$ and $G_{2}$, in \cite{L-S:verlinde} where it also shown that the space of sections $H^{0}(\M,{\cal{L}}^{\ell})$ may be identified to the space of conformal blocks $B_{G,X}(\ell;p;0)$ (see (\ref{def:conf-blocks}) for its definition). Now, once the generator of the Picard group is known in the exceptional cases, this identification is also valid in general, as well what happens when one considers additionally parabolic structures as we did in \cite{L-S:verlinde} (theorems 1.1 and 1.2). \par\hskip 1truecm\relax In fact, as we will see, to prove theorem \ref{th:pic} for the exceptional groups it is enough to prove the existence of the 60-th root of ${\cal{D}}_{\varpi_{8}}$ on $\MEeight$. This will be deduced from the splitting of a certain central extension, which in turn will follow from the fact that $B_{E_8,X}(1;p;0)$ is one dimensional in any genus $g$ as predicted by the Verlinde formula. However, in our particulier case we don't need the Verlinde formula in order to prove the last statement: it will follow directly from the decomposition formulas. \par\hskip 1truecm\relax Suppose $g(X)\geq 2$. For the coarse moduli spaces $\Mod$ of semi-stable $G$-bundles, we will see that the roots of the determinant bundle of theorem \ref{th:pic} do only exist on the open subset of regularly stable $G$-bundles which, as shown in \cite{B-L-S:picard}, has as consequence the following: \begin{th}{Theorem}\label{th:local_factoriality} Let $G$ be semi-simple and $\tau\in\pi_{1}(G)$. Then $\Mod^{\tau}$ is locally factorial if and only if $G$ is special in the sens of Serre. \end{th} \par\hskip 1truecm\relax Note that $\mathop{\rm dim}\nolimits H^{0}(\MEeight,{\cal{L}})=\mathop{\rm dim}\nolimits B_{E_{8},X}(1;p;0)=1$ has the somehow surprising consequence that the stack $\MEeight$ and (for $g(X)\geq 2$) the normal variety $\ModEeight$ have a {\em canonical} hypersurface. \par\hskip 1truecm\relax I would like to thank C. Teleman for pointing out that a reference I used in a previous version of this paper was incomplete and mention his preprint \cite{Te:BWB}, which contains a different, topological approach to theorem \ref{th:pic}. \section{Conformal Blocks} \subsection{\em Affine Lie algebras. }\label{subsec:Lie-general-set-up} Let $\g$ be a simple finite dimensional Lie algebra of rank $r$ over $\comp$. Let $P$ be the weight lattice, $P_{+}$ be the subset of dominant weights and $(\varpi_{i})_{i=1,\dots,r}$ be the fundamental weights. Given a dominant weight $\lambda$, we denote $L({\lambda})$ the associated simple $\g$-module with highest weight $\lambda$. Finally $(\,,\,)$ will be the Cartan-Killing form normalized such that for the highest root $\theta$ we have $(\theta,\theta)=2$. Let $\Lg=\g\otimes_{\comp}\comp((z))$ be the {\it loop algebra} of $\g$ and $\Lgh$ be the central extension of $\Lg$ \begin{formula}\label{form:cent_ext} 0\efl{}{}\comp\efl{}{}\Lgh\efl{}{}\Lg\lra0 \end{formula} defined by the $2$-cocycle $(X\otimes f,Y\otimes g)\mapsto (X,Y)\mathop{\rm Res}\nolimits_{0}(gdf).$ \par\hskip 1truecm\relax Fix an integer $\ell$. Call a representation of $\Lgh$ of level $\ell$ if the center acts by multiplication by $\ell$. The theory of affine Lie algebras affirms that the irreducible and integrable representations of $\Lgh$ are classified by the dominant weights belonging to $P_{\ell}=\{\lambda\in P_{+}/(\lambda,\theta)\leq\ell\}$. For $\lambda\in P_{\ell}$, denote ${\cal{H}}_{\ell}(\lambda)$ the associated representation. \subsection{\em Definition of conformal blocks. }\label{subsec:conf-blocks} Fix an integer (the level) $\ell\geq0$. Let $(X,\ul{p})$ be an $n$-pointed stable curve (we denote $\ul{p}=(p_{1},\dots,p_{n})$) and suppose that the points are labeled by $\ul{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in P_{\ell}^{n}$ respectively. Choose a non-singular point $p\in X$ and a local coodinate $z$ at $p$. Let $X^{*}=X\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\{p\}$ and $\LgX$ be the Lie algebra $\g\otimes{\cal{O}}(X^{*})$. We have a morphism on Lie algebras $\LgX\ra\Lg$ by associating to $X\otimes f$ the element $X\otimes\hat{f}$, where $\hat{f}$ is the Laurent developpement of $f$ at $p$. By the residue theorem, the restriction to $\LgX$ of the central extension (\ref{form:cent_ext}) splits and we may see $\LgX$ as a Lie subalgebra of $\Lgh$. In particuler, the $\Lgh$-module ${\cal{H}}_{\ell}(0)$ may be seen as a $\LgX$-module. In addition, we may consider the $\g$-modules $L(\lambda_{i})$ as a $\LgX$-modules by evaluation at $p_{i}$. The vector space of conformal blocks is defined as follows: \begin{formula}\label{def:conf-blocks} B_{G,X}(\ell;\ul{p};\ul\lambda)=[{\cal{H}}_{\ell}(0)\otimes_{\comp} L(\lambda_{1})\otimes_{\comp}\dots_{\comp}L(\lambda_{n})]_{\LgX} \end{formula} where $[]_{\LgX}$ means that we take co-invariants. It is known (\cite{TUY:conf-field} or \cite{So:NB794}, 2.5.1) that these vector spaces are finite-dimensional. Important properties are as follows: \par\hskip 1truecm\relax $a)$ $\mathop{\rm dim}\nolimits B_{G,\proj_{1}}(\ell;p_{1};0)=1$ \par\hskip 1truecm\relax $b)$ If one adds a non-singular point $q\in X$, then the spaces $B_{G,X}(\ell;\ul{p};\ul\lambda)$ and $B_{G,X}(\ell;\ul{p},q;\ul\lambda,0)$ are canonically isomorphic (\cite{So:NB794}, 2.3.2). \par\hskip 1truecm\relax $c)$ Suppose $X$ is singular in $c$ and let $\widetilde{X}\ra X$ be a partial desingularization of $c$. Let $a$ and $b$ be the points of $\widetilde{X}$ over $c$. Then there is a canonical isomorphism $$\bigoplus_{\mu\in P_{\ell}} B_{G,X}(\ell;\ul{p},a,b;\ul\lambda,\mu,\mu^{*})\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} B_{G,X}(\ell;\ul{p};\ul\lambda)$$ \par\hskip 1truecm\relax $d)$ The dimension of $B_{G,X}(\ell;\ul{p};\ul\lambda)$ does not change when $(X;\ul{p})$ varies in the stack of $n$-pointed stable curves ${\goth{M}}_{g,n}$ (\cite{TUY:conf-field}). \subsection{\em Application: }\label{subsec:E_8} Consider the case of $G=E_8$ and level $1$ and remark that $P_{1}$ contains {\em only} the trivial representation. In order to calculate $B_{E_8,X}(\ell;p;0)$, one reduces to $\proj_{1}$ with points labeled with the trivial representation using $c)$ and $d)$, then it follows from $b)$ and $a)$ that it is one-dimensional. \section{The Picard group of $\M$} \subsection{} We recall the description of $\mathop{\rm Pic}\nolimits(\M)$ of \cite{L-S:verlinde}, which uses as main tool the {\it uniformization} theorem which I now recall. Let $\LG$ be the loop group $G\bigl(\comp((z))\bigr)$, seen as an ind-scheme over $\comp$, $\LGp$ the sub-group scheme $G\bigl(\comp[[z]]\bigr)$ and $\Q=\LG/\LGp$ be the infinite Grassmannian, which is a direct limit of projective integral varieties ($loc.\, cit.$). Finally let $\LGX$ be the sub-ind group $G({\cal{O}}(X^{*}))$ of $\LG$. The uniformization theorem (\cite{L-S:verlinde}, 1.3) states that there is a canonical isomorphism of stacks $\LGX\bk\Q\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\M$ and moreover that $\Q\ra\M$ is a $\LGX$-bundle. \par\hskip 1truecm\relax Let $\mathop{\rm Pic}\nolimits_{\LGX}(\Q)$ be the group of $\LGX$-linearized line bundles on $\Q$. Recall that a $\LGX$-linearization of the line bundle ${\scr{L}}$ on $\Q$ is an isomorphism $m^{*}{\scr{L}}\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} pr_{2}^{*}{\scr{L}}$, where $m:\LGX\times\Q\ra\Q$ is the action of $\LGX$ on $\Q$, satisfying the usual cocycle condition. It follows from the uniformization theorem that $$\mathop{\rm Pic}\nolimits(\M)\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\mathop{\rm Pic}\nolimits_{\LGX}(\Q),$$ hence in order to understand $\mathop{\rm Pic}\nolimits(\M)$ it suffices to understand $\mathop{\rm Pic}\nolimits_{\LGX}(\Q)$. The Picard group of $\Q$ itself is infinite cyclic; let me recall how its positive generator may be defined in terms of central extensions of $\LG$. \subsection{} If ${\cal{H}}$ is an (infinite) dimensional vector space over $\comp$, we define the $\comp$-space $\mathop{\rm End}\nolimits({\cal{H}})$ by $R\mapsto\mathop{\rm End}\nolimits({\cal{H}}\otimes_{\comp}R)$, the $\comp$-group $GL({\cal{H}})$ as the group of its units and $PGL({\cal{H}})$ by $GL({\cal{H}})/G_{m}$. The $\comp$-group $\LG$ acts on $\Lg$ by the adjoint action which is extended to $\Lgh$ by the following formula: $$\mathop{\rm Ad}\nolimits(\gamma).(\alpha^{\prime},s)=\bigl(\mathop{\rm Ad}\nolimits(\gamma).\alpha^{\prime}, s+\mathop{\rm Res}\nolimits_{z=0}(\gamma^{-1}\frac{d}{dz}\gamma,\alpha^{\prime})\bigr)$$ where $\gamma\in\LG(R)$, $\alpha=(\alpha^{\prime},s)\in\Lgh(R)$ and $(\,,\,)$ is the $R((z))$-bilinear extension of the Cartan-Killing form. The main tool we use is that if $\bar\pi:\Lgh\ra\mathop{\rm End}\nolimits({\cal{H}})$ is an integral highest weight representation, then for $R$ a $\comp$-algebra and $\gamma\in\LG(R)$ there is, locally over $\mathop{\rm Spec}(R)$, an automorphism $u_{\gamma}$ of ${\cal{H}}_{R}={\cal{H}}\otimes_{\comp}R$, unique up to $R^{*}$, such that \begin{formula}\label{form:Faltings} \begin{diagram} {\cal{H}}&\efl{\bar\pi(\alpha)}{}&{\cal{H}}\\ \sfl{u_{\gamma}}{}&&\sfl{}{u_{\gamma}}\\ {\cal{H}}&\efl{\bar\pi(\mathop{\rm Ad}\nolimits(\gamma).\alpha)}{}&{\cal{H}} \end{diagram} \end{formula} is commutative for any $\alpha\in\Lgh(R)$ (\cite{L-S:verlinde}, Prop. 4.3). \par\hskip 1truecm\relax By the above, the representation $\bar\pi$ may be ``integrated" to a (unique) {\it algebraic} projective representation of $\LG$, {\it i.e.\/}\ that there is a morphism of $\comp$-groups $\pi:\LG\ra PGL({\cal{H}})$ whose derivate coincides with $\bar\pi$ up to homothety. Indeed, thanks to the unicity property the automorphisms $u$ associated locally to $\gamma$ glue together to define an element $\pi(\gamma)\in PGL({\cal{H}})(R)$ and still because of the unicity property, $\pi$ defines a morphism of $\comp$-groups. The assertion on the derivative is consequence of (\ref{form:Faltings}). We apply this to the basic representation ${\cal{H}}_{1}(0)$ of $\Lgh$. Consider the central extension \begin{formula}\label{gl(H)-ext} 1\efl{}{} G_{m}\efl{}{} GL({\cal{H}}_{1}(0))\efl{}{} PGL({\cal{H}}_{1}(0))\efl{}{} 1. \end{formula} The pull back of (\ref{gl(H)-ext}) to $\LG$ defines a central extension to which we refer as the {\it canonical} central extension of $\LG$: \begin{formula}\label{can-ext} 1\efl{}{} G_{m}\efl{}{}\LGh\efl{}{}\LG\efl{}{} 1 \end{formula} A basic fact is that the extension (\ref{can-ext}) splits canonically over $\LGp$ (\cite{L-S:verlinde}, 4.9), hence we may define a line bundle on the homogeneous space $\Q=\widehat\LG/\widehat\LGp$ via the character $G_m\times\LGp\ra G_m$ defined by the first projection. Then this line bundle generates $\mathop{\rm Pic}\nolimits(\Q)$ (\cite{L-S:verlinde}, 4.11); we denote by ${\cal{O}}_{\Q}(1)$ its dual. \subsection{} By (\cite{L-S:verlinde}, 6.2) the forgetful morphism $\mathop{\rm Pic}\nolimits_{\LGX}(\Q)\ra\mathop{\rm Pic}\nolimits(\Q)$ is injective, and moreover ($loc.\,cit.$, 6.4), the line bundle ${\cal{O}}_{\Q}(1)$ admits a $\LGX$-linearization if and only if the restriction of the central extension (\ref{can-ext}) to $\LGX$ splits. It is shown in \cite{L-S:verlinde} that this is indeed the case for classical $G$ and $G_2$ by directly constructing line bundles on $\M$ which pull back to ${\cal{O}}_{\Q}(1).$ In one case the existence of the splitting can be proved directly: \begin{th}{Proposition} The restriction of the central extension (\ref{can-ext}) to $\LGX$ splits for $G=E_8$. \end{th} {\it Proof:} Let ${\cal{H}}={\cal{H}}_{1}(0)$. It suffices to show that the representation $\bar\pi:\LgX\ra\mathop{\rm End}\nolimits({\cal{H}})$ integrates to an algebraic representation $\pi:\LGX\ra GL({\cal{H}})$, which in turn will follow from the fact that in the case $\gamma\in\LGX(R)$ we can {\it normalize} the automorphism $u_{\gamma}$ of (\ref{form:Faltings}). Indeed, the commutativity of (\ref{form:Faltings}) shows that coinvariants are mapped to coinvariants under $u_{\gamma}$. For ${\goth{g}}=e_{8}$, $\ell=1$ and $\lambda=0$, we know by (\ref{subsec:E_8}) that these spaces are $1$-dimensional, hence we may choose $u_{\gamma}$ (in a unique way) such that it induces the identity on coinvariants. \cqfd \begin{th}{Corollary} Suppose $G=F_4,E_6,E_7$ or $E_8$. There is a line bundle ${\cal{L}}$ on $\M$ such that the pullback to $\Q$ is ${\cal{O}}_{\Q}(1)$. \end{th} {Proof:} For $E_8$, this follows from the above proposition. Now consider the well known tower of natural inclusions \begin{formula}\label{tower} F_{4}\rInto^{\alpha}_{} E_6\rInto^{\beta}_{} E_7\rInto^{\gamma}_{} E_8. \end{formula} On the level of Picard groups we deduce $$\begin{diagram}[silent] \mathop{\rm Pic}\nolimits(\QEeight)&\rTo^{\tilde f_\alpha^{*}}_{}&\mathop{\rm Pic}\nolimits(\QEseven)& \rTo^{\tilde f_\beta^{*}}_{}&\mathop{\rm Pic}\nolimits(\QEsix)&\rTo^{\tilde f_\gamma^{*}}_{}&\mathop{\rm Pic}\nolimits(\QFfour)\cr \nfl{\pi_{E_8}^{*}}{}&&\nfl{\pi_{E_7}^{*}}{}&&\nfl{\pi_{E_6}^{*}}{}&& \nfl{\pi_{F_4}^{*}}{}\cr \mathop{\rm Pic}\nolimits(\MEeight)&\rTo^{f_\alpha^{*}}_{}&\mathop{\rm Pic}\nolimits(\MEseven)& \rTo^{f_\beta^{*}}_{}&\mathop{\rm Pic}\nolimits(\MEsix)&\rTo^{f_\gamma^{*}}_{}&\mathop{\rm Pic}\nolimits(\MFfour)\cr \nfl{f_{\varpi_{8}}^{*}}{}&&&&&&\ruTo(12.8,2.2)_{f_{\varpi_{8|F_4}}^{*}}\cr \mathop{\rm Pic}\nolimits(\MSLTF8) \end{diagram} $$ The Dynkin index of the representation $\varpi_{8}$ of $E_8$ is 60, and an easy calculation shows that $\varpi_{8|F_4}=14\,\comp\oplus \varpi_{1}\oplus 7\,\varpi_{4}$, hence is equally of Dynkin index 60 (\cite{K-N:picard},\cite{L-S:verlinde}, 2.3). By the Kumar-Narasimhan-Ramanathan lemma (\cite{L-S:verlinde}, 6.8) the determinant bundle ${\cal{D}}$ pulls back, to ${\cal{O}}_{\QEeight}(60)$ via $\pi_{E_8}\circ f_{\varpi_{8}}^{*}$ and to ${\cal{O}}_{\QFfour}(60)$ via $\pi_{F_4}\circ f_{\varpi_{8|F_4}}^{*}$. If follows that $\tilde f_\alpha^{*},\tilde f_\beta^{*}$ and $\tilde f_\gamma^{*}$ are isomorphisms and that the pullback of the line bundle ${\cal{L}}$ on $\MEeight$ under $f_\alpha^{*}$ (resp. $f_\beta^{*}\circ f_\alpha^{*}$, $f_\gamma^{*}\circ f_\beta^{*}\circ f_\alpha^{*}$) pulls back to ${\cal{O}}_{\QEseven}(1)$ (resp. ${\cal{O}}_{\QEsix}(1)$, ${\cal{O}}_{\QFfour}(1))$ \cqfd \subsection{Proof of theorem \ref{th:local_factoriality}: } According to (\cite{B-L-S:picard}, 13) it remains to prove that $\Mod$ is not locally factorial for $G=F_4,E_6,E_7$ or $E_8$. In order to see this we consider again the tower (\ref{tower}) with additionally the natural inclusion $\mathop{\rm Spin}\nolimits_{8}\rInto^{}F_4$. Again the restriction of the representation $\varpi_{8}$ of $E_8$ to $\mathop{\rm Spin}\nolimits_{8}$ has Dynkin index $60$, hence if the generator of $\mathop{\rm Pic}\nolimits(\M)$ would exist on $\Mod$, then the Pfaffian bundle would exist on $\ModSpinEight$, which is not the case (\cite{B-L-S:picard}, 8.2). But the generators exist on the open subset of regularly stable bundles, as the center of $G$ acts trivally on the fibers by construction (we started with the trivial representation) and then the arguments of (\cite{B-L-S:picard}, 13) apply. \cqfd {\eightpoint

9408

alg-geom/9408007

en

https://arxiv.org/abs/alg-geom/9408007

alg-geom/9408007

Caryn Werner

Caryn Werner

A surface of general type with \( p_g =q =0, K^2 =1 \)

13 pages, AMS-LaTex version 1.1

We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain singularities, resolving these, and taking the double cover branched along the resulting smooth curve.

alg-geom

\section{Introduction} In this paper we construct a minimal surface \(X\) of general type with \(\rm{{p}_{g}}=0,\rm{q}=0, {K}^{2}=1,\) and \( \operatorname{Tors} X \cong \Bbb{Z}/{2}\). In \cite{Ca}, Campedelli noted that if a degree ten plane curve could be found having certain singularities, a double plane construction would yield a surface with $\rm{{p}_{g}} = \rm{q} =0.$ In \cite{OP}, Oort and Peters construct such a double plane and compute the torsion group of their surface to be $\Bbb{Z}/{2}$; however we will show that the torson group is actually $\Bbb{Z}/{4}$. (Weng Lin has also constructed surfaces with $\rm{{p}_{g}}=\rm{q}=0,K^{2} =1$ using double covers, but I do not believe his results are published.) For minimal surfaces of general type with $\rm{{p}_{g}}=\rm{q}=0$ and ${K}^{2}=1$ it is known that \( \left| \operatorname{Tors} X \right| \leq 5.\) (See for example \cite{Do}.) By writing down generators for the pluricanonical rings, Reid \cite{R} has described surfaces with torsion of order three, four, and five. Barlow \cite{Ba1,Ba2} has constructed surfaces with torsion of order two and four, as well as a simply connected surface. The double plane constructions we call {\em Campedelli surfaces}, while {\em numerical Godeaux surfaces} are minimal surfaces of general type with the invariants $\rm{p_{g}}=\rm{q}=0, {K}^{2}=1.$ Here \(\rm{{p}_{g}} = \dim \operatorname{H}^{0}\left( X,\cal{O}_{X}\left( {K} \right)\right) = \dim \operatorname{H}^{2} \left(X, \cal{O}_{X} \right), \, \rm{q} = \dim \operatorname{H}^{1} \left( X, \cal{O}_{X} \right), \) and \( {K}^{2} = {K} \cdot {K} \) is the self-intersection number of the canonical class \({K}.\) Write $h^{i}\left(D\right)= \dim_{\Bbb{C}} H^{i}\left(X,\cal{O}_{X}\left(D\right)\right)$ for $D$ a divisor on $X,$ \( \operatorname{Tors} X\) for the torsion subgroup of the Picard group of $X,$ \( \equiv\) to represent linear equivalence of divisors, and \(\left| D \right|\) for the complete linear system of a divisor class $D.$ \section{The double plane construction.} Let $D$ be a degree ten plane curve with an ordinary order four point at $p,$ five infinitely near triple points at $p_{1}, \dots, p_{5},$ and no other singularities. An infinitely near triple point refers to a triple point which remains of order three after the plane is blown up at this point, so that all three tangent directions of $D$ coincide. We assume that each triple point becomes ordinary after one blow up. Assume further that the six singular points do not lie on a conic, and that the system of plane quartics with double point at $p$ and through each $p_{i}$ with the same tangent direction as $D$ is exactly a pencil. Let \( \sigma_{1} :{Y}_{1} \rightarrow \Bbb{P}^{2} \) be the blowup of \( \Bbb{P}^{2} \) at ${p},$ and let ${E}=\sigma_{1}^{-1}({p})$ be the exceptional curve on ${Y}_{1}.$ The total transform of ${D}$ is \( \sigma_{1}^{\ast} \left( D \right) = \Bar{{D}} + 4 {E},\) where \( \Bar{{D}} \) is the proper transform of $D$. Set \( {D}_{1}=\Bar{{D}} = \sigma_{1}^{\ast} \left( D \right) -4{E}.\) Now let \( \sigma_{2} : {Y}_{2} \rightarrow {Y}_{1} \) be the blowup of ${Y}_{1}$ at \( {p}_{1}, \dots, {p}_{5}. \) With \( {E}_{i} =\sigma_{2}^{-1} \left( {p}_{i} \right), \) the total transform of \( {D}_{1} \) is \( \Bar{{D}_{1}} + 3 \displaystyle{\sum_{1}^{5} {E}_{i},} \) where $\Bar{{D}_{1}}$ is the proper transform of $D_{1}.$ Set \[ {D}_{2} = \Bar{{D}_{1}} + \sum_{1}^{5} {E}_{i} \equiv \sigma_{2}^{\ast}\left( {D}_{1} \right) -2 \sum_{1}^{5} {E}_{i}, \] which is the reduced divisor consisting of the proper transform of the degree ten curve, together with the five exceptional curves $E_{i}.$ As each ${p}_{i}$ is an infinitely near triple point of the original branch curve, ${D}_{2}$ has an order four point on each ${E}_{i}.$ Let \( \sigma_{3} : {Y}_{3} \rightarrow {Y}_{2} \) be the blowup of each of these quadruple points, and let \( {F}_{1}, \dots, {F}_{5} \) be the corresponding exceptional divisors. We will write ${E}_{i}$ to denote both the exceptional curve on $Y_{2}$ and its proper transform on $Y_{3}$ ( and similarly for ${E}$) so that \( \sigma_{3}^{\ast} \left( {E}_{i} \right) = {E}_{i} + {F}_{i}. \) The total transform of ${D}_{2}$ is \( \displaystyle{\Bar{{D}_{2}} + 4 \sum_{1}^{5} {F}_{i},} \) where $\Bar{{D}_{2}}$ is the proper transform of $D_{2};$ set \[ \begin{array}{cl} {B} &= \Bar{{D}_{2}} \\ &\equiv \sigma_{3}^{\ast} \left( {D}_{2} \right) -4 \sum {F}_{i} \\ &= \sigma_{3}^{\ast} \left( \sigma_{2}^{\ast} \left( \sigma_{1}^{\ast} \left( {D} \right) -4 {E} \right) -2 \sum {E}_{i} \right) -4 \sum {F}_{i} \\ &= \sigma^{\ast} \left( {D} \right) -4 {E} -2 \sum {E}_{i} -6 \sum {F}_{i} \end{array} \] where $\sigma = \sigma_{1} \circ \sigma_{2} \circ \sigma_{3}.$ If $H$ represents the pullback of the hyperplane class in \( \Bbb{P}^{2},\) then \[ {B} \equiv 10H -4{E} -2 \sum_{1}^{5}{E}_{i} -6\sum_{1}^{5}{F}_{i} = 2 \cal{L} \] where \[ \cal{L} \equiv 5H -2E-\sum {E_{i}} -3 \sum{F_{i}}; \] \( {B} \) is now a non-singular even curve on the surface \( {Y}_{3} . \) The canonical divisor on \({Y}_{3}\) is \[ {K}_{{Y}_{3}} \equiv \sigma^{\ast}\left(K_{\Bbb{P}^{2}}\right) +{E} +\sum_{1}^{5} {E}_{i} +2\sum_{1}^{5}{F}_{i} \equiv -3H +{E} +\sum_{1}^{5} {E}_{i} +2\sum_{1}^{5} {F}_{i} .\] Let \( \pi : {X} \rightarrow {Y}_{3} \) be the double cover of ${Y}_{3}$ branched at \( {B}. \) Then \[ \begin{array}{ll} {K}_{{X}} &\equiv \pi^{\ast} \left( {K}_{{Y}_{3}} + \cal{L} \right) \\ &\equiv \pi^{\ast} \left( 2 {H} - {E} - \sum {F}_{i} \right) \end{array} \] and \( {{K}_{X}}^{2} = 2 {\left[ 2 {H} -{E} -\sum {F}_{i} \right]}^{2} = 2 \left( 4 -1 +5\left(-1 \right) \right) = -4. \) Since each ${E}_{i}$ is part of the branch locus and \( {{E}_{i}}^{2} = -2 \) on \( Y_{3}, {\left[ \pi^{-1} \left( {E}_{i} \right) \right] }^{2} = -1.\) Let \({X} \rightarrow \tilde{{X}} \) be the map contracting these five \( \left( -1 \right) \) curves. Then \[ \left( {K}_{\tilde{{X}}} \right)^{2} = \left( {K_{X}}\right)^{2} +5 = 1.\] \begin{prop} \( \tilde{\bf{X}} \) is a minimal surface of general type with \( \rm{p_{g}} =0, \rm{q} =0,\mbox{and} \;{K}^{2} =\chi = 1.\) \end{prop} Since \(\tilde{\bf{X}}\) is obtained from \(\bf{X}\) by blowing down five exceptional curves, we can compute \( \rm{{p}_{g}}\) and \(\rm{q}\) for the surface \(\bf{X}.\) To compute these invariants, we will use the following. \vskip .25 cm \noindent{\bf Projection formula.} \begin{sl} Let \( \pi: X \rightarrow Y\) be a double cover branched along a smooth curve $B \equiv 2\cal{L}.$ For any divisor ${\cal A}$ on $Y,$ \[ \pi_{\ast}\cal{O}_{X}\left( \pi^{\ast}\cal{A}\right) \cong \cal{O}_{Y}\left(\cal{A}\right) \oplus \cal{O}_{Y}\left( \cal{A}-\cal{L}\right). \] \end{sl} In particular, \[ \pi_{\ast} \cal{O}_{X}\left( nK_{X}\right) \cong \cal{O}_{Y}\left( nK_{Y}+n\cal{L}\right) \oplus \cal{O}_{Y}\left( nK_{Y} + \left(n-1\right)\cal{L}\right). \] Therefore in our example, \[ \operatorname{H}^{0}\left(\cal{O}_{{X}}\left( {K}_{X} \right)\right) \cong \operatorname{H}^{0}\left(\cal{O}_{{{Y}_{3}}} \left( {K}_{{{Y}_{3}}}+\cal{L} \right)\right) \oplus \operatorname{H}^{0}\left(\cal{O}_{{Y}_{3}} \left( {K}_{{{Y}_{3}}} \right)\right), \] so that \[ p_{g} \left( {X} \right)= h^{0}\left( \cal{O}_{X}\left({K_{X}} \right) \right) = h^{0} \left( {K}_{{{Y}_{3}}} + \cal{L} \right) + p_{g} \left( {{Y}_{3}} \right). \] Since \( p_{g} \left( {{Y}_{3}} \right) = p_{g} \left( \Bbb{P}^{2} \right) =0, \) \( p_{g} \left( {X} \right) = h^{0} \left(K_{{Y}_{3}}+\cal{L}\right).\) The space $\operatorname{H}^{0}\left( K_{{Y}_{3}}+\cal{L}\right)$ corresponds to the the linear system \(\left| 2 {H} - {E} -\sum {F}_{i} \right| \) of conics through \( {p}, {p}_{1}, \dots, {p}_{5}, \) so \( p_{g} \left( {X} \right) = 0.\) Also \[ \operatorname{H}^{0} \left( \cal{O}_{{X}} \left( 2 {K}_{{X}} \right) \right) = \operatorname{H}^{0} \left( \cal{O}_{{{Y}_{3}}} \left( 2 {K}_{{{Y}_{3}}} + 2 \cal{L} \right) \right) \oplus \operatorname{H}^{0} \left( \cal{O}_{{{Y}_{3}}} \left( 2 {K}_{{{Y}_{3}}} + \cal{L} \right) \right). \] Since \( \displaystyle{ 2K_{{Y}_{3}} + \cal{L} \equiv -H +\sum_{1}^{5}{\left( E_{i} +F_{i}\right)}, } \) \( \operatorname{H}^{0}\left( 2K_{{Y}_{3}} +\cal{L}\right) =0.\) The divisor $\sum E_i$ is a fixed part of the linear system \[ \left| 2 {K}_{{{Y}_{3}}} + 2 \cal{L} \right| = \left| 4 {H} - 2 {E} -2 \sum_{1}^{5} {F}_{i} \right| \] since $E_i \cdot F_i =1$ and $E_i^2 =-2$; the difference \( \left| 4H-2E-2\sum F_i -\sum E_i \right| \) corresponds to quartics in \( \Bbb{P}^{2} \) with a double point at \( \it{p} ,\) through each \( \it{p}_{i} \) with the same tangent direction as the branch curve. By assumption this system is a pencil, thus \[ \rm{P}_{2} = h^{0}\left( 2 {K}_{{X}} \right) = \dim \operatorname{H}^{0} \left( \cal{O}_{{X}} \left( 2 {K}_{{X}} \right) \right) = 2. \] Suppose $S$ is the minimal model of $\tilde{X};$ then $\rm{P}_{2}\left( S \right) = 2$ and ${K_{S}}^{2} \geq {K_{\tilde{X}}}^{2} =1,$ so $S$ is of general type (see for example \cite{BPV}). But $2=\rm{P}_{2} = \chi + K_{S}^{2} =1 +K_{S}^{2},$ so $K_{S}^{2} =K_{\tilde{X}}^{2}$ and $S=\tilde{X}.$ Thus \( \tilde{X} \) is minimal of general type with $K^2 =1$; since $q \leq p_g $ \cite[\S{3}.1,lemma 3]{Do}, \( p_{g} = q =0 \). \section{The branch curve {D}} \newcommand{F_{\phi}}{F_{\phi}} \newcommand{p,p_{1},\dots,p_{5}}{p,p_{1},\dots,p_{5}} To construct a plane curve of degree ten with the necessary singularities, we will find an octic and a conic as follows. We wish to find an octic $C$ with one order four point, one infinitely near triple point, and four tacnodes, where a tacnode refers to a double point which remains double after one blowup. Furthermore we want these tacnodes to lie on a conic $Q$ with the same tangent direction, so that the octic and conic will still intersect after the plane is blown up at these points. Let $F$ be a homogeneous polynomial of degree eight in three variables defining an octic ${C}$ in $\Bbb{P}^{2}.$ After imposing an order four point at ${p} = [1:0:0]$ and an infinitely near triple point at ${p}_{1} = [0:1:0],$ $F$ has $23$ free coefficients. Let \[ \begin{array}{cl} \gamma : \Bbb{P}^{1} &\rightarrow {Q} \\ \left[ s : t \right] &\rightarrow \left[ as^2+bst+ct^2 : ds^2+est+ft^2 : gs^2+hst+it^2 \right] \end{array} \] be a parametrization of a conic \( {Q} \) in \( \Bbb{P}^{2}, \) where \( a,b,c,d,e,f,g,h,i \) are variables over $\Bbb{C}.$ Set \[ \begin{array}{cl} {p}_{2} &= \gamma([0:1]) \\ {p}_{3} &= \gamma([1:0]) \\ {p}_{4} &= \gamma([1:1]) \\ {p}_{5} &= \gamma([-1:1]). \end{array} \] The condition that $F$ have a double point at $p_{i}$ can be expressed by requiring the three partial derivatives of $F$ at $p_{i}$ to vanish, thus a double point is three linear conditions on the coefficients of $F;$ a tacnode at a given point with a designated tangent direction puts six conditions on $F,$ while a cusp at a given point with a given tangent direction is five linear conditions on the coefficients. If we impose tacnodes tangent to $Q$ on the octic at \( {p}_{2} \) and \( {p}_{3} \), this gives twelve linear relations on the coefficients of $F.$ Imposing cusps tangent to $Q$ at \( {p}_{4} \) and \( {p}_{5} \) gives ten more relations; solving these gives an octic whose coefficients are polynomials in \( a,b,c,d,e,f,g,h,i .\) Imposing the conditions that \( {p}_{4} \) and \( {p}_{5} \) be tacnodes of \( {C} \) gives two more linear relations in the coefficients of $F,$ and therefore two higher degree polynomials in \( a,b,c,d,e,f,g,h,i .\) In solving these two relations for \(a,b,c,d,e,f,g,h,\mbox{and} \, i\) we hope to obtain an irreducible polynomial \(F\) over \( \Bbb{C}, \) and thus an octic plane curve as desired. We use Maple to compute the equations for these conditions on $F,$ and to find the coefficients. Let \( \left\{ \rm{A}_{j} \right\}_{1}^{22} \) be the equations corresponding to these conditions on \( F; \) the \( \rm{A}_{j} \) are linear in the coefficients of $F.$ Form the matrix {\bf M} generated by the \( \rm{A}_{j} \) where {\bf M}\(_{i,j} \) is the coefficient in \( \rm{A}_{j} \) of the \( ith \) coefficient of \(F. \) Then {\bf M} is a $22 \times 23$ matrix, and if we set {\bf D}\(_{j}\) to be the determinant of the matrix obtained from {\bf M} be deleting the $j$th column, we have {\bf M}\(((-1)^{j}\){\bf D}\(_{j}) =0, \) so that setting the $jth$ coefficient of $F$ to be $(-1)^{j}${\bf D}$_{j}$ gives the desired octic. In order for Maple to compute these determinants quickly enough, we first set \(a=e=g=i=1\) and \(b=d=h=0\) in the parametrization of ${Q}.$ This reduces the number of free parameters in this problem to two, namely $c$ and $f;$ since we will end up imposing two non-linear conditions on the remaining parameters, there is still the possibility of a non-degenerate solution. After finding the determinants {\bf D}$_{j}$, the coefficients of $F$ become polynomials in $c$ and $f.$ Imposing the final two conditions on the octic, Maple finds several degenerate solutions, where the octic splits into several curves of smaller degree, and thus has more singularities, and also a solution for $c$ and $f$ giving an octic which we will show has the desired properties. The branch curve $D$ is defined by the equations for the conic and the octic, which are both polynomials in three variables over \( \Bbb{Z} \left[ \alpha, \beta, \delta \right] \) where \[ \begin{array}{ll} \alpha &= \sqrt{17} \\ \beta &= \sqrt{21 +5 \sqrt{17}} \\ \delta &= \sqrt{5 + \sqrt{17}}. \end{array} \] The polynomial defining the conic $\cal{Q},$ which is given parametrically by $\gamma,$ is \[ \begin{array}{l} \left( 9\,\alpha \beta +90\,\alpha +81\, \beta +234\, \right) x^{2} +\left( +176\,\alpha \beta +1568\,\alpha +1200\, \beta+5920 \right) y^{2} \\ + \left( 57\,\alpha \beta +258\, \alpha +129\, \beta +1170 \right) z^{2} +\left( -48\,\alpha \beta \delta -168\, \alpha \delta -48\, \beta \delta -936\,\delta \right) xy \\ +\left( -66\, \alpha \beta -348\,\alpha -210\,\beta -1404 \right) xz +\left( 48\, \alpha \beta \delta +168\, \alpha \delta +48\,\beta \delta +936\, \delta \right) yz \end{array} \] The octic ${C}$ is defined by $F=0$ where $F$ is \[ \begin{array}{c} 24\,\left (14408408592\,x^{4}y^{2}z+50076004923\,x^{4}z^{3}+ 14182182144\,x^{3}y^{4} \right. \\ \left. +219953469600\,x^{3}y^{2}z^{2}-363210576777\,x^ {3}z^{4}-1093337332608\,x^{2}y^{2}z^{3} \right. \\ \left. +831133690121\,x^{2}z^{5} + 858975454416\,xy^{2}z^{4}-772939669603\,xz^{6} \right. \\ \left. +254940551336\,z^{7} \right )y \alpha\beta\delta \\ +\left (-72389196288\,x^{4}y^{4}-3335393797632\,x^{4}y^{2} z^{2}-1065820046526\,x^{4}z^{4} \right. \\ \left. -8342111361024\,x^{3}y^{4}z + 3945428471808\,x^{3}y^{2}z^{3} +10184161263912\,x^{3}z^{5} \right. \\ \left. + 20168534212608\,x^{2}y^{4}z^{2}+53110876008192\,x^{2}y^{2}z^{4}- 32270723397636\,x^{2}z^{6} \right. \\ \left. -100932292129536\,xy^{2}z^{5}+38252243189640 \,xz^{7}+47211381447168\,y^{2}z^{6} \right. \\ \left. -15099861009390\,z^{8}\right ) \alpha \beta + \end{array} \] \[ \begin{array}{c} 144\,\left (15490159728\,x^{4}y^{2}z+53840161671\,x^{4}z^{3}+ 15251365120\,x^{3}y^{4} \right. \\ \left. +236488232416\,x^{3}y^{2}z^{2}-390512591333\,x^ {3}z^{4}-1175521621376\,x^{2}y^{2}z^{3} \right. \\ \left. +893608694925\,x^{2}z^{5} + 923543229232\,xy^{2}z^{4}-831040262535\,xz^{6} \right. \\ \left. +274103997272\,z^{7} \right )y \alpha \delta \\ +24\,\left (59398585488\,x^{4}y^{2}z+206468708787\,x^{4}z^{ 3}+58496365824\,x^{3}y^{4} \right. \\ \left. +906899335584\,x^{3}y^{2}z^{2}-1497555836337 \,x^{3}z^{4}-4507944789888\,x^{2}y^{2}z^{3} \right. \\ \left. +3426852351793\,x^{2}z^{5} + 3541646868816\,xy^{2}z^{4}-3186912029723\,xz^{6} \right. \\ \left. +1051146805480\,z^{7} \right )y \beta \delta \\ +\left (-466877917440\,x^{4}y^{4}-21516582641184\,x^{4}y^{2 }z^{2}-6875617032333\,x^{4}z^{4} \right. \\ \left. -53815549731840\,x^{3}y^{4}z +25451977456512\,x^{3}y^{2}z^{3}+65698123816692\,x^{3}z^{5} \right. \\ \left. + 130107481479168\,x^{2}y^{4}z^{2} +342618718898784\,x^{2}y^{2}z^{4}- 208178748305934\,x^{2}z^{6} \right. \\ \left. -651115201689280\,xy^{2}z^{5} + 246765593291124\,xz^{7}+304561087975168\,y^{2}z^{6} \right. \\ \left. -97409351769549\,z^ {8}\right ) \alpha \\ +\left (-298344909312\,x^{4}y^{4}-13752106145280\,x^{4}y^{ 2}z^{2}-4394491299054\,x^{4}z^{4} \right. \\ \left. -34395989170176\,x^{3}y^{4}z + 16267404201984\,x^{3}y^{2}z^{3} +41990375439720\,x^{3}z^{5} \right. \\ \left. + 83157067186176\,x^{2}y^{4}z^{2}+218981688444672\,x^{2}y^{2}z^{4}- 133055599121316\,x^{2}z^{6} \right. \\ \left. -416154499902208\,xy^{2}z^{5}+ 157718037119688\,xz^{7}+194657513400832\,y^{2}z^{6} \right. \\ \left. -62258322139038\,z^ {8}\right ) \beta \\ +48\,\left (191605550544\,x^{4}y^{2}z+665965983645\,x^{4}z ^{3}+188641373952\,x^{3}y^{4} \right. \\ \left. +2925195141792\,x^{3}y^{2}z^{2} -4830373865031\,x^{3}z^{4}-14540399432832\,x^{2}y^{2}z^{3} \right. \\ \left. + 11053328983807\,x^{2}z^{5} +11423598740496\,xy^{2}z^{4}-10279400307101 \,xz^{6} \right. \\ \left. +3390479204680\,z^{7}\right )y \delta \\ -1925078503680\,x^{4}y^{4}- 88715185482528\,x^{4}y^{2}z^{2}-28348893570645\,x^{4}z^{4} \\ - 221886790124544\,x^{3}y^{4}z+104941198124928\,x^{3}y^{2}z^{3}+ 270880301999124\,x^{3}z^{5} \\ +536446841591808\,x^{2}y^{4}z^{2}+ 1412653204386144\,x^{2}y^{2}z^{4}-858342969385662\,x^{2}z^{6} \\ - 2684616751584448\,xy^{2}z^{5} +1017440607056532\,xz^{7}+ 1255737534555904\,y^{2}z^{6} \\-401629046099349\,z^{8}. \end{array} \] The resulting singular points of the branch curve are \[ \begin{array}{ll} p &= [1:0:0] \\ p_{1} &= [0:1:0] \\ p_{2} &= [{{10 +{4\,\alpha}+{4\,\beta}}}:{ 3{\delta}}:6] \\ p_{3} &= [1:0:1] \\ p_{4} &= [{{ {16} +{4\,\alpha}+{4\,\beta}}}:{3{\delta}}+6:12] \\ p_{5} &= [{{16} +{4\,\alpha}+{4\,\beta}}:{ 3{\delta}}-6:12]. \end{array} \] We need to check that the branch curve $D$ has no singularities outside the set $\left\{ p,p_{1},\dots,p_{5} \right\}.$ Since $F$ is a polynomial over the complex numbers, Maple is unable to quickly check that the octic has no other singularities, so we use Macaulay to check the smoothness of ${C}$ outside of the set \( \left\{p,p_{1},\dots,p_{5} \right\}.\) As Macaulay only makes computations over finite fields, we first find a prime number $P$ where $\alpha,\beta,\mbox{and} \, \delta$ exist mod $P,$ so that we can map $F$ to a polynomial over $\Bbb{Z}/{P}.$ To check that ${C}$ has no singularities other than at the points \( {p}, {p}_{1}, \dots, {p}_{5}, \) consider the map \( \phi: \Bbb{Z}[\alpha,\beta,\delta] \rightarrow \Bbb{Z}/30047 \) given by sending \[ \begin{array}{ll} \alpha &\rightarrow 20452 \\ \beta &\rightarrow 6941 \\ \delta &\rightarrow 27962; \end{array} \] mapping $F$ to $F_{\phi}$ we obtain \[ \begin{array}{c} 24082\,x^{4}y^{4}+3438\,x^{4}y^{3}z+4775\,x^{4}y^{2}z^{2}+29499\,x^{4} yz^{3}+12698\,x^{4}z^{4} \\ +29927\,x^{3}y^{5}+14121\,x^{3}y^{4}z+17243\,x ^{3}y^{3}z^{2}+3139\,x^{3}y^{2}z^{3}+8704\,x^{3}yz^{4}+80\,x^{3}z^{5} \\ + 28712\,x^{2}y^{4}z^{2}+10654\,x^{2}y^{3}z^{3}+12817\,x^{2}y^{2}z^{4}+ 8239\,x^{2}yz^{5}+5515\,x^{2}z^{6} \\ +28759\,xy^{3}z^{4}+7372\,xy^{2}z^{5 }+19696\,xyz^{6}+28079\,xz^{7} \\ +1944\,y^{2}z^{6}+24003\,yz^{7}+13722\,z^{8}. \end{array} \] \begin{claim} $F_{\phi}$ has no singularities other than at $p,p_{1},\dots,p_{5}.$ \end{claim} First we have Macaulay compute $\operatorname{Jac} F_{\phi},$ the Jacobian ideal of the octic generated by \(\displaystyle{ {{\partial{F_{\phi}}} \over {\partial{x}}}, {{\partial{F_{\phi}}} \over {\partial{y}}}, \mbox{and} {{\partial{F_{\phi}}} \over {\partial{z}}}, }\) and the ideal $\cal{I}$ associated to the points \( {p},{p}_{1},{p}_{2},p_{3},p_{4},p_{5}.\) Since the zeros of \( \operatorname{Jac}\left( F_{\phi} \right) \) are precisely the singular points of the octic, the zeros of the saturation of \( \operatorname{Jac} \left( F_{\phi} \right) \) by \( \cal{I} \) are any singularites other than at the zeros of $\cal{I}.$ Macaulay computes \[ \left( \operatorname{Jac}\left( {F_{\phi}} \right) : \cal{I}^{\infty} \right) = \left\{ g : g \cal{I}^{n} \subset \operatorname{Jac} F_{\phi} \, \mbox{for some n} \right\} = \left( 1 \right), \] thus there are no zeros of $\operatorname{Jac} F_{\phi}$ other than at the points \(p, p_{1},\dots,p_{5}\) and therefore no other singularities of $C_{\phi}.$ \begin{claim} To check that $F$ has no singularities outside the set $\left\{ p,p_{1}, \dots, p_{5} \right\},$ it suffices to check this for the polynomial $F_{\phi}$ over \( \Bbb{Z}/{30047}.\) \end{claim} It is easy to check, using Maple, that $C_{\phi}$ has an ordinary quadruple point at $p$, and after one blow up, the triple point at $p_{1}$ and the double points at $p_{2},\dots,p_{5}$ become ordinary. Since $C$ maps to $C_{\phi}$, the same is true for the singularities on $C$. Maple is not reliable about completely factoring polynomials in many variables; hence Maple cannot check directly that $C$ is irreducible. Hence we fall back on a more case-by-case analysis. Maple {\em can} check that a given polynomial divides another; and so one can use Maple to conclude that $Q$ is not a component of $C$. Next note that since $\deg Q = 2$ and $\deg C = 8$, $Q \cdot C =16$. We know that $C$ and $Q$ meet four times at each $p_{i}, i=2,\dots,5$; thus $Q$ cannot meet any component of $C$ at any other point. We check that none of the tangent lines to $C$ at any $p_i$ are contained in $C$. If any other line was a component of $C$, say $C=\ell G$, then $\left( G \cdot Q \right) =14$; however $G$ must meet $Q$ four times each at $p_2,\dots,p_5$, so no line can be contained in $C$. Suppose a conic $G$ is a component of $C$. Then $G$ must meet $Q$ at two of the four points, say $p_{i}$ and $p_{j}$, with the proper tangent directions, to multiplicity two. {Case 1.} If $C$ breaks up into $G$ and an irreducible sextic $S$, then $S$ must have at least a triple point at $p$ and tacnodes at $p_{k},p_{l}$ for $l,k \neq i,j$; since there is no conic through $p_{1},p_{i},p_{j}$ with the required tangent directions, $S$ must have an infinitely near triple point at $p_{1}$. But these conditions would drop the genus of $S$ by $13$, while an irreducible degree six curve has genus $10$, so no such sextic exists. {Case 2.} If $C$ breaks up into two conics $G$ and $H$ and a degree four part, then $G$ meets $Q$ at $p_{i},p_{j}$, $H$ meets $Q$ at $p_{k},p_{l}$, so neither conic can pass through $p_{1}$. Therefore the degree four part of $C$ would have to have an infinitely near triple point, which is impossible (even for a reducible quartic). {Case 3.} If $C$ is composed of a conic $G$ and two cubics $S_{1},S_{2}$, then one of the cubics must have a tacnode at $p_{1}$, which is impossible. Thus the octic $C$ cannot contain either a line or a conic as a component. We can conclude therefore that if $C$ does split, it splits into at most two components (of degrees $3$ and $5$ or $4$ and $4$). Suppose $C$ is composed of a cubic $G$ and a quintic $S$, both of which are irreducible. The arithmetic genus of $S$ is six, and $S$ must have at least a double point at $p$ and a tacnode at $p_{1}$, which together drop the genus by three. Since $Q \cdot G =6$, $G$ must meet $Q$ at three of the $p_{i}$, thus $S$ must have a tacnode along $Q$ (at the fourth point) which drops the genus by two more. Thus $S$ can have exactly a double point at $p$ and a tacnode at $p_{1}$, and $G$ must have a double point at $p$ and pass through $p_{1}$ and three of the $p_{i}$ with the necessary tangent directions. But no such cubics exist, as can be checked using Maple; (this gives $11$ linear conditions on the cubic, and Maple checks that this linear system has no solutions). Next, suppose $C$ is composed of two irreducible quartics $G$ and $S$. Then one of the quartics, say $G$, must have a tacnode at $p_{1}$ and pass through $p$. Also $G$ must meet $Q$ along $p_{2},\dots,p_{5}$. But these are all linear conditions on the quartic, and again Maple can be used to check that there are no such quartics. Thus the octic $C$ is irreducible. Since $C$ is irreducible, we can compute the arithmetic genus to be $\left( \begin{array}{c} 7 \\ 2 \end{array} \right) = 21;$ after blowing up a point of multiplicity $n,$ the genus of the proper transform goes down by $\left( \begin{array}{c} n \\ 2 \end{array} \right).$ After resolving the singularities of $C$ at $p,p_{1},\dots,p_{5},$ the resulting curve has genus equal to \[ 21 - \left( \begin{array}{c} 4 \\ 2 \end{array} \right) - 2 \left( \begin{array}{c} 3 \\2 \end{array} \right) -8 \left( \begin{array}{c} 2 \\ 2 \end{array} \right) = 1, \] thus $C$ can have at most one more singularity of multiplicity two. We will now prove that $C$ has no other singularities than the known ones at $p$ and $p_1,\ldots,p_5$. The curve $C$ is defined over the field $K={\bf Q}(\alpha,\beta,\delta)$, as is its strict transform $\bar C$ after resolving the singularities at $p,p_1,\dots,p_5$. Suppose that $\bar C$ is singular; since it can have at most one singularity, the coordinates of this singular point are then invariant by the action of the Galois group of the algebraic closure of $K$ over $K$, hence lie in $K$. Thus the normalization $\tilde C$ of the curve $\bar C$ is defined over $K$. Since the genus of $\tilde C$ is $0$, its anti-canonical map induces an isomorphism, defined over $K$, onto a smooth conic in ${\bf P}^2_K$. Since the curve $\tilde C$ has a rational point over $K$ (namely the eighth point of intersection of the line $z=0$ with the curve $C$: this line meets $C$ four times at $p$,three times at $p_1$, and then once at a point with coordinates in $K$), the projection from this point yields an isomorphism defined over $K$ between $\tilde C$ and ${\bf P}^1_K$. By composing with the map $\tilde C\rightarrow C$ (also defined over $K$), we obtain a parametrization $\psi:{\bf P}^1_K\rightarrow C$ defined over $K$; by clearing denominators we can take $\psi$ to be defined over ${\Bbb Z} \left[ \alpha, \beta, \delta \right]$. Since ${\Bbb Z} \left[ \alpha,\beta, \delta \right]$ maps to ${\Bbb Z}/{30047}$, we get a map ${\Bbb P}^{1}_{{\Bbb Z}/{30047}} \rightarrow C_{\phi}$. Thus $C_{\phi}$ is rational over ${\Bbb Z}/{30047}$, so the genus is zero and the genus of $C_{\phi}$ over the algebraic closure of ${\Bbb Z}/{30047}$ is also zero. But Macaulay can and does check that $C_{\phi}$ has no other singularities in the algebraic closure of the finite field; so $\bar{C_{\phi}}$ is smooth and its genus must be one (using the genus formula, which is essentially adjunction). Therefore the genus of $\bar{C}$ must be one as well, which gives a contradiction. Hence $C$ can have no other singularities. We can also use Maple to check that the system of quartics with a double point at $p,$ through each $p_i$ with the necessary tangent direction is a pencil; thus $C$ and $Q$ give a degree ten curve as needed. \section{The torsion group of \( \tilde{X} \)} The following lemma will show that the torsion group is non-trivial. \begin{lemma}(Beauville \cite{B}) Let \( Y\) be a smooth surface with \(\operatorname{Tors}(\operatorname{Pic}(Y)) =0,\) \( \left\{ C_{i} \right\}_{i \in I} \) a collection of smooth disjoint curves on \( Y ,\) and \( \pi : X \rightarrow Y \) a connected double cover branched along \( \cup_{i \in I}C_{i}.\) Define a map \[ \varphi : {\Bbb{Z}/{2}}^{I} \rightarrow \operatorname{Pic} Y \otimes \Bbb{Z}/{2} \] by sending \( \sum n_{i} C_{i} \, \) to its class in \( \operatorname{Pic} Y. \) If \( e = \sum_{i \in I} C_{i} \), then the group $\operatorname{Pic}_{2} X$ of $2-$ torsion elements in \( \operatorname{Pic} X\) is isomorphic to \(\rm{ker}\left( \varphi \right) / \left(\Bbb{Z}/{2}\right)e.\) \end{lemma} If \( \sum_{i \in J} C_{i} \equiv 2A\) for some divisor \(A, \) where $J$ is a subset of $I,$ then the map from \( \rm{ker} \left( \varphi \right) \) to the $2-$torsion elements in \( \operatorname{Pic} X \) sends \( \sum_{i \in J} C_{i} \) to \( \sum_{i \in J} \pi^{-1} \left( C_{i} \right) - \pi^{\ast} \left(A \right) ; \) for components $C_{i}$ of the branch locus \[ 2 \pi^{-1} \left( C_{i} \right) \equiv \pi^{\ast} \left( C_{i} \right) , \] so that \( \sum_{i \in J} \pi^{-1} \left( C_{i} \right) - \pi^{\ast} \left( A \right) \) is in \( \operatorname{Pic}_{2} \left( X \right) .\) Let $\bar{Q}$ be the strict transform of $Q$ on $Y_3$. Since $\bar{Q} +\sum E_{i}$ is a sum of components of the branch locus and \(\bar{Q} +\sum E_{i} \equiv 2 \left( H-\sum F_{i} \right) \), the lemma shows that the divisor \[ \pi^{-1} \left( \bar{Q} +\sum_{2}^{5} {E}_{i} \right) - \pi^{\ast} \left( H - \sum_{2}^{5} {F}_{i} \right) \] has order two in \( \operatorname{Pic} \left( {X} \right) .\) Thus \( \operatorname{Tors} \left({X}\right) \) is non-trivial. For numerical Godeaux surfaces, the torsion group has order less than or equal to five, and it is known that $\Bbb{Z}/{2} \oplus \Bbb{Z}/{2}$ does not occur. (See \cite{Do}.) To determine whether $\operatorname{Tors} X$ is \( \Bbb{Z}/{2} \) or \( \Bbb{Z}/{4} ,\) we use a base point lemma due to Miyaoka \cite{M}: for a minimal Godeaux surface, the number of base points of $\left| 3 {K} \right| $ is equal to \[ \# \left\{ T \in \operatorname{Pic} X \, : \, T \neq -T\right\}/2. \] Thus if $\, \left| 3 {K} \right|$ has no base points, the torsion group is \( \Bbb{Z}/2 \). Write $\epsilon : X \rightarrow \tilde{X}$ for the map contracting the $\pi^{-1}\left( E_i \right)$. Then $3K_X \equiv \epsilon^{\ast}\left( 3K_{\tilde{X}} \right) +3\sum \pi^{-1}\left(E_i \right)$. To compute $\left| 3 {K}_{X} \right|,$ first consider the system $\left| 3K_{Y_3}+3\cal{L} \right|$. The divisor $2\sum E_i$ is fixed in this system; the difference $\left| 6H-3E-2\sum E_i -3 \sum F_i \right|$ is the pencil of sextics with a triple point at $p$ and double points at each $p_i$ with one tangent direction coinciding with the branch curve. Set $M= \pi^{\ast}\left( 6H-3E-2\sum E_i -3 \sum F_i \right)$; we have $\epsilon^{\ast}\left( 3K_{\tilde{X}} \right) \equiv M+ \sum \pi^{-1} \left( E_i \right)$, so any base point must either lie on $\sum \pi^{-1} \left( E_i \right)$ or be a base point of $\left| M \right|$. We use Maple to find two sextics in $M$ and their two points of intersection. These two points do not lie on $Q$ or $C$, so there is no base point of $\left| M \right|$ on the branch curve. Since $3K_X \equiv \pi^{-1}\left( B \right) + \pi^{\ast} \left( H-E \right) + 2 \sum \pi^{-1} \left(E_i \right)$, we also have $\epsilon^{\ast}\left( 3K_{\tilde{X}} \right) \equiv \pi^{-1} \left( B \right) +\pi^{\ast}\left( H-E \right) -\sum \pi^{-1}\left( E_i \right) $, so any base point must lie on the branch curve, away from the divisor $\sum \pi^{-1} \left( E_i \right)$. Therefore there are no base points of the tricanonical system. {}From the Miyaoka lemma, this shows that $\operatorname{Tors} X \cong \Bbb{Z}/{2}.$ \section{The Oort and Peters example} In \cite{OP}, Oort and Peters construct a branch curve $B$ from two conics ${Q}_{1},{Q}_{2}$ and two cubics ${C}_{1},{C}_{2}$ where \[ \begin{array}{ll} {Q}_{1} &= y^2+2x^2-2xy-5xz+2yz+3z^{2} \\ {Q}_{2} &= y^2+2x^2+2xy-5xz-2yz+3z^2 \\ {C}_{1} &= y^{2}z+x^3-4x^{2}z+3xz^2 \\ {C}_{2} &= 2y^{2}z-xy^2+4x^{2}z-12xz^{2}+9z^3. \end{array} \] We have \[ \begin{array}{ll} ( {Q}_{1} \cdot { Q}_{2}) &= {P} + 3 { P}_{1} \\ ( {Q}_{1} \cdot { {C}}_{1}) &= 2( {{P}}_{1} + {P}_{2} + {P}_{3} ) \\ ({ {Q}}_{1} \cdot { {C}}_{2}) &= 2({{P}} + {P}_{2} + {P}_{3} ) \\ ({ {Q}}_{2} \cdot { {C}}_{1}) &= 2( {{P}}_{1} + {P}_{4} + {P}_{5} ) \\ ({ {Q}}_{2} \cdot { {C}}_{2}) &= 2({{P}}_{4} + {P}_{5} +{P}) \\ ( { {C}}_{1} \cdot { C}_{2}) &= 2({ {P}}_{2} +{P}_{3} +{P}_{4} +{P}_{5} ) + \infty \end{array} \] where \[ \begin{array}{ll} {P} &= [{{3} \over {2}}:0:1] \\ {P}_{1} &= [1:0:1] \\ {P}_{2} &= [ {{3+i \sqrt{3}}\over{2}} : {{3+i \sqrt{3}}\over{2}} : 1] \\ {P}_{3} &= [ {{3-i \sqrt{3}}\over{2}} : {{3-i \sqrt{3}}\over{2}} : 1] \\ {P}_{4} &= [ {{3+i \sqrt{3}}\over{2}} : {{-3-i \sqrt{3}}\over{2}} : 1] \\ {P}_{5} &= [ {{3-i \sqrt{3}}\over{2}} : {{-3+i \sqrt{3}}\over{2}} : 1]\\ \infty &= [0:1:0]. \end{array} \] In this case the branch curve has two extra ordinary double points, one at $\infty,$ and the other which occurs on the second blowup above ${P}_{1},$ since ${Q}_{1}$ and ${Q}_{2}$ intersect with multiplicity three at this point. However these double points do not affect the invariants of the double plane $Z$ constructed. Write \( \pi : Z \rightarrow Y\) for the double cover, where $Y$ is the blowup of the plane resolving the singularities of the branch curve; although the double points do not affect the computations, we will blow them up to obtain a smooth branch divisor ${B}$ with \[ {B} =2 \cal{L} \equiv 10H -4 {E} -2\sum_{1}^{5} {E}_{i} -6 \sum_{1}^{5} {F}_{i} -8 {G}_{1} -2 {E}_{6} \] where we use the notation for the exceptional curves as above, with ${G}_{1}$ being the divisor lying above the extra double point on ${F}_{1}$ and ${E}_{6}$ the exceptional divisor above $\infty.$ Let $\tilde{Z}$ be the minimal surface obtained from $Z$ by blowing down the $E_i$. Note that we have the following equivalences of divisors: \[ \begin{array}{cl} \bar{Q_1} &\equiv 2H-E-E_1-E_2-E_3-2F_1-2F_2-2F_3-3G_1 \\ \bar{Q_2} &\equiv 2H-E-E_1-E_4-E_5-2F_1-2F_4-2F_5-3G_1 \\ \bar{C_1} &\equiv 3H-\sum_{1}^{5} E_i -2\sum_{1}^{5} F_i -2G_1 -E_6\\ \bar{C_2} &\equiv 3H -2E- \sum_{2}^{5} E_i -2 \sum_{2}^{5} F_i -E_6 \\ K_Y &\equiv -3H +E+\sum_{1}^{5} E_i+2\sum_{1}^{5} F_i +3G_1 +E_6 \\ \cal{L} &\equiv 5H -2E-\sum E_i-3\sum F_i -4 G_1 -E_6. \end{array} \] Let ${\cal{L}}_1 =2H-E -E_1-2F_1-\sum_{2}^{5} F_i-3G_1$ and $B_1 =2 {\cal{L}}_1 \equiv Q_1 +Q_2 +\sum_{2}^{5} E_i$; set ${\cal{L}}_2 = {\cal{L}} -{\cal{L}}_1$ and $B_2 =2{\cal{L}}_2 \equiv C_1+C_2+E_1$. It follows from Beauville's lemma that \[ T =\pi^{-1}\left(B_1 \right) -\pi^{\ast} \left( {\cal{L}}_1 \right) \equiv -\pi^{-1}\left( B_2 \right) +\pi^{\ast} \left( {\cal{L}}_2 \right) \] is of order two, thus $\operatorname{Tors} \tilde{Z}$ is either $\Bbb{Z}/2$ or $\Bbb{Z}/4$. We will show that $\operatorname{Tors} \tilde{Z} \cong \Bbb{Z}/{4}.$ Note that it was previously believed that $\operatorname{Tors} \tilde{Z} \cong \Bbb{Z}/{2}$ (\cite{Do,OP}). Oort and Peters use the base point lemma of Miyaoka to argue that $\operatorname{Tors} \tilde{Z}$ is $\Bbb{Z}/{2}$; however they miss a base point of the system $\left| 3K_{\tilde{Z}} \right|$ in their computation. Again if $\epsilon : Z \rightarrow \tilde{Z}$ is the map from $Z$ to its minimal model, we have $$\epsilon^{\ast} \left(3K_{\tilde{Z}} \right) \equiv M + \sum \pi^{-1} \left( E_i \right) \equiv \pi^{-1} \left( B \right) +\pi^{\ast} \left( H-E \right) -\sum \pi^{-1} \left( E_i \right),$$ where $M= \pi^{\ast} \left(3K_Y +3{\cal{L}} -2\sum E_i\right) $. Thus any base point must lie on $\pi^{-1} \left( B \right) -\sum \pi^{-1} \left( E_i \right)$ and be a base point of $\left| M \right|$. The divisors $\bar{Q_1} +\bar{Q_2} +\bar{Q} +F_1$ and $\bar{\ell} +\bar{C_2} +\bar{\tilde{Q}}$ are in $\left| M \right|$, where $\bar{Q} \equiv 2H-E-\sum_{2}^{5} \left(E_i +F_i \right)$ is the proper transform of the conic \(Q = 2x^{2}-9xz+y^{2}+9z^{2}\) through $P,P_2,\dots,P_5,$ $\bar{\ell} \equiv H-E-E_1-F_1-G_1$ is the proper transform of the line $y=0$, and $\bar{\tilde{Q}} \equiv 2H -\sum_{1}^{5} E_i -2F_1 -\sum_{2}^{5} F_i -2G_1$ is the proper transform of the conic \(\tilde{Q} = 3xz-3z^{2}-y^{2} \) through each $P_i$ where the tangent direction at $P_1$ coincides with that of the branch curve. The point $[3:0:1]$ lies on the curves ${Q}, \ell, \mbox{and} \, {C}_{1},$ and therefore is a base point of $\left| \epsilon^{\ast} \left(3K_Z \right) \right|$. It follows from Miyaoka's result that $\operatorname{Tors} \tilde{Z}$ is $\Bbb{Z}/{4}.$ In \cite{Do}, Dolgachev assumes that there exists an order four divisor on $Z$ and gets a contradiction after finding a fixed part of the pencil $\left| 2 {K}_{Z} \right|.$ However his computation of generators for $\left| 2 {K}_{Z} \right|$ is incorrect. We find divisors in the system \[ \begin{array}{ll} \left| \epsilon^{\ast}\left( 2K_{\tilde{Z}} \right) \right| &= \left| 2K_Z-2\sum \pi^{-1}\left(E_i \right) \right| \\ &= \left| \pi^{\ast}\left(2K_Y +2\cal{L} -\sum E_i \right) \right| \\ &= \left| \pi^{\ast}\left(4H-2E-\sum E_i -2\sum F_i -2G_1\right) \right|. \end{array} \] This pencil has generators \( y_0=\bar{{Q}_{1}} +\bar{{Q}_{2}} +2{F}_{1}+4{G}_{1}+E_1 \) and \( y_1=\bar{{C}_{2}} + \bar{\tilde{\ell}} +2E_6, \) where $\tilde{\ell}$ is the line tangent to the branch curve at $P_1$; thus there is no fixed part to this system. We can also check that $\tilde{Z}$ has order four torsion by calculating the bicanonical system of a double cover of $\tilde{Z}.$ Form the double cover $S$ of $\tilde{Z}$ branched over $2{T} \equiv 0,$ \( \rho : S \rightarrow \tilde{Z}. \) Since there is no ramification, $\rho$ is \'etale over $\tilde{Z}$. Also $K_{S} \equiv \rho^{\ast} \left( K_{Z}+T\right)$ and ${K_{S}}^{2} = 2.$ We have already found two sections $y_0$ and $y_1$ in $\operatorname{H}^{0} \left( 2K_{\tilde{Z}} \right)$, and hence two sections $\rho^{\ast}\left( y_0 \right)$ and $\rho^{\ast}\left( y_1 \right)$ in $\operatorname{H}^{0} \left( 2K_S \right)$. Since $\rho^{\ast}\left( T \right) \equiv 0$, we also have \[ 2K_S \equiv \rho^{\ast}\bigl(\pi^{-1}\left(B_1 \right) +\pi^{\ast}(2H-E-\sum_{2}^{5} \left( E_i +F_i \right) +G_1 ) \bigr) \] and \[ 2K_S \equiv \rho^{\ast}\left(\pi^{-1}\left(B_2 \right) +\pi^{\ast}\left( H-E-E_1 -F_1-G_1+E_6 \right) \right). \] We have seen that the proper transform $\bar{Q}$ of the conic $Q$ is in the linear system \( \left| \pi^{\ast}\left(2H-E-\sum_{2}^{5} \left( E_i + F_i \right) \right) \right| \) and the proper transform of the line $\ell$ is in \( \left| \pi^{\ast}\left( H-E -F_1-G_1 \right) \right| \); set $y_2 = \pi^{-1}\left( B_1 \right) +\pi^{\ast} \left( \bar{Q} +G_1 \right)$ and $y_3 = \pi^{-1}\left( B_2 \right) + \pi^{\ast}\left( \bar{\ell} + E_6 \right)$. We have \( \left(y_{0}-2y_{1} \right)^{2} -y_{2}^{2} +4y_{3}^{2} = 0 \). This gives a quadratic relation among the four elements of $\operatorname{H}^{0}(2{K}_{{S}});$ in fact we obtain a quadric cone as the bicanonical image of ${S}$. By \cite{CD}, if the bicanonical image is a cone then $\operatorname{Tors} S \cong \Bbb{Z}/{2}$ and $\pi_{1} \left( S \right) \cong \Bbb{Z}/{2}.$ Since $S$ is a covering space of $\tilde{Z}$ of degree two, $\left[ \pi_{1}\left(\tilde{Z}\right) : \pi_{1}\left(S\right)\right] =2.$ Thus $\pi_{1} \left( \tilde{Z} \right)$ is abelian of order four and $\pi_{1} \left( \tilde{Z} \right) \cong \operatorname{Tors} \left(\tilde{Z} \right) \cong \Bbb{Z}/{4}$.

9408

alg-geom/9408001

en

https://arxiv.org/abs/alg-geom/9408001

alg-geom/9408001

Daniel Huybrechts

Lothar Goettsche, Daniel Huybrechts

Hodge numbers of moduli spaces of stable bundles on K3 surfaces

12 pages, latex

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is: Theorem: Let $X$ be a K3 surface, $L$ a primitive big and nef line bundle and $H$ a generic polarization. If the moduli space of rank two $H$ semi-stable torsion-free sheaves with determiant $L$ and second Chern class $c_2$ has at least dimension 10 then its Hodge numbers coincide with those of the Hilbert scheme of $l:=2c_2-\frac{L^2}{2}-3$ points on $X$.

alg-geom

\section{A special case} In this section we prove the theorem in the case that ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$ and $c_2=\frac{L^2}{2}+3$. {\small\subsection{ The birational correspondence to the Hilbert scheme}} Throughout this section we will assume that the Picard group is generated by an ample line bundle $L$, i.e. ${\rm Pic }(X)=\hbox{\sym \char '132} \cdot L$. Under this assumption a torsion-free sheaf with determinant $L$ is $\mu-$stable if and only if it is $\mu-$semi-stable. For the convenience of the reader we recall the stability condition for framed modules (\cite{HL}): Let $\delta=\delta_1\cdot n+\delta_0$, $D\in {\rm Pic }(X)$ and $E$ be a torsion-free rank two sheaf. A framed module $(E,\alpha)$ consists of $E$ and a non-trivial homomorphism $\alpha:E\to D$. It is (semi-)stable if $P_{{\rm Ker }(\alpha)}(\leq)P_E/2-\delta/2$ and for all rank one subsheaves $M\subset E$ the inequality $P_M(\leq)P_E/2+\delta/2$ holds.\\ In fact, any semi-stable framed module is torsion-free. In \cite{HL} it was shown that there exists a coarse projective moduli space of semi-stable framed modules. \begin{lemma} Let $D=L$ and $0<\delta_1<L^2$. Then a framed module $(E,\alpha)$ is $\mu-$stable if and only if $E$ is $\mu-$stable. The moduli space $\overline M_H(L,c_2,D,\delta)$ is independent of the specific $\delta$ in this range. \end{lemma} {\it Proof:} Let $(E,\alpha)$ be semi-stable, then $\mu(M)\leq L^2/2+\delta_1/2$ for all $M=L^{\otimes n}\otimes I_Z\subset E$. Thus $nL^2\leq L^2/2+\delta_1<L^2$, i.e. $n<1$. Hence $E$ is $\mu-$stable. If $E$ is $\mu-$stable, then $(E,\alpha)$ is $\mu-$stable if $\mu({\rm Ker }(\alpha))<L^2/2-\delta_1/2$. But writing ${\rm Ker }(\alpha)=L^{\otimes n}\otimes I_Z$ and using the stabilty of $E$ we conclude $n<1$. Hence $\mu({\rm Ker }(\alpha))\leq0<L^2/2-\delta_1/2$. The second statement follows immediately.\hspace*{\fill}\hbox{$\Box$} Henceforth $\delta$ is chosen as in the lemma. Note that as for sheaves also for framed modules $\mu-$stability is equivalent to $\mu-$semi-stability. It can also be shown that both moduli spaces $ M(L,c_2,L,\delta)$ and $\overline M(L,c_2)$ are fine. The universality property of the moduli space induces a morphism $$\overline\varphi:\overline M(L,c_2,L,\delta)\longrightarrow\overline M(L,c_2).$$ Note that by the previous lemma the fibre of $\overline\varphi$ over $[E]$ is isomorphic to $\hbox{\sym \char '120}({\rm Hom }(E,L))$. \begin{lemma}\label{phisur} If $c_2\leq L^2/2+3$, then $\varphi$ is surjective. \end{lemma} {\it Proof:} It suffices to show that for a $\mu-$stable vector bundle $E$ there is always a non-trivial homomorphism $E\to L$. Since ${\rm Hom }(E,L)=H^0(X,E)$ and $H^2(X,E)\cong H^0(X,E^*)=0$ by the stability of $E$, the Riemann-Roch-Hirzebruch formula $\chi(E)=L^2/2-c_2+4$ shows that under the assumption $h^0(X,E)>0$.\hspace*{\fill}\hbox{$\Box$} \begin{lemma}\label{defN} Let $N(L,c_2,L,\delta)$ be the set of all $(E,\alpha)\in\overline {M}(L,c_2,L,\delta)$ such that ${\rm Ker }(\alpha)\cong{\cal O}_X$. It is a closed subset, which contains all stable pairs $(E,\alpha)$ with $E$ locally free. \end{lemma} {\it Proof:} If $E$ is locally free, then ${\rm Ker }(\alpha)$ has to be locally free. By stability it is thus isomorphic to ${\cal O}_X$. $N(L,c_2,L,\delta)$ is closed, since ${\rm Ker }(\alpha)\cong{\cal O}_X$ if and only if $length({\rm Coker }(\alpha))=c_2$, i.e. if the length is maximal; this is a closed condition.\hspace*{\fill}\hbox{$\Box$} Under the condition of \ref{phisur} we have a surjective morphism$$\varphi:{N(L,c_2,L,\delta)}\longrightarrow\overline M(L,c_2).$$ By \ref{defN} any framed module $(E,\alpha)\in {N(L,c_2,L,\delta)}$ sits in an extension $$\sesq{{\cal O}_X}{}{E}{\alpha}{I_Z\otimes L},$$ where $I_Z$ is the ideal sheaf of a codimension two cycle of length $c_2$. Thus we can define a morphism $$\psi: {N(L,c_2,L,\delta)}\longrightarrow Hilb^{c_2}(X)$$ by mapping $(E,\alpha)$ to $[{\rm Coker }(\alpha)]$. \begin{lemma}, If $c_2\geq L^2/2+3$, $\psi$ is surjective. \end{lemma} {\it Proof:} It is enough to show that ${\rm Ext }^1(I_Z\otimes L,{\cal O}_X)\not=0$ for all $Z\in Hilb^{c_2}(X)$. By the assumption $h^0(X,L|_Z)-h^0(X,L)\geq1$. Thus $h^1(L\otimes I_Z)\geq 1$. Now use ${\rm Ext }^1(I_Z\otimes L,{\cal O}_X)\cong H^1(X,I_Z\otimes L)^*$.\hspace*{\fill}\hbox{$\Box$} We have seen that any $(E,\alpha)\in {N(L,c,L,\delta)}$ induces an exact sequence $$\sesq{{\cal O}_X}{}{E}{\alpha}{I_Z\otimes L}.$$ Conversely, any section $s\in H^0(X,E)$ of $E\in\overline M(L,c)$ gives a homomorphism $\alpha:E\to L$ with ${\rm Ker }(\alpha)\cong{\cal O}_X$. Thus the fibre of $\varphi: {N(L,c,L,\delta)} \to \overline M(L,c)$ over $[E]$ is isomorphic to $\hbox{\sym \char '120}(H^0(X,E))$. In fact, $ {N(L,c,L,\delta)}$ can be identified with Le Potier's moduli space of coherent systems of rank one \cite{LP}. The picture we get in the case $c_2=c:=L^2/2+3$ is described by the following diagram. $$\begin{array}{rlcrl} && {N(L,c,L,\delta)}&&\\ &\varphi\swarrow&&\searrow\psi&\\ {}~~~~\overline{M}(L,c)&&&&Hilb^{c}(X)~~~~\\ \end{array}$$ Both morphisms $\varphi$ and $\psi$ are birational. This is due to the fact that for the generic $[Z]\in Hilb^{c}(X)$ the restriction map $H^0(X,L)\to H^0(X,L_Z)$ is injective and hence $h^1(X,I_Z \otimes L)=1$. This shows that $\psi$ is generically an isomorphism. Since the fibres of $\varphi$ are connected and both spaces are of the same dimension, also $\varphi$ is birational. Note that in particular the moduli space $\overline{M}(L,c)$ is irreducible.\\ Results about birationality of certain moduli spaces and corresponding Hilbert schemes have been known for some time, e.g. Zuo has shown that $M_H({\cal O}_X,n^2H^2+3)$ is birational to $Hilb^{2n^2H^2+3}(X)$ (cf. \cite{Z}). The moduli spaces of framed modules make this relation more explicit. They are used in the next section to show that the Hodge numbers of the moduli space and the Hilbert scheme coincide.\\ {\small\subsection{Comparison of the Hodge numbers}} First, we recall the notion of virtual Hodge polynomials \cite{D}, \cite{Ch}.\\ For any quasi-projective variety $X$ there exists a polynomial $e(X,x,y)$ with the following properties:\\ {\it i)} If $X$ is smooth and projective then $$e(X,x,y)=h(X,-x,-y):=\sum_{p,q}(-1)^{p+q}h^{p,q}(X)x^py^q.$$ {\it ii)} If $Y\subset X$ is Zariski closed and $U$ its complement then $$e(X,x,y)=e(Y,x,y)+e(U,x,y).$$ {\it iii)} If $X\to Y$ is a Zariski locally trivial fibre bundle with fibre $F$ then $$e(X,x,y)=e(Y,x,y)\cdot e(F,x,y).$$ {\it iv)} If $X\to Y$ is a bijective morphism then $e(X,x,y)=e(Y,x,y)$. In particular, if $$\begin{array}{ccccc} &&Z&&\\ &\swarrow&&\searrow\\ X&&&&Y\\ \end{array}$$ is a diagram of quasi-projective varieties, where $Z\to X$ and $Z\to Y$ admit a bijective morphism to a $\hbox{\sym \char '120}_n-$ bundle over $X$, resp. $Y$, then $e(X,x,y)\cdot e(\hbox{\sym \char '120}_n,x,y)=e(Z,x,y)=e(Y,x,y)\cdot e(\hbox{\sym \char '120}_n,x,y)$. Hence $e(X,x,y)=e(Y,x,y)$.\\ The idea to prove that $\overline{M}(L,c)$ and $Hilb^{c}(X)$, with $ c:=\frac{L^2}{2}+3$, have the same Hilbert polynomial is to stratify both by locally closed subsets $\overline{M}(L,c)_k$ and $Hilb^c(X)_k$ such that the birational correspondence given by the moduli space of framed modules induces $\hbox{\sym \char '120}_{k-1}-$bundles ${N(L,c,L,\delta)}_k \to \overline{M}(L,c)_k$ and ${N(L,c,L,\delta)}_k \to Hilb^c(X)_k$. One concludes $e(\overline{M}(L,c)_k,x,y)=e(Hilb^c(X)_k,x,y)$ and hence $e(\overline{M}(L,c),x,y)=\sum_k e(\overline{M}(L,c)_k,x,y)=\sum_k e(Hilb^c(X)_k,x,y)=e(Hilb^c(X),x,y)$. We first define the stratification. \begin{definition} $Hilb^c(X)_k:=\{[Z]\in Hilb^c(X)~|~h^1(X,I_Z\otimes L)=k\}$\\ ${N(L,c,L,\delta)}_k:=\psi^{-1}(Hilb^c(X)_k)$\\ $\overline{M}(L,c)_k:=\varphi({N(L,c,L,\delta)}_k)$ \end{definition} Using the universal subscheme ${\cal Z}\subset X\times Hilb^c(X)$ with the two projections $p$ and $q$ to $X$ and $Hilb^c(X)$, resp., and the semi-continuity applied to the sheaf $I_{\cal Z}\otimes p^*(L)$ and the projection $q$ it is easy to see that this defines a stratification into locally closed subschemes. All strata are given the reduced induced structure. We want to show that both morphisms $${N(L,c,L,\delta)}_k\to Hilb^c(X)_k$$ and $${N(L,c,L,\delta)}_k\to \overline{M}(L,c)_k$$ admit a bijective morphism to a $\hbox{\sym \char '120}_{k-1}-$bundle over the base. In fact, they are $\hbox{\sym \char '120}_{k-1}-$bundles, but by property {\it iv)} of the virtual Hodge polynomials we only need the bijectivity. \begin{definition} Let ${\cal A}_k:=\pi_*({\cal E}_k)$, where $\pi:{N(L,c,L,\delta)}_k\times X\to {N(L,c,L,\delta)}_k$ denotes the projection and ${\cal E}_k$ is the restriction of the universal sheaf ${\cal E}$, and let ${\cal B}_k:={\cal E}xt^1_q((I_{\cal Z})_k\otimes p^*(L),{\cal O}_X)$ be the relative Ext-sheaf, where $(I_{\cal Z})_k$ denotes the restriction of $I_{\cal Z}$ to $ Hilb^c(X)_k\times X$. \end{definition} \begin{lemma} ${\cal A}_k$ and ${\cal B}_k$ are locally free sheaves on $\overline{M}(L,c)_k$ and $Hilb^c(X)_k$, resp., and compatible with base change, i.e. ${\cal A}_k([E])\cong H^0(X,E)$ and ${\cal B}_k([Z])\cong {\rm Ext }^1(I_Z\otimes L,{\cal O}_X)$. \end{lemma} {\it Proof:} By definition and using Serre-duality we see that $Hilb^c(X)_k=\{Z|\dim{\rm Ext }^1(I_Z\otimes L,{\cal O}_X)=k\}$ and that it is reduced. Thus the claim for ${\cal B}_k$ follows immediately from the base change theorem for global Ext-groups \cite{BPS}. In order to prove the assertion for ${\cal A}_k$ it suffices to show that $\overline M(L,c)_k=\{E\,|\,h^0(X,E)=k\}$. Consider the exact sequence $$\sesq{{\cal O}_X}{}{E}{}{I_Z\otimes L}.$$ Then $E\in \overline M(L,c)_k$ if and only if $h^1(X,I_Z\otimes L)=k$ if and only if $h^0(X,I_Z\otimes L)=k-1$ if and only if $h^0(X,E)=k$.\hspace*{\fill}\hbox{$\Box$} The kernel of the universal framed module on ${N(L,c,L,\delta)}\times X$ restricted to ${N(L,c,L,\delta)}_k$ induces a morphism to $\hbox{\sym \char '120}({\cal A}_k)$ which is obviously bijective. Analogously, by the universality of $\hbox{\sym \char '120}({\cal B}_k)$ (cf. \cite{La}) the universal framed module over ${N(L,c,L,\delta)}\times X$ completed to an exact sequence and restricted to the stratum induces a bijective morphism of ${N(L,c,L,\delta)}_k$ to $\hbox{\sym \char '120}({\cal B}_k)$. We summarize: \begin{proposition} If $X$ is a K3 surface with ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$, $L$ ample and $c_2=L^2/2+3$, then $h^{p,q}(\overline{M}(L,c_2))=h^{p,q}(Hilb^{c_2}(X))$. \hspace*{\fill}\hbox{$\Box$} \end{proposition} Both manifolds $\overline{M}(L,c_2)$ and $Hilb^{c_2}(X)$ are symplectic. One might conjecture that in general two birational symplectic manifolds have the same Hodge numbers or even isomorphic Hodge structures, but we don't know how to prove this. \section{The general case} By deforming the underlying K3 surface the proof of the theorem is reduced to the case considered in section 1. \subsection{Deformation of K3 surfaces} The following statements about the existence of certain deformations of a given K3 surface will be needed. {\bf 2.1.1} {\it Let $X$ be a K3 surface, $L\in{\rm Pic }(X)$ a primitive nef and big line bundle. Then there exists a smooth connected family ${\cal X}\longrightarrow S$ of K3 surfaces and a line bundle ${\cal L}$ on ${\cal X}$ such that: $\cdot$ ${\cal X}_0\cong X$ and ${\cal L}_0\cong L$. $\cdot$ ${\rm Pic }({\cal X}_t)=\hbox{\sym \char '132}\cdot{\cal L}_t$ for all $t\not=0$ (${\cal L}_t$ is automatically ample).} {\it Proof:} The moduli space of primitive pseudo-polarized K3 surfaces is irreducible (\cite{B2}). Since any even lattice of index $(1,\rho-1)$ with $\rho\leq10$ can be realized as a Picard group of a K3 surface (\cite{Ni},\cite{Mor}) the generic pseudo-polarized K3 surface has Picard group $\hbox{\sym \char '132}$.\hspace*{\fill}\hbox{$\Box$} {\bf 2.1.2} {\it Let $X$ be a K3 surface whose Picard group is generated by an ample line bundle $L$, i.e. ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$. Furthermore, let $d\geq 5$ be an integer. Then there exists a smooth connected family ${\cal X}\longrightarrow S$ of K3 surfaces and a line bundle ${\cal L}$ on ${\cal X}$ such that: $\cdot$ $({\cal X}_{t_0},{\cal L}_{t_0})\cong(X,L)$ for some point $t_0\in S\setminus\{0\}$. $\cdot$ ${\rm Pic }({\cal X}_t)\cong \hbox{\sym \char '132}\cdot{\cal L}_t$ for all $t\not=0$. $\cdot$ ${\rm Pic }({\cal X}_0)=\hbox{\sym \char '132}\cdot{\cal L}_0\oplus\hbox{\sym \char '132} \cdot D$, where $D$ is represented by a smooth rational curve, both line bundles ${\cal L}_0$ and ${\cal L}_0(2D)$ are ample and primitive and the intersection matrix is} $$\left(\begin{array}{cc}L^2&d\\ d&-2\ \end{array}\right)$$ {\it Proof:} Again we use the irreducibility of the moduli space of primitive polarized K3 surfaces. The existence of a triple $({\cal X}_0,{\cal L}_0,D)$ with ample ${\cal L}_0$, smooth rational $D$ and the given intersection form was shown by Oguiso \cite{Og}. It remains to show that ${\cal L}_0(2D)$ is ample. Obviously, ${\cal L}_0(2D)$ is big and for any irreducible curve $C\not=D$ the strict inequality $({\cal L}_0(2D)).C>0$ holds. The assumption on $d$ implies $({\cal L}_0(2D)).D>0$. Note that the extra assumption $L^2\geq4$ in \cite{Og} is only needed for the very ampleness of ${\cal L}_0$ which we will not use.\hspace*{\fill}\hbox{$\Box$} {\bf 2.1.3(a)} {\it Let $X$ be a K3 surface whose Picard group is generated by an ample line bundle $L$, i.e. ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$. If $L^2>2$ there exists a smooth connected family ${\cal X}\longrightarrow S$ of K3 surfaces and a line bundle ${\cal L}$ on ${\cal X}$ such that: $\cdot$ $({\cal X}_{t_0},{\cal L}_{t_0})\cong(X,L)$ for some point $t_0\in S\setminus\{0\}$. $\cdot$ ${\rm Pic }({\cal X}_t)\cong \hbox{\sym \char '132}\cdot{\cal L}_t$ for all $t\not=0$. $\cdot$ ${\rm Pic }({\cal X}_0)=\hbox{\sym \char '132}\cdot{\cal L}_0\oplus\hbox{\sym \char '132} \cdot D$, where both line bundles ${\cal L}_0$ and ${\cal L}_0(2D)$ are ample and primitive and the intersection matrix is} $$\left(\begin{array}{cc}L^2&1\\ 1&0\ \end{array}\right)$$ {\bf 2.1.3(b)} {\it If we assume that $L^2>6$ we have the same result as in (a) with ``${\cal L}_0(2D)$ is ample'' replaced by ``${\cal L}_0(-2D)$ is ample''.} {\it Proof:} For both parts we need to prove the existence of a triple $(X_0,H,D)$ with ample and primitive $H$ and $H(2D)$, such that $D^2=0$, $H.D=1$ and $H^2=2n>2$ for given $n$. By the results of Nikulin we can find a K3 surface with this intersection form. It remains to show that $H$ and $H(2D)$ can be chosen ample. We can assume that $H\in{\cal C}^+$, i.e. $H$ is in the positive component of the positive cone (if necessary change $(H,D)$ to $(-H,-D)$). We check that $H$ is not orthogonal to any (-2) class, i.e. for any $\delta:=aH+bD$ ($a,b\in\hbox{\sym \char '132}$) with $\delta^2=a^2H^2+2ab=-2$ we have $H.\delta\not=0$. If $H$ were orthogonal to $\delta$ this would imply that $aH^2+b=0$. Hence $-a^2H^2=-2$ which contradicts $H^2>2$. Thus $H$ is contained in a chamber. Since the Weyl group $W_{X_0}$, which is generated by the reflection on the walls, acts transitively on the set of chambers, we find $\sigma\in W_{X_0}$ such that $\sigma(H)$ is contained in the chamber $\{w\in{\cal C}^+|w\delta>0 {\rm ~for~all~effective~(-2)~classes~}\delta\}$. Applying $\sigma$ to $(H,D)$ we can in fact assume that $H$ is contained in this chamber. On a K3 surface the effective divisors are generated by the effective (-2) classes and $\overline{{\cal C}^+}\setminus\{0\}$. On both sets $H$ is positive. Thus $H$ is ample. In order to prove that also $H(2D)$ is ample we show that $D$ is effective and irreducible. This follows from the Riemann-Roch-Hirzebruch formula $\chi({\cal O}(D))=2$, which implies $D$ or $-D$ effective, and $H.D=1$. Thus $C.D\geq0$ for any curve $C$. Thus $H(2D).C>0$. Since $H(2D)$ is big we conclude that $H(2D)$ is ample. To prove (a) we choose $H^2:=L^2$ and use the irreducibility of the moduli space to show that $(X,L)$ degenerates to $(X_0,H)$. Defining ${\cal L}_0:=H$ this proves (a). In order to prove (b) we fix $(H(2D))^2:=L^2$ and let $(X,L)$ degenerate to $(X_0,H(2D))$. The assumption on $H$ translates to $L^2>6$. With ${\cal L}_0:=H(2D)$ we obtain (b). \hspace*{\fill}\hbox{$\Box$} \subsection{Deformation of the moduli space} We start out with the following \begin{lemma}\label{def} Let $E$ be a simple vector bundle on a K3 surface such that $L:=det(E)$ is big. The joint deformations of $E$ and $X$ are unobstructed, i.e. $Def(E,X)$ is smooth. Moreover, $Def(E,X)\to Def(X)$ and $Def(L,X)\to Def(X)$ have the same image. \end{lemma} {\it Proof:} The infinitesimal deformations of a bundle $E$ together with its underlying manifold $X$ are paramatrized by $H^1(X,{\cal D}_0^1(E))$, where ${\cal D}_0^1(E)$ is the sheaf of differential operators of order $\leq1$ with scalar symbol. The obstructions are elements in the second cohomology of this sheaf. Using the symbol map we have a short exact sequence $$\sesq{{\cal E} nd(E)}{}{{\cal D}_0^1(E)}{}{{\cal T}_X}.$$ Its long exact cohomology sequence $$H^1(X,{\cal D}_0^1(E))\to H^1(X,{\cal T}_X)\to H^2(X,{\cal E} nd(E))\to H^2(X,{\cal D}_0^1(E))\to 0$$ compares the deformations of $E$, $X$, and $(E,X)$. In particular, if $E$ is simple the trace homomorphism $H^2(X,{\cal E} nd(E))\to H^2(X,{\cal O}_X)$ is bijective and the composition with the boundary map $H^1(X,{\cal T}_X)\to H^2(X,{\cal E} nd(E))$ is the cup-product with $c_1(E)$. Since there is exactly one direction in which a big and nef line bundle $L$ cannot be deformed with $X$ the cup-product with $c_1(L)=c_1(E)$ is surjective. Thus $H^1(X,{\cal D}_0^1(E))\to H^1(X,{\cal T}_X)$ is onto the algebraic deformations of $X$ and $H^2(X,{\cal D}_0^1(E))$ vanishes.\hspace*{\fill}\hbox{$\Box$} The following lemma will be needed for the next proposition. Its proof is quite similar to what we will use to prove the theorem. \begin{lemma}\label{irr} If ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$, then $\overline M(L,c_2)$ is irreducible for $\dim\overline M(L,c_2)=4c_2-L^2-6>8$. \end{lemma} {\it Proof:} {\it 1st step:} First, we show that $\overline M_H(L,\frac{L^2}{2}+3)$ is irreducible whenever $L$ is an ample line bundle on a K3 surface.\\ By a result of \cite{Q} the moduli spaces $\overline M_H(L,\frac{L^2}{2}+3)$ and $\overline M_L(L,\frac{L^2}{2}+3)$ are birational. In particular, the number of irreducible components is the same. We consider a deformation as in 2.1.1. The corresponding family of moduli spaces $\overline M_{{\cal L}_t}({\cal L}_t,\frac{L^2}{2}+3)$ is proper and by lemma \ref{def} every stable bundle on $X$ can be deformed to a stable bundle on any nearby fibre. This shows that $\overline M_{{\cal L}_0}({\cal L}_0,\frac{L^2}{2}+3)$ has as many irreducible components as $\overline M_{{\cal L}_{t\not=0}}({\cal L}_{t\not=0},\frac{L^2}{2}+3)$, which is irreducible.\\ {\it 2nd step:} Assume $e:=c_2-\frac{L^2}{2}-3>0$ and $L^2>2$. We apply 2.1.3(a). By the same arguments as above we obtain that the number of irreducible components of $\overline M(L,c_2)$ is at most the number of irreducible components of $\overline M_{{\cal L}_0}({\cal L}_0,c_2)$. Again using \cite{Q} we know that $\overline M_{{\cal L}_0}({\cal L}_0,c_2)$ is birational to $\overline M_{{\cal L}_0(2D)}({\cal L}_0,c_2)$. The $\mu$-stable part of the latter is isomorphic to the $\mu$-stable part of $\overline M_{{\cal L}_0(2D)}({\cal L}_0(2D),c_2+1)$. We have $({\cal L}_0(2D))^2=L^2+4>2$ and $c_2+1-\frac{({\cal L}_0(2D))^2}{2}-3=e-1$. Therefore we obtain by induction over $e$ and step 1 that $\overline M_{{\cal L}_0(2D)}({\cal L}_0(2D),c_2+1)$ is irreducible. Since the locally free $\mu$-stable sheaves are dense in the moduli spaces, this accomplishes the proof in this case.\\ {\it 3rd step:} Here we assume that $e:=c_2-\frac{L^2}{2}-3<0$. By assumption $4c_2-L^2-6\geq10$. Hence $c_2\geq6$ and $L^2>6$. Now we apply 2.1.3(b). The same arguments as in the previous step show that the number of irreducible components of $\overline M(L,c_2)$ is at most that of $\overline M_{{\cal L}_0(-2D)}({\cal L}_0(-2D),c_2-1)$. Since $c_2-1-\frac{({\cal L}_0(-2D))^2}{2}-3=e+1$, we can use induction over $-e$ and step 1 to show the irreducibility in this case.\\ {\it 4th step:} It remains to consider the case $L^2=2$. Here we apply 2.1.2 with $d=5$. As above we conclude that the number of irreducible components of $\overline M(L,c_2)$ is at most that of $\overline M_{{\cal L}_0(2D)}({\cal L}_0(2D),c_2+3)$. Since $({\cal L}_0(2D))^2=L^2+20-8=14$ we can conclude by step 2 or 3. \hspace*{\fill}\hbox{$\Box$} Mukai seems to know that all moduli spaces of rank two bundles on a K3 surface are irreducible (\cite{Mu2}, p.\ 157). Since we could not find a proof of this in the literature we decided to include the above lemma. Let $X$ be a K3 surface and $L$ a line bundle on $X$. For any $c_2$ there exists a coarse moduli space $\overline{M}_s(L,c_2)$ of simple sheaves of rank two with determinant $L$ and second Chern class $c_2$. $\overline{M}_s(L,c_2)$ can be realized as a non-separated algebraic space (\cite{AK}, \cite{KO}). For any polarization $H$ such that $H-$semi-stabilty implies $H-$stability the projective manifold $\overline{M}_H(L,c_2)$ is an open subset of $\overline{M}_s(L,c_2)$. Note that in the case that ${\rm Pic }(X)=\hbox{\sym \char '132}\cdot L$ and $H=L$ any simple vector bundle is in fact slope stable. For sheaves the situation is more complicated. Now let $({\cal X},{\cal L})\longrightarrow S$ be a family of K3 surfaces with a line bundle ${\cal L}$ on ${\cal X}$ over a smooth curve $S$. By \cite{AK}, \cite{KO} there exists a relative moduli space of simple sheaves, i.e. there exists an algebraic space $\overline{{\cal M}}_s({\cal L},c_2)$ and a morphism from it to $S$ such that the fibre over a point $t\in S$ is isomorphic to $\overline{M}_s({\cal L}_t,c_2)$. By a result of Mukai the fibres are smooth \cite{Mu1}. Lemma \ref{irr} shows that for a family $({\cal X},{\cal L})\longrightarrow S$ both $\overline{\cal M}_s({\cal L},c_2)$ and $\overline{\cal M}_s({\cal L},c_2)\longrightarrow S$ are smooth (at least over the locally free sheaves).\\ For the following we want to assume that ${\rm Pic }({\cal X}_t)\cong\hbox{\sym \char '132}\cdot{\cal L}_t$ for $t\not=0$ and ${\cal L}_t^2>0$. To shorten notation we denote by $Z^*\longrightarrow S^*$ the restriction of a family $Z\longrightarrow S$ to $S^*:=S\setminus\{0\}$. \begin{proposition}\label{defofmod} Assume that $\overline M_{{\cal L}_t}({\cal L}_t,c_2)$ is irreducible for $t\not=0$. Then for any generic ample $H\in {\rm Pic }({\cal X}_0)$ there exists a smooth proper family $Z\longrightarrow S$ of projective manifolds such that $Z^*\longrightarrow S^*$ has fibres $\overline{M}_{{\cal L}_t}({\cal L}_t,c_2)$ and the fibre over $0$ is isomorphic to $\overline{M}_{H}({\cal L}_0,c_2)$. (``The moduli spaces for different $H$ cannot be separated'') \end{proposition} {\it Proof:} By $\overline{{\cal M}}({\cal L},c_2)^*\to S^*$ we denote the family of the moduli spaces $\overline{M}({\cal L}_t,c_2)$. It is proper over $S^*$ and the fibres are smooth and irreducible.\\ {\it Claim:} If $[E]\in \overline M_s({\cal L}_0,c_2)$ is a point in the closure $T_0$ of $\overline{\cal M}_s({\cal L},c_2)^*\setminus\overline{{\cal M}}({\cal L},c_2)^*$ in $\overline{\cal M}_s({\cal L},c_2)$, then $E$ is not semi-stable with respect to any polarization $H$: Semi-continuity shows that a point $E$ in the closure has a subsheaf of rank one with determinant $ {\cal L}_0^{\otimes n}$ with $n>0$. Hence it is not semi-stable with respect to any polarization.\\ The set $T_1$ of simple sheaves $[E]\in\overline M_s({\cal L}_0,c_2)$ which are not stable with respect to $H$ is a closed subset of $\overline{\cal M}_s({\cal L},c_2)$. We define $Z$ to be the complement of the union of $T_0$ and $T_1$ in $\overline{\cal M}_s({\cal L},c_2)$. It is an open subset of $\overline{{\cal M}}_s({\cal L},c_2)$. The fibres meet the requirements of the assertion.\\ {\it Claim:} $Z$ is separated: Any simple sheaf on any of the fibres ${\cal X}_t$ can also be regarded as a simple coherent sheaf on the complex space ${\cal X}$. Thus $Z$ is a subspace of the space of all simple sheaves on ${\cal X}$. In order to show that two points are separated in $Z$ it suffices to separate them in the bigger space. Now we apply the criterion of \cite{KO} which says that if two simple coherent sheaves are not separated then there exists a non-trivial homomorphism between them. Since any two sheaves parametrized by $Z$ are either supported on different fibres or stable with respect to the same polarization, this is excluded.\\ Thus $Z$ is a separated with compact irreducible fibres over $S^*$. Take a locally free $E\in \overline M_H(L,c_2)$ and consider a neighbourhood of it in $\overline{{\cal M}}_s({\cal L},c_2)$. By the arguments above this neighbourhood contains locally free simple sheaves on all the nearby fibres. Hence we can assume that all these sheaves on ${\cal X}_{t\not=0}$ are stable, since ${\rm Pic }({\cal X}_t)=\hbox{\sym \char '132}\cdot{\cal L}_t$ for $t\not=0$. This implies the connectedness of $Z$. Thus $Z\longrightarrow S$ is proper and smooth.\hspace*{\fill}\hbox{$\Box$} {\bf Proof of the theorem:} {\it i)} We first show that the result of section 1 generalizes to the case where we drop the assumption that $L$ generates ${\rm Pic }(X)$. This is done as follows. By applying \ref{defofmod} to a deformation of the type 2.1.1 one sees that $\overline M_{{\cal L}_t}({\cal L}_t,\frac{{\cal L}_t^2}{2}+3)$ is a deformation of $\overline M_H(L,\frac{L^2}{2}+3)$ for generic $H$. Since Hodge numbers are invariant under deformations, both spaces have the same Hodge numbers. Those of the second were compared in section 1 with the Hodge numbers of the appropriate Hilbert scheme. By the same trick we can always reduce to the case where the Picard group is generated by $L$, in particular we can assume that $L$ is ample. \\ {\it ii)} By applying \ref{defofmod} to a deformation of type 2.1.2, 2.1.3(a) or 2.1.3(b) we see that $\overline M(L,c_2)$ is a deformation of $\overline M_H({\cal L}_0,c_2)$ for generic $H$. Since $\mu-$stabilty does not change under twisting by line bundles we have $\overline M_H({\cal L}_0,c_2)\cong \overline M_H({\cal L}_0(2D),c_2+{\cal L}_0.D+D^2)$ (or $\overline M_H({\cal L}_0(-2D),c_2-{\cal L}_0.D+D^2)$ in case 2.1.3(b)). The proof of lemma \ref{irr} shows that by applying 2.1.2, 2.1.3(a) and 2.1.3(b) repeatedly we can reduce to the situation of {\it i)}, i.e. $c_2=\frac{L^2}{2}+3$. \hspace*{\fill}\hbox{$\Box$} \begin{corollary} Let $X$ be an arbitrary K3 surface and $L$ a primitive big and nef line bundle. As long as a polarization $H$ does not lie on any wall, all deformation invariants, e.g. Hodge- and Betti numbers, of $\overline{M}_H(L,c_2)$ are independent of $H$.\hspace*{\fill}\hbox{$\Box$} \end{corollary} For similar results compare \cite{G}. {\footnotesize

9408

alg-geom/9408002

en

https://arxiv.org/abs/alg-geom/9408002

alg-geom/9408002

Luca Barbieri-Viale

Luca Barbieri-Viale

${\cal H}$-cohomologies versus algebraic cycles

51 pages, LaTeX 2.09

Math. Nachr. 184 (1997), 5-57

Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincar\'e duality theories are obtained. We assume given a (twisted) cohomology theory $H^*$ having a cup product structure and we let consider the ${\cal H}$-cohomology functor $X\leadsto H^{\#}_{Zar}(X,{\cal H}^*)$ where ${\cal H}^*$ is the Zariski sheaf associated to $H^*$. We show that the ${\cal H}$-cohomology rings generalize the classical ``intersection rings'' obtained via rational or algebraic equivalences. Several basic properties e.g.\, Gysin maps, projection formula and projective bundle decomposition, of ${\cal H}$-cohomology are obtained. We therefore obtain, for $X$ smooth, Chern classes $c_{p,i} : K_i(X) \to H^{p-i}(X,{\cal H}^p)$ from the Quillen $K$-theory to ${\cal H}$-cohomologies according with Gillet and Grothendieck. We finally obtain the ``blow-up formula'' $$H^p(X',{\cal H}^q) \cong H^p(X,{\cal H}^q)\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i})$$ where $X'$ is the blow-up of $X$ smooth, along a closed smooth subset $Z$ of pure codimension $c$. Singular cohomology of associated analityc space, \'etale cohomology, de Rham and Deligne-Beilinson cohomologies are examples for this setting.

alg-geom

\section{Introduction} After Quillen's proof of the Gersten conjecture (see \cite{Q}), for algebraic regular schemes, a natural approach to the theory of algebraic cycles appears to be by dealing with the ``formalism'' associated to (local) higher $K$-theory, as it is manifestly expressed by the work of Bloch and Gillet (cf. \cite{BL}, \cite{GIN}). As a matter of fact a more general and flexbile setting has been exploited by Bloch and Ogus (see \cite{BO}) by axiomatic methods.\\ The aim of this paper is to going further with this axiomatic method in order to obtain a ``global intersection theory'' (in the Grothendieck sense \cite{GI}) directly from a given ``cohomology theory''. To this aim we will assume given a (twisted) cohomology theory $H^*$ and we let consider the ${\cal H}$-cohomology functor $$X\leadsto H^{\#}_{Zar}(X,{\cal H}^*\p{(\cdot)})$$ where ${\cal H}^*\p{(\cdot)}$ is the Zariski sheaf associated to $H^*$. By dealing with a cup-product structure on $H^*$ we are granted of a product in ${\cal H}$-cohomology; by arguing with the cap-product structure we are able to obtain a cap-product between algebraic cycles and ${\cal H}$-cohomology classes, for $Y$ and $Z$ closed subschemes of $X$ ($\mbox{$\Lambda$}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, H^0({\rm point})$) $$\p{\cap} :C_{n}(Y;\mbox{$\Lambda$})\otimes H^p_Z(X,{\cal H}^p\p{(p)}) \to C_{n-p}(Y\cap Z;\mbox{$\Lambda$})$$ where $C_{*}(-;\mbox{$\Lambda$})$ is the ``${\cal H}$-homology theory'' given by the hypercohomology of the complexes of $E^1$-terms of the niveau spectral sequence.\\ If $X$ is smooth of pure dimension $d$, by capping with the ``fundamental cycle'' $[X]\in C_d(X;\mbox{$\Lambda$})$ we have a ``Poincar\'e duality'' isomorphism $$[X]\p{\cap}- : H^p_Z(X,{\cal H}^p\p{(p)}) \cong C_{d-p}(Z;\mbox{$\Lambda$})$$ Thus the ``${\cal H}$-cycle class'' $\eta (Z)\in H^p_Z(X,{\cal H}^p\p{(p)})$ is defined by $[X]\p{\cap}\eta (Z) = [Z]$, for $i: Z\hookrightarrow X$ a closed subscheme of pure codimension $p$ in $X$. By capping with the ${\cal H}$-cycle class we do obtain Gysin maps for algebraic cycles i.e. maps $i^!:C_{n}(X;\mbox{$\Lambda$})\to C_{n-p}(Z;\mbox{$\Lambda$})$. Furthermore, the ${\cal H}$-cycle classes are compatible with the intersection of cycles (when existing!) so that the ${\cal H}$-cohomology rings generalize the classical ``intersection rings'' obtained via rational or algebraic equivalences (cf. \cite{GIN} for the ${\cal K}$-cohomology).\\ The covariant property of the niveau spectral sequence grant us of ``${\cal H}$-Gysin maps'' $$f_*: H^{\#}_{f^{-1}(Z)}(Y,{\cal H}^*\p{(\cdot)}) \to H^{\#+\rho}_{Z}(X,{\cal H}^{*+\rho}\p{(\cdot+\rho)})$$ associated with a proper morphism $f:Y\to X$ of relative dimension $\rho$ between smooth schemes. The corresponding projection formula holds. By the homotopy property of $H^*$ we are obtaining homotopy and Dold-Thom decomposition for ${\cal H}$-cohomologies. By observing that the canonical cycle map for line bundles $c\ell : {\rm Pic}\, (X) \to H^2(X,\p{1})$ has always its image contained in the subgroup of the locally trivial cohomology classes i.e. $H^1(X,{\cal H}^1\p{(1)})$ by the coniveau spectral sequence, we are able to construct Chern classes in ${\cal H}$-cohomologies according with Gillet and Grothendieck (see \cite{GIL} and \cite{GC}) $$c_{p,i} : K_i^Z(X) \to H^{p-i}_Z(X,{\cal H}^p\p{(p)})$$ where $Z$ is any closed subset of $X$ smooth. These yield Riemann-Roch theorems and, notably, Chern classes in $H^{2*}(-,\p{*})$ by composition with the cycle map $H^*(-,{\cal H}^*\p{(*)})\to H^{2*}(-,\p{*})$ canonically induced by the coniveau spectral sequence.\\ At last, an immediate application of this setting is the ``blow-up formula'' $$H^p(X',{\cal H}^q\p{(r)}) \cong H^p(X,{\cal H}^q\p{(r)})\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i}\p{(r-1-i)})$$ where $X'$ is the blow-up of $X$ smooth, along a closed smooth subset $Z$ of pure codimension $c$. Remarkably the formula is obtained by no use of ``self-intersection'' nor ``formule-clef'' (used by the redundant arguments made in \cite[Expos\'e VII]{SGA5} for \'etale cohomology or Chow groups).\\ The paper is organized by adding structure to the assumed Bloch-Ogus cohomology to proving the claimed results. The common cohomologies (e.g. \'etale, de Rham or Deligne-Beilinson cohomology) are examples for this setting as explained in the Appendix. Some of these results are already been used by the author for applications (see \cite{BV1},\cite{BV2}); as our second main goal is that this ``formalism'' can be used for applications to birational geometry and algebraic cycles.\\ I would like to thank H.Gillet and R.W.Thomason for some corroborating conversations on these topics. \section{Preliminaries} Let $\cat{V}_k$ be the category of schemes of finite type over a fixed ground field $k$; usually, an object of $\cat{V}_k$ is called `algebraic scheme'. Let $\cat{V}^2_k$ be the category whose objects are pairs $(X,Z)$ where $X$ is an algebraic scheme and $Z$ is a closed subscheme of $X$; morphisms in $\cat{V}^2_k$ are fibre products in $\cat{V}_k\ .$ In the following we will consider a contravariant functor $$(X,Z) \leadsto H_Z^*(X,\cdot)$$ from $\cat{V}^2_k$ to \Z -bigraded abelian groups.\\ We need to assume, at least, that the above functor gives rise to a `Poincar\'e duality theory with supports' as setted out by Bloch and Ogus \cite[1.1-1.3 and 7.1.2]{BO}; one can also consider $K_i^Z(X)$ the relative Quillen $K$-theory \cite{Q} (see \cite[Def. 2.13]{GIL}). For $X\in \cat{V}_k$ we denote $H^*(X,\cdot)$ for $H_X^*(X,\cdot)\ .$ \subsection{Bloch-Ogus theory}\label{inter1} For the sake of notation we recall some facts by \cite{BO}. Togheter with the cohomology theory $H^*(\ ,\cdot)$ is given an homological functor $H_{\star}(\ ,\dagger)$ covariant for proper morphisms in $\cat{V}_k$ and a pairing: $$\p{\cap_{X,Z}}: H_l(X,\p{m})\otimes H_Z^r(X,\p{s})\to H_{l-r}(Z,\p{m-s})$$ having a `projection formula'. For $f$ a proper map let $f_!$ denote the induced map on homology. It is also assumed the existence of a `fundamental class' $\eta_X\in H_{2d}(X,\p{d})$ , $d = {\rm dim} X$ , such that $f_!(\eta_X)=[K(X):K(Y)]\cdot \eta_Y$ if $f:X\to Y$ is proper and ${\rm dim} X ={\rm dim} Y$ (cf.~\cite[7.1.2]{BO}). For $X$ smooth of dimension $d$ $$\eta_X\p{\cap_{X,Z}}- : H_Z^{2d-i}(X,\p{d-j})\by{\simeq} H_{i}(Z,\p{j})$$ is an isomorphism (`Poincar\'e duality') suitably compatible with restrictions (cf.~\cite[1.4]{BO}). For $Z\subseteq T\subseteq X$ such that $Z$ and $T$ are closed in $X$ there is a long exact sequence (see \cite[1.1.1]{BO}) \B{equation}\label{loc} \cdots \to H_Z^i(X,\cdot)\to H_T^i(X,\cdot)\to H_{T-Z}^i(X-Z,\cdot)\to H_Z^{i+1}(X,\cdot)\to \cdots \E{equation} suitably contravariant; moreover, for $X$ smooth and irreducible of dimension $d$ , the following exact sequence ( $h+i=2d\ ,\ \dagger + \cdot =d$ ): \B{equation}\label{hloc} \cdots \to H_h(Z,\dagger)\to H_h(T,\dagger)\to H_h(T-Z,\dagger)\to H_{h-1}(Z,\dagger)\to \cdots \E{equation} is the corresponding Poincar\'e dual of the above (cf. \cite[6.1 k)]{JA}). \subsection{Gersten or arithmetic resolution} Let $Z^p(X)= \{Z\subset X \mbox{: closed of}\ {\rm codim}_XZ\geq p\}$, ordered by inclusion, and let define $$H^i_{Z^p(X)}(X,\cdot) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{\pp{Z\in Z^p(X)}} H_Z^i(X,\cdot) $$ and for $x\in X$ $$H^i(x,\cdot) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{\pp{ U open \subset \overline{\{ x\} }}} H^i(U,\cdot)$$ Taking the direct limit of the exact sequences (\ref{loc}) (over the pairs $Z\subseteq T$ with $Z\in Z^{p+1}(X)$ and $T\in Z^p(X)$) and using `local purity' (cf. \cite[Prop.3.9]{BO}) on $X$ smooth over $k$ perfect, one obtains long exact sequences \B{equation}\label{limloc} H^i_{Z^{p+1}(X)}(X,\p{j})\to H^i_{Z^p(X)}(X,\p{j})\to \coprod_{x\in X^p}^{} H^{i-2p}(x,\p{j-p}) \to H^{i+1}_{Z^{p+1}(X)}(X,\p{j}) \E{equation} where $X^p$ is the set of points whose closure has codimension $p$ in $X$. Furthermore, if $f:X \to Y$ is a flat morphism and $Z\in Z^p(Y)$ then we have $ f^{-1}(Z)\in Z^p(X)$; thus the sequence in (\ref{limloc}) yields a sequence of presheaves for the Zariski topology. Let $H^*_{Z^p,X}\p{(\cdot)}$ denote the Zariski presheaf $U \leadsto H^*_{Z^p(U)}(U,\cdot)$ on $X$ and let $a: \cat{P}(X_{Zar}) \to \cat{S}(X_{Zar})$ be the associated sheaf exact functor. Denote $$aH^*_{Z^p,X}\p{(\cdot)}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, {\cal H}^*_{Z^p,X}\p{(\cdot)}$$ The presheaf $H^*_{Z^0,X}\p{(\cdot)}$ is just the functor $H^*\p{(\cdot)}$ on $X_{Zar}$ and so one has ${\cal H}^*_{Z^0,X}\p{(\cdot)} = {\cal H}^*_X\p{(\cdot)}\ .$ One of the main results of \cite{BO} is in proving the vanishing of the map $${\cal H}^*_{Z^{p+1},X}\p{(\cdot)} \to {\cal H}^*_{Z^p,X}\p{(\cdot)}$$ for all $p \geq 0$ . From this vanishing, sheafifying the sequence (\ref{limloc}), one has the following exact sequences of sheaves on $X$ smooth over $k$ perfect: \B{equation}\label{shortloc} 0\to {\cal H}^i_{Z^{p},X}\p{(j)}\to \coprod_{x\in X^p}^{} i_x H^{i-2p}(x,\p{j-p}) \to {\cal H}^{i+1}_{Z^{p+1},X}\p{(j)}\to 0 \E{equation} where: for $A$ an abelian group and $x\in X$ we let $i_xA$ denote the constant sheaf $A$ on $\overline{\{ x\}}$ extended by zero to all $X$. Patching toghether the above short exact sequences we do get a resolution of the sheaf ${\cal H}^i_X\p{(j)}$ (`arithmetic resolution' in \cite[Theor.4.2]{BO}): $$ 0\to {\cal H}^i_X\p{(j)}\to \coprod_{x\in X^0}^{} i_x H^{i}(x,\p{j}) \to \coprod_{x\in X^1}^{} i_x H^{i-1}(x,\p{j-1}) \to \cdots $$ \B{rmk} The assumption of $k$ perfect is unnecessary (cf. \ref{app}.1) if $H^*(\ ,\cdot)$ is the \'etale theory (namely $H^i(X,\p{j}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, H^i(\mbox{$X_{\acute{e}t}$},\mu_{\nu}^{\otimes j})$ where $\mu_{\nu}$ is the \'etale sheaf of $\nu^{\rm th}$ root of unity and $\nu$ is any positive integer prime to char($k$) ). \E{rmk} \subsection{Quillen $K$-theory}\label{inter2} Let $X\leadsto K_p(X)$ be the Quillen $K$-functor associated with the exact category of vector bundles on any scheme $X$ (see \cite{Q}). For a fixed $X$ we let ${\cal K}_{p}({\cal O}_X)$ be the associated Zariski sheaf on $X$. For any noetherian separated scheme $X$ we have a complex of flasque sheaves (`Gersten's complex') $${\cal I}^{\mbox{\Large $\cdot $}}_{q,X}: \coprod_{x\in X^0}^{} i_x K_{q}(k(x)) \to \coprod_{x\in X^1}^{} i_x K_{q-1}(k(x)) \to \cdots $$ conjecturally exact if $X$ is regular, being proved exact by Quillen \cite{Q} if $X$ is regular and essentially of finite type over a field. By tensoring coherent modules with locally free sheaves and sheafifying one has a pairing of complexes of sheaves for $p,q\geq 0$ (see \cite[p.276-277]{GIL}): $$\cap : {\cal K}_{p}({\cal O}_X)\otimes {\cal I}^{\mbox{\Large $\cdot $}}_{q,X}\to {\cal I}^{\mbox{\Large $\cdot $}}_{p+q,X}$$ By capping with $1_X$ = the `fundamental class' i.e. the identity section of the constant sheaf $\Z \cong {\cal I}^{\mbox{\Large $\cdot $}}_{0,X}$ we get an augmentation $$\cap 1_X : {\cal K}_{p}({\cal O}_X) \to {\cal I}^{\mbox{\Large $\cdot $}}_{p,X}$$ which is a quasi-isomorphism if $X$ is regular essentially of finite type over a field (conjecturally for all regular schemes.) If $f:X\to Y$ is a proper morphism between biequidimensional schemes and $r={\rm dim}\,Y - {\rm dim}\,X$ then there is an induced map of complexes $f_{!}: f_*{\cal I}^{\mbox{\Large $\cdot $}}_{q,X} \to {\cal I}^{\mbox{\Large $\cdot $}}_{q+r,Y}[r]$ (which takes the elements of $K_*(k(x))$ to $K_*(k(f(x)))$ if dim $\bar{x}$ = dim $\overline{f(x)}$ and takes them to zero otherwise) and a commutative diagram (see the `projection formula' \cite[p.411]{GIN}): \B{equation}\B{array}{c}\label{kpr} \hspace{20pt}f_*{\cal K}_{p}({\cal O}_X)\otimes f_*{\cal I}^{\mbox{\Large $\cdot $}}_{q,X}\\ \p{f^{\natural}\otimes id}\nearrow \ \hspace{40pt}\ \searrow \p{\cap_{f_*}} \\ {\cal K}_{p}({\cal O}_Y)\otimes f_*{\cal I}^{\mbox{\Large $\cdot $}}_{q,X}\hspace{60pt} f_*{\cal I}^{\mbox{\Large $\cdot $}}_{p+q,X}\\ \p{id\otimes f_{!}}\downarrow \ \hspace{105pt} \ \downarrow \p{f_{!}} \\ {\cal K}_{p}({\cal O}_Y)\otimes {\cal I}^{\mbox{\Large $\cdot $}}_{q+r,Y}[r] \hspace{10pt} \longby{\cap} \hspace{10pt} {\cal I}^{\mbox{\Large $\cdot $}}_{p+q+r,Y}[r] \E{array} \E{equation} Thus the formula: $f_!(f^{\natural}(\tau) \cap_{f_*}\sigma) = \tau\cap f_!(\sigma ) $ for all sections $\tau \p{\otimes} \sigma$ of the complex of sheaves ${\cal K}_{p}({\cal O}_Y)\otimes f_*{\cal I}^{\mbox{\Large $\cdot $}}_{q,X}$\ . \section{Invariance} Let $X\in \cat{V}_k$ be an algebraic scheme; in the following we will assume $X$ smooth and the ground field $k$ perfect. We moreover assume given a cohomological functor $H_Z^*(X,\cdot)$ satisfying the list of axioms \cite[1.1--1.3]{BO} and the assumption \cite[7.1.2]{BO}. We are going to consider a morphism $f:X\to Y$ from $X$ as above to $Y\in \cat{V}_k$ tacitly assuming $Y$ to be smooth.\\ Let X be in $\cat{V}_k$ and let ${\cal H}^i_X\p{(j)}$ (resp. ${\cal K}_{i}({\cal O}_X)$) denote the sheaf on $X$, for the Zariski topology, associated to the presheaf $U \leadsto H^i(U, \p{j})$ (resp. $U \leadsto K_i(\mbox{vector bundles on}\, U)$). If $f:X\to Y$ is any morphism in $\cat{V}_k$ then there are maps $f^{\sharp}: {\cal H}^i_Y\p{(j)} \to f_*{\cal H}^i_X\p{(j)}$ (resp. $f^{\natural}: {\cal K}_{i}({\cal O}_Y) \to f_*{\cal K}_{i}({\cal O}_X)$). \B{teor} {\rm (Invariance)} Let $f:X\to Y$ be a proper birational morphism in $\cat{V}_k$. For $X$ and $Y$ smooths over $k$ perfect then $f^{\sharp}$ yields the isomorphism $${\cal H}^i_Y\p{(j)}\cong f_*{\cal H}^i_X\p{(j)}$$ for all integers $i$ and $j$. For $X$ and $Y$ regular algebraic schemes $f^{\natural}$ induces the isomorphism $$ {\cal K}_{i}({\cal O}_Y)\cong f_*{\cal K}_{i}({\cal O}_X)$$ for all $i\geq 0$.\\ Hence there are isomorphisms $$H^0(X,{\cal H}^i_X\p{(j)})\cong H^0(Y,{\cal H}^i_Y\p{(j)})$$ and $$ H^0(X,{\cal K}_{i}({\cal O}_X))\cong H^0(Y,{\cal K}_{i}({\cal O}_Y))$$ \E{teor} \B{rmk} By the recent results of M. Spivakovsky the problem of `the elimination of points of indeterminacy' appears to be solved also in positive characteristic. Thus, by the Theorem~1, the groups $H^0(X,{\cal H}^{*}_X\p{(\cdot)})$ and $H^0(X,{\cal K}_{*}({\cal O}_X))$ are birational invariants of $X$ smooth and proper (cf. \cite{CO}). \E{rmk} The proof of the Theorem~1 is quite natural and easy after a sheaf form of the projection formula (cf. Lemma~\ref{chiave} and (\ref{hpr})). We will first give an explicit description of the map $f^{\sharp}$. \subsection{Functoriality}\label{rip1} If we are given a morphism $f:X\to Y$ then, for any open $V$ subset of $Y$, there is a homomorphism $H^*(V,\cdot) \to H^*(f^{-1}(V),\cdot)$ induced by $f$ simply because $H^*(\ ,\cdot)$ is a contravariant functor; thus, with the notation previously introduced, we get indeed a map $$f^H:H^*_Y\p{(\cdot)} \to f_*H^*_X\p{(\cdot)}$$ of presheaves on $Y$. Moreover there is a canonical map $f_*H^*_X\p{(\cdot)} \to f_*{\cal H}^*_X\p{(\cdot)}$ induced by sheafification and direct image. Hence, taking the associated sheaves, one get \B{eqnarray}\label{map} {\cal H}^*_Y\p{(\cdot)} & \by{af^H} & af_*H^*_X\p{(\cdot)} \nonumber \\ & & \ \downarrow \p{f_a} \\ & & f_*{\cal H}^*_X\p{(\cdot)} \nonumber \E{eqnarray} Thus the map $f^{\sharp}: {\cal H}^*_Y\p{(\cdot)} \to f_*{\cal H}^*_X\p{(\cdot)}$ is defined to be the composite of $af^H$ and $f_a$ as above. \subsection{Key Lemma}\label{svol1} In the following, till the end of this subsection, we will let $f:X\to Y$ be a proper birational morphism between smooth algebraic schemes over a perfect field. Our goal is to prove that $f^{\sharp }$ is an isomorphism of sheaves.\\[1pt] {\it Step 1}.\, We can reduce to proving Theorem~1 for irreducible schemes because if not then, from smoothness, the irreducible components coincide with the connected components and $f$ maps components to components; hence, if $X_0$ and $Y_0$ are components such that $f:X_0\to Y_0$ and $y\in Y_0$ then there are isomorphisms on the stalk $({\cal H}^*_Y\p{(\cdot)})_y\cong ({\cal H}^*_{Y_0}\p{(\cdot)})_y$ and $(f_*{\cal H}^*_X\p{(\cdot)})_y\cong (f_*{\cal H}^*_{X_0}\p{(\cdot)})_y$ If $X$ is irreducible and $K(X)$ is the function field of $X$ then we denote $$H^*(K(X),\cdot) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{U open\subset X} H^*(U,\cdot).$$ $H^*(K(X),\cdot)$ is canonically contravariant and birationally invariant. Hence $f$ induces an isomorphism $H^*(K(Y),\cdot)\cong H^*(K(X),\cdot).$ So, we moreover assume $X$ and $Y$ irreducibles.\\[1pt] {\it Step 2}.\, Case ${\cal H}^0\p{(\cdot)}.$ Assume that the cohomology theory is concentrated in positive degrees {\it i.e.\/}\ $H^i\p{(\cdot)}=0$ if $i<0$; hence the arithmetic resolution yields an isomorphism ${\cal H}^0_X\p{(\cdot)}\cong i_XH^0(K(X),\cdot)$ = the constant sheaf $H^0(K(X),\cdot)$ on $X$. The same holds on $Y$. Since $f$ has connected fibres (`Zariski main theorem') then ${\cal H}^0_Y\p{(\cdot)}\cong f_*{\cal H}^0_X\p{(\cdot)}\ .$ The non bounded case is considered below.\\[1pt] {\it Step 3}.\, So, associated to $f:X\to Y$, by (\ref{shortloc}) and (\ref{map}), we can construct a diagram \B{equation} \label{diagramma chiave} \begin{array}{ccccccccc} {0}&{\to}&{{\cal H}^i_{Y}\p{(j)}}&{\to}&{i_YH^i(K(Y),\p{j})} & {\to}& {{\cal H}^{i+1}_{Z^{1},Y}\p{(j)}} &{\to}&{0} \\ & &{\p{f^{\sharp}}\downarrow \ \ } & &{\cong\downarrow\;\;} & & {\downarrow \ \ \p{f^{\sharp}_{Z^1}}} & &\\ {0}&{\to}&{f_*{\cal H}^i_{X}\p{(j)}}&{ \to}&{f_*(i_XH^i(K(X),\p{j}))}&{\to}& {f_*({\cal H}^{i+1}_{Z^{1},X}\p{(j)})}& & \end{array} \E{equation} where the right most vertical arrow (it will be seen explicitly below) is defined by commutativity of the left hand square. (Note: because $f$ has connected fibres then the middle vertical map is an isomorphism. The commutativity is straightforward.)\\ {}From the above diagram one can see that $f^{\sharp}: {\cal H}^*_Y\p{(\cdot)} \to f_*{\cal H}^*_X\p{(\cdot)}$ is injective. Because of $f$ proper, and the arithmetic resolution is covariant for proper maps, we do aim to get the following commuative diagram \B{displaymath}\B{array}{ccccccc} 0 &\to & {\cal H}^{i}_{Y}\p{(j)} & \to &{i_YH^i(K(Y),\p{j}) }&\to &{\displaystyle \coprod_{y\in Y^1}^{} i_y H^{i-1}(y,\p{j-1})}\\ & &\p{f_{\sharp}} \uparrow \ \ & & \cong \uparrow \ \ & & \uparrow \\ 0 & \to & f_*({\cal H}^{i}_{X}\p{(j)}) &\to & f_*(i_XH^i(K(X),\p{j})) &\to & f_*({\displaystyle \coprod_{x\in X^1}^{} i_x H^{i-1}(x,\p{j-1})}) \E{array}\E{displaymath} where $f_{\sharp}$ is an injection. The Theorem~1 is obtained by proving: $f_{\sharp}\p{\circ }f^{\sharp}=id$ as a consequence of the projection formula. Indeed we have: \B{lemma}\label{chiave} Let $f:X\to Y$ be a proper birational morphism between irreducible algebraic smooth schemes. Then there are maps of sheaves ($k=0,1$): $$f^{\sharp}_{Z^k}:{\cal H}^{i}_{Z^{k},Y}\p{(j)} \to f_*({\cal H}^{i}_{Z^{k},X}\p{(j)})$$ and $$f_{\sharp}^{Z^k}:f_*({\cal H}^{i}_{Z^{k},X}\p{(j)}) \to {\cal H}^{i}_{Z^{k},Y}\p{(j)}$$ such that $$f_{\sharp}^{Z^k}\p{\circ }f^{\sharp}_{Z^k}=id$$ \E{lemma} (Remind: $f_{\sharp}^{Z^0} =f_{\sharp}$ and $f^{\sharp}_{Z^0}=f^{\sharp}$.)\\[2pt] \B{proof} We will follow the framework given by Grothendieck in \cite[III.9.2]{GR}.\\ Note that $f(X)$ is closed and dense in $Y$ irreducible: $f(X)=Y$. For all $Z\in Z^k(Y)$ so that $f^{-1}(Z)\in Z^k(X)$ ($k=0,1$) we have a map $H^{i}_{Z}(Y,\cdot)\to H^{i}_{f^{-1}(Z)}(X,\cdot)$ and since $f$ is a proper morphism between smooth schemes we have also maps $H^{i}_{f^{-1}(Z)}(X,\cdot)\to H^{i}_{Z}(Y,\cdot)$ for all $Z\in Z^k(Y)$. We then have: \B{slemma}\label{prfor} The composition: $$H^{i}_{Z}(Y,\cdot) \by{f^{\star}} H^{i}_{f^{-1}(Z)}(X,\cdot) \by{f_{\star}} H^{i}_{Z}(Y,\cdot)$$ is the identity. \E{slemma} \B{proof} Let $H_*(\ ,\dagger)$ denote the `twin' homology theory and consider the pairing: $$\p{\cap_{Y,Z}}: H_l(Y,\p{m})\otimes H_Z^r(Y,\p{s})\to H_{l-r}(Z,\p{m-s})$$ Denote $f_{!}: H_*(f^{-1}(T),\dagger) \to H_*(T,\dagger)$ the homomorphisms induced, by covariance, from the proper maps $f^{-1}(T) \to T $ for every closed subset $T$ of $Y$. Let $f^{\star}: H^*_{Z}(Y,\cdot) \to H^{*}_{f^{-1}(Z)}(X,\cdot)$ be the map given by contravariancy. Because of \cite[Axiom 1.3.3]{BO} we have the projection formula: $$f_!(x\p{\cap_{X,f^{-1}(Z)}}f^{\star}(y))=f_!(x)\p{ \cap_{Y,Z}}y$$ for every $x\in H_l(X,m)$ and $y\in H_Z^r(Y,s)$. Let $\eta_X$ denote the fundamental class in $H_{2d}(X,d)$ where $d= {\rm dim} X$. Because of \cite[7.1.2]{BO} and $f$ proper birational we get: $f_!(\eta_X)=\eta_Y$. Thus the projection formula yields the equation: $$f_!(\eta_X\p{\cap_{X,f^{-1}(Z)}}f^{\star}(y))= \eta_Y\p{\cap_{Y,Z}}y$$ By Poincar\'e duality \cite[1.3.5]{BO} the cap product with the fundamental class is an isomorphism; we define $$f_{\star}(z) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, (\eta_Y\p{\cap_{Y,Z}}-)^{-1}\p{\circ} f_!(\eta_X\p{\cap_{X,f^{-1}(Z)}}z)$$ for all $z\in H^{*}_{f^{-1}(Z)}(X,\cdot)$. Thus: $f_{\star}\p{\circ} f^{\star}=1.$ \E{proof} Taking the direct limit of the concerned maps over $Z\in Z^k(Y)$ (note: because $f$ is closed the direct system $\{ f^{-1}(Z)\, :\, Z\in Z^1(Y)\}$ is cofinal in $Z^1(X)$) we have that the composition $$H^{i}_{Z^k(Y)}(Y,\cdot)\by{f^{\star}_{Z^k}} H^{i}_{Z^k(X)}(X,\cdot) \by{f_{\star}^{Z^k}}H^{i}_{Z^k(Y)}(Y,\cdot)$$ is the identity as a consequence of the Sublemma~\ref{prfor} and limit arguments (the compatibilities are given by \cite[1.1.2~and~1.2.4]{BO}).\\ Because of~\cite[1.2.2~and~1.4]{BO} the maps $f_{\star}^{Z^k}$ are natural trasformations of Zariski presheaves $H^*_{Z^k,Y}(\cdot) \to f_*H^*_{Z^k,X}(\cdot)$ on $Y$. Thus, taking the associated sheaves, we have that: $${\cal H}^{i}_{Z^{k},Y}\p{(j)} \by{af^{\star}_{Z^k}} af_*(H^{i}_{Z^{k},X}\p{(j)}) \by{af_{\star}^{Z^k}} {\cal H}^{i}_{Z^{k},Y}\p{(j)} $$ is the identity. Now it sufficies to make up a commutative diagram as follows \B{equation}\B{array}{ccc}\label{bravo} {\cal H}^{i}_{Z^{k},Y}\p{(j)}\ \by{af^{\star}_{Z^k}} & af_*(H^{i}_{Z^{k},X}\p{(j)}) & \by{af_{\star}^{Z^k}} \ {\cal H}^{i}_{Z^{k},Y}\p{(j)} \\ \p{f^{\sharp}_{Z^k}}\searrow & \ \downarrow \p{f_a^{Z^k}}& \nearrow \p{f_{\sharp}^{Z^k}} \\ & f_*({\cal H}^{i}_{Z^{k},X}\p{(j)}) & \E{array}\E{equation} {}From (\ref{bravo}) we then have: $$f_{\sharp}^{Z^k}\p{\circ}f^{\sharp}_{Z^k}= f_{\sharp}^{Z^k}\p{\circ}\,f_a^{Z^k}\p{\circ}\, af^{\star}_{Z^k}= af_{\star}^{Z^k}\p{\circ}\,af^{\star}_{Z^k}=id$$ as claimed. Indeed $f^{\sharp}_{Z^k}$ is simply defined by composition; since $f$ is proper, dim$X$ = dim$Y$ and the arithmetic resolution is covariant for proper maps: $f_{\sharp}^{Z^k}$ is obtained, e.g. $f_{\sharp}^{Z^1}$ from the commutativity and the exactness of the following: \B{displaymath}\B{array}{ccccccc} 0 &\to & {\cal H}^{i+1}_{Z^{1},Y}\p{(j)} & \to &{\displaystyle \coprod_{y\in Y^1}^{} i_y H^{i-1}(y,\p{j-1}) }&\to &{\displaystyle \coprod_{y\in Y^2}^{} i_y H^{i-2}(y,\p{j-2})}\\ & & & & \uparrow & & \uparrow \\ 0 & \to & f_*({\cal H}^{i+1}_{Z^{1},X}\p{(j)}) &\to & f_*({\displaystyle \coprod_{x\in X^1}^{} i_x H^{i-1}(x,\p{j-1})}) &\to & f_*({\displaystyle \coprod_{x\in X^2}^{} i_x H^{i-2}(x,\p{j-2})}) \E{array}\E{displaymath} The proof of the Lemma~\ref{chiave} is complete. \E{proof} \subsection{Proof of the Invariance Theorem}\label{H-coda} To summarize the proof: if $k=0,1$ and $Z\in Z^k(Y)$ then $f^{-1}(Z)\in Z^k(X)$; we have a splitting between long exact sequences (cf.~(\ref{loc})) $$ \begin{array}{ccccccccc} {\cdots}&{\to}&{H^i(Y,\cdot)}&{\to}&{H^i(Y- Z,\cdot)} & {\to}& {H^{i+1}_{Z}(Y,\cdot)} &{\to}&{\cdots}\\ & &{\downarrow \ \uparrow} & &{\downarrow\ \uparrow} & & {\downarrow \ \uparrow} & &\\ {\cdots}&{\to}&{H^i(X,\cdot)}&{\to}&{H^i(X- f^{-1}(Z),\cdot)} & {\to}& {H^{i+1}_{f^{-1}(Z)}(X,\cdot)} &{\to}&{\cdots} \end{array} $$ Taking the direct limit of the concerned diagram over $Z\in Z^1(Y)$ we do get $$ \begin{array}{ccccccccc} {\cdots}&{\to}&{H^i(Y,\cdot)}&{\to}&{H^i(K(Y),\cdot)} & {\to}& {H^{i+1}_{Z^1(Y)}(Y,\cdot)} &{\to}&{\cdots}\\ & &{\downarrow \ \uparrow} & &{\cong\downarrow\;\;\ \uparrow} & & {\downarrow \ \uparrow} & &\\ {\cdots}&{\to}&{H^i(X,\cdot)}&{\to}&{H^i(K(X),\cdot)} & {\to}& {H^{i+1}_{Z^1(X)}(X,\cdot)} &{\to}&{\cdots} \end{array} $$ Thus, taking the associated sheaves, we have: $$ \begin{array}{ccccccccc} {\cdots}&{\by{zero}}&{{\cal H}^i_{Y}\p{(\cdot)}}&{\to}&{ i_YH^i(K(Y),\p{\cdot})} & {\to}& {{\cal H}^{i+1}_{Z^{1},Y}\p{(\cdot)}} &{\by{zero}}&{\cdots}\\ & &{\downarrow\ \uparrow} & &{\cong\downarrow\;\;\ \uparrow} & & {\downarrow \ \uparrow} & &\\ {\cdots}&{\to}&{af_*H^i_{X}\p{(\cdot)}}&{\to}&{af_*H^i(K(X),\cdot)} & {\to}& {af_*(H^{i+1}_{Z^{1},X}\p{(\cdot)})} &{\to}&{\cdots} \end{array} $$ and furthermore $$ \begin{array}{ccccccccc} {\cdots}&{\to}&{af_*H^i_{X}\p{(\cdot)}}&{\to}&{af_*H^i(K(X), \cdot)} & {\to}& {af_*(H^{i+1}_{Z^{1},X}\p{(\cdot)})} &{\to}&{\cdots}\\ & &{\downarrow} & &{\cong\downarrow\;\;} & & {\downarrow} & &\\ {0}&{\to}&{f_*({\cal H}^i_{X}\p{(\cdot)}})&{\to}&{ f_*(i_XH^i(K(X),\p{\cdot})} & {\to}& {f_*({\cal H}^{i+1}_{Z^{1},X}\p{(\cdot)})} &{\to}&{\cdots} \end{array} $$ One then obtain by patching the diagram (\ref{diagramma chiave}). Now because of Lemma~\ref{chiave} and (\ref{diagramma chiave}) we do get the first claimed isomorphism $f^{\sharp}: {\cal H}^*_Y\p{(\cdot)} \by{\simeq} f_*{\cal H}^*_X\p{(\cdot)}$\,. \subsection{Proving the $K$-theory case}\label{svol2} We now consider the Quillen $K$-theory of vector bundles. The proof of Theorem~1 is the analogous of the previous one by using the Gillet's projection formula and the Gersten's conjecture.\\ To prove the Theorem~1, arguing as in \S\ref{svol1}, we can assume $X$ and $Y$ irreducibles, $r=0$ and $f_{!}: f_*{\cal I}^{\mbox{\Large $\cdot $}}_{0,X} \by{ \simeq} {\cal I}^{\mbox{\Large $\cdot $}}_{0,Y}$ given by $f_{!}(1_X)=1_Y$ (because $K(X)\cong K(Y)$). Hence, by (\ref{kpr}), we obtain $$\B{array}{ccccc} & &f_*{\cal K}_{p}({\cal O}_X) & &\\ &\p{f^{\natural}}\nearrow \ & &\ \searrow \p{\cap_{f_*}1_X}& \\ {\cal K}_{p}({\cal O}_Y)& & & & f_*{\cal I}^{\mbox{\Large $\cdot $}}_{p,X}\\ \mbox{\large $\parallel$} & & & & \ \downarrow \p{f_{!}} \\ {\cal K}_{p}({\cal O}_Y) & & \longby{\cap 1_Y} & & {\cal I}^{\mbox{\Large $\cdot $}}_{p,Y} \E{array}$$ One defines $f_{\natural}:f_*{\cal K}_{p}({\cal O}_X) \to{\cal K}_{p}({\cal O}_Y)$ in the derived category, as follows: $$f_{\natural} \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, (\ \cap 1_Y)^{-1}\p{\circ} f_{!} \p{\circ} (\ \cap_{f_*}1_X)$$ Thus: $f_{\natural}\p{\circ}f^{\natural}=id$ . (Note: the map $\cap_{f_*}1_X$ is not a quasi-isomorphism in general, but it induces an isomorphism on homology in degree zero because $f_*$ is left exact; indeed, taking $h^0$= the zero homology of a complex, we have the commutative diagram of sheaves $$\B{array}{ccccc} & &f_*{\cal K}_{p}({\cal O}_X) & &\\ &\p{f^{\natural}}\nearrow \ & &\ \ \searrow \p{\simeq} & \\ {\cal K}_{p}({\cal O}_Y)& & & & h^0(f_*{\cal I}^{\mbox{\Large $\cdot $}}_{p,X})\\ \mbox{\large $\parallel$} & & & & \ \downarrow \p{f_{!}} \\ {\cal K}_{p}({\cal O}_Y) & & \longby{\simeq} & & h^0({\cal I}^{\mbox{\Large $\cdot $}}_{p,Y}) \E{array}$$ and $f_{\natural}\p{\circ}f^{\natural}=id$ between sheaves on $Y$ .)\\ Associated to $f:X\to Y$ proper birational morphism between regular (irreducible) algebraic schemes, we have \B{displaymath}\B{array}{ccc} {\cal K}_{p}({\cal O}_Y) & \hookrightarrow & i_YK_p(K(Y)) \\ \p{f_{\natural}} \uparrow \downarrow \p{f^{\natural}} & & \cong\downarrow\;\;\uparrow \ \\ f_*({\cal K}_{p}({\cal O}_X)) & \hookrightarrow & f_*(i_XK_p(K(X))) \E{array}\E{displaymath} so that by the same argument as in \S\ref{H-coda} we do get the second claimed isomorphism $f^{\natural}: {\cal K}_{p}({\cal O}_Y) \by{\simeq} f_*{\cal K}_{p}({\cal O}_X)$\,. \B{rmk} Assuming the Gersten's conjecture and applying the above argument one can see that $f^{\natural}$ is an isomorphism if $f$ is a proper birational morphism between regular biequidimensional schemes. \E{rmk} \section{Homotopy and proto-decomposition} We maintain the notations and the assumed `cohomology theory' introduced in the previous Section (see \S1). Let $\P^n_X$ be the scheme $X\times _{k}{\rm Proj}\, k [t_0,\ldots,t_n]$; let $\pi_n : \P^n_X\to X$ denote the canonical projection on $X$ smooth and equidimensional in $\cat{V}_k$ and assume $k$ perfect. For any couple of non-negative integers $n\geq m$ let $j_{(n,m)}$ denote the `Gysin homomorphism'(see \S\ref{cad1} below) $$H^{p}(\P^m_X,{\cal H}^{q}\p{(j)})\to H^{p+n-m}(\P^n_X,{\cal H}^{q+n-m}\p{(j+n-m)})$$ given by the smooth pair $(\P^n_X,\P^m_X)$ of pure codimension $n-m$\,; if $m\geq l$ is another such couple, i.e. $(\P^m_X,\P^l_X)$ is a pair, then $j_{(n,l)} = j_{(n,m)}\p{\circ}j_{(m,l)}$ . Let $\A^1_X$ denote the scheme $X\otimes _{k}k [t]$ and assume that the cohomology theory satisfies the following.\\[1pt] {\bf Homotopy property}. {\em Let $X$ be an algebraic smooth scheme. The natural morphism $\pi: \A^1_X \to X$ induces an isomorphism $$\pi^*: H^*(X,\cdot) \by{\simeq} H^*(\A^1_X,\cdot)$$ by pulling-back along $\pi$.}\\[1pt] For ${\cal E}$ a locally free sheaf on $X$, ${\rm rank}\, {\cal E} = n+1$ and $\pi:{\bf V}({\cal E}) \to X$ the associated vector bundle, we then get the isomorphism (see \S\ref{minom} below) $$H^p_Z(X,{\cal H}^q\p{(j)})\cong H^p_{\pi^{-1}(Z)}({\bf V}({\cal E}),{\cal H}^q\p{(j)})$$ pulling back along $\pi$ where $Z\subseteq X$ is any closed subset. Furthermore, it is now possible to prove the following Dold-Thom type decomposition. \B{teor} {\rm (Proto-decomposition)} Let $X$ be algebraic, equidimensional and smooth over a perfect field. Assuming the homotopy property above then there is an isomorphism $$H^p(\P^n_X,{\cal H}^q_X\p{(j)})\cong \bigoplus_{i=0}^{n} H^{p-i}(X,{\cal H}^{q-i}_X\p{(j-i)})$$ where every $x\in H^p(\P^n_X,{\cal H}^q_X\p{(j)})$ is written as $$\pi^*_n(x_{n})+j_{(n,n-1)}\pi^*_{n-1}(x_{n-1})+\cdots +j_{(n,1)}\pi^*_1(x_{1})+j_{(n,0)}(x_0)$$ for $x_{n-i}\in H^{p-i}(X,{\cal H}^{q-i}_X\p{(j-i)})$ and $i=0,\ldots ,n$ . \E{teor} \B{rmk} Note that for ${\cal E}$ a locally free sheaf on $X$ , ${\rm rank}\, {\cal E} = n+1$, we will obtain the decomposition of $H^p(\P({\cal E}),{\cal H}^q_X\p{(j)})$ in Scholium~\ref{Edeco}. \E{rmk} Before proving the Theorem~2 we need the following results. \subsection{Gysin maps for ${\cal H}$-cohomologies}\label{cad1} The category $\cat{V}^2_k$ is the category of pairs of algebraic schemes over a perfect field $k$ . \B{lemma}{\rm (Purity) } If $(X,Z)$ is a pure smooth pair in $\cat{V}^2_k$, ${\rm codim}_XZ=c$, then $H^p_Z(X,{\cal H}^q_X\p{(j)})$ is canonically isomorphic to $H^{p-c}(Z,{\cal H}^{q-c}_Z\p{(j-c)})$. \label{purity}\E{lemma} \B{proof} Let ${\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}$ denote the arithmetic resolution of the sheaf ${\cal H}^q\p{(j)}$ on $X$ (resp. on $Z$) and denote $H^0(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, {\rm R}_q^{\mbox{\Large $\cdot $}}(X)\p{(j)}$ (resp. ${\rm R}_q^{\mbox{\Large $\cdot $}}(Z)\p{(j)}$ ). Then: \B{slemma}\label{sup} For $Z \subset X$ of pure codimension c : $$H^0_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}) \cong {\rm R}_{q-c}^{\mbox{\Large $\cdot $}}(Z)\p{(j-c)}[-c]$$ \E{slemma} \B{proof} Straightforward. \E{proof} Since ${\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}$ is a bounded complex (graded by codimension) of flasque sheaves the hypercohomology spectral sequence ($h^n(C^{\mbox{\Large $\cdot $}}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,$ the n$^{th}$ homology group of a complex $C^{\mbox{\Large $\cdot $}}$) $$'E^{r,s}_2 = h^r(H^s_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)})) \Rightarrow \H^{r+s}_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)})$$ degenerates to isomorphisms $$ h^p(H^0_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)})) \cong \H^p_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)})$$ Taking account of the Sublemma~\ref{sup}, because of the (flasque) arithmetic resolutions, we do get a chain of isomorphisms $$\begin{array}{rcl} H^p_Z(X,{\cal H}^q_X\p{(j)}) & \cong & \H^p_Z(X, {\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}) \\ & \cong & h^p(H^0_Z(X,{\cal R}_q^{\mbox{\Large $\cdot $}}\p{(j)}))\\ & \cong & h^p({\rm R}_{q-c}^{\mbox{\Large $\cdot $} -c}(Z)\p{(j-c)}) \\ & \cong & H^{p-c}(Z,{\cal H}^{q-c}_Z\p{(j-c)}) \end{array} $$ The proof of the Lemma~\ref{purity} is complete. \E{proof} \B{schol}{\rm (Gysin map) } \label{Gysin} Let $(X,Z)\in \cat{V}^2_k$ be a smooth pair of pure codimension $c$. There is an homomorphism $$j_{\pp{(X,Z)}} :H^{p}(Z,{\cal H}^{q}_Z\p{(j)})\to H^{p+c}(X,{\cal H}^{q+c}_X\p{(j+c)})$$ such that if $(Z,T)$ is another smooth pair then $j_{\pp{(X,T)}} = j_{\pp{(X,Z)}}\p{\circ}j_{\pp{(Z,T)}}$ . \E{schol} \B{proof} The map $j_{\pp{(X,Z)}}$ is induced on cohomology by the composition in the derived category (by sheafifying the isomorphism in the Sublemma~\ref{sup} and using the arithmetic resolutions) $$j_*{\cal H}^{q}_Z\p{(j)}\by{\simeq}j_*{\cal R}_{q,Z}^{\mbox{\Large $\cdot $}}\p{(j)} \by{\simeq}{\bf \Gamma}_Z{\cal R}_{q+c,X}^{\mbox{\Large $\cdot $}}\p{(j+c)}[c] \to {\cal R}_{q+c,X}^{\mbox{\Large $\cdot $}}\p{(j+c)}[c] \by{\simeq} {\cal H}^{q+c}_X\p{(j+c)}[c]$$ where $j:Z\hookrightarrow X$ and ${\bf \Gamma}_Z$ are the sections supported in $Z$ . The compability simply follows by considering the resolutions ${\cal R}^{\mbox{\Large $\cdot $}}$ and observing that a global section of ${\cal R}^{\mbox{\Large $\cdot $}}$ on $T$ can be seen as a section of ${\cal R}^{\mbox{\Large $\cdot $}}$ on $Z$ supported in $T$, shifted by the codimension of $T$ in $Z$, etc\ldots , as a section of ${\cal R}^{\mbox{\Large $\cdot $}}$ on $X$ shifted by ${\rm codim}_TZ + {\rm codim}_ZX = {\rm codim}_TX$. \E{proof} \subsection{Homotopy for ${\cal H}$-cohomologies}\label{minom} For $X\in \cat{V}_k$ smooth over $k$ perfect, we recall (see \cite[6.3]{BO}) that exists a spectral sequence (`coniveau') $$E^{p,q}_2 = H^p(X,{\cal H}^q\p{(\cdot)}) \Rightarrow H^{p+q}(X,\cdot)$$ \B{lemma}{\rm (Homotopy) } If the functor $H^*(\ ,\cdot)$ has the homotopy property then the functor $H^{\#}_{Zar}(\ ,{\cal H}^*\p{(\cdot)})$ has the homotopy property. \label{homotopy} \E{lemma} \B{proof} Let $\pi : \A^1_X \to X$ be the given structural flat morphism over $X$ smooth. We will show that $\pi^*: H^{\#}_{T}(X,{\cal H}^*\p{(\cdot)})\cong H^{\#}_{\pi^{-1}(T)}(\A^1_X,{\cal H}^*\p{(\cdot)})$ for any closed subscheme $T\subseteq X$. The proof is divided in two steps.\\ First step: reducing to the function field case $\A^1_{K} \to K$ . This is done using a trick by Quillen \cite[Prop.4.1]{Q}. Associated to $\pi$ and $Z\subset T\subset X$ closed subsets, $U =X-Z$ , we have a map of long exact sequences: $$ \begin{array}{ccccccccc} \cdots & \to & H^p_T(X,{\cal H}^*\p{(\cdot)}) & \to & H^p_{T\cap U}(U,{\cal H}^*\p{(\cdot)}) & \to & H^{p+1}_Z(X,{\cal H}^*\p{(\cdot)}) & \to & \cdots \\ & &{\downarrow} & &{\downarrow} & & {\downarrow} & &\\ {\cdots}&{\to}& H^p_{\pi^{-1}(T)}(\A^1_X,{\cal H}^*\p{(\cdot)}) &{\to}& H^p_{\pi^{-1}(T\cap U)}(\A^1_U,{\cal H}^*\p{(\cdot)}) & {\to}& H^{p+1}_{\pi^{-1}(Z)}(\A^1_X,{\cal H}^*\p{(\cdot)}) &{\to}&{\cdots} \end{array} $$ By the five lemma the induced (middle vertical) homomorphisms $$H^p_T(X,{\cal H}^*\p{(\cdot)}) \to H^p_{\pi^{-1}(T)}(\A^1_X,{\cal H}^*\p{(\cdot)})$$ are isomorphisms (all $p$) if the others vertical maps are. Using noetherian induction we can assume $H^p_Z(X,{\cal H}^*\p{(\cdot)}) \to H^p_{\pi^{-1}(Z)}(\A^1_X,{\cal H}^*\p{(\cdot)})$ to be an isomorphism for all closed subsets $Z\neq T$ and all $p\geq 0$. We can also suppose $X$ irreducible. Taking the direct limit over all closed proper subschemes $Z$ of $T$ we can also assume that $T$ is integral of codimension $t$. Thus by local purity we are left to show that \B{eqnarray}\label{field} \limdir{U = X-Z} H^p_{T\cap U}(U,{\cal H}^*\p{(\cdot)}) & \cong & H^{p-t}({\rm Spec}\, (K(T)),{\cal H}^{*-t}\p{(\cdot-t)}) \nonumber \\ & &\ \ \ \ \ \ \downarrow \\ \limdir{U = X-Z} H^p_{\pi^{-1}(T\cap U)}(\A^1_U,{\cal H}^*\p{(\cdot)}) & \cong & H^{p-t}(\A^1_{K(T)},{\cal H}^{*-t}\p{(\cdot-t)}) \nonumber \E{eqnarray} is an isomorphism for all $p$. (Note: the horizontal isomorphisms in (\ref{field}) are obtained by continuity of the arithmetic resolution of the sheaf ${\cal H}^*\p{(\cdot)}$ ).\\ Second step: proving the function field case $K=K(X)$. Having defined $$H^*(\A^1_{K},\cdot) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{U\subset X} H^*(\A^1_{U},\cdot)$$ by continuity of the coniveau spectral sequence we do have $$E^{p,q}_2 = H^p(\A^1_{K},{\cal H}^q\p{(\cdot)}) \Rightarrow H^{p+q}(\A^1_{K},\cdot)$$ and $E^{p,q}_2 =0$ if $p>1={\rm dim}\, \A^1_{K}$, thus all the differentials are zero which just yields short exact sequences $$0 \to H^1(\A^1_{K},{\cal H}^{q-1}\p{(\cdot)})\to H^q(\A^1_{K},\cdot)\to H^0(\A^1_{K},{\cal H}^q\p{(\cdot)}) \to 0 $$ Associated to the flat map $\A^1_K \to K$ we have a commutative diagram \B{equation}\begin{array}{ccccccccc}\label{split} 0 & \to & H^1(\A^1_{K},{\cal H}^{q-1}\p{(\cdot)}) & \to & H^q(\A^1_{K},\cdot) & \to & H^0(\A^1_{K},{\cal H}^q\p{(\cdot)}) & \to & 0\\ & & & &{\cong\uparrow\;\;} & & {\downarrow \ \uparrow} & &\\ & & & & H^q(K,\cdot) & \by{\simeq} & H^0(K,{\cal H}^q\p{(\cdot)}) & & \end{array} \E{equation} where: because $H^*(\ ,\cdot)$ has the homotopy property then the middle vertical map is an isomorphism while the arrow $H^0(\A^1_{K},{\cal H}^q\p{(\cdot)})\to H^q(K,\cdot)$ is the evaluation at any $K$-rational point (cf. \cite[Proof of 2.5]{BR}). From (\ref{split}) it follows that $H^1(\A^1_{K},{\cal H}^{q-1}\p{(\cdot)}) = 0$ hence the required isomorphism in (\ref{field}) is given by: $ H^0(K,{\cal H}^q\p{(\cdot)}) \cong H^0(\A^1_{K},{\cal H}^q\p{(\cdot)})$ \E{proof} \B{schol}\label{hind} Let $\A^n_X$ denote the n$^{th}$ affine space over $X$ smooth (i.e. the scheme $X\otimes _{k}k [t_1,\ldots,t_n]$ ) and let $\pi : \A^n_X \to X$ be the natural projection. Then $\pi$ induces an isomorphism $$H^{\#}_{\pi^{-1}(Z)}(\A^n_X,{\cal H}^*\p{(\cdot)})\cong H^{\#}_Z(X,{\cal H}^*\p{(\cdot)})$$ \E{schol} \B{proof} By induction from the Lemma~\ref{homotopy}. \E{proof} \B{cor}\label{Ehomo} Let ${\cal E}$ be a locally free sheaf on $X$ smooth and $\pi:{\bf V}({\cal E}) \to X$ the associated vector bundle, we then get the isomorphism $$\pi^*: H^p_Z(X,{\cal H}^q\p{(j)})\cong H^p_{\pi^{-1}(Z)}({\bf V}({\cal E}),{\cal H}^q\p{(j)})$$ \E{cor} \B{proof} By reduction to open Zariski neighborhoods on which ${\cal E}$ is free and noetherian induction (cf. the proof of Lemma~\ref{homotopy}). \E{proof} \subsection{Proof of the proto-decomposition Theorem}\label{svol3} The proof of Theorem~2 is by induction on $n$. For $n=0$, $\P^0_X \cong X$, hence the induction starts: one consider an `hyperplane at infinity' $\infty$ in $\P_X^n$ so that $\infty \cong \P_X^{n-1}$ and $\P_X^n - \infty\cong\A_X^n$. There is a standard long exact sequence of Zariski cohomology groups $$H^{p-1}(\A_X^n,{\cal H}^q\p{(j)}) \to H^{p}_{\infty}(\P_X^n, {\cal H}^q\p{(j)}) \to H^{p}(\P_X^n,{\cal H}^q\p{(j)}) \to H^{p}(\A_X^n,{\cal H}^q\p{(j)})$$ Let $\pi_n :\P_X^n\to X$ be the projection. By homotopy (see \S\ref{minom}) the restriction $\pi_n \mid_{\pp{\A_X^n}} : \A_X^n \to X$ induces a splitting of the previous long exact sequence, given by the commutative square $$\begin{array}{ccc} H^{p}(\P_X^n,{\cal H}^q\p{(j)})&\to &H^{p}(\A_X^n,{\cal H}^q\p{(j)})\\ \p{\pi_n^*}\uparrow & &\uparrow \cong \ \\ H^{p}( X,{\cal H}^q\p{(j)})& = &H^{p}( X,{\cal H}^q\p{(j)}) \end{array} $$ By the purity Lemma~\ref{purity} we have an isomorphism $$H^{p}_{\infty}(\P_X^n,{\cal H}^q\p{(j)})\cong H^{p-1}(\P_X^{n-1},{\cal H}^{q-1}\p{(j-1)})$$ and, by composing, the Gysin map (see \S\ref{cad1}) $$j_{(n,n-1)}:H^{p-1}(\P^{n-1}_X,{\cal H}^{q-1}\p{(j-1)})\to H^{p}(\P^n_X,{\cal H}^{q}\p{(j)})$$ So one can split the long exact sequence into short exact sequences $$0 \to H^{p-1}(\P^{n-1}_X,{\cal H}^{q-1}\p{(j-1)}) \by{j_{(n,n-1)}} H^{p}(\P^n_X,{\cal H}^{q}\p{(j)}) \stackrel{\stackrel{\pi_n^*}{\leftarrow}}{\to} H^{p}(X,{\cal H}^q\p{(j)}) \to 0 $$ Thus we do get the formula: $$H^p(\P _X^n,{\cal H}^q\p{(j)})\cong H^{p}( X,{\cal H}^q\p{(j)}) \oplus H^{p-1}(\P_X^{n-1},{\cal H}^{q-1}\p{(j-1)})$$ which does the induction's step: an element $x\in H^p(\P^n_X,{\cal H}^q_X\p{(j)})$ is written as $\pi^*_n(x_n)+j_{(n,n-1)}(x')$ for $x_n\in H^{p}(X,{\cal H}^{q}_X\p{(j)})$ and $x'\in H^{p-1}(\P_X^{n-1},{\cal H}^{q-1}\p{(j-1)})$; this last $x'$ because of the inductive hypothesis is written as $$\pi^*_{n-1}(x_{n-1})+j_{(n-1,n-2)}\pi^*_{n-2}(x_{n-2})+ \cdots +j_{(n-1,1)}\pi^*_1(x_{1})+j_{(n-1,0)}(x_0)$$ for $x_{n-1-i}\in H^{p-1-i}(X,{\cal H}^{q-1-i}_X\p{(j-1-i)})$ and $i=0,\ldots ,n-1$ and because of the compatibility of the Gysin homomorphisms (see Scholium~\ref{Gysin}) applying $j_{(n,n-1)}$ we are done. \section{Cap products} This section is devoted to construct a cap product between algebraic cycles and ${\cal H}$-cohomology classes. \subsection{Sophisticated Poincar\'e duality theories} Let assume we are given a cohomology theory $H^*(\p{\cdot})$ and a homology theory $H_*(\p{\cdot})$ on $\cat{V}_k$ satisfying the Bloch-Ogus axioms \cite[1.1-1.2]{BO}. Furthermore, we let assume the existence of a {\it sophisticated}\, cap-product with supports i.e. for all $(X,Z),(X,Y)\in\cat{V}^2_k$ a pairing $$\p{\cap_{Y,Z}}: H_n(Y,\p{m})\otimes H_Z^q(X,\p{s})\to H_{n-q}(Y\cap Z, \p{m-s})$$ which satisfies the following axioms: \B{description} \item[A1] $\p{\cap}$ is natural with respect to \'etale maps (or just open Zariski immersions according with \cite[1.4.2]{BO}) of pairs in $\cat{V}^2_k$. \item[A2] If $(X,T),(T,Z)$ are pairs in $\cat{V}^2_k$ then the following diagram \B{displaymath}\B{array}{ccc} H_n(Z,\p{m})\otimes H^q(X,\p{s}) & \longby{\cap} & H_{n-q}(Z,\p{m-s})\\ \downarrow & & \downarrow \\ H_n(T,\p{m})\otimes H^q(X,\p{s}) & \longby{\cap} & H_{n-q}(T,\p{m-s}) \E{array}\E{displaymath} commutes. \item[A3] For $(X,T),(T,Z)$ pairs in $\cat{V}^2_k$ let $U=X-Z$ and let $j: U \to X$ be the inclusion. Let denote $\partial: H_n(T\cap U,\cdot)\to H_{n-1}(Z,\cdot)$ the boundary map in the long exact sequence (\ref{hloc}) of homology groups. Then the following diagram \B{displaymath}\B{array}{c} \ \hspace{20pt} H_n(T\cap U,\cdot)\otimes H^q(U,\cdot) \\ \p{id\otimes j^*}\nearrow \hspace{50pt} \searrow\p{\cap}\\H_n(T\cap U,\cdot)\otimes H^q(X,\cdot)\hspace{40pt} H_{n-q}(T\cap U,\cdot) \\ \p{\partial\otimes id}\downarrow \hspace{97pt} \downarrow \p{\partial} \\ H_{n-1}(Z,\cdot)\otimes H^q(X,\cdot) \hspace{10pt} \longby{\cap} \hspace{10pt} H_{n-q-1}(Z,\cdot)\E{array} \E{displaymath} commutes i.e. we have the equation: \B{equation} \partial (y\p{\cap}j^*(x))=\partial (y)\p{\cap}x \E{equation} for $y\in H_n(T\cap U,\cdot)$ and $x\in H^q(X,\cdot)$. \item[A4 {\it (Projection Formula)}] Let $f:X'\to X$ be a proper morphism in $\cat{V}_k$. For $(X,Y)$ and $(X,Z)$ let $Y'=f^{-1}(Y)$ and $Z'=f^{-1}(Z)$. The following diagram $$\B{array}{c} \hspace{20pt} H_{n}(Y',\p{m})\otimes H^{q}_{Z'}(X',\p{s})\\ \p{id\otimes f^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cap} \\H_{n}(Y',\p{m})\otimes H^{q}_Z(X,\p{s})\hspace{60pt} H_{n-q}(Y'\cap Z',\p{m-s})\\ \p{f_{*}\otimes id}\downarrow \hspace{121pt}\downarrow \p{f_{*}} \\ H_{n}(Y,\p{m})\otimes H^{q}_Z(X,\p{s})\hspace{10pt} \longby{\cap} \hspace{10pt}H_{n-q}(Y\cap Z,\p{m-s}) \E{array}$$ commutes. \E{description} By the way, for $(H^*,H_*)$ as above, we have the following `projection formula': \B{schol}\label{A4} Let $f:Y\to X$ be a proper morphism in $\cat{V}_k$. Let $T$ be any closed subset of $Y$ and let $f(T)=Z$. Then the following diagram \B{equation}\B{array}{c} \ \hspace{20pt} H_{n}(T,\p{m})\otimes H^q(Y,\p{s}) \\ \p{id\otimes f^*}\nearrow \hspace{50pt} \searrow\p{\cap}\\ H_{n}(T,\p{m})\otimes H^q(X,\p{s})\hspace{40pt} H_{n-q}(T,\p{m-s}) \\ \p{f_{*}\otimes id}\downarrow \hspace{105pt} \downarrow \p{f_{*}} \\H_{n}(Z,\p{m})\otimes H^q(X,\p{s}) \hspace{10pt} \longby{\cap} \hspace{10pt} H_{n-q}(Z,\p{m-s}) \E{array} \E{equation} commutes. \E{schol} \B{proof} This is a simple consequence of A4 by observing that $T\hookrightarrow f^{-1}(Z)$. \E{proof} \B{defi} We will say that $(H^*,H_*)$ is a {\it sophisticated} Poincar\'e duality theory with supports if the axioms A1--A4 are satisfied and Poincar\'e duality holds i.e. the Bloch-Ogus axioms \cite[1.3.4-5 and 7.1.2]{BO} are satisfied (see \S\ref{inter1}). \E{defi} \subsection{${\cal H}$-cap product}\label{H-cap} Associated with the homology theory $H_*$, for $X\in\cat{V}_k$ possibly singular, we have a niveau spectral sequence (cf. \cite[Prop.3.7]{BO}) $$E_{a,b}^1 = \coprod_{x\in X_a}^{} H_{a+b}(x,\p{\cdot}) \Rightarrow H_{a+b}(X,\cdot)$$ which is covariant for proper morphisms and contravariant for \'etale maps. Let denote ${\rm Q}^n_{\mbox{\Large $\cdot $}}(X)\p{(m)}$ the (homological) complex $E_{\mbox{\Large $\cdot $},n}^1\p{(m)}$. \B{prop} \label{hpairing} Let $H^*$ and $H_*$ be cohomological and homological functors satisfying the axioms {\rm A1--A3} above. For $X\in\cat{V}_k$ there is a pairing of complexes $${\rm Q}^n_{\mbox{\Large $\cdot $}}(X)\p{(m)}\otimes H^q(X,\p{s})\to {\rm Q}^{n-q}_{\mbox{\Large $\cdot $}}(X)\p{(m-s)}$$ contravariant w.r.t. \'etale maps. \E{prop} \B{proof} Let $Z\subset T\subseteq X$ be closed subsets of $X$, dim$T\leq a$, dim$Z\leq a-1$ and let $U=X-Z$; thus by restriction to $U$ and cap-product we do have a pairing associated to such pairs $Z\subseteq T$: $$H_i(T\cap U,\p{j})\otimes H^q(X,\p{s}) \to H_{i-q}(T\cap U,\p{j-s}) $$ i.e. $t\p{\otimes}x \leadsto t\p{\cap}j^*(x)$ where $j: U\hookrightarrow X$. By taking the direct limit over such pairs (this makes sense because of A1--A2) we do have a pairing $$\coprod_{x\in X_a}^{} H_{n+a}(x,\p{m})\otimes H^q(X,\p{s}) \to \coprod_{x\in X_a}^{} H_{n-q+a}(x,\p{m-s})$$ We need to check compatibility with the differentials of ${\rm Q}^n_{\mbox{\Large $\cdot $}}(X)\p{(m)}$. Because of A2 we have a pairing $H_{i}(Z_{a},\p{j})\otimes H^q(X,\p{s})\to H_{i-q}(Z_{a},\p{j-s})$ where $H_{i}(Z_{a},\p{j}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{\pp{T\subset X}} H_i(T,\p{j})$, $T$ as above. Because of A3 and limit arguments the following diagram \B{displaymath} \begin{array}{ccc} {\displaystyle\coprod_{x\in X_a}^{} H_{i}(x,\p{j})\otimes H^q(X,\p{s})} & \longby{\partial\otimes id} & H_{i-1}(Z_{a-1},\p{j}) \otimes H^q(X,\p{s})\\ {\downarrow \p{\cap}} & &{\downarrow \p{\cap} }\\ {\displaystyle\coprod_{x\in X_a}^{} H_{i-q}(x,\p{j-s})} &\to& H_{i-q-1}(Z_{a-1},\p{j-s}) \end{array} \E{displaymath} commutes (indeed $ \partial (t\p{\cap}j^*(x))=\partial (t)\p{\cap}x$ ) and A1 implies that the following \B{displaymath} \begin{array}{ccc} H_{i-1}(Z_{a-1},\p{j}) \otimes H^q(X,\p{s}) &\to & {\displaystyle\coprod_{x\in X_{a-1}}^{} H_{i-1}(x,\p{j}) \otimes H^q(X,\p{s}) }\\ {\downarrow \p{\cap}} & &{\downarrow \p{\cap}}\\ H_{i-q-1}(Z_{a-1},\p{j-s})&\to &{\displaystyle \coprod_{x\in X_{a-1}}^{} H_{i-q-1}(x,\p{j-s})} \end{array} \E{displaymath} commutes. By construction, the differential is the composition of $${\rm Q}^n_{a}(X)\p{(m)}\to H_{n+a-1}(Z_{a-1},\p{m})\to{\rm Q}^n_{a-1}(X)\p{(m)}.$$ Thus the result. \E{proof} \B{defi} For $X\in\cat{V}_k$, by taking associated sheaves for the Zariski topology of the pairing above, we get a pairing $$\p{\cap}_{{\cal H}}:{{\cal Q}}^n_{\mbox{\Large $\cdot $} ,X}\p{(m)}\otimes {\cal H}^q_X\p{(s)}\to {{\cal Q}}^{n-q}_{\mbox{\Large $\cdot $} ,X}\p{(m-s)}$$ which we call {\it ${\cal H}$-cap-product} on $X$. \E{defi} \subsection{Projection formula} Let $f:Y\to X$ be a proper morphism in $\cat{V}_k$. We do have a map of niveau spectral sequences $$E_{p,q}^1\p{(r)}(Y) \to E_{p,q}^1\p{(r)}(X)$$ which takes $y\in Y_{p}$ to $f(y)$ if dim $\overline{\{f(y)\}} =$ dim $\overline{\{y\}}$ zero otherwise and maps $H_i(y)$ to $H_i(f(y))$. Thus by sheafifying it for the Zariski topology we obtain a map $$f_{\sharp}: f_*{{\cal Q}}^n_{\mbox{\Large $\cdot $} ,Y}\p{(m)} \to {{\cal Q}}^n_{\mbox{\Large $\cdot $} ,X}\p{(m)}$$ of complexes of sheaves on $X$. \B{prop} The following diagram: \B{equation} \B{array}{c}\label{hpr} \hspace{10pt} f_*{{\cal Q}}^n_{\mbox{\Large $\cdot $} ,Y}\p{(m)}\otimes f_*{\cal H}^{q}\p{(s)}\\ \p{id\otimes f^{\sharp}}\nearrow \ \hspace{40pt} \searrow \p{f_*\cap_{{{\cal H}}}} \\ f_*{{\cal Q}}^n_{\mbox{\Large $\cdot $} ,Y}\p{(m)}\otimes {\cal H}^{q}\p{(s)}\hspace{60pt} f_*{{\cal Q}}^{n-p}_{\mbox{\Large $\cdot $} ,Y}\p{(m-s)}\\ \p{f_{\sharp}\otimes id}\downarrow \hspace{105pt}\downarrow \p{f_{\sharp}} \\ {{\cal Q}}^n_{\mbox{\Large $\cdot $} ,X}\p{(m)}\otimes {\cal H}^{q}\p{(s)} \hspace{10pt} \longby{\cap_{{\cal H}}} \hspace{10pt}{{\cal Q}}^{n-q}_{\mbox{\Large $\cdot $} ,X}\p{(m-s)} \E{array} \E{equation} commutes. \E{prop} \B{proof} The commutative diagram above will be otbained from the following: $$ \B{array}{c} \hspace{20pt} {\rm Q}^n_{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(m)}\otimes H^{q}(f^{-1}(U),\p{s})\\ \p{id\otimes f^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cap} \\ {\rm Q}^n_{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(m)}\otimes H^{q}(U,\p{s})\hspace{60pt} {\rm Q}^{n-q}_{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(m-s)}\\ \p{f_{*} \otimes id}\downarrow \hspace{105pt}\downarrow \p{f_{*}} \\ {\rm Q}^{n}_{\mbox{\Large $\cdot $}}(U)\p{(m)}\otimes H^{q}(U,\p{s}) \hspace{10pt} \longby{\cap} \hspace{10pt} {\rm Q}^{n-q}_{\mbox{\Large $\cdot $}}(U)\p{(m-s)} \E{array} $$ where $U\subset X$ is any Zariski open subset of $X$, by taking associated sheaves on $X_{Zar}$.\\ Moreover it sufficies to prove the case of $U=X$.\\ Let $\overline{\{y\}}\subset Y$ such that $y\in Y_{p}$ and $f(y)\in X_{p}$. The maps $$f_{!,y}: \limdir{\pp{V\subset Y}}H_*({\overline{\{y\}}\cap V}) \to \limdir{\pp{U\subset X}}H_*({\overline{\{f(y)\}}\cap U})$$ are defined by mapping the elements of $H_*({\overline{\{y\}}\cap (Y-T)})$ by restriction and the induced maps $$f_{!,y}: H_*({\overline{\{y\}}\cap (Y-f^{-1}(Z))}) \to H_*({\overline{\{f(y)\}}\cap (X-Z)})$$ where $Z=f(T)$ (Note: compabilities are ensured by \cite[1.2.2]{BO}). By the definition of the pairing in Proposition \ref{hpairing} we are left to show that the following diagram \B{equation}\B{array}{c} \label{hopen} \ \hspace{20pt} H_{*}(\overline{\{y\}}\cap V,\cdot)\otimes H^q(V,\p{s}) \\ \p{id\otimes f^*}\nearrow \hspace{50pt} \searrow\p{\cap}\\ H_{*}(\overline{\{y\}}\cap V,\cdot) \otimes H^q(U,\p{s})\hspace{40pt} H_{*-q}(\overline{\{y\}}\cap V,\cdot \p{-s}) \\ \p{f_{!,y}\otimes id}\downarrow \hspace{105pt} \downarrow \p{f_{!,y}} \\ H_{*}(\overline{\{f(y)\}}\cap U,\cdot)\otimes H^q(U,\cdot) \hspace{10pt} \longby{\cap} \hspace{10pt} H_{*-q}(\overline{\{f(y)\}}\cap U,\cdot \p{-s}) \E{array} \E{equation} commutes where $f^{-1}(U) = V$. Since $f(\overline{\{y\}}\cap f^{-1}(U))= \overline{\{f(y)\}}\cap U$ the diagram (\ref{hopen}) commutes because of the projection formula in the Scholium~\ref{A4}. \E{proof} \B{lemma} If $i: Z\hookrightarrow X$ is a closed embedding then the canonical map $i_{\sharp}: i_*{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \to {{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)}$ admits a factorisation by a quasi-isomorphism $i_Z: i_*{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \by{\sim}{\bf \Gamma}_Z{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)}$ and the natural map ${\bf \Gamma}_Z{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \hookrightarrow {{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)}$ \E{lemma} \B{proof} Clear (cf. the proof of the Scholium~3.3). \E{proof} \B{cor} For $i: Z\hookrightarrow X$ as above the following diagram \B{equation} \B{array}{c}\label{hipr} \hspace{10pt} i_*{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \otimes i_*{\cal H}^{q}\p{(s)}\\ \p{id\otimes i^{\sharp}}\nearrow \ \hspace{40pt} \searrow \p{i_*\cap_{{{\cal H}}}} \\ i_*{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \otimes {\cal H}^{q}\p{(s)}\hspace{60pt} i_*{{\cal Q}}^{n-q}_{\mbox{\Large $\cdot $}}\p{(m-s)} \\ \p{i_{Z}\otimes id}\downarrow\wr \hspace{105pt}\wr\downarrow \p{i_{Z}} \\ {\bf \Gamma}_Z{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)} \otimes {\cal H}^{q}\p{(s)} \hspace{10pt} \longby{\cap_{{\cal H}}} \hspace{10pt}{\bf \Gamma}_Z{{\cal Q}}^{n-q}_{\mbox{\Large $\cdot $}}\p{(m-s)} \E{array} \E{equation} commutes. \E{cor} \B{proof} This follows by the factorisation of $i_{\sharp}$ and the projection formula (cf. the Lemma and the Proposition above). \E{proof} \subsection{Algebraic cycles}\label{algcyc} We are now going to consider the cycle group naturally involved with a fixed theory $(H^*,H_*)$ on $\cat{V}_k$. To this aim we need to assume a `dimension axiom' (cf. \cite[7.1.1]{BO}). Let assume that our cohomology theory $H^*$ takes values in a fixed category of $\mbox{$\Lambda$}$-modules where $\mbox{$\Lambda$} = H^0(k,\p{0})$ is a commutative ring with 1.\\ \B{defi} We will say that $(H^*,H_*)$ satisfies the {\it dimension axiom}\, when the following properties \B{description} \item[A5] If dim$X\leq d$ then $H_i(X,\p{m})=0$ for $i>2d$. \item[A6] If $X$ is irreducible then the canonical map $\lambda^*: \mbox{$\Lambda$} \by{\sim} H^0(X,\p{0})$, induced by the structural morphism $\lambda: X\to k$, is an isomorphism. \E{description} hold for any $X\in \cat{V}_k$. We will say that $(H^*,H_*)$ satisfies the {\it point axiom}\, if the properties A5-A6 above just hold locally, at the generic point of any integral subvariety of each $X\in \cat{V}_k$. \E{defi} \B{rmk} Clearly the dimension axiom implies the point axiom.\E{rmk} Thus: if $X$ is reduced and $\Sigma$ is its singular locus then $$H_{i}(X,\p{m}) \cong H_{i}(X-\Sigma ,\p{m})$$ for $i\geq 2d$ by applying A5 to the long exact sequence of homology groups. In particular, if $X$ is integral of dimension $d$ then by A6 $$H_{2d}(X,\p{d}) \cong H_{2d}(X-\Sigma ,\p{d}) \cong H^0(X-\Sigma ,\p{0}) \cong \mbox{$\Lambda$} $$ Regarding ${{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)}$ as a (cohomological) complex of flasque sheaves graded by negative degrees we do have $$ \H^{-p}(X,{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(m)})\cong \frac{ {\rm ker} (\coprod_{x\in X_p}^{} H_{n+p}(x,\p{m}) \to \coprod_{x\in X_{p-1}}^{} H_{n+p-1}(x,\p{m}) )}{{\rm im} (\coprod_{x\in X_{p+1}}^{} H_{n+p+1}(x,\p{m}) \to \coprod_{x\in X_p}^{} H_{n+p}(x,\p{m}))}$$ In particular, for $n=p=m$ and the `dimension axiom' above we do have $$\H^{-n}(X,{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(n)})\cong {\rm coker} (\coprod_{x\in X_{n+1}}^{} H_{2n+1}(x,\p{n}) \to \coprod_{x\in X_n}^{} \mbox{$\Lambda$})$$ where ${\displaystyle\coprod_{x\in X_n}^{} \mbox{$\Lambda$}}$ is the $\mbox{$\Lambda$}$-module of algebraic cycles of dimension $n$ in $X$.\\ \B{defi} Let $(H^*,H_*)$ be a theory satisfying the point axiom. We will denote $$C_n(X;\mbox{$\Lambda$})\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \H^{-n}(X,{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(n)})$$ the corresponding group of $n$-dimensional algebraic $\mbox{$\Lambda$}$-cycles modulo the equivalence relation given by $$ {\rm im}(\coprod_{x\in X_{n+1}}^{} H_{2n+1}(x,\p{n})\to \coprod_{x\in X_n}^{} \mbox{$\Lambda$})$$ the image of the differential of the niveau spectral sequence. \E{defi} \hfill\\ For $i: Z\hookrightarrow X$ a closed embedding we clearly do have an isomorphism $$C_n(Z;\mbox{$\Lambda$}) \cong \H^{-n}_Z(X,{{\cal Q}}^n_{\mbox{\Large $\cdot $}}\p{(n)}).$$ Thus, by taking hypercohomology with supports, the ${\cal H}$-cap-product yields a cap product \B{equation} C_n(Z;\mbox{$\Lambda$}) \otimes H^p_Y(X,{\cal H}^p\p{(p)})\to C_{n-p}(Z\cap Y;\mbox{$\Lambda$}) \E{equation} Because of the projection formula (\ref{hpr}) this cap product will have a projection formula as well. \section{Cup products} We are now going to show that the cup-product in cohomology give us a nice intersection theory for ${\cal H}$-cohomology. To this aim we will assume the algebraic schemes in $\cat{V}_k$ to be equidimensionals and $k$ to be a perfect field. The main results hold true just for non-singular varieties nevertheless we will not assume this hypothesis a priori. \subsection{Multiplicative Poincar\'e duality theories} Let assume we are given a twisted cohomology theory $H^*(\p{\cdot})$ on $\cat{V}_k$ in the sense of Bloch-Ogus \cite[1.1]{BO}. Furthermore, we want to assume the existence of a cup-product i.e. for all $(X,Z),(X,Y)\in\cat{V}^2_k$ an associative anticommutative pairing $$\p{\cup_{Y,Z}}: H_Y^p(X,\p{r})\otimes H_Z^q(X,\p{s})\to H^{p+q}_{Y\cap Z}(X,\p{r+s})$$ which satisfies the following axioms: \B{description} \item[$\forall$1] $\p{\cup}$ is natural with respect to pairs in $\cat{V}^2_k$ \item[$\forall$2] If $(X,T),(T,Z)$ are pairs in $\cat{V}^2_k$ then the following diagram \B{displaymath}\B{array}{ccc} H_Z^p(X,\p{r})\otimes H^q(X,\p{s}) & \longby{\cup} & H^{p+q}_{Z}(X,\p{r+s})\\ \downarrow & & \downarrow \\ H_T^p(X,\p{r})\otimes H^q(X,\p{s}) & \longby{\cup} & H^{p+q}_{T}(X,\p{r+s}) \E{array}\E{displaymath} commutes. \item[$\forall$3] For $(X,T),(T,Z)$ pairs in $\cat{V}^2_k$ let $U=X-Z$ and let $j: U \to X$ be the inclusion. Let denote $\partial: H_{T\cap U}^p(U,\cdot)\to H_Z^{p+1}(X,\cdot)$ the boundary map in the long exact sequence (\ref{loc}) of cohomology with supports. Then the following diagram \B{displaymath}\B{array}{c} \ \hspace{20pt} H_{T\cap U}^p(U,\cdot)\otimes H^q(U,\cdot) \\ \p{id\otimes j^*}\nearrow \hspace{50pt} \searrow\p{\cup}\\ H_{T\cap U}^{p}(U,\cdot)\otimes H^q(X,\cdot)\hspace{40pt} H_{T\cap U}^{p+q}(U,\cdot) \\ \p{\partial\otimes id}\downarrow \hspace{97pt} \downarrow \p{\partial} \\ H_Z^{p+1}(X,\cdot)\otimes H^q(X,\cdot) \hspace{10pt} \longby{\cup} \hspace{10pt} H_Z^{p+q+1}(X,\cdot) \E{array} \E{displaymath} commutes i.e. we have the equation: \B{equation} \partial (y\p{\cup}j^*(x))=\partial (y)\p{\cup}x \E{equation} for $y\in H_{T\cap U}^{p}(U,\cdot)$ and $x\in H^q(X,\cdot)$. \E{description} \B{defi} A twisted cohomology theory with supports $H^*$ has a {\it cup-product}\, if there is a pairing $\p{\cup_{Y,Z}}: H_Y^p(X,\p{r})\otimes H_Z^q(X,\p{s})\to H^{p+q}_{Y\cap Z}(X,\p{r+s})$ which satisfies the axioms $\forall$1--$\forall$3 listed above. \E{defi} \hfill\\[4pt] If furthermore $(H^*,H_*)$ is a Poincar\'e duality theory we let assume that the following compatibility between cap and cup products holds: \B{description} \item[$\forall$4] For $X$ smooth of dimension $d$ let $\eta_X\in H_{2d}(X,\p{d})$ be the fundamental class. Then the following diagram, where $q+j=2d, s+n=d$, \B{displaymath}\B{array}{ccc} H^q(X,\p{s})\otimes H_Z^p(X,\p{r}) & \longby{\cup} & H^{p+q}_{Z}(X,\p{r+s})\\ \p{\eta_X\cap -\otimes id} \downarrow & & \downarrow\p{\eta_X\cap -} \\H_j(X,\p{n}) \otimes H_Z^p(X,\p{r}) & \longby{\cap} & H_{j-p}(Z,\p{n-r}) \E{array}\E{displaymath} commutes i.e. we have the equation: \B{equation}\label{cap=cup} (\eta_X\p{\cap}x)\p{\cap}z= \eta_X\p{\cap}(x\p{\cup}z) \E{equation} for $x\in H^q(X,\p{s})$ and $z\in H_Z^p(X,\p{r})$. \E{description} \hfill\\[4pt] Let $f:Y\to X$ be a proper map of {\it smooth}\, equidimensional algebraic schemes. Let dim $X$ = $\delta$ and dim $Y$ = $d$. Let $\rho =\delta - d$. For $Z\subseteq X$ a closed subset there are maps $$f_{!}: H_{2d-i}(f^{-1}(Z),\p{d-i})\to H_{2d-i}(Z,\p{d-i})$$ Because of Poincar\'e duality $f_{!}$ induces a Gysin map $$f_{*}:H^{i}_{f^{-1}(Z)}(Y,\p{i})\to H^{i+2\rho}_{Z}(X,\p{i+\rho})$$ which is uniquely determined by the equation \B{equation}\label{cov} f_!(\eta_Y\p{\cap}y)=\eta_X\p{\cap}f_*(y) \E{equation} for $y\in H^{i}_{f^{-1}(Z)}(Y,\p{i})$. Thus $H^*$ is a covariant functor w.r.t. proper maps of pairs $(X,Z)$ where $Z$ is a closed subset of $X$ smooth: indeed $(1_X)_*= id$ because of $(1_X)_!=id$ and $(f\p{\circ}g)_* = f_*\p{\circ} g_*$ because of $(f\p{\circ}g)_!(\eta\p{\cap}-)=f_!(g_!(\eta\p{\cap}-))= f_!(\eta\p{\cap}g_*(-))=\eta\p{\cap}f_*(g_*(-))$.\\ The projection formula by \cite{BO} w.r.t. the cap product give us, via $\forall$4, the following projection formula: $$\B{array}{c} \hspace{20pt} H^{p}(Y,\p{r})\otimes H^{q}_{f^{-1}(Z)}(Y,\p{s})\\ \p{id\otimes f^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cup} \\ H^{p}(Y,\p{r})\otimes H^{q}_Z(X,\p{s})\hspace{60pt} H^{p+q}_{f^{-1}(Z)}(Y,\p{r+s})\\ \p{f_{*}\otimes id}\downarrow \hspace{121pt}\downarrow \p{f_{*}} \\ H^{p+2\rho}(X,\p{r+\rho})\otimes H^{q}_Z(X,\p{s})\hspace{10pt} \longby{\cup} \hspace{10pt} H^{p+q+2\rho}_{Z}(X,\p{r+s+\rho}) \E{array}$$ Indeed we have: \B{center} \parbox{3in}{$f_!(\eta_Y\p{\cap}(y\p{\cup}f^*(x)))=$ \hfill by (\ref{cap=cup})\\ $=f_!((\eta_Y\p{\cap}y)\p{\cap}f^*(x)) =$\hfill proj. form. for $\p{\cap}$\\ $=f_!(\eta_Y\p{\cap}y)\p{\cap}x=$\hfill by (\ref{cov})\\ $=(\eta_X\p{\cap}f_*(y))\p{\cap}x=$ \hfill by (\ref{cap=cup})\\ $=\eta_X\p{\cap}(f_*(y)\p{\cup}x).$} \E{center} Because of the lack of symmetry of the projection formula stated above we need to assume the following ``projection formula with supports''. \B{description} \item[$\forall$5] For $f:Y\to X$, $Z$ and $\rho$ as above, the following: $$\B{array}{c} \hspace{20pt} H^{p}_{f^{-1}(Z)}(Y,\p{r})\otimes H^{q}(Y,\p{s})\\ \p{id\otimes f^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cup} \\ H^{p}_{f^{-1}(Z)}(Y,\p{r})\otimes H^{q}(X,\p{s})\hspace{60pt} H^{p+q}_{f^{-1}(Z)}(Y,\p{r+s})\\ \p{f_{*}\otimes id}\downarrow \hspace{121pt}\downarrow \p{f_{*}} \\ H^{p+2\rho}_{Z}(X,\p{r+\rho})\otimes H^{q}(X,\p{s})\hspace{10pt} \longby{\cup} \hspace{10pt} H^{p+q+2\rho}_{Z}(X,\p{r+s+\rho}) \E{array}$$ commutes, i.e. we have the equation $f_*(y\p{\cup}f^*(x))=f_*(y)\p{\cup}x$ for $x\in H^{q}(X,\p{s})$ and $y\in H^{p}_{f^{-1}(Z)}(Y,\p{r})$ \E{description} As a matter of fact, in order to obtain the projection formula (\ref{chpr}) we just need the following apparently weaker but almost equivalent form of $\forall$5. \B{description} \item[$\forall$5'] Let $i:Z\hookrightarrow X$ be a smooth pair of pure codimension $c$. Then: $$\B{array}{c} \hspace{20pt} H^{p}(Z,\p{r})\otimes H^{q}(Z,\p{s})\\ \p{id\otimes i^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cup} \\ H^{p}(Z,\p{r})\otimes H^{q}(X,\p{s})\hspace{60pt} H^{p+q}(Z,\p{r+s})\\ \p{i_{*}\otimes id}\downarrow\wr \hspace{121pt}\wr\downarrow \p{i_{*}} \\ H^{p+2c}_{Z}(X,\p{r+c})\otimes H^{q}(X,\p{s})\hspace{10pt} \longby{\cup} \hspace{10pt} H^{p+q+2c}_{Z}(X,\p{r+s+c}) \E{array}$$ commutes. \E{description} By the way $\forall$5 implies $\forall$5'. (Convention: the purity isomorphism $i_*$ is induced by the identity on $Z$.) \B{schol} Let $i: Z\hookrightarrow X$ be a smooth pair. Then the following square $$\begin{array}{ccc} H^{p}_Z(X,\p{r})\otimes H^{q}(X,\p{s}) &\longby{\cup} & H^{p+q}_Z(X,\p{r+s})\\ \p{\eta_X\cap -\otimes i^*}\downarrow & &\downarrow \wr\ \p{\eta_X\cap -}\\ H_{2d-p}(Z,\p{d-r})\otimes H^{q}(Z,\p{s}) & \longby{\cap}& H_{2d-p-q}(Z,\p{d-r-s}) \end{array} $$ commutes, i.e. we have the following formula: \B{equation}\label{rest} \eta_X \p{\cap}(z\p{\cup}x) = (\eta_X \p{\cap}z)\p{\cap}i^*(x) \E{equation} for $z\in H^{p}_Z(X,\p{r})$ and $x\in H^{q}(X,\p{s})$, {\rm if and only if $\forall$5'} holds. \E{schol} \B{proof} Let $c$ be the codimension of $Z\subset X$ and $d=$ dim $X$.\\ Let assume that $\forall$5' holds. Since we do have the purity isomorphism $i_*:H^{p-2c}(Z,\p{r-c})\by{\cong} H^{p}_Z(X,\p{r})$ there is an element $\zeta\in H^{p-2c}(Z,\p{r-c})$ such that $i_*(\zeta)=z$. Because of $\forall$5' the equation (\ref{rest}) is obtained by showing the following equality \B{equation}\label{zeta} \eta_X\p{\cap}i_*(\zeta\p{\cup}i^*(x))= (\eta_X\p{\cap}i_*(\zeta))\p{\cap}i^*(x) \E{equation} Note: $\eta_Z\p{\cap}- :H^{p-2c}(Z,\p{r-c}) \by{\cong} H_{2d-p}(Z,\p{d-r})$ and $\eta_Z\p{\cap}- = \eta_X\p{\cap}i_*(-)$ by the definition of $i_*$; thus the right-hand side in the equation (\ref{zeta}) above becomes \B{equation}\label{eta} (\eta_Z\p{\cap}\zeta)\p{\cap}i^*(x) \E{equation} and the left-hand side becomes $$ \eta_Z\p{\cap}(\zeta\p{\cup}i^*(x)) $$ which by $\forall$4 (see the equation (\ref{cap=cup})) is exactly the same of (\ref{eta}).\\ Conversely, if (\ref{rest}) holds we then have \B{center} \parbox{3in}{$\eta_X\p{\cap}(i_*(\zeta)\p{\cup}x)=$\hfill by (\ref{rest})\\ $=(\eta_X\p{\cap}i_*(\zeta))\p{\cap}i^*(x)=$\hfill by def. of $i_*$\\ $=(\eta_Z\p{\cap}\zeta)\p{\cap}i^*(x)=$\hfill by (\ref{cap=cup})\\ $=\eta_Z\p{\cap}(\zeta\p{\cup}i^*(x))=$\hfill by def. of $i_*$\\ $=\eta_X\p{\cap}i_*(\zeta\p{\cup}i^*(x)).$} \E{center} Since $\eta_X\p{\cap}-$ is an isomorphism we can erase it from the left of the resulting equation. \E{proof} \B{rmk} As we will see below (cf. Lemma \ref{P5}): the formula (\ref{rest}), by restriction and the projection formula w.r.t. cap-product, allow us to deduce $\forall$5 for $X$, $Y$, $Z$ and $f^{-1}(Z)$ smooth. Moreover, by assuming the formula (\ref{rest}) holds for $Z$ possibly singular we can as well obtain $\forall$5. \E{rmk} \B{defi} Let $(H^*,H_*)$ be a Poincar\'e duality theory with supports and let assume that $H^*$ has a cup-product (so that $\forall$1--$\forall$3 are satisfied). We will say that $(H^*,H_*)$ is {\it multiplicative}\, if the axioms $\forall$4 and $\forall$5' (or the strong form $\forall$5) are satisfied. \E{defi} \subsection{${\cal H}$-cup products}\label{H-cup} Let $H^*$ be a twisted cohomology theory with supports on $\cat{V}_k$. Let $X\in \cat{V}_k$ be equidimensional but possibly singular. Applying the exact couple method to the exact sequence (\ref{limloc}) $$ H^i_{Z^{p+1}(X)}(X,\p{j})\to H^i_{Z^p(X)}(X,\p{j})\to \coprod_{x\in X^p}^{} H^{i}_x(X,\p{j}) \to H^{i+1}_{Z^{p+1}(X)}(X,\p{j}) $$ where: $H^{i}_x(X,\p{j}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{\pp{U\subset X}} H^i_{\overline{\{x\}}\cap U}(U,\p{j})$, we do get the coniveau spectral sequence $$E^{p,q}_1 =\coprod_{x\in X^p}^{} H^{q+p}_x(X,\p{\cdot}) \Rightarrow H^{p+q}(X,\cdot)$$ Let denote ${\rm R}_q^{\mbox{\Large $\cdot $}}(X)\p{(r)}$ the corresponding Gersten type complexes $E^{\mbox{\Large $\cdot $},q}_1$. \B{prop} \label{pairing}Let $H^*$ be a twisted cohomology theory with supports and cup-product on $\cat{V}_k$. For $X\in\cat{V}_k$ there is a pairing of complexes $$ {\rm R}_q^{\mbox{\Large $\cdot $}}(X)\p{(r)}\otimes H^n(X,\p{s})\to {\rm R}_{q+n}^{\mbox{\Large $\cdot $}}(X)\p{(r+s)}$$ contravariant w.r.t. flat maps. \E{prop} \B{proof} Let $Z\subseteq T\subseteq X$ with $Z\in Z^{p+1}(X)$ and $T\in Z^p(X)$ and let $U=X-Z$; thus by restriction to $U$ and cup-product we do have a pairing associated to such pairs $Z\subseteq T$: $$H_{T\cap U}^i(U,\p{r})\otimes H^n(X,\p{s}) \to H_{T\cap U}^{i+n}(U,\p{r+s}) $$ i.e. $t\p{\otimes}x \leadsto t\p{\cup}j^*(x)$ where $j: U\hookrightarrow X$. By taking the direct limit over such pairs (this makes sense because of $\forall$1--$\forall$2) we do have a pairing $$\coprod_{x\in X^p}^{} H^{q+p}_x(X,\p{r})\otimes H^n(X,\p{s}) \to \coprod_{x\in X^p}^{} H^{q+n+p}_x(X,\p{r+s})$$ In order to check compatibility with the differentials of ${\rm R}_q^{\mbox{\Large $\cdot $}}(X)\p{(r)}$, because of $\forall$2 we have a pairing $H^{i}_{Z^{p}(X)}(X,\p{j})\otimes H^n(X,\p{s})\to H^{i+n}_{Z^{p}(X)}(X,\p{j+s})$ and, by construction, the differential is the composition of $${\rm R}_q^{p}(X)\p{(r)}\to H^{q+p+1}_{Z^{p+1}(X)}(X,\p{r})\to{\rm R}_q^{p+1}(X)\p{(r)}$$ we can argue as in the proof of the Proposition~\ref{hpairing} via $\forall$3 and limit arguments. \E{proof} \B{defi} For $X\in\cat{V}_k$, by taking associated sheaves for the Zariski topology of the pairing above, we get a cap-product pairing $${{\cal R}}_q^{\mbox{\Large $\cdot $}}(X)\p{(r)}\otimes {\cal H}^p_X\p{(s)}\to {{\cal R}}_{p+q}^{\mbox{\Large $\cdot $}}(X)\p{(r+s)}$$ By sheafifying the cup-product we do have a product $$\p{\cup}_{{\cal H}}:{\cal H}^p_X\p{(r)} {\otimes} {\cal H}^q_X\p{(s)}\to {\cal H}^{p+q}_X\p{(r+s)}$$ which we call {\it ${\cal H}$-cup-product} on $X$. \E{defi} \B{rmk} We list several expected compabilities.\\ \B{enumerate} \item The ${\cal H}$-products above are compatible via the canonical augmentations ${\cal H}^p_X\p{(r)}\to {{\cal R}}_p^{\mbox{\Large $\cdot $}}(X)\p{(r)}.$ But, if $X$ is singular the augmentations are not quasi-isomorphisms. \item Let suppose the existence of an external product $$\p{\times}: H_Z^p(X,\p{r})\otimes H_T^q(Y,\p{s})\to H^{p+q}_{Z\times T}(X\times Y,\p{r+s})$$ functorial on $\cat{V}^2_k$ i.e. we have the equation \B{equation}(f\p{\times} g)^*(x\p{\times}y) = f^*(x)\p{\times}g^*(y) \E{equation} for $f$ and $g$ maps of pairs. Thus, by composing with the diagonal $\Delta :(X,Z\p{\cap}T) \to (X\times X, Z\times T)$, we obtain a cup-product satisfying the axiom $\forall$1.\\ In this case the pairing defined in the Proposition~\ref{pairing} can be obtained as follows $$H_{T\cap U}^i(U,\p{r})\otimes H^n(X,\p{s}) \ni t\p{\otimes}x \leadsto \Delta^*(1\p{\times}j)^*(t \p{\times}x)$$ where $j: U\hookrightarrow X$. \item On a smooth variety $X$, after $\forall$4, we have that the pairing defined in Proposition~\ref{pairing} is Poincar\'e dual of the pairing defined as follows: let $j: U\hookrightarrow X$, if $\xi\in H_*(X,\cdot)$ and $t\in H^*_{\overline{\{p\}}\cap U}(U,\cdot)$ then $\xi\p{\otimes}t \leadsto j^*(x)\p{\cap}t$. In fact: $\xi=\eta_X\p{\cap}x$ for some $x\in H^*(X,\cdot)$ and we then have: \B{center} \parbox{3in}{ $\eta_U\p{\cap}(j^{*}(x)\p{\cup}t)=$\hfill by $\forall$4\\ $=(\eta_U\p{\cap}j^{*}(x))\p{\cap}t=$\hfill \cite[1.3.2-4]{BO}\\ $=j^{*}(\eta_X\p{\cap}x)\p{\cap}t).$} \E{center} \item On a smooth variety $X$, after the equation (\ref{rest}), we have that the pairing defined in the Proposition~\ref{pairing} can be obtained by restriction as follows. Let $t\in H^*_{\overline{\{p\}}\cap U}(U,\cdot)$ and $x\in H^*(X,\cdot)$ where $X$ is smooth and $\overline{\{p\}}\cap U \subset U$ is a smooth pair. Thus $\eta_U\p{\cap}(t\p{\cup}j^*(x))$ is equal to $(\eta_U\p{\cap}t)\p{\cap}i^*(x)$ where $i: \overline{\{p\}}\cap U \hookrightarrow X$. \E{enumerate}\E{rmk} \subsection{${\cal H}$-Gysin maps}\label{H-Gys} Let $(H^*,H_*)$ be a Poincar\'e duality theory with supports. For $X\in\cat{V}_k$ we have a niveau spectral sequence $E_{a,b}^1 \Rightarrow H_{a+b}(X,\cdot)$ which is covariant for proper morphisms. For $k$ perfect and $X$ smooth equidimensional, $d=$ dim $X$, by local purity, we do have that $H^{q+p}_x(X,\p{r})\cong H^{q-p}(x,\p{r-p})$ if $x\in X^p$ and isomorphisms $$E_{d-p,d-q}^1\p{(d-r)} =\coprod_{x\in X_{d-p}}^{} H_{2d-p-q}(x,\p{d-r}) \cong \coprod_{x\in X^p}^{} H^{q-p}(x,\p{r})\cong E^{p,q}_1\p{(r)}$$ \B{lemma}\label{co=ni}If $X\in \cat{V}_k$ is smooth of pure dimension $d$ then $E_{d-p,d-q}^1\p{(d-r)}\cong E^{p,q}_1\p{(r)}$ is an isomorphism of spectral sequences which is natural w.r.t. \'etale maps. \E{lemma} \B{proof} This is a consequence of the above once we have identified (via Poincar\'e duality) the long exact sequence of cohomology with supports with the corresponding long exact sequence of homology groups. \E{proof} Let $f:Y\to X$ be a proper morphism between $k$-algebraic schemes where dim~$X$ = $\delta$ and dim $Y$ = $d$. Let $\rho =\delta - d$. Since $E^1\p{(\cdot)}$ is covariant w.r.t. proper maps we do have a map of niveau spectral sequences $$E_{d-p,d-q}^1\p{(d-r)}(Y) \to E_{d-p,d-q}^1\p{(d-r)}(X)$$ If $X$ and $Y$ are {\it smooth}\, and equidimensionals, then, via the Lemma \ref{co=ni}, we get a map of coniveau spectral sequences as follows: $$E^{p,q}_1\p{(r)}(Y)\cong E_{d-p,d-q}^1\p{(d-r)}(Y) \to E_{d-p,d-q}^1\p{(d-r)}(X)\cong E^{p+\rho ,q+\rho }_1\p{(r+\rho )}(X)$$ \B{defi} For $f:Y\to X$ as above we will call the induced map of complexes ${\rm R}_q^{\mbox{\Large $\cdot $}}(Y)\p{(r)}\to {\rm R}_{q+\rho}^{\mbox{\Large $\cdot $}}(X)\p{(r+\rho)}[\rho]$ the {\it global Gysin map}. By taking associated sheaves we have the {\it local} Gysin map $$f_{\flat}: f_*{{\cal R}}_q^{\mbox{\Large $\cdot $}}(Y)\p{(r)}\to {{\cal R}}_{q+\rho}^{\mbox{\Large $\cdot $}}(X)\p{(r+\rho)}[\rho].$$ In the derived category we do have the ${\cal H}$-Gysin map $${\bf R}f_{\flat}: {\bf R}f_*{\cal H}^q_Y\p{(r)} \to {\cal H}^{q+\rho}_X\p{(r+\rho)}[\rho].$$ \E{defi} \B{rmk} For $i: Z\hookrightarrow X$ a (smooth) pair of pure codimension $c$ we do have the isomorphism (cf. \S\ref{cad1}) $$i_Z: i_*{\cal R}_{q,Z}^{\mbox{\Large $\cdot $}}\p{(r)} \by{\simeq}{\bf \Gamma}_Z{\cal R}_{q+c,X}^{\mbox{\Large $\cdot $}}\p{(r+c)}[c].$$ Thus, as it is easily seen, the local Gysin map $i_{\flat}$ is obtained by composition of $i_Z$ with the canonical map ${\bf\Gamma}_Z{\cal R}_{q+c,X}^{\mbox{\Large $\cdot $}}\p{(r+c)}[c] \hookrightarrow {\cal R}_{q+c,X}^{\mbox{\Large $\cdot $}}\p{(r+c)}[c]$ (see the proof of Scholium~\ref{Gysin}). \E{rmk} \subsection{Projection formula} Let $f:Y\to X$ be a proper morphism between smooth equidimensional algebraic schemes over a perfect field $k$. Let dim $X$ = $\delta$ and dim $Y$ = $d$. Let $\rho =\delta - d$. \B{prop} For $f:Y\to X$ as above we have the following commutative diagram: \B{equation} \B{array}{c}\label{chpr} \hspace{10pt} f_*{\cal R}^{\mbox{\Large $\cdot $}}_{q,X}\p{(r)}\otimes f_*{\cal H}^{p}\p{(s)}\\ \p{id\otimes f^{\sharp}}\nearrow \ \hspace{40pt} \searrow \p{f_*\cap_{{{\cal H}}}} \\ f_*{\cal R}^{\mbox{\Large $\cdot $}}_{q,X}\p{(r)}\otimes {\cal H}^{p}\p{(s)}\hspace{60pt} f_*{\cal R}^{\mbox{\Large $\cdot $}}_{p+q,X}\p{(r+s)}\\ \p{f_{\flat}\otimes id}\downarrow \hspace{105pt}\downarrow \p{f_{\flat}} \\ {\cal R}^{\mbox{\Large $\cdot $}}_{q+\rho ,X}\p{(r+\rho )}[\rho ]\otimes {\cal H}^{p}\p{(s)} \hspace{10pt} \longby{\cap_{{\cal H}}} \hspace{10pt}{\cal R}^{\mbox{\Large $\cdot $}}_{p+q+\rho,X}\p{(r+s+\rho)}[\rho] \E{array} \E{equation} \E{prop} \B{proof} The commutative diagram above will be otbained from the following: $$ \B{array}{c} \hspace{20pt} {\rm R}_q^{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(r)}\otimes H^{p}(f^{-1}(U),\p{s})\\ \p{id\otimes f^{*}}\nearrow \ \hspace{40pt} \searrow \p{\cap} \\ {\rm R}_q^{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(r)}\otimes H^{p}(U,\p{s})\hspace{60pt} {\rm R}_{p+q}^{\mbox{\Large $\cdot $}}(f^{-1}(U))\p{(r+s)}\\ \p{f_{\flat} \otimes id}\downarrow \hspace{105pt}\downarrow \p{f_{\flat}} \\ {\rm R}_{q+\rho}^{\mbox{\Large $\cdot $}}(U)\p{(r+\rho)}[\rho]\otimes H^{p}(U,\p{s}) \hspace{10pt} \longby{\cap} \hspace{10pt} {\rm R}_{p+q+\rho}^{\mbox{\Large $\cdot $}}(U)\p{(r+s+\rho)}[\rho] \E{array} $$ where $U\subset X$ is any Zariski open subset of $X$, by taking associated sheaves on $X_{Zar}$.\\ Moreover it sufficies to prove the case of $U=X$.\\ Let $\overline{\{y\}}\subset Y$ such that $y\in Y^{c}$ and $f(y)\in X^{c+\rho}$. The Gysin map (cf. \S\ref{H-Gys}) $$f_{\flat,y}: \limdir{\pp{V\subset Y}}H^*_{\overline{\{y\}}\cap V}(V) \to \limdir{\pp{U\subset X}}H^*_{\overline{\{f(y)\}}\cap U}(U)$$ is the Poincar\'e dual of $f_{!,y}: \limdir{\pp{V\subset Y}}H_*({\overline{\{y\}}\cap V}) \to \limdir{\pp{U\subset X}}H_*({\overline{\{f(y)\}}\cap U})$ (see the proof of (\ref{hpr})). By the definition of the pairing in Proposition \ref{pairing} we are left to show that the following diagram \B{equation}\B{array}{c} \label{open} \ \hspace{20pt} H_{\overline{\{y\}}\cap V}^p(V,\cdot)\otimes H^q(V,\cdot) \\ \p{id\otimes f^*}\nearrow \hspace{50pt} \searrow\p{\cup}\\ H_{\overline{\{y\}}\cap V}^{p}(V,\cdot)\otimes H^q(U,\cdot)\hspace{40pt} H_{\overline{\{y\}}\cap V}^{p+q}(V,\cdot) \\ \p{f_{\flat}\otimes id}\downarrow \hspace{105pt} \downarrow \p{f_{\flat}} \\ H_{\overline{\{f(y)\}}\cap U}^{p+2\rho}(U,\cdot +\rho)\otimes H^q(U,\cdot) \hspace{10pt} \longby{\cup} \hspace{10pt} H_{\overline{\{f(y)\}}\cap U}^{p+q+2\rho}(U,\cdot +\rho) \E{array} \E{equation} commutes where $f^{-1}(U) = V$. By shrinking the open sets involved we may assume that $\overline{\{y\}}\cap V \subset V$ and $\overline{\{f(y)\}}\cap U \subset U$ are smooth pairs. Since $f(\overline{\{y\}}\cap f^{-1}(U))= \overline{\{f(y)\}}\cap U$ the diagram (\ref{open}) commutes because of the following Lemma. \E{proof} \B{lemma}\label{P5} Let $f:Y\to X$ and $\rho$ as above. Let $T$ be a closed subset of $Y$, $f(T)=Z$ and let assume that $T\hookrightarrow Y$ and $Z \hookrightarrow X$ are smooth pairs. Then the following diagram \B{equation}\B{array}{c} \ \hspace{20pt} H_{T}^p(Y,\p{r})\otimes H^q(Y,\p{s}) \\ \p{id\otimes f^*}\nearrow \hspace{50pt} \searrow\p{\cup}\\ H_{T}^{p}(Y,\p{r})\otimes H^q(X,\p{s})\hspace{40pt} H_{T}^{p+q}(Y,\p{r+s}) \\ \p{f_{\flat}\otimes id}\downarrow \hspace{105pt} \downarrow \p{f_{\flat}} \\ H_{Z}^{p+2\rho}(X,\p{r +\rho})\otimes H^q(X,\p{s}) \hspace{10pt} \longby{\cup} \hspace{10pt} H_{Z}^{p+q+2\rho}(X,\p{r+s+\rho}) \E{array} \E{equation} commutes. \E{lemma} \B{proof} We will give two proofs.\\ {\it First proof.}\, Let assume the axiom $\forall$5 (in which case we do not need the smoothness of $T$ and $Z$). Let $i: T \hookrightarrow f^{-1}(Z)$ so that $f\mid_{T} =f \p{\circ} i$ hence $f\mid_{T,!} =f_!\p{\circ} i_!$ and $f_{\flat} =f_*\p{\circ} i_{\diamond}$ where $i_{\diamond}: H^*_T(Y,\cdot) \to H^*_{f^{-1}(Z)}(Y\cdot)$ is the canonical map. Thus, for $y\in H^*_T(Y,\cdot)$ and $x\in H^*(X,\cdot)$ \B{center} \parbox{3in}{$f_{\flat}(y\p{\cup}f^*(x))=$ \\ $=f_{*}\p{\circ}i_{\diamond}(y\p{\cup}f^*(x))=$\hfill by $\forall$2\\ $=f_{*}(i_{\diamond}(y)\p{\cup}f^*(x))=$\hfill by $\forall$5\\ $=f_{*}(i_{\diamond}(y))\p{\cup}x)=$\\ $=f_{\flat}(y)\p{\cup}x.$} \E{center} \noindent{\it Second proof.}\, Just assuming the axiom $\forall$5' we can prove the Lemma as follows. Let $k:T\hookrightarrow Y$, $i:Z\hookrightarrow X$ and $f\mid_T: T \to Z$. Thus we have: $k^*f^*=(f\mid_{T})^*i^*$. Let $y\in H_{T}^{p}(Y,\p{r})$ and $x\in H^q(X,\p{s})$. We have: \B{center} \parbox{5in}{$\eta_X\p{\cap}f_{\flat}(y\p{\cup}f^*(x))=$ \hfill by def. of $f_{\flat}$\\ $=(f\mid_{T})_!(\eta_Y\p{\cap}(y\p{\cup}f^*(x)))=$\hfill by (\ref{rest})\\ $=(f\mid_{T})_!((\eta_Y\p{\cap}y)\p{\cap}k^*f^*(x))=$\ \\ $=(f\mid_{T})_!((\eta_Y\p{\cap}y)\p{\cap} (f\mid_{T})^*i^*(x))=$\hfill by \cite[1.3.3]{BO}\\ $=(f\mid_{T})_!(\eta_Y\p{\cap}y)\p{\cap}i^*(x)=$\hfill by def. of $f_{\flat}$\\ $=(\eta_X\p{\cap}f_{\flat}(y))\p{\cap}i^*(x)=$\hfill by (\ref{rest})\\ $=\eta_X\p{\cap}(f_{\flat}(y)\p{\cup}x)$.} \E{center} Since $\eta_X\p{\cap}-$ is an isomorphism we conclude. \E{proof} \B{cor} For $i: Z\hookrightarrow X$ a smooth pair of pure codimension $c$ the following diagram \B{equation} \B{array}{c}\label{ipr} \hspace{10pt} i_*{\cal R}^{\mbox{\Large $\cdot $}}_{q,Z}\p{(r)}\otimes i_*{\cal H}^{p}\p{(s)}\\ \p{id\otimes i^{\sharp}}\nearrow \ \hspace{40pt} \searrow \p{i_*\cap_{{{\cal H}}}} \\ i_*{\cal R}^{\mbox{\Large $\cdot $}}_{q,Z}\p{(r)}\otimes {\cal H}^{p}\p{(s)}\hspace{60pt} i_*{\cal R}^{\mbox{\Large $\cdot $}}_{p+q,Z}\p{(r+s)}\\ \p{i_{Z}\otimes id}\downarrow\wr \hspace{105pt}\wr\downarrow \p{i_{Z}} \\ {\bf \Gamma}_Z{\cal R}^{\mbox{\Large $\cdot $}}_{q+c,X}\p{(r+c)}[c]\otimes {\cal H}^{p}\p{(s)} \hspace{10pt} \longby{\cap_{{\cal H}}} \hspace{10pt}{\bf \Gamma}_Z{\cal R}^{\mbox{\Large $\cdot $}}_{p+q+c,X}\p{(r+s+c)}[c] \E{array} \E{equation} commutes. \E{cor} \B{proof} This follows by the factorisation of $i_{\flat}$ (cf. the Remark at the end of \S\ref{H-Gys}) and the Proposition above. \E{proof} \subsection{${\cal H}$-cohomology ring}\label{H-ring} Let $(H^*,H_*)$ be a multiplicative Poincar\'e duality theory with supports. Let suppose that our cohomology theory $H^*$ takes values in a fixed category of $\mbox{$\Lambda$}$-modules where $\mbox{$\Lambda$} = H^0(k,\p{0})$ is a commutative ring with 1; we assume that the bigraded $\mbox{$\Lambda$}$-module $\bigoplus_{q,r}^{} H^q(X,\p{r})$ has a $\mbox{$\Lambda$}$-algebra structure via the cup-product pairing e.g. a canonical isomorphism of rings $H^0(X,\p{0})\cong \mbox{$\Lambda$}$ if $X$ is irreducible.\\ Let $$A(X) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \bigoplus_{p,q,r}^{} H^p(X,{\cal H}^q\p{(r)}).$$ Then $X \leadsto A(X)$ is a contravariant functor on $\cat{V}_k$. If $f:Y \to X$ is a proper map of relative dimension $\rho$ then the ${\cal H}$-Gysin maps ${\bf R}f_*{\cal H}^q\p{(r)}\to {\cal H}^{q+\rho}\p{(r+\rho)}[\rho]$ induce direct image $A(Y) \to A(X)$ (a map of degree $\rho$) so that $A$ is a covariant functor w.r.t. proper maps of smooth varieties. {}From the ${\cal H}$-cup-product pairing by taking cohomology we have an external pairing $$\p{\times}:A(X)\otimes_{\mbox{$\Lambda$}}A(Y) \to A(X\times Y)$$ which is associative and anticommutative (can be made commutative by using the trick in \cite{GIN}). In particular, let consider the functor $$X \leadsto \bigoplus_{p}^{} H^p(X,{\cal H}^p\p{(p)})\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, A_{diag}(X).$$ If ${\cal H}^0(\p{(0)})$ is identified with the flasque sheaf $\coprod_{X^0}^{} \mbox{$\Lambda$}$ (e.g. by assuming the `dimension axiom') we then have an augmentation $\varepsilon :A_{diag}(X)\to\mbox{$\Lambda$}$ where $X$ is irreducible and $\varepsilon^0 : H^0(X,{\cal H}^0\p{(0)})\cong \mbox{$\Lambda$}$ zero otherwise.\\ Let denote $f_*$ and $f^*$ ``direct and inverse'' images. We have the following formulas (cf. \cite[I.1--I.9]{GI}): \B{eqnarray} (f\p{\times}g)^*(-\p{\times}\cdot)& =& f^*(-)\p{\times}g^*(\cdot)\nonumber \\ (f\p{\times}g)_*(-\p{\times}\cdot) &=&f_*(-)\p{\times}g_*(\cdot)\nonumber \\ \varepsilon (f^*(\cdot))&=&\varepsilon (\cdot) \nonumber \\ \varepsilon (-\p{\times}\cdot)&=&\varepsilon (-)\varepsilon (\cdot) \E{eqnarray} Furthermore, for $X={\rm Spec} k$ we have that $\varepsilon : A_{diag}(k)\cong \mbox{$\Lambda$}$: let $e$ be the unique element such that $\varepsilon (e) =1$. We have the formula: $e\p{\times}-=-\p{\times}e=-$.\\ Let $\lambda :X\to {\rm Spec} k$ be the structural map and let denote $\lambda^*(e)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, 1_X$. This is equal to $\varepsilon^{-1}(1)$ on $X$ irreducible. For $x\in A_{diag}(X)$ and $y\in A_{diag}(Y)$ we have \B{eqnarray} \label{cross} x\p{\times}1_Y&=&p_1^*(x)\nonumber\\ 1_X\p{\times}y&=&p_2^*(y) \E{eqnarray} where $p_1$ and $p_2$ are the first and the second projections of $X\times Y$ on its factors.\\ By composing the external product $\p{\times}:A_{diag}(X)\otimes_{\mbox{$\Lambda$}}A_{diag}(X) \to A_{diag}(X\times X)$ with the diagonal $\Delta^*_X: A_{diag}(X\times X)\to A_{diag}(X)$ we do get a product $x\p{\otimes}x\prime \leadsto xx\prime$ in $A_{diag}(X)$ making it an associative anticommutative algebra with identity $1_X$. The homomorphism $\varepsilon : A_{diag}(X)\to \mbox{$\Lambda$}$ is an homomorphism of unitary $\mbox{$\Lambda$}$-algebras. For $f:X\to Y$ the map $f^*: A_{diag}(Y)\to A_{diag}(X)$ is a homomorphism of $\mbox{$\Lambda$}$-algebras. The external product is a homomorphism of augmented $\mbox{$\Lambda$}$-algebras. This last fact, via the equations (\ref{cross}), give us the formula \B{equation} x\p{\times}y=p_1^*(x)p_2^*(y) \E{equation} for $x\in A_{diag}(X)$ and $y\in A_{diag}(Y)$.\\ For $f:X\to Y$ a proper map of smooth varieties over $k$, $A_{diag}$ satisfies the `projection formula' as a consequence of the projection formula (\ref{chpr}). Furthermore, if $f$ is surjective of relative dimension $\rho$ over $Y$ irreducible we have a canonical map $$\int_{X/Y}: H^{-\rho}(X,{\cal H}^{-\rho}\p{(-\rho)})\to \mbox{$\Lambda$}$$ and its extension by zero $\int : A_{diag}(X)\to\mbox{$\Lambda$}$, both defined by composition of the ${\cal H}$-Gysin map $f_*$ and the augmentation $\varepsilon$. In particular $$\int_{X/k}: H^{d}(X,{\cal H}^{d}\p{(d)})\to \mbox{$\Lambda$}$$ for any $X$ proper smooth $d$-dimensional variety; for any map $f$ between $X$ and $Y$ proper smooth varieties we have $$\int_{Y/k}f_* = \int_{X/k}$$ Finally, if the proper map has a section $fs=1$ then $\int$ is a surjection; this is the case of $X/k$ having a $k$-rational point. \section{Intersection theory} Since we are going to deal with Poincar\'e duality theories which are `sophisticated' {\it and}\, `multiplicatives' we need to arrange the axioms in order to be not redundant. This arrangement will yields the notion of `duality theory {\it appropriate}\, for algebraic cycles' or for short `appropriate duality theory'. We will show that the ${\cal H}$-cohomology rings associated with such a theory reproduce the classical intersection rings. Roughly speaking, the denomination `appropriate duality' is the corresponding cohomological version of `relation d'\'equivalence ad\'equate' introduced by P.Samuel (see \cite{SAM}) for algebraic cycles. \subsection{Axiomatic menuet} Let $H^*$ be a cohomology theory and let $H_*$ be a homology theory (as defined by \cite[1.1 and 1.2]{BO}). Let assume that the pair $(H^*,H_*)$ yields a sophisticated Poincar\'e duality which satisfies the dimension axiom; furthermore we assume the existence of an associative anticommutative functorial cup product pairing $$ H_Y^p(X,\p{r})\otimes H_Z^q(X,\p{s})\to H^{p+q}_{Y\cap Z}(X,\p{r+s})$$ where $\mbox{$\Lambda$} = H^0(k,\p{0})$ is a commutative ring with 1 and the bigraded $\mbox{$\Lambda$}$-module $\bigoplus_{q,r}^{} H^q(X,\p{r})$ has a $\mbox{$\Lambda$}$-algebra structure via the cup-product pairing.\\[3pt] \B{defi} An {\it appropriate}\, duality theory is a pair $(H^*,H_*)$ as above such that the sophisticated cap product is compatible with the cup product via Poincar\`e duality i.e. the following diagram, where $q+j=2d, s+n=d$ and $X$ is smooth \B{displaymath}\B{array}{ccc} H^q_Y(X,\p{s})\otimes H_Z^p(X,\p{r}) & \longby{\cup} & H^{p+q}_{Y\cap Z}(X,\p{r+s})\\ \p{\eta_X\cap -\otimes id} \downarrow & & \downarrow\p{\eta_X\cap -} \\H_j(Y,\p{n}) \otimes H_Z^p(X,\p{r}) & \longby{\cap} & H_{j-p}(Y\cap Z,\p{n-r}) \E{array}\E{displaymath} commutes. In particular, the fundamental class $\eta_X\in H_{2d}(X\p{d})$ corresponds to the unit $1\in\mbox{$\Lambda$}\cong H^0(X,\p{0})$ in the $\mbox{$\Lambda$}$-algebra structure. \E{defi} \hfill\\[4pt] Let $(H^*,H_*)$ be an appropriate duality theory on $\cat{V}_k$. Then, by adopting the same notation of \S\ref{algcyc}, $$\cat{V}_k\ni Y\leadsto C_{n,m}(Y;\mbox{$\Lambda$}\p{(s)})\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,\H^{-n}(Y,{{\cal Q}}^m_{\mbox{\Large $\cdot $}}\p{(s)})$$ is a covariant functor for proper morphisms in $\cat{V}_k$. It is a presheaf for the \'etale topology (or just for the Zariski topology, depending with the homology theory). On the other hand we have a contravariant functor $$(X,Y)\leadsto H^p_Y(X,{\cal H}^q\p{(r)})$$ which yields the ${\cal H}$-cohomology ring with the properties stated in \S\ref{H-ring}. Indeed, on a smooth scheme $X$ of pure dimension $d$, these two functors are related via the duality isomorphism $${{\cal Q}}^{d-q}_{-\mbox{\Large $\cdot $}}\p{(d-r)}[d]\cong {{\cal R}}_q^{\mbox{\Large $\cdot $}}\p{(r)}$$ and this isomorphism is compatible with the ${\cal H}$-cap and ${\cal H}$-cup products (cf. \S\ref{H-cap} and \S\ref{H-cup}); by construction this duality isomorphism identifies Gysin maps (cf. \S\ref{H-Gys}) and projection formulas (cf. (\ref{hpr}) with (\ref{chpr})). Thus we do have a canonical cap-product associated with pairs $(X,Y)$ and $(X,Z)$ $$C_{n,m}(Y;\mbox{$\Lambda$}\p{(s)})\otimes H^p_Z(X,{\cal H}^q\p{(r)}) \to C_{n-p,m-q}(Y\cap Z;\mbox{$\Lambda$}\p{(s-r)})$$ and a corresponding projection formula. We have indeed a canonical ``trace map'' on $X$ irreducible $${{\cal Q}}^{d}_{-\mbox{\Large $\cdot $}}\p{(d)}[d]\by{\sim} \mbox{$\Lambda$} $$ yielding a global section $[X]\in C_d(X;\mbox{$\Lambda$})$; by capping with this ``fundamental class'' $[X]$ we get the quasi-isomorphism $${\cal H}^q\p{(r)}\by{\cap [X]} {{\cal Q}}^{d-q}_{-\mbox{\Large $\cdot $}}\p{(d-r)}[d]$$ By taking hypercohomology with support on $Z$ (a closed equidimensional subscheme) we do get the ``duality'' isomorphism \B{equation} \p{\cap}[X]: H^p_Z(X,{\cal H}^q\p{(r)}) \by{\sim} C_{d-p,d-q}(Z;\mbox{$\Lambda$}\p{(d-r)}). \E{equation} Conversely: for $i:Z\hookrightarrow X$ we have a quasi-isomorphism $$i_*{{\cal Q}}^{d-q}_{-\mbox{\Large $\cdot $}}\p{(d-r)}[d]\by{\sim} {\bf \Gamma}_Z{\cal H}^q\p{(r)}$$ hence the canonical isomorphism \B{equation} \eta : C_{d-p,d-q}(Z;\mbox{$\Lambda$}\p{(d-r)})\by{\sim} H^p_Z(X,{\cal H}^q\p{(r)}) \E{equation} In particular: \B{schol}\label{prs} We have a commutative diagram \B{displaymath} \B{array}{c}\hspace{10pt} C_{d-p}(Z;\mbox{$\Lambda$})\otimes H^q_{Y\cap Z}(Z,{\cal H}^q\p{(q)})\\ \p{id\otimes i^*}\nearrow \ \hspace{40pt} \searrow \p{H^*(\cap_{{{\cal H}}})} \\ C_{d-p}(Z;\mbox{$\Lambda$})\otimes H^q_{Y}(X,{\cal H}^q\p{(q)})\hspace{60pt} C_{d-p-q}(Y\cap Z;\mbox{$\Lambda$})\\ \p{\eta\otimes id}\downarrow \hspace{105pt}\downarrow \p{\eta} \\ H^p_{Z}(X,{\cal H}^p\p{(p)})\otimes H^q_{Y}(X,{\cal H}^q\p{(q)})\hspace{10pt} \longby{H^*(\cup_{{\cal H}})} \hspace{10pt}H^{p+q}_{Y\cap Z}(X,{\cal H}^{p+q}\p{(p+q)}) \E{array}\E{displaymath} \E{schol} \B{proof} This is a consequence of the commutative diagram (\ref{hipr}) and the compatibilities between ${\cal H}$-products. \E{proof} \subsection{${\cal H}$-cycle classes} \label{H-cycle} We mantain the notations and the assumptions of the previous section. Let $Z\subset X$ be a prime cycle of dimension $d-c$. Then $$C_{d-c}(Z;\mbox{$\Lambda$}) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, C_{d-c,d-c}(Z;\mbox{$\Lambda$}\p{(d-c)})=H_{2d-2c}(K(Z),\p{d-c}) \cong\mbox{$\Lambda$}$$ by the `dimension axiom' and we do have a cycle class \B{equation} \eta (Z) \in H^{c}_Z(X,{\cal H}^{c}\p{(c)}) \E{equation} where: $[Z]\in H_{2d-2c}(K(Z),\p{d-c})$ is obtained by restriction of the fundamental class $\eta_Z\in H_{2d-2c}(Z,\p{d-c})$ to the generic point and we have $$\mbox{$\Lambda$}\ni 1 \leadsto [Z] \leadsto \eta(1)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \eta (Z) \in H^{c}_Z(X,{\cal H}^{c}\p{(c)}).$$ Furthermore, by capping with the fundamental class $[X]$, we find the formula \B{equation}\label{inv} [Z]=\eta (Z)\p{\cap}[X] \E{equation} In particular the cycle class $\eta (Z)$ is independent from the imbedding of $Z$ as a subvariety and it is functorial w.r.t. \'etale maps. \B{lemma}\label{etaext} For $Y$ and $Z$ prime cycles of codimension $p$ and $q$ in $X$ smooth we have $$\eta(Y\p{\times}Z) =\eta (Y) \p{\times}\eta (Z) \in H^{p+q}_{Y\times Z}(X\times X,{\cal H}^{p+q}\p{(p+q)}).$$ \E{lemma} \B{proof} By standard sheaf theory the external product is obtained by using flasque resolutions (see \cite[6.2.1]{GO}). Thus via the canonical quasi-isomorphisms ${\cal H}\p{()} \cong{\cal Q}_{\mbox{\Large $\cdot $}}\p{()}$ we do have a commutative diagram \B{displaymath}\B{array}{ccc} H_Y^p(X,{\cal H}^p\p{(p)})\otimes H_Z^q(X,{\cal H}^q\p{(q)}) & \longby{\times} & H^{p+q}_{Y\times Z}(X\times X,{\cal H}^{p+q}\p{(p+q)})\\ \downarrow\wr& & \downarrow\wr \\ C_{d-p}(Y;\mbox{$\Lambda$})\otimes C_{d-q}(Z;\mbox{$\Lambda$}) & \longby{\times} & C_{d-p-q}(Y\times Z;\mbox{$\Lambda$}) \E{array}\E{displaymath} To conclude one would see that the bottom arrow is in fact the external product of cycles: this last claim is clear because the external products are homomorphisms of $\mbox{$\Lambda$}$-algebras. (Note: $C_{dim?}(?;\mbox{$\Lambda$})\cong \mbox{$\Lambda$}$ by the dimension axiom). \E{proof} Let $\Delta :X\to X\times X$ be the diagonal embedding and let $\Delta (X)$ be the diagonal cycle on $X\times X$. Let $Y$ and $Z$ be prime cycles of codimension $p$ and $q$ in $X$ smooth such that $Y\cap Z$ is of pure codimension $p+q$. For $d=$dim$X$ we then have \B{displaymath}\B{array}{ccc} H_{Y\cap Z}^{p+q}(X,{\cal H}^{p+q}\p{(p+q)}) & \longby{\Delta_{\mbox{\Large $\cdot $}}} & H^{p+q+d}_{(Y\times Z)\cap \Delta (X)}(X\times X,{\cal H}^{p+q+d}\p{(p+q+d)})\\ \p{\eta}\ \uparrow\wr& & \uparrow\wr\ \p{\eta} \\ C_{d-p-q}(Y\cap Z;\mbox{$\Lambda$}) & \longby{\Delta_*} & C_{d-p-q}((Y\times Z)\cap \Delta (X);\mbox{$\Lambda$}) \E{array}\E{displaymath} where $\Delta_{\mbox{\Large $\cdot $}}$ is obtained by making the diagram commutative and $\Delta_*$ is induced by the isomorphism $\Delta : Y\cap Z \to (Y\times Z)\cap \Delta (X)$. Thus the formula \B{equation}\label{etapoint} \Delta_{\mbox{\Large $\cdot $}}(\eta (\ ))=\eta(\Delta_*(\ )) \E{equation} \B{lemma} We have the following formula: \B{equation}\label{etadiag} \Delta_{\mbox{\Large $\cdot $}}\Delta^*(\eta(Y\times Z))=\eta(Y\times Z)\eta (\Delta (X)) \E{equation} \E{lemma} \B{proof} The formula is a consequence of the projection formula in the Scholium~\ref{prs} (cf. (\ref{hipr}) and (\ref{ipr}) ) applied to the diagonal embedding $\Delta$ by taking ${\cal H}$-cohomology with supports on $Y\times Z$ and compatibility of the ${\cal H}$-products with the canonical augmentations. \E{proof} \subsection{Intersection of cycles}\label{int-cyc} Let now assume that our cycle group $C_{*}(X;\mbox{$\Lambda$} )$ has an intersection product satisfying the classical properties (cf. \cite{FU}): local nature of the intersection multiplicity, normalization and reduction to the diagonal for $X$ a smooth projective variety over a field $k$.\\ Moreover, for a pair $(X,D)$ where $X\in\cat{V}_k$ and $D$ is a Cartier divisor on $X$ we let assume the existence of a homomorphism (cf. Definition~\ref{line} in \S\ref{H-chern} below) $$c\ell:H^1_D(X,{\cal O}^*_X)\otimes \mbox{$\Lambda$}\to H^1_D(X,{\cal H}^1\p{(1)})$$ such that \B{description} \item[{\it (i)}] $c\ell$ is a natural trasformation of contravariant functors w.r.t. morphisms $f:X'\to X$ such that $f^{-1}(D)$ is a divisor on $X'$; \item[{\it (ii)}] $c\ell$ is compatible with the cap-products in the sense that the following \B{displaymath}\B{array}{ccc} H_{D}^{1}(X,{\cal O}^*_X)\otimes \mbox{$\Lambda$} & \by{c\ell} & H^1_D(X,{\cal H}^1\p{(1)}) \\ \p{\cap [X]}\ \downarrow& & \downarrow\ \p{\cap [X]} \\ CH_{d-1}(D;\mbox{$\Lambda$}) & = & C_{d-1}(D;\mbox{$\Lambda$}) \E{array}\E{displaymath} commutes, where $d=$dim$X$. \E{description} \B{rmk} The map $\p{\cap [X]}: H_{D}^{1}(X,{\cal O}^*_X)\to CH_{d-1}(D)$ is given by the cap-product in ${\cal K}$-cohomology with the canonical cycle $[X]\in CH_{d}(X)$; when applyed to the cycle class of the Cartier divisor yields just the associated Weil divisor (cf. \cite[\S2]{GIN}). By the way, if $X$ is non-singular then $c\ell$ is an isomorphism. \E{rmk} Thus we can prove the following key lemma. \B{lemma}\label{intdiv} Let $X$ be smooth. Let $D$ be a principle effective Cartier divisor and let $i: Z\hookrightarrow X$ be a closed integral subscheme of codimension $c$ in $X$ such that $Z\cap D$ is a divisor on $Z$. Then the following \B{displaymath}\B{array}{ccc} H^1_D(X,{\cal H}^1\p{(1)}) & \by{i^*} & H^1_{Z\cap D}(Z,{\cal H}^1\p{(1)}) \\ \p{\eta}\ \uparrow& & \downarrow\ \p{\cap [Z]} \\ C_{d-1}(D;\mbox{$\Lambda$}) & \by{i^*} & C_{d-c-1}(Z\cap D;\mbox{$\Lambda$}) \E{array}\E{displaymath} commutes i.e. we have the following formula \B{equation} D\mbox{\Large $\cdot $} Z = i^*\eta(D)\p{\cap} [Z] \E{equation} \E{lemma} \B{proof} The claimed commutative diagram is obtained by the corresponding one for the Picard groups (cf. \cite[\S2]{GIN}). Let denote $\bar D\in H_{D}^{1}(X,{\cal O}^*_X)\otimes \mbox{$\Lambda$}$ the canonical class of the Cartier divisor: thus $\bar D\p{\cap} [X] = [D]\in CH_{d-1}(D;\mbox{$\Lambda$})$; since $X$ is non-singular, $[D] \leadsto \bar D$ under the isomorphism $CH_{d-1}(D;\mbox{$\Lambda$})\cong H_{D}^{1}(X,{\cal O}^*_X)\otimes\mbox{$\Lambda$}$ and $[D] \leadsto \eta(D)$ under the isomorphism $CH_{d-1}(D;\mbox{$\Lambda$})\cong H^1_D(X,{\cal H}^1\p{(1)}).$\\ We then have \B{center} \parbox{3in}{$i^*\eta(D)\p{\cap} [Z] =$\hfill by {\it (ii)}\\ $=i^*c\ell(\bar D)\p{\cap} [Z] =$\hfill by {\it (i)}\\ $=c\ell(i^*\bar D)\p{\cap} [Z] =$\hfill by \cite[\S2]{GIN}\\ $=i^*(D) =$\hfill\\ $=D\mbox{\Large $\cdot $} Z$} \E{center} where the last equality is just the normalization property of the intersection theory. \E{proof} \B{teor} With the above assumptions and notations, let $Y$ and $Z$ be prime cycles of codimension $p$ and $q$ on $X$ smooth which intersect properly. Then $$ \eta (Y)\eta (Z) = \eta (Y\mbox{\Large $\cdot $} Z) \in H^{p+q}_{Z\cap Y}(X,{\cal H}^{p+q}\p{(p+q)})$$ \E{teor} \B{proof} The proof is similar to that of the ``uniqueness of the intersection theory'' and it consists of 3 steps.\\ {\it Step 1.\, (Intersection with divisors).} Let $Y=D\hookrightarrow X$ be a principle Cartier divisor. Let $i:Z\hookrightarrow X$. Then \B{center} \parbox{3in}{$\eta (D)\eta (Z)=$\hfill by Scholium~\ref{prs}\\$\eta (i^*\eta (D) \p{\cap} [Z])=$\hfill by Lemma~\ref{intdiv}\\$\eta (D\mbox{\Large $\cdot $} Z).$} \E{center} {\it Step 2.\, (Intersection with smooth subvarieties).} Let assume $Y$ to be smooth. Since we can reduce to open Zariski neighborhoods of the generic points of $Y\cap Z$ we may assume that $X$ is affine and $Y=V(f_1,\ldots ,f_p)$ where $\{f_1,\ldots ,f_p\}$ is a regular sequence. Thus: $Y={\displaystyle\cap_{i=1}^{p} D_i}$ where $D_i =V(f_i)$ and \B{center} \parbox{3in}{$\eta (Y)\eta (Z) =$ \\ $=\eta (D_1\cdots D_p)\eta (Z)=$\\ $=\eta (D_1)\eta (D_2 \cdots D_p)\eta (Z)=\ldots$\\ $\ldots=\eta (D_1)\cdots \eta (D_{p-1})\eta (D_p\mbox{\Large $\cdot $} Z)= \ldots$\\ $\ldots =\eta (Y\mbox{\Large $\cdot $} Z)$} \E{center} by iterative application of Step 1.\\ {\it Step 3.\, (Reduction to the intersection with the diagonal).} We prove the general case as follows: \B{center} \parbox{3in}{$\Delta_{\mbox{\Large $\cdot $}}(\eta (Y)\eta (Z)) =$\hfill by definition\\$=\Delta_{\mbox{\Large $\cdot $}}\Delta^*(\eta (Y)\times \eta (Z))=$\hfill by Lemma~\ref{etaext}\\ $=\Delta_{\mbox{\Large $\cdot $}}\Delta^*(\eta (Y\times Z))=$\hfill by the formula (\ref{etadiag})\\$=\eta (Y\times Z)\eta (\Delta (X))=$\hfill by Step 2\\$=\eta (Y\times Z\mbox{\Large $\cdot $} \Delta (X))=$\hfill int. with the diag.\\$=\eta (\Delta_*(Y\mbox{\Large $\cdot $} Z)=$\hfill by the formula (\ref{etapoint})\\ $=\Delta_{\mbox{\Large $\cdot $}}(\eta (Y\mbox{\Large $\cdot $} Z))$} \E{center} Since $\Delta_{\mbox{\Large $\cdot $}}$ is an isomorphism we conclude. \E{proof} \B{cor} If $X$ is smooth of pure dimension $d$ then the graded isomorphism $$\eta : \bigoplus C_{d-p}(X;\mbox{$\Lambda$}) \cong \bigoplus H^{p}(X,{\cal H}^{p}\p{(p)})$$ is a $\mbox{$\Lambda$}$-algebra isomorphism. \E{cor} \section{Chern classes and blow-ups} Let $X$ be a variety i.e. $X\in \cat{V}_k$ reduced and equidimensional over a perfect field $k$, which admits a closed imbedding in a smooth variety; such varieties are usually called {\it imbeddable}. The existence of ${\cal H}$-cap-products grant us to construct Gysin maps for the functor $C_{*}(-;\mbox{$\Lambda$})$ associated with such imbeddings. By using the results from \S3--\S6 we construct Chern classes in ${\cal H}$-cohomologies. Furthermore, we are able to obtain the nice decomposition formula for the ${\cal H}$-cohomology of blow-ups generalising the classical one for Chow groups. \subsection{Gysin maps for algebraic cycles} Let $(H^*,H_*)$ be an appropriate duality. We consider an imbeddable variety $X$ with a fixed ambient smooth variety $Y$. Let $i:X\hookrightarrow Y$ be a closed imbedding of pure codimension $c$. Thus we have a ${\cal H}$-cycle class $\eta (X)\in H^c_X(Y,{\cal H}^c\p{(c)})$ and the corresponding Gysin maps $$i^!:C_{n}(Y;\mbox{$\Lambda$})\to C_{n-c}(X;\mbox{$\Lambda$})$$ are defined as follows: \B{equation}y\leadsto y\p{\cap}\eta (X)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, i^!(y) \E{equation} Thus $i^!(Y) = [X]$ because of $[Y]\p{\cap}\eta (X) =[X]$ by the definition of ${\cal H}$-cycle classes.\\[4pt] \B{rmk} Actually we got ``Gysin maps'' $i^!$ for imbeddings $i:X\hookrightarrow Y$ where $Y$ is just imbeddable in $V$ smooth, by capping with the ${\cal H}$-cycle class of $X$ in $V$. This operation will take a cycle of codimension $p$ on $Y$ to a cycle on $X$ of codimension $p$ plus the codimension of $Y$ in $V$. \E{rmk} Let denote $i_!:C_*(X)\to C_*(Y)$ the canonical map induced by $i$. Since $Y$ is smooth we do have the following equation \B{equation} \eta i_! = i_{\diamond}\eta \E{equation} where $i_{\diamond}: H^*_X(Y,{\cal H}^*)\to H^*(Y,{\cal H}^*)$ is the standard map. Let denote $i^*: H^*(Y,{\cal H}^*)\to H^*(X,{\cal H}^*)$. Let consider the intersection product of cycles induced by the ${\cal H}$-cohomology ring, according with \S\ref{H-ring}. \B{prop} The operation $i^!$ is functorial and compatible with \'etale pull-backs. We have the self-intersection property: \B{equation} i^!i_! (X) = X\mbox{\Large $\cdot $} X \E{equation} If $i:X\hookrightarrow Y$ is a smooth pair we then have $$ i^! = i^*\eta \p{\cap} [X]$$ and $i^!$ is a ring homomorphism; there is a projection formula $$i_!(x\mbox{\Large $\cdot $} i^!(y))=i_!(x)\mbox{\Large $\cdot $} y$$ for cycles $x$ and $y$ on $X$ and $Y$ respectively. \E{prop} \B{proof} Compatibilities are easy to check. The self-intersection property is obtained as follows: \B{center} \parbox{3in}{$i^!(i_!(x))=i_!(x)\p{\cap}\eta (X)=$\hfill proj. form.\\$=x\p{\cap}i^*\eta (X)$} \E{center} whence, by taking $x=[X]$, we have $$[X]\p{\cap}i^*\eta (X)=X\mbox{\Large $\cdot $} X $$ This last equation holds because of the Scholium~\ref{prs}, giving us the following $$\eta ([X]\p{\cap}i^*\eta (X))=\eta (X)\eta (X)$$ where $\eta (X)\eta (X)=\eta (X\mbox{\Large $\cdot $} X)$ (see Theorem~3) and $\eta $ is an isomorphism.\\ The other equation is given by the following commutative diagram $$\begin{array}{ccc} H^{*}(Y,{\cal H}^*)\otimes H^{c}_X(Y,{\cal H}^c\p{(c)}) &\longby{\cdot} & H^{*+c}_X(Y,{\cal H}^{*+c}\p{(*+c)})\\ \p{i^*\otimes \cap [Y]}\downarrow & &\downarrow \wr\ \p{\cap [Y]}\\ H^{*}(X,{\cal H}^*)\otimes C_{d-c}(X;\mbox{$\Lambda$} ) & \longby{\cap}& C_{d-c-*}(X;\mbox{$\Lambda$} ) \end{array} $$ which is obtained by the Scholium~\ref{prs} (cf. the formula (\ref{rest}) ). The projection formula is obtained from the projection formula w.r.t. the ${\cal H}$-product (cf. the Scholium~\ref{prs} and the Theorem~3). \E{proof} By using the contravariant structure of ${\cal H}$-cohomologies we can construct `refined' Gysin maps $f^!$ for algebraic cycles between imbeddable varieties. \subsection{Grothendieck-Gillet axioms for Chern classes} \label{H-chern} The way to obtain a theory of Chern classes in ${\cal H}$-cohomologies and the corresponding Riemann-Roch Theorems will be to show that the cohomology theory $H^{*}_Z(X,{\cal H}^*\p{(*)})$ and the homology theory $C_{*,*}(-;\mbox{$\Lambda$}\p{(*)})$ satisfy the list of axioms in \cite[Definition 1.1 -- 1.2]{GIL}. We are going to consider $X\in\cat{V}_k$ smooth over a perfect field. We also assume that $\mbox{$\Lambda$}$ (constant sheaf for the Zariski topology) has finite weak global dimension (see \cite[Definition 2.6.2]{KS}) in order to consider tensor products $-\stackrel{L}{\otimes}_{\mbox{$\Lambda$}}-$ in the derived category. \B{defi}\label{line} We let say that a natural transformation $$c\ell :{\rm Pic}\, (X)\otimes\mbox{$\Lambda$} \to H^1(X,{\cal H}^1\p{(1)})\subset H^2(X,\p{1})$$ of contravariant functors is a {\it cycle class map for line bundles} if $c\ell$ localizes satisfiyng the properties {\it (i) -- (ii)}\, stated in \S\ref{int-cyc}. Therefore $c\ell$ is compatible via the local triviality property \cite[1.5]{BO}, with the map obtained mapping a prime Weil divisor $i: D\hookrightarrow X$ to the Poincar\'e dual of the direct image under $i$ of the fundamental class $\eta_D$. \E{defi} \B{teor} Let $(H^*,H_*)$ be an appropriate Poincar\'e duality on $\cat{V}_k$ for a perfect field $k$, with values in a fixed category of $\mbox{$\Lambda$}$-modules such that $H^*$ satisfies the homotopy property and there is a cycle class map for line bundles. Then there is a theory of Chern classes $$c_{p,i} : K_i^Z(X) \to H^{p-i}_Z(X,{\cal H}^p\p{(p)})$$ associated with any closed $Z$ in $X\in\cat{V}_k$ smooth. \E{teor} \B{proof} With the notations of \cite{GIL} we let $\oplus\underline \Gamma^*\p{(p)}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,\oplus {\cal H}^p\p{(p)}$ be the graded sheaf with the ${\cal H}$-cup-product (according with \cite[Definition 1.1]{GIL} and \S\ref{H-ring}) defining our cohomology theory ring on the category $\cat{V}_k$. We let define the homology as $$H_i(X,\Gamma \p{(j)})\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \H^{-i}(X,{{\cal Q}}^{j}_{\mbox{\Large $\cdot $}}\p{(j)})=C_{i,j}(X;\mbox{$\Lambda$}\p{(j)})$$ which is covariant w.r.t. proper morphisms and a presheaf for the \'etale topoloy by \cite[3.7]{BO}; the compatibility \cite[1.2.(i)]{GIL} is ensured by the compatibility \cite[1.2.2]{BO} and limits arguments (cf. \S~4.3). The functorial long exact sequence of homology, for a pair $i:Y\hookrightarrow X$, is obtained via the hypercohomology long exact sequence with supports $$\H^{-i}_Y(X,{{\cal Q}}^{j}_{\mbox{\Large $\cdot $}}\p{(j)})\to \H^{-i}(X,{{\cal Q}}^{j}_{\mbox{\Large $\cdot $}}\p{(j)})\to \H^{-i}(X-Y,{{\cal Q}}^{j}_{\mbox{\Large $\cdot $}}\p{(j)})$$ since $\Gamma_Y{{\cal Q}}^{j}_{\mbox{\Large $\cdot $},X}\p{(j)}\cong i_*{{\cal Q}}^{j}_{\mbox{\Large $\cdot $},Y}\p{(j)}$ and Lemma~4.6 (i.e. \cite[1.2.(ii)]{GIL} holds). The cap product structure is given by the ${\cal H}$-cap-product and all the properties required by \cite[1.2.(iii) -- (viii)]{GIL} are easily seen by using the results of \S4 and \S5. The homotopy property \cite[1.2.(ix)]{GIL} is our Lemma~3.4. Thus we are left to show the following classical Dold-Thom decomposition (see \cite[1.2.(x)--(xi)]{GIL}). \E{proof} \B{schol}\label{Edeco}{\rm (Decomposition)} Let ${\cal E}$ be a locally free sheaf, ${\rm rank}\, {\cal E} = n+1$, on $X$ smooth. Let $\pi: P\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, {\bf P}({\cal E}) \to X$ be the corresponding projective bundle. For ${\cal O}_P(1)\in {\rm Pic}\, P$ let $$\xi \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, c\ell ({\cal O}_P(1)) \in H^1(P,{\cal H}^1\p{(1)})$$ we then have $$\oplus \pi^*( )\p{\cup}\xi^i: \bigoplus_{i=0}^{n} H^{p-i}(X,{\cal H}^{q-i}_X\p{(r-i)})\cong H^p(P,{\cal H}^q_P\p{(r)})$$ Furthermore: \B{equation}\label{trace} \int_{P/X}^{}\xi^n =1_{\mbox{$\Lambda$}} \E{equation} \E{schol} \B{proof} The element $\xi\in H^1(P,{\cal H}^1\p{(1)})$ defines a map $\mbox{$\Lambda$} \to {\cal H}^1\p{(1)}[1]$ in the derived category $\cat{D}(P_{Zar};\mbox{$\Lambda$})$ of complexes of sheaves of $\mbox{$\Lambda$}$-modules on the Zariski site. By cup-product we have a map $\xi^i : \mbox{$\Lambda$} [-i] \to {\cal H}^i\p{(i)}$ hence a map ${\bf R}\xi^i :\mbox{$\Lambda$} [-i] \to {\bf R}\pi_*{\cal H}^i\p{(i)}$ in $\cat{D}(X_{Zar};\mbox{$\Lambda$})$. On the other hand we have the canonical map ${\bf R}\pi^{\sharp} :{\cal H}^{q-i}\p{(r-i)} \to {\bf R}\pi_*{\cal H}^{q-i}\p{(r-i)}$ induced by contravariancy (cf. \S 2.1). By taking ${\bf R}\pi^{\sharp} \stackrel{L}{\otimes} {\bf R}\xi^i$ we obtain the maps $${\cal H}^{q-i}\p{(r-i)}[-i]\to {\bf R}\pi_*{\cal H}^{q-i}\p{(r-i)} \stackrel{L}{\otimes}{\bf R}\pi_*{\cal H}^i\p{(i)}$$ Since the ${\cal H}$-cup-product yields products (cf. \cite[Expos\'e \S2.1]{SGA5}) $${\bf R}\pi_*{\cal H}^{q-i}\p{(r-i)} \stackrel{L}{\otimes}{\bf R}\pi_*{\cal H}^i\p{(i)}\to {\bf R}\pi_*{\cal H}^q\p{(r)}$$ by composing we do get the maps ${\cal H}^{q-i}\p{(r-i)}[-i]\to {\bf R}\pi_*{\cal H}^q\p{(r)}$ whence the map $$\gamma : \bigoplus_{i=0}^{n}{\cal H}^{q-i}\p{(r-i)}[-i]\to {\bf R}\pi_*{\cal H}^q\p{(r)}$$ in the derived category $\cat{D}(X_{Zar};\mbox{$\Lambda$})$. Now that $\gamma$ is defined the claimed decomposition will follows by proving that $\gamma$ is a quasi-isomorphism because of the Leray spectral sequence $$H^p(X,{\bf R}\pi_*{\cal H}^q\p{(r)})\cong H^p(P,{\cal H}^q\p{(r)})$$ In order to show that $\gamma$ is a quasi-isomorphism we are left to show the isomorphisms of groups $$(\gamma^p)_x : \bigoplus_{i=0}^{n}H^{p-i}({\rm Spec}{\cal O}_{X,x},{\cal H}^{q-i}\p{(r-i)})\cong (R^p\pi_*{\cal H}^q\p{(r)})_x$$ for all $x\in X$ and $p\geq 0$. By continuity of the arithmetic resolutions the stalks $(R^p\pi_*{\cal H}^q\p{(r)})_x$ are computed by $H^{p}(\P^n_{{\cal O}_{X,x}},{\cal H}^{q}\p{(r)})$; we need the following compatibility: \B{lemma} Let $U\subset X$ be an open Zariski neighborhood of $x$ on which ${\cal E}$ is free. Let $\xi\in H^1_{\infty}(\P_U^n,{\cal H}^1\p{(1)})$ be the restriction of the tautological divisor, where $i:\infty \cong \P_U^{n-1}\hookrightarrow \P_U^n$ is a hyperplane at infinity. Then (with the notation of \S3.3) $$j_{(n,n-1)}\pi^*_{n-1} = \pi^*_{n}\p{\cup}\xi$$ equality between maps from $H^{p-1}(U,{\cal H}^{q-1}\p{(r-1)})$ to $H^p(\P_U^n,{\cal H}^q\p{(r)})$. \E{lemma} \B{proof} Note that $\pi^*_{n-1}=i^*\pi^*_{n}$. The purity isomorphism $$H^{p-1}(\infty ,{\cal H}^{q-1}\p{(r-1)})\cong H^p_{\infty}(\P_U^n,{\cal H}^q\p{(r)})$$ is obtained as $\eta (- \p{\cap} [\infty])$ and $\eta (\infty ) = \xi$ (because of the compabilities of the cycle class $c\ell$) thus we have \B{center} \parbox{3in}{$\eta (\pi^*_{n-1}\p{\cap}[\infty])=$\hfill\\ $=\eta (i^*\pi^*_{n}\p{\cap}[\infty])=$\hfill Scholium~6.2\\ $=\eta (\pi^*_{n}\p{\cup}\eta (\infty)\p{\cap}[\P^n_U])=$\hfill\\ $=\pi^*_{n}\p{\cup}\eta (\infty) =$ \hfill\\ $=\pi^*_{n}\p{\cup}\xi$ \hfill as elements in $H^p_{\infty}(\P_U^n,{\cal H}^q\p{(r)})$} \E{center} and, by definition of $j_{(n,n-1)}$, the image of it under $H^p_{\infty}(\P_U^n,{\cal H}^q\p{(r)})\to H^p(\P_U^n,{\cal H}^q\p{(r)})$ yields the claimed equation. \E{proof} Thus: $(\gamma^p)_x$ is clearly an isomorphism by reduction to open Zariski neighborhoods on which ${\cal E}$ is free, arguing as in \S3.3 via the Lemma above and induction on the rank of ${\cal E}$.\\ Let show the equation (\ref{trace}). By the definition of the ${\cal H}$-Gysin map we have that $\pi_*: H^n(P,{\cal H}^n\p{(n)})\to H^0(X,{\cal H}^0\p{(0)})$ is obtained (via the Leray spectral sequence) by composition with $${\bf R}\pi_{\flat}:{\bf R}\pi_*{\cal H}^n\p{(n)}\to {\cal H}^0\p{(0)}[-n]$$ in $\cat{D}(X_{Zar};\mbox{$\Lambda$})$. Thus (see \S\ref{H-ring} ) we have to prove that the composition of $$\mbox{$\Lambda$} [-n] \longby{{\bf R}\xi^n}{\bf R}\pi_*{\cal H}^n\p{(n)}\longby{{\bf R}\pi_{\flat}}{\cal H}^0\p{(0)}[-n]\cong\mbox{$\Lambda$} [-n]$$ is the identity. Arguing as above we are reduced to show the equation (\ref{trace}) for $\pi:\P^n_{{\cal O}_{X,x}}\to {\rm Spec} {\cal O}_{X,x}$. By the projection formula $$\pi_*(\pi^*(1)\p{\cup}\xi^n)=\pi_*(\xi^n)$$ we are left to show that $\pi_*$ is the inverse of the ``decomposition'' isomorphism $$\pi^*( )\p{\cup}\xi^n : H^0({\rm Spec} {\cal O}_{X,x},{\cal H}^0\p{(0)}) \to H^n(\P^n_{{\cal O}_{X,x}}, {\cal H}^n\p{(n)})$$ By choosing a $k$-rational point of $\P^n_k$ we get a proper section $\sigma$ of $\pi$. With the notation above: $\sigma_* = j_{(n,0)}\pi^*_{0}$ and by the Lemma we have $$\pi^*(1)\p{\cup}\xi^n = \sigma_*(1)$$ By applying $\pi_*$ to the latter and taking the image of it under the canonical augmentation $H^0({\rm Spec} {\cal O}_{X,x},{\cal H}^0\p{(0)})\cong \mbox{$\Lambda$}$ we do obtain the claimed formula. \E{proof} \B{rmk} After Grothendieck-Verdier, this `decomposition argument' is quite standard. See \cite[Expos\'e VII]{SGA5} for \'etale cohomology and \cite[Theor.8.2]{GIL} or \cite{SH} for the $K$-theory. \E{rmk} Let $$A(-) = \bigoplus_{p,q,r}^{} H^p(-,{\cal H}^q\p{(r)})$$ be the ${\cal H}$-cohomology ring functor. \B{cor} Let $\pi :\P({\cal E})\to X$ be as above, ${\rm rank}\, {\cal E} = n+1$. Then $$\pi^* : A(X) \to A(\P({\cal E}))$$ is an injective homomorphism of unitary $\mbox{$\Lambda$}$-algebras and the elements $$1,\xi , \ldots , \xi^n$$ generate freely $A(\P({\cal E}))$ as $A(X)$-module. Furthermore $$\pi_* : A(\P({\cal E})) \to A(X)$$ is a surjective homomorphism of $A(X)$-modules (having degree $-n$). \E{cor} \B{proof} The statement is clear after \S\ref{H-ring} and the Scholium above. For example, $\pi_*(\xi^i)=0$ for $i=0,\ldots ,n-1$ but $\pi_*(\xi^n)=1$ by (\ref{trace}) whence the linear independence of $1,\xi , \ldots , \xi^n$ can be seen as follows: let suppose that $$\pi^*(x_0)+ \cdots +\pi^*(x_n)\p{\cup}\xi^n =0$$ then by applying $\pi_*$ and the projection formula we get $x_n=0$ thus $$\pi^*(x_0)\p{\cup}\xi+ \cdots +\pi^*(x_{n-1})\p{\cup}\xi^n =0$$ and the same argument gives $x_{n-1}=0$ and so on. Again: $\pi_*$ is a surjection because of $$\pi_*(\pi^*(\dag )\p{\cup}\xi^n)=\pi_*(\xi^n)\p{\cup}\dag =\dag$$ \E{proof} \B{rmk} By the prescription of \cite{GC} we therefore obtain Chern classes $c_p : K_0(X) \to H^p(X,{\cal H}^p\p{(p)})$ satisfying the equation $$\xi^{n}+\pi^*c_{1}(E)\xi^{n-1}+\cdots +\pi^*c_n(E)=0$$ for $E$ a vector bundle of rank $n$. By \cite{GIL} we have that $c_p$ is just $c_{p,0}$. \E{rmk} \subsection{Variation on the invariance theme} \label{finite} Let consider a sophisticated Poincar\'e duality theory $(H^*,H_*)$ satisfying the point axiom. Let consider $f:X\to Y$ a proper dominant morphism between connected smooth schemes in $\cat{V}_k$. If dim$X$ = dim$Y$ then $K(X)$ is a finite field extension of $K(Y)$; let ${\rm deg} f=[K(X):K(Y)]$ be its degree. Following the proof of (\ref{prfor}) we have that the composition $$H^{*}(Y,\cdot) \by{f^{\star}} H^{*}(X,\cdot) \by{f_{\star}} H^{*}(Y,\cdot)$$ is the multiplication by ${\rm deg} f$, as a consequence of the projection formula and our assumption that $f_!(\eta_X)={\rm deg} f\cdot\eta_Y$ (cf. \S\ref{inter1}). Thus: \B{prop}\label{mult} The composition of $$H^p_Z(Y,{\cal H}^{q}\p{(r)}) \by{f^{*}} H^p_{f^{-1}(Z)}(X,{\cal H}^{q}\p{(r)}) \by{f_{*}} H^p_Z(Y,{\cal H}^{q}\p{(r)}) $$ is the multiplication by ${\rm deg} f$. \E{prop} \B{proof} Let $d=$dim$X=$dim$Y$. Then the projection formula (\ref{hpr}) looks $$\B{array}{c} \hspace{10pt} f_*{{\cal Q}}^d_{\mbox{\Large $\cdot $}}\p{(d)}\otimes f_*{\cal H}^{q}\p{(r)}\\ \p{id\otimes f^{\sharp}}\nearrow \ \hspace{40pt} \searrow \p{f_*\cap_{{{\cal H}}}} \\ f_*{{\cal Q}}^d_{\mbox{\Large $\cdot $}}\p{(d)}\otimes {\cal H}^{q}\p{(r)}\hspace{60pt} f_*{{\cal Q}}^{d-q}_{\mbox{\Large $\cdot $}}\p{(d-r)}\\ \p{f_{\sharp}\otimes id}\downarrow \hspace{105pt}\downarrow \p{f_{\sharp}} \\ {{\cal Q}}^d_{\mbox{\Large $\cdot $}}\p{(d)}\otimes {\cal H}^{q}\p{(r)} \hspace{10pt} \longby{\cap_{{{\cal H}}}} \hspace{10pt}{{\cal Q}}^{d-q}_{\mbox{\Large $\cdot $}}\p{(d-r)} \E{array}$$ By the dimension axiom the complex ${{\cal Q}}^d_{\mbox{\Large $\cdot $}}\p{(d)}$ is concentrated in degree $d$ and its hypercohomology $C_{d,d}(X,\mbox{$\Lambda$}\p{(d)})$ has a natural global section $[X]$ corresponding to the fundamental class $\eta_X\in H_{2d}(X,\p{(d)})$. The same holds on $Y$ and $[X]\leadsto {\rm deg}f [Y]$ under $f_*$. Thus by taking cohomology with supports we have the result. \E{proof} \B{rmk} The same argument applies to the $K$-theory by using the projection formula in (\ref{kpr}). \E{rmk} \B{lemma}\label{birsplit} Let $f:X'\to X$ be a proper birational morphism between smooth varieties; let $i: Z\hookrightarrow X$ and $i':Z'=f^{-1}(Z)\hookrightarrow X'$ be closed subschemes such that $f: X'-Z' \cong X-Z$. Then we have splitting short exact sequences $$0\to H^p_Z(X,{\cal H}^q\p{(r)}) \by{u} H^p(X,{\cal H}^q\p{(r)})\oplus H^p_{Z'}(X',{\cal H}^q\p{(r)}) \by{v} H^p(X',{\cal H}^q\p{(r)})\to 0$$ where: $$ u=\left( \begin{array}{c} {i_{\diamond}}\\{f^*} \end{array} \right) $$ and $$v= (f^*,-i'_{\diamond})$$ The left splitting of $u$ is given by $u':(0,f_*)$. \E{lemma} \B{proof} Let consider the following maps of long exact sequences $$ \begin{array}{ccccccccc} \cdots & \to & H^p_{Z'}(X',{\cal H}^q\p{(r)}) & {\by{i'_{\diamond}}}& H^p(X',{\cal H}^q\p{(r)}) & \to & H^{p}(X'-Z',{\cal H}^q\p{(r)}) & \to & \cdots \\ & &{\p{f_*}\downarrow\uparrow\p{f^*}} & &{\downarrow\uparrow} & & {\downarrow\wr} & &\\ {\cdots}&{\to}& H^p_Z(X,{\cal H}^q\p{(r)}) &{\by{i_{\diamond}}}& H^p(X,{\cal H}^q\p{(r)}) & {\to}& H^{p}(X-Z,{\cal H}^q\p{(r)}) &{\to}&{\cdots} \end{array} $$ Since deg$f=1$ by the Proposition above $f_*f^*=1$; thus the corresponding Mayer-Vietoris exact sequence splits (because the boundary is zero) in short exact sequences as claimed. \E{proof} \B{schol} Let $X$, $X'$, $Z$ and $Z'$ be as above and pure dimensional. We have isomorphisms ($d=$dim$X$) $$C_{d-p}(X;\mbox{$\Lambda$})\oplus C_{d-p}(Z';\mbox{$\Lambda$}) \by{\simeq} C_{d-p}(Z;\mbox{$\Lambda$})\oplus C_{d-p}(X';\mbox{$\Lambda$})$$ given by the matrix $$\left( \B{array}{cc} 0 & f_! \\ f^! & -i'_! \E{array}\right) $$ and, for $Z$ and $Z'$ smooth of codimension $c$ and $c'$: $$H^{p}(X,{\cal H}^{q}\p{(r)})\oplus H^{p-c'}(Z',{\cal H}^{q-c'}\p{(r-c')}) \by{\simeq} H^{p-c}(Z,{\cal H}^{q-c}\p{(r-c)})\oplus H^{p}(X',{\cal H}^{q}\p{(r)})$$ given by the matrix $$\left(\B{array}{cc} 0 & f_* \\ f^* & -j_{\pp{(X',Z')}} \E{array}\right) $$ where $j_{\pp{(X',Z')}}$ is the Gysin map in ${\cal H}$-cohomology (cf. Scholium~\ref{Gysin}, \S\ref{H-Gys}). \E{schol} \B{proof} By the Lemma~\ref{birsplit} and purity. \E{proof} \subsection{Blowing-up} Let $(H^*,H_*)$ be an appropriate Poincar\'e duality such that $H^*$ satisfies the homotopy property and there is a cycle class map for line bundles (cf. \S\ref{H-chern}).\\ Let $f:X'\to X$ be the blow up of a smooth subvariety $Z$ of codimension $c\geq 2$ in a smooth variety $X$ of dimension $d$. Thus the exceptional divisor is the projective bundle over $Z$ given by $\P ({\cal N})$ where ${\cal N}$ is the normal sheaf, locally free of rank $c$. \B{prop}\label{blow-up} For the blow-up $X'$ of $X$ along $Z$ as above we have the following canonical formulas $$H^p(X',{\cal H}^q\p{(r)}) \cong H^p(X,{\cal H}^q\p{(r)})\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i}\p{(r-1-i)})$$ and in particular $$C_{n}(X';\mbox{$\Lambda$}) \cong C_{n}(X;\mbox{$\Lambda$}) \oplus \bigoplus_{i=0}^{c-2} C_{n-c+1+i}(Z;\mbox{$\Lambda$})$$ \E{prop} \B{proof} Since $f:Z'\to Z$ is a proper morphism, between smooth varieties, having relative dimension $1-c$ we have a push-forward (see \S\ref{H-Gys}) $$f_*: H^{p-1}(Z',{\cal H}^{q-1} \p{(r-1)}) \to H^{p-c}(Z,{\cal H}^{q-c}\p{(r-c)})$$ Because of purity $f^*: H^p_Z(X,{\cal H}^q\p{(r)}) \hookrightarrow H^p_{Z'}(X',{\cal H}^q\p{(r)})$ induces a map $$f^!: H^{p-c}(Z,{\cal H}^{q-c}\p{(r-c)}) \hookrightarrow H^{p-1}(Z',{\cal H}^{q-1}\p{(r-1)})$$ as well. By the Lemma~\ref{birsplit} we have a splitting exact sequence $$0\to H^{p-c}(Z,{\cal H}^{q-c}\p{(r-c)}) \by{u} H^p(X,{\cal H}^q\p{(r)})\oplus H^{p-1}(Z',{\cal H}^{q-1}\p{(r-1)}) \by{v} H^p(X',{\cal H}^q\p{(r)})\to 0$$ with left splitting $u'= (0,f_*)$. Let consider the projector $\pi =uu'$; we then have $\pi u=u$, $v\pi =0$ thus $v$ restricts to an isomorphism $$v: {\rm ker} \pi \cong H^p(X',{\cal H}^q\p{(r)})$$ Now $\pi (x, z') = u (f_*(z')) = (j_{\pp{(X,Z)}}(f_*(z')), f^!f_*(z'))=0$ if and only if $f_*(z')=0$ (because $f^!$ is injective). Thus we have $$ {\rm ker} \pi = H^p(X,{\cal H}^q\p{(r)})\oplus {\rm ker} f_*$$ Since $Z' =\P ({\cal N})$ and $f_{\mid Z'}$ is the standard projection then, by the Dold-Thom decomposition (see Scholium~\ref{Edeco}), we have an exact sequence $$0\to\bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i}\p{(r-1-i)}) \by{\xi^i f^*} H^{p-1}(Z',{\cal H}^{q-1} \p{(r-1)}) \by{f_*} H^{p-c}(Z,{\cal H}^{q-c} \p{(r-c)})\to 0 $$ where $\xi$ is the tautological divisor, hence: $${\rm ker} \pi \cong H^p(X,{\cal H}^q\p{(r)}) \oplus\bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i}\p{(r-1-i)})$$ and the claimed isomorphisms are easily obtained. \E{proof} \B{rmk} For the ${\cal K}$-cohomology the same proof applies yielding the formula $$H^p(X',{\cal K}_q) \cong H^p(X,{\cal K}_q)\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal K}_{q-1-i})$$ \E{rmk}

9408

alg-geom/9408006

en

https://arxiv.org/abs/alg-geom/9408006

alg-geom/9408006

Serge M. L'vovsky

S.L'vovsky

On Landsberg's criterion for complete intersections

4 pages, LaTeX 2.09

In his preprint ``Differential-Geometric Characterizations of Complete Intersections'' (alg-geom/9407002), J.M.Landsberg introduces an elementary characterization of complete intersections. The proof of this criterion uses the method of moving frames. The aim of this note is to present an elementary proof of Landsberg's criterion that is valid over any ground field.

alg-geom

\section*{Introduction} In his preprint~\cite{Lan}, J.M.~Landsberg introduces an elementary characterization of complete intersections (Proposition~1.2 in \cite{Lan}). The proof of this proposition uses the method of moving frames. The aim of this note is to present an elementary proof of Landsberg's criterion that is valid over any ground field. \section{Notation and statement of results} Let $k$ be an algebraically closed field and ${\bf P}^N= \mathop{\rm Proj}\nolimits k[T_0,\ldots,T_N]$ the $N$-dimensional projective space over $k$. If $F$ is a homogeneous polynomial in $T_0,\ldots ,T_N$, we will denote by $Z(F)\subset {\bf P}^N$ the hypersurface defined by $F$. If $F$ is a homogeneous polynomial and $x=(x_0:\ldots:x_N)\in \PP^N$, put $d_x F=\left(\partial F/\partial T_0(z),\ldots, \partial F/\partial T_N(z)\right)\in k^{N+1}$ (actually $d_x F$ depends on the choice of homogeneous coordinates for $x$; this abuse of notation should not lead to confusion). If $x\in X$, where $X\subset\PP^N$ is a projective variety, then $T_xX\subset \PP^N$ denotes the embedded Zariski tangent space to $X$ at $x$. If $X\subset {\bf P}^N$ is a projective variety, then its ideal sheaf will be denoted by $\idsheaf X\subset \O_{{\bf P}^N}$ and its homogeneous ideal by $I_X\subset k[T_0, \ldots, T_N]$. We will say that a hypersurface $Y=Z(F)$ {\em trivially contains $X$\/} iff $F=\sum G_iF_i$, where $G_i$'s and $F_i$'s are homogeneous polynomials, $F_i$ vanish on $X$ for all $i$, and $\deg F_i<\deg F$ for all $i$. If $Y$ trivially contains $X$, then $Y\supset X$. We will say that a hypersurface $W$ {\em non-trivially contains $X$\/} iff $W$ contains $X$, but not trivially. The following proposition is a slight reformulation of Landsberg's criterion (cf.\ \cite[Proposition 1.2]{Lan}): \begin{prop} For a projective variety $X\subset {\bf P}^N$, the following conditions are equivalent: \begin{itemize} \item[(i)] $X$ is a complete intersection. \item[(ii)] There exists a smooth point $x\in X$ having the following property: any hypersurface $W\subset {\bf P}^N$ that non-trivially contains $X$ must be smooth at $x$. \item[(iii)] For any smooth point $x\in X$ and any hypersurface $W$ that non-trivially contains $X$, $W$ is smooth at $x$. \item[(iv)] For any smooth point $x\in X$ and any hypersurface $W$ that non-trivially contains $X$, $T_xW$ cannot contain an intersection $\bigcap_i T_xW_i$, where each $W_i$ is a hypersurface s.t.\ $W_i \supset X$ and $\deg W_i<\deg W$ (it is understood that the intersection of an empty family of tangent spaces is the entire $\PP^N$). \end{itemize} \end{prop} \section{Proofs} For the sequel we need two lemmas. \begin{lemma}\label{subst} Let $F_1,\ldots,F_r$ be homogeneous polynomials over $k$ in $T_0,\ldots, T_N$. Assume that $x=(x_0:\ldots:x_N)\in \PP^N$ is their common zero and that the vectors $d_xF_1,\ldots, d_xF_r$ are linearly dependent. Then one of the following alternatives holds: \begin{enumerate} \item There is $j\in [1;r]$ s.t.\ $F_j$ belongs to the ideal in $k[T_0,\ldots,T_N]$ generated by $F_i$'s with $i\ne j$. \item There are homogeneous polynomials $\tilde F_0,\ldots, \tilde F_N$ s.t.\ the ideals $(F_0,\ldots,F_N)$ and $(\tilde F_0,\ldots, \tilde F_N)$ coincide, $\deg \tilde F_i=\deg F_i$ for all $i$, and $d_x \tilde F_j=0$ for some $j$. \end{enumerate} \end{lemma} {\bf Proof.} Let the shortest linear relation among $d_xF_j$'s have the form $$ \lambda_1d_xF_1+\cdots+\lambda_sd_xF_s=0, $$ where $\lambda_j\ne 0$ for all $j$. Reordering $F_j$'s if necessary, we may assume that $\deg F_1\le \deg F_2\le \cdots\le \deg F_s$. Let $t$ be such a number that $\deg F_t=\deg F_s$ and $\deg F_{t-1} <\deg F_s$ (if $\deg F_1=\deg F_s$, set $t=1$). If the polynomials $F_t,\ldots,F_s$ are linearly dependent, then it is clear that one of them lies in the ideal generated by the others and there is nothing more to prove. Assume from now on that $F_t, F_{t+1},\ldots, F_s$ are linearly independent. Then there exists an index $j\in[t;s]$ and numbers $\mu_i$, where $i \in [t;s]$ s.t.\ \begin{equation}\label{G:def} F_j=\sum_{i\in [t;s]\setminus \{j\}}\mu_i F_i+ \mu_j(\lambda_t F_t+\cdots+\lambda_s F_s). \end{equation} For each $i\in[1;t-1]$, choose a homogeneous polynomial $G_i$ s.t.\ $\deg G_i= \deg F_s-\deg F_i$ and $G_i(x_0,\ldots, x_N)=\lambda_i$, and set \begin{equation}\label{tilde:def} \tilde F_j=\sum_{i<t}G_iF_i+\sum_{i\ge t}\lambda_i F_i. \end{equation} If $\tilde F_j=0$, then $F_s\in (F_1,\ldots, F_{s-1})$ and the first alternative holds. Otherwise, $\deg \tilde F_j=\deg F_j$, $d_x\tilde F_j=0$ by virtue of (\ref{tilde:def}), and it follows from (\ref{G:def}) and (\ref{tilde:def}) that $$ F_j=\sum_{i\in [t;s]\setminus \{j\}}\mu_i F_i +\mu_j \tilde F_j -\mu_j\sum_{i<t}G_i F_i, $$ whence $(F_1,\ldots,F_{j-1}, \tilde F_j, F_{j+1},\ldots, F_s)=(F_1,\ldots,F_s)$. Hence in this case the second alternative holds, and we are done. The second lemma belongs to folklore. To state this lemma, let us introduce some notation. Denote by $\S$ the set of sequences of non-negative integers $\delta=(\delta_1,\delta_2,\ldots)$ s.t.\ $\delta_M=0$ for all $M\gg 0$. If $\delta,\eta\in \S$, we will write~$\delta \succ \eta$ iff there is an integer $i$ s.t.\ $\delta_i >\eta_i$ and $\delta_j=\eta_j$ for all $j>i$. \begin{lemma}\label{folk} Any sequence $\delta_1 \succ \delta_2 \succ\cdots$ must terminate. \end{lemma} \noindent {\bf Proof.} For any $\delta\in \S$, set $n(\delta)=\max\{j:\delta_j\ne 0\}$, $\ell(\delta)=\delta_{n(\delta)}> 0$. If $\delta\succ \eta$ and $n(\delta)=n(\eta)$, then $\ell(\delta)\ge \ell(\eta)$. Let us prove the lemma by induction on $n(\delta_1)$. If $n(\delta_1)\le 1$, the result is evident. Assuming that the lemma is true whenever $n(\delta_1)< m$, suppose that there is an infinite sequence $\delta_1 \succ \delta_2 \succ\cdots$ with $n(\delta_1)=m$. If $n(\delta_j)<n(\delta_1)$ for some $j$, we arrive at a contradiction by the induction hypothesis. Hence, $n(\delta_j) =n(\delta_1) =m$ for all $j$ and $\ell(\delta_1)\ge \ell(\delta_2)\ge\cdots >0$. Thus there exists an integer $N$ s.t\ $\ell(\delta_j)$ is connstant for $j\ge N$. For any $j\ge N$, denote by $\delta'_j\in \S$ a sequence that is obtained from $\delta_j$ by replacing its last positive term by zero. It is clear that $\delta'_N \succ \delta'_{N+1} \succ\cdots$, and this sequence is infinite by our assumption. This is again impossible by the induction hypothesis since $n(\delta'_j)< n(\delta_j)=m$, whence the lemma. \smallskip \par\addvspace{\smallskipamount of $(ii)\Rightarrow (i)$. Put $a=N-\dim X$. Let $(F_1,\ldots,F_r)$ be a system of (homogeneous) generators of $I_X$. To any such system assign a sequence $\delta(F_1,\ldots, F_r) \in \S$, where $\delta(F_1, \ldots,F_r)_i = \#\{j\in [1;r]:\deg F_j=i\}$. I claim that \begin{quote} if $r>a$, then $I_X=(\Phi_1,\ldots,\Phi_s)$, where $\Phi_i$'s are homogeneous polynomials s.t.\ $\delta(F_1,\ldots,F_r)\succ \delta(\Phi_1,\ldots, \Phi_s)$. \end{quote} To prove this claim, observe that $d_xF_1,\ldots,d_xF_r$ are linearly dependent since $X$ is smooth at $x$ and $r>\mathop{\rm codim}\nolimits X$. Now Lemma~\ref{subst} implies that either one of the $F_j$'s (say, $F_1$) can be removed without affecting $I_X$, or $I_X=(\tilde F_1,\ldots,\tilde F_r)$, where $\deg \tilde F_j=\deg F_j$ for all $j$ and $d_x \tilde F_j=0$ for some $j$. In the first case, the required $\Phi_1,\ldots,\Phi_s$ can be obtained by merely removing $F_1$; in the second case, hypothesis~$(ii)$ shows that $\tilde F_j=\sum_{i=1}^t G_i\Psi_i$, where $\Psi_i\in I_X$ and $\deg\Psi_i < \deg \tilde F_j$ for all $j$. Replacing $\tilde F_j$ by $\Psi_1,\ldots, \Psi_t$ in the sequence $\tilde F_1,\ldots,\tilde F_r$ and putting $s=r+t-1$, we obtain a sequence $\Phi_1,\ldots,\Phi_s$ s.t.\ $I_X =(\Phi_1,\ldots, \Phi_s)$ and $\delta(\tilde F_1,\ldots,\tilde F_r)\succ \delta(\Phi_1,\ldots, \Phi_s)$. Since the degrees of $\tilde F_j$'s and $F_j$'s are the same, this means that $\delta(F_1, \ldots,F_r)\succ \delta(\Phi_1,\ldots, \Phi_s)$ as well, and the claim is proved. Now we can finish the proof as follows. If $r=a$, then $X$ is the complete intersection of $Z(F_1),\ldots,Z(F_r)$ and there is nothing to prove. If $r>a$, then by virtue of our claim we can replace the system of generators $F_1,\ldots,F_r$ by $\Phi_1,\ldots, \Phi_s$. Let us iterate this process. By virtue of Lemma~\ref{folk} this process must terminate and by virtue of our claim this is possible only when we have found a system of exactly $a$ generators of the ideal $I_X$. This means that $X$ is a complete intersection, thus completing our proof. \smallskip \par\addvspace{\smallskipamount of $(iv)\Rightarrow (iii)\Rightarrow (ii)$. Trivial. \par\addvspace{\smallskipamount of $(i)\Rightarrow (iv)$. Let $X$ be a complete intersection of the hypersurfaces $Z(F_1), \ldots, Z(F_a)$. Assume that a hypersurface $W=Z(F)$, with $F$ irreducible, non-trivially contains $X$ and that $x=(x_0:\ldots :x_N)\in \PP^N$ is a smooth point of $X$; set $m=\deg F$. Since $Z(F)\supset X$ and $X$ is a complete intersection of the $Z(F_i)$'s, we see that \begin{equation}\label{expr} F=\sum G_iF_i; \end{equation} since $W$ contains $X$ non-trivially, at least some of the $G_j$'s must be non-zero constants. Reordering $F_j$'s if necessary, we may assume that $G_j$ is a constant (hence, $\deg F_j=m$) iff $1\le j\le s$. Taking $d_x$ of the both parts of (\ref{expr}), we see that \begin{equation}\label{diffls} d_xF=\sum_{i=1}^a c_i d_x F_i,\qquad \mbox{where $c_i\ne 0$ for some $i\in [1;s]$.} \end{equation} On the other hand, assume that $W_i=Z(B_i)$ with irreducible $B_i$'s. Then the hypothesis implies that $d_x F$ is a linear combination of $d_xB_j$'s, and the fact that $X$ is a complete intersection of $Z(F_t)$'s and $Z(B_j)\supset X$ implies that, for each $j$, there is a relation \begin{equation}\label{expr'} B_j=\sum_{t>s} G_{jt}F_t \end{equation} (it suffices to sum only over $t>s$ since for $t\le s$ we have $\deg F_t= \deg W > \deg B_j$). If we take $d_x$ of both parts of (\ref{expr'}), we see that, for each $j$, $d_xB_j$ is a linear combination of $d_x F_t$'s with $t>s$. Hence $d_xF$ is also a linear combination of $d_x F_t$'s with $t>s$. Taking into account (\ref{diffls}) we see that $d_xF_i$'s are linearly dependent. This is, however, impossible since $x$ is a smooth point of the comlete intersection of $Z(F_j)$'s. This contradiction completes the proof.

9408

alg-geom/9408008

en

https://arxiv.org/abs/alg-geom/9408008

alg-geom/9408008

Robert W. Berger

Robert W. Berger

Various Notions of Associated Prime Ideals

27 pages, AMS-LaTeX 1.1

Three notions of associated prime ideals, which are equivalent in the noetherian case but differ in the non notherian case, are discussed. Examples illustrate the scope of the notions.

alg-geom

\section*{Introduction} In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal $\frak p$ for a module $M$ is that $\frak p$ is the annihilator of an element of $M$. In \cite{Bourbaki-Alg-Comm-4} \S 1 exercise 17 a generalization of this notion is given. $\frak p$ is called weakly associated (faiblement associ\'e) to $M$ if $\frak p$ is minimal in the set of the prime ideals containing the annihilator of an element of $M$ (see Definition \ref{def-Ass-essential-first-kind}). In this paper a further generalization of this notion will be given (Definition \ref{def-assprim}). We use ideas of Krull \cite{Krull-Ringe-ohne-Endlichkeit}.\\ As long as the modules are noetherian all these definitions are equivalent. But for non noetherian $R$-modules this is no longer true, even if the ring $R$ is noetherian.\\ In this paper we give a selfcontained introduction to the various concepts and discuss their relation with the support and the radical of a module. Then we illustrate by examples the scope of the notions. For a comprehensive introduction the theory we refer to the now classic lecture notes \cite{Serre-Alg-loc} of Serre and to \cite{Bourbaki-Alg-Comm-4}. Another extensive exposition of the general theory with many examples was given by Stefan Mittelbach in \cite{Mittelbach-Dipl}. \\ Throughout this paper ``ring'' always denotes a commutative ring with unit element denoted by $1$. If $M$ is an $R$-module we always assume that $1\cdot x=x$ for all $x\in M$.\\ In the first section we recall some basic definitions and facts from ``additive ideal theory''. \section{Primary Decomposition} \subsection{Primary and Coprimary Modules. Primary Decomposition.} Let $R$ be a ring, $M$ an $R$-module. \begin{defn} A submodule $F$ of $M$ is called ``indecomposable in $M$'' iff from $F=F_1\cap F_2$,\quad $F_1,F_2$ submodules of $M$, it follows that $F_1=F$ or $F_2=F$. \end{defn} \begin{rem} \label{zero-indcomp} Obviously $F$ is indecomposable in $M$ iff $(0)$ is indecomposable in $M/F$.\\ It is well known that in a noetherian $R$-Module every submodule is can be written as an intersection of finitely many indecomposable submodules. \end{rem} \begin{defn} An element $\xi\in R$ is called a ``zero divisor for M'', iff there exists an element $0\ne x\in M$ with $\xi\cdot x=0$.\\ An element $\xi\in R$ is called ``nilpotent for $M$'', iff for every $y\in M$ there exists a natural number $n=n(y)$ with $\xi^n\cdot y=0$. (In \cite{Bourbaki-Alg-Comm-4} these elements are called {\em presque nilpotent}.) \end{defn} \begin{rem} \label{rem:zerodivisorsmultclosed}\strut \begin{enumerate} \item The set of all non zero divisors for $M$ is a multiplicatively closed subset of $R$. \item The set of all nilpotent elements for $M$ is an ideal of $R$. \item If $M\ne(0)$ every nilpotent element for $M$ is also a zero divisor for $M$. \end{enumerate} \end{rem} \begin{defn}(\cite{Serre-Alg-loc}) $M$ is called ``coprimary'', iff $M\ne(0)$ and every zero divisor for $M$ is nilpotent for $M$. \end{defn} \begin{rem} \label{submodule-coprimary-is coprimary} From Remark \ref{rem:zerodivisorsmultclosed} it follows that the zero divisors for a coprimary module $M$ form an ideal $\ne R$, whose complement is multiplicatively closed, i.e. a {\em prime ideal} $\frak p$.\\ We say that ``$M$ is $\frak p$-coprimary''.\\ Obviously every non zero submodule of a $\frak p$-coprimary module is again $\frak p$-coprimary. \end{rem} \begin{ex} \label{R-mod-p-is-coprimary} Let $V$ be a cyclic $R$-module whose annihilator $\operatorname{Ann}_R(V)$ is a prime ideal $\frak p$. Then $V$ is $\frak p$-coprimary. \end{ex} \begin{pf} By definition we have $V=R\cdot x\cong R/\frak p$. So each element of $\frak p$ is nilpotent for $V$. On the other hand every zero divisor for $R/\frak p$ lies in $\frak p$, since $\frak p$ is a prime ideal. \end{pf} \begin{prop}[Noether] \label{indcomp-coprim} Let $M\ne(0)$ be a noetherian $R$-module, $(0)$ indecomposable in $M$. Then $M$ is coprimary. \end{prop} \begin{pf} (indirect) If $M$ was not coprimary, there would be a non nilpotent zero divisor $\rho\in R$. Let $M_i:=\{x\mid x\in M,\quad \rho^i\cdot x=0\}$. $M_i$ is a submodule of $M$ and $M_i\subseteq M_{i+1}$. Since $M$ is noetherian there exists an $n\in\Bbb N$ with $M_n=M_{n+1}$. Further we have $M_1\ne(0)$ because $\rho$ is a zero divisor for $M$ and $\rho^n M\ne(0)$ since $\rho$ is not nilpotent for $M$. But $M_1\cap\rho^n M\stackrel{!}{=}(0)$ (and so $(0)$ would be decomposable in $M$).\\ Proof: If $x\in M_1\cap\rho^n M$ then $\rho x=0$ and $x=\rho^n y$ with an $y\in M$. $\Rightarrow \rho^{n+1}y=0$ $\Rightarrow y\in M_{n+1}$. But $M_{n+1}=M_n$ and therefore $x=\rho^n y=0$. \end{pf} If $M$ is non noetherian then Proposition \ref{indcomp-coprim} does not hold. See Example \ref{0-indcomp-not-coprim}. \begin{defn} A submodule $N$ of $M$ is called ``$\frak p$-primary (or primary for $\frak p$) in $M$'', iff $M/N$ is $\frak p$-coprimary. \end{defn} Immediately from the definitions follows: \begin{rem}\strut \begin{enumerate} \item $M$ is coprimary $\Longleftrightarrow$ $(0)$ is $\frak p$-primary in $M$. \item Let $F$ and $N$ be submodules of $M$ with $M\supseteq F\supseteq N$. Then:\\ $F$ is $\frak p$-primary in $M$ $\Longleftrightarrow$ $F/N$ is $\frak p$-primary in $M/N$. \item If $M=R$ and $N=\frak q$ an ideal in $R$ then $\frak q$ is primary in $R$ iff $\frak q$ is what is called a primary ideal. \end{enumerate} \end{rem} With these notions Proposition \ref{indcomp-coprim} can be reformulated as: \begin{cor} \label{indecomp-primary} Let $N$ be a proper submodule of $M$ which is indecomposable in $M$ and $M/N$ noetherian. Then $N$ is primary in $M$. \end{cor} The converse of Corollary \ref{indecomp-primary} is not true but: \begin{prop} \label{intersection-for-same-primideal} Let $N_1,\dots,N_r$ be $\frak p$-primary submodules of $M$.\\ Then $\bigcap\limits_{i=1}^r N_i$ is also $\frak p$-primary in $M$. \end{prop} \begin{pf} 1) Every element of $\frak p$ is nilpotent for $N:=\bigcap\limits_{i=1}^r N_i$:\\ If $\rho\in\frak p$ and $x\in M$ then for for every $i$ there is an $n_i\in\Bbb N$ with $\rho^{n_i}\cdot x\in N_i$. Then with $n:=\max\{n_1,\dots,n_r\}$ we have $\rho^n\cdot x\in N$.\\ 2) Every zero divisor for $M/N$ is an element of $\frak p$:\\ Let $\rho\in R$ be a zero divisor for $M/N$. Then there exists an $x\in M\setminus N$ with $\rho\cdot x\in N$. Then $x\notin N_{i_0}$ for an $i_0$ and $\rho\cdot x\in N_{i_0}$. Then $\rho$ is a zero divisor for $M/N_{i_0}$ and therefore $\rho\in\frak p$. \end{pf} \begin{defn} \label{primary-decomp} Let $N$ be a submodule of $M$. A decomposition $$ N=\bigcap\limits_{i=1}^r F_i,\qquad \text{where the } F_i \text{ are } \frak p_i \text{-primary in }M $$ is called a ``primary decomposition (or representation) of $N$ in $M$''.\\ The $\frak p_i$ are called the ``prime ideals belonging to the primary decomposition''\\ A primary decomposition of $N$ in $M$ is called ``reduced'' (or ``irredundant'') or a ``normal decomposition (or representation) of $N$ in $M$'' if the following two conditions hold: \begin{enumerate} \item \label{unequal-primes} $i\ne k\Longrightarrow\frak p_i\ne\frak p_k$. \item \label{superfluous} No $F_i$ contains the intersection of the others \end{enumerate} \end{defn} \begin{rem} \label{normal-repr-exists} From a primary decomposition one can obtain a normal representation:\\ Group together the $F_i$ which are primary for the same prime ideal $\frak p$ and take their intersection. By Proposition \ref{intersection-for-same-primideal} this is again a $\frak p$-primary submodule of $M$. This take care of condition (\ref{unequal-primes}) of Definition \ref{primary-decomp}. Then omit a primary submodule that contains the intersection of the others. Proceed until also condition (\ref{superfluous}) of Definition \ref{primary-decomp} is satisfied. \end{rem} \begin{ex} \label{coprimary-normal} In a $\frak p$-coprimary $R$-module $M$ the submodule $(0)$ has the normal representation $(0)=(0)$, since $(0)$ is a $\frak p$-primary submodule of $M$. \end{ex} From Remark \ref{zero-indcomp}, Corollary \ref{indecomp-primary} together with Remark \ref{normal-repr-exists} we obtain the well known \begin{prop} If $M$ is noetherian then every proper submodul $N$ of $M$ has a normal representation in $M$. \end{prop} \subsection{Quotient Modules and \boldmath $S$-Components.} Let $M$ be an $R$-module, $S$ a multiplicatively closed subset of $R$ with $1\in S$, $$ \psi:M\longrightarrow M_S $$ the canonical homomorphism of M into the quotient module of $M$ with respect to $S$. \begin{notation} For an $R_S$-submodule $U$ of $M_S$ we denote by $U\cap M$ the full inverse image of $U$ under $\psi$: $$ U\cap M:=\psi^{-1}(U). $$ $U\cap M$ obviously is an $R$-submodule of $M$.\\[1ex] For an $R$-submodule $N$ of $M$ we denote by $R_S\cdot N$ the $R_S$-submodule of $M_S$ generated by $\psi(N)$: $$ R_S\cdot N:=R_S\cdot\psi(N)=\left\{\frac{x}{s}\mid x\in N, s\in S\right\} \subseteq M_S $$ For a subset $A\subseteq M$ we define $A_S:=\{\frac{x}{s}\mid x\in A, s\in S\}\subseteq M_S$. \end{notation} \begin{rem} Since forming the quotient module is an exact functor we can identify the submodule $R_S\cdot N$ of $M_S$ with the quotient module $N_S$. Also the notation $A_S$ for a set $A$ is compatible with the notation $A_S$ if $A$ is an $R$-module. \end{rem} \begin{prop} \label{up-and down} \strut \begin{enumerate} \item For each $R_S$-submodule $U$ of $M_S$ we have $$ R_S\cdot(U\cap M)= U. $$ \item For each $R$-submodule $N$ of $M$ we have $$ (R_S\cdot N)\cap M=\{x\mid x\in M, \text{ exists } s\in S \text{ with } s\cdot x\in N \} \supseteq N $$ \end{enumerate} \end{prop} \begin{pf} (1) Clearly $R_S\cdot(U\cap M)\subseteq U$. Now let $\frac{x}{s}\in U$ with $s\in S$. Then $\psi(x)=\frac{x}{1}\in U$ and therefore $\frac{x}{t}=\frac{1}{t}\cdot\psi(x)\in U$. \halign{&$#$\hfil\cr (2)\quad&(N_S)\cap M&=\psi^{-1}(R_S\cdot\psi(N))\cr &&=\{x\mid x\in M,\ \frac{x}{1}\in N_S\}\cr &&=\{x\mid x\in M,\text{ ex. }s'\in S,\ y\in N \text{ with } \frac{x}{1}=\frac{y}{s'} \text{ in }M_S \}\cr &&=\{x\mid x\in M,\text{ ex. }s',t\in S,\ y\in N\text{ with } \underbrace{ts'}_{=s\in S}\cdot x=t\cdot y\text{ in }M\}\cr &&=\{x\mid x\in M,\text{ ex. }s\in S\text{ with }s\cdot x\in N\} \qquad\qquad\qed\cr } \renewcommand{\qed}{} \end{pf} The operation of extending a submodule of $M$ to $M_S$ and then restricting it back to $M$ plays an important role. Therefore an extra name is introduced: \begin{defn} For any $R$-submodule $N$ of $M$ and any multiplicatively closed subset $S$ of $R$ we define $$ S^M(N):=(R_S\cdot N)\cap M $$ the ``$S$-component of $N$ in $M$''.\ If no confusion can arise we will also write $S(N)$ instead of $S^M(N)$. \end{defn} The basic properties of these operations are summarized in the following Proposition, the proof of which is immediate: \begin{prop} \label{basics-for-S-component} Let $S,T$ be multiplicatively closed subsets of $R$; $N, \widetilde N, N_i$ submodules of $M$. Then: \begin{enumerate} \item \label{RSN=RN} $R_S\cdot S(N)=R_S\cdot N$ \item $S(N)\supseteq N$ \item \label{T-bigger-S} $T\supseteq S\Longrightarrow S(T(N))=T(N)=T(S(N))$. Especially $S(S(N))=S(N)$ \item $N\subseteq \widetilde N\Longrightarrow S(N)\subseteq S(\widetilde N)$ \item \label{RS-ext-intersect} $R_S\cdot(N\cap \widetilde N)=R_S\cdot N\cap R_S\cdot \widetilde N$ \item \label{S-comp-intersect} $S(N\cap \widetilde N)=S(N)\cap S(\widetilde N)$ \item \label{RS-ext-union} $\left(\bigcup\limits_i N_i\right)_S=\bigcup\limits_i (N_i)_S$ for arbitrary unions.\\ (But in general these are only sets, not modules\,!) \item \label{RS-ext-sum} $R_S\cdot\sum\limits_{i} N_i=\sum\limits_{i} R_S\cdot N_i$ for arbitrary sums. \item $S(N+\widetilde N)\supseteq S(N)+S(\widetilde N)$,\\ but equality does not hold in general (Example \ref{S-comp-non-add}). \item \label{S-comp-factor} $S^{M/N}(0)=S^M(N)/N$ \item If $\frak a$ is an ideal of $R$ then $R_S\cdot(\frak a\cdot N)=(R_S\cdot\frak a)\cdot(R_S\cdot N)$ \item \label{S-comp-M-M'} Let $R':=R/Ann_R(M)$ and $\phi:R\rightarrow R'$ the canonical homomorphism. Then $M$ and $N$ have natural structures as $R'$-modules. We denote these by $M'$ and $N'$. Let $S$ be a multiplicatively closed subset of $R$. Then $S':=\phi(S)$ is a multiplicatively closed subset of $R'$ and\\ ${S'}^{M'}(N')=S^M(N)$ as $R$-modules and as $R'$-modules. \end{enumerate} \end{prop} \begin{ex}[$S(N+\widetilde N)\nsubseteq S(N)+S(\widetilde N)$] \label{S-comp-non-add} \strut\\ $R:=M:=k[X,Y]$ polynomial ring in $X,Y$ over a field $k$,\\ $N:=R\cdot X$,\quad$\widetilde N:=R\cdot Y$\\ $S:=\{(X+Y)^\nu\mid \nu=0,1,2,\dots\}$.\\ Then $S(N)=N$, $S(\widetilde N)=\widetilde N$, but $S(N+\widetilde N)=R\nsubseteq R\cdot X+R\cdot Y$. \end{ex} An immediate consequence of Proposition \ref{basics-for-S-component} is the following: \begin{prop} The map $$ M_S\supseteq U\mapsto U\cap M\subseteq M $$ is an order preserving isomorphism between the lattice of the $R_S$-submodules of $M_S$ and the lattice of those $R$-submodules of $M$ which are $S$-components.\\ (The order is defined by ``$\,\supseteq\,$'' and the lattice operations are ``$\,\cap\,$'',\ ``$\,+\,$'' in $M_S$ and $``\,\cap\,$'', \ ``$\,S(\quad+\quad)\,$'' in $M$.) \end{prop} \begin{cor} If $M$ is a noetherian $R$-module then $M_S$ is a noetherian $R_S$-module. \end{cor} There is a close connection between the primary submodules of $M$ and $M_S$: \begin{prop} \label{primary-up-down} There is a one-to-one correspondence $$ \begin{array}{rcrclcl} M&\supset& N&\rightmapsto&R_S\cdot N&\subset&M_S\\ M&\supset& U\cap M&\leftmapsto&U&\subset&M_S\\ \end{array} $$ between the primary submodules $U$ of $M_S$ and those primary submodules $N$ of $M$ whose prime ideals $\frak p$ don't intersect with $S$, i.e. $\frak p\cap S=\emptyset$:\\ More precisely: \begin{enumerate} \item If $U$ is a $\frak P$-primary submodule of $M_S$ then $U\cap M$ is a $\frak p:=\frak P\cap R$-primary submodule of $M$, $\frak p\cap S=\emptyset$ and $R_S\cdot (U\cap M)=U$. \item \label{up-P-primary} If $N$ is a $\frak p$-primary submodule of $M$ and $\frak p\cap S=\emptyset$ then $R_S\cdot N$ is a\\ $\frak P:=R_S\cdot\frak p$-primary submodule of $M_S$ and $R_S\cdot N\cap M=N$. \item \label{S-comp-of-primary} If $N$ is a $\frak p$-primary submodule of $M$ then: $$ S^M(N)=\left\{ \begin{array}{ll} N&\text{for }\frak p\cap S=\emptyset\\ M&\text{for }\frak p\cap S\ne\emptyset \end{array} \right. $$ Consequently $$ R_S\cdot N=M_S\quad\text{for }\frak p\cap S\ne\emptyset. $$ \end{enumerate} \end{prop} \begin{pf} (1) Let $U\subset M_S$ be $\frak P$-primary in $M_S$,\quad $N:=U\cap M$ and $r\in R$ an arbitrary zero divisor for $M/N$. Then there is a $x\in M\setminus N$ with $r\cdot x\in N$. It follows that $\frac{r}{1}\cdot\frac{x}{1}\in R_S\cdot N=U$ (see Proposition \ref{up-and down}). But $\frac{x}{1}\notin U$, because else $x\in U\cap M=N$. Therefore $\frac{r}{1}$ is a zero divisor for $M_S/U$, then $\frac{r}{1}\in\frak P$ and so $r\in\frak P\cap R=\frak p$. Further every element $r\in\frak p$ is nilpotent for $M$: Let $x\in M$ arbitrary. Then there is a $n\in\Bbb N$ with $\left(\frac{r}{1}\right)^n\cdot\frac{x}{1}\in R_S\cdot N=U$ because $\frac{r}{1}\in\frak P$ is nilpotent for $M_S/U$, and so $r^n\cdot x\in U\cap M=N$: \quad $U\cap M$ is $\frak P\cap R$-primary. \\ (2) and (3): Let $N$ be $\frak p$-primary in $M$.\\ $1^{st}$ case: $\frak p\cap S\ne\emptyset$. Then there is an $s\in\frak p\cap S$. $s$ is nilpotent for $M/N$, i.e. for each $x\in M$ there is an $n\in\Bbb N$ with $s^n\cdot x\in N$ and $s^n\in S$ since $S$ is multiplicatively closed. Therefore by definition $x\in S^M(N)$ and so $S^M(N)=M$.\\ $2^{nd}$ case: $\frak p\cap S=\emptyset$. Then $\frak P:=R_S\cdot\frak p$ is a prime ideal of $R_S$. We claim that $R_S\cdot N$ is $\frak P$-primary in $M_S$ and $S^M(N)=N$.\\ Proof: First we show $S(N)=N$.\quad $S(N)\supseteq N$ is always true (Proposition \ref{basics-for-S-component}). To show the other inclusion let $x\in S(N)$ be arbitrary. Then there is an $s\in S$ with $s\cdot x\in N$. If $x\notin N$ then $s$ would be a zero divisor for $M/N$ and therefore $s\in \frak p\cap S=\emptyset$~! So we have that $S(N)=N$.\\ Now we show that $M_S/(R_S\cdot N)$ is $\frak P$-coprimary:\\ Let $\frac{r}{s}$ be an arbitrary zero divisor for $M_S/N_S$. There is an $\frac{x}{t}\in M_S\setminus N_S$ with $\frac{r}{s}\cdot\frac{x}{t}\in N_S$. $\Rightarrow\frac{r\cdot x}{1}\in N_S\Rightarrow r\cdot x\in N_S\cap M =S(N)=N$. But $x\notin N$, because else $\frac{x}{t}\in N_S$. So $r$ is a zero divisor for $M/N$ $\Rightarrow r\in\frak p$ $\Rightarrow \frac{r}{s}\in R_S\cdot\frak p=\frak P$.\\ Further each element of $\frak P$ is nilpotent for $M_S/N_S$ because each element of $\frak p$ is nilpotent for $M/N$.\\ Finally: If $\frak p\cap S\ne\emptyset$ we have $N_S=R_S\cdot N=R_S\cdot S(N)=R_S\cdot M=M_S$. \end{pf} \begin{prop} \label{prim-decomp-S-comp} Let $N$ be a proper submodule of $M$, $S$ a multiplicatively closed subset of $R$, and $$ N=\bigcap_{i\in I} N_i $$ a normal (resp. primary) decomposition of $N$ in $M$, where the $N_i$ are $\frak p_i$-primary in $M$.\\ Let $$ I':=\{i\mid i\in I,\quad \frak p_i\cap S=\emptyset\} $$ Then $$ S(N)=\bigcap_{i\in I'}N_i $$ is a normal (resp. primary) decomposition of $S(N)$ in $M$ and $$ N_S=\bigcap_{i\in I'}(N_i)_S $$ is a normal (resp. primary) decomposition of $N_S$ in $M_S$. \end{prop} \begin{pf} \baselineskip=1.3\baselineskip From Proposition \ref{primary-up-down} (\ref{S-comp-of-primary}) we know that $$ S(N_i)=\left\{ \begin{array}{ll} N_i&\text{ for } i\in I'\\ M &\text{ for } i\in I\setminus I' \end{array} \right. $$ It then follows from Proposition \ref{basics-for-S-component}, (\ref{S-comp-intersect}) that $S(N)=\bigcap\limits_{i\in I'}S(N_i)\cap\bigcap\limits_{k\in I\setminus I'} S(N_k)$\\ $=\bigcap\limits_{i\in I'}N_i$ is a primary decomposition of $S(N)$ in $M$.\\ Clearly, if $\bigcap\limits_{i\in I}N_i$ is irredundant then $\bigcap\limits_{i\in I'}N_i$ is irredundant too, so from a normal decomposition of $N$ in $M$ one obtains a normal representation of $S(N)$ in $M$.\\[1ex] Further from Proposition \ref{basics-for-S-component} (\ref{RS-ext-intersect}) and Proposition \ref{primary-up-down} (\ref{up-P-primary}) and (\ref{S-comp-of-primary}) we have\\ $ N_S=\bigcap\limits_{i\in I'}(N_i)_S\cap \bigcap\limits_{k\in I\setminus I'}(N_k)_S= \bigcap\limits_{i\in I'}(N_i)_S$, where the $(N_i)_S$ are $\frak P_i:=R_S\cdot\frak p_i$-primary in $M_S$. \\ If $\frak p_i\ne \frak p_k$ then $\frak P_i\ne\frak P_k$, and if one of the $(N_i)_S$ could be omitted in the representation $\bigcap\limits_{i\in I'}(N_i)_S$ then the $N_i$ could be omitted in the representation $S(N)=\bigcap\limits_{i\in I'}N_i$.\\ So, if $N=\bigcap\limits_{i\in I}N_i$ was a normal representation then so is $N_S=\bigcap\limits_{i\in I'} (N_i)_S$.\qed \renewcommand{\qed}{} \end{pf} \subsection{Uniqueness Theorems} \begin{prop} Let $N=\bigcap\limits_{i\in I}N_i$ be a normal representation of $N$ in $M$,\ $N_i$ primary for $\frak p_i$ in $M$.\\ The $N_i$, whose prime ideals $\frak p_i$ are minimal in the set of all $\frak p_i,\ i\in I$, are uniquely determined by $N$ and $M$ (i.e. they belong to any normal representation of $N$ in $M$). \end{prop} \begin{pf} Let $I=\{1,\dots,r\}$ ,\ $\frak p_1$ minimal among the $\{\frak p_1,\dots,\frak p_r\}$, and $S:=\complement\frak p_1:=R\setminus\frak p_1$. Then for all $i\ne 1$ we have $\frak p_i\cap S\ne\emptyset$, because else $\frak p_i\subseteq\frak p_1$ and therefore $\frak p_i=\frak p_1$ because of the minimality of $\frak p_1$, but by definition of a normal representation $\frak p_i\ne\frak p_k$ for $i\ne k$. Proposition \ref{prim-decomp-S-comp} yields: $S(N)=N_1$.\\ Let $N=\bigcap\limits_{j\in J}F_k$ be a second normal decomposition of $N$ in $M$,\ $F_j$ primary for $\frak q_j$. Again by Proposition \ref{prim-decomp-S-comp} we get $S(N)=\bigcap\limits_{j\in J'}F_j$ with\\ $J'=\{j\mid j\in J,\ \frak q_j\cap S=\emptyset\} =\{j\mid j\in J,\frak q_j\subseteq\frak p_1\}$.\\ $N_1$ is $\frak p_1$-primary in $M$, therefore for each $x\in M$ and each $p\in \frak p_1$ there is a $\nu=\nu(p,x)\in\Bbb N$ with $p^\nu\cdot x\in N_1\subseteq F_j$ for all $j\in J'$. Then $p\in\frak q_j$, since $F_j$ is $\frak q_j$-primary in $M$, and so finally $\frak p_1\subseteq \frak q_j$ for all $j\in J'$.\\ On the other hand by definition of $J'$ we have $\frak q_j\subseteq\frak p_1$, and so $\frak q_j=\frak p_1$ for all $j\in J'$. Therefore $J=\{j_0 \}$ contains exactly one element $j_0$,\ ($\frak q_{j_0}=\frak p_1$) and therefore $F_{j_0}=S(N)=N_1$, i.e. $N_1$ belongs also to the second normal decomposition of $N$ in $M$. \end{pf} For the next Proposition we need the following facts about prime ideals: \begin{prop} \label{union-of-prime-ideals} {\shape{n}\selectfont(\cite{Serre-Alg-loc}, Chap I, prop.2)} Let $\frak a$ be an ideal and $\frak p_1,\dots\frak p_r$ finitely many prime ideals in $R$. Then $$ \frak a\subseteq\bigcup_{i=1}^r\frak p_i \Longleftrightarrow \text{There exists an } i_0\in\{1,\dots r\} \text{ with }\frak a\subseteq\frak p_{i_0} $$ \end{prop} \begin{cor} \label{union-finitely-many-primes-ideal} If $\frak p_1,\dots,\frak p_r$ are finitely many prime ideals of $R$ such that $\frak a:=\bigcup\limits_{i=1}^r\frak p_i$ is an ideal of $R$ then there exists an $i_0\in\{1,\dots,r\}$ with $\frak a=\frak p_{i_0}$. \end{cor} \begin{pf} By Proposition \ref{union-of-prime-ideals} there is an $i_0\in\{1,\dots,r\}$ with $\frak p_{i_0}\subseteq\bigcup\limits_{i=1}^r\frak p_i =\frak a\subseteq\frak p_{i_0}$ and so $\frak a=\frak p_{i_0}$. \end{pf} Another immediate consequence of Proposition \ref{union-of-prime-ideals} is: \begin{rem} \label{a-in-a-notin-pi} Let $\frak a$ be an arbitrary ideal, $\frak p_1,\dots,\frak p_r$ prime ideals in $R$, and $\frak a\supsetneq \frak p_i$ for $i=1,\dots,r$.\\ Then there exists an $a\in\frak a$ with $a\notin\frak p_i$ for all $i=1,\dots,r$. \end{rem} \begin{pf} Otherwise $\frak a$ would be contained in the union of the $\frak p_i$ and therefore in one of the $\frak p_i$. \end{pf} \begin{prop} \label{complement-prime-ideal-not-in-primary-decomp} Let $\frak p$ be a prime ideal of $R$ and $N=\bigcap\limits_{i=1}^r N_i$ a primary decomposition of $N$ in $M$, $N_i$ primary for $\frak p_i$.\\ Assume that $\frak p\ne\frak p_i$ for all $i=1,\dots,r$.\\ Choose an $a\in\frak p$ such that $a\notin\frak p_i$ for all $\frak p_i$ with $\frak p_i\subseteq\frak p$\\ {\em (such an $a$ exists by Remark \ref{a-in-a-notin-pi})}\\ Let $S:=\complement\frak p$ and $T:=S\cdot\{a^\nu\mid \nu=0,1,2,\dots\}$.\\ Then $$ T\supsetneq S\text{\quad but\quad} T(N)=S(N). $$ \end{prop} \begin{pf} Since the $\frak p_i$ are prime ideals we have:\\ $\frak p_i\cap T=\emptyset\Longleftrightarrow \frak p_i\cap S=\emptyset \text{ and } a\notin \frak p_i \Longleftrightarrow \frak p_i\subseteq\frak p \text{ and } a\notin \frak p_i$.\\ By the choice of $a$ the condition $a\notin\frak p_i$ automatically holds for all $\frak p_i\subseteq\frak p$ and so we get:\\ $\frak p_i\cap T=\emptyset\Longleftrightarrow \frak p_i\cap S=\emptyset$.\\ Together with Proposition \ref{prim-decomp-S-comp} we obtain $T(N)=\bigcap\limits_{\frak p_i\cap T=\emptyset}N_i = \bigcap\limits_{\frak p_i\cap S=\emptyset}N_i =S(N)$. On the other hand $T\supsetneq S$ since $a\in T\setminus S$. \end{pf} The situation is quite different for a prime ideal that occurs in a normal decomposition: \begin{prop} \label{complement-prime-ideal-in-normal-decomp} Let $N=\bigcap\limits_{i=1}^r N_i$ be a normal representation of $N$ in $M$,\\ $N_i$ primary for $\frak p_i$, and let $\frak p\in\{\frak p_1,\dots,\frak p_r\}$,\quad $S:=\complement\frak p$. Then: \begin{gather*} \text{For any multiplicatively closed subset }T\supsetneq S \text{ of }R\\ T(N)\supsetneq S(N). \end{gather*} \end{prop} \begin{pf} Let $\frak p=\frak p_{i_0}$. Then by Proposition \ref{prim-decomp-S-comp} $S(N)=\bigcap\limits_{i\in I'} N_i$ with $I'=\{i\mid \frak p_i\subseteq\frak p \}\ni i_0$ is a normal decomposition of $S(N)$.\\ Now let $T\supsetneq S$ be an arbitrary multiplicatively closed set bigger than $S$. Then $T\cap\frak p_{i_0}\ne\emptyset$ and therefore $T(N)=\bigcap\limits_{i\in I''}N_i$ with $I''=\{i\mid\frak p_i\cap T=\emptyset\}\subsetneq I'$, since $i_0\in I'\setminus I''$. It follows that $T(N)\supsetneq S(N)$ because $S(N)=\bigcap\limits_{i\in I'}N_i$ is a normal representation and therefore $N_{i_0}$ cannot be omitted. \end{pf} From Propositions \ref{complement-prime-ideal-not-in-primary-decomp} and \ref{complement-prime-ideal-in-normal-decomp} one obtains: \begin{cor} \label{charact-prime-ideal-in normal-decomp} {\shape{n}\selectfont(Compare \cite{Krull-Ringe-ohne-Endlichkeit}, Satz 12.)} Let $\frak p$ be a prime ideal of $R$ and $N$ a proper submodule of $M$.\\ $\frak p$ belongs to every normal representation of $N$ in $M$ if and only if for any multiplicatively closed subset $T$ of $R$ with $T\supsetneq\complement\frak p$ one has $T(N)\ne\complement\frak p(N)$ (i.e. $T(N)\supsetneq\complement\frak p(N)$). \end{cor} Since Corollary \ref{charact-prime-ideal-in normal-decomp} gives a characterization of the prime ideals that belong to an arbitrary normal representation independently of that decomposition one obtains \begin{cor} The set of prime ideals belonging to a normal representation of $N$ in $M$ depends only on $N$ and $M$ and not on the representation. \end{cor} \section{Associated and Essential Prime Ideals} We would like to define the ``associated'' prime ideals of a module $M$ as those that belong to a normal representation of $(0)$ in $M$. But for non noetherian modules such a decomposition may not exist. Nevertheless we can use the characterization given in Corollary \ref{charact-prime-ideal-in normal-decomp} which makes sense also in the non noetherian case (compare \cite{Krull-Ringe-ohne-Endlichkeit}, Definition on page 742): \begin{defn} \label{def-assprim} Let $M$ be an arbitrary $R$-module. A prime ideal $\frak p$ of $R$ is called an ``associated prime ideal of $M$'' iff for any multiplicatively closed subset $T$ of $R$ with $T\supsetneq\complement\frak p$ $$ T^M((0))\supsetneq\complement\frak p^M((0))\ . $$ The set of all prime ideals associated to $M$ is denoted by $\operatorname{Ass}(M)$.\\ If $N$ is a proper submodule of $M$ then the associated prime ideals of $M/N$ are called the ``essential prime ideals for $N$ in $M$''. \end{defn} \begin{rem} \label{ess-and-mult-closed} $\frak p$ is essential for $N$ in $M$ iff for any multiplicatively closed subset $T$ of $R$ with $T\supsetneq\complement\frak p$ one has $T^M(N)\supsetneq\complement\frak p^M(N)$. \end{rem} \begin{pf} By Proposition \ref{basics-for-S-component} (\ref{S-comp-factor}) we have\\ $T^{M/N}((0))=T^M(N)/N$ and $\complement\frak p^{M/N}((0))=\complement\frak p^M(N)/N$.\\ Consequently $T^{M/N}((0))\supsetneq\complement\frak p^{M/N}((0)) \Longleftrightarrow T^M(N)\supsetneq\complement\frak p^M(N)$. \end{pf} Immediately from the definition together with Corollary \ref{charact-prime-ideal-in normal-decomp} follows: \begin{rem} \label{essential-primes-for-normal-decomp} If there exists a primary decomposition of $N$ in $M$ (e.g. if $M/N$ is noetherian) then $\frak p$ is essential for $N$ in $M$ iff $\frak p$ belongs to a normal representation of $N$ in $M$.\\ In this case there are only {\em finitely many} essential prime ideals for $N$ in $M$. \end{rem} Since in a $\frak p$-coprimary module $(0)=(0)$ is a normal representation of $(0)$ in $M$ one has: \begin{rem} \label{coprimary-Ass} If $M$ is $\frak p$-coprimary then $\operatorname{Ass}(M)=\{\frak p\}$\\ (The converse is also true: Corollary \ref{coprimary-iff-Assp}.) \end{rem} \begin{rem}\strut \label{AssM-and-M'} \begin{enumerate} \item \label{AssM-AnnM} Each $\frak p\in\operatorname{Ass}(M)$ contains $\operatorname{Ann}_R(M)$. \item \label{AssM-AssM'} Let $R':=R/\operatorname{Ann}_R(M)$ and $\phi:R\rightarrow R'$ the canonical homomorphism. Then $M$ can be regarded as an $R'$-module $M'$ in a natural way. There is a one-one correspondence between $\operatorname{Ass}(M)$ and $\operatorname{Ass}(M')$, given by $$ \operatorname{Ass}(M)\ni\frak p\longmapsto \phi(\frak p)\in\operatorname{Ass}(M') $$ \end{enumerate} \end{rem} \begin{pf} (\ref{AssM-AnnM}) If there was an $s\in\operatorname{Ann}_R(M)$ but $s\notin\frak p$ we would obtain $\complement\frak p((0))=M$ and therefore also $T((0))=M=\complement\frak p((0))$ for all $T\supsetneq\complement\frak p$, which means that $\frak p\notin\operatorname{Ass}(M)$.\\ (\ref{AssM-AssM'}) Since the prime ideals of $R'$ are in one-to-one correspondence under $\phi$ with those prime ideals of $R$ that contains $\operatorname{Ann}_R(M)$, and by (\ref{AssM-AnnM}) we know that the elements of $\operatorname{Ass}(M)$ contain $\operatorname{Ann}_R(M)$, we only need to show that $\frak p\in \operatorname{Ass}(M)$ iff $\frak p':=\phi(\frak p)\in\operatorname{Ass}(M')$. Obviously $\phi\left(\complement\frak p\right)=\complement\frak p'$. Let $T\supsetneq\complement\frak p$. Then $T':=\phi(T)\supsetneq\complement\frak p'$ and vice versa. On the other hand we know from \ref{basics-for-S-component} (\ref{S-comp-M-M'}) that $T^M((0))={T'}^{M'}((0))$ and also ${\complement\frak p}^M((0))={\complement\frak p'}^{M'}((0)$. Then it follows immediately from Definition \ref{def-assprim} that $\frak p\in\operatorname{Ass}(M)\Leftrightarrow\frak p'\in\operatorname{Ass}(M')$. \end{pf} \begin{prop} \label{zero-divisors-in-Mp} $\frak p\in\operatorname{Ass}(M)$ iff each element of $\frak p\cdot R_{\frak p}$ is a zero divisor for $M_{\frak p}$. \end{prop} \begin{pf} $\underline{\Rightarrow:}$ Let $\frak p\in\operatorname{Ass}(M)$, $p\in\frak p$, $s\in\complement\frak p$, and $T:=\complement\frak p\cdot\{p^\nu\mid \nu=0,1,\dots\}$. Then $T\supsetneq\complement\frak p$ and consequently $T((0))\supsetneq\complement\frak p((0))$, because of $\frak p\in\operatorname{Ass}(M)$. Therefore there is an $x\in M$, $x\notin\complement\frak p((0))$ with $x\in T(0)$, i.e. there are $s'\in\complement\frak p$ and $\nu\in\Bbb N$ such that $s'\cdot p^\nu\cdot x=0$. ($\nu>0$ since $x\notin\complement\frak p((0))$.) Then $\frac{p^\nu}{1}\cdot\frac{x}{1}=0$ in $M_{\frak p}$ and therefore also $(\frac{p}{s})^\nu\cdot\frac{x}{1}=0$ in $M_{\frak p}$. But $\frac{x}{1}\ne0$ since $x\notin\complement\frak p(0)$ and so $(\frac{p}{s})^\nu$ and therefore also $\frac{p}{s}$ is a zero divisor for $M_{\frak p}$.\\ $\underline{\Leftarrow:}$ Let $\frak p$ be a prime ideal of $R$ such that each element of $\frak p\cdot R_{\frak p}$ is a zero divisor for $M_{\frak p}$, and let $T$ be a multiplicatively closed set with $T\supsetneq\complement\frak p$. Then there exists a $p\in T\cap \frak p$. By assumption $\frac{p}{1}$ is a zero divisor for $M_{\frak p}$. So there are $x\in M$ and $s\in\complement\frak p$ such that $\frac{x}{s}\ne0$ but $\frac{p}{1}\cdot\frac{x}{s}=0$ in $M_{\frak p}$. Then there is an $s'\in \complement\frak p$ with $s'\cdot p\cdot x=0$ in $M$. By definition of $T$ we have $s'\cdot p\in T$ and therefore $x\in T((0))$. But $x\notin\complement\frak p$, since otherwise $\frac{x}{s}=0$ in $M_{\frak p}$ contrary to our assumptions. So we get $T((0))\supsetneq\complement\frak p((0))$ and therefore $\frak p\in\operatorname{Ass}(M)$. \end{pf} Since there are no zero divisors for the zero module we get: \begin{cor} \label{p-Ass-Mp-not-0} $\frak p\in\operatorname{Ass}(M)\Longrightarrow M_{\frak p}\ne (0)$ \end{cor} \begin{cor} \label{p-in Ass-is-zero-divisor} $\frak p\in\operatorname{Ass}(M)\Longrightarrow$ Each element of $\frak p$ is a zero divisor for $M$. (For the converse see Theorem \ref{ass-zero-divisors}.) \end{cor} \begin{pf} If $p\in\frak p$ then by Proposition \ref{zero-divisors-in-Mp} $\frac{p}{1}$ is a zero divisor for $M_{\frak p}$. Therefore there is an $x\in M$ with $\frac{x}{1}\ne0$ and $\frac{p\cdot x}{1}=0$ in $M_{\frak p}$. Then there is an $s\in\complement\frak p$ with $p\cdot(s\cdot x)=0$ in $M$. But $s\cdot x\ne0$, since else $\frac{x}{1}=\frac{s\cdot x}{s}=0$ in $M_{\frak p}$. So $p$ is a zero divisor for $M$. \end{pf} \begin{cor} \label{Ass-up-down} Let $S$ be a multiplicatively closed subset of $R$ and $\frak P$ a prime ideal of $R_S$. Then $$ \frak P\in\operatorname{Ass}(M_S)\Longleftrightarrow \frak P\cap R\in\operatorname{Ass}(M) $$ \end{cor} \begin{pf} Let $\frak p:=\frak P\cap R$. Then $R_{\frak p}=\left(R_S\right)_{\frak P}$,\quad $M_{\frak p}=\left(M_S\right)_{\frak P}$,\quad $\frak p\cdot R_{\frak p}=\frak P\cdot\left(R_S\right)_{\frak P}$.\\ Therefore by Proposition \ref{zero-divisors-in-Mp} $\frak P\in\operatorname{Ass}(M_S)\Longleftrightarrow$ each element of $\frak P\cdot\left(R_S\right)_{\frak P} =\frak p\cdot R_{\frak p}$ is a zero divisor for $\left(M_S\right)_{\frak P}=M_{\frak p}\Longleftrightarrow \frak p\in\operatorname{Ass}(M)$. \end{pf} \begin{cor} \label{Ass-submodule} If $N$ is a submodule of $M$ then $\operatorname{Ass}(N)\subseteq\operatorname{Ass}(M)$. \end{cor} \begin{pf} Let $\frak p\in\operatorname{Ass}(N)$. By Proposition \ref{zero-divisors-in-Mp} each element of $\frak p\cdot R_{\frak p}$ is a zero divisior for $N_{\frak p}\subseteq M_{\frak p}$ and therefore also for $M_{\frak p}$. Again by Proposition \ref{zero-divisors-in-Mp} we get $\frak p\in\operatorname{Ass}(M)$. \end{pf} \begin{prop} \label{min-prime-ideal-of Ann Rpx} Let $\frak p$ be a prime ideal of $R$, $x\in M$. The following conditions are equivalent: \begin{enumerate} \item \label{min-prime-over-ann} $\frak p$ is minimal among the prime ideals containing $\operatorname{Ann}_R(x)$. \item \label{nilpotent-in-Rp} $\frac{x}{1}\ne0$, and each element of $\frak p\cdot R_{\frak p}$ is nilpotent for $R_{\frak p}\cdot x$. \item \label{xRp-coprimary} $R_{\frak p}\cdot x$ is $\frak p\cdot R_{\frak p}$-coprimary. \end{enumerate} \end{prop} \begin{pf} (\ref{min-prime-over-ann}) $\Rightarrow$ (\ref{nilpotent-in-Rp}): $\frac{x}{1}\ne0$ in $M_{\frak p}$, because else there would be a $s\in\complement\frak p$ with $s\cdot x=0$ in $M$. Then $s\in\operatorname{Ann}_R(x)\subseteq\frak p$, which contradicts $s\in\complement\frak p$.\\ Now let $p$ be an arbitrary element of $\frak p$, and $S:=\complement\frak p\cdot\{p^\nu\mid\nu=0,1,\dots\}$. We show that $\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)=R_S$:\quad Otherwise there would exist a prime ideal $\frak P'$ of $R_S$ with $\frak P'\supseteq\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)$. Let $\frak p':=\frak P'\cap R$. Then $\frak p'\supseteq \operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)\cap R\supseteq\operatorname{Ann}(x)$ and $\frak p'\cap S=\emptyset$, since also $\frak p'\cap\complement\frak p=\emptyset$, i.e. $\frak p'\subseteq\frak p$. Now by hypothesis $\frak p$ is minimal among the prime ideals containing $\operatorname{Ann}_R(x)$ and therefore $\frak p'=\frak p$. It would follow that $p\in S\cap\frak p'$, contradicting $\frak p'\cap S=\emptyset$. So $\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)=R_S$ and consequently $\frac{x}{1}=0$ in $M_S$. Therefore there exist $\nu\in\Bbb N$ and $s\in\complement\frak p$ with $s\cdot p^\nu\cdot x=0$ in $M$. If follows that $\left(\frac{p}{1}\right)^\nu\cdot\frac{x}{1}=0$ in $M_{\frak p}$. Since $p$ was an arbitrary element of $\frak p$ we have shown that each element of $\frac{\frak p}{1}$ and therefore also each element of $\frak p\cdot R_{\frak p}$ is nilpotent for $R_{\frak p}\cdot x$. \\[1ex] (\ref{nilpotent-in-Rp}) $\Rightarrow$ (\ref{xRp-coprimary}): By assumption each element of $\frak p\cdot R_{\frak p}$ is nilpotent for $R_{\frak p}\cdot x$ and therefore a zero divisor for $R_{\frak p}\cdot x$, because $R_{\frak p}\cdot x\ne(0)$. Since the elements of $R_{\frak p}\setminus\frak p\cdot R_{\frak p}$ are the units of $R_{\frak p}$, $\frak p\cdot R_{\frak p}$ is the set of all zero divisors for $R_{\frak p}\cdot x$. Consequently $R_{\frak p}\cdot x$ is $\frak p\cdot R_{\frak p}$-coprimary. \\[1ex] (\ref{xRp-coprimary}) $\Rightarrow$ (\ref{min-prime-over-ann}): $\frak p\supseteq\operatorname{Ann}(x)$, because else there would exist an $s\in R\setminus\frak p$ with $s\cdot x=0$ and therefore $\frac{x}{1}=0$ in $M_{\frak p}$ which cannot be since $R_{\frak p}\cdot x$ is coprimary and therefore $\ne (0)$.\\ $\frak p$ is minimal among the prime ideals containing $\operatorname{Ann}(x)$: Let $\frak p'$ be a prime ideal with $\frak p\supseteq\frak p'\supseteq\operatorname{Ann}(x)$ and $p\in\frak p$. Then by assumption $\frac{p}{1}$ is nilpotent for $R_{\frak p}\cdot x$. Then there exist $s\in R\setminus\frak p$ and $\nu\in\Bbb N$ with $s\cdot p^\nu\in\operatorname{Ann}(x)\subseteq\frak p'$. But $s\notin\frak p\supseteq\frak p'$. It follows that $p\in\frak p'$ and therefore $\frak p=\frak p'$. \end{pf} \begin{cor} \label{min-primes-Ann-Ass} Let $0\ne x\in M$ be an arbitrary element of $M$ and $\frak p$ minimal among the prime ideals containing $\operatorname{Ann}(x)$. Then $\frak p\in\operatorname{Ass}(M)$. \end{cor} \begin{pf} By Proposition \ref{min-prime-ideal-of Ann Rpx} $\frac{x}{1}\ne0$ in $M_{\frak p}$ and each element of $\frak p\cdot R_{\frak p}$ is nilpotent for $R_{\frak p}\cdot x$ and therefore a zero divisor for $R_{\frak p}\cdot x$ and hence also for $M_{\frak p}$. From Proposition \ref{zero-divisors-in-Mp} then follows that $\frak p\in\operatorname{Ass}(M)$. \end{pf} \begin{defn} \label{def-Ass-essential-first-kind} \strut \begin{enumerate} \item \label{def-Ass1} A prime ideal $\frak p$ of $R$ is called ``associated of the first kind to $M$'' iff there exists an $x\in M$ such that $\frak p$ is minimal among all prime ideals that contain $\operatorname{Ann}_R(x)$: $$ \operatorname{Ass}_1(M):=\{\frak p\mid \frak p \text{ associated of the first kind to $M$}\} $$ \item \label{def-essential1} A prime ideal $\frak p$ of $R$ is called ``essential of the first kind for $N$ in $M$'' iff $\frak p$ is associated of the first kind to $M/N$ . \end{enumerate} \end{defn} \begin{rem} \label{Ass1-rems} \strut \begin{enumerate} \item In \cite{Bourbaki-Alg-Comm-4} \S 1 exercise 17 the prime ideals which we call associated of the first kind to $M$ are called ``faiblement associ\'e \`a $M$''. \item \label{Ass1-in-Ass} $\operatorname{Ass}_1(M)\subseteq\operatorname{Ass}(M)$, but in general equality does not hold (Example \ref{Ass-not-Ass1}). \item \label{M-not-zero-Ass1} $M\ne(0)\Longleftrightarrow\operatorname{Ass}_1(M)\ne\emptyset \Longleftrightarrow\operatorname{Ass}(M)\ne\emptyset$. \item \label{essential-primes-for-ideal} Let $\frak a$ be a proper ideal in $R$. The prime ideals which are minimal in the set of all prime ideals containing $\frak a$ are essential of the first kind for $\frak a$ in $R$. \item \label{Ass1M-Ass1M'} Let $R':=R/\operatorname{Ann}_R(M)$ and $\phi:R\rightarrow R'$ the canonical homomorphism. Then $M$ can be regarded as an $R'$-module $M'$ in a natural way. There is a one-to-one correspondence between $\operatorname{Ass}_1(M)$ and $\operatorname{Ass}_1(M')$, given by $$ \operatorname{Ass}_1(M)\ni\frak p\longmapsto \phi(\frak p)\in\operatorname{Ass}_1(M') $$ \end{enumerate} \end{rem} \begin{pf} (\ref{Ass1-in-Ass}) follows from Corollary \ref{min-primes-Ann-Ass}. \\[1ex] (\ref{M-not-zero-Ass1}) $M\ne(0)\Rightarrow$ Ex. $0\ne x\in M\Rightarrow\operatorname{Ann}_R(x)\ne R \Rightarrow$ Ex. prime ideal $\frak p\supseteq\operatorname{Ann}_R(x)$ and therefore there also exists a prime ideal $\frak p'$ which is minimal among the prime ideals containing $\operatorname{Ann}_R(x)$. By definition $\frak p'\in\operatorname{Ass}_1(M)\subseteq\operatorname{Ass}(M)$. \\ Conversely: If $\operatorname{Ass}(M)\ne\emptyset$ then $M\ne(0)$ (and then also $\operatorname{Ass}_1(M)\ne\emptyset$ as was just shown), since by Corollary \ref{p-Ass-Mp-not-0} one even has $M_{\frak p}\ne(0)$ for all $\frak p\in\operatorname{Ass}(M)$. \\[1ex] (\ref{essential-primes-for-ideal}) For the residue class $\bar1\in R/\frak a$ we have $\operatorname{Ann}_R(\bar1)=\frak a$. Therefore the prime ideals $\frak p$ which are minimal among the prime ideals containing $\frak a$ belong to $\operatorname{Ass}_1(R/\frak a)$, which means that $\frak p$ is essential of the first kind for $\frak a$ in $R$. \\[1ex] (\ref{Ass1M-Ass1M'}) This follows immediately from the fact that for each $x\in M$ we have\\ $\operatorname{Ann}_{R'}(\phi(x))=\phi\left(\operatorname{Ann}_R(x)\right)$. (See also the proof of Remark \ref{AssM-and-M'} (\ref{AssM-AssM'}) .) \end{pf} With respect to quotient modules the elements of $\operatorname{Ass}_1(M)$ behave similar to those of $\operatorname{Ass}(M)$ (Corollary \ref{Ass-up-down}): \begin{prop} \label{Ass1-up-down} Let $S$ be a multiplicatively closed subset of $R$ and $\frak P$ a prime ideal of $R_S$. Then $$ \frak P\in\operatorname{Ass}_1(M_S)\Longleftrightarrow \frak P\cap R\in\operatorname{Ass}_1(M) $$ \end{prop} \begin{pf} We use the same notations as in the proof of Corollary \ref{Ass-up-down}.\\ $\frak P\in\operatorname{Ass}_1(M)\Leftrightarrow$ there exists $x\in M$ such that $\frak P$ is minimal among the prime ideals containing $\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)$. By propositon \ref{min-prime-ideal-of Ann Rpx} this means that $\frac{x}{1}\ne0$ and each element of $\frak P\cdot\left(R_S\right)_{\frak P}= \frak p\cdot R_{\frak p}$ is nilpotent for $\left(R_S\right)_{\frak P}\cdot\frac{x}{1}=R_{\frak p}\cdot x$, which again by Proposition \ref{min-prime-ideal-of Ann Rpx}, is equivalent to $\frak p$ being minimal among the prime ideals of $R$ containing $\operatorname{Ann}_R(x)$, i.e. $\frak p\in\operatorname{Ass}_1(M)$. \end{pf} \begin{prop} \label{union-of-Ass1-primes} \begin{enumerate} \item \label{each-p-is-union} Each $\frak p\in\operatorname{Ass}(M)$ is the union of certain\\ $\frak p'\in\operatorname{Ass}_1(M)$. More exactly: $$ \frak p=\bigcup_{\frak P'\in\operatorname{Ass}_1(M_{\frak p})}\frak P'\cap R $$ \item \label{finite-union-of Ass1} If $\frak p'_1,\dots,\frak p'_r\in\operatorname{Ass}_1(M)$ are {\em finitely many} prime ideals of $\operatorname{Ass}_1(M)$ such that $\bigcup\limits_i\frak p'_i=:\frak p$ is a prime ideal, then $\frak p$ is equal to one of the $\frak p'_i$ and therefore $\frak p\in\operatorname{Ass}_1(M)$. \item \label{if-union-is-prime} If $\frak p_i\in\operatorname{Ass}(M)$ and $\bigcup\limits_i\frak p_i=:\frak p$ is a prime ideal then $\frak p\in\operatorname{Ass}(M)$. \end{enumerate} \end{prop} \begin{pf} (\ref{each-p-is-union}) Let $\frak p\in\operatorname{Ass}(M)$ and $\frak P'\in\operatorname{Ass}_1\left(M_{\frak p}\right)$. Then $\frak P'\cap R\subseteq\frak p$, and by Proposition \ref{Ass1-up-down} $\frak P'\cap R\in\operatorname{Ass}_1(M)$. Therefore $\frak p\supseteq \bigcup\limits_{\frak P'\in\operatorname{Ass}_1\left(M_{\frak p}\right)} \frak P'\cap R$.\\ Conversely: Let $p\in\frak p$. Then by Proposition \ref{zero-divisors-in-Mp}\quad $\frac{p}{1}$ is a zero divisor for $M_{\frak p}$. Therefore there exists $0\ne\frac{x}{1}\in M_{\frak p}$ with $\frac{p}{1}\in \operatorname{Ann}_{R_{\frak p}}\left(R_{\frak p}\cdot\frac{x}{1}\right)$. Let $\frak P'_x$ be minimal among the prime ideals containing $\operatorname{Ann}_{R_{\frak p}}\left(R_{\frak p}\cdot\frac{x}{1}\right)$. Then by Definition \ref{def-Ass-essential-first-kind}\ \ $\frak P'_x\in\operatorname{Ass}_1(M_{\frak p})$ and $p\in\frak P'_x\cap R$. Therefore $\frak p\subseteq \bigcup\limits_{\frak P'\in\operatorname{Ass}_1\left(M_{\frak p}\right)} \frak P'\cap R$. \\[1ex] (\ref{finite-union-of Ass1}) This is an immediate consequence of Corollary \ref{union-finitely-many-primes-ideal}. \\[1ex] (\ref{if-union-is-prime}) Let $\frak p=\bigcup\limits_i\frak p_i$ with $\frak p_i\in\operatorname{Ass}(M)$ be a prime ideal. Then $\frak p\supseteq\frak p_i$ and therefore $\frak P_i:=R_{\frak p}\cdot\frak p_i$ is a prime ideal of $R_{\frak p}$ and $M_{\frak p_i}=\left(M_{\frak p}\right)_{\frak P_i}$. Now let $p\in\frak p$ be an arbitrary element. We show that $\frac{p}{1}$ is a zero divisor of $M_{\frak p}$:\\ Since $p\in\frak p_{i_0}$ for some $i_0$ and $\frak p_{i_0}\in\operatorname{Ass}(M)$ it follows from Proposition \ref{zero-divisors-in-Mp} that $\frac{p}{1}\in R_{\frak p_{i_0}}$ is a zero divisor for $M_{\frak p_{i_0}}$. Then there is an $x\in M$ with $\frac{x}{1}\ne0$ in $M_{\frak p_{i_0}}$ but $\frac{p}{1}\cdot\frac{x}{1}=0$ in $M_{\frak p_{i_0}}$. Then there is a $\rho\in R_{\frak p}\setminus\frak P_{i_0}$ with $\rho\cdot\frac{p}{1}\cdot\frac{x}{1}=0$ in $M_{\frak p}$, since $M_{\frak p_{i_0}}=\left(M_{\frak p}\right)_{\frak P_{i_0}}$. But $\rho\cdot\frac{x}{1}\ne0$ in $M_{\frak p}$, because else $\frac{x}{1}=0$ in $M_{\frak p_{i_0}}=\left(M_{\frak p}\right)_{\frak P_{i_0}}$. Consequently $\frac{p}{1}\in R_{\frak p}$ is a zero divisor for $M_{\frak p}$. Proposition \ref{zero-divisors-in-Mp} then yields $\frak p\in\operatorname{Ass}(M)$. \end{pf} An immediate consequence of Proposition \ref{union-of-Ass1-primes} (\ref{each-p-is-union}) and (\ref{finite-union-of Ass1}) is: \newpage \begin{cor}\strut \begin{enumerate} \item Each $\frak p\in\operatorname{Ass}(M)$ contains a\/ $\frak p'\in\operatorname{Ass}_1(M)$. Therefore all minimal elements of $\operatorname{Ass}(M)$ (if there are any) lie in $\operatorname{Ass}_1(M)$. \item If $\operatorname{Ass}_1(M)$ is a finite set then $\operatorname{Ass}(M)=\operatorname{Ass}_1(M)$. (E.g. if $M$ is noetherian or if $(0)$ has a primary decomposition.) \end{enumerate} \end{cor} \begin{thm} \label{ass-zero-divisors} $$ \bigcup\limits_{\frak p'\in\operatorname{Ass}_1(M)}\frak p'= \bigcup\limits_{\frak p\in\operatorname{Ass}(M)}\frak p= \{r\mid r\in R,\ r\text{ zero divisor for }M \}. $$ \end{thm} \begin{pf} From $\operatorname{Ass}_1(M)\subseteq\operatorname{Ass}(M)$ follows $\bigcup\limits_{\frak p'\in\operatorname{Ass}_1(M)}\frak p'\subseteq \bigcup\limits_{\frak p\in\operatorname{Ass}(M)}\frak p$, and because of Proposition \ref{union-of-Ass1-primes} (\ref{each-p-is-union}) one has also the converse inclusion. That shows the first equality.\\ To show the second equality we only need to show that every zero divisor for $M$ lies in a $\frak p\in\operatorname{Ass}(M)$, because the other inclusion follows from Corollary \ref{p-in Ass-is-zero-divisor}. Let $r\in R$ be a zero divisor for $M$. Then there exists an $x\in M$, $x\ne0$ with $r\in\operatorname{Ann}_R(x)$. Since $\operatorname{Ann}_R(x)\ne R$ there exist prime ideals in $R$ containing $\operatorname{Ann}_R(x)$. A minimal element among these primes belongs to $\operatorname{Ass}(M)$ by Corollary \ref{min-primes-Ann-Ass} and contains $r$. \end{pf} Since $\operatorname{Ass}(M)$ describes the zero divisors for $M$ it is plausible that there is also connection with the annihilators of submodules of $N$: \begin{prop} \label{ess-Ann-Ass-M-prim-decomp-0} Let $N\ne(0)$ be a finitely generated submodule of $M$, then \begin{enumerate} \item \label{ess-Ann-N-Ass-M} The essential prime ideals for $\operatorname{Ann}_R(N)$ in $R$ belong to $\operatorname{Ass}(M)$ \item \label{ess1-Ann-N-Ass1-M} The essential prime ideals of the first kind for $\operatorname{Ann}_R(N)$ in $R$\\ belong to $\operatorname{Ass}_1(M)$ \item \label{primary-decomp-0-decomp-Ann} If there exists a primary decomposition of $(0)$ in $M$ then there is also a primary decomposition of $\operatorname{Ann}_R(N)$ in $R$: \end{enumerate} \end{prop} \begin{pf} (\ref{ess-Ann-N-Ass-M}): Let $\frak p$ be essential for $\operatorname{Ann}_R(N)$ in $R$ and $T$ a multiplicatively closed subset of $R$ with $T\supsetneq\complement\frak p$. Then by Remark \ref{ess-and-mult-closed}\ \ $T^R\left(\operatorname{Ann}_R(N)\right)\supsetneq {\complement\frak p}^R\left(\operatorname{Ann}_R(N)\right)$. We will show that $R_{\frak p}\cdot T^R\left(\operatorname{Ann}_R(N)\right) \supsetneq R_{\frak p}\cdot{\complement\frak p}^R\left(\operatorname{Ann}_R(N)\right)$:\\ Trivially ``$\supseteq$'' holds. To show the inequality first remark that by Proposition \ref{basics-for-S-component} (\ref{T-bigger-S}) $T^R\left(\operatorname{Ann}_R(N)\right)= {\complement\frak p}^R\left(T^R(\operatorname{Ann}_R(N))\right)$ is a $\complement\frak p$-component because of $T\supseteq\complement\frak p$. Therefore from $R_{\frak p}\cdot T^R\left(\operatorname{Ann}_R(N)\right) = R_{\frak p}\cdot{\complement\frak p}^R\left(\operatorname{Ann}_R(N)\right)$ by taking the inverse images in $R$ one would obtain (using Proposition \ref{basics-for-S-component} (\ref{RSN=RN}) and the definition of the $\complement\frak p$-components): $T^R\left(\operatorname{Ann}_R(N)\right)= R_{\frak p}\cdot T^R\left(\operatorname{Ann}_R(N)\right)\cap R= R_{\frak p}\cdot{\complement\frak p}^R\left(\operatorname{Ann}_R(N)\right)\cap R= {\complement\frak p}^R\left(\operatorname{Ann}_R(N)\right)$, contradiction.\\ Further, again by Proposition \ref{basics-for-S-component} (\ref{RSN=RN}), we know that $R_{\frak p}\cdot\complement\frak p\left(\operatorname{Ann}_R(N)\right)= R_{\frak p}\cdot\operatorname{Ann}_R(N)$, and $R_{\frak p}\cdot\operatorname{Ann}_R(N)=\operatorname{Ann}_{R_{\frak p}}(N_{\frak p})$ , because $N$ is finitely generated (Remark \ref{transporteurs} (\ref{transp-quotmodule})), so that finally $R_{\frak p}\cdot T^R\left(\operatorname{Ann}_R(N)\right)\supsetneq \operatorname{Ann}_{R_{\frak p}}(N_{\frak p})$.\\ Let $r\in T^R\left(\operatorname{Ann}_R(N)\right)$ with $\frac{r}{1}\in R_{\frak p}\cdot T^R\left(\operatorname{Ann}_R(N)\right)\setminus \operatorname{Ann}_{R_{\frak p}}(N_{\frak p})$, i.e. there is an $x\in N$ with $\frac{r}{1}\cdot\frac{x}{1}\ne(0)$ in $N_{\frak p}\subseteq M_{\frak p}$ and so $r\cdot x\notin{\complement\frak p}^M(0)$. But $\frac{r}{1}\in R_T\cdot T^R\left(\operatorname{Ann}_R(N)\right)= R_T\cdot\operatorname{Ann}_R(N)=\operatorname{Ann}_{R_T}(N_T)$ since $N$ is finitely generated (Remark \ref{transporteurs} (\ref{transp-quotmodule})). Therefore $\frac{r}{1}\cdot\frac{x}{1}=0$ in $N_T\subseteq M_T$ and so $r\cdot x\in T^M((0))$.\\ So we have shown that for any $T\supsetneq\complement\frak p$ we have $T^M((0))\supsetneq{\complement\frak p}^M((0))$ and consequently $\frak p\in\operatorname{Ass}(M)$. \\[1ex] (\ref{ess1-Ann-N-Ass1-M}): Now let $\frak p$ be essential of the first kind for $\operatorname{Ann}_R(N)$ in $R$,\\ i.e. $\frak p\in\operatorname{Ass}_1(R/\operatorname{Ann}_R(N))$. By definition there is an $r\in R$ such that $\frak p$ is minimal among the prime ideals containing the annihilator $\operatorname{Ann}_R(\bar r)$ of the residue class $\bar r$ of $r$ mod $\operatorname{Ann}_R(N)$. Then by Proposition \ref{min-prime-ideal-of Ann Rpx} \quad $\frac{\bar r}{1}\ne 0$, and for each $\frac{p}{1}\in\frak p\cdot R_{\frak p}$ there exists a $\nu\in\Bbb N$ with $\frac{p^\nu\cdot r}{s^\nu}\in \left(\operatorname{Ann}_R(N)\right)_{\frak p}=\operatorname{Ann}_{R_{\frak p}}(N_{\frak p})$, since $N$ is finitely generated (Remark \ref{transporteurs} (\ref{transp-quotmodule})). Then there is an $x\in N$ with $\frac{r\cdot x}{1}\ne0$ in $N_{\frak p}\subseteq M_{\frak p}$, but for each $\frac{p}{s}\in\frak p\cdot R_{\frak p}$ there is a $\nu\in\Bbb N$ with $\left(\frac{p}{s}\right)^\nu\cdot\frac{r\cdot x}{1}=0$ in $M_{\frak p}$, showing that each element of $\frak p\cdot R_{\frak p}$ is nilpotent for $R_{\frak p}\cdot(r\cdot x)$. Proposition \ref{min-prime-ideal-of Ann Rpx} then yields that $\frak p$ is minimal among the prime ideals containing $\operatorname{Ann}_R(r\cdot x)$ and therefore $\frak p\in\operatorname{Ass}_1(M)$. \\[1ex] (\ref{primary-decomp-0-decomp-Ann}): Assume that there is a primary decomposition $(0)=\bigcap\limits_{i=1}^n F_i$, with $F_i$ primary in $M$. Then by Remark \ref{transporteurs} $\operatorname{Ann}_R(N)=\left((0):N\right)=\bigcap\limits_{i=1}^n\left(F_i:N\right)$ and\\ $ (F_i:N)= \left\{\begin{array}{ll} \text{ primary in } R&\text{ if }F_i\nsupseteq N\\ R &\text{ if }F_i\supseteq N \end{array}\right., $ because $N$ is finitely generated. Hence $\operatorname{Ann}_R(N)$ has a primary decomposition in $R$. \end{pf} \begin{rem} If $N$ is not finitely generated Proposition \ref{ess-Ann-Ass-M-prim-decomp-0} is not true as is shown in Example \ref{0-primary-decomp-Ann-not ass} \end{rem} In the proof of Proposition \ref{ess-Ann-Ass-M-prim-decomp-0} we used the following \begin{defn} \label{transpdefs} Let $N$ and $U$ be subsets of the $R$-module $M$ $$ (N:U):=\{r\mid r\in R,\ r\cdot U\subseteq N\} $$ \end{defn} \begin{rem} \label{transporteurs} Let $N$ and $U$ be subsets of $M$. \begin{enumerate} \item \label{submodule-ideal} If $N$ is a $R$-submodule of $M$ then $(N:U)$ is an ideal of $R$ and\\ $(N:U)=(N:\langle U\rangle)$, where $\langle U\rangle$ denotes the $R$-module generated by $U$. \item \label{Ann-transp} $\operatorname{Ann}_R(U)=\left((0):U\right)$ \item \label{intersect-transp} For arbitrary intersections we have $$ \left(\left(\bigcap\limits_{i\in I}F_i\right):U\right)= \bigcap\limits_{i\in I}\left(F_i:U\right) $$ \item \label{primary-transp} Let $F$ be $\frak p$-primary in $M$ and $U$ a finitely generated submodule of $M$. Then $$ (F:U)\text{ is }\left\{ \begin{array}{ll} \frak p\text{-primary in }R &\text{ if }F\nsupseteq U\\ =R &\text{ if }F\supseteq U \end{array} \right. $$ \item \label{transp-quotmodule} Let $S$ be a multiplicatively closed subset of $R$,\\ $N$, $U$ submodules of $M$. Then $$ \left(N_S:U_S\right)\supseteq R_S\cdot\left(N:U\right)\ . $$ If $U$ is finitely generated equality holds.\\ Especially: If $U$ is finitely generated then $\operatorname{Ann}_{R_S}(U_S)=R_S\cdot\operatorname{Ann}_R(U)$ \end{enumerate} \end{rem} \begin{pf} (\ref{submodule-ideal}), (\ref{Ann-transp}), (\ref{intersect-transp}) are obvious. \\[1ex] (\ref{primary-transp}): Trivially $(F:U)=R$ if $F\supseteq U$.\\ Let $F\nsupseteq U$. Then $1\notin(F:U)$, i.e. $(F:U)\subsetneq R$. By hypothesis $F$ is $\frak p$-primary in $M$. Then for each $p\in\frak p$ and each $x\in U$ there is a $\nu\in\Bbb N$ with $p^\nu\cdot x\in F$. Now by hypothesis $U$ is finitely generated. Given a $p\in\frak p$ we define $n$ to the maximum of the $\nu_i$ such that for a finite set of generators $x_i$ of $U$ we have $p^{\nu_i}\cdot x_i\in F$. Then by (\ref{submodule-ideal})\ $p^n\cdot U\subseteq F$, which means $p^n\cdot R\subseteq(F:U)$. It follows that each element of $\frak p$ is nilpotent for $R/(F:U)$. On the other hand, if $r\in R$ is any zero divisor for $R/(F:U)$ there exists a $y\in R\setminus(F:U)$ with $r\cdot y\in (F:U)$, i.e. $y\cdot U\nsubseteq F$, but $r\cdot y\cdot U\subseteq F$. Therefore there is a $z\in y\cdot U$, $z\notin F$ with $r\cdot z\in r\cdot y\cdot U\subseteq F$. So $r$ is also a zero divisor for $M/F$ and hence $r\in\frak p$ since $F$ is $\frak p$-primary in $M$, and we have shown that $(F:U)$ is $\frak p$-primary in $R$. \\[1ex] (\ref{transp-quotmodule}): From $r\cdot U\subseteq N$ follows $\frac{r}{s}\cdot U_S\subseteq N_S$ for all $s\in S$, and therefore\linebreak $R_S\cdot(N:U)\subseteq\left(N_S:U_S\right)$. \\[1ex] Now let $U$ be finitely generated, $U=\langle x_1,\dots,x_r\rangle$, and let $\frac{r}{s}\in\left(N_S:U_S\right)$ be arbitrary. Then for $i=1,\dots r$\quad $\frac{r}{s}\cdot\frac{x_i}{1}\in N_S$, i.e. for all $i=1,\dots r$ there exists an $s_i\in S$ with $s_i\cdot r\cdot x_i\in N$. Define $s':=s_1\cdots s_r$. Then $s'\cdot r\cdot x_i\in N$ for all $i$ and therefore $s'\cdot r\in (N:U)$. It follows $\frac{r}{1}\in R_S\cdot(N:U)$ and therefore also $\frac{r}{s}\in R_S\cdot(N:U)$. \end{pf} \begin{rem} Without the assumption that $U$ is finitely generated the conclusion of Remark \ref{transporteurs} (\ref{primary-transp}) may be false as can be seen from Example \ref{0-primary-decomp-Ann-not ass} with $F:=(0)$, $U:=M$. Then $(0)$ is $\frak p$-primary in $M$, but $(F:U)=\operatorname{Ann}_R(M)=(0)$ is not $\frak p$-primary in $R$. \end{rem} \vspace{2ex}\noindent In the classical case of noetherian modules one defines the associated prime ideals of $M$ as those prime ideals which are annihilators of elements of $M$ and not just minimal elements in the set of all prime ideals containing the annihilator of an element. Here we will denote the set of these prime ideals by $\operatorname{Ass}_0(M)$: \begin{defn} $$ \operatorname{Ass}_0(M):=\{\frak p\mid \frak p {\size{10}{12pt}\selectfont\text{ prime ideal of $R$ such that there is an $x\in R$ with }}\frak p=\operatorname{Ann}_R(x)\}. $$ \end{defn} \begin{rem} \label{Ass0-rems} \strut \begin{enumerate} \item Clearly by definition $\operatorname{Ass}_0(M)\subseteq\operatorname{Ass}_1(M)$. But in general equality does not hold. (See Example \ref{Ass=Ass1-not-Ass0}.) \item If $M$ is noetherian, then $\operatorname{Ass}(M)=\operatorname{Ass}_1(M)=\operatorname{Ass}_0(M)$. (See Theorem \ref{Ass-M-noetherian}.) In the non noetherian case $\operatorname{Ass}_0(M)$ is not very useful. For instance it may happen that $\operatorname{Ass}_0(M)=\emptyset$ although there exists a primary decomposition of $(0)$ in $M$. (See Example \ref{Ass=Ass1-not-Ass0}.) \item \label{Ass0M-Ass0M'} Let $R':=R/\operatorname{Ann}_R(M)$ and $\phi:R\rightarrow R'$ the canonical homomorphism. Then $M$ can be regarded as an $R'$-module $M'$ in a natural way. There is a one-to-one correspondence between $\operatorname{Ass}_0(M)$ and $\operatorname{Ass}_0(M')$, given by\hfil $\operatorname{Ass}_0(M)\ni\frak p\longmapsto \phi(\frak p)\in\operatorname{Ass}_0(M')$\\ (See the proof of Remark \ref{Ass1-rems} (\ref{Ass1M-Ass1M'}) .) \end{enumerate} \end{rem} \begin{thm} \label{Ass-M-noetherian} Let $M$ be an $R$-module and $\frak p$ a prime ideal of $R$. \begin{enumerate} \item \label{p-coprim-p-Ass} If there exists a $\frak p$-coprimary submodule $U$ of $M$, then $\frak p\in\operatorname{Ass}_1(M)$. \item \label{0-primary-decomp-cyclic-p-primary} If $\frak p\in\operatorname{Ass}(M)$ and $(0)$ has a primary decomposition in $M$ then there exists a cyclic $\frak p$-coprimary submodule $U$ of $M$. \item \label{0-primary-decomp-cyclic-p-primary-finite-p} If there exists a primary decomposition of $(0)$ in $M$ and $\frak p$ is finitely generated then: $\frak p\in \operatorname{Ass}(M)\Longleftrightarrow \frak p\in\operatorname{Ass}_0(M)$. Especially, if $M$ and $R$ are both noetherian then $\operatorname{Ass}(M)=\operatorname{Ass}_1(M)=\operatorname{Ass}_0(M)$ \item \label{R-noetherian-Ass1=Ass0} If $R$ is noetherian then $\operatorname{Ass}_1(M)=\operatorname{Ass}_0(M)$ \item \label{M-noetherian-Ass=Ass0} If $M$ is noetherian then $\operatorname{Ass}(M)=\operatorname{Ass}_1(M)=\operatorname{Ass}_0(M)$ \end{enumerate} \end{thm} \begin{pf} (\ref{p-coprim-p-Ass}) Let $U$ be a $\frak p$-coprimary submodule of $M$. Then by definition $U\ne(0)$. Choose an $0\ne x\in U$. By Remark \ref{submodule-coprimary-is coprimary} $R\cdot x$ is $\frak p$-coprimary and therefore by Proposition \ref{primary-up-down} $R_{\frak p}\cdot x$ is $\frak p\cdot R_{\frak p}$-coprimary. By Proposition \ref{min-prime-ideal-of Ann Rpx} and the definition of $\operatorname{Ass}_1(M)$ we get $\frak p\in \operatorname{Ass}_1(M)$. \\[1ex] (\ref{0-primary-decomp-cyclic-p-primary}) By hypothesis there exists a primary and therefore also a normal decomposition of $(0)$ in $M$:\\ $(0)=\bigcap\limits_{i=1}^n F_i$ with $F_i$ being $\frak p_i$-primary in $M$. By Remark \ref{essential-primes-for-normal-decomp} $\operatorname{Ass}(M)=\{\frak p_1,\dots,\frak p_n\}$. Without loss of generality we may assume that $\frak p=\frak p_1$. We distinguish two cases:\\ 1$^{st}$ case: $n=1$. Then $(0)=F_1$ is $\frak p$-primary in $M$, i.e. $M$ is $\frak p$-coprimary, especially $M\ne(0)$. Let $0\ne x\in M$. Then by Remark \ref{submodule-coprimary-is coprimary} $U:=R\cdot x$ is a cyclic $\frak p$-coprimary submodule of $M$.\\ 2$^{nd}$ case: $n\ge 2$. Let $\widetilde U:=\bigcap\limits_{i=2}^n F_i$ Then $\widetilde U\ne(0)$, since $(0)=F_1\cap\dots\cap F_n$ is reduced. We show that $\widetilde U$ is $\frak p$-coprimary:\\ Since $F_1$ is $\frak p$-primary in $M$, for each $p\in\frak p$ and each $x\in M$ there exists a $\nu\in\Bbb N$ with $p^\nu\cdot x\in F_1$. Applying this to an $x\in\widetilde U$ we get $p^\nu\cdot x\in F_1\cap\widetilde U=(0)$. Therefore each element of $\frak p$ is nilpotent for $\widetilde U$.\\ Now let $r$ be an arbitrary zero divisor for $\widetilde U$. Then there is an $0\ne x\in\widetilde U$ with $r\cdot x=0$. We have $x\notin F_1$, since else $x\in F_1\cap\widetilde U=(0)$, but $r\cdot x=0\in F_1$. Therefore $r$ is a zero divisor for $M/F_1$, hence $r\in\frak p$. It follows that $\widetilde U$ is $\frak p$-coprimary. Then any $0\ne x\in\widetilde U$ generates a cyclic $\frak p$-coprimary submodule of $M$. \\[1ex] (\ref{0-primary-decomp-cyclic-p-primary-finite-p}) Since $\operatorname{Ass}(M)\supseteq\operatorname{Ass}_1(M)\supseteq\operatorname{Ass}_0(M)$ all we have to show is that for each $\frak p\in\operatorname{Ass}(M)$ there exists a cyclic submodule of $M$ whose annihilator is $\frak p$. By (\ref{0-primary-decomp-cyclic-p-primary}) we have a cyclic $\frak p$-coprimary submodule $U$ of $M$. Since each element of $\frak p$ is nilpotent for $U$ and $\frak p$ and $U$ are finitely generated there exists a $\nu\in\Bbb N$ with $\frak p^\nu\cdot U=(0)$. Choose $\nu$ minimal with that property, then $\frak p^{\nu-1}\cdot U\ne(0)$. Let $0\ne y\in\frak p^{\nu-1}\cdot U$. Then $\frak p\cdot y=0$ and every $r\in R$ with $r\cdot y=0$ lies in $\frak p$ since $R\cdot y$ is $\frak p$-coprimary as a submodule of $U$. Therefore $\operatorname{Ann}(y)=\frak p$. \\[1ex] (\ref{R-noetherian-Ass1=Ass0}) Let $R$ be noetherian. Since $\operatorname{Ass}_1(M)\supseteq\operatorname{Ass}_0(M)$ all we have to show is that each $\frak p\in\operatorname{Ass}_1(M)$ belongs to $\operatorname{Ass}_0(M)$:\\ For $\frak p\in\operatorname{Ass}_1(M)$ there exists an $x\in M$ such that $\frak p$ is minimal among the prime ideals containing $\frak a:=\operatorname{Ann}_R(x)$. Let $V:=R\cdot x\subseteq M$. Then by definition $\frak p\in\operatorname{Ass}_1(V)$. But $V\cong R/\frak a$ is a noetherian $R$-module since $R$ is noetherian. By~(\ref{0-primary-decomp-cyclic-p-primary-finite-p}) we then get $\frak p\in\operatorname{Ass}_0(V)$, i.e. there is an element $y\in V$ with $\frak p=\operatorname{Ann}_R(y)$. But since $y\in V\subseteq M$ we obtain $\frak p\in\operatorname{Ass}_0(M)$. \\[1ex] (\ref{M-noetherian-Ass=Ass0}) Let $M$ be noetherian, $\frak a:=\operatorname{Ann}_R(M)$, and $R':=R/\frak a$. Then $M$ can be regarded in a natural way as an $R'$-module $M'$ and by \cite{Nagata-Local-Rings} Corollary (3.17) $R'$ is a noetherian ring. Therefore by (\ref{0-primary-decomp-cyclic-p-primary-finite-p}) we have $\operatorname{Ass}(M')=\operatorname{Ass}_1(M')=\operatorname{Ass}_0(M')$ as an $R'$-module. But then the same equality holds for $M$ as an $R$-module, because of the one-to-one correspondence between the respective $\operatorname{Ass}$. (See Remarks \ref{AssM-and-M'}, \ref{Ass1-rems}, \ref{Ass0-rems}.) \end{pf} \begin{cor} \label{Ass12-exactq-sequ} Let $0\rightarrow N\rightarrow M\rightarrow L\rightarrow 0$ be an exact sequence of $R$-modules. \begin{enumerate} \item \label{Ass1-exact-sequ} $\operatorname{Ass}_1(N)\subseteq\operatorname{Ass}_1(M)\subseteq\operatorname{Ass}_1(N)\cup\operatorname{Ass}_1(L)$. \item \label{Ass0-exact-sequ} $\operatorname{Ass}_0(N)\subseteq\operatorname{Ass}_0(M)\subseteq\operatorname{Ass}_0(N)\cup\operatorname{Ass}_0(L)$. \end{enumerate} \end{cor} \begin{pf} To simplify the notation we may assume that $N\subseteq M$ and $L=M/N$.\\ Since the annihilator of an element $x\in N$ is the same as the annihilator of $x$ regarded as an element of $M$ it is trivial that $\operatorname{Ass}_1(N)\subseteq\operatorname{Ass}_1(M)$ and $\operatorname{Ass}_0(N)\subseteq\operatorname{Ass}_0(M)$. So all we have to show is that each $\frak p\in\operatorname{Ass}_i(M)$ lies in $\operatorname{Ass}_i(N)$ or in $\operatorname{Ass}_i(M/N)$ for $i=0,1$. \\[1ex] (\ref{Ass1-exact-sequ}): If $\frak p\in\operatorname{Ass}_1(M)$ then by Proposition \ref{min-prime-ideal-of Ann Rpx} there is an $x\in M$ such that $R_{\frak p}\cdot x$ is coprimary for $\frak p\cdot R_{\frak p}$.\\ $1^{st}$ case: $R_{\frak p}\cdot x\cap N_{\frak p}=(0)$. Then $R_{\frak p}\cdot x\cong R_{\frak p}\cdot x+N_{\frak p}/N_{\frak p} \subseteq(M/N)_{\frak p}$, which shows that $(M/N)_{\frak p}$ contains a $\frak p\cdot R_{\frak p}$-coprimary submodule. Then by Theorem \ref{Ass-M-noetherian}, (\ref{p-coprim-p-Ass})\ $\frak p\cdot R_{\frak p}\in\operatorname{Ass}_1\left((M/N)_{\frak p}\right)$ and then by Proposition \ref{Ass1-up-down}\ $\frak p\in\operatorname{Ass}_1(M/N)$.\\ $2^{nd}$ case: $U:=R_{\frak p}\cdot x\cap N_{\frak p}\ne(0)$. Then by Proposition \ref{submodule-coprimary-is coprimary} $U$ is $\frak p\cdot R_{\frak p}$-coprimary as a submodule $\ne (0)$ of $R_{\frak p}\cdot x$. Since $U\subseteq N_{\frak p}$ it follows from Theorem \ref{Ass-M-noetherian}, (\ref{p-coprim-p-Ass})\ that $\frak p\cdot R_{\frak p}\in\operatorname{Ass}_1(N_{\frak p})$ and then by Proposition \ref{Ass1-up-down}\ $\frak p\in\operatorname{Ass}_1(N)$. \\[1ex] (\ref{Ass0-exact-sequ}): If $\frak p\in\operatorname{Ass}_0(M)$ there exists a submodule $U=R\cdot x\cong R/\frak p$ of $M$.\\ $1^{st}$ case: $U\cap N=(0)$. Then $U\cong U+N/N\subseteq M/N$ and therefore $\frak p\in\operatorname{Ass}_0(M/N)$.\\ $2^{nd}$ case: $U\cap N\ne (0)$. Let $0\ne y\in U\cap N$. Since $U\cong R/\frak p$ and $\frak p$ is a prime ideal, $\operatorname{Ann}_R(y)=\frak p$, and therefore $\frak p\in\operatorname{Ass}_0(N)$. \end{pf} \begin{rem} While $\operatorname{Ass}(N)\subseteq\operatorname{Ass}(M)$ by Corollary \ref{Ass-submodule}, in general\\ $\operatorname{Ass}(M)\nsubseteq\operatorname{Ass}(N)\cup\operatorname{Ass}(L)$, even if the exact sequence splits,\\ as is shown in Example \ref{Ass-exact-sequ-false}. \end{rem} \section{The Support of a Module} \begin{defn} Let $M$ be an $R$-module. $$ \operatorname{Supp}(M):=\{\frak p\mid\frak p\text{ prime ideal of }M \text{ with }M_{\frak p}\ne0\} $$ \end{defn} We summarize the basic properties of $\operatorname{Supp}$: \begin{rem} \label{Supp-basics} \strut \begin{enumerate} \item \label{M-not-0-Supp-not-empty} $M\ne0\Longleftrightarrow \operatorname{Supp}(M)\ne\emptyset$ \item \label{Supp-exact-sequence} If $0\rightarrow N\rightarrow M\rightarrow L\rightarrow 0$ is an exact sequence of $R$-modules then $\operatorname{Supp}(M)=\operatorname{Supp}(N)\cup\operatorname{Supp}(L)$. \item \label{Supp-bigger-ideal} If $\frak p\in\operatorname{Supp}(M)$ and $\frak p'$ is a prime ideal with $\frak p'\supseteq\frak p$ then $\frak p'\in\operatorname{Supp}(M)$. \item \label{Supp-and-Ann} $\frak p\in\operatorname{Supp}(M)\Longrightarrow\frak p\supseteq\operatorname{Ann}_R(M)$\\ (but the converse is not true in general: There may be prime ideals containing $\operatorname{Ann}_R(M)$ which do not belong to $\operatorname{Supp}(M)$ as is shows in Example \ref{Ann-not-in-Supp}.)\\ If $M$ is finitely generated then each $\frak p$ containing $\operatorname{Ann}_R(M)$ also belongs to $\operatorname{Supp}(M)$. \item \label{Supp-of-sums} $\operatorname{Supp}\sum\limits_{i\in I} N_i=\bigcup\limits_{i\in I}\operatorname{Supp}(N_i)$\\ for arbitrary families of submodules $N_i$ of $M$. \end{enumerate} \end{rem} \begin{pf} (\ref{M-not-0-Supp-not-empty}) Trivially if $M=(0)$ the $M_{\frak p}=(0)$ for all $\frak p$.\\ Conversely: If $M\ne(0)$ let $0\ne x\in M$. Then $\operatorname{Ann}_R(x)\subsetneq R$, and so there is a prime ideal $\frak p$ with $\operatorname{Ann}_R(x)\subseteq\frak p$. Then $0\ne\frac{x}{1}\in M_{\frak p}$ and therefore $\frak p\in\operatorname{Supp}(M)$. \\[1ex] (\ref{Supp-exact-sequence}) For each $\frak p$ the sequence $0\rightarrow N_{\frak p}\rightarrow M_{\frak p}\rightarrow L_{\frak p} \rightarrow 0$ is exact. Therefore $M_{\frak p}\ne(0)$ iff $N_{\frak p}\ne(0)$ or $L_{\frak p}\ne(0)$. \\[1ex] (\ref{Supp-bigger-ideal}) Because of $M_{\frak p}=\left(M_{\frak p'}\right)_{\frak p\cdot R_{\frak p'}}$ from $M_{\frak p}\ne(0)$ follows $M_{\frak p'}\ne(0)$. \\[1ex] (\ref{Supp-and-Ann}) If $\frak p\nsupseteq\operatorname{Ann}_R(M)$ there is an $s\in\complement\frak p \cap\operatorname{Ann}_R(M)$, so that $s\cdot M=(0)$ and therefore $M_{\frak p}=(0)$. For the converse if $M$ is finitely generated see \ref{Supp-and-Ass} \\[1ex] (\ref{Supp-of-sums}) This follows immediately from the fact that $M_{\frak p}=\sum\limits_{i\in I}{N_i}_{\frak p}$ \end{pf} There is a close connection between the support of a module and its associated prime ideals: \begin{prop} \label{Supp-and-Ass} Let $M$ be an $R$-module and $\frak p$ a prime ideal of $R$. Consider the following conditions: \begin{enumerate} \item \label{p-in-Supp} $\frak p\in\operatorname{Supp}(M)$ \item \label{p-Ass-in-p} $\frak p$ contains a prime ideal of $\operatorname{Ass}(M)$. \item \label{p-ess-for submod} $\frak p$ is essential for a submodule of $M$. \item \label{p-ess1-for submod} $\frak p$ is essential of the first kind for a submodule of $M$. \item \label{p-contains-Ann} $\frak p\supseteq\operatorname{Ann}_R(M)$. \end{enumerate} Then {\fontshape{n}\selectfont(\ref{p-in-Supp})--(\ref{p-ess1-for submod})} are equivalent, and {\fontshape{n}\selectfont(\ref{p-contains-Ann})} follows from them.\\ If $M$ is finitely generated then {\fontshape{n}\selectfont(\ref{p-in-Supp})--(\ref{p-contains-Ann})} are equivalent. \end{prop} \begin{pf} (\ref{p-in-Supp}) $\Rightarrow$ (\ref{p-Ass-in-p}): Let $\frak p\in\operatorname{Supp}(M)$. Then $M_{\frak p}\ne(0)$ and therefore $\operatorname{Ass}(M_{\frak p})\ne\emptyset$ by Remark \ref{Ass1-rems} (\ref{M-not-zero-Ass1}). Let $\frak P'\in\operatorname{Ass}(M_{\frak p})$ and $\frak p':=\frak P'\cap R$. Then $\frak p'\in\operatorname{Ass}(M)$ by Corollary \ref{Ass-up-down} and $\frak p'\subseteq\frak p$. \\[1ex] (\ref{p-Ass-in-p}) $\Rightarrow$ (\ref{p-in-Supp}): Let $\frak p'\in\operatorname{Ass}(M)$ and $\frak p'\subseteq\frak p$. Then by Corollary \ref{p-Ass-Mp-not-0} $M_{\frak p'}\ne(0)$. But then a fortiori $M_{\frak p}\ne(0)$. \\[1ex] (\ref{p-in-Supp}) $\Rightarrow$ (\ref{p-ess1-for submod}): Let $\frak p\in\operatorname{Supp}(M)$. Then $M_{\frak p}\ne(0)$. Let $0\ne y\in M_{\frak p}$, and $\widetilde N:=\frak p\cdot R_{\frak p}\cdot y$. Then $y\notin\widetilde N$ by Krull-Nakayama. Let $N:=\widetilde N\cap M$, and therefore $\widetilde N=N_{\frak p}$. We will show that $\frak p\in\operatorname{Ass}_1(M/N)$:\\ Because of Proposition \ref{Ass1-up-down} it is enough to show that $\frak p\cdot R_{\frak p}\in\operatorname{Ass}_1\left((M/N)_{\frak p}\right)= \operatorname{Ass}_1(M_{\frak p}/\widetilde N)$:\\ If we denote by $\bar y$ the residue class of $y$ mod $\widetilde N$ we have $\bar y\ne0$, since $y\notin\widetilde N$, but $\frak p\cdot R_{\frak p}\cdot\bar y=0$ and therefore $\frak p\cdot R_{\frak p}\subseteq\operatorname{Ann}_{R_{\frak p}}(\bar y)$. But $\frak p\cdot R_{\frak p}$ is the maximal ideal of $R_{\frak p}$ and so equality holds. That means that $\frak p\cdot R_{\frak p}\in\operatorname{Ass}_0(M_{\frak p}/\widetilde N) \subseteq\operatorname{Ass}_1(M_{\frak p}/\widetilde N)$. \\[1ex] (\ref{p-ess1-for submod}) $\Rightarrow$ (\ref{p-ess-for submod}) is trivial since $\operatorname{Ass}_1(M/N)\subseteq\operatorname{Ass}(M/N)$. \\[1ex] (\ref{p-ess-for submod}) $\Rightarrow$ (\ref{p-in-Supp}): If $\frak p\in\operatorname{Ass}(M/N)$ then by Corollary \ref{p-Ass-Mp-not-0} $\left(M/N\right)_{\frak p}\ne(0)$. Because of $\left(M/N\right)_{\frak p}=M_{\frak p}/N_{\frak p}$ we have a fortiori $M_{\frak p}\ne (0)$. \\[1ex] (\ref{p-in-Supp}) $\Rightarrow$ (\ref{p-contains-Ann}) holds by Remark \ref{Supp-basics} (\ref{Supp-and-Ann}).\\ Now let $M$ be finitely generated. Then we show\\ (\ref{p-contains-Ann}) $\Rightarrow$ (\ref{p-in-Supp}): Let $\frak p\supseteq\operatorname{Ann}_R(M)$. For a finitely generated $R$-module one has\\ $\operatorname{Ann}_{R_{\frak p}}(M_{\frak p})=R_{\frak p}\cdot\operatorname{Ann}_R(M)$ (while in general only $\operatorname{Ann}_{R_{\frak p}}(M_{\frak p})\supseteq R_{\frak p}\cdot\operatorname{Ann}_R(M)$ (Remark \ref{transporteurs} (\ref{transp-quotmodule}))). Therefore $\operatorname{Ann}_{R_{\frak p}}(M_{\frak p})\subseteq \frak p\cdot R_{\frak p}\ne R_{\frak p}$ and therefore\\ $M_{\frak p}\ne(0)$. \end{pf} From (\ref{p-Ass-in-p})$\Rightarrow$(\ref{p-in-Supp}) we get: \begin{cor} \label{Ass-in-Supp} $$ \operatorname{Ass}(M)\subseteq\operatorname{Supp}(M) $$ \end{cor} \begin{cor} \label{Supp-R-mod-a} Let $\frak a$ be an ideal of $R$. Then $$ \operatorname{Supp}(R/\frak a)=\{\frak p\mid \frak p\supseteq\frak a\}. $$ \end{cor} \begin{pf} $\frak a=\operatorname{Ann}_R(R/\frak a)$ and $M:=R/\frak a$ is a finitely generated $R$-module. Proposition \ref{Supp-and-Ass} (\ref{p-in-Supp}) $\Leftrightarrow$ (\ref{p-contains-Ann}) gives the corollary. \end{pf} \begin{cor} \label{min-in-Ass-and-Supp} \strut \begin{enumerate} \item \label{min-Supp-min-Ass} $\frak p'$ is minimal in $\operatorname{Supp}(M)\Leftrightarrow\frak p'$ is minimal in $\operatorname{Ass}(M)$. \item \label{M-finite-ex-min-prime} If $M$ is finitely generated each $\frak p\in\operatorname{Supp}(M)$ contains a minimal\\ $\frak p'\in\operatorname{Supp}(M)$.\\ If $M$ is not finitely generated there may be no minimal elements in $\operatorname{Supp}(M)$. (See Example \ref{no-minimal-primes-in-Supp}.) \end{enumerate} \end{cor} \begin{pf} (\ref{min-Supp-min-Ass}): Let $\frak p'$ be minimal in $\operatorname{Supp}(M)$. Then by Proposition \ref{Supp-and-Ass} (\ref{p-in-Supp}) $\Rightarrow$ (\ref{p-Ass-in-p}) there is a $\frak p''\in\operatorname{Ass}(M)$ with $\frak p'\supseteq\frak p''$, which by Corollary \ref{Ass-in-Supp} lies in $\operatorname{Supp}(M)$ and therefore $\frak p'=\frak p''$, because of the minimality of $\frak p'$. So we get $\frak p'\in\operatorname{Ass}(M)$ and $\frak p'$ is also minimal in $\operatorname{Ass}(M)$, because of $\operatorname{Ass}(M)\subseteq\operatorname{Supp}(M)$.\\ Conversely, by the same arguments we see that a minimal prime ideal of $\operatorname{Ass}(M)$ is also minimal in $\operatorname{Supp}(M)$. \\[1ex] (\ref{M-finite-ex-min-prime}): By Proposition \ref{Supp-and-Ass}\ $\operatorname{Supp}(M)=\{\frak p\mid\frak p\supseteq\operatorname{Ann}_R(M)\}$. Now each $\frak p\supseteq\operatorname{Ann}_R(M)$ contains a $\frak p'\supseteq\operatorname{Ann}_R(M)$, which is minimal among the prime ideals containing $\operatorname{Ann}_R(M)$ and therefore minimal in $\operatorname{Supp}(M)$. \end{pf} \section{The Radical of a Submodule} \begin{defn} Let $N$ be a proper submodule of $M$. We define the ``radical of $N$ in $M$'' as $$ \frak r_M(N):=\{r\mid r\in R,\ r \text{ nilpotent for } M/N\}. $$ \end{defn} \begin{rem} \label{basics-radical} \strut \begin{enumerate} \item $\frak r_M(N)$ is an ideal of $R$. \item \label{radN-rad0} $\frak r_M(N)=\frak r_{M/N}((0))$. \item $\frak r_M((0))\supseteq\operatorname{Ann}_R(M)$. \item If $\frak a$ is an ideal of $R$ then $\frak r_R(\frak a)\supseteq\frak a$. \item $\frak r_R((0))=\{\text{nilpotent elements of R}\}$ is the ``nil-radical'' of $R$. \end{enumerate} \end{rem} \begin{prop} \label{nilpotents-intersection-of Ass} $$ \frak r_M(N)=\bigcap\limits_{\frak p\in\operatorname{Supp}(M/N)}\frak p =\bigcap\limits_{\frak p\in\operatorname{Ass}(M/N)}\frak p =\bigcap\limits_{\frak p\in\operatorname{Ass}_1(M/N)}\frak p $$ \end{prop} \begin{pf} Since $\operatorname{Ass}_1(M/N)\subseteq\operatorname{Ass}(M/N)$ and each $\frak p\in\operatorname{Ass}(M/N)$ contains a $\frak p'\in\operatorname{Ass}_1(M/N)$ one has $\bigcap\limits_{\frak p\in\operatorname{Ass}(M/N)}\frak p =\bigcap\limits_{\frak p\in\operatorname{Ass}_1(M/N)}\frak p$. \\[1ex] Because of Corollary \ref{Ass-in-Supp} we have $\operatorname{Ass}(M/N)\subseteq\operatorname{Supp}(M/N)$ and each\\ $\frak p\in\operatorname{Supp}(M/N)$ contains a $\frak p'\in\operatorname{Ass}(M/N)$.\\ Therefore $\bigcap\limits_{\frak p\in\operatorname{Supp}(M/N)}\frak p =\bigcap\limits_{\frak p\in\operatorname{Ass}(M/N)}\frak p$. \\[1ex] Because of Remark \ref{basics-radical} (\ref{radN-rad0}) we may now assume that $N=(0)$. Then $M\ne(0)$. Let $r\in\frak r_M((0))$. Then for each $x\in M$ there is a $\nu\in\Bbb N$ with $r^\nu\cdot x=0$ and therefore $M_S=(0)$ for each multiplicatively closed set $S$ which contains $r$. But for each $\frak p\in\operatorname{Supp}(M)$ by definition $M_{\complement\frak p}\ne (0)$ and therefore $r\in\frak p$ for all $\frak p\in\operatorname{Supp}(M)$, i.e. $\frak r_M((0))\subseteq\bigcap\limits_{\frak p\in\operatorname{Supp}(M)}\frak p$.\\ Conversely: Let $r\in\bigcap\limits_{\frak p\in\operatorname{Supp}(M)}\frak p$ and let $S:=\{r^\nu\mid\nu=0,1,2,\dots\}$. We will show that $M_S=(0)$, which means that for each $x\in M$ there is a $\nu\in\Bbb N$ with $r^\nu\cdot x=0$ and so $r\in\frak r_M((0))$:\\ $1^{st}$ case: $R_S=(0)$. Then also $M_S=(0)$, since we always assume $M$ to be a unitary $R$-module and in $R=(0)$ the $0$ is the unit element.\\ $2^{nd}$ case: $R_S\ne(0)$, hence $1\ne0$ in $R_S$. If $M_S\ne(0)$ there would be an $x\in M$ with $\frac{x}{1}\ne0$ in $M_S$, hence $\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)\ne R_S$. Then there would be a prime ideal $\frak P$ of $R_S$ with $\frak P\supseteq\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)$. Let $\frak p:=\frak P\cap R$. Then $\frak p\cap S=\emptyset$, hence $r\notin\frak p$.\\ But $M_{\frak p}=\left(M_S\right)_{\frak P}\ne(0)$, since $\frak P\supseteq\operatorname{Ann}_{R_S}\left(\frac{x}{1}\right)$. Therefore $\frak p\in\operatorname{Supp}(M)$ and so $r\in\frak p$, contradiction! \end{pf} \begin{cor} \label{coprimary-iff-Assp} $$ \operatorname{Ass}(M)=\{\frak p\}\Longleftrightarrow M\text{ is }\frak p\text{-coprimary}. $$ \end{cor} \begin{pf} If $M$ is $\frak p$-coprimary then $\operatorname{Ass}(M)=\{\frak p\}$ by Remark \ref{coprimary-Ass}.\\ Conversely: If $\operatorname{Ass}(M)=\{\frak p\}$ then by Theorem \ref{ass-zero-divisors} $\frak p$ is the set of all zero divisors for $M$ and by Proposition \ref{nilpotents-intersection-of Ass} each element of $\frak p$ is nilpotent for $M$. \end{pf} \begin{cor} $\frak p\in\operatorname{Supp}(M)\Longrightarrow\frak p\supseteq\frak r_M((0))$\\ But in general the converse is not true (Example \ref{p-rad-not-Supp}).\\ If $M$ is finitely generated, then the converse holds. \end{cor} \begin{pf} ``$\Longrightarrow$'' holds because of Proposition \ref{nilpotents-intersection-of Ass}.\\ ``$\Longleftarrow$'': Let $\frak p\supseteq\frak r_M((0))$ then $\frak p\supseteq\operatorname{Ann}_R(M)$ because of Remark \ref{basics-radical}. If $M$ is finitely generated then also $\frak p\in\operatorname{Supp}(M)$ by Proposition \ref{Supp-and-Ass}. \end{pf} \begin{cor} Let $N$ be a proper submodule of $M$ and $M/N$ finitely generated or $\operatorname{Ass}(M/N)$ finite. (E.g. if there exists a primary decomposition of $N$ in $M$.) Then $$ \frak r_M(N)=\bigcap\limits_{\vbox{\hsize1.8cm\noindent \size{8}{8pt}\selectfont $\frak p$ minimal in\\ $\operatorname{Ass}(M/N)$}} \frak p $$ \end{cor} \begin{pf} We will show that in both cases each $\frak p\in\operatorname{Ass}(M/N)$ contains a $\frak p'$ which is minimal in $\operatorname{Ass}(M/N)$. (Therefore one can restrict the intersection $\frak r_M(N)=\bigcap\limits_{\frak p\in\operatorname{Ass}(M/N)}\frak p$ to the minimal elements of $\operatorname{Ass}(M/N)$.): If $\operatorname{Ass}(M/N)$ is finite this is trivial. If $M/N$ is finitely generated then by Corollary \ref{min-in-Ass-and-Supp} (\ref{M-finite-ex-min-prime}) each $\frak p\in\operatorname{Ass}(M/N)$ contains a minimal element $\frak p'$ of $\operatorname{Supp}(M/N)$, and by Corollary \ref{min-in-Ass-and-Supp} (\ref{min-Supp-min-Ass}) this is also minimal in $\operatorname{Ass}(M/N)$. \end{pf} \begin{cor} If $\frak a$ is an ideal of $R$ then $$ \frak r_R(\frak a)=\bigcap\limits_{\frak p\supseteq\frak a}\frak p= \bigcap\limits_{\vbox{\hsize1.8cm\noindent \size{8}{8pt}\selectfont $\frak p$ minimal \\ containing $\frak a$}} \frak p $$ \end{cor} \begin{pf} By Corollary \ref{Supp-R-mod-a} $\operatorname{Supp}(R/\frak a)=\{\frak p\mid\frak p\supseteq\frak a\}$ and therefore by \ref{nilpotents-intersection-of Ass} $\frak r_R(\frak a)=\bigcap\limits_{\frak p\supseteq\frak a}\frak p$. Since each $\frak p\supseteq\frak a$ contains a $\frak p'$ which is minimal among the prime ideals containing $\frak a$ one can restrict the intersection to the minimal ones among the prime ideals. \end{pf} \begin{cor} If $R$ is ``reduced'' (i.e. $\frak r_R((0))=(0)$ ) then $$ \{\text{zero divisors of }R\}=\bigcap\limits_{\vbox{\hsize2.1cm\noindent \size{8}{8pt}\selectfont $\frak p$ minimal\\ prime ideal of $R$}}\frak p $$ \end{cor} \begin{pf} By Remark \ref{Ass1-rems} (\ref{essential-primes-for-ideal}) the minimal prime ideals of $R$ belong to\\ $\operatorname{Ass}(R/(0))=\operatorname{Ass}(R)$, and therefore by Theorem \ref{ass-zero-divisors} all of their elements are zero divisors for $R$.\\ Conversely: If $r$ is a zero divisor for $R$ there is an $s\in R$, $s\ne0$ with $r\cdot s=0\in\frak p$ for all minimal prime ideals of $R$. But since by hypothesis $\bigcap\limits_{\vbox{\hsize2.1cm\noindent \shape{it}\size{8}{8pt}\selectfont $\frak p$ minimal\\ prime ideal of $R$}}\frak p=\frak r_R((0))=(0)$ there is a minimal prime ideal $\frak p$ of $R$ with $s\notin\frak p$ and therefore $r\in\frak p$. \end{pf} \section{The (Counter-)Examples} \begin{ex}[\bf\boldmath $(0)$ indecomposable in $M$ but $M$ not coprimary] \label{0-indcomp-not-coprim} Let\\ $R$ be a rank 2 discrete valuation ring in the sense of Krull \cite{Krull-Bewertung} with valuation $\nu$ and value group $\Bbb Z\times\Bbb Z$ (lexicographically ordered).\\ $R$ has three prime ideals $\frak P_2\supset\frak P_1\supset (0)$. Let $\pi_1,\pi_2\in R$ be elements with $\nu(\pi_2)=(0,1)$ and $\nu(\pi_1)=(1,0)$. Then $\frak P_2=R\cdot\pi_2$ is a principal ideal, but $\frak P_1$ is not finitely generated. A generating set for $\frak P_1$ is $\{\pi_1/\pi_2^i\mid i=0,1,2,\dots\}$. {\fontshape{n}\selectfont (Compare also \cite{Berger-Modul-diskret-ganz-Diff}).}\\ Let $M:=R/R\cdot\pi_1$.\\ We claim that \begin{enumerate} \item $(0)$ is indecomposable in $M$ (i.e. $R\cdot\pi_1$ is indecomposable in $R$), but \item $M$ is {\em not} coprimary. \end{enumerate} \end{ex} \begin{pf} (1) Let $R\cdot\pi_1=\frak a\cap\frak b$ with ideals $\frak a,\frak b$ of $R$. If $R\pi_1\subsetneq\frak a$ and $R\pi_1\subsetneq\frak b$ then an element of value $<\nu(\pi_1)=(1,0)$ must be contained in $\frak a$ and in $\frak b$. Among these values $(1,-1)=\nu(\pi_1/\pi_2)$ is the biggest. Since $R$ is a valuation ring $\pi_1/\pi_2\in\frak a\cap\frak b=R\pi_1$. This cannot happen because the values of all elements of $R\pi_1$ are $\ge\nu(\pi_1)=(1,0)$. It follows that $R\pi_1$ is indecomposable in $R$.\\ (2) $\pi_1/\pi_2\notin R\pi_1$ but $\pi_2\cdot(\pi_1/\pi_2)=\pi_1\in R\pi_1$ $\Rightarrow \pi_2$ is a zero divisor for $M$.\quad But $\pi_2$ is not nilpotent for $M$; because for all $i\in\Bbb N$ we have $\pi_2^i\cdot 1\notin R\pi_1$, since $\nu(\pi_2^i)=(0,i)<(1,0)$. Therefore $M$ is not coprimary. \end{pf} \begin{ex}[\boldmath$\operatorname{Ass}(M)\supsetneq\operatorname{Ass}_1(M)$] \label{Ass-not-Ass1} Let\\ $R:=k[X_1,X_2,\dots]$ polynomial ring in countably many indeterminates over a field $k$,\\ $\frak p_i:=(X_1,X_2,\dots,X_i)$\\ $\frak p:=(X_1,X_2,X_3,\dots)$,\\ $M:=\bigoplus\limits_{i=1}^\infty R/\frak p_i = \bigoplus\limits_{i=1}^\infty R\cdot e_i$ with $e_i:= 1+\frak p_i\in R/\frak p_i$.\\ $\frak p_i$ and $\frak p$ are prime ideals,\\ $\frak p_i=\operatorname{Ann}_R(e_i)$ and therefore $\frak p_i\in\operatorname{Ass}_0(M)\subseteq\operatorname{Ass}_1(M)$ for $i=1,2,\dots$.\\ $\frak p=\bigcup\limits_{i=1}^\infty\frak p_i$ and therefore $\frak p\in\operatorname{Ass}(M)$ by Proposition \ref{union-of-Ass1-primes} (\ref{if-union-is-prime}).\\ But $\frak p\notin\operatorname{Ass}_1(M)$. \end{ex} \begin{pf} We have to show that $\frak p$ is not minimal in the set of all prime ideals containing the annihilator of an element of $M$:\\ Let $0\ne y\in M$ be arbitrary. Then there exists an $n\in\Bbb N$ with $y\in\bigoplus\limits_{i=1}^n R/\frak p_i$. Let $r\in\operatorname{Ann}_R(y)$ be an arbitrary element of $\operatorname{Ann}_R(y)$.\\ We show that $r\in\frak p_n$:\\ $y=\sum\limits_{i=1}^n\xi_i\cdot e_i$ with $\xi_i\in R$.\\ $r\cdot y=0\Rightarrow r\cdot\xi_i\cdot e_i=0 \Rightarrow r\cdot\xi_i\in\frak p_i$ for $i=1,\dots n$. But for an $i_0\in\{1,\dots,n\}$ we have $\xi_{i_0}\notin\frak p_{i_0}$ because else $y=0$. Then $r\in\frak p_{i_0}\subseteq\frak p_n$.\\ It follows that $\operatorname{Ann}_R(y)\subseteq\frak p_n\subsetneq\frak p$ and therefore $\frak p$ is not minimal among the prime ideals containing $\operatorname{Ann}_R(y)$. \end{pf} One can even find a cyclic $R$-module $M$ with $\operatorname{Ass}(M)\ne\operatorname{Ass}_1(M)$: \begin{ex}[\boldmath$M$ cyclic and $\operatorname{Ass}(M)\supsetneq\operatorname{Ass}_1(M)$] Let\\ $R':=k[X_1,Y_1,X_2,Y_2,\dots]$ the polynomial ring in the countably many independent indeterminates $X_i,Y_i$, $i=1,2,\dots$ over a field $k$,\\ $\frak a':=\left(X_1\cdot Y_1,Y_1^2,X_2\cdot Y_2,Y_2^2,\dots\right)$ ideal in $R'$,\\ $M:=R:=R'/\frak a'=k[x_1,y_1,x_2,y_2,\dots]$, where the $x_i,y_i$ denote the residue classes of the $X_i,Y_i$ mod $\frak a'$,\\ $\frak p:=(x_1,y_1,x_2,y_2,\dots)\subset R$.\\ Then $\frak p\in\operatorname{Ass}(M)$ but $\frak p\notin\operatorname{Ass}_1(M)$. \end{ex} \begin{pf} (1) The set $$ A:=\left\{x_1^{\nu_1}\cdot y_1^{\epsilon_1}\cdots x_n^{\nu_n}\cdot y_n^{\epsilon_n}\ \big|\ n\in\Bbb N,\ \nu_i\in\Bbb N_0,\ \epsilon_i\in\{0,1\},\ \epsilon_i=0 \text{ if }\nu_i>0\right\} $$ is a basis for $R$ as a $k$-vector space:\\ Obviously any polynomial of $R'$ can be reduced mod $\frak a'$ to a linear combination of monomials $X_1^{\nu_1}\cdot~Y_1^{\epsilon_1}\cdots% X_n^{\nu_n}\cdot~Y_n^{\epsilon_n}$ with $n\in\Bbb N,\ \nu_i\in\Bbb N_0,\ \epsilon_i\in\{0,1\},\ \epsilon_i=0 \text{ if }\nu_i>0$ and coefficients in $k$. Therefore $A$ is a set of generators for $R$ as a $k$-vector space.\\ On the other hand one sees that by definition of $\frak a'$ every monomial of an element of $\frak a'$ contains an $X_i\cdot Y_i$ or an $Y_i^2$, while a linear combination of the monomials $X_1^{\nu_1}\cdot~Y_1^{\epsilon_1}\cdots% X_n^{\nu_n}\cdot~Y_n^{\epsilon_n}$ with $n\in\Bbb N,\ \nu_i\in\Bbb N_0,\ \epsilon_i\in\{0,1\},\ \epsilon_i=0 \text{ if }\nu_i>0$ and coefficients in $k$ never contains these products. Therefore the elements of the set $A$ are also linearely independent over $k$. \\[1ex] (2) No element of $R\setminus\frak p$ is a zero divisor of $R$:\\ Let $F\in R'$ and $F=\sum_{i=0}^n F_i$ its decomposition into homogeneous polynomials (with respect to the total degree, all $X_i,Y_i$ having degree $1$). Then $F$ represents an element of $R\setminus\frak p$ modulo $\frak a'$ iff $F_0\ne 0$, and $F\in\frak a'$ iff all $F_i\in\frak a'$ since $\frak a'$ is generated by monomials (homogeneous elements).\\ Now let $T=\sum_{i=1}^n T_i\in R'$ represent an element $t\in R\setminus\frak p$ and let $Z=\sum_{i=1}^m Z_i\in R'$ represent an arbitrary element $z\in R$ with $t\cdot z=0$, i.e. $T\cdot Z\in\frak a'$. We may assume that $T_0=1$.\\ $T\cdot Z=\sum\limits_\lambda \left(\sum\limits_{\nu+\mu=\lambda}T_\mu\cdot Z_\nu\right)$ is a homogeneous decomposition. Therefore by the preceding remark $\sum\limits_{\nu+\mu=\lambda}T_\mu\cdot Z_\nu\in\frak a'$ for all $\lambda$. We show by induction on $\lambda$ that all $Z_i\in\frak a'$ and therefore $z=0$:\\ $\lambda=0$:\quad $Z_0\in\frak a'\cap k$=(0).\\ Now let $Z_0,\dots,Z_n\in\frak a'$. Then from $Z_{n+1}+Z_n\cdot T_1+\dots+Z_0\cdot T^{n+1}\in\frak a'$ we obtain $Z_{n+1}\in\frak a'$. \\[1ex] (3) From (2) we see that the canonical homomorphism $M\rightarrow M_{\frak p}$ is injective and so $\frac{y_1}{1}\cdots\frac{y_n}{1}\ne0$ in $M_{\frak p}$ for all $n=1,2,\dots$, because $y_1\cdots y_n\ne0$ in $M$ for all $n$ as elements of a $k$-basis of $M$ according to (1). \\[1ex] (4) We now proof that $\frak p\in\operatorname{Ass}(M)$ by showing that each element of $\frak p\cdot R_{\frak p}$ is a zero divisor for $M_{\frak p}$ (Proposition \ref{zero-divisors-in-Mp}): Let $p\in\frak p$ be arbitrary. Then $p$ can be written as $p=\sum\limits_{\nu_i,\mu_i\ge1}a_{\nu_1,\dots,\nu_n,\mu_1,\dots,\mu_n} \cdot x_1^{\nu_1}\cdots x_n^{\nu_n}\cdot y_1^{\mu_1}\cdots y_n^{\mu_n}$ with $a_{\nu_1,\dots,\nu_n,\mu_1,\dots,\mu_n}\in k$. Since each summand $\ne0$ contains at least one $x_i$ or $y_i$ with $i\in\{1,\dots n\}$, and since $x_i\cdot y_i=0$ and $y_i\cdot y_i=0$, we obtain $p\cdot y_1\cdots y_n=0$ in $M$ and therefore $\frac{p}{s}\cdot(\frac{y_1}{1}\cdots\frac{y_n}{1})=0$ in $M_{\frak p}$ for each element $s\in R\setminus\frak p$. By (3) $\frac{y_1}{1}\cdots\frac{y_n}{1}\ne0$ in $M_{\frak p}$, and therefore $\frac{p}{s}$ is a zero divisor for $M_{\frak p}$. \\[1ex] (5) $\frak p\notin\operatorname{Ass}_1(M)$:\\ Proof (indirect): If $\frak p$ was minimal among the prime ideals containing $\operatorname{Ann}_R(z)$ for a $z\in M$ then by Proposition \ref{min-prime-ideal-of Ann Rpx} each element of $\frak p\cdot R_{\frak p}$ would be nilpotent for $R_{\frak p}\cdot z$. According to (1) $z$ has a representation by the $k$-basis $A$:\\ $z=\sum a_{\nu_1,\dots,\nu_n,\epsilon_1,\dots,\epsilon_n}\cdot x_1^{\nu_1}\cdot y_1^{\epsilon_1}\cdots x_n^{\nu_n}\cdot y_n^{\epsilon_n}$ with $\nu_i\in\Bbb N_0$,\ $\epsilon_i\in\{0,1\}$,\ $\epsilon_1=0$ if $\nu_i>0$, and $a_{\nu_1,\dots,\nu_n,\epsilon_1,\dots,\epsilon_c}\in k$, not all of them $=0$.\\ Let $m$ be a natural number $m>n$ and $\lambda\in\Bbb N$ arbitrary.\\ Then $x_m^{\lambda}\cdot z= \sum a_{\nu_1,\dots,\nu_n,\epsilon_1,\dots,\epsilon_c}\cdot x_1^{\nu_1}\cdot y_1^{\epsilon_1}\cdots x_n^{\nu_n}\cdot y_n^{\epsilon_n}\cdot x_m^\lambda$ is again a representation of $x_m^\lambda\cdot z$ by the basis $A$ and therefore $x_m^\lambda\cdot z\ne0$, since not all of the coefficients are $0$. By (3) it follows that $(\frac{x_m}{1})^\lambda\cdot\frac{z}{1}\ne0$ in $M_{\frak p}$. But $x_m\in\frak p$ and so $\frac{x_m}{1}$ is an element of $\frak p\cdot R_{\frak p}$ which is not nilpotent for $R_{\frak p}\cdot z$. \end{pf} \begin{ex}[{\boldmath$\operatorname{Ass}(M)=\operatorname{Ass}_1(M)\supsetneq\operatorname{Ass}_0(M) =\emptyset$}]% \label{Ass=Ass1-not-Ass0} $M$ cyclic and $(0)$ has a primary decomposition in $M$: Let\\ $R$ a valuation ring with value group $\Gamma=\Bbb Q$ or $\Gamma=\Bbb R$ (rank one, non discrete),\\ $\nu$ the (additive) valuation of $R$,\\ $\frak P=\{z\mid z\in R,\ \nu(z)>0\}$ the maximal ideal of $R$,\\ $\frak a=\{z\mid z\in R,\ \nu(z)\ge 1\}$,\\ $M:=R/\frak a$.\\We show that: \begin{enumerate} \item \label{zero-primary-dec} $(0)$ is $\frak P$-primary in $M$. (Therefore $(0)$ has as primary decomposition in $M$.) \item \label{noAnn} There is no $x\in M$ with $\frak P=\operatorname{Ann}_R(x)$. \end{enumerate} It follows from {\shape{n}\selectfont(\ref{zero-primary-dec})} by Remark \ref{coprimary-Ass} that $\operatorname{Ass}(M)=\{\frak P\}$ and from {\shape{n}\selectfont(\ref{noAnn})} by definition that $\frak P\notin\operatorname{Ass}_0(M)$ and therefore $\operatorname{Ass}_0(M)=\emptyset$ since $\operatorname{Ass}_0(M)\subseteq\operatorname{Ass}(M)$. \end{ex} \begin{pf} (\ref{zero-primary-dec}): Let $p\in\frak P$ and $a\in\frak a$. Then there is an $n\in\Bbb N$ with $n\cdot\nu(p)\ge\nu(a)$, hence $p^n\in R\cdot a\subseteq\frak a$ and therefore $p^n\cdot M=(0)$, i.e. each element of $\frak P$ is nilpotent for $M$. But since $\frak P$ is the maximal ideal of $R$ all zero divisors for $M$ lie in $\frak P$ and so it follows that $(0)$ is $\frak P$-primary in $M$. \\[1ex] (\ref{noAnn}): If there was an $x\in M$ with $\operatorname{Ann}_R(x)=\frak P$ there would be a representative $z\in R\setminus\frak a$ with $p\cdot z\in\frak a$ for all $p\in\frak P$, i.e. $\nu(z)+\nu(p)\ge1$ for all $p\in\frak P$. Now $\Gamma\supseteq\Bbb Q$, and therefore for each $n\in\Bbb N$ there is a $p_n\in\frak P$ with $\nu(p_n)=\frac{1}{n}$. Then $\nu(z)+\frac{1}{n}\ge1$ for all $n\in\Bbb N$ and hence $\nu(z)\ge 1$, i.e. $z\in\frak a$ against our assumption. \end{pf} \begin{ex}[{\size{11}{12pt}\selectfont\boldmath $R$ noetherian (local), $\operatorname{Ass}(M)\supsetneq\operatorname{Ass}_1(M)$($=\operatorname{Ass}_0(M)$)}] \label{R-noeth-Ass-not-Ass1}\strut\\ Let $R:=k[X,Y]_{(X,Y)}$ localization of the polynomial ring in $X$ and $Y$ over a field $k$,\\ $\cal P:=\{R\cdot p\mid p\in R,\ R\cdot p\text{ prime ideal of }R\}$,\\ $M:=\bigoplus\limits_{R\cdot p\in\cal P}R/R\cdot p$.\\ Then\\ $\operatorname{Ass}_1(M)=\{R\cdot p\mid R\cdot p\in\cal P\}$,\\ $\frak m:=(X,Y)\in\operatorname{Ass}(M)\setminus\operatorname{Ass}_1(M)$ \end{ex} \begin{pf} Denote $U_p:=R/R\cdot p = R\cdot e_p$. Then $\operatorname{Ann}_R(e_p)=R\cdot p$, hence $R\cdot p\in\operatorname{Ass}_0(M)\subseteq\operatorname{Ass}_1(M)$.\\ Conversely let $0\ne\xi\in M$ arbitrary and $z\in\operatorname{Ann}_R(\xi)$. $\xi=\sum\xi_p\cdot e_p$ with $\xi_{p_0}\cdot e_{p_0}\ne0$ for some $R\cdot p_0\in\cal P$, i.e. $\xi_{p_0}\notin R\cdot p$. Now from $z\cdot \xi=0$ we get $z\cdot \xi_{p_0}\cdot e_{p_0}=0$, i.e. $z\cdot\xi_{p_0}\in R\cdot p_0$ and therefore $z\in R\cdot p_0$ since $\xi_{p_0}\notin R\cdot p_0$ and $R\cdot p_0$ is a prime ideal. It follows that $\operatorname{Ann}_R(\xi)\subseteq R\cdot p_0$. Since the only non principal prime ideal of $R$ is $\frak m$, which contains all the $R\cdot p$, it follows that the minimal elements among the prime ideals containing the annihilator of an element of $M$ are the principal prime ideals $R\cdot p$. So we obtain that $\operatorname{Ass}_1(M)=\{R\cdot p\mid R\cdot p\in\cal P\}$.\\ Then $\frak m\notin\operatorname{Ass}_1(M)$. But $\frak m\in\operatorname{Ass}(M)$:\\ In view of Proposition \ref{zero-divisors-in-Mp} we only must show that each element of $\frak m$ is a zero divisor for $M$ ($R_{\frak m}=R$ !). To show this let $z\in\frak m$ be arbitrary. $R$ being a UFD there is a prime element $p$ and a $z_1\in R$ with $z=z_1\cdot p$. But then $z\cdot e_p=0$ and therefore $z$ is a zero divisor for $M$. (Another way of showing $\frak m\in\operatorname{Ass}(M)$ would be to use Proposition \ref{union-of-Ass1-primes} (\ref{if-union-is-prime}).) \end{pf} \begin{ex}[\boldmath$\operatorname{Ass}(N\oplus L)\nsubseteq\operatorname{Ass}(N)\cup\operatorname{Ass}(L)$] \label{Ass-exact-sequ-false}\strut\\ Let $R$,\ $\frak m$, and $M$ be the same as in Example \ref{R-noeth-Ass-not-Ass1},\\ $N:=R/R\cdot X$,\\ $L:=\bigoplus\limits_{\size{8}{6pt}\selectfont \begin{array}{c} R\cdot p\in\cal P\\ R\cdot p\ne R\cdot X \end{array}}% \hbox to 6mm{\hss$R/R\cdot p$}$. \\[1ex] Then we have by definition\\ $M=N\oplus L$ and therefore we have the splitting exact sequence\\ $0\rightarrow N\rightarrow M\rightarrow L\rightarrow 0$ \\[1ex] We show \begin{enumerate} \item \label{m-in-AssM} $\frak m\in\operatorname{Ass}(M)$ \item \label{m-notin-AssN} $\frak m\notin\operatorname{Ass}(N)$ \item \label{m-notin-AssL} $\frak m\notin\operatorname{Ass}(L)$ \end{enumerate} \end{ex} \begin{pf} (\ref{m-in-AssM}): This was already shown in Example \ref{R-noeth-Ass-not-Ass1}. \\[1ex] (\ref{m-notin-AssN}): Since $X$ is a prime element of $R$ and $N=R/R\cdot X$ we see that $N$ is $R\cdot X$-coprimary and so $\operatorname{Ass}(M)=\{R\cdot X\}\not\ni\frak m$. \\[1ex] (\ref{m-notin-AssL}): By definition of $L$ it is obvious that $X$ is not a zero divisor for $L$ and therefore $\frak m\notin\operatorname{Ass}(L)$, because $X\in\frak m$. \end{pf} \begin{ex} [{\bf\boldmath$\frak p\supseteq\frak r_M((0))$ but $\frak p\notin\operatorname{Supp}(M)$}] \label{p-rad-not-Supp}Let\\ $R:=\Bbb Z$,\\ $M:=\bigoplus \limits_{\size{8}{9pt}\selectfont \begin{array}{c} 0\ne (p)\\ \text{ prime ideal} \end{array}}% \hbox to.3cm{\hss$\Bbb Z/(p)$}$.\quad Then\\ $\operatorname{Supp}(M)=\{(p)\mid 0\ne p\text{ prime element in }\Bbb Z\}$, but\\ $\frak r_M((0))=(0)$. \\[1ex] So $\frak p:=(0)\supseteq\frak r_M((0))$, but $\frak p\notin\operatorname{Supp}(M)$. \end{ex} \begin{pf} $(0)\notin\operatorname{Supp}(M)$, because $M$ is a torsion module and $R_{(0)}=\Bbb Q$ is a field. Therefore $M_{(0)}=(0)$.\\ $\frak r_M((0))=(0)$: For each $0\ne n\in\Bbb Z$ there is a prime element $p$ with $p\nmid n$ and therefore $p\nmid n^\nu$ for all $\nu\in\Bbb N$. So $n^\nu\cdot e_p\ne0$ for all $\nu\in\Bbb N$, with $e_p:=1+(p)$ in the summand $\Bbb Z/(p)$ of $M$. Therefore $n\notin\frak r_M((0))$, i.e. $\frak r_M((0))=(0)$. \end{pf} \begin{ex}[\bf\boldmath$\frak p\supseteq\operatorname{Ann}_R(M)$, but $\frak p\notin\operatorname{Supp}(M)$] \label{Ann-not-in-Supp}\strut\\ Let: $R:=\Bbb Z$,\quad $M:=\Bbb Q/\Bbb Z$.\\ Then $\operatorname{Ann}_R(M)=(0)$, but $(0)\notin\operatorname{Supp}(M)$ since $R_{(0)}=\Bbb Q$ is a field and $M$ is a torsion module. \end{ex} \begin{ex} [{\bf\boldmath No minimal elements in $\operatorname{Supp}(M)$}] \label{no-minimal-primes-in-Supp} Let\\ $R:=k[X_1,X_2,\dots]$ a polynomial ring in countably many indeterminates over a field $k$,\\ $\frak p_i:=(X_i,X_{i+1},\dots)$,\\ $M:=\bigoplus\limits_{i=1}^\infty R/\frak p_i$. \\[1ex] There are no minimal elements in $\operatorname{Supp}(M)$ \end{ex} \begin{pf} Let $M_i:=R/\frak p_i$. By Corollary \ref{Supp-R-mod-a} we have $\operatorname{Supp}(M_i)=\{\frak p\mid\frak p\supseteq\frak p_i\}$. Further by Remark \ref{Supp-basics} (\ref{Supp-of-sums}) $\operatorname{Supp}(M)=\bigcup\limits_{i=1}^\infty\operatorname{Supp}(M_i)$. For every $\frak p\in\operatorname{Supp}(M)$ there is an $i_0$ with $\frak p\in\operatorname{Supp}(M_{i_0})$ and therefore $\frak p\supseteq\frak p_{i_0}\supsetneq\frak p_{i_0+1} \supsetneq\dots$ and all the $\frak p_i\in\operatorname{Supp}(M)$. So obviously $\frak p$ is not minimal in $\operatorname{Supp}(M)$. \end{pf} \begin{ex}[{\size{11}{12pt}\selectfont\bf\boldmath Essential prime ideals for $\operatorname{Ann}_R(M)$ not in $\operatorname{Ass}(M)$}] \label{0-primary-decomp-Ann-not ass}\strut\\ Let $R$ a rank one discrete valuation ring,\quad $\frak p$ the maximal ideal of $R$,\\ $M:=\operatorname{Quot}(R)/R$.\\ $\operatorname{Ann}_R(M)=(0)$, a prime ideal of $R$.\\ Then\\ $\operatorname{Ann}_R(M)$ has a primary decomposition and $(0)$ is essential for $\operatorname{Ann}_R(M)$,\\ but\\ $M$ is $\frak p$-coprimary, because each element of $\frak p$ is nilpotent for $M$ while the elements of $R\setminus\frak p$ are units of $R$. Then $\operatorname{Ass}(M)=\{\frak p\}\not\ni(0)$ (Remark \ref{coprimary-Ass}). \end{ex}

9301

alg-geom/9301003

en

https://arxiv.org/abs/alg-geom/9301003

alg-geom/9301003

Marc Coppens and Takao Kato

Non-trivial Linear Systems on Smooth Plane Curves

15 pages, LaTeX 2.09

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and $r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let $g^r_n$ be a base point free very special non-trivial complete linear system on $C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b \le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is best possible.

alg-geom

\section{Introduction} Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. In \cite{noether}, while studying space curves, Max Noether considered the following question. For $n\in{\bbb Z}_{\ge 1}$ find $\ell (n)\in{\bbb Z}_{\ge 0}$ such that there exists a linear system $g^{\ell (n)}_n$ on $C$ but no linear system $g^{\ell (n)+1}_n$ and classify those linear systems $g^{\ell (n)}_n$ on $C$. The arguments given by Noether in the answer to this question contained a gap. In \cite{cil} C.~Ciliberto gave a complete proof for Noether's claim using different arguments. In \cite{harts1} R.~Hartshorne completed Noether's original arguments by solving the problem also for integral (not necessarily smooth) plane curves (see Remark \ref{rem:1}). The linear systems $g^{\ell (n)}_n$ are either non-special, or special but not very special, or very special but trivial. By a very special (resp. trivial) linear system on a smooth plane curve $C$ we mean: \vs 2 {\raggedright {\sc Definition}. } {\it A linear system $g^r_n$ on $C$ is very special if $r\ge 1$ and $\dim |K_C-g^r_n|\ge 1$. $($Here $K_C$ is a canonical divisor on $C)$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and $r=\fracd{m^2+3m}{2}-(md-n)$. A complete very special linear system $g^r_n$ on $C$ is trivial if its associated base point free linear system is trivial.} \vs 2 In this paper, we consider the following question. For $n\in{\bbb Z}_{\ge 1}$ find $r(n)$ such that there exists a very special non-trivial complete linear system $g^{r(n)}_n$ on $C$ but no such linear system $g^{r(n)+1}_n$. Our main result is the following: \vs 2 {\raggedright {\sc Theorem}. } {\it Let $g^r_n$ be a base point free very special non-trivial complete linear system on $C$. Write $r=\fracd{(x+1)(x+2)}{2}-\beta$ with $x, \beta$ integers satisfying $x\ge 1, 0\le\beta\le x$. Then $$ n\ge n(r):=(d-3)(x+3)-\beta . $$} \vs 2 This theorem only concerns linear systems of dimension $r\ge 2$. But 1-dimensional linear systems are studied in \cite{c3}. From these results one finds that $C$ has no non-trivial very special linear system of dimension 1 if $d\le 5$ and for $d\ge 6$, $C$ has non-trivial very special complete linear systems $g^1_{3d-9}$ but no such linear system $g^1_n$ with $n<3d-9$. The proof of this theorem is also effective for case $r=1$ if one modifies it little bit. Concerning the original problem one can make the following observation. For $x\ge d-2$ one has $r>g(C)$ and of course $C$ has no non-trivial very special linear systems $g^r_n$. For $x\le d-3$ one has $(d-3)(x+2)\le (d-3)(x+3)-x$. So, if the bound $n(r)$ is sharp, then also the bound $r(n)$ can be found. Concerning the sharpness of the bound $n(r)$, we prove it in case ${\rm char}(k)=0$ for $x\le d-6$. In case ${\rm char}(k)\ne 0$ we prove that there exists smooth plane curves of degree $d$ with a very special non-trivial $g^r_{n(r)}$ in case $x\le d-6$. Finally for the case $x>d-6$ we prove that there exist no base point free complete very special non-trivial linear systems of dimension $r$ on $C$. Hence, at least in case ${\rm char}(k)=0$ the numbers $r(n)$ are determined. \vs 2 {\raggedright{\sc Some Notations}} \vs 1 We write ${\bbb P}_a$ to denote the space of effective divisors of degree $a$ on ${\bbb P}^2$. If ${\bbb P}$ is a linear subspace of some ${\bbb P}_a$ then we write ${\bbb P} .C$ for the linear system on $C$ obtained by intersection with divisors ${\mit \Gamma}\in{\bbb P}$ not containing $C$. We write $F({\bbb P} .C)$ for the fixed point divisor of ${\bbb P} .C$ and $f({\bbb P} .C)$ for the associated base point free linear system on $C$, so $f({\bbb P} .C)=\{ D-F({\bbb P} .C):D\in{\bbb P} .C\}$. If $Z$ is a 0-dimensional subscheme of ${\bbb P}^2$ then ${\bbb P}_a(-Z)$ is the subspace of divisors $D\in{\bbb P}_a$ with $Z\subset D$. \setcounter{equation}{0} \section{Bound on the degree of non-trivial linear systems} A complete linear system $g^r_n$ on a smooth curve $C$ is called very special if $r\ge 1$ and $\dim |K_C-g^r_n|\ge 1$. From now on, $C$ is a smooth plane curve of degree $d$ and $g^r_n$ is a very special base point free linear system on $C$ with $r\ge 2$. \vs 1 \begin{defi} $g^r_n$ is called a trivial linear system on $C$ if there exists an integer $m\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and $r=\fracd{m^2+3m}{2}-(md-n)$. \label{def:trivial} \end{defi} \vs 2 \begin{thm} Write $r=\fracd{(x+1)(x+2)}{2}-\beta$ with $x, \beta$ integers satisfying $x\ge 1, 0\le\beta\le x$. If $g^r_n$ is not trivial, then $$ n\ge n(r):=(d-3)(x+3)-\beta . $$ \label{thm:1} \end{thm} \vs 1 \begin{rem} {\rm In the proof of this theorem we are going to make use of the main result of Hartshorne \cite{harts1} which describes the linear systems on $C$ of maximal dimension with respect to their degrees. The result is as follows:} Let $g^r_n$ be a linear system on $C$ $($not necessarily very special$)$. Write $g(C)=\fracd{(d-1)(d-2)}{2}$. {\rm i)} If $n>d(d-3)$ then $r=n-g$ $($the non-special case$)$ {\rm ii)} If $n\le d(d-3)$ then write $n=kd-e$ with $0\le k\le d-3, 0\le e<d$, one has \vs 1 $\left\{ \begin{array}{ll} r\le\fracd{(k-1)(k+2)}{2} & {\rm if\ }e>k+1\\ r\le\fracd{k(k+3)}{2}-e & {\rm if\ }e\le k+1. \end{array}\right.$ \vs 1 {\rm Hartshorne also gives a description for the case one has equality. This theorem (a claim originally stated by M. Noether with an incomplete proof) is also proved in \cite{cil}. However, Hartshorne also proves the theorem for integral plane curves using the concept of generalized linear systems on Gorenstein curves. We need this more general result in the proof of Theorem \ref{thm:1}} \label{rem:1} \end{rem} \vs 2 {\it Proof of Theorem\/} \ref{thm:1}. Assume $g^r_n=rg^1_{n/r}$ and $n<(x+3)(d-3)-\beta$. Noting $2r=(x+1)(x+2)-2\beta\ge x^2+x+2\ge x+3$, we have $\fracd{(x+3)(d-3)-\beta}{r}< 2(d-2)$. Hence, $g^1_{n/r}=|g^2_d-P|$ for some $P\in C$. Since $\dim |rg^1_{n/r}|=r$, certainly $\dim |2g^1_{n/r}|=2$. But $\dim|2g^2_d-2P|=3$. A contradiction. Since $g^r_n$ is special, there exist an integer $1\le m\le d-3$ and a linear system ${\bbb P}\subset{\bbb P}_m$ such that $g^r_n=f({\bbb P}.C)$ and ${\bbb P}$ has no fixed components. In Lemma \ref{lem:1} we are going to prove that, because $g^r_n$ is not a multiple of a pencil, a general element ${\mit \Gamma}$ of ${\bbb P}$ is irreducible. Now, for each element ${\mit \Gamma}'$ of ${\bbb P}$ we have $F({\bbb P}.C)\subset{\mit \Gamma}'$ (inclusion of subschemes of ${\bbb P}^2$). In particular $F({\bbb P}.C)\subset{\mit \Gamma}\cap{\mit \Gamma}'$. This remark is known in the literature as Namba's lemma. As a subscheme of ${\mit \Gamma}$, $F({\bbb P}.C)$ is an effective generalized divisor on ${\mit \Gamma}$ (terminology of \cite{harts1}). We find that for each ${\mit \Gamma}'\in{\bbb P}$ with ${\mit \Gamma}'\ne{\mit \Gamma}$ the residual of $F({\bbb P}.C)$ in ${\mit \Gamma}\cap{\mit \Gamma}'$ (we denote it by ${\mit \Gamma}\cap{\mit \Gamma}'-F({\bbb P}.C)$) is an element of the generalized complete linear system on ${\mit \Gamma}$ associated to ${\cal O}_{{\mit \Gamma}}(m-F({\bbb P}.C))$. Hence, we obtain a generalized linear system $g^{r-1}_{m^2-(md-n)}$ on ${\mit \Gamma}$. Now we are going to apply Hartshorne's theorem (Remark \ref{rem:1}) to this $g^{r-1}_{m^2-(md-n)}$ on ${\mit \Gamma}$. Since $g^r_n$ is non-trivial on $C$, we know that $r>\fracd{m^2+3m}{2}-(md-n)$. If $m^2-(md-n)>m(m-3)$, then i) in Remark \ref{rem:1} implies $r-1\le m^2+n-md-\fracd{(m-1)(m-2)}{2}$ so $r\le\fracd{m^2+3m}{2}-(md-n)$, a contradiction. So $m^2-(md-n)\le m(m-3)$ and we apply ii) in Remark \ref{rem:1}. We find $x\le m-3$ and $m^2+n-md\ge mx-\beta$, so $n\ge\varphi (m):=-m^2+m(d+x)-\beta$. Since $x+3\le m\le d-3$, we find $n\ge\varphi (x+3)=\varphi (d-3)=(d-3)(x+3)-\beta=n(r)$. This completes the proof of the theorem except for the proof of Lemma \ref{lem:1}. \vs 2 \begin{lem} Let $C$ be a smooth plane curve of degree $d$ and let $g^r_n$ be a base point free complete linear system on $C$ which is not a multiple of a one-dimensional linear system. Suppose there exists a linear system ${\bbb P}\subset{\bbb P}_e$ without fixed component for some $e\le d-1$ such that $g^r_n=f({\bbb P}.C)$. Then the general element of ${\bbb P}$ is irreducible. \label{lem:1} \end{lem} \def\underline{e}{\underline{e}} \def\underline{m}{\underline{m}} \vs 2 {\it Proof\/}. Let $F=F({\bbb P} .C)=\sum_{j=1}^sn_jP_j$ with $n_j\ge 1$ and $P_i\ne P\j$ for $i\ne j$. For $t\in{\bbb Z}_{\ge 1}, \underline{e} = (e_1,\dots ,e_t)\in ({\bbb Z}_{\ge 1})^t$ with $\sum_{i=1}^te_i=e$ and $\underline{m} =[m_{ij}]_{1\le i\le t,1\le j\le s}$, let $$ V(t,\underline{e} ,\underline{m} )=\{ {\mit \Gamma}_1+\cdots +{\mit \Gamma}_t:{\mit \Gamma}_i\in{\bbb P}_{e_i}{\rm\ is\ irreducible\ and\ }i({\mit \Gamma}_i,C;P_j)=m_{ij}\}. $$ It is not so difficult to prove that this defines a stratification of ${\bbb P}_e$ by means of locally closed subspaces. Since ${\bbb P}$ is irreducible there is a unique triple $(t_0,\underline{e}_0, \underline{m}_0)$ such that ${\bbb P}\cap V(t_0,\underline{e}_0,\underline{m}_0)$ is an open non-empty subset of ${\bbb P}$. In particular, ${\bbb P}\subset \{ {\mit \Gamma}_1+\cdots +{\mit \Gamma}_{t_0}:{\mit \Gamma}_i\in{\bbb P}_{e_{0i}}{\rm\ and\ } i({\mit \Gamma}_i,C;P_j)\ge m_{0ij}\}$. We need to prove that $t_0=1$, so assume that $t_0>1$. Let forget the subscript $0$ from now on. Let $F_i=\sum_{j=1}^sm_{ij}P_j\subset C$. For each $D\in g^r_n$ there exists ${\mit \Gamma} ={\mit \Gamma}_1+\dots +{\mit \Gamma}_t$ with ${\mit \Gamma}_i\in{\bbb P}_i(-F_i)$ and $D={\mit \Gamma} .C-(F_1+\dots +F_s)=\sum_{j=1}^t({\mit \Gamma}_i.C-F_i)$. Writing $D_i={\mit \Gamma}_i.C-F_i\in|e_ig^2_d-F_i|$ we find $D=\sum_{i=1}^tD_i$. Suppose for some $1\le i\le t$ we have $\dim|e_ig^2_d-F_i|=0$. If ${\mit \Gamma}'$ and ${\mit \Gamma}''$ are in ${\bbb P}_i(-F_i)$ then ${\mit \Gamma}'.C={\mit \Gamma}''.C$, but $e_i<d$ so ${\mit \Gamma}'={\mit \Gamma}''$. This implies that ${\bbb P}_i(-F_i)=\{{\mit \Gamma}_0\}$, but then ${\mit \Gamma}_0$ is a fixed component of ${\bbb P}$, a contradiction. Hence, for $1\le i\le t$, we have $\dim|e_ig^2_d-F_i|\ge 1$. Now, let $L_i$ be the irreducible sheaf on $C$ associated to $|e_ig^2_d-F_i|$ and let $L$ be the irreducible sheaf on $C$ associated to $g^r_n$. Then $L=L_1\otimes\cdots\otimes L_t$ and we find that the natural map $$ H^0(C,L_1)\otimes\cdots\otimes H^0(C,L_t)\to H^0(C,L) $$ is surjective, while $\dim H^0(C,L_i)\ge 2$ for $1\le i\le t$. From \cite[Corollary 5.2]{eisen}, it follows that $g^r_n$ is a multiple of a pencil. But this is a contradiction. \vs 2 \begin{rem} {\rm In \cite{mez} one makes a classification of linear systems on smooth plane curves for which $r$ is one less than the maximal dimension with respect to the degree. In that paper one uses arguments like in \cite{cil}. That classification is completely contained in our Theorem \ref{thm:1}} \label{rem:2} \end{rem} \vs 2 \begin{rem} {\rm If $r\ge n-\fracd{(d-1)(d-2)}{2}+d-1$ then $\dim |K_C-g^r_n|\ge d-2$. Hence, $|K_C-g^2_d-g^r_n|=|(d-4)g^2_d-g^r_n|\ne\emptyset$. So in this case we can assume $m\le d-4$ in the proof of Theorem \ref{thm:1}. Then, in the proof of Theorem \ref{thm:1}, using Bertini's theorem, we can prove that, for $D\in g^r_n$ general there exists an irreducible curve ${\mit \Gamma}$ of degree $d-3$ with ${\mit \Gamma} .C\ge D$ (see argument in \cite[p.384]{harts1}). So we don't need the proof of Lemma \ref{lem:1} under that assumption on $r$.} \label{rem:25} \end{rem} \vs 2 \begin{rem} {\rm In that case $n=n(r)$, we find $m=x+3$ and $m^2+n-md=mx-\beta$. So the generalized linear system $g^{r-1}_{m^2+n-md}=g^{r-1}_{mx-\beta}$ on ${\mit \Gamma}$ is of maximal dimension with respect to its degree. The description of those linear systems in \cite{harts1} implies that it is induced by a family of plane curves of degree $x$ containing some subspace $E\subset{\mit \Gamma}$ of length $\beta$. Writing $Z=F({\bbb P} .C)\subset{\mit \Gamma}$ we find $|{\bbb P}_m.{\mit \Gamma} -Z|=|{\bbb P}_x.{\mit \Gamma} -E|$ and so $Z\in |{\bbb P}_3.{\mit \Gamma} +E|$. In order to find non-trivial $g^r_{n(r)}$'s it is interesting to find for a smooth curve $C$ of degree $d$, a curve ${\mit \Gamma}$ of degree $m$ and a curve ${\mit \Gamma}'$ of degree 3 such that ${\mit \Gamma}\cap{\mit \Gamma}'\subset C$. We discuss this in \S 3. First we solve the following postulation problem: let ${\mit \Gamma}'$ be the union of 3 distinct lines $L_1, L_2, L_3$ and let $D_i$ be an effective divisor of degree $a$ on $L_i$ with $D_i\cap (L_j\cup L_k)=\emptyset$ for $\{ i,j,k\} =\{ 1,2,3\}$. Give necessary and sufficient conditions for the existence of a smooth curve ${\mit \Gamma}$ of degree $a$ such that ${\mit \Gamma} .L_i=D_i$ for $i=1,2,3$.} \label{rem:3} \end{rem} \setcounter{equation}{0} \def\binom#1#2{{#1\choose#2}} \section{Carnot's theorem} We begin with pointing out the following elementary fact: \vs 1 \begin{lem} Let $m(\ge 4)$ and $m_j\ (j=1,\dots ,\ell)$ be positive integers satisfying $\displaystyle\sum^{\ell}_{j=1}m_j=m$ and let ${\mit \Phi} (X)=\displaystyle\sum^m_{j=1}a_jX^j$ be a non-zero polynomial of degree at most $m$. If ${\mit \Phi} (X)$ is divisible by $(X-X_j)^{m_j}$ for $\ell$ distinct values $X_1,\dots ,X_{\ell}$, then the ratio $a_0:\cdots :a_m$ is uniquely determined. In particular, $a_m\ne 0, a_0=(-1)^ma_m\displaystyle\prod^{\ell}_{j=1}X^{m_j}_j$ and $a_{m-1}=-a_m\displaystyle\sum^{\ell}_{j=1}m_jX_j$. \label{lem:31} \end{lem} \vs 1 Using this, we have the following Carnot's theorem and infinitesimal Carnot's theorems. Another generalization of this theorem is given by Thas et al. \cite{thas} (see also \cite{ver}). They call this B.~Segre's generalization of Menelaus' theorem. \vs 1 \begin{lem}{\rm (Carnot, cf. \cite[Proposition 1.8]{ver}, \cite[p.219]{koe})} Let $L_1, L_2, L_3$ be lines in ${\bbb P}^2$ so that $L_1\cap L_2\cap L_3=\emptyset$ and let $D_i=\displaystyle\sum^{\ell_i}_{j=1}m_{ij}P_{ij},\ (\displaystyle\sum^{\ell_i}_{j=1}m_{ij}=m, i=1,2,3)$ be an effective divisor on $L_i$ such that $D_{i_1}\cap L_{i_2}=\emptyset$ if $i_1\ne i_2$. Assume $(x:y:z)$ is a coordinate system on ${\bbb P}^2$ such that $L_1, L_2, L_3$ correspond to the coordinate axes $x=0, y=0, z=0$, respectively. Let the coordinate of $P_{ij}$ be given by $(x_{ij}:y_{ij}:z_{ij})$ $($of course $x_{1j}=y_{2j}=z_{3j}=0)$. Then, there exists a curve ${\mit \Gamma}$ not containing any one of the lines $L_1,L_2,L_3$ of degree $m$ satisfying $({\mit \Gamma} .L_i)=D_i$ $(i=1,2,3)$ if and only if \begin{equation} \prod^{\ell_1}_{j=1}\left(\frac{y_{1j}}{z_{1j}}\right)^{m_{1j}} \prod^{\ell_2}_{j=1}\left(\frac{z_{2j}}{x_{2j}}\right)^{m_{2j}} \prod^{\ell_3}_{j=1}\left(\frac{x_{3j}}{y_{3j}}\right)^{m_{3j}} =(-1)^m. \label{eq:31} \end{equation} \label{lem:car1} \end{lem} \vs 1 {\it Proof\/}. Assume there exists a curve ${\mit \Gamma}$ of degree $m$ not containing any one of the lines $L_1,L_2,L_3$ satisfying $({\mit \Gamma} .L_i)=D_i$ $(i=1,2,3)$. Such a curve is given by ${\mit \Phi} (x,y,z)=\displaystyle\sum_{i+j+k=m}a_{ijk}x^iy^jz^k=0$. In this description, if $i({\mit \Gamma} ,L_1;P_{1j})=m_{1j}$, then ${\mit \Phi} (0,y,z)$ is divisible by $(y_{1j}z-z_{1j}y)^{m_{1j}}$. This implies $a_{00m}=(-1)^ma_{0m0}\displaystyle\prod^{\ell_1}_{j=1} \left(\fracd{y_{1j}}{z_{1j}}\right)^{m_{1j}}$. Similarly, we have $a_{m00}=(-1)^ma_{00m}\displaystyle\prod^{\ell_2}_{j=1} \left(\fracd{z_{2j}}{x_{2j}}\right)^{m_{2j}}$ and $a_{0m0}=(-1)^ma_{m00}\displaystyle\prod^{\ell_3}_{j=1} \left(\fracd{x_{3j}}{y_{3j}}\right)^{m_{3j}}$. Since ${\mit \Gamma}$ does not contain an intersection point $L_{i_1}\cap L_{i_2}$ for $i_1\ne i_2$, we have $a_{m00}a_{0m0}a_{00m}\ne 0$. Hence, we have (\ref{eq:31}). For the converse, take $a_{m00}=1$. Then, by (\ref{eq:31}) we can determine $a_{ijk}$ so that ${\mit \Gamma}$ has the desired property. This completes the proof. \vs 2 Next, we see two infinitesimal cases i.~e. case $D_{i_1}\cap L_{i_2}\ne\emptyset$ and case $L_1\cap L_2\cap L_3\ne\emptyset$. \vs 2 \begin{lem} Let $L_1, L_2, L_3$ be lines in ${\bbb P}^2$ so that $L_1\cap L_2\cap L_3=\emptyset$ and let $D_i=\displaystyle\sum^{\ell_i}_{j=1}m_{ij}P_{ij}, (\displaystyle\sum^{\ell_i}_{j=1}m_{ij}=m)$ be an effective divisor on $L_i$ such that $m_{11}=m_{21}=1, P_{11}=P_{21}=L_1\cap L_2$ and $D_i\cap L_3=D_3\cap L_i=\emptyset$ $(i=1,2)$. Let $(x:y:z)$ and $(x_{ij}:y_{ij}:z_{ij})$ be as in Lemma {\rm \ref{lem:car1}}. Then, there exists a curve ${\mit \Gamma}$, not containing any one of the lines $L_1, L_2, L_3$, of degree $m$ such that $({\mit \Gamma} .L_i)=D_i$ $(i=1,2,3)$ and whose tangent line $T$ at $P_{11}=(0:0:1)$ is given by $\alpha x+y=0$ $(\alpha\ne 0)$ if and only if \begin{equation} \alpha \prod^{\ell_1}_{j=2}\left(\frac{y_{1j}}{z_{1j}}\right)^{m_{1j}} \prod^{\ell_2}_{j=2}\left(\frac{z_{2j}}{x_{2j}}\right)^{m_{2j}} \prod^{\ell_3}_{j=1}\left(\frac{x_{3j}}{y_{3j}}\right)^{m_{3j}} =(-1)^m. \label{eq:32} \end{equation} \label{lem:car2} \end{lem} \vs 1 {\it Proof\/}. We use the same notation as in the proof of Lemma \ref{lem:car1}. Assume there exists a curve ${\mit \Gamma}$ having the desired property. The condition that the tangent line at $P_{11}$ is given by $\alpha x+y=0$ implies that ${\mit \Phi} (0,0,1)=0$ and the linear term of ${\mit \Phi} (x,y,1)$ is divisible by $\alpha x+y$. Hence, $a_{00m}=0$ and $a_{10,m-1}-\alpha a_{01,m-1}=0$. As in the proof of the previous lemma, we have $a_{m00}a_{0m0}\ne 0, a_{01,m-1}\ne 0$, $a_{0m0}=(-1)^ma_{m00}\displaystyle\prod^{\ell_3}_{j=1} \left(\fracd{x_{3j}}{y_{3j}}\right)^{m_{3j}}$, $a_{01,m-1}=(-1)^{m-1}a_{0m0}\displaystyle\prod^{\ell_1}_{j=2} \left(\fracd{y_{1j}}{z_{1j}}\right)^{m_{1j}}$ and $\alpha a_{01,m-1}= a_{10,m-1}=(-1)^{m-1}a_{m00}\displaystyle\prod^{\ell_2}_{j=2}\left( \fracd{x_{2j}}{z_{2j}}\right)^{m_{2j}}$. Hence, we have (\ref{eq:32}). Similar to the proof of the previous lemma, we have the converse. \vs 2 \begin{lem} Let $L_1, L_2, L_3$ be lines in ${\bbb P}^2$ so that $L_1\cap L_2\cap L_3\ne\emptyset$ and let $D_i=\displaystyle\sum^{\ell_i}_{j=1}m_{ij}P_{ij}, (\displaystyle\sum^{\ell_i}_{j=1}m_{ij}=m)$ be an effective divisor on $L_i$ such that $D_i\cap L_j=\emptyset$ if $i\ne j$. Let $(x:y:z)$ be a coordinate system on ${\bbb P}^2$ such that $L_1, L_2, L_3$ correspond to the line $y-z=0, y=0, z=0$, respectively. Let the coordinate of $P_{ij}$ be given by $(x_{ij}:y_{ij}:z_{ij})$ $($of course $y_{1j}=z_{1j}, y_{2j}=0, z_{3j}=0)$. Then, there exists a curve ${\mit \Gamma}$ of degree $d$ not containing any one of the lines $L_1,L_2,L_3$ such that $({\mit \Gamma} .L_i)=D_i$ $(i=1,2,3)$ if and only if \begin{equation} \sum^{\ell_1}_{j=1}m_{1j}\frac{x_{1j}}{y_{1j}}- \sum^{\ell_2}_{j=1}m_{2j}\frac{x_{2j}}{z_{2j}}- \sum^{\ell_3}_{j=1}m_{3j}\frac{x_{3j}}{y_{3j}}=0. \label{eq:33} \end{equation} \label{lem:car3} \end{lem} \vs 1 {\it Proof\/}. Again, we use the same notation as in the proof of Lemma \ref{lem:car1}. Assume there exists a curve ${\mit \Gamma}$ having the desired property. Since ${\mit \Gamma}$ does not contain $L_2\cap L_3$, we have $a_{m00}\ne 0$. By Lemma \ref{lem:31}, $$ a_{m-1,10}=-\sum^{\ell_3}_{j=1}m_{3j}\frac{x_{3j}}{y_{3j}}a_{m00}\quad {\rm and}\quad a_{m-1,01}=-\sum^{\ell_2}_{j=1}m_{2j}\frac{x_{2j}}{z_{2j}}a_{m00}. $$ To consider the condition on $L_1$, we take the coordinate system $(\xi :\eta :\zeta )$ with $\xi =x, \eta =y, \zeta =z-y$. Put $$ {\mit \Psi} (\xi ,\eta ,\zeta )={\mit \Phi} (\xi ,\eta ,\eta +\zeta )=\sum_{i+j+k=m}a_{ijk} \xi^i\eta^j(\zeta +\eta )^k=\sum_{i+j+k=m}b_{ijk}\xi^i\eta^j\zeta^k. $$ Then, $b_{m00}=a_{m00}$ and $b_{m-1,10}=a_{m-1,10}+a_{m-1,01}$. In this description, if $i({\mit \Gamma} , L_1;P_{1j})=m_{1j}$, then ${\mit \Psi} (\xi ,\eta ,0)$ is divisible by $(\xi_{1j}\eta -\eta_{1j}\xi )^{m_{1j}}$. This implies that $b_{m-1,10}=-\displaystyle\sum^{\ell_1}_{j=1}m_{1j} \fracd{x_{1j}}{y_{1j}}b_{m00}$. Then, we have (\ref{eq:33}). For the converse, noting that if $a_{m00}\ne 0$ then ${\mit \Gamma}$ does not contains $L_i$ $(i=1,2,3)$ as a component, we can find a ${\mit \Gamma}$ having the desired property. \vs 2 \begin{rem} {\rm In each of the lemmas \ref{lem:car1}, \ref{lem:car2} and \ref{lem:car3}, if (\ref{eq:31}) (resp. (\ref{eq:32}), (\ref{eq:33})) holds, we can find a smooth curve ${\mit \Gamma}$ of degree $m$ with ${\mit \Gamma} .L_i=D_i$ for $i=1,2,3$. Indeed, let ${\bbb P}\subset{\bbb P}_m$ be the linear system of divisors ${\mit \Gamma}$ of degree $m$ on ${\bbb P}^2$ satisfying, as schemes, $D_i\subset{\mit \Gamma}\cap L_i$. Clearly $L_1+L_2+L_3+{\bbb P}_{m-3}\subset{\bbb P}$ and we proved that $U=\{{\mit \Gamma}\in{\bbb P}:{\mit \Gamma}{\rm \ does\ not\ contain\ any\ of\ the\ lines\ }L_1, L_2, L_3\}$ is a non-empty open subset of ${\bbb P}$. Because of Bertini's theorem, for ${\mit \Gamma}\in U$ we have $L_i\cap{\mit \Gamma} =\{ P_{i1},\dots ,P_{i\ell_i}\}$. Consider the linear system ${\bbb P}'=\{{\mit \Gamma}\cap{\bbb P}^2\backslash (L_1\cup L_2 \cup L_3):{\mit \Gamma}\in{\bbb P}\}$ on $M={\bbb P}^2\backslash (L_1\cup L_2\cup L_3)$. Since ${\mit \Gamma}\cap M\in{\bbb P}'$ for any ${\mit \Gamma}\in{\bbb P}_{m-3}$, ${\bbb P}'$ separates tangent directions and points on $M$. Because of Bertini's theorem in arbitrary characteristics (see \cite{kleiman}), we find that a general element ${\mit \Gamma}$ of ${\bbb P}'$ is smooth. So, a general element ${\mit \Gamma}$ of ${\bbb P}$ satisfies ${\rm Sing}({\mit \Gamma} ) \subset\{ P_{ij}:i=1,2,3{\rm\ and\ }1\le j\le\ell\}$. But, using ${\mit \Gamma}'\in{\bbb P}_{m-3}$ suited we find ${\mit \Gamma} ={\mit \Gamma}'+L_1+L_2+L_3$ is smooth at $P_{ij}$, except for the case $P_{11}=P_{21}$ in Lemma \ref{lem:car2}. In that case, however, if ${\mit \Gamma}\in U$ then ${\mit \Gamma}$ is smooth at $P_{11}$ because of Bezout's theorem. This completes the proof of the remark. (For Bertini's theorem in arbitrary characteristics, one can also use \cite{greco}.)} \label{rem:33} \end{rem} \setcounter{equation}{0} \section{Sharpness of the bound} The next proposition implies that it is enough to solve the postulation problem mentioned in Remark \ref{rem:3} in order to prove sharpness for the bound $n(r)$ in Theorem \ref{thm:1}. \vs 2 \begin{prop} Let $C$ be a smooth plane curve of degree $d$ and let $r=\fracd{(x+1)(x+2)}{2}-\beta$ with $x, \beta\in{\bbb Z}$ satisfying $4\le x+3\le d-3, 0\le\beta\le x$. Let $n=n(r)=(d-3)(x+3)-\beta$. Suppose there exists a smooth curve ${\mit \Gamma}$ of degree $m=x+3$, a curve ${\mit \Gamma}'$ of degree $3$ and an effective divisor $E$ of degree $\beta$ on ${\mit \Gamma}$ such that $Z\subset C$, where the divisor $Z=({\mit \Gamma}\cap{\mit \Gamma}' )+E$ on ${\mit \Gamma}$ is considered as a closed subscheme of ${\bbb P}^2$. Then $|mg^2_d-Z|$ is a non-trivial $g^r_n$ on $C$. \label{prop:non-trivial} \end{prop} \vs 1 {\it Proof\/}. We write $E=P_1+\cdots +P_{\beta}$. Let $L_1,\dots ,L_{\beta}$ be general lines through $P_1,\dots ,P_{\beta}$, resp., and let $L_{\beta +1} \dots ,L_x$ be general lines in ${\bbb P}^2$. If $P\in C$, then we write $\mu_P(Z)$ for the multiplicity of $Z$ at $P$. \vs{05} i) $|mg^2_d-Z|$ is base point free. Suppose $P$ is a base point for $|mg^2_d-Z|$. Since ${\mit \Gamma} .C-Z\in|mg^2_d-Z|$ one finds $P+Z\le{\mit \Gamma} .C$, hence $i({\mit \Gamma} ,C;P)>\mu_P(Z)\ge i({\mit \Gamma} ,{\mit \Gamma}';P)$. Also $({\mit \Gamma}'+\sum_{i=1}^xL_i).C-Z\in|mg^2_d-Z|$, hence $P\in({\mit \Gamma}'+\sum_{i=1}^xL_i).C-Z=({\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma} )+(\sum_{i=1}^xL_i.C -E)$ (sum of two effective divisors). Since $P\not\in\sum_{i=1}^xL_i.C-E$, we find $P\in{\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma}$. This implies $i({\mit \Gamma}',C;P)>i({\mit \Gamma} ,{\mit \Gamma}';P)$. But $i({\mit \Gamma} ,{\mit \Gamma}';P)\ge\min (i({\mit \Gamma} ,C;P),i({\mit \Gamma}',C;P))$ (so called Namba's lemma), hence we have a contradiction. \vs{05} ii) $\dim (|mg^2_d-Z|)\ge r$. Indeed, $({\mit \Gamma}'+{\bbb P}_x(-E)).C-Z\subset|mg^2_d-Z|$ and $\dim ({\mit \Gamma}'+{\bbb P}_x(-E))= \fracd{(x+1)(x+2)}{2}-\beta -1$. But also ${\mit \Gamma} .C-Z\subset|mg^2_d-Z|$ while ${\mit \Gamma} .C\not\in ({\mit \Gamma}'+{\bbb P}_x(-E)).C$. This proves the claim. \vs{05} iii) $\dim (|mg^2_d-Z|)=r$. If $\dim (|mg^2_d-Z|)>r$ then on ${\mit \Gamma}$ it induces a linear system $g^{r'}_{mx-\beta}$ with $r'\ge r$. But Hartshorne's theorem (see 1.3) implies that this is impossible. iv) $|mg^2_d-Z|$ is not trivial. First of all, $|mg^2_d-Z|$ is very special. Indeed $(d-3-m)g^2_d+Z\subset |K_C-(mg^2_d-Z)|$. If $d-3=m$ then from the Riemann-Roch theorem, one finds $\dim|Z|=1$. Suppose $|mg^2_d-Z|$ would be trivial, i.~e. $|mg^2_d-Z|=|kg^2_d-F|$ with $r=\fracd{k^2+3k}{2}-(dk-n)$. Since $g^r_n$ is very special, one has $k<d-3$. Consider $D=({\mit \Gamma}'.C-{\mit \Gamma}\cap{\mit \Gamma}')+(\sum_{i=1}^xL_i.C-E)$ as in step i). There exists $\gamma\in{\bbb P}_k(-F)$ with $\gamma .C=D+F$. Because of Bezout's theorem one has $\gamma =\gamma'+L_1+\cdots +L_x$. If $P\in E$ then $P\not\in{\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma}$, otherwise both $i({\mit \Gamma}',C;P)>i({\mit \Gamma}',{\mit \Gamma} ;P)$ and $i({\mit \Gamma} ,C;P)>i({\mit \Gamma}',{\mit \Gamma} ;P)$, a contradiction to Namba's lemma. This implies $\gamma'.C\ge{\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma}$. Once more from Namba's lemma, we obtain ${\mit \Gamma}'.\gamma'\ge{\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma}$ and so $$ \deg ({\mit \Gamma}'.\gamma')=3(k-x)\ge\deg ({\mit \Gamma}'.C-{\mit \Gamma}'\cap{\mit \Gamma} )=3(d-x-3), $$ a contradiction to $k<d-3$. \vs 2 \begin{cor} Let $C$ be a smooth plane curve of degree $d$. Assume there exists a plane curve ${\mit \Gamma}'$ of degree $3$ and a smooth plane curve ${\mit \Gamma}$ of degree $a\ (4\le a\le d-6)$ such that ${\mit \Gamma}\cap{\mit \Gamma}'\subset C$ $($as schemes$)$. Then $C$ posseses a non-trivial linear system $g^r_n$ for $r=\fracd{(a-2)(a-1)}{2}-\beta, 0\le\beta\le a-3$ and $n=n(r)=a(d-3)-\beta$. \label{cor:1} \end{cor} \vs 1 {\it Proof\/}. ${\mit \Gamma} .C={\mit \Gamma}\cap{\mit \Gamma}'+D$ for some effective divisor $D$ of degree $a(d-3)$ on $C$. But $a(d-3)\ge d-3\ge a+3$, so we can choose an effective divisor $E\subset D$ of degree $\beta$ and then one has to apply Proposition \ref{prop:non-trivial} to $|ag^2_d-Z|$ for $Z={\mit \Gamma}\cap{\mit \Gamma}'+E$. \vs 2 \begin{const} {\rm Fix ${\mit \Gamma}'\in{\bbb P}_3, {\mit \Gamma}\in{\bbb P}_a\ (a\ge 4)$ general and look at ${\bbb P}_{a+\varepsilon}(-{\mit \Gamma}\cap{\mit \Gamma}'), (\varepsilon\ge 1)$. Clearly ${\mit \Gamma}'+{\bbb P}_{a+\varepsilon -3} \subset{\bbb P}_{a+\varepsilon}(-{\mit \Gamma}\cap{\mit \Gamma}'), {\mit \Gamma} +{\bbb P}_{\varepsilon}\subset{\bbb P}_{a+\varepsilon} (-{\mit \Gamma}\cap{\mit \Gamma}')$. Take $P\in{\bbb P}^2\backslash ({\mit \Gamma}\cap{\mit \Gamma}')$. If $P\not\in{\mit \Gamma}'$ then using ${\mit \Gamma}'+{\bbb P}_{a+\varepsilon -3}$ one can separate tangent vectors at $P$. If $P\not\in{\mit \Gamma}$ then one uses ${\mit \Gamma} +{\bbb P}_{\varepsilon}$. Kleiman's Bertini theorem \cite{kleiman} in arbitrary characteristics implies that a general element $C\in{\bbb P}_{a+\varepsilon}(-{\mit \Gamma}\cap{\mit \Gamma}')$ is smooth outside ${\mit \Gamma}\cap{\mit \Gamma}'$. But if ${\mit \Gamma}''\in{\bbb P}_{\varepsilon}$ is general then ${\mit \Gamma} +{\mit \Gamma}''$ is smooth on ${\mit \Gamma}\cap{\mit \Gamma}'$. This implies that a general element $C\in{\bbb P}_{a+\varepsilon}(-{\mit \Gamma}\cap{\mit \Gamma}')$ is smooth. This proves that, for each $d, 1\le x\le d-6, 0\le\beta\le x$, there exists a smooth plane curve $C$ of degree $d$ possesing a non-trivial $g^r_{n(r)}$ with $r=\fracd{(x+1)(x+2)}{2}-\beta$ and $n(r)=(d-3)(x+3)-\beta$.} \label{const} \end{const} \vs 2 In trying to prove this statement for all smooth plane curves of degree $d$, we only succeeded in case ${\rm char}(k)=0$. This is the following theorem. \vs 2 \begin{thm} Let $C$ be a smooth plane curve of degree $d$ over an algebraically closed field of characteristic zero. Let $d>m\ge 4$. There exists ${\mit \Gamma}'\in{\bbb P}_3$ and a smooth ${\mit \Gamma}\in{\bbb P}_m$ such that, as schemes, ${\mit \Gamma}\cap{\mit \Gamma}'\subset C$. \end{thm} \vs 1 {\it Proof\/}. Fix two general lines $L_1, L_2$ in ${\bbb P}^2$, let $S=L_1\cap L_2$. We may assume neither $L_1$ nor $L_2$ is a tangent line of $C$ and $S\not\in C$. Choose points $P_{11},\dots ,P_{1m}$ on $C\cap L_1$ and $P_{21},\dots ,P_{2m}$ on $C\cap L_2$. Choose a general point $S'$ in ${\bbb P}^2\backslash C\cup L_1\cup L_2$. The pencil of lines in ${\bbb P}^2$ through $S'$ induces a base point free $g^1_d$ on $C$. Because $S'$ is general we have: \begin{itemize} \item If $Q$ is a ramification point of $g^1_d$ then the associated divisor looks like $2Q+E$ with $Q\not\in E$ and $E$ consists of $d-2$ different points (here we use characteristic zero). \item If $Q\in L_i\cap C$ $(i=1,2)$ then $Q$ is not a ramification of $g^1_d$. The associated divisor is $Q+E$ with $E\cap (L_1\cup L_2)=\emptyset$. \item The line $SS'$ is not a tangent line of $C$. \end{itemize} On the symmetric product $C^{(m)}$ we consider $V=\{ E\in C^{(m)}:{\rm there\ exists\ }D\in g^1_d\ {\rm with\ }E\le D\}$. In terminology of \cite{c2} it is the set $V^1_k(g^1_d)$ and we consider $V$ with its natural scheme structure. From Chapter 2 in {\it loc.~cit.}, it follows that $V$ is a smooth curve. Let $D_0\in g^1_d$ corresponding to the line $SS'$ and let $V_0=\{ E\in V:E\le D_0\}$. We define a map $\psi :V\backslash V_0\to{\bbb P}^1$ as follows. Associated to $E\in V\backslash V_0$ there is a line $L$ through $S'$. Write $E=P_{31}+\cdots +P_{3m}$. We distinguish 3 possibilities: \vs{05} i) $E\cap (L_1\cup L_2)=\emptyset$. Choose coordinates $x,y,z$ on ${\bbb P}^2$ such that $L_1, L_2, L$ corresponds to the coordinate axes $x=0, y=0, z=0$, respectively. Let $(x_{ij}:y_{ij}:z_{ij})$ be the coordinates of $P_{ij}$ $(i=1,2,3;1\le j\le m)$. Then $$ \psi (E)= \prod^m_{j=1}\left(\frac{y_{1j}}{z_{1j}}\right) \prod^m_{j=1}\left(\frac{z_{2j}}{x_{2j}}\right) \prod^m_{j=1}\left(\frac{x_{3j}}{y_{3j}}\right). $$ As long as we take $L_1, L_2, L$ as axes, this value is independent of the coordinates. \vs{05} ii) $E\cap (L_1\cup L_2)\ne\emptyset$. Say $P_{11}=P_{31}\in E\cap (L_1\cup L_2)$. Choose coordinates as before and let $\alpha x+z=0$ be the equation of the tangent line to $C$ at $P_{11}$ $(\alpha\ne 0)$. Then $$ \psi (E)=\alpha \prod^m_{j=2}\left(\frac{y_{1j}}{z_{1j}}\right) \prod^m_{j=1}\left(\frac{z_{2j}}{x_{2j}}\right) \prod^m_{j=2}\left(\frac{x_{3j}}{y_{3j}}\right). $$ Again taking $L_1, L_2, L$ as axes, this value is independent of the coordinates. (Of course this is a function to ${\bbb C}$ and we consider ${\bbb P}^1={\bbb C}\cup\{\infty\}$.) \vs{05} iii) If $\psi (E)$ is not defined in ${\bbb C}$ then $\psi (E)=\infty$. \vs{05} For $E\in V_0$, we define $\psi (E)=(-1)^m$ \vs{05} This map is a holomorphic map. Indeed, fixing coordinates $(x:y:z)$ such that $L_1, L_2$ corresponds to $x=0,y=0$, resp. and $S'=(1:1:0)$, we can write $z-\gamma (x-y)=0$ for the pencil of lines through $S'$ (except for $SS'$). If $E\in V\backslash V_0$ and $E$ is a part of a divisor of $g^1_d$ consisting of $d$ different points, then $\gamma$ is a local coordinate of $V$ at $E$. In case i) we write down $\psi$ locally as a holomorphic function in $\gamma$. It is easy to check that $\psi$ is continuous at $E$ in case ii). For $E\in V_0$, write $\beta z+(x-y)=0$ for the pencil of lines through $S'$ close to $SS'$. Let $(x_{3j}:x_{3j}:z_{3j})$ be the coordinates at the points $P_{3j}$ of $E$. For $E'\in V$ close to $E$ we have $E'=\sum_{j=1}^mP'_{3j}$ and coordinates $(x'_{3j}:\beta z'_{3j}+x'_{3j}:z'_{3j})$ at $P'_{3j}$. Here we can assume that $x'_{3j}=x'_{3j}(\beta ), z'_{3j}=z'_{3j}(\beta )$ are holomorphic functions in $\beta$ (local coordinate at $V$ in $E$) and $x_{3j}=x'_{3j}(0), z_{3j}=z'_{3j}(0)$. Choose new coordinates $\xi =\beta z+x-y, \eta =y, \zeta =x$. The coordinates of $P_{1j}$ are $(0:y_{1j}:\beta z_{1j}+y_{1j})$, of $P_{2j}$ are $(x_{2j}:0:\beta z_{2j}+x_{2j})$, of $P'_{3j}$ are $(x'_{3j}:\beta z'_{3j}+x'{3j}:0)$. We find \begin{eqnarray} \psi (E') & = & \prod^m_{j=1}\frac{y_{1j}}{\beta z_{1j}-y_{1j}} \prod^m_{j=1}\frac{\beta z_{2j}+x_{2j}}{x_{2j}} \prod^m_{j=1}\frac{x'_{3j}}{\beta z'_{3j}+x'_{3j}} \label{eq:v0} \\ & = & (-1)^m-\left( \sum^m_{j=1}\frac{z_{3j}}{x_{3j}}- \sum^m_{j=1}\frac{z_{1j}}{y_{1j}}- \sum^m_{j=1}\frac{z_{2j}}{x_{2j}}\right)\beta +o(\beta ).\nonumber \end{eqnarray} Hence, $\psi$ is continuous at $E$. Since $V$ is smooth and $\psi$ is continuous on $V$ and holomorphic except for a finite number of points, $\psi$ is a holomorphic map $V\to{\bbb P}^1$. At some component of $V$, $\psi$ is not constant. Indeed, look at a fibre $2Q+E\in g^1_d$ with $E\in C^{(d-2)}$. Take a close fibre $P_1+P_2+E'$ with $P_1, P_2$ close to $Q$. Choose $F\le E'$ with $\deg(F)=m-1$ and consider $P_1+F\in V$. Let $W$ be the irreducible component of $V$ containing $P_1+F$. Using monodromy one finds $P_2+F\in W$. But clearly $\psi (P_1+F)\ne\psi (P_2+F)$, hence $\psi :W\to{\bbb P}^1$ is a covering. In particular $\psi^{-1}((-1)^m)\ne\emptyset$. If for some $E\in W\backslash V_0$ we have $\psi (E)=(-1)^m$ then the theorem follows from Lemmas \ref{lem:car1}, \ref{lem:car2} and Remark \ref{rem:33}. So, we have to take a closer look to $\psi$ at $V_0$. By the equation (\ref{eq:v0}), if $E\in V_0$ is not a simple zero of $\psi -(-1)^m$ then the theorem follows from Lemma \ref{lem:car3} and Remark \ref{rem:33}. Suppose that each zero of $\psi -(-1)^m$ belonging to $V_0$ is simple. Then $\psi -(-1)^m$ has exactly $\binom{d}{m}$ zeros at those points. Now we look at zeros of $\psi$ on $V\backslash V_0$. The number of zeros is finite. For case i) there is none. For case ii) we have two possiblities. If $E\in V$ corresponds to a line $L$ through $S'$ containing one of the points $P_{21},\dots ,P_{2m}$ but $E\cap (L_1\cup L_2)=\emptyset$. There are $m\binom{d-1}{m}$ such possibilities. If $E\in V$ corresponds to a line $L$ through $S'$ not containing any of the points $P_{11},\dots ,P_{1m}$ but $E\cap L_1\ne\emptyset$. There are $(d-m)\binom{d-1}{m-1}$ such possibilities. So, on the components of $V$ where $\psi$ is not constant, $\psi$ has at least $m\binom{d-1}{m}+(d-m)\binom{d-1}{m-1}$ zeros. But this number is greater than $\binom{d}{m}$, so $\psi -(-1)^m$ has a zero on $V\backslash V_0$. This completes the proof of the theorem. \vs 2 \begin{rem} {\rm In order to obtain the bound $r(n)$ mentioned in the introduction, we have to prove that $C$ possesses no base point free very special non-trivial linear systems $g^r_n$ with $r\ge\fracd{(d-4)(d-3)}{2}-(d-5)$ (i.~e. $x\ge d-5$). In the introduction we already noticed that $x\le d-3$. Assume $g^r_n$ is a very special non-trivial linear system. {}From Theorem \ref{thm:1} we find $n\ge n((d-4)(d-5)/2-(d-5))=d^2-6d+11$. But then $\deg (K_C-g^r_n)\le (d-1)(d-2)-2-(d^2-6d+11)=3d-11$. However, very special linear systems $g^s_m$ of degree $m\le 3d-11$ are trivial. So, the associated base point free linear system $g^s_m$ of $|K_C-g^r_n|$ is of type $|ag^2_d-E|$ with $a\le d-4$, $E$ effective and $\dim|K_C-g^r_n|=\fracd{a^2+3a}{2}-\deg E$. If $E\ne\emptyset$, then for $P\in E$ one has $\dim|ag^2_d-E+P|>\dim|ag^2_d-E|$, so $\dim|g^r_n-P|=r$. This implies that $g^r_n$ has a base point, hence $E=\emptyset$. But then, $g^r_n=|(d-3-a)g^2_d-F|$ and since $\dim|ag^2_d+F|=\dim|ag^2_d|$, we have $r=\fracd{(d-3-a)^2+3(d-3-a)}{2}-\deg F$. This is a contradiction to the fact that $g^r_n$ would be non-trivial.} \end{rem} \vs 1 \begin{rem} {\rm It would be interesting to find an answer to the following questions: For which values of $n$ do there exist non-trivial base point free very special linear systems $g^r_n$ on a (general) smooth plane curve. Classify those linear systems and study $W^r_n$ on $J(C)$. More concretely, is the subscheme of $W^r_{n(r)}$ corresponding to non-trivial linear systems irreducible ? What are the dimension of those irreducible components ? And so on. } \end{rem}

9301

alg-geom/9301006

en

https://arxiv.org/abs/alg-geom/9301006

alg-geom/9301006

Sheldon Katz

Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror Symmetry

12 pages, LaTeX (Replaced version corrects an error in the formula for bundle $B'$ on page 5, and changes the order of some entries in tables 2 and 3 for compatibility with the associated computer file)

OSU Math 1992-3

Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy certain incidence properties are calculated, and shown to agree with the numbers predicted by Greene, Morrison, and Plesser using mirror symmetry in every instance. This increases the number of verified predictions from 3 to 65. Calculations are performed using the Maple package {\sc schubert} written by Katz and Str{\o}mme.

alg-geom

\section*{} Recently, mirror symmetry, a phenomenon in superstring theory, has been used to give tentative calculations of several numbers in algebraic geometry \nolinebreak\footnote{See the papers in \cite{yau} for general background on mirror symmetry.}. This yields predictions for the number of rational curves of any degree $d$ on general Calabi-Yau hypersurfaces in $\P^4\ \cite{cogp},\ \P(2,1^4),\ \P(4,1^4)$, and $\P(5,2,1^3)$ \cite{fontper,kt,morp-f}. The techniques used in the calculation rely on manipulations of path integrals which have not yet been put on a rigorous mathematical footing. On the other hand, there is currently no prospect of calculating most of these numbers by algebraic geometry. Until this point, three of these numbers have been verified, all for the quintic hypersurface in $\P^4$: the number of lines (2875) was known classically, the number of conics (609250) was calculated in \cite{finite}, and the number of twisted cubics (317206375) was found recently by Ellingsrud and Str{\o}mme \cite{escub}. Even more recently \cite{gmp}, higher dimensional mirror symmetry has been used to predict the number of rational curves on Calabi-Yau hypersurfaces in higher dimensional projective spaces which meet 3 linear subspaces of certain dimensions. Again, there is no known way to calculate these using algebraic geometry. The purpose of this paper is to verify some of these numbers in low degree, giving more evidence for the validity of mirror symmetry. In \S 1, the number of weighted lines in a weighted sextic in $\P(2,1^4)$ is calculated, as well as the number of weighted lines in a weighted octic in $\P(4,1^4)$. In \S 2, the number of lines on Calabi-Yau hypersurfaces of dimension up to 10 which satisfy certain incidence properties is calculated. In \S 3, the number of conics on these same Calabi-Yau hypersurfaces satisfying the same incidence properties is calculated. These numbers are closely related to the Gromov-Witten invariants defined in \cite{wittsm,morht,gmp}; and it is {\em these} numbers that are recorded here. In all instances, the calculations agree with those predicted by mirror symmetry. Thus the number of verified predictions has increased from 3 to 65. There are two parts to all of these calculations. The first part is to express the desired numbers in terms of the standard constructs of intersection theory. The second part is to evaluate the number using the Maple package {\sc schubert}\ \cite{schub} (although the number of weighted lines in a weighted octic in $\P(4,1^4)$ was first found via classical enumerative geometry, using a classical enumerative formula). The short {\sc schubert}\ code is not included here, but is available upon request. While it is checked that the data being enumerated is finite, no attempt has been made here to check that the multiplicities are 1. All enumeration takes multiplicities into consideration. This suffices for comparison to the numbers arising in physics, since the Feynman path integrals would take account of any multiplicities greater than 1 as well. Some of the Gromov-Witten invariants were computed in \cite{gmp} using an intriguing relation between the various invariants. These relations arise in conformal field theory. A mathematical proof of the relations for the invariants corresponding to lines is sketched here. It is appropriate to point out the recent work of Libgober and Teitelbaum \cite{ltcy}, who have apparently correctly guessed the mirror manifold of complete intersection Calabi-Yau threefolds in an ordinary projective space. Their conjectured mirrors yield predictions for the numbers of rational curves. The predicted number of lines coincides with the results of a calculation done by Libgober 20 years ago, and the predicted number of conics coincides with the results of an unpublished calculation done by Str\o mme and Van Straten in 1990. I'd like to thank D.~Morrison for helpful suggestions and conversations, and for encouraging me to write this paper. I'd also like to thank S.~Kleiman for his suggestions which have improved the manuscript. \section{Weighted projective spaces and their Grassmannians} Let $\P(k,1^n)$ denote an $n$-dimensional weighted projective space with first coordinate having weight $k>1$, all other coordinates having weight 1. Thus $\P(k,1^n)$ consists of all non-zero $(n+1)$-tuples $(x_0,\ldots,x_n)$, with $(x_0,\ldots,x_n)$ identified with $(\l^kx_0,\l x_1,\ldots,\l x_n)$ for any $\l\neq 0$. Note that $\P(k,1^n)$ is smooth outside the singular point $p=(1,0,\ldots,0)$. There is a natural rational projection map $\pi:\P(k,1^n) ---> \P^{n-1}$ defined outside $p$ given by omitting the first coordinate. Let $X$ be a weighted hypersurface of weight $d$. Assume $p\not\in X$ (this implies that $k|d$, and that the monomial $x_0^{d/k}$ occurs in an equation for $X$). It is further assumed that $X$ is smooth. The general weighted hypersurface whose weight is a multiple of $k$ is an example of such an $X$. \bigskip\noindent {\bf Definition} A {\em weighted $r$-plane in $\P(k,1^n)$} is the image of a section of $\pi$ over an $r$-plane in $\P^{n-1}$. \bigskip Note that weighted $r$-planes do not contain $p$. Let $P$ be a weighted $r$-plane, with $L$ its image in $\P^{n-1}$. Let $(q_0,\ldots ,q_r)$ be any homogeneous coordinates on $L\simeq\P^r$. Then $P$ may be thought of as the image of $L$ via the mapping $x_0=f_k(q_0,\ldots,q_r),\ x_i=l_i(q_0,\ldots,q_r)$, where $f_k$ is a form of degree $k$, and the $l_i$ are all linear. Once $L$ is fixed, we may fix in mind a choice of the $q_i$ and $l_i$. The moduli space of weighted $r$-planes can be represented (and compactified) as follows. Conventions have been chosen to be consistent with those in {\sc schubert}\ \cite{schub}. Let $G=G(r+1,n)$ be the Grassmannian of $r$-dimensional linear subspaces $L$ of $\P^{n-1}$. Let $V$ be the $n$-dimensional vector space of linear forms on $\P^{n-1}$. This identifies $\P^{n-1}$ with $\P(V)=Proj(S^*V)$. $G$ is then the space of $r+1$ dimensional quotients of $V$ (since the space of linear forms restricted to $L$ is an $r+1$-dimensional quotient of $V$). Let $Q$ be the universal rank $r+1$ quotient bundle on $G$. The equation $x_0-f_k(q_0,\ldots,q_r)=0$ which describes a section of $\pi$ over an $r$-plane may be identified with a section $s$ of the bundle $\mbox{${\bf C}$}\oplus S^k(Q)$, where \mbox{${\bf C}$}\ denotes the trivial bundle. A scalar multiple of this section would correspond to the equation $ax_0-af_k(q_0,\ldots,q_r)=0$, which defines the same weighted $r$-plane. Note that (up to scalar) $s$ does not depend on any of the choices which have been made. So $M=\P(\mbox{${\bf C}$}\oplus S^k(Q)^*)$ gives a compactification of the space of weighted $r$-planes. Here, $\P(E)$ denotes the space of rank 1 quotients of the fibers of the bundle $E$; hence the need for dualizing in defining $M$. In the sequel, we will also refer to $M^o\subset M$, the open subset which corresponds to the actual weighted $\P^1$'s, in other words, $M^o=M-\P(S^kQ^*)$, where $\P(S^kQ^*)$ is included in $M$ via the map induced by the natural projection $\mbox{${\bf C}$}\oplus S^kQ^*\to S^kQ^*$. \section{Lines on weighted hypersurface Calabi-Yau threefolds} \label{weight-section} Now let $k>1$, and let $X\subset\P(k,1^4)$ be a smooth weighted hypersurface of weight $k+4$ with $(1,0,0,0,0)\not\in X$. As has been noted in the previous section, this implies that $k|k+4$, which in turn implies that $k=2$ or $k=4$. The weight $k+4$ has been chosen to ensure that $X$ is Calabi-Yau, i.e. that $X$ has trivial canonical bundle. The rational projection map $\pi$ restricts to a morphism $\pi:X\to\P^3$. This is a 3-1 cover for $k=2$, and a 2-1 cover for $k=4$. The goal of this section is to enumerate the weighted $\P^1$'s contained in $X$. Let us first consider the case $k=4$. Then an equation for $X$ has the form \begin{equation} \label{hyper} F=ax_0^2+g_4(x_1,\ldots,x_4)x_0+g_8(x_1,\ldots,x_4)=0, \end{equation} where $a\in\mbox{${\bf C}$}$ and $g_i$ has degree $i$ for $i=4$ or 8. Such an equation naturally induces a section $s$ of the bundle $\mbox{${\bf C}$}\oplus S^4Q\oplus S^8Q$. Consider a point $C\in M^o$. We abuse notation by allowing $C$ to also denote the corresponding curve. Let $(q_0,q_1)$ be homogeneous coordinates on $\P^1$. Identifying $C$ with $\P^1$, we may describe $C$ by equations of the form \begin{equation} \label{param} x_0=f_4(q_0,q_1),\ x_i=l_i(q_0,q_1). \end{equation} The equation $x_0-f_4(q_0,q_1)=0$ and its multiples for varying $C$ form the tautological subbundle $\O_{\P}(-1)\subset\mbox{${\bf C}$}\oplus S^4Q$ on $\P=\P(\mbox{${\bf C}$}\oplus S^4Q^*)$. $C$ is contained in $X$ if and only if an equation for $X$, when pulled back to $\P^1$ via a parametrization of $C$, vanishes. Substituting from the second of equations~(\ref{param}) into (\ref{hyper}), it is seen that this happens if and only if $ax_0^2+g_4(l_1(q_0,q_1),\ldots,l_4(q_0,q_1))x_0+ g_8(l_1(q_0,q_1),\ldots,l_4(q_0,q_1))$ is a multiple of $x_0-f_4(q_0,q_1)$. Multiplication induces an inclusion of bundles $$(\mbox{${\bf C}$}\oplus S^4Q)\otimes \O_{\P}(-1)\hookrightarrow \mbox{${\bf C}$}\oplus S^4Q\oplus S^8Q.$$ Putting all this together, we see that $C\subset X$ if and only the section ${\bar s}$ of $$B=(\mbox{${\bf C}$}\oplus S^4Q\oplus S^8Q) /((\mbox{${\bf C}$}\oplus S^4Q)\otimes \O_{\P}(-1))$$ induced by $s$ vanishes at $C$. Note that if $C\in M-M^o$, then $C$ corresponds to a curve defined by equations of the form $f_4(q_0,q_1)=0,\ x_i=l_i(q_0,q_1)$. Since such a curve would contain $p$, it follows that $C$ is not in the zero locus of ${\bar s}$. Also note that $\dim(M)=rank(B)=9$; so one expects finitely many zeros of such a section; hence finitely many weighted $\P^1$'s. It is easy to prove that this is indeed the case for general $X$. The actual number is the degree of $c_9(B)$. This may be calculated by standard techniques in intersection theory \cite{fintthy} and the calculation may be implemented via {\sc schubert}\ \cite{schub}. The case $k=2$ is similar. Changing the meaning of the notation in the obvious manner, one must consider $M=\P(\mbox{${\bf C}$}\oplus S^2Q^*)$, and calculate the degree of $c_7(B')$, where $$B'=(\mbox{${\bf C}$}\oplus S^2Q\oplus S^4Q\oplus S^6Q) /((\mbox{${\bf C}$}\oplus S^2Q\oplus S^4Q)\otimes\O_{\P}(-1)).$$ Combining these with the well-known number of lines on a quintic threefold, the calculation of some examples considered in \cite{fontper,kt,morp-f} via mirror symmetry may be verified. The results are displayed in table~\ref{table1}. \begin{table}\begin{center}\begin{tabular}{|ccc|} \hline Ambient space & Weighted degree & Number of lines \\ \hline $\P(1^5)$ & 5 & 2875 \\ $\P(2,1^4)$ & 6 & 7884 \\ $\P(4,1^4)$ & 8 & 29504 \\ \hline \end{tabular}\end{center}\caption{The number of lines.} \label{table1}\end{table} \bigskip\noindent {\bf Problem:} Verify the predictions of mirror symmetry for weighted $\P^1$'s in a weight 10 hypersurface in $\P(5,2,1^3)$. Also, verify the predictions of mirror symmetry for weighted conics on the weighted hypersurfaces considered in this section. \bigskip\noindent {\em Remark:} The family of weighted conics on the general weighted octic in $\P(4,1^4)$ is positive dimensional (independently observed by Koll{\'a}r); hence part of the problem in this case is to systematically assign numbers to positive dimensional families. This can be defined as the number of such curves that remain almost holomorphic under a general almost complex deformation; but it is desirable to give a purely algebraic description. \section{Lines on higher dimensional varieties} In this section and the next, we consider rational curves on the generic Calabi-Yau hypersurface $X$ in $P^{k+1}$. This is a hypersurface of dimension $k$ and degree $k+2$. For $k>3$, there will be infinitely many lines and conics contained in $X$. But there will only be finitely many lines or conics which satisfy certain incidence properties with fixed linear subspaces. Since the normal bundle $N$ of $C$ in $P^{k+1}$ has degree $-2$, one expects that for general $X$ and any $C\subset X$, $N\simeq\O\oplus\ldots\oplus\O \oplus\O(-1)\oplus\O(-1)$ (with $k-3\ \O$'s). Since $h^0(N)=k-3$ and $h^1(N)=0$ in this case, the scheme of rational curves on $X$ is expected to have dimension $k-3$. For each $i$, let $L_i\subset\P^{k+1}$ denote a general linear subspace of codimension $i$. Pick positive integers $a,\ b,\ c$ such that $a+b+c=k$. Following \cite{wittsm,morht,gmp}, define an invariant $n^a_b(d)$ of $X$ as the number of holomorphic immersions $f:\P^1\to X$ with $f(\P^1)$ of degree $d$ such that $f(0)\in L_a,\ f(1)\in L_b,\linebreak f(\infty)\in L_c$. These numbers, called ``Gromov-Witten invariants'' in \cite{morht}, are expected to be finite. Note that the value of $c$ is implicit in the notation $n^a_b(d)$ by virtue of the equation $a+b+c=k$. These invariants are essentially the same as the number of reduced, irreducible rational curves of degree $d$ in $X$ which meet each of $L_a,\ L_b$, and $L_c$. The $n^a_b(d)$ differ from the corresponding numbers of curves by one factor of $d$ for each of the indices $a,b$, or $c$ equal to 1 (since $C$ meets a general $L_1\ d$ times). There is no difference for lines; and for conics, we will see that in the calculation of the number of conics satisfying the required incidence properties, the Gromov-Witten invariants arise naturally. So the Gromov-Witten invariants will be calculated and tabulated, while the numbers of rational curves follow immediately by division by the appropriate power of $d$, if necessary. In the remainder of this section, we specialize to $d=1$, i.e. lines. A theorem of Barth-van de Ven \cite{bv} states that the Fano variety of lines on a degree $l$ hypersurface $X \subset \P^n$ is smooth of dimension $2n-l-3$ for generic $X$ when $l+3 \le 2n$. Applied in the present context of $X_{k+2}\subset\P^{k+1}$, we find that the variety of lines must be smooth of dimension $k-3$ whenever $k\ge3$. {}From this, standard techniques show that a general $X$ contains finitely many lines which meet each of $L_a,\ L_b$, and $L_c$. So we can calculate the Gromov-Witten invariants by using the Schubert calculus. The lines are parametrized by the Grassmannian $G(2,k+2)$. The class of lines meeting $L_a$ is the Schubert cycle \s{a-1}; similarly for $L_b$ and $L_c$. Let $Q$ be the rank 2 universal quotient bundle on $G$. Since the class of the variety of lines on $X$ is represented by $c_{k+3}(S^{k+2}Q)$ and dimensions work out correctly, the answer is the degree of $c_{k+3}S^{k+2}Q\cdot\s{a-1}\cdot\s{b-1}\cdot\s{c-1}$. These may be easily worked out as integers using {\sc schubert}. The answers obtained are displayed in table~\ref{table3}. \bigskip The original predictions for the numbers found in \cite{gmp} resulted from a two-step process arising from mirror symmetry and conformal field theory. First, the $n^1_b(d)$ are found, followed by what amounts to an expression for any $n^a_b(d)$ in terms of the various $n^1_{b'}(d')$ for $d'\le d$. Most of these expressions remain a mathematical mystery at present. However, the case $d=1$ can be established mathematically as follows. \begin{fact} Let $X$ be any Calabi-Yau manifold of dimension $k$ in any projective space. Define $n^a_b(1)$ as above. Assume that there are finitely many lines in $X$ satisfying each of the respective incidence conditions needed to define the $n^a_b(1)$. Then $$n^i_j(1)=\sum_{l=0}^{j-1}n^1_{i+l}(1)-\sum_{l=1}^{j-1}n^1_l(1).$$ \end{fact} \bigskip\noindent {\em Proof (sketch).} Follows immediately by intersecting the cycle class (in the appropriate Grassmannian) of the scheme of lines in $X$ with the identity $$\s{i-1}\s{j-1}\s{k-i-j-1}=\sum_{l=0}^{j-1}\s{i+l-1}\s{k-i-l-2}- \sum_{l=1}^{j-1}\s{l-1}\s{k-l-2},$$ an identity which can be proven by a few applications of Pieri's formula. \begin{table}\small\begin{center}\begin{tabular}{|c|c|} \hline $k$ & $n^a_b(1)$\\ \hline 3 & $n^1_1(1)=2875$ \\ \hline 4 & $n^1_1(1)=60480$ \\ \hline 5 & $n^1_1(1)=1009792,\ n^1_2(1)=1707797$ \\ \hline 6 & $n^1_1(1)=15984640,\ n^1_2(1)=37502976,\ n^2_2(1)=59021312$ \\ \hline 7 & $n^1_1(1)=253490796,\ n^1_2(1)=763954092,\ n^1_3(1)=1069047153$ \\ \cline{2-2} & $n^2_2(1)=1579510449$ \\ \hline 8 & $n^1_1(1)=4120776000,\ n^1_2(1)=15274952000,\ n^1_3(1)=27768048000$ \\ \cline{2-2} & $n^2_2(1)=38922224000,\ n^2_3(1)=51415320000$ \\ \hline 9 & $n^1_1(1)=69407571816,\ n^1_2(1)=307393401172,\ n^1_3(1)=695221679878$ \\ \cline{2-2} & $n^1_4(1)=905702054829,\ n^2_2(1)=933207509234,\ n^2_3(1)=1531516162891$ \\ \cline{2-2} & $n^3_3(1)=1919344441597$ \\ \hline 10 & $n^1_1(1)=1217507106816,\ n^1_2(1)=6306655500288$ \\ \cline{2-2} & $n^1_3(1)=17225362851840,\ n^1_4(1)=28015971489792$ \\ \cline{2-2} & $n^2_2(1)=22314511245312,\ n^2_3(1)=44023827234816$ \\ \cline{2-2} & $n^2_4(1)=54814435872768,\ n^3_3(1)=65733143224320$ \\ \hline \end{tabular}\end{center}\caption{Gromov-Witten invariants for lines.} \label{table3}\end{table} \section{Conics on higher dimensional varieties} It can easily be shown that if $X$ is a general hypersurface of degree $k+2$ in $\P^{k+1}$, then the variety of conics on $X$ has the expected dimension $k-3$. Standard techniques show that given positive integers $a,b,c$ with $a+b+c=k$, there will be a finite number of conics in $X$ which meet each of $L_a,\ L_b$, and $L_c$. Thus the Gromov-Witten invariants are finite. They will be calculated here; the answers obtained are displayed in table~\ref{table4}. \begin{table}\footnotesize\begin{center}\begin{tabular}{|c|c|} \hline $k$& $n^a_b(2)$ \\ \hline 3 & $n^1_1(2)=4874000$ \\ \hline 4 & $n^1_1(2)=1763536320$ \\ \hline 5 & $n^1_1(2)=488959144352,\ n^1_2(2)=1021575491286$ \\ \hline 6 & $n^1_1(2)=133588638826496,\ n^1_2(2)=448681408315392 \ n^2_2(2)=821654025830400$ \\ \hline 7 & $n^1_1(2)=39031273362637440,\ n^1_2(2)=187554590257349088$ \\ \cline{2-2} & $n^1_3(2)=312074852318965368,\ n^2_2(2)=506855012110118424$ \\ \hline 8 & $n^1_1(2)=12607965435718224000,\ n^1_2(2)=80684596772238448000$ \\ \cline{2-2} & $n^1_3(2)=200581960800610752000,\ n^2_2(2)=295035175517918176000$ \\ \cline{2-2} & $n^2_3(2)=444475303469701680000$\\ \hline 9 & $n^1_1(2)=4565325719860021608624,\ n^1_2(2)=37005001823802188657624$ \\ \cline{2-2} & $n^1_3(2)=127922335050535174614916,\ n^1_4(2)=193693669320390878077186$ \\ \cline{2-2} & $n^2_2(2)=173901546566279203106468,\ n^2_3(2)=364629304647788940660824$ \\ \cline{2-2} & $n^3_3(2)=498705676383823268404990$ \\ \hline 10 & $n^1_1(2)=1861791822397620935737344,\ n^1_2(2)=18415607624138339954786304$ \\ \cline{2-2} & $n^1_3(2)=83885220561474498867757056,\ n^1_4(2)=179982840924749584358866944$ \\ \cline{2-2} & $n^2_2(2)=107227899142191919158312960,\ n^2_3(2)=297755098999730079369412608$ \\ \cline{2-2} & $n^2_4(2)=417950364467570984815214592,\ n^3_3(2)=527556832251612742800359424$ \\ \hline \end{tabular}\end{center}\caption{Gromov-Witten invariants for conics.} \label{table4}\end{table} We start with the well known description of the moduli space of conics in $\P^{k+1}=\P(V)$, where $V$ is a $k+2$-dimensional vector space. To describe a conic, we first describe the 2 plane it spans, and then choose a quadric in that 2-plane (up to scalar). So let $G=G(3,V)$ be the Grassmannian of 2-planes in $\P(V)$ (that is, of rank 3 quotients of $V$), and let $Q$ be the universal rank 3 quotient bundle of linear forms on the varying subspace. Then the moduli space of conics is $M=\P(S^2Q^*)$. Following the reasoning in section~\ref{weight-section} (or \cite{finite}), the scheme of conics on $X$ is given by the locus over which a certain section of $F=S^{k+2}Q/(S^kQ \otimes\O_{\P}(-1))$ vanishes. Here $\O_{\P}(1)$ is the tautological sheaf on $\P(S^2Q^*)$. Since $F$ has rank $2k+5$, the conics on $X$ are represented by $c_{2k+5}(F)$. It remains to find the condition that a conic $C$ meets $L_a$. One way to find this is to consider the moduli space ${\cal M}$ of pointed conics, i.e. pairs $(p,C)$, with $C$ a conic, $p\in C$. This may easily be constructed as a bundle over $\P^{k+1}$, with fiber over $p\in \P^{k+1}$ being the set of conics containing $p$. We start by constructing the moduli space of pointed 2-planes as follows. Consider the tautological exact sequence on $\P^{k+1}$: $$0\to K\to V_{\P^{k+1}}\to \O(1)\to 0,$$ \noindent where $V_{\P^{k+1}}$ is a trivial bundle of rank $k+2$ on $\P^{k+1}$ (more generally, $E_Y$ will stand for the pullback of $E$ to $Y$, the morphism used for the pullback assumed to be clear in context). Let $H=G(2,K)$ be the Grassmannian of rank 2 quotients of $K$, \mbox{${\cal Q}$}\ its universal rank 2 quotient, and ${\cal S}\subset K_H$ the universal subbundle. These fit into the exact sequence $$0\to{\cal S}\to K_H\to \mbox{${\cal Q}$}\to 0$$ of sheaves on $H$. The natural quotient $V_H\to V_H/{\cal S}$ induces a map $H\to G$ by the universal property of the Grassmannian; since $V_H\to \O(1)_H$ clearly factors through $V_H/{\cal S}$, it is easy to see that $H$ may be identified with the space of pointed 2-planes in $\P(V)$. Here $\O(1)$ denotes the tautological sheaf on $\P(V)$ as before. The conics containing $p$ globalize to a rank 5 bundle $W$ on $H$. This bundle is in fact the the kernel of the natural map $S^2(V_H/{\cal S})\to\O(2)_H$. Then the moduli space $M'$ of pointed conics may be seen to be $\P(W^*)$. Let $\O_W(1)$ be its tautological bundle. Let $h=c_1(\O(1)_{M'})$. Consider the natural morphism $f:M'\to M$. The variety of conics meeting $L^a$ is represented by the class $f_*(h^a)$ for $a>1$. Note that for $a=1$, $f_*(h)=2$. This factor exactly gives the factor needed to give the Gromov-Witten invariants rather than the number of conics meeting three linear subspaces. So the Gromov-Witten invariants are given by the formula $n^a_b(2)=\int_Mc_{2k+5}(Q)f_*(h^a)f_*(h^b)f_*(h^c)$, which is valid since the dimensions work out correctly. To compute these as numbers using {\sc schubert}, everything is clear, except the description of the morphism $f$. But this may be described merely by knowing the pullbacks $f^*(Q)$ and $f^*(\O_{\P}(1))$. However, from the above description and the universal properties, this is just $V_H/{\cal S}$ and $\O_W(1)$. {\sc schubert}\ takes care of the rest.

9301

alg-geom/9301007

en

https://arxiv.org/abs/alg-geom/9301007

alg-geom/9301007

Zhi-Jie Chen

Zhi-Jie Chen

Bounds of automorphism groups of genus 2 fibrations

30 pages, LaTeX2.09

For a complex surface of general type with a relatively minimal genus 2 fibration, the bounds of the orders of the automorphism group of the fibration, of its abelian subgroups and of its cyclic subgroups are determined as linear functions of $c^2_1$. Most of them are the best.

alg-geom

\section{Preliminaries} The surfaces with genus 2 pencils have been largely studied by many authors. The facts we needed in this paper mostly appeared in [3, 6, 9, 10]. In particular, Xiao's book \cite{X3} gave a systematic description of the properties of genus 2 fibrations which are just what we needed here. Unfortunately, this book has not been translated into English yet, hence it is not available for most readers. For this reason, we will recall some materials in this section. Let $f:S\longrightarrow C$ be a relatively minimal fibration of genus 2, $\omega_{S/C}=\omega_S\otimes f^*\omega_C^\vee$ the relative canonical sheaf of $f$. For a sufficiently ample invertible sheaf $\cal L$ on $C$, the natural morphism $f^*(f_*\omega_{S/C}\otimes{\cal L})\longrightarrow \omega_{S/C} \otimes f^*{\cal L}$ defines a natural map $\Phi$: \begin{center} \begin{picture}(75,25)(0,-25) \multiput(5,-2.5)(1.5,0){18}{\line(1,0){1}} \put(32,-2.5){\vector(1,0){3}} \put(2.5,-5){\vector(1,-1){15}} \put(37.5,-5){\vector(-1,-1){15}} \put(20,-22){\makebox(0,0)[t]{$C$}} \put(2.5,-4){\makebox(0,0)[b]{$S$}} \put(36,-4){\makebox(0,0)[lb]{$P=\mbox{\mib P}\,(f_*\omega_{S/C}\otimes {\cal L})$}} \put(20,-1.5){\makebox(0,0)[b]{$\Phi$}} \put(7,-10.5){\makebox(0,0)[tr]{$f$}} \put(32,-11.5){\makebox(0,0)[tl]{$\pi$}} \end{picture} \end{center} $\Phi$ is called a relatively canonical map. By a succession of blow-ups, we can obtain the following commutative diagram: \begin{center} \begin{picture}(40,49)(0,-25) \multiput(5,-2.5)(1.5,0){18}{\line(1,0){1}} \put(32,-2.5){\vector(1,0){3}} \put(2.5,-5){\vector(1,-1){15}} \put(37.5,-5){\vector(-1,-1){15}} \put(20,-22){\makebox(0,0)[t]{$C$}} \put(2.5,-4){\makebox(0,0)[b]{$S$}} \put(37.5,-4){\makebox(0,0)[b]{$P$}} \put(20,-1.5){\makebox(0,0)[b]{$\Phi$}} \put(7,-10.5){\makebox(0,0)[tr]{$f$}} \put(32,-11.5){\makebox(0,0)[tl]{$\pi$}} \put(5,18.5){\vector(1,0){30}} \multiput(2.5,16)(35,0){2}{\vector(0,-1){16}} \put(2.5,17){\makebox(0,0)[b]{$\tilde{S}$}} \put(37.5,17){\makebox(0,0)[b]{$\tilde{P}$}} \put(1.5,8){\makebox(0,0)[r]{$\rho$}} \put(38.5,8){\makebox(0,0)[l]{$\psi$}} \put(20,19.5){\makebox(0,0)[b]{$\tilde{\theta}$}} \end{picture} \end{center} where $\rho$ and $\psi$ are compositions of finitely many blow-ups, $\tilde{\theta}$ is a double cover. Then we get the branch loci $\tilde R$ on $\tilde P$ and $R$ on $P$ such that $\tilde R$ is the minimal even resolution of $R$ (i.e. the canonical resolution of double cover). If $\cal L$ is sufficiently ample, then all the singularities of $R$ must be located in one of the 6 types of singular fibers defined by Horikawa \cite{H1}---0), I), II), III), IV) or V). $P$ is a relatively minimal ruled surface. We denote a section which has the least self-intersection by $C_0$ such that $C_0^2=-e$. We will use $F$ to denote both the fiber of $f$ or of $\pi$. A singular point of the branch locus will be called {\it negligible} if this point itself and all its infinitely near points are double points or triple points with at least 2 different tangents. By minimal even resolution, the inverse image of a negligible singular point is composed of $(-2)$-curves. All other singular points are called {\it non-negligible}. The singular fiber of type 0) in the classification of Horikawa is nothing else but the fiber which does not contain any non-negligible singular points. The minimal even resolution $\psi:\tilde P\longrightarrow P$ can be decomposed into $\tilde{\psi}:\tilde P\longrightarrow \hat P$ followed by $\hat{\psi}: \hat{P}\longrightarrow P$, where $\tilde{\psi}$ and $\hat{\psi}$ are composed respectively of negligible and non-negligible blow-ups. The image of $\tilde{R} $ in $\hat{P}$ is denoted by $\hat{R}$. If we take away all the isolated vertical $(-2)$-curves from the reduced divisor $\hat{R}$, we get a new reduced divisor $\hat{R}_p$, which is called the {\it principal part} of the branch locus $\hat{R}$. Then for any fiber $F$ of $\pi:P\longrightarrow C$, the second and third {\it singularity index} of $F$, $s_2(F)$, $s_3(F)$, will be defined as follows: If $R$ has no quadruple singularities on $F$, then $s_3(F)$ equals the number of $(3\rightarrow3)$ type singularities of $R$ on $F$. Otherwise $s_3(F)$ equals the number of $(3\rightarrow3)$ type singularities of $R$ on $F$ plus 1. Hence $s_3(F)=0$ if and only if $R$ has no non-negligible singularities on $F$. Let $\phi:\hat{R}_p\longrightarrow C$ be the natural projection induced by $\pi\circ\hat{\psi}:\hat{P}\longrightarrow C$. Then the second singularity index $s_2(F)$ of $F$ will be the ramification index of the divisor $\hat{R}_p$ on $f(F)$ with respect to the projection $\phi$. If $\hat{R}_p$ has singularities (which must be negligible) on $F$, the singularity index $s_2(F)$ can be calculated as follows. For a smooth point $p\in\hat{R}_p\cap F$, the ramification index of $\phi$ at $p$ can be defined as for an ordinary smooth curve. If $p\in\hat{R}_p\cap F$ is a singular point of $\hat{R}_p$, then the ramification index of $\phi$ at $p$ will be defined as the sum of ramification indices of the normalization of $\hat{R}_p$ at the preimage of $p$ with respect to its projection to $C$ plus the double of the influence to the arithmetic genus of $\hat{R}_p$ during its normalization at the singular point $p$. If the normalization of $\hat{R}_p$ contains an isolated vertical component $E$, then the contribution of $E$ to the ramification index of $\phi$ is equal to $2g(E)-2$. As there are finite number of fibers $F$ with $s_i(F)\ne0$, we define the $i$-th {\it singularity index} of $f$, $s_i(f)$, to be the sum of $s_i(F)$ for all fibers, when $i=2$, 3. If we take away from the branch locus $R$ all the fibers $F$ with odd $s_3(F)$, we obtain a divisor $R_p$ which is called the {\it principal part} of $R$. Suppose that $$R_p\sim -3K_{P/C}+nF,$$ where $K_{P/C}$ is the relative canonical divisor of $\pi$ and $\sim$ represents numerical equivalence. With these definitions, the formula computing the relative invariants of a genus 2 fibration can be stated as follows. \begin{Theorem}[Xiao \cite{X3}]\label{ThmX} Let $f:S\longrightarrow C$ be a relatively minimal fibration of genus 2. Then $$K_{S/C}^2=K_S^2-8(g(C)-1)=\frac15s_2(f)+\frac75s_3(f)=2n-s_3(f),$$ $$\chi_f=\chi({\cal O}_S)-(g(C)-1)=\frac1{10}s_2(f)+\frac15s_3(f)=n-s_3(f).$$ \end{Theorem} \section{Local cases} We begin with a local fibration $f:S_{\Delta}\longrightarrow\Delta$ where $f$ is an analytic mapping onto the unit disk $\Delta$, $S_{\Delta}$ is a 2-dimensional analytic smooth manifold and the fibers of $f$ are projective curves. We assume that the fiber of the zero is singular and all the fibers over $\Delta^*=\Delta-\{0\}$ are smooth curves of genus 2. Similarly, we have a commutative diagram \begin{center} \begin{picture}(40,49)(0,-25) \multiput(5,-2.5)(1.5,0){18}{\line(1,0){1}} \put(32,-2.5){\vector(1,0){3}} \put(2.5,-5){\vector(1,-1){15}} \put(37.5,-5){\vector(-1,-1){15}} \put(20,-22){\makebox(0,0)[t]{$\Delta$}} \put(2.5,-4){\makebox(0,0)[b]{$S_\Delta$}} \put(38.5,-4){\makebox(0,0)[b]{$P_\Delta$}} \put(20,-1.5){\makebox(0,0)[b]{$\Phi$}} \put(7,-10.5){\makebox(0,0)[tr]{$f$}} \put(32,-11.5){\makebox(0,0)[tl]{$\pi$}} \put(5,18.5){\vector(1,0){30}} \multiput(2.5,16)(35,0){2}{\vector(0,-1){16}} \put(2.5,17){\makebox(0,0)[b]{$\tilde{S}_\Delta$}} \put(37.5,17){\makebox(0,0)[b]{$\tilde{P}_\Delta$}} \put(1.5,8){\makebox(0,0)[r]{$\rho$}} \put(38.5,8){\makebox(0,0)[l]{$\psi$}} \put(20,19.5){\makebox(0,0)[b]{$\tilde{\theta}$}} \end{picture} \end{center} Denote the branch locus in $P_\Delta$ by $R_\Delta$. We also denote the horizontal part of $R_\Delta$ by $R'_\Delta$, that is, $$R'_\Delta=\left\{\begin{array}{lll} R_\Delta-F_0,&\qquad&\mbox{if $R_\Delta$ contains $F_0$,}\\ R_\Delta&&\mbox{otherwise.} \end{array}\right.$$ Let $F_0=\pi^{-1}(0)$, $F_t=\pi^{-1}(t)$, $t\in\Delta^*$, and $K_\Delta=\{\tilde{\sigma}\in\mbox{Aut} (S_\Delta)|f\circ\tilde{\sigma}=f\}$. Any automorphism $\tilde{\sigma}\in K_\Delta$ induces an automorphism $\sigma$ of $P_\Delta$ satisfying $\pi\circ\sigma =\pi$ and $\sigma(R_\Delta)=R_\Delta$. We denote the image of $K_\Delta$ by $\bar{K}_\Delta\subseteq \mbox{Aut }P_\Delta$, then $$|K_\Delta|=2|\bar{K}_\Delta|.$$ Note that any finite automorphism subgroup of {\mib P\,}$^1$ must be one of the following: \begin{center} \begin{tabular}{llcl} $G\subseteq \mbox{Aut({\mib P\,}$^1$)}$&&$|G|$&Number of points in an orbit\\ Cyclic group&$Z_n$&$n$&1, $n$\\ Dihedral group&$D_{2n}$&$2n$&2, $n$, $2n$\\ Tetrahedral group&$T_{12}$&12&4, 6, 12\\ Octahedral group&$O_{24}$&24&6, 8, 12, 24\\ Icosahedral group&$I_{60}$&60&12, 20, 30, 60 \end{tabular} \end{center} For any $\sigma\in\bar{K}_\Delta$, its restriction to $F_t\cong \mbox{\mib P\,}^1$, $\sigma|_{F_t}$, must preserve the set of 6 points contained in $F_t\cap R_\Delta$. Hence $\bar{K}_\Delta$ can be isomorphic to the following groups: $O_{24}$, $T_{12}$, $D_{12}$, $D_6$, $Z_6$, $Z_5$, $D_4$, $Z_4$, $Z_3$, $Z_2$ and 1. \begin{Lemma}\label{L1} If $\bar{K}_\Delta\cong O_{24}$, $T_{12}$ or $D_{12}$, then $F_0$ is contained in $R_\Delta$ and $R_\Delta$ has 6 ordinary double points on $F_0$. In this case, we have $s_2(F_0)=10$, $s_3(F_0)=0$. \end{Lemma} \noindent{\bf Proof.} Since $\bar{K}_\Delta\cong O_{24}$, $T_{12}$ or $D_{12}$, $R_\Delta\cap F_t$ ($t\in \Delta^*$) consists respectively of 6 vertices of a regular octahedron, of 6 points corresponding to the centers of edges of a regular tetrahedron, or of sixth roots of unit. These 6 horizontal branches of $R_\Delta$ cannot intersect when $t\rightarrow 0$. But by assumption, $R_{\Delta}$ must have some singularities, so $F_0$ is contained in $R_\Delta$. Since $R_\Delta$ does not contain non-negligible singularities, one has $s_3(F_0)=0$ and $R_\Delta=\hat{R}_\Delta=(\hat{R} _\Delta )_p$. On $F_0$, $R_\Delta$ has 6 ordinary double points, the influence of each double point to the arithmetic genus of $R_\Delta$ during its normalization is equal to 1. The preimage of $F_0$ in the normalization of $R_\Delta$ is a smooth vertical rational curve which does not meet any other branches, so its contribution to the index $s_2(F_0)$ is equal to $-2$. Therefore $s_2(F_0)=2\times 6+(-2)=10$.\hspace{\fill} $\Box$ \vspace{5mm} We list the following useful lemmas, the proof is evident. Since local equations are used for calculation of singularity indices, they are given in simplified form, omitting some higher order terms. All the non-negligible singularities here are canonical, i.e. defined by Horikawa. \begin{Lemma}\label{L2} If $\bar{K}_\Delta\cong D_6$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, (1) The equation of $R'_\Delta$ is $(x^3-t^k)(t^kx^3-1)$, $k>0$. In this case, $s_3(F_0)=0$ implies $s_2(F_0)\ge4$. (2) The equation of $R'_\Delta$ is $(x^3-1)^2-t^k(x^3+1)^2$, $k>0$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge3$. \end{Lemma} \begin{Lemma}\label{L3} If $\bar{K}_\Delta\cong Z_6$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, the equation of $R'_\Delta$ is $x^6-t^k$, $1\le k\le3$. If $k=3$, it has a non-negligible singularity and $s_3(F_0)=1$, $s_2(F_0)=3$. Otherwise $s_2(F_0)\ge5$. \end{Lemma} \begin{Lemma}\label{L4} If $\bar{K}_\Delta\cong Z_5$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, (1) The equation of $R'_\Delta$ is $x(x^5-t^k)$, $k=1$, $2$. In this case, $s_3(F_0)=0$, $s_2(F_0)\ge6$. (2) The equation of $R'_\Delta$ is $x(t^kx^5-1)$, $k=1$, $2$. In this case, $s_3(F_0)=0$, $s_2(F_0)\ge4$. \end{Lemma} \begin{Lemma}\label{L5} If $\bar{K}_\Delta\cong D_4$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, (1) The equation of $R'_\Delta$ is $(x^2-1)((x-1)^2-t^k(x+1) ^2) (t^k(x-1)^2-(x+1)^2)$, $k>0$. In this case, $s_3(F_0)=0$ implies $s_2(F_0)\ge 6$. (2) The equation of $R'_\Delta$ is $(x^2-1)(x^2-t^k)(t^kx^2-1)$, $k>0$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge2$. \end{Lemma} \begin{Lemma}\label{L6} If $\bar{K}_\Delta\cong Z_4$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, the equation of $R'_\Delta$ is $x(x^4-t^k)$, $k=1$, $2$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge5$. \end{Lemma} \begin{Lemma}\label{L7} If $\bar{K}_\Delta\cong Z_3$ and $R'_\Delta$ is not \'etale over $\Delta$, then up to a coordinates transformation, (1) The equation of $R'_\Delta$ is $(x^3-t^{k_1})(t^{k_2}x^3- a(t))$, $k_1, k_2>0$, $a(0)\ne 0$. In this case, $s_3(F_0)=0$ implies $s_2(F_0)\ge 4$. (2) The equation of $R'_\Delta$ is $x^6+a(t)x^3+t^k$, $1\le k\le 3$. In this case, $s_3(F_0)=0$ implies $s_2(F_0)\ge5$. (3) The equation of $R'_\Delta$ is $(x^3-b-t^{k_1})(x^3-b- t^{k_2}a(t))$, $k_1, k_2>0$, $a(0)\ne0$ and $b\ne0$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge 6$. (4) The equation of $R'_\Delta$ is $(x^3-t^k)(x^3-a(t))$, $1\le k\le3$, $a(0)\ne0$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge2$. (5) The equation of $R'_\Delta$ is $((x-b)^2-t^ka(t))((x-b\omega)^2- \omega^2t^k a(t))((x-b\omega^2)^2-\omega t^ka(t))$, $k>0$, $a(0)\ne0$, $b\ne0$, $\omega=\exp(2\pi i/3)$. In this case, we have $s_3(F_0)=0$, $s_2(F_0)\ge 3$. \end{Lemma} We summarize the results of Lemmas \ref{L2} through \ref{L7} in the following table. Here we assume that $R'_\Delta$ has only negligible singularities or ramifications on $F_0$. $$\begin{array}{lccc} \bar{K}_\Delta&|K_\Delta|&\multicolumn{1}{c}{s_2(F_0)} &\multicolumn{1}{c}{|K_\Delta|/s_2(F_0)}\\ D_6&12&\ge3&\le4\\ Z_6&12&\ge5&\le2.4\\ Z_5&10&\ge4&\le2.5\\ D_4&8&\ge2&\le4\\ Z_4&8&\ge5&\le1.6\\ Z_3&6&\ge2&\le3\\ Z_2&4&\ge1&\le4\\ 1&2&\ge1&\le2 \end{array}$$ \begin{Lemma}\label{L8} If $R'_\Delta$ has only negligible singularities or ramifications on $F_0$, then $|K_\Delta|/s_2(F_0)\le4$. Moreover, if $\bar K_{\Delta}\cong Z_6$, $Z_5$, $Z_4$ or 1, then $|K_\Delta|/s_2(F_0)$ $\le2.5$. \end{Lemma} \section{Bounds of automorphism groups} Let $G=\mbox{Aut}(f)$ be the automorphism group of the fibration of genus two $f:S\longrightarrow C$. Then we have an exact sequence $$\begin{array}{ccccccccc} 1&\longrightarrow &K&\longrightarrow &G&\longrightarrow &H&\longrightarrow& 1,\\ &&&&(\tilde{\sigma},\sigma)&\mapsto&\sigma& \end{array} $$ where $H\subseteq \mbox{Aut}(C)$, $K=\{(\tilde{\sigma}, \mbox{id}) \in G\}=\{\tilde{\sigma}\in\mbox{Aut}(S)|f\circ \tilde{\sigma} =f\}$. Thus $$|G|=|K||H|.$$ The elements of $H$ are often regarded as transformations of the fibers of $f$ or $\pi$. \begin{Proposition}\label{P1} If $f:S\longrightarrow C$ is a relatively minimal fibration of genus 2 with $g(C)\ge2$, then $$|\mbox{\rm Aut}(f)|\le504K^2_S.$$ \end{Proposition} \noindent{\bf Proof.} Since $|K|\le48$, $|H|\le|\mbox{Aut}(C)|\le84(g(C)-1)$, we have $$|G|=|K||H|\le4032(g(C)-1).$$ On the other hand, $K^2_{S/C}\ge0$ and the equality holds if and only if $f$ is locally trivial. Hence $$K_S^2\ge8(g-1)(g(C)-1)=8(g(C)-1),$$ and $$|G|\le504K_S^2.$$ \vspace{-8mm}\hspace*{\fill}$\Box$ \begin{Proposition}\label{P2} If $f:S\longrightarrow C$ is a relatively minimal fibration of genus 2 which is not locally trivial with $g(C)\ge2$, then $$|\mbox{\rm Aut}(f)|\le126K_S^2.$$ \end{Proposition} \noindent{\bf Proof.} Let $R'$ denote the horizontal part of the branch locus $R$. If $R'$ is not \'etale over $C$, then by Lemmas of Section 2, we have $|K|\le12$. As $|H|\le84 (g(C)-1)\le 10.5K_S^2$, $$|G|\le12|H|\le126K_S^2.$$ Now assume that $R'$ is \'etale. Since $f$ is not locally trivial, we must have $K_{S/C}^2>0$, i.e. either $s_3(f)>0$ or $s_2(f)>0$. So $R$ must contain some fiber $F_0$. By Lemma~\ref{L1}, $s_3(F_0)=0$, $s_2(F_0)=10$. Let $p=f(F_0)$, $n=|H|$. Since $H$ is a subgroup of Aut$(C)$, $H$ determines a finite morphism $\tau:C \longrightarrow X=C/H$. Denote the ramification index of $p\in C$ with respect to $\tau$ by $r$ and the other ramification indices by $r_i$. Then the Hurwitz's theorem implies that $$2g(C)-2=n(2g(X)-2)+n\sum\left(1-\frac1{r_i}\right).$$ As the $H$-orbit of the point $p$ has $n/r$ points, this implies that $s_2(f)\ge10n/r$. Hence \begin{eqnarray*} K_S^2&\ge&\dfrac15s_2(f)+8(g(C)-1)=\dfrac{2n}r +4n\left[2g(X) -2+ \sum \left(1-\dfrac1{r_i}\right)\right]\\ &=&4n\left[2g(X)-2+\dfrac1{2r}+\sum\left(1-\dfrac1{r_i}\right)\right]. \end{eqnarray*} It is not difficult to see that the expression $2g(X)-2+1/2r+ \sum(1- 1/r_i)$ reaches its minimal value $2/21$ (under the condition $2g(X)-2+ \sum(1-1/r_i)>0$) when $g(X)=0$, $r_1=2$, $r_2=3$, and $r=r_3=7$. That is $$K_S^2\ge\frac8{21}n=\frac8{21}|H|.$$ Thus $$|G|\le48|H|\le126K_S^2.$$ \vspace{-8mm} \hspace*{\fill}$\Box$ \vspace{5mm} \noindent{\bf Remark.} It is not difficult to see that if $f$ is not locally trivial with $g(C)\ge2$ and $|\mbox{Aut}(f)|=126K_S^2$, then $|\mbox{Aut}(C)|=84(g(C)-1)$, $|\mbox{Aut}(F)|=48$ for any smooth fiber $F$ and Aut$(f)\cong \mbox{Aut}(C)\times\mbox{Aut}(F)$. We will give an example later. In this case, the fibration $f$ is of constant moduli and $S/\mbox{Aut}(f)\cong{\mib F}_1$. \begin{Lemma}\label{L31} Let $S$ be a surface of general type which has a relatively minimal genus 2 fibration $f:S\longrightarrow C$. If the third singularity index $s_3(f)\ne0$, then $$|\mbox{\rm Aut}(f)|\le \frac{60}7rK^2_{S/C},$$ where $$r=\min_{\mbox{${\scriptstyle s_3(F)\ne0}$}}|\mbox {\rm Stab}_Hf(F)|,$$ $\mbox{\rm Stab}_Hf(F)$ is the stabilizer of $f(F)$ in $H$. \end{Lemma} \noindent{\bf Proof.} Let $F_0$ be a singular fiber such that $s_3(F_0)\ne 0$ and $r=|\mbox{Stab}_H f(F_0)|$. Then $$K^2_{S/C}\ge\frac75s_3(f)\ge\frac{7s_3(F_0)}{5r}|H|,$$ and we get $$|G|=|K||H|\le\frac{r}{s_3(F_0)}\cdot\frac{60}7K^2_{S/C} \le\frac{60}7rK^2_{K/C}.$$ \vspace{-8mm}\hspace*{\fill}$\Box$ \begin{Lemma}\label{L32} Let $S$ be a surface of general type which has a relatively minimal genus 2 fibration $f:S\longrightarrow C$. If the horizontal part $R'$ of the branch locus $R$ is not \'etale and has only negligible singularities or ramifications, then $$|\mbox{\rm Aut}(f)|\le20rK^2_{S/C},$$ where $$r=\min_{\mbox{\scriptsize ${\scriptstyle F}$ singular fiber}}|\mbox {\rm Stab}_Hf(F)|.$$ \end{Lemma} \noindent{\bf Proof.} Let $F_0$ be a singular fiber with $r=|\mbox{Stab}_Hf(F_0)|$. Since here $$K^2_{S/C}\ge\frac15s_2(f)\ge\frac{s_2(F_0)}{5r}|H|,$$ we have $$|G|=|K||H|\le\frac{r|K|}{s_2(F_0)}\cdot5K^2_{S/C}\le 20rK^2_{S/C},$$ by Lemma \ref{L8}.\hspace{\fill}$\Box$ \begin{Lemma}\label{L33} Let $S$ be a surface of general type which has a relatively minimal genus 2 fibration $f:S\longrightarrow C$. If the horizontal part $R'$ of the branch locus $R$ is \'etale, then $$|\mbox{\rm Aut}(f)|\le24rK^2_{S/C},$$ where $$r=\min_{\mbox{\scriptsize ${\scriptstyle F}$ singular fiber}}|\mbox {\rm Stab}_Hf(F)|.$$ \end{Lemma} \noindent{\bf Proof.} Let $F_0$ be a singular fiber with $r=|\mbox{Stab}_Hf(F_0)|$. By assumption, we have $s_2(F_0)=10$. Hence $$|G|=|K||H|\le\frac{r|K|}{s_2(F_0)}\cdot5K^2_{S/C}\le 24rK^2_{S/C}.$$ \vspace{-9.5mm}\hspace*{\fill}$\Box$ \vspace{5mm} Let $\bar{K}$ denote the subgroup in Aut$(P)$ which is induced by $K$. If $\sigma\in\bar{K}$, then $\pi\circ\sigma=\pi$ and $\sigma(R)=R$. Let $K_1$ be a cyclic subgroup of order $m$ of $\bar{K}$, $Q=P/K_1$ be the quotient surface. Then $Q$ is a ruled surface. We have a commutative diagram \begin{center} \begin{picture}(40,25)(0,-25) \put(5,-2.5){\vector(1,0){30}} \put(2.5,-5){\vector(1,-1){15}} \put(37.5,-5){\vector(-1,-1){15}} \put(20,-22){\makebox(0,0)[t]{$C$}} \put(2.5,-4){\makebox(0,0)[b]{$P$}} \put(37.7,-3.5){\makebox(0,0)[b]{$Q$}} \put(20,-1.5){\makebox(0,0)[b]{$\alpha$}} \put(7,-11.5){\makebox(0,0)[tr]{$\pi$}} \put(32,-11.5){\makebox(0,0)[tl]{$\pi'$}} \end{picture} \end{center} Let $C_0$ and $C_\infty\sim C_0+eF$ be the reduced ramification divisors of $K_1$. Let $C'_0$ be a section of $\pi'$ with the least self-intersection ${C'_0}^2=-e'$, $F'$ be a general fiber of $\pi'$. Then $\alpha^*C'_0=mC_0$, $\alpha^*C'_\infty=mC_\infty$, $\alpha^* F'=F$ and $e'=me$. Let $D=\alpha(R')$, $C'=C'_0+C'_\infty$ be the branch locus. Then $C'\sim2C'_0+e'F'\sim -K_{Q/C}$. \begin{Lemma}\label{L34} Assume $\bar{K}\cong D_6$. If $R'$ is not \'etale and has only negligible singularities or ramifications, then $f$ has more than one $H$-orbit of singular fibers. \end{Lemma} \noindent{\bf Proof.} Let $K_1$ be the unique cyclic subgroup of order 3 of $\bar{K}$. There are 2 types of singular fibers as listed in Lemma \ref{L2}. Let $F_0$ be a singular fiber. Then the local equations of $D$ near $F_0$ are (1) $(x-t^k)(t^kx-1)$, $k\le3$, (2) $(x-1)^2-t^k(x+1)^2$, $k>0$. In case (1), $D$ meets $C'$ at 2 points in $F_0$. In case (2), $D$ does not meet $C'$ in $F_0$. If all the singular fibers of $f$ is of type (1), then $D$ is an \'etale cover of $C$. This means that $a=e'$, $C'\sim D$. Hence $DC'=0$ which is impossible because $D$ and $C'$ meet in $F_0$. If all the singular fibers of $f$ is of type (2), then $DC'=0$. Hence $D\sim C'$ and $D(D+K_{Q/C})=0$. This means that $D$ is \'etale over $C$. A contradiction.\hspace*{\fill}$\Box$ \begin{Lemma}\label{L35} Assume $\bar{K}\cong D_4$. If $R'$ is not \'etale, then $f$ has more than one $H$-orbit of singular fibers. If $H$ is cyclic and $g(C)=0$, then $$|\mbox{\rm Aut}(f)|\le 12.5K^2_{S/C}.$$ \end{Lemma} \noindent{\bf Proof.} In this case, there are 4 sections in $P$ which do not meet each other. Hence $e=0$. $R'$ contains 2 of these sections and denoted by $C_0$ and $C_\infty$. Let $K_1$ be a cyclic subgroup of $\bar K$ with $C_0$ and $C_\infty$ as ramifications. Assume that there is only one $H$-orbit of singular fibers. If these singular fibers are all of type (1) in Lemma \ref{L5}, then the local equation of $D=\alpha(R'-C_0-C_\infty)$ is $(x-t^k)(t^kx-1)$, namely, $D$ is \'etale. Therefore $D\sim 2C'_0$, $DC'_0=DC'_\infty=0$, a contradiction. If the singular fibers are of type (2) in Lemma \ref{L5}, then $D$ does not meet $C'_0$ and $C'_\infty$. Hence $D\sim 2C'_0$, $D^2=0$, a contradiction. That implies there are at least 2 $H$-orbits. Now suppose $H$ is cyclic. Let $h=|H|$. We call an $H$-orbit {\it big} if it contains $h$ fibers. If there is a big $H$-orbit whose singular fibers are of type (1), then $s_2(F_0)\ge6$, so $|G|\le (20/3)K^2_{S/C}$. If $|G|>(20/3)K^2_{S/C}$, then the singular fibers in a big $H$-orbit must be of type (2) with $k\le2$. Let $F_2$ and $F_3$ denote 2 fibers fixed by $H$. Then at least one of them is of type (1). The structure of types (1) and (2) implies that the normalization of $D=\alpha(R'-C_0-C_\infty)$ is \'etale with respect to $\pi'$. Hence $D$ must decompose into 2 isomorphic sections $D_1$ and $D_2$, $D_1\sim D_2\sim C'_0+aF'$. Since both $D_1$ and $D_2$ meet $C'_0$ and $C'_\infty$, $F_2$ and $F_3$ are all singular of type (1). Since $D_1D_2=2a=kh$, $D_1C'_0=a=kh/2$. Hence the local equation of $R'$ near $F_2$ or $F_3$ is $(x^2-1)((x-1)^2-t^{kh/2}(x+1)^2)(t^{kh/2}(x-1)^2-(x+1)^2)$. When $h\ge6$, these are non-negligible singularities. If $F_i$ ($i=2$, 3) is singular fiber of type I), $s_3(F_i)=2[(kh-2)/8]+1\ge(kh-1)/4$. If $F_i$ is of type II), $s_3(F_i)=2[kh/8]\ge(kh-6)/4$. So $$K^2_{S/C}\ge \frac15\times 2\times h+\frac75\times\frac{h-6}4\times2=\frac{11}{10}h-\frac{21}5.$$ $$|G|=8h\le\frac{80}{11}(K^2_{S/C}+\frac{21}5)<12.5K^2_{S/C}.$$ If there are more than one big $H$-orbits, it can be similarly shown that $|G|\le 12.5K^2_{S/C}$.\hspace*{\fill}$\Box$ \begin{Lemma}\label{L36} Assume $\bar{K}\cong Z_3$. If $R'$ is not \'etale and has only negligible singularities or ramifications and $f$ has only one $H$-orbit of singular fibers, then $$|\mbox{\rm Aut}(f)|\le 6rK^2_{S/C}.$$ where $$r=\min_{\mbox{\scriptsize ${\scriptstyle F}$ singular fiber}}|\mbox {\rm Stab}_Hf(F)|.$$ \end{Lemma} \noindent{\bf Proof.} Let $K_1=\bar K$. If the singular fibers are of types (1) or (4) in Lemma \ref{L7}, then $D\sim 2C'_0+aF'$ is \'etale. $D(K_{Q/C}+D)=0$ implies $a=e'$. Hence $D(C'_0+C'_\infty)=0$, a contradiction. If the singular fiber $F_0$ is of type (5) with $k=1$, then $D$ is irreducible and smooth near $F_0$. This implies $DC'_\infty\ne0$, a contradiction. Therefore $s_2(F_0)\ge5$ for any singular fiber $F_0$. So $|G|\le6rK^2_{S/C}$.\hspace{\fill}$\Box$ \begin{Lemma}\label{L37} Assume $\bar{K}\cong Z_2$. If $R'$ is not \'etale and $f$ has only one $H$-orbit of singular fibers, then $$|\mbox{\rm Aut}(f)|\le 5rK^2_{S/C}.$$ where $$r=\min_{\mbox{\scriptsize ${\scriptstyle F}$ singular fiber}}|\mbox {\rm Stab}_Hf(F)|.$$ \end{Lemma} \noindent{\bf Proof.} Let $F_0$ be a singular fiber. $|G|>5rK^2_{S/C}$ implies $s_2(F_0)\le3$. We distinguish between 2 cases. {\it Case I.\/} $R'$ contains $C_0$ and $C_\infty$. Then the local equation of $R'$ near $F_0$ must be (1) $x(x^2-t)(x^2-a(t))$, $a(0)\ne 0$, $s_2(F_0)=3$; (2) $x((x^2-a^2)^2-t)$, $a\ne0$, $s_2(F_0)=2$. Let $D=\alpha (R'-C_0-C_\infty)\sim 2C'_0+aF'$. If all the singular fibers are of type (1), then $D$ is \'etale. This is impossible. If the singular fibers are of type (2), then $D$ is irreducible and does not meet $C'$. This is impossible. {\it Case II.\/} $R'$ does not contain $C_0$ and $C_\infty$. Then the local equation of $R'$ may be (1) $(x^2-t)(x^2-a(t))(x^2-b(t))$, $a(0)b(0)\ne0$, $a(0)\ne b(0)$, $s_2(F_0)=1$; (2) $(x^2-t)(ta(t)x^2-1)(x^2-b(t))$, $a(0)b(0)\ne0$, $s_2(F_0)=2$; (3) $((x^2-a^2)^2-t)(x^2-b(t))$, $ab(0)\ne0$, $s_2(F_0)=2$; (4) $((x^2-a^2)^2-t)(x^2 -tb(t))$, $b(0)\ne0$, $s_2(F_0)=3$. Let $D=\alpha(R')\sim 3C'_0+aF'$. If $F_0$ is of type (1) or (2), then $D$ is \'etale and smooth. $D$ must be decomposed into 3 disjoint components. This means $e'=0$, a contradiction. If $F_0$ is of type (3) or (4), then $D$ is smooth. The ramification index $D(D+K_{Q/C})=4a-6e'=|H|/r$. Hence $DC'=2a-3e'=|H|/2r$. This is a contradiction because we have $DC'=0$ for type (3) and $DC'=|H|/r$ for type (4). \hspace{\fill}$\Box$ \begin{Proposition}\label{P3} If $S$ is a minimal surface of general type which has a genus 2 fibration $f:S\longrightarrow C$ with $g(C)=1$, then $$|\mbox{\rm Aut}(f)|\le144K_S^2.$$ \end{Proposition} \noindent{\bf Proof.} In this case, we have $$K_S^2=K_{S/C}^2=\frac15s_2(f)+\frac75s_3(f)>0.$$ Thus either $s_3(f)>0$ or $s_2(f)>0$. Let $j(C)$ be the $j$-invariant of the elliptic curve $C$. Let $m$ denote the number of points contained in a smallest $H$-orbit of $C$. Since $H$ is a finite subgroup of Aut$(C)$, we have $$m=\left\{\begin{array}{lll} |H|/2&\qquad&\mbox{if $j(C)\ne0$, 1728,}\\ |H|/4&&\mbox{if $j(C)=1728$,}\\ |H|/6&&\mbox{if $j(C)=0$.} \end{array}\right.$$ Since $r\le6$, by Lemma \ref{L31}, \ref{L32} and \ref{L33}, the conclusion is immediate.\hspace{\fill}$\Box$ \begin{Proposition}\label{P4} If $S$ is a surface of general type which has a relatively minimal fibration of genus two $f:S\longrightarrow C$ with $g(C)=0$, then $$|\mbox{\rm Aut}(f)|\le120(K_S^2+8).$$ Moreover, we have $$|\mbox{\rm Aut}(f)|\le48(K_S^2+8)$$ for $K^2_S\ge33$, and when $K^2_S\le32$, there are only 4 exceptions. \end{Proposition} \noindent{\bf Proof.} In this case, we have $$K_S^2+8=K_{S/C}^2=\frac15s_2(f)+\frac75s_3(f)>0.$$ Hence either $s_3(f)>0$ or $s_2(f)>0$. {\it Case I.\/} Assume that $R'$ is \'etale over $C$. If $r\le5$, then by Lemma \ref{L33} $$|G|\le24rK^2_{S/C}\le120(K^2_S+8).$$ If $r\ge6$, then $H$ must be cyclic or dihedral group. In this case, there are at most 2 singular fibers. Hence $K^2_{S/C}\le4$ by Theorem \ref{ThmX}. This means $S$ is not of general type[10, Theorem 4.2.5, p.90]. {\it Case II.\/} Assume that $R'$ is not \'etale. Then $f$ is a fibration of variable moduli. Hence $f$ must contain more than 2 singular fibers (\cite{Beauville}). This implies $r\le5$. The conclusion is obtained by Lemmas \ref{L31} and \ref{L32}. In the preceding argument, we can see that $|G|\le48(K^2_S+8)$ holds if $r\le2$. If $|G|>48(K^2_S+8)$, we must have $r>3$. Then $H$ is one of $T_{12}$, $O_{24}$ or $I_{60}$. If $f$ has more than one $H$-orbit of singular fibers, then \begin{eqnarray*} \frac{K^2_{S/C}}{|G|}&\ge&\frac1{5r}\left(\frac{s_2(F_0)}{|K|}+\frac{7s_3(F_0)} {|K|}\right)+\frac1{5r_1}\left(\frac{s_2(F_1)}{|K|}+\frac{7s_3(F_1)}{|K|} \right)\\ &\ge&\frac1{25}\times\frac14+\frac1{20}\times\frac14=\frac9{400}>\frac1{48}. \end{eqnarray*} Therefore $f$ has only one $H$-orbit. If the singular fibers has non-negligible singularities, then by Lemma \ref{L31}, $|G|\le (60/7)rK^2_{S/C}\le(300/7)K^2_{S/C} <48K^2_{S/C}$. Suppose that the horizontal part $R'$ of the branch locus has only negligible singularities or ramifications, then by Lemmas \ref{L34}, \ref{L35}, \ref{L36} and \ref{L37}, we have $$|G|\le12.5rK^2_{S/C}.$$ Thus $|G|>48K^2_{S/C}$ implies that $r\ge4$ and $\bar K$ is $Z_6$ or $Z_5$. If $\bar K\cong Z_6$, then $r=5$ and $H\cong I_{60}$. To ensure $|G|>48K_{S/C}^2$, we have $s_2(F_0)=5$, i.e. $R=R'\sim -3K_{P/C}+nF$ is a smooth irreducible divisor. As a multiple cover on $C$, the ramification index of $R$ is equal to $R(R+K_{P/C})=12n$. On the other hand, this ramification index is equal to $5\times(60/5)=60$, i.e. $n=5$. But $2n=10=K^2_{S/C}\ne s_2(f)/5=12$, a contradiction. If $\bar K\cong Z_5$, then $|G|>48K^2_{S/C}$ implies $s_2(F_0)=4$. In this case $R=R'= C_0+R_1$ where $R_1\sim 5C_0+(n+3e)F$ is an smooth irreducible divisor and $R_1C_0=0$, i.e. $n=2e$. Computing the ramification index of $R_1$ we get $R_1(R_1+K_{P/C})=10n=4|H|/r$. This implies $5r||H|$, a contradiction. Hence $|G|>48(K^2_{S}+8)$ implies that $R'$ is \'etale over $C$. There are only finite number of possibilities. We list the possible fibrations with $|G|>48(K^2_S+8)$ as follows. $$\begin{array}{cccccc} H&r&|G|&K^2_S&|K|/(K^2_S+8)&|K|/K^2_S\\ I_{60}&5&2880&16&120&180\\ I_{60}&3&2880&32&72&90\\ O_{24}&4&1152&4&96&288\\ O_{24}&3&1152&8&72&144 \end{array}$$ In the Section \ref{FS} we will show their existence.\hspace{\fill}$\Box$ \begin{Corollary}\label{Cor} If $S$ is a minimal surface of general type which has a genus 2 fibration $f:S\longrightarrow C$ with $g(C)=0$, then $$|\mbox{\rm Aut}(f)|\le288K^2_S.$$ \end{Corollary} \noindent{\bf Proof.} If $K^2_S\ge2$, then $$48(K^2_S+8)<288K^2_S.$$ By Proposition \ref{P4} we need only check the 4 exceptional examples. That fact leads to the inequality $|G|\le288K^2_S$.\hspace{\fill}$\Box$ \section{Abelian automorphism groups} Let $G\subseteq\mbox{Aut}(f)$ be an abelian group. Then it is well known that $|K|\le12$. \begin{Proposition}[{[7, Lemma 8]}]\label{AP1} Let $f:S\longrightarrow C$ be a relatively minimal fibration of genus 2 with $g(C)\ge2$, $G$ is an abelian automorphism group of $S$, then $$|G|\le6K^2_S+96.$$ \end{Proposition} Let $\bar G\subseteq\mbox{Aut}(P)$ be the induced automorphism group of a commutative group $G$, then $$1\longrightarrow \bar K\longrightarrow \bar G\longrightarrow H \longrightarrow 1.$$ \begin{Lemma}\label{Z3} Assume that $\bar K\cong Z_3$, $g(C)=0$. Let $p\in C$ be a fixed point of the cyclic group $H$, $F=\pi^{-1}(p)$. If there is a $\bar K|_F$-orbit containing 3 points in $F$, then $$s_2(F)\ge3|H|.$$ \end{Lemma} \noindent{\bf Proof.} Since $p$ is a fixed point of $H$, the induced action of $\bar G$ on $F$ forms a commutative subgroup $\bar G|_F\subseteq \mbox{Aut}(F)\cong \mbox{Aut({\mib P}\,$^1$)}$. Since $\bar G|_F$ stabilize this $\bar K|_F$-orbit, $\bar G|_F=\bar K|_F\cong Z_3$, i.e. $H|_F=1$. Hence the local equation of $R'$ near $F$ has the form $f(x^3,t^h)$ where $h=|H|$. Or explicitely, the local equation of $R'$ are (3) $(x^3-b-t^{k_1h}a_1(t^h))(x^3-b-t^{k_2h}a_2(t^h))$; (5) $((x-b)^2 -t^{kh}a(t^h))((x-b\omega)^2-\omega^2t^{kh}a(t^h)) ((x-b\omega ^2)^2-\omega t^{kh}a(t^h))$, $b\ne0$. Thus $s_2(F)\ge3h=3|H|$. \hspace{\fill}$\Box$ \begin{Proposition}\label{AP2} If $S$ is a surface of general type which has a relatively minimal fibration of genus two $f:S\longrightarrow C$ with $g(C)\le1$, $G$ is an abelian automorphism group of $f$, then $$|G|\le12.5(K^2_S+8).$$ \end{Proposition} \noindent{\bf Proof.} It is well known that $H$ must be a cyclic group or a dihedral group $D_4\cong Z_2\oplus Z_2$. If $g(C)=1$ and that $H$ does not act freely on $C$, then $|H|\le6$. Hence $|G|\le72<12.5(K^2_S+8)$. If $g(C)=0$ and $H\cong D_4$, then $|G|\le48$, the claim holds too. So we can assume that $H$ is a cyclic group and that there exists a singular fiber $F_0$ with $|\mbox{Stab}_Hf(F_0)|=1$. {\it Case I.\/} Suppose that the horizontal part $R'$ of the branch locus $R$ is \'etale over $C$. Then $$|G|\le6K^2_{S/C}.$$ {\it Case II.\/} Suppose that $R'$ is not \'etale. If there is a big $H$-orbit with $s_3(F_0)\ne0$, then $$K^2_{S/C}\ge\frac75s_3(f)\ge\frac75|H|,$$ so $$|G|\le\frac{60}7K^2_{S/C}<12.5(K^2_S+8).$$ Now suppose that on the big $H$-orbits $R'$ has only negligible singularities or ramifications. If $\bar K\cong Z_6$, $Z_5$, $Z_4$ or 1, then by Lemma \ref{L8}, we have $$|G|\le\frac{|K|}{s_2(F_0)}\cdot 5K^2_{S/C}\le 12.5K^2_{S/C}\le 12.5(K^2_S+8).$$ Suppose that $\bar K\cong D_4$, $Z_3$ or $Z_2$ and that $|G|>12.5(K^2_S+8)$. Then Lemmas \ref{L35}, \ref{L36} and \ref{L37} implies that $f$ must have more than one $H$-orbit of singular fibers. To ensure $|G|>12.5(K^2_S+8)$, $f$ cannot have more than one big $H$-orbits. Thus we have $g(C)=0$. Lemma \ref{L35} excludes the case of $\bar K\cong D_4$. If $\bar K\cong Z_3$, then $s_2(F_0)\le2$. Hence $F_0$ must be of type (4) of Lemma \ref{L7} with $k=1$. Taking $K_1=\bar K$ we construct the quotient surface $Q=P/K_1$ as in \S3. Then $D=\alpha(R')$ is \'etale near $F_0$. But $D$ cannot be \'etale. Hence at least one of the $H$-stabilized fibers $F_2$ or $F_3$ is of type (2) $k=1$ or type (5) $k=1$. Lemma \ref{Z3} excludes the case of type (5). Suppose one of the $F_i$ is of type (2). Then $D\sim 2C'_0+aF'$ is irreducible and smooth. As a smooth double cover of $C\cong\mbox{\mib P\/}^1$, the ramification index of $D$ is at least 2. So $F_2$ and $F_3$ are all of type (2). Then $DC'=D(D+K_{Q/C})=2(a-e')=2$, a contradiction. If $\bar K\cong Z_2$, then $s_2(F_0)=1$. Hence the local equation of $R'$ near $F_0$ is $(x^2-t)(x^2-a(t))(x^2-b(t))$, $a(0)b(0)\ne0$, $a(0)\ne b(0)$. So $D=\alpha(R')$ is \'etale near $F_0$. If $F_2$ and $F_3$ have no ramifications, the $D$ can be decomposed into 3 components $D_i\sim C'_0+a_iF'$, $i=1$, 2, 3. These 3 components must meet each other on $F_2$ and $F_3$. So there exists at least one point on $F_i$ where 3 components intersect. The local eqution of $R'$ will be $(x^4+a(t)x^2+t^2)(x^2-t^2b(t))$. But as $D_i(C'_\infty -C'_0)=e'$, we will have $|H|\le1$. If $F_2$ or $F_3$ has ramifications, the equation of $R'$ near $F_i$ must be (1) $x^6-t$; (2) $(x^4-t)(t^ka(t)x^2-1)$, $a(0)\ne0$; (3) $((x^2-a^2)^2-t)(x^2-t^k)$, $a\ne0$; (4) $((x^2-a^2)^2-t)(x^2-b(t))$, $b(t)\ne0$. If $F_2$ is of type (1), then $D$ is irreducible and smooth. As a smooth triple cover of $C\cong \mbox{\mib P\/}^1$, the ramification index of $D$ is at least 4. Hence $F_3$ is of type (1) as well. Let $D\sim 3C'_0+aF'$. Then $2DC'=D(D+K_{Q/C})=4$, impossible. If $F_2$ is of type (2), then $D$ is smooth and cannot be irreducible. $D$ has 2 components $D_1\sim2C'_0+aF'$ and $D_2\sim 2C'_0+bF'$. By the same argument, we have $D_1C'=D_1(D_1+K_{Q/C})+2$. Hence $D_1C'_0=0$ and $D_1D_2=0$. It is impossible. \hspace*{\fill}$\Box$ \vspace{5mm} Suppose that $G$ is a cyclic automorphism group of $f$. Similarly, there is an exact sequence $$1\longrightarrow K\stackrel{\alpha}{\longrightarrow} G\stackrel{\beta}{\longrightarrow} H\longrightarrow 1$$ where $H\subseteq\mbox{Aut}(C)$, $K=\{(\tilde\sigma, \mbox{id})\in G\}$. It is known that $|K|\le10$. \begin{Lemma}\label{AG3-1} Suppose that $f:S\longrightarrow C$ is a fibration and that $G$ is a cyclic automorphism group of $f$. If there exists a point $p\in C$ such that (1) $\sigma|_{f^{-1}(p)}\in K|_{f^{-1}(p)}$, for $\sigma\in G$ and $\sigma$ stabilize $f^{-1}(p)$; (2) $K\longrightarrow\mbox{\rm Aut}(f^{-1}(p))$ is injective. Then $|K|$ and $|\mbox{\rm Stab}_H(p)|$ are coprime. \end{Lemma} \noindent{\bf Proof.} Let $H_1=\mbox{Stab}_Hp$, $F=f^{-1}(p)$. Let $h=|H_1|$, $k=|K|$, $d=(h,k)$. Assume that $\sigma$ is a generator of $\beta^{-1}(H_1)$. Then $\beta((\sigma^{k/d})^h)=1$ implies $\sigma^{hk/d}\in K$. On the other hand, since $\sigma|_F\in K|_F$ by (1), we obtain $(\sigma^{h/d})^k|_F=\mbox{id}_F$. Thus $\sigma^{kh/d}=1$ by (2). This is impossible.\hspace{\fill}$\Box$ \begin{Proposition}\label{AP3} If $S$ is a surface of general type which has a relatively minimal fibration of genus two $f:S\longrightarrow C$ with $g(C)=1$, $G$ is a cyclic automorphism group of $f$, then $$|G|\le5K^2_S$$ for $K^2_S\ge 12$. \end{Proposition} \noindent{\bf Proof.} If $H$ does not act freely on $C$, then $|H|\le 6$. Hence $|G|\le60$ and the conclusion holds. Therefore we will assume $H$ acts freely afterwards. So $G\cong K\times H$ and $G$ is cyclic if and only if $(|K|,|H|)=1$. We will discuss case by case. {\it Case I.\/} Suppose that the horizontal part $R'$ of the branch locus $R$ is \'etale over $C$. There exists a singular fiber $F_0$ with $|\mbox{Stab}_Hf(F_0)|=1$. It is not difficult to show that in this case $$|G|\le5K_S^2.$$ {\it Case II.\/} Suppose that $R'$ is not \'etale. ({\it a\/}) $\bar K\cong Z_5$. Let $F_0$ be a singular fiber. The local equation of $R'$ near $F_0$ is (1) $x(x^5-t^k)$ or (2) $x(t^kx^5-1)$, $k=1$, 2. We construct the quotient surface $Q=P/\bar K$ as in Section 3. $R'$ must contain one of the section $C_0$ or $C_\infty$. We take this section away from $R'$, get a reduced divisor $R_1$ with $R_1F=5$. Let $D=\alpha(R_1)$, then $D\sim C'_0+aF'$. Since $DC'_0=0$, $a=e'=5e$. Thus $R_1\sim 5C_0+5eF$ and $R_1C_\infty=5e$. Since the intersection number of $R_1$ and $F$ on the fiber $F_0$ is equal to $k\le2$, the number of singular fibers must be a multiple of 5. But $|H|$ can not be divided by 5, hence the singular fibers are located in different $H$-orbits. This means $|G|\le5K^2_{S}$. ({\it b\/}) $\bar K\cong Z_4$. The local equation of $R'$ near a singular fiber $F_0$ is $x(x^4-t^k)$, $k=1$, 2. We use the same construction as in case ({\it a}). Then $R'$ must contain $C_0$ and $C_\infty$. Let $R_1=R'-C_0-C_\infty$, $D=\alpha(R_1)$. Then $D\sim C'_0+e'F'$. Similarly we deduce $R_1C_\infty=4e$. Since $|H|$ cannot be even, there are more than one singular $H$-orbits. So $|G|\le5K^2_{S}$. ({\it c\/}) $\bar K\cong Z_3$. If $f$ has only one $H$-orbit of singular fibers and that $|G|>5K^2_S$, then $s_2(F_0)=5$, namely, the local equation of $R'$ is $x^6+a(t)x^3+t$. Constructing the quotient surface $Q=P/\bar K$, $D=\alpha(R')\sim 2C'_0+aF'$ is a smooth irreducible curve and $r\ne|H|$. Since $DC'_0=0$, $DC'_\infty=|H|$, we get $a=e'=3e=|H|$, i.e. $(|H|,|K|)=3$, a contradiction. ({\it d\/}) $\bar K\cong Z_2$. Lemma \ref{L37} ensures $|K|\le5K^2_S$. ({\it e\/}) $\bar K=1$. If $s_2(F_0)\ge2$, then $|G|\le5K^2_{S/C}$. If $s_2(F_0)=1$, there is only one situation, i.e. the local equation of $R'$ near $F_0$ is $(x^2-t)(x-a_1(t))(x-a_2(t)) (x-a_3(t))(x-a_4(t))(x-a_5(t))$, $a_i(0)\ne0$. Suppose that there is only one singular $H$-orbit. Then $R'$ is a smooth 6-tuple cover of $C$. The contribution of each singular fiber to the ramification index equals 1. By Hurwitz formula, $$2g(R')-2=6(2g(C)-2)+|H|.$$ So $|H|$ is even, a contradiction. \hspace{\fill}$\Box$ \begin{Proposition}\label{genus0} If $S$ is a surface of general type which has a relatively minimal fibration of genus two $f:S\longrightarrow C$ with $g(C)=0$, $G$ is a cyclic automorphism group of $f$, then $$|G|\le12.5K^2_S+90.$$ \end{Proposition} \noindent{\bf Proof.} If $R'$ is \'etale, we have $|G|\le5K^2_{S/C}$. If there is a singular fiber in a big $H$-orbit with $s_3(F)>0$, then $|G|\le(50/7)K^2_{S/C}$. Now assume that $R'$ has only negligible singularities or ramifications in big $H$-orbits. If $\bar K\cong Z_4$ or 1, we have $|G|\le10K^2_{S/C}$ by Lemma \ref{L8}. When $\bar K\cong Z_3$ or $Z_2$, if $f$ has only one $H$-orbit of singular fibers, then Lemmas \ref{L36} and \ref{L37} ensure $|G|\le6K^2_{S/C}$. Otherwise, by the proof of Proposition \ref{AP2}, $f$ has at least 2 big $H$-orbits of singular fibers, hence $|G|\le10K^2_{S/C}$. It remains the case of $\bar K\cong Z_5$. The proof of Proposition \ref{AP3} tells us that if $f$ has only one big $H$-orbit of singular fibers, then $f$ has another singular fiber which is stabilized by $H$. By Lemma \ref{L4}, we have $$K^2_{S/C}\ge \frac45(|H|+1),$$ so $$|G|+10|H|\le12.5K^2_{S/C}-10=12.5K^2_S+90.$$ \vspace{-8mm}\hspace*{\fill}$\Box$ \vspace{5mm} When $g(C)\ge 2$, we need the following lemma on the order of some automorphisms of a curve. The proof of the lemma is just a slight modification of that of the theorem of Wiman\cite{W}. For the convenience of the reader, we include its proof here which is a modified copy of the version given in [8, Lemma B]. \begin{Lemma}\label{AL1} Let $H$ be a cyclic group of automorphisms of a curve $C$ of genus $g\ge2$ such that the order of $|\mbox{Stab}_H(p)|$ is odd for any $p\in C$. Then $$|H|\le3g+3.$$ \end{Lemma} \noindent{\bf Proof.} Let $x$ be a non-zero element in $H$ with maximal number of fixed points, $H'$ the subgroup of $H$ generated by elements fixing all fixed points of $x$, $n$ the number of fixed elements of $x$, $k$ the order of $H'$. Then $k$ must be odd. Let $C'=C/H'$, $g'=g(C')$, and let $\Sigma$ be the image of the set of fixed points of $H'$ on $C'$. We have \begin{equation}\label{eq} 2g-2=2kg'-2k+n(k-1). \end{equation} and the quotient group $H''=H/H'$ is a cyclic group of automorphisms of $C'$ which satisfies the same condition imposed on $H$, i.e. $|\mbox{Stab}_{H''} (p)|$ is odd for any $p\in C'$. If $n=0$, then $g'\ge2$ and $|H|\le g-1$. If $n=2$, then because every non-zero element of $H''$ induces a non-trivial translation on $\Sigma$, we must have $|H''|\le2$, so $|H|\le2k$, then $|H|\le2g$ by (\ref{eq})(note that $g'\ne0$ in this case). So we may assume $n\ge3$. Suppose $g'=1$, $H''$ acts freely on $C'$. Considering the induced action $H''$ on $\Sigma$, we see that $|H''|\le n$. So (\ref{eq}) gives $|H|\le 2g+n-2$. On the other hand, since $k\ge3$, (\ref{eq}) also gives $n\le g-1$, therefore $$|H|\le3g-3$$ in this case. Suppose $g'=1$, $H''$ does not act freely on $C'$, then $H''$ has a fixed point. By assumption, $|H''|$ must be odd. This implies $|H''|\le3$. So (\ref{eq}) gives $$|H|\le2g+1.$$ Now suppose that $C'$ is a rational curve. Then the action of $H''$ has exactly 2 fixed points. So $|H''|$ must be odd. If one of these two points is in $\Sigma$, then $|H''|\le n-1$ in view of the action of $H''$ on $\Sigma$. Since $|H''|$ is odd, we have $n\ge4$. So $$|H|\le3g+3.$$ Suppose that $\Sigma$ and the two fixed points, $\xi$, $\eta$ of $H''$ are disjoint. Let $H_1\subset H$ be the stabilizer of a point in the inverse image of $\xi$. Then $[H:H_1]=k$. As the stabilizer of a point in the inverse image of $\eta$ is also of index $k$ in $H$, we see that any non-zero element in $H_1$ fixes exactly $2k$ points, i.e., the inverse image of $\xi$ and $\eta$. Now we can replace $H'$ by $H_1$ and repeat the arguments above (note that the only conditions we used are that non-trivial elements in $H'$ have same fixed point set and that $H/H'$ acts faithfully on $\Sigma$). But then $\Sigma$ is composed of two orbits of $H''$, so $|H''|\le n/2$, whereby $$|H|\le\frac32g+3$$ by (\ref{eq}). At last we use induction on $g$. Suppose that $g'\ge2$ and $|H''|\le3g'+3$. (\ref{eq}) gives $$3g+3-(n-4)\frac{3(g-g')}{2g'-2+n}\ge|H|.$$ If $n\ge4$, we have done. If $n=3$, by assumption, we must have $|H''|\le3$. Therefore $$|H|\le\frac{3(2g+1)}{2g'+1}\le\frac35(2g+1) \le3g+3.$$ \vspace{-8mm}\hspace*{\fill}$\Box$ \begin{Proposition}\label{AP4} If $f:S\longrightarrow C$ is a relatively minimal fibration of genus 2 with $g(C)\ge2$, $G$ is a cyclic automorphism group of $f$, then $$|G|\le5K^2_S+30$$ for $K^2_S\ge48$. \end{Proposition} \noindent{\bf Proof.} (1) Assume that $|H|=4g(C)+2$ and $|K|=10$. Let $g=g(C)$. By the theorem of Wiman (see the version given in [8, Lemma B]), $C$ is a cyclic cover of {\mib P\,}$^1$ with ramification index $r_1=2$, $r_2=2g+1$, $r_3=4g+2$ or $r_1=3$, $r_2=6$, $r_3=(4g+2)/3$. In fact, these $r_i$ are the orders of Stab$_H(p)$ for $p\in C$. Since $Z_{10}$ is a maximal cyclic automorphism subgroup of a smooth curve of genus 2, by Lemma \ref{AG3-1} we have $(|\mbox{Stab}_H(p)|,|K|)=1$ if $f^{-1}(p)$ is a smooth fiber. But in case 1, $r_1$ and $r_3$ are even, in case 2, $r_2$ and $r_3$ are even. So $f$ has at least $(2g+10)/3$ singular fibers. By Lemma \ref{L4}, we have $s_2(F)\ge4$ for a singular fiber $F$. Hence $$K^2_S-8(g-1)=K^2_{S/C}\ge\frac45 \cdot\frac{2g+10}3=\frac{8(g+5)}{15},$$ $$|G|=10|H|=40g+20\le \frac{75}{16}K^2_S+45\le5K^2_S+30$$ when $K^2_S\ge48$. If $|K|\le8$ and $|K|$ is even, then by Lemma \ref{AL1} there exist points $p\in C$ with $(|\mbox{Stab}_H(p)|,2)\ne1$. Hence $K^2_S-8(g-1)=K^2_{S/C}\ge1$ and $$|G|\le8|H|=32g+16\le 4K^2_S+44\le5K^2_S+30$$ when $K^2_S\ge14$. If $|K|$ is odd, then $|K|\le5$. The inequality is immediate. (2) Assume that $|H|$ is odd. By Lemma \ref{AL1}, we have $$|H|\le3g+3.$$ So $$|G|\le10|H|\le30g+30\le\frac{15}4K^2_S+60\le5K^2_S+30$$ when $K^2_S\ge24$. (3) Assume that $|H|$ is even and $|H|<4g+2$. If $|K|=10$, $f$ must have more than one singular fibers by Lemma \ref{L4}. So $K^2_S-8(g-1) =K^2_{S/C}\ge2$. We get $$|G|=10|H|\le40g\le5K^2_S+30.$$ If $|K|\le8$, it is not difficult to obtain this inequality.\hspace*{\fill}$\Box$ \vspace{5mm} It seems that this bound is not the best. In Section 5 we will give an example to show there are infinitely many fibrations which has an automorphism with order $3.75K^2_S+60$. \section{Examples}\label{FS} \begin{Example}\label{Ex1} Fibration with $|G|=50K^2_S$. \end{Example} Let $C$ be a Hurwitz curve, i.e. $|\mbox{Aut}(C)|=84(g(C)-1)$, $F$ be a curve of genus 2 with $|\mbox{Aut}(F)|=48$. Let $S=C\times F$, $f=p_1:S\longrightarrow C$. Then $K^2_S=8(g(C)-1)$, Aut$(f)\cong\mbox{Aut}(C)\times\mbox{Aut}(F)$, $$|\mbox{Aut}(f)|=|\mbox{Aut}(C)|\cdot|\mbox{Aut}(F)|=504K^2_S.$$ \begin{Example}\label{Ex2} Fibrations with $|G|=126K^2_S$ which is not locally trivial. \end{Example} Let $F=\mbox{\mib P\,}^1$. Let $p_1=0$, $p_2=\infty$, $p_3=1$, $p_4=\sqrt{-1}$, $p_5=-1$, $p_6=-\sqrt{-1}$ be 6 points on $F$. Let $C$ be a Hurwitz curve. Then $C$ has an $H$-orbit $\{ q_1,\dots,q_m\}$ which contains $m=12(g(C)-1)$ points. Let $P=C\times F$. Taking $R=p^*_1(q_1+\dots+ q_m)+p_2^*(p_1+\dots+p_6)$ as the branch locus, we construct a double cover of $P$. After desingularization, we get a smooth surface $S$ with a genus 2 fibration $f:S\longrightarrow C$. By computation, we obtain $K^2_S=32(g(C)-1)$, $|G|=48\times 84(g(C)-1)=126K^2_S$. \begin{Example}\label{Ex3} Fibrations with $|G|=144K^2_S$ and $g(C)=1$. \end{Example} Let $F$ and $p_1,\dots,p_6$ as Example \ref{Ex2}. Let $C$ be an elliptic curve with $j$-invariant $j(C)=0$. Fix a $q_1\in C$, then the order of the group of automorphisms Aut$(C,q_1)$ of $C$ leaving $q_1$ fixed is equal to 6. Let $H_1\cong Z_m\oplus Z_m$ be a subgroup of translations of Aut$(C)$. Take an extension subgroup $H_1\subset H\subset \mbox{Aut}(C)$ such that $H/H_1\cong \mbox{Aut}(C,q_1)$. Then $|H|=6m^2$. Let $q_1,\dots,q_{m^2}$ be the orbit of $q_1$ under $H$. Let $P=C\times F$. Using $R=p^*_1(q_1+\dots+q_{m^2})+p_2^*(p_1+ \dots+p_6)$ as the branch locus, we construct a double cover of $P$. After desingularization, we get a smooth surface $S$ with a genus 2 fibration $f:S\longrightarrow C$. By computation, we get $K^2_S=2m^2$. On the other hand, $|K|=48$ gives $|G|=288m^2=144K^2_S$. \begin{Example}\label{Ex4} Rational fibration with $|G|=120(K^2_S+8)$. \end{Example} Let $F$ and $p_1,\dots ,p_6$ as Example~\ref{Ex2}. Let $C=\mbox{\mib P\,}^1$, $q_1,\dots,q_{12}$ be the 12 vertices of an icosahedron. Let $P=C\times F$. Taking $R=p_1^*(q_1+\dots +q_{12})+p_2^*(p_1+\dots +p_6)$ as the branch locus, we can construct a double cover of $P$. After desingularization, we obtain a genus 2 fibration $f:S\longrightarrow C$ with $K^2_S=16$, $|H|=60$, $|K|=48$, $|G|=2880=120(K^2_S+8)$. \begin{Example}\label{Ex5} Rational fibrations with $|G|=48(K^2_S+8)$. \end{Example} Let $F$ and $p_1,\dots,p_6$ as Example~\ref{Ex2}. Let $C=\mbox{\mib P\,}^1$, $q_1,\dots,q_m$ be the $m$-th roots of unit. Then use the same construction as Example~\ref{Ex2}, we obtain a genus 2 fibration with $K^2_S=2(m-4)$, $|K|=48$, $|H|=2m$, $|G|=96m=48(K^2_S+8)$. \begin{Example}\label{Ex6} Exceptional rational fibrations listed in the proof of Proposition \ref{P4}. \end{Example} Using the same construction as Example~\ref{Ex2}, take $q_1,\dots,q_{20}$ as the 20 vertices of a dodecahedron. We get a fibration with $K^2_S=32$, $|G|=2880=90K^2_S$. If we take $q_1,\dots,q_{6}$ as the 6 vertices of an octahedron, we get a fibration with $K^2_S=4$, $|G|=1152=288K^2_S$. If we take $q_1,\dots,q_{8}$ as the 8 vertices of a cube, we get a fibration with $K^2_S=8$, $|G|=1152=144K^2_S$. \begin{Example}\label{Ex7} Fibrations the order of whose abelian automorphism group is $12.5(K^2_S+8)$. \end{Example} Let $x_0$, \dots, $x_{2m}$, $x_{2m+1}$ be the homogeneous coordinates in $\mbox{\mib P\,}^{2m+1}$, $\mbox{\mib P\,}^{2m}$ be the hyperplane defined by $x_{2m+1}=0$. Let $\phi:t \mapsto (1,t,\dots,t^{2m},0)$ be a $2m$-uple embedding of $\mbox{\mib P\,}^1$ in $\mbox{\mib P\,}^{2m}$ and denote its image by $Y$. Then $Y$ is a rational normal curve of degree $2m$. Let $X$ be the cone over $Y$ in $\mbox{\mib P\,}^{2m+1}$ with vertex $P_0(0,0,\dots,0,1)$. Denote $\eta=\exp(2\pi i/10m)$. Then the automorphism $\sigma:(x_0,\dots,x_{2m+1}) \mapsto(x_0, x_1\eta,\dots,x_{2m}\eta^{2m},x_{2m+1})$ of $\mbox{\mib P\,}^{2m+1}$ is of order $10m$. The automorphism $\tau:(x_0,\dots,x_{2m+1}) \mapsto(x_0, \dots,x_{2m},x_{2m+1}\eta^{2m})$ of $\mbox{\mib P\,}^{2m+1}$ is of order $5$. The cone $X$ is stabilized by these automorphisms $\sigma$ and $\tau$. Take a hypersurface $H$ defined by $x^5_0+x_{2m}^5+x_{2m+1}^5$ which is also stabilized by $\sigma$ and $\tau$. Moreover, $P_0\not\in H$. Now blowing up the cone $X$ at the vertex $P_0$ we get a Hirzebruch surface $P=F_{2m}$ which has an automorphism $\tilde\sigma$ of order $10m$ induced by $\sigma$ and an automorphism $\tilde\tau$ of order 5 induced by $\tau$. The pull-back of the intersection $H\cap X$ is a smooth divisor $R_1$ on $P$ which is linearly equivalent to $5C_0+10mF$. Taking $R=R_1+C_0\equiv 6C_0+10mF$ which is a smooth even divisor and stabilized under $\tilde\sigma$ and $\tilde\tau$, as the branch locus, we can construct a double cover $S$ of $P$ which has a natural genus 2 fibration $f:S\longrightarrow \mbox{\mib P\,}^1$. Since $K_P\equiv -2C_0-(2m+2)F$, $K_S^2=2(K_P+R/2)^2=8(m-1)$. The pull-back of $\tilde\sigma$ on $S$ can generate a cyclic automorphism subgroup $H$ of order $10m$. The pull-back of $\tilde\tau$ on $S$ together with the hyperelliptic involution of the fibration $f$ generates a cyclic automorphism subgroup $K\cong Z_{10}$. As $H$ and $K$ commute, $G=KH\cong Z_{10}\oplus Z_{10m}$ is an abelian automorphism group of $f$ with order $|G|=100m=12.5(K^2_S+8)$. \begin{Example}\label{Ex8+} Rational fibrations which has an automorphism with order $12.5K^2_S+90$. \end{Example} Let $x_0$, \dots, $x_{2m}$, $x_{2m+1}$ be the homogeneous coordinates in $\mbox{\mib P\,}^{2m+1}$, $\mbox{\mib P\,}^{2m}$ be the hyperplane defined by $x_{2m+1}=0$. Let $\phi:t \mapsto (1,t,\dots,t^{2m},0)$ be a $2m$-uple embedding of $\mbox{\mib P\,}^1$ in $\mbox{\mib P\,}^{2m}$ and denote its image by $Y$. Then $Y$ is a rational normal curve of degree $2m$. Let $X$ be the cone over $Y$ in $\mbox{\mib P\,}^{2m+1}$ with vertex $P_0(0,0,\dots,0,1)$. Denote $\eta=\exp(2\pi i/(50m-5))$. Then the automorphism $\sigma:(x_0,\dots,x_{2m+1}) \mapsto(x_0, x_1\eta^5,\dots,x_{2m}\eta^{10m},x_{2m+1}\eta)$ of $\mbox{\mib P\,}^{2m+1}$ is of order $50m-5$. The cone $X$ is stabilized by this automorphism $\sigma$. Take a hypersurface $H$ defined by $x_0^4x_1+x^5_{2m}+x_{2m+1}^5$ which is also stabilized by $\sigma$ and $P_0\not\in H$. Now blowing up the cone $X$ at the vertex $P_0$ we get a Hirzebruch surface $P=F_{2m}$ which has an automorphism $\tilde\sigma$ of order $50m-5$ induced by $\sigma$. The pull-back of the intersection $H\cap X$ is a smooth divisor $R_1$ on $P$ which is linearly equivalent to $5C_0+10mF$. Taking $R=R_1+C_0\equiv 6C_0+10mF$ which is a smooth even divisor and stabilized under $\tilde\sigma$, as the branch locus, we can construct a double cover $S$ of $P$ which has a natural genus 2 fibration $f:S\longrightarrow \mbox{\mib P\,}^1$. Since $K_P\equiv -2C_0-(2m+2)F$, $K_S^2=2(K_P+R/2)^2=8(m-1)$. The pull-back of $\tilde\sigma$ on $S$ can generate a cyclic automorphism group $G_1$ of order $50m-5$. Since $|G_1|$ is odd, $G_1$ and the hyperelliptic involution of the fibration $f$ generate a cyclic automorphism group $G$ of $S$. Therefore $|G|=100m-10=12.5K_S^2+90$. \begin{Example}\label{Ex8} Fibrations which has an automorphism with order $5K^2_S$. \end{Example} Let $F=\mbox{\mib P\,}^1$. Let $p_1=0$, $p_k=\exp(2k\pi i/5)$, $k=1,\dots,5$, be 6 points in $F$. Let $C$ be an elliptic curve, $q_1,\dots,q_m$ be an orbit of a cyclic translation group $H\subseteq \mbox{Aut}(C)$ of order $m$ where $m$ is an odd prime different from 5. Then using the same construction as Example \ref{Ex2}, we obtain a genus 2 fibration with $K^2_S=2m$, $K\cong Z_{10}$. Let $G=K\times H\cong Z_{10m}$. $|G|=10m=5K^2_S$. \begin{Example}\label{Ex9} Fibrations which has an automorphism with order $3.75K^2_S+60$. \end{Example} Let $p_0=0$, $p_k=\exp(2k\pi i/3)$, $k=1$, 2, 3 be 4 points in $C'=\mbox{\mib P\,}^1$. For any odd prime $m\ne 3$, 5, taking $D=p_0+p_1+(m-1)p_2+(m-1)p_3$ as a branch locus, we can construct a $m$-cyclic cover $\sigma:C\longrightarrow C'$. Then $g(C)=m-1$. $H''=\{x\mapsto x\exp(2k\pi i/3)|k=1,2,3\}\cong Z_3$ is a cyclic automorphism group of $C'$ which stabilizes the set $\{p_0,p_1,p_2,p_3\}$. On the other hand, the Galois group $H'$ of the $m$-cyclic cover $\sigma$ is isomorphic to $Z_m$. We can obtain an extension $$1\longrightarrow H'\longrightarrow H\longrightarrow H''\longrightarrow 1$$ such that $Z_{3m}\cong H\subseteq\mbox{Aut}(C)$. Let $q_0=0$, $q_k=\exp(2k\pi i/5)$, $k=1,\dots,5$, be 6 points in $F\cong \mbox{\mib P\,}^1$. Let $P=C\times F$. Take $R=p_2^*(q_0+q_1+\dots +q_5)$ as branch locus, we can construct a double cover $\theta:S\longrightarrow P$ which is also a genus 2 fibration $f=p_1\circ\theta:S\longrightarrow C$. $F$ has a cyclic automorphism group $K_1=\{y\mapsto y\exp(2k\pi i/5)|k=1,\dots,5\}\cong Z_5$ which stabilizes the set $\{q_0,\dots,q_5\}$ and can be lift to $P$. It is not difficult to see that we can get $K\cong Z_{10}$ by adding the involution of the double cover. Then $G=K\times H\cong Z_{30m}$ is a cyclic automorphism group of $f$ which satisfies $$|G|=30m=30(g(C)+1)=\frac{15}4K^2_S+60,$$ because $K^2_S=8(g(C)-1)$.

9301

alg-geom/9301004

en

https://arxiv.org/abs/alg-geom/9301004

alg-geom/9301004

Sorin Popescu

A. Aure, W. Decker, K. Hulek, S. Popescu, K. Ranestad

The Geometry of Bielliptic Surfaces in P^4

28 pages. AMSLaTeX 1.1

In 1988 Serrano \cite{Ser}, using Reider's method, discovered a minimal bielliptic surface in $\PP^4$. Actually he showed that there is a unique family of such surfaces and that they have degree 10 and sectional genus 6. In this paper we describe, among other things, the geometry of the embedding of the minimal bielliptic surfaces. A consequence of this description will be the existence of smooth nonminimal bielliptic surfaces of degree 15 in $\PP^4$. We also explain how to construct the degree 15 surfaces with the help of the quadro-cubic Cremona transformation of $\PP^4$. Finally, we remark that the quintic elliptic scroll and the abelian and bielliptic surfaces of degree 10 and 15 are essentially the only smooth irregular surfaces known in $\PP^4$ (all others can be derived via finite morphisms $\PP^4\to\PP^4 $).

alg-geom

\section{Heisenberg invariants on $\Bbb P^2$} Here we collect some well-known facts about invariants of the Schr\"odinger representation of $H_3$, the Heisenberg group of level $3$. Let $x_0,x_1,x_2$ be a basis of $\mathrm H^\circ(\cal O_{\Bbb P^2}(1))$ and consider the dual of the Schr\"odinger representation of $H_3$ on $V=\mathrm H^\circ(\cal O_{\Bbb P^2}(1))$ given by \begin{equation}\label{(1)} \begin{aligned} \sigma_3(x_i)&=x_{i-1}\\ \tau_3(x_i)&=\varepsilon_3^{-i}x_i\quad(\varepsilon_3=e^{2\pi i/3}) \end{aligned} \end{equation} where $i$ is counted modulo 3 and $\sigma_3$ and $\tau_3$ generate $H_3$. Note that \begin{equation}\label{(1a)} [\sigma_3,\tau_3]=\varepsilon_3^{-1}\cdot\operatorname{id}, \end{equation} hence $H_3$ is a central extension \[ 1 \to \mu_3 \to H_3 \to {\Bbb Z}_3\times{\Bbb Z}_3 \to 1. \] The induced representation on $\mathrm H^\circ(\cal O_{\Bbb P_2}(3))$ decomposes into characters since $\sigma_3$ and $\tau_3$ commute on the third symmetric power of $\mathrm H^\circ(\cal O_{\Bbb P^2}(1))$. By $(a,b)$ we denote the character where $\sigma_3$ (resp.\ $\tau_3$) acts by $\varepsilon_3^a$ (resp.\ $\varepsilon_3^b$)). Here again $a,b$ have to be taken modulo 3. There is a pencil of invariant polynomials, called the Hesse pencil, spanned by \[ x_0^3+x_1^3+x_2^3,\quad x_0x_1x_2 \] and eight invariant polynomials corresponding to the eight non-trivial characters: \begin{align*} F_{(1,0)}&\colon\quad x_0^3+\varepsilon_3x_1^3+\varepsilon_3^2x_2^3\\ F_{(2,0)}&\colon\quad x_0^3+\varepsilon_3^2x_1^3+\varepsilon_3x_2^3\\ F_{(0,1)}&\colon\quad x_0x_1^2+x_1x_2^2+x_2x_0^2\\ F_{(1,1)}&\colon\quad x_0x_1^2+\varepsilon_3x_1x_2^2+\varepsilon_3^2x_2x_0^2\\ F_{(2,1)}&\colon\quad x_0x_1^2+\varepsilon_3^2x_1x_2^2+\varepsilon_3x_2x_0^2\\ F_{(0,2)}&\colon\quad x_0^2x_1+x_1^2x_2+x_2^2x_0\\ F_{(1,2)}&\colon\quad x_0^2x_1+\varepsilon_3x_1^2x_2+\varepsilon_3^2x_2^2x_0\\ F_{(2,2)}&\colon\quad x_0x_1^2+\varepsilon_3^2x_1x_2^2+\varepsilon_3x_2x_0^2 \end{align*} where $F_{(a,b)}$ denotes the curve defined by the corresponding polynomial. On each smooth member of the Hesse pencil the group acts by translation of $3$-torsion points. There are precisely four singular members, namely: \begin{align*} T_{(0,1)}&\colon\quad x_0x_1x_2\\ T_{(1,1)}&\colon\quad(x_0+\varepsilon_3^2x_1+\varepsilon_3^2x_2) (x_0+x_1+\varepsilon_3x_2) (x_0+\varepsilon_3x_1+x_2)\\ T_{(1,0)}&\colon\quad(x_0+\varepsilon_3x_1+\varepsilon_3^2x_2) (x_0+\varepsilon_3^2x_1+\varepsilon_3x_2) (x_0+x_1+x_2)\\ T_{(1,2)}&\colon\quad(x_0+x_1+\varepsilon_3^2x_2) (x_0+\varepsilon_3x_1+\varepsilon_3x_2) (x_0+\varepsilon_3^2x_1+x_2) \end{align*} which equal $x_0^3+x_1^3+x_2^3+\lambda x_0x_1x_3$ for $\lambda=\infty,-3\varepsilon_3^2,-3,-3\varepsilon_3$. For each $(i,j)$ the subgroup of order 3 which is generated by $\sigma_3^i\tau_3^j$ fixes the vertices of the triangle $T{(i,j)}$. Consider the involution \[ \iota_3\colon(x_0,x_1,x_2)\mapsto(x_0,x_2,x_1). \] This involution leaves each member of the Hesse pencil invariant. In fact choosing $(0,1,-1)$ as the origin it acts as $x\mapsto-x$ on smooth members. The nontrivial characters come in pairs since $\iota_3(F_{(a,b)})=F_{(-a,-b)}$. \begin{Lemma}\label{L1} The curves $F_{(a,b)}$ are Fermat curves, and $H_3$ acts on each of them with translation by a 3-torsion point and multiplication by $\varepsilon_3$. \end{Lemma} \begin{Proof} Let $\eta^3=\varepsilon_3$, $\mu^3=\frac19$. Then \begin{align*} x_0^3+\varepsilon_3x_1^3+\varepsilon_3^2x_2^3 & = x_0^3+(\eta x_1)^3+(\eta^2x_2)^3\\ x_0^2x_1+x_1^2x_2+x_2^2x_0 & = (\eta\mu(x_0+\varepsilon_3^2x_1+\varepsilon_3x_2))^3 +(\eta^2\mu(x_0+\varepsilon_3x_1+\varepsilon_3^2x_2))^3\\ &\qquad+(\mu(x_0+x_1+x_2))^3\\ x_0^2x_1+\varepsilon_3x_1^2x_3+\varepsilon_3^2x_2^2x_0 & = (\eta\mu(x_0+\varepsilon_3^2x_1+\varepsilon_3^2x_2))^3 +(\eta^2\mu(x_0+\varepsilon_3x_1+x_2))^3\\ &\qquad+(\mu(x_0+x_1+\varepsilon_3x_2))^3\\ x_0^2x_1+\varepsilon_3^2x_1^2x_2+\varepsilon_3x_2^2x_0 & = (\eta\mu(x_0+\varepsilon_3^2x_1+x_2))^3 +(\eta^2\mu(x_0+\varepsilon_3x_1+\varepsilon_3x_2))^3\\ &\qquad+(\mu(x_0+x_1+\varepsilon_3^2x_2))^3. \end{align*} Since $\iota_3(F_{(a,b)})=F_{(-a,-b)}$ all curves are Fermat curves. The Fermat curve $F_{(a,b)}$ intersects each of the triangles $T_{(i,j)}$, $(i,j)\ne\pm(a,b)$ in its vertices so $H_3$ has three subgroups of order 3 with 3 fixed points on $F_{(a,b)}$. The fourth subgroup has no fixed points, hence $H_3$ acts as stated. \end{Proof} \section{Threefolds containing bielliptic surfaces}\label{Par1} In this part we will construct a $\Bbb P^2$-bundle over an elliptic curve $E$, and a $\Bbb P^1$-bundle over the symmetric product $S^2E$ of the elliptic curve containing bielliptic surfaces. Choose a smooth element of the Hesse pencil \[ E=E_\lambda=\{x_0^3+x_1^3+x_2^3+\lambda x_0x_1x_2=0\} \] where $\lambda\ne\infty,-3,-3\varepsilon_3,-3\varepsilon_3^2$. We choose the inflection point $p_0=(0,1,-1)$ to be the origin of $E$. Let $\xi_0,\xi_1,\xi_2$ be a dual basis of $x_0,x_1,x_2\in V=\Gamma(\cal O_{\Bbb P^2}(1))$. The induced action of $H_3$ is given by \begin{equation}\label{(2)} \begin{aligned} \sigma_3(\xi_i) &= \xi_{i-1}\\ \tau_3(\xi_i) &= \varepsilon_3^i\xi_i. \end{aligned} \end{equation} Note that in this case \eqref{(2)} implies \begin{equation}\label{(2a)} [\sigma_3,\tau_3]=\varepsilon_3\cdot\operatorname{id}. \end{equation} Next consider the line bundle $\cal O_E(15p_0)$. Let $y_0,\ldots,y_{14}$ be a basis of $\mathrm H^\circ(\cal O_E(15p_0))$ such that $H_{15}$, the Heisenberg group of level 15, acts in the standard way, i.e., by \begin{equation}\label{(3)} \begin{aligned} \sigma_{15}(y_i) &= y_{i-1}\\ \tau_{15}(y_i) &= \varepsilon_{15}^{-i}y_i \quad (\varepsilon_{15}=e^{2\pi i/15}). \end{aligned} \end{equation} {}From \eqref{(3)} it follows that \begin{equation}\label{(3a)} [\sigma_{15}^5,\tau_{15}^5]=\varepsilon_{15}^{-10}\cdot\operatorname{id} =\varepsilon_3\cdot\operatorname{id}. \end{equation} Hence identifying $\sigma_{15}^5$ with $\sigma_3$ and $\tau_{15}^5$ with $\tau_3$ we get an isomorphism of the subgroup of $H_{15}$ generated by $\sigma_{15}^5$ and $\tau_{15}^5$ with $H_3\subset\operatorname{SL}(V\spcheck)$, where the latter inclusion is given by the Schr\"odinger representation. Since $y_0,\ldots,y_{14}$ generate $\cal O_E(15p_0)$ this gives an action of $H_3$ on the line bundle $\cal O_E(15p_0)$ itself. Hence we can consider the natural action of $H_3\times H_3$ on the rank 3 bundle $W_E=\cal O_E(15p_0)\otimes V$. Let $\Delta$ be the diagonal of $H_3\times H_3$. Then $\Delta\cong H_3$ and \begin{align*} \sigma_3(y_i\otimes x_j) &= y_{i-5}\otimes x_{j-1}\\ \tau_3(y_i\otimes x_j) &= \varepsilon_3^{-i-j}y_i\otimes x_j. \end{align*} It follows from \eqref{(1a)} and \eqref{(3a)} that the centre of $H_3$ acts trivially on $W_E$. Hence the quotient \[ \cal E_E=W_E\spcheck/\Delta \] is a rank 3 vector bundle over \[ E/\Bbb Z_3\times\Bbb Z_3=E. \] \begin{Lemma}\label{L2} \rom{(i)}$\cal E_E$ is stable of degree -5. \par\noindent\rom{(ii)}$\det\cal E_E=\cal O_E(-5p_0)$ \end{Lemma} \begin{Proof} (i) Since $\deg W_E=45$, the degree of $\cal E_E$ is clearly -5. Now assume that $\cal F\subset\cal E_E\spcheck$ is a subbundle of rank $r$ ($r=1,2$) and degree $d$ contradicting semistability, i.e., $d/r>5/3$. Then ${\cal F}$ pulls back to a subbundle $\cal F'\subset W_E$ of degree $9d>15r$. This implies that $\cal F'\otimes\cal O_E(-15p_0)$ and hence $W_E\otimes\cal O_E(-15p_0)$ has a nonconstant section, a contradiction. \par\noindent(ii) $y_0y_5y_{10}$ is a section of $\det W_E=\cal O_E(45p_0)$ which is invariant under the induced action of $\Delta$ on $\det W_E$. It defines an invariant divisor on $E$ whose image in the quotient is a divisor linearly equivalent to $5p_0$. \end{Proof} Let us now look at the corresponding action of $\Bbb Z_3\times\Bbb Z_3$ on the trivial projective bundle $E\times\Bbb P^2=E\times\Bbb P(V\spcheck )$. Its quotient is a $\Bbb P^2$-bundle \[ \Bbb P^2_E=\Bbb P(\cal E_E) \] where we use the geometric projective bundle. By the above lemma $\Bbb P^2_E$ is the unique indecomposable $\Bbb P^1$-bundle over $E$ with invariant $e=-1$ \cite[V. theorem 2.15]{Ha}. We consider the quotient map \[ \pi\colon E\times\Bbb P^2\to\Bbb P_E^2. \] Clearly this map is unramified and we can use $\pi$ to compute the cohomology of line bundles on $\Bbb P_E^2$. This was done in \cite{CC} for the dual bundle $\cal E_E\spcheck$. We are particularly interested in line bundles numerically equivalent to the anticanonical bundle. The Picard group $\Bbb P_E^2$ is generated by the tautological bundle $\cal O_{\Bbb P_E^2}(1)$ and the pullback of the Picard group on $E$. The pullback of any line bundle on $\Bbb P_E^2$ to $E\times\Bbb P^2$ is the tensor product of a line bundle on $E$ and a line bundle on $\Bbb P^2$. \begin{Lemma}[Catanese, Ciliberto]\label{L3} If $\cal O_{\Bbb P_E^2}(L)$ is numerically equivalent to the anticanonical bundle $\cal O_{\Bbb P_E^2}(-K)$ and $h^\circ(\cal O_{\Bbb P^2_E}(L))>0$ then either $L\equiv-K$ in which case $h^\circ(\cal O_{\Bbb P_E^2}(L))=2$ or $3L\equiv-3K$ and $L\not\equiv-K$ in which case $h^\circ(\cal O_{\Bbb P_E^2}(L))=1$. Moreover there are 8 nonisomorphic bundles of the latter kind corresponding to the nontrivial characters of $\Bbb Z_3\times\Bbb Z_3$. \end{Lemma} \begin{Proof} Let $L=-K+\rho$ where $\rho$ is the pullback of a degree 0 line bundle on $E$. Since $\pi$ is unramified $\pi^*(-K+\rho)=-K_{E\times\Bbb P_2}+\rho'$ where $\rho'$ also has degree 0. This bundle can only have sections when $\rho'=0$. Since \[ \cal O_{\Bbb P_E^2}(-K+\rho)\subset\pi_*\pi^*\cal O_{\Bbb P_E^2}(-K+\rho) =\pi_*(-K_{E\times\Bbb P_2}) \] we are left to consider the decomposition \[ \pi_*(-K_{E\times\Bbb P_2}) =\bigoplus_{{\cal X}\in(\Bbb Z_3\times\Bbb Z_3)\spcheck}(-K+L_{\cal X}) \] where $L_{\cal X}$ is the torsion bundle associated to the character ${\cal X}$. I.e., $L_{\cal X}$ is a torsion bundle of degree 0 on $E$. Hence $L$ is of the form stated. The sections of $L$ are given by the sections of $\cal O_{E\times\Bbb P^2}(-K_{E\times\Bbb P_2})$ associated to the character ${\cal X}$. By what we have said in the previous paragraph the dimension of these sections is 2 if ${\cal X}$ is trivial and 1 otherwise. \end{Proof} \begin{Lemma}\label{L4} In the pencil $|-K|$ the singular members are four singular scrolls while the smooth members are abelian surfaces among which one is isomorphic to $E\times E$. The divisors $-K+L_{\cal X}$, ${\cal X}$ nontrivial, are smooth bielliptic surfaces. \end{Lemma} \begin{Proof} The divisors $\pi^*(-K)$ and $\pi^*(-K+L_{\cal X})$ are $E\times E_{\lambda'}$ where $E_{\lambda'}$ is a member of the Hesse pencil and $E\times F_{(a,b)}$, respectively. On each of these surfaces the $\Bbb Z_3\times\Bbb Z_3$-action is the one described in the previous paragraph. When $E_{\lambda'}$ is smooth then the 9 base points of the Hesse pencil form a subgroup of the product, so the quotient is abelian. In particular when $\lambda'=\lambda$ we get $E\times E/\Bbb Z_3\times\Bbb Z_3$ where $\Bbb Z_3\times\Bbb Z_3$ acts diagonally. It is easy to see that this quotient is again isomorphic to $E\times E$ (we shall soon discuss this in more detail). When $E_{\lambda'}$ is a triangle then the surface upstairs is the union of three scrolls, whose quotient downstairs is irreducible since the group acts transitively on the edges of the triangles. Finally on $E\times F_{(a,b)}$ the group acts with translation on the first factor and with translation and multiplication on the second factor. Hence the quotient is bielliptic. \end{Proof} We want to describe the intersection of the abelian and bielliptic surfaces with the special abelian surface $A_0\cong E\times E$ described above. By $A_K$ we'll denote the general abelian surface in $|-K|$. Let $T_K$ be the singular scrolls in $|-K|$ and $B_{(a,b)}$ the bielliptic surfaces. Let us first consider the structure of the abelian and bielliptic surfaces. Each of them has an elliptic fibration over $E$ whose fibres are the plane cubic curves $E_{\lambda'}$, $F_{(a,b)}$ respectively. For the abelian surfaces the elliptic fibration over $E_{\lambda'}$ upstairs remains an elliptic fibration over $E_{\lambda'}/\Bbb Z_3\times\Bbb Z_3=E_{\lambda'}$ downstairs, the fibres being isomorphic to $E$. Upstairs the intersection of $E\times E$ and $E\times E_{\lambda'}$ is 9 translates of the curve $E$ over the 9 base points of the Hesse pencil. On the quotient these translates are mapped to the same curve isomorphic to $E$, which in turn is a member of the fibration over $E_{\lambda'}$ described above. For the bielliptic surfaces the elliptic fibration over $F_{(a,b)}$ upstairs is mapped to an elliptic fibration over $F_{(a,b)}/\Bbb Z_3\times\Bbb Z_3\cong\Bbb P^1$ downstairs, i.e., a pencil. The intersection upstairs with $E\times E$ is 9 translates by the group of $E$ over the 9 points of intersection $E_\lambda\cap F_{(a,b)}$ which are mapped to the same curve $A_0\cap B_{(a,b)}$ downstairs. This is a member of the pencil over $\Bbb P^1$. The intersection $A_K\cap B_{(a,b)}$ downstairs is linearly equivalent to and different from $A_0\cap B_{(a,b)}$, so $|-K|$ restricts to $B_{(a,b)}$ to give the pencil described above. In particular the three triple fibres of this pencil are the intersections $T_K\cap B_{(a,b)}$ for the scrolls $T_K$ coming from triangles $T_{(i,j)}$ with $(i,j)\ne\pm(a,b)$. Our next aim is to describe the intersection of $A_K$, resp.\ of $B_{(a,b)}$ with $A_0$ more arithmetically. We look at the map \[ \begin{pmatrix}\phantom{-}3&\phantom{-}0\\ -2& -1 \end{pmatrix}\colon \left\{\begin{aligned}E\times E &\to E\times E\\ (q_1,q_2) &\mapsto (3q_1,-2q_1-q_2). \end{aligned}\right. \] The kernel of this map is the group $E^{(3)}$ of $3$-torsion points of $E$ embedded diagonally into $E\times E$. Hence the above map induces an isomorphism \[ A_0=(E\times E)/\Bbb Z_3\times\Bbb Z_3\cong E\times E. \] Whenever we shall refer to $A_0$ as a product it will be via this isomorphism. Note that the curve $\{(q,-2q);\ q\in E\}$ goes 9$:$1 onto the first factor and that $\{(0,-q);\ q\in E\}$ is mapped isomorphically onto the second factor. Moreover the curve $\{(q,-5q);\ q \in E\}$ goes 9$:$1 onto the diagonal and $\{(q,q);\ q \in E\}$ is mapped 9$:$1 onto the antidiagonal of $E\times E$. Finally we consider the map given by $\left(\begin{smallmatrix}-2&-1\\ -5&2\end{smallmatrix}\right)$ upstairs. One checks immediately that this induces an endomorphism downstairs, and that this endomorphism is $\left(\begin{smallmatrix}0&3\\3&0\end{smallmatrix}\right)$, i.e., 3 times the standard involution interchanging the factors of $E\times E$. \begin{Lemma}\label{L5} Let $\Delta_E$ be the diagonal in $A_0=E\times E$. \noindent\rom{(i)} The curve $A_K\cap E\times E$ is \[ \{(q,r);\ 3r+2q=0\} \] and $A_K\cap \Delta_E$ consists of the 25 points \[ \{(p,p);\ 5p=0\}. \] \noindent\rom{(ii)} The curves $B_{(a,b)}\cap E\times E$ are \[ \{(q,r);\ 3r+2q=-\tau_{(a,b)}\} \] and $B_{(a,b)}\cap \Delta_E$ are the sets of points \[ \{(p,p);\ 5p=-\tau_{(a,b)}\}, \] where $0\ne\tau_{(a,b)}$, $3\tau_{(a,b)}=0$. \end{Lemma} \begin{Proof} (i) Upstairs $E\times E_\lambda\cap E\times E=\{(q,\tau_3);\ q\in E, \ 3\tau_3=0\}$. The image of this set downstairs is $\{(3q,-2q-\tau_3); \ q\in E\}$ which is the curve described. The second part follows immediately. \noindent (ii) Similarly $E\times F_{(a,b)}\cap E\times E=\{(q,\tau_9); \ q\in E,\ 3\tau_9=\tau_{(a,b)}\}$ where the $\tau_{(a,b)}$ are the 3-torsion points on $E$. Downstairs this is $\{(3q,-2q-\tau_9);\ q\in E\}$ which gives the claim. \end{Proof} At this point we want to return to the product $E\times\Bbb P^2$. Let $p,q$ be the projections onto $E$ and $\Bbb P^2$. Let $\cal O_E(15p_0)\boxtimes\cal O_{\Bbb P^2}(1)=p^*\cal O_E(15p_0)\otimes q^*\cal O_{\Bbb P^2}(1)$. The centre of the diagonal $\Delta\subset H_3\times H_3$ acts trivially on this line bundle which, therefore, descends to a line bundle $\cal L$ on $\Bbb P_E^2$. \begin{Proposition}\label{P6} \rom{(i)} $h^\circ(\cal L)=5$\par \noindent\rom{(ii)} The following sections are invariant under $\Delta$, hence define a basis of $\mathrm H^\circ(\cal L)$: \begin{align*} s_0 &= y_{ 0}\otimes x_0+y_{ 5}\otimes x_1+y_{10}\otimes x_2\\ s_1 &= y_{ 3}\otimes x_0+y_{ 8}\otimes x_1+y_{13}\otimes x_2\\ s_2 &= y_{ 6}\otimes x_0+y_{11}\otimes x_1+y_{ 1}\otimes x_2\\ s_3 &= y_{ 9}\otimes x_0+y_{14}\otimes x_1+y_{ 4}\otimes x_2\\ s_4 &= y_{12}\otimes x_0+y_{ 2}\otimes x_1+y_{ 7}\otimes x_2 \end{align*} \end{Proposition} \begin{Proof} (i) Clearly \[ \mathrm H^\circ(\cal O_E(15p_0)\boxtimes\cal O_{\Bbb P^2}(1)) = \mathrm H^\circ(\cal O_E(15p_0))\otimes \mathrm H^\circ(\cal O_{\Bbb P^2}(1)). \] As an $H_3$-module \[ \mathrm H^\circ(\cal O_E(15p_0)) = 5 V\spcheck. \] This can be seen by looking at the subspaces spanned by $(y_0,y_5,y_{10})$, $(y_3,y_8,y_{13})$, $(y_6,y_{11},y_1)$, $(y_9,y_{14},y_4)$, $(y_{12},y_2,y_7)$. Hence as an $H_3$-module \[ \mathrm H^\circ(\cal O_E(15p_0)\boxtimes\cal O_{\Bbb P^2}(1)) = 5V\spcheck\otimes V = 5(\bigoplus_{{\cal X}\in(\Bbb Z_3\times\Bbb Z_3)\spcheck}V_{\cal X}). \] I.e. we have 5 invariant sections, and thus $h^\circ(\cal L)=5$. \noindent(ii) It is straightforward to check that the $s_i$ are invariant under $H_3$. \end{Proof} Next we consider the subgroup of $H_{15}$ spanned by $\sigma_{15}^3, \tau_{15}^3$. From \eqref{(3)} \[ [\sigma_{15}^3,\tau_{15}^3] = \varepsilon_{15}^{-9}\cdot\operatorname{id} = \varepsilon_5^{-2}\cdot\operatorname{id} \quad (\varepsilon_5=e^{2\pi i/5}). \] Hence mapping $\sigma_{15}^3$ to $\sigma_5$ and $\tau_{15}^3$ to $\tau_5$ we can identify this subgroup with $H_5\subset\operatorname{SL}(\Bbb C^5)$, where this inclusion is given by the representation which arises from the Schr\"odinger representation of the Heisenberg group $H_5$ of level 5 by replacing $\varepsilon$ by $\varepsilon^2$. Now let $H_5$ act on $E\times\Bbb P^2$ where the action on the second factor is trivial. Then $H_5$ acts on $\cal O_E(15p_0)\boxtimes\cal O_{\Bbb P^2}(1)$ and it is straightforward to check that this action commutes with $H_3$. Hence we get an action of $H_5$ on $\cal L$. \begin{Proposition}\label{P7} The action of $H_5$ on $\mathrm H^\circ(\cal L)$ is given by \[ \sigma_5(s_i)=s_{i-1},\quad\tau_5(s_i)=\varepsilon_5^{-2i}s_i. \] \end{Proposition} \begin{Proof} Straightforward calculation. \end{Proof} We have involutions on $E$ (given by $x\mapsto-x$) and on $\Bbb P^2$ (given by $\iota_3(x_i)=x_{-i}$). Hence we have an involution $\iota$ on $E\times\Bbb P^2$. This lifts to an involution on $\cal O_E(15p_0)\boxtimes\cal O_{\Bbb P^2}(1)$ where it acts on sections by \begin{equation}\label{(4)} \iota(y_i\otimes x_i) = y_{-i}\otimes x_{-i}. \end{equation} This involution does not commute with $H_3$, but we have an action of a semi-direct product $H_3\rtimes\langle\iota\rangle$. In the quotient this defines an involution on $\Bbb P_E^2$ and on $\cal L$. Note that on $A_0=E\times E$ this is given by $\left(\begin{smallmatrix}-1&\phantom{-}0\\ \phantom{-}0&-1\end{smallmatrix}\right)$. \begin{Proposition}\label{P8} $\iota$ acts on $\mathrm H^\circ(\cal L)$ by \[ \iota(s_i)=s_{-i}. \] \end{Proposition} \begin{Proof} Immediately from \eqref{(4)}. \end{Proof} Finally we remark that we really have an action of $(\Bbb Z_3\times\Bbb Z_3)^2$ on $E\times\Bbb P^2$ and that $\Bbb P_E^2$ was constructed by taking the quotient with respect to the diagonal. Hence we have still got an action of $\Bbb Z_3\times\Bbb Z_3$ on $\Bbb P_E^2$ which on every fibre of $\Bbb P_E^2$ lifts to the Schr\"odinger representation of $H_3$. Let $\Delta_E$ be the diagonal in $A_0=E\times E$. We can consider the blow-up \[ \rho\colon U\to\Bbb P_E^2 \] along $\Delta_E$. Since $\Delta_E$ is a section of $\Bbb P_E^2$ the variety $U$ has the structure of an $\Sigma^1$-bundle over $E$. Here $\Sigma^1$ denotes the $\Bbb P^1$-bundle over $\Bbb P^1$ with $e=-1$. \begin{Lemma}\label{L9} $U$ has the structure of a $\Bbb P^1$-bundle over $S^2E$. \end{Lemma} \begin{Proof} Let $E_\Delta$ be the exceptional surface over $\Delta_E$ and $B=\cal O_{\Bbb P_E^2}(1)$. By $F$ we denote the class of a fibre of $\Bbb P_E^2$. For $\beta$ sufficiently large $|B-E_\Delta+\beta F|$ is base point free. This linear system maps each $\Sigma^1$ to a $\Bbb P^1$-bundle over a scroll over $E$, and it remains to determine this scroll. To do this we look at $A_0=E\times E$. The map given by $|B-E_\Delta+\beta F|$ restricted to a curve $\{q\}\times E$ is nothing but projection of the plane cubic $E\subset\Bbb P^2$ from the point $q$. Hence we get an involution on $E\times E$ whose branch locus is the curve $\Delta'=\{(q,t)\in E\times E;\ 2t+q=0\}$. $\Delta'$ is the image of $E\to E\times E$, $q\mapsto\left(\begin{smallmatrix}2q\\-q \end{smallmatrix}\right)$. The isomorphism of $E\times E$ given by the matrix $\left(\begin{smallmatrix}-1&-1\\ \phantom{-}0&\phantom{-}1\end{smallmatrix}\right)$ maps $\Delta'$ to the diagonal and the curves $\{q\}\times E$ to the translates of the antidiagonal $\{(q,-q);\ q\in E\}$. Under this isomorphism the above involution becomes the standard involution given by interchanging the factors. This proves that the scroll in question is indeed $S^2E$. We have indeed a (locally trivial) $\Bbb P^1$-bundle by \cite[V.4.1]{BPV} and the remark after this. \end{Proof} In view of the above lemma we shall change our notation and write \[ \Bbb P_{S^2E}^1=U. \] The strict transform of $A_0=E\times E$ under the map $\rho\colon\Bbb P_{S^2E}^1\to\Bbb P_E^2$ is again $E\times E$. The other abelian surfaces $A_K$ in $\Bbb P_E^2$ are blown up in the points \[ \Delta_E\cap(A_K\cap A_0) = \{q;\ 5q=0\}, \] i.e., in 25 distinct points (cf. lemma \ref{L5}). Similarly the surfaces $B_{(a,b)}$ are blown up in 25 distinct points by $\rho$. \section{Two quintic hypersurfaces in $\Bbb P^4$}\label{Par2} Let $E$ be an elliptic normal curve of degree 5 in $\Bbb P^4$, embedded by the linear system $|5p_0|$, where $p_0 \in E$ is the origin which we have chosen before. We assume that $E$ is invariant under the action of $H_5$ given by its Schr\"odinger representation. In this paragraph we will describe geometrically the following diagram \unitlength1.2pt \[ \begin{picture}(70,50) \put(0,40){$\Bbb P^1_{S^2E}$} \put(32,43){$^\rho$} \put(60,40){$\Bbb P_E^2$} \put(12,43){\vector(1,0){46}} \put(20,31){$^{f_1}$} \put(12,36){\vector(2,-1){20}} \put(33,20){$\cal F$} \put(-1,21){$^{\pi_1}$} \put(67,21){$^{\pi_2}$} \put(6,36){\vector(0,-1){28}} \put(66,36){\vector(0, -1){28}} \put(18,13.5){$^{p_1}$} \put(50,13){$^{p_2}$} \put(31,18){\vector(-2,-1){20}} \put(42,18){\vector(2,-1){20}} \put(2,0){$V_1$} \put(62,0){$V_2$.} \end{picture} \] Here $V_1$ and $V_2$ are quintic hypersurfaces in $\Bbb P^4$: $V_1$ is the secant variety to $E$ and $V_2$ is ruled in an elliptic family of planes. $V_2$ is singular (set theoretically) along an elliptic quintic scroll whose trisecant variety it is. $\Bbb P^1_{S^2E}$, $\Bbb P_E^2$ and $\rho$ are as in \ref{Par1}. Via $\pi_1$ and $\pi_2$ they are minimal desingularizations of $V_1$ and $V_2$. The varieties above are tied together via an incidence variety $\cal F$ consisting of pairs $(p,Q)$ where $Q$ is a singular quadric containing $E$ and $p\in\operatorname{Sing}Q$. The morphisms to $V_1$ and $V_2$ are projections to the first and second factor respectively. Hence both hypersurfaces can be described in terms of the 3-dimensional family of singular quadrics through $E$. Furthermore one can make the identifications $${\Bbb P}^1_{S^2E} = \{(p,\{e_1,e_2\}) | e_1, e_2\in E, p\in L_{\langle e_1, e_2\rangle}\} =\{(p,W_p) | p\in V_1, W_p=\mathrm H^\circ (I_{E\cup \{p\}}(2))\}$$ where $L_{\langle e_1,e_2\rangle}$ is the secant line through $e_1,e_2$. The second projection for these incidence varieties is the map to $S^2E\subset{\Bbb P(\mathrm H^\circ(I_E(2))^{\spcheck})}$. Thus ${\Bbb P}^1_{S^2E}$ is the graph of the quadro cubic Cremona transformation restricted to the secant variety $V_1$ of $E$. One can also show that $V_2\subset {\Bbb P(\mathrm H^\circ(I_E(2)))}$ in this setting is the natural dual to $S^2E\subset{\Bbb P(\mathrm H^\circ(I_E(2))^{\spcheck})}$. Since this will not be essential for our argument we will omit the proofs. Some of the results we collect here are also contained in \cite{EL}, \cite{Hu}, \cite{d'Al}. Let $E$ be as above. It is well known that $h^\circ(I_E(2))=5$ and that a basis for the space of quadrics through $E$ is given by \[ Q_i = x_i^2+ax_{i+2}x_{i+3}-\frac1ax_{i+1}x_{i+4}\quad(i\in\Bbb Z_5). \] Here $a\in\Bbb C\cup\{\infty\}$ and five such quadrics define a smooth elliptic curve if and only if $a$ is not a vertex of the icosahedron, i.e., $a\ne0,\infty,\varepsilon_5^k(\varepsilon_5^2+\varepsilon_5^3), \varepsilon_5^k(\varepsilon_5+\varepsilon_5^4)$, $k=0,\ldots,4$ ($\varepsilon_5=e^{2\pi i/5}$). \begin{Definition}\label{D10} \rom{(i)} For $y\in\Bbb P^4$ let $M(y)$ be the symmetric 5$\times$5-matrix \[ M(y):=(y_{i+j}z_{i-j})\quad0\le i,j\le4 \] where $z\in\Bbb P^4$, $z_i=z_{-i}$ and $z_0=2$, $z_1=a$, $z_2=-\frac1a$. \noindent\rom{(ii)} Let $\cal F$ denote the incidence variety \[ \cal F:=\{(x,y)\in\Bbb P^4\times\Bbb P^4;\ M(y)\tr x=0\} \] and let $V_1$ and $V_2$ denote the images under the first and second projections $p_1$ and $p_2$ of $\cal F$ to the respective $\Bbb P^4$'s. \noindent\rom{(iii)} Let $M'(x)$ be the 5$\times$5-matrix defined by \[ M'(x)\tr y=M(y)\tr x. \] \end{Definition} \begin{Remark} This set-up was also considered in \cite{A} in the case of a general point $z\in\Bbb P^4$. Here we have chosen a special point, namely one that lies on the conic section invariant under the icosahedral group $A_5$ on the Bring plane $z_i=z_{-i}$; $i=0,\ldots,4$ (see \cite{BHM}). This conic can be identified naturally with the modular curve of level 5, $X(5)$, which is in 1$:$1-correspondence with $H_5$-invariantly embedded elliptic quintics in $\Bbb P^4$. Under this identification $z$ corresponds to the curve $E$ we have started with. The matrix $M$ was first considered by Moore. \end{Remark} \begin{Proposition}\label{P12} \rom{(i)} The set of quadrics $\{xM(y)\tr x;\ y\in\Bbb P^4\}$ is $\Bbb P(\mathrm H^\circ(I_E(2)))$. \noindent\rom{(ii)} $\cal F$ can be identified with the incidence variety of pairs $(p,Q)$ where $Q$ is a singular quadric through $E$ and $p\in\operatorname{Sing}Q$. \noindent\rom{(iii)} $V_1$ and $V_2$ are Heisenberg invariant quintic hypersurfaces in $\Bbb P^4$. \end{Proposition} \begin{Proof} (i) It is easily checked that \[ 2Q_{3i}(x)=xM(e_i)\tr x\quad(i=0,\ldots,4). \] \noindent(ii) This follows immediately since $\cal F$ is given by $M(y)\tr x=0$. \noindent(iii) Since $M(y)\tr x=0$ is equivalent to $M'(x)\tr y=0$ it follows that $V_1$ and $V_2$ are given by the quintic equations $\operatorname{det}M'(x)=0$ resp.\ $\operatorname{det}M(y)=0$. Because both $M'(x)$ and $M(y)$ are invariant under $H_5$, up to an even number of permutations of rows and columns, these equations are $H_5$-invariant. Since $H_5$ has no characters on $\mathrm H^\circ(\cal O_{\Bbb P_4}(n))$, $n<5$ it follows that $V_1$ and $V_2$ are in fact reduced of degree 5. \end{Proof} \begin{Remark}\label{R13} Since the general singular quadric through $E$ has rank 4, it follows that the projection $\cal F\to V_2$ is generically finite and hence $\cal F$ is also of dimension 3. \end{Remark} \begin{Corollary}\label{C14} \rom{(i)} $V_1$ is the locus of singular points of the singular quadrics through $E$. \noindent\rom{(ii)} $V_1=\operatorname{Sec}E$. \end{Corollary} \begin{Proof} (i) is obvious. \noindent\rom{(ii)} If $p\in\operatorname{Sec}E\backslash E$ then projection to $\Bbb P^3$ from $p$ maps $E$ to a nodal quintic curve in $\Bbb P^3$, which always lies on a quadric surface. Hence $p$ lies on a quadric cone through $E$. This implies $\operatorname{Sec}E\subset V_1$ and since both hypersurfaces have degree 5, the claim follows. \end{Proof} \begin{Corollary}\label{C15} $V_2$ is the discriminant locus of the family of quadrics through $E$. \end{Corollary} \begin{Proof} Clear. \end{Proof} The mapping $p_1$ (resp.\ $p_2$) is a ``small resolution'' of $V_1$ (resp.\ $V_2$), i.e., a (singular) point where $M'(x)$ (resp.\ $M(y)$) has rank 3 is replaced by a $\Bbb P^1$. One of our aims is to describe the rank 3 loci. For $M(y)$ the corresponding $\Bbb P^1$s yield the locus of the singular lines. We shall come back to this later. For $M'(x)$ it is simpler. \begin{Proposition}\label{P16} \rom{(i)} The quintic hypersurface $V_1 = \operatorname{Sec} E$ is singular precisely at $E$ where it has multiplicity 3. \noindent\rom{(ii)} The curve $E$ is exactly the locus where rank $M'(x) = 3$. \end{Proposition} \begin{Proof} We shall first prove that the multiplicity of $\operatorname{Sec} E$ along $E$ is three. This was already known to Segre \cite{Seg}, \cite{Sem}. Here we reproduce his proof. We consider a point $p \in E$ and choose a general line $l$ through $p$. We can assume that $l$ meets $\operatorname{Sec} E$ transversally at a finite number of smooth points outside $p$. Since secants and tangents of $E$ do not meet outside $E$ (see \cite[Lemma IV.11]{Hu}) every such point of intersection lies on a unique secant or tangent of $E$. On the other hand projection from a general line $l$ maps $E$ to a plane curve of degree 4 which, by the genus formula, must have 2 nodes. Hence $l$ intersects $\operatorname{Sec} E$ in precisely 2 points (counted properly) outside $p$, and it follows that the multiplicity of $\operatorname{Sec} E$ along $E$ is 3. Now assume that a singularity $x$ of $\operatorname{Sec} E$ exists outside $E$. Let $l$ be a line through $x$ which meets $E$ in a point $p$, but is neither a secant nor a tangent line of $E$. (Such a line exists since $x$ lies on at most one secant or tangent --- see above). Using \cite[Proposition IV.4.6]{Hu} we can also assume that $l$ is not a singular line of a rank 3 quadric through $E$. It follows that $l$ is not contained in $\operatorname{Sec} E$. The latter would only be possible if projection from $l$ defines a 2:1 map onto a conic, but this implies that $l$ is the vertex of a rank 3 quadric. Since the multiplicity of $\operatorname{Sec} E$ along $E$ is 3 and since $x$ was assumed to be singular, it follows that the intersection of $l$ and $\operatorname{Sec} E$ consists precisely of the two points $x$ and $p$. Projection from $l$ now gives a curve of degree 4 and genus 1 in ${\Bbb P}^2$ with exactly one singular point, given by the unique secant or tangent of $E$ through $x$. On the other hand, we can project from $x$ first. In this case $E$ is mapped to a quintic curve $E^{\prime}$ in ${{\Bbb P}^3}$ with one singularity and arithmetic genus 2, which lies on a unique quadric surface $Q'\subset{\Bbb P}^3$. We have two possibilities \noindent\rom{(1)} $Q'$ is a smooth quadric. In this case $E'$ is a divisor on $Q'$ of bidegree $(2,3)$. Then projection from a general point on $E'$ (which corresponds to a general choice of the point $p \in E$) projects $E'$ to a plane quartic with 2 different singularities, a contradiction to what we have found above. \noindent\rom{(2)} $Q'$ is a quadric cone. In this case $E'$ contains the vertex of this cone as a smooth point and meets every ruling of $Q'$ in 2 points outside the vertex. Again projection from a general point on $E'$ gives a quartic plane curve with two different singularities and we have arrived at the same contradiction as above. It follows that $\operatorname{Sec} E$ has no singularities outside $E$. \noindent\rom{(ii)} The locus where rank $M'(x)$ has rank $\le 3$ is contained in $\operatorname{Sing} V_1$. On the other hand if $p \in E$, then projection from $p$ gives a smooth quartic elliptic curve in ${\Bbb P}^3$ which lies on a pencil of quadrics. Hence $E$ lies precisely on a pencil of quadric cones with vertex $p$. It follows that \[ M(y) \tr{p} = M'(p) \tr{y} \] for $y$ in some (linear) ${\Bbb P}^1$, and $M'(p)$ has rank exactly 3. \end{Proof} \begin{Remark}\label{R17} One shows easily that \[ M'(x)=\fracwithdelims(){\partial Q_{3j}}{\partial x_i}_{0\le i,j\le4}. \] \end{Remark} \medskip We consider the natural desingularization \[ \widetilde V_1 := \{(p,\{e_1,e_2\})\in\Bbb P^4\times S^2E; \ p\in L_{\langle e_1,e_2\rangle}\} \] of $V_1=\operatorname{Sec}(E)$, where $L_{\langle e_1,e_2\rangle}$ is the secant line through $e_1,e_2$. Projection onto $S^2E$ gives $\widetilde V_1$ a structure of a $\Bbb P^1$-bundle over the surface $S^2E$. Let $\pi_1$ denote projection to the first factor. Then $\pi_1$ contracts the divisor \[ D_1:=\{(p,\{e_1,e_2\});\ p\in\{e_1,e_2\}\}. \] We have an isomorphism \[ \psi_1\colon \left\{\begin{aligned} D_1 &\cong E\times E\\ (p,\{e_1,e_2\}) &\mapsto (p,e_1+e_2). \end{aligned}\right. \] Next we consider the natural composition \[ \widetilde V_1\to S^2E\to E \] where the map $S^2E\to E$ maps $\{e_1,e_2\}$ to $e_1+e_2$. The fibre of this map over a point $e\in E$ is the surface \[ \{(p,\{e_1,e_2\});\ e_1+e_2=e,\ p\in L_{\langle e_1,e_2\rangle}\}. \] This is a ruled surface over the curve \[ E/\kappa\cong\Bbb P^1 \] where $\kappa$ is the involution on $E$ given by $\kappa(q)=-q+e$. Via $\pi_1$ this is a smooth, rational ruled surface in $\Bbb P^4$, i.e., a cubic scroll. As an abstract surface this is $\Bbb P^2$ blown up in a point, or equivalently the Hirzebruch surface $\Sigma^1$. In this way $\widetilde V_1$ acquires the structure of a $\Sigma^1$-fibration over $E$. We denote the fibre of this fibration over a point $e\in E$ by $\Sigma_e^1$. We shall often identify $\Sigma_e^1$ with $\pi_1(\Sigma_e^1)$. Thus we can write \[ \widetilde V_1=\{(p,e);\ p\in \Sigma_e^1\subset V_1\}. \] We will use this notation in the sequel. \begin{Proposition}\label{P18} The map $\pi_1$ defines an isomorphism from $\widetilde V_1\sm D_1$ with $\operatorname{Sec} E \sm E$. It contracts $D_1$ to the curve $E$ and its differential has rank 2 at every point of $D_1$. \end{Proposition} \begin{Proof} Since every point on $\operatorname{Sec} E \sm E$ lies on a unique secant or tangent of $E$ the map from $\widetilde V_1 \sm D_1$ to $\operatorname{Sec} E \sm E$ is bijective. Since both are smooth, it is an isomorphism. We have already seen that $\pi_1$ contracts $D_1$ to the curve $E$. Hence the differential of $\pi_1$ along $D_1$ has rank at most 2. On the other hand consider the fibres $\Sigma_e^1$ of the map $\widetilde V_1 \to E$. Via the map $\pi_1$ they are embedded into ${\Bbb P}^4$, and hence the differential of $\pi_1$ has rank at least 2 at every point of $\widetilde V_1$. \end{Proof} Now we return to the cubic scroll $\Sigma_e^1\subset\Bbb P^4$. Since $\Sigma_e^1$ is the degeneration locus of a 2$\times$3 matrix with linear coefficients, it follows that there is a $\Bbb P^2$ of quadrics containing $\Sigma_e^1$. All of these quadrics are singular. Geometrically they arise as follows: Projection from $p\in\Sigma_e^1$ maps $\Sigma_e^1$ to a quadric in $\Bbb P^3$. Then take the cone over the quadric in $\Bbb P^3$. Note that the quadric surface in $\Bbb P^3$ is singular if and only if $p$ is on the exceptional line in $\Sigma_e^1$. In this case the corresponding quadric hypersurface is singular along the exceptional line in $\Sigma_e^1$. Finally it follows easily from $V_1=\operatorname{Sec}E$ that every singular quadric through $E$ arises in the way described above. We define \[ \widetilde V_2 := \{(Q_e,e);\ e\in E, \ \text{$Q_e$ is a quadric through $\Sigma_e^1$}\}. \] Via the obvious map $\widetilde V_2\to E$ this carries the structure of a $\Bbb P^2$-bundle. For $p\in\Sigma_e^1$ we denote by $Q_e=Q_e(p)$ the unique quadric through $\Sigma_e^1$ which is singular at $p$. This enables us to define the following maps: \[ f_1\colon\left\{\begin{aligned}\widetilde V_1 &\to \cal F\\ (p,e) &\mapsto (p,Q_e) \end{aligned}\right. \] where $(p,e)$ stands for the point $p\in\Sigma_e^1$, and \[ f_2\colon\left\{\begin{aligned}\widetilde V_1 &\to \widetilde V_2\\ (p,e) &\mapsto (Q_e,e). \end{aligned}\right. \] In this way we get a commutative diagram \unitlength1.2pt \begin{equation}\label{(D)} \begin{picture}(70,50) \put(2,40){$\widetilde V_1$} \put(32,43){$^{f_2}$} \put(62,40){$\widetilde V_2$} \put(12,43){\vector(1,0){46}} \put(20,31){$^{f_1}$} \put(12,36){\vector(2,-1){20}} \put(33,20){$\cal F$} \put(-1,21){$^{\pi_1}$} \put(67,21){$^{\pi_2}$} \put(6,36){\vector(0,-1){28}} \put(66,36){\vector(0, -1){28}} \put(18,13.5){$^{p_1}$} \put(50,13){$^{p_2}$} \put(31,18){\vector(-2,-1){20}} \put(42,18){\vector(2,-1){20}} \put(2,0){$V_1$} \put(62,0){$V_2$.} \end{picture} \end{equation} Moreover it follows from our geometric discussion that $f_2$ contracts precisely the divisor \[ X:=\{(p,e);\ p\in\ \text{exceptional line in $\Sigma_e^1$}\}. \] In other words $f_2$ is the blowing down map from the $\Sigma^1$-bundle $\widetilde V_1$ to the $\Bbb P^2$-bundle $\widetilde V_2$. Furthermore $X$ is an elliptic ruled surface and $\pi_1(X)$ is the locus of singular lines. We now return to the divisor $D_1\cong E\times E$ in $\widetilde V_1$. \begin{Lemma}\label{L18} \rom{(i)} $f_1$ is an isomorphism outside $D_1$. \noindent\rom{(ii)} $f_1(p,e)=f_1(p',e')$ if and only if $p'=p\in E$ and $e+e'=-p$. \end{Lemma} \begin{Proof} \rom{(i)} This follows from proposition \ref{P16} \rom{(i)}. \noindent\rom{(ii)} If $f_1(p,e) = f_1(p',e')$ then clearly $p= p'$ by construction of the map $f_1$. Now $Q_e = Q_{e'}$ means that $Q_e$ is a singular quadric with vertex $p\in E$ containing both $\Sigma_e^1$ and $\Sigma_{e'}^1$. $\Sigma_e^1$ and $\Sigma_{e'}^1$ are determined by the families of planes in $Q_e$: A plane intersects $E$ in two points besides $p$, defining a line in the ruling of the scroll. If $L_{\langle e_1,e_2\rangle}\subset \Sigma_e^1$ and $L_{\langle e_1',e_2'\rangle}\subset\Sigma_{e'}^1$ then $e_1,e_2,e_1',e_2'$ and $p$ are contained in a $\Bbb P^3$, hence \[ e_1+e_2+e_1'+e_2'+p=0. \] So \[ e+e'=-p. \] The converse is analogous. \end{Proof} {}From this lemma it follows that $f_1$ restricts to $D_1\cong E\times E$ as the quotient map to $E\times E/\iota'$, where $\iota'$ is the involution $\iota'(p,e)=(p,-p-e)$. The curve $\Delta':=\{(-2e,e);\ e\in E\}$ is pointwise fixed under $\iota'$, while $\iota'$ acts as the standard involution on the curve $(\Delta')^-:=\{(0,e);\ e\in E\}$. Consider the change of coordinates (compare the proof of lemma \ref{L9}): \[ \psi_2\colon \left\{\begin{aligned}E\times E &\to E\times E\\ (p,e) &\mapsto (p+e,-e). \end{aligned}\right. \] This maps $\Delta'$ to the diagonal $\Delta=\{(e,e);\ e\in E\}$ and $(\Delta')^-$ to the antidiagonal $\Delta^-=\{(e,-e);\ e\in E\}$. Moreover $\iota'$ becomes the involution $\tilde\iota$ interchanging the two factors. From now on we shall identify $D_1$ with $E\times E$ via the isomorphism $\psi:=\psi_2\circ\psi_1$. Finally we denote by $\bar\Delta$ the image of the diagonal $\Delta$ in $S^2E=E\times E/\tilde\iota$. \begin{Proposition}\label{P19} The exceptional divisor $X\subset\Sigma_E^1$ intersects $D_1=E\times E$ in the diagonal $\Delta$. \end{Proposition} \begin{Proof} The involution $\iota$ (resp.\ $\iota'$) is induced by a switching of cubic scrolls in a singular quadric. A fixed scroll is precisely the unique scroll containing $E$ in a rank 3 quadric. \end{Proof} \begin{Corollary}\label{C20} $\cal F$ is singular along an elliptic scroll $S^2E$. The scroll $f_1(X)$ intersects $S^2E$ along $\bar\Delta$. \end{Corollary} \begin{Proposition}\label{P21} The singular scroll $\pi_1(X)$ has degree 15. The curve $\Delta$ is mapped 4$:$1 to $E$ by $\pi_1$. \end{Proposition} \begin{Proof} For the first part see \cite[prop.\ IV.4.7]{Hu}. The second statement follows since $\Delta'\to E$ is given by $(-2e,e)\mapsto-2e$. It also follows since the pencil of quadrics with vertex $p\in E$ contains 4 rank 3 quadrics (see the proof of proposition \ref{P16}). \end{Proof} We now turn our attention to the quintic hypersurface $V_2$. \begin{Proposition}\label{P22} Restricted to $D_1=E\times E$ the blowing down map $f_2$ is an isomorphism. \end{Proposition} \begin{Proof} Fix some $e\in E$. Then the exceptional line in $\Sigma_e^1$ and the curve $E$ intersect transversally. \end{Proof} \begin{Proposition}\label{P23} \rom{(i)} The quintic hypersurface $V_2$ is ruled by an elliptic family of planes. \noindent\rom{(ii)} The map $\pi_2$ restricted to $D_1 \subset \widetilde V_2$ maps $D_1 \cong E {\times} E$ surjectively 2:1 onto a quintic elliptic scroll $S^2 E$. The scroll $S^2 E$ parametrizes those quadrics which are singular at a point of $E$. The quintic hypersurface $V_2$ is the trisecant scroll of $S^2 E$. It is singular exactly at $S^2 E$ (set theoretically). \noindent\rom{(iii)} Via $\pi_2$ the diagonal $\Delta \subset D_1 \cong E {\times} E$ is mapped to a degree 10 curve $\bar\Delta$ in ${\Bbb P}^4$. The curve $\bar\Delta$ parametrizes the rank 3 quadrics through $E$. \noindent\rom{(iv)} The map $\pi_2$ gives an isomorphism of $\widetilde V_2 \sm D_1$ with $V_2 \sm S^2 E$. \noindent\rom{(v)} The rank of the differential of $\pi_2$ is 3 everywhere with the exception of $\Delta$ where it is 2. \end{Proposition} \begin{Proof} \rom{(i)} The fibre of $\widetilde V_2$ over a point $e \in E$ is mapped to the net of quadrics through the scroll $\Sigma_e^1$. \noindent\rom{(ii)} We have already seen that $f_1$ restricted to $D_1 \cong E {\times} E$ factors through $S^2 E$. Hence using diagram \eqref{(D)} the same must be true for $\pi_2$. The map from $S^2E$ to ${\Bbb P}^4$ given by $\pi_2$ is injective, which means that the image has degree at least 5. The ruling of $S^2E$ over a point $p\in E$ is mapped to the pencil of quadrics through $E$ which are singular at $p$. Now intersect $V_2$ with a general plane. Since $V_2$ is singular on the image of $S^2E$, this intersection is a plane curve with at least 5 singular points. Since the map from $\widetilde V_2 \sm D_1$ to $V_2 \sm S^2 E$ is bijective, this curve dominates the elliptic base curve of $\widetilde V_2$, and therefore, by the genus formula, it cannot have more then 5 singular points. Thus $\pi_2(S^2E)$ has degree 5 and, by the same argument, $V_2$ has no singularities outside $\pi_2(S^2E)$. If $C_0$ is a section of $S^2E$ with $C_0^2=1$ and $F$ is a fibre, then the map from $S^2E$ to ${\Bbb P}^4$ is given by the linear system $|C_0+2F|$. In fact, by $H_5-$invariance, the map is given by the complete linear system, in which case it is well known to be an embedding. We therefore identify $S^2E$ with its image. It remains to show that $V_2$ is the trisecant scroll of $S^2E$. Now, the curve $C_0$ moves in an elliptic family on $S^2E$, so each member is a plane cubic curve. Thus the planes of these curves are part of the trisecant scroll of $S^2E$. Since $S^2E$ is the singular part of $V_2$, each such trisecant is contained in $V_2$ by Bezout. But the planes of the trisecant scroll cannot dominate the elliptic base curve, hence these planes must coincide with the elliptic family of planes of $V_2$. \noindent\rom{(iii)} The curve $\Delta$ is the branch locus of the map $ E {\times} E \to S^2 E \subset {\Bbb P}^4 $. It is well known that this is mapped to a curve $\bar\Delta$ of degree 10 in ${\Bbb P}^4$ (in fact the class of $\bar\Delta$ on $S^2 E$ is $4 C_0 - 2 F$ and the assertion follows from $ (4 C_0 - 2 F) (C_0 + 2 F) = 10 $). The assertion that $\bar\Delta$ parametrizes the rank 3 quadrics through $E$ follows from the description of the map $p_2$ and proposition \ref{P19}. \noindent\rom{(iv)} We have already seen that the map from $\widetilde V_2 \sm D_1$ to $V_2 \sm S^2 E$ is bijective. Since both sets are smooth, the claim follows. \noindent\rom{(v)} By \rom{(iv)} the rank of $d\pi_2$ is 3 outside $D_1$. Since the fibres of $\widetilde V_2$ are mapped to planes in ${\Bbb P}^4$, it follows that the rank of the differential is at least 2 everywhere. Since $\Delta$ is the branch locus of the map $E {\times} E \to S^2 E$ the rank of $d \pi_2$ cannot be 3 along $\Delta$. It remains to prove that the rank of the differential is 3 on $D_1 \sm \Delta$. Let $x$ be a point on $D_1 \sm \Delta$ and let $E_x$ be the elliptic curve through $x$ which is mapped to a ruling of $S^2 E$. The differential of $\pi_2$ restricted to $E_x$ is 2 at $x$. Hence it is enough to see that the ruling $L_x = \pi_2(E_x)$ and the plane ${\Bbb P}^2_x$ which is the image of the fibre of $\widetilde V_2$ containing $x$ meet transversally. For this it is enough to show that $L_x$ is not contained in ${\Bbb P}^2_x$. But the intersection of ${\Bbb P}_x^2$ with $S^2 E$ is a smooth plane cubic and does not contain a line. \end{Proof} \begin{Remark}\label{R24} A general symmetric 5$\times$5 matrix with linear coefficients has rank 3 along a curve of degree 20. \end{Remark} \bigskip We are now in a position to connect the geometric approach of this paragraph with the abstract approach from \ref{Par1}. To do this, recall the sections $s_0,\ldots,s_4$ of $\cal L$ from proposition \ref{P6}. \begin{Proposition}\label{P25} There is an isomorphism $\widetilde V_2\cong\Bbb P_E^2$ such that the map $\pi_2$ is given by $s_0,\ldots,s_4$. \end{Proposition} \begin{Proof} The argument has two parts. First we identify $\widetilde V_2$ with ${\Bbb P}(N_E(-2))$, where $N_E(-2)$ is the twisted normal bundle of the elliptic curve $E\subset {{\Bbb P}^4}$. Afterwards we show that ${\Bbb P}(N_E(-2))\cong {\Bbb P_E^2}$ and in fact also the existence of an $H_5$-isomorphism between ${\cal O}_{{\Bbb P}(N_E(-2))}(1)$ and ${\cal O}_{{\Bbb P}({\cal E})}(1)$. We use the basis of $\mathrm H^\circ(I_E(2))$ given by \[ Q_i=x_i^2+ax_{i+2}x_{i+3}-\frac1ax_{i+1}x_{i+4}\quad(i\in\Bbb Z_5). \] The natural map \[ \mathrm H^\circ(I_E(2))\otimes\cal O_E\stackrel{\alpha}{\to}N_E^*(2) \] is surjective and there is an exact sequence \[ 0\to K\stackrel{\beta}{\to}\mathrm H^\circ(I_E(2))\otimes\cal O_E(-1) \stackrel{A}{\to}\mathrm H^\circ(I_E(2))\otimes\cal O_E \stackrel{\alpha}{\to}N_E^*(2)\to0 \] with \[ A=\begin{pmatrix} 0 & ax_4 & -x_3 & x_2 & -ax_1 \\ -ax_4 & 0 & ax_2 & -x_1 & x_0 \\ x_3 & -ax_2 & 0 & ax_0 & -x_4 \\ -x_2 & x_1 & -ax_0 & 0 & ax_3 \\ ax_1 & -x_0 & x_4 & -ax_3 & 0 \end{pmatrix} \] \medskip (see \cite[p.\ 68]{Hu}). Dualising this sequence we get \[ \begin{CD} \mathrm H^\circ(I_E(2))\spcheck\otimes\cal O_E @>(\tr A)=-A(1)>> \mathrm H^\circ(I_E(2))\spcheck\otimes\cal O_E(1) @>\tr\beta>> K^* @>>> 0. \end{CD} \] Hence \[ K^*\cong N_E^*(3), \] i.e., \[ K\cong N_E(-3). \] We want now to describe the map \[ \begin{CD} \Bbb P(N_E(-2))&\hookrightarrow&\Bbb P(\mathrm H^\circ(I_E(2)))\times E\\ &\searrow &\downarrow \\ & &{\Bbb P}(\mathrm H^\circ(I_E(2))) \end{CD} \] where the horizontal map is given by the inclusion \[ N_E(-2)\stackrel{\beta(1)}{\hookrightarrow}\mathrm H^\circ(I_E(2)) \otimes\cal O_E. \] We first want to identify the subbundle \[ \Bbb P(N_E(-2))= \{(p,Q);\ p\in E,\ Q\in\operatorname{Im}\beta(1)|_p\} \subset\Bbb P(\mathrm H^\circ(I_E(2))\times E. \] \begin{Claim}\label{P25-Claim1} $Q\in\operatorname{Im}\beta(1)|_p$ if and only if $Q$ contains the unique cubic scroll containing the secant $L_{\langle o,-p\rangle}$. \end{Claim} \begin{ProofwCaption}{Proof of the claim} Consider the matrix \[ M'=\left(\frac{\partial Q_{3j}}{\partial x_i}\right)_{i,j}\quad (i,j\in\Bbb Z_5) \] from remark \ref{R17}. By proposition \ref{P16} \rom{(ii)} this has rank $3$ on $E$. One easily checks that the entries of $A\tr M'$ are all elements of $\mathrm H^\circ(I_E(2))$. Since $A$ has rank $2$ on $E$ the sequence \[ \mathrm H^\circ(I_E(2))\otimes\cal O(-1)\stackrel{\tr M'}{\to} \mathrm H^\circ(I_E(2))\otimes\cal O_E\stackrel{A}{\to} \mathrm H^\circ(I_E(2))\otimes\cal O_E(1) \] is exact. Therefore \[ \operatorname{Im}\beta(1)=\operatorname{Im}\tr M'. \] Since there is a net of quadrics through a cubic scroll it suffices to show that any quadric in the image of $\tr M'(p)$ contains the scroll. For this it suffices to show that the secant lines $L_{\langle o,-p\rangle}$ and $L_{\langle\eta_5,\eta_{5-p}\rangle}$, where $\eta_5$ is a non-zero 5-torsion point, are contained in $Q$ (recall that the cubic scroll in question is the union of all secants $L_{\langle q,r\rangle}$ with $q+r=-p$). If $Q$ contains two secants in the scroll it must contain the scroll by Bezout. Now $\operatorname{Im}\tr M'(p)$ is spanned by the elements \[ M'_i(p)\begin{pmatrix} Q_0 \\ \vdots \\ Q_4 \end{pmatrix} \] where $M'_i(p)$ is the $i$-th row of $M'$ evaluated at $p$. The origin has coordinates $(0,a,-1,1,-a)$ and we can take $\eta_5$ to be $(a,-1,1,-a,0)$. If $p$ has coordinates $(x_0,\ldots,x_4)$ then $-p$ has coordinates $(x_0,x_4,x_3,x_2,x_1)$ and $-p-\eta_5$ has coordinates $(x_4,x_3,x_2,x_1,x_0)$. Evaluating the quadrics in $\operatorname {Im}\tr M'(p)$ on the secant lines one gets quadrics in the coordinates $x_i$ which vanish on $E$. This proves the claim. \noqed \end{ProofwCaption} This shows that $\widetilde V_2=\Bbb P(N_E(-2))$ and that the map to $\Bbb P^4$ is given by $\cal O_{\Bbb P(N_E(-2))}(1)$. By \cite[proposition V.1.2]{Hu} the twisted normal bundle $N_E(-2)$ is indecomposable with $c_1(N_E(-2))=\cal O_E(-1)$. By Atiyah's classification \cite{At} $N_E(-2)\cong\cal E_E$ and in particular $\Bbb P(N_E(-2))\cong \Bbb P^2_E$. Both bundles $N_E(-2)$ and $\cal E_E$ come with an $H_5$-action which covers the same action on $E$. Since $N_E(-2)$ resp.\ $\cal E_E$ are stable, and hence simple, the two $H_5$-actions on $N_E(-2)$ and $\cal E_E$ differ at most by a character. But since the induced actions on the respective determinants coincide, this character must be trivial. By construction $\cal L =\cal O_{\Bbb P(\cal E_E)}(1)$ (As a check note that the representation of $H_5$ on both $\mathrm H^0(\cal L)$ (see proposition \ref{P7}) and on $ \mathrm H^0(\cal O_{\Bbb P(\mathrm H^0(I_E(2)))} (1)) = \mathrm H^0(I_E(2))\spcheck $ are in each case derived from the Schr\"odinger representation by replacing $\varepsilon$ by $\varepsilon^2$). In any case the above argument shows that we have an $H_5$-isomorphism between $\cal O_{\Bbb P(N_E(-2))}(1)$ and $\cal O_{\Bbb P(\cal E)}(1)$ and we are done. \end{Proof} \begin{Remark}\label{R27} We have seen in proposition \ref{P23} (ii) that $A_0=E\times E$ is mapped 2$:$1 by $\pi_2$ onto an elliptic quintic scroll. Since the abelian surfaces $A_K$ and the bielliptic surfaces $B_{(a,b)}$ are numerically equivalent to $A_0$ on $\widetilde V_2$ these surfaces must be mapped to surfaces in ${\Bbb P}^4$ of degree 10. \end{Remark} \begin{Proposition}\label{P26} There is an isomorphism $\widetilde V_1\cong\Bbb P^1_{S^2E}$ such that $f_2$ is identified with $\rho$. \end{Proposition} \begin{Proof} $f_2\colon\widetilde V_1\to\widetilde V_2$ is the blow up of $\widetilde V_2$ in the diagonal of $D_1$ after we have identified $D_1$ with $E\times E$ via the isomorphism $\psi$. Recall also that the map $\pi_2$ is bijective outside $D_1$ and that $\pi_2$ restricted to $E\times E$ is a 2$:$1 branched covering onto its image whose branch locus is the diagonal of $E\times E$. The map $\rho\colon\Bbb P^1_{S^2E}\to\Bbb P^2_E$ is the blow up of $\Bbb P^2_E$ along the diagonal of $A_0=E\times E$. In view of our identification of $\pi_2$ with the map given by $s_0,\ldots,s_4$ it is enough to prove the following: The map $(s_0:\ldots:s_4)$ restricted to $A_0=E\times E$ is a 2$:$1 branched covering with branch locus the diagonal. But this is easy to see. Recall that the curve $\{(q,q)\in E\times E\}\subset E\times\Bbb P^2$ is mapped 9$:$1 to the antidiagonal $\{(q,-q);\ q \in E\}$ in $A_0$. By construction of $\cal L$ this shows that the degree of $\cal L$ restricted to the antidiagonal, and hence all its translates, is 2. Moreover the degree of $\cal L$ restricted to $A_0$ is 10. It is well known that then $A_0$ is mapped 2$:$1 onto a quintic elliptic scroll with branch locus the diagonal (e.g. see \cite{HL}). \end{Proof} We are now ready to prove that the maps $\pi_1$ and $\pi_2$ give rise to abelian and bielliptic surfaces of degree 15, resp.\ 10. Before we do this, we recall from lemma 5 that \[ A_K \cap A_0 = \{(q,r) \in E \times E;\ 3 r + 2 q = 0\} \] resp. \[ B_{(a,b)} \cap A_0 = \{(q,r) \in E \times E;\ 3 r + 2 q = - \tau_{(a,b)}\}. \] We set \[ E_K = A_K \cap A_0, \quad E_{(a,b)} = B_{(a,b)} \cap A_0. \] Moreover, we consider the following curves on $E {\times} E$: \[ \Delta_p^- = \{(e,-e + p),\ e \in E\}. \] Under the quotient map $E {\times} E \to S^2 E$ these curves are mapped to the rulings of the ${\Bbb P}^1$-bundle $S^2 E$. \begin{Lemma}\label{L27} The curves $E_K$, resp.\ $E_{(a,b)}$, intersect the curves $\Delta_p^-$ transversally in one point. \end{Lemma} \begin{Proof} A point $(e,-e + p)$ lies on $E_K$ if and only if $-e + 3 p = 0$, i.e., $e = 3 p$. Both curves are elliptic curves. Two curves on an abelian surface which do not coincide, meet transversally. The claim for the curve $E_{(a,b)}$ is proved in exactly the same way. \end{Proof} \begin{Theorem}\label{T28} \rom{(i)} Let $A_K$ be a smooth element different from $A_0$ in the pencil $|-K|$ on ${\Bbb P}^2_E \cong \widetilde V_2$. Then $\pi_2$ embeds $A_K$ as a smooth abelian surface of degree 10. \noindent\rom{(ii)} The bielliptic surfaces $B_{(a,b)}$ are also embedded as surfaces of degree 10 by the map $\pi_2$. \end{Theorem} \begin{Proof} \rom{(i)} By proposition \ref{P23} the map $\pi_2$ is an isomorphism outside $D_1 \cong A_0$. Hence it is sufficient to consider the intersection $E_K = A_K \cap A_0$. We first claim that $\pi_2$ restricted to $E_K$ is injective. The map $\pi_2$ identifies points $(q,r)$ and $(r,q)$. Assume that two such points lie on $E_K$. This implies that \[ 3 r + 2 q = 0, \quad 3 q + 2 r = 0. \] Subtracting these two equations from each other gives $q = r$, and hence $(q,r) = (r,q)$. Finally we have to check that the differential of $\pi_2$ restricted to $A_K$ is injective at the 25 points $E_K \cap \Delta$. For this recall that the kernel of $d\pi_2$ along $\Delta$ is given by the directions defined by the curves $\Delta_p^-$. The claim follows, therefore, from lemma \ref{L27}. The degree of the embedded surfaces is 10 by remark \ref{R27}. \noindent\rom{(ii)} The same proof goes through for the surfaces $B_{(a,b)}$. \end{Proof} \begin{Remarks}\label{R29} \rom{(i)} The pencil $|-K|$ contains 4 singular elements corresponding to the 4 triangles in the Hesse pencil. These surfaces are mapped to translation scrolls of quintic elliptic curves where the translation parameter is a non-zero 3-torsion point. \noindent\rom{(ii)} All abelian surfaces $A_K$ are isogeneous to a product. Hence this construction does not give the general abelian surface in ${\Bbb P}^4$. On the other hand we get all minimal bielliptic abelian surfaces in ${\Bbb P}^4$ in this way (up to a change of coordinates). \noindent\rom{(iii)} The abelian surfaces in ${\Bbb P}^4$ which are of the form $E {\times} F / {\Bbb Z}_3 {\times} {\Bbb Z}_3$ were studied by Barth and Moore in \cite{BM}. By their work the pencils which we have constructed above, are tangents to the rational sextic curve $C_6$ in the space of Horrocks-Mumford surfaces which parametrizes Horrocks-Mumford surfaces which are double structures on elliptic quintic scrolls. \noindent\rom{(iv)} The involution $\iota$ of proposition \ref{P8} induces the involution $x \mapsto -x$ on the surfaces $A_K$. The surfaces $B_{(a,b)}$ are identified pairwise, more precisely $\iota B_{(a,b)} = B_{(-a,-b)}$. This follows since the 8 characters $F_{(a,b)}$ are identified in this way by the Heisenberg involution on ${\Bbb P}^2$. \end{Remarks} \begin{Theorem}\label{T30} \rom{(i)} Let $A_K$ be a smooth element different from $A_0$ in the pencil $|-K|$ on ${\Bbb P}^2_E \cong \widetilde V_2$. Then the map $\pi_1$ embeds $\tilde A_K$ as a smooth non-minimal abelian surface of degree 15 in ${\Bbb P}^4$. \noindent\rom{(ii)} The surfaces $\tilde B_{(a,b)}$ are embedded by $\pi_1$ as smooth bielliptic surfaces of degree 15. \end{Theorem} \begin{Proof} \rom{(i)} Again it is enough to look at the intersection of $\tilde A_K$ with $A_0$ on ${\Bbb P}^1_{S^2 E} \cong \widetilde V_1$. The curves $\Delta_p^-$ are contracted by $\pi_1$. These are the only tangent directions which are in the kernel of the differential of $\pi_1$. Hence our claim follows again from lemma \ref{L27}. The double point formula reads \[ d^2 = 10 d + 5 H K + K^2 - e. \] In our case $H K = 25$, $K^2 = -25$ and $e = 25$. This leads to \[ d(d - 10) = 75 \] and the only positive solution is $d = 15$. \noindent\rom{(ii)} The claim about the bielliptic surfaces can be proved in exactly the same way. \end{Proof} \begin{Remark}\label{R31} The degree of the surfaces $\tilde A_K$, resp.\ $\tilde B_{(a,b)}$, can also be computed by studying the linear system which maps $\widetilde V_1$ to ${\Bbb P}^4$. We shall come back to this. \end{Remark} \bigskip We shall now turn our attention to the quintic hypersurfaces which contain the surfaces $A_K$, $B_{(a,b)}$, $\tilde A_K$ and $\tilde B_{(a,b)}$. \begin{Proposition}\label{P33} The bielliptic surfaces $B_{(a,b)}$ lie on a unique quintic hypersurface, namely $V_2$. \end{Proposition} \begin{Proof} We consider the elliptic quintic scroll $S^2E\subset V_2\subset {\Bbb P}^4$. Recall that $S^2 E$ is the quotient of $E \times E$ by the involution which interchanges the two factors. Let $C_{p_0}$ be the section of $S^2 E$ which is the image of $\{p_0\} \times E$, resp.\ $E \times \{p_0\}$ in $S^2 E$ where $p_0$ is the origin of $E$ which we have chosen before. Note that the normal bundle of $C_{p_0}$ in $S^2 E$ is the degree 1 line bundle which is given by the origin. Let $F_{p_0}$ be the fibre over the origin of the map $S^2 E \to E$, $\{q_1,q_2\} \mapsto q_1 + q_2$. Moreover let $H$ be the hyperplane section of $S^2 E \subset {\Bbb P}^4$. It follows immediately from our choice of the line bundle $\cal L$ in \ref{Par1} that \begin{equation}\label{(9)} H \sim C_{p_0} + 2 F_{p_0}. \end{equation} Next we consider the intersection $C_{(a,b)} = B_{(a,b)} \cap S^2 E$. The curve $C_{(a,b)}$ is by lemma \ref{L27} a section of $S^2E$. Now $E\times E\to S^2E$ is ramified along the diagonal and maps the curve $E_{(a,b)}$ isomorphically to $C_{(a,b)}$ so, combining with lemma \ref{L5}\rom{(ii)}, the intersection of $C_{(a,b)}$ with the diagonal is twice the set $\{(p,p);\ 5p=-\tau_{(a,b)}\}$. Thus $C_{(a,b)} \equiv C_{p_0}+12F$. In fact the intersection with the diagonal goes by the map $S^2E\to E$ to $\{4p;\ 5p=-\tau_{(a,b)}\}$, which summed up is $\tau_{(a,b)}$. Therefore \begin{equation}\label{(10)} C_{(a,b)} \sim C_{p_0} + 11 F_{p_0} + F_{(a,b)} \end{equation} where $F_{(a,b)}$ is the fibre over the 3-torsion point $\tau_{(a,b)}$. Finally recall that \begin{equation}\label{(11)} K \sim - 2 C_{p_0} + F_{p_0}. \end{equation} Let $Q$ be a quintic containing $B_{(a,b)}$. We first claim that $Q$ must contain $S^2 E$. In order to see this look at the exact sequence \[ 0 \to \cal O_{S^2 E}(5 H - C_{(a,b)}) \to \cal O_{S^2 E}(5 H) \to \cal O_{C_{(a,b)}}(5 H) \to 0. \] It follows from formulas \eqref{(9)}, \eqref{(10)}, and \eqref{(11)} that \begin{equation}\label{(12)} 5 H - C_{(a,b)} \sim 4 C_{p_0} - F_{p_0} - F_{(a,b)} \sim -2 K + (F_{p_0} - F_{(a,b)}). \end{equation} It is well known that $h^0(\cal O_{S^2 E}(-2 K + F_p - F_q)) = 0$ unless $2p = 2q$ (cf. \cite{CC}). Hence $Q$ must contain $S^2 E$. We next claim that $Q = V_2$. In order to see this, consider a plane on $V_2$, i.e., a trisecant plane of $S^2 E$. Both $S^2 E$ and $B_{(a,b)}$ intersect such a plane in different irreducible cubic curves. Since $Q$ has degree 5 it must contain this plane and hence $V_2$. By reasons of degree this implies $Q = V_2$. \end{Proof} \begin{Remark}\label{R34} The surfaces $A_K$ lie on 3 independent quintics. This follows e.g.\ from the fact that $A_K$ is the zero-scheme of a section $s$ of the Horrocks-Mumford bundle $F$ (where we normalize $F$ such that $c_1(F) = 5$, $c_2(F) = 10$). In other words there is an exact sequence \[ 0 \to \cal O_{{\Bbb P}_4} \stackrel{s}{\to} F \to I_{A_K}(5) \to 0. \] The claim follows from $h^\circ(F) = 4$ (see \cite{HM}). \end{Remark} \medskip Note that if we replace $B_{(a,b)}$ by a surface $A_K$ in the proof of proposition \ref{P33} we obtain $-2 K$ in formula \eqref{(11)}. Since $ h^\circ(\cal O_{S^2 E}(-2 K)) = 2 $, this gives rise to two more elements in $ \mathrm H^\circ(\cal O_{S^2 E}(-2 K)) $ which vanish along $C_K = A_K \cap S^2 E$. These can be lifted to $H_5$-invariant quintics in ${\Bbb P}^4$. It is then easy to check that any $H_5$-invariant quintic which contains $C_K$ must contain $A_K$. (Look at the intersection of the quintic with the cubic curves on $A_K$. Unless the quintic contains these curves, this would split up into two $H_5$-orbits of length 9 and 6 resp., a contradiction). In this way one can also prove the existence of 3 independent quintics through the surfaces $A_K$. We now turn our attention to the degree 15 surfaces. \begin{Proposition}\label{P35} \rom{(i)} The non-minimal abelian surfaces $\tilde A_K$ lie on exactly three quintic hypersurfaces. They are linked $(5,5)$ to translation scrolls. \par\noindent\rom{(ii)} The non-minimal bielliptic surfaces $\tilde B_{(a,b)}$ lie on a unique quintic hypersurface, namely $V_1$. \end{Proposition} \begin{Proof} \rom{(i)} Let $\bar H_1$ be the hyperplane section on ${\Bbb P}^1_{S^2 E}$ given by the map $\pi_1\colon {\Bbb P}^1_{S^2 E} \to V_1 \subset {\Bbb P}^4$. Let $\bar C_{p_0}$, resp.\ $\bar F_{p_0}$ be the fibres over the curves $C_{p_0}$, resp.\ $F_{p_0}$ in $S^2 E$ with respect to the map ${\Bbb P}^1_{S^2 E} \to S^2 E$. The classes $\bar H_1$, $\bar C_{p_0}$ and $\bar F_{p_0}$ generate the Neron-Severi group of ${\Bbb P}^1_{S^2 E}$. Under $\pi_1$ the surface $\bar F_{p_0}$ is mapped isomorphically to a cubic scroll, while $\bar C_{p_0}$ is mapped birationally to the cone over an elliptic curve of degree 4 in ${\Bbb P}^3$ with vertex in the origin $p_0$ of the elliptic curve in ${\Bbb P}^4$ for which $V_1$ is the secant variety. Thus we get the following intersection numbers: $\bar H_1^3 = 5$, $\bar H_1^2 \bar C_{p_0} = 4$, $\bar H_1^2 \bar F_{p_0} = 3$, $\bar H_1 \bar C_{p_0} \bar F_{p_0} = \bar H_1 \bar C_{p_0}^2 = 1$ and $ \bar C_{p_0}^2 \bar F_{p_0} = \bar C_{p_0} \bar F_{p_0}^2 = \bar H_1 \bar F_{p_0}^2 = \bar F_{p_0}^3 = \bar C_{p_0}^3 = 0 $. The exceptional divisor $X = E_\Delta$ is a section of the ${\Bbb P}^1$-bundle ${\Bbb P}^1_{S^2 E}$. Under $\pi_1$ it is mapped birationally onto a ruled surface of degree 15. From this information it is easy to compute that numerically $X \equiv \bar H_1 - 2 \bar C_{p_0} + 6 \bar F_{p_0}$. It is also straightforward to check that for the canonical divisor $K \equiv -2 \bar H_1 + \bar C_{p_0} + 2 \bar F_{p_0}$. In fact we claim that these equalities are also true with respect to linear equivalence. It is enough to check this on a section of the composite projection ${\Bbb P}^1_{S^2 E} \to S^2 E \to E$. We consider the curve $D = X \cap D_1 = X \cap (E \times E)$. On $E \times E$ this is the diagonal by proposition \ref{P19}. Using the projection ${\Bbb P}^1_{S^2 E} \to S^2 E$ we can identify the section $X$ with $S^2 E$. Then by \cite[lemma IV.4.4]{Hu} we have $D \sim C_{p_0} + 12 f_{p_0}$ on $X$. Since $X$ is mapped to a ruled surface of degree 15 we have $\bar H_1|_X \equiv C_{p_0} + 7 F_{p_0}$. Since $D$ is mapped by multiplication with $-2$ four to one onto the elliptic curve $E$ and since $E$ is embedded by $|5 p_0|$ we have in fact that this equality also holds with respect to linear equivalence. We also note that $X$ restricted to $D$ is trivial. This follows since $X$ restricted to $A_0$ is $D$ and since the normal bundle of $D$ in $A_0$ is trivial. Hence in order to check that $ X \sim \bar H_1 - 2 \bar C_{p_0} + 6 \bar F_{p_0} $ it is enough to prove that the restriction of $\bar H_1 - 2 \bar C_{p_0} + 6 \bar F_{p_0}$ to $D$ is trivial. This follows from \[ (-C_{p_0} + 13 F_{p_0}) (C_{p_0} + 12 p_0) \sim 0 \] which has to be read as an equality of divisors on $D \cong E$. Next we want to prove that \[ K \sim -2 \bar H_1 + \bar C_{p_0} + 2 \bar F_{p_0}. \] We know that $(K + X)|_D \sim K_X|_D \sim -25 p_0$. The first is the adjunction formula, the second follows from \[ (-2 C_{p_0} + F_{p_0}) (C_{p_0} + 12 F_{p_0}) \sim -25 p_0. \] Since $X|_D \sim 0$ it is now enough to show that $-2 \bar H_1 + \bar C_{p_0} + 2 \bar F_{p_0}$ restricted to $D$ is linearly equivalent to $-25 p_0$. This follows from \[ (-C_{p_0} - 12 F_{p_0}) ( C_{p_0} + 12 F_{p_0}) \sim -25 p_0. \] Hence we have proved that \begin{equation}\label{(12a)} \bar X \sim \bar H_1 - 2 \bar C_{p_0} + 6 \bar F_{p_0}, \quad K \sim -2 \bar H_1 + \bar C_{p_0} + 2 \bar F_{p_0}. \end{equation} Since $A_K$ is anticanonical on ${\Bbb P}^2_E$ we get that $ \tilde A_K \sim -K + X \sim 3 \bar H_1 - 3 \bar C_{p_0} + 4 \bar F_{p_0} $. Furthermore $-K \sim D_1$, and $D_1$ is contracted under $\pi_1$ to the curve $E$. Since twice the anticanonical divisor on $S^2 E$ moves in a pencil it follows that also $4 \bar C_{p_0} - 2 \bar F_{p_0}$ moves in a pencil on ${\Bbb P}^1_{S^2 E}$. Note that \begin{equation}\label{(13)} 5 \bar H_1 \sim -K + \tilde A_K + (4 \bar C_{p_0} - 2 \bar F_{p_0}). \end{equation} The members of the pencil $(4 \bar C_{p_0} - 2 \bar F_{p_0})$ on ${\Bbb P}^1_{S^2 E}$ are mapped to the translation scrolls of $E$. Take such a translation scroll (which is not a quintic elliptic scroll). Then it is a Horrocks-Mumford surface (cf. \cite{Hu2}, \cite{BHM}) and hence lies on three quintics of which $V_1$ is one. We can choose a pencil of quintics through such a scroll which does not contain $V_1$. All these quintics contain $E$. They cut out a pencil of residual surfaces and it follows from \eqref{(13)} that this is just the pencil formed by the surfaces $\tilde A_K$. Hence every such surface is linked $(5,5)$ to a translation scroll $S$. Now consider the well known liaison sequence (cf.\ \cite{PS}): \[ 0 \to I_{S \cup \tilde A_K}(5) \to I_{\tilde A_K}(5) \to \omega_S \to 0. \] We have $h^\circ(I_{S \cup \tilde A_K}(5)) = 2$, $h^1(I_{S \cup \tilde A_K}(5)) = 0$. Since $S$ is a a Horrocks-Mumford surface $\omega_S = \cal O_S$ and hence $h^\circ(\omega_S) = 1$. It follows that $h^\circ(I_{\tilde A_K}(5)) = 3$. \noindent\rom{(ii)} Now consider a bielliptic surface $\tilde B_{(a,b)}$. If $\tilde B_{(a,b)}$ lies on two quintics, it would be linked to a surface $T$ in the numerical equivalence class of $4 \bar C_{p_0} - 2 \bar F_{p_0}$. Since $3 \tilde B_{(a,b)}$ is linearly equivalent to $3 \tilde A_K$, we must have that $3 T$ is linearly equivalent to $3 (4 \bar C_{p_0} - 2 \bar F_{p_0})$ while $T$ is not linearly equivalent to $4 \bar C_{p_0} - 2 \bar F_{p_0}$. But in this numerical equivalence class the only effective divisors are the pencil $(4 \bar C_{p_0} - 2 \bar F_{p_0})$ and the three divisors $4 \bar C_{p} - \bar F_{p_0} - \bar F_\tau$ where $\tau$ is a non-trivial 2-torsion point. \end{Proof} \begin{Remark}\label{R36} At this point we would like to say a few more words about liaison. As said before, the space $\Gamma(F)$ of sections of the Horrocks-Mumford bundle has dimension 4. The three-dimensional space ${\Bbb P} \Gamma = {\Bbb P}(\Gamma(F))$ parametrizes the Horrocks-Mumford (\rom{HM}) surfaces $X_s = \{s = 0\}$ where $0 \ne s \in \Gamma(F)$. The space of Heisenberg invariant quintics is related to $\Gamma(F)$ via the isomorphism \[ \Lambda^2 \Gamma(F) \cong \Gamma_{H_5}(\cal O_{{\Bbb P}^4}(5)) \] given by the natural map $ \Lambda^2 \Gamma(F) \to \Gamma(\Lambda^2 F) = \Gamma(\cal O_{{\Bbb P}^4}(5)) $. Set \[ {\Bbb P}_{H_5}^5 = {\Bbb P}(\Gamma_{H_5}(\cal O_{{\Bbb P}^4}(5))) \cong {\Bbb P}(\Lambda^2 \Gamma(F)). \] In ${\Bbb P}_{H_5}^5$ we consider the Pl\"ucker quadric $G = G(1,3)$ of decomposable tensors. If $X_{s_0} = \{s_0 = 0\}$ is a \rom{HM}-surface, then \[ \Gamma(I_{X_{s_0}}(5)) = \{s_0 \wedge s;\ s \in \Gamma(F)\}. \] This defines a ${\Bbb P}^2$ of decomposable tensors. In $G = G(1,3)$ this is an $\alpha$-plane, i.e., a plane of lines through one point. In this way we get a bijection between ${\Bbb P}\Gamma$ and the set of all $\alpha$-planes in $G$. Now consider a line in an $\alpha$-plane, i.e., a pencil of quintics spanned by quintics of the form $s_0 \wedge s_1$ and $s_0 \wedge s_2$. This give rise to a complete intersection \[ Y_1 \cap Y_2 = X_{s_0} \cup X' \] where $X'$ is of degree 15. By the liaison sequence which we have already used before, we find that $h^\circ(I_{X'}(5)) = 3$. The space of quintics is spanned by $s_0 \wedge s_1$, $s_0 \wedge s_2$ and $s_1 \wedge s_2$. To see that $s_1 \wedge s_2$ is contained in this space consider $X' \sm X_{s_0}$. At these points $s_0$ does not vanish and $s_1$ and $s_2$ are linearly dependent of $s_0$. It follows that $s_1 \wedge s_2 = 0$. Hence all quintics through $X'$ are in particular $H_5$-invariant and the space of these quintics is a $\beta$-plane in $G$, i.e., a ${\Bbb P}^2$ of lines which lie in a fixed plane in ${\Bbb P}^3$. Let $\tilde A_K$ be one of our non-minimal degree 15 abelian surfaces. By proposition \ref{P35} the surface $\tilde A_K$ is linked to a translation scroll. Hence it defines a $\beta$-plane in $G$. For every line in this plane there exists exactly one $\alpha$-plane intersecting the $\beta$-plane in this line. Hence every such line gives rise to liaison with an \rom{HM}-surface. In this way $\tilde A_K$ is linked to a 2-dimensional family of \rom{HM}-surfaces. This 2-dimensional family is parametrized by a linear ${\Bbb P}^2$ in ${\Bbb P} \Gamma$. Since the singular \rom{HM}-surfaces form an irreducible surface of degree 10 \cite{BM} in ${\Bbb P} \Gamma$, it follows in particular that $\tilde A_K$ is linked to smooth abelian surfaces. \end{Remark} \medskip We want to conclude this paragraph with a short discussion of the 6-secants of the surfaces $A_K$ and $B_{(a,b)}$. The 6-secants of the surfaces $A_K$ are exactly the 25 Horrocks-Mumford lines \cite{HM}. The 6-secant formula from \cite{L} shows that the surfaces $B_{(a,b)}$ either also have 25 6-secants or infinitely many. In fact there are exactly 25 6-secants and we shall now describe them. First note that every 6-secant of $B_{(a,b)}$ must lie in one of the planes of $V_2$ (it must be contained in $V_2$ by reasons of degree and since the base of the bundle ${\Bbb P}^2_E$ is elliptic, it must be in one of the fibres). Now fix a point $e \in E$ and let $f=f_{2}\! \! \shortmid_{\Sigma^{1}_{e}}$ be the blowing down map $\Sigma_e^1 \to {\Bbb P}_e^2$. If we interpret $\Sigma_e^1$ as a cubic scroll in ${\Bbb P}^4$, then $\Sigma_e^1$ consists of the secants of the elliptic quintic curve $E$ joining points $p$ and $q$ with $p + q = e$. We denote this curve by $E_e \subset \Sigma_e^1$. The map $f\colon \Sigma_e^1 \to {\Bbb P}_e^2 \subset {\Bbb P}^4$ is given by the linear system $|X_e + l|$ where $l$ are the fibres of the ${\Bbb P}^1$-bundle $\Sigma_e^1$ and $X_e$ is the exceptional line. By what we have said before, we have $l \cap E_e = \{p,q\}$ with $p + q = e$. One also computes easily that $X_e \cap E_e = \{-2 e\}$ where we consider $-2 e$ as a point on $E_e$. It follows that the map $f$ is given by the linear system $(-2 e) + e + p_0 \sim 5 p_0 - 2 e_0$ on $E_e$ where $2 e_0 = e$. The composite $\pi_{2} \circ f_{2}$ maps both curves $E_{e}$ and $\iota' E_{e} \subset E\times E$ onto the same plane cubic curve in ${\Bbb P}^{4}$. When considering the change of coordinates $\psi_2$ followed by the involution interchanging the factors on $E\times E$, $E_{e}$ is mapped to the first factor, while $\iota 'E_{e}$ is mapped to the second factor. Therefore, by lemma \ref{L5} (ii), the intersection $\tilde B_{(a,b)} \cap \iota' (E_{e})$ consists of the points $\{(p,-p-e);\ 2p=e-\tau_{(a,b)}\}$. These have the same image on the scroll as the points $p$ on $E_{e}$ with $2p=e-\tau_{(a,b)}$. These are the points $p_i = e_0 + \tau_{(a,b)} + \tau_i$, where the $\tau_i$, $i=0,1,2,3$, are the 2-torsion points with $\tau_0 = p_0$. The points $f(p_i)$, $i=1,2,3$, are collinear if and only if $\sum_{i = 1}^3 p_i + 2 e_0 = p_0$ which is equivalent to $5 e_0 = p_0$, i.e., $e_0$ is a 5-torsion point on $E$. In this case we also get $e_0 = -4 e_0 = -2 e \in X_e \cap E_e$. Hence for $e$ a 5-torsion point on $E$ the line through these 3 points intersects $B_{(a,b)}$ in addition in 3 points on its plane cubic in ${\Bbb P}_e^2$, hence is a 6-secant. The above discussion also shows that there are only finitely many 6-secants, and hence we have found them all. \section{Cremona transformations}\label{Par3} In this paragraph we want to explain how the abelian, resp.\ bielliptic surfaces of degree 15 can be constructed via Cremona transformations. It is known that the quadrics through an elliptic quintic curve $E \subset {\Bbb P}^4$ define a Cremona transformation $\Phi\colon {\Bbb P}^4 \mathrel{\dashrightarrow} {\Bbb P}^4$ \cite{Sem}. Via $\Phi$ the secant variety of $E$ is mapped to a quintic elliptic scroll $S^2 E$ in ${\Bbb P}^4$. The exceptional locus of $\Phi$ is mapped to the trisecant variety of this scroll. The cubics through the quintic elliptic scroll $S^2 E$ define the inverse of $\Phi$. Under $\Phi^{-1}$ the trisecant variety of $S^2 E$ is mapped to $E$, while the exceptional divisor is mapped to the secant variety of $E$. We refer to \cite{Sem} for details on the geometry of this transformation. We consider the scroll $S^2 E$ whose trisecant variety equals $V_2$. On $S^2 E$ there exist three elliptic 2-sections $E_i$, $i=1,2,3$, such that $S^2 E$ is the translation scroll of $E_i$ defined by a non-zero 2-torsion point $p_i$. Let $\widetilde V_2 \cong {\Bbb P}^2_E$ be the desingularization of $V_2$. Then the curves $ \Delta_i = \{(e + p_i,e);\ e \in E\} \subset A_0 \cong E \times E $ are mapped 2$:$1 to the elliptic curves $E_i$. Let $\Delta_E \subset A_0 \cong E \times E$ be the diagonal. Let $\bar H_2$ be the line bundle on $\widetilde V_2$ given by the map $\pi_2\colon \widetilde V_2 \to V_2 \subset {\Bbb P}^4$. Next consider the isomorphism \[ \begin{aligned} \phi_i\colon E \times {\Bbb P}^2 &\to E \times {\Bbb P}^2 \\ (e,x) &\mapsto (e + p_i,x). \end{aligned} \] This commutes with the diagonal action of ${\Bbb Z}_3 \times {\Bbb Z}_3$ on $E \times {\Bbb P}^2$ and hence induces an isomorphism \[ \tilde\phi_i\colon \widetilde V_2 \to \widetilde V_2 \] which maps $\Delta_E$ to $\Delta_i$. Note also that $\tilde\phi_i$ maps the surfaces $A_K$, resp.\ $B_{(a,b)}$ to themselves. Recall that $\widetilde V_1$ is the blow-up of $\widetilde V_2$ along $\Delta_E$. Let $\widetilde V_1^{(i)}$ be the blow-up of $\widetilde V_2$ along $\Delta_i$. Then $\tilde\phi_i$ induces an isomorphism \[ \bar\phi_i\colon \widetilde V_1 \to \widetilde V_1^{(i)} \] such that the diagram \[ \begin{CD} \widetilde V_1 @>\bar\phi_i>> \widetilde V_1^{(i)} \\ @V\rho VV @VV\rho^{(i)}V \\ \widetilde V_2 @>\tilde\phi_i>> \widetilde V_2 \end{CD} \] commutes where the vertical maps are the blowing down maps. $X$ is the exceptional locus of $\rho$. Let $X_i$ be the exceptional loci of the maps $\rho^{(i)}$. By abuse of notation we denote the pullback of $\bar H_2$ by $\rho$ also by $\bar H_2$. We denote the pullback of $\bar H_2$ by $\rho^{(i)}$ by $\bar H_2^{(i)}$. The Cremona transformation defined by the quadrics through $E_i$ gives rise to the linear system $|2\bar H_2^{(i)} - X_i|$ on $\widetilde V_1^{(i)}$. Note that \[ (\bar\phi_i)^* (2\bar H_2^{(i)} - X_i) \sim \rho^*(\tilde\phi_i)^*(2 \bar H_2) - X \sim 2 \bar H_2 - X. \] The latter follows since $\bar H_2$ restricted to $\Delta_E$ has degree 10 and this implies that translation by a 2-torsion point leaves the linear equivalence class invariant. \begin{Proposition}\label{P37} $\bar H_1 \sim 2 \bar H_2 - X$. \end{Proposition} \begin{Proof} We first claim that $\bar H_1 \equiv 2 \bar H_2 - X$. The N\'eron-Severi group of $\widetilde V_1$ is generated by $\bar H_2$, $X$ and $\Sigma^1$ where $\Sigma^1$ denotes the class of a fibre of $\widetilde V_1 \to E$. Restriction to such a fibre implies immediately that $\bar H_1 \equiv \alpha \Sigma^1 + 2 \bar H_2 - X$. To compute $\alpha$ we use $\bar H_1^3 = 5$. Since $(\Sigma^1)^2 = 0$ this implies \begin{equation}\label{(14)} 5 = 9 \alpha + (2 \bar H_2 - X)^3. \end{equation} Now $\bar H_2^2\cdot X=0$, and $\bar H_2\cdot X^2=-10$, since $X$ is blown down to the diagonal $\Delta_E\subset A_0$ and $\Delta_E\cdot\bar H_2=10$. On the other hand $X^3=-25$ from our computations in the proof of proposition \ref{P35}, thus $\alpha =0$ as claimed. In order to prove the proposition it is now enough to consider the restriction of $\bar H_1$, resp.\ $2 \bar H_2 - X$ to the section $D$ which we have already used in the proof of proposition \ref{P35}. Via $\rho$ the curves $D$ and $\Delta_E$ are identified. We know that $\bar H_1$ restricted to $D$ is linearly equivalent to $20 p_0$. On the other hand $\bar H_2$ restricted to $\Delta_E$ is linearly equivalent to $10 p_0$ (this can be seen e.g.\ by using proposition \ref{P19} and the construction of the line bundle $\cal L$). We also have already seen in the proof of proposition \ref{P35} that the restriction of $X$ to $D$ is trivial. This proves the proposition. \end{Proof} \begin{Corollary}\label{C38} The map $\pi_1\colon \widetilde V_1 \to V_1$ is given by the complete linear system $|\bar H_1| = |2 \bar H_2 - X|$. \end{Corollary} \begin{Proof} We have to show that the (affine) dimension of the linear system $|2 \bar H_2 - X|$ is five. We consider the exact sequence \[ 0 \to \cal O_{\widetilde V_1}(2 \bar H_2 - A_0 - X) \to \cal O_{\widetilde V_1}(2 \bar H_2 - X) \to \cal O_{A_0}(2 \bar H_2 - \Delta_E) \to 0. \] Since $(2H_2-A_0-X)\cdot H_2\cdot\Sigma^1 =-1$ it follows that $h^\circ( \cal O_{\widetilde V_1}(2 \bar H_2 - A_0 - X)) =0$. Hence we have the inclusion \[ 0 \to \mathrm H^\circ(\cal O_{\widetilde V_1}(2 \bar H_2 - X)) \to \mathrm H^\circ(\cal O_{A_0}(2 \bar H_2 - \Delta_E)). \] The linear system $|2 \bar H_2 - \Delta_E|$ restricted to $A_0$ has degree 0 on the curves $\{(p,-p + e);\ p \in E\}$ and degree 5 on the curves $\{(e,p);\ p \in E\}$. It follows that $h^\circ(\cal O_{A_0}(2 \bar H_2 - \Delta_E)) = 5$ and this proves the corollary. \end{Proof} \begin{Corollary}\label{C39} The hyperplane bundle of $\tilde A_K$ is of the form $2 H' - \sum_{i = 1}^{25} E_i$ where $H'$ is a polarization of type $(1,5)$ on the minimal model of $\tilde A_K$. \end{Corollary} \begin{Proof} Immediately from proposition \ref{P37}. \end{Proof}

9303

alg-geom/9303005

en

https://arxiv.org/abs/alg-geom/9303005

alg-geom/9303005

Roberto Paoletti

Roberto Paoletti

Free pencils on divisors

18 pages, amslatex

Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument given here is based on Bogomolov's theorem on an n-dimensional variety, and on its recent adaptations to the setting of prime charachterstic (due to Shepherd-Barron and Moriwaki).

alg-geom

\section{\bf {Introduction}} In algebraic geometry, it is rather typical that the embedding of a variety $Y$ in another variety $X$ forces strong constraints on the existence of free linear series on $Y$. For example, a classical result in plane curve theory states that the gonality of a smooth plane curve of degree $d$ is $d-1$ (\cite{acgh}). It is then natural to look for general statements of this flavor. One particular case, which is quite well understood, is the one where $Y$ is a divisor in $X$. This problem has been studied by several researchers. In particular, a wide range of situations is dealt with by the following result of Sommese (\cite{so:amp}): \begin{thm}{(Sommese)} Let $Y\subset X$ be an irreducible smooth ample divisor and let $\phi :Y@>>>B$ be a morphism onto another projective manifold. If $\dim (Y)\ge \dim (B)+2$ then $\phi$ extends to a morphism $\psi :X@>>>B$. \end{thm} Serrano (\cite{se:ext}) then studied the case where $B$ is a smooth curve and $\dim (X)=2$ or $3$. Namely, he proved the two following theorems: \begin{thm}{(Serrano)} Let $C$ be an irreducible smooth curve contained in a smooth surface $S$. Suppose that there exists a morphism $\phi :C@>>>\bold P^1$ of degree $d$. If $C^2>(d+1)^2$, then there exists a morphism $\psi :S@>>>\bold P^1$ extending $\phi$. \end{thm} \noindent and \begin{thm}{(Serrano)} Let $X$ be a smooth projective threefold, and let $S\subset X$ be a smooth very ample surface. Let $\phi :S@>>>\bold P^1$ be a morphism with connected fibers. Let $g(F)$ be the arithmetic genus of a fiber and set $d=F\cdot S$. If $S^3>(d+1)^2$ and $dim H^0(X,\cal O_X(S))\ge 3d+3+2g(F)$, then $\phi$ extends to a morphism $\psi :X@>>>\bold P^1$. \end{thm} Actually, Serrano proves more, in the sense that he shows how these statements imply analogous ones with $\bold P^1$ replaced by a general smooth curve $B$, and he can also replace the above numerical conditions by weaker ones if $S+K_X$ is a numerically even divisor. His argument is based on Miyaoka's vanishing theorem combined with a refinement of Bompieri's method. Furthermore, Serrano applies the above results and methods to the study of the ampleness of the adjoint divisor. \bigskip On the other hand, in a celebrated theorem Reider \cite{re:vbls} has shown how adjunction problems on surfaces can be exaustively studied using vector bundle methods. His argument is based on an application of Bogomolov's instability theorem. Furthermore, Reider himself has also given a proof along these lines of a statement close to Serrano's theorem for surfaces (\cite{re:app}). Also in light of Serrano's result for threefolds, it is therefore reasonable to expect that methods of this type should be applicable to obtain some more general statement about the extension of linear series on a divisor. Our result in this direction is the following: \begin{thm} ($char(k)=0$) Let $X$ be a smooth projective $n$-fold, and let $Y\subset X$ be a reduced irreducible divisor. If $n\ge 3$ assume that $Y$ is ample, and if $n=2$ assume that $Y^2>0$ (so that in particular it is at least nef). Let $\phi :Y@>>>\bold P^1$ be a morphism, and let $F$ denote the numerical class of a fiber. \noindent (i) If $$F\cdot Y^{n-2}<\sqrt {Y^n}-1,$$ then there exists a morphism $\psi :X@>>>\bold P^1$ extending $\phi$. Furthermore, the restriction $$H^0(X,\psi ^{*}\cal O_{\bold P^1}(1))@>>> H^0(Y,\phi ^{*}\cal O_{\bold P^1}(1))$$ is injective. In particular, $\psi$ is linearly normal if $\phi$ is. \noindent (ii) If $$F\cdot Y^{n-2}=\sqrt {Y^n}-1$$ and $Y^n\neq 4$, then either there exists an extension $\psi :X@>>>\bold P^1$ of $\phi$, or else we can find an effective divisor $D$ on $X$ such that $(D\cdot Y^{n-1})^2=(D^2\cdot Y^{n-2})Y^n$ and $D\cdot Y^{n-1}=\sqrt {Y^n}$, and an inclusion $$\phi ^{*}\cal O_{\bold P^1}(1)\subset \cal O_Y(D).$$ \end{thm} When applied to $n=2$ and $n=3$, this gives the above statements of Serrano. However, the hypothesis are weaker, because we are not requiring $Y$ to be smooth and we don't need the assumption about the number of sections of $\cal O_X(Y)$. Besides, we don't require $Y$ to be very ample (unlike Serrano's statement for $n=3$). We have furthermore a description of what happens in the boundary situation; for example, $(d+1)^2=C^2$ is the case of a minimal pencil on a smooth plane curve (of degree $d+1$). With respect to Reider's result on surfaces, the assumption that $Y^2\ge 19$ and that $Y$ be smooth is not necessary. The above furthermore shows that conclusion (a) in Proposition 2.15 of \cite{re:app} always occurs for $f<\sqrt {Y^2}-1$ (just take the Stein factorization of $X@>>>\bold P^1$), and therefore the other possibilities can only occur in the boundary case (ii). This in turn gives more information about this case. For example, comparison with Reider's theorem shows that when $f=\sqrt {Y^2}-1$, under the additional hypothesis that the curve $Y$ be smooth and $Y^2\ge 19$, if $\cal O_X(D)$ is not base point free then it has exactly one base point. As to $n\ge 4$, this is clearly weaker than Sommese's result except that we are not requiring $Y$ to be smooth. The argument provides a direct geometric construction of the extension, as follows. If we let $A=\phi ^{*}\cal O_{\bold P^1}(1)$, $V=\phi ^{*}H^0(\bold P^1,\cal O_{\bold P^1}(1))$, we can define a rank two vector bundle $\cal F$ by the exactness of the sequence $$0@>>>\cal F@>>>V\otimes \cal O_X@>>>A@>>>0.$$ In light of Bogomolov's instability theorem on a $n$-dimensional variety, the given numerical assumption implies that $\cal F$ is Bogomolov unstable with respect to $Y$, and so we have a saturated destabilzing line bundle $\cal L\subset \cal F$. Then $\cal L=\cal O_X(-D)$ for some effective divisor on $X$, and hence we are reduced to arguing that the numerology forces $D$ to move in a base point free pencil. Using the relative version of the Harder-Narashiman filtration (\cite{fl}) this concrete description can be adapted to families of morphisms, and one can also prove a more general statement about morphisms to arbitrary smooth curves. \bigskip Finally, using recent results of Moriwaki concerning a version of the Bogomolov-Gieseker inequality in prime charachteristic, the above statements can be generalized to varieties defined over a field of charachteristic $p$. \bigskip This paper covers part of the content of my Phd thesis at UCLA. I want to thank Robert Lazarsfeld, my advisor, for introducing me to Algebraic Geometry and taking continuous interest in my progress. I am also endebted to a number of people for valuable comments and discussions; among them D. Gieseker, M. Green and especially A. Moriwaki. \section{\bf {Instability of rank two bundles}} In this section we collect some statements about instability of rank two vector bundles on a smooth projective manifold. References in this direction are, for example, \cite{bo:st}, \cite{gi} and \cite{mi:cc}. For the statements in charachteristic $p$, we shall be using results from \cite{mo:fpb}. Let us first assume $char(k)=0$. We shall keep this convention until otherwise stated. The basic result is given by the Bogomolov-Gieseker inequality for semistable bundles: \begin{thm} Let $S$ be a smooth projective surface, and let $\cal E$ be a rank two vector bundle on $X$ with Chern classes $c_1(\cal E)$ and $c_2(\cal E)$. If $H$ is any polarization and $\cal E$ is $H$-semistable, then $c_1(\cal E)^2-4c_2(\cal E)\le 0$. \end{thm} \begin{defn} Let $X$ be any projective $n$-dimensional manifold and let $\cal E$ be a rank two vector bundle on $X$. Let $c_i(\cal E)\in A^i(X)$ be the Chern classes of $\cal E$, $i=1$ and $2$. Define the {\it discriminant of $\cal E$} as $$\Delta (\cal E)=c_1(\cal E)^2-4c_2(\cal E)\in A^2(X).$$ \label{defn:discr} \end{defn} \begin{lem} Let $X$ be a smooth projective $n$-fold and fix a polarization $H$ on $X$. Consider a rank two vector bundle $\cal E$ on $X$ which is $H$-unstable. Suppose that $\cal L_1,\cal L_2 \subset \cal E$ are line bundles and set $e=deg_H(\cal E)$, $l_1=deg_H(\cal L_1)$ and $l_2=deg_H(\cal L_2)$. Suppose that $2l_i>e$, $i=1,2$ and that $\cal L_2$ is saturated in $\cal E$. Then $\cal L_1\subset \cal L_2$. \end{lem} {\it Proof.} Set $l=min\{l_1,l_2\}$. By assumption, we have $2l>e$. Let $$\cal Q=:\cal E/\cal L_2.$$ Then $\cal Q$ is a torsion free sheaf on $X$. If $\cal L_1\not\subset \cal L_2$, then the induced morphism $\cal L_1@>>>\cal Q$ is not identically zero, and therefore it is generically nonzero. This implies that the obvious morphism of vector bundles $$\cal L_1\oplus \cal L_2@>>>\cal E$$ is generically surjective. Hence the line bundle $\wedge ^2\cal E\otimes \cal L_1^{-1}\otimes \cal L_2^{-1}$ is effective, and therefore $$e\ge l_1+l_2\ge 2l,$$ a contradiction. $\sharp$ \bigskip \begin{rem} The above argument still works if $2l_1\ge e$. \end{rem} \begin{cor} Let $\cal E$ be an $H$-unstable rank two vector bundle on $X$. If $\cal L\subset \cal E$ is a saturated destabilizing line bundle, then it is the maximal $H$-destabilizing line bundle of $\cal E$. $\cal L$ contains any $H$-destabilizing line bundle of $\cal E$. \end{cor} \begin{defn} Let $S$ be a smooth projective surface, and let $N^1(S)$ be the vector space of all numerical equivalence classes of divisors on $S$. The positive cone $K^{+}(S)\subset N^1(S)$ is described by the equations $D^2>0$ and $D\cdot H>0$ for some (and hence for all) polarizations $H$ on $S$. \end{defn} If we apply this to the situation of Bogomolov's theorem, we have \begin{cor} Let $S$ be a smooth projective surface and let $\cal E$ be a rank two vector bundle on $S$ with $\Delta (\cal E)>0$. Then there exists a sequence $$0@>>>A@>>>\cal E@>>>\cal B\otimes \cal J_Z@>>>0,$$ where $Z\subset S$ is local complete intersection codimension two subscheme and $A$ and $B$ are line bundles on $S$ such that $A-B\in K^{+}(S)$. Furthermore, $A$ is the maximal destabilizing line bundle of $\cal E$ with respect to any polarization on $X$. \label{cor:devissage2} \end{cor} {\it Proof.} Fix any polarization $H$ on $S$. Since $\cal E$ is $H$-unstable, there is an exact sequence $$0@>>>A@>>>\cal E@>>>B\otimes \cal J_Z@>>>0$$ where $A$ is the maximal destabilizing subsheaf of $\cal E$, and in particular it is saturated. Hence we have $(A-B)\cdot H>0$. On the other hand, since $c_1(\cal E)=A+B$ and $c_2(\cal E)=A\cdot B+[Z]$, we also have $$0<\Delta (\cal E)=(A+B)^2-4A\cdot B-4deg([Z])\le (A-B)^2.$$ This implies $A-B\in K^{+}(S)$, and therefore $A$ strictly destabilizes $\cal E$ with respect to any polarization on $S$. On the other hand, being saturated, it then has to be the maximal destabilizing subsheaf of $\cal E$ with respect to any polarization on $X$. $\sharp$ \bigskip We now want to generalize the above results to higher dimensional varieties. We start by recalling the following fundamental result of Mumford-Mehta-Ramanathan (cfr \cite{mi:cc}): \begin{thm} Let $X$ be a smooth projective $n$-fold, and let $H$ be a polarization on $X$. Suppose that $\cal E$ is a vector bundle on $X$. If $m\gg 0$, and $Y\in |mH|$ is general, then the maximal destabilizing subsheaf of $\cal E|_Y$ is the restriction to $Y$ of the maximal destabilizing subsheaf of $\cal E$. \label{thm:mumera} \end{thm} \begin{rem} By the {\it maximal destabilizing subsheaf} of $\cal E$ one means the first term $\cal E_1$ of the Harder-Narashiman filtration of $\cal E$. If $\cal E$ is semistable, $\cal E_1=\cal E$. \end{rem} We generalize definition 1.3 as follows: \begin{defn} Let $X$ be a smooth projective $n$-fold, and let $H$ be a polarization on $X$. Denote by $N^1(X)$ the vector space of all numerical equivalence classes of divisors on $X$. Then the {\it $H$-positive cone} $K^{+}(X,H)\subset N^1(X)$ is described by the equations $D^2\cdot H^{n-2}>0$ and $D\cdot H^{n-1}>0$. Note that this implies $D\cdot H^{n-2}\cdot L>0$ for any other polarization $L$ on $X$. \end{defn} We then have: \begin{thm} Let $X$ be a smooth projective $n$-fold and let $H$ be a fixed polarization on $X$. Consider a rank two vector bundle $\cal E$ on $X$ of discriminant $\Delta (\cal E)$. If $\Delta (\cal E)\cdot H^{n-2}>0$, then there exists an exact sequence $$0@>>>\cal A@>>>\cal E@>>>\cal B\otimes \cal J_W@>>>0$$ where $W\subset X$ is a (possibly empty) codimension two local complete intersection subscheme, and $\cal A$ and $\cal B$ are line bundles on $X$ such that $\cal A-\cal B\in K^{+}(X,H)$. \label{thm:main} \end{thm} {\it Proof.} For $n=2$, this is the content of Corollary \ref{cor:devissage2}. For $n\ge 3$, let $V\in |mH|$ be general, with $m\gg 0$. We may assume that $V$ is a smooth irreducible surface, and that the maximal $H$-destabilizing subsheaf of $\cal E|_S$ is the restriction of the maximal $H$-destabilizing subsheaf of $\cal E$ (Theorem \ref{thm:mumera}). By the hypothesis, $$\Delta (\cal E|_S)=\Delta (\cal E)\cdot mH>0.$$ Therefore, by induction $\cal E|_V$ is Bogomolov-unstable with respect to $H|_V$, and so there exists an exact sequence $$0@>>> A@>>>\cal E@>>> B\otimes \cal J_Z@>>>0,$$ satisfying the conclusions of theorem 1.1. Furthermore, by the above there is $\cal A\subset \cal E$ such that $\cal A|_S=A$. Being normal of rank one, $\cal A$ is a line bundle. $\sharp$ \bigskip \begin{rem} Note the inequality $(\cal A-\cal B)^2 \cdot H^{n-2}\ge \Delta (\cal E)\cdot H^{n-2}$. \label{rem:sat} \end{rem} \begin{defn} Let $X$ be a smooth $n$-dimensional projective variety, and let $H$ be an line bundle on $X$. Consider a rank two vector bundle $\cal E$ on $X$. We shall say that $\cal E$ is {\it Bogomolov-unstable} with respect to $H$ if there exists a line bundle $\cal L\subset \cal E$ such that $2c_1(\cal L)-c_1(\cal E)\in K^{+}(X,H)$. Hence Theorem \ref{thm:main} can be rephrased by saying that if $\Delta (\cal E)\cdot H^{n-2}>0$, then $\cal E$ is Bogomolov-unstable with respect to $H$. \label{defn:bogunst} \end{defn} Let us now come to the case of positive charachteristic. The basic result is given here by Moriwaki's generalization of the Bogomolov-Gieseker inequality (\cite{mo:fpb}). Before stating his theorem, we need the following: \begin{defn} Let $X$ be a smooth projective $n$-fold and let $H$ be an ample line bundle on $X$. Let $\cal E$ be a rank two vector bundle on $X$. We say that $\cal E$ is weakly $\mu$-semistable w.r.t. $H$ if for any proper subsheaf $\cal F\subset \cal E$ there exists an ample divisor $D$ on $X$ such that $\mu (\cal F,H,D)\le \mu (\cal E,H,D)$, where for a sheaf $\cal G$ we set $\mu (\cal G,H,D)=:\dfrac {c_1(\cal G) \cdot H^{n-2}\cdot D} {rank(\cal G)}$. \label{defn:mu} \end{defn} \begin{rem} In any charachteristic, if $\cal E$ is Bogomolov-unstable w.r.t. $H$ (definition \ref{defn:bogunst}), then it is not $\mu$-semistable. On the other hand, if $\cal E$ is not $\mu$-semistable w.r.t. $H$ and $\Delta (\cal E)\cdot H^{n-2} >0$, then it is necessarily Bogomolov-unstable w.r.t. $H$. \label{rem:mub} \end{rem} \begin{defn} Let $X$ be a smooth projective $n$-fold, and let $H$ be an ample line bundle on $X$, and let $Nef(X)\subset N^1(X)$ denote the nef cone of $X$. Set $$\sigma (H)=inf_{D\in Nef(X)}\Big \{\frac{(K_X\cdot D\cdot H^{n-2})^2} {D^2\cdot H^{n-2}}\Big \}.$$ We agree to take the above ratio equal to $\infty$ when the denumerator vanishes. \label{defn:sigma} \end{defn} \begin{thm} (Moriwaki) Let $X$ be a smooth projective $n$-fold over an algebraically closed field of charachteristic $p>0$. Assume that $X$ is not uniruled. Let $H$ be a polarization on $X$, and let $\cal E$ be a rank two vector bundle on $X$. Suppose that for all $0\le i<r$ the Frobenius pull-back $\cal E^{(i)}$ of $\cal E$ is weakly $\mu$-semistable with respect to $H$. Then we have $$\Delta (\cal E)\cdot H^{n-2}\le \frac {\sigma (H)}{(p^r-1)^2}.$$ \label{thm:mor} \end{thm} Furthermore, Moriwaki proves the following powerful restriction lemma: \begin{lem} ($char (k)\ge 0$) Let $X$ be a smooth projective $n$-fold, and let $H$ be a very ample line bundle on $X$. Suppose that $\cal E$ is a rank two vector bundle on $X$, which is weakly $\mu$-semistable w.r.t. $H$. Then for a general $Y\in |H|$ the restriction $\cal E|_Y$ is weakly $\mu$-semistable w.r.t. $H|_Y$. \label{lem:A2} \end{lem} \begin{defn} Let $X$ be a smooth projective $n$-fold, and let $H$ be an ample line bundle on $X$. Define $$\beta (H)=inf_{D\in Nef(X)}\Big \{\frac {(D\cdot (H+K_X)\cdot H^{n-2})^2}{D^2\cdot H^{n-2}}\Big \}.$$ \label{defn:beta} \end{defn} \begin{cor} Let $X$ be a smooth projective $n$-fold, with $n\ge 3$ on an algebraically closed field, and let $H$ be a very ample line bundle on $X$. Suppose that the general $Y\in |H|$ is not uniruled. Let $\cal E$ be a rank two vector bundle on $X$ such that $$\Delta (\cal E)\cdot H^{n-2}>\dfrac {\beta (H)}{(p-1)^2}.$$ Then $\cal E$ is Bogomolov-unstable with respect to $H$, i.e. there exists an exact sequence $$0@>>>\cal A@>>>\cal E@>>>\cal B\otimes \cal J_Z@>>>0,$$ where $\cal A$ and $\cal B$ are line bundles on $X$, $Z\subset X$ is a codimension two local complete intersection and $\cal A-\cal B\in K^{+}(X,H)$. \label{cor:devissagep} \end{cor} \begin{rem} Although this is an immediate application of Moriwaki's theorem \ref{thm:mor}, it is phrased in a way that makes it applicable to uniruled varieties. Furthermore, observe that if $k$ is an uncountable algebraically closed field and $X$ is a smooth non-uniruled projective variety over $k$ with a very ample line bundle $H$ on it, the general element of $|H|$ is not uniruled either. \label{claim:uniruled} In fact, since $k$ is uncountable, a variety $X$ over $k$ is uniruled if and only if through a general point of $X$ there passes a rational curve (\cite{mm}). But a general point in a general divisor of a very ample linear series is a general point of $X$. \end{rem} {\it Proof.} By Lemma \ref{lem:A2}, it is sufficient to show that for general $Y\in |H|$ the restriction $\cal E|_Y$ is Bogomolov-unstable with respect to $H|_Y$. It is easy to deduce this fact from theorem \ref{thm:mor} and the definition of $\beta (H)$. $\sharp$ \bigskip \begin{cor} Let $\cal E$ be a rank two vector bundle on $\bold P^r_k$, where $k$ is an algebraically closed field of charachteristic $p$. If $\Delta (\cal E)>0$, then $\cal E$ is unstable. \end{cor} {\it Proof.} For $r=2$, this is well-known. For $r\ge 3$, we apply corollary \ref{cor:devissagep} taking the very ample line bundle in the statement to be $\cal O_{\bold P^3}(4)$, so that $\beta (H)=0$. We can also proceed inductively from the case $r=2$ by applying Lemma \ref{lem:A2}. $\sharp$ \bigskip \section{\bf {Extension Of Pencils}}\label{section:ext} Let $Y\subset X$ be an inclusion of projective varieties, and let $|L|$ be a base point free pencil on $Y$. It is natural to look for conditions under which $|L|$ extends to $X$, in the spirit of the results of Sommese, Serrano and Reider (\cite{re:app}, \cite{so:amp}, \cite{se:ext}). Our main result is the following: \begin{thm} ($char(k)=0$) Let $X$ be a smooth projective $n$-fold, $n\ge 2$, and let $Y\subset X$ be a reduced irreducible divisor. If $n\ge 3$, assume that $Y$ is ample, and if $n=2$ that $Y^2>0$ (so that in particular it is nef). Suppose given a morphism $\phi :Y@>>>\bold P^1$ and let $F$ denote the numerical class of a fiber of $\phi$. (i) Suppose that $$F\cdot Y^{n-2}<\sqrt {Y^n}-1.$$ Then there exists a morphism $\psi :X@>>>\bold P^1$ extending $\phi$, and such that $$H^0(X,\psi ^{*}\cal O_{\bold P^1}(1))\hookrightarrow H^0(Y,\phi ^{*}\cal O_{\bold P^1}(1)).$$ In particular, if $\phi$ is linearly complete, then so is $\psi$. (ii) Suppose $F\cdot Y^{n-2}=\sqrt {Y^n}-1$ and $Y^n\neq 4$. Then either $\phi ^{*}\cal O_{\bold P^1}(1)$ extends to a base-point free pencil on $X$, or else there exists an effective divisor $D$ on $X$ such that \noindent (a) the following equalities hold: $$(D^2\cdot Y^{n-2})Y^n=(D\cdot Y^{n-1})^2$$ and $$D\cdot Y^{n-1}=\sqrt {Y^n}.$$ \noindent (b) there is an inclusion $$\phi ^{*} \cal O_{\bold P^1}(1)\subset \cal O_Y(D).$$ \label{thm:ext} \end{thm} \begin{rem} If $Y$ is ample, the equalities in (a) of (ii) can be phrased as follows. If $S\subset X$ is a smooth complete intersection of $n-2$ divisor equivalent to multiples of $Y$, then $$D-\frac 1{\sqrt {Y^n}}Y\in Ker\{N(X)@>>>N(S)\}.$$ If $n=2$, this is just saying that $D\equiv _n\frac 1{\sqrt {Y^n}}Y$. \end{rem} {\it Proof.} Set \begin{equation} A=:\phi ^{*}\cal O_{\bold P^1}(1) \label{eq:A} \end{equation} and let \begin{equation} V\subset H^0(Y,A) \label{eq:V} \end{equation} be the pencil associated to $\phi$, i.e. $V=\phi ^{*}H^0(\bold P^1,\cal O_{\bold P^1}(1))$. Define a sheaf $\cal F$ on $X$ by the exactness of the sequence \begin{equation} 0@>>>\cal F@>>>V\otimes \cal O_X@>>>A@>>>0. \label{eq:F} \end{equation} Then $\cal F$ is a rank two vector bundle with Chern classes $c_1(\cal F)=-Y$ and $c_2(\cal F)=[A]$, where $Y$ denotes the divisor class on $Y$ of an element of the pencil $|A|$. In particular, $[A]$ is represented by a fiber $F$ of $\phi$. Therefore the discriminant of $\cal F$ (definition \ref{defn:discr}) is \begin{equation} \Delta (\cal F)=Y^2-4[A] \label{eq:Delta} \end{equation} and so \begin{equation} \Delta (\cal F)\cdot Y^{n-2}=Y^n-4F\cdot Y^{n-2}. \label{eq:Delta1} \end{equation} It is easy to check that \begin{equation} \sqrt {Y^n}-1\le \frac {Y^n}4 \label{eqn:easy} \end{equation} and by assumption we then have in particular that $\Delta (\cal F)\cdot Y^{n-2}>0$, and therefore $\cal F$ is Bogomolov-unstable with respect to $Y$ (definition \ref{defn:bogunst}). Hence there exists a saturated invertible subsheaf $$\cal L\subset \cal F$$ which is the maximal destabilizing subsheaf of $\cal F$ with respect to $(Y,\cdots,Y,L)$, for any ample divisor $L$ on $X$ (theorem \ref{thm:main}). Since $\cal L\subset \cal F\subset \cal O_X^2$, we can write $$\cal L=\cal O_X(-D)$$ for some effective divisor $D$ on $X$. The instability condition then reads \begin{equation} (Y-2D)\cdot Y^{n-1}\ge 0, \label{eq:inst} \end{equation} with strict inequality holding if $Y$ is ample. Furthermore, using the fact that $\cal L$ is saturated one can see that \begin{equation} (Y-2D)^2\cdot Y^{n-2}\ge \Delta (\cal F)\cdot Y^{n-2} \label{eq:inst1} \end{equation} (see remark \ref{rem:sat}) and if we set $f=:F\cdot Y^{n-2}$ this can be rewritten as \begin{equation} f\ge D\cdot Y^{n-1}-D^2\cdot Y^{n-2}. \label{eq:inst2} \end{equation} By assumption, we have $f<\sqrt {Y^n}-1$ and together with (\ref{eq:inst2}) this gives $$D^2\cdot Y^{n-2}-1> D\cdot Y^{n-1}-\sqrt {Y^n}.$$ Applying the Hodge Index Theorem, we then get \begin{equation} \frac {(D\cdot Y^{n-1})^2}{Y^n} -1 >D\cdot Y^{n-1}-\sqrt {Y^n}. \label{eq:hit} \end{equation} \begin{claim} $\cal L$ is saturated in $\cal O_X^2$. \end{claim} {\it Proof.} If not, there would exist an inclusion $\cal O_X(Y-D)\subset \cal O_X^2$ (here we use the fact that $Y$ is reduced and irreducible) and therefore we should have $$(D-Y)\cdot Y^{n-1}\ge 0.$$ Together with (\ref{eq:inst}), this would imply $Y^n\le 0$, a contradiction. $\sharp$ \bigskip Hence we have an exact sequence of the form \begin{equation} 0@>>>\cal O_X(-D)@>>>\cal O_X^2@>>>\cal O_X(D) \otimes \cal J_Z@>>>0, \label{eq:sat} \end{equation} where $Z\subset X$ is a codimension two local complete intersection. Computing $c_2(\cal O_X^2)=0$ from the above sequence we then get $$D^2=[Z]$$ (equivalently, one might just observe that $Z$ is the complete intersection of the two sections of $\cal O_X(D)$ coming from the above sequence). Therefore under the assumptions of the theorem either $Z=\emptyset$, or else $D^2\cdot Y^{n-2}>0$. \begin{lem} $Z=\emptyset$ \end{lem} {\it Proof.} Suppose, otherwise, that $D^2\cdot Y^{n-2}>0$. In this case the Hodge Index Theorem yields $$(D\cdot Y^{n-1})^2\ge (D^2\cdot Y^{n-2})Y^n\ge Y^n$$ and therefore $$D\cdot Y^{n-1}\ge \sqrt {Y^n}.$$ Therefore the right hand side of (\ref{eq:hit}) is nonnegative. We can rewrite (\ref{eq:hit}) as $$\frac {(D\cdot Y^{n-1})^2}{Y^n}-1> D\cdot Y^{n-1}-\sqrt {Y^n}=Y^n\{\frac {D\cdot Y^{n-1}}{Y^n}- \frac 1{\sqrt {Y^n}}\}.$$ Let us now make use of the destabilizing condition $Y^n\ge 2D\cdot Y^{n-1}$: we obtain $$ \frac {(D\cdot Y^{n-1})^2}{Y^n}-1> 2\frac {(D\cdot Y^{n-1})^2}{Y^n}-2\frac {D\cdot Y^{n-1}}{\sqrt Y^n}$$ and this leads to $0>\Big \{ \dfrac {D\cdot Y^{n-1}}{\sqrt {Y^n}}-1\Big \}^2$, absurd. $\sharp$ \bigskip Since $Z=\emptyset$, $\cal O_X(-D)@>>>\cal O_X^2$ never drops rank, and therefore neither does $\cal O_X(-D)@>>>\cal F$. Hence we have a commutative diagram \begin{equation} \CD @. @. 0 @. 0 @. @. \\ @. @. @VVV @VVV @. \\ 0@>>>\cal O_X(-D)@>>>\cal F@>>>\cal O_X(D-Y)@>>>0 \\ @. @| @VVV @VVV @. \\ 0@>>>\cal O_X(-D)@>>>V\otimes \cal O_X@>>>\cal O_X(D)@>>>0 \\ @. @. @VVV @VVV @. \\ @. @. A @= \cal O_Y(D) @. @. \\ @. @. @VVV @VVV @. \\ @. @. 0 @. 0 @. @. \endCD \label{eq:bigcd} \end{equation} from which we see that $$A\simeq \cal O_Y(D).$$ Furthermore, since $$(D-Y)\cdot Y^{n-1}\le 2D\cdot Y^{n-1}-Y^n<0$$ by the destabilizing condition (\ref{eq:inst}), we have $H^0(X,\cal O_X(D-Y))=0$ and therefore an injection $$H^0(X,\cal O_X(D))\hookrightarrow H^0(Y,\cal O_Y(D-Y)).$$ Since $\cal O_X(D)$ is a quotient of $\cal O_X^2$, it is globally generated and $V$ gives a base point free pencil of sections of $\cal O_X(D)$. $\sharp$ \bigskip {\it Proof of (ii)} Suppose now that $F\cdot Y^{n-2}=\sqrt {Y^n}-1$. It is easy to see that if $Y^n\neq 4$ then the inequality (\ref{eqn:easy}) is strict, and therefore $\cal F$ is still Bogomolov unstable with respect to $Y$. Arguing exactly as in the proof of the previous lemma we get: \begin{lem} Either $Z=\emptyset$, or else the following equalities hold: $$(D\cdot Y^{n-1})^2=(D^2\cdot Y^{n-2})Y^n$$ and $$D\cdot Y^{n-1}=\sqrt {Y^n}.$$ \end{lem} To complete the argument, observe that a variant of the commutative diagram (\ref{eq:bigcd}) gives the exact sequence $$0@>>>\cal O_X(D-Y)\otimes \cal J_W@>>> \cal O_X(D)\otimes \cal J_Z@>>>\cal O_Y(D)\otimes \cal I_{Z\cap Y}@>>>0$$ where $\cal I$ denotes an ideal sheaf on $Y$. Therefore we also get an isomorphism $A\simeq \cal O_Y(D)\otimes \cal I_{Z\cap Y}.$ $\sharp$ \bigskip \begin{cor} (Serrano) Let $S$ be a smooth projective surface and let $C\subset S$ be an irreducible smooth curve with $C^2>0$. Then either $$gon(C)\ge \sqrt {C^2}-1$$ or else for every minimal pencil $A$ on $C$ there exists a base point free pencil $\cal O_S(D)$ on $S$ such that $$A\simeq \cal O_C(D).$$ \end{cor} \begin{cor} Let $C\subset \bold P^2$ be a smooth curve of degree $d$. Then $$gon(C)= d-1.$$ Furthermore, any base point free pencil on $C$ is given by projecting through a point of $C$. \label{cor:plane} \end{cor} {\it Proof.} The bound in the theorem gives $gon(C)\ge d-1$. On the other hand projecting from a point of $C$ shows that equality must hold. Let $A$ be any minimal pencil on $C$. We may assume that $d>2$. We are then in the boundary situation $f=\sqrt {C^2}-1$ (case (ii) of Theorem \ref{thm:ext}, $n=2$). Hence we must have an inclusion $A\subset \cal O_C(H)$, which shows that $A$ has the form \begin{equation} A=\cal O_C(H-P) \label{eq:proj} \end{equation} for some $P\in C$. But (\ref{eq:proj}) is saying exactly that $A$ is the pull back of the hyperplane bundle on $\bold P^1$ under the morphism given by projection from $P$. Hence all the minimal pencils are obtained in this way. $\sharp$ \bigskip \begin{exmp} Let us apply the Theorem to the gonality of Castelnuovo extremal curves in $\bold P^3$. If $C$ has even degree $d=2a$, then $C$ is the complete intersection of a quadric $S$ and an hypersurface of degree $a$. Suppose that $S$ is smooth. Then either $gon(C)\ge \sqrt {C\cdot _SC}-1=\sqrt 2a-1$, or else a minimal pencil is induced by a base point free pencil on $S$. $C$ on $S\simeq \bold P^1\times \bold P^1$ is a curve of type $(a,a)$, and restriction to it of the two rulings gives two pencils of degree $a=\frac d2$, which is the well-known answer. The argument is the same for even degree. \end{exmp} \begin{exmp} For an example with $n=3$, let $S\subset \bold P^3$ be a smooth surface of degree $s$ containing a line $L$, and let $\phi :S@>>>\bold P^1$ be induced by projection from $L$. Then a straighforward computation shows that $f>s\sqrt s-1$. \end{exmp} We now give an application to singular plane curves. \begin{cor} Let $C\subset \bold P^2$ be a reduced irreducible curve of degree $d$, and suppose that the only singularities of $C$ are ordinary singular points $P_1,\cdots,P_k$ of multiplicities $m_1,\cdots,m_k$, respectively. Let $m=max\{m_i\}$ and denote by $\tilde C$ the normalization of $C$. Suppose that $d^2>\sum _im_i^2$. Then $$gon(\tilde C)\ge min\Big \{\sqrt {d^2-\sum _im_i^2}-1,d-\sqrt {\sum _im_i^2} \Big \}.$$ \end{cor} {\it Proof.} Let $$f:X@>>>\bold P^2$$ be the blow up of $\bold P^2$ at $P_1,\cdots,P_k$, $$E_i=f^{-1}P_i$$ for $i=1,\cdots,k$ be the exceptional divisors,and let $\tilde C\subset X$ be the proper transform of $C$. Then $\tilde C$ is an irreducible smooth curve and $$\tilde C\in |dH-\sum _{i=1}^km_iE_i|.$$ Therefore we have $$\tilde C^2=d^2-\sum _{i=1}^km_i^2>0$$ by assumption, and the hypothesis of the theorem are satisfied. Hence either $$gon(\tilde C)\ge \sqrt {\tilde C^2}-1,$$ or else there exists an effective divisor $D$ on $X$ moving in a base point free pencil and inducing a minimal pencil on $\tilde C$. We may then assume that $D$ has the form $$D=xH-\sum _ia_iE_i$$ with $x>0$ and all the $a_i\ge 0$. The condition $D^2=0$ then gives $$x=\sqrt {\sum _ia_i^2}.$$ Hence $$D\cdot \tilde C=xd-\sum _ia_im_i\ge xd-\sqrt {\sum _ia_i^2}\sqrt { \sum _im_i^2}= $$ $$=xd-x\sqrt {\sum _im_i^2}\ge d-\sqrt {\sum _im_i^2}.$$ The statement follows. $\sharp$ \bigskip \begin{exmp} Let us consider for example the case of a reduced irreducible plane curve $C\subset \bold P^2$ whose only singularities are nodes $P_1,\cdots,P_{\delta}$. Suppose that $$4\delta <d^2.$$ Then by the Corollary $$gon (\tilde C)\ge min\{\sqrt {d^2-4\delta}-1,d-2\sqrt {\delta}\}.$$ For example, if we also assume that $$\delta <d-2$$ then $$\sqrt{d^2-4\delta}-1>d-3,$$ and for any effective divisor $D=xH-\sum _ia_iE_i$ with $D^2=0$ it is easy to see that $D\cdot \tilde C\ge d-2$. Since projecting from a node gives a pencil of degree $d-2$, we then have $$gon(\tilde C)=d-2.$$ \end{exmp} We now show how theorem 2.1 applies to families of morphisms. \begin{prop} Let $X$ be a smooth projective $n$-fold and $Y\subset X$ be a reduced irreducible divisor in $X$. Suppose that $Y$ is ample when $n>2$ and that $Y^2>0$ when $n=2$. Let $\Phi :Y\times B@>>>\bold P^1$ be a family of morphisms with $B$ smooth and set $\phi _b=\Phi |_{Y\times \{b\}}$. Denote by $F$ the numerical class of a fiber of $\phi _b$ (it is independent of $b\in B$), and suppose that $$F\cdot Y^{n-2}<\sqrt {Y^n}-1.$$ Then there exists a nonempty open subset $T\subset B$ and a morphism $$\Psi :X\times T@>>>\bold P^1$$ that restricts to $\Phi$ on $Y\times T$. \label{prop:rel} \end{prop} {\it Proof.} Let $$A=\Phi ^{*}\cal O_{\bold P^1}(1)$$ and $$V=:\Phi ^{*}H^0(\bold P^1, \cal O_{\bold P^1}(1)).$$ Then we can define a rank two vector bundle on the smooth variety $X\times B$ in the usual guise, by the exactness of the sequence \begin{equation} 0@>>>\cal F@>>>V\otimes \cal O_{X\times B}@>>>A@>>>0. \label{eq:Frel} \end{equation} For $b\in B$ let us set $X_b=X\times \{b\}$ and $A_b=A|_{X_b}$. Then we have $\cal F|_{X_b}\simeq \cal F|_b$, where $\cal F_b=:Ker\{V\otimes \cal O_{X_b}@>>>A_b\}$. Then $\cal F$ can be seen as a family of vector bundles on $X$, with Chern classes $c_1(\cal F)=-Y$ and $c_2(\cal F)= [A_b]$. As in the proof of the Theorem, these vector bundles are Bogomolov unstable with respect to $Y$. Let $\cal L_b\subset \cal F_b$ be the maximal destabilizing line bundle of $\cal F_b$. By the construction in Theorem \ref{thm:ext}, the morphisms $\psi _b$ are associated to base point free pencils of sections of $\cal L_b^{-1}$ induced by $V$. Therefore, the proposition will follow once we show that the line bundles $\cal L_b$ can be glued to a line bundle $\cal L \subset \cal F|_{X\times T}$ on some open subset $X\times T$. In fact, we have: \begin{claim} For some nonempty open subset $T\subset B$ there exists a line bundle $\cal L\subset F|_{X\times T}$ such that $\cal L$ restricts to $\cal L_b$ on $X_b$, for each $b\in T$. \end{claim} {\it Proof} This follows from the relative version of the Harder-Narashiman filtration introduced in \cite{fhs} and \cite{fl}. $\sharp$ \bigskip This proves the statement of the proposition. $\sharp$ \bigskip In his paper (\cite{se:ext}) Serrano expressed his results about extensions in terms of morphisms to arbitrary smooth curves. It seems in order to give here a corresponding generalization of theorem 2.1. \begin{defn} Let $B$ be a smooth curve. We shall denote by $s(B)$ the smallest degree of a nondegenerate plane birational model of $B$, i.e. the smallest $k$ for which $B$ has a birational $g^2_k$. Nondegenerate only means that we agree to take $s(\bold P^1)=2$. \end{defn} \begin{cor} Let $X$ and $Y$ satisfy the hypothesis of the theorem, and let $\phi :Y@>>>B$ be a morphism to a smooth curve. Denote by $F$ the numerical class of a fiber of $\phi$, and suppose that $$(s(B)-1)F\cdot Y^{n-2}<\sqrt {Y^n}-1.$$ Then there exists a morphism $\psi :X@>>>B$ extending $\phi$. \end{cor} {\it Proof.} We adapt the argument in \cite{se:ext}, Lemma 3.2. Let $f:B@>>>G\subset \bold P^2$ be a plane birational model of $B$, of degree $s=s(B)$, and let $B^{*}\subset B$ be the inverse image of the smooth locus of $G$. For $b\in B^{*}$, let $\pi _b:B@>>>\bold P^1$ be the projection from $f(b)$; $\pi _b$ is a morphism of degree $s-1$. Consider the composition $\phi _b=\pi _b\circ \phi:Y@>>>\bold P^1$. A fiber of $\phi _b$ is numerically equivalent to a sum of $s-1$ fibers of $\phi$, and the numerical hypothesis then imply, by theorem \ref{thm:ext}, that there exist extensions $\psi _b:X@>>>\bold P^1$. By Proposition \ref{prop:rel}, we can find a nonempty open subset $T\subset B$ and a morphism $$\Psi :X\times T@>>>\bold P^1$$ extending the morphism $$\Phi :Y\times T@>>>\bold P^1$$ given by $\Phi (y,b)=\phi _b (y)$. {}From this one sees that, if $$X@>\gamma _b>>\Delta _b@>g_b>>\bold P^1$$ is the Stein factorization of $\phi _b$, then $\Delta _b\simeq \Delta$ for some fixed curve $\Delta$ and all the morphisms $\gamma _b$ can be identified. Consider the morphism $$h=(\gamma |_Y,\phi):Y@>>>\Delta \times \bold P^1.$$ It is easy to see that $\pi _1:h(Y)@>>>\Delta$ is an isomorphism. Hence we can define $\psi =\pi _2\circ \pi _1 ^{-1}\circ \gamma$. $\sharp$ Let us consider now the case of prime charachteristic. We give the corresponding version of Theorem \ref{thm:ext}. \begin{thm} Let $k$ be an algebraically closed field of charachteristic $p$, and let $X$ be a smooth projective n-fold over $k$. Let $Y\subset X$ be a reduced irreducible divisor, and suppose that there exists a morphism $\phi :Y@>>>\bold P^1$; let $F$ denote the numerical class of a fiber of $\phi$. Then $\phi$ can be extended to a morphism $\psi :X@>>>\bold P^1$ in the following situations: (i) $char(k)\neq 2,3$, $n=2$, $Y^2>0$, $deg(F)<\sqrt {Y^2}-1$, $X$ not of general type. (ii) $n=2$, $Y^2>0$, $X$ is not uniruled and $$deg(F)<min \Big \{\sqrt {Y^2}-1,\frac 14Y^2-\frac 1{(p-1)^2} \sigma_S\Big \}.$$ (iii) $n\ge 3$, $X$ is not uniruled and there exists a ample line bundle $H$ on $X$, such that $Y\equiv lH$ and $$F\cdot Y^{n-2}<min\Big \{\sqrt {Y^n}-1, \frac 14Y^n-\frac {l^{n-2}}{(p-1)^2}\sigma (H)\Big \}.$$ (iv) $n\ge 3$ and there exists a very ample line bundle $H$ on $X$ such that $Y\equiv lH$, the general $Z\in |H|$ is not uniruled and $$F\cdot Y^{n-2}<min \Big \{\sqrt {Y^n}-1, \frac 14Y^n-\frac {l^{n-2}}{(p-1)^2} \beta (H)\Big \}.$$ \label{thm:extp} \end{thm} \begin{rem} The definitions of $\sigma (H)$ and $\beta (H)$ are given in section 2 (definitions \ref{defn:sigma} and \ref{defn:beta}). If $n=2$, $\sigma $ does not depend on $H$, and we denote it by $\sigma _S$. \end{rem} {\it Proof.} As to (i), that $\phi$ does not extend means that $\Delta (\cal F)>0$ (cfr eq. (\ref{eq:F})), but $\cal F$ is not Bogomolov-unstable. That this forces $X$ to be of general type is the content of Theorem 7 of \cite{sb}. For the other statements, the argument is exactly the same as in the charachteristic zero case, the extra assumptions being needed to apply the results about unstable rank two bundles from section 2 (e.g., Corollary \ref{cor:devissagep}). $\sharp$ \bigskip \begin{rem} For $n=2$, we have to assume that $S$ is not uniruled to apply the chrarachteristic $p$ version of Bogomolov's theorem. However, in the case of $\bold P^2$ Bogomolov's theorem still holds (\cite{schw}). We can therefore still argue as in Corollary \ref{cor:plane} to deduce the classical statement about the gonality of plane curves. \end{rem} Theorem \ref{thm:extp} can be strengthened as follows (see Theorem \ref{thm:mor}). \begin{thm} Let notation be as in Theorem \ref{thm:extp}, and suppose that $F\cdot Y^{n-2}<\sqrt{Y^n}-1$. Assume that $X$ is not uniruled. Let $r$ be the smallest positive integer such that $$F\cdot Y^{n-2}< min\Big \{\sqrt{Y^n}-1, \frac {Y^n}4-\frac {l^{n-2}}{(p^r-1)^2}\sigma (H)\Big \}.$$ Then if $X^{\prime}\supset Y^{\prime}$ denote the $(r-1)$-th Frobenius pull-backs of the varieties $X$ and $Y$, there exists $\psi :X^{\prime}@>>>\bold P^1$ extending the induced morphism $\phi ^{\prime}:Y^{\prime} @>>>\bold P^1$. \end{thm}

9303

alg-geom/9303004

en

https://arxiv.org/abs/alg-geom/9303004

alg-geom/9303004

Ron Donagi

Ron Donagi and Loring W. Tu

Theta Functions for $\SL(n)$ versus $\GL(n)$

10 pages, Latex

Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM (n,L)$, the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle $L$ over $C$. Let $\theta_F$ and $\theta$ be theta bundles over these two moduli spaces. We prove a simple formula relating their spaces of sections: if $h=\gcd (n,d)$ is the greatest common divisor of $n$ and $d$, and $L\in \Pic ^d(C)$, then $$\dim H^0(\SM (n,L), \theta^k) \cdot k^g=\dim H^0(\M(n,d),\theta_F^k)\cdot h^g.$$ We also formulate a conjectural duality between these two types of spaces of sections.

alg-geom

\section{Theta bundles} \label{bundles} We recall here the definitions of the theta bundles on a fixed-determinant moduli space and on a full moduli space. Our definitions are slightly different from but equivalent to those in \cite{drezet-narasimhan}. For $L \in {\rm Pic} ^d (C)$, the Picard group of ${\cal SM} := {\cal SM} (n,L)$ is ${\Bbb Z}$ and the theta bundle $\theta$ on ${\cal SM}$ is the positive generator of ${\rm Pic} ({\cal SM})$. When $n$ and $d$ are such that $\chi (E) = 0$ for $E \in {\cal M} (n,d)$, i. e., when $d=(g-1)n$, there is a natural divisor $\Theta \subset {\cal M} (n,n(g-1)) $: $$\Theta = \mbox{closure of } \{ E \mbox{ stable in } {\cal M}(n,n(g-1))\ | \ h^0 (E) \ne 0 \}.$$ The theta bundle $\theta$ over ${\cal M} (n, n(g-1))$ is the line bundle corresponding to this divisor. We say that a semistable bundle $F$ is {\it complementary} to another bundle $E$ if $\chi (E\otimes F) = 0$. We also say that $F$ is {\it complementary} to ${\cal M} (n,d)$ if $\chi (E \otimes F) = 0$ for any $E \in {\cal M} (n,d)$. It follows easily from the Riemann-Roch theorem that if $E \in {\cal M} (n,d)$, $h=\gcd (n,d), n= h \bar{n}$, and $d= h\bar{d}$, then $F$ has rank $n_F$ and degree $d_F$, where $$n_F = k\bar{n} \quad \mbox{ and }\quad d_F = k(\bar{n} (g-1) - \bar{d})$$ for some positive integer $k$. If $F$ is complementary to ${\cal M}(n,d)$, let $$\tau _F : {\cal M} (n,d) \to {\cal M} (nn_F, nn_F (g-1))$$ be the map $$E \mapsto E \otimes F.$$ Pulling back the theta bundle $\theta$ from ${\cal M} (nn_F, nn_F (g-1))$ via $\tau_F$ gives a line bundle $\theta_F := \tau_F^* \theta$ over ${\cal M}(n,d)$. (This bundle may or may not correspond to a divisor in ${\cal M}(n,d)$.) Let $\det: {\cal M}(n,d) \to J_d(C)$ be the determinant map. When ${\rm \ rk\, } F$ is the minimal possible: ${\rm \ rk\, } F = \bar{n} =n/h$, then $\theta_F$ is called a {\it theta bundle over} ${\cal M} (n,d)$; otherwise, it is a multiple of a theta bundle. Indeed, we extract from \cite{drezet-narasimhan} the formula: \begin{prop} \label{thetaF} Let $F$ and $F_0$ be two bundles complementary to ${\cal M} (n,d)$. If ${\rm \ rk\, } F = a {\rm \ rk\, } F_0$, then $$\theta_F \simeq \theta_{F_0}^{\otimes a} \otimes \det{}^*(\det F \otimes (\det F_0)^{-a}),$$ where we employ the usual identification of ${\rm Pic}^0(C)$ with ${\rm Pic}^0 (J_0)$. \end{prop} In particular, $\theta_F$ depends only on ${\rm \ rk\, } F$ and $\det F$. If $\theta_{F}$ is a theta bundle on ${\cal M} (n,d)$, then for any $L \in {\rm Pic}^d(C)$, $\theta_{F}$ restricts to the theta bundle on ${\cal SM} (n,L)$. \section{A Galois covering} \label{covering} Let $\tau : Y \to X$ be a covering of varieties, by which we mean a finite \'etale morphism. A {\it deck transformation} of the covering is an automorphism $\phi : Y \to Y$ that commutes with $\tau$. The covering is said to be {\it Galois} if the group of deck transformations acts transitively (hence simply transitively) on a general fiber of the covering. Denote by $J= {\rm Pic} ^0 (C)$ the group of isomorphism classes of line bundles of degree 0 on the curve $C$, and $G = T_n$ the subgroup of torsion points of order $n$. Fix $L \in {\rm Pic}^d(C)$ and let ${\cal SM} = {\cal SM} (n,L)$, $J=J_0(C)$, and ${\cal M} = {\cal M}(n,d)$. Recall that the tensor product map \begin{eqnarray*} \tau : {\cal SM} \times J &\to& {\cal M}\\ (E, M) &\mapsto& E \otimes M \end{eqnarray*} gives an $n^{2g}$-sheeted \'etale map (\cite{teixidor-tu}, Prop. 8). The group $G=T_n$ acts on ${\cal SM} \times J$ by $$N.(E, M) = (E\otimes N^{-1}, N\otimes M).$$ It is easy to see that $G$ is the group of deck transformations of the covering $\tau$ and that it acts transitively on every fiber. Therefore, $\tau: {\cal SM} \times J \to {\cal M}$ is a Galois covering. \begin{prop} \label{directimage} If $\tau: Y \to X$ is a Galois covering with finite abelian Galois group $G$, then $\tau_*{\cal O}_Y$ is a vector bundle on $X$ which decomposes into a direct sum of line bundles indexed by the characters of $G$: $$\tau_*{\cal O}_Y = \sum_{\lambda \in \hat G} L_{\lambda},$$ where $\hat G$ is the character group of $G$. \end{prop} \medskip \noindent {\sc Proof}. Write ${\cal O}= {\cal O}_Y$. The fiber of $\tau_*{\cal O}$ at a point $x\in X$ is naturally a complex vector space with basis $\tau^{-1}(x)$. Hence, $\tau_*{\cal O}$ is a vector bundle over $X$. The action of $G$ on $\tau^{-1}(x)$ induces a representation of $G$ on $(\tau_*{\cal O})(x)$ equivalent to the regular representation. Because $G$ is a finite abelian group, this representation of $G$ decomposes into a direct sum of one-dimensional representations indexed by the characters of $G$: $$(\tau_*{\cal O})(x) = \sum_{\lambda \in \hat G} L_{\lambda}(x).$$ Thus, for every $\lambda \in \hat G$, we obtain a line bundle $L_\lambda$ on $X$ such such $\tau_*{\cal O} = \sum_{\lambda} L_\lambda$. \quad\quad$\Box$ \section{Pullbacks} \label{pullbacks} We consider the tensor product map \begin{eqnarray*} \tau: {\cal SM} (n_1, L_1) \times {\cal M} (n_2, d_2) &\to& {\cal M}(n_1n_2, n_1d_2+n_2d_1)\\ (E_1, E_2) &\mapsto& E_1 \otimes E_2, \end{eqnarray*} where $d_1 = \deg L_1$. For simplicity, in this section we write ${\cal SM}_1 = {\cal SM} (n_1, L_1)$, ${\cal M}_2 = {\cal M} (n_2, d_2)$, and ${\cal M}_{12} = {\cal M}(n_1n_2, n_1d_2 + n_2d_1)$. \begin{prop} \label{pullbackform} Let $F=F_{12}$ be a bundle on $C$ complementary to ${\cal M}_{12}$. Then $$\tau^*\theta_F \simeq \theta^c \boxtimes \theta_{E_1 \otimes F}$$ for any $E_1 \in {\cal SM} (n_1, L_1)$, where $$c:={ {n_2 {\rm \ rk\, } F}\over {{\rm \ rk\, } F_1}} = {{n_2{\rm \ rk\, } F} \over {n_1/\gcd (n_1,d_1)}}$$ and $F_1$ is a minimal complementary bundle to $E_1$. \end{prop} \medskip \noindent {\bf Proof.} For $E_2 \in {\cal M}(n_2,d_2)$, let $$\tau_{E_2} : {\cal SM}_1 \to {\cal M}_{12}$$ be tensoring with $E_2$. Then $$(\tau^*\theta_F)|_{{\cal SM} \times \{E_2\}} = \tau_{E_2}^*\theta_F = \tau_{E_2}^*\tau_F^*\theta = \tau_{E_2 \otimes F}^* \theta = \theta^c,$$ where by Proposition \ref{thetaF} \begin{eqnarray*} c&=& {\rm \ rk\, } (E_2 \otimes F)/ {\rm \ rk\, } F_1 \\ &=& {{n_2 {\rm \ rk\, } F}\over {n_1/\gcd(n_1,d_1)}}. \end{eqnarray*} Similarly, \begin{eqnarray*} (\tau^*\theta_F)|_{\{E_1\}\times{\cal M}_2} &=& \tau_{E_1}^*\theta_F = \tau_{E_1}^*\tau_F^*\theta \\ &=& \tau_{E_1\otimes F}^*\theta = \theta_{E_1\otimes F}. \end{eqnarray*} Note that the bundle $\theta_{E_1\otimes F}$ depends only on ${\rm \ rk\, } (E_1\otimes F) = n_1{\rm \ rk\, } F$ and $\det (E_1 \otimes F) = L_1^{{\rm \ rk\, } F} \otimes (\det F)^{n_1}$. Hence, both $(\tau^*\theta_F)|_{{\cal SM}_1 \times\{E_2\}}$ and $(\tau^*\theta_F)|_{\{E_1\} \times {\cal M}_2}$ are independent of $E_1$ and $E_2$. By the seesaw theorem, $$\tau^*\theta_F \simeq \theta^c \boxtimes \theta_{E_1\otimes F}.$$ \rightline{$\Box$} \begin{cor} \label{jacobian} Let $L \in {\rm Pic}^d(C)$ and $$\tau: {\cal SM} (n,L) \times J_0 \to {\cal M}(n,d)$$ be the tensor product map. Suppose $F$ is a minimal complementary bundle to ${\cal M} (n,d)$. Choose $N \in {\rm Pic}^{g-1}(C)$ to be a line bundle such that $N^n = L \otimes (\det F)^h$, where $h=\gcd (n,d)$. Then $$\tau^* \theta_F = \theta \boxtimes \theta_N^{n^2/h}.$$ \end{cor} \medskip \noindent {\bf Proof.} Apply the Proposition with ${\rm \ rk\, } F = n/h$ and $n_1 = n, d_1=d, n_2=1, d_2=0$. Then $c=1$. By Proposition \ref{thetaF}, \begin{eqnarray*} \theta_{E_1\otimes F} &=& \theta_N^{n^2/h} \otimes \det{}^* (\det (E_1 \otimes F)\otimes N^{-n^2/h}) \\ &=& \theta_N^{n^2/h}. \end{eqnarray*} \rightline{$\Box$} \section{Proof of Theorem 1} \label{proof} We apply the Leray spectral sequence to compute the cohomology of $\tau^*\theta_F^k$ on the total space of the covering $\tau : {\cal SM} \times J \to {\cal M}$ of Section \ref{covering}. Recall that ${\cal SM} = {\cal SM} (n,d)$, $J=J_0$, and ${\cal M} = {\cal M}(n,d)$. Because the fibers of $\tau$ are 0-dimensional, the spectral sequence degenerates at the $E_2$-term and we have \begin{equation} \label{a} H^0({\cal SM} \times J , \tau^*\theta_F^k ) = H^0({\cal M}, \tau_*\tau^*\theta_F^k). \end{equation} By Cor. \ref{jacobian} and the K\"unneth formula, the left-hand side of (\ref{a}) is \begin{eqnarray*} H^0({\cal SM} \times J, \tau^*\theta_F^k))&=& H^0({\cal SM} \times J, \theta^k \boxtimes \theta_N^{kn^2/h}))\\ &=& H^0({\cal SM}, \theta^k)\otimes H^0(J, \theta_N^{kn^2/h}). \end{eqnarray*} By the Riemann-Roch theorem for an abelian variety, $$h^0(J, \theta_N^{kn^2/h}) = (kn^2/h)^g.$$ So the left-hand side of (\ref{a}) has dimension \begin{equation} \label{left} h^0({\cal SM}, \theta^k) \cdot (kn^2/h)^g. \end{equation} Next we look at the right-hand side of (\ref{a}). By the projection formula and Prop. \ref{directimage}, \begin{eqnarray*} \tau_*\tau^*\theta_F^k &=& \theta_F^k \otimes \tau_*{\cal O} \\ &=& \theta_F^k \otimes \sum_{\lambda \in {\hat G}} L_{\lambda}\\ &=& \sum_{\lambda \in {\hat G}} \theta_F^k \otimes L_{\lambda}. \end{eqnarray*} Our goal now is to show that for any character $\lambda \in \hat G$, \begin{equation} \label{b} H^0({\cal M}, \theta_F^k \otimes L_\lambda) \simeq H^0 ({\cal M}, \theta_F^k). \end{equation} This will follow from two lemmas. \begin{lem} \label{L} The line bundle $L_{\lambda}$ on ${\cal M}$ is the pullback under $\det : {\cal M} \to J_d$ of some line bundle $N_{\lambda}$ of degree 0 on $J_d := {\rm Pic} ^d(C)$. \end{lem} \begin{lem} \label{independence} For $F$ a vector bundle as above, $k$ a positive integer, and $M$ a line bundle of degree 0 over $C$, $$H^0 ({\cal M}, \theta_{F\otimes M}^k) \simeq H^0({\cal M}, \theta_F^k ).$$ \end{lem} Assuming these two lemmas, let's prove (\ref{b}). By Proposition \ref{thetaF}, $$\theta_{F\otimes M} = \theta_F \otimes \det {}^* M^{n_F};$$ hence, $$\theta_{F\otimes M}^k = \theta_F^k \otimes \det {}^* M^{n_Fk}.$$ If $L_{\lambda}= \det{}^*N_\lambda$, and we choose a root $M= N_{\lambda}^{1/(n_Fk)}$, then $$\theta_F^k \otimes L_{\lambda} = \theta_F^k \otimes \det{}^*N_{\lambda} = \theta_{F\otimes M}^k.$$ Equation (\ref{b}) then follows from Lemma \ref{independence}. \medskip \noindent {\sc Proof of Lemma} \ref{L}. Define $\alpha : {\cal SM} \times J \to J$ to be the projection onto the second factor, $\beta : {\cal M} \to J$ to be the composite of $\det : {\cal M} \to J_d$ followed by multiplication by $L^{-1}: J_d \to J$, and $\rho: J \to J$ to be the $n$-th tensor power map. Then there is a commutative diagram $$\begin{array}{rcccl} \; & {\cal SM} \times J & \stackrel{\tau}{\to} & {\cal M} &\; \\ & & & & \\ \alpha & \downarrow & \; &\downarrow &\beta \\ & & & & \\ \; & J & \stackrel{\rho}{\to} & J. &\; \end{array}$$ Furthermore, in the map $\alpha: {\cal SM} \times J \to J$ we let $G= T_n$ act on $J$ by $$N.M = N \otimes M, \quad\quad M \in J,$$ and in the map $\beta: {\cal M} \to J$ we let $G$ act trivially on both ${\cal M}$ and $J$. Then all the maps in the commutative diagram above are $G$-morphisms. By the push-pull formula (\cite{hartshorne}, Ch. III, Prop. 9.3, p. 255), $$ \tau_*\alpha^*{\cal O}_J = \beta^*\rho_*{\cal O}_J .$$ By Proposition \ref{directimage}, $\rho_*{\cal O}_J$ is a direct sum of line bundles $V_\lambda$ on $J$, where $\lambda \in \hat G$. In fact, these $V_\lambda$ are precisely the $n-$torsion bundles in $J$; in particular, their degrees are zero. If $\tau_{L^{-1}} : J_d \to J$ is multiplication by the line bundle $L^{-1}$, we set $N_\lambda := \tau_{L^{-1}}^* V_\lambda$. Then \begin{eqnarray*} \tau_*{\cal O}_{{\cal SM} \times J} &=& \beta^*\sum_{\lambda \in \hat G} V_{\lambda} \\ &=& \det{}^*\tau_{L^{-1}}^* \sum V_\lambda \\ &=& \sum \det{}^*N_{\lambda}. \end{eqnarray*} By Prop. \ref{directimage}, $\tau_*{\cal O}_{{\cal SM} \times J} = \sum L_\lambda$. Since both $L_\lambda$ and $\det{}^* N_\lambda$ are eigenbundles of $\tau_*{\cal O}_{{\cal SM} \times J}$ corresponding to the character $\lambda \in \hat G$, $$L_\lambda = \det{}^* N_\lambda.$$ \hfill $\Box$ \medskip \noindent {\sc Proof of Lemma} \ref{independence}. Tensoring with $M\in J_0(C)$ gives an automorphism \begin{eqnarray*} \tau_M : {\cal M} &\to& {\cal M} \\ E &\mapsto& E\otimes M, \end{eqnarray*} under which $$\theta_{F\otimes M} = \tau_M^*\theta_F .$$ Hence, $$\theta_{F\otimes M}^k = \tau_M^*(\theta_F^k)$$ and the lemma follows. \quad\quad\quad $\Box$ \medskip Returning now to Eq. (\ref{a}), its right-hand side is \begin{eqnarray*} H^0({\cal M}, \tau_*\tau^*\theta_F^k) &=& \sum _{\lambda \in \hat G} H^0({\cal M}, \theta_F^k \otimes L_\lambda ) \\ &\simeq& \sum _{\lambda \in \hat G} H^0 ({\cal M}, \theta_F^k),\quad\quad\mbox{ (by (\ref{b}))} \end{eqnarray*} which has dimension $$h^0({\cal M} ,\theta_F^k) \cdot n^{2g}.$$ By (\ref{left}) the left-hand side of Eq. (\ref{a}) has dimension $$h^0({\cal SM}, \theta^k)\cdot (kn^2/h)^{g}.$$ Equating these two expressions gives $$h^0({\cal M}, \theta_F^k) = h^0({\cal SM},\theta^k) \cdot ({k\over h})^g.$$ This completes the proof of Theorem \ref{formula}. \section{A conjectural duality} \label{duality} As in the Introduction we start with integers $\bar{n}$, $\bar{d}$, $h$, $k$ such that $\bar{n}$, $h$, $k$ are positive and $\gcd (\bar{n}, \bar{d})=1$. Take $$n_1=h\bar{n} ,\ d_1=h\bar{d},\ n_2=k\bar{n},\ d_2=k(\bar{n}(g-1)-\bar{d}), \ {\rm and }\ L_1\in {\rm Pic}^{d_1}(C).$$ The tensor product induces a map $$\tau: {\cal SM} (n_1, L_1) \times {\cal M}(n_2, d_2) \to {\cal M} (n_1n_2, n_1n_2(g-1)).$$ As before, write ${\cal SM}_1 = {\cal SM} (n_1, L_1)$, ${\cal M}_2= {\cal M}(n_2,d_2)$, and ${\cal M}_{12} ={\cal M}(n_1n_2, n_1n_2(g-1))$. Let $F_2=F$ and $F_{12}={\cal O}$ be complementary to ${\cal M}_2$ and ${\cal M}_{12}$ respectively. By the pullback formula (Proposition \ref{pullbackform}) $$\tau^*\theta_{{\cal O}} = \theta^{{n_2}/\bar{n}} \boxtimes \theta_{E_1}.$$ But by Proposition \ref{thetaF}, $$\theta_{E_1} = \theta_F^h \otimes \det{}^*(L\otimes (\det F)^{-h}).$$ If $L= (\det F)^h$, then $\theta_{E_1} = \theta_F^h$ and $$\tau^*\theta_{{\cal O}} = \theta ^k \boxtimes \theta_F^h.$$ By the K\"unneth formula, $$H^0({\cal SM}_1 \times {\cal M}_2, \tau^*\theta_{{\cal O}})= H^0({\cal SM}_1, \theta^k)\otimes H^0({\cal M}_2, \theta_F^h).$$ In \cite{beauville-narasimhan-ramanan} it is shown that up to a constant, $\theta_{{\cal O}}$ has a unique section $s$ over ${\cal M}_{12}$. Then $\tau^*s$ is a section of $H^0({\cal SM}_1 \times {\cal M}_2 , \tau^*\theta_{{\cal O}})$ and therefore induces a natural map \begin{equation} \label{dual} H^0({\cal SM}_1, \theta^k)^{\vee} \to H^0({\cal M}_2, \theta_F^h). \end{equation} We conjecture that this natural map is an isomorphism. Among the evidence for the duality (\ref{dual}), we cite the following. \begin{enumerate} \item[i)] (Rank 1 bundles) The results of \cite{beauville-narasimhan-ramanan} that $$H^0({\cal SM} (n,{\cal O}), \theta )^{\vee} \simeq H^0({\cal M} (1,g-1), \theta_{{\cal O}}^n) \quad {\rm and } \quad H^0({\cal M} (n,n(g-1)), \theta_{{\cal O}} ) = {\Bbb C},$$ are special cases of (\ref{dual}), for $(n_2,d_2)=(1,g-1)$ and $(n_1,d_1) =(1,0)$ respectively. \item[ii)] (Consistency with Theorem \ref{formula}) Given a triple of integers $(n_1, d_1, k)$, we define $h, \bar{n}, \bar{d}$ by $$h=\gcd (n_1,d_1), n_1=h\bar{n}, d_1=h\bar{d}$$ and let $n_2, d_2$ be as before: $$n_2=k\bar{n}, d_2=k(\bar{n}(g-1)-\bar{d}).$$ Assuming $n_1$ and $k$ to be positive, it is easy to check that the function $$(n_1,d_1,k) \mapsto (n_2, d_2, h)$$ is an involution. Write $v(n,d,k)=h^0({\cal M}(n,d), \theta_F^k)$ and $s(n,d,k)=h^0({\cal SM}(n,d), \theta^k)$. Then Theorem \ref{formula} assumes the form \begin{equation} \label{one} v(n,d,k) \cdot h^g = s(n,d,k)\cdot k^g. \end{equation} The duality (\ref{dual}) implies that there is an equality of dimensions \begin{equation} \label{two} s(n_1,d_1, k) = v(n_2, d_2, h). \end{equation} Because $(n_1,d_1,k) \mapsto (n_2,d_2,h)$ is an involution, it follows that \begin{equation} \label{three} s(n_2,d_2, h) = v(n_1, d_1, k). \end{equation} Putting (\ref{one}), (\ref{two}), and (\ref{three}) together, we get $$v(n_2,d_2,h)k^g = s(n_2,d_2, h)h^g,$$ which is Theorem \ref{formula} again. \item[iii)] (Elliptic curves) We keep the notation above, specialized to the case of a curve $C$ of genus $g=1$: $$n_1 = h\bar{n}, d_1 = h\bar{d}, n_2= k\bar{n}, d_2= -k\bar{d}.$$ Set $C' := {\rm Pic}^{\bar{d}} (C) $. The map sending a line bundle to its dual gives an isomorphism $C'\simeq {\rm Pic}^{-\bar{d}}(C)$. If $L \in {\rm Pic} ^{\bar{d}}(C)$, viewed as a line bundle on $C$, we let $\ell$ be the corresponding point in $C'$, and ${\cal O} _{C'}(\ell)$ the associated line bundle of degree 1 on the curve $C'$. There is a natural map $$\gamma : {\rm Pic}^{h\bar{d}} (C) \to {\rm Pic} ^h (C')$$ which sends $L:= L_1\otimes \cdots \otimes L_h \in {\rm Pic}^{h\bar{d}}(C)$ to $L':= {\cal O}_{C'} (\ell _1 + \cdots +\ell_h )$, where $L_i \in {\rm Pic}^{\bar{d}}(C)$ corresponds to the point $\ell _i \in C'$. \end{enumerate} From \cite{atiyah} and \cite{tu} we see that there are natural identifications $${\cal M}(h\bar{n}, h\bar{d}) \simeq S^h{\cal M} (\bar{n},\bar{d}) \simeq S^h {\rm Pic}^{\bar{d}}(C) = S^hC'$$ and $${\cal M}(k\bar{n}, -k\bar{d}) \simeq S^k{\cal M} (\bar{n},-\bar{d}) \simeq S^k {\rm Pic}^{-\bar{d}}(C) \simeq S^kC'.$$ Furthermore, there is a commutative diagram $$\begin{array}{rcccl} \; & {\cal M} (h\bar{n},h\bar{d}) & \stackrel{\sim}{\to} & S^hC' &\; \\ & & & & \\ \det & \downarrow & \; &\downarrow &\alpha \\ & & & & \\ \; & {\rm Pic}^{h\bar{d}}(C)& \stackrel{\gamma}{\to} & {\rm Pic}^h(C'). &\; \end{array}$$ Since the fiber of the Abel-Jacobi map $\alpha: S^hC' \to {\rm Pic}^h(C')$ above $L'$ is the projective space ${\Bbb P} H^0(C',L')$, it follows that there is a natural identification $${\cal SM} (h\bar{n}, L) \simeq {\Bbb P} H^0 (C', L').$$ Since the theta bundle is the positive generator of ${\cal SM} (h\bar{n}, L)$, it is the hyperplane bundle. For $F\in {\cal M} (\bar{n}, \bar{d})$, let $q\in C'$ be the point corresponding to the line bundle $Q:= \det F \in {\rm Pic}^{\bar{d}}(C)$. Then \begin{eqnarray*} H^0({\cal SM}(h\bar{n}, (\det F)^h), \theta^k) &\simeq& H^0 ({\Bbb P} H^0(C', {\cal O}_{C'}(hq)), {\cal O}(k))\\ &=& S^kH^0(C', {\cal O}_{C'} (hq))^{\vee}. \end{eqnarray*} Recall that each point $q \in C'$ determines a divisor $X_q$ on the symmetric product $S^kC'$: $$X_q := \{ q+D \ | \ D\in S^{k-1}C'\}.$$ The proof of Theorem 6 in \cite{tu} actually shows that if $F \in {\cal M} (\bar{n},-\bar{d})$, then under the identification ${\cal M} (k\bar{n}, -k\bar{d}) \simeq S^kC'$, the theta bundle $\theta_F$ corresponds to the bundle associated to the divisor $X_q$ on $S^kC'$, where $q$ is the point corresponding to $\det F \in {\rm Pic}^{\bar{d}}$. Therefore, by the calculation of the cohomology of a symmetric product in \cite{tu} \begin{eqnarray*} H^0 ({\cal M} (k\bar{n}, -k\bar{d}), \theta_F^h) &=& H^0 (S^kC', {\cal O} (h X_q))\\ &=& S^k H^0 (C', {\cal O} (hq)). \end{eqnarray*} So the two spaces $H^0({\cal SM}(h\bar{n}, (\det F)^h), \theta^k)$ and $H^0 ({\cal M} (k\bar{n}, -k\bar{d}), \theta_F^h)$ are naturally dual to each other. \begin{enumerate} \item[iv)] (Degree 0 bundles) Consider the moduli space ${\cal SM}(n,0)$ of rank $n$ and degree $0$ bundles. In this case, $$n_1=n,\ d_1=0,\ h=\gcd(n,0)=n,\ n_2=k,\ d_2=k(g-1).$$ So the conjectural duality is $$H^0({\cal SM}(n,{\cal O}), \theta^k)^{\vee} \simeq H^0({\cal M}(k,k(g-1)),\theta_{{\cal O}}^n).$$ Because ${\cal M} (k,k(g-1))$ is isomorphic to ${\cal M} (k,0)$ (though noncanonically), it follows that in the notation of ii) $$s(n,0,k) = v(k,0, n).$$ According to R. Bott and A. Szenes, this equality follows from Verlinde's formula. \end{enumerate}